PARAMETER ESTIMATION OF STOCHASTIC INTEREST RATE MODELS AND APPLICATIONS..™) BOND PRICING by A. L. ANANTHANARAYANAN B. Tech. (Hons), I.I.T., Kharagpur, India, 1967 ^THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Commerce & Business Administration We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1978 A. L. Ananthanarayanan In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or. by his represenjtWtVve'sv • I t; ;i s~ understood "that copy i ng- or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ] • The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 11 ABSTRACT 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 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 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 S Boss., 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 111 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 iv 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. , i 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. Research Supervisor Dr. Eduardo S. Schwartz V TABLE OF CONTENTS CHAPTER PAGE 1. INTRODUCTION . , • . .V..»>.•• «v. ..... , • • • - r. • • ? • 1 Preamble 1 Contingent Claims Valuation of Bonds: A Brief Review 2 Canadian Retractables/ Extendibles in 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 Brief Review of Published Research in Related Areas 28 Maximum Likelihood (M.L.) Method of Estimation 31 The Simple Linearization Approximation ....... 34 The Transition Probability Density Method .... 35 The Steady State or Stationary Density Method 36 The Phillips 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 Capital Gains TaX . Wr. .. 151 . The "Moving Average" Model ................... 152 Tests of Market Efficiency ................... 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. Classification of Singular Boundary Behaviour for the Cases * = 1/2 & 1 187 2. Details of the Estimation Procedure for the Linearized Model . ».•.•..../.^. ..... ., 191 3. Solution to the Forward Equation for <* = 1 195 4. Solution to the Forward Equation for <X = 0 with no Restriction at the Origin ...................... 204 5. Derivation of the Stationary (or Steady State) Densities 205 6. Details of the Phillips Approach to Estimation ..... 209 vii 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 Probability 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 viii LIST 0? TABLES Table Page I Comparison of Retractables/Extendibles with Other Forms of Debt in 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 crz by Different Methods for 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 Serially Dependent/Independent Samples ................ 59 VII Comparison of Results of Estimation Using Weekly and Daily Data {^-Y^ known) 60 VIII Theoretical Sensitivity of Pure Discount Bond Prices to Errors in m 63 IX ° Theoretical Sensitivity of Pure Discount Bond Prices to Errors in ^ 64 X Theoretical sensitivity of Pure Discount Bond Prices to Errors in <rv .. ,..vr.«..«•. 65 XI Sensitivity of Pure Discount Bond Prices to Distribution of Estimated Interest Rate Process Parameters ( r, = /V2)