UBC Theses and Dissertations

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UBC Theses and Dissertations

The capital asset pricing model and the probability of bankruptcy: theory and empirical tests. Turnbull, Stuart McLean, 1974

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THE C A P I T A L A S S E T P R I C I N G MODEL AND THE PROBABILITY OF THEORY AND E M P I R I C A L  BANKRUPTCY:  TESTS  by  STUART McLEAN B.Sc, A.R.C.S., M.Sc, D.I.C.,  Royal Royal Royal Royal  College College College College  A THESIS  of of of of  TURNBULL  Science, Science, Science, Science,  London London London London  University, University, University, University,  SUBMITTED I N P A R T I A L F U L F I L M E N T  THE REQUIREMENTS FOR THE DEGREE DOCTOR OF  OF  PHILOSOPHY  in the Faculty of Commerce a n d B u s i n e s s A d m i n i s t r a t i o n  We a c c e p t  this  to the required  thesis  as  conforming  standard  THE UNIVERSITY OF April,  BRITISH 10  74  COLUMBIA  OF  1969 1969 1970 1970  In  presenting  an  advanced  the I  Library  further  for  degree shall  agree  scholarly  by  his  of  this  this  thesis  in  at  University  the  make  that  it  written  p u r p o s e s may  for  financial  is  of  of  Columbia,  British  available  for  for extensive  be g r a n t e d  It  fulfilment  by  the  shall  not  of  Commerce & B u s i n e s s  The University of B r i t i s h V a n c o u v e r 8, C a n a d a  A p r i l 16th. 1974  Columbia  requirements  copying of  I  aqree  and  be  copying or allowed  Administration  f  tha  study.  this  thesis  Head o f my D e p a r t m e n t  understood that  gain  the  reference  permission.  Department  Date  freely  permission  representatives. thesis  partial  nr  publication  without  my  A B S T R A C T  Empirical i s misspecified. than p r e d i c t e d  e v i d e n c e shows t h a t  t h e C a p i t a l A s s e t P r i c i n g Model  S e c u r i t i e s o f low s y s t e m a t i c r i s k c o n s i s t e n t l y  by t h e model, t h e r e v e r s e b e i n g t r u e  systematic r i s k .  W h i l s t t h e r e l a t i o n s h i p between e x - p o s t r e t u r n s and s y s t e -  s i g n i f i c a n t l y d i f f e r e n t from t h e i r t h e o r e t i c v a l u e s .  but  c o e f f i c i e n t s are  V a r i o u s attempts t o  t h e o r e t i c a l l y t h e causes o f t h e m i s s p e c i f i c a t i o n  have been e x p l o r e d ,  f a i l t o p r o v i d e an adequate e x p l a n a t i o n o f a l l t h e o b s e r v e d d e f i c i e n c i e s . The  d i s s e r t a t i o n examines how t h e mechanism o f b a n k r u p t c y  the  structure  the  thesis i s that  of returns for corporate f i n a n c i a l assets.  Using stochastic  natural  The h y p o t h e s i s o f  after abstracting  c o n t r o l t h e o r y , a two v a r i a b l e  c o n t i n u o u s t i m e analogue o f t h e CAPM i s d e r i v e d .  i s associated  affects  t h e p r o b a b i l i t y o f b a n k r u p t c y a c r o s s s e c u r i t i e s and a c r o s s  time i s r e f l e c t e d i n t h e r e s i d u a l r e t u r n  the  e a r n more  f o r s e c u r i t i e s o f high  m a t i c r i s k appears t o be l i n e a r , t h e e s t i m a t e d r e g r e s s i o n  explain  (CAPM)  with the p r o b a b i l i t y o f bankruptcy.  extended form o f  The second  A d i s c r e t e time ex-  p o s t f o r m u l a t i o n o f t h e model i s used t o t e s t e m p i r i c a l l y  the hypothesis.  b e i n g a b l e t o measure t h e p r o b a b i l i t y o f b a n k r u p t c y .  A model f o r m u l a t e d i n terms o f a f i r m ' s n a l l y or externally,  variable  The model p r o v i d e s a  e x p l a n a t i o n o f t h e d e f i c i e n c i e s o f t h e CAPM.  This necessitates  from t h e market.  a b i l i t y t o r a i s e funds, e i t h e r  inter-  t o c o v e r f i x e d c h a r g e s i s d e v e l o p e d , and t h e p r o b a b i l i t y  o f b a n k r u p t c y e s t i m a t e d u s i n g t h e maximum l i k e l i h o o d m e t h o d o l o g i e s o f l o g i t analysis  and p r o b i t  analysis.  The a b i l i t y o f t h e model t o p r e d i c t  iii  bankruptcy  iv  is  t e n t e d on a s e c o n d a r y  sample o f bankrupt  firms.  o b t a i n e d w i t h t h e model p r e d i c t i n g b a n k r u p t c y , f i v e years before the a c t u a l Using  Excellent  results  f o r some f i r m s , f o u r o r  occurrence.  a p o o l i n g o f time  s e r i e s and  cross s e c t i o n data to  mate the c o e f f i c i e n t s o f the r e g r e s s i o n e q u a t i o n r e p r e s e n t i n g the sis,  evidence  i s found  i n d i c a t i n g t h a t bankruptcy  o f common s t o c k r e t u r n s .  are  estihypothe-  i s an e x p l a n a t o r y  factor  TABLE OF CONTENTS Page LIST OF TABLES  vii  LIST OF FIGURES  viii  CHAPTER I.  INTRODUCTION  1  Hypothesis Importance Organization of Thesis II.  III..  IV.  A CRITICAL SURVEY OF THE  5  . . . . .  6  ^° RELEVANT FINANCIAL LITERATURE . . . .  12  E m p i r i c a l S t u d i e s on Bankruptcy C o s t s t o Bankruptcy Bankruptcy and S t o c k Market P r i c e s P r e d i c t i o n o f Bankruptcy C a p i t a l A s s e t P r i c i n g Model., F o u n d a t i o n s o f t h e CAPM C r o s s - S e c t i o n a l T e s t s o f t h e Model Time S e r i e s T e s t s o f t h e Model T h e o r e t i c a l E x t e n s i o n s t o t h e CAPM Summary Hypothesis o f the T h e s i s E m p i r i c a l T e s t i n g o f the Hypothesis  13 13 15 18 24 25 27 29 31 37  PROBABILITY OF BANKRUPTCY  43  Theory Ex-post F o r m u l a t i o n P r e d i c t i v e Model . . . S t a t i s t i c a l Methodology T e s t i n g o f t h e Model Summary  44 48 57 60  AN  EXTENSION OF THE  . . . . .  3  8  3  9  6  5  66  CAPITAL ASSET PRICING MODEL:  BANKRUPTCY .  F o u n d a t i o n s o f Model P r i c e Dynamics S t a t e Space D e s c r i p t i o n and the Budget C o n s t r a i n t The E q u a t i o n o f O p t i m a l i t y : The Demand F u n c t i o n s f o r A s s e t s  v  6  7  .  8  70 5 87 94  vi  Chapter  Page Bankruptcy and S t r u c t u r e o f R e t u r n s S t o c h a s t i c Changes i n t h e P r o b a b i l i t y  9  o f Bankruptcy  110  Summary V.  1  2  0  EMPIRICAL RESULTS  122  E s t i m a t i o n o f t h e P r o b a b i l i t y o f Bankruptcy S t a t i s t i c a l Methodology Data Predictive A b i l i t y Results A l t e r n a t i v e Model Stationarity P r e d i c t i v e Model Summary Testing o f Hypothesis Methodology P o o l i n g o f Time S e r i e s and C r o s s S e c t i o n a l D a t a Aggregation. . . . . . Data Empirical Results . Use o f P o r t f o l i o s I n d i v i d u a l S e c u r i t y Data Random Sample Effect of Asset Size Adjustment o f Time P e r i o d  124 I  Cross S e c t i o n S t u d i e s . . . Changes i n t h e P r o b a b i l i t y o f B a n k r u p t c y Summary VI.  9  3  1  . . .  3  I 1  .  3  3  3  5  I I  l  0  i  4  0  4  3  ^ 150 I 152 152 l ^ I l ^ 4  5  0  3  5  6  8  1  5  I  8  6  4  1  6  4  1  6  6  1  6  8  170 . .  ±  ,  J  1  7  5  SUMMARY  177  Conclusion . „ Further Research  177 179  APPENDIX >" A.  MATHEMATICAL DERIVATION OF THE  B.  NAMES OF BANKRUPT FIRMS  BIBLIOGRAPHY  RESULTS IN CHAPTER IV  182 218  2  2  2  LIST OF TABLES  TABLE  Page  4.1  The P r o b a b i l i t y o f O c c u r r e n c e o f D i f f e r e n t  States  89  5.1  Number o f Bankrupt and Non-Bankrupt F i r m s i n Data Sample . . . 134  5.2  Estimation of C o e f f i c i e n t s  5.3  C l a s s i f i c a t i o n o f O r i g i n a l Data Sample by G e n e r a l M o d e l . . . . 138  5.4  Predictive  5.5  Estimation o f Coefficients A l t e r n a t i v e Model  f o r a G e n e r a l Model  136  ^.39  A b i l i t y o f G e n e r a l Model and T e s t f o r S t a t i o n a r i t y :  141  5.6  C l a s s i f i c a t i o n o f O r i g i n a l D a t a Sample:  5.7  Predictive  5.8  Estimation o f C o e f f i c i e n t s P r e d i c t i v e Model  Ability of Alternative  Alternative  Model . .  Model  142 144  and T e s t f o r S t a t i o n a r i t y : 146  5.9  C l a s s i f i c a t i o n o f O r i g i n a l D a t a Sample:  5.10  Predictive  5.11  V a l u e s o f C o e f f i c i e n t s Used t o E s t i m a t e t h e P r o b a b i l i t y of Bankruptcy  157  Average Y e a r l y V a l u e s o f t h e P r o b a b i l i t y o f B a n k r u p t c y . . . .  159  5.12 5.13  5.14  Predictive  Model. . .  A b i l i t y o f Model  148 149  P o r t f o l i o D a t a : P o o l i n g o f Time S e r i e s and C r o s s S e c t i o n . . P r o b a b i l i t y o f B a n k r u p t c y E s t i m a t e d U s i n g Market V a l u e s o f Corporate V a r i a b l e s  160  P o r t f o l i o D a t a : P o o l i n g o f Time S e r i e s and C r o s s S e c t i o n . P r o b a b i l i t y o f B a n k r u p t c y E s t i m a t e d U s i n g Book V a l u e s o f Corporate Variables . . .  162  5.15  Random Sample:  165  5.16  P o o l i n g o f Time and C r o s s D a t a on Groups o f F i r m s S o r t e d by Asset Size  P o o l i n g o f Time S e r i e s  vii  and C r o s s S e c t i o n D a t a .  1  6  7  viii  TABLE  Page  5.17  Adjustment o f Time P e r i o d  169  5.18  C r o s s S e c t i o n Study  172  5.19  Differences i n the P r o b a b i l i t y  o f Bankruptcy  174  LIST OF FIGURES  FIGURE 4.1  Page The E f f e c t o f B a n k r u p t c y Upon t h e C a p i t a l Market L i n e .  ix  . . .  1  0  8  A C K N O W L E D G E M E N T S  The  author i s indebted  h e l p f u l suggestions, ability.  t o h i s s u p e r v i s o r , Dr. R. White, f o r h i s  acumen, c o n t r i b u t i o n o f i d e a s , and c o n s t a n t  I am g r a t e f u l t o Dr. J . G. Cragg f o r p r o v i d i n g a d v i c e ,  t r a t i n g i n s i g h t , and encouragement. Dr.  I thank D r . M. J . Brennan f o r  and c r i t i c i s m t h r o u g h o u t t h e whole p e r i o d o f my  at the U n i v e r s i t y o f B r i t i s h  pene-  The h e l p o f Dr. S. L. Brumelle,  J . Mao, and Dr. A. Kraus i s n o t e d .  help, advice,  avail-  sojourn  Columbia.  I am t h a n k f u l t o M i s s Susan Aizenman f o r c o f f e e , T i a M a r i a , ( l a t k e s ) , t r a n s p o r t a t i o n and t h e u n f o r g e t t a b l e , of  how she t y p e d  the t h e s i s .  many g r a d u a t e s t u d e n t  Finally,  i f nightmarish,  food  memory  I acknowledge t h e k i n d n e s s o f my  f r i e n d s , e s p e c i a l l y R. F i s h e r and B. MacDonald, t h e  Department o f H i s t o r y , U.B.C., f o r p r o v i d i n g a s m a l l o a s i s o f i n t e l l e c t u a l s t i m u l a t i o n i n an o t h e r w i s e  barren  desert.  x  But even at the very grave I trust  the time shall  When over malice The good will  3  come to be  over wrong,  win its  victory.  ( B o r i s P a s t e r n a k , January,  1959)  CHAPTER I  INTRODUCTION  One o f t h e c e n t r a l i s s u e s i n t h e t h e o r y  of finance  i s the r e l a t i o n -  s h i p between r i s k and r e t u r n demanded by i n v e s t o r s i n s e c u r i t i e s . C a p i t a l Asset  P r i c i n g Model  (CAPM) p r o v i d e s  The  such a t h e o r e t i c a l r e l a t i o n s h i p  between t h e e x p e c t e d r a t e o f r e t u r n and t h e e x p e c t e d r i s k o f an a s s e t  under  1 c o n d i t i o n s o f market e q u i l i b r i u m .  Explicitly,  t h e model s t a t e s t h a t t h e  e x p e c t e d one p e r i o d r e t u r n f o r a s e c u r i t y i s a l i n e a r f u n c t i o n o f i t s s y s t e m a t i c r i s k , which i s a measure o f t h e r e s p o n s i v e n e s s o f t h e s e c u r i t y ' s r e t u r n t o changes i n t h e r e t u r n on t h e market as a whole. In a d d i t i o n t o p r o v i d i n g  i n s i g h t s i n t o the functioning of c a p i t a l  markets, t h e CAPM h o l d s c o n s i d e r a b l e  p r o m i s e a s an o p e r a t i o n a l  tool.  Proposed  a p p l i c a t i o n s have ranged from s t r a t e g i e s f o r s e c u r i t y s e l e c t i o n , e s t i m a t i o n o f the c o s t o f c a p i t a l , measurement o f i n v e s t m e n t p e r f o r m a n c e , and e s t a b l i s h i n g a structure o f managerial fees. gree o f confidence,  B e f o r e t h e model c a n be implemented w i t h any de-  i t i s n e c e s s a r y t o i n v e s t i g a t e t h e t h e o r e t i c a l and e m p i r i -  c a l v a l i d i t y o f t h e model. One i m p o r t a n t a s s u m p t i o n u n d e r l y i n g  t h e model i s t h a t o f p e r f e c t c a p i -  2 t a i markets, distress.  i n p a r t i c u l a r , t h e absence o f c o s t s  The s i t u a t i o n i n which t h e f i r m ' s T h e o r e t i c a l l y t h e model a p p l i e s o n l y  associated with  income b e f o r e t o a l l equity  financial  i n t e r e s t and t a x e s financed  firms.  2 A p e r f e c t c a p i t a l market i s one where a l l p a r t i c i p a n t s a r e p r i c e t a k e r s , have e q u a l and c o s t l e s s access t o a l l i n f o r m a t i o n , and t h e r e a r e no transaction costs or taxes. 1  2  is  l e s s t h a n i t s f i x e d commitments, f o r example i n t e r e s t payments on d e b t ,  i s termed a c o n d i t i o n o f " f i n a n c i a l d i s t r e s s . "  In such a case the  must c u r t a i l d i v i d e n d s , o r i n v e s t m e n t , o r o b t a i n a c a p i t a l example by s e l l i n g  s t o c k ) t o meet i t s f i x e d commitments.  uation of f i n a n c i a l distress because  firm  inflow (for The  extreme  i s when t h e f i r m i s f o r c e d t o d e c l a r e  sit-  bankruptcy  i t i s u n a b l e t o meet the c l a i m s o f i t s c r e d i t o r s . In t h e t h e o r y o f c o r p o r a t e f i n a n c e t h e c o n c e p t s o f b a n k r u p t c y  and  f i n a n c i a l d i s t r e s s a r e i m p o r t a n t f a c t o r s i n d e t e r m i n i n g the o p t i m a l v a l u e o f the f i r m .  In the absence o f c o r p o r a t e t a x e s and d e f a u l t , M o d i g l i a n i  M i l l e r ^ have shown t h a t t h e v a l u e o f the f i r m i s i n d e p e n d e n t o f the mix and e q u i t y , g i v e n the assumptions e x p e c t a t i o n s and t h e a b i l i t y  o f p e r f e c t c a p i t a l markets,  and of debt  homogeneity o f  of the i n v e s t o r to r e p l i c a t e across a l l s t a t e s  o f n a t u r e the income o b t a i n e d by i n v e r t i n g i n an u n l e v e r e d f i r m .  In a  sub-  4 sequent paper M o d i g l i a n i and M i l l e r  r e l a x the assumption  t a x , i n t r o d u c i n g a market i m p e r f e c t i o n . t h e government f o r u s i n g debt and,  The  f i r m r e c e i v e s a t a x s u b s i d y from  as d e b t i s r i s k l e s s ,  t h e f i r m t o use as much debt as p o s s i b l e ;  o f no c o r p o r a t e  i t i s advantageous  for  c a p i t a l s t r u c t u r e i s relevant to the  v a l u e o f the f i r m . Such a c o n c l u s i o n has l i t t l e p r a c t i c a l a p p e a l f o r c o r p o r a t e debt i s not, i n g e n e r a l , r i s k l e s s .  I n c r e a s i n g t h e l e v e l o f debt i n c r e a s e s t h e  charges and t h e p r o b a b i l i t y t h a t the f i r m w i l l  " M o d i g l i a n i , F . and M i l l e r , M. and t h e Theory o f Investment," American (June, 1 9 5 8 ) , pp. 2 6 1 - 2 9 7 .  fixed  n o t be a b l e t o meet i t s f i n a n c i a l  "The C o s t o f C a p i t a l , C o r p o r a t e F i n a n c e , Economic Review, V o l . X L V I I I , No. 3 ,  4 M o d i g l i a n i , F . and M i l l e r , M. " C o r p o r a t e Income Taxes and the C o s t of C a p i t a l : A C o r r e c t i o n , " American Economic Review, V o l . L I I I (June, 1 9 6 3 ) , pp. 4 3 3 - 4 4 3 .  3  obligations. be  T h e r e f o r e , a second market i m p e r f e c t i o n , b a n k r u p t c y ,  incorporated.  must a l s o  By assuming t h e r e a r e no a s s o c i a t e d p e n a l t i e s or  resource 5  c o s t s t o bankruptcy  and t h a t i n v e s t o r s have l i m i t e d l i a b i l i t y ,  shown t h a t b a n k r u p t c y of  has  In t h e  case  has no e f f e c t upon t h e v a l u e o f t h e f i r m .  no c o r p o r a t e t a x , c a p i t a l s t r u c t u r e i s i r r e l e v a n t ,  porate  Stiglitz  taxes, i t i s s t i l l  and  i f there are  advantageous f o r the f i r m t o use  Bankruptcy i s a " t e c h n i c a l i t y ; " t h e b a n k r u p t f i r m can be  cor-  as much as p o s s i b l e .  r e p l a c e d by a  new  f i r m and s i n c e no r e s o u r c e s have been expended t h e r e a r e no economic l o s s e s . The  a s s e t s o f the bankrupt f i r m a r e s t i l l  t h a t t h e management ( c o n t r o l ) and group o f p e o p l e holders).  intact;  ownership a r e i n t h e hands o f a d i f f e r e n t  (namely, the d e b t h o l d e r s i n s t e a d o f the o r i g i n a l  equity  7  W i t h i n t h e same framework S t i l g l i t z  has  assumption o f homogeneity o f e x p e c t a t i o n s and has al  the o n l y d i f f e r e n c e i s  assumption t h a t s h o r t s e l l i n g  attempted t o r e l a x t h e shown, g i v e n t h e a d d i t i o n -  i s not allowed, t h a t bankruptcy  does  t h e v a l u e o f t h e f i r m , even though t h e r e a r e no r e s o u r c e c o s t s t o However, i f the assumption o f no  short s e l l i n g  American  S t i g l i t z , J . E. Economic Review,  bankruptcy.  i s r e l a x e d , i t can be shown  that S t i g l i t z ' s conclusion i s , i n general, i n v a l i d .  5  affect  Short s e l l i n g  allows  the  "A R e - E x a m i n a t i o n o f t h e M o d i g l i a n i M i l l e r Theorem," V o l . LIX, No. 5 (December, 1969), pp. 784-793.  F o r a f u r t h e r d i s c u s s i o n see M o d i g l i a n i , F. and M i l l e r , M., "Reply to Heins and S p r e n k l e , " American Economic Review, V o l . , L I X , No. 4 (September, 1969), pp. 592-595. 6  Debt has a f i x e d s e t o f c l a i m s on the f i r m p l u s the r i g h t t o o v e r " t h e f i r m i n t h e event t h e f i r m does n o t meet i t s o b l i g a t i o n s .  "take  S t i g l i t z , J . E. "Some A s p e c t s o f the Pure Theory o f C o r p o r a t e F i n a n c e : Bankruptcy and Take-Over," Bell Journal of Economics and Management Science, Vol. 3, No. 2 (Autumn, 1972), pp. 450-402.  4  i n v e s t o r the a b i l i t y t o r e p l i c a t e a c r o s s a l l s t a t e s o f n a t u r e t a i n e d by i n v e s t i n g i n an u n l e v e r e d  the income  f i r m , g i v e n the assumptions o f  ob-  limited  liability. In r e a l i t y , t h e r e a r e c o s t s a s s o c i a t e d w i t h f i n a n c i a l d i s t r e s s . t h e event o f bankruptcy  r e a l r e s o u r c e s a r e consumed.  proceedings i n v o l v e c o s t s — t u r n o v e r o f employees due  That  In  i s , bankruptcy  legal fees, trustee fees, administration fees, t o u n c e r t a i n t y , a n d l o s s o f customers due  t a i n t y as t o whether the f i r m w i l l be a b l e t o f u l f i l l  contracts.  to  uncer-  Kraus  and  9  Litzenburger  have f o r m a l l y i n t r o d u c e d t h e tax advantage o f d e b t  and  banki  r u p t c y p e n a l t i e s i n t o a s t a t e p r e f e r e n c e framework.  They have shown t h a t  t h e market v a l u e o f a l e v e r e d f i r m i s e q u a l t o t h e u n l e v e r e d market v a l u e , p l u s t h e c o r p o r a t e tax r a t e times t h e market v a l u e o f the f i r m ' s debt, the complement o f the c o r p o r a t e tax r a t e t i m e s t h e p r e s e n t v a l u e o f costs.  less  bankruptcy  Thus t h e r e i s a t r a d e - o f f between t h e e f f e c t s o f t h e two market  fections:  c o r p o r a t e t a x and  c o s t s t o bankruptcy,  i m p l y i n g an o p t i m a l  imper-  capital  structure. Under t h e more g e n e r a l c o n c e p t  of f i n a n c i a l d i s t r e s s ,  i t has  been  argued t h a t t h e r e a r e a d d i t i o n a l c o s t s a s s o c i a t e d w i t h changes i n f i n a n c i n g and  investment  strategies. ^ 1  I f t h e r e i s a r e a l d i s t i n c t p o s s i b i l i t y o f bank-  r u p t c y , p o t e n t i a l i n v e s t o r s w i l l demand a premium t o compensate f o r the a s s o c i a t e d w i t h bankruptcy.  The  l e v e r e d f i r m m i g h t be  risks  i n a poor b a r g a i n i n g  p o s i t i o n and have t o o f f e r h i g h e r r e t u r n s t o s u p p l i e r s o f a d d i t i o n a l  capital.  Kraus, A. and L i t z e n b e r g , R. "A S t a t e - P r e f e r e n c e Model o f O p t i m a l F i n a n c i a l Leverage," Journal of Finance, V o l . XXVIII, No. 3 (September, 1973),  pp. 911-922.  tal  Robichek, A. and Myers, S. S t r u c t u r e , " Journal of Financial  1966), pp. 1-35.  "Problems i n the Theory o f O p t i m a l C a p i Analysis, V o l . 1 (June,  and Quantitative  5  The very  underwriting  c o s t s o f a common s t o c k i s s u e i n such a s i t u a t i o n  l a r g e , i f not p r o h i b i t i v e .  The e x i s t e n c e  might be  o f c r e d i t r a t i o n i n g o r con-  s t r a i n t s on t h e i n v e s t m e n t p o l i c i e s o f many i n s t i t u t i o n s might f u r t h e r weaken t h e b a r g a i n i n g p o s i t i o n o f t h e f i r m by r e d u c i n g p o t e n t i a l s u p p l i e r s o f funds.  and i n v e s t m e n t s t r a t e g i e s t h u s  This implies the existence of a d d i t i o n a l  market i m p e r f e c t i o n s b e s i d e s t h a t o f b a n k r u p t c y . should,  I n a p e r f e c t market a f i r m  a t a p r i c e , be a b l e t o s a t i s f y i t s c a p i t a l  are non-price then these  restrictions inhibiting  a r e market  among  Loan c o n t r a c t s u n d e r t a k e n by t h e f i r m m i g h t  impose c o n s t r a i n t s on t h e f i r m ' s f i n a n c i a l l i m i t i n g i t s freedom o f a c t i o n .  competition  requirements.  I f there  the firm's a b i l i t y to r a i s e  capital,  imperfections.  Hypothesis In summary, t h e p o t e n t i a l c o s t s a s s o c i a t e d w i t h a r e thought t o be a n i m p o r t a n t  f a c t o r i n determining  Assuming t h a t i n v e s t o r s a r e r i s k a v e r s e , bear the r i s k s a s s o c i a t e d with  Unfortunately,  The d i f f i c u l t y  a concept l i k e f i n a n c i a l d i s t r e s s , which should  a continuum, p r e c l u d e s  i t s use i n t h i s t h e s i s .  o f mathematibe t r e a t e d as  C o n s e q u e n t l y , t h e extreme  s i t u a t i o n , bankruptcy, i s the o p e r a t i o n a l concept used.  I t i s t h e sharp  t i n c t i o n between f a i l u r e and n o n - f a i l u r e t h a t a v a i l s i t s e l f matical analysis.  an i n d e x  What i s a v a i l a b l e , however, i s t h e  c l a s s i f i c a t i o n o f f i r m s as f a i l e d o r n o n - f a i l e d . l a l l y modelling  the value o f a f i r m .  t h e y w i l l demand a r i s k premium t o  financial distress.  of financial d i s t r e s s i s not available.  financial distress  dis-  r e a d i l y t o mathe-  6  The hypothesis o f the t h e s i s i s that d i f f e r e n c e s i n the p r o b a b i l i t y of bankruptcy across s e c u r i t i e s and across time are r e f l e c t e d i n the r e s i d ual return a f t e r abstracting from the market. The e x p l i c i t o b j e c t i v e s o f the t h e s i s are; 1, To analyse t h e o r e t i c a l l y how t h e mechanism of bankruptcy a f f e c t s the s t r u c t u r e o f returns f o r corporate f i n a n c i a l assets. 2, To quantify the determinants o f bankruptcy} to a r r i v e a t a £igure which can be i d e n t i f i e d as the p r o b a b i l i t y o f bankruptcy.  thesis.  3,  To t e s t e m p i r i c a l l y the hypothesis o f t h e  From the t h e o r e t i c a l a n a l y s i s a two v a r i a b l e rooaiel i s derived, t h e second v a r i a b l e being associates "with the p r o b a b i l i t y o f banltomptcy.  Tnhe  a b i l i t y t o measnare the p r o b a b i l i t y o f bankruptcy implies that the hypothesis of the t h e s i s , as represented by t h e two v a r i a b l e model, can be e m p i r i c a l l y tested.  Importance The e m p i r i c a l work o f B e a v e r ^ and W e s t e r f i e l d ^ has o f f e r e d same evidence vhich suggests that impending bankruptcy does appear t o a f f e c t the structure o f returns on common stocks.  The behaviownr o f ex-post rettorns,  a f t e r abstracting from the market, f o r 'common stocks o f firms that event u a l l y went bankrupt, are s i g n i f i c a n t l y d i f f e r e n t from those o f healthy  11 Beaver, W. H,, "Market P r i c e s , F i n a n c i a l R a t i o s , tion of Failure,"  Journal  of Accounting  Research  a  and t h e P r e d i c  V o l . 4 '(Autumn, 19&8J,  pp. 179-192. 12 W c s f c e r f i e l d , R., "The Assessment o f M a r k e t R i s k and C o r p o r a t e F a i l u r e , " U n i v e r s i t y o f P e n n s y l v a n i a , Wharton S c h o o l o f F i n a n c e a n d Commerce, August 1970 { u n p u b l i s h e d ) .  7  f i r m s f o r t h e same time p e r i o d .  These f i n d i n g s , i f c o r r e c t , have  important  i m p l i c a t i o n s a s t o t h e s i g n i f i c a n c e o f t h e e f f e c t s o f b a n k r u p t c y upon t h e  13 s t r u c t u r e o f common s t o c k  returns.  Fisher,  i n an e m p i r i c a l  study, a d -  vanced t h e h y p o t h e s i s t h a t t h e r i s k o f d e f a u l t and m a r k e t a b i l i t y a f f e c t the r i s k premium on c o r p o r a t e  bonds.  A l l o f these e m p i r i c a l  studies  suffer  from t h e l a c k o f any t h e o r e t i c framework w i t h i n which t o i n v e s t i g a t e how the mechanism o f b a n k r u p t c y a f f e c t s t h e s t r u c t u r e o f r e t u r n s financial  on  corporate  assets.  A number o f r e c e n t  s t u d i e s have c o n c l u d e d t h a t t h e C a p i t a l  Asset  14 P r i c i n g Model  (CAPM) i s m i s s p e c i f i e d .  term i s n o n - s t a t i o n a r y , systematic  I t i s found t h a t t h e i n t e r c e p t  c o n s i s t e n t l y negative  f o r s e c u r i t i e s with  r i s k and p o s i t i v e f o r s e c u r i t i e s w i t h low s y s t e m a t i c  high  risk.  The  l a c k o f e m p i r i c a l f i t can be a t t r i b u t e d t o t h e f a c t t h a t e i t h e r t h e model i s c o r r e c t and t h e d i f f i c u l t y  i s one o f measurement, o r t h a t t h e model i s i n -  c o r r e c t and must be extended t o i n c l u d e a d d i t i o n a l v a r i a b l e s . In t h e f i r s t  c a s e , measurement e r r o r s may r e s u l t e i t h e r i n making  the t r a n s i t i o n from a n ex-ante t o an e x - p o s t f o r m u l a t i o n errors i nvariables. formulation  The t r a n s f o r m a t i o n  Journal  Fisher,  of Political  on any s e c u r i t y  by a market model; t h a t i s , t h e r e t u r n on a s e c u r i t y i s  a l i n e a r f u n c t i o n o f a market f a c t o r . A J  o f t h e e x - a n t e model t o an e x - p o s t  i s based upon t h e assumption t h a t t h e r e t u r n s  can be r e p r e s e n t e d  o r because o f  Thus any t e s t o f t h e e x - p o s t formu-  L. "Determinants o f R i s k Premiums on C o r p o r a t e Bonds," Economy, V o l . L X V I I , No. 3 (June, 1959), pp. 217-237.  14  Some r e c e n t s t u d i e s a r e B l a c k , F., J e n s e n , M. C. and S c h o l e s , M., C a p i t a l A s s e t P r i c i n g Model: Some E m p i r i c a l T e s t s , " p u b l i s h e d i n Studies in the Theory of Capital Markets, e d i t e d by J e n s e n , M. (New York: P r a e g e r , 1972); and Blume, M. and F r i e n d , I . , "A New Look a t t h e C a p i t a l A s s e t T r i c i n g Model," Journal of Finance, V o l . XXVII, No. 1 (March, 1973), pp. 19-34. "The  8  l a t i o n i s a j o i n t t e s t o f t h e CAPM and market model.  Errors i n variables  might a r i s e t h r o u g h measurement e r r o r s i n t h e e s t i m a t i o n o f t h e i n d i v i d u a l b e t a f a c t o r s o r i f t h e market  factor i s incorrectly  specified;  t h e market  f a c t o r i s supposed t o measure t h e r e t u r n on a l l a s s e t s and n o t s i m p l y t h e r e t u r n o n t h e New York Stock Exchange.  A n o t h e r problem which might  vilify  t h e r e s u l t s o f any i n v e s t i g a t i o n i s t h a t o f t h e skewness o f t h e d i s t r i b u t i o n s of ex-post returns. In t h e second c a s e , a number o f s t u d i e s have attempted t o r e l a x t h e v a r i o u s assumptions u n d e r l y i n g t h e model. d e v e l o p e d b y B l a c k ^ and M e r t o n . ^ 1  Two v a r i a b l e models have been  The second v a r i a b l e i n t h e B l a c k v e r s i o n ,  which h a s been termed t h e z e r o b e t a f a c t o r , a r i s e s from r e l a x i n g t h e assumption o f i n v e s t o r s b e i n g a b l e t o borrow and l e n d a t t h e r i s k f r e e r a t e o f i n t e r e s t . The  second v a r i a b l e i n Merton's model i s t h e r e s u l t o f r e l a x i n g t h e assump-  t i o n o f a c o n s t a n t investment o p p o r t u n i t y s e t and r e f l e c t s i n v e s t o r s ' t o hedge a g a i n s t s u c h changes.  N e i t h e r model p r o v i d e s an adequate  of a l l t h e o b s e r v e d d e f i c i e n c i e s o f t h e CAPM.  attempts  explanation  There i s n o t h i n g i n t h e B l a c k  f o r m u l a t i o n t o suggest t h a t t h e second f a c t o r i s n o n - s t a t i o n a r y , and t h e Merton model does n o t e x p l a i n why impending b a n k r u p t c y a f f e c t s t h e r e s i d u a l r e t u r n o f common s t o c k s a f t e r a b s t r a c t i n g from t h e market.  B l a c k , F . " C a p i t a l Market E q u i l i b r i u m With R e s t r i c t e d of Business, V o l . 4 5 , No. 3 ( J u l y , 1 9 7 2 ) , pp. 4 4 4 - 4 5 5 . i : >  Journal  Borrowing,"  16  Merton, R. C. "A Dynamic G e n e r a l E q u i l i b r i u m Model o f t h e A s s e t Market and i t s A p p l i c a t i o n t o t h e P r i c i n g o f t h e C a p i t a l S t r u c t u r e o f t h e F i r m , " M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , S l o a n S c h o o l o f Management, December, 1 9 7 0 .  9  The the  p r i m a r y f o c u s o f t h e t h e s i s i s t o extend the f o r m u l a t i o n s o f  CAPM n o t from t h e v i e w p o i n t o f r e s t r i c t i o n s upon t h e i n v e s t o r ,  considering  t h e impact o f b a n k r u p t c y upon t h e s t r u c t u r e  v a r i a b l e model i s d e r i v e d , p r o b a b i l i t y o f bankruptcy.  of returns.  t h e second v a r i a b l e b e i n g a s s o c i a t e d  b u t by A two  with the  An e s s e n t i a l s t e p t o e m p i r i c a l l y t e s t i n g such  a model i s t h e development o f an o p e r a t i o n a l  measure o f t h e p r o b a b i l i t y  o f bankruptcy f o r a f i r m . Many o f t h e s t u d i e s d e a l i n g  with the p r e d i c t i o n o f bankruptcy  17 have c o n c e n t r a t e d upon t h e i n f o r m a t i o n a l The  c o n t e n t o f a c c o u n t i n g numbers.  hypothesis being that there i s a d i f f e r e n c e  a c c o u n t i n g d a t a , between f a i l e d and n o n - f a i l e d differences discriminant firms  e x i s t , models c a n be c o n s t r u c t e d ,  firms usually  a s measured by  and t h a t g i v e n employing  these  multiple  a n a l y s i s , t o d e t e r m i n e i f a f i r m s h o u l d b e l o n g t o a group o f  having t h e c h a r a c t e r i s t i c s o f a f a i l e d  having the c h a r a c t e r i s t i c s o f a n o n - f a i l e d has  i nprofile,  f i r m o r t o a group o f f i r m s  firm.  The main  been t o c l a s s i f y a f i r m i n t o one o f t h e s e two g r o u p s .  been made t o c o n s t r u c t  consideration No attempt has  a t h e o r y o f t h e d e t e r m i n a n t s o f b a n k r u p t c y o r t o mea-  sure the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t . The studies  approach proposed i n the t h e s i s  i n a t l e a s t two ways.  It identifies  i s an e x t e n s i o n o f t h e p r e v i o u s , a set of variables  t h a t c a n be  """'See, f o r example, Beaver, W. H. " F i n a n c i a l R a t i o s as P r e d i c t o r s o f F a i l u r e , " Empirical Research in Accounting: Selected Studies, supplement t o Journal of Accounting Research (1966), pp. 77-111; Altman, E. I . " F i n a n c i a l R a t i o s , D i s c r i m i n a n t A n a l y s i s and t h e P r e d i c t i o n o f C o r p o r a t e B a n k r u p t c y , " Journal of Finance, V o l . X X I I I , No. 4 (September, 1 9 6 8 ) , pp. 589-609; and Edmister, R. 0. "An E m p i r i c a l T e s t o f F i n a n c i a l R a t i o A n a l y s i s f o r S m a l l B u s i n e s s F a i l u r e P r e d i c t i o n , " Journal of Financial and Quantitative Analysis, V o l . 7 (March, 1 9 7 2 ) , pp. 1477-1493.  10  used i n p r e d i c t i n g b a n k r u p t c y and i t i n t r o d u c e s  a new methodology t o t h i s  f i e l d w i t h which t o measure t h e p r o b a b i l i t y o f b a n k r u p t c y , t h a t o f l o g i t a n a l y s i s and p r o b i t To the  analysis.  e m p i r i c a l l y t e s t the h y p o t h e s i s o f t h e t h e s i s , as r e p r e s e n t e d by  derived  two v a r i a b l e model, n e c e s s i t a t e s  o l o g y t o be employed.  consideration  The two v a r i a b l e model d e s c r i b e s  o f t h e method-  a relationship for  a s e c u r i t y a t a s p e c i f i c time between i t s c o n d i t i o n a l e x p e c t e d r e t u r n , t h e s e c u r i t y ' s s y s t e m a t i c r i s k , and the. p r o b a b i l i t y o f b a n k r u p t c y . t i o n a l approach t o t h e t e s t i n g o f t h e CAPM i s t o c o n s t r u c t s e c u r i t i e s so as t o reduce measurement e r r o r s small  number o f p o r t f o l i o s .  For empirical  s i o n s o f t h e CAPM, e l a b o r a t e have been u t i l i z e d .  The t r a d i -  portfolios of  and t o t e s t the model u s i n g  a  t e s t i n g o f models which a r e e x t e n -  forms o f t h e same t y p e o f a g g r e g a t i o n p r o c e d u r e  However, t h e s e p r o c e d u r e s make e x t e n s i v e use o f e s t i -  mated p a r a m e t e r s and i t i s n o t c l e a r how t h e a g g r e g a t e d e f f e c t o f measurement e r r o r s The  i n these parameters a f f e c t s the f i n a l estimated c o e f f i c i e n t s .  thesis introduces  a new methodology t o t h e t e s t i n g o f two v a r i a b l e  models, t h a t o f p o o l i n g  Organization  time s e r i e s and c r o s s  survey o f the r e l e v a n t  f i n a n c i a l l i t e r a t u r e and i t s impor-  tance t o t h e t h e s i s i s g i v e n i n C h a p t e r I I . topics of empirical  studies  the u n d e r l y i n g  of d e f i c i e n c y  assumptions o f t h e r e v i e w e d l i t e r a t u r e ,  and t h e c o n t r i b u t i o n s  terms o f ex-ante v a r i a b l e s . that  The s u r v e y , which c o v e r s t h e  i n b a n k r u p t c y and t h e C a p i t a l A s s e t P r i c i n g Model,  Chapter I I I d e s c r i b e s  requires  data.  of Thesis  A critical  stresses  section  the a r e a s  o f the t h e s i s .  a model f o r t h e p r o b a b i l i t y o f b a n k r u p t c y i n To use t h e model f o r e m p i r i c a l  the ex-ante v a r i a b l e s be r e p l a c e d  estimation  by e x - p o s t s u r r o g a t e s .  A  11  g e n e r a l f o r m u l a t i o n i n terms o f e x p l a n a t o r y primary  v a r i a b l e s i s developed.  f o c u s i s upon p r e d i c t i o n o f b a n k r u p t c y , a second f o r m u l a t i o n  market v a l u e s o f a p p r o p r i a t e c o r p o r a t e v a r i a b l e s i s c o n s t r u c t e d . t i c a l methodology to e s t i m a t e described. be  The  the C a p i t a l  o p p o r t u n i t y s e t , and  t i o n s f o r the i n d i v i d u a l a r e d e s c r i b e d . o f o p t i m a l i t y i s then d e r i v e d . models a r e p r e s e n t e d .  The  Due  The  general  The  framework  g e n e r a l form o f the  t o the c o m p l e x i t y  the e v e n t  nature  equation  o f the g e n e r a l a n a l y s i s ,  of bankruptcy.  investment From  this  v a r i a b l e model, w h i c h w i l l be e m p i r i c a l l y t e s t e d , i s d e r i v e d .  second model assumes t h a t t h e p r o b a b i l i t y o f b a n k r u p t c y changes  t i c a l l y over  can  t h e b e h a v i o r i a l assump-  f i r s t model assumes t h a t the  o p p o r t u n i t y s e t i s a l t e r e d o n l y by a n a l y s i s a two  statis-  Asset  f i n a n c i a l a s s e t s a v a i l a b l e , t h e i r p r i c e dynamics, the  o f changes i n the i n v e s t m e n t  The  The  chapter.  P r i c i n g Model t o i n c o r p o r a t e b a n k r u p t c y i s g i v e n . the  using  d e t a i l s o f t h r e e d i f f e r e n t methods by which the model  t e s t e d a r e g i v e n i n t h e l a s t p a r t o f the  detailing  the  the c o e f f i c i e n t s o f the p r o x y v a r i a b l e s i s  In C h a p t e r IV the t h e o r e t i c a n a l y s i s e x t e n d i n g  two  As  stochas-  time.  C h a p t e r V d e s c r i b e s t h e e m p i r i c a l work o f the t h e s i s . p a r t o f the c h a p t e r d e s c r i b e s the d a t a , the r e s u l t s , and model t o e s t i m a t e  the p r o b a b i l i t y o f b a n k r u p t c y .  The  The  first  the t e s t i n g o f  r e s u l t s o f t h i s work  are used i n the second p a r t o f the c h a p t e r , w h i c h d e s c r i b e s the t e s t i n g the h y p o t h e s i s r e s u l t s are  o f the t h e s i s .  The  the  d a t a , s t a t i s t i c a l methodology, and  of  the  presented.  C h a p t e r VI summarizes the main f i n d i n g s o f the topics for further research that a r i s e  thesis.  from the t h e s i s i s g i v e n .  A list  of  CHAPTER I I A CRITICAL SURVEY OF THE RELEVANT FINANCIAL LITERATURE  In t h i s c h a p t e r a r e v i e w o f t h e f i n a n c i a l l i t e r a t u r e t h a t i s r e l e vant t o the t h e s i s i s given.  The r e l a t i o n s h i p o f the reviewed  t o t h e t h e s i s , i t s importance, t h e u n d e r l y i n g cations are described.  literature  assumptions and t h e i r i m p l i -  I t i s demonstrated t h a t t h e r e  are areas o f d e f i c i e n c y  i n the l i t e r a t u r e and t h e c o n t r i b u t i o n s o f the t h e s i s i n e r a d i c a t i n g t h e s e deficiencies stated.  Two t o p i c s a r e d i s c u s s e d ,  on b a n k r u p t c y and t h e C a p i t a l A s s e t The questions:  P r i c i n g Model  (CAPM).  are there  c o s t s t o b a n k r u p t c y ; does impending b a n k r u p t c y  The q u e s t i o n s  o f the existence  affect  o f c o s t s t o b a n k r u p t c y and t h e  a s s e t s a r e o f prime importance t o t h e f o r m u l a t i o n The l a s t q u e s t i o n ,  three  f i n a n c i a l a s s e t s ; and can b a n k r u p t c y be  e f f e c t s o f b a n k r u p t c y upon t h e s t r u c t u r e o f r e t u r n s  thesis.  studies  e m p i r i c a l s t u d i e s on b a n k r u p t c y t h a t a r e r e v i e w e d a d d r e s s  the p r i c e b e h a v i o u r o f c o r p o r a t e predicted.  those o f e m p i r i c a l  f o r corporate  financial  o f the hypothesis o f the  that o f p r e d i c t i o n o f bankruptcy, i s r e l e v a n t  t o t h e t e s t i n g o f t h e h y p o t h e s i s o f the t h e s i s , where i t i s n e c e s s a r y t o measure the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t . The  second t o p i c reviewed i s t h e CAPM.  The b a s i c model and t h e  e m p i r i c a l evidence i n d i c a t i n g that i t i s m i s s p e c i f i e d i s d e s c r i b e d .  The  i m p l i c a t i o n s o f t h e m i s s p e c i f i c a t i o n and some o f t h e t h e o r e t i c a l attempts t o e x p l a i n i t s causes a r e r e p o r t e d .  Incorporating  some o f t h e f i n d i n g s o f the  e m p i r i c a l s t u d i e s on b a n k r u p t c y and the n a t u r e o f the m i s s p e c i f i c a t i o n o f the CAPM, t h e h y p o t h e s i s o f the t h e s i s i s p r e s e n t e d .  12  To t e s t e m p i r i c a l l y  13  the  h y p o t h e s i s a two v a r i a b l e model i s used, t h e second v a r i a b l e b e i n g t h e  p r o b a b i l i t y o f a f i r m going bankrupt. it  i s n e c e s s a r y t o determine  B e f o r e t h e h y p o t h e s i s can be t e s t e d ,  the s t a t i s t i c a l  methodology t o u s e .  Existing  methods and t h e i r d e f i c i e n c i e s a r e d e s c r i b e d and an a l t e r n a t i v e methodology advanced,  t h a t o f p o o l i n g time s e r i e s and c r o s s s e c t i o n d a t a .  E m p i r i c a l S t u d i e s on B a n k r u p t c y The e m p i r i c a l s t u d i e s on b a n k r u p t c y which a r e r e v i e w e d p e r t a i n t o t h r e e a r e a s : t h e e x i s t e n c e o f c o s t s t o b a n k r u p t c y , t h e e f f e c t s o f impending b a n k r u p t c y upon t h e p r i c e b e h a v i o u r o f c o r p o r a t e f i n a n c i a l a s s e t s , and t h e p r e d i c t i o n o f bankruptcy.  The f i r s t  two t o p i c s combined w i t h some o f t h e  f i n d i n g s from t h e e m p i r i c a l e v i d e n c e on t h e m i s s p e c i f i c a t i o n o f t h e CAPM c o n t r i b u t e t o the f o r m u l a t i o n o f the hypothesis o f the t h e s i s .  The l a s t  t o p i c , t h a t o f p r e d i c t i o n o f bankruptcy, i s r e l e v a n t t o the e m p i r i c a l of  testing  t h e h y p o t h e s i s o f t h e t h e s i s , where i t i s n e c e s s a r y t o measure t h e p r o b a -  b i l i t y o f a f i r m going bankrupt.  Costs t o Bankruptcy I f t h e r e were no c o s t s t o b a n k r u p t c y t h e n i t would be a " t e c h n i c a l i t y " ; the  bankrupt  f i r m c o u l d be r e p l a c e d by a new f i r m and s i n c e no r e s o u r c e s have  been expended t h e r e a r e no economic l o s s e s .  However, i f t h e r e a r e c o s t s t o  b a n k r u p t c y then t h i s i s no l o n g e r t r u e and i t i m p l i e s t h a t c a p i t a l  structure  w i l l be r e l e v a n t t o t h e v a l u a t i o n o f t h e f i r m . Baxter  Journal  1  has c o n s i d e r e d t h e e x i s t e n c e o f c o s t s t o c o r p o r a t e b a n k r u p t c y  B a x t e r , N. D., "Leverage, R i s k o f R u i n , and the C o s t o f C a p i t a l , " of Finance, V o l . XXII, No. 3 (September, 1967), pp. 395-403.  14  and  t h e i r e f f e c t s upon c a p i t a l c o s t s .  His  thesis i s that increased  enhances the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t and t r a t i o n and  legal costs  incurred during  i n e f f i c i e n c i e s which are m a n i f e s t due the v a l u e on  o f the  firm.  the e a r n i n g s and  were a d v e r s e l y  t h a t the  reorganization,  added  as w e l l as  the  s a l e s o f a s m a l l sample o f f i r m s and  found t h a t  a r e c o s t s t o b a n k r u p t c y and  that excessive  would reduce the v a l u e  o f the  o f o n l y the  Consideration  e f f e c t s o f b a n k r u p t c y can be p o t e n t i a l l y m i s l e a d i n g  discounting  for t h e i r opportunity  t o do  was leverage  immediate  for i t i s possible  or a f t e r r e o r g a n i z a t i o n  f o r t h e o r i g i n a l common s t o c k h o l d e r s  they  This  i n t e r p r e t e d that there  e i t h e r during  reduces  immediate e f f e c t s o f b a n k r u p t c y  a f f e c t e d a f t e r the d e c l a r a t i o n o f b a n k r u p t c y .  a firm's operations  adminis-  operating  t o the b a n k r u p t c y c o n d i t i o n ,  Baxter analyzed  firm.  leverage  exceedingly  t o improve  for  and  well, inclusive of  costs.  2  Altman, d i a t e but  i n a more g e n e r a l  a l s o the  a n a l y s i s , considered  l o n g term e f f e c t s o f b a n k r u p t c y .  were c o n f i r m e d , though some e v i d e n c e was  not o n l y  The  findings of  found t h a t p r o h i b i t e d the  statement t h a t i n v e s t o r s always s u f f e r because o f b a n k r u p t c y . d a t a showed t h a t b a n k r u p t f i r m s ' e q u i t y on  average can be  The  immeBaxter  general empirical  expected to  i n b a n k r u p t c y , though a number o f b a n k r u p t c y r e o r g a n i z a t i o n s favorable  the  fall  resulted i n  o v e r a l l performance.  The  e v i d e n c e o f the e x i s t e n c e  o f the  costs  to bankruptcy  t h a t c a p i t a l s t r u c t u r e w i l l a f f e c t the v a l u a t i o n o f the l a t e d t o t h i s i s the q u e s t i o n the s t r u c t u r e o f r e t u r n s on  o f how  corporate  firm.  implies  Closely  re-  b a n k r u p t c y a f f e c t s the mechanism o f financial  assets.  2  and pp.  Altman, E. I . , "Corporate Bankruptcy P o t e n t x a l , Share V a l u a t i o n , " Journal of Finance, V o l . XXIV, No. 887-900.  S t o c k h o l d e r Returns 5 (December, 19G9),  15  Bankruptcy and Stock Market P r i c e s The  greater  the p r o b a b i l i t y o f a f i r m going bankrupt the g r e a t e r  w i l l be t h e ex-ante e x p e c t e d r a t e o f r e t u r n t h a t r i s k - a v e r s e quire.. adjust  Each p e r i o d  investors w i l l  investors r e -  reassess  t h e c o n d i t i o n o f t h e f i r m and  t h e market p r i c e o f t h e common s t o c k  such t h a t t h e e x - a n t e r a t e o f  r e t u r n would c o n t i n u e t o be commensurate w i t h t h e h i g h e r  risk.  time t h e p r o b a b i l i t y o f t h e f i r m g o i n g b a n k r u p t i s g r e a t e r  I f a t any  than expected,  t h e r e w i l l be a downward adjustment o f t h e market p r i c e and t h e e x - p o s t r e t u r n w i l l be l e s s t h a n t h e ex-ante e x p e c t e d r a t e o f r e t u r n .  I t i s not  p o s s i b l e t o make any statement about t h e d i f f e r e n c e i n e x - p o s t r e t u r n s f o r healthy  f i r m s and f o r those f i r m s t h a t f a i l .  The d i r e c t i o n and magnitude o f  any d i f f e r e n c e w i l l depend upon t h e s i z e and d i r e c t i o n o f change i n t h e p r o b a b i l i t y o f the f i r m going bankrupt. Beaver^ considered  what e f f e c t s impending f a i l u r e had upon t h e p r i c e  b e h a v i o u r o f common s t o c k p r i c e s .  A c r u d e market model was u s e d .  The  r e s i d u a l between t h e e x - p o s t r e t u r n and t h e comparable F i s h e r L i n k  Relative,  which i s an average r a t e o f r e t u r n on a l l f i r m s l i s t e d on t h e New York S t o c k Exchange, was c a l c u l a t e d on an a n n u a l b a s i s f o r a group o f f a i l e d  f i r m s up  t o f i v e y e a r s p r i o r t o f a i l u r e and f o r a group o f n o n - f a i l e d f i r m s . s u l t s were t h a t t h e median r a t e s o f r e t u r n f o r f a i l e d than those o f n o n - f a i l e d f i r m s  f i r m s were p o o r e r  f o r f i v e y e a r s p r i o r t o a c t u a l f a i l u r e and  the d i f f e r e n c e between t h e median v a l u e s Beaver a l s o d e r i v e d  The r e -  increased  as f a i l u r e  approached.  t e s t s t o a s s e s s t h e f a i l u r e p r e d i c t i v e power o f r a t e s o f  r e t u r n measures and s e v e r a l f i n a n c i a l r a t i o s t h a t had been i n a  previous  3  Beaver, W. H., "Market P r i c e s , F i n a n c i a l R a t i o s , and t h e P r e d i c t i o n o f F a i l u r e , " Journal of Accounting Research, V o l . A (Autumn, 1 9 6 8 ) , pp.  179-192.  16  study.  Univariate  t e s t s showed t h a t i n v e s t o r s f o r e c a s t f a i l u r e  any  o f the r a t i o s used, w i t h the  the  f a i l u r e p r e d i c t i o n t o the d a t e o f f a i l u r e b e i n g  o f r e t u r n measure. f a c t t h a t the  T h i s was  informational  source i n v e s t o r s use  average l e n g t h o f time from the y e a r  i n t e r p r e t e d by  2.45  the p o s s i b i l i t y o f  of  y e a r s f o r the  Beaver as r e c o g n i t i o n o f  content of accounting  to detect  sooner than  numbers i s not  rate  the  the  only  failure.  4  Westerfield,  u s i n g monthly d a t a and  the market model d e v e l o p e d  Sharpe^ examined the b e h a v i o u r o f r e s i d u a l r e t u r n s , a f t e r a b s t r a c t i n g the market, f o r f a i l e d f i r m s up  to s i x years p r i o r to f a i l u r e .  The  t o 72 p r i o r t o f a i l u r e .  For  from  market  model parameters f o r the i n d i v i d u a l s e c u r i t i e s were e s t i m a t e d u s i n g f o r the months 120  the months 71 t o 0  data the  r e s i d u a l o f the r e a l i z e d r e t u r n minus the ex-ante e x p e c t e d r e t u r n were culated.  U s i n g two  performance measures, i t was  by  cal-  found t h a t the market began  t o b i d down the market p r i c e f i v e y e a r s p r i o r t o f a i l u r e , w i t h a r a p i d d e t e r i o r a t i o n o c c u r r i n g i n the y e a r subsequent t o f a i l u r e . t r a s t t o the t h a n one  f i n d i n g s o f Beaver and  e r r o r prone.  b a s e d upon i n d i c e s o f market  W e s t e r f i e l d a l s o examined the  s h i p between the  systematic  hypothesis being  t h a t f i r m s whose common e q u i t y e x h i b i t h i g h  r i s k measure and  the  rate of f a i l u r e ,  r i s k w i t h market movements ( h i g h b e t a s ) e x p e r i e n c e a h i g h e r than those a s s e s s e d as low  risk  i s i n con-  s u g g e s t s t h a t f o r e c a s t s o f f a i l u r e more  y e a r p r i o r t o when f a i l u r e o c c u r s ,  performance, w i l l be  This  (low b e t a s ) .  G i v e n the  relationthe  systematic rate of  failure  l i m i t a t i o n s of  the  4  W e s t e r f i e l d , R., "The Assessment o f Market R i s k and C o r p o r a t e F a i l u r e , " U n i v e r s i t y o f P e n n s y l v a n i a , Wharton S c h o o l o f F i n a n c e , August, 1970 (unpublished). 5  S h a r p e , W. , "A S i m p l i f i e d Model f o r P o r t f o l i o A n a l y s i s , " No. 9 (January, 1963), pp. 277-293.  ment Science,  Manage-  17  estimation  t e c h n i q u e and  average h i g h  risk  the  small  sample s i z e ,  the  r e s u l t s showed t h a t  firms experience f a i l u r e at a greater  r a t e than low  on  risk  firms. A d e f i c i e n c y o f t h i s study i s i t s r e l i a n c e upon the  Sharpe  market  7  model.  The  work o f B l a c k ,  J e n s e n , and  Scholes  has  shown t h a t the r e a l i z e d  r e t u r n f o r h i g h b e t a s e c u r i t i e s i s c o n s i s t e n t l y l o w e r than t h a t p r e d i c t e d the market model used by  Westerfield.  This implies  t h a t the r e s i d u a l between  the r e a l i z e d r e t u r n and  the e s t i m a t e d r e t u r n w i l l be b i a s e d  p r o b l e m assumes g r e a t e r  importance when a c c o u n t i s t a k e n o f  downwards.  are b i a s e d  Thus the  two  The  Westerfield's  f i n d i n g s that suggests t h a t i t i s high beta s e c u r i t i e s t h a t tend to more o f t e n t h a n low b e t a s e c u r i t i e s .  by  fail  performance measures u s e d  towards i n d i c a t i n g a d e t e r i o r a t i o n e a r l i e r than when i t a c t u a l l y  occurs. A common f i n d i n g o f t h e s e s t u d i e s i s t h a t the market u n d e r e s t i m a t e d the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t .  constantly  From an  ex-ante  v i e w p o i n t the e x p e c t e d r a t e o f r e t u r n , c o n d i t i o n a l upon no  bankruptcy,  should  so t o compensate  increase  i f the p r o b a b i l i t y o f b a n k r u p t c y i n c r e a s e s  r i s k a v e r s e i n v e s t o r s f o r the the e x - p o s t r e t u r n s h o u l d  increased  increase.  r i s k , but  I f during  t h i s does n o t  a p a r t i c u l a r period  b i l i t y o f a f i r m g o i n g b a n k r u p t unexpected i n c r e a s e s , flected  i n a lower p r i c e a t the end  the b e g i n n i n g o f the p e r i o d and  the  imply  that  the  proba-  t h e n t h i s w i l l be  o f the p e r i o d than had  been e x p e c t e d  r e a l i z e d return w i l l decrease.  reat  The  Ibid.  6  7 B l a c k , F., J e n s e n , M. C , and S c h o l e s , M. , "The C a p i t a l A s s e t P r i c i n g Model:.Some E m p i r i c a l T e s t s , " p u b l i s h e d i n Studies in the Theory of Capital Markets, e d i t e d by J e n s e n , M. (New York: P r a e g e r , 1 9 7 2 ) .  i  18  l o n g e r t h e time p e r i o d becomes f o r e m p i r i c a l  the greater studies.  the p o t e n t i a l s e r i o u s n e s s t h i s problem  I t does, however, s t r e s s t h e need t o c o n s i d e r  the  r a m i f i c a t i o n s o f s t o c h a s t i c changes i n t h e p r o b a b i l i t y o f b a n k r u p t c y i n  any  t h e o r e t i c i n v e s t i g a t i o n s o f t h e e f f e c t s o f b a n k r u p t c y upon t h e s t r u c t u r e  of returns  f o r corporate f i n a n c i a l assets.  Both s t u d i e s  offer empirical  e v i d e n c e which s u g g e s t s t h a t  b e h a v i o u r o f common s t o c k s i s a f f e c t e d by impending b a n k r u p t c y . c o r r e c t , then i t i s p e r t i n e n t  I f this i s  t o e n q u i r e what i s t h e nexus between t h e  market p r i c i n g p r o c e s s and c o r p o r a t e f a i l u r e . is  the p r i c e  The p u r s u i t o f t h i s q u e s t i o n  the primary focus o f the t h e s i s .  P r e d i c t i o n o f Bankruptcy To t e s t e m p i r i c a l l y  t h e h y p o t h e s i s o f t h e t h e s i s a two v a r i a b l e model,  which i s an extended form o f t h e CAPM, i s u s e d .  The second v a r i a b l e i s t h e  p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t and t h u s i t i s n e c e s s a r y t o be a b l e t o estimate t h i s The has  quantity.  primary focus o f previous studies  been on t h e i n f o r m a t i o n a l  ratios.  These s t u d i e s  on t h e p r e d i c t i o n o f b a n k r u p t c y  c o n t e n t o f a c c o u n t i n g s t a t e m e n t s and f i n a n c i a l  have advanced t h e h y p o t h e s i s t h a t t h e r e i s a d i f -  f e r e n c e i n p r o f i l e , as measured by a c c o u n t i n g d a t a , between f a i l e d f a i l e d firms  and t h i s d i f f e r e n c e  that i s , i t i s possible two  using  can be u t i l i z e d as an a i d t o p r e d i c t i o n ;  accounting data to a l l o c a t e firms  groups: f a i l e d and n o n - f a i l e d .  univariate ability.  form.  Single  T h i s h y p o t h e s i s was f i r s t  f i n a n c i a l r a t i o s were t e s t e d  However, u n i v a r i a t e  and non-  analysis  t o one o f  used i n a  fortheir predictive  can be p o t e n t i a l l y m i s l e a d i n g f o r  f a i l u r e depends upon many d i f f e r e n t f a c t o r s .  Consequently, a m u l t i v a r i a t e  19  approach  t o the p r e d i c t i o n o f b a n k r u p t c y has been d e v e l o p e d . Beaver  used a u n i v a r i a t e approach  t o s e l e c t from a sample o f  f i n a n c i a l r a t i o s the one most a b l e t o c o r r e c t l y p r e d i c t the f a i l u r e of a firm.  F a i l u r e was  d e f i n e d t o o c c u r when a f i r m was  f i n a n c i a l o b l i g a t i o n when t h e y matured.  thirty  status  unable t o pay i t s  O p e r a t i o n a l l y , a f i r m was  identified  as f a i l e d when one o f the f o l l o w i n g e v e n t s o c c u r r e d : b a n k r u p t c y , bond d e f a u l t , an overdrawn bank a c c o u n t , o r non-payment o f a p r e f e r r e d s t o c k g dividend.  Such a d e f i n i t i o n i s v e r y b r o a d and i s more i n k e e p i n g w i t h the  concept o f f i n a n c i a l d i s t r e s s .  To c l a s s i f y a l l t h o s e c a t e g o r i e s under  one  group, t h a t o f f a i l u r e , w i l l r e s u l t i n i n e f f i c i e n t e s t i m a t i o n , as n o t a l l the i n f o r m a t i o n i s b e i n g u t i l i z e d .  Presumably,  f o r a f i r m t o d e f a u l t on i t s  bond payments i s a more s e r i o u s e v e n t t h a n f o r a f i r m t o o m i t payment on a preferred dividend.  By c l a s s i f y i n g t h e s e two e v e n t s under t h e one group  n o t take t h i s f a c t i n t o a c c o u n t .  A n o t h e r problem  does  i s t h a t many f i r m s o m i t  d i v i d e n d s f o r r e a s o n s o t h e r t h a n t h a t caused by impending  failure  and  thus  t o c l a s s i f y t h e s e f i r m s as f a i l e d r e s u l t s i n a m i s c l a s s i f i c a t i o n . Beaver  found t h a t the r a t i o o f c a s h f l o w t o t o t a l d e b t was  b e i n g a b l e t o c o r r e c t l y p r e d i c t the f a i l u r e  s t a t u s o f f i r m s and  a b i l i t y existed f o r at l e a s t f i v e years before f a i l u r e . of a single f i n a n c i a l r a t i o f a i l e d and  best at  that  this  Thus on the b a s i s  f i r m s were a l l o c a t e d t o one o f two  groups:  non-failed.  U n i v a r i a t e a n a l y s i s i s s u s c e p t i b l e t o f a u l t y i n t e r p r e t a t i o n and i s potentially misleading. Beaver, W.  H.,  F o r i n s t a n c e , a f i r m whose c a p i t a l  " F i n a n c i a l R a t i o s as P r e d i c t o r s o f F a i l u r e , " supplement t o Journal  Empirical Research in Accounting: Selected Studies, of Accounting Research (1966), pp. 77-111. 9  Ibid.,  p. 71.  structure  20  contains  a large proportion  However, because o f v e r y not be c o n s i d e r e d  o f debt may be r e g a r d e d as a p o t e n t i a l b a n k r u p t .  low v a r i a b i l i t y  serious.  I n an attempt t o a v o i d  m u l t i v a r i a t e approach c o n s i d e r i n g oped.  This involves being  important i n d e t e c t i n g  able  t h e s i t u a t i o n may  t h i s type o f problem a  s e v e r a l f i n a n c i a l r a t i o s has been  devel-  t o d e t e r m i n e which f i n a n c i a l r a t i o s a r e  f u t u r e b a n k r u p t c y , what w e i g h t s s h o u l d  t o the s e l e c t e d r a t i o s and how s h o u l d The  i n i t s cash flow,  be  attached  t h e w e i g h t s be o b j e c t i v e l y e s t a b l i s h e d .  methodology t h a t i s u s u a l l y used i s t h a t o f m u l t i p l e  discriminant  analysis. Multiple discriminant  analysis  (MDA) i s a s t a t i s t i c a l  technique 10  used t o c l a s s i f y an o b s e r v a t i o n The  t o one o f two m u t u a l l y e x c l u s i v e  b a s i s o f the technique i s t o c o n s t r u c t  l i n e a r combination o f explanatory  a discriminant  groups.  f u n c t i o n from a  v a r i a b l e s , the weights being  d e t e r m i n e d by  minimizing the expected c o s t o f m i s c l a s s i f i c a t i o n . To use MDA i t i s n e c e s s a r y t o d e f i n e groups.  t h e two m u t u a l l y  exclusive  F o r p r e d i c t i o n o f b a n k r u p t c y t h e two groups a r e d e f i n e d  r u p t and non-bankrupt.  Data a r e c o l l e c t e d f o r t h e f i r m s  t o be bank-  i n t h e two g r o u p s ;  MDA then attempts t o d e r i v e a l i n e a r c o m b i n a t i o n o f t h e c h a r a c t e r i s t i c s ( f i n a n c i a l r a t i o s ) which d i s c r i m i n a t e s between t h e two groups so a s t o minimize the c o s t o f m i s c l a s s i f y i n g a f i r m . One  o f t h e advantages o f MDA i s t h a t i t i s c a p a b l e o f c o n s i d e r i n g an  e n t i r e p r o f i l e o f c h a r a c t e r i s t i c s common t o t h e r e l e v a n t the  f i r m s , as w e l l as  i n t e r a c t i o n o f these p r o p e r t i e s , and t o combine them i n a s i n g l e  ^MDA can be extended t o t h e g e n e r a l case o f many m u t u a l l y e x c l u s i v e g r o u p s . A good i n t r o d u c t i o n t o MDA i s g i v e n i n A n d e r s o n , T. W., An Introduction to Multivariate Statistical Analysis (New York: John W i l e y & Sons, 1 9 5 7 ) .  21  discriminant to MDA.  function.  There a r e ,  however, a number of p o t e n t i a l  I t i s n e c e s s a r y to make some s p e c i f i c a t i o n about the  e x p e c t e d c o s t s o f m i s c l a s s i f i c a t i o n and probability groups.  distributions  possible  known.  The  the  Altman  1 1  conditional  form o f the  probability  the  o f an  p r i m a r y f o c u s o f MDA  the  different  c o s t s o f m i s c l a s s i f i c a t i o n nor  t o a p a r t i c u l a r group w i t h o u t measuring the belonging to that  p r o p e r t i e s of  Without knowledge o f  to c a l c u l a t e  a p a r t i c u l a r group.  parameters and  d e s c r i b e the  In g e n e r a l , n e i t h e r the  p r o b a b i l i t i e s are i s not  that  the  drawbacks  the  prior  prior distributions  it  observation belonging  i s to a l l o c a t e probability  of  an  the  to  observation observation  group. developed a m u l t i v a r i a t e  r u p t c y u s i n g MDA.  A  f i r m was  prediction  of  bank-  c l a s s i f i e d as b a n k r u p t i f i t f i l e d a bank-  r u p t c y p e t i t i o n under C h a p t e r X o f the T h i s d e f i n i t i o n a v o i d s the  approach t o the  N a t i o n a l Bankruptcy Act  (U.S.A.).  p r o b l e m o f m i s c l a s s i f i c a t i o n t h a t was  encountered  12  in  the  Beaver s t u d y .  a discriminant function  From an  initial  composed o f  determined.  c l a s s i f y 94  per  dard of  or  c e n t o f the  s u c c e s s on  combination o f  The  two  twenty-five f i n a n c i a l  five financial ratios  liquidity, p r o f i t a b i l i t y , productivity, a b i l i t y was  set of  f i n a n c i a l r i s k , and  discriminant function initial  d a t a sample and  secondary samples.  five financial ratios  Thus on  was  characterizing sales  able to  generating  correctly  to achieve a h i g h the  b a s i s of a  f i r m s were a l l o t t e d as  ratios  either  stan-  linear bankrupt  non-bankrupt.  Altman, E. I . , " F i n a n c i a l R a t i o s , D i s c r i m i n a n t A n a l y s i s and P r e d i c t i o n ' o f C o r p o r a t e B a n k r u p t c y , " Journal of Finance, V o l . X X I I I , 4 (September, 1968), pp. 589-609. Beaver, " F i n a n c i a l  Ratios,"  loc.  cit.  the No.  22  Using  a d i f f e r e n t d a t a sample, Deakin  13  replicated  the Alt-man "* s t u d i e s o b t a i n i n g s i m i l a r c o n c l u s i o n s , but 1  the Beaver  14  then p r o c e e d e d  c l a i m t h a t the p r o b a b i l i t y o f group membership c o u l d be d e r i v e d u s i n g statistic  having  a chi-square  d i s t r i b u t i o n , the number o f degrees of  and to  a freedom  16 equaling  the number o f v a r i a b l e s used i n the MDA.  p r o b a b i l i t y o f a s s i g n i n g an o b s e r v a t i o n c a l c u l a t e d i n t h i s context with  T h i s i s wrong.  The  t o a p a r t i c u l a r group can o n l y  the knowledge o f the p r i o r  be  distributions,  17 but  these  a r e unknown.  The  statistic  used by D e a k i n i s p r o b a b l y  a  test  18 for  the n u l l h y p o t h e s i s  t h a t the two  group means a r e  identical.  19 Accounting  d a t a v a r i a b l e s have been used by F i s h e r  study o f the d e t e r m i n a n t s o f the r i s k premiums on c o r p o r a t e hypothesized  The  m a r k e t a b i l i t y o f the bond was  a s i n g l e v a r i a b l e , the market v a l u e o f a l l the the r i s k o f d e f a u l t was  ficient  bonds.  It  was  t h a t the r i s k premium depended upon the m a r k e t a b i l i t y and  r i s k o f d e f a u l t o f the bond.  and  i n an e m p i r i c a l  the  estimated  firm's p u b l i c l y traded  by  bonds,  assumed t o depend upon t h r e e v a r i a b l e s : the  o f v a r i a t i o n o f the f i r m ' s n e t income, the p e r i o d o f s o l v e n c y ,  coefand  13 D e a k i n , E. B., "A D i s c r i m i n a n t A n a l y s i s o f P r e d i c t o r s o f B u s i n e s s F a i l u r e , " Journal of Accounting Research, V o l . 10, No. 1 ( S p r i n g , 1 9 7 2 ) , pp.  167-179. ^Beaver,  " F i n a n c i a l R a t i o s , " loc.  cit.  ^ A l t m a n , " F i n a n c i a l R a t i o s , " loc.  cit.  16 D e a k i n , op.  cit.,  p.  175.  17 statistic,  18  p. 56  As D e a k i n f a i l s to g i v e c l e a r d e f i n i t i o n s of the terms used i n the i t i s d i f f i c u l t t o make any p o s i t i v e s t a t e m e n t . A d e s c r i p t i o n of t h i s s t a t i s t i c  1 9  Journal  i s given  i n Anderson, op.  F i s h e r , L., "Determinants o f R i s k Premiums on C o r p o r a t e of P o l i t i c a l Economy, V o l . L X V l l , No. 3 (June, 1 9 5 9 ) , pp.  ext.,  Bonds," 217-237  23  the r a t i o o f the market v a l u e From t h i s f o r m u l a t i o n  of equity  t o the par value  o f the f i r m ' s  a p p r o x i m a t e l y 70 p e r c e n t o f the v a r i a n c e  pendent v a r i a b l e c o u l d be e x p l a i n e d . c i e n c i e s i n the study.  debt.  o f t h e de-  There a r e , however, a number o f d e f i -  I t i s n o t c l e a r t h a t the m a r k e t a b i l i t y o f a bond  can be e s t i m a t e d by a s i n g l e v a r i a b l e , o r t h a t t h e r i s k o f d e f a u l t can be d e t e r m i n e d by a f u n c t i o n o f t h r e e v a r i a b l e s .  Without f u r t h e r i n v e s t i g a t i o n  o f t h e v a l i d i t y o f t h e s e measures, t h e i n t e r p r e t a t i o n o f t h e f i n d i n g s o f the s t u d y a r e j e o p a r d i z e d . Whilst  t h i s study b r e a k s away from t h e s t r i c t use o f f i n a n c i a l  ratios, i ts t i l l  r e l i e s upon t h e i n f o r m a t i o n a l  content of  accounting  numbers, a c r i t i c i s m t h a t can be a p p l i e d t o a l l t h e s t u d i e s p e r t a i n i n g t o the p r e d i c t i o n o f f a i l u r e .  A c c o u n t i n g d a t a r e f l e c t s t h e consequences o f  past a c t i o n s , w h i l s t f o r p r e d i c t i o n i t i s not the past but the future i s relevant.  The f o c u s o f p r e v i o u s  that  s t u d i e s has been t o a l l o c a t e f i r m s , on  the b a s i s o f f i n a n c i a l r a t i o s , t o one o f two g r o u p s : f a i l e d o r n o n - f a i l e d . No attempt has been made t o o f f e r e i t h e r a t h e o r y f a i l u r e o r t o measure t h e p r o b a b i l i t y o f a f i r m  o f the determinants o f failing.  A c o n t r i b u t i o n o f t h e t h e s i s i s t h e development o f a model t o d e t e r mine t h e p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t .  A t any p o i n t  i n time  this  depends upon t h e f i r m ' s a b i l i t y t o r a i s e f u n d s , e i t h e r i n t e r n a l l y o r e x t e r n a l l y , t o cover f i x e d charges.  To use t h e model t o e m p i r i c a l l y e s t i m a t e  the p r o b a b i l i t y o f b a n k r u p t c y r e q u i r e s p l a c e d by e x - p o s t s u r r o g a t e s .  This necessitates  s t a t i s t i c a l methodology t o u t i l i z e ex-post surrogates.  t h a t the ex-ante v a r i a b l e s be r e -  i n estimating  Multiple discriminant  consideration  o f the  the c o e f f i c i e n t s o f the  a n a l y s i s can n o t be used, f o r i t  24  i s p r i m a r i l y designed  t o a l l o c a t e an o b s e r v a t i o n t o a p a r t i c u l a r group w i t h -  out measuring the p r o b a b i l i t y o f the o b s e r v a t i o n b e l o n g i n g t o t h a t group. The  t h e s i s i n t r o d u c e s a new  methodology t o e s t i m a t e the c o e f f i c i e n t s and  p r o b a b i l i t y o f a f i r m g o i n g bankrupt,  t h a t o f l o g i t a n a l y s i s and  the  probit  analysis. To t e s t e m p i r i c a l l y the h y p o t h e s i s o f the t h e s i s , i t i s o n l y s a r y t o be a b l e t o measure the p r o b a b i l i t y o f b a n k r u p t c y i t s determinants;  t h a t i s , the p r i m a r y  b i l i t y o f bankruptcy nants.  and not t o e x p l a i n  f o c u s i s upon p r e d i c t i n g the  proba-  and not t o advance a complete t h e o r y o f i t s d e t e r m i -  Consequently,  two  f o r m u l a t i o n s o f the model are d e r i v e d .  c o n c e n t r a t e s on b e i n g a b l e t o e x p l a i n the d e t e r m i n a n t s w h i l s t the second  neces-  i s developed  first  bankruptcy,  solely for i t s predictive a b i l i t y using  market v a l u e s f o r a p p r o p r i a t e c o r p o r a t e v a r i a b l e s . the model w i l l be used  of  The  Both f o r m u l a t i o n s o f  i n the t e s t i n g o f the h y p o t h e s i s .  C a p i t a l A s s e t P r i c i n g Model The The  second  t o p i c reviewed  i s the C a p i t a l A s s e t P r i c i n g Model  t h e o r e t i c f o u n d a t i o n s o f the Model and the e m p i r i c a l e v i d e n c e which  cates t h a t i t i s m i s s p e c i f i e d are d e s c r i b e d . to  (CAPM).  Some o f the v a r i o u s  indi-  attempts  e x p l a i n the c a u s e ( s ) o f the m i s s p e c i f i c a t i o n a r e d i s c u s s e d and an  alter-  n a t i v e e x p l a n a t i o n , which forms the h y p o t h e s i s o f the t h e s i s , i s advanced. To t e s t e m p i r i c a l l y  the h y p o t h e s i s a two v a r i a b l e model i s used,  v a r i a b l e b e i n g the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t . h y p o t h e s i s can be t e s t e d , i t i s n e c e s s a r y t o determine utilize.  E x i s t i n g t e c h n i q u e s and  a l t e r n a t i v e methodology  presented.  Before  the  second  the  the methodology t o  t h e i r d e f i c i e n c i e s are d e s c r i b e d and  an  25  F o u n d a t i o n s o f the CAPM The  CAPM d e s c r i b e s  a l i n e a r r e l a t i o n s h i p between t h e e q u i l i b r i u m  e x p e c t e d r e t u r n on an a s s e t and i t s s y s t e m a t i c the  asset's  covariance  r i s k , which i s a measure o f  w i t h t h e market p o r t f o l i o .  The market p o r t f o l i o i s  composed o f an i n v e s t m e n t i n e v e r y r i s k y a s s e t o u t s t a n d i n g its  t o t a l value.  t e x t by T r e y n o r ,  The CAPM was o r i g i n a l l y f o r m u l a t e d 20  Sharpe  21  i n a mean-variance c o n -  and l a t e r c l a r i f i e d by L i n t n e r  In the development o f t h e model i t i s assumed t h a t :  i n proportion to  22  23 and M o s s i n .  (a) a l l i n v e s t o r s a r e  s i n g l e p e r i o d e x p e c t e d u t i l i t y o f t e r m i n a l w e a l t h m a x i m i z e r s who choose among a l t e r n a t i v e p o r t f o l i o s on t h e b a s i s o f mean and v a r i a n c e ;  (b) a l l  i n v e s t o r s can borrow o r l e n d an u n l i m i t e d amount a t an e x o g e n o u s l y r i s k f r e e r a t e o f i n t e r e s t and t h e r e any  assets;  a r e no r e s t r i c t i o n s on s h o r t s a l e s o f  (c) a l l i n v e s t o r s have i d e n t i c a l  means, v a r i a n c e s ,  and c o v a r i a n c e s  subjective estimates o f the  o f r e t u r n among a l l a s s e t s ;  are p e r f e c t l y d i v i s i b l e and t h e r e a r e no t r a n s a c t i o n c o s t s ; taxes; are  given  (f) a l l investors are p r i c e takers;  (d) a l l a s s e t s  (e) t h e r e  a r e no  and (g) t h e q u a n t i t i e s o f a s s e t s  given. The  model may be s t a t e d i n t h e m a t h e m a t i c a l form E  (R ) =  B  [E ( R ^ ] ,  [2.1]  20 T r e y n o r , J . , "Towards a Theory o f Market V a l u e o f R i s k y A s s e t s " (unpublished memorandum, 1961). 21 Sharpe, W. F., " C a p i t a l A s s e t P r i c e s : A T h e o r y o f Market E q u i l i b r i u m Under C o n d i t i o n s o f R i s k , " Journal of Finance, V o l . XIX, No. 3 (September, 1964), pp. 425-442. 22 L i n t n e r , J . , "The V a l u a t i o n o f R i s k A s s e t s and the S e l e c t i o n o f R i s k y Investments i n S t o c k P o r t f o l i o s and C a p i t a l Budgets," Review of Economics and Statistics, V o l . X L V l l ( F e b r u a r y , 1965), pp. 13-37. 23 M o s s i n , J . , " E q u i l i b r i u m i n a C a p i t a l A s s e t Market, Econometrzca, Vol. 34, No. 4 (October, 1966), pp. 760-703.  26  where, i f P.. (t) denotes t h e p r i c e o f t h e j * " * a s s e t a t t h e end o f t h e p e r i o d , 1  E E  ( R  [ P . ( t ) ] - P. ( t - l )  Y(t-l)  ) = J  3  r  F  = e x p e c t e d e x c e s s r e t u r n on t h e j * ^ r^ E  asset;  1  = the r i s k l e s s rate o f i n t e r e s t ; = e x p e c t e d e x c e s s r e t u r n on a 'market p o r t f o l i o '  c o n s i s t i n g o f an i n v e s t m e n t i n e v e r y  asset outstanding  i n proportion to i t s  value; and  B_. = c o v (R_. , R^) / v a r (R^) th = the 'systematic' r i s k o f the j  The  asset.  above r e l a t i o n s t a t e s t h a t t h e e x p e c t e d e x c e s s r e t u r n on any a s s e t i s  d i r e c t l y proportional to i t s systematic o then equation  = E (R  risk.  ) - B E  [2.1] i m p l i e s t h a t a f o r e v e r y  I f a_. i s d e f i n e d a s (R^),  [2.2]  asset i s zero.  E m p i r i c a l t e s t s o f t h e CAPM have been b a s e d upon e x - p o s t d a t a . transformation  o f t h e ex-ante model t o an e x - p o s t f o r m u l a t i o n  The  i s b a s e d upon  the assumption t h a t t h e r e t u r n on any s e c u r i t y can be r e p r e s e n t e d  by a  market model; t h a t i s , t h e r e t u r n on a s e c u r i t y i s a l i n e a r f u n c t i o n o f a market f a c t o r .  Thus any t e s t o f t h e e x - p o s t f o r m u l a t i o n  t h e CAPM and t h e market model.  Using  i s a joint test-of  t h e market model o r i g i n a l l y p r o p o s e d  by M a r k o w i t z ^ and extended by S h a r p e ^ and F a m a ^ t h e e x - p o s t  formulation  M a r k o w i t z , H., Portfolio Selection: Efficient Diversification of Cowles F o u n d a t i o n Monograph No. 16 (New York: John W i l e y and Sons, 1959). 2 4  Investments, 25  Sharpe, "A S i m p l i f i e d Model," loc. cit  or  Journal  Fama, E . , " R i s k , R e t u r n and E q u i l i b r i u m : Some C l a r i f y i n g of Finance, V o l . X X I I I , No. 4 (March, 1960), pp. 29-40.  Comments,"  27  o f t h e CAPM, as r e p r e s e n t e d by E q u a t i o n  [2.1] can be w r i t t e n  R . = 3 . R + e . D : M ] where e. i s a n o r m a l l y d i s t r i b u t e d 3  [2.3]  z e r o mean random v a r i a b l e .  I f assets  a r e p r i c e d a c c o r d i n g t o t h e CAPM then a j o i n t t e s t o f t h e CAPM and the market model c a n be o b t a i n e d by a d d i n g an i n t e r c e p t a_. t o [2.3] and subs c r i p t i n g each o f t h e v a r i a b l e s by t , r e p r e s e n t i n g t i m e , t o o b t a i n V  =  a  j  +  3  j V  +  e j  t  '  [  2  -  4  ]  which i s a r e g r e s s i o n e q u a t i o n , t h e n u l l h y p o t h e s i s b e i n g t h a t t h e i n t e r c e p t term  {a.}, i s z e r o f o r a l l a s s e t s . 3  C r o s s S e c t i o n a l T e s t s o f t h e Model For c r o s s s e c t i o n a l t e s t s the procedure  used  i s t o estimate the  cross sectional regression R\ = v„ +  Vi  [2.5]  3^ + e_.,  where 8j i s o b t a i n e d from t h e r e g r e s s i o n o f a time s e r i e s o f i n d i v i d u a l s e c u r i t y r e t u r n s on an i n d e x used n u l l hypothesis i s that v  Q  as a p r o x y  = 0, and  = (R  f o r t h e market p o r t f o l i o . M  - r  p  The  ) , where R^ i s t h e average  r e t u r n on t h e market index o v e r t h e time p e r i o d , and r  i s u s u a l l y taken t o  be t h e y i e l d t o m a t u r i t y o f a government bond w i t h t h e same m a t u r i t y as t h e l e n g t h o f t h e time p e r i o d under  examination.  27 E v i d e n c e p r e s e n t e d by Douglas,  who r e g r e s s e d t h e r e t u r n s on a l a r g e  c r o s s s e c t i o n a l sample o f common s t o c k s on t h e i r own v a r i a n c e and on t h e i r c o v a r i a n c e w i t h an index c o n s t r u c t e d from t h e sample, found  t h a t t h e model  27 Douglas, G., "Risk i n t h e E q u i t y Markets: An E m p i r i c a l A p p r a i s a l o f Market E f f i c i e n c y , " Yale Economic Essays, V o l . 9 ( S p r i n g , 1 9 6 9 ) , pp. 3-45.  28  d i d not p r o v i d e a complete d e s c r i p t i o n o f s e c u r i t y r e t u r n s . s e p a r a t e f i v e y e a r p e r i o d s from 1926 was  t o 1960,  the average  For  seven  realized return  s i g n i f i c a n t l y p o s i t i v e l y r e l a t e d t o the v a r i a n c e o f the  security's  r e t u r n s over t i m e , b u t not t o t h e i r c o v a r i a n c e w i t h the i n d e x o f r e t u r n s . These r e s u l t s appear t o be i n c o n f l i c t w i t h the r e l a t i o n g i v e n by  [2.1] f o r  the v a r i a n c e term s h o u l d have a c o e f f i c i e n t o f z e r o . Douglas a l s o summarizes some u n p u b l i s h e d r e s u l t s o f L i n t n e r ' s t h a t a l s o appear t o be  inconsistent with Eouation [2.1].  L i n t n e r estimates  E q u a t i o n [ 2 . 5 ] , f o r a c r o s s s e c t i o n o f s e c u r i t i e s , adding the v a r i a n c e o f the r e s i d u a l s from the time Equation [2.4].  s e r i e s r e g r e s s i o n s g i v e n by  T h i s e x t r a v a r i a n c e s h o u l d have no e x p l a n a t o r y power  thus i t s c o e f f i c i e n t s h o u l d be z e r o .  0  was  g r e a t e r than z e r o and  and  In L i n t n e r ' s t e s t s i t d i d n o t , the  e f f i c i e n t on t h e r e s i d u a l v a r i a n c e b e i n g p o s i t i v e v  an e x t r a v a r i a b l e :  much l e s s than  and  significant.  co-  Also,  (R^ - r ) .  28 M i l l e r and  Scholes  r e p l i c a t e d t h e L i n t n e r s t u d y on a d i f f e r e n t  body o f d a t a o b t a i n i n g the same g e n e r a l r e s u l t s . f i c a t i o n may  The  a r i s e n o t because t h e model i s wrong, b u t due  o f measuring t h e d i f f e r e n t v a r i a b l e s . (a) f a i l u r e t o a c c o u n t  adequately  t o h e t e r o s c e d a s t i c i t y , may However, i t was  to the  difficulty  F o r example, the b i a s e s i n t r o d u c e d by  f o r the r i s k l e s s r a t e o f i n t e r e s t ,  p o s s i b l e n o n - l i n e a r i t y i n the r i s k - r e t u r n r e l a t i o n , o r  was  source o f the m i s s p e c i -  (b)  (c) d i s t o r t i o n s  be the cause o f the D o u g l a s - L i n t n e r  due  finding.  shown t h a t t h e s e e r r o r s c o u l d n o t produce such r e s u l t s .  a l s o demonstrated t h a t w h i l s t measurement e r r o r s i n the r i s k  It  variable  28 M i l l e r , M. and S c h o l e s , M., "Rates o f Return i n R e l a t i o n t o R i s k : A Re-examination o f Some Recent F i n d i n g s , " p r i n t e d i n Studies in the Theory of Capital Markets e d i t e d by J e n s e n , M. (Mew York: P r a e g e r , 1972) . 3  29  {8j},  and the c o r r e l a t i o n between t h e v a r i a n c e  time s e r i e s r e g r e s s i o n s  o f t h e r e s i d u a l s from t h e  and t h e e s t i m a t e s o f t h e r i s k v a l u e  t r i b u t e s u b s t a n t i a l l y t o the Douglas-Lintner  con-  from t h e model.  A  a c c o u n t f o r s u c h r e s u l t s i s t h e p r e s e n c e o f skewness i n  the p r o b a b i l i t y d i s t r i b u t i o n s o f e x - p o s t r e t u r n s . able  could  r e s u l t s , they were n o t s u f -  f i c i e n t t o account f o r a l l the observed d e v i a t i o n s problem which c o u l d  (8^},  M i l l e r and S c h o l e s were  t o show t h a t skewness e f f e c t s c o u l d cause s e r i o u s d i f f i c u l t i e s and  t h a t combined w i t h t h e measurement e r r o r s i n t h e r i s k v a r i a b l e c o u l d i n p r i n c i p l e cause t h e D o u g l a s - L i n t n e r not  results.  Whilst  t h e i r a n a l y s i s does  imply t h e complete r e j e c t i o n o f t h e D o u g l a s - L i n t n e r  show t h a t t h e y must be t r e a t e d w i t h c a u t i o n  r e s u l t s , i t does  i n view o f t h e e c o n o m e t r i c  dif-  f i c u l t i e s i n t e s t i n g t h e model.  Time S e r i e s T e s t s o f t h e Model 29 Black,  J e n s e n , and S c h o l e s  a time s e r i e s p r o c e d u r e . regression using  R j  t  [2.4]; t h a t i s , - °j  zero.  Whilst  information  +  e  j V  +  e j  t  '  [  t h a t t h e i n t e r c e p t term, a., i s z e r o  Thus a d i r e c t t e s t can be o b t a i n e d some time p e r i o d  have t e s t e d t h e CAPM by u s i n g  The model c a n be t e s t e d b y r u n n i n g a time s e r i e s  the equation  the n u l l h y p o t h e s i s b e i n g  (B-J-S)  by e s t i m a t i n g  2  -  4  1  f o r a l l assets.  [2.4] f o r a s e c u r i t y o v e r  and t e s t i n g t o see i f a_. i s s i g n i f i c a n t l y d i f f e r e n t from  t h i s t e s t i s simple,  i ti s inefficient  on o n l y a s i n g l e s e c u r i t y .  i n that i t u t i l i z e s  To overcome t h i s p r o b l e m B-J-S  29 B l a c k , F., J e n s e n , M., and S c h o l e s , M., "The C a p i t a l A s s e t P r i c i n g Model: Some E m p i r i c a l T e s t s , " p r i n t e d i n Studies in the Theory of Capital Markets, e d i t e d by J e n s e n , M. (New York: P r a e g e r , 1972) .  30  p e r f o r m t h e i r t e s t s on p o r t f o l i o r e t u r n s  o v e r the p e r i o d  to 1965,  so as  systematic r i s k .  t h e i r t e s t s t o t e n p o r t f o l i o s , which c o n -  t a i n e d a l l s e c u r i t i e s on t h a t the For  low  the New  i n t e r c e p t term was r i s k s e c u r i t i e s the  s e c u r i t i e s i t was  negative.  dispersion  York S t o c k Exchange.  d i r e c t l y r e l a t e d t o the i n t e r c e p t term was There was  The  of  where  the p o r t f o l i o s a r e c o n s t r u c t e d They a p p l i e d  t o maximize the  1931  results  their  indicated  systematic r i s k  p o s i t i v e , and  level.  for high r i s k  substantial indication that  the  i n t e r c e p t terms f o r the d i f f e r e n t p o r t f o l i o s were n o n - s t a t i o n a r y , e s p e c i a l l y f o r s e c u r i t i e s whose l e v e l o f s y s t e m a t i c r i s k was  d i f f e r e n t from  S i m i l a r l y f i n d i n g s have a l s o been o b t a i n e d i n a r e c e n t  s t u d y by  unity. Blume  and  30  Friend. B-J-S  go  on t o demonstrate t h a t the p r o c e s s g e n e r a t i n g the  i n d i v i d u a l s e c u r i t i e s can be  r  <\»  where r  Zt  r e p r e s e n t s the  jt  described "  (1  "V  by  r  Zt  a two  +  most o f the  duced by measurement e r r o r s nation  o f the  cross  v a r i a b l e model o f the  j Mt r  +  G  j t »  r e t u r n on what t h e y c a l l the  o t h e r lower c a s e r ' s i n d i c a t e t o t a l r e t u r n s . which e l i m i n a t e s  B  one  £3_^3" i n c r o s s  tionship i s highly  l i n e a r , but  randomly from p e r i o d 30  f i v e months was  to p e r i o d  b o t h the and  6 ]  the  intro-  s e c t i o n a l t e s t s , an  exami-  conducted. intercepts  are o f t e n  '  w i t h the b i a s e s  s e c t i o n a l r e l a t i o n s h i p s between r i s k and  hundred and  firm  "beta f a c t o r ' and  seventeen s u b p e r i o d s o f l e n g t h s t w e n t y - f o u r months and length  t 2  on  Using a grouping procedure  d i f f i c u l t i e s associated i n the  return  four  return  for  subperiods  I t appears t h a t the and  negative.  slopes B-J-S  of rela-  fluctuate argue t h e s e  • • Blume, M. and F r i e n d , I . , "A New Look a t the C a p i t a l A s s e t P r i c i n g Model," Journal of Finance, V o l . XXVIII, No. 1 (March, 1973), pp. 19-34.  31  n o n - s t a t i o n a r i t i e s are c o n s i s t e n t w i t h the r e t u r n g e n e r a t i n g described  by  [ 2 . 6 ] , which i m p l i e s t h a t the  s e c t i o n a l r e g r e s s i o n s w i l l be denote sample means over the Since  r  w i l l a l s o be  and  i n t e r c e p t and  mechanism  slope  i n the  cross  ( r ^ - r^) r e s p e c t i v e l y , where the  time p e r i o d c o v e r e d by  a random v a r i a b l e , e q u a t i o n  the c r o s s  bars  sections.  [ 2 . 6 ] i s consistent  with  the o b s e r v e d e m p i r i c a l r e s u l t s . The  e v i d e n c e seems t o i n d i c a t e t h a t the CAPM does n o t p r o v i d e  adequate d e s c r i p t i o n o f the p r o c e s s g e n e r a t i n g documentation o f n o n - s t a t i o n a r i t y and f a c t o r imply t h a t the model must be  the  an  common s t o c k r e t u r n s .  existence  The  of a t l e a s t another  extended t o i n c l u d e a d d i t i o n a l  variables.  T h e o r e t i c a l E x t e n s i o n s o f the CAPM The  majority  o f the assumptions u n d e r l y i n g  the model v i o l a t e t o  some degree the c o n d i t i o n s o b s e r v e d i n p r a c t i c e .  A number o f r e c e n t  have attempted t o r e l a x v a r i o u s  to incorporate  the c o m p l e x i t i e s cated  assumptions so as  o f c a p i t a l markets i n t o the model.  The  The  some o f  r e s u l t s have  t h a t the b a s i c s t r u c t u r e o f t h e model i s r e m a r k a b l y r o b u s t  t i o n s o f these  studies  to  indi-  viola-  assumptions.  assumption t h a t i n v e s t o r s a r e  t e r m i n a l w e a l t h maximizers i s v e r y  s i n g l e p e r i o d expected u t i l i t y  restrictive  and  may  n o t be  an  of  accurate  32  description of investors * behaviour.  Fama  has  i n v e s t i g a t e d the  conditions  31 An e x c e l l e n t s u r v e y a r t i c l e d e s c r i b i n g even more e v i d e n c e about the m i s s p e c i f i c a t i o n of the CAPM i s g i v e n i n J e n s e n , M., " C a p i t a l M a r k e t s : Theory and E v i d e n c e , " Bell Journal of Economics and Management Science, V o l . 3,  No.  2  (Autumn, 1 9 7 2 ) , pp.  357-398.  32 Fama, E.,  can Economic  "Multiperiod Consumption—Investment Decisions," Review, V o l . L X , No. 1 (March, 1 9 7 0 ) , pp. 1 6 3 - 1 7 4 .  Ameri-  32  f o r the v a l i d i t y o f such an assumption.  Arguing t h a t i n v e s t o r ' s problem  i s more a c c u r a t e l y s t a t e d as the m a x i m i z a t i o n o f the e x p e c t e d  lifetime  u t i l i t y o f consumption and t e r m i n a l w e a l t h , i t i s demonstrated t h a t t h e s i n g l e p e r i o d CAPM can be j u s t i f i e d if  i n the context  of a multiperiod  t h e i n v e s t o r behaves as i f f u t u r e consumption and i n v e s t m e n t  are g i v e n and t h a t t a s t e s a r e n o t s t a t e dependent.  problem  opportunities  Thus even though t h e i n -  v e s t o r must s o l v e a m u l t i p e r i o d p r o b l e m t o a r r i v e a t t h e o p t i m a l  current  d e c i s i o n s , t h e s e d e c i s i o n s a r e i n d i s t i n g u i s h a b l e from t h o s e o f a r i s k s i n g l e p e r i o d expected u t i l i t y  averse  o f t e r m i n a l wealth maximizer.  These f i n d i n g s have i m p o r t a n t i m p l i c a t i o n s f o r t h e c o n d i t i o n s must be s a t i s f i e d i f t h e CAPM i s t o be e m p i r i c a l l y t e s t e d .  which  Previous  e m p i r i c a l s t u d i e s have u t i l i z e d e x - p o s t d a t a e x t e n d i n g o v e r many time periods  and have t a c i t l y assumed t h e v a l i d i t y o f t h e model i n a m u l t i p e r i o d  context.  B u t t a s t e s do change o v e r time and t h u s one o f t h e assumptions  n e c e s s a r y t o use t h e model o v e r e x t e n d e d time p e r i o d s the c o n c l u s i o n s  o f such s t u d i e s a r e j e o p a r d i z e d .  I t i s assumed i n t h e CAPM t h a t a l l a s s e t s t h a t i s , a l l a s s e t s a r e m a r k e t a b l e and t h e r e There a r e many a s s e t s example, c l a i m s  are p e r f e c t l y l i q u i d ;  a r e no t r a n s a c t i o n  costs.  f o r which t h i s assumption i s n o t a p p l i c a b l e .  on l a b o u r  For  income o r s o c i a l s e c u r i t y payments a r e c l a i m s  t h a t can n o t be s o l d i n c a p i t a l m a r k e t s . what e f f e c t n o n - m a r k e t a b i l i t y case o f o n l y  i s v i o l a t e d and hence  of assets  Thus i t i s p e r t i n e n t t o e n q u i r e has upon t h e CAPM.  For the s p e c i a l  two types o f a s s e t s , p e r f e c t l y l i q u i d and p e r f e c t l y n o n - l i q u i d ,  33 Mayers  d e r i v e s a simple expression  between t h e e x p e c t e d r a t e o f r e t u r n on  33 Mayers, D., "Non-Marketable A s s e t s and C a p i t a l Market Under U n c e r t a i n t y , " p r i n t e d i n Studies in the Theory of Capital e d i t e d by J e n s e n , M. (New York: P r a e g e r , 1972).  Equilibrium  Markets,  33  any  a s s e t and  i t s covariance  r i s k i n terms o f market parameters and  demon-  s t r a t e s t h a t the b a s i c i m p l i c a t i o n s o f the model are not weakened i n major r e s p e c t by  the e x i s t e n c e  o f non-marketable  There a r e , however, many a s s e t s  any  assets.  t h a t can  not be d e s c r i b e d  as  being  e i t h e r p e r f e c t l y l i q u i d o r p e r f e c t l y n o n - l i q u i d , r e a l e s t a t e , second hand automobiles, being concept but  p o s s i b l e examples.  i s a continuum.  To  M a r k e t a b i l i t y i s not  i n v e s t i g a t e the  f u l l e f f e c t s of  upon the CAPM r e q u i r e s t h a t i t s d e t e r m i n a n t s be known and o f m a r k e t a b i l i t y o f an a s s e t can be measured. Mayers' study are general  asset.  i m p o r t a n t , the  a dichotomous  Whilst  s t u d y does not  marketability  t h a t the  "degree"  the r e s u l t s o f  address i t s e l f  the  t o the more  and  d i f f i c u l t p r o b l e m o f t r e a t i n g m a r k e t a b i l i t y as a continuum.  One  o f the assumptions o f the CAPM i s the e x i s t e n c e  In the p r e s e n c e o f u n c e r t a i n t y  about the  of a  riskless  l e v e l of future p r i c e s  as c o n t r a c t s a r e n o t denominated i n r e a l terms, the assumption o f the  and  exis-  34  t e n c e o f such an a s s e t  i s tenuous.  Black  has  shown, under assumptions  i d e n t i c a l t o t h o s e o f the CAPM, t h a t i f a r i s k l e s s a s s e t , o r b o r r o w i n g  or  l e n d i n g o p p o r t u n i t i e s do n o t e x i s t , t h e n i n e q u i l i b r i u m the p o r t f o l i o s o f all  i n v e s t o r s c o n s i s t o f a l i n e a r c o m b i n a t i o n o f two  being  the market p o r t f o l i o and  zero covariance p o r t f o l i o has  the o t h e r  a p o r t f o l i o whose r e t u r n s  w i t h the market p o r t f o l i o and  been termed the  (r.) = 3  has  (1 - 3.) 3  (r)  This  Black demonstrates  a s s e t w i l l be  F. (r_) + 3. E Z ]  one  have  minimum v a r i a n c e .  "zero b e t a " p o r t f o l i o .  t h a t i n e q u i l i b r i u m the e x p e c t e d r e t u r n on any E  basic portfolios,  M  given  by  ,  34  Black,  Journal  F.,  " C a p i t a l Market E q u i l i b r i u m With R e s t r i c t e d B o r r o w i n g , " 3 ( J u l y , 1972), pp. 445-455.  of Business, V o l . 45, No.  34  where E (r ) i s t h e e x p e c t e d r e t u r n on t h e z e r o b e t a p o r t f o l i o and t h e o t h e r v a r i a b l e s a r e as p r e v i o u s l y  defined.  Whilst  t h i s i s a two f a c t o r 35  model and b e a r s a c l o s e r e l a t i o n s h i p t o t h e model s u g g e s t e d by B-J-S, there  i s nothing i n the formulation  t o s u g g e s t t h a t t h e second  factor  i s n o t c o n s t a n t , and thus i t c a n n o t a d e q u a t e l y e x p l a i n a l l t h e o b s e r v e d e m p i r i c a l d e f i c i e n c i e s o f t h e CAPM. The  CAPM, w h i c h i s f o r m u l a t e d i n a d i s c r e t e time framework, r e s t s  upon t h e assumption t h a t t h e r e  a r e no t r a n s a c t i o n c o s t s .  Whilst  the formula-  t i o n o f a d i s c r e t e time model i s c o n v e n i e n t f o r e m p i r i c a l work, i t s t h e o retic  j u s t i f i c a t i o n i s questionable.  p r e f e r t o have t h e o p t i o n be  t o trade  The r a t i o n a l i n v e s t o r would always  any i n s t a n t o f t i m e — a t no c o s t — t h e n  to  r e s t r i c t e d t o t r a d i n g a t d i s c r e t e time i n t e r v a l s . The  usual  reason given  t r a n s a c t i o n c o s t s do e x i s t . spacings o f non-specified  f o r t h e d i s c r e t e time f o r m u l a t i o n  i s that  However, t h e approach i s t o t a k e e q u a l time  length.  I f transaction costs  a r e t o be i n c l u d e d ,  the  i n t e r v a l between t r a d i n g p e r i o d s  w i l l become a v a r i a b l e d e p e n d i n g upon  the  s i z e o f t r a n s a c t i o n c o s t s , changes i n t h e p r i c e s o f s e c u r i t i e s , i n i t i a l  w e a l t h d i s t r i b u t i o n , and e x p e c t a t i o n s . also r a i s e s the question  The i n c l u s i o n o f t r a n s a c t i o n  o f the existence  costs  o f an e q u i l i b r i u m .  36 Merton the  has d e r i v e d  a c o n t i n u o u s time v e r s i o n o f t h e CAPM, a l t h o u g h  symbols, which appear t h e same, have a d i f f e r e n t i n t e r p r e t a t i o n , b e i n g  e x p r e s s e d i n terms o f i n s t a n t a n e o u s r a t e s o f r e t u r n .  Four e x t r a  assumptions  35 B l a c k , J e n s e n , and S c h o l e s , loc.  cut.  36 Merton, R. C , "A Dynamic G e n e r a l E q u i l i b r i u m Model o f t h e A s s e t Market and I t s A p p l i c a t i o n t o t h e P r i c i n g o f t h e C a p i t a l S t r u c t u r e o f t h e F i r m , " M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , S l o a n S c h o o l o f Management, December, 1970.  35  are r e q u i r e d : t r a d i n g t a k e s p l a c e c o n t i n u o u s l y , which f o l l o w s d i r e c t l y the assumption  o f no t r a n s a c t i o n c o s t s ; a c o n s t a n t i n v e s t m e n t  s e t , t h a t i s , the means, v a r i a n c e s and  from  opportunity  c o v a r i a n c e s t h a t d e s c r i b e the  charac-  t e r i s t i c s o f the d i f f e r e n t a s s e t s a r e c o n s t a n t ; o n l y l o c a l changes i n the s t a t e v a r i a b l e s o f the p r o c e s s a r e a l l o w e d , which r u l e s o u t P a r e t o - L e v y P o i s s o n type p r o c e s s e s ; and  i n v e s t o r s a c t so as t o maximize the  u t i l i t y of l i f e t i m e consumption and t e r m i n a l w e a l t h .  expected  Representing  dynamics o f s e c u r i t i e s by Weiner p r o c e s s e s the c o n t i n u o u s  or  the p r i c e  time analogue t o  CAPM i s d e r i v e d : a. - r = B. (cc — r ) . j j M ' ;  th  where a. i s the i n s t a n t a n e o u s r a t e o f r e t u r n on t h e j asset; r i s the 3 instantaneous r i s k f r e e rate of i n t e r e s t ; a i s the instantaneous r a t e o f M r e t u r n on the market p o r t f o l i o ; and  B. = o" . / a ^ ; and j jM M w  o. jM w  i s the  instan-  taneous c o v a r i a n c e o f the r e t u r n on t h e j ^ a s s e t w i t h the r e t u r n on 1  the  market p o r t f o l i o . 37 I f any o f the t h r e e c o n d i t i o n s o u t l i n e d by Fama s i n g l e period maximization  o f expected  utility  to j u s t i f y  of terminal wealth  the  assump-  t i o n o f the CAPM a r e v i o l a t e d , then the s i m p l e s t r u c t u r e o f the model w i l l p r o b a b l y be d e s t r o y e d .  F o r example, i f the i n v e s t m e n t  changes, then t h i s w i l l ,  d e c i s i o n o f the i n v e s t o r .  i n g e n e r a l , a f f e c t the 38 Merton,  opportunity set  consumption-investment  i n the c o n t e x t o f a c o n t i n u o u s  time  framework, 37 d e m o n s t r a t e s t h a t changes i n the i n v e s t m e n t o p p o r t u n i t y s e t do Fama, " M u l t i p e r i o d Consumption," too. cit. 38 Merton, R. C , "An I n t e r t e m p o r a l C a p i t a l A s s e t P r i c i n g Model," Working Paper 588-72, M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , S l o a n S c h o o l o f Management, F e b r u a r y , 1972.  36  a f f e c t the can be  s t r u c t u r e of the model.  c h a r a c t e r i z e d by changes i n a s i n g l e i n s t r u m e n t a l  less interest r a t e — a can be  Under the assumption t h a t t h e s e changes  three  v a r i a b l e model i s d e r i v e d .  The  third  i n t e r p r e t e d as the r e s u l t o f i n v e s t o r s h e d g i n g a g a i n s t  o f f u t u r e u n f o r e s e e n changes i n the the  variable—the  i n v e s t o r ' s p o r t f o l i o w i l l be  riskless interest rate.  a linear conbination  the r i s k l e s s a s s e t , the market p o r t f o l i o , and  variable  the e f f e c t s In  of three  a portfolio  risk-  equilibrium, portfolios:  (or a s s e t )  which  i s perfectly negatively  c o r r e l a t e d w i t h changes i n the r i s k l e s s i n t e r e s t  rate.  s t a t e d i n the  The model can be  °j . i  A  =  r  F  +  X l  (a  M  "  r )  _ ^jM " ^ j N ^NM : i - p NM  form:  *  +  2  (0t  N  "  '  r )  ,  2  8.  .  *2  jN 1  -  -  8.  jM  "  B  NM  ,  P NM Z  a. B.  ^ a  jk  - ^ ,  =  °kK  i s the i n s t a n t a n e o u s c o v a r i a n c e  between the  J  j  th and k th  i s the i n s t a n t a n e o u s c o r r e l a t i o n between the r e t u r n s and  the a s s e t N, which i s n e g a t i v e l y  Merton argues t h a t the  b e t a a s s e t s and  negative  s i g n o f A2  f o r high beta assets.  8C i d e n t i c a l t o t h a t o f - r — , where C i s the or  f u n c t i o n o f y i e l d s and  the market p o r t f o l i o  c o r r e l a t e d w i t h changes i n the  interest rate.  Macro-economic t h e o r y  on  sign for  (a  a g g r e g a t e consumption  u s u a l l y assumes t h a t a g g r e g a t e s a v i n g 3C t h e r e f o r e - r — < 0, or  riskless  w i l l be p o s i t i v e f o r  The  which i m p l i e s  p,,„ KM  a s s e t s ; and  that  - r) i s function.  i s an (a  low  N  increasing  - r) w i l l  be  37  positive.  Thus, Merton c o n c l u d e s  empirical  t h a t t h e model i s a t l e a s t c o n s i s t e n t w i t h  evidence.  Summary Accumulating  e m p i r i c a l evidence  i n d i c a t e s t h a t t h e CAPM does n o t  p r o v i d e an adequate d e s c r i p t i o n o f t h e mechanism g e n e r a t i n g common s t o c k r e -  39 turns.  The time  s e r i e s work o f B l a c k , J e n s e n ,  and S c h o l e s ,  has shown t h a t  a s s e t s w i t h h i g h l e v e l s o f s y s t e m a t i c r i s k c o n s i s t e n t l y e a r n l e s s than  that  p r e d i c t e d by t h e model, w h i l s t a s s e t s w i t h low l e v e l s o f s y s t e m a t i c r i s k more than t h a t p r e d i c t e d .  Even though t h e r e appears t o be a l i n e a r  s h i p between a s e c u r i t y ' s e x - p o s t  s y s t e m a t i c r i s k have been i n v e r s e l y r e l a t e d . that the data i n d i c a t e t h a t the expected s e n t e d by a l i n e a r two f a c t o r model.  B l a c k , J e n s e n and S c h o l e s  argue  The second f a c t o r , which they c a l l t h e  identified.  The works o f W e s t e r f i e l d  41  and Beaver  have shown t h a t p r i c e b e -  h a v i o u r o f common s t o c k s i s a f f e c t e d b y impending b a n k r u p t c y . the expected  r e t u r n and  r e t u r n on a s e c u r i t y c a n be r e p r e -  40  ante v i e w p o i n t ,  relation-  r e t u r n and i t s s y s t e m a t i c r i s k , t h e  r e l a t i o n s h i p i s n o n - s t a t i o n a r y ; o v e r v a r i o u s time p e r i o d s e x - p o s t  beta factor, i s not e x p l i c i t l y  earn  From an e x -  r a t e o f r e t u r n , c o n d i t i o n a l upon no  bankruptcy,  s h o u l d i n c r e a s e i f t h e r e i s an i n c r e a s e i n t h e p r o b a b i l i t y o f b a n k r u p t c y t o compensate r i s k a v e r s e  i n v e s t o r s f o r the extra r i s k .  Black, Jensen,  W i t h i n t h e framework  and S c h o l e s , loo. oit.  40 W e s t e r f i e l d , R., "The Assessment o f Market R i s k and C o r p o r a t e F a i l u r e , " U n i v e r s i t y o f P e n n s y l v a n i a , Wharton S c h o o l o f F i n a n c e , August 1970 ( u n p u b l i s h e d ) . 4 1  Beaver,  "Market P r i c e s , "  loo. oit.  38  o f t h e CAPM t h i s i m p l i e s systematic  that the s i n g l e explanatory  r i s k , which i s a measure o f t h e c o v a r i a n c e  with the returns o f a l l other  s e c u r i t i e s , should  d i c t s t h e assumption o f a c o n s t a n t  v a r i a b l e , that of o f the f i r m ' s  increase.  investment opportunity  But t h i s set.  bankruptcy.  attempts have been made t o e x p l a i n t h e s e d e f i c i e n c i e s . The  effects of non-marketability  o f assets, the non-existence o f a r i s k l e s s  a s s e t , and r e s t r i c t i o n s upon t h e i n v e s t o r ' s a b i l i t y been e x p l o r e d ,  contra-  Thus t h e  CAPM can n o t e x p l a i n why p r i c e b e h a v i o u r i s a f f e c t e d by impending Various  return  though t h e s e f a i l  the o b s e r v e d d e f i c i e n c i e s .  to provide  t o borrow o r l e n d have  an adequate e x p l a n a t i o n  of a l l  The e f f e c t s o f changes i n t h e i n v e s t m e n t o p p o r -  t u n i t y s e t have been shown t o i m p l y t h e e x i s t e n c e  o f a three  v a r i a b l e model  which i s a t l e a s t t h e o r e t i c a l l y c o n s i s t e n t w i t h t h e e m p i r i c a l f i n d i n g s o f  42 Black,  J e n s e n , and S c h o l e s ,  though i t does n o t p r o v i d e  an e x p l a n a t i o n o f  why impending b a n k r u p t c y a f f e c t s t h e r e s i d u a l r e t u r n b e h a v i o u r , a f t e r a b s t r a c t i n g from t h e market, o f common  stocks.  Hypothesis o f the Thesis I f the p r o b a b i l i t y o f bankruptcy f o r a f i r m i n c r e a s e s ,  then the  e x p e c t e d r e t u r n , c o n d i t i o n a l upon no b a n k r u p t c y , which r i s k a v e r s e require w i l l  increase  t o compensate f o r t h e e x t r a r i s k .  investors  A t any p o i n t i n  time t h e p r o b a b i l i t y o f b a n k r u p t c y f o r a f i r m i s a f u n c t i o n o f i t s a b i l i t y to r a i s e funds, e i t h e r i n t e r n a l l y o r e x t e r n a l l y , t o cover f i x e d charges.  As  c o n d i t i o n s w i t h i n t h e f i r m and t h e economy change o v e r t i m e , so w i l l t h e firm's a b i l i t y  t o r a i s e f u n d s , which i m p l i e s t h a t t h e p r o b a b i l i t y o f t h e  f i r m going bankrupt w i l l a l s o vary  across  time.  42 Black,  J e n s e n , and S c h o l e s ,  loc. cit.  This w i l l d i r e c t l y a f f e c t  39  the expected r a t e o f r e t u r n which i n v e s t o r s r e q u i r e on the f i r m ' s  financial  assets. The  t h e s i s gives a theoretic explanation  r u p t c y upon t h e s t r u c t u r e o f c o r p o r a t e  o f t h e e f f e c t s o f bank-  f i n a n c i a l assets.  The h y p o t h e s i s o f  the t h e s i s i s t h a t d i f f e r e n c e s i n the p r o b a b i l i t y o f b a n k r u p t c y s e c u r i t i e s and a c r o s s  across  time a r e r e f l e c t e d i n t h e r e s i d u a l r e t u r n a f t e r  a b s t r a c t i n g from t h e market.  E m p i r i c a l T e s t i n g o f the Hypothesis From t h e t h e o r e t i c a l a n a l y s i s a two v a r i a b l e model d e s c r i b i n g t h e s t r u c t u r e o f common s t o c k r e t u r n s  i s derived.  The model i s an e x t e n s i o n o f  the CAPM and i s o f t h e form ex. = r  p  + A. +  3. ( a - r M  p  - x) ,  where a_. i s t h e i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on t h e j  th  a s s e t ; a i s t h e i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on t h e M w  market p o r t f o l i o ; r ^ i s t h e i n s t a n t a n e o u s r i s k f r e e r a t e o f i n t e r e s t ; X_. i s th the r a t e o f p r o b a b i l i t y o f b a n k r u p t c y f o r t h e j averaqe o f t h e {A.}; and 3. = o . / c , a. b e i n g j j jM MM jM th t i o n a l covariance pirically i s used.  of the j  asset; x I  necessary.  First,  a weighted  the instantaneous  a s s e t w i t h t h e market p o r t f o l i o .  t h e h y p o t h e s i s a d i s c r e t e time, e x - p o s t f o r m u l a t i o n However, b e f o r e  s  condi-  To t e s t emo f t h e model  t e s t i n g t h e h y p o t h e s i s two p r e l i m i n a r y  steps are  the p r o b a b i l i t y o f a f i r m going bankrupt over a given  p e r i o d needs t o be e s t i m a t e d ; and second, a c h o i c e  o f methodology t o employ  when t e s t i n g t h e h y p o t h e s i s must be made. The  p r o b a b i l i t y o f b a n k r u p t c y f o r a f i r m depends upon i t s a b i l i t y  40  to  r a i s e funds, e i t h e r i n t e r n a l l y o r e x t e r n a l l y , t o cover f i x e d charges.  model d e s c r i b i n g t h e d e t e r m i n a n t s o f t h e f i r m ' s a b i l i t y constructed.  t o r a i s e funds i s  The p r i m a r y f o c u s i s t h e p r e d i c t i o n and e s t i m a t i o n o f t h e  p r o b a b i l i t y o f b a n k r u p t c y , a s opposed theory.  to constructing a f u l l explanatory  The c o e f f i c i e n t s o f t h e model a r e d e t e r m i n e d u s i n g l o g i t  and p r o b i t  A  analysis  analysis.  A methodology t o t e s t a two v a r i a b l e model has been d e v e l o p e d by  43 B l a c k and S c h o l e s .  I n an attempt t o examine t h e e f f e c t s o f d i v i d e n d s on  common s t o c k p r i c e s an a d h o c two v a r i a b l e e x t e n s i o n o f t h e CAPM has been advanced.  The model i s o f t h e form 6. - 5 E  1  0  +  V  l  (-^  j  M  c  a s s e t ; 5^ i s t h e e x p e c t e d d i v i d e n d y i e l d on t h e market p o r t f o l i o ;  $. = c o v ( r . , r ) / v a r ( r ) ; v 3 j M M of  -) +  6  [E ( r ) - v ] , th. where E (r.) i s t h e e x p e c t e d r e t u r n f o r t h e j a s s e t ; E (r ) i s t h e e x p e c t e d 3 M r e t u r n on t h e market p o r t f o l i o ; 6^ i s t h e e x p e c t e d d i v i d e n d y i e l d f o r t h e j*"*  (r.) = v  0  i s a constant t o account f o r the e x i s t e n c e  a b e t a f a c t o r ; and \>\ i s a c o n s t a n t .  The h y p o t h e s i s i s t h a t t h e r e s i d u a l  r e t u r n on a s e c u r i t y , a f t e r a b s t r a c t i n g from t h e market f a c t o r , can be e x p l a i n e d by t h e d i v i d e n d y i e l d . ficient  V i s h o u l d be non-zero  I f t h e h y p o t h e s i s i s t r u e , then t h e c o e f -  and s t a t i s t i c a l l y  significant.  A c r o s s s e c t i o n a l a n a l y s i s , which would u t i l i z e all  s e c u r i t i e s , i s r u l e d o u t because  by e r r o r s i n v a r i a b l e s .  t h e i n f o r m a t i o n on  o f the econometric d i f f i c u l t i e s  Thus, a time s e r i e s approach  i s used.  caused  The method-  o l o g y i s t o c o n s t r u c t a p o r t f o l i o such t h a t i t s e x p e c t e d r e t u r n i s v i and 43 B l a c k , F. and S c h o l e s , M., " D i v i d e d Y i e l d s and Common S t o c k R e t u r n s : A New Methodology," F i n a n c i a l Note No. 19B, M a s s a c h u s e t t s I n s t i t u t e o f Technology, S l o a n S c h o o l o f Management, August, 1971.  41  the p o r t f o l i o t o have minimum v a r i a n c e .  To s o l v e the e q u a t i o n s  the f i r s t o r d e r c o n d i t i o n s r e q u i r e s knowledge o f the b e t a expected  dividend yields,  and  the v a r i a n c e - c o v a r i a n c e  are unknown and must be e s t i m a t e d . aggregation  dividend y i e l d .  These p o r t f o l i o s  The  To r e d u c e measurement e r r o r s a method o f  a r e then  returns.  Securities  t r e a t e d as s e c u r i t i e s  d i v i d e n d y i e l d , and  Given  are  and  variance-covariance  b e t a c o e f f i c i e n t f o r a p o r t f o l i o i s d e t e r m i n e d by  i t s r e t u r n on the market r e t u r n , a f t e r both  the  a l l o f which  on t h e b a s i s o f t h e i r e s t i m a t e d b e t a c o e f f i c i e n t  beta c o e f f i c i e n t s , expected estimated.  coefficients,  matrix,  i s used t o form a s m a l l number o f p o r t f o l i o s .  assigned to p o r t f o l i o s  representing  these estimates,  s u b t r a c t i n g the i n t e r e s t the f i n a l p o r t f o l i o  and  their matrix  regressing  r a t e from  i s constructed f o r  d i f f e r e n t time p e r i o d s . There a r e a number o f major d e f i c i e n c i e s  w i t h t h i s methodology.  The  e s t i m a t i o n o f the b e t a c o e f f i c i e n t s n e g l e c t i n g t h e d i v i d e n d y i e l d , i m p l i e s t h a t there i s a m i s s i n g v a r i a b l e s problem, which w i l l estimated  coefficients.  This w i l l  cause b i a s i n the  a f f e c t the e s t i m a t i o n o f t h e  c o v a r i a n c e m a t r i x , which u s e s the e s t i m a t e d b e t a c o e f f i c i e n t s . t a i l e d knowledge o f how affect  the  varianceWithout  de-  t h e measurement e r r o r s o f the d i f f e r e n t v a r i a b l e s  f i n a l estimated  coefficients,  the a p p l i c a b i l i t y o f t h e method-  ology i s questionable. The  t h e s i s i n t r o d u c e s a new  methodology t o t h e t e s t i n g  v a r i a b l e models, t h a t o f p o o l i n g time s e r i e s ex-post  and  f o r m u l a t i o n o f t h e model used t o t e s t  the c o n s t a n t  term and  r u p t c y are n o t  firm  of  cross section data.  the h y p o t h e s i s  o f the  the c o e f f i c i e n t m u l t i p l y i n g the p r o b a b i l i t y  specific.  The  time s e r i e s  two  data  In  the  thesis,  o f bank-  for a l l individual  42  securities  a r e combined t o e s t i m a t e  t h e s e two  taneously estimating t h e . f i r m s p e c i f i c beta  c o e f f i c i e n t s , whilst simul-  coefficients.  CHAPTER I I I  PROBABILITY OF BANKRUPTCY  Bankruptcy  i n a single period context occurs i f ,  a t the terminal  p o i n t , t h e income o f t h e f i r m i s l e s s t h a n i t s f i x e d o b l i g a t i o n s . m u l t i - p e r i o d s e t t i n g such a d e f i n i t i o n  In a  i s n o t a p p r o p r i a t e , f o r i n an on-  g o i n g f i r m income c a n be l e s s t h a n t h e o b l i g a t i o n s o f t h e f i r m and y e t the f i r m i s n o t bankrupt;  i t can s i m p l y borrow more.  A t any p o i n t i n time,  t h e p r o b a b i l i t y o f a f i r m g o i n g bankrupt depends upon i t s a b i l i t y t o r a i s e funds, e i t h e r i n t e r n a l l y o r e x t e r n a l l y , t o c o v e r f i x e d c h a r g e s .  A firm  t h a t f a i l s t o c o v e r these f i x e d c h a r g e s i s s a i d t o be b a n k r u p t .  This  d e f i n i t i o n o f bankruptcy i s t h e b a s i c c o n s t r u c t i n f o r m u l a t i n g a model o f the p r o b a b i l i t y o f b a n k r u p t c y . The p r o b a b i l i t y o f a f i r m g o i n g bankrupt depends n o t o n l y upon i t s c u r r e n t l e v e l o f e a r n i n g s b u t a l s o i t s a b i l i t y t o r a i s e f u n d s , which i s subsumed i n i t s f u t u r e e a r n i n g power.  However, such v a r i a b l e s a r e ex-ante  i n n a t u r e and can n o t be d i r e c t l y o b s e r v e d .  To use a model t o e m p i r i c a l l y  e s t i m a t e the p r o b a b i l i t y o f b a n k r u p t c y r e q u i r e s t h a t the ex-ante be r e p l a c e d by e x - p o s t s u r r o g a t e s .  variables  The c o n s t r u c t i o n o f a model t o e s t i m a t e  the p r o b a b i l i t y o f b a n k r u p t c y i n terms o f e x - p o s t v a r i a b l e s i s d e s c r i b e d i n the f i r s t p a r t o f t h e c h a p t e r .  As t h e p r i m a r y f o c u s i s upon t h e p r e d i c t i o n  o f bankruptcy and n o t t o advance a complete  theory o f i t s determinants, a  second f o r m u l a t i o n u t i l i z i n g market v a l u e s f o r a p p r o p r i a t e c o r p o r a t e v a r i ables i s developed. When t h e ex-ante d e t e r m i n a n t s a r e r e p l a c e d by e x - p o s t s u r r o g a t e s , i t  43  44  i s necessary to estimate ables.  The  the  relative  c o n t r i b u t i o n o f the d i f f e r e n t  s t a t i s t i c a l methodology t o e s t i m a t e  proxy v a r i a b l e s i s d e s c r i b e d  i n the  the  vari-  c o e f f i c i e n t s of  second p a r t o f the  the  chapter.  As the p r o b a b i l i t y o f b a n k r u p t c y can n o t be o b s e r v e d , d i r e c t t e s t s on  the models are n o t p o s s i b l e .  are s p e c i f i e d must be  on  Thus, the main check on how  their predictive ability.  d i f f e r e n t methods by which the models can be p a r t o f the  The  w e l l the  d e t a i l s of  t e s t e d are given  models  three  i n the  last  chapter.  Theory I n a s i n g l e p e r i o d model a d e f i n i t i o n o f b a n k r u p t c y p r e s e n t s problem; a t the t e r m i n a l p o i n t i f the income o f the  f i r m i s l e s s than i t s  f i x e d o b l i g a t i o n s , then a s t a t e o f bankruptcy i s d e c l a r e d . o f the p e r i o d t o e s t i m a t e  the p r o b a b i l i t y o f the  terminal point i s equivalent to estimating f u t u r e income b e i n g period.  Pr where P r  (B) = P r  f i r m going bankrupt a t  the p r o b a b i l i t y o f the  (FI - FC  be  e x p r e s s e d i n the  of  the  firm's the  form  [3.1]  < 0),  FC  i s the  f i x e d c h a r g e s a t the  term on the r i g h t hand s i d e o f the above e x p r e s s i o n  o f the f i r m ' s income n e t o f a l l f i x e d c h a r g e s b e i n g In a m u l t i - p e r i o d c o n t e x t  a f i r m ' s income can be  more.  Apart  from b o r r o w i n g t h e r e  terminal  point.  i s the p r o b a b i l i t y  l e s s than  f i x e d o b l i g a t i o n s w i t h o u t b a n k r u p t c y o c c u r r i n g ; f o r the row  beginning  (B) i s the p r o b a b i l i t y o f b a n k r u p t c y a t the t e r m i n a l p o i n t ; F I i s  the f i r m ' s f u t u r e income; and The  A t the  l e s s t h a n the f i x e d o b l i g a t i o n s a t the end  I n m a t h e m a t i c a l n o t a t i o n t h i s may  no  zero.  l e s s than i t s f i r m can  simply  bor-  are many o t h e r means by w h i c h a f i r m  45  may be a b l e  t o obtain e x t r a sources o f funds: issuance  t i o n of trade  of equity,  utiliza-  c r e d i t , s e l l i n g o f a s s e t s , o r r e d u c t i o n o f i n v e s t m e n t programs  b e i n g p o s s i b l e examples.  The a b i l i t y o f a f i r m t o u t i l i z e  these  different  s o u r c e s depends e x t e n s i v e l y upon i t s s i z e , t h e n a t u r e o f i t s t e c h n o l o g y , f u t u r e p r o s p e c t s , m a n a g e r i a l a b i l i t y , and t h e p r e v a i l i n g and e x p e c t e d e c o nomic c o n d i t i o n s . Donaldson  1  p r o p o s e d t h r e e b r o a d c a t e g o r i e s o f funds a f i r m may  l i z e : uncommitted r e s e r v e s , r e d u c t i o n o f p l a n n e d o u t f l o w s , of assets.  Uncommitted r e s e r v e s  (cash, v e r y  l i q u i d a s s e t s ) , trade c r e d i t , negotiable  l o n g term d e b t , and i s s u a n c e  and l i q u i d a t i o n  e n t a i l s such f a c t o r s as i n s t a n t  of equity.  uti-  reserves  reserves, addition of  Reduction o f planned outflows  v o l v e s t h e r e v i s i n g o f e x i s t i n g commitments on o u t f l o w s  in-  o f funds; t h a t i s ,  the p o s s i b l e r e d u c t i o n o f i n v e s t m e n t programs and g e n e r a l  a u s t e r i t y measures.  L i q u i d a t i o n o f a s s e t s i s e i t h e r t h e s e l l i n g o f some o f t h e f i r m ' s a s s e t s , o r i n t h e extreme c a s e ,  t h e shutdown o f t h e f i r m .  F o r t h e s m a l l f i r m t h e number o f a l t e r n a t i v e s may n o t be as g r e a t . I t s a b i l i t y t o o b t a i n a commercial c r e d i t l o a n d u r i n g c r e d i t c o n d i t i o n s may be v e r y r e s t r i c t e d .  a period of tight  Due t o t h e h i g h  issue costs, i t  2 may n o t have a c c e s s  t o the e q u i t y markets.  e r a l reduction o f planned outflows ability  I t s c a p a c i t y t o c o n d u c t a gen-  may be v e r y  s m a l l , as m i g h t be i t s  t o engage i n t h e l i q u i d a t i o n o f a s s e t s .  ^Donaldson, G., " S t r a t e g y f o r F i n a n c i a l E m e r g e n c i e s , " Harvard ness Review, V o l . 47 (November-December, 1 9 6 9 ) , pp. 67-79.  Busi-  2 F o r an i n t r o d u c t o r y d i s c u s s i o n o f some o f the l i m i t i n g f a c t o r s see Duesenberry, J . S., " C r i t e r i a f o r J u d g i n g t h e Performance o f C a p i t a l M a r k e t s , " r e p r i n t e d i n Elements of Investment, e d i t e d by Wu, II. K. and Zakon, A. J . (New York: H o l t , R i n e h a r t and Winston, I n c . , 1 9 6 5 ) .  46  An borrow.  i m p o r t a n t s o u r c e o f funds f o r f i r m s  derives  But t h i s a b i l i t y depends upon t h e w i l l i n g n e s s  tions to lend.  from t h e a b i l i t y t o of financial institu-  Among t h e major f i n a n c i a l i n s t i t u t i o n s banks have been  a c t i v e l y engaged i n e x t e n d i n g c r e d i t t o b u s i n e s s e s .  J a f f e e and M o d i g l i a n i ^  have d e v e l o p e d a s i m p l e model t o d e t e r m i n e t h e r a t i o n a l i t y and e x t e n t o f c r e d i t r a t i o n i n g i n a commercial l o a n market.  The b a s i s o f t h e model i s t h e  d e r i v a t i o n o f t h e bank's s u p p l y c u r v e f o r l o a n s b a s e d upon t h e assumption t h a t banks a c t t o maximize e x p e c t e d p r o f i t s and from c o n s i d e r i n g demand f u n c t i o n  for a loan.  I t i s shown t h a t  the firm's  i f a bank i s a d i s c r i m i n a t i n g  m o n o p o l i s t f r e e t o charge each customer a d i f f e r e n t r a t e , then c r e d i t rationing w i l l not occur. number o f r i s k c l a s s e s  However, i f banks d i v i d e f i r m s  into a  small  and charge each c l a s s a d i f f e r e n t r a t e , t h e n i n  g e n e r a l i t w i l l be o p t i m a l f o r t h e bank t o r a t i o n c r e d i t .  The e x c e p t i o n t o  t h i s b e i n g i f t h e f i r m i s c l a s s i f i e d as r i s k f r e e , f o r t h e n i t i s u n p r o f i t able  f o r t h e bank t o l i m i t c r e d i t .  that for a firm not c l a s s i f i e d  The e x i s t e n c e  of credit rationing  as r i s k f r e e , t h e r e i s a l i m i t t o t h e amount  t h a t i t can borrow, which i s dependent upon t h e b a n k i n g s t r u c t u r e s t a t e o f t h e economy.  Whilst  implies  and t h e  t h e assumption i s made t h a t a f i r m borrows  from a bank, t h e model i s r e a d i l y a p p l i c a b l e  t o other types o f f i n a n c i a l i n -  stitutions. In a m u l t i - p e r i o d  context the p r o b a b i l i t y o f a f i r m going bankrupt  i s determined by i t s a b i l i t y t o cover f i x e d charges e i t h e r w i t h i t s cash flow o r by r a i s i n g f u n d s .  Thus t h e p r o b a b i l i t y o f b a n k r u p t c y can be  repre-  s e n t e d i n t h e form 3  J a f f e e , D. M. and M o d i g l i a n i , F., "A Theory and T e s t o f C r e d i t R a t i o n i n g , " American Economic Review, V o l . L I X , No. 5 (December, 1 9 6 9 ) , p p . 850-872.  47  Pr  (B ) = PR ( F I fc  fc  - FC  fc  + MB  fc  + AS  fc  < 0)  [3.2]  where P r (B^) i s t h e p r o b a b i l i t y o f t h e e v e n t o f b a n k r u p t c y end o f p e r i o d t ; F I ^ i s t h e f i r m ' s f u t u r e income; FC MB  fc  i s t h e maximum amount t h e f i r m c o u l d borrow; AS  fc  fc  occurring a t the  i s the fixed i s a l l other  charges; alternative  s o u r c e s o f f u n d s ; and t h e time s u f f i x , t , i s used t o denote t h a t t h e v a r i a b l e s a r e v a l u e d a t t h e end o f t h e p e r i o d t . The represent  first  two terms on t h e r i g h t hand s i d e o f t h e above e x p r e s s i o n  t h e f i r m ' s f u t u r e income n e t o f a l l f i x e d c h a r g e s .  The magnitude  and c h a r a c t e r i s t i c o f t h i s term w i l l depend upon t h e f i r m ' s f i n a n c i a l  struc-  t u r e and t h e type and s t a t e o f t h e p r o d u c t and r e s o u r c e markets i n w h i c h i t d e a l s ; t h a t i s , the competitiveness o f the markets, t h e i r c y c l i c a l and e x t e r n a l f a c t o r s .  behaviour,  F o r example, i f t h e r e i s economic r e c e s s i o n t h e f i r m ' s  p r o d u c t and r e s o u r c e markets may be a f f e c t e d , thus c a u s i n g changes i n i t s net cash flow.  The f i r m ' s p r o d u c t d i v e r s i f i c a t i o n , and i t s t e c h n o l o g y  also influence i t s a b i l i t y  to stabilize  will  i t s cash flows a g a i n s t c y c l i c a l  b e h a v i o u r and e x t e r n a l f a c t o r s . The borrow.  t h i r d term r e p r e s e n t s t h e maximum amount t h a t t h e f i r m c o u l d  T h i s depends upon t h e b a n k i n g  s t r u c t u r e o f t h e economy, and t h e  r i s k c h a r a c t e r i s t i c s o f t h e f i r m , a s p e r c e i v e d by a bank.  Over t i m e , as  economic c o n d i t i o n s i n t h e economy and t h e r i s k c h a r a c t e r i s t i c s o f t h e f i r m change, so w i l l  t h e amount o f c r e d i t r a t i o n i n g and thus t h e b o r r o w i n g  power  of the f i r m . The  l a s t term r e p r e s e n t s t h e t o t a l o f a l l o t h e r a l t e r n a t i v e  sources  4 o f funds a f i r m may u t i l i z e .  The n a t u r e o f such s o u r c e s , which Donaldson  4 Donaldson,  Loo. cvt.  48  describes i n d e t a i l ,  c o n s i s t s o f t h r e e b r o a d c a t e g o r i e s : uncommitted r e -  serves, reduction o f planned outflows, a v a i l a b i l i t y o f these n o l o g y , and f u t u r e The it  sources  and l i q u i d a t i o n o f a s s e t s .  The  depends upon t h e type o f f i r m , i t s s i z e ,  tech-  prospects.  f i r m ' s a b i l i t y t o r a i s e funds i s d e s c r i b e d b y t h e maximum amount  c o u l d borrow and a l l o t h e r  not independent.  a l t e r n a t i v e sources.  These q u a n t i t i e s a r e  A f i r m may be a b l e t o borrow u s i n g an a s s e t as c o l l a t e r a l ,  o r i t may i s s u e a d e b e n t u r e w i t h a n e g a t i v e p l e d g e c l a u s e p r o h i b i t i n g i t from p l e d g i n g t h e a s s e t t o o t h e r c r e d i t o r s . The  I t w i l l n o t be a b l e t o do b o t h .  d i f f e r e n t means b y which a f i r m may be a b l e t o u t i l i z e a l t e r n a t i v e  sources  o f funds a r e a l s o n o t i n d e p e n d e n t .  represents  I f a f i r m i s s u e s d e b t , then  this  a c l a i m a g a i n s t f u t u r e e a r n i n g s , w h i c h may i n h i b i t i t s a b i l i t y  to issue equity. The  a v a i l a b i l i t y o f the d i f f e r e n t sources  by w h i c h a f i r m may be  a b l e t o r a i s e funds a r e dependent upon c e r t a i n common f a c t o r s : t h e e x i s t i n g f i n a n c i a l s t r u c t u r e o f the f i r m , i t s operating  c h a r a c t e r i s t i c s , and t h e  future prospects  I f the future prospects f o r  o f t h e f i r m and t h e economy.  the f i r m a r e p o o r , then t h i s may have a d e c r e m e n t a l e f f e c t upon i t s a b i l i t y t o borrow, t o i s s u e d e b t , o r e q u i t y . v a r i o u s sources  p r o h i b i t s unique e m p i r i c a l i d e n t i f i c a t i o n o f t h e r e l a t i v e  contribution o f the underlying r a i s e funds v i a p a r t i c u l a r Ex-Post  T h i s i n t e r d e p e n d e n c e between the.,  f a c t o r s which d e t e r m i n e a f i r m ' s a b i l i t y t o  sources.  Formulation To use t h e model t o e m p i r i c a l l y e s t i m a t e  t h e p r o b a b i l i t y o f bank-  r u p t c y r e q u i r e s t h e ex-ante v a r i a b l e s be r e p l a c e d by e x - p o s t  surrogates.  49  However, r e a l i z e d v a l u e s  o f the maximum amount the  t o t a l o f a l l a l t e r n a t i v e s o u r c e s are not v a r i a b l e s must be  constructed.  which c o n t r i b u t e t o the  f i r m c o u l d borrow o r  r e a d i l y observable  and  so p r o x y  T h i s r e q u i r e s t h a t the u n d e r l y i n g  f i r m ' s a b i l i t y t o r a i s e funds be  the  factors  identified  and  measured. The  v a r i a b l e s determining  i n expression d a t a w i l l be f i r m s and  [ 3 . 2 ] , are i n terms o f d o l l a r amounts. u s e d , the v a r i a b l e s are n o t  so w i l l be dominated by  among f i r m s would be ficients. the  the p r o b a b i l i t y o f b a n k r u p t c y , as  To  adjusted  scale effects.  As  stated  cross s e c t i o n a l  for differences i n size  of  Very l a r g e s c a l e e f f e c t s  expected to l e a d to i n e f f i c i e n t e s t i m a t i o n of  coef-  a v o i d t h i s , the p r o b a b i l i t y o f b a n k r u p t c y can be w r i t t e n i n  form Fl (B. ) = Pr  Pr  (—  * where A  - FC  MB  AS  —4  - + t-1  < 0) , A  i s the book v a l u e o f the  fc  +  [3.3]  t-1  f i r m ' s a s s e t s a t the  Thus the p r o b a b i l i t y o f a f i r m g o i n g  b a n k r u p t a t the end  start of period o f time p e r i o d  t. t  depends upon i t s f u t u r e c a s h f l o w n e t o f a l l f i x e d c h a r g e s p e r u n i t o f a s s e t s and  the  Whilst  t o t a l amount o f funds t h a t i t c o u l d r a i s e p e r u n i t o f the ex-ante v a l u e s  are not observable,  assets.  r e a l i z e d values  of  the f i r m ' s cash f l o w n e t o f a l l f i x e d c h a r g e s are r e a d i l y a v a i l a b l e and be u s e d t o form an e x - p o s t s u r r o g a t e . time and value  then the e s t i m a t e d  o f the  The  e x - p o s t d a t a are  regression equation  used t o p r e d i c t the  f i r m ' s cash f l o w n e t o f a l l f i x e d c h a r g e s .  d i v i d e d by the book v a l u e o f the as the e x - p o s t s u r r o g a t e .  regressed  f i r m ' s t o t a l a s s e t s and  This value the  can  against future  i s then  r e s u l t a n t used  50  N e i t h e r ex-ante n o r r e a l i z e d v a l u e s o f the maximum amount t h a t a f i r m c o u l d borrow a r e o b s e r v a b l e .  In any p a r t i c u l a r p e r i o d  only that  quan-  t i t y o f debt t h e f i r m a c t u a l l y borrowed from a bank can be r e a d i l y  deter-  mined, b u t such an amount need n o t n e c e s s a r i l y  the f i r m  could  have borrowed.  Jaffee  and M o d i g l i a n i ^  be t h e maximum t h a t  have shown under s i m p l i s t i c  assumptions t h a t t h e maximum amount a f i r m can borrow t o f i n a n c e ment p r o j e c t  i s g i v e n by  L =  1 1 + r. i  where L i s t h e maximum amount w h i c h t h e bank w i l l  r a t e o f i n t e r e s t t h e bank c h a r g e s t o f i r m s F  ( ) i s t h e bank's s u b j e c t i v e  evaluation  t r i b u t i o n o f t h e outcome o f t h e p r o j e c t ; rate.  From an o p e r a t i o n a l  applied,  an i n v e s t -  lend  the firm; r . i s the i th  assigned t o the i  risk class;  o f the cumulative p r o b a b i l i t y d i s and p i s t h e bank's o p p o r t u n i t y  v i e w p o i n t t h e above e q u a t i o n c a n n o t be d i r e c t l y  as many o f t h e terms c a n n o t be o b s e r v e d .  v a l u e f o r i t shows how t h e bank's o p p o r t u n i t y r a t e  However,  i t i s still of  and thus c r e d i t  a f f e c t s t h e maximum amount a f i r m can borrow.  Also,  increases,  g r e a t e r decrease i n the maxi-  t h e n t h e r e w i l l be a p r o p o r t i o n a l l y  mum amount t h e f i r m c a n borrow, i m p l y i n g a n o n - l i n e a r  i fcredit  rationing  rationing  relationship.  The a b i l i t y o f t h e f i r m t o borrow w i l l depend upon t h e amount o f credit rationing quantity  o f d e b t which i t c a n u t i l i z e .  rationing, the  i n t h e economy, i t s c u r r e n t  l e v e l o f d e b t , and t h e o p t i m a l  The g r e a t e r t h e amount o f c r e d i t  t h e l e s s t h e r i s k y f i r m w i l l be a b l e t o borrow.  f i r m t o borrow w i l l be enhanced t h e l a r g e r Jaffee  and M o d i g l i a n i ,  loo. oit.  The a b i l i t y o f  the d i f f e r e n c e  between t h e  51  optimal  l e v e l and the c u r r e n t l e v e l o f d e b t .  t h i s d i f f e r e n c e i s t h e book v a l u e  An approximate measure o f  o f n e t worth, w h i c h can be i n t u i t e d as  d e s c r i b i n g t h a t p a r t o f t h e f i r m ' s a s s e t s n o t f i n a n c e d by d e b t .  The p r o x y  v a r i a b l e used f o r t h e ex-ante maximum amount t h e f i r m c o u l d borrow p e r u n i t asset for period t i s .book v a l u e o f n e t w o r t h a t t - l , , ( ) exp (-CR \ - l t  . ), 1  where A. , i s t h e book v a l u e o f t h e f i r m ' s a s s e t s a t time t - l ; t-l the amount o f c r e d i t r a t i o n i n g a t time t - l .  and CR^ .. t - l  is  The s m a l l e r t h e book v a l u e o f  n e t worth r e l a t i v e t o t h e f i r m ' s t o t a l a s s e t s , t h e l e s s t h e f i r m w i l l be a b l e t o borrow.  The f u n c t i o n a l form o f dependence on c r e d i t r a t i o n i n g i s  used t o a c c o u n t f o r t h e n o n - l i n e a r r e l a t i o n s h i p between c r e d i t r a t i o n i n g and  t h e amount a f i r m c a n borrow. The  f i n a l determinant o f the p r o b a b i l i t y o f bankruptcy i s the t o t a l 6  of a l l other a l t e r n a t i v e sources  o f funds a f i r m may u t i l i z e .  As Donaldson  o u t l i n e d , t h i s i s dependent upon t h r e e b r o a d c a t e g o r i e s : uncommitted r e serves, reduction of planned outflows,  and l i q u i d a t i o n o f a s s e t s .  There  are many f a c t o r s w h i c h a f f e c t t h e a g g r e g a t e t o t a l o f funds t h a t c a n be obtained  from these  pects, business  different  risk,  sources: o p e r a t i o n a l e f f i c i e n c y ,  f i n a n c i a l r i s k b e i n g o f prime i m p o r t a n c e .  dence o f t h e f i r m ' s a b i l i t y  t o r a i s e funds,  The depen-  The g r e a t e r  c i e n c y o f t h e f i r m t h e more a b l e i t i s t o cope w i t h r e d u c t i o n s o r t o undertake the l i q u i d a t i o n o f a s s e t s .  ^Donaldson, loc. cit.  pros-  either internally or externally,  upon o p e r a t i o n a l e f f i c i e n c y a r i s e s f o r two r e a s o n s .  outflows,  future  the e f f i -  i n planned  F o r the p o t e n t i a l  52  investor  the more e f f i c i e n t the  investment p r o p o s i t i o n . its  The  f i r m t h e n the more a t t r a c t i v e i t i s as  future  a b i l i t y to r a i s e external  prospects of  sources of funds.  the  very d i f f i c u l t it the  t o a t t r a c t c a p i t a l , as  remains w i t h i n  the  industry.  f i r m a r i s i n g from the  to i t s debt c a p a c i t y . and  firm's  i t s future  The  more v a r i a b l e  the  low  c a s h f l o w , the  debt f i n a n c i n g .  financing.  t o t a l of a l t e r n a t i v e sources, a l i n e a r function used as a p r o x y j the  The  be  bleak i f  o v e r a l l r i s k to pertains  g r e a t e r the  risk  Financial risk arises lower the f i n a n c i a l cash flows,  To measure the of these four  the  aggregate attributes  is  variable.  O p e r a t i o n a l e f f i c i e n c y s h o u l d be  firm's  p r o s p e c t s w i l l be  u t i l i z a t i o n o f d e b t or s t a b l e  more a b l e i t i s t o a t t r a c t e x t e r n a l  in-  obsolescence, i t w i l l  v a r i a b i l i t y o f i t s o p e r a t i n g income, and  a b i l i t y to cover f i x e d charges.  r i s k , e i t h e r because o f  f i r m i s i n an  B u s i n e s s r i s k measures the  thus r e s t r i c t s i t s a b i l i t y t o use  from the  firm d i r e c t l y affect  I f the  d u s t r y which i s d e c l i n i n g because o f t e c h n o l o g i c a l  an  assets,  abstracting  from tax  a measure o f the p r o d u c t i v i t y or  leverage f a c t o r s .  c i a l r a t i o s have been used as p r o x y v a r i a b l e s .  P i n c h e s and  Various Mingo  7  use  of finannet  g  income d i v i d e d by t i v e r a t i o s : net t o t a l debt. abstract  t o t a l assets,  w h i l s t Beaver  income t o s a l e s , n e t  income t o n e t w o r t h , and  However, a l l o f t h e s e measures are  from the  e f f e c t s o f the  suggests three other  firm's  d e f i c i e n t , as  f i n a n c i a l structure  net  alternaincome t o  t h e y do  and  not  thus a r e  not  7 P i n c h e s , G. E. and Mingo, K. A., "A M u l t i v a r i a t e t r i a l Bond R a t i n g s , " Journal of Finance, V o l . XXVIII, No. pp. 1 - 1 8 .  A n a l y s i s o f Indus1 (March, 1 9 7 3 ) ,  8  Beaver, W.  H.,  " F i n a n c i a l R a t i o s as P r e d i c t o r s o f F a i l u r e , " Empirisupplement to Journal of  cal Research in Accounting: Selected Studies, Accounting Research ( 1 9 6 6 ) , pp. 7 7 - 1 1 1 .  53  a c c u r a t e measures o f t h e u t i l i z a t i o n o f t h e f i r m ' s a s s e t s .  An a l t e r n a t i o n  9 f o r m u l a t i o n by Altman  using earnings  t o t a l assets avoids t h i s d e f i c i e n c y .  before  i n t e r e s t and t a x e s  d i v i d e d by  I t a b s t r a c t s from t a x o r l e v e r a g e  f a c t o r s , and i s a measure o f t h e f i r m ' s e a r n i n g power.  This formulation i s  used i n t h e t h e s i s . To a t t a i n t h e f u t u r e p r o s p e c t s  o f t h e f i r m and thus i t s a b i l i t y t o  a t t r a c t c a p i t a l r e q u i r e s measuring t h e p r o f i t a b i l i t y o f t h e f i r m ' s investment o p p o r t u n i t i e s , t h e i r s i z e and d u r a t i o n . are n o t d i r e c t l y o b s e r v a b l e . t o t h e same problem, f o c u s e d  future  A l l o f these q u a n t i t i e s  M i l l e r and M o d i g l i a n i ^ a d d r e s s i n g 1  themselves  upon t h e most t r a c t a b l e component, t h e l e v e l o f  investment o p p o r t u n i t i e s , as an o v e r a l l measure o f growth and f u t u r e pects.  pros-  F o r an e m p i r i c a l e s t i m a t o r o f t h e l e v e l o f i n v e s t m e n t o p p o r t u n i t i e s  p e r u n i t a s s e t , a l i n e a r f i v e y e a r growth r a t e o f t o t a l a s s e t s i s used. T h i s measure i s used i n t h e t h e s i s . Business  r i s k d e s c r i b e s t h e r i s k t o t h e f i r m t h a t a r i s e s from t h e  v a r i a b i l i t y o f i t s o p e r a t i n g income, a b s t r a c t i n g from t a x o r l e v e r a g e tors .  fac-  The debt c a p a c i t y o f t h e f i r m depends upon t h e v a r i a b i l i t y o f i t s  cash f l o w and thus b u s i n e s s  r i s k : t h e more r e s p o n s i v e  t o changes i n t h e economy, t h e lower t h e o p t i m a l f i r m can use.  Van H o m e  1 1  t h e f i r m ' s cash  flow  amount o f debt which t h e  uses t h e c o e f f i c i e n t o f v a r i a t i o n o f o p e r a t i n g  9 Altman, E . I . , " F i n a n c i a l R a t i o s , D i s c r i m i n a n t A n a l y s i s and t h e P r e d i c t i o n o f C o r p o r a t e Bankruptcy," Journal of Finance, V o l . X X I I I , No. 4 (September, 1968), pp. 589-609. M i l l e r , M. and M o d i g l i a n i , F., "Some E s t i m a t e s o f t h e C o s t o f C a p i t a l t o t h e E l e c t r i c U t i l i t y I n d u s t r y , 1954-1957," American Economic 1 0  Review,  V o l . LVI, No. 3 (June, 1 9 6 6 ) , pp. 333-391.  "Van Home, J . , Financial P r e n t i c e H a l l I n c . , 1972). 1;1  Management and Policy  (New J e r s e y :  54  income t o measure b u s i n e s s r i s k .  However, t h i s does n o t d i r e c t l y measure  the r e s p o n s i v e n e s s o f the f i r m ' s cash f l o w t o changes i n the economy. a l t e r n a t i v e f o r m u l a t i o n , and one  t h a t i s used  i n the t h e s i s ,  An  i s t o measure  b u s i n e s s r i s k by the a b s o l u t e v a l u e o f the p r o p o r t i o n a l change i n s a l e s t o the p r o p o r t i o n a l change i n g r o s s n a t i o n a l p r o d u c t ; the more r e s p o n s i v e to changes i n the economy the g r e a t e r the b u s i n e s s  sales  risk.  F i n a n c i a l r i s k i s a measure o f the f i r m ' s a b i l i t y t o c o v e r i t s f i s c a l charges.  The  greater i t s a b i l i t y ,  the lower the f i n a n c i a l r i s k and  a b l e i t s h o u l d be t o a t t r a c t e x t e r n a l f i n a n c i n g .  the more  To measure f i n a n c i a l  risk  12 Altman  s u g g e s t s two p o s s i b l e f i n a n c i a l r a t i o s : market o f e q u i t y d i v i d e d  the book v a l u e o f t o t a l d e b t , and the book v a l u e o f n e t worth d i v i d e d by book v a l u e o f t o t a l d e b t .  Both r a t i o s a r e d e f i c i e n t ,  f o r t h e y do n o t  s a r i l y take account o f a l l f i x e d c h a r g e s which the f i r m must meet. o f the book v a l u e o f n e t worth does n o t measure the f i r m ' s a b i l i t y f i x e d charges.  The a b i l i t y o f a f i r m t o c o v e r i t s f i x e d charges  depends upon i t s f u t u r e c a s h f l o w , i t s v a r i a b i l i t y , and charges which i t c o v e r s .  Norton ^ 1  by the  neces-  The  use  to cover  primarily  the t o t a l o f  fixed  uses t h e c o e f f i c i e n t o f v a r i a t i o n o f t h e  f i r m ' s p a s t income over and above the amount o f f i x e d c h a r g e s .  T h i s measure  i s d e f i c i e n t f o r the f i r m ' s a b i l i t y t o meet f i x e d c h a r g e s depends upon i t s f u t u r e income as opposed t o p a s t income.  The p r o x y v a r i a b l e used i n t h e  t h e s i s i s the d i f f e r e n c e between the f i r m ' s f i x e d charges  and i t s f u t u r e  c a s h f l o w , the d i f f e r e n c e b e i n g d i v i d e d by the s t a n d a r d d e v i a t i o n o f t h e  12  .  Altman, loc.  cit.  13 N o r t o n , J . , "The  Theory  o f Loan C r e d i t i n R e l a t i o n t o C o r p o r a t i o n  Economics," Publications of the American Economic Association, V o l . V (1904), pp. 278-300.  3rd s e r . ,  55  f u t u r e cash f l o w . its  The  f i x e d charges  and  s m a l l e r the v a r i a b l e , the more a b l e the the lower  the f i n a n c i a l r i s k .  The  firm to  cover  f i r m ' s f u t u r e cash  f l o w i s e s t i m a t e d by r e g r e s s i n g r e a l i z e d v a l u e s o f i t s o p e r a t i n g income a g a i n s t time and future value.  then the e s t i m a t e d r e g r e s s i o n e q u a t i o n used  The  square  r o o t o f the r e s i d u a l sum  o f squares  e s t i m a t e o f the s t a n d a r d d e v i a t i o n o f the f i r m ' s f u t u r e c a s h Combining t h e p r o x y  to p r e d i c t i s used  the  as  an  flow.  v a r i a b l e s f o r the f i r m ' s f u t u r e c a s h f l o w n e t  o f a l l f i x e d c h a r g e s p e r u n i t a s s e t s , the maximum amount i t c o u l d borrow p e r u n i t a s s e t s , and  the t o t a l o f a l l other a l t e r n a t i v e  sources per  unit  assets, gives F I —FC t t t-1  A  MB +  AS  t t-1  t _ g g t-1 °  ^estimated f u t u r e cash flow net of a l l f i x e d t-1  +  +  A  A  „  +3 ( 2  .book v a l u e o f n e t worth a t t - l . j ) exp t-1  , (-CR  A  + 3 (.earning b e f o r e i n t e r e s t and 0  3  A  tax a t  p 5  +  g  _  i  t - l .)  t-1 assets)  1 p r o p o r t i o n a l change i n s a l e s i . ' p r o p o r t i o n a l change i n GNP ' f i x e d charges  6  . ) t  + 8 4 ( f i v e y e a r l i n e a r growth r a t e i n t o t a l +  charges^  A  at t - l - estimated  f u t u r e cash  estimated standard d e v i a t i o n of f u t u r e cash  flow^ flow [3.4]  +e, where 8 o » 3 i , . - . / 8 s a r e unknown c o e f f i c i e n t s ;  and  e i s a z e r o mean random v a r -  i a b l e e r r o r term, which i s assumed t o be o f u n i t v a r i a n c e and u n c o r r e l a t e d between f i r m s . The  c o e f f i c i e n t s i n the above e q u a t i o n r e p r e s e n t the r e l a t i v e  b u t i o n s o f the d i f f e r e n t u n d e r l y i n g f a c t o r s t o the aggregate  contri-  t o t a l o f net  56  funds a v a i l a b l e t o the f i r m .  The  c o e f f i c i e n t s can n o t be e s t i m a t e d by  re-  g r e s s i o n , as t h e dependent v a r i a b l e i s an ex-ante q u a n t i t y which can n o t measured o r o b s e r v e d .  Substituting  f o r the p r o b a b i l i t y o f b a n k r u p t c y ~  »  -  n  «  E q u a t i o n [3.4] i n t o the e x p r e s s i o n  Pr (B ) = P r [ e < Bo+Bi ( Z  A  .book v a l u e o f n e t worth a t t - 1 , j: ) exp t-1  . (-CR  A  „ +g  .earnings b e f o r e i n t e r e s t and (  3  A  g "*  +  g  1  t a x e s a t t-1^ ;  t-1 assets)  j p r o p o r t i o n a l change i n s a l e s 1^ p r o p o r t i o n a l change i n GNP ' ^fixed charges  6  -—)  ) C  + 8 4 ( f i v e y e a r l i n e a r growth r a t e f o r t o t a l +  charges.  t-1  +8 < 2  [3.3]  gives  .estimated f u t u r e cash f l o w n e t o f a l l f i x e d  „  be  a t t - 1 - estimated f u t u r e cash  flow^1  estimated standard d e v i a t i o n of f u t u r e cash flow  where t h e c o e f f i c i e n t s have been r e d e f i n e d t o i n c l u d e the minus s i g n . p r o b a b i l i t y o f t h e e v e n t o f the f i r m g o i n g b a n k r u p t  ' I-3.5J The  a t t h e end o f p e r i o d t  i s the p r o b a b i l i t y o f t h e random v a r i a b l e e r r o r term minus t h e summation o f the u n d e r l y i n g f a c t o r s which c o n t r i b u t e t o t h e n e t t o t a l o f funds t o the f i r m b e i n g l e s s than z e r o .  Apart  from the random e r r o r term, a l l t h e  v a r i a b l e s on t h e r i g h t hand s i d e o f the e q u a t i o n a r e e x - p o s t measured.  The  s i g n s o f the c o e f f i c i e n t s can be d e t e r m i n e d  considerations.  and  and  from  can  be  theoretic  U s i n g a c e t e r i s p a r i b u s argument, the g r e a t e r the f i r m ' s  f u t u r e cash f l o w n e t o f a l l f i x e d c h a r g e s , borrow, the lower  available  and  the amount which i t c o u l d  the p r o b a b i l i t y o f b a n k r u p t c y .  82 s h o u l d be n e g a t i v e .  Thus the c o e f f i c i e n t s  S i m i l a r l y f o r t h e c o e f f i c i e n t s 83 and  Bi+, as  g r e a t e r the e f f i c i e n c y o f the f i r m , the b e t t e r i t s f u t u r e p r o s p e c t s , the  Bi the  57  more a b l e i t i s t o r a i s e e x t r a s h o u l d be p o s i t i v e ;  sources o f funds.  the g r e a t e r t h e b u s i n e s s r i s k , as measured by t h e v a r i -  a b i l i t y o f t h e f i r m ' s cash f l o w s , and t h e l a r g e r l e s s able the f i r m t o a t t r a c t extra of  The c o e f f i c i e n t s 85 and 86  the f i n a n c i a l r i s k , the  c a p i t a l and t h e g r e a t e r t h e p r o b a b i l i t y  bankruptcy.  Predictive  Model  A complete model f o r t h e p r o b a b i l i t y o f b a n k r u p t c y s h o u l d all  the i n t e r a c t i o n s  between t h e d i f f e r e n t f a c t o r s .  The f i r m ' s a b i l i t y t o  borrow o r t o i s s u e d e b t i s dependent upon i t s debt c a p a c i t y . capacity  describe  But debt  i s dependent upon t h e p r o b a b i l i t y o f b a n k r u p t c y and thus t h e r e i s  a circularity.  The a b i l i t y  t o use a p a r t i c u l a r s o u r c e o f funds i s dependent  upon t h e u t i l i z a t i o n o f o t h e r s o u r c e s .  I f a firm issues  d e b t , t h i s may have  a d e c r e m e n t a l e f f e c t upon i t s a b i l i t y t o borrow from a bank o r t o i s s u e equity.  Due t o t h e complex i n t e r a c t i o n o f t h e u n d e r l y i n g f a c t o r s  d i f f i c u l t y o f measuring t h e i r magnitude and a v a i l a b i l i t y , a second t i o n u s i n g market v a l u e s f o r t h e a p p r o p r i a t e v a r i a b l e s use  can  t o measure such q u a n t i t i e s  The  constructing  as t h e maximum amount t h e f i r m  borrow and t h e t o t a l o f a l l o t h e r a l t e r n a t i v e The  formula-  i s developed.  o f market v a l u e s c i r c u m v e n t s many o f t h e d i f f i c u l t i e s o f  proxy v a r i a b l e s  and t h e  sources.  p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t depends upon i t s f u t u r e  in-  come n e t o f a l l f i x e d c h a r g e s , t h e maximum amount i t can borrow and a l l other alternative  sources o f funds.  To use t h e model t o e m p i r i c a l l y  the p r o b a b i l i t y o f b a n k r u p t c y r e q u i r e s by  ex-post  surrogates.  that  the ex-ante v a r i a b l e s  estimate  be r e p l a c e d  58  F o r the  firm's future cash  f l o w net o f a l l f i x e d c h a r g e s the same  p r o x y v a r i a b l e , as p r e v i o u s l y d e f i n e d i s u s e d ; t h a t i s , r e a l i z e d v a l u e s the f i r m ' s cash and  flow n e t o f a l l f i x e d c h a r g e s are r e g r e s s e d  then the e s t i m a t e d  o f the  f i r m ' s cash  regression equation  as the e x - p o s t The  used t o p r e d i c t the  flow net o f a l l f i x e d charges.  d i v i d e d by t h e book v a l u e o f the  against  This value  f i r m ' s t o t a l a s s e t s and  a b i l i t y o f the  future  value  i s then  the r e s u l t a n t used  f i r m t o borrow depends upon the amount o f  o f debt which i t can u t i l i z e .  The  the o p t i m a l  and  level  current l e v e l of debt.  measure t h i s d i f f e r e n c e r e q u i r e s t h a t the d e b t c a p a c i t y o f the  the  To  f i r m be known.  However, debt c a p a c i t y depends upon t h e p r o b a b i l i t y o f b a n k r u p t c y , t h a t the e x p l a n a t o r y  credit  f i r m ' s a b i l i t y t o borrow i s enhanced  l a r g e r the d i f f e r e n c e between the o p t i m a l  implying  v a r i a b l e i s a f u n c t i o n o f the dependent v a r i a b l e .  market v a l u e o f e q u i t y f o r a f i r m , w h i c h r e f l e c t s the p r o b a b i l i t y b a n k r u p t c y , i s a measure o f i t s b o r r o w i n g a b i l i t y .  For a given  The  of  level  a s s e t s , the g r e a t e r the market v a l u e o f e q u i t y , the more a b l e the The  time  surrogate.  r a t i o n i n g i n t h e economy, i t s c u r r e n t l e v e l o f d e b t and  borrow.  of  of  f i r m to  p r o x y v a r i a b l e used t o measure the ex-ante maximum amount the  f i r m c o u l d borrow p e r u n i t o f a s s e t s f o r p e r i o d t i s .market v a l u e o f e q u i t y a t t - 1 . ( * ~ ) exp A  where A  fc  t-1  , „„ (-CR t  , ) , 1  ^ i s the book v a l u e o f the f i r m ' s a s s e t s a t time t - 1 ; and  the amount o f c r e d i t r a t i o n i n g a t time The  t o t a l of a l l other  CR  fc  ^ is  t-1.  a l t e r n a t i v e sources  depends upon t h r e e b r o a d c a t e g o r i e s : uncommitted  o f funds f o r the  firm  reserves, reduction  of  59  planned  outflows,  and  the l i q u i d a t i o n o f a s s e t s .  utilize  these d i f f e r e n t  The  a b i l i t y o f the  firm  to  s o u r c e s p r i m a r i l y depends upon i t s o p e r a t i o n a l  e f f i c i e n c y , future prospects, business  r i s k and  financial risk.  A variable  which s y n t h e s i z e s these d i v e r s e q u a n t i t i e s i s the market v a l u e o f e q u i t y . U s i n g a c e t e r i s p a r i b u s argument, the more e f f i c i e n t l y a s s e t s , o r the b r i g h t e r i t s f u t u r e p r o s p e c t s , value of i t s equity.  the f i r m u t i l i z e s i t s  the g r e a t e r i s the market  S i m i l a r l y , the lower the b u s i n e s s  o f the f i r m , t h e g r e a t e r i t ' s market v a l u e o f e q u i t y .  and  financial  For a given l e v e l  a s s e t s , the g r e a t e r the market v a l u e o f e q u i t y the more a b l e i s the generate  and a t t r a c t e x t r a s o u r c e s o f Thus, the e x - p o s t  surrogates  risk of  firm  to  f o r the ex-ante n e t a g g r e g a t e t o t a l  of  funds.  funds a v a i l a b l e t o the f i r m , can be w r i t t e n : F I —FC t t  —  A  t-1  MB  + — A  AS t  t-1  —  + A  t  t-1  .estimated  = Yo+Yl (  f u t u r e cash flow net of a l l f i x e d  £  A  +  Y  2  A  + Y 3  (  1  H  )  ,  e x p  .  (_  t-1  )  CR t _ 1  .market v a l u e o f e q u i t y a t  *  A  t-l.  )  t-1  [3.6]  +n, where  Yo,Yl#Y2/  an<  ^ Y3  a  r  e  unknown c o e f f i c i e n t s ;  and  n i s a z e r o mean random  v a r i a b l e e r r o r term, w h i c h i s assumed t o be o f u n i t v a r i a n c e and r e c t e d between f i r m s .  S u b s t i t u t i n g Equation  f o r the p r o b a b i l i t y o f b a n k r u p t c y g i v e s  )  t-1  .market v a l u e o f e q u i t y a t t - l . (  charges.  [3.6]  uncor-  i n t o the e x p r e s s i o n  [3.3]  60  ^ ) = Pr[n  .estimated < Yo+Yl (  r  Pr(D  f u t u r e cash  f l o w net o f a l l f i x e d r \ - l  t  .market v a l u e o f e q u i t y a t t - 1 . + Y 2  (  .  +Y3(  I  a A  charges, —)  ,  ) exp  (-CR  t-1  ) t  _  i  .market v a l u e o f e q u i t y a t t - 1 . , J ~ )], t-1  [3.7]  1  A  where the c o e f f i c i e n t s Yo,Yl/Y2» minus s i g n .  Apart  a n  d  Y3 have been r e d e f i n e d t o absorb  from t h e random v a r i a b l e , a l l the terms on t h e r i g h t hand  s i d e o f the above e q u a t i o n  are ex-post  and  can be measured.  The  signs of  the c o e f f i c i e n t s can be d e t e r m i n e d from t h e o r e t i c c o n s i d e r a t i o n s . c e t e r i s p a r i b u s argument, the g r e a t e r the e s t i m a t e d all  f i x e d charges,  the  Using  f u t u r e cash flow net  the s m a l l e r i s the p r o b a b i l i t y o f b a n k r u p t c y , and  the c o e f f i c i e n t y i s h o u l d be n e g a t i v e .  The  s m a l l e r the amount o f  and  to r a i s e  funds, e i t h e r i n t e r n a l l y o r e x t e r n a l l y , t o c o v e r  the lower the p r o b a b i l i t y o f b a n k r u p t c y .  Y 3 s h o u l d be  of  thus  credit  r a t i o n i n g , o r the g r e a t e r the market v a l u e o f e q u i t y , the more a b l e t h e is  a  fixed  firm  charges  Hence, t h e c o e f f i c i e n t s y 2  and  negative.  S t a t i s t i c a l Methodology The  p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t has been f o r m u l a t e d i n '  terms o f two models, as r e p r e s e n t e d by g e n e r a l s t r u c t u r e o f these P  where X ~^- l -  r  (  B  t j  Equations  [3.5]  f o r m u l a t i o n s i s o f the  I ^t-l,j  )  =  P  r  (  . i s a v e c t o r o f the j ij  "  th  < i '^t-l,j  and  [3.7].  The  form  )  [ 3  I X  .) i s the  probability  th the event  t h a t the j  8 ]  f i r m ' s a t t r i b u t e s measured a t time t - 1 ;  a i s a v e c t o r o f unknown c o e f f i c i e n t s ; P r ( B of  -  f i r m goes b a n k r u p t d u r i n g p e r i o d t , g i v e n  the  61  vector of a t t r i b u t e s  , .; and -t-l,D  term which i s assumed t o have firms;  that  i s , E(e.)  £. i s a z e r o mean random v a r i a b l e D unit  = 0, v a r  (e.)  3 all  j and  [3.8]  k.  observed.  strained  t o the  , and  cov  (e.,e  J  i n d e p e n d e n t among j f  ) = 0,  for  k  for s t a t i s t i c a l estimation.  the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t , can  As  k,  a number o f s p e c i a l c h a r a c t e r i s t i c s about E q u a t i o n  which have i m p o r t a n t i m p l i c a t i o n s  d i r e c t l y measured; t h a t be  = a  t o be  D  There are  dependent v a r i a b l e ,  not  v a r i a n c e and  error  i s , the  ex-ante v a l u e o r  the dependent v a r i a b l e  i n t e r v a l zero-one.  The  the  The  not  be  r e a l i z e d values  i s a probability,  can  i t i s con-  prime f o c u s i s t o d e r i v e some form  o f t e c h n i q u e t o e s t i m a t e the p r o b a b i l i t y o f b a n k r u p t c y , c o n s t r a i n i n g  the  e s t i m a t e t o a zero-one i n t e r v a l .  the  relative contribution b o t h s i d e s o f the are  related  o f the  The  c o e f f i c i e n t s , a_, which measure  d i f f e r e n t a t t r i b u t e s , are  e q u a t i o n t h e r e are  unknown.  unknown q u a n t i t i e s  Thus,  which, i n g e n e r a l ,  i n a n o n - l i n e a r manner.  W h i l s t the o b s e r v e d , a t any  ex-ante p r o b a b i l i t i e s o f a f i r m g o i n g b a n k r u p t can point  i n time a f i r m i s e i t h e r b a n k r u p t o r n o t  not  c o l l e c t i n g d a t a f o r a random sample o f b a n k r u p t and  b a n k r u p t f i r m s the  c o e f f i c i e n t s , cx, can  <Z  tj  I  Z  . =  c o e f f i c i e n t s can  e f f i c i e n t estimators.  The  may  or  larger  than one  e s t i m a t e d by p o s i t i n g  model  Z  i s an  [3.9] indicator  function  fl; <  i f j t h f i r m b a n k r u p t a t time t ;  (o,  otherwise.  be  the  non-*  + C  =  where t, i s a random d i s t u r b a n c e ; and  be  be  e s t i m a t e d by  systematic part  regression, of  be  bankrupt.  T h i s suggests t h a t  Though the  on  the  defined  t h e y w i l l not  r i g h t hand s i d e , a'  s m a l l e r than z e r o , whereas  takes only  by  be X^_, two  .  62  (0 and 1) which means t h a t the d i s t u r b a n c e term,  values  g i v e n X^ , . ^t-l/D  can  take o n l y two v a l u e s : - a ' X , . and 1-a X^ , .. I f ? i s t o have an e x p e c t e d — —t-1, j — -^t-1,3 v a l u e o f z e r o f o r a l l v a l u e s o f X^ , ., i t must t a k e t h e former v a l u e w i t h -t-1,j p r o b a b i l i t y l-ct_' X . and the l a t t e r w i t h p r o b a b i l i t y cx' X .. But t 1, j t 1, ] 1  a_* X  . can be n e g a t i v e o r l a r g e r than one.  mation procedure  t o ensure  There i s n o t h i n g i n t h e e s t i -  t h a t the e s t i m a t e d v a l u e s o f the dependent  variable  , . . 14 are c o n s t r a i n e d . The dependent v a r i a b l e o f E q u a t i o n which can n o t be o b s e r v e d ,  [3.8]  w h i l s t the e x - p o s t  i s an ex-ante p r o b a b i l i t y  v a r i a b l e s on t h e r i g h t hand  s i d e o f the e q u a t i o n c a n r e a d i l y be measured.  Using t h i s property, the coef-  f i c i e n t s , <x, c a n be e s t i m a t e d u s i n g maximum l i k e l i h o o d . sample o f f i r m s a t time and  the remainder n-n'  t and suppose t h a t t h e f i r s t non-bankrupt.  C o n s i d e r a random  n' f i r m s are  bankrupt  The l o g a r i t h m i c l i k e l i h o o d f u n c t i o n  can t h e n be w r i t t e n  n' E j=l where P r ( B  n l o g Pr(B. .|x. .) " " ' t  3  t  1  3  +  Z j=n'+l  t  .|)T .) i s d e f i n e d by E q u a t i o n —"t—l, ]  the parameters a_.  By d i f f e r e n t i a t i n g  and e q u a t i n g t h e f i r s t  k  log [l-Pr(B  [3.10]  )],  [3.10]  j  [3.8]  and i s thus a f u n c t i o n o f  w i t h r e s p e c t t o these parameters  d e r i v a t i o n t o zero, a set o f non-linear equations are  o b t a i n e d and can be s o l v e d i t e r a t i v e l y .  F o r p r a c t i c a l a p p l i c a t i o n the u s e o f  maximum l i k e l i h o o d r e q u i r e s t h a t a p a r t i c u l a r form  f o r the p r o b a b i l i t y  b u t i o n be assumed; t h a t i s , the p r o b a b i l i t y d i s t r i b u t i o n o f t h e random  distrivari-  'Vi  a b l e , £, i n E q u a t i o n  [ 3 . 8 ] must be s p e c i f i e d .  14 F o r a more e x t e n s i v e d i s c u s s i o n o f t h e e c o n o m e t r i c problems, see G o l d b e r g e r , A . S., Economic Theory (New York: John Wiley & Sons, 1964), pp. 248-255.  63  Two  estimation  logit analysis. ^ 1  p r o c e d u r e s are used i n the  The  thesis: probit analysis  e s s e n t i a l d i f f e r e n c e between the two  the e x p l i c i t form of the p r o b a b i l i t y d i s t r i b u t i o n s .  procedures i s  For p r o b i t a n a l y s i s  normal p r o b a b i l i t y d i s t r i b u t i o n i s assumed, w h i l s t f o r l o g i t a n a l y s i s distribution is logistic.  The  and  a  the  l o g i s t i c d i s t r i b u t i o n i s very s i m i l a r to  normal d i s t r i b u t i o n , b e i n g s l i g h t l y  f a t t e r i n the  t a i l s and  the  more c e n t r a l i z e d  16 about the mean. 17 Probit analysis  can be d e f i n e d  z e r o mean, u n i t v a r i a n c e than o r e q u a l t o the F (p), where  as f o l l o w s : i f the p r o b a b i l i t y o f  n o r m a l l y d i s t r i b u t e d random v a r i a b l e b e i n g  a  less  s c a l a r produce ot'X i s p, then the p r o b i t o f a_'X_ i s  1  1 F (t) = - — / /2TT  t  and  p = Pr(Y  1 (- — u ) 2  exp  du,  . v a'x)  <  [3.11]  = F(a'X) , given  t h a t Y ~ N(0,1).  m e t e r s , ex, and expression  "^An  Econometrics  The  unknowns i n the  the p r o b a b i l i t y p.  [3.10],the l o g a r i t h m i c  introductory (New  York:  Equation  formulation  [3.11] can be  likelihood function,  discussion  are the s e t o f p a r a -  i s qiven  substituted  giving  i n T h e i l , H.,  John W i l e y & Sons, 1971), pp.  into  Principles  of  628-635.  16 F o r a more d e t a i l e d d i s c u s s i o n see Winsor, C. P., "A Comparison o f C e r t a i n Symmetrical Growth C u r v e s , " Journal of the Washington Academy of Science, V o l . 22, No. 4 (February, 1932), pp. 73-84. 17 F o r a g e n e r a l d i s c u s s i o n and a p p l i c a t i o n s o f p r o b i t a n a l y s i s see F i n n e y , D. J . , Probit Analysis (Cambridge: Cambridge U n i v e r s i t y P r e s s , 1971), 3rd e d i t i o n ; and Cragg, J . G., "Some S t a t i s t i c a l Models f o r L i m i t e d Dependent V a r i a b l e s With A p p l i c a t i o n t o the Demand f o r Durable Goods," Econometrica, V o l . 39, No. 5 (September, 1971), pp. 829-844.  64  n £ l o g F(a'X .) + E =1 -t"-L/3 j=n'+l  log [l-F(a'X  [3.12]  )], , J  3 where  F  (a'X,. , .) =  and the parameters  /  exp ( —  u ) du,  can be e s t i m a t e d by s o l v i n g t h e s e t o f f i r s t  t i o n s o b t a i n e d by d i f f e r e n t i a t i n g t h e l i k e l i h o o d  order condi-  function.  18 Logit analysis  can be d e f i n e d i n a s i m i l a r manner.  b i l i t y o f a f i r m going bankrupt v a r i a b l e , which has a l o g i s t i c  I f the p r o b a -  i s e q u a l t o the p r o b a b i l i t y , p , o f a random d i s t r i b u t i o n , b e i n g l e s s than o r equal t o the  s c a l a r p r o d u c t cx'X_, t h a t i s , p = P r (Z < cx_'X)  [3.13]  1 + exp (-a'X)' where Z i s a random v a r i a b l e h a v i n g a l o g i s t i c d i s t r i b u t i o n , of  then the l o g i t  a'X i s  log  = a'X.  A g a i n t h e unknowns i n t h e f o r m u l a t i o n a r e t h e s e t o f p a r a m e t e r s , probability p. parameters  By s u b s t i t u t i n g E q u a t i o n  a_ can be e s t i m a t e d by maximum  [3.13]  a_, and t h e  into expression [ 3 . 1 0 ] ,  the  likelihood.  To e m p i r i c a l l y e s t i m a t e t h e p a r a m e t e r s ,  a_, t h e l i k e l i h o o d  function  18 For an i n t r o d u c t o r y d i s c u s s i o n t o l o g i t a n a l y s i s and f o r a p p l i c a t i o n s see Berkson, J . , " A p p l i c a t i o n s o f L o g i s t i c F u n c t i o n s t o B i o - A s s a y , " Journal of the American Statistical Association, V o l . 39 ( 1 9 4 4 ) , pp. 3 5 7 3 6 5 ; and B a x t e r , N. D. and Cragg, J . G., " C o r p o r a t e C h o i c e Among Long-Term F i n a n c i a l Instruments," The Review of Economics and Statistics, Vol. LII, No. 3 (August, 1 9 7 0 ) , pp. 2 2 5 - 2 3 5 .  65  must be the  constructed  by  t a k i n g a random sample o f f i r m s and  f i r m s as b a n k r u p t or not b a n k r u p t .  sample a v o i d s  selection bias.  bankrupt i s s m a l l , a v e r y a representative  As  the  The  then  procedure of using  classifying a random  average p r o b a b i l i t y o f a f i r m g o i n g  l a r g e random sample must be  c o l l e c t i o n of bankrupt f i r m s .  t a k e n so as t o  obtain  In p r a c t i c e , a common p r o -  cedure i s t o c o l l e c t d a t a f o r a l l b a n k r u p t f i r m s o v e r a s p e c i f i e d time p e r i o d and  then to c o l l e c t a random sample o f non-bankrupt f i r m s .  t o determine how  many f i r m s  number chosen s h o u l d o f a l l f i r m s and  be  the  should  be  included  i n the  same as t h a t o b t a i n e d  by  I t i s necessary  sample.  Ideally,  the  t a k i n g a random sample  t h e n c l a s s i f y i n g them as b a n k r u p t o r n o t b a n k r u p t .  t o determine the r e q u i r e d sample s i z e e n t a i l s e s t i m a t i n g  the  Thus,  average  value  o f the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t .  T e s t i n g o f the  Model  As t h e p r o b a b i l i t y o f b a n k r u p t c y can n o t be on  the models are n o t p o s s i b l e .  observed, d i r e c t t e s t s  T h i s i m p l i e s t h a t the magnitude o f  b i a s or measurement e r r o r i n the e s t i m a t e s can n o t be the main check on how dictive ability.  w e l l the models a r e  T h e r e are  The  s p e c i f i e d must be  on  Thus,  their  t h r e e methods by which the models can be  From t h e o r e t i c c o n s i d e r a t i o n s determined and  determined.  any  the  compared t o those o b t a i n e d  s i g n o f the p a r a m e t e r s can from the e m p i r i c a l  number o f e s t i m a t e d p a r a m e t e r s w i t h the  i n t o the s p e c i f i c a t i o n o f the model and  the  pretested. be  estimation.  correct sign provides  insight  a c c u r a c y o f the p r o x y v a r i a b l e s  a t measuring the ex-ante q u a n t i t i e s . I f the model i s c o m p l e t e l y s p e c i f i e d so as t o measure a l l the  66  d i f f e r e n t a t t r i b u t e s o f the f i r m s i n the d a t a sample used t o e s t i m a t e the parameters, t h e n i t s h o u l d be a b l e t o c o r r e c t l y i d e n t i f y the b a n k r u p t non-bankrupt  f i r m s i n the sample.  and  The c l a s s i f i c a t i o n a b i l i t y p r o v i d e s  i n f o r m a t i o n about the model's s p e c i f i c a t i o n and the number o f common d e t e r minants o f b a n k r u p t c y . The g e n e r a l i t y o f the model and i t s o v e r a l l independence  o f the  p e c u l i a r i t i e s o f the d a t a sample used t o e s t i m a t e the p a r a m e t e r s , can be t e s t e d by examining i t s p r e d i c t i v e a b i l i t y on a s e t o f b a n k r u p t f i r m s n o t used i n the o r i g i n a l sample.  By e s t i m a t i n g the p r o b a b i l i t y o f b a n k r u p t c y  o v e r s e v e r a l time p e r i o d s f o r f i r m s i n the new  sample p r o v i d e s a demon-  s t r a t i o n o f the model's p r e d i c t i v e a b i l i t y t o d i s c e r n a f i r m ' s p a t h t o bankruptcy.  Summary In a m u l t i p e r i o d c o n t e x t a f i r m ' s income can be l e s s t h a n i t s o b l i g a t i o n s and y e t i t i s n o t b a n k r u p t ; i t can s i m p l y borrow more.  The  p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t depends upon i t s a b i l i t y t o r a i s e funds, e i t h e r i n t e r n a l l y or e x t e r n a l l y , t o cover f i x e d charges. t h a t f a i l s t o c o v e r t h e s e c h a r g e s i s s a i d t o be b a n k r u p t .  A  firm  From t h i s  defi-  n i t i o n , a model f o r the p r o b a b i l i t y o f b a n k r u p t c y i s c o n s t r u c t e d i n terms o f ex-ante v a r i a b l e s .  To use the model t o e m p i r i c a l l y e s t i m a t e the  p r o b a b i l i t y o f b a n k r u p t c y an e x - p o s t f o r m u l a t i o n u s i n g p r o x y v a r i a b l e s i s developed. opposed  As the p r i m a r y f o c u s i s upon the p r e d i c t i o n o f b a n k r u p t c y , as  t o a d v a n c i n g a complete  t h e o r y o f the d e t e r m i n a n t s o f b a n k r u p t c y , a  second f o r m u l a t i o n u s i n g market v a l u e s o f a p p r o p r i a t e c o r p o r a t e v a r i a b l e s  67  i s constructed.  To empirically estimate the c o e f f i c i e n t s of the models, a  s t a t i s t i c a l methodology employing p r o b i t analysis and l o g i t analysis i s used. are  Three d i f f e r e n t methods to check the p r e d i c t i v e a b i l i t y of the models  described.  CHAPTER IV AN  EXTENSION OF THE  CAPITAL ASSET PRICING MODEL:  BANKRUPTCY  The p r i c e b e h a v i o u r of common s t o c k s i s a f f e c t e d by impending bankruptcy.  As a f i r m p r o g r e s s e s  towards b a n k r u p t c y i t s ex-post  r e t u r n s , when  compared to those o f f i r m s t h a t d i d n o t go b a n k r u p t d u r i n g the same p e r i o d appear t o be  significantly different  i n behaviour.  E m p i r i c a l evidence  in-  d i c a t e s t h a t the CAPM does n o t p r o v i d e an adequate d e s c r i p t i o n o f the mechanism g e n e r a t i n g common s t o c k r e t u r n s .  I t i s found  that assets with  high  l e v e l s o f s y s t e m a t i c r i s k c o n s i s t e n t l y e a r n l e s s t h a n t h a t p r e d i c t e d by the model, t h e r e v e r s e b e i n g risk.  t r u e f o r a s s e t s w i t h low  systematic  W h i l s t t h e r e appears t o be a l i n e a r r e l a t i o n s h i p between ex-post  t u r n s and  systematic r i s k ,  The d a t a i n d i c a t e t h a t the expected by a l i n e a r two  re-?  i t i s not c o n s t a n t w i t h b o t h t h e i n t e r c e p t and  s l o p e f l u c t u a t i n g randomly from p e r i o d t o p e r i o d and  fined.  levels of  are o f t e n negative.  r e t u r n on a s e c u r i t y can be  f a c t o r model, the second f a c t o r not b e i n g  explicitly  V a r i o u s attempts have been made t o p r o v i d e a t h e o r e t i c a l  f o r the e x i s t e n c e o f a second f a c t o r . a s s e t s , changes i n the investment  The  represented de-  explanation  e f f e c t s of non-marketability  o p p o r t u n i t y s e t , the n o n - e x i s t e n c e  of  of a  r i s k l e s s a s s e t , and r e s t r i c t i o n s upon the i n v e s t o r ' s a b i l i t y t o borrow o r l e n d have been e x p l o r e d , all  the observed  though f a i l  t o p r o v i d e an adequate e x p l a n a t i o n o f  d e f i c i e n c i e s o f the CAPM and why  impending b a n k r u p t c y  affects  the r e s i d u a l r e t u r n , a f t e r a b s t r a c t i n g from the market, o f common s t o c k s . i\  primary  focus of the t h e s i s i s to extend  CAPM n o t from the v i e w p o i n t  the f o r m u l a t i o n o f  o f r e s t r i c t i o n s upon the i n v e s t o r , but by 68  the  con-  69  s i d e r i n g t h e impact o f b a n k r u p t c y upon t h e s t r u c t u r e o f r e t u r n s f o r c o r porate f i n a n c i a l  assets.  A model, f o r m u l a t e d i n c o n t i n u o u s time, c o n s i d e r s t h e i n v e s t m e n t consumption time u t i l i t y the  d e c i s i o n o f an i n d i v i d u a l a c t i n g t o maximize t h e e x p e c t e d o f consumption  i n d i v i d u a l must  and t e r m i n a l w e a l t h .  decide  vest i n financial assets.  life-  A t each i n s t a n t i n time  t h e p o r t i o n s o f w e a l t h t o consume and t o i n I t i s assumed t h a t c o r p o r a t i o n s i s s u e b o t h d e b t  and e q u i t y a s f i n a n c i a l a s s e t s and t h a t a t each p o i n t i n time t h e r e i s a p r o b a b i l i t y t h a t t h e f i r m w i l l go b a n k r u p t t h e n e x t i n s t a n t .  I f bankruptcy  o c c u r s i t i s assumed t h a t e q u i t y h o l d e r s s u f f e r a 1 0 0 p e r c e n t l o s s , w h i l s t bondholders r e c e i v e a non-negative l i q u i d a t i n g  premium.  When t h e i n v e s t m e n t o p p o r t u n i t y s e t i s a l t e r e d o n l y by t h e e v e n t of  bankruptcy, a two v a r i a b l e model i s d e r i v e d which d e s c r i b e s t h e e x p e c t e d  r e t u r n , c o n d i t i o n a l upon no b a n k r u p t c y , f o r a f i r m ' s common s t o c k i n terms of  i t s s y s t e m a t i c r i s k and a v a r i a b l e a s s o c i a t e d w i t h t h e p r o b a b i l i t y o f t h e  f i r m going bankrupt. in ing  F o r t h e g e n e r a l c a s e when t h e r e a r e s t o c h a s t i c changes  t h e p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t , a d d i t i o n a l terms a r i s e i n v e s t o r s ' attempts t o hedge a g a i n s t unexpected  reflect-  changes.  The f o u n d a t i o n s o f t h e model a r e s e t o u t i n t h e f i r s t p a r t o f t h e chapter.  The major assumptions  o f t h e model, t h e n a t u r e o f t h e f i n a n c i a l  a s s e t s a v a i l a b l e and t h e i r p r i c e dynamics a r e d e s c r i b e d . of  t h e e q u a t i o n o f o p t i m a l i t y and the system o f e q u a t i o n s d e s c r i b i n g t h e  f i r s t order maximization conditions are derived. general a n a l y s i s , a d d i t i o n a l the  The g e n e r a l form  structure i s injected  second p a r t o f t h e c h a p t e r t h e e q u i l i b r i u m  Due t o t h e c o m p l e x i t y o f t h e into the analysis.  instantaneous expected  In rates  70  of r e t u r n , c o n d i t i o n a l upon no b a n k r u p t c y , f o r bonds and given  the a s s u m p t i o n t h a t the  bankruptcy.  The  investment o p p o r t u n i t y  f i n a l p a r t o f the c h a p t e r c o n s i d e r s  e q u i t y are  derived  s e t i s o n l y change by. the  t i c changes i n the p r o b a b i l i t y o f b a n k r u p t c y upon the  e f f e c t of  stochas-  s t r u c t u r e of  returns.  As much o f the a n a l y s i s i s o f a h i g h l y m a t h e m a t i c a l n a t u r e , an attempt  has  been made t o r e l e g a t e as much o f the mathematics t o Appendix A, w h i l s t  still  maintaining  c o n t i n u i t y i n the  chapter.  F o u n d a t i o n s o f Model F o r an turn  1  i n v e s t o r t o buy  the bonds o f a f i r m the  must compensate t h e i n v e s t o r f o r t h e r i s k t h a t the  r u p t and  firm w i l l  go  f o r t h e r i s k o f a c a p i t a l l o s s which m i g h t r e s u l t i f t h e r e  unexpected change i n the g e n e r a l i t y of bankruptcy. the v a l u e o f the  I f bankruptcy occurs,  i t i s assumed the  Intuitively,  bank-  i s an  firm i s liquidated,  f i r m ' s a s s e t s , n e t o f t a x minus t h e d i r e c t c o s t s  bondholder i s s u b j e c t  re-  l e v e l o f i n t e r e s t r a t e s , o r i n the p r o b a b i l -  w i t h b a n k r u p t c y , i s d i s t r i b u t e d on a p r o - r a t a the  expected r a t e of  associated  b a s i s t o bondholders.  Thus,  t o the r i s k o f a d i r e c t l o s s i f b a n k r u p t c y  occurs.  the e x p e c t e d r a t e o f r e t u r n on a f i r m ' s bonds t h a t a p o t e n t i a l  investor requires w i l l  be a f u n c t i o n o f the r i s k f r e e i n t e r e s t r a t e , the  a b i l i t y o f b a n k r u p t c y , the e x p e c t e d l o s s i f b a n k r u p t c y o c c u r s ,  and  the  prob-  expected  2 change i n t h e g e n e r a l  l e v e l of i n t e r e s t r a t e s .  ^ S t i g l i t z u s e s the term 'nominal r a t e o f i n t e r e s t . ' S t i g l i t z , J . , "A Re-Examination o f the M o d i g l i a n i - M i l l e r Theorem," American Economic Review,  V o l . 59, No. 2 (1969), pp. 786-793. 2 F o r an i n t r o d u c t o r y d i s c u s s i o n see F i s h e r , L., "Determinants o f R i s k Premiums on C o r p o r a t e Bonds," Journal of P o l i t i c a l Economy, V o l . LXXVII, No. 3, (June, 1959), pp. 217-237.  71  To buy t h e s h a r e s o f a f i r m , an i n v e s t o r buys a c l a i m t o a v a r i a b l e cash flow.  From t h e t r a d i t i o n a l f o r m u l a t i o n o f t h e c a p i t a l a s s e t p r i c i n g  model, as g i v e n by Sharpe,  T r e y n o r , and M o s s i n , t h e  f o r a f i r m ' s e q u i t y depends upon t h e r i s k  expected  rate of return  f r e e i n t e r e s t r a t e , and i t s l e v e l  o f s y s t e m a t i c r i s k , which i s a measure o f t h e c o v a r i a n c e o f t h e r e t u r n on t h e f i r m ' s s t o c k w i h t h e r e t u r n on t h e market p o r t f o l i o .  Such a f o r m u l a t i o n does  n o t e x p l i c i t l y c o n s i d e r b a n k r u p t c y o r changes i n t h e i n v e s t m e n t o p p o r t u n i t y set.  I f these f a c t o r s are c o n s i d e r e d , then they w i l l  expected r a t e o f r e t u r n which p o t e n t i a l i n v e s t o r s  i n general a f f e c t the  require.  In order t o d e r i v e t h e e q u i l i b r i u m expected r e t u r n s f o r a f i r m ' s bonds and e q u i t y , t h e demand f u n c t i o n s f o r t h e d i f f e r e n t f i n a n c i a l a s s e t s f o r an i n d i v i d u a l a r e o b t a i n e d .  I t i s assumed t h a t a t time t , t h e r e a r e n  f i r m s , each f i r m h a v i n g a s i m p l e c a p i t a l s t r u c t u r e o f one t y p e o f d e b t and one t y p e of common s t o c k .  I t i s a l s o assumed t h a t t h e r e i s a r i s k l e s s a s s e t ,  so t h a t t h e r e a r e 2n+l f i n a n c i a l a s s e t s , which a r e assumed t o be t r a d e d i n a p e r f e c t c a p i t a l market  (with b a n k r u p t c y ) .  I t i s important t o note t h a t t h e  bonds o f d i f f e r e n t f i r m s a r e t r e a t e d a s d i f f e r e n t f i n a n c i a l a s s e t s . t a l s t r u c t u r e was i r r e l e v a n t , t h e n t h i s would n o t be n e c e s s a r y .  If capi-  But, as c a p i -  4 t a l i s r e l e v a n t t h e n , as S t i g l i t z  c o r r e c t l y observes, the presence  o f bankrupt-  c y c r e a t e s a new s e c u r i t y .  "Sharpe, W., " C a p i t a l A s s e t P r i c e s : A Theory o f Market E q u i l i b r i u m Under C o n d i t i o n s o f R i s k , " J o u r n a l of Finance, V o l . XIX, No. 3 (September, 1964), pp. 425-442; T r e y n o r , J . "Towards A Theory o f Market V a l u e o f R i s k y A s s e t s , " u n p u b l i s h e d memorandum (1961); M o s s i n , J . , " E q u i l i b r i u m I n A C a p i t a l A s s e t Market," Econometrica V o l . 34, No. 4 (October, 1966), pp. 468-783. f  t  4 S t i g l i t z , J . , "Some A s p e c t s o f t h e Pure Theory o f C o r p o r a t e Bankruptcy and Take O v e r s , " Bell Journal of Economics and Management V o l . 3, No. 2 (Autumn, 1972), p p . 458-482.  Finance: Science t  72  It  i s assumed t h a t the c a p i t a l market i s s t r u c t u r e d such t h a t  5  Al.  a l l a s s e t s have l i m i t e d l i a b i l i t y ;  A2.  t h e r e are no t r a n s a c t i o n c o s t s ( e x c l u d i n g b a n k r u p t c y ) , p e r s o n a l t a x e s , o r problems w i t h i n d i v i s i b i l i t i e s o f assets;  A3.  t h e r e a r e s u f f i c i e n t number o f i n v e s t o r s w i t h compara b l e w e a l t h l e v e l s so t h a t each i n v e s t o r can buy o r s e l l as much o f an a s s e t w i t h o u t a f f e c t i n g the market p r i c e ;  A4.  the c a p i t a l market i s always i n e q u i l i b r i u m ; t h a t i s , t h e r e i s no t r a d i n g a t n o n - e q u i l i b r i u m p r i c e s ;  A5.  t h e r e e x i s t s an exchange market f o r b o r r o w i n g and a t the same r a t e o f i n t e r e s t ;  A6.  s h o r t s a l e s o f a l l a s s e t s , w i t h f u l l use i s allowed;  A7.  t h e r e a r e homogeneous e x p e c t a t i o n s among i n v e s t o r s about the f u t u r e p r o s p e c t s o f each f i n a n c i a l a s s e t ;  A8.  bonds a r e , i n g e n e r a l , r i s k y . I f a f i r m goes bankrupt, t h e n i t i s l i q u i d a t e d ; t h a t i s , i t i s assumed t h a t the f i r m ceases t o e x i s t ;  A9.  trading i n assets takes place continuously  Assumptions A l t o A7  a r e the  standard  o f the  lending  proceeds,  i n time.  p e r f e c t market  (ex-  c l u d i n g bankruptcy) assumptions, which have been e x t e n s i v e l y d i s c u s s e d the f i n a n c i a l l i t e r a t u r e .  Assumption A8  a s s e t p r i c i n g model assumptions. p r e s e n c e of b a n k r u p t c y .  i s a d e p a r t u r e from t h e  I t a r i s e s due  standard  to t h e c o n d i t i o n s o f  I f a f i r m cannot meet i t s o b l i g a t i o n s and  in  the  i s unable  F o r an a l t e r n a t i v e d i s c u s s i o n o f t h e s e assumptions, see Merton, R.C., "An I n t e r t e m p o r a l C a p i t a l A s s e t P r i c i n g Model," Working Paper 588-72, S l o a n School o f Management, M a s s a c h u s e t t s I n s t i t u t e o f Technology (February, 1972). 3  A good r e f e r e n c e  dence," Bell  Journal  (Autumn, 1972), pp.  i s Jensen, M.,  of Economics 458-482.  " C a p i t a l Markets:  and Management Science,  Theory and  V o l . 3, No.  Evi-  2,  73  t o secure e x t r a i s declared. is,  finance,  It will  then i t w i l l  be assumed t h a t a s t a t e o f b a n k r u p t c y  a l s o be assumed t h a t the f i r m c e a s e s t o e x i s t ;  the f i r m i s l i q u i d a t e d , the p o s s i b l i t y o f the f i r m u n d e r g o i n g  z a t i o n being  c o s t s , no i n d i v i s i b i l i t i e s ,  then i n v e s t o r s  v i s e t h e i r p o r t f o l i o s a t any time. indivisibilities,  time which,  reorgani-  neglected.  Assumption A9 f o l l o w s d i r e c t l y from Assumption A2.  i s advanced.  that  should p r e f e r  In r e a l i t y  there  I f there  t o be a b l e  Usually,  to r e -  are transaction  and i t i s f o r t h e s e r e a s o n t h a t a d i s c r e t e time t h e approach i s t o take e q u a l l y  a r e no  costs,  formulation  spaced i n t e r v a l s o f  though c o n v e n i e n t from an e m p i r i c a l v i e w p o i n t , i s t h e o r e t i c a l l y  unsatisfactory.  The t r a d i n g  intervals will,  i n general,  be s t o c h a s t i c and  o f a n o n - c o n s t a n t l e n g t h , d e p e n d i n g upon t h e t y p e s o f s e c u r i t i e s a v a i l a b l e , the s i z e and n a t u r e o f p r i c e changes, t r a n s a c t i o n c o s t s , and f u t u r e  expec-  7  tations. It will  be assumed t h a t t h e i n d i v i d u a l a c t s i n a manner  the e x p e c t e d l i f e t i m e u t i l i t y o f consumption and t e r m i n a l the k k i n d i v i d u a l a c t s fc  Max E { /  subject  k o  U (C (s),s]ds k  k  + B F [ W ( T ) ,T ] } k  t o an i n i t i a l w e a l t h c o n s t r a i n t , where  expected v a l u e o p e r a t o r ,  wealth; t h a t i s ,  so t h a t T  q  t o maximize  k  k  represents  (4.1)  the c o n d i t i o n a l  c o n d i t i o n a l on t h e f a c t t h a t a l l s t a t e v a r i a b l e s a t  time t a r e known; C ^ t s ) i s the i n d i v i d u a l ' s i n s t a n t a n e o u s consumption a t time s; U, [ C ( ) , ] s  s  i s the i n d i v i d u a l ' s u t i l i t y f u n c t i o n , which i s assumed t o be a  F o r a f u r t h e r d i s c u s s i o n o f t h i s p o i n t , see pp. 46 o f Merton, R. C , "A Dynamic G e n e r a l E q u i l i b r i u m Model o f t h e A s s e t Market and i t s A p p l i c a t i o n t o the P r i c i n g o f the C a p i t a l S t r u c t u r e o f the F i r m , " M a s s a c h u s e t t s I n s t i t u t e of Technology, S l o a n S c h o o l o f Management, O c t o b e r , 1970.  74  s t r i c t l y concave von  a steady concave  Neumann- M o r g e n s t e r n u t i l i t y  'bequest' o r u t i l i t y  t h e date o f d e a t h o f the k*"*  1  the  above f o r m u l a t i o n If  of terminal  individual.  function;  BF  JC  [W(T  wealth function;  I t s h o u l d be  X  ), T  K.  J is  and  is  noted t h a t i m p l i c i t  i s the assumption o f an a d d i t i v e u t i l i t y  function.  in  Q  c e r t a i n assumptions are made about the p r i c e dynamics o f the  stock  9 and bond p r i c e s , t h e n by o p t i m a l consumption and t h u s the  the  technique of s t o c h a s t i c c o n t r o l theory,  i n v e s t m e n t r u l e s f o r the  i n d i v i d u a l ' s demand f u n c t i o n s  Hence, by a g g r e g a t i n g a c r o s s  o f z e r o e x c e s s demand, t h e n the r e t u r n can be  equilibrium  using  the  equilibrium  and  assets. condition  instantaneous expected r a t e s  of  determined.  Such a problem has  been c o n s i d e r e d  by b o t h S a m u e l s o n ^ and  the former t r e a t i n g t h e d i s c r e t e time c a s e and case.  i n d i v i d u a l can be d e r i v e d  f o r the d i f f e r e n t f i n a n c i a l  i n d i v i d u a l s and  the  1  the  latter  the  1 1  c o n t i n u o u s time  Both, however, t r e a t e d c a p i t a l s t r u c t u r e a s i r r e l e v a n t and  sidered a l l equity  Merton,  thus c o n -  firms.  g F o r the c a s e o f m u l t i p l i c a t i v e u t i l i t y f u n c t i o n s , see Pye, G., " L i f e t i m e P o r t f o l i o S e l e c t i o n i n C o n t i n u o u s Time f o r A M u l t i p l i c a t i v e C l a s s o f U t i l i t y F u n c t i o n s , " American Economic Review, V o l . L X I I I , No. 5 (December, 1973), pp. 1013-1016.  g A d e s c r i p t i o n o f t h i s t e c h n i q u e i s g i v e n i n Bellman, F., P r i n c e t o n U n i v e r s i t y P r e s s , 1957).  Dynamic  Programming ( P r i n c e t o n , N.J.:  S a m u e l s o n , P. A., " L i f e t i m e P o r t f o l i o S e l e c t i o n by Dynamic S t o c h a s t i c Programming," Review of Economics and Statistics, V o l . L I , No. 3 (August, 1969), pp. 239-246. 10  M e r t o n , R. C , "Lifetime P o r t f o l i o Selection Uncertainty: The Cont i n u o u s Case," Review of Economics and Statistics, V o l . L i , No. 3 (August, 1969), pp. 247-257. 1 1  75  P r i c e Dynamics I t i s proposed t o r e p r e s e n t t h e p r i c e movements o f a f i r m ' s e q u i t y bonds by  stochastic differential  equations.  As  i t i s not intended  and  to present  a r i g o r o u s d e r i v a t i o n o f the e q u a t i o n s , t h e i n t e r e s t e d r e a d e r s h o u l d r e f e r t o ~12 t h e paper by I t o , i n which the t h e o r y o f s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s was  f i r s t advanced i n 1951.  equations t o describe  13  The  a p p l i c a t i o n of s t o c h a s t i c d i f f e r e n t i a l  t h e p r i c e dynamics o f bonds and  common s t o c k s has  been  14 e x t e n s i v e l y u t i l i z e d by Merton. investment  f o r the d e r i v a t i o n o f o p t i m a l consumption  rules.  The  assumptions made about t h e p r i c e dynamics o f the s t o c k and  p r i c e s are v e r y i m p o r t a n t  bond  f o r t h e y have d i r e c t b e a r i n g upon the d e r i v e d f o r m  o f t h e e x p r e s s i o n s f o r the e q u i l i b r i u m i n s t a n t a n e o u s  expected  rates of return.  A complete d e s c r i p t i o n o f t h e dynamics would r e q u i r e a s p e c i f i c a t i o n o f s u p p l y s i d e o f the f i r m ; t h a t i s , t o r e l a t e t h e r e a l a s s e t s and  the  the produc-  t i o n f u n c t i o n o f t h e f i r m t o the p r i c e dynamics o f t h e f i r m ' s s t o c k s and t r a d e d i n t h e c a p i t a l markets.  bonds  F o r example, t o s p e c i f y t h e p r i c e dynamics f o r  the f i r m ' s s t o c k s r e q u i r e s some assumption about t h e d i v i d e n d p o l i c y o f firm.  and  the  I f i n t h e e q u i t y p r i c e e q u a t i o n d i v i d e n d s a r e t r e a t e d as a random  v a r i a b l e , t h e n i n o r d e r t o have a c l o s e d system o f e q u a t i o n s  i t i s necessary  12 I t o , K.,  American  "On  Mathematical  S t o c h a s t i c D i f f e r e n t i a l E q u a t i o n s , " Memoirs  Society,  No. 4 (1951), pp.  of  the  1-51.  13 Other r e f e r e n c e s a r e I t 6 , K., and McKean, H. P., Diffusion Processes and Their Sample Paths (New York: Academic P r e s s , 1964); and Kushner, H. J . , Introduction to Stochastic Control (New York: H o l t , R i n e h a r t & Winston, I n c . , 1971) . 14 Merton, R. C., "Optimal Consumption and P o r t f o l i o Rules In A Cont i n u o u s Time Model," Journal of Economic Theory, V o l . 3 (1971), pp. 373-413.  76  to  s p e c i f y an e q u a t i o n d e s c r i b i n g t h e d i v i d e n d payments over t i m e .  u n l e s s some assumption  i s made about  However,  t h e form o f the e q u a t i o n a t the o u t s e t ,  such a d e s c r i p t i o n would imply a s p e c i f i c a t i o n o f t h e f i r m ' s b e h a v i o u r i n d e t e r m i n i n g i t s i n v e s t m e n t and f i n a n c i a l p o l i c i e s o v e r some f u t u r e  time  horizon. I t w i l l be assumed t h a t t h e s u p p l y s i d e o f t h e f i r m i s f i x e d taken as g i v e n .  and  D i v i d e n d s can e i t h e r be t r e a t e d by assuming t h a t t h e  firm  does n o t make a c t u a l d i v i d e n d payments, b u t i s s u e s o r r e p u r c h a s e s i t s own s h a r e s i n t h e market, o r by assuming a t the o u t s e t a form o f an e q u a t i o n t h a t d e s c r i b e s d i v i d e n d behaviour over time. From assumption  A9  t r a d i n g t a k e s p l a c e c o n t i n u o u s l y i n time and  thus  any r e p r e s e n t a t i o n o f t h e p r i c e dynamics o f a f i r m ' s s t o c k s o r bonds s h o u l d a l s o be i n a c o n t i n u o u s t i m e framework. are  p a i d on a d i s c r e t e t i m e b a s i s .  In p r a c t i c e coupons o r d i v i d e n d s  D i s c r e t e payments o f e i t h e r coupons o r  d i v i d e n d s r e p r e s e n t s a major t h e o r e t i c a l problem for  i n c o n t i n u o u s time models,  i t d e s t r o y s t h e symmetry o f t h e r e p r e s e n t a t i o n and  p o s s i b l e t o have compact  i t i s no l o n g e r  distributions. *' 1  Symmetry i s i m p o r t a n t f o r i t o f f e r s a tremendous for  simplification  b o t h the s p e c i f i c a t i o n o f the p r i c e e q u a t i o n s and f o r t h e d e r i v a t i o n o f  optimality conditions.  I f symmetry i s not p r e s e r v e d then i t would be  s a r y t o i d e n t i f y the t i m i n g o f t h e asymmetric  neces-  e v e n t s and t o keep t r a c k o f  T h i s i s s u e i s d i s c u s s e d i n Merton, R. C , "An I n t e r t e m p o r a l C a p i t a l A s s e t P r i c i n g Model," Working Paper No. 588-72, M a s s a c h u s e t t s I n s t i t u t e o f Technology, S l o a n S c h o o l o f Management, F e b r u a r y , 1972. ^ A s i m p l e e x p l a n a t i o n o f compact d i s t r i b u t i o n s i s g i v e n i n Samuelson, P. A., "The Fundamental A p p r o x i m a t i o n Theorem o f P o r t f o l i o A n a l y s i s i n Terms o f Means, V a r i a n c e , and H i g h e r Moments," Review of Economic Studies, V o l . 37 (October, 1970), pp. 537-542.  77  how f a r ahead i n time t h e s e e v e n t s w i l l t i o n p r o c e s s becomes f a r more  occur.  Thus t h e whole  complicated.  Compact d i s t r i b u t i o n s a r e i m p o r t a n t because f o r s m a l l v a l s the uncertainty nates the a n a l y s i s  neither  specifica-  "works o u t "  time  (that i s , zero variance)  (that i s , i n f i n i t e v a r i a n c e ) .  intern o r domi-  Some p r o p e r t i e s o f compact 17  d i s t r i b u t i o n s and t h e i r u s e f u l n e s s  are given  G i v e n t h e assumption A2 o f z e r o son why a f i r m s h o u l d basis.  i n Samuelson.  t r a n s a c t i o n costs, there  n o t pay a coupon o r a d i v i d e n d  i s no r e a -  on a c o n t i n u o u s t i m e  One c o u l d always imagine t h e f i r m p a y i n g a c o n t i n u o u s coupon o r d i v i -  dend t o a t r u s t e e who would d i s t r i b u t e t h e coupon o r d i v i d e n d  on a d i s c r e t e  time b a s i s i n t h e name o f t h e company, a s i n p r a c t i c e b o t h coupon and d i v i dends a r e p a i d on a d i s c r e t e t i m e b a s i s . I f a f i r m goes b a n k r u p t , t h e n i t i s assumed t h a t i t i s l i q u i d a t e d , the p o s s i b i l i t y o f r e o r g a n i z a t i o n being ceive, on a pro r a t a b a s i s , the value d i r e c t costs associated w i l l receive The  neglected.  The b o n d h o l d e r w i l l r e -  o f the f i r m , n e t o f taxes,  with bankruptcy.  minus t h e  I t i s assumed t h a t t h e s h a r e h o l d e r  nothing. assumption t h a t t h e f i r m undergoes l i q u i d a t i o n , and n o t r e o r g a n i -  zation avoids  two d i f f i c u l t p r o b l e m s :  both w h i l s t i t i s being  reorganized  the v a l u a t i o n of a firm's s e c u r i t i e s  and a f t e r r e o r g a n i z a t i o n ;  and a  represen-  t a t i o n i n c o n t i n u o u s time o f t h e p r i c e dynamics t h a t i s symmetric and o f a compact d i s t r i b u t i o n a l form d u r i n g  these p e r i o d s .  T h i s i s j u s t one  facet o f  the much b r o a d e r problem t h a t t h e r e a r e no t h e o r i e s o f t h e f i r m t h a t p e r t a i n  Samuelson, P. A.  nal of Financial  " G e n e r a l P r o o f That D i v e r s i f i c a t i o n Pays," Analysis, V o l . 2 (March, 1967), pp.  and Quantitative  Jour-  1-13.  78  to the state of bankruptcy and l i q u i d a t i o n . It i s assumed that when an investor buys the bonds of a firm a subj e c t i v e evaluation i s made of not only the return to be expected i f the firm does not go bankrupt, but also the return to be expected i f the f i r m does go bankrupt;  that i s , the investor form a subjective p r o b a b i l i t y d i s t r i b u t i o n  of the l i q u i d a t i o n value of the f i r m (on a pro-rata basis) i f bankruptcy occurred.  If bankruptcy does not occur i t i s assumed that the p r i c e of the bond at the end of the period can be represented by the equation  bj(t+h) = bj (t)  (1+r^h ) - g^h  + b_. (t)yJh  y.. ( t ) ,  (4.  j = 1, 2...,n, where, f o r the j  firm,  bj(t+h) represents  the p r i c e of a bond a t time  t+h;  r . represents the instantaneous c o n d i t i o n a l expected r a t e of r e turn f o r the firm's bonds, c o n d i t i o n a l on the f a c t that the firm does not go bankrupt; g. represents the instantaneous c o n d i t i o n a l coupon rate on ^ firm's bonds;  the  2  represents  the instantaneous c o n d i t i o n a l  variance;  and y.(t) represents a zero mean, u n i t variance, purely random process; that i s , y(t) and y(t+s), s > 0 are independent and i d e n t i c a l l y d i s t r i b u t e d gaussian random v a r i a b l e s . 3  I m p l i c i t i n the above formulations assumptions.  are a number of very important  F i r s t , the maturity of the bond has been neglected.  been assumed that the bonds are p e r p e t u i t i e s .  I t has  An a l t e r n a t i v e approach  would be to assume that a l l bonds had a long, but f i n i t e maturity a l l bonds matured a f t e r the . I n r H v i r l i i A i • s death.  such t h a t '  79  There a r e two major r e a s o n s f o r making such an assumption bonds b e i n g p e r p e t u i t i e s .  about  the  I f i t i s assumed t h a t t h e bonds had f i n i t e matur-  i t y and matured w i t h i n t h e l i f e t i m e o f the i n d i v i d u a l , n e c e s s a r y t o s p e c i f y t h e new  i t would t h e n be  f i n a n c i n g the f i r m undertakes.  pend upon t h e i n v e s t m e n t o p p o r t u n i t i e s which upon t h e g e n e r a l economic c o n d i t i o n s .  This w i l l  de-  f a c e the f i r m a t t h a t time  The b a s i c problem  and  i s to construat  a r e p r e s e n t a t i o n i n c o n t i n u o u s t i m e t o d e s c r i b e t h e p r i c e movements o f t h e f i r m ' s bonds t a k i n g i n t o a c c o u n t t h e p o i n t o f d i s c o n t i n u i t y which m i g h t at  the p o i n t i n time when", the bonds mature and the f i r m makes a new  issue.  C l o s e l y r e l a t e d t o t h i s i s t h e q u e s t i o n o f symmetry.  bonds mature and the f i r m makes a new  occur  debt  If a firm's  bond i s s u e w i t h d i f f e r e n t terms  to  t h o s e t h a t have j u s t matured, t h e n i t i s n e c e s s a r y t o keep t r a c k o f t h i s event i n d e t e r m i n i n g t h e i n v e s t o r ' s o p t i m a l c o n t i n g e n t s t r a t e g i e s . the is  Hence,  whole s p e c i f i c a t i o n p r o c e s s becomes f a r more c o m p l i c a t e d when symmetry broken. The second assumption  a random element term.  about t h e f o r m u l a t i o n i s t h e i n c l u s i o n o f  I f i t i s not p r e s e n t then, g i v e n a g e n e r a l  equili-  b r i u m s t a t e , t h e i n d i v i d u a l would know w i t h c e r t a i n t y what the p r i c e o f t h e bond would be a t the end o f t h e p e r i o d .  Unexpected  be i n t h e g e n e r a l l e v e l o f i n t e r e s t r a t e s , or the  changes,  whether t h e y  the p r o b a b i l i t y o f b a n k r u p t c y ,  g e n e r a l economic c o n d i t i o n s , a f f e c t t h e f u t u r e p r i c e o f t h e bond and random element  i s added i n an attempt t o c a t c h t h e s e e f f e c t s .  thus  I t should  be noted t h a t i f the bond had a f i n i t e m a t u r i t y , t h e n i t would be n e c e s s a r y for  the v a r i a n c e term o f the random element  t o be a f u n c t i o n o f the time t o  80  maturity. the  T h i s i s because a t m a t u r i t y  f i r m has  thus be  not d e f a u l t e d ,  w i l l be  independent o f t h e  o f the  the p r i n c i p a l v a l u e  i n f l u e n c e of f u t u r e  If bankruptcy occurs, the end  the v a l u e  bond, g i v e n  that  o f the bond and  will  expectations.  i t i s assumed t h a t the p r i c e o f the bond a t  o f the p e r i o d can be r e p r e s e n t e d  by  the  expression  b..(t+h) = A ( t + h ) - 9_.(t+h),  (4.3)  j  j = 1, where A_. (t+h)  2,...,n,  represents  the  a n t i c i p a t e d l i q u i d a t i o n value  per  number o f  til bonds o u t s t a n d i n g  f o r the  0..(t+h) r e p r e s e n t s per  j  f i r m i f bankruptcy occurred  I t i s assumed t h a t t h e  f o r the  d a t i o n value o f the  j  firm.  t h a t b a n k r u p t c y has  f i r m w i l l depend upon t h e  a t the  end  o f the  f i r m g i v e n t h a t b a n k r u p t c y has  expected general  occurred.  by t h e  term  liqui-  economic c o n state,  investor's subjective evaluation of  the  the  B e f o r e a l i q u i d a t i o n premium  l e g a l fees, trustee fees, referee fees, administrative These a r e r e p r e s e n t e d  The  the  I t i s assumed t h a t A_. (t+h)  be p a i d t o t h e b o n d h o l d e r s , the d i r e c t c o s t s a s s o c i a t e d  paid.  occurred.  o f t h e p e r i o d , as w e l l as upon the  t y p e and m a r k e t a b i l i t y o f t h e f i r m ' s a s s e t s . the mean v a l u e  t+h,  i n v e s t o r forms a s u b j e c t i v e e v a l u a t i o n o f  l i q u i d a t i o n value of the f i r m given  represents  and  the d i r e c t c o s t s a s s o c i a t e d w i t h b a n k r u p t c y a t time  number o f bonds o u t s t a n d i n g ,  ditions prevalent  a t t i m e t+h;  0_. (t+h) .  with  can  bankrupty—  costs —  must  be  Thus, the amount which  the b o n d h o l d e r e x p e c t s t o r e c e i v e , on a p r o - r a t a b a s i s , i s t h u s  b.(t+h) = Max[A^(t+h) - e ^ ( t + h ) , 0]  (4.4)  81  For  expositional simplicity,  i n the  i n g the p r i c e dynamics, E q u a t i o n that Equation  (4.4)  is strictly  formulation  (4.3)  o f the g e n e r a l  equations d e s c r i b -  i s used, w h i l s t i t i s f o r m a l l y  recognized  correct.  til Hence, the p r i c e dynamics o f t h e  j  f i r m ' s bonds can be  represented  by  *'b (t) (1+r j  b.(t+h)  h)  - g^h  + b . ( t ) yVb  y.. ( t ) ;  i f no  default,  = < A  I  (t+h)  - 9.(t+h)  j  ; i f default,  3  j = 1,  2,...,n.  (4.5)  I t i s assumed t h a t the e v e n t o f b a n k r u p t c y can f o l l o w i n g type o f s t o c h a s t i c  process:  be  described  by  the  18  th Pr{j  f i r m not  going bankrupt i n  (t,t+h] } = 1-X_. ( t ) h  and  (4.6) Pr{j j  1/  =  where Xj ( t ) h can be during the p e r i o d  t h  f i r m going bankrupt i n 2,...,n  (t,t+h]} =  X^(t)h,  f  i n t e r p r e t e d as the p r o b a b i l i t y o f b a n k r u p t c y f o r f i r m j  (t,t+h].  I t s determinants are e x t e n s i v e l y discussed  in  Chapter I I I . I t i s assumed t h a t the e v e n t o f b a n k r u p t c y f o r one a f f e c t other  firms.  but o n l y a t t h e The  i t i s very  simple t o r e l a x t h i s  c o s t o f g r e a t l y i n c r e a s i n g t h e c o m p l e x i t y o f the  very small gain  is given  Conceptually,  f i r m does  i n generality of derived  A more r i g o r o u s d e v i a t i o n u s i n g i n Appendix A.  not  assumption,  notation.  r e s u l t s does n o t w a r r a n t t h i s  Poisson  probability distributions  cost.  82  D e f i n e a random v a r i a b l e i n d i c a t o r f u n c t i o n , can t a k e o n l y two v a l u e s , z e r o and one,  P r  [ I . ( t , t + h ) =0]  = 1 -  3  I.. (t,t+h)  which  such t h a t  X.(t)h, (4.7)  3  and P r [ I j ( t , t + h ) = 1] =  3 The  (t)h,  1, 2,...,n.  i n d i c a t o r f u n c t i o n d e s c r i b e s the s t a t u s o f t h e f i r m a t t h e end o f the  p e r i o d ; t h a t i s , i t i n d i c a t e s whether t h e f i r m has gone bankrupt Equation  (4.5)  can be w r i t t e n i n the  bj(t+h)  I.(t,t+h)  i s zero.  [bj (t) (1+r^.h) - g..h + b.. (t) yVh  +  [Aj(t+h) -  does o c c u r  - I..(t,t+h)]  (t,t+h], then  then I.(t,t+h)  the l i m i t , as h tends t o z e r o , i t can be shown t h a t : db.(t) = :  y.(t)][l  0^(t+h)]I(t,t+h)  does n o t o c c u r t o f i r m j i n the  I f bankruptcy  not.  form  =  I f t h e event o f b a n k r u p t c y  or  [ b . ( t ) r . - g 1dt + b . ( t ) y . d Z . - { b . ( t ) : : j u D D 3  e q u a l one.  19  [A.(t+dt) 3  - 6.(t+dt)]}dq.. : u  j = 1,2,...,n, where dq^ j  (4.8)  i s a P o i s s o n p r o c e s s c h a r a c t e r i z i n g the event o f bankruptcy  f i r m ; and dZ.  In  f o r the  i s a s t a n d a r d Gaussian-Wiener p r o c e s s .  th The p r i c e dynamics o f the j  f i r m ' s e q u i t y w i l l be a f f e c t e d by what  happens t o the f i r m ' s bonds, t h a t i s , i f d e f a u l t o c c u r s the v a l u e o f e q u i t y , 19  Appendix A, E q u a t i o n  (A.2).  83  by assumption, w i l l be expressed  i n the  zero.  I t i s assumed t h a t the p r i c e dynamics can  be  form  p.(t)(l+a.h) - f.h + p.(t)c./hY . ( t ) ; i f no d e f a u l t , 3 3 3 3 3 n+3 P j  (4.9)  (t+h) = 0  ; i f default,  j =  1,  . . ,n  2,.  where th p . ( t ) r e p r e s e n t s the market p r i c e o f a share f o r the j firm a t time t ; tt. r e p r e s e n t s t h e i n s t a n t a n e o u s c o n d i t i o n a l expected r a t e o f r e t u r n f o r the j f i r m ' s e q u i t y , c o n d i t i o n a l on the f a c t t h a t t h e f i r m does n o t go b a n k r u p t ; J  J  f c  f. 3  r e p r e s e n t s the i n s t a n t a n e o u s f o r the j firm;  conditional dividend rate  represents the instantaneous the j firm;  conditional variance f o r  t  h  2 a J  t  h  and Y n  , . ( t ) r e p r e s e n t s a z e r o mean, u n i t v a r i a n c e , p u r e l y random p r o cess, that i s , Y (t) and Y . ( t + s ) , s > 0 , are i n d e p e n d e n t and i d e n t i c a l l y d i s t r i b u t e d g a u s s i a n random v a r i a b l e s . 3  n +  I f i t i s assumed t h a t t h e r e i s no p o s s i b i l i t y o f b a n k r u p t c y ,  then  t h e e q u a t i o n becomes  Ap.(t)  ,^ Pj(t) 3  =  (a.-6.)h + a.A  where 6^ i s the i n s t a n t a n e o u s to zero,  then  3D  3  Y  n+3  dividend y i e l d .  (t),  Taking  t h e l i m i t as h  tends  84  dp.(t)  —3  p.(t)  =  (a -6 ) d t + a.dZ  ...  j  .,  (4.10)  D n+u  j  where dZ r e p r e s e n t s a standard Gaussian-Wiener p r o c e s s . n+j  I f i t i s assumed  t h a t a , 6 and a a r e c o n s t a n t , then p r i c e s w i l l be s t a t i o n a r i l y and l o g j j 3 normally  distributed. In  20  general, Equation  p.. (t+h) = [  P j  (4.9) c a n b e w r i t t e n i n t h e form  (t) (1+a^h) - f j h + p . ( t ) a . / h Y  I f t h e f i r m does n o t go b a n k r u p t equal z e r o .  I f i t does go bankrupt,  v a l u e o f e q u i t y w i l l be z e r o .  shown t h a t :  i nthe interval  n + j  (t)][l  (t,t+h] t h e n I (t,t+h)  t h e n I (t,t+h) w i l l  In the l i m i t  - l..(t,t+h)l.  will  e q u a l one and t h e  as h t e n d s t o z e r o , i t c a n be  2 1  d p ^ ( t ) = [p_. {t)<Xj-f J d t + p.. ( t ) o \ d Z  n + j  - p_.(t)dq_..  (4.11)  I t i s i n s t r u c t i v e t o c o n s i d e r what a r e t h e d i f f e r e n c e s between t h e common s t o c k and p e r p e t u i t i e s o f a f i r m and t o e n q u i r e how t h e s e are r e f l e c t e d i n the s e t o f Equations  (4.8) and (4.11) t h a t a r e used  model t o d e s c r i b e t h e p r i c e b e h a v i o u r o f bonds and common s t o c k . holder purchases  i n the  A bond-  a c l a i m t o a f i x e d s e r i e s o f payments, w h i l s t t h e common  stockholder purchases  a c l a i m t o a v a r i a b l e cash flow.  T h e r e i s mutual  i n t e r a c t i o n between t h e p r i c e o f t h e two f i n a n c i a l a s s e t s . an unexpected  differences  change i n t h e p r o b a b i l i t y o f b a n k r u p t c y  will  F o r example, a f f e c t the p r i c e  F o r a more g e n e r a l d e r i v a t i o n , see Merton, R. C. "Optimum Consumpt i o n and P o r t f o l i o Rules i n a C o n t i n u o u s Time Model," Journal of Economic  Theory, V o l . 3 (1971), pp. 373-413. 21 Appendix A, E q u a t i o n  (A.3).  85  of both f i n a n c i a l a s s e t s .  I n t h e e v e n t o f b a n k r u p t c y , t h e b o n d h o l d e r has  p r i o r c l a i m t o the assets o f t h e f i r m . In t h e model the p r i o r i t y o f c l a i m o f t h e b o n d h o l d e r i n t h e e v e n t o f b a n k r u p t c y i s r e f l e c t e d i n t h e d i f f e r e n c e between t h e n a t u r e o f t h e c o e f f i c i e n t s o f t h e dq.. term. ium,  The b o n d h o l d e r might r e c e i v e a l i q u i d a t i o n prem-  w h i l s t t h e common s t o c k h o l d e r  s u f f e r s a hundred p e r c e n t  t e r a c t i o n between t h e two s e t s o f a s s e t s , i s r e p r e s e n t e d between t h e two Gaussian-Wiener p r o c e s s e s dZ^ and s e n t e d i s t h e d i f f e r e n c e between t h e c l a i m s payments.  d  z n  +  loss.  The i n -  by t h e c o r r e l a t i o n  j •  What i s n o t r e p r e -  t o a s e r i e s o f f i x e d and v a r i a b l e  An attempt c a n be made t o r e c t i f y this'.by r e i n t e r p r e t i n g t h e i n s t a n -  taneous c o n d i t i o n a l e x p e c t e d d i v i d e n d  r a t e , and t h e n t o s p e c i f y some t y p e o f  p r o c e s s d e s c r i b i n g how t h e i n s t a n t a n e o u s c o n d i t i o n a l d i v i d e n d r a t e m i g h t 22 change o v e r t i m e . Over t i m e changes i n e x p e c t a t i o n s a b l e t o t h e i n v e s t o r changes. i n the expectations the o p p o r t u n i t y  T h a t i s , new i n f o r m a t i o n  set avail-  may cause a r e v i s i o n  about t h e b e h a v i o u r o f d i f f e r e n t f i n a n c i a l a s s e t s .  s e t i s constantly  the s t r u c t u r e o f r e t u r n s . i n expectations,  o c c u r as t h e i n f o r m a t i o n  If  c h a n g i n g , then t h i s w i l l p r o b a b l y a f f e c t  I n an attempt t o a n a l y z e t h e e f f e c t s o f changes  i t i s assumed t h a t changes i n t h e o p p o r t u n i t y  d e s c r i b e d by t h e f o l l o w i n g s e t o f s t o c h a s t i c d i f f e r e n t i a l  s e t c a n be  equations:  I t i s assumed a t t h e o u t s e t t h a t i t i s p o s s i b l e t o s p e c i f y such an equation. I t i s r e c o g n i z e d t h a t t h i s b r u s h e s o v e r t h e major p r o b l e m t h a t t o j u s t i f y t h e s p e c i f i c a t i o n o f t h e e q u a t i o n would r e q u i r e a complete d e s c r i p t i o n o f the supply side o f the f i r m .  86  da. 3 d  d  °j r  =  =  F . (a ., t) dt + G . (a ., t) dQ., 3D 3 3 D n j  F  (  a  +  j  -  F  2n  j' ( r  + j  t  )  j'  d  t  +  G  n j  (  a  +  t ) d t  2n  + j  3n  + j  + G  j' ( r  t  )  d  Q  n  D'  t ) d Q  D'  t ) d Q  +  j  '  2n j'  ( 4  +  *  1 2 )  and d  Y  j  =  3n j  F  +  j =1,  2,.  ( Y  j'  t ) d t  + G  ( Y  3n j' +  . ,,n,  where F( ), and G( ) are s p e c i f i e d functions, and dQ represents Gaussian-Wiener process.  The f i r s t two  p r i c e dynamics o f equity.  standard  equations describe the changes to the  The f i r s t equation shows how  the instantaneous con-  d i t i o n a l expected return i s affected and the second equation how eous conditanal standard  a  deviation i s affected.  the  instantan-  The l a s t two equations describe  the changes t o the p r i c e dynamics o f the bonds. In  a s i m i l a r fashion, i t i s assumed the p r o b a b i l i t y of bankruptcy  i s also stochastic over time, and the investor attempts to take i n t o consideration the e f f e c t s of such changes; that i s  D  ~  F  *  (4  . .(X.,t)dt + G,. .(X ,t)dO. ., 4n+D D 5n+j j ' ' Mn+j'  D l/2,...,n =  This implies that X..(t)h, as defined i n Equation (4.6), should be r e i n t e r preted as the mean value of the p r o b a b i l i t y of bankruptcy f o r the j i n the i n t e r v a l  I  (t, t+At].  13)  l^.J-j;  t  h  firm  87  Analogous t o the above r e a s o n i n g , changes i n the to  the  equation describing  instantaneous c o n d i t i o n a l expected d i v i d e n d  the  r a t e i s assumed  be  df  An  j W j' =  f  t)dt+  WV  t)dQ  -  (4  5n r +  14)  i n s t a n t a n e o u s r i s k l e s s a s s e t means t h a t a t each i n s t a n c e o f  each i n v e s t o r knows w i t h c e r t a i n t y the s t a n t by h o l d i n g  the a s s e t .  time,  r a t e of r e t u r n , r , over the next i n -  However, the f u t u r e v a l u e s  o f r a r e n o t known  til with c e r t a i n t y . the  By  c o n v e n t i o n the  instantaneous r i s k l e s s asset.  (2n+l)  f i n a n c i a l a s s e t i s taken t o  Hence, the p r i c e dynamics can be  be  described  by „ " * 2n+1  P  and  =  rdt,  (4.15)  ( t )  i t i s assumed t h a t changes i n t h e r a t e o f r e t u r n c a n be dr = F  where m =  State  m  by  ( r , t ) d t + G (r,t)dQ_, m Tti  (4.16)  6n+l.  Space D e s c r i p t i o n and  the Budget  Before proceeding to d e r i v e equations,  described  a b r i e f d i g r e s s i o n on  p r i c e b e h a v i o u r o f e q u i t y and f i r m has d e f a u l t e d w i l l  help  the  Constraint  the budget c o n s t r a i n t and i m p l i c a t i o n s o f the  the  optimality  f a c t that both  bonds a r e i n f l u e n c e d by whether o r n o t to c l a r i f y  the d e r i v a t i o n o f the  a  equations.  the  88  If there are n-firms i n existence are  2  n  a t time t , t h e n a t time t+h  p o s s i b l e s t a t e s , where a s t a t e i s d e f i n e d  firms, l i s t i n g  those f i r m s which have d e f a u l t e d  ence a t time t+h.  there  as a d e s c r i p t i o n o f the and  t h o s e which a r e  in exist-  G i v e n the assumption t h a t t h e e v e n t o f b a n k r u p t c y by  f i r m has  no a f f e c t upon the r e m a i n i n g f i r m s , t h e n i t i s o n l y n e c e s s a r y  consider  ( n + l ) o f the p o s s i b l e s t a t e s , f o r the o t h e r  i t y of order  a r e shown i n T a b l e 4.1. (t,t+h]  one to  s t a t e s have a p r o b a b i l -  h and make no c o n t r i b u t i o n when a l i m i t i n g p r o c e s s i s u s e d .  C o n s i d e r t h r e e f i r m s , A,  val  n  The  B,  and  C.  The  e i g h t s t a t e s of the  system  p r o b a b i l i t y t h a t no f i r m s d e f a u l t i n t h e  inter-  equals  (1-A.jh) (1-X h) ( 1 - A h ) 3  2  3  = 1 -  n.h  + 0 (h)  /  where X ^ h e q u a l s t h e p r o b a b i l i t y o f b a n k r u p t c y f o r f i r m A i n the (t,t+h].  The  p r o b a b i l i t y t h a t one  the o t h e r s do  not,  From T a b l e 4.1,  going bankrupt  and  equals  X j h d  -  =  +  X  h  X h) 2  (1  -  X h) 3  0(h).  i t i s c l e a r that only  Thus o n l y those s t a t e s where one considered.  f i r m , say f i r m A,  interval  s t a t e s 1,  2,  or l e s s bankruptcies  T h i s g r e a t l y s i m p l i f i e s the  analysis.  3 and  4 are  occurred  important. need t o  be  TABLE 4.1  THE PROBABILITY OF OCCURRENCE OF DIFFERENT STATES  STATES  FIRMS A  1  NB  B  PROBABILITY OF OCCURRENCE  C  NB  NB  3 1- E X.h + 0(h) 3=1  3  2  B  NB  NB  X^h + 0(h)  3  NB  B  NB  X h + 0(h)  4  NB  NB  B  X h + 0(h)  5  B  B  NB  0(h)  6  B  NB  B  0(h)  7  NB  B  B  0(h)  8  B  B  B  0(h)  B = bankruptcy NB = b a n k r u p t c y 0(h) = o f o r d e r h  2  3  o c c u r r e d i n (t,t+hj d i d n o t o c c u r i n (t,t+h]  90  To d e r i v e  the budget e q u a t i o n i t i s n e c e s s a r y t o examine the  c r e t e time f o r m u l a t i o n  o f the model and  tend to zero) t o o b t a i n the  W(t)  then to take l i m i t s  c o n t i n u o u s time f o r m u l a t i o n .  (that i s , l e t h Let  represent time t - ;  the  i n v e s t o r ' s t a n g i b l e wealth  C(t) represent time t;  the  i n v e s t o r ' s consumption r a t e  y ( t ) r e p r e s e n t the i n v e s t o r ' s wage income be s t o c h a s t i c ) a t time t ;  dis-  at  at  (may  and I(t) represent  the  i n v e s t o r ' s i n v e s t m e n t a t time  t+  where s u p e r s c r i p t s d e n o t i n g t h e i n v e s t o r ' s i d e n t i f i c a t i o n have been dropped, except when r e q u i r e d  for c l a r i t y .  The  amount t h a t an  i n d i v i d u a l can  a t time t+ depends upon c u r r e n t t a n g i b l e w e a l t h , wage income, and  invest  planned  consumption; t h a t i s , I ( t ) = W(t)  -  [C(t) - y ( t ) ] h .  .  Let N. (t) r e p r e s e n t t h e number o f bonds o f f i r m j p u r c h a s e d d u r i n g p e r i o d (t,t+h]; N  N  , . ( t ) r e p r e s e n t t h e number o f s h a r e s o f f i r m j p u r chased d u r i n g (t,t+h];  2n+1^  r e p r e s e n t the number o f s h a r e s o f the i n s t a n t aneous r i s k l e s s a s s e t p u r c h a s e d d u r i n g ( t , t + h j ;  and w^(t)  r e p r e s e n t ^ t h e f r a c t i o n o f the i n v e s t m e n t i n the k asset during (t,t+h],  invested  (4.17)  91  j = 1,  2, . .  k = 1,  2,.  .,n,  and . .,2n+l.  Hence, the i n d i v i d u a l ' s i n v e s t m e n t can be r e p r e s e n t e d  I(t) -  The  EN 3=1  ( t ) b (t)  i n d i v i d u a l ' s w e a l t h a t the  p r i c e o f the bonds and  ?N 3=1  +  J  end  shares,  by  ( t ) p (t)  N  +  2 n + 1  p  2 n + 1  (t).  J  o f t h e p e r i o d , t+h,  the coupons and  w i l l depend upon  dividends  r e c e i v e d , and  s t a t e o f the system; t h a t i s , upon which f i r m s t h a t went b a n k r u p t i n interval  the the  the  (t,t+h]. Suppose t h a t no  t h e w e a l t h a t the end  o f the  n E N.(t) j-1 3  W(t+h) =  +  f i r m s went b a n k r u p t i n the  N  2n l  (  t  )  +  i n t e r v a l can be  [b.(t+h) + g.h] 3  3  P n l 2  (  I  (  t  )  t  represented  n E N j=l  W  (  then  by  , . (t) [p. (t+h)  n + 3  +  f.h] 3  3  >  b.(t+h)+g.h  .^ 3  (t,t+h],  +  n =  +  interval  t  )  [  'b.(t)  P  "I  n +  I  (  t  , j  ;  i  V j  p.(t+h)+f.h (  t  )  [  3 P  j  (t)  1  1  P(t+h) 2n+l 2n+l* ' p ( t ) 2n+l  + KtJw,.,, (t)  I t w i l l be assumed t h a t a l l income i s d e r i v e d a s s e t s ; t h a t i s , y ( t ) = 0, which i m p l i e s  I(t)  = W(t)  -  that  C(t)h.  from i n v e s t m e n t i n the  financial  92  Thus t h e change i n w e a l t h can be e x p r e s s e d i n t h e form n  b.(t+h) + g.h - b (t)  w(t+h)- W(t) = (w(t) - C ( t ) h } { E w.(t) [-3 ITTt)— j=l j n p (t+h) + f ,h - p . (t) + E w . (t) [ -2 -1— 3 _ ] + rh} - C ( t ) h , j=l j r  h  ]  3  r  1 1 + 3  p  h  ( t )  2n+l E w.(t) = 1 have been u s e d . j=l (4.8) and (4.11) i n t o t h e above e x p r e s s i o n s g i v e s  where E q u a t i o n  (4.15) and t h e r e l a t i o n  Substituting  3  Equations  n W(t+h) - W(t) = {W(t) - C ( t ) h } { E w. (t) [ ( r ^ - r ) h + Y ^ d Z j D 1 1 1 j  +  j  n E w =  1  n+:  =  1  (t) [ (a .-r)h  3  + a.dZ  3  .] + rh} - C ( t ) h + 0 ( h ) .  n+n  I A  (4.18  Let AW(t)  = W(t+h) - W ( t ) ,  t h e n t h e expected v a l u e o f AW(t) c o n d i t i o n a l on t h e f a c t t h a t no b a n k r u p t c i e s occurred i n the i n t e r v a l  (t,t+h] i s  n [AW(t) ] = [W(t) - C ( t ) h ] [ E w . ( b ) ( r . - r ) +  E  j=l  t  3  n E w  j-1  3  n  +  .(a.-r) + r] h 3  3  - C ( t ) h + 0(h) ,  (4.19)  and 2  E.[AW(t t  = W(t) r 1  n +  E j=l  n n E E w.(t)Y•.w.(t • -i • i D ] ii J 3=i 1=1 i]==li  n n + 2 E E w,(t)y•P..o.w . ( t ) . - , . , D D J i i n+i J J 3=1 i3= =l J 1=1  h E w j-1  (t)a..w  n + : )  3  1  n  +  .(t)I  (4.20)  h + 0(h),  1  when p.. i s t h e i n s t a n t a n e o u s c o n d i t i o n a l c o r r e l a t i o n between dZ. and dZ 31 3 n+i til Suppose t h a t t h e j  f i r m goes b a n k r u p t i n t h e i n t e r v a l  w i l l a f f e c t the investors holding the j  t  h  (t,t+h].  This  f i r m ' s bonds and e q u i t y , t h e bond-  h o l d e r might r e c e i v e a l i q u i d a t i o n premium, w h i l s t t h e e q u i t y h o l d e r  will  93  suffer  a hundred 'per c e n t l o s s .  val, conditional  Hence, t h e w e a l t h a t t h e end o f t h e i n t e r -  on t h e f a c t t h a t t h e j  th  firm  has gone b a n k r u p t , i s  n b.(t+h)+g.h A. (t+h) - 9 . (t+h) W(t+h) = {W(t) - C ( t ) h } { E w. (t) [ — . . ] + w. (t) [ -3 -3 ] i=l i ' j' ' 1  1  +  3  n p. (t+h) + f .h E w . (t) [— rn —1 . , n+i p. (t) 1=1 i  Substituting  .P + w_ . (t) 2n+l  •  x  (t+h)  p  )  -A;  (t) 2n+l  Equations  (4.8), (4.11) and (4.15) i n t o t h e above e q u a t i o n s 2n+l and u s i n g t h e r e l a t i o n E w.(t) = 1, g i v e s j=l n A.(t+h) - 8.(t+h) W(t+h) - W(t) = {W(t) - C ( t ) h } { E w. (t) [ ( r . - r ) h + y.dZ.] + w. (t) [-3 ——r— i 1 1 3 h j (t) 3  3  1  i  =  1  n +  w_^ (t) [ ( a . - r ) h . , n+i l i=l s  4  + 6\dZ ] - w _,_. (t) + rh} + C ( t ) h + 0 ( h ) . l n+i n+3 ± 1  A. (t+h) - 6.(t+h) - W^V^  [  b-ttf  J  "  2  1  1  " n+j W  ( t ) >  n + (w(t) - C ( t ) h } { E w. (t) [ ( r . - r ) h + Y - d Z j i=l i*j  +  n E w ,. ( t ) [ ( a . - r ) h + a.dZ ,.] + rh} . , n+i l I n+i i=l i*)  - C(t)h{l  A.(t+h) - . 9 . ( t + h ) + w. (t) r ~* ... - 1] - w ( t ) } + 0(h) 3 b.(t) ' n+3 3 3  w  1  The f i r s t term i n t h e above e x p r e s s i o n , t h a t i s , A  w(t){w. (t) [ - J 3  (t+h) - 8.(t+h)  r—jrr  1  b_.(t)  " 1 1  -w . ( t ) } n+3  (4.21)  1  94  can be i n t e r p r e t e d as t h e l o s s t o t h e i n d i v i d u a l who i n v e s t e d  i n the j  s e c u r i t i e s , g i v e n t h a t t h e f i r m goes bankrupt i n the i n t e r v a l  (t,t+h].  i n d i v i d u a l p a i d b_. (t) f o r t h e bond a t t h e b e g i n n i n g  r  firm's  n  I f the  o f the period, then a t the  end o f the p e r i o d t h e i n d i v i d u a l w i l l r e c e i v e a l i q u i d a t i n g premium o f A., (t+h) 6_.(t+h), which i s l e s s than t h e i n i t i a l amount p a i d f o r t h e bond, g i v e n 6  (t+h) i s g r e a t e r than z e r o .  Pj(t)  a t the beginning  Similarly,  f o r equity,  that  i f the i n d i v i d u a l paid  o f t h e p e r i o d , t h e n t h e i n d i v i d u a l w i l l s u f f e r a hundred  p e r c e n t l o s s a t t h e end o f t h e p e r i o d .  Note, however, t h a t as s h o r t  is  t o g i v e the gains  allowed,  -  t h e argument c a n be r e v e r s e d  selling  t h a t o c c u r when t h e  f i r m goes b a n k r u p t .  The  Equation The  of Optimality:  The Demand F u n c t i o n s  f o r Assets  i n d i v i d u a l i s assumed t o a c t i n such a manner so as t o maximize  the e x p e c t e d l i f e t i m e u t i l i t y r e w r i t i n g Equation  o f consumption and t e r m i n a l w e a l t h ; t h a t i s ,  (4.1)  T Max E {/ U [ C ( S ) , s ] d s + BF[W(T),T]}, o o s u b j e c t t o an and  i n i t i a l w e a l t h c o n s t r a i n t , t h e budget  where t h e s u p e r s c r i p t s d e n o t i n g  (4.1)  c o n s t r a i n t , and C(s)  >_ 0,  t h e i n v e s t o r ' s i d e n t i t y have been dropped,  except when r e q u i r e d f o r c l a r i t y . Define T J[W(t),a,a,r,Y,f,X,r ~ where at  ,t,s(t)] F  = Max E {/ {C,w} fc  U[C(s),s]ds  + BF[W(T),T]}  s(t) i s a s t a t e v e c t o r w h i c h d e s c r i b e s what f i r m s a r e s t i l l  time t ; 2/°_f r,Y,»  (4.22)  fc  f_, and X_ a r e v e c t o r s which d e s c r i b e t h e v a l u e s  i n existence a t time t o f  {a . },{a.},{r.},[y.},{f.} and {X.} r e s p e c t i v e l y ; and r 3 3 3 j 3 3 F r a t e o f r e t u r n a t time t o n t h e i n s t a n t a n e o u s J,  i s t h e v a l u e o f the  r i s k l e s s asset.  i s c a l l e d the derived u t i l i t y o f V7ealth.  The f u n c t i o n ,  I t s arguments a r e t h e s t a t e  v a r i a b l e which, a t time t, an i n d i v i d u a l knows.  The problem f a c i n g t h e  i n d i v i d u a l i s t o choose v a l u e s  f o r {C,w} which maximizes t h e e x p r e s s i o n on  t h e r i g h t hand s i d e o f (4.22);  that i s , the decision variables are the rate  o f consumption C, and t h e p r o p o r t i o n f i n a n c i a l a s s e t s {w.}.  o f wealth t o i n v e s t i n the d i f f e r e n t  T h i s may be a c h i e v e d  u s i n g t h e Bellman p r i n c i p l e o f  . 23 optimality. 24 I t c a n b e shown who a c t s a c c o r d i n g contingent  t h a t t h e o p t i m a l i t y c o n d i t i o n s f o r an i n d i v i d u a l  t o Equation  (4.1) i n d e t e r m i n i n g  t h e consumption-investment  s t r a t e g y a t each p o i n t i n time, a r e m  0 = Max  (U[C(t),t] + J + E F . J .  {C,w}  j=l  t  n  (t)  + J„(W(t) [ E w. W j - i3  3  (r.-r) + 3 j  3  n  =  1  2  j ^ j ^ D i V ^  '^WW^  tt n  +  E  E w . (t)a. .w  j-1 i - 1  n  +  3  3  1  n  +  E  1  m +  EG.V..G.J..+  i - 1 j-1  2  1 3  3  1 3  j. x  (o.-r) + r ] -  C(t)}  3  j  W 1  j  ( t ) Y  j  P  i i n i d  j  W  ( t )  +  2  (t)]W(tr m  n  E  E W(t)w.  i - 1 j-1  3  3  (t)y.n. 1  3  1  l  .G.J  W  n  E E W(t)w .(t)a.n. ^..G.J... .... n+3 3 i»n+3 l iW 1=1 3=1  n +  (t) 3  1  mm + —  +  E w n+  E j=l  X.(t){j[W., V ( t ) , 3  3  t, S.]  -  J[W(t), V(t),  t, S(t)}),  (4.23)  3  23 F o r a f o r m a l statement o f t h i s p r i n c i p l e , see pp. 15 o f Bellman, R. E. and D r e y f u s , S. E., Applied Dynamic Programming ( P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y P r e s s , 1962). See  Appendix A, E q u a t i o n  (A.17).  96  subject  t o t h e boundary c o n d i t i o n J [ W ( T ) , V ( T ) , T , S ( T ) ] = BF[W(T),T], and  where s u b s c r i p t s on t h e f u n c t i o n J [ W ( t ) , V ( t ) , t , S ( t ) ] tions;  S . i s a state vector 3  th denoting that the i  denote p a r t i a l  f i r m no l o n g e r  deriva-  e x i s t s ; a...  i j  i s t h e i n s t a n t a n e o u s c o n d i t i o n a l c o v a r i a n c e between dZ. and dZ.; p.. i n s t a n t a n e o u s c o n d i t i o n a l c o r r e l a t i o n c o e f f i c i e n t between dZ  i s the  and dZ , .; v. .  1  13  n+3  i s t h e i n s t a n t a n e o u s c o n d i t i o n a l c o r r e l a t i o n c o e f f i c i e n t between dQ. and dQ.;  i  3  n. . i s t h e i n s t a n t a n e o u s c o n d i t i o n a l c o r r e l a t i o n c o e f f i c i e n t between dQ. and  13  dZ.; 3  1  and W. i s d e f i n e d 3  25  by A.  Wj(t)  = W(t){l + w  j  (t)[  - 9 . ( t  ) - 1] "  w n  +  j  (  t  )  }  '  ( 4  '  2 4 )  th which c a n be i n t e r p r e t e d as t h e new w e a l t h p o s i t i o n a f t e r t h e j  f i r m has gone  bankrupt. Equation  (4.23) i s a p a r t i a l d i f f e r e n t i a l e q u a t i o n which d e s c r i b e s t h e  behaviour o f t h e derived  utility  function.  As such, t h e r e  elements i n t h e e q u a t i o n ; i t i s c o m p l e t e l y d e t e r m i n i s t i c . derived u t i l i t y  a r e no s t o c h a s t i c The v a l u e o f t h e  f u n c t i o n depends upon t h e consumption r a t e and t h e amount o f  investment i n t h e d i f f e r e n t f i n a n c i a l a s s e t s  that are available.  v a l u e s o f t h e s e d e c i s i o n v a r i a b l e s t h a t maximize t h e d e r i v e d  The o p t i m a l  utility  function  a t each p o i n t i n t i m e , c a n be d e t e r m i n e d by s o l v i n g t h e s e t o f e q u a t i o n s which describe  the f i r s t order conditions  f o r a maximum.  The assumptions about t h e  form o f t h e u t i l i t y and  f u n c t i o n and t h e b e q u e s t f u n c t i o n e n s u r e t h a t a maximum, 26 n o t a minimum, i s o b t a i n e d .  25 i s given  See Appendix A, E q u a t i o n s (A.15) t o (A.20), where a d e t a i l e d f o r t h e reasons motivating t h i s d e f i n i t i o n .  discussion  26 (New  York:  A p r o o f i s g i v e n i n Kushner, H. J . , Academic P r e s s , 1967). !  Stochastic  Stability  and Control,  97  The Equation  (2n+l) f i r s t o r d e r c o n d i t i o n s a r e o b t a i n e d by f i r s t  (4.23) w i t h r e s p e c t t o t h e r a t e o f consumption  differentiating  27  0 = U [C(t) ,t] - J ; c w then by d i f f e r e n t i a t i n g . . 28 , . , equities; that i s , ^  0 = (a - r ) J  + w  (4.25)  (4.23) w i t h r e s p e c t t o t h e amount o f investment i n w  -, n  +  j J  :  W(t)[ ? a w i = l „..  (t)  +  ? y P i=l  a.w.  (  t  )  ]  (  4  .  2  6  )  J  m  j  i , 2,. • •,n;  =  and f i n a l l y ,  differentiating  (4.23) w i t h r e s p e c t t o t h e amount o f i n v e s t m e n t  29 i n bonds;  t h a t i s , {w..}:  n n 0 = (r . - r ) J + W(t) [ Z y . .w. (t) + Z y .p . .a. ..(t)]J._. l W . ,31 i . , 3 13 i n + i WW 1=1 1=1 m 7A. - 9 . +  ^ n . - G . J ^ - X.[.l - - 1 - ^  J [W.,V(t),t,S.],  (4.27)  w  j ~ 1, 2,. « *,n« Equation  (4.25) i s t h e i n t e r t e m p o r a l e n v e l o p e  condition:  the marginal  u t i l i t y o f c u r r e n t consumption e q u a l s t h e m a r g i n a l d e r i v e d u t i l i t y  o f wealth.  Equation  i n the  (4.26) d e s c r i b e s a system o f e q u a t i o n s f o r t h e i n v e s t m e n t  commonstocks a v a i l a b l e a n d E q u a t i o n  (4.27) d e s c r i b e s a s i m i l a r  See Appendix A, E q u a t i o n  (A.18)  See Appendix A, E q u a t i o n  (A.19).  See Appendix A, E q u a t i o n  (A.20).  system f o r bonds.  98  The two systems a r e n o t independent.  I f a f i r m goes bankrupt, t h e event o f  bankruptcy w i l l a f f e c t b o t h t h e v a l u e o f i t s bonds and e q u i t y . a f f e c t t h e v a l u e o f t h e f i r m , f o r example unexpected ity  o f bankruptcy, w i l l be r e f l e c t e d  f i r m ' s bonds and e q u i t y .  Events t h a t  changes i n t h e p r o b a b i l -  i n changes i n b o t h t h e v a l u e o f t h e  The l a c k o f independence  between t h e two  systems  i m p l i e s t h a t the two s e t s o f e q u a t i o n s must be s o l v e d s i m u l t a n e o u s l y . s o l u t i o n w i l l n o t , however, be easy t o o b t a i n because linear. which the  o f t h e f a c t t h a t b a n k r u p t c y causes a d i s c o n t i n u i t y i n  t h i s l e v e l o f g e n e r a l i t y , l i t t l e i n s i g h t i n t o the i m p l i c a t i o n s o f  s e t o f e q u a t i o n s c a n be g a i n e d .  simplifying,  assumptions  I t i s proposed t o add some f u r t h e r , and  t o r e s t r i c t t h e s t r u c t u r e o f the o p p o r t u n i t y s e t .  Two models w i l l be c o n s i d e r e d . the  {j[W_.,V(t) ,t,s^] }  wealth of the i n d i v i d u a l . At  the  t h e e q u a t i o n s a r e non-  The n o n - l i n e a r i t y r e s u l t s from t h e p r e s e n c e o f t h e terms  i s a consequence  A  The f i r s t  i s a s i m p l e model i n which  investment o p p o r t u n i t y s e t i s assumed o n l y t o be a l t e r e d by t h e e v e n t o f  bankruptcy; t h e p r o b a b i l i t y o f b a n k r u p t c y i s assumed n o t t o change s t o c h a s t i c a l l y over time. simplified.  The e q u a t i o n s t h a t d e s c r i b e t h e bond p r i c e dynamics a r e a l s o  W h i l s t such a l e v e l o f s i m p l i c i t y  i s unrealistic,  i t does a f f o r d  p e n e t r a t i n g i n s i g h t i n t o how t h e mechanism o f b a n k r u p t c y a f f e c t s t h e s t r u c t u r e of  returns.  The second model r e l a x e s t h e assumption  that the p r o b a b i l i t y o f  b a n k r u p t c y f o r a f i r m does n o t change s t o c h a s t i c a l l y o v e r t i m e . i s more r e a l i s t i c level of insight.  The framework  t h a n t h a t o f t h e f i r s t model, b u t does n o t o f f e r  t h e same  99  Bankruptcy and S t r u c t u r e o f R e t u r n s By r e d u c i n g  t h e l e v e l o f g e n e r a l i t y o f the formulation  enables  greater  i n s i g h t i n t o t h e impact o f b a n k r u p t c y upon t h e mechanism d e s c r i b i n g t h e s t r u c ture o f returns. {a,a,r,y,f,X,r_}  I t i s assumed t h a t t h e o p p o r t u n i t y i s d e t e r m i n i s t i c ; t h a t i s , there  F  s e t c h a r a c t e r i z e d by  a r e no s t o c h a s t i c changes  i n t h e s e parameters so t h a t t h e i n d i v i d u a l knows w i t h c e r t a i n t y t h e i r future values.  I t i s f u r t h e r assumed t h a t t h e r e  i s no s t o c h a s t i c element  t o t h e c o n d i t i o n a l e q u a t i o n d e s c r i b i n g t h e p r i c e dynamics o f bonds, t h a t i s , Equation  (4.5)  becomes b. (t) (1+r^.h) - g^.h; b..(t+h)  i f no d e f a u l t  = <  (4.28) A.(t+h) - 6.(t+h) ; i f d e f a u l t , 3 3  j =1/  2,. . .,n  W h i l s t t h e absence o f a s t o c h a s t i c element term i s an o v e r s i m p l i f i c a t i o n , i t does imply t h a t t h e r e w i l l be no m u t u a l l y i n t e r a c t i o n between bonds and common s t o c k s a p a r t from t h e d i r e c t e f f e c t o f b a n k r u p t c y . be  no i n t e r a c t i o n terms i n t h e e x p r e s s i o n s f o r t h e e q u i l i b r i u m r a t e s  t u r n f o r bonds and common From E q u a t i o n  Max  t  + J (w(t)[ w  ]_ 2 m J  l  A  \  J=l  n  I w.(t)(r.-r) +  J  {c,w} +  1  . \ n+j 1-1  (  t  )  a  ji  w n + i  <t)Jw(t)  E ^  n  +  (t) ( a ^ r )  j  +  r  ]  _  c  (  t  )  }  3  n  w  of r e -  (4.23) t h e e q u a t i o n o f o p t i m a l i t y becomes  (U[C(t),t] + J  n  will  stock.  n 0 =  Thus, t h e r e  2  +  n z A {J[W j=l  J  J  t , S ] - .Trw,t,S(t)]>) J  (4.29)  100  and t h e f i r s t o r d e r c o n d i t i o n s a r e , a f t e r 0 = U  simplification"^  [C(t),t] - J , w  c  (4.30)  r. - r (a. - r - - 3 ) j + ( t ) [ E a. .w (t)] J , 3 L. W . . ] i n+i WW j i=l  (4.31)  (r.-r)J -A.L.J(W.,t), 3 W 3 3 W 3  (4.32)  0 =  W  J  J  and 0= 3  =  1»  2,.  • .,n,  where f o r e x p o s i t i o n a l s i m p l i c i t y  L. i s d e f i n e d  by  A. - 9 L  j •  1  -  ~\d>  which can be g i v e n an i n t u i t i v e meaning.  Suppose an i n d i v i d u a l p u r c h a s e d a  bond f o r b..(t) and t h e f i r m went b a n k r u p t . (A. - 8.) 3 D  and thus s u f f e r a l o s s o f  measure o f t h a t l o s s .  [b. 3  I f L_. e q u a l s ohe,  The  i n d i v i d u a l would  (A. - 9 . ) ] . 3 3  receive  L. i s a p e r c e n t a g e 3  the i n d i v i d u a l s u f f e r s a  hundred  per cent l o s s . Equation  (4.30) i s  u t i l i t y o f consumption tion  the i n t e r t e m p o r a l envelope c o n d i t i o n : marginal  equals the marginal d e r i v e d u t i l i t y of wealth.  (4.31) d e s c r i b e s a l i n e a r system o f e q u a t i o n s from w h i c h t h e demand f u n c -  t i o n s f o r e q u i t y can be d e t e r m i n e d the  demand f u n c t i o n s f o r bond.  i n d e p e n d e n t l y from the d i r e c t  T h i s independence  influence of  i s a consequence  assumption o f h a v i n g no random element term i n t h e bond e q u a t i o n . of  Equa-  o f the The  system  E q u a t i o n s ( 4 . 3 2 ) , t h a t d e s c r i b e t h e demand f u n c t i o n s f o r bonds a r e non-  linear  and t h u s , i n g e n e r a l , i t w i l l  be d i f f i c u l t t o o b t a i n an e x a c t  solution.  The n o n - l i n e a r i t y r e s u l t s from the d i s c o n t i n u i t i e s i n w e a l t h t h a t a r e caused the  event o f b a n k r u p t c y .  "*°See Appendix  A, E q u a t i o n s  (A.26),  (A.27) and  (A.20)  by  101  C o n s i d e r f i r s t the demand f u n c t i o n s (4.32). cit  As  the  solution.  equations are non-linear, There are a t l e a s t two  s t r u c t u r e i n t o the dual's u t i l i t y  formulation  f u n c t i o n and  general  individuals.  The  Equation  i t i s d i f f i c u l t t o o b t a i n an  alternatives.  then attempt t o s o l v e the Whilst  The  first  and  i s t o put more indivi-  system o f e q u a t i o n s  by  e x p l i c i t forms f o r  even i f they c o u l d be  intractability  the  obtained,  s p e c i f i c assumed form f o r the u t i l i t y f u n c t i o n s  l a c k o f g e n e r a l i t y and  expli-  t h i s method might produce a s o l u t i o n ,  I t w i l l be d i f f i c u l t t o d e r i v e  equilibrium rates of return,  w i l l depend upon the  by  by assuming a p a r t i c u l a r form f o r the  a numerical i t e r a t i v e procedure. i t w i l l be a t a c o s t .  f o r bonds, d e s c r i b e d  they  for  the  o f t h i s method i s a  serious d i s t r a c t i o n to i t s u t i l i z a t i o n . The  second a l t e r n a t i v e i s t o make an a p p r o x i m a t i o n so as t o o b t a i n  l i n e a r system.  The  a p p r o x i m a t i o n e v o l v e s around t h e assumption t h a t i t i s  p o s s i b l e t o expand the d e r i v a t i o n s of the d e r i v e d l o r ' s s e r i e s and  to neglect  J (W(t)[l w  = J 3  w  1,  =  For a quadratic cases of u t i l i t y  q u a d r a t i v e and  - w_.(t)L. - w  function in a  h i g h e r o r d e r terms;  (t)],  31  Tay-  that i s ,  t} T  2,.  utility  .  »,n» f u n c t i o n t h i s approximation i s exact, w h i l s t f o r other  functions,  f o r example the  constant r e l a t i v e r i s k  aversion  be v e r y good, depending upon the n u m e r i c a l v a l u e s  o f the parameters o f the u t i l i t y  32  n + j  utility  [W(t),t] - W ( t ) [ w . ( t ) L . + w ( t ) ] j [w(t),t], 3 3 ""'"3 WW  c l a s s , the a p p r o x i m a t i o n can  31  a  See  Appendix A,  For  a f u l l discussion,  function.  Equation  32  (A.24).  see Appendix A,  Equations  (A.52) t o  (A.62)  102  Using t h i s approximation,  Equation  (4.32) c a n be e x p r e s s e d  i n the  form °  =  (  3  =  1/ 2,. .  r  j"  r  _  X  j j L  )  J  W  +  X  j j ' L  I  W  J  (  t  >  L  J  +  W  n+j  ( t ) 1 W ( t ) J  WW'  ( 4  "  3 3 )  .,n. 33  From t h i s e q u a t i o n i t c a n be shown  the ^equilibrium instantaneous  conditional  expected r a t e s o f r e t u r n a r e X ,L.[N.b.(t)L. + N .p.(t)] r . = r + X.L. + ( T T - r - y X . 3 3 3 3 3 ^2-2 } 3 3 3 n E X. ( t ) X . L . [N.b. (t)L. + N , .p. ( t ) ] . , 1 1 1 1 1 3 n+i^i 1=1 j = 1, 2,. . ,,n, (4.34) where N^ i s t h e t o t a l number o f bonds o u t s t a n d i n g f o r t h e j th t h e t o t a l number o f s h a r e s o u t s t a n d i n g f o r t h e j  f i r m ; N^+j i s  firm; X^(t) i s the propor-  th t i o n o f t h e t o t a l market v a l u e o f t h e j  f i r m ' s bonds t o t h e t o t a l market  v a l u e o f a l l bonds; ir i s t h e i n s t a n t a n e o u s c o n d i t i o n a l expected  r e t u r n on t h e  bond market, d e f i n e d by n TT =  E X . (t)r.; i=l  1  (4.35)  1  and y c a n be i n t e r p r e t e d a s a weighted of  bankruptcy  sum o f t h e e x p e c t e d  l o s s i n t h e event  d e f i n e d by Y =  n EX.(t)L.X..  i-1  Equation  1  1  (4.36)  1  (4.34) c a n be i n t e r p r e t e d a s t h e i n s t a n t a n e o u s  c o n d i t i o n a l ex-  p e c t e d r a t e o f r e t u r n f o r a f i r m ' s bonds e q u a l s t h e sum o f t h e r i s k f r e e r a t e o f r e t u r n , t h e expected second  term, X  33  ,  l o s s i n t h e event o f b a n k r u p t c y ,  c a n be i n t u i t e d  See Appendix A, E q u a t i o n  as t h e expected  (A.45).  and a market term.  l o s s i f bankruptcy  The  occurs.  103  I t i s composed o f two terms: the  t h e r a t e o f t h e p r o b a b i l i t y o f b a n k r u p t c y and  c o n d i t i o n a l e x p e c t e d l o s s , c o n d i t i o n a l upon t h e e v e n t o f b a n k r u p t c y .  magnitude o f t h i s term w i l l be dependent firm's  assets  upon t h e e x p e c t e d n e t v a l u e o f t h e  a f t e r t h e event o f b a n k r u p t c y .  t o o c c u r between t h e s e two t e r m s :  The  I t i s possible  fora  trade-off  f o r example, t h e r a t e o f p r o b a b i l i t y o f  b a n k r u p t c y m i g h t be l a r g e w h i l s t t h e e x p e c t e d l o s s i n t h e event o f b a n k r u p t c y might be s m a l l . 34 The second term, ^ j j / L  c  a  n  D  e  compared t o t h e f o r m u l a t i o n  who h y p o t h e s i z e d t h a t t h e r i s k premium on a bond i s a f u n c t i o n the p r o b a b i l i t y o f d e f a u l t and t h e m a r k e t a b i l i t y sideration i s not relevant the  o f a bond.  of Fisher, o f two terms:  This  latter  i n t h e p r e s e n t c o n t e x t g i v e n t h e assumptions  s t r u c t u r e o f the c a p i t a l markets.  conabout  F i s h e r d i d n o t , however, d i r e c t l y con-  s i d e r t h e impact upon t h e r i s k premium o f t h e e x p e c t e d l o s s t h a t might  occur  i n the event o f bankruptcy. The t h i r d term c a n be i n t e r p r e t e d as a market f a c t o r . o f two elements:  the f i r s t ,  (n-r-y),  I t i s composed,  can be i n t u i t e d a s t h e i n s t a n t a n e o u s  c o n d i t i o n a l e x p e c t e d r i s k premium on t h e bond market, and t h e second i s a p o s i t i v e weighting f a c t o r . I t i s i n s t r u c t i v e t o examine t h e i n s t a n t a n e o u s c o n d i t i o n a l rate o f return  expected  f o r a bond f o r t h e two c a s e s o f when an i n d i v i d u a l does n o t  s u f f e r a l o s s i n t h e event o f b a n k r u p t c y , L_.=0, and when t h e p r o b a b i l i t y o f •bankruptcy i s z e r o .  I n b o t h c a s e s , the r e o u i r e d  rate  of return  i s the r i s k  f r e e r a t e , as would be e x p e c t e d .  Journal  F i s h e r , L., "Determinants o f R i s k Premiums on C o r p o r a t e Bonds," L X V I I , N O . 3 (June, 1959); pp. 217-237.  of P o l i t i c a l Economy, V o l .  104  The  demand f u n c t i o n s f o r e q u i t y a r e d e s c r i b e d by E q u a t i o n  T h i s system o f e q u a t i o n s i s l i n e a r a n d independent o f t h e d i r e c t  (4.31). influence  of the demand f u n c t i o n s f o r bonds; t h a t i s , i t does n o t c o n t a i n terms {w_.(t)}.  like  T h i s i s a consequence o f t h e a s s u m p t i o n about t h e p r i c e dynamics  o f bonds d e s c r i b e d by E q u a t i o n  (4.28) which does n o t c o n t a i n a random element  term t h a t would have p r e c i p i t a t e d i n t e r a c t i o n between t h e two s e t s o f demand f u n c t i o n s f o r bonds and e q u i t y . 35 From E q u a t i o n  (4.31) i t c a n be shown  that the equilibrium instan-  taneous c o n d i t i o n a l e x p e c t e d r a t e s o f r e t u r n c a n be e x p r e s s e d i n t h e form  3  =  l f  = g.(u-r-x),  r ^. - r  a . - r 3  j  (4.37)  3  2,. • .,n,  where u i s t h e i n s t a n t a n e o u s  c o n d i t i o n a l e x p e c t e d r e t u r n on t h e market,  d e f i n e d by y =  n Z Y. ( t ) a . , j=l 3  (4.38)  3  Yj (t) b e i n g t h e p r o p o r t i o n o f t h e market v a l u e o f t h e j*"*  1  firm's equity t o the  t o t a l market v a l u e o f a l l e q u i t y ; x i s d e f i n e d by  X =  n r .-r £ Y. (t) {-} ); j=l  and  8. i s t h e i n s t a n t a n e o u s 3  3  L  (4.39)  3  c o n d i t i o n a l covariance  of the return o f the j  f i r m ' s e q u i t y w i t h t h e e q u i t y market, d i v i d e d by t h e i n s t a n t a n e o u s v a r i a n c e o f t h e r e t u r n on t h e market, d e f i n e d by  35  See  Appendix A, E q u a t i o n  36 From E q u a t i o n  conditional  36  (A.39).  (4.37) i t c a n be shown t h a t  " I Y . ( t ) B . = 1. j=l 3  3  th  105  n E Y.(t)a.. i=l B.j = n n Z Z Y.(t)a..Y.(t) j-1 i - 1 1  3  1  3  3  1  (4.40)  1  B. i s c a l l e d t h e b e t a f a c t o r f o r t h e j  Greater i n s i g h t i n t o the s i g r.-r (4.37) can be g a i n e d by e l i m i n a t i n g t h e term ( - ^ — ) .  3  n i f i c a n c e of Equation  firm.  Li ,  T h i s can be a c h i e v e d by u s i n g t h e e x p r e s s i o n s f o r t h e i n s t a n t a n e o u s expected r a t e o f r e t u r n f o r t h e j  th  f i r m ' s bonds.  This gives  3 conditional  37  N.b . ( t ) L . + N .p. ( t ) a.-r-A. = ( y - r - x X B .+A .[-3-3 3 SJU ]}, 3 D D D n n M(t) .Z. .Z Y. (t) a . . Y. (t) i = 1 2 n 3=1 i = l D Di i  where M(t) i s t h e t o t a l market v a l u e o f a l l e q u i t y . Bj g i v e n by E q u a t i o n ( 4 . 4 0 ) ,  Equation  iHZE3>  Using the d e f i n i t i o n of  (4.41) can be w r i t t e n n  a.-r-A. j 3  (4.41)  N.b. ( t ) L .  •( Z o..Y.(t)+A.[-J-3 31 l D n  n n Z Z Y.(t)a..Y.(t) .... 3 31 l 3=1 1=1  3  + N  .p. (t)  ^  ]}  _  E N ,p.(t) . , n+1 1 i=l  I f t h e r e a r e a l a r g e number o f f i r m s , t h e l a s t t e r m on t h e r i g h t hand s i d e o f t h e above e x p r e s s i o n c a n be n e g l e c t e d , a s i t i s o f o r d e r 2/n, where n i s t h e number o f f i r m s .  Hence, t h e e x p r e s s i o n f o r t h e i n s t a n t a n e o u s c o n d i t i o n a l  ex-  p e c t e d r a t e o f r e t u r n f o r t h e j * " * f i r m ' s e q u i t y can be w r i t t e n 1  a -r-A j  j  = B ' (u-r-x)  (4.42)  37  I t i s n o t o b v i o u s from E q u a t i o n (4.34) how E q u a t i o n (4.41) i s d e r i v e d . As shown i n Appendix A, E q u a t i o n (A.46), an a l t e r n a t i v e form o f E q u a t i o n (4.34) can be d e v e l o p e d . T h i s a l t e r n a t i v e form has been used t o e l i m i n a t e r . - r from J E q u a t i o n (4.37). L. D  106  The  e x p r e s s i o n on t h e l e f t  hand s i d e o f the above e q u a t i o n can  i n t e r p r e t e d as t h e i n s t a n t a n e o u s e x p e c t e d  r i s k premium, as opposed t o the  i n s t a n t a n e o u s c o n d i t i o n expected r i s k premium f o r t h e j can be  i d e n t i f i e d as a weighted  average  be  th  firm.  The  term  — x  o f the r a t e o f t h e p r o b a b i l i t y o f bank-  38 ruptcy for a l l firms.  Hence, t h e e x p r e s s i o n ( u - r ~ x ) can be i n t e r p r e t e d  the i n s t a n t a n e o u s e x p e c t e d market r i s k premium, and be w r i t t e n i n the  thus E q u a t i o n  (4.42)  may  form  E(R..) - r = SjEECR^) - r l , where E ( R J  as  (4.43)  i s t h e i n s t a n t a n e o u s e x p e c t e d r a t e o f r e t u r n on t h e e q u i t y o f the  th j  firm.) and E (R^) i s the i n s t a n t a n e o u s e x p e c t e d  market.  T h i s r e s u l t i s analogous  p r o b a b i l i t y o f bankruptcy  t o t h a t d e r i v e d by Merton,  Equation  to  i f the  identical.  e m p i r i c a l p o i n t o f view t h e above r e s u l t s ,  i n the instantaneous c o n d i t i o n a l expected  (4.42) o r the i n s t a n t a n e o u s e x p e c t e d  important.  and  f o r a l l f i r m s i s zero, the r e s u l t s are  Both from a t h e o r e t i c a l and whether they be e x p r e s s e d  r a t e o f r e t u r n on t h e e q u i t y 39  form o f E q u a t i o n  valid  time  f o r t h e case when  i s e x p l i c i t l y c o n s i d e r e d , p r o v i d e d the i n s t a n t a n e o u s e x p e c t e d  of  (4.43), a r e  T h e o r e t i c a l l y , t h e r e s u l t s show t h a t t h e c o n t i n u o u s  the c a p i t a l a s s e t p r i m a r y model i s s t i l l  form  analogy bankruptcy  rates of return  a r e used and n o t t h e i n s t a n t a n e o u s e x p e c t e d r a t e s o f r e t u r n c o n d i t i o n a l upon 38  From the d e f i n i t i o n o f the i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r e t u r n the market, see (4.38), E q u a t i o n (4.42) i m p l i e s t h a t n X = £ Y (t)X j=l T h i s i s not, however, a d e f i n i t i o n a l i d e n t i t y . I t i s the r e s u l t o f the a p p r o x i m a t i o n made i n d e r i v i n g E q u a t i o n (4.42) and, as s u c h , i s i t s e l f an a p p r o x i m a t i o n . See Appendix A, E q u a t i o n (A.50) f o r a f u l l e r d i s c u s s i o n . on  3  3 9  ing  M e r t o n , R. C ,  "An  3  I n t e r t e m p o r a l C a p i t a l A s s e t P r i c i n g Model," Work-  Paper 588-72, M a s s a c h u s e t t s  ment, F e b r u a r y ,  1972.  I n s t i t u t e of Technology,  S l o a n S c h o o l o f Manage-  107  no b a n k r u p t c y .  In the t r a d i t i o n a l c a p i t a l  a s s e t p r i c i n g model, where  capital  s t r u c t u r e i s assumed t o be i r r e l e v a n t and b a n k r u p t c y i s t o t a l l y i g n o r e d , such a d i s t i n c t i o n i s n o t n e c e s s a r y .  B u t i t i s t h i s d i s t i n c t i o n t h a t makes  the r e s u l t i m p o r t a n t from an e m p i r i c a l v i e w p o i n t . a s s e t p r i c i n g model, t h e assumption  In t e s t i n g the c a p i t a l  i s made t h a t i t i s p o s s i b l e t o go from  an ex-ante t o an ex-post f o r m u l a t i o n and t o use r e a l i z e d  returns.  Plowever,  from t h e way e m p i r i c a l t e s t s a r e c o n d u c t e d , t h e r e a l i z e d r e t u r n s a r e p r o x i e s f o r t h e e x p e c t e d r e t u r n , c o n d i t i o n a l upon no b a n k r u p t c y ; t h a t i s , p r o x i e s f o r t h e terms {cO fication  and n o t { E ( R J } .  they a r e  Thus, t h e r e i s a b a s i c  misspeci-  error. The e f f e c t o f t h i s e r r o r c a n be d e m o n s t r a t e d ,  as shown i n F i g u r e  4.1.  40 Merton  has shown t h a t t h e t r a d i t i o n a l c a p i t a l a s s e t p r i c i n g model f o r c o n -  t i n u o u s time i s o f t h e form •<*j » r  + B.. ( u - r ) ,  which i s denoted i n t h e f i g u r e b y CAPM. the p a r t i c u l a r f i r m , t h a t i s ,  (4.44) If A  was c o n s t a n t and i n d e p e n d e n t o f  A_. = A, f o r a l l j , t h e n E q u a t i o n (4.42) becomes  0^ = r + A + B ..(u-r-A).  (4.45)  T h i s i s r e p r e s e n t e d i n F i g u r e 4.1 by t h e l i n e d e n o t e d b y CAPM'.  The l i n e i s  l i n e a r and f l a t t e r t h a n t h e l i n e CAPM due t o t h e p r e s e n c e o f t h e term A.  How-  e v e r , t h e r a t e o f p r o b a b i l i t y o f b a n k r u p t c y does v a r y a c r o s s f i r m s and t h u s , i n g e n e r a l , t h e r e w i l l be a n o n - l i n e a r r e l a t i o n s h i p between a and 0.  If i t i s  41 assumed t h a t as B i n c r e a s e s , A i n c r e a s e s ,  -r1_ -J  40 41  t h e n a l i n e o f t h e form denoted by  Merton,  Iota,  1  W e s t e r f i e l d has p r e s e n t e d some e v i d e n c e j u s t i f y i n g t h i s assumption. W e s t e r f i e l d , R., "The Assessment o f Market R i s k and C o r p o r a t e F a i l u r e , " U n i v e r s i t y o f P e n n s y l v a n i a , Wharton S c h o o l o f F i n a n c e , August, 1970 ( u n p u b l i s h e d ) .  108  FIGURE 4 . 1  THE  EFFECT OF BANKRUPTCY UPON THE CAPITAL MARKET LINE  Instantaneous C o n d i t i o n a l Expected Return CAPM  Beta C o e f f i c i e n t ,  CAPM  d e n o t e s t h e e q u a t i o n a . = r + 3.(u - r ) 3 3  CAPM' denotes t h e e q u a t i o n a.. = r + X+ 3^(u - r - x ) CAPM" d e n o t e s t h e e q u a t i o n a . = r + A . + 3 . (u - r - x ) 3 3 3  3  109  CAPM" w i l l be o b t a i n e d . Such a c o n c l u s i o n i s v e r y i m p o r t a n t when viewed recent empirical findings.  i n the l i g h t o f  The c u r v e CAPM" i s d e r i v e d from t h e e q u a t i o n  (4.42) which c a n be r e w r i t t e n i n t h e form  a . - r = (6. - 0.x) + B-.(u-r),  3  3  t  n  e  (4.46)  3  1^ *2. f • • * / n •  ^ I f x i s °f  3  same o r d e r a s X, t h e n f o r s m a l l v a l u e s o f 8(6<1) t h e f i r s t  term  on t h e r i g h t hand s i d e o f t h e above e q u a t i o n w i l l be p o s i t i v e , w h i l s t f o r l a r g e v a l u e s o f 8(8>1) i t w i l l be n e g a t i v e . a p a r t i c u l a r i n s t a n t i n time.  Equation  (4.46) d e s c r i b e s a r e l a t i o n a t  T h e r e i s no a priori  v a r i o u s f a c t o r s i n t h i s r e l a t i o n w i l l remain  r e a s o n t o suppose t h a t t h e  constant over time.  t h e r a t e o f t h e p r o b a b i l i t y o f b a n k r u p t c y may change because rationing. constant.  F o r example,  o f a severe  I f t h i s i s s o , t h e n t h e r e i s no r e a s o n f o r t h e c u r v e CAPM" t o be Both c o n c l u s i o n s a r e c o n s i s t e n t w i t h t h e t i m e s e r i e s r e s u l t s o f  B l a c k , J e n s e n and S c h o l e s  . .  .  42  who found t h a t t h e i n t e r c e p t term o f t h e c a p i t a l  43  a s s e t p r i c i n g model i s n o n - s t a t i o n a r y  and f o r a time s e r i e s r e g r e s s e d o v e r a  30 y e a r p e r i o d , t h e i n t e r c e p t was c o n s i s t e n t l y n e g a t i v e f o r h i g h r i s k folios of  credit  (B>1) and p o s i t i v e f o r low r i s k p o r t f o l i o s  t h e model i n l i g h t  (B<1).  port-  The m i s s p e c i f i c a t i o n  o f t h e above d i s c u s s i o n might a l s o e x p l a i n t h e n e g a t i v e  42 B l a c k , F., Jensen, M.C, and S c h o l e s , M., "The C a p i t a l A s s e t P r i c i n g Model: Some E m p i r i c a l T e s t s , " p r i n t e d i n Studies in The Theory of Capital Markets (Ed.) Jensen, M.C. (New York: P r a e g e r , 1 9 7 2 ) .  43  *v>  ^  The time s e r i e s r e g r e s s i o n was o f t h e form R •= a . + B,R..^ + e.. . ]t •) j Mt i t where R. i s t h e ex-post excess r e t u r n on t h e market p o r t f o l i o o v e r t h e same p e r i o d , a n d e _ a random e r r o r term. fc  3  3t  110  r e l a t i o n s h i p found o v e r c e r t a i n p e r i o d s between average  monthly r e t u r n s and  44 systematic  risk.  The model, as r e p r e s e n t e d by E q u a t i o n  (4.42), forms t h e b a s i s f o r  the e m p i r i c a l t e s t i n g o f t h e h y p o t h e s i s o f t h e t h e s i s . l a t i o n o f t h e equation i s used. u t i l i z i n g t h e work d e v e l o p e d  A d i s c r e t e time  The p r o b a b i l i t y o f b a n k r u p t c y  formu-  i s estimated  i n Chapter I I I .  S t o c h a s t i c Changes i n t h e P r o b a b i l i t y o f  Bankruptcy  I n t h e model j u s t c o n s i d e r e d knowledge o f how t h e mechanism o f bankr u p t c y a f f e c t e d t h e s t r u c t u r e o f r e t u r n s was g a i n e d u s i n g a s i m p l e model i n w h i c h t h e i n v e s t m e n t o p p o r t u n i t y s e t d i d n o t change s t o c h a s t i c a l l y . assumption  i s restrictive:  Such an  t h e random a r r i v a l o f hew i n f o r m a t i o n and t h e r e -  assessment o f e x i s t i n g i n v e s t m e n t  o p p o r t u n i t i e s may cause t h e i n v e s t m e n t  t u n i t y s e t t o be a l t e r e d w i t h t h e i m p l i c a t i o n t h a t t h e e x p e c t e d  oppor-  rate of return  r e q u i r e d by p o t e n t i a l i n v e s t o r s w i l l a l s o change. I f t h e investment o p p o r t u n i t y s e t i s n o t constant, then t h i s  invalidates  one o f t h e c o n d i t i o n s f o r t h e c a p i t a l a s s e t p r i c i n g model t o be a p p l i c a b l e f o r use  i n a multi-period context.  p o r t f o l i o behavior of a r a t i o n a l when t h e r e i s a c h a n g i n g  Such a c o n c l u s i o n i s h a r d l y s u r p r i s i n g , i n v e s t o r would n o t be e x p e c t e d  investment  f o r the  t o be t h e same  o p p o r t u n i t y s e t i n s t e a d o f a c o n s t a n t one.  45 Merton  has demonstrated t h a t changes i n t h e i n v e s t m e n t  a f f e c t t h e s t r u c t u r e o f common s t o c k r e t u r n s .  o p p o r t u n i t y s e t do  Under t h e assumption  that a l l  changes can be c h a r a c t e r i z e d by changes i n a s i n g l e i n s t r u m e n t a l v a r i a b l e  See  45  B l a c k , Jensen,and S c h o l e s ,  loc. cit.  Merton, "An I n t e r t e m p o r a l C a p i t a l A s s e t P r i c i n g Model," op. cit.,  p. 38.  —  Ill  the r i s k l e s s i n t e r e s t r a t e —  a two f a c t o r model i s d e r i v e d .  f a c t o r c a n be i n t e r p r e t e d a s the r e s u l t o f i n v e s t o r s hedging e f f e c t s of f u t u r e unforeseen  changes i n t h e r i s k l e s s  Changes i n t h e investment  going bankrupt.  against the  interest  rate.  o p p o r t u n i t y s e t c a n be caused  changes i n t h e p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t . come and i t s a b i l i t y  The s e c o n d  t o borrow change o v e r t i m e ,  by s t o c h a s t i c  As t h e f i r m ' s f u t u r e i n -  so w i l l  the p r o b a b i l i t y o f i t  Some o f t h e s e changes w i l l be e x p e c t e d ,  and t h e i r  signifi-  c a n c e w i l l a l r e a d y be d i s c o u n t e d i n t h e p r i c e o f t h e f i r m ' s f i n a n c i a l a s s e t s . However, o t h e r changes w i l l  be u n e x p e c t e d and w i l l  a f f e c t the p r i c e o f the  f i r m ' s f i n a n c i a l a s s e t s and t h e r e a l i z e d r e t u r n o f i n v e s t o r s .  The p e r t i n e n t  q u e s t i o n t o ask i s how t h e s t o c h a s t i c n a t u r e o f t h e changes i n t h e r a t e o f the p r o b a b i l i t y o f b a n k r u p t c y assets?  affect the  structure of returns f o r f i n a n c i a l  One method t o a n a l y z e t h i s p r o b l e m i s t o r e p r e s e n t  g e n e r a t i n g t h e s e s t o c h a s t i c changes b y a s p e c i f i e d p r o c e s s . assumed t h a t t h e mechanism c a n be r e p r e s e n t e d  t h e mechanism I t w i l l be  i n t h e form  dX.(t) = F. (X . , t ) d t + G. (X . ,t)dQ., 3  3  3  =  3  3  3  (4.47)  3  1/ 2,. • .,n,  where dQ r e p r e s e n t s a s t a n d a r d G a u s s i a n - W i e n e r p r o c e s s . s h o u l d be compared t o E q u a t i o n The  e q u a t i o n d e s c r i b i n g t h e p r i c e dynamics o f a f i r m ' s bonds w i l l be which r e f l e c t s  p r i c e given t h a t d e f a u l t has not occurred.  the form, r e w r i t i n g E q u a t i o n  db  equation  (4.13).  assumed t o c o n t a i n a random element term, of  The above  the uncertainty  T h i s c a n be r e p r e s e n t e d i n  (4.8)  (t) = [b. ( t ) r -g ] d t + b  (t)Y.dZ.-(b.(t) -  [A.(t+dt)-9.(t+dt)]}dq.  (4.8) j  = 1, 2,. . . ,n.  112  The  i n c l u s i o n o f t h e random element  term, dZ , d e s c r i b i n g t h e p r i c e dynamics  of bonds w i l l d e t e r m i n e t h e degree o f r e s p o n s e upon t h e s t r u c t u r e o f e x p e c t e d r a t e s o f r e t u r n c a u s e d by t h e s t o c h a s t i c n a t u r e o f t h e changes i n t h e r a t e o f the p r o b a b i l i t y o f b a n k r u p t c y . for  It will  a l s o r e s u l t i n t h e demand  bonds and e q u i t y b e i n g d i r e c t l y c o r r e l a t e d , a s e v e n t s w i l l  functions  affect  both  types of assets. The  e q u a t i o n o f o p t i m a l i t y c a n be s i m p l y d e r i v e d  case c o n s i d e r e d  i n Equation  (4.23) and c a n be w r i t t e n i n t h e form  0 = Max ( U [ C ( t ) , t ] + J fc {c,w} + {W(t)[  +  j=l  3  j  2  j  n  =  j  n E =  =  1  n E w ^. (t) (ct.-r) + r ] - C ( t ) } j n+3 w D  n E w.(t)y..w.(t) 3 3i x  n + 2 E  1  j  =  1  i  n E w.(t)y.p..a.w . ( t ) 3 3 3i i n+i* T  =  1  P  ;  V j ( t , c , j l , V i < t ) I J w w  1=1 3=1 n E W(t)w (t)a.n. ^.G.J. n+3 3 i/n+3 l iW  r7  1  n E j=l  +  i  =  3  n  Ai-i  +  1  AC  n EF.J.  n E w.(t)(r.-r) + j=l 3 3  . , n + ;rW(t) [ E 2  from t h e g e n e r a l  i  =  1  X .{j[W.,Mt),t,S.] 3  3  3  - J[W(t) ,X (t) , t , S ( t ) ] }) ~  (4.48)  s u b j e c t t o t h e boundary c o n d i t i o n J[W (T), A_(T) ,T,S (T) ] = BF(W(T),T], set  of first  o r d e r m a x i m i z a t i o n - c o n d i t i o n s a r e , a f t e r some m a n i p u l a t i o n ,  o = U [ C ( t ) ,t] - J c  46  See Appendix  i  and t h e  A, E q u a t i o n s  w  ,  (A.64),  -  (A.65),  (A.66) and  (A.67).  (4.49)  113  r. -r  y . p . . a.  n  (4.50)  •)J.  iw'  and n  n WW  n (4.51)  Equation  (4.49) i s t h e i n t e r t e m p o r a l e n v e l o p e  condition:  u t i l i t y o f consumption e q u a l s t h e m a r g i n a l d e r i v e d u t i l i t y tion  (4.50) d e s c r i b e s a system  o f wealth.  Equa-  o f n l i n e a r e q u a t i o n s i n terms o f t h e demand  f u n c t i o n s f o r bonds and e q u i t y . of  marginal  The d i r e c t dependence between t h e two s e t s  demand f u n c t i o n s a r i s e s from t h e c o r r e l a t i o n o f t h e p r i c e dynamics f o r  the f i n a n c i a l a s s e t s .  T h i s e q u a t i o n s h o u l d be compared t o E q u a t i o n  A p a r t from t h e c o r r e l a t i o n terms, terms,  Equation  (4.31).  (4.50) c o n t a i n s an e x t r a s e t o f  { j . }, t h a t a r e a d i r e c t r e s u l t o f t h e s t o c h a s t i c n a t u r e o f changes i n i w  the r a t e of t h e p r o b a b i l i t y o f bankruptcy.  The s i g n i f i c a n c e o f t h e s e  terms  w i l l become v e r y a p p a r e n t when t h e i n d i v i d u a l demand f u n c t i o n s a r e d e t e r m i n e d . Equation  (4.51) d e s c r i b e s a n o n - l i n e a r system  o f n equations  i n terms o f t h e  demand f u n c t i o n s f o r bonds and e q u i t y , t h e n o n - l i n e a r i t y a r i s i n g  from t h e d i s -  c o n t i n u i t i e s i n w e a l t h t h a t a r e caused by t h e e v e n t o f b a n k r u p t c y .  I t also  c o n t a i n s t h e e x t r a s e t o f terms {J..,}, and s h o u l d be compared t o E q u a t i o n iw 47 (4.32). As i n t h e l a s t s e c t i o n , i t w i l l be assumed t h a t E q u a t i o n (4.51)  47  See E q u a t i o n s  (4.32) and  (4.33).  114  can  be approximated t o g i v e  a l i n e a r system: n n 0 = ( r . - r - X . (t)L.)J„ + W(t) [ E y ..w. (t) + E y .p. .w . 3 3 3 W . x x . X3 n+i =  + A (t)L.[w.. (t)L.. + w  n + j  1 D  =  1 3  WW  (tJlWtt)^  n (4.52)  Hence, E q u a t i o n s and  (4.50) and (4.52) d e s c r i b e  t h u s i t i s p o s s i b l e t o determine an e x p l i c i t s o l u t i o n f o r t h e demand  functions  f o r bonds and e q u i t y .  i s c l e a r that.the  From t h e s t r u c t u r e o f t h e s e e q u a t i o n s , i t  form o f such a s o l u t i o n w i l l be i n v o l v e d ,  l a r g e number o f t e r m s . the  a system o f 2n l i n e a r e q u a t i o n s  containing  a  The c o m p l e x i t y o f t h e s o l u t i o n a r i s e s because o f  c o v a r i a n c e terms and t h e terms t h a t r e f l e c t t h e e f f e c t s o f t h e s t o c h a s -  t i c n a t u r e o f t h e changes i n t h e r a t e o f t h e p r o b a b i l i t y o f b a n k r u p t c y . I t c a n be shown  49  t h a t t h e demand f u n c t i o n s  th f o r bonds f o r t h e k  i n d i v i d u a l can be e x p r e s s e d i n t h e f o l l o w i n g form, u s i n g  matrix  notation,  (4.53) and  t h e demand f u n c t i o n s  for equities,  (4.54) where  WW  See  Appendix A, E q u a t i o n  See  the whole o f t h e l a s t  (A.68). s e c t i o n o f Appendix A.  115  {w } . = W ( t ) w . ( t ) ; k  k  k  {w }. = W(t) w ( t ) k  k  —2 3  k  n+3  ;  r . - r  2  {al . = a . - r -D J {C}.  j  - r . - r - X.L.;  i i  Y  {D } . . = Y . p . n  -1  31  f—;  . 0 .  3 13 l  Lj  { D > . . = A.L. + Y.P. . 0 -2 31 33 '3 13 i { D , } . . = a.n. .G.- j -3 31 3 i,n+3 i • Y  . ;  n  J  i j  H  L  {D.} .  ] l] l  Y -P . {D  G  . = Y.n. . G . ;  31  - 4  i j  . =a.. -12} .31 31 1 9  - J - *  Lj  .0". 3  ^  2 {  °21>ji " j i Y  +  * j V  5i  - (2a! " E a S - i k i ) " '  £  - °12 - S . ! ^ ) "  1  and (  2  1  '  As t h e b a s i c s t r u c t u r e o f t h e two demand f u n c t i o n s Equations one.  (4.53) and (4.54) i s i d e n t i c a l ,  C o n s i d e r t h e demand f u n c t i o n s  i twill  consideration of  bankruptcy.  to discuss  f o r equity,- r e p r e s e n t e d  The f u n c t i o n i s e s s e n t i a l l y composed o f two p a r t s : given a constant  suffice  represented  investment o p p o r t u n i t y  by just  by E q u a t i o n  (4.54)  t h e demand t h a t a r i s e s  s e t , and t h e demand t h a t a r i s e s from  o f t h e e f f e c t s o f s t o c h a s t i c changes i n t h e r a t e o f p r o b a b i l i t y  116  k The f i r s t  -1  p a r t , H^E^ (a - D-^D^iSJ c a n , p e r h a p s , be more e a s i l y  ted i f w r i t t e n i n a s c a l a r H ^ C a - D ^ c ) } .  interpre-  form: - Hj[ E ^ . . ( ^ . f p , 1  _  i  x  ?  2  D. (r.-r-X L.)], i  i  i=l  j = 1, 2,. . .,n where  and  '^i^'ji -  2  V  tli  The f i r s t  t e r m i n t h e above e x p r e s s i o n  represents  the demand f o r the j  firm's  th e q u i t y based upon the i n s t a n t a n e o u s  expected r a t e o f r e t u r n f o r the i  e q u i t y , and t h e second term r e p r e s e n t s demand f o r t h e i  ^ f i r m ' s bond.  a s u b s t i t u t i o n term a r i s i n g from t h e  The p r e s e n c e o f s u c h a term i s t o be e x p e c t e d , 50 k  f o r bonds a r e s u b s t i t u t e s f o r e q u i t y . and  i s the usual preference  c u r r e n t and f u t u r e form:  The f a c t o r  i s strictly positive  f a c t o r r e f l e c t i n g t h e i n d i v i d u a l ' s d e s i r e between  consumption.  The second p a r t , ^  (E  <D,  —2 —3  3  firm's  =  l f  -1 k (D^-D^^^D^) H_  - DD'JD  )H*}  —1—21-4 —2  2,.  c a n a l s o be w r i t t e n i n a s c a l a r  2  =  Z E.. H* . , 3 j i 2i 1=1 •  . .,n,  where ^ 2 ^ 3 - ^ 0 " ^ ) }  n  31  3  They c o u l d a l s o be complements.  3  i  f  (4.55)  117  and  i/3  The  =  2, • • ., n.  1/  term can be d i r e c t l y a t t r i b u t e d t o t h e e f f e c t s o f the  s t o c h a s t i c nature  o f the changes i n the r a t e o f the p r o b a b i l i t y o f b a n k r u p t c y , and preted  as an attempt t o hedge a g a i n s t  J  I t can be  shown^  interthat  k  a  c  such changes.  can be  3\.  >  3c  <  aw* 5  3  Thus i f  3c^  TT?—  <  firm's equity. n preference  —  0 and  The  1/  „E..  3  33  2, • • ., n. < 0,  then the  form o f e x p r e s s i o n  terms o f t h e  i n v e s t o r w i l l demand l e s s o f t h e  th  (4.55) i s i m p o r t a n t f o r i t c o n t a i n s  i n d i v i d u a l , {H  .}.  T h i s i m p l i e s t h a t i f the  equili-  brium i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r a t e s o f r e t u r n f o r bonds and a r e t o be d e t e r m i n e d f r e e o f any  j  preference  terms, t h e n t h e  terms {H  equity }  and  ^3 H  1  must be  eliminated.  G i v e n the t o d e r i v e the  form o f the demand f u n c t i o n , i t i s c l e a r t h a t any  expressions  f o r the  e q u i l i b r i u m instantaneous c o n d i t i o n a l  p e c t e d r a t e s o f r e t u r n w i l l be d i f f i c u l t due  t o the  nate t h e s e p r e f e r e n c e  (n+1)  preference  terms.  Whilst  j u s t i f y the  See  Appendix A,  Equation  See  Appendix A,  Equations  effort.  (A.22). (A.79) and  (A.81).  there  common s t o c k ,  i t i s possible to  terms, the r e s u l t i n g c o m p l e x i t y and  s i g h t t h a t r e s u l t s , does not  ex-  f a c t t h a t not o n l y a r e  c o r r e l a t i o n terms r e s u l t i n g from the p r e s e n c e o f the bonds and 52 a l s o because o f t h e  attempt  general  but  elimi-  lack of i n -  118  Some i n s i g h t can be g a i n e d by assuming t h a t the s t o c h a s t i c changes i n the r a t e o f p r o b a b i l i t y o f b a n k r u p t c y f o r one f i r m a c t s as an i n s t r u m e n t a l iable, characterizing a l l For  var-  t h e changes i n the investment o p p o r t u n i t y s e t .  convenience, c a l l t h i s f i r m the n  th  firm.  I t can be shown  instantaneous c o n d i t i o n a l expected r a t e o f r e t u r n  f o r the j  th  53  that the  firm's  equity  i s o f t h e form  a  V  , nM ix~ nX jH, , . - e =( ^ J-> (y - r - x - £„>  r  g  J —  - r -  5  5  q  j  a 6 , - 6 .,a ( ? ) (a O  +  M  m  X  3 X  M  n  x  j = 1, 2,.  . .,n-l,  n  r -r - r - -S_ L n  where M ( t )  =  ! Vj j p  j  N y. D  ( t  )  P  =  =  ( t ) ;  1  .p . (t) " j J ; M(t)  ?^( ) .; a  t  j=l  3  3  n r ,-r = E Y. (t) (-J );  X  j-1  J  3  5 j i - <P_  " 2.i2^i22> i'  12  o.„ = 3  a 2 = M  j  n E ?..Y.(t); i=l 3  1  n E Y.(t)a. ; n ]M  See Appendix A, E q u a t i o n  M  (A.86).  e  ), M  (4.56)  119  e. 3  «•! nX  6  =  £  e..(r.  j l3  1  - r - X.L.);  11  1  =  «a.h .G T 'n,n+n n  MY - 3=1 .Vj  ( t ) J  l i L^.l A 3  £  +  e  i i i n , i Y  J  1  n  n , ; L  G  n  n  i=l  V  j=l  and 0 *  2 a 6,-6,a . nM mX nX M '  =  A s i m i l a r e x p r e s s i o n c a n be o b t a i n e d f o r b o n d s . The complex n a t u r e o f E q u a t i o n The l e f t  5 4  (4.56) makes i n t e r p r e t a t i o n  difficult.  hand s i d e o f t h e e q u a t i o n can be i n t u i t e d a s t h e i n s t a n t a n e o u s ex-  pected rate o f return.  The f i r s t  as t h e instantaneous expected tor s p e c i f i c t o the j  t  h  firm.  excess  the Equation  a  c a n be  interpreted  r e t u r n on t h e market m u l t i p l i e d by a f a c -  The s e c o n d  o f t h e changes i n t h e i n v e s t m e n t be i n t e r p r e t e d  term on t h e r i g h t hand s i d e  term a r i s e s from t h e s t o c h a s t i c  opportunity set.  a s meaning t h a t t h e n  Suppose  nature  = 0, w h i c h c a n  f i r m i s u n c o r r e l a t e d w i t h t h e market,  (4.56) t h e n s i m p l i f i e s t o t h e form  -r-ll-e  = B (a-r-x-e ) M  +  3  [(^) nX  j = 1, 2,. . .,n, where  ^ j=  ^  V  See Appendix A, E q u a t i o n  (A.91).  ia^rnX  IS-l  - c ^ ,  n  (4.57)  120  The  above e q u a t i o n i s i n a form t h a t can be more e a s i l y compared t o  t h e t r a d i t i o n a l c a p i t a l a s s e t p r i c i n g model g i v e n by E q u a t i o n  (4.44) and t o  t h e more g e n e r a l form which c o n s i d e r s b a n k r u p t c y g i v e n a c o n s t a n t opportunity s e t , expressed  by E q u a t i o n  (4.42).  g e n e r a l remarks t h a t a p p l i e d t o E q u a t i o n  difference:  Equation  s t o c h a s t i c nature  I t i s c l e a r t h a t t h e same  (4.42) a p p l y t o E q u a t i o n  though perhaps w i t h even g r e a t e r emphasis: s t a t i o n a r y r e l a t i o n s h i p between a and B..  (a  - r -  "  (4.57),  t h e r e i s a n o n - l i n e a r , nonT h e r e i s , however, one major  (4.57) c o n t a i n s an e x t r a v a r i a b l e t h a t a r i s e s from t h e  o f t h e changes i n t h e i n v e s t m e n t  opportunity  set.  n i t u d e and s i g n o f t h i s v a r i a b l e w i l l depend upon t h e p a r t i c u l a r If  investment  - e ) i s p o s i t i v e and  approximately  The mag-  security.  equal t o  /  n nX nX t h e n f o r h i g h b e t a s e c u r i t i e s t h e term w i l l be n e g a t i v e , w h i l s t p o s i t i v e for  low b e t a s e c u r i t i e s .  T h i s o f f e r s an e x p l a n a t i o n o f t h e e m p i r i c a l f i n d -  55 i n g s o f B l a c k , J e n s e n and S c h o l e s , e n t l y earned  less  who found  that high beta  stocks c o n s i s t -  t h a n t h a t p r e d i c t e d by t h e CAPM, w h i l s t t h e r e v e r s e  b e i n g t r u e f o r low b e t a  stocks.  Summary The  primary  focus of the chapter  CAPM n o t from t h e v i e w p o i n t  i s t o extend  the formulation o f the  o f r e s t r i c t i o n s upon t h e i n v e s t o r , b u t  by c o n -  s i d e r i n g t h e impact o f b a n k r u p t c y upon t h e s t r u c t u r e o f r e t u r n s f o r c o r p o r a t e financial assets.  55  B l a c k , et al.  t  op. cit.  121  Two models a r e d e v e l o p e d . investment o p p o r t u n i t y  In t h e f i r s t model i t i s assumed t h a t t h e  s e t i s o n l y a l t e r e d by t h e e v e n t o f b a n k r u p t c y .  A  s i m p l i f i e d form o f s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e p r i c e dynamics o f bonds i s used so a s t o a b s t r a c t from i n t e r a c t i o n between bonds and  common s t o c k .  F o r common s t o c k  the instantaneous c o n d i t i o n a l expected  rate o f return i s a l i n e a r function o f i t s systematic  r i s k and a second  i a b l e which i s a s s o c i a t e d w i t h the p r o b a b i l i t y o f bankruptcy. t i e s o f the derived expression  provide  an e x p l a n a t i o n  f i n d i n g s o f t h e d e f i c i e n c y o f t h e CAPM.  The p r o p e r -  f o r recent  empirical  The model i s used a s a b a s i s f o r  e m p i r i c a l l y t e s t i n g t h e hypothesis o f the t h e s i s . bonds depends upon two v a r i a b l e s :  var-  the f i r s t  The r i s k premium f o r  term i s t h e product o f t h e  p r o b a b i l i t y o f b a n k r u p t c y and t h e l i q u i d a t i n g d i v i d e n d ,  and second term i s  a type o f market f a c t o r . The  second model examines t h e e f f e c t s o f s t o c h a s t i c changes i n t h e  p r o b a b i l i t y o f b a n k r u p t c y upon t h e s t r u c t u r e o f r e t u r n s . o f s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n i s used t o d e s c r i b e o f bonds, s o t h a t t h e r e assets:  form  t h e p r i c e dynamics  i s i n t e r a c t i o n between t h e two s e t s o f f i n a n c i a l  common s t o c k and bonds.  common s t o c k  A general  I t i s f o u n d t h a t t h e demand f u n c t i o n s f o r  c o n t a i n an e x t r a s e t o f terms, r e f l e c t i n g  investor's  attempts  t o hedge a g a i n s t unexpected changes i n t h e p r o b a b i l i t i e s o f b a n k r u p t c y f o r the d i f f e r e n t f i r m s .  Due t o t h e complex n a t u r e o f t h e demand f u n c t i o n s , a  s i m p l i f y i n g assumption i s made by u s i n g a s i n g l e i n s t r u m e n t a l c h a r a c t e r i z e a l l t h e changes.  variable to  I t i s found t h a t t h e i n s t a n t a n e o u s c o n d i t i o n a l  expected r a t e o f r e t u r n f o r common s t o c k  contains  an e x t r a  i n v e s t o r s ' a t t e m p t s t o hedge a g a i n s t u n e x p e c t e d changes.  term  reflecting  CHAPTER V  EMPIRICAL RESULTS  In t h i s c h a p t e r t h e e m p i r i c a l r e s u l t s o f t h e t h e s i s a r e p r e s e n t e d . It describes the estimation  o f a model t o d e t e r m i n e t h e p r o b a b i l i t y o f a  f i r m g o i n g b a n k r u p t and t h e e m p i r i c a l t e s t i n g o f t h e h y p o t h e s i s o f t h e thesis using  annual  data.  I f the p r o b a b i l i t y o f bankruptcy f o r a f i r m increases,  t h e n t h e ex-  p e c t e d r e t u r n , c o n d i t i o n a l upon no b a n k r u p t c y , which r i s k a v e r s e require, w i l l  increase  t o compensate f o r t h e e x t r a r i s k .  investors  A t any p o i n t i n  time t h e p r o b a b i l i t y o f b a n k r u p t c y f o r a f i r m i s a f u n c t i o n o f i t s a b i l i t y to r a i s e funds, e i t h e r i n t e r n a l l y o r l e x t e r n a l l y , t o cover f i x e d charges. As  c o n d i t i o n s w i t h i n t h e f i r m and t h e economy change o v e r t i m e , so w i l l t h e  f i r m ' s a b i l i t y t o r a i s e f u n d s , and t h u s t h e p r o b a b i l i t y o f b a n k r u p t c y , w h i c h may a f f e c t t h e e x p e c t e d r e t u r n w h i c h i n v e s t o r s r e q u i r e on t h e f i r m ' s c i a l assets.  The h y p o t h e s i s o f t h e t h e s i s i s t h a t d i f f e r e n c e s i n t h e p r o b a b i l -  i t y o f bankruptcy across ual  finan-  s e c u r i t i e s and a c r o s s  time a r e r e f l e c t e d i n t h e r e s i d -  r e t u r n a f t e r a b s t r a c t i n g from t h e market.  From t h e t h e o r e t i c a l a n a l y s i s g i v e n  i n C h a p t e r IV, a two v a r i a b l e  model d e s c r i b i n g t h e expected r a t e o f r e t u r n on a f i r m ' s common s t o c k i s derived.  The model i s o f t h e form  "  r  =  A  j  +  VM a  "  r  " *  ) f  j = 1, 2,...,N,  where  i s t h e i n s t a n t a n e o u s c o n d i t i o n a l expected r a t e o f r e t u r n on t h e j * " *  122  1  123  asset; a  M  i s the  i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on  market p o r t f o l i o ;  r i s the  i n s t a n t a n e o u s r i s k f r e e r a t e o f i n t e r e s t ; A.  the r a t e o f p r o b a b i l i t y o f b a n k r u p t c y f o r the j average o f the  {A.}  and  3  the  8. = a /o j JM  th  ,, a.., b e i n g  MM  — x i s a weighted  asset;  the  3M  is  D  instantaneous c o n d i t i o n -  th a l covariance formulation  o f the  j  a s s e t w i t h the market p o r t f o l i o .  o f t h e model i s a p p r o x i m a t e l y g i v e n E(r.) - r  = X.  p  A d i s c r e t e time  by  + B.[E(r ) - r M  p  - J , X  where E(r..) i s t h e c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on the i s the c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on p r o b a b i l i t y o f b a n k r u p t c y f o r the average o f the  {X.};  v a r ( r ) , cov(r.,r ) being M j M  asset  asset;  T h i s i m p l i e s a t r a n s i t i o n from an  the ex-ante f o r m u l a t i o n  and  Thus any  r M  o f the  j  t  n  asset with  the the  ex-ante t o an e x - p o s t formu-  e m p i r i c a l t e s t i s a j o i n t examination o f  the market model.  The  e x - p o s t form o f the model  is  R  where R R^  t  3 *-  jt  =  V  o  +  Vjt  +  3  j  ( R  Mt  " t>  i s the r e a l i z e d excess r e t u r n f o r the  X  j  +  th  i s t h e r e a l i z e d e x c e s s market r e t u r n f o r p e r i o d  t u r b a n c e term; and represented  v  )  8. = c o v ( r . , r ) / 3 3 M  To t e s t e m p i r i c a l l y the h y p o t h e s i s an e x - p o s t form o f  l a t i o n u s i n g a market model.  (  E  f o r the p e r i o d ; x i s a w e i g h t e d  the c o n d i t i o n a l c o v a r i a n c e  w  model i s u s e d .  th  th  the market p o r t f o l i o ; X_. i s t h e  r„ i s t h e r i s k f r e e r a t e o f i n t e r e s t ; and F  • j  market p o r t f o l i o .  j  j  V  asset during t; u  period  t;  i s a random d i s -  and v . a r e c o n s t a n t s . The h y p o t h e s i s o f the t h e s i s i s o 1 by t e s t i n g i f the c o e f f i c i e n t , v ^ , i s p o s i t i v e .  124  To and  test empirically  market p o r t f o l i o a r e  p r o b a b i l i t y o f the  required.  satisfied,  found which s u p p o r t s the  i n the  the  h y p o t h e s i s can  h y p o t h e s i s o f the  be  thesis;  p r o b a b i l i t y o f b a n k r u p t c y a c r o s s s e c u r i t i e s and  f l e c t e d i n the In the  residual return f i r s t part  bankruptcy i s p r e s e n t e d . d e t e r m i n a t i o n o f the  In t h e  empirically  The  h y p o t h e s i s and  the  The  d a t a and  is,  The  Evidence differences  a c r o s s time a r e market.  estimation of  d e s c r i b e d , as w e l l F i n a l l y , the  chapter the regression  the p r o b a b i l i t y  as  the  empirical  results  h y p o t h e s i s o f the  thesis  is  the  regression are  data presented.  externally,  i s a function  by  + MB  t  + AS  t  its ability  bankrupt.  the p r o b a b i l i t y o f b a n k r u p t c y can  t  of  to cover f i x e d charges.  f a i l s t o c o v e r t h e s e f i x e d c h a r g e s i s s a i d t o be  = Pr (FNF.  are  Bankruptcy  i n t e r n a l l y or  t  of  the means used  equation representing  p r o b a b i l i t y of bankruptcy f o r a f i r m  A  re-  r e s u l t s u s i n g aggregated p o r t f o l i o data  P r o b a b i l i t y of  Chapter I I I , i t i s shown t h a t  the  Once t h e s e  s t a t i s t i c a l methodology used t o e s t i m a t e the  t o r a i s e funds, e i t h e r firm that  that  from t h e  then r e s u l t s u s i n g i n d i v i d u a l s e c u r i t y  E s t i m a t i o n o f the The  o f the  tested.  firm  s t a t i s t i c a l methodology used i n  ability.  form o f the  c o e f f i c i e n t s are d e s c r i b e d . g i v e n f i r s t and  c h a p t e r the  c o e f f i c i e n t s are  remaining p a r t  tested.  after abstracting  o f the  t o t e s t t h e model's p r e d i c t i v e given.  I t i s a l s o n e c e s s a r y t o know how  f i r m g o i n g bankrupt over d i f f e r e n t p e r i o d s .  data requirements are is  the model, e x - p o s t e x c e s s r e t u r n s f o r the  < 0) ,  be  A  From  represented  125  where A  i s the p r o b a b i l i t y o f the f i r m going bankrupt  s t a t e o f the f i r m a t year t - l ; all  i n year t , g i v e n the  FNF^ i s t h e f i r m ' s f u t u r e c a s h f l o w n e t o f  f i x e d c h a r g e s a t y e a r t ; MB^ i s t h e maximum amount t h e f i r m can borrow  f o r t h e year t ; and AS  fc  r e p r e s e n t s a l l o t h e r s o u r c e s o f funds a v a i l a b l e t o  the f i r m a t y e a r t.^"  I f the f u t u r e cash flow  net o f a l l f i x e d  charges  the maximum amount t h e f i r m can borrow and a l l o t h e r a l t e r n a t i v e  plus  sources  o f funds i s n e g a t i v e , t h e n t h e f i r m i s s a i d t o be b a n k r u p t . The v a r i a b l e s d e t e r m i n i n g t h e p r o b a b i l i t y o f b a n k r u p t c y , i n t h e above e x p r e s s i o n , a r e i n terms o f d o l l a r amounts. d a t a w i l l be used, o f f i r m s and so  as s t a t e d  As c r o s s s e c t i o n a l  the v a r i a b l e s are not adjusted f o r d i f f e r e n c e s  may be dominated by s c a l e e f f e c t s .  i n the size  Very large scale  effects  among f i r m s would be e x p e c t e d t o l e a d t o i n e f f i c i e n t e s t i m a t i o n o f c o e f f i cients.  To a v o i d t h i s , t h e p r o b a b i l i t y o f b a n k r u p t c y  c a n be w r i t t e n i n t h e  form  A  = •  P  FNF r ( - l r  A  t-1  +  r  A  MB i -  +  r  AS ^ - < 0),  t-1 t-1 A  where A^ ^ i s t h e book v a l u e o f t h e f i r m ' s a s s e t s i n t h e y e a r t - l . the p r o b a b i l i t y a t year t - l  o f a f i r m g o i n g bankrupt  Thus  i n t h e y e a r t depends  upon i t s f u t u r e c a s h f l o w n e t o f a l l f i x e d c h a r g e s p e r u n i t o f a s s e t s and t h e t o t a l amount o f funds t h a t i t c o u l d r a i s e p e r u n i t o f a s s e t s . To e s t i m a t e e m p i r i c a l l y t h e p r o b a b i l i t y o f b a n k r u p t c y the ex-ante v a r i a b l e s be r e p l a c e d by ex-post  surrogates.  firm's f u t u r e cash flow net o f a l l f i x e d charges  requires that  An e s t i m a t e o f t h e  i s o b t a i n e d by r e g r e s s i n g  * I n C h a p t e r I I I a f u l l d i s c u s s i o n o f t h e model f o r t h e p r o b a b i l i t y o f bankruptcy . and s t a t i s t i c a l methodology i s g i v e n .  i  126  n e t income a g a i n s t time o v e r a f i v e y e a r p e r i o d and t h e n u s i n g t h e e s t i mated r e g r e s s i o n e q u a t i o n t o p r e d i c t t h e n e x t y e a r ' s v a l u e . i s t h e n d i v i d e d by t h e c u r r e n t book v a l u e o f t h e f i r m ' s t o t a l o p e r a t i o n a l d e f i n i t i o n o f n e t income used a t i n g and n o n - o p e r a t i n g  The e s t i m a t e assets.  The  i s income a f t e r d e d u c t i n g a l l o p e r -  income and expenses and m i n o r i t y i n t e r e s t b u t b e f o r e  p r e f e r r e d and common d i v i d e n d s . The maximum amount a f i r m c o u l d borrow i s e s t i m a t e d by a m u l t i p l i c a t i v e f u n c t i o n o f t h e amount o f c r e d i t r a t i o n i n g and t h a t p a r t o f t h e f i r m n o t f i n a n c e d b y debt,  as measured by t h e book v a l u e o f n e t worth. The f u n c -  t i o n i s d i v i d e d by t h e a s s e t s i z e o f t h e f i r m .  The f i n a l  form o f t h e f u n c t i o n  being .book v a l u e o f n e t worth a t t - 1 . , ( _ ) exp (-CR A  where A CR  fc  t-1  Z  . )  ~  ^ i s t h e book v a l u e o f t h e f i r m ' s t o t a l a s s e t s a t time t - 1 ; and  F C  ^ i s t h e amount o f c r e d i t r a t i o n i n g a t time t - 1 .  Credit rationing i s  2 measured u s i n g a p r o x y v a r i a b l e d e v e l o p e d it  by J a f f e e .  I n i t s s i m p l e s t form  i s t h e r a t i o o f t h e amount o f l o a n s g r a n t e d a t t h e prime r a t e t o t h e  t o t a l amount o f l o a n s g r a n t e d .  Due t o s e v e r a l d a t a problems t h e d a t a a r e  smoothed and s e a s o n a l l y a d j u s t e d t o have a mean o f z e r o and a s t a n d a r d deviation of unity.  To o b t a i n a p o s s i b l y more complete measure o f c r e d i t  r a t i o n i n g a p r i n c i p a l component t e c h n i q u e  i s used  t o combine f o u r d i f f e r e n t  3  s e r i e s , the r e s u l t s being g i v e n i n J a f f e e .  The v a r i a b l e CR^ ^ i s measured  2 J a f f e e , D. Credit Rationing John W i l e y & Sons, Inc., 1 9 7 1 ) . 3  Jaffee,  Ibid.,  pp. 1 0 1 - 1 0 3 .  and the Commercial  Loan Market  (New York:  127  u s i n g a l i n e a r average o f the p a s t  four quarters  o f the  first  principal  component. I t i s assumed t h a t a l l o t h e r  a l t e r n a t i v e s o u r c e s o f f u n d s can  measured by a l i n e a r f u n c t i o n o f e f f i c i e n c y , growth, b u s i n e s s r i s k financial risk. e s t and  E f f i c i e n c y i s e s t i m a t e d by u s i n g  t a x e s d i v i d e d by  the book v a l u e  of t o t a l assets;  y e a r l i n e a r average growth r a t e i n a s s e t s ; value GNP;  and  f i n a n c i a l r i s k by  f i v e y e a r p e r i o d and  the r e s i d u a l sum of operating before  the  An  by r e g r e s s i n g o p e r a t i n g then using  income i s n e t  The  standard  brokerage f e e s , a d v e r t i s i n g c o s t s , ex-post formulation  < 6  Q  the  e s t i m a t e o f the  firm's  of  future  income a g a i n s t t i m e o v e r a  d e v i a t i o n i s e s t i m a t e d by The  interest,  f i x e d c h a r g e s i t i s a l l i n t e r e s t expense, the  = Pr[£  change i n  standard  operational  s a l e s l e s s c o s t o f s a l e s and  the a m o r t i z a t i o n  five  absolute  the estimated r e g r e s s i o n equation to  deducting depreciation, amortization,  The  a  d i v i d e d by the  o f s q u a r e s from the r e g r e s s i o n .  c o u n t o r premium and  X  b u s i n e s s r i s k by  the d i f f e r e n c e b e i n g  d i c t t h e next y e a r ' s v a l u e .  inter-  the d i f f e r e n c e between f i x e d c h a r g e s and  d e v i a t i o n o f the f i r m ' s f u t u r e c a s h f l o w . cash flow i s obtained  and  growth by  o f the p r o p o r t i o n a l change i n s a l e s t o t h e p r o p o r t i o n a l  f i r m ' s future cash flow,  For  earnings before  be  operating taxes,  and  amortization  expenses(that i s ,  preusing  definition expenses dividends.  o f debt  dis-  underwriting,  etc.).  o f t h e model can be  written  e s t i m a t e d f u t u r e c a s h f l o w n e t o f a l l f i x e d c h a r g e s f o r time t + 0 (•  book v a l u e  o f net worth a t t - l  •)exp(-CR  t-l  128  „  + 3  .earnings b e f o r e i n t e r e s t and t a x e s a t t - 1 . (  a  _  )  t - i  J  + 8^ ( f i v e year l i n e a r growth r a t e f o r t o t a l  assets)  + g . 1 p r o p o r t i o n a l change i n s a l e s 1 5 ' p r o p o r t i o n a l change i n GNP '  +  where X  g ^ f i x e d charges a t t - 1 - e s t i m a t e d f u t u r e c a s h f l o w f o r time 6 estimated standard d e v i a t i o n o f f u t u r e cash flow  t.j  i s t h e p r o b a b i l i t y o f t h e f i r m going bankrupt i n year t , g i v e n the  s t a t e o f t h e f i r m a t y e a r t - 1 ; e i s a z e r o mean random v a r i a b l e e r r o r term, which i s assumed t o be o f u n i t v a r i a n c e and u n c o r r e l a t e d between f i r m s ; 8 , o a r e c o e f f i c i e n t s which a r e t o be e s t i m a t e d ; and A i s t h e book i D t~1 value o f the f i r m ' s t o t a l a s s e t s a t time t - 1 . The prime f o c u s i s t o be a b l e t o p r e d i c t t h e p r o b a b i l i t y o f a f i r m g o i n g bankrupt and n o t t o c o n s t r u c t a complete g e n e r a l t h e o r y . due  Consequently,  t o t h e complex i n t e r a c t i o n o f t h e u n d e r l y i n g f a c t o r s and t h e d i f f i c u l t y  o f d e v e l o p i n g an a c c u r a t e e m p i r i c a l r e p r e s e n t a t i o n o f t h e d e t e r m i n a n t s o f bankruptcy,  a second f o r m u l a t i o n u s i n g market v a l u e s f o r t h e a p p r o p r i a t e  corporate v a r i a b l e s i s developed.  The use o f market v a l u e s c i r c u m v e n t s  o f t h e d i f f i c u l t i e s o f c o n s t r u c t i n g proxy  v a r i a b l e s t o measure such  many  quanti-  t i e s as t h e maximum amount t h e f i r m c o u l d borrow and t h e t o t a l o f a l l o t h e r alternative  sources.  F o r the f i r m ' s f u t u r e c a s h f l o w n e t o f a l l f i x e d c h a r g e s ,  t h e same  proxy v a r i a b l e , a s p r e v i o u s l y d e f i n e d , i s used; t h a t i s , r e a l i z e d v a l u e s o f the f i r m ' s cash flow n e t o f a l l f i x e d charges  a r e r e g r e s s e d a g a i n s t time  over  .,. ^.  129  a f i v e year p e r i o d and then t h e e s t i m a t e d t h e next y e a r ' s v a l u e o f t h e f i r m ' s cash value  r e g r e s s i o n equation  flow net o f a l l f i x e d charges.  i s t h e n d i v i d e d by t h e book v a l u e o f t h e f i r m ' s t o t a l  r e s u l t a n t used a s t h e e x - p o s t The  used t o p r e d i c t This  a s s e t s and t h e  surrogate.  p r o x y v a r i a b l e used t o measure t h e ex-ante maximum amount t h e  f i r m c o u l d borrow p e r u n i t o f a s s e t s f o r y e a r t i s .market v a l u e o f e q u i t y at t-l, , . ( a * ) exp (-CR ), A  where A  fc  t-1  ^ i s t h e book v a l u e o f t h e f i r m ' s a s s e t s a t time t - l ;  i s t h e amount o f c r e d i t r a t i o n i n g a t t i m e The  fc  ^  t-l.  t o t a l o f a l l other a l t e r n a t i v e sources  may u t i l i z e depends upon t h r e e b r o a d c a t e g o r i e s : duction o f planned outflows,  and CR  o f funds which t h e f i r m uncommitted r e s e r v e s , r e -  and t h e l i q u i d a t i o n o f a s s e t s .  A variable  which s y n t h e s i z e s t h e s e d i v e r s e q u a n t i t i e s i s t h e market v a l u e o f e q u i t y . Thus, t h e second f o r m u l a t i o n o f t h e model i s o f t h e form X  t  = PrU < Y  ,estimated  Q  + Y (  f u t u r e cash  ->  .market v a l u e o f e q u i t y a t time t - l . A ^  + y < 2  +  Y  7  t  -  )  1  ^ i s t h e book v a l u e o f t h e f i r m ' s t o t a l a s s e t s ;  u n c o r r e l a t e d between f i r m s ; and Y / Y ^ » V ^ ' "^3 Q  estimated.  , -  . i  1  , „. K  t-1  random v a r i a b l e e r r o r term, which i s assumed t o be o f  be  t  ,market v a l u e o f e q u i t y a t t i m e t - l , , A  where A  f l o w net o f a l l f i x e d c h a r g e s f o r time t.  X  a  r  e  e i s a z e r o mean unit  v a r i a n c e and  c o e f f i c i e n t s which a r e t o  130  Statistical The by  Methodology (5.1)  c o e f f i c i e n t s i n Equations  regression  a s t h e dependent v a r i a b l e  e f f i c i e n t s c a n be e s t i m a t e d by u s i n g dom sample o f f i r m s  and ( 5 . 2 )  i s unobservable.  maximum l i k e l i h o o d .  a t time t and suppose the f i r s t  and  t h e r e m a i n d e r n-n  can  t h e n be w r i t t e n  non-bankrupt.  1  However, t h e c o Consider a ran-  n' f i r m s  The l o g a r i t h m i c  n' Z j=l  cannot be e s t i m a t e d  are bankrupt  likelihood  function  n l o g Pr(B ^  |x t - 1  ) + '3  Z j=n'+l  l o g [ l - Pr(B Z  J  Z  |x ~ '  )], 3  (5.3) where P r ( B _ . | x t  ^ _.) i s t h e p r o b a b i l i t y o f t h e j * " * f i r m g o i n g b a n k r u p t i n 1  t  y e a r t , g i v e n a s e t o f a t t r i b u t e s measured a t y e a r t - 1 . either  (5.1) o r (5.2) i n t o  (5.3) and d i f f e r e n t i a t i n g w i t h r e s p e c t  coefficients, a setof  non-linear  maximization conditions  a r e o b t a i n e d and c a n be s o l v e d  Two e s t i m a t i o n  By s u b s t i t u t i n g  equations representing  the f i r s t  logit analysis.  the  e x p l i c i t form o f t h e p r o b a b i l i t y d i s t r i b u t i o n s .  order  iteratively.  p r o c e d u r e s a r e used i n t h e t h e s i s : p r o b i t  and  t o the  analysis  The e s s e n t i a l d i f f e r e n c e between t h e two p r o c e d u r e s i s  normal p r o b a b i l i t y d i s t r i b u t i o n i s assumed, w h i l s t  For probit analysis a forlogit  a n a l y s i s the 4  distribution i s logistic.  A computer programme d e v e l o p e d by Cragg  i s used  t o d e t e r m i n e t h e e s t i m a t e s o f t h e c o e f f i c i e n t s f o r t h e two p r o c e d u r e s .  Cragg, J . G. "Programs f o r M u l t i p l e P r o b i t and L o g i t A n a l y s i s and E x t e n s i o n s t o Them," mimeographed, U n i v e r s i t y o f B r i t i s h Columbia, 1968.  131  Data To  estimate  t h e p a r a m e t e r s o f the model, t h e l i k e l i h o o d f u n c t i o n  must be c o n s t r u c t e d by t a k i n g a random sample o f f i r m s and the f i r m s as b a n k r u p t o r n o t b a n k r u p t . sample and age  classifying  p r o c e d u r e o f u s i n g a random  the f i r m s a v o i d s  selection bias.  p r o b a b i l i t y of a f i r m going bankrupt i s small,a very  must be Due  then c l a s s i f y i n g  The  then  As the  aver-  l a r g e random sample  t a k e n so as t o o b t a i n a r e p r e s e n t a t i v e c o l l e c t i o n o f b a n k r u p t f i r m s .  t o the problem o f c o l l e c t i n g d a t a  f o r such a l a r g e sample an a l t e r n a t i v e  i  procedure i s used.  Instead  o f t a k i n g a random sample o f f i r m s and  then  f y i n g t h e f i r m s as b a n k r u p t o r non-bankrupt, d a t a a r e c o l l e c t e d f o r a o f b a n k r u p t f i r m s and For the  10 year p e r i o d 1960-1969 a l i s t  sources  Dunn and  t h e sample. data age  A list  f o r these  o f 34  t i o n o f the  fact  The  criteria  f r a u d ; and  S e c u r i t y and  f i r m s used i s g i v e n  No  corporate  i n Appendix B.  shells The  t h o s e t h a t a r e t r a d e d on  Stock Exchange r a r e l y go b a n k r u p t , t h e o b v i o u s e x c e p t i o n b e i n g If a l a r g e f i r m i s i n d i f f i c u l t i e s  the  f i r m must have been  i t i s e i t h e r acquired  entered  corporate The  m i l l i o n v/hich i s q u i t e s m a l l , a  that large firms, l i k e  Ex-  o f d a t a be a v a i l a b l e ;  i n Moody's I n d u s t r i a l Manual.  f i r m s i s $11  The  Street  used t o s e l e c t  (c) t h e  or service for s a l e .  firms are obtainable  asset s i z e of these  the  (a) a t l e a s t t h r e e y e a r s  (b) the b a n k r u p t c y i s n o t caused by a c t u a l l y o f f e r i n g a product  are:  Moody's I n d u s t r i a l Manual, W a l l  Bradstreet.  sample o f b a n k r u p t f i r m s a r e :  of firms that declared  i n t h e U.S.A. i s c o m p i l e d .  used i n the c o m p i l a t i o n  change Commission A n n u a l R e p o r t s , J o u r n a l Index, and  universe  t h e n f o r a u n i v e r s e o f non-bankrupt f i r m s .  b a n k r u p t c y u n d e r t h e B a n k r u p t c y A c t o f 1938 f o u r main d a t a  classi-  aver-  reflec-  the New  York  Penn C e n t r a l .  by a n o t h e r f i r m o r  132  a merger  occurs. To o b t a i n a r e p r e s e n t a t i v e  random sample o f non-bankrupt  r e q u i r e s d e t e r m i n i n g the number o f f i r m s t o be s e l e c t e d .  firms,  Ideally,  the  number chosen s h o u l d be the same as t h a t o b t a i n e d by t a k i n g a random sample o f a l l f i r m s and then c l a s s i f y i n g them as b a n k r u p t o r n o t b a n k r u p t .  As  the  s i z e o f t h e sample o f bankrupt f i r m s i s a l r e a d y known, t h e n t h e s i z e o f t h e sample o f non-bankrupt  f i r m s s h o u l d be such t h a t the number o f  non-bankrupt  t o bankrupt f i r m s a p p r o x i m a t e l y e q u a l s t h e a v e r a g e p r o b a b i l i t y o f n o t g o i n g bankrupt t o t h e average p r o b a b i l i t y o f b a n k r u p t c y . the whole u n i v e r s e used.  of bankrupt firms s a t i s f y i n g  Thus, t o o b t a i n a r e p r e s e n t a t i v e  t a k i n g t h e whole u n i v e r s e ,  F o r t h e 10 y e a r p e r i o d  the s e l e c t i o n c r i t e r i a i s  sample o f non-bankrupt  firms  f o r the 10 y e a r p e r i o d , o f non-bankrupt  s a t i s f y the s e l e c t i o n c r i t e r i a  from t h e same p o p u l a t i o n  requires  firms  that  which t h e b a n k r u p t  firms are s e l e c t e d . As t h e a v e r a g e p r o b a b i l i t y o f b a n k r u p t c y i s q u i t e s m a l l , t h i s  im-  p l i e s t h a t t h e s i z e o f t h e sample o f non-bankrupt  f i r m s w i l l be l a r g e .  d a t a f o r the b a n k r u p t f i r m s  from Moody's  Manual;  i s manually c o l l e c t e d  t h u s t o o b t a i n d a t a f o r a l a r g e sample o f non-bankrupt  t h e same s o u r c e would be p r o h i b i t i v e l y time consuming. problem t h e non-bankrupt f i r m s a r e sampled t a i n e d on the Compustat  property  firms  using  To a v o i d t h i s s e v e r e  from the u n i v e r s e  the p o p u l a t i o n  of firms  con-  of bankrupt f i r m s i s  a l l f i r m s a r e l e s s t h a n $200 m i l l i o n  i s used t o d e f i n e a p o p u l a t i o n  firms are s e l e c t e d .  Industrial  File.  A characteristic that describes t h a t of asset s i z e :  The  in size.  o f f i r m s from which  The sample o f non-bankrupt  the  This  non-bankrupt  i s o b t a i n e d by s e l e c t i n g ,  on  133  a year by y e a r b a s i s f o r the 10 than $200 m i l l i o n . i s shown i n T a b l e  The 5.1.  year p e r i o d ,  a l l firms with  number o f f i r m s s e l e c t e d each y e a r f o r b o t h F o r the y e a r 1960  there  f i r m s i n the non-bankrupt s e t .  f l e c t s the  the  fact that, given  selection criteria,  i s a r a r e event.  By  Predictive  This simply  re-  b a n k r u p t c y f o r the  t a k i n g a random sample o f  from the d e f i n e d p o p u l a t i o n o v e r a 10 y e a r p e r i o d , t h e r e t h a t f o r a p a r t i c u l a r y e a r the  samples  i s a n u l l s e t of bankrupt  f i r m s , w h i l s t t h e r e a r e 237  of firms considered  asset size less  i s no  s e t of bankrupt firms i s not  type  firms  guarantee  empty.  Ability  As t h e p r o b a b i l i t y o f b a n k r u p t c y c a n n o t be o b s e r v e d , d i r e c t t e s t s on the models are not p o s s i b l e .  T h i s i m p l i e s t h a t t h e magnitude o f  b i a s o r measurement e r r o r i n t h e e s t i m a t e s t h e main check on how ability.  w e l l the models a r e  s p e c i f i e d must be  Thus,  their predictive  T h r e e methods a r e u s e d t o t e s t t h e models. From t h e o r e t i c c o n s i d e r a t i o n s the  d e t e r m i n e d and  compared t o t h o s e o b t a i n e d  number o f e s t i m a t e d the  c a n n o t be d e t e r m i n e d .  any  parameters with  s p e c i f i c a t i o n o f t h e model and  s i g n s o f the p a r a m e t e r s can  be  from e m p i r i c a l e s t i m a t i o n .  The  the c o r r e c t sign provides the a c c u r a c y  insight into  of the proxy v a r i a b l e s at  measuring the ex-ante q u a n t i t i e s . I f the model i s c o m p l e t e l y  specified  so as t o measure a l l the  ent a t t r i b u t e s o f the f i r m s i n the sample u s e d t o e s t i m a t e  the  then i t s h o u l d be a b l e t o c o r r e c t l y i d e n t i f y t h e bankrupt and f i r m s i n the  sample.  The  c l a s s i f i c a t i o n a b i l i t y provides  the model's s p e c i f i c a t i o n and  differ-  parameters, non-bankrupt  information  about  the number o f common d e t e r m i n a n t s o f b a n k r u p t c y .  TABLE 5.1 NUMBER OF BANKRUPT AND NON-BANKRUPT FIRMS IN DATA SAMPLE  Y E A R  BANKRUPT  NON-BANKRUPT  1960  1961  1962  1963  1964  1965  1966  0  5  8  3  2  3  4  237  253  254  265  262  248  224  1967  .  3  208  1968  1969  TOTAL  3  3  34  ' 168  150  2303  135  The  g e n e r a l i t y o f the model and i t s o v e r a l l  p e c u l i a r i t i e s o f t h e data  sample used t o e s t i m a t e  independence o f t h e  t h e p a r a m e t e r s , can be  t e s t e d by examining t h e p r e d i c t v e a b i l i t y on a s e t o f b a n k r u p t f i r m s n o t used i n t h e o r i g i n a l one is  sample.  A list  o f firms that vent  b a n k r u p t i n 1970,  y e a r a f t e r t h e end o f t h e p e r i o d used f o r e s t i m a t i n g t h e c o e f f i c i e n t s compiled  a list  u s i n g t h e S e c u r i t y and Exchange Commission A n n u a l R e p o r t s and  o f b a n k r u p t f i r m s g i v e n i n a n a r t i c l e by Altman.^  By e s t i m a t i n g t h e  p r o b a b i l i t y o f b a n k r u p t c y o v e r s e v e r a l time p e r i o d s f o r f i r m s i n t h e new sample p r o v i d e s  a d e m o n s t r a t i o n o f t h e model's p r e d i c t i v e a b i l i t y t o d i s -  cern a firm's path  to bankruptcy.  Results The  p r o b a b i l i t y o f bankruptcy i s estimated  l a t i o n g i v e n by E q u a t i o n Table  5.2.  (5.1).  The e s t i m a t e d  u s i n g t h e ex-post  c o e f f i c i e n t s a r e shown i n  With t h e x c e p t i o n o f growth and b u s i n e s s  have t h e c o r r e c t s i g n .  formu-  risk,  a l l t h e parameters  The v a r i a b l e s r e p r e s e n t i n g t h e maximum amount t h e  f i r m c o u l d borrow and e f f i c i e n c y , a r e t h e o n l y two t h a t a r e n o t s t a t i s t i c a l l y 6 significant.  The R-squared  values  f o r l o g i t a n a l y s i s and p r o b i t a n a l y s i s a r e  h i g h and, a s e x p e c t e d , a l m o s t  identical.  Altman, E. 1972), pp. 718-721.  Journal  3  "Reply,"  of Finance, V o l . XXVII, No. 3 (June,  ^ F o r t h e maximum l i k e l i h o o d a n a l y s i s , R-squared i s d e f i n e d by R-squared = { l - e x p [ 2 ( L  -L )/T]}/{l-exp[2(L  -L )/T} max where L i s t h e maximum o f t h e l o g a r i t h m i c l i k e l i h o o d f u n c t i o n u s i n g o n l y a c o n s t a n t , L i s t h e maximum u s i n g a l l v a r i a b l e s and L i s t h e maximum possible. T i s t h e t o t a l number o f o b s e r v a t i o n s . 0  «  \i  w  TABLE 5.2 ESTIMATION OF COEFFICIENTS FOR A GENERAL MODEL DESCRIPTION  FUTURE CASH FLOW NET ALL FIXED CHARGES  Notation  6  1  S  Expected Sign  -  Logit Analysis  -10.313 (- 2.651)  Probit Analysis  MAXIMUM AMOUNT THAT CAN BE BORROWED  t i l  "5-455 „ (-2.734)  EFFICIENCY  2  6 3  -  -  GROWTH  6  4  BUSINESS RISK  6  -  FINANCIAL RISK  RSQUARED  e  5  +  CONSTANT  LOGARITHM OF MAXIMUM LIKELIHOOD FUNCTION  o  +  -0.824 (-1.405)  -6.901 (-1.885)  2.502 (4.126)  (-2.326)  °-275 „ (2.435)  -3.219 (-6.836)  0.58  -76.092  -0.438 (-1.617)  -3.069 (-1.599)  1.2307^ (3.921)  -0-0772,„ (-2.308)  0-116 (2.014)  -1.663 (-7.809)  0.59  -75.286  (Figures i n brackets are t - s t a t i s t i c s * ** ***  OF VARIABLES  s t a t i s t i c a l l y s i g n i f i c a n t a t 0.1% s t a t i s t i c a l l y s i g n i f i c a n t a t 2.0% s t a t i s t i c a l l y s i g n i f i c a n t a t 5.0%  t  t  t  137  The a b i l i t y of the model to c l a s s i f y c o r r e c t l y the bankrupt  and  non-bankrupt firms i n the o r i g i n a l data sample i s shown i n Table 5.3.  A  bankrupt firm i s c l a s s i f i e d as non-bankrupt i f i t s estimated p r o b a b i l i t y of bankruptcy i s less than the average value of the estimated p r o b a b i l i t y of bankruptcy f o r the whole data sample. as bankrupt  A non-bankrupt firm i s c l a s s i f i e d  i f i t s estimated p r o b a b i l i t y of bankruptcy i s greater than the  average value of the estimate p r o b a b i l i t y of bankruptcy f o r the whole data sample.  This c r i t e r i o n i s used throughout.  For the c o e f f i c i e n t s estimated  using l o g i t analysis, the model c o r r e c t l y i d e n t i f i e s over 91 per cent of the non-bankrupt firms and over 94 per cent of the bankrupt  firms.  For p r o b i t  analysis, the model c o r r e c t l y i d e n t i f i e s over 90 per cent of the non-bankrupt firms and over 94 per cent of the bankrupt firms.  The small discrepancies i n  the two sets of r e s u l t s a r i s e from the differences between the l o g i s t i c and normal p r o b a b i l i t y d i s t r i b u t i o n s .  The hypothesis that the model's c l a s s i f i -  cation a b i l i t y i s due to a purely random process can be rejected with a proba b i l i t y of over 99 per cent. The model's a b i l i t y to predict bankruptcy  i s shown i n Table  5.4.  The p r o b a b i l i t y of bankruptcy i s estimated over as many periods as a v a i l a b l e data permit, f o r a group of eight firms that declared bankruptcy  i n 1970.  For  Uniservices Incorporated the model p r e d i c t s f a i l u r e four years before the date of bankruptcy and f o r Bishop Industries three years.  Bankruptcy i s  predicted two years i n advance f o r G. F. Industries, and Roberts Company, and one year f o r V i s u a l E l e c t r o n i c s and Dolly Madison Incorporated.  The  model t o t a l l y f a i l s f o r Century Geophysical Incorporated, giving a p r o b a b i l i t y of zero one year before bankruptcy.  The f a i l u r e of the model can be a t t r i -  138 TABLE 5.3 CLASSIFICATION OF ORIGINAL DATA SAMPLE BY GENERAL MODEL  ACTUAL OUTCOME  LOGIT ANALYSIS  PREDICTED OUTCOME  BANKRUPT  NON-BANKRUPT  T O T A L S  BANKRUPT  NON-BANKRUPT  TOTALS  AVERAGE VALUE OF PROBABILITY  32  191  223  0.434  2  2078  2080  0.009  34  2269  2303  Type one e r r o r == p r o b a b i l i t y [of a f i r m i 191/2269 = .084  bankrupt|non-bankrupt]  Type two e r r o r == p r o b a b i l i t y [of a f i r m non-bankrupt|bankrupt] 2/34 = 0.059  ACTUAL OUTCOME  PROBIT ANALYSIS  PREDICTED OUTCOME  BANKRUPT  NON-BANKRUPT  TOTALS  AVERAGE VALUE OF PROBABILITY  BANKRUPT  32  203  235  0.413  NON-BANKRUPT  2  2066  2068  0.009  34  2269  2303  T O T A L S  Type one e r r o r = 0.098 Type two e r r o r = 0.059  TABLE 5.4 PREDICTIVE ABILITY OF GENERAL MODEL  \  NAME OF FIRM  1961  1962  1963  1964  UNISERVICES INC.  ROBERTS COMPANY  CENTURY GEOPHYSICAL INCORPORATED  0.1164 0.1133  1969  1966  1967  1968  .0.0005 0.0001  0.151* 0.169  0.737* 0.707  0.468*  Sept., 1968  1970  0.492  Nov., 1968  Feb., 1970  June,  1970  March, 1970  June, 1970  June,  1970  Sept.,  June, 1970  0.0007 0.0002  0.108 0.135  0.0009 0.0002  0.0014 0.0006  0.0014 0.0007  0.004 0.003  0.052* 0.067  0.010 0.008  0.106 0.108  0.147 0.191  0.045 0.058  0.0128 0.015  0.004 0.002  0.002 0.001  0.04 0.057  0.0 0.0  0.002 0.0  0.044* 0.063  VISUAL ELECTRONICS  G.F. INDUSTRIES  0.021  0.008 0.009  0.343  0.366*  0.832*  DOLLY MADISON INC.  0.004  0.0034  0.012 0.015  0.003 0.001  °- ** 0.036  FARRINGTON MANUFACTURING COMPANY BISHOP INDUSTRIES  0.025  0.373 0.337  0.991 0.986  0.994 0.994  0.235 0.258  DATE OF BANKRUPTCY  1965  0.046 0.0G3  1970  LAST DATE OF DATA  1969  0.998* 0.999  0.740  03  1969  1969  0.056 0.067  0.012 0.012  0.005 0.004  0.001 0.001  0.004  0.343* 0.330  Dec., 1969  1970  0.0 0.0  0.015 0.016  0.003 0.002  0.017* 0.020  0.982* 0.967  0.968* 0.951  Oct., 1969  Oct., 1970  Average value o f p r o b a b i l i t y of bankruptcy - 0.014  *  0.005  indicates probability above average value.  140  buted and  t o t h e i n c o r r e c t s i g n s o f the growth and b u s i n e s s r i s k  the l a r g e v a r i a b i l i t y  coefficients,  o f the f i r m ' s cash f l o w n e t o f a l l f i x e d  charges.  A l t e r n a t i v e Model The  i n c o r r e c t s i g n o f two  poor p r e d i c t i v e a b i l i t y ,  X  distracts  T h i s p r e c i p i t a t e d development o f a model o f  form _ p [ e < g +g  ^estimated f u t u r e c a s h f l o w net o f a l l f i x e d charges  r  t  coefficients,  and h i g h c o r r e l a t i o n among t h e v a r i a b l e s ,  from the a p p e a l o f t h e model. the  statistically significant  "  0  f o r time  t  V i  1  + B ( b ° ° * v a l u e o f n e t worth a t 2  +  a ^ i ^ charges a t t-1 - e s t i m a t e d f u t u r e cash f l o w f o r time t... 3 estimated standard d e v i a t i o n of f u t u r e cash flows ' x  e  <  (5.3) where t h e c o e f f i c i e n t  8^ measures t h e s i g n i f i c a n c e o f f i n a n c i a l r i s k .  e s t i m a t e d v a l u e s o f the c o e f f i c i e n t s a r e shown i n Table 5.5.  A l l the  The coeffi-  c i e n t s e s t i m a t e d o v e r t h e 10 y e a r p e r i o d have the c o r r e c t s i g n and a r e  statis-  tically  probit  significant.  The R-squared v a l u e s f o r b o t h l o g i t a n a l y s i s and  a n a l y s i s are high. The T a b l e 5.6.  a b i l i t y o f t h e model t o c l a s s i f y t h e o r i g i n a l d a t a i s g i v e n i n F o r t h e c o e f f i c i e n t s c a l c u l a t e d u s i n g l o g i t a n a l y s i s , the model  c o r r e c t l y c l a s s i f i e s over the bankrupt  firms.  89 p e r c e n t o f the non-bankrupt and 91 p e r c e n t o f  For p r o b i t a n a l y s i s s i m i l a r r e s u l t s are obtained.  h y p o t h e s i s t h a t t h e model's c l a s s i f i c a t i o n a b i l i t y  i s due  p r o c e s s can be r e j e c t e d w i t h a p r o b a b i l i t y o f over 99 p e r  The  t o a p u r e l y random cent.  TABLE 5.5  ESTIMATION OF COEFFICIENTS AND TEST FOR STATIONARITY ALTERNATIVE MODEL  METHODOLOGY  FUTURE CASH FLOW NET OF ALL FIXED CHARGES  TIME  NOTATION  6  MAXIMUM AMOUNT THAT CAN BE BORROWED  1  B  FINANCIAL RISK  2  6  CONSTANT  RSQUARED  LOGARITHM OF MAXIMUM LIKELIHOOD FUNCTION ( l o g L) e  STATIONARITY TEST  -2 [(log L. + ei  3  log L ) e  EXPECTED SIGN  2 l o g L ] -v. x  +  -13.091 1960-1969 (- 4.182)  LOGIT  -1.084 *  (-1.973)  e  0.330 *  (3.155)  2  3  4  -2.769 *  (-6.767)  *  0.54  -84.490  5.68 ANALYSIS  -  7.697  -3.372  0.389  -1.365  (- 1.562)  (-2.183)  (2.654)  (-1.517)  -15.059  -0.026  0.268  -3.239  1960-1964  0.63  significant  1965-1969 (- 3.399)  -  6.639  1960-1969 (- 4.153)  PROBIT  (-0.038)  (1.827)  -0.549 *  (-2.219)  (2.702)  0.46  -46.422  0.53  -86.072  at  10% l e v e l  (-6.285)  0.1388 *  Not  -37.227  -1.470 *  (-8.031)  *  6.84 Not  ANALYSIS  1960-1964  3.684  -1.716  0.180  -0.774 0.64  (-1.543)  (-2.483)  -8.087  -0.017  0.101  -1.729  (-3.513)  (-0.05)  (1.486)  (-7.12)  (2.3952)  -36.236  significant  (-1.908) at  1965-1969  0.46  (figures i n brackets are t - s t a t i s t i c s ) s t a t i s t i c a l l y s i g n i f i c a n t a t 5% o r l e s s .  -46.305  10% l e v e l  142 TABLE 5.6 CLASSIFICATION OF ORIGINAL DATA SAMPLE:  ACTUAL OUTCOME  LOGIT ANALYSIS  PREDICTED OUTCOME  ALTERNATIVE MODEL  BANKRUPT  NON-BANKRUPT  TOTALS  AVERAGE VALUE OF PROBABILITY  31  249  280  0.396  NON-BANKRUPT  3  2020  2023  0.009  TOTALS  34  2269  2303  BANKRUPT  Type one e r r o r = p r o b a b i l i t y = 249/2269 = 0.11  [of a f i r m  Type two e r r o r = p r o b a b i l i t y = 3/34 = 0.089  [of a f i r m  non-bankrupt|bankrupt]  ACTUAL OUTCOME  PROBIT ANALYSIS  PREDICTED OUTCOME  bankrupt|non-bankrupt]  BANKRUPT  NON-BANKRUPT  TOTALS  AVERAGE VALUE OF PROBABILITY  BANKRUPT  31  261  292  0.372  NON-BANKRUPT  3  2008  2011  0.009  TOTALS  34  2269  2303  Type one e r r o r = 0.115  Type two e r r o r = 0.089  143  The p r e d i c t i v e a b i l i t y o f the model i s demonstrated Failure i s predicted five  y e a r s i n advance f o r B i s h o p  years f o r Uniservices Incorporated. tries  bankruptcy  i s p r e d i c t e d two  f i r m s , except V i s u a l E l e c t r o n i c s ,  i n Table  5.7.  I n d u s t r i e s and  four  F o r R o b e r t s Company and G. F.  y e a r s i n advance and a one  f o r the  Indus-  remaining  year p r e d i c t i o n i s given.  This i n -  c l u d e s Century G e o p h y s i c a l I n c o r p o r a t e d f o r w h i c h t h e g e n e r a l model, ..as r e p r e s e n t e d by E q u a t i o n  (5.1), f a i l e d  year before bankruptcy.  by g i v i n g a p r o b a b i l i t y o f z e r o  F o r V i s u a l E l e c t r o n i c s t h e model p r e d i c t s a p r o b -  a b i l i t y o f a p p r o x i m a t e l y 90 p e r c e n t o f i t g o i n g bankrupt fore i t actually Due  one  t h r e e months be-  failed.  t o t h e model's good c l a s s i f i c a t i o n  and p r e d i c t i v e  ability,  and the e s t i m a t e d c o e f f i c i e n t s h a v i n g t h e c o r r e c t s i g n , i t i s u s e d i n t h e second p a r t o f t h e e m p i r i c a l work t o e s t i m a t e the p r o b a b i l i t y o f i n t h e t e s t i n g o f t h e h y p o t h e s i s o f the  bankruptcy  thesis.  Stationarity The  c o e f f i c i e n t s o f t h e model a r e e s t i m a t e d o v e r a 10 y e a r p e r i o d .  As the model i s t o be used  f o r e s t i m a t i o n purposes,  amine i t s s t a t i o n a r i t y ; t h a t i s , o v e r d i f f e r e n t still  an a c c u r a t e e s t i m a t o r o f t h e p r o b a b i l i t y o f b a n k r u p t c y .  (1965-1969).  The  ex-  s u b - p e r i o d s i s t h e model  n o n - s t a t i o n a r i t y the d a t a sample i s s p l i t i n t o two and  i t i s necessary to  time p e r i o d s  To t e s t f o r (1960-1964)  c o e f f i c i e n t s o f t h e model a r e e s t i m a t e d o v e r t h e  p e r i o d s and the l o g a r i t h m o f the maximum l i k e l i h o o d f u n c t i o n d e t e r m i n e d . asymptotic t e s t  to the  f o r s t a t i o n a r i t y i s g i v e n by  subAn  7  F o r p r o o f , see C h a p t e r X o f Mood, A. and G r a y h i l l , F. Theory of Statistics (New York: M c G r a w - H i l l , 1963).  Introduction  TABLE 5.7 PREDICTIVE ABILITY OF ALTERNATIVE MODEL  LAST DATE OF NAME OF FIRM  1960  1961  1962  1963  1964  1965  0.0007 0.0003  UNISERVICES INC.  1966  1967  1968  0.116* 0.134  0.816* 0.753  0.637* 0.549  1969  1970  D  A  T  A  DATE OF BANKRUPTCY  Sept., 1968  1970  Nov., 1968  Feb., 1970  June, 1969  1970  March, 1970  June, 1970  *  ROBERTS COMPANY  CENTURY GEOPHYSICAL INC.  0.576 0.463  0.031 0.043  0.002 0.001  0.049 0.062  0.013 0.013  0.005 0.003  0.0027 0.0019  0.0025 0.0016  0.019* 0.022  0.168 0.169  0.018 0.019  0.463 0.436  0.052 0.066  0.065 0.083  0.008 0.008  0.004 0.003  0.005 0.004  0.0291* 0.035*  0.0001 0.0  0.01 0.01  VISUAL ELECTRONICS  0.929* 0.882  G. F. INDUSTRIES  0.015 0.018  0.005 0.005  0.095* 0.109  0.468* 0.416  June, 1969  1970  DOLLY MADISON. INC.  0.004 0.004  0.012 0.013  0.008 0.008  0.05* 0.06  Sept., 1969  June, 1970  0.012 0.013  0.008 0.009  0.002 0.001  0.244* 0.250  Dec., 1969  1970  0.911* 0.855  0.956* 0.925*  Oct., 1969  Oct., 1970  FARRINGTON MANUFACTURING COMPANY  BISHOP INDUSTRIES  0.774 0.672  0.987 0.975  0.99 0.99  0.464 0.432  0.069 0.082  0.012 0.013  0.0003 0.019* 0.020 0.0  0.0121* 0.028* 0.034 0.012  Average v a l u e o f p r o b a b i l i t y o f bankruptcy - 0.014 * " I n d i c a t e s p r o b a b i l i t y above average v a l u e .  145  -2 { [ l o g  (L ) + l o g (L )] - l o g (L)} * e l e 2 e  where L i s t h e maximum l i k e l i h o o d  y}, x  f u n c t i o n f o r t h e whole p e r i o d ;  and 2  are  t h e maximum l i k e l i h o o d  chi-square d i s t r i b u t e d  f u n c t i o n s f o r t h e two s u b - p e r i o d s ; and x  random v a r i a b l e  on k d e g r e e s o f f u n c t i o n ,  I  v  s  a  k being the  number o f c o e f f i c i e n t s t o be e s t i m a t e d i n t h e model. The  results  for the stationarity  t e s t a r e shown i n T a b l e 5.5  While  t h e r e i s some v a r i a b i l i t y i n t h e c o e f f i c i e n t s , t h e y a l l have t h e c o r r e c t and  are,  i n general, s t a t i s t i c a l l y  b i t analysis,  significant.  For l o g i t analysis  sign  and p r o -  t h e r e i s no e v i d e n c e o f n o n - s t a t i o n a r i t y a t t h e 10 p e r c e n t  level.  Predictive  Model  As  the primary focus i s the p r e d i c t i o n  and e s t i m a t i o n o f t h e p r o b a b i l -  i t y o f b a n k r u p t c y , a second f o r m u l a t i o n o f t h e model u s i n g market v a l u e s o f the  appropriate corporate variables  Equation  estimation o f the c o e f f i c i e n t s  g i v e n i n T a b l e 5.8.  For l o g i t analysis  and a t e s t  f o r s t a t i o n a r i t y are  a l l the coefficients  have t h e c o r r e c t  f u t u r e c a s h f l o w n e t o f a l l f i x e d c h a r g e s and t h e c o n s t a n t b e i n g  tically significant. alternative  In p r o b i t  analysis,  s o u r c e s o f funds i s i n c o r r e c t ,  c a l l y i n s i g n i f i c a n t , as i s the c o e f f i c i e n t c o u l d borrow. In t h e t e s t cients  T h i s i s represented by  (5.2). The  sign,  i s developed.  the sign  o f the variable  though t h e c o e f f i c i e n t  representing is statisti-  f o r t h e maximum amount t h a t t h e f i r m  The R-squared i s h i g h f o r b o t h l o g i t  for stationarity  statis-  there i s f l u c t u a t i o n  analysis  and p r o b i t  analysis.  i n the signs o f the c o e f f i -  r e p r e s e n t i n g t h e maximum amount t h a t t h e f i r m  c o u l d borrow and a l t e r -  TABLE 5.8  ESTIMATION OF COEFFICIENTS AND TEST FOR STATIONARITYj PREDICTIVE MODEL  METHODOLOGY  TIME  FUTURE CASH FLOW NET OF ALL FIXED CHARGES  MAXIMUM AMOUNT THAT CAN BE BORROWED  ALTERNATIVE SOURCES  CONSTANT  RSQUARED  LOGARITHM OF MAXIMUM LIKELIHOOD FUNCTION  STATIONARITY TEST  *  NOTATION Y  l  Y  2 _  EXPECTED SIGN  -23.006 1960-1969 (- 9.411)  LOGIT  -0.270 *  vV0.638)  Y  3 _  -0.163 (-0.293)  -3.848  n (-9.942)  0.48  -95.31  3.28 ANALYSIS  -23.425  -1.169  1.286  -4.120  (1.139)  (-7.108)  0.54  1960-1964 (- 6.156)  .  (-1.068)  -45.727  Not at  -23.099  0.176  -0.979  -3.403  (- 6.912)  (0.326)  (-1.219)  (-6.591)  -0.121  +0.060  -2.044  (-0.656)  ( 0.271)  (-12.599)*  -11.033  -0.481  0.589  - 2.133  (- 6.856)  (-1.042)  ( 1.282)  (- 9.39)  -11.295  0.079  -0.299  -1.845  (- 6.864)  (0.313)  (-0.897)  (-8.16)  1965-1969  -11.025 PROBIT  ANALYSIS  1960-1969 (- 9.871)*  1960-1964  1965-1969  (figures i n brackets are t - s t a t i s t i c s )  'statistically  s i g n i f i c a n t a t t h e 0.1% l e v e l .  0.44  -47.95  0.49  -94.223  -44.94  0.44  -47.65  10* l e v e l  3.27 Not  0.54  significant  at  significant 10% l e v e l  CT.  147  n a t i v e s o u r c e s o f funds, though t h e s e c o e f f i c i e n t s a r e not s t a t i s t i c a l l y nificant.  There i s no e v i d e n c e o f n o n - s t a t i o n a r i t y a t t h e 10 p e r  sig-  cent  level. The  a b i l i t y o f t h e model t o c o r r e c t l y c l a s s i f y t h e o r i g i n a l  i s shown i n T a b l e 5.9.  F o r t h e c o e f f i c i e n t s e s t i m a t e d by l o g i t  data  analysis,  t h e model c o r r e c t l y c l a s s i f i e s o v e r 91 p e r c e n t o f the non-bankrupt and p e r c e n t o f t h e bankrupt  firms.  F o r p r o b i t a n a l y s i s , o v e r 90 p e r c e n t o f  t h e non-bankrupt and 88 p e r c e n t o f :the b a n k r u p t fied.  85  firms are c o r r e c t l y  identi-  The h y p o t h e s i s t h a t t h e model's c l a s s i f i c a t i o n a b i l i t y i s due  p u r e l y random p r o c e s s c a n be r e j e c t e d w i t h a p r o b a b i l i t y of o v e r 99  to a per  cent. The p r e d i c t i o n a b i l i t y o f t h e model i s d e m o n s t r a t e d i n T a b l e F o r U n i s e r v i c e s Incoproated bankruptcy and  f o r G. F. I n d u s t r i e s and B i s h o p  given.  Bankruptcy  i s p r e d i c t e d one  5.10.  i s p r e d i c t e d f o u r y e a r s i n advance  I n d u s t r i e s a two  year p r e d i c t i o n i s  y e a r i n advance f o r C e n t u r y  Geophysi-  c a l I n c o r p o r a t e d , D o l l y Madison I n c o r p o r a t e d , and F a r r i n g t o n M a n u f a c t u r i n g Company.  F o r V i s u a l E l e c t r o n i c s t h e model p r e d i c t s a 98 p e r c e n t chance  o f i t f a i l i n g t h r e e months b e f o r e i t went b a n k r u p t . r u p t c y f o r Roberts Company does n o t exceed  The  t h e average  p r o b a b i l i t y o f bank-  v a l u e , though f o r t h e  l a s t y e a r f o r which d a t a are a v a i l a b l e t h e p r o b a b i l i t y o f b a n k r u p t c y increase.  Comparing T a b l e 5.10  t o T a b l e 5.7,  a b i l i t y o f the model does n o t exceed  i t i s seen t h a t t h e  does  predictive  t h a t o f the a l t e r n a t i v e f o r m u l a t i o n  u s i n g book v a l u e s f o r t h e a p p r o p r i a t e c o r p o r a t e v a r i a b l e s . t i o n of the poor q u a l i t y o f t h e a v a i l a b l e market d a t a .  This i s a  reflec-  148  TABLE 5.9 CLASSIFICATION OF ORIGINAL DATA SAMPLE:  LOGIT ANALYSIS  PREDICTED OUTCOME BANKRUPT  PREDICTIVE MODEL  ACTUAL OUTCOME  BANKRUPT  NON-BANKRUPT  TOTALS  AVERAGE VALUE OF PROBABILITY  29  201  230  0.391  NON-BANKRUPT  5  2068  2073  0.009  TOTALS  34  2269  2303  Type one e r r o r = p r o b a b i l i t y  [of a f i r m  bankrupt|non-bankrupt]  = 201/2269 = 0.089 Type two e r r o r = p r o b a b i l i t y  [of a f i r m  non-bankrupt|bankrupt]  = 5/34 = .147  PROBIT ANALYSIS  PREDICTED OUTCOME  BANKRUPT  ACTUAL OUTCOME  BANKRUPT  NON-BANKRUPT  TOTALS  AVERAGE VALUE OF PROBABILITY  30  225  255  0.372  NON-BANKRUPT  4  2044  2267  0.009  TOTALS  34  2269  2303  Type one e r r o r = 0.099  Type two e r r o r = 0.117  TABLE 5.10  PREDICTIVE ABILITY OP MODEL  NAME OF FIRM  1960  1961  1962  1963  1964  UNISERVIES INC.  1965  1966  0.001  0.067*  0.973*^  0.906*  0.001  0.080  0.942  0.847  19G8  1970  0.010  0.078  0.009  0.003  0.002  0.001  0.008  Nov.,  Fob.,  0.065  0.008  0.087  0.008  0.001  0.001  0.001  0.008  1968  1970  0.152  0.041  0.008  0.927  0.112  0.071  0.001  0.0007  0.003  0.012^  June,  0.179  0.077  0.013  0.867  0.1]9  0.089  0.001  0.0004  0.003  0.014  1969  1970  0.001  0.006  0.987*  March,  Jur.e,  0.001  0.007  0.974  1970  1970  G. F . ItJDUSTRIES  DOLLY MADISON INCORPORATED  BISHOP INDUSTRIES  Sept.,  0  VISUAL ELECTRONICS  FARRINGTON KA: ."U F A CTU RING COMPANY  1970  1969  DATE CF BANKFUPTCY  0.056  ROBERTS COMPANY  CEtrnjRY GEOPHYSICAL INCORPORATED  1968  1967  LAST DATE OF DATA .  0.011  0.002  0.033*  0.2(33,*  Juno,  0.012  0.002  0.072  0.33  19f>9  1970  0.004  0.009  0.005  0.028  Sept.,  June,  0.003  0.009  0.005  0.033  1969  1970  * 4  0.398  0.982  0.993  0.44  0.012  0.004  0.006  0.004  0.004  0.09*„  Doc.,  0.591  0.983  0.992  0.484  0.024  0.004  0.009  0.010  0.013  0.189  19i'9  0.0004  0.006  0.005  0.012  0.974*  0.994*  Oct.,  0.0  0.006  0.004  0.014  0.963  0.993  1969  Average v a l u e o f p r o b a b i l i t y o f bankruptcy •  0.014  * i n d i c a t e s t h a t t h e p r o b a b i l i t y i s above the average v a l u e  1970  Oct., 1970  ^  150  Summary To e s t i m a t e the p r o b a b i l i t y o f a f i r m g o i n g b a n k r u p t  over a g i v e n  p e r i o d two models, one u s i n g book v a l u e s and t h e o t h e r market v a l u e s o f t h e a p p r o p r i a t e c o r p o r a t e v a r i a b l e s , have been c o n s t r u c t e d .  The models  have been t e s t e d f o r n o n - s t a t i o n a r i t y and p r e d i c t i v e a b i l i t y . t h a t t h e r e i s no e v i d e n c e o f n o n - s t a t i o n a r i t y .  I t i s found  Both models have d e m o n s t r a t e d  good c l a s s i f i c a t i o n and p r e d i c t i v e a b i l i t y , b e i n g a b l e t o f o r e c a s t f o r some f i r m s , f o u r o r f i v e y e a r s b e f o r e t h e a c t u a l  bankruptcy,  occurrence.  T e s t i n g o f Hypothesis In C h a p t e r  IV i t i s shown t h a t when t h e i n v e s t m e n t  i s changed o n l y b y t h e event o f b a n k r u p t c y , taneous c o n d i t i o n a l expected  opportunity set  then i n e q u i l i b r i u m t h e i n s t a n -  r a t e o f r e t u r n , c o n d i t i o n a l upon no  bankruptcy,  on common s t o c k i s r e p r e s e n t e d by t h e e x p r e s s i o n  (5.4)  a. - r = A + 6.(a - r - x ) , D D D M where a_. i s t h e i n s t a n t a n e o u s  c o n d i t i o n a l expected  r a t e o f r e t u r n on t h e  th j  a s s e t ; ct^ i s t h e i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d  the market p o r t f o l i o ;  r i s the instantaneous r i s k  i s the r a t e o f p r o b a b i l i t y of bankruptcy average ^  o f t h e { A . } and 8. = a. /a , a. ] ] jM MM jM  c o v a r i a n c e o f t h e j*"*  1  free rate of interest; A ^  th — f o r the j asset; x i s a being t h e instantaneous  a s s e t w i t h t h e market p o r t f o l i o .  a b l e f o r d i s c r e t e time i n t e r v a l s , a model f o r m u l a t e d not be t e s t e d d i r e c t l y . involves integrating  r a t e o f r e t u r n on  To f o r m u l a t e a d i s c r e t e time  weighted conditional  As d a t a a r e o n l y  i n continuous analogy  the c o n d i t i o n a l p r i c e d i s t r i b u t i o n s and  time  availcan-  o f t h e model using the  151  equilibrium expression mately given  (5.4).  approxi-  by  E(r.)  where E ( r . ) j  A d i s c r e t e time form o f the model i s  i s the  - r  + B.[E(r )  = X.  p  - r  M  p  -  x  l ,  c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on  (5.5)  the  j  th  asset;  i s t h e c o n d i t i o n a l expected r a t e of r e t u r n on t h e market p o r t f o l i o ; the p r o b a b i l i t y o f the  j  th  f i r m going bankrupt during  weighted average o f t h e {X.};  r  i s the  B. = c o v ( r . , r ), c o v ( r . , r ) b e i n g 3 3 M D M  X_. i s  the p e r i o d ; x i s  r i s k free rate of i n t e r e s t ;  the c o n d i t i o n a l covariance  E(r„) M  a  and  of the j  th  asset  w i t h t h e market p o r t f o l i o . To t e s t e m p i r i c a l l y t h e h y p o t h e s i s an e x - p o s t form o f the model i s used.  T h i s i m p l i e s a t r a n s i t i o n from an  u s i n g a market model.  Thus any  the ex-ante f o r m u l a t i o n  and  ex-ante t o an e x - p o s t  formulation  e m p i r i c a l t e s t i s a j o i n t examination of  the market model.  The  ex-post form o f  the  model i s  R  where R^  i s the  t  i s the s t a n t s ; and  jt  =  V  o  Vjt  +  +  e  j  (  R  Mt-  r e a l i z e d excess r e t u r n f o r the  V j  +  th  U  j t '  i s a z e r o mean random d i s t u r b a n c e  '  6 )  asset during period t;  r e a l i z e d e x c e s s market r e t u r n f o r p e r i o d t ; V u.  ( 5  term.  q  and  are  con-  I t i s assumed t h a t  3*R  M  and  u. are n o r m a l l y d i s t r i b u t e d random v a r i a b l e s and ] The  h y p o t h e s i s o f the  bankrupcy a c r o s s  t h e s i s i s t h a t d i f f e r e n c e s i n the p r o b a b i l i t y o f  s e c u r i t i e s and  across  t u r n a f t e r a b s t r a c t i n g from the market. t h e s i s i s represented  by  uncorrelated.  estimating  the  time are r e f l e c t e d i n the  residual re-  Thus an e m p i r i c a l t e s t o f the c o e f f i c i e n t s of Equation  (5.6).  hypoIf  152  the c o e f f i c i e n t , v^, o f the p r o b a b i l i t y o f b a n k r u p t c y  is positive,  then  this  o f f e r s c o n f i r m a t i o n o f t h e v a l i d i t y o f the h y p o t h e s i s .  Methodology The  c o n s t a n t term,  o f bankruptcy 3^,  v , and o  the c o e f f i c i e n t , v., o f the i  a r e n o t f i r m dependent, i n c o n t r a s t t o the b e t a  which i s f i r m s p e c i f i c .  probability coefficient,  This s t r u c t u r a l property of Equation  (5.6)  is  u t i l i z e d by t h e methodology employed t o t e s t the h y p o t h e s i s .  P o o l i n g o f Time S e r i e s and C r o s s S e c t i o n a l The methodology used  Data  i s t h a t o f p o o l i n g t h e time s e r i e s and c r o s s  g sectional data.  Time s e r i e s d a t a f o r i n d i v i d u a l s e c u r i t i e s a r e p o o l e d to-  g e t h e r t o e s t i m a t e the two : common c o e f f i c i e n t s , w h i l s t s i m u l t a n e o u s l y e s t i mating  t h e f i r m s p e c i f i c b e t a c o e f f i c i e n t s . Thus, f o r two  r e g r e s s i o n equation i s o f the R  1  t  i  •  1  •  i  1  i  i  " ll"  R  1T  R  21  =  i i i  R  .  r> MT  1  0  i  t  •  1  •  i  t  •  X  2T  0  the  form U  *1  -  1  1  2T  -  ii  x  securities  U R  M1  '  X  l  11  IT  u21  2T  A d i s c u s s i o n o f t h i s methodology i s g i v e n i n Kuh, E. "The V a l i d i t y o f C r o s s - S e c t i o n a l l y E s t i m a t e d B e h a v i o u r E q u a t i o n s i n Time S e r i e s A p p l i c a t i o n s , " Econometvica, V o l . 27 ( A p r i l , 1959), pp. 197-214; and B a l e s t r a , P. and N e r l o v e , M., " P o o l i n g C r o s s S e c t i o n and Time S e r i e s Data i n the E s t i m a t i o n o f a Dynamic Model: The Demand f o r N a t u r a l Gas," Econometvica, V o l . 34 ( J u l y , 1966), pp. 585-612.  i  153  where t h e s u f f i x T d e n o t e s  t h e number o f time p e r i o d s .  case can be w r i t t e n i n t h e m a t r i x  Y = X 0  form  (5.7)  + u,  where Y i s a (NxT) v e c t o r ; X a [ (NxT) efficients;  and u_ a  number o f s e c u r i t i e s .  The g e n e r a l N s e c u r i t y  x  (N+2)] matrix,- 0_ a  (N+2) v e c t o r o f c o -  (NxT) v e c t o r o f random d i s t u r b a n c e terms,  N b e i n g the  I t i s assumed t h a t  E(u  ) = 0,  (5.8)  and (5.9)  j , k = 1, 2,...,N,  The v a l i d i t y o f t h e l a s t assumption a c r o s s time and between s e c u r i t i e s .  i s tenuous.  I t i m p l i e s zero  I f i t i s not s a t i s f i e d ,  correlation  then i t i m p l i e s  t h a t t h e e s t i m a t e d c o e f f i c i e n t s w h i l s t b e i n g u n b i a s e d and c o n s i s t e n t , w i l l n o t be minimum v a r i a n c e o r , i n g e n e r a l , a s y m p t o t i c a l l y e f f i c i e n t . 0_, a r e e s t i m a t e d b y o r d i n a r y l e a s t s q u a r e s . s t r u c t u r e o f t h e X m a t r i x c a n be u t i l i z e d  The c o e f f i c i e n t s ,  F o r l a r g e d a t a samples,  t o reduce c o m p u t a t i o n a l  the s p e c i a l  difficulties.  Aggregation In o r d e r t o be a b l e t o e s t i m a t e t h e c o e f f i c i e n t s o f E q u a t i o n i s n e c e s s a r y t o know t h e p r o b a b i l i t y o f b a n k r u p t c y time p e r i o d s .  As the p r o b a b i l i t y o f b a n k r u p t c y  (5.6), i t  f o r every s e c u r i t y over a l l  cannot be d i r e c t l y  observed,  154  it  i s e s t i m a t e d u s i n g the models d e v e l o p e d  i n the f i r s t  p a r t of t h i s  chapter.  T h i s i m p l i e s t h a t t h e r e w i l l be e r r o r s i n the measurement o f the  variable,  which w i l l cause the e s t i m a t i o n s of t h e c o e f f i c i e n t s o f E q u a t i o n  (5.6)  biased. tion;  The  e f f e c t s o f t h e s e measurement e r r o r s can be reduced  t h a t i s , the r e l a t i o n s h i p  of  the d a t a and  V  and  q  (5.6)  can be aggregated  t h e mean v a l u e s o f t h e v a r i a b l e s u s e d .  aggrega-  over c e r t a i n As  a r e n o t f i r m s p e c i f i c , t h e same c o e f f i c i e n t s w i l l  a t e f o r the a g g r e g a t e d  by  the  t o be  subsets  coefficients  still  be a p p r o p r i -  relationship.  To i n c r e a s e t h e magnitude i n t h e changes i n t h e p r o b a b i l i t y o f bankr u p t c y over t i m e , s e c u r i t i e s a r e a s s i g n e d i n t o p o r t f o l i o s on the b a s i s o f the value o f bankruptcy  f o r the p r e v i o u s y e a r .  v a r i a n c e , w h i l s t maximizing of  bankruptcy.  The  the w i t h i n  t h e between p o r t f o l i o v a r i a n c e f o r t h e  probability  number o f p o r t f o l i o s i s v a r i e d , e n a b l i n g e x a m i n a t i o n  p o s s i b l e a g g r e g a t i o n e f f e c t s . The which had  T h i s minimizes  f i r s t p o r t f o l i o contains the  of  securities  the lowest p r o b a b i l i t i e s , w h i l s t the l a s t p o r t f o l i o c o n t a i n s the  s e c u r i t i e s which had  the l a r g e s t p r o b a b i l i t i e s .  This assignment process i s  r e p e a t e d on a y e a r by y e a r b a s i s ,  so t h a t , i n g e n e r a l , t h e c o m p o s i t i o n  each p o r t f o l i o changes a n n u a l l y .  The  use o f t h e  previous year's  of  probability  v a l u e as a c r i t e r i o n f o r a s s i g n m e n t a v o i d s s e l e c t i o n b i a s i n t h e c o n s t r u c t i o n of  the  portfolios. The  average  v a l u e o f the p r o b a b i l i t y o f b a n k r u p t c y ,  and  the  average  r a t e of r e t u r n f o r s e c u r i t i e s i n each p o r t f o l i o a r e c a l c u l a t e d on a y e a r l y b a s i s o v e r the whole time p e r i o d ; t h a t i s , t h e average  r a t e o f r e t u r n f o r the  p o r t f o l i o i s d e f i n e d t o be R  pt  = — N  p  E R. , . „ jt' ieS P J  J  (5.1  155  where S i s the s e t o f s e c u r i t i e s P number o f s e c u r i t i e s .  c o n t a i n e d i n t h e p o r t f o l i o , and N  The average p r o b a b i l i t y  of bankruptcy  P  the  i s defined to  be 1_  =  Pt  Substituting  E X jeS P  P  E q u a t i o n (5.6) i n t o  tion  (5.11)  (5.10) g i v e s  R ^ = V + i - E v X. + pt o N . _ 1 jt P D£S P  As t h e c o e f f i c i e n t  .  3  N  i s not firm  — N  .E e.(R -X)+ j e S j Mt t p p A  J  —  N  E . p ;jeS  u.,.. jt  (5.12)  P  s p e c i f i c , t h e n u s i n g E q u a t i o n (5.11), Equa-  (5.12) becomes  V  =  p  neS  v  o  +  v  i  x P  t  +  V'WV  +  V'  ( 5  -  1 3 )  where  c  J  P  and u  1  pt  = — N  r E u. • . „ jt P  that i s , the c o e f f i c i e n t s V  q  and  are s t i l l  r e l a t i o n , and a r e n o t weighted a v e r a g e s . mated by p o o l i n g t h e t i m e s e r i e s u s i n g o r d i n a r y l e a s t squares  !  a p p r o p r i a t e f o r the aggregate  The c o e f f i c i e n t s  and c r o s s s e c t i o n  regression.  o f (5.13) a r e e s t i -  a g g r e g a t e d d a t a and t h e n  156  Data To t e s t e m p i r i c a l l y t h e model c o r p o r a t e d a t a a r e r e q u i r e d t o mate the p r o b a b i l i t y o f b a n k r u p t c y The  File  and p r i c e d a t a f o r t h e r a t e s of r e t u r n .  d a t a s e t c o n s i s t s of a l l f i r m s common t o t h e Compustat F i l e and  s i t y of Chicago Center  f o r Research  dividend information f o r a l l s e c u r i t i e s l i s t e d 1926  adjusted p r i c e  on the New  - June, 1970.  York S t o c k  t i n u o u s c o r p o r a t e d a t a f o r t h e y e a r s 1955  set.  Univer-  t o 1969.  f i r m s w i t h 10 y e a r s o f a n n u a l  and  Exchange  A f i r m i s i n c l u d e d i n the  d a t a s e t i f i t has c o n t i n u o u s p r i c e d a t a f o r the y e a r s 1959  t o t a l o f 360  the  i n S e c u r i t y P r i c e s Monthly P r i c e R e l a t i v e  (CRSP), which c o n t a i n s monthly p r i c e , d i v i d e n d and  (NYSE) i n the p e r i o d January,  esti-  t o 1969,  and  Using t h i s c r i t e r i o n  cona  data are contained i n the data  The monthly r e t u r n s on t h e market p o r t f o l i o a r e d e f i n e d as the r e t u r n s  which would have been earned  on a p o r t f o l i o c o n s i s t i n g o f an e q u a l  investment  i n e v e r y s e c u r i t y l i s t e d on the NYSE a t t h e b e g i n n i n g o f each month.  The  r i s k f r e e i n t e r e s t r a t e i s d e f i n e d as t h e 30 day r a t e on U n i t e d S t a t e s Treasury  Bills. F o r each s e c u r i t y t h e annual  excess  r a t e o f r e t u r n and the  annual  excess market r a t e o f r e t u r n a r e c a l c u l a t e d from t h e s t a r t o f t h e f i r m ' s f i s c a l year f o r the 10 y e a r p e r i o d 1960-1969. mated u s i n g t h e two The  second uses book v a l u e . The  p r o b a b i l i t y o f bankruptcy  f o r m u l a t i o n s r e p r e s e n t e d by the e x p r e s s i o n s  f i r s t expression u t i l i z e s  T a b l e 5.11.  The  (5.2)  and  (5.3).  market v a l u e s o f c o r p o r a t e v a r i a b l e s , w h i l s t  the  The v a l u e s o f the e s t i m a t e d c o e f f i c i e n t s are shown i n  c o e f f i c i e n t s e s t i m a t e d u s i n g p r o b i t a n a l y s i s f o r the  model u t i l i z i n g market v a l u e s o f c o r p o r a t e v a r i a b l e s , the c o e f f i c i e n t s had  is esti-  t h e wrong  sign.  a r e not used,  first  as one  of  TABLE 5.11  VALUES OF COEFFICIENTS USED TO ESTIMATE THE PROBABILITY OF BANKRUPTCY  M E T H O D O L O G Y  MARKET VALUES OF CORPORATE VARIABLES  Logit  BOOK VALUES OF CORPORATE VARIABLES  V A R I A B L E S  Y  Q  -3.848  B  Q  Y  x  -23.006  &^  Y  ^  2  3  -0.270  -0.163  3  63  2  Logit  -2.769  -13.091  -1.084  0.330  Probit  -1.470  - 6.639  -0.549  0.139  158  The  use  o f the two  formulations r e s u l t s i n a noticeable difference  i n t h e e s t i m a t e d lvalues o f the p r o b a b i l i t y o f b a n k r u p t c y , on average estimates  from t h e model u s i n g market v a l u e s o f c o r p o r a t e v a r i a b l e s [ e x p r e s s i o n  (5.2) ] a r e g r e a t e r and  have more v a r i a b i l i t y  than  those o b t a i n e d  a l t e r n a t i v e f o r m u l a t i o n u s i n g book v a l u e s o f c o r p o r a t e v a r i a b l e s (5.3) ] .  This difference in v a r i a b i l i t y  the r e g r e s s i o n c o e f f i c i e n t s i n Equation The  the  i s reflected (5.6)  from  the  [expression  i n the e s t i m a t i o n o f  when t e s t i n g  the  hypothesis.  average y e a r l y v a l u e s o f the p r o b a b i l i t y o f b a n k r u p t c y a r e g i v e n i n  Table  5.12,  which c l e a r l y demonstrates the d i f f e r e n c e i n magnitude  and  variability.  Empirical  Results  The v a l i d i t y o f t h e h y p o t h e s i s c o e f f i c i e n t s o f Equation this offers confirmation.  (5.6); The  i s t e s t e d by examining t h e  i f the c o e f f i c i e n t , v^,  i s positive  r e s u l t s a r e s e t o u t i n two  t h e s i s i s t e s t e d u s i n g a g g r e g a t e d s e c u r i t y d a t a , and  estimated  parts.  then  The  hypo-  then i n d i v i d u a l  security  data.  Use  of  Portfolios The  i n d i v i d u a l s e c u r i t y data  i s a g g r e g a t e d i n t o p o r t f o l i o s on  the  b a s i s o f t h e p r o b a b i l i t y o f b a n k r u p t c y f o r t h e p a s t y e a r , as p r e v i o u s l y plained.  U s i n g 360  s e c u r i t i e s , p o r t f o l i o s c o n t a i n i n g 12,  t i e s over a nine year p e r i o d are c o n s t r u c t e d . ruptcy estimated  by the market v a l u e  a r e shown i n T a b l e  5.13.  The  72  securi-  F o r the p r o b a b i l i t y o f bank-  formation of Equation  coefficient,  36 and  ex-  (5.2), the  results  v , f o r the p r o b a b i l i t y o f bank-  159  TABLE 5.12  AVERAGE YEARLY VALUES OF THE PROBABILITY OF BANKRUPTCY  MARKET VALUE FORMULATION  YEAR  BOOK VALUE FORMULATION  LOGIT  LOGIT  PROBIT  1960  0.00589  0.00279  .00260  1961  0.00909  0.00384  .00374  1962  0.01034  0.00337  .00316  1963  0.00563  0.00293  .00273  1964  0.00276  0.00036  .00026  1965  0.00330  0.00039  .00033  1966  0.00443  0.00048  .00041  1967  0.00430  0.00071  .00064  1968  0.00673  0.00304  .00295  1969  0.00781  0.00276  .00274  i  160  TABLE 5.13  PORTFOLIO DATA:  POOLING OF TIME SERIES AND CROSS SECTION  PROBABILITY OF BANKRUPTCY ESTIMATED USING MARKET VALUES OF CORPORATE VARIABLES  NUMBER OF PORTFOLIOS  NUMBER OF SECURITIES IN EACH PORTFOLIO  NUMBER OF OBSERVATIONS  COEFFICIENT V  COEFFICIENT V.  4.601 (39.082)  R-SQUARED  0.755  72  45  0.059 (44.141)  10  36  90  0.093 (16.393)  2.593 . (6.297)  0.678  30  12  270  0.115 . (17.695)  1.175 ** (3.162)  0.575  (Figures i n brackets are t - s t a t i s t i c s )  * statistically ** s t a t i s t i c a l l y  s i g n i f i c a n t a t t h e 0.1 p e r c e n t  level  significant  level  a t t h e 0.2 p e r c e n t  161  ruptcy i s p o s i t i v e ,  statistically  significant  the l e v e l o f a g g r e g a t i o n i n c r e a s e s . and  statistically  significant.  As  The  i n c r e a s e s i n magnitude as  c o n s t a n t term, v ^ ,  expected,  when t h e l e v e l o f a g g r e g a t i o n i n c r e a s e s . as a g g r e g a t i o n r e d u c e s  and  i t decreases  The v a l u e o f R-squared i n c r e a s e s  can be e s t i m a t e d by E q u a t i o n  u s i n g book v a l u e s f o r c o r p o r a t e v a r i a b l e s .  5.14).  The  c o e f f i c i e n t , v^,  f o r two  of the p o r t f o l i o s .  are reported  f o r the p r o b a b i l i t y of bankruptcy  statistically  significant,  i n magnitude as t h e l e v e l o f a g g r e g a t i o n i n c r e a s e s . i n T a b l e 5.13.  The  i s positive  The  significant.  and  decreases  It i s consistently  v a l u e o f R-squared  larger  i n c r e a s e s as  the l e v e l o f a g g r e g a t i o n i n c r e a s e s , though i t i s u n i f o r m l y lower v a l u e s o f R-squared g i v e n i n T a b l e  similar  (see T a b l e  In a l l c a s e s i t i s n o t s t a t i s t i c a l l y  c o n s t a n t term, v , i s p o s i t i v e , o  (5.3)  As t h e r e s u l t s are v e r y  f o r the l o g i t and p r o b i t e s t i m a t e s , o n l y t h e former  than the c o n s t a n t term  i n magnitude  the c r o s s s e c t i o n a l n a t u r e o f t h e d a t a .  The p r o b a b i l i t y o f b a n k r u p t c y  The  is positive  than  the  5.13.  d i f f e r e n c e s i n T a b l e s 5.13  and  5.14  i r e s u l t d i r e c t l y from  method o f e s t i m a t i n g t h e p r o b a b i l i t y o f b a n k r u p t c y .  the  As shown i n T a b l e  the v a l u e s o b t a i n e d from t h e f o r m u l a t i o n g i v e n by E q u a t i o n  (5.2), which  5.12, utilizes  market v a l u e s o f c o r p o r a t e v a r i a b l e s , a r e l a r g e r and have g r e a t e r v a r i a b i l i t y t h a n those g i v e n by E q u a t i o n  (5.3) u s i n g book v a l u e s .  F o r many f i r m s the  book v a l u e f o r m u l a t i o n g i v e s a z e r o v a l u e f o r the p r o b a b i l i t y o f which causes  severe econometric  problems because o f i l l - c o n d i t i o n e d  W h i l s t a g g r e g a t i o n m i t i g a t e s t h i s p r o b l e m i t does, however, a l l variance. efficient,  This w i l l  bankruptcy,  reduce  matrices. the  imply t h a t t o o b t a i n r e l i a b l e e s t i m a t e s o f the  v , w i l l require l a r g e data  samples.  overco-  162  T A B L E  P O R T F O L I O  D A T A :  P O O L I N G  P R O B A B I L I T Y U S I N G  N U M B E R N U M B E R  I N  O F  V A L U E S  T I M E  S E R I E S  B A N K R U P T C Y  A N D C R O S S  S E C T I O N  E S T I M A T E D  O F C O R P O R A T E  V A R I A B L E S  O F  S E C U R I T I E S  O F  B O O K  O F  5.14  E A C H  N U M B E R O F O B S E R V A T I O N S  C O E F F I C I E N T V  C O E F F I C I E N T R - S Q U A R E D  P O R T F O L I O S  P O R T F O L I O  5  72  45  0.089 ^ (13.039)  0.924 (1.65)  10  36  90  °- * (16.20)  -0.266 (0.587)  0.651  30  12  270  °- * (18.59)  0.419 (1.342)  0.558  o  1 1 4  1 2 1  V  l  (figures i n brackets are t - s t a t i s t i c s ) * s t a t i s t i c a l l y s i g n i f i c a n t a t t h e 0.1 p e r c e n t  level  0.732  163  The  c o e f f i c i e n t , v^, f o r the p r o b a b i l i t y o f bankruptcy i s , i n g e n e r a l  p o s i t i v e and s t a t i s t i c a l l y  significant.  v a l i d i t y of the hypothesis;  This provides  some e v i d e n c e  of the  that i s , bankruptcy i s a c o n t r i b u t o r y f a c t o r t o  t h e s t r u c t u r e o f common s t o c k r e t u r n s . I f t h e r e i s a m i s s i n g v a r i a b l e which i s n o t e x p l a i n e d by e i t h e r t h e p r o b a b i l i t y o f b a n k r u p t c y o r t h e c o v a r i a n c e w i t h t h e market, t h e n t h i s bias the estimates ing  o f the c o e f f i c i e n t s .  I f t h e data a r e aggregated, t h e p o o l -  o f time s e r i e s and c r o s s s e c t i o n a l d a t a becomes more l i k e  an a g g r e g a t e d  time s e r i e s , and t h u s t h e v a r i a n c e o f t h e m i s s i n g v a r i a b l e w i l l t-statistics  o f t h e c o e f f i c i e n t s as w e l l as c a u s i n g b i a s .  be e x a c e r b a t e d The  will  increase the  The s i t u a t i o n  will  i f the m i s s i n g v a r i a b l e i s n o n - s t a t i o n a r y .  c o n s t a n t term i n a r e g r e s s i o n e q u a t i o n p i c k s up i n d i v i d u a l  not a c c o u n t e d f o r b y t h e exogenous v a r i a b l e s .  effects  The p o o l i n g o f t i m e s e r i e s and  c r o s s - s e c t i o n d a t a c o n s t r a i n s t h e c o n s t a n t t e r m t o be t h e same f o r a l l s e c u r i ties  ( p o r t f o l i o s ) , and t h u s  i f there are i n d i v i d u a l security e f f e c t s  t h i s w i l l cause t h e e s t i m a t e d  present,  c o e f f i c i e n t s t o be b i a s e d .  To examine t h e s e p o s s i b i l i t i e s dummy v a r i a b l e s c o u l d be i n t r o d u c e d t o represent s e c u r i t y  (portfolio) specific  by t h e exogenous v a r i a b l e s .  effects not e x p l i c i t l y  accounted f o r  However, dummy v a r i a b l e s w i l l be h i g h l y c o r r e -  l a t e d w i t h the p r o b a b i l i t y o f b a n k r u p t c y .  T h i s a r i s e s because t h e w i t h i n  p o r t f o l i o v a r i a n c e a c r o s s t i m e f o r t h e p r o b a b i l i t y of b a n k r u p t c y w i l l be m i n i mized and thus t h e dummy v a r i a b l e w i l l  a c t as" a p r o x y f o r i t .  The r e s u l t i n g  m u l t i c o l l i n e a r i t y w i l l cause a l o s s o f p r e c i s i o n o f e s t i m a t i o n , as s p e c i f i c e s t i m a t e s may have v e r y l a r g e sampling confirmed  t h i s and c o n s e q u e n t l y  errors.  E x p l o r a t o r y e m p i r i c a l work  t h i s l i n e o f i n v e s t i g a t i o n was n o t p u r s u e d .  164  The  l i m i t e d size of the data set —  annual data — aggregated  360  s e c u r i t i e s w i t h 10 y e a r s  of  r e s t r i c t s t h e scope o f methods t o t e s t the h y p o t h e s i s u s i n g  data.  By f o r m i n g p o r t f o l i o s on t h e b a s i s o f t h e p a s t v a l u e o f  the p r o b a b i l i t y of bankruptcy  reduces  t h e time span o f d a t a a v a i l a b l e  one  y e a r , and a g g r e g a t i o n a c r o s s s e c u r i t i e s d e c r e a s e s  set  even f u r t h e r .  are conducted  Due  to these l i m i t a t i o n s ,  using individual security  by  the s i z e o f the data  f u r t h e r t e s t s on t h e  hypothesis  data.  I n d i v i d u a l S e c u r i t y Data The use o f i n d i v i d u a l s e c u r i t y d a t a f a c i l i t a t e s a l a r g e number o f o b s e r v a t i o n s .  Off-setting this  t h e advantage o f  i s the disadvantage  the low d i s c r i m i n a t o r y power o f t h e exogenous v a r i a b l e s due measurement.  of  to errors i n  T h i s i s e s p e c i a l l y i m p o r t a n t when c o n s i d e r a t i o n i s t a k e n  t h e l a r g e e r r o r s i n measurement when e s t i m a t i n g t h e p r o b a b i l i t y o f  of  bankruptcy.  T h i s i m p l i e s t h a t a l l the e s t i m a t e d r e g r e s s i o n c o e f f i c i e n t s w i l l be b i a s e d . Thus the r e s u l t s u s i n g u n a g g r e g a t e d d a t a w i l l  n o t be e x p e c t e d  same r e l i a b i l i t y as t h o s e o b t a i n e d u s i n g a g g r e g a t e d F i v e t e s t s a r e conducted  u s i n g unaggregated d a t a .  t o draw a random sample o f f i r m s and t o p e r f o r m s e r i e s and c r o s s s e c t i o n a l d a t a . i o u s problems which may efficients. of  The  portfolio The  to offer  the  data. first  test i s  a r e g r e s s i o n p o o l i n g time  T h r e e o f the r e m a i n i n g t e s t s examine v a r -  cause b i a s i n the e s t i m a t i o n o f t h e r e g r e s s i o n c o -  f i n a l t e s t c o n s i d e r s t h e e f f e c t o f changes i n the  probability  c a n k r u p t c y upon t h e s t r u c t u r e o f e x - p o s t r e t u r n s .  Random Sample From t h e u n i v e r s e o f 360  securities,  a random sample o f 100  i s chosen,  165  TABLE 5.15  RANDOM SAMPLE:  DATA SET FOR PROBABILITY  POOLING OF TIME SERIES AND CROSS SECTION DATA  METHODOLOGY FOR PROBABILITY  COEFFICIENT V  LOGIT BOOK  0.115 * ( 1 0  VALUES  MARKET VALUES  COEFFICIENT V.  '  5 4 1 )  -0.820 (  - ' 1  R-SQUARED  0.303  NUMBER OF OBSERVATIONS  1,000  3 6 6 )  PROBIT  0.114 (10.504)  -0.661 (-1.147)  0.303  1,000  LOGIT  0.116 (10.142)  -0.911 (-1.151)  0.304  1,000  t  #  (Figures i n brackets are t - s t a t i s t i c s ) * s t a t i s t i c a l l y significant  a t the 0.1 per cent l e v e l  166  g i v i n g a t o t a l o f 1,000  observations.  mated u s i n g t h e market v a l u e formulation of Equation  The  formulation  (5.3).  The  s e c t i o n a l d a t a a r e shown i n T a b l e  p r o b a b i l i t y of bankruptcy i s e s t i -  of E q u a t i o n  (5.2)  and  r e s u l t s o f p o o l i n g time s e r i e s and  5.15.  The  f o r the p r o b a b i l i t y o f bankruptcy i s negative,  significant.  The  constant  term, v  cross  r e s u l t s are s i m i l a r f o r the  f e r e n t methods o f e s t i m a t i n g t h e p r o b a b i l i t y o f b a n k r u p t c y . v^,  t h e book v a l u e  The  coefficient,  though s t a t i s t i c a l l y  , i s p o s i t i v e and  dif-  statistically  in-  signifi-  o cant.  The  v a l u e o f R-squared i s q u i t e l a r g e c o n s i d e r i n g t h e s t r o n g  s e c t i o n a l nature The  cross  o f the r e g r e s s i o n d a t a .  n e g a t i v i t y o f t h e c o e f f i c i e n t , v^,  and  the  lack of d i f f e r e n c e  i n the r e s u l t s when t h e p r o b a b i l i t y o f b a n k r u p t c y i s e s t i m a t e d  u s i n g market  v a l u e s o f c o r p o r a t e v a r i a b l e s o r book v a l u e s , r e f l e c t s t h e e f f e c t s o f e r r o r s i n t h e measurement o f t h e v a r i a b l e and served  i n Tables  folios  reduces the v a r i a n c e o f the estimate  and  (5.13) and  (5.14).  stands  i n c o n t r a s t to the r e s u l t s  Aggregation  o f the data  The  c o e f f i c i e n t s used t o e s t i m a t e  To t e s t t h e h y p o t h e s i s  f i r m s i n the data s e t i s not r e s t r i c t e d . c o e f f i c i e n t s used t o estimate i t s measurement w i l l  the p r o b a b i l i t y of bankruptcy  s e t o f f i r m s c h a r a c t e r i z e d by h a v i n g  of l e s s t h a n $200 m i l l i o n .  groups and  coefficients.  Size  determined u s i n g a data  possibility,  set into port-  o f the p r o b a b i l i t y of bankruptcy  r e s u l t s i n more r e l i a b l e e s t i m a t i o n o f t h e r e g r e s s i o n  E f f e c t o f Asset  ob-  the asset  an a s s e t size of  are  size the  I f t h e r e are s c a l e e f f e c t s i n the  the p r o b a b i l i t y o f b a n k r u p t c y , then e r r o r s i n  i n c r e a s e when a p p l i e d t o l a r g e f i r m s .  the u n i v e r s e o f 360  f i r m s i s s o r t e d by  a p o o l i n g o f time s e r i e s and  To examine t h i s  asset size  c r o s s s e c t i o n data  into three  f o r the  firms  167  TABLE 5.16  POOLING OF TIME SERIES AND CROSS DATA ON GROUPS OF FIRMS SORTED BY ASSET SIZE  DATA SET FOR PROBABILITY  BOOK VALUES  MARKET VALUES  ASSET SIZE  COEFFICIENT V  COEFFICIENT R-SQUARED  $(0 - 200)m.  0.117 (11.841)  -0.998 . (-2.712)  0.358  1,270  $ (200-500)m.  0.119 ( 9.654)  -0.216 (-1.116)  0.287  1,010  $500m. and o v e r  0.084 . (10.691)  -0.300 (-0.302)  0.281  1,320  $(0 - 200)m.  0.118 (11.645)  -1.124 ** (-2.425)  0.359  1,270  $(200-500)m.  0.119 ( 9.531)  -0.147 (-0.621)  0.287  1,010  $500m. and o v e r  0.076 ( 8.676)  1.936 ( 1.767)  0.285  1,320  (Figures i n brackets are t - s t a t i s t i c s ) * **  NUMBER OF OBSERVATIONS  statistically  significant  a t t h e 0.1 p e r c e n t  level.  statistically  s i g n i f i c a n t a t t h e 2.0 p e r c e n t  level.  168  w i t h i n each group conducted. p r o b a b i l i t y o f bankruptcy  As t h e r e s u l t s  f o r the estimates o f the  u s i n g t h e book v a l u e f o r m u l a t i o n o f E q u a t i o n  (5.3) when i t s c o e f f i c i e n t s a r e e s t i m a t e d by e i t h e r bit  a n a l y s i s a r e v e r y s i m i l a r , o n l y t h e former  l o g i t analysis or pro-  are reported  (see T a b l e  5.16). There does appear t o be some e v i d e n c e a r e a f f e c t e d by a s s e t s i z e .  that the regression estimates  The c o n s t a n t term, v ^ , f o r l a r g e f i r m s i s r e -  duced i n magnitude when compared t o t h e v a l u e f o r s m a l l and i n t e r m e d i a t e size firms.  The c o e f f i c i e n t , v^, f o r t h e p r o b a b i l i t y o f b a n k r u p t c y ,  t i v e and s t a t i s t i c a l l y  significant  i s nega-  f o r f i r m s o f a s s e t s i z e o f l e s s than $200  million.  Adjustment o f Time P e r i o d The p r o b a b i l i t y o f b a n k r u p t c y  i s estimated using corporate data  that r e f l e c t s the state o f the firm a t the beginning o f i t s f i s c a l The  year.  annual r a t e s o f r e t u r n d a t a f o r a f i r m a r e c a l c u l a t e d o v e r t h e 12  month p e r i o d s t a r t i n g a t t h e b e g i n n i n g o f the f i r m ' s f i s c a l y e a r , and t h e n t h e two d a t a s e t s combined and used t o e s t i m a t e t h e r e g r e s s i o n c o e f f i c i e n t s . However,  a f i r m ' s annual  r e p o r t i s u s u a l l y announced two o r t h r e e months  a f t e r t h e end o f t h e f i s c a l y e a r , and t h u s as t h e p r o b a b i l i t y o f b a n k r u p t c y i s e s t i m a t e d u s i n g c o r p o r a t e d a t a which, t o t h e average be a v a i l a b l e from a f i r m ' s annual  investor, w i l l  r e p o r t , i t may n o t be used  only  to refer to  the 12 month p e r i o d b e g i n n i n g a t t h e s t a r t o f t h e f i r m ' s f i s c a l  year.  Hence, t h e c a l c u l a t e d a n n u a l r a t e o f r e t u r n and t h e e s t i m a t e o f t h e p r o b a b i l i t y of bankruptcy,  which a r e used as p r o x i e s f o r t h e v a l u e s an i n v e s t o r ex-  169  TABLE  5.17  ADJUSTMENT OF TIME  NUMBER OF MONTHS AFTER BEGINNING OF FISCAL YEAR  0  COEFFICIENT V o  PERIOD  COEFFICIENT  0.117 (11.841)  #  V  R-SQUARED  l  NUMBER OF OBSERVATIONS  -0.998 ^ (-2.712)  0.358  1,270  i  2  0.104 (10.541)  -0.800 (-1.504)  0.271  1,260  3  0.105 (10.559)  -0.873 (-1.631)  0.273  1,260  4  0.089 ( 9.401)  -0.520 (-0.990)  0.302  1,260  t  t  #  (Figures i n brackets are t - s t a t i s t i c s )  * statistically  significant  a t t h e 0.1 p e r c e n t  level.  statistically significant  a t t h e 2.0 p e r c e n t  level.  **  170  p e c t s , may  n o t be  synchronized.  This p o s s i b i l i t y  t h e a n n u a l r a t e o f r e t u r n from a p o i n t two, the beginning The  of a firm's f i s c a l  t h r e e , and  r e g r e s s i o n c o e f f i c i e n t s are estimated  As  by one  5.16,  The  s i z e o f the  from t h e f o r m u l a t i o n  z i n g market v a l u e s o f c o r p o r a t e v a r i a b l e s a r e r e p o r t e d .  data  The  the utili-  results  are  5.17.  In T a b l e  5.17  t h e r e does appear t o be  c o e f f i c i e n t , v^,  some d i f f e r e n c e between t h e  results.  The  negative,  i n c r e a s e s i n v a l u e and becomes s t a t i s t i c a l l y  f o r the p r o b a b i l i t y of bankruptcy, w h i l s t  the r a t e of r e t u r n data are s h i f t e d forward v , i s reduced  de-  requirements.  the f i n d i n g s a r e v e r y s i m i l a r f o r d i f f e r e n t methods o f e s t i m a t i n g  shown i n T a b l e  firms  w h i c h i n d i c a t e d some  f i r m because o f t h e e x t r a p r i c e d a t a  p r o b a b i l i t y o f bankruptcy, o n l y those o b t a i n e d  two  i n s i g n i f i c a n t when  months.  i n magnitude though remains s t a t i s t i c a l l y  v a l u e o f R squared set  set of  This choice of data set i s moti-  i n view of t h e f i n d i n g g i v e n i n T a b l e  i s reduced  f o u r months a f t e r  using a data  pendence o f r e g r e s s i o n c o e f f i c i e n t s upon a s s e t s i z e . set  calculating  year.  w i t h a s s e t s i z e l e s s t h a n $200 m i l l i o n . vated  i s examined by  i s reduced. F u r t h e r s h i f t s  The  constant  significant.  term, The  i n the r a t e o f r e t u r n d a t a  produce o n l y minor d i f f e r e n c e s .  Cross  Section  Studies  In a r e g r e s s i o n w i t h two w i t h e r r o r , then  i t s estimated  i n d e p e n d e n t v a r i a b l e s i f one  i s measured  c o e f f i c i e n t w i l l be b i a s e d downwards, w h i l s t  the c o n s t a n t term w i l l be b i a s e d upwards.  I f b o t h v a r i a b l e s a r e measured  w i t h e r r o r no comment can be made about the d i r e c t i o n o f t h e r e s u l t i n g However, i f an i n s t r u m e n t a l v a r i a b l e i s s u b s t i t u t e d f o r one  biases.  o f the v a r i a b l e s  171  and  t h e measurement e r r o r s i n t h e i n s t r u m e n t a l v a r i a b l e a r e independent o f  the measurement e r r o r s i n t h e r e m a i n i n g the remaining  v a r i a b l e , then  v a r i a b l e w i l l be b i a s e d downwards.  the c o e f f i c i e n t o f  I t i s not possible to  make any comment about t h e d i r e c t i o n o f b i a s i n t h e c o n s t a n t To  term.  e x p l o r e t h e p o s s i b l e consequences o f measurement e r r o r s i n t h e  v a r i a b l e s , a s e r i e s of cross s e c t i o n s t u d i e s a r e conducted. b e t a c o e f f i c i e n t i s r e p l a c e d by an i n s t r u m e n t a l v a r i a b l e . s e c u r i t i e s an i n s t r u m e n t a l b e t a c o e f f i c i e n t i s e s t i m a t e d the e x - p o s t ,  excess  return against the r e a l i z e d  The f i r m ' s For individual  by r e g r e s s i n g  r e t u r n on t h e market u s i n g  t h e p a s t f i v e y e a r s o f monthly p r i c e d a t a p r i o r t o t h e p e r i o d o f t h e c r o s s s e c t i o n a l study.  These e s t i m a t e s  a r e s u b s t i t u t e d i n t o t h e r e g r e s s i o n equa-  tions  R  j  =  v  o  +  V j  +  V  j  +  u  j  ( 5  -  j = 1, 2,...,N, where R. i s t h e a n n u a l e x c e s s  r e t u r n f o r t h e j*"*  1  a s s e t ; X_, i s t h e p r o b a b i l i t y  th of the j  * a s s e t g o i n g b a n k r u p t o v e r a p e r i o d o f one y e a r ;  mental beta c o e f f i c i e n t ;  u . i s a random d i s t u r b a n c e 3  are r e g r e s s i o n c o e f f i c i e n t s .  B.. i s t h e i n s t r u -  term; and v , v, and v . o 1 2  I f t h e measurement e r r o r s i n t h e b e t a  coeffi-  c i e n t s a r e i n d e p e n d e n t o f t h e measurement e r r o r s i n t h e p r o b a b i l i t y o f bankr u p t c y then t h e e s t i m a t e which underestimates  of coefficient  the true value.  duce t h e d a t a s e t t o 318 f i r m s . using Equation Using  w i l l o n l y be s u b j e c t t o a b i a s  The e x t r a p r i c e d a t a r e q u i r e m e n t s r e -  The p r o b a b i l i t y o f b a n k r u p t c y i s e s t i m a t e d  (5.2), which u t i l i z e s market v a l u e s o f c o r p o r a t e v a r i a b l e s .  annual d a t a  five cross section regressions f o r d i f f e r e n t periods of  time a r e c o n d u c t e d .  The r e s u l t s a r e g i v e n  i n T a b l e 5.18.  172  TABLE 5 . 1 8 CROSS SECTION STUDY  COEFFICIENT TIME  COEFFICIENT  COEFFICIENT  V.  1959-60  0.179 . (4.417)  1961-62  TO.139 (-0.555)  (  -0.750 (-0.851)  (-2.481)  0.155 , 3.422)  -0.186  1967-68  -0.110  -0.016  (  ANNUAL EXCESS RETURN ON THE MARKET  -0.210 . (-4.081)  (-0.419)  1963-64  1965-66  -3.579 (-1.630)  3  -0.061  0.111  •0.156  0.031  0.125  0.002  -0.069  0.019  0.237  0.015  1.302)  0.001  0.128  -0.930  R-SQUARED  (  **  T  5.843)  (-3.466)'  (-1.182)  0.245  -0.386  -0.083  (5.655)  (-0.887)  (-7.438)  T  (Figures i n brackets are t - s t a t i s t i c s ) * s t a t i s t i c a l l y s i g n i f i c a n t a t the 0 . 1 per cent  level.  statistically  level.  **  3  t-statistics  s i g n i f i c a n t a t the 2.0 p e r cent a r e taken about t h e annual  excess  r e t u r n on the market.  173  c o e f f i c i e n t , v^,  The negative,  f o r the p r o b a b i l i t y o f b a n k r u p t c y i s c o n s i s t e n t l y  though s t a t i s t i c a l l y  is generally s t a t i s t i c a l l y  s i g n i f i c a n t and  value o f the c o e f f i c i e n t folio.  insignificant  constant  fluctuates in sign.  value  The  term,  V , Q  theoretic  i s compared t o i t s t h e o r e t i c v a l u e  found t h a t , i n g e n e r a l , t h e y a r e d i f f e r e n t , Two  The  i s the annual e x c e s s r e t u r n on the market p o r t -  When the e s t i m a t e d  significant.  from z e r o .  o f the estimates  the d i f f e r e n c e being  i t is  statistically  have t h e wrong s i g n .  Changes i n t h e P r o b a b i l i t y o f B a n k r u p t c y The method o f p o r t f o l i o c o n s t r u c t i o n used t o i n c r e a s e t h e between p o r t f o l i o variance  o f changes i n the p r o b a b i l i t y o f b a n k r u p t c y and  t i v e r e s u l t s derived, data  suggests that s i m i l a r r e s u l t s u s i n g i n d i v i d u a l s e c u r i t y  s h o u l d be o b t a i n e d  by c o n s i d e r i n g the e f f e c t s o f changes i n t h e p r o b a b i l -  i t y o f b a n k r u p t c y ; t h a t i s , by  R.. = 3t  V  o  j = 1, t  the p o s i -  ,  + v.X .. 1 jt  c o n s i d e r i n g a r e l a t i o n s h i p o f the  (\  _\  1 + R. (R  -Y)  form  + U...  (5.15)  2/ • • • f Nf  = 2, 3 • • • i T / f  where R., i s the a n n u a l e x c e s s r a t e o f r e t u r n f o r y e a r t f o r t h e j 3t til i s the p r o b a b i l i t y o f t h e  j  a s s e t g o i n g b a n k r u p t d u r i n g y e a r t ; R^  market e x c e s s r e t u r n ; x+. i s a w e i g h t e d average o f t h e p r o b a b i l i t i e s ; random d i s t u r b a n c e The  term; and  o f 360  f i r m s and  s i z e i n t o three groups.  s e c t i o n data  v^, v^,  and  -f6^J a r e r e g r e s s i o n  r e l a t i o n s h i p i s examined by f i r s t  f i r m s from a u n i v e r s e by a s s e t  v^,  i s performed on  t h e n by  The  3t  i s the  t  u.  is a  coefficients.  t a k i n g a random sample o f s o r t i n g the u n i v e r s e  A regression pooling  each group.  asset; X  100  of  firms  time s e r i e s and  cross  p r o b a b i l i t y of bankruptcy i s  174  TABLE 5.19 DIFFERENCES IN THE PROBABILITY OF BANKRUPTCY  NUMBER OF OBSERVATIONS  DATA SET  900  RANDOM  $ (0-200)m.  0.132 (10.497)'  COEFFICIENT  COEFFICIENT  -3.073 ^ (-3.206)  0.291  -0.548 (-0.838)  -1.649 ^ (-2.306)  0.350  0.271  0.138 (12.287)  909  0.142, (10.39)  0.300 (1.1375)  -0.841 ^ (-3.111)  0.078 ( 7.310)  6.887 (4.932)  -10.520 ( 5.637)  1,188  R-SQUARED  0.433 (0.471)  1,143  $(200-500)ra.  $500m. and o v e r  COEFFICIENT  (Figures i n brackets are t - s t a t i s t i c s ) * statistically  s i g n i f i c a n t a t t h e 0.1 p e r c e n t  level.  statistically  s i g n i f i c a n t a t t h e 2.0 p e r c e n t  level.  **  #  0.282  175  estimated using Equation variables. The  (5.2), which u t i l i z e s market v a l u e s o f c o r p o r a t e  The r e s u l t s a r e shown i n T a b l e  5.19.  c o e f f i c i e n t , v^, f o r t h e p r o b a b i l i t y o f b a n k r u p t c y  i s now  t i v e , with the exception f o r the group of f i r m s o f a s s e t s i z e l e s s $200 m i l l i o n .  F o r f i r m s o f a s s e t s i z e $500 m i l l i o n and o v e r ,  is statistically  significant.  posi-  than  the c o e f f i c i e n t  The c o n s t a n t term, v , i s u n i f o r m l y  positive  and  statistically  significant.  The c o e f f i c i e n t , v^, i s c o n s i s t e n t l y  and  statistically  significant.  The r e s u l t s s h o u l d be compared t o t h o s e  i n T a b l e 5.16.  given  I t i s seen t h a t t h e s i g n o f t h e c o e f f i c i e n t , v^, w i t h one  e x c e p t i o n , i s now r e v e r s e d . though  negative  F o r f i r m s o f a s s e t s i z e l e s s t h a n $200 m i l l i o n ,  t h e c o e f f i c i e n t , v^, i s s t i l l  n e g a t i v e , i t i s no l o n g e r  statistically  significant.  Summary The h y p o t h e s e s o f t h e t h e s i s i s t h a t d i f f e r e n c e s i n t h e p r o b a b i l i t y of  bankruptcy  a c r o s s s e c u r i t i e s and a c r o s s t i m e a r e r e f l e c t e d  r e t u r n o f common s t o c k s a f t e r a b s t r a c t i n g from t h e market. i s t e s t e d u s i n g i n d i v i d u a l and a g g r e g a t e d  s e c u r i t y data.  i n the residual  The  hypothesis  The use o f i n d i v i -  d u a l s e c u r i t y d a t a p r o v i d e a l a r g e data s e t w i t h which t o t e s t the h y p o t h e s i s . However, t h e b i a s e s t h a t r e s u l t from t h e e r r o r s i n t h e measurement o f t h e p r o b a b i l i t y o f bankruptcy  p r e s e n t s e r i o u s econometric  lems can be m i t i g a t e d by a g g r e g a t i n g  problems.  S e c u r i t i e s are assigned  p o r t f o l i o s on t h e b a s i s o f p a s t v a l u e s o f t h e i r p r o b a b i l i t y o f t h e aggregated  prob-  t h e d a t a t o reduce t h e v a r i a n c e o f t h e  e s t i m a t e s o f the p r o b a b i l i t y o f b a n k r u p t c y .  and  These  into  bankruptcy,  p o r t f o l i o d a t a t r e a t e d as r e p r e s e n t a t i v e s e c u r i t i e s .  The  176  use o f a g g r e g a t i o n does, however, reduce h i b i t s extensive Evidence aggregated  i s found v e r i f y i n g the h y p o t h e s i s of the t h e s i s .  p o r t f o l i o d a t a and  estimating the p r o b a b i l i t y  of corporate v a r i a b l e s ,  the evidence  s e c u r i t y d a t a does n o t g i v e any c l e a r  of bankruptcy  with  i s e s t i m a t e d u s i n g book v a l u e s  i s inconclusive.  Use  of  individual  i n d i c a t i o n o f t h e v a l i d i t y o f the  The major d i f f i c u l t y appears  measurement o f t h e p r o b a b i l i t y o f  Using  c o n f i r m a t i o n o f t h e h y p o t h e s i s i s ob-  When the p r o b a b i l i t y o f b a n k r u p t c y  hypotheses.  pro-  testing.  market v a l u e s o f c o r p o r a t e v a r i a b l e s , tained.  the s i z e o f the d a t a s e t and  t o be due  bankruptcy.  to the e r r o r s  i n the  CHAPTER VI SUMMARY  The purpose o f t h i s c h a p t e r i s t o summarize the main c o n c l u s i o n s o f the  t h e s i s and t o d e s c r i b e the a r e a s o f f u r t h e r r e s e a r c h t h a t a r i s e from i t .  Conclusions The impact o f b a n k r u p t c y upon the s t r u c t u r e o f r e t u r n s f o r c o r p o r a t e f i n a n c i a l a s s e t s i s i n v e s t i g a t e d from a t h e o r e t i c a l and e m p i r i c a l v i e w p o i n t . A model, f o r m u l a t e d i n c o n t i n u o u s t i m e , c o n s i d e r s the  investment-consumption  d e c i s i o n o f an i n d i v i d u a l a c t i n g t o maximize the e x p e c t e d l i f e t i m e of  consumption  and t e r m i n a l w e a l t h .  A t each i n s t a n t i n time the  utility  individual  must d e c i d e the p o r t i o n s o f w e a l t h t o consume and t o i n v e s t i n f i n a n c i a l assets.  I t i s assumed t h a t a f i r m i s s u e s b o t h bonds and common'stock as  f i n a n c i a l a s s e t s and t h a t a t each p o i n t i n time t h e r e i s a p r o b a b i l i t y the  f i r m w i l l go b a n k r u p t the n e x t i n s t a n t .  I f bankruptcy occurs i t i s  assumed t h a t e q u i t y h o l d e r s s u f f e r a hundred p e r c e n t l o s s , and r e c e i v e a non-negative l i q u i d a t i n g d i v i d e n d .  that  From t h i s  bondholders  f o r m u l a t i o n the  e q u i l i b r i u m e x p e c t e d r a t e s o f r e t u r n on the d i f f e r e n t f i n a n c i a l  assets—  bonds and common s t o c k s — a r e d e t e r m i n e d u s i n g s t o c h a s t i c c o n t r o l t h e o r y . For  bonds o f i n f i n i t e m a t u r i t y , the e q u i l i b r i u m e x p e c t e d  excess  r a t e o f r e t u r n , c o n d i t i o n a l upon no b a n k r u p t c y , i s a l i n e a r f u n c t i o n o f  two  v a r i a b l e s : a market v a r i a b l e and a v a r i a b l e which i s the p r o d u c t o f the p r o b a b i l i t y o f b a n k r u p t c y and the e x p e c t e d l o s s i f b a n k r u p t c y o c c u r s .  This  r e s u l t d e m o n s t r a t e s , on a t h e o r e t i c a l b a s i s , some o f the d e t e r m i n a n t s o f the  177  178  r i s k premium on a bond and i s d i r e c t l y amiable t o e m p i r i c a l t e s t i n g . F o r common s t o c k s  a two v a r i a b l e e x p r e s s i o n  f o r the e q u i l i b r i u m  e x p e c t e d r a t e o f r e t u r n , c o n d i t i o n a l upon no b a n k r u p t c y , i s d e r i v e d . expression Asset  i s an extended form o f t h e c o n t i n u o u s time a n a l o g y t o t h e C a p i t a l  P r i c i n g Model  (CAPM), t h e second v a r i a b l e b e i n g  p r o b a b i l i t y o f bankruptcy.  returns.  associated with the  This r e s u l t i s important.  bankruptcy i s a c o n t r i b u t o r y stock  The  I t provides  I t demonstrates  that  f a c t o r i n d e s c r i b i n g t h e s t r u c t u r e o f common a theoretic explanation  o f the recent  empirical  f i n d i n g s i n d i c a t i n g t h a t t h e CAPM i s m i s s p e c i f i e d , and o f f e r s a n a t u r a l i n t e r p r e t a t i o n o f t h e "beta f a c t o r . " d i s c r e t e time f o r m u l a t i o n  This r e s u l t i s empirically tested.  o f t h e model i s u s e d .  p r o b a b i l i t y o f bankruptcy across  A  The h y p o t h e s i s i s t h a t t h e  s e c u r i t i e s and a c r o s s  time i s r e f l e c t e d i n  the r e s i d u a l r e t u r n a f t e r a b s t r a c t i n g from t h e market. To  t e s t t h e h y p o t h e s i s i t i s n e c e s s a r y t o be a b l e  p r o b a b i l i t y o f bankruptcy.  E x i s t i n g e m p i r i c a l work on b a n k r u p t c y h a s n o t  a d d r e s s e d t h i s problem, b u t has c o n c e n t r a t e d c l a s s i f y firms  t o measure t h e  on c o n s t r u c t i n g models t o  i n t o one o f two g r o u p s : b a n k r u p t o r n o t b a n k r u p t .  A model  f o r t h e p r o b a b i l i t y o f b a n k r u p t c y i n terms o f a f i r m ' s a b i l i t y t o r a i s e funds, e i t h e r i n t e r n a l l y o r e x t e r n a l l y , t o cover f i x e d charges i s cons t r u c t e d , and an e x - p o s t f o r m u l a t i o n developed. and  i n terms o f measurable q u a n t i t i e s  The p r o b a b i l i t y o f b a n k r u p t c y i s e s t i m a t e d u s i n g p r o b i t  logit analysis.  The a b i l i t y o f t h e model t o p r e d i c t b a n k r u p t c y i s  t e s t e d on a secondary sample o f b a n k r u p t f i r m s .  The r e s u l t s a r e v e r y  w i t h t h e model p r e d i c t i n g b a n k r u p t c y , f o r some f i r m s , f o u r o r f i v e before  analysis  the a c t u a l occurrence.  good  years  179  The  h y p o t h e s i s i s t e s t e d u s i n g i n d i v i d u a l and a g g r e g a t e d U.S.A.  annual s e c u r i t y d a t a f o r t h e 10 y e a r p e r i o d  1960 t o 1969. A new method-  o l o g y t o t h e t e s t i n g o f two v a r i a b l e extended p o o l i n g time s e r i e s and c r o s s s e r i e s data f o r a l l i n d i v i d u a l  forms o f t h e CAPM, t h a t o f  s e c t i o n a l data, i s introduced. securities  ( p o r t f o l i o s ) a r e combined t o  e s t i m a t e t h e c o e f f i c i e n t s t h a t a r e common t o a l l s e c u r i t i e s whilst simultaneously estimating the firm efficients.  The time  (portfolios),  (hypothesis) s p e c i f i c beta c o -  E v i d e n c e i s found v e r i f y i n g t h e h y p o t h e s i s o f t h e t h e s i s ;  t h a t i s , b a n k r u p t c y i s an e x p l a n a t o r y f a c t o r o f t h e s t r u c t u r e o f c o r p o r ate  financial  assets.  Hence, t h e e x p l i c i t  objectives o f the t h e s i s :  (a) t o a n a l y z e  t h e o r e t i c a l l y how t h e mechanism o f b a n k r u p t c y a f f e c t s t h e s t r u c t u r e o f r e turns f o r corporate f i n a n c i a l assets;  (b) t o q u a n t i f y t h e d e t e r m i n a n t s o f  b a n k r u p t c y , and t o a r r i v e a t a f i g u r e which  can be i d e n t i f i e d  as the prob-  a b i l i t y o f b a n k r u p t c y ; and (c) t o t e s t e m p i r i c a l l y t h e h y p o t h e s i s o f t h e thesis, are successfully achieved.  Future  Research The t h e s i s d e m o n s t r a t e s  t h e need  t o d e v e l o p a complete e x p l a n a t o r y  theory o f the p r o b a b i l i t y o f bankruptcy. the  Such a t h e o r y needs t o c o n s i d e r ,  f a c t o r s t h a t determine a f i r m ' s a b i l i t y  t o r a i s e funds and t h e i n t e r -  dependence between s o u r c e s ; t h a t i s , t h e e f f e c t o f u s i n g one s o u r c e on t h e a b i l i t y to u t i l i z e  other sources o f funds.  Development o f such a t h e o r y  e n t a i l s c o n s i d e r a t i o n o f the broader question o f v a l u a t i o n . a firm's a b i l i t y  F o r example,  t o i s s u e d e b t depends upon i t s d e b t c a p a c i t y , b u t t h e con-  c e p t o f debt c a p a c i t y i s i n t r i n s i c a l l y  r e l a t e d t o t h e v a l u e o f t h e f i r m and  180  to the p r o b a b i l i t y o f bankruptcy. s i s requires the  a description  Thus t h e r e i s c i r c u l a r i t y .  of the d i f f e r e n t a t t r i b u t e s  Any a n a l y -  which d e t e r m i n e  p r o b a b i l i t y o f b a n k r u p t c y and t h e n a s p e c i f i c a t i o n o f t h e d e t e r m i n a n t s  of t h e a t t r i b u t e s .  T h i s l e a d s t o a system o f s i m u l t a n e o u s e q u a t i o n s  which,  i n g e n e r a l , w i l l be n o n - l i n e a r . The  development o f a complete e x p l a n a t o r y t h e o r y o f t h e p r o b a b i l i t y  of b a n k r u p t c y i s i m p o r t a n t n o t o n l y from a t h e o r e t i c a l v i e w p o i n t , b u t a l s o because o f t h e many p r a c t i c a l a p p l i c a t i o n s For  t o w h i c h i t c a n be  example, i t c a n be u t i l i z e d i n b u s i n e s s l o a n e v a l u a t i o n ,  t e r n a l management. commercial  loans —  For business loan evaluation  —  t h e same form o f methodology can be u t i l i z e d  i t can be used f o r a c c o u n t s r e c e i v a b l e  p r o b a b i l i t y o f a customer d e f a u l t i n g uses i n v o l v e  and f o r i n -  consumer l o a n s o r  mine t h e p r o b a b i l i t y o f an i n d i v i d u a l o r f i r m d e f a u l t i n g larly,  applied.  to deter-  on a l o a n .  Simi-  management t o e s t i m a t e t h e  on payment.  O t h e r i n t e r n a l management  d e t e r m i n i n g t h e e f f e c t s o f d i f f e r e n t i n v e s t m e n t and f i n a n c i a l  mixes upon t h e p r o b a b i l i t y o f t h e f i r m g o i n g b a n k r u p t . A second area o f r e s e a r c h i s t o c o n s i d e r the e f f e c t s o f the market i m p e r f e c t i o n o f f i n a n c i a l d i s t r e s s upon t h e C a p i t a l Model.  This necessitates  d i s t r e s s and d e s c r i b i n g  defining  Asset  Pricing  d e f i n i t i o n of f i n a n c i a l  t h e consequences o f f i n a n c i a l d i s t r e s s upon t h e  market v a l u e o f t h e f i r m . can  an o p e r a t i o n a l  introducing  From t h i s b a s i s  the e q u i l i b r i u m  rate  of return  be d e t e r m i n e d . A t h i r d a r e a o f r e s e a r c h i s t h e d e t e r m i n a t i o n o f t h e r i s k premium  on a bond.  The t h e s i s p r o v i d e s a t h e o r e t i c  framework w i t h i n  which t o  a n a l y z e t h i s problem and r e s u l t s which a r e d i r e c t l y a m i a b l e t o  empirical  verification.  Further  development  o f t h i s work needs t o c o n s i d e r  e f f e c t s o f such f a c t o r s as c a l l a b i l i t y , term  structure.  maturity,  marketability  the  and  APPENDIX A  MATHEMATICAL DERIVATION OF THE RESULTS IN CHAPTER IV  The purpose o f t h i s appendix i s t o e x p l a i n i n g r e a t e r d e t a i l t h e mathematical d e r i v a t i o n o f the r e s u l t s presented d i x s h o u l d be r e a d  i n C h a p t e r IV.  i n c o n j u n c t i o n w i t h C h a p t e r IV, as i n some c a s e s  which have been g i v e n s a t i s f a c t o r y e x p l a n a t i o n i n t h e c h a p t e r t h e appendix w i t h o u t The  The appenresults  a r e used i n  further explanation.  f i r s t p a r t o f t h e appendix d e s c r i b e s t h e d e r i v a t i o n o f t h e  s t o c h a s t i c d i f f e r e n t i a l equations, f i r s t order maximization  the e q u a t i o n o f o p t i m a l i t y , and t h e  conditions.  o b t a i n i n g a s o l u t i o n t o these  A d i s c u s s i o n o f the d i f f i c u l t i e s i n  equations  and t h e a p p r o x i m a t i o n s  that are  used t o o b t a i n a l i n e a r system i s g i v e n . In t h e second p a r t t h e e q u i l i b r i u m i n s t a n t a n e o u s r a t e s o f r e t u r n , g i v e n t h e assumption o f a c o n s t a n t set, are derived.  The a p p r o x i m a t i o n  t i o n s i s examined i n g r e a t e r  c o n d i t i o n a l expected  investment  opportunity  used t o d e r i v e a l i n e a r system o f equa-  detail.  In t h e t h i r d and f i n a l p a r t o f t h e appendix, t h e g e n e r a l c a s e o f s t o c h a s t i c changes i n t h e r a t e o f t h e p r o b a b i l i t y o f b a n k r u p t c y i s c o n s i d e r e d . The  demand f u n c t i o n s f o r t h e f i n a n c i a l a s s e t s a r e d e r i v e d .  considered  A s p e c i a l case i s  i n which t h e s t o c h a s t i c changes i n t h e r a t e o f t h e p r o b a b i l i t y  o f b a n k r u p t c y f o r one f i r m a c t s a s an i n s t r u m e n t a l v a r i a b l e f o r a l l o t h e r changes.  For t h i s case,  the instantaneous  return are derived.  182  c o n d i t i o n a l expected  rates of  183  General  Formulation A d e r i v a t i o n o f the s t o c h a s t i c d i f f e r e n t i a l  equations  describing  the p r i c e dynamics o f t h e f i n a n c i a l a s s e t s , the e q u a t i o n o f o p t i m a l i t y , and  the f i r s t  order maximization  the d i f f i c u l t i e s  conditions i s given.  A d i s c u s s i o n of  i n o b t a i n i n g a s o l u t i o n t o t h e s e e q u a t i o n s and  mations t h a t a r e used t o o b t a i n a l i n e a r system  the a p p r o x i -  i s presented.  P r i c e Dynamics An i n f o r m a t i o n d e r i v a t i o n o f t h e s e e q u a t i o n s i s p r e s e n t e d i n Chapter  IV.  I t i s proposed  d e r i v a t i o n o f the equations.  t o g i v e , on a s l i g h t l y more r i g o r o u s b a s i s , a The  two  approaches  a r e , however, e q u i v a l e n t  as t h e time i n t e r v a l t e n d s t o z e r o . I t w i l l be assumed t h a t t h e e v e n t o f b a n k r u p t c y process.  follows a Poisson  A P o i s s o n p r o c e s s i s a c o n t i n u o u s time p r o c e s s w i t h a d i s c r e t e  s t a t e space;  t h a t i s , one  i n the v a r i a b l e s . ^ * 2 (t,t+h].  interval  where t h e r e a r e d i s c r e t e o r d i s c o n t i n u o u s changes  L e t N(t,t+h) d e n o t e t h e number o f e v e n t s Then t h e P o i s s o n p r o c e s s i s d e f i n e d as  Pr[N(t,t+h)  = 0] = 1 - \ ( t ) h + 0 ( h ) ,  Pr[N(t,t+h)  =1]  Pr[N(t,t+h)  > 1] = 0 ( h ) ,  i n the  time  follows:  = A(t)h + 0(h),  and  Miller,  *For a good i n t r o d u c t i o n t o P o i s s o n p r o c e s s e s , see Cox, D. R. H. D., Theory of Stochastic Processes (London: Methuen s Co.  and L t d . , 1968).  2 The b r a c k e t n o t a t i o n i s d e f i n e d a s : i f an element x b e l o n g s t o the i n t e r v a l ( t , t + h ] , t h e n t < x <_t+h; t h a t i s , the p o i n t t i s e x c l u d e d f r o m consideration.  184  where P r ( ) means t h e p r o b a b i l i t y o f ; and  0(h)  i s the a s y m p t o t i c  order  sym-  b o l d e f i n e d by  f ( h ) i s 0(h)  i f lim [f(h)/h] = h-*o  To a f i r s t o r d e r a p p r o x i m a t i o n ,  0.  A ( t ) h , can be i n t e r p r e t e d as  the  3  p r o b a b i l i t y o f bankruptcy  o c c u r r i n g i n the i n t e r v a l  assumed t h a t P o i s s o n p r o c e s s e s event o f b a n k r u p t c y it  f o r one  It i s  f o r d i f f e r e n t f i r m s a r e independent;  f i r m does not a f f e c t o t h e r f i r m s .  i s v e r y s i m p l e t o r e l a x t h i s assumption,  c r e a s i n g the complexity of n o t a t i o n s . o f r e s u l t s does n o t w a r r a n t  (t,t+h].  the  Conceptually,  b u t a t the c o s t o f g r e a t l y i n -  The v e r y s m a l l g a i n i n g e n e r a l i t y  t h e b u r d e n o f u s i n g an even more complex  form  of notation. I t i s perhaps not i n t u i t i v e l y  c l e a r t h a t t h e event o f  bankruptcy  c a n be r e p r e s e n t e d by s u c h a p r o c e s s , as i t i s p o s s i b l e f o r N(t,t+h) t o equal, .  f i v e , w h i l s t the event o f bankruptcy  once i n t h i s model.  However,  f o r a f i r m can o n l y occur  t h e p r o b a b i l i t y o f N(t,t+h) e q u a l l i n g  five  i s o f o r d e r h o r l e s s and thus i n t h e l i m i t as h tends t o z e r o , i s z e r o . As t h e whole f o r m u l a t i o n i s i n c o n t i n u o u s utilized,  t i m e , and  a l i m i t i n g process i s  t h e n t h e r e p r e s e n t a t i o n o f the event o f b a n k r u p t c y  process i s p e r f e c t l y v a l i d .  The advantage o f u s i n g a P o i s s o n  by a P o i s s o n distribution  l i e s i n i t s c o n t i n u i t y o v e r the time domain. th From E q u a t i o n  (4.5)  the p r i c e dynamics o f t h e j  f i r m ' s bonds  can be d e s c r i b e d by 3 I t i s i m p o r t a n t t o r e a l i z e t h a t A ( t ) i s not a p r o b a b i l i t y , b u t a p r o b a b i l i t y r a t e ; i t i s the p r o b a b i l i t y p e r u n i t i n t e r v a l o f t i m e . The l e n g t h o f the i n t e r v a l i s a r b i t r a r y ; f o r example, i t may be a day, a month, o r a y e a r .  185  b j (t) (1+r^.h) - g_.h + b_. (t) Y VhY.. ( t ) ; no b. (t+h) =  < ^A.. (t+h)  3 — 1/ The  - 9.. (t+h)  b..(t+h) = +  form  [bj (t) (l+r_.h) - g^h  + b^ (t) Y V h Y ^ (t) ] [1-N.. ( t , t+h) ]  [Aj(t+h) - 0 j ( t + h ) N j ( t , t + h ) .  I f the event o f bankruptcy i s zero.  ; default,  2, • . ., n«  above e q u a t i o n can be w r i t t e n i n t h e  then Nj(t,t+h)  default,  (A.l)  does n o t o c c u r t o f i r m j i n t h e i n t e r v a l  I f bankruptcy  (t,t+h],  does o c c u r t h e n N.. (t,t+h) e q u a l s  one.  D e f i n e a s t o c h a s t i c p r o c e s s , Z ( t ) , by  Z(t+h) = Z ( t ) + Y ( t ) / h , where Z ( t ) i s a s t o c h a s t i c p r o c e s s w i t h independent  increments.  The  limit  as h tends t o z e r o o f Z ( t + h ) - Z ( t ) d e s c r i b e s a Wiener p r o c e s s , o r Brownian  4 motion.  In t h e t e r m i n o l o g y o f s t o c h a s t i c d i f f e r e n t i a l  equations  dZ(t) = Y ( t ) / d t . In  the l i m i t as h tends t o z e r o , E q u a t i o n  the form: db.(t) =  [ b . ( t ) r . - g . ] d t + b . ( t ) Y . d Z . - {b . (t) - [A . ( t ) - 6 . (t) ] }dq .  j = 1, 2,. where dq^ j  t  h  ( A . l ) can be w r i t t e n i n  (A.2)  . .,n,  i s a P o i s s o n p r o c e s s c h a r a c t e r i z i n g t h e event o f b a n k r u p t c y  for.the  firm.  4  F o r a g e n e r a l d i s c u s s i o n , see Cox  i  and M i l l e r ,  op.  ext.,  pp.  205-208,  186  In  a s i m i l a r manner t h e p r i c e dynamics f o r e q u i t y can be d e r i v e d .  From E q u a t i o n  (4.9), i t c a n be w r i t t e n i n t h e form  p.(t+h) = [ p . ( t ) ( l + a . h ) - f . h + . ( ) a . / h Y 3 J 3 3 3 3 p  In  the l i m i t  as h tends  t  n+  3  (t)l[1-N.(t,t+h)] 3  t o z e r o , t h e above e q u a t i o n becomes  dpj(t) = t  P  j  (t)ct..-f ] d t + p ^ t ) j ^  n  j  +  - Pj(t)dq  (A.3)  j f  3 — 1/ 2f» « •/n•  The  Equation o f Optimality: From E q u a t i o n  J[W(t) , a , 0 , r , Y , f ,X,r  The Demand F u n c t i o n s f o r A s s e t s  (4.22), t h e d e r i v e d u t i l i t y  .t,S(t)]  =  f u n c t i o n i s d e f i n e d as  Max E. { / j J [ C ( s ) , S ] d S + BF[W(T),T]},  (A.4)  {c,w} subject t o a wealth utility  c o n s t r a i n t , budget c o n s t r a i n t , and C ( s ) >_ 0.  The d e r i v e d  f u n c t i o n , J , c a n be w r i t t e n i n a more compact and c o n v e n i e n t  form. 5 C o n s i d e r t h e s e t o f e q u a t i o n s d e s c r i b i n g how t h e o p p o r t u n i t y s e t changes: da . = F . (a . , t ) d t 3  3  da.=F 3  dr. 3  =  F  =  dX . 3 = df . 3  5  =  F  See Equations  F  3  3  3  .(a.,t)dt+G . (a . ,t)dQ , ., n+3 j n+3 3 n+D  2n+j  F  + G . (oi., t ) d Q .,  3-  (  r  3 n + j 4n+j  5n+j  (  j'  t  )  d  V  t  )  d  (  ( X  f  (4.12),  +  G  2n+j  ( r  j'  t ) d Q  2n+j'  +  G  3n+j  ( Y  3'  t ) d Q  3n+j'  G  4n+j  ( X  j'  t ) d Q  4n+j'  t  +  j'  j'  t  t ) d t  t  )  d  t  (A.5)  + G  (4.13),  5  n  +  j  (f t)dQ r  (4.14) and  5  n  +  j  ,  (4.16).  187  and dr  F  = F  m  (r , t ) d t + G F  m  (r , t ) d Q , * ro  j = 1, 2,. . .,n, and m = 6n+l.  Define a  dV  =  (mxl) v e c t o r dV by  (do,,..., da ,do- ,...,da 1  X  n  ,dr , . . . , d r .dy.,.•.,dy n  x  n  x  .dX  ,dX ,...,  u  x  , d f d f n 1  n  ,dr ) m  (A. 6) Hence, dV = F d t + GdQ, where F i s a (mxl) v e c t o r such t h a t dp' =  such t h a t F' =  (dQ,,...,dQ ) and G i s a 1 m  d i a g o n a l and z e r o s e l s e w h e r e . (A.7) w i t h  (A.7) 0 ^ , . . . ^ ) , dQ i s a  (mxl) v e c t o r  (mxm) m a t r i x w i t h elements a l o n g t h e  G i v e n the d e f i n i t i o n  (A.5) t h e elements o f F and G c a n be  ( A . 6 ) , t h e n by  comparing  identified.  T h e r e f o r e , E q u a t i o n (A.4) c a n be w r i t t e n i n t h e form J[W(t),V(t),t,S(t)]  =  Max E { / ^ U [ C ( x ) , x ] d x + BF[W(T),T]}, {c,w}  s u b j e c t t o a w e a l t h c o n s t r a i n t , a budget  J[W(t),V(t),t,S(t)]  where S(t+h)  c o n s t r a i n t , and C ( x ) >^ 0.  t+h > E {/ U[C(x),x]dx + fc  for integrals,  Thus,  J[W(t+h),V(t+h),t+h,S(t+h)]},  i s a state description vector describing  By u s i n g t h e mean v a l u e theorem  (A.8)  fc  t h e system a t t i m e t+h.  the f i r s t  term on t h e r i g h t  hand s i d e o f t h e above e q u a t i o n can be a p p r o x i m a t e d t o  U [ C ( t ) , t ] h + 0(h) .  6  S e e C h a p t e r 7 o f D r e y f u s , S. E . , Dynamic Programming (Mew York: Academic P r e s s , 1965).  lus of Variation  and the Calcu-  188  Hence, J[W(t),V(t),t,S(t)]  ~ U[C(t),t]h  + 0(h) + E {j[W(t+h) ,V(t+h), t+h, S (t+h)] } (A.9)  In o r d e r t o e v a l u a t e the second term on t h e r i g h t hand s i d e o f Equation it  (A.9), a c o n d i t i o n a l e x p e c t a t i o n  i s only  tem,  (n+1) o f t h e 2  n  A t t i m e (t+h)  p o s s i b l e states o f the sys-  t h e p r o b a b i l i t y o f o c c u r r e n c e o f t h e s e s t a t e s b e i n g known.  particular the  necessary t o consider  argument i s u s e d .  Given a  s t a t e , t h e e x p e c t e d v a l u e o f t h e random v a r i a b l e , c o n d i t i o n a l upon  s t a t e i s c a l c u l a t e d and t h e n t h e u n c o n d i t i o n a l  expected value determined.  7 Mathematically,  the argument c a n be r e p r e s e n t e d a s :  s i o n a l random v a r i a b l e , t h e c o n d i t i o n a l e x p e c t a t i o n  i f (X,Y) i s a two dimen-  of X f o r a given  Y=y^ i s  d e f i n e d by CO  E(XJY=y.) =  and  E x.P 1=1  the expected value of X i s  E(X)  = E [ E ( X | Y = y )] . V  n E X.h + 0(h)]  W i t h p r o b a b i l i t y [1 (t,t+h].  no d e f a u l t s  occur i n the i n t e r -  ' 8 Thus, c o n d i t i o n a l upon t h e e v e n t , t h e b u d g e t c o n s t r a i n t i s 1  val  (X=x.|Y=y.),  =  1  1  n - C ( t ) h } { E w. (t) [ ( r . - r ) h + y.dZ.]  W(t+h) - W(t) = ( w ( t )  j=l n +  j  E w  n +  =  1  D  3  3  . (t) [ ( c t . - r ) h + a.dZ  3  3  n+j  3  3  .] + rh} - C ( t ) h  + 0(h)  (A.10)  7 ity  F o r a n i n t r o d u c t o r y d i s c u s s i o n , see Meyer, P. L., Introductory and Statistical Application (Massachusetts: A d d i s o n - W e s l e y , 1965).  Q See  Equation  (4.18).  Probabil-  189  The e x p e c t e d v a l u e o f t h e change i n w e a l t h , c o n d i t i o n a l upon no d e f a u l t s , i s  E  n = [ w (t) -C (t) h] [ E w . ( t ) ( r . - r ) j-1  (W (t+h) -W(t) | no d e f a u l t s ] fc  3  3  n  +  3  . (t) (ct.-r)+r] h 3  - C ( t ) h + 0(h) ,  and  E {[W(t+h)-W(t)] | n o 2  h n [W(t) [ Z Z w j=l i = l  =  ( t ) y w. (t) + 2 3 l  3  n +  1  n Z  n Z w. (t)Y.p . . 0 3 ] ] H  1  j  =  i  =  1  n-1  1  n  +  1  Expanding t h e d e r i v e d about t h e p o i n t  ( t ) ] h + o (h) .  utility  (A.12)  f u n c t i o n J[W(t+h),V(t+h),t+h,S(t+h)]  [ W ( t ) , V ( t ) , t , S ( t ) ] , and t a k i n g  e v e n t o f no d e f a u l t s ,  expected values,  w  w  n  1  i  f  W 1  D  " n+j W  m  ( t ) a  0  (  t  )  Y  D i i W  ji n+i< W  m  (  t  )  +  D  :  Ew j=l  3  ^  See E q u a t i o n s  t ) ] W ( t ) 2 h  n  . (t) (a .-r)+r] +  3  -C(t)}h  3  j . V ^ V j i ^ n + i ^  +  =•  (t)a.n. .G.J h 3 *->n 3 i iw  (4.19) and  ^Ji  n  m  m n Z Z W(t)w i=l j=l  +  9  j=l  3  +  n  A  +  f  upon  n  + J h + J{W(t)[ Zw.(t)(r.-r) j=l  J  conditional  gives  n J[W(t),V(t),t,S(t)]  ^  ( t )  n+i  n  Z Z w (t)a w j = l i = i "-> J  +  (A.11)  defaults}  2  the  n + Z w j=l  9  +  (4.20)  + 0(h) '' u u l  /* (A.13)  190  where S(t+h) = S ( t ) , as no f i r m s have d e f a u l t e d ;  v  i s the instantaneous  c o n d i t i o n a l c o r r e l a t i o n c o e f f i c i e n t between dQ^ and dC\ , c o n d i t i o n a l the  f a c t t h a t d e f a u l t has n o t o c c u r r e d t o e i t h e r t h e x"^ o r j***  1  firm;  upon n„  i s t h e i n s t a n t a n e o u s c o n d i t i o n a l c o r r e l a t i o n c o e f f i c i e n t between do. and dZ.;  i J  J  = |^J[W(t),V(t) , t , S ( t ) ] ;  f c  w  =  j  =  WW J 3 A  J  =  ij  fw  J  [  w  t  )  ,  2  v  (  t  '  )  f c  '  S ( t ) 1  •  /  = a i^7 [w(t) v(t),t,s(t)] 9  J  v  /  ;  3  and  JW  (  ^2_ J t W ( t ) V ( t ) , t , S ( t ) ] ; 9W 3 3V. J[W(t) ,V(t) , t , S ( t ) ] ;  J  J  =  J  gy-^-^fwct)  ,v(t).,t,s(t)].  j  With p r o b a b i l i t y X_.(t)h + 0 ( h ) t h e e v e n t t h a t t h e j  f i r m goes bank-  r u p t and no o t h e r f i r m s go b a n k r u p t i n t h e i n t e r v a l (t,t+h] o c c u r s . event o f b a n k r u p t c y n o t o n l y also i t s equity  3  a f f e c t s the j  p r i c e behaviour.  Conditional  The  f i r m ' s bond p r i c e b e h a v i o u r , b u t upon t h i s event, t h e change i n  w e a l t h i s now o f the f o r m ^ A. W(t+h) - W(t) = W(t){w. (t) [-3  (t+h) - 0.(t+h) ? - 1] - w . (t)} 3 b j (t) n+3 n n + {W(t)-C(t)h}{ Z w. (t) [ ( r . - r ) h + y.dz.] + Z w . (t) [ (a . - r ) h + a.dZ .] . , i i l l . , n+i l I n+i i=l i=l i^j  10 • See E q u a t i o n  (4.21).  u  i ^ j  + rh}  191  A. (t+h) [-2  C ( t ) h { l + w. (t)  - 0.(t+h) - 1] - w  b (t) 4  The d e r i v e d u t i l i t y f i r m went b a n k r u p t ,  (t)} + 0 ( h ) .  n+3  (A.14)  f u n c t i o n , c o n d i t i o n a l on t h e e v e n t t h a t the j  i t w i l l be o f t h e form J[W(t+h),  V ( t + h ) , t+h,  th  S_.] where  th 5^ i s a s t a t e v e c t o r a t time exists.  (t+h) d e n o t i n g  As b e f o r e , t h e d e r i v e d u t i l i t y  t h a t the j  f i r m no l o n g e r  f u n c t i o n i s expanded and  t i o n a l e x p e c t e d v a l u e s t a k e n . However, u n l i k e E q u a t i o n  the c o n d i -  (A.13) w e a l t h W(t+h)  i s n o t expanded about  t h e p o i n t W(t), b u t about the p o i n t A. (t+h) - 9.(t+h) V7(t){l + (t) [-1 b. (t) 3 W  =  which d e f i n e s  (t),  ( t ) . The  the change i n w e a l t h . w e a l t h f u n c t i o n and  j  (A.15)  reason f o r t h i s  i s t o p r e s e r v e the compactness o f  The e v e n t o f b a n k r u p t c y c a u s e s a d i s c o n t i n u i t y i n t h e  thus i t no l o n g e r becomes p o s s i b l e t o r e p r e s e n t t h e  changes i n w e a l t h by a summation o f compact d i s t r i b u t i o n s . of  The p r o p e r t y  compactness i s v e r y i m p o r t a n t f o r i t enables' many o f t h e terms i n t h e  T a y l o r e x p a n s i o n o f the d e r i v e d u t i l i t y l i m i t i n g p r o c e s s i s used. derived u t i l i t y  f u n c t i o n t o be n e g l e c t e d  when a  I t i s p o s s i b l e t o p r e s e r v e such a p r o p e r t y .  The  f u n c t i o n i s expanded i n a T a y l o r ' s s e r i e s and t h e n i t s e x p e c t e d th  v a l u e i s t a k e n c o n d i t i o n a l upon the e v e n t t h a t the j Utilizing  bankrupt.  t h i s c o n d i t i o n a l argument, the change i n w e a l t h t h a t r e s u l t s  from the b a n k r u p t c y bonds and e q u i t y .  1  Thus any o t h e r changes i n w e a l t h r e s u l t i n g assets.can s t i l l  compact d i s t r i b u t i o n s , b u t now  s t e a d o f W(t).  solely  o f the j * " * f i r m i s known f o r a g i v e n i n v e s t m e n t i n i t s  ment i n the o t h e r f i n a n c i a l of  f i r m went  from the  be r e p r e s e n t e d by the  c e n t r e d around  invest-  summation  the w e a l t h p o s i t i o n W ^ ( t ) , i n -  T h e r e a r e , however, a number o f o t h e r i m p o r t a n t  ramifications  192 t h a t r e s u l t from the d i s c o n t i n u i t i e s t h a t o c c u r due r u p t c y , as w i l l become q u i c k l y The  apparent.  c o n d i t i o n a l expected  J[W(t+h), V ( t + h ) ,  t+h,  t o the e v e n t o f bank-  v a l u e o f the d e r i v e d u t i l i t y f u n c t i o n  S..] i s t h u s  J[W_.(t),V(t),t,S_J  S u b s t i t u t i n g Equations  + terms o f o r d e r a t most h.  (A.13) and  (A.16) i n t o E q u a t i o n  (A.9)  (A.16)  and  t a k i n g the  u n c o n d i t i o n a l expected v a l u e , g i v e s .  J[W(t),V(t) t,S(t)] /  = Max (U[C (t) , t ] h + {c,w}  0(h)  n +  E {A . ( t ) h + 0(h)}{j[W. ( t ) , V ( t ) , t , S . ] + terms o f o r d e r a t j=l most h} 3  3  3  m + { l - . E A . ( t ) h + 0 ( h ) } { j [ W ( t ) , V ( t ) ,t,S] + J h + E J.F.h 3=1 j-i r n  3  t  n  n  + J ( W ( t ) [ Ew_.(t) ( w  .2 w  +  +  3=1  1=1  n  n  C  r j  -r) +  t ) Y  w  (  t  t w  )  E E w j-1 i = l  . (t)o..w +  3  3  1  n  +  m m 2  3  +  3  1  1  3  1  i  =  n + j  ? 3=1  2  (t) ( o ^ - r ) + r ] - C ( t ) } h  (t) p Y  i=l  , ( t ) ] W(t)  3  E l  . ( t ) a .n. ' 3  x  n +  h  n  E W(t)w. ( t ) Y . n . .G.J * 3  j  =  1  3  1 3  1  n  E E w(t)w i=l j=l  a.w .(t)  J  1  m  E E G.\)..G.J..h + j-1 i = l m  +  J  n  +  3  _,_.G.Jh + 0 ( h ) } ) . 3  1  1  W  h  193  Simplifying/  and  t a k i n g the  l i m i t as h tends t o z e r o ,  p a r t i a l d i f f e r e n t i a l equation  g i v e s the  f o r the d e r i v e d u t i l i t y  fundamental  function:  m 0 = Max {c,w}  (U[C(t),t] + J  +  E F.J. 3=1  *  3  3  n n + J.,{w(t) [ E w . ( t ) ( r . - r ) + Ew j - l j - l W  3  +  3  3=1  1=1  n E j-l  n E w i=l  n  n  1  m +  E i  =  1  n E j=l  +  3  i=l  J  _ ( t ) a . .w +  3  3  1  n  1 3  3  . (t)]W(t) +  1  m n + E E W (t) w, (t) y . n. .G . J... i-lj-1  + r- E E G.V..G.J... • i=l j - l 2  3  3=1  n  m  . (t) (a .-r) + r ] - C (t)} +  1 3  3  Y  3  1  3  1  l  W  n j  E W(t)w =  1  ,.(t)a.h. .G.J,__ n+3 3 i,n+3 i iW  X (t){j[W  ,V(t),t,S ] - Jrw(t),V(t),t,S(t)]}),  3  (A.17)  3  s u b j e c t t o t h e boundary c o n d i t i o n s J [ W ( t ) ,V(T) ,T,S (T) ] = BF[W(T),T]„  The tion  2n+l f i r s t o r d e r m a x i m i z a t i o n c o n d i t i o n s a r e d e r i v e d  (A.17) by  first  differentiating  0 = U [C(t),t] c  - J  with  w  r e s p e c t t o the r a t e o f  from Equaconsumption:  (A. 18)  where U [C(t),t] c  = ~[C(t),t];  t h e n by d i f f e r e n t i a t i n g w i t h in  common s t o c k ,  {w  .(t)}, n+3  r e s p e c t t o the p r o p o r t i o n o f wealth to i n v e s t  194  n 0  =  J  w( .T W 3 a  r )  +  J  T7T,t  WW  n  E o . .w i  =  31 n+i  1  n o n G J - X J E j i,n+j i i w i=l  +  ]  I  , . (t)]W(t) n+i  3  . .,n;  (A.19)  t o the p r o p o r t i o n o f wealth t o i n v e s t  bonds'{Wj(t)}:  0 =  n n (r.-r)J + W(t) [ EYi .w. (t) + E y .p . .o.s (t) ] J D W . , ] i n l n+i WW 1=1 1=1 m  A  ^ V i j V i H ""j j = 1, 2,.  For  '] ] i  ]  and f i n a l l y , d i f f e r e n t i a t i n g w i t h r e s p e c t in  E Y . P . .a.w  [w , v ( t ) , t , s . ] ,  2  j = 1, 2,.  (t) +  1 1  "  (  t  )  =  ]  l^tW./VCt)^^.],  P  (A.20)  define A.(t)  3  b.(t)  . .,n,  expositional simplicity  L  (t) - 0 . (t)  -  1  h  =  Substituting Equation  1  - 8 (t)  b.(t)  ]  '  2,• • • / n •  (A.20) i n t o  (A.19) so as t o e l i m i n a t e J [ W ^ , V ( t ) , t,S J W  gives: .-r n " y .P. .a. 0 = [ ( o . - r ) - - L - ] J + W ( t ) [ Ea..w (t) ,Jt w (t) D L.(t) W -,31 n+i . , -L.(t) n+i D i=l 1=1 D r  7  +  E 0 p 1=1 3  3  1  11  Y 1  1  w (t) E i=l  Y,..w.(t) ' j i " i ^ L.(t) ' l J  Y.n, 1=1  J  i=l  .G,J (t)  WW  r  X 1  195  A t t h i s p o i n t i t i s perhaps worth making a s m a l l d i g r e s s i o n d e r i v e two the  i n e q u a l i t i e s t h a t w i l l be u s e f u l l a t e r .  From E q u a t i o n  to  (A.18)  following r e l a t i o n s h i p s hold.  0 = u  o = u  [C(t>,t]|£  c c  - J  dW  <-t-  [c(t),t]|^- - j ;  j "2 Hence, u s i n g  1/  —  rw(t),V(t),t,S(t)],  [w(t) v(t),t,s(t)] f  f  3  2,• • •,n#  Equation J  WW  (A,18), we  w  u  ww  u  J  have the i n e q u a l i t i e s :  c  cc%  w  > 0  (A.22)  and 3C 3V.  J. =  J ™  _ _1J 3C 3W  i o -  '  <  j ~ l,2,*«»,n«  The (A.19).  demand f u n c t i o n s f o r bonds can be d e r i v e d  from the  I n t h e i r s t a t e d form t h e e q u a t i o n s a r e n o n - l i n e a r  i t w i l l be d i f f i c u l t t o o b t a i n an e x p l i c i t alternatives.  The  first  thus i n  i n d i v i d u a l ' s u t i l i t y f u n c t i o n and  system o f e q u a t i o n s by  a numerical i t e r a t i v e  d i f f i c u l t t o d e r i v e e x p l i c i t forms f o r t h e g e n e r a l even i f t h e s e c o u l d be o b t a i n e d ,  The  two  by  then  procedure, It w i l l  e q u i l i b r i u m r a t e s of  t h e y w i l l depend upon the  form f o r the u t i l i t y f u n c t i o n s f o r i n d i v i d u a l s .  general  There a r e a t l e a s t  w h i l s t t h i s method might produce a s o l u t i o n , i t w i l l be a t a c o s t .  and  Equations  i s t o p u t more s t r u c t u r e i n t o the f o r m u l a t i o n  assuming a p a r t i c u l a r form f o r the attempt t o s o l v e the  solution.  and  set of  be  return,  s p e c i f i c assumed  lack of g e n e r a l i t y ,  and  the c o m p l e x i t y o f t h i s method i s a s e r i o u s d i s t r a c t i o n t o i t s u t i l i z a t i o n . The two  second a l t e r n a t i v e i s t o o b t a i n an approximate s o l u t i o n by making  assumptions.  F i r s t , the  i n d i v i d u a l i s i n d i f f e r e n t as  i f there  are n o r  196  (n-1)  f i r m s i n e x i s t e n c e a t t i m e t such  that  J [ W ( t ) , V ( t ) , t , S ( t ) ] = J[W(t),V(t),t,S..] , 3 and  =  (A.23)  1/2/..., n,  a l l derivations are equal.  I f the number o f f i r m s i n e x i s t e n c e , n, i s  ' l a r g e ' , then such an a s s u m p t i o n seems i n t u i t i v e l y q u i t e r e a s o n a b l e . second assumption i n v o l v e s t h e a b i l i t y utility  The  t o expand t h e d e r i v a t i o n s o f the d e r i v e d  f u n c t i o n and t o n e g l e c t t h e q u a d r a t i c and h i g h e r power e x p a n s i o n terms,  that i s J [W(t){l + w  -  W  j  A.-8. ? - 11 - w  (t)  < t  n + j  (t)}, V ( t ) , t , S(t)]  J [W(t),V(t),t,S(t)] - W(t){w,(t)[l W j  -  M  •i -  i  + w  b . (t)  o  (t)}  n+3  J w w  [W(t),V(t),t, (t)] S  3  (  A  '  2  4  )  The e r r o r i n t r o d u c e d by making t h i s assumption w i l l be examined i n t h e next s e c t i o n . Given t h a t these and  (A.24) i n t o  assumption h o l d , t h e n s u b s t i t u t i n g E q u a t i o n s  (A.20) g i v e s  n n 0 = J „ ( r . - r ) + J „,[ E y . . w . ( t ) + £Y.P..O\W W j WW j i i i=l T  i  - X. ( t ) L . ( t ) J  3  3  (A.23)  W  +  =  3  3  1  3  1  n  +  n . (t)]W(t) + T, y.n. .G.J. i=l  1  3  1  . ( t ) L . (t) [w. ( t ) L . (t) + w  3  3  3  n+3  .(t)]W(t)J  1  3  1  l  W  (A.25)  WW  r 7 t 7  j ~ I f 2;#••^n» Hence, E q u a t i o n s tions,  linear  I  (A.21) and  (A.25) g i v e a s e t o f s i m u l t a n e o u s equa-  i n t h e demand f u n c t i o n s f o r the f i n a n c i a l  assets  {w.}. 3  197  Constant  Investment O p p o r t u n i t y S e t Suppose t h a t  t h e investment  f_,r > i s c o n s t a n t and t h a t  Yj = 0 f o r a l l j .  F  Equations  (A.18),  (A.19),  o p p o r t u n i t y s e t c h a r a c t e r i z e d by  (A.20),  (A.21) and  The f i r s t  {a,a_,r,y,X_,  order conditions, using  (A.23) become  0 = U [C(t),t] - J , c  0 =  (A.26)  w  (a  - r -  W(t)[ ? a  +  3  J  1=1  w  (t)]J ,  (A.27)  J  and 0  =  j"  ( r  "2  —  r ) J  1/  Using the approximation  w  VJVV^'  -  *  (A  28)  2 $ • » »§r z\» d e s c r i b e d by e x p r e s s i o n  (A.24),  then E q u a t i o n  (A.28)  becomes 0 =  ( r . - r - X 3  From E q u a t i o n  (A.27),  .L.)J 3 3  (t)W(t) = 1 1 +  (-  1  w T TT T  WW 1 = 1/ 2,««.,n, where {A..} IT  + X.L.[w.(t)L. 3 3 3 3  + w .(t)]W(t)J . n+3 WW  n  r  E A., (a. - r • T D 3 1  3=1  •" - J ), L. r  From E q u a t i o n  (A.30)  3  a r e t h e elements o f t h e i n v e r s e o f the i n s t a n t a n e o u s  variance-covariance matrix.  (A.29)  the demand f u n c t i o n f o r e q u i t y can be d e r i v e d : J  w  w  conditional  (A.29) t h e demand f u n c t i o n f o r bonds  can be w r i t t e n i n t h e form w (- —) J WW J  [w. ( t ) L . + w 3 3 3 n +  3 — 1/ 2,...,n.  (t)]W(t) =  r 7 I 7  r  ,  _  expected  " D  (A.31)  3 3  To d e r i v e the e x p r e s s i o n s f o r the e q u i l i b r i u m al  r  (rVX .L.  instantaneous c o n d i t i o n -  r a t e s o f r e t u r n , c o n s i d e r t h e demand f u n c t i o n s f o r t h e k  individual.  198  Equation  (A.30) c a n be w r i t t e n i n t h e form r.-r o  W (t)J = -[ — ^ ] j * k  - r j  j 3  k  n E a..w i 3  i  =  1  n  (t), +  (A.32)  1  1/ 2,.../n,  =  where t h e s u p e r s c r i p t k i s used t o i d e n t i f y t h e p a r t i c u l a r i n d i v i d u a l . f o r e , s u b s t i t u t i n g f o r {w r.-r  I  -J—)  (a. - r -  3  J*  Z (-  L. 3  . ( t ) } and summing a c r o s s a l l i n d i v i d u a l s , n+i  . . k  =  1  Jk ww  1  =  1  k  =  1  . (t) i s t h e o p t i m a l number o f n+i  f i r m t h a t t h e k*"* i n d i v i d u a l i n v e s t s i n .  1  gives  n I Z Z a..p. ( t ) N * . ( t ) , . .. , , 31 i n+i  =  where I i s t h e t o t a l number o f i n v e s t o r s ; shares o f the i * " *  There-  1  e q u i l i b r i a c o n d i t i o n s a r e used, t h a t i s , a l l m a r k e t s c l e a r ,  I f the general then  I - N ^ . (t) =»_,.., k=l n + i n+i E  k  1  *• til where N . ' i s t h e t o t a l number o f s h a r e s o u t s t a n d i n g f o r t h e i firm. n+i k / j~*\ Z , W. ..p.(t)N^. (a. - r <?—) Z (- -—) = Z ;ji*i n+i j k=l WW i=l r  n  a  3  L  v  (A.33)  J  The above e q u a t i o n c a n be e x p r e s s e d t r a d i t i o n a l c a p i t a l a s s e t p r i c i n g model. equilibrium  u  Hence,  i n a form comparable t o t h e  D e f i n e M(t) t o be the t o t a l market  value o f a l l e q u i t i e s :  M(t)  =  n ' EN .p. ( t ) ; . , n+i 1 1=1  (A.34) til  and  Y ^ ( t ) t o be t h e p e r c e n t a g e  t o t h e t o t a l market v a l u e :  o f the e q u i l i b r i u m v a l u e o f the i  firm's equity  199  V ' = 4f?lr t}  -  (A 35)  J  D e f i n e t h e e q u i l i b r i u m i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on t h e e q u i t y market t o be u =  M u l t i p l y i n g Equation tions  (A.34),  n E  j  =  (A.35)  - r D  For  (A.36)  summing o v e r a l l j  (A.33) by Y_. ( t ) , and  (A.36)  n [y  a . Y. (t) 3 3  1  gives  r.-r  E Y. (t) M • i 3 L. =l  I  )]  D  expositional simplicity,  J  n  E (- — - ) „k k=l  n  = M (t) E E Y. ( t ) a . . Y. (t) . , . , 1 1 1 1  1=1  define  3  (A.37)  (A.37)  3=1  n r.-r = E Y. (t)(-2=--). j=l j  X(t)  Substituting Equation  and s u b s t i t u t i n g Equa-  (A.38)  L  into Equation  (A.33)  and u s i n g  (A.38)  gives  r.-r a  - r - - J — 3 ^  = 8. (u - r - x) / 3  (A.39)-  where a..Y.(t)  E 3  n  n  E  E Y.: ( t ) a . .Y. (t)  i=l j=l  1  1  3  3  j = 1/ 2,...,n. To d e r i v e  the e q u i l i b r i u m instantaneous expected r a t e s  of returns  th f o r bonds, E q u a t i o n k  (A.31)  f o r the k  i n d i v i d u a l c a n be w r i t t e n  k  J  [N?(t)b ( t ) L . + i r . ( t ) . (t)] = P  i  l  l  n+i  i  w  r  -~  r  (--~-)(r-h-_k  ww  J  A. L. 1  1  i n t h e form  200  where N ( t ) i s t h e o p t i m a l  number o f bonds o f t h e i  k  f c l 1  f i r m t h a t the k  v i d u a l i n v e s t s i n . Summing o v e r a l l i n d i v i d u a l s and u s i n g  t h  the general  indiequili-  brium c o n d i t i o n t h a t a l l markets c l e a r s ; t h a t i s ,  I E N. (t) = N. , k=l k  1  x  where the  —  1  If  2f«»«/iif  i s t h e t o t a l number of. bonds o u t s t a n d i n g  firm,  gives  equation r.-r  N.b.(t)L. + N .p.(t) = i l l n+i I  Define M  a  .  A.L  1  I J 1) ^ (, , k ww  k  (T^—'1  )  R  0  =  ,  1  n Z Kb  3  1=1  .  (A.40)  T  k  =  1  J  ( t ) t o be t h e t o t a l market e q u i l i b r i u m v a l u e  M (t)  and  f o r the i  o f a l l bonds:  (t);  X ^ ( t ) t o be t h e p e r c e n t a g e o f t h e e q u i l i b r i u m v a l u e  bonds t o t h e t o t a l market v a l u e  (A.41)  o f the i  th  firm's  o f a l l bonds, t h a t i s ,  N.b.(t) M  4* ~  Define  If  B  (A.\42)  ( t )  2f««»fXl#  t h e e q u i l i b r i u m i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on  the bond market t o be n E X. ( t ) r . i=l  ir(t) =  1  (A.43)  1  M u l t i p l y i n g Equation  (A.40) by X . ( t ) and summing o v e r a l l i g i v e s k W " (rc-r-Y) E (- -r-) = E . ( t ) A . L . [N.b. (t)L. + N .p.(t)], k=l J i=l " - i l l i n+i i WW 1  J  k  0  1  (A.44)  201  where Y  E X . (t)L.A.. i=l  =  Substituting  Equation  (A.44) i n t o E q u a t i o n  (A.40)  1  gives r . - r - A.L. = {A .L . [N.b. ( t ) L .+N  3  3 3  3 3  3 3  3  . . (t)] }{  * "  D  n+j j  ~  r  }  y  n  p  E X.(t)A.L.[N.b.(t)L.+N .p.(t)] . , l l i I i l n+i i i=l  j = 1, 2,...,n.' Equation  (A.45  (A.45) d e s c r i b e s t h e  t u r n f o r t h e bonds o f t h e j An  equilibrium  th  instantaneous expected r a t e o f r e -  firm.  a l t e r n a t i v e and u s e f u l d e r i v a t i o n o f E q u a t i o n  Consider Equations  (A.38)*  (A.39) a n d (A.40), t h u s by u s i n g  t h e term  E () from Equation * ww = 1  r  3  - r - XL  =  l f  these equations i t  ^  1  i s possible t o eliminate  (A.45) i s p o s s i b l e .  - XL  (A.40); t h a t i s  J  (y- r -*)  N.b.(t)L. + N \ "  .p.(t) 3 ]  3  (A.46  2,...,n.  The  u s e f u l n e s s o f t h i s a l t e r n a t i v e d e r i v a t i o n i s d e m o n s t r a t e d when i t i s u s e d r.-r t o e l i m i n a t e (-*!—) from E q u a t i o n (A.39), w h i c h d e s c r i b e s t h e e q u i l i b r i u m i n 3 stantaneous c o n d i t i o n a l Equation  expected r a t e s o f r e t u r n  (A.46) i n t o E q u a t i o n _  _  r  X  j  (A.39)  f o r equity.  Substituting  gives  = (y - r - x) {6  + A [•  3  j  N.b.(t)L. + N ^ 3  !  L  i  M(t)  E E Y.a. .Y. i-1 j-1 1  Substituting  p.(t) J >  J  n  1  3  3  t h e e x p r e s s i o n f o r 8.., g i v e s  n a  \  j- - j = J r  X  (u - r -Y) — — — n n E E Y o .Y i=l j=l i  N.b. ( t ) L . + N {E a .. Y. + A.f.J-3 ] ii 3  3  j[  E  i  =  N  1  n +  .p. (t) 3  ( t )  n+i*i  v  3  i •> 1*  202  If  t h e r e a r e a l a r g e number o f f i r m s , t h e l a s t term on t h e r i g h t hand s i d e o f  the above e x p r e s s i o n number o f f i r m s .  c a n be n e g l e c t e d ,  Hence, t h e e x p r e s s i o n  as i t i s o f o r d e r  2/n, where n i s t h e  f o r the instantaneous c o n d i t i o n a l  e x p e c t e d r a t e o f r e t u r n c a n be w r i t t e n  - r - A.. = 8 (p - r - x ) ,  (A.47)  3 ~~ 1, 2,...,n«. B e f o r e p r o c e e d i n g t o i n v e s t i g a t e the v a l i d i t y o f t h e a p p r o x i m a t i o n o f being  a b l e t o expand t h e f i r s t d e r i v a t i v e s o f t h e d e r i v e d u t i l i t y  neglect quadratic (A.39),  and h i g h e r  (A.36) and  order  (A.47) i n o r d e r  f u n c t i o n and  terms, i t i s w o r t h r e c o n s i d e r i n g to derive  two i d e n t i t i e s .  Equations  From E q u a t i o n  (A.36), t h e e q u i l i b r i u m i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d r a t e o f r e t u r n on the e q u i t y m a r k e t i s d e f i n e d  t o be  n y =  E a.Y.(t) j=l  Substituting Equation  3  (A.39),  n y = r E Y 3=1 which  3  gives n --»~ E Y. (t) (-} ) + r  (t) +  j=l  3  3  L  n  (\i - r - x)  j  Z Y (t) 8 ,  j - l  3  3  implies  E Y 3=1  (t)  = 1,  3  and n E Y.(t)8. = 1 j=l 3  (A.48)  3  as one would e x p e c t , g i v e n t h e d e f i n i t i o n a l forms o f t h e v a r i o u s q u a n t i t i e s . If  Equation  (A.47) i s s u b s t i t u t e d i n t o  (A.36), t h e n  203  n u = r E Y. (t) + j - l  n E Y. (t)X . + j = l  3  which i m p l i e s , u s i n g E q u a t i o n  Note, however, t h a t E q u a t i o n  3  (A.49)  3  (A.49) i s n o t , u n l i k e  expression  (A.48), a d e f i -  I t i s t h e r e s u l t o f t h e a p p r o x i m a t i o n made i n d e r i v i n g  (A.47), a r e s u l t w h i c h depends Equation  3  (A.48), t h a t  3  (1/n).  (u - r - x ) E Y. (t) 8.,  3  n E Y.(t)X j-l  X =  nitional identity.  3  upon t h e a b i l i t y t o n e g l e c t terms o f o r d e r  (A.49) i s o b t a i n e d from a w e i g h t e d summation o f terms d e s -  c r i b e d by E q u a t i o n ' (A.47) and t h u s n e g l e c t s the summation o f t h e terms t h a t a r e c o n s i d e r e d t o be o f n e g l i g i b l e s i g n i f i c a n c e .  I t i s n o t , however,  clear  t h a t t h e sum o f t h e s e terms c a n be n e g l e c t e d . C o n s i d e r t h e l e f t hand s i d e o f E q u a t i o n and  (A.49).  From E q u a t i o n s  (A.36) n r.-r E Y. (t) )  X =  j=l  J  3  J  n E Y. ( t ) X . +  =  3  3  n E Y. (t)X . [N.b. ( t ) L . . , 3 3 3 3 3 (y-r-x) -^  +N.p.(t)] n+3 3  n  n  M(t) E E Y. (t) a . . Y. (t) j=l i-1 3  Hence, E q u a t i o n  (A.49) n e g l e c t s t h e l a s t t e r m o f E q u a t i o n n E {[ ._,  (y-r-x) 3  (A.50)  3  3  3  1  1  (A.50), t h a t i s  Y.(t)X. N.b.(t)L. + N .p.(t) 2 ] l-^-J 3 P ]> n n M(t; E E Y.(t)0..Y.(t) j - l i-1 3  n +  3  It  (A.38)  3  1  (A.51)  3  1  i s n o t c l e a r t h a t t h i s term c a n be n e g l e c t e d , w h i c h i s s i m p l y a r e f l e c t i o n  o f the f a c t t h a t t h e summation o f terms, w h i c h a r e i n d i v i d u a l l y n e g l i g i b l e , need n o t i t s e l f  be n e g l i g i b l e .  T h i s does n o t imply t h a t E q u a t i o n  (A.47) i s  204  wrong, o n l y t h a t i t i s d e r i v e d by n e g l e c t i n g a term of o r d e r Equations i t was  (A.39) and  (A.45) were d e r i v e d under the assumption  p o s s i b l e t o expand i n a T a y l o r ' s s e r i e s t h e  derived u t i l i t y  f u n c t i o n and  higher order  because o f t h e r e s u l t i n g  i t would be o f some c o m f o r t t o d e t e r m i n e t h e magnitude o f t h e  mation.  One  approach t o t h i s equation  t o a t t e m p t t o o b t a i n an e x a c t  s o l u t i o n obtained  terms, linearapproxi-  individual's utility  s o l u t i o n and  by assuming t h e v a l i d i t y o f the  Assume a c o n s t a n t  the  i s t o p u t more s t r u c t u r e i n t o t h e  f o r m u l a t i o n by assuming a p a r t i c u l a r form f o r t h e t i o n and  that  f i r s t d e r i v a t i v e of  t o n e g l e c t q u a d r a t i c and  w h i l s t t h i s assumption i s v e r y c o n v e n i e n t ity,  (1/n).  func-  t h e n compare i t t o  the  approximation.  r e l a t i v e risk aversion u t i l i t y  function defined  by U[C(t),t] «='£-e~ ,  (A. 52)  p t  where p and conditions  v are p o s i t i v e constants can be  and  V  1  ( a  0=  order  (A.53)  w  n  - r - -J-HJ j  + W(t)[I a i=l  w  n +  .(t)]  J w w  ,  (A.54)  J  r  (A.55)  TT  2,...,n,  of optimality  r Max {— i \ v ic,w}  v  n  e~  + J_. + J„{ t W  n - C}  first  (r . - r ) J - X .L.J (W.,t), 3 W 3 3 W 3  j = l ,  0=  system o f  - J ,  P t  r ,-r 0 =  the equation  The  written  0 = C " e"  and  1.  v  + i j J ( t ) '  E i=l  n  W(t) [ £ w. (t) (r .-r) + , 3 3 3=1  J  n I w j - l  E .  w  3=1  , . (ot .-r) + r ]  n+3  3  n (t)a. v 3  (t) 3  E X j=l  [j(w ,t) - J(W,t]} (A.56)  205  s u b j e c t t o t h e boundary c o n d i t i o n J[W(T),T] = B F [ W ( T ) T ] . /  system o f e q u a t i o n s t a k e as a t r i a l  J[W(t),t] = B ( t ) e ~  Substituting  (A.57) i n t o  P  (A.54) and  To s o l v e  this  solution^" " 3  -—  (A.57)  (A.55) g i v e s  ^  0 =  r.-r (a - r - - f — ) + 3 L. 3  n (v-1) Z o.,w .(t), . , 31 n+i 1=1  0 =  ( r . - r ) - X.L.(1 - w.L. - w . ) 3 3 3 3 3 n+3  (A.58)  and V  _  (A.59)  1  3 ~ I f 2,..«,n.  Equation  (A.59) c a n be w r i t t e n i n t h e f o r m  r  j  _  1  r  w.L. + w, = 1 - (-rV") 3 3 n+3 j j  <-> A 60  t h a t f a c i l i t a t e s d i r e c t comparison w i t h E q u a t i o n assuming t h e v a l i d i t y o f t h e a p p r o x i m a t i o n .  (A.31), w h i c h was  Equation  derived  (A.31) can be w r i t t e n  i n t h e form w.L. + w = l n n+3 J  and s u b s t i t u t i n g  J  J r.-r ( r ~ - ) (1 - r^r-)' WJ ' A.L, WW 33  (A.61)  (A.54) g i v e s  Vj  +  W  n  +  j  • ' ^  ( 1  "  (  A  *  3  1  3 3 The p e r t i n e n t q u e s t i o n i s how good an a p p r o x i m a t i o n i s (A.31') t o Equation  (A.60)?  1  -  (1  +  The r i g h t hand s i d e o f (A.60) can be w r i t t e n as  -  1  — r - =  A  j L j  i- -) 3  v-1  ,  )  206  =  (^)  (1  " -j^JT") j j  (^IJ7 - 1 ) ( 1 V  " xV" j j  )2+  h  i  9  h  Hence, the degree o f a p p r o x i m a t i o n depends upon the  e  r  O  R  D  E  R  T  ability  E  R  M  S  -  (A.62)  to n e g l e c t  the  terms 1  1  R  - — ( — - ) (—-  but  t h i s w i l l depend upon the v a l u e  w i l l be  very  good, b u t For  mations w i l l be (A.53),  one  will  _  R  o f v.  higher  +  For  small  drop o f f as v i n c r e a s e s  class of u t i l i t y functions,  exact,  ( A . 5 4 ) and  -  - 1 ) ( 1 - -r-^lF—) 3 3  as  can  be  shown by  order  v the  approximation  i n magnitude.  the q u a d r a t i c ,  the  approx—.  s u b s t i t u t i n g i n Equations  (A.52),  (A.55).  T h i s work o f f e r s some encouragement to use I t has  been shown t h a t f o r one  exact,  w h i l s t f o r the c l a s s o f constant  t i o n s the a p p r o x i m a t i o n can be meter o f the  terms,  the  approximation.  t y p e o f u t i l i t y f u n c t i o n the  very  approximation i s  r e l a t i v e r i s k aversion  utilities  good, d e p e n d i n g upon the v a l u e  func-  o f the  para-  function.  S t o c h a s t i c Changes i n t h e Previously, changes i n the  Rate o f the P r o b a b i l i t y o f  i t had  been assumed t h a t t h e r e  investment opportunity  p r i c e dynamics had  Bankruptcy  been s i m p l i f i e d  The  so t h a t t h e r e  was  no  direct  common s t o c k  a r e now  relaxed  so t h a t e f f e c t s o f s t o c h a s t i c changes i n the  ability  o f b a n k r u p t c y upon the derived  c u r r e n t w e a l t h , W(t),  from t h a t o f b a n k r u p t c y .  s t r u c t u r e of returns  u t i l i t y function w i l l  a vector  describing  the  stochastic  e q u a t i o n d e s c r i b i n g the  tween bonds and  The  apart  set.  were no  can  be  bond  i n t e r a c t i o n be-  Both assumptions r a t e o f the  prob-  analyzed.  depend upon the i n d i v i d u a l ' s current values  of  the  rates  of  207  the p r o b a b i l i t y o f b a n k r u p t c y f o r the d i f f e r e n t  f i r m s , A_(t) , a s t a t e v e c t o r  d e s c r i b i n g which f i r m s a r e c u r r e n t l y i n e x i s t e n c e , S ( t ) , and the d e r i v e d u t i l i t y To  analyze  f u n c t i o n may  be w r i t t e n i n the  Thus  form J [ W ( t ) , £_(t), t ,  the e f f e c t s o f s t o c h a s t i c changes i n the  ruptcy i t i s necessary  time t .  S(t)J.  r a t e o f p r o b a b i l i t y o f bank-  t o assume a form t h a t d e s c r i b e s the s t o c h a s t i c n a t u r e  of  th t h e mechanism g e n e r a t i n g  t h e changes; f o r t h e  j  f i r m i t i s assumed t h a t  d A ^ ( t ) = F..(A_.,t)dt + G_. (A_.,t)dQ.., 3  3., 2,...,n,  =  which should be compared t o the g e n e r a l Equation  (A.63)  system o f e q u a t i o n s  described  by  (A.5). The  equation  d e s c r i b e d by E q u a t i o n  0 =  o f o p t i m a l i t y c a n be d e r i v e d f r o m t h e g e n e r a l  (A.17); t h a t i s  Max {C,w}  (U[C(t),t] + J  +  fc  n E F.J. j=l 3  n + J „ { w ( t ) [ Z w. (t) ( r . - r ) + W • , 3 3 3=1  ' +  n  ^rrr^t 2 WW  2  n 'E  3  ]i i  i-1  +  3  3  1  n E =  1  1  j  1 3  n E =  3  1 3  W(t)w 1  Z  . 3=1 n  "(t)a..w  n  n E G.v. .G.J. . +  i  n  n  n Z w  i - 1 j-1  (t) ( a . - r ) + r ] 3  3=1  n  n Z  +  n Z w • n+3  J  -. . . . 3=1 1=1  j-1 1  3  £ W. ( t ) Y . .W. (t) + 2  E  +  + —  case  C(t)}  n Z W . (t) Y . P . . 0" . W  . - 3 1=1  3 J i i  J  n+i  . (t)  (t)]W(t) n  +  1  n E  n E W(t)w. (t)Y .n. .G.J.  i - 1 j-1  3  3  1  3  1  l  W  .(t)a.n. .G.J. n+3 3 i,n+3 i lW  n +  E j=l  A . (t) {J[W. ,A_(t) , t , S.] 3  3  3  - J[W(t),  \{t) , t , S ( t ) ] } )  (A.64)  208  s u b j e c t t o t h e boundary c o n d i t i o n J[W(T), A ( T ) , T , S ( T ) ]  = BF[W(T),T].  The  set of f i r s t order maximization c o n d i t i o n s are, a f t e r  some m a n i p u l a t i o n ,  0 = U [C(t),t] - J , c  -  0  (  a  (A.65)  w  j -  E^>W J  -  R  "  +  (  t  )  I  3  n E i-1  t  )  - A^^r^W^  n o.n. .,.G.J. E 3 i.n+3 i iW .  „ WW  = 1  Y.n. -G.J". , j  ow  (t)  tJ  3  3  1=1  E ' j ii ' ' i=l L. Y. .w. (t)  1=1  +  (  1=1  E o.p ..y.w. (t) -  A  . V j i V i  1 3  1  M  • (A.66)  1 W  and 0=  (r-r)J  K  +  W  ( t ) [ ?  Y  p  1=1  + .  E  Equation  +  (A.65) i s the i n t e r t e m p o r a l e n v e l o p e  (t) ]  (A.67)  condition.  Equation  t h e e q u a t i o n o f o p t i m a l i t y by t h e p r o p o r -  t i o n o f w e a l t h t o i n v e s t i n e q u i t y and E q u a t i o n  linear.  w J  Yjn^G^J^ - X . l t j L . ^ t W . ^ f t J / t ^ . } ,  (A.66) i s d e r i v e d by d i f f e r e n t i a t i n g  t i o n f o r bonds.  z \ 1=1  J  (A.67) i s the e q u i v a l e n t equa-  As i n the p r e v i o u s s e c t i o n , t h i s  system  o f e q u a t i o n s i s non-  I t i s assumed t h a t i t i s p o s s i b l e t o r e p l a c e the system o f n o n - l i n e a r  e q u a t i o n s by an approximate  12  See E q u a t i o n s  linear  (A.18),  system;  (A.20) and  that i s , Equation  (A.21).  (A.67) c a n  be  209 approximated  by  o = (r. - r - X (t)L.)J 3 D D W  +  Vt)L.[„ t)L.  n n + w(t) [ Z y. .w. (t) + Z y p. .a.w ] ii ] i ] 1 n+i i  ^Vj  j (  ( t )  l  =  1  W ( t , J  i  =  1  . (t) ] J WW  WW  n +  Z _ , i = l j i j i iW  (A.68)  T h e r e f o r e , Equations (A66) and (A.68) d e s c r i b e two c o r r e l a t e d l i n e a r systems o f e q u a t i o n s fromwhich i t i s p o s s i b l e t o d e t e r m i n e t h e demand f u n c t i o n s f o r bonds and e q u i t y and t h e n d e r i v e the e q u i l i b r i u m expected r a t e s o f r e t u r n . it  instantaneous c o n d i t i o n a l  W h i l s t such a d e r i v a t i o n  i s , unfortunately, mathematically very tedious.  s o l u t i o n a r i s e s from a l l t h e c o v a r i a n c e terms. which r e f l e c t  i s c o n c e p t u a l l y simple, The c o m p l e x i t y o f t h e  I t i s , however, t h e s e terms  t h e e f f e c t s o f b a n k r u p t c y upon t h e s t r u c t u r e o f r e t u r n s .  Define the f o l l o w i n g  {a}, = --j .  (nxl) v e c t o r s  _ ! L. i r  a. - r j  3  r \ {c}..  » = r . - r - X . L . ;  {v^L  = W(t)w_.(t);  x  - 3  3  {w }. = W ( t ) w 2  3 3  n + j  (t);  and {U}. = J.„;  - 3  3W  j = 1, 2, n, and t h e f o l l o w i n g  (nxn) m a t r i c e s  ' (A.69)  210  {£.}. . = Y.P. .cr.; —3 i ] i i ] l -4 13  ' ll] l  (-5O .13.~ a.n. . .G.; 3 i,n+3 I =  {L}..=  - 13  L.6..;  3 13  and  {A},. = X.6..1 -  i/j  i  ID =  1/  (A.70)  ID  2,...,n,  where  ID  i;  i=jr  0;  There, i n m a t r i x n o t a t i o n t h e s e t o f e q u a t i o n d e s c r i b e d by  (A.66)  can be w r i t t e n i n t h e form  0  -  J w  •  +  (=5 "  li" ^)" 1  (A.71)  and Equation(A.68) i n t h e form 0 = J__c + E,U  + ^ww^l  + £3^2  +  'i i L 2 l  +  L  T h i s s e t o f e q u a t i o n s may be w r i t t e n i n a more compact H  l2-  +  P-3^2  =  £l^l  +  P-12^2^  ^  (A.72)  form: (A.73)  and -21-1  ^2-2'  (A.74)  211  where  P_i = I  3  - L  P. = I  2  -  12  L\.  L L hJ  P_  - Ii  P_  = A. L +  2-3  " £3 "  D. •-4  = E ; —4  Hl  = i -  21  2  +  E^;  ^  _  1  - WW  and  = - fw_  2  ww  J  •  J  ?  >.,  T  {H >  l4  I  0,  3 — I f 2f«.#,n» Thus t h e demand f u n c t i o n s f o r e q u i t y a r e  £2  =  H  i=2<2. "  ^2 2.  +  (  3  and f o r bonds  ^  = i2.!(£ - 2.2.i2l> H  2  +  -S j & x W  =^24 - 222. E3^2 12  ( A  ( A  -  7 5 )  -  7 6 )  where  5j. - (2a! "  2 2.& ) 2  1  C 1  »  and  =2 - ( D - S ^ k , ) " . 12  1  To d e r i v e t h e e x p r e s s i o n  f o r the e q u i l i b r i u m instantaneous  rate of return f o r e q u i t i e s , consider the form  Equation  expected  (A.75) w h i c h can be w r i t t e n i n  212  th where t h e s u p e r s c r i p t k d e n o t e s t h e k duals,  individual.  Summing o v e r a l l i n d i v i -  t h e a g g r e g a t e demand f u n c t i o n s f o r e q u i t y c a n b e d e r i v e d  s t a n t a n e o u s market e q u i l i b r i u m c o n d i t i o n rused. d e s c r i b i n g the aggregate supply j = 1, 2,...,n.  (nxl) v e c t o r  t h a t i s , {ASE}.. = N ^ ^ p ^ (t) ,  Hence, ASE  which i m p l i e s  f o r equity;  L e t ASE be a  and t h e i n -  =  A  E  J  2  ( a  -  D ^ c  )  +  -  D  ^  J  D  ^  ,  that » "  j ^ S S E  +  D  l 2  jjc  -  (D  3  - D ^ J ^ )  i -A , 2  (A.78)  where 1,  1  A, =  1  E H" H* k=l  = -  -  k  1  1  E "  '=1  4WW / W  and  •  WW  K  F o r bonds a s i m i l a r r e l a t i o n c a n be d e t e r m i n e d .  From E q u a t i o n  (A.77)  th the demand f u n c t i o n s f o r t h e k  and  i n d i v i d u a l c a n be w r i t t e n i n t h e form  (c - D ^ a )  + E  x  - ^ D ^ g ,  (A.79)  thus C = — E" — A^ —1  1  ASB + D D ^ a —2—12—  where {ASB}. = N  (t), 3 3 b  j = 1/  2,...,n.  (D - D D ~ \ D ) | - A . —4 —4—12—3 A^ —2  (A.80)  213  Equations  (A.78) and (A.80) d e s c r i b e  the e q u i l i b r i u m  instantaneous  c o n d i t i o n a l e x p e c t e d r a t e s o f r e t u r n f o r e q u i t y and bonds, r e s p e c t i v e l y . The  e q u a t i o n s a r e not,  however, i n d e p e n d e n t o f t h e p r e f e r e n c e  i n d i v i d u a l s due t o t h e p r e s e n c e o f t h e terms A 1 Whilst general  i t i s possible to eliminate  structure of  and {A»}., j = 1, 2,...,n. ^3  t h e s e terms, t h e r e s u l t i n g c o m p l e x i t y and  l a c k o f i n s i g h t t h a t r e s u l t s does n o t w a r r a n t t h e e f f e c t .  s i g n t can be g a i n e d by assuming t h a t , e x c e p t f o r one f i r m , t h e r e  Some i n a r e no  s t o c h a s t i c changes i n t h e r a t e o f t h e p r o b a b i l i t y o f b a n k r u p t c y ; t h i s i s equivalent  t o assuming t h a t t h e s t o c h a s t i c changes i n t h e r a t e o f t h e p r o b -  a b i l i t y of b a n k r u p t c y f o r t h i s one f i r m a c t as an i n s t r u m e n t a l  variable. For  til convenience, c a l l  t h i s f i r m the n  firm.  F o r t h i s case,  i s now a s c a l a r  q u a n t i t y and Z^ and Z^ w i l l be (nxl) v e c t o r s , which i m p l i e s w i l l be (nxl) v e c t o r s . L e t M(t) r e p r e s e n t  t h e t o t a l market v a l u e  t h a t p_ and 3  o f a l l equity  " ^ V j P j ^ '  M ( t )  th and  Yj(t)  represent  the proportion  o f the t o t a l value  o f the j  firm's  equity  t o t h e t o t a l market v a l u e o f a l l e q u i t y ; t h a t i s Y_. ( t ) = N ^ ^ p ^ (t)/M (t). Define  a  tuting  into  (nxl) v e c t o r Y such t h a t Equation  (A.* 78)  ^X^j  j(t)*j  =  1» 2,...,n.  Thus, s u b s t i -  A  ^ ( £+^ V '  +  <A  be w r i t t e n in t h e s c a l a r form r  a  v  gives A  A^3 w h i c h may  =  3 " " r  -f  r  M  n  A  " A 7iIlSi i Y  ( t )  2  Y  iX  Z e  i=l  3 1  1  * A ^ j X " LT- . f ^ U  n +  n  (r. - r - X L ),  +  81)  214  where  5  " ° 1 2 -^l°2l! {  3i  e. .  a  •  ^  {  }  3 i ' ,  = cr ,n .G 3 n,n+3 n  jx  = Y.n  .G  3 n,3 n  .  1/ 2/«•«^ n»  j  M u l t i p l y i n g t h e above e q u a t i o n by Y ^ ( t ) and summing o v e r j g i v e s  ~  V  ^ 2 V ^ j i ! * ' " AT^iX " MA ]=1 1=1 1  " * " k1  r  1  +  n  n  E  E  1  Y  +  3=1  .1=1 V j ^ V ^  Y . ( t ) e . . ( r . - r - A.L.),  j-1 i - 1  3  3 1  (A.82)  1  where  a  n =  mA  E j  =  Y. ( t ) o \ . , 3  1  3*  and h  j-1  MX  3  L  j  The p r e f e r e n c e s t r u c t u r e c a n be removed from E q u a t i o n  (A.82).  th e q u i t y o f the n  f i r m must s a t i s f y E q u a t i o n r  n  r  r  -r  L  A " A^  n  (A.81), t h a t i s  1  V i i=l  (  t  l  Y ,  A^  n  X  L  l  n y i  n  i  ( r .  - r - A.L.)^  n  . . ni'iA n  1  =  1  The  1  215  which c a n be w r i t t e n i n t h e form r  a  - r  n  -  r  n  , - e n  L n  A„ 2 a , - — 6 ,, nM A. nX 1 1  M —  =  A,  (A.83)  where £  =  n  n E e . ( r , - r - A .L.) . ni 1 i i ' i=l  and 5 = c , - - ^ - + E G . ., nX nX L . , n i IX n i=l Y  From E q u a t i o n  (A.83) we have t h e r e l a t i o n s h i p M "  W  " X "  R  £ M  ^2  A7°M  *  "  A7 MA' 6  (  A  '  8  4  )  where e  =  n E j-l  n E Y.(t)G..(r. - r i-1 i 3  3  1  1  n 6  MA= MA- MA a  Y  n  =  +  A.L.); i  W^lfiX'  -j=l i = l  J  J  and  al  =  M  n E j-l  n E Y i-1  (t)s..Y.(t).  3  3  1  1  ,M , 2, (A.83) and (A.84) i t i s p o s s i b l e t o s o l v e f o r (~) and (-p-) and t h e l l A  From E q u a t i o n s  4  A  r e s u l t s substituted into Equation  A  (A.81) t o d e t e r m i n e t h e i n s t a n t a n e o u s  condi-  t i o n a l expected r a t e o f r e t u r n V  °k " " — r  , kM MA- k A ° M , Q > n " P  r  5  6  =  (  ( a  V " n> r  t  "k E  r  " —  e  'a 6. , - 6 , o, , nM kA nX kM. , . ( ) (u - r - x " e ) , M  k = 1,  f  2,...,n-l.  ., . (A.85) o c  216  where Q = o 5 , * nM mX  - 6 ,o . nX M 2  A s i m i l a r r e s u l t h o l d s f o r t h e bond e q u a t i o n w r i t t e n i n t h e s c a l a r form, u s i n g E q u a t i o n M  (A.80) w h i c h may be  (A.41),  n  A_  n  Y  (A.86)  n . r.-r + E e .. (a. - r i — ) , i-1 i 1  Let  3  1  1  L  n r.-r _ e. = E . c .. (a. - r - — — ) , 1] i=i i 1  3  1  1  L  and n  1 JA £  =  Y  i A  Y  j A - *1=1 l ii c  l a  h ~ IT" ' l 0  j = 1, 2,...,n, and thus E q u a t i o n  (A.86) may b e w r i t t e n  r. - r - X . L . D  3  - ~ .  E i  e  i i=l  3  e  j i  X  i  (  t  )  " A71 3X'  (A.87)  3  1  3 = 1, 2,»««,n« M u l t i p l y i n g E q u a t i o n (A.87) by X.. (t) and summing o v e r j g i v e s , u s i n g  Equations  (A.42) and (A.44) M 11  "  r  "  Y  - 1 M E  =  Af M Y  A " A75MA'  ( A  '  8 8 )  217  where 2 Y  n £ j-l  =  M  n E X . (t) E . X . ( t ) , i-1 3  1  3  1  1  and 6.  =  MA M  n S X. (t)6 . j 3 DA =  1  The i n s t a n t a n e o u s c o n d i t i o n a l e x p e c t e d bonds must a l s o s a t i s f y E q u a t i o n  r  - r - X L n n n  r a t e o f r e t u r n f o r t h e n*"* f i r m ' s 1  (A.87); t h a t i s ,  -  e  1 n  M -2. „ A, nM Jt  =  Y  -  A ^ .6 . , A. 1 nX' 1  (A.89)  where n  Y  nM n M  -  £ ,S .x ( t ) . . , x ni i i=l B (—) l M  Equations  (A.88) and  (A.89) can be used  to eliminate  A  (A.87) t o g i v e  r  r . - r - X.L. - , e . =  3  3 J I 3  6 Y — A n'jM ( *—  Q  6  2 and (—) l A  Y^~  1 jX'M, . , )( r - r - X L T  n  1  nn  , ! j X n M " l n^jM, , (—J —>—) (TT - r - y Q " 5  Y  5  x  j - If 2 , . . . , n - l , where Q  l  =  Y  nM  1 MX 5  "  l nX M6  Y  i n Equation  A  -  .e  .  I n '  . e ), IH'  . (A.90) o  o  v  APPENDIX B NAMES OF BANKRUPT FIRMS  NAME  DATE OF BANKRUPTCY  A t l a s Sewing C e n t e r  1962  Avien  1964  Incorporated  Barcalo Manufacturing  Company  1965  Barchris Construction Corporation  1962  Betteringer Corporation  1961  B i s h o p O i l Company  1961  Bowl-Mor Company  1966  Buckner I n d u s t r i e s  Incorporated  1967  Davega S t o r e s C o r p o r a t i o n  1962  Dejay S t o r e s  1962  Incorporated  D i l b e r t ' s Q u a l i t y Supermarkets  Incorporated  1962  E r i e Forge and S t e e l C o r p o r a t i o n  1969  Fashion  Tree  1968  Gilbert  (A.C.) Company  1967  Goebel Brewing  1964  Grayson-Robinson S t o r e s  Incorporated  1962  G r e a t Western P r o d u c e r s  Incorporated  1965  Guidance Technology  Incorporated  1962  I n t e r n a t i o n a l O i l and Gas C o r p o r a t i o n  1965  Keystone A l l o y s  1966  Marrud  Company  Incorporated  1966  218  219  DATE OF BANKRUPTCY  NAME McCandless C o r p o r a t i o n  1968  Muskegon Motor S p e c i a l t i e s Company-  1961  National Video Corporation  1969  Okalta  1961  Oils  1966  Polycast Corporation P r e c i s i o n R a d i a t i o n Instruments P r e m i e r Albums  Incorporated  Industries Incorporated  U n i t e d S t a t e s Chemical M i l l i n g Vinco Webcor  Corporation Incorporated  Yuba C o n s o l i d a t e d I n d u s t r i e s  1963 1968  Incorporated  P u e r t o R i c o Brewing Company Trans-United  Incorporated  Corporation  1969 1963 1962 1963 1967 1961  B I B L I O G R A P H Y  Altman, E. I . " F i n a n c i a l R a t i o s , D i s c r i m i n a n t A n a l y s i s and the P r e d i c t i o n o f C o r p o r a t e Bankruptcy," Journal of Finance, V o l . X X I I I , No. 4 (September, 1968), pp. 589-609. • " C o r p o r a t e B a n k r u p t c y P o t e n t i a l , S t o c k h o l d e r Returns and Share V a l u a t i o n , " Journal of Finance, V o l . XXIV, No. 5 (December, 1969) pp. 887-900.  1970), pp.  "A R e p l y , " Journal 1169-1172.  Corporate Heath & Co., T1967. r\ *~ —t 1972), pp.  of Finance,  Bankruptcy  "A R e p l y , " Journal 718-721.  in America.  of Finance,  V o l . XXV,  No.  5  (December,  L e x i n g t o n , Mass.:  V o l . XXVII, No.  3  D.  C.  (June,  . " P r e d i c t i n g R a i l r o a d B a n k r u p t c i e s i n America," Bell Journal of Economics and Management Science, V o l . 4, No. 1 ( S p r i n g , 1973), pp. 184-211. Anderson, T. W. An Introduction York: John W i l e y & Sons,  to Multivariate 1957.  Statistical  Analysis.  A s h f o r d , J . R. and Sowden, R. R. " M u l t i - V a r i a t e P r o b i t A n a l y s i s , " V o l . 26 (1970), pp. 535-547.  New  Biometrics,,  B a c h e l i e r , L. 'Theory o f S p e c u l a t i o n , ' r e p r i n t e d i n The Random Character of Stock Market Prices, e d i t e d by C o o t n e r , P. Cambridge, Mass.: M.I.T. P r e s s , 1971. B a l e s t r a , P. and N e r l o v e , M. " P o o l i n g C r o s s S e c t i o n and Time S e r i e s Data i n the E s t i m a t i o n o f a Dynamic M o d e l : The Demand f o r N a t u r a l Gas," Econometrica, V o l . 34, No. 3 ( J u l y , 1966), pp. 585-612. B a x t e r , N. D. of Finance, 2_  "Leverage, R i s k o f R u i n , and the C o s t o f C a p i t a l , " V o l . X X I I , No. 3 (September, 1967), pp. 395-403.  Journal  , and Cragg, J . G. " C o r p o r a t e C h o i c e Among Long Term F i n a n c i n g Instruments," Review of Economics and Statistics, V o l . L I I , No. 3 (August, 1970), pp. 225-235.  Beaver, W. H. " F i n a n c i a l R a t i o s as P r e d i c t o r s o f F a i l u r e , " Empirical Research in Accounting: Selected Studies, supplement t o Journal of Accounting Research (1966), pp. 77-111.  220  221  Beaver, W. H. "Market P r i c e s , F i n a n c i a l R a t i o s and t h e P r e d i c t i o n o f F a i l u r e , " Journal of Accounting Research, V o l . 4 (Autumn, 1968), pp, 179-192. Bellman, R. Dynamic Programming. s i t y P r e s s , 1957.  P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r -  , and D r e y f u s , S. Applied Programming. P r i n c e t o n U n i v e r s i t y P r e s s , 1962  P r i n c e t o n , N.J.:  Berkson, J . " A p p l i c a t i o n o f t h e L o g i s t i c F u n c t i o n t o B i o - A s s a y , " Journal of the American Statistical Association, V o l . 39, pp. 357-365. Bierman, H. J . , and Thomas, J . "Ruin C o n s i d e r a t i o n s and Debt I s s u a n c e , " Journal of Financial and Quantitative Analysis, V o l . 7, No. 1 (January, 1972), p p . 1361-1378. B l a c k , F . " C o r p o r a t e Investment D e c i s i o n s , " Financial Note No. 28, Massac h u s e t t s I n s t i t u t e o f T e c h n o l o g y , S l o a n S c h o o l o f Management, May, 1969. . " C a p i t a l M a r k e t E q u i l i b r i u m and R e s t r i c t e d B o r r o w i n g , " of Business, V o l . 45, No. 3 ( J u l y , 1972), p p . 444-455.  Journal  and S c h o l e s , M. " D i v i d e n d Y i e l d s and Common S t o c k R e t u r n s : A New Methodology," Financial Note No. 19B, M a s s a c h u s e t t s I n s t i t u t e " o f Technology, S l o a n S c h o o l o f Management, August, 1971. ——'. ' • "The P r i c i n g o f O p t i o n s and C o r p o r a t e L i a b i l i t i e s , " Working Paper No. 16F, M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , S l o a n S c h o o l of'Management, May, 1972. , Jensen,  M. C. and S c h o l e s , M.  "The C a p i t a l A s s e t P r i c i n-"3g Model: . •- - .  Some E m p i r i c a l T e s t s , " p r i n t e d i n Studies in the Theory of Capital kets, e d i t e d by J e n s e n , M. C. New York:. P r a e g e r , 1972.  Mar-  Blume, M., and F r i e n d , I . "A New Look a t t h e C a p i t a l P r i c i n g Model," nal of'Finance, V o l . XXVII, No. 1 (March, 1973), pp. 17-34.  Jour-  Brennan, M. J . " C a p i t a l A s s e t P r i c i n g and t h e S t r u c t u r e o f S e c u r i t y Ret u r n s , " Working Paper, U n i v e r s i t y o f B r i t i s h Columbia, May, 1971.  Journal  "An Approach t o t h e V a l u a t i o n o f U n c e r t a i n Income Streams," of Finance, V o l . XXVIII, No. 3 (June, 1973), p p . 661-674  Chow, G. C. " T e s t s o f E q u a l i t y Between S e t s o f C o e f f i c i e n t s i n Two L i n e a r R e g r e s s i o n s , " Econometrica, V o l . 28, No. 3 ( J u l y , 1960), pp. 591-605. C o t t n e r , P. ( E d i t o r ) . The Random Character of Stock b r i d g e , Mass: : M.I.T. P r e s s , J u l y , 1970. , D. R., and M i l l e r , M a c m i l l a n , 1968.  H. D.  The Theory  Market  of Stochastic  Prices.  Processes.  Cam-  London:  222  Cragg, J . G. "Programs f o r M u l t i p l e P r o b i t and L o g i t A n a l y s i s and E x t e n s i o n s t o Them," Mimeographed, U n i v e r s i t y o f B r i t i s h Columbia, 1968. "Some S t a t i s t i c a l Models f o r L i m i t e d Dependent V a r i a b l e s w i t h A p p l i c a t i o n t o t h e Demand f o r D u r a b l e Goods," Econometrica, Vol. 39, No. 5 (September, 1971), pp. 829-844. , and B a x t e r , N. D. "The I s s u i n g o f Long-Term C o r p o r a t e S e c u r i t i e s , " Journal of P o l i t i c a l Economy, V o l . 78 (1970), pp. 1310-1324. , and U h l e r , R . S . "The Demand f o r A u t o m o b i l e s , " Canadian nal of Economics, V o l . I l l (August, 1970), pp. 386-406.  Jour-  Dake, J . L. "Comment: An E m p i r i c a l T e s t o f F i n a n c i a l R a t i o A n a l y s i s , " Journal of Financial and Quantitative Analysis, V o l . 7, No. 2 (March, 1972) , pp. 1495-1497. D e a k i n , E . B. "A D i s c r i m i n a n t A n a l y s i s o f P r e d i c t o r s o f B u s i n e s s F a i l u r e , " Journal of Accounting Research, V o l . 10, No. 1 ( S p r i n g , 1972), pp. 167179. Donaldson, G.  Business  Vol.  47  Corporate  Debt Capacity.  Boston:  Harvard,  1961.  " F i n a n c i a l G o a l s — Management V e r s u s S t o c k h o l d e r s , " Review, V o l . 41 (May-June, 1963), pp. 116-129. . " S t r a t e g y f o r F i n a n c i a l E m e r g e n c i e s , " Harvard (November-December, 1969), pp. 67-79.  D r e y f u s , S. E . Dynamic Programming York: Academic P r e s s , 1965.  and  the Calculus  Harvard  Business  of Variation.  Review,  New  Duesenberry, J . S. " C r i t e r i a f o r J u d g i n g t h e Performance o f C a p i t a l Mark e t s , " r e p r i n t e d i n the Elements of Investment, e d i t e d by Wu, H. K., and Zakon, A. J . New York: H o l t , R i n e h a r t , and W i n s t o n , 1965. E d m i s t e r , R. 0. "An E m p i r i c a l T e s t o f F i n a n c i a l R a t i o A n a l y s i s f o r S m a l l B u s i n e s s F a i l u r e P r e d i c t i o n , " Journal of Financial and Quantitative Analysis, V o l . 7, No. 7 (March, 1972), pp. 1477-1493. E l t o n , E . J . and G r u b e r , M. J . "Dynamic Programming A p p l i c a t i o n s i n F i n a n c e , " Journal of Finance, V o l . XXVI, No. 7 (May, 1971), pp. 473-505. Fama., E. F. " R i s k , R e t u r n and E q u i l i b r i u m : Some C l a r i f y i n g Comments," Journal of Finance, V o l . X X I I I , No. 4 (March, 1968), pp. 29-40.  Economic  " M u l t i p e r i o d Consumption - Investment D e c i s i o n s , " American Review, V o l . LX, No. 1 (March, 1970), pp. 163-174.  " E f f i c i e n t C a p i t a l M a r k e t s : A Review o f Theory and E m p i r i c a l Work," Journal of Finance, V o l . XXV, No. 5 (May, 1970), pp. 383-417.  223  Fama, E. F., and M i l l e r , M. H. R i n e h a r t & Winston, 1972. F i n n e y , D. J . 1964.  1971.  Statistical  Probit  The  Methods  Analysis.  Theory  of Finance.  in Biological  Cambridge:  Assay.  Cambridge  New  York:  Holt,  London:  Griffin,  University  Press,  F i s h e r , L. "Determinants o f R i s k Premiums on C o r p o r a t e Bonds," Journal of P o l i t i c a l Economy, V o l . L X V I I , No. 3 (June, 1959), pp. 217-237. G o l d b e r g e r , A. S. 1964.  Econometric  Theory.  New  York:  J o h n W i l e y & Sons,  Gordon, M. J . "Towards A Theory o f F i n a n c i a l D i s t r e s s , " Journal Finance, V o l . XXVI, No. 7 (May, 1971), pp. 347-356.  of  Hakansson, N. H. "Optimal Investment and Consumption S t r a t e g i e s Under R i s k F o r a C l a s s o f U t i l i t y F u n c t i o n s , " Econometrica, V o l . 38, No. 5, (September, 1970), pp. 587-607. Hamada, R. S. " P o r t f o l i o A n a l y s i s , M a r k e t E q u i l i b r i u m and C o r p o r a t e F i n a n c e , " Journal of Finance, V o l . XXIV, No. 5 (March, 1969), pp. 13-31. . "The E f f e c t o f the F i r m ' s C a p i t a l S t r u c t u r e on t h e S y s t e m a t i c R i s k o f Common S t o c k s , " Journal of Finance, V o l . XXVII, No. 2, (May, 1972), pp. 435-451. Hanna, M. " C o r p o r a t e B a n k r u p t c y P o t e n t i a l , S t o c k h o l d e r R e t u r n s and Share Valuation: Comment," Journal of Finance, V o l . XXVII, No. 3 (June, 1972), pp. 711-717. H i r s h l e i f e r , J . Investment, Interest P r e n t i c e - H a l l I n c . , 1970. Howard, R. A. Dynamic Programming M.I.T. P r e s s , 1960. Ito,  and  Capital.  and Markov  Englewood  Processes.  C l i f f s , N.J.:  Cambridge,  K. "On s t o c h a s t i c D i f f e r e n t i a l E q u a t i o n s , " Memoirs matical Society, No. 4 (1951), pp. 1-50.  of American  , and McKean, H. P. York: Academic P r e s s ,  Sample  Diffusion 1964.  J a f f e e , D. M. Credit Rationing and John W i l e y & Sons, I n c . , 1971.  American 872.  Processes  and  the Commercial  Their  Loan Market.  Mass.:  Mathe-  Paths.  New  New  York:  , and M o d i g l i a n i , F. "A Theory and T e s t o f C r e d i t R a t i o n i n g , " Economic Review, V o l . LIX, No. 5 (December, 1969), pp. 850-  224  Jensen, M. C. " C a p i t a l Markets : Theory and E v i d e n c e , " Bell Journal Economics and Management Science, V o l . 3, No. 2 (Autumn, 1972), 357-398.  York:  . (Editor). Studies P r a e g e r , 1972.  in the Theory  of Capital  of pp.  Markets.  New  Johnson, C. G. " R a t i o A n a l y s i s and the P r e d i c t i o n o f F i r m F a i l u r e , " Journal of Finance, V o l . XXV, No. 5 (December, 1970), pp. 1166-1168. Johnson, K. H., and Lyon, H. L . " E x p e r i m e n t a l E v i d e n c e on Combining C r o s s S e c t i o n and Time S e r i e s I n f o r m a t i o n , " Review of Economics and Statistics, V o l . LV, No. 4 (November, 1973), pp. 465-474. Johnson, T. " Q u a l i t a t i v e and L i m i t e d Dependent V a r i a b l e s : In Economic Rel a t i o n s h i p s , " Econometrica, V o l . 40, No. 3 (May, 1972), pp. 455-461. Keran, M. W. " E x p e c t a t i o n s , Money, and t h e S t o c k M a r k e t , " Federal Bank of St. Louis (January, 1971), pp. 16-31.  Reserve  Kraus, A., and L i t z e n b e r g e r , R. "A S t a t e - P r e f e r e n c e Model o f O p t i m a l F i n a n c i a l L e v e r a g e , " Journal of Finance, V o l . X X V I I I , No. 3 (September, 1973), pp. 911-922. Kuh, E . "The V a l i d i t y o f C r o s s - S e c t i o n a l l y E s t i m a t e d B e h a v i o r E q u a t i o n s i n Time S e r i e s A p p l i c a t i o n s , " Econometrica, V o l . 27 ( A p r i l , 1959), pp. 197-214. Kushner, H. J . Stochastic P r e s s , 1967.  Stability  and  . Introduction to Stochastic R i n e h a r t & Winston, 1971.  Control.  Control.  New  York:  New  York:  Academic  Holt,  L i n t n e r , J . "The V a l u a t i o n o f R i s k A s s e t s and S e l e c t i o n o f R i s k y I n v e s t ments i n S t o c k P o r t f o l i o and C a p i t a l B u d g e t s , " Review of Economics and Statistics, V o l . 47, No. 1 ( F e b r u a r y , 1965), pp. 13-37. Maddala, G. S. "The Use o f V a r i a n c e Components Models i n P o o l i n g C r o s s S e c t i o n and Time S e r i e s D a t a , " Econometrica, V o l . 39, No. 2 (March, 1971), pp. 341-357. M a n d e l b r o t , B. "The V a l u a t i o n o f C e r t a i n S p e c u l a t i v e P r i c e s , " Journal Business, V o l . 34, No. 4 (1963), pp. 394-419. Markowitz, H. M. Portfolio Selection : Efficient ment. New Y o r k : John W i l e y & Sons, 1959.  Diversification  of  of  Invest-  Mayers, D. "Non-Marketable A s s e t s and C a p i t a l Market E q u i l i b r i u m Under Unc e r t a i n t y , " p r i n t e d i n Studies in the Theory of Capital Markets, edited by J e n s e n , M. New York : P r a e g e r , 1972.  225  Merton, R. C. " L i f e t i m e P o r t f o l i o S e l e c t i o n Under U n c e r t a i n t y : The Cont i n u o u s Case," Review of Economics and Statistics, V o l . L I , No. 3, (August, 1969), pp. 247-257. . "A Dynamic G e n e r a l E q u i l i b r i u m Model o f the A s s e t Market and i t s A p p l i c a t i o n t o t h e P r i c i n g o f the C a p i t a l S t r u c t u r e o f t h e F i r m , " Working Taper No. 497-90, M a s s a c h u s e t t s I n s t i t u t e o f T e c h nology, S l o a n S c h o o l o f Management, December, 1970. . "Optimum Consumption and P o r t f o l i o R u l e s i n a C o n t i n u o u s Time Model," Journal of Economic Theory, V o l . 3 (December, 1971), pp. 373-413. . "An I n t e r t e m p o r a l C a p i t a l A s s e t P r i c i n g Model," Working Paper 588-72, M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , S l o a n S c h o o l o f Management, F e b r u a r y , 1972. . "An A n a l y t i c D e r i v a t i o n o f t h e E f f i c i e n t P o r t f o l i o F r o n t i e r , " Journal of Financial and Quantitative Analysis, V o l . 7, No. 4 (September, 1972), pp. 1851-1872. . "Theory o f R a t i o n a l O p t i o n P r i c i n g , " Bell Journal of Economics and Management Science, V o l . 4, No. 1 ( S p r i n g , 1973), pp. 141-183. Meyer, P. L. Introductory Probability Mass.: Addison-Wesley, 1965.  and  Statistical  Applications.  Meyer, P. A. and P i f e r , H. W. " P r e d i c t i o n o f Bank F a i l u r e , " Journal Finance, V o l . XXV, No. 4 (September, 1970), pp. 853-868.  Reading,  of  M i l l e r , M. H., and M o d i g l i a n i , F . " D i v i d e n d P o l i c y , Growth, and t h e V a l u a t i o n o f S h a r e s , " Journal of Business, V o l . 34, No. 4 (October, 1961), pp. 411-433. . "Some E s t i m a t e s o f t h e C o s t o f C a p i t a l t o the E l e c t r i c U t i l i t y I n d u s t r y , 1954-1957," American Economic Review, V o l . L V I , No. 3 (June, 1966), pp. 333-391. M i l l e r , M. H., and O r r , D. "A Model o f t h e Demand f o r Money By F i r m s , " Quarterly Journal of Economics, V o l . LXXX (August, 1966), pp. 413-435. , and S c h o l e s , M. "Rates o f R e t u r n i n R e l a t i o n t o R i s k : A Re-examination o f Some Recent F i n d i n g s , " p r i n t e d i n Studies in the Theory of Capital Markets, e d i t e d by J e n s e n , M. New York: P r a e g e r , 1972. Modigliani, F., and M i l l e r M. H. "The C o s t o f C a p i t a l , C o r p o r a t e F i n a n c e , and t h e T h e o r y o f Investment," American Economic Review, V o l . X L V I I I , No. 3 (June, 1958), pp. 261-297.  226  M o d i g l i a n i , F., and M i l l e r , M. H. " C o r p o r a t e Income Taxes and t h e C o s t o f C a p i t a l : A C o r r e c t i o n , " American Economic Review, V o l . L I I I , No. 3, (June, 1963), pp. 433-443. , Economic  Review,  . " R e p l y t o Heins and S p r e n k l e , " American V o l . LIX, No. 4 (September, 1969), pp. 592-595.  Mood, A., and G r a y b i l l , F . Introduction York: M c G r a w - H i l l , 1963.  to the Theory  of Statistics.  M o s s i n , J . " E q u i l i b r i u m i n a C a p i t a l A s s e t Market," Econometrica, No. 4 (October, 1966), pp. 768-783.  New  V o l . 34,  . " S e c u r i t y P r i c i n g and Investment C r i t e r i a i n C o m p e t i t i v e Markets,;" American Economic Review, V o l . LIX, No. 5 (December, 1969), pp. 749-756. Myers, S. C. "On t h e I n t e r a c t i o n s o f C o r p o r a t e F i n a n c i n g and Investment D e c i s i o n s and t h e Weighted Average C o s t o f C a p i t a l , " M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , S l o a n :.School o f Management, ( u n p u b l i s h e d ' and u n d a t e d ) . Nemhauser, G. L . Introduction W i l e y S Sons, 1966.  to Dynamic  Programming.  New  York:  John  N e r l o v e , M. " F u r t h e r E v i d e n c e on t h e E s t i m a t i o n o f Dynamic Economic Rel a t i o n s From a Time S e r i e s o f C r o s s S e c t i o n s , " Econometrica, V o l . 39, No. 2 (March, 1971), pp. 359-382. N o r t o n , J . "The T h e o r y o f Loan C r e d i t i n R e l a t i o n t o C o r p o r a t i o n Economics," Publications of the American Economic Association, 3 r d s e r . V o l . V (1904), pp. 278-300. Osborne, M. F. "Brownian M o t i o n i n t h e S t o c k M a r k e t , " Operations V o l . 7 ( M a r c h - A p r i l , 1959), pp. 145-173.  Research,  P i n c h e r , G. E., and Mingo, K. A. "A M u l t i v a r i a t e A n a l y s i s o f I n d u s t r i a l Bond R a t i n g s , " Journal of Finance, V o l . X X V I I I , No. 1 (March, 1973), pp. 1-18. P r a t t , J . W. " R i s k A v e r s i o n i n t h e S m a l l and t h e L a r g e , " Econometrica, 32, No. 1 (January, 1964), pp. 122-136.  Vol.  Pye, G. " L i f e t i m e P o r t f o l i o S e l e c t i o n i n C o n t i n u o u s Time f o r a M u l t i p l i c a t i v e C l a s s o f U t i l i t y F u n c t i o n s , " American Economic Review, V o l . L X I I I , No. 5 (December, 1973), pp. 1013-1016. P y l e , D. H. and Turnovsky, S. J . " S a f e t y - F i r s t and E x p e c t e d U t i l i t y z a t i o n i n Mean-Standard D e v i a t i o n P o r t f o l i o A n a l y s i s , " Review of and Statistics, V o l . L I I , No. 1 (February, 1970), pp. 75-81.  MaximiEconomics  ^  227  Robichek, A., and Myers, S. "Problems i n t h e Theory o f O p t i m a l C a p i t a l S t r u c t u r e , " Journal of Financial and Quantitative Analysis, V o l . 1, (June, 1966), pp. 1-35. R o l l , R. " B i a s i n F i t t i n g t h e Sharpe Model t o Time S e r i e s D a t a , " Journal of Financial and Quantitative Analysis, V o l . 4 (September, 1969), pp. 271-289. Samuelson, P. A. " G e n e r a l P r o o f t h a t D i v e r s i f i c a t i o n Pays," Journal of Financial and Quantitative Analysis, V o l . 2 (March, 1967), pp. 1-13. . " L i f e t i m e P o r t f o l i o S e l e c t i o n by Dynamic S t o c h a s t i c Programming," Review of Economics and Statistics, V o l . L i , No. 3, (August, 1969), pp. 239-246.  Analysis Economic Sharpe, W. Science,  . "The Fundamental A p p r o x i m a t i o n Theorem o f P o r t f o l i o i n Terms o f Means, V a r i a n c e s and H i g h e r Moments," Review of Studies, V o l . 37 (October, 1970), p p . 537-542. "A S i m p l i f i e d Model f o r P o r t f o l i o A n a l y s i s , " V o l . 9 (January, 1963), pp. 277-293.  Management  . "Capital Asset Prices: A T h e o r y o f Market E q u i l i b r i u m Under C o n d i t i o n s o f R i s k , " Journal of Finance, V o l . XIX, No. 3 (September, 1964), p p . 425-442. Smith, V. L . " D e f a u l t R i s k , S c a l e and t h e Homemade L e v e r a g e Theorem," American Economic Review^ V o l . L X I I , No. 2 (March, 1972), pp. 66-76. S t i g l i t z , J . E . "A R e - e x a m i n a t i o n o f t h e M o d i g l i a n i M i l l e r Theorem," American Economic Review, V o l . L I X , No. 5 (December, 1969), pp. 784-795. . "Some A s p e c t s o f t h e Pure T h e o r y o f C o r p o r a t e F i n a n c e : Bankruptcy and Take O v e r s , " Bell Journal of Economics and Management Science, V o l . 3, No. 2 (Autumn, 1972), p p . 458-492. T a t s u o k a , M. M. Psychological  Multivariate Research.  Analysis: New York:  Techniques for Educational John W i l e y & Sons, 1971.  and  T e l s e r , L . G. " I t e r a t i v e E s t i m a t i o n o f a S e t o f L i n e a r R e g r e s s i o n Equat i o n s , " Journal of the American Statistical Association, V o l . 59 (September, 1964), pp. 845-862. T h i e l , H.  Principles  of Econometrics.  New York:  John W i l e y & Sons, 1971.  T i n s l e y , P. A. " C a p i t a l S t r u c t u r e , P r e c a u t i o n a r y B a l a n c e s , and V a l u a t i o n o f t h e F i r m ; The Problem o f F i n a n c i a l R i s k , " Journal of Financial and Quantitative Analysis, V o l . 5 (March, 1970), pp. 33-62.  228  T r e y n o r , J . "Towards A Theory o f Market V a l u e o f R i s k y A s s e t , " memorandum, 1961.  unpublished  . " I m p l i c a t i o n s f o r the Theory o f F i n a n c e , " u n p u b l i s h e d memorandum, 1961. Van  Home, J . C. Financial Management and Policy. P r e n t i c e - H a l l , I n c . , 1972.  Englewood C l i f f s , N.J.:  W a l t e r s , J . E . "Avoidance o f F i n a n c i a l D i s t r e s s , " U n i v e r s i t y u n p u b l i s h e d and undated. W e i l , R. L. Economic  of Pennsylvania,  " R e a l i z e d I n t e r e s t Rates and Bondholders' R e t u r n s , " Review, V o l . LX, No. 3 (June, 1970), pp. 502-511.  American  W e s t e r f i e l d , R. "The Assessment o f Market R i s k and C o r p o r a t e F a i l u r e . " U n i v e r s i t y o f P e n n s y l v a n i a , Wharton S c h o o l o f F i n a n c e and Commerce, U n p u b l i s h e d (August, 1970). Whalen, E . L. "A R a t i o n a l i s a t i o n o f the P r e c u a t i o n a r y Demand f o r Cash," Quarterly Journal of Economics, V o l . LXXX (May, 1966), pp. 314-324. Winsor, C. P. "A Comparison o f C e r t a i n Symmetrical Growth C u r v e s , " Journal of the Washington Academy of Sciences, V o l . 22, No. 4 (February, 1932), pp. 73-84.  i  

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