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Parameter estimation of stochastic interest rate models and applications to bond pricing Ananthanarayanan, A. L. 1980

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PARAMETER ESTIMATION OF STOCHASTIC INTEREST RATE MODELS AND APPLICATIONS..™) BOND PRICING by A. L. ANANTHANARAYANAN B. Tech. (Hons), I.I.T., Kharagpur, I n d i a , 1967  ^ T H E S I S SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Commerce & Business A d m i n i s t r a t i o n We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May 1978 A. L. Ananthanarayanan  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  by h i s of  this  written  at  the U n i v e r s i t y  make  that  in p a r t i a l  it  freely  permission  purposes  may  fulfilment of of  British  available  for  for extensive  be g r a n t e d  by  ;  thesis  for  financial  gain  permission.  of  University  ] of  •  British  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Columbia  shall  the  not  requirements  Columbia,  I  agree  r e f e r e n c e and copying  t h e Head o f  r e p r e s e n j t W t V v e ' s v • I t; i s~ u n d e r s t o o d " t h a t  Department The  thesis  of  be a l l o w e d  that  study.  this  thesis  my D e p a r t m e n t  c o p y i ng- o r  for  or.  publication  without  my  11  ABSTRACT  A p a r t i a l equilibrium on  the idea o f contingent  by  Black  &  Scholes.,  v a l u a t i o n model f o r a s e c u r i t y , based claims  The  Herton, who showed how the liability by  instruments.  treating  as  first  model was c o n s i d e r a b l y approach  Valuation  them  instantaneously  a n a l y s i s , was  could  be  developed extended by  used  to  value  models f o r d e f a u l t - f r e e bonds,  contingent  upon  the  value  r i s k f r e e i n t e r e s t r a t e , have been  of  developed  the by  6 Boss, Brennan 6 Schwartz , Vasicek and R i c h a r d s .  Cox,Ingersoll  There has, however, not been much a t t e n t i o n d i r e c t e d towards the empirical bonds.  testing  of  This research  attention i s confined Central  to  these  i s an  valuation attempt  in  arriving  at  any  interest  rate  process,  made  since  the  upon i t . These bond models,  exogenous t o them. subject  to  since  some  valuation  the  The choice  bond  process  model  interest  value  riskless  are  partial  i s assumed as  o f the i n t e r e s t r a t e  process  is  r e s t r i c t i o n s on i t s behaviour which a r e rates.  The  interest  adopted i n t h i s study i s a g e n e r a l i z a t i o n o f that  used by Vasicek and C o x , I n g e r s o l l  S Boss., The p r o p e r t i e s o f the  chosen mathematical model a r e i n v e s t i g a t e d t o a s c e r t a i n it  of bond  i s t r e a t e d as  models rate  Our  bonds.  instantaneously  based on expected p r o p e r t i e s o f i n t e r e s t rate  direction.  equilibrium  i s t h e assumption about the  equilibrium  that  to r e t r a c t a b l e and e x t e n d i b l e  valuation  contingent  models o f d e f a u l t - f r e e  whether  conforms t o those expected o f an i n t e r e s t r a t e process based  on economic reasoning. We go on t o develop a l t e r n a t e e s t i m a t i o n  methods  f o r the  111  parameters  of  the  interest  r e a l i z a t i o n of the p r o c e s s . based  on  rate One  process,  "exact" method and  approximations are o u t l i n e d .  "exact" method i s processes  not  included  available in  the  using  to  of  two  on a others  I t i s observed that the the  complete  continuous  family  time  s p e c i f i c a t i o n assumed t o model i n t e r e s t r a t e s . properties  data  of  stochastic  The  asymptotic  the e s t i m a t o r s from the " e x a c t " method are known  from the e x i s t i n g l i t e r a t u r e .  However, s i n c e we would  have  to  adopt  one o f the approximate methods, we need t o know something  about  the  properties  approaches.,  This  of  could  the not  Monte C a r l o study i s conducted.  estimators  based  on  these  be d e r i v e d a n a l y t i c a l l y and The r e s u l t s  seem  to  so a  indicate  t h a t the p r o p e r t i e s o f t h e e s t i m a t o r s from the three methods are not  very d i f f e r e n t . The y i e l d t o maturity on 91-day Canadian F e d e r a l Government  Treasury b i l l s , the  on the date of i s s u e , i s chosen as the proxy f o r  instantaneously r i s k f r e e i n t e r e s t rate.  The impact of using  such a proxy i s b r i e f l y i n v e s t i g a t e d and found to be  negligible.  The bond sample chosen i s the complete i s s u e s o f r e t r a c t a b l e and extendible 20  bonds  made by the Government of Canada.  i s s u e s between January 1959  prices  on  and  October  1975,  There were and  weekly  a l l these bonds are a v a i l a b l e i n the Bank o f Canada  Review . To a r r i v e  at  the  final  bond  valuation  equation,  assumptions are made about the term s t r u c t u r e of i n t e r e s t This  study  first  assumes  a  form  of  the  some rates.  pure e x p e c t a t i o n s  h y p o t h e s i s and i t i s shown t h a t the performance of the model p r e d i c t i n g market  p r i c e movements, i s c o n s i d e r a b l y improved  in when  iv we  assume  a  s p e c i f i c form o f t e r m / l i q u i d i t y p r e f e r e n c e on the  part o f i n v e s t o r s .  I n c o r p o r a t i n g t a x e s i n t o the  i n s i m i l a r improvements. e f f i c i e n t to information  model  results  The h y p o t h e s i s t h a t the bond market i s contained  i n these models i s t e s t e d and  not r e j e c t e d . , i  Finally,  an  ad hoc r e g r e s s i o n based model i s developed t o  serve as a bench mark f o r partial  equilibrium  evaluating  models.  perform a t l e a s t as w e l l as  the  the  performance  of  the  I t i s observed that these models ad  hoc  model,  and  could  be  improved by r e l a x i n g some of the r e s t r i c t i v e assumptions made.  Research S u p e r v i s o r Dr. Eduardo S. S c h w a r t z  V  TABLE OF CONTENTS CHAPTER  PAGE  1. INTRODUCTION .  . , • . . V . . » > . • • « v .  . . .....  . , • • • -r. • • ? •  Preamble Contingent Claims V a l u a t i o n of Bonds: A B r i e f Review Canadian Retractables/ Extendibles i n Perspective O u t l i n e of the T h e s i s .. ...  1  2. THE PRICING THEORY OF DEFAULT FREE BONDS i . . . . . . . . .  10  r  Determinants of Bond Value The B a s i c Bond V a l u a t i o n Equation ............ Boundary Conditions f o r Retractable/ E x t e n d i b l e Bonds I n c o r p o r a t i n g Taxes i n t o the Model 3. THE INTEREST RATE PROCESS  4. ESTIMATING THE INTEREST RATE PROCESS PARAMETERS . ... ... ........ ....... ... ... .... ......,, . Brief Review of Published Research i n Related Areas Maximum Likelihood (M.L.) Method of Estimation The Simple L i n e a r i z a t i o n Approximation ....... The T r a n s i t i o n P r o b a b i l i t y Density Method .... The Steady S t a t e or S t a t i o n a r y Density Method The P h i l l i p s Approximation Method ............ OF  THE  DIFFERENT  4 7 • •  10 13 16 20 22  P r o p e r t i e s of I n t e r e s t Rate Processes ........ The I n t e r e s t Rate Process .................... Interest Rate Process Behaviour at S i n g u l a r Boundaries  5. COMPARISON METHODS  2  22 25 26 28 28 31 34 35 36 41  ESTIMATING  The Method of Comparison .................. ... Generating an "exact" Sequence f o r the Square Root Process .......................... , R e s u l t s of Monte C a r l o Simulations f o r the o( =i/x(known) Case R e s u l t s of Monte C a r l o Simulations f o r the <* On known Case., ............................. The R e l a t i o n Between the I n t e r e s t Rate Process Parameters  H4 44 45 48 71 79  1  vi  6. THE INTEREST RATE AND BOND PRICE DATA . ............  88  The Short Term R i s k l e s s I n t e r e s t Rate ......... Price S e r i e s on Retractable/Extendinle Bonds .. . P r i c e S e r i e s on Ordinary Pederal Bonds .......  88 91 96  7. EMPIRICAL TESTING OF BOND VALUATION MODELS  97  Estimated Parameters F o r The I n t e r e s t Rate Process S o l v i n g t h e Bond V a l u a t i o n Equation Bond V a l u a t i o n Under t h e Pure Expectations Model . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . • . . Estimating the Liquidity/Term Premium Paramters .................................... Bond Valuation Under t h e L i q u i d i t y / t e r m Premium (LIQP) Model Bond V a l u a t i o n With Revenue Taxes ............ Bond V a l u a t i o n I n c o r p o r a t i n g C a p i t a l Gains  97 101 106 129 104 148  . Wr. .. 151 . The "Moving Average" Model ................... 152 T e s t s o f Market E f f i c i e n c y ................... 157 Comparison o f Current Models with a "Naive" Model ..,.. • • * • 169 TaX  8. SUMMARY AND CONCLUSIONS ............ ................ , 174 Summary Of The T h e s i s Conclusions And D i r e c t i o n s For Further Research ...........................•......... BIBLIOGRAPHY  .. .  174 177 181  APPENDIX 1. C l a s s i f i c a t i o n o f S i n g u l a r f o r the Cases * = 1/2 & 1  Boundary  Behaviour 187  2. D e t a i l s o f t h e E s t i m a t i o n Procedure L i n e a r i z e d Model . ».•.•..../.^.  f o r the ..... .,  3. S o l u t i o n t o the Forward Equation f o r <* =  1  191 195  4. S o l u t i o n t o the Forward Equation f o r <X = 0 with no R e s t r i c t i o n a t the O r i g i n ......................  204  5. D e r i v a t i o n o f t h e Densities  205  Stationary  (or Steady  State)  6. D e t a i l s o f the P h i l l i p s Approach t o E s t i m a t i o n  ..... 209  vii  7. D e t a i l s o f E s t i m a t i n g (known) Case  Procedure  f o r <*=  1/2 213  8. A n a l y s i s o f E f f e c t of Measurement E r r o r s of Data ............,.................................  221  9. An Approximate Estimate o f the Asymptotic Correlation Matrix Between I n t e r e s t Bate Process Parameters ................................  223  10. Maximum L i k e l i h o o d E s t i m a t i o n of Parameters {m. fx, <r. ok) Osing t h e Steady S t a t e P r o b a b i l i t y Density Approach ..................................  228  11. E f f e c t on Bond V a l u a t i o n o f Using t h e Y e i l d to Maturity on a 91-day Pure Discount Bond I n s t e a d o f the Instantaneously R i s k f r e e Bate of Interest i...  239  t  viii LIST 0? TABLES Table  Page  I  Comparison of R e t r a c t a b l e s / E x t e n d i b l e s with Other Forms o f Debt i n Canada  II  Estimate o f m by D i f f e r e n t Methods for rf-i/j. (known) Case ............................  51  Estimate of /A, by dU'/a. (known) Case  52  III  D i f f e r e n t Methods  for  Estimate o f c r by D i f f e r e n t Methods for 0(^.1/2. (known) Case ............................  IV  6  z  V  Estimate  o f Infer'  by D i f f e r e n t Methods  53  for  (known) Case ............................  54  Comparison of Monte C a r l o Results on Parameter Estimation Using Serially Dependent/Independent Samples ................  59  VII  Comparison o f R e s u l t s o f E s t i m a t i o n Weekly and D a i l y Data {^-Y^ known)  60  VIII  Theoretical Bond P r i c e s  Sensitivity of to Errors i n m  Pure  Theoretical  Sensitivity  Pure  Bond P r i c e s  to Errors  Theoretical Bond P r i c e s  sensitivity o f Pure Discount t o E r r o r s i n <r .. , . . v r . « . . « • .  0U1/2.  VI  IX X XI  XII  XIII  XIV  XV  °  of  Using  Discount 63 Discount  i n^  64 v  65  Sensitivity o f Pure Discount Bond P r i c e s t o Distribution of Estimated Interest Rate Process Parameters ( r, = /V2)  67  Sensitivity of Pure Discount Bond P r i c e s t o Distribution o f Estimated Interest Rate Process Parameters ( r, = )  68  Sensitivity o f Pure Discount Bond P r i c e s to Distribution of Estimated Interest Rate Process Parameters ( r, = 2^)  69  Comparison of Bond Price Sensitivity to the Use o f D a i l y vs Weekly Data i n the Estimation of Interest Rate Process Parameters ( = j/^)  70  E s t i m a t i o n o f Parameters  73  for  Unknown Case  XVI  Comparison of Daily vs Weekly  Parameters Estimated Using Data f o r the Unknown Case  75  XVII  Theoretical Sensitivity of Pure Discount Bond P r i c e s to Errors i n <* { <r Has Hot Been 'Corrected* to R e f l e c t the E r r o r i n <* )  77  Theoretical Sensitivity of Pure Discount Bond P r i c e s t o E r r o r s i n (<r Has Been •Corrected* a c c o r d i n g t o the Value) .......  78  XVIII  2  2  of  XIX  Details of Data Sample R e t r a c t a b l e / E x t e n d a b l e Bonds .........  XX  Details of Data Sample of S t r a i g h t Coupon Bonds ........................................  95  XXI  Comparison of Model and Market P r i c e s Bond: .4% Jan. 1, 1963 (R1) ...................  109  XXII  Comparison of Model and Market P r i c e s Bond: 5'/i % Oct. 1, 1960 (E1) .................  110  XXIII  Comparison of Model and Market P r i c e s Bond: 5Ki % Oct. 1, 1962 <E2) .....• • • •  111  XXIV  Comparison of Model and Market P r i c e s Bond: 5/2. % Dec. 15, 1964 (E3)  112  XXV  Comparison of Model and Market P r i c e s Bond: 5Vo. % A p r i l 1, 1963 (E4)  113  XXVI  Comparison of Model and Market P r i c e s Bond: 6% A p r i l 1, 1971 (E5) ..................  114  XXVII  Comparison  of Model  and Market P r i c e s  Bond: 6/4. % Dec. . 1, 1973  (E6) ............. ....  94  115  XXVIII  Comparison of Model and Market P r i c e s Bond: VJi\ % . A p r i l 1, 1974 <E7)  116  XXIX  Comparison of Model and Market P r i c e s Bond: 8% Oct. 1, 1974 (E8)  117  XXX  Comparison of Model and Market P r i c e s Bond: 7% % Dec. 15, 1975 (E9)  118  XXXI  Comparison of Model and Market P r i c e s Bond: 6'/i| Aug. 1, 1976 (E10)  119  XXXII  Comparison of Model and Market Bond: 7% J u l y 1, 1977 (E11)  120  Prices  X  XXXIII XXXIV XXXV XXXVI XXXVII XXXVIII XXXIX XL XLI XLII XLIII XLIV  Comparison of Model and Market P r i c e s Bond: 1% % Oct. 1, 1978 (E12)  121  Comparison of Model and Market P r i c e s Bond: 1'A % Dec. 1, 1980 (E13)  122  Comparison of Model and Market P r i c e s Bond: 1% A p r i l 1, 1979 (E14) .................  123  Comparison of Model and Market P r i c e s Bond: 9^ % A p r i l 1, 1978 (E15) ...............  124  Comparison of Model and Market P r i c e s Bond: 9J4j % Feb. 1, 1977 (E16) ................  125  Comparison of Model and Market P r i c e s Bond: 7/£ 31 Oct. 1, 1979 (E17) ................  126  Comparison of Model and Market P r i c e s Bond: 9% Feb. 1, 1978 (E18) ..................  127  Comparison of Model and Market P r i c e s Bond: 9% Oct. 1, 1980 (E19) ...................  128  Comparison of Mean E r r o r For A l l Bond Across D i f f e r e n t Models .............................  145  Comparison of Betas & Market 6 Model P r i c e s  146  Correlation  Between  T h e o r e t i c a l S e n s i t i v i t y o f Pure Discount P r i c e s t o E r r o r s i n K,  Bond  T h e o r e t i c a l S e n s i t i v i t y o f Pure Discount Bond Prices t o Errors i n K ......................  156  Return on Zero Investment P o r t f o l i o Based on Constant Long P o s i t i o n i n Bond ..........  159  Return on Zero Set Investment P o r t f o l i o Using a Strategy Based on Returns to Similar P o r t f o l i o From a Constant Long Position i n the Generic Bond .........,.....  161  Return on Zero Investment P o r t f o l i o Based on Varying P o s i t i o n i n Bond ................  162  Results of Estimation  171  2  XLV XLVI  XLVII XLVIII XLIX  155  Yield  Eguation  Coefficient  Comparison of Model and Market P r i c e s Summary Over A l l Bonds .......................  147  Comparison of Returns to the Zero Investment Hedge P o r t f o l i o by Using Market vs. Model P r i c e s f o r t h e S t r a i g h t Bond .... Return on Zero Net Investment Portfolio (Based on a Constant Long P o s i t i o n i n the Generic Bond) by Aggregating Over A l l Bonds .  /  XI1  LIST OF FIGURES Figure 1  Page P l o t of T r a n s i t i o n Density F u n c t i o n (6 Cumulative Probability) f o r = at D i f f e r e n t r Values ...«..«.««»»• • •••••••••»•• ••••••••• ; e  2  P l o t s of t h e S e n s i t i v i t y of the T r a n s i t i o n Density F u n c t i o n to Changes i n tr and <* ......  85  P l o t s of t h e S e n s i t i v i t y o f the T r a n s i t i o n D e n s i t y F u n c t i o n t o Canges i n m a t D i f f e r e n t r Values .••.............. .• •;•» .•... ...........  ^6  Normal P r o b a b i l i t y P l o t of Resultant Error Vector from t h e E s t i m a t i o n o f Liquidity/Term Premium Prameters ............................  139  1  3  0  4  219  5  P l o t o f L i q u i d i t y Premium vs Time t o M a t u r i t y on Pure Discount Bonds Corresponding t o E s t i mated Parameters ............................,141  6  Plot of Term S t r u c t u r e Curve ( Y i e l d to M a t u r i t y vs Time t o M a t u r i t y on Pure Discount Bonds) Corresponding to Estimated Parameters a t D i f f e r e n t Values o f r,  142  P l o t o f Term S t r u c t u r e Curve t o Show P o s s i b l e "Humped" Shape f o r C e r t a i n v Values .........  143  Plots o f Model vs Market P r i c e s For Bond E4 : C a p i t a l Gains Tax (25%): Model, and o f D i s t r i b u t i o n o f Hedge P o r t f o l i o Returns ......  167  Plots o f Model vs Market P r i c e s For Bond E7 : C a p i t a l Gains Tax (25%) Model, and o f D i s t r i b u t i o n o f Hedge P o r t f o l i o Returns ......  168  7  0  8  9  ACKNOWLEDGEMENTS  I  would  completing  like  this  individuals..  to  share  dissertation  Professors  my  supervisor.  constant  Dr.  source  t h i s research topic.,  encouragement.,  towards  diffusion  considerable help i n my  understanding  processes.  read d r a f t s of my p r o p o s a l and aspects  pertaining  to  and  of  the  singular  with s e v e r a l other  Schwartz was a  provided  stages  for  Eduardo  Dr. John A. Petkau early  credit  Michael J . Brennan  Eduardo S. Schwartz suggested As  the  of  Dr. M. Puterman clarified  diffusion  certain  equations.  As  members o f my committee, P r o f e s s o r s Alan Kraus and R o l f Banz p a i n s t a k i n g l y read e a r l y d r a f t s of report,  and  improvement.  this  have c o n s i d e r a b l y c o n t r i b u t e d to i t s Professor  special  mention.  towards  the  Phelim P. Boyle  Apart  substance  merits  from  his contribution  and  style  of  this  d i s s e r t a t i o n , i t was h i s warm f r i e n d s h i p and moral support  that  kept  me  going  through  the rough  periods. I  cannot  Dr. Kent M. Brothers Every  part  statistics from  his  contributed  of and  this  sufficiently  f o r h i s h e l p and guidance. research  numerical  advice.  thank  pertaining  to  methods have b e n e f i t e d  Dr. Shelby  Brumelle  has  immensely t o t h e r e s e a r c h c u l m i n a t i n g  in  this  report.  He  was  always  available for  c o n s u l t a t i o n s , and i t i s to him t h a t I owe much of my understanding David have  of Markov  Emanuel,  helped  me  Hav  at  dissertation.  help  S b l a n k i and Gordon S i c k  various  t h e Bank  i n putting  of  the  in  initial  Kari  but  of  Boyle  helped  data c o l l e c t i o n , and Kent Wada  also  and i t s  with t h e p l o t s and t y p i n g the  t e x t i n t o t h e computer.  Seline  Carmen de S i l v a d i d an e x c e l l e n t tables  and appendices,  of  dissertation.  this  was  the data on  helped not only with t h e data c o l l e c t i o n punching  this local  Canada,  together  r e t r a c t a b l e / e x t e n d i b l e bonds. with  stages  Mr. Wayne Deans,  r e p r e s e n t a t i v e of immense  processes.  Gunawardene  and  job of t y p i n g t h e  as w e l l as t h e f i r s t  draft  1 CHAPTER 1 :  1,1  INTRODUCTION  Preamble The  a p p l i c a t i o n o f contingent  equilibrium  valuation  models  claims  analysis  f o r corporate continuing  has  in  actively  literature. estimation  This  investigated  study  addresses  of a particular  the  the  of  estimated  interest  current  problem  of  is and  finance empirical  s t o c h a s t i c s p e c i f i c a t i o n o f the spot  i n t e r e s t r a t e , and then goes on t o e v a l u a t e t h e model  derive  liabilities  p r e s e n t l y an area o f c o n s i d e r a b l e and been  to  retractable/extendible i n t e r e s t r a t e process,  bond  efficacy  of  a  v a l u a t i o n , based on the  in  pricing  of  Black  Canadian  Federal  Scholes  [ 7 ] and  Government i s s u e s . In  the  seminal  works  &  Merton [ 4 7 ] , the p r i n c i p a l focus was on a r r i v i n g valuation  models  f o r put and c a l l  at c l o s e d  o p t i o n s on c o r p o r a t e  equity.  Both the works c i t e d above d i d p o i n t out i n c o n c l u s i o n t h a t approach  could  liabilities  be  used  directly  to  of the f i r m .  in  other  the  corporate  by t r e a t i n g i n d i v i d u a l s e c u r i t i e s w i t h i n the c a p i t a l  s t r u c t u r e as " o p t i o n s " o r "contingent  corporate  value  form  c l a i m s " on t h e t o t a l  Herton [ 4 6 ] a l s o d e r i v e s v a l u a t i o n  bonds.  the area of  equations  value for  Smith [ 6 5 ] provides a good review of the work  option  pricing,  valuation of r e l a t e d s e c u r i t i e s .  and  i t s application  t o the  2 1  •  2  Contingent Claims V a l u a t i o n The  application  valuation [9],  of Bonds: & B r i e f Review  of  the  option  p r i c i n g approach t o bond  was extended by Black & Cox [ 5 ] ,  and  I n g e r s o l l [37],  such  &  various  types  of  bond  i t f a i l s t o meet some s t a n d a r d .  effect  of  subordination  They  further  among bonds, i e .  effect  dividend  of  restrictions  payments.  Both  on  look  firm,  to  presented  a  valuation So  among the finally  the f i n a n c i n g of i n t e r e s t and  Brennan  &  Schwartz  [9]  and  convertible  p r o v i s i o n s , the p r i n c i p a l d i f f e r e n c e  being that I n g e r s o l l was concerned with a r r i v i n g solutions  a t the  and  I n g e r s o l l [ 3 7 ] addressed t h e v a l u a t i o n o f c o r p o r a t e bonds with and without c a l l  holders  o f the f i r m  hierarchy  debt h o l d e r s , t o c l a i m s on t h e value o f the the  indenture  as s a f e t y convenants, whereby t h e bond  have the r i g h t t o bankrupt o r f o r c e a r e o r g a n i z a t i o n if  Schwartz  Black & Cox extended t h e a n a l y s i s o f  Merton [ 4 8 ] , t o i n c o r p o r a t e provisions  Brennan  at a n a l y t i c a l  the v a l u a t i o n problem, whereas Brennan & Schwartz general  numerical  algorithm  for  solving  the  equations. f a r , the  underlying  asset  emphasis  was  on c o r p o r a t e bonds, where the  was the value o f the f i r m . , The works  referred  to  above t r e a t e d t h e i n t e r e s t r a t e as non s t o c h a s t i c - constant  and  known with c e r t a i n t y over the period  area  that  of the bond.  The  was addressed was the p r i c i n g o f d e f a u l t f r e e bonds.  These s e c u r i t i e s , ( g e n e r a l l y Government bonds o f v a r i o u s were  valued by t r e a t i n g them as " c o n t i n g e n t "  the spot i n t e r e s t r a t e , along with  suitable  the  rates.  term  [10,12],  next  structure Cox,  of  Ingersoll  interest S  Ross [ 1 6 ] ,  types)  upon the course o f assumptions Brennan  about  & Schwartz  Vasicek[72],  and  3 Bichard [ 5 8 ] ,  have  all  addressed  the problem of d e f a u l t f r e e  bond v a l u a t i o n i n the o p t i o n p r i c i n g framework. works  of  Brennan  apart from  the  6 Schwartz ( c i t e d above), the r e s t p r i m a r i l y  d e a l t with the v a l u a t i o n of pure d i s c o u n t bonds, so as t o a r r i v e at c l o s e d form expressions Brennan  S  of  -  discount  as  a  savings,  -  all  conditions.  function  solely  retractable,  They  feature also  equation.  of  the  being  the  short  term  show t h a t v a r i o u s  types  extendible,  callable  the  or  equation,  associated  boundary  present a numerical a l g o r i t h m to s o l v e  the v a l u a t i o n equations.  In t h e i r l a t e r paper [ 1 2 ] , they  posit  value of the d e f a u l t - f r e e bond as a f u n c t i o n of the time to  maturity short  and two  term  &s can done  valuation stochastic interest  related interest  riskless  i n t e r e s t process  been  structure  f o l l o w the same p a r t i a l d i f f e r e n t i a l  distinguishing  the  term  r a t e and time to maturity, and  bonds  the  the  Schwartz, i n t h e i r e a r l i e r paper [ 1 0 ] , represent  d e f a u l t f r e e bond interest  for  interest  rate rate  processes and  very  the  very  l o n g term  ( y i e l d s on a c o n s o l bond).,  be seen from the f o r e g o i n g , on  the  -  the  theoretical  equations  under  properties rates.  In  of  considerable  front,  varying  ie,,  addition,  developing  assumptions  i n t e r e s t r a t e s and numerical  work  has bond  about  the  term s t r u c t u r e of  methods  have  been  developed t o s o l v e r a t h e r g e n e r a l forms o f the r e s u l t a n t p r i c i n g formulae.  However, to date, t h e r e have been few  of  models.  these  contingent on  Host  of  published  the e m p i r i c a l work i n the area  claims a n a l y s i s , has been on the market  corporate  equity  Scholes [ 6 ] , and  ,  tests  (to c i t e the important  G a l a i [ 2 9 ] ) , except f o r  for  options  papers: Black  Ingersoll [38],  of  &  which  4 is  an  application  shares,  of  option  pricing  analysis  t o dual  and Brennan & Schwartz [12]» who value Canadian  fund  Federal  Government coupon bonds. The  aim  o f t h i s r e s e a r c h i s t o conduct an e m p i r i c a l  of contingent  claims  analysis  on  retractable  and  study  extendible  bonds of t h e Government of Canada.,  1. 3  Canadian R e t r a c t a b l e s / E x t e n d i b j e s i n P e r s p e c t i v e : ftn e x t e n d i b l e i s a medium t o long term debt o b l i g a t i o n t h a t gives  the  instrument, 5k %,  holder  extend  the  term  of  the  at a predetermined coupon r a t e . ,  For  example,  the  October  the  option  to  1st, 1962, maturity  October, 1959.  e x t e n d i b l e was i s s u e d on 1st  I t was exchangeable on o r before June 1st,  i n t o 5%,%, October 1 s t , 1975 bonds. was  extendible  Thus the 3 year i n t i a l  an  e a r l i e r maturity.  There  a r e very are  e x t e n d i b l e bond. put  option.  two  valuation  option  option  to  theory,  the  two  similar. ways  in  which  to  view a r e t r a c t a b l e o r  I t may be viewed as a long  term  bond  with  a  The e x e r c i s e p r i c e i n t h i s s i t u a t i o n i s the value  Of t h e long term bond, and the payoff i s the The  A  Both from the p r a c t i c a l investment  p o i n t of view, and with r e s p e c t t o instruments  bond  i n t o a 16 year bond, at the h o l d e r ' s o p t i o n .  r e t r a c t a b l e , on t h e other hand, g i v e s the h o l d e r the elect  1962  i s exerciseable  short  term  bond.  on t h e e x t e n s i o n / r e t r a c t i o n date.  A l t e r n a t i v e l y , the r e t r a c t a b l e o r e x t e n d i b l e may be viewed as short  term  bond  with a c a l l o p t i o n .  a  From t h i s p o i n t o f view,  the e x e r c i s e p r i c e i s the value of the short term bond, and  the  5 payoff i s the long term bond. Extendibles Canadian  and  scene i n 1959  January 1 s t , 1963 retractable 1961  and  retractables  on  with the  appeared*  on  the  F e d e r a l Government i s s u e of  (maturity date) r e t r a c t a b l e bonds, which any  this  by  was  giving  the  only  3  months  prior  retractable  H%, were  i n t e r e s t payment date between January  January 1st, 1962  (Incidentally,  first  1st,  notice.  i s s u e d by  the  Government of Canada). While t h e r e were a d d i t i o n a l Government  i n the mid  more widely  i n the high  issues  made  by  s i x t i e s , these instruments interest  rate  period  the  have been used since  Table I g i v e s some numbers t o p l a c e r e t r a c t a b l e s and in  perspective  vis-a-vis  major i s s u e r of government.  other  retractable/extendible  columns  in  Table  current l i a b i l i t i e s ,  to  long term debt. the  compared though  medium  I  Federal  extendibles C l e a r l y , the  is  the  Federal  outstanding,  treasury b i l l s , e t c . ) ,  The  as w e l l  to  long  4.536  represent  of  total  as  R e t r a c t a b l e s and e x t e n d i b l e s belong  retractables  approximately  governments.  i n c l u d e very s h o r t term debt, ( i e . ,  term maturity c l a s s , and  with the other debt i n that  instruments  bonds  1969/70.  and e x t e n d i b l e s appear to be i n c r e a s i n g over time,  both with the P r o v i n c i a l and  to  of debt.  F u r t h e r , as a p r o p o r t i o n of t o t a l debt  retractables  debt  forms  Federal  and the  class  extendibles total  alone.  medium strictly  so should Thus  constitute  Provincial  debt,  a l a r g e r p r o p o r t i o n of the medium and  be  even only these long  * Information obtained from a p u b l i c a t i o n of M/S Mood Gundy L t d . on retractable/extendible bonds, l i s t i n g a l l outstanding F e d e r a l / P r o v i n c i a l / c o r p o r a t e i s s u e s as of January 15th, 1975.  TABLE I  COMPARISON OF RETRACTABLES/EXIENDABLES WITH OTHER FORMS OF DEBT IN NOTES OH TABLE I O/S as on 31st March 1975 Z Tot.Debt. Ret/Ext.  B r i t i s h Columbia Alberta Manitoba New Brunswick  O/S as on 31st Marc'i 1976 Z Tot.Debt Ret/Ext.  Ret/Ext as on 31st March 1977  -  3845  -  50  5093  0.98  50  128  3031  4.22  128  3578  3.58  128  58  2473  2.34  58  2884  2.01  58 61  51  1199  5.08  61  1665  3.66  -  -  182  Newfoundland  161  1504  10.70  182  Ontario  225  13397  1.90  675  16760  4.02  675  9.00  10  P.E. Island Quebec Saskatchewan  10  98  10.20  10  111  734  8403  8,73  808  8391  9.63  983  70  912  7.67  70  14.31  5850  38299  15.27  6250  -  2315  -  -  2503  -  816  -  Total P r o v i n c i a l Federal  4825  Corporate  1902  Total  8104  33700  -  10207  10970  a) A l l figures are i n millions of dollars b) The t o t a l debt includes a l l bonds, b i l l s and notes, Issued by by the Provincial government, as well as a l l debt guaranteed by the Provinces. c) Likewise, the retractables/extendables included i n each Provinces' a/c (as well as i n the Federal a/c), including issues guaranteed by the Provinces as well, d) No figure of aggregate corporate debt was included as the same was not readily available, e) The total Federal debt figures were taken from the Bank of Canada Review. For the Provinces, the same were from the Public Accounts. f) The public accounts for Newfoundland as of 31st March 1976 were .not readily available.  7 term  debt.  In gross amounts, i n c l u d i n g c o r p o r a t e i s s u e s ,  t o t a l about $10  billion.  Apart from s i z e of outstanding  another f a c t o r c o n t r i b u t e s retractable attached  and  valuation and  extendible  t o the  conventional  ordinary  methods  i n the  extendible  option  O u t l i n e of th.e  the  ad  interest  bonds.  bond.  This makes  hoc,  and  in  the  issues,  study  These bonds have an their  option by  amenable  to  Clearly, retractable  bonds are i n t e r e s t i n g i n s t r u m e n t s , and a d e t a i l e d order.  Thesis  Chapter 2 develops the  basic  bond  valuation  equation  terms of the  parameters o f the l o c a l i n t e r e s t r a t e process.  appropriate  boundary  r e t r a c t a b l e and incorporating i n t o the approach  I n g e r s o l l [38]} The rate  conditions  extendible  relevant  the The  The  p r i c i n g of approach  account  for  outlined.  taxes  (along  An  to  approximate  the  lines  of  i s a l s o presented.  s t o c h a s t i c s p e c i f i c a t i o n of  the  short  term  process i s c e n t r a l t o the bond v a l u a t i o n model.  this  in  d i f f e r e n t assumptions about t e r m / l i q u i d i t y premia  addresses the d e s i r a b l e p r o p e r t i e s t h a t any of  to  bonds are d e r i v e d .  v a l u a t i o n model i s b r i e f l y to  of  valuation  particularly  p r i c i n g framework.  study of them i s q u i t e i n  1•4  to  they  process should possess.  Chapter 3  mathematical  model  A s p e c i f i c d i f f u s i o n equation  i s suggested t o model i n t e r e s t r a t e s , and s p e c i f i c a t i o n are  interest  the p r o p e r t i e s of t h i s  investigated.  Having s p e c i f i e d the form o f the i n t e r e s t r a t e process, next problem i s t h a t o f e s t i m a t i n g  i t s parameters, given  a realization  Methods  of  the  process,  for  data  estimating  the on the  8 parameters  are  examined  in  review  the  existing  literature  of  parameters  of  Markov  and  Chapter  1.  S t a r t i n g with a b r i e f  on  the  estimation  d i f f u s i o n processes, t h r e e d i f f e r e n t  methods o f e s t i m a t i n g the parameters are proposed. of  the  of  e s t i m a t i o n procedure  The  details  f o r each o f these methods are a l s o  presented., Chapter  5 i s devoted  o f e s t i m a t i o n proposed C a r l o methods are  to the comparison of the three methods  i n the p r e v i o u s chapter.  used  to  examine  the  For t h i s , Monte  distribution  of  the  estimated parameters by each method, under d i f f e r e n t c o n d i t i o n s , as  part  of  the comparison of the three methods, the e f f e c t of  the estimated d i s t r i b u t i o n of parameters on bond also  briefly  investigated  concludes  with  is  s i n c e our primary concern i s t o use  the e s t i m a t e s t o value r e t r a c t a b l e and chapter  valuation,  a  brief  between the estimated parameters,  extendible  look  a t the  bonds.  The  inter-relations  as w e l l as the  way  in  which  they a f f e c t the i n t e r e s t r a t e process. Details and  about the data sample on short term i n t e r e s t  bond p r i c e s are given i n Chapter 6.  empirical  t e s t s of the models developed  with the bond v a l u a t i o n model based hypothesis.  We  term/liquidity preference premium  then  premium.  on  incorporate The  Chapter  7  rates  reports  i n Chapter  2.  the  expectations  a  estimation  pure  We  the start  specific  form  of  investor  the  of  parameters i n the assumed form of the t e r m / l i g u i d i t y  expression  is  addressed  and  estimates  of  these  parameters,  based on a sample o f n o n - c a l l a b l e coupon bonds, are  presented.  These  valuation  model and  estimates  are  incorporated  in  the  the r e s u l t a n t bond values are compared  bond with  9 market p r i c e s .  The  incorporating  taxes  effect (both  taxes), i s investigated. the  returns  to  on  the  revenue  Tests  bond  valuation  taxes  o f market  and  priced  model  required  gains on  a zero-investment p o r t f o l i o are conducted.  In  models  to  identify  an approach based  F i n a l l y , an ad hoc, r e g r e s s i o n  (the "naive"  developed.  of  based  bonds i s a l s o i n v e s t i g a t e d using  on G a l a i [ 2 9 ] .  capital  efficiency  t h i s s e c t i o n , t h e a b i l i t y o f the d i f f e r e n t over  model  based  valuation  model) f o r r e t r a c t a b l e s and e x t e n d i b l e s i s  Using t h e sample o f n o n - c a l l a b l e c o e f f i c i e n t s f o r the " n a i v e "  are e s t i m a t e d . , The performance  predicting  bond  is  briefly  bonds,  the  model o f r e t r a c t a b l e s and  extendibles  prices  coupon  of  this  model  in  compared  with t h a t o f the  with  summary  models developed e a r l i e r i n Chapter 2.» The study concludes i n Chapter 8 principal  results,  and  some  remarks  a  about the c h o i c e  of  of the  s t o c h a s t i c s p e c i f i c a t i o n f o r t h e i n t e r e s t r a t e process, as as  about  research  the model o f bond v a l u a t i o n . .  the  well  Suggestions f o r f u r t h e r  i n r e l a t e d areas conclude t h e study.  10 CHAPTER 2: THE  2.1  PRICING THEORY OF  DEFAULT FREE BONDS  Determinants of Bond Value The  approach to the  v a l u a t i o n of r e t r a c t a b l e and  bonds w i l l c l o s e l y f o l l o w Schwartz [10].. the  present  the  method  set  B a s i c a l l y , the value of any  value  of  i t s p r i n c i p a l and  out  extendible  in  Brennan  6  d e f a u l t f r e e bond i s coupon payments.  The  f u t u r e cash flows are known with c e r t a i n t y , once the coupon r a t e and  time to maturity  flows,  what  ( i e . the choice  required  to  Knowing  arrive  at  is  the  short  value  reasoning,  we  value  A  natural  could  evaluate  Following  this  c o u l d j u s t i f y the assumption t h a t the be represented  term i n t e r e s t r a t e and  i s some u n c e r t a i n t y spot  present  over a l l p o s s i b l e f u t u r e sample paths of  a d e f a u l t f r e e bond may short  cash  factor.  t h a t i n t e r e s t r a t e s are s t o c h a s t i c , we  present  future  term i n t e r e s t r a t e . . In a model where we  i n t e r e s t r a t e , over the terra o f the bond. of  the  their  bond value) i s a s u i t a b l e discount  recognize the  is  are s p e c i f i e d .  as  a  a s s o c i a t e d with  the  line  p r i c e of  function  the time t o maturity.  of  Since  assessment  the  of  the there  future  r a t e s , i n a market where r i s k averse i n v e s t o r s e x i s t , term  premia enter  the v a l u a t i o n equation v i a the s p e c i f i c  assumptions  made about the term s t r u c t u r e of i n t e r e s t r a t e s . To model the f u t u r e course of the spot assume path and future  interest  rate,  we  t h a t i t i s a s t o c h a s t i c process with a continuous sample Markov p r o p e r t i e s .  Under the  development of the spot  Markov  r a t e process,  assumption,  (given i t s present  value) i s independent of the past development t h a t the present  level.  the  Processes that are Markov and  has  led  to  c o n t i n u o u s are  11 called  d i f f u s i o n p r o c e s s e s , and f o r the one dimensional case  can  i n g e n e r a l be d e s c r i b e d by a s t o c h a s t i c d i f f e r e n t i a l equation of the  form  dr  where  bCr.t} <&>  -f  dl  b ( r , t ) , and a ( r , t ) r e p r e s e n t the instantaneous d r i f t 2  v a r i a n c e r e s p e c t i v e l y of the p r o c e s s , stochastic  element  present, t h e r e i s generality  of  and  is  nothing the  and  that  both  b(r,t)  to  above  and  be  stochastic  a(r,t)  2  particular  as  gained  is  the  by  For  restricting  differential  must may  at  least not  be  equation be be  noted known,  stochastic  family of processes, when we address the later  the the  He s h a l l however r e s t r i c t our a t t e n t i o n  r a t e process i n g r e a t e r d e t a i l  and  driving  H(0,dt).  However, i t may  d e t e r m i n i s t i c f u n c t i o n s of time - they f u n c t i o n s of t i m e . ;  dz  distributed  governing the i n t e r e s t r a t e process.  a  (2.1).,  to  interest  on.  The main competing t h e o r i e s about  the  term  structure  of  i n t e r e s t r a t e s are a)  the pure e x p e c t a t i o n s h y p o t h e s i s  In the standard o p t i o n v a l u a t i o n framework, t h e r e i s no r e s t r i c t i o n on the instantaneous d r i f t term of the u n d e r l y i n g a s s e t (the s t o c k ) , i e . t h a t i t should be n o n - s t o c h a s t i c . T h i s is because, the f i n a l p a r a b o l i c p a r t i a l d i f f e r e n t i a l equation governing the o p t i o n v a l u e does not c o n t a i n the d r i f t term. For the bond, t h e corresponding partial differential equation i s equation (2.9). The i n s t a n t a n e o u s d r i f t of the i n t e r e s t r a t e process (the u n d e r l y i n g a s s e t being the pure d i s c o u n t bond due to mature the next i n s t a n t ) e n t e r s the v a l u a t i o n equation. I f e i t h e r b ( r , t ) o r a ( r , t ) i n equation (2.1) were s t o c h a s t i c , then the v a l u a t i o n equation would no longer be an o r d i n a r y second order p a r a b o l i c p a r t i a l d i f f e r e n t i a l equation. 2  12 b)  the term or l i q u i d i t y premium hypothesis  c)  the  market  segmentation  (or  preferred  habitat)  hypothesis. , The adopt  d e f i n i t i o n of the pure expectations  we  i s that the instantaneous expected r e t u r n on bonds of a l l  maturities  is  the  same .  This  3  n e u t r a l i t y " on the p a r t  of  The  over  some  s o r t of " r i s k  the  instantaneous  bonds of a l l m a t u r i t i e s .  second hypothesis argues t h a t concern over f l u c t u a t i o n s  wealth  long  implies  investors  holding period returns across  in  hypothesis that  causes  i n v e s t o r s to demand a " l i q u i d i t y " premium on  term bonds over those of s h o r t e r maturity.  hand,  concern  term premiums  over that  On  the  other  f l u c t u a t i o n s i n income l e a d s to a case f o r would  obviously  have  just,  the  opposite  pattern., The  market  segmentation h y p o t h e s i s proposes that  d i f f e r e n t m a t u r i t i e s are t o t a l l y not  substitutable.  of  interest  rates,  d i f f e r e n t instruments, and  T h i s would r e q u i r e that the at  any  bonds of  point  term  thus  structure  i n time, be d e f i n e d by  the  What f o l l o w s i s based on Cox, I n g e r s o l l & Ross [ 1 6 ] , In the existing literature, the pure e x p e c t a t i o n s hypothesis i s c h a r a c t e r i z e d by one o f the f o l l o w i n g p r o p o s i t i o n s : 1) Implied forward r a t e s are e q u a l to expected future spot rates 2) The yield to maturity from holding a long term bond i s equal t o the y i e l d from r o l l i n g over a s e r i e s of s h o r t term bonds 3) The expected r e t u r n over the next h o l d i n g p e r i o d from bonds of a l l m a t u r i t i e s i s equal Under c e r t a i n t y , a l l t h r e e forms are equivalent., With u n c e r t a i n i t y , however, Cox, I n g e r s o l l S Ross have shown t h a t the first two p r o p o s i t i o n s are c o n s i s t e n t with each other, but not with the t h i r d . Hore s p e c i f i c a l l y , i f the term s t r u c t u r e is unbiased i n the sense of the f i r s t two p r o p o s i t i o n s , then the i n s t a n t a n e o u s expected r a t e of r e t u r n on any bond must exceed the spot r a t e . 3  13 supply  and  demand f o r each of the number of m a t u r i t i e s  existing  i n the market at t h a t time. Most s t u d i e s of the term s t r u c t u r e of i n t e r e s t r a t e s i n the option  pricing  Ingersoll S considered  framework,  Ross [ 1 6 ] ,  {  Brennan  Vasicek [72]  Brennan  &  Schwartz  [12],  a form of the  i n t r o d u c i n g two  f a c t o r s i n the maturity  the  end,  first  rates  two  into  and  the  have  structure  long term maturity,,  bond  [58]),  Cox, have  tried  to  market segmentation h y p o t h e s i s ,  hypotheses about the term the  Richard  [10];  or t e r m / l i q u i d i t y premium  operationalize  short  Schwartz  and  only the pure e x p e c t a t i o n s  assumptions.  &  valuation  the  very  Only i n c o r p o r a t i o n o f  structure  models  -  by  of  i s considered  interest in this  study.  2.2  The  Basic, Bond Valuation, Equation Let us represent  by B ( r t ) , #  which pays $1 at maturity;  the value of an  where r i s the spot  ordinary  riskless interest  r a t e , and X the time to m a t u r i t y .  S i m i l a r l y , l e t the value  retractable  be  or  extendible  g e n e r a l i t y , l e t B ( r , l ) pay Then, using I t o ' s L e n a  bond  G(r/£).  a coupon* c, , and  (McKean [45])  and e q u a t i o n  the s t r a i g h t bond B, and  G,  the  are  governed  by  following  of a  For purposes o f  G(r,T) a coupon  i n t e r e s t r a t e process,  bond  (2.1)  z  f o r the  the g e n e r i c  stochastic  c»  bond  differential  equations (SDE) :  * For ease of computation i n a continuous time framework, we assume that t h e s e are continuous coupons. A c o n t i n u o u s coupon of c means a coupon payment of c d o l l a r s per u n i t of time per bond.. As pointed out i n Chapter 6, t h i s assumption i s q u i t e reasonable.  14  (2.2)  where  b=b(r,t)  and  a=a(r,t), first  and  subscripts  denote  derivatives;  B, i s t h e  with r e s p e c t  to i t s f i r s t , argument - the spot r i s k l e s s  partial  p a r t i a l d e r i v a t i v e of the bond p r i c e interest  rate, etc. The  spot  riskless  y i e l d to maturity  on  interest  generic  i s , by d e f i n i t i o n ,  a d e f a u l t f r e e discount  tike next i n s t a n t i n time., The the  rate  bond due  to  mature  r e t u r n on a l l three a s s e t s , v i z . ,  bond> the s t r a i g h t bond and  the short  term i n t e r e s t  instrument, have the same s t o c h a s t i c element d r i v i n g them ie.,  they  are  all  perfectly  l e n d i n g at the i n s t a n t a n e o u s l y possible  (and  all  the  model h e l d ) , a zero net s  using the  riskless  correlated.  asset.  %  of  other assumptions of the investment p o r t f o l i o  d o l l a r s i n B and The  If  r i s k l e s s rate  above three s e c u r i t i e s .  d o l l a r s i n G, x  the  (dz);  borrowing interest  were  option p r i c i n g  could  be  formed  Consider an investment of x  i  = - (x,  *-x ) z  and  dollars in  r e t u r n on such a p o r t f o l i o i s given  x, the  by  The perfect market assumption i s implied with a l l the attendant properties of unlimited borrowing/lending at the riskless rate by a l l i n v e s t o r s , no margin reguirments on short s a l e s and immediate f u l l availability of proceeds of short selling and ability t o t r a d e every i n s t a n t a t c u r r e n t p r i c e s , and f i n a l l y the absence of a l l taxes., 3  15  Rewriting  equation  (2.2)  as  (2.4)  ft  we can  rewrite(2.3)  as  (2.5)  We  can  see  from  equation  (2.5)  t h a t a l l u n c e r t a i n t y from  r e t u r n on the zero investment p o r t f o l i o would be we  choose  x,  and  x  eliminated  the if  such t h a t the c o e f f i c i e n t of dz i s zero,  2  ie.,  X  z  <%_  •=  A r b i t r a g e would now investment  to  g i v e s the b a s i c v a l u a t i o n  (/VVQ-  point  in  time.  -  zero.  (2.6)  .J__  *i  d r i v e the c e r t a i n r e t u r n  portfolio  T h i s expression  =  on  the  S u b s t i t u t i n g (2.6)  zero  net  into  (2.5)  equation.  „  (A  + C  '/B)  -  (2.7)  r  has to h o l d f o r bonds of a l l m a t u r i t i e s I t i s the f a m i l i a r expression  per u n i t o f r i s k on each s e c u r i t y (see Cox  at  any  of excess r e t u r n  £ Ross [ 17 ]).  We  may  16 r e p r e s e n t t h e p r i c e o f i n s t a n t a n e o u s standard (r,t) ,  noting  maturity,  t h a t , whereas  deviation risk  by  i s independent o f the time t o  i t may change over time and with the spot r a t e .  This  gives  (/W<0~  where \{t,t t)  -  instantaneous  maturity T . partial  »  return  d i f f e r e n t i a l equation  u  in  equation  equilibrium,  (2.9).  conditions  at  (2.8).  time t on a bond with time t o  S u b s t i t u t i n g f o r jl^ and 0£ from  -la<q + Cb-^)(5 Thus,  UT.t.t)  r e p r e s e n t s the term or l i q u i d i t y premium, i e . the  0  excess  R  that  (  the  f o r t h e bond p r i c e  -T<5+C -<q L  any  (2.4), y i e l d s  bond  =0  t  follows  What  distinguishes  each  has  the  them,  to satisfy.  (2.9)  are  same v a l u a t i o n the  boundary  ( T h i s r e s u l t was f i r s t  demonstrated by Brennan & Schwartz [ 10 ] ) .  2. 3  Boundary, C o n d i t i o n s f o r R e t r a c t a b l e / E x t e n d j b l e Bonds Let  us  now  consider  the  boundary  conditions  that  the  g e n e r i c bond has to s a t i s f y . a)  Terminal  value  at  maturity:  aspect, t h e p r i n c i p a l o f $1 i s guaranteed irrespective  From the d e f a u l t f r e e at  maturity.  Thus  o f the c u r r e n t i n t e r e s t r a t e a t maturity, the bond  value e q u a l s i t s face value, i e . ,  17  G (r,o) = 1 b)  (2.10a)  Retraction/extension  f e a t u r e : Here, we s h a l l  t h r e e types of r e t r a c t i o n / e x t e n s i o n appropriate equation  features  and  consider  develop  the  boundary c o n d i t i o n s a p p l i c a b l e to the bond v a l u a t i o n  i n each case;  i)  the r e t r a c t i o n / e x t e n s i o n option has t o be  exercised  at a s i n g l e point i n time. ii)  the o p t i o n may  iii)  the option t o r e t r a c t / e x t e n d may  period  be e x e r c i s e d over a p e r i o d of time. be e x e r c i s e d over a  o f time, but even i f the d e c i s i o n i s t o r e t r a c t ,  t o extend, i n the case of an e x t e n d i b l e )  the f a c e value  (or not of $1 i s  a v a i l a b l e only on a f i x e d f u t u r e date beyond the f i n a l  exercise  date. These t h r e e cases may  diagramatically  r—  1  VkrruxL  The  first  TQ.( I  a n <  wh<n cf>tum  case  *  above  c o i n c i d e at one  not a f i x e d point beyond  %i  and  € t  as:  1  1  Sh«t  k ^ 01  would correspond to the s i t u a t i o n where  is  t  represented  point.  For the second  , but could be any  depending upon the bond holder's  case, t  5  p o i n t between  choice.  To d e r i v e the boundary c o n d i t i o n f o r each case, i t would be h e l p f u l t o c o n s i d e r an example, a 5% coupon bond, which he may  Consider t h a t an i n v e s t o r extend on  f o r a 6% coupon bond maturing January investor 1975  does not choose t o accept  maturity,  the o l d bond may  (say) January 1st,  1st,1975.  the new  1970  case  the  bond of January  1st,  be cashed i n f o r $1  In  holds  on  January  18 1st,  1970.  C l e a r l y on any day p r i o r to January 1 s t , 1970, the  h o l d e r of the short bond has a European c a l l Jananuary now  1st,  represent  January  expiry date). to  the  1975, Let ^  decision  on  and  by  represent  e  point,  the i n s t a n t i n time  and te r e p r e s e n t  ie.  (the option just  prior  the i n s t a n t i n time  Then we have  S C T X ) - Max [ $C*X*) ,  1  (2.10d.1)  The c o n d i t i o n above i m p l i e s that the bond v a l u e , cashed  6%  L e t us  date of the long bond,  le. , January 1 s t , 1970  j u s t a f t e r the d e c i s i o n p o i n t .  not  the  1975 bond with an e x e r c i s e p r i c e of $1.  by t = 0, the maturity  1st,  option  i f the bond  i n at t h e d e c i s i o n p o i n t , i s continuous across  is that  p o i n t i n time. In c a s e , however, the o p t i o n t o extend c o u l d over  a  period  of  time,  ( c a s e ( i i ) ) , condition  rather  than  at  a  be  exercised  point  in  time  (2.10d.1) would be a l t e r e d as:  (2.10d.2)  Here the f i r s t c o n d i t i o n i s extension  option  the holder  during  i s i n f o r c e , the value  below by the par value as  that  of $1.  T h i s i s the  has an American o p t i o n .  the  period  the  of the bond i s bounded arbitrage  condition  Further, s i n c e i t has to  be continuous across t h e e x p i r y point of the o p t i o n ,  we have the  second c o n d i t i o n , as before. For  actual  bonds  in  the  market,  case  ( i i i )i s  the  19 representative  case.  The  option  to  extend/retract  may  e x e r c i s e a b l e over a 3 to 6 month p e r i o d , but, even i f the were e x e r c i s e d , the par value further  6  to  1969,  generally  12 months l a t e r .  January 1st, 1970 1st,  is  and  we  r_  and  may  as  cj  available  1st,  1969  and  now  T«i  represent  October  1st,  as  1969.  October 1 s t , 1969;  a v a i l a b l e only on January exercise  boundary c o n d i t i o n there  <qCT,'E«0 +  '  only  1st, 1970.  (say)  time,  It is clearly  optimal  so we  to  have the  <qtr,t l) t  H(T r ;-t ) (  t  €  <2.10d.3)  s  bond.  been  e x p l i c i t l y recognize  July  as:  M<W^  term  ~  the p r i n c i p a l of $1 i s  the value of the long  short  s  between  In the c o n d i t i o n above, G r e p r e s e n t s The  a  C l e a r l y , i f the  option at the l a s t p o i n t , fe2. , and  the  option  Going back t o our example, X  i n v e s t o r decides t o choose the s h o r t bond a t any July  be  bond  has  represented  t h a t the coupon of the two  bonds  term  by H, could  to be  different. c)  Value  the previous value  are  considerations,  at  investigated developing  the  r=0  we  very  the in  the  i n t e r e s t r a t e process  closely  require  Whether t h i s  conditions  at  the i n t e r e s t r a t e boundaries: We  s e c t i o n t h a t the  process  negative.  at  interest  requires  interest next  related.  the  rate  rates  to  imposition  We  oo  conditions  till  later.  imposed  on  the  bond  value  economic  remain of  therefore  For the present,  the bond  non-  specific  (r=0 and co ) i s  c o n d i t i o n s that the bond value process  and  and  From  boundaries  chapter.  know from  we  has to  postpone satisfy  j u s t note that  process  at  the  20 boundaries with the  of  the  interest  behaviour  of  r a t e process  the  interest  should be c o n s i s t e n t  rate  process  at  these  boundaries. In  general,  attendant  the  differential  boundary  retractable/extendable  equations  conditions) bonds,  cannot  (along with the  governing be  solved  Numerical f i n i t e d i f f e r e n c e methods w i l l be used equations.  The  general  procedure  2.4  analytically. to  solve  the  i s t o develop the s o l u t i o n  r e c u r s i v e l y backwards from the boundaries, known.  the  where t h e s o l u t i o n i s  T h i s i s addressed f u r t h e r i n Chapter 7.  I n c o r p o r a t i n q Taxes i n t o t h e Model So f a r the model has been developed on the assumption of no taxes, e i t h e r on revenues gains.  (coupons  and  interest)  or  capital  He c o u l d attempt t o i n c o r p o r a t e taxes i n t o the v a l u a t i o n  equation,  along  t h e l i n e s of I n g e r s o l l [ 3 8 ] , but the f o l l o w i n g  assumptions need t o be made e x p l i c i t : a)  Taxes at  are assumed  a fixed rate.  "average" used  payable on a continuous This  implies  that  basis  there  t a x r a t e over a l l i n v e s t o r s t h a t  in  the model.  that interest  payable  and  i s some could  be  The  assumption f u r t h e r i m p l i e s  on  all  borrowings  i s tax  deductible. b)  A l l capital  gains are t r e a t e d as taxed  at the c a p i t a l  gains tax r a t e , and payable c o n t i n u o u s l y . capital  gains  taxes  actually realized  are  by a s a l e .  paid  only  In  when  reality, gains a r e  F u r t h e r , any c a p i t a l  over a p e r i o d o f l e s s than 91 days i s t r e a t e d  gain  f o r tax  21 purposes cannot The  a revenue item.  make t h i s  assumptions may  question or  as  I n our model however, we  distinction . 6  be r e s t r i c t i v e , but i t i s an  empirical  as to whether i t i s b e t t e r t o i g n o r e taxes a l t o g e t h e r ,  i n c o r p o r a t e them i n t o the v a l u a t i o n equation with the c u r r e n t  assumptions - a q u e s t i o n t h a t i s addressed Let  later.  us r e p r e s e n t by R, the r a t e of taxes on revenues  T, the r a t e of taxes on c a p i t a l g a i n s . investment  The  p o r t f o l i o , as given i n equation  r e t u r n on (2.3)  and  the  by  zero  i s modified to  The same a n a l y s i s as before l e a d s t o the v a l u a t i o n equation  which l e a d s to the f o l l o w i n g p a r t i a l d i f f e r e n t i a l equation  iaV0<$„ + [bG-T)-a{]5, +  The  0-R)(c ~f^) 2  0-T)^  2  0  (2.11)  boundary c o n d i t i o n a s s o c i a t e d with t h i s equation are e x a c t l y  those a s s o c i a t e d with the p r e v i o u s equation  (2.9).  * T h i s assumption i s r e q u i r e d to ensure an unique equilibrium bond value. Given our continuous time hedging approach to v a l u a t i o n , c a p i t a l g a i n s as per the e x i s t i n g tax laws a r e never applicable. Capital g a i n s taxes do e x i s t , and are accepted as one of the determinants of i n v e s t o r s c h o i c e among available securities. The present approach i s one way of i n c o r p o r a t i n g t h i s r e a l i t y i n t o our model.  22 CHAPTER 3 : THE INTEREST RATE PROCESS  3.1  P r o p e r t i e s o f I n t e r e s t Rate Processes In  the  previous  chapter,  we  left  s p e c i f i c a t i o n o f the i n t e r e s t r a t e process form.  Lacking  a  well  developed  theory  u n c e r t a i n t y t o s p e c i f y a f u n c t i o n a l form process,  (the only  work  a)  a  very  of  f o r the  stochastic general  growth  under  interest  rate  addressing the problem appears t o be  Herton [ 4 9 ] ) , we are l e f t t o draw s a t i s f y some very broad  in  the  upon  functional  forms  that  criteria . 7  I n t e r e s t r a t e s should never become n e g a t i v e , as h o l d i n g wealth  in  the  form  of  cash  dominates such  a  situation. b)  An i n t e r e s t r a t e process should tendency,  i e . , one  possess  some  central  would not expect t h e spot r a t e of  i n t e r e s t t o r i s e t o some high l e v e l , and yet be e q u a l l y l i k e l y t o go f u r t h e r up, as move downwards. c)  Preferably,  the  probability  of  process  should  be  such  that  the  t h e i n t e r e s t r a t e reaching e i t h e r zero  or i n f i n i t y i s i d e n t i c a l l y n i l . d)  Mathematical  tractability.  To ensure t h a t i n t e r e s t r a t e s do not c o u l d adopt one o f two a)  7  8  make  become  negative,  we  approaches:  r=0 a s i n g u l a r boundary  8  with p o s i t i v e d r i f t , i e .  These c r i t e r i a are drawn from I n g e r s o l l £ 3 9 ] .  By d e f i n i t i o n , t h e d i f f u s i o n process as d e f i n e d by equation (2.1) has s i n g u l a r boundaries wherever b(r,t)->°o o r a{r,t)->0.  23  b(0,t)  >  0;  a(0,t) = 0.  T h i s i m p l i e s t h a t once the  i n t e r e s t r a t e reaches z e r o ,  it  changes  only  in  one  d i r e c t i o n ; upwards, b)  r e s t r i c t the process to remain non negative by an a r t i f i c i a l  The second barrier  at  barrier at  r=0.  approach i s more s t r a i g h t forward,  r=0  ensures  that  imposing  A reflecting  the i n t e r e s t r a t e never becomes  negative, and f u r t h e r , i t never remains a t z e r o , except infinitesimal  instant.  However,  once  r  reaches  for  an  zero  the  d i r e c t i o n of i t s change the next i n s t a n t i s known with c e r t a i n t y - s i n c e r cannot it  become negative  can only i n c r e a s e .  arbitrage  T h i s would  opportunity;  efficiency arbitrage  in  a  a  appear  to  present  barrier) a  clear  s i t u a t i o n not c o n s i s t e n t with market  continuous  profit  (due t o the r e f l e c t i n g  opportunity  time  framework.  However,  no  need e x i s t i f the bond v a l u a t i o n  model i s made t o s a t i s f y s u i t a b l e boundary c o n d i t i o n s a t r = 0 Though  it  may  zero,  it  .  seem counter i n t u i t i v e , even i f b (0,t) > 0  and a(0,t)=0, i t does not reaches  9  will  ensure  that  if  the  interest  rate  l e a v e i t and enter the p o s i t i v e r e g i o n  again.  The  behaviour  of the  cannot  be  inferred  by  process  intuition  at alone.  a  singular  boundary  Thus i f we chose a  » We have from equation (2.2): (dB/B) =[ (B, b - B ^ a Z B J / B j d t + (aB /B)dz. At r=0, B i s not z e r o , and i s f i n i t e . Further, s i n c e the i n t e r e s t r a t e process and the bond value process have to be perfectly c o r r e l a t e d , the bond value should a l s o have a r e f l e c t i n q b a r r i e r a t r=0. From the standard r e f l e c t i n g b a r r i e r c o n d i t i o n (see Cox S H i l l e r [ 15 ]) , t h i s r e q u i r e s t h a t B, =0. The instantaneous r e t u r n t o h o l d i n g the bond thus becomes c e r t a i n , as B =0 reduces the c o e f f i c i e n t of dz to z e r o . To ensure that no a r b i t r a g e o p p o r t u n i t y e x i s t s at r=0, the c e r t a i n r e t u r n t o holding the bond should also be zero. Thus we require (  (  24 functional  form  investigate  that  the  has  a  singular  boundary,  we  must  behaviour of the process a t the s i n g u l a r  more r i g o r o u s l y ,  before we can judge the a c c e p t a b i l i t y  1 0  point  of  the  f u n c t i o n a l form of the s t o c h a s t i c s p e c i f i c a t i o n . F e l l e r [25]  s t u d i e d the problem o f c h a r a c t e r i z i n g  the  behaviour of a d i f f u s i o n process at i t s s i n g u l a r boundaries,  by  the  method  readable  has  of  semigroups.  exposition  K e i l s o n [ 41 ]).....  simplified  and somewhat more  of  Feller*s  work  Broadly  speaking  the behaviour of a d i f f u s i o n  process a t a s i n g u l a r the  (A  boundary c o u l d  may  be  be c h a r a c t e r i z e d  found  as one  in  of  following: a)  Natural: from  The  any  boundary  starting  i s inaccessible  point  in  the  in f i n i t e  interior.  It  i n t e r e s t i n g t o note t h a t a n a t u r a l boundary can be inaccessible  and  time is both  absorbing ( i e . as i n the case of the  lognormal process, where zero i s both i n a c c e s s i b l e  and  absorbing). b)  Exit:  the  boundary i s a c c e s s i b l e  once  the  process  reaches  the  i n f i n i t e time boundary,  it  and is  absorbed. c)  Entrance:  in finite  time  from t h e i n t e r i o r , but i f the process s t a r t e d from  the  boundary, finite d)  the  it  boundary i s i n a c c e s s i b l e  would  leave  and  e n t e r the i n t e r i o r i n  time.  Regular: the s i n g u l a r boundary i s  accessible,  and  we  io From economic considerations, i t i s undesireable to have r = 0 as an absorbing boundary, i e . once the i n t e r e s t rate reaches zero, i t never leaves i t .  25 can  further  there  specify  ( i e . absorbing,  s u i t a b l e boundary  3.2  The I n t e r e s t  the  behaviour i t should  reflecting,  e t c . ) by  exhibit imposing  conditions.  Rate Process  Keeping t h e above requirements i n mind, l e t us c o n s i d e r the following  stochastic specification.  &i~ ^ First  note  that  the  Though  tractability The r>  the  realism  (  3.D  are not time dependent.  interest  rate  process  over  i s l o s t , considerable a n a l y t i c a l  has been gained.,  process has the mean r e v e r t i n g  (< jx) ,  the  deterministic the  some  di  parameters  T h i s assumes s t a t i o n a r i t y o f time.  + <rr*  m(jii--r) dt  drift  is  negative  property,  (positive),  movement o f t h e i n t e r e s t r a t e  c e n t r a l tendency.  adjustment towards s t o c h a s t i c part  .  because  this,  that  the  i s always towards J U L  The parameter m c o n t r o l s  To see  so  when  consider  t h e speed o f  only  the non-  o f t h e process f o r t h e moment:  dl~  ~  - m c£t  On i n t e g r a t i o n we have  which the  shows  distance  t h a t t h e l a r g e r m, the more r a p i d the r e d u c t i o n of o f the c u r r e n t  value o f r from the o v e r a l l mean ^ ,  26 f o r a given time i n t e r v a l & . Looking at t h e s t o c h a s t i c term, singular  boundary .  Further,  11  we  we  find  want  the  introducing the  3.3  family  variance  term  not  two f r e e parameters o f the i n t e r e s t r a t e  only  a  the  r=0  of  Again r , but  ( cr , <A) , adds t o the r i c h n e s s o f process.  Boundaries  i s a s i n g u l a r boundary, we need t o i n v e s t i g a t e  behaviour o f t h e process a t r=0 (as well as a t r= <*>) ,  i s s e t out i n Appendix 1 . as  is a  result.  function  i n t e r e s t Rate Process Behaviour jit S i n g u l a r Since  r=0  <A >0, as n e g a t i v e <A  makes a ( r , t ) -> °o as r->0, which i s an u n d e s i r a b l e making  that  The r e s u l t s may be b r i e f l y  This  summarized  follows: 1)  The  process  extensively a)  t o <A = 1/2 has been  corresponding  studied  by F e l l e r [ 2 3 ] and h i s r e s u l t s are  For a l l parameter values,  r= oo i s an  inaccessible  boundary. ,,. b)  At  r=0; when m,/^>0, the boundary can be e i t h e r an  absorbing or When 2mu„ 2)  In  case  natural 3)  reflecting  barrier  2mjx  < <r . 2  £ r , r=0 i s an entrance boundary., 2  c< =1,  we  find  that  both  r=0 and r= <£> a r e  boundaries.  I t was not p o s s i b l e t o i n v e s t i g a t e t h e behaviour a t the s i n g u l a r boundary f o r a r b i t r a r y necessary Appendix 1) .  1 1  when  integrals  could  values not  be  of  c*  as the  evaluated  (see  By c o n t i n u i t y o f behaviour, we c o n j e c t u r e  r= oO i s a l s o a s i n g u l a r boundary.  27 that as in  crt reduces, and  relation  parameter  to  space  <r , 2  2mcorrespondingly  increases  there w i l l e x i s t , a r e g i o n i n  where  r = 0  is  not  an  the  absorbing  boundary . 12  The boundary behaviour of the process f o r values of o( # ^ or 1 is currently being further investigated jointly with Kent B r o t h e r s and David Emanuel. The preliminary r e s u l t s seem to i n d i c a t e t h a t <A = '/z i s the only i n t e r e s t i n g case, where we can have e i t h e r an accessible or inaccessible boundary at r=0, depending upon the values o f the parameters. The i n d i c a t i o n s are that f o r 0< < '/, r=0 i s always a c c e s s i b l e , and f o r ' / < ° < < 1 , r=0 i s always i n a c c e s s i b l e . 1 2  a  x  28  CHAPTER 4: ESTIMATING THE  4. 1  INTER EST RATE PROCESS PARAMETERS  B r i e f Review of P u b l i s h e d Research - i n • Related Areas •• The i n t e r e s t r a t e process s p e c i f i e d i n the previous chapter has  a  continuous  sample  path  over time.  r e c o r d of i t s r e a l i z a t i o n only at d i s c r e t e say d a i l y ' o r weekly o b s e r v a t i o n s . address  is  the  following:  However, we intervals  the  parameters  frn  r  Given  a  what procedure  jx*(T  r <A*\  corresponding  independent  random  a  for obtaining  rate  sequence  of  3.1). realizations  v a r i a b l e s which are i d e n t i c a l l y  estimators  for  P#  or  ?  ,  been  researchers  generalized ( f o r an  Billingsley [3,4]). transition  the  stochastic  extensive For  probabilities  i n such a way  to  1 3  survey  Markov  the  processes  an  , methods with  These methods by  several  literature with  see  stationary  t h e s e g e n e r a l i z a t i o n s are c a r r i e d  that the Markov k e r n e l now  probability  on  respectively,  processes of  of  distributed  , which depends  d e s i r a b l e l a r g e sample p r o p e r t i e s are w e l l known. have  process  to estimate  Q- ranging over a parameter space &  parameter  now  to our s t o c h a s t i c  (equation  a c c o r d i n g to some p r o b a b i l i t y measure P^ unknown  time,  s e t of data p o i n t s ( r ^ ,  does one adopt  s p e c i f i c a t i o n of the p r e v i o u s chapter I n g e n e r a l , when we have  in  The problem t h a t we s h a l l  t=1,...T)> which are o b s e r v a t i o n s on the i n t e r e s t at d i s c r e t e i n t e r v a l s ,  have a  p l a y s the same r o l e  measure i n the case of independent  out as  identically  * I f we r e p r e s e n t the t r a n s i t i o n p r o b a b i l i t y by P (r")> t \z ',s) , t>s, then s-tationarity of the t r a n s i t i o n p r o b a b i l i t y r e q u i r e s t h a t P ( r , t | r , s ) = P ( r ^ , u | r , v ) f o r a l l (u-v) = (t-s) . T h i s i s the time homogeneity c o n d i t i o n . 3  t  s  t  5  v  s  29 d i s t r i b u t e d random v a r i a b l e s . shows  that  maximum  In  likelihood  approach e x h i b i t almost  particular, estimates  Billingsley  based  on the above  a l l the p r o p e r t i e s of s i m i l a r  estimates  i n the independent random v a r i a b l e case.  (See a l s o Roussas  for  estimators  properties  of  maximum  likelihood  [3]  {59]  f o r Markov  processes with d i s c r e t e time and s t a t e s p a c e ) . 1 4  Much of the l i t e r a t u r e (ie.  continuous  on s t a t i s t i c s of d i f f u s i o n  time s t o c h a s t i c processes)  processes  has addressed  i s c a l l e d the problem of o p t i m a l n o n - l i n e a r f i l t r a t i o n .  what  This i s  i n the area of e l e c t r i c a l communications, where we have a s i g n a l (a  s t o c h a s t i c process)  which i s unobservable.  What i s observed  however, i s a " d i s t o r t e d " t r a n s f o r m a t i o n of the s i g n a l , and it  i n f e r e n c e s are to be made about the u n d e r l y i n g s i g n a l .  i s a l a r g e body of l i t e r a t u r e ;  Though  c i t e d above estimation previous likelihood diffusion  that  there has  is a  There  papers of p a r t i c u l a r i n t e r e s t  S i r j a e v [ 64], Ganssler [ 3 0 ] and some therein.  from  of  the  references  nothing s p e c i f i c i n the  direct  bearing  on  the  are  cited  literature problem  of  of parameters of the d i f f u s i o n process s e t up i n the chapter,  Sirjaev £64]  estimators process  asymptotically  are  of  parameters  biased  unbiased).  proves  in  that  the  i n the d r i f t small  term of  samples  any  (though  He shows that o b t a i n i n g c l o s e d form  e x p r e s s i o n s f o r the s m a l l sample b i a s f o r g e n e r a l forms d i f f u s i o n equation  maximum  i s a very d i f f i c u l t  problem.  of  the  I t appears t h a t  K e n d a l l 6 S t u a r t [ 4 2 ] have a l s o shown that the ML estimators are consistent though generally biased. The asymptotic ' normality of the e s t i m a t o r s i s a l s o shown by Anderson S Gocdman [ 1 ] . Lee, Judge & Z e l l n e r [ 4 3 ] provide good coverage of the area of e m p i r i c a l e s t i m a t i o n f o r the d i s c r e t e s t a t e space process. 1 4  30 Novikov [52] has i n v e s t i g a t e d the e s t i m a t i o n i n the  of the  parameter  process  dx = - X-xdt + dz  and  X  found the r e s u l t i n g b i a s i n This  general  is  Omstein-Uhlenbeck'-  r a t e process.  stochastic  processes  distribution shortly),  (we  Ganssler which  the  density  distribution  set of o b s e r v a t i o n s  conjunction  do  with  have  stationary complete kernel. on  unique  to  function set  instead the"  pointed  distribution identification  c o n t a i n any  Finally,  out may, of  of  up the of  the  joint  the  stationary  l i k e l i h o o d of a  Markov  kernel;,  of  parameters  problem f a c i n g one  last  diffusion  equation  of  by in all  [30]  general, the  of  of  parameter estimates.  Ganssler  that  not  It  using  lead  is s e  S Miller  [15]  the  to  parameters i n the  the  Markov  literature  d i f f u s i o n e q u a t i o n s does not upon  us.  area that was  the  in  minimum-distance-method  l i t e r a t u r e d e a l i n g with g e n e t i c s .  Cox  case of  s p e c i f i c r e s u l t s t h a t could be brought t o bear  the e s t i m a t i o n  the  stationary  In c o n c l u s i o n , i t appears t h a t the e x i s t i n g  estimation  e  a  as  chapter f o r  [30] shows t h a t i n the  Wclfowitz [73,74], l e a d s to c o n s i s t e n t however,  nowhere  s h a l l say more about s t a t i o n a r y d i s t r i b u t i o n s  using  probability  was,  process,  5  as the process o u t l i n e d i n the previous  interest  given  the  form  briefly  surveyed was  F e l l e r [24] i n d i c a t e d that (3.1)  with c{ =J4.  resulted  the a by  31 taking  the  discrete  appropriate  time  continuous  birth  and  time l i m i t s .  death  I t was  process  to  its  therefore f e l t  that  t h e r e could p o s s i b l y have been some e m p i r i c a l work on the parameters of b i r t h and which  could  be  Unfortunately, i n a continuous interest  are  brought  death to  processes,  bear  upon  the  joint  4. 2  the  time framework. Immel [ 36 "J  and  The  only  two  transition  probability  Both the  f o r the M.L.  B i l l i n g s l e y f3,4]  of  f u n c t i o n f o r s e t t i n g up  the  p r o p e r t i e s of M.L.  others,  estimates  a)  The  b)  They are c o n s i s t e n t .  c)  The  the  of the Hessian  asymptotic  matrix joint  in  As pointed  the out  asymptotic  can be b r i e f l y s t a t e d as f o l l o w s :  i s a c o n s i s t e n t estimate covariance  support  desirable  e s t i m a t o r s are a s y m p t o t i c a l l y  inverse  process.  Estimation:.  approach to e s t i m a t i o n . and  any  approach  From the above, we see t h a t there i s some literature  of  address the  l i k e l i h o o d f u n c t i o n , given a r e a l i z a t i o n of the  (M.L. ) Method of  problem  papers  Darwin f" 181.  of  problem.  none o f the p u b l i s h e d works addressed the  Maximum L i k e l i h o o d  by  results  our s p e c i f i c  d i s c r e t e parameters case only, but they adopt using  estimating  of  of  unbiased.  matrix the  the  with s i g n s  asymptotic parameters,  distribution  of  reversed variance-  where  the  the  estimated  parameters i s m u l t i v a r i a t e normal.  Given  a  sequence  (r^. ,t=1,.,.T)  short term i n t e r e s t r a t e , the j o i n t  of  observations  likelihood function  on  the  can  be  32  set  up as T  s  t((Uo.\8)  P( r\ j r , , 0 )  where  d e n s i t y , and P initial of  P^el-^.e).  Tl  represents  (r, ) i s the  0  point  of the sample. $  need  to  be  e x p r e s s i o n s i n (4.1) a)  the  corresponding  about  the  .  probability  This  is  is  Two  density  quite  assumed  valid,  i n the p r e v i o u s chapter t h a t  equation  the  :  homogeneous.  assumption  to  [ yn , jx., o~~, °^ ]  case  The t r a n s i t i o n p r o b a b i l i t y d e n s i t y time  probability  here r e p r e s e n t s the parameters  our  noted  (4.1)  transition  probability  the d i f f u s i o n process - i n  points  P.OVj  modelling  to  be  given the  the  diffusion  the i n t e r e s t r a t e process d i s p l a y s  no e x p l i c i t time dependence of the c o e f f i c i e n t s . . b)  The  implication  equally as  in  is  that  the  spaced over time. economic  data  observations  fr^}  are  T h i s poses no r e a l problem,  observations  are  generally  eguispaced. The  joint  likelihood  corresponding to the i n i t i a l further  analysis.  In  of point  general,  forward t o drop the e x p r e s s i o n point i n the j o i n t a)  Hhen  we  the  which  contains poses  several  the  term  problems  with  arguments may  corresponding  to  the  be  put  starting  l i k e l i h o o d of the data: have  a  reasonably  c o n t r i b u t i o n of the i n i t i a l negligible  data  large point  data may  be  sample, the considered  i n comparison t o t h e r e s t of the p o i n t s and  33  may  be dropped  estimation  (see  theory  Billingsley [3]).  In f a c t a l l  the  r e s u l t s are asymptotic r e s u l t s , and  l a r g e sample s i z e s are i m p l i c i t l y assumed. b)  I t i s not  uncommon i n s e v e r a l s i t u a t i o n s to  estimators Following  as  strictly  conditional  such an approach, we  estimators  upon the  could  are c o n d i t i o n a l upon the  argue initial  thus a t t r i b u t e a p r o b a b i l i t y of 1 t o that c)  F i n a l l y , Zellner C 1  reasons  76  the  probability  corresponding  independent of & .  Since  !  estimating  &  ,  distribution initial In  view  (4.1).  To  of the  the  it  our  can  [V  is  be  that  we  to  T|  easily  unaffected  that  the  point,  and  assume that is is  shown by  sample.  point.  may  interest  the  totally only  in  that  the  dropping  the  point.  set up the  ascertain  of  1 6  treat  above arguments, we joint  likelihood  s h a l l drop P  0  function,  t r a n s i t i o n p r o b a b i l i t y density  we  (r )  from  (  need  to  f o r the d i f f u s i o n  process  *  E(dz)  =  0  and  E (dz ) 2  = dt.  (4. 2)  cr r - dz  In general d z . i s assumed to be a  Gauss-Weiner process, i e .  z e l l n e r * s reasoning i s f o r the analysis a u t o r e g r e s s i v e systems i n a Bayesian framework. 1 6  of  first  order  34 4. 3  The  Simple The  Linearization  s p e c i f i c a t i o n of equation  e s t i m a t i o n procedure,  if  we  frequency  now  (4.2)  by l i n e a r i z i n g  (discrete) differences.  and  A£j3roximation  Thus we  suggests a very  simple  the d i f f e r e n t i a l s t o  finite  have  At  choose our u n i t of time such t h a t  =  1 (the  of the o b s e r v a t i o n s on r) we have  where ' Y| /\> N (0,1) . In the l i m i t as characterization  At  0,  the  of the d i f f u s i o n equation  However, the f u r t h e r apart  the  greater  extent  the  approximation chapter.  For  approximation i e , , we  approximation  error.  The  as  the  on error  r  are, due  the  to t h i s  i s i n v e s t i g a t e d by Monte C a r l o methods i n the the (4.4),  have a l i n e a r  present, closely  however,  a  (4.2) becomes exact.  observations of  (4.4)  we  see  next  that  the  resembles a r e g r e s s i o n equation,  r e g r e s s i o n of r^  have a h e t e r o s c e d a s t i c e r r o r term.  on  Thus we  r ^ " ,  wherein  we  have (4, 5)  Given (4.1).  the data, we can now  set up the l i k e l i h o o d f u n c t i o n as i n  The d e t a i l s of the e s t i m a t i o n procedure  Appendix 2.  are s e t  out  in  35 4.4  The T r a n s i t i o n P r o b a b i l i t y  The  exact  probability (eguation  approach  density  4.1).  It  corresponding  to  equations  that  satisfy.  These  Kolmogorov or to  the  Thus, equation  for  to  set  is  known  in  well  every  diffusion  transition  are  the  equation the  (FP) is  up t h e  equation,  probability  forward  the  transition  the  diffusion  1 8  transition function  theory  that  there  exist  density  has  equation  equation.  1 7  the  likelihood  probability  diffusion'equation  our case o f  and  two to the  The  solution  density  function  .  given  by  (4.3)  the  FP  is  F = P ^ r vt)r  function.  ascertain  K o l m o g o r o v backward  Fokker-Plank  to  be t o  it  -JL^()t-T)FJ where  would  Method  and u s e  the  FP  corresponding  Density  To  0  , 6 )  solve  ^JL[<^FJ ^ is  this  the  transition  parabolic  2£  (4.6)  probability  partial  density  differential  The e x i s t e n c e o f u n i q u e s o l u t i o n s t o the forward (FP) ' and backward e q u a t i o n s depends upon t h e d r i f t and v a r i a n c e t e r m s o f t h e d i f f u s i o n e q u a t i o n s a t i s f y i n g some continuity requirements (see Friedman [ 2 8 ] ) . More specifically, i t i s reguired that t h e y be bounded and u n i f o r m l y L i p s c h i t z c o n t i n u o u s i n (r,t) in compact s u b s e t s o f R x [ 0 , T ] , and f u r t h e r , t h a t t h e v a r i a n c e be s t r i c t l y n o n - n e g a t i v e o v e r t h e whole d o m a i n . 1  7  r  I t i s a w e l l known r e s u l t (see F e l l e r [ 2 6 ] ) t h a t the s o l u t i o n t o t h e FP e q u a t i o n a l s o s a t i s f i e s t h e backward e g u a t i o n , except in rare situations where t h e s o l u t i o n i s not u n i q u e . I t has been O b s e r v e d i n t h e l i t e r a t u r e t h a t t h e s o l u t i o n a l s o possesses the properties of a probability density function, i e . the function is s t r i c t l y n o n n e g a t i v e o v e r t h e s t a t e s p a c e , and i t s i n t e g r a l over the state space <1 (these are the ChapmanKolmogorov conditions). If the equality is satisfied, the s o l u t i o n t o t h e FP and backward equation is unique, but in general, different diffusion processes may s a t i s f y t h e same f o r w a r d and backward e g u a t i o n s . 1  8  36  equation, infinity  we  ( i f r=0  are r e q u i r e d of the  need  to  and  on the b a s i s of our  conditions  studied  the  approach  i n v e s t i g a t i o n o f the  f o r g e n e r a l values of^ oC  (4.6)  solution  =0,  I f we  the  In  When  boundary.  corresponding to  suggested  Appendix 3.  case  to  . the  cK =1,  where  straight  pointed  out  t h a t the  the  f o r the  origin  is  no  longer  The  u n r e s t r i c t e d process  virtually  (and  by Vasicek [ 7 2 ] ,  arbitrarily  small,  mass so  detailed  the  an  a  singular  that  the  solution  to  o r i g i n i s quite  (where a p o s i t i v e  in  solution  is  Appendix 4).  parameters c o u l d  below  origin  As  be chosen such could  be  made  f o r a l l p r a c t i c a l purposes, r=0  is  inaccessible.  Steady State o r S t a t i o n a r y We  on  a n  i n t e r e s t r a t e s to remain non-negative,  forward  probability  <A~ - K^*  based  p r o b a b i l i t y of negative i n t e r e s t r a t e s e x i t s ) , the rather  has  Goel 6 Richter-Dyn [ 3 3 ] , i s sketched i n  reguire  but  form s o l u t i o n  case  FP equation with a r e f l e c t i n g b a r r i e r at the  can  probability foregoing values  as  behaviour  F e l l e r [23]  need to impose a r e f l e c t i n g b a r r i e r at r=0.  complicated,  The  and  i n a c c e s i b l e boundaries)  t h e r e appears t o be no c l o s e d  approximate s o l u t i o n  4.5  at r=0  process at these s i n g u l a r boundaries.,  f o r equation  the  boundary  i n f i n i t y are not  Unfortunately,  we  impose  of  stationary transition  see  that  density  the  exponent  density  solving  may  some g e n e r a l i t y  not  for always  i n the o\ :. , He  i n t o the  probability  Density Method  joint  density.  model, could  the  exact  be  possible, ie,,  transition except  restricting  however, s u b s t i t u t e  likelihood  instead  Ganssler [30]  has  of  by the the the  shown that  using  t h i s approach i n conjunction,  estimation  method  of  with  the  Wolf owitz [ 73,74 ],  minimum  leads  parameter e s t i m a t e s , which are a s y m p t o t i c a l l y The the  stationary  limit  interval  probability d i s t r i b u t i o n  observation  tends  to'oO  to c o n s i s t e n t  unbiased., is, in  1 9  of the t r a n s i t i o n p r o b a b i l i t y d e n s i t y , between  distance  .,  a  sense,  where the time It  could  be  represented as  The  existence  is  usually  of an unique steady s t a t e p r o b a b i l i t y d i s t r i b u t i o n assured  when  we  have  a  process that has a time  homogeneous t r a n s i t i o n p r o b a b i l i t y d i s t r i b u t i o n . singular  diffusion  processes,when  Further,  we r u l e out those ranges of  parameters where one o f the s i n g u l a r boundaries a c t s as an barrier,  we  function  e x i t boundary.  exit  ensure that the s t a t i o n a r y d i s t r i b u t i o n i s not the  t r i v i a l P(r) = 0 over the complete s t a t e delta  for  concentrating  with  a  Dirac  a l l the p r o b a b i l i t y mass at the  Thus the s t a t i o n a r y  s o l u t i o n to the FP equation  space,  density  (0.6) by s e t t i n g  is  given  by  the  :r? - 0; OX,  (0.7)  The s t a t i o n a r y p r o b a b i l i t y d i s t r i b u t i o n e x i s t s _ b n l y f o r time homogeneous processes. Another way of r e p r e s e n t i n g the s t a t i o n a r y d i s t r i b u t i o n c o u l d be as f o l l o w s ; Given t h a t the diffusion process has a t t a i n e d i t s steady s t a t e , the s t a t i o n a r y p r o b a b i l i t y d i s t r i b u t i o n then gives the p r o b a b i l i t y o f f i n d i n g the process a t any p a r t i c u l a r p o i n t (or i n t e r v a l ) i n the s t a t e space at any i n s t a n t . 1 9  38 the  solution  Richter-Dyn  to  which  can  shown o f t h e  form  (see  Goel &  [33])  POO  -  JL  (Pf ^  L J  where C i s  d e t e r m i n e d by t h e  represents  integration  Appendix  5  the  It. i s  of  state  where  Si.  space.  details the  (4. 8)  \P(r)dr = 1,  condition  over the  gives  density.  I  Hb\-i[  2  stationary  be  of  evaluation  of  the  when  we  (-4.9A)  and  form  P(f+0 r tel . 1 1 - l L *  J  1+A  it-  where  A, 1 2 c ^ .  take  the  (4.9b)  =  It  -  limit  is  as  respectively.  continuous i n  .  also  shown i n A p p e n d i x  A,-? 0 Thus  or the  -1  in  (4. 9c) ,  steady  5  that  we g e t  state  density  is  39 Given up t h e (4.9), net  realization  and  asymptotic  estimate to  be any  the  i n the  next  crudely  r a t i o n a l i z e d as  resemble b)  If the  the  However,  be t h a t  stationary  of is  of  data  shall  does  to  the  look  at  approach  distribution the that  data  the  may  be  we  are as  should  approach.  2 0  sufficiently sample  were i n d e p e n d e n t , , to  set  treating if  might  .  up  would be e x a c t .  independence the  we have a of  points  Lack  of  if  distribution  random v a r i a b l e s  validity  literature  the  distribution  dependent of  There  on some l i m i t e d Monte C a r l o s i m u l a t i o n  the  stationary  objection  distribution  by ML methods.  We  set  follows:  the sequence  likelihood  stationary  estimators.  could  sample,  the  T) , we p r o p o s e t o  i n the e x i s t i n g  chapter.  argument  large  ...  parameters  such  based  results  One  using  reference  properties of  properties,  a)  (r^. , t = 1 ,  likelihood function  appear  these  a  a  t h e y were hopefully  the  using joint  The c r u c i a l sequence  of  independent. not a l t e r  T h i s may be t r e a t e d  as  if  the we  T h i s r a t i o n a l i z a t i o n c a n be m o t i v a t e d by t h e f o l l o w i n g r e s u l t for Markov p r o c e s s e s (see C i n l a r f 13 ] ) . Consider a continuous t i m e , d i s c r e t e s t a t e s p a c e Markov p r o c e s s which has a s t a t i o n a r y distribution. L e t o b s e r v a t i o n s be made on this process, such that the time interval between o b s e r v a t i o n s i s e x p o n e n t i a l l y distributed. The s e q u e n c e o f observations then represents a discrete time Markov p r o c e s s . It can be shown that this d i s c r e t e t i m e p r o c e s s has t h e same stationary distribution as the continuous time process from w h i c h t h e o b s e r v a t i o n s were taken. Rs t h e number o f observations goes to infinity, the distribution of the sample observations approaches the stationary distribution. The e x p o n e n t i a l sampling scheme was required to ensure that all points on the half real line r e p r e s e n t i n g the t i m e a x i s , were e q u a l l y l i k e l y to be chosen. The e x t e n s i o n o f t h i s r e s u l t t o c o n t i n u o u s s t a t e s p a c e p r o c e s s e s can be found in Dynkin £ 2 1 ] . , We have used equispaced observations, and that should introduce bias, which we c o n j e c t u r e s h o u l d r e d u c e a s t h e number o f o b s e r v a t i o n s i n c r e a s e . 2  0  40 are u s i n g a " b i a s e d " approach ; the depending  upon  how  extent  of  c l o s e the s u c c e s s i v e  "bias"  observations  are. Finally parameters  the steady m  and  s t a t e approach cannot i d e n t i f y  cr2*  separately cr  -  estimated .  Both m and  dimensions.  Thus, using the steady  2 1  approach, we  should  only  two  t h e i r r a t i o can  have time u n i t s as  not expect to  the  part  of  be  their  s t a t e (or time independent)  be  able  to  identify  these  parameters s e p a r a t e l y . To summarize the v a r i o u s aspects of methods, we a)  The  may  the  three  estimating  note the f o l l o w i n g :  transition  p r o b a b i l i t y d e n s i t y approach to s e t t i n g  up the l i k e l i h o o d of the data i s  exact,  but  its  use  r e q u i r e s that we g r e a t l y r e s t r i c t the g e n e r a l i t y of the ds = '/^ o r , i f we choose  model - e i t h e r s e t to  r e c o n c i l e having  a p o s i t i v e p r o b a b i l i t y of  r a t e s becoming n e g a t i v e .  In case o ( = i , we  approximate s o l u t i o n t o the FP equation, i s quite i n t r a c t a b l e f o r estimation b)  The  stationary  i n d e n t i f y m and Further,  probability cr "" 2  o(=0,  -  interest  have only  and  even  an  that  approach  only  their  cannot ratio.  when the data p o i n t s are near each other,  l i k e l i h o o d f u n c t i o n i s probably  have  purposes.  density  separately  we  f a r from exact, as  the the  i n d i v i d u a l o b s e r v a t i o n s are not independent. c)  This Ganssler 2 1  The  simple  linearization  method  (or  normal  approximation) i s very t r a c t a b l e , and  the  closer  our  was [30].  the  results  in  expected  on  the  basis  of  41 data In  points,  the r e a l  how  The  Phillips  approach  to  [2],  Consider  differential  observations  c h a p t e r we of which  B are  [60],  the  system  matrices,  D  is  white n o i s e d i s t u r b a n c e v e c t o r .  Phillips of  stochastic by  (among and  linear  stochastic  fa  (4.10)  the  differential  I s a v e c t o r o f e x o g e n o u s v a r i a b l e s , and  (see. Sargan  term  assume t h a t  E[  of  other  [55,56,57],  4  The  then  in  £(t) ] = 0 and  ^ (t)  solution  to  is a  (4.10)  (4.11)  0  i s a stochastic  ^ ( t ) i s Gaussian we  operator  [60] f o r proof)  0  exists,  This  one  been a d v o c a t e d  -- A |(t| + bzC-t)  pure  Z (t)  last  be.  outline  parameters  has  Sargan  ,  The  can  to  6.  can  the  ct/cit  satisfies  limitations  equations  D^(tl  where A and  are  approximation.  Method  e q u a t i o n s , (SDE)  Hymer [ 7 5 ] .  the  estimation  Bergstrom  i n the  i s d i s c u s s e d i n Chapter  conclude t h i s the  differential others)  spaced  Approximatipn  B e f o r e we  the e r r o r  w o r l d , however, t h e r e  closely  limitation  4.6  the l e s s  can  replace  N(0,_Q.), and the  last  integral, that term  the by  and  i f we  integral  f ( t ) , where  42  o  Thus, we have' £,(t) though^JT-  may  diagonal  problem  be noted  even  been d i a g o n a l , _ Q _ * w i l l have non-zero o f f  have  of  back t o  (4.11), i n the g e n e r a l case where Z (t)  exogenous  in  the  manageable.  In  variables,  way  of  the  the  reducing  special  first  (4.11)  case  where  integral to  and  (4.11) s u i t a b l y reduced.  is  Z(t)  is be  a  poses a  something  d e t e r m i n i s t i c f u n c t i o n of time, the i n t e g r a t i o n can through  that  elements.  Going vector  m ' N{0,]|;*) , and i t may  a  more known  carried  However, when Z(t) i s a l s o  s t o c h a s t i c , no exact equation system can be o b t a i n e d , e q u i v a l e n t to  the  SDE  approximation (4,11) may formula [t,  to  system  (4,10).  P h i l l i p s [57]  method, whereby the i n t e g r a l of Z(t)  be reduced express  (t-h) ] , [ A p p e n d i x  presents in  an  eguation  using a three point Lagrange i n t e r p o l a t i o n Z(t)  as  a  polynomial  in  the  interval  6 presents more d e t a i l s on the a d a p t a t i o n  of t h i s approach t o the SDE  (4.3), which i s  model.] Using t h i s method reduces  (4.11) t o  cur  interest  rate  43  Where  the  E*s  are f u n c t i o n s cf a,  has shown t h a t the approximation superior (4.5)) .  to  the  Phillips  2 2  ;  approximation order  of  discrete (op  (4.13) to  approximation,  cit)  points  scheme l e a d s t o bias  0(h ). 3  But  B and  in  h.  Phillips  the  SDE  (op c i t )  (4.10)  is  (4.4)  and  (equations  out  that  the  proposed  the  parameters  of  i n case Z ( t ) i s not d i f f e r e n t i a b l e at a  countable set of p o i n t s on the r e a l l i n e , the b i a s i s l a r g e r of the order  0(h).  required  get  met.  to  In  the  our  improved  case,  the  regularity  and  condition  e s t i m a t o r s by t h i s approach are not  We s h a l l t h e r e f o r e not pursue t h i s approach f u r t h e r .  In the case where Z(t) i s s t o c h a s t i c , P h i l l i p s requires rather extended d i f f e r e n t i a b i l i t y c o n d i t i o n s on Z ( t ) . Now, i n g e n e r a l , we know t h a t though d i f f u s i o n processes have continuous sample paths, they are nowhere d i f f e r e n t i a b l e . So, the r e g u l a r i t y requirements are q u i t e steep. , P h i l l i p s p o i n t out the superiority vanishes when t h e r e g u l a r i t y requirements are not met. F u r t h e r , as can be seen from Appendix 6, due to presence of r* in the v a r i a n c e element, the r e s u l t a n t equation corresponding t o (4. 13) i s r a t h e r i n v o l v e d . Some attempt was made to estimate the parameters u s i n g the P h i l l i p s (and even the r e l a t i v e l y s i m p l e r Sargan approximation), but non l i n e a r methods to estimate parameters from the l o g l i k e l i h o o d f u n c t i o n s did not r e s u l t i n much success. 2  2  44 CHAPTER 5: COMPARISON OF THE DIFFERENT ESTIMATING METHODS  5.1  The Method o f Comparison In t h i s chapter, a l i m i t e d attempt relative  merits  of  i s made t o  compare  the  the d i f f e r e n t approaches t o e s t i m a t i n g the  parameters o f t h e i n t e r e s t r a t e process  outlined  in  the  last  chapter: a)  The T r a n s i t i o n P r o b a b i l i t y Density Method (TRP)  b)  The Steady  State P r o b a b i l i t y Density Method (SS)  c)  The Simple  L i n e a r i z a t i o n Method (SL)  The  method  adopted  discrete realiztions set  generate  ( a l l eguispaced)  G = (S,^X , (f , d )  parameters from  i s to  Then  with  using  each  a l a r g e sequence o f a  known  parameter  method we e s t i m a t e the  t h i s generated data base, using s e v e r a l samples.  We then look at the d i s t r i b u t i o n of t h e parameters estimated the d i f f e r e n t methods, using v a r y i n q sample Data  f o r t h e s i m u l a t i o n s was generated  by  sizes. ='/x because  using  t h i s i s t h e one case where the t r a n s i t i o n p r o b a b i l i t y d e n s i t y i s known e x a c t l y , and q u i t e t r a c t a b l e .  The r e s t of the  parameters  were chosen by a p p l y i n g t h e TRP method corresponding to o( = '/a. on actual  weekly  interest  r a t e s over the past 18 y e a r s .  year subperiods were taken, each. the  and  {m,^U.,CT)  were  Three 6  estimated  on  The average o f these three e s t i m a t e s was used to generate synthetic  data.  actual interest  data  estimating  methods  The was may  reason that  f o r choosinq  the r e l a t i v e  (m,^t,cr) merits  from  of the  be a f u n c t i o n of t h e parameter v a l u e s .  S i n c e an e x t e n s i v e comparison of the Monte C a r l o r e s u l t s was not done, (mainly due t o  the  large  computing  cost  involved)  we  45 confined  our  attention  to  the neighbourhood of the parameter  values of i n t e r e s t to us. The (for  a  aim of the Honte C a r l o s i m u l a t i o n s particular  sample behaviour  parameter value  is  to  investigate  of the process)  of each of the e s t i m a t o r s .  He  the  small  look f o r answers  t o the f o l l o w i n g q u e s t i o n s : 1)  are the e s t i m a t o r s unbiased  2)  Do they  appear  to  thouqh they may 3)  What  is  the  be  i n s m a l l samples?  asymptotically  unbiased  even  be biased i n s m a l l samples? relative  efficiency  of  the  different  estimators? 4)  Which  estimator  approaches  the  asymptotic  values  fastest? 5)  For  a  qiven  frequency  365  of the data, does i n c r e a s i n g the  o f o b s e r v a t i o n l e a d to any  estimators? usinq  spread  improvement i n the  S p e c i f i c a l l y , i s t h e r e any daily  observations  improvement  rather  in  than 52 weekly  points?  5.2  Generating  an "exact" Sequence f o r the Square Boot  Having chosen the parameter s e t qenerate  synthetically,  T h i s i s very important observed the  first  step  corresponding  of  as we should  estimation.  be a b l e t o a s s e r t  The t r a n s i t i o n  t o the A = J/j. case  ( F e l l e r [23 ] ) :  is  to  d i s c r e t e r e a l i z a t i o n that i s exact.  b i a s i n the parameters e s t i m a t e d , i s a r e s u l t  method  qiven by  a  , the  Process  (the  square  that  any  purely of  probability density root  process)  is  46  j  •o  J V\fo V  (5.1)  w = exp(mt) and 1 ^ (.) i s the modified B e s s e l f u n c t i o n of  where  order k, and i s d e f i n e d by  One  way  uniform  to  generate  r ,  given  0 and r _ , i s t o generate a fc  ( r e c t a n g u l a r ) random v a r i a b l e p on [ 0 , 1 ]  r_j. = C-» (p)  where  C  f u n c t i o n corresponding  is  the  cumulative  t o F(.) i n equation  i n v e r t e d , there would be no problem. structure  of  (5.1).  However  Substituting  <$ = Im^/v1  (S~1) «J; n=2&, and n i s i n t e g r a l , then =:  then  probability  (S-0  If  and  for  set  density  I f C c o u l d be the  special  [ 5 . 1 ] , Boyle [ 8 ] has developed a s o l u t i o n using a  d i f f e r e n t approach.  where  (  (5.3)  l(  where deqrees  * of  s  *^e a o n - c e n t r a l  freedom,  "X  and  chi-squared  from  without  the  central  actual  n  2$ i s  such t h a t  much l o s s o f g e n e r a l i t y ( s i n c e the value of 8 interest  chi-squared  3 0  data was l a r g e ) .  variates  Generating  a non-  random v a r i a b l e i s q u i t e s t r a i g h t f o r w a r d .  (Fishman £ 2 7 ] has d e t a i l e d i n s t r u c t i o n s stochastic  with  i s t h e n o n - c e n t r a l i t y parameter.  Now, we can e a s i l y choose our parameter s e t $ integral,  density  corresponding  probability distributions).  on to  the a  generation  wide  variety  T h i s method was adopted  using  of of the  parameter v a l u e s :  jX  and  r  0. 09517 '/o/iwk  S = -\%\5_.  a sequence o f weekly i n t e r e s t r a t e s (100,000 weeks lonq)  was  generated,,  One way t o generate a n o n - c e n t r a l c h i - s q u a r e random (Y) with (n*1) degrees of freedom, would be:  3 0  Y = Z  2  variable  • £x?  where t h e a r e N{0,1) and Z i s N ( A # 1 ) r ^ being the nonc e n t r a l i t y parameter of t h e c h i - s q u a r e . This r e q u i r e s the qeneration o f (n+1) Gaussian random v a r i a t e s . Another approach is based on the e q u i v a l e n c e of t h e c h i - s q u a r e and Gamma distributions. Osing t h i s approach Y = Z  2  -2 £ l o q ( 0 . ) Ul  where Y and Z are as before, but the U,; • s a r e u n i f o r m ( r e c t a n q u l a r ) on (0,1). T h i s r e q u i r e s only (n/21) random v a r i a t e q e n e r a t i o n s f o r each c h i - s q u a r e v a r i a t e .  48  R e s u l t s of Monte C a r l o S i m u l a t i o n s f o r the <A = '/? (known! Case To  start  s i n c e we  with,  we  want t o compare a l l t h r e e methods,  do not have a s o l u t i o n t o the FP equation f o r a r b i t r a r y  <7x, we have t o assume cV i s known and equal to Yi . °^=/ 2 * /  and  we  When we assume  know the t r a n s i t i o n p r o b a b i l i t y d e n s i t y , and  thus  can  compare a l l t h r e e e s t i m a t i n g methods, based on the p r o p e r t i e s of the  estimated  n = 100, 200  250,  parameters. 500  and  945.  Four  sample  sizes  For the n = 100 and  s i m u l a t i o n s each were performed, i e . , 200  were  estimated.  simulations procedure  as  the  n = 500  and  each were performed., The  200  s e t s of  n = 945  3 1  cases,  (m,yW-,cr)  cases,  100  d e t a i l s of t h e e s t i m a t i o n and  -Yt-w  7.  s i m u l a t i o n s f o r the n=100 case  follows.,  synthetically  used :  n = 250  f o r t h e parameters when d\ i s assumed known  are i n Appendix The  For  were  From  the  generated  s u c c e s s i v e b l o c k s o f 100  sequence  weekly  of  data  (say) were performed 100,000  on  points  interest  rates,  p o i n t s were taken., Using each b l o c k of  100  data p o i n t s , one  s e t of parameters  (m,/^,<r ) was  By  performing  e s t i m a t i o n on 200  s u c c e s s i v e b l o c k s , we  200  estimates of the parameters.  these  200  the  parameter e s t i m a t e s  of the parameter estimate) deviation, SD  .  If  and the  in  the  Monte  of  2  The standard  reported Carlo  deviation  (which r e p r e s e n t t h e  i s c a l l e d the  estimated.  Monte  get  across  distribution  Carlo  standard  simulation results i s c a l l e d  distribution  of  the  estimated  Data was generated u s i n g 1 week as the u n i t of time. Thus the s e l e c t e d sample s i z e s correspond t o 2,5,10 and 18 years of weekly data. Actually n=945 was chosen as t h a t was the e x a c t number of weekly data points on the s h o r t term i n t e r e s t rate between January 1st, 1969 and December 31st, 1976. 3 1  49 paramters  were Gaussian,  as i n d i c a t e d by asymptotic  mean and standard d e v i a t i o n should convey about t h e d i s t r i b u t i o n . Carlo  distribution  theory, the  a l l the  information  To cover the p o s s i b i l i t y t h a t the Monte  might  not  be  exactly  Gaussian  , t h e 10  p e r c e n t i l e and 90 p e r c e n t i l e values are a l s o r e p o r t e d .  Purther,  c o r r e s p o n d i n g t o each s i m u l a t i o n , we not only parameter estimates,  get  summary  asymptotic  results  reported,  SD^  so  3 2  standard d e v i a t i o n computed f o r each t r i a l .  the  median  was  r e p r e s e n t a t i o n of l o c a t i o n . Method  theory .  reported  In  r e f e r s t o the mean o f the  cases t h e mean SD^, i s very high due t o a few and  set of  but a l s o a s e t o f estimates o f the standard  d e v i a t i o n o f the parameters, based on asymptotic the  one  In  extreme  instead,  as  F i n a l l y , the Steady  an  some  values , 3 3  alternate  State  Density  cannot i d e n t i f y the paramters m and ( T s e p a r a t e l y - only 2  a composite  (2m/ ) i s e s t i m a t e d .  the  methods,  three  2  <r  the  To be able t o compare  across  value of (2m/cr ) was computed i n each 2  I f L=log of t h e j o i n t l i k e l i h o o d f u n c t i o n (corresponding to a given s e t o f data p o i n t s ) , then the matrix of second p a r t i a l d e r i v a t i v e s o f L with r e s p e c t t o t h e paramters, a t t h e maximum of L, may be c a l l e d t h e Hessian Matrix. The i n v e r s e o f the Hessian matrix with s i g n s reversed i s an e s t i m a t e of the v a r i a n c e - c o v a r i a n c e matrix o f t h e estimated parameters, based on asymptotic theory (see B i l l i g s l e y [ 3 ] ) . The standard d e v i a t i o n s are the square r o o t s o f t h e d i a g o n a l elements of t h e v a r i a n c e c o v a r i a n c e matrix. 3 2  Extreme values do not n e c e s s a r i l y imply that t h e s e a r e nonrepresentative the Monte Carlo method gives a representation of the t r u e d i s t r i b u t i o n . . However, i n t h e TRP method, n o n l i n e a r o p t i m i z t i o n r o u t i n e s had t o be used to f i n d the parameter s e t t h a t maximizes the l i k e l i h o o d f u n c t i o n . I n such r o u t i n e s , convergence i s assumed to have been! a t t a i n e d when the r e l a t i v e change i n t h e parameter values between s u c c e s s i v e iterations i s less than a s p e c i f i e d accuracy l e v e l . I f the l i k e l i h o o d f u n c t i o n i s very peaked, then i t s second derivative can change a l o t around the optimum p o i n t . T h i s c o u l d l e a d to extreme values of SD- . 3 3  case f o r the other two its  distribution  methods, and  are  also  the  summary  tabulated.  statistics  Tables I I  of  through  present summary s t a t i s t i c s on t h e d i s t r i b u t i o n of the  V  estimated  paramters. From the t a b u l a t e d r e s u l t s , the f o l l o w i n g broad can  conclusions  be drawn: 1)  There  is  methods  little in  or  the  no  difference  estimated  means  of  a c r o s s the three the  parameter  distributions, 2)  The  dispersion  measured by S D proxy  of  the  (which  m c  parameter  could  f o r the asymptotic  be  treated  deviation,  a  measure  as  the asymptotic  of  the  However, i f SD^ asymptotic  The  SL method g r o s s l y o v e r e s t i m a t e s  v a r i a n c e , (SD-  method appears to perform SD;,  value  s i z e s g r e a t e r than 3)  The  i s much l a r g e r than S D  quite  c l o s e to SD,„  W  )  The the  f o r sample  C  500. unbiased  u  even  in  samples - a t l e a s t f o r the number of s i m u l a t i o n s  performed. SS  is  it.  rather well - i n fact  parameters jx and <r appear to be  small  is  standard  whereas the SS method g r o s s l y underestimates  median  good  t h e r e i s a f a i r amount of d i f f e r e n c e a c r o s s  the three methods.  TBP  a  as  standard d e v i a t i o n ) i s almost  i d e n t i c a l a c r o s s the three methods. evaluated as  distribution  However, the parameter m (or 2mA-  method)  overestimated  2  is  biased  in  small  samples.  in  the  It  is  by a l l methods, and the extent of b i a s i s  n e a r l y the same a c r o s s the roughly i n v e r s e l y  three  methods  p r o p o r t i o n a l to n ) .  (and  seems  TABLE I I ESTIMATE OF m BY DIFFERENT METHODS FOR a = h (KNOWN) CASE TRUE VALUE m = 0.0077617  METHOD  n=100  n=250  n=500  n=945  Simple L i n e a r i z a t i o n Method  Mean 10% Median(50%) 90% SDmc SDi Trials  0.05884 0.01328 0.04891 0.11053 0.04458 2.58618 200  0.02694 0.00776 0.02205 0.05294 0.02176 0.10596 200  0.01559 0.00599 0.01372 0.02490 0.01024 0.60140 ,100  0.01211 0.00659 0.01060 0.08179 0.00500 0.39186 100  Transition Probability Method  Mean 10% Median(50%) 90% SDmc SD.i Trials  0.06205 0.01348 0.05002 0.11646 0.04874 0.02682 200  0.02773 0.00781 0.02204 0.05524 0.02298 0.06244* 199  0.01577 0.00596 0.01382 0.02538 0.01051 0.00506 100  0.01219 0.00663 0.01078 0.01903 0.00510 0.00268 100  Density  * The mean i s h i g h , but t h e median was 0.00586 and 90% i l e was 0. 01797. on page -. Fora d e s c r i p t i o n o f SD and SD- see t e x t page mc i  See  footnote  in  text  1  cn  TABLE 111 ESTIMATE OF U BY DIFFERENT METHODS FOR a = % (KNOWN) CASE TRUE VALUE u = 0 .09517  METHOD  Simple L i n e a r i z a t i o n Method -  Transition Probability Method  Steady S t a t e Method  Density  Density  >  n=100  n=250  n=500  n=945  Mean 10% Median(50%) 90% SDmc SDi* Trials  0.06803 0.05945 0.08919 0.12481 0.23261 0.20554 200  0.09114 0.06439 0.09213 0.12402 0.05958 0.36961 200  0.09091 0.07137 0.09006 0.11522 0.01866 0.50491 100  0.09371 0.07701 0.09390 0.10948 0.01275 0.41076 100  Mean 10% Median(50%) 90% SDmc SDi Trials  0.11303 0.06194 0.08984 0.12807 0.21429 0.02841 200  0.09801 0.06487 0.09251 0.12555 0.05050 • 0.07511 199  0.10410 0.07409 0.09077 0.11607 0.11204 0.09782 100  0.09370 0.07698 0.09390 0.10949 0.01275 0.01272  Mean 10% Median(5 0%) 90% SDmc SDi Trials  0.09266 0.06562 0.09007 0.11926 0.02193 0.00095 200  0.09359 0.07066 0.09117 0.11661 0.01792 0.00087 199  0.09349 0.07423 0.09205 0.11382 0.01454 0.00078 100  0.09358 0.07858 0.09334 0.10853 0.01207 0.00061 100  * The S D i f i g u r e s a r e n o t t h e means b u t medians. The mean S D i was v e r y h i g h due to a few exceptionally high values. The mean o f S D i ranged from 5.305x10^ f o r n=100 to 163.51 f o r n=500, and 0.568 f o r n=945. The i n d i c a t i o n i s t h a t , even t h e SL method, SD^^ can have extreme v a l u e s . cn  TABLE IV ESTIMATE OF O  2  BY DIFFERENT METHODS FOR g = h (KNOW) CASE TRUE VALUE 0  (All  Transition. Probability Method  - 4  f i g u r e s i n t h eT a b l e have been M u l t i p l i e d by a f a c t o r of 1 0 ) 4  IffiTHOD  Simple L i n e a r i z a t i o n Method  = 0.78427 x 1 0  n=100  n=250  n=500  n=945  Mean 10% Median(50%) 90% SDmc SDi Trials  0.78414 0.65251 0.77073 0.91753 0.11073  0.79242 0.69203 0.79157 0.88499 0.07316  -  -  0.79721 0.73421 0.79707 0.87423 0.05619  0.79443 0.75482 0.78882 0.83654 0.03402  200  200  100  100  Mean 10% Median(50%) 90% SDmc SDi Trials  0.82021 0.66768 0.81653 0.97041 0.13434 0.14156 200  0.81073 0.70148 0.81141 0.90429 0.07720 0.17047 199  0.80820 0.74135 0.80820 0.88491 0.05714 0.06016 100  0.80324 0.76374 0.79685 0.84169 0.03439 0.07423 100  ,..  —  _  TABLE V ESTIMATE OF 2m/o --  METHOD  2  BY DIFFERENT METHODS FOR a = h (KNOWN) CASE TRUE VALUE 2m/a = 194.389 2  n=100  n=250  n=500  n=945  Mean 10% Median(50%) 90% SDmc Trials  1547.22 319.51 1235.32 3129.81 1225.81 200  687.87 196.49 557.23 1342.18 573.54 200  393.72 144.13 327.99 671.70 266.59 100  305.90 169.00 263.69 469.88 128.77 100  Transition Probability Method  Mean 10% Median (50%) 90% SDmc Trials  1494.77 331.29 1216.86 2963.03 1121.05 200  679.22199.66 ' 550.43 1317.36 540.30 199  390.06 143.22 328.85 669.54 260.03 100  303.83 164.43 265.43 467.26 127.32 100  Steady S t a t e D e n s i t y Method  Mean 10% Median(50%) 9,0% SDmc SDi Trials  1482.59 449.55 1256.65 2926.93 1067.60 209.80 200  677.70 242.43 528.45 1281.06 506.45 60.70 199  390.93 153.33 330.29 640.97 247.17 24.78 100  302.11 158.44 260.19 481.75 129.21 13.94 100  Simple Method  Linearization  55 The  consistent  overestication  of  m  (or 2m/<r i n the  consideration.  To  an  2  method) needs  some  anticipated,  based  Novikov [ 5 2 ] ,  He do f i n d t h a t as the sample s i z e i n c r e a s e s , a l l  methods  reduced  show  conjectured  that  propose a form  on  bias.  Based  of  on  was  S i r j a e v [64]  and  this,  it  could  be  L e t us  f o r the b i a s as f o l l o w s :  m  +  i s the estimate of m  represents  results  this  t h e b i a s a s y m p t o t i c a l l y goes t o zero.  rn  where m^  the  extent,  SS  its  true  value;  r e s u l t s f o r n=100,250,500 and the b i a s s t r u c t u r e proposed  _JL_  (5.4)  using  a  sample  of  size  c and d are c o n s t a n t s . 94 5, the value of  above was  d  n,  m~  Using the  that  fits**  estimated as 1<d<1.1  .  Based on these r e s u l t s , the sample s i z e s r e q u i r e d t o reduce the  bias  36090. process  on  estimate  To assume t h a t are  unreasonable. how  the  important  constant  the over  of  m t o 10% i s 4450, and  parameters such  of  the  to  111s  interest  rate  l a r q e time p e r i o d s , would be  The n a t u r a l q u e s t i o n to ask t h e r e f o r e  would  i s i t t o get an accurate estimate of m?  be;  For our  * For the 4 values of n, we have { £ - ro) from the Monte C a r l o results (where the mean o f the Monte C a r l o s i m u l a t i o n was taken as ). The crude method adopted was t o choose a value o f d, and corresponding to t h a t value, compute the values of c using equation (5.4). T h i s was done on the f o u r means of the Monte C a r l o v a l u e s o f m. The a p p r o p r i a t e n e s s of d was decided by observing the computed v a l u e s of c. I f the values of c d i d not exhibit a t r e n d from n=100 t o n=945, i t was assumed t h a t c was beinq observed with a random e r r o r . This f i t t i n q approach was t r i e d on the estimated m v a l u e s by S.L. and TRP methods. d=1.1 appears to qive the best f i t , and the c o r r e s p o n d i n g value of c i s approximately 8.0. 3  h  purposes,  the d e c i d i n g c r i t e r i o n must be  the  bond v a l u a t i o n , f o r a given e r r o r i n m.  error  caused  in  This i s investigated i n  a subsequent s e c t i o n o f t h i s c h a p t e r . accepting an  the  intuitive  occurrence.  f a c t t h a t m w i l l be overestimated,  reason  that  Consider  the  could  be  diagram  used  the mean of the process. process  to  to " s t r a y " away from i t s mean.  subperiods  parameters u s i n g c h a p t e r , we by  the  of  one  reversion  Thus, the higher m i s , the l e s s  above, l e t the 4 segments (represented refer  process.  be r e c a l l e d t h a t m r e p r e s e n t s the speed  the  this  t)oi-(A. Z  ])o>ij<?c  is  explain  below, which i s supposed to  r e p r e s e n t one r e a l i z a t i o n of the i n t e r e s t  I t may  to  there i s  by  of  the t o t a l sample.  of  the  methods  data  In the 1  I f we  proposed  lines.  m  diagram 4)  estimate the in  the  last  as shown  In sub p e r i o d 1 (Data 1), the p r o c e s s i s  seen as moving upwards and then somewhat s t a b i l i z i n g . o v e r e s t i m a t e s jUo .  likely  through  might expect JU. i n each case t o be estimated  broken  to  is  also  overestimated,  as the  Thus y~\ process  57 appears t o be moving r a p i d l y towards the In  subperiod  t h e process mean. process  level,  i s obviously  jtf.7,  perceived  does  (^2.) v  not s t r a y away from the p e r c e i v e d mean  reasoning f o r t h e overestimate of m, but underestimate  in  subperiod  3,  i s exactly  as  that  proposed  1 : the mean being p e r c e i v e d i s jXi, and t h e pulled  as  Here a g a i n , m w i l l be h i g h l y overestimated as  The  being  (/*i).  mean  2, t h e i n t e r e s t r a t e process remains more or l e s s  constant around a s i n g l e  the  perceived  toward  of pL  f o r subperiod  process  i t due to a high value of m.  is  rapidly  Finally, i n  subperiod 4, the process mean i s probably p e r c e i v e d at ju^ , but here  m  will  not be as h i g h l y overestimated as i n t h e p r e v i o u s  t h r e e subperiods. either  side  S i n c e the process appears  t o wander a b i t  o f the p e r c e i v e d mean, a lower value o f m (than i n  p r e v i o u s cases) would be estimated.  From the above, we see t h a t  yu-is sometimes overestimated, and at other times On average, However,  to  i t s estimate  in  might  be  expected  underestimated.  to  be  unbiased.  almost every s i t u a t i o n , m c o u l d be over-estimated.  I f now, the complete data were employed, i t i s easy to see why JLL might be q u i t e a c c u r a t e l y estimated. data  convey  the  Furthermore,  the  complete  i n f o r m a t i o n t h a t t h e process c o u l d s t r a y away  from the mean f o r r a t h e r long s p e l l s , which i n d i c a t e s  a  weaker  f o r c e p u l l i n g towards t h e mean - m would be estimated nearer i t s t r u e value., Before  we  present  further  r e s u l t s on the s i m u l a t i o n s , a  minor m e t h o d o l o g i c a l point needs t o be employed  f o r the  clarified.  One  method  s i m u l a t i o n s was t o take s u c c e s s i v e b l o c k s o f  o b s e r v a t i o n s from the long sequence t h a t had been generated.  The  objection  to  this  approach  synthetically could  be  that  58 s u c c e s s i v e t r i a l s were not this  strictly  o b j e c t i o n , f o r t h e n=945 case  of primary i n t e r e s t sample),  100  generated.  independent.  t o us, as t h a t i s the l e n g t h of  "independent"  The s t a r t i n g  which  in  samples  of  size  Steady S t a t e method "dependent"  and  of  SL  the  i n f e r e n c e s from  were  does  on  the  purpose,  700  "daily"  process  be  that  are the  alter  (keeping constant the spread  and  the  e q u i v a l e n t r e s u l t s corresponding t o  over  improvement.  To compare, parameters were  observations,  the  whether using more frequent  " d a i l y " o b s e r v a t i o n s were generated  parameter v a l u e s .  for  materially  time of the t o t a l o b s e r v a t i o n s ) l e a d s t o any this  35  The  experiments.  The next point i n v e s t i g a t e d was observations  not  to  was  reasonable  samples case, and the r e s u l t s  samples  Monte C a r l o  were  100 samples  compared  The c o n c l u s i o n appears  "dependent"  each  (a  one  actual  i s a gamma d i s t r i b u t i o n .  method  and "independent"  presented i n Table VI. use  case  our  945  point f o r each of these  this  counter  (which happens t o be the  randomly chosen from the s t a t i o n a r y d i s t r i b u t i o n approach),  To  For  f o r the same  estimated  using  r e s u l t s compared with the weekly  observations.  The  r e s u l t s are presented i n Table V I I . Comparison  among  the  3  methods  shows  p e r c e p t i b l e improvement i n the mean of the distributions,  but  estimated  (as expected) the d i s p e r s i o n reduces  " d a i l y " observations.  Thus i t appears  of  data,  collecting  t h a t there i s no  daily  parameter by using  that the i n c r e a s e d e f f o r t  pays o f f by lower v a r i a n c e s on  the  The TEP method was not investigated, as it was computationally expensive. Since the o b j e c t i v e i s only to get an i d e a of the e f f e c t , i t was felt t h a t the extra cost was unnecessary.... 3 5  TABLE VI COMPARISON OF MONTE CARLO RESULTS ON PARAMETER ESTIMATION USING SERIALLY DEPENDENT/INDEPENDENT SAMPLES (SAMPLE SIZE n= 945, a^j KNOWN)  METHOD  Simple Linearization Method  Steady S t a t e Density Method  2m/a (194.389) DEPNDT INDEP  M (0.09517) DEPNDT INDEP  m (0.007162) DEPNDT INDEP  a ( 0 .7 8 4 3 x l 0 ~ ) DEPNDT INDEP  0.09371 0.07701 0.09390 0.10948 0.01275 0.41076 100  0.09404 0.08186 0.09246 0.10853 0.01093 0.36074 100  0.01211 0.00659 0.01060 0.01879 0.00500 0.39186 100' •  0.79443 0.75482 0.78882 0.83654 0.03402  0.78438 0.73083 0.78380 0.82981 0.03806  100  100  0.09358 0.07858 0.09334 0.10853 0.01207 0.00061 100  0.09398 0.08010 0.09292 0.10887 0.01088 0.00060 100  2  Mean 10% 50% 90% SDmc SDi Trials  305.90, 169.00 263.69 469.88 128.77  324.56 154.67 300.35 536.65 159.80  100 -  100  Mean 10% 50% 90% SDmc SDi Trials  302.11 158.44 260.19 481.75 129.21 13.94 100  320.29 175.30 281.99 534.57 149.10 14.78 100  -  0.01267 0.00597 0.01172 0.02168 0.00618 0.40281 100  2  4  _  inn I!Sr i / t h e r e s u l t s o f u s i n g a sequence o f b l o c k s (n=94S) o f d a t a p o i n t s from t h e 1UU,000 l o n g sequence o f s y n t h e t i c d a t a g e n e r a t e d f o r t h e Monte C a r l o s i m u l a t i o n s . P  ,  r e p r e S e n t S  t  h  e  e  p  r  e  s  e  n  t  S  "INDEP" r e p r e s e n t s r e s u l t s o f u s i n g " i n d e p e n d e n t " samples (see t e x t , page  for details).  on  TABLE VII COMPARISON OF RESULTS OF ESTIMATION USING WEEKLY & DAILY DATA ( a = h KNOWN) (For Weekly Results n<=100, and For Dally Results n=700)  2mlO  METHOD  DAILY  WEEKLY  200  0.11303 0.06194 0.08984 0.12807 0.21429 0.00534 200  0.80413 0.74359 0.80324 0.87379 0.04565 0.04723 100  1482.59 449.55 1256.65 2926.93 1067.60 209.80 200  0.09088 0.06706 0.08987 0.11629 0.01934 0.00038 100  0.09266 0.06562 0.09007 0.11926 0.02193 0.00095 200  200  Mean 10% Median(50%) 90% SDmc SDi* Trials  1281.26 274.95 1100.19 2396.25 1008.27  1494.77 331.29 1216.86 2963.03 1121.05  100  represent  DAILY  0.10397 0.06194 0.09013 0.12699 0.09201 0.00589 100  100  Reported f i g u r e s  WEEKLY  0.80031 0.73697 0.79943 0.87289 0.04648  1547.22 319.51 1235.32 3129.81 1225.81  1278.87 Mean 298.54 10% Median(50%) 1132.24 2240.06 90% 922.98 SDmc 68.41 SDi 100 Trials  DAILY  0.06803 0.05945 0.08919 0.12481 0.23261 0.20554 200  1276.42 276.04 1097.32 2416.77 1040.43  Steady State Density Method  111  0  0.10178 0.06192 0.09012 0.12695 0.07533 2.21445 100  Simple Mean Linearization 10% Median(50%) Method 90% SDmc SDi Trials Transition. Probability Density Method  *  V  1  (True value:194.389) (True value:0.09517) (True value: 0.7843x10-4) (True value:0.007162)  m e d i a n s o f SDj^ a n d n o t  t h e mean.  -  WEEKLY  DAILY  WEEKLY  0.78414 0.65251 0.77073 0.91753 0.11073 200  0.05037 0.01088 0.04206 0.09479 0.03979 6.32839 100  0.05884 0.01328 0.04891 0.11053 0.04459 2.58618 200  0.82021 0.66768 0.81653 0.97041 1.13434 0.10974 200  0.05111 0.01098 0.04237 0.09523 0.03963 0.01007 100  0.06205 0.01348 0.05002 0.11646 0.04874 0.01548 200  -  61 parameter estimates. parameters  for  However, our  interest i s  valuation.  Therefore,  bond  in  in  dispersion  of  estimated  bond  addressed l a t e r on i n t h i s s e c t i o n . be noted when we interest  rate  process  would be of the 1)  following  reduction  This  data  question  point on  Even  needs to  the  actual  measurement e r r o r s w i l l occur.  They  types: at  which  some  the mean of a  bid  and  specific  Quoted  ask  rates  price,  ie.  if  the  daily  rate  were  based  on  specific  transactions, a l l transactions  would not  be e x a c t l y  hours  would not  be  apart  daily be  is  market p r i c e s .  as r e q u i r e d  3)  comparable  occured, would be a v a i l a b l e .  are g e n e r a l l y not  dispersion  However, one  Normally no exact d a i l y r a t e transaction  2)  -  that  i n the  values.  attempt to c o l l e c t d a i l y  these  the question  needs to be answered i s whether t h i s r e d u c t i o n of parameter extimates would t r a n s l a t e t o  using  i e . , observations to s i m p l i f y  our  estimation  24  equi-spaced  process.  data, the r e l a t i v e magnitude of t h i s e r r o r  In could  high.,  Due  to the presence  daily  of  week-ends  s e r i e s of i n t e r e s t r a t e s has  c o r r e s p o n d i n g weekly s e r i e s . holiday,  as  over  a  d a i l y data s e r i e s and that  and  such  holidays,  more " h o l e s " than a  Every  time  there  is  weekend, c o n t i n u i t y i s l o s t we  occurences  have a are  less  the  gap.  It  likely  is in  a  in a  obvious a weekly  series. A l l the above f a c t o r s would tend to d i m i n i s h daily  series.  In  Appendix  8  we  outline  a  the value of a very  brief  62 investigation  of  the  impact o f a s p e c i f i c form o f measurement  e r r o r . ,• F i n a l l y we look a t the impact  of the  distribution  parameter estimates on the v a l u a t i o n o f pure This  i s crucial,  as  our  primary  the impact  There i s l i t t l e  reason  bonds *. 3  i n t e r e s t i n e s t i m a t i n g the  parameters i s t o use them t o value bonds. investigate  discount  of the  For  simplicity,  we  on the v a l u a t i o n of pure d i s c o u n t bonds. to  believe  that  the  results  on  the  v a l u a t i o n of other types of bonds should be any d i f f e r e n t , s i n c e a  coupon bond, f o r example, may be thought  o f as a p o r t f o l i o o f  d i s c o u n t bonds o f v a r y i n g maturity. Tables VIII t o X present the " t h e o r e t i c a l "  sensitivity  pure d i s c o u n t bond v a l u e s t o e r r o r s i n t h e paramter values.  of The  e x p r e s s i o n " t h e o r e t i c a l " s e n s i t i v i t y i s used only t o d i s t i n g u i s h these  results  from  those  w i l l be presented s h o r t l y . changes  in  the  called "empirical" s e n s i t i v i t y "Theoretical" sensitivity refers  v a l u e of bonds due t o a c e r t a i n f i x e d  e r r o r i n one parameter a t a time, while " e m p i r i c a l " refers  to  estimated  the  distribution  37  from  to  l e v e l of  sensitivity  o f bond v a l u e s r e s u l t i n g  j o i n t d i s t r i b u t i o n o f the parameters  Carlo experiments .,  that  from the  the  We can draw the f o l l o w i n g i n f e r e n c e s  Monte from  * The value of a discount bond was computed using I n g e r s o l l s [39] solution. The procedure adopted i s as f o l l o w s . C o n s i d e r t h e Monte C a r l o s i m u l a t i o n f o r the n=945 ( known) case. Here, we have generated 100 e s t i m a t e s of the parameter s e t (m, p., <r ) • Corresponding t o each estimated parameter s e t , we can compute the value o f a pure d i s c o u n t bond (for d i f f e r e n t times t o maturity and c u r r e n t v a l u e o f r) . Thus using the 100 e s t i m a t e s of (m, /JL , a~ ), we get 100 bond values ( f o r each chosen maturity and c u r r e n t v a l u e o f r ) . T h i s r e p r e s e n t s t h e " e m p i r i c a l " distribution of bond v a l u e s r e s u l t i n g from the estimated j o i n t d i s t r i b u t i o n o f the parameters. 3  ,  3 7  2  z  TABLE VIII THEORETICAL  SENSITIVITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN m  ERROR IN m  CURRENT INTEREST  r  =2u  100%  50%  25%  10%  0%  %  TIME TO MATURITY IN YEARS  BOND PRICE  BOND PRICE  ERROR  BOND PRICE  ERROR  BOND PRICE  ERROR  BOND PRICE  ERROR  1 3 5 7 10  97.13 90.01 82.38 74.99 64.86  97.09 89.82 82.09 74.65 64.51  -0.0378 -0.2024 -0.3506 -0.4504 -0.5332  97.04 89.57 81.71 74.22 64.09  -0.0928 -0.4796 -0.8094 -1.0202 -1.1865  96.96 89.21 81.20 73.67 63.56  -0.1797 -0.8808 -1.4304 -1.7568 -1.9985  96.80 88.65 80.48 72.94 62.90  -0.3376 -1.5046 -2.3003 -2.7264 -3.0207  1 3 5 7 10  95.17 86.22 78.13 70.80 61.09  95.17 86.33 78.12 70.79 61.07  -0.0000 -0.0014 -0.0060 -0.0132 -0.0271  95.17 86.22 78.12 70.78 61.05  -0.0001 -0.0034 -0.0135 -0.0294 -0.0590  95.17 86.22 78.11 70.77 61.03  -0.0002 -0.0061 -0.0233 -0.0492 -0.0962  95.17 86.21 78.10 70.75 61.00  -0.0003 -0.0100 -0.0360 -0.0730 -0.1381  1 3 5 7 10  91.38 79.12 70.27 63.12 54.19  91.45 79.44 70.76 63.67 54.73  0.0756 0.4017 0.6869 0.8669 0.9928  91.55 79.88 71.40 64.37 55.40  0.1856 0.9560 1.5974 1.9820 2.2346  91.71 80.52 72.28 65.30 56.26  0.3599 1.7666 2.8513 3.4555 3.8197  92.00 81.53 73.54 66.56 57.38  0.6777 3.0477 4.6514 5.4529 5.8867  %  %  (Tl  CO.  T A B L E IX THEORETICAL  S E N S I T I V I T Y OF PURE DISCOUNT- BOND P R I C E S TO ERRORS  IN  y  ERROR IN VI  -25% CURRENT INTEREST  r=y/2  r=y  r=2y  BOND  %  BOND  %  BOND  BOND  %  BOND  %  PRICE  ERROR  PRICE  ERROR  PRICE  ' PRICE  ERROR  PRICE  ERROR  1  97.34  0.22  97.17  0.04  97.13  97.09  -0.04  96.92  -0.22  3  91.42  1.57  90.29  90.01  89.73  -0.31  88.61  -1.55  5-  85.33  3.58  82.96  0.31 0.71  82.38  81.80  -0.70  79.53  -3.46  7  79.43  5.93  75.86  1.16  74.99  74.13  -1.15  -5.60  10  71.20  9.78  66.08  1.88  64.86  63.66  -1.85  70.79 59.08  1  95.38  0.22  95.21  0.04  95.17  95.13  -0.04  94.96  -0.22  3  87.57  1.57  0.31  86.22  3.58  0.71  78.13  -0.70  84.89 75.42  -1.55  80.93  85.95 77.58  -0.31  5  86.49 78.68  7  75.00  5.93  71.62  1.16  70.80  69.99  -1.15  66.84  -5.60  10  67.06  9.78  62.24  1.88  61.09  59.96  -1.85  55.65  -8.91  1  91.58  0.22  91.42  0.04  91.38  91.34  -0.04  91.18  -0.22  3  80.36  1.57  79.37  0.31  79.12  78.88  -0.31  77.90  -1.55  5  72.79  3.58  70.77  0.71  70.27  69.78  -0.70  67.84  -3.46  7  66.87  5.93  63.85  1.16  63.12  62.40  -1.15  59.59  -5.60  10  59.49  9.78  55.21  1.88  54.19  53.19  -1.85  49.37  -8.91  T I M E -TO MATURITY IN  +25%  +5%  0%  -5%  YEARS  -8.91  -3.46  CTl  TABLE  X  T H E O R E T I C A L S E N S I T I V I T Y OF PURE DISCOUNT BOND P R I C E S TO ERRORS IN a-  ERROR  CURRENT INTEREST  r=y/2  r= . p  r=2p  0%  -5%  -25%  INO  BOND PRICE  BOND  %  ERROR  PRICE  ERROR  0.0000 0.0007 0.0022  97.13 90.01  0.0002  82.38  97.13 90.01 82.38  0.0109  74.99 64.86  0.0044  -0.0084  74.99 64.86  82.39 75.00  0.0084  64.89  BOND  %  BOND  %  BOND  PRICE  ERROR  PRICE  ERROR  PRICE  1  97.13  -0.0002  3  90.00 82.37  -0.0033  97.13 90.01 82.38  -0.0000 -0.0007 -0.0022  97.13 90.01  74.97  -0.0220 -0.0422  74.98  -0.0044  64.85  T I M E TO MATURITY IN YEARS  5  -0.0109  +25%  +5%  '  %  0.0033 0.0219 0.0420  7 10  64.83  1  95.17  95.17 86.22  95.17  -0.0010  86.22  95.17 86.22  0.0001 0.0010  95.17  86.22  -0.0003 -0.0052  -0.0001  3 5  78.12  -0.0154  78.13  -0.0031  0.0031  78.14  70.78  70.80 61.08  -0.0057 -0.0101  70.81  61.06  -0.0285 -0.0506  78.13 70.80  78.13  7 10  61.09  61.09  0.0057 0.0101  70.82 61.12  1  91.38  -0.0006  91.38  -0.0001  91.38  91.38  0.0006  -0.0090  79.12  -0.0018  79.12  79.12  5  79.11 70.26  0.0001 0.0018  91.38  3  -0.0242  70.27 63.12  0.0048  -0.0415 -0.0674  -0.0048 -0.0083  70.28  63.10 54.16  70.27 63.12  0.0090 0.0241  7 10  63.13  54.19  -.0.0135  54.19  54.20  0.0083 0.0135  79.13 70.29 63.15 54.23  86.23  0.0003 0.0052 0.0153 0.0284 0.0504  0.0414 0.0672  the  results: 1)  Bond  values  are s e n s i t i v e t o jx , the mean l e v e l o f the  interest rate. e r r o r s i n <r 2)  Errors  2  The s e n s i t i v i t y to m  have h a r d l y any impact  is  much  less  on bond values.  i n /A- cause e r r o r s i n bond v a l u e s which i n c r e a s e  as the time t o maturity of the bond i n c r e a s e s ,  the c u r r e n t l e v e l of the i n t e r e s t r a t e has no e f f e c t  on  For example, o v e r e s t i m a t i n g jx  by  the  amount of e r r o r .  5% causes the 10 year d i s c o u n t bond to by  1.85%  be  undervalued  i r r e s p e c t i v e of whether the c u r r e n t l e v e l of  i n t e r e s t r a t e i s at 3)  or  2^.  E r r o r s i n m cause e r r o r s i n bond values which with  the maturity of t h e bond.  increase  Furthermore,  the e r r o r  i n the bond v a l u e depends on the c u r r e n t l e v e l interest  He now  levelJJ»  the d i s t r i b u t i o n of t h e estimated parameters. T a b l e s XI  to  XIII.  parameters  estimated  However, t h e r e are  by  interesting  any  values  These r e s u l t s are  of  results  the when  almost  identical,  three methods . 38  we  compare  distribution  of bond p r i c e s u s i n g • "weekly**, versus " d a i l y "  Surprisingly,  (as can be seen  standard 50%  deviation (SD  or  more  using  W C  from Table XI?)  even  ) of the parameters was  "daily"  data  to  The r e s u l t s are e x a c t l y as  expected: the d i s t r i b u t i o n o f bond values i s using  the  .  look a t t h e s e n s i t i v i t y of d i s c o u n t bond  in  of  r a t e - more a c c u r a t e l y , on i t s d e v i a t i o n from  the mean i n t e r e s t  presented  whereas  (see  though  always  Table V I I ) ,  the data. the  reduced similar  For the SS method, r r was taken from the SL method. Dsing t h i s cr- ; m was computed from the parameter {2m/<r ) estimated f o r the SS method. 3 8  z  2  2  TABLE XI SENSITIVITY OF PURE DISCOUNT BpND PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS" ( C u r r e n t v a l u e o f i n t e r e s t r a t e = >ju) MATURITY(YRS) TRUE VALUE  1 97.13  3 90.01  5 82.38  7 74.99  10 64.86  Simple Linearization Method  Mean SDmc 10% Median 90%  96.972 0.251 96.599 97.013 97.238  89.417 1.163 87.753 89.516 90.710  81.638 1.999 78.906 81.790 83.936  74.290 2.695 70.783 74.469 77.497  64.364 3.505 60.077 64.439 68.649  Transition Probability D e n s i t y Method  Mean SDmc 10% Median 90%  96.969 0.253 96.595 97.015 97.241  89.408 1.170 87.750 89.491 90.714  81.627 2.008 78.891 81''. 763 83.942  74.279 2.705 70.752 74.435 77.505  64.355 3.515 60.047 64.496 88.659  Steady S t a t e D e n s i t y Method  Mean SDmc 10% Median 90%  96.979 0.246 96.576 97.019 97.246  89.450 1.133 87.726 89.547 90.748  81.691 1.929 79.251 81.720 83.988  74.355 2.583 70.720 74.436 77.522  64.435 3.336 59.975 64.488 69.011  NOTE: - The I n t e r e s t r a t e parameters (m, u,a) have been e s t i m a t e d f o r the a=*5(knovra) case u s i n g 945 o b s e r v a t i o n s on the i n t e r e s t r a t e . 100 such s i m u l a t i o n s were performed, and d i s t r i b u t i o n o f bond p r i c e s r e p r e s e n t s the bond v a l u e c o r r e s p o n d i n g to each of those parameter e s t i m a t e s . - True v a l u e of bond c o r r e s p o n d s i n t e r e s t process parameters.  t o the bond p r i c e c o r r e s p o n d i n g  t o the • t r u e u n d e r l y i n g  TABLE X I I SENSITIVITY OF PURE DISCOUNT BOND PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS ( C u r r e n t v a l u e o f i n t e r e s t r a t e = u)  MATURITY(YRS) ' 1 TRUE VALUE 95.17  3 86.22  5 78.13  7 70.80  10 61.09  Simple Linearization Method  Mean SDmc 10% Median 90%  95.193 0.160 95.005 95.184 95.373  86.335 0.898 85.174 86.274 87.410  78.348 1.708 76.036 78.212 80.451  71.122 2.426 67.765 70.906 74.088  61.537 3.266 56.951 61.207 65.700  Transition Probability D e n s i t y Method  Mean. SDmc 10% Median 90%  95.193 0.161 95.004 95.184 95.373  86.335 0.900 85.169 86.275 87.413  78.348 1.711, 76.029 78.213 80.451  71.122 2.429 67.757 70.906 74.091  61.537 3.270 56.943 61.207 65.711  Steady S t a t e D e n s i t y Method  Mean SDmc 10% Median 90%  95.193 0.150 95.002 95.191 95.362  86.340 0.842 85.318 86.324 87.298  78.358 1.601 76.494 78.329 80.237  71.137 2.275 68.339 71.092 73.986  61.557 3.065 57.575 61.485 65.460  NOTE:  Refer  t o comments on T a b l e XI f o r more d e t a i l s .  TABLE  XIII  SENSITIVITY OF PURE DISCOUNT BONDS PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS ( C u r r e n t v a l u e of i n t e r e s t r a t e = 2y)  MATURITY(YRS) TRUE VALUE  1 91.38  3 79.12  5 70.27  7 63.12  10 54.19  Simple Linearization Method  Mean SDmc 10% Median 90%  91.732 0.402 91.283 91.670 92.278  80.504 1.604 78.565 80.493 82.550  72.194 2.440 69.115 72.411 75.173  65.227 3.007 61.277 65.410 68.962  56.293 3.585 51.532 56.441 60.776  Transition. Probability D e n s i t y Method  Mean SDmc 10% Median 90%  91.738 0.407 91.289 91.685 92.293  80.520 1.615 78.553 80.525 82.596  72.213 2.449 69 .'091 72.454 75.235-  65.245 3.013 61.262 65.370 69.013  56.310 3.587 51.518 56.441 60.783  Steady S t a t e D e n s i t y Method  Mean SDmc 10% Median 90%  91.721 0.402 91.235 91.658 92.284  80.458 1.591 78.452 80.494 82.480  72.129 2.395 69.123 72.214 75.075  65.156 2.919 61.637 65.439 68.579  56.227 3.435 51.956 56.657 60.352  NOTE:  R e f e r t o comments on T a b l e  XI  f o r more d e t a i l s .  TABLE COMPARISION OF BOND P R I C E DATA*  IN  Mean mc 10%  S D  50% 90%  96.178  88.021  87.782  80.227  80.020  73.095  0.917  2.623  2.957  4.333  4.707  5.869  95.204  94.931  84.514  83.797  74.323  73.950  64.624  96.336  96.345  87.952  88.087  80.080  80.488  97.204  97.320  91.388  91.682  85.901  86.286  64.86  72.963  63.599  63.595  7.754  7.832  65.108  53.087  53.796  72.970  73.200  63.325  63.843  80.742  81.110  73.424  73.978  6.160 "  61.09  70.80  78.13  86 . 2 2  WEEKLY  DAILY  7 4 . 99  0.773  95.17  T=10 WEEKLY  DAILY  96.312  TRUE V A L U E  62.266  95.269  95.250  86.563  86.553  78.667  78.709  71.516  71.627  62.060  0.617  0.743  2.402  2.639  4.125  4.288  5.637  5.641  7.393  7.167  94.463  94.290  82.886  83.139  72.666  73.086  63.680  64.053  52.240  52.437  50%  95.256  95.289  86.545  86.721  78.606  78.967  71.396  71.918  611.801  62.529  90%  96.042  96.149  89.538  89.830  83.595  84.211  78.319  78.854  70.636  71.548  Mean SD  io?  c  Mean SDmc 10% 50% 90%  *  n=100  59.957  93.208  93.429  83.779  84.198  75.768  76.264  68.661  69.202  59.389  1.235  1.215  3.676  3.509  5.638  5.273  7.148  6.612  8.647  7.993  91.538  91.844  79.529  79.161  69.113  68.345  59.861  59.475  48.238  48.305  93.315  93.447  83.912  84.566  76.422  76.981  69.314  70.116  59.892  60.806  94.639  94.966  88.096  88.228  81.769  82.272  76.779  77.042  68.938  69.681  The I n p u t p a r a m e t e r Method:  54.19  63.12  70.27  79.12  91.38  TRUE V A L U E  r=2y  WEEKLY  8 2 . 38  90 .01  WEEKLY  PARAMETERS(«=^s)  T=7  DAILY  WEEKLY  DAILY  WEEKLY  OF D A I L Y VS  T=5  T= 3  97.13  TRUE VALUE  S E N S I T I V I T Y TO THE USE  THE E S T I M A T I O N OF I N T E R E S T RATE PROCESS  T=l year DAILY  XIV  for  estimates  weekly  were  estimates  the r e s u l t s  of  a n d n=700 f o r  estimation daily  using  the T r a n s i t i o n  Probability  Density  estimates.  O  71  decreases  in  dispersion  of  bond  value  distributions  appear to r e s u l t - the r e d u c t i o n i n the bond value truly  marginal.  behaviour l i e s i n particularly  m  bond p r i c e s we the  parameter  correlation  between  know t h a t o v e r e s t i m a t i n g y/I f now  the estimates  or  this  result  estimates  is  3 9  .  (The  in  we  have  -  s e n s i t i v i t y of underestimates  of m and ^  are n e g a t i v e l y  offsetting  effects  between m and^x.  correlation  addressed  toward  the  favour  of  between end  of  using  the this  weekly;  r a t e data.  R e s u l t s of Monte C a r l o S i m u l a t i o n s  far  parameters  m  Thus a n e g a t i v e c o r r e l a t i o n  r a t h e r than d a i l y i n t e r e s t  So  the  Based on the t h e o r e t i c a l  c h a p t e r ) . , T h i s i s more evidence  5.4  is  the  bond v a l u a t i o n .  could explain  variance  e x p l a n a t i o n f o r t h i s seemingly anomalous  c o r r e l a t e d , then, to some e x t e n t , they have on  not  The  and^.  bond value.  do  only  f o r the <x Unknown Case.  compared  the d i f f e r e n t  methods under the assumption t h a t the value  of  cA  estimation  were  known.  For the j o i n t e s t i m a t i o n of a l l the parameters  (m »/A. , cr , crt ) , we  can  SS method, as  only  compare  the  transition probability i s not known. model  have  The been  method, i n Appendix  SL  method  and  d e n s i t y corresponding  d e t a i l s o f parameter set  the  t o general <* values  estimation  out i n Appendix 2 and  the  in  f o r the SS  the  SL  density  10.  The e f f e c t may be understood more i n t u i t i v e l y by c o n s i d e r i n g the r e t u r n on a p o r t f o l i o o f 2 n e g a t i v e l y c o r r e l a t e d s e c u r i t i e s . Increasing the variance on the returns of the individual s e c u r i t i e s need not cause p r o p o r t i o n a l i n c r e a s e s i n the v a r i a n c e on the r e t u r n of the p o r t f o l i o . 3 9  72  For t h e SL method, the n=500 and 945 cases (100  trials  each),  e s t i m a t e d , as  the  additional  insights  for  sample  other  estimated  were  estimated  but f o r the SS method, o n l y n=945 case was computation  cost  was  very  high,  and  no  seemed forthcoming by doing t h e e s t i m a t i o n sizes.  parameter  The  summary  distributions  are  statistics  for  the  presented i n Table XV.  The f o l l o w i n g remarks about the r e s u l t s are i n o r d e r : 1)  Comparing  the  S.L. method, cA= y (known)  estimates  in  the c*  of  m  y*  and  from  unknown case with those i n the  case, we f i n d t h a t the r e s u l t i n g  v  distributions  are  almost  something  about  the  estimated  parameters.  identical.  This indicates  interrelationship The  parameter  between  correlation  points  uncorrelated 2)  to  the  possibility  t h a t m and  2  does  not  appear  d i s p e r s i o n seems l a r g e , p a r t i c u l a r l y oV = y-2_ (known)  case.  v a r i a n c e o f the process i s r f z  when cA i s f r e e t o adopt has  to  adjust  biased  are  but  the  when compared with  and o< .  2  2  this  The reason f o r t h i s i s the  c l o s e r e l a t i o n s h i p between <r  cr  but  the  with cX .  The estimate o f t r  the  the  between  parameters i s d i s c u s s e d i n the next s e c t i o n , result  the  U  The i n s t a n t a n e o u s  and,  understandably,  a range o f v a l u e s , the value of accordingly,  f o r a given data  sample,, 3)  The e s t i m a t e of jx by e i t h e r the SL  or  the  SS  method  appears the same. 4)  The  estimate  of  o(  by both methods appears  though the SL e s t i m a t e has a lower  dispersion.  unbiased,  TABLE XV ESTIMATION OF PARAMETERS FOR a UNKNOWN CASE  METHOD  (0.09517)  Simple Linearization  o \ J  Method  n  LO  c  m  ^jCT*  c  Steady S t a t e Density Method  (0.50)  (194.389)  (0.0077617)  639.87 88.97 349.92 1545.86 979.71  0.01556 0.00603 0.01392 0.02491 0.01020 0.00728  1.2145 0.1925 0.7559 2.0897 1.9348 0.6172  0.01212 0.00657 0.01084 0.01884 0.00503 0.00478  0.8859 0.3589 0.7823 1.4744 0.51171 0.4064  Mean 10% Median(50%) 90% SDmc SDi*  0.08876 0.07138 0.09007 0.11521 0.04530 0.00929  0.49385 0.22481 0.48476 0.71715 0.19318 0.16915  Mean .10% Median(50%) 90% SDmc SDi  0.09371 0.07703 0.09390 . 0.10949 0.01276 0.00800  0.49204 0.34414 0.50129 0.62975 0.11480 0.11141  366.34 129.71 295.00" 659.36 297.43  Mean 10% Median(50%) 90%  0.09167 0.08863 0.09211 0.11142 0.01022 0.00061  0.56049 0.04242 0.49884 1.40816 0.42979 0.10516  1036.18 14.3348 210.624 6626.37 2173.70 80.8525**  SDmc SDi  -  -  The S D i f i g u r e r e p o r t e d i s t h e median o f the SDi from each t r i a l n o t the mean. ** T h i s i s the median - t h e mean S D i  w  - The f i g u r e s  i n the a  2  a  s  625.458 i n t h i s c a s e .  column have been m u l t i p l i e d by 10^  (0.78427x10  )  74 5)  Comparing method  the  appears  and d i s p e r s i o n . measure  of  reasonable.  composite to  parameter  (2ra/<f ),  t h e SL  2  g i v e e s t i m a t e s having a lower  bias  However, t h e median (which i s a l s o  location),  of  the  SS  a  estimate  i s very  I t seems t h a t t h e SS method has a  tendency  to produce extreme e s t i m a t e s * . , 0  6)  Using SD,;  as a measure of the t r u e asymptotic  o f the parameters we see t h a t , i n t h e appears t o s a t i s f y in  that  SL  variance  case,  t h e asymptotic sample s i z e  n-945  criteria,  SD^ f o r a l l parameters i s very c l o s e to S D . n c  For s m a l l e r sample s i z e s  (see n=500), SD^ i s an  under  estimate o f the asymptotic standard d e v i a t i o n . For  the  sake  of  completeness,  we present i n Table XVI a  comparison o f t h e d i s t r i b u t i o n o f t h e estimated " d a i l y " versus "weekly" data. was used., the  other  soon  F o r t h i s case, o n l y the SL method  The only parameter of i n t e r e s t here i s o( . parameters,  the d i s p e r s i o n o f shall  parameters using  the  As  with  the improvement i s only with r e s p e c t t o estimated  parameter  distribution.  We  see whether t h i s improvement i n accuracy makes any  s i g n i f i c a n t d i f f e r e n c e t o t h e bond value. Before we conclude t h i s s e c t i o n , we present some r e s u l t s on the s e n s i t i v i t y of t h e pure discount bond value t o v a r i a t i o n s i n *° The SS method i s based on the assumption that t h e s t a t i o n a r y density i s not the t r i v i a l P(r) .= 0, which o b t a i n s when e i t h e r singular boundary i s absorbing f o r some parameter v a l u e s . Whenever the n o n - l i n e a r search procedure (to i d e n t i f y the maximum o f t h e j o i n t l i k e l i h o o d f u n c t i o n ) takes on parameter values which correspond t o an absorbing barrier at either s i n g u l a r boundary, the SS method breaks down., I f t h e range of the parameter space where we get absorbing b a r r i e r s were known, a c o n s t r a i n e d maximization could be done. T h i s however i s not the case. The breakdown o f the SS method i n some parameter ranges causes these extreme values.  TABLE  XVI  COMPARISON OF PARAMETERS ESTIMATED USING DAILY vs WEEKLY DATA FOR THE ct UNKNOWN CASE n=500 FOR WEEKLY & n=35O0 FOR DAILY  METHOD (0.09517) Simple Linearization Method  o >-( o  >-i <  n II  a a  Simple Linearization Method  ^  o  w  II  The f i g u r e s i n t h e a  2  Mean 10% Median(50%) 90% SDmc SDi  Mean 10% Median(50%) 90% SDmc SDi  2m/o (194.389)  a (0.50)  (0.0077617)  (0.78427xl0 ) -4  458.80 118.91 330.48 969.65 405.28  0.00907  0.48586 0.38203 0.48552 0.57420 0.07084 0.06507  0.01534 0.00515 0.01315 0.02946 0.00956 0.00Z25  0.7903 0.4487 0.7399 1.1368 0.2605 0.2200  0.08876 0.07138 0.09007 0.11521 0.04530 0.00929  0.49385 0.22481 0.48476 0.71715 0.19318 0.16915  639.87 88.97 349.92 1545.86 979.81  0.01556 0.00603 0.01392 0.02491 0.01020 0.00728  1.2146 0.1925 0.7559 2.0897 1.9348 0.6172  0.09729 0.07797 0.09441 0.11430  column have been m u l t i p l i e d by 1 0 , 4  cn  o(.  In t h i s context, only t h e " t h e o r e t i c a l " s e n s i t i v i t y  are  presented  additional  in  T a b l e s XVII and XVIII.  information  could  "empirical" sensitivity. discount  seen t h a t i n c r e a s i n g Table X  ^decreases  the  bond  value.  bond  values  S i n c e r has a numerical  on  the  We  i t s e l f by a  estimates  cr2  (assuming  that  there  v a r i a n c e ) . , Let  us  high cK .,  and  a~  a  1.0;  has  2  the  this close  positive  is  correspondingly no  represent  corresponding  to  an  corresponding  to  any  assumption, we would  by  correlation  The q u e s t i o n of the addressed  in  the  f o r the p r e s e n t , l e t us accept t h a t i f <?< i s would  z  caused  expect  between the estimated parameters i s  o~  of  would  relation  overestimated,  is  C l e a r l y , decreasing  the  but,  be  Comparison  value l e s s than  between  section  can  and so a l s o the i n s t a n t a n e o u s v a r i a n c e  v a r i a n c e . ,.  relationship to r e f l e c t  next  It  2  2o  effect  the  on bond value by v a r y i n g cr ) , shows t h a t  the i n t e r e s t r a t e : cr2r V.  same  presenting  1  i n c r e a s i n g o( decreases of  by  f i x e d at t h e i r t r u e v a l u e s * .  {effect  i n <r2.  gained  cA about the value Vi. , with the  the same d i r e c t i o n o f e f f e c t on decrease  I t was f e l t t h a t no  In T a b l e XVII we present the e f f e c t on  bond v a l u e s of v a r y i n g  other parameters kept  with  be  results  bias by  <T , 0  z  cX  value  over  estimated  identifying the  of y , and  o\ value other  in  be  true  the t o t a l  value  o f er  <r^ as the value of cr  estimated.  Given  our  expect  * I t may be n o t i c e d t h a t t h e 0% e r r o r bond p r i c e i n T a b l e XVII and XVIII i s s l i g h t l y d i f f e r e n t from t h a t i n T a b l e s V I I I , IX and X. T h i s i s because, the values i n t h a t column i n Tables XVII and XVIII have been computed using a f i n i t e d i f f e r e n c i n g method to s o l v e the bond equation. T h i s was done, as what we want t o present i s the e f f e c t of v a r i a t i o n s i n <A , and f i l t e r out the e f f e c t due t o the s o l u t i o n technigue employed. l  TABLE THEORETICAL SENSITIVITY (02  HAS  OF PURE DISCOUNT BOND P R I C E S TO ERRORS IN  NOT BEEN  'CORRECTED'  ERROR IN  TO R E F L E C T T H E ERROR IN  a*  a)  a  0%  -5%  -25%  XVII  +25%  +5%  CURRENT  T I M E TO  BOND  %  INTEREST  MATURITY  PRICE  ERROR  BOND PRICE  ERROR  PRICE  PRICE  ERROR  PRICE  ERROR  1  96.96  0.0053  96.95  0.0004  96.95  96.95  -0.0003  96.95  -0.0008  3  89.40  0.0862  89.33  0.0074  89.32  89.32  -0.0051  89.31  -0.0138  5  81.66  0.2572  81.47  0.0227  81.45  81.44  -0.0158  81.42  -0.0434  7  74.37  0.4812  74.04  0.0431  74.01  73.99  -0.0303  73.95  -0.0837  10  64.50  0.8634  64.00  0.0785  63.95  63.92  -0.0554  63.85  -0.1541  1  95.18  0.0063  95.18  0.0006  95.18  95.17  -0.0004  95.17  -0.0011  3  86.31  0.0963  .86.23  0.0087  86.23  86.22  -0.0062  86.21  -0.0173  5  78.35  0.2788  78.16  0.0254  78.14  78.12  -0.0180  78.10  -0.0507  7  71.18  0.5125  70.85  0.0470  70.81  70.79  -0.0334  70.75  -0.0941  10  61.66  0.9040  61.15  0.0835  61.10  61.07  -0.0593  61.00  -0.1672  1  91.23  0.0117  .91.22  0.0011  91.22  91.22  -0.0008  91.21  -0.0022  3  78.56  0.1685  78.44  0.0157  78.43  78.42  -0.0113  78.41  -0.0322  5  69.62  0.4502  69.34  0.0424  69.31  69.29  -0.0305  69.25  -0.0872  7  62.57  0.7656  62.14  0.0724  62.09  62.06  -0.0521  62.00  -0.1490  10  53.88  1.2288  53.29  0.1164  53.22  53.18  -0.0836  53.10  -0.2389  IN  r=y/2  r=2y  BOND  **  BOND  %  BOND  %  YEARS  The o t h e r parameters  See  %  f o o t n o t e XXI  .  of  the  Interest  rate  process  assume  their  true values.  TABLE XVII1 -HFORFTICAL SENSITIVITY  PHBF. DISCOUNT BOND PRICES TO ERRORS IN &_  ERROR IN a  CURRENT INTEREST  r = u/2  r =u  r =2u  **  ERROR  BOND PRICE  ERROR  BOND PRICE  ERROR  96.95 89.33 81.46 74.02 63.97  0.0002 0.0026 0.0076 0.0140 0.0248  96.95 89.32 81.45 74.01 63.95  96.95 89.32 81.45 74.00 63.94  -0.0001 -0.0023 -0.0067 -0.0124 -0.0220  96.95 89.32 81.43 73.98 63.89  -0.0006 -0.0090 -0.0265 -0.0493 -0.0879  0.0011 0.0166 0.0486 0.0898 0.1592  95.18 86.23 78.14 70.82 61.12  0.0002 0.0026 0.0076 0.0141 0.0250  95.18 86.23 78.14 70.81 61.10  95.18 86.23 78.13 70.81 61.09  -0.0002 -0.0023 -0.0068 -0.0126 -0.0223  95.17 86.22 78.12 70.78 61.05  -0.0006 -0.0093 -0.0273 -0.0505 -0.0897  0.0019 0.0269 0.0724 0.1245 0.2031  91.22 78.44 69.32 62.10 53.24  0.0003 0.0044 0.0119 0.0203 0.0330  91.22 78.43 69.31 62.09 53.22  91.22 78.43 69.30 62.08 53.21  -0.0003 -0.0040 -0.0108 -0.0184 -0.0298  91.22 78.42 69.28 62.04 53.16  -0.0012 -0.0167 -0.0447 -0.0762 -0.1226  % ERROR  BOND PRICE  1 3 5 7 10  96.59 89.34 81.49 74.08 64.05  0.0011 0.0169 0.0490 0.0901 0.1591  1 3 5 ' 7 10  95.18 86.24 78.17 70.88 61.20  1 3 5 7 10  91.22 78.45 69.36 62.17 53.33  * The o t h e r  %  BOND ' PRICE  BOND PRICE  TIME TO MATURITY IN YEARS  +25%  +5%  0%  -5%  -25%  parameters o f the I n t e r e s t r a t e p r o c e s s  assume t h e i r  %  true values.  See f o o t n o t e XXL  Co  79  <J  Thus  now  C  varied.  Clearly,  average the  represents  the  "corrected"  value  of cr as c< i s  <T v a r i e s with r , but we c o u l d assume t h a t C  r i s expected  sensitivity  t o remain around  of  discount  bond  .  Table XVIII  on  presents  values t o v a r i a t i o n i n  where <r has been " c o r r e c t e d " as i n d i c a t e d above.  As  ,  expected,  " c o r r e c t i o n " has reduced t h e e f f e c t of a v a r i a t i o n i n o( on  this  discount bond v a l u e s .  However, what i s more  important  is  the  f a c t t h a t the net e f f e c t i s s m a l l .  5  •  5  The R e l a t i o n Between the I n t e r e s t Rate Process Finally,  before  concluding  look a t the r e l a t i o n s h i p between There  are  two  Parameters  t h i s c h a p t e r , we take a b r i e f the  parameters  <T , oS) •  (m,  c l o s e l y i n t e r c o n n e c t e d p o i n t s from which we  may  view t h i s r e l a t i o n s h i p ; a)  What i s the expected c o r r e l a t i o n between the values of these parameters,  b)  In  what  the  interconnected  characteristics  of  estimated  given a data sample?  way the  do these parameters a l t e r interest  rate  process  dynamics? One  way  to  calculate  the  estimated  during  n=945  case  correlation*  try  correlation the  displayed 2  t o answer the f i r s t matrix  between  simulation. close  to  q u e s t i o n would be to the  parameters  Since the SL method f o r the  asymptotic  behaviour,  between the parameters f o r that case was  the  computed  * For the n=945 case, we performed 100 s i m u l a t i o n s and so generated 100 e s t i m a t e s of the parameters (m , JX , <r , ). The t a b l e r e p r e s e n t s the c o r r e l a t i o n between values. 2  and found t o be rv\  2  He (which  0.2081  0.0375  0.1725  0.0875  can is  see as  correlations  expected),  but  apart  from  that  correlated any  other  appear to be q u i t e s m a l l .  correlation  Appendix  0.9339  cr*" and o\ are almost p e r f e c t l y  that  Another approximate the  (T^  -0.0 207  jX  cr  /A-  (and q u i t e ad hoc) method of e s t i m a t i n g  matrix  between  the  parameters i s s e t out i n  9, based on t h a t method, the c o r r e l a t i o n matrix i s •u cr  jiK  -0.1582  There  0.0  0.0  0.0  0.0  of  the  asymptotic c o r r e l a t i o n m a t r i c e s i n broad q u a l i t a t i v e terms.  The  second  is  agreement  0.9877  between  the  two  estimates  e s t i m a t e (based on the approach presented i n Appendix  i m p l i e s t h a t t h e parameters i n t h e v a r i a n c e and d r i f t the This  diffusion would  similarity  equation  explain  the  of  are t o t a l l y independent o f each o t h e r . earlier  observation,  namely,  the  of the d i s t r i b u t i o n s of estimated v a l u e s of m and  between the cL^'/z.  known case and the  S.L.  The  method.  anticipated  terms  9)  two  important  cA  unknown  case,  in  the  c h a r a c t e r i s t i c s t h a t were  are borne out i n both c a s e s , i e .  81 1)  fa  and • JJ^  are  anticipated,  negatively  based  on  correlated.  their  This  was  combined e f f e c t on bond  values. ( T and o\ a r e very  2)  highly p o s i t i v e l y correlated.  2  Further  i n s i g h t s i n t o the n a t u r e o f the i n t e r - r e l a t i o n s h i p s  among the parameters can be gained by which  each  process,  i s contained  function.  at  a l l information in  the  about  To i n v e s t i g a t e how i t i s a l t e r e d  Parameters  f ( »V<r ) =2  >  °\ h \r^c l  virtually  a  way  in  the  process  t r a n s i t i o n p r o b a b i l i t y density by  parameters, we c o n s i d e r the f o l l o w i n g parameter  On  the  of them a f f e c t s the i n t e r e s t r a t e process dynamics. ,  For a d i f f u s i o n dynamics  looking  Set 1  Set 2  460.098  311.398  0.06904  0.06905  0.36202  0.43333  1314.92  1314.71  changing  the  values:  p a r t i c u l a r data sample these two parameter s e t s gave i d e n t i c a l values  function,  using  the  performing r o u t i n e  SS  f o r t h e l o g o f the method.  This  joint  likelihood  s i t u a t i o n arose  while  p r e l i m i n a r y t r i a l s with t h e SS method f o r the  °<7\ unknown case.  I t i s w e l l known  r o u t i n e s provide  no guarantee t h a t convergence t o a optimum w i l l  occur.  Further,  the  the  nonlinear  even i f convergence i s obtained,  sure whether t h e p o i n t i s a investigate  that  local  or  a  global  one i s never optimum..  To  behaviour of t h e p a r t i c u l a r f u n c t i o n a l form o f  l i k e l i h o o d f u n c t i o n on some data samples,  generated  optimization  sequence)  (chosen  d i f f e r e n t available nonlinear  from  the  optimization  82  methods  were  applied  to  see  whether  (using  different  algorithms), a)  convergence  was  always  to  the  same  point  parameter space, i r r e s p e c t i v e of the s t a r t i n g values, b)  whether  was  algorithm)  the  parameter  speed  of  convergence  differed  across  algorithms.  found  was  the  and  different It  in  that  the q u i c k e s t  which converqence was  the guasi-Newton method by f a r , and  obtained,  (the F l e t c h e r  in qeneral,  appeared  to  be  the the  point to "global"  optimum..,, We  have  corresponding identical.  a  case  where the s t a t i o n a r y p r o b a b i l i t y d e n s i t y  to  very  different  T h i s was  distribution  parameter  values  is  f u r t h e r v e r i f i e d by p l o t t i n g the s t a t i o n a r y  corresponding  to  these parameter values, and  d e n s i t y f u n c t i o n s were seen to v i r t u a l l y c o i n c i d e .  This  t h a t , given a data sample, the SS method  be  identify one more  an  may  not  unique parameter s e t t h a t f i t s i t - i t may  or more e q u i v a l e n t relevent  almost  points in  question,  the  however,  parameter is  whether  implies able  to  identify  space* . 3  the  the  The  transition  * An attempt was made t o f i n d out whether, correspondinq to this data sample , the two "optimum" parameter s e t s represented two independent "peaks". To i n v e s t i q a t e t h i s , a c l o s e mesh q r i d (50x50) was p l a c e d over the ( , <k ) space, between (fi,<*i) and (•P-j,#<*i.) , keeping /A- constant (as was ., The l o g l i k e l i h o o d f u n c t i o n was evaluated over the q r i d ,to see i f t h e r e was a " d i p " between the peaks., I t was observed t h a t t h e r e was a c t u a l l y a plateau i n the likelihood function., This i n d i c a t e d that the 2 "optima" were l o c a t e d p u r e l y as a matter of roundinq e r r o r i n the numerical process., In other data samples, the plateau was not so marked. The i d e n t i f i c a t i o n of an unique optimum using the SS method appears to be, i n p a r t , a f u n c t i o n o f the data sample. T h i s would a l s o e x p l a i n some of the extreme values obtained durinq the s i m u l a t i o n s . 3  83  p r o b a b i l i t y d e n s i t y c o r r e s p o n d i n g t o these parameter very  different.  If  the  transition  probability  c o r r e s p o n d i n g t o d i f f e r e n t parameter values were the  parameters  values  very  is  density similar,  would not be o r t h o g o n a l to each other and  there  would be a l o s s of r i c h n e s s i n t h e present model. The  t r a n s i t i o n p r o b a b i l i t y d e n s i t y f u n c t i o n i s the  to the Fokker-Planck  e q u a t i o n , which we  solve  values  for  general  method was the  of ck .  have not  Thus a f i n i t e  employed to s o l v e the FP equation.  exercise  was  to  try  and  see  require the  could  be  made  almost  to  able  to  differencing objective  of  the  transition  the  above  identical**.  9e  two also  a s t a t i s t i c t o measure the " c l o s e n e s s " to each other of  two  transition  "matchinq"  criterion  probability was  that  the  area  of  density  to minimize  between the computed t r a n s i t i o n found  The  whether  p r o b a b i l i t y d e n s i t y f u n c t i o n s corresponding parameter s e t s  been  solution  density  non-overlap  functions.  the area of  non-overlap,  functions* . 5  between  The  It  was  the t r a n s i t i o n  p r o b a b i l i t y d e n s i t y f u n c t i o n s corresponding to the two  parameter  ** The approach was as f o l l o w s . Parameter s e t 1 was used as the b a s i s and f>, (=460.098) was a r b i t r a r i l y s p l i t i n t o reasonable values o f m, and <TJ Z. The transition probability density function f o r ©i = (m, , <r , <*,) was evaluated. Then, keeping ^» A?c o n s t a n t , d i f f e r e n t values of vx and c^. were chosen, and the transition probability function for 02.= ( n *M f <T r^%) was e v a l u a t e d . The o b j e c t i v e was t o match the transition probability d e n s i t i e s c o r r e s p o n d i n g to the two d i f f e r e n t parameter sets.„ 2  x  1  a  n  d  z  1  2  z  u  2  84  s e t s c o u l d be brought down t o about 7%, when  r *f^at  for  r = f*-* .  However  6  0  t h e s e "matched" parameter values, the area of  0  overlap increases greatly.  This i s as expected - what  informative  to which the shape of the  is  the extent  non-  is  more  transition  p r o b a b i l i t y d e n s i t y i s changed by a p r o p o r t i o n a l change i n parameter.  This i s p i c t o r i a l l y  T h i s i s an i n d i c a t i o n that the function  is  not  represented transition  i n Figures  each  2 and  probability  3.  density  very s i m i l a r f o r d i f f e r e n t parameter values  given a data sample, we c o u l d expect an unique parameter s e t maximize the l i k e l i h o o d As  expected,  density function, changes  the  chanqes variance,  both  cA and  <r  a f f e c t the d i s p e r s i o n of the  2  a{ more so than element  <r .  has  particularly  since  r  transition  is  in  m,  qreater  when.r=/^  f  speed  of  effect  I t can be  mean.  on  f o r chanqes  produce hardly any  r e v e r s i o n to the  2  whereas o{ the  seen  that  change.  When r *J^  t  s h i f t the l o c a t i o n because of the skewing  in  the  t o changes i n /x ,  i n c r e a s e s i n m make the f u n c t i o n s l i g h t l y more peaked the  a~  because  always f a r from u n i t y i n  d e n s i t y corresponding  as t h i s a f f e c t s only the l o c a t i o n . changes  a  p i c t u r e s are presented  probability  is  multiplicatively,  which  No  This  2  exponent,  numerical v a l u e .  to  function.  variance  the  -  effect  large However  as  m  is  changes i n m of  the  mean  The area of o v e r l a p i s given by J^abs[ F (r K„,e)-F (r j-r^J ] dr, where Y{T\r p) represents the transition p r o b a b i l i t y density f u n c t i o n corresponding to parameter s e t 0 . i t may be noted that the area under e i t h e r t r a n s i t i o n d e n s i t y f u n c t i o n adds up t o 1.0. Thus the area of non-overlap i n d i c a t e s d i r e c t l y the fraction of t o t a l area under each curve f o r which the two f u n c t i o n s do not match. e  ** The transition p r o b a b i l i t y density is represented as F (rj. , 1 1 r ,9) . Thus, i t i s a f u n c t i o n of r and t as w e l l as the parameter s e t 0 ={m, p- o~, d\) • Here t was chosen equal to 1 week. 0  e  f  85  FIGURE. 2 Sensitivity of the Transition Probability Density Function to Change In a  'Sensitivity of the Transition Probability Density Function to Chance i n o  FIGURE 3  87  r e v e r s i o n property (which i s s i m i l a r t o changing  jx a very s m a l l  amount).... To  summarize,  parameters  in  respectively.  i t appears t h a t  determining m  has  a  a f f e c t s o n l y d i s p e r s i o n . ,.  the  ^  and  location  ck are the important and  dispersion,  marginal e f f e c t oh both, whereas c r  2  88  CHAPTER 6: THE  6.1  The  IHTJRjgST PATE AND  BOND PRICE DATA  Short Term R i s k l e s s I n t e r e s t Rate  By d e f i n i t i o n , the s h o r t term (instantaneous) r i s k less, r a t e of  r e t u r n i s the y i e l d t o maturity on a  bond, such  maturing a  security  available Chapter  the  i n any  default  next i n s t a n t i n time.  does  not  case).  exist  all  unobservable  require  entity.  that A  (and  is  we  suitable  know  i n the  something  proxy  for  about  the  treated  as  Federal  totally  payment on maturity. Government  bonds  default  (a)  outstanding  are  91 days, which b r i n g s us to two what time to maturity may  T r e a s u r y b i l l s are not markets  in  Canada.,  and e x p l a i n t h i s .  this  s h o r t term  bills  banks,and  respect pure  could  to  be  principal  discount  Federal  Apart  minimum  very  from  maturity  c l o s e l y r e l a t e d matters, v i z .  be t r e a t e d as " i n s t a n t a n e o u s "  and  of measurement be? traded  in  secondary  A few c o n j e c t u r e s c o u l d be put forward To s t a r t with , a widespread demand  outstanding  16% were held by the  they  Treasury B i l l s .  very a c t i v e l y  not appear t o have developed. Treasury  with  as  markets, these have a  (b) what should our frequency  try  free  bonds,  However, the only  q u o t a t i o n s i n secondary of  Government  in  preceeding  r i s k l e s s r a t e of i n t e r e s t would be the y i e l d t o maturity on short maturity  not  bond v a l u a t i o n models developed  2, and the e s t i m a t i o n theory developed  chapters,  discount  In a c t u a l p r a c t i c e ,  continuously  The  free  Bank  to  does  Of the t o t a l Government o f Canada over the l a s t s e v e r a l years, about  of  Canada,  74%  by  the  1% by the Government of Canada accounts,  chartered  with only  9%  89  accounted f o r by a l l institutions Neufeld  and  [53]).  holders,  as  the  other  individuals  Chartered they  are  to  non  have  constantly  financial  obtained  always  been  from  principal  i n need of very secure  Since there are only  short  five  major  banks i n Canada, the number of a c t i v e p a r t i c i p a n t s i s  g r e a t l y reduced. law  and  (figures  banks  term investment o p p o r t u n i t i e s . chartered  financial  Furthermore, Canadian banks  are  required  maintain secondary reserves a t p r e s c r i b e d l e v e l s ,  which  tends to reduce t r a d i n g i n short term government s e c u r i t i e s . the-U.S., deep due  , however, the Treasury b i l l  market i s very a c t i v e  In and  to the f o l l o w i n g f a c t o r s :  a)  Banks do not  b)  There are very trading  have t o maintain secondary many  The  more  i n the market  as opposed to the c)  by  U.S.  commercial  (due t o the  reserves. banks  =  actively  u n i t banking system,  branch banking system of Canada).  d o l l a r i s a major reserve c u r r e n c y  as  well  as the denomination of a l a r g e p o r t i o n of i n t e r n a t i o n a l trade. and  Thus, s e v e r a l f o r e i g n i n v e s t o r s  government)  enter  the  short  (both  corporate  term U.S.  dollar  denominated bond market. These f a c t o r s c o u l d  explain  the  relative  secondary markets f o r Canadian Treasury Given  the  present  state,  it  is  inactivity  in  bill. to  be  expected  that  t r a n s a c t i o n s p r i c e s i n secondary markets, would be d i f f i c u l t  to  obtain.  in  No  record  of  sale  prices  for  secondary markets, were a v a i l a b l e e i t h e r with or  from the Bank of Canada.  o f the data,  Treasury security  From c o n s i d e r a t i o n s  (and keeping i n mind  that  we  bills  dealers  of  reliability  require  equispaced  90 observations) the  91-day  i t was  f e l t that t r e a t i n g the y i e l d t o maturity  Treasury b i l l s , o n the date of i s s u e , as a proxy f o r  the short term i n t e r e s t r a t e  best  purposes the term s t r u c t u r e over such s h o r t  91  day  actual  are  pure d i s c o u n t ( i i ) the  transaction  bond may  to  use Treasury b i l l  i s no guarantee that we can actual  transaction  maturity  on  observation weekly and in  the  (which  The  could  Other considered, money  1)  for  as  we  did there on  the y i e l d  to  term  The  error  the  There was  no  could  obtained.  our  short  term  matter of frequency of  Treasury b i l l s a r e  Given  the  source  issued  reported of  this  rate  were  daily  call  reliable. short  term  interest  loan r a t e and  the  s e v e r a l problems on account of which  t o be dropped from s e r i o u s  be  If  average s a l e p r i c e , are  Review.  were  the  settled.  such as the interbank There  treated  quotes.  e f f e c t o f using  asset,  the data are very  rate.  they had  Canada  proxies  be  the  based  bond as a proxy f o r the short  the y i e l d s , based on  information,  yield  '  maturing)  of  the  i n v e s t i g a t e d i n Appendix 11.  i s automatically  Bank  maturities  i s computed  Having chosen the 91-day Treasury b i l l as (instantaneously  all  c o n s i s t e n t l y get y i e l d s computed  a 91-day discount  appears to be s m a l l . ,  for  p r i c e s from secondary markets,  prices.  interest rate i s b r i e f l y  so t h a t  y i e l d t o maturity  prices  (i)  The  be assumed e q u a l t o  e q u i l i b r i u m p r i c e s ) , r a t h e r than Jbased on want  that  be t r e a t e d as v i r t u a l l y f l a t ,  instantaneous r a t e on  choice  alternative.  practical  the  this  the  advantages  as 91 days may  of  was  distinct  on  on  reliable  consideration:  source  from  where  these  data  91 2)  Most  money  market d e a l e r s c o u l d only  r a t e s with a r a t h e r l a r g e spread. the  b i d and  ask  Even  Taking the  place.  r a t e s , no s e r i e s on them almost  20  of  actual  i f i t were p o s s i b l e to get some data on the  back  ask  mean  r a t e s c o u l d be meaningless i f no  t r a n s a c t i o n s took 3)  give b i d and  years*  could 7  r e t r a c t a b l e / e x t e n d i b l e was  -  be  the  other  constructed time  going  when the  first  i s s u e d by the Government  of  a l o t of " n o i s e " i n them, which  has  Canada. 4)  These  rates  little  or nothing  For  example,  have  t o do  with changes  in  bond  prices.  they are s t r o n g l y i n f l u e n c e d by the  of very s h o r t term c a p i t a l between the U.S.  and  flow  Canada  ( c a l l e d "weekend money").  6.2  P r i c e S e r i e s on B e t r a c t a b 1 e/ Ex t e n d i b 1 e Bonds  In  the  corporate  Canadian market, t h e r e are F e d e r a l , P r o v i n c i a l and  (including  retractable  and  the  issues  extendible  bonds  of  the  outstanding.  F e d e r a l bonds, weekly p r i c e s are reported Review.  Due  to  the  chartered For  i n the Bank of  banks) a l l the Canada  l a r g e volume of each of these i s s u e s  and  t h e i r m a r k e t a b i l i t y , an a c t i v e secondary market e x i s t s f o r them. The  p r i c e s reported  i n the Bank of Canada Review a r e , more o f t e n  than otherwise, average a c t u a l  transaction  prices,  at  midday  * Bid and ask p r i c e s on d a i l y c a l l money r a t e s were a v a i l a b l e going back about 18 months from the present.. The d e a l e r s do not keep them on r e c o r d f o r l o n g . The spread between the bid and ask r a t e s was around 0.2% t o 0.4% on an annualized b a s i s . 7  92 every  Thursday.  In  the  case o f the P r o v i n c i a l and c o r p o r a t e  bonds, however, the i s s u e s are much s m a l l e r and very i n number.  many  more  The problems a s s o c i a t e d with p u t t i n g together a data  base on P r o v i n c i a l and c o r p o r a t e r e t r a c t a b l e s / e x t e e d i b l e s may be summarized as f o l l o w s : 1)  There  are  very many i s s u e s o u t s t a n d i n g but not widely  traded, so that a continuous s e r i e s o f even b i d and  ask  prices  i s not a v a i l a b l e . 2)  Even  when  available,  (quoted  i n the F i n a n c i a l  what i s i n d i c a t e d are b i d and ask p r i c e s (with  large  There i s no guarantee that i f t r a n s a c t i o n s took  p l a c e they  Post)  spreads). would  be between those p r i c e s ; i e . , the quotes do not always r e p r e s e n t f i r m commitments to t r a n s a c t . 3) not  ,,  The a v a i l a b l e data on P r o v i n c i a l and c o r p o r a t e bonds a r e  compatible  The  prices  with t h e data on the s h o r t term i n t e r e s t r a t e .  N  quoted  i n the  Financial  Post a r e F r i d a y c l o s i n g  v a l u e s , whereas the Bank of Canada Beview  observations  on the  s h o r t term i n t e r e s t r a t e are Thursday mid-morning p r i c e s . model  prices  f o r the bonds  Thus,  (using t h e models of Chapter  2),  would be Thursday mid-morning p r i c e s , whereas t h e data on market p r i c e s would be F r i d a y c l o s i n g values. not  strictly  evaluate  Consequently,  we  could  the performance of the model i n v a l u i n g  these bonds. 4)  Whereas t h e F e d e r a l bonds  Provincial continuous is  and  corporate  bonds  a r e very are  not.  actively  The assumption of  t r a d i n g o p p o r t u n i t y , upon which t h e model  violated.  shows up when  The impact we  compare  traded,  is  on bond p r i c e s seems n o n t r i v i a l . yields  on  Federal  and  based, This  comparable  93 Provincial level. {on  bonds,  where  default  risk  is  of n e a r l y  the same  The y i e l d d i f f e r e n c e on some i s s u e s i s as high an  annualized  basis).  This  i s an  as  0.5%  indication  that  m a r k e t a b i l i t y o f the bonds i s an important determinant of value.  Therefore,  inappropriate 5)  the  interest  models developed i n Chapter 2 would be  f o r v a l u i n g P r o v i n c i a l and c o r p o r a t e  Corporate bonds rate  risk.  have The  default  theory  risk,  issues.  over  developed  and  r,  the  together  value  of  functions  the f i r m , and time t o maturity.  a data s e r i e s f o r the  value  of  a  above  i n the e x i s t i n g  l i t e r a t u r e f o r v a l u i n g such bonds i s t o t r e a t them as of  their  firm  Putting  has  several  obvious problems. Since  complete data on a l l F e d e r a l  i s s u e d t o date were a v a i l a b l e , i t was  retractable/extendibles  decided  to  confine  our  a t t e n t i o n to them alone - t o the e x c l u s i o n o f the P r o v i n c i a l and corporate  issues.  Table XIX  gives  r e t r a c t a b l e / e x t e n d i b l e s forming our such  issues  by  the  Government  some  details  sample.  of  It  Canada.  and  extending  to  of  the exchange or r e t r a c t i o n date.  In  exchange  date,  i n d i c a t i o n i s t h a t the s h o r t bond was p r e f e r r e d to the long  bond by the majority study,  these  bonds  of the i n v e s t o r s .  E2,  E3 and E4, o b s e r v a t i o n s  date.  For the purpose of  I t may be noted that f o r H1,  cease even before  The matter was i n v e s t i g a t e d by t h e  the  Bank  this  have been named B1, E1 through E19 - B f o r  r e t r a c t a b l e and E f o r e x t e n d i b l e .  of  the  have been date  cases where data were a v a i l a b l e beyond the l a s t the  includes a l l  Data  c o l l e c t e d f o r each bond s t a r t i n g w i t h i n a week of issue,  on a l l the  the o p t i o n  local  o f Canada, and i t appears t h a t ,  expiry  representative  ( f o r some unknown  TABLE XIX DETAILS  O FDATA  SAMPLE  SHORT BOND Coupon Maturity  BOND  LONG BOND Maturity Coupon  Rl  Jan.1,1963  4.00  Oct.1,75  5.50  Oct.1,60  5.50  . Oct.1,75  5.50  Oct.1,62  El E2  O FRETRACTABLE/EXTENDIBLE  BONDS  DATA ISSUE DATE  OPTION PERIOD  FROM  AVAILABLE TO  t  R e t r a c t a b l e on any i n t e r e s t Jan.1,59 date between Jan.1,1961 and Jan 1, 1962 g i v i n g 3 months n o t i c e  Jan.7,59  Jan.27,60  56"  On o r before June 30,60  Oct.1,59  Oct.7,59  May  25,60  34  5.50  On o r b e f o r e June 30,62  Oct.1,59  Oct.7,59  Oct.25,61  108  E3  Dec.15,71  5.50  Dec.15,64  5.50  On or b e f o r e June 15,64  Dec.15,59  Dec.16,59  Oct.25,61  98  E4  Apr.1,76  5.50  Apr.1,63  5.50  On o r b e f o r e Dec.31,62  Feb.15,60  Feb.17,60  Oct.25,61  89  E5  Oct.1,93  6.00  Apr.1,71  6.00  On o r b e f o r e Dec.1,70  Oct.1,67  Oct.4,67  Mar.3,71  179  E6  Dec.1,94  6.25  Dec.1,73  6.25  On or b e f o r e Dec.1,72  Dec.1,67  Dec.6,67  Nov.7,73  310  E7 '  Apr.1,84  7.50  Apr.1,74  7.25  Apr.1,73 to  Sept.30,73 .  Apr.1,69  Apr.2,69  Dec.5,73  245  E8  Oct. 1,86  8.00  Oct.1,74  8.00  On o r before  Apr.1,74  Oct.1,69  Oct.1,69  Sep.25,74  261  E9  Dec.15,85  8.00  Dec.15,75  7.25  Dec.15,74 t o June 14,75  Aug.15,70  Aug.19,70  Nov.26,75  278  E10  Aug.1,81  7.25  Aug.1,76  6.25  Aug.1,75 to Jan.31,76  Aug.1,71  Aug.4,71  July  Ell  J u l y 1,82  7.50  J u l y 1,77  7.00  J u l y 1,76 t o Dec.31,76  J u l y 1,72  J u l y 5,72  June 29,77 263  E12  Dec.15,85  8.00  Oct.1,78  7.75  Oct.1,77 to Mar.31,78  Oct.1,73  Oct.3,73  Nov.9,77  215  E13  Dec.1,87  8.00 .  Dec.1,80  7.50  Dec.1,79 to May 31,80  Dec.1,73  Dec.5,73  Nov.9,77  207  E14  Apr.1,84  8.00  Apr.1,79  7.00  Apr.1,78  Apr  a,  74  Apr.3,74  Nov.9,77  189  E15  Apr.1,84  9.25  April,78  9.25  On or b e f o r e  Oct.1,74  Oct.2,74  Nov.9,77  162  E16  Feb.1,82  9.25  Feb.1,77  9.25  On or b e f o r e No.1,76  June 15,74  June 19,74  Jan.12,77  137  E17  Oct.1,84  8.75  Oct.1,79  7.50  Jan.1,79 to June  J u l y 1,75  J u l y 2,75  Nov.9,77  125  E18  Feb. 1,80  9.00  Feb.1,78  9.00  On o r b e f o r e  Oct.1,75  Oct.1,75  Nov.9,77  112  E19  Oct.1,85  9.50  Oct.1,80  9.00  Jan. 1,80  Oct.1,75  Oct.1,75  Nov.9,77  112  - A l l i s s u e s a r e byGovernment o f Canada. The above Government o f Canada. - Source o f data was Bank o f Canada.  to Sep.30,78 Jan.1,78  29,79.  Oct.31,77  to Jan."30, 80  sample c o n s t i t u t e s the t o t a l  28,76 260  sample on r e t r a c t a b l e s / e x t e n d i b l e s i s s u e d by the  95  TABLE XX DETAILS OF DATA SAMPLE OF STRAIGHT COUPON BONDS  DATA COLLECTED To  BOND  Coupon & Maturity  From  #  Fl  4%% Dec 1, 1962  Jun 1, 1960  Aug 1, 1962  114  F2  4%% Sep 1, 1972  Oct 7, 1959  Aug 2, 1972  670  F3  5%% Oct 1, 1975 J u l 6, 1960  Sep 10, 1975  793  FA  4% Dec 1, 1964  Aug 2, 1961  Sep 30, 1964  166  F5  4% Dec 1, 1963  Dec 21, 1960  J u l 31, 1963  137  F6  5h% Apr 1, 1976  Apr 3, 1963  Mar 24, 1976  678  F7  5% Jan 1, 1971  Oct 4, 1967  Oct 21, 1970  160  F8  5 3/4% Sep 1,1992  Oct 4, 1967  Nov 9, 1977  528  F9  5^5% Dec 1, 1974  Oct 2, 1968  Oct 2, 1974  314  F10  5% J u l 1, 1970  Dec 6, 1967  May 6, 1970  127  Fll  5% Oct 1, 1973  Dec 6, 1967  Sep 26, 1973  304  F12  5 3/4% Jan 1, 1985  Apr 2, 1969  Nov 9, 1977  450  F13  7% Jun 15, 1974  Apr 2, 1969  Jun 5, 1974  271  F14  5% Oct 1, 1987  Oct 1, 1969  May 5, 1971  84  F15  5% Jun 1, 1988  Jan8, 1969  Nov 9, 1977  462  F16  5h% Aug 1, 1980  Aug 1, 1962  Nov 9, 1977  798  F17  5% Oct 1, 1968  Oct 2, 1963  Sep 11, 1968  259  F18  3 3/4% Sep 1, 1965  Jan 7, 1959  Aug 25, 1965  347  - The l a s t column represent the number of weekly data points f o r which data was c o l l e c t e d . - Source of data was Bank of Canada Review.  96 reason) the data on these bonds f o r the  remaining  period  were  not a v a i l a b l e .  6.3  P r i c e S e r i e s on Ordinary  Apart  from  Federal  Bonds  the p r i c e s e r i e s on a l l r e t r a c t a b l e / e x t e n d i b l e  bonds, p r i c e s o f o r d i n a r y  (non-callable)  coupon bonds*  are a l s o  8  required f o r a)  estimating  the  utility-dependent  aggregate  liquidity  premium parameters b)  conducting t e s t s o f market e f f i c i e n c y and  To  capture  every  as  much  information  rates  effort  (between  represented. s t r a i g h t bond  model  during  as p o s s i b l e on the term  the  period  1/2  1959  t o the  was made t o choose the bond sample such  t h a t , at every i n s t a n t i n time, a t l e a s t 5 p o i n t s structure  on  market p r i c e s of the r e t r a c t a b l e / e x t e n d i b l e bonds.  structure of i n t e r e s t present,  based  year  and  18  on  the  term  y e a r s t o maturity) were  Table XX i n d i c a t e s some d e t a i l s on the  sample  of  data.  * The reason f o r s p e c i f i c a l l y choosing n o n - c a l l a b l e bonds i s for computational convenience i n the e s t i m a t i o n of the l i q u i d i t y / t e r m premium parameters. , T h i s will become evident when we address that problem i n the next chapter. 8  97 CHAPTER 7: EMPIRICAL TESTING OF  7•1  Estimated To  BOND VALUATION MODELS  Parameters For The I n t e r e s t Rate Process estimate  the  instantaneously  risk free interest  process parameters (m,/*-, (T ck)  the weekly s e r i e s  maturity  b i l l s was  t  on  91-day  Treasury  p o i n t s s t a r t i n g from January were  7 t h , 1959,  used i n the e s t i m a t i o n .  to e s t i m a t e cA ;  The  SS and  used.,  ^ ^  to  weekly  data 1977,  I n i t i a l l y , the primary o b j e c t  SL methods were used  SS Method  (=2m/cr2)  990  yield  t o December 2 1 s t ,  d a t a , and the estimated parameter v a l u e s were* Parameters  of  rate  on  the  was  total  9  SL Method  8183.48  1655.75x105  0.9974x10-3  0.1334x10-2  0.4938  -0.2195 0.2174x10-2  cr^ The  -  negative  reasonable  <V for  value an  0.2626x10-*°  estimated  interest  by  the  SL  r a t e process.  method  The  is  not  e s t i m a t e o f a-  2  has, t h e r e f o r e , c o r r e s p o n d i n g l y decreased. To i n v e s t i g a t e f u r t h e r , the t o t a l data sample into  two  subperiods  (each c o n s i s t i n g of 495  was  divided  data p o i n t s ) ,  the parameters were r e - e s t i m t e d u s i n g the SS and the SL  and  methods  * The SS method was r e s t a r t e d at d i f f e r e n t parameter v a l u e s , but the non l i n e a r o p t i m i z a t i o n a l g o r i t h m used (Fletch guasiNewton method) always converged t o the above parameter v a l u e s . This appears to i n d i c a t e t h a t these c o - o r d i n a t e s uniquely maximize the j o i n t l i k e l i h o o d o f the given d a t a , i n a parameter range t h a t appears r e a s o n a b l e f o r an i n t e r e s t r a t e process. 9  98  on  these two s u b p e r i o d s .  The estimated  parameter values are as  follows: SS Method  SL Method  Parameters  SPJ  SJM  SP2  f> (=2m/cr )  627x10+7  188x10+'  5149.5  2195.0  jA.  1.152  1.3510  0.7884  1.2080  -0.1247  -0.0676  0.4032  0.4030  ra (x10 )  0.3451  0.1698  a- (x10*)  0.0011  0.0018  Even i n the two  subperiods,  2  X103  cA 2  2  negative  <A v a l u e .  value of o<^0.4  .  i s as e x p e c t e d . 5 0  t h e SL  consistently  estimate of jx by t h e SS method i s ,  e s t i m a t i o n procedure.  method y~ * last  s  increasing  estimate.  t h e process  1, which  This  over the past s e v e r a l y e a r s .  could  however, be  always  attributed  I n the SL method, Jx i s the value i s moving  Thus  when  lower  t o the towards  t o s t a b i l i z e , whereas i n the SS  the mean o f the sample p o i n t s  footnote).  a  T h i s i s because i n t e r e s t r a t e s have been more  The  which  estimates  The value of jx i s lower i n subperiod  less  t h e SL  method  The SS method, however, g i v e s a c o n s i s t e n t  or  than  SP2  interest  (as pointed rates  i n the  are r i s i n g , j x as  estimated by t h e SL method would be higher than t h e SS  estimate  so I t might be i n t e r e s t i n g t o r e c a l l from Appendix 7 t h a t the estimate o f jx f o r the SS method f o r (A = </z. was t h e mean o f t h e data p o i n t s . T h i s was because the SS d e n s i t y was t h e Gamma density. The a c t u a l mean of the data p o i n t s f o r the two subperiods i s 0.7884 x 10-3 and 0.1206 x 10~ respectively, which very c l o s e l y t a l l i e s with / A as estimated by t h e SS method f o r the two subperiods. Thus the SS d e n s i t y f o r g e n e r a l c* may be looked upon as a " g e n e r a l i z e d " Gamma d e n s i t y . 2  99  over the same p e r i o d . Since  the e s t i m t e of ch (and t h e r e f o r e c r as well) from the 2  SL method was u n a c c e p t a b l e s i , we estimates  from  the SS method.  t o t a l data was very c l o s e to T h i s has t o the f o l l o w i n g a)  The  consider  transition  estimation  the  parameter  Now, the e s t i m a t e o f <A from the  Yz.  .  He may t h e r e f o r e assume  ^ - ' / T - .  advantages: probability  f o r As'/z., and so  only  the  d e n s i t y f u n c t i o n i s known  "exact"  approach  to  parameter  f o r the i n t e r e s t r a t e process model can be  employed. b)  Considerable  simplification  is  achieved  in  the  e s t i m a t i o n o f t h e i n v e s t o r u t i l i t y dependent parameters i n the p a r t i c u l a r f u n c t i o n a l form o f the t e r m / l i q u i d i t y premium s t r u c t u r e t h a t we adopt Further,  the  adjustment  the a n a l y s i s i n Chapter valuation  i s quite  5, we  in know  neqliqible.  l a t e r on. ,  i s very s m a l l , and based on that  the  impact  Assuming t<= %,  over t h e 2 subperiods.  Purely f o r comparison,  bond  the parameters  jx , <r ) were estimated over the complete p e r i o d , as 2  on  well  as  a l l three methods  were used, and the r e s u l t i n g parameter e s t i m a t e s are as f o l l o w s :  s i As p o i n t e d out i n Chapter 3, neqative cA v a l u e s imply t h a t the i n s t a n t a n e o u s v a r i a n c e o f the i n t e r e s t r a t e process approaches oo as r approches z e r o . Such a model o f the i n t e r e s t r a t e process i s u n r e a l i s t i c , and t h e r e f o r e unacceptable,.  100  a) T o t a l  SL Method  SS Method  TRP Method  Parameters  Data:  f> (=2m/ -)  73 04.8  8183.0  924 4. 8  /Mx 103)  1.2930  0.9974  1.2320  <r {x 10»)  0.6905  0.6 886  m{x 102)  0.2522  0.3183  1  (r  u  b) Subperiod 1: 10993.9  20730.0  14099.9  103)  1.0314  0.7884  0.9771  cr{x 10*)  0.9518  0.9522  rn(x 102)  0. 52 32  0.6713  ^(X  c) Subperiod 2: 67 05.2  8564.0  7824.6  103)  1.3753  1.2070  1.3530  cMx 10*)  0. 4322  0.4 266  102)  0.1449  0.1669  /X.(X  ^  As  (X  expected,  t h e parameter  v a l u e s estimated assuming  are almost i d e n t i c a l t o those based on  the  SS  method  ^=Yi.  with  a  g e n e r a l (A . Thus,  we  assume  as  the  parameters  of the i n t e r e s t  process those e s t i m a t e d u s i n g the TBP method over data s e t , i e . ,  0\ =0.5  CT =0.690494x10-* 2  m=0.25221x10 f o r a l l f u r t h e r analyses on bond  =0.12934x10-* valuation.  the  rate  complete  101  7.2  S o l v i n g t h e Bond V a l u a t i o n The  basic  equation  was  Chapter 2; the p a r t i a l d i f f e r e n t i a l equation  was  L  market  ,  b=b(r),  rate  process  b (r) =m (jjL -r) . form of  We  c£(r,t).  expectations equation  In  correspondinq  start  with,  (PEXP) ,  have  let  instantaneous risk.  For  a (r) —•o~J~r  us  whereby  <q, - T<=j  may  Chapter 2 t o maturity  be s o l v e d we  -V  Cj.  consider <^>=0.  - <q z  is  -  the and  the  pure  T h i s reduces  the  ~-  =0  (  parabolic p a r t i a l the  bond  boundary c o n d i t i o n  t h a t correspondinq  I"  paqe  I ff  the maturity correspondinq  r e t r a c t i o n / e x t e n s i o n o p t i o n may  to  (7.2a)  ^ ^ C f , ^ ) ^ ^ ^ ^ ) ]  2,  value  They are  the l o n q m a t u r i t y ,  (see diaqram i n Chapter  yield  date v a l u e , and  <$L+,0)  SC^O  to  developed  the r e t r a c t i o n / e x t e n s i o n f e a t u r e .  represents  (2.9)  the  s u i t a b l e boundary c o n d i t i o n s , t h i s  d i f f e r e n t i a l equation  £ =0  - O  2  standard-deviation we  in  need t o make some assumption about the  -4 T n o ( ^ - T T )  (|  where  - <q  developed  to  -1_0-V^  G (r,£).  z  represents  chosen,  also To  <^  of  hypothesis  (2.9)  By imposinq  and  price  -+ c  -  (  perceived  interest  valuation  -v- ( b - acj>) (q  .I a, <q„  where a?a (r)  bond  Equation  i s the s h o r t for  more  (7.2b)  maturity,  details),  t o the l a s t date when the  be e x e r c i s e d ,  To  recoqnize  the  7  >  1  )  102 p o s s i b i l i t y t h a t the coupon on t h e long and different,  we have r e p r e s e n t e d  short bonds c o u l d be  (on the B.H.S.of equation  7.2b)  the lonq bond by G and the s h o r t bond by H. I t was conditions behaviour These  also at  noted v=0  and  in  2  that  further  boundary  oO would be r e q u i r e d , depending upon the  of the i n t e r e s t  boundary  Chapter  rate on  the  at  bond  these value  boundaries. process  (if  required) would have t o be c o n s i s t e n t with those imposed on  the  interest  conditions  process  r a t e process at t h e correspondinq boundaries.  i n t e r e s t r a t e process having the parameter v a l u e s i n the p r e v i o u s s e c t i o n , both r=0 and  so  cX) are n a t u r a l boundaries,  should be a b l e to s o l v e the d i f f e r e n t i a l equation  u s i n g the c o n d i t i o n s (7.2a) and technique  (7.2b).  However,  employed r e q u i r e s f u r t h e r assumptions  the  (7.1)  solution  (as w i l l become  shortly). The  s o l u t i o n technique w i l l be the standard i m p l i c i t  d i f f e r e n c i n q approach Brennan  S  (see McCraken 6 Dorn [ 4 4 ] , Schwartz  Schwartz [ 10 ]) .  are approximated  where  ,  and  £63], (7.1)  by d i f f e r e n c e e q u a t i o n s , y i e l d i n q  I,  (tl-0  Wi, a r e known, and  h and k are the d i s c r e t e increments i n to  finite  The d i f f e r e n t i a l s i n equation  'iz  time  estimated  no boundary c o n d i t i o n s need be imposed a t these p o i n t s .  Thus, we  clear  and  as  For the  maturity  respectively.  the  interest  rate  and  I t must be noted here t h a t j  103 i n c r e a s e s as we move away from the maturity date. time  step just p r i o r to m a t u r i t y ,  ^_,  Thus, at  (which i s the value of  the bond on the maturity date) i s known from c o n d i t i o n s When  we  X =0,  the system of eguations  equations j.  adopt  in  the  (7.2a).  a r e c u r s i v e method f o r s o l v i n g f o r G ( r , T )  (n*1)  (7.3)  therefore  unknowns (G;  represents  ,i=0,...n),  To be a b l e t o s o l v e f o r Gc j' , we need two (  From economic reasoning we  know  that  as  from (n-1)  at any time  step  more eguations. interest  rates  approach cO , bond values approach zero, i e .  T h i s o b s e r v a t i o n y i e l d s one  - o  <kj The  above  equation  absorbinq b o u n d a r y " . interest behaviour. at  process  more equation t o our system* i e .  could  j . . ,  hold  ....  < -*'  ™  7  strictly  only i f r= «0 were an  However, f o r the parameter values of  as estimated, r=  does not e x h i b i t  As time t o maturity i n c r e a s e s , bond value  the  maturity.  interest  r a t e may  probability  r e t u r n to reasonable l e v e l s b e f o r e  In a s t r i c t sense, when r= °o i s i n a c c e s s i b l e t h e r e i s  no meaning t o a s s i g n i n g a value Equation  absorbinq increases  hiqh i n t e r e s t r a t e v a l u e s , as t h e r e i s a p o s i t i v e  that  the  (7.4)  may  to  the  bond  at  that  point.  be looked upon as a l i m i t i n g v a l u e , and i n  -Referring to Appendix 1, a s i n g u l a r boundary i s i n a c c e s s i b l e in finite time i f the i n t e g r a l s of h,(r) a n d h (r) are unbounded. In case however, the i n t e g r a l o f 7T(r) were finite (with the i n t e g r a l s of h,(r) and b ( r ) being i n f i n i t e ) then the b a r r i e r would be both i n a c c e s s i b l e and absorbing (see Goel 6 Richter-Dyn [33]). For our process, the i n t e g r a l of 7T(r) i s unbounded, and so r= °° i s i n a c c e s s i b l e and not absorbing. In case IT (r) were i n t e g r a b l e , equation (7.4) would be s t r i c t l y valid. a  t  104 that l i g h t i s v a l i d . The  f i n a l eguation comes from the  valuation  equation  as  r  approaches  s i m i l a r t o t h e one used t o o b t a i n biases ., S 3  because  The  previous  ( i n numerical  Thus, bond values numerical  behaviour zero.  value) r and jx  lead  was  to  valid  over  This  i s not  true  at  r = °o  at  r = 0.  in  At r=0, we (7.1) i s  t h e t o t a l s t a t e space o f r , i t i s v a l i d as we make  a r b i t r a r i l y c l o s e approaches t o r=0. limit  serious  become very small q u i t e r a p i d l y as r r i s e s  value.  bond  are very c l o s e to zero.  t h e r e f o r e adopt a c o n t i n u i t y arqument: s i n c e equation valid  the  An approximation  (7.4) would  approximation  of  exists,  and  approximate  Thus, we assume  that  the  i t by s e t t i n q r=0 i n (7.1) t o  yield  r Y ^ . O T ) - Giji&.r)  +  =o  S t r i c t l y , we are assuminq that t h e f o l l o w i n g  (  limits  7.5)  exist  and  are as shown:  S 3 An e q u i v a l e n t assumption at r=0 t o t h a t a t r-e° i s t h a t o f an absorbinq b a r r i e r at r=0. T h i s would imply ( f o r a pure discount bond) B(0,T ) •= 1. The l a r q e r t h e f o r c e of mean r e v e r s i o n , the q r e a t e r t h e e r r o r due to such an assumption.  105  The  assumptions  seem  reasonable^*.  Thus,  we now  equations i n (n+1) unknowns, a t any value of  j.  ;  have  The  (n+1)  solution  procedure i s s t r a i g h t f orwardss . A s m a l l d e t a i l needs to be differencing  approach  used.  upper  of  To  limit  indicated  above,  <A , the  highlighted  the  variable  solution  the  value of r . new  condition  of  finite  procedure  has t o be bounded.  approach would be t o c o n s i d e r only a f i n i t e impose  the  Here, the s t a t e v a r i a b l e r has an  implement  state  about  segment  of  r,  equation (7.4), a t a s u i t a b l e  A b e t t e r approach however, i s t o t r a n s f o r m r  state  variable  which  is  bounded.  One  Consider  and  finite to  a the  transformation  * Ingersoll [ 3 9 ] has s o l v e d f o r the pure d i s c o u n t bond c o r r e s p o n d i n q t o the process where = Using h i s r e s u l t , we have _ i_ . s  n  B, , «\\\-  Hct)-^pC-A-c)](a  .  \  -- B . / 6  Since B i s f i n i t e as r approaches z e r o , both B and B are f i n i t e as r approaches zero. Be c o n j e c t u r e t h a t i n t r o d u c t i o n of a c o n t i n u o u s coupon and a boundary c o n d i t i o n of the form (7.2b), would not a l t e r the behaviour of the d e r i v a t i v e s of bond value as r->0. For further details see McCraken 6 Dorn [ 4 4 ] or Schwartz [ 6 3 ] . B r i e f l y , i t i s not necessary to i n v e r t an [ (n + 1) x (n + 1) ] matrix t o a r r i v e a t the s o l u t i o n v e c t o r a t each time s t e p . Osinq the equations (7.4) and (7.5) reduces the system o f equations i n (7.3) to a t r i d i a g o n a l system. A simple s o l u t i o n method i s available, which r e q u i r e s subtracting a suitable m u l t i p l e o f each eguation from the precedinq equation i n the system. 5  S  106  S  cO>r>0.  where 0<s< 1 a c c o r d i n g as c o n d i t i o n s can now variable.  be expressed  Brennan  6  The equation and the  i n terms of  Schwartz  s,  [12 j  the  have  boundary  new  state  adopted  the  trasformation 5  Here n can be any number so chosen t h a t a l a r g e p o r t i o n range of s i s i n the r e l e v a n t range of r . n=5,  the  i n t e r v a l r=(0% t o 20%)  This allows for interest  corresponded not  accuracy  For  our purpose,  t o s=0.65.  equally  divided;  v a r i a b l e was )  was  corresponding were t r i e d , and  7.3  ie., h  divided  corresponding t o r = to  the  relevant  range  n was chosen such t h a t  ,  the  grid  size  on the  to  the  reasonable  limits).  of r=^  i n t o 500 equal s t e p s , the  {0, /V3)  i n t o 300  r=(3yiA.,oo)  choice  of  was state  The range of s corresponding  into  range  200  number  to  of  steps,and the range of  s s  s t e p s . . S e v e r a l schemes  the s o l u t i o n v e c t o r of bond values was  sensitive  0.5).  F u r t h e r , the whole range of s(0,1)  not kept c o n s t a n t .  r= { JX./3,3JX,  in  the  To c l a r i f y , i f we s e t  corresponds t o s= (1.0 to  greater  rates.  of  of  grid  not  too  points (within  Bond V a l u a t i o n Under the Pure E x p e c t a t i o n s Model The bond  b a s i c p a r t i a l d i f f e r e n t i a l equation  (p.d.e)  v a l u a t i o n under the pure e x p e c t a t i o n s h y p o t h e s i s  obtained by s e t t i n g  |=0  i n equation  (2.9) of Chapter  governing (PEXP) i s 2.  This  107  was  developed  in  the  r e t r a c t a b l e / e x t e n d i b l e bonds T a b l e XIX)  were  valued  previous <E1 t o E19)  using  the  section. in  methods  our of  all sample  the  20 (see  previous  section. Before we proceed continuous  with f u r t h e r a n a l y s i s , the assumption  coupon payments on bonds needs to be j u s t i f i e d .  F e d e r a l bonds pay coupon semi-annually, and so  of All  coupon  payments  from the Government are d i s c r e t e .  However,  quoted  bond p r i c e s always exclude the coupon i n t e r e s t ,  i e . , the  buyer  o f the bond pays the s e l l e r the agreed purchase  price for  to  the  bondholders  the bond p l u s the accumulated  p r o p o r t i o n a l coupon from  coupon date to the t r a n s a c t i o n date.  the  last  T h i s arrangement i s almost  e q u i v a l e n t t o continuous coupon payments to the h o l d e r * . 5  To  compare  model  prices  with market p r i c e s , an approach  alonq the l i n e s o f I n q e r s o l l [ 3 8 ] was adopted. error  The mean  square  (MSE) may be computed as  MSE where G.» The MSE  -  (7.6,  and G^ a r e , r e s p e c t i v e l y , the market and model p r i c e s . (or i t s square r o o t  (RMSE)) i s b r o a d l y i n d i c a t i v e o f the  l a c k o f f i t between the model and t h e market p r i c e s . , F u r t h e r , a simple  reqression  decomposition  o f market p r i c e s on model p r i c e s permits the  of the MSE i n t o three component  parts.  Consider  * The difference between continuous coupons and this arranqement i s t h a t t h e h o l d e r gets no i n t e r e s t on the coupon, and l o s e s the compoudinq e f f e c t , i e . the " i n t e r e s t on i n t e r e s t " . , I t can be c l e a r l y seen t h a t t h i s omission i s very s m a l l , and can s a f e l y be i q n o r e d . s  108 the  regression &  =  4 ft ?  *  + ec  c  <7.7)  then T  Z  T  -J where  G*  prices.  and  G  stand  i-l  f o r the means of the market and model  The t h r e e component  1)  I  p a r t s may be i d e n t i f i e d as  The part due t o b i a s -  attributable  to  a  difference  between t h e mean l e v e l s . 2)  The  part  due  t o ^ #1,  responsiveness ( f < 1 ) o f  i e . , under the  model  ((3 >1) to  or  over  market  price  movements. 3} The  The part due t o r e s i d u a l e r r o r .  results  of  t h e r e g r e s s i o n and t h e e r r o r decomposition f o r  the model based on t h e pure e x p e c t a t i o n s presented  in  T a b l e s XXI  through  XL  hypothesis in  column  (PEXP) 1.  are  Cursory  examination c l e a r l y r e v e a l s that the predominant element of the HSE  across  a l l bonds i s b i a s .  T h i s i s a l s o i n d i c a t e d by n o t i n g  t h a t , f o r t h e PEXP model, t h e mean e r r o r [ which i s _ L ^ (G^* - G-) ] is  c o n s i s t e n t l y negative  the model o v e r p r i c e s expected model. some  yield  f o r a l l bonds.  t h e bonds, which i m p l i e s that the  on the bonds i s higher  One p o s s i b l e explanation liquidity  of longer  The i n d i c a t i o n i s t h a t  o r term premium  than instantaneous  markets  than that assumed  i s that  the  market  i n the  requires  i n the expected r e t u r n on bonds  maturity.  TABLE XXI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 4% JAN.l, 1963 (Rl)  MODEL  PURE EXP.  LIQ. PREM.  REV.TAX* (50%)  REV.TAX* (25%)  C.G. TAX** (10%)  C.G. TAX** (20%)  0.755  0.705  0.697  0.701  0.692  0.682  RMSE  0.812  2.554  0.419  1.350  1.916  2.652  MEAN ERROR  0.361  2.458  -0.005  1.253  1.831  2.572  ESTIMATED SLOPE  0.515  0.534  1.016  0.695  0.630  0.566  (S.E. OF SLOPE)  0.039  0.046  0.090  0.061  0.056  0.052  46.912  46.572  -1.568  30.523  37.074  43.618  3.861  4.424  8.783  5.877  5.407  4.942  DUE TO BIAS  0.197  0.926  0.000  0.861  0.913  0.940  8/1  0.582  0.047  0.017  0.041  0.036  0.032  RES.VARIANCE  0.219  0.026  0.982  0.097  0.049  0.026  MISSPEC ERROR  0.514  6.352  0.003  1.646  3.489  6.845  RESID.ERROR  0.144  0.174  0.173  0.176  0.181  R  2  EST.INTERCEPT (S.E. OF INTR) FRACTION OF ERROR  * "*  The Revenue Tax models The C a p i t a l G a i n s  incorporate the  «  0.188  l i q u i d i t y premium a s s u m p t i o n .  Tax model i n c o r p o r a t e the  l i q u i d i t y premium a s s u m p t i o n ,  as  well  as  a  Revenue Tax at  25%.  o  TABLE COMPARISON  XXII  OE MODEL AND MARKET P R I C E S BOND  : 5h%  OCT.l,  1960  ( A L L MODELS)  (El)  LIQ. PREM.  REV.TAX  REV.TAX  C.G.TAX  C.G.TAX  EXP.  (50%)  (25%)  (10%)  (20%)  R'  0.700  0.668  0.667  0.667  0.664  0.661  0.520  RMSE  3.123  0.751  0.656  0.694  0.753  0.837  0.505  -1.877  0.670  0.634  0.653  0.699  0.763  0.393  SLOPE  0.087  0.439  0.870  0.583  0.508  0.435  0.454  SLOPE)  0.009  0.053  0.107  0.071  0.062  0.054  0.075  91.609  56.691  13.553  42.279  49.826  57.092  55.033.  1.022  5.387  10.698  7.162  6.273  5.411  7.614  DUE TO B I A S  0.361  0.794  0.934  0.885  0.861  0.830  0.606  6 * 1  0.636  0.156  0.000  0.056  0.088  0.128  0.235  RES.VARIANCE  0.002  0.049  0.065  0.058  0.049  0.040  0.158  M I S S P E C ERROR  9.731  0.537  0.403  0.454  0.539  0.672  0.214  RESID.  0.025  0.028  0.028  0.028  0.028  0.028  0.040  PURE  MODEL  MEAN ERROR ESTIMATED (S.E. EST.  OF  INTERCEPT  (S.E.  OF  FRACTION  INTR) OF ERROR  ERROR  See  footnote  in Table  XXI  .  "NAIVE"  TABLE COMPARISON  XXIII  OF MODEL AND MARKET P R I C E S BOND  : 5h O C T . l ,  1962  ( A L L MODELS)  (E2)  C.G.TAX  C.G.TAX  "NAIVE"  REV.TAX  REV.TAX  EXP.  LIQ. PREM.  (50%)  (25%)  R'  0.792  0.798  0.796  0.797  0.802  0.807  0.686  RMSE  4.729  1.854  2.305  0.797  0.802  0.807  0.686  -4.223  1.636  2.181  1.910  1.949  2.020  0.826  SLOPE  0.393  0.697  1.347  0.914  0.813  0.714  0.509  SLOPE)  0.019  0.033  0.065  0.044  0.038  0.033  0.033  60.660  32.236  -32.742  10.571  20.720  30.741  50.781  2.083  3.418  6.614  4.484  3.928  3.396  3.395  0.797  0.778  0.895  0.889  0.880  0.854  0.249  0.182  0.093  0.020  0.002  0.020  0.057  0.499  VARIANCE  0.020  0.127  0.083  0.107  0.099  0.087  0.250  M I S S P E C ERROR  21.914  2.999  4.869  3.662  3.885  4.354  2.050  0.453  0.438  0.443  0.441  0.430  0.419  0.684  PURE  MODEL  MEAN ERROR ESTIMATED (S.E.  OF  EST.INTERCEPT (S.E.  OF I N T R )  FRACTION  OF ERROR  DUE TO B I A S B  *  RES.  1  RESID.ERROR  -  See  footnote in  Table  XXI  (10%)  (20%)  TABLE COMPARISON  XXIV  AND MARKET P R I C E S ( A L L MODELS) BOND : 5*2% D E C . 1 5 , 1964 ( E 3 )  OF MODEL  C.G.TAX  REV.TAX  C.G.TAX  "NAIVE"  REV.TAX  EXP.  LIQ. PREM.  0.756  0.846  0.851  0.848  0.856  0.865  0.704  5.739  2.115  2.698  2.336  2.534  2.858  1.885  -5.513  1.851  2.441  2.157  2.378  2.708  -0.545  SLOPE  0.639  0.812  1.530  1.052  0.954  0.858  0.596  SLOPE)  0.036  0.035  0.064  0.045  0.039  0.034  0.039  INTERCEPT  33.585  20.817  -50.919  -3.087  6.966  16.939  41.262  OF I N T R )  3.997  3.553  6.526  4.542  3.986  3.449  4.062  DUE TO B I A S  0.922  0.766  0.818  0.852  0.881  0.897  0.083  S  0.037  0.051  0.072  0.000  0.000  0.014  0.473  VARIANCE  0.039  0.182  0.108  0.147  0.118  0.087  0.442  M I S S P E C ERROR  31.650  3.658  6.490  4.655  5.662  7.451  1.983  1.294  0.816  0.789  0.803  0.761  0.717  1.573  PURE  MODEL  R  2  RMSE MEAN ERROR ESTIMATED (S.E.  OF  EST. (S.E.  (50%)  F R A C T I O N OF ERROR  5  s  RES.  1  RESID.  See  ERROR ,  footnote  in Table  XXI.  (10%)  (25%)  -  (20%)  TABLE  XXV  COMPARISON OF MODEL AND MARKET P R I C E S ( A L L BOND : 5h A P R I L 1, 1963 ( E 4 )  MODEL  PURE  LIQ.  EXP.  PREM.  0.558  0.651  5.051  1.654  -4.771  1.460  0.392  REV.TAX (50%)  REV.TAX (25%)  MODELS)  C.G.TAX  C.G.TAX  (10%)  (20%)  "NAIVE"  0.652  0.667  0.683  0.488  2.087  2.074  2.096  1.621  2.450  1.953  1.946  1.967  0.895  0.794  1.550  1.046  0.945  0.843  0.037  0.062  0.120  0.081  0.071  0.061  0.050  61.017  22.381  -53.110  -2.792  7.434  17.889  56.297  4.025  6.328  12.152  8.270  7.219  6.210  '5.140  DUE TO B I A S  0.892  0.779  0.900  0.874  0.879  0.880  0.305  B *  0.080  0.022  0.018  0.000  0.000  0.007  0.391  0.026  0.198  0.081  0.124  0.120  0.112  0.302  24.828  2.194  6.124  3.813  3.784  3.901  1.832  0.688  0.543  0.539  0.541  0.517  0.493  0.796  R  2  RMSE MEAN  ERROR  ESTIMATED (S.E.  OF  SLOPE  -  SLOPE)  EST.INTERCEPT (S.E.  OF  FRACTION  RES.  INTR) OF  VARIANCE ERROR  RESID.ERROR  See  footnote  •  2.581  ERROR  1  MISSPEC  0.653  in  Table  XXI  0.459 s  TABLE  XXVI  COMPARISON OF MODEL AND MARKET P R I C E S BOND  PURE  MODEL  EXP.  R  C.G.TAX  REV.TAX  (10%) .  (25%)  C.G.TAX  MOV.  (20%)  AVG.  0.401  0.378  0.710  0.349  1.462  2.291  0.907  1.544  1.931  2.444  0.584  1.661  -1.040  1.937  0.430  1.201  1.587  2.081  -0.181  0.141  SLOPE  0.490  0.406  0.745  0.519  0.447  0.379  0.861  0.290  SLOPE)  0.024  0.036"  0.069  0.047  0.042  0.037  0.043  0.030  OF INTR)  .  49.773  59.387  25.484  48.096  55.267  62.102  13.481  70.116  2.416  3.497  6.859  4.620  4.145  3.669  4.254  3.054  0.506 .  0.715  0.224  0.605  0.675  0.725  0.096  0.007  0.359  0.176  0.053  0.150  0.163  0.169  0.048  0.755  0.134  0.108  0.721  0.243  0.161  0.104  0.855  0.237  1.851  4.681  0.229  1.804  3.129  5.347  0.049  2.104  0.287  0.568  0.594  0.581  0.603  0.626  0.292  0.656  F R A C T I O N OF ERROR DUE TO B I A S 6  *  RES.  1  VARIANCE  M I S S P E C ERROR RESID.ERROR  See  "NAIVE'  0.423  EST.INTERCEPT (S.E.  (50%)  ( A L L MODELS)  (E5)  0.410  MEAN ERROR  OF  REV.TAX  1971  0.436  RMSE  (S.E.  A P R I L 1,  0.714  2  ESTIMATED  LIQ. PREM.  : 6%  f o o t n o t e i n Table XXI.  TABLE  XXVII  COMPARISON OF MODEL AND MARKET P R I C E S BOND  : 6VDEC.1,  1973  ( A L L MODELS)  (E6)  C.G.TAX  C.G.TAX  MOV.  (10%)  (20%)  AVG.  "NAIVE"  0.737  0.827  0.749  2.815  3.499  1.568  5.018  1.435  1.878  2.448  -0.906  -3.084  1.046  0.729  0.650  0.574  0.951  0.420  0.020  0.037  0.025  0.023  0.021  0.026  0.015  54.559  43.623  -3.828  27.809  35.798  43.555  3.932  56.080  1.455  1.949  3.718  2.538  2.294  2.053  2.696  1.542  0.674  0.400  0.184  0.385  0.445  0.489  0.334  0.377  0.282  0.380  0.001  0.178  0.252  0.309  0.005  0.528  RES.VARIANCE  0.042  0.218  0.813  0.436  0.302  0.201  0.660  0.093  M I S S P E C ERROR  45.561  8.172  0.546  3.017  5.529  9.779  0.836  22.828  2.041  2.285  2.383  2.332  2.395  2.466  1.625  2.353  R  REV.TAX  EXP.  0.782  0.756  0.746  0.751  0.745  6.899  3.234  1.711  2.313  -5.667  2.046  0.736  SLOPE  0.424  0.571  SLOPE)  0.013  2  RMSE MEAN ERROR ESTIMATED (S.E.  OF  EST.  INTERCEPT  (S.E.  OF  INTR)  F R A C T I O N OF ERROR DUE TO B I A S  6  *  1  RESID.ERROR  See  REV.TAX  LIQ. PREM.  PURE  MODEL  footnote in  Table  XXI.  (50%)  (25%)  TABLE  XXVIII  COMPARISON OF MODEL AND MARKET —  BOND  :  Tk A P R I L 1,  PRICES  1974  ( A L L MODELS)  (E7)"  SE  S&.  "  0.783  0.759  0.772  0.769  0.764  0.759  0.759  0.071  15.209  3.758  3.892  6.441  6.256  5.947  4.789  11.734  -14.184  -1.544  -3.590  -5.917  -5.571  -5.030  -3.868  -7.191  SLOPE  0.313  0.431  0.765  0.525  0.488  0.454  0.491  0.237  SLOPE)  0.010  0.015  0.027  0.018  0.017  0.016  0.018  0.056  65.557  57.229  21.116  45.229  49.384  53.287  49.952  75.655  1.253  1.649  2.873  2.032  1.911  1.794  1.920  6.127  0.869  0.168  0.850  0.843  0.793  0.715  0.652  0.375  0.123  0.702  0.035  0.113  0.161  0.233  0.267  0.273  VARIANCE  0.007  0.129  0.113  0.042  0.045  0.051  0.079  0.350  M I S S P E C ERROR  229.686  12.304  13.427  39.748  37.355  33.552  21.110  89.432  1.640  1.824  1.725  1.747  1.785  1.824  1.828  48.268  R  2  RMSE MEAN ERROR ESTIMATED (S.E.  OF  EST.  INTERCEPT  (S.E.  OF  FRACTION  INTR) OF ERROR  DUE TO B I A S 1 RES.  RESID.ERROR  See  footnote  in  Table  XXI.  c  •  1i«  SS:  "NAIVE"  T A B L E XXIX COMPARISON OF MODEL AND MARKET P R I C E S BOND : 8% O C T . l ,  MODEL  LIQ. PREM.  REV.TAX (50%)  C.G.TAX (10%)  C.G.TAX (20%)  "NAIVE"  0.750  0.728  0 . 732  0.730  0.730  0.730  0.729  0.682  19.505  4.926  1. 719  2.976  3.039  3.099  3.903  10.338  -18.620  -3.227  0 . 427  -1.476  -1.252  -0.931  -2.630  -8.011  SLOPE  0,312  0.426  0 . 791  0.548  0.522  0.499  0.508  0.278  SLOPE)  0.011  0.017  0 . 031  0.021  0.020  0.019  0.020  0.012  INTERCEPT  66.140  58.682  2 2 . 158  46.510  49.348  51.932  50.174  73.329  OF INTR)  1.457  1.839  3. 261  2.312  2.199  2.096  2.175  1.405  0.911  0.429  0 . 061  0.245  0.169  0.090  0.453  0.600  0.082  0.473  0 . 146  0.487  0.574  0.664  0.390  0.373  VARIANCE  0.005  0.097  0 . 791  0.266  0.255  0.245  0.155  0.026  M I S S P E C ERROR  378.277  21.894  0 . 615  6.498  6.881  7.248  12.865  104.095  2.185  2.376  2. 341  2.358  2.357  2.357  2.372  2.785  2  RMSE MEAN ERROR ESTIMATED (S.E.  OF  EST. (S.E.  F R A C T I O N OF ERROR DUE TO B I A S • B *  1  RES.  RESID.ERROR  See  footnote in Table  XXI.  REV.TAX (25%)  ( A L L MODELS)  (E8)  MOV. AVG.  R  PURE EXP.  1974  TABLE  XXX  COMPARISON OF MODEL AND MARKET P R I C E S BOND :  REV.TAX (50%)  REV.TAX (25%)  C.G.TAX  0.728  0.709  0.723  0.722  0.721  17.237  4.216  4.659  7.680  -15.831  -2.514  -4.244  SLOPE  0.284  0.467  SLOPE)  0.010  RMSE ERROR  ESTIMATED (S.E. EST.  OF  INTERCEPT  (S.E.  OF  • FRACTION  INTR)  *  RES.  VARIANCE '  ERROR  RESID.ERROR  See  footnote  in  MOV. "NAIVE"  0.718  0.707  0.680  7.519  7.192  2.757  9.480  -7.044  -6.765  -6.282  -1.188  -7.280  0.716  0.502  0.478  0.457  0.583  0.304  0.018  0.027  0.019  0.018  0.018  0.023  0.013  69.093  53.508  26.048  47.631  50.368  52.896  42.094  69.323  1.299  1.989  2.989  2.154  2.056  1.968  2.457  1.449  0.843  0.355  0.829  0.841  0.809  0.762  0.185  0.589  0.147  0.488  0.048  0.114  0.143  0.185  0.448  0.376  0.008  0.155  0.121  0.044  0.047  0.051  0.366  0.033  294.562  15.012  19.076  56.353  53.884  49.050  4.818  86.825  .2.585  2.768  2.633  2.641  2.658  2.679  2.782  3.045  '  (10%)  (20%)  j  1  MISSPEC  C.G.TAX  AVG. .  OF ERROR  DUE TO B I A S S  ( A L L MODELS)  (E9)  LIQ. PREM.  2  MEAN  1975  EXP.  PURE  MODEL  R  Ih'i D E C . 1 5 ,  Table  XXI.  TABLE  XXXI  COMPARISON OF MODEL AND MARKET BOND  : 6% A U G . l ,  REV.TAX  PRICES  1976  ( A L L MODELS)  (E10)  C.G.TAX  REV.TAX  C.G.TAX •  "NAIVE"  MOV.  PURE  LIQ.  EXP.  PREM .  0.519  0.588  0.540  0.540  0.548  0.557  0.559  0.516  RMSE  . 9.065  2.563  3.087  4.105  3.887  3.592  3.175  5.721  MEAN ERROR  -7.152  -0.272  -2.468  -3.140  -2.768  -2.251  -1.959  SLOPE  0.245  0.475  0.664  0.458  0.446  0.438  0.482  0.298  SLOPE)  0.015  0.026  0.040  0.027  0.026  0.025  0.028  0.018  73.003  51.834  31.565  52.210  53.584  54.703  50.313  68.404  1.650  2.595  4.091  2.838  2.716  2.602  2.844  1.952  0.622  0.011  0.638  0.585  0.507  0.392  0.380  0.413  0.343  0.626  0.081  0.257  0.319  0.408  0.366  0.501  0.033  0.362  0.279  0.157  0.173  0.198  0.253  0.085  79.409  4.190  6.870  14.194  12.497  10.341  7.531  29.936  2.781  2.381  2.662  2.658  2.618  2.566  2.551  2.799  MODEL  R  2  ESTIMATED (S.E.  OF  EST.  INTERCEPT  (S.E.  OF  FRACTION  INTR)  *  1  RES.  VARIANCE  MISSPEC  ERROR  RESID.ERROR  See  (10%)  (25%)  O F ERROR  DUE TO B I A S  0  (50%)  footnote  in  T a b l e XXI  .  .  (20%)  AVG.  -  -3.677  TABLE  XXXII  COMPARISON OF MODEL AND MARKET BOND:  MODEL  7% J u l y  PRICES(ALL  1977  LIQ.  C.G.TAX  C.G.TAX  EXP.  PREM.  (50%)  (25%)  (10%)  (20%)  AVG.  0.552  0.649  0.589  0.590  0.602  0.615  0.542  0.538  8.246  2.930  2.783  3.877  3.736  3.560  6.374  5.143  -5.799  0.477  -2.069  -2.610  -2.240  -1.730  -5.159  -2.498  SLOPE  0.226  0.411  0.590  0.408  0.396  . 0.386  0.324  0.279  OF S L O P E )  0.013  0.019  0.032  0.022  0.021  0.019  0.019  0.016  INTERCEPT  75.459  58.570  39.454  57.633  58.998  60.261  65.376  70.839  OF I N T R )  1.401  1.950  3.254  2.263  2.136  2.012  2.032  1.717  DUE TO B I A S  0.494  0.026  0.552  0.453  0.359  0.236  0.655  0.235  6  0.472  0.769  0.181  0.410  0.497  0.611  0.288  0.676  VARIANCE  0.033  0.204  0.265  0.136  0.142  0.151  0.056  0.087  M I S S P E C ERROR  65.755  6.830  5.694  12.979  11.965  10.749  38.344  24.141  2.244  1.754  2.055  2.052  1.993  1.926  2.292  2.310  2  RMSE MEAN ERROR ESTIMATED (S.E. EST. (S.E.  F R A C T I O N OF  RESID.  See  MOV.  "NAIVE"  ERROR  tl  RES.  REV.TAX  MODELS)  (Ell)  PURE  R  REV.TAX  1,  ERROR  footnote  in  Table  XXI.  '  .  TABLE XXXIII COMPARISON O F MODEL.AND BOND:  ( A L L MODELS)  (E12)  "NAIVE"  REV.TAX  C.G.TAX  C.G.TAX  MOV.  (50%)  (25%)  (10%)  (20%)  AVG.  0.691  0.814  0.699  0.707  0.723  0.742  0.653  0.827  6.260  2.476  2.293  2.924  2.468  1.931  6.359  0.956  -5.675  2.187  -1.938  -2.567  -2.017  -1.282  -6.115  0.048  SLOPE  0.443  0.755  1.024  0.726  0.702  0.682  0.610  0.897  OF S L O P E )  0.020  0.024  0.045  0.031  0.029  0.027  0.030  0.028  INTERCEPT  53.101  26.045  -4.431  25.455  28.254  30.846  35.147  10.291  OF I N T R )  2.136  2.410  4.677  3.271  3.026  2.780  3.222  2.796  0.780  0.712  0.771  0.668  0.441  0.924  0.002  0.068  0.000  0.057  0.104  0.213  0.032  0.054  0.151  0.286  0.171  0.227  0.345  0.042  0.942  5.204  3.754  7.087  4.709  2.440  38.713  0.052  0.929  1.505  1.463  1.383  1.289  1.734  0.863  l  RMSE MEAN ERROR ESTIMATED (S.E.  (S.E.  FRACTION  OF ERROR  DUE TO B I A S 6 * 1  0.821 0.138  RES.VARIANCE M I S S P E C ERROR RESID.  See  1978  REV.TAX  EXP.  EST.  MARKET P R I C E S Oct.l,  LIQ. PREM.  PURE  MODEL  K  7 3/4%  ERROR  0.039 37.646 1.543  f o o t n o t e i n T a b l e XXI •  TABLE  XXXIV  COMPARISON OF MODEL AND MARKET BOND:  MODEL  R  2  RMSE MEAN ERROR ESTIMATED (S.E.OF  SLOPE  SLOPE)  EST.INTERCEPT (S.E.  OF I N T R )  FRACTION  t  1  RES.VARIANCE M I S S P E C ERROR . RESID.ERROR  See  REV.TAX  footnote in  Table  1980  P R I C E S ( A L L MODELS) (E13)  PURE  LIQ.  C.G.TAX  C.G.TAX  EXP.  PREM.  (50%)  (25%)  (10%)  (20%)  AVG.  0.638  0.744  0.648  0.656  0.671  0.686  0.617  0.761  7.479  3.077  3.941  4.215  3.525  2.691  7.019  1.555  -7.005  2.600  -3.436  -3.804  -3.035  -2.018  -6.739  0.316  0.571  0.837  1.328  0.942  0.900  0.858  0.843  0.929  0.029  0.034  0.068  0.047  0.043  0.040  0.046  0.036  37.987  18.135  -36.699  2.104  7.005  12.094  9.617  7.170  3.149  3.263  6.904  4.835  4.438  4.047  4.850  3.543  0.877  0.713  0.759  0.814  0.741  0.562'  0.92175  0.041  0.060  0.027  0.023  0.000  0.005  0.022  0.003  0.012  0.062  0.259  0.216  0.185  0.253  0.415  0.074  0.946  52.480  7.017  12.168  14.482  9.276  4.235  45.602  0.130  3.469  2.453  3.367  3.291  3.153  3.008  3.671  2.288  OF ERROR  DUE TO B I A S 6  7h D e c . l ,  XXI.  REV.TAX  .  MOV.  "NAIVE"  TABLE  XXXV  COMPARISON OF MODEL AND MARKET P R I C E S ( A L L MODELS) BOND: 7% A P R I L 1, 1979 ( E 1 4 )  MOV.  "NAIVE"  PURE  LIQ.  REV.TAX  REV.TAX  C.G.TAX  C.G.TAX  EXP.  PREM.  (50%)  (25%)  (10%)  (20%)  R2  0.661  0 . 808  0.702  0.706  0.726  0.746  0.690  0.835  RMSE  4.537  2. 861-  3.000  2.737  2.158  1.551  3.116  1.097  -4.135  2 . 591  -2.513  -2.323  -1.649  -0.769  -2.743  0.190  SLOPE  0.673  0 . 872  1.522  1.062  1.001  0.930  0.974  0.977  SLOPE)  0.035  0 . 030  0.072  0.049  0.044  0.039  0.047  0.031  INTERCEPT  28.979  1 4 . 671  -54.673  -8.526  -1.837  5.156  -0.190  2.407  OF I N T R )  3.565  2 . 934  7.213  4.980  4.442  3.918  4.755  3.077  DUE TO B I A S  0.830  0 . 819  0.701  0.720  0.579  0.246  0.773  0.030  0.052  0 . 014  0.063  0.000  0.002  0.005  0.000  0.002  B  VARIANCE  0.116  0 . 166  0.234  0.278  0.417  0.748  0.225  0.967  18.184  6. 828  6.891  5.404  2.710  0.604  7.518  0.039  M I S S P E C ERROR  2.403  1. 361  2.113  2.088  1.946  1.801  2.195  1.166  MODEL  MEAN ERROR ESTIMATED (S.E.  OF  EST. (S.E.  F R A C T I O N OF ERROR  * 1  RES.  RESID.ERROR  See  f o o t n o t e on Table XXI .  AVG.  .  TABLE  XXXVI  COMPARISON OF MODEL AND MARKET P R I C E S BOND:  EST.  OF  C.G.TAX  C.G.TAX  MOV.  EXP.  (25%)  (10%)  (20%)  AVG-  0.725  0.754  0.750  0.752  0.756  0.760  0.741  0.757  6.868  2.280  3.192  2.557  2.744  3.020  1.519  2.390  -6.422  1.662  2.998  2.264  2.446  2.719  0.658  -0.949  0.623  1.116  0.787  0.747  0.707  0.495  0.460  0.694  SLOPE  0.028  0.050  0.035  0.033  0.031  0.022  0.022  0.032  SLOPE)  53.645  40.568  -8.860  24.069  28.373  32.583  32.506  2.896  5.169  3.653  3.427  3.206  3.367  2.339  2.494  0.531  0.881  0.884  0.794  0.810  0.187  0.157  0.874  0.246  0.003  0.038  0.052  0.065  0.286  0.642  0.098  0.222  0.115  0.177  0.152  0.124  0.525  0.199  0.027  4.044  9.019  5.376  6.383  7.992  1.096  4.575  45.877  1.155  1.172  1.162  1.146  1.128  1.212  1.140  1.293  INTR)  F R A C T I O N OF ERROR DUE TO B I A S 6  J* 1  RES.  VARIANCE  MISSPEC  "NAIVE"  REV JTAX  (50%)  INTERCEPT  (S.E.  ( A L L MODELS)  (E15)  REV.TAX  MEAN ERROR  OF  1978  PREM.  RMSE  (S.E.  1,  LIQ.  2  ESTIMATED  APRIL  PURE  MODEL  R  9k%  ERROR  RESID.  ERROR  See  footnote  in  Table  XXI  .  52.524  TABLE  COMPARISON O F MODEL AND MARKET BOND:  MODFT  9k F E B . l .  P R I C E S ( A L L MODELS) (E16)  PURE  LIQ.  C.G.TAX  C.G.TAX  MOV.  EXP.  PREM.  (50%)  (25%)  (10%)  (20%)  AVG.  0.681  0.763  0.758  0.761  0.772  0.783  0.678  0.752  5.661  2.017  2.537  2.129  2.256  2.448  2.450  2.291  -5.107  1.405  2.212 '  1.772  1.922  2.138  -1.822  SLOPE  0.488  0.701  1.273  0.892  0.855  0.819  0.673  0.563  SLOPE)  0.029  0.034  0.064  0.044  0.041  0.038  0.041  0.028  50.396  31.851  -25.464  12.743  16.605  20.401  32.532  44.409  3.238  3.558  6.501  4.539  4.215  3.903  4.362  3.027  DUE TO B I A S  0.814  0.485  0.760  0.692  0.725  0.762  0.552  0.258  B *  0.130  0.187  0.028  0.011  0.022  0.033  0.146  0.476  VARIANCE  0.055  0.326  0.211-  0.296  0.251  0.203  0.301  0.265  M I S S P E C ERROR  30.256  2.738  5.080  3.193  3.810  4.775  4.197  3.856  1.791  1.329  1.359  1.343  1.282  1.219  1.809  1.394  R  2  RMSE MEAN ERROR ESTIMATED (S.E. EST.  OF  INTERCEPT  (S.E.  OF  FRACTION  INTR) OF ERROR  1  RES.  RESID.ERROR  See  footnote  in  Table  XXI•  REV.TAX  1977  XXXVII  REV.  TAX  • "NAIVE"  '  -1.164  .  TABLE  COMPARISON OF MODEL AND MARKET BOND  : 7h%  OCT.l,  XXXVIII  P R I C E S ( A L L MODELS)  1979  (E17)  "NAIVE'  REV.TAX.  REV.TAX  C.G.TAX  C.G.TAX  MOV.  EXP.  LIQ. PREM.  (50%)  (25%)  (10%)  (20%)  AVG.  0.570  0.700  0.581  0.594  0.611  0.630  0.726  0.714  5.700  2.598  2.942  3.384  2.912  2.319  2.922  1.174  -5.383  2.160  -2.673  -3.130  -2.594  -1.855  2.685  -0.094  SLOPE  0.053  0.610  1.095  0.763  0.709  0.655  0.729  0.535  0.039  0.035  0.083  0.056  0.050  0.045  0.040  0.030  SLOPE)  46.370  39.767  -12.404  20.960  26.857  32.784  28.706  45.823  INTERCEPT  4.086  3.468  8.482  5.780  5.148-  4.537  3.860  3.011  OF I N T R )  0.891  0.691  0.825  0.855  0.793  0.640  0.844  0.003  0.060  0.148  0.000  0.016  0.041  0.113  0.040  0.647  0.047  0.159  0.173  0.127  0.164  0.246  0.114  0.349  VARIANCE  30.956  5.674  7.154  9.993  7.084  4.051  7.560  1.913  M I S S P E C ERROR  1.543  1.077  1.506  1.459  1.395  1.327  0.982  1.027  PURE  MODEL  R  2  RMSE MEAN ERROR ESTIMATED (S.E.  OF  EST. (S.E.  FRACTION  OF ERROR  DUE TO B I A S B  *  RES.  1  RESID.  See  ERROR  footnote in  T a b l e XXI  .  TABLE  XXXIX  COMPARISON O F MODEL AND MARKET P R I C E S BOND  : 9% F E B . l ,  ( A L L MODELS)  1978 (E18)  MOV.  "NAIVE"  C.G.TAX  C.G.TAX  (25%)  (10%)  (20%)  AVG.  0.776  0.777  0.785  0.794  0.809  0.740  2.740  2.493  2.600  2.806  3.083  3.390  0.757  -1.016  2.670  2.452  2.560  2.764  3.034  3.355  0.436  SLOPE  0.590  0.640  1.234  0.838  0.762  0.685  0.737  0.650  OF S L O P E )  0.050  0.032  0.063  0.043  0.038  0.033  0.034  0.037  INTERCEPT  41.278  38.450  -20.968  18.634  26.395  34.280  29.299  36.017  OF I N T R )  5.186  3.280  6.351  4.304  3.810  3.330  3.402  3.769  DUE TO B I A S  0.642  0.949  0.967  0.969  0.970  0.968  0.979  0.332  6 *  0.134  0.026  0.003  0.003  0.007  0.014  0.007  0.298  RES.VARIANCE  0.222  0.024  0.029  0.026  0.022  0.017  0.013  0.369  MISSPEC  1.247  7.329  6.035  6.579  7.704  9.340  11.340  0.361  0.358  0.181  0.182  0.181  0.174  0.168  0.155  0.211  MODEL  R  2  RMSE MEAN ERROR ESTIMATED (S.E. EST. (S.E.  PURE EXP.  LIQ. PREM.  REV.TAX  0.561  0.777  1.267  (50%)  F R A C T I O N O F ERROR  1  RESID.  ERROR ERROR  See  footnote  i n Table  XXI  REV.TAX  TABLE XL COMPARISON OF MODEL AND MARKET P R I C E S BOND  MODEL  PURE  LIQ.  EXP.  PREM.  : 9% O C T . l ,  1980  ( A L L MODELS)  (E19)  REV.TAX  REV.TAX  C.G.TAX  C.G.TAX  MOV.  (50%)  (25%)  (10%)  (20%)  AVG.  "NAIVE"  R  0.584  0.646  0.543  0.566  0.581  0.596  .0.663  0.699  RMSE  7.260  2.427  1.878  3.591  3.444  3.174  4.281  2.193  -7.037  1.859  -1.352  -3.351  -3.181  -2.857  4.077  0.322  0.528  0.586  1.189  0.819  0.751  0.686  0.688  0.455  0.042  0.041  0.103  0.068  0.060  0.053  0.046  0.028  45.410  44.148  -21.347  16.099  23.431  30.724  35.287  56.886  4.704  4.207  10.904  7.309  6.501  5.731  4.651  2,942  0.586  0.518  0.870  0.853  0.810  0.906  0.021  0.195  0.009  0.006  0.018  0.043  0.026  0.750  0.218  0.471  0.122  0.128  0.145  0.066  0.227  4.606  1.865  11.316  10.337  8.604  17.108  3.716  1.287  1.662  1.579  1.526  1.470  1.225  1.096  MEAN ERROR ESTIMATED (S.E.  SLOPE  OF S L O P E )  EST.INTERCEPT (S.E.  OF INTR)  FRACTION  OF ERROR  DUE TO B I A S  0.939  8 * 1 RES.  0.031 VARIANCE  0.028  M I S S P E C ERROR  51.199  RESID.ERROR  See  footnote  1.516  in Table  XXI  r-o CO  129  7.4  E s t i m a t i n g the L i q u i d i t y / T e r m Premium In Chapter  2, we had as the b a s i c bond v a l u a t i o n e q u a t i o n  + SL) ~T = S A(r»t,T)  where  investors. now  Paramters  =  AC^/r)  i s t h e instantaneous excess r e t u r n expected by  Under the PEXP model, we had  make  assumptions  the l i n e s of independent  t,  set  A-(r,t, X )=0.  We  about aggregate i n v e s t o r behaviour along  Ingersoll [39], of  t2.8)  First,  ie., i t is  time  we  assume  that  homogeneous.  ^  is  Second, we  assume  (TAR . 3>(1~) = which y i e l d s  - k?.r  (see Chapter 2, equation (2.8))  \(f,V)  Vasicek [ 7 2 ] = constant.  . _cfe, + ^ T ) _ ^ _  and  Brennan  6  may  find  _(7.9)  Schwartz  [1.0] both  T h i s i s a statement about the p r i c e  excess r e t u r n ) o f i n s t a n t a n e o u s standard One  (7.8)  the  assumption  (in terms of  deviation  <|> = constant  risk  more  intuitively  comprehensible than t h e assumption i n (7.8)., However, be  shown  shortly,  assume  as  will  the assumption o f equation (7.8) l e a d s t o a  simple s t r u c t u r e f o r t h e form  of A  .  Much  of  the  existinq  l i t e r a t u r e on t h e term s t r u c t u r e of i n t e r e s t r a t e s addresses the form  and determinants of A- .  I t w i l l be shown that  (7.8) l e a d s  130 to  a  form  for  X  that  i s consistent  with  the  existing  literature. Ingersoll(op  c i t ) points  (and assuming t h e i n t e r e s t assumed  here),  g i v e n by  r  b(i70  the  rate  r  j^'  2  of  the  form  j  i'  I - Her) £  J  is  (7.10)  k  ]/<r  2  2  = [ 1>(m«-A) 0-e-  the  ratio  Ar  (7.10)  (B, /B)  )/2A]-»  that  B,/B =  [1-H(T)e-^  i s independent  f u n c t i o n o f time t o m a t u r i t y . as  be  » (m«2 + 2 cr )^ "  I t can be seen from ie.  to  * |  =  2  H(f)  process  Y) m'/A-'t -v ^ T  = [ m» - (m» z + 2o- ) A  t h a t under t h i s assumption  v a l u e of t h e pure d i s c o u n t bond B ( r , T )  "1 UCt)J  where m» = (m-k^ ) l  out  of  r  ]=q(f ),  and s t r i c t l y a  This implies that the  i n d i c a t e d by t h e r e l a t i o n s h i p i n equation  r  choice  of  (7.8) l e a d s t o  an e x p r e s s i o n f o r the l i q u i d i t y / t e r m premium as  (7.11)  As p o i n t e d out by I n g e r s o l l [ 3 9 ] , f o r ( k, • k^ r) > 0, the term premium i s a p o s i t i v e , (k  (  r)<0,  concave  function,  i s n e g a t i v e , d e c r e a s i n g and convex.  the usual p r o p e r t i e s Further,  increasing  associated  with  the  liquidity  and f o r These a r e premium.  f o r a qiven m a t u r i t y , the r e l a t i o n between A- and r as  qiven by equation  (7.11), i s c o n s i s t e n t with some of the popular  131  assumptions about t e r m / l i q u i d i t y a)  a constant term (set k^ =0) .  premia, i e . ,  premium independent  of  interest  rates  T h i s would s p e c i f y t h a t the expected  rate  of r e t u r n on a qiven maturity o f bonds be a constant i n excess o f t h e instantaneous i n t e r e s t b)  term  premiums  proportional  rate.  t o the i n t e r e s t r a t e (set  k, - 0 ) . T h i s would s p e c i f y t h a t the r e t u r n on a maturity  of  bonds  be  instantaneous i n t e r e s t c)  a  constant  ratio  term premia t h a t a r e p o s i t i v e as lonq as i n t e r e s t r a t e s  value  (see Van  and k  %  Home [ 7 1 ] ) .  negative  above  and  X.  that  T h i s o b t a i n s when k, >0  <0.  Probably the most compelling reason f o r choosing t h e as i n equations  forms f o r  (7.8) and (7.9) i s t h a t i t permits a  simple method o f e s t i m a t i n g the parameters k  (  we  to the  rate.  are below a t h r e s h o l d l e v e l , and  O  qiven  and  k  r  because  have a c l o s e d form s o l u t i o n f o r the pure d i s c o u n t bond under  t h i s assumption  .  The p r i c e  of  a  bond  paying  a  continuous  coupon may be represented by  r  where  B(.,.)  r e p r e s e n t s the p r i c e of a pure d i s c o u n t bond, and  i s as qiven by equation  (7.10).  Given a sample of market p r i c e s  on s t r a i g h t coupon bonds, one method o f e s t i m a t i n g k would  be  to  minimize  some  measure  (  and  k  2  o f d e v i a t i o n between the  market and model p r i c e s over the data sample.  Corresponding  to  132 any  choice  interest rate maturity, coupon  ky  of  and  process,  and  bond  coupon can  be  e.  and P^  computed  a  ,e  +  ) =0  problem  and k^ of  eguation  to  straight  (7. 12) .  The  5 7  €i,  17.13)  market and  for i#j.  in  time  was  the  coefficient  r e g r e s s i o n framework .  model p r i c e s ,  I t may  be  noted  present estimation  and  scenario in  is  and that k  t  the  a non-linear  Throughout, we adopt maximum l i k e l i h o o d  ss  (ML)  using  PL  rate,  non - l i n e a r f u n c t i o n of the parameters k,  Thus, e s t i m a t i n g k, standard  c  interest  model p r i c e of any  considered  =  2  is  the  are r e s p e c t i v e l y  r-J N(0, ( T ) ; Cov(e  P  current  rate),  P.'  1  (and g i v e n the parameters o f the  the  s i m p l e s t model t h a t was  where P  k^  methods f o r parameter e s t i m a t i o n .  In t h i s s i t u a t i o n  least  squares e s t i m a t i o n = M L ( a s y m p t o t i c a l l y ) . However s i n c e P^»  and  are  strictly  positive,  it  was  c o n s i d e r e d more a p p r o p r i a t e to assume a model of the form  P(r,^,c) can be evaluated very easily by numerical integration. Due t o the smooth shape of the f u n c t i o n B ( r , T ) with r e s p e c t to X , a simple 4 p o i n t quadrature method gave very accurate r e s u l t s . To check the accuracy f o r a sample case, the coupon bond p r i c e was evaluated using up to a 64 p o i n t adaptive quadrature and the i n c r e a s e d accuracy was n e g l i g i b l e . I t may be noted t h a t i n any approach t o e s t i m a t i n g k, and kj_ , model prices of the t o t a l bond sample would have to be e v a l u a t e d s e v e r a l hundred times. Even with the present assumptions, estimating k, and k i s computationally q u i t e expensive. However, i f were not , (or zero) and i f we d i d not assume A ( r , t , f ) t o have the form as an equation (7.9), the bond model prices correspoding to each (k| , k^. ) v a l u e would have to be o b t a i n e d by finite d i f f e r e n c e methods., That would mean a computation expense more than j u s t p r o h i b i t i v e ! s 7  L  G o l d f e l d 6 Quandt [34 J present a problem., 5 8  good  introduction  to  the  133  -+ -6c  where  the  assumptions  parameters k| the  and k  x  on  e^  are  as  before.  were estimated by both models above,  parameter e s t i m a t e s were h a r d l y (Eqn.  In  exactly  (7.14)  The and  different *: 5  7.13)  (Eqn. 7. 14)  k,  0.3113x10-5  0.3093x10-5  k^  -0.1581x10-2  -0.1548x10-2  both models above, t h e r e s i d u a l v e c t o r  has been assumed  t o e x h i b i t n e i t h e r a u t o c o r r e l a t i o n , nor h e t e r o s c e d a s t i c i t y . linear are  models  unbiased, even where the r e s i d u a l c o v a r i a n c e  covariance non-linear unbiased for and  i t i s w e l l known that t h e e s t i m a t e d  whether  the  estimated  i n s m a l l samples i s not known when Si£  heteroscedasticity,  the  the f o l l o w i n g r e g r e s s i o n  t h a t var(e•  coefficients  i s - Q - ^ c r " J : the  matrix o f t h e e s t i m a t e d parameters i s biased. setting,  In  parameters O- - I.  TO  5  r e s i d u a l v e c t o r e^  was performed:  a  are test  was r e t r i e v e d  (The h y p o t h e s i s  ) i s a f u n c t i o n o f time t o matruity  In  was  of P .) 60  The s t a n d a r d e r r o r s o f the estimates, based on asymptotic theory ( i e . , by i n v e r t i n g t h e F i s h e r I n f o r m a t i o n matrix) a r e not r e p o r t e d , as t h e i r values was very different across the two models. This was i n v e s t i g a t e d f u r t h e r and found to be due t o numerical inaccuracy i n e v a l u a t i n g the second d e r i v a t i v e of the j o i n t l i k e l i h o o d f u n c t i o n near t h e optimal p o i n t , 5 9  P i s a function of r and ? . H e t e r o s c e d a s t i c i t y as a f u n c t i o n only o f X was c o n s i d e r e d . Understandably, i t could have a l s o been a f u n c t i o n of r . However, t h i s was not c o n s i d e r e d , as the v a r i a b i l i t y o f T over t h e sample was much larger than t h a t o f r . I t was t h e r e f o r e f e l t t h a t most o f the h e t e r o s c e d a s t i c i t y c o u l d be e x p l a i n e d by T alone. 6 0  134  loo^{^ )  a  1  -t- b Lft(Vi)  where T =time t o maturity o f the i™  + IA)O  (7.15)  data p o i n t .  t  I f b=0, then we cannot r e j e c t the hypothesis t h a t the exhibit  homoscedasticity .  from equation static  This  61  was  was  done f o r the r e s i d u a l s  (7.14) and b was estimated at 1.06.  This  seems  to  residuals  2.091,  indicate  and  its t  t h a t there i s no  compellinq reason to suspect h e t e r o s c e d a s t i c i t y to be p r e s e n t . T e s t i n q f o r a u t o c o r r e l a t i o n among the r e s i d u a l s i s complicated  matter..  a  more  There are two types of e r r o r c o r r e l a t i o n s  to consider. 1)  S e r i a l c o r r e l a t i o n w i t h i n each bond a c r o s s time. ,  2)  Contemporaneous  correlation  across  bonds,  at  any  i n s t a n t o f time. It  must  be  remembered  consists  of  time  correlation  of  series  residuals  consecutive r e s i d u a l s reasonable  to  that  expect  of  the  on  18  refers each  the  o r d i n a r y coupon bond sample different to  the  bond.  It  errors  across  bonds. correlation  Serial between  i s , however, a l l bonds,  also at  a  The more " c o r r e c t " method o f t e s t i n g f o r h e t e r o s c e d a s t i c i t y would be to do a " c o n s t r a i n e d " and " u n c o n s t r a i n e d " estimation, and then perform a l i k e l i h o o d r a t i o t e s t . Under t h e c o n s t r a i n e d e s t i m a t i o n JL i s assumed = cr 1 and i n the unconstrained JL- i s d i a q o n a l with elements o ^ ^ a ? ^ . The r e s t of the approach i s to s e t up the l i k e l i h o o d f u n c t i o n as 2  where p (e^ ) ~ N (0, JL) . For our case the sample s i z e was 6662 data p o i n t s on bonds, and doinq this would have been computationally expensive. Thus the more ad hoc approach was taken. T h i s method o f hypothesis testinq on b, i s a l s o dependent upon w,- b e i n g i . i . d and normally d i s t r i b u t e d .  135 p a r t i c u l a r p o i n t i n time t o be c o r r e l a t e d .  Since each bond  s e r i e s s t a r t s and ends a t a d i f f e r e n t point i n time of d i f f e r e n t would  be  length) , accounting  a  horrendous  data  (and each i s  f o r contemporaneous c o r r e l a t i o n  task.  Considering  the  difficulties  i n v o l v e d , i t was decided t o leave t h e problem of contemporaneous correlation  in  abeyance,  but  tackle  the  serial  correlation  problem. When we c o n s i d e r s e r i a l c o r r e l a t i o n matrix  SL  of  the  residual  vector  only,  the  i s block  covariance  diagonal  s t r u c t u r e , with the r e p r e s e n t a t i v e matrix having the usual as  in form  when we have f i r s t - o r d e r a u t o c o r r e l a t i o n , i e , J l = ( Si-c) where  J l j , i s the matrix a l o n g the d i a g o n a l f o r bond i ,  f  f'  f  -Hi (Tc  f  f  of  the  r  form I  and i s  f  \  (7.16)  x- "ft )  7-i  We could f u r t h e r assume that  "f i s egual a c r o s s a l l bonds.  This  s i m p l i f i e d s t r u c t u r e makes i t c o m p u t a t i o n a l l y much e a s i e r t o s e t up t h e l i k e l i h o o d f u n c t i o n o f the r e s i d u a l s and thereby the  parameters.  serial  correlation  heteroscedasticity and (k,  ML ,k  t  methods ,-f,a,b).  What  was  (using of  the  were  estimate  a c t u a l l y done was t h a t , a l o n g the  model  of  equation  with  7. 14) ,  form d i s c u s s e d e a r l i e r was assumed, employed  to  estimate  I t was c o m p u t a t i o n a l l y  no c o n s t r a i n e d e s t i m a t i o n was performed,  jointly  very expensive and so  (to do l i k e l i h o o d  ratio  136 t e s t s f o r t e s t i n g hypotheses on any of the parameters). of the j o i n t  likelihood function  L  = -L^\SL\  where e i s t h e column and  e^/v/  N(  0  cr- ), 2  (  convergence  quasi-Newton  (7.17)  v e c t o r o f r e s i d u a l s , e* i s i t s t r a n s p o s e , with  was  Q~  2 c  not  algorithm  parameter values w e r e  = aT^ and Corr  for  (e  t  ^  (  ) = f  The r e s u l t of the e s t i m a t i o n attained  in  maximizing  and was  60 i t e r a t i o n s using a L.  The  intermediate  6 2  k,  = 0.3916 x 10-s  k  = -0.2144 x  f  = 0.0097  a  * 0.1394 x 10-ft  b  =  r  was  - i«'Jl e  i s c o n s t a n t across a l l bonds. that  The l o g  10~  2  1.586  The g r a d i e n t s on k,  and k  L  indicated  that  r e q u i r e both v a l u e s t o move towards zero.  the  optimum  would  The broad c o n c l u s i o n s  that can be a r r i v e d a t , based on the r e s u l t s , a r e : a)  The  estimates  of  k,  and  kj_ based on the model of  equation  (7.14) are probably not  the  model  assuminq  very  different  autocorrelation  from and  h e t e r o s c e d a s t i c i t y of t h e e r r o r v e c t o r . b)  The s e r i a l c o r r e l a t i o n  coefficient  (f)  between  the  » a p p a r e n t l y , the converqence r a t e i s very slow. The CPU time used f o r t h i s p a r t i a l converqence run was 5000 seconds on an IBH 370/168. , Since the computational c o s t was extremely hiqh and no a d d i t i o n a l i n s i q h t s appeared to be l i k e l y by restarting the s e a r c h f o r the optimum a n d q o i n q on u n t i l converqence was a t t a i n e d , the matter was not pursued f u r t h e r . 2  137  r e s i d u a l s appears t o be c l o s e t o zero. c)  A statistically  significant  l e v e l of h e t e r o s c e d a s t i c i t y  does not seem t o e x i s t . The of  f i n a l q u e s t i o n that was c o n s i d e r e d under the e s t i m a t i o n  the  parameters  k  and  v  k  x  ,  was  the  validity  of the  assumption o f n o r m a l i t y of t h e r e s i d u a l s -  after  approach here i s based on t h i s assumption.  The approach t h a t we  adopt  in  testing  f o r normality  p r o b a b i l i t y graphing. the behaviour variable  with  variable Z = (u-/^)/ linear  transformation  straight detect  /A r  line.  prices.  If  and  variance  u  is  a  cr , 2  Gaussian the  random  standardized  w i l l be u n i t normal., Since Z i s  just  (  order,  a  o f u, t h e graph o f Z a g a i n s t u i s j u s t a to  from n o r m a l i t y i n t h e d i s t r i b u t i o n o f u. I f  u^ i-=1.,H a r e N sample v a l u e s of ascending  (or departures therefrom) i s  The r e l a t i o n s h i p between Z and u can be used  departures  HL  Fama [ 2 2 ] uses t h i s approach i n examining  o f stock mean  a l l , the  the  then a p a r t i c u l a r u  variable L  u  arranged  i s an estimate  in  o f the f  f r a c t i l e o f t h e d i s t r i b u t i o n o f u, where the value of f i s g i v e n by  63  As p o i n t e d out i n Fama [ 2 2 ] , t h i s p a r t i c u l a r convention f o r estimating f i s only one of many t h a t are a v a i l a b l e . Other popular conventions a r e i / ( N + 1), (i-3/8)/(N«- ) and ( i )/N. All four techniques give reasonable estimates of t h e f r a c t i l e s and, f o r t h e l a r g e sample t h a t we have, i t makes l i t t l e d i f f e r e n c e which s p e c i f i c convention i s chosen.  6  3  138  Now  the  exact value of Z f o r the f f r a c t i l e of the u n i t normal  d i s t r i b u t i o n can cumulative  easily  normal.  be  obtained  Computer  by  inverting  the  unit  r o u t i n e s are a v a i l a b l e f o r t h i s .  I f u i s a Gaussian random v a r i a b l e , then a graph of  the  values of u a g a i n s t the values o f Z d e r i v e d from the  theoretical  unit  normal  cumulative  straight line.  There may,  linearity  to  linearity  due  distribution of course,  sampling  error.  function be  some  If  the  are extreme, however, the Gaussian  d i s t r i b u t i o n of u should be  sample  should  be  a  departure  from  departures  from  hypothesis f o r  the  questioned.  The normal p r o b a b i l i t y p l o t of the r e s i d u a l s from the model of  equation  (7.14) i s presented  i n F i g u r e 4.  I n s p e c t i o n of the  p l o t i n d i c a t e s t h a t t h e d i s t r i b u t i o n o f the r e s i d u a l s i s t h i n n e r than the normal at the t a i l s , In  fact,  it  could  be  that  and a l s o more peaked a t the we  have  a  mixture  of  mode. normal  d i s t r i b u t i o n s with i d e n t i c a l means but d i f f e r i n g v a r i a n c e s (or  more) c o r r e s p o n d i n g to the t a i l s ;  and  c o r r e s p o n d i n g t o the peak at the mean. of  another  (or  others)  This c o u l d be t h e r e s u l t  h e t e r o s c e d a s t i c i t y of the form we considered e a r l i e r  not f i n d s t a t i s t i c a l l y s i g n i f i c a n t ) .  P o s s i b l y i f we had  (but d i d adopted  the more " c o r r e c t " method of t e s t i n g f o r h e t e r o s c e d a s t i c i t y footnote  54)  significant not  is  at  we  might  level. present  have  observed  i t  an unresolved i s s u e . account  (in  the  (see  at a s t a t i s t i c a l l y  Thus, whether h e t e r o s c e d a s t i c i t y  that even t a k i n g i t i n t o  one  exists  However, we model  of  or  did f i n d equation  139  FIGURE 4  NORMAL PROBRBILITY PLOT OF RE5ULTRNT ERROR FROM THE ESTIMATION OF LIQUIDITY/TERM PREMIUM PRRAMETER5 [ERROR = LOG (MARKET PR/MODEL PR)] K] 5. K2 BR5ED ON DRTR JRN 59 - NOV 77  3.71 ^  :  0.194  140 (7.17)),  d i d not  k  k  and  (  and  k  .  z  seem to a l t e r m a t e r i a l l y the p o i n t e s t i m a t e s of  He may  t h e r e f o r e assume t h a t our e s t i m a t e s o f  based on the model of equation  t  (7.14) are s a t i s f a c t o r y .  To get a b e t t e r f e e l f o r the numerical values k  ,  %  the  l i q u i d i t y / t e r m premium f u n c t i o n  time to m a t u r i t y , interest  rate  for (see  different  values  F i g u r e 5).  The  A  was  of  of  the term s t r u c t u r e by  RC*,*) where  B(r,t )  the  R ( r , t ) , then we  i s the pure discount  of  the  instantaneous  may  be of i n t e r e s t t o note t h a t when  maturity.. this  positive  When r=-k,  does  not  the  If  /k  and  r=  ,  increasing  , A =0  2  of the c u r r e n t '/^JX.  the  , to  2JUL  value .  function  of  for a l l maturities.  time t o Obviously  is  downward  sloping.  when r=-k,  /k  I n g e r s o l l [ 39 ]  %  , the has  term  pointed  form  of  (as i n equation  As  structure  out  that  term s t r u c t u r e c o r r e s p o n d i n g t o t h i s i n t e r e s t r a t e process, assumed  this  expectations  l i q u i d i t y / t e r m premium hypotheses models c o i n c i d e .  be seen from F i g u r e 6,  It  term/liquidity  imply a f l a t term s t r u c t u r e - only that at  can  the  we  F i g u r e 6 shows  value o f r , the term s t r u c t u r e curves f o r the pure and  7.  have  bond value.  s h o r t term i n t e r e s t r a t e v a r y i n g from  a  against  - -1 ^ [ e c - r , r ) j  6  is  and  term s t r u c t u r e curves were  the shape of the term s t r u c t u r e * at values  premium  k,  plotted  a l s o p l o t t e d and these are presented i n F i g u r e s 6 and represent  k,  the and  (7.8)), c o u l d have a  *•* The value of HISF i n the f i g u r e corresponds t o the limiting value of R ( r , T ) as T -><* . From the term s t r u c t u r e e q u a t i o n , t h i s i s given by (2m ' ,/<r ) (A-m«) where A= (m* *2 <r j— , and 1  m= ,  (m-k^  ) and  fA*={m  2  +k,  ) /m*.  2  .  z  FIGURE 5  LIQUIDITY PREMIfl V5 TIME TO MATURITY ON DISCOUNT BONDS K l = 0.309 X 10 XX -5 K2 = -0.154 X 10 XX -2 Kl  l K2 BRSED ON DRTR JRN 59 - NOV 77  _  4.99  142 FIGURE 6  YIELD TO MATURITY V5 TIME TO MATURITY ON DISCOUNT BONDS K l - 0.309 X 10 XX - 5 K2 =-0.154 X 10 XX -2 Kl t K2 BRSEQ ON BOND DATR JRN 59 - NOV 77 15.oa..  2.73 _ d  $  iO  J5  20 TIME  25  i0  TO M A T U R I T Y  i5 IN  YEARS  4)  45  40  143  FIGURE 7  YIELD TO MATURITY VS TIME TO MATURITY ON DISCOUNT BONDS Kl  - 0 . 3 0 9 X 10 XX - 5  K2 = - 0 . 1 5 4 X 10 XX - 2  Kl t K2 BRSE0 ON BOND 0R1R JRN 59 - NOV 77  B.Al  d  S  iO  i5  20  25 30 i5 TIME 10 HRTURITY IN YEARS  40  45  40  144  humped shape, but t h a t ( f o r hump  would  be  very  reasonable  small.  This  parameter  is  values)  the  borne out i n F i g u r e 7.  Before comparing F i g u r e s 6 and 7, care must be taken to note l a r g e d i f f e r e n c e i n t h e s c a l e along, the Y-axis between the  7.  5  the  two.  Bond V a l u a t i o n Under the L i q u i d i t y / t e r m Premium ILIfiPL Model Having  estimated  parameters  that  requirements,  we  the  aggregate  determine can  their  proceed  investor  liguidity/term  to  value  our  r e t r a c t a b l e / e x t e n d i b l e bonds, with t h i s assumption The  p.d.e. governing  (cf.  e q u a t i o n 7.1)  l(rV<5  M  JJL'  where i ' and boundary case.  the  bond  we now  are  conditions  as  in  exactly  Model p r i c e s were computed f o r  results  of  regressing  the  market  presented i n column 2 of Tables XXI the mean e r r o r under  the  (defined e a r l i e r )  PEXP  bonds E7 t o E10). bonds,  sample  of  incorporated. altered  have  defined  remain  premia  p r i c e i s only s l i g h t l y  ^V"^)^' - ^ ^ ^ ^ - ^  -i  preference  model,  i s now  For  purposes  =  equation the  (7.18)  °  (7.10).  The  same as f o r the PEXP  a l l 20  bonds,  and  the  p r i c e s on model p r i c e s are through  XL.  &s  expected,  which was c o n s i s t e n t l y more o f t e n p o s i t i v e of  quick  negative  (except f o r  comparison  across  T a b l e XLI p r e s e n t s the mean e r r o r f o r a l l 20 bonds using  the d i f f e r e n t models, and T a b l e XLII  presents  results  from r e g r e s s i n g t h e market  on P , the s l o p e c o e f f i c i e n t  similar  summary  p r i c e on the model p r i c e s as w e l l as the c o r r e l a t i o n between the model and  market p r i c e s . , Comparing  the  results  of  the  LIQP  TABLE XLI COMPARISON OF MEAN ERROR FOR ALL BOND ACROSS DIFFERENT MODELS  BOND  PORE EXP.  LIQ. PREM.  REV.TAX (50%)  REV.TAX '(25%)  C.G.TAX (10Z)  C.G.TAX (25)  MOV. AVG.  Rl  0.36  2.45  -0.00  1.25  1.83  2.57  El  -1.87  0.67  0.63  0.65  0.70  0.76  E2  -4.22  1.63  2.18  1.91  1.95  2.02  E3  -5.51  1.85  2.44  2.15  2.37  2.70 1.96  -  '' NAIVE"  0.39 0.82 -0.55 0.89  E4  -4.77  1.46  2.45  1.95  1.94  E5  -1.04  1.93  0.43  1.20  1.58  2.08  -0.18  0.14  -0.91  -3.08  E6 E7  -5.66 -14.1  2.04  0.73  1.43  1.87  2.44  -1.54  -3.59  -5.91  -5.57  -5.03  -3.87  -7.19 -8.01  E8  -18.62  -3.22  0.42  -1.47  -1.25  -0.93  -2.63  E9  -15.83  -2.51  -4.24  -7.04  -6.76  -6.28  -1.18  -7.28  E10  -7.15  -0.27  -2.46  -3.14  -2.76  -2.25  -1.96  -3.68  Ell  -5.79  0.47  -2.06  -2.61  -2.24  -1.73  -5.16  -2.50  E12  -5.67  2.18  -1.93  -2.56  -2.01  -1.28  -6.11  0.05  -3.80  -3.03  -2.01  -6.74  0.31  E13  -7.00  2.60  -3.43  EU  -4.13  2.59  -2.51  -2.32  -1.64  -0.77  -2.74  0.19  2.26  '2.44  2.72  0.66  -0.95  1.77  1.92  2.13  -1.82  -1.16  2.68  -0.09  3.36  0.44  5.15  0.89  E15  -6.42  1.66  2.99  E16  -5.10  1.40  2.21  E17  -5.38  2.16  -2.67  -3.13  -2.59  -1.85  E18  -1.01  2.67  2.45  2.56  2.76  3.03  E19  -5.96  2.93  0.28  -2.27  -2.11  -1.78  v  a 8  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  Q  O  Q  O  O  O  O  O  O  O  O  O  O  O  1 -*  s  8  S ^- > > " »<  -*  O  O  O  O  O  O  O  c i O er*>o* 'c-a ' CO i o CO r - ct w i ^c S o GD c  coc^ooaD<or*»r-co O O O O O O O O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  3  II11II IIiIIiI  <  i  <  1 i  §  I 3  E  s s  I 11g! o  i  '  d  o  3 5  o  o  d  o  O  O  o  O  o  O  o  '  O  o  O  d  d  o  d  o  d  d  d  3  d  I i II I1 2 s-l i ! I I II ! 1  1 s i i s s s s i l l l l l l s l l l l l  1  3 5 £  j  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d d  I 1HIsIIsi! 11I|§ §1 I I  sH  s ns  s I 1H Ii§!1g!|§isg o  d  d  o  o  o  = o  §  d  o  o -HO  d  d  d  d  d  _  o  d  d  d  u*  d  H d  d  d  d  d  d  d  d  d  d  d  d  d  -  d  d  n  d  z d  s  d d  m§m d d  s s s s s l s s s c s s 8 s i s S s 5  o  s  o  o  *: ,  o  s H  o  o  o  o  o  o  d  o  '  d  o  o  o  '  o  '  S 2 S K K 2 S s o - « o o o o o * - 4 - * - » 4 _ ^ ^  o  r  s  2s  S S 2 C S W  '  r  j ^  s §I Ii ! ! I I s; I I I I I ! i i i g  B  !  d  d  5  ! ISII gIiISS  d  1 I d  d  3 d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  d  § S 2 § 1 d  d  d  d  d  d  d  d  I  1  =  III 1 d  d  d  .  I3g 1II 1S IH I2 g§gIS j  d  d  d  1 § § SH  d  d  d  d  d  d  d  d  d  d  d  d  d  d d  I  ! ssa 3g5I§s §I|s  a a B o a a s a s s . a a - a a a a a a a a  g  f  ?  146  TABLE XLIX COMPARISON OF MODEL AND  MARKET PRICES  (SUMMARY BASED ON A L L BONDS IN THE  (ALL MODELS) SAMPLE)  PURE EXP.  LIQ. PREM.  REV.TAX (50%)  REV.TAX (25%)  C.G.TAX (10%)  C.G.TAX (20%)  0.391  0.491  0.306  0.311  0.332  0.357  0.254  0.371  RMSE  10.253  3.944  3.781  4.611  4.513  4.412  4.965  4.346  MEAN ERROR  -7.570  0.778  -0.905  -1.621  -1.258  -0.751  • -2.075  -0.841  ESTIMATED SLOPE  0.301  0.546  0.678  0.479  0.478  0.477  0.469  0.444  ( S . E . OF SLOPE)  0.006  0.009  0.017  0.012  0.011  0.011  0.015  0.010  68.183  46.170  31.876  51.725  52.057  52.360  52.520  55.716  0.718  0.978  1.825  1.285  1.216  1.144  1.546  1.042  0.545  0.038  0.057  0.123  0.077  0.029  0.174  0.037  0.352  0.383  0.085  0.304  0.343  0.387  0.250  0.462  0.102  0.577  0.857  0.572  0.579  0.583  0.574  94.386  6.573  2.035  9.096  8.576  8.114'  10.748  8.986  12.264  12.170  11.797  MODEL  R2  EST.INTERCEPT ( S . E . OF  INTR)  FRACTION OF ERROR DUE  TO  ^  j_  RES.  BIAS  VARIANCE  MISSPEC ERROR RESID.ERROR  11.360  MOV. AVG.  "NAIVE"  .  0.500  10.483  9.445  14.168  9.448  -1^  148 i  model with the PEXP r e s u l t s we c o u l d i n f e r t h a t : a)  Whereas  the  PEXP  model  consistently  bonds, the LIQP model tends undervalue  them.  This  (more o f t e n  is  o v e r v a l u e s the than  indicated  not)  to  by t h e g r e a t e r  number of p o s i t i v e mean e r r o r f i g u r e s i n Table XLI. b)  The s l o p e c o e f f i c i e n t o f measure model  regression  is  over-responsive  model).  of  the  (since  market  (7.7)  If f  o f r e l a t i v e responsiveness.  responsiveness  j£=1.  the  measures  with  LIQP model l e a d s t o  a  <1, then the  respect  I d e a l l y , we would r e q u i r e a model The  is  the  to  that  the gives  values c o n s i s t e n t l y  c l o s e r t o 1 than the PEXP model, and may, t h e r e f o r e , be regarded as an improvement over the PEXP model. Be would s u r e l y expect the LIQP model model, interest  to  outperform  rates.  a  qlobal  measure  Tables XXI  to  XL  regression  of  equation  all  PEXP  as i t c o n t a i n s more i n f o r m a t i o n on the term s t r u c t u r e of  To enable one t o compare the d i f f e r e n t bonds,  the  is  that  desireable.  models  aqqreqates For  the  this  (7.7) was performed  across a l l results  purpose,  of the  by p o o l i n g data of  the bonds, and t h e r e s u l t s a r e presented i n T a b l e XLIX.,  Be  now i n v e s t i g a t e the impact on the model of i n c o r p o r a t i n g t a x e s .  7,6  Bond V a l u a t i o n With Revenue Taxes In  t h i s s e c t i o n , we look a t the e f f e c t of i n c l u d i n g i n the  model taxes on coupons and i n t e r e s t , but not on In  Chapter  2,  we  developed  the  capital  p.d.e. governing  v a l u a t i o n under s p e c i f i c assumptions about  the  way  gains.  the  bond  taxes  are  149  applicable be  a  {see  gross  however,  equation 2,11).  The assumptions d i d appear t o  over-simplification  remains;  of  reality.  a r e we b e t t e r o f f without  The  question,  i n c o r p o r a t i n g taxes  i n t o the model? I n c l u s i o n of revenue taxes i n the bond  valuation  equation  has two opposing  influences.  F i r s t , the coupon y i e l d i s reduced  from  c(1-R)dt/G,  where  cdt/G  to  revenue tax r a t e . the  c  Thus, the net gain  bond i s reduced,  is  the coupon and R the  (or b e n e f i t )  and so i t s value i s lowered.  hand, the r a t e of r e t u r n on the i n s t a n t a n e o u s l y is  also  reduced  from  from  rdt  to  r(1-R)dt,  i n s t a n t a n e o u s r i s k l e s s r a t e of i n t e r e s t .  owning  On the other  riskless where  r  T h i s has the  asset is  opposite  e f f e c t on the bond v a l u e - i t pushes up the bond p r i c e . the  Whether  net e f f e c t of these two f o r c e s pushes the model p r i c e up or  down i s not a p r i o r i I t was Ideally,  apparent.  not c l e a r what value of  it  should  represent  R  the  decided  r a t e was comparing  to  try  to  use  marginal  r e p r e s e n t a t i v e i n v e s t o r , . S i n c e no one was  the  both 8=25% and 50%.  and  tax  f i g u r e was  it  (See a l s o T a b l e  results  of  previous  two  XLIX).  with the r e s u l t s of the LIQP model, we note that  t h e mean e r r o r value  (which was  to under-valuation of  model  negative  20  over  r a t e o f the  cases are a l s o  reported i n the same t a b l e s as the r e s u l t s of the  Comparing  model.  available,  The  model p r i c e s f o r these two  models, i e . . Tables XXI to XL.  the  The value o f the tax  kept constant over the whole p e r i o d . , market  in  the  almost c o n s i s t e n t l y p o s i t i v e  price)  bonds.  is This  equally seems  i n t r o d u c t i o n of revenue taxes has pushed up model  positive to  and  imply prices  due  that -  at  150  least  f o r t h i s sample of bonds.  Comparing the  , we f i n d  that  i n c r e a s i n g R (from z e r o  i n the LIQP model to 25% and  then  increases  consistently.  pushes |J  ^  considerably |3  keeps  almost  above 1.0 i n s e v e r a l  below  1.0  Using  cases,  R=50%  whereas  more o f t e n than otherwise.  i n d i c a t e t h a t an a p p r o p r i a t e  50%)  using This  E=2S%  seems t o  revenue tax r a t e i s between the two  figures., So f a r , we have been  comparing  across  models  using  two  measures.  The  a)  The mean e r r o r as a measurement o f b i a s  b)  The value o f  term  as a measure of  "responsiveness"  is  "responsiveness".  supposed  movement o f the two p r i c e s - the market However,  we  aspects: drops  should  recognize  that  d i r e c t i o n and magnitude.  to  and  measure the j o i n t the  joint  model  movement  To c l a r i f y , i f  price. has  market  two price  from one week t o the next, and so does model p r i c e , there  i s p e r f e c t harmony between the two with r e s p e c t t o d i r e c t i o n movement.  But i f market p r i c e drops by 500, whereas model p r i c e  by $1, then the model i s o v e r - r e a c t i n g low  ^  of  value).  We know that  ^  (which would show up i n a  can be expressed i n terms o f the  c o r r e l a t i o n between the independent  (market p r i c e ) and dependent  (model p r i c e ) v a r i a b l e s of the r e g r e s s i o n as  f,  where  S  mfet  and  -  f  s ^  between  J  ^  L  represent  market and model p r i c e s correlation  •  (  the standard  respectively,  the  two.  Now  and we  measure of d i r e c t i o n a l co-movement, whereas  -  1  9  '  d e v i a t i o n of the  -f  represents  5  can  7  see t h a t  the  ratio  f of  the is a the  151  standard  deviations  breakdown o f ^ change  in  increasing  is a  measure  of  the  e n a b l e s us t o see which  the  value  of p - . ,  the t a x r a t e  From  (or even  magnitude.  aspect  has  T a b l e XLII  including  l e d to  we  it in  This a  see that the  first  place) does not improve t h e c o r r e l a t i o n between market and model prices  -  i t i s t h e magnitude f a c t o r that i s a f f e c t e d .  introducing volatility  7•7  revenue taxes h e l p s  in  fine  tuning  the  relative  of model and market p r i c e movements.  Bond Valuation- I n c o r p p r a t i n g Having  introduced  C a p i t a l Gains Tax  revenue taxes i n t o the model i n the l a s t  s e c t i o n , we proceed t o see the e f f e c t on model p r i c e vis-a-vis  market p r i c e s , when we i n c o r p o r a t e  tax i n t o the v a l u a t i o n model.  Here a g a i n ,  what  we  want  to  investigate  behaviour,  c a p i t a l g a i n s (CG)  the  assumptions that appear s i m p l i s t i c (as pointed but  Thus,  approach  makes  out i n Chapter 2)  i s whether  there  i s any  improvement i n the p r e d i c t i v e power of the model. The  e f f e c t on model  unambiguous.  The  prices  of  but  on t h e mean e r r o r i s c l e a r  a  CG tax r a t e of 10% and 20%.  constant at 25%, as that are  i t s introduction.  fill  20 bonds were  results  XL.  (See a l s o Table XLIX).  The  (The revenue tax was kept far).  presented i n columns 5 and 6 o f T a b l e s XXI t o  Rs expected, model p r i c e s a r e c o n s i s t e n t l y are  is  valued  appeared t o be the best model so  The  taxes  taxes  ( i t i s expected t o i n c r e a s e ) ,  the e f f e c t on ^ , i s not obvious,  using  CG  b e n e f i t s t o owning t h e bond are reduced, and  so the model p r i c e s w i l l decrease with effect  introducing  introduced.  This  lower  when  CG  i s r e f l e c t e d i n t h e v a l u e o f the  152  mean e r r o r - the p o s i t i v e value,  and  the  values  negative  have  increased  ones have reduced  and  the  CG  Tax  models).  In  absolute  i n absolute  (The comparisons a r e between the r e s u l t s of the model,  in  25%  almost  value.  Rev.  Tax  a l l t h e bonds,  i n t r o d u c i n g CG taxes m a r g i n a l l y improves the c o r r e l a t i o n between model and market p r i c e s but, i n a l l cases, down.  This  i m p l i e s t h a t ( S ^ /S^^)  goes up (see eguation The (ie.  volatility S ^ mo  the  ^  values  go  goes down by more than  (7.19)), r e s u l t i n g  in  lower  ^  ^  values.,  o f t h e model p r i c e s thus c o n s i s t e n t l y i n c r e a s e s  i n c r e a s e s ) with CG taxes.  By a p p r o p r i a t e l y  choosing  revenue and CG tax v a l u e s , we can achieve both an improvement i n •f and t h e s l o p e .  7  •  8  The ffHovinq Average" Hodel From  our  analysis  i n the l a s t two s e c t i o n s , we f i n d  i n c o r p o r a t i n g taxes i n t o the tuning"  effect  model  leads  mainly  to  o f our o b j e c t i v e s , we a r e attempting p r i c e s , u s i n g broadly t h r e e the  "fine  i n onr attempt t o match market and model p r i c e s  on our sample o f r e t r a c t a b l e and e x t e n d i b l e bonds.  a)  a  that  correlation  Taking  t o match model  and  stock market  measures: as  a  measure  of  joint  directional  movement b)  the p> c o e f f i c i e n t as a measure of equal movement  amplitude  of  153 c)  t h e mean e r r o r as a measure o f b i a s . , 6 5  We n o t i c e d to  that use o f the l i q u i d i t y  substantial  Incorporating  improvements  in  premium  all  hypothesis.-led  three  measures.  taxes l e d to improvement on t h e f i r s t  two measures  of model performance.  However, by using revenue and CG taxes t o  improve  measures  the  model's  of co-movement with the market,  c o n t r o l on the extent of b i a s was foregone t o some draw  a  extent.  To  crude analogy with the macro-economic p o l i c y problem o f  matching " t o o l s and t a r g e t s " , t a c k l e the bias.  we  need  some  other  In our case, t o o l s a r e c r e a t e d  "tool"  to  by r e l a x i n g our  p r i o r assumptions t o match r e a l i t y . In  the  analysis  i n Chapter 5, we found t h a t the i n t e r e s t  r a t e process parameter ^ ' had the bond values. lead  most  C e t e r i s paribus , increasing  t o an across-the-board decrease  I t was f e l t , t h e r e f o r e , t h a t the  interest  the  p r i n c i p a l source o f b i a s .  In  i n bond  time  would values.  homogeneity  this  section,  we  adopt  an  assumption.  most elegant approach to the problem t o date  i s t h a t o f Brennan & Schwartz [ 1 2 ] , a function  (increase)  on  r a t e process parameters ( p a r t i c u l a r l y ^ - ) was  approximate method o f r e l a x i n g that the  impact  (decreasing) y.  the assumption o f  of  Probably  significant  who s e t up t h e bond p r i c e as  of both the s h o r t term i n t e r e s t r a t e  and a long  term  i n t e r e s t r a t e , where these two r a t e s f o l l o w c o r r e l a t e d d i f f u s i o n  " i t may be argued that r o o t mean square e r r o r (RMSE) i s a better measure o f o v e r a l l e r r o r . From the r e s u l t s presented i n Tables XXI t o XL, i t may be seen t h a t the ranking o f each bond a c r o s s models using e i t h e r mean e r r o r or BASE i s v i r t u a l l y identical. Thus, none o f the c o n c l u s i o n s would be a l t e r e d by using RMSE r a t h e r than mean e r r o r .  154  processes.  They  take  the  value  i n t e r e s t r a t e as the value of ^ rate  process.  However,  of  the  current  f o r the s h o r t  long  term  term  interest  there a r e s e v e r a l problems a s s o c i a t e d  with the e s t i m a t i o n o f parameters o f such j o i n t process, as w e l l as with t h e s o l u t i o n o f the p.d.e. f o r bond v a l u a t i o n , which a r e beyond t h e scope o f t h i s study. as of  I n s t e a d , what we do i s to  the value of jx f o r each bond, (R1 t o E19) the average the s h o r t term i n t e r e s t r a t e i n t h e two  prior  to  the  date  of  constant f o r t h e l i f e  issue.^  of  that  years  T h i s value of ^ bond.  The  take value  immediately i s maintained  results  of  this  approach are presented i n Tables XXVI through XL f o r bonds E5 t o E19. There  does not appear t o be any s i g n i f i c a n t improvement i n  the f i t between market and model p r i c e s from t h i s approach. ) and c o r r e l a t i o n s move a l i t t l e , direction;  but  not  l i k e w i s e w i t h t h e mean e r r o r .  that t h i s approximation  in  any  The  particular  Thus, we may  conclude  o f the n o n - s t a t i o n a r i t y of jx over  time  does not appear t o improve our r e s u l t s . So  f a r , we  have  not  looked  a t the s e n s i t i v i t y of bond  values t o the l i q u i d i t y premium parameters. and  k<3_  directly  The parameters  k,  a f f e c t m and jx (as shown i n equation 7. 10) ,  a l t e r i n q them as f o l l o w s : ra» =  (m-k^. )  JX'  {mjx +k, ) /m'  -  Tables XLIII and XLIV present discount bonds t o e r r o r s i n k noted  that  bond  values  the (  price  and k  x  sensitivity  respectively.  of  pure  I t may be  do not appear t o be very s e n s i t i v e t o  chanqes i n these parameters.  However, v a r i a t i o n s a c r o s s time i n  TABLE THEORETICAL  S E N S I T I V I T Y O F PURE DISCOUNT BOND P R I C E S TO ERRORS I N Kj^  ERROR I N  INTEREST  +25%  +5% BOND  %  %  BOND  %  BOND  BOND  ERROR  PRICE  ERROR  PRICE  PRICE  ERROR  PRICE  ERROR  96.96 89.05 80.63 72.61 61.85  0.0896 0.6138 1.3539 2.1805 3.4843  96.89 88.62 79.77 71.37 60.18  0.0179 0.1225 0.2693 0.4323 0.6873  96.87 88.51 79.55 71.07 59.77  96.85 88.40 79.34 70.76 59.36  -0.0179 -0.1223 -0.2686 -0.4305 -0.6827  96.78  -0.0895 -0.6101 -1.3358 -2.1339 -3.3670  95.07 85.64 76.97 69.13 58.81  0.0896 0.6138 1.3539 2.1805 3.4843  95.01 85.22 76.15 67.95 57.22  0.0179 0.1225 0.2693 0.4323 0.6873  94.99 85.12 75.94 67.65 56.83  94.97 85.01 75.74 67.36 56.44  -0.0179 -0.1223 -0.2686 -0.4305 -0.6827  94.90 84.60 74.93 .66.21 54.91  -0.0895-  1 3 5 7  91.42  10  53.16  0.0896 0.6138 1.3539 2.1805 3.4843  91.35 78.81 69.39 61.58 51.72  0.0179 0.1225 0.2693 0.4323 0.6873  91.34 78.71 69.21 61.31 51.37  91.32 78.62 69.02 61.05 51.02  -0.0179 -0.1223 -0.2686 -0.4305 -0.6827  91.25 78.23 68.28 60.00 49.64  -0.0895 -0.6101 -1.3358 -2.1339 -3.3670  3 5 7 10 1 3 5 7 10  r=2y  x  PRICE  T I M E TO MATURITY IN YEARS  1 Vi/2  K  0%  -5%  -25% CURRENT  XLIII  BOND  79.20 70.14 62.65  87.97 78.49 69.55 57.76  -0.6101 -1.3358 -2.1339 -3.3670  cn cn  TABLE THEORETICAL  S E N S I T I V I T Y OF PURE DISCOUNT  ERROR IN  12  r=y  r=y  r=2  U  \  IN  K,  K.  +25%  +5%  %  BOND  BOND  BOND  %  BOND  %  PRICE  ERROR  PRICE  ERROR  PRICE  PRICE  ERROR  PRICE  ERROR  1  96.85  -0.0258  96.87  -0.0051  96.87  96.88  0.0051  96.90  0.0255  3  88.31  -0.2238  88.47  88.51  88.55  0.0440  88.70  0.2178  5  79.10  -0.5643  79.47  -0.0443 -0.1112  79.55  79.64  0.1104  79.99  0.5441  7  70.37  -0.9843  70.93  -0.1936  71.07  71.20  0.1920  71.74  0.9445  10  58.76  -1.6802  59.57  -0.3302  59.77  59.96  0.3273  60.73  1.6086  1  94.95  -0.0440  94.98  -0.0088  94.99  95.00  0.0087  95.03  0.0435  3 5  84.85  85.06  85.17  75.94  76.05  0.0620 0.1397  85.38  75.84  -0.0624 -0.1407  85.12  75.40  -0.3154 -0.7141  76.47  0.3067 0.6881  T I M E TO MATURITY IN  BOND P R I C E S TO ERRORS  0%  -5%  •25% CURRENT INTEREST  XLIV  BOND  %  YEARS  7  66.86  -1.1675  67.50  -0.2296  67.65  67.81  0.2277  68.41  1.1200  10 .  55.75  -1.8845  56.61  -0.3704  56.83  57.03  0.3672  57.85  1.8051  1  91.26  -0.0805  91.32  -0.0160  91.34  91.35  0.0160  91.41  0.0796  3  78.32  -0.4983  78.64  -0.0986  78.71  78.79  0.0980  0.4847  5  68.51  -1.0131  69.07  -0.1997  69.21  69.34  0.1982  79.10 69.88  7  60.37  -1.5329  61.13  -0.3016  61.31  61.49  0.2992  62.21  0.9769 1.4721  10  50.19  -2.2919  51.13  -0.4508  51.37  51.60  0.4471  52.50  2.1990  157  these parameters could account f o r a reasonable bias  between e x i s t i n g model and  amount  of  market p r i c e s , as the e x t e n t  the of  b i a s i n percentage terms i s a l s o g u i t e s m a l l .  7.9  T e s t s o f Market We  Efficiency  proceed  to  test  retractable/extendible models.  In  Chapter 2,  bonds  deriving we  used  the a  short  on  any  efficiency to  bond  the  bond,  and  market  contained  valuation  argument,  formed by going  other  of  information  basic  hedging  investment p o r t f o l i o was bond,  the  on  finally  in  a zero  net  the  generic  making up  d i f f e r e n c e by borrowing or i n v e s t i n g i n the s h o r t term asset.. given  The  dollar  amounts  i n the  eguation  wherein  long  for  the  riskless  t o be i n v e s t e d i n each a s s e t were  as:  where  and  x,  =  d o l l a r investment i n g e n e r i c bond  x  =  d o l l a r investment i n any other bond  G represents  the g e n e r i c bond p r i c e (with  G  (  d e r i v a t i v e with r e s p e c t to the i n t e r e s t rate) and any  other bond  (with B  (  the i n t e r e s t r a t e ) . - (x^  + x •).-... 2  The  (F1  p r i c e of  investment i n  the  riskless  asset  For each of the 20 bonds (.81 t o E19) , we  t o F18), and  r e s p e c t t o r on  B the  each  partial  i t s p a r t i a l d e r i v a t i v e with r e s p e c t to  based on each model.. He a l s o have bonds  its  prices  on  is  have G,  straight  coupon  p a r t i a l d e r i v a t i v e s of those bonds with  date  were  computed  assuming  that  the  158  v a l u a t i o n equation f o r coupon bonds, equation In  our  the beqinninq  first  (7.12), was v a l i d .  t e s t of market e f f i c i e n c y , we assume t h a t a t  of each p e r i o d  (which i s a week i n our case, as we  have weekly bond d a t a ) , we qo lonq on the g e n e r i c one  bond at the market p r i c e (x =G). (  assume  a  short  we then compute  period,  we  and  over  proceed  Table XLV  on  and  ?  At the  end  of  assume t h a t we l i q u i d a t e t h i s p o r t f o l i o a t the  t h e n - e x i s t i n g market p r i c e s , portfolio  x  p o s i t i o n i n a s t a i g h t bond, and t h e balance i s  made up by an investment i n the r i s k l e s s asset. the  bond by buyinq  the  one p e r i o d .  until  presents  and  the  the  end  mean  compute  the  return  Be then form a new p o r t f o l i o , of  and  the data standard  on  each  clear  indication  the  hedge  p o r t f o l i o s are i n s i g n i f i c a n t l y  bond.  deviation  r e t u r n s on these hedges f o r each bond and f o r each i s that  to t h e  o f the  model.  The  r e t u r n s t o the zero-investment d i f f e r e n t from z e r o * .  It  appears t h a t we cannot r e j e c t the h y p o t h e s i s that t h e market  is  efficient  to information  contained  6  i n the models.  An a l t e r n a t i v e s t r a t e g y was a l s o adopted f o r t e s t i n g market efficiency. the  above  strategy bond  I t was observed t h a t t h e hedge p o r t f o l i o r e t u r n s on test  were  highly  serially correlated.  t e s t e d was t o assume a long  position  in  only i f the p o r t f o l i o return i n the previous  on a constant  The second the  period  generic (based  long p o s i t i o n i n t h e g e n e r i c bond) was p o s i t i v e  -  Hypothesis t e s t i n g was based on the t - s t a t i s t i c , which assumes that the returns t o t h e hedge p o r t f o l i o are normally distributed. Thorpe [ 6 8 ] has shown t h a t i n the o p t i o n pricing framework the hedge portfolio returns a r e not normally d i s t r i b u t e d . , T h i s need not be cause f o r concern, as the t - t e s t is q u i t e robust t o reasonable departures from normality. The d i s t r i b u t i o n of the hedqe p o r t f o l i o returns i s very briefly' i n v e s t i q a t e d toward t h e end o f t h i s s e c t i o n . 6 6  TABLE XLV RETURN ON ZERO INVESTMENT PORTFOLIO BASED ON CONSTANT LONG POSITION IN BOND (Results for a l l models)  BOND  PURE EXP.  LIQ.PREM.  REV. TAX(50a:)  REV.TAX(25%)  C.G.TAX(IOX)  C.G.TAX (20Z)  R 1  0.0286 (0.2784)  0.0210 (0.2478)  -0.0012 (0.1890)  0.0102 (0.2108)  0.0133 (0.2203)  0.0171 (0.2331)  l  -0.0676 (0.4558)  0.0368 (0.1852)  0.0440 (0.1800)  0.0404 (0.1804)  0.0392 (0.1980)  0.0378 (0.1833)  -0.0059 (0.3804)  0.0506 (0.280)  0.0668 (0.2961)  0.0586 (0.2837)  0.0562 (0.2816)  0.0533 (0.2801)  E  E  2  MOV. AVG.  -  3  0.0288 (0.3399)  0.0695 (0.2982)  0.0968 (0.3462)  0.0829 (0.3145)  0.0079 (0.3087)  0.0751 (0.3032)  E  4  -0.0093 (0.3395)  0.0515 (0.2291)  0.0720 (0.2675)  0.0671 (0.2436)  0.0587 (0.2383)  0.0551 (0.2331)  E  5  -0.0022 (0.3748)  0.0054 (0.1722)  -0.0007 (0.1728)  0.0024 (0.1652)  0.0032 (0.1659)  0.0042 (0.1683)  0.0037 (0.1663)  E  6  -0.1011 (0.4254)  0.0068 (0.5235)  0.0169 (0.3560)  0.0119 (0.4254)  0.0103 (0.4522)  0.0084 (0.4869)  0.0069 (0.5419)  F  7  -0.0453 (0.6514  -0.0067 (0.3868)  0.0089 (0.3119)  -0.0052 (0.3654)  -0.0072 (0.3790)  -0.0091 (0.3938)  -0.0045 (0.3710)  E 8  -0.0453 (0.7945)  0.0020 (0.4821)  0.0282 (0.3760)  0.0147 (0.4127)  0.0131 (0.4173)  0.0094 (0.4189)  0.0096 (0.4337)  E  9  -0.0207 (0.7338)  0.0112 (0.3547)  0.0143 (0.3138)  0.0055 (0.3681)  0.0058 (0.3783)  0.0066 (0.3885)  0.0144 (0.3087)  E  10  0.0058 (0.5242)  -0.0019 (0.2968)  0.0068 (0.2455)  0.0057 (0.2747)  0.0045 (0.2835)  0.0030 (0.2962)  0.0028 (0.2732)  11  0.0110 (0.3628)  0.0034 (0.2759)  0.0043 (0.2970)  0.0061 (0.2828)  0.0062 (.02825)  0.0060 (0.2828)  0.0037 (0.3069)  E  12  0.0042 (0.6296)  0.0047 (0.4063  -0.0019 (0.3577)  -0.0007 (0.4106)  0.0000 (0.4204)  0.0009 (0.4307)  -0.0035 (0.4626)  E  13  0.0048 (0.4381)  0.0016 (0.3558)  -0.0067 (0.3975)  -0.0041 (0.3600)  -0.0036 (0.3579)  -0.0031 (0.3568)  -0.0039 (0.3572)  E  14  0.0014 (0.4031)  0.0024 (0.2784)  0.0010 (0.2753)  0.0006 (0.2750)  -0.0007 (0.2792)  0.0009 (0.2848)  0.0000 (0.2846)  E  15  0.0088 (0.4816)  0.0231 (0.4037)  0.0318 (0.4351)  0.0271 (0.4056)  0.0266 (0.4034)  0.0260 (0.4022)  0.0234 (0.4003)  E  16  0.0164 (0.5054)  0.0227 (0.4195)  0.0242 (0.4412)  0.0233 (0.1463)  0.0233 (0.4155)  0.0232 (0.4158)  0.0196 (0.4144  E  17  -0.0222 (0.4739  -0.0148 (0.3905)  -0.0124 (0.3810)  -0.0163 (0.3861)  -0.0169 (0.3901)  -0.0176 (0.3953)  -0.0121 (0.3785)  E  18  0.0008 (0.2369)  0.0120 (0.2209)  0.0220 (0.2459)  0.0169 (0.2279)  0.0158 (0.2252)  0.0146 (0.2229)  0.0173 (0.2278)  E  19  -0.0319 (0.4985)  -0.0056 (0.4221  0.0178 (0.4102)  0.0016 (0.4116)  -0.0015 (0.4152)  -0.0049 (0.4203)  0.0066 (0.4083)  E  E  _  -  160  if  negative,  a  s h o r t p o s i t i o n was assumed i n G, and t h e hedge  p o s i t i o n formed a c c o r d i n g l y .  T h i s s t r a t e g y was t e s t e d  models, but only the r e s u l t s f o r the pure e x p e c t a t i o n model  a r e presented  all  hypothesis  i n Table XLVI, because the r e s u l t s are very  s i m i l a r f o r a l l the other models. our previous  for  There i s no reason  to  alter  conclusion.  The t h i r d t e s t was t o see i f the model was able t o i d e n t i f y over-  and  underpriced  Galai[29])  This  model p r i c e i s lower  on a t e s t i n  return,  over  the  hedge  portfolio  could  say  i d e n t i f y overpriced/underpriced and  bond i f  bonds  that  the  bonds.  on  this  long p o s i t i o n i n  model  i s able  The r e s u l t s of t h i s  i s presented  again, t h e mean r e t u r n appears t o be  based  s i g n i f i c a n t i n c r e a s e i n the  t h e s t r a t e g y of a constant  the g e n e r i c bond, we  a l l models  generic  (higher) than the market p r i c e a t t h a t  strategy resulted i n a s t a t i s t i c a l l y  for  (based  (short) p o s i t i o n i n t h e  point.. I f t h e r e t u r n on  mean  test  i s q u i t e s i m i l a r to t h e previous ones, only that each  p e r i o d we take a long its  bonds.  i n Table XLVII.  insignificantly  to test Here  different  from z e r o , based on a t - t e s t . The  results  o f the p r e v i o u s t h r e e t e s t s were based on the  r e t u r n s t o hedge p o r t f o l i o s , using one bond a t a time. felt  that  i f the  each p e r i o d  was  It  was  hedge r e t u r n s over a l l bonds o u t s t a n d i n g i n  considered  (along  the  lines  of  Brennan  S  Schwartz [ 11 ] ) , t h e aggregation  might l e a d t o a r e d u c t i o n i n the  variance  the hedge p o r t f o l i o and thereby  improve overcome  of  the  returns  the s t a t i s t i c a l the  problem  to  significance of  of  the  heteroscedasticity  returns. caused  by  To the  d i f f e r e n t numbers of hedge p o r t f o l i o s i n each p e r i o d , t h e d o l l a r  161  TABLE XLVI RETURN ON ZERO NET INVESTMENT PORTFOLIO USING A STRATEGY BASED ON RETURNS TO SIMILAR PORTFOLIO FROM A CONSTANT LONG POSITION IN THE GENERIC BOND. ( R e s u l t s f o r PEXP model o n l y )  BOND  Mean S t d . Dev. o f Return ( $ ) R e t u r n  t - Stat  Rl  -0.0412  2.777  -0.149  El  -0.0808  0.454  -0.178  E2  0.0208  0.380  0.055  E3  0.0491  0.338  0.145  E4  0.0206  0.339  0.061  E5  0.0531  0.372  0.143  E6  0.0556  0.153  0.036  E7  0.0612  0.650  0.094  E8  -0.0639.  0.793  -0.081  E9  0.0961  0.728  0.132  E10  0.0159  0.524  0.032  EH  -0.0099  0.363  -0.028  El 2  -0.0693  0.626  -0.111  E13  -0.0840  0.430  -0.195  E14  -0.0281  0.402  -0.070  E15  -0.0078  0.482  -0.016  E16  0.0042  0.506  0.008  E17  -0.1990 .  0.430  -0.462  E18  -0.0415  0.233  -0.178  E19  -0.1470  0.477  -0.307  TABLE XLVII RETURN ON ZERO INVESTMENT PORTFOLIO BASED ON VARYING POSITION IN BOND (Results for a l l models)  BOND  PURE EXP.  LIQ. PREM.  REV.TAX(50Z)  REV.TAX(25Z)  C.G.TAX(10%)  C.G.TAX(20%)  MOV.AVG.  R 1  -0.0636 (0.2724)  -0.0210 (0.2478)  0.0020 (0.1890  -0.0102 (0.2108)  -0.0103 (0.2203)  -0.0173 (0.2331)  **  E  1  -0.0848 (0.4558)  -0.0368 (0.1852)  -0.0440 (0.1800)  -0.0404 (0.1804)  -0.0392 (0.1814)  -0.0378 (0.1833)  -  2  -0.0095 (0.3804)  -0.0449 (0.2814)  -0.0668 (0.2961)  -0.0473 (0.2858)  -0.0453 (0.2836)  -0.04 70 (0.2812)  -  3  0.0288 (0.3399)  -0.0790 (0.2988)  -0.0882 (0.3485)  -0.0822 (0.3147)  -0.0788 (0.3088)  -0.0751 (0.3032)  -  4  -0.0093 (0.3395)  -0.0429 (0.2309)  -0.0625 (0.2700)  -0.0502 (0.2463)  -0.0477 (0.2408)  -0.0457 (0.2352)  -  E 5  -0.0223 (0.3748  -0.0060 (0.1722)  0.0024 (0.1728)  -0.0016 (0.1652)  -0.0029 (0.1659)  -0.0042 (0.1683)  0.0029 (.16632)  E 6  -0.0433 (0.4823)  0.0027 (0.5236)  -0.0095 (0.3563)  -0.0106 (0.4254)  -0.0092 (0.4522)  -0.0051 (0.4870)  0.0080 (.54194)  E  7  -0.0453 (0.6514)  -0.0042 (0.3868)  0.0110 (0.3119)  -0.0052 (0.3654)  -0.0094 (0.3789)  -0.0133 (0.3937)  -0.0172 (0.3706)  E  8  -0.0435 (0.7549)  -0.0185 (0.4817)  -0.0263 (0.3761)  0.0055 (0.4129  0.0104 (0.4173)  0.0040 (0.4190)  -0.0094 (0.4337)  9  -0.0207 (0.7338)  0.0063 (0.3549)  0.0129 (0.3139)  0.0055 (0.3681)  0.0058 (0.3783)  0.0229 (0.3879)  0.0115 (0.3088)  10  0.0045 (0.5242)  -0.0039 (0.2968)  -0.0143 (0.2452)  -0.0023 (0.2747)  -0.0055 (0.2835)  -0.0106 (0.2960)  -0.0091 (0.2730)  11  0.0142 (0.3627)  -0.0030 (0.2759)  -0.0098 (0.2968)  0.0096 (0.2827)  0.0219 (0.0281)  0.0102 (0.2827)  -0.0037 (0.3069  12  0.0042 (0.6296)  -0.0170 (0.4060  -0.0024 (0.3577)  -0.0072 (0.4106)  0.0003 (0.4204)  0.0321 (0.4295)  -0.0058 (0.4626)  13  0.0048 (0.4381)  -0.0219 (0.3552)  -0.0175 (0.3972)  -0.0041 (0.3600)  T0.0093 (0.3577)  -0.0182 (0.3564)  -0.0039 (0.3572)  14  0.0014 (0.4031)  -0.0047 (0.2784)  -0.0033 <0.2753)  0.0006 (0.2750  -0.0056 (0.2792)  0.0076 (0.2847)  0.0004 (0.2846)  E  E  E  E E E E E  E  E  0.0088 (0.4816)  -0.0167 (0.4040)  -0.0335 40.4349)  -0.0303 (0.4054)  -0.0221 (0.4037)  -0.0322 (0.4017)  -0.0338 (0.3996)  16  0.0164 (0.5055)  0.0086 (0.4201)  -0.0313 (0.4408)  -0.0177 (0.4166)  -0.0286 (0.4151)  -0.0240 (0.4158)  0.0142 (0.4146)  17  -0.0222 (0.4739)  -O.0310 (0.3895)  -0.0124 (0.3810)  -0.0163 (0.3861)  -0.0169 (0.3901)  -0.0204 (0.3952)  0.0121 (0.3785)  18  -0.0171 (0.2363)  -0.0120 (0.2209)  -0.0220 (0.2539)  -0.0169 (0.2279  -0.0158 (0.2252)  -0.0146 (0.2229)  -0.0173 (0.2278)  -0.1421 (0.4764)  -0.0948 10.4061)  -0.0076 (0.4025  -0.0744 (0.3986)  -0.0781 (0.4020)  -0.0938 (0.4042)  -0.0540 (0.3981)  15  E E E  E 19  163  r e t u r n i n each p e r i o d was the  number  of  weighted by 1/JIT,  period.  The  of  the  as  formed)  i n Table L I .  its  (which  standard  The  in mean  deviation,  same order of magnitude as i n the case o f the  r e s u l t s i n Tables XL?  to XLVII - aggregation  statistically  significant  increased  r e s u l t was  unexpected.  The  not  portfolios  r e s u l t s are presented  d o l l a r r e t u r n per p e r i o d , as w e l l remained  represents  r e t r a c t a b l e / e x t e n d i b l e bonds o u t s t a n d i n g  t h e r e f o r e represent the number o f hedge each  where N  has  not l e d t o  any  p r o f i t opportunity.  This  movement of bond p r i c e s e x h i b i t s  high contemporaneous c o r r e l a t i o n , s o t h a t the r e t u r n s t o the investment hedge p o r t f o l i o s would a l s o be Thus,  aggregating  l e a d to any  a c r o s s bonds a t any  likewise  correlated.,  i n s t a n t i n time would  s i g n i f i c a n t r e d u c t i o n i n the d i s p e r s i o n  t o the hedge p o r t f o l i o .  of  not  so  In the case of o p t i o n s on common s t o c k ,  on a s i m i l a r  In forming dollars  stocks  high, which c o u l d l e a d to v a r i a n c e r e d u c t i o n due  aggregation  in  generic  bond,  d o l l a r s i n another bond, where x  investment  the s t r a t e g y was L  was  given  of  was  the market p r i c e of the s t r a i g h t bond.  bond  want to observe  offers  arbitrage  f o r other f a c t o r s .  whether  the  The  use  above  I t could reasoning  retractable/extendible  p r o f i t opportunities, after  When we  x^  by  be argued t h a t model p r i c e s should be used f o r B. i s t h a t we  x,  to i n v e s t  In the t e s t s performed so f a r , t h e value of B used i n the expression  to  test.  the hedge p o r t f o l i o s , f o r an  the  not  returns  however, the contemporaneous c o r r e l a t i o n a c r o s s d i f f e r e n t is  zero  controlling  market p r i c e f o r B, due  to  the  TABLE L I RETURN ON ZERO NET INVESTMENT PORTFOLIO (BASED ON A CONSTANT LONG POSITION IN THE GENERIC BOND) BY AGGREGATING  OVER ALL BONDS  ( R e s u l t s f o r a l l models)  Mean R e t u r n ($)  Model  S t d . Dev. of R e t u r n  t-Stat  PEXP  -0.0197  0.955  -0.021  LIQP  0.0228  0.476  0.048  REV.TAX (50%)  0.0352  0.486  0.072  REV.TAX (25%)  0.0270  0.451  0.060  CG.TAX (10%)  0.0258  0.457  0.056  CG.TAX (20%)  0.0242  0.466  0.052  MOV.AVG.  0.0113  0.484  0.023  Notes:  1)  The above t e s t , by a g g r e g a t i n g over a l l k i n d s o u t s t a n d i n g i n e v e r y p e r i o d , was a l s o performed on t h e o t h e r two market e f f i c i e n c y t e s t s . The r e s u l t s a r e not r e p o r t e d as they a r e v e r y s i m i l a r t o t h e ones above.  2)  The models have been l i s t e d i n the t a b l e u s i n g t h e a b b r e v i a t i o n s used i n t h e t e x t and i n e a r l i e r t a b l e s .  165  valuation  error  maintained  which i n c r e a s e s the variance of the  zero  in  investment  B,  the  hedge  c o r r e c t hedge p r o p o r t i o n s are  portfolio.  of e r r o r - i t i s a  retractable/extendible  bond.  all  pure  three  r e l a t e d t e s t s reported above were repeated,  for  test  market  of  different  from those obtained  of  evaluate  x  to  p o r t f o l i o returns.  ,  %  as  well  To i n d i c a t e  from using market and  market e f f i c i e n c y , the mean and investment  the  efficiency  generic  The  by using  used  r e s u l t s were market  price  as f o r e v a l u a t i n g the hedge  the  degree  of  similarity  of  model p r i c e s of B i n the t e s t s of standard  d e v i a t i o n of  the  zero  hedge p o r t f o l i o r e t u r n f o r the C a p i t a l Gains Tax  model(using a s t r a t e g y bond)  are  of  a  constant  presented  8,  ( f o r each i n d i v i d u a l  the r e t a c t a b i e / e x t e n d i b l e bonds.  h a r d l y any  results  the  aggregated over a l l bonds) f o r each of the models  valuing  B  to  By using model values o f  there i s no other source  bond and  returns  not  long  position  i n Table L.., I t was  in  f e l t that  f u r t h e r i n f o r m a t i o n would be conveyed by p r e s e n t i n g the  20% the no  complete  r e s u l t s a c r o s s a l l models f o r a l l three hedging s t r a t e g i e s . F i n a l l y , the p o r t f o l i o hedge  returns  p o s i t i o n ) were t e s t e d f o r normality  graphing  approach  necessary,  as  outlined the  earlier..  t-statistic  ( i e . mean/standard d e v i a t i o n ) was 0.1  (on  and  that  should  be  Figures 8  and  9, we  investment  using the  probability  of  the  mean and  not mean  strictly return  from  sample cases.,  variance.  of  i n s i g n i f i c a n t i n most  d i s t r i b u t i o n s appear t o have more mass at the d i s t r i b u t i o n of equal  was  almost always of the order  departures  present two  zero  This  statistically  s i t u a t i o n s even with resonable  the  normality. In g e n e r a l ,  In the  mean than a normal  TABLE L COMPARISON OF RETURNS TO THE ZERO INVESTMENT HEDGE PORTFOLIO BY USING MARKET VS. MODEL PRICES FOR THE STRAIGHT BOND  USING MODEL P R I C E S STRAIGHT  BOND  USING MARKET P R I C E S  FOR  FOR  S T R A I G H T BOND  BOND  MEAN  STD.DEV  0.047  0.0171  0.233  0.073  0.349  0.180  0.379  0.183  0.207  0.0622  0.517  0.120  0.0534  0.280  0.191  E3  0.0831  0.664  0.125  0.0752  0.303  0.248  E4  0.0631  0.474  0.133  0.0551  0.233  0.237  E5  -0.0019  0.220  -0.009  0.0042  0.168  0.025  E6  0.0010  0.334  0.003  0.0084  0.487  0.017  E7  0.0125  0.425  0.029  -0.0091  0.394  -0.023  E8  0.0193  0.452  0.043  0.0095  0.419  0.023  E9  0.0106  0.567  0.019  0.0067  0.389  0.017  ElO  0.0058  0.505  0.011  0.0030  0.296  0.010  Ell  0.0089  0.686  0.013  0.0061  0.283  0.021  E12  -0.0034  0.343  -0.010  0.0010  0.431  0.002  E13  -0.0075  0.482  -0.016  -0.0031  0.357  -0.009  E14  -0.0032  0.375  -0.009  0.0009  0.285  0.003  E15  0.0200  0.509  0.039  0.0260  0.402  0.065  E16  0.0281  0.511  0.055  0.0232  0.416  0.056  E17  -0.0257  0.484  -0.053  -0.0176  0.395  -0.045  E18  0.0149  0.289  0.051  0.0147  0.223  0.066  -0.0152  0.588  -0.026  -0.0049  0.420  -0.012  0.0265  0.786  0.034  0.0242  0.466  0.052  MEAN  STD.DEV  Rl  0.0135  0.287  El  0.0627  E2  El9 Aggregate  - t-STAT  NOTES: 1) The above r e t u r n s correspond g e n e r i c bond s t r a t e g y .  t o u s i n g the constant  t-STAT  l o n g p o s i t i o n i n the  2) The model used f o r the v a l u a t i o n of the g e n e r i c bond was the C a p i t a l Gains 20% model.  167  FIGURE 8 C0MPRRI5QN OF MARKET 8. MODEL PRICES  (MODEL  BDJU5T1NG FOR CRPITRL GRINS TRX) BONOi E4s 5.50J RPR ] 1963  DISTRIBUTION O F fCDEE PORTFOLIO RETURNS "MHEDGE BRSED ON VRRm'G POSITION IN BOND :IKDD£L R D J U 5 T I N G FOR tRPITRL GRINS TRX) KWDB\-.5.5DJ «P» 4 1963  •-NORMRL PROBABILITY PLOT OF HEDGE PORTFOLIO RETURNS "HEDGE BRSED ON VRRTING POSITION IN BOND (MODEL RD JUST ING FOR CRPITRL GRINS TRX) 501 BOHDi f4-*- I * ™ l ^ 1  Jtlli -0.45 « 10 » -I •SmntVi 0.335  -O.OflS " VPLUt tr HECSE  !«  HU3GE K1URH IK  96  FIGURE 9  168  CQMPRRISDN OF MARKET I MODEL PRICES (MODEL ADJUSTING FOR CRP1TRL GRINS TRX) BDNDi E7s 7.25X RPR 19 1974  MRRKET PRlCEl  HOOtL PRItti DO00DD0  DISTRIBUTION OF HEDGE PORTFOLIO RETURNS HEDGE BR5ED ON VRRY1NG POSITION IN BOND (MODEL BDJUSTING FOR CRPITRL GRINS TAX) BONOftt T.75I RPR ]9 1974  KR.1l -0.13 X JO KX -I SIOKVi 0.J93  NORHRL PROBRBILITY PLOT OF HEDGE PORTFOLIO RETURNS HEDGE BRSEO ON VRRY1NG POSITION IN BOND IHODEL ADJUSTING FOR CRPITRL GRIN5 TRX! BOHDt I7» 7.25/ RPR 19 1974  169  7.10  Comparison of Current Models with a "Naive"  Model  Before we conclude our a n a l y s i s on bond p r i c e s , we bench  mark  against  which  models i n v a l u i n g bonds. valuation  to  compare  To t h i s end,  need  the performance of our we  develop  an  ad  hoc  model - which we s h a l l r e f e r t o as the " n a i v e " model.  I t i s based on an approach suggested  i n Dipchand S Banrahan [ 9 ] .  Based on a r e g r e s s i o n equation f o r the y i e l d curve developed Bell  a  Canada*s  maturity  Bond  by  Research D i v i s i o n , we compute the y i e l d t o  on each e x t e n d i b l e *  7  i n our sample.  each p o i n t i n time, we estimate two  -  one  correspondinq t o each of the a l t e r n a t i v e m a t u r i t i e s .  Usinq  each  yield,  continuous  we  discount  the  future  yields  coupons  to  For each bond, a t maturity  (assuming  coupon payments) and the p r i n c i p a l , and thus f i n d the v a l u e s the set  long  and short bonds.  The  of  p r i c e of the e x t e n d i b l e i s then  to the higher of the long and s h o r t bonds, a t every  point i n  time. B e l l Canada's y i e l d curve r e g r e s s i o n model  where Y having  was  r e p r e s e n t s the y i e l d t o maturity a t time t on  t  X  t  months to maturity.  For our study, we modified  model s l i g h t l y t o i n c l u d e i n the r e g r e s s i o n equation v a l u e of the short term i n t e r e s t r a t e . inclusion  should  improve  the  a  f i t of  I t was the  felt model.  bond the  the c u r r e n t that  this  Thus, the  The r e t r a c t a b l e B1 was not priced according to the naive model because i t had s e v e r a l r e t r a c t i o n dates. T h i s makes i t complicated to p r i c e , and i t was f e l t t h a t dropping one case should not a f f e c t the comparison. 6 7  170  r e g r e s s i o n model used to determine y i e l d s  Y  a  t  The  t  - v a ^  + aX<. 3  next problem was  purpose,  the  c^X + <* \ + t  4  5  t o determine the  future payoffs.  instant  Thus we  \  can  =  is  CL^Xf.  coefficients.  s t r a i g h t coupon bond sample was  p r i c e of a bond at any  where  -V  was  the  used.  present  this  The  market of  its  write  c*~* dtt  +  i s the time to maturity, and  bond i s $100.  For  value  loo  e  r  y i s the y i e l d to maturity at time t , c i s the  coupon, X  (7.20)  the f a c e  continuous  value  of  the  This gives  (7.21)  Using the  eguation  other parameters.  bonds and  (7.21) above, we  at  each  T h i s was  p o i n t i n time.  of  the  regression  C o n s i s t e n t with the experience from  2  most of the equations  the whole sample). total  The  6 8  ,  given B  %  done f o r a l l 18 s t r a i g h t Then, f o r each of the  f o r the whole sample, r e g r e s s i o n  results  R  can s o l v e f o r y  are  (7.20) was reported  coupon  18 bonds,  performed. in  Table  ;  regression  over 0.80  (except f o r F9  coefficients  sample were used to p r i c e each of the  based  The  XLVIII.  o f Dipchand S Hanrahan [ 1 9 ] , was  and  on  the and the  19 e x t e n d i b l e bonds  * A numerical a l g o r i t h m t h a t s o l v e s f o r the zeros of n o n l i n e a r equations was used. The s t a r t i n g value s u p p l i e d i n the search for a root was the c u r r e n t value of r . I t can e a s i l y be shown t h a t equation (7.21) has only one r o o t . 8  171  TABLE  XLVIII  RESULTS OF YIELD EQUATION COEFFICIENT ESTDIATION FOR "NAIVE" MODEL (Yeild = a  BOND  a^lO  2 a  2  x  + a r  a «10  2  3  3  t  + a^T + a ^ / i  «4 1.0 X  2  Fl  0.6510 (16.42)  0.4857 (14.96)  0.8323 (5.13)  -0.0434 (-4.13)  F2  0.0776 (13.53)  0.4636 (36.70)  -0.0241 (-11.61)  0.0652 (12.71)  F3  0.0z82 (15.42)  0.5863 (52.52)  -0.0057 (-3.10)  0.0210 (4.57)  F4  0.0531 (6.41)  0.8941 (43.28)  0.1026 (3.90)  -0.1273 (-3.14)  F5  -0.0658 (-1.31)  0.9979 (31.38)  -0.3540 (-2.87)  F6  0.0882 (20.78)  0.59S2  0.3142  0.8066  (20.24)  (20.71)  -  + ajT  + a T  2  a *10  6  5  5  -0.3163 (-6.72) 0.00151 (7.76) -0.0001 (-0.69) -0.0414 (5.69)  3  + a logT ) 7  a xl0  9  6  a *10  2  7  R  2  7.8739 (7.89)  0.8427 (3.01)  0.9418  -0.0046 (-4.03)  -0.0940 (-11.85)  0.9295  - 0.0032 (3.90)  -0.0347 (4.97)  0.8977  1.0565 (7.62)  0.1129 (2.61)  0.9342  0.4993 (2.44)  0.1211 (3.89)  -2.7779 (-4.84)  -0.5007 (2.09)  0.9270  0.0098 (5.31)  -0.0105 (-2.63)  -0.0018 (-9.12)  0.0140 (10.76)  0.0004 (0.07)  0.8916  0.0918 (1.33)  -0.0486 (-0.45)  -0.0546 (-2.94)  1.4785 (4.28)  -0.0668 (-0.58)  0.9513  (21.87)  251.09 (3.20)  -172.87 (-3.26)  -0.3991 (-3.06)  0.5026 (2.92) .  870.01 (3.33)  0.9004  -0.0803 (-1.17)  0.9587 (6.86)  0.2523 (3.24)  -0.4556 (-3.23)  -0.0460 (-3.11)  0.5171 (2.94)  0.5555 (3.15)  0.4990  0.0738  0.5713 (15.66)  0.1367 (2.33)  -0.2001 (-2.54)  -0.0483 (-2.33)  1.2322 (2.49)  0 .2116 (2.89)  0.9635  (3.50)  0.6947 (23.19)  -0.0081 (-1.27)  0.0097 (1.01)  0.0022 (1.46)  -0.0338 (-1.54)  -0.0027 (-0.31)  0.9015  F12  371.99 (4.45)  0.4623 (25.37)  -8.6736 (-4.83)  42.659 (4.69)  0.2689 (5.13)  -0.6572 (-5.47)  -153.04 (-4.56)  0.8614  F13  0.0896 (9.19)  0.8474 (15.63)  0.0677 (4.13)  -0.0877 (-3.76)  -0.0189 (-4.37)  0.2772 (4.36)  0.0695 (3.21)  0.8695  F14  4195.5  -0.2159 (-2.62)  80.383 (0.79)  -213.34 (-0.62)  -3.1702 (-0.81)  7.0379 (0.79)  -437.32 (-0.35)  0.8404  F7  -2704.9  F8  (-3.38) F9  F10  (3.09)  0.0289  Fll  (0.70)  (47.67)  0.3566  F15  2031.73 (5.35)  0.3441 (19.14)  -30.575 (-5.59  175.98 (5.50)  0.6934 (5.76)  -1.2435 (-5.93)  -739.53 (-5.42)  0.8772  F16  -0.2189 (-0.25)  0.4323 (43.25)  0.0579 (1.90)  -0.1699 (-1.28)  -0.0038 (-3.33)  0.0148 (4.68)  0.3186 (0.77)  0.9306  F17  -0.0081 (-1.13)  0.7408 (25.08)  -0.0447 (-5.25)  0.0672 (4.95)  0.0109 (5.50)  -0.1579 (-5.46)  -0.0598 (-4.32)  0.8818  El 8  0.0154 (4.22)  0.4883 (30.39)  -0.0371 (-11.96)  0.0599 (12.20)  0.0080 (12.41)  -0.1020 (-12.74)  -0.0580 (-12.18)  0.8750  TOTAL  0.0509 (16.81)  0.7049 (138.12)  -0.0067 (-12.39)  0.0184 (11.78)  0.0005 (14.20)  -0.0019 (-15.34)  -0.0271 (-9.80)  0.7679  -  Figures i n parenthesis are the t s t a t i s t i c for the estimated c o e f f i c i e n t  J  172  i n our sample. these  model  The r e s u l t s of r e g r e s s i n g the market prices  are  reported  r e s u l t s from the summary run by bonds  is  in  Table XLIX.  i n d i c a t e s t h a t the n a i v e  superiority  Tables XXII to XL.  aggregating  over  A c u r s o r y examination model  comparison t o t h e o t h e r models. the  in  of  the  performs  three  criteria  used  to  all  on The  the  19  of the r e s u l t s  reasonably  well,  in  C l o s e r s c r u t i n y however r e v e a l s ,-, more  rigorous  r e t r a c t a b l e / e x t e n d i b l e bond v a l u a t i o n developed The  prices  models  in  this  of  study.  e v a l u a t e the performance of each  model were: 1)  c o r r e l a t i o n between market and model v a l u e s  2)  s l o p e of the r e g r e s s i o n of market and  3)  mean e r r o r  model p r i c e s  (or RMSE) as a measure of b i a s .  Comparing the r e s u l t s of the C a p i t a l Gains Tax 20% model (column 6 i n T a b l e s XXI the  naive  model,  outperforms  it  is  the naive model  consistently.  Looking  with  respect  to  seen on  that the  the first  is  the  simple  (the  model  counts  The root  almost  pooling a l l  observation  square  However, as  is  borne  correlation of  the  pointed  out  mechanism  c o r r e l a t i o n and the slope c o e f f i c i e n t .  in  to  the  improve  S i n c e the o b j e c t i v e  of the present study i s more one of d e s c r i p t i o n , r a t h e r than "fitting"  R-  s e c t i o n s ; a l t e r i n g the Revenue Tax r a t e and the C a p i t a l  Gains Tax r a t e , p r o v i d e s a " f i n e t u n i n g " the  two  2.0%  c o r r e l a t i o n c o e f f i c i e n t ) i s marginally  s u p e r i o r i n the naive model. earlier  Tax  slope c o e f f i c i e n t .  between market and model p r i c e s squared  CG  a t the summary r e s u l t s from  the  20%)  to XL and Table XLIX) with t h a t of  20 bonds (Table XLIX), we see t h a t above out  (CG Tax  the best model, no f u r t h e r attempt  was  of  made t o f i n d a  173  s e t of t a x r a t e s t h a t a c t u a l l y correlations the  provided  consistently  over t h a t of the naive model.  F i n a l l y , l o o k i n g at  bias measures, we see from t h e summary r e s u l t s i n T a b l e XLTX  t h a t t h e mean e r r o r i s lower f o r the CG Tax 20% the  P.MSE  i s lower f o r t h e naive model.  model,  split  -  the  naive model performs b e t t e r  the CG Tax 20% model. increase  However, we  note  almost  even  j u s t as many times as that  i f we  were  to  the CG Tax r a t e used i n the model, t h i s would l e a d not  only t o a r e d u c t i o n i n the c o r r e l a t i o n .  i n t h e mean e r r o r but a l s o t o an improvement Thus, i t would be f a i r  t o say t h a t , even i n  t h e i r present s t a t e , t h e p a r t i a l e q u i l i b r i u m Chapter  whereas  Comparison of the mean  e r r o r over i n d i v i d u a l bonds (Table XLI), we see an  2  are superior  retractable/extendible  i n several  models developed i n  respects  in  naive  improvement.  model  -  predicting  bond p r i c e movements, when compared  a naive model o f a reasonable l e v e l of complexity,and the  improved  a r e amenable  to  considerable  -  with  unlike further  174  CHAPTER 8: SUMMARY AHD CONCLOSIONS  8. 1  Summary Of The T h e s i s  The c u r r e n t r e s e a r c h can be d i v i d e d i n t o three broad a r e a s ; 1)  choosing  an  appropriate  specification of 2)  continuous  time  stochastic  t o model the i n s t a n t a n e o u s r i s k l e s s r a t e  interest.  i d e n t i f y i n g methods t o estimate the parameters of a  model,  given  interest  rate  a  discrete  process,  time  and  such  r e a l i z a t i o n o f the  comparing  the  relative  e f f i c i e n c i e s of t h e d i f f e r e n t e s t i m a t i n g methods. 3)  developing  and e m p i r i c a l l y t e s t i n g a model f o r v a l u i n g  d e f a u l t - f r e e r e t r a c t a b l e and e x t e n d i b l e bonds. Chapter mathematical process.  3 addresses t h e problem o f choosing model  f o r the  In the absence  reasoning  of  and mathematical  A mean-reverting  any  as t h a t adopted  literature,  (see Vasicek [ 7 2 ] ,  with a more  general  eguations adopted are  both  study. process  special  economic  were the o n l y c r i t e r i a . having a d r i f t  by others i n t h e  existing  C o x , I n g e r s o l l & Boss [16]) but element.  Thus,  the  diffusion  cases  o f t h e more g e n e r a l form  of the assumed form of the  used i n t h i s  interest  rate  i t s s i n g u l a r boundaries i s i n v e s t i g a t e d , t o ensure  t h a t i t s behaviour properties  guidelines,  by Vasicek [ 7 2 ] and Cox, I n g e r s o l 6 Ross [ 1 6 ] ,  The behaviour at  formal  tractability  variance  appropriate  s h o r t term r i s k l e s s i n t e r e s t r a t e  d i f f u s i o n process was suggested,  term o f t h e same form  an  at  these  attributable  to  points an  i s consistent  interest  rate  with  process  the from  175  economic  reasoning.  Three a l t e r n a t e methods a r e proposed i n Chapter 4 estimation  of  t h e parameters o f t h e i n t e r e s t r a t e process, and  t h e i r r e l a t i v e merits and weaknesses a r e pointed them a r e maximum l i k e l i h o o d methods. density  out.  A l l of  The T r a n s i t i o n P r o b a b i l i t y  method i s exact, but the t r a n s i t i o n  probability density  i s not known f o r a l l parameter values o f the Its  f o r the  proposed  process.  use would r e q u i r e c u r t a i l i n g t h e g e n e r a l i t y of the i n t e r e s t  r a t e process density based  model.  approach on  to  C a r l o methods  Ho  analytical  method  to  Monte the  employed.  Carlo  Chapter 5  simulations  estimators,  parameter e s t i m a t i o n .  using  to the  and,  (b)  the  resultant  The r e s u l t s i n d i c a t e t h a t with  rather  similar  presents arrive  three  The c r i t e r i a  across the t h r e e methods was (a) the b i a s and  estimators  be  be  of  prices.  could  had  distribution  estimators  State  the e s t i m a t o r s o f the parameters - Monte  the  of  (the Steady  compare  of  methods  methods  and the Simple L i n e a r i z a t i o n method) are both  approximations.  developed  results  The other two  the  at the alternate  used t o compare  variance  of the  b i a s and v a r i a n c e on bond  a l l three properties,  methods  produce  and so a r e q u i t e  comparable. P a r t i a l e q u i l i b r i u m v a l u a t i o n models based pricing  approach  on  the  were developed i n Chapter 2, f o r very  general  s t o c h a s t i c s p e c i f i c a t i o n s of the  interest  valuation  from the e a r l i e r works o f Cox,  Ingersol  models &  Vasicek [ 7 2 ] . when  the  draw  heavily  Boss [ 1 6 ] ,  Brennan  6  rate  option  Schwartz  process.  [10,12],  The  and  The performance o f models developed i n Chapter 2, interest  rate  process  of  the  chosen  form  is  176  incorporated, bonds  was  in  pricing  a  sample  t e s t e d i n Chapter 7.  complete s e t of  of  The bond sample chosen was the  retractable/extendible  Canada.  retractable/extendible  Government  of  bond i s s u e d  i n January 1959 and 19  bonds  The sample c o n s i s t e d  October 1959 and October 1975.  issued  by the  o f one r e t r a c t a b l e  extendibles  issued  between  weekly data on market p r i c e s f o r  t h i s s e t was c o l l e c t e d from the Bank o f Canada Review . Model  prices  based  about t h e term s t r u c t u r e investors  were  on of  the  pure e x p e c t a t i o n s  interest  consistently  rates  on  hypothesis  the  part  of  higher than a c t u a l market p r i c e s .  When a p r o v i s i o n was made f o r a t e r m / l i q u i d i t y  premium  i n the  term s t r u c t u r e o f i n t e r e s t r a t e s , model p r i c e s were more i n l i n e with  market  prices.  Incorporating  revenue  taxes ( t a x e s on  i n t e r e s t payments and on coupon r e c e i p t s ) and then c a p i t a l taxes, market the  improved the  in  predicting  p r i c e movements,, To s e r v e as a benchmark f o r  evaluating  performance  valuation extendible models  performance  of  formula bonds.  performed  of  the  model,  was  developed  I t was found atleast  an  that  as  well  the  model  gains  ad to  hoc price  the  regression-based the  partial  sample  of  equilibrium  as the ad hoc model - with  f u r t h e r r e f i n e m e n t s the e q u i l i b r i u m models c o u l d dominate the ad hoc  model. F i n a l l y , t h e e f f i c i e n c y o f t h e bond market  contained  i n t h e models was t e s t e d .  to  information  The approach was t o s e t up  a zero n e t investment  hedge  portfolio  retractable/extendible  bond,  t h e short term i n t e r e s t r a t e , and  any  other  opportunities  bond, were  and  observing  available.  by  whether  investing  any  The r e s u l t s i n d i c a t e d  i n the  arbitrage that the  177  market was  consistently efficient to  information  contained  in  these models.  8*2  Conclusions The  And  D i r e c t i o n s For F u r t h e r  interest  Research  r a t e process proposed i n Chapter 3 i s o f  the  form  The  processes used i n e a r l i e r s t u d i e s were s p e c i a l cases of  above  process..  Thus, Vasicek's  process corresonds to  whereas Richards and  Cox,  I n g e r s o l l S Ross both use the  having  The  results  cK - 1/2.  in  Chapter 5  i n c r e a s i n g the g e n e r a l i t y o f the model free  parameter  (o()  i n the  very  rate  that  bond  fact  though  process are  t h a t the  other  found that  the  above  resulting  cr  extra  and  2  from  most s e n s i t i v e t o the What i s  parameters  bond v a l u a t i o n .  This  model  s a t i s f a c t o r y to p o r t r a y far  I t was  values  the o v e r a l l mean of the process.  impact on  an  that  c\  process  almost t o t a l l y s u b s t i t u t a b l e .  I t appears  the  indicate  including  0,  process  h i g h l y c o r r e l a t e d and t h e i r i n f l u e n c e on the  dynamics was  interest  d\ =  variance element does not m a t e r i a l l y  e n r i c h the f a m i l y of processes. were  by  the  of  is  more  the  parameter jx interesting  ( mn , <r , d.) have very v  an  indication  interest  rates  above  may  is  little  that,  even  be  quite  the i n t e r e s t r a t e dynamice jper se  ,  as  as bond v a l u a t i o n i s concerned we have only a one-parameter  process.  This  alternative process,  clearly  stochastic  indicates  the  specifications  where more than one  need for  to the  look  for  i n t e r e s t rate  parameter has a s i g n i f i c a n t  impact  178  on bond v a l u a t i o n . . The  assumption  of  homogeneity  over time of the i n t e r e s t  r a t e process parameters appears r e s t r i c t i v e . however, to  afford  mathematical  The c o n s t r a i n t i s ,  tractability,  both  estimation  of  valuation.  The approach of Brennan S Schwartz [ 1 2 ]  rate  model  in  to setting  = 1 and making  (they s e t yjL as the long term i n t e r e s t r a t e ) , where r  the  text,  such  of t h e parameters,  resultant  valuation  allowing  of  the  and  equation.  might w e l l be worthwhile.  As  pointed  processes pose a d d i t i o n a l problems i n  estimation  parameter  to  In the framework o f the  and jx f o l l o w c o r r e l a t e d j o i n t d i f f u s i o n processes. out  appears  o f t h i s t h e s i s , t h e i r model f o r the s h o r t  term i n t e r e s t r a t e i s eguivalent stochastic  the  the parameters o f the process as w e l l as i n bond  be one e l e g a n t s o l u t i o n t o the problem. interest  for  even  more  rate  solving  the  However, the a d d i t i o n a l e f f o r t  We have seen that ^  interest  in  process  in  i s the  critical  bond v a l u a t i o n ;  i t t o be s t o c h a s t i c should l e a d to improved  congruence  between model and market p r i c e s . The  term  s t r u c t u r e of i n t e r e s t r a t e s p l a y s a p i v o t a l r o l e  i n the v a l u a t i o n o f d e f a u l t - f r e e bonds.  In the approach of  the  t h e s i s , we attempted t o p r e d i c t the complete term s t r u c t u r e from a  knowledge o f the i n s t a n t a n e o u s i n t e r e s t r a t e .  ambitious. attempt knowing  to  The approach  Brennan  &  Schwartz  [12]  is  the two extreme p o i n t s - the i n s t a n t a n e o u s and t h e  of  an  p r e d i c t the term s t r u c t u r e , a t any i n s t a n t i n time,  long term y e i l d s . model  of  This i s rather  Thus, i t would be reasonable t o expect t h a t a  retractable/extendible  state variables  very  (the s h o r t  term  bond and  valuation the  long  based on term  two  interest  179  rates)  and  time  to  maturity should give s i g n i f i c a n t l y b e t t e r  results. It i s  evident  literature  from  presented  needs to be done i n  the  in  Chapter  the  area  v a l u a t i o n models developed present addressed  brief 1  of  survey  of  the  existing  t h a t a f a i r amount of work  empirical  testing  of  bond  i n t h e option p r i c i n g framework.  t h e s i s i s one step i n t h a t d i r e c t i o n .  However,  only the v a l u a t i o n o f d e f a u l t - f r e e bonds.  The  we have  The  whole  area of c o r p o r a t e bonds {where a p o s i t i v e p r o b a b i l i t y o f d e f a u l t exists) developed problem  has  not  been  tackled.  i n the l i t e r a t u r e , but of  The v a l u a t i o n theory has been empirical  choosing some observable proxy  testing  poses  the  f o r the value o f the  f i r m , as t h i s i s a r e q u i r e d i n p u t t o the bond  valuation  model.  T h i s would be a f r u i t f u l d i r e c t i o n f o r f u t u r e r e s e a r c h . Finally, arrivinq  there  is  considerable  at c l o s e d form or  s t r u c t u r e equation.  interest  analytical  at  solutions  present i n  to  the  term  Vasicek [ 7 2 ] and Cox, I n g e r s o l l 6 Ross [ 1 6 ]  have two d i f f e r e n t s t o c h a s t i c s p e c i f i c a t i o n s t o model the course of  the  qeneral, valuation  instantaneously be  shown  equation  that  riskless the  closely  rate  o f r e t u r n . , I t can, i n  resultinq  resembles  pure  discount  the Kolmogorov backward  eguation qoverninq t h e d i f f u s i o n equation chosen interest  r a t e process.  be transformed  model  backward  i n t o a s i m i l a r forward e q u a t i o n ,  out i n Appendix 3, the forward into  to  the  I t i s a l s o w e l l known t h a t , i n g e n e r a l ,  by a s u i t a b l e r e d e f i n i t i o n o f v a r i a b l e s , the may  bond  equation  could  be  equation as p o i n t e d  transformed  the time homogeneous Schroedinger wave eguation of quantum  physics.  This  equation  has  been  very  widely  studied  and  solutions  for  rather  general  forms have been obtained.  might be an i n t e r e s t i n g d i r e c t i o n f o r r e s e a r c h e r s i n t e r e s t e d analytical  solutions  alternate stochastic  to  the  term  structure  models f o r the i n t e r e s t r a t e  eguation process.  This in for  181 BIBLIOGRAPHY  1  Anderson T. fl. 6 Goodman L. A. (1957), "Statistical Inference About Markov Chains", The Annals of Mathematical S t a t i s t i c s , 28.  2  Bergstrom A.R. (1966) "Nonrecrusive Models as D i s c r e t e Approximations to Systems of Stochastic Differential Equations", Econometrica. V o l . 3ft, No. . 1.  3  B i l l i n g s l e y P. (1961) , " S t a t i s t i c a l I n f e r e n c e s f o r Markov Processes", The U n i v e r s i t y o f Chicago P r e s s .  4  Billingsley P. (1961), "Statistical Methods Chains", Annals o f Math. S t a t . , V o l . 32.  5  Black P. S Cox J.C. 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C.E. ,(1972), "Econometric E s t i m a t i o n o f S t o c h a s t i c D i f f e r e n t i a l Equation Systems", Econometrica, Vol. , 40, No. 3, pp. 565.  187  APPENDIX - 1 C l a s s i f i c a t i o n o f s i n g u l a r boundary behaviour f o r the cases a=%,l. We have as our d i f f u s i o n equation dr  = b(r)dt  +  Al  a ( r ) dZ  where b (r) = m(y-r) and  (Al.l)  a() = a r 2  (AOi.la)  2 a  r  The type o f behaviour a t the s i n g u l a r boundary i s determined by the i n t e g r a b i l i t y o f the f o l l o w i n g two f u n c t i o n s fr  h (r) x  [a (s)  = 7T(r) ro  h (r) 2  = [a(r)  TT  -1  IT  (s) ]  ds  1  (A1.2a)  fr  (r) ]  TT  (s) dS  (A1.2b)  Over the i n t e r v a l I E [r #r ] , where r  i s any i n t e r i o r  Q  point  of the s t a t e space o f the p r o c e s s , and r i s the boundary (r might be i n f i n i t e i n the case o f no " b u i l t i n " f i n i t e The f u n c t i o n  'k(r) i n equations rr  TT  (r)  = exp {-2  boundary)  (A1.2) i s d e f i n e d by  [b ( s ) / a ( s ) ] d s }  (A1.2c)  When both h^ and h^ are i n t e g r a b l e over I , the boundary i s c a l l e d a r e g u l a r boundary and by imposing s u i t a b l e boundary conditions,  the behaviour can be e i t h e r r e f l e c t i n g o r a b s o r b i n g .  When h-^ i s i n t e g r a b l e over I , but h  2  i s not, the boundary i s  c a l l e d an e x i t boundary and i t acts as an absorbing boundary. When h-^ i s not i n t e g r a b l e over I , but h  2  boundary i s c a l l e d an entrance boundary.  i s , the An extrance boundary  i s i n a c c e s s i b l e from i n s i d e the open i n t e r v a l ( r , f ) , but any Q  188  p r o b a b i l i t y assigned to i t i n i t i a l l y ,  flows i n t o the open  interval.  integrable  When both h-^ and h.^  a  the boundary  r  e  n  o  t  i s c a l l e d a n a t u r a l boundary.  This  over I , boundary  i s i n a c c e s s i b l e from i n s i d e the open i n t e r v a l , and any p r o b a b i l i t y assigned to i t i n i t i a l l y I t can be shown t h a t  i s trapped t h e r e  forever.  (see K e i l s o n [41])  rr h-j_(s)ds  =  M ^ ( r | r ) = average time t o reach r s t a r t i n g Q  r  from r  0  Q  (r  Q  i s a r e f l e c t i n g boundary)  r h (s)ds 2  r  =  M ^ ( r | r ) = average time t o reach r Q  o  Q  starting  from r (r i s a r e f l e c t i n g boundary)  This p r o v i d e s the i n t u i t i o n behind the s i n g u l a r boundary  classi-  fication. For the caseoof a = l , we have by s u b s t i t u t i o n from into  (A1.2c), and performing the r e q u i r e d TT (r)  = r ^ exp(3u/r)  and f u r t h e r from  3 = 2m/a  (Al.la)  integration  2  (A1.2)  r exp(3u/r) B  h, (r) =  x" exp(-3y/x) d-x.;6=(2+3)  2  1  >0  r  o rr  6  h,(r) = r " =exp(-3y/r)  x$ exp(3y/x)dx r  o  Performing the i n t e g r a t i o n f o r h^(r) gives  ...(A1.2d)  189  ey/r. h^r)  =•;[  a^re a  2  (By)  hj(s)ds'  Clearly  7 o the  3ya  2  1» ^  —  'Constant  a  of  integralton  approaches  infinity  as  r tends  s e c o n d and t h i r d  terras i n  (Al.. 3)'  (Al.3) to  r  infinity  due t o  We n e e d o n l y e v a l u a t e  the  the  (A1.3)  to  last look  two t e r m s at  the  of  first  constants]  a  ra  If get  ^  integrals  as  r tends  clearly  to  zero.  finite.  Here,  Thus we  [We may c o n v e n i e n t l y  need  drop  (3y/s se  we now make t h e (all  are  term o n l y .  multiplicative  h-^sjds  integral  above.  (A1.3a)  d s  substitution were  1/y  obtained  = s,  we c a n i n t e g r a t e  from G r a d s h t e y n  and  and  Ryzhik[35])  y3y h (s)ds 1  where  E  i (.)  is  2  •«  +  (A1.3b)  8yEi(eyy)  y-  the  1/a  exponential  unbounded as Y a p p r o a c h e s  integral.  infinity.  Now  Thus h ^ ( r )  at both boundaries,  which c l e a r l y  implies  both boundaries  inaccessible,  or entrance  depending  upon  Making the in  equation h (s)ds 2  are  rr  h„(s)ds,  substitution (A1.2b) =  r~  1/z  as  (A1.3b)  r tends  to  = x i n the  that,  zero  is  is  clearly  unbounded  either  boundaries, and  expression  infinity. for  h (r) 2  gives exp(-i.S/r)  !;  < '±z  a  2  1/a  6  exp ( y g z ) d z  dr  (A1.4)  1.90  S i n c e we a r e o n l y at the  interested  the boundaries, integrals  h  2  we c a n w i t h o u t  f o r 8=1  ie  any l o s s  6=3. T h i s  a .exp(-Bu/s)  (s)d£  i n the behaviour  2  •  2  of generality  .0u. ,  +  2a ^  aV "  integral as  approaches  r approaches  infinity,  second term a l o n e . at  infinity  ^3 ds  . . (A1.4a)  i n the expression  as r a p p r o a c h e s  the i n t e g r a l  Thus t h e i n t e g r a l  both boundaries.  2  loT^l  s  terms  evaluate  (B-u.) exp (-Bu/s)  ;  Ei(3y/s)  Due t o t h e s e c o n d and t h i r d  integral  gives  1  +  of the  above,  zero.  the  Further,  i s unbounded,  due t o t h e  o f h ( r ) i s unbounded 2  Thus b o t h r=0 and r=°° a r e n a t u r a l  boundaries. For boundaries  the case has been  u=h, t h e b e h a v i o u r s t u d i e d by F e l l e r [ 1 8 ]  at the singular . In b r i e f ,  his  results are: 1)  T=y°° i s a n a t u r a l  2)  a t r = 0 , the boundary b e h a v i o u r parameter a) the  boundary depends  upon t h e  values.  i f m i s negative boundary  ( o r r a t h e r my were  i s an e x i t  negative),  boundary  b)  2 i f 0< 2my <a , t h e n we have e i t h e r an a b s o r b i n g  or  a reflecting  c)  2 i f 2my>a , we have  barrier. an e n t r a n c e  boundary.  APPENDIX 2  Details  of the E s t i m a t i o n Procedure  The  SDE  Model  governing the d i f f u s i o n process i s dx = m ( y - x ) d t  We  f o r the L i n e a r i z e d  c a n r e p l a c e dx  =  + ax dz  (A2.1)  a  ( t+l~ t ^ x  x  a  n  d  x  E  f  x  I  f  w  further  e  choose our u n i t o f time e q u a l t o the d i s c r e t i z a t i o n i n t e r v a l we  have x  where n  t  t+l  m 1 X >  ^ N(0,1).  P(x  We  =  t + 1  can t h e r e f o r e  , 1 „ , as L = 2 E l o g x  ^  +  1 - m  ^  x  t  +  a  x  t  E q u a t i o n (A2.2)  x ,8)  a  t ( A 2 . 2 )  n  implies  * N[{my+(l-m)x },a x 2  t  t  s e t up t h e l i k e l i h o o d  2 a t  ] . . . (A2.2a)  function  (logs  taken)  , [x.^v--my-(1-m) x. ] l ^ r t+1 t - -1 ^ z{ ^ j—} 2  2a  0  -  n , o j log c -  J  2  2  x  t (A2.3)  From t h e f o r m o f e q u a t i o n ( A 2 . 3 ) , i t c a n c l e a r l y t h e m and is  y e n t e r o n l y i n the l a s t  exactly  the l e a s t L w.r.t.  the r e s i d u a l  variance  summation t e r m ,  term.  s q u a r e s e s t i m a t e s g i v e n a. a  2  and  setting  •n  ,  2 2  a  Thus m and Further  „  (2a )a  2  1  f {  [ * t + l - my-(l-m)x ] t  ^5a  seen which y are  just  differentiating  to zero gives  1 2  be  2  _  or 1 n  "  r E  T t+l" x  ~ " 2a  m y  ( 1  {  m ) x  t  (A2.4)  3  x,  The  structure  procedure  of  a ,  m and y s u g g e s t a s i m p l e  2  for estimating  a)  Pick a starting  b)  U s i n g OLS A2.2,  after  a .  parameters.  of  a = squares)  to  equation  d i v i d i n g t h r o u g h by x ^. , we c a n 0  is  m and y a r e c)  value  Equation It  2  the  (ordinary Least  m and y . of  all  iterative  (A2.4)  then gives  w e l l known t h a t  an  estimate  estimate  our estimates  of  inefficient  Evaluate  8L/9a  and p i c k  the  for  the present parameter  next  a to  attempt  setting  values  9L/3a  = 0  where [x ^  = -  Zlogx  t  + 2-  - my-(l-m)x ] logx — t  Z{  a2  X  The of  next a?  question i s  (dx)  the  2  Squaring  a reasonable  = a x 2  we now r e p l a c e unit of  y  time  2 t  ~-  (A2.1)  (dz) ,  2 a  dz * N  2  differentials as b e f o r e ,  a  o  2  x  ? ot  2  x  we  we  ? 2  1 }  starting  }  value  problem using  have  (0,dt)  by d i f f e r e n c e s , have  2  t  T h i s c a n be done by b r e a k i n g down t h e  an a p p r o x i m a t i o n .  If  how t o p i c k  2  and c h o o s i n g  193  where  y  Let  z  fc  f(z)  =  Ax  E  y  =  - i -  =  t  2  z"  fc  t+J  /(a x  2  x )  (x __-  t  )  2 a  <\, X(^)  exp  1 / 2  a  (-z/2)  n  d  s  o  w  have  e  dz  /2TT  Suppose we now s e t terms o f  the  y's  (data)  Taking  logs  =  up t h e  we  joint  l i k e l i h o o d of  the  data  in  have  T  n  n t=l  — /2T  (y .a .x. 2  2a  2  )  _ i x  / n /  Y+.  1  exp  — a x  2  and d r o p p i n g a d d i t i v e  2  ) 2 a  c o n s t a n t s .-gives V 2^ ( ,— t— ) 2a t 2  L = -  iT l,__ og z  a_ 2 9  i1 E „ log z  (y\ ) ,2 „x . ^2a, r r  -  a  = 1•  2  (A2.5)  y  X  3L 97  2  T " l a  =  2  1 2 7 ^ "  +  ^t 7^"« t 2  E (  ) =  0  X  which gives  a  1 — *T  =  2  t E ( —— ) . 2a t Y  2  (A2.6)  X  and  ^  The a p p r o a c h i s to  reduce  Finally, mate o f  I  =  to  the  C  (  iterate  9L/9a = 0 . g o i n g back  E  It to  asymptotic  ^  2  x^T  between  of  X )  the  l  (A2.6)  i s - found t h a t  o  ^ t x  and  2  ]  ••••  (A2.7),  convergance  is  (  variance  2  '  by:  )  fast.  an e s t i inverting  chosen o p t i m a l p o i n t .  the h e s s i a n m a t r i x are g i v e n  7  as  very  r- c o v a r i a n c e m a t r i x by the  A  so  o r i g i n a l p r o b l e m , we c a n g e t  the F i s h e r i n f o r m a t i o n m a t r i x at The e l e m e n t s  "  194  3 L 3(a )  _ "  2  2  {x  3 L 3a2 Z  la  a2  3_fL 3y2  a2  3 L 3 a 3m 2  3 L 3a3y  -  2  -  m ^  2  E {  3 L 3 a 30 "  T^fyy  3 L 3m3y.  1 a  Z  Z  2  3m3a^,  t  y  r  E  l  (  -t • ^  1J-x  u  )  X  t  1  9  9  ~ - •  2  log  EHa(x)}  1 x  t  2  *  fc  3 L "•• _ _ m 3y3c '' " (a2)2 2  2  x 2 fc  t  2  a  m(H- t)  a(x)  I  a  t  x 2  log  x  a(x)(y-x  ( 2) 2  X  2  2a t  2  log x  t  2  ,  2  2  a(x)  i  T  -my-(1-m)x }  x  2  2  x 2a  1  E {a(x)  2  2  •v 2a  ( r t ) x ToT  2  2  -T-" 2 (a )  * t  2  x  3 L 3m  a  t + 1  2  , a(x) , "x 2a J  ) J  2  >;  a(x)  x  t + 1  -my-(1-m)x  f c  195 APPENDIX - 3 Solution  to the forward equation  The SDE f o r t h e d i f f u s i o n  for a = 1  process  dx = m ( y - x ) d t  + axdz  (A3.1)  and t h e t r a n s i t i o n a l p r o b a b i l i t y  f f = ~ 4  [m(u-x)P] +  density  \-Ii8x  with  is  [ « r 2  x  2  q  We c a n t r a n s f o r m e q n (A3.2)  t o t h e form  -  \  by u s i n g  the  -St"  =  A  3  >  2  )  6 ( X - X  Q  )  . . .  (A3.2a)  (A3.3)  3z  a(z)  =[m(y-x)  (trs) -"-ds  (ax)P(x|  We s h a l l  therefore  equation  (A3.3)  - j  |  (a x ) ] (a x ) 2  x  2  2  2  ...  x , t ) | x = x(z) g (Jz {  q)  2Q>  concentrate  (A3.4a)  =  (A3.4c)  <5(Z-ZQ)  on a s o l u t i o n to the transformed  a n d once we s o l v e  f o r g , we c a n r e t r i e v e  P  (A3.4b) .  Using the standard s e p a r a t i o n g(z  |-'z t ) f  =  .  (A3.4b)  Q  with i n i t i a l condition  using  (  x:  z(x)  g(z,|'z ,t)=  9 1 •  z  ] _ .  substitutions r  0  [*< '  p  2  the i n i t i a l c o n d i t i o n P ( x | X , 0 ) =  H  follows^ t h e FP e q n  of variables  Q(z)e~  E t / 2  we g e t (A3.5)  E q u a t i o n (A3.„3)now r e d u c e s  to  t h e .eigen v a l u e  problem  2 -  |  [a(z)Q]  + EQ  =  0  (A3.6)  dz where t h e  b o u n d a r y c o n d i t i o n s on Q a r e g i v e n by  conditions  on g t h r o u g h eqn  (A3.5)  We c a n f u r t h e r t r a n s f o r m eqn  Q(z)  the  = i|.(z)  (A3.6)  [ T T ( Z ) ]  where  by  1  /  substituting  (A3.7a)  2  •rz TT  (z)  and t h e n we have e q n  = exp  {-  (A3.6)  a  2  U)  (A3.7b)  de.}  as  ,2. 5-1 dz  +  [E -  U(z)  =  |§  U(z)]  ijj  =  0  ((A3.7c)  2  where  +  The b o u n d a r y c o n d i t i o n s on ditions  on P t h r o u g h  For our p r o c e s s , accessible  a  are got  (A3.4B), A 3 . 5 ) ,  we have  the  two  (A3.7d)  2  from t h e b o u n d a r y c o n (A3.7a)  (A3.7b)  s i n g u l a r b o u n d a r i e s as i n -  i.e. p (x  0 , »)  + 0  (A3.8)  We now have by o u r e a r l i e r d e f i n i t i o n o f  z  Thus f o r  and  (  x  )  =  1  0 £ x < <*> we have  Further using  to  -  Z(x)  x  » < Z<  (A3.9)  00  (A3.4b) and a s s u m i n g t h a t  P(x)  ->  0  faster  t h a n x -* °° (.and u s i n g e q n (A. 3.8)  4- ±  g(z  Equation  00  )  =  0  (A3. 8a)  =  0  (A3.8b)  (A3.5) so g i v e s us Q (z -> ±  00)  form o f e q n ( A 3 . 7 c ) a s  We now p r o c e e d t o g e t t h e f u n c t i o n a l in  t h e f o r m o f t h e t i m e homogeneous  mechanics.  a ( z ) . = [m(y-x)  =  From  \ 2a x] 4  -  — ax  2  (m + §—) x ] -^x-  my ax  <? a  x = e  a  -az  a^ (z)  da dz  Schroedinger  [my -  (A3.9) we have  From  we have  my e  z  +  7> z which  gives  ( £ +§ ) a 2  •az  ( A 3 . 7 d we have U(z) ',T T /  \  / —  -  my e  my e  -az , +  - a z , , m-y, 2 + ( — -) a  my a  e  -az ~  .(..'ii + °) a 2  r  my  equation  it  is  o f wave  my e  +  (  —)  (e  , —az  2  1) n  v  -  ^ y  +  . mu.2  _ a Z  1) +  -az  +  ( HLbL )  (z)  = a  (e~° 2  x  l)  + a  2  2 (my) ( 1 2  2  :  f- ) 2my  {  (  J- - ^ )  - l + | 1 ,2 y 2my  _ !  - my  }  e  2  CTZ  +  a  3  my , 2 a  a.  2\\W- ) 2  - ^ ' f )  <1  y ^ a '  Substituting  dj>  y  -  gives  U  where  tl--  ^  a  which  a z  2my^  +  ,  + e  2e-  2  {  into equation  (E>-a ) 3  dz  Now by a s u i t a b l e  -  a  { 2  1  change  1  — a. *-  of  1  my  2my  y  -  ;  (A3.7c)  1  we  have  ,, ..- -a-2, 2  -az  (1-  e  -)  - e  I-i  I  v a r i a b l e we want  .  ^  to  _  0  „  transform  (A3  199  eqn  (A3.10)  to  the  following  form  2 + { E * - c(e 1  d? where  IJJ( 5 -> ± »)  3, E  and where parameters £  Then  2 d_j,  Further  a  .a  2  )  } i|i = 0  (A3.11a)  to  be c h o s e n  i n terms o f  the  (A3.10) (A3.11c)  2 d_J_ dz  (A3, l i d )  =  2  the  1(A3 .10) a^  ?  (z-z*)  2  taking  equation  2e  (A3, l i b )  and c a r e  =  d'S,  -  = 0  of equation  Let  C  i  2  s e c o n d t e r m i n the. s q u a r e  we  bracket  of  have  ,, „ -az. , -2az. (l-2e + e ) -  e  -az  * and s u b s t i t u t e  -az = -(£+az ) which g i v e s a e * _ a a„ = a. + -±—* 2e M + — zr ) 1 2az* az* az* e e 2e n  (A3.11)  0  Comparing w i t h to  choose  z*  ^,C=  the  corresponding part of  so as  a  x  e  to  -2az*  satisfy -az*  = a e  , +  x  • =  and d r o p p i n g t h e  equation,(A3.11)  trivial  , (a  x  , +  a  52"  e  2 . -az* 2- ) e  solution e  =; 6 we  have  we  want  or z* =  2a iog  x  (A3.12)  (  Q  Thus we n o t e  t h a t we c a n t r a n s f o r m eqn (A3.10)  form e q n (9)  by t h e f o l l o w i n g „ £  where  z* i s given  _  to the  substitutions  , * a ( z - z*)  (A3.11c)  x  by e q n (A3.12) 3  = a  (A3.12a)  2  E' = ( E - a + a ) 3  and  =  The p o i n t the  of a l l this  (A3.12b)  1  (A3.12c)  e~ * 2oz  a  ±  effort  Schrfidinger equation  i s t h a t e q n (A3.11)  f o r a diatomic molecule  Morse p o t e n t i a l  - a n e q u a t i o n w h i c h h a s been  quantum p h y s i c s  literature  a l s o Morse  marsh  [  ] , Dunham [ 2?  boundaries,  are n o t s t r i c t l y  [  i n the  ].(See  ]) .  ?  U ( 5 ) [by c o m p a r i n g e q n (A3.11a) w i t h e q n time  homogeneous  i s n o t always i n f i n i t e . J) t h a t  with a  TJ(£) = c (e~ -2e~ )->• 0 i . e . a t  (A.3 .7c)which i s t h e b a s i c equation]  [  just  studied  by T r i s c h k a & Salwen  [ '.. ] , S c h r o e d i n g e r  At t h e boundary infinite  is  This  t h e s e t o f eigenvalues discrete,  and t h e r e  Schrodinger  implies  (see T i t c h -  o f equation  i s a continuous  (A3.11a) interval  201 of  eigenvalues  as  well.  The d i s c r e t e , are  given  E  -  where  [x]  c  [  i _  is  £ /c  the  (n+|)]  less  ing normalized eigen  U)  M u n  =  [  0< n <  2  functions  o f eqn  (A3.1la)  ] as  [ r -  |  / c  the  ]  (A3.13)  number x i . e . ,  than or equal to x.  1 r  eigenvalues  i n t e g r a l part of  integral  * n  the  i n T r i s c h k a & Salwan  = _  largest  region of  are given  the  The c o r r e s p o n d -  by  (q-2n-l) e~  2  u / 2  F  (u)  n  (A3.14)  n  where  and  q ^  =  u  =  £Q i s  7 exp(-az*)  q exp  got  U  from t h e  Q  (A3.14a)  - O  (A3.14b)  initial  condition  (A3.2a)  suitably  transformed.  and  P (u)  M  where  =  n  J  .2 1 n = n! r(.)  (x)  l ^  T  r  i r(q-2n-l) (  is =  ( J )  i  the {  q  2  n  (J)  =  I T T  ^  i r i  ....(A3.14c)  )  '  . * ( A 3 . : 1 4 d ) .  gamma f u n c t i o n and  1  x(x+l)  The s o l u t i o n as  ;  i n eqn  if n = 0 (x+n-1) i f  (A3.5)  is  (x)  n  is  defined  as (A3.14e)  n > 1  thus  g i v e n by  (for  the  202 discrete  portion).  g(z  where Q the  z ,t)  is  n  got  constants  setting  =  Q  t=0,  E a n  from ip  n  general  Q  solution is  g(zfz ,t) o  =  n  E  n  t  /  (A3.15)  2  ffrc rom the  which gives  =  e  n  using equations  a n ' aa r e g o t  °n  the  Q  n  (A3.7a) (A3.7b)  i n i t i a l condition  , and  (A3.2a),  and  us  0  ( z  )  7r  ( z  0  :-•  }  now g i v e n  u(z )  £ Q (z)  0  n  (  A 3  -  1 5 a  )  by Q (z ) m  exp  Q  (-E^  t/2) (A3.16)  + continuous  spectra  contribution  The c o n t r i b u t i o n o f  the continuous p a r t  (see  h a s n o t been s o l v e d  i n closed  is  the  form, but  Goel et  al  known t o be  [,.  )  of  form as u n d e r  F (E' , x)exp  {  (z-e ) ] z  j  E*t  > °X  (A3.16a)  0 where the  the  r e l a t i o n between  f u n c t i o n F depends  1  and E i s  upon c o n f l u e n t  Without p u r s u r i n g t h i s ing  E  l i n e of  given  by  (A3.12b)  hypergeometrie  analysis  further  and  functions. the  follow-  comments may be made: a)  I t a p p e a r s t h a t an i m p o r t a n t c h a r a c t e r i s t i c o f e x p r e s s i o n (A3.16a) i s t h a t i t d e c a y s v e r y r a p i d l y w i t h  t i m e ( t ) , so t h a t by an a p p r o p r i a t e c h o i c e o f i t may be n e g l i g i b l y s m a l l , and c o n v e n i e n t l y dropped.  t,  The t r a n s i t i o n a l d e n s i t y i s r a t h e r cumbersome and may n o t be m e a n i n g f u l l y t r a c t a b l e from t h e p o i n t o f v i e w o f p a r a m e t e r e s t i m a t i o n by ML methods.  APPENDIX 4 S o l u t i o n o f t h e F o k k e r - P l a n c y E q u a t i o n f o r g=0 W i t h No Restriction at Origin. The SDE f o r t h e d i f f u s i o n dx = m ( y - x )  If  process  with  a=0 i s  d t + adz  (A4.1)  we now make t h e s u b s t i t u t i o n  - y = u - x , we g e t  dy = - my d t + adz  (A4.2)  which i s the s t r a i g h t has  a transitional  P(y y , 0 ) o  = [2rrV ] 2  Ornstein-Uhlenbeck  probability  1 / 2  exp  density  {-|[{y-y e 0  _ r n t  p r o c e s s and  g i v e n by  }/V] } 2  (A4.3)  where  V  =  g2m  The s o l u t i o n t o substituting  (l-e  - 2  ^)  (A4.1 i s t h e r e f o r e  y = x-u ; y = x -y. Q  Q  s i m p l y g o t by  APPENDIX 5  Derivation of  the  Stationary  (or Steady  We have o u r d i f f u s i o n p r o c e s s dx = m ( v i - x ) d t w h i c h has t h e  by  ax 'dz"  (A5.1)  dt +  = m(y-x)  /a\(x)  dz  ; a.(x)  (A5.1a)  = a x 2  2 0 1  The FP e q u a t i o n c o r r e s p o n d i n g t o A 5 . 1 a )  -  |  x  densities  form  dx = b (x)  where b;(x)  +  defined  State)  |b(x)F]  where F = F(x-1  +  [ a(x)F]  \ Jl  XQ,t,6)  is  the  is  (A5.1b)  =  transitional probability  density. The s t e a d y got  state  by s e t t i n g  P (x  8)  density  is  8 F / 3 t = 0,  the  and i s  [a/(x) TT (x) ]  =  s o l u t i o n to of  where  TT(S)  = exp  and Q i n d i c a t e s  [-2  form  -1  [a, (s)rr(s)] !  the  (A5.1b)  -l  •(A5.1c) ds  b (r) d r ]  (AS.ld)-  a.(r)  i n t e g r a t i o n over  the  total  state  space  of  x  206  For our process  TT  (A5.1)  (x)  we have  =  x  exp(Bx)  x  exp'(By/x)  2m  ;  n2  o=l/2  TT ( X )  a=l  TT  (x) a-y 1/2 ,1  = exp [  ex  l+X  e y x -]  i+x  ;  X=l-2a  (A5.2)  F o r a = 1/2  we have  [b(x)  •1 _  TT(X)]  1  x  By-1  exp(-ex) ey  \  and  2  x  3  -  exp(-ex)  1  ey-i  = Ifi-  _  (e)  (e)  ey  exp(-ex)  r(3y)  (A5.2a)  P(x) a'=l/2  For  a = 1 we have  - 6  [b(x)TT(x)]~-  and  K 0U  (edx)  x"  6  L  = ^  exp(-ey/x)  exp(-ey/x)dx  = \ U  2  ;  (By)  6 =  ( 3 + 1 )  (2+e)  r(B+l)  which  gives  >3P) r(e+l) 3 + 1  P(x)  l  a=l  Finally  for  the  general  case  x  -(2+6) e x p ( - 6 y / x )  of  1/2,  1 we  •l+A [b(x)  TT  (x) ] = x  exp[  1 _ X  3  x  w h i c h c a n n o t be r e a d i l y i n t e g r a t e d , steady  state  (A5.2b)  have  B:yx  l +A  -]  ;  A=l-2a  and so we have  for  the  density  x  P(x)  A-l  ~-i  a^l/2,1  Byx [ *\  exp  r  M  •-x  ex  l+X'  eyy^  r  exp[  l+X  eyi+x n  -p- -  , ] dy (A5.2c)  Finally, steady  it  w o u l d be o f  state  functional  forms  (A5.2a) (i.e.  Now d r o p p i n g t h e we c a n w r i t e  P  f o r us t o v e r i f y  p r o b a b i l i t y d e n s i t y - ( A 5 . 2c) ,  and 1 r e s p e c t i v e l y  (A5.2c)  interest  T-,  A  «i  x  A-l  and  (A5.2b)  X -»- 0 ;  denominator the  exp  density  r  [  reduces  that to  the  the  as a a p p r o a c h e s  1/2  -1)  (which i s function  eyx ^—  1L  A  and m u l t i p l y i n g a n d d i v i d i n g b y exp  l+A ex l+A  a constant)  from  as  ]  (gy/A)  (A5.3)  gives  208  Mow  x - 1 _ A  exp(Alogx)-1 X  1 2 X log x + 2 A"(log  x)  2  -i  -+  Now a s X •+ 0  ( i . e . a -> 1/2)  00  E  clearly  x -l X  Lt A+0  A  *  a -> 1/2 A 0  P(x)  (A5.3a)  = log x  r  -i  £ (A l o g x ) A n=3 nij;  ±  x^  y  exp(-$x)  1  w h i c h h a s t h e same k e r n e l as  (A5.2a).  To show t h e same s o r t o f c o n t i n u i t y f o r t h e a = l c a s e we can w r i t e  P °= A  (A5.3)  as  B {x 1+A  l}  1 + X  exp [  and a s i n (A5.3a)  x  Lt  X-y-1  a s a -> 1  above  l+A_  P,  (  2  +  B  )  1 we g e t  ex  P  [-  ] X  A  w h i c h h a s t h e same form as  (1+A)  = log x  1+A  « x"  ;  X  1  —T-T^  Thus t a k i n g l i m i t s a s A -> -  A-l  i  J  (A5.2b)  0 and  n  APPENDIX ~ 6 D e t a i l s o f the P h i l l i p ' s Approach t o E s t i m a t i o n The  stochastic d i f f e r e n t i a l equation  (s.d.e.)  governing  the i n t e r e s t r a t e p r o c e s s i s dx = m(yU-x) d t + ax" dz I t i s necessary  (A  T h i s can be done by a t r a n s f o r m a t i o n  L e t the t r a n s f o r m a t i o n be y = f ( x ) where the  f u n c t i o n a l form o f f ( . ) i s unknown.By: Ito s  Lemma we  1  dy = f mf x (y-x) + % f xx a x 2  L  we now  6.1)  t o t r a n s f o r m the above s.d.e. so as t o e l i m i n a t e  x from the v a r i a n c e element. of v a r i a b l e s .  ...  choose  2 a  ] dt +  J  fx a x  a  have  dz  f ( x ) so t h a t  a  f  x  =  1  (or any  which on i n t e g r a t i o n  gives  1-a  y  constant)  =  -=r_——  1-a-  for  =  log x  for  a  ^  1  aC^j\  P r o c e e d i n g w i t h the a ^ l case (as i t i s the more g e n e r a l we  get by  substitution -a  dy =  [myx  "*"  -a  - m(l-a) -= 1-a  2-1 Jgaa x ] dt + ~ J  adz  form),  I f we  now  form o f may  set u  =  x  ; v = x  we  get the  equivalent  (A 6 . 1 ) , i n a f o r m where t h e P h i l l i p ' s  be a p p l i e d .  approach  Thus  2 dy = m ( a - l ) y ( t ) + my  u ( t ) - haa  S i n c e " 0U.T o b j e c t i v e a t p r e s e n t let  us p r o c e e d a h e a d  dy =  If  now  we  [-(m/2)y +  Phillips  [ ] exogenous  discrete  time e q u i v a l e n t  where  t  =  = exp  E  l^t-1  +  E  E =  ( m y - a / 4 ) [ (2/m ) exp  E =  (my-a /4)[-(2/m )  3  4  2  to  U  2  +  (A6.2)  E  3 t-l U  gives-  ...(A6.2)  u as e q u i v a l e n t  Then  to  an a p p r o x i m a t e  is  +  E  4 t-2 U  V  +  (  A  6  "  3  )  exp(-m/2)(1-4/m)+(2/m )(m -3m+4)] 3  3  2  This  (-m/2)  (my-a /4)[(2/m ) 2  expositional,- -  2  can t r e a t  2 t  K (t)  ( m y - a ) 2 ] d t + adz 4 y  variable.  E = 2  i s purely  f u r t h e r assuming. a=h.  s e t 2/y = u , we  Y  vr(t) +  3  exp  2  (-m/2)(8-m ) +  (8/m )(m-2)]  (-m/2)(m+4) -  (2/m )(m-4)]  2  3  3  (A6.4)  211 and  n  Thus t h e  ^ N[0,  t  log  2  2  likelyhood  L = -  where w  (a /m)(l-exp(-m))]  |  function  log w  -  2  (A6.5)  E  (a /m)(l-exp(-m))  =  is  ~ E n.• t=l T  and E =  2  2  It  may be  t  noted  that  the  we r e q u i r e If  the  degrees  lagged  values  t i m e between  small  i n magnitude.  terms  of  However,  we f i n d  is  regression  also  as  fails  a  2  data  sets,  optimization  It  therefore  The d i f f e r e n c e  the  it  small, m is  Then w  between  Phillips[  ] is  Stochastic  differential  the  and d r o p  a . 2  approach  residual overdetermined.  fails.  Constrained  L  found t h a t  regression  directly. using  convergence  drop f u r t h e r  approaches  very minor.  =  also  E's.  routines, to  -  2  regression  to maximize  was  decided  D y(t)  y is  approach  enters  linear was  m, and t h e  only approach i s  On sample  very  using a d i r e c t  Thus we f i n d  a .  The d i r e c t  Thus t h e  is  We c a n t h e n expand e x p ( - m ) ,  that  as  (A6.3).  in  observations  determines 2  r e d u c e d by 2  f r e e d o m have  s e c o n d o r d e r and h i g h e r .  -uniquely variance  of  We have  of the  was  + B z(t)  not  obtained.  investigation.  Sargan [  ] and  s o l u t i o n to  equation  Ay(t)  s t a n d a r d non  + E(t)  the  as  shown i n S a r g a n  [ ] h  y(t)  = e^ytt-h)  +  e  B z(t-s)ds  e  +  E(t-s)  (A6.6)  0  Both approaches  approximate  on t h e  r.h.s.  series  expansion of  third the  of  (A6.6)  z(t-s)  differentiability  more d i r e c t  whereas  =  t~  Z  t-h  about (Clearly  of  z(t)).  derivatives  a p p r o a c h and  2  z'(t)  the  i n t e g r a n d i n the  s=0, the  and d r o p p i n g t e r m s  They d i f f e r of  formula  of  approximation hinges  z(t).  on  o n l y i n the  way  Sargan adopts  the  sets 2Z  z"(t)  =  t-h 2h  + Z  t-2h  2  P h i l l i p s u s e s t h e more i n v o l v e d L a g r a n g e  interpolation  second term  by a p o l y n o m i a l i n s by a T a y l o r  and h i g h e r o r d e r .  they approximate  the  ds  (see  Conte  de B o o r  [ J).  three  points  213 APPENDIX - 7 D e t a i l s o f Estimating Procedure  f o r a = 1/2  T h i s a p p e n d i x o u t l i n e s t h e method parameters known.  (m,y,a), f o r the case  The d i f f u s i o n  equation  (Known) Case  adopted  to estimate  where a = 1/2  i s given  the  i s assumed  by  dx = m ( y - x ) d t + cr/x dz a)  Simple  (A7.1)  l i n e a r i z a t i o n method:  differential  equation  Approximating the  (A7.1) by a d i f f e r e n c e e x p r e s s i o n  gives (x ~  ^ _ )  t  t  where e and Y  = m(y  1  - x  ^ N(0,1).  t  )  t - 1  + ^ _i a  e  x  t  (A7.2)  t  D i v i d i n g t h r o u g h by  v / x t  _^  rearranging.-> t e r m s g i v e s t  %  =  X  l t  +  ( 1  "  m )  X  2t  +  t  n  (  A  7  *  3  )  where y  = x //x^_  t  t  ;  1  x  = l//x^_  l t  ; x  1  =  2 t  / x ^  and n \^  Now  t  =ae  i n equation  t  ^N(0,a  (A7.1),  2  )  ^  the d z s are i n t e r t e m p o r a l  independent, which i m p l i e s t h a t E ( n Thus of  (A7.3) i s t h e s t a n d a r d  n ,)  t  =0  t  for a l l t ' ^ t.  regression equation  the parameters i s e q u i v a l e n t t o l e a s t  Thus t h e l o g o f t h e j o i n t l i k e l i h o o d given  •  1  and ML  squares  estimates.  of T observations i s  by T  ^ ^  v  estimation  -L E  2 t  I  l  n  _l.  = I y 2my  2  + m y 2  (1-m)  Ex  2  E  x  i  x t  2 t  +  2t~  (1-m) 2  (  1  -  m  )  2  Ex E x  2 t  -  2t t Y  4 2*^ I x ^ '' (A7.4)  214  Setting  =  Z x 2  M, = ly  j _ - '< M i _ t  2  =  E x  lt 2t X  ' 22 M  .Y. ; JVL = Ex„,y. It t ^y 2t- t  =  E x  2t  ; M = Ey yy t  ' 2  J  r  We have the f i r s t order conditions as | ^ = 2my M 2  - 2(l-m) M ^ - 2 p JA^ + 2 P M - 4my M ^  11  12  | ^ = 2m y ^  - 2m ^  2  Setting  (A7.5b) m =  + 2 M^  (A7.5a)  + 2m(l-m) M ^  (A7.5b)  equal to zero gives  ^ ^2 yM^- M ^  (A7.6a)  Substituting the above u n i t s (A7.5a)  v  .  and setting  3L/  9 m  = 0 gives  \ « 1 2 - * K % - M . _M_  2  (M  12  " \2%  (fl7  + M..M-  +  ^1%  -  "  NL-M-i )  W  i  The Fisher information matrix corresponding to the present sample i s I and i t s elements are  where 6 ^ = m and  0  (by i n v o k i n g a s y m p t o t i c  = y,  2  and a r e t h e M . L .  For the present case we have 8 L 2  4VM  3m2 3fL Y 3y 9£L_  9m9y  =  L 2  - 2y M 2  n  - 2M  22  - 2m M ^  = 4m M  2  1 2  + 2M  l y  - 2^2 -  toM^  results)  estimates  .  6b)  215  This us to to estimate of o f the the variance-covariance matrix o ff the This enables enables us the estimates based on asymptotic theory.  b)  Steady State Density method: to  The steady state density corresponding  a = 1/2 i s (see Appendix 5)  F(x) = ( ^ . ey-i. x  (  _  BX)  (A7  .  7)  .(By)  r  The j o i n t l i k e l i h o o d o f the data f o r t h i s  approach i s  T  a =.n, i=l  F(x.) i  Taking logs and s e t t i n g  lo,g (I) = L we have  L = Tgu l o g B + (B'y- 1) E l o g x, - 6 Z x ~ T l o g [r(By)] i  (A7.8)  where r ( . ) i s the Gamma function. The f i r s t order conditions corresponding to maximizing L are 3T  TT— 9y  = TB l o g B + Ba - TB^ (By) ' = 0  (A7.9a)  1  3T  i=- = Ty l o g B + Ty + y a - b - TyiJ; (By) = 0  (A7.9b)  dp  where a = E l o g x^ and b = Ex^ and  i s the p s i function i . e . the  f i r s t derivative of log [ F ( . ) ] . Equating (A7.9a) and (A7.9b), and observing that  i s a single valued  function for p o s i t i v e arguments y i e l d s y = /T b  (A7.10a)  To estimate B, we need to solve the following equation (got by substituting (A7.10a) i n e i t h e r o f equations iJi(By) = l o g B + a / T  (A7.9) (A7.10b)  Since ip (.) and l o g (.) are monotone increasing functions o f t h e i r arguments, we are guaranteed unique s o l u t i o n to (A7.10b). To get an estimate o f the asymptotic variance-covariance matrix of the parameters, we need to evaluate the Hessian of L a t the neximum.  |^  Thus  = - T 3 r(8v)  (A7.ll)  2  =  TP2 [ i  -  (A7.11b)  (Bu)]  3 L j r g p = T l o g 3 + T + a - Tip (3y) - T 3 y ^ ' ,(3y) 2  (A7.11c)  where ip' (.) i s the digamma function. C l e a r l y the optimum i s a maximum, as the diagonal elements  (equations  A7.11a and A7.11b) are negative. Equation (A7.10b) was solved for 3 , by a numerical routine (DRZFUN i n the UBCrNLE routines) which evaluate the zeros- - of nonlinear v  equations.  The p s i function has been coded and i s a v a i l a b l e i n the UBC  programme l i b r a r y . was used.  For the digaimia function, f i r s t a s e r i e s expansion  However t h i s was not s a t i s f a c t o r y ,  as truncation (even after  a large number o f terms) resulted i n sizeable e r r o r s i n the function value, which was detected as the diagonal elements o f the hessian matrix sometimes became p o s i t i v e .  An  asymtotic expansion (for large arguments)  was very satisfactory f o r the parameter values o f our problem. c)  Transition., . P r o b a b i l i t y Density Method:  The t r a n s i t i o n  density corresponding to a = 1/2 i s given by  probability  217  F(x |x ,8,t) = {2m/a (w-l)} . exp [-{2m (x+wx )}/{a 2  Q  • • ( — )'"' i wx o  °  x  I' 1 ^ 2mp a  2  (w-1)}J.  [4m^5wx / a (w-1)] o 2  _ ^  2  (A7.12a) where w = exp (mt) and  is  the m o d i f i e d B e s s e l  function of  o r d e r k.  The l i k e l i h o o d function i s therefore T-l . '* = £ F(x |x ,0) . P (x ) ±  where  p  s s  («)  i + 1  g s  i  i s the steady state density  large, the contribution o f P ( . ) c e  (A7.12b)  1  (A7.7) .  In general, when Ti  is  may be considered very small compared to the  other terms, and so may be dropped from the l i k e l i h o o d function. The log l i v e l i h o o d function i s not further tractable a n a l y t i c a l l y as expressions f o r (9L/96^) require derivatives of the Bessel function with respect to i t s order, (for a r b i t r a r y p o s i t i v e orders) which are problematic. The approach towards parameter estimation has to be d i r e c t iraximization o f the log of (A7.12b).  For t h i s purpose, the Fletcher algorithm using a q u a s i -  Newton method was used.  In general, i t was found that convergence was obtained  to a reasonable degree of accuracy within 15 i t e r a t i o n s , given s t a r t i n g values for the parameters as the r e s u l t s of the simple l i n e a r i z a t i o n model. In small sample t r i a l s , to ascertain whether convergence i s to a l o c a l o r a "global" maximum, very d i f f e r e n t s t a r t i n g values were given.  Without f a i l i n  a l l cases, the convergence took longer, but the f i n a l maximum value parameters were unchanged.  The term "global" has been set within quotes, as there i s  no rigorous guarantee that the maximum obtained i s t r u l y global without much  more extensive t e s t i n g . A word about the numerical evaluation of the density function (A7.12a). The modified Bessel function could not be evaluated i n a s t r a i g h t forward manner, using the series expansion.  This was because, f o r large values  of the order and/or argument, the s e r i e s was very slow to converge.  To  overcome t h i s , the expression was s p l i t up as F(x |t ,9) t  •=  0  f(x ,x ,0) t  . exp (-g(x))  o  This was more successful as exp (-g(x)).  1^  . 1^  (g(x))  (A7.  (g(x)) converged more r a p i d l y .  However, for large 6, t h i s method was very expensive computationally. Thus, an asymptotic expansion along the l i n e s of Giver [66] whenever 6 was greater than 20.  This was very e f f i c i e n t .  was used, The r e l a t i v e  accuracy o f the asymptotic values as compared to the more exact expression (A7.13) was tested by a c t u a l l y evaluating the density function (A7.12a) for a given parameter set 6, and several values o f X  q  ranging from near  0% to 30%, by the two methods and computing i t s f i r s t two moments.  These  were compared with the exact values o f the moments, which are given by (see Cox, I n g e r s o l l & Ross [13]). &. = r e 1 o = r  M  2  -m , ,-. -m . + y (1-e ) (  °  ) (e" - e" ™ ) + y m  2  ( ^ -  ) (1 -  e~ ) m  2  2m2  m  where JXL, i s a c e n t r a l moment.  The asymptotic expansion performed very  w e l l , as may be seen from the tabulation i n Figure  1  Just to show the shape of the t r a n s i t i o n a l p r o b a b i l i t y density function i n equation (A7.12a), Figure  1  was prepared.  What i s i n t e r e s t i n g to note  i s that, for the parameter set used, y - 5% per annum, and when the current i n t e r e s t i s a t or above y, the t r a n s i t i o n a l density function  FIGURE 1 P l o t o f T r a n s i t i o n P r o b a b i l i t y D e n s i t y F u n c t i o n (& C u m u l a t i v e P r o b a b i l i t y ) f o r a = 1/2 a t D i f f e r e n t r Values n  Comparison of Theoretical Mean & Etd. Deviation of Density Function In Eqn. (A7.12a) With That Computed Using An Assyirptotic Expansion For The Modified Bessel Function.  Mean of r  t  1.0 2.0 3.0 5.5 7.0 9.0 10.0 12.0  r  Theoretical 1.955 2.787 3.612 5.675 6.912 8 .562 9 . 388 11.038  t +  ^/  r  t  «  Std. dev of  9  Numerical 1.962 2.777 3.593 5.651 6.887 8.534 9.338 10.961  r  t + 1  Theoretical  A '6 t  Numerical  1.126 1.421 1.664 2.155 2.402 2.697 2.832 3.086  NOTES: - A l l figures are i n percent per annun - 6 i s the parameter set {m, y,c } and are the values used in the Monte Carlo simulations. 2  1.116 1.411 1.650 2.151 2.405 2.708 2.848 3.119  does not appear too skewed from the normal density.  This could imply  that the simple l i n e a r i z a t i o n of the d i f f u s i o n equation, Gaussian  (which assumes  t r a n s i t i o n p r o b a b i l i t i e s ) may not perform too badly.  F i n a l l y the second  derivatives of the log l i k e l i h o o d function  were computed numerically,  (the quasi-Newton method evaluates numerical  second derivatives a t every iteration) and these were used to evaluate the asymptotic variance-covariance  matrix of the estimated parameters.  APPENDIX 8  A n a l y s i s Of E f f e c t  The a n a l y s i s  Of Measurement E r r o r s  here  assumes  combined e f f e c t  of  error process.  The f o r m u l a t i o n o f  follows: follows  = m(y-i)  where we o b s e r v e  the  square  i  dt  r  n is  have t o L e t us  r as  data  is  the  problem runs  true  interest  as  rate  (i)  2  The r a t i o n a l e  i  (A8,l)  (For t h i s as  the  purpose of  / T  Dif ferenciating  I  have  this  section  n  i.e.  (A8.2)  noise.  To be  able  a p a r t i c u l a r form o f  as  analysis,  expositional).  w i t h an e r r o r  dz  ; E(n)  2  behi n d " t h i s  zero  dz,  Vafi  to proceed  some a d d i t i o n a l s t r u c t u r e  at  dn = a  the  further,  on t h e error  structure  =0  form i s  ..(A8:3) that  it  ensures  that  2 ) =  (A8.2.)  dt  = m(y-r)  a n d s u b s t i t u t i n g (A8.1)  + oj/I'dx,  and  (A8.3)  2  + o  2  ^  dz  Cov  (dz  2  dt  2 2.. (o +a )r d t + 2a i a 1  the error T  i + 0.  = m(p-i)  E(dr  we  problem.  d r = d i + dn  where E (dr)  the  and a s u p e r i m p o s e d  + n  white  impose  +  root process,  = i  look  goes t o  the  with error.  p r e s e n t moment i s  L e t us o b s e r v e  where  that  observed  process  di  at  the  true process  We may b e l i e v e the  used the  the  that  On Data :  2  pr dt  [since  1  we  get  -  (a  = a  2  + a  + 2  2  r dt  2  0 l  a p) r  where  a  == ( a  2  Thus we c a n r e p r e s e n t t h e p r o c e s s  d r =:  m(y-r)  the  interest  rate process.  r  2  + a  +  2  2a^p)  as  dz  3  exactly  of  of  dt + a / r  which i s  dt  2  (  same form as e q u a t i o n  :A8  -4)  (A8V1) -,the tr;'u'e' ;  C l e a r l y , we c a n n o t i d e n t i f y  a , a  2  p .  Further,  i f we i g n o r e t h e e r r o r i n measurement  e r r o r does  exist),  t h e n a | as an e s t i m a t e  over or under e s t i m a t e d  (a  2  + 2o  l  a  2  according  p) |  of  (when an  a , is 2  either  as  0  2a  1  This  implies that  e v e n when p = 0,  w i t h the t r u e i n t e r e s t In t h i s error  error  and t r u e  value)  structure,  as  easily  2  is  i s uncorrelated  over estimated  l o n g as we assume t h a t  i n t e r e s t process  the v a r i a n c e term, the p r e s e n t is  a  (the e r r o r  have t h e analysis  v e r i f i e d by c a r r y i n g t h e  by  o . 2  both  same a e x p o n e n t holds i n toto.  algebra  through.  the in This  223 APPENDIX 9 AN APPROXIMATE ESTIMATE OF THE . ASYMPTOTIC CORRELATION MATRIX BETWEEN INTEREST RATE PROCESS PARAMETERS In the case of ML estimation when we have independent random variables, a widely known result i s that, the asymptotic covariance matrix of the estimated parameters i s got by inverting the H e s s i a n matrix (with signs reversed on the elements) where the H e s s i a n m a t r i x i s the matrix of second p a r t i a l s of the logarithm of the j o i n t l i k e l i h o o d function with respect to the parameters. and Quandt [  (see T h e i l  [  ]  t  Goldfeld  ] ) . This r e s u l t uses the property of ML estimators whereby  3^  ,  E (—  9L 2  )=  30 90j  (A9.1)  90 90,  i  i  Where L = l o g l i k e l i h o o d function of the data, 0 i s the vector of parameters, 0 i s the ML estimate of 6 , and E i s the expectations Thus i n general, i f we know 0  operator.  (the true value), then we can  compute the assymptotic covariance matrix of 0 as T1  Cov(0)  =  -E  (A9.2)  ' 90^0^  Further i f we represent by L ^ \ the j o i n t l i k e l i h o o d of n data points, n  we "can approximate  a .W 2  I D  ,2 (1) - i  3  "  E q u a t i o n (A9.3) i s v a l i d s t r i c t l y o n l y f o r independent random v a r i a b l e s . We hope t h a t t h e " b i a s " due t o dependence o f t h e sequence does n o t a l t e r the b a s i c n a t u r e o f t h e a n a l y s i s t o f o l l o w v e r y much.  The p o i n t t o be n o t e d h e r e i s t h a t when we asymptotic  compute the  c o r r e l a t i o n m a t r i x from t h e c o v a r i a n c e m a t r i x , i t i s  o b v i o u s l y i m m a t e r i a l whether we use the e x p e c t a t i o n over n  data  p o i n t s o r even 1 d a t a p o i n t .  Let I r e p r e s e n t the F i s h e r I n f o r m a t i o n m a t r i x . Then  I =  r*  E  )  2h  86.80.  J  and f u r t h e r  30.30.  1  fetaeT  J  0  (  t'  r  r  0  l  e  )  -  P^fr,. r ,0 ) . P ( r 6) dr d r l t o' ss^ o t y  J  0  IA9.4)  where Prp(-) r e p r e s e n t s t h e t r a n s i t i o n a l p r o b a b i l i t y d e n s i t y and the s t a t i o n a r y p r o b a b i l i t y d e n s i t y .  S i n c e we want t o e v a l u a t e  c o r r e l a t i o n m a t r i x over a l l parameters ( i n c l u d i n g ot P (.)  t o be normal - which i s the case i n the SL  T  T h i s now  P (r T  we  P  s s  (0  the  c o u l d assume  approximation.  gives  t  r  f f i  e) a :  {my  N  r  N.[ ( a  r 0  + ( l - m ) r }, o J  )>  °  2 r 0  ]  a r o  (A9.5)  i P (^ol ) s s  By  _ -  e  ;  .  r(By)  By-i  .exp (-Pro)  /°  (A9.6)  have a l l been set out i n 96,30, Appendix,2. Substituting (A9.5) and (A9.6) into (A9.4); noting that one  Where B = 2m/a . Expressions for 2  of the integrals i s now from -°° to +  00  due to the normal density approxi-  mation; and further that P  r  r  t  p  (* |r ,e)dr =  T  0  t  1  t  T( tl o.8) t= r  r  d r  a  ( o> r  {r  t  - a(r )}  P ( r |r ,6) r ' =  {r  t  - a(r )}  P (r |r ,6) dr^-  Q  0  t  T  gxves E (  9*L -) 9a9ta.  =  0  )  =  0  =  0  =  0  E ( 9a9y E 9m9cT  E (  9L 2  9y9a"  -)  t  t  Q  0  d  t  2  226 (The d e f i n i t e i n t e g r a l s a r e from G r a d s h t e i n  By-2  m  y.r  8p m  $ Rzyhik [35[)  2  exp(-3r ) dr ; y o  Q  Q  3  C (3u-l) E(4^ = 9m  ^  " o r  By  -  ) 2  r  o  3 y  "  2  e x  P  <~ r ) B  d r  o  o  l  By-2  E(  9m9y  my  "  }  ^- o r  }  r  o  6 u  "  2  «p(-e^ >  d r  0  o  m Q:  (9a ) /  (Bu-D  2(a ) 2  /  f  9L 2  E (  9  ) = - 2y  2  00  (log r ) . r 2  Q  3 y _ 1 Q  . exp(-3r ) dr o  Q  9a  = ^2 { i^(gy) - log e r + 1 ( 2 , BU-D} Where  §(z,q) i s the Riemann's Zeta function =  I" n  (  7 ) and q+n  p s i function.  i s the  227  roo  9a9a  l o g  a  r  o ' o  * P( "^ o  r  e x  r  r  )  d r  o  0  - - V  1  2  n+  2a  where  n = {  i i  +(1  - s i r * -  s  < v ^ -  1  (3y) - l o g 3 )  The h e s s i a n m a t r i x o f t h e l o g l i k e l i h o o d f u n c t i o n has a b l o c k d i a g o n a l form, w i t h t h e two o f f d i a g o n a l ( 2 x 2 ) m a t r i c i e s b e i n g zero ( t h e o r d e r o f t h e  2 parameters i s assumed  {m,y,a , a } ) . T h i s means t h a t t h e i n v e r s e o f t h e  hessian matrix i s the matrix w i t h the i n d i v i d u a l blocks inverted.  This  tells  us t h a t we can i n f e r t h e s i g n o f t h e c o r r e l a t i o n between m and y, and ,  2 a  and  a.  These a r e e x a c t l y t h e same as t h e c o r r e s p o n d i n g  of L ( w i t h t h e s i g n r e v e r s e d ) . Thus we expect Cor &  (m,u)  < 0  Cor (a ,a) > 0 2  The c o r r e l a t i o n m a t r i x i s p r e s e n t e d  i n t h e main t e x t .  cross d e r i v a t i v e s  )  ]  A P P E N D I X <- 1 0  Maximum L i k e l i h o o d E s t i m a t i o n o f t h e Parameters {m,u,a,a} U s i n g t h e Steady S t a t e P r o b a b i l i t y D e n s i t y Approach  The s t e a d y s t a t e p r o b a b i l i t y d e n s i t y f u n c t i o n c o r r e s p o n d i n g t o g e n e r a l a v a l u e s ( i e a ^ 1/2, 1) i s x  A-l  exp  Bux  Bx 1+A (A10.1)  P(x) Buy A-l y exp A Q  X  0  x ^ '''exp [ a ( x ) ]  1+A By 1+A  dy  ^ ^ B = 2m/a  &  a  ( ) & suitably defined x  an  D  D The j o i n t l o g l i k e l i h o o d f u n c t i o n o f n o b s e r v a t i o n s i s n n L = ( A - l ) I l o g x i + E a(x ) i=l 1=1 ±  (A10.2)  + n log D  The l o g l i k e l i h o o d f u n c t i o n i s n o t t r a c t a b l e a n a l y t i c a l l y f o r purposes of e s t i m a t i n g i t s maximum w i t h r e s p e c t t o t h e parameter, and so o n l y n o n l i n e a r o p t i m i z a t i o n methods must be employed.  However, i t was found  t h a t methods t h a t used n u m e r i c a l d e r i v a t i v e s ( l i k e any m o d i f i c a t i o n o f the Newton method) l e d t o problems due t o t h e c o m p l i c a t e d way i n w h i c h the parameters e n t e r t h e l i k e l i h o o d f u n c t i o n .  Expressions f o r the f i r s t  and second d e r i v a t i v e o f L w i t h r e f e r e n c e t o t h e parameters were  229 derived as under n 3L =  9u  D  i=l  n  3L 3m  i=l  where  a(z)  A a(x-i)  i  i=l  D  x  {  33  x  3D 3A  ±  3yxi X  +  2  1  A  l o g  x  i "  3yxj  A  A  *  l A  3yz A  l+A -  D&x A .  iT " l+A'  l+A ,  0  g  -*  x  -3yz  3 z  A  z  l+A  33)  (  6  3z^ l+A 2  I  a 3  3^ 3y  2  2  z  2  b(z)  exp  2  = I  333y  dz  +  (  9 D /  D  2  9 D /  3D/3A  , . 2 ~ (l+A)'  (3D/3y) '  2  3 D/  333y  dz  b(z)  b(z) =  2  where  N  m  . exp  3 D/3y =  a(z)  }exp  where 6 = 1 + 1/A ;  3L ly?  dz  3D/3m  3  L  3D  _ ? . . log  where  3z  _3D 3y.  where  3L 3A  3D/3y  2  3y)  logz A  3 z z  a+A)  6  J  " dz  230  where  3A3u  my  o  3XJ  = Z  3 i log  A  . U + b ( z ) } bz  b(z)  exp  A'  x  _  x  ±  . —  - D —  ~  1/A  ,00  3D 2  where  3z exp  3A3u  — X  log z)  ?  I  3X33  3D/g 2  = I  X  • X  +  z  (l+X)  2  2  (1+A)  X  2  Ux-j  (3D/ )  g  3 ^  ^3  A  J  1/A  z  dz  — 5 —  33  where  b(z)  _y  Z  r  =  + 0*  2  1  3  ae  +  T2  b(z)  exp  log  X-L +  dz  l+X x. l (l+X) 2  (3D/33QD/3A)  D  2  l+X  3 D/ 2  Ti+I)" s lo  x  ±  3A33  D  231  where 2  9 D 3A83  2Uz  b(z)  jexp  ' Pi < A :  (l+A)  2  gux^log x  3L 2  3A<  =Z  px.  log X.  1  °  1  (1+A)  +  log  A  px.  —  (log x )  2  2  (  +  3D/  D  Z  A  A  L +  b(z)  \ dz  2  23yx  2  2px_  ±  Bux^^ l o g x  A  ±  A'  3  1+A L  ^ — (1+A)  1+A 3A D  A  +  1+A  o  2  9 D /  :  (1+A)  2  '  A(1+A)  (1+A)  Bux^Clog'jCj)  •' •  A'  (1+2A)  gz^  3z^  4.  12  +  J  1+A  px^  + — - — = — log x (1+A)'  3A ) 2  2  where exp tb(z)]  8A  2  2 33jz  (  >  A (l+A)  3 z log z  c- K  I >  3z (1+4A+3A ) 2  6  A (1+A) 2  +  3z  6  A (1+A) 3  log z , , _ 3 z j _  ..A (1+A) 3  4  +  A(l+X)  )  2  y  (  . 3 2 (  X  + 4 x 3 )  2  3yz  +  3z (1+A)'  _ g |lo 8  8  z)  A (1+A) A"(1+A)  b  6  4  l°g z  ^  +  3  -  3yz  log z  3z .A  2  (1+A)  dz  232  Most of the derivatives of the i n t e g r a l D with reference to the parameters appear very imposing.  Since evaluating the second  derivative of L would require .numerical evaluation of these i n t e g r a l s , i t was f e l t necessary that these functions  (ie 3D/3y ; 3 D/3y33 etc)  be examined further to ascertain whether they are "smooth and well behaved" for purposes of numerical integration.  The objective of the investigation  N  may be stated as:  a) To evaluate the integrand and i t s slope as the variable approaches i t s l i m i t s (O, ) 00  b) To try and infer the shape of the functions from the information i n (a) above If we represent  the integrand  i n the derivatives (both f i r s t  and second) of D with reference to the parameters {3,A,U} i n general as f ( z ) , then the table below outlines the p r i n c i p a l r e s u l t s  Limit of 3f/3z  Limit of f ( z ) z-*0  z  0 0  0  z  3D/3y  +0  +0  B/X  3D/33  +00  - 0  y M  3D/3A  1/A  3 D/3y  2  + 0 +0  +0  3 D/333y  +0  - 0  l M  3 D/3X3y  -0  2  2  2  - 0  2  2  +0  Not investigated  +0  2  z->°°  3(3y-2)M  -0 2  3  + 0  + o  -0 , A< -1 3 D/33 2  2  +0  +0  +0  -0  233  A few c l a r i f y i n g comments are i n order: a)  The expression -0 and +0 indicate that the function approaches zero from the negative and positive directions respectively.  b)  the l i m i t s indicated are v a l i d only, given the parameter values i e they do not represent the l i m i t s as, say, A -*• 0. It i s anyway shown that A •*• 0 and -1,  represent s p e c i a l  cases (see Appendix 5) . c)  the behaviour of 9D/9A was not a n a l y t i c a l l y examined with 2  2  reference to i t s slope at the l i m i t s , nor was 9 D/9A , as the functional forms were rather complex. The indications from the analysis are that the area of the i n t e g r a l may not l i e e n t i r e l y either i n the f i r s t or fourth quadrant, but partly in both i n some cases.  To investigate  further the shape of each of these  functions, and also to see what proportion of the t o t a l area l i e s i n either quadrant for a broad range of parameter values, the  functions  were numerically evaluated and p l o t t e d . The conclusions were that for a l l p r a c t i c a l purposes a l l the area was i n either the f i r s t or fourth quadrant.  A l l the functions were unimodal.  The importance of this  information becomes clearer when we address the problem of numerical integration of these functions. In general, given a function that can be evaluated over the whole range of integration,  ( i e . there exist no d i s c o n t i n u i t i e s  etc)  evaluating the i n t e g r a l using a quadrature (or even the more powerful adaptive quadrature) method, i s a t r i v i a l matter. problems that we face,  To see the s p e c i a l  l e t us address the problem of evaluating  the  seemingly innocuous i n t e g r a l D.  D  Buy  A-1  =  y  We have  By  exp  l+A dy  l+X  W i t h a change o f v a r i a b l e s we c a n t r a n s f o r m D as under  Let  z = y  D =  X  dz  e x  Buz  P  X  =  A-1 Xy dy  Bz  l+X -r dz  l+X  j o f ( z ) dz  [ I n p a s s i n g i t may be noted t h a t i t h e l i m i t s o f i n t e g r a t i o n have t o be i n t e r c h a n g e d f o r X<o]  JHo i d e n t i f y t h e mode of f ( z ) , we s e t i t s f i r s t d e r i v a t i v e t o zero, which g i v e s 1/X f (z) =  j  ( j  ^-) exp  Byz - Bz X  " l+X  l+X  -t  =  0  1/X =  f ( z ) . B(  (A10.3)  R u l i n g o u t t h e a l t e r n a t i v e t h a t f ( z ) = 0 a t t h e mode g i v e s t h e mode a  z = y  The integrand i n D i s c l e a r l y unimodal. therefore see that from o to p \  the f i r s t term i n the exponential  dominates, and as z increases beyond u \ and sends f ( z )  0.  Looking at f ( z ) we  the second term overtakes,  The point here i s that at u^;  f (z) i s very large  ( p a r t i c u l a r l y when 3u i s moderately large and X i s near zero i e . a - 1/2). In the computer, this gives a f l o a t i n g point overflow. To overcome this problem, we multiply the probability density function (A10.1) by exp(-p) in the numerator and denominator.  This reduces the integrant f ( z ) to  l+X f (z) =  j  ^P X  • . B u z 3z •IT "  l+X  - P  (A10.4)  and everytime D has to be evaluated p may be chosen such that f ( z ) at the mode i s a reasonable number.  The approach poses no problem even  when we evaluate the derivatives of L; as we always have D i n denominator with a derivative of D with reference to (B,u,X} i n the numerator, and the same adjustment works there. The next point i s that the mode jump a l l over the half r e a l l i n e as X goes from p o s i t i v e to negative. order 0.1.  I f X ranged from +1.0  to -1.0;  As u becomes smaller, the range increases.  In our problem u i s of the ranges from 0.1 to 10.0. That by i t s e l f should not  cause any concern, but when coupled with the fact f ( z ) happens to be a very spiked function, ( i e i t s t o t a l mass i s concentrated over a very small range) poses some problems.  A l l numerical integration algorithms  require that we provide the l i m i t s of integration.  Since the mode moves  a l o t , we may be tempted to provide a large range (say 0 to 100).  236  However, due to the s p i k e d 'nature of f ( z ) , i t s value i s very close to zero over a l l but a very small segment of this range. integration algorithms  The numerical  value f(z) at a set of points over the range, and  very l i k e l y finds the value of f ( z ) at a l l those points very close to zero, and returns the value of the i n t e g r a l as 0.  This i s because the  t o t a l area may l i e over a small f r a c t i o n of the distance between any two of the points at which f(z) was evaluated.  To be able to value the  l i k e l i h o o d function (A10.2), with any accuracy, the i n t e g r a l D has to be accurately computed.  The problem therefore b o i l s down to one of finding  reasonable integration l i m i t s for D. Given that the integrand f(z) of D (eqn A10.4) i s unimodal suggests a straight forward approach to getting the required integration limits.  Let z  f ( z ) ; zi < z  m  be the mode and  m  and z  2  the two i n f l e c t i o n points of  I f now we represent the l i m i t s of integration by z  < z£.  and z" ', (z'' < z" ) then we can choose k^ and k  z  '  z  n  =  z  =  m " z  k  such that  l (m ~ l ) z  + k  m  2  2  z  (z - z ) 2  m  where z'' and z"+ are required to s a t i s f y some c r i t e r i a l i k e (say) f(z*)/f(z ) m  and  40 f ( z " ) / f ( z ) < 10" " m  or some other such small value. Thus,locating the i n f l e c t i o n points should solve our problems. The second derivative of f(z) i s got by d i f f e r e n c i a t i n g (A10.3)  %/  237  f,  (z) = — - X  exp 2  Buz  Bz  1+1/X -, X  ~x  ^  i/x-i  z  .1/X, 3 + ( u ^ - " ) . £ . ( u ^l"A )v :l  "  v  v  X  S e t t i n g t h e above to z e r o , n o t i n g t h a t f ( z ) ^ 0 a t the i n f l e c t i o n  points  gives  + 3".(y - z  - z Multiplying  z  through by  Y  z  1/X, )  and  - 3 (uz - zY^2 ')"  =  0  s u b s t i t u t i n g y = 1+1/X  =  =  gives  0  (The f u n c t i o n a l form c l e a r l y suggests t h a t the above e q u a t i o n has An  i t e r a t i v e method to s o l v e f o r the r o o t s of the above e q u a t i o n  two  roots)  isngot  from a f i r s t o r d e r T a y l o r s e r i e s expansion.  g(z)  M z?  -  3 (uz -  ^ z')2  (A10.5)  Then z<n+I  =  z  n  (A10.6)  g^(z ) n  Where z +^ n  i s the s o l u t i o n to (A10.5) at the  (n+l)*"* i t e r a t i o n . 1  g e n e r a l , the scheme above should converge q u i t e r a p i d l y . was  In  However, i t  found t h a t f o r some parameter v a l u e s , the scheme tended to converge  always towards the same r o o t I t was  therefore necessary  and u s i n g them, and  ( i e . the second s o l u t i o n was  not  to f i n d an approximate s o l u t i o n to  (A10.6) a r r i v e a t more a c c u r a t e v a l u e s of  obtainable) (A10.5), the  238  i n f l e c t i o n points. •u .  For t h i s we expand TX using a Binomial s e r i e s , about  This gives  J  (z-u ) +  u  A  =  =  y ' (l  -  ¥  AY  1  T  A  (A10.7)  A  +  •y  Y Plugging equation (A10.7) for z' into (A10.5) and setting y = (z - y ) / y A  A  we get  -i  y  A Y  ( l - y) - 6 Y  yA+1  (y+l)  -  yAy  which can be reduced to  l - Yy  - y  2  3U-Y)  =  2  0 1/2  or  =  - Y ±' { Y +43 2  (1-Y) }  '26 (1 - y )  2  2  and that gives us the approximate solution.  (I^YY)  2  =  0  239 APPJNDXX I I  E f f e c t on bond v a l u a t i o n of using the y i e l d t o maturity a 91-day pure d i s c o u n t  bond i n s t e a d o f the i n s t a n t a n e o u s l y  on risk-  f r e e r a t e of i n t e r e s t * The  b a s i c assumption of t h e bond v a l u a t i o n model i s t h a t i t  i s a function of the instantaneously time  to  maturity,  i n t e r e s t r a t e i s the discount  bond  due  By d e f i n i t i o n , the i n s t a n t a n e o u s r i s k  free  yield  pure  to  longer time t o maturity, rate,  would  to  maturity  on  a  riskless  mature the next i n s t a n t i n time.  using the y i e l d t o maturity  free  r i s k f r e e i n t e r e s t r a t e and  on  a  riskless  bond  Thus,  which  has  as a proxy f o r t h e i n s t a n t a n e o u s l y  bias  the  bond v a l u a t i o n .  a  risk  T h i s b i a s can be  broken down i n t o two p a r t s : 1}  The estimated parameters o f t h e i n t e r e s t  rate  process  (m,^A , ( p , d ) a r e biased because we have estimated them 2  from  a process which i s not the instantaneous  r a t e process.,  T h i s b i a s e s the  bond  interest  valuation,  which  uses these parameters a s input. 2)  In  the  bond  valuation  equation,  instead  instantaneous i n t e r e s t r a t e a proxy i s used, biases the bond To  analyse  of and  the this  value.  the nature o f these b i a s e s , l e t us assume that  the t r u e model of t h e i n t e r e s t r a t e process i s given by  IT  7n(>--0dt  + cr-f^f  Then I n g e r s o l l [ 3 9 ] has a s o l u t i o n f o r the o n  a  pare discount  (A11.1)  yield  to  bond having time t o maturity  maturity t , and  240  c u r r e n t v a l u e of instantaneous i n t e r e s t r a t e r , as  RC^rt") = - ^ J ^ ^ £ ^ ^ H < r ) J 4 ^ 1 - H e r ) *  For a given value of t , equation  (A11.2) may  y  (A11.2)  be r e p r e s e n t e d as  , a(T) 4- bCt).f Since  we  are  interested  in  (A 11.3)  a f i x e d value of f =91 days, the  c o e f f i c i e n t s i n equation  (A11.3) may be t r e a t e d  Thus  by R, t h e y i e l d t o maturity on a 91 day  if  we  represent  as  parameters.  pure d i s c o u n t bond, we have  From  (A11. i») we have the s.d.e. f o r B as dR = bdr _ (A11.5)  241  The  first  t h i n g t o be noted  process of the  i s t h a t the assumption  jU^ *  {bfkJr<h  ) and  i s incorrect.  estimates of jk^ and <r^ (we need not c o n s i d e r m^ on  equation  s t a r t with.  (A11.6) i n s t e a d o f  However,the e r r o r due  investigate analytically!.  to  as i t  0^ , as e s t i m a t e s of ^  this  is  and (T r e s p e c t i v e l y ?  using jx^ and  To q u a n t i f y ^  let  us  compute a  and  we are i n t e r e s t e d i n the e r r o r s i n the neighbourhood  o f the parameter range we have estimated f o r the  Thus, we  to  L e t us, t h e r e f o r e , o n l y c o n s i d e r the  - 2  process,  Thus,  i s = m) ,  complicated  use numerical v a l u e s f o r (m,yt, a ) , so t h a t we may Since  a  ( A l l . 5) are i n c o r r e c t to  r e l a t i v e l y s i m p l e r q u e s t i o n : what i s the e r r o r from  b.  is  form  where  based  that R  we may  use those v a l u e s themselves  interest  rate  t o compute a and  b.  use m = 0.002522 jj- — 0.001293 <r = 0.690494 x z  10-*  The extent of the e r r o r can be e a s i l y i n v e s t i g a t e d by Monte C a r l o methods. We c o u l d expect the e r r o r to be q u i t e s m a l l due t o the nearness of (R-a) t o R. . T h i s i s because a R (and s i n c e R~]x ,in relative terms, a a 0) and thus assuming t h a t the d i f f u s i o n equation governing B has a s i n g u l a r boundary a t R = 0, (as i n equation A l t . 6) i n s t e a d o f a t R = a, (as i n eguation A11.5) should have only a marginal e f f e c t on the parameters. Further, i t appears t h a t the principal effect of the approximation, i s on the v a r i a n c e element, i e . , <r . Given t h a t a i s p o s i t i v e (R-a) < R . Thus, t o compensate, from equation (A11.6) would be l e s s than <T from equation (A 11.5). I t can be shown t h a t b 4 1,and so a\ i s already an underestimate of <r . Using equation (A11.6) i n s t e a d of (A11.5), would l e a d to a f u r t h e r downward b i a s on the estimate of r . 1  R  ft  242 which g i v e s us values f o r a and b a = 0. 001637|A,  b = 0.998343 This implies that JUfc =  (r2  0.99  9981  = 0. 998343 <r~  The e r r o r s i n assuming t h a t JA^ i s approximately  jX  and  is  2  approximately <r 2 are n e g l i g i b l e . It  must  be  noted  expressions f o r expression  of  and  the  only under the structure  a  b  the  above  based  yield  pure  of  that  on  rates.  we  from as  an  That equation i s v a l i d  hypothesis If  stems  (a 11.2).  equation  to maturity.  expectations  interest  conclusion  about  assume  the  term  liguidity/-term  premium o f the form  which i s what we have prices,  used  Ingersoll [39]  has  in  subsequent  modelling  shown that equation  of  bond  (A11.2) h o l d s ,  but with m and jUL r e d e f i n e d as m* and jx. * and given by m*  =  (m-k_2.)  y~ = {mjx * k, )/m* 1  Thus equation using m» and section  (A11.4) holds; with,a  and  b  JK* . , He had estimated k, and k  sui,taj>ly 2  7.3) k, = 0.3093 x 10-s k  2  = -0. 1548 x  10-2  Osing these values g i v e f o r a and b the v a l u e s a =  0.01705/v  redefined  as (see Chapter  7,  243  b = 0.98837 Comparison  with  the e a r l i e r values of a and b shows t h a t 8  i s a poorer proxy f o r r ~ which i s as now  expected.  These  now  values  give jU^ * 1.00542/^ 0^2 = 0 . 98837 cr *  We  see  that  underestimate The  JUR i s  overestimate  of jx , • and  o f <r*. The extent of the  direction  resultant  an  of  the  bias  b i a s on both jx and <r  bond value i s biased downward.  quite  a r e such  z  i s an  2  small. that the  Bowever, the e f f e c t  the bond v a l u e due t o such o r d e r s of e r r o r s w i l l be n e g l i g i b l e .  is  <j^  on  the  parameters  (This i s i n d i c a t e d from the T a b l e s IX and X  on s e n s i t i v i t y of discount bond values to e r r o r s i n jUL and Hext,  we  on  consider  the  effect  on  i n s t e a d o f r , i n the v a l u a t i o n equation. i n r , by s u b s t i t u t i n g fi i n s t e a d of r may  0-  2  bond value by u s i n g H The  proportional error  be represented  as  {b-\)  +  <r)  (A11.7)  T  The  error i s clearly  Since,  on  an  dependent  average,  on  the  current  the i n t e r e s t r a t e i s expected  around jx , l e t us c o n s i d e r the e r r o r a t z -jx.  _ r V  T  JT^JX  *  have  +  of  r.  t o remain  Thus  (b-i)  fx  S u b s t i t u t i n g the values of a and premium model we  value  b, based on the  liquidity/terra  The  percent  error  i n the v a l u e of a pure d i s c o u n t bond due to  the above e r r o r i n r may  Where  be represented as  i s t h e bond value e l a s t i c i t y with r e s p e c t to r .  I f we  r e p r e s e n t the d i s c o u n t bond value by B we have YI  .  db  r  where the second e q u a l i t y comes value  of  the  pure  discount  - b^T  from  the  (611.8)  expression  for  the  bond as given i n I n g e r s o l l [ 3 9 ] ,  Thus Percent e r r o r i n bond value = 0.542 x <-brf )  which f o r r = It  seems  works out to 0.009% - a t r u l y n e g l i g i b l e  error.  reasonable to expect t h a t a t other values of r around  ^U., the e r r o r i s a l s o o f s i m i l a r orders of magnitude. I t may t h e r e f o r e be concluded  t h a t the e r r o r due to the use  of the y i e l d t o maturity on a 91-day d i s c o u n t bond f o r the i n s t a n t a n e f r e e i n t e r e s t r a t e , i s minimal..  as  a  proxy  

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