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

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THE CAPITAL ASSET PRICING MODEL AND THE PROBABILITY OF BANKRUPTCY: THEORY AND EMPIRICAL TESTS by STUART McLEAN TURNBULL B . S c , R o y a l C o l l e g e o f S c i e n c e , London U n i v e r s i t y , 1969 A.R.C.S., R o y a l C o l l e g e o f S c i e n c e , London U n i v e r s i t y , 1969 M . S c , R o y a l C o l l e g e o f S c i e n c e , London U n i v e r s i t y , 1970 D.I.C., R o y a l C o l l e g e o f S c i e n c e , London U n i v e r s i t y , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e F a c u l t y o f Commerce and 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 t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA A p r i l , 10 74 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I aq ree tha the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department nr by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . Department o f Commerce & Business A d m i n i s t r a t i o n The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date A p r i l 16th. 1974 A B S T R A C T Empirical evidence shows that the C a p i t a l Asset P r i c i n g Model (CAPM) i s m i s s p e c i f i e d . S e c u r i t i e s of low systematic r i s k c o n s i s t e n t l y earn more than predicted by the model, the reverse being true f o r s e c u r i t i e s of high systematic r i s k . Whilst the r e l a t i o n s h i p between ex-post returns and sy s t e -matic r i s k appears to be l i n e a r , the estimated re g r e s s i o n c o e f f i c i e n t s are 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 values. Various attempts to explain t h e o r e t i c a l l y the causes o f the m i s s p e c i f i c a t i o n have been explored, but f a i l to provide an adequate explanation of a l l the observed 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 the mechanism of bankruptcy a f f e c t s the 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. The hypothesis of the t h e s i s i s that 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 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 the market. Using s t o c h a s t i c c o n t r o l theory, a two v a r i a b l e extended form o f the continuous time analogue of the CAPM i s derived. The second v a r i a b l e i s associated with the p r o b a b i l i t y of bankruptcy. The model provides a natural explanation o f the d e f i c i e n c i e s of the CAPM. A d i s c r e t e time ex-post formulation of the model i s used to t e s t e m p i r i c a l l y the hypothesis. This n e c e s s i t a t e s being able to measure the p r o b a b i l i t y of bankruptcy. A model formulated i n terms of a firm'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 or e x t e r n a l l y , to cover f i x e d charges i s developed, and the p r o b a b i l i t y of bankruptcy estimated using the maximum l i k e l i h o o d methodologies of 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 . The a b i l i t y of the model to p r e d i c t bankruptcy i i i i v i s tented on a secondary sample of bankrupt firms. E x c e l l e n t r e s u l t s are obtained with the model p r e d i c t i n g bankruptcy, f o r some firms, four or f i v e years before the a c t u a l occurrence. Using a p o o l i n g of time s e r i e s and cross s e c t i o n data to e s t i -mate the c o e f f i c i e n t s of the r e g r e s s i o n equation representing the hypothe-s i s , evidence i s found i n d i c a t i n g that bankruptcy i s an explanatory f a c t o r of common stock returns. TABLE OF CONTENTS Page LIST OF TABLES v i i LIST OF FIGURES v i i i CHAPTER I. INTRODUCTION 1 Hypothesis 5 Importance . . . . . 6 Organization o f T h e s i s ^° I I . A CRITICAL SURVEY OF THE RELEVANT FINANCIAL LITERATURE . . . . 12 Empirical Studies on Bankruptcy 13 Costs t o Bankruptcy 13 Bankruptcy and Stock Market P r i c e s 15 P r e d i c t i o n o f Bankruptcy 18 C a p i t a l Asset P r i c i n g Model., 24 Foundations o f the CAPM 25 Cross-Sectional Tests of the Model 27 Time Series Tests of the Model 29 T h e o r e t i c a l Extensions to the CAPM 31 Summary 37 Hypothesis o f the Thesis 3 8 Empirical T e s t i n g o f the Hypothesis 3 9 I I I . . PROBABILITY OF BANKRUPTCY 43 Theory 44 Ex-post Formulation 48 P r e d i c t i v e Model . . . . . . . . 57 S t a t i s t i c a l Methodology 60 Testing o f the Model 6 5 Summary 66 IV. AN EXTENSION OF THE CAPITAL ASSET PRICING MODEL: BANKRUPTCY . 6 8 Foundations o f Model 70 Price Dynamics 7 5 State Space D e s c r i p t i o n and the Budget Co n s t r a i n t 87 The Equation o f Optimality: The Demand Functions f o r Assets . 94 v v i Chapter Page Bankruptcy and Structure of Returns 9 9 Stochastic Changes i n the P r o b a b i l i t y of Bankruptcy 110 Summary 1 2 0 V. EMPIRICAL RESULTS 122 Estimation of the P r o b a b i l i t y of Bankruptcy 124 S t a t i s t i c a l Methodology I 3 0 Data 1 3 i P r e d i c t i v e A b i l i t y . . . I 3 3 Results 1 3 5 A l t e r n a t i v e Model . I 4 0 S t a t i o n a r i t y I 4 3 P r e d i c t i v e Model l 4 ^ Summary 150 Testing of Hypothesis I 5 0 Methodology 152 Pooling o f Time S e r i e s and Cross S e c t i o n a l Data 152 Aggregation. . . . . . l ^ 3 Data I 5 6 Empirical Results . l ^ 8 Use of P o r t f o l i o s 1 5 8 Individual S e c u r i t y Data I 6 4 Random Sample 1 6 4 E f f e c t o f Asset Size 1 6 6 Adjustment o f Time Period 1 6 8 170 Cross Section Studies... . Changes i n the P r o b a b i l i t y of Bankruptcy ± , J Summary . 1 7 5 VI. SUMMARY 177 Conclusion . „ 177 Further Research 179 APPENDIX >" A. MATHEMATICAL DERIVATION OF THE RESULTS IN CHAPTER IV 182 B. NAMES OF BANKRUPT FIRMS 218 BIBLIOGRAPHY 2 2 2 LIST OF TABLES TABLE Page 4.1 The P r o b a b i l i t y of Occurrence of D i f f e r e n t States 89 5.1 Number o f Bankrupt and Non-Bankrupt Firms i n Data Sample . . . 134 5.2 Estimation of C o e f f i c i e n t s f o r a General Model 136 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 General Model. . . . 138 5.4 P r e d i c t i v e A b i l i t y o f General Model ^.39 5.5 Estimation o f C o e f f i c i e n t s and Test f o r S t a t i o n a r i t y : A l t e r n a t i v e Model 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 Data Sample: A l t e r n a t i v e Model . . 142 5.7 P r e d i c t i v e A b i l i t y of A l t e r n a t i v e Model 144 5.8 Estimation o f C o e f f i c i e n t s and Test f o r S t a t i o n a r i t y : P r e d i c t i v e Model 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 Data Sample: P r e d i c t i v e Model. . . 148 5.10 P r e d i c t i v e A b i l i t y o f Model 149 5.11 Values of C o e f f i c i e n t s Used t o Estimate the P r o b a b i l i t y of Bankruptcy 157 5.12 Average Yearly Values o f the P r o b a b i l i t y of Bankruptcy . . . . 159 5.13 P o r t f o l i o Data: Pooling o f Time Series and Cross S e c t i o n . . P r o b a b i l i t y of Bankruptcy Estimated Using Market Values of Corporate V a r i a b l e s 160 5.14 P o r t f o l i o Data: Pooling of Time S e r i e s and Cross S e c t i o n . P r o b a b i l i t y of Bankruptcy Estimated Using Book Values of Corporate V a r i a b l e s . . . 162 5.15 Random Sample: Pooling of Time Ser i e s and Cross Section Data. 165 5.16 Pooling of Time and Cross Data on Groups of Firms Sorted by Asset Size 1 6 7 v i i v i i i TABLE Page 5.17 Adjustment of Time Period 169 5.18 Cross Section Study 172 5.19 Differences i n the P r o b a b i l i t y of Bankruptcy 174 LIST OF FIGURES FIGURE Page 4 . 1 The E f f e c t of Bankruptcy Upon the C a p i t a l Market Line. . . . 1 0 8 i x A C K N O W L E D G E M E N T S The author i s indebted to h i s supervisor, Dr. R. White, f o r h i s h e l p f u l suggestions, acumen, c o n t r i b u t i o n of ideas, and constant a v a i l -a b i l i t y . I am g r a t e f u l to Dr. J . G. Cragg f o r providing advice, pene-t r a t i n g i n s i g h t , and encouragement. The help of Dr. S. L. Brumelle, Dr. J . Mao, and Dr. A. Kraus i s noted. I thank Dr. M. J . Brennan f o r help, advice, and c r i t i c i s m throughout the whole period of my sojourn at the U n i v e r s i t y of B r i t i s h Columbia. I am thankful to Miss Susan Aizenman f o r coffee, T i a Maria, food (l a t k e s ) , t r a n s p o r t a t i o n and the unforgettable, i f nightmarish, memory of how she typed the t h e s i s . F i n a l l y , I acknowledge the kindness of my many graduate student f r i e n d s , e s p e c i a l l y R. Fisher and B. MacDonald, the Department of History, U.B.C., f o r providing a small oasis of i n t e l l e c t u a l s timulation i n an otherwise barren desert. x But even at the very grave I trust the time shall come to be When over malice3 over wrong, The good will win its victory. (Boris Pasternak, January, 1959) CHAPTER I INTRODUCTION One of the c e n t r a l issues i n the theory of finance i s the r e l a t i o n -ship between r i s k and return demanded by inve s t o r s i n s e c u r i t i e s . The C a p i t a l Asset P r i c i n g Model (CAPM) provides 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 the expected rate of r e t u r n and the expected r i s k of an asset under 1 conditions of market e q u i l i b r i u m . E x p l i c i t l y , the model stat e s t h a t the expected one period return 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 of i t s s y s t e -matic r i s k , which i s a measure of the responsiveness of the s e c u r i t y ' s r e t u r n to changes i n the return on the market as a whole. In a d d i t i o n to pr o v i d i n g i n s i g h t s i n t o the fu n c t i o n i n g of c a p i t a l markets, the CAPM holds considerable promise as an o p e r a t i o n a l t o o l . 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 , estimation of the cost of c a p i t a l , measurement of investment performance, and e s t a b l i s h i n g a structure of managerial fees. Before the model can be implemented with any de-gree o f confidence, i t i s necessary t o i n v e s t i g a t e the t h e o r e t i c a l and empiri-c a l v a l i d i t y of the model. One important assumption underlying the model i s that of p e r f e c t c a p i -2 t a i markets, i n p a r t i c u l a r , the absence of costs associated with f i n a n c i a l d i s t r e s s . The s i t u a t i o n i n which the firm's income before i n t e r e s t and taxes T h e o r e t i c a l l y the model a p p l i e s only to a l l equity 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 are p r i c e takers, have equal and c o s t l e s s access to a l l information, and there are no tra n s a c t i o n costs or taxes. 1 2 i s l e s s than i t s f i x e d commitments, f o r example i n t e r e s t payments on debt, i s termed a c o n d i t i o n of " f i n a n c i a l d i s t r e s s . " In such a case the firm must c u r t a i l dividends, or investment, or obtain a c a p i t a l inflow (for example by s e l l i n g stock) to meet i t s f i x e d commitments. The extreme s i t -uation of f i n a n c i a l d i s t r e s s i s when the f i r m i s forced to declare bankruptcy because i t i s unable to meet the claims of i t s c r e d i t o r s . In the theory of corporate finance the concepts of bankruptcy and f i n a n c i a l d i s t r e s s are important f a c t o r s i n determining the optimal value of the f i r m . In the absence of corporate taxes and d e f a u l t , M o d i g l i a n i and M i l l e r ^ have shown that the value of the f i r m i s independent of the mix of debt and equity, given the assumptions of p e r f e c t c a p i t a l markets, homogeneity of expectations and the a b i l i t y 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 nature the income obtained by i n v e r t i n g i n an unlevered 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 of no corporate tax, introducing a market imperfection. The f i r m r e c e i v e s a tax subsidy from the government f o r using debt and, as debt i s r i s k l e s s , i t i s advantageous f o r the fi r m to use as much debt as p o s s i b l e ; c a p i t a l s t r u c t u r e i s relevant to the value o f the f i r m . Such a conclusion has l i t t l e p r a c t i c a l appeal f o r corporate debt i s not, i n general, r i s k l e s s . Increasing the l e v e l of debt increases the f i x e d charges and the p r o b a b i l i t y that the f i r m w i l l not be able to meet i t s f i n a n c i a l " M o d i g l i a n i , F . and M i l l e r , M. "The Cost of C a p i t a l , Corporate Finance, and the Theory of Investment," American Economic Review, V o l . XLVIII, No. 3 , (June, 1 9 5 8 ) , pp. 2 6 1 - 2 9 7 . 4 M o d i g l i a n i , F . and M i l l e r , M. "Corporate Income Taxes and the Cost 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 . LIII (June, 1 9 6 3 ) , pp. 4 3 3 - 4 4 3 . 3 o b l i g a t i o n s . Therefore, a second market imperfection, bankruptcy, must a l s o be incorporated. By assuming there are no associated p e n a l t i e s or resource 5 costs to bankruptcy and that 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 , S t i g l i t z has shown that bankruptcy has no e f f e c t upon the value of the f i r m . In the case of no corporate tax, 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 , and i f there are cor-porate taxes, i t i s s t i l l advantageous f o r the firm to use as much as p o s s i b l e . Bankruptcy i s a " t e c h n i c a l i t y ; " the bankrupt fi r m can be replaced by a new f i r m and since no resources have been expended there are no economic l o s s e s . The assets of the bankrupt f i r m are s t i l l i n t a c t ; the only d i f f e r e n c e i s that the management (control) and ownership are i n the hands of a d i f f e r e n t group of people (namely, the debt holders instead of the o r i g i n a l equity 7 h o l d e r s ) . Within the same framework S t i l g l i t z has attempted to r e l a x the assumption of homogeneity of expectations and has shown, given the a d d i t i o n -a l assumption t h a t short s e l l i n g i s not allowed, t h a t bankruptcy does a f f e c t the value of the f i r m , even though there are no resource costs to bankruptcy. However, i f the assumption o f no short s e l l i n g 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 . Short s e l l i n g allows the 5 S t i g l i t z , J. E. "A Re-Examination of the M o d i g l i a n i M i l l e r Theorem," American Economic Review, V o l . LIX, No. 5 (December, 1969), pp. 784-793. 6 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 Sprenkle," American Economic Review, Vol.,LIX, No. 4 (September, 1969), pp. 592-595. Debt has a f i x e d set o f claims on the f i r m plus the r i g h t to "take over" the firm i n the event the firm does not meet i t s o b l i g a t i o n s . S t i g l i t z , J. E. "Some Aspects of the Pure Theory of Corporate Finance: Bankruptcy and Take-Over," Bell Journal of Economics and Management Science, V o l . 3, No. 2 (Autumn, 1972), pp. 450-402. 4 investor the a b i l i t y t o r e p l i c a t e across a l l states of nature the income ob-tained by i n v e s t i n g i n an unlevered firm, given the assumptions of l i m i t e d l i a b i l i t y . In r e a l i t y , there are costs associated with f i n a n c i a l d i s t r e s s . In the event of bankruptcy r e a l resources are consumed. That i s , bankruptcy proceedings involve costs — l e g a l fees, t r u s t e e fees, administration fees, turnover of employees due to uncertainty,and loss of customers due to uncer-t a i n t y as to whether the f i r m w i l l be able t o f u l f i l l c o n t r a c t s . Kraus and 9 Litzenburger have formally introduced the tax advantage of debt and bank-i ruptcy p e n a l t i e s i n t o a state preference framework. They have shown that the market value of a levered f i r m i s equal t o the unlevered market value, plus the corporate tax r a t e times the market value of the firm's debt, l e s s the complement of the corporate tax rate times the present value of bankruptcy c o s t s . Thus there i s a t r a d e - o f f between the e f f e c t s of the two market imper-f e c t i o n s : corporate tax and costs t o bankruptcy, implying an optimal c a p i t a l s t r u c t u r e . Under the more general concept of f i n a n c i a l d i s t r e s s , i t has been argued that there are a d d i t i o n a l costs associated with changes i n f i n a n c i n g and investment s t r a t e g i e s . 1 ^ I f there 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 of bank-ruptcy, p o t e n t i a l investors w i l l demand a premium to compensate f o r the r i s k s associated with bankruptcy. The levered f i r m might be i n a poor bargaining p o s i t i o n and have to o f f e r higher returns to s u p p l i e r s of a d d i t i o n a l c a p i t a l . Kraus, A. and Litzenberg, R. "A State-Preference Model of Optimal F i n a n c i a l Leverage," Journal of Finance, V o l . XXVIII, No. 3 (September, 1973), pp. 911-922. Robichek, A. and Myers, S. "Problems i n the Theory of Optimal Capi-t a l Structure," Journal of Financial and Quantitative Analysis, V o l . 1 (June, 1966), pp. 1-35. 5 The underwriting costs of a common stock issue i n such a s i t u a t i o n might be very large, i f not p r o h i b i t i v e . The existence of c r e d i t r a t i o n i n g or con-s t r a i n t s on the investment p o l i c i e s of many i n s t i t u t i o n s might f u r t h e r weaken the bargaining p o s i t i o n of the f i r m by reducing competition among p o t e n t i a l s u p p l i e r s of funds. Loan contracts undertaken by the f i r m might impose constraints on the firm's f i n a n c i a l and investment s t r a t e g i e s thus l i m i t i n g i t s freedom of a c t i o n . This implies the existence of a d d i t i o n a l market imperfections besides that of bankruptcy. In a p e r f e c t market a f i r m should, at a p r i c e , be able to s a t i s f y i t s c a p i t a l requirements. I f there are non-price r e s t r i c t i o n s i n h i b i t i n g the firm's a b i l i t y to r a i s e c a p i t a l , then these are market imperfections. Hypothesis In summary, the p o t e n t i a l costs associated with f i n a n c i a l d i s t r e s s are thought to be an important f a c t o r i n determining the value of a f i r m . Assuming that i n v e s t o r s are r i s k averse, they w i l l demand a r i s k premium to bear the r i s k s associated with f i n a n c i a l d i s t r e s s . Unfortunately, an index of f i n a n c i a l d i s t r e s s i s not a v a i l a b l e . What i s a v a i l a b l e , however, i s the c l a s s i f i c a t i o n of firms as f a i l e d or n o n - f a i l e d . The d i f f i c u l t y of mathemati-l a l l y modelling 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 be treated as a continuum, precludes i t s use i n t h i s t h e s i s . Consequently, the extreme s i t u a t i o n , bankruptcy, i s the op e r a t i o n a l concept used. I t i s the sharp d i s -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 that a v a i l s i t s e l f r e a d i l y to mathe-matical a n a l y s i s . 6 The hypothesis of the thesis i s that differences in the probability of bankruptcy across securities and across time are reflected i n the resi d -ual return after abstracting from the market. The e x p l i c i t objectives of the thesis are; 1, To analyse theoretically how the mechanism of bankruptcy affects the structure of returns for corporate financial assets. 2, To quantify the determinants of bankruptcy} to arrive at a £igure which can be ident i f i e d as the probabil-i t y of bankruptcy. 3 , To test empirically the hypothesis of the thesis. From the theoretical analysis a two variable rooaiel i s derived, the second variable being associates "with the probability of banltomptcy. Tnhe a b i l i t y to measnare the probability of bankruptcy implies that the hypothesis of the thesis, as represented by the two variable model, can be empirically tested. Importance The empirical work of Beaver^ and West e r f i e l d ^ has offered same evidence vhich suggests that impending bankruptcy does appear to affect the structure of returns on common stocks. The behaviownr of ex-post rettorns, after abstracting from the market, for 'common stocks of firms that even-tually went bankrupt, are significantly different from those of healthy 11 Beaver, W. H,, "Market P r i c e s , F i n a n c i a l Ratios, and the Predic t i o n of F a i l u r e , " Journal of Accounting Researcha V o l . 4 '(Autumn, 19&8J, pp. 179-192. 12 Wcsfcerfield, R., "The Assessment of Market Risk and Corporate F a i l u r e , " U n i v e r s i t y o f Pennsylvania, Wharton School o f Finance and Commerce, August 1970 {unpublished). 7 firms f o r the same time period. These f i n d i n g s , i f c o r r e c t , have important im p l i c a t i o n s as to the s i g n i f i c a n c e of the e f f e c t s of bankruptcy upon the 13 structure of common stock returns. F i s h e r , i n an e m p i r i c a l study, ad-vanced the hypothesis that the r i s k of 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 corporate bonds. A l l of these e m p i r i c a l studies s u f f e r from the lack of any t h e o r e t i c framework wit h i n which t o i n v e s t i g a t e how the mechanism o f bankruptcy a f f e c t s the str u c t u r e of returns on corporate f i n a n c i a l assets. A number of recent studies have concluded t h a t the 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 . I t i s found that the i n t e r c e p t term i s non-stationary, 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 high systematic 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 with low systematic r i s k . The lack of empirical f i t can be a t t r i b u t e d t o the f a c t that e i t h e r the model i s co r r e c t and the d i f f i c u l t y i s one of measurement, or that the model i s i n -c o r r e c t and must be extended to include a d d i t i o n a l v a r i a b l e s . In the f i r s t case, 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 an ex-ante to an ex-post formulation or because of err o r s i n v a r i a b l e s . The transformation o f the ex-ante model to an ex-post formulation i s based upon the assumption that the returns on any s e c u r i t y can be represented by a market model; tha t i s , the re 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 of a market f a c t o r . Thus any t e s t of the ex-post formu-A J F i s h e r , L. "Determinants of Risk Premiums on Corporate Bonds," Journal of Political Economy, V o l . LXVII, No. 3 (June, 1959), pp. 217-237. 1 4 Some recent studies are Black, F., Jensen, M. C. and Scholes, M., "The C a p i t a l Asset P r i c i n g Model: Some Empirical Tests," published i n Studies in the Theory of Capital Markets, edited by Jensen, M. (New York: Praeger, 1972); and Blume, M. and Friend, I., "A New Look at the C a p i t a l Asset T r i c i n g Model," Journal of Finance, V o l . XXVII, No. 1 (March, 1973), pp. 19-34. 8 l a t i o n i s a j o i n t t e s t of the CAPM and market model. Errors i n v a r i a b l e s might a r i s e through measurement e r r o r s i n the estimation of the i n d i v i d u a l beta f a c t o r s or i f the market f a c t o r i s i n c o r r e c t l y s p e c i f i e d ; the market f a c t o r i s supposed to measure the return on a l l assets and not simply the return on the New York Stock Exchange. Another problem which might v i l i f y the r e s u l t s of any i n v e s t i g a t i o n i s that of the skewness of the d i s t r i b u -t i o n s of ex-post returns. In the second case, a number of studies have attempted to rela x the various assumptions underlying the model. Two v a r i a b l e models have been developed by B l a c k 1 ^ and Mer t o n . ^ The second v a r i a b l e i n the Black v e r s i o n , which has been termed the zero beta f a c t o r , a r i s e s from r e l a x i n g the assumption of investors being able to borrow and lend at the r i s k free rate of 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 the r e s u l t of r e l a x i n g the assump-t i o n of a constant investment opportunity set and r e f l e c t s investors' attempts to hedge against such changes. Neither model provides an adequate explanation of a l l the observed d e f i c i e n c i e s of the CAPM. There i s nothing i n the Black formulation to suggest t h a t the second f a c t o r i s non-stationary, and the Merton model does not explain why impending bankruptcy a f f e c t s the r e s i d u a l return of common stocks a f t e r a b s t r a c t i n g from the market. i : >Black, 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 Borrowing," Journal of Business, V o l . 45, No. 3 (July, 1972), pp. 444-455. 16 Merton, R. C. "A Dynamic General Eq u i l i b r i u m Model of the Asset Market and i t s A p p l i c a t i o n to the P r i c i n g of the C a p i t a l Structure of the Firm," Massachusetts I n s t i t u t e of Technology, Sloan School of Management, December, 1970. 9 The primary focus of the t h e s i s i s to extend the formulations of the CAPM not from the viewpoint of r e s t r i c t i o n s upon the investor, but by considering the impact of bankruptcy upon the s t r u c t u r e of returns. A two va r i a b l e model i s derived, the second v a r i a b l e being associated with the p r o b a b i l i t y of bankruptcy. An e s s e n t i a l step to e m p i r i c a l l y t e s t i n g such a model i s the development of an o p e r a t i o n a l measure of the p r o b a b i l i t y o f bankruptcy f o r a f i r m . Many of the studies d e a l i n g with the p r e d i c t i o n of bankruptcy 17 have concentrated upon the informational content of accounting numbers. The hypothesis being that there 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 accounting data, between f a i l e d and n o n - f a i l e d firms and that given these d i f f e r e n c e s e x i s t , models can be constructed, u s u a l l y employing m u l t i p l e discriminant a n a l y s i s , t o determine i f a f i r m should belong to a group of firms having the c h a r a c t e r i s t i c s of a f a i l e d f i r m or to a group of firms having the c h a r a c t e r i s t i c s of a n o n - f a i l e d f i r m . The main c o n s i d e r a t i o n has been t o c l a s s i f y a f i r m i n t o one of these two groups. No attempt has been made to construct a theory of the determinants of bankruptcy or to mea-sure the p r o b a b i l i t y of a f i r m going bankrupt. The approach proposed i n the the s i s i s an extension of the previous, studies i n at l e a s t two ways. I t i d e n t i f i e s a set of v a r i a b l e s that can be """'See, f o r example, Beaver, W. H. " F i n a n c i a l Ratios as Pr e d i c t o r s of F a i l u r e , " Empirical Research in Accounting: Selected Studies, supplement to Journal of Accounting Research (1966), pp. 77-111; Altman, E. I. " F i n a n c i a l Ratios, Discriminant Analysis and the P r e d i c t i o n of Corporate Bankruptcy," Journal of Finance, V o l . XXIII, No. 4 (September, 1968), pp. 589-609; and Edmister, R. 0. "An Empirical Test of F i n a n c i a l Ratio A n a l y s i s f o r Small Business F a i l u r e P r e d i c t i o n , " Journal of Financial and Quantitative Analy-sis, V o l . 7 (March, 1972), pp. 1477-1493. 10 used i n p r e d i c t i n g bankruptcy and i t introduces a new methodology to t h i s f i e l d with which to measure the p r o b a b i l i t y of bankruptcy, that of 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 . To e m p i r i c a l l y t e s t the hypothesis of the t h e s i s , as represented by the derived two v a r i a b l e model, necessitates consideration of the method-ology to be employed. The two v a r i a b l e model describes a r e l a t i o n s h i p f o r a s e c u r i t y at 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 expected r e t u r n , the se c u r i t y ' s systematic r i s k , and the. p r o b a b i l i t y of bankruptcy. The t r a d i -t i o n a l approach to the t e s t i n g of the CAPM i s to construct p o r t f o l i o s of s e c u r i t i e s so as to reduce measurement e r r o r s and to t e s t the model using a small number of p o r t f o l i o s . For e m p i r i c a l t e s t i n g of models which are exten-sions o f the CAPM, elaborate forms of the same type of aggregation procedure have been u t i l i z e d . However, these procedures make extensive use of e s t i -mated parameters and i t i s not c l e a r how the aggregated e f f e c t of measure-ment err o r s 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 . The t h e s i s introduces a new methodology to the t e s t i n g of two v a r i a b l e models, that o f po o l i n g time s e r i e s and cross s e c t i o n data. Organization of Thesis A c r i t i c a l survey of the rele 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 to the t h e s i s i s given i n Chapter I I . The survey, which covers the to p i c s of emp i r i c a l studies i n bankruptcy and the C a p i t a l Asset P r i c i n g Model, stresses the underlying assumptions of the reviewed l i t e r a t u r e , the areas of d e f i c i e n c y and the con t r i b u t i o n s of the t h e s i s . Chapter I I I describes a model f or the p r o b a b i l i t y of bankruptcy i n terms of ex-ante v a r i a b l e s . To use the model f or e m p i r i c a l estimation requires that the ex-ante v a r i a b l e s be replaced by ex-post surrogates. A 11 general formulation i n terms of explanatory v a r i a b l e s i s developed. As the primary focus i s upon p r e d i c t i o n of bankruptcy, a second formulation using market values of appropriate corporate v a r i a b l e s i s constructed. The s t a t i s -t i c a l methodology to estimate the c o e f f i c i e n t s of the proxy v a r i a b l e s i s described. The d e t a i l s of three d i f f e r e n t methods by which the model can be tested are given i n the l a s t p a r t of the chapter. In Chapter IV the t h e o r e t i c a n a l y s i s extending the C a p i t a l Asset P r i c i n g Model to incorporate bankruptcy i s given. The general framework d e t a i l i n g the f i n a n c i a l assets a v a i l a b l e , t h e i r p r i c e dynamics, the nature of changes i n the investment opportunity s e t , and the b e h a v i o r i a l assump-tio n s f o r the i n d i v i d u a l are described. The general form of the equation o f o p t i m a l i t y i s then derived. Due to the complexity of the general a n a l y s i s , two models are presented. The f i r s t model assumes that the investment opportunity set i s a l t e r e d only by the event of bankruptcy. From t h i s a n a l y s i s a two v a r i a b l e model, which w i l l be e m p i r i c a l l y tested, i s derived. The second model assumes that the p r o b a b i l i t y of bankruptcy changes stochas-t i c a l l y over time. Chapter V describes the e m p i r i c a l work of the t h e s i s . The f i r s t p a r t of the chapter describes the data, the r e s u l t s , and the t e s t i n g of the model to estimate the p r o b a b i l i t y of bankruptcy. The r e s u l t s of t h i s work are used i n the second part of the chapter, which describes the t e s t i n g of the hypothesis of the t h e s i s . The data, s t a t i s t i c a l methodology, and the r e s u l t s are presented. Chapter VI summarizes the main f i n d i n g s of the t h e s i s . A l i s t of topics f o r f u r t h e r research that a r i s e from the t h e s i s i s given. CHAPTER II A CRITICAL SURVEY OF THE RELEVANT FINANCIAL LITERATURE In t h i s chapter a review of the f i n a n c i a l l i t e r a t u r e that i s r e l e -vant to the t h e s i s i s given. The r e l a t i o n s h i p of the reviewed l i t e r a t u r e to the t h e s i s , i t s importance, the underlying assumptions and t h e i r i m p l i -cations are described. I t i s demonstrated that there are areas of d e f i c i e n c y i n the l i t e r a t u r e and the contributions of the t h e s i s i n e r a d i c a t i n g these d e f i c i e n c i e s stated. Two t o p i c s are discussed, those of e m p i r i c a l studies on bankruptcy and the C a p i t a l Asset P r i c i n g Model (CAPM). The empirical studies on bankruptcy that are reviewed address three questions: are there costs to bankruptcy; does impending bankruptcy a f f e c t the p r i c e behaviour of corporate f i n a n c i a l assets; and can bankruptcy be predicted. The questions of the existence of costs to bankruptcy and the e f f e c t s of bankruptcy upon the structure of returns f o r corporate f i n a n c i a l assets are of prime importance to the formulation of the hypothesis of the t h e s i s . The l a s t question, that of p r e d i c t i o n of bankruptcy, i s relevant to the t e s t i n g of the hypothesis of the t h e s i s , where i t i s necessary to measure the p r o b a b i l i t y of a fir m going bankrupt. The second t o p i c reviewed i s the CAPM. The ba s i c model and the empirical 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 described. The implications of the m i s s p e c i f i c a t i o n and some of the t h e o r e t i c a l attempts to explain i t s causes are reported. Incorporating some of the findi n g s of the empirical studies on bankruptcy and the nature of the m i s s p e c i f i c a t i o n of the CAPM, the hypothesis of the t h e s i s i s presented. To t e s t e m p i r i c a l l y 12 13 the hypothesis a two v a r i a b l e model i s used, the second v a r i a b l e being the p r o b a b i l i t y of a fir m going bankrupt. Before the hypothesis can be tested, i t i s necessary to determine the s t a t i s t i c a l methodology to use. E x i s t i n g methods and t h e i r d e f i c i e n c i e s are described and an a l t e r n a t i v e methodology advanced, that of pooling time s e r i e s and cross s e c t i o n data. Empirical Studies on Bankruptcy The e m p i r i c a l studies on bankruptcy which are reviewed p e r t a i n to three areas: the existence of costs to bankruptcy, the e f f e c t s of impending bankruptcy upon the p r i c e behaviour o f corporate f i n a n c i a l a ssets, and the p r e d i c t i o n of bankruptcy. The f i r s t two t o p i c s combined with some of the fin d i n g s from the e m p i r i c a l evidence on the m i s s p e c i f i c a t i o n of the CAPM contribute to the formulation of the hypothesis of the t h e s i s . The l a s t t o p i c , that of p r e d i c t i o n of bankruptcy, i s rel e v a n t to the e m p i r i c a l t e s t i n g of the hypothesis of the t h e s i s , where i t i s necessary to measure the proba-b i l i t y of a f i r m going bankrupt. Costs to Bankruptcy I f there were no costs to bankruptcy then i t would be a " t e c h n i c a l i t y " ; the bankrupt f i r m could be replaced by a new fir m and since no resources have been expended there are no economic l o s s e s . However, i f there are costs to bankruptcy then t h i s i s no longer true and i t implies that c a p i t a l s t r u c t u r e w i l l be relevant to the v a l u a t i o n of the f i r m . B a x t e r 1 has considered the existence of costs to corporate bankruptcy Baxter, N. D., "Leverage, Risk of Ruin, and the Cost of C a p i t a l , " Journal 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 costs. His t h e s i s i s that increased leverage enhances the p r o b a b i l i t y of a f i r m going bankrupt and that the added adminis-t r a t i o n and l e g a l costs incurred during r e o r g a n i z a t i o n , as w e l l as operating i n e f f i c i e n c i e s which are manifest due to the bankruptcy c o n d i t i o n , reduces the value of the f i r m . Baxter analyzed the immediate e f f e c t s of bankruptcy on the earnings and sales of a small sample of firms and found that they were adversely 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 of bankruptcy. This was i n t e r p r e t e d that there are costs to bankruptcy and that excessive leverage would reduce the value of the f i r m . Consideration of only the immediate e f f e c t s of bankruptcy can be p o t e n t i a l l y misleading f o r i t i s p o s s i b l e f o r a firm's operations e i t h e r during or a f t e r r e o r g a n i z a t i o n to improve and f o r the o r i g i n a l common stockholders to do exceedingly w e l l , i n c l u s i v e o f discounting f o r t h e i r opportunity c o s t s . 2 Altman, i n a more general a n a l y s i s , considered not only the imme-di a t e but also the long term e f f e c t s of bankruptcy. The f i n d i n g s of Baxter were confirmed, though some evidence was found that p r o h i b i t e d the general statement that i n v e s t o r s always s u f f e r because of bankruptcy. The e m p i r i c a l data showed that bankrupt firms' equity on average can be expected to f a l l i n bankruptcy, though a number of bankruptcy reorganizations r e s u l t e d i n favorable o v e r a l l performance. The evidence of the existence of the costs to bankruptcy implies that 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 of the f i r m . C l o s e l y r e -l a t e d to t h i s i s the question of how bankruptcy a f f e c t s the mechanism of the structure of returns on corporate f i n a n c i a l a s s e t s . 2 Altman, E. I., "Corporate Bankruptcy Potentxal, Stockholder Returns and Share Valuation," Journal of Finance, V o l . XXIV, No. 5 (December, 19G9), pp. 887-900. 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 of a firm going bankrupt the greater w i l l be the ex-ante expected rate of return that risk-averse investors r e -quire.. Each perio d investors w i l l reassess the condit i o n of the firm and adjust the market p r i c e of the common stock such that the ex-ante r a t e of return would continue to be commensurate with the higher r i s k . I f at any time the p r o b a b i l i t y of the f i r m going bankrupt i s greater than expected, there w i l l be a downward adjustment of the market p r i c e and the ex-post return w i l l be l e s s than the ex-ante expected rate of r e t u r n . I t i s not pos s i b l e to make any statement about the d i f f e r e n c e i n ex-post returns f o r healthy firms and f o r those firms that f a i l . The d i r e c t i o n and magnitude of any d i f f e r e n c e w i l l depend upon the s i z e and d i r e c t i o n of change i n the p r o b a b i l i t y of 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 the p r i c e behaviour of common stock p r i c e s . A crude market model was used. The r e s i d u a l between the ex-post r e t u r n and the comparable F i s h e r Link R e l a t i v e , which i s an average rate of return on a l l firms l i s t e d on the New York Stock Exchange, was c a l c u l a t e d on an annual b a s i s f o r a group of f a i l e d firms up to f i v e years p r i o r t o f a i l u r e and f o r a group of n o n - f a i l e d f i r m s . The r e -s u l t s were that the median rates of return f o r f a i l e d firms were poorer than those of no n - f a i l e d firms f o r f i v e years p r i o r to ac t u a l f a i l u r e and the d i f f e r e n c e between the median values increased as f a i l u r e approached. Beaver also derived t e s t s to assess the f a i l u r e p r e d i c t i v e power of rates of return measures and several f i n a n c i a l r a t i o s that 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 Ratios, and the P r e d i c t i o n of F a i l u r e , " Journal of Accounting Research, V o l . A (Autumn, 1968), pp. 179-192. 16 study. Univariate t e s t s showed that i n v e s t o r s f o r e c a s t f a i l u r e sooner than any of the r a t i o s used, with the average length of time from the year of the f a i l u r e p r e d i c t i o n to the date of f a i l u r e being 2.45 years f o r the rate of return measure. This was i n t e r p r e t e d by Beaver as r e c o g n i t i o n of the f a c t that the informational content of accounting numbers i s not the only source inv e s t o r s use to detect the p o s s i b i l i t y of f a i l u r e . 4 W e s t e r f i e l d , using monthly data and the market model developed by Sharpe^ examined the behaviour of r e s i d u a l returns, a f t e r a b s t r a c t i n g from the market, f o r f a i l e d firms up to s i x years p r i o r to f a i l u r e . The 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 estimated using data f o r the months 120 to 72 p r i o r to f a i l u r e . For the months 71 to 0 the r e s i d u a l of the r e a l i z e d return minus the ex-ante expected return were c a l -c u lated. Using two performance measures, i t was found that the market began to b i d down the market p r i c e f i v e years p r i o r to f a i l u r e , with a r a p i d d e t e r i o r a t i o n occurring i n the year subsequent to f a i l u r e . This i s i n con-t r a s t to the f i n d i n g s of Beaver and suggests that f o r e c a s t s of f a i l u r e more than one year p r i o r to when f a i l u r e occurs, based upon i n d i c e s of market performance, w i l l be e r r o r prone. W e s t e r f i e l d also examined the r e l a t i o n -ship between the systematic r i s k measure and the rate of f a i l u r e , the hypothesis being that firms whose common equity e x h i b i t high systematic r i s k with market movements (high betas) experience a higher rate of f a i l u r e than those assessed as low r i s k (low b e t a s ) . Given the 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 of Market Risk and Corporate F a i l u r e , " U n i v e r s i t y of Pennsylvania, Wharton School of Finance, August, 1970 (unpublished). 5Sharpe, 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 , " Manage-ment Science, No. 9 (January, 1963), pp. 277-293. 17 estimation technique and the small sample s i z e , the r e s u l t s showed that on average high r i s k firms experience f a i l u r e at a greater rate than low r i s k firms. A d e f i c i e n c y of 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 of Black, Jensen, and Scholes has shown that the r e a l i z e d return f o r high beta s e c u r i t i e s i s c o n s i s t e n t l y lower than that p r e d i c t e d by the market model used by W e s t e r f i e l d . This implies that the r e s i d u a l between the r e a l i z e d return and the estimated return w i l l be biased downwards. The problem assumes greater importance when account i s taken of Westerfield's f i n d i n g s that suggests that i t i s high beta s e c u r i t i e s that tend to f a i l more often than low beta s e c u r i t i e s . Thus the two performance measures used are biased 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 of these studies i s that the market constantly underestimated the p r o b a b i l i t y of a f i r m going bankrupt. From an ex-ante viewpoint the expected rate of return, c o n d i t i o n a l upon no bankruptcy, should increase i f the p r o b a b i l i t y of bankruptcy increases so to compensate r i s k averse in v e s t o r s f o r the increased r i s k , but t h i s does not imply that the ex-post r e t u r n should increase. I f during a p a r t i c u l a r p e r i o d the proba-b i l i t y of a f i r m going bankrupt unexpected increases, then t h i s w i l l be r e -f l e c t e d i n a lower p r i c e at the end of the p e r i o d than had been expected at the beginning of the p e r i o d and the r e a l i z e d return w i l l decrease. The 6Ibid. 7 Black, F., Jensen, M. C , and Scholes, M. , "The C a p i t a l Asset P r i c -ing Model:.Some Empirical T e s t s , " published i n Studies in the Theory of Capital Markets, edited by Jensen, M. (New York: Praeger, 1 9 7 2 ) . i 18 longer the time period the greater the p o t e n t i a l seriousness t h i s problem becomes for empirical studies. I t does, however, s t r e s s the need to consider the r a m i f i c a t i o n s of s t o c h a s t i c changes i n the p r o b a b i l i t y of bankruptcy i n any th e o r e t i c i n v e s t i g a t i o n s of the e f f e c t s of bankruptcy upon the stru c t u r e of returns f o r corporate f i n a n c i a l assets. Both studies o f f e r e m p i r i c a l evidence which suggests that the p r i c e behaviour of common stocks i s a f f e c t e d by impending bankruptcy. I f t h i s i s co r r e c t , then i t i s p e r t i n e n t to enquire what i s the nexus between the market p r i c i n g process and corporate f a i l u r e . The p u r s u i t of t h i s question i s the primary focus of the t h e s i s . P r e d i c t i o n of Bankruptcy To t e s t e m p i r i c a l l y the hypothesis of the t h e s i s a two v a r i a b l e model, which i s an extended form of the CAPM, i s used. The second v a r i a b l e i s the p r o b a b i l i t y of a fir m going bankrupt and thus i t i s necessary to be able to estimate t h i s quantity. The primary focus of previous studies on the p r e d i c t i o n of bankruptcy has been on the informational content of accounting statements and f i n a n c i a l r a t i o s . These studies have advanced the hypothesis that there i s a d i f -ference i n p r o f i l e , as measured by accounting data, between f a i l e d and non-f a i l e d firms and t h i s d i f f e r e n c e can be u t i l i z e d as an a i d to p r e d i c t i o n ; that i s , i t i s poss i b l e using accounting data to a l l o c a t e firms to one of two groups: f a i l e d and n o n - f a i l e d . This hypothesis was f i r s t used i n a univ a r i a t e form. Single f i n a n c i a l r a t i o s were tested for t h e i r p r e d i c t i v e a b i l i t y . However, un i v a r i a t e a n a l y s i s can be p o t e n t i a l l y misleading 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 to the p r e d i c t i o n of bankruptcy has been developed. Beaver used a u n i v a r i a t e approach to s e l e c t from a sample of t h i r t y f i n a n c i a l r a t i o s the one most able to c o r r e c t l y p r e d i c t the f a i l u r e status of a f i r m . F a i l u r e was defined to occur when a fi r m was unable to pay i t s f i n a n c i a l o b l i g a t i o n when they matured. Opera t i o n a l l y , a f i r m was i d e n t i f i e d as f a i l e d when one of the following events occurred: bankruptcy, bond d e f a u l t , an overdrawn bank account, or non-payment of a p r e f e r r e d stock g dividend. Such a d e f i n i t i o n i s very broad and i s more i n keeping with the concept of 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 those categories under one group, that of 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 estimation, as not a l l the information i s being u t i l i z e d . Presumably, f o r a f i r m to d e f a u l t on i t s bond payments i s a more serious event than f o r a f i r m to omit payment on a p r e f e r r e d dividend. By c l a s s i f y i n g these two events under the one group does not take t h i s f a c t i n t o account. Another problem i s that many firms omit dividends f o r reasons other than that caused by impending f a i l u r e and thus to c l a s s i f y these firms 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 that the r a t i o of cash flow to t o t a l debt was best at being able to c o r r e c t l y p r e d i c t the f a i l u r e status of firms and that t h i s a b i l i t y e x i s t e d f o r at l e a s t f i v e years before f a i l u r e . Thus on the bas i s of a s i n g l e f i n a n c i a l r a t i o firms were a l l o c a t e d to one of two groups: f a i l e d and n o n - f a i l e d . Univariate a n a l y s i s i s s u s c e p t i b l e to f a u l t y i n t e r p r e t a t i o n and i s p o t e n t i a l l y misleading. For instance, a fi r m whose c a p i t a l s t r u c t u r e Beaver, W. H., " F i n a n c i a l Ratios as P r e d i c t o r s of F a i l u r e , " Empirical Research in Accounting: Selected Studies, supplement to Journal of Accounting Research (1966), pp. 77-111. 9 Ibid., p. 71. 20 contains a large proportion of debt may be regarded as a p o t e n t i a l bankrupt. However, because of very low v a r i a b i l i t y i n i t s cash flow, the s i t u a t i o n may not be considered serious. In an attempt to avoid t h i s type of problem a mu l t i v a r i a t e approach considering s e v e r a l f i n a n c i a l r a t i o s has been d e v e l -oped. This involves being able to determine which f i n a n c i a l r a t i o s are important i n detecting future bankruptcy, what weights should be attached to the selected r a t i o s and how should the weights be o b j e c t i v e l y e s t a b l i s h e d . The methodology that i s u s u a l l y used i s that of m u l t i p l e d i s c r i m i n a n t a n a l y s i s . M u l t i p l e discriminant a n a l y s i s (MDA) i s a s t a t i s t i c a l technique 10 used to c l a s s i f y an observation to one of two mutually e x c l u s i v e groups. The b a s i s of the technique i s to construct a di s c r i m i n a n t f u n c t i o n from a l i n e a r combination of explanatory v a r i a b l e s , the weights being determined by minimizing the expected cost of m i s c l a s s i f i c a t i o n . To use MDA i t i s necessary to define the two mutually e x c l u s i v e groups. For p r e d i c t i o n of bankruptcy the two groups are defined to be bank-rupt and non-bankrupt. Data are c o l l e c t e d f o r the firms i n the two groups; MDA then attempts to derive a l i n e a r combination of the 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 the two groups so as to minimize the cost of m i s c l a s s i f y i n g a f i r m . One o f the advantages of MDA i s that i t i s capable of considering an e n t i r e p r o f i l e of c h a r a c t e r i s t i c s common to the relevant firms, as we l l as the i n t e r a c t i o n of these p r o p e r t i e s , and to combine them i n a s i n g l e ^MDA can be extended to the general case of many mutually exclusive groups. A good i n t r o d u c t i o n to MDA i s given i n Anderson, T. W., An Introduc-tion to Multivariate Statistical Analysis (New York: John Wiley & Sons, 1957). 21 discriminant fun c t i o n . There are, however, a number of p o t e n t i a l drawbacks to MDA. I t i s necessary to make some s p e c i f i c a t i o n about the c o n d i t i o n a l expected costs of m i s c l a s s i f i c a t i o n and the parameters and 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 that describe the p r o p e r t i e s of the d i f f e r e n t groups. In general, neither the costs of m i s c l a s s i f i c a t i o n nor the p r i o r p r o b a b i l i t i e s are known. Without knowledge of the p r i o r d i s t r i b u t i o n s i t i s not p o s s i b l e to c a l c u l a t e the p r o b a b i l i t y of an observation belonging to a p a r t i c u l a r group. The primary focus of MDA i s to a l l o c a t e an observation to a p a r t i c u l a r group without measuring the p r o b a b i l i t y of the observation belonging to that group. A l t m a n 1 1 developed a m u l t i v a r i a t e approach to the p r e d i c t i o n of bank-ruptcy using MDA. A f i r m was c l a s s i f i e d as bankrupt i f i t f i l e d a bank-ruptcy p e t i t i o n under Chapter X of the National Bankruptcy Act (U.S.A.). This d e f i n i t i o n avoids the problem of m i s c l a s s i f i c a t i o n that was encountered 1 2 i n the Beaver study. From an i n i t i a l set of twenty-five f i n a n c i a l r a t i o s a discriminant function composed of f i v e f i n a n c i a l r a t i o s c h a r a c t e r i z i n g l i q u i d i t y , p r o f i t a b i l i t y , p r o d u c t i v i t y , f i n a n c i a l r i s k , and s a l e s generating a b i l i t y was determined. The d i s c r i m i n a n t function was able to c o r r e c t l y c l a s s i f y 94 per cent of the i n i t i a l data sample and to achieve a high stan-dard of success on two secondary samples. Thus on the b a s i s of a l i n e a r combination of f i v e f i n a n c i a l r a t i o s firms were a l l o t t e d as e i t h e r bankrupt or non-bankrupt. Altman, E. I., " F i n a n c i a l Ratios, Discriminant A n a l y s i s and the P r e d i c t i o n ' o f Corporate Bankruptcy," Journal of Finance, V o l . XXIII, No. 4 (September, 1968), pp. 589-609. Beaver, " F i n a n c i a l Ratios," loc. cit. 22 13 14 Using a d i f f e r e n t data sample, Deakin r e p l i c a t e d the Beaver and the Alt-man1"* studies obtaining s i m i l a r conclusions, but then proceeded to claim that the p r o b a b i l i t y of group membership could be derived using a s t a t i s t i c having a chi-square d i s t r i b u t i o n , the number of degrees of freedom 16 equaling the number of v a r i a b l e s used i n the MDA. This i s wrong. The p r o b a b i l i t y of assigning an observation to a p a r t i c u l a r group can only be c a l c u l a t e d i n t h i s context with the knowledge of the p r i o r d i s t r i b u t i o n s , 17 but these are unknown. The s t a t i s t i c used by Deakin i s probably a t e s t 18 f o r the n u l l hypothesis that the two group means are i d e n t i c a l . 19 Accounting data v a r i a b l e s have been used by F i s h e r i n an e m p i r i c a l study of the determinants of the r i s k premiums on corporate bonds. I t was hypothesized that the r i s k premium depended upon the m a r k e t a b i l i t y and the r i s k of d e f a u l t of the bond. The m a r k e t a b i l i t y of the bond was estimated by a s i n g l e v a r i a b l e , the market value of a l l the firm's p u b l i c l y traded bonds, and the r i s k of d e f a u l t was assumed to depend upon three v a r i a b l e s : the coef-f i c i e n t of v a r i a t i o n of the firm's net income, the p e r i o d of solvency, and 13 Deakin, E. B., "A Discriminant A n a l y s i s of P r e d i c t o r s of Business F a i l u r e , " Journal of Accounting Research, V o l . 10, No. 1 (Spring, 1972), pp. 167-179. ^ B e a v e r , " F i n a n c i a l Ratios," loc. cit. ^Altman, " F i n a n c i a l Ratios," loc. cit. 16 Deakin, op. cit., p. 175. 17 As Deakin f a i l s to give c l e a r d e f i n i t i o n s of the terms used i n the s t a t i s t i c , i t i s d i f f i c u l t to make any p o s i t i v e statement. p. 56 Journal of P o l i t i c a l Economy, V o l . L X V l l , No. 3 (June, 1959), pp. 217-237 18 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 i s given i n Anderson, op. ext., 1 9 F i s h e r , L., "Determinants of Risk Premiums on Corporate Bonds," 23 the r a t i o of the market value of equity to the par value of the firm's debt. From t h i s formulation approximately 70 per cent of the variance of the de-pendent v a r i a b l e could be explained. There are, however, a number of d e f i -c i e n c i e s i n the study. I t i s not c l e a r that the m a r k e t a b i l i t y of a bond can be estimated by a s i n g l e v a r i a b l e , or that the r i s k of d e f a u l t can be determined by a fu n c t i o n o f three 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 of the v a l i d i t y of these measures, the i n t e r p r e t a t i o n of the f i n d i n g s of the study are jeopardized. Whilst t h i s study breaks away from the s t r i c t use of f i n a n c i a l r a t i o s , i t s t i l l r e l i e s upon the informational content of accounting numbers, a c r i t i c i s m that can be ap p l i e d to a l l the studies p e r t a i n i n g to the p r e d i c t i o n of f a i l u r e . Accounting data r e f l e c t s the consequences of 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 that i s relevant. The focus of previous studies has been to a l l o c a t e firms, on the b a s i s of f i n a n c i a l r a t i o s , to one of two groups: f a i l e d or n o n - f a i l e d . No attempt has been made to o f f e r e i t h e r a theory of the determinants of f a i l u r e or to measure the p r o b a b i l i t y of a f i r m f a i l i n g . A c o n t r i b u t i o n of the t h e s i s i s the development of a model to deter-mine the p r o b a b i l i t y of a fir m going bankrupt. At any poin t i n time t h i s depends upon the firm'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 or exter-n a l l y , to cover f i x e d charges. To use the model to e m p i r i c a l l y estimate the p r o b a b i l i t y of bankruptcy requires that the ex-ante v a r i a b l e s be r e -placed by ex-post surrogates. This necessitates consideration of the s t a t i s t i c a l methodology to u t i l i z e i n estimating the c o e f f i c i e n t s of the ex-post surrogates. M u l t i p l e discriminant a n a l y s i s can not be used, for i t 24 i s p r i m a r i l y designed to a l l o c a t e an observation to a p a r t i c u l a r group with-out measuring the p r o b a b i l i t y of the observation belonging to that group. The t h e s i s introduces a new methodology to estimate the c o e f f i c i e n t s and the p r o b a b i l i t y of a firm going bankrupt, that of 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 . To t e s t e m p i r i c a l l y the hypothesis of the t h e s i s , i t i s only neces-sary to be able to measure the p r o b a b i l i t y of bankruptcy and not to explain i t s determinants; that i s , the primary focus i s upon p r e d i c t i n g the proba-b i l i t y of bankruptcy and not to advance a complete theory of i t s determi-nants. Consequently, two formulations of the model are derived. The f i r s t concentrates on being able to explain the determinants of bankruptcy, w h i l s t the second i s developed s o l e l y f o r i t s p r e d i c t i v e a b i l i t y using market values f o r appropriate corporate v a r i a b l e s . Both formulations of the model w i l l be used i n the t e s t i n g of the hypothesis. C a p i t a l Asset P r i c i n g Model The second t o p i c reviewed i s the C a p i t a l Asset P r i c i n g Model (CAPM). The t h e o r e t i c foundations of the Model and the empirical evidence which i n d i -cates that i t i s m i s s p e c i f i e d are described. Some of the various attempts to explain the cause(s) of the m i s s p e c i f i c a t i o n are discussed and an a l t e r -native explanation, which forms the hypothesis of 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 hypothesis a two v a r i a b l e model i s used, the second v a r i a b l e being the p r o b a b i l i t y of a firm going bankrupt. Before the hypothesis can be tested, i t i s necessary to determine the methodology to u t i l i z e . E x i s t i n g techniques and t h e i r d e f i c i e n c i e s are described and an a l t e r n a t i v e methodology presented. 25 Foundations of the CAPM The CAPM describes a l i n e a r r e l a t i o n s h i p between the e q u i l i b r i u m expected return on an asset and i t s systematic r i s k , which i s a measure of the asset's covariance with the market p o r t f o l i o . The market p o r t f o l i o i s composed of an investment i n every r i s k y asset outstanding i n proportion to i t s t o t a l value. The CAPM was o r i g i n a l l y formulated i n a mean-variance con-20 21 22 23 text by Treynor, Sharpe and l a t e r c l a r i f i e d by Lint n e r and Mossin. In the development of the model i t i s assumed tha t : (a) a l l i n v e s t o r s are sin g l e period expected u t i l i t y of terminal wealth maximizers 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 the ba s i s of mean and variance; (b) a l l investors can borrow or lend an unli m i t e d amount at an exogenously given r i s k free r a t e of i n t e r e s t and there are no r e s t r i c t i o n s on short sales of any assets; (c) a l l i n v e s t o r s have i d e n t i c a l subjective estimates of the means, variances, and covariances of return among a l l assets; (d) a l l assets are p e r f e c t l y d i v i s i b l e and there are no t r a n s a c t i o n costs; (e) there are no taxes; (f) a l l investors are p r i c e takers; and (g) the q u a n t i t i e s of assets are given. The model may be stated i n the mathematical form E (R ) = B [E ( R ^ ] , [2.1] 20 Treynor, J., "Towards a Theory of Market Value of Risky Assets" (unpublished memorandum, 1961). 21 Sharpe, W. F., " C a p i t a l Asset P r i c e s : A Theory of Market E q u i l i b -rium Under Conditions of Risk," Journal of Finance, V o l . XIX, No. 3 (Sep-tember, 1964), pp. 425-442. 22 L i n t n e r , J., "The Valuation of Risk Assets and the S e l e c t i o n of Risky Investments i n Stock 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 (February, 1965), pp. 13-37. 23 Mossin, J . , "Equilibrium i n a C a p i t a l Asset Market, Econometrzca, V o l . 34, No. 4 (October, 1966), pp. 760-703. 26 where, i f P.. (t) denotes the p r i c e of the j*"* 1 asset at the end of the per i o d , E [ P . ( t ) ] - P. ( t - l ) E ( RJ) = Y ( t - l ) 3 r F = expected excess return on the j * ^ 1 asset; r ^ = the r i s k l e s s r a t e of i n t e r e s t ; E = expected excess 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 of an investment i n every asset outstanding i n proportion to i t s value; and B_. = cov (R_. , R^) / var (R^) th = the 'systematic' r i s k of the j asset. The above r e l a t i o n states that the expected excess r e t u r n on any asset i s d i r e c t l y p r o p o r t i o n a l to i t s systematic r i s k . I f a_. i s defined as o = E (R ) - B E (R^), [2.2] then equation [2.1] implies that a f o r every asset i s zero. Empirical t e s t s of the CAPM have been based upon ex-post data. The transformation of the ex-ante model to an ex-post formulation i s based upon the assumption that the return on any s e c u r i t y can be represented by a market model; that i s , the return 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 of a market f a c t o r . Thus any t e s t of the ex-post formulation i s a j o i n t t e s t - o f the CAPM and the market model. Using the market model o r i g i n a l l y proposed by Markowitz^ and extended by S h a r p e ^ and Fama^ the ex-post formulation 2 4Markowitz, H., Portfolio Selection: Efficient Diversification of Investments, Cowles Foundation Monograph No. 16 (New York: John Wiley and Sons, 1959). 25 Sharpe, "A S i m p l i f i e d Model," loc. cit o r Fama, E., "Risk, Return 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 . XXIII, No. 4 (March, 1960), pp. 29-40. 27 of the CAPM, as represented by Equation [2.1] can be w r i t t e n R . = 3 . R + e . [2.3] D : M ] where e. i s a normally d i s t r i b u t e d zero mean random v a r i a b l e . I f assets 3 are p r i c e d according to the CAPM then a j o i n t t e s t of the CAPM and the market model can be obtained by adding an i n t e r c e p t a_. to [2.3] and sub-s c r i p t i n g each of the v a r i a b l e s by t , representing time, to obtain V = a j + 3 j V + e j t ' [ 2 - 4 ] which i s a regression equation, the n u l l hypothesis being that the i n t e r c e p t term {a.}, i s zero f o r a l l assets. 3 Cross Sectional Tests of the Model For cross s e c t i o n a l t e s t s the procedure used i s to estimate the cross s e c t i o n a l regression R\ = v„ + Vi 3^ + e_., [2.5] where 8j i s obtained from the regression of a time s e r i e s of i n d i v i d u a l s e c u r i t y returns on an index used as a proxy f o r the market p o r t f o l i o . The n u l l hypothesis i s that v Q = 0, and = (R M - r p ) , where R^ i s the average return on the market index over the time p e r i o d , and r i s u s u a l l y taken to be the y i e l d to maturity of a government bond with the same maturity as the length of the time period under examination. 27 Evidence presented by Douglas, who regressed the returns on a large cross s e c t i o n a l sample of common stocks on t h e i r own variance and on t h e i r covariance with an index constructed from the sample, found that the model 27 Douglas, G., "Risk i n the Equity Markets: An Empirical A p p r a i s a l of Market E f f i c i e n c y , " Yale Economic Essays, V o l . 9 (Spring, 1969), pp. 3-45. 28 d i d not provide a complete d e s c r i p t i o n of s e c u r i t y r eturns. For seven separate f i v e year periods from 1926 to 1960, the average r e a l i z e d return was 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 to the variance of the s e c u r i t y ' s returns over time, but not to t h e i r covariance with the index of returns. These r e s u l t s appear to be i n c o n f l i c t with the r e l a t i o n given by [2.1] f o r the variance term should have a c o e f f i c i e n t of zero. Douglas a l s o summarizes some unpublished r e s u l t s of L i n t n e r ' s that a l s o appear to be i n c o n s i s t e n t with Eouation [2.1]. L i n t n e r estimates Equation [2.5], f o r a cross s e c t i o n of s e c u r i t i e s , adding an extra v a r i a b l e : the variance of the r e s i d u a l s from the time s e r i e s regressions given by Equation [2.4]. This extra variance should have no explanatory power and thus i t s c o e f f i c i e n t should be zero. In L i n t n e r ' s t e s t s i t d i d not, the co-e f f i c i e n t on the r e s i d u a l variance being p o s i t i v e and s i g n i f i c a n t . A lso, v 0 was greater than zero and much l e s s than (R^ - r ). 28 M i l l e r and Scholes r e p l i c a t e d the L i n t n e r study on a d i f f e r e n t body of data obtaining the same general r e s u l t s . The source of the misspeci-f i c a t i o n may a r i s e not because the model i s wrong, but due to the d i f f i c u l t y of measuring the d i f f e r e n t v a r i a b l e s . For example, the biases introduced by (a) f a i l u r e to account adequately f o r the r i s k l e s s r a t e of i n t e r e s t , (b) po 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 , or (c) d i s t o r t i o n s due to h e t e r o s c e d a s t i c i t y , may be the cause of the Douglas-Lintner f i n d i n g . However, i t was shown that these e r r o r s could not produce such r e s u l t s . I t was also demonstrated that w h i l s t measurement e r r o r s i n the r i s k v a r i a b l e 28 M i l l e r , M. and Scholes, M., "Rates of Return i n R e l a t i o n to Risk: A Re-examination of Some Recent Findings," p r i n t e d i n Studies in the Theory of Capital Markets3 edited by Jensen, M. (Mew York: Praeger, 1972) . 29 {8j}, and the c o r r e l a t i o n between the variance of the r e s i d u a l s from the time series regressions and the estimates of the r i s k value (8^}, could con-t r i b u t e s u b s t a n t i a l l y to the Douglas-Lintner r e s u l t s , they were not suf-f i c i e n t to account f o r a l l the observed d e v i a t i o n s from the model. A problem which could account f o r such r e s u l t s i s the presence of 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 of ex-post returns. M i l l e r and Scholes were able to show that skewness e f f e c t s could cause serious d i f f i c u l t i e s and that combined with the measurement e r r o r s i n the r i s k v a r i a b l e could i n p r i n c i p l e cause the Douglas-Lintner r e s u l t s . Whilst t h e i r a n a l y s i s does not imply the complete r e j e c t i o n of the Douglas-Lintner r e s u l t s , i t does show that they must be treated with caution i n view of the econometric d i f -f i c u l t i e s i n t e s t i n g the model. Time Series Tests o f the Model 29 Black, Jensen, and Scholes (B-J-S) have teste d the CAPM by using a time s e r i e s procedure. The model can be tested by running a time s e r i e s regression using the equation [2.4]; t h a t i s , Rj t - °j + e j V + e j t ' [ 2 - 4 1 the n u l l hypothesis being that the i n t e r c e p t term, a., i s zero f o r a l l assets. Thus a d i r e c t t e s t can be obtained by estimating [2.4] f o r a s e c u r i t y over some time period and t e s t i n g to 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 zero. Whilst t h i s t e s t i s simple, i t i s i n e f f i c i e n t i n that i t u t i l i z e s information on only a s i n g l e s e c u r i t y . To overcome t h i s problem B-J-S 29 Black, F., Jensen, M., and Scholes, M., "The C a p i t a l Asset P r i c i n g Model: Some Empirical Tests," p r i n t e d i n Studies in the Theory of Capital Markets, edited by Jensen, M. (New York: Praeger, 1972) . 30 perform t h e i r t e s t s on p o r t f o l i o returns over the period 1931 to 1965, where the p o r t f o l i o s are constructed so as to maximize the d i s p e r s i o n of t h e i r systematic r i s k . They ap p l i e d t h e i r t e s t s to ten p o r t f o l i o s , which con-tained a l l s e c u r i t i e s on the New York Stock Exchange. The r e s u l t s i n d i c a t e d that the i n t e r c e p t term was d i r e c t l y r e l a t e d to the systematic r i s k l e v e l . For low r i s k s e c u r i t i e s the i n t e r c e p t term was p o s i t i v e , and f o r high r i s k s e c u r i t i e s i t was negative. There was s u b s t a n t i a l i n d i c a t i o n 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 non-stationary, 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 of systematic r i s k was d i f f e r e n t from u n i t y . S i m i l a r l y f i n d i n g s have a l s o been obtained i n a recent study by Blume and 3 0 F r i e n d . B-J-S go on to demonstrate that the process generating the return on i n d i v i d u a l s e c u r i t i e s can be described by a two v a r i a b l e model of the f i r m r j t " ( 1 " V r Z t + B j rMt + G j t » t 2 ' 6 ] <\» where r represents the r e t u r n on what they c a l l the "beta f a c t o r ' and the Zt other lower case r's i n d i c a t e t o t a l r e t u r n s . Using a grouping procedure which eliminates most of the d i f f i c u l t i e s associated with the biases i n t r o -duced by measurement err o r s i n the £3_^ 3" i n cross s e c t i o n a l t e s t s , an exami-nation of the cross 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 return f o r seventeen subperiods of lengths twenty-four months and four subperiods of length one hundred and f i v e months was conducted. I t appears that the r e l a -t i o n s h i p i s h i g h l y l i n e a r , but both the i n t e r c e p t s and slopes f l u c t u a t e randomly from period to p e r i o d and are often negative. B-J-S argue these 30 • • Blume, M. and F r i e n d , I., "A New Look at the C a p i t a l Asset P r i c i n g Model," Journal of Finance, V o l . XXVIII, No. 1 (March, 1973), pp. 19-34. 3 1 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 with the r e t u r n generating mechanism described by [ 2 . 6 ] , which implies that the i n t e r c e p t and slope i n the cross s e c t i o n a l regressions w i l l be and ( r ^ - r^) r e s p e c t i v e l y , where the bars denote sample means over the time p e r i o d covered by the cross s e c t i o n s . Since r w i l l a lso be a random v a r i a b l e , equation [ 2 . 6 ] i s c o n s i s t e n t with the observed e m p i r i c a l r e s u l t s . The evidence seems to i n d i c a t e that the CAPM does not provide an adequate d e s c r i p t i o n of the process generating common stock r e t u r n s . The documentation of n o n - s t a t i o n a r i t y and the existence of at l e a s t another f a c t o r imply that the model must be extended to include a d d i t i o n a l v a r i a b l e s . T h e o r e t i c a l Extensions of the CAPM The majority of the assumptions underlying the model v i o l a t e to some degree the conditions observed i n p r a c t i c e . A number of recent s t u d i e s have attempted to r e l a x various assumptions so as to incorporate some of the complexities of c a p i t a l markets i n t o the model. The r e s u l t s have i n d i -cated that the b a s i c s t r u c t u r e of the model i s remarkably robust to v i o l a -t i o n s of these assumptions. The assumption that i n v e s t o r s are s i n g l e p e r i o d expected u t i l i t y o f terminal wealth maximizers i s very r e s t r i c t i v e and may not be an accurate 32 d e s c r i p t i o n o f 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 survey a r t i c l e d e s c r i b i n g even more evidence about the m i s s p e c i f i c a t i o n of the CAPM i s given i n Jensen, M., " C a p i t a l Markets: Theory and Evidence," Bell Journal of Economics and Management Science, V o l . 3, No. 2 (Autumn, 1 9 7 2 ) , pp. 3 5 7 - 3 9 8 . 32 Fama, E., "Multiperiod Consumption—Investment Decisions," Ameri-can Economic Review, V o l . L X , No. 1 (March, 1 9 7 0 ) , pp. 1 6 3 - 1 7 4 . 32 for the v a l i d i t y of such an assumption. Arguing that i n v e s t o r ' s problem i s more accurately stated as the maximization of the expected l i f e t i m e u t i l i t y of consumption and terminal wealth, i t i s demonstrated that the sin g l e period CAPM can be j u s t i f i e d i n the context of a multipe r i o d problem i f the investor behaves as i f future consumption and investment o p p o r t u n i t i e s are given and that tastes are not state dependent. Thus even though the i n -vestor must solve a multip e r i o d problem to a r r i v e at the optimal current d e c i s i o n s , these d e c i s i o n s are i n d i s t i n g u i s h a b l e from those of a r i s k averse s i n g l e period expected u t i l i t y of terminal wealth maximizer. These find i n g s have important i m p l i c a t i o n s f o r the conditions which must be s a t i s f i e d i f the CAPM i s to be e m p i r i c a l l y tested. Previous empirical studies have u t i l i z e d ex-post data extending over many time periods and have t a c i t l y assumed the v a l i d i t y o f the model i n a m u l t i p e r i o d context. But tastes do change over time and thus one of the assumptions necessary to use the model over extended time periods i s v i o l a t e d and hence the conclusions of such studies are jeopardized. I t i s assumed i n the CAPM that a l l assets are p e r f e c t l y l i q u i d ; that i s , a l l assets are marketable and there are no t r a n s a c t i o n c o s t s . There are many assets f o r which t h i s assumption i s not a p p l i c a b l e . For example, claims on labour income or s o c i a l s e c u r i t y payments are claims that can not be sold i n c a p i t a l markets. Thus i t i s p e r t i n e n t to enquire what e f f e c t non-marketability of assets has upon the CAPM. For the s p e c i a l case of only two types of assets, 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 derives a simple expression between the expected rate of return on 33 Mayers, D., "Non-Marketable Assets and C a p i t a l Market E q u i l i b r i u m Under Uncertainty," p r i n t e d i n Studies in the Theory of Capital Markets, edited by Jensen, M. (New York: Praeger, 1972). 3 3 any asset and i t s covariance r i s k i n terms of market parameters and demon-st r a t e s that the b a s i c i m p l i c a t i o n s of the model are not weakened i n any major respect by the existence of non-marketable assets. There are, however, many assets that can not be described as being e i t h e r p e r f e c t l y l i q u i d or p e r f e c t l y n o n - l i q u i d , r e a l estate, second hand automobiles, being p o s s i b l e examples. M a r k e t a b i l i t y i s not a dichotomous concept but i s a continuum. To i n v e s t i g a t e the f u l l e f f e c t s of m a r k e t a b i l i t y upon the CAPM requires that i t s determinants be known and that the "degree" of m a r k e t a b i l i t y of an asset can be measured. Whilst the r e s u l t s of the Mayers' study are important, the study does not address i t s e l f to the more general and d i f f i c u l t problem of 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 of the assumptions of the CAPM i s the existence of a r i s k l e s s asset. In the presence of uncertainty about the l e v e l of future p r i c e s and as contracts are not denominated i n r e a l terms, the assumption of the e x i s -34 tence of such an asset i s tenuous. Black has shown, under assumptions i d e n t i c a l to those of the CAPM, that i f a r i s k l e s s asset, or borrowing or lending opportunities do not e x i s t , then i n e q u i l i b r i u m the p o r t f o l i o s of a l l investors c o n s i s t of a l i n e a r combination of two b a s i c p o r t f o l i o s , one being the market p o r t f o l i o and the other a p o r t f o l i o whose returns have zero covariance with the market p o r t f o l i o and has minimum variance. This p o r t f o l i o has been termed the "zero beta" p o r t f o l i o . Black demonstrates that i n e q u i l i b r i u m the expected return on any asset w i l l be given by E (r.) = (1 - 3.) F. (r_) + 3. E ( r ) , 3 3 Z ] M 34 Black, 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 Borrowing," Journal of Business, V o l . 45, No. 3 (July, 1972), pp. 445-455. 34 where E (r ) i s the expected return on the zero beta p o r t f o l i o and the other v a r i a b l e s are as pre 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 bears a close r e l a t i o n s h i p to the model suggested by B-J-S, there i s nothing i n the formulation to suggest that the second f a c t o r i s not constant, and thus i t can not adequately e x p l a i n a l l the observed empirical d e f i c i e n c i e s of the CAPM. The CAPM, which i s formulated i n a d i s c r e t e time framework, r e s t s upon the assumption that there are no t r a n s a c t i o n c o s t s . Whilst the formula-t i o n of a d i s c r e t e time model i s convenient f o r e m p i r i c a l work, i t s theo-r e t i c j u s t i f i c a t i o n i s questionable. The r a t i o n a l i n v e s t o r would always pr e f e r to have the option to trade any i n s t a n t of t i m e — a t no c o s t — t h e n to be r e s t r i c t e d to t r a d i n g at d i s c r e t e time i n t e r v a l s . The usual reason given f o r the d i s c r e t e time formulation i s that transaction costs do e x i s t . However, the approach i s to take equal time spacings of non-specified length. I f t r a n s a c t i o n costs are to be included, the i n t e r v a l between t r a d i n g periods w i l l become a v a r i a b l e depending upon the s i z e of t r a n s a c t i o n c o s t s , changes i n the p r i c e s of s e c u r i t i e s , i n i t i a l wealth d i s t r i b u t i o n , and expectations. The i n c l u s i o n of t r a n s a c t i o n costs also r a i s e s the question of the existence of an e q u i l i b r i u m . 36 Merton has derived a continuous time ve r s i o n of the CAPM, although the symbols, which appear the 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 , being expressed i n terms of instantaneous r a t e s of re t u r n . Four e x t r a assumptions 35 Black, Jensen, and Scholes, loc. cut. 36 Merton, R. C , "A Dynamic General E q u i l i b r i u m Model of the Asset Market and Its A p p l i c a t i o n to the P r i c i n g of the C a p i t a l Structure of the Firm," Massachusetts I n s t i t u t e of Technology, Sloan School of Management, December, 1970. 35 are required: trading takes place continuously, which follows d i r e c t l y from the assumption of no t r a n s a c t i o n costs; a constant investment opportunity set, that i s , the means, variances and covariances that describe the charac-t e r i s t i c s of the d i f f e r e n t assets are constant; only l o c a l changes i n the state variables of the process are allowed, which r u l e s out Pareto-Levy or Poisson type processes; and investors act so as to maximize the expected u t i l i t y of l i f e t i m e consumption and terminal wealth. Representing the p r i c e dynamics of s e c u r i t i e s by Weiner processes the continuous time analogue to CAPM i s derived: a. - r = B. (cc — r) . j j M ; ' t h where a. i s the instantaneous rate of return on the 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 rate of M return on the market p o r t f o l i o ; and B. = o" . w / a ^ ; and o. w i s the i n s t a n -j jM M jM taneous covariance of the return on the j ^ 1 asset with the r e t u r n on the market p o r t f o l i o . 37 If any of the three conditions o u t l i n e d by Fama to j u s t i f y the s i n g l e period maximization of expected u t i l i t y of terminal wealth assump-t i o n of the CAPM are v i o l a t e d , then the simple s t r u c t u r e of the model w i l l probably be destroyed. For example, i f the investment opportunity set changes, then t h i s w i l l , i n general, a f f e c t the consumption-investment 38 d e c i s i o n of the i n v e s t o r . Merton, i n the context of a continuous time framework, demonstrates that changes i n the investment opportunity set do 37 Fama, "Multiperiod Consumption," too. cit. 38 Merton, R. C , "An Intertemporal C a p i t a l Asset P r i c i n g Model," Working Paper 588-72, Massachusetts I n s t i t u t e of Technology, Sloan School of Management, February, 1972. 3 6 a f f e c t the structure of the model. Under the assumption that these changes can be characterized by changes i n a s i n g l e instrumental v a r i a b l e — t h e r i s k -l e s s i n t e r e s t r a t e — a three v a r i a b l e model i s derived. The t h i r d v a r i a b l e can be i n t e r p r e t e d as the r e s u l t of i n v e s t o r s hedging against the e f f e c t s of future unforeseen changes i n the r i s k l e s s i n t e r e s t r a t e . In e q u i l i b r i u m , the investor's p o r t f o l i o w i l l be a l i n e a r conbination of three p o r t f o l i o s : the r i s k l e s s asset, the market p o r t f o l i o , and a p o r t f o l i o (or asset) which i s p e r f e c t l y negatively c o r r e l a t e d with changes i n the r i s k l e s s i n t e r e s t r a t e . The model can be stated i n the form: °j = r F + X l ( aM " r ) + * 2 ( 0 tN " r ) ' . _ ^jM " ^jN N^M , A i - : i - p 2 NM 8 . - 8 . B . jN jM NM , *2 " 1 - P Z NM a. B. = - ^ , ^ °kK th th a i s the instantaneous covariance between the j and k assets; and p,,„ j k J KM i s the instantaneous c o r r e l a t i o n between the returns on the market p o r t f o l i o and the asset N, which i s negatively c o r r e l a t e d with changes i n the r i s k l e s s i n t e r e s t r a te. Merton argues that the sign of A2 w i l l be p o s i t i v e f o r low beta assets and negative f o r high beta assets. The sign f o r (a - r) i s 8C i d e n t i c a l to that of - r — , where C i s the aggregate consumption f u n c t i o n . or Macro-economic theory usually assumes that aggregate saving i s an i n c r e a s i n g 3C function of y i e l d s and therefore - r — < 0, which implies that (a - r) w i l l be or N 37 p o s i t i v e . Thus, Merton concludes that the model i s at l e a s t c onsistent with empirical evidence. Summary Accumulating e m p i r i c a l evidence i n d i c a t e s that the CAPM does not provide an adequate d e s c r i p t i o n of the mechanism generating common stock r e -39 turns. The time s e r i e s work of Black, Jensen, and Scholes, has shown that assets with high l e v e l s of systematic r i s k c o n s i s t e n t l y earn l e s s than t h a t predicted by the model, w h i l s t assets with low l e v e l s of systematic r i s k earn more than that p r e d i c t e d . Even though there appears to be a l i n e a r r e l a t i o n -ship between a s e c u r i t y ' s ex-post return and i t s systematic r i s k , the r e l a t i o n s h i p i s non-stationary; over various time periods ex-post r e t u r n and systematic r i s k have been i n v e r s e l y r e l a t e d . Black, Jensen and Scholes argue that the data i n d i c a t e that the expected return on a s e c u r i t y can be repre-sented by a l i n e a r two f a c t o r model. The second f a c t o r , which they c a l l the beta f a c t o r , i s not e x p l i c i t l y i d e n t i f i e d . 4 0 4 1 The works o f We s t e r f i e l d and Beaver have shown that p r i c e be-haviour of common stocks i s a f f e c t e d by impending bankruptcy. From an ex-ante viewpoint, the expected rate of return, c o n d i t i o n a l upon no bankruptcy, should increase i f there i s an increase i n the p r o b a b i l i t y of bankruptcy to compensate r i s k averse investors f o r the extra r i s k . Within the framework Black, Jensen, and Scholes, loo. oit. 40 W e s t e r f i e l d , R., "The Assessment of Market Risk and Corporate F a i l u r e , " U n i v e r s i t y of Pennsylvania, Wharton School of Finance, August 1970 (unpublished). 4 1 B e a v e r , "Market P r i c e s , " loo. oit. 38 of the CAPM t h i s implies that the s i n g l e explanatory v a r i a b l e , that of systematic r i s k , which i s a measure of the covariance of the firm's return with the returns of a l l other s e c u r i t i e s , should increase. But t h i s contra-d i c t s the assumption of a constant investment opportunity s e t . Thus the CAPM can not explain why p r i c e behaviour i s a f f e c t e d by impending bankruptcy. Various attempts have been made to explain these d e f i c i e n c i e s . The e f f e c t s of non-marketability of assets, the non-existence of a r i s k l e s s asset, and r e s t r i c t i o n s upon the inv e s t o r ' s a b i l i t y to borrow or lend have been explored, though these f a i l to provide an adequate explanation of a l l the observed d e f i c i e n c i e s . The e f f e c t s of changes i n the investment oppor-t u n i t y set have been shown to imply the existence of a three v a r i a b l e model which i s at 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 with the em p i r i c a l f i n d i n g s of 42 Black, Jensen, and Scholes, though i t does not provide an explanation of why impending bankruptcy a f f e c t s the r e s i d u a l return behaviour, a f t e r abstracting from the market, of common stocks. Hypothesis of the Thesis I f the p r o b a b i l i t y o f bankruptcy f o r a fir m increases, then the expected return, c o n d i t i o n a l upon no bankruptcy, which r i s k averse in v e s t o r s require w i l l increase to compensate f o r the extra r i s k . At any po i n t i n time the p r o b a b i l i t y of bankruptcy f o r a fir m i s a fun c t i o n of 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 or e x t e r n a l l y , to cover f i x e d charges. As conditions w i t h i n the f i r m and the economy change over time, so w i l l the firm's a b i l i t y to r a i s e funds, which implies that the p r o b a b i l i t y of the firm going bankrupt w i l l a l s o vary across time. This w i l l d i r e c t l y a f f e c t 42 Black, Jensen, and Scholes, loc. cit. 39 the expected rate of return which investors require on the firm's f i n a n c i a l assets. The t h e s i s gives a t h e o r e t i c explanation of the e f f e c t s of bank-ruptcy upon the structure of corporate f i n a n c i a l assets. The hypothesis of 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 u a l return a f t e r abstracting from the market. Empirical Testing of the Hypothesis 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 model des c r i b i n g the structure of common stock returns i s derived. The model i s an extension of the CAPM and i s of the form ex. = r p + A. + 3. ( a M - r p - x) , th where a_. i s the instantaneous c o n d i t i o n a l expected rate of return on the j asset; a w i s the instantaneous c o n d i t i o n a l expected rate of return on the M market p o r t f o l i o ; r ^ i s the instantaneous r i s k free rate of i n t e r e s t ; X_. i s th the rate of p r o b a b i l i t y of bankruptcy f o r the j asset; x I s a weighted averaqe of the {A.}; and 3. = o . / c , a. being the instantaneous condi-j j jM MM jM th t i o n a l covariance of the j asset with the market p o r t f o l i o . To t e s t em-p i r i c a l l y the hypothesis a d i s c r e t e time, ex-post formulation of the model i s used. However, before t e s t i n g the hypothesis two preliminary steps are necessary. F i r s t , the p r o b a b i l i t y of a fir m going bankrupt over a given period needs to be estimated; and second, a choice of methodology to employ when t e s t i n g the hypothesis must be made. The p r o b a b i l i t y of bankruptcy f o r a fir 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 or e x t e r n a l l y , to cover f i x e d charges. A model de s c r i b i n g the determinants of the firm's a b i l i t y to r a i s e funds i s constructed. The primary focus i s the p r e d i c t i o n and estimation of the p r o b a b i l i t y of bankruptcy, as opposed to constructing a f u l l explanatory theory. The c o e f f i c i e n t s of the model are determined using 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 methodology to t e s t a two v a r i a b l e model has been developed by 43 Black and Scholes. In an attempt to examine the e f f e c t s of dividends on common stock p r i c e s an ad hoc two v a r i a b l e extension o f the CAPM has been advanced. The model i s of the form 6. - 5 E (r.) = v 0 + V l (-^ -) + 6j [E ( r M ) - v c ] , th. where E (r.) i s the expected r e t u r n f o r the j asset; E (r ) i s the expected 3 M return on the market p o r t f o l i o ; 6^ i s the expected dividend y i e l d f o r the j*"* 1 asset; 5^ i s the expected dividend y i e l d on the market p o r t f o l i o ; $. = cov ( r . , r ) / var (r ) ; v 0 i s a constant to account f o r the existence 3 j M M of a beta f a c t o r ; and \>\ i s a constant. The hypothesis i s that the r e s i d u a l r eturn 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 the market f a c t o r , can be explained by the dividend y i e l d . I f the hypothesis i s true, then the coef-f i c i e n t Vi should be non-zero and 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 cross 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 the information on a l l s e c u r i t i e s , i s r u l e d out because of the econometric d i f f i c u l t i e s caused by e r r o r s i n v a r i a b l e s . Thus, a time s e r i e s approach i s used. The method-ology i s to construct a p o r t f o l i o such that i t s expected return i s v i and 43 Black, F. and Scholes, M., "Divided Y i e l d s and Common Stock Returns: A New Methodology," F i n a n c i a l Note No. 19B, Massachusetts I n s t i t u t e of Technology, Sloan School of Management, August, 1971. 41 the p o r t f o l i o to have minimum variance. To solve the equations representing the f i r s t order conditions requires knowledge of the beta c o e f f i c i e n t s , the expected dividend y i e l d s , and the variance-covariance matrix, a l l of which are unknown and must be estimated. To reduce measurement e r r o r s a method of aggregation i s used to form a small number of p o r t f o l i o s . S e c u r i t i e s are assigned to p o r t f o l i o s on the b a s i s of t h e i r estimated beta c o e f f i c i e n t and dividend y i e l d . These p o r t f o l i o s are then treated as s e c u r i t i e s and t h e i r beta c o e f f i c i e n t s , expected dividend y i e l d , and variance-covariance matrix estimated. The beta 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 determined by regressing i t s return on the market r e t u r n , a f t e r s u b t r a c t i n g the i n t e r e s t r a t e from both returns. Given these estimates, the f i n a l p o r t f o l i o i s constructed f o r d i f f e r e n t time peri o d s . There are a number of major d e f i c i e n c i e s with t h i s methodology. The estimation of the beta c o e f f i c i e n t s n e g l e c t i n g the dividend y i e l d , implies that there i s a missing v a r i a b l e s problem, which w i l l cause b i a s i n the estimated c o e f f i c i e n t s . This w i l l a f f e c t the estimation of the variance-covariance matrix, which uses the estimated beta c o e f f i c i e n t s . Without de-t a i l e d knowledge of how the measurement e r r o r s of the d i f f e r e n t v a r i a b l e s a f f e c t the f i n a l estimated c o e f f i c i e n t s , the a p p l i c a b i l i t y of the method-ology i s questionable. The t h e s i s introduces a new methodology to the t e s t i n g of two v a r i a b l e models, that of pooling time s e r i e s and cross s e c t i o n data. In the ex-post formulation of the model used to t e s t the hypothesis of the t h e s i s , the constant term and 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 of bank-ruptcy are not fi r m s p e c i f i c . The time s e r i e s data for a l l i n d i v i d u a l 42 s e c u r i t i e s are combined to estimate these two c o e f f i c i e n t s , w h i l s t simul-taneously estimating the.firm s p e c i f i c beta c o e f f i c i e n t s . CHAPTER III PROBABILITY OF BANKRUPTCY Bankruptcy i n a sing l e period context occurs i f , a t the terminal point, the income of the firm 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 . In a multi-period s e t t i n g such a d e f i n i t i o n i s not appropriate, f o r i n an on-going fi r m income can be l e s s than the o b l i g a t i o n s of the firm and yet the f i r m i s not bankrupt; i t can simply borrow more. At any point i n time, the p r o b a b i l i t y of a firm going bankrupt depends upon 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 or ex t e r n a l l y , to cover f i x e d charges. A fir m that f a i l s to cover these f i x e d charges i s said to be bankrupt. This d e f i n i t i o n of bankruptcy i s the basic construct i n formulating a model of the p r o b a b i l i t y of bankruptcy. The p r o b a b i l i t y of a fir m going bankrupt depends not only upon i t s current l e v e l of earnings but also i t s a b i l i t y to r a i s e funds, which i s subsumed i n i t s future earning power. However, such v a r i a b l e s are ex-ante i n nature and can not be d i r e c t l y observed. To use a model to e m p i r i c a l l y estimate the p r o b a b i l i t y of bankruptcy requires that the ex-ante v a r i a b l e s be replaced by ex-post surrogates. The construction of a model to estimate the p r o b a b i l i t y of bankruptcy i n terms of ex-post v a r i a b l e s i s described i n the f i r s t p a rt of the chapter. As the primary focus i s upon the p r e d i c t i o n of bankruptcy and not to advance a complete theory of i t s determinants, a second formulation u t i l i z i n g market values for appropriate corporate v a r i -ables i s developed. When the ex-ante determinants are replaced by ex-post surrogates, i t 43 44 i s necessary to estimate the r e l a t i v e c o n t r i b u t i o n of the d i f f e r e n t v a r i -ables. The s t a t i s t i c a l methodology to estimate the c o e f f i c i e n t s of the proxy v a r i a b l e s i s described i n the second p a r t of the chapter. As the p r o b a b i l i t y of bankruptcy can not be observed, d i r e c t t e s t s on the models are not p o s s i b l e . Thus, the main check on how w e l l the models are s p e c i f i e d must be on t h e i r p r e d i c t i v e a b i l i t y . The d e t a i l s of three d i f f e r e n t methods by which the models can be tested are given i n the l a s t p a r t of the chapter. Theory In 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 of bankruptcy presents no problem; at the terminal p o i n t i f the income of 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 state of bankruptcy i s declared. At the beginning of the p e r i o d to estimate the p r o b a b i l i t y of the f i r m going bankrupt at the terminal p o i n t i s equivalent to estimating the p r o b a b i l i t y o f the firm's future income being l e s s than the f i x e d o b l i g a t i o n s at the end of the p e r i o d . In mathematical notation t h i s may be expressed i n the form Pr (B) = Pr (FI - FC < 0), [ 3 . 1 ] where Pr (B) i s the p r o b a b i l i t y of bankruptcy at the terminal p o i n t ; FI i s the firm's future income; and FC i s the f i x e d charges at the terminal p o i n t . The term on the r i g h t hand side of the above expression i s the p r o b a b i l i t y of the firm's income net o f a l l f i x e d charges being l e s s than zero. In a multi-period context a firm's income can be l e s s than i t s f i x e d o b l i g a t i o n s without bankruptcy occurring; f o r the f i r m can simply bor-row more. Apart from borrowing there are many other means by which a f i r m 45 may be able to obtain extra sources of funds: issuance of equity, u t i l i z a -t i o n of trade c r e d i t , s e l l i n g of assets, or reduction of investment programs being p o s s i b l e examples. The a b i l i t y of a firm to u t i l i z e these d i f f e r e n t sources depends extensively upon i t s s i z e , the nature of i t s technology, future prospects, managerial a b i l i t y , and the p r e v a i l i n g and expected eco-nomic c o n d i t i o n s . Donaldson 1 proposed three broad categories of funds a f i r m may u t i -l i z e : uncommitted reserves, reduction of planned outflows, and l i q u i d a t i o n of assets. Uncommitted reserves e n t a i l s such f a c t o r s as i n s t a n t reserves (cash, very l i q u i d a s s e t s ) , trade c r e d i t , negotiable reserves, a d d i t i o n of long term debt, and issuance of equity. Reduction of planned outflows i n -volves the r e v i s i n g of e x i s t i n g commitments on outflows of funds; that i s , the p o s s i b l e reduction of investment programs and general a u s t e r i t y measures. L i q u i d a t i o n of assets i s e i t h e r the s e l l i n g o f some of the firm's a s s e t s , or i n the extreme case, the shutdown of the f i r m . For the small f i r m the number of a l t e r n a t i v e s may not be as great. I t s a b i l i t y to obtain a commercial c r e d i t loan during a p e r i o d of t i g h t c r e d i t conditions may be very r e s t r i c t e d . Due to the high issue c o s t s , i t 2 may not have access to the equity markets. I t s capacity to conduct a gen-e r a l reduction of planned outflows may be very small, as might be i t s a b i l i t y to engage i n the l i q u i d a t i o n of assets. ^Donaldson, G., "Strategy f o r F i n a n c i a l Emergencies," Harvard Busi-ness Review, V o l . 47 (November-December, 1969), pp. 67-79. 2 For an introductory d i s c u s s i o n of some of 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 Judging the Performance of C a p i t a l Markets," re 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: Holt, Rinehart and Winston, Inc., 1965). 46 An important source of funds f o r firms derives from the a b i l i t y to borrow. But t h i s a b i l i t y depends upon the w i l l i n g n e s s of f i n a n c i a l i n s t i t u -t i o n s to lend. Among the 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 extending c r e d i t to businesses. J a f f e e and M o d i g l i a n i ^ have developed a simple model to determine the r a t i o n a l i t y and extent of c r e d i t r a t i o n i n g i n a commercial loan market. The b a s i s of the model i s the d e r i v a t i o n of the bank's supply curve f o r loans based upon the assumption that banks a c t to maximize expected p r o f i t s and from considering the firm's demand function f o r a loan. I t i s shown that i f a bank i s a d i s c r i m i n a t i n g monopolist free to charge each customer a d i f f e r e n t r a t e , then c r e d i t r a t i o n i n g w i l l not occur. However, i f banks d i v i d e firms i n t o a small number of r i s k c l a s s e s and charge each c l a s s a d i f f e r e n t r a t e , then i n general i t w i l l be optimal f o r the bank to r a t i o n c r e d i t . The exception to t h i s being i f the 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 then i t i s u n p r o f i t -able f o r the bank to l i m i t c r e d i t . The existence of c r e d i t r a t i o n i n g implies that f o r a f i r m not c l a s s i f i e d as r i s k f r e e , there i s a l i m i t to the amount that i t can borrow, which i s dependent upon the banking s t r u c t u r e and the state of the economy. Whilst the assumption i s made that a f i r m borrows from a bank, the model i s r e a d i l y a p p l i c a b l e to other types of f i n a n c i a l i n -s t i t u t i o n s . In a mu l t i - p e r i o d context the p r o b a b i l i t y of a f i r m going bankrupt i s determined by i t s a b i l i t y to cover f i x e d charges e i t h e r with i t s cash flow or by r a i s i n g funds. Thus the p r o b a b i l i t y of bankruptcy can be repre-sented i n the 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 Test of C r e d i t Rationing," American Economic Review, V o l . LIX, No. 5 (December, 1969), pp. 850-872. 47 Pr (Bfc) = PR (FI f c - FC f c + MBfc + ASfc < 0) [3.2] where Pr (B^) i s the p r o b a b i l i t y of the event of bankruptcy occurring at the end of period t ; F I ^ i s the firm's future income; FC f c i s the f i x e d charges; MBfc i s the maximum amount the f i r m could borrow; AS f c i s a l l other a l t e r n a t i v e sources of funds; and the time s u f f i x , t , i s used to denote that the v a r i -ables are valued at the end of the pe r i o d t . The f i r s t two terms on the r i g h t hand side o f the above expression represent the firm's future income net of a l l f i x e d charges. The magnitude and c h a r a c t e r i s t i c of t h i s term w i l l depend upon the firm's f i n a n c i a l s t r u c -ture and the type and state of the product and resource markets i n which i t deals; that i s , the competitiveness of the markets, t h e i r c y c l i c a l behaviour, and external f a c t o r s . For example, i f there i s economic r e c e s s i o n the firm's product and resource markets may be a f f e c t e d , thus causing changes i n i t s net cash flow. The firm's product d i v e r s i f i c a t i o n , and i t s technology w i l l a l so influence i t s a b i l i t y to s t a b i l i z e i t s cash flows against c y c l i c a l behaviour and ex t e r n a l f a c t o r s . The t h i r d term represents the maximum amount that the f i r m could borrow. This depends upon the banking s t r u c t u r e of the economy, and the r i s k c h a r a c t e r i s t i c s of the f i r m , as perceived by a bank. Over time, as economic conditions i n the economy and the r i s k c h a r a c t e r i s t i c s of the f i r m change, so w i l l the amount of c r e d i t r a t i o n i n g and thus the borrowing power of the f i r m . The l a s t term represents the t o t a l of a l l other a l t e r n a t i v e sources 4 of funds a f i r m may u t i l i z e . The nature of such sources, which Donaldson 4 Donaldson, Loo. cvt. 48 describes i n d e t a i l , c o n s i s t s of three broad categories: uncommitted r e -serves, reduction of planned outflows, and l i q u i d a t i o n of assets. The a v a i l a b i l i t y of these sources depends upon the type of fir m , i t s s i z e , tech-nology, and future prospects. The firm's a b i l i t y to r a i s e funds i s described by the maximum amount i t could borrow and a l l other a l t e r n a t i v e sources. These q u a n t i t i e s are not independent. A firm may be able to borrow using an asset as c o l l a t e r a l , or i t may issue a debenture with a negative pledge clause p r o h i b i t i n g i t from pledging the asset to other c r e d i t o r s . I t w i l l not be able to do both. The d i f f e r e n t means by which a f i r m may be able to u t i l i z e a l t e r n a t i v e sources of funds are also not independent. I f a f i r m issues debt, then t h i s represents a claim against future earnings, which 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 of the d i f f e r e n t sources by which a f i r m may be able to r a i s e funds are dependent upon c e r t a i n common f a c t o r s : the 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 of 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 the future prospects of the f i r m and the economy. I f the future prospects f o r the f i r m are poor, then t h i s may have a decremental e f f e c t upon i t s a b i l i t y to borrow, to issue debt, or equity. This interdependence between the., various 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 of the r e l a t i v e c o n t r i b u t i o n of the underlying f a c t o r s which determine a firm's a b i l i t y to ra i s e funds v i a p a r t i c u l a r sources. Ex-Post Formulation To use the model to e m p i r i c a l l y estimate the p r o b a b i l i t y of bank-ruptcy requires the ex-ante v a r i a b l e s be replaced by ex-post surrogates. 49 However, r e a l i z e d values of the maximum amount the f i r m could borrow or the t o t a l of a l l a l t e r n a t i v e sources are not r e a d i l y observable and so proxy v a r i a b l e s must be constructed. This requires that the underlying f a c t o r s which contribute to the firm's a b i l i t y to r a i s e funds be i d e n t i f i e d and measured. The v a r i a b l e s determining the p r o b a b i l i t y of bankruptcy, as stated i n expression [3.2], are i n terms of d o l l a r amounts. As cross s e c t i o n a l data w i l l be used, 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 i n s i z e o f firms and so w i l l be dominated by scale e f f e c t s . Very large scale e f f e c t s among firms would be expected to lead to i n e f f i c i e n t estimation of coef-f i c i e n t s . To avoid t h i s , the p r o b a b i l i t y of bankruptcy can be w r i t t e n i n the form F l - FC MB + AS Pr (B. ) = Pr (— - + — 4 < 0) , [3.3] * t-1 A t - 1 where A f c i s the book value of the firm's assets at the s t a r t of p e r i o d t . Thus the p r o b a b i l i t y of a f i r m going bankrupt at the end of time peri o d t depends upon i t s future cash flow net of a l l f i x e d charges per u n i t of assets and the t o t a l amount of funds that i t could r a i s e per u n i t o f assets. Whilst the ex-ante values are not observable, r e a l i z e d values of the firm's cash flow net of a l l f i x e d charges are r e a d i l y a v a i l a b l e and can be used to form an ex-post surrogate. The ex-post data are regressed against time and then the estimated regression equation used to p r e d i c t the future value of the firm's cash flow net of a l l f i x e d charges. This value i s then d i v i d e d by the book value of the firm's t o t a l assets and the r e s u l t a n t used as the ex-post surrogate. 50 Neither ex-ante nor r e a l i z e d values of the maximum amount that a firm could borrow are observable. 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 of debt the 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 d e t e r-mined, but such an amount need not n e c e s s a r i l y be the maximum that the f i r m could have borrowed. J a f f e e and M o d i g l i a n i ^ have shown under s i m p l i s t i c assumptions that the maximum amount a f i r m can borrow to finance an i n v e s t -ment p r o j e c t i s given by where L i s the maximum amount which the bank w i l l lend the fi r m ; r. i s the i t h r a t e of i n t e r e s t the bank charges to firms assigned to the i r i s k c l a s s ; F ( ) i s the bank's su b j e c t i v e evaluation of the cumulative 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 outcome of the p r o j e c t ; and p i s the bank's opportunity r a t e . From an o p e r a t i o n a l viewpoint the above equation can not be d i r e c t l y applied, as many of the terms can not be observed. However, i t i s s t i l l o f value f o r i t shows how the bank's opportunity rate and thus c r e d i t r a t i o n i n g a f f e c t s the maximum amount a f i r m can borrow. A l s o , i f c r e d i t r a t i o n i n g increases, then there w i l l be a p r o p o r t i o n a l l y greater decrease i n the maxi-mum amount the f i r m can borrow, implying a non-linear r e l a t i o n s h i p . The a b i l i t y of the f i r m to borrow w i l l depend upon the amount of c r e d i t r a t i o n i n g i n the economy, i t s current l e v e l of debt, and the optimal quantity of debt which i t can u t i l i z e . The greater the amount of c r e d i t r a t i o n i n g , the l e s s the r i s k y f i r m w i l l be able to borrow. The a b i l i t y of the f i r m to borrow w i l l be enhanced the la r g e r the d i f f e r e n c e between the L = 1 + r . 1 i J a f f e e and M o d i g l i a n i , loo. oit. 51 optimal l e v e l and the current l e v e l of debt. An approximate measure of t h i s d i f f e r e n c e i s the book value of net worth, which can be i n t u i t e d as des c r i b i n g that part of the firm's assets not financed by debt. The proxy v a r i a b l e used f o r the ex-ante maximum amount the f i r m could borrow per u n i t asset f o r p e r i o d t i s .book value of net worth at t - l , , . ( ) exp (-CR ), \ - l t 1 where A. , i s the book value of the firm's assets at time t - l ; and CR^ .. i s t - l t - l the amount of c r e d i t r a t i o n i n g at time t - l . The smaller the book value of net worth r e l a t i v e to the firm's t o t a l assets, the l e s s the f i r m w i l l be able to borrow. The f u n c t i o n a l form of dependence on c r e d i t r a t i o n i n g i s used to account f o r the non-linear 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 the amount a f i r m can 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 of 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 three broad 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 of assets. There are many f a c t o r s which a f f e c t the aggregate t o t a l of funds that can be obtained from these d i f f e r e n t sources: o p e r a t i o n a l e f f i c i e n c y , future pros-pects, business r i s k , f i n a n c i a l r i s k being of prime importance. The depen-dence of the firm'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 or e x t e r n a l l y , upon ope 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 reasons. The greater the e f f i -ciency of the fir m the more able i t i s to cope with reductions i n planned outflows, or to undertake the l i q u i d a t i o n of assets. For the p o t e n t i a l ^Donaldson, loc. cit. 52 investor the more e f f i c i e n t the f i r m then the more a t t r a c t i v e i t i s as an investment p r o p o s i t i o n . The future prospects of the f i r m d i r e c t l y a f f e c t i t s a b i l i t y to r a i s e external sources of funds. I f the f i r m i s i n an i n -dustry which i s d e c l i n i n g because of t e c h n o l o g i c a l obsolescence, i t w i l l be very d i f f i c u l t to a t t r a c t c a p i t a l , as i t s future prospects w i l l be bleak i f i t remains wit h i n the industry. Business r i s k measures the o v e r a l l r i s k to the f i r m a r i s i n g from the v a r i a b i l i t y of i t s operating income, and p e r t a i n s to i t s debt capacity. The more v a r i a b l e the cash flow, the greater the r i s k and thus r e s t r i c t s i t s a b i l i t y to use debt f i n a n c i n g . F i n a n c i a l r i s k a r i s e s from the firm's a b i l i t y to cover f i x e d charges. The lower the f i n a n c i a l r i s k , e i t h e r because of low u t i l i z a t i o n of debt or stable cash flows, the more able i t i s to a t t r a c t e x t e r n a l f i n a n c i n g . To measure the aggregate 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 of these four a t t r i b u t e s i s used as a proxy v a r i a b l e . j Operational e f f i c i e n c y should be a measure of the p r o d u c t i v i t y of the firm's assets, a b s t r a c t i n g from tax or leverage f a c t o r s . Various f i n a n -7 c i a l r a t i o s have been used as proxy v a r i a b l e s . Pinches and Mingo use net g income d i v i d e d by t o t a l assets, w h i l s t Beaver suggests three other a l t e r n a -t i v e r a t i o s : net income to s a l e s , net income to net worth, and net income to t o t a l debt. However, a l l of these measures are d e f i c i e n t , as they do not abstract from the e f f e c t s of the firm's f i n a n c i a l s t r u c t u r e and thus are not 7 Pinches, 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 of Indus-t r i a l Bond Ratings," Journal of Finance, V o l . XXVIII, No. 1 (March, 1 9 7 3 ) , pp. 1 - 1 8 . 8 Beaver, W. H., " F i n a n c i a l Ratios as P r e d i c t o r s of F a i l u r e , " Empiri-cal Research in Accounting: Selected Studies, supplement to Journal of Accounting Research ( 1 9 6 6 ) , pp. 7 7 - 1 1 1 . 53 accurate measures of the u t i l i z a t i o n of the firm's assets. An a l t e r n a t i o n 9 formulation by Altman using earnings before i n t e r e s t and taxes divided by t o t a l assets avoids t h i s d e f i c i e n c y . I t abstracts from tax or leverage f a c t o r s , and i s a measure of the firm's earning power. This formulation i s used i n the t h e s i s . To a t t a i n the future prospects of the fir m and thus i t s a b i l i t y to a t t r a c t c a p i t a l requires measuring the p r o f i t a b i l i t y of the firm's future investment opportunities, t h e i r s i z e and duration. A l l of these q u a n t i t i e s are not d i r e c t l y observable. M i l l e r and M o d i g l i a n i 1 ^ addressing themselves to the same problem, focused upon the most t r a c t a b l e component, the l e v e l of investment op p o r t u n i t i e s , as an o v e r a l l measure of growth and future pros-pects. For an empirical estimator of the l e v e l of investment opportunities per u n i t asset, a l i n e a r f i v e year growth rate of t o t a l assets i s used. This measure i s used i n the t h e s i s . Business r i s k describes the r i s k to the f i r m that a r i s e s from the v a r i a b i l i t y of i t s operating income, abstracting from tax or leverage f a c -tors . The debt capacity of the fir m depends upon the v a r i a b i l i t y of i t s cash flow and thus business r i s k : the more responsive the firm's cash flow to changes i n the economy, the lower the optimal amount of debt which the firm can use. Van Home 1 1 uses the c o e f f i c i e n t of v a r i a t i o n of operating 9 Altman, E. I., " F i n a n c i a l Ratios, Discriminant Analysis and the Pr e d i c t i o n of Corporate Bankruptcy," Journal of Finance, V o l . XXIII, No. 4 (September, 1968), pp. 589-609. 1 0 M i l l e r , M. and Mo d i g l i a n i , F., "Some Estimates of the Cost of C a p i t a l to the E l e c t r i c U t i l i t y Industry, 1954-1957," American Economic Review, V o l . LVI, No. 3 (June, 1966), pp. 333-391. 1;1"Van Home, J . , Financial Management and Policy (New Jersey: Prentice H a l l Inc., 1972). 54 income to measure business r i s k . However, t h i s does not d i r e c t l y measure the responsiveness of the firm's cash flow to changes i n the economy. An a l t e r n a t i v e formulation, and one that i s used i n the t h e s i s , i s to measure business r i s k by the absolute value of the p r o p o r t i o n a l change i n sales to the p r o p o r t i o n a l change i n gross n a t i o n a l product; the more responsive sales to changes i n the economy the greater the business r i s k . F i n a n c i a l r i s k i s a measure of the firm's a b i l i t y to cover 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 the more able i t should be to a t t r a c t e x ternal f i n a n c i n g . To measure f i n a n c i a l r i s k 12 Altman suggests 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 of equity d i v i d e d by the book value of t o t a l debt, and the book value of net worth d i v i d e d by the book value of t o t a l debt. Both r a t i o s are d e f i c i e n t , f o r they do not neces-s a r i l y take account of a l l f i x e d charges which the f i r m must meet. The use of the book value of net worth does not measure the firm's a b i l i t y to cover f i x e d charges. The a b i l i t y of a f i r m to cover i t s f i x e d charges p r i m a r i l y depends upon i t s future cash flow, i t s v a r i a b i l i t y , and the t o t a l of f i x e d charges which i t covers. Norton 1^ uses the c o e f f i c i e n t of v a r i a t i o n of the firm's past income over and above the amount of f i x e d charges. This measure i s d e f i c i e n t f o r the firm's a b i l i t y to meet f i x e d charges depends upon i t s future income as opposed to past income. The proxy v a r i a b l e used i n the t h e s i s i s the d i f f e r e n c e between the firm's f i x e d charges and i t s future cash flow, the d i f f e r e n c e being d i v i d e d by the standard d e v i a t i o n of the 12 . Altman, loc. cit. 13 Norton, J . , "The Theory of Loan C r e d i t i n Relation to Corporation Economics," Publications of the American Economic Association, 3rd ser., V o l . V (1904), pp. 278-300. 55 future cash flow. The smaller the v a r i a b l e , the more able the firm to cover i t s f i x e d charges and the lower the f i n a n c i a l r i s k . The firm's future cash flow i s estimated by regressing r e a l i z e d values of i t s operating income against time and then the estimated regression equation used to p r e d i c t the future value. The square root of the r e s i d u a l sum of squares i s used as an estimate of the standard d e v i a t i o n of the firm's future cash flow. Combining the proxy v a r i a b l e s f o r the firm's future cash flow net of a l l f i x e d charges per u n i t assets, the maximum amount i t could borrow per u n i t assets, and the t o t a l of a l l other a l t e r n a t i v e sources per u n i t assets, gives FI —FC MB AS t t + t + t _ g + g ^estimated future cash flow net of a l l f i x e d charges^ A t - 1 A t - 1 A t - 1 ° A t - 1 „ .book value of net worth at t - l . , . +32( j ) exp (-CR ) A t - 1 t _ i 0 .earning before i n t e r e s t and tax at t - l . +3 3 ( ) A t - 1 + 8 4 ( f i v e year l i n e a r growth rate i n t o t a l assets) + p 1 p r o p o r t i o n a l change i n s a l e s i . 5 'proportional change i n GNP ' + g f i x e d charges at t - l - estimated future cash flow^ 6 estimated standard d e v i a t i o n of future cash flow +e, [ 3 . 4 ] where 8o»3i , . - . / 8 s are unknown c o e f f i c i e n t s ; and e i s a zero mean random var-i a b l e e r r o r term, which i s assumed to be of u n i t variance and uncorrelated between firms. The c o e f f i c i e n t s i n the above equation represent the r e l a t i v e c o n t r i -butions of the d i f f e r e n t underlying f a c t o r s to the aggregate t o t a l of net 56 funds a v a i l a b l e to the f i r m . The c o e f f i c i e n t s can not be estimated by r e -gression, as the dependent v a r i a b l e i s an ex-ante quantity which can not be measured or observed. S u b s t i t u t i n g Equation [3.4] i n t o the expression [3.3] f o r the p r o b a b i l i t y of bankruptcy gives ~ » - n « .estimated future cash flow net of a l l f i x e d charges. Pr (B ) = Pr[e < Bo+Bi ( -—) Z A t - 1 „ .book value of net worth at t - 1 , . +8 2 < j: ) exp (-CR ) A t - 1 C 1 „ .earnings before i n t e r e s t and taxes at t-1^ +g 3 ( ; A t - 1 +84(five year l i n e a r growth rate f o r t o t a l assets) + g jp 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 ' + g ^fixed charges a t t - 1 - estimated future cash flow^1 6 estimated standard d e v i a t i o n of future cash flow ' I-3.5J where the c o e f f i c i e n t s have been redefined to include the minus s i g n . The p r o b a b i l i t y of the event of the f i r m going bankrupt at the end of p e r i o d t i s the p r o b a b i l i t y of the random v a r i a b l e e r r o r term minus the summation o f the underlying f a c t o r s which contribute to the net t o t a l of funds a v a i l a b l e to the f i r m being l e s s than zero. Apart from the random e r r o r term, a l l the v a r i a b l e s on the r i g h t hand side of the equation 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 determined from t h e o r e t i c considerations. Using a c e t e r i s paribus argument, the greater the firm's future cash flow net of a l l f i x e d charges, and the amount which i t could borrow, the lower the p r o b a b i l i t y of bankruptcy. Thus the c o e f f i c i e n t s Bi and 82 should be negative. S i m i l a r l y f o r the c o e f f i c i e n t s 83 and Bi+, as the greater the e f f i c i e n c y of the f i r m , the b e t t e r i t s future prospects, the 57 more able i t i s to r a i s e extra sources of funds. The c o e f f i c i e n t s 85 and 86 should be p o s i t i v e ; the greater the business r i s k , as measured by the v a r i -a b i l i t y of the firm's cash flows, and the l a r g e r the f i n a n c i a l r i s k , the l e s s able the f i r m to a t t r a c t extra c a p i t a l and the greater the p r o b a b i l i t y of bankruptcy. P r e d i c t i v e Model A complete model f o r the p r o b a b i l i t y of bankruptcy should describe a l l the i n t e r a c t i o n s between the d i f f e r e n t f a c t o r s . The firm's a b i l i t y to borrow or to issue debt i s dependent upon i t s debt cap a c i t y . But debt capacity i s dependent upon the p r o b a b i l i t y of bankruptcy and thus there i s a c i r c u l a r i t y . The a b i l i t y to use a p a r t i c u l a r source of funds i s dependent upon the u t i l i z a t i o n of other sources. I f a f i r m issues debt, t h i s may have a decremental e f f e c t upon i t s a b i l i t y to borrow from a bank or to issue equity. Due to the complex i n t e r a c t i o n of the underlying f a c t o r s and the d i f f i c u l t y of measuring t h e i r magnitude and a v a i l a b i l i t y , a second formula-t i o n using market values f o r the appropriate v a r i a b l e s i s developed. The use o f market values circumvents many of the d i f f i c u l t i e s of con s t r u c t i n g proxy v a r i a b l e s to measure such q u a n t i t i e s as the maximum amount the f i r m can borrow and the t o t a l of a l l other a l t e r n a t i v e sources. The p r o b a b i l i t y of a f i r m going bankrupt depends upon i t s future i n -come net of a l l f i x e d charges, the maximum amount i t can borrow and a l l other a l t e r n a t i v e sources of funds. To use the model to e m p i r i c a l l y estimate the p r o b a b i l i t y of bankruptcy requires that the ex-ante v a r i a b l e s be replaced by ex-post surrogates. 58 For the firm's future cash flow net of a l l f i x e d charges the same proxy v a r i a b l e , as p r e v i o u s l y defined i s used; that i s , r e a l i z e d values of the firm's cash flow net of a l l f i x e d charges are regressed against time and then the estimated regression equation used to p r e d i c t the future value of the firm's cash flow net of a l l f i x e d charges. This value i s then d i v i d e d by the book value of the firm's t o t a l assets and the r e s u l t a n t used as the ex-post surrogate. The a b i l i t y of the f i r m to borrow depends upon the amount of c r e d i t r a t i o n i n g i n the economy, i t s current l e v e l of debt and the optimal l e v e l of debt which i t can u t i l i z e . The firm's a b i l i t y to borrow i s enhanced the larger the d i f f e r e n c e between the optimal and current l e v e l of debt. To measure t h i s d i f f e r e n c e requires that the debt capacity of the f i r m be known. However, debt ca p a c i t y depends upon the p r o b a b i l i t y o f bankruptcy, implying that the explanatory v a r i a b l e i s a f u n c t i o n of the dependent v a r i a b l e . The market value of equity f o r a f i r m , which r e f l e c t s the p r o b a b i l i t y of bankruptcy, i s a measure of i t s borrowing a b i l i t y . For a given l e v e l of assets, the greater the market value of equity, the more able the f i r m to borrow. The proxy v a r i a b l e used to measure the ex-ante maximum amount the fi r m could borrow per u n i t of assets f o r p e r i o d t i s .market value of equity at t-1. , „„ , ( - * ~ ) exp (-CR ) , A t - 1 t 1 where A f c ^ i s the book value of the firm's assets at time t-1; and CRfc ^ i s the amount of c r e d i t r a t i o n i n g at time t-1. The t o t a l of a l l other a l t e r n a t i v e sources of funds f o r the f i r m depends upon three broad categories: uncommitted reserves, reduction of 59 planned outflows, and the l i q u i d a t i o n of assets. The a b i l i t y of the f i r m to u t i l i z e these d i f f e r e n t sources 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 f i n a n c i a l r i s k . A v a r i a b l e which synthesizes these diverse q u a n t i t i e s i s the market value of equity. Using a c e t e r i s paribus argument, the more e f f i c i e n t l y the f i r m u t i l i z e s i t s assets, or the b r i g h t e r i t s future prospects, the greater i s the market value of i t s equity. S i m i l a r l y , the lower the business and f i n a n c i a l r i s k of the firm, the greater i t ' s market value of equity. For a given l e v e l of assets, the greater the market value of equity the more able i s the f i r m to generate and a t t r a c t e x t r a sources o f funds. Thus, the ex-post surrogates f o r the ex-ante net aggregate t o t a l of funds a v a i l a b l e to the f i r m , can be w r i t t e n : FI —FC MB AS t t t t .estimated future cash flow net of a l l f i x e d charges. — + — + — = Yo+Yl ( £ ) A t - 1 A t - 1 A t - 1 A t - 1 .market value of equity at t - l . , . + Y 2 ( H 1 ) e x p (_CR ) A t - 1 t _ 1 .market value of equity at t - l . + Y 3 ( * ) A t - 1 +n, [3.6] where Yo,Yl#Y2/ a n <^ Y3 a r e unknown c o e f f i c i e n t s ; and n i s a zero mean random v a r i a b l e e r r o r term, which i s assumed to be of u n i t variance and uncor-r e c t e d between firms. S u b s t i t u t i n g Equation [3.6] i n t o the expression [3.3] f o r the p r o b a b i l i t y of bankruptcy gives 60 r ^ .estimated future cash flow net of a l l f i x e d charges, Pr(D ) = Pr[n < Yo+Yl ( r — ) t \ - l .market value of equity at t-1. , + Y 2 ( . a I ) exp (-CR ) A t - 1 t _ i .market value of equity at t - 1 . , +Y3( J ~ 1 ) ] , [3.7] A t - 1 where the c o e f f i c i e n t s Yo,Yl/Y2» a n d Y3 have been redefined to absorb the minus s i g n . Apart from the random v a r i a b l e , a l l the terms on the r i g h t hand side of the above equation 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 determined from t h e o r e t i c c o n s i d e r a t i o n s . Using a c e t e r i s paribus argument, the greater the estimated future cash flow net of a l l f i x e d charges, the smaller i s the p r o b a b i l i t y of bankruptcy, and thus the c o e f f i c i e n t y i should be negative. The smaller the amount of c r e d i t r a t i o n i n g , or the greater the market value of equity, the more able the f i r m i s to 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 , to cover f i x e d charges and the lower the p r o b a b i l i t y of bankruptcy. Hence, the c o e f f i c i e n t s y 2 and Y 3 should be 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 of a f i r m going bankrupt has been formulated in' terms of two models, as represented by Equations [3.5] and [3.7]. The general structure of these formulations i s of the form P r ( B t j I ^ t - l , j ) = P r ( " < i ' ^ t - l , j ) [ 3 - 8 ] th where X . i s a vector of the j firm's a t t r i b u t e s measured at time t-1; ~ ^ - - l i j a i s a vector of unknown c o e f f i c i e n t s ; Pr(B I X .) i s the p r o b a b i l i t y th of the event that the j f i r m goes bankrupt during period t , given the 61 vector of a t t r i b u t e s , .; and £. i s a zero mean random v a r i a b l e e r r o r - t - l , D D term which i s assumed to have u n i t variance and to be independent among firms; that i s , E(e.) = 0, var (e.) = a , and cov (e.,e ) = 0, j f k, f o r 3 D J k a l l j and k. There are a number of s p e c i a l c h a r a c t e r i s t i c s about Equation [3.8] which have important i m p l i c a t i o n s f o r s t a t i s t i c a l estimation. The dependent v a r i a b l e , the p r o b a b i l i t y of a f i r m going bankrupt, can not be d i r e c t l y measured; that i s , the ex-ante value or the r e a l i z e d values can not be observed. As the dependent v a r i a b l e i s a p r o b a b i l i t y , i t i s con-s t r a i n e d to the i n t e r v a l zero-one. The prime focus i s to derive some form of technique to estimate the p r o b a b i l i t y of bankruptcy, c o n s t r a i n i n g the estimate to a zero-one i n t e r v a l . The c o e f f i c i e n t s , a_, which measure the r e l a t i v e c o n t r i b u t i o n of the d i f f e r e n t a t t r i b u t e s , are unknown. Thus, on both sides of the equation there are unknown q u a n t i t i e s which, i n general, are r e l a t e d i n a non-linear manner. Whilst the ex-ante p r o b a b i l i t i e s of a f i r m going bankrupt can not be observed, at any p o i n t i n time a f i r m i s e i t h e r bankrupt or not bankrupt. This suggests that c o l l e c t i n g data f o r a random sample of bankrupt and non-* bankrupt firms the c o e f f i c i e n t s , cx, can be estimated by p o s i t i n g the model <Z t j I = + C [3.9] where t, i s a random disturbance; and Z i s an i n d i c a t o r f u n c t i o n defined by f l ; i f j t h f i r m bankrupt at time t ; Z . = < (o, otherwise. Though the c o e f f i c i e n t s can be estimated by r e g r e s s i o n , they w i l l not be e f f i c i e n t estimators. The systematic p a r t of the r i g h t hand s i d e , a' X^_, . may be larger than one or smaller than zero, whereas takes only two 62 values (0 and 1) which means that the disturbance term, given X^ , . can ^ t - l / D take only two values: -a' X , . and 1-a 1 X^ , .. I f ? i s to have an expected — — t - 1 , j — -^t-1,3 value of zero f o r a l l values of X^ , ., i t must take the former value with - 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 with p r o b a b i l i t y cx' X .. t 1, j t 1, ] But a_* X . can be negative or l a r g e r than one. There i s nothing i n the e s t i -mation procedure to ensure that the estimated values of the dependent v a r i a b l e , . . 14 are constrained. The dependent v a r i a b l e of Equation [3.8] i s an ex-ante p r o b a b i l i t y which can not be observed, w h i l s t the ex-post v a r i a b l e s on the r i g h t hand side of the equation can 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, can be estimated using maximum l i k e l i h o o d . Consider a random sample of firms at time t and suppose that the f i r s t n' firms are bankrupt and the remainder n-n' non-bankrupt. The logarithmic l i k e l i h o o d f u n c t i o n can then be w r i t t e n n' n E log Pr(B. .|x. .) + Z l o g [ l - P r ( B k ) ] , [3.10] j = l t 3 " t " 1 ' 3 j=n'+l t j where Pr(B .|)T .) i s defined by Equation [3.8] and i s thus a f u n c t i o n of — " t — l , ] the parameters a_. By d i f f e r e n t i a t i n g [3.10] with respect to these parameters and equating the f i r s t d e r i v a t i o n to zero, a set of non-linear equations are obtained and can be solved i t e r a t i v e l y . For p r a c t i c a l a p p l i c a t i o n the use of maximum l i k e l i h o o d r e q u i r e s that 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 d i s t r i -bution be assumed; that 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 of the random v a r i -'Vi able, £, i n Equation [3.8] must be s p e c i f i e d . 14 For a more extensive d i s c u s s i o n of the econometric problems, see Goldberger, A . S., Economic Theory (New York: John Wiley & Sons, 1964), pp. 248-255. 63 Two estimation procedures are used i n the t h e s i s : p r o b i t a n a l y s i s and l o g i t a n a l y s i s . 1 ^ The e s s e n t i a l d i f f e r e n c e between the two procedures i s 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 . For p r o b i t a n a l y s i s a 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 alysis the d i s t r i b u t i o n i s l o g i s t i c . 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 the normal d i s t r i b u t i o n , being s l i g h t l y f a t t e r i n the t a i l s and more c e n t r a l i z e d 16 about the mean. 17 Probit a n a l y s i s can be defined as follows: i f the p r o b a b i l i t y of a zero mean, u n i t variance normally d i s t r i b u t e d random v a r i a b l e being l e s s than or equal to the s c a l a r produce ot'X i s p, then the p r o b i t of a_'X_ i s F 1 (p), where 1 t 1 2 F (t) = - — / exp (- — u ) du, /2TT . v [3.11] and p = Pr(Y < a'x) = F(a'X) , given that Y ~ N(0,1). The unknowns i n the formulation are the set of para-meters, ex, and the p r o b a b i l i t y p. Equation [3.11] can be s u b s t i t u t e d i n t o expression [3.10],the logarithmic l i k e l i h o o d function, g i v i n g "^An introductory d i s c u s s i o n i s qiven i n T h e i l , H., Principles of Econometrics (New York: John Wiley & Sons, 1971), pp. 628-635. 16 For 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 of C e r t a i n Symmetrical Growth Curves," Journal of the Washington Academy of Science, V o l . 22, No. 4 (February, 1932), pp. 73-84. 17 For a general d i s c u s s i o n and a p p l i c a t i o n s of p r o b i t a n a l y s i s see Finney, D. J . , Probit Analysis (Cambridge: Cambridge U n i v e r s i t y Press, 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 for Limited Dependent Variables With A p p l i c a t i o n to the Demand for Durable Goods," Econometrica, V o l . 39, No. 5 (September, 1971), pp. 829-844. 64 n [ 3 . 1 2 ] £ log F(a'X .) + E log [l-F(a'X ) ] , =1 -t"-L/3 j=n'+l , J 3 where F (a'X,. , .) = / exp (— u ) du, and the parameters can be estimated by s o l v i n g the set of f i r s t order condi-t i o n s obtained by d i f f e r e n t i a t i n g the l i k e l i h o o d f u n c t i o n . 18 Logit a n a l y s i s can be defined i n a s i m i l a r manner. I f the proba-b i l i t y of a f i r m going bankrupt i s equal to the p r o b a b i l i t y , p, of a random v a r i a b l e , which has a l o g i s t i c d i s t r i b u t i o n , being l e s s than or equal to the s c a l a r product cx'X_, that i s , p = Pr (Z < cx_'X) [ 3 . 1 3 ] 1 + exp (-a'X)' where Z is a random v a r i a b l e having a l o g i s t i c d i s t r i b u t i o n , then the l o g i t of a'X i s log = a'X. Again the unknowns i n the formulation are the set of parameters, a_, and the p r o b a b i l i t y p. By s u b s t i t u t i n g Equation [ 3 . 1 3 ] i n t o expression [ 3 . 1 0 ] , the parameters a_ can be estimated by maximum l i k e l i h o o d . To e m p i r i c a l l y estimate the parameters, a_, the l i k e l i h o o d f unction 18 For an introductory d i s c u s s i o n to 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 . , "Applications of L o g i s t i c Functions to Bio-Assay," Journal of the American Statistical Association, V o l . 39 ( 1 9 4 4 ) , pp. 3 5 7 -3 6 5 ; and Baxter, N. D. and Cragg, J . G., "Corporate Choice Among Long-Term F i n a n c i a l Instruments," The Review of Economics and Statistics, V o l . L I I , No. 3 (August, 1 9 7 0 ) , pp. 2 2 5 - 2 3 5 . 65 must be constructed by taking a random sample of firms and then c l a s s i f y i n g the firms as bankrupt or not bankrupt. The procedure of using a random sample avoids s e l e c t i o n b i a s . As the average p r o b a b i l i t y of a f i r m going bankrupt i s small, a very large random sample must be taken so as to obtain a representative c o l l e c t i o n of bankrupt firms. In p r a c t i c e , a common pro-cedure i s to c o l l e c t data f o r a l l bankrupt firms over 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 of non-bankrupt f i r m s . I t i s necessary to determine how many firms should be included i n the sample. I d e a l l y , the number chosen should be the same as that obtained by taking a random sample of a l l firms and then c l a s s i f y i n g them as bankrupt or not bankrupt. Thus, to determine the required sample s i z e e n t a i l s estimating the average value of the p r o b a b i l i t y of a f i r m going bankrupt. Testing of the Model As the p r o b a b i l i t y of bankruptcy can not be observed, d i r e c t t e s t s on the models are not p o s s i b l e . This implies that the magnitude o f any b i a s or measurement e r r o r i n the estimates can not be determined. Thus, the main check on how w e l l the models are s p e c i f i e d must be on t h e i r pre-d i c t i v e a b i l i t y . There are three methods by which the models can be t e s t e d . From t h e o r e t i c considerations the s i g n of the parameters can be determined and compared to those obtained from the e m p i r i c a l estimation. The number of estimated parameters with the c o r r e c t s i g n provides i n s i g h t i n t o the s p e c i f i c a t i o n of the model and the accuracy 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 completely s p e c i f i e d so as to 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 of the firms i n the data sample used to estimate the parameters, then i t should be able to c o r r e c t l y i d e n t i f y the bankrupt and non-bankrupt firms 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 information about the model's s p e c i f i c a t i o n and the number of common deter-minants of bankruptcy. The g e n e r a l i t y of the model and i t s o v e r a l l independence of the p e c u l i a r i t i e s of the data sample used to estimate the parameters, can be tested by examining i t s p r e d i c t i v e a b i l i t y on a set of bankrupt firms not used i n the o r i g i n a l sample. By estimating the p r o b a b i l i t y of bankruptcy over several time periods f o r firms i n the new sample provides a demon-s t r a t i o n of the model's p r e d i c t i v e a b i l i t y to d i s c e r n a firm's path to bankruptcy. Summary In a multiperiod context a firm's income can be l e s s than i t s o b l i g a t i o n s and yet i t i s not bankrupt; i t can simply borrow more. The p r o b a b i l i t y of a firm going bankrupt depends upon 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 or e x t e r n a l l y , to cover f i x e d charges. A f i r m that f a i l s to cover these charges i s s a i d to be bankrupt. From t h i s d e f i -n i t i o n , a model f o r the p r o b a b i l i t y of bankruptcy i s constructed i n terms of ex-ante v a r i a b l e s . To use the model to e m p i r i c a l l y estimate the p r o b a b i l i t y of bankruptcy an ex-post formulation using proxy v a r i a b l e s i s developed. As the primary focus i s upon the p r e d i c t i o n of bankruptcy, as opposed to advancing a complete theory of the determinants of bankruptcy, a second formulation using market values of appropriate corporate v a r i a b l e s 67 is constructed. To empirically estimate the coefficients of the models, a s t a t i s t i c a l methodology employing probit analysis and l o g i t analysis i s used. Three different methods to check the predictive a b i l i t y of the models are described. CHAPTER IV AN EXTENSION OF THE CAPITAL ASSET PRICING MODEL: BANKRUPTCY The p r i c e behaviour of common stocks i s a f f e c t e d by impending bank-ruptcy. As a f i r m progresses towards bankruptcy i t s ex-post returns, when compared to those of firms that d i d not go bankrupt during the same period appear to be s i g n i f i c a n t l y d i f f e r e n t i n behaviour. Empirical evidence i n -dicates that the CAPM does not provide an adequate d e s c r i p t i o n of the mech-anism generating common stock returns. I t i s found that assets with high l e v e l s of systematic r i s k c o n s i s t e n t l y earn le s s than that predicted by the model, the reverse being true f o r assets with low l e v e l s of systematic r i s k . Whilst there appears to be a l i n e a r r e l a t i o n s h i p between ex-post re-? turns and systematic r i s k , i t i s not constant with both the i n t e r c e p t and slope f l u c t u a t i n g randomly from period to period and are often negative. The data i n d i c a t e that the expected return on a s e c u r i t y can be represented by a l i n e a r two f a c t o r model, the second f a c t o r not being e x p l i c i t l y de-f i n e d . Various attempts have been made to provide a t h e o r e t i c a l explanation f o r the existence of a second f a c t o r . The e f f e c t s of non-marketability of assets, changes i n the investment opportunity set, the non-existence of a r i s k l e s s asset, and r e s t r i c t i o n s upon the investor's a b i l i t y to borrow or lend have been explored, though f a i l to provide an adequate explanation of a l l the observed d e f i c i e n c i e s of the CAPM and why impending bankruptcy a f f e c t s the r e s i d u a l return, a f t e r abstracting from the market, of common stocks. i\ primary focus of the t h e s i s i s to extend the formulation of the CAPM not from the viewpoint of r e s t r i c t i o n s upon the investor, but by con-6 8 69 s i d e r i n g the impact of bankruptcy upon the stru c t u r e of returns f o r c o r -porate f i n a n c i a l assets. A model, formulated i n continuous time, considers the investment-consumption d e c i s i o n of an i n d i v i d u a l a c t i n g to maximize the expected l i f e -time u t i l i t y of consumption and terminal wealth. At each i n s t a n t i n time the i n d i v i d u a l must decide the portions of wealth to consume and to i n -v e s t i n f i n a n c i a l a s s e t s . I t i s assumed that corporations issue both debt and equity as f i n a n c i a l assets and that a t each p o i n t i n time there i s a p r o b a b i l i t y that the f i r m w i l l go bankrupt the next i n s t a n t . If bankruptcy occurs i t i s assumed that equity holders s u f f e r a 100 per cent l o s s , w h i l s t bond-holders r e c e i v e a non-negative l i q u i d a t i n g premium. When the investment opportunity s e t i s a l t e r e d only by the event of bankruptcy, a two v a r i a b l e model i s derived which describes the expected return, c o n d i t i o n a l upon no bankruptcy, f o r a firm's common stock i n terms of i t s systematic r i s k and a v a r i a b l e associated with the p r o b a b i l i t y of the f i r m going bankrupt. For the general case when there are s t o c h a s t i c changes i n the p r o b a b i l i t y of a f i r m going bankrupt, a d d i t i o n a l terms a r i s e r e f l e c t -ing i n vestors' attempts to hedge against unexpected changes. The foundations of the model are set out i n the f i r s t p a r t of the chapter. The major assumptions of the model, the nature of the f i n a n c i a l assets a v a i l a b l e and t h e i r p r i c e dynamics are described. The general form of the equation of o p t i m a l i t y and the system of equations d e s c r i b i n g the f i r s t order maximization conditions are derived. Due to the complexity of the general a n a l y s i s , a d d i t i o n a l s t r u c t u r e i s i n j e c t e d i n t o the a n a l y s i s . In the second p a r t of the chapter the eq u i l i b r i u m instantaneous expected rates 70 of return, c o n d i t i o n a l upon no bankruptcy, f o r bonds and equity are derived given the assumption that the investment opportunity set i s only change by. bankruptcy. The f i n a l part of the chapter considers the e f f e c t of stochas-t i c changes i n the p r o b a b i l i t y of bankruptcy upon the s t r u c t u r e of returns. As much of the a n a l y s i s i s of a h i g h l y mathematical nature, an attempt has been made to relegate as much of the mathematics to Appendix A, w h i l s t s t i l l maintaining c o n t i n u i t y i n the chapter. Foundations of Model For an investor to buy the bonds of a f i r m the expected r a t e of r e -t u r n 1 must compensate the investor f o r the r i s k that the f i r m w i l l go bank-rupt and f o r the r i s k of a c a p i t a l l o s s which might r e s u l t i f there i s an unexpected change i n the general l e v e l of 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 -i t y of bankruptcy. I f bankruptcy occurs, i t i s assumed the f i r m i s l i q u i d a t e d , the value of the firm's assets, net of tax minus the d i r e c t costs associated with bankruptcy, i s d i s t r i b u t e d on a pro-rata b a s i s t o bondholders. Thus, the bondholder i s subject to the r i s k of a d i r e c t l o s s i f bankruptcy occurs. I n t u i t i v e l y , the expected r a t e of r e t u r n on a firm's bonds tha 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 of the r i s k f r e e i n t e r e s t r a t e , the prob-a b i l i t y of bankruptcy, the expected l o s s i f bankruptcy occurs, and the expected 2 change i n the general 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 uses the term 'nominal r a t e of i n t e r e s t . ' S t i g l i t z , J., "A Re-Examination of 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 For an introductory d i s c u s s i o n see F i s h e r , L., "Determinants of Risk Premiums on Corporate 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 the shares of a fir m , an in v e s t o r buys a claim to a v a r i a b l e cash flow. From the t r a d i t i o n a l formulation of the c a p i t a l asset p r i c i n g model, as given by Sharpe, Treynor, and M o s s i n , t h e expected rate of re t u r n f o r a firm's equity depends upon the r i s k free i n t e r e s t r a t e , and i t s l e v e l of systematic r i s k , which i s a measure of the covariance of the re t u r n on the firm's stock wih the return on the market p o r t f o l i o . Such a formulation does not e x p l i c i t l y consider bankruptcy or changes i n the investment opportunity se t . I f these f a c t o r s are considered, then they w i l l i n general a f f e c t the expected rate of 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 r e q u i r e . In order to derive the e q u i l i b r i u m expected returns f o r a firm's bonds and equity, the demand functions f o r the d i f f e r e n t f i n a n c i a l assets f o r an i n d i v i d u a l are obtained. I t i s assumed that a t time t , there are n firms, each f i r m having a simple c a p i t a l s t r u c t u r e of one type of debt and one type of common stock. I t i s al s o assumed that there i s a r i s k l e s s asset, so th a t there are 2n+l f i n a n c i a l a s sets, which are assumed to be traded i n a perfect c a p i t a l market (with bankruptcy). I t i s important to note that the bonds of d i f f e r e n t firms are treated as d i f f e r e n t f i n a n c i a l a s s e t s . I f c a p i -t a l structure was i r r e l e v a n t , then t h i s would not be necessary. But, as c a p i -4 t a l i s relevant then, as S t i g l i t z c o r r e c t l y observes, the presence of bankrupt-cy creates a new s e c u r i t y . "Sharpe, W., " C a p i t a l Asset P r i c e s : A Theory of Market E q u i l i b r i u m Under Conditions of Risk,"Journal of Finance, V o l . XIX, No. 3 (September, 1964), pp. 425-442; Treynor, J . f "Towards A Theory of Market Value of Risky Assets," unpublished memorandum (1961); Mossin, J . , "E q u i l i b r i u m In A C a p i t a l Asset Market," Econometricat V o l . 34, No. 4 (October, 1966), pp. 468-783. 4 S t i g l i t z , J . , "Some Aspects of the Pure Theory of Corporate Finance: Bankruptcy and Take Overs," Bell Journal of Economics and Management Sciencet V o l . 3, No. 2 (Autumn, 1972), pp. 458-482. 72 5 I t i s assumed that the c a p i t a l market i s structured such that A l . a l l assets have l i m i t e d l i a b i l i t y ; A2. there are no t r a n s a c t i o n costs (excluding bankruptcy), personal taxes, or problems with i n d i v i s i b i l i t i e s of assets; A3. there are s u f f i c i e n t number of investors with compar-able wealth l e v e l s so that each investor can buy or s e l l as much of an asset without 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 equilibrium; that i s , there i s no trading a t non-equilibrium p r i c e s ; A5. there e x i s t s an exchange market f o r borrowing and lending at the same r a t e of i n t e r e s t ; A6. short sales of a l l assets, with f u l l use of the proceeds, i s allowed; A7. there are homogeneous expectations among investors about the future prospects of each f i n a n c i a l asset; A8. bonds are, i n general, r i s k y . I f a f i r m goes bankrupt, then 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 that the f i r m ceases t o e x i s t ; A9. trading i n assets takes place continuously i n time. Assumptions A l to A7 are the standard p e r f e c t market (ex-c l u d i n g bankruptcy) assumptions, which have been extensively discussed i n the f i n a n c i a l l i t e r a t u r e . Assumption A8 i s a departure from the standard asset p r i c i n g model assumptions. I t a r i s e s due to the conditions of the presence of bankruptcy. If a f i r m cannot meet i t s o b l i g a t i o n s and i s unable 3 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 of these assumptions, see Merton, R.C., "An Intertemporal C a p i t a l Asset P r i c i n g Model," Working Paper 588-72, Sloan School of Management, Massachusetts I n s t i t u t e of Technology (February, 1972). A good reference i s Jensen, M., " C a p i t a l Markets: Theory and E v i -dence," Bell Journal of Economics and Management Science, V o l . 3, No. 2, (Autumn, 1972), pp. 458-482. 73 to secure extra finance, then i t w i l l be assumed that a state of bankruptcy i s declared. I t w i l l a l s o be assumed that the f i r m ceases to e x i s t ; that i s , the firm i s l i q u i d a t e d , the p o s s i b l i t y of the f i r m undergoing reo r g a n i -zation being neglected. Assumption A9 follows d i r e c t l y from Assumption A2. If there are no costs, no i n d i v i s i b i l i t i e s , then inv e s t o r s should p r e f e r to be able to r e -v i s e t h e i r p o r t f o l i o s at any time. In r e a l i t y there are t r a n s a c t i o n c o s t s , i n d i v i s i b i l i t i e s , and i t i s f o r these reason that a d i s c r e t e time formulation i s advanced. Usually, the approach i s to take equally spaced i n t e r v a l s of time which, though convenient from an em p i r i c a l viewpoint, i s t h e o r e t i c a l l y u n s a t i s f a c t o r y . The trad i n g i n t e r v a l s w i l l , i n general, be s t o c h a s t i c and of a non-constant length, depending upon the types of 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 nature of p r i c e changes, t r a n s a c t i o n c o s t s , and future expec-7 t a t i o n s . I t w i l l be assumed that the i n d i v i d u a l acts i n a manner to maximize the expected l i f e t i m e u t i l i t y of consumption and terminal wealth; t h a t i s , the k f ck i n d i v i d u a l acts so tha t T k Max E q { / o U k ( C k ( s ) , s ] d s + BF k[W(T k) ,T k] } (4.1) subject to an i n i t i a l wealth c o n s t r a i n t , where represents the c o n d i t i o n a l expected value operator, c o n d i t i o n a l on the f a c t that a l l state v a r i a b l e s at time t are known; C^ts) i s the i n d i v i d u a l ' s instantaneous consumption at 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 to be a For a f u r t h e r d i s c u s s i o n of t h i s point, see pp. 46 of Merton, R. C , "A Dynamic General Eq u i l i b r i u m Model of the Asset Market and i t s A p p l i c a t i o n to the P r i c i n g of the C a p i t a l Structure of the Firm," Massachusetts I n s t i t u t e of Technology, Sloan School of Management, October, 1970. 74 s t r i c t l y concave von Neumann- Morgenstern u t i l i t y f u nction; BF [W(T ), T J i s JC X K. a steady concave 'bequest' or u t i l i t y of terminal wealth function; and i s the date of death of the k*"*1 i n d i v i d u a l . I t should be noted that i m p l i c i t i n Q the above formulation i s the assumption of an a d d i t i v e u t i l i t y f u n c t i o n . If c e r t a i n assumptions are made about the p r i c e dynamics of the stock 9 and bond p r i c e s , then by the technique of s t o c h a s t i c c o n t r o l theory, the optimal consumption and investment r u l e s f o r the i n d i v i d u a l can be derived and thus the i n d i v i d u a l ' s demand functions f o r the d i f f e r e n t f i n a n c i a l a s sets. Hence, by aggregating across i n d i v i d u a l s and u s i n g the e q u i l i b r i u m c o n d i t i o n of zero excess demand, then the e q u i l i b r i u m instantaneous expected r a t e s of return can be determined. Such a problem has been considered by both Samuelson 1^ and M e r t o n , 1 1 the former t r e a t i n g the d i s c r e t e time case and the l a t t e r the continuous time case. Both, however, tr e a t e d c a p i t a l s t r u c t u r e as i r r e l e v a n t and thus con-sidered a l l equity firms. g For the case of m u l t i p l i c a t i v e u t i l i t y functions, see Pye, G., "Lifetime P o r t f o l i o S e l e c t i o n i n Continuous Time f o r A M u l t i p l i c a t i v e Class of U t i l i t y Functions," American Economic Review, V o l . LXIII, 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 technique i s given i n Bellman, F., Dynamic Programming (Princeton, N.J.: Princeton U n i v e r s i t y Press, 1957). 1 0Samuelson, P. A., "Lifetime P o r t f o l i o S e l e c t i o n by Dynamic Stochas-t i c Programming," Review of Economics and Statistics, V o l . LI, No. 3 (August, 1969), pp. 239-246. 1 1Merton, R. C , "Lifetime P o r t f o l i o S e l e c t i o n Uncertainty: The Con-tinuous Case," Review of Economics and Statistics, V o l . L i , No. 3 (August, 1969), pp. 247-257. 75 Price Dynamics I t i s proposed to represent the p r i c e movements of a firm's equity and bonds by s t o c h a s t i c d i f f e r e n t i a l equations. As i t i s not intended to present a rigorous d e r i v a t i o n of the equations, the i n t e r e s t e d reader should r e f e r to ~12 the paper by Ito, i n which the theory of s t o c h a s t i c d i f f e r e n t i a l equations 13 was f i r s t advanced i n 1951. 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 equations to describe the p r i c e dynamics of bonds and common stocks has been 14 extensively u t i l i z e d by Merton. f o r the d e r i v a t i o n of optimal consumption and investment r u l e s . The assumptions made about the p r i c e dynamics of the stock and bond p r i c e s are very important f o r they have d i r e c t bearing upon the derived form o f the expressions f o r the e q u i l i b r i u m instantaneous expected rates of r e t u r n . A complete d e s c r i p t i o n o f the 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 the supply side of the firm; t h a t i s , to r e l a t e the r e a l assets and the produc-t i o n f u n c t i o n of the f i r m to the p r i c e dynamics of the firm's stocks and bonds traded i n the c a p i t a l markets. For example, to s p e c i f y the p r i c e dynamics f o r the firm's stocks requires some assumption about the dividend p o l i c y of the f i r m . I f i n the equity p r i c e equation dividends are t r e a t e d as a random v a r i a b l e , then i n order to have a closed system o f equations i t i s necessary 12 Ito, K., "On Stochastic D i f f e r e n t i a l Equations," Memoirs of the American Mathematical Society, No. 4 (1951), pp. 1-51. 13 Other references are It6, K., and McKean, H. P., Diffusion Processes and Their Sample Paths (New York: Academic Press, 1964); and Kushner, H. J . , Introduction to Stochastic Control (New York: Holt, Rinehart & Winston, Inc., 1971) . 14 Merton, R. C., "Optimal Consumption and P o r t f o l i o Rules In A Con-tinuous Time Model," Journal of Economic Theory, V o l . 3 (1971), pp. 373-413. 76 to s p e c i f y an equation d e s c r i b i n g the dividend payments over time. However, unless some assumption i s made about the form of the equation at the outset, 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 of the firm's behaviour i n determining i t s investment and f i n a n c i a l p o l i c i e s over some future time horizon. I t w i l l be assumed that the supply side of the f i r m i s f i x e d and taken as given. Dividends can e i t h e r be t r e a t e d by assuming tha t the fi r m does not make a c t u a l dividend payments, but issues or repurchases i t s own shares i n the market, or by assuming at the outset a form of an equation tha t describes dividend behaviour over time. From assumption A9 trading takes place continuously i n time and thus any representation of the p r i c e dynamics of a firm's stocks or bonds should a l s o be i n a continuous time framework. In p r a c t i c e coupons or dividends are paid on a d i s c r e t e time b a s i s . D i s c r e t e payments of e i t h e r coupons or dividends represents a major t h e o r e t i c a l problem i n continuous time models, f o r i t destroys the symmetry of the representation and i t i s no longer p o s s i b l e to have compact d i s t r i b u t i o n s . 1 * ' Symmetry i s important f o r i t o f f e r s a tremendous s i m p l i f i c a t i o n f o r both the s p e c i f i c a t i o n of the p r i c e equations and f o r the d e r i v a t i o n of o p t i m a l i t y c o n d i t i o n s . I f symmetry i s not preserved then i t would be neces-sary to i d e n t i f y the timing of the asymmetric events and t o keep track of This issue i s discussed i n Merton, R. C , "An Intertemporal C a p i t a l Asset P r i c i n g Model," Working Paper No. 588-72, Massachusetts I n s t i t u t e of Technology, Sloan School of Management, February, 1972. ^ A simple explanation of compact d i s t r i b u t i o n s i s given i n Samuelson, P. A., "The Fundamental Approximation Theorem o f P o r t f o l i o A n a l y s i s i n Terms of Means, Variance, and Higher Moments," Review of Economic Studies, V o l . 37 (Octo-ber, 1970), pp. 537-542. 77 how f a r ahead i n time these events w i l l occur. Thus the whole s p e c i f i c a -t i o n process becomes f a r more complicated. Compact d i s t r i b u t i o n s are important because f o r small time i n t e r -v a l s the uncertainty neither "works out" (that i s , zero variance) nor domi-nates the a n a l y s i s (that i s , i n f i n i t e v a r i a n c e ) . Some pr o p e r t i e s of compact 17 d i s t r i b u t i o n s and t h e i r usefulness are given i n Samuelson. Given the assumption A2 of zero t r a n s a c t i o n costs, there i s no r e a -son why a f i r m should not pay a coupon or a dividend on a continuous time b a s i s . One could always imagine the f i r m paying a continuous coupon or d i v i -dend to a trustee who would d i s t r i b u t e the coupon or dividend on a d i s c r e t e time bas i s i n the name of the company, as i n p r a c t i c e both coupon and d i v i -dends are paid on a d i s c r e t e time b a s i s . I f a f i r m goes bankrupt, then i t i s assumed that 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 of reo r g a n i z a t i o n being neglected. The bondholder w i l l r e -ceive, on a pro r a t a b a s i s , the value of the f i r m , net of taxes, minus the d i r e c t costs associated with bankruptcy. I t i s assumed t h a t the shareholder w i l l r e c e ive nothing. The assumption that the f i r m undergoes l i q u i d a t i o n , and not re o r g a n i -zation avoids two d i f f i c u l t problems: the v a l u a t i o n of a firm's s e c u r i t i e s both w h i l s t i t i s being reorganized 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 continuous time of the p r i c e dynamics tha t i s symmetric and of a compact d i s t r i b u t i o n a l form during these periods. This i s j u s t one fac e t o f the much broader problem that there are no theories of the f i r m that p e r t a i n Samuelson, P. A. "General Proof That D i v e r s i f i c a t i o n Pays," Jour-nal of Financial and Quantitative Analysis, V o l . 2 (March, 1967), pp. 1-13. 78 to the state of bankruptcy and liquidation. It is assumed that when an investor buys the bonds of a firm a sub-jective evaluation is 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 firm does go bankrupt; that i s , the investor form a subjective probability distribution of the liquidation value of the firm (on a pro-rata basis) i f bankruptcy occurred. If bankruptcy does not occur i t i s assumed that the price 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, for the j firm, bj(t+h) represents the price of a bond at time t+h; r. represents the instantaneous conditional expected rate of re-turn for the firm's bonds, conditional on the fact that the firm does not go bankrupt; g. represents the instantaneous conditional coupon rate on the ^ firm's bonds; 2 represents the instantaneous conditional variance; and y.(t) represents a zero mean, unit variance, purely random process; 3 that i s , y(t) and y(t+s), s > 0 are independent and identi-c a l l y distributed gaussian random variables. Implicit in the above formulations are a number of very important assumptions. F i r s t , the maturity of the bond has been neglected. It has been assumed that the bonds are perpetuities. An alternative approach would be to assume that a l l bonds had a long, but f i n i t e maturity such that' a l l bonds matured after the . InrHvir l i iAi • s death. 79 There are two major reasons for making such an assumption about the bonds being p e r p e t u i t i e s . I f i t i s assumed that the bonds had f i n i t e matur-i t y and matured within the l i f e t i m e of the i n d i v i d u a l , i t would then be necessary to s p e c i f y the new f i n a n c i n g the f i r m undertakes. This w i l l de-pend upon the investment o p p o r t u n i t i e s which face the f i r m at that time and upon the general economic c o n d i t i o n s . The b a s i c problem i s to construat a representation i n continuous time to describe the p r i c e movements of the firm's bonds taking i n t o account the p o i n t of d i s c o n t i n u i t y which might occur a t the po i n t i n time when", the bonds mature and the f i r m makes a new debt issue. C l o s e l y r e l a t e d t o t h i s i s the question of symmetry. I f a firm's bonds mature and the f i r m makes a new bond issue with d i f f e r e n t terms to those th a t have j u s t matured, then i t i s necessary to keep track o f t h i s event i n determining the invest o r ' s optimal contingent s t r a t e g i e s . Hence, the whole s p e c i f i c a t i o n process becomes f a r more complicated when symmetry i s broken. The second assumption about the formulation i s the i n c l u s i o n of a random element term. I f i t i s not present then, given a general e q u i l i -brium state, the i n d i v i d u a l would know with c e r t a i n t y what the p r i c e of the bond would be a t the end o f the p e r i o d . Unexpected changes, whether they be i n the general l e v e l of i n t e r e s t r a t e s , the p r o b a b i l i t y of bankruptcy, or general economic c o n d i t i o n s , a f f e c t the future p r i c e of the bond and thus the random element i s added i n an attempt to catch these e f f e c t s . I t should be noted that i f the bond had a f i n i t e maturity, then i t would be necessary f o r the variance term o f the random element to be a f u n c t i o n of the time to 80 maturity. This i s because at maturity the value of the bond, given that the f i r m has not defaulted, w i l l be the p r i n c i p a l value of the bond and w i l l thus be independent of the influence of future expectations. If bankruptcy occurs, i t i s assumed that the p r i c e of the bond a t the end of the period can be represented by the expression b..(t+h) =A j(t+h) - 9_.(t+h), (4.3) j = 1, 2,...,n, where A_. (t+h) 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 of til bonds outstanding for the j f i r m i f bankruptcy occurred at time t+h; and 0..(t+h) represents the d i r e c t costs associated with bankruptcy a t time t+h, per number of bonds outstanding, f o r the j f i r m . I t i s assumed that the investor forms a s u b j e c t i v e evaluation of the l i q u i d a t i o n value of the f i r m given that bankruptcy has occurred. The l i q u i -d a t i o n value o f the f i r m w i l l depend upon the expected general economic con-d i t i o n s prevalent a t the end of the period, as w e l l as upon the s t a t e , the type and m a r k e t a b i l i t y of the firm's assets. I t i s assumed t h a t A_. (t+h) represents the mean value of the investor's s u b j e c t i v e evaluation of the f i r m given t h a t bankruptcy has occurred. Before a l i q u i d a t i o n premium can be paid to the bondholders, the d i r e c t costs associated with b a n k r u p t y — l e g a l fees, t r u s t e e fees, referee fees, a d m i n i s t r a t i v e costs — must be paid. These are represented by the term 0_. (t+h) . Thus, the amount which the bondholder expects to r e c e i v e , on a pro-rata b a s i s , i s thus b.(t+h) = Max[A^(t+h) - e^(t+h), 0] (4.4) 81 For e x p o s i t i o n a l s i m p l i c i t y , i n the formulation of the general equations d e s c r i b -ing the p r i c e dynamics, Equation (4.3) i s used, w h i l s t i t i s formally recognized that Equation (4.4) i s s t r i c t l y c o r r e c t . til Hence, the p r i c e dynamics of the j firm's bonds can be represented by *'b j(t) (1+r h) - g^h + b.(t) yVb y.. ( t ) ; i f no d e f a u l t , b.(t+h) = < A (t+h) - 9.(t+h) ; i f d e f a u l t , I j 3 j = 1, 2,...,n. (4.5) I t i s assumed that the event of bankruptcy can be described by the 18 f o l l o w i n g type o f s t o c h a s t i c process: t h Pr{j f i r m not going bankrupt i n (t,t+h] } = 1-X_. (t)h and (4.6) P r { j t h f i r m going bankrupt i n (t,t+h]} = X ^ ( t ) h , j = 1/ 2,...,nf where Xj (t)h can be 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 bankruptcy f o r f i r m j during the p e r i o d (t,t+h]. I t s determinants are e x t e n s i v e l y discussed i n Chapter I I I . I t i s assumed that the event of bankruptcy f o r one f i r m does not a f f e c t other firms. Conceptually, i t i s very simple to r e l a x t h i s assumption, but only at the cost of g r e a t l y i n c r e a s i n g the complexity of the notation. The very small gain i n g e n e r a l i t y of derived r e s u l t s does not warrant t h i s c o s t . A more rigorous d e v i a t i o n using Poisson p r o b a b i l i t y d i s t r i b u t i o n s i s given i n Appendix A. 82 Define a random v a r i a b l e i n d i c a t o r function, I.. (t,t+h) which can take only two values, zero and one, such that P r [ I . ( t , t + h ) =0] = 1 - X . ( t ) h , 3 3 (4 .7) and Pr[Ij(t,t+h) = 1] = (t)h, 3 1, 2,...,n. The i n d i c a t o r function describes the status of the f i r m a t the end of the period; that i s , i t i n d i c a t e s whether the f i r m has gone bankrupt or not. Equation (4.5) can be w r i t t e n i n the form bj(t+h) = [bj (t) (1+r^.h) - g..h + b.. (t) yVh y . ( t ) ] [ l - I..(t,t+h)] + [Aj(t+h) - 0^(t+h)]I(t,t+h) If the event of bankruptcy does not occur to f i r m j i n the (t,t+h], then I.(t,t+h) i s zero. If bankruptcy does occur then I.(t,t+h) equal one. In 19 the l i m i t , as h tends to zero, i t can be shown that: db.(t) = [ b . ( t ) r . - g 1dt + b.(t)y.dZ. - {b.(t) - [A.(t+dt) - 6.(t+dt)]}dq.. : : : j u D D 3 3 : u j = 1,2,...,n, (4.8) where dq^ i s a Poisson process c h a r a c t e r i z i n g the event of bankruptcy f o r the j firm; and dZ. i s a standard Gaussian-Wiener process. th The p r i c e dynamics of the j firm's equity w i l l be a f f e c t e d by what happens to the firm's bonds, that i s , i f d e f a u l t occurs the value of equity, 19 Appendix A, Equation (A.2). 83 by assumption, w i l l be zero. I t i s assumed tha t the p r i c e dynamics can be expressed i n the form where and P j ( t + h ) = 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 0 ; i f d e f a u l t , j = 1 , 2 , . . . ,n t h p.(t) represents the market p r i c e of a share f o r the j f i r m J a t time t ; tt. represents the instantaneous c o n d i t i o n a l expected r a t e of J r e t u r n f o r the j f c firm's equity, c o n d i t i o n a l on the f a c t t h a t the f i r m does not go bankrupt; f . represents the instantaneous c o n d i t i o n a l dividend r a t e 3 f o r the j t h f i r m ; 2 a represents the instantaneous c o n d i t i o n a l variance f o r J the j t h firm; (4.9) Y ,.(t) represents a zero mean, u n i t variance, purely random pro-n 3 cess, that i s , Y (t) and Y n + . ( 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 . If i t i s assumed tha t there i s no p o s s i b i l i t y of bankruptcy, then the equation becomes Ap.(t) 3 ,^ = (a.-6.)h + a.A Y ( t ) , P j ( t ) 3 D 3 n+3 where 6^ i s the instantaneous dividend y i e l d . Taking the l i m i t as h tends to zero, then 84 dp.(t) —3 = (a -6 )dt + a.dZ ., (4.10) p.(t) ... j j D n+u where dZ represents a standard Gaussian-Wiener process. I f i t i s assumed n+j that a , 6 and a are constant, 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 20 normally d i s t r i b u t e d . In general, Equation (4.9) can be w r i t t e n i n the form p.. (t+h) = [ P j (t) (1+a^h) - f j h + p. (t)a./h Y n + j ( t ) ] [ l - l . . ( t , t + h ) l . If the f i r m does not go bankrupt i n the i n t e r v a l (t,t+h] then I (t,t+h) w i l l equal zero. I f i t does go bankrupt, then I (t,t+h) w i l l equal one and the value o f equity w i l l be zero. In the l i m i t as h tends to zero, i t can be shown t h a t : 2 1 dp^(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 to consider what are the d i f f e r e n c e s between the common stock and p e r p e t u i t i e s of a f i r m and to enquire how these d i f f e r e n c e s are r e f l e c t e d i n the set of Equations (4.8) and (4.11) that are used i n the model to describe the p r i c e behaviour of bonds and common stock. A bond-holder purchases a claim t o a f i x e d s e r i e s of payments, w h i l s t the common stockholder purchases a claim t o a v a r i a b l e cash flow. There i s mutual i n t e r a c t i o n between the p r i c e of the two f i n a n c i a l a s s ets. For example, an unexpected change i n the p r o b a b i l i t y of bankruptcy w i l l a f f e c t the p r i c e For a more general d e r i v a t i o n , see Merton, R. C. "Optimum Consump-t i o n and P o r t f o l i o Rules i n a Continuous Time Model," Journal of Economic Theory, V o l . 3 (1971), pp. 373-413. 21 Appendix A, Equation (A.3). 85 of both f i n a n c i a l assets. In the event of bankruptcy, the bondholder has p r i o r c l a i m to the assets of the fi r m . In the model the p r i o r i t y of claim of the bondholder i n the event of bankruptcy i s r e f l e c t e d i n the d i f f e r e n c e between the nature of the co-e f f i c i e n t s of the dq.. term. The bondholder might receive a l i q u i d a t i o n prem-ium, w h i l s t the common stockholder s u f f e r s a hundred per cent l o s s . The i n -t e r a c t i o n between the two sets of assets, i s represented by the c o r r e l a t i o n between the two Gaussian-Wiener processes dZ^ and d z n + j • What i s not repre-sented i s the d i f f e r e n c e between the claims t o a s e r i e s of f i x e d and v a r i a b l e payments. An attempt can 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 the i n s t a n -taneous c o n d i t i o n a l expected dividend r a t e , and then t o s p e c i f y some type of process d e s c r i b i n g how the instantaneous c o n d i t i o n a l d i v i d e n d r a t e might 2 2 change over time. Over time changes i n expectations occur as the information s e t a v a i l -able to the i n v e s t o r changes. That i s , new information may cause a r e v i s i o n i n the expectations about the behaviour of d i f f e r e n t f i n a n c i a l assets. I f the opportunity s e t i s constantly changing, then t h i s w i l l probably a f f e c t the structure of returns. In an attempt to analyze the e f f e c t s of changes i n expectations, i t i s assumed that changes i n the opportunity s e t can be described by the f o l l o w i n g set of s t o c h a s t i c d i f f e r e n t i a l equations: I t i s assumed at the outset that i t i s p o s s i b l e to s p e c i f y such an equation. I t i s recognized that t h i s brushes over the major problem that to j u s t i f y the s p e c i f i c a t i o n o f the equation would re q u i r e a complete d e s c r i p -t i o n of the supply side of the f i r m . 86 da. = F . (a ., t) dt + G . (a ., t) dQ., 3 3 D 3 3 D d°j = F n + j ( a j ' t ) d t + G n + j ( a j ' t ) d Q n + j ' d r j - F 2 n + j ( r j ' t ) d t + G 2 n + j ( r D ' t ) d Q 2 n + j ' ( 4 * 1 2 ) and d Y j = F 3 n + j ( Y j ' t ) d t + G 3 n + j ( Y D ' t ) d Q 3 n + j ' j =1, 2,. . ,,n, where F( ), and G( ) are specified functions, and dQ represents a standard Gaussian-Wiener process. The f i r s t two equations describe the changes to the price dynamics of equity. The f i r s t equation shows how the instantaneous con-ditional expected return i s affected and the second equation how the instantan-eous conditanal standard deviation i s affected. The last two equations describe the changes to the price dynamics of the bonds. In a similar fashion, i t i s assumed the probability of bankruptcy is also stochastic over time, and the investor attempts to take into con-sideration the effects of such changes; that i s ~ F * . .(X.,t)dt + G,. .(X ,t)dO. ., (4 13) D 4n+D D 5n+j j ' ' Mn+j' l^.J-j; D = l / 2 , . . . , n This implies that X..(t)h, as defined i n Equation (4.6), should be reinter-preted as the mean value of the probability of bankruptcy for the j t h firm in the interval (t, t+At]. I 87 Analogous to the above reasoning, the equation d e s c r i b i n g the changes i n the instantaneous c o n d i t i o n a l expected dividend rate i s assumed to be dfj = Wfj't)dt + WV t ) d Q5n +r (4-14) An instantaneous r i s k l e s s asset means t h a t at each instance of time, each investor knows with c e r t a i n t y the rate of return, r , over the next i n -stant by holding the asset. However, the f u t u r e values of r are not known til with c e r t a i n t y . By convention the (2n+l) f i n a n c i a l asset i s taken to be the instantaneous r i s k l e s s asset. Hence, the p r i c e dynamics can be described by „ " * = r d t , ( 4 . 1 5 ) P 2 n + 1 ( t ) and i t i s assumed that changes i n the rate of r e t u r n can be described by dr = F ( r , t ) d t + G (r,t)dQ_, ( 4 . 1 6 ) m m Tti where m = 6n+l. State Space D e s c r i p t i o n and the Budget Con s t r a i n t Before proceeding to derive the budget c o n s t r a i n t and the o p t i m a l i t y equations, a b r i e f d i g r e s s i o n on the i m p l i c a t i o n s of the f a c t that both the p r i c e behaviour of equity and bonds are i n f l u e n c e d by whether or not a firm has defaulted w i l l help to c l a r i f y the d e r i v a t i o n of the equations. 88 If there are n-firms i n existence at time t, then at time t+h there are 2 n p o s s i b l e s t a t e s , where a state i s defined as a d e s c r i p t i o n of the n firms, l i s t i n g those firms which have defaulted and those which are i n e x i s t -ence at time t+h. Given the assumption tha t the event of bankruptcy by one f i r m has no a f f e c t upon the remaining firms, then i t i s only necessary to consider (n+l)of the p o s s i b l e s t a t e s , f o r the other s t a t e s have a p r o b a b i l -i t y of order 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 process i s used. Consider three firms, A, B, and C. The eight states of the system are shown i n Table 4.1. The p r o b a b i l i t y t h a t no firms d e f a u l t i n the i n t e r -v a l ( t , t + h ] equals (1-A.jh) (1-X2h) (1 - A 3 h ) 3 = 1 - n . h + 0 (h) / where X ^ h equals the p r o b a b i l i t y of bankruptcy f o r f i r m A i n the i n t e r v a l ( t , t + h ] . The p r o b a b i l i t y that one firm, say f i r m A, going bankrupt and the others do not, equals X j h d - X 2 h ) (1 - X 3 h ) = X h + 0 ( h ) . From Table 4.1, i t i s c l e a r t h a t only s t a t e s 1, 2, 3 and 4 are important. Thus only those states where one or l e s s bankruptcies occurred need to be considered. This g r e a t l y s i m p l i f i e s the a n a l y s i s . TABLE 4.1 THE PROBABILITY OF OCCURRENCE OF DIFFERENT STATES STATES FIRMS PROBABILITY OF OCCURRENCE A B C 3 1 NB NB NB 1- E X.h + 0(h) 3=1 3 2 B NB NB X^h + 0(h) 3 NB B NB X 2h + 0(h) 4 NB NB B X 3 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 NB 0(h) = bankruptcy occurred i n (t,t+hj = bankruptcy d i d not occur i n (t,t+h] = of order h 90 To derive the budget equation i t i s necessary to examine the d i s -c r e t e time formulation of the model and then to take l i m i t s (that i s , l e t h tend to zero) to obtain the continuous time formulation. Let W(t) represent the i n v e s t o r ' s t a n g i b l e wealth a t time t - ; C(t) represent the i n v e s t o r ' s consumption rate a t time t; y(t) represent the investor's wage income (may be stochastic) a t time t; and I ( t ) represent the investor's investment a t time t+ where s u p e r s c r i p t s denoting the 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 required f o r c l a r i t y . The amount that an i n d i v i d u a l can i n v e s t at time t+ depends upon current t a n g i b l e wealth, wage income, and planned consumption; tha t i s , I( t ) = W(t) - [C(t) - y ( t ) ] h . . (4.17) Let N. (t) represent the number of bonds o f f i r m j purchased during p e r i o d (t,t+h]; N ,.(t) represent the number of shares of f i r m j pur-chased during (t,t+h]; N 2 n + 1 ^ represent the number of shares of the i n s t a n t -and aneous r i s k l e s s asset purchased during (t,t+hj; w^(t) represent^the f r a c t i o n of the investment invested i n the k asset during (t,t+h], 91 j = 1 , 2, . . .,n, and k = 1 , 2,. . .,2n+l. Hence, the i n d i v i d u a l ' s investment can be represented by I(t) - E N (t)b (t) + ? N (t)p (t) + N 2 n + 1 p 2 n + 1 ( t ) . 3=1 J 3=1 J The i n d i v i d u a l ' s wealth at the end of the period, t+h, w i l l depend upon the p r i c e of the bonds and shares, the coupons and dividends received, and the state of the system; t h a t i s , upon which firms t h a t went bankrupt i n the i n t e r v a l (t,t+h]. Suppose tha t no firms went bankrupt i n the i n t e r v a l (t,t+h], then the wealth a t the end of the i n t e r v a l can be represented by n n W(t+h) = E N.(t) [b.(t+h) + g.h] + E N , . (t) [p. (t+h) + f.h] j-1 3 3 3 j = l n + 3 3 3 + N 2 n + l ( t ) P 2 n + l ( t > n b.(t+h)+g.h n p.(t+h)+f.h = I ( t ) . ^ W 3 ( t ) [ ' b . ( t ) P "I + I ( t , j ; i V j ( t ) [ 3 P j ( t ) 1 1 + KtJw,.,, (t) P(t+h) 2n+l 2n+l* ' p(t) 2n+l It w i l l be assumed that a l l income i s derived from investment i n the f i n a n c i a l assets; that i s , y(t) = 0, which implies t h a t I ( t ) = W(t) - C ( t ) h . 92 Thus the change i n wealth can be expressed i n the form n b.(t+h) + g.h - b (t) w(t+h)- W(t) = (w(t) - C(t)h}{ E w.(t) [-3 ITTt)— r h ] j=l 3 j n p (t+h) + f ,h - p . (t) + E w . (t) [ -2 - 1 — 3 _ r h ] + rh} - C(t)h, j = l 1 1 + 3 p j ( t ) 2n+l where Equation (4.15) and the r e l a t i o n E w.(t) = 1 have been used. S u b s t i t u t i n g j= l 3 Equations (4.8) and (4.11) in t o the above expressions gives n W(t+h) - W(t) = {W(t) - C(t)h}{ E w. (t) [ ( r ^ - r ) h + Y^dZj j = 1 D 1 1 1 n + E w (t) [ (a .-r)h + a.dZ .] + rh} - C ( t ) h + 0(h). I A j = 1 n+: 3 3 n+n (4.18 Let AW(t) = W(t+h) - W(t), then the expected value of AW(t) c o n d i t i o n a l on the f a c t that no bankruptcies occurred i n the i n t e r v a l (t,t+h] i s n n E [AW(t) ] = [W(t) - C(t ) h ] [ E w.(b)(r.-r) + E w .(a.-r) + r] h t j = l 3 3 j-1 n + 3 3 - C ( t ) h + 0(h) , (4.19) and 2 n n n n E.[AW(t = W(t) r E E w.(t)Y•.w.(t + 2 E E w,(t)y•P. .o.w .(t) t 1 • -i • i D ] i i . - , . , D D J i i n+i 1=1 i = l J 1=1 i = l J J J 3=i ]=i 3=1 3= n h + E E w (t)a..w .(t)I h + 0(h), j=l j-1 n + : ) 3 1 n + 1 (4.20) when p.. 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 between dZ. and dZ 31 3 n+i ti l Suppose that the j firm goes bankrupt i n the i n t e r v a l (t,t+h]. This w i l l a f f e c t the investors holding the j t h firm's bonds and equity, the bond-holder might receive a l i q u i d a t i o n premium, w h i l s t the equity holder w i l l 93 s u f f e r a hundred 'per cent l o s s . Hence, the wealth a t the end of the i n t e r -t h v a l , c o n d i t i o n a l on the f a c t that the j fir m has gone bankrupt, 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) [ — . . 1 ] + w. (t) [ -3 -3 ] i = l 1 i ' 3 j ' ' n p. (t+h) + f .h . P (t+h) + E w . (t) [— r n —1 + w_ x . (t) • -A; ) . , n+i p. (t) 2n+l p (t) 1=1 i 2n+l Substituting Equations (4.8), (4.11) and (4.15) i n t o the above equations 2n+l and using the r e l a t i o n E w.(t) = 1, gives j = l 3 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 — 3 - 1 i = 1 i 1 1 1 3 hj (t) n + s w_ 4^ (t) [ (a.-r)h + 6\dZ ± 1 ] - w _,_. (t) + rh} + C ( t ) h + 0(h). i = l . , n+i l l n+i n+3 A. (t+h) - 6.(t+h) - W^V^ [ J b - t t f 2 " 1 1 " W n + j ( 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 * ) A.(t+h) -.9.(t+h) - C ( t ) h { l + w. (t) r ~* w ... 3 - 1] - w (t)} + 0(h) 3 1 b.(t) ' n+3 3 The f i r s t term i n the above expression, that i s , A (t+h) - 8.(t+h) w(t){w. (t) [ - J r—jrr1 " 11 -w .(t)} 3 b_.(t) n+3 (4.21) 94 can be interpreted as the loss to the i n d i v i d u a l who invested i n the j r n firm's s e c u r i t i e s , given that the fir m goes bankrupt i n the i n t e r v a l (t,t+h]. If the i n d i v i d u a l paid b_. (t) f o r the bond at the beginning of the period, then at the end of the period the i n d i v i d u a l w i l l receive a l i q u i d a t i n g premium of A., (t+h) -6_.(t+h), which i s les s than the i n i t i a l amount paid f o r the bond, given that 6 (t+h) i s greater than zero. S i m i l a r l y , f o r equity, i f the i n d i v i d u a l paid P j ( t ) a t the beginning o f the period, then the i n d i v i d u a l w i l l s u f f e r a hundred per cent loss a t the end of the period. Note, however, that as short s e l l i n g i s allowed, the argument can be reversed t o g i v e the gains t h a t occur when the firm goes bankrupt. The Equation of Optimality: The Demand Functions f o r Assets The i n d i v i d u a l i s assumed to act i n such a manner so as to maximize the expected l i f e t i m e u t i l i t y of consumption and terminal wealth; that i s , r e w r i t i n g Equation (4.1) T Max E {/ U[C(S),s]ds + BF[W(T),T]}, (4.1) o o subject to an i n i t i a l wealth co n s t r a i n t , the budget c o n s t r a i n t , and C(s) >_ 0, and where the superscripts denoting the investor's i d e n t i t y have been dropped, except when required f o r c l a r i t y . Define T J[W(t),a,a,r,Y,f,X,r , t , s ( t ) ] = Max E {/ U[C(s),s]ds + BF[W(T),T]} (4.22) ~ F {C,w} fc fc where s(t) i s a state vector which describes what firms are s t i l l i n existence at time t; 2/°_f r,Y,» f_, and X_ are vectors which describe the values at time t of {a . },{a.},{r.},[y.},{f.} and {X.} r e s p e c t i v e l y ; and r i s the value of the 3 3 3 j 3 3 F ra t e o f return a t time t on the instantaneous r i s k l e s s asset. The functio n , J, i s c a l l e d the derived u t i l i t y of V7ealth. I t s arguments are the st 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 the i n d i v i d u a l i s to choose values f o r {C,w} which maximizes the expression on the r i g h t hand sid e o f (4.22); t h a t i s , the d e c i s i o n v a r i a b l e s are the r a t e of consumption C, and the proportion of wealth to in v e s t i n the d i f f e r e n t f i n a n c i a l assets {w.}. Thi s may be achieved u s i n g the Bellman p r i n c i p l e of . 23 o p t i m a l i t y . 24 I t can be shown t h a t the 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 who acts according to Equation (4.1) i n determining the consumption-investment contingent strategy at each p o i n t i n time, are m 0 = Max (U[C(t),t] + J + E F.J. {C,w} t j = l 3 3 n n + J„(W(t) [ E w. (t) (r.-r) + E w (t) (o.-r) + r] - C(t)} W j - i 3 3 j = 1 n+ 3 3 ' ^ W W ^ j ^ j ^ D i V ^ + 2 j x. j 1 W j ( t ) Y j P j i d i W n + i ( t ) tt n 2 + E E w . (t)a. .w (t)]W(tr j-1 i-1 n + 3 3 1 n + 1 m m m n + — E EG.V..G.J..+ E E W(t)w. (t)y.n. .G.J 2 i-1 j-1 1 1 3 3 1 3 i-1 j-1 3 3 1 3 1 l W m n + E E W(t)w .(t)a.n. ^..G.J... . . . . n+3 3 i»n+3 l iW 1=1 3=1 n + E X.(t){j[W., V ( t ) , t, S.] - J[W(t), V(t), t, S(t)}), (4.23) j= l 3 3 3 23 For a formal statement of t h i s p r i n c i p l e , see pp. 15 of Bellman, R. E. and Dreyfus, S. E., Applied Dynamic Programming (Princeton, N.J.: Princeton U n i v e r s i t y Press, 1962). See Appendix A, Equation (A.17). 9 6 subject to the boundary c o n d i t i o n J[W(T),V(T),T,S(T)] = BF[W(T),T], and where subscripts on the function J [ W ( t ) , V ( t ) , t , S ( t ) ] denote p a r t i a l d e r i v a -th tions; S . i s a state vector denoting th a t the i fir m no longer e x i s t s ; a... 3 i j i s the instantaneous c o n d i t i o n a l covariance between dZ. and dZ.; p.. 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 dZ and dZ , .; v. . 1 n+3 13 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 dQ.; i 3 n. . 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 13 1 25 dZ.; and W. i s defined by 3 3 A. - 9 . Wj(t) = W(t){l + wj ( t ) [ ( t ) - 1] " w n + j ( t ) } ' ( 4 ' 2 4 ) t h which can be int e r p r e t e d as the new wealth p o s i t i o n a f t e r the 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 equation which describes the behaviour of the derived u t i l i t y f u n c t i o n . As such, there are no s t o c h a s t i c elements i n the equation; i t i s completely d e t e r m i n i s t i c . The value of the derived u t i l i t y f u n c t i o n depends upon the consumption r a t e and the amount of investment i n the d i f f e r e n t f i n a n c i a l assets t h a t are a v a i l a b l e . The optimal values of these d e c i s i o n v a r i a b l e s that maximize the derived u t i l i t y f u n c t i o n at each poi n t i n time, can be determined by so l v i n g the set of equations which describe the f i r s t order c o n d i t i o n s f o r a maximum. The assumptions about the form of the u t i l i t y f u n c t i o n and the bequest f u n c t i o n ensure that a maximum, 26 and not a minimum, i s obtained. 25 See Appendix A, Equations (A.15) to (A.20), where a d e t a i l e d d i s c u s s i o n i s given f o r the reasons motivating t h i s d e f i n i t i o n . 26 A proof i s given i n Kushner, H. J . , Stochastic Stability and Control, (New York: Academic Press, 1967). ! 97 The (2n+l) f i r s t order c o n d i t i o n s are obtained by f i r s t d i f f e r e n t i a t i n g 27 Equation (4.23) with respect to the r a t e of consumption 0 = U [C(t) ,t] - J ; (4.25) c w then by d i f f e r e n t i a t i n g (4.23) with respect to the amount of investment i n . . 28 , . , -, eq u i t i e s ; that i s , ^ w n + j J : 0 = (a - r ) J w + W(t)[ ? a w (t) + ? y P a.w. ( t ) ] ( 4 . 2 6 ) i = l „.. i = l J m j = i , 2,. • •,n; and f i n a l l y , d i f f e r e n t i a t i n g (4.23) with respect to the amount of investment 29 i n bonds; that 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[W.,V(t),t,S.], (4.27) j ~ 1, 2,. « *,n« Equation (4.25) i s the intertemporal envelope c o n d i t i o n : the marginal u t i l i t y of current consumption equals the marginal derived u t i l i t y of wealth. Equation (4.26) describes a system of equations f o r the investment i n the commonstocks availableand Equation (4.27) describes a s i m i l a r system f o r bonds. See Appendix A, See Appendix A, See Appendix A, Equation (A.18) Equation (A.19). Equation (A.20). 98 The two systems are not independent. I f a f i r m goes bankrupt, the event of bankruptcy w i l l a f f e c t both the value of i t s bonds and equity. Events that a f f e c t the value of the firm, f o r example unexpected changes i n the p r o b a b i l -i t y of bankruptcy, w i l l be r e f l e c t e d i n changes i n both the value of the firm's bonds and equity. The lack of independence between the two systems implies that the two sets of equations must be solved simultaneously. A s o l u t i o n w i l l not, however, be easy to obtain because the equations are non-l i n e a r . The n o n - l i n e a r i t y r e s u l t s from the presence of the terms {j[W_.,V(t) ,t,s^] } which i s a consequence of the f a c t that bankruptcy causes a d i s c o n t i n u i t y i n the wealth of the i n d i v i d u a l . At t h i s l e v e l of g e n e r a l i t y , l i t t l e i n s i g h t into the imp l i c a t i o n s of the set of equations can be gained. I t i s proposed to add some fur t h e r , and s i m p l i f y i n g , assumptions to r e s t r i c t the structure of the opportunity set. Two models w i l l be considered. The f i r s t i s a simple model i n which the investment opportunity s et i s assumed only to be a l t e r e d by the event of bankruptcy; the p r o b a b i l i t y of bankruptcy i s assumed not to change s t o c h a s t i -c a l l y over time. The equations that describe the bond p r i c e dynamics are also s i m p l i f i e d . Whilst such a l e v e l of s i m p l i c i t y i s u n r e a l i s t i c , i t does a f f o r d penetrating i n s i g h t into how the mechanism of bankruptcy a f f e c t s the structure of returns. The second model relaxes the assumption that the p r o b a b i l i t y of bankruptcy f o r a f i r m does not change s t o c h a s t i c a l l y over time. The framework i s more r e a l i s t i c than that of the f i r s t model, but does not o f f e r the same l e v e l of i n s i g h t . 99 Bankruptcy and Structure of Returns By reducing the l e v e l of g e n e r a l i t y of the formulation enables greater i n s i g h t into the impact of bankruptcy upon the mechanism d e s c r i b i n g the s t r u c -ture o f ret u r n s . I t i s assumed that the opportunity set c h a r a c t e r i z e d by {a,a,r,y,f,X,r_} i s d e t e r m i n i s t i c ; t h a t i s , there are no s t o c h a s t i c changes F i n these parameters so that the i n d i v i d u a l knows with c e r t a i n t y t h e i r future values. I t i s furth e r assumed that there i s no s t o c h a s t i c element to the c o n d i t i o n a l equation d e s c r i b i n g the p r i c e dynamics of bonds, t h a t i s , Equation (4.5) becomes b. (t) (1+r^.h) - g^.h; i f no d e f a u l t b..(t+h) = < (4.28) A.(t+h) - 6.(t+h) ; i f d e f a u l t , 3 3 j = 1 / 2,. . .,n Whilst the absence of 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 th a t there w i l l be no mutually i n t e r a c t i o n between bonds and common stocks apart from the d i r e c t e f f e c t of bankruptcy. Thus, there w i l l be no i n t e r a c t i o n terms i n the expressions f o r the eq u i l i b r i u m r a t e s of r e -turn f o r bonds and common stock. From Equation (4.23) the equation o f o p t i m a l i t y becomes n n 0 = Max (U[C(t),t] + J t + J w ( w ( t ) [ I w . ( t ) ( r . - r ) + E ^ n + j (t) ( a ^ r ) + r ] _ c ( t ) } {c,w} J 1 3 ]_ n n n + 2  J m l A \ . \ w n + j ( t ) a j i w n + i < t ) J w ( t ) 2 + z A {J[W t,S ] - .Trw,t,S(t)]>) J = l 1-1 j = l J J J (4.29) 100 and the f i r s t order conditions are, a f t e r s i m p l i f i c a t i o n " ^ 0 = U [ C ( t ) , t ] - J , (4.30) c w r . - r 0 = (a. - r - -3- ) j + W ( t ) [ E a. .w (t)] J , (4.31) 3 L. W . . ] i n+i WW J j i = l J and 0 = ( r . - r ) J -A.L.J(W.,t), (4.32) 3 W 3 3 W 3 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 defined by A. - 9 L j • 1 - ~ \ d > which can be given an i n t u i t i v e meaning. Suppose an i n d i v i d u a l purchased a bond f o r b..(t) and the f i r m went bankrupt. The i n d i v i d u a l would r e c e i v e (A. - 8.) and thus s u f f e r a l o s s of [b. - (A. - 9 . ) ] . L. i s a percentage 3 D 3 3 3 3 measure of tha t l o s s . I f L_. equals ohe, 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 the intertemporal envelope condition: marginal u t i l i t y of consumption equals the marginal derived u t i l i t y of wealth. Equa-t i o n (4.31) describes a l i n e a r system of equations from which the demand func-t i o n s f o r equity can be determined independently from the d i r e c t i n f l u e n c e of the demand functions f o r bond. This independence i s a consequence of the assumption of having no random element term i n the bond equation. The system of Equations(4.32), that describe the demand functions f o r bonds are non-l i n e a r and thus, i n general, i t w i l l be d i f f i c u l t to o b t a i n an exact s o l u t i o n . 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 wealth that are caused by the event of bankruptcy. "*°See Appendix A, Equations (A.26), (A.27) and (A.20) 101 Consider f i r s t the demand functions f o r bonds, described by Equation (4.32). As the equations are non-linear, i t i s d i f f i c u l t to obtain an e x p l i -c i t s o l u t i o n . There are a t l e a s t two a l t e r n a t i v e s . The f i r s t i s to put more structure into the formulation by assuming a p a r t i c u l a r form f o r the i n d i v i -dual's u t i l i t y function and then attempt to solve the system of equations by a numerical i t e r a t i v e procedure. Whilst t h i s method might produce a s o l u t i o n , i t w i l l be at a cost. I t w i l l be d i f f i c u l t to derive e x p l i c i t forms f o r the general equilibrium rates of return, and even i f they could be obtained, they w i l l depend upon the s p e c i f i c assumed form f o r the u t i l i t y functions f o r the i n d i v i d u a l s . The lack of g e n e r a l i t y and i n t r a c t a b i l i t y of 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 to make an approximation so as to obtain a l i n e a r system. The approximation evolves around the assumption that i t i s p o s s i b l e to expand the d e r i v a t i o n s of the derived u t i l i t y f u n c t i o n i n a Tay-31 l o r ' s s e r i e s and to neglect quadrative and higher order terms; that i s , J w ( W ( t ) [ l - w_.(t)L. - w n + j ( t ) ] , t} = J [W(t),t] - W(t)[w.(t)L. + w ( t ) ] j T [ w ( t ) , t ] , w 3 3 ""'"3 WW 3 = 1, 2,. . »,n» For a quadratic u t i l i t y function t h i s approximation i s exact, w h i l s t f o r other cases of u t i l i t y functions, for example the constant r e l a t i v e r i s k aversion c l a s s , the approximation can be very good, depending upon the numerical values 32 of the parameters of the u t i l i t y f u n c t i o n . 31 See Appendix A, Equation (A.24). 32 For a f u l l d i s c u s s i o n , see Appendix A, Equations (A.52) to (A.62) 102 Using t h i s approximation, Equation (4.32) can be expressed i n the form ° = ( r j " r _ X j L j ) J W + X j L j ' I W J ( t > L J + W n + j ( t ) 1 W ( t ) J W W ' ( 4 " 3 3 ) 3 = 1/ 2,. . .,n. 33 From t h i s equation i t can be shown the ^ equilibrium instantaneous c o n d i t i o n a l expected r a t e s of r e t u r n are X ,L.[N.b.(t)L. + N .p.(t)] r . = r + X.L. + (TT-r-yX. 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 the t o t a l number of bonds outstanding f o r the j firm; N^+j i s th the t o t a l number of shares outstanding f o r the j firm; X^(t) i s the propor-th t i o n of the t o t a l market value of the j firm's bonds to the t o t a l market value of a l l bonds; ir i s the instantaneous c o n d i t i o n a l expected r e t u r n on the bond market, defined by n TT = E X . ( t ) r . ; (4.35) i = l 1 1 and y can be in t e r p r e t e d as a weighted sum of the expected l o s s i n the event of bankruptcy defined by n Y = E X . ( t ) L . X . . (4.36) i-1 1 1 1 Equation (4.34) can be in t e r p r e t e d as the instantaneous c o n d i t i o n a l ex-pected rate of return f o r a firm's bonds equals the sum of the r i s k f r e e r a t e of return, the expected los s i n the event of bankruptcy, and a market term. The second term, X , can be i n t u i t e d as the expected l o s s i f bankruptcy occurs. 3 3 See Appendix A, Equation (A.45). 103 It i s composed of two terms: the r a t e of the p r o b a b i l i t y of bankruptcy and the c o n d i t i o n a l expected l o s s , c o n d i t i o n a l upon the event of bankruptcy. The magnitude o f t h i s term w i l l be dependent upon the expected net value of the firm's assets a f t e r the event of bankruptcy. I t i s p o s s i b l e f o r a tr a d e - o f f to occur between these two terms: f o r example, the r a t e of p r o b a b i l i t y of bankruptcy might be la r g e w h i l s t the expected l o s s i n the event of bankruptcy might be small. 34 The second term, ^ j L j / c a n D e compared t o the formulation of F i s h e r , who hypothesized that the r i s k premium on a bond i s a fu n c t i o n of two terms: the p r o b a b i l i t y of d e f a u l t and the m a r k e t a b i l i t y of a bond. This l a t t e r con-s i d e r a t i o n i s not rele v a n t i n the present context given the assumptions about the s t r u c t u r e of the c a p i t a l markets. F i s h e r d i d not, however, d i r e c t l y con-s i d e r the impact upon the r i s k premium of the expected l o s s that might occur i n the event o f bankruptcy. The t h i r d term can be i n t e r p r e t e d as a market f a c t o r . I t i s composed, of two elements: the f i r s t , (n-r-y), can be i n t u i t e d as the instantaneous c o n d i t i o n a l expected r i s k premium on the bond market, and the 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 the instantaneous c o n d i t i o n a l expected rate o f re t u r n f o r a bond f o r the two cases o f when an i n d i v i d u a l does not su f f e r a l o s s i n the event of bankruptcy, L_.=0, and when the p r o b a b i l i t y of •bankruptcy i s zero. In both cases, the reouired rate of return i s the r i s k free rate, as would be expected. F i s h e r , L., "Determinants of Risk Premiums on Corporate Bonds," Journal of P o l i t i c a l Economy, V o l . LXV I I, NO. 3 (June, 1959); pp. 217-237. 104 The demand functions f o r equity are described by Equation (4.31). This system of equations i s l i n e a r and independent o f the d i r e c t i n f l u e n c e of the demand functions f o r bonds; that i s , i t does not contain terms l i k e {w_.(t)}. Th i s i s a consequence of the assumption about the p r i c e dynamics of bonds described by Equation (4.28) which does not contain a random element term that 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 the two sets of demand functions f o r bonds and equity. 35 From Equation (4.31) i t can be shown that the e q u i l i b r i u m i n s t a n -taneous c o n d i t i o n a l expected rates of re t u r n can be expressed i n the form r .-r a . - r ^ = g.(u-r-x) , (4.37) 3 j 3 3 = l f 2,. • .,n, where u i s the instantaneous c o n d i t i o n a l expected r e t u r n on the market, defined by n y = Z Y. ( t ) a . , (4.38) j = l 3 3 Yj (t) being the proportion of the market value of the j*"* 1 firm's equity to the t o t a l market value o f a l l equity; x i s defined by n r .-r X = £ Y. (t) {-} ); (4.39) j = l 3 L3 t h and 8. i s the instantaneous c o n d i t i o n a l covariance of the return of the j 3 firm's equity with the equity market, di v i d e d by the instantaneous c o n d i t i o n a l 3 6 variance of the re t u r n on the market, defined by 35 See Appendix A, Equation (A.39). 36 " From Equation (4.37) i t can be shown that I Y . ( t ) B . = 1. j = l 3 3 105 n E Y . ( t ) a . . i = l 1 3 1 B. = j n n (4.40) Z Z Y.(t)a..Y.(t) j-1 i - 1 3 3 1 1 B. i s c a l l e d the beta f a c t o r f o r the j f i r m . Greater i n s i g h t i n t o the s i g -3 r . - r n i f i c a n c e of Equation (4.37) can be gained by e l i m i n a t i n g the term ( - ^ — ) . Li , 3 This can be achieved by using the expressions f o r the instantaneous c o n d i t i o n a l t h 37 expected rate of return f o r the j firm's bonds. This gives N.b . (t ) L . + N .p. (t) a.-r-A. = ( y - r - x X B .+A .[-3-3 3 S J U ]}, (4.41) 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 the t o t a l market value of a l l equity. Using the d e f i n i t i o n of Bj given by Equation(4.40), Equation (4.41) can be w r i t t e n n N.b. (t ) L . + N .p. (t) a.-r-A. - iHZE3> •( Z o..Y.(t)+A.[-J-3 3 ^ ]} j 3 n n 31 l D n _ Z Z Y.(t)a..Y.(t) E N ,p.(t) . . . . 3 31 l . , n+1 1 3=1 1=1 i = l I f there are a large number o f firms, the l a s t term on the r i g h t hand side of the above expression can be neglected, as i t i s o f order 2/n, where n i s the number o f firms. Hence, the expression f o r the instantaneous c o n d i t i o n a l ex-pected rate of r e t u r n f o r the j * " * 1 firm's equity can be w r i t t e n a j - r - A j = B ' (u-r-x) (4.42) 37 I t i s not obvious from Equation (4.34) how Equation (4.41) i s derived. As shown i n Appendix A, Equation (A.46), an a l t e r n a t i v e form of Equation (4.34) can be developed. This a l t e r n a t i v e form has been used to eliminate Equation (4.37). r .-r J L. D from 106 The expression on the l e f t hand side of the above equation can be inte r p r e t e d as the instantaneous expected r i s k premium, as opposed to the th — instantaneous c o n d i t i o n expected r i s k premium f o r the j f i r m . The term x can be i d e n t i f i e d as a weighted average of the r a t e of the p r o b a b i l i t y of bank-38 ruptcy for a l l firms. Hence, the expression ( u - r ~ x ) can be i n t e r p r e t e d as the instantaneous expected market r i s k premium, and thus Equation (4.42) may be written i n the form E(R..) - r = SjEECR )^ - r l , (4.43) where E ( R J i s the instantaneous expected r a t e of r e t u r n on the equity of the th j firm.) and E (R^) i s the instantaneous expected r a t e of r e t u r n on the equity 39 market. This r e s u l t i s analogous t o t h a t derived by Merton, and i f the p r o b a b i l i t y of bankruptcy f o r a l l firms i s zero, the r e s u l t s are i d e n t i c a l . Both from a t h e o r e t i c a l and e m p i r i c a l p o i n t o f view the above r e s u l t s , whether they be expressed i n the instantaneous c o n d i t i o n a l expected form o f Equation (4.42) or the instantaneous expected form o f Equation (4.43), are important. T h e o r e t i c a l l y , the r e s u l t s show that the continuous time analogy to the c a p i t a l asset primary model i s s t i l l v a l i d f o r the case when bankruptcy i s e x p l i c i t l y considered, provided the instantaneous expected rates of r e t u r n are used and not the instantaneous expected r a t e s of 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 of the instantaneous c o n d i t i o n a l expected r e t u r n on the market, see (4.38), Equation (4.42) implies th a t n X = £ Y (t)X j=l 3 3 This 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 of the approxi-mation made i n d e r i v i n g Equation (4.42) and, as such, i s i t s e l f an approximation. See Appendix A, Equation (A.50) f o r a f u l l e r d i s c u s s i o n . 3 9Merton, R. C , "An Intertemporal C a p i t a l Asset P r i c i n g Model," Work-ing Paper 588-72, Massachusetts I n s t i t u t e of Technology, Sloan School of Manage-ment, February, 1972. 107 no bankruptcy. In the t r a d i t i o n a l c a p i t a l asset p r i c i n g model, where c a p i t a l structure i s assumed to be i r r e l e v a n t and bankruptcy i s t o t a l l y ignored, such a d i s t i n c t i o n i s not necessary. But 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 important from an emp i r i c a l viewpoint. In t e s t i n g the c a p i t a l asset p r i c i n g model, the assumption i s made that i t i s p o s s i b l e to go from an ex-ante to an ex-post formulation and to use r e a l i z e d returns. Plowever, from the way empirical t e s t s are conducted, the r e a l i z e d returns are proxies f o r the expected return, c o n d i t i o n a l upon no bankruptcy; t h a t i s , they are proxies f o r the terms {cO and not { E ( R J } . Thus, there i s a b a s i c misspeci-f i c a t i o n e r r o r . The e f f e c t of t h i s e r r o r can be demonstrated, as shown i n Figure 4.1. 40 Merton has shown that 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 f o r con-tinuous time i s of the form •<*j » r + B.. (u-r), (4.44) which i s denoted i n the f i g u r e by CAPM. I f A was constant and independent o f the p a r t i c u l a r firm, t h a t i s , A_. = A, f o r a l l j , then Equation (4.42) becomes 0^ = r + A + B ..(u-r-A). (4.45) This i s represented i n Figure 4.1 by the l i n e denoted by CAPM'. The l i n e i s l i n e a r and f l a t t e r than the l i n e CAPM due to the presence of the term A. How-ever, the rate of p r o b a b i l i t y of bankruptcy does vary across firms and thus, i n general, there w i l l be a non-linear r e l a t i o n s h i p between a and 0. I f i t i s 41 assumed that as B increases, A increases, then a l i n e of the form denoted by 40 -r1_ -J Merton, Iota, 41 1 Westerfield has presented some evidence j u s t i f y i n g t h i s assumption. Westerfield, R., "The Assessment of Market Risk and Corporate F a i l u r e , " Uni-v e r s i t y of Pennsylvania, Wharton School of Finance, August, 1970 (unpublished). 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 , 3 CAPM denotes the equation a. = r + 3.(u - r) 3 3 CAPM' denotes the equation a.. = r + X+ 3^(u - r - x) CAPM" denotes the equation a. = r + A . + 3 . (u - r - x) 3 3 3 109 CAPM" w i l l be obtained. Such a conclusion i s very important when viewed i n the l i g h t of recent e m p i r i c a l f i n d i n g s . The curve CAPM" i s derived from the equation (4.42) which can be re w r i t t e n i n the form a. - r = (6. - 0.x) + B-.(u-r), (4.46) 3 3 3 3 ^ 1^ *2. f • • * / n • If x i s °f t n e same order as X, then f o r small values of 8(6<1) the f i r s t term on the r i g h t hand side of the above equation w i l l be p o s i t i v e , w h i l s t f o r large values of 8(8>1) i t w i l l be negative. Equation (4.46) describes a r e l a t i o n a t a p a r t i c u l a r i n s t a n t i n time. There i s no a priori reason to suppose tha t the various 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 constant over time. For example, the rate of the p r o b a b i l i t y of bankruptcy may change because of a severe c r e d i t r a t i o n i n g . I f t h i s i s so, then there i s no reason f o r the curve CAPM" to be constant. Both conclusions are c o n s i s t e n t with the time s e r i e s r e s u l t s o f 42 Black, Jensen and Scholes who found th a t the i n t e r c e p t term of the c a p i t a l . . . 43 asset p r i c i n g model i s non-stationary and f o r a time s e r i e s regressed over a 30 year period, the i n t e r c e p t was c o n s i s t e n t l y negative f o r high r i s k p o r t -f o l i o s (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 (B<1). The m i s s p e c i f i c a t i o n of the model i n l i g h t of the above d i s c u s s i o n might a l s o explain the negative 42 Black, F., Jensen, M.C, and Scholes, M., "The C a p i t a l Asset P r i c i n g Model: Some Empirical Tests," p r i n t e d i n Studies in The Theory of Capital Mar-kets (Ed.) Jensen, M.C. (New York: Praeger, 1972). 43 *v> ^ The time s e r i e s r egression was of the form R •= a. + B,R..^  + e.. . ] t •) j Mt i t where R.fc i s the ex-post excess return on the market p o r t f o l i o over the same period, 3and e_ a random e r r o r term. 3t 110 r e l a t i o n s h i p found over c e r t a i n periods between average monthly returns and 44 systematic r i s k . The model, as represented by Equation (4.42), forms the b a s i s f o r the empirical t e s t i n g o f the hypothesis of the t h e s i s . A d i s c r e t e time formu-l a t i o n of the equation i s used. The p r o b a b i l i t y of bankruptcy i s estimated u t i l i z i n g the work developed i n Chapter I I I . Stochastic Changes i n the P r o b a b i l i t y o f Bankruptcy In the model j u s t considered knowledge of how the mechanism of bank-ruptcy a f f e c t e d the s t r u c t u r e of returns was gained using a simple model i n which the investment opportunity set d i d not change s t o c h a s t i c a l l y . Such an assumption i s r e s t r i c t i v e : the random a r r i v a l of hew information and the r e -assessment o f e x i s t i n g investment o p p o r t u n i t i e s may cause the investment oppor-t u n i t y set t o be a l t e r e d with the i m p l i c a t i o n t h a t the expected rate of re t u r n required 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 the investment opportunity s e t i s not constant, then t h i s i n v a l i d a t e s one of the conditions f o r the c a p i t a l asset 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. Such a conclusion i s hardly s u r p r i s i n g , f o r the p o r t f o l i o behavior of a r a t i o n a l i n v e s t o r would not be expected to be the same when there i s a changing investment opportunity set instead of a constant one. 45 Merton has demonstrated t h a t changes i n the investment opportunity set do a f f e c t the str u c t u r e of common stock returns. Under the assumption that a l l changes can be cha r a c t e r i z e d by changes i n a s i n g l e instrumental v a r i a b l e — See Black, Jensen,and Scholes, loc. cit. 45 Merton, "An Intertemporal C a p i t a l Asset P r i c i n g Model," op. cit., p. 38. I l l the r i s k l e s s i n t e r e s t rate — a two f a c t o r model i s derived. The second f a c t o r can be in t e r p r e t e d as the r e s u l t of i n v e s t o r s hedging against the e f f e c t s of future unforeseen changes i n the r i s k l e s s i n t e r e s t r a t e . Changes i n the investment opportunity set can be caused by s t o c h a s t i c changes i n the p r o b a b i l i t y of a f i r m going bankrupt. As the firm's future i n -come and i t s a b i l i t y to borrow change over time, so w i l l the p r o b a b i l i t y of i t going bankrupt. Some of these changes w i l l be expected, and t h e i r s i g n i f i -cance w i l l already be discounted i n the p r i c e o f the firm's f i n a n c i a l a s s e t s . However, other changes w i l l be unexpected and w i l l a f f e c t the p r i c e of the firm's f i n a n c i a l assets and the r e a l i z e d r e t u r n of i n v e s t o r s . The p e r t i n e n t question to ask i s how the s t o c h a s t i c nature o f the changes i n the rate of the p r o b a b i l i t y of bankruptcy a f f e c t the s t r u c t u r e of returns f o r f i n a n c i a l assets? One method to analyze t h i s problem i s to represent the mechanism generating these s t o c h a s t i c changes by a s p e c i f i e d process. I t w i l l be assumed that the mechanism can be represented i n the form dX.(t) = F. (X . , t ) d t + G. (X . ,t)dQ., (4.47) 3 3 3 3 3 3 3 = 1/ 2,. • .,n, where dQ represents a standard Gaussian-Wiener process. The above equation should be compared to Equation (4.13). The equation d e s c r i b i n g the p r i c e dynamics o f a firm's bonds w i l l be assumed t o contain a random element term, which r e f l e c t s the uncertainty of p r i c e given t h a t d e f a u l t has not occurred. T h i s can be represented i n the form, r e w r i t i n g Equation (4.8) db (t) = [b. ( t ) r -g ]dt + 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 of the random element term, dZ , d e s c r i b i n g the p r i c e dynamics of bonds w i l l determine the degree of response upon the s t r u c t u r e of expected rates of return caused by the s t o c h a s t i c nature of the changes i n the rate of the p r o b a b i l i t y o f bankruptcy. I t w i l l also r e s u l t i n the demand functions f o r bonds and equity being d i r e c t l y c o r r e l a t e d , as events w i l l a f f e c t both types of assets. The equation o f o p t i m a l i t y can be simply derived from the general AC case considered i n Equation (4.23) and can be w r i t t e n i n the form n 0 = Max (U[C(t),t] + J + E F.J. {c,w} fc j=l 3 3 n n + {W(t)[ E w. ( t ) ( r . - r ) + E w ^. (t) (ct.-r) + r] - C ( t ) } j j=l 3 3 j = 1 n+3 D w . , 2 n n n n + ;rW(t) [ E E w.(t)y..w.(t) + 2 E E w.(t)y.p..a.w .(t) 2 j = 1 i = 1 3 3 i x j = 1 i = 1 3 T 3 P 3 i i n + i * ; n n A i - i V j ( t , c , j l , V i < t ) I J w w 1=1 3=1 n n + E E W(t)w (t)a.n. ^.G.J. r 7 j = 1 i = 1 n+3 3 i/n+3 l iW n + E X .{j[W.,Mt),t,S.] - J[W(t) ,X (t) , t , S ( t ) ] }) (4.48) j = l 3 3 3 ~ subject t o the boundary cond i t i o n J[W (T), A_(T) ,T,S (T) ] = BF(W(T),T], and the set of f i r s t order maximization -conditions are, a f t e r some manipulation, o = U c [ C ( t ) ,t] - J w , - (4.49) 46 See Appendix A, Equations (A.64), (A.65), (A.66) and (A.67). i 113 r . - r n y . p . . a. •)J. iw' (4.50) and n n WW n (4.51) Equation (4.49) i s the intertemporal envelope c o n d i t i o n : marginal u t i l i t y of consumption equals the marginal derived u t i l i t y of wealth. Equa-t i o n (4.50) describes a system of n l i n e a r equations i n terms of the demand functions f o r bonds and equity. The d i r e c t dependence between the two sets of demand functions a r i s e s from the c o r r e l a t i o n of the 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 . This equation should be compared to Equation (4.31). Apart from the c o r r e l a t i o n terms, Equation (4.50) contains an extra s e t of terms, { j . }, that are a d i r e c t r e s u l t of the s t o c h a s t i c nature of changes i n i w the rate of the p r o b a b i l i t y of bankruptcy. The s i g n i f i c a n c e o f these terms w i l l become very apparent when the i n d i v i d u a l demand functions are determined. Equation (4.51) describes a non-linear system o f n equations i n terms of the demand functions f o r bonds and equity, the n o n - l i n e a r i t y a r i s i n g from the d i s -c o n t i n u i t i e s i n wealth that are caused by the event of bankruptcy. I t al s o contains the extra set of terms {J..,}, and should be compared to Equation iw 47 (4.32). As i n the l a s t s e c t i o n , i t w i l l be assumed that Equation (4.51) 47 See Equations (4.32) and (4.33). 114 can be approximated to give 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 . = 1 Dx x . = 1 3 X3 n+i WW + A (t)L.[w.. (t)L.. + w n + j ( t J l W t t ) ^ n (4.52) Hence, Equations (4.50) and (4.52) describe a system o f 2n l i n e a r equations and thus i t i s p o s s i b l e to determine an e x p l i c i t s o l u t i o n f o r the demand functions f o r bonds and equity. From the s t r u c t u r e of these equations, i t i s c l e a r that.the form of such a s o l u t i o n w i l l be involved, containing a large number of terms. The complexity of the s o l u t i o n a r i s e s because of the covariance terms and the terms tha t r e f l e c t the e f f e c t s of the stochas-t i c nature of the changes i n the r a t e o f the p r o b a b i l i t y of bankruptcy. 49 t h I t can be shown that the demand functions f o r bonds f o r the k i n d i v i d u a l can be expressed i n the f o l l o w i n g form, using matrix notation, (4.53) and the demand functions f o r e q u i t i e s , (4.54) where WW See Appendix A, Equation (A.68). See the whole of the l a s t s e c t i o n of Appendix A. 115 and {wk} . = W(t) kw.(t) k; {wk}. = W(t) kw ( t ) k ; —2 3 n+3 r . - r {al . = a. - r 2 -D J j {C}. - r . - r - X.L.; Y i i {Dn } . . = Y .p . . 0 . f—; -1 31 3 13 l L j {D> . . = A.L. + Y.P. . 0 . ; -2 31 3 3 '3 13 i {D,}.. = a.n. . G . - Y j n i j G i -3 31 3 i,n+3 i • J L H j { D . } . . = Y.n. . G . ; - 4 31 ] l ] l Y -P . . 0 " . { D 1 9 } . . =a.. - J - * 3 ^ -12 31 31 L j 2 { ° 2 1 > j i " Y j i + * j V 5i - (2a! " EaS - ik i ) " 1 ' £ 2 - (°12 - S . ! ^ ) " 1 ' As the ba s i c structure of the two demand functions represented by Equations (4.53) and (4.54) i s i d e n t i c a l , i t w i l l s u f f i c e to discuss j u s t one. Consider the demand functions f o r equity,- represented by Equation (4.54) The function i s e s s e n t i a l l y composed of two p a r t s : the demand that a r i s e s given a constant investment opportunity set, and the demand that a r i s e s from consideration of the e f f e c t s of s t o c h a s t i c changes i n the rate of p r o b a b i l i t y of bankruptcy. 116 k -1 The f i r s t p a r t , H^E^ (a - D-^D^iSJ can, perhaps, be more e a s i l y i n t e r p r e -ted i f w r i t t e n i n a s c a l a r form: H ^ C a - D ^ c ) } . - Hj[ E ^ . . ( ^ . f p , _ ? 2 D . i ( r . - r - X i L . ) ] , 1 x i i = l j = 1, 2,. . .,n where and ' ^ i ^ ' j i - 2 V t l i The f i r s t term i n the above expression represents the demand f o r the j firm's t h equity based upon the instantaneous expected r a t e of r e t u r n f o r the i firm's equity, and the second term represents a s u b s t i t u t i o n term a r i s i n g from the demand f o r the i ^ firm's bond. The presence o f such a term i s to be expected, 50 k f o r bonds are substitutes f o r equity. The f a c t o r i s s t r i c t l y p o s i t i v e and i s the usual preference f a c t o r r e f l e c t i n g the i n d i v i d u a l ' s d e s i r e between current and future consumption. -1 k The second part, ^ (D^-D^^^D^) H_2 can a l s o be w r i t t e n i n a s c a l a r form: where (E <D, - D D ' J D )H*} = Z E.. H* (4.55) —2 —3 —1—21-4 —2 . , 3 j i 2i 1=1 • 3 = l f 2,. . .,n, ^ 2 ^ 3 - ^ 0 " ^ ) } n 31 3 3 i f They could a l s o be complements. 117 and i / 3 = 1/ 2, • • ., n. The term can be d i r e c t l y a t t r i b u t e d to the e f f e c t s of the s t o c h a s t i c nature of the changes i n the r a t e of the p r o b a b i l i t y of bankruptcy, and can be i n t e r -preted as an attempt t o hedge against such changes. I t can be shown^ t h a t a k c 3\. > J 3c < aw5* 3 — 1/ 2, • • ., n. 3c^ t h Thus i f T T ? — < 0 and „E.. < 0, then the i n v e s t o r w i l l demand l e s s of the j 3 33 firm's equity. The form of expression (4.55) i s important f o r i t contains n preference terms of the i n d i v i d u a l , {H .}. T h i s implies that i f the e q u i l i -brium instantaneous c o n d i t i o n a l expected r a t e s o f r e t u r n f o r bonds and equity are to be determined f r e e o f any preference terms, then the terms {H } and ^3 H 1 must be eliminated. Given the form of the demand fu n c t i o n , i t i s c l e a r that any attempt t o derive the 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 ex-pected r a t e s of r e t u r n w i l l be d i f f i c u l t due to the f a c t that not only are there c o r r e l a t i o n terms r e s u l t i n g from the presence o f the bonds and common stock, but 52 a l s o because of the (n+1) preference terms. Whilst i t i s p o s s i b l e to e l i m i -nate these preference terms, the r e s u l t i n g complexity and general lack of i n -s i g h t that r e s u l t s , does not j u s t i f y the e f f o r t . See Appendix A, Equation (A.22). See Appendix A, Equations (A.79) and (A.81). 118 Some i n s i g h t can be gained by assuming that the s t o c h a s t i c changes i n the rate of p r o b a b i l i t y of bankruptcy f o r one fi r m acts as an instrumental var-i a b l e , c h a r a c t e r i z i n g a l l the changes i n the investment opportunity set. th 5 3 For convenience, c a l l t h i s f i r m the n fi r m . I t can be shown that the th instantaneous c o n d i t i o n a l expected r a t e of return f o r the j firm's equity i s of the form where V r , g n M 5 i x ~ 5nX qjH, , - . a - r - J — - e =( ^ J-> (y - r - x - £„> j a 6 , - 6 .,a r - r + ( ? M m X n 3 X M ) (a - r - - S _ - e ), (4.56) O n L M x n j = 1, 2,. . . , n - l , M ( t ) = j ! 1 V j p j ( t ) ; N .p . (t) y . ( t ) = " j J ; D M(t) P = ? ^ ( t ) a . ; j = l 3 3 n r ,-r X = E Y. (t) (-J ); j-1 3 J 5 j i - <P_12 " 2.i2^i22>ji' n o.„ = E ?..Y.(t); 3 i = l 3 1 n a 2 = E Y . ( t ) a . M ; M n ]M See Appendix A, Equation (A.86). 119 e. = £ e..(r. - r - X.L.); 3 j = l 3 1 1 1 1 «•! « a . h .G - l i ^ l A + £ e i i Y i n n , i G n nX T 'n,n+n n L. J 1 n , ; L n 3 i = l 6MY - . V j ( t ) V 3=1 J j = l and 2 0 = a 6 , - 6 , a . * nM mX nX M ' A s i m i l a r expression can be obtained f o r b o n d s . 5 4 The complex nature o f Equation (4.56) makes i n t e r p r e t a t i o n d i f f i c u l t . The l e f t hand s i d e of the equation can be i n t u i t e d as the instantaneous ex-pected rate o f r e t u r n . The f i r s t term on the r i g h t hand side can be i n t e r p r e t e d as the instantaneous expected excess r e t u r n on the market m u l t i p l i e d by a f a c -t o r s p e c i f i c to the j t h f i r m . The second term a r i s e s from the s t o c h a s t i c nature o f the changes i n the investment opportunity s e t . Suppose = 0, which can be inte r p r e t e d as meaning t h a t the n f i r m i s uncorrelated with the market, the Equation (4.56) then s i m p l i f i e s to the form a - r - l l - e = B (a-r-x-e M) + [ ( ^ ) - ia^r- IS-l - c ^ , 3 nX nX n j = 1, 2,. . .,n, (4.57) where ^ ^ j = V See Appendix A, Equation (A.91). 120 The above equation i s i n a form that can be more e a s i l y compared to the t r a d i t i o n a l c a p i t a l asset p r i c i n g model given by Equation (4.44) and to the more general form which considers bankruptcy given a constant investment opportunity set, expressed by Equation (4.42). I t i s c l e a r that the same general remarks that a p p l i e d to Equation (4.42) apply to Equation (4.57), though perhaps with even greater emphasis: there i s a non-linear, non-sta t i o n a r y r e l a t i o n s h i p between a and B.. There i s , however, one major d i f f e r e n c e : Equation (4.57) contains an extra v a r i a b l e that a r i s e s from the stochas t i c nature of the changes i n the investment opportunity set. The mag-nitude 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 the p a r t i c u l a r s e c u r i t y . I f (a - r - " - e ) i s p o s i t i v e and approximately equal to / n nX nX then f o r high beta s e c u r i t i e s the term w i l l be negative, w h i l s t p o s i t i v e f o r low beta s e c u r i t i e s . T h i s o f f e r s an explanation of the e m p i r i c a l f i n d -55 ings of Black, Jensen and Scholes, who found that high beta stocks c o n s i s t -e n t l y earned l e s s than that p r e d i c t e d by the CAPM, w h i l s t the reverse being true f o r low beta stocks. Summary The primary focus of the chapter i s to extend the formulation of the CAPM not from the viewpoint of r e s t r i c t i o n s upon the in v e s t o r , but by con-s i d e r i n g the impact of bankruptcy upon the s t r u c t u r e of returns f o r corporate f i n a n c i a l a s s ets. 55 Black, et al.t op. cit. 1 2 1 Two models are developed. In the f i r s t model i t i s assumed that the investment opportunity set i s only a l t e r e d by the event of bankruptcy. A s i m p l i f i e d form of sto c h a s t i c d i f f e r e n t i a l equations d e s c r i b i n g the p r i c e dynamics of bonds i s used so as to abstract from i n t e r a c t i o n between bonds and common stock. For common stock the instantaneous c o n d i t i o n a l expected rate of return i s a l i n e a r f u n c t i o n of i t s systematic r i s k and a second var-i a b l e which i s associated with the p r o b a b i l i t y of bankruptcy. The proper-t i e s of the derived expression provide an explanation f o r recent e m p i r i c a l f i n d i n g s of the d e f i c i e n c y of the CAPM. The model i s used as a ba s i s f o r em p i r i c a l l y t e s t i n g the hypothesis of the t h e s i s . The r i s k premium f o r bonds depends upon two v a r i a b l e s : the f i r s t term i s the product of the p r o b a b i l i t y of bankruptcy and the l i q u i d a t i n g dividend, and second term i s a type of market f a c t o r . The second model examines the e f f e c t s of s t o c h a s t i c changes i n the p r o b a b i l i t y of bankruptcy upon the st r u c t u r e o f re t u r n s . A general form of s t o c h a s t i c d i f f e r e n t i a l equation i s used to describe the p r i c e dynamics of bonds, so that there i s i n t e r a c t i o n between the two sets of f i n a n c i a l assets: common stock and bonds. I t i s found that the demand functions f o r common stock contain an extra set of terms, r e f l e c t i n g i n v e s t o r ' s attempts to hedge against unexpected changes i n the p r o b a b i l i t i e s of bankruptcy f o r the d i f f e r e n t firms. Due to the complex nature of the demand functions, a s i m p l i f y i n g assumption i s made by using a s i n g l e instrumental v a r i a b l e to characterize a l l the changes. I t i s found that the instantaneous c o n d i t i o n a l expected rate of return f o r common stock contains an extra term r e f l e c t i n g investors' attempts to hedge against unexpected changes. CHAPTER V EMPIRICAL RESULTS In t h i s chapter the emp i r i c a l r e s u l t s of the t h e s i s are presented. I t describes the estimation of a model to determine the p r o b a b i l i t y of a fi r m going bankrupt and the empirical t e s t i n g o f the hypothesis of the th e s i s using annual data. I f the p r o b a b i l i t y of bankruptcy f o r a f i r m increases, then the ex-pected return, c o n d i t i o n a l upon no bankruptcy, which r i s k averse in v e s t o r s require, w i l l increase to compensate f o r the extra r i s k . At any point i n time the p r o b a b i l i t y of bankruptcy f o r a f i r m i s a fun c t i o n of 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 or lexternally, t o cover f i x e d charges. As conditions w i t h i n the f i r m and the economy change over time, so w i l l the firm's a b i l i t y t o r a i s e funds, and thus the p r o b a b i l i t y of bankruptcy,which may a f f e c t the expected r e t u r n which i n v e s t o r s require on the firm's f i n a n -c i a l assets. The hypothesis of 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 -u a l return a f t e r a b s t r a c t i n g from the market. From the t h e o r e t i c a l a n a l y s i s given i n Chapter IV, a two v a r i a b l e model des c r i b i n g the expected r a t e of re t u r n on a firm's common stock i s derived. The model i s of the form " r = A j + VaM " r " * ) f j = 1, 2,...,N, where i s the instantaneous c o n d i t i o n a l expected rate of r e t u r n on the j*"* 1 122 123 asset; a i s the instantaneous c o n d i t i o n a l expected rate of return on the M market p o r t f o l i o ; r i s the instantaneous r i s k f r e e rate of i n t e r e s t ; A. i s D th — the rate of p r o b a b i l i t y of bankruptcy f o r the j asset; x i s a weighted average of the {A.} and 8. = a /o ,, a.., being the instantaneous c o n d i t i o n -3 j J M M M 3M t h a l covariance o f the j asset with the market p o r t f o l i o . A d i s c r e t e time formulation of the model i s approximately given by E(r.) - r p = X. + B.[E(r M) - r p - X J , th where E(r..) i s the c o n d i t i o n a l expected rate o f r e t u r n on the j asset; E ( r M ) i s the c o n d i t i o n a l expected r a t e of r e t u r n on the market p o r t f o l i o ; X_. i s the t h p r o b a b i l i t y of bankruptcy f o r the j asset f o r the period; x i s a weighted average of the {X.}; r„ i s the r i s k f r e e rate o f i n t e r e s t ; and 8. = c o v ( r . , r ) / • j F 3 3 M v a r ( r w ) , c o v ( r . , r ) being the c o n d i t i o n a l covariance o f the j t n asset with the M j M 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 the hypothesis an ex-post form of the model i s used. This implies a t r a n s i t i o n from an ex-ante to an ex-post formu-l a t i o n using a market model. Thus any e m p i r i c a l t e s t i s a j o i n t examination of the ex-ante formulation and the market model. The ex-post form of the model i s R j t = V o + V j t + 3 j ( R M t " Xt> + V th where R i s the r e a l i z e d excess r e t u r n f o r the j asset during period t; 3 *-R^t i s the r e a l i z e d excess market r e t u r n for p e r i o d t; u i s a random d i s -turbance term; and v and v. are constants. The hypothesis of the t h e s i s i s o 1 represented 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 t e s t e m p i r i c a l l y the model, ex-post excess returns f o r the f i r m and market p o r t f o l i o are r e q u i r e d . I t i s a l s o necessary to know how the p r o b a b i l i t y of the f i r m going bankrupt over d i f f e r e n t periods. Once these data requirements are s a t i s f i e d , the hypothesis can be tested. Evidence i s found which supports the hypothesis of the t h e s i s ; that i s , 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 u a l return a f t e r a b s t r a c t i n g from the market. In the f i r s t p a r t of the chapter the estimation of the p r o b a b i l i t y of bankruptcy i s presented. The data and s t a t i s t i c a l methodology used i n the determination of the c o e f f i c i e n t s are described, as w e l l as the means used to t e s t the model's p r e d i c t i v e a b i l i t y . F i n a l l y , the e m p i r i c a l r e s u l t s are given. In the remaining p a r t of the chapter the hypothesis of the t h e s i s i s e m p i r i c a l l y t e s t e d . The form o f the r e g r e s s i o n equation representing the hypothesis and the s t a t i s t i c a l methodology used to estimate the r e g r e s s i o n c o e f f i c i e n t s are described. The r e s u l t s using aggregated p o r t f o l i o data are given f i r s t and then r e s u l t s using i n d i v i d u a l s e c u r i t y data presented. Estimation o f the P r o b a b i l i t y of Bankruptcy The p r o b a b i l i t y of bankruptcy f o r a f i r m i s a f u n c t i o n of 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 or e x t e r n a l l y , to cover f i x e d charges. A f i r m t h a t f a i l s to cover these f i x e d charges i s s a i d to be bankrupt. From Chapter I I I , i t i s shown that the p r o b a b i l i t y of bankruptcy can be represented by A = Pr (FNF. + MB + AS < 0) , t t t t 125 where A i s the p r o b a b i l i t y of the f i r m going bankrupt i n year t, given the state of the firm a t year t - l ; FNF^ i s the firm's future cash flow net of a l l f i x e d charges at year t; MB^ i s the maximum amount the f i r m can borrow f o r the year t; and ASfc represents a l l other sources of funds a v a i l a b l e to the f i r m a t year t.^" I f the future cash flow net of a l l f i x e d charges plus the maximum amount the f i r m can borrow and a l l other a l t e r n a t i v e sources of funds i s negative, then the f i r m i s sa i d to be bankrupt. The v a r i a b l e s determining the p r o b a b i l i t y of bankruptcy, as stated i n the above expression, are i n terms of d o l l a r amounts. As cross s e c t i o n a l data w i l l be used, 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 i n the s i z e of firms and so may be dominated by scale e f f e c t s . Very large scale e f f e c t s among firms would be expected t o lead t o i n e f f i c i e n t estimation o f c o e f f i -c i e n t s . To avoid t h i s , the p r o b a b i l i t y of bankruptcy can be w r i t t e n i n the form FNF MB AS A = P r ( r - l + r i - + r ^ - < 0), • A t - 1 A t - 1 A t - 1 where A^ ^ i s the book value of the firm's assets i n the year t - l . Thus the p r o b a b i l i t y at year t - l o f a f i r m going bankrupt i n the year t depends upon i t s future cash flow net o f a l l f i x e d charges per u n i t of assets and the t o t a l amount of funds t h a t i t could r a i s e per u n i t of assets. To estimate e m p i r i c a l l y the p r o b a b i l i t y of bankruptcy requires that the ex-ante v a r i a b l e s be replaced by ex-post surrogates. An estimate of the firm's future cash flow net of a l l f i x e d charges i s obtained by regressing *In Chapter I I I a f u l l d i s c u s s i o n of the model f o r the p r o b a b i l i t y of bankruptcy . and s t a t i s t i c a l methodology i s given. i 126 net income against time over a f i v e year period and then using the e s t i -mated regression equation to p r e d i c t the next year's value. The estimate i s then d i v i d e d by the current book value of the firm's t o t a l assets. The operational d e f i n i t i o n of net income used i s income a f t e r deducting a l l oper-a t i n g and non-operating income and expenses and minority i n t e r e s t but before p r e f e r r e d and common dividends. The maximum amount a f i r m could borrow i s estimated by a m u l t i p l i -c a t i v e f u nction of the amount of c r e d i t r a t i o n i n g and that p a r t of the f i r m not financed by debt, as measured by the book value of net worth. The func-t i o n i s d i v i d e d by the asset s i z e of the f i r m . The f i n a l form of the f u n c t i o n being .book value of net worth a t t-1. , . ( _ ) exp (-CR ) At-1 Z ~ where A F C ^ i s the book value of the firm's t o t a l assets at time t - 1 ; and CRfc ^ i s the amount of c r e d i t r a t i o n i n g a t time t - 1 . C r e d i t r a t i o n i n g i s 2 measured using a proxy v a r i a b l e developed by J a f f e e . In i t s simplest form i t i s the r a t i o of the amount of loans granted a t the prime rate to the t o t a l amount of loans granted. Due to seve r a l data problems the data are smoothed and seasonally adjusted to have a mean of zero and a standard d e v i a t i o n of u n i t y . 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 technique i s used to combine four 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 given 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 and the Commercial Loan Market (New York: John Wiley & Sons, Inc., 1 9 7 1 ) . 3 J a f f e e , Ibid., pp. 1 0 1 - 1 0 3 . 127 using a l i n e a r average of the past four quarters of the f i r s t p r i n c i p a l component. measured by a l i n e a r f u n c t i o n of e f f i c i e n c y , growth, business r i s k and f i n a n c i a l r i s k . E f f i c i e n c y i s estimated by using earnings before i n t e r -est and taxes d i v i d e d by the book value of t o t a l assets; growth by a f i v e year l i n e a r average growth rate i n assets; business r i s k by the absolute value of the p r o p o r t i o n a l change i n sales t o the p r o p o r t i o n a l change i n GNP; and f i n a n c i a l r i s k by the d i f f e r e n c e between f i x e d charges and the firm's future cash flow, the d i f f e r e n c e being d i v i d e d by the standard d e v i a t i o n of the firm's future cash flow. An estimate of the firm's future cash flow i s obtained by r e g r e s s i n g operating income against time over a f i v e year period and then using the estimated r e g r e s s i o n equation to pre-d i c t the next year's value. The standard d e v i a t i o n i s estimated by using the r e s i d u a l sum of squares from the r e g r e s s i o n . The o p e r a t i o n a l d e f i n i t i o n of operating income i s net sales l e s s cost of s ales and operating expenses before deducting d e p r e c i a t i o n , amortization, i n t e r e s t , taxes, and dividends. For f i x e d charges i t i s a l l i n t e r e s t expense, the amortization of debt d i s -count or premium and the amortization of expenses(that i s , underwriting, brokerage fees, a d v e r t i s i n g costs, e t c . ) . I t i s assumed that a l l other a l t e r n a t i v e sources of funds can be The ex-post formulation of the model can be w r i t t e n X = Pr[£ < 6 Q + 0 (• estimated future cash flow net of a l l f i x e d charges f o r time t book value of net worth a t t - l •)exp(-CR t - l 128 „ .earnings before i n t e r e s t and taxes at t-1. + 3 ( a _ ) J t - i + 8^ ( f i v e year l i n e a r growth rate 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 sales 1 5 'proportional change i n GNP ' + g ^ f i x e d charges a t t-1 - estimated future cash flow f o r time t . j .,. ^. 6 estimated standard d e v i a t i o n of future cash flow where X i s the p r o b a b i l i t y o f the f i r m going bankrupt i n year t , given the state of the f i r m a t year t-1; e i s a zero mean random v a r i a b l e e r r o r term, which i s assumed t o be of u n i t variance and uncorrelated between firms; 8 , o are c o e f f i c i e n t s which are to be estimated; and A i s the book i D t~1 value of the firm's t o t a l assets at time t-1. The prime focus i s to be able to p r e d i c t the p r o b a b i l i t y of a f i r m going bankrupt and not to construct a complete general theory. Consequently, due to the complex i n t e r a c t i o n of the underlying f a c t o r s and the d i f f i c u l t y of developing an accurate empirical representation of the determinants of bankruptcy, a second formulation using market values f o r the appropriate corporate v a r i a b l e s i s developed. The use o f market values circumvents many of the d i f f i c u l t i e s of con s t r u c t i n g proxy v a r i a b l e s to measure such quanti-t i e s as the maximum amount the f i r m could borrow and the t o t a l of a l l other a l t e r n a t i v e sources. For the firm's future cash flow net of a l l f i x e d charges, the same proxy v a r i a b l e , as pr e v i o u s l y defined, i s used; that i s , r e a l i z e d values of the firm's cash flow net o f a l l f i x e d charges are regressed against time over 129 a f i v e year period and then the estimated r e g r e s s i o n equation used to p r e d i c t the next year's value of the firm's cash flow net of a l l f i x e d charges. This value i s then d i v i d e d by the book value of the firm's t o t a l assets and the re s u l t a n t used as the ex-post surrogate. The proxy v a r i a b l e used to measure the ex-ante maximum amount the firm could borrow per u n i t of assets f o r year t i s .market value of equity a t t - l , , . ( a * ) exp (-CR ), A t - 1 where Afc ^ i s the book value of the firm's assets a t time t - l ; and CRfc ^ i s the amount of c r e d i t r a t i o n i n g at time t - l . The t o t a l of a l l other a l t e r n a t i v e sources of funds which the f i r m may u t i l i z e depends upon three broad c a t e g o r i e s : uncommitted reserves, r e -duction of planned outflows, and the l i q u i d a t i o n o f as s e t s . A v a r i a b l e which synthesizes these diverse q u a n t i t i e s i s the market value of equity. Thus, the second formulation of the model i s of the form ,estimated future cash flow net of a l l f i x e d charges f o r time t. X t = P r U < Y Q + Y X( -> .market value of equity at time t - l . , . + y2 < A t - 1 7 ) ^ t - i 1 + Y ,market value of equity at time t - l , , , K „. A t - 1 where A ^ i s the book value of the firm's t o t a l assets; e i s a zero mean random v a r i a b l e error term, which i s assumed to be o f u n i t variance and uncorrelated between firms; and Y Q / Y ^ » V ^ ' "^3 a r e c o e f f i c i e n t s which are to be estimated. 130 S t a t i s t i c a l Methodology The c o e f f i c i e n t s i n Equations ( 5 . 1 ) and ( 5 . 2 ) cannot be estimated by regression as the dependent v a r i a b l e i s unobservable. However, the co-e f f i c i e n t s can be estimated by using maximum l i k e l i h o o d . Consider a ran-dom sample of firms at time t and suppose the f i r s t n' firms are bankrupt and the remainder n-n 1 non-bankrupt. The logarithmic l i k e l i h o o d f u n c t i o n can then be w r i t t e n n' n Z l o g Pr(B |x ) + Z l o g [ l - Pr(B |x )], j = l ^ t - 1 ' 3 j=n'+l Z J Z ~ ' 3 (5.3) where Pr(B t_.|x t ^ _.) i s the p r o b a b i l i t y of the j*"* 1 f i r m going bankrupt i n year t , given a set of a t t r i b u t e s measured a t year t-1. By s u b s t i t u t i n g e i t h e r (5.1) or (5.2) i n t o (5.3) and d i f f e r e n t i a t i n g with respect t o the c o e f f i c i e n t s , a s e t o f non-linear equations representing the f i r s t order maximization conditions are obtained and can be solved i t e r a t i v e l y . Two estimation procedures are used i n the t h e s i s : p r o b i t a n a l y s i s and l o g i t a n a l y s i s . The e s s e n t i a l d i f f e r e n c e between the two procedures i s 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 . For p r o b i t a n a l y s i s a 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 the 4 d i s t r i b u t i o n i s l o g i s t i c . A computer programme developed by Cragg i s used to determine the estimates of the c o e f f i c i e n t s f o r the two procedures. Cragg, J . G. "Programs f o r M u l t i p l e P r o b i t and Log i t A n a l y s i s and Extensions to Them," mimeographed, U n i v e r s i t y of B r i t i s h Columbia, 1 9 6 8 . 131 Data To estimate the parameters of the model, the l i k e l i h o o d f u n c t i o n must be constructed by taking a random sample of firms and then c l a s s i f y i n g the firms as bankrupt or not bankrupt. The procedure of using a random sample and then c l a s s i f y i n g the firms avoids s e l e c t i o n b i a s . As the aver-age p r o b a b i l i t y of a f i r m going bankrupt i s small,a very large random sample must be taken so as to obtain a representative c o l l e c t i o n of bankrupt firms. Due to the problem o f c o l l e c t i n g data f o r such a large sample an a l t e r n a t i v e i procedure i s used. Instead of taking a random sample of firms and then c l a s s i -f y i n g the firms as bankrupt or non-bankrupt, data are c o l l e c t e d f o r a universe of bankrupt firms and then f o r a universe o f non-bankrupt firms. For the 10 year p e r i o d 1960-1969 a l i s t of firms t h a t declared bankruptcy under the Bankruptcy Act of 1938 i n the U.S.A. i s compiled. The four main data sources used i n the compilation are: the S e c u r i t y and Ex-change Commission Annual Reports, Moody's I n d u s t r i a l Manual, Wall S t r e e t Journal Index, and Dunn and Bradstreet. The c r i t e r i a used t o s e l e c t the sample of bankrupt firms are: (a) a t l e a s t three years of data be a v a i l a b l e ; (b) the bankruptcy i s not caused by fraud; and (c) the f i r m must have been a c t u a l l y o f f e r i n g a product or s e r v i c e f o r s a l e . No corporate s h e l l s entered the sample. A l i s t of 34 firms used i s given i n Appendix B. The corporate data for these firms are obtainable i n Moody's I n d u s t r i a l Manual. The aver-age asset s i z e of these firms i s $11 m i l l i o n v/hich i s quite small, a r e f l e c -t i o n of the f a c t that large firms, l i k e those t h a t are traded on the New York Stock Exchange r a r e l y go bankrupt, the obvious exception being Penn C e n t r a l . If a large f i r m i s i n d i f f i c u l t i e s i t i s e i t h e r acquired by another f i r m or 132 a merger occurs. To obtain a representative random sample of non-bankrupt firms, requires determining the number of firms to be s e l e c t e d . I d e a l l y , the number chosen should be the same as that obtained by taking a random sample of a l l firms and then c l a s s i f y i n g them as bankrupt or not bankrupt. As the s i z e of the sample of bankrupt firms i s already known, then the s i z e of the sample of non-bankrupt firms should be such that the number of non-bankrupt to bankrupt firms approximately equals the average p r o b a b i l i t y of not going bankrupt to the average p r o b a b i l i t y of bankruptcy. For the 10 year p e r i o d the whole universe of bankrupt firms s a t i s f y i n g the s e l e c t i o n c r i t e r i a i s used. Thus, to obtain a representative sample of non-bankrupt firms requires taking the whole universe, f o r the 10 year period, of non-bankrupt firms that 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 the same population which the bankrupt firms are s e l e c t e d . As the average p r o b a b i l i t y of bankruptcy i s q u i t e small, t h i s im-p l i e s that the s i z e of the sample of non-bankrupt firms w i l l be l a r g e . The data f o r the bankrupt firms i s manually c o l l e c t e d from Moody's I n d u s t r i a l Manual; thus t o obtain data f o r a large sample of non-bankrupt firms using the same source would be p r o h i b i t i v e l y time consuming. To avoid t h i s severe problem the non-bankrupt firms are sampled from the universe of firms con-tained on the Compustat F i l e . A c h a r a c t e r i s t i c that describes the population of bankrupt firms i s tha t of asset s i z e : a l l firms are l e s s than $200 m i l l i o n i n s i z e . This property i s used t o define a population of firms from which the non-bankrupt firms are select e d . The sample of non-bankrupt i s obtained by s e l e c t i n g , on 133 a year by year basis f o r the 10 year period, a l l firms with asset s i z e l e s s than $200 m i l l i o n . The number of firms s e l e c t e d each year f o r both samples i s shown i n Table 5.1. For the year 1960 there i s a n u l l set of bankrupt firms, w h i l s t there are 237 firms i n the non-bankrupt set. This simply r e -f l e c t s the f a c t that, given the s e l e c t i o n c r i t e r i a , bankruptcy f o r the type of firms considered i s a rare event. By taking a random sample of firms from the defined population over a 10 year period, there i s no guarantee that f o r a p a r t i c u l a r year the set of bankrupt firms i s not empty. P r e d i c t i v e A b i l i t y As the p r o b a b i l i t y of bankruptcy cannot be observed, 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 the magnitude of any b i a s or measurement e r r o r i n the estimates cannot be determined. Thus, the main check on how w e l l the models are s p e c i f i e d must be t h e i r p r e d i c t i v e a b i l i t y . Three methods are used to t e s t the models. From t h e o r e t i c considerations the signs of the parameters can be determined and compared t o those obtained from e m p i r i c a l estimation. The number of estimated parameters with the c o r r e c t sign provides i n s i g h t i n t o the s p e c i f i c a t i o n of the model and the accuracy 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 . If the model i s completely s p e c i f i e d so as t o measure a l l the d i f f e r -ent a t t r i b u t e s o f the firms i n the sample used t o estimate the parameters, then i t should be able to c o r r e c t l y i d e n t i f y the bankrupt and non-bankrupt firms 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 information about the model's s p e c i f i c a t i o n and the number of common determinants of bankruptcy. TABLE 5.1 NUMBER OF BANKRUPT AND NON-BANKRUPT FIRMS IN DATA SAMPLE Y E A R 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 TOTAL BANKRUPT 0 5 8 3 2 3 4 . 3 3 3 34 NON-BANKRUPT 237 253 254 265 262 248 224 208 ' 168 150 2303 135 The g e n e r a l i t y of the model and i t s o v e r a l l independence of the p e c u l i a r i t i e s of the data sample used to estimate the parameters, can be tested by examining the pred i c t v e a b i l i t y on a set of bankrupt firms not used i n the o r i g i n a l sample. A l i s t o f firms that vent bankrupt i n 1970, one year a f t e r the end of the pe r i o d used f o r estimating the c o e f f i c i e n t s i s compiled using the Security and Exchange Commission Annual Reports and a l i s t of bankrupt firms given i n an a r t i c l e by Altman.^ By estimating the p r o b a b i l i t y of bankruptcy over se v e r a l time periods f o r firms i n the new sample provides a demonstration of the model's p r e d i c t i v e a b i l i t y to d i s -cern a firm's path to bankruptcy. Results The p r o b a b i l i t y of bankruptcy i s estimated using the ex-post formu-l a t i o n given by Equation (5.1). The estimated c o e f f i c i e n t s are shown i n Table 5.2. With th exception o f growth and business r i s k , a l l the parameters have the c o r r e c t s i g n . The v a r i a b l e s representing the maximum amount the f i r m could borrow and e f f i c i e n c y , are the only two that are not s t a t i s t i c a l l y 6 s i g n i f i c a n t . 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 are high and, as expected, almost i d e n t i c a l . 3Altman, E. "Reply," Journal of Finance, V o l . XXVII, No. 3 (June, 1972), pp. 718-721. ^For the maximum l i k e l i h o o d a n a l y s i s , R-squared i s defined by R-squared = {l-exp[2(L -L 0)/T]}/{l-exp[2(L -L )/T} « \i w max where L i s the maximum o f the lo 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 using only a constant, L i s the maximum using a l l v a r i a b l e s and L i s the maximum po s s i b l e . T i s the t o t a l number of observations. TABLE 5.2 ESTIMATION OF COEFFICIENTS FOR A GENERAL MODEL DESCRIPTION OF VARIABLES FUTURE CASH FLOW NET ALL FIXED CHARGES MAXIMUM AMOUNT THAT CAN BE BORROWED EFFICIENCY GROWTH BUSINESS RISK FINANCIAL RISK CONSTANT R-SQUARED LOGARITHM OF MAXIMUM LIKELIHOOD FUNCTION Notation 61 S2 6 3 64 65 e o Expected Sign - - - - + + Logit Analysis -10.313 t i l (- 2.651) -0.824 (-1.405) -6.901 (-1.885) 2.502 t (4.126) (-2.326) °-275 „ (2.435) -3.219 t (-6.836) 0.58 -76.092 Probit Analysis "5-455 „ (-2.734) -0.438 (-1.617) -3.069 (-1.599) 1.2307^ (3.921) -0-0772,„ (-2.308) 0-116 (2.014) -1.663 t (-7.809) 0.59 -75.286 (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 at 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 at 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 at 5.0% 137 The a b i l i t y of the model to cla s s i f y correctly the bankrupt and non-bankrupt firms in the original data sample is shown in Table 5.3. A bankrupt firm i s cl a s s i f i e d as non-bankrupt i f i t s estimated probability of bankruptcy i s less than the average value of the estimated probability of bankruptcy for the whole data sample. A non-bankrupt firm i s classified as bankrupt i f i t s estimated probability of bankruptcy is greater than the average value of the estimate probability of bankruptcy for the whole data sample. This criterion i s used throughout. For the coefficients estimated using logit analysis, the model correctly identifies over 91 per cent of the non-bankrupt firms and over 94 per cent of the bankrupt firms. For probit analysis, the model correctly identifies 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 results arise from the differences between the l o g i s t i c and normal probability distributions. 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 prob-a 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 in Table 5.4. The probability of bankruptcy is estimated over as many periods as available data permit, for a group of eight firms that declared bankruptcy in 1970. For Uniservices Incorporated the model predicts failure four years before the date of bankruptcy and for Bishop Industries three years. Bankruptcy i s predicted two years in advance for G. F. Industries, and Roberts Company, and one year for Visual Electronics and Dolly Madison Incorporated. The model totall y f a i l s for Century Geophysical Incorporated, giving a probability of zero one year before bankruptcy. The failure of the model can be a t t r i -138 TABLE 5.3 CLASSIFICATION OF ORIGINAL DATA SAMPLE BY GENERAL MODEL LOGIT ANALYSIS ACTUAL OUTCOME PREDICTED OUTCOME BANKRUPT NON-BANKRUPT TOTALS AVERAGE VALUE OF PROBABILITY BANKRUPT 32 191 223 0.434 NON-BANKRUPT 2 2078 2080 0.009 T O T A L S 34 2269 2303 Type one err o r = = p r o b a b i l i t y i 191/2269 = .084 [of a fi r m 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 PROBIT ANALYSIS ACTUAL OUTCOME PREDICTED OUTCOME BANKRUPT NON-BANKRUPT TOTALS AVERAGE VALUE OF PROBABILITY BANKRUPT 32 203 235 0.413 NON-BANKRUPT 2 2066 2068 0.009 T O T A L S 34 2269 2303 Type one e r r o r = Type two erro r = 0.098 0.059 TABLE 5.4 PREDICTIVE ABILITY OF GENERAL MODEL \ NAME OF FIRM 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 LAST DATE OF DATA DATE OF BANK-RUPTCY UNISERVICES INC. .0.0005 0.0001 0.151* 0.169 0.737* 0.707 0.468* 0.492 Sept., 1968 1970 ROBERTS COMPANY 0.046 0.0G3 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 Nov., 1968 Feb., 1970 CENTURY GEOPHYSICAL INCORPORATED 0.1164 0.1133 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 June, 1969 1970 VISUAL ELECTRONICS 0.002 0.0 0.044* 0.063 0.998* 0.999 March, 1970 June, 1970 G.F. INDUSTRIES 0.021 0.025 0.008 0.009 0.366* 0.343 0 .832* 0.740 June, 1969 1970 DOLLY MADISON INC. 0.004 0.0034 0.012 0.015 0.003 0.001 °- 0 3** 0.036 Sept., 1969 June, 1970 FARRINGTON MANUFACTURING COMPANY 0.373 0.337 0.991 0.986 0.994 0.994 0.235 0.258 0.056 0.067 0.012 0.012 0.005 0.004 0.001 0.001 0.004 0.005 0.343* 0.330 Dec., 1969 1970 BISHOP INDUSTRIES 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 of probability of bankruptcy - 0.014 * indicates probability above average value. 140 buted to the i n c o r r e c t signs of the growth and business r i s k c o e f f i c i e n t s , and the large v a r i a b i l i t y of the firm's cash flow net of 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 sign of two s t a t i s t i c a l l y s i g n i f i c a n t c o e f f i c i e n t s , poor p r e d i c t i v e a b i l i t y , and high c o r r e l a t i o n among the v a r i a b l e s , d i s t r a c t s from the appeal of the model. This p r e c i p i t a t e d development of a model of the form X _ p r[e<g +g ^estimated future cash flow net of a l l f i x e d charges f o r time t t " 0 1 V i + B 2(b°°* value of net worth at + a ^ i x e < ^ charges at t-1 - estimated future cash flow f o r time t... 3 estimated standard d e v i a t i o n of future cash flows ' (5 .3) where the c o e f f i c i e n t 8^ measures the s i g n i f i c a n c e of f i n a n c i a l r i s k . The estimated values of the c o e f f i c i e n t s are shown i n Table 5.5. A l l the c o e f f i -c i e n t s estimated over the 10 year period have the c o r r e c t sign and are s t a t i s -t i c a l l y s i g n i f i c a n t . The R-squared values f o r both 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 are high. The a b i l i t y of the model t o c l a s s i f y the o r i g i n a l data i s given i n Table 5.6. For the c o e f f i c i e n t s c a l c u l a t e d using 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 89 per cent of the non-bankrupt and 91 per cent of the bankrupt firms. 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. The hypothesis that the 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 to a purely random process can be r e j e c t e d with a p r o b a b i l i t y of over 99 per cent. TABLE 5.5 ESTIMATION OF COEFFICIENTS AND TEST FOR STATIONARITY ALTERNATIVE MODEL FUTURE MAXIMUM LOGARITHM CASH FLOW AMOUNT OF MAXIMUM NET OF ALL THAT LIKELIHOOD FIXED CAN BE FINANCIAL R- FUNCTION STATIONARITY METHODOLOGY TIME CHARGES BORROWED RISK CONSTANT SQUARED ( l o g e L) TEST NOTATION 61 B2 63 -2 [(log L. + e i l o g e L 2 ) -EXPECTED 2 SIGN + l o g e L 3 ] -v. x 4 -13.091 -1.084 0.330 -2.769 1960-1969 * * * * 0.54 -84.490 (- 4.182) (-1.973) (3.155) (-6.767) LOGIT 5.68 ANALYSIS - 7.697 -3.372 0.389 -1.365 1960-1964 0.63 -37.227 Not (- 1.562) (-2.183) (2.654) (-1.517) s i g n i f i c a n t -15.059 -0.026 0.268 -3.239 at 10% l e v e l 1965-1969 0.46 -46.422 (- 3.399) (-0.038) (1.827) (-6.285) - 6.639 -0.549 0.1388 -1.470 1960-1969 * * * * 0.53 -86.072 6.84 PROBIT (- 4.153) (-2.219) (2.702) (-8.031) Not - 3.684 -1.716 0.180 -0.774 ANALYSIS 0.64 -36.236 s i g n i f i c a n t 1960-1964 (-1.543) (-2.483) (2.3952) (-1.908) at 10% l e v e l -8.087 -0.017 0.101 -1.729 1965-1969 0.46 -46.305 (-3.513) (-0.05) (1.486) (-7.12) (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 at 5% or l e s s . 142 TABLE 5.6 CLASSIFICATION OF ORIGINAL DATA SAMPLE: ALTERNATIVE MODEL LOGIT ANALYSIS ACTUAL OUTCOME PREDICTED OUTCOME BANKRUPT NON-BANKRUPT TOTALS AVERAGE VALUE OF PROBABILITY BANKRUPT 31 249 280 0.396 NON-BANKRUPT 3 2020 2023 0.009 TOTALS 34 2269 2303 Type one erro r = p r o b a b i l i t y [of a f i r m bankrupt|non-bankrupt] = 249/2269 = 0.11 Type two erro r = p r o b a b i l i t y [of a fir m non-bankrupt|bankrupt] = 3/34 = 0.089 PROBIT ANALYSIS ACTUAL OUTCOME AVERAGE VALUE PREDICTED OUTCOME BANKRUPT NON-BANKRUPT TOTALS OF PROBABILITY BANKRUPT 31 261 292 0.372 NON-BANKRUPT 3 2008 2011 0.009 TOTALS 34 2269 2303 Type one error = 0.115 Type two error = 0.089 143 The p r e d i c t i v e a b i l i t y of the model i s demonstrated i n Table 5.7. F a i l u r e i s predicted f i v e years i n advance f o r Bishop Industries and four years for Uniservices Incorporated. For Roberts Company and G. F. Indus-t r i e s bankruptcy i s predicted two years i n advance and f o r the remaining firms, except V i s u a l E l e c t r o n i c s , a one year p r e d i c t i o n i s given. This i n -cludes Century Geophysical Incorporated f o r which the general model, ..as represented by Equation (5.1), f a i l e d by g i v i n g a p r o b a b i l i t y o f zero one year before bankruptcy. For V i s u a l E l e c t r o n i c s the model p r e d i c t s a prob-a b i l i t y of approximately 90 per cent of i t going bankrupt three months be-fore i t a c t u a l l y f a i l e d . Due to the 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 a b i l i t y , and the estimated c o e f f i c i e n t s having the c o r r e c t s i g n , i t i s used i n the second part of the e m p i r i c a l work to estimate the p r o b a b i l i t y of bankruptcy i n the t e s t i n g o f the hypothesis of the t h e s i s . S t a t i o n a r i t y The c o e f f i c i e n t s of the model are estimated over a 10 year p e r i o d . As the model i s t o be used f o r estimation purposes, i t i s necessary to ex-amine i t s s t a t i o n a r i t y ; that i s , over d i f f e r e n t sub-periods i s the model s t i l l an accurate estimator of the p r o b a b i l i t y of bankruptcy. To t e s t f o r n o n - s t a t i o n a r i t y the data sample i s s p l i t i n t o two time periods (1960-1964) and (1965-1969). The c o e f f i c i e n t s of the model are estimated over the sub-periods and the logarithm o f the maximum l i k e l i h o o d f u n c t i o n determined. An 7 asymptotic t e s t f o r s t a t i o n a r i t y i s given by For proof, see Chapter X of Mood, A. and G r a y h i l l , F. Introduction to the Theory of Statistics (New York: McGraw-Hill, 1963). TABLE 5.7 PREDICTIVE ABILITY OF ALTERNATIVE MODEL LAST DATE DATE OF OF BANK-NAME OF FIRM 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 D A T A RUPTCY UNISERVICES INC. 0.0007 0.0003 0.116* 0.134 0.816* 0.753 * 0.637* 0.549 Sept., 1968 1970 ROBERTS COMPANY 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 Nov., 1968 Feb., 1970 CENTURY GEOPHYSICAL INC. 0.576 0.463 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* June, 1969 1970 VISUAL ELECTRONICS 0.0001 0.0 0.01 0.01 0.929* March, 0.882 1970 June, 1970 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 FARRINGTON MANUFACTURING COMPANY 0.774 0.672 0.987 0.975 0.99 0.99 0.464 0.432 0.069 0.082 0.012 0.013 0.012 0.013 0.008 0.009 0.002 0.001 0.244* 0.250 Dec., 1969 1970 BISHOP INDUSTRIES 0.0003 0.0 0.019* 0.020 0.0121* 0.012 0.028* 0.034 0.911* 0.855 0.956* 0.925* Oct., 1969 Oct., 1970 Average value o f p r o b a b i l i t y of bankruptcy - 0.014 * " Indicates p r o b a b i l i t y above average value. 145 -2 { [ l o g (L ) + log (L )] - log (L)} * y}, e l e 2 e x where L i s the maximum l i k e l i h o o d function f o r the whole period; and 2 are the maximum l i k e l i h o o d functions f o r the two sub-periods; and x v I s a chi-square d i s t r i b u t e d random v a r i a b l e on k degrees of function, k being the number of c o e f f i c i e n t s to be estimated i n the model. The r e s u l t s f o r the s t a t i o n a r i t y t e s t are shown i n Table 5.5 While there i s some v a r i a b i l i t y i n the c o e f f i c i e n t s , they a l l have the c o r r e c t s i g n and are, i n general, s t a t i s t i c a l l y s i g n i f i c a n t . For l o g i t a n a l y s i s and pro-b i t a n a l y s i s , there i s no evidence o f n o n - s t a t i o n a r i t y at the 10 per cent l e v e l . P r e d i c t i v e Model As the primary focus i s the p r e d i c t i o n and estimation of the p r o b a b i l -i t y o f bankruptcy, a second formulation of the model using market values of the appropriate corporate v a r i a b l e s i s developed. Th i s i s represented by Equation (5.2). The estimation of the c o e f f i c i e n t s and a t e s t f o r s t a t i o n a r i t y are given i n Table 5.8. For l o g i t a n a l y s i s a l l the c o e f f i c i e n t s have the c o r r e c t sign, future cash flow net o f a l l f i x e d charges and the constant being s t a t i s -t i c a l l y s i g n i f i c a n t . In p r o b i t a n a l y s i s , the sign of the v a r i a b l e representing a l t e r n a t i v e sources of funds i s i n c o r r e c t , though the c o e f f i c i e n t i s s t a t i s t i -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 f o r the maximum amount that the fir m could borrow. The R-squared i s high f o r both 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 . In the t e s t f o r s t a t i o n a r i t y there i s f l u c t u a t i o n i n the signs of the c o e f f i -c i e n t s representing the maximum amount that the f i r m could borrow and a l t e r -TABLE 5.8 ESTIMATION OF COEFFICIENTS AND TEST FOR STATIONARITYj PREDICTIVE MODEL MAXIMUM FUTURE AMOUNT LOGARITHM CASH FLOW THAT OF MAXIMUM NET OF ALL CAN BE ALTERNATIVE R- LIKELIHOOD STATIONARITY FIXED BORROWED SOURCES CONSTANT SQUARED FUNCTION TEST METHODOLOGY TIME CHARGES * NOTATION Y l Y2 Y3 EXPECTED _ _ SIGN 1960-1969 -23.006 * -0.270 -0.163 -3.848 n 0.48 -95.31 LOGIT (- 9.411) vV0.638) (-0.293) (-9.942) 3.28 Not s i g n i f i c a n t at 10* l e v e l ANALYSIS 1960-1964 -23.425 (- 6.156) -1.169 . (-1.068) 1.286 (1.139) -4.120 (-7.108) 0.54 -45.727 -23.099 0.176 -0.979 -3.403 1965-1969 (- 6.912) (0.326) (-1.219) (-6.591) 0.44 -47.95 -11.025 -0.121 +0.060 -2.044 PROBIT 1960-1969 (- 9.871)* (-0.656) ( 0.271) (-12.599)* 0.49 -94.223 3.27 ANALYSIS 1960-1964 -11.033 (- 6.856) -0.481 (-1.042) 0.589 ( 1.282) - 2.133 (- 9.39) 0.54 -44.94 Not s i g n i f i c a n t at 10% l e v e l -11.295 0.079 -0.299 -1.845 1965-1969 (- 6.864) (0.313) (-0.897) (-8.16) 0.44 -47.65 CT. (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 at the 0.1% l e v e l . 147 native sources of funds, though these c o e f f i c i e n t s are not s t a t i s t i c a l l y s i g -n i f i c a n t . There i s no evidence of n o n - s t a t i o n a r i t y at the 10 per cent l e v e l . The a b i l i t y of the model to c o r r e c t l y c l a s s i f y the o r i g i n a l data i s shown i n Table 5.9. For the c o e f f i c i e n t s estimated by 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 91 per cent of the non-bankrupt and 85 per cent of the bankrupt firms. For p r o b i t a n a l y s i s , over 90 per cent o f the non-bankrupt and 88 per cent of :the bankrupt firms are c o r r e c t l y i d e n t i -f i e d . The hypothesis t h a t the 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 to a purely random process can be r e j e c t e d with a p r o b a b i l i t y of over 99 per cent. The p r e d i c t i o n a b i l i t y of the model i s demonstrated i n Table 5.10. For Uniservices Incoproated bankruptcy i s p r e d i c t e d four years i n advance and f o r G. F. Industries and Bishop 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 given. Bankruptcy i s p r e d i c t e d one year i n advance f o r Century Geophysi-c a l Incorporated, D o l l y Madison Incorporated, and F a r r i n g t o n Manufacturing Company. For V i s u a l E l e c t r o n i c s the model p r e d i c t s a 98 per cent chance o f i t f a i l i n g three months before i t went bankrupt. The p r o b a b i l i t y of bank-ruptcy f o r Roberts Company does not exceed the average value, though f o r the l a s t year f o r which data are a v a i l a b l e the p r o b a b i l i t y of bankruptcy does increase. Comparing Table 5.10 to Table 5.7, i t i s seen that the p r e d i c t i v e a b i l i t y of the model does not exceed that of the a l t e r n a t i v e formulation using book values f o r the appropriate corporate v a r i a b l e s . T h i s i s a r e f l e c -t i o n of the poor q u a l i t y of the a v a i l a b l e market data. 148 TABLE 5.9 CLASSIFICATION OF ORIGINAL DATA SAMPLE: PREDICTIVE MODEL LOGIT ANALYSIS ACTUAL OUTCOME AVERAGE VALUE PREDICTED OUTCOME BANKRUPT NON-BANKRUPT TOTALS OF PROBABILITY BANKRUPT 29 201 230 0.391 NON-BANKRUPT 5 2068 2073 0.009 TOTALS 34 2269 2303 Type one er 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 ACTUAL OUTCOME PREDICTED OUTCOME BANKRUPT NON-BANKRUPT TOTALS AVERAGE VALUE OF PROBABILITY BANKRUPT 30 225 255 0.372 NON-BANKRUPT 4 2044 2267 0.009 TOTALS 34 2269 2303 Type one er r o r = 0.099 Type two erro r = 0.117 TABLE 5.10 PREDICTIVE ABILITY OP MODEL LAST DATE DATE CF OF BANK-NAME OF FIRM 1 9 6 0 1 9 6 1 1 9 6 2 1 9 6 3 1 9 6 4 1 9 6 5 1 9 6 6 1 9 6 7 1 9 6 8 1 9 6 9 1 9 7 0 DATA . FUPTCY UNISERVIES INC. 0 . 0 0 1 0 . 0 0 1 0 . 0 6 7 * 0 . 0 8 0 0 . 9 7 3 * ^ 0 . 9 4 2 0 0 . 9 0 6 * 0 . 8 4 7 Sept., 19G8 1 9 7 0 ROBERTS COMPANY 0 . 0 5 6 0 . 0 6 5 0 . 0 1 0 0 . 0 0 8 0 . 0 7 8 0 . 0 8 7 0 . 0 0 9 0 . 0 0 8 0 . 0 0 3 0 . 0 0 1 0 . 0 0 2 0 . 0 0 1 0 . 0 0 1 0 . 0 0 1 0 . 0 0 8 0 . 0 0 8 Nov., 1 9 6 8 Fob., 1 9 7 0 CEtrnjRY GEOPHYSICAL INCORPORATED 0 . 1 5 2 0 . 1 7 9 0 . 0 4 1 0 . 0 7 7 0 . 0 0 8 0 . 0 1 3 0 . 9 2 7 0 . 8 6 7 0 . 1 1 2 0 . 1 ] 9 0 . 0 7 1 0 . 0 8 9 0 . 0 0 1 0 . 0 0 1 0 . 0 0 0 7 0 . 0 0 0 4 0 . 0 0 3 0 . 0 0 3 0 . 0 1 2 ^ 0 . 0 1 4 June, 1 9 6 9 1 9 7 0 VISUAL ELECTRONICS 0 . 0 0 1 0 . 0 0 1 0 . 0 0 6 0 . 0 0 7 0 . 9 8 7 * 0 . 9 7 4 March, 1 9 7 0 Jur.e, 1 9 7 0 G. F. ItJDUSTRIES 0 . 0 1 1 0 . 0 1 2 0 . 0 0 2 0 . 0 0 2 0 . 0 3 3 * 0 . 0 7 2 0.2(33,* 0 . 3 3 Juno, 19f>9 1 9 7 0 DOLLY MADISON INCORPORATED 0 . 0 0 4 0 . 0 0 3 0 . 0 0 9 0 . 0 0 9 0 . 0 0 5 0 . 0 0 5 * 0 . 0 2 8 4 0 . 0 3 3 Sept., 1 9 6 9 June, 1 9 7 0 FARRINGTON KA: ."U F A CTU RING COMPANY 0 . 3 9 8 0 . 5 9 1 0 . 9 8 2 0 . 9 8 3 0 . 9 9 3 0 . 9 9 2 0 . 4 4 0 . 4 8 4 0 . 0 1 2 0 . 0 2 4 0 . 0 0 4 0 . 0 0 4 0 . 0 0 6 0 . 0 0 9 0 . 0 0 4 0 . 0 1 0 0 . 0 0 4 0 . 0 1 3 0 . 0 9 * „ 0 . 1 8 9 Doc., 1 9 i ' 9 1 9 7 0 BISHOP INDUSTRIES 0 . 0 0 0 4 0 . 0 0 . 0 0 6 0 . 0 0 6 0 . 0 0 5 0 . 0 0 4 0 . 0 1 2 0 . 0 1 4 0 . 9 7 4 * 0 . 9 6 3 0 . 9 9 4 * 0 . 9 9 3 Oct., 1 9 6 9 Oct., 1 9 7 0 Average value of p r o b a b i l i t y of bankruptcy • 0.014 ^ * ind i c a t e s that the p r o b a b i l i t y i s above the average value 150 Summary To estimate the p r o b a b i l i t y o f a f i r m going bankrupt over a given period two models, one using book values and the other market values of the appropriate corporate v a r i a b l e s , have been constructed. The models have been tested 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 . I t i s found that there i s no evidence of n o n - s t a t i o n a r i t y . Both models have demonstrated 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 , being able to fo r e c a s t bankruptcy, f o r some fi r m s , four or f i v e years before the ac t u a l occurrence. Te s t i n g of Hypothesis In Chapter IV i t i s shown that when the investment opportunity set i s changed only by the event of bankruptcy, then i n e q u i l i b r i u m the i n s t a n -taneous c o n d i t i o n a l expected r a t e of return, c o n d i t i o n a l upon no bankruptcy, on common stock i s represented by the expression a. - r = A + 6.(a - r - x ) , (5 .4) D D D M where a_. i s the instantaneous c o n d i t i o n a l expected r a t e of re t u r n on the th j asset; ct^ i s the instantaneous c o n d i t i o n a l expected r a t e of re t u r n on the market p o r t f o l i o ; r i s the instantaneous r i s k free rate of i n t e r e s t ; A ^ th — i s the rate of p r o b a b i l i t y of bankruptcy f o r the j asset; x i s a weighted average of the { A . } and 8. = a. /a , a. being the instantaneous c o n d i t i o n a l ^ ] ] jM MM jM covariance of the j*"* 1 asset with the market p o r t f o l i o . As data are only a v a i l -able f o r d i s c r e t e time i n t e r v a l s , a model formulated i n continuous time can-not be tested d i r e c t l y . To formulate a d i s c r e t e time analogy of the model involves i n t e g r a t i n g 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 using the 151 equilibrium expression (5.4). A d i s c r e t e time form of the model i s approxi-mately given by E(r.) - r p = X. + B . [ E ( r M ) - r p - x l , (5.5) th where E(r.) i s the c o n d i t i o n a l expected rate of return on the j asset; E(r„) j M i s the c o n d i t i o n a l expected rate of return on the market p o r t f o l i o ; X_. i s t h the p r o b a b i l i t y of the j f i r m going bankrupt during the period; x i s a weighted average of the {X.}; r i s the r i s k free rate of i n t e r e s t ; and th B. = cov(r.,r ), c o v ( r . , r ) being the c o n d i t i o n a l covariance of the j asset 3 3 M D M with the 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 the hypothesis an ex-post form of the model i s used. This implies a t r a n s i t i o n from an ex-ante to an ex-post formulation using a market model. Thus any empirical t e s t i s a j o i n t examination of the ex-ante formulation and the market model. The ex-post form of the model i s R j t = V o + Vjt + e j ( R M t - V + U j t ' ( 5 ' 6 ) t h where R^t i s the r e a l i z e d excess return for the j asset during p e r i o d t; i s the r e a l i z e d excess market return f o r period t; V q and are con-stants; and u. i s a zero mean random disturbance term. I t i s assumed that 3*-R and u. are normally d i s t r i b u t e d random v a r i a b l e s and uncorrelated. M ] The hypothesis of 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 bankrupcy 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 u a l r e -turn a f t e r a b s t r a c t i n g from the market. Thus an empirical t e s t of the hypo-t h e s i s i s represented by estimating the c o e f f i c i e n t s of Equation (5.6). If 152 the c o e f f i c i e n t , v^, of the p r o b a b i l i t y of bankruptcy i s p o s i t i v e , then t h i s o f f e r s confirmation of the v a l i d i t y of the hypothesis. Methodology The constant term, v , and the c o e f f i c i e n t , v., of the p r o b a b i l i t y o i of bankruptcy are not f i r m dependent, i n contrast to the beta c o e f f i c i e n t , 3^, which i s fi r m s p e c i f i c . This s t r u c t u r a l property of Equation (5.6) i s u t i l i z e d by the methodology employed t o t e s t the hypothesis. Pooling of Time Series and Cross S e c t i o n a l Data The methodology used i s that of p o o l i n g the time s e r i e s and cross g s e c t i o n a l data. Time s e r i e s data f o r i n d i v i d u a l s e c u r i t i e s are pooled to-gether to estimate the two : common c o e f f i c i e n t s , w h i l s t simultaneously e s t i -mating the firm s p e c i f i c beta c o e f f i c i e n t s . Thus, f o r two s e c u r i t i e s the regression equation i s of the form " R l l " 1 x i i - *1 t i • 1 • i 1 i i R1T = 1 r> MT -R21 i 1 i 0 t i • 1 • i . i t • R 2T 1 X2T 0 RM1 ' X l U 11 U u IT 21 2T A d i s c u s s i o n of t h i s methodology i s given i n Kuh, E. "The V a l i d i t y of C r o s s - S e c t i o n a l l y Estimated Behaviour Equations i n Time Series 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 Nerlove, M., "Pooling Cross Section and Time Series Data i n the Estimation of a Dynamic Model: The Demand for Natural Gas," Econometvica, V o l . 34 (July, 1966), pp. 585-612. i 153 where the s u f f i x T denotes the number of time periods. The general N s e c u r i t y case can be w r i t t e n i n the matrix form where Y i s a (NxT) vector; X a [ (NxT) x (N+2)] matrix,- 0_ a (N+2) vector of co-e f f i c i e n t s ; and u_ a (NxT) vector o f random disturbance terms, N being the number of s e c u r i t i e s . I t i s assumed that j , k = 1, 2,...,N, The v a l i d i t y of the l a s t assumption i s tenuous. I t implies zero c o r r e l a t i o n across time and between s e c u r i t i e s . I f i t i s not s a t i s f i e d , then i t implies t h a t the estimated c o e f f i c i e n t s w h i l s t being unbiased and con s i s t e n t , w i l l not be minimum variance or, i n general, a s y m p t o t i c a l l y e f f i c i e n t . The c o e f f i c i e n t s , 0_, are estimated by ordinary l e a s t squares. For large data samples, the s p e c i a l s t r u c t u r e of the X matrix can be u t i l i z e d to reduce computational d i f f i c u l t i e s . Aggregation In order to be able t o estimate the c o e f f i c i e n t s of Equation (5.6), i t i s necessary to know the p r o b a b i l i t y of bankruptcy f o r every s e c u r i t y over a l l time periods. As the p r o b a b i l i t y of bankruptcy cannot be d i r e c t l y observed, Y = X 0 + u, (5.7) E(u ) = 0, (5.8) and (5.9) 154 i t i s estimated using the models developed i n the f i r s t p a rt of t h i s chapter. This implies t h a t there w i l l be e r r o r s i n the measurement of the v a r i a b l e , which w i l l cause the estimations of the c o e f f i c i e n t s of Equation (5.6) to be biased. The e f f e c t s of these measurement err o r s can be reduced by aggrega-t i o n ; that i s , the r e l a t i o n s h i p (5.6) can be aggregated over c e r t a i n subsets of the data and the mean values of the v a r i a b l e s used. As the c o e f f i c i e n t s V q and are not f i r m s p e c i f i c , the same c o e f f i c i e n t s w i l l s t i l l be a p p r opri-ate f o r the aggregated r e l a t i o n s h i p . To increase the magnitude i n the changes i n the p r o b a b i l i t y of bank-ruptcy over time, s e c u r i t i e s are assigned i n t o p o r t f o l i o s on the b a s i s of the value of bankruptcy f o r the previous year. This minimizes the w i t h i n variance, w h i l s t maximizing the between p o r t f o l i o variance f o r the p r o b a b i l i t y of bankruptcy. The number o f p o r t f o l i o s i s v a r i e d , enabling examination of p o s s i b l e aggregation e f f e c t s . The f i r s t p o r t f o l i o contains the s e c u r i t i e s which had 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 contains 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 repeated on a year by year b a s i s , so that, i n general, the composition of each p o r t f o l i o changes annually. The use of the previous year's p r o b a b i l i t y value as a c r i t e r i o n f o r assignment avoids s e l e c t i o n bias i n the c o n s t r u c t i o n of the p o r t f o l i o s . The average value of the p r o b a b i l i t y of bankruptcy, and the average rate of return 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 are c a l c u l a t e d on a y e a r l y basis over the whole time period; that i s , the average r a t e of r e t u r n f o r the p o r t f o l i o i s defined t o be R = — E R. , (5.1 pt N . „ j t ' p ieS J  J P 155 where S i s the set o f s e c u r i t i e s contained i n the p o r t f o l i o , and N the P P number of s e c u r i t i e s . The average p r o b a b i l i t y of bankruptcy i s defined to be = 1_ E X . (5.11) Pt N jeS 3 P P Substituting Equation (5.6) i n t o (5.10) gives R ^ = V + i - E v X . + — .E e.(R - X ) + — E u.,.. (5.12) pt o N . _ 1 j t N jeS j Mt A t N . j t P D£S p p J p ;jeS P P As the c o e f f i c i e n t i s not fir m s p e c i f i c , then using Equation (5.11), Equa-t i o n (5.12) becomes V = vo + v i x P t + V ' W V + V ' ( 5 - 1 3 ) where c p neS J P and 1 r u = — E u. • pt N . „ 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 appropriate f o r the aggregate r e l a t i o n , and are not weighted averages. The c o e f f i c i e n t s of (5.13) are e s t i -mated by pooling the time s e r i e s and cross s e c t i o n aggregated data and then using ordinary l e a s t squares r e g r e s s i o n . ! 156 Data To t e s t e m p i r i c a l l y the model corporate data are required to e s t i -mate the p r o b a b i l i t y of bankruptcy and p r i c e data f o r the r a t e s of r e t u r n . The data set c o n s i s t s of a l l firms common to the Compustat F i l e and the Univer-s i t y of Chicago Center f o r Research 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 F i l e (CRSP), which contains monthly p r i c e , dividend and adjusted p r i c e and dividend information f o r a l l s e c u r i t i e s l i s t e d on the New York Stock Exchange (NYSE) i n the period January, 1926 - June, 1970. A f i r m i s included i n the data set i f i t has continuous p r i c e data f o r the years 1959 to 1969, and con-tinuous corporate data f o r the years 1955 to 1969. Using t h i s c r i t e r i o n a t o t a l of 360 firms with 10 years of annual data are contained i n the data set. The monthly returns on the market p o r t f o l i o are defined as the returns 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 of an equal investment i n every s e c u r i t y l i s t e d on the NYSE a t the beginning of each month. The r i s k f r e e i n t e r e s t rate i s defined as the 30 day r a t e on United States Treasury B i l l s . For each s e c u r i t y the annual excess rate of r e t u r n and the annual excess market r a t e of r e t u r n are c a l c u l a t e d from the s t a r t of the firm's f i s -c a l year f o r the 10 year p e r i o d 1960-1969. The p r o b a b i l i t y of bankruptcy i s e s t i -mated using the two formulations represented by the expressions (5.2) and (5.3). The f i r s t expression u t i l i z e s market values of corporate v a r i a b l e s , w h i l s t the second uses book value. The values of the estimated c o e f f i c i e n t s are shown i n Table 5.11. The c o e f f i c i e n t s estimated using p r o b i t a n a l y s i s f o r the f i r s t model u t i l i z i n g market values of corporate v a r i a b l e s , are not used, as one of the c o e f f i c i e n t s had the wrong s i g n . TABLE 5.11 VALUES OF COEFFICIENTS USED TO ESTIMATE THE PROBABILITY OF BANKRUPTCY M E T H O D O L O G Y V A R I A B L E S MARKET VALUES OF CORPORATE VARIABLES Y Q Y x Y 2 ^ 3 L o g i t -3.848 -23.006 -0.270 -0.163 BOOK VALUES OF CORPORATE VARIABLES B Q &^ 3 2 63 L o g i t -2.769 -13.091 -1.084 0.330 P r o b i t -1.470 - 6.639 -0.549 0.139 158 The use of the two formulations r e s u l t s i n a noticeable d i f f e r e n c e i n the estimated lvalues of the p r o b a b i l i t y of bankruptcy, on average the estimates from the model using market values of corporate v a r i a b l e s [expression (5.2) ] are greater and have more v a r i a b i l i t y than those obtained from the a l t e r n a t i v e formulation using book values o f corporate v a r i a b l e s [expression (5.3) ]. This d i f f e r e n c e i n v a r i a b i l i t y i s r e f l e c t e d i n the estimation of the regression c o e f f i c i e n t s i n Equation (5.6) when t e s t i n g the hypothesis. The average y e a r l y values of the p r o b a b i l i t y of bankruptcy are given 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 v a r i a b i l i t y . E m p i r i cal Results The v a l i d i t y of the hypothesis i s t e s t e d by examining the estimated c o e f f i c i e n t s o f Equation (5.6); 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 then t h i s o f f e r s confirmation. The r e s u l t s are set out i n two p a r t s . The hypo-th e s i s i s t e s t e d using aggregated s e c u r i t y data, and then i n d i v i d u a l s e c u r i t y data. Use of P o r t f o l i o s The i n d i v i d u a l s e c u r i t y data i s aggregated i n t o p o r t f o l i o s on the basis of the p r o b a b i l i t y of bankruptcy f o r the past year, as p r e v i o u s l y ex-plai n e d . Using 360 s e c u r i t i e s , p o r t f o l i o s containing 12, 36 and 72 s e c u r i -t i e s over a nine year p e r i o d are constructed. For the p r o b a b i l i t y of bank-ruptcy estimated by the market value formation of Equation (5.2), the r e s u l t s are shown i n Table 5.13. 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 of bank-159 TABLE 5.12 AVERAGE YEARLY VALUES OF THE PROBABILITY OF BANKRUPTCY YEAR MARKET VALUE FORMULATION LOGIT BOOK VALUE FORMULATION LOGIT PROBIT 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 0.00589 0.00909 0.01034 0.00563 0.00276 0.00330 0.00443 0.00430 0.00673 0.00781 0.00279 0.00384 0.00337 0.00293 0.00036 0.00039 0.00048 0.00071 0.00304 0.00276 .00260 .00374 .00316 .00273 .00026 .00033 .00041 .00064 .00295 .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 NUMBER SECURITIES NUMBER OF IN EACH OF COEFFICIENT COEFFICIENT PORTFOLIOS PORTFOLIO OBSERVATIONS V V. R-SQUARED 72 45 0.059 (44.141) 4.601 (39.082) 0.755 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 ) * 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 l e v e l ** 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 0.2 per cent l e v e l 161 ruptcy i s p o s i t i v e , 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 increases i n magnitude as the l e v e l o f aggregation increases. The constant term, v ^ , i s p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t . As expected, i t decreases i n magnitude when the l e v e l of aggregation increases. The value of R-squared increases as aggregation reduces the cross s e c t i o n a l nature of the data. The p r o b a b i l i t y of bankruptcy can be estimated by Equation (5.3) using book values f o r corporate v a r i a b l e s . As the r e s u l t s are very s i m i l a r f o r the l o g i t and p r o b i t estimates, only the former are reported (see Table 5.14). 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 of bankruptcy i s p o s i t i v e f o r two of the p o r t f o l i o s . In a l l cases i t i s not s t a t i s t i c a l l y s i g n i f i c a n t . The constant term, v , i s p o s i t i v e , 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 decreases o i n magnitude as the l e v e l of aggregation increases. I t i s c o n s i s t e n t l y l a r g e r than the constant term i n Table 5.13. The value of R-squared increases as the l e v e l of aggregation increases, though i t i s uniformly lower than the values of R-squared given i n Table 5.13. The d i f f e r e n c e s i n Tables 5.13 and 5.14 i r e s u l t d i r e c t l y from the method of estimating the p r o b a b i l i t y o f bankruptcy. As shown i n Table 5.12, the values obtained from the formulation given by Equation (5.2), which u t i l i z e s market values o f corporate v a r i a b l e s , are l a r g e r and have greater v a r i a b i l i t y than those given by Equation (5.3) using book values. For many firms the book value formulation gives a zero value f o r the p r o b a b i l i t y of bankruptcy, which causes severe econometric problems because of i l l - c o n d i t i o n e d matrices. Whilst aggregation mitigates t h i s problem i t does, however, reduce the over-a l l variance. This w i l l imply that t o obtain r e l i a b l e estimates of the co-e f f i c i e n t , v , w i l l require large data samples. 162 T A B L E 5.14 P O R T F O L I O D A T A : P O O L I N G O F T I M E S E R I E S A N D 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 B O O K 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 N U M B E R O F N U M B E R S E C U R I T I E S N U M B E R O F I N E A C H O F C O E F F I C I E N T C O E F F I C I E N T P O R T F O L I O S P O R T F O L I O O B S E R V A T I O N S V o V l R - S Q U A R E D 5 72 45 0.089 ^ 0.924 0.732 (13.039) (1.65) 10 36 90 ° - 1 1 4 * -0.266 0.651 (16.20) (0.587) 30 12 270 ° - 1 2 1 * 0.419 0.558 (18.59) (1.342) (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 l e v e l 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 of bankruptcy i s , i n general p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t . This provides some evidence of the v a l i d i t y of the hypothesis; that i s , bankruptcy i s a contr i b u t o r y f a c t o r to the structure o f common stock r e t u r n s . If there i s a missing v a r i a b l e which i s not explained by e i t h e r the p r o b a b i l i t y of bankruptcy or the covariance with the market, then t h i s w i l l b ias the estimates of the c o e f f i c i e n t s . I f the data are aggregated, the p o o l -ing of time s e r i e s and cross s e c t i o n a l data becomes more l i k e an aggregated time s e r i e s , and thus the variance of the missing v a r i a b l e w i l l increase the t - s t a t i s t i c s o f the c o e f f i c i e n t s as w e l l as causing b i a s . The s i t u a t i o n w i l l be exacerbated i f the missing v a r i a b l e i s non-stationary. The constant term i n a re g r e s s i o n equation p i c k s up i n d i v i d u a l e f f e c t s not accounted f o r by the exogenous v a r i a b l e s . The po o l i n g of time s e r i e s and cro s s - s e c t i o n data c o n s t r a i n s the constant term t o be the same f o r a l l s e c u r i -t i e s ( p o r t f o l i o s ) , and thus i f there are i n d i v i d u a l s e c u r i t y e f f e c t s present, t h i s w i l l cause the estimated c o e f f i c i e n t s to be biased. To examine these p o s s i b i l i t i e s dummy v a r i a b l e s could be introduced to represent s e c u r i t y ( p o r t f o l i o ) s p e c i f i c e f f e c t s not e x p l i c i t l y accounted f o r by the exogenous v a r i a b l e s . However, dummy v a r i a b l e s w i l l be hi g h l y c o r r e -l a t e d with the p r o b a b i l i t y of bankruptcy. This a r i s e s because the with i n p o r t f o l i o variance across time f o r the p r o b a b i l i t y of bankruptcy w i l l be mini-mized and thus the dummy v a r i a b l e w i l l act as" a proxy 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 loss of p r e c i s i o n of estimation, as s p e c i f i c estimates may have very large sampling e r r o r s . Exploratory empirical work confirmed t h i s and consequently t h i s l i n e of i n v e s t i g a t i o n was not pursued. 164 The l i m i t e d s i z e of the data set — 360 s e c u r i t i e s with 10 years of annual data — r e s t r i c t s the scope of methods to t e s t the hypothesis using aggregated data. By forming p o r t f o l i o s on the basis of the past value of the p r o b a b i l i t y of bankruptcy reduces the time span of data a v a i l a b l e by one year, and aggregation across s e c u r i t i e s decreases the s i z e of the data set even f u r t h e r . Due to these l i m i t a t i o n s , f u r t h e r t e s t s on the hypothesis are conducted using i n d i v i d u a l s e c u r i t y data. Individual S e c u r i t y Data The use of i n d i v i d u a l s e c u r i t y data f a c i l i t a t e s the advantage of a large number of observations. O f f - s e t t i n g t h i s i s the disadvantage of the low d i s c r i m i n a t o r y power of the exogenous v a r i a b l e s due to errors i n measurement. T h i s i s e s p e c i a l l y important when co n s i d e r a t i o n i s taken of the large e r r o r s i n measurement when estimating the p r o b a b i l i t y of bankruptcy. This implies that a l l the estimated 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 biased. Thus the r e s u l t s u s i n g unaggregated data w i l l not be expected t o o f f e r the same r e l i a b i l i t y as those obtained using aggregated p o r t f o l i o data. F i v e t e s t s are conducted u s i n g unaggregated data. The f i r s t t e s t i s to draw a random sample of firms and to perform a regression p o o l i n g time s e r i e s and cross s e c t i o n a l data. Three o f the remaining t e s t s examine var-ious problems which may cause b i a s i n the estimation of the regression co-e f f i c i e n t s . The f i n a l t e s t considers the e f f e c t of changes i n the p r o b a b i l i t y of cankruptcy upon the structure o f ex-post r e t u r n s . Random Sample From the universe of 360 s e c u r i t i e s , a random sample of 100 i s chosen, 165 TABLE 5.15 RANDOM SAMPLE: POOLING OF TIME SERIES AND CROSS SECTION DATA DATA SET METHODOLOGY NUMBER FOR FOR COEFFICIENT COEFFICIENT OF PROBABILITY PROBABILITY V V. R-SQUARED OBSERVATIONS LOGIT 0.115 * -0.820 0.303 1,000 BOOK ( 1 0 ' 5 4 1 ) ( - 1 ' 3 6 6 ) VALUES PROBIT 0.114 t -0.661 0.303 1,000 (10.504) (-1.147) MARKET LOGIT (10.142) (-1.151) VALUES 0.116 # -0.911 0.304 1,000 (Figures in 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 at the 0.1 per cent level 166 g i v i n g a t o t a l of 1,000 observations. The p r o b a b i l i t y of bankruptcy i s e s t i -mated using the market value formulation of Equation (5.2) and the book value formulation of Equation (5.3). The r e s u l t s of pooling time s e r i e s and cross s e c t i o n a l data are shown i n Table 5.15. The r e s u l t s are s i m i l a r f o r the d i f -f e r e n t methods of estimating the p r o b a b i l i t y of bankruptcy. 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 of bankruptcy i s negative, though s t a t i s t i c a l l y i n -s i g n i f i c a n t . The constant term, v , i s p o s i t i v e and s t a t i s t i c a l l y s i g n i f i -o cant. The value of R-squared i s quite large considering the strong cross s e c t i o n a l nature o f the r e g r e s s i o n data. The n e g a t i v i t y of the 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 the p r o b a b i l i t y of bankruptcy i s estimated using market values of corporate v a r i a b l e s or book values, r e f l e c t s the e f f e c t s of e r r o r s i n the measurement of the v a r i a b l e and stands i n c o n t r a s t to the r e s u l t s ob-served i n Tables (5.13) and (5.14). Aggregation o f the data s e t i n t o p o r t -f o l i o s reduces the variance of the estimate of the p r o b a b i l i t y of bankruptcy and r e s u l t s i n more r e l i a b l e estimation of the regression c o e f f i c i e n t s . E f f e c t o f Asset Size The c o e f f i c i e n t s used to estimate the p r o b a b i l i t y of bankruptcy are determined using a data set of firms c h a r a c t e r i z e d by having an asset s i z e of l e s s than $200 m i l l i o n . To t e s t the hypothesis the asset s i z e of the firms i n the data set i s not r e s t r i c t e d . I f there are scale e f f e c t s i n the c o e f f i c i e n t s used to estimate the p r o b a b i l i t y of bankruptcy, then e r r o r s i n i t s measurement w i l l increase when app l i e d to large firms. To examine t h i s p o s s i b i l i t y , the universe of 360 firms i s sorted by asset s i z e i n t o three groups and a pooling of time s e r i e s and cross s e c t i o n data 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 ASSET SIZE COEFFICIENT V COEFFICIENT NUMBER OF R-SQUARED OBSERVATIONS BOOK VALUES $(0 - 200)m. $ (200-500)m. $500m. and over 0.117 (11.841) 0.119 ( 9.654) 0.084 . (10.691) -0.998 . (-2.712) -0.216 (-1.116) -0.300 (-0.302) 0.358 0.287 0.281 1,270 1,010 1,320 MARKET VALUES $(0 - 200)m. $(200-500)m. $500m. and over 0.118 (11.645) 0.119 ( 9.531) 0.076 ( 8.676) -1.124 (-2.425) -0.147 (-0.621) 1.936 ( 1.767) ** 0.359 0.287 0.285 1,270 1,010 1,320 (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 at the 0.1 per cent l e v e l . ** s t a t i s t i c a l l y s i g n i f i c a n t at the 2.0 per cent l e v e l . 168 within each group conducted. As the r e s u l t s f o r the estimates of the p r o b a b i l i t y of bankruptcy using the book value formulation of Equation (5.3) when i t s c o e f f i c i e n t s are estimated by e i t h e r l o g i t a n a l y s i s or pro-b i t analysis are very s i m i l a r , only the former are reported (see Table 5.16). There does appear to be some evidence that the regression estimates are a f f e c t e d by asset s i z e . The constant term, v^, f o r large firms i s r e -duced i n magnitude when compared to the value f o r small and intermediate s i z e firms. 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 of bankruptcy, i s nega-t i v e and s t a t i s t i c a l l y s i g n i f i c a n t f o r firms of asset s i z e of l e s s than $200 m i l l i o n . Adjustment of Time Period The p r o b a b i l i t y of bankruptcy i s estimated using corporate data that r e f l e c t s the st a t e of the f i r m at the beginning of i t s f i s c a l year. The annual r a t e s o f return data f o r a f i r m are c a l c u l a t e d over the 12 month perio d s t a r t i n g a t the beginning of the firm's f i s c a l year, and then the two data sets combined and used t o estimate the regression c o e f f i c i e n t s . However, a firm's annual report i s u s u a l l y announced two or three months a f t e r the end of the f i s c a l year, and thus as the p r o b a b i l i t y of bankruptcy i s estimated using corporate data which, to the average in v e s t o r , w i l l only be a v a i l a b l e from a firm's annual report, i t may not be used to r e f e r to the 12 month perio d beginning a t the s t a r t of the firm's f i s c a l year. Hence, the c a l c u l a t e d annual rate of r e t u r n and the estimate of the p r o b a b i l -i t y of bankruptcy, which are used as proxies f o r the values an investor ex-169 TABLE 5.17 ADJUSTMENT OF TIME PERIOD NUMBER OF MONTHS AFTER BEGINNING OF FISCAL YEAR COEFFICIENT V o COEFFICIENT V l R-SQUARED NUMBER OF OBSERVATIONS 0 0.117 # -0.998 ^ 0.358 1,270 (11.841) i (-2.712) 2 0.104 t -0.800 0.271 1,260 (10.541) (-1.504) 3 0.105 t -0.873 0.273 1,260 (10.559) (-1.631) 4 0.089 # -0.520 0.302 1,260 ( 9.401) (-0.990) (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 at the 0.1 per cent l e v e l . ** 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 2.0 per cent l e v e l . 170 pects, may not be synchronized. This p o s s i b i l i t y i s examined by c a l c u l a t i n g the annual rate of r e t u r n from a p o i n t two, three, and four months a f t e r the beginning of a firm's f i s c a l year. The regression c o e f f i c i e n t s are estimated using a data set of firms with asset si z e l e s s than $200 m i l l i o n . This choice of data set i s moti-vated i n view of the f i n d i n g given i n Table 5.16, which i n d i c a t e d some de-pendence of regression c o e f f i c i e n t s upon asset s i z e . The s i z e o f the data set i s reduced by one f i r m because of the extra p r i c e data requirements. As the f i n d i n g s are very s i m i l a r f o r d i f f e r e n t methods of estimating the p r o b a b i l i t y of bankruptcy, only those obtained from the formulation u t i l i -z i n g market values of corporate v a r i a b l e s are reported. The r e s u l t s are shown i n Table 5.17. In Table 5.17 there does appear to be some d i f f e r e n c e between the r e s u l t s . 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 of bankruptcy, w h i l s t negative, increases i n value and becomes s t a t i s t i c a l l y i n s i g n i f i c a n t when the rate of r e t u r n data are s h i f t e d forward two months. The constant term, v , i s reduced i n magnitude though remains s t a t i s t i c a l l y s i g n i f i c a n t . The value of R squared i s reduced. Further s h i f t s i n the rate o f r e t u r n data set produce only minor d i f f e r e n c e s . Cross Section Studies In a regression with two independent v a r i a b l e s i f one i s measured with error, then i t s estimated c o e f f i c i e n t w i l l be biased downwards, w h i l s t the constant term w i l l be biased upwards. If both v a r i a b l e s are measured with error no comment can be made about the d i r e c t i o n of the r e s u l t i n g biases. However, i f an instrumental v a r i a b l e i s substituted f o r one of the v a r i a b l e s 171 and the measurement er r o r s i n the instrumental v a r i a b l e are independent of the measurement er r o r s i n the remaining v a r i a b l e , then the c o e f f i c i e n t of the remaining v a r i a b l e w i l l be biased downwards. I t i s not pos s i b l e to make any comment about the d i r e c t i o n of b i a s i n the constant term. To explore the p o s s i b l e consequences of measurement er r o r s i n the 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 studies are conducted. The firm's beta c o e f f i c i e n t i s replaced by an instrumental v a r i a b l e . For i n d i v i d u a l s e c u r i t i e s an instrumental beta c o e f f i c i e n t i s estimated by regressing the ex-post, excess return against the r e a l i z e d r e turn on the market using the past f i v e years of monthly p r i c e data p r i o r t o the pe r i o d o f the cross s e c t i o n a l study. These estimates are s u b s t i t u t e d i n t o the reg r e s s i o n equa-t i o n s R j = v o + V j + V j + u j ( 5 -j = 1, 2,...,N, where R. i s the annual excess r e t u r n f o r the j*"* 1 asset; X_, i s the p r o b a b i l i t y t h * of the j asset going bankrupt over a p e r i o d o f one year; B.. i s the i n s t r u -mental beta c o e f f i c i e n t ; u. i s a random disturbance term; and v , v, and v. 3 o 1 2 are regression c o e f f i c i e n t s . I f the measurement e r r o r s i n the beta c o e f f i -c i e n t s are independent of the measurement e r r o r s i n the p r o b a b i l i t y of bank-ruptcy then the estimate of c o e f f i c i e n t w i l l only be subject to a b i a s which underestimates the true value. The ex t r a p r i c e data requirements r e -duce the data set to 318 firms. The p r o b a b i l i t y of bankruptcy i s estimated using Equation (5.2), which u t i l i z e s market values of corporate v a r i a b l e s . Using annual data f i v e cross s e c t i o n regressions f o r d i f f e r e n t periods of time are conducted. The r e s u l t s are given i n Table 5.18. 1 7 2 TABLE 5 . 1 8 CROSS SECTION STUDY TIME ANNUAL COEFFICIENT COEFFICIENT COEFFICIENT 3 EXCESS RETURN V. ON THE MARKET R-SQUARED 1 9 5 9 - 6 0 0 . 1 7 9 . ( 4 . 4 1 7 ) - 3 . 5 7 9 (-1.630) - 0 . 2 1 0 . (-4.081) - 0 . 0 6 1 0 . 1 1 1 1 9 6 1 - 6 2 - 0 . 0 1 6 (-0.419) T O . 1 3 9 (-0.555) - 0 . 1 1 0 ( 1 . 3 0 2 ) •0.156 0 . 0 3 1 1 9 6 3 - 6 4 0 . 1 5 5 , ( 3 . 4 2 2 ) - 0 . 7 5 0 (-0.851) 0 . 0 0 1 (-2.481) ** 0 . 1 2 5 0 . 0 0 2 1 9 6 5 - 6 6 - 0 . 1 8 6 ( - 3 . 4 6 6 ) ' - 0 . 9 3 0 (-1.182) 0 . 1 2 8 T ( 5 . 8 4 3 ) - 0 . 0 6 9 0 . 0 1 9 1 9 6 7 - 6 8 0 . 2 4 5 ( 5 . 6 5 5 ) - 0 . 3 8 6 (-0.887) - 0 . 0 8 3 T (-7.438) 0 . 2 3 7 0 . 0 1 5 (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 l e v e l . ** s t a t i s t i c a l l y s i g n i f i c a n t at the 2.0 per cent l e v e l . 3 t - s t a t i s t i c s are taken about the annual excess return on the market. 173 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 of bankruptcy i s c o n s i s t e n t l y negative, though s t a t i s t i c a l l y i n s i g n i f i c a n t from zero. The constant term, V Q , i s g e n e r a l l y 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 f l u c t u a t e s i n sign. The t h e o r e t i c value of the c o e f f i c i e n t i s the annual excess r e t u r n on the market po r t -f o l i o . When the estimated value i s compared to i t s t h e o r e t i c value i t i s found that, i n general, they are d i f f e r e n t , the d i f f e r e n c e being s t a t i s t i c a l l y s i g n i f i c a n t . Two o f the estimates have the wrong s i g n . Changes i n the P r o b a b i l i t y of Bankruptcy The method of p o r t f o l i o c o n s t r u c t i o n used to increase the between p o r t f o l i o variance of changes i n the p r o b a b i l i t y of bankruptcy and the p o s i -t i v e r e s u l t s derived, suggests that s i m i l a r r e s u l t s using i n d i v i d u a l s e c u r i t y data should be obtained by considering the e f f e c t s of changes i n the p r o b a b i l -i t y of bankruptcy; t h a t i s , by considering a r e l a t i o n s h i p of the form , (\ _\ 1 + R. (R - Y ) + U... (5.15) R.. = 3t V o + v.X .. 1 j t j = 1, 2/ • ••f Nf t = 2, 3 f •••iT/ where R., i s the annual excess r a t e o f r e t u r n f o r year t f o r the j asset; X 3t 3t t i l i s the p r o b a b i l i t y o f the j asset going bankrupt during year t ; R^t i s the market excess return; x+. i s a weighted average o f the p r o b a b i l i t i e s ; u. i s a random disturbance term; and v^, v^, v^, and -f6^J are r e g r e s s i o n c o e f f i c i e n t s . The r e l a t i o n s h i p i s examined by f i r s t taking a random sample o f 100 firms from a universe of 360 firms and then by s o r t i n g the universe of firms by asset s i z e i n t o three groups. A r e g r e s s i o n pooling time s e r i e s and cross s e c t i o n data i s performed on each group. The 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 DATA SET NUMBER OF OBSERVATIONS COEFFICIENT COEFFICIENT COEFFICIENT R-SQUARED RANDOM 900 0.132 (10.497)' 0.433 (0.471) -3.073 ^ 0.291 (-3.206) $ (0-200)m. 1,143 0.138 (12.287) -0.548 (-0.838) -1.649 ^ 0.350 (-2.306) $(200-500)ra. 909 0.142, (10.39) 0.300 (1.1375) -0.841 ^ 0.271 (-3.111) $500m. and over 1,188 0.078 6.887 ( 7.310) (4.932) -10.520 # 0.282 ( 5.637) (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 l e v e l . ** s t a t i s t i c a l l y s i g n i f i c a n t at the 2.0 per cent l e v e l . 175 estimated using Equation (5.2), which u t i l i z e s market values of corporate v a r i a b l e s . The r e s u l t s are shown i n Table 5.19. 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 of bankruptcy i s now p o s i -t i v e , with the exception f o r the group of firms of asset s i z e l e s s than $200 m i l l i o n . For firms of asset s i z e $500 m i l l i o n and over, the c o e f f i c i e n t i 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 . The constant term, v , i s uniformly p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t . 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 negative and s t a t i s t i c a l l y s i g n i f i c a n t . The r e s u l t s should be compared to those given i n Table 5.16. I t i s seen that the sign of the c o e f f i c i e n t , v^, with one exception, i s now reversed. For firms of asset s i z e l e s s than $200 m i l l i o n , though the c o e f f i c i e n t , v^, i s s t i l l negative, i t i s no longer s t a t i s t i c a l l y s i g n i f i c a n t . Summary The hypotheses 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 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 u a l return of common stocks a f t e r a b s t r a c t i n g from the market. The hypothesis i s tested using i n d i v i d u a l and aggregated s e c u r i t y data. The use of i n d i v i -dual s e c u r i t y data provide a l a r g e data set with which to t e s t the hypothesis. However, the biases t h a t r e s u l t from the e r r o r s i n the measurement of the p r o b a b i l i t y of bankruptcy present serious econometric problems. These prob-lems can be mitigated by aggregating the data to reduce the variance of the estimates of the p r o b a b i l i t y of bankruptcy. S e c u r i t i e s are assigned i n t o p o r t f o l i o s on the basis of past values of t h e i r p r o b a b i l i t y of bankruptcy, and the aggregated p o r t f o l i o data treated as representative s e c u r i t i e s . The 1 7 6 use of aggregation does, however, reduce the s i z e of the data set and pro-h i b i t s extensive t e s t i n g . Evidence i s found v e r i f y i n g the hypothesis of the t h e s i s . Using aggregated p o r t f o l i o data and estimating the p r o b a b i l i t y of bankruptcy with market values of corporate v a r i a b l e s , confirmation of the hypothesis i s ob-tained. When the p r o b a b i l i t y of bankruptcy i s estimated using book values of corporate v a r i a b l e s , the evidence i s i n c o n c l u s i v e . Use of i n d i v i d u a l s e c u r i t y data does not give any c l e a r i n d i c a t i o n of the v a l i d i t y of the hypotheses. The major d i f f i c u l t y appears to be due to the e r r o r s i n the measurement of the p r o b a b i l i t y o f bankruptcy. CHAPTER VI SUMMARY The purpose of t h i s chapter i s to summarize the main conclusions of the t h e s i s and to describe the areas of f u r t h e r research that a r i s e from i t . Conclusions The impact of bankruptcy upon the 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 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 viewpoint. A model, formulated i n continuous time, considers the investment-consumption d e c i s i o n of an i n d i v i d u a l a c t i n g to maximize the expected l i f e t i m e u t i l i t y of consumption and terminal wealth. At each i n s t a n t i n time the i n d i v i d u a l must decide the portions of wealth to consume and to i n v e s t i n f i n a n c i a l assets. I t i s assumed that a f i r m i ssues both bonds and common'stock as f i n a n c i a l assets and that at each p o i n t i n time there i s a p r o b a b i l i t y t h a t the f i r m w i l l go bankrupt the next i n s t a n t . I f bankruptcy occurs i t i s assumed that equity holders s u f f e r a hundred per cent l o s s , and bondholders receive a non-negative l i q u i d a t i n g dividend. From t h i s formulation the e q u i l i b r i u m expected rates of return on the d i f f e r e n t f i n a n c i a l a s s e t s — bonds and common s t o c k s — a r e determined using s t o c h a s t i c c o n t r o l theory. For bonds of i n f i n i t e maturity, the e q u i l i b r i u m expected excess rate of return, c o n d i t i o n a l upon no bankruptcy, i s a l i n e a r f u n c t i o n of 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 product of the p r o b a b i l i t y of bankruptcy and the expected l o s s i f bankruptcy occurs. This r e s u l t demonstrates, on a t h e o r e t i c a l b a s i s , some of the determinants of the 177 178 r i s k premium on a bond and i s d i r e c t l y amiable to em p i r i c a l t e s t i n g . For common stocks a two v a r i a b l e expression f o r the e q u i l i b r i u m expected rate of return, c o n d i t i o n a l upon no bankruptcy, i s derived. The expression i s an extended form of the continuous time analogy to the C a p i t a l Asset P r i c i n g Model (CAPM), the second v a r i a b l e being associated with the p r o b a b i l i t y of bankruptcy. This r e s u l t i s important. I t demonstrates that bankruptcy i s a contributory f a c t o r i n d e s c r i b i n g the str u c t u r e of common stock returns. I t provides a t h e o r e t i c explanation of the recent e m p i r i c a l f i n d i n g s i n d i c a t i n g that the 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 of the "beta f a c t o r . " This r e s u l t i s e m p i r i c a l l y tested. A d i s c r e t e time formulation of the model i s used. The hypothesis i s that 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 i s r e f l e c t e d i n the r e s i d u a l return a f t e r a b s t r a c t i n g from the market. To t e s t the hypothesis i t i s necessary to be able to measure the 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 bankruptcy has not addressed t h i s problem, but has concentrated on constructing models to c l a s s i f y firms i n t o one of two groups: bankrupt or not bankrupt. A model for the p r o b a b i l i t y of bankruptcy i n terms of a firm'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 or e x t e r n a l l y , to cover f i x e d charges i s con-structed, and an ex-post formulation i n terms of measurable q u a n t i t i e s developed. The p r o b a b i l i t y of bankruptcy i s estimated using p r o b i t a n a l y s i s and l o g i t a n a l y s i s . The a b i l i t y of the model to p r e d i c t bankruptcy i s tested on a secondary sample of bankrupt firms. The r e s u l t s are very good with the model p r e d i c t i n g bankruptcy, f o r some firms, four or f i v e years before the ac t u a l occurrence. 179 The hypothesis i s tested u s i n g i n d i v i d u a l and aggregated U.S.A. annual s e c u r i t y data f o r the 10 year period 1960 to 1969. A new method-ology to the t e s t i n g of two v a r i a b l e extended forms of the CAPM, that of pooling time s e r i e s and cross s e c t i o n a l data, i s introduced. The time se r i e s data f o r a l l i n d i v i d u a l s e c u r i t i e s ( p o r t f o l i o s ) are combined to estimate the c o e f f i c i e n t s t h a t are common to a l l s e c u r i t i e s ( p o r t f o l i o s ) , w h i l s t simultaneously estimating the f i r m (hypothesis) s p e c i f i c beta co-e f f i c i e n t s . Evidence i s found v e r i f y i n g the hypothesis of the t h e s i s ; that i s , bankruptcy i s an explanatory f a c t o r of the structure of corpor-ate f i n a n c i a l a s s e t s . Hence, the e x p l i c i t o b j e c t i v e s of the t h e s i s : (a) to analyze t h e o r e t i c a l l y how the mechanism of bankruptcy a f f e c t s the structure of r e -turns f o r corporate f i n a n c i a l assets; (b) t o qu a n t i f y the determinants of bankruptcy, and to a r r i v e at 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 of bankruptcy; and (c) to t e s t e m p i r i c a l l y the hypothesis of the t h e s i s , are s u c c e s s f u l l y achieved. Future Research The t h e s i s demonstrates the need t o develop a complete explanatory theory of the p r o b a b i l i t y of bankruptcy. Such a theory needs t o consider, the f a c t o r s that determine a firm's a b i l i t y t o r a i s e funds and the i n t e r -dependence between sources; that i s , the e f f e c t of using one source on the a b i l i t y to u t i l i z e other sources of funds. Development of such a theory e n t a i l s c o nsideration of the broader question of v a l u a t i o n . For example, a firm's a b i l i t y t o issue debt depends upon i t s debt capacity, but the con-cept of debt capacity i s i n t r i n s i c a l l y r e l a t e d to the value of the firm and 180 to the p r o b a b i l i t y o f bankruptcy. Thus there i s c i r c u l a r i t y . Any analy-s i s requires a d e s c r i p t i o n of the d i f f e r e n t a t t r i b u t e s which determine the p r o b a b i l i t y of bankruptcy and then a s p e c i f i c a t i o n of the determinants of the a t t r i b u t e s . This leads to a system of simultaneous equations which, i n general, w i l l be non-linear. The development of a complete explanatory theory of the p r o b a b i l i t y of bankruptcy i s important not only from a t h e o r e t i c a l viewpoint, but al s o because of the many p r a c t i c a l a p p l i c a t i o n s to which i t can be a p p l i e d . For example, i t can be u t i l i z e d i n business loan evaluation, and f o r i n -t e r n a l management. For business loan evaluation — consumer loans or commercial loans — the same form of methodology can be u t i l i z e d to deter-mine the p r o b a b i l i t y o f an i n d i v i d u a l or fir m d e f a u l t i n g on a loan. Simi-l a r l y , i t can be used f o r accounts r e c e i v a b l e management to estimate the p r o b a b i l i t y of a customer d e f a u l t i n g on payment. Other i n t e r n a l management uses involve determining the e f f e c t s of d i f f e r e n t investment and f i n a n c i a l mixes upon the p r o b a b i l i t y of the f i r m going bankrupt. A second area of research i s to consider the e f f e c t s o f intr o d u c i n g the market imperfection of f i n a n c i a l d i s t r e s s upon the C a p i t a l Asset P r i c i n g Model. Th i s ne c e s s i t a t e s d e f i n i n g an op e r a t i o n a l d e f i n i t i o n of f i n a n c i a l d i s t r e s s and d e s c r i b i n g the consequences of f i n a n c i a l d i s t r e s s upon the market value of the f i r m . From t h i s b a s i s the e q u i l i b r i u m r a t e of return can be determined. A t h i r d area of research i s the determination of the r i s k premium on a bond. The th e s i s provides a t h e o r e t i c framework w i t h i n which to analyze t h i s problem and r e s u l t s which are d i r e c t l y amiable to empirical v e r i f i c a t i o n . Further development of t h i s work needs to consider the e f f e c t s of such f a c t o r s as c a l l a b i l i t y , maturity, m a r k e t a b i l i t y and term structure. APPENDIX A MATHEMATICAL DERIVATION OF THE RESULTS IN CHAPTER IV The purpose of t h i s appendix i s to explain i n greater d e t a i l the mathematical d e r i v a t i o n of the r e s u l t s presented i n Chapter IV. The appen-dix should be read i n conjunction with Chapter IV, as i n some cases r e s u l t s which have been given s a t i s f a c t o r y explanation i n the chapter are used i n the appendix without f u r t h e r explanation. The f i r s t p a r t of the appendix describes the d e r i v a t i o n of the sto c h a s t i c d i f f e r e n t i a l equations, the equation of o p t i m a l i t y , and the f i r s t order maximization c o n d i t i o n s . A d i s c u s s i o n of the d i f f i c u l t i e s i n obtaining a s o l u t i o n to these equations and the approximations t h a t are used to obtain a l i n e a r system i s given. In the second part the eq u i l i b r i u m instantaneous c o n d i t i o n a l expected r a t e s of return, given the assumption of a constant investment opportunity set, are derived. The approximation used to derive a l i n e a r system of equa-t i o n s i s examined i n greater d e t a i l . In the t h i r d and f i n a l p a rt of the appendix, the general case of st o c h a s t i c changes i n the r a t e of the p r o b a b i l i t y of bankruptcy i s considered. The demand functions f o r the f i n a n c i a l assets are derived. A s p e c i a l case i s considered i n which the s t o c h a s t i c changes i n the r a t e of the p r o b a b i l i t y of bankruptcy f o r one fir m acts as an instrumental v a r i a b l e f o r a l l other changes. For t h i s case, the instantaneous c o n d i t i o n a l expected r a t e s of ret u r n are der i v e d . 182 183 General Formulation A d e r i v a t i o n of the s t o c h a s t i c d i f f e r e n t i a l equations d e s c r i b i n g the p r i c e dynamics of the f i n a n c i a l assets, the equation of o p t i m a l i t y , and the f i r s t order maximization conditions i s given. A d i s c u s s i o n of the d i f f i c u l t i e s i n obtaining a s o l u t i o n to these equations and the approxi-mations that are used to o b t a i n a l i n e a r system i s presented. P r i c e Dynamics An information d e r i v a t i o n o f these equations i s presented i n Chapter IV. I t i s proposed to give, on a s l i g h t l y more rigorous b a s i s , a d e r i v a t i o n o f the equations. The two approaches are, however, equivalent as the time i n t e r v a l tends to zero. I t w i l l be assumed that the event of bankruptcy follows a Poisson process. A Poisson process i s a continuous time process with a d i s c r e t e state space; th a t i s , one where there are d i s c r e t e or discontinuous changes i n the variables.^* Let N(t,t+h) denote the number of events i n the time 2 i n t e r v a l (t,t+h]. Then the Poisson process i s defined as f o l l o w s : Pr[N(t,t+h) = 0] = 1 - \ (t)h + 0(h), Pr[N(t,t+h) =1] = A ( t ) h + 0(h), and Pr[N(t,t+h) > 1] = 0(h), *For a good i n t r o d u c t i o n to Poisson processes, see Cox, D. R. and M i l l e r , H. D., Theory of Stochastic Processes (London: Methuen s Co. Ltd., 1968). 2 The bracket notation i s defined as: i f an element x belongs to the i n t e r v a l (t,t+h], then t < x <_t+h; that i s , the p o i n t t i s excluded from consideration. 184 where Pr( ) means the p r o b a b i l i t y of; and 0(h) i s the asymptotic order sym-bol defined by f(h) i s 0(h) i f l i m [f(h)/h] = 0. h-*o To a f i r s t order approximation, 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 of bankruptcy occurring i n the i n t e r v a l (t,t+h]. I t i s assumed that Poisson processes f o r d i f f e r e n t firms are independent; the event of bankruptcy f o r one f i r m does not a f f e c t other f i r m s . Conceptually, i t i s very simple to re l a x t h i s assumption, but at the cost of g r e a t l y i n -creasing the complexity of notations. The very small gain i n g e n e r a l i t y of r e s u l t s does not warrant the burden of using 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 the event of bankruptcy can be represented by such a process, as i t i s p o s s i b l e f o r N(t,t+h) to equal, . f i v e , w h i l s t the event of bankruptcy for a f i r m can only occur once i n t h i s model. However, the p r o b a b i l i t y of N(t,t+h) e q u a l l i n g f i v e i s of order h or l e s s and thus i n the l i m i t as h tends t o zero, i s zero. As the whole formulation i s i n continuous time, and a l i m i t i n g process i s u t i l i z e d , then the representation of the event of bankruptcy by a Poisson process i s p e r f e c t l y v a l i d . The advantage of using a Poisson d i s t r i b u t i o n l i e s i n i t s c o n t i n u i t y over the time domain. th From Equation (4.5) the p r i c e dynamics of the j firm's bonds can be described by 3 I t i s important to r e a l i z e that A(t) i s not a p r o b a b i l i t y , but a p r o b a b i l i t y rate; i t i s the p r o b a b i l i t y per u n i t i n t e r v a l of time. The length of 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, or a year. 185 bj (t) (1+r^.h) - g_.h + b_. (t) Y VhY.. ( t ) ; no d e f a u l t , b. (t+h) = < A^.. (t+h) - 9.. (t+h) ; d e f a u l t , 3 — 1/ 2, • . ., n« The above equation can be w r i t t e n i n the form b..(t+h) = [bj (t) (l+r_.h) - g^h + b^ (t) Y VhY^ (t) ] [1-N.. (t, t+h) ] + [Aj(t+h) - 0j(t+h)Nj(t,t+h). (A.l) If the event of bankruptcy does not occur to f i r m j i n the i n t e r v a l (t,t+h], then Nj(t,t+h) i s zero. If bankruptcy does occur then N.. (t,t+h) equals one. Define a s t o c h a s t i c process, 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 process with independent increments. The l i m i t as h tends t o zero of Z(t+h)-Z(t) describes a Wiener process, o r Brownian 4 motion. In the terminology of 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 to zero, Equation (A.l) can be w r i t t e n i n the form: db.(t) = [ b . ( t ) r . - g . ] d t + b.(t)Y.dZ. - {b . (t) - [A . (t)-6 . (t) ] }dq . (A.2) j = 1, 2,. . .,n, where dq^ i s a Poisson process c h a r a c t e r i z i n g the event of bankruptcy for.the j t h f i r m . 4 For a general d i s c u s s i o n , see Cox and M i l l e r , op. ext., pp. 205-208, i 1 8 6 In a s i m i l a r manner the p r i c e dynamics f or equity can be derived. From Equation (4.9), i t can be w r i t t e n i n the form p.(t+h) = [p.(t)(l+a.h)-f.h+ p . ( t ) a . / h Y (t)l[1-N.(t,t+h)] 3 J 3 3 3 3 n+3 3 In the l i m i t as h tends to zero, the above equation becomes dpj(t) = t P j (t)ct..-f ] d t + p ^ t ) j ^ n + j - P j ( t ) d q j f (A.3) 3 — 1/ 2f» « •/n• The Equation of Optimality: The Demand Functions f o r Assets From Equation (4.22), the derived u t i l i t y f u n c t i o n i s defined as J[W(t) , a , 0 , r , Y , f ,X,r .t,S(t)] = Max E. {/jJ[C(s),S]dS + BF[W(T),T]}, (A.4) {c,w} subject to a wealth 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 derived u t i l i t y f unction, J , can be w r i t t e n i n a more compact and convenient form. 5 Consider the s e t o f equations d e s c r i b i n g how the opportunity s e t changes: da . = F . (a . , t ) d t + G . (oi., t)dQ ., 3 3 3- 3 3 3 d a . = F . ( a . , t ) d t + G . (a . ,t)dQ , ., 3 n+3 j n+3 3 n+D dr. 3 = F 2 n + j ( r j ' t ) d t + G 2 n + j ( r j ' t ) d Q 2 n + j ' = F 3 n + j ( V t ) d t + G 3 n + j ( Y 3 ' t ) d Q 3 n + j ' (A.5) dX . 3 = F 4 n + j ( X j ' t ) d t + G 4 n + j ( X j ' t ) d Q 4 n + j ' df . 3 = F 5 n + j ( f j ' t ) d t + G 5 n + j ( f r t ) d Q 5 n + j , 5See Equations (4.12), (4.13), (4.14) and (4.16). 187 and dr = F (r , t ) d t + G (r , t ) d Q , F m F m * ro j = 1, 2,. . .,n, and m = 6n+l. Define a (mxl) vector dV by dV = (do,,..., da ,do- ,...,da ,dr ,...,dr .dy.,.•.,dy ,dX ,..., 1 n X n x n x u x .dX , d f d f ,dr ) n 1 n m (A. 6) Hence, dV = Fdt + GdQ, (A.7) where F i s a (mxl) vector such that F' = 0 ^ , . . . ^ ) , dQ i s a (mxl) vector such th a t dp' = (dQ,,...,dQ ) and G i s a (mxm) matrix with elements along the 1 m diagonal and zeros elsewhere. Given the d e f i n i t i o n (A.6), then by comparing (A.7) with (A.5) the elements of F and G can be i d e n t i f i e d . Therefore, Equation (A.4) can be w r i t t e n i n the form J [ W ( t ) , V ( t ) , t , S ( t ) ] = Max E f c{/^U[C(x),x]dx + BF[W(T),T]}, (A.8) {c,w} subject to a wealth c o n s t r a i n t , a budget c o n s t r a i n t , and C(x) >^  0. Thus, t+h J [ W ( t ) , V ( t ) , t , S ( t ) ] > E f c{/ U[C(x),x]dx + J[W(t+h),V(t+h),t+h,S(t+h)]}, where S(t+h) i s a state d e s c r i p t i o n vector d e s c r i b i n g the system at time t+h. By using the mean value theorem f or i n t e g r a l s , the f i r s t term on the r i g h t hand side of the above equation can be approximated to U [ C ( t ) , t ] h + 0(h) . 6See Chapter 7 of Dreyfus, S. E., Dynamic Programming and the Calcu-lus of Variation (Mew York: Academic Press, 1965). 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 order to evaluate the second term on the r i g h t hand side of Equation (A.9), a c o n d i t i o n a l expectation argument i s used. At time (t+h) i t i s only necessary to consider (n+1) of the 2 n p o s s i b l e s t a t e s of the sys-tem, the p r o b a b i l i t y of occurrence of these states being known. Given a p a r t i c u l a r s t a t e , the expected value of the random v a r i a b l e , c o n d i t i o n a l upon the state i s c a l c u l a t e d and then the unconditional expected value determined. 7 Mathematically, the argument can be represented as: i f (X,Y) i s a two dimen-s i o n a l random v a r i a b l e , the c o n d i t i o n a l expectation of X f o r a given Y=y^ i s defined by C O E(XJY=y.) = E x.P (X=x.|Y=y.), 1=1 and the expected value of X i s E(X) = E V[E(X|Y = y )] . n With p r o b a b i l i t y [1 - E X.h + 0(h)] no d e f a u l t s occur i n the i n t e r -1 = 1 1 ' 8 v a l (t,t+h]. Thus, c o n d i t i o n a l upon the event, the budget c o n s t r a i n t i s n W(t+h) - W(t) = (w(t) - C(t)h}{ E w. (t) [ ( r . - r ) h + y.dZ.] j=l 3 3 3 3 n + E w . (t) [ ( c t.-r)h + a.dZ .] + rh} - C ( t ) h + 0(h) j = 1 n +D 3 3 n+j (A.10) 7 For an introductory d i s c u s s i o n , see Meyer, P. L., Introductory Probabil-ity and Statistical Application (Massachusetts: Addison-Wesley, 1965). Q See Equation (4.18). 189 The expected value of the change in wealth, c o n d i t i o n a l upon no d e f a u l t s , i s 9 n n E (W (t+h) -W(t) | no defaults] = [ w (t) -C (t) h] [ E w . ( t ) ( r . - r ) + Z w . (t) (ct.-r)+r] h fc j-1 3 3 j = l n + 3 3 - C(t)h+ 0(h) , (A.11) and E {[W(t+h)-W(t)] 2|no defaults} 2 h n n n ( t ) = [W(t) [ Z Z w (t)y w. (t) + 2 Z Z w. (t)Y.p . . 0 j = l i = l 3 3 1 1 j = l i = 1 3 ] ] H n+i n n + Z Z w (t)a w ( t ) ] h + o (h) . (A.12) j = l i = i n-1"-> J 1 n + 1 Expanding the derived u t i l i t y f u n c t i o n J[W(t+h),V(t+h),t+h,S(t+h)] about the p o i n t [ W ( t ) , V ( t ) , t , S ( t ) ] , and taking expected values, c o n d i t i o n a l upon the event of no d e f a u l t s , gives n n J [ W ( t ) , V ( t ) , t , S ( t ) ] + J h + J { W ( t ) [ Z w . ( t ) ( r . - r ) + E w . (t) (a .-r)+r] -C(t)}h j = l 3 3 j = l n + 3 3 + ^ J w w f j = l i f 1 W D ( t ) Y D i W i ( t ) + ^ j . V ^ V j i ^ n + i ^ D: n n + A " W n + j ( t ) a j i W n + i < t ) ] W ( t ) 2 h + =• ^Ji 1 m m m n m n + Z Z W(t)w (t)a.n. .G.J h + 0(h) /* i = l j = l 3 *->n+3 i iw u u l ' ' (A.13) 9 0 See Equations (4.19) and (4.20) 190 where S(t+h) = S ( t ) , as no firms have defaulted; 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 upon the f a c t that d e f a u l t has not occurred to e i t h e r the x"^ or j*** 1 firm; n „ 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 do. and dZ.; i 3 J f c = |^J[W(t),V(t) , t , S ( t ) ] ; Jw = f w J [ w ( t ) , v ( t ) ' f c ' S ( t ) 1 • j = ^2_2 J t W ( t ) / V ( t ) , t , S ( t ) ] ; WW 9W JA = 3 J[W(t) ,V(t) , t , S ( t ) ] ; 3 3V. and J i j = 9 a v i ^ 7 J [ w ( t ) / v ( t ) , t , s ( t ) ] ; J 3 JJW = gy-^-^fwct) , v ( t ) . , t , s ( t ) ] . J j With p r o b a b i l i t y X_.(t)h + 0(h) the event t h a t the j f i r m goes bank-rupt and no other firms go bankrupt i n the i n t e r v a l (t,t+h] occurs. The event of bankruptcy not only a f f e c t s the j firm's bond p r i c e behaviour, but also i t s equity p r i c e behaviour. C o n d i t i o n a l upon t h i s event, the change i n wealth i s now of the f o r m ^ A. (t+h) - 0.(t+h) W(t+h) - W(t) = W(t){w. (t) [-3 u ? - 1] - w . (t)} 3 bj (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 i ^ j + rh} 10 • See Equation (4.21). 191 C ( t ) h { l + w. (t) [-2 A. (t+h) - 0.(t+h) b 4 (t) - 1] - w (t)} + 0(h). n+3 (A.14) The derived u t i l i t y f u n c t i o n , c o n d i t i o n a l on the event that the j th f i r m went bankrupt, i t w i l l be of the form J[W(t+h), V(t+h), t+h, S_.] where t h 5^ i s a state vector at time (t+h) denoting that the j f i r m no longer e x i s t s . As before, the derived u t i l i t y f u n c t i o n i s expanded and the condi-t i o n a l expected values taken. However, u n l i k e Equation (A.13) wealth W(t+h) i s not expanded about the p o i n t W(t), but about the p o i n t which defines ( t ) . The reason f o r t h i s i s to preserve the compactness of the change i n wealth. The event o f bankruptcy causes a d i s c o n t i n u i t y i n the wealth f u n c t i o n and thus i t no longer becomes p o s s i b l e to represent the changes i n wealth by a summation o f compact d i s t r i b u t i o n s . The property of compactness i s very important f o r i t enables' many of the terms i n the Taylor expansion of the derived u t i l i t y f u n c t i o n to be neglected when a l i m i t i n g process i s used. I t i s p o s s i b l e to preserve such a property. The derived u t i l i t y f u n c t i o n i s expanded i n a Taylor's s e r i e s and then i t s expected t h value i s taken c o n d i t i o n a l upon the event that the j f i r m went bankrupt. U t i l i z i n g t h i s c o n d i t i o n a l argument, the change i n wealth that r e s u l t s s o l e l y from the bankruptcy of the j*"* 1 f i r m i s known f o r a given investment i n i t s bonds and equity. Thus any other changes i n wealth r e s u l t i n g from the i n v e s t -ment i n the other f i n a n c i a l assets.can s t i l l be represented by the summation of compact d i s t r i b u t i o n s , but now centred around the wealth p o s i t i o n W^(t), i n -stead of W(t). There are, however, a number of other important r a m i f i c a t i o n s V7(t){l + W j (t) [-1 = ( t ) , A. (t+h) - 9.(t+h) b. (t) 3 (A.15) 192 that r e s u l t from the d i s c o n t i n u i t i e s that occur due to the event of bank-ruptcy, as w i l l become q u i c k l y apparent. The c o n d i t i o n a l expected value of the derived u t i l i t y f u n ction J[W(t+h), V(t+h), t+h, S..] i s thus J[W_.(t),V(t),t,S_J + terms of order a t most h. (A.16) Subs t i t u t i n g Equations (A.13) and (A.16) i n t o Equation (A.9) and taking the unconditional expected value, g i v e s . J [ W ( t ) , V ( t ) / t , S ( t ) ] = Max (U[C (t) , t ] h + 0(h) {c,w} n + E {A . (t)h + 0(h)}{j[W. ( t ) , V ( t ) , t , S . ] + terms of order a t j = l 3 3 3 most h} n m + {l-.E A . ( t ) h + 0(h)}{j[W(t),V(t) ,t,S] + J h + E J.F.h 3=1 3 t j - i 3 r n n + J w ( W ( t ) [ Ew_.(t) ( r j - r ) + t w n + j (t) (o^-r) + r] - C(t)}h + .2 w C t ) Y w ( t ) + 2 ? ( t ) Y p a.w n +.(t) 3=1 1=1 J 3=1 i = l J n n + E E w . (t)o..w , ( t ) ] W(t) h j-1 i = l n + 3 3 1 n + 1 m m m n + E E G.\)..G.J..h + E E W(t)w. (t)Y.n. .G.J h 2 j-1 i = l 3 3 1 1 3 1 i = l j = 1 3 3 1 3 1 * W m n + E E w(t)w . (t)a .n. _,_.G.Jh + 0(h)}). i = l j = l 3 3 x ' 3 1 1 1 9 3 Simplifying/ and taking the l i m i t as h tends to zero, gives the fundamental p a r t i a l d i f f e r e n t i a l equation f o r the derived u t i l i t y f u n c t i o n : m 0 = Max (U[C(t),t] + J + E F . J . {c,w} * 3=1 3 3 n n + J.,{w(t) [ E w . ( t ) ( r . - r ) + E w . (t) (a .-r) + r] - C (t)} W j - l 3 3 j - l n + 3 3 3=1 1=1 3=1 i = l J n n _ + E E w ( t ) a . .w . (t)]W(t) j - l i = l n + 3 3 1 n + 1 m n m n + r- E E G.V..G.J... + E E W (t) w, (t) y . n. .G . J... • 2 i = l j - l 1 1 3 3 1 3 i - l j - 1 3 Y 3 1 3 1 l W m n + E E W(t)w ,.(t)a.h. .G.J,__ i = 1 j = 1 n+3 3 i,n+3 i iW n + E X (t){j[W ,V(t),t,S ] - J r w ( t ) , V ( t ) , t , S ( t ) ] } ) , (A.17) j = l 3 3 subject t o the boundary conditions J[W(t) ,V(T) ,T,S (T) ] = BF[W(T),T]„ The 2n+l f i r s t order maximization conditions are derived from Equa-t i o n (A.17) by f i r s t d i f f e r e n t i a t i n g with respect to the r a t e of consumption: 0 = U c [ C ( t ) , t ] - J w (A. 18) where U c [ C ( t ) , t ] = ~ [ C ( t ) , t ] ; then by d i f f e r e n t i a t i n g with respect to the p r o p o r t i o n o f wealth to i n v e s t i n common stock, {w .(t)}, n+3 194 n n 0 = J w ( a . T r ) + J T 7 T , t E o . . w (t) + E Y.P . .a.w , . (t)]W(t) W 3 WW i = 1 31 n+i '] ] i I n+i + 2 o n G J - X J [w , v ( t ) , t , s . ] , j i,n+j i iw ] ] 3 n E i = l j = 1, 2,. . .,n; (A.19) and f i n a l l y , d i f f e r e n t i a t i n g with respect to the pr o p o r t i o n of wealth to i n v e s t i n bonds' {Wj(t)}: n n 0 = ( r . - r ) J + W(t) [ EYi .w. (t) + E y .p . .o.s (t) ] J D W . , ] in l n+i WW 1=1 1=1 m A (t) - 0 . (t) ^ V i j V i H " " j 1 1 " b . ( t ) P l ^ t W . / V C t ) ^ ^ . ] , j = 1, 2,. . .,n, (A.20) For e x p o s i t i o n a l s i m p l i c i t y d e f i n e A.(t) - 8 (t) L 3 ( t ) = 1 - 1 b . ( t ) ] ' ] = h 2,• • • / n • Substituting Equation (A.20) i n t o (A.19) so as to eliminate J W[W^,V(t), t,S J g i v e s : r .-r n " y .P. .a. 0 = [(o.-r) - - L - ] J + W(t)[ Ea..w (t) - 7 r X 1 , J t w (t) D L.(t) W -,31 n+i . , - L . ( t ) n+i D i = l 1=1 D 1 1 Y,..w.(t) + E 0 p Y w (t) - E ' j i " i ^ 1=1 3 3 1 1 1 i = l L . ( t ) l J ' WW Y.n, .G,J 1=1 J i = l (t) 195 At t h i s p o i n t i t i s perhaps worth making a small d i g r e s s i o n to derive two 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 Equation (A.18) the following r e l a t i o n s h i p s hold. 0 = u [C(t>,t]|£ - J r w ( t ) , V ( t ) , t , S ( t ) ] , <-t- dW WW o = u c c [ c ( t ) , t ] | ^ - - j ; [ w ( t ) f v ( t ) , t , s ( t ) ] f j 3 "2 — 1/ 2,• • •,n# Hence, using Equation (A,18), we have the i n e q u a l i t i e s : Jw u c > 0 (A.22) and Jww u c c % w 3C J . 3V. = _ _1J i o J 3C - ' ™ 3W < j ~ l,2,*«»,n« The demand functions f o r bonds can be derived from the set of Equations (A.19). In t h e i r stated form the equations are non-linear and thus i n general 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 s o l u t i o n . There are at l e a s t two a l t e r n a t i v e s . The f i r s t i s t o put more s t r u c t u r e i n t o the formulation by assuming a p a r t i c u l a r form f o r 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 and then attempt to solve the system of equations by a numerical i t e r a t i v e procedure, whilst 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 . I t w i l l be d i f f i c u l t to derive e x p l i c i t forms f o r the general e q u i l i b r i u m r a t e s of return, and even i f these could be obtained, they w i l l depend upon the s p e c i f i c assumed form f o r the u t i l i t y functions f o r i n d i v i d u a l s . The lack of g e n e r a l i t y , and the complexity of 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 to obtain an approximate s o l u t i o n by making two 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 or 196 (n-1) firms i n existence a t time t such that J [ W ( t ) , V ( t ) , t , S ( t ) ] = J[W(t),V(t),t,S..] , (A.23) 3 = 1/2/..., n, and a l l d e r i v a t i o n s are equal. I f the number of firms i n existence, n, i s 'large', then such an assumption seems i n t u i t i v e l y q u i t e reasonable. The second assumption involves the a b i l i t y to expand the d e r i v a t i o n s of the derived u t i l i t y function and to neglect the quadratic and higher power expansion terms, that i s A.-8. J w [ W ( t ) { l + W j (t) < t ? - 11 - w n + j (t)}, V ( t ) , t , S(t)] - J M [ W ( t ) , V ( t ) , t , S ( t ) ] - W(t){w,(t)[l - + w ( t ) } J w w [ W ( t ) , V ( t ) , t , S ( t ) ] W j b. (t) n+3 •i - i o 3 ( A ' 2 4 ) The e r r o r introduced by making t h i s assumption w i l l be examined i n the next s e c t i o n . Given th a t these assumption hold, then s u b s t i t u t i n g Equations (A.23) and (A.24) i n t o (A.20) gi v e s n n n 0 = J„ (r.-r) + J „,[ Ey..w.(t) + £ Y . P . . O \ W . (t)]W(t) + T, y.n. .G.J. W j WW i = 1 T j i i i = l 3 1 3 1 n + 1 i = l 3 1 3 1 l W - X. (t)L. ( t ) J + . ( t ) L . (t) [w. (t)L. (t) + w . ( t ) ] W ( t ) J r 7 t 7 (A.25) 3 3 W 3 3 3 3 n+3 WW j ~ If 2;#••^n» Hence, Equations (A.21) and (A.25) give a s e t of simultaneous equa-t i o n s , l i n e a r i n the demand functions f o r the f i n a n c i a l assets {w.}. 3 I 197 Constant Investment Opportunity Set Suppose that the investment opportunity set characterized by {a,a_,r,y,X_, f_,rF> i s constant and that Yj = 0 f o r a l l j . The f i r s t order conditions, using Equations (A.18), (A.19), (A.20), (A.21) and (A.23) become 0 = U c [ C ( t ) , t ] - J w , (A.26) 0 = (a - r - + W(t)[ ? a w ( t ) ] J , (A.27) J 3 1=1 J and 0 = ( r j " r ) J w - VJVV^' (A*28) "2 — 1/ 2 $ • » »§r z\» Using the approximation described by expression (A.24), then Equation (A.28) becomes 0 = ( r . - r - X .L.)J + X.L.[w.(t)L. + w .(t)]W(t)J . (A.29) 3 3 3 w 3 3 3 3 n+3 WW From Equation (A.27), the demand f u n c t i o n f o r equity can be derived: Jw n r • " r w (t)W(t) = (- E A., (a. - r - - J ), (A.30) 1 1 + 1 T TT T • T 1D 3 L . WW 3=1 3 1 = 1/ 2,««.,n, where {A..} are the elements of the inverse of the instantaneous c o n d i t i o n a l IT variance-covariance matrix. From Equation (A.29) the demand function f o r bonds can be written i n the form Jw r , _ r [w. (t)L. + w (t)]W(t) = (- —) (rV- " D (A.31) 3 3 n + 3 J r 7 I 7 X .L. WW 3 3 3 — 1/ 2,...,n. To derive the expressions f o r the equilibrium instantaneous c o n d i t i o n -a l expected rates of return, consider the demand functions f o r the k i n d i v i d u a l . 198 Equation (A.30) can be w r i t t e n i n the form r . - r W k ( t ) J k n o - r - = -[ — ^ ] E a..w ( t ) , (A.32) j j j * i = i 3 1 n + 1 3 = 1/ 2,.../n, where the superscript k i s used to i d e n t i f y the p a r t i c u l a r i n d i v i d u a l . There-f o r e , s u b s t i t u t i n g f o r {w .(t)} and summing across a l l i n d i v i d u a l s , gives n+i r . - r I J * n I (a. - r - -J—) Z (- = Z Z a..p. (t)N* . ( t ) , 3 L. . . Jk . .. , , 31 i n+i 3 k = 1 ww 1 = 1 k = 1 where I i s the t o t a l number of in v e s t o r s ; . (t) i s the optimal number of n+i shares of the i * " * 1 f i r m t h a t the k*"*1 i n d i v i d u a l i n v e s t s i n . I f the general e q u i l i b r i a c o n d i t i o n s are used, th a t i s , a l l markets c l e a r , then I - E 1N k^. (t) =»_,.., k=l n+i n+i *• t i l where N . ' i s the t o t a l number of shares outstanding f o r the i f i r m . Hence, n+i k / r j ~ * \ Z , W. n a . . p . ( t ) N ^ . (A.33) (a. - r <?—) Z (- -—) = Z u ; j i * i v n+i 3 L j k=l JWW i = l The above equation can be expressed i n a form comparable to the t r a d i t i o n a l c a p i t a l asset p r i c i n g model. Define M(t) to be the t o t a l market equilibrium value of a l l e q u i t i e s : n ' M(t) = E N .p. ( t ) ; (A.34) . , n+i 1 1=1 ti l and Y^(t) to be the percentage of the e q u i l i b r i u m value of the i firm's equity to the t o t a l market value: 199 V t } ' = J4f?lr (A-35) Define the eq u i l i b r i u m instantaneous c o n d i t i o n a l expected rate of return on the equity market to be n u = E a . Y. (t) (A.36) j = 1 3 3 M u l t i p l y i n g Equation (A.33) by Y_. ( t ) , summing over a l l j and s u b s t i t u t i n g Equa-t i o n s (A.34), (A.35) and (A.36) g i v e s n r . - r I J n n [y - r - E Y. (t) M )] E (- —-) = M (t) E E Y. ( t ) a . . Y. (t) (A.37) • i 3 L . „k . , . , 1 1 1 1 D = l D k=l 1=1 3=1 For e x p o s i t i o n a l s i m p l i c i t y , d e f i n e n r . - r X(t) = E Y. (t)(-2=--). (A.38) j = l 3 L j Substituting Equation (A.37) i n t o Equation (A.33) and using (A.38) gives r.-r a - r - - J — = 8. (u - r - x) / (A.39)-3 ^ 3 where E a..Y.(t) 3 n n E E Y.: (t)a. .Y. (t) i = l j = l 1 1 3 3 j = 1/ 2,...,n. To derive the eq u i l i b r i u m instantaneous expected rates of returns th for bonds, Equation (A.31) f o r the k i n d i v i d u a l can be w r i t t e n i n the form k k Jw r - ~ r [N?(t)b (t)L. + i r . ( t ) P . (t)] = ( - - ~ - ) ( r - h - -i l l n+i i _k A. L. Jww 1 1 200 where N k(t) i s the optimal number of bonds of the i f c l 1 f i r m that the k t h i n d i -v i d u a l invests i n . Summing over a l l i n d i v i d u a l s and using the general e q u i l i -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 k E N. (t) = N. , k=l 1 1 x — I f 2 f « » « / i i f where i s the t o t a l number of. bonds outstanding f o r the i fi r m , gives the equation r . - r I J k N.b.(t)L. + N .p.(t) = ( T ^ — ' - 1) ^ (- ) (A.40) i l l n+i I A . L . , , Tk 1 1 k = 1 Jww Define M (t) to be the t o t a l market e q u i l i b r i u m value of a l l bonds: a n M R(t) = Z K b ( t ) ; (A.41) 0 , 1 3 . 1=1 t h and X^(t) to be the percentage of the e q u i l i b r i u m value o f the i firm's bonds to the t o t a l market value of a l l bonds, tha t i s , N.b.(t) M B ( t ) (A.\42) 4* ~ I f 2f« « » f X l # Define the e q u i l i b r i u m instantaneous c o n d i t i o n a l expected r a t e of r e t u r n on the bond market t o be n ir(t) = E X. ( t ) r . (A.43) i = l 1 1 M u l t i p l y i n g Equation (A.40) by X.(t) and summing over a l l i gives k 1 JW " (rc-r-Y) E (- -r-) = E . ( t ) A . L . [N.b. (t)L. + N . p . ( t ) ] , (A.44) k=l J k i = l 1 " - i l l i n+i i WW 0 201 where Y = E X . (t)L.A.. S u b s t i t u t i n g Equation (A.44) in t o Equation (A.40) i = l 1 gives r . - r - A.L. = {A .L . [N.b. (t)L .+N . D. (t)] }{ * " r ~ y } 3 3 3 3 3 3 3 3 n+jpj n 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.' (A.45 Equation (A.45) describesthe e q u i l i b r i u m instantaneous expected r a t e of r e -t h turn f o r the bonds of the j f i r m . An 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 of Equation (A.45) i s p o s s i b l e . Consider Equations (A.38)* (A.39) and (A.40), thus by using these equations i t 1 ^ i s p o s s i b l e t o eliminate the term E (- ) from Equation (A.40); t h a t i s * = 1 Jww N.b.(t)L. + N .p.(t) r - r - X L - X L (y- r -*) \ " 3 3 ] (A.46 3 = l f 2,...,n. The usefulness 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 demonstrated when i t i s used r . - r to eliminate (-*!—) from Equation (A.39), which describes the 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 expected ra t e s of r e t u r n f o r equity. S u b s t i t u t i n g Equation (A.46) i n t o Equation (A.39) giv e s N.b.(t)L. + N p.(t) _ r _ X j = (y - r - x) {6 + A j [• 3 3 n ^ ! L i J J > M(t) E E Y.a. .Y. i-1 j-1 1 1 3 3 S u b s t i t u t i n g the expression f o r 8.., gives n \ (u - r -Y) N.b. (t)L. + N .p. (t) a j - r - X j = — — - — {E a .. Y. + A.f.J-3 3 n + 3 3 i •> J n n ] i i 3 1* E E Y io j [.Y E N ( t ) i = l j = l i = 1 n + i * i v 202 If there are a large number of firms, the l a s t term on the r i g h t hand side of the above expression can be neglected, as i t i s of order 2/n, where n i s the number of firms. Hence, the expression f o r the instantaneous c o n d i t i o n a l expected rate of r e t u r n can be w r i t t e n - r - A.. = 8 (p - r - x) , (A.47) 3 ~~ 1, 2,...,n«. Before proceeding to i n v e s t i g a t e the v a l i d i t y of the approximation of being able to expand the f i r s t d e r i v a t i v e s of the derived u t i l i t y f u n c t i o n and neglect quadratic and higher order terms, i t i s worth rec o n s i d e r i n g Equations (A.39), (A.36) and (A.47) i n order t o derive two i d e n t i t i e s . From Equation (A.36), the e q u i l i b r i u m 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 on the equity market i s defined t o be n y = E a.Y.(t) j= l 3 3 S u b s t i t u t i n g Equation (A.39), g i v e s n n --»~ r n y = r E Y (t) + E Y. (t) (-} ) + (\i - r - x) Z Y (t) 8 , 3=1 3 j = l 3 L j j - l 3 3 which implies E Y (t) = 1, 3=1 3 and n E Y.(t)8. = 1 (A.48) j = l 3 3 as one would expect, given the d e f i n i t i o n a l forms of the various q u a n t i t i e s . If Equation (A.47) i s sub s t i t u t e d i n t o (A.36), then 203 n n u = r E Y. (t) + E Y. (t)X . + (u - r - x)E Y. (t) 8., j - l 3 j = l 3 3 3 3 which implies, using Equation (A.48), that n X = E Y.(t)X (A.49) j - l 3 3 Note, however, tha t Equation (A.49) i s not, u n l i k e expression (A.48), 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 approximation made i n d e r i v i n g (A.47), a r e s u l t which depends upon the a b i l i t y to neglect terms of order (1/n). Equation (A.49) i s obtained from a weighted summation o f terms des-c r i b e d by Equation' (A.47) and thus neglects the summation of the terms tha t are considered to be of 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 not, however, c l e a r t h a t the sum of these terms can be neglected. Consider the l e f t hand side o f Equation (A.49). From Equations (A.38) and (A.36) n r . - r X = E Y. (t) ) j = l 3 J J n E Y. (t)X . [N.b. ( t ) L . + N . p . ( t ) ] n . , 3 3 3 3 3 n+3 3 = E Y. (t)X. + ( y - r - x ) 3 - ^ (A.50) 3 3 n n 3 M(t) E E Y. (t) a . . Y. (t) j = l i - 1 3 3 1 1 Hence, Equation (A.49) neglects the l a s t term o f Equation (A.50), t h a t i s n Y.(t)X. N.b.(t)L. + N .p.(t) (y-r-x) E {[ 3 - 2 ] l-^-J 3 n + P 3 ]> (A.51) ._, n n M(t; 3 E E Y.(t)0..Y.(t) j - l i - 1 3 3 1 1 I t i s not c l e a r t h a t t h i s term can be neglected, which i s simply a r e f l e c t i o n of the f a c t that the summation of terms, which are i n d i v i d u a l l y n e g l i g i b l e , need not i t s e l f be n e g l i g i b l e . This does not imply t h a t Equation (A.47) i s 204 wrong, only t h a t i t i s derived by neglecting a term of order (1/n). Equations (A.39) and (A.45) were derived under the assumption th a t i t was p o s s i b l e to expand i n a Taylor's s e r i e s the f i r s t d e r i v a t i v e of the derived u t i l i t y f u n c t i o n and t o neglect quadratic and higher order terms, wh i l s t t h i s assumption i s very convenient because of the r e s u l t i n g l i n e a r -i t y , i t would be of some comfort to determine the magnitude of the approxi-mation. One approach to t h i s equation i s to put more s t r u c t u r e i n t o the formulation by assuming a p a r t i c u l a r form f o r the i n d i v i d u a l ' s u t i l i t y func-t i o n and to attempt to obtain an exact s o l u t i o n and then compare i t to the s o l u t i o n obtained by assuming the v a l i d i t y of the approximation. Assume a constant r e l a t i v e r i s k aversion u t i l i t y f u n c t i o n defined by U[C(t),t] «='£-e~ p t, (A. 52) where p and v are p o s i t i v e constants and v 1. The system o f f i r s t order conditions can be w r i t t e n 0 = C V " 1 e " P t - J w , (A.53) r ,-r n 0 = ( a - r - -J-HJ + W(t)[I a w n + . ( t ) ] J w w , (A.54) j i = l J 0 = (r . - r ) J r - X .L.J T T(W.,t), (A.55) 3 W 3 3 W 3 j = l , 2,...,n, and the equation of o p t i m a l i t y r v n n 0 = Max { — e~ + J_. + J„{ W(t) [ £ w. (t) (r .-r) + E w , . (ot .-r) + r] i \ v t W , 3 3 . n+3 3 ic,w} 3=1 J 3=1 n n n - C} + i j J ( t ) ' E I w ( t ) a . v (t) E X [j(w ,t) - J(W,t]} i = l j - l 3 3 j = l (A.56) 205 subject to the boundary co n d i t i o n J[W(T),T] = BF[W(T) /T]. To solve t h i s system of equations take as a t r i a l solution^" 3" J[W(t),t] = B ( t ) e ~ P - — (A.57) Su b s t i t u t i n g (A.57) in t o (A.54) and (A.55) gives ^ r . - r n 0 = (a - r - - f — ) + (v-1) Z o.,w . ( t ) , (A.58) 3 L. . , 31 n+i 3 1=1 and 0 = (r.-r) - X.L.(1 - w.L. - w . ) V _ 1 (A.59) 3 3 3 3 3 n+3 3 ~ If 2,..«,n. Equation (A.59) can be w r i t t e n i n the form r j _ r 1 w.L. + w, = 1 - (-rV") <A-60> 3 3 n+3 j j t h a t f a c i l i t a t e s d i r e c t comparison with Equation (A.31), which was derived assuming the v a l i d i t y o f the approximation. Equation (A.31) can be w r i t t e n i n the form J r . - r w.L. + w = ( r ~ - ) (1 - r^r-)' (A.61) l n n+3 WJ ' A.L, J J WW 3 3 and s u b s t i t u t i n g (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 question i s how good an approximation i s (A.31') t o Equation (A.60)? The r i g h t hand side of (A.60) can be w r i t t e n as 1 - (1 + - 1 — r - = i- 3-) v-1 A j L j 206 = ( ^ ) ( 1 " -j^ JT") - (^IJ7 - 1 ) ( 1 " xV")2 + h i 9 h e r O R D E R T E R M S - ( A . 6 2 ) j j V j j Hence, the degree of approximation depends upon the a b i l i t y to neglect the terms 1 1 R - _ R - — ( — - ) ( — - - 1 ) ( 1 - -r-^ lF—) + higher order terms, 3 3 but t h i s w i l l depend upon the value of v. For small v the approximation w i l l be very good, but w i l l drop o f f as v increases i n magnitude. For one c l a s s of u t i l i t y f u nctions, the quadratic, the approx—. mations w i l l be exact, as can be shown by s u b s t i t u t i n g i n Equations ( A . 5 2 ) , ( A . 5 3 ) , ( A . 5 4 ) and ( A . 5 5 ) . T h i s work o f f e r s some encouragement to use the approximation. I t has been shown that f o r one type of u t i l i t y f u n c t i o n the approximation i s exact, w h i l s t f o r the c l a s s o f constant r e l a t i v e r i s k aversion u t i l i t i e s func-t i o n s the approximation can be very good, depending upon the value of the para-meter o f the f u n c t i o n . Stochastic Changes i n the Rate of the P r o b a b i l i t y of Bankruptcy Pr e v i o u s l y , i t had been assumed that there were no s t o c h a s t i c changes i n the investment opportunity s e t . The equation d e s c r i b i n g the bond p r i c e dynamics had been s i m p l i f i e d so t h a t there was no d i r e c t i n t e r a c t i o n be-tween bonds and common stock apart from that of bankruptcy. Both assumptions are now relaxed so that e f f e c t s of s t o c h a s t i c changes i n the rate of the prob-a b i l i t y of bankruptcy upon the s t r u c t u r e of returns can be analyzed. The derived u t i l i t y f u n c t i o n w i l l depend upon the i n d i v i d u a l ' s current wealth, W(t), a vector describing the current values of the rates of 207 the p r o b a b i l i t y of bankruptcy f o r the d i f f e r e n t firms, A_(t) , a state vector d e s c r i b i n g which firms are c u r r e n t l y i n existence, S ( t ) , and time t. Thus the derived u t i l i t y f u n ction may be w r i t t e n i n the form J[W(t), £_(t), t , S ( t ) J . To analyze the e f f e c t s of s t o c h a s t i c changes i n the rate of p r o b a b i l i t y of bank-ruptcy i t i s necessary to assume a form that describes the s t o c h a s t i c nature of th the mechanism generating the changes; f o r the j f i r m i t i s assumed t h a t dA^(t) = F..(A_.,t)dt + G_. (A_.,t)dQ.., (A.63) 3 = 3., 2,...,n, which should be compared to the general system of equations described by Equation (A.5). The equation o f o p t i m a l i t y can be derived from the general case described by Equation (A.17); that i s n 0 = Max (U[C(t),t] + J + E F . J . {C,w} fc j = l 3 3 n n + J„{w(t) [ Z w. (t) (r.-r) + Z w (t) (a.-r) + r] - C(t)} W • , 3 3 • n n+3 3 3=1 J 3=1 ' n n n n + ^rrr^t E £ W. ( t ) Y . .W. (t) + 2 Z Z W . (t) Y . P . . 0" . W . (t) 2 WW -. . . . 3 ] i i . n . - 3 3 J i i n+i 3=1 1=1 3=1 1=1 J n n + Z Z w "(t)a..w (t)]W(t) j- 1 i - 1 n + 3 3 1 n + 1 1 n n n n + — ' E E G.v. .G.J. . + E E W(t)w. (t)Y .n. .G.J. 2 i - 1 j-1 1 1 3 3 1 3 i - 1 j- 1 3 3 1 3 1 l W n n + E E W(t)w .(t)a.n. .G.J. i = 1 j = 1 n+3 3 i,n+3 i lW n + E A . (t) {J[W. ,A_(t) , t , S.] - J[W(t), \{t) , t, S(t)]}) (A.64) j=l 3 3 3 208 subject to the 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 conditions are, a f t e r some manipulation, 0 = U c [ C ( t ) , t ] - J w , (A.65) 0 - ( a j - R - E^ >JW + " ( t ) I . V j i V i ( t ) - A^^r^W^ 3 1=1 1=1 3 Y. .w. (t) A E o.p ..y.w. (t) - E ' j i i ' ' „ 1=1 i = l L. WW and n n Y.n. -G.J".M + E o.n. .,.G.J.tJ - E 3 1 3 1 1 W , • (A.66) i-1 3 i.n+3 i iW . = 1 j 0 = ( r - r ) J K + W ( t ) [ ? Y p o w (t) + z \ w (t) ] 1=1 J 1=1 J + . E Yjn^G^J^ - X . l t j L . ^ t W . ^ f t J / t ^ . } , (A.67) Equation (A.65) i s the intertemporal envelope c o n d i t i o n . Equation (A.66) i s derived by d i f f e r e n t i a t i n g the equation of o p t i m a l i t y by the propor-t i o n of wealth to invest i n equity and Equation (A.67) i s the equivalent equa-t i o n f o r bonds. As i n the previous s e c t i o n , t h i s system of equations i s non-l i n e a r . I t i s assumed t h a t i t i s p o s s i b l e to r e p l a c e the system of non-linear equations by an approximate l i n e a r system; th a t i s , Equation (A.67) can be 12 See Equations (A.18), (A.20) and (A.21). 209 approximated by n n o = (r. - r - X ( t ) L . ) J + w(t) [ Z y. .w. (t) + Z y p. .a.w . (t) ] J 3 D D W i = 1 ] i i i = 1 ] i ] 1 n+i WW + V t ) L . [ „ j ( t ) L . ^ V j ( t ) l W ( t , J W W n + Z _ , (A.68) i = l j i j i iW Therefore, Equations (A66) and (A.68) describe two c o r r e l a t e d l i n e a r systems o f equations fromwhich i t i s p o s s i b l e to determine the demand functions f o r bonds and equity and then d e r i v e the eq u i l i b r i u m instantaneous c o n d i t i o n a l expected rates of r e t u r n . Whilst such a d e r i v a t i o n i s conceptually simple, i t i s , unfortunately, mathematically very tedious. The complexity of the s o l u t i o n a r i s e s from a l l the covariance terms. I t i s , however, these terms which r e f l e c t the e f f e c t s of bankruptcy upon the s t r u c t u r e of r e t u r n s . Define the f o l l o w i n g (nxl) vectors {a}, = a. - r _ ! i r - - j . j L. ' 3 r \ x » (A.69) {c}.. = r . - r - X . L . ; - 3 3 3 3 { v ^ L = W(t)w_.(t); {w2}. = W ( t ) w n + j ( t ) ; and {U}. = J.„; - 3 3W j = 1, 2, n, and the following (nxn) matrices 210 {£.}. . = Y.P. .cr.; —3 i ] i i ] l -4 13 ' l l ] l ( O . . = a.n. . .G.; -5 13 ~ 3 i,n+3 I {L}..= L.6..; - 13 3 13 and {A},. = X.6..1 - I D i I D i / j = 1/ 2,...,n, (A.70) where I D i ; i=jr 0; There, i n matrix notation the set o f equation described by (A.66) can be written i n the form 0 - J w • + (=5 " l i " 1 ^ ) " (A.71) and Equation(A.68) i n the form 0 = J__c + E,U + ^ w w ^ l + £ 3 ^ 2 + 'i i L 2 l + L ^ (A.72) This set of equations may be w r i t t e n i n a more compact form: H l 2 - + P-3^ 2 = £l^l + P-12^ 2^  and (A.73) -21-1 ^2-2' (A.74) 211 where P_i = I 3 - L P.12 = I 2 - L\. P_21 - I i + L L hJ P_2 = A. L + E^; 2-3 " £3 " ^ _ 1 l 4 ? and D. = E ; •-4 —4 H l = T i - >., - WW {H 2> = - fw_ I 0, J Jww • 3 — If 2f«.#,n» Thus the demand functions f o r equity are and f o r bonds £2 = Hi=2<2. " + ^2(2.3 - S j & x W ( A - 7 5 ) ^ = Hi2.!(£ - 2.2 .i2l> + =^24 - 222.12E3^2 ( A - 7 6 ) where and 5j. - (2a! " 2 2 2 . & 1 ) C 1 » =2 - ( D 1 2 - S^k, ) " 1 . To d e r i v e the expression f o r the e q u i l i b r i u m instantaneous expected rate of return f o r e q u i t i e s , consider Equation (A.75) which can be w r i t t e n i n the form 212 th where the su p e r s c r i p t k denotes the k i n d i v i d u a l . Summing over a l l i n d i v i -duals, the aggregate demand functions f o r equity can bederived and the i n -stantaneous market e q u i l i b r i u m c o n d i t i o n rused. Let ASE be a (nxl) vector d e s c r i b i n g the aggregate supply f o r equity; that i s , {ASE}.. = N^^p^ (t) , j = 1, 2,...,n. Hence, ASE = A J E 2 ( a - D ^ c ) + - D ^ J D ^ , which implies t h a t » " j ^ S S E + D l 2 j j c - (D 3 - D ^ J ^ ) i - A 2, (A.78) where 1 1, 1 H* - " W A, = E " = E -k=l 1 k 1 '=1 4 / WW and • K WW For bonds a s i m i l a r r e l a t i o n can be determined. From Equation (A.77) th the demand functions f o r the k i n d i v i d u a l can be w r i t t e n i n the form - (c - D ^ a ) + E x - ^ D ^ g , (A.79) and thus C = — E " 1 ASB + D D ^ a - (D - D D ~ \ D ) | - A . (A.80) — A^ —1 —2—12— —4 —4—12—3 A^ —2 where {ASB}. = N b ( t ) , 3 3 j = 1/ 2,...,n. 213 Equations (A.78) and (A.80) describe the e q u i l i b r i u m instantaneous c o n d i t i o n a l expected r a t e s of r e t u r n f o r equity and bonds, r e s p e c t i v e l y . The equations are not, however, independent of the preference s t r u c t u r e of i n d i v i d u a l s due to the presence of the terms A and {A»}., j = 1, 2,...,n. 1 ^ 3 Whilst i t i s p o s s i b l e to eliminate these terms, the r e s u l t i n g complexity and general lack of i n s i g h t t h a t r e s u l t s does not warrant the e f f e c t . Some i n -s i g n t can be gained by assuming that, except f o r one f i r m , there are no sto c h a s t i c changes i n the rate of the p r o b a b i l i t y of bankruptcy; t h i s i s equivalent to assuming that the s t o c h a s t i c changes i n the ra t e of the prob-a b i l i t y of bankruptcy f o r t h i s one f i r m act as an instrumental v a r i a b l e . For til convenience, c a l l t h i s f i r m the n f i r m . For t h i s case, i s now a s c a l a r quantity and Z^ and Z^ w i l l be (nxl) v e c t o r s , which implies that p_3 and w i l l be (nxl) v e c t o r s . Let M(t) represent the t o t a l market value o f a l l equity M ( t ) " ^ V j P j ^ ' t h and Yj(t) represent the proportion of the t o t a l value of the j firm's equity t o the t o t a l market value of a l l equity; t h a t i s Y_. (t) = N^^p^ (t)/M (t). Define a (nxl) vector Y such tha t ^X j^ = v j ( t )* j = 1» 2,...,n. Thus, s u b s t i -t u t i n g into Equation (A.* 78) gives A A A ^ 3 + ^ ( £ + ^  V ' <A-81) which may be w r i t t e n in the s c a l a r form r - f r M n A 2 Y i X n 1 a 3 " r " " A 7 i I l S i Y i ( t ) * A ^ j X " LT- + . f ^ U n + Z e (r. - r - X L ), i=l 3 1 214 where 5 3 i " {°12 -^l°2l! e . . • { ^ } 3 i ' a j x = cr ,n . G , 3 n,n+3 n = Y.n . G . 3 n,3 n j 1/ 2/«•«^ n» M u l t i p l y i n g the above equation by Y^(t) and summing over j gives V ~ r " * " k ^ 2 V ^ j i 1 ! 1 * ' " AT^iX " YMA + . V j ^ V ^ 1 1 ]=1 1=1 1 3=1 1=1 n n + E E Y . ( t ) e . . ( r . - r - A.L.), (A.82) j-1 i-1 3 3 1 1 where n a = E Y. (t)o\. , mA j = 1 3 3* and h MX j-1 3 L j The preference s t r u c t u r e can be removed from Equation (A.82). The t h equity of the n f i r m must s a t i s f y Equation (A.81), t h a t i s r - r A Y , n n r r L " A^ V i ( t l A^ n X L . . n i ' i A n 1 i = l l n 1 = 1 n i y n i ( r . - r - A.L.)^ 215 which can be wri t t e n i n the form r - r , A„ n M 2 a - r - e = — a , - — 6 , , (A.83) n L n A, nM A . nX n 1 1 where n and £ = E e . ( r , - r - A .L.) n . n i 1 i i ' i = l 5 = c , - - ^ - + E G . Y., nX nX L . , n i I X n i = l From Equation (A.83) we have the r e l a t i o n s h i p M ^2 W " R " X " £ M * A7°M " A 7 6 M A ' ( A ' 8 4 ) where n n e = E E Y.(t)G..(r. - r - A.L.); j - l i - 1 3 3 1 1 i i n n 6 M A = a M A - Y M A + = W^lfiX' -j=l i = l J J and n n al = E E Y ( t ) s . . Y . ( t ) . M j - l i - 1 3 3 1 1 ,M 4 , A2, From Equations (A.83) and (A.84) i t i s p o s s i b l e t o solve f o r (~) and (-p-) and the A l A l r e s u l t s substituted i n t o Equation (A.81) t o determine the instantaneous condi-t i o n a l expected rate of r e t u r n V r ,PkM5MA- 6kA°M t , V r °k " r " — " Ek = ( Q > ( a n " r " — " en> 'a 6. , - 6 , o, , nM kA nX kM. , - . ., o c . ( ) (u - r - x " e M ) , (A.85) k = 1, 2,...,n-l. f 216 where Q = o 5 , - 6 , o 2 . * nM mX nX M A s i m i l a r r e s u l t holds f o r the bond equation (A.80) which may be wri t t e n i n the s c a l a r form, using Equation (A.41), M n A_ n Y n . r . - r + E e .. (a. - r i — ) , i - 1 1 3 1 1 L i (A.86) Let and n r . - r _ e. = E . c .. (a. - r - — — ) , 1 ] i = i 1 3 1 1 L i n Y i A 1 £ J A = Y j A - * l c i i l a h ~ IT" 0' 1=1 l j = 1, 2,...,n, and thus Equation (A.86) may be w r i t t e n r . - r - X . L . - ~ e . E i e j i X i ( t ) " A71 33X' (A.87) D 3 3 i i = l 1 3 = 1, 2,»««,n« M u l t i p l y i n g Equation (A.87) by X.. (t) and summing over j gives, using Equations (A.42) and (A.44) M A 1 1 " r " Y - 1EM = A f Y M " A75MA' ( A ' 8 8 ) 217 where 2 n n Y M = £ E X . (t) E . X . ( t ) , j - l i - 1 3 1 3 1 1 and n 6M. = S X. (t)6 . MA j = 1 3 DA The instantaneous c o n d i t i o n a l expected rate of r e t u r n f o r the n*"*1 firm's bonds must also s a t i s f y Equation (A.87); that i s , M A r - r - X L - e = -2. Y „ - ^ .6 . , (A.89) n n n 1 n A, nM A. 1 nX' Jt 1 where n Y n M - £ ,S .x ( t ) . nM . , x n i i i = l M B A 2 Equations (A.88) and (A.89) can be used to eliminate (—) and (—) i n Equation A l A l (A.87) to give 6 Y — 6 Y^~ r A n'jM 1 jX'M, . , T . r . - r - X.L. - , e . = ( * — ) ( r - r - X L - .e 3 3 J I 3 Q 1 n n n I n ' ,! 5jX YnM " l 5 n^jM, , . . o o v (—J —>—) (TT - r - y - e ), (A.90) Q x " I H ' j - If 2,...,n-l, where Q l = YnM 15MX " l 6nX YM-APPENDIX B NAMES OF BANKRUPT FIRMS NAME DATE OF BANKRUPTCY At l a s Sewing Center 1962 Avien Incorporated 1964 Barcalo Manufacturing Company 1965 Barchris Construction Corporation 1962 Betteringer Corporation 1961 Bishop O i l Company 1961 Bowl-Mor Company 1966 Buckner Industries Incorporated 1967 Davega Stores Corporation 1962 Dejay Stores Incorporated 1962 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 Corporation 1969 Fashion Tree 1968 G i l b e r t (A.C.) Company 1967 Goebel Brewing 1964 Grayson-Robinson Stores Incorporated 1962 Great Western Producers Incorporated 1965 Guidance Technology Incorporated 1962 International O i l and Gas Corporation 1965 Keystone A l l o y s Company 1966 Marrud Incorporated 1966 218 219 NAME McCandless Corporation Muskegon Motor S p e c i a l t i e s Company-National Video Corporation Okalta O i l s Polycast Corporation P r e c i s i o n Radiation Instruments Incorporated Premier Albums Incorporated Puerto Rico Brewing Company Incorporated Trans-United Industries Incorporated United States Chemical M i l l i n g Corporation Vinco Corporation Webcor Incorporated Yuba Consolidated Industries DATE OF BANKRUPTCY 1968 1961 1969 1961 1966 1963 1968 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 Ratios, Discriminant A n a l y s i s and the P r e d i c t i o n of Corporate Bankruptcy," Journal of Finance, V o l . 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