PARAMETER ESTIMATION OF STOCHASTIC INTEREST RATE MODELS AND APPLICATIONS..™) BOND PRICING by A. L. ANANTHANARAYANAN B. Tech. (Hons), I.I.T., Kharagpur, India, 1967 ^THES IS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Commerce & Business Administrat ion We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1978 A. L. Ananthanarayanan In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I ag ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or . by h i s represenjtWtVve'sv • I t; ; i s~ u n d e r s t o o d " tha t copy i ng- o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f ] • The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 11 ABSTRACT A p a r t i a l equilibrium valuation model for a security, based on the idea of contingent claims analysis, was f i r s t developed by Black & Scholes., The model was considerably extended by Herton, who showed how the approach could be used to value l i a b i l i t y instruments. Valuation models for default-free bonds, by treating them as contingent upon the value of the instantaneously r i s k f r e e i n t e r e s t rate, have been developed by Cox,Ingersoll 6 Boss, Brennan 6 Schwartz , Vasicek and Richards. There has, however, not been much attention directed towards the empirical testing of these valuation models of default-free bonds. This research i s an attempt i n that d i r e c t i o n . Our attention i s confined to retractable and extendible bonds. Central to a r r i v i n g at any equilibrium model of bond valuation i s the assumption about the instantaneously r i s k l e s s i n t e r e s t rate process, since the bond value i s treated as contingent upon i t . These bond valuation models are p a r t i a l equilibrium models, since the i n t e r e s t rate i s assumed as exogenous to them. The choice of the i n t e r e s t rate process i s made subject to some r e s t r i c t i o n s on i t s behaviour which are based on expected properties of intere s t rates. The intere s t rate process adopted in t h i s study i s a generalization of that used by Vasicek and Cox,Ingersoll S Boss., The properties of the chosen mathematical model are investigated to ascertain whether i t conforms to those expected of an i n t e r e s t rate process based on economic reasoning. We go on to develop alternate estimation methods for the 111 parameters of the i n t e r e s t rate process, using data on a r e a l i z a t i o n of the process. One "exact" method and two others based on approximations are outlined. I t i s observed that the "exact" method i s not available to the complete family of processes included in the continuous time stochastic s p e c i f i c a t i o n assumed to model i n t e r e s t rates. The asymptotic properties of the estimators from the "exact" method are known from the e x i s t i n g l i t e r a t u r e . However, since we would have to adopt one of the approximate methods, we need to know something about the properties of the estimators based on these approaches., This could not be derived a n a l y t i c a l l y and so a Monte Carlo study i s conducted. The r e s u l t s seem to indicate that the properties of the estimators from the three methods are not very d i f f e r e n t . The y i e l d to maturity on 91-day Canadian Federal Government Treasury b i l l s , on the date of issue, i s chosen as the proxy for the 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 is b r i e f l y investigated and found to be n e g l i g i b l e . The bond sample chosen i s the complete issues of retractable and extendible bonds made by the Government of Canada. There were 20 issues between January 1959 and October 1975, and weekly prices on a l l these bonds are available i n the Bank of Canada Review . To a r r i v e at the f i n a l bond valuation equation, some assumptions are made about the term structure of i n t e r e s t rates. This study f i r s t assumes a form of the pure expectations hypothesis and i t i s shown that the performance of the model i n predicting market price movements, i s considerably improved when iv we assume a s p e c i f i c form of term/liquidity preference on the part of investors. Incorporating taxes into the model r e s u l t s i n s i m i l a r improvements. The hypothesis that the bond market i s e f f i c i e n t to information contained in these models i s tested and not rejected. , i F i n a l l y , an ad hoc regression based model i s developed to serve as a bench mark for evaluating the performance of the p a r t i a l equilibrium models. I t i s observed that these models perform atleast as well as the ad hoc model, and could be improved by relaxing 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. Schwar tz V TABLE OF CONTENTS CHAPTER PAGE 1. INTRODUCTION . . , • . . V . . » > . • • « v . . . ..... . , • • • - r . • • ? • 1 Preamble 1 Contingent Claims Valuation of Bonds: A Bri e f Review 2 Canadian Retractables/ Extendibles i n Perspective 4 Outline of the Thesis . . r . . . 7 • • 2. THE PRICING THEORY OF DEFAULT FREE BONDS i . . . . . . . . . 10 Determinants of Bond Value 10 The Basic Bond Valuation Equation ............ 13 Boundary Conditions for Retractable/ Extendible Bonds 16 Incorporating Taxes into the Model 20 3. THE INTEREST RATE PROCESS 22 Properties of Interest Rate Processes ........ 22 The Interest Rate Process .................... 25 Interest Rate Process Behaviour at Singular Boundaries 26 4. ESTIMATING THE INTEREST RATE PROCESS PARAMETERS . ... ... ........ ....... ... ... .... ......,, . 28 Br i e f Review of Published Research i n Related Areas 28 Maximum Likelihood (M.L.) Method of Estimation 31 The Simple Lin e a r i z a t i o n Approximation ....... 34 The Transition Probability Density Method .... 35 The Steady State or Stationary Density Method 36 The P h i l l i p s Approximation Method ............ 41 5. COMPARISON OF THE DIFFERENT ESTIMATING METHODS H4 The Method of Comparison .................. ... 44 Generating an "exact" Sequence for the Square Root Process .......................... , 45 Results of Monte Carlo Simulations for the o( =i/x(known) Case 48 Results of Monte Carlo Simulations for the <* On known Case., ............................. 71 The Relation Between the Interest Rate Process Parameters 79 vi 6. THE INTEREST RATE AND BOND PRICE DATA . ............ 88 The Short Term Riskless Interest Rate ......... 88 Price Series on Retractable/Extendinle Bonds .. . 91 Price Series on Ordinary Pederal Bonds ....... 96 7. EMPIRICAL TESTING OF BOND VALUATION MODELS 97 Estimated Parameters For The Interest Rate Process 97 Solving the Bond Valuation Equation 101 Bond Valuation Under the Pure Expectations Model ...............,..................•.. 106 Estimating the Liquidity/Term Premium Paramters .................................... 129 Bond Valuation Under the Liquidity/term Premium (LIQP) Model 104 Bond Valuation With Revenue Taxes ............ 148 Bond Valuation Incorporating C a p i t a l Gains TaX . Wr. .. 151 . The "Moving Average" Model ................... 152 Tests of Market E f f i c i e n c y ................... 157 Comparison of Current Models with a "Naive" Model ..,.. • • * • 169 8. SUMMARY AND CONCLUSIONS ............ ................ , 174 Summary Of The Thesis 174 Conclusions And Directions For Further Research ...........................•......... 177 BIBLIOGRAPHY . . . 181 APPENDIX 1. C l a s s i f i c a t i o n of Singular Boundary Behaviour for the Cases * = 1/2 & 1 187 2. Details of the Estimation Procedure f o r the Linearized Model . ».•.•..../.^ . ..... ., 191 3. Solution to the Forward Equation for <* = 1 195 4. Solution to the Forward Equation for <X = 0 with no R e s t r i c t i o n at the Origin ...................... 204 5. Derivation of the Stationary (or Steady State) Densities 205 6. Details of the P h i l l i p s Approach to Estimation ..... 209 v i i 7. Details of Estimating Procedure for <*= 1/2 (known) Case 213 8. Analysis of Effect of Measurement Errors of Data ............,................................. 221 9. An Approximate Estimate of the Asymptotic Correlation Matrix Between Interest Bate Process Parameters ................................ 223 10. Maximum Likelihood Estimation of Parameters {m. fx, t<r. ok) Osing the Steady State Probab i l i t y Density Approach .................................. 228 11. Effect on Bond Valuation of Using the Yeild to Maturity on a 91-day Pure Discount Bond Instead of the Instantaneously Riskfree Bate of Interest i . . . 239 v i i i LIST 0? TABLES Table Page I Comparison of Retractables/Extendibles with Other Forms of Debt i n Canada 6 II Estimate of m by Different Methods for rf-i/j. (known) Case ............................ 51 III Estimate of /A, by Different Methods for dU'/a. (known) Case 52 IV Estimate of c r z by Different Methods f o r 0(^.1/2. (known) Case ............................ 53 V Estimate of Infer' by Different Methods for 0U1/2. (known) Case ............................ 54 VI Comparison of Monte Carlo Results on Parameter Estimation Using S e r i a l l y Dependent/Independent Samples ................ 59 VII Comparison of Results of Estimation Using Weekly and Daily Data {^-Y^ known) 60 VIII Theoretical S e n s i t i v i t y of Pure Discount Bond Prices to Errors i n m 63 IX ° Theoretical S e n s i t i v i t y of Pure Discount Bond Prices to Errors i n ^ 64 X Theoretical s e n s i t i v i t y of Pure Discount Bond Prices to Errors i n <rv .. ,..vr.«..«•. 65 XI S e n s i t i v i t y of Pure Discount Bond Prices to Di s t r i b u t i o n of Estimated Interest Rate Process Parameters ( r, = /V2) 67 XII S e n s i t i v i t y of Pure Discount Bond Prices to Dist r i b u t i o n of Estimated Interest Rate Process Parameters ( r, = ) 68 XIII S e n s i t i v i t y of Pure Discount Bond Prices to Dis t r i b u t i o n of Estimated Interest Rate Process Parameters ( r, = 2^) 69 XIV Comparison of Bond Price S e n s i t i v i t y to the Use of Daily vs Weekly Data i n the Estimation of Interest Rate Process Parameters ( = j/^ ) 70 XV Estimation of Parameters for Unknown Case 73 XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI XXVII XXVIII XXIX XXX XXXI XXXII Comparison of Parameters Estimated Using Daily vs Weekly Data for the Unknown Case Theoretical S e n s i t i v i t y of Pure Discount Bond Prices to Errors i n <* { <r2 Has Hot Been 'Corrected* to Reflect the Error i n <* ) Theoretical S e n s i t i v i t y of Pure Discount Bond Prices to Errors i n (<r2 Has Been •Corrected* according to the Value) ....... Details of Data Sample Retractable/Extendable Bonds ......... of Details of Data Sample of Straight Coupon Bonds ........................................ Comparison of Model and Market Prices Bond: .4% Jan. 1, 1963 (R1) ................... Comparison of Model and Market Prices Bond: 5'/i % Oct. 1, 1960 (E1) ................. Comparison of Model and Market Prices Bond: 5Ki % Oct. 1, 1962 <E2) ..... • • • • Comparison of Model and Market Prices Bond: 5/2. % Dec. 15, 1964 (E3) Comparison of Model and Market Prices Bond: 5Vo. % A p r i l 1, 1963 (E4) Comparison of Model and Market Prices Bond: 6% A p r i l 1, 1971 (E5) .................. Comparison of Model and Market Prices Bond: 6/4. % Dec. . 1, 1973 (E6) ............. .... Comparison of Model and Market Prices Bond: VJi\ % .April 1, 1974 <E7) Comparison of Model and Market Prices Bond: 8% Oct. 1, 1974 (E8) Comparison of Model and Market Prices Bond: 7% % Dec. 15, 1975 (E9) Comparison of Model and Market Prices Bond: 6'/i| Aug. 1, 1976 (E10) Comparison of Model and Market Prices Bond: 7% July 1, 1977 (E11) 75 77 78 94 95 109 110 111 112 113 114 115 116 117 118 119 120 X XXXIII Comparison of Model and Market Prices Bond: 1% % Oct. 1, 1978 (E12) 121 XXXIV Comparison of Model and Market Prices Bond: 1'A % Dec. 1, 1980 (E13) 122 XXXV Comparison of Model and Market Prices Bond: 1% A p r i l 1, 1979 (E14) ................. 123 XXXVI Comparison of Model and Market Prices Bond: 9^ % A p r i l 1, 1978 (E15) ............... 124 XXXVII Comparison of Model and Market Prices Bond: 9J4j % Feb. 1, 1977 (E16) ................ 125 XXXVIII Comparison of Model and Market Prices Bond: 7/£ 31 Oct. 1, 1979 (E17) ................ 126 XXXIX Comparison of Model and Market Prices Bond: 9% Feb. 1, 1978 (E18) .................. 127 XL Comparison of Model and Market Prices Bond: 9% Oct. 1, 1980 (E19) ................... 128 XLI Comparison of Mean Error For A l l Bond Across Different Models ............................. 145 XLII Comparison of Betas & Correlation Between Market 6 Model Prices 146 XLIII Theoretical S e n s i t i v i t y of Pure Discount Bond Prices to Errors in K, 155 XLIV Theoretical S e n s i t i v i t y of Pure Discount Bond Prices to Errors in K2 ...................... 156 XLV Return on Zero Investment P o r t f o l i o Based on Constant Long Position i n Bond .......... 159 XLVI 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 XLVII Return on Zero Investment P o r t f o l i o Based on Varying Position i n Bond ................ 162 XLVIII Results of Yield Eguation C o e f f i c i e n t Estimation 171 XLIX Comparison of Model and Market Prices 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 Prices f o r the Straight Bond .... Return on Zero Net Investment P o r t f o l i o (Based on a Constant Long Position i n the Generic Bond) by Aggregating Over A l l Bonds . / XI1 LIST OF FIGURES Figure Page 1 Plot of Transition Density Function (6 Cumul-ative Probability) f o r = at Different r e Values ...«..«.««»»• • •••••••••»•• ••••••••• ; 219 2 Plots of the S e n s i t i v i t y of the Tr a n s i t i o n Density Function to Changes in tr1 and <* ...... 85 3 Plots of the S e n s i t i v i t y of the Transition Density Function to Canges i n m at Different r 0 Values .••.............. .• •;•» .•... ........... 6^ 4 Normal Probability Plot of Resultant Error Vector from the Estimation of Liquidity/Term Premium Prameters ............................ 139 5 Plot of L i q u i d i t y Premium vs Time to Maturity on Pure Discount Bonds Corresponding to E s t i -mated Parameters ............................,141 6 Plot of Term Structure Curve (Yield to Maturity vs Time to Maturity on Pure Discount Bonds) Corresponding to Estimated Parameters at Different Values of r, 142 7 Plot of Term Structure Curve to Show Possible "Humped" Shape for Certain v0 Values ......... 143 8 Plots of Model vs Market Prices For Bond E4 : Capital Gains Tax (25%): Model, and of Dis t r i b u t i o n of Hedge P o r t f o l i o Returns ...... 167 9 Plots of Model vs Market Prices For Bond E7 : Capital Gains Tax (25%) Model, and of Distribution of Hedge P o r t f o l i o Returns ...... 168 ACKNOWLEDGEMENTS I would l i k e to share the c r e d i t for completing t h i s dissertation with several other i n d i v i d u a l s . . Professors Michael J. Brennan and Eduardo S. Schwartz suggested t h i s research topic., As my supervisor. Dr. Eduardo Schwartz was a constant source of encouragement., Dr. John A. Petkau provided considerable help i n the early stages towards my understanding of singular d i f f u s i o n processes. Dr. M. Puterman read drafts of my proposal and c l a r i f i e d c e r t a i n aspects pertaining to d i f f u s i o n equations. As members of my committee, Professors Alan Kraus and Rolf Banz painstakingly read early drafts of thi s report, and have considerably contributed to i t s improvement. Professor Phelim P. Boyle merits s p e c i a l mention. Apart from his contribution towards the substance and s t y l e of t h i s d i s s e r t a t i o n , i t was his warm friendship and moral support that kept me going through the rough periods. I cannot s u f f i c i e n t l y thank Dr. Kent M. Brothers for his help and guidance. Every part of t h i s research pertaining to s t a t i s t i c s and numerical methods have benefited from his advice. Dr. Shelby Brumelle has contributed immensely to the research culminating i n t h i s report. He was always available for consultations, and i t i s to him that I owe much of my understanding of Markov processes. David Emanuel, Hav Sblanki and Gordon Sick have helped me at various stages i n t h i s d i s s e r t a t i o n . Mr. Wayne Deans, l o c a l representative of the Bank of Canada, was of immense help i n putting together the data on retractable/extendible bonds. Kari Boyle helped with the i n i t i a l data c o l l e c t i o n , and Kent Wada helped not only with the data c o l l e c t i o n and i t s punching but also with the plots and typing the text into the computer. Seline Gunawardene and Carmen de Si l v a did an excellent job of typing the tables and appendices, as well as the f i r s t draft of t h i s d i s s e r t a t i o n . 1 CHAPTER 1 : INTRODUCTION 1 , 1 Preamble The application of contingent claims analysis to derive equilibrium valuation models for corporate l i a b i l i t i e s i s presently an area of considerable and continuing i n t e r e s t and has been actively investigated in the current finance l i t e r a t u r e . This study addresses the problem of empirical estimation of a pa r t i c u l a r stochastic s p e c i f i c a t i o n of the spot i n t e r e s t rate, and then goes on to evaluate the e f f i c a c y of a model of retractable/extendible bond valuation, based on the estimated interest rate process, i n pricing Canadian Federal Government issues. In the seminal works of Black & Scholes [7] and Merton [47], the p r i n c i p a l focus was on a r r i v i n g at closed form valuation models for put and c a l l options on corporate equity. Both the works ci t e d above did point out i n conclusion that the approach could be used d i r e c t l y to value other corporate l i a b i l i t i e s by treating i n d i v i d u a l s e c u r i t i e s within the c a p i t a l structure as "options" or "contingent claims" on the t o t a l value of the firm. Herton [46] also derives valuation equations for corporate bonds. Smith [65] provides a good review of the work in the area of option p r i c i n g , and i t s application to the valuation of related s e c u r i t i e s . 2 1 • 2 Contingent Claims Valuation of Bonds: & Brief Review The application of the option pricing approach to bond valuation was extended by Black & Cox [ 5 ] , Brennan & Schwartz [9] , and Ingersoll [37], Black & Cox extended the analysis of Merton [48], to incorporate various types of bond indenture provisions such as safety convenants, whereby the bond holders have the ri g h t to bankrupt or force a reorganization of the firm i f i t f a i l s to meet some standard. They further look at the ef f e c t of subordination among bonds, i e . hierarchy among the debt holders, t o claims on the value of the firm, and f i n a l l y the e f f e c t of r e s t r i c t i o n s on the financing of i n t e r e s t and dividend payments. Both Brennan & Schwartz [9] and Ingersoll [37] addressed the valuation of corporate convertible bonds with and without c a l l provisions, the p r i n c i p a l difference being that Ingersoll was concerned with a r r i v i n g at a n a l y t i c a l solutions to the valuation problem, whereas Brennan & Schwartz presented a general numerical algorithm f o r solving the valuation equations. So f a r , the emphasis was on corporate bonds, where the underlying asset was the value of the firm., The works referred to above treated the intere s t rate as non stochastic - constant and known with certainty over the period of the bond. The next area that was addressed was the p r i c i n g of default free bonds. These s e c u r i t i e s , (generally Government bonds of various types) were valued by treating them as "contingent" upon the course of the spot inte r e s t rate, along with suitable assumptions about the term structure of i n t e r e s t rates. Brennan & Schwartz [10,12], Cox, Ingersoll S Ross [16], V a s i c e k [ 7 2 ] , and 3 Bichard [58], have a l l addressed the problem of default free bond valuation in the option p r i c i n g framework. apart from the works of Brennan 6 Schwartz (cited above), the rest primarily dealt with the valuation of pure discount bonds, so as to a r r i v e at closed form expressions for the term structure equation. Brennan S Schwartz, in t h e i r e a r l i e r paper [10], represent the default free bond as a function s o l e l y of the short term in t e r e s t rate and time to maturity, and show that various types of bonds - savings, retractable, extendible, c a l l a b l e or discount - a l l follow the same p a r t i a l d i f f e r e n t i a l equation, the distinguishing feature being the associated boundary conditions. They also present a numerical algorithm to solve the valuation equations. In t h e i r l a t e r paper [12], they posit the value of the default-free bond as a function of the time to maturity and two related i n t e r e s t rate processes - the very short term r i s k l e s s i n t e r e s t rate and the very long term i n t e r e s t process (yields on a consol bond)., &s can be seen from the foregoing, considerable work has been done on the t h e o r e t i c a l front, i e , , developing bond valuation equations under varying assumptions about the stochastic properties of i n t e r e s t rates and term structure of i n t e r e s t rates. In addition, numerical methods have been developed to solve rather general forms of the resultant p r i c i n g formulae. However, to date, there have been few published tests of these models. Host of the empirical work in the area of contingent claims analysis, has been on the market for options on corporate equity , (to c i t e the important papers: Black & Scholes [ 6 ] , and Galai [29]), except for I n g e r s o l l [38], which 4 i s an application of option pricing analysis to dual fund shares, and Brennan & Schwartz [12]» who value Canadian Federal Government coupon bonds. The aim of t h i s research i s to conduct an empirical study of contingent claims analysis on retractable and extendible bonds of the Government of Canada., 1. 3 Canadian Retractables/Extendibjes i n Perspective: ftn extendible i s a medium to long term debt obligation that gives the holder the option to extend the term of the instrument, at a predetermined coupon rate., For example, the 5k %, October 1st, 1962, maturity extendible was issued on 1st October, 1959. I t was exchangeable on or before June 1st, 1962 into 5%,%, October 1st, 1975 bonds. Thus the 3 year i n t i a l bond was extendible into a 16 year bond, at the holder's option. A retractable, on the other hand, gives the holder the option to ele c t an e a r l i e r maturity. Both from the p r a c t i c a l investment point of view, and with respect to valuation theory, the two instruments are very s i m i l a r . There are two ways i n which to view a retractable or extendible bond. It may be viewed as a long term bond with a put option. The exercise price i n t h i s s i t u a t i o n i s the value Of the long term bond, and the payoff i s the short term bond. The option i s exerciseable on the extension/retraction date. A l t e r n a t i v e l y , the retractable or extendible may be viewed as a short term bond with a c a l l option. From t h i s point of view, the exercise price i s the value of the short term bond, and the 5 payoff i s the long term bond. Extendibles and retractables f i r s t appeared* on the Canadian scene i n 1959 with the Federal Government issue of H%, January 1st, 1963 (maturity date) retractable bonds, which were retractable on any interest payment date between January 1st, 1961 and January 1st, 1962 by giving 3 months prior notice. (Incidentally, t h i s was the only retractable issued by the Government of Canada). While there were addit i o n a l issues made by the Federal Government i n the mid s i x t i e s , these instruments have been used more widely i n the high i n t e r e s t rate period since 1969/70. Table I gives some numbers to place retractables and extendibles i n perspective v i s - a - v i s other forms of debt. Clearly, the major issuer of retractable/extendible bonds i s the Federal government. Further, as a proportion of t o t a l debt outstanding, retractables and extendibles appear to be increasing over time, both with the P r o v i n c i a l and Federal governments. The t o t a l debt columns i n Table I include very short term debt, ( i e . , current l i a b i l i t i e s , treasury b i l l s , e t c . ) , as well as medium to long term debt. Retractables and extendibles belong s t r i c t l y to the medium to long term maturity c l a s s , and so should be compared with the other debt i n that class alone. Thus even though retractables and extendibles constitute only approximately 4.536 of the t o t a l P r o v i n c i a l debt, these instruments represent a larger proportion of the medium and long * Information obtained from a publication of M/S Mood Gundy Ltd. on retractable/extendible bonds, l i s t i n g a l l outstanding Federal/Provincial/corporate issues as of January 15th, 1975. TABLE I COMPARISON OF RETRACTABLES/EXIENDABLES WITH OTHER FORMS OF DEBT IN O/S as on Ret/Ext. 31st March Tot.Debt. 1975 Z O/S as on Ret/Ext. 31st Marc' Tot.Debt i 1976 Z Ret/Ext as on 31st March 1977 Br i t i sh Columbia - 3845 - 50 5093 0.98 50 Alberta 128 3031 4.22 128 3578 3.58 128 Manitoba 58 2473 2.34 58 2884 2.01 58 New Brunswick 51 1199 5.08 61 1665 3.66 61 Newfoundland 161 1504 10.70 182 - - 182 Ontario 225 13397 1.90 675 16760 4.02 675 P.E. Island 10 98 10.20 10 111 9.00 10 Quebec 734 8403 8,73 808 8391 9.63 983 Saskatchewan - 816 - 70 912 7.67 70 Total Provincial Federal 4825 33700 14.31 5850 38299 15.27 6250 Corporate 1902 - - 2315 - - 2503 Total 8104 10207 10970 NOTES OH TABLE I a) A l l figures are in millions of dollars b) The total 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 in each Provinces' a/c (as well as in 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, including corporate issues, they t o t a l about $10 b i l l i o n . Apart from s i z e of outstanding issues, another factor contributes to the interest in the study of retractable and extendible bonds. These bonds have an option attached to the ordinary bond. This makes the i r valuation by conventional methods ad hoc, and p a r t i c u l a r l y amenable to valuation in the option p r i c i n g framework. Clearly, retractable and extendible bonds are i n t e r e s t i n g instruments, and a detailed study of them i s quite i n order. 1•4 Outline of th.e Thesis Chapter 2 develops the basic bond valuation equation i n terms of the parameters of the l o c a l i n t e r e s t rate process. The appropriate boundary conditions relevant to the p r i c i n g of retractable and extendible bonds are derived. The approach to incorporating 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 into the valuation model i s b r i e f l y outlined. An approximate approach to account for taxes (along the l i n e s of Ingersoll [38]} i s also presented. The stochastic s p e c i f i c a t i o n of the short term interest rate process i s central to the bond valuation model. Chapter 3 addresses the desirable properties that any mathematical model of t h i s process should possess. A s p e c i f i c d i f f u s i o n equation i s suggested to model interest rates, and the properties of t h i s s p e c i f i c a t i o n are investigated. Having s p e c i f i e d the form of the i n t e r e s t rate process, the next problem i s that of estimating i t s parameters, given data on a r e a l i z a t i o n of the process, Methods for estimating the 8 parameters are examined i n Chapter 1 . Starting with a b r i e f review of the e x i s t i n g l i t e r a t u r e on the estimation of parameters of Markov and d i f f u s i o n processes, three d i f f e r e n t methods of estimating the parameters are proposed. The d e t a i l s of the estimation procedure for each of these methods are also presented., Chapter 5 i s devoted to the comparison of the three methods of estimation proposed i n the previous chapter. For t h i s , Monte Carlo methods are used to examine the d i s t r i b u t i o n of the estimated parameters by each method, under d i f f e r e n t conditions, 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 valuation, i s also b r i e f l y investigated since our primary concern i s to use the estimates to value retractable and extendible bonds. The chapter concludes with a b r i e f look at the i n t e r - r e l a t i o n s between the estimated parameters, as well as the way i n which they af f e c t the inte r e s t rate process. Details about the data sample on short term i n t e r e s t rates and bond prices are given in Chapter 6. Chapter 7 reports the empirical tests of the models developed i n Chapter 2. We s t a r t with the bond valuation model based on the pure expectations hypothesis. We then incorporate a s p e c i f i c form of ter m / l i q u i d i t y premium. The estimation of the investor preference parameters i n the assumed form of the te r m / l i g u i d i t y premium expression i s addressed and estimates of these parameters, based on a sample of non-callable coupon bonds, are presented. These estimates are incorporated i n the bond valuation model and the resultant bond values are compared with 9 market prices. The e f f e c t on the bond valuation model of incorporating taxes (both revenue taxes and c a p i t a l gains taxes), i s investigated. Tests of market e f f i c i e n c y based on the returns to a zero-investment p o r t f o l i o are conducted. In t h i s section, the a b i l i t y of the di f f e r e n t models to i d e n t i f y over priced bonds i s also investigated using an approach based on Galai [29]. F i n a l l y , an ad hoc, regression based valuation model (the "naive" model) for retractables and extendibles i s developed. Using the sample of non-callable coupon bonds, the required c o e f f i c i e n t s for the "naive" model of retractables and extendibles are estimated., The performance of t h i s model i n predicting bond prices i s b r i e f l y compared with that of the models developed e a r l i e r i n Chapter 2.» The study concludes in Chapter 8 with a summary of the pr i n c i p a l r e s u l t s , and some remarks about the choice of the stochastic s p e c i f i c a t i o n for the in t e r e s t rate process, as well as about the model of bond valuation.. Suggestions for further research i n related areas conclude the study. 10 CHAPTER 2: THE PRICING THEORY OF DEFAULT FREE BONDS 2.1 Determinants of Bond Value The approach to the valuation of retractable and extendible bonds w i l l closely follow the method set out i n Brennan 6 Schwartz [10].. B a s i c a l l y , the value of any default free bond i s the present value of i t s p r i n c i p a l and coupon payments. The future cash flows are known with certainty, once the coupon rate and time to maturity are s p e c i f i e d . Knowing the future cash flows, what i s required to a r r i v e at t h e i r present value (ie. the bond value) i s a s u i t a b l e discount factor. A natural choice i s the short term i n t e r e s t rate.. In a model where we recognize that i n t e r e s t rates are stochastic, we could evaluate the present value over a l l possible future sample paths of the i n t e r e s t rate, over the terra of the bond. Following t h i s l i n e of reasoning, we could j u s t i f y the assumption that the price of a default free bond may be represented as a function of the short term interest rate and the time to maturity. Since there i s some uncertainty associated with the assessment of future spot rates, i n a market where r i s k averse investors e x i s t , term premia enter the valuation equation via the s p e c i f i c assumptions made about the term structure of interest rates. To model the future course of the spot i n t e r e s t rate, we assume that i t i s a stochastic process with a continuous sample path and Markov properties. Under the Markov assumption, the future development of the spot rate process, (given i t s present value) i s independent of the past development that has led to the present l e v e l . Processes that are Markov and continuous are 11 c a l l e d d i f f u s i o n processes, and for the one dimensional case can i n general be described by a stochastic d i f f e r e n t i a l equation of the form dr bCr.t} <&> -f dl (2.1)., where b ( r , t ) , and a 2 ( r , t ) represent the instantaneous d r i f t and variance respectively of the process, and dz i s the driving stochastic element and i s distributed as H(0,dt). For the present, there i s nothing to be gained by r e s t r i c t i n g the generality of the above stochastic d i f f e r e n t i a l equation governing the i n t e r e s t rate process. However, i t may be noted that both b(r,t) and a(r,t) must at l e a s t be known, deterministic functions of time - they may not be stochastic functions of time 2.; He s h a l l however r e s t r i c t our attention to a p a r t i c u l a r family of processes, when we address the inte r e s t rate process in greater d e t a i l l a t e r on. The main competing theories about the term structure of in t e r e s t rates are a) the pure expectations hypothesis 2 In the standard option valuation framework, there 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 underlying asset (the stock), i e . that i t should be non-stochastic. This i s because, the f i n a l parabolic p a r t i a l d i f f e r e n t i a l equation governing the option value does not contain the d r i f t term. For the bond, the corresponding p a r t i a l d i f f e r e n t i a l equation i s equation (2.9). The instantaneous d r i f t of the i n t e r e s t rate process (the underlying asset being the pure discount bond due to mature the next instant) enters the valuation equation. I f either b(r,t) or a(r,t) i n equation (2.1) were stochastic, then the valuation equation would no longer be an ordinary second order parabolic p a r t i a l d i f f e r e n t i a l equation. 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 d e f i n i t i o n of the pure expectations hypothesis that we adopt i s that the instantaneous expected return on bonds of a l l maturities i s the same3. This implies some sort of " r i s k n e u t r a l i t y " on the part of investors over the instantaneous holding period returns across bonds of a l l maturities. The second hypothesis argues that concern over fluctuations i n wealth causes investors to demand a " l i q u i d i t y " premium on long term bonds over those of shorter maturity. On the other hand, concern over fluctuations i n income leads to a case for term premiums that would obviously have just, the opposite pattern., The market segmentation hypothesis proposes that bonds of di f f e r e n t maturities are t o t a l l y d i f f e r e n t instruments, and thus not substitutable. This would require that the term structure of int e r e s t rates, at any point i n time, be defined by the 3 What follows i s based on Cox, Ingersoll & Ross [16], In the ex i s t i n g l i t e r a t u r e , the pure expectations hypothesis i s characterized by one of the following propositions: 1) Implied forward rates are equal to expected future spot rates 2) The y i e l d to maturity from holding a long term bond i s equal to the y i e l d from r o l l i n g over a series of short term bonds 3) The expected return over the next holding period from bonds of a l l maturities i s equal Under certainty, a l l three forms are equivalent., With uncertainity, however, Cox, Ingersoll S Ross have shown that the f i r s t two propositions are consistent 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 structure i s unbiased in the sense of the f i r s t two propositions, then the instantaneous expected rate of return on any bond must exceed the spot rate. 13 supply and demand for each of the number of maturities existing i n the market at that time. Most studies of the term structure of i n t e r e s t rates i n the option p r i c i n g framework, { Brennan & Schwartz [10]; Cox, Ingersoll S Ross [16], Vasicek [72] and Richard [58]), have considered only the pure expectations or term/liquidity premium assumptions. Brennan & Schwartz [12], have t r i e d to operationalize a form of the market segmentation hypothesis, by introducing two factors i n the maturity structure - the very short end, and the long term maturity,, Only incorporation of the f i r s t two hypotheses about the term structure of i n t e r e s t rates i n t o the bond valuation models i s considered in t h i s study. 2.2 The Basic, Bond Valuation, Equation Let us represent by B ( r # t ) , the value of an ordinary bond which pays $1 at maturity; where r i s the spot r i s k l e s s i n t e r e s t rate, and X the time to maturity. S i m i l a r l y , l e t the value of a retractable or extendible bond be G(r/£). For purposes of generality, l e t B(r,l) pay a coupon* c, , and G(r,T) a coupon cz» Then, using Ito's L e n a (McKean [45]) and equation (2.1) for the i n t e r e s t rate process, the straight bond B, and the generic bond G, are governed by the following stochastic d i f f e r e n t i a l equations (SDE) : * For ease of computation i n a continuous time framework, we assume that these are continuous coupons. A continuous coupon of c means a coupon payment of c d o l l a r s per unit of time per bond.. As pointed out i n Chapter 6, t h i s assumption i s quite reasonable. 14 (2.2) where b=b(r,t) and a=a(r,t), and subscripts denote p a r t i a l derivatives; B, i s the f i r s t p a r t i a l derivative of the bond price with respect to i t s f i r s t , argument - the spot r i s k l e s s i n t e r e s t rate, etc. The spot r i s k l e s s i n t e r e s t rate i s , by d e f i n i t i o n , the y i e l d to maturity on a default free discount bond due to mature tike next instant in time., The return on a l l three assets, v i z . , the generic bond> the straight bond and the short term inte r e s t instrument, have the same stochastic element d r i v i n g them (dz); i e . , they are a l l perfectly correlated. I f borrowing and lending at the instantaneously r i s k l e s s rate of i n t e r e s t were possible (and a l l the other assumptions of the option p r i c i n g model h e l d s ) , a zero net investment p o r t f o l i o could be formed using the above three s e c u r i t i e s . Consider an investment of x, d o l l a r s i n G, x% dollars in B and xi = - (x, *-xz) d o l l a r s i n the r i s k l e s s asset. The return on such a p o r t f o l i o i s given by 3 The perfect market assumption i s implied with a l l the attendant properties of unlimited borrowing/lending at the r i s k l e s s rate by a l l investors, no margin reguirments on short sales and immediate f u l l a v a i l a b i l i t y of proceeds of short s e l l i n g and a b i l i t y to trade every instant at current prices, and f i n a l l y the absence of a l l taxes., 15 Rewriting equation (2.2) as ft (2.4) we can rewrite(2.3) as (2.5) We can see from equation (2.5) that a l l uncertainty from the return on the zero investment p o r t f o l i o would be eliminated i f we choose x, and x 2 such that the c o e f f i c i e n t of dz i s zero, i e . , X z •= <%_ = - * i . J _ _ (2.6) Arbitrage would now drive the certain return on the zero net investment p o r t f o l i o to zero. Substituting (2.6) i n t o (2.5) gives the basic valuation equation. ( / V V Q - „ ( A + C ' /B ) - r (2.7) This expression has to hold f o r bonds of a l l maturities at any point i n time. I t i s the f a m i l i a r expression of excess return per unit of r i s k on each security (see Cox £ Ross [ 17 ]). We may 16 represent the price of instantaneous standard deviation risk by (r,t) , noting that, whereas i s independent of the time to maturity, i t may change over time and with the spot rate. This gives ( / W < 0 ~ R - » UT.t.t) (2.8). where \{t,t0t) represents the term or l i q u i d i t y premium, i e . the excess instantaneous return at time t on a bond with time to maturity T . Substituting for jl^ and 0£ from (2.4), yields the p a r t i a l d i f f e r e n t i a l equation for the bond price -la<qu + Cb-^)(5( -T<5+CL-<qt =0 (2.9) Thus, in equilibrium, any bond follows the same valuation equation (2.9). What distinguishes them, are the boundary conditions that each has to s a t i s f y . (This r e s u l t was f i r s t demonstrated by Brennan & Schwartz [ 10 ]). 2. 3 Boundary, Conditions for Retractable/Extendjble Bonds Let us now consider the boundary conditions that the generic bond has to s a t i s f y . a) Terminal value at maturity: From the default free aspect, the p r i n c i p a l of $1 i s guaranteed at maturity. Thus i r r e s p e c t i v e of the current i n t e r e s t rate at maturity, the bond value equals i t s face value, i e . , 17 G (r,o) = 1 (2.10a) b) Retraction/extension feature: Here, we s h a l l consider three types of retraction/extension features and develop the appropriate boundary conditions applicable to the bond valuation equation i n each case; i) the retraction/extension option has to be exercised at a single point i n time. i i ) the option may be exercised over a period of time. i i i ) the option to retract/extend may be exercised over a period of time, but even i f the decision i s to retract, (or not to extend, i n the case of an extendible) the face value of $1 i s available only on a f i x e d future date beyond the f i n a l exercise date. These three cases may diagramatically represented as: r — 1 1 1 VkrruxL wh<n cf>tum Sh«t k 0 1^ The f i r s t case above would correspond to the s i t u a t i o n where TQ.( I a n <* coincide at one point. For the second case, t 5 i s not a fixed point beyond , but could be any point between %i and t € t depending upon the bond holder's choice. To derive the boundary condition for each case, i t would be h e l p f u l to consider an example, Consider that an investor holds a 5% coupon bond, which he may extend on (say) January 1st, 1970 f o r a 6% coupon bond maturing January 1st,1975. In case the investor does not choose to accept the new bond of January 1st, 1975 maturity, the old bond may be cashed i n f o r $1 on January 18 1st, 1970. Clearly on any day prior to January 1st, 1970, the holder of the short bond has a European c a l l option on the 6% Jananuary 1st, 1975 bond with an exercise price of $1. Let us now represent by t = 0, the maturity date of the long bond, i e . January 1st, 1975, and by le. , January 1st, 1970 (the option expiry date). Let ^ e represent the instant in time just prior to the decision point, and te represent the instant i n time just a f t e r the decision point. Then we have S C T X ) - Max [ $C*X*) , 1 (2.10d.1) The condition above implies that the bond value, i f the bond i s not cashed i n at the decision point, i s continuous across that point in time. In case, however, the option to extend could be exercised over a period of time, rather than at a point in time ( c a s e ( i i ) ) , condition (2.10d.1) would be altered as: (2.10d.2) Here the f i r s t condition i s that during the period the extension option i s in force, the value of the bond i s bounded below by the par value of $1. This i s the arbitrage condition as the holder has an American option. Further, since i t has to be continuous across the expiry point of the option, we have the second condition, as before. For actual bonds i n the market, case ( i i i ) i s the 19 representative case. The option to extend/retract may be exerciseable over a 3 to 6 month period, but, even i f the option were exercised, the par value i s generally available only a further 6 to 12 months l a t e r . Going back to our example, Xs ~ January 1st, 1970 and we may now represent T«i as (say) July 1st, 1969, and rcj_ as October 1st, 1969. C l e a r l y , i f the investor decides to choose the short bond at any time, between July 1st, 1969 and October 1st, 1969; the p r i n c i p a l of $1 i s available only on January 1st, 1970. I t i s c l e a r l y optimal to exercise the option at the l a s t point, fe2. , and so we have the boundary condition there as: <qCT,'E«+0 ' M<W^ <qtr,ttl) t H(T(r€;-ts) <2.10d.3) In the condition above, G represents the value of the long term bond. The short term bond has been represented by H, to e x p l i c i t l y recognize that the coupon of the two bonds could be d i f f e r e n t . c) Value at the i n t e r e s t rate boundaries: We know from the previous section that the interest rate process and the bond value process are very c l o s e l y related. From economic considerations, we require i n t e r e s t rates to remain non-negative. Whether t h i s requires the imposition of s p e c i f i c conditions at the i n t e r e s t rate boundaries (r=0 and co ) i s investigated i n the next chapter. We therefore postpone developing conditions that the bond value process has to s a t i s f y at r=0 and oo t i l l l a t e r . For the present, we just note that the conditions imposed on the bond value process at the 20 boundaries of the i n t e r e s t rate process should be consistent with the behaviour of the int e r e s t rate process at these boundaries. In general, the d i f f e r e n t i a l equations (along with the attendant boundary conditions) governing the retractable/extendable bonds, cannot be solved a n a l y t i c a l l y . Numerical f i n i t e difference methods w i l l be used to solve the equations. The general procedure i s to develop the solution recursively backwards from the boundaries, where the solution i s known. This i s addressed further i n Chapter 7. 2.4 Incorporatinq Taxes into the Model So f a r the model has been developed on the assumption of no taxes, either on revenues (coupons and interest) or c a p i t a l gains. He could attempt to incorporate taxes into the valuation equation, along the l i n e s of Inger s o l l [38], but the following assumptions need to be made e x p l i c i t : a) Taxes are assumed payable on a continuous basis and at a fixed rate. This implies that there i s some "average" tax rate over a l l investors that could be used in the model. The assumption further implies that i n t e r e s t payable on a l l borrowings i s tax deductible. b) A l l c a p i t a l gains are treated as taxed at the c a p i t a l gains tax rate, and payable continuously. In r e a l i t y , c a p i t a l gains taxes are paid only when gains are actually realized by a sale. Further, any c a p i t a l gain over a period of les s than 91 days i s treated for tax 21 purposes as a revenue item. In our model however, we cannot make t h i s d i s t i n c t i o n 6 . The assumptions may be r e s t r i c t i v e , but i t i s an empirical question as to whether i t i s better to ignore taxes altogether, or incorporate them into the valuation equation with the current assumptions - a question that i s addressed l a t e r . Let us represent by R, the rate of taxes on revenues and by T, the rate of taxes on c a p i t a l gains. The return on the zero investment p o r t f o l i o , as given i n equation (2.3) i s modified to The same analysis as before leads to the valuation equation which leads to the following 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, + 0 - R ) ( c 2 ~ f ^ ) - 0 - T ) ^ 2 0 (2 .11) The boundary condition associated with t h i s equation are exactly those associated with the previous equation (2.9). * This assumption i s required to ensure an unique equilibrium bond value. Given our continuous time hedging approach to valuation, c a p i t a l gains as per the existing tax laws are never applicable. Capital gains taxes do ex i s t , and are accepted as one of the determinants of investors choice among available s e c u r i t i e s . The present approach i s one way of incorporating t h i s r e a l i t y into our model. 22 CHAPTER 3: THE INTEREST RATE PROCESS 3.1 Properties of Interest Rate Processes In the previous chapter, we l e f t the stochastic s p e c i f i c a t i o n of the in t e r e s t rate process in a very general form. Lacking a well developed theory of growth under uncertainty to specify a functional form for the int e r e s t rate process, (the only work addressing the problem appears to be Herton [ 4 9 ] ) , we are l e f t to draw upon functional forms that s a t i s f y some very broad c r i t e r i a 7 . a) Interest rates should never become negative, as holding wealth i n the form of cash dominates such a si t u a t i o n . b) An interest rate process should possess some cen t r a l tendency, i e . , one would not expect the spot rate of interes t to r i s e to some high l e v e l , and yet be equally l i k e l y to go further up, as move downwards. c) Preferably, the process should be such that the probability of the intere s t rate reaching either 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 t r a c t a b i l i t y . To ensure that inte r e s t rates do not become negative, we could adopt one of two approaches: a) make r=0 a singular boundary 8 with pos i t i v e d r i f t , i e . 7 These c r i t e r i a are drawn from Ingersoll £ 3 9 ] . 8 By d e f i n i t i o n , the d i f f u s i o n process as defined by equation (2.1) has singular boundaries wherever b(r,t)->°o or a{r,t)->0. 23 b(0,t) > 0; a(0,t) = 0. This implies that once the inter e s t rate reaches zero, i t changes only i n 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 imposing an a r t i f i c i a l barrier at r=0. The second approach i s more straight forward, A r e f l e c t i n g barrier at r=0 ensures that the interest rate never becomes negative, and further, i t never remains at zero, except for an i n f i n i t e s i m a l instant. However, once r reaches zero the direction of i t s change the next instant i s known with certainty - since r cannot become negative (due to the r e f l e c t i n g barrier) i t can only increase. This would appear to present a cl e a r arbitrage opportunity; a s i t u a t i o n not consistent with market e f f i c i e n c y in a continuous time framework. However, no arbitrage p r o f i t opportunity need exi s t i f the bond valuation model i s made to s a t i s f y suitable boundary conditions at r=0 9 . Though i t may 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 ensure that i f the i n t e r e s t rate reaches zero, i t w i l l leave i t and enter the posi t i v e region again. The behaviour of the process at a singular boundary cannot be infe r r e d by i n t u i t i o n alone. 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 zero, and i s f i n i t e . Further, since the interest rate process and the bond value process have to be perfectly correlated, the bond value should also have a r e f l e c t i n q barrier at r=0. From the standard r e f l e c t i n g b a r r i e r condition (see Cox S H i l l e r [ 15 ]) , t h i s requires that B, =0. The instantaneous return to holding the bond thus becomes ce r t a i n , as B(=0 reduces the c o e f f i c i e n t of dz to zero. To ensure that no arbitrage opportunity e x i s t s at r=0, the c e r t a i n return to holding the bond should also be zero. Thus we require 24 f u n c t i o n a l form that has a singular boundary, we must investigate the behaviour of the process at the singular point more rigorously, before we can judge the a c c e p t a b i l i t y 1 0 of the functional form of the stochastic s p e c i f i c a t i o n . F e l l e r [25] has studied the problem of characterizing the behaviour of a d i f f u s i o n process at i t s singular boundaries, by the method of semigroups. (A s i m p l i f i e d and somewhat more readable exposition of F e l l e r * s work may be found i n Keilson [ 41 ])..... Broadly speaking the behaviour of a d i f f u s i o n process at a singular boundary could be characterized as one of the following: a) Natural: The boundary i s inaccessible in f i n i t e time from any s t a r t i n g point i n the i n t e r i o r . I t i s inter e s t i n g to note that a natural boundary can be both inaccessible and absorbing ( i e . as i n the case of the lognormal process, where zero i s both inaccessible and absorbing). b) E x i t : the boundary i s accessible i n f i n i t e time and once the process reaches the boundary, i t i s absorbed. c) Entrance: the boundary i s inaccessible i n f i n i t e time from the i n t e r i o r , but i f the process started from the boundary, i t would leave and enter the i n t e r i o r i n f i n i t e time. d) Regular: the singular boundary i s accessible, and we io From economic considerations, i t is undesireable to have r = 0 as an absorbing boundary, ie. once the interest rate reaches zero, i t never leaves i t . 25 can further specify the behaviour i t should exhibit there (ie. absorbing, r e f l e c t i n g , etc.) by imposing suitable boundary conditions. 3.2 The Interest Rate Process Keeping the above requirements in mind, l e t us consider the following stochastic s p e c i f i c a t i o n . &i~ ^ m(jii--r) dt + <rr* di (3.D F i r s t note that the parameters are not time dependent. This assumes s t a t i o n a r i t y of the int e r e s t rate process over time. Though some realism i s l o s t , considerable a n a l y t i c a l t r a c t a b i l i t y has been gained., The process has the mean reverting property, because when r> (< jx) , the d r i f t i s negative ( p o s i t i v e ) , so that the deterministic movement of the intere s t rate i s always towards JUL the central tendency. The parameter m controls the speed of adjustment towards . To see t h i s , consider only the non-stochastic part of the process f o r the moment: dl~ ~ - m c£t On integration we have which shows that the larger m, the more rapid the reduction of the distance of the current value of 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 the stochastic term, we f i n d that r=0 i s a singular boundary 1 1. Further, we want <A >0, as negative <A makes a (r,t) -> °o as r->0, which i s an undesirable r e s u l t . Again making the variance term not only a function of r, but introducing two free parameters ( cr , <A) , adds to the richness of the family of the in t e r e s t rate process. 3.3 interest Rate Process Behaviour jit Singular Boundaries Since r=0 i s a singular boundary, we need to investigate the behaviour of the process at r=0 (as well as at r= <*>) , This i s set out i n Appendix 1. The r e s u l t s may be b r i e f l y summarized as follows: 1) The process corresponding to <A = 1/2 has been studied extensively by F e l l e r [23] and his r e s u l t s are a) For a l l parameter values, r= oo i s an inaccessible boundary. ,. b) At r=0; when m,/^ >0, the boundary can be either an absorbing or r e f l e c t i n g b a r r i e r when 2mjx < <r2. When 2mu„ £ r 2 , r=0 i s an entrance boundary., 2) In case c< =1, we f i n d that both r=0 and r= <£> are natural boundaries. 3) It was not possible to investigate the behaviour at the singular boundary for a r b i t r a r y values of c* as the necessary inte g r a l s could not be evaluated (see Appendix 1) . By continuity of behaviour, we conjecture 1 1 r= oO i s also a singular boundary. 27 that as crt reduces, and 2 m c o r r e s p o n d i n g l y increases in r e l a t i o n to <r 2 , there w i l l exist, a region i n the parameter space where r = 0 i s not an absorbing boundary 1 2. 1 2 The boundary behaviour of the process for values of o( # ^ or 1 i s currently being further investigated j o i n t l y with Kent Brothers and David Emanuel. The preliminary r e s u l t s seem to i n d i c a t e that <A = '/z i s the only i n t e r e s t i n g case, where we can have either an accessible or inaccessible boundary at r=0, depending upon the values of the parameters. The indications are that for 0< < '/a, r=0 i s always accessible, and for ' / x < ° < < 1 , r=0 i s always inaccessible. 28 CHAPTER 4: ESTIMATING THE INTER EST RATE PROCESS PARAMETERS 4. 1 Brief Review of Published Research - i n • Related Areas • The inte r e s t rate process specified i n the previous chapter has a continuous sample path over time. However, we have a record of i t s r e a l i z a t i o n only at discrete i n t e r v a l s i n time, say daily'or weekly observations. The problem that we s h a l l now address i s the following: Given a set of data points (r^ , t=1,...T)> which are observations on the i n t e r e s t rate process at discrete i n t e r v a l s , what procedure does one adopt to estimate the parameters frn r jx*(T r <A*\ corresponding to our stochastic s p e c i f i c a t i o n of the previous chapter (equation 3.1). In general, when we have a sequence of r e a l i z a t i o n s of independent random variables which are i d e n t i c a l l y d istributed according to some probability measure P^ , which depends on an unknown parameter Q- ranging over a parameter space & , methods for obtaining estimators for P# or ? , respectively, with desirable large sample properties are well known. These methods have been generalized to stochastic processes by several researchers (for an extensive survey of the l i t e r a t u r e see B i l l i n g s l e y [3,4]). For Markov processes with stationary t r a n s i t i o n p r o b a b i l i t i e s 1 3 these generalizations are carried out i n such a way that the Markov kernel now plays the same role as the pr o b a b i l i t y measure i n the case of independent i d e n t i c a l l y * 3 If we represent the t r a n s i t i o n probability by P (rt")> t \zs ',s) , t>s, then s-tationarity of s the t r a n s i t i o n p r o b a b i l i t y requires that P ( r t , t | r 5 , s ) = P(r^,u|r v,v) for a l l (u-v) = (t-s) . This i s the time homogeneity condition. 29 d i s t r i b u t e d random variables. In p a r t i c u l a r , B i l l i n g s l e y [3] shows that maximum like l i h o o d estimates based on the above approach exhibit almost a l l the properties of s i m i l a r estimates i n the independent random variable case. (See also Roussas {59] f o r properties of maximum lik e l i h o o d estimators for Markov processes with discrete time and state s p a c e 1 4 ) . Much of the l i t e r a t u r e on s t a t i s t i c s of d i f f u s i o n processes (ie. continuous time stochastic processes) has addressed what i s c a l l e d the problem of optimal non-linear f i l t r a t i o n . This i s i n the area of e l e c t r i c a l communications, where we have a sig n a l (a stochastic process) which i s unobservable. What i s observed however, i s a "distorted" transformation of the s i g n a l , and from i t inferences are to be made about the underlying sig n a l . There i s a large body of l i t e r a t u r e ; papers of p a r t i c u l a r i n t e r e s t are Sirjaev [ 64], Ganssler [30] and some of the references cited therein. Though there i s nothing s p e c i f i c i n the l i t e r a t u r e c i t e d above that has a d i r e c t bearing on the problem of estimation of parameters of the d i f f u s i o n process set up in the previous chapter, Sirjaev £64] proves that the maximum lik e l i h o o d estimators of parameters in the d r i f t term of any d i f f u s i o n process are biased in small samples (though asymptotically unbiased). He shows that obtaining closed form expressions for the small sample bias for general forms of the d i f f u s i o n equation i s a very d i f f i c u l t problem. I t appears that 1 4 Kendall 6 Stuart [42] have also shown that the ML estimators are consistent though generally biased. The asymptotic ' normality of the estimators i s also shown by Anderson S Gocdman [1]. Lee, Judge & Zellner [43] provide good coverage of the area of empirical estimation for the discrete state space process. 30 Novikov [52] has investigated the estimation of the parameter i n the process dx = - X-xdt + dz and found the r e s u l t i n g bias i n X This i s the Omstein-Uhlenbeck'- 5 process, nowhere as general as the process outlined in the previous chapter for the interest rate process. Ganssler [30] shows that i n the case of stochastic processes which do have a unique stationary d i s t r i b u t i o n (we s h a l l say more about stationary d i s t r i b u t i o n s s h o r t l y ) , using the density function of the stationary pr o b a b i l i t y d i s t r i b u t i o n to set up the joint l i k e l i h o o d of a given set of observations instead of the Markov kernel;, i n conjunction with the" minimum-distance-method of Wclfowitz [73,74], leads to consistent parameter estimates. I t was, however, pointed out by Ganssler [30] that using the stationary d i s t r i b u t i o n may, in general, not lead to the complete i d e n t i f i c a t i o n of a l l the parameters i n the Markov kernel. In conclusion, i t appears that the e x i s t i n g l i t e r a t u r e on estimation of parameters of diffusion equations does not contain any s p e c i f i c r e s u l t s that could be brought to bear upon the estimation problem facing us. F i n a l l y , one l a s t area that was b r i e f l y surveyed was the l i t e r a t u r e dealing with genetics. F e l l e r [24] indicated that a d i f f u s i o n equation of the form (3.1) with c{ =J4. resulted by is s ee Cox S M i l l e r [15] 31 taking the discrete time b i r t h and death process to i t s appropriate continuous time l i m i t s . It was therefore f e l t that there could possibly have been some empirical work on estimating the parameters of b i r t h and death processes, the r e s u l t s of which could be brought to bear upon our s p e c i f i c problem. Unfortunately, none of the published works addressed the problem i n a continuous time framework. The only two papers of any i n t e r e s t are Immel [ 36 "J and Darwin f" 181. Both address the discrete parameters case only, but they adopt the approach of using the t r a n s i t i o n probability function for s e t t i n g up the j o i n t l i k e l i h o o d function, given a r e a l i z a t i o n of the process. 4. 2 Maximum Likelihood (M.L. ) Method of Estimation:. From the above, we see that there i s some support i n the l i t e r a t u r e for the M.L. approach to estimation. As pointed out by B i l l i n g s l e y f3,4] and others, the desirable asymptotic properties of M.L. estimates can be b r i e f l y stated as follows: a) The estimators are asymptotically unbiased. b) They are consistent. c) The inverse of the Hessian matrix with signs reversed i s a consistent estimate of the asymptotic variance-covariance matrix of the parameters, where the asymptotic joint d i s t r i b u t i o n of the estimated parameters i s multivariate normal. Given a sequence (r^. ,t=1,.,.T) of observations on the short term i n t e r e s t rate, the j o i n t likelihood function can be 3 2 set up as T t((Uo.\8) s T l P ^ e l - ^ . e ) . P.OVj (4.1) where P( r\ j r , , 0 ) represents the t r a n s i t i o n probability density, and P0 (r, ) i s the p r o b a b i l i t y corresponding to the i n i t i a l point of the sample. $ here represents the parameters of the d i f f u s i o n process - i n our case [ yn , jx., o~~, °^ ] . Two points need to be noted about the pr o b a b i l i t y density expressions i n (4.1) : a) The t r a n s i t i o n probability density i s assumed to be time homogeneous. This is quite v a l i d , given the assumption i n the previous chapter that the d i f f u s i o n equation modelling the interest rate process displays 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 i s that the observations fr^} are equally spaced over time. This poses no r e a l problem, as i n economic data observations are generally eguispaced. The joint l i k e l i h o o d of the data contains the term corresponding to the i n i t i a l point which poses problems with further analysis. In general, several arguments may be put forward to drop the expression corresponding to the sta r t i n g point in the joint l i k e l i h o o d of the data: a) Hhen we have a reasonably large data sample, the contribution of the i n i t i a l point may be considered n e g l i g i b l e in comparison to the rest of the points and 33 may be dropped (see B i l l i n g s l e y [ 3 ] ) . In fact a l l the estimation theory results are asymptotic r e s u l t s , and large sample sizes are i m p l i c i t l y assumed. b) I t i s not uncommon in several s i t u a t i o n s to treat the estimators as s t r i c t l y conditional upon the sample. Following such an approach, we could argue that the estimators are conditional upon the i n i t i a l point, and thus at t r i b u t e a probability of 1 to that point. c) F i n a l l y , Zellner C 7 61 r easons 1 6 that we may assume that the probability corresponding to T | i s t o t a l l y independent of &! . Since our interest i s only i n estimating & , i t can be e a s i l y shown that the d i s t r i b u t i o n of [V i s unaffected by dropping the i n i t i a l point. In view of the above arguments, we s h a l l drop P0 (r () from ( 4 . 1 ) . To set up the joint l i k e l i h o o d function, we need to ascertain the t r a n s i t i o n probability density for the d i f f u s i o n process E(dz) = 0 and E (dz 2) = dt. In general dz.is assumed to be a * cr r - dz (4. 2) Gauss-Weiner process, i e . 1 6 z e l l n e r * s reasoning i s for the analysis of f i r s t order autoregressive systems i n a Bayesian framework. 3 4 4. 3 The Simple Linearization A£j3roximation The s p e c i f i c a t i o n of equation (4.2) suggests a very simple estimation procedure, by l i n e a r i z i n g the d i f f e r e n t i a l s to f i n i t e (discrete) differences. Thus we have and i f we now choose our unit of time such that A t = 1 (the frequency of the observations on r) we have where ' Y| /\> N (0,1) . In the l i m i t as At 0, the approximation (4.4) as a characterization of the d i f f u s i o n equation (4.2) becomes exact. However, the further apart the observations on r are, the greater the error. The extent of the error due to t h i s approximation i s investigated by Monte Carlo methods i n the next chapter. For the present, however, we see that the approximation (4.4), closely resembles a regression equation, i e , , we have a l i n e a r regression of r^ on r ^ " , wherein we have a heteroscedastic error term. Thus we have (4, 5) Given the data, we can now set up the l i k e l i h o o d function as i n (4.1). The d e t a i l s of the estimation procedure are set out in Appendix 2. 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 D e n s i t y Method The e x a c t a p p r o a c h would be t o a s c e r t a i n t h e 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 use i t 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 ( e g u a t i o n 4 . 1 ) . I t i s w e l l known i n p r o b a b i l i t y t h e o r y t h a t c o r r e s p o n d i n g t o e v e r y d i f f u s i o n e q u a t i o n , t h e r e e x i s t two e q u a t i o n s t h a t 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 has to s a t i s f y . These a r e the Kolmogorov backward e q u a t i o n and t h e Kolmogorov or F o k k e r - P l a n k (FP) f o r w a r d e q u a t i o n . The s o l u t i o n t o t h e FP e q u a t i o n i s t h e 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 c o r r e s p o n d i n g to t h e d i f f u s i o n ' e q u a t i o n 1 7 1 8 . T h u s , f o r our c a s e o f t h e d i f f u s i o n g i v e n by (4.3) the FP e q u a t i o n i s -JL^()t -T)FJ ^JL[<^FJ ^ 2 £ (4 .6) t h e 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 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 1 7 The e x i s t e n c e o f unique s o l u t i o n s t o t h e f o r w a r d (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 terms 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 c o n t i n u i t y r e q u i r e m e n t s (see F r i e d m a n [ 2 8 ] ) . More s p e c i f i c a l l y , i t i s r e g u i r e d t h a t 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 ) i n compact s u b s e t s o f R 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 over the whole domain. 1 8 I t i s a w e l l known r e s u l t (see F e l l e r [26]) 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 , e x c e p t i n r a r e s i t u a t i o n s 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 Observed i n t h e l i t e r a t u r e t h a t the s o l u t i o n a l s o p o s s e s s e s the p r o p e r t i e s o f a 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 e . the f u n c t i o n i s s t r i c t l y n o n n e g a t i v e o v e r the s t a t e s p a c e , and i t s i n t e g r a l over t h e s t a t e s p a c e <1 (these a r e the Chapman-Kolmogorov c o n d i t i o n s ) . I f the e q u a l i t y i s s a t i s f i e d , t h e s o l u t i o n to t h e FP and backward e q u a t i o n i s u n i q u e , but i n g e n e r a l , d i f f e r e n t d i f f u s i o n p r o c e s s e s may s a t i s f y the same f o r w a r d and backward e g u a t i o n s . where F = P^r vt)r 0 , 6 ) i s f u n c t i o n . To s o l v e t h i s 36 equation, we need to impose boundary conditions at r=0 and i n f i n i t y ( i f r=0 and i n f i n i t y are not inaccesible boundaries) as are required on the basis of our investigation of the behaviour of the process at these singular boundaries., Unfortunately, there appears to be no closed form solution for equation (4.6) for general values of^ oC . F e l l e r [23] has studied the solution corresponding to the case <A~ - K *^ a n approximate solution to the case where cK =1, based on an approach suggested In Goel 6 Richter-Dyn [33], i s sketched in Appendix 3. When =0, the o r i g i n i s no longer a singular boundary. I f we reguire interest rates to remain non-negative, we need to impose a r e f l e c t i n g barrier at r=0. The solution to the FP equation with a r e f l e c t i n g b a r r i e r at the o r i g i n i s quite complicated, but for the unrestricted process (where a positive probability of negative i n t e r e s t rates e x i t s ) , the solution i s rather straight forward (and detailed i n Appendix 4). As pointed out by Vasicek [72], the parameters could be chosen such that the probability mass below the o r i g i n could be made a r b i t r a r i l y small, so that for a l l p r a c t i c a l purposes, r=0 i s v i r t u a l l y inaccessible. 4.5 The Steady State or Stationary Density Method We can see that solving for the exact t r a n s i t i o n p r o b a b i l i t y density may not always be possible, except by foregoing some generality in the model, i e , , r e s t r i c t i n g the values of the exponent o\ :. , He could however, substitute the stationary density into the joint l i k e l i h o o d instead of the t r a n s i t i o n p r o b a b i l i t y density. Ganssler [30] has shown that using t h i s approach i n conjunction, with the minimum distance estimation method of Wolf owitz [ 73,74 ], leads to consistent parameter estimates, which are asymptotically unbiased., The stationary p r o b a b i l i t y d i s t r i b u t i o n 1 9 i s , i n a sense, the l i m i t of the t r a n s i t i o n probability density, where the time i n t e r v a l between observation tends to'oO ., I t could be represented as The existence of an unique steady state probability d i s t r i b u t i o n i s usually assured when we have a process that has a time homogeneous t r a n s i t i o n probability d i s t r i b u t i o n . Further, for singular d i f f u s i o n processes,when we rule out those ranges of parameters where one of the singular boundaries acts as an e x i t b a r r i e r , we ensure that the stationary 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 state space, with a Dirac delta function concentrating a l l the probability mass at the exit boundary. Thus the stationary density i s given by the solution to the FP equation (0.6) by setting :r? - 0; OX, (0.7) 1 9 The stationary p r o b a b i l i t y d i s t r i b u t i o n exists_bnly f o r time homogeneous processes. Another way of representing the stationary d i s t r i b u t i o n could be as follows; Given that the di f f u s i o n process has attained i t s steady state, the stationary 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 probability of finding the process at any pa r t i c u l a r point (or interval) i n the state space at any instant. 38 t h e s o l u t i o n to which can be shown o f t h e form (see Goel & R i c h t e r - D y n [33]) POO - J L Hb\-i[ I (Pf2^ L J (4. 8) where C i s d e t e r m i n e d by the c o n d i t i o n \ P ( r ) d r = 1 , where Si. r e p r e s e n t s i n t e g r a t i o n o v e r t h e s t a t e s p a c e . Appendix 5 g i v e s t h e d e t a i l s o f e v a l u a t i o n o f t h e s t a t i o n a r y d e n s i t y . It. i s o f the form P(f+0 r tel . 1 1 - l L * 1+A J i t -where A, = 1 -2c^. I t i s a l s o shown i n Appendix 5 t h a t when we t a k e the l i m i t as A,-? 0 o r - 1 i n (4. 9c) , we g e t (-4.9A) and (4.9b) r e s p e c t i v e l y . Thus t h e s t e a d y s t a t e d e n s i t y i s c o n t i n u o u s i n . 39 G i v e n a r e a l i z a t i o n (r^. , t=1, . . . T) , we p r o p o s e t o s e t up the l i k e l i h o o d f u n c t i o n u s 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 ( 4 . 9 ) , and e s t i m a t e t h e p a r a m e t e r s by ML methods. T h e r e does net appear t o be any r e f e r e n c e i n t h e e x i s t i n g l i t e r a t u r e t o the a s y m p t o t i c p r o p e r t i e s o f such e s t i m a t o r s . We s h a l l l o o k a t t h e s e p r o p e r t i e s , based 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 r e s u l t s i n t h e next c h a p t e r . However, the a p p r o a c h may be c r u d e l y r a t i o n a l i z e d as f o l l o w s : a) One argument c o u l d be t h a t i f we have a s u f f i c i e n t l y l a r g e s a m p l e , the d i s t r i b u t i o n of the sample might r e s e m b l e the s t a t i o n a r y d i s t r i b u t i o n 2 0 . b) I f t h e sequence o f d a t a p o i n t s were i n d e p e n d e n t , , u s i n g t h e s t a t i o n a r y d i s t r i b u t i o n to s e t up t h e j o i n t l i k e l i h o o d o f t h e d a t a would be e x a c t . The c r u c i a l o b j e c t i o n i s t h a t we a r e t r e a t i n g a sequence o f dependent random v a r i a b l e s as i f t h e y were i n d e p e n d e n t . Lack o f i n d e p e n d e n c e s h o u l d h o p e f u l l y n o t a l t e r t h e v a l i d i t y of the a p p r o a c h . T h i s may be t r e a t e d as i f we 2 0 T h i s r a t i o n a l i z a t i o n can 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 f o r Markov p r o c e s s e s (see C i n l a r f 13 ] ) . C o n s i d e r a c o n t i n u o u s 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 d i s t r i b u t i o n . L e t o b s e r v a t i o n s be made on t h i s p r o c e s s , such t h a t t h e t i m e i n t e r v a l 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 d i s t r i b u t e d . The sequence o f o b s e r v a t i o n s then r e p r e s e n t s a d i s c r e t e t ime Markov p r o c e s s . I t can be shown t h a t t h i s d i s c r e t e t i m e p r o c e s s has the same s t a t i o n a r y d i s t r i b u t i o n as the c o n t i n u o u s t i m e p r o c e s s from which t h e o b s e r v a t i o n s were t a k e n . Rs t h e number of o b s e r v a t i o n s goes t o i n f i n i t y , t h e d i s t r i b u t i o n o f t h e sample o b s e r v a t i o n s a p p r o a c h e s the s t a t i o n a r y d i s t r i b u t i o n . The e x p o n e n t i a l s a m p l i n g scheme was r e q u i r e d to e n s u r e t h a t a l l p o i n t s on t h e h a l f r e a l l i n e 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 t o be c h o s e n . 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 i n D y n k i n £ 2 1 ] . , We have used e q u i s p a c e d o b s e r v a t i o n s , and t h a t s h o u l d i n t r o d u c e b i a s , which we c o n j e c t u r e s h o u l d reduce as 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 . 40 are using a "biased" approach ; the extent of "bias" depending upon how close the successive observations are. F i n a l l y the steady state approach cannot i d e n t i f y the two parameters m and cr2* separately - only t h e i r r a t i o can be estimated 2 1. Both m and cr have time units as part of their dimensions. Thus, using the steady state (or time independent) approach, we should not expect to be able to i d e n t i f y these parameters separately. To summarize the various aspects of the three estimating methods, we may note the following: a) The t r a n s i t i o n probability density approach to setting up the l i k e l i h o o d of the data i s exact, but i t s use requires that we greatly r e s t r i c t the generality of the model - either set ds = '/^ or, i f we choose o(=0, we have to reconcile having a positive probability of interest rates becoming negative. In case o(=i, we have only an approximate solution to the FP equation, and even that i s quite i n t r a c t a b l e for estimation purposes. b) The stationary probability density approach cannot indentify m and cr2"" separately - only their r a t i o . Further, when the data points are near each other, the lik e l i h o o d function i s probably far from exact, as the i n d i v i d u a l observations are not independent. c) The simple l i n e a r i z a t i o n method (or normal approximation) i s very tractable, and the closer our 2 1 This was expected on the basis of the r e s u l t s i n Ganssler [30]. 41 data p o i n t s , the l e s s the e r r o r i n the approximation. In the r e a l world, however, there are l i m i t a t i o n s to how c l o s e l y spaced the o b s e r v a t i o n s can be. T h i s l i m i t a t i o n i s d i s c u s s e d i n Chapter 6. 4.6 The P h i l l i p s Approximatipn Method Before we conclude t h i s chapter we can o u t l i n e one other approach to the e s t i m a t i o n of the parameters of s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s , (SDE) which has been advocated by (among others) Bergstrom [ 2 ] , Sargan [ 6 0 ] , P h i l l i p s [55,56,57], and Hymer [ 7 5 ] . Consider the system o f l i n e a r s t o c h a s t i c d i f f e r e n t i a l equations D^(tl - A |(t| + bzC-t) 4 fa (4.10) where A and B are m a t r i c e s , D i s the d i f f e r e n t i a l operator ct/cit , Z (t) Is a v e c t o r of exogenous v a r i a b l e s , and ^ (t) i s a pure white n o i s e d i s t u r b a n c e v e c t o r . The s o l u t i o n to (4.10) s a t i s f i e s (see. Sargan [60] f o r proof) 0 0 The l a s t term i n (4.11) i s a s t o c h a s t i c i n t e g r a l , and i f we assume that ^ ( t ) i s Gaussian N(0,_Q.), and t h a t the i n t e g r a l e x i s t s , then we can r e p l a c e the l a s t term by f ( t ) , where E[ £(t) ] = 0 and 42 o Thus, we have' £,(t) m ' N{0,]|;*) , and i t may be noted that even though^JT- may have been diagonal,_ Q _ * w i l l have non-zero off diagonal elements. Going back to (4.11), i n the general case where Z (t) i s a vector of exogenous variables, the f i r s t i n t e g r a l poses a problem i n the way of reducing (4.11) to something more manageable. In the s p e c i a l case where Z(t) i s a known deterministic function of time, the integration can be carried through and (4.11) suitably reduced. However, when Z(t) i s also stochastic, no exact equation system can be obtained, equivalent to the SDE system (4,10). P h i l l i p s [57] presents an approximation method, whereby the i n t e g r a l of Z(t) i n eguation (4,11) may be reduced using a three point Lagrange in t e r p o l a t i o n formula to express Z(t) as a polynomial i n the i n t e r v a l [ t , (t-h) ],[Appendix 6 presents more d e t a i l s on the adaptation of t h i s approach to the SDE (4.3), which i s cur i n t e r e s t rate model.] Using t h i s method reduces (4.11) to 43 Where the E*s are functions cf a, B and h. P h i l l i p s (op c i t ) has shown that the approximation (4.13) to the SDE (4.10) i s superior to the discrete approximation, (equations (4.4) and ( 4 . 5 ) ) 2 2 . ; P h i l l i p s (op c i t ) points out that the proposed approximation scheme leads to bias in the parameters of the order of 0(h 3). But in 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 points on the r e a l l i n e , the bias i s larger and of the order 0(h). In our case, the r e g u l a r i t y condition required to get improved estimators by t h i s approach are not met. We s h a l l therefore not pursue this approach further. 2 2 In the case where Z(t) i s stochastic, 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 conditions on Z ( t ) . Now, i n general, we know that 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 regu l a r i t y requirements are quite steep. , P h i l l i p s point out the superiority vanishes when the reg u l a r i t y requirements are not met. Further, as can be seen from Appendix 6, due to presence of r * i n the variance element, the resultant equation corresponding to (4. 13) i s rather involved. Some attempt was made to estimate the parameters using the P h i l l i p s (and even the r e l a t i v e l y simpler Sargan approximation), but non l i n e a r methods to estimate parameters from the log likelihood functions did not r e s u l t in much success. 44 CHAPTER 5: COMPARISON OF THE DIFFERENT ESTIMATING METHODS 5.1 The Method of Comparison In t h i s chapter, a limited attempt i s made to compare the r e l a t i v e merits of the d i f f e r e n t approaches to estimating the parameters of the int e r e s t rate process outlined i n the l a s t chapter: a) The Transition Probability Density Method (TRP) b) The Steady State Probability Density Method (SS) c) The Simple Li n e a r i z a t i o n Method (SL) The method adopted i s to generate a large sequence of discrete r e a l i z t i o n s ( a l l eguispaced) using a known parameter set G = (S,^X , (f , d ) Then with each method we estimate the parameters from this generated data base, using several samples. We then look at the d i s t r i b u t i o n of the parameters estimated by the d i f f e r e n t methods, using varyinq sample size s . Data for the simulations was generated using ='/x because t h i s i s the 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 density i s known exactly, and quite tractable. The rest of the parameters were chosen by applying the TRP method corresponding to o( = '/a. on actual weekly in t e r e s t rates over the past 18 years. Three 6 year subperiods were taken, and {m,^ U.,CT) were estimated on each. The average of these three estimates was used to generate the synthetic data. The reason for choosinq (m,^t,cr) from actual interest data was that the r e l a t i v e merits of the estimating methods may be a function of the parameter values. Since an extensive comparison of the Monte Carlo r e s u l t s was not done, (mainly due to the large computing cost involved) we 45 confined our attention to the neighbourhood of the parameter values of inte r e s t to us. The aim of the Honte Carlo simulations i s to investigate (for a p a r t i c u l a r parameter value of the process) the small sample behaviour of each of the estimators. He look for answers to the following questions: 1) are the estimators unbiased i n small samples? 2) Do they appear to be asymptotically unbiased even thouqh they may be biased i n small samples? 3) What i s the r e l a t i v e e f f i c i e n c y of the d i f f e r e n t estimators? 4) Which estimator approaches the asymptotic values fastest? 5) For a qiven spread of the data, does increasing the frequency of observation lead to any improvement in the estimators? S p e c i f i c a l l y , i s there any improvement i n usinq 365 daily observations rather than 52 weekly points? 5 . 2 Generating an "exact" Sequence f o r the Square Boot Process Having chosen the parameter set , the f i r s t step i s to qenerate s y n t h e t i c a l l y , a d i s c r e t e r e a l i z a t i o n that i s exact. This i s very important as we should be able to assert that any observed bias i n the parameters estimated, i s a r e s u l t purely of the method of estimation. The t r a n s i t i o n probability density corresponding to the A = J/ j . case (the square root process) i s qiven by (Feller [23 ] ) : 46 JVV\fo •o j (5 .1) where w = exp(mt) and 1 ^ (.) i s the modified Bessel function of order k, and i s defined by One way to generate r , given 0 and r f c_ ( , i s to generate a uniform (rectangular) random variable p on [0,1] and then set r_j. = C-» (p) where C i s the cumulative probability density function corresponding to F(.) i n equation (5.1). I f C could be inverted, there would be no problem. However f o r the sp e c i a l structure of [5.1], Boyle [8] has developed a solution using a di f f e r e n t approach. Substituting (S-0 where <$ = Im^/v1 I f (S~1)=:«J; n=2&, and n i s i n t e g r a l , then (5.3) where l( * s *^e aon-central chi-squared density with n deqrees of freedom, and "X i s the non-centrality parameter. Now, we can ea s i l y choose our parameter set $ such that 2$ i s i n t e g r a l , without much loss of generality (since the value of 8 from the actual i n t e r e s t data was lar g e ) . Generating a non-central chi-squared 3 0 random variable i s quite straightforward. (Fishman £27] has detailed i n s t r u c t i o n s on the generation of stochastic variates corresponding to a wide variety of proba b i l i t y d i s t r i b u t i o n s ) . This method was adopted using the parameter values: jX r 0. 09517 '/o/iwk S = -\%\5_. and a sequence of weekly i n t e r e s t rates (100,000 weeks lonq) was generated,, 3 0 One way to generate a non-central chi-square random variable (Y) with (n*1) degrees of freedom, would be: Y = Z 2 • £ x ? where the are N{0,1) and Z i s N(A#1) r ^ being the non-c e n t r a l i t y parameter of the chi-square. This requires the qeneration of (n+1) Gaussian random variates. Another approach i s based on the equivalence of the chi-square and Gamma di s t r i b u t i o n s . Osing t h i s approach Y = Z 2 -2 £ loq(0. ) U l where Y and Z are as before, but the U,; • s are uniform(rectanqular) on (0,1). This requires only ( n / 2 1 ) random variate qenerations f o r each chi-square variate. 48 Results of Monte Carlo Simulations for the <A = '/? (known! Case To s t a r t with, we want to compare a l l three methods, and since we do not have a solution to the FP equation f o r a r b i t r a r y <7x, we have to assume cV i s known and equal to Yi . When we assume °^=/ /2 * we know the t r a n s i t i o n probability density, and thus can compare a l l three estimating methods, based on the properties of the estimated parameters. Four sample sizes were u s e d 3 1 : n = 100, 250, 500 and 945. For the n = 100 and n = 250 cases, 200 simulations each were performed, i e . , 200 sets of (m,yW-,cr) were estimated. For the n = 500 and n = 945 cases, 100 simulations each were performed., The d e t a i l s of the estimation procedure for the parameters when d\ i s assumed known and -Yt-w are in Appendix 7. The 200 simulations for the n=100 case (say) were performed as follows., From the sequence of 100,000 points of s y n t h e t i c a l l y generated weekly data on i n t e r e s t rates, successive blocks of 100 points were taken., Using each block of 100 data points, one set of parameters (m,/^,<r2) was estimated. By performing the estimation on 200 successive blocks, we get 200 estimates of the parameters. The standard deviation across these 200 parameter estimates (which represent the d i s t r i b u t i o n of the parameter estimate) i s c a l l e d the Monte Carlo standard deviation, and i n the reported simulation r e s u l t s i s c a l l e d SD . If the Monte Carlo d i s t r i b u t i o n of the estimated 3 1 Data was generated using 1 week as the unit of time. Thus the selected sample sizes correspond to 2,5,10 and 18 years of weekly data. Actually n=945 was chosen as that was the exact number of weekly data points on the short term i n t e r e s t rate between January 1st, 1969 and December 31st, 1976. 49 paramters were Gaussian, as indicated by asymptotic theory, the mean and standard deviation should convey a l l the information about the d i s t r i b u t i o n . To cover the p o s s i b i l i t y that the Monte Carlo d i s t r i b u t i o n might not be exactly Gaussian , the 10 percentile and 90 percentile values are also reported. Purther, corresponding to each simulation, we not only get one set of parameter estimates, but also a set of estimates of the standard deviation of the parameters, based on asymptotic t h e o r y 3 2 . In the summary r e s u l t s reported, SD^ refers to the mean of the asymptotic standard deviation computed for each t r i a l . In some cases the mean SD^ , i s very high due to a few extreme v a l u e s 3 3 , and so the median was reported instead, as an alternate representation of locati o n . F i n a l l y , the Steady State Density Method cannot i d e n t i f y the paramters m and ( T 2 separately - only a composite (2m/<r2) i s estimated. To be able to compare across the three methods, the value of (2m/cr2) was computed i n each 3 2 I f L=log of the j o i n t l i k e l i h o o d function (corresponding to a given set of data points), then the matrix of second p a r t i a l derivatives of L with respect to the paramters, at the maximum of L, may be c a l l e d the Hessian Matrix. The inverse of the Hessian matrix with signs reversed i s an estimate of the variance-covariance matrix of the estimated parameters, based on asymptotic theory (see B i l l i g s l e y [ 3 ] ) . The standard deviations are the square roots of the diagonal elements of the variance-covariance matrix. 3 3 Extreme values do not necessarily imply that these are nonrepresentative - the Monte Carlo method gives a representation of the true d i s t r i b u t i o n . . However, i n the TRP method, nonlinear optimiztion routines had to be used to f i n d the parameter set that maximizes the l i k e l i h o o d function. In such routines, convergence i s assumed to have been! attained when the r e l a t i v e change i n the parameter values between successive i t e r a t i o n s i s less than a s p e c i f i e d accuracy l e v e l . I f the li k e l i h o o d function i s very peaked, then i t s second derivative can change a l o t around the optimum point. This could lead to extreme values of SD- . case for the other two methods, and the summary s t a t i s t i c s of i t s d i s t r i b u t i o n are also tabulated. Tables II through V present summary s t a t i s t i c s on the d i s t r i b u t i o n of the estimated paramters. From the tabulated re s u l t s , the following broad conclusions can be drawn: 1) There i s l i t t l e or no difference across the three methods in the estimated means of the parameter d i s t r i b u t i o n s , 2) The dispersion of the parameter d i s t r i b u t i o n as measured by SD m c (which could be treated as a good proxy for the asymptotic standard deviation) i s almost i d e n t i c a l across the three methods. However, i f SD^ i s evaluated as a measure of the asymptotic standard deviation, there i s a f a i r amount of difference across the three methods. The SL method grossly overestimates the asymptotic variance, (SD- i s much larger than S D W ) whereas the SS method grossly underestimates i t . The TBP method appears to perform rather well - i n fact the median SD;, value i s quite close to SD,„C f o r sample s i z e s greater than 500. 3) The parameters jx and <ru appear to be unbiased even i n small samples - at l e a s t for the number of simulations performed. However, the parameter m (or 2mA-2 in the SS method) i s biased i n small samples. I t i s overestimated by a l l methods, and the extent of bias i s nearly the same across the three methods (and seems roughly inversely proportional to n). TABLE II 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 Mean 0.05884 0.02694 0.01559 0.01211 Method 10% 0.01328 0.00776 0.00599 0.00659 Median(50%) 0.04891 0.02205 0.01372 0.01060 90% 0.11053 0.05294 0.02490 0.08179 SDmc 0.04458 0.02176 0.01024 0.00500 SDi 2.58618 0.10596 0.60140 0.39186 T r i a l s 200 200 ,100 100 T r a n s i t i o n Mean 0.06205 0.02773 0.01577 0.01219 P r o b a b i l i t y Density 10% 0.01348 0.00781 0.00596 0.00663 Method Median(50%) 0.05002 0.02204 0.01382 0.01078 90% 0.11646 0.05524 0.02538 0.01903 SDmc 0.04874 0.02298 0.01051 0.00510 SD.i 0.02682 0.06244* 0.00506 0.00268 T r i a l s 200 199 100 100 * The mean i s high, on page -. For but the median was 0.00586 and a d e s c r i p t i o n o f SD and SD-1 mc i 90% i l e was 0. see t e x t page 01797. See footnote i n t e x t c n TABLE 111 ESTIMATE OF U BY DIFFERENT METHODS FOR a = % (KNOWN) CASE TRUE VALUE u = 0 .09517 > METHOD n=100 n=250 n=500 n=945 Simple L i n e a r i z a t i o n Mean 0.06803 0.09114 0.09091 0.09371 Method 10% 0.05945 0.06439 0.07137 0.07701 Median(50%) 0.08919 0.09213 0.09006 0.09390 90% 0.12481 0.12402 0.11522 0.10948 - SDmc 0.23261 0.05958 0.01866 0.01275 SDi* 0.20554 0.36961 0.50491 0.41076 T r i a l s 200 200 100 100 T r a n s i t i o n Mean 0.11303 0.09801 0.10410 0.09370 P r o b a b i l i t y Density 10% 0.06194 0.06487 0.07409 0.07698 Method Median(50%) 0.08984 0.09251 0.09077 0.09390 90% 0.12807 0.12555 0.11607 0.10949 SDmc 0.21429 0.05050 • 0.11204 0.01275 SDi 0.02841 0.07511 0.09782 0.01272 T r i a l s 200 199 100 Steady State Density Mean 0.09266 0.09359 0.09349 0.09358 Method 10% 0.06562 0.07066 0.07423 0.07858 Median(5 0%) 0.09007 0.09117 0.09205 0.09334 90% 0.11926 0.11661 0.11382 0.10853 SDmc 0.02193 0.01792 0.01454 0.01207 SDi 0.00095 0.00087 0.00078 0.00061 T r i a l s 200 199 100 100 * The SDi f i g u r e s are not the means but medians. The mean SDi was very high due to a few e x c e p t i o n a l l y high values. The mean of SDi 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 the SL method, SD^ ^ can have extreme values. c n TABLE IV ESTIMATE OF O2 BY DIFFERENT METHODS FOR g = h (KNOW) CASE TRUE VALUE 0 = 0.78427 x 1 0 - 4 ( A l l f i g u r e s i n the Table have been M u l t i p l i e d by a factor of 10 4) IffiTHOD n=100 n=250 n=500 n=945 Simple L i n e a r i z a t i o n Mean 0.78414 0.79242 0.79721 0.79443 Method 10% 0.65251 0.69203 0.73421 0.75482 Median(50%) 0.77073 0.79157 0.79707 0.78882 90% 0.91753 0.88499 0.87423 0.83654 SDmc 0.11073 0.07316 0.05619 0.03402 SDi - - ,.. — _ T r i a l s 200 200 100 100 T r a n s i t i o n . Mean 0.82021 0.81073 0.80820 0.80324 P r o b a b i l i t y 10% 0.66768 0.70148 0.74135 0.76374 Method Median(50%) 0.81653 0.81141 0.80820 0.79685 90% 0.97041 0.90429 0.88491 0.84169 SDmc 0.13434 0.07720 0.05714 0.03439 SDi 0.14156 0.17047 0.06016 0.07423 T r i a l s 200 199 100 100 TABLE V ESTIMATE OF 2m/o 2 BY DIFFERENT METHODS FOR a = h (KNOWN) CASE -- TRUE VALUE 2m/a 2 = 194.389 METHOD n=100 n=250 n=500 n=945 Simple L i n e a r i z a t i o n Mean 1547.22 687.87 393.72 305.90 Method 10% 319.51 196.49 144.13 169.00 Median(50%) 1235.32 557.23 327.99 263.69 90% 3129.81 1342.18 671.70 469.88 SDmc 1225.81 573.54 266.59 128.77 T r i a l s 200 200 100 100 T r a n s i t i o n Mean 1494.77 679.22- 390.06 303.83 P r o b a b i l i t y 10% 331.29 199.66 ' 143.22 164.43 Method Median (50%) 1216.86 550.43 328.85 265.43 90% 2963.03 1317.36 669.54 467.26 SDmc 1121.05 540.30 260.03 127.32 T r i a l s 200 199 100 100 Steady State Density Mean 1482.59 677.70 390.93 302.11 Method 10% 449.55 242.43 153.33 158.44 Median(50%) 1256.65 528.45 330.29 260.19 9,0% 2926.93 1281.06 640.97 481.75 SDmc 1067.60 506.45 247.17 129.21 SDi 209.80 60.70 24.78 13.94 T r i a l s 200 199 100 100 55 The consistent overestication of m (or 2m/<r2 in the SS method) needs some consideration. To an extent, t h i s was anticipated, based on the r e s u l t s of Sirjaev [64] and Novikov [52], He do f i n d that as the sample s i z e increases, a l l methods show reduced bias. Based on t h i s , i t could be conjectured that the bias asymptotically goes to zero. Let us propose a form for the bias as follows: rn m + _JL_ (5.4) where m^ i s the estimate of m using a sample of si z e n, m~ represents i t s true value; c and d are constants. Using the r e s u l t s for n=100,250,500 and 94 5, the value of d that f i t s * * the bias structure proposed above was estimated as 1<d<1.1 . Based on these r e s u l t s , the sample sizes required to reduce the bias on the estimate of m to 10% i s 4450, and to 111s 36090. To assume that the parameters of the i n t e r e s t rate process are constant over such larqe time periods, would be unreasonable. The natural question to ask therefore would be; how important i s i t to get an accurate estimate of m? For our 3 * For the 4 values of n, we have { £ h - ro) from the Monte Carlo r e s u l t s (where the mean of the Monte Carlo simulation was taken as ). The crude method adopted was to choose a value of d, and corresponding to that value, compute the values of c using equation (5.4). This was done on the four means of the Monte Carlo values of m. The appropriateness of d was decided by observing the computed values of c. I f the values of c did not exhibit a trend from n=100 to n=945, i t was assumed that c was beinq observed with a random error. This f i t t i n q approach was t r i e d on the estimated m values by S.L. and TRP methods. d=1.1 appears to qive the best f i t , and the corresponding value of c i s approximately 8.0. purposes, the deciding c r i t e r i o n must be the error caused i n bond valuation, for a given error i n m. This i s investigated i n a subsequent section of t h i s chapter. accepting the fact that m w i l l be overestimated, there i s an i n t u i t i v e reason that could be used to explain t h i s occurrence. Consider the diagram below, which i s supposed to represent one r e a l i z a t i o n of the interest process. ])o>ij<?c t)oi-(A. Z I t may be r e c a l l e d that m represents the speed of reversion to the mean of the process. Thus, the higher m i s , the l e s s l i k e l y i s the process to "stray" away from i t s mean. In the diagram above, l e t the 4 segments (represented by data 1 through 4) refer to subperiods of the t o t a l sample. I f we estimate the parameters using one of the methods proposed i n the l a s t chapter, we might expect JU. i n each case to be estimated as shown by the broken l i n e s . In sub period 1 (Data 1), the process i s seen as moving upwards and then somewhat s t a b i l i z i n g . Thus y~\ overestimates jUo . m i s also overestimated, as the process 57 appears to be moving rapid l y towards the perceived mean ( / * i ) . In subperiod 2, the interest rate process remains more or less constant around a single l e v e l , j tf .7, i s obviously perceived as the process mean. Here again, m w i l l be highly overestimated as the process does not stray away from the perceived mean (^2.) v The reasoning for the overestimate of m, but underestimate of pL i n subperiod 3, i s exactly as that proposed for subperiod 1 : the mean being perceived i s jXi, and the process i s rapidl y being pulled toward i t due to a high value of m. F i n a l l y , i n subperiod 4, the process mean i s probably perceived at ju^ , but here m w i l l not be as highly overestimated as i n the previous three subperiods. Since the process appears to wander a b i t to either side of the perceived mean, a lower value of m (than i n previous cases) would be estimated. From the above, we see that yu-is sometimes overestimated, and at other times underestimated. On average, i t s estimate might be expected to be unbiased. However, i n almost every s i t u a t i o n , m could be over-estimated. I f now, the complete data were employed, i t i s easy to see why JLL might be quite accurately estimated. Furthermore, the complete data convey the information that the process could stray away from the mean for rather long s p e l l s , which indicates a weaker force p u l l i n g towards the mean - m would be estimated nearer i t s true value., Before we present further r e s u l t s on the simulations, a minor methodological point needs to be c l a r i f i e d . The method employed for the simulations was to take successive blocks of observations from the long sequence that had been s y n t h e t i c a l l y generated. One objection to t h i s approach could be that 58 successive t r i a l s were not s t r i c t l y independent. To counter t h i s objection, for the n=945 case (which happens to be the one of primary i n t e r e s t to us, as that i s the length of our actual sample), 100 "independent" samples of si z e 945 each were generated. The s t a r t i n g point f o r each of these 100 samples was randomly chosen from the stationary d i s t r i b u t i o n (a reasonable approach), which in t h i s case i s a gamma d i s t r i b u t i o n . The Steady State method and SL method were compared 3 5 for the "dependent" and "independent" samples case, and the r e s u l t s are presented i n Table VI. The conclusion appears to be that the use of the "dependent" samples does not materially a l t e r inferences from Monte Carlo experiments. The next point investigated was whether using more frequent observations on the process (keeping constant the spread over time of the t o t a l observations) leads to any improvement. For t h i s purpose, " d a i l y " observations were generated f o r the same parameter values. To compare, parameters were estimated using 700 " d a i l y " observations, and the r e s u l t s compared with the equivalent r e s u l t s corresponding to weekly observations. The r e s u l t s are presented in Table VII. Comparison among the 3 methods shows that there i s no perceptible improvement i n the mean of the estimated parameter d i s t r i b u t i o n s , but (as expected) the dispersion reduces by using " d a i l y " observations. Thus i t appears that the increased e f f o r t of c o l l e c t i n g daily data, pays o f f by lower variances on the 3 5 The TEP method was not investigated, as i t was computationally expensive. Since the objective i s only to get an idea of the e f f e c t , i t was f e l t that the extra cost was unnecessary.... TABLE VI COMPARISON OF MONTE CARLO RESULTS ON PARAMETER ESTIMATION USING SERIALLY DEPENDENT/INDEPENDENT SAMPLES (SAMPLE SIZE n= 945, a^j KNOWN) METHOD 2m/a 2 DEPNDT (194.389) INDEP M (0.09517) DEPNDT INDEP m (0.007162) DEPNDT INDEP a 2 (0. DEPNDT 7843xl0~ 4) INDEP Simple Mean 305.90, 324.56 0.09371 0.09404 0.01211 0.01267 0.79443 0.78438 L i n e a r i z a t i o n 10% 169.00 154.67 0.07701 0.08186 0.00659 0.00597 0.75482 0.73083 Method 50% 263.69 300.35 0.09390 0.09246 0.01060 0.01172 0.78882 0.78380 90% 469.88 536.65 0.10948 0.10853 0.01879 0.02168 0.83654 0.82981 SDmc 128.77 159.80 0.01275 0.01093 0.00500 0.00618 0.03402 0.03806 SDi - - 0.41076 0.36074 0.39186 0.40281 _ T r i a l s 100 100 100 100 100' • 100 100 100 Steady St a t e Mean 302.11 320.29 0.09358 0.09398 Density 10% 158.44 175.30 0.07858 0.08010 Method 50% 260.19 281.99 0.09334 0.09292 90% 481.75 534.57 0.10853 0.10887 SDmc 129.21 149.10 0.01207 0.01088 SDi 13.94 14.78 0.00061 0.00060 T r i a l s 100 100 100 100 i n n P I ! S r , i r e p r e S e n t S t h e / e p r e s e n t S the r e s u l t s o f using a sequence o f blocks (n=94S) o f data points from the 1UU,000 long sequence of s y n t h e t i c data generated f o r the Monte Carlo s i m u l a t i o n s . "INDEP" represents r e s u l t s of using "independent" samples (see t e x t , page f o r d e t a i l s ) . 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) METHOD 2mlO1 V 0 (True value:194.389) (True value:0.09517) (True value: 111 0.7843x10-4) (True value:0.007162) Simple Mean Linearization 10% Method DAILY 1276.42 Transition. Probability Density Method Steady State Density Method 276.04 Median(50%) 1097.32 90% 2416.77 SDmc 1040.43 SDi Trials 100 Mean 1281.26 10% 274.95 Median(50%) 1100.19 90% 2396.25 SDmc 1008.27 SDi* Trials 100 Mean 10% Median(50%) 90% SDmc SDi Trials 1278.87 298.54 1132.24 2240.06 922.98 68.41 100 WEEKLY 1547.22 319.51 1235.32 3129.81 1225.81 200 1494.77 331.29 1216.86 2963.03 1121.05 200 1482.59 449.55 1256.65 2926.93 1067.60 209.80 200 DAILY 0.10178 0.06192 0.09012 0.12695 0.07533 2.21445 100 0.10397 0.06194 0.09013 0.12699 0.09201 0.00589 100 0.09088 0.06706 0.08987 0.11629 0.01934 0.00038 100 WEEKLY DAILY WEEKLY DAILY WEEKLY 0.06803 0.80031 0.78414 0.05037 0.05884 0.05945 0.73697 0.65251 0.01088 0.01328 0.08919 0.79943 0.77073 0.04206 0.04891 0.12481 0.87289 0.91753 0.09479 0.11053 0.23261 0.04648 0.11073 0.03979 0.04459 0.20554 - - 6.32839 2.58618 200 200 100 200 0.11303 0.80413 0.82021 0.05111 0.06205 0.06194 0.74359 0.66768 0.01098 0.01348 0.08984 0.80324 0.81653 0.04237 0.05002 0.12807 0.87379 0.97041 0.09523 0.11646 0.21429 0.04565 1.13434 0.03963 0.04874 0.00534 0.04723 0.10974 0.01007 0.01548 200 100 200 100 200 0.09266 0.06562 0.09007 0.11926 0.02193 0.00095 200 * R e p o r t e d f i g u r e s r e p r e s e n t med ians o f SDj^ and n o t t h e mean. 61 parameter estimates. However, our interest i s i n using these parameters for bond valuation. Therefore, the question that needs to be answered i s whether t h i s reduction i n the dispersion of parameter extimates would translate to comparable reduction i n dispersion of estimated bond values. This question i s addressed l a t e r on i n t h i s section. However, one point needs to be noted when we attempt to c o l l e c t d a i l y data on the actual i n t e r e s t rate process - measurement errors w i l l occur. They would be of the following types: 1) Normally no exact daily rate at which some s p e c i f i c transaction occured, would be available. Quoted rates are generally the mean of a bid and ask price, i e . not market prices. 2) Even i f the d a i l y rate were based on s p e c i f i c transactions, a l l transactions would not be exactly 24 hours apart i e . , observations would not be equi-spaced as required to simplify our estimation process. In da i l y data, the r e l a t i v e magnitude of t h i s error could be high., 3) Due to the presence of week-ends and holidays, the d a i l y series of i n t e r e s t rates has more "holes" than a corresponding weekly series. Every time there i s a holiday, as over a weekend, continuity i s l o s t in a da i l y data series and we have a gap. I t i s obvious that such occurences are less l i k e l y i n a weekly serie s . A l l the above factors would tend to diminish the value of a daily s e r i e s . In Appendix 8 we outline a very b r i e f 62 inve s t i g a t i o n of the impact of a s p e c i f i c form of measurement error. ,• F i n a l l y we look at the impact of the d i s t r i b u t i o n of the parameter estimates on the valuation of pure discount bonds 3*. This i s c r u c i a l , as our primary i n t e r e s t i n estimating the parameters i s to use them to value bonds. For s i m p l i c i t y , we investigate the impact on the valuation of pure discount bonds. There i s l i t t l e reason to believe that the results on the valuation of other types of bonds should be any d i f f e r e n t , since a coupon bond, for example, may be thought of as a p o r t f o l i o of discount bonds of varying maturity. Tables VIII to X present the " 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 values to errors i n the paramter values. The expression " 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 to distinguish these r e s u l t s from those c a l l e d "empirical" s e n s i t i v i t y that w i l l be presented s h o r t l y . "Theoretical" s e n s i t i v i t y r e f e r s to changes i n the value of bonds due to a certain fixed l e v e l of error i n one parameter at a time, while "empirical" s e n s i t i v i t y refers to the d i s t r i b u t i o n of bond values r e s u l t i n g from the estimated joint d i s t r i b u t i o n of the parameters from the Monte Carlo experiments 3 7., We can draw the following inferences from 3 * The value of a discount bond was computed using I n g e r s o l l , s [39] solution. 3 7 The procedure adopted i s as follows. Consider the Monte Carlo simulation for the n=945 ( known) case. Here, we have generated 100 estimates of the parameter set (m, p., <r2) • Corresponding to each estimated parameter set, we can compute the value of a pure discount bond (for d i f f e r e n t times to maturity and current value of r) . Thus using the 100 estimates of (m, /JL , a~z), we get 100 bond values (for each chosen maturity and current value of r ) . This represents the "empirical" d i s t r i b u t i o n of bond values 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 of the parameters. TABLE VIII THEORETICAL SENSITIVITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN m ERROR IN m 0% 10% 25% 50% 100% CURRENT INTEREST TIME TO MATURITY IN YEARS BOND PRICE BOND PRICE % ERROR BOND PRICE % ERROR BOND PRICE % ERROR BOND PRICE ERROR r=2u 1 97.13 97.09 -0.0378 3 90.01 89.82 -0.2024 5 82.38 82.09 -0.3506 7 74.99 74.65 -0.4504 10 64.86 64.51 -0.5332 1 95.17 95.17 -0.0000 3 86.22 86.33 -0.0014 5 78.13 78.12 -0.0060 7 70.80 70.79 -0.0132 10 61.09 61.07 -0.0271 1 91.38 91.45 0.0756 3 79.12 79.44 0.4017 5 70.27 70.76 0.6869 7 63.12 63.67 0.8669 10 54.19 54.73 0.9928 97.04 -0.0928 96.96 -0.1797 96.80 -0.3376 89.57 -0.4796 89.21 -0.8808 88.65 -1.5046 81.71 -0.8094 81.20 -1.4304 80.48 -2.3003 74.22 -1.0202 73.67 -1.7568 72.94 -2.7264 64.09 -1.1865 63.56 -1.9985 62.90 -3.0207 95.17 -0.0001 95.17 -0.0002 95.17 -0.0003 86.22 -0.0034 86.22 -0.0061 86.21 -0.0100 78.12 -0.0135 78.11 -0.0233 78.10 -0.0360 70.78 -0.0294 70.77 -0.0492 70.75 -0.0730 61.05 -0.0590 61.03 -0.0962 61.00 -0.1381 91.55 0.1856 91.71 0.3599 92.00 0.6777 79.88 0.9560 80.52 1.7666 81.53 3.0477 71.40 1.5974 72.28 2.8513 73.54 4.6514 64.37 1.9820 65.30 3.4555 66.56 5.4529 55.40 2.2346 56.26 3.8197 57.38 5.8867 (Tl CO. TABLE IX THEORETICAL SENS IT IV ITY OF PURE DISCOUNT- BOND PRICES TO ERRORS IN y ERROR IN VI - 2 5 % - 5 % 0% +5% +25% CURRENT TIME -TO BOND % BOND % BOND BOND % BOND % INTEREST MATURITY PRICE ERROR PRICE ERROR PRICE ' PRICE ERROR PRICE ERROR IN YEARS 1 97 .34 0 .22 97 .17 0.04 97.13 97.09 - 0 . 0 4 96.92 - 0 . 2 2 3 91 .42 1.57 90 .29 0 .31 90.01 89 .73 - 0 . 3 1 88.61 - 1 . 5 5 r=y/2 5- 85.33 3.58 82 .96 0.71 82.38 81.80 - 0 . 7 0 79.53 - 3 . 4 6 7 79.43 5.93 75.86 1.16 74.99 74.13 - 1 . 1 5 70.79 - 5 . 6 0 10 71 .20 9 .78 66 .08 1.88 64.86 63 .66 - 1 . 8 5 59 .08 - 8 . 9 1 1 95 .38 0.22 95.21 0.04 95 .17 95 .13 - 0 . 0 4 94 .96 - 0 . 2 2 3 87.57 1.57 86 .49 0 .31 86.22 85.95 - 0 . 3 1 84.89 - 1 . 5 5 r=y 5 80.93 3.58 78.68 0.71 78.13 77.58 - 0 . 7 0 75.42 - 3 . 4 6 7 75.00 5 .93 71.62 1.16 70.80 69.99 - 1 . 1 5 66.84 - 5 . 6 0 10 67 .06 9 .78 62.24 1.88 61.09 59 .96 - 1 . 8 5 55.65 - 8 . 9 1 1 91 .58 0 .22 91 .42 0.04 91.38 91 .34 - 0 . 0 4 91 .18 - 0 . 2 2 3 80 .36 1.57 79.37 0.31 79.12 78.88 - 0 . 3 1 77.90 - 1 . 5 5 r=2y 5 72.79 3.58 70.77 0.71 70.27 69 .78 - 0 . 7 0 67 .84 - 3 . 4 6 7 66.87 5.93 63.85 1.16 63.12 62 .40 - 1 . 1 5 59.59 - 5 . 6 0 10 59 .49 9.78 55 .21 1.88 54.19 53.19 - 1 . 8 5 49 .37 - 8 . 9 1 CTl TABLE X THEORETICAL SENS IT IV ITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN a-ERROR INO - 2 5 % - 5 % 0% ' +5% +25% CURRENT TIME TO BOND % BOND % BOND BOND % BOND % INTEREST MATURITY PR ICE ERROR PRICE ERROR PRICE PRICE ERROR PRICE ERROR IN YEARS 1 9 7 . 1 3 - 0 . 0 0 0 2 97 .13 - 0 . 0 0 0 0 97 .13 97 .13 0 .0000 97 .13 0.0002 3 9 0 . 0 0 - 0 . 0 0 3 3 90 .01 - 0 . 0 0 0 7 90.01 90.01 0 .0007 90.01 0 .0033 r=y/2 5 82 .37 - 0 . 0 1 0 9 82 .38 - 0 . 0 0 2 2 82.38 82.38 0 .0022 82.39 0.0109 7 74 .97 - 0 . 0 2 2 0 74 .98 - 0 . 0 0 4 4 74.99 74.99 0.0044 75.00 0.0219 10 64 .83 - 0 . 0 4 2 2 64.85 - 0 . 0 0 8 4 64.86 64.86 0 .0084 64 .89 0 .0420 1 95 .17 - 0 . 0 0 0 3 95.17 - 0 . 0 0 0 1 95.17 95.17 0 .0001 95.17 0 .0003 3 86 .22 - 0 . 0 0 5 2 86 .22 - 0 . 0 0 1 0 86.22 86.22 0 .0010 86.23 0.0052 r = p . 5 78 .12 - 0 . 0 1 5 4 78.13 - 0 . 0 0 3 1 78 .13 78.13 0.0031 78.14 0 .0153 7 70 .78 - 0 . 0 2 8 5 70 .80 - 0 . 0 0 5 7 70.80 70.81 0.0057 70.82 0 .0284 10 61 .06 - 0 . 0 5 0 6 61 .08 - 0 . 0 1 0 1 61.09 61.09 0.0101 61 .12 0.0504 1 91 .38 - 0 . 0 0 0 6 91 .38 - 0 . 0 0 0 1 91.38 91 .38 0.0001 91.38 0.0006 r=2p 3 79.11 - 0 . 0 0 9 0 79 .12 - 0 . 0 0 1 8 79.12 79.12 0 .0018 79.13 0 .0090 5 70 .26 - 0 . 0 2 4 2 70.27 - 0 . 0 0 4 8 70.27 70.28 0.0048 70.29 0.0241 7 6 3 . 1 0 - 0 . 0 4 1 5 63.12 - 0 . 0 0 8 3 63.12 63 .13 0 .0083 63.15 0.0414 10 54 .16 - 0 . 0 6 7 4 54.19 -.0.0135 54.19 54.20 0 .0135 54 .23 0 .0672 the r e s u l t s : 1) Bond values are s e n s i t i v e to jx , the mean l e v e l of the in t e r e s t rate. The s e n s i t i v i t y to m i s much l e s s errors i n <r 2 have hardly any impact on bond values. 2) Errors i n /A- cause errors i n bond values which increase as the time to maturity of the bond increases, whereas the current l e v e l of the i n t e r e s t rate has no e f f e c t on the amount of error. For example, overestimating jx by 5% causes the 10 year discount bond to be undervalued by 1.85% irresp e c t i v e of whether the current l e v e l of in t e r e s t rate i s at or 2^. 3) Errors i n m cause errors i n bond values which increase with the maturity of the bond. Furthermore, the error in the bond value depends on the current l e v e l of the inter e s t rate - more accurately, on i t s deviation from the mean in t e r e s t l e v e l J J » . He now look at the s e n s i t i v i t y of discount bond values to the d i s t r i b u t i o n of the estimated parameters. These r e s u l t s are presented i n Tables XI to XIII. The results are exactly as expected: the d i s t r i b u t i o n of bond values i s almost i d e n t i c a l , using parameters estimated by any of the three methods 3 8. However, there are i n t e r e s t i n g r e s u l t s when we compare the d i s t r i b u t i o n of bond prices using • "weekly**, versus " d a i l y " data. Surprisingly, (as can be seen from Table XI?) even though the standard deviation (SD W C ) of the parameters was always reduced 50% or more using " d a i l y " data (see Table VII), s i m i l a r 3 8 For the SS method, r r z was taken from the SL method. Dsing t h i s cr-2; m was computed from the parameter {2m/<r2) estimated for the SS method. TABLE XI SENSITIVITY OF PURE DISCOUNT BpND PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS" (Current value of 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 Mean 96.972 89.417 81.638 74.290 64.364 L i n e a r i z a t i o n SDmc 0.251 1.163 1.999 2.695 3.505 Method 10% 96.599 87.753 78.906 70.783 60.077 Median 97.013 89.516 81.790 74.469 64.439 90% 97.238 90.710 83.936 77.497 68.649 T r a n s i t i o n Mean 96.969 89.408 81.627 74.279 64.355 P r o b a b i l i t y SDmc 0.253 1.170 2.008 2.705 3.515 Density Method 10% 96.595 87.750 78.891 70.752 60.047 Median 97.015 89.491 81''. 763 74.435 64.496 90% 97.241 90.714 83.942 77.505 88.659 Steady State Mean 96.979 89.450 81.691 74.355 64.435 Density Method SDmc 0.246 1.133 1.929 2.583 3.336 10% 96.576 87.726 79.251 70.720 59.975 Median 97.019 89.547 81.720 74.436 64.488 90% 97.246 90.748 83.988 77.522 69.011 NOTE: - The I n t e r e s t r a t e parameters (m, u,a) have been estimated f o r the a=*5(knovra) case using 945 observations on the i n t e r e s t r a t e . 100 such simulations were performed, and d i s t r i b u t i o n of bond p r i c e s represents the bond value corresponding to each of those parameter estimates. - True value of bond corresponds to the bond p r i c e corresponding to the • true u n d e r l y i n g i n t e r e s t process parameters. TABLE XII SENSITIVITY OF PURE DISCOUNT BOND PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS (Current value of 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 L i n e a r i z a t i o n 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 T r a n s i t i o n P r o b a b i l i t y Density 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 State Density 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 to comments on Table 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 (Current value of i n t e r e s t r a t e = 2y) MATURITY(YRS) 1 TRUE VALUE 91.38 3 79.12 5 70.27 7 63.12 10 54.19 Simple L i n e a r i z a t i o n 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 T r a n s i t i o n . P r o b a b i l i t y Density 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 State Density 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: Refer to comments on Table XI f o r more d e t a i l s . TABLE XIV COMPARISION OF BOND PRICE SENSITIVITY TO THE USE OF DAILY VS WEEKLY DATA* IN THE ESTIMATION OF INTEREST RATE PROCESS PARAMETERS(«=^s ) r=2y T= l y e a r T= 3 T=5 T=7 T=10 DAILY WEEKLY DAILY WEEKLY DAILY WEEKLY DAILY WEEKLY DAILY WEEKLY TRUE VALUE 97 .13 90 .01 82. 38 74. 99 64.86 Mean S D m c 10% 50% 90% 96 .312 0 .773 95 .204 96.336 97 .204 96 .178 0 .917 94 .931 96 .345 9 7 . 3 2 0 88 .021 2.623 84.514 87.952 91 .388 87.782 2.957 83.797 88.087 91.682 80.227 4 .333 74.323 80.080 85.901 80.020 4.707 73.950 80 .488 86.286 73.095 5.869 64.624 72.970 80.742 72.963 6.160 " 65.108 73.200 81 .110 63.599 7.754 53.087 63.325 73.424 63.595 7.832 53.796 63.843 73.978 TRUE VALUE 9 5 . 1 7 86 .22 78.13 70.80 61 .09 Mean SD i o ? c 50% 90% 95 .269 0 .617 94 .463 9 5 . 2 5 6 96.042 95 .250 0 .743 94 .290 9 5 . 2 8 9 96 .149 86 .563 2.402 82.886 86 .545 89.538 86.553 2.639 83.139 86.721 89.830 78.667 4.125 72.666 78.606 83.595 78.709 4.288 73.086 78.967 84.211 71.516 5.637 63 .680 71.396 78.319 71.627 5.641 64.053 71.918 78.854 62.060 7.393 52.240 611.801 70.636 62.266 7.167 52 .437 62 .529 71.548 TRUE VALUE 9 1 . 3 8 79.12 70.27 63.12 54.19 Mean SDmc 10% 50% 90% 93 .208 1.235 91 .538 93 .315 94.639 93 .429 1.215 91 .844 9 3 . 4 4 7 9 4 . 9 6 6 83 .779 3.676 79.529 83 .912 88 .096 84.198 3.509 79.161 84.566 88.228 75.768 5.638 69 .113 76.422 81.769 76.264 5.273 68 .345 76.981 82.272 68.661 7.148 59.861 69.314 76.779 69.202 6.612 59.475 70.116 77.042 59.389 8.647 48.238 59.892 68.938 59 .957 7.993 48 .305 60 .806 69.681 * The I npu t p a r a m e t e r e s t i m a t e s were t h e r e s u l t s o f e s t i m a t i o n u s i n g t h e 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 M e t h o d : n=100 f o r w e e k l y e s t i m a t e s and n=700 f o r d a i l y e s t i m a t e s . O 71 decreases i n dispersion of bond value d i s t r i b u t i o n s do not appear to r e s u l t - the reduction i n the bond value variance i s t r u l y marginal. The explanation f o r t h i s seemingly anomalous behaviour l i e s i n the c o r r e l a t i o n between the parameters -p a r t i c u l a r l y m and^. Based on the t h e o r e t i c a l s e n s i t i v i t y of bond prices we know that overestimating y/- or m underestimates the bond value. I f now the estimates of m and ^ are negatively correlated, then, to some extent, they have o f f s e t t i n g e f f e c t s on bond valuation. Thus a negative c o r r e l a t i o n between m and^x. could explain t h i s r e s u l t 3 9 . (The c o r r e l a t i o n between the parameter estimates i s addressed toward the end of t h i s chapter)., This i s more evidence i n favour of using weekly; rather than daily i n t e r e s t rate data. 5.4 Results of Monte Carlo Simulations for the <x Unknown Case. So far we have only compared the d i f f e r e n t estimation methods under the assumption that the value of cA were known. For the j o i n t estimation of a l l the parameters (m »/A. , cr , crt ) , we can only compare the SL method and the SS method, as the t r a n s i t i o n probability density corresponding to general <* values i s not known. The d e t a i l s of parameter estimation in the SL model have been set out i n Appendix 2 and for the SS density method, i n Appendix 10. 3 9 The e f f e c t may be understood more i n t u i t i v e l y by considering the return on a p o r t f o l i o of 2 negatively correlated s e c u r i t i e s . Increasing the variance on the returns of the i n d i v i d u a l s e c u r i t i e s need not cause proportional increases i n the variance on the return of the p o r t f o l i o . 72 For the SL method, the n=500 and 945 cases were estimated (100 t r i a l s each), but f o r the SS method, only n=945 case was estimated, as the computation cost was very high, and no additional insights seemed forthcoming by doing the estimation for other sample s i z e s . The summary s t a t i s t i c s for the estimated parameter d i s t r i b u t i o n s are presented i n Table XV. The following remarks about the r e s u l t s are i n order: 1) Comparing the estimates of m and y* from the S.L. method, i n the c* unknown case with those i n the cA= yv(known) case, we f i n d that the resulting parameter di s t r i b u t i o n s are almost i d e n t i c a l . This indicates something about the i n t e r r e l a t i o n s h i p between the estimated parameters. The c o r r e l a t i o n between the parameters i s discussed i n the next section, but t h i s result points to the p o s s i b i l i t y that m and are uncorrelated with cX . 2) The estimate of tr 2 does not appear biased but the dispersion seems large, p a r t i c u l a r l y when compared with the oV = y-2_ (known) case. The reason for t h i s i s the close r e l a t i o n s h i p between <r2 and o< . The instantaneous variance of the process i s rzfU and, understandably, when cA i s free to adopt a range of values, the value of cr 2 has to adjust accordingly, for a given data sample,, 3) The estimate of jx by either the SL or the SS method appears the same. 4) The estimate of o( by both methods appears unbiased, though the SL estimate has a lower dispersion. TABLE XV ESTIMATION OF PARAMETERS FOR a UNKNOWN CASE METHOD (0.09517) (0.50) (194.389) (0.0077617) (0.78427x10 ) Simple Mean 0.08876 0.49385 639.87 0.01556 1.2145 10% 0.07138 0.22481 88.97 0.00603 0.1925 L i n e a r i z a t i o n o Median(50%) 0.09007 0.48476 349.92 0.01392 0.7559 \ J LO 90% 0.11521 0.71715 1545.86 0.02491 2.0897 Method n c SDmc 0.04530 0.19318 979.71 0.01020 1.9348 SDi* 0.00929 0.16915 - 0.00728 0.6172 Mean 0.09371 0.49204 366.34 0.01212 0.8859 .10% 0.07703 0.34414 129.71 0.00657 0.3589 Median(50%) 0.09390 . 0.50129 295.00" - 0.01084 0.7823 m ^j- 90% 0.10949 0.62975 659.36 0.01884 1.4744 CT* SDmc 0.01276 0.11480 297.43 0.00503 0.51171 c SDi 0.00800 0.11141 - 0.00478 0.4064 Steady State Mean 0.09167 0.56049 1036.18 10% 0.08863 0.04242 14.3348 Density Median(50%) 0.09211 0.49884 210.624 90% 0.11142 1.40816 6626.37 Method SDmc 0.01022 0.42979 2173.70 SDi 0.00061 0.10516 80.8525** The SDi f i g u r e reported i s the median of the SDi from each t r i a l not the mean. ** This i s the median - the mean SDi w a s 625.458 i n t h i s case. - The f i g u r e s i n the a 2 column have been m u l t i p l i e d by 10^ 74 5) Comparing the composite parameter (2ra/<f2), the SL method appears to give estimates having a lower bias and dispersion. However, the median (which i s also a measure of location), of the SS estimate i s very reasonable. I t seems that the SS method has a tendency to produce extreme estimates* 0., 6) Using SD,; as a measure of the true asymptotic variance of the parameters we see that, i n the SL case, n-945 appears to s a t i s f y the asymptotic sample siz e c r i t e r i a , i n that SD^ for a l l parameters i s very close to SD n c. For smaller sample sizes (see n=500), SD^ i s an under estimate of the asymptotic standard deviation. For the sake of completeness, we present i n Table XVI a comparison of the d i s t r i b u t i o n of the estimated parameters using " d a i l y " versus "weekly" data. For t h i s case, only the SL method was used., The only parameter of int e r e s t here i s o( . As with the other parameters, the improvement i s only with respect to the dispersion of the estimated parameter d i s t r i b u t i o n . We s h a l l soon see whether t h i s improvement i n accuracy makes any s i g n i f i c a n t difference to the bond value. Before we conclude t h i s section, we present some r e s u l t s on the s e n s i t i v i t y of the pure discount bond value to variations i n *° The SS method i s based on the assumption that the stationary density i s not the t r i v i a l P(r) .= 0, which obtains when either singular boundary i s absorbing f o r some parameter values. Whenever the non-linear search procedure (to i d e n t i f y the maximum of the joint l i k e l i h o o d function) takes on parameter values which correspond to an absorbing ba r r i e r at either singular boundary, the SS method breaks down., If the range of the parameter space where we get absorbing b a r r i e r s were known, a constrained maximization could be done. This however i s not the case. The breakdown of 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) a (0.50) 2m/o (194.389) (0.0077617) (0.78427xl0 - 4) Simple L i n e a r i z a t i o n Method Simple L i n e a r i z a t i o n Method o >-( o >-i n < II a a ^ o w II Mean 10% Median(50%) 90% SDmc SDi Mean 10% Median(50%) 90% SDmc SDi 0.09729 0.07797 0.09441 0.11430 0.00907 0.08876 0.07138 0.09007 0.11521 0.04530 0.00929 0.48586 0.38203 0.48552 0.57420 0.07084 0.06507 0.49385 0.22481 0.48476 0.71715 0.19318 0.16915 458.80 118.91 330.48 969.65 405.28 639.87 88.97 349.92 1545.86 979.81 0.01534 0.00515 0.01315 0.02946 0.00956 0.00Z25 0.01556 0.00603 0.01392 0.02491 0.01020 0.00728 0.7903 0.4487 0.7399 1.1368 0.2605 0.2200 1.2146 0.1925 0.7559 2.0897 1.9348 0.6172 The f i g u r e s i n the a 2 column have been m u l t i p l i e d by 10 4, cn o(. In t h i s context, only the " t h e o r e t i c a l " s e n s i t i v i t y r e s u l t s are presented in Tables XVII and XVIII. It was f e l t that no addit i o n a l information could be gained by presenting the "empirical" s e n s i t i v i t y . In Table XVII we present the e f f e c t on discount bond values of varying cA about the value Vi. , with the other parameters kept fixed at t h e i r true values* 1. It can be seen that increasing ^decreases the bond value. Comparison with Table X {effect on bond value by varying cr2) , shows that the same dire c t i o n of e f f e c t on bond values i s caused by a decrease i n < r 2 . Since r has a numerical value l e s s than 1.0; increasing o( decreases and so also the instantaneous variance of the interest rate: cr2r 2 oV. Clearly, decreasing a~ 2 has the same ef f e c t on the variance. ,. We would expect t h i s close rel a t i o n s h i p to r e f l e c t i t s e l f by a high posi t i v e c o r r e l a t i o n between the estimates of c r 2 and cK ., The question of the rela t i o n between the estimated parameters i s addressed i n the next section but, f o r the present, l e t us accept that i f <?< i s overestimated, o~z would correspondingly be over estimated (assuming that there i s no bias i n i d e n t i f y i n g the t o t a l variance)., Let us represent by <T0 , the true value of er corresponding to an o\ value of y z , and <r^ as the value of cr corresponding to any other cX value estimated. Given our assumption, we would expect * l I t may be noticed that the 0% error bond price i n Table XVII and XVIII i s s l i g h t l y d i f f e r e n t from that i n Tables VIII, IX and X. This i s because, the values i n that column i n Tables XVII and XVIII have been computed using a f i n i t e d ifferencing method to solve the bond equation. This was done, as what we want to present i s the e f f e c t of variations i n <A , and f i l t e r out the ef f e c t due to the solution technigue employed. TABLE XVI I THEORETICAL SENSIT IV ITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN a * (02 HAS NOT BEEN 'CORRECTED' TO REFLECT THE ERROR IN a) ERROR IN a - 2 5 % - 5 % 0% +5% +25% CURRENT INTEREST TIME TO MATURITY IN YEARS BOND PR ICE % ERROR BOND PRICE % ERROR BOND * * PRICE BOND PRICE % ERROR BOND PRICE % ERROR r=y/2 1 3 5 7 10 9 6 . 9 6 8 9 . 4 0 8 1 . 6 6 74 .37 6 4 . 5 0 0 . 0 0 5 3 0 .0862 0 .2572 0 .4812 0 .8634 96 .95 89 .33 81 .47 74.04 64 .00 0 .0004 0.0074 0.0227 0.0431 0.0785 96 .95 89 .32 81 .45 74.01 63.95 96.95 89.32 81.44 73.99 63.92 -0 .0003 -0 .0051 -0 .0158 -0 .0303 -0 .0554 9 6 . 9 5 89 .31 81 .42 73.95 63 .85 - 0 . 0 0 0 8 - 0 . 0 1 3 8 -0 .0434 - 0 . 0 8 3 7 -0 .1541 1 3 5 7 10 9 5 . 1 8 86 .31 78 .35 71 .18 61 .66 0 . 0 0 6 3 0 .0963 0 .2788 0 .5125 0 .9040 95 .18 .86.23 78.16 70.85 61.15 0 .0006 0.0087 0.0254 0 .0470 0.0835 95 .18 86 .23 78.14 70.81 61 .10 95.17 86.22 78.12 70.79 61.07 - 0 . 0 0 0 4 -0 .0062 - 0 . 0 1 8 0 - 0 . 0 3 3 4 - 0 . 0 5 9 3 95 .17 86 .21 78 .10 70 .75 61 .00 - 0 .0011 - 0 . 0 1 7 3 - 0 . 0 5 0 7 -0 .0941 -0 .1672 r=2y 1 3 5 7 10 9 1 . 2 3 78 .56 69 .62 6 2 . 5 7 5 3 . 8 8 0 .0117 0 .1685 0 .4502 0 .7656 1.2288 .91.22 78 .44 69.34 62.14 53 .29 0.0011 0.0157 0.0424 0.0724 0.1164 91 .22 78 .43 69 .31 62 .09 53 .22 91.22 78.42 69.29 62.06 53 .18 -0 .0008 -0 .0113 -0 .0305 -0 .0521 -0 .0836 91 .21 78.41 69.25 62 .00 53 .10 - 0 . 0 0 2 2 - 0 . 0 3 2 2 - 0 . 0 8 7 2 - 0 . 1 4 9 0 - 0 . 2 3 8 9 The o t h e r p a r a m e t e r s o f t h e 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 t r u e See f o o t n o t e XXI . v a l u e s . TABLE XVII1 -HFORFTICAL SENSITIVITY PHBF. DISCOUNT BOND PRICES TO ERRORS IN &_ ERROR IN a CURRENT INTEREST r = u/2 r =u r =2u TIME TO MATURITY IN YEARS 1 3 5 7 10 1 3 5 ' 7 10 1 3 5 7 10 -25% BOND % PRICE ERROR 96.59 89.34 81.49 74.08 64.05 95.18 86.24 78.17 70.88 61.20 91.22 78.45 69.36 62.17 53.33 0.0011 0.0169 0.0490 0.0901 0.1591 0.0011 0.0166 0.0486 0.0898 0.1592 0.0019 0.0269 0.0724 0.1245 0.2031 -5% BOND 0% BOND ' PRICE ERROR PRICE 96.95 0.0002 96.95 89.33 0.0026 89.32 81.46 0.0076 81.45 74.02 0.0140 74.01 63.97 0.0248 63.95 95.18 0.0002 95.18 86.23 0.0026 86.23 78.14 0.0076 78.14 70.82 0.0141 70.81 61.12 0.0250 61.10 91.22 0.0003 91.22 78.44 0.0044 78.43 69.32 0.0119 69.31 62.10 0.0203 62.09 53.24 0.0330 53.22 +5% BOND PRICE 96.95 89.32 81.45 74.00 63.94 95.18 86.23 78.13 70.81 61.09 91.22 78.43 69.30 62.08 53.21 % ERROR -0.0001 -0.0023 -0.0067 -0.0124 -0.0220 -0.0002 -0.0023 -0.0068 -0.0126 -0.0223 -0.0003 -0.0040 -0.0108 -0.0184 -0.0298 +25% BOND PRICE 96.95 89.32 81.43 73.98 63.89 95.17 86.22 78.12 70.78 61.05 91.22 78.42 69.28 62.04 53.16 % ERROR -0.0006 -0.0090 -0.0265 -0.0493 -0.0879 -0.0006 -0.0093 -0.0273 -0.0505 -0.0897 -0.0012 -0.0167 -0.0447 -0.0762 -0.1226 * The other parameters of the Interest rate process assume their true values. ** See footnote XXL Co 79 Thus <JC now represents the "corrected" value of cr as c< i s varied. C l e a r l y , <TC varies with r, but we could assume that on average r i s expected to remain around . Table XVIII presents the s e n s i t i v i t y of discount bond values to v a r i a t i o n i n , where <r has been "corrected" as indicated above. As expected, t h i s "correction" has reduced the e f f e c t of a variation in o( on discount bond values. However, what i s more important i s the fact that the net e f f e c t i s small. 5 • 5 The Relation Between the Interest Rate Process Parameters F i n a l l y , before concluding t h i s chapter, we take a b r i e f look at the rel a t i o n s h i p between the parameters (m, <T , oS) • There are two cl o s e l y interconnected points 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 correlation between the estimated values of these parameters, given a data sample? b) In what interconnected way do these parameters a l t e r the c h a r a c t e r i s t i c s of the interest rate process dynamics? One way to try to answer the f i r s t question would be to calc u l a t e the correlation matrix between the parameters estimated during the simulation. Since the SL method fo r the n=945 case displayed close to asymptotic behaviour, the c o r r e l a t i o n * 2 between the parameters for that case was computed * 2 For the n=945 case, we performed 100 simulations and so generated 100 estimates of the parameters (m , JX , <r , ). The table represents the correlation between values. and found to be rv\ /A- (T^ jX -0.0 207 cr 2 0.2081 0.0375 0.1725 0.0875 0.9339 He can see that cr*" and o\ are almost per f e c t l y correlated (which i s as expected), but apart from that any other corr e l a t i o n s appear to be quite small. Another approximate (and quite ad hoc) method of estimating the c o r r e l a t i o n matrix between the parameters i s set out i n Appendix 9, based on that method, the c o r r e l a t i o n matrix i s •u jiK cr 0.0 0.0 0.9877 There i s agreement between the two estimates of the asymptotic c o r r e l a t i o n matrices i n broad q u a l i t a t i v e terms. The second estimate (based on the approach presented i n Appendix 9) implies that the parameters i n the variance and d r i f t terms of the d i f f u s i o n equation are t o t a l l y independent of each other. This would explain the e a r l i e r observation, namely, the s i m i l a r i t y of the d i s t r i b u t i o n s of estimated values of m and between the cL^'/z. known case and the cA unknown case, i n the S.L. method. The two important c h a r a c t e r i s t i c s that were anticipated are borne out i n both cases, i e . -0.1582 0.0 0.0 81 1) fa and • JJ^ are negatively correlated. This was anticipated, based on t h e i r combined e f f e c t on bond values. 2) ( T 2 and o\ are very highly p o s i t i v e l y correlated. Further insights into the nature of 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 looking at the way i n which each of them aff e c t s the i n t e r e s t rate process dynamics. , For a d i f f u s i o n process, a l l information about the process dynamics i s contained in the t r a n s i t i o n probability density function. To investigate how i t i s altered by changing the parameters, we consider the following parameter values: Parameters Set 1 Set 2 f ( = 2»V<r >) 460.098 311.398 0.06904 0.06905 °\ 0.36202 0.43333 hl\r^c 1314.92 1314.71 On a particular data sample these two parameter sets gave v i r t u a l l y i d e n t i c a l values for the log of the j o i n t l i k e l i h o o d function, using the SS method. This s i t u a t i o n arose while performing routine preliminary t r i a l s with the SS method for the °<7\ unknown case. It i s well known that nonlinear optimization routines provide no guarantee that convergence to a optimum w i l l occur. Further, even i f convergence i s obtained, one i s never sure whether the point i s a l o c a l or a global optimum.. To investigate the behaviour of the p a r t i c u l a r f u n c t i o n a l form of the l i k e l i h o o d function on some data samples, (chosen from the generated sequence) dif f e r e n t available nonlinear optimization 8 2 methods were applied to see whether (using d i f f e r e n t algorithms), a) convergence was always to the same point i n the parameter space, i r r e s p e c t i v e of the s t a r t i n g parameter values, and b) whether the speed of convergence di f f e r e d across d i f f e r e n t algorithms. It was found that the guasi-Newton method (the Fletcher algorithm) was the quickest by f a r , and in qeneral, the point to which converqence was obtained, appeared to be the "global" optimum..,, We have a case where the stationary p r o b a b i l i t y density corresponding to very d i f f e r e n t parameter values i s almost i d e n t i c a l . This was further v e r i f i e d by p l o t t i n g the stationary d i s t r i b u t i o n corresponding to these parameter values, and the density functions were seen to v i r t u a l l y coincide. This implies that, given a data sample, the SS method may not be able to i d e n t i f y an unique parameter set that f i t s i t - i t may i d e n t i f y one or more equivalent points in the parameter space* 3. The more relevent question, however, i s whether the t r a n s i t i o n * 3 An attempt was made to find out whether, correspondinq to t h i s data sample , the two "optimum" parameter sets represented two independent "peaks". To investiqate t h i s , a close mesh qrid (50x50) was placed 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 function was evaluated over the q r i d ,to see i f there was a "dip" between the peaks., It was observed that there was actually a plateau i n the l i k e l i h o o d function., This indicated that the 2 "optima" were located purely as a matter of roundinq error in 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 part, a function of the data sample. This would also explain some of the extreme values obtained durinq the simulations. 83 probability density corresponding to these parameter values i s very d i f f e r e n t . I f the t r a n s i t i o n probability density corresponding to d i f f e r e n t parameter values were very s i m i l a r , the parameters would not be orthogonal to each other and there would be a loss of richness i n the present model. The t r a n s i t i o n probability density function i s the solution to the Fokker-Planck equation, which we have not been able to solve for general values of ck . Thus a f i n i t e differencing method was employed to solve the FP equation. The objective of the exercise was to try and see whether the t r a n s i t i o n p robability density functions corresponding to the above two parameter sets could be made almost i d e n t i c a l * * . 9e also require a s t a t i s t i c to measure the "closeness" to each other of the two t r a n s i t i o n p r o b a b i l i t y density functions. The "matchinq" c r i t e r i o n was to minimize the area of non-overlap, between the computed t r a n s i t i o n density f u n c t i o n s * 5 . It was found that the area of non-overlap between the t r a n s i t i o n p robability density functions corresponding to the two parameter ** The approach was as follows. Parameter set 1 was used as the basis 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 of m, and <TJ Z. The t r a n s i t i o n probability density function f o r ©i = (m, , <rx 2 , <*,) was evaluated. Then, keeping ^» A1?- a n d constant,different values of vxz and c^.2 were chosen, and the t r a n s i t i o n probability function for 02.= (n 2 *M1f <Tuzr^%) was evaluated. The objective was to match the t r a n s i t i o n probability densities corresponding to the two dif f e r e n t parameter sets.„ 84 sets could be brought down to about 7%, for r 0 = f*-* 6. However when r0*f^at these "matched" parameter values, the area of non-overlap increases greatly. This i s as expected - what i s more informative i s the extent to which the shape of the t r a n s i t i o n probability density i s changed by a proportional change in each parameter. This is p i c t o r i a l l y represented in Figures 2 and 3. This i s an i n d i c a t i o n that the t r a n s i t i o n probability density function i s not very s i m i l a r for d i f f e r e n t parameter values -given a data sample, we could expect an unique parameter set to maximize the l i k e l i h o o d function. As expected, both cA and <r 2 a f f e c t the dispersion of the density function, a{ more so than <r 2 . This i s because a~ 2 changes the variance element m u l t i p l i c a t i v e l y , whereas o{ chanqes the exponent, which has a qreater effect on the variance, p a r t i c u l a r l y since r i s always far from unity i n numerical value. No pictures are presented f o r chanqes in the t r a n s i t i o n probability density corresponding to changes i n /x , as t h i s affects only the l o c a t i o n . It can be seen that large changes i n m, when.r=/^ f produce hardly any change. However increases i n m make the function s l i g h t l y more peaked as m i s the speed of reversion to the mean. When r *J^t changes i n m s h i f t the location because of the skewing e f f e c t of the mean The area of overlap i s given by J^abs[ F (r K„,e)-F (r j - r^J ] dr, where Y{T\rep) represents the t r a n s i t i o n p r obability density function corresponding to parameter set 0 . i t may be noted that the area under either t r a n s i t i o n density function adds up to 1.0. Thus the area of non-overlap indicates d i r e c t l y the f r a c t i o n of t o t a l area under each curve f o r which the two functions do not match. ** The t r a n s i t i o n p r o b a b i l i t y density i s represented as F (rj. , 1 1r 0 ,9) . Thus, i t i s a function of r e and t as well as the parameter set 0 ={m, p-f o~, d\) • Here t was chosen equal to 1 week. 85 F I G U R E . 2 Sensitivity of the Transition Probability Density Function to Change In a 'Sensitivity of the Transition Probability Density Function to Chance in o F I G U R E 3 87 reversion property (which i s s i m i l a r to changing jx a very small amount).... To summarize, i t appears that ^ and ck are the important parameters i n determining the location and dispersion, respectively. m has a marginal ef f e c t oh both, whereas c r 2 a f f e c t s only dispersion. ,. 88 CHAPTER 6: THE IHTJRjgST PATE AND BOND PRICE DATA 6.1 The Short Term Riskless Interest Rate By d e f i n i t i o n , the short term (instantaneous) risk less, rate of return i s the y i e l d to maturity on a default free discount bond, maturing the next instant i n time. In actual practice, such a security does not continuously ex i s t (and i s not available in any case). The bond valuation models developed i n Chapter 2, and the estimation theory developed i n the preceeding chapters, a l l require that we know something about t h i s unobservable e n t i t y . A suitable proxy for the short term r i s k l e s s rate of i n t e r e s t would be the yield to maturity on very short maturity Federal Government bonds, as they could be treated as t o t a l l y default free with respect to p r i n c i p a l payment on maturity. However, the only pure discount Federal Government bonds outstanding are Treasury B i l l s . Apart from quotations i n secondary markets, these have a minimum maturity of 91 days, which brings us to two cl o s e l y related matters, v i z . (a) what time to maturity may be treated as "instantaneous" and (b) what should our frequency of measurement be? Treasury b i l l s are not very actively traded i n secondary markets i n Canada., A few conjectures could be put forward to try and explain t h i s . To s t a r t with , a widespread demand does not appear to have developed. Of the t o t a l Government of Canada Treasury b i l l s outstanding over the l a s t several years, about 16% were held by the Bank of Canada, 74% by the chartered banks,and 1% by the Government of Canada accounts, with only 9% 89 accounted for by a l l the other f i n a n c i a l and non f i n a n c i a l i n s t i t u t i o n s and i n d i v i d u a l s (figures obtained from Neufeld [53]). Chartered banks have always been p r i n c i p a l holders, as they are constantly i n need of very secure short term investment opportunities. Since there are only f i v e major chartered banks i n Canada, the number of active participants i s greatly reduced. Furthermore, Canadian banks are required by law to maintain secondary reserves at prescribed l e v e l s , which tends to reduce trading i n short term government s e c u r i t i e s . In the-U.S., , however, the Treasury b i l l market i s very active and deep due to the following f a c t o r s : a) Banks do not have to maintain secondary reserves. = b) There are very many more commercial banks a c t i v e l y trading i n the market (due to the unit banking system, as opposed to the branch banking system of Canada). c) The U.S. d o l l a r i s a major reserve currency as well as the denomination of a large portion of i n t e r n a t i o n a l trade. Thus, several foreign investors (both corporate and government) enter the short term U.S. d o l l a r denominated bond market. These factors could explain the r e l a t i v e i n a c t i v i t y i n secondary markets for Canadian Treasury b i l l . Given the present state, i t i s to be expected that transactions prices i n secondary markets, would be d i f f i c u l t to obtain. No record of sale prices for Treasury b i l l s i n secondary markets, were av a i l a b l e either with security dealers or from the Bank of Canada. From considerations of r e l i a b i l i t y of the data, (and keeping in mind that we require equispaced 90 observations) i t was f e l t that treating the y i e l d to maturity on the 91-day Treasury b i l l s , o n the date of issue, as a proxy for the short term i n t e r e s t rate was the best a l t e r n a t i v e . The d i s t i n c t advantages of t h i s choice are that (i) for a l l p r a c t i c a l purposes the term structure over such short maturities as 91 days may be treated as v i r t u a l l y f l a t , so that the y i e l d on the 91 day pure discount bond may be assumed equal to the instantaneous rate ( i i ) the y i e l d to maturity i s computed based on actual transaction prices (which could be treated as equilibrium prices), rather than Jbased on quotes. I f we did want to use Treasury b i l l prices from secondary markets, there i s no guarantee that we can consistently get y i e l d s computed on actual transaction prices. The e f f e c t of using the y i e l d to maturity on a 91-day discount bond as a proxy f o r the short term i n t e r e s t rate i s b r i e f l y investigated in Appendix 11. The error appears to be small., ' Having chosen the 91-day Treasury b i l l as our short term (instantaneously maturing) asset, the matter of frequency of observation i s automatically s e t t l e d . Treasury b i l l s are issued weekly and the y i e l d s , based on average sale price, are reported i n the Bank of Canada Review. Given the source of t h i s information, the data are very r e l i a b l e . Other proxies for the short term interest rate were considered, such as the interbank loan rate and the d a i l y c a l l money rate. There were several problems on account of which they had to be dropped from serious consideration: 1) There was no r e l i a b l e source from where these data could be obtained. 91 2) Most money market dealers could only give bid and ask rates with a rather large spread. Taking the mean of the bid and ask rates could be meaningless i f no actual transactions took place. 3) Even i f i t were possible to get some data on the other rates, no s e r i e s on them could be constructed going back almost 20 years* 7 - the time when the f i r s t retractable/extendible was issued by the Government of Canada. 4) These rates have a l o t of "noise" i n them, which has l i t t l e or nothing to do with changes i n bond prices. For example, they are strongly influenced by the flow of very short term c a p i t a l between the U.S. and Canada (called "weekend money"). 6.2 Price Series on B e t r a c t a b 1 e/ Ex t e n d i b 1 e Bonds In the Canadian market, there are Federal, P r o v i n c i a l and corporate (including the issues of the chartered banks) retractable and extendible bonds outstanding. For a l l the Federal bonds, weekly prices are reported i n the Bank of Canada Review. Due to the large volume of each of these issues and t h e i r marketability, an active secondary market e x i s t s for them. The prices reported i n the Bank of Canada Review are, more often than otherwise, average actual transaction prices, at midday * 7 Bid and ask prices on daily c a l l money rates were available going back about 18 months from the present.. The dealers do not keep them on record f o r long. The spread between the bid and ask rates was around 0.2% to 0.4% on an annualized basis. 92 every Thursday. In the case of the Pr o v i n c i a l and corporate bonds, however, the issues are much smaller and very many more i n number. The problems associated with putting together a data base on Pr o v i n c i a l and corporate retractables/exteedibles may be summarized as follows: 1) There are very many issues outstanding but not widely traded, so that a continuous s e r i e s of even bid and ask prices i s not avai l a b l e . 2) Even when a v a i l a b l e , (quoted in the Fina n c i a l Post) what i s indicated are bid and ask prices (with large spreads). There i s no guarantee that i f transactions took place they would be between those prices; i e . , the quotes do not always represent firm commitments to transact. 3) The available data on Provincial and corporate bonds are not , ,compatible N with the data on the short term i n t e r e s t rate . The prices quoted i n the Financial Post are Friday closing values, whereas the Bank of Canada Beview observations on the short term in t e r e s t rate are Thursday mid-morning prices . Thus, model prices for the bonds (using the models of Chapter 2 ) , would be Thursday mid-morning prices, whereas the data on market prices would be Friday closing values. Consequently, we could not s t r i c t l y evaluate the performance of the model i n valuing these bonds. 4) Whereas the Federal bonds are very a c t i v e l y traded, P r o v i n c i a l and corporate bonds are not. The assumption of continuous trading opportunity, upon which the model i s based, i s violated. The impact on bond prices seems n o n t r i v i a l . This shows up when we compare yi e l d s on Federal and comparable 9 3 P r o v i n c i a l bonds, where default r i s k i s of nearly the same l e v e l . The y i e l d difference on some issues i s as high as 0.5% {on an annualized b a s i s ) . This i s an indication that marketability of the bonds i s an important determinant of the i r value. Therefore, the models developed i n Chapter 2 would be inappropriate for valuing P r o v i n c i a l and corporate issues. 5) Corporate bonds have default r i s k , over and above intere s t rate r i s k . The theory developed i n the existing l i t e r a t u r e for valuing such bonds i s to treat them as functions of r, the value of the firm, and time to maturity. Putting together a data series f o r the value of a firm has several obvious problems. Since complete data on a l l Federal retractable/extendibles issued to date were a v a i l a b l e , i t was decided to confine our attention to them alone - to the exclusion of the P r o v i n c i a l and corporate issues. Table XIX gives some d e t a i l s on a l l the retractable/extendibles forming our sample. I t includes a l l such issues by the Government of Canada. Data have been colle c t e d for each bond s t a r t i n g within a week of the date of issue, and extending to the exchange or r e t r a c t i o n date. In cases where data were available beyond the l a s t exchange date, the i n d i c a t i o n i s that the short bond was preferred to the long bond by the majority of the investors. For the purpose of t h i s study, these bonds have been named B1, E1 through E19 - B for retractable and E for extendible. I t may be noted that for H1, E2, E3 and E4, observations cease even before the option expiry date. The matter was investigated by the l o c a l representative of the Bank of Canada, and i t appears that, (for some unknown TABLE XIX D E T A I L S O F D A T A S A M P L E O F R E T R A C T A B L E / E X T E N D I B L E B O N D S BOND LONG BOND Maturity Coupon SHORT Maturity BOND Coupon OPTION PERIOD ISSUE DATE DATA AVAILABLE FROM TO t Rl Jan.1,1963 4.00 Retractable on any interest date between Jan.1,1961 and Jan 1, 1962 giving 3 months notice Jan.1,59 Jan.7,59 Jan.27,60 56" E l Oct.1,75 5.50 Oct.1,60 5.50 On or before June 30,60 Oct.1,59 Oct.7,59 May 25,60 34 E2 . Oct.1,75 5.50 Oct.1,62 5.50 On or before 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 before 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 or before 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 or before 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 before 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 or 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 to 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 28,76 260 E l l July 1,82 7.50 July 1,77 7.00 July 1,76 to Dec.31,76 July 1,72 July 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 to Sep.30,78 Apr a , 74 Apr.3,74 Nov.9,77 189 E15 Apr.1,84 9.25 April,78 9.25 On or before Jan.1,78 Oct.1,74 Oct.2,74 Nov.9,77 162 E16 Feb.1,82 9.25 Feb.1,77 9.25 On or before 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 29,79. July 1,75 July 2,75 Nov.9,77 125 E18 Feb. 1,80 9.00 Feb.1,78 9.00 On or before Oct.31,77 Oct.1,75 Oct.1,75 Nov.9,77 112 E19 Oct.1,85 9.50 Oct.1,80 9.00 Jan. 1,80 to Jan."30, 80 Oct.1,75 Oct.1,75 Nov.9,77 112 - A l l issues are by Government of Canada. The above sample constitutes the to t a l sample on retractables/extendibles issued by the Government of Canada. - Source of data was Bank of Canada. 95 TABLE XX DETAILS OF DATA SAMPLE OF STRAIGHT COUPON BONDS DATA COLLECTED BOND Coupon & Maturity From To # F l 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 Jul 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 Jul 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% Jul 1, 1970 Dec 6, 1967 May 6, 1970 127 F l l 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 last column represent the number of weekly data points for which data was collected. - Source of data was Bank of Canada Review. 96 reason) the data on these bonds for the remaining period were not available. 6.3 Price Series on Ordinary Federal Bonds Apart from the price series on a l l retractable/extendible bonds, prices of ordinary (non-callable) coupon bonds* 8 are also required for a) estimating the utility-dependent aggregate l i q u i d i t y premium parameters b) conducting tests of market e f f i c i e n c y based on model and market prices of the retractable/extendible bonds. To capture as much information as possible on the term structure of interest rates during the period 1959 to the present, every e f f o r t was made to choose the bond sample such that, at every instant i n time, at least 5 points on the term structure (between 1/2 year and 18 years to maturity) were represented. Table XX indicates some d e t a i l s on the sample of straight bond data. * 8 The reason for s p e c i f i c a l l y choosing non-callable bonds i s for computational convenience i n the estimation of the li q u i d i t y / t e r m premium parameters. , This w i l l become evident when we address that problem i n the next chapter. 97 CHAPTER 7: EMPIRICAL TESTING OF BOND VALUATION MODELS 7•1 Estimated Parameters For The Interest Rate Process To estimate the instantaneously r i s k free i n t e r e s t rate process parameters (m,/*-, (Ttck) the weekly s e r i e s of y i e l d to maturity on 91-day Treasury b i l l s was used., 990 weekly data points s t a r t i n g from January 7th, 1959, to December 21st, 1977, were used i n the estimation. I n i t i a l l y , the primary object was to estimate cA ; The SS and SL methods were used on the t o t a l data, and the estimated parameter values were*9 Parameters SS Method SL Method (=2m/cr2) 8183.48 1655.75x105 ^ 0.9974x10-3 0.1334x10-2 ^ 0.4938 -0.2195 0.2174x10-2 cr^ - 0.2626x10-*° The negative <V value estimated by the SL method i s not reasonable for an i n t e r e s t rate process. The estimate of a-2 has, therefore, correspondingly decreased. To investigate further, the t o t a l data sample was divided into two subperiods (each consisting of 495 data points), and the parameters were re-estimted using the SS and the SL methods * 9 The SS method was restarted at d i f f e r e n t parameter values, but the non l i n e a r optimization algorithm used (Fletch guasi-Newton method) always converged to the above parameter values. This appears to indicate that these co-ordinates uniquely maximize the j o i n t l i k e l i h o o d of the given data, i n a parameter range that appears reasonable for an i n t e r e s t rate process. 98 on these two subperiods. The estimated parameter values are as follows: Parameters f> (=2m/cr 2) jA. X 1 0 3 cA ra (x10 2) a- 2 (x10*) Even i n the two subperiods, the SL method estimates a negative <A value. The SS method, however, gives a consistent value of o<^ 0.4 . The value of jx i s lower i n subperiod 1, which i s as expected 5 0. This i s because in t e r e s t rates have been more or less consistently increasing over the past several years. The estimate of jx by the SS method i s , however, always lower than the SL estimate. This could be attributed to the estimation procedure. In the SL method, Jx i s the value towards which the process i s moving to s t a b i l i z e , whereas i n the SS method y~ * s the mean of the sample points (as pointed i n the l a s t footnote). Thus when i n t e r e s t rates are r i s i n g , j x as estimated by the SL method would be higher than the SS estimate so I t might be interesting to r e c a l l from Appendix 7 that the estimate of jx for the SS method for (A = </z. was the mean of the data points. This was because the SS density was the Gamma density. The actual mean of the data points for the two subperiods i s 0.7884 x 10-3 and 0.1206 x 10~ 2 respectively, which very closely t a l l i e s with / A as estimated by the SS method for the two subperiods. Thus the SS density f o r general c* may be looked upon as a "generalized" Gamma density. SL Method SPJ SP2 627x10+7 188x10+' 1.152 1.3510 -0.1247 -0.0676 0.3451 0.1698 0.0011 0.0018 SS Method SJM SP2 5149.5 2195.0 0.7884 1.2080 0.4032 0.4030 99 over the same period. Since the estimte of ch (and therefore cr 2 as well) from the SL method was unacceptablesi, we consider only the parameter estimates from the SS method. Now, the estimate of <A from the t o t a l data was very close to Yz. . He may therefore assume ^ - ' / T - . This has to the following advantages: a) The t r a n s i t i o n p r o b a b i l i t y density function i s known for As'/z., and so the "exact" approach to parameter estimation f o r the int e r e s t rate process model can be employed. b) Considerable s i m p l i f i c a t i o n i s achieved in the estimation of the investor u t i l i t y dependent parameters i n the pa r t i c u l a r functional form of the term/liquidity premium structure that we adopt l a t e r on. , Further, the adjustment i n i s very small, and based on the analysis i n Chapter 5, we know that the impact on bond valuation i s quite n e q l i q i b l e . Assuming t<= %, the parameters jx , <r 2 ) were estimated over the complete period, as well as over the 2 subperiods. Purely f o r comparison, a l l three methods were used, and the r e s u l t i n g parameter estimates are as follows: s i As pointed out i n Chapter 3, neqative cA values imply that the instantaneous variance of the intere s t rate process approaches oo as r approches zero. Such a model of the int e r e s t rate process i s u n r e a l i s t i c , and therefore unacceptable,. 100 Parameters TRP Method SS Method SL Method a) Total Data: f> (=2m/(r1-) /Mx 103) <ru{x 10») m{x 102) b) Subperiod 1: ^ ( X 103) cr{x 10*) rn(x 102) c) Subperiod 2: /X.(X 103) cMx 10*) ^ (X 102) 73 04.8 1.2930 0.6905 0.2522 10993.9 1.0314 0.9518 0. 52 32 67 05.2 1.3753 0. 4322 0.1449 8183.0 0.9974 20730.0 0.7884 8564.0 1.2070 924 4. 8 1.2320 0.6 886 0.3183 14099.9 0.9771 0.9522 0.6713 7824.6 1.3530 0.4 266 0.1669 As expected, the parameter values estimated assuming ^=Yi. are almost i d e n t i c a l to those based on the SS method with a general (A . Thus, we assume as the parameters of the i n t e r e s t rate process those estimated using the TBP method over the complete data set, i e . , 0\ =0.5 CT2=0.690494x10-* m=0.25221x10 =0.12934x10-* fo r a l l further analyses on bond valuation. 101 7.2 Solving the Bond Valuation Equation The basic bond valuation equation was developed i n Chapter 2; the p a r t i a l d i f f e r e n t i a l equation was .I a,L <q„ -v- ( b - acj>) (q( - -+ c z - <q2 - O (2.9) where a?a (r) , b=b(r), and <^ represents the instantaneous market perceived price of standard-deviation r i s k . For the in t e r e s t rate process chosen, we have a (r) —•o~J~r and b (r) =m (jjL -r) . We also need to make some assumption about the form of c£(r,t). To sta r t with, l e t us consider the pure expectations hypothesis (PEXP) , whereby <^ >=0. This reduces equation (2.9) to -1_0-V (^| -4 T n o ( ^ - T T ) <q, - T<=j -V Cj. - <qz = 0 ( 7 > 1 ) By imposinq suitable boundary conditions, t h i s parabolic p a r t i a l d i f f e r e n t i a l equation may be solved to y i e l d the bond value G ( r , £ ) . In Chapter 2 we developed the boundary condition correspondinq to maturity date value, and that correspondinq to the retraction/extension feature. They are <$L+,0) ~- I" (7.2a) S C ^ O - ^ ^ C f , ^ ) ^ ^ ^ ^ ) ] (7.2b) where £ =0 i s the lonq maturity, I i s the short maturity, (see diaqram in Chapter 2, paqe ff for more d e t a i l s ) , represents the maturity correspondinq to the l a s t date when the retraction/extension option may be exercised, To recoqnize the 102 p o s s i b i l i t y that the coupon on the long and short bonds could be d i f f e r e n t , we have represented (on the B.H.S.of equation 7.2b) the lonq bond by G and the short bond by H. It was also noted in Chapter 2 that further boundary conditions at v=0 and oO would be required, depending upon the behaviour of the i n t e r e s t rate process at these boundaries. These boundary conditions on the bond value process ( i f required) would have to be consistent with those imposed on the i n t e r e s t rate process at the correspondinq boundaries. For the i n t e r e s t rate process having the parameter values as estimated i n the previous section, both r=0 and cX) are natural boundaries, and so no boundary conditions need be imposed at these points. Thus, we should be able to solve the d i f f e r e n t i a l equation (7.1) using the conditions (7.2a) and (7.2b). However, the solution technique employed requires further assumptions (as w i l l become clear s h o r t l y ) . The solution technique w i l l be the standard i m p l i c i t f i n i t e differencinq approach (see McCraken 6 Dorn [44], Schwartz £63], Brennan S Schwartz [ 10 ]) . The d i f f e r e n t i a l s i n equation (7.1) are approximated by difference equations, y i e l d i n q ' i z I, (tl-0 where , and Wi, are known, and h and k are the discrete increments i n the i n t e r e s t rate and time to maturity respectively. It must be noted here that j 103 increases as we move away from the maturity date. Thus, at the time step just p r i o r to maturity, _^, (which i s the value of the bond on the maturity date) i s known from conditions (7.2a). When we adopt a recursive method for solving f o r G(r,T) from X =0, the system of eguations (7.3) therefore represents (n-1) equations i n (n*1) unknowns (G; ,i=0,...n), at any time step j . To be able to solve for Gc(j' , we need two more eguations. From economic reasoning we know that as interest rates approach cO , bond values approach zero, i e . This observation y i e l d s one more equation to our system* i e . < k j - o j . . , . . . . ™ < 7 -* ' The above equation could hold s t r i c t l y only i f r= «0 were an absorbinq boundary". However, for the parameter values of the in t e r e s t process as estimated, r= does not exhibit absorbinq behaviour. As time to maturity increases, bond value increases at hiqh i n t e r e s t rate values, as there i s a pos i t i v e probability that the i n t e r e s t rate may return to reasonable l e v e l s before maturity. In a s t r i c t sense, when r= °o i s inaccessible there i s no meaning to assigning a value to the bond at that point. Equation (7.4) may be looked upon as a l i m i t i n g value, and in -Referring to Appendix 1, a singular boundary i s inaccessible i n f i n i t e time i f the integrals of h,(r) a n d a h (r) are unbounded. In case however, the i n t e g r a l of 7T(r) were f i n i t e (with the integrals of h,(r) and b t(r) being i n f i n i t e ) then the barrier would be both inaccessible 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 inaccessible and not absorbing. In case IT (r) were integrable, equation (7.4) would be s t r i c t l y v a l i d . 104 that l i g h t i s v a l i d . The f i n a l eguation comes from the behaviour of the bond valuation equation as r approaches zero. An approximation s i m i l a r to the one used to obtain (7.4) would lead to serious b i a s e s S 3 . , The previous approximation was valid at r = °o because (in numerical value) r and jx are very close to zero. Thus, bond values become very small quite rapidly as r r i s e s i n numerical value. This i s not true at r = 0. At r=0, we therefore adopt a continuity arqument: since equation (7.1) i s v a l i d over the t o t a l state space of r , i t i s v a l i d as we make a r b i t r a r i l y close approaches to r=0. Thus, we assume that the l i m i t e x i s t s , and approximate i t by settinq r=0 in (7.1) to y i e l d r Y ^ . O T ) - Giji&.r) + = o (7.5) S t r i c t l y , we are assuminq that the following l i m i t s e x i s t and are as shown: S 3 An equivalent assumption at r=0 to that at r-e° i s that of an absorbinq barrier at r=0. This would imply (for a pure discount bond) B(0,T ) •= 1. The larqer the force of mean reversion, the qreater the error due to such an assumption. 105 The assumptions seem reasonable^*. Thus, we now have (n+1) equations in (n+1) unknowns, at any value of j . ; The solution procedure i s s t r a i g h t f orwardss . A small d e t a i l needs to be highlighted about the f i n i t e differencing approach used. Here, the state variable r has an upper l i m i t of <A , To implement the solution procedure indicated above, the state variable has to be bounded. One approach would be to consider only a f i n i t e segment of r, and impose the condition of equation (7.4), at a suitable f i n i t e value of r. A better approach however, i s to transform r to a new state variable which i s bounded. Consider the transformation s * Ingersoll [39] has solved f o r the pure discount bond correspondinq to the process where = Using h i s r e s u l t , we have _ n i_ . B, , «\\\- Hct ) - ^ p C -A - c ) ] ( a . \ -- B . / 6 Since B i s f i n i t e as r approaches zero, both B and B are f i n i t e as r approaches zero. Be conjecture that introduction of a continuous coupon and a boundary condition of the form (7.2b), would not a l t e r the behaviour of the derivatives of bond value as r->0. 5 S For further d e t a i l s see McCraken 6 Dorn [44] or Schwartz [63]. B r i e f l y , i t i s not necessary to in v e r t an [ (n + 1) x (n + 1) ] matrix to arrive at the solution vector at each time step. Osinq the equations (7.4) and (7.5) reduces the system of equations i n (7.3) to a t r i d i a g o n a l system. A simple solution method i s a v a i l a b l e , which requires subtracting a suitable multiple of each eguation from the precedinq equation in the system. 106 S where 0<s< 1 according as cO>r>0. The equation and the boundary conditions can now be expressed in terms of s, the new state variable. Brennan 6 Schwartz [12 j have adopted the trasformation Here n can be any number so chosen that a large portion of the range of s i s i n the relevant range of r. To c l a r i f y , i f we set n=5, the i n t e r v a l r=(0% to 20%) corresponds to s= (1.0 to 0.5). This allows for greater accuracy i n the relevant range of in t e r e s t rates. For our purpose, n was chosen such that r = ^ corresponded to s=0.65. Further, the whole range of s(0,1) was not equally divided; i e . , h , the grid s i z e on the state variable was not kept constant. The range of s corresponding to r= { JX./3,3JX, ) was divided into 500 equal steps, the range of s corresponding to r = {0, /V3) into 300 steps,and the range of s corresponding to r=(3yiA.,oo) i n t o 200 steps.. Several schemes were t r i e d , and the solution vector of bond values was not too se n s i t i v e to the choice of number of grid points (within reasonable l i m i t s ) . 7.3 Bond Valuation Under the Pure Expectations Model The basic p a r t i a l d i f f e r e n t i a l equation (p.d.e) governing bond valuation under the pure expectations hypothesis (PEXP) i s obtained by setting |=0 i n equation (2.9) of Chapter 2. This 5 107 was developed in the previous section. a l l 20 retractable/extendible bonds <E1 to E19) i n our sample (see Table XIX) were valued using the methods of the previous section. Before we proceed with further analysis, the assumption of continuous coupon payments on bonds needs to be j u s t i f i e d . A l l Federal bonds pay coupon semi-annually, and so coupon payments to the bondholders from the Government are discrete. However, quoted bond prices always exclude the coupon i n t e r e s t , i e . , the buyer of the bond pays the s e l l e r the agreed purchase price for the bond plus the accumulated proportional coupon from the l a s t coupon date to the transaction date. This arrangement i s almost equivalent to continuous coupon payments to the holder 5*. To compare model prices with market prices, an approach alonq the l i n e s of Inqer s o l l [38] was adopted. The mean square error (MSE) may be computed as MSE - ( 7 . 6 , where G.» and G^ are, respectively, the market and model prices. The MSE (or i t s square root (RMSE)) i s broadly i n d i c a t i v e of the lack of f i t between the model and the market prices., Further, a simple reqression of market prices on model prices permits the decomposition of the MSE into three component parts. Consider s * The difference between continuous coupons and t h i s arranqement i s that the holder gets no i n t e r e s t on the coupon, and loses the compoudinq e f f e c t , i e . the "interest on i n t e r e s t " . , I t can be c l e a r l y seen that t h i s omission i s very small, and can safely be iqnored. 108 the regression & = * 4 ft ? c + ec <7.7) then T Z T -J I i-l where G* and G stand for the means of the market and model prices. The three component parts may be i d e n t i f i e d as 1) The part due to bias - attributable to a difference between the mean l e v e l s . 2) The part due to ^ #1, i e . , under ((3 >1) or over responsiveness ( f < 1 ) of the model to market price movements. 3} The part due to re s i d u a l error. The re s u l t s of the regression and the error decomposition for the model based on the pure expectations hypothesis (PEXP) are presented i n Tables XXI through XL i n column 1. Cursory examination c l e a r l y reveals that the predominant element of the HSE across a l l bonds i s bias. This i s also indicated by noting that, for the PEXP model, the mean error [ which i s _ L ^ (G^ * - G-) ] i s consistently negative for a l l bonds. The i n d i c a t i o n i s that the model overprices the bonds, which implies that the markets expected yi e l d on the bonds i s higher than that assumed in the model. One possible explanation i s that the market requires some l i q u i d i t y or term premium i n the expected return on bonds of longer than instantaneous maturity. TABLE XXI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 4% JAN.l, 1963 (Rl) MODEL PURE LIQ. REV.TAX* REV.TAX* C.G. TAX** C.G. TAX** EXP. PREM. (50%) (25%) (10%) (20%) R2 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 EST.INTERCEPT 46.912 46.572 -1.568 30.523 37.074 43.618 (S.E. OF INTR) 3.861 4.424 8.783 5.877 5.407 4.942 FRACTION OF ERROR 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 « 0.188 * The Revenue Tax mode l s i n c o r p o r a t e t h e l i q u i d i t y premium a s s u m p t i o n . " * The C a p i t a l G a i n s Tax model i n c o r p o r a t e t h e l i q u i d i t y premium a s s u m p t i o n , as w e l l as a Revenue Tax a t 25%. o TABLE XXII COMPARISON OE MODEL AND MARKET PRICES (ALL MODELS) BOND : 5h% O C T . l , 1960 ( E l ) MODEL PURE EXP. L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) " N A I V E " R ' RMSE MEAN ERROR ESTIMATED SLOPE ( S . E . OF SLOPE) EST . INTERCEPT ( S . E . OF INTR) FRACTION OF ERROR DUE TO BIAS 6 * 1 RES.VARIANCE MISSPEC ERROR RESID. ERROR 0 .700 3 .123 - 1 . 8 7 7 0 .087 0 .009 91 .609 1.022 0 .361 0 .636 0 .002 9 .731 0 .025 0 .668 0 .751 0 .670 0.439 0 .053 56.691 5.387 0 .794 0 .156 0.049 0 .537 0.028 0 .667 0 .656 0.634 0 .870 0 .107 13.553 10 .698 . 0 .934 0.000 0.065 0 .403 0.028 0 .667 0 .694 0 .653 0 .583 0.071 42.279 7.162 0.885 0.056 0 .058 0.454 0 .028 0.664 0.753 0.699 0.508 0.062 49 .826 6.273 0.861 0.088 0.049 0.539 0.028 0.661 0.837 0.763 0.435 0.054 57.092 5.411 0.830 0 .128 0.040 0.672 0.028 0 .520 0 .505 0 .393 0 .454 0 .075 55 .033 . 7.614 0 .606 0 . 2 3 5 0 .158 0 .214 0 .040 See f o o t n o t e i n T a b l e XXI TABLE XXI I I COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 5h O C T . l , 1962 (E2) MODEL PURE EXP. L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) " N A I V E " R ' RMSE MEAN ERROR ESTIMATED SLOPE ( S . E . OF SLOPE) EST. INTERCEPT ( S . E . OF INTR) FRACTION OF ERROR DUE TO BIAS B * 1 RES. VARIANCE MISSPEC ERROR RESID.ERROR 0 .792 4 .729 - 4 . 2 2 3 0 .393 0.019 6 0 . 6 6 0 2.083 0 .797 0 .182 0 . 0 2 0 21 .914 0 .453 0 .798 1.854 1.636 0 .697 0 .033 32 .236 3.418 0 .778 0 .093 0 .127 2.999 0 .438 0 .796 2.305 2.181 1.347 0.065 -32 .742 6.614 0.895 0 .020 0.083 4 .869 0.443 0 .797 0.797 1.910 0.914 0.044 10.571 4 .484 0.889 0.002 0.107 3.662 0.441 0.802 0 .802 1.949 0.813 0 .038 20.720 3.928 0.880 0 .020 0.099 3.885 0 .430 0.807 0.807 2.020 0.714 0.033 30.741 3.396 0.854 0.057 0.087 4.354 0.419 0 .686 0.686 0 .826 0.509 0 .033 50.781 3.395 0.249 0 .499 0 .250 2.050 0.684 - See f o o t n o t e i n T a b l e XXI TABLE XXIV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 5*2% DEC .15 , 1964 (E3) MODEL PURE L I Q . REV.TAX REV.TAX C .G.TAX C .G.TAX "NA IVE " EXP . PREM. (50%) (25%) (10%) (20%) R 2 0 .756 0 .846 0.851 0.848 0 .856 0.865 0.704 RMSE 5.739 2.115 2.698 2.336 2.534 2.858 1.885 MEAN ERROR - 5 . 5 1 3 1.851 2.441 2.157 - 2.378 2.708 - 0 . 5 4 5 ESTIMATED SLOPE 0 .639 0 .812 1.530 1.052 0.954 0.858 0 .596 ( S . E . OF SLOPE) 0 .036 0 .035 0.064 0 .045 0.039 0.034 0.039 E S T . INTERCEPT 33 .585 20 .817 - 5 0 . 9 1 9 - 3 . 0 8 7 6.966 16.939 41.262 ( S . E . OF INTR) 3.997 3.553 6.526 4.542 3.986 3.449 4.062 FRACTION OF ERROR 0.897 0 .083 DUE TO BIAS 0 .922 0.766 0 .818 0.852 0.881 S 5s 1 0 .037 0 .051 0.072 0 .000 0 .000 0.014 0 .473 RES. VARIANCE 0 .039 0 .182 0.108 0 .147 0 .118 0.087 0 .442 MISSPEC ERROR 31 .650 3.658 6.490 4.655 5.662 7.451 1.983 RES ID. ERROR , 1.294 0.816 0.789 0.803 0.761 0.717 1.573 See f o o t n o t e i n T a b l e XX I . TABLE XXV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 5h APRIL 1, 1963 (E4) MODEL PURE EXP . L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) "NA IVE " R 2 0 .558 0 .651 0.653 0.652 0.667 0.683 0 .488 RMSE 5.051 1.654 • 2.581 2.087 2.074 2.096 1.621 MEAN ERROR - 4 . 7 7 1 1.460 2.450 1.953 1.946 1.967 0.895 ESTIMATED SLOPE - 0 .392 0 .794 1.550 1.046 0.945 0.843 s 0 .459 ( S . E . OF SLOPE) 0 . 0 3 7 0 .062 0 .120 0.081 0.071 0.061 0 .050 EST. INTERCEPT 61 .017 22 .381 - 5 3 . 1 1 0 - 2 . 7 9 2 7.434 17.889 56 .297 ( S . E . OF INTR) 4 . 0 2 5 6.328 12.152 8.270 7.219 6.210 ' 5 . 1 4 0 FRACTION OF ERROR 0 .305 DUE TO BIAS 0 .892 0.779 0.900 0 .874 0.879 0.880 B * 1 0 .080 0 .022 0 .018 0.000 0.000 0.007 0 .391 RES. VARIANCE 0 .026 0 .198 0.081 0.124 0.120 0.112 0 .302 MISSPEC ERROR 24 .828 2.194 6.124 3.813 3.784 3.901 1.832 RESID.ERROR 0 .688 0 .543 0.539 0.541 0.517 0.493 0 .796 See f o o t n o t e i n T a b l e XXI TABLE XXVI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 6% APRIL 1, 1971 (E5) MODEL PURE EXP. L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) . C.G.TAX (20%) MOV. AVG. "NAIVE ' R 2 0.714 0 .436 0.410 0 .423 0.401 0.378 0 .710 0.349 RMSE 1.462 2.291 0.907 1.544 1.931 2.444 0 .584 1.661 MEAN ERROR - 1 . 0 4 0 1.937 0.430 1.201 . 1.587 2.081 - 0 . 1 8 1 0.141 ESTIMATED SLOPE 0 .490 0 .406 0.745 0.519 0 .447 0.379 0.861 0 .290 ( S . E . OF SLOPE) 0.024 0 . 0 3 6 " 0.069 0.047 0.042 0.037 0.043 0 .030 EST. INTERCEPT 49 .773 59 .387 25.484 48.096 55.267 62.102 13 .481 70.116 ( S . E . OF INTR) 2.416 3.497 6.859 4 .620 4 .145 3.669 4 .254 3.054 FRACTION OF ERROR 0 .096 0 .007 DUE TO BIAS 0.506 . 0 .715 0.224 0.605 0.675 0.725 6 * 1 0.359 0 .176 0.053 0.150 0 .163 0.169 0 .048 0 .755 RES. VARIANCE 0.134 0 .108 0.721 0.243 0.161 0.104 0 .855 0 .237 MISSPEC ERROR 1.851 4.681 0.229 1.804 3.129 5.347 0.049 2.104 RESID.ERROR 0 .287 0 .568 0.594 0.581 0 .603 0.626 0.292 0 .656 See footnote i n Table XXI. TABLE XXVII COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 6 V D E C . 1 , 1973 (E6) MODEL PURE EXP. L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) MOV. AVG. " N A I V E " R 2 0.782 0 .756 0 .746 0.751 0.745 0 .737 0 .827 0 .749 RMSE 6.899 3.234 1.711 2.313 2.815 3.499 1.568 5.018 MEAN ERROR - 5 . 6 6 7 2.046 0 .736 1.435 1.878 2.448 - 0 . 9 0 6 - 3 . 0 8 4 ESTIMATED SLOPE 0 .424 0 .571 1.046 0.729 0.650 0 .574 0 .951 0 .420 ( S . E . OF SLOPE) 0 .013 0 .020 0.037 0.025 0.023 0.021 0 .026 0.015 EST . INTERCEPT 54 .559 43 .623 - 3 . 8 2 8 27 .809 35.798 43 .555 3.932 56 .080 ( S . E . OF INTR) 1.455 1.949 3.718 2.538 2.294 2.053 2.696 1.542 FRACTION OF ERROR DUE TO BIAS 0.674 0 .400 0 .184 0 .385 0.445 0 .489 0 .334 0 .377 6 * 1 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 MISSPEC ERROR 45.561 8.172 0 .546 3.017 5.529 9.779 0 .836 22 .828 RESID.ERROR 2.041 2.285 2.383 2.332 2.395 2.466 1.625 2.353 See f o o t n o t e i n T a b l e XX I . TABLE XXVI I I COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) — BOND : Tk APRIL 1, 1974 (E7)" SE S&. " c 1 i « SS: "NAIVE" R 2 0 .783 0 .759 0.772 0.769 0.764 0.759 0.759 0 .071 RMSE 15 .209 3.758 3.892 6.441 6.256 5.947 4.789 11 .734 MEAN ERROR - 1 4 . 1 8 4 - 1 . 5 4 4 - 3 . 5 9 0 - 5 . 9 1 7 - 5 . 5 7 1 - 5 . 0 3 0 - 3 . 8 6 8 - 7 . 1 9 1 ESTIMATED SLOPE 0 .313 0.431 0.765 0.525 0.488 0.454 0.491 0 .237 ( S . E . OF SLOPE) 0 . 0 1 0 0.015 0 .027 0.018 0.017 0.016 0.018 0 .056 E S T . INTERCEPT 65 .557 57 .229 21.116 45 .229 49 .384 53.287 49.952 75.655 ( S . E . OF INTR) 1.253 1.649 2.873 • 2.032 1.911 1.794 1.920 6 .127 FRACTION OF ERROR DUE TO BIAS 0 .869 0 .168 0 .850 0 .843 0.793 0.715 0.652 0 .375 1 0 .123 0 .702 0.035 0.113 0.161 0.233 0.267 0 . 2 7 3 RES. VARIANCE 0 .007 0 .129 0 .113 0.042 0 .045 0.051 0.079 0 . 3 5 0 MISSPEC ERROR 229.686 12.304 13.427 39.748 37.355 33.552 21.110 89.432 RESID.ERROR 1.640 1.824 1.725 1.747 1.785 1.824 1.828 48 .268 See f o o t n o t e i n T a b l e XX I . TABLE XXIX COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 8% O C T . l , 1974 (E8) MODEL PURE L I Q . REV.TAX REV.TAX C .G .TAX C .G .TAX MOV. EXP. PREM. (50%) (25%) (10%) (20%) AVG. " N A I V E " 2 R 0 .750 0.728 0 . 732 0 .730 0 .730 0 .730 0.729 0 .682 RMSE 19.505 4 .926 1. 719 2.976 3.039 3.099 3.903 10.338 MEAN ERROR - 1 8 . 6 2 0 - 3 . 2 2 7 0. 427 - 1 . 4 7 6 - 1 . 2 5 2 - 0 . 9 3 1 - 2 . 6 3 0 - 8 . 0 1 1 ESTIMATED SLOPE 0,312 0 .426 0 . 791 0 .548 0.522 0 .499 0.508 0.278 ( S . E . OF SLOPE) 0 .011 0 .017 0. 031 0.021 0 .020 0.019 0.020 0.012 E S T . INTERCEPT 66.140 58 .682 22. 158 46.510 49 .348 51.932 50.174 73.329 ( S . E . OF INTR) 1.457 1.839 3. 261 2.312 2.199 2.096 2.175 1.405 FRACTION OF ERROR 0.453 0 .600 DUE TO BIAS 0.911 0 .429 0 . 061 0.245 0 .169 0 .090 • B * 1 0 .082 0 .473 0 . 146 0.487 0.574 0 .664 0.390 0 .373 RES. VARIANCE 0 .005 0 .097 0 . 791 0.266 0.255 0.245 0.155 0 .026 MISSPEC ERROR 378.277 21.894 0 . 615 6.498 6.881 7.248 12.865 104.095 RESID.ERROR 2.185 2.376 2. 341 2.358 2.357 2.357 2.372 2.785 See f o o t n o t e i n T a b l e X X I . TABLE XXX COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : Ih'i DEC. 15 , 1975 (E9) MODEL PURE EXP. L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) MOV. AVG. . " N A I V E " R 2 0.728 0 .709 0.723 0.722 0.721 0.718 0.707 0 .680 RMSE 17 .237 4 .216 4.659 7.680 7.519 7.192 2.757 9 .480 MEAN ERROR - 1 5 . 8 3 1 - 2 . 5 1 4 ' - 4 . 2 4 4 - 7 . 0 4 4 - 6 . 7 6 5 - 6 . 2 8 2 - 1 . 1 8 8 - 7 . 2 8 0 ESTIMATED SLOPE 0 .284 0 .467 0.716 0 .502 0 .478 0.457 0.583 0.304 ( S . E . OF SLOPE) 0 .010 0 .018 0.027 0.019 0 .018 0.018 0.023 0 .013 EST . INTERCEPT 69.093 53 .508 26.048 47.631 50.368 52.896 42.094 69 .323 ( S . E . OF INTR) 1.299 1.989 2.989 2.154 2.056 1.968 2.457 1.449 • FRACTION OF ERROR DUE TO BIAS 0.843 0.355 j 0.829 0.841 0.809 0.762 0.185 0.589 S * 1 0.147 0 .488 0.048 0.114 0.143 0.185 0.448 0 .376 RES. VARIANCE 0.008 0 .155 0.121 0.044 0.047 0 .051 0.366 0 .033 MISSPEC ERROR ' 294.562 15 .012 19.076 56.353 53.884 49 .050 4.818 86.825 RESID.ERROR .2 .585 2.768 2.633 2.641 2.658 2.679 2.782 3.045 See f o o t n o t e i n T a b l e XXI. TABLE XXXI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 6% A U G . l , 1976 (E10) MODEL PURE L I Q . REV.TAX REV.TAX C .G.TAX C .G .TAX • MOV. "NA IVE " EXP . PREM . (50%) (25%) (10%) (20%) AVG. R 2 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 . 1 5 2 - 0 . 2 7 2 - 2 . 4 6 8 - 3 . 1 4 0 - 2 . 7 6 8 - 2 . 2 5 1 - 1 . 9 5 9 - - 3 . 6 7 7 ESTIMATED SLOPE 0 .245 0 .475 0.664 0 .458 0 .446 0 .438 0.482 0 .298 ( S . E . OF SLOPE) 0 .015 0 .026 0 .040 0.027 0.026 0.025 0.028 0.018 EST . INTERCEPT 73.003 51 .834 31.565 52.210 53.584 54.703 50.313 68.404 ( S . E . OF INTR) 1.650 2.595 4.091 2.838 2.716 2.602 2.844 1.952 FRACTION OF ERROR 0.380 0 .413 DUE TO BIAS 0 .622 0.011 0 .638 0.585 0 .507 0.392 0 * 1 0 .343 0 .626 0.081 0.257 0.319 0 .408 0.366 0.501 RES. VARIANCE 0 .033 0 .362 0.279 0 .157 0 .173 0.198 0.253 0.085 MISSPEC ERROR 79.409 4 .190 6.870 14.194 12.497 10.341 7.531 29.936 RESID.ERROR 2.781 2.381 2.662 2.658 . 2.618 2.566 2.551 2.799 See f o o t n o t e i n T a b l e XXI . TABLE XXXII COMPARISON OF MODEL AND MARKET P R I C E S ( A L L MODELS) BOND: 7% J u l y 1, 1977 ( E l l ) MODEL PURE L I Q . REV.TAX REV.TAX C .G.TAX C .G.TAX MOV. "NA IVE " R RMSE EXP. PREM. (50%) (25%) (10%) (20%) AVG. 2 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 MEAN ERROR - 5 . 7 9 9 0 .477 - 2 . 0 6 9 - 2 . 6 1 0 - 2 . 2 4 0 - 1 . 7 3 0 - 5 . 1 5 9 - 2 . 4 9 8 ESTIMATED SLOPE 0 .226 0 .411 0 .590 0 .408 0 .396 . 0 .386 0.324 0 . 2 7 9 ( S . E . OF SLOPE) 0 .013 0.019 0.032 0 .022 0.021 0 .019 0.019 0 .016 E S T . INTERCEPT 75.459 58 .570 39.454 57.633 58 .998 60.261 65.376 70 .839 ( S . E . OF INTR) 1.401 1.950 3.254 2.263 2.136 2.012 2.032 1.717 FRACTION OF ERROR DUE TO BIAS 0 .494 0 .026 ' . 0 .552 0 .453 0.359 0 .236 0.655 0.235 6 tl 0 .472 0 . 7 6 9 0.181 0 .410 0 .497 0 .611 0.288 0 .676 RES . VARIANCE 0.033 0.204 0.265 0.136 0.142 0 .151 0.056 0 .087 MISSPEC ERROR 65.755 6.830 5.694 12 .979 11.965 10 .749 38.344 24 .141 RES ID. ERROR 2.244 1.754 2.055 2.052 1.993 1.926 2.292 2.310 See f o o t n o t e i n T a b l e XX I . TABLE XXXI I I COMPARISON OF MODEL.AND MARKET PRICES (ALL MODELS) BOND: 7 3/4% O c t . l , 1978 (E12) MODEL PURE EXP . L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) MOV. AVG. " N A I V E " Kl 0 .691 RMSE 6 .260 MEAN ERROR - 5 . 6 7 5 ESTIMATED SLOPE 0 .443 ( S . E . OF SLOPE) 0 . 0 2 0 E S T . INTERCEPT 53 .101 ( S . E . OF INTR) 2.136 FRACTION OF ERROR DUE TO BIAS 0 .821 6 * 1 0 .138 RES.VARIANCE 0.039 MISSPEC ERROR 37 .646 RESID. ERROR 1.543 0 .814 2.476 2.187 0.755 0 .024 26.045 2.410 0 . 7 8 0 0 .068 0.151 5.204 0.929 0 .699 2.293 - 1 . 9 3 8 1.024 0 .045 - 4 . 4 3 1 4 .677 0 .712 0 .000 0 .286 3.754 1.505 0.707 2.924 - 2 . 5 6 7 0 .726 0.031 25.455 3.271 0.771 0 .057 0.171 7.087 1.463 0 .723 2.468 - 2 . 0 1 7 0.702 0.029 28.254 3.026 0.668 0.104 0 .227 4 .709 1.383 0.742 1.931 - 1 . 2 8 2 0 .682 0 .027 30.846 2.780 0.441 0.213 0 .345 2.440 1.289 0.653 6.359 - 6 . 1 1 5 0.610 0.030 35.147 3.222 0.924 0.032 0.042 38.713 1.734 0 .827 0 .956 0 .048 0 .897 0 .028 10 .291 2.796 0 .002 0 .054 0 .942 0 .052 0 . 8 6 3 See f o o t n o t e i n T a b l e XXI • TABLE XXXIV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 7h D e c . l , 1980 (E13) MODEL PURE L I Q . REV.TAX REV.TAX C .G .TAX C .G .TAX MOV. "NA IVE " EXP . PREM. (50%) (25%) (10%) (20%) AVG. 2 R 0 .638 0 .744 0 .648 0 .656 0.671 0 .686 0.617 0 .761 RMSE 7.479 3.077 3.941 4 .215 3.525 2.691 7.019 1.555 MEAN ERROR - 7 . 0 0 5 2.600 - 3 . 4 3 6 - 3 . 8 0 4 - 3 . 0 3 5 - 2 . 0 1 8 - 6 . 7 3 9 0 .316 ESTIMATED SLOPE 0 .571 0 .837 1.328 0.942 0.900 0 .858 0 .843 0.929 ( S . E . O F SLOPE) 0 .029 0 .034 0 .068 0 .047 0.043 0 .040 0.046 0 .036 EST. INTERCEPT 37 .987 18.135 - 3 6 . 6 9 9 2.104 7.005 12.094 9.617 7.170 ( S . E . OF INTR) 3.149 3.263 6.904 4 .835 4.438 4.047 4 .850 3.543 FRACTION OF ERROR DUE TO BIAS 0 .877 0 .713 0.759 0 .814 0.741 0 . 5 6 2 ' 0 .92175 0.041 6 t 1 0 . 0 6 0 0 .027 0.023 0 .000 0.005 0.022 0.003 0 .012 RES.VARIANCE 0 .062 0 .259 0 .216 0.185 0.253 0.415 0 .074 0 .946 MISSPEC ERROR . 52 .480 7.017 12.168 14.482 . 9.276 4.235 45.602 0 .130 RESID.ERROR 3.469 2.453 3.367 3.291 3.153 3.008 3.671 2.288 See f o o t n o t e i n T a b l e XX I . TABLE XXXV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 7% APRIL 1, 1979 (E14) MODEL PURE L I Q . REV.TAX REV.TAX C .G .TAX C .G .TAX MOV. "NA IVE " EXP . PREM. (50%) (25%) (10%) (20%) AVG. 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 MEAN ERROR - 4 . 1 3 5 2. 591 - 2 . 5 1 3 - 2 . 3 2 3 - 1 . 6 4 9 - 0 . 7 6 9 - 2 . 7 4 3 0 .190 ESTIMATED SLOPE 0 . 6 7 3 0 . 872 1.522 1.062 1.001 0 . 9 3 0 0.974 0 .977 ( S . E . OF SLOPE) 0 .035 0 . 030 0 .072 0 .049 0.044 0.039 0.047 0.031 EST . INTERCEPT 28 .979 14 . 671 - 5 4 . 6 7 3 - 8 . 5 2 6 - 1 . 8 3 7 5.156 - 0 . 1 9 0 2.407 ( S . E . OF INTR) 3.565 2. 934 7.213 4 .980 4 .442 3.918 4.755 3.077 FRACTION OF ERROR DUE TO BIAS 0 . 8 3 0 0 . 819 0.701 0.720 0.579 0.246 0.773 0 .030 B * 1 0 .052 0 . 014 0 .063 0 .000 0.002 0.005 0.000 0 .002 RES. VARIANCE 0 .116 0. 166 0 .234 0 .278 0.417 0 .748 . 0 .225 0 .967 MISSPEC ERROR 18 .184 6. 828 6.891 5.404 2.710 0 .604 7.518 0 .039 RESID.ERROR 2.403 1. 361 2.113 2.088 1.946 1.801 2.195 1.166 See footnote on Table XXI . TABLE XXXVI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 9k% APRIL 1, 1978 (E15) MODEL PURE EXP. L I Q . PREM. REV.TAX (50%) REV JTAX (25%) C .G .TAX (10%) C .G .TAX (20%) MOV. AVG-"NA IVE " R 2 0 .725 0 .754 0 .750 0.752 0.756 0 .760 0.741 0.757 RMSE 6.868 2.280 3.192 2.557 2.744 3.020 1.519 2.390 MEAN ERROR - 6 . 4 2 2 1.662 2.998 2.264 2.446 2.719 0 .658 - 0 . 9 4 9 ESTIMATED SLOPE 0 .460 0 .623 1.116 0.787 0 .747 0 .707 0.694 0.495 ( S . E . OF SLOPE) 0 .022 0 .028 0 .050 0.035 0.033 . 0 .031 0 .032 0.022 EST . INTERCEPT 53 .645 40 .568 - 8 . 8 6 0 24.069 28.373 32.583 32.506 52.524 ( S . E . OF INTR) 2.494 2.896 5.169 3.653 3.427 3.206 3.367 2.339 FRACTION OF ERROR DUE TO BIAS 0 .874 0.531 0 .881 0.884 0 .794 0 .810 0.187 0.157 6 J* 1 0 .098 0 .246 0 .003 0.038 0.052 0 .065 0.286 0 .642 RES. VARIANCE 0 .027 0 .222 0 .115 0.177 0.152 0 .124 0.525 0 .199 MISSPEC ERROR 45.877 4 .044 9.019 5.376 6.383 7.992 1.096 4.575 RESID. ERROR 1.293 1.155 1.172 1.162 1.146 1.128 1.212 1.140 See f o o t n o t e i n T a b l e XXI TABLE XXXVII COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 9k F E B . l . 1977 (E16 ) MODFT PURE L I Q . REV.TAX REV. TAX C .G.TAX C .G.TAX MOV. • "NA IVE " EXP . PREM. (50%) (25%) (10%) (20%) AVG. R 2 0.681 0 .763 0.758 0 .761 0.772 0.783 0 .678 0 .752 RMSE 5.661 2 .017 2.537 2.129 2.256 2.448 2.450 2.291 MEAN ERROR - 5 . 1 0 7 1.405 2.212 ' 1.772 1.922 2.138 - 1 . 8 2 2 ' - 1 . 1 6 4 ESTIMATED SLOPE 0 .488 0 .701 1.273 0 .892 0.855 0.819 0 .673 0 .563 ( S . E . OF SLOPE) 0 .029 0 .034 0.064 0.044 0.041 0.038 0.041 0 .028 EST . INTERCEPT 50.396 31 .851 - 2 5 . 4 6 4 12.743 16.605 20.401 32.532 44 .409 ( S . E . OF INTR) 3.238 3.558 6.501 4 .539 4.215 3.903 4.362 3.027 . FRACTION OF ERROR DUE TO BIAS 0 .814 0 .485 0 .760 0.692 0.725 0.762 0.552 0 .258 B * 1 0 .130 0 .187 0 .028 0.011 0.022 0.033 0.146 0 .476 RES. VARIANCE 0.055 0 .326 0 . 2 1 1 - 0 .296 0.251 0 .203 0.301 0 .265 MISSPEC ERROR 30.256 2 .738 5.080 3.193 3.810 4.775 4.197 3.856 RESID.ERROR 1.791 1.329 1.359 1.343 1.282 1.219 1.809 1.394 See f o o t n o t e i n T a b l e XXI• TABLE XXXVI I I COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 7h% O C T . l , 1979 (E17) MODEL PURE EXP . L I Q . PREM. REV.TAX. (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) MOV. AVG. "NAIVE ' 2 R 0 .570 0 .700 0.581 0.594 0.611 0.630 0.726 0 .714 RMSE 5.700 2.598 2.942 3.384 2.912 2.319 2.922 1.174 MEAN ERROR - 5 . 3 8 3 2.160 - 2 . 6 7 3 - 3 . 1 3 0 - 2 . 5 9 4 - 1 . 8 5 5 2.685 - 0 . 0 9 4 ESTIMATED SLOPE 0 .053 0 .610 1.095 0 .763 0.709 0.655 0.729 0.535 ( S . E . OF SLOPE) 0 .039 0 .035 0.083 0.056 0 .050 0.045 0.040 0 .030 EST . INTERCEPT 46 .370 39 .767 - 1 2 . 4 0 4 20.960 26.857 32.784 28.706 45 .823 ( S . E . OF INTR) 4 .086 3.468 8.482 5.780 5 .148 - 4.537 3.860 3.011 FRACTION OF ERROR DUE TO BIAS 0 .891 0.691 0.825 0.855 0.793 . 0 .640 0.844 0 . 0 0 3 B * 1 0 . 0 6 0 0 .148 0.000 0 .016 0 .041 0 .113 0.040 0 .647 RES. VARIANCE 0 .047 0 .159 0.173 0.127 0 .164 0.246 0.114 0 .349 MISSPEC ERROR 30.956 5.674 7.154 9.993 7.084 4.051 7.560 1.913 RES ID. ERROR 1.543 1.077 1.506 1.459 1.395 1.327 0.982 1.027 See f o o t n o t e i n T a b l e XXI TABLE XXXIX COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 9% F E B . l , 1978 (E18) MODEL PURE L I Q . REV.TAX REV.TAX C .G.TAX C .G .TAX MOV. "NA IVE " EXP . PREM. (50%) (25%) (10%) (20%) AVG. R 2 0.561 0.777 0 .776 0.777 0.785 0 .794 0.809 0.740 RMSE 1.267 2.740 2.493 2.600 2.806 3.083 3.390 0.757 MEAN ERROR - 1 . 0 1 6 2.670 2.452 2.560 2.764 3.034 3.355 0.436 ESTIMATED SLOPE 0 . 5 9 0 0 .640 1.234 0 .838 0.762 0.685 0 .737 0.650 ( S . E . OF SLOPE) 0 . 0 5 0 0.032 0 .063 0.043 0.038 0 .033 0 .034 0.037 E S T . INTERCEPT 4 1 . 2 7 8 38 .450 - 2 0 . 9 6 8 18.634 26.395 34.280 29.299 36.017 ( S . E . OF INTR) 5.186 3.280 6.351 4.304 3.810 3.330 3.402 3.769 FRACTION OF ERROR 0.968 0.979 0.332 DUE TO BIAS 0 .642 0.949 0 .967 0.969 0.970 6 * 1 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 ERROR 1.247 7.329 6.035 6.579 7.704 9.340 11.340 0.361 RES ID. ERROR 0 .358 0 .181 0.182 0.181 0.174 0 .168 0.155 0.211 See f o o t n o t e i n T a b l e XXI TABLE XL COMPARISON OF MODEL AND MARKET P R I C E S (ALL MODELS) BOND : 9% O C T . l , 1980 (E19) MODEL PURE EXP. L I Q . PREM. REV.TAX (50%) REV.TAX (25%) C .G .TAX (10%) C .G .TAX (20%) MOV. AVG. "NA IVE " R 0 .584 RMSE 7.260 MEAN ERROR - 7 . 0 3 7 ESTIMATED SLOPE 0 .528 ( S . E . OF SLOPE) 0 .042 EST. INTERCEPT 45 .410 ( S . E . OF INTR) 4.704 FRACTION OF ERROR DUE TO BIAS 0.939 8 * 1 0 .031 RES . VARIANCE 0 .028 MISSPEC ERROR 51.199 RESID.ERROR 1.516 0 .646 2.427 1.859 0.586 0.041 44 .148 4 .207 0.586 0.195 0.218 4 .606 1.287 0 .543 1.878 - 1 . 3 5 2 1.189 0.103 - 2 1 . 3 4 7 10.904 0.518 0.009 0.471 1.865 1.662 0.566 3.591 - 3 . 3 5 1 0.819 0 .068 16.099 7.309 0.870 0.006 0.122 11.316 1.579 0.581 3.444 - 3 . 1 8 1 0.751 0.060 23.431 6.501 0.853 0.018 0.128 10.337 1.526 0 .596 3.174 - 2 . 8 5 7 0.686 0.053 30.724 5.731 0 .810 0 .043 0.145 8.604 1.470 .0 .663 4.281 4.077 0.688 0.046 35.287 4.651 0.906 0.026 0.066 17.108 1.225 0.699 2.193 0.322 0.455 0.028 56.886 2,942 0.021 0.750 0.227 3.716 1.096 See f o o t n o t e i n T a b l e XXI r - o CO 129 7.4 Estimating the Liquidity/Term Premium Paramters In Chapter 2, we had as the basic bond valuation equation + SL) ~T = S = AC^/r) t2.8) where A(r»t ,T ) i s the instantaneous excess return expected by investors. Under the PEXP model, we had set A-(r,t, X )=0. We now make assumptions about aggregate investor behaviour along the l i n e s of Inger s o l l [39], F i r s t , we assume that ^ i s independent of t, i e . , i t i s time homogeneous. Second, we assume (TAR . 3>(1~) = - k?.r (7.8) which yie l d s (see Chapter 2, equation (2.8)) \(f,V) . _cfe, + ^ T ) _ ^ _ _(7.9) Vasicek [72] and Brennan 6 Schwartz [1.0] both assume = constant. This i s a statement about the price (in terms of excess return) of instantaneous standard deviation r i s k One may fi n d the assumption <|> = constant more i n t u i t i v e l y comprehensible than the assumption i n (7.8)., However, as w i l l be shown shortly, the assumption of equation (7.8) leads to a simple structure for the form of A . Much of the exi s t i n q l i t e r a t u r e on the term structure of i n t e r e s t rates addresses the form and determinants of A- . I t w i l l be shown that (7.8) leads 130 to a form for X that i s consistent with the existing l i t e r a t u r e . Ingersoll(op c i t ) points out that under t h i s assumption (and assuming the in t e r e s t rate process to be of the form assumed here), the value of the pure discount bond B(r, T ) i s given by r "1 r i' b(i70 - UCt)J Y) m'/A-'t -v ^ Tj I - Her) £ J (7.10) where m» = (m-k^ ) l j^ ' = * k | ] / < r 2 = [ m» - (m» z + 2o- 2) A » (m«2 + 2 cr2)^2" H ( f ) = [ 1>(m«-A) 0-e-Ar )/2A]-» I t can be seen from (7.10) that B,/B = [ 1 - H ( T ) e - ^ r ]=q(f ), i e . the r a t i o (B, /B) i s independent of r and s t r i c t l y a function of time to maturity. This implies that the choice of as indicated by the rel a t i o n s h i p i n equation (7.8) leads to an expression for the liquidity/term premium as (7.11) As pointed out by Ing e r s o l l [39], for ( k, • k^ r) > 0, the term premium i s a pos i t i v e , increasing concave function, and for (k ( r)<0, i s negative, decreasing and convex. These are the usual properties associated with the l i q u i d i t y premium. Further, f o r a qiven maturity, the re l a t i o n between A- and r as qiven by equation (7.11), i s consistent with some of the popular 131 assumptions about term/liquidity premia, i e . , a) a constant term premium independent of in t e r e s t rates (set k^ =0) . This would specify that the expected rate of return on a qiven maturity of bonds be a constant in excess of the instantaneous i n t e r e s t rate. b) term premiums proportional to the int e r e s t rate (set k, -0). This would specify that the return on a qiven maturity of bonds be a constant r a t i o to the instantaneous i n t e r e s t rate. c) term premia that are posit i v e as lonq as i n t e r e s t rates are below a threshold l e v e l , and negative above that value (see Van Home [71]). This obtains when k, >0 and k% <0. Probably the most compelling reason for choosing the forms for O and X. as i n equations (7.8) and (7.9) i s that i t permits a simple method of estimating the parameters k( and k r because we have a closed form solution for the pure discount bond under t h i s assumption . The price of a bond paying a continuous coupon may be represented by r where B(.,.) represents the price of a pure discount bond, and i s as qiven by equation (7.10). Given a sample of market prices on s t r a i g h t coupon bonds, one method of estimating k( and k 2 would be to minimize some measure of deviation between the market and model prices over the data sample. Corresponding to 132 any choice of ky and k^ (and given the parameters of the i n t e r e s t rate process, the current i n t e r e s t rate, time to maturity, and coupon r a t e ) , the model price of any straight coupon bond can be computed using eguation (7. 12) 5 7 . The simplest model that was considered was P.' = PL + € i , 17.13) where P 1 and P^ are respectively market and model prices, and e. r-J N(0, ( T 2 ) ; Cov(e c ,e ) =0 f o r i#j. It may be noted that P i s a non - l i n e a r function of the parameters k, and k t Thus, estimating k, and k^ i n the present scenario i s the standard problem of c o e f f i c i e n t estimation i n a non-linear regression framework s s. Throughout, we adopt maximum l i k e l i h o o d (ML) methods f o r parameter estimation. In t h i s s i t u a t i o n l e a s t squares estimation = ML(asymptotically). However since P^» and are s t r i c t l y p o s i t i v e , i t was considered more appropriate to assume a model of the form s 7 P ( r , ^ , c ) can be evaluated very e a s i l y by numerical integration. Due to the smooth shape of the function B(r,T ) with respect to X , a simple 4 point quadrature method gave very accurate r e s u l t s . To check the accuracy for a sample case, the coupon bond price was evaluated using up to a 64 point adaptive quadrature and the increased accuracy was n e g l i g i b l e . I t may be noted that i n any approach to estimating k, and kj_ , model prices of the t o t a l bond sample would have to be evaluated several hundred times. Even with the present assumptions, estimating k, and k L i s computationally quite expensive. However, i f were not , (or zero) and i f we did not assume A ( r , t , f ) to have the form as an equation (7.9), the bond model prices correspoding to each (k| , k^ . ) value would have to be obtained by f i n i t e difference methods., That would mean a computation expense more than just p r o h i b i t i v e ! 5 8 Goldfeld 6 Quandt [34 J present a good introduction to the problem., 133 -+ -6c (7.14) where the assumptions on e^ are exactly as before. The parameters k| and k x were estimated by both models above, and the parameter estimates were hardly d i f f e r e n t 5 * : (Eqn. 7.13) (Eqn. 7. 14) k, 0.3113x10-5 0.3093x10-5 k^ -0.1581x10-2 -0.1548x10-2 In both models above, the residual vector has been assumed to exhibit neither autocorrelation, nor heteroscedasticity. In l i n e a r models i t i s well known that the estimated c o e f f i c i e n t s are unbiased, even where the residual covariance i s - Q - ^ c r " J : the covariance matrix of the estimated parameters i s biased. In a non-linear s e t t i n g , whether the estimated parameters are unbiased in small samples i s not known when Si£ O-5- I. TO test for heteroscedasticity, the re s i d u a l vector e^ was retrieved and the following regression was performed: (The hypothesis was that var(e• ) i s a function of time to matruity of P 6 0.) 5 9 The standard errors of the estimates, based on asymptotic theory ( i e . , by inv e r t i n g the Fisher Information matrix) are not reported, as t h e i r values was very d i f f e r e n t across the two models. This was investigated further and found to be due to numerical inaccuracy i n evaluating the second derivative of the j o i n t l i k e l i h o o d function near the optimal point, 6 0 P i s a function of r and ? . Heteroscedasticity as a function only of X was considered. Understandably, i t could have also been a function of r. However, t h i s was not considered, as the v a r i a b i l i t y of T over the sample was much larger than that of r. I t was therefore f e l t that most of the heteroscedasticity could be explained by T alone. 134 loo^{^1) a -t- b Lft(Vi) + IA)O (7.15) where T t=time to maturity of the i™ data point. I f b=0, then we cannot r e j e c t the hypothesis that the residuals exhibit homoscedasticity 6 1. This was done for the residuals from equation (7.14) and b was estimated at 2.091, and i t s t s t a t i c was 1.06. This seems to indicate that there i s no compellinq reason to suspect heteroscedasticity to be present. Testinq f o r autocorrelation among the residuals i s a more complicated matter.. There are two types of error correlations to consider. 1) S e r i a l correlation within each bond across time. , 2) Contemporaneous c o r r e l a t i o n across bonds, at any instant of time. I t must be remembered that the ordinary coupon bond sample consists of time series on 18 d i f f e r e n t bonds. S e r i a l c o r r e l a t i o n of residuals refers to the c o r r e l a t i o n between consecutive residuals of each bond. I t i s , however, also reasonable to expect the errors across a l l bonds, at a The more "correct" method of t e s t i n g for heteroscedasticity would be to do a "constrained" and "unconstrained" estimation, and then perform a l i k e l i h o o d r a t i o test. Under the constrained estimation JL i s assumed = cr 21 and in the unconstrained JL- i s diaqonal with elements o^^a?^ . The rest of the approach i s to set up the likelihood function as where p (e^ ) ~ N (0, JL) . For our case the sample size was 6662 data points on bonds, and doinq t h i s would have been computationally expensive. Thus the more ad hoc approach was taken. This method of hypothesis testinq on b, i s also dependent upon w,- being i . i . d and normally di s t r i b u t e d . 135 p a r t i c u l a r point i n time to be correlated. Since each bond data series s t a r t s and ends at a di f f e r e n t point in time (and each i s of different length) , accounting for contemporaneous c o r r e l a t i o n would be a horrendous task. Considering the d i f f i c u l t i e s involved, i t was decided to leave the problem of contemporaneous co r r e l a t i o n i n abeyance, but tackle the s e r i a l c o r r e l a t i o n problem. When we consider s e r i a l c o r r e l a t i o n only, the covariance matrix SL of the re s i d u a l vector i s block diagonal i n structure, with the representative matrix having the usual form as when we have f i r s t - o r d e r autocorrelation, i e , J l = ( Si-c) where Jl j , i s the matrix along the diagonal f o r bond i , and i s of the form -Hi (Tc x- "ft ) I f f ' f f r f f \ 7-i (7.16) We could further assume that "f i s egual across a l l bonds. This s i m p l i f i e d structure makes i t computationally much easier to set up the l i k e l i h o o d function of the residuals and thereby estimate the parameters. What was actually done was that, along with s e r i a l c o r r e l a t i o n (using the model of equation 7. 14) , heteroscedasticity of the form discussed e a r l i e r was assumed, and ML methods were employed to estimate j o i n t l y (k, ,k t ,-f,a,b). I t was computationally very expensive and so no constrained estimation was performed, (to do l i k e l i h o o d r a t i o 136 tests for testing hypotheses on any of the parameters). The log of the j o i n t l i k e l i h o o d function was L = -L^\SL\ - i « ' J l e (7.17) where e i s the column vector of residuals, e* i s i t s transpose, and e^ /v/ N( 0 ( c r - 2 ) , with Q ~ c 2 = aT^ and Corr (et ^ ( ) = f and i s constant across a l l bonds. The r e s u l t of the estimation was that convergence was not attained i n 60 i t e r a t i o n s using a quasi-Newton algorithm for maximizing L. The intermediate parameter values were 6 2 k, = 0.3916 x 10-s kr = -0.2144 x 10~ 2 f = 0.0097 a * 0.1394 x 10-ft b = 1.586 The gradients on k, and k L indicated that the optimum would require both values to move towards zero. The broad conclusions that can be arrived at, based on the r e s u l t s , are: a) The estimates of k, and kj_ based on the model of equation (7.14) are probably not very d i f f e r e n t from the model assuminq autocorrelation and heteroscedasticity of the error vector. b) The s e r i a l c o r r e l a t i o n c o e f f i c i e n t ( f ) between the » 2 apparently, the converqence rate i s very slow. The CPU time used for t h i s p a r t i a l converqence run was 5000 seconds on an IBH 370/168. , Since the computational cost was extremely hiqh and no additional insiqhts appeared to be l i k e l y by r e s t a r t i n g the search for the optimum andqoinq on u n t i l converqence was attained, the matter was not pursued further. 137 residuals appears to be close to zero. c) A s t a t i s t i c a l l y s i g n i f i c a n t l e v e l of heteroscedasticity does not seem to exi s t . The f i n a l question that was considered under the estimation of the parameters k v and k x , was the v a l i d i t y of the assumption of normality of the residuals - a f t e r a l l , the HL approach here i s based on t h i s assumption. The approach that we adopt in testing for normality (or departures therefrom) i s probability graphing. Fama [22] uses t h i s approach i n examining the behaviour of stock prices. If u i s a Gaussian random variable with mean /A and variance c r 2 , the standardized variable Z = (u- / ^ ) / r w i l l be unit normal., Since Z i s just a l i n e a r transformation of u, the graph of Z against u i s just a straight l i n e . The relationship between Z and u can be used to detect departures from normality i n the d i s t r i b u t i o n of u. I f u^(i-=1.,H are N sample values of the variable u arranged i n ascending order, then a par t i c u l a r u L i s an estimate of the f f r a c t i l e of the d i s t r i b u t i o n of u, where the value of f i s given 6 3 As pointed out i n Fama [22], t h i s p a r t i c u l a r convention for estimating f i s only one of many that are available. Other popular conventions are i/(N + 1), (i-3/8)/(N«- ) and ( i - )/N. A l l four techniques give reasonable estimates of the f r a c t i l e s and, for the large sample that we have, i t makes l i t t l e difference which s p e c i f i c convention i s chosen. by 63 138 Now the exact value of Z for the f f r a c t i l e of the unit normal d i s t r i b u t i o n can e a s i l y be obtained by i n v e r t i n g the unit cumulative normal. Computer routines are available for t h i s . I f u i s a Gaussian random variable, then a graph of the sample values of u against the values of Z derived from the t h e o r e t i c a l unit normal cumulative d i s t r i b u t i o n function should be a straight l i n e . There may, of course, be some departure from l i n e a r i t y due to sampling error. I f the departures from l i n e a r i t y are extreme, however, the Gaussian hypothesis for the d i s t r i b u t i o n of u should be questioned. The normal probability plot of the residuals from the model of equation (7.14) i s presented in Figure 4. Inspection of the plot indicates that the d i s t r i b u t i o n of the residuals i s thinner than the normal at the t a i l s , and also more peaked at the mode. In f a c t , i t could be that we have a mixture of 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 variances - one (or more) corresponding to the t a i l s ; and another (or others) corresponding to the peak at the mean. This could be the r e s u l t of heteroscedasticity of the form we considered e a r l i e r (but did not find s t a t i s t i c a l l y s i g n i f i c a n t ) . Possibly i f we had adopted the more "correct" method of testing f o r heteroscedasticity (see footnote 54) we might have observed i t at a s t a t i s t i c a l l y s i g n i f i c a n t l e v e l . Thus, whether heteroscedasticity e x i s t s or not i s at present an unresolved issue. However, we did find that even taking i t into account (in the model of 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)), did not seem to a l t e r materially the point estimates of k( and kz . He may therefore assume that our estimates of k, and k t based on the model of equation (7.14) are s a t i s f a c t o r y . To get a better f e e l for the numerical values of k, and k% , the l i q u i d i t y / t e r m premium function A was plotted against time to maturity, f o r different values of the instantaneous interest rate (see Figure 5). The term structure curves were also plotted and these are presented in Figures 6 and 7. I f we represent the term structure by R(r,t ), then we have R C * , * ) - -1 ^ [ e c - r , r ) j where B ( r , t ) i s the pure discount bond value. Figure 6 shows the shape of the term s t r u c t u r e 6 * at values of the current value of the short term i n t e r e s t rate varying from '/^JX. , to 2JUL . I t may be of i n t e r e s t to note that when r= , the term/liquidity premium i s a positive and increasing function of time to maturity.. When r=-k, /k 2 , A =0 for a l l maturities. Obviously t h i s does not imply a f l a t term structure - only that at t h i s value of r , the term structure curves f o r the pure expectations and the l i q u i d i t y / t e r m premium hypotheses models coincide. As can be seen from Figure 6, when r=-k, /k% , the term structure i s downward sloping. Ingersoll[ 39 ] has pointed out that the term structure corresponding to t h i s i n t e r e s t rate process, and the assumed form of (as i n equation (7.8)), could have a *•* The value of HISF i n the figure corresponds to the l i m i t i n g value of R ( r , T ) as T -><* . From the term structure equation, t h i s i s given by (2m1 ' ,/<r 2) (A-m«) where A= (m*2*2 <r zj— , and m,= (m-k^ ) and fA*={m +k, ) /m*. . 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 $ i O J5 20 25 i0 i5 4) 45 40 T I M E TO M A T U R I T Y IN Y E A R S 143 FIGURE 7 YIELD TO MATURITY VS TIME TO MATURITY ON DISCOUNT BONDS K l - 0.309 X 10 XX -5 K2 = -0 .154 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 40 45 40 TIME 10 HRTURITY IN YEARS 144 humped shape, but that (for reasonable parameter values) the hump would be very small. This i s borne out i n Figure 7. Before comparing Figures 6 and 7, care must be taken to note the large difference i n the scale along, the Y-axis between the two. 7. 5 Bond Valuation Under the Liquidity/term Premium ILIfiPL Model Having estimated the aggregate investor preference parameters that determine t h e i r l i g u i d i t y / t e r m premia requirements, we can proceed to value our sample of retractable/extendible bonds, with t h i s assumption incorporated. The p.d.e. governing the bond price i s only s l i g h t l y altered (cf. equation 7.1) we now have l(rV<5M -i ^ V " ^ ) ^ ' - ^ ^ ^ ^ - ^ = ° (7.18) where i ' and JJL' are as defined i n equation (7.10). The boundary conditions remain exactly the same as for the PEXP case. Model prices were computed for a l l 20 bonds, and the r e s u l t s of regressing the market prices on model prices are presented i n column 2 of Tables XXI through XL. &s expected, the mean error (defined e a r l i e r ) which was consistently negative under the PEXP model, i s now more often positive (except for bonds E7 to E10). For purposes of quick comparison across bonds, Table XLI presents the mean error for a l l 20 bonds using the d i f f e r e n t models, and Table XLII presents s i m i l a r summary resu l t s on P , the slope c o e f f i c i e n t from regressing the market price on the model prices as well as the c o r r e l a t i o n between the model and market prices., Comparing the r e s u l t s of the LIQP TABLE XLI COMPARISON OF MEAN ERROR FOR ALL BOND ACROSS DIFFERENT MODELS BOND PORE LIQ. REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. '' NAIVE" EXP. PREM. (50%) '(25%) (10Z) (25) AVG. Rl 0.36 2.45 -0.00 1.25 1.83 2.57 - -E l -1.87 0.67 0.63 0.65 0.70 0.76 - 0.39 E2 -4.22 1.63 2.18 1.91 1.95 2.02 - 0.82 E3 -5.51 1.85 2.44 2.15 2.37 2.70 - -0.55 E4 -4.77 1.46 2.45 1.95 1.94 1.96 - 0.89 E5 -1.04 1.93 0.43 1.20 1.58 2.08 -0.18 0.14 E6 -5.66 2.04 0.73 1.43 1.87 2.44 -0.91 -3.08 E7 -14.1 -1.54 -3.59 -5.91 -5.57 -5.03 -3.87 -7.19 E8 -18.62 -3.22 0.42 -1.47 -1.25 -0.93 -2.63 -8.01 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 E l l -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 E13 -7.00 2.60 -3.43 -3.80 -3.03 -2.01 -6.74 0.31 E U -4.13 2.59 -2.51 -2.32 -1.64 -0.77 -2.74 0.19 E15 -6.42 1.66 2.99 2.26 '2.44 2.72 0.66 -0.95 E16 -5.10 1.40 2.21 1.77 1.92 2.13 -1.82 -1.16 E17 -5.38 2.16 -2.67 -3.13 -2.59 -1.85 2.68 -0.09 E18 -1.01 2.67 2.45 2.56 2.76 3.03 3.36 0.44 E19 -5.96 2.93 0.28 -2.27 -2.11 -1.78 5.15 0.89 v 146 a 8 1 s 8 < i < i 1 s § 1 I 3 s 1 E 3 5 £ j § g 1 B ! 1 O O O O O 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 O O -* S -^ >"» < -* c i O e o ' - ' i o r - t w i ^ S c o c ^ o o a D < o r * » r - c o r*>* ca CO CO co GD cc O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O I 1 1 g ! 3 I I 1 1 I I I I i I I i I 3 o ' d o o o d o o o o ' o d d o d o d d d d i 3 5 I i I I I 1 2 s - l i ! I I I I ! 1 s i i s s s s i l l l l l l s l l l l l d d d d d d d d d d d d d d d d d d d d I 1 H I s I I s i ! 1 1 I | § § 1 I I s I 1 H I i § ! 1 g ! | § i s g n s o d d d d d d d d d d d d d d d d d d d s H u * H s n s z m § m o o o _ o d d d d d d d d - d d d d d = s s s s s l s s s c s s 8 s i s S s 5 o o o o o o o o o o o o d o ' d o o o ' o ' o ' o - s *: s S S 2 C S S 2 S K K 2 S s 2 s -HO , H W o - « o o o o o * - 4 - * r - » 4 _ ^ ^ r j ^ s I s § I I i ! ! I I s ; I I I I I ! i i i 1 d d d d d d d d d d d d d d d d d d d d = 5 ! I S I I g I i I S S § S 2 § 1 I I I 1 d d d d d d d d d d d d d d d d d d d d . I 3 I 3 g 1 I I 1 S I H I 2 g § g I S j d d d d d d d d d d d d d d d d d d d d g § § S H I ! s s a 3 g 5 I § s § I | s f a a B o a a s a s s . a a - a a a a a a a a ? TABLE XLIX COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) (SUMMARY BASED ON ALL BONDS IN THE SAMPLE) PURE LIQ. REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. "NAIVE" MODEL EXP. PREM. (50%) (25%) (10%) (20%) AVG. (S.E. OF INTR) FRACTION OF ERROR DUE TO BIAS RES. VARIANCE MISSPEC ERROR RESID.ERROR R2 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 . 5 7 0 0 .778 - 0 . 9 0 5 - 1 . 6 2 1 - 1 . 2 5 8 - 0 . 7 5 1 • - 2 . 0 7 5 - 0 . 8 4 1 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 EST.INTERCEPT 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 ^ j_ 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 . 0 .500 94.386 6.573 2.035 9.096 8.576 8 . 1 1 4 ' 10.483 9.445 10.748 8.986 12.264 12.170 11.797 11.360 14.168 9.448 -1^ 148 i model with the PEXP res u l t s we could i n f e r that: a) Whereas the PEXP model consistently overvalues the bonds, the LIQP model tends (more often than not) to undervalue them. This i s indicated by the greater number of p o s i t i v e mean error figures i n Table XLI. b) The slope c o e f f i c i e n t of the regression (7.7) i s a measure of r e l a t i v e responsiveness. I f f <1, then the model i s over-responsive (since measures the responsiveness of the market with respect to the model). Ideally, we would require a model that gives j£=1. The LIQP model leads to values consistently closer to 1 than the PEXP model, and may, therefore, be regarded as an improvement over the PEXP model. Be would surely expect the LIQP model to outperform the PEXP model, as i t contains more information on the term structure of in t e r e s t rates. To enable one to compare the different models across a l l bonds, a qlobal measure that aqqreqates the r e s u l t s of Tables XXI to XL i s desireable. For t h i s purpose, the regression of equation (7.7) was performed by pooling data of a l l the bonds, and the results are presented i n Table XLIX., Be now investigate the impact on the model of incorporating taxes. 7,6 Bond Valuation With Revenue Taxes In t h i s section, we look at the e f f e c t of including i n the model taxes on coupons and i n t e r e s t , but not on c a p i t a l gains. In Chapter 2, we developed the p.d.e. governing the bond valuation under s p e c i f i c assumptions about the way taxes are 149 applicable {see equation 2,11). The assumptions did appear to be a gross o v e r - s i m p l i f i c a t i o n of r e a l i t y . The question, however, remains; are we better off without incorporating taxes into the model? Inclusion 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 cdt/G to c(1-R)dt/G, where c i s the coupon and R the revenue tax rate. Thus, the net gain (or benefit) from owning the bond i s reduced, and so i t s value i s lowered. On the other hand, the rate of return on the instantaneously r i s k l e s s asset i s also reduced from rdt to r(1-R)dt, where r i s the instantaneous r i s k l e s s rate of in t e r e s t . This has the opposite e f f e c t on the bond value - i t pushes up the bond price. Whether the net e f f e c t of these two forces pushes the model price up or down i s not a p r i o r i apparent. I t was not clear what value of R to use i n the model. Ide a l l y , i t should represent the marginal tax rate of the representative investor,. Since no one figure was a v a i l a b l e , i t was decided to try both 8=25% and 50%. The value of the tax rate was kept constant over the whole period., The r e s u l t s of comparing market and model prices for these two cases are also reported in the same tables as the re s u l t s of the previous two models, i e . . Tables XXI to XL. (See also Table XLIX). Comparing with the r e s u l t s of the LIQP model, we note that the mean error value (which was almost consistently p o s i t i v e due to under-valuation of model price) i s equally posi t i v e and negative over the 20 bonds. This seems to imply that introduction of revenue taxes has pushed up model prices - at 150 least for t h i s sample of bonds. Comparing the , we f i n d that increasing R (from zero in the LIQP model to 25% and then 50%) increases ^ almost consistently. Using R=50% pushes |J considerably above 1.0 i n several cases, whereas using E=2S% keeps |3 below 1.0 more often than otherwise. This seems to indicate that an appropriate revenue tax rate i s between the two figures., So f a r , we have been comparing across models using two measures. a) The mean error as a measurement of bias b) The value of as a measure of "responsiveness". The term "responsiveness" i s supposed to measure the j o i n t movement of the two prices - the market and the model price. However, we should recognize that joint movement has two aspects: d i r e c t i o n and magnitude. To c l a r i f y , i f market price drops from one week to the next, and so does model price, there i s perfect harmony between the two with respect to d i r e c t i o n of movement. But i f market price drops by 500, whereas model price by $1, then the model i s over-reacting (which would show up i n a low ^ value). We know that ^ can be expressed i n terms of the c o r r e l a t i o n between the independent (market price) and dependent (model price) variables of the regression as f, - f • J ^ L ( 7 - 1 9 ' where S m f e t and s ^ represent the standard deviation of the market and model prices respectively, and -f5 represents the co r r e l a t i o n between the two. Now we can see that f i s a measure of d i r e c t i o n a l co-movement, whereas the r a t i o of the 151 standard deviations i s a measure of the magnitude. This breakdown of ^ enables us to see which aspect has l e d to a change in the value of p - . , From Table XLII we see that increasing the tax rate (or even including i t i n the f i r s t place) does not improve the c o r r e l a t i o n between market and model prices - i t i s the magnitude factor that i s affected. Thus, introducing revenue taxes helps in fine tuning the r e l a t i v e v o l a t i l i t y of model and market price movements. 7 • 7 Bond Valuation- Incorpprating C a p i t a l Gains Tax Having introduced revenue taxes into the model in the l a s t section, we proceed to see the e f f e c t on model price behaviour, vi s - a - v i s market prices, when we incorporate c a p i t a l gains (CG) tax into the valuation model. Here again, the approach makes assumptions that appear s i m p l i s t i c (as pointed out i n Chapter 2) but what we want to investigate i s whether there i s any improvement i n the predictive power of the model. The effect on model prices of introducing CG taxes i s unambiguous. The benefits to owning the bond are reduced, and so the model prices w i l l decrease with i t s introduction. The ef f e c t on the mean error i s clear ( i t i s expected to increase) , but the e f f e c t on ^ , i s not obvious, fill 20 bonds were valued using a CG tax rate of 10% and 20%. (The revenue tax was kept constant at 25%, as that appeared to be the best model so f a r ) . The r e s u l t s are presented i n columns 5 and 6 of Tables XXI to XL. (See also Table XLIX). Rs expected, model prices are consistently lower when CG taxes are introduced. This i s reflected i n the value of the 152 mean error - the positive values have increased i n absolute value, and the negative ones have reduced in absolute value. (The comparisons are between the results of the 25% Rev. Tax model, and the CG Tax models). In almost a l l the bonds, introducing CG taxes marginally improves the c o r r e l a t i o n between model and market prices but, i n a l l cases, the ^ values go down. This implies that ( S ^ /S^^) goes down by more than ^ goes up (see eguation (7.19)), r e s u l t i n g i n lower ^ values., The v o l a t i l i t y of the model prices thus c o n s i s t e n t l y increases (ie. S m o^ increases) with CG taxes. By appropriately choosing revenue and CG tax values, we can achieve both an improvement i n •f and the slope. 7 • 8 The ffHovinq Average" Hodel From our analysis i n the l a s t two sections, we fi n d that incorporating taxes into the model leads mainly to a " f i n e tuning" effect i n onr attempt to match market and model prices on our sample of retractable and extendible bonds. Taking stock of our objectives, we are attempting to match model and market prices, using broadly three measures: a) the co r r e l a t i o n as a measure of j o i n t d i r e c t i o n a l movement b) the p> c o e f f i c i e n t as a measure of equal amplitude of movement 153 c) the mean error as a measure of b i a s 6 5 . , We noticed that use of the l i q u i d i t y premium hypothesis.-led to substantial improvements i n a l l three measures. Incorporating taxes l e d to improvement on the f i r s t two measures of model performance. However, by using revenue and CG taxes to improve the model's measures of co-movement with the market, control on the extent of bias was foregone to some extent. To draw a crude analogy with the macro-economic policy problem of matching "tools and targets", we need some other " t o o l " to tackle the bias. In our case, too l s are created by relaxing our prio r assumptions to match r e a l i t y . In the analysis i n Chapter 5, we found that the intere s t rate process parameter ^ ' had the most s i g n i f i c a n t impact on bond values. Ceteris paribus , increasing (decreasing) y. would lead to an across-the-board decrease (increase) i n bond values. I t was f e l t , t h e r e f o r e , that the assumption of time homogeneity of the intere s t rate process parameters ( p a r t i c u l a r l y ^ - ) was the p r i n c i p a l source of bias. In t h i s section, we adopt an approximate method of relaxing that assumption. Probably the most elegant approach to the problem to date i s that of Brennan & Schwartz [12], who set up the bond price as a function of both the short term i n t e r e s t rate and a long term int e r e s t rate, where these two rates follow correlated d i f f u s i o n " i t may be argued that root mean square error (RMSE) i s a better measure of o v e r a l l error. From the r e s u l t s presented i n Tables XXI to XL, i t may be seen that the ranking of each bond across models using either mean error or BASE i s v i r t u a l l y i d e n t i c a l . Thus, none of the conclusions would be altered by using RMSE rather than mean error. 154 processes. They take the value of the current long term i n t e r e s t rate as the value of ^ for the short term inte r e s t rate process. However, there are several problems associated with the estimation of parameters of such j o i n t process, as well as with the solution of the p.d.e. for bond valuation, which are beyond the scope of t h i s study. Instead, what we do i s to take as the value of jx f o r each bond, (R1 to E19) the average value of the short term i n t e r e s t rate i n the two years immediately p r i o r to the date of issue.^ This value of ^ i s maintained constant for the l i f e of that bond. The r e s u l t s of this approach are presented i n Tables XXVI through XL fo r bonds E5 to E19. There does not appear to be any s i g n i f i c a n t improvement i n the f i t between market and model prices from t h i s approach. The ) and correlations move a l i t t l e , but not i n any pa r t i c u l a r d i r e c t i o n ; likewise with the mean error. Thus, we may conclude that this approximation of the non-stationarity of jx over time does not appear to improve our r e s u l t s . So f a r , we have not looked at the s e n s i t i v i t y of bond values to the l i q u i d i t y premium parameters. The parameters k, and k<3_ d i r e c t l y 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 follows: ra» = (m-k^ . ) JX' - {mjx +k, ) /m' Tables XLIII and XLIV present the price s e n s i t i v i t y of pure discount bonds to errors in k( and k x respectively. I t may be noted that bond values do not appear to be very s e n s i t i v e to chanqes i n these parameters. However, variations across time i n TABLE XL I I I THEORETICAL SENS IT IV ITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN Kj^ ERROR IN K x -25% -5% 0% +5% +25% CURRENT INTEREST TIME TO MATURITY IN YEARS BOND PRICE % ERROR BOND PRICE % ERROR BOND PRICE BOND PRICE ERROR BOND PRICE % ERROR Vi/2 r=2y 1 3 5 7 10 1 3 5 7 10 1 3 5 7 10 96 .96 89 .05 80 .63 72 .61 61 .85 95 .07 85 .64 76 .97 69 .13 5 8 . 8 1 91 .42 79 .20 70 .14 62 .65 53 .16 0 .0896 0 .6138 1.3539 2.1805 3.4843 0 .0896 0 .6138 1.3539 2.1805 3.4843 0.0896 0 .6138 1.3539 2.1805 3.4843 96 .89 88 .62 79.77 71.37 60 .18 95 .01 85.22 76.15 67 .95 57 .22 91 .35 78 .81 69.39 61 .58 51 .72 0 .0179 0.1225 0.2693 0.4323 0 .6873 0.0179 0 .1225 0 .2693 0.4323 0 .6873 0.0179 0.1225 0 .2693 0 .4323 0 .6873 96 .87 88.51 79.55 71.07 59 .77 94.99 85 .12 75 .94 67.65 56 .83 91 .34 78.71 69.21 61 .31 51 .37 96.85 88.40 79.34 70.76 59.36 94 .97 85.01 75.74 67.36 56.44 91 .32 78.62 69.02 61.05 51.02 - 0 . 0 1 7 9 - 0 . 1 2 2 3 - 0 . 2 6 8 6 - 0 . 4 3 0 5 - 0 . 6 8 2 7 - 0 . 0 1 7 9 - 0 . 1 2 2 3 - 0 . 2 6 8 6 - 0 . 4 3 0 5 - 0 . 6 8 2 7 - 0 . 0 1 7 9 - 0 . 1 2 2 3 - 0 . 2 6 8 6 - 0 . 4 3 0 5 - 0 . 6 8 2 7 96 .78 87.97 78.49 69.55 57.76 94 .90 84 .60 74.93 .66.21 54.91 91.25 78.23 68 .28 60.00 49 .64 - 0 .0895 -0 .6101 - 1 . 3 3 5 8 - 2 .1339 - 3 . 3 6 7 0 - 0 . 0 8 9 5 --0 .6101 -1 .3358 -2 .1339 - 3 .3670 -0 .0895 -0 .6101 -1 .3358 - 2 .1339 - 3 .3670 cn cn TABLE XLIV THEORETICAL SENS IT IV ITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN K, ERROR IN K. CURRENT INTEREST r=y 12 r=y r = 2 U •25% - 5 % 0% +5% +25% TIME TO BOND % BOND % BOND BOND % BOND % MATURITY PRICE ERROR PRICE ERROR PRICE PRICE ERROR PRICE ERROR IN YEARS 1 96 .85 - 0 . 0 2 5 8 96.87 - 0 . 0 0 5 1 96 .87 96 .88 0 .0051 96 .90 0 .0255 3 88 .31 - 0 . 2 2 3 8 88.47 - 0 . 0 4 4 3 88.51 88.55 0 .0440 88.70 0 .2178 5 79 .10 - 0 . 5 6 4 3 79.47 - 0 . 1 1 1 2 79.55 79 .64 0.1104 79.99 0.5441 7 70.37 - 0 . 9 8 4 3 70 .93 - 0 . 1 9 3 6 71.07 71.20 0 .1920 71.74 0 .9445 10 58 .76 - 1 . 6 8 0 2 59 .57 - 0 . 3 3 0 2 59 .77 59 .96 0 .3273 60.73 1.6086 1 94 .95 - 0 . 0 4 4 0 94 .98 - 0 . 0 0 8 8 94.99 95 .00 0 .0087 95 .03 0 .0435 3 84 .85 - 0 . 3 1 5 4 85 .06 - 0 . 0 6 2 4 85 .12 85 .17 0.0620 85 .38 0 .3067 5 75 .40 - 0 . 7 1 4 1 75.84 - 0 . 1 4 0 7 75.94 76.05 0 .1397 76.47 0 .6881 7 66 .86 - 1 . 1 6 7 5 67 .50 - 0 . 2 2 9 6 67.65 67.81 0.2277 68.41 1.1200 10 . 55 .75 - 1 . 8 8 4 5 56.61 - 0 . 3 7 0 4 56 .83 57 .03 0.3672 57.85 1.8051 1 91 .26 - 0 . 0 8 0 5 91 .32 - 0 . 0 1 6 0 91 .34 91 .35 0 .0160 91 .41 0 .0796 3 78.32 - 0 . 4 9 8 3 78.64 - 0 . 0 9 8 6 78.71 78.79 0 .0980 79.10 0 .4847 5 68 .51 - 1 . 0 1 3 1 69 .07 - 0 . 1 9 9 7 69.21 69.34 0 .1982 69.88 0 .9769 7 60 .37 - 1 . 5 3 2 9 61 .13 - 0 . 3 0 1 6 61.31 61 .49 0 .2992 62.21 1.4721 10 50 .19 - 2 . 2 9 1 9 51 .13 - 0 . 4 5 0 8 51.37 51 .60 0.4471 52.50 2 .1990 \ 157 these parameters could account f o r a reasonable amount of the bias between ex i s t i n g model and market prices, as the extent of bias i n percentage terms i s also guite small. 7.9 Tests of Market E f f i c i e n c y We proceed to test the e f f i c i e n c y of the market for retractable/extendible bonds to information contained i n the models. In deriving the basic bond valuation eguation i n Chapter 2, we used a hedging argument, wherein a zero net investment p o r t f o l i o was formed by going long on the generic bond, short on any other bond, and f i n a l l y making up the difference by borrowing or investing i n the short term r i s k l e s s asset.. The dollar amounts to be invested i n each asset were given as: where x, = d o l l a r investment i n generic bond x = d o l l a r investment in any other bond and G represents the generic bond price (with G( i t s p a r t i a l derivative with respect to the i n t e r e s t rate) and B the price of any other bond (with B( i t s p a r t i a l derivative with respect to the i n t e r e s t r a t e ) . The investment in the r i s k l e s s asset i s - (x^ + x 2 •).-... For each of the 20 bonds (.81 to E19) , we have G, based on each model.. He also have prices on straight coupon bonds (F1 to F18), and p a r t i a l derivatives of those bonds with respect to r on each date were computed assuming that the 158 valuation equation for coupon bonds, equation (7.12), was v a l i d . In our f i r s t test of market e f f i c i e n c y , we assume that at the beqinninq of each period (which i s a week i n our case, as we have weekly bond data), we qo lonq on the generic bond by buyinq one bond at the market price (x( =G). we then compute x ? and assume a short position i n a staight bond, and the balance i s made up by an investment i n the r i s k l e s s asset. At the end of the period, we assume that we liq u i d a t e t h i s p o r t f o l i o at the then-existing market prices, and compute the return to the p o r t f o l i o over the one period. Be then form a new p o r t f o l i o , and proceed on u n t i l the end of the data on each bond. Table XLV presents the mean and standard deviation of the returns on these hedges for each bond and for each model. The clear i n d i c a t i o n i s that the returns to the zero-investment 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 d i f f e r e n t from zero 6*. I t appears that we cannot r e j e c t the hypothesis that the market i s e f f i c i e n t to information contained in the models. An a l t e r n a t i v e strategy was also adopted for t e s t i n g market e f f i c i e n c y . It was observed that the hedge p o r t f o l i o returns on the above test were highly s e r i a l l y correlated. The second strategy tested was to assume a long position i n the generic bond only i f the p o r t f o l i o return i n the previous period (based on a constant long p o s i t i o n in the generic bond) was positive -6 6 Hypothesis testing was based on the t - s t a t i s t i c , which assumes that the returns to the hedge p o r t f o l i o are normally distributed. Thorpe [68] has shown that i n the option p r i c i n g framework the hedge p o r t f o l i o returns are not normally distributed., This need not be cause for concern, as the t - t e s t i s quite robust to 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 b r i e f l y ' investiqated toward the end of t h i s section. 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) MOV. AVG. 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) -E 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) -E 2 -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 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) E10 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) E 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) E12 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) E13 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) E14 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) E15 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) E16 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 E17 -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) E18 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) E19 -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) 160 i f negative, a short position was assumed in G, and the hedge position formed accordingly. This strategy was tested for a l l models, but only the r e s u l t s for the pure expectation hypothesis model are presented i n Table XLVI, because the results are very s i m i l a r for a l l the other models. There i s no reason to a l t e r our previous conclusion. The t h i r d test was to see i f the model was able to i d e n t i f y over- and underpriced bonds. This test (based on a test i n Galai[29]) i s quite s i m i l a r to the previous ones, only that each period we take a long (short) position i n the generic bond i f i t s model price i s lower (higher) than the market price at that point.. I f the return on the hedge p o r t f o l i o based on this strategy resulted i n a s t a t i s t i c a l l y s i g n i f i c a n t increase i n the mean return, over the strategy of a constant long position i n the generic bond, we could say that the model i s able to i d e n t i f y overpriced/underpriced bonds. The r e s u l t s of t h i s test for a l l models and bonds i s presented i n Table XLVII. Here again, the mean return appears to be i n s i g n i f i c a n t l y d i f f e r e n t from zero, based on a t - t e s t . The r e s u l t s of the previous three tests were based on the returns to hedge p o r t f o l i o s , using one bond at a time. I t was f e l t that i f the hedge returns over a l l bonds outstanding i n each period was considered (along the l i n e s of Brennan S Schwartz [ 11 ]), the aggregation might lead to a reduction i n the variance of the returns to the hedge p o r t f o l i o and thereby improve the s t a t i s t i c a l s i g n i f i c a n c e of the returns. To overcome the problem of heteroscedasticity caused by the dif f e r e n t numbers of hedge p o r t f o l i o s i n each period, the 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. (Results f o r PEXP model only) BOND Mean Std. Dev. of t - Stat Return ( $ ) R e t u r n R l -0.0412 2.777 -0.149 E l -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 E H -0.0099 0.363 -0.028 E l 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) -E 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) -E 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) -E 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) E 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) E 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) E 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 E 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) E 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) E14 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 15 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) E 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) E 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) E 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) E 19 -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) 163 return i n each period was weighted by 1/JIT, where N represents the number of retractable/extendible bonds outstanding (which therefore represent the number of hedge p o r t f o l i o s formed) in each period. The r e s u l t s are presented in Table LI. The mean dol l a r return per period, as well as i t s standard deviation, remained of the same order of magnitude as i n the case of the r e s u l t s i n Tables XL? to XLVII - aggregation has not led to any s t a t i s t i c a l l y s i g n i f i c a n t increased p r o f i t opportunity. This r e s u l t was not unexpected. The movement of bond prices exhibits high contemporaneous correlation,so that the returns to the zero investment hedge p o r t f o l i o s would also be likewise correlated., Thus, aggregating across bonds at any instant in time would not lead to any s i g n i f i c a n t reduction i n the dispersion of returns to the hedge p o r t f o l i o . In the case of options on common stock, however, the contemporaneous co r r e l a t i o n across d i f f e r e n t stocks i s not so high, which could lead to variance reduction due to aggregation on a s i m i l a r te s t . In forming the hedge p o r t f o l i o s , for an investment of x, d o l l a r s i n the generic bond, the strategy was to i n v e s t x^ d o l l a r s i n another bond, where x L was given by In the tests performed so f a r , the value of B used i n the above expression was the market price of the s t r a i g h t bond. I t could be argued that model prices should be used f o r B. The reasoning i s that we want to observe whether the retractable/extendible bond offers arbitrage p r o f i t opportunities, a f t e r c o n t r o l l i n g for other factors. When we use market price for 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 (Results f o r a l l models) Model Mean Return ($) Std. Dev. of Return t - S t a t 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 aggregating over a l l kinds outstanding i n every p e r i o d , was a l s o performed on the other 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 are not reported as they are very s i m i l a r to the ones above. 2) The models have been l i s t e d i n the t a b l e using the abb r e v i a t i o n s used i n the text and i n e a r l i e r t a b l e s . 165 valuation error i n B, the correct hedge proportions are not maintained which increases the variance of the returns to the zero investment hedge p o r t f o l i o . By using model values of 8, there i s no other source of error - i t i s a pure test of the retractable/extendible bond. a l l three market e f f i c i e n c y related t e s t s reported above were repeated, (for each i n d i v i d u a l bond and aggregated over a l l bonds) for each of the models used for valuing the retactabie/extendible bonds. The r e s u l t s were hardly any different from those obtained by using market p r i c e of B to evaluate x% , as well as for evaluating the hedge p o r t f o l i o returns. To indicate the degree of s i m i l a r i t y of r e s u l t s from using market and model prices of B in the tests of market e f f i c i e n c y , the mean and standard deviation of the zero investment hedge p o r t f o l i o return for the C a p i t a l Gains Tax 20% model(using a strategy of a constant long position i n the generic bond) are presented i n Table L.., It was f e l t that no further information would be conveyed by presenting the complete r e s u l t s across a l l models f o r a l l three hedging strategies. F i n a l l y , the p o r t f o l i o returns (on the zero investment hedge position) were tested for normality using the probability graphing approach outlined e a r l i e r . . This was not s t r i c t l y necessary, as the t - s t a t i s t i c of the mean return (ie. mean/standard deviation) was almost always of the order of 0.1 and that should be s t a t i s t i c a l l y i n s i g n i f i c a n t i n most sit u a t i o n s even with resonable departures from normality. In Figures 8 and 9, we present two sample cases., In general, the d i s t r i b u t i o n s appear to have more mass at the mean than a normal d i s t r i b u t i o n of equal mean and variance. TABLE L COMPARISON OF RETURNS TO THE ZERO INVESTMENT HEDGE PORTFOLIO BY USING MARKET VS. MODEL PRICES FOR THE STRAIGHT BOND USING MODEL PRICES FOR STRAIGHT BOND USING MARKET PRICES FOR STRAIGHT BOND BOND MEAN STD.DEV - t-STAT MEAN STD.DEV t-STAT Rl E l E2 E3 E4 E5 E6 E7 E8 E9 ElO E l l E12 E13 E14 E15 E16 E17 E18 E l 9 Aggregate 0.0135 0.0627 0.0622 0.0831 0.0631 -0.0019 0.0010 0.0125 0.0193 0.0106 0.0058 0.0089 -0.0034 -0.0075 -0.0032 0.0200 0.0281 -0.0257 0.0149 -0.0152 0.0265 0.287 0.349 0.517 0.664 0.474 0.220 0.334 0.425 0.452 0.567 0.505 0.686 0.343 0.482 0.375 0.509 0.511 0.484 0.289 0.588 0.786 0.047 0.180 0.120 0.125 0.133 -0.009 0.003 0.029 0.043 0.019 0.011 0.013 -0.010 -0.016 -0.009 0.039 0.055 -0.053 0.051 -0.026 0.034 0.0171 0.379 0.0534 0.0752 0.0551 0.0042 0.0084 -0.0091 0.0095 0.0067 0.0030 0.0061 0.0010 -0.0031 0.0009 0.0260 0.0232 -0.0176 0.0147 -0.0049 0.0242 0.233 0.183 0.280 0.303 0.233 0.168 0.487 0.394 0.419 0.389 0.296 0.283 0.431 0.357 0.285 0.402 0.416 0.395 0.223 0.420 0.466 0.073 0.207 0.191 0.248 0.237 0.025 0.017 -0.023 0.023 0.017 0.010 0.021 0.002 -0.009 0.003 0.065 0.056 -0.045 0.066 -0.012 0.052 NOTES: 1) The above returns correspond to using the constant long p o s i t i o n i n the generic bond strategy. 2) The model used for the valuation of the generic bond was the C a p i t a l Gains 20% model. FIGURE 8 167 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 RDJU5TING 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) BOHDi f4-*-501 I *™ 1 l 9 6 ^ Jtll i -0.45 « 10 » -I •SmntVi 0.335 -O.OflS " VPLUt tr HECSE !« HU3GE K1URH IK FIGURE 9 1 6 8 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 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 KR.1l -0.13 X JO KX -I SIOKVi 0.J93 169 7.10 Comparison of Current Models with a "Naive" Model Before we conclude our analysis on bond prices, we need a bench mark against which to compare the performance of our models i n valuing bonds. To t h i s end, we develop an ad hoc valuation model - which we s h a l l refer to as the "naive" model. I t i s based on an approach suggested i n Dipchand S Banrahan [9]. Based on a regression equation f o r the y i e l d curve developed by B e l l Canada*s Bond Research D i v i s i o n , we compute the y i e l d to maturity on each extendible* 7 i n our sample. For each bond, at each point i n time, we estimate two yields to maturity - one correspondinq to each of the al t e r n a t i v e maturities. Usinq each y i e l d , we discount the future coupons (assuming continuous coupon payments) and the p r i n c i p a l , and thus f i n d the values of the long and short bonds. The price of the extendible i s then set to the higher of the long and short bonds, at every point i n time. B e l l Canada's y i e l d curve regression model was where Y t represents the y i e l d t o maturity at time t on a bond having X t months to maturity. For our study, we modified the model s l i g h t l y to include in the regression equation the current value of the short term interest rate. I t was f e l t that t h i s i n c l u s i o n should improve the f i t of the model. Thus, the 6 7 The retractable B1 was not priced according to the naive model because i t had several retraction dates. This makes i t complicated to price, and i t was f e l t that dropping one case should not a f f e c t the comparison. 170 regression model used to determine yields was Yt at - v a ^ + a3X<. -V c^ Xt + <*5 \ + 4 CL^Xf. (7.20) The next problem was to determine the c o e f f i c i e n t s . For t h i s purpose, the s t r a i g h t coupon bond sample was used. The market price of a bond at any instant i s the present value of i t s future payoffs. Thus we can write \ = c*~* dtt + loo e r where y i s the y i e l d to maturity at time t, c i s the continuous coupon, X i s the time to maturity, and the face value of the bond i s $100. This gives (7.21) Using eguation (7.21) above, we can solve f o r y 6 8 , given B% and the other parameters. This was done f o r a l l 18 straight coupon bonds at each point in time. Then, for each of the 18 bonds, and for the whole sample, regression (7.20) was performed. ; The r e s u l t s of the regression are reported i n Table XLVIII. Consistent with the experience of Dipchand S Hanrahan [19], the R 2 from most of the equations was over 0.80 (except f o r F9 and the whole sample). The regression c o e f f i c i e n t s based on the t o t a l sample were used to price each of the 19 extendible bonds * 8 A numerical algorithm that solves for the zeros of nonlinear equations was used. The s t a r t i n g value supplied i n the search for a root was the current value of r. I t can e a s i l y be shown that equation (7.21) has only one root. 171 TABLE XLVIII RESULTS OF YIELD EQUATION COEFFICIENT ESTDIATION FOR "NAIVE" MODEL (Yeild = a x + a 2 r t + a^T + a ^ / i - + a j T 2 + a 6 T 3 + a 7 l o g T ) BOND a ^ l O 2 a 2 a 3 «103 «4X1.02 a 5 *10 5 a 6 x l0 9 a 7 *10 2 R 2 F l 0.6510 (16.42) 0.4857 (14.96) 0.8323 (5.13) -0.0434 (-4.13) -0.3163 (-6.72) 7.8739 (7.89) 0.8427 (3.01) 0.9418 F2 0.0776 (13.53) 0.4636 (36.70) -0.0241 (-11.61) 0.0652 (12.71) 0.00151 (7.76) -0.0046 (-4.03) -0.0940 (-11.85) 0.9295 F3 0.0z82 (15.42) 0.5863 (52.52) -0.0057 (-3.10) 0.0210 (4.57) -0.0001 (-0.69) - 0.0032 (3.90) -0.0347 (4.97) 0.8977 F4 0.0531 (6.41) 0.8941 (43.28) 0.1026 (3.90) -0.1273 (-3.14) -0.0414 (5.69) 1.0565 (7.62) 0.1129 (2.61) 0.9342 F5 -0.0658 (-1.31) 0.9979 (31.38) -0.3540 (-2.87) 0.4993 (2.44) 0.1211 (3.89) -2.7779 (-4.84) -0.5007 (2.09) 0.9270 F6 0.0882 (20.78) 0.59S2 (47.67) 0.0098 (5.31) -0.0105 (-2.63) -0.0018 (-9.12) 0.0140 (10.76) 0.0004 (0.07) 0.8916 F7 0.3142 (20.24) 0.8066 (20.71) 0.0918 (1.33) -0.0486 (-0.45) -0.0546 (-2.94) 1.4785 (4.28) -0.0668 (-0.58) 0.9513 F8 -2704.9 (-3.38) 0.3566 (21.87) 251.09 (3.20) -172.87 (-3.26) -0.3991 (-3.06) 0.5026 (2.92) . 870.01 (3.33) 0.9004 F9 -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 F10 0.0738 (3.09) 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 F l l 0.0289 (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 (0.70) -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 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 in parenthesis are the t s tat i s t ic for the estimated coefficient J 172 i n our sample. The r e s u l t s of regressing the market prices on these model prices are reported i n Tables XXII to XL. The r e s u l t s from the summary run by aggregating over a l l the 19 bonds i s i n Table XLIX. A cursory examination of the r e s u l t s indicates that the naive model performs reasonably well, i n comparison to the other models. Closer scrutiny however reveals the superiority of the ,-, more rigorous models of retractable/extendible bond valuation developed i n t h i s study. The three c r i t e r i a used to evaluate the performance of each model were: 1) c o r r e l a t i o n between market and model values 2) slope of the regression of market and model prices 3) mean error (or RMSE) as a measure of bias. Comparing the r e s u l t s of the C a p i t a l Gains Tax 20% (CG Tax 20%) model (column 6 i n Tables XXI to XL and Table XLIX) with that of the naive model, i t i s seen that the CG Tax 2.0% model outperforms the naive model on the f i r s t two counts almost consistently. Looking at the summary res u l t s from pooling a l l 20 bonds (Table XLIX), we see that above observation i s borne out with respect to the slope c o e f f i c i e n t . The c o r r e l a t i o n between market and model prices (the square root of the R-squared i s the simple 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 superior i n the naive model. However, as pointed out i n the e a r l i e r sections; a l t e r i n g the Revenue Tax rate and the C a p i t a l Gains Tax rate, provides a " f i n e tuning" mechanism to improve the 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 . Since the objective of the present study i s more one of description, rather than of " f i t t i n g " the best model, no further attempt was made to f i n d a 173 set of tax rates that actually provided consistently improved correlations over that of the naive model. F i n a l l y , looking at the bias measures, we see from the summary r e s u l t s i n Table XLTX that the mean error i s lower for the CG Tax 20% model, whereas the P.MSE i s lower f o r the naive model. Comparison of the mean error over i n d i v i d u a l bonds (Table XLI), we see an almost even s p l i t - the naive model performs better just as many times as the CG Tax 20% model. However, we note that i f we were to increase the CG Tax rate used i n the model, t h i s would lead not only to a reduction i n the mean error but also to an improvement i n the c o r r e l a t i o n . Thus, i t would be f a i r to say that, even i n t h e i r present state, the p a r t i a l equilibrium models developed i n Chapter 2 are superior i n several respects i n predicting retractable/extendible bond price movements, when compared with a naive model of a reasonable l e v e l of complexity,and - unlike the naive model - are amenable to considerable further improvement. 174 CHAPTER 8: SUMMARY AHD CONCLOSIONS 8. 1 Summary Of The Thesis The current research can be divided into three broad areas; 1) choosing an appropriate continuous time stochastic s p e c i f i c a t i o n to model the instantaneous r i s k l e s s rate of interest. 2) i d e n t i f y i n g methods to estimate the parameters of such a model, given a discrete time r e a l i z a t i o n of the interest rate process, and comparing the r e l a t i v e e f f i c i e n c i e s of the d i f f e r e n t estimating methods. 3) developing and empirically testing a model for valuing default-free retractable and extendible bonds. Chapter 3 addresses the problem of choosing an appropriate mathematical model for the short term r i s k l e s s i n t e r e s t rate process. In the absence of any formal guidelines, economic reasoning and mathematical t r a c t a b i l i t y were the only c r i t e r i a . A mean-reverting d i f f u s i o n process was suggested, having a d r i f t term of the same form as that adopted by others i n the existing l i t e r a t u r e , (see Vasicek [72], Cox,Ingersoll & Boss [16]) but with a more general variance element. Thus, the d i f f u s i o n eguations adopted by Vasicek [72] and Cox, Ingersol 6 Ross [16], are both s p e c i a l cases of the more general form used in t h i s study. The behaviour of the assumed form of the i n t e r e s t rate process at i t s singular boundaries i s investigated, to ensure that i t s behaviour at these points i s consistent with the properties attributable to an interest rate process from 175 economic reasoning. Three alternate methods are proposed in Chapter 4 for the estimation of the parameters of the intere s t rate process, and th e i r r e l a t i v e merits and weaknesses are pointed out. A l l of them are maximum li k e l i h o o d methods. The Transition Probability density method i s exact, but the t r a n s i t i o n probability density i s not known for a l l parameter values of the proposed process. I t s use would require c u r t a i l i n g the generality of the interest rate process model. The other two methods (the Steady State density approach and the Simple Li n e a r i z a t i o n method) are both based on approximations. Ho a n a l y t i c a l method could be developed to compare the estimators of the parameters - Monte Carlo methods had to be employed. Chapter 5 presents the res u l t s of the Monte Carlo simulations to arrive at the d i s t r i b u t i o n of the estimators, using the three alternate methods of parameter estimation. The c r i t e r i a used to compare across the three methods was (a) the bias and variance of the estimators and, (b) the resultant bias and variance on bond prices. The results indicate that a l l three methods produce estimators with rather s i m i l a r properties, and so are quite comparable. P a r t i a l equilibrium valuation models based on the option pri c i n g approach were developed i n Chapter 2, f o r very general stochastic s p e c i f i c a t i o n s of the i n t e r e s t rate process. The valuation models draw heavily from the e a r l i e r works of Cox, Ingersol & Boss [16], Brennan 6 Schwartz [10,12], and Vasicek [72]. The performance of models developed i n Chapter 2, when the intere s t rate process of the chosen form i s 176 incorporated, i n p r i c i n g a sample of retractable/extendible bonds was tested i n Chapter 7. The bond sample chosen was the complete set of retractable/extendible bonds issued by the Government of Canada. The sample consisted of one retractable bond issued i n January 1959 and 19 extendibles issued between October 1959 and October 1975. weekly data on market prices for t h i s set was collected from the Bank of Canada Review . Model prices based on the pure expectations hypothesis about the term structure of intere s t rates on the part of investors were consistently higher than actual market prices. When a provision was made for a term/liquidity premium i n the term structure of in t e r e s t rates, model prices were more i n l i n e with market prices. Incorporating revenue taxes ( t a x e s on inte r e s t payments and on coupon receipts) and then c a p i t a l gains taxes, improved the performance of the model i n predicting market p r i c e movements,, To serve as a benchmark for evaluating the performance of the model, an ad hoc regression-based valuation formula was developed to price the sample of extendible bonds. I t was found that the p a r t i a l equilibrium models performed a t l e a s t as well as the ad hoc model - with further refinements the equilibrium models could dominate the ad hoc model. F i n a l l y , the e f f i c i e n c y of the bond market to information contained i n the models was tested. The approach was to set up a zero net investment hedge p o r t f o l i o by investing in the retractable/extendible bond, the short term i n t e r e s t rate, and any other bond, and observing whether any arbitrage opportunities were a v a i l a b l e . The r e s u l t s indicated that the 177 market was consistently e f f i c i e n t to information contained i n these models. 8*2 Conclusions And Directions For Further Research The i n t e r e s t rate process proposed in Chapter 3 i s of the form The processes used i n e a r l i e r studies were s p e c i a l cases of the above process.. Thus, Vasicek's process corresonds to d\ = 0, whereas Richards and Cox, Ingersoll S Ross both use the process having cK - 1/2. The r e s u l t s i n Chapter 5 indicate that increasing the generality of the model by including an extra free parameter (o() i n the variance element does not materially enrich the family of processes. It was found that cr2 and c\ were very highly correlated and t h e i r influence on the process dynamics was almost t o t a l l y substitutable. It appears that bond values r e s u l t i n g from the above in t e r e s t rate process are most sensi t i v e to the parameter jx the o v e r a l l mean of the process. What i s more in t e r e s t i n g i s the fact that the other parameters ( mn , <rv, d.) have very l i t t l e impact on bond valuation. This i s an i n d i c a t i o n that, even though the above model of i n t e r e s t rates may be quite s a t i s f a c t o r y to portray the i n t e r e s t rate dynamice jper se , as far as bond valuation i s concerned we have only a one-parameter process. This c l e a r l y indicates the need to look for alternative stochastic s p e c i f i c a t i o n s for the i n t e r e s t rate process, where more than one parameter has a s i g n i f i c a n t impact 178 on bond valuation.. The assumption of homogeneity over time of the i n t e r e s t rate process parameters appears r e s t r i c t i v e . The constraint i s , however, to afford mathematical t r a c t a b i l i t y , both for the estimation of the parameters of the process as well as i n bond valuation. The approach of Brennan S Schwartz [12] appears to be one elegant solution to the problem. In the framework of the in t e r e s t rate model of t h i s t h e s i s , their model for the short term i n t e r e s t rate i s eguivalent to setting = 1 and making stochastic (they set yjL as the long term i n t e r e s t r a t e ) , where r and jx follow correlated j o i n t d i f f u s i o n processes. As pointed out i n the text, such processes pose additional problems i n estimation of the parameters, and even more i n solving the resultant valuation equation. However, the additional e f f o r t might well be worthwhile. We have seen that ^ i s the c r i t i c a l parameter of the i n t e r e s t rate process i n bond valuation; allowing i t to be stochastic should lead to improved congruence between model and market prices. The term structure of i n t e r e s t rates plays a p i v o t a l role i n the valuation of default-free bonds. In the approach of the thesis, we attempted to predict the complete term structure from a knowledge of the instantaneous i n t e r e s t rate. This i s rather ambitious. The approach of Brennan & Schwartz [12] i s an attempt to predict the term structure, at any instant i n time, knowing the two extreme points - the instantaneous and the very long term y e i l d s . Thus, i t would be reasonable to expect that a model of retractable/extendible bond valuation based on two state variables (the short term and the long term interest 179 rates) and time to maturity should give s i g n i f i c a n t l y better r e s u l t s . I t i s evident from the br i e f survey of the exis t i n g l i t e r a t u r e presented i n Chapter 1 that a f a i r amount of work needs to be done i n the area of empirical testing of bond valuation models developed i n the option pricing framework. The present thesis i s one step in that d i r e c t i o n . However, we have addressed only the valuation of default-free bonds. The whole area of corporate bonds {where a positive p r o b a b i l i t y of default exists) has not been tackled. The valuation theory has been developed i n the l i t e r a t u r e , but empirical testing poses the problem of choosing some observable proxy for the value of the firm, as t h i s i s a required input to the bond valuation model. This would be a f r u i t f u l d i r e c t i o n for future research. F i n a l l y , there i s considerable inte r e s t at present i n ar r i v i n q at closed form or a n a l y t i c a l solutions to the term structure equation. Vasicek [72] and Cox, Inger s o l l 6 Ross [16] have two d i f f e r e n t stochastic s p e c i f i c a t i o n s to model the course of the instantaneously r i s k l e s s rate of return., I t can, i n qeneral, be shown that the re s u l t i n q pure discount bond valuation equation c l o s e l y resembles the Kolmogorov backward eguation qoverninq the d i f f u s i o n equation chosen to model the int e r e s t rate process. I t i s also well known that, i n general, by a suitable r e d e f i n i t i o n of variables, the backward equation may be transformed into a s i m i l a r forward equation, as pointed out i n Appendix 3, the forward equation could be transformed into 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. This might be an in t e r e s t i n g d i r e c t i o n f o r researchers interested i n a n a l y t i c a l solutions to the term structure eguation f o r alternate stochastic models for the inter e s t rate process. 181 BIBLIOGRAPHY 1 Anderson T. fl. 6 Goodman L. A. (1957), " S t a t i s t i c a l 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 Discrete Approximations to Systems of Stochastic D i f f e r e n t i a l Equations", Econometrica. 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We have as our d i f f u s i o n equation dr = b ( r ) d t + Al a(r) dZ (Al.l) where b (r) = m(y-r) and a ( r) = a 2 r 2 a (AOi.la) The type of behaviour at the singular boundary i s determined by the i n t e g r a b i l i t y of the following two functions h x ( r ) = 7T(r) f r r o [a (s) IT (s) ] 1 ds h 2 ( r ) = [a(r) TT (r) ] -1 f r TT (s) dS (A1.2a) (A1.2b) Over the i n t e r v a l I E [r #r ], where r Q i s any i n t e r i o r point of the state space of the process, and r i s the boundary (r might be i n f i n i t e i n the case of no " b u i l t i n " f i n i t e boundary) The function 'k(r) i n equations (A1.2) i s defined by r r TT (r) = exp {-2 [b (s)/a(s)]ds} (A1.2c) When both h^ and h^ are integrable over I, the boundary i s c a l l e d a regular boundary and by imposing suitable boundary conditions, the behaviour can be either r e f l e c t i n g or absorbing. When h-^ i s integrable over I, but h 2 i s not, the boundary i s ca l l e d an e x i t boundary and i t acts as an absorbing boundary. When h-^ i s not integrable over I, but h 2 i s , the boundary i s c a l l e d an entrance boundary. An extrance boundary i s inaccessible from inside the open i n t e r v a l ( r Q , f ) , but any 188 r 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 into the open i n t e r v a l . When both h-^ and h.^ a r e n o t integrable over I, the boundary i s c a l l e d a natural boundary. This boundary i s inaccessible from inside the open i n t e r v a l , and any prob a b i l i t y assigned to i t i n i t i a l l y i s trapped there forever. It can be shown that (see Keilson [41]) r r h-j_(s)ds = M^(r|r Q) = average time to reach r s t a r t i n g 0 from r Q ( r Q i s a r e f l e c t i n g boundary) r h 2(s)ds = M^(r Q|r) = average time to reach r Q s t a r t i n g r o from r (r i s a r e f l e c t i n g boundary) This provides the i n t u i t i o n behind the singular boundary c l a s s i -f i c a t i o n . For the caseoof a=l, we have by substitution from (Al.la) into (A1.2c), and performing the required integration TT (r) = r^ exp ( 3 u/r) 3 = 2m/a2 and further from (A1.2) h, (r) = r Bexp ( 3u/r) 1 2 x" exp(-3y/x) d-x.;6=(2+3) >0 r o h,(r) = 6 r " =exp(-3y/r) r r x$ exp(3y/x)dx ...(A1.2d) r o Performing the integration for h^(r) gives 189 ey/r. h ^ r ) =•;[ a^re a 2 ( B y ) 2 3 y a — 1 » a^ ' C o n s t a n t o f i n t e g r a l t o n ( A l . 3 ) C l e a r l y h j ( s ) d s ' a p p r o a c h e s i n f i n i t y as r t e n d s t o 7 r o i n f i n i t y due t o t h e s e c o n d and t h i r d terras i n (Al.. 3)' above. We need o n l y e v a l u a t e t h e i n t e g r a l as r t e n d s t o z e r o . H e r e , t h e l a s t two terms o f (A1.3) a r e c l e a r l y f i n i t e . Thus we need t o l o o k a t t h e f i r s t t e r m o n l y . [We may c o n v e n i e n t l y d r o p m u l t i p l i c a t i v e c o n s t a n t s ] a ra h - ^ s j d s ^ (3y/s se d s (A1.3a) I f we now make t h e s u b s t i t u t i o n 1/y = s , we can i n t e g r a t e and g e t ( a l l i n t e g r a l s were o b t a i n e d from G r a d s h t e y n and R y z h i k [ 3 5 ] ) h 1 ( s ) d s •« y3y 2 +8yEi(eyy) y-(A1.3b) 1/a where E i (.) i s t h e e x p o n e n t i a l i n t e g r a l . Now (A1.3b) i s c l e a r l y unbounded as Y a p p r o a c h e s i n f i n i t y . Thus h ^ ( r ) i s unbounded a t b o t h b o u n d a r i e s , w h i c h c l e a r l y i m p l i e s t h a t , e i t h e r b o t h b o u n d a r i e s a r e i n a c c e s s i b l e , o r e n t r a n c e b o u n d a r i e s , r r d e p e n d i n g upon h „ ( s ) d s , as r t e n d s t o z e r o and i n f i n i t y . Making t h e s u b s t i t u t i o n 1/z = x i n t h e e x p r e s s i o n f o r h 2 ( r ) i n e q u a t i o n (A1.2b) g i v e s h 2 ( s ) d s = r ~ e x p ( - i . S / r ) !; a2 < '±z 6 exp (ygz)dz 1/a d r (A1.4) 1.90 S i n c e we a r e o n l y i n t e r e s t e d i n t h e b e h a v i o u r o f t h e i n t e g r a l a t t h e b o u n d a r i e s , we can w i t h o u t any l o s s o f g e n e r a l i t y e v a l u a t e t h e i n t e g r a l s f o r 8=1 i e 6=3. T h i s g i v e s h 2 (s)d£ a2.exp(-Bu/s) + • 1 + .0;u. , (B-u.) 2exp (-Bu/s) a V 2 " 2a ^ s loT^l ^3 E i ( 3 y / s ) ds . . (A1.4a) Due t o t h e s e c o n d and t h i r d terms i n t h e e x p r e s s i o n a b o v e , t h e i n t e g r a l a p p r o a c h e s i n f i n i t y as r a p p r o a c h e s z e r o . F u r t h e r , as r a p p r o a c h e s i n f i n i t y , t h e i n t e g r a l i s unbounded, due t o t h e s e c o n d term a l o n e . Thus the i n t e g r a l o f h 2 ( r ) i s unbounded a t b o t h b o u n d a r i e s . Thus b o t h r=0 and r=°° a r e n a t u r a l b o u n d a r i e s . F o r the c a s e u=h, t h e b e h a v i o u r a t t h e s i n g u l a r b o u n d a r i e s has been s t u d i e d by F e l l e r [ 1 8 ] . I n b r i e f , h i s r e s u l t s a r e : 1) T=y°° i s a n a t u r a l boundary 2) a t r = 0 , the boundary b e h a v i o u r depends upon the p a r a m e t e r v a l u e s . a) i f m i s n e g a t i v e (or r a t h e r my were n e g a t i v e ) , t h e boundary i s an e x i t boundary 2 -b) 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 o r a r e f l e c t i n g b a r r i e r . 2 c) i f 2my>a , we have an e n t r a n c e b o u n d a r y . APPENDIX 2 D e t a i l s of the E s t i m a t i o n Procedure f o r the L i n e a r i z e d Model The SDE governing the d i f f u s i o n process i s dx = m(y-x)dt + ax adz (A2.1) We can r e p l a c e dx = ( x t + l ~ x t ^ a n d x E x f I f w e f u r t h e r choose our u n i t of time equal to 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 t + l = m 1 X > + ^ 1 - m ^ x t + a x t a n t ( A 2 . 2 ) where n t ^ N(0,1). Equation (A2.2) i m p l i e s P ( x t + 1 x t,8) * N[{my+(l-m)x t},a 2x t 2 a] . . . (A2.2a) We can t h e r e f o r e s e t up the l i k e l i h o o d f u n c t i o n (logs taken) , 0 , [x.^v--my-(1-m) x. ] 2 1 „ , 2a n , o l ^ r t+1 t J - -1 as L = 2 E l o g x - j l o g c 2 - ^ 2z{ ^ j - —} x t (A2.3) From the form of e q u a t i o n (A2.3), i t can c l e a r l y be seen the m and y e n t e r o n l y i n the l a s t summation term, which i s e x a c t l y the r e s i d u a l v a r i a n c e term. Thus m and y are j u s t the l e a s t squares estimates g i v e n a. F u r t h e r d i f f e r e n t i a t i n g L w.r.t. a 2 and s e t t i n g to zero g i v e s •n , 1 „ f [ * t + l - m y - ( l - m ) x t ] 2 _ 2 a2 ( 2 a 2 ) a 2 1 { ^ 5 a o r 1 r T x t + l " m y ~ ( 1 " m ) x t 3 " n E { 2a x, (A2.4) The s t r u c t u r e o f a 2 , m and y s u g g e s t a s i m p l e i t e r a t i v e p r o c e d u r e f o r e s t i m a t i n g a l l the p a r a m e t e r s . a) P i c k a s t a r t i n g v a l u e o f a = b) U s i n g OLS ( o r d i n a r y L e a s t squares) t o e q u a t i o n A 2 . 2 , a f t e r d i v i d i n g t h r o u g h by x0 .^ , we can e s t i m a t e m and y . E q u a t i o n (A2.4) t h e n g i v e s an e s t i m a t e o f a 2 . I t i s w e l l known t h a t o u r e s t i m a t e s o f m and y a r e i n e f f i c i e n t c) E v a l u a t e 8L/9a f o r t h e p r e s e n t p a r a m e t e r v a l u e s and p i c k the n e x t a t o a t t e m p t s e t t i n g 9L/3a = 0 where [x - m y - ( l - m ) x t ] 2 l o g x 2 ^ = - Z l o g x t + 2- a 2 Z{ — } X t The n e x t q u e s t i o n i s how t o p i c k a r e a s o n a b l e s t a r t i n g v a l u e o f a? T h i s can be done by b r e a k i n g down the p r o b l e m u s i n g an a p p r o x i m a t i o n . S q u a r i n g (A2.1) we have ( d x ) 2 = a 2 x 2 a ( d z ) 2 , dz * N ( 0 , d t ) I f we now r e p l a c e d i f f e r e n t i a l s by d i f f e r e n c e s , and c h o o s i n g the u n i t o f t ime as b e f o r e , we have 2 o ? ot ? y t ~- a 2 x 2 x 2 1 } 193 where y f c = Ax t = (x t + J__- x f c ) L e t z E y 2 / ( a 2 x t 2 a ) <\, X(^) a n d s o w e have f ( z ) = - i - z " 1 / 2 exp ( - z / 2 ) dz / 2 T T Suppose we now s e t up t h e j o i n t l i k e l i h o o d o f the d a t a i n terms o f the y ' s we have T n 2a _ i / n 1 Y+. (data) = n — ( y 2 . a 2 . x . ) x / exp — ) t = l /2T 2 a 2 x 2 a T a k i n g l o g s and d r o p p i n g a d d i t i v e c o n s t a n t s .-gives V 2 T ,__ _9 1 „ ,2 „ 2a, 1 ^ , y t L = - i l o g a 2 - i E l o g (y\ x . ^ a ) - =• 2 2 ( — — ) (A2.5) z z r r 2a X t 3L T 1 ^ t 2 9 7 2 = " l a 2 + 2 7 ^ " E ( 7 ^ " « ) = 0 X t 1 Y t 2 w h i c h g i v e s a 2 = — E ( — — ) (A2.6) *T . 2a X t and ^ = I E C ( ^ 2 x ^ T " X ) l o ^ x t 2 ] • • • • ( A 2 ' 7 ) The a p p r o a c h i s t o i t e r a t e between (A2.6) and ( A 2 . 7 ) , so as t o reduce 9L/9a = 0 . I t i s - found t h a t c o n v e r g a n c e i s v e r y f a s t . F i n a l l y , g o i n g back t o t h e o r i g i n a l p r o b l e m , we can g e t an e s t i -mate o f t h e a s y m p t o t i c v a r i a n c e r- c o v a r i a n c e m a t r i x 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 m a t r i x a t t h e chosen o p t i m a l p o i n t . The e l e m e n t s o f the h e s s i a n m a t r i x a r e g i v e n b y : 194 3 2 L _ -T-" 3 ( a 2 ) 2 " 2 ( a 2 ) 2 3 Z L 3 a 2 l a 2 * t { x t + 1 - m y - ( 1 - m ) x t } l o g x t 2 , 2 •v 2a X t 3 2 L 3 m 2 ( r x t ) 2 a 2 x ToT 3_fL 3 y 2 a 2 1 x 2a 3 2 L 3 a 3m ( 1 J - x - t ) 1 9 9 X t 2 - 2 E {a(x) u • ^ 2 ~ - • >; a(x) x t + 1 - m y - ( 1 - m ) x f c 3 2 L 3 a 3 y m - ^ 2 E { a(x) l o g xfc2 3 Z L 3 a 30 " T ^ f y y E H a ( x ) } 2 l o g xfc2 x t 2 a 3 Z L 3m3y. 1 y r 1 a(x) a 2 E l x t 2 * m ( H - x t ) a 2 T i a ( x ) ( y - x ) 3 m 3 a ^ , ( a 2 ) 2 I 2 a J t 3 2 L "•• _ _ m , a(x) , 3 y 3 c 2 ' ' " ( a 2 ) 2 2 "x 2aJ 195 APPENDIX - 3 S o l u t i o n to the f o r w a r d e q u a t i o n f o r a = 1 The SDE f o r the d i f f u s i o n p r o c e s s i s 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 d e n s i t y follows^ t h e FP eqn f f = ~ 4 [m(u-x )P] + \ - I i - [ « r 2 x 2 p ] _ . ( A 3 > 2 ) 8 x 2 w i t h the i n i t i a l c o n d i t i o n P ( x | X q , 0 ) = 6 ( X - X Q ) . . . (A3.2a) We can t r a n s f o r m eqn (A3.2) t o t h e form H - -St" [ *< z ' 9 1 • \ (A3.3) 3 z by u s i n g the s u b s t i t u t i o n s rx: z(x) = (trs) -"-ds a ( z ) = [ m ( y - x ) - j | x ( a 2 x 2 ) ] ( a 2 x 2 ) . . . (A3.4a) . g ( z , | ' z 0 , t ) = ( a x ) P ( x | x Q , t ) | x = x(z) (A3.4b) w i t h i n i t i a l c o n d i t i o n g (Jz { 2 Q > q ) = < 5 ( Z - Z Q ) (A3.4c) We s h a l l t h e r e f o r e c o n c e n t r a t e on a s o l u t i o n t o the t r a n s f o r m e d e q u a t i o n (A3.3) and once we s o l v e f o r g , we can r e t r i e v e P u s i n g (A3.4b) . U s i n g the s t a n d a r d s e p a r a t i o n o f v a r i a b l e s we g e t g(z |-'z f t ) = Q ( z ) e ~ E t / 2 (A3.5) E q u a t i o n (A3.„3)now r e d u c e s t o the .eigen v a l u e p r o b l e m 2 - | [a(z)Q] + EQ = 0 (A3.6) dz where t h e boundary c o n d i t i o n s on Q a r e g i v e n by t h e c o n d i t i o n s on g t h r o u g h eqn (A3.5) We can f u r t h e r t r a n s f o r m eqn (A3.6) by s u b s t i t u t i n g Q(z) = i|.(z) [ T T ( Z ) ] 1 / 2 (A3.7a) where •rz a U) de.} (A3.7b) TT (z) = exp {- 2 and t h e n we have eqn (A3.6) as ,2. 5 - 1 + [E - U ( z ) ] ijj = 0 ((A3.7c) d z 2 where U(z) = | § + a 2 (A3.7d) The boundary c o n d i t i o n s on a r e g o t from t h e boundary con-d i t i o n s on P t h r o u g h (A3.4B), A3.5 ) , (A3.7a) and (A3.7b) F o r o u r p r o c e s s , we have t h e two s i n g u l a r b o u n d a r i e s as i n -a c c e s s i b l e 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(x) z ( x ) = 1 to x (A3.9) Thus f o r 0 £ x < <*> we have - » < Z < 0 0 F u r t h e r u s i n g (A3.4b) and assuming t h a t P(x) -> 0 f a s t e r t h a n x -* °° (.and u s i n g eqn (A. 3.8) we have g ( z 4- ± 00) = 0 (A3. 8a) E q u a t i o n (A3.5) so g i v e s us Q (z -> ± 00) = 0 (A3.8b) 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 form o f eqn ( A 3 . 7 c ) a s i t i s i n the form o f the t ime homogeneous S c h r o e d i n g e r e q u a t i o n o f wave m e c h a n i c s . a ( z ) . = [m(y-x) - \ 2 a 2 x ] — 4 ax = [my - (m + §—) x ] -^x-my ax < ? + 7 > a z From (A3.9) we have x = e a z w h i c h g i v e s a^ (z) - a z ( £ + § ) a 2 da dz my e •az From (A3.7d we have T T / \ / — - a z , U(z) ',- - my e + - a z , , m-y, 2 my e + ( —r-) a - a z my e .(..'ii + °) a ~ a 2 my my e + ( — ) ( e _ a Z - 1 ) 2 + 2 e - a z t l - - - : f - ) y 2my + ^ y + 2my^ ^ . mu.2 , —az n v 2 , - a z 1) + e 2 (my) 2 ( 1 - J- - ^ ) - my + ( HLbL ) 2 { ( - l + | 1 ,2 _ ! } a y 2my which g i v e s U (z) = a x ( e ~ ° 2 - l ) 2 + a 2 e CTZ + a 3 where a. my , 2 a 2\\W- ) 2 <1 - ^ ' f ) y 1 my ^ a ' { y 2my ; - 1 S u b s t i t u t i n g i n t o e q u a t i o n (A3.7c) we have dj> dz (E>-a3) - a 2 { 1 ,, ..- -a-2, 2 - a z I-i . _ „ 1 — (1 - e -) - e I ^ 0 a. *-(A3 Now by a s u i t a b l e change o f v a r i a b l e we want t o t r a n s f o r m 199 eqn (A3.10) t o the f o l l o w i n g form 2 + { E 1 * - c ( e C - 2e ? ) } i|i = 0 (A3.11a) d ? 2 i where IJJ( 5 -> ± ») = 0 (A3, l i b ) and where 3 , E and c are t o be c h o s e n i n terms o f t h e p a r a m e t e r s o f e q u a t i o n (A3.10) L e t £ = a ( z - z * ) (A3.11c) 2 2 Then d_j, . a 2 = d_J_ (A3, l i d ) d'S,2 d z 2 F u r t h e r t a k i n g t h e s e c o n d t e r m i n the. square b r a c k e t o f e q u a t i o n 1(A3 .10) we have ,, „ -az. , - 2 a z . - a z a^ ( l - 2 e + e ) - e * and s u b s t i t u t e - a z = - ( £ + a z ) w h i c h g i v e s a n e * _ a a„ = a. + - ± — * - 2e M + — zr ) (A3.11) 1 2az* az* 0 az* e e 2e Comparing w i t h the c o r r e s p o n d i n g p a r t o f e q u a t i o n , ( A 3 . 1 1 ) we want t o choose z* so as t o s a t i s f y -^ - 2 a z * - a z * , 2 e , C = a x e = a x e + 5 " • , , a 2 . - a z * = ( a x + 2 - ) e and d r o p p i n g t h e t r i v i a l s o l u t i o n e =; 6 we have o r z * = i o g Q ( 2 a x (A3.12) Thus we note t h a t we can t r a n s f o r m eqn (A3.10) t o the form eqn (9) by t h e f o l l o w i n g s u b s t i t u t i o n s „ _ , * x (A3.11c) £ a(z - z*) where z* i s g i v e n by eqn (A3.12) 3 = a 2 (A3.12a) E' = ( E - a 3 + a 1 ) (A3.12b) and = a ± e~2oz* (A3.12c) The p o i n t o f a l l t h i s e f f o r t i s t h a t eqn (A3.11) i s j u s t t h e S c h r f i d i n g e r e q u a t i o n f o r a d i a t o m i c m o l e c u l e w i t h a Morse p o t e n t i a l - an e q u a t i o n w h i c h has been s t u d i e d i n the quantum p h y s i c s l i t e r a t u r e by T r i s c h k a & Salwen [ ] . ( S e e a l s o Morse [ '.. ] , S c h r o e d i n g e r [ ] , Dunham [ ]) . A t t h e boundary TJ(£) = c (e~ 2 ? -2e~ ? ) ->• 0 i . e . a t i n f i n i t e b o u n d a r i e s , U(5) [by c o m p a r i n g eqn (A3.11a) w i t h eqn (A.3 .7c)which i s the b a s i c t ime homogeneous S c h r o d i n g e r e q u a t i o n ] i s n o t always i n f i n i t e . T h i s i m p l i e s (see T i t c h -marsh [ J) t h a t the s e t o f eigenvalues o f e q u a t i o n (A3.11a) are n o t s t r i c t l y d i s c r e t e , and t h e r e i s a c o n t i n u o u s i n t e r v a l 201 o f e i g e n v a l u e s as w e l l . The d i s c r e t e , r e g i o n o f the e i g e n v a l u e s o f eqn ( A 3 . 1 l a ) a r e g i v e n i n T r i s c h k a & Salwan [ ] as E - = _ c [ i _ £ ( n + | ) ] 2 0< n < [ / c r - | ] (A3.13) / c where [x] i s the i n t e g r a l p a r t o f the number x i . e . , the l a r g e s t i n t e g r a l l e s s t h a n o r e q u a l t o x . The c o r r e s p o n d -i n g n o r m a l i z e d e i g e n f u n c t i o n s a r e g i v e n by 1 ( q - 2 n - l ) * U) = M u 2 e ~ u / 2 F n (u) (A3.14) r n n n where q = 7 (A3.14a) ^ e x p ( - a z * ) u = q exp U Q - O (A3.14b) and £Q i s got from the i n i t i a l c o n d i t i o n (A3.2a) s u i t a b l y t r a n s f o r m e d . and P n ( u ) = J i ( J ) T l ^ r ; (J) = I T T ^ i r i . . . . ( A 3 . 1 4 c ) .2 1 ( q - 2 n ) i M n = n ! r ( q - 2 n - l ) ' . * ( A 3 . : 1 4 d ) . where r ( . ) i s the gamma f u n c t i o n and ( x ) n i s d e f i n e d as (x) = { 1 i f n = 0 (A3.14e) x(x+l) (x+n-1) i f n > 1 The s o l u t i o n as i n eqn (A3.5) i s t h u s g i v e n by ( f o r the 202 d i s c r e t e p o r t i o n ) . g ( z z Q , t ) = E a n Q n e E n t / 2 (A3.15) n where Q n i s g o t from ip n u s i n g e q u a t i o n s (A3.7a) (A3.7b) , and the c o n s t a n t s a a r e g o t frc n s e t t i n g t=0, w h i c h g i v e s us ' om the i n i t i a l c o n d i t i o n ( A 3 . 2 a ) , and ° n = Q n ( z 0 ) 7r ( z 0 } : - • ( A 3 - 1 5 a ) the g e n e r a l s o l u t i o n i s now g i v e n by g ( z f z o , t ) = u ( z 0 ) £ Q n ( z ) Q m ( z Q ) exp ( - E ^ t / 2 ) (A3.16) + c o n t i n u o u s s p e c t r a c o n t r i b u t i o n The c o n t r i b u t i o n o f t h e c o n t i n u o u s p a r t (see G o e l e t a l [,. ) has n o t been s o l v e d i n c l o s e d f o r m , b u t i s known t o be o f t h e form as u n d e r F ( E ' , x ) e x p { ( z - e z ) ] - j E * t > ° X (A3.16a) 0 where the r e l a t i o n between E 1 and E i s g i v e n by (A3.12b) and t h e f u n c t i o n F depends upon c o n f l u e n t h y p e r g e o m e t r i e f u n c t i o n s . W i t h o u t p u r s u r i n g t h i s l i n e o f a n a l y s i s f u r t h e r the f o l l o w -i n g 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 t , 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 d r o p p e d . 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 iew 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 the 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 R e s t r i c t i o n a t O r i g i n . The SDE f o r the d i f f u s i o n p r o c e s s w i t h a=0 i s dx = m(y -x) d t + adz (A4.1) I f we now make the 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) w h i c h i s the s t r a i g h t O r n s t e i n - U h l e n b e c k p r o c e s s and has a 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 g i v e n by P(y y o , 0 ) = [2rrV 2] 1 / 2 e x p { - | [ { y - y 0 e _ r n t } / V ] 2 } (A4.3) where V = g- ( l - e - 2 ^ ) 2m The s o l u t i o n t o (A4.1 i s t h e r e f o r e s i m p l y g o t by s u b s t i t u t i n g y = x - u ; y Q = x Q - y . APPENDIX 5 D e r i v a t i o n o f the S t a t i o n a r y (or Steady S t a t e ) d e n s i t i e s We have o u r d i f f u s i o n p r o c e s s d e f i n e d by dx = m(vi-x)dt + ax 'dz" (A5.1) w h i c h has t h e form dx = b (x) d t + /a\(x) dz (A5.1a) where b;(x) = m(y-x) ; a.(x) = a 2 x 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 A5.1a) i s - | x | b ( x ) F ] + \ Jl [ a ( x ) F ] = (A5.1b) where F = F(x-1 X Q , t , 6 ) i 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 . The s t e a d y s t a t e d e n s i t y i s the s o l u t i o n t o (A5.1b) g o t by s e t t i n g 8 F / 3 t = 0, and i s o f the form P (x 8) = [a/(x) TT (x) ] -1 [a, !(s)rr(s)] - l ds •(A5.1c) where T T ( S ) = exp [-2 b (r) a.(r) dr] (AS.ld)-and Q i n d i c a t e s i n t e g r a t i o n o v e r the t o t a l s t a t e space o f x F o r o u r p r o c e s s (A5.1) we have 206 TT (x) o=l/2 = x exp(Bx) ; 2m n2 TT ( X ) a= l x exp ' (By/x) TT (x) a-y 1/2 ,1 = exp [ ex l+X i+x e y x -] ; X=l-2a (A5.2) F o r a = 1/2 we have and [b(x) T T ( X ) ] •1 _ 1 B y - 1 x e x p ( - e x ) ey \ 2 x 3 - 1 e x p ( - e x ) = I f i -_ (e) ey e y - i (e) e x p ( - e x ) ( e d x ) r ( 3 y ) P(x) a'=l/2 (A5.2a) F o r a = 1 we have - 6 [ b ( x ) T T ( x ) ] ~ - L = ^ e x p ( - e y / x ) ; 6 = (2+e) and K x " 6 e x p ( - e y / x ) d x = \ 2 ( B y ) ( 3 + 1 ) r ( B + l ) U U 0 w h i c h g i v e s P(x) a= l > 3 P ) 3 + 1 -(2+6) l r ( e + l ) x e x p ( - 6 y / x ) (A5.2b) F i n a l l y f o r the g e n e r a l c a s e o f 1/2, 1 we have [b(x) TT (x) ] = x 1 _ X exp[ 3 x •l+A B:yx l + A -] ; A=l-2a 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 , and so we have f o r t h e s t e a d y s t a t e d e n s i t y P(x) x A - l r Byx x exp [ M *\ •--ex l+X l+X' a ^ l / 2 , 1 ~-i r eyy^ eyi+xn , exp[ - p - - ] dy (A5.2c) F i n a l l y , i t would be o f i n t e r e s t f o r us t o v e r i f y t h a t t h e 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 - ( A 5 . 2c) , r e d u c e s t o t h e f u n c t i o n a l forms (A5.2a) and (A5.2b) as a a p p r o a c h e s 1/2 and 1 r e s p e c t i v e l y ( i . e . X -»- 0 ; -1) Now d r o p p i n g the d e n o m i n a t o r (which i s a c o n s t a n t ) from (A5.2c) we can w r i t e t h e d e n s i t y f u n c t i o n as T-, A - l r eyx P «i x exp [ 1 L^— A A ex l+A l+A ] (A5.3) and m u l t i p l y i n g and d i v i d i n g by exp (gy/A) g i v e s 208 Mow x - 1 _ e x p ( A l o g x ) - 1 A X 1 2 2 X l o g x + 2 A " ( l o g x) -i 00 -i -+ ± E £ (A l o g x ) n A n=3 nij; Now as X •+ 0 ( i . e . a -> 1/2) c l e a r l y x X - l L t r = l o g x A+0 A (A5.3a) P(x) a -> 1/2 A 0 * x ^ y 1 e x p ( - $ x ) w h i c h has the same k e r n e l as ( A 5 . 2 a ) . To show the same s o r t o f c o n t i n u i t y f o r the a=l case we can w r i t e (A5.3) as PA°= exp [ B { x 1 + X - l } i A - l 1+A J X and as i n (A5.3a) above as a -> 1 ; (1+A) 0 and x l + A _ 1 L t — T - T^ = l o g x X-y-1 1+A Thus t a k i n g l i m i t s as A -> - 1 we g e t P, « x " ( 2 + B ) e x P [- ] A X w h i c h has t h e same form as (A5.2b) APPENDIX ~ 6 D e t a i l s of the P h i l l i p ' s Approach to E s t i m a t i o n The s t o c h a s t i c 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 process i s dx = m(yU-x) dt + ax" dz ... (A 6.1) I t i s necessary to transform the above s.d.e. so as to e l i m i n a t e x from the variance element. This can be done by a transformation of v a r i a b l e s . Let the transformation be y = f(x) where the f u n c t i o n a l form of f ( . ) i s unknown.By: Ito1s Lemma we have dy = f mf (y-x) + % f a 2 x 2 a ] dt + f a x a dz L x xx J x we now choose f(x) so t h a t a f x = 1 (or any constant) which on i n t e g r a t i o n gives 1-a y = -=r_—— f o r a ^ 1 1-a-= log x f o r aC^j\ Proceeding w i t h the a ^ l case (as i t i s the more general form), we get by s u b s t i t u t i o n -a "*"-a 2 - 1 dy = [myx - m(l-a) -= Jgaa x ] dt + adz 1-a ~ J I f we now s e t u = x ; v = x we get the e q u i v a l e n t form of (A 6 . 1 ) , i n a form where the P h i l l i p ' s approach may be a p p l i e d . Thus 2 dy = m ( a - l ) y ( t ) + my u(t) - haa vr(t) + K (t) Since" 0U.T o b j e c t i v e a t pres e n t i s p u r e l y e x p o s i t i o n a l , - -l e t us proceed ahead f u r t h e r assuming. a=h. T h i s g i v e s -dy = [-(m/2)y + (my-a 2)2 ] d t + adz ...(A6.2) 4 y I f now we s e t 2/y = u, we can t r e a t u as e q u i v a l e n t t o P h i l l i p s [ ] exogenous v a r i a b l e . Then an approximate d i s c r e t e time e q u i v a l e n t t o (A6.2) i s Y t = E l ^ t - 1 + E 2 U t + E 3 U t - l + E 4 U t - 2 + V ( A 6 " 3 ) where = exp (-m/2) E 2= (my-a 2/4)[(2/m 2) exp(-m/2)(1-4/m)+(2/m 3)(m 2-3m+4)] E 3= (my-a 2/4) [ (2/m 3) exp (-m/2)(8-m2) + (8/m 3)(m-2)] E 4= (my-a 2/4)[-(2/m 3) exp (-m/2)(m+4) - (2/m 3)(m-4)] (A6.4) 211 and n t ^ N [ 0 , ( a 2 / m ) ( l - e x p ( - m ) ) ] Thus the l o g l i k e l y h o o d f u n c t i o n i s L = - | l o g w 2 - E (A6.5) 2 T ~ 2 where w = ( a 2 / m ) ( l - e x p ( - m ) ) and E = E n . • I t may be t = l t-n o t e d t h a t t h e d e g r e e s o f freedom have r e d u c e d by 2 as we r e q u i r e l a g g e d v a l u e s i n ( A 6 . 3 ) . I f t h e t i m e between o b s e r v a t i o n s i s v e r y s m a l l , m i s a l s o s m a l l i n m a g n i t u d e . We can t h e n expand e x p ( - m ) , and drop terms o f second o r d e r and h i g h e r . Then w 2 - a 2 . However, we f i n d t h a t u s i n g a d i r e c t r e g r e s s i o n a p p r o a c h - u n i q u e l y d e t e r m i n e s m, and the r e s i d u a l 2 v a r i a n c e i s a . Thus we f i n d y i s o v e r d e t e r m i n e d . The d i r e c t r e g r e s s i o n a p p r o a c h f a i l s . C o n s t r a i n e d r e g r e s s i o n a l s o f a i l s as a 2 e n t e r s t h e E ' s . Thus t h e o n l y a p p r o a c h i s t o maximize L d i r e c t l y . On sample d a t a s e t s , i t was f o u n d t h a t u s i n g s t a n d a r d non l i n e a r o p t i m i z a t i o n r o u t i n e s , c o n v e r g e n c e was n o t o b t a i n e d . I t was t h e r e f o r e d e c i d e d t o d r o p f u r t h e r i n v e s t i g a t i o n . The d i f f e r e n c e between the a p p r o a c h e s o f S a r g a n [ ] and P h i l l i p s [ ] i s v e r y m i n o r . We have the s o l u t i o n t o the S t o c h a s t i c d i f f e r e n t i a l e q u a t i o n D y ( t ) = A y ( t ) + B z ( t ) + E ( t ) as shown i n S a r g a n [ ] y ( t ) = e ^ y t t - h ) + e B z ( t - s ) d s + h e E ( t - s ) ds 0 (A6.6) B o t h a p p r o a c h e s a p p r o x i m a t e the i n t e g r a n d i n the s e c o n d term on t h e r . h . s . o f (A6.6) by a p o l y n o m i a l i n s by a T a y l o r s e r i e s e x p a n s i o n o f z ( t - s ) about s=0, and d r o p p i n g terms o f t h i r d and h i g h e r o r d e r . ( C l e a r l y t h e a p p r o x i m a t i o n h i n g e s on the d i f f e r e n t i a b i l i t y o f z ( t ) ) . They d i f f e r o n l y i n the way t h e y a p p r o x i m a t e t h e d e r i v a t i v e s o f z ( t ) . S a r g a n a d o p t s t h e more d i r e c t a p p r o a c h and s e t s z ' ( t ) = 2 t ~ Z t - h z " ( t ) = 2Z + Z t - h t - 2 h 2 h 2 whereas P h i l l i p s uses the more i n v o l v e d Lagrange t h r e e p o i n t s i n t e r p o l a t i o n f o r m u l a (see Conte de Boor [ J ) . 213 APPENDIX - 7 D e t a i l s o f E s t i m a t i n g Procedure f o r a = 1/2 (Known) Case T h i s appendix o u t l i n e s the method adopted to estimate the parameters (m,y,a), f o r the case where a = 1/2 i s assumed known. The d i f f u s i o n equation i s given by dx = m(y-x)dt + cr/x dz (A7.1) a) Simple l i n e a r i z a t i o n method: Approximating the d i f f e r e n t i a l equation (A7.1) by a d i f f e r e n c e e x p r e s s i o n g i v e s ( x t ~ ^ t_ 1) = m(y - x t - 1 ) + a^xt_i e t (A7.2) where e t ^ N(0,1). D i v i d i n g through by v / x t _ ^ and rearranging.-> terms g i v e s Y t = % X l t + ( 1 " m ) X 2 t + n t ( A 7 * 3 ) where y t = x t / / x ^ _ 1 ; x l t = l / / x ^ _ 1 ; x 2 t = / x ^ and n t = a e t ^ N ( 0 , a 2 ) ^ \^ Now i n equation (A7.1), the d z 1 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 t n t,) = 0 f o r a l l t ' ^ t . Thus (A7.3) i s the standard r e g r e s s i o n equation and ML e s t i m a t i o n o f the parameters i s e q u i v a l e n t to l e a s t squares e s t i m a t e s . Thus the l o g o f the j o i n t l i k e l i h o o d o f T o b s e r v a t i o n s i s given by T 4 ^ -L E t I l n 2 = I y 2 + m 2y 2 E x 2 t + (1-m) 2 E x 2 t - 2*^ I x ^ v^ _l. 2my (1-m) E x i t x 2 t ~ 2 ( 1 - m ) E x 2 t Y t '' (A7.4) 214 Setting = Z x 2 j _ t - '< Mi_ 2 = E x l t X 2 t ' M22 = E x 2 t ' M, = .Y. ; JVL = Ex„,y. ; M = Ey 2 l y I t t ^y 2t- rt yy Jt We have the f i r s t order conditions as | ^ = 2my2 M11 - 2(l-m) M ^ - 2 p JA^ + 2 P M 1 2 - 4my M ^ + 2 M ^ (A7.5a) | ^ = 2m2y ^ - 2m ^ + 2m(l-m) M ^ (A7.5b) Setting (A7.5b) equal to zero gives m = ^ ^ 2 (A7.6a) y M ^ - M ^ Substituting the above units (A7.5a) and setting 3 L / 9 m = 0 g i v e s v . \ « 1 2 - * K % (fl7.6b) 2 - M. _M_ + M . . M - - N L - M - i ) ( M12 " \2% + ^ 1 % " W i The Fisher information matrix corresponding to the present sample i s I and i t s elements are (by i n v o k i n g a s y m p t o t i c r e s u l t s ) where 6 ^ = m and 0 2 = y, and a r e the M . L . e s t i m a t e s For the present case we have 8 2 L 3m2 3 f L 3y 9£L_ 9m9y 4 V M L 2 - 2 y 2 M n - 2 M 2 2 Y = - 2m2 M ^ = 4m M 1 2 + 2 M l y - 2^2 - toM^ This enables us to estimate of the variance-covariance matrix of the 215 testimates based on asymptotic theory. b) Steady State Density method: The steady state density corresponding to a = 1/2 i s (see Appendix 5) F(x) = ( ^ .xey-i. (_ B X ) (A7.7) r.(By) The j o i n t l ikel ihood of the data f o r t h i s a p p r o a c h i s T a =.n, F(x.) i=l i Taking logs and s e t t i n g lo,g (I) = L we have L = Tgu log B + (B'y- 1) E log x, - 6 Z x i ~ T log [r(By)] (A7.8) where r ( . ) i s the Gamma function. The f i r s t order conditions corresponding to maximizing L are 3T TT— = TB log B + Ba - TB^ (By)1' = 0 (A7.9a) 9y 3T i=- = Ty log B + Ty + y a - b - TyiJ; (By) = 0 (A7.9b) dp where a = E log 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 positive arguments yields y = b / T (A7.10a) To estimate B, we need to solve the following equation (got by substituting (A7.10a) i n either of equations (A7.9) iJi(By) = l o g B + a / T (A7.10b) Since ip (.) and log (.) are monotone increasing functions of their arguments, we are guaranteed unique solution 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 at the neximum. Thus | ^ = - T3 2 r(8v) (A7.ll) = T P 2 [ i - (Bu)] (A7.11b) 3 2L j r gp = T log 3 + T + a - Tip (3y) - T3y^ ' ,(3y) (A7.11c) where ip' (.) i s the digamma function. Clearly 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- -vof nonlinear equations. The p s i function has been coded and i s available i n the UBC programme l i b r a r y . For the digaimia function, f i r s t a series expansion was used. However t h i s was not satisfactory, as truncation (even after a large number of terms) resulted i n sizeable errors i n the function value, which was detected as the diagonal elements of the hessian matrix some-times became posit ive. An asymtotic expansion (for large arguments) was very satisfactory for the parameter values of our problem. c) Transition., . Probability Density Method: The transit ion probabil ity density corresponding to a = 1/2 i s given by 217 F(x |x ,8,t) = {2m/a2(w-l)} . exp [-{2m (x+wxQ)}/{a2 (w-1)}J. •x • ( — )'"' i ° I' [4m 5^wx / a 2 (w-1)] wx 1 o o ^ 2mp _ ^ a 2 (A7.12a) where w = exp (mt) and i s the m o d i f i e d B e s s e l f u n c t i o n o f o r d e r k. The l ikel ihood function i s therefore T - l . ' * = ± £ F ( x i + 1 | x i , 0 ) . P g s ( x 1 ) (A7.12b) where p s s ( « ) i s the steady state density (A7.7) . In general, when Ti i s large, the contribution of P c e ( . ) may be considered very small compared to the other terms, and so may be dropped from the l ikel ihood function. The log l ivel ihood function i s not further tractable a n a l y t i c a l l y as expressions for (9L/96^) require derivatives of the Bessel function with respect to i t s order, (for arbitrary positive orders) which are problematic. The approach towards parameter estimation has to be d i r e c t iraximization of the log of (A7.12b). For t h i s purpose, the Fletcher algorithm using a quasi-Newton method was used. In general, i t was found that convergence was obtained to a reasonable degree of accuracy within 15 i terat ions, given starting values for the parameters as the results 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 or a "global" maximum, very different starting 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 testing. A word about the numerical evaluation of the density function (A7.12a). The modified Bessel function could not be evaluated i n a straight forward manner, using the series expansion. This was because, for large values of the order and/or argument, the series was very slow to converge. To overcome t h i s , the expression was s p l i t up as F ( x t | t 0 , 9 ) •= f ( x t , x o , 0 ) . exp (-g(x)) . 1^ (g(x)) (A7. This was more successful as exp (-g(x)). 1^ (g(x)) converged more rapidly. However, for large 6, this method was very expensive computationally. Thus, an asymptotic expansion along the l ines of Giver [66] was used, whenever 6 was greater than 20. This was very e f f i c i e n t . The r e l a t i v e accuracy of the asymptotic values as compared to the more exact expression (A7.13) was tested by actually evaluating the density function (A7.12a) for a given parameter set 6, and several values of 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 of the moments, which are given by (see Cox, Ingersoll & Ross [13]). -m , ,-. -m . &. = r e + y (1-e ) 1 o M = r ( ) (e" m - e"2™ ) + y ( ^ - ) (1 - e ~ m ) 2 2 ° m 2m2 where JXL, i s a central 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 probabil i ty density function i n equation (A7.12a), Figure 1 was prepared. What i s interesting to note i s that, for the parameter set used, y - 5% per annum, and when the current interest i s a t or above y, the transit ional 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 n Va lue s 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. r t Mean of r t + ^ / r t « 9 Std. dev of r t + 1 A t ' 6 Theoretical Numerical Theoretical Numerical 1.0 1.955 1.962 1.126 1.116 2.0 2.787 2.777 1.421 1.411 3.0 3.612 3.593 1.664 1.650 5.5 5.675 5.651 2.155 2.151 7.0 6.912 6.887 2.402 2.405 9.0 8 .562 8.534 2.697 2.708 10.0 9 . 388 9.338 2.832 2.848 12.0 11.038 10.961 3.086 3.119 NOTES: - A l l figures are in percent per annun - 6 is the parameter set {m, y,c2} and are the values used in the Monte Carlo simulations. does not appear too skewed from the normal density. This could imply that the simple l inearizat ion of the diffusion equation, (which assumes Gaussian transit ion probabilities) may not perform too badly. F i n a l l y the second derivatives of the log l ikelihood function were computed numerically, (the quasi-Newton method evaluates numerical second derivatives at 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 Of Measurement E r r o r s On Data : The a n a l y s i s h e r e assumes t h a t t h e o b s e r v e d d a t a i s the combined e f f e c t o f the t r u e p r o c e s s and a s u p e r i m p o s e d e r r o r p r o c e s s . The f o r m u l a t i o n o f the p r o b l e m r u n s as f o l l o w s : We may b e l i e v e t h a t t h e t r u e i n t e r e s t r a t e ( i ) f o l l o w s t h e p r o c e s s d i = m ( y - i ) d t + Vafi d z , ( A 8 , l ) where we o b s e r v e i w i t h e r r o r . (For t h i s a n a l y s i s , I have used the square r o o t p r o c e s s , as t h e p u r p o s e o f t h i s s e c t i o n a t the p r e s e n t moment i s e x p o s i t i o n a l ) . L e t us o b s e r v e r as i w i t h an e r r o r n i . e . r = i + n (A8.2) where n i s w h i t e n o i s e . To be a b l e t o p r o c e e d f u r t h e r , we have t o impose some a d d i t i o n a l s t r u c t u r e on t h e p r o b l e m . L e t us l o o k a t a p a r t i c u l a r form o f the e r r o r s t r u c t u r e dn = a 2 / T d z 2 ; E(n) = 0 . . ( A 8 : 3 ) The r a t i o n a l e behind" t h i s form i s t h a t i t e n s u r e s t h a t t h e T e r r o r goes t o z e r o as i + 0. D i f f e r e n c i a t i n g (A8.2.) and s u b s t i t u t i n g (A8.1) and (A8.3) we get d r = d i + dn = m ( p - i ) d t + o j / I ' d x , + o 2 ^ d z 2 where E (dr) = m ( y - r ) d t 2 2 2.. E ( d r ) = (o1+a2)r d t + 2a i a 2 p r d t [ s i n c e Cov (dz1 - ( a 2 + a 2 + 2 0 l a 2 p ) r d t = a 2 r d t where a 2 == ( a 2 + a 2 + 2 a ^ p ) Thus we c a n r e p r e s e n t t h e p r o c e s s r as d r =: m ( y - r ) d t + a 3 / r dz ( : A 8 -4) w h i c h i s e x a c t l y o f the same form as e q u a t i o n (A8V1) ;-,the tr;'u'e' i n t e r e s t r a t e p r o c e s s . 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 o f p . F u r t h e r , i f we i g n o r e the e r r o r i n measurement (when an e r r o r does e x i s t ) , t h e n a | as an e s t i m a t e o f a 2 , i s e i t h e r o v e r o r under e s t i m a t e d a c c o r d i n g as ( a 2 + 2ol a 2 p) | 0 2a1 T h i s i m p l i e s t h a t even when p = 0, (the e r r o r i s u n c o r r e l a t e d w i t h t h e t r u e i n t e r e s t v a l u e ) a 2 i s o v e r e s t i m a t e d by o 2 . I n t h i s e r r o r s t r u c t u r e , as l o n g as we assume t h a t b o t h t h e e r r o r and t r u e i n t e r e s t p r o c e s s have t h e same a exponent i n t h e v a r i a n c e t e r m , t h e p r e s e n t a n a l y s i s h o l d s i n t o t o . T h i s i s e a s i l y v e r i f i e d by c a r r y i n g t h e a l g e b r a t h r o u g h . 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 is that, the asymptotic covariance matrix of the estimated parameters is got by inverting the Hessian matrix (with signs reversed on the elements) where the Hessian matrix is the matrix of second partials of the logarithm of the joint likelihood function with respect to the parameters. (see T h e i l [ ] t Goldfeld and Quandt [ ]). This result uses the property of ML estimators whereby 3 ^ , 9 2L E ( — ) = 30 i90j 90 i90, (A9.1) Where L = log likelihood function of the data, 0 is the vector of parameters, 0 is the ML estimate of 6 , and E is the expectations operator. Thus in general, i f we know 0 (the true value), then we can compute the assymptotic covariance matrix of 0 as T 1 Cov(0) = -E ' 90^0^ (A9.2) Further i f we represent by L ^ n \ the joint likelihood of n data points, we "can approximate a 2 . W ,2 (1) I D - i 3 " Equation (A9.3) i s v a l i d s t r i c t l y only f o r independent random v a r i a b l e s . We hope that the " b i a s " due to dependence o f the sequence does not a l t e r the b a s i c nature o f the a n a l y s i s to f o l l o w very much. The p o i n t to be noted here i s that when we compute the asymptotic c o r r e l a t i o n m a t r i x from the covariance matrix, i t i s o b v i o u s l y immaterial whether we use the expectation over n data p o i n t s or even 1 data p o i n t . Let I represent the F i s h e r Information matrix. Then I = and f u r t h e r E r * 2 h ) 86.80. J 1 30.30. J 0 0 f e t a e T ( r t ' r 0 l e ) - P^fr,. r ,0 ) . P (r 6) dr dr ly t o' s s ^ o J t IA9.4) where Prp(-) represents the 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 P s s ( 0 the s t a t i o n a r y p r o b a b i l i t y density. Since we want to evaluate the c o r r e l a t i o n matrix over a l l parameters ( i n c l u d i n g ot we could assume P T(.) to be normal - which i s the case i n the SL approximation. This now gives P T ( r t r f f i e ) : a N {my + (l-m)r }, a r r J o o (A9.5) N.[ a ( r 0 ) > ° 2 r 0] i _ B y B y - i P s s ( ^ o l e ) - ; . .exp (-Pro) (A9.6) r ( B y ) / ° Where B = 2m/a2. Expressions for 96,30, have a l l been set out i n Appendix,2. Substituting (A9.5) and (A9.6) into (A9.4); noting that one of the integrals i s now from -°° to +00 due to the normal density approxi-mation; and further that r P T ( * t | r 0 , e ) d r t = 1 r t p T ( r t l r o . 8 ) d r t = a ( r o> { r t - a(r Q )} P t ( r t |r Q ,6) d r t ' = { r t - a(r 0 )} P T ( r t | r 0 , 6 ) d r ^ -2 gxves E ( 9*L 9a9ta. -) = 0 E ( 9a9y ) = 0 E 9m9cT = 0 E ( 9 2 L 9y9a" -) = 0 226 (The d e f i n i t e i n t e g r a l s are from Gradshtein $ Rzyhik [35[) 8p m By-2 y . r Q e x p ( - 3 r o ) dr Q ; y m2 3 C ( 3 u - l ) E ( 4 ^ = 9m ^ " r o ) 2 r o 3 y " 2 e xP <~ Br o) d r o By B y - 2 - l E (9m9y } " my ^ - r o } r o 6 u " 2 « p ( - e ^ 0 > d r o m Q: (Bu-D ( 9 a / ) / 2 ( a 2 ) 2 9 2L E ( 9 ) = - 2 y 9a f00 (log r Q ) 2 . r Q 3 y _ 1 . exp ( - 3 r o ) dr Q = ^2 { i^ (gy) - log er + 1 ( 2 , BU-D} Where §(z,q) i s the Riemann's Zeta function = In" ( 7) and is the q+n psi function. 227 roo l o g r o ' r o * e x P ( r " ^ r o ) d r o 0 - - V 1 2 n + i i + ( 1 - s i r * - s < v ^ - 1 ) ] 2a where n = { (3y) - log3) The hessian matrix of the l o g l i k e l i h o o d f u n c t i o n has a bl o c k diagonal form, w i t h the two o f f diagonal ( 2 x 2 ) m a t r i c i e s being zero (the order of the 2 parameters i s assumed {m,y,a ,a}). This means that the i n v e r s e of the hessian matrix i s the matrix w i t h the i n d i v i d u a l b l o c k s i n v e r t e d . T h i s t e l l s us that we can i n f e r the s i g n of the c o r r e l a t i o n between m and y, and , 2 a and a. These are e x a c t l y the same as the corresponding cross d e r i v a t i v e s of L (wi t h the s i g n reversed). Thus we expect Cor (m,u) < 0 & Cor (a2,a) > 0 The c o r r e l a t i o n matrix i s presented i n the main t e x t . 9a9a a A P P E N D I X <- 10 Maximum L i k e l i h o o d E s t i m a t i o n of the Parameters {m,u,a,a} Using the Steady State P r o b a b i l i t y D e nsity Approach The steady 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 corresponding to general a values ( i e a ^ 1/2, 1) i s Bux Bx P(x) A - l x exp 1+A A - l y exp Q X 0 1+A Buy By A 1+A dy (A10.1) x^ '''exp [ a(x) ] D ^ ^ & a ( x ) a n& D s u i t a b l y B = 2m/a defined 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 of n observations i s n n L = (A-l) I l o g x i + E a(x±) + n l o g D i = l 1=1 (A10.2) The l o g l i k e l i h o o d f u n c t i o n i s not 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 respect to the parameter, and so only 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 that methods that used numerical 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 of the Newton method) l e d to problems due to the complicated way i n which the parameters enter the 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 of L w i t h reference to the parameters were 229 derived as under n 3L 9 u = 2 i=l 3D/3y D where _3D 3y. 3z A a(z) dz 3L 3m n i i=l L 3 a(x-i) 3D/3m D where 3D 33 x { } e x p a(z) dz 3L _ ? 3A . . i=l log x ± + l+A A &x T A N A DA . 3 y x i m 3 y x j - i , , . X l o g x i " * " 0 g x-* ~ l+A' l+A 2 (l+A)' 3D/3A where 3D 3A 1 . exp where 6 = 1 + 1/A ; b(z) = b(z) 3 y z l A 3z^ - 3 y z 3 z 6 logz 3 z 6 " Az l+A A z a+A)J A l+A dz 3 2L l y ? = I 3 2 D /3y 2 (3D/3y) 2 ' where 3 ^ 3y 2 a 2 2 3 z exp b(z) dz 3 3 3 y = I 3 2 D / 3 3 3 y + ( 9 D / 3 3 ) ( 9 D / 3 y ) D 2 230 where my o A ' exp b(z) . U+b(z)} bz 3A3u = Z 3X J A 3 x i log x ± _ . — - D — ~ where 3 2D ,00 3A3u 3 z exp b(z) 2 + 0 * A J X 1/A 1/A _y + z z • X 2 (l+X) 2 (1+A) — log z) X Z dz ? 33 = I r 3D/g g 2 (3D/ a e) — 5 — + T2 where 3 ^ ^ 3 2 X 1 3 exp b(z) dz 3X33 = I Ux-j log X-L + l+X x. l (l+X) 2 l+X Ti+I)" l o s x± 3 2 D/ 3A33 D (3D/33QD/3A) D 2 231 where 2 9 D 3A83 j e x p b(z) 2Uz + (1+2A) (1+A): A(1+A) ' A L + b(z) ' Pi 1 2 4. 3 z ^ gz^ log Z <: A 2 (l+A)2 (1+A) A 2 \ dz 32L = Z 3A< gux^log x Bux^Clog ' jCj) 2 2 3 y x ± A Bux^^ log x ± •' • + + A' A A3 A' px. l o g X . 1 ° 1 (1+A)2 o 1+A 1+A 1+A px. 2 2px_L px^ — (log x ) ^ — + —-—=— log x 1+A (1+A)J (1+A)' 9 D / 3 A 2 D + ( 3 D / 3A 2 ) D 2 where 8A2 exp tb(z)] ( 2 + 33jz ^ l°g z . ( 3 X 2 + 4 x 3 ) _ g 8|lo 8 z) I >3 >4 Ab(l+A) A (1+A-) "(1+ ) 3z6(1+4A+3A2) 3 z 6 l o g z A2(1+A)4 A3(1+A)2 c- K 3 y z + 3 z 6 log z , + , _ 3 z j _ ) y ( ..A 3 (1+A) A(l+X)2 3 y z + 3 z (1+A)' 3 z log z .A2 (1+A) dz 232 Most of the derivatives of the integral D with reference to the parameters appear very imposing. Since evaluating the second derivative of L would require .numerical evaluation of these integrals, 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 may be stated as: N a) To evaluate the integrand and it s slope as the variable approaches its limits (O,00) b) To try and infer the shape of the functions from the information in (a) above If we represent the integrand in the derivatives (both f i r s t and second) of D with reference to the parameters {3,A,U} in general as f(z), then the table below outlines the principal results Limit of f(z) Limit of 3f/3z z - * 0 z 0 0 z 0 z->°° 3D/3y +0 +0 B / X 2 - 0 3D/33 +00 - 0 y M 2 +0 3D/3A 1 /A 2 + 0 Not investigated 32D/3y2 +0 +0 +0 - 0 32D/333y +0 - 0 l M 2 + 0 32D/3X3y - 0 3(3y-2)M 3 + o -0 , A< -1 32D/332 +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 imits indicated are v a l i d only, given the parameter values ie they do not represent the l imits as, say, A -*• 0. It i s anyway shown that A •*• 0 and -1, represent special cases (see Appendix 5) . c) the behaviour of 9D/9A was not analytical ly 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 integral may not l i e entirely either in 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 total area l i e s i n either quadrant for a broad range of parameter values, the functions were numerically evaluated and plotted. 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, ( ie. there exist no discontinuities etc) evaluating the integral using a quadrature (or even the more powerful adaptive quadrature) method, is a t r i v i a l matter. To see the special problems that we face, l e t us address the problem of evaluating the seemingly innocuous i n t e g r a l D. We have D = A-1 y exp B u y By l+A l+X dy With a change of v a r i a b l e s we can transform D as under Let z = y A-1 dz = Xy dy D = X e x P l+X -r Buz Bz X l+X dz j o f ( z ) dz [In passing i t may be noted thatithe l i m i t s of i n t e g r a t i o n have to be interchanged f o r X<o] JHo i d e n t i f y the mode of f ( z ) , we set i t s f i r s t d e r i v a t i v e to zero, which gives 1/X f (z) = j ( j ^-) exp l+X -t Byz - Bz X " l+X = 0 = f ( z ) . B( 1/X (A10.3) R u l i n g out the a l t e r n a t i v e that f ( z ) = 0 at the mode gives the mode a z = y The integrand in D i s clearly unimodal. Looking at f(z) we therefore see that from o to p \ the f i r s t term in the exponential dominates, and as z increases beyond u \ the second term overtakes, and sends f(z) 0. The point here is that at u^; f (z) i s very large (particularly when 3u i s moderately large and X is near zero i e . a - 1/2). In the computer, this gives a floating 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 (A10.4) f (z) = j ^ XP • . B u z 3z •IT " l+X - P 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} in the numerator, and the same adjustment works there. The next point i s that the mode jump a l l over the half real line as X goes from positive to negative. In our problem u i s of the order 0.1. If X ranged from +1.0 to -1.0; ranges from 0.1 to 10.0. As u becomes smaller, the range increases. 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, (ie i t s total mass i s concentrated over a very small range) poses some problems. A l l numerical integration algorithms require that we provide the limits of integration. Since the mode moves a lot, we may be tempted to provide a large range (say 0 to 100). 236 However, due to the spiked '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. The numerical integration algorithms 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 integral as 0. This i s because the total area may l i e over a small fraction of the distance between any two of the points at which f(z) was evaluated. To be able to value the likelihood function (A10.2), with any accuracy, the integral D has to be accurately computed. The problem therefore boils down to one of finding reasonable integration limits for D. Given that the integrand f(z) of D (eqn A10.4) is unimodal suggests a straight forward approach to getting the required integration limits. Let z m be the mode and and z 2 the two inflection points of f( z ) ; zi < z m < z£. If now we represent the limits of integration by z%/ and z" ', (z'' < z" ) then we can choose k^ and k 2 such that z' = zm " k l (zm ~ z l ) z n = z m + k 2 ( z 2 - z m) where z'' and z"+ are required to satisfy some c r i t e r i a like (say) 40 f( z * ) / f ( z m ) and f(z" )/f(z m) < 10" " or some other such small value. Thus,locating the inflection points should solve our problems. The second derivative of f(z) is got by differenciating (A10.3) 237 f, (z) = — exp - X 2 Buz Bz ~x 1+1/X -, i / x - i X .1 /X, 3 X l A v l ^ z " + ( u ^ - " v ) . £ . ( u ^ " v ) : Setting the above to zero, noting that f ( z ) ^ 0 at the i n f l e c t i o n points gives 1 / X , - z + 3".(y - z ) = 0 M u l t i p l y i n g through by z and s u b s t i t u t i n g y = 1+1/X gives z Y - 3 (uz - z ' ) " = 0 (The f u n c t i o n a l form c l e a r l y suggests that the above equation has two roots) An i t e r a t i v e method to solve for the roots of the above equation isngot from a f i r s t order Taylor s e r i e s expansion. Y^2 = g(z) M z ? - 3 (uz - z') ^2 (A10.5) Then z< n+I = z -n g^(z n) (A10.6) Where z n+^ i s the s o l u t i o n to (A10.5) at the (n+l)*"*1 i t e r a t i o n . In general, the scheme above should converge quite r a p i d l y . However, i t was found that for some parameter values, the scheme tended to converge always towards the same root ( i e . the second s o l u t i o n was not obtainable) I t was therefore necessary to f i n d an approximate s o l u t i o n to (A10.5), and using them, and (A10.6) a r r i v e at more accurate values of the 238 inflection points. For this we expand TX using a Binomial series, about •u . This gives J = (z-u A) + u A T = y ' ( l - ¥ AY 1 + A •y (A10.7) Y Plugging equation (A10.7) for z' into (A10.5) and setting y = (z - y A ) / y A we get y A Y ( l - Y y) - 6 y A + 1 (y+l) - y A y ( I ^YY ) - i 2 = 0 which can be reduced to or l - Yy - y 2 3 U - Y ) 2 = 0 = - Y ±' { Y2+43 (1-Y)2} 1/2 '26 (1 - y ) 2 and that gives us the approximate solution. 239 APPJNDXX II Effect on bond valuation of using the y i e l d to maturity on a 91-day pure discount bond instead of the instantaneously r i s k -free rate of interest* The basic assumption of the bond valuation model i s that i t i s a function of the instantaneously r i s k free i n t e r e s t rate and time to maturity, By d e f i n i t i o n , the instantaneous r i s k free i n t e r e s t rate i s the y i e l d to maturity on a r i s k l e s s pure discount bond due to mature the next instant in time. Thus, using the y i e l d to maturity on a r i s k l e s s bond which has a longer time to maturity, as a proxy for theinstantaneously r i s k free rate, would bias the bond valuation. This bias can be broken down int o two parts: 1} The estimated parameters of the i n t e r e s t rate process (m,^ A , ( p 2 , d ) are biased because we have estimated them from a process which i s not the instantaneous interest rate process., This biases the bond valuation, which uses these parameters as input. 2) In the bond valuation equation, instead of the instantaneous i n t e r e s t rate a proxy i s used, and t h i s biases the bond value. To analyse the nature of these biases, l e t us assume that the true model of the interest rate process i s given by IT 7n(>--0dt + c r - f ^ f (A11.1) Then Ingersoll [39] has a solution for the y i e l d to maturity o n a pare discount bond having time to maturity t , and 240 current value of instantaneous i n t e r e s t rate r , as RC^rt") = - ^ J ^ ^ £ ^ ^ H < r ) J 4 ^ 1 - H e r ) * y (A11.2) For a given value of t , equation (A11.2) may be represented as , a(T) 4- bCt).f (A 11 .3 ) Since we are interested i n a fixed 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 treated as parameters. Thus i f we represent by R, the y i e l d to maturity on a 91 day pure discount bond, we have From (A11. i») we have the s.d.e. for B as dR = bdr _ (A11.5) 241 The f i r s t thing to be noted i s that the assumption that R i s a process of the form where j U ^ * {bfkJr<h ) and i s in c o r r e c t . Thus, estimates of jk^ and <r^ (we need not consider m^ as i t i s = m) , based on equation (A11.6) instead of ( A l l . 5) are incorrect to s t a r t with. However,the error due to t h i s i s complicated to investigate a n a l y t i c a l l y ! . Let us, therefore, only consider the r e l a t i v e l y simpler question: what i s the error from using jx^ and 0^ , as estimates of ^ and (T respectively? To quantify^ l e t us use numerical values for (m,yt, a - 2 ) , so that we may compute a and b. Since we are interested in the errors i n the neighbourhood of the parameter range we have estimated for the i n t e r e s t rate process, we may use those values themselves to compute a and b. Thus, we use m = 0.002522 jj- — 0.001293 <rz = 0.690494 x 10-* 1 The extent of the error can be e a s i l y investigated by Monte Carlo methods. We could expect the error to be quite small due to the nearness of (R-a) to R. . This i s because a R (and since R~]x ,in r e l a t i v e terms, a a 0) and thus assuming that the d i f f u s i o n equation governing B has a singular boundary at R = 0, (as in equation Alt. 6) instead of at 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 that the p r i n c i p a l e f f e c t of the approximation, i s on the variance element, i e . , <rR . Given that a i s posi t i v e (R-a) < R . Thus, to compensate, from equation (A11.6) would be le s s than <Tft from equation (A 11.5). I t can be shown that b 4 1,and so a\ i s already an underestimate of <r . Using equation (A11.6) instead of (A11.5), would lead to a further downward bias on the estimate of r . 242 which gives us values for a and b a = 0. 001637|A, b = 0.998343 This implies that JUfc = 0.99 9981 ( r 2 = 0. 998343 <r~ The errors i n assuming that JA^ i s approximately jX and 2 i s approximately <r2 are ne g l i g i b l e . I t must be noted that the above conclusion stems from expressions for a and b based on equation (a 11.2). as an expression of the y i e l d to maturity. That equation i s v a l i d only under the pure expectations hypothesis about the term structure of i n t e r e s t rates. I f we assume li g u i d i t y / - t e r m premium of the form which i s what we have used i n subsequent modelling of bond prices, Ingersoll [39] has shown that equation (A11.2) holds, but with m and jUL redefined as m* and jx. * and given by m* = (m-k_2.) y~1 = {mjx * k, )/m* Thus equation (A11.4) holds; with,a and b sui,taj>ly redefined using m» and JK* . , He had estimated k, and k 2 as (see Chapter 7, section 7.3) k, = 0.3093 x 10-s k 2 = -0. 1548 x 10-2 Osing these values give for a and b the values a = 0.01705/v 2 4 3 b = 0.98837 Comparison with the e a r l i e r values of a and b shows that 8 now i s a poorer proxy for r ~ which i s as expected. These values now give jU^ * 1.00542/^ 0^2 = 0 . 98837 cr * We see that JUR i s an overestimate of jx , • and <j^2 i s an underestimate of <r*. The extent of the bias i s quite small. The d i r e c t i o n of the bias on both jx and <rz are such that the resultant bond value i s biased downward. Bowever, the e f f e c t on the bond value due to such orders of errors on the parameters w i l l be n e g l i g i b l e . (This i s indicated from the Tables IX and X on s e n s i t i v i t y of discount bond values to errors i n jUL and <r2) Hext, we consider the e f f e c t on bond value by using H instead of r , in the valuation equation. The proportional error i n r , by substituting fi instead of r may be represented as 0- + {b-\) (A11.7) T The error i s c l e a r l y dependent on the current value of r . Since, on an average, the i n t e r e s t rate i s expected to remain around jx , l e t us consider the error at z -jx. Thus _ r - * + ( b - i ) V T JT^JX fx Substituting the values of a and b, based on the liquidity/terra premium model we have The percent error i n the value of a pure discount bond due to the above error in r may be represented as Where i s the bond value e l a s t i c i t y with respect to r. I f we represent the discount bond value by B we have YI . db r - b^T (611.8) where the second equality comes from the expression for the value of the pure discount bond as given i n Ingersoll [39], Thus Percent error i n bond value = 0.542 x <-brf ) which for r = works out to 0.009% - a t r u l y n e g l i g i b l e e r r o r . I t seems reasonable to expect that at other values of r around ^U., the error i s also of s i m i l a r orders of magnitude. I t may therefore be concluded that the error due to the use of the yi e l d to maturity on a 91-day discount bond as a proxy for the instantane free int e r e s t rate, i s minimal..
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Parameter estimation of stochastic interest rate models and applications to bond pricing Ananthanarayanan, A. L. 1980
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Title | Parameter estimation of stochastic interest rate models and applications to bond pricing |
Creator |
Ananthanarayanan, A. L. |
Publisher | University of British Columbia |
Date Issued | 1980 |
Description | A partial equilibrium valuation model for a security, based on the idea of contingent claims analysis, was first developed by Black & Scholes. The model was considerably extended by Herton, who showed how the approach could be used to value liability instruments. Valuation models for default-free bonds, by treating them as contingent upon the value of the instantaneously riskfree interest rate, have been developed by Cox, Ingersoll & Ross, Brennan & Schwartz, Vasicek and Richards. There has, however, not been much attention directed towards the empirical testing of these valuation models of default-free bonds. This research is an attempt in that direction. Our attention is confined to retractable and extendible bonds. Central to arriving at any equilibrium model of bond valuation is the assumption about the instantaneously riskless interest rate process, since the bond value is treated as contingent upon it. These bond valuation models are partial equilibrium models, since the interest rate is assumed as exogenous to them. The choice of the interest rate process is made subject to some restrictions on its behaviour which are based on expected properties of interest rates. The interest rate process adopted in this study is a generalization of that used by Vasicek and Cox, Ingersoll & Ross., The properties of the chosen mathematical model are investigated to ascertain whether it conforms to those expected of an interest rate process based on economic reasoning. We go on to develop alternate estimation methods for the parameters of the interest rate process, using data on a realization of the process. One "exact" method and two others based on approximations are outlined. It is observed that the "exact" method is not available to the complete family of processes included in the continuous time stochastic specification assumed to model interest rates. The asymptotic properties of the estimators from the "exact" method are known from the existing literature. However, since we would have to adopt one of the approximate methods, we need to know something about the properties of the estimators based on these approaches. This could not be derived analytically and so a Monte Carlo study is conducted. The results seem to indicate that the properties of the estimators from the three methods are not very different. The yield to maturity on 91-day Canadian Federal Government Treasury bills, on the date of issue, is chosen as the proxy for the instantaneously riskfree interest rate. The impact of using such a proxy is briefly investigated and found to be negligible. The bond sample chosen is the complete issues of retractable and extendible bonds made by the Government of Canada. There were 20 issues between January 1959 and October 1975, and weekly prices on all these bonds are available in the Bank of Canada Review. To arrive at the final bond valuation equation, some assumptions are made about the term structure of interest rates. This study first assumes a form of the pure expectations hypothesis and it is shown that the performance of the model in predicting market price movements, is considerably improved when we assume a specific form of term/liquidity preference on the part of investors. Incorporating taxes into the model results in similar improvements. The hypothesis that the bond market is efficient to information contained in these models is tested and not rejected. Finally, an ad hoc regression based model is developed to serve as a bench mark for evaluating the performance of the partial equilibrium models. It is observed that these models perform atleast as well as the ad hoc model, and could be improved by relaxing some of the restrictive assumptions made. |
Subject |
Bonds -- Prices |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0095018 |
URI | http://hdl.handle.net/2429/22286 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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