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 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 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 XII Sensitivity of Pure Discount Bond Prices to Distribution of Estimated Interest Rate Process Parameters ( r, = ) 68 XIII Sensitivity of Pure Discount Bond Prices to Distribution of Estimated Interest Rate Process Parameters ( r, = 2^) 69 XIV Comparison of Bond Price Sensitivity to the Use of Daily vs Weekly Data in the Estimation of Interest Rate Process Parameters (= j/^) 70 XV Estimation of Parameters for Unknown Case 73 XVI Comparison of Parameters Estimated Using Daily vs Weekly Data for the Unknown Case 75 XVII Theoretical Sensitivity of Pure Discount Bond Prices to Errors in <* { XVIII Theoretical Sensitivity of Pure Discount Bond Prices to Errors in ( XIX Details of Data Sample of Retractable/Extendable Bonds ...... 94 XX Details of Data Sample of Straight Coupon Bonds ...... 95 XXI Comparison of Model and Market Prices Bond: .4% Jan. 1, 1963 (R1) ...... 109 XXII Comparison of Model and Market Prices Bond: 5'/i % Oct. 1, 1960 (E1) ...... 110 XXIII Comparison of Model and Market Prices Bond: 5Ki % Oct. 1, 1962 XXIV Comparison of Model and Market Prices Bond: 5/2. % Dec. 15, 1964 (E3) 112 XXV Comparison of Model and Market Prices Bond: 5Vo. % April 1, 1963 (E4) 113 XXVI Comparison of Model and Market Prices Bond: 6% April 1, 1971 (E5) ...... 114 XXVII Comparison of Model and Market Prices 115 Bond: 6/4. % Dec. . 1, 1973 (E6) ...... XXVIII Comparison of Model and Market Prices Bond: VJi\ % .April 1, 1974 XXIX Comparison of Model and Market Prices Bond: 8% Oct. 1, 1974 (E8) 117 XXX Comparison of Model and Market Prices Bond: 7% % Dec. 15, 1975 (E9) 118 XXXI Comparison of Model and Market Prices Bond: 6'/i| Aug. 1, 1976 (E10) 119 XXXII Comparison of Model and Market Prices Bond: 7% July 1, 1977 (E11) 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% April 1, 1979 (E14) ...... 123 XXXVI Comparison of Model and Market Prices Bond: 9^ % April 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 All Bond Across Different Models ...... 145 XLII Comparison of Betas & Correlation Between Market 6 Model Prices 146 XLIII Theoretical Sensitivity of Pure Discount Bond Prices to Errors in K, 155 XLIV Theoretical Sensitivity of Pure Discount Bond Prices to Errors in K2 ...... 156 XLV Return on Zero Investment Portfolio Based on Constant Long Position in Bond ...... 159 XLVI Return on Zero Set Investment Portfolio Using a Strategy Based on Returns to Similar Portfolio From a Constant Long Position in the Generic Bond ...... ,..... 161 XLVII Return on Zero Investment Portfolio Based on Varying Position in Bond ...... 162 XLVIII Results of Yield Eguation Coefficient Estimation 171 XLIX Comparison of Model and Market Prices Summary Over All Bonds ...... 147 Comparison of Returns to the Zero Investment Hedge Portfolio by Using Market vs. Model Prices for the Straight Bond .... Return on Zero Net Investment Portfolio (Based on a Constant Long Position in the Generic Bond) by Aggregating Over All Bonds . / XI1 LIST OF FIGURES Figure Page 1 Plot of Transition Density Function (6 Cumul• ative Probability) for = at Different re Values ...«..«.««»»• • •••••••••»•• ••••••••• ; 219 2 Plots of the Sensitivity of the Transition Density Function to Changes in tr1 and <* ...... 85 3 Plots of the Sensitivity of the Transition Density Function to Canges in m at Different r0 Values .••...... • •;•» .•...... ^6 4 Normal Probability Plot of Resultant Error Vector from the Estimation of Liquidity/Term Premium Prameters ...... 139 5 Plot of Liquidity Premium vs Time to Maturity on Pure Discount Bonds Corresponding to Esti• 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 Distribution of Hedge Portfolio Returns ...... 167 9 Plots of Model vs Market Prices For Bond E7 : Capital Gains Tax (25%) Model, and of Distribution of Hedge Portfolio Returns ...... 168 ACKNOWLEDGEMENTS I would like to share the credit for completing this dissertation with several other individuals.. Professors Michael J. Brennan and Eduardo S. Schwartz suggested this research topic., As my supervisor. Dr. Eduardo Schwartz was a constant source of encouragement., Dr. John A. Petkau provided considerable help in the early stages towards my understanding of singular diffusion processes. Dr. M. Puterman read drafts of my proposal and clarified certain aspects pertaining to diffusion equations. As members of my committee, Professors Alan Kraus and Rolf Banz painstakingly read early drafts of this report, and have considerably contributed to its improvement. Professor Phelim P. Boyle merits special mention. Apart from his contribution towards the substance and style of this dissertation, it was his warm friendship and moral support that kept me going through the rough periods. I cannot sufficiently thank Dr. Kent M. Brothers for his help and guidance. Every part of this research pertaining to statistics and numerical methods have benefited from his advice. Dr. Shelby Brumelle has contributed immensely to the research culminating in this report. He was always available for consultations, and it is 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 in this dissertation. Mr. Wayne Deans, local representative of the Bank of Canada, was of immense help in putting together the data on retractable/extendible bonds. Kari Boyle helped with the initial data collection, and Kent Wada helped not only with the data collection and its punching but also with the plots and typing the text into the computer. Seline Gunawardene and Carmen de Silva did an excellent job of typing the tables and appendices, as well as the first draft of this dissertation. 1 CHAPTER 1: INTRODUCTION 1,1 Preamble The application of contingent claims analysis to derive equilibrium valuation models for corporate liabilities is presently an area of considerable and continuing interest and has been actively investigated in the current finance literature. This study addresses the problem of empirical estimation of a particular stochastic specification of the spot interest rate, and then goes on to evaluate the efficacy of a model of retractable/extendible bond valuation, based on the estimated interest rate process, in pricing Canadian Federal Government issues. In the seminal works of Black & Scholes [7] and Merton [47], the principal focus was on arriving at closed form valuation models for put and call options on corporate equity. Both the works cited above did point out in conclusion that the approach could be used directly to value other corporate liabilities by treating individual securities within the capital structure as "options" or "contingent claims" on the total 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 pricing, and its application to the valuation of related securities. 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 right to bankrupt or force a reorganization of the firm if it fails to meet some standard. They further look at the effect of subordination among bonds, ie. hierarchy among the debt holders, to claims on the value of the firm, and finally the effect of restrictions on the financing of interest and dividend payments. Both Brennan & Schwartz [9] and Ingersoll [37] addressed the valuation of corporate convertible bonds with and without call provisions, the principal difference being that Ingersoll was concerned with arriving at analytical solutions to the valuation problem, whereas Brennan & Schwartz presented a general numerical algorithm for solving the valuation equations. So far, the emphasis was on corporate bonds, where the underlying asset was the value of the firm., The works referred to above treated the interest rate as non stochastic - constant and known with certainty over the period of the bond. The next area that was addressed was the pricing of default free bonds. These securities, (generally Government bonds of various types) were valued by treating them as "contingent" upon the course of the spot interest rate, along with suitable assumptions about the term structure of interest rates. Brennan & Schwartz [10,12], Cox, Ingersoll S Ross [16], Vasicek[72], and 3 Bichard [58], have all addressed the problem of default free bond valuation in the option pricing 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 arrive at closed form expressions for the term structure equation. Brennan S Schwartz, in their earlier paper [10], represent the default free bond as a function solely of the short term interest rate and time to maturity, and show that various types of bonds - savings, retractable, extendible, callable or discount - all follow the same partial differential equation, the distinguishing feature being the associated boundary conditions. They also present a numerical algorithm to solve the valuation equations. In their later paper [12], they posit the value of the default-free bond as a function of the time to maturity and two related interest rate processes - the very short term riskless interest rate and the very long term interest process (yields on a consol bond)., &s can be seen from the foregoing, considerable work has been done on the theoretical front, ie,, developing bond valuation equations under varying assumptions about the stochastic properties of interest rates and term structure of interest rates. In addition, numerical methods have been developed to solve rather general forms of the resultant pricing 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 cite the important papers: Black & Scholes [6], and Galai [29]), except for Ingersoll [38], which 4 is an application of option pricing analysis to dual fund shares, and Brennan & Schwartz [12]» who value Canadian Federal Government coupon bonds. The aim of this research is to conduct an empirical study of contingent claims analysis on retractable and extendible bonds of the Government of Canada., 1. 3 Canadian Retractables/Extendibjes in Perspective: ftn extendible is 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. It was exchangeable on or before June 1st, 1962 into 5%,%, October 1st, 1975 bonds. Thus the 3 year intial 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 elect an earlier maturity. Both from the practical investment point of view, and with respect to valuation theory, the two instruments are very similar. There are two ways in which to view a retractable or extendible bond. It may be viewed as a long term bond with a put option. The exercise price in this situation is the value Of the long term bond, and the payoff is the short term bond. The option is exerciseable on the extension/retraction date. Alternatively, the retractable or extendible may be viewed as a short term bond with a call option. From this point of view, the exercise price is the value of the short term bond, and the 5 payoff is the long term bond. Extendibles and retractables first appeared* on the Canadian scene in 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, this was the only retractable issued by the Government of Canada). While there were additional issues made by the Federal Government in the mid sixties, these instruments have been used more widely in the high interest rate period since 1969/70. Table I gives some numbers to place retractables and extendibles in perspective vis-a-vis other forms of debt. Clearly, the major issuer of retractable/extendible bonds is the Federal government. Further, as a proportion of total debt outstanding, retractables and extendibles appear to be increasing over time, both with the Provincial and Federal governments. The total debt columns in Table I include very short term debt, (ie., current liabilities, treasury bills, etc.), as well as medium to long term debt. Retractables and extendibles belong strictly to the medium to long term maturity class, and so should be compared with the other debt in that class alone. Thus even though retractables and extendibles constitute only approximately 4.536 of the total Provincial 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, listing all outstanding Federal/Provincial/corporate issues as of January 15th, 1975. TABLE I COMPARISON OF RETRACTABLES/EXIENDABLES WITH OTHER FORMS OF DEBT IN NOTES OH TABLE I Ret/Ext as O/S as on 31st March 1975 O/S as on 31st Marc' i 1976 Z on 31st March Ret/Ext. Tot.Debt. Z Ret/Ext. Tot.Debt 1977 a) All figures are in millions of dollars 50 b) The total debt includes all bonds, bills and notes, Issued by - 3845 - 50 5093 0.98 British Columbia by the Provincial government, as well as all debt guaranteed by 3578 3.58 128 the Provinces. Alberta 128 3031 4.22 128 58 c) Likewise, the retractables/extendables included in 58 2473 2.34 58 2884 2.01 Manitoba each Provinces' a/c (as well as in the Federal a/c), 1665 3.66 61 including issues guaranteed by the Provinces as well, New Brunswick 51 1199 5.08 61 182 d) No figure of aggregate corporate debt was included as the 161 1504 10.70 182 Newfoundland - - same was not readily available, 4.02 675 225 13397 1.90 675 16760 Ontario e) The total Federal debt figures were taken from the 10 Bank of Canada Review. For the Provinces, the same 10 98 10.20 10 111 9.00 P.E. Island were from the Public Accounts. 983 734 8403 8,73 808 8391 9.63 Quebec f) The public accounts for Newfoundland as of 31st March 912 7.67 70 1976 were .not readily available. Saskatchewan - 816 - 70 Total Provincial 38299 15.27 6250 Federal 4825 33700 14.31 5850 2503 Corporate 1902 - - 2315 - - 10207 10970 Total 8104 7 term debt. In gross amounts, including corporate issues, they total about $10 billion. Apart from size 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 their valuation by conventional methods ad hoc, and particularly amenable to valuation in the option pricing framework. Clearly, retractable and extendible bonds are interesting instruments, and a detailed study of them is quite in order. 1•4 Outline of th.e Thesis Chapter 2 develops the basic bond valuation equation in terms of the parameters of the local interest rate process. The appropriate boundary conditions relevant to the pricing of retractable and extendible bonds are derived. The approach to incorporating different assumptions about term/liquidity premia into the valuation model is briefly outlined. An approximate approach to account for taxes (along the lines of Ingersoll [38]} is also presented. The stochastic specification of the short term interest rate process is central to the bond valuation model. Chapter 3 addresses the desirable properties that any mathematical model of this process should possess. A specific diffusion equation is suggested to model interest rates, and the properties of this specification are investigated. Having specified the form of the interest rate process, the next problem is that of estimating its parameters, given data on a realization of the process, Methods for estimating the 8 parameters are examined in Chapter 1. Starting with a brief review of the existing literature on the estimation of parameters of Markov and diffusion processes, three different methods of estimating the parameters are proposed. The details of the estimation procedure for each of these methods are also presented., Chapter 5 is devoted to the comparison of the three methods of estimation proposed in the previous chapter. For this, Monte Carlo methods are used to examine the distribution of the estimated parameters by each method, under different conditions, as part of the comparison of the three methods, the effect of the estimated distribution of parameters on bond valuation, is also briefly investigated since our primary concern is to use the estimates to value retractable and extendible bonds. The chapter concludes with a brief look at the inter-relations between the estimated parameters, as well as the way in which they affect the interest rate process. Details about the data sample on short term interest rates and bond prices are given in Chapter 6. Chapter 7 reports the empirical tests of the models developed in Chapter 2. We start with the bond valuation model based on the pure expectations hypothesis. We then incorporate a specific form of term/liquidity premium. The estimation of the investor preference parameters in the assumed form of the term/liguidity premium expression is addressed and estimates of these parameters, based on a sample of non-callable coupon bonds, are presented. These estimates are incorporated in the bond valuation model and the resultant bond values are compared with 9 market prices. The effect on the bond valuation model of incorporating taxes (both revenue taxes and capital gains taxes), is investigated. Tests of market efficiency based on the returns to a zero-investment portfolio are conducted. In this section, the ability of the different models to identify over priced bonds is also investigated using an approach based on Galai [29]. Finally, an ad hoc, regression based valuation model (the "naive" model) for retractables and extendibles is developed. Using the sample of non-callable coupon bonds, the required coefficients for the "naive" model of retractables and extendibles are estimated., The performance of this model in predicting bond prices is briefly compared with that of the models developed earlier in Chapter 2.» The study concludes in Chapter 8 with a summary of the principal results, and some remarks about the choice of the stochastic specification for the interest rate process, as well as about the model of bond valuation.. Suggestions for further research in 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 will closely follow the method set out in Brennan 6 Schwartz [10].. Basically, the value of any default free bond is the present value of its principal and coupon payments. The future cash flows are known with certainty, once the coupon rate and time to maturity are specified. Knowing the future cash flows, what is required to arrive at their present value (ie. the bond value) is a suitable discount factor. A natural choice is the short term interest rate.. In a model where we recognize that interest rates are stochastic, we could evaluate the present value over all possible future sample paths of the interest rate, over the terra of the bond. Following this line of reasoning, we could justify 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 is some uncertainty associated with the assessment of future spot rates, in a market where risk averse investors exist, term premia enter the valuation equation via the specific assumptions made about the term structure of interest rates. To model the future course of the spot interest rate, we assume that it is a stochastic process with a continuous sample path and Markov properties. Under the Markov assumption, the future development of the spot rate process, (given its present value) is independent of the past development that has led to the present level. Processes that are Markov and continuous are 11 called diffusion processes, and for the one dimensional case can in general be described by a stochastic differential equation of the form dr bCr.t} <&> -f dl (2.1)., where b(r,t), and a2(r,t) represent the instantaneous drift and variance respectively of the process, and dz is the driving stochastic element and is distributed as H(0,dt). For the present, there is nothing to be gained by restricting the generality of the above stochastic differential equation governing the interest rate process. However, it may be noted that both b(r,t) and a(r,t) must at least be known, deterministic functions of time - they may not be stochastic functions of time2.; He shall however restrict our attention to a particular family of processes, when we address the interest rate process in greater detail later on. The main competing theories about the term structure of interest rates are a) the pure expectations hypothesis 2 In the standard option valuation framework, there is no restriction on the instantaneous drift term of the underlying asset (the stock), ie. that it should be non-stochastic. This is because, the final parabolic partial differential equation governing the option value does not contain the drift term. For the bond, the corresponding partial differential equation is equation (2.9). The instantaneous drift of the interest rate process (the underlying asset being the pure discount bond due to mature the next instant) enters the valuation equation. If either b(r,t) or a(r,t) in equation (2.1) were stochastic, then the valuation equation would no longer be an ordinary second order parabolic partial differential equation. 12 b) the term or liquidity premium hypothesis c) the market segmentation (or preferred habitat) hypothesis. , The definition of the pure expectations hypothesis that we adopt is that the instantaneous expected return on bonds of all maturities is the same3. This implies some sort of "risk neutrality" on the part of investors over the instantaneous holding period returns across bonds of all maturities. The second hypothesis argues that concern over fluctuations in wealth causes investors to demand a "liquidity" premium on long term bonds over those of shorter maturity. On the other hand, concern over fluctuations in income leads to a case for term premiums that would obviously have just, the opposite pattern., The market segmentation hypothesis proposes that bonds of different maturities are totally different instruments, and thus not substitutable. This would require that the term structure of interest rates, at any point in time, be defined by the 3 What follows is based on Cox, Ingersoll & Ross [16], In the existing literature, the pure expectations hypothesis is characterized by one of the following propositions: 1) Implied forward rates are equal to expected future spot rates 2) The yield to maturity from holding a long term bond is equal to the yield from rolling over a series of short term bonds 3) The expected return over the next holding period from bonds of all maturities is equal Under certainty, all three forms are equivalent., With uncertainity, however, Cox, Ingersoll S Ross have shown that the first two propositions are consistent with each other, but not with the third. Hore specifically, if the term structure is unbiased in the sense of the first 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 in the market at that time. Most studies of the term structure of interest rates in the option pricing 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 tried to operationalize a form of the market segmentation hypothesis, by introducing two factors in the maturity structure - the very short end, and the long term maturity,, Only incorporation of the first two hypotheses about the term structure of interest rates into the bond valuation models is considered in this 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 is the spot riskless interest rate, and X the time to maturity. Similarly, let the value of a retractable or extendible bond be G(r/£). For purposes of generality, let B(r,l) pay a coupon* c, , and G(r,T) a coupon cz» Then, using Ito's Lena (McKean [45]) and equation (2.1) for the interest rate process, the straight bond B, and the generic bond G, are governed by the following stochastic differential equations (SDE) : * For ease of computation in a continuous time framework, we assume that these are continuous coupons. A continuous coupon of c means a coupon payment of c dollars per unit of time per bond.. As pointed out in Chapter 6, this assumption is quite reasonable. 14 (2.2) where b=b(r,t) and a=a(r,t), and subscripts denote partial derivatives; B, is the first partial derivative of the bond price with respect to its first, argument - the spot riskless interest rate, etc. The spot riskless interest rate is, by definition, the yield to maturity on a default free discount bond due to mature tike next instant in time., The return on all three assets, viz., the generic bond> the straight bond and the short term interest instrument, have the same stochastic element driving them (dz); ie., they are all perfectly correlated. If borrowing and lending at the instantaneously riskless rate of interest were possible (and all the other assumptions of the option pricing model helds), a zero net investment portfolio could be formed using the above three securities. Consider an investment of x, dollars in G, x% dollars in B and xi = - (x, *-xz) dollars in the riskless asset. The return on such a portfolio is given by 3 The perfect market assumption is implied with all the attendant properties of unlimited borrowing/lending at the riskless rate by all investors, no margin reguirments on short sales and immediate full availability of proceeds of short selling and ability to trade every instant at current prices, and finally the absence of all 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 all uncertainty from the return on the zero investment portfolio would be eliminated if we choose x, and x2 such that the coefficient of dz is zero, ie., Xz •= <%_ = - *i . J__ (2.6) Arbitrage would now drive the certain return on the zero net investment portfolio to zero. Substituting (2.6) into (2.5) gives the basic valuation equation. (/VVQ- „ (A+C'/B) - r (2.7) This expression has to hold for bonds of all maturities at any point in time. It is the familiar expression of excess return per unit of risk on each security (see Cox £ Ross [ 17 ]). We may 16 represent the price of instantaneous standard deviation risk by (r,t) , noting that, whereas is independent of the time to maturity, it may change over time and with the spot rate. This gives R (/W<0~ - » UT.t.t) (2.8). where \{t,t0t) represents the term or liquidity premium, ie. 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 partial differential equation for the bond price ( -la Thus, in equilibrium, any bond follows the same valuation equation (2.9). What distinguishes them, are the boundary conditions that each has to satisfy. (This result was first 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 satisfy. a) Terminal value at maturity: From the default free aspect, the principal of $1 is guaranteed at maturity. Thus irrespective of the current interest rate at maturity, the bond value equals its face value, ie., 17 G (r,o) = 1 (2.10a) b) Retraction/extension feature: Here, we shall consider three types of retraction/extension features and develop the appropriate boundary conditions applicable to the bond valuation equation in each case; i) the retraction/extension option has to be exercised at a single point in time. ii) the option may be exercised over a period of time. iii) the option to retract/extend may be exercised over a period of time, but even if the decision is to retract, (or not to extend, in the case of an extendible) the face value of $1 is available only on a fixed future date beyond the final exercise date. These three cases may diagramatically represented as: r— 1 1 1 VkrruxL wh The first case above would correspond to the situation where an< TQ.( I * coincide at one point. For the second case, t5 is 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, it would be helpful to consider an example, Consider that an investor holds a 5% coupon bond, which he may extend on (say) January 1st, 1970 for 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 in for $1 on January 18 1st, 1970. Clearly on any day prior to January 1st, 1970, the holder of the short bond has a European call 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, ie. 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 in time just after the decision point. Then we have SCTX) - Max [ $C*X*) ,1 (2.10d.1) The condition above implies that the bond value, if the bond is not cashed in at the decision point, is 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 (case(ii)), condition (2.10d.1) would be altered as: (2.10d.2) Here the first condition is that during the period the extension option is in force, the value of the bond is bounded below by the par value of $1. This is the arbitrage condition as the holder has an American option. Further, since it has to be continuous across the expiry point of the option, we have the second condition, as before. For actual bonds in the market, case (iii) is the 19 representative case. The option to extend/retract may be exerciseable over a 3 to 6 month period, but, even if the option were exercised, the par value is generally available only a further 6 to 12 months later. 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. Clearly, if the investor decides to choose the short bond at any time, between July 1st, 1969 and October 1st, 1969; the principal of $1 is available only on January 1st, 1970. It is clearly optimal to exercise the option at the last point, fe2. , and so we have the boundary condition there as: + In the condition above, G represents the value of the long term bond. The short term bond has been represented by H, to explicitly recognize that the coupon of the two bonds could be different. c) Value at the interest rate boundaries: We know from the previous section that the interest rate process and the bond value process are very closely related. From economic considerations, we require interest rates to remain non- negative. Whether this requires the imposition of specific conditions at the interest rate boundaries (r=0 and co ) is investigated in the next chapter. We therefore postpone developing conditions that the bond value process has to satisfy at r=0 and oo till later. For the present, we just note that the conditions imposed on the bond value process at the 20 boundaries of the interest rate process should be consistent with the behaviour of the interest rate process at these boundaries. In general, the differential equations (along with the attendant boundary conditions) governing the retractable/extendable bonds, cannot be solved analytically. Numerical finite difference methods will be used to solve the equations. The general procedure is to develop the solution recursively backwards from the boundaries, where the solution is known. This is addressed further in Chapter 7. 2.4 Incorporatinq Taxes into the Model So far the model has been developed on the assumption of no taxes, either on revenues (coupons and interest) or capital gains. He could attempt to incorporate taxes into the valuation equation, along the lines of Ingersoll [38], but the following assumptions need to be made explicit: a) Taxes are assumed payable on a continuous basis and at a fixed rate. This implies that there is some "average" tax rate over all investors that could be used in the model. The assumption further implies that interest payable on all borrowings is tax deductible. b) All capital gains are treated as taxed at the capital gains tax rate, and payable continuously. In reality, capital gains taxes are paid only when gains are actually realized by a sale. Further, any capital gain over a period of less than 91 days is treated for tax 21 purposes as a revenue item. In our model however, we cannot make this distinction6. The assumptions may be restrictive, but it is an empirical question as to whether it is better to ignore taxes altogether, or incorporate them into the valuation equation with the current assumptions - a question that is addressed later. Let us represent by R, the rate of taxes on revenues and by T, the rate of taxes on capital gains. The return on the zero investment portfolio, as given in equation (2.3) is modified to The same analysis as before leads to the valuation equation which leads to the following partial differential equation iaV0<$„ + [bG-T)-a{]5, + 0-R)(c2~f^) - 0-T)^2 0 (2.11) The boundary condition associated with this equation are exactly those associated with the previous equation (2.9). * This assumption is required to ensure an unique equilibrium bond value. Given our continuous time hedging approach to valuation, capital gains as per the existing tax laws are never applicable. Capital gains taxes do exist, and are accepted as one of the determinants of investors choice among available securities. The present approach is one way of incorporating this reality into our model. 22 CHAPTER 3: THE INTEREST RATE PROCESS 3.1 Properties of Interest Rate Processes In the previous chapter, we left the stochastic specification of the interest rate process in a very general form. Lacking a well developed theory of growth under uncertainty to specify a functional form for the interest rate process, (the only work addressing the problem appears to be Herton [49]), we are left to draw upon functional forms that satisfy some very broad criteria7. a) Interest rates should never become negative, as holding wealth in the form of cash dominates such a situation. b) An interest rate process should possess some central tendency, ie., one would not expect the spot rate of interest to rise to some high level, and yet be equally likely to go further up, as move downwards. c) Preferably, the process should be such that the probability of the interest rate reaching either zero or infinity is identically nil. d) Mathematical tractability. To ensure that interest rates do not become negative, we could adopt one of two approaches: a) make r=0 a singular boundary8 with positive drift, ie. 7 These criteria are drawn from Ingersoll £39]. 8 By definition, the diffusion 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 interest rate reaches zero, it changes only in one direction; upwards, b) restrict the process to remain non negative by imposing an artificial barrier at r=0. The second approach is more straight forward, A reflecting barrier at r=0 ensures that the interest rate never becomes negative, and further, it never remains at zero, except for an infinitesimal instant. However, once r reaches zero the direction of its change the next instant is known with certainty - since r cannot become negative (due to the reflecting barrier) it can only increase. This would appear to present a clear arbitrage opportunity; a situation not consistent with market efficiency in a continuous time framework. However, no arbitrage profit opportunity need exist if the bond valuation model is made to satisfy suitable boundary conditions at r=09 . Though it may seem counter intuitive, even if b (0,t) > 0 and a(0,t)=0, it does not ensure that if the interest rate reaches zero, it will leave it and enter the positive region again. The behaviour of the process at a singular boundary cannot be inferred by intuition alone. Thus if we chose a » We have from equation (2.2): (dB/B) =[ (B, b-B^aZBJ/Bjdt + (aB(/B)dz. At r=0, B is not zero, and is finite. Further, since the interest rate process and the bond value process have to be perfectly correlated, the bond value should also have a reflectinq barrier at r=0. From the standard reflecting barrier condition (see Cox S Hiller[ 15 ]) , this requires that B, =0. The instantaneous return to holding the bond thus becomes certain, as B(=0 reduces the coefficient of dz to zero. To ensure that no arbitrage opportunity exists at r=0, the certain return to holding the bond should also be zero. Thus we require 24 functional 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 acceptability10 of the functional form of the stochastic specification. Feller [25] has studied the problem of characterizing the behaviour of a diffusion process at its singular boundaries, by the method of semigroups. (A simplified and somewhat more readable exposition of Feller*s work may be found in Keilson [ 41 ])..... Broadly speaking the behaviour of a diffusion process at a singular boundary could be characterized as one of the following: a) Natural: The boundary is inaccessible in finite time from any starting point in the interior. It is interesting to note that a natural boundary can be both inaccessible and absorbing (ie. as in the case of the lognormal process, where zero is both inaccessible and absorbing). b) Exit: the boundary is accessible in finite time and once the process reaches the boundary, it is absorbed. c) Entrance: the boundary is inaccessible in finite time from the interior, but if the process started from the boundary, it would leave and enter the interior in finite time. d) Regular: the singular boundary is accessible, and we io From economic considerations, it is undesireable to have r = 0 as an absorbing boundary, ie. once the interest rate reaches zero, it never leaves it. 25 can further specify the behaviour it should exhibit there (ie. absorbing, reflecting, etc.) by imposing suitable boundary conditions. 3.2 The Interest Rate Process Keeping the above requirements in mind, let us consider the following stochastic specification. &i~ ^ m(jii--r) dt + First note that the parameters are not time dependent. This assumes stationarity of the interest rate process over time. Though some realism is lost, considerable analytical tractability has been gained., The process has the mean reverting property, because when r> (< jx) , the drift is negative (positive), so that the deterministic movement of the interest rate is always towards JUL the central tendency. The parameter m controls the speed of adjustment towards . To see this, consider only the non- stochastic part of the process for 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 overall mean ^ , 26 for a given time interval & . Looking at the stochastic term, we find that r=0 is a singular boundary11. Further, we want 0, as negative makes a (r,t) -> °o as r->0, which is an undesirable result. Again making the variance term not only a function of r, but introducing two free parameters ( cr , the family of the interest rate process. 3.3 interest Rate Process Behaviour jit Singular Boundaries Since r=0 is a singular boundary, we need to investigate the behaviour of the process at r=0 (as well as at r= <*>) , This is set out in Appendix 1. The results may be briefly summarized as follows: 1) The process corresponding to extensively by Feller [23] and his results are a) For all parameter values, r= oo is an inaccessible boundary. ,,. b) At r=0; when m,/^>0, the boundary can be either an absorbing or reflecting barrier when 2mjx < When 2mu„ £ r2, r=0 is an entrance boundary., 2) In case c< =1, we find that both r=0 and r= <£> are natural boundaries. 3) It was not possible to investigate the behaviour at the singular boundary for arbitrary values of c* as the necessary integrals could not be evaluated (see Appendix 1) . By continuity of behaviour, we conjecture 11 r= oO is also a singular boundary. 27 that as crt reduces, and 2mcorrespondingly increases in relation to parameter space where r = 0 is not an absorbing boundary12. 12 The boundary behaviour of the process for values of o( # ^ or 1 is currently being further investigated jointly with Kent Brothers and David Emanuel. The preliminary results seem to indicate that CHAPTER 4: ESTIMATING THE INTER EST RATE PROCESS PARAMETERS 4. 1 Brief Review of Published Research - in • Related Areas •• The interest rate process specified in the previous chapter has a continuous sample path over time. However, we have a record of its realization only at discrete intervals in time, say daily'or weekly observations. The problem that we shall now address is the following: Given a set of data points (r^ , t=1,...T)> which are observations on the interest rate process at discrete intervals, what procedure does one adopt to estimate the parameters frn r jx*(T r specification of the previous chapter (equation 3.1). In general, when we have a sequence of realizations of independent random variables which are identically distributed 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 literature see Billingsley [3,4]). For Markov processes with stationary transition probabilities13 these generalizations are carried out in such a way that the Markov kernel now plays the same role as the probability measure in the case of independent identically 3 * If we represent the transition probability by P (rt")> t \zs ',s) , t>s, then s-tationarity ofs the transition probability requires that P(rt,t|r5,s) = P(r^,u|rv,v) for all (u-v) = (t-s) . This is the time homogeneity condition. 29 distributed random variables. In particular, Billingsley [3] shows that maximum likelihood estimates based on the above approach exhibit almost all the properties of similar estimates in the independent random variable case. (See also Roussas {59] for properties of maximum likelihood estimators for Markov processes with discrete time and state space14). Much of the literature on statistics of diffusion processes (ie. continuous time stochastic processes) has addressed what is called the problem of optimal non-linear filtration. This is in the area of electrical communications, where we have a signal (a stochastic process) which is unobservable. What is observed however, is a "distorted" transformation of the signal, and from it inferences are to be made about the underlying signal. There is a large body of literature; papers of particular interest are Sirjaev [ 64], Ganssler [30] and some of the references cited therein. Though there is nothing specific in the literature cited above that has a direct bearing on the problem of estimation of parameters of the diffusion process set up in the previous chapter, Sirjaev £64] proves that the maximum likelihood estimators of parameters in the drift term of any diffusion 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 diffusion equation is a very difficult problem. It appears that 14 Kendall 6 Stuart [42] have also shown that the ML estimators are consistent though generally biased. The asymptotic ' normality of the estimators is 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 in the process dx = - X-xdt + dz and found the resulting bias in X This is 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 in the case of stochastic processes which do have a unique stationary distribution (we shall say more about stationary distributions shortly), using the density function of the stationary probability distribution to set up the joint likelihood of a given set of observations instead of the Markov kernel;, in conjunction with the" minimum-distance-method of Wclfowitz [73,74], leads to consistent parameter estimates. It was, however, pointed out by Ganssler [30] that using the stationary distribution may, in general, not lead to the complete identification of all the parameters in the Markov kernel. In conclusion, it appears that the existing literature on estimation of parameters of diffusion equations does not contain any specific results that could be brought to bear upon the estimation problem facing us. Finally, one last area that was briefly surveyed was the literature dealing with genetics. Feller [24] indicated that a diffusion equation of the form (3.1) with c{ =J4. resulted by is see Cox S Miller [15] 31 taking the discrete time birth and death process to its appropriate continuous time limits. It was therefore felt that there could possibly have been some empirical work on estimating the parameters of birth and death processes, the results of which could be brought to bear upon our specific problem. Unfortunately, none of the published works addressed the problem in a continuous time framework. The only two papers of any interest are Immel [ 36 "J and Darwin f" 181. Both address the discrete parameters case only, but they adopt the approach of using the transition probability function for setting up the joint likelihood function, given a realization of the process. 4. 2 Maximum Likelihood (M.L. ) Method of Estimation:. From the above, we see that there is some support in the literature for the M.L. approach to estimation. As pointed out by Billingsley f3,4] and others, the desirable asymptotic properties of M.L. estimates can be briefly stated as follows: a) The estimators are asymptotically unbiased. b) They are consistent. c) The inverse of the Hessian matrix with signs reversed is a consistent estimate of the asymptotic variance- covariance matrix of the parameters, where the asymptotic joint distribution of the estimated parameters is multivariate normal. Given a sequence (r^. ,t=1,.,.T) of observations on the short term interest rate, the joint likelihood function can be 32 set up as T t((Uo.\8) s Tl P^el-^.e). P.OVj (4.1) where P( r\ jr , , 0 ) represents the transition probability density, and P0 (r, ) is the probability corresponding to the initial point of the sample. $ here represents the parameters of the diffusion process - in our case [ yn , jx., o~~, °^ ] . Two points need to be noted about the probability density expressions in (4.1) : a) The transition probability density is assumed to be time homogeneous. This is quite valid, given the assumption in the previous chapter that the diffusion equation modelling the interest rate process displays no explicit time dependence of the coefficients.. b) The implication is that the observations fr^} are equally spaced over time. This poses no real problem, as in economic data observations are generally eguispaced. The joint likelihood of the data contains the term corresponding to the initial point which poses problems with further analysis. In general, several arguments may be put forward to drop the expression corresponding to the starting point in the joint likelihood of the data: a) Hhen we have a reasonably large data sample, the contribution of the initial point may be considered negligible in comparison to the rest of the points and 33 may be dropped (see Billingsley [3]). In fact all the estimation theory results are asymptotic results, and large sample sizes are implicitly assumed. b) It is not uncommon in several situations to treat the estimators as strictly conditional upon the sample. Following such an approach, we could argue that the estimators are conditional upon the initial point, and thus attribute a probability of 1 to that point. c) Finally, Zellner C761 reasons16 that we may assume that the probability corresponding to T| is totally independent of &! . Since our interest is only in estimating & , it can be easily shown that the distribution of [V is unaffected by dropping the initial point. In view of the above arguments, we shall drop P0 (r() from (4.1). To set up the joint likelihood function, we need to ascertain the transition probability density for the diffusion process * cr r - dz (4. 2) E(dz) = 0 and E (dz2) = dt. In general dz.is assumed to be a Gauss-Weiner process, ie. 16 zellner*s reasoning is for the analysis of first order autoregressive systems in a Bayesian framework. 34 4. 3 The Simple Linearization A£j3roximation The specification of equation (4.2) suggests a very simple estimation procedure, by linearizing the differentials to finite (discrete) differences. Thus we have and if we now choose our unit of time such that At = 1 (the frequency of the observations on r) we have where ' Y| /\> N (0,1) . In the limit as At 0, the approximation (4.4) as a characterization of the diffusion 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 this approximation is investigated by Monte Carlo methods in the next chapter. For the present, however, we see that the approximation (4.4), closely resembles a regression equation, ie, , we have a linear 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 likelihood function as in (4.1). The details of the estimation procedure are set out in Appendix 2. 35 4.4 The Transition Probability Density Method The exact approach would be to ascertain the transition probability density and use it to set up the likelihood function (eguation 4.1). It is well known in probability theory that corresponding to every diffusion equation, there exist two equations that the transition probability density has to satisfy. These are the Kolmogorov backward equation and the Kolmogorov or Fokker-Plank (FP) forward equation. The solution to the FP equation is the transition density function corresponding to the diffusion'equation17 18. Thus, for our case of the diffusion given by (4.3) the FP equation is -JL^()t-T)FJ ^JL[<^FJ ^ 2£ (4.6) where F = P^r vt)r0 , 6 ) is the transition probability density function. To solve this parabolic partial differential 17 The existence of unique solutions to the forward (FP) ' and backward equations depends upon the drift and variance terms of the diffusion equation satisfying some continuity requirements (see Friedman [28]). More specifically, it is reguired that they be bounded and uniformly Lipschitz continuous in (r,t) in compact subsets of Rrx [0,T], and further, that the variance be strictly non-negative over the whole domain. 18 It is a well known result (see Feller [26]) that the solution to the FP equation also satisfies the backward eguation, except in rare situations where the solution is not unique. It has been Observed in the literature that the solution also possesses the properties of a probability density function, ie. the function is strictly nonnegative over the state space, and its integral over the state space <1 (these are the Chapman- Kolmogorov conditions). If the equality is satisfied, the solution to the FP and backward equation is unique, but in general, different diffusion processes may satisfy the same forward and backward eguations. 36 equation, we need to impose boundary conditions at r=0 and infinity (if r=0 and infinity 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 . Feller [23] has studied the solution corresponding to the case approximate solution to the case where cK =1, based on an approach suggested In Goel 6 Richter-Dyn [33], is sketched in Appendix 3. When =0, the origin is no longer a singular boundary. If we reguire interest rates to remain non-negative, we need to impose a reflecting barrier at r=0. The solution to the FP equation with a reflecting barrier at the origin is quite complicated, but for the unrestricted process (where a positive probability of negative interest rates exits), the solution is rather straight forward (and detailed in Appendix 4). As pointed out by Vasicek [72], the parameters could be chosen such that the probability mass below the origin could be made arbitrarily small, so that for all practical purposes, r=0 is virtually inaccessible. 4.5 The Steady State or Stationary Density Method We can see that solving for the exact transition probability density may not always be possible, except by foregoing some generality in the model, ie,, restricting the values of the exponent o\ :. , He could however, substitute the stationary density into the joint likelihood instead of the transition probability density. Ganssler [30] has shown that using this approach in conjunction, with the minimum distance estimation method of Wolf owitz [ 73,74 ], leads to consistent parameter estimates, which are asymptotically unbiased., The stationary probability distribution19 is, in a sense, the limit of the transition probability density, where the time interval between observation tends to'oO ., It could be represented as The existence of an unique steady state probability distribution is usually assured when we have a process that has a time homogeneous transition probability distribution. Further, for singular diffusion processes,when we rule out those ranges of parameters where one of the singular boundaries acts as an exit barrier, we ensure that the stationary distribution is not the trivial P(r) = 0 over the complete state space, with a Dirac delta function concentrating all the probability mass at the exit boundary. Thus the stationary density is given by the solution to the FP equation (0.6) by setting :r? - 0; OX, (0.7) 19 The stationary probability distribution exists_bnly for time homogeneous processes. Another way of representing the stationary distribution could be as follows; Given that the diffusion process has attained its steady state, the stationary probability distribution then gives the probability of finding the process at any particular point (or interval) in the state space at any instant. 38 the solution to which can be shown of the form (see Goel & Richter-Dyn [33]) POO - JL Hb\-i[ I (4. 8) (Pf2^ L J where C is determined by the condition \P(r)dr = 1, where Si. represents integration over the state space. Appendix 5 gives the details of evaluation of the stationary density. It. is of the form P(f+0 r tel . 11- l L * 1+A J it- where A,= 1-2c^. It is also shown in Appendix 5 that when we take the limit as A,-? 0 or -1 in (4. 9c) , we get (-4.9A) and (4.9b) respectively. Thus the steady state density is continuous in . 39 Given a realization (r^. , t=1, ... T) , we propose to set up the likelihood function using the stationary distribution (4.9), and estimate the parameters by ML methods. There does net appear to be any reference in the existing literature to the asymptotic properties of such estimators. We shall look at these properties, based on some limited Monte Carlo simulation results in the next chapter. However, the approach may be crudely rationalized as follows: a) One argument could be that if we have a sufficiently large sample, the distribution of the sample might resemble the stationary distribution20. b) If the sequence of data points were independent,, using the stationary distribution to set up the joint likelihood of the data would be exact. The crucial objection is that we are treating a sequence of dependent random variables as if they were independent. Lack of independence should hopefully not alter the validity of the approach. This may be treated as if we 20 This rationalization can be motivated by the following result for Markov processes (see Cinlar f 13 ]). Consider a continuous time, discrete state space Markov process which has a stationary distribution. Let observations be made on this process, such that the time interval between observations is exponentially distributed. The sequence of observations then represents a discrete time Markov process. It can be shown that this discrete time process has the same stationary distribution as the continuous time process from which the observations were taken. Rs the number of observations goes to infinity, the distribution of the sample observations approaches the stationary distribution. The exponential sampling scheme was required to ensure that all points on the half real line representing the time axis, were equally likely to be chosen. The extension of this result to continuous state space processes can be found in Dynkin £21]., We have used equispaced observations, and that should introduce bias, which we conjecture should reduce as the number of observations increase. 40 are using a "biased" approach ; the extent of "bias" depending upon how close the successive observations are. Finally the steady state approach cannot identify the two parameters m and cr2* separately - only their ratio can be estimated21. 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 identify these parameters separately. To summarize the various aspects of the three estimating methods, we may note the following: a) The transition probability density approach to setting up the likelihood of the data is exact, but its use requires that we greatly restrict the generality of the model - either set ds = '/^ or, if 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 is quite intractable for estimation purposes. b) The stationary probability density approach cannot indentify m and cr2"" separately - only their ratio. Further, when the data points are near each other, the likelihood function is probably far from exact, as the individual observations are not independent. c) The simple linearization method (or normal approximation) is very tractable, and the closer our 21 This was expected on the basis of the results in Ganssler [30]. 41 data points, the less the error in the approximation. In the real world, however, there are limitations to how closely spaced the observations can be. This limitation is discussed in Chapter 6. 4.6 The Phillips Approximatipn Method Before we conclude this chapter we can outline one other approach to the estimation of the parameters of stochastic differential equations, (SDE) which has been advocated by (among others) Bergstrom [2], Sargan [60], Phillips [55,56,57], and Hymer [75]. Consider the system of linear stochastic differential equations D^(tl -- A |(t| + bzC-t) 4 fa (4.10) where A and B are matrices, D is the differential operator ct/cit , Z (t) Is a vector of exogenous variables, and ^ (t) is a pure white noise disturbance vector. The solution to (4.10) satisfies (see. Sargan [60] for proof) 0 0 The last term in (4.11) is a stochastic integral, and if we assume that ^(t) is Gaussian N(0,_Q.), and that the integral exists, then we can replace the last term by f(t), where E[ £(t) ] = 0 and 42 o Thus, we have' £,(t) m ' N{0,]|;*) , and it may be noted that even though^JT- may have been diagonal,_Q_* will have non-zero off diagonal elements. Going back to (4.11), in the general case where Z (t) is a vector of exogenous variables, the first integral poses a problem in the way of reducing (4.11) to something more manageable. In the special case where Z(t) is a known deterministic function of time, the integration can be carried through and (4.11) suitably reduced. However, when Z(t) is also stochastic, no exact equation system can be obtained, equivalent to the SDE system (4,10). Phillips [57] presents an approximation method, whereby the integral of Z(t) in eguation (4,11) may be reduced using a three point Lagrange interpolation formula to express Z(t) as a polynomial in the interval [t, (t-h) ],[Appendix 6 presents more details on the adaptation of this approach to the SDE (4.3), which is cur interest rate model.] Using this method reduces (4.11) to 43 Where the E*s are functions cf a, B and h. Phillips (op cit) has shown that the approximation (4.13) to the SDE (4.10) is superior to the discrete approximation, (equations (4.4) and 22 (4.5)) .; Phillips (op cit) points out that the proposed approximation scheme leads to bias in the parameters of the order of 0(h3). But in case Z(t) is not differentiable at a countable set of points on the real line, the bias is larger and of the order 0(h). In our case, the regularity condition required to get improved estimators by this approach are not met. We shall therefore not pursue this approach further. 22 In the case where Z(t) is stochastic, Phillips requires rather extended differentiability conditions on Z (t). Now, in general, we know that though diffusion processes have continuous sample paths, they are nowhere differentiable. So, the regularity requirements are quite steep. , Phillips point out the superiority vanishes when the regularity requirements are not met. Further, as can be seen from Appendix 6, due to presence of r* in the variance element, the resultant equation corresponding to (4. 13) is rather involved. Some attempt was made to estimate the parameters using the Phillips (and even the relatively simpler Sargan approximation), but non linear methods to estimate parameters from the log likelihood functions did not result in much success. 44 CHAPTER 5: COMPARISON OF THE DIFFERENT ESTIMATING METHODS 5.1 The Method of Comparison In this chapter, a limited attempt is made to compare the relative merits of the different approaches to estimating the parameters of the interest rate process outlined in the last chapter: a) The Transition Probability Density Method (TRP) b) The Steady State Probability Density Method (SS) c) The Simple Linearization Method (SL) The method adopted is to generate a large sequence of discrete realiztions (all 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 distribution of the parameters estimated by the different methods, using varyinq sample sizes. Data for the simulations was generated using ='/x because this is the one case where the transition probability density is known exactly, and quite tractable. The rest of the parameters were chosen by applying the TRP method corresponding to o( = '/a. on actual weekly interest 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 relative merits of the estimating methods may be a function of the parameter values. Since an extensive comparison of the Monte Carlo results was not done, (mainly due to the large computing cost involved) we 45 confined our attention to the neighbourhood of the parameter values of interest to us. The aim of the Honte Carlo simulations is to investigate (for a particular 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 in small samples? 2) Do they appear to be asymptotically unbiased even thouqh they may be biased in small samples? 3) What is the relative efficiency of the different 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? Specifically, is there any improvement in usinq 365 daily observations rather than 52 weekly points? 5.2 Generating an "exact" Sequence for the Square Boot Process Having chosen the parameter set , the first step is to qenerate synthetically, a discrete realization that is exact. This is very important as we should be able to assert that any observed bias in the parameters estimated, is a result purely of the method of estimation. The transition probability density corresponding to the A = J/j. case (the square root process) is qiven by (Feller [23 ]): 46 •o j (5.1) JVV\fo where w = exp(mt) and 1^ (.) is the modified Bessel function of order k, and is defined by One way to generate r , given 0 and rfc_( , is to generate a uniform (rectangular) random variable p on [0,1] and then set r_j. = C-» (p) where C is the cumulative probability density function corresponding to F(.) in equation (5.1). If C could be inverted, there would be no problem. However for the special structure of [5.1], Boyle [8] has developed a solution using a different approach. Substituting (S-0 where <$ = Im^/v1 If (S~1)=:«J; n=2&, and n is integral, then (5.3) where l( *s *^e aon-central chi-squared density with n deqrees of freedom, and "X is the non-centrality parameter. Now, we can easily choose our parameter set $ such that 2$ is integral, without much loss of generality (since the value of 8 from the actual interest data was large). Generating a non- central chi-squared30 random variable is quite straightforward. (Fishman £27] has detailed instructions on the generation of stochastic variates corresponding to a wide variety of probability distributions). This method was adopted using the parameter values: jX r 0. 09517 '/o/iwk S = -\%\5_. and a sequence of weekly interest rates (100,000 weeks lonq) was generated,, 30 One way to generate a non-central chi-square random variable (Y) with (n*1) degrees of freedom, would be: Y = Z2 • £x? where the are N{0,1) and Z is N(A#1) r ^ being the non- centrality parameter of the chi-square. This requires the qeneration of (n+1) Gaussian random variates. Another approach is based on the equivalence of the chi-square and Gamma distributions. Osing this approach Y = Z2 -2 £ loq(0. ) Ul where Y and Z are as before, but the U,; • s are uniform(rectanqular) on (0,1). This requires only (n/21) random variate qenerations for each chi-square variate. 48 Results of Monte Carlo Simulations for the To start with, we want to compare all three methods, and since we do not have a solution to the FP equation for arbitrary <7x, we have to assume cV is known and equal to Yi . When we assume °^=//2 * we know the transition probability density, and thus can compare all three estimating methods, based on the properties of the estimated parameters. Four sample sizes were used31: n = 100, 250, 500 and 945. For the n = 100 and n = 250 cases, 200 simulations each were performed, ie. , 200 sets of (m,yW-,cr) were estimated. For the n = 500 and n = 945 cases, 100 simulations each were performed., The details of the estimation procedure for the parameters when d\ is 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 synthetically generated weekly data on interest rates, successive blocks of 100 points were taken., Using each block of 100 data points, one set of parameters (m,/^, 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 distribution of the parameter estimate) is called the Monte Carlo standard deviation, and in the reported simulation results is called SD . If the Monte Carlo distribution of the estimated 31 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 interest 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 all the information about the distribution. To cover the possibility that the Monte Carlo distribution 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 theory32. In the summary results reported, SD^ refers to the mean of the asymptotic standard deviation computed for each trial. In some cases the mean SD^, is very high due to a few extreme values33, and so the median was reported instead, as an alternate representation of location. Finally, the Steady State Density Method cannot identify the paramters m and (T2 separately - only 2 a composite (2m/ 32 If L=log of the joint likelihood function (corresponding to a given set of data points), then the matrix of second partial derivatives of L with respect to the paramters, at the maximum of L, may be called the Hessian Matrix. The inverse of the Hessian matrix with signs reversed is an estimate of the variance-covariance matrix of the estimated parameters, based on asymptotic theory (see Billigsley [3]). The standard deviations are the square roots of the diagonal elements of the variance- covariance matrix. 33 Extreme values do not necessarily imply that these are nonrepresentative - the Monte Carlo method gives a representation of the true distribution.. However, in the TRP method, nonlinear optimiztion routines had to be used to find the parameter set that maximizes the likelihood function. In such routines, convergence is assumed to have been! attained when the relative change in the parameter values between successive iterations is less than a specified accuracy level. If the likelihood function is very peaked, then its second derivative can change a lot around the optimum point. This could lead to extreme values of SD- . case for the other two methods, and the summary statistics of its distribution are also tabulated. Tables II through V present summary statistics on the distribution of the estimated paramters. From the tabulated results, the following broad conclusions can be drawn: 1) There is little or no difference across the three methods in the estimated means of the parameter distributions, 2) The dispersion of the parameter distribution as measured by SDmc (which could be treated as a good proxy for the asymptotic standard deviation) is almost identical across the three methods. However, if SD^ is evaluated as a measure of the asymptotic standard deviation, there is a fair amount of difference across the three methods. The SL method grossly overestimates the asymptotic variance, (SD- is much larger than SDW ) whereas the SS method grossly underestimates it. The TBP method appears to perform rather well - in fact the median SD;, value is quite close to SD,„C for sample sizes greater than 500. 3) The parameters jx and small samples - at least for the number of simulations performed. However, the parameter m (or 2mA-2 in the SS method) is biased in small samples. It is overestimated by all methods, and the extent of bias is 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 Linearization 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 Trials 200 200 ,100 100 Transition Mean 0.06205 0.02773 0.01577 0.01219 Probability 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 Trials 200 199 100 100 * The mean is high, but the median was 0.00586 and 90% ile was 0. 01797. See footnote in text on page -. For a descriptio1 n of SD and SD- see text page mc i cn 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 Linearization 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 Trials 200 200 100 100 Transition Mean 0.11303 0.09801 0.10410 0.09370 Probability 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 Trials 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 Trials 200 199 100 100 * The SDi figures are not the means but medians. The mean SDi was very high due to a few exceptionally high values. The mean of SDi ranged from 5.305x10^ for n=100 to 163.51 for n=500, and 0.568 for n=945. The indication is that, even the SL method, SD^^ can have extreme values. cn TABLE IV ESTIMATE OF O2 BY DIFFERENT METHODS FOR g = h (KNOW) CASE TRUE VALUE 0 = 0.78427 x 10-4 (All figures in theTabl e have been Multiplied by a factor of 104) IffiTHOD n=100 n=250 n=500 n=945 Simple Linearization 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 - - ,.. — _ Trials 200 200 100 100 Transition. Mean 0.82021 0.81073 0.80820 0.80324 Probability 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 Trials 200 199 100 100 TABLE V ESTIMATE OF 2m/o2 BY DIFFERENT METHODS FOR a = h (KNOWN) CASE -- TRUE VALUE 2m/a2 = 194.389 METHOD n=100 n=250 n=500 n=945 Simple Linearization 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 Trials 200 200 100 100 Transition Mean 1494.77 679.22- 390.06 303.83 Probability 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 Trials 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 Trials 200 199 100 100 55 The consistent overestication of m (or 2m/ Novikov [52], He do find that as the sample size increases, all methods show reduced bias. Based on this, it 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^ is the estimate of m using a sample of size n, m~ represents its true value; c and d are constants. Using the results for n=100,250,500 and 94 5, the value of d that fits** the bias structure proposed above was estimated as 1 Based on these results, the sample sizes required to reduce the bias on the estimate of m to 10% is 4450, and to 111s 36090. To assume that the parameters of the interest rate process are constant over such larqe time periods, would be unreasonable. The natural question to ask therefore would be; how important is it to get an accurate estimate of m? For our 3 * For the 4 values of n, we have { £h - ro) from the Monte Carlo results (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. If the values of c did not exhibit a trend from n=100 to n=945, it was assumed that c was beinq observed with a random error. This fittinq approach was tried on the estimated m values by S.L. and TRP methods. d=1.1 appears to qive the best fit, and the corresponding value of c is approximately 8.0. purposes, the deciding criterion must be the error caused in bond valuation, for a given error in m. This is investigated in a subsequent section of this chapter. accepting the fact that m will be overestimated, there is an intuitive reason that could be used to explain this occurrence. Consider the diagram below, which is supposed to represent one realization of the interest process. ])o>ij It may be recalled that m represents the speed of reversion to the mean of the process. Thus, the higher m is, the less likely is the process to "stray" away from its mean. In the diagram above, let the 4 segments (represented by data 1 through 4) refer to subperiods of the total sample. If we estimate the parameters using one of the methods proposed in the last chapter, we might expect JU. in each case to be estimated as shown by the broken lines. In sub period 1 (Data 1), the process is seen as moving upwards and then somewhat stabilizing. Thus y~\ overestimates jUo . m is also overestimated, as the process 57 appears to be moving rapidly towards the perceived mean (/*i). In subperiod 2, the interest rate process remains more or less constant around a single level, jtf.7, is obviously perceived as the process mean. Here again, m will 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 in subperiod 3, is exactly as that proposed for subperiod 1 : the mean being perceived is jXi, and the process is rapidly being pulled toward it due to a high value of m. Finally, in subperiod 4, the process mean is probably perceived at ju^ , but here m will not be as highly overestimated as in the previous three subperiods. Since the process appears to wander a bit to either side of the perceived mean, a lower value of m (than in previous cases) would be estimated. From the above, we see that yu-is sometimes overestimated, and at other times underestimated. On average, its estimate might be expected to be unbiased. However, in almost every situation, m could be over-estimated. If now, the complete data were employed, it is 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 spells, which indicates a weaker force pulling towards the mean - m would be estimated nearer its true value., Before we present further results on the simulations, a minor methodological point needs to be clarified. The method employed for the simulations was to take successive blocks of observations from the long sequence that had been synthetically generated. One objection to this approach could be that 58 successive trials were not strictly independent. To counter this objection, for the n=945 case (which happens to be the one of primary interest to us, as that is the length of our actual sample), 100 "independent" samples of size 945 each were generated. The starting point for each of these 100 samples was randomly chosen from the stationary distribution (a reasonable approach), which in this case is a gamma distribution. The Steady State method and SL method were compared35 for the "dependent" and "independent" samples case, and the results are presented in Table VI. The conclusion appears to be that the use of the "dependent" samples does not materially alter 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 total observations) leads to any improvement. For this purpose, "daily" observations were generated for the same parameter values. To compare, parameters were estimated using 700 "daily" observations, and the results compared with the equivalent results corresponding to weekly observations. The results are presented in Table VII. Comparison among the 3 methods shows that there is no perceptible improvement in the mean of the estimated parameter distributions, but (as expected) the dispersion reduces by using "daily" observations. Thus it appears that the increased effort of collecting daily data, pays off by lower variances on the 35 The TEP method was not investigated, as it was computationally expensive. Since the objective is only to get an idea of the effect, it was felt 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) 2 METHOD 2m/a 2 4 (194.389) M (0.09517) m (0.007162) a (0. 7843xl0~ ) DEPNDT INDEP DEPNDT INDEP DEPNDT INDEP DEPNDT INDEP Simple Mean 305.90, 324.56 0.09371 0.09404 0.01211 0.01267 0.79443 0.78438 Linearization 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 _ Trials 100 100 100 100 100' • 100 100 100 Steady State 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 Trials 100 100 100 100 P inn I!Sr,irepreSentS the/epresentS the results of using a sequence of blocks (n=94S) of data points from the 1UU,000 long sequence of synthetic data generated for the Monte Carlo simulations. "INDEP" represents results of using "independent" samples (see text, page for details) . on TABLE VII COMPARISON OF RESULTS OF ESTIMATION USING WEEKLY & DAILY DATA (a = h KNOWN) (For Weekly Results n<=100, and For Dally Results n=700) 111 METHOD 2mlO1 V 0 (True value:194.389) (True value:0.09517) (True value: 0.7843x10-4) (True value:0.007162) WEEKLY DAILY WEEKLY DAILY WEEKLY DAILY WEEKLY DAILY 0.80031 0.78414 0.05037 0.05884 Simple Mean 1276.42 1547.22 0.10178 0.06803 0.05945 0.73697 0.65251 0.01088 0.01328 Linearization 10% 276.04 319.51 0.06192 0.77073 0.04206 0.04891 Median(50%) 1097.32 1235.32 0.09012 0.08919 0.79943 Method 0.91753 0.09479 0.11053 90% 2416.77 3129.81 0.12695 0.12481 0.87289 0.11073 0.03979 0.04459 SDmc 1040.43 1225.81 0.07533 0.23261 0.04648 6.32839 2.58618 SDi 2.21445 0.20554 - 20-0 100 200 Trials 100 200 100 200 0.82021 0.05111 0.06205 Mean 1281.26 1494.77 0.10397 0.11303 0.80413 Transition. 0.01098 0.01348 10% 274.95 331.29 0.06194 0.06194 0.74359 0.66768 Probability 0.04237 0.05002 Median(50%) 1100.19 1216.86 0.09013 0.08984 0.80324 0.81653 Density 0.09523 0.11646 90% 2396.25 2963.03 0.12699 0.12807 0.87379 0.97041 Method 0.03963 0.04874 SDmc 1008.27 1121.05 0.09201 0.21429 0.04565 1.13434 0.01007 0.01548 SDi* 0.00589 0.00534 0.04723 0.10974 100 200 Trials 100 200 100 200 100 200 0.09266 Steady State Mean 1278.87 1482.59 0.09088 0.06562 Density 10% 298.54 449.55 0.06706 Method Median(50%) 1132.24 1256.65 0.08987 0.09007 90% 2240.06 2926.93 0.11629 0.11926 SDmc 922.98 1067.60 0.01934 0.02193 SDi 68.41 209.80 0.00038 0.00095 Trials 100 200 100 200 * Reported figures represent medians of SDj^ and not the mean. 61 parameter estimates. However, our interest is in using these parameters for bond valuation. Therefore, the question that needs to be answered is whether this reduction in the dispersion of parameter extimates would translate to comparable reduction in dispersion of estimated bond values. This question is addressed later on in this section. However, one point needs to be noted when we attempt to collect daily data on the actual interest rate process - measurement errors will occur. They would be of the following types: 1) Normally no exact daily rate at which some specific transaction occured, would be available. Quoted rates are generally the mean of a bid and ask price, ie. not market prices. 2) Even if the daily rate were based on specific transactions, all transactions would not be exactly 24 hours apart ie., observations would not be equi-spaced as required to simplify our estimation process. In daily data, the relative magnitude of this error could be high., 3) Due to the presence of week-ends and holidays, the daily series of interest rates has more "holes" than a corresponding weekly series. Every time there is a holiday, as over a weekend, continuity is lost in a daily data series and we have a gap. It is obvious that such occurences are less likely in a weekly series. All the above factors would tend to diminish the value of a daily series. In Appendix 8 we outline a very brief 62 investigation of the impact of a specific form of measurement error. ,• Finally we look at the impact of the distribution of the parameter estimates on the valuation of pure discount bonds3*. This is crucial, as our primary interest in estimating the parameters is to use them to value bonds. For simplicity, we investigate the impact on the valuation of pure discount bonds. There is little reason to believe that the results on the valuation of other types of bonds should be any different, since a coupon bond, for example, may be thought of as a portfolio of discount bonds of varying maturity. Tables VIII to X present the "theoretical" sensitivity of pure discount bond values to errors in the paramter values. The expression "theoretical" sensitivity is used only to distinguish these results from those called "empirical" sensitivity that will be presented shortly. "Theoretical" sensitivity refers to changes in the value of bonds due to a certain fixed level of error in one parameter at a time, while "empirical" sensitivity refers to the distribution of bond values resulting from the estimated joint distribution of the parameters from the Monte Carlo experiments37., We can draw the following inferences from 3* The value of a discount bond was computed using Ingersoll,s [39] solution. 37 The procedure adopted is 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., THEORETICAL SENSITIVITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN m ERROR IN m 0% 10% 25% 50% 100% CURRENT TIME TO BOND BOND % BOND % BOND % BOND ERROR INTEREST MATURITY PRICE PRICE ERROR PRICE ERROR PRICE ERROR PRICE IN YEARS 1 97.13 97.09 -0.0378 97.04 -0.0928 96.96 -0.1797 96.80 -0.3376 3 90.01 89.82 -0.2024 89.57 -0.4796 89.21 -0.8808 88.65 -1.5046 5 82.38 82.09 -0.3506 81.71 -0.8094 81.20 -1.4304 80.48 -2.3003 7 74.99 74.65 -0.4504 74.22 -1.0202 73.67 -1.7568 72.94 -2.7264 10 64.86 64.51 -0.5332 64.09 -1.1865 63.56 -1.9985 62.90 -3.0207 1 95.17 95.17 -0.0000 95.17 -0.0001 95.17 -0.0002 95.17 -0.0003 3 86.22 86.33 -0.0014 86.22 -0.0034 86.22 -0.0061 86.21 -0.0100 5 78.13 78.12 -0.0060 78.12 -0.0135 78.11 -0.0233 78.10 -0.0360 7 70.80 70.79 -0.0132 70.78 -0.0294 70.77 -0.0492 70.75 -0.0730 10 61.09 61.07 -0.0271 61.05 -0.0590 61.03 -0.0962 61.00 -0.1381 1 91.38 91.45 0.0756 91.55 0.1856 91.71 0.3599 92.00 0.6777 3 79.12 79.44 0.4017 79.88 0.9560 80.52 1.7666 81.53 3.0477 r=2u 5 70.27 70.76 0.6869 71.40 1.5974 72.28 2.8513 73.54 4.6514 7 63.12 63.67 0.8669 64.37 1.9820 65.30 3.4555 66.56 5.4529 10 54.19 54.73 0.9928 55.40 2.2346 56.26 3.8197 57.38 5.8867 (Tl CO. TABLE IX THEORETICAL SENSITIVITY OF PURE DISCOUNT- BOND PRICES TO ERRORS IN y ERROR IN VI -25% -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 -0.04 96.92 -0.22 3 91.42 1.57 90.29 0.31 90.01 89.73 -0.31 88.61 -1.55 r=y/2 5- 85.33 3.58 82.96 0.71 82.38 81.80 -0.70 79.53 -3.46 7 79.43 5.93 75.86 1.16 74.99 74.13 -1.15 70.79 -5.60 10 71.20 9.78 66.08 1.88 64.86 63.66 -1.85 59.08 -8.91 1 95.38 0.22 95.21 0.04 95.17 95.13 -0.04 94.96 -0.22 3 87.57 1.57 86.49 0.31 86.22 85.95 -0.31 84.89 -1.55 r=y 5 80.93 3.58 78.68 0.71 78.13 77.58 -0.70 75.42 -3.46 7 75.00 5.93 71.62 1.16 70.80 69.99 -1.15 66.84 -5.60 10 67.06 9.78 62.24 1.88 61.09 59.96 -1.85 55.65 -8.91 1 91.58 0.22 91.42 0.04 91.38 91.34 -0.04 91.18 -0.22 3 80.36 1.57 79.37 0.31 79.12 78.88 -0.31 77.90 -1.55 r=2y 5 72.79 3.58 70.77 0.71 70.27 69.78 -0.70 67.84 -3.46 7 66.87 5.93 63.85 1.16 63.12 62.40 -1.15 59.59 -5.60 10 59.49 9.78 55.21 1.88 54.19 53.19 -1.85 49.37 -8.91 CTl TABLE X THEORETICAL SENSITIVITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN a- ERROR INO -25% -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 97.13 -0.0002 97.13 -0.0000 97.13 97.13 0.0000 97.13 0.0002 3 90.00 -0.0033 90.01 -0.0007 90.01 90.01 0.0007 90.01 0.0033 r=y/2 5 82.37 -0.0109 82.38 -0.0022 82.38 82.38 0.0022 82.39 0.0109 7 74.97 -0.0220 74.98 -0.0044 74.99 74.99 0.0044 75.00 0.0219 10 64.83 -0.0422 64.85 -0.0084 64.86 64.86 0.0084 64.89 0.0420 1 95.17 -0.0003 95.17 -0.0001 95.17 95.17 0.0001 95.17 0.0003 0.0052 3 86.22 -0.0052 86.22 -0.0010 86.22 86.22 0.0010 86.23 r=p. 5 78.12 -0.0154 78.13 -0.0031 78.13 78.13 0.0031 78.14 0.0153 0.0284 7 70.78 -0.0285 70.80 -0.0057 70.80 70.81 0.0057 70.82 10 61.06 -0.0506 61.08 -0.0101 61.09 61.09 0.0101 61.12 0.0504 1 91.38 -0.0006 91.38 -0.0001 91.38 91.38 0.0001 91.38 0.0006 3 79.11 -0.0090 79.12 -0.0018 79.12 79.12 0.0018 79.13 0.0090 r=2p 0.0241 5 70.26 -0.0242 70.27 -0.0048 70.27 70.28 0.0048 70.29 7 63.10 -0.0415 63.12 -0.0083 63.12 63.13 0.0083 63.15 0.0414 0.0672 10 54.16 -0.0674 54.19 -.0.0135 54.19 54.20 0.0135 54.23 the results: 1) Bond values are sensitive to jx , the mean level of the interest rate. The sensitivity to m is much less errors in 2) Errors in /A- cause errors in bond values which increase as the time to maturity of the bond increases, whereas the current level of the interest rate has no effect on the amount of error. For example, overestimating jx by 5% causes the 10 year discount bond to be undervalued by 1.85% irrespective of whether the current level of interest rate is at or 2^. 3) Errors in m cause errors in bond values which increase with the maturity of the bond. Furthermore, the error in the bond value depends on the current level of the interest rate - more accurately, on its deviation from the mean interest levelJJ» . He now look at the sensitivity of discount bond values to the distribution of the estimated parameters. These results are presented in Tables XI to XIII. The results are exactly as expected: the distribution of bond values is almost identical, using parameters estimated by any of the three methods38. However, there are interesting results when we compare the distribution of bond prices using • "weekly**, versus "daily" data. Surprisingly, (as can be seen from Table XI?) even though the standard deviation (SDWC ) of the parameters was always reduced 50% or more using "daily" data (see Table VII), similar 38 For the SS method, rrz was taken from the SL method. Dsing this cr-2; m was computed from the parameter {2m/ SENSITIVITY OF PURE DISCOUNT BpND PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS" (Current value of interest rate = >ju) MATURITY(YRS) 1 3 5 7 10 TRUE VALUE 97.13 90.01 82.38 74.99 64.86 Simple Mean 96.972 89.417 81.638 74.290 64.364 Linearization 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 Transition Mean 96.969 89.408 81.627 74.279 64.355 Probability 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 Interest rate parameters (m, u,a) have been estimated for the a=*5(knovra) case using 945 observations on the interest rate. 100 such simulations were performed, and distribution of bond prices represents the bond value corresponding to each of those parameter estimates. - True value of bond corresponds to the bond price corresponding to the • true underlying interest process parameters. TABLE XII SENSITIVITY OF PURE DISCOUNT BOND PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS (Current value of interest rate = u) MATURITY(YRS) ' 1 3 5 7 10 TRUE VALUE 95.17 86.22 78.13 70.80 61.09 Simple Mean 95.193 86.335 78.348 71.122 61.537 Linearization SDmc 0.160 0.898 1.708 2.426 3.266 Method 10% 95.005 85.174 76.036 67.765 56.951 Median 95.184 86.274 78.212 70.906 61.207 90% 95.373 87.410 80.451 74.088 65.700 Transition Mean. 95.193 86.335 78.348 71.122 61.537 Probability SDmc 0.161 0.900 1.711, 2.429 3.270 Density Method 10% 95.004 85.169 76.029 67.757 56.943 Median 95.184 86.275 78.213 70.906 61.207 90% 95.373 87.413 80.451 74.091 65.711 Steady State Mean 95.193 86.340 78.358 71.137 61.557 Density Method SDmc 0.150 0.842 1.601 2.275 3.065 10% 95.002 85.318 76.494 68.339 57.575 Median 95.191 86.324 78.329 71.092 61.485 90% 95.362 87.298 80.237 73.986 65.460 NOTE: Refer to comments on Table XI for more details. TABLE XIII SENSITIVITY OF PURE DISCOUNT BONDS PRICES TO DISTRIBUTION OF ESTIMATED INTEREST RATE PROCESS PARAMETERS (Current value of interest rate = 2y) MATURITY(YRS) 1 3 5 7 10 TRUE VALUE 91.38 79.12 70.27 63.12 54.19 Simple Mean 91.732 80.504 72.194 65.227 56.293 Linearization SDmc 0.402 1.604 2.440 3.007 3.585 Method 10% 91.283 78.565 69.115 61.277 51.532 Median 91.670 80.493 72.411 65.410 56.441 90% 92.278 82.550 75.173 68.962 60.776 Transition. Mean 91.738 80.520 72.213 65.245 56.310 Probability SDmc 0.407 1.615 2.449 3.013 3.587 Density Method 10% 91.289 78.553 69 .'091 61.262 51.518 Median 91.685 80.525 72.454 65.370 56.441 90% 92.293 82.596 75.235- 69.013 60.783 Steady State Mean 91.721 80.458 72.129 65.156 56.227 Density Method SDmc 0.402 1.591 2.395 2.919 3.435 10% 91.235 78.452 69.123 61.637 51.956 Median 91.658 80.494 72.214 65.439 56.657 90% 92.284 82.480 75.075 68.579 60.352 NOTE: Refer to comments on Table XI for more details. TABLE XIV COMPARISION OF BOND PRICE SENSITIVITY TO THE USE OF DAILY VS WEEKLY DATA* IN THE ESTIMATION OF INTEREST RATE PROCESS PARAMETERS(«=^s) T=l year T= 3 T=5 T=7 T=10 WEEKLY DAILY WEEKLY DAILY WEEKLY DAILY WEEKLY DAILY WEEKLY DAILY TRUE VALUE 97.13 90 .01 82. 38 74. 99 64.86 63.595 Mean 96.312 96.178 88.021 87.782 80.227 80.020 73.095 72.963 63.599 0.773 0.917 2.623 2.957 4.333 4.707 5.869 6.160 " 7.754 7.832 SD mc 74.323 73.950 64.624 65.108 53.087 53.796 10% 95.204 94.931 84.514 83.797 80.080 80.488 72.970 73.200 63.325 63.843 50% 96.336 96.345 87.952 88.087 91.388 91.682 85.901 86.286 80.742 81.110 73.424 73.978 90% 97.204 97.320 61.09 TRUE VALUE 95.17 86 .22 78.13 70.80 71.516 71.627 62.060 62.266 Mean 95.269 95.250 86.563 86.553 78.667 78.709 5.637 5.641 7.393 7.167 SD 0.617 0.743 2.402 2.639 4.125 4.288 52.437 io?c 94.463 94.290 82.886 83.139 72.666 73.086 63.680 64.053 52.240 78.967 71.396 71.918 611.801 62.529 50% 95.256 95.289 86.545 86.721 78.606 83.595 84.211 78.319 78.854 70.636 71.548 90% 96.042 96.149 89.538 89.830 TRUE VALUE 91.38 79.12 70.27 63.12 54.19 69.202 59.389 59.957 Mean 93.208 93.429 83.779 84.198 75.768 76.264 68.661 1.235 1.215 3.676 3.509 5.638 5.273 7.148 6.612 8.647 7.993 SDmc 79.529 79.161 69.113 68.345 59.861 59.475 48.238 48.305 r=2y 10% 91.538 91.844 84.566 76.422 76.981 69.314 70.116 59.892 60.806 50% 93.315 93.447 83.912 76.779 77.042 68.938 69.681 90% 94.639 94.966 88.096 88.228 81.769 82.272 * The Input parameter estimates were the results of estimation using the Transition Probability Density Method: n=100 for weekly estimates and n=700 for daily estimates. O 71 decreases in dispersion of bond value distributions do not appear to result - the reduction in the bond value variance is truly marginal. The explanation for this seemingly anomalous behaviour lies in the correlation between the parameters - particularly m and^. Based on the theoretical sensitivity of bond prices we know that overestimating y/- or m underestimates the bond value. If now the estimates of m and ^ are negatively correlated, then, to some extent, they have offsetting effects on bond valuation. Thus a negative correlation between m and^x. could explain this result39. (The correlation between the parameter estimates is addressed toward the end of this chapter)., This is more evidence in favour of using weekly; rather than daily interest rate data. 5.4 Results of Monte Carlo Simulations for the So far we have only compared the different estimation methods under the assumption that the value of cA were known. For the joint estimation of all the parameters (m »/A. , cr , crt ) , we can only compare the SL method and the SS method, as the transition probability density corresponding to general <* values is not known. The details of parameter estimation in the SL model have been set out in Appendix 2 and for the SS density method, in Appendix 10. 39 The effect may be understood more intuitively by considering the return on a portfolio of 2 negatively correlated securities. Increasing the variance on the returns of the individual securities need not cause proportional increases in the variance on the return of the portfolio. 72 For the SL method, the n=500 and 945 cases were estimated (100 trials each), but for 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 sizes. The summary statistics for the estimated parameter distributions are presented in Table XV. The following remarks about the results are in order: 1) Comparing the estimates of m and y* from the S.L. method, in the c* unknown case with those in the cA= yv(known) case, we find that the resulting parameter distributions are almost identical. This indicates something about the interrelationship between the estimated parameters. The correlation between the parameters is discussed in the next section, but this result points to the possibility that m and are uncorrelated with cX . 2) The estimate of tr2 does not appear biased but the dispersion seems large, particularly when compared with the oV = y-2_ (known) case. The reason for this is the close relationship between variance of the process is rzfU and, understandably, when cA is free to adopt a range of values, the value of cr2 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 Linearization o Median(50%) 0.09007 0.48476 349.92 0.01392 0.7559 \ J 90% 0.11521 0.71715 1545.86 0.02491 2.0897 LnO Method 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 ^jm- 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 figure reported is the median of the SDi from each trial not the mean. ** This is the median - the mean SDiwa s 625.458 in this case. - The figures in the a2 column have been multiplied by 10^ 74 5) Comparing the composite parameter (2ra/ method appears to give estimates having a lower bias and dispersion. However, the median (which is also a measure of location), of the SS estimate is very reasonable. It 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, in the SL case, n-945 appears to satisfy the asymptotic sample size criteria, in that SD^ for all parameters is very close to SDnc. For smaller sample sizes (see n=500), SD^ is an under estimate of the asymptotic standard deviation. For the sake of completeness, we present in Table XVI a comparison of the distribution of the estimated parameters using "daily" versus "weekly" data. For this case, only the SL method was used., The only parameter of interest here is o( . As with the other parameters, the improvement is only with respect to the dispersion of the estimated parameter distribution. We shall soon see whether this improvement in accuracy makes any significant difference to the bond value. Before we conclude this section, we present some results on the sensitivity of the pure discount bond value to variations in *° The SS method is based on the assumption that the stationary density is not the trivial P(r) .= 0, which obtains when either singular boundary is absorbing for some parameter values. Whenever the non-linear search procedure (to identify the maximum of the joint likelihood function) takes on parameter values which correspond to an absorbing barrier at either singular boundary, the SS method breaks down., If the range of the parameter space where we get absorbing barriers were known, a constrained maximization could be done. This however is not the case. The breakdown of the SS method in 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 a 2m/o (0.09517) (0.50) (194.389) (0.0077617) (0.78427xl0-4) Simple Mean 0.09729 0.48586 458.80 0.01534 0.7903 Linearization o 10% 0.07797 0.38203 >-( o 118.91 0.00515 0.4487 Method >-i n Median(50%) 0.09441 0.48552 330.48 0.01315 0.7399 < II 90% 0.11430 0.57420 969.65 1.1368 a a 0.02946 SDmc 0.07084 405.28 0.00956 0.2605 SDi 0.00907 0.06507 0.00Z25 0.2200 Simple Mean 0.08876 0.49385 639.87 0.01556 1.2146 Linearization 10% 0.07138 0.22481 88.97 0.00603 0.1925 Method Median(50%) 0.09007 0.48476 349.92 ^ o 0.01392 0.7559 90% 0.11521 0.71715 1545.86 0.02491 2.0897 w II SDmc 0.04530 0.19318 979.81 0.01020 1.9348 SDi 0.00929 0.16915 0.00728 0.6172 The figures in the a2 column have been multiplied by 104, cn o(. In this context, only the "theoretical" sensitivity results are presented in Tables XVII and XVIII. It was felt that no additional information could be gained by presenting the "empirical" sensitivity. In Table XVII we present the effect on discount bond values of varying cA about the value Vi. , with the other parameters kept fixed at their 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 direction of effect on bond values is caused by a decrease in (assuming that there is no bias in identifying the total variance)., Let us represent by corresponding to an o\ value of yz , and *l It may be noticed that the 0% error bond price in Table XVII and XVIII is slightly different from that in Tables VIII, IX and X. This is because, the values in that column in Tables XVII and XVIII have been computed using a finite differencing method to solve the bond equation. This was done, as what we want to present is the effect of variations in THEORETICAL SENSITIVITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN a* (02 HAS NOT BEEN 'CORRECTED' TO REFLECT THE ERROR IN a) ERROR IN a -25% -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 96.96 0.0053 96.95 0.0004 96.95 96.95 -0.0003 96.95 -0.0008 3 89.40 0.0862 89.33 0.0074 89.32 89.32 -0.0051 89.31 -0.0138 r=y/2 5 81.66 0.2572 81.47 0.0227 81.45 81.44 -0.0158 81.42 -0.0434 7 74.37 0.4812 74.04 0.0431 74.01 73.99 -0.0303 73.95 -0.0837 10 64.50 0.8634 64.00 0.0785 63.95 63.92 -0.0554 63.85 -0.1541 1 95.18 0.0063 95.18 0.0006 95.18 95.17 -0.0004 95.17 -0.0011 3 86.31 0.0963 .86.23 0.0087 86.23 86.22 -0.0062 86.21 -0.0173 5 78.35 0.2788 78.16 0.0254 78.14 78.12 -0.0180 78.10 -0.0507 7 71.18 0.5125 70.85 0.0470 70.81 70.79 -0.0334 70.75 -0.0941 10 61.66 0.9040 61.15 0.0835 61.10 61.07 -0.0593 61.00 -0.1672 1 91.23 0.0117 .91.22 0.0011 91.22 91.22 -0.0008 91.21 -0.0022 3 78.56 0.1685 78.44 0.0157 78.43 78.42 -0.0113 78.41 -0.0322 5 0.0424 69.31 69.29 -0.0305 69.25 -0.0872 r=2y 69.62 0.4502 69.34 7 62.57 0.7656 62.14 0.0724 62.09 62.06 -0.0521 62.00 -0.1490 10 53.88 1.2288 53.29 0.1164 53.22 53.18 -0.0836 53.10 -0.2389 The other parameters of the Interest rate process assume their true values. See footnote XXI . TABLE XVII1 -HFORFTICAL SENSITIVITY PHBF. DISCOUNT BOND PRICES TO ERRORS IN &_ ERROR IN a +25% 0% +5% -25% -5% BOND % BOND % BOND BOND ' CURRENT TIME TO BOND % PRICE ERROR PRICE ERROR PRICE ERROR PRICE INTEREST MATURITY PRICE ERROR IN YEARS -0.0001 96.95 -0.0006 0.0002 96.95 96.95 1 96.59 0.0011 96.95 89.32 -0.0090 89.32 89.32 -0.0023 0.0169 89.33 0.0026 3 89.34 81.45 -0.0067 81.43 -0.0265 0.0490 81.46 0.0076 81.45 r = u/2 5 81.49 74.00 -0.0124 73.98 -0.0493 74.02 0.0140 74.01 7 74.08 0.0901 -0.0220 63.89 -0.0879 0.0248 63.95 63.94 10 64.05 0.1591 63.97 -0.0002 95.17 -0.0006 95.18 0.0002 95.18 95.18 1 95.18 0.0011 -0.0023 86.22 -0.0093 86.23 0.0026 86.23 86.23 3 86.24 0.0166 78.12 -0.0273 78.14 78.13 -0.0068 78.17 0.0486 78.14 0.0076 -0.0505 r =u 5 70.81 -0.0126 70.78 0.0898 70.82 0.0141 70.81 ' 7 70.88 -0.0223 61.05 -0.0897 61.12 0.0250 61.10 61.09 10 61.20 0.1592 -0.0012 91.22 -0.0003 91.22 91.22 0.0003 91.22 1 91.22 0.0019 -0.0040 78.42 -0.0167 0.0044 78.43 78.43 78.45 0.0269 78.44 69.28 -0.0447 3 69.31 69.30 -0.0108 0.0724 69.32 0.0119 -0.0762 r =2u 5 69.36 62.08 -0.0184 62.04 62.10 0.0203 62.09 7 62.17 0.1245 -0.0298 53.16 -0.1226 0.0330 53.22 53.21 10 53.33 0.2031 53.24 * The other parameters of the Interest rate process assume their true values. ** See footnote XXL Co 79 Thus varied. Clearly, average r is expected to remain around . Table XVIII presents the sensitivity of discount bond values to variation in , where this "correction" has reduced the effect of a variation in o( on discount bond values. However, what is more important is the fact that the net effect is small. 5•5 The Relation Between the Interest Rate Process Parameters Finally, before concluding this chapter, we take a brief look at the relationship between the parameters (m, There are two closely interconnected points from which we may view this relationship; a) What is the expected correlation between the estimated values of these parameters, given a data sample? b) In what interconnected way do these parameters alter the characteristics of the interest rate process dynamics? One way to try to answer the first question would be to calculate the correlation matrix between the parameters estimated during the simulation. Since the SL method for the n=945 case displayed close to asymptotic behaviour, the correlation*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 , rv\ /A- (T^ jX -0.0 207 cr2 0.2081 0.0375 0.1725 0.0875 0.9339 He can see that cr*" and o\ are almost perfectly correlated (which is as expected), but apart from that any other correlations appear to be quite small. Another approximate (and quite ad hoc) method of estimating the correlation matrix between the parameters is set out in Appendix 9, based on that method, the correlation matrix is •u jiK cr -0.1582 0.0 0.0 0.0 0.0 0.9877 There is agreement between the two estimates of the asymptotic correlation matrices in broad qualitative terms. The second estimate (based on the approach presented in Appendix 9) implies that the parameters in the variance and drift terms of the diffusion equation are totally independent of each other. This would explain the earlier observation, namely, the similarity of the distributions of estimated values of m and between the cL^'/z. known case and the cA unknown case, in the S.L. method. The two important characteristics that were anticipated are borne out in both cases, ie. 81 1) fa and • JJ^ are negatively correlated. This was anticipated, based on their combined effect on bond values. 2) (T2 and o\ are very highly positively correlated. Further insights into the nature of the inter-relationships among the parameters can be gained by looking at the way in which each of them affects the interest rate process dynamics. , For a diffusion process, all information about the process dynamics is contained in the transition probability density function. To investigate how it is altered by changing the parameters, we consider the following parameter values: Parameters Set 1 Set 2 f (=2»V 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 virtually identical values for the log of the joint likelihood function, using the SS method. This situation arose while performing routine preliminary trials with the SS method for the °<7\ unknown case. It is well known that nonlinear optimization routines provide no guarantee that convergence to a optimum will occur. Further, even if convergence is obtained, one is never sure whether the point is a local or a global optimum.. To investigate the behaviour of the particular functional form of the likelihood function on some data samples, (chosen from the generated sequence) different available nonlinear optimization 82 methods were applied to see whether (using different algorithms), a) convergence was always to the same point in the parameter space, irrespective of the starting parameter values, and b) whether the speed of convergence differed across different algorithms. It was found that the guasi-Newton method (the Fletcher algorithm) was the quickest by far, and in qeneral, the point to which converqence was obtained, appeared to be the "global" optimum..,, We have a case where the stationary probability density corresponding to very different parameter values is almost identical. This was further verified by plotting the stationary distribution corresponding to these parameter values, and the density functions were seen to virtually coincide. This implies that, given a data sample, the SS method may not be able to identify an unique parameter set that fits it - it may identify one or more equivalent points in the parameter space*3. The more relevent question, however, is whether the transition *3 An attempt was made to find out whether, correspondinq to this data sample , the two "optimum" parameter sets represented two independent "peaks". To investiqate this, a close mesh qrid (50x50) was placed over the (, The transition probability density function is the solution to the Fokker-Planck equation, which we have not been able to solve for general values of ck . Thus a finite differencing method was employed to solve the FP equation. The objective of the exercise was to try and see whether the transition probability density functions corresponding to the above two parameter sets could be made almost identical**. 9e also require a statistic to measure the "closeness" to each other of the two transition probability density functions. The "matchinq" criterion was to minimize the area of non-overlap, between the computed transition density functions*5. It was found that the area of non-overlap between the transition probability 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 arbitrarily split into reasonable values of m, and 6 sets could be brought down to about 7%, for r0 = f*-* . However when r0*f^at these "matched" parameter values, the area of non- overlap increases greatly. This is as expected - what is more informative is the extent to which the shape of the transition probability density is changed by a proportional change in each parameter. This is pictorially represented in Figures 2 and 3. This is an indication that the transition probability density function is not very similar for different parameter values - given a data sample, we could expect an unique parameter set to maximize the likelihood function. As expected, both cA and density function, a{ more so than changes the variance element multiplicatively, whereas o{ chanqes the exponent, which has a qreater effect on the variance, particularly since r is always far from unity in numerical value. No pictures are presented for chanqes in the transition probability density corresponding to changes in /x , as this affects only the location. It can be seen that large changes in m, when.r=/^f produce hardly any change. However increases in m make the function slightly more peaked as m is the speed of reversion to the mean. When r *J^t changes in m shift the location because of the skewing effect of the mean The area of overlap is given by J^abs[ F (r K„,e)-F (r j-r^J ] dr, where Y{T\rep) represents the transition probability density function corresponding to parameter set 0 . it may be noted that the area under either transition density function adds up to 1.0. Thus the area of non-overlap indicates directly the fraction of total area under each curve for which the two functions do not match. ** The transition probability density is represented as F (rj. ,11r0 ,9) . Thus, it is a function of re and t as well as the parameter set 0 ={m, p-f o~, d\) • Here t was chosen equal to 1 week. 85 FIGURE. 2 Sensitivity of the Transition Probability Density Function to Change In a 'Sensitivity of the Transition Probability Density Function to Chance in o FIGURE 3 87 reversion property (which is similar to changing jx a very small amount).... To summarize, it appears that ^ and ck are the important parameters in determining the location and dispersion, respectively. m has a marginal effect oh both, whereas cr2 affects only dispersion. ,. 88 CHAPTER 6: THE IHTJRjgST PATE AND BOND PRICE DATA 6.1 The Short Term Riskless Interest Rate By definition, the short term (instantaneous) risk less, rate of return is the yield to maturity on a default free discount bond, maturing the next instant in time. In actual practice, such a security does not continuously exist (and is not available in any case). The bond valuation models developed in Chapter 2, and the estimation theory developed in the preceeding chapters, all require that we know something about this unobservable entity. A suitable proxy for the short term riskless rate of interest would be the yield to maturity on very short maturity Federal Government bonds, as they could be treated as totally default free with respect to principal payment on maturity. However, the only pure discount Federal Government bonds outstanding are Treasury Bills. Apart from quotations in secondary markets, these have a minimum maturity of 91 days, which brings us to two closely related matters, viz. (a) what time to maturity may be treated as "instantaneous" and (b) what should our frequency of measurement be? Treasury bills are not very actively traded in secondary markets in Canada., A few conjectures could be put forward to try and explain this. To start with , a widespread demand does not appear to have developed. Of the total Government of Canada Treasury bills outstanding over the last 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 all the other financial and non financial institutions and individuals (figures obtained from Neufeld [53]). Chartered banks have always been principal holders, as they are constantly in need of very secure short term investment opportunities. Since there are only five major chartered banks in Canada, the number of active participants is greatly reduced. Furthermore, Canadian banks are required by law to maintain secondary reserves at prescribed levels, which tends to reduce trading in short term government securities. In the-U.S., , however, the Treasury bill market is very active and deep due to the following factors: a) Banks do not have to maintain secondary reserves.= b) There are very many more commercial banks actively trading in the market (due to the unit banking system, as opposed to the branch banking system of Canada). c) The U.S. dollar is a major reserve currency as well as the denomination of a large portion of international trade. Thus, several foreign investors (both corporate and government) enter the short term U.S. dollar denominated bond market. These factors could explain the relative inactivity in secondary markets for Canadian Treasury bill. Given the present state, it is to be expected that transactions prices in secondary markets, would be difficult to obtain. No record of sale prices for Treasury bills in secondary markets, were available either with security dealers or from the Bank of Canada. From considerations of reliability of the data, (and keeping in mind that we require equispaced 90 observations) it was felt that treating the yield to maturity on the 91-day Treasury bills,on the date of issue, as a proxy for the short term interest rate was the best alternative. The distinct advantages of this choice are that (i) for all practical purposes the term structure over such short maturities as 91 days may be treated as virtually flat, so that the yield on the 91 day pure discount bond may be assumed equal to the instantaneous rate (ii) the yield to maturity is computed based on actual transaction prices (which could be treated as equilibrium prices), rather than Jbased on quotes. If we did want to use Treasury bill prices from secondary markets, there is no guarantee that we can consistently get yields computed on actual transaction prices. The effect of using the yield to maturity on a 91-day discount bond as a proxy for the short term interest rate is briefly investigated in Appendix 11. The error appears to be small., ' Having chosen the 91-day Treasury bill as our short term (instantaneously maturing) asset, the matter of frequency of observation is automatically settled. Treasury bills are issued weekly and the yields, based on average sale price, are reported in the Bank of Canada Review. Given the source of this information, the data are very reliable. Other proxies for the short term interest rate were considered, such as the interbank loan rate and the daily call money rate. There were several problems on account of which they had to be dropped from serious consideration: 1) There was no reliable 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 if no actual transactions took place. 3) Even if it were possible to get some data on the other rates, no series on them could be constructed going back almost 20 years*7 - the time when the first retractable/extendible was issued by the Government of Canada. 4) These rates have a lot of "noise" in them, which has little or nothing to do with changes in bond prices. For example, they are strongly influenced by the flow of very short term capital 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, Provincial and corporate (including the issues of the chartered banks) retractable and extendible bonds outstanding. For all the Federal bonds, weekly prices are reported in the Bank of Canada Review. Due to the large volume of each of these issues and their marketability, an active secondary market exists for them. The prices reported in the Bank of Canada Review are, more often than otherwise, average actual transaction prices, at midday *7 Bid and ask prices on daily call money rates were available going back about 18 months from the present.. The dealers do not keep them on record for 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 Provincial and corporate bonds, however, the issues are much smaller and very many more in number. The problems associated with putting together a data base on Provincial and corporate retractables/exteedibles may be summarized as follows: 1) There are very many issues outstanding but not widely traded, so that a continuous series of even bid and ask prices is not available. 2) Even when available, (quoted in the Financial Post) what is indicated are bid and ask prices (with large spreads). There is no guarantee that if transactions took place they would be between those prices; ie., the quotes do not always represent firm commitments to transact. 3) The available data on Provincial and corporate bonds are not ,,compatibleN with the data on the short term interest rate . The prices quoted in the Financial Post are Friday closing values, whereas the Bank of Canada Beview observations on the short term interest 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 strictly evaluate the performance of the model in valuing these bonds. 4) Whereas the Federal bonds are very actively traded, Provincial and corporate bonds are not. The assumption of continuous trading opportunity, upon which the model is based, is violated. The impact on bond prices seems nontrivial. This shows up when we compare yields on Federal and comparable 93 Provincial bonds, where default risk is of nearly the same level. The yield difference on some issues is as high as 0.5% {on an annualized basis). This is an indication that marketability of the bonds is an important determinant of their value. Therefore, the models developed in Chapter 2 would be inappropriate for valuing Provincial and corporate issues. 5) Corporate bonds have default risk, over and above interest rate risk. The theory developed in the existing literature for valuing such bonds is to treat them as functions of r, the value of the firm, and time to maturity. Putting together a data series for the value of a firm has several obvious problems. Since complete data on all Federal retractable/extendibles issued to date were available, it was decided to confine our attention to them alone - to the exclusion of the Provincial and corporate issues. Table XIX gives some details on all the retractable/extendibles forming our sample. It includes all such issues by the Government of Canada. Data have been collected for each bond starting within a week of the date of issue, and extending to the exchange or retraction date. In cases where data were available beyond the last exchange date, the indication is that the short bond was preferred to the long bond by the majority of the investors. For the purpose of this study, these bonds have been named B1, E1 through E19 - B for retractable and E for extendible. It may be noted that for H1, E2, E3 and E4, observations cease even before the option expiry date. The matter was investigated by the local representative of the Bank of Canada, and it appears that, (for some unknown TABLE XIX DETAILS OF DATA SAMPLE OF RETRACTABLE/EXTENDIBLE BONDS BOND DATA AVAILABLE BOND LONG BOND SHORT Maturity Coupon Maturity Coupon OPTION PERIOD ISSUE DATE FROM TO t Rl Jan.1,1963 4.00 Retractable on any interest Jan.1,59 Jan.7,59 Jan.27,60 56" date between Jan.1,1961 and Jan 1, 1962 giving 3 months notice May 25,60 34 El Oct.1,75 5.50 Oct.1,60 5.50 On or before June 30,60 Oct.1,59 Oct.7,59 Oct.25,61 108 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 98 E3 Dec.15,71 5.50 Dec.15,64 5.50 On or before June 15,64 Dec.15,59 Dec.16,59 Feb.17,60 Oct.25,61 89 E4 Apr.1,76 5.50 Apr.1,63 5.50 On or before Dec.31,62 Feb.15,60 Oct.4,67 Mar.3,71 179 E5 Oct.1,93 6.00 Apr.1,71 6.00 On or before Dec.1,70 Oct.1,67 Dec.6,67 Nov.7,73 310 E6 Dec.1,94 6.25 Dec.1,73 6.25 On or before Dec.1,72 Dec.1,67 Apr.2,69 Dec.5,73 245 E7 ' Apr.1,84 7.50 Apr.1,74 7.25 Apr.1,73 to Sept.30,73 . Apr.1,69 Oct.1,69 Sep.25,74 261 E8 Oct. 1,86 8.00 Oct.1,74 8.00 On or before Apr.1,74 Oct.1,69 Aug.15,70 Aug.19,70 Nov.26,75 278 E9 Dec.15,85 8.00 Dec.15,75 7.25 Dec.15,74 to June 14,75 Aug.1,71 Aug.4,71 July 28,76 260 E10 Aug.1,81 7.25 Aug.1,76 6.25 Aug.1,75 to Jan.31,76 July 5,72 June 29,77 263 Ell July 1,82 7.50 July 1,77 7.00 July 1,76 to Dec.31,76 July 1,72 Oct.3,73 Nov.9,77 215 E12 Dec.15,85 8.00 Oct.1,78 7.75 Oct.1,77 to Mar.31,78 Oct.1,73 Dec.5,73 Nov.9,77 207 E13 Dec.1,87 8.00 . Dec.1,80 7.50 Dec.1,79 to May 31,80 Dec.1,73 Apr.3,74 Nov.9,77 189 E14 Apr.1,84 8.00 Apr.1,79 7.00 Apr.1,78 to Sep.30,78 Apr a, 74 Oct.2,74 Nov.9,77 162 E15 Apr.1,84 9.25 April,78 9.25 On or before Jan.1,78 Oct.1,74 June 19,74 Jan.12,77 137 E16 Feb.1,82 9.25 Feb.1,77 9.25 On or before No.1,76 June 15,74 Nov.9,77 125 E17 Oct.1,84 8.75 Oct.1,79 7.50 Jan.1,79 to June 29,79. July 1,75 July 2,75 Oct.1,75 Nov.9,77 112 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 - All issues are byGovernmen t of Canada. The above sample constitutes the total 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 # Fl 4%% Dec 1, 1962 Jun 1, 1960 Aug 1, 1962 114 F2 4%% Sep 1, 1972 Oct 7, 1959 Aug 2, 1972 670 F3 5%% Oct 1, 1975 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 Fll 5% Oct 1, 1973 Dec 6, 1967 Sep 26, 1973 304 F12 5 3/4% Jan 1, 1985 Apr 2, 1969 Nov 9, 1977 450 F13 7% Jun 15, 1974 Apr 2, 1969 Jun 5, 1974 271 F14 5% Oct 1, 1987 Oct 1, 1969 May 5, 1971 84 F15 5% Jun 1, 1988 Jan8, 1969 Nov 9, 1977 462 F16 5h% Aug 1, 1980 Aug 1, 1962 Nov 9, 1977 798 F17 5% Oct 1, 1968 Oct 2, 1963 Sep 11, 1968 259 F18 3 3/4% Sep 1, 1965 Jan 7, 1959 Aug 25, 1965 347 - The 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 all retractable/extendible bonds, prices of ordinary (non-callable) coupon bonds*8 are also required for a) estimating the utility-dependent aggregate liquidity premium parameters b) conducting tests of market efficiency 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 effort was made to choose the bond sample such that, at every instant in time, at least 5 points on the term structure (between 1/2 year and 18 years to maturity) were represented. Table XX indicates some details on the sample of straight bond data. *8 The reason for specifically choosing non-callable bonds is for computational convenience in the estimation of the liquidity/term premium parameters. , This will become evident when we address that problem in 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 risk free interest rate process parameters (m,/*-, (Ttck) the weekly series of yield to maturity on 91-day Treasury bills was used., 990 weekly data points starting from January 7th, 1959, to December 21st, 1977, were used in the estimation. Initially, the primary object was to estimate cA ; The SS and SL methods were used on the total 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 reasonable for an interest rate process. The estimate of a-2 has, therefore, correspondingly decreased. To investigate further, the total 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 different parameter values, but the non linear 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 joint likelihood of the given data, in a parameter range that appears reasonable for an interest rate process. 98 on these two subperiods. The estimated parameter values are as follows: Parameters SL Method SS Method SPJ SP2 SJM SP2 f> (=2m/cr2) 627x10+7 188x10+' 5149.5 2195.0 jA. X103 1.152 1.3510 0.7884 1.2080 cA -0.1247 -0.0676 0.4032 0.4030 ra (x102) 0.3451 0.1698 a- 2 (x10*) 0.0011 0.0018 Even in the two subperiods, the SL method estimates a negative The estimate of jx by the SS method is, however, always lower than the SL estimate. This could be attributed to the estimation procedure. In the SL method, Jx is the value towards which the process is moving to stabilize, whereas in the SS method y~ *s the mean of the sample points (as pointed in the last footnote). Thus when interest rates are rising,jx as estimated by the SL method would be higher than the SS estimate so It might be interesting to recall from Appendix 7 that the estimate of jx for the SS method for (A = Since the estimte of ch (and therefore cr2 as well) from the SL method was unacceptablesi, we consider only the parameter estimates from the SS method. Now, the estimate of This has to the following advantages: a) The transition probability density function is known for As'/z., and so the "exact" approach to parameter estimation for the interest rate process model can be employed. b) Considerable simplification is achieved in the estimation of the investor utility dependent parameters in the particular functional form of the term/liquidity premium structure that we adopt later on. , Further, the adjustment in is very small, and based on the analysis in Chapter 5, we know that the impact on bond valuation is quite neqliqible. Assuming t<= %, the parameters jx , si As pointed out in Chapter 3, neqative cA values imply that the instantaneous variance of the interest rate process approaches oo as r approches zero. Such a model of the interest rate process is unrealistic, and therefore unacceptable,. 100 Parameters TRP Method SS Method SL Method a) Total Data: 1 f> (=2m/(r -) 73 04.8 8183.0 924 4. 8 /Mx 103) 1.2930 0.9974 1.2320 m{x 102) 0.2522 0.3183 b) Subperiod 1: 10993.9 20730.0 14099.9 ^(X 103) 1.0314 0.7884 0.9771 cr{x 10*) 0.9518 0.9522 rn(x 102) 0. 52 32 0.6713 c) Subperiod 2: 67 05.2 8564.0 7824.6 /X.(X 103) 1.3753 1.2070 1.3530 cMx 10*) 0. 4322 0.4 266 ^ (X 102) 0.1449 0.1669 As expected, the parameter values estimated assuming ^=Yi. are almost identical to those based on the SS method with a general (A . Thus, we assume as the parameters of the interest rate process those estimated using the TBP method over the complete data set, ie., 0\ =0.5 CT2=0.690494x10-* m=0.25221x10 =0.12934x10-* for all further analyses on bond valuation. 101 7.2 Solving the Bond Valuation Equation The basic bond valuation equation was developed in Chapter 2; the partial differential equation was L .I a, where a?a (r) , b=b(r), and <^ represents the instantaneous market perceived price of standard-deviation risk. For the interest 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 start with, let us consider the pure expectations hypothesis (PEXP) , whereby <^>=0. This reduces equation (2.9) to Tno(^-TT) -V =0 (7>1) -1_0-V^(| -4 By imposinq suitable boundary conditions, this parabolic partial differential equation may be solved to yield 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) SC^O - ^^Cf,^)^^^^)] (7.2b) where £ =0 is the lonq maturity, I is the short maturity, (see diaqram in Chapter 2, paqe ff for more details), represents the maturity correspondinq to the last date when the retraction/extension option may be exercised, To recoqnize the 102 possibility that the coupon on the long and short bonds could be different, 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 interest rate process at these boundaries. These boundary conditions on the bond value process (if required) would have to be consistent with those imposed on the interest rate process at the correspondinq boundaries. For the interest rate process having the parameter values as estimated in 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 differential equation (7.1) using the conditions (7.2a) and (7.2b). However, the solution technique employed requires further assumptions (as will become clear shortly). The solution technique will be the standard implicit finite differencinq approach (see McCraken 6 Dorn [44], Schwartz £63], Brennan S Schwartz [ 10 ]) . The differentials in equation (7.1) are approximated by difference equations, yieldinq 'iz I, (tl-0 where , and Wi, are known, and h and k are the discrete increments in the interest 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 prior to maturity, ^_, (which is the value of the bond on the maturity date) is known from conditions (7.2a). When we adopt a recursive method for solving for G(r,T) from X =0, the system of eguations (7.3) therefore represents (n-1) equations in (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, ie. This observation yields one more equation to our system* ie. The above equation could hold strictly only if r= «0 were an absorbinq boundary". However, for the parameter values of the interest process as estimated, r= does not exhibit absorbinq behaviour. As time to maturity increases, bond value increases at hiqh interest rate values, as there is a positive probability that the interest rate may return to reasonable levels before maturity. In a strict sense, when r= °o is inaccessible there is no meaning to assigning a value to the bond at that point. Equation (7.4) may be looked upon as a limiting value, and in -Referring to Appendix 1, a singular boundary is inaccessible in finite time if the integrals of h,(r) andah (r) are unbounded. In case however, the integral of 7T(r) were finite (with the integrals of h,(r) and bt(r) being infinite) then the barrier would be both inaccessible and absorbing (see Goel 6 Richter-Dyn [33]). For our process, the integral of 7T(r) is unbounded, and so r= °° is inaccessible and not absorbing. In case IT (r) were integrable, equation (7.4) would be strictly valid. 104 that light is valid. The final eguation comes from the behaviour of the bond valuation equation as r approaches zero. An approximation similar to the one used to obtain (7.4) would lead to serious biasesS3., 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 rises in numerical value. This is not true at r = 0. At r=0, we therefore adopt a continuity arqument: since equation (7.1) is valid over the total state space of r, it is valid as we make arbitrarily close approaches to r=0. Thus, we assume that the limit exists, and approximate it by settinq r=0 in (7.1) to yield rY^.OT) - Giji&.r) + = o (7.5) Strictly, we are assuminq that the following limits exist and are as shown: S3 An equivalent assumption at r=0 to that at r-e° is 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 is straightf orwardss . A small detail needs to be highlighted about the finite differencing approach used. Here, the state variable r has an upper limit of s* Ingersoll [39] has solved for the pure discount bond correspondinq to the process where = Using his result, we have _ n i_ . B, , «\\\- Hct)-^pC-A-c)](a . \ -- B./6 Since B is finite as r approaches zero, both B and B are finite as r approaches zero. Be conjecture that introduction of a continuous coupon and a boundary condition of the form (7.2b), would not alter the behaviour of the derivatives of bond value as r->0. 5S For further details see McCraken 6 Dorn [44] or Schwartz [63]. Briefly, it is not necessary to invert 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 in (7.3) to a tridiagonal system. A simple solution method is available, which requires subtracting a suitable multiple of each eguation from the precedinq equation in the system. 106 S where 0 conditions can now be expressed in terms of s, the new state variable. Brennan 6 Schwartz [12 j have adopted the trasformation 5 Here n can be any number so chosen that a large portion of the range of s is in the relevant range of r. To clarify, if we set n=5, the interval r=(0% to 20%) corresponds to s= (1.0 to 0.5). This allows for greater accuracy in the relevant range of interest 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; ie., h , the grid size 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) into 200 steps.. Several schemes were tried, and the solution vector of bond values was not too sensitive to the choice of number of grid points (within reasonable limits). 7.3 Bond Valuation Under the Pure Expectations Model The basic partial differential equation (p.d.e) governing bond valuation under the pure expectations hypothesis (PEXP) is obtained by setting |=0 in equation (2.9) of Chapter 2. This 107 was developed in the previous section. all 20 retractable/extendible bonds 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 justified. All 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 interest, ie., the buyer of the bond pays the seller the agreed purchase price for the bond plus the accumulated proportional coupon from the last coupon date to the transaction date. This arrangement is almost equivalent to continuous coupon payments to the holder5*. To compare model prices with market prices, an approach alonq the lines of Inqersoll [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 its square root (RMSE)) is broadly indicative of the lack of fit 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 this arranqement is that the holder gets no interest on the coupon, and loses the compoudinq effect, ie. the "interest on interest"., It can be clearly seen that this omission is 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 identified as 1) The part due to bias - attributable to a difference between the mean levels. 2) The part due to ^ #1, ie., under ((3 >1) or over responsiveness (f<1) of the model to market price movements. 3} The part due to residual error. The results of the regression and the error decomposition for the model based on the pure expectations hypothesis (PEXP) are presented in Tables XXI through XL in column 1. Cursory examination clearly reveals that the predominant element of the HSE across all bonds is bias. This is also indicated by noting that, for the PEXP model, the mean error [ which is _L^ (G^* - G-) ] is consistently negative for all bonds. The indication is that the model overprices the bonds, which implies that the markets expected yield on the bonds is higher than that assumed in the model. One possible explanation is that the market requires some liquidity or term premium in 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 models incorporate the liquidity premium assumption. "* The Capital Gains Tax model incorporate the liquidity premium assumption, as well as a Revenue Tax at 25%. o TABLE XXII COMPARISON OE MODEL AND MARKET PRICES (ALL MODELS) BOND : 5h% OCT.l, 1960 (El) REV.TAX C.G.TAX C.G.TAX MODEL PURE LIQ. REV.TAX (20%) "NAIVE" EXP. PREM. (50%) (25%) (10%) 0.664 0.661 0.520 R' 0.700 0.668 0.667 0.667 0.753 0.837 0.505 RMSE 3.123 0.751 0.656 0.694 0.699 0.763 0.393 MEAN ERROR -1.877 0.670 0.634 0.653 0.583 0.508 0.435 0.454 ESTIMATED SLOPE 0.087 0.439 0.870 0.071 0.062 0.054 0.075 (S.E. OF SLOPE) 0.009 0.053 0.107 42.279 49.826 57.092 55.033. EST. INTERCEPT 91.609 56.691 13.553 7.162 6.273 5.411 7.614 (S.E. OF INTR) 1.022 5.387 10.698 FRACTION OF ERROR 0.861 0.830 0.606 DUE TO BIAS 0.361 0.794 . 0.934 0.885 0.000 0.056 0.088 0.128 0.235 6*1 0.636 0.156 0.049 0.040 0.158 RES.VARIANCE 0.002 0.049 0.065 0.058 0.539 0.672 0.214 MISSPEC ERROR 9.731 0.537 0.403 0.454 0.028 0.028 0.028 0.040 RESID. ERROR 0.025 0.028 0.028 See footnote in Table XXI TABLE XXIII COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 5h OCT.l, 1962 (E2) REV.TAX C.G.TAX C.G.TAX "NAIVE" MODEL PURE LIQ. REV.TAX (10%) (20%) EXP. PREM. (50%) (25%) 0.798 0.796 0.797 0.802 0.807 0.686 R' 0.792 1.854 2.305 0.797 0.802 0.807 0.686 RMSE 4.729 1.636 2.181 1.910 1.949 2.020 0.826 MEAN ERROR -4.223 0.697 1.347 0.914 0.813 0.714 0.509 ESTIMATED SLOPE 0.393 0.065 0.044 0.038 0.033 0.033 (S.E. OF SLOPE) 0.019 0.033 32.236 -32.742 10.571 20.720 30.741 50.781 EST.INTERCEPT 60.660 6.614 4.484 3.928 3.396 3.395 (S.E. OF INTR) 2.083 3.418 FRACTION OF ERROR 0.895 0.889 0.880 0.854 0.249 DUE TO BIAS 0.797 0.778 0.057 0.499 0.182 0.093 0.020 0.002 0.020 B * 1 0.127 0.083 0.107 0.099 0.087 0.250 RES. VARIANCE 0.020 2.999 4.869 3.662 3.885 4.354 2.050 MISSPEC ERROR 21.914 0.438 0.443 0.441 0.430 0.419 0.684 RESID.ERROR 0.453 - See footnote in Table XXI TABLE XXIV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 5*2% DEC.15, 1964 (E3) C.G.TAX C.G.TAX "NAIVE" MODEL PURE LIQ. REV.TAX REV.TAX EXP. PREM. (50%) (25%) (10%) (20%) 0.704 2 0.856 0.865 R 0.756 0.846 0.851 0.848 2.534 2.858 1.885 RMSE 5.739 2.115 2.698 2.336 2.708 -0.545 MEAN ERROR -5.513 1.851 2.441 2.157 - 2.378 0.954 0.858 0.596 ESTIMATED SLOPE 0.639 0.812 1.530 1.052 0.034 0.039 (S.E. OF SLOPE) 0.036 0.035 0.064 0.045 0.039 16.939 41.262 EST. INTERCEPT 33.585 20.817 -50.919 -3.087 6.966 3.449 4.062 (S.E. OF INTR) 3.997 3.553 6.526 4.542 3.986 FRACTION OF ERROR 0.881 0.897 0.083 DUE TO BIAS 0.922 0.766 0.818 0.852 0.000 0.000 0.014 0.473 s 0.051 0.072 S 5 1 0.037 0.087 0.442 RES. VARIANCE 0.039 0.182 0.108 0.147 0.118 7.451 1.983 MISSPEC ERROR 31.650 3.658 6.490 4.655 5.662 0.761 0.717 1.573 RESID. ERROR , 1.294 0.816 0.789 0.803 See footnote in Table XXI. TABLE XXV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 5h APRIL 1, 1963 (E4) C.G.TAX C.G.TAX "NAIVE" MODEL PURE LIQ. REV.TAX REV.TAX EXP. PREM. (50%) (25%) (10%) (20%) 2 0.488 R 0.558 0.651 0.653 0.652 0.667 0.683 1.621 RMSE 5.051 1.654 • 2.581 2.087 2.074 2.096 MEAN ERROR -4.771 1.460 2.450 1.953 1.946 1.967 0.895 0.459 ESTIMATED SLOPE - 0.392 0.794 1.550 1.046 0.945 0.843 s 0.050 (S.E. OF SLOPE) 0.037 0.062 0.120 0.081 0.071 0.061 56.297 EST.INTERCEPT 61.017 22.381 -53.110 -2.792 7.434 17.889 '5.140 (S.E. OF INTR) 4.025 6.328 12.152 8.270 7.219 6.210 FRACTION OF ERROR 0.880 0.305 DUE TO BIAS 0.892 0.779 0.900 0.874 0.879 0.000 0.007 0.391 B * 1 0.080 0.022 0.018 0.000 0.112 0.302 RES. VARIANCE 0.026 0.198 0.081 0.124 0.120 1.832 MISSPEC ERROR 24.828 2.194 6.124 3.813 3.784 3.901 0.493 0.796 RESID.ERROR 0.688 0.543 0.539 0.541 0.517 See footnote in Table XXI TABLE XXVI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 6% APRIL 1, 1971 (E5) REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. MODEL PURE LIQ. (10%) . (20%) AVG. "NAIVE' EXP. PREM. (50%) (25%) 2 0.710 0.349 R 0.714 0.436 0.410 0.423 0.401 0.378 0.584 1.661 RMSE 1.462 2.291 0.907 1.544 1.931 2.444 0.141 MEAN ERROR -1.040 1.937 0.430 1.201 . 1.587 2.081 -0.181 0.861 0.290 ESTIMATED SLOPE 0.490 0.406 0.745 0.519 0.447 0.379 0.043 0.030 (S.E. OF SLOPE) 0.024 0.036" 0.069 0.047 0.042 0.037 13.481 70.116 EST.INTERCEPT 49.773 59.387 25.484 48.096 55.267 62.102 4.254 3.054 (S.E. OF INTR) 2.416 3.497 6.859 4.620 4.145 3.669 FRACTION OF ERROR 0.096 0.007 DUE TO BIAS 0.506 . 0.715 0.224 0.605 0.675 0.725 0.150 0.163 0.169 0.048 0.755 6 * 1 0.359 0.176 0.053 0.237 RES. VARIANCE 0.134 0.108 0.721 0.243 0.161 0.104 0.855 5.347 0.049 2.104 MISSPEC ERROR 1.851 4.681 0.229 1.804 3.129 0.292 0.656 RESID.ERROR 0.287 0.568 0.594 0.581 0.603 0.626 See footnote in Table XXI. TABLE XXVII COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 6VDEC.1, 1973 (E6) REV.TAX C.G.TAX C.G.TAX MOV. MODEL PURE LIQ. REV.TAX (20%) AVG. "NAIVE" EXP. PREM. (50%) (25%) (10%) 0.751 0.745 0.737 0.827 0.749 2 0.782 0.756 0.746 R 2.313 2.815 3.499 1.568 5.018 RMSE 6.899 3.234 1.711 1.435 1.878 2.448 -0.906 -3.084 MEAN ERROR -5.667 2.046 0.736 0.729 0.650 0.574 0.951 0.420 ESTIMATED SLOPE 0.424 0.571 1.046 0.025 0.023 0.021 0.026 0.015 (S.E. OF SLOPE) 0.013 0.020 0.037 35.798 43.555 3.932 56.080 EST. INTERCEPT 54.559 43.623 -3.828 27.809 2.538 2.294 2.053 2.696 1.542 (S.E. OF INTR) 1.455 1.949 3.718 FRACTION OF ERROR 0.445 0.489 0.334 0.377 DUE TO BIAS 0.674 0.400 0.184 0.385 0.005 0.528 0.380 0.001 0.178 0.252 0.309 6 * 1 0.282 0.302 0.201 0.660 0.093 RES.VARIANCE 0.042 0.218 0.813 0.436 22.828 0.546 3.017 5.529 9.779 0.836 MISSPEC ERROR 45.561 8.172 2.395 2.466 1.625 2.353 RESID.ERROR 2.041 2.285 2.383 2.332 See footnote in Table XXI. TABLE XXVIII COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) — BOND : Tk APRIL 1, 1974 (E7)" SE S&. " c1i« SS: "NAIVE" 0.759 0.759 0.071 2 0.769 0.764 R 0.783 0.759 0.772 11.734 6.441 6.256 5.947 4.789 RMSE 15.209 3.758 3.892 -5.917 -5.571 -5.030 -3.868 -7.191 MEAN ERROR -14.184 -1.544 -3.590 0.488 0.454 0.491 0.237 ESTIMATED SLOPE 0.313 0.431 0.765 0.525 0.056 0.018 0.017 0.016 0.018 (S.E. OF SLOPE) 0.010 0.015 0.027 49.384 53.287 49.952 75.655 EST. INTERCEPT 65.557 57.229 21.116 45.229 6.127 2.032 1.911 1.794 1.920 (S.E. OF INTR) 1.253 1.649 2.873 • FRACTION OF ERROR 0.375 0.843 0.793 0.715 0.652 DUE TO BIAS 0.869 0.168 0.850 0.267 0.273 0.035 0.113 0.161 0.233 1 0.123 0.702 0.045 0.051 0.079 0.350 RES. VARIANCE 0.007 0.129 0.113 0.042 89.432 39.748 37.355 33.552 21.110 MISSPEC ERROR 229.686 12.304 13.427 1.785 1.824 1.828 48.268 RESID.ERROR 1.640 1.824 1.725 1.747 See footnote in Table XXI. TABLE XXIX COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 8% OCT.l, 1974 (E8) MODEL PURE LIQ. REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. EXP. PREM. (50%) (25%) (10%) (20%) AVG. "NAIVE" 2 0.730 0.730 0.729 0.682 R 0.750 0.728 0. 732 0.730 3.099 3.903 10.338 RMSE 19.505 4.926 1. 719 2.976 3.039 -0.931 -2.630 -8.011 MEAN ERROR -18.620 -3.227 0. 427 -1.476 -1.252 0.499 0.508 0.278 ESTIMATED SLOPE 0,312 0.426 0. 791 0.548 0.522 0.020 0.019 0.020 0.012 (S.E. OF SLOPE) 0.011 0.017 0. 031 0.021 49.348 51.932 50.174 73.329 EST. INTERCEPT 66.140 58.682 22. 158 46.510 2.096 2.175 1.405 (S.E. OF INTR) 1.457 1.839 3. 261 2.312 2.199 FRACTION OF ERROR 0.090 0.453 0.600 DUE TO BIAS 0.911 0.429 0. 061 0.245 0.169 0.487 0.574 0.664 0.390 0.373 • B * 1 0.082 0.473 0. 146 0.245 0.155 0.026 RES. VARIANCE 0.005 0.097 0. 791 0.266 0.255 104.095 21.894 0. 615 6.498 6.881 7.248 12.865 MISSPEC ERROR 378.277 2.357 2.372 2.785 RESID.ERROR 2.185 2.376 2. 341 2.358 2.357 See footnote in Table XXI. TABLE XXX COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : Ih'i DEC. 15, 1975 (E9) MODEL PURE LIQ. REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. EXP. PREM. (50%) (25%) (10%) (20%) AVG. . "NAIVE" R2 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 -15.831 -2.514 ' -4.244 -7.044 -6.765 -6.282 -1.188 -7.280 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 j DUE TO BIAS 0.843 0.355 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 footnote in Table XXI. TABLE XXXI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 6% AUG.l, 1976 (E10) REV.TAX C.G.TAX C.G.TAX • MOV. "NAIVE" MODEL PURE LIQ. REV.TAX (20%) AVG. EXP. PREM . (50%) (25%) (10%) 2 0.557 0.559 0.516 R 0.519 0.588 0.540 0.540 0.548 3.592 3.175 5.721 RMSE . 9.065 2.563 3.087 4.105 3.887 - -3.677 MEAN ERROR -7.152 -0.272 -2.468 -3.140 -2.768 -2.251 -1.959 0.482 0.298 ESTIMATED SLOPE 0.245 0.475 0.664 0.458 0.446 0.438 0.028 0.018 (S.E. OF SLOPE) 0.015 0.026 0.040 0.027 0.026 0.025 50.313 68.404 EST. INTERCEPT 73.003 51.834 31.565 52.210 53.584 54.703 2.844 1.952 (S.E. OF INTR) 1.650 2.595 4.091 2.838 2.716 2.602 FRACTION OF ERROR 0.392 0.380 0.413 DUE TO BIAS 0.622 0.011 0.638 0.585 0.507 0.257 0.319 0.408 0.366 0.501 0 * 1 0.343 0.626 0.081 0.253 0.085 RES. VARIANCE 0.033 0.362 0.279 0.157 0.173 0.198 12.497 10.341 7.531 29.936 MISSPEC ERROR 79.409 4.190 6.870 14.194 2.551 2.799 RESID.ERROR 2.781 2.381 2.662 2.658 . 2.618 2.566 See footnote in Table XXI . TABLE XXXII COMPARISON OF MODEL AND MARKET PRICES(ALL MODELS) BOND: 7% July 1, 1977 (Ell) MODEL PURE LIQ. REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. "NAIVE" EXP. PREM. (50%) (25%) (10%) (20%) AVG. 2 R 0.552 0.649 0.589 0.590 0.602 0.615 0.542 0.538 RMSE 8.246 2.930 2.783 3.877 3.736 3.560 6.374 5.143 MEAN ERROR -5.799 0.477 -2.069 -2.610 -2.240 -1.730 -5.159 -2.498 ESTIMATED SLOPE 0.226 0.411 0.590 0.408 0.396 . 0.386 0.324 0.279 (S.E. OF SLOPE) 0.013 0.019 0.032 0.022 0.021 0.019 0.019 0.016 EST. 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 0.769 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 RESID. ERROR 2.244 1.754 2.055 2.052 1.993 1.926 2.292 2.310 See footnote in Table XXI. TABLE XXXIII COMPARISON OF MODEL.AND MARKET PRICES (ALL MODELS) BOND: 7 3/4% Oct.l, 1978 (E12) REV.TAX C.G.TAX C.G.TAX MOV. "NAIVE" MODEL PURE LIQ. REV.TAX (20%) EXP. PREM. (50%) (25%) (10%) AVG. 0.707 0.723 0.742 0.653 0.827 Kl 0.691 0.814 0.699 2.924 2.468 1.931 6.359 0.956 RMSE 6.260 2.476 2.293 -2.567 -2.017 -1.282 -6.115 0.048 MEAN ERROR -5.675 2.187 -1.938 0.726 0.702 0.682 0.610 0.897 ESTIMATED SLOPE 0.443 0.755 1.024 0.031 0.029 0.027 0.030 0.028 (S.E. OF SLOPE) 0.020 0.024 0.045 28.254 30.846 35.147 10.291 EST. INTERCEPT 53.101 26.045 -4.431 25.455 3.026 2.780 3.222 2.796 (S.E. OF INTR) 2.136 2.410 4.677 3.271 FRACTION OF ERROR 0.924 0.002 0.780 0.712 0.771 0.668 0.441 DUE TO BIAS 0.821 0.032 0.054 0.068 0.000 0.057 0.104 0.213 6*1 0.138 0.042 0.942 0.151 0.286 0.171 0.227 0.345 RES.VARIANCE 0.039 38.713 0.052 5.204 3.754 7.087 4.709 2.440 MISSPEC ERROR 37.646 1.734 0.863 0.929 1.505 1.463 1.383 1.289 RESID. ERROR 1.543 See footnote in Table XXI • TABLE XXXIV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 7h Dec.l, 1980 (E13) MODEL PURE LIQ. REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. "NAIVE" EXP. PREM. (50%) (25%) (10%) (20%) AVG. 2 0.617 0.761 0.638 0.744 0.648 0.656 0.671 0.686 R 4.215 3.525 2.691 7.019 1.555 RMSE 7.479 3.077 3.941 -3.035 -2.018 -6.739 0.316 MEAN ERROR -7.005 2.600 -3.436 -3.804 0.900 0.858 0.843 0.929 ESTIMATED SLOPE 0.571 0.837 1.328 0.942 0.043 0.040 0.046 0.036 (S.E.OF SLOPE) 0.029 0.034 0.068 0.047 12.094 9.617 7.170 EST.INTERCEPT 37.987 18.135 -36.699 2.104 7.005 4.438 4.047 4.850 3.543 (S.E. OF INTR) 3.149 3.263 6.904 4.835 FRACTION OF ERROR 0.741 0.562' 0.92175 0.041 DUE TO BIAS 0.877 0.713 0.759 0.814 0.000 0.005 0.022 0.003 0.012 6 t 1 0.060 0.027 0.023 0.415 0.074 0.946 RES.VARIANCE 0.062 0.259 0.216 0.185 0.253 12.168 14.482 . 9.276 4.235 45.602 0.130 MISSPEC ERROR . 52.480 7.017 3.153 3.008 3.671 2.288 RESID.ERROR 3.469 2.453 3.367 3.291 See footnote in Table XXI. TABLE XXXV COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 7% APRIL 1, 1979 (E14) REV.TAX C.G.TAX C.G.TAX MOV. "NAIVE" MODEL PURE LIQ. REV.TAX (20%) AVG. EXP. PREM. (50%) (25%) (10%) 0.706 0.726 0.746 0.690 0.835 R2 0.661 0. 808 0.702 2.737 2.158 1.551 3.116 1.097 RMSE 4.537 2. 861- 3.000 -2.323 -1.649 -0.769 -2.743 0.190 MEAN ERROR -4.135 2. 591 -2.513 1.062 1.001 0.930 0.974 0.977 ESTIMATED SLOPE 0.673 0. 872 1.522 0.049 0.044 0.039 0.047 0.031 (S.E. OF SLOPE) 0.035 0. 030 0.072 -8.526 -1.837 5.156 -0.190 2.407 EST. INTERCEPT 28.979 14. 671 -54.673 4.980 4.442 3.918 4.755 3.077 (S.E. OF INTR) 3.565 2. 934 7.213 FRACTION OF ERROR 0.720 0.579 0.246 0.773 0.030 DUE TO BIAS 0.830 0. 819 0.701 0.000 0.002 014 0.063 0.000 0.002 0.005 B *1 0.052 0. 0.278 0.417 0.748 . 0.225 0.967 RES. VARIANCE 0.116 0. 166 0.234 7.518 0.039 828 6.891 5.404 2.710 0.604 MISSPEC ERROR 18.184 6. 2.088 1.946 1.801 2.195 1.166 RESID.ERROR 2.403 1. 361 2.113 See footnote on Table XXI . TABLE XXXVI COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 9k% APRIL 1, 1978 (E15) "NAIVE" REV JTAX C.G.TAX C.G.TAX MOV. MODEL PURE LIQ. REV.TAX (25%) (10%) (20%) AVG- EXP. PREM. (50%) 0.756 0.760 0.741 0.757 0.754 0.750 0.752 R2 0.725 2.744 3.020 1.519 2.390 2.280 3.192 2.557 RMSE 6.868 0.658 -0.949 2.998 2.264 2.446 2.719 MEAN ERROR -6.422 1.662 0.707 0.694 0.495 1.116 0.787 0.747 ESTIMATED SLOPE 0.460 0.623 0.032 0.022 0.050 0.035 0.033 . 0.031 (S.E. OF SLOPE) 0.022 0.028 32.583 32.506 52.524 -8.860 24.069 28.373 EST. INTERCEPT 53.645 40.568 3.206 3.367 2.339 5.169 3.653 3.427 (S.E. OF INTR) 2.494 2.896 FRACTION OF ERROR 0.810 0.187 0.157 0.881 0.884 0.794 DUE TO BIAS 0.874 0.531 0.642 0.052 0.065 0.286 0.246 0.003 0.038 6 J* 1 0.098 0.124 0.525 0.199 0.115 0.177 0.152 RES. VARIANCE 0.027 0.222 7.992 1.096 4.575 9.019 5.376 6.383 MISSPEC ERROR 45.877 4.044 1.128 1.212 1.140 1.172 1.162 1.146 RESID. ERROR 1.293 1.155 See footnote in Table XXI TABLE XXXVII COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND: 9k FEB.l. 1977 (E16) MODFT PURE LIQ. REV.TAX REV. TAX C.G.TAX C.G.TAX MOV. • "NAIVE" EXP. PREM. (50%) (25%) (10%) (20%) AVG. 0.761 0.772 0.783 0.678 0.752 R2 0.681 0.763 0.758 2.129 2.256 2.448 2.450 2.291 RMSE 5.661 2.017 2.537 1.772 1.922 2.138 -1.822 ' -1.164 MEAN ERROR -5.107 1.405 2.212 ' 0.892 0.855 0.819 0.673 0.563 ESTIMATED SLOPE 0.488 0.701 1.273 0.044 0.041 0.038 0.041 0.028 (S.E. OF SLOPE) 0.029 0.034 0.064 16.605 20.401 32.532 44.409 EST. INTERCEPT 50.396 31.851 -25.464 12.743 4.215 3.903 4.362 3.027 . (S.E. OF INTR) 3.238 3.558 6.501 4.539 FRACTION OF ERROR 0.692 0.725 0.762 0.552 0.258 DUE TO BIAS 0.814 0.485 0.760 0.011 0.022 0.033 0.146 0.476 B * 1 0.130 0.187 0.028 0.251 0.203 0.301 0.265 RES. VARIANCE 0.055 0.326 0.211- 0.296 3.810 4.775 4.197 3.856 MISSPEC ERROR 30.256 2.738 5.080 3.193 1.343 1.282 1.219 1.809 1.394 RESID.ERROR 1.791 1.329 1.359 See footnote in Table XXI• TABLE XXXVIII COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 7h% OCT.l, 1979 (E17) MOV. "NAIVE' REV.TAX. REV.TAX C.G.TAX C.G.TAX MODEL PURE LIQ. (10%) (20%) AVG. EXP. PREM. (50%) (25%) 2 0.630 0.726 0.714 0.700 0.581 0.594 0.611 R 0.570 2.922 1.174 2.942 3.384 2.912 2.319 RMSE 5.700 2.598 2.685 -0.094 -2.673 -3.130 -2.594 -1.855 MEAN ERROR -5.383 2.160 0.535 0.763 0.709 0.655 0.729 ESTIMATED SLOPE 0.053 0.610 1.095 0.040 0.030 0.083 0.056 0.050 0.045 (S.E. OF SLOPE) 0.039 0.035 28.706 45.823 -12.404 20.960 26.857 32.784 EST. INTERCEPT 46.370 39.767 3.860 3.011 8.482 5.780 5.148- 4.537 (S.E. OF INTR) 4.086 3.468 FRACTION OF ERROR 0.844 0.003 0.825 0.855 0.793 . 0.640 DUE TO BIAS 0.891 0.691 0.040 0.647 0.000 0.016 0.041 0.113 B * 1 0.060 0.148 0.114 0.349 0.173 0.127 0.164 0.246 RES. VARIANCE 0.047 0.159 7.560 1.913 7.154 9.993 7.084 4.051 MISSPEC ERROR 30.956 5.674 0.982 1.027 1.506 1.459 1.395 1.327 RESID. ERROR 1.543 1.077 See footnote in Table XXI TABLE XXXIX COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 9% FEB.l, 1978 (E18) C.G.TAX MOV. "NAIVE" MODEL PURE LIQ. REV.TAX REV.TAX C.G.TAX EXP. PREM. (50%) (25%) (10%) (20%) AVG. 0.740 2 0.785 0.794 0.809 R 0.561 0.777 0.776 0.777 2.806 3.083 3.390 0.757 RMSE 1.267 2.740 2.493 2.600 3.034 3.355 0.436 MEAN ERROR -1.016 2.670 2.452 2.560 2.764 0.685 0.737 0.650 ESTIMATED SLOPE 0.590 0.640 1.234 0.838 0.762 0.033 0.034 0.037 (S.E. OF SLOPE) 0.050 0.032 0.063 0.043 0.038 26.395 34.280 29.299 36.017 EST. INTERCEPT 41.278 38.450 -20.968 18.634 3.330 3.402 3.769 (S.E. OF INTR) 5.186 3.280 6.351 4.304 3.810 FRACTION OF ERROR 0.968 0.979 0.332 DUE TO BIAS 0.642 0.949 0.967 0.969 0.970 0.007 0.298 6 * 1 0.134 0.026 0.003 0.003 0.007 0.014 0.017 0.013 0.369 RES.VARIANCE 0.222 0.024 0.029 0.026 0.022 9.340 11.340 0.361 MISSPEC ERROR 1.247 7.329 6.035 6.579 7.704 0.168 0.155 0.211 RESID. ERROR 0.358 0.181 0.182 0.181 0.174 See footnote in Table XXI TABLE XL COMPARISON OF MODEL AND MARKET PRICES (ALL MODELS) BOND : 9% OCT.l, 1980 (E19) MODEL PURE LIQ. REV.TAX REV.TAX C.G.TAX C.G.TAX MOV. "NAIVE" EXP. PREM. (50%) (25%) (10%) (20%) AVG. R 0.584 0.646 0.543 0.566 0.581 0.596 .0.663 0.699 4.281 RMSE 7.260 2.427 1.878 3.591 3.444 3.174 2.193 4.077 MEAN ERROR -7.037 1.859 -1.352 -3.351 -3.181 -2.857 0.322 0.686 0.688 ESTIMATED SLOPE 0.528 0.586 1.189 0.819 0.751 0.455 0.053 0.046 (S.E. OF SLOPE) 0.042 0.041 0.103 0.068 0.060 0.028 30.724 35.287 EST.INTERCEPT 45.410 44.148 -21.347 16.099 23.431 56.886 4.651 (S.E. OF INTR) 4.704 4.207 10.904 7.309 6.501 5.731 2,942 FRACTION OF ERROR 0.021 0.586 0.518 0.870 0.853 0.810 0.906 DUE TO BIAS 0.939 0.750 0.195 0.009 0.006 0.018 0.043 0.026 8*1 0.031 0.227 0.218 0.471 0.122 0.128 0.145 0.066 RES. VARIANCE 0.028 3.716 4.606 1.865 11.316 10.337 8.604 17.108 MISSPEC ERROR 51.199 1.096 1.287 1.662 1.579 1.526 1.470 1.225 RESID.ERROR 1.516 See footnote in Table 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) is 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 lines of Ingersoll [39], First, we assume that ^ is independent of t, ie., it is time homogeneous. Second, we assume (TAR . 3>(1~) = - k?.r (7.8) which yields (see Chapter 2, equation (2.8)) \(f,V) . _cfe, + ^T)_^_ _(7.9) Vasicek [72] and Brennan 6 Schwartz [1.0] both assume = constant. This is a statement about the price (in terms of excess return) of instantaneous standard deviation risk One may find the assumption <|> = constant more intuitively comprehensible than the assumption in (7.8)., However, as will be shown shortly, the assumption of equation (7.8) leads to a simple structure for the form of A . Much of the existinq literature on the term structure of interest rates addresses the form and determinants of A- . It will be shown that (7.8) leads 130 to a form for X that is consistent with the existing literature. Ingersoll(op cit) points out that under this assumption (and assuming the interest rate process to be of the form assumed here), the value of the pure discount bond B(r, T ) is given by r "1 r i' b(i70 - UCt)J Y) m'/A-'t -v ^ T j I - Her) £ J (7.10) where m» = (m-k^ )l j^' = *k| 2 = [ m» - (m» z + 2o- ) ]/ A » (m«2 + 2 cr2)^2" H(f) = [ 1>(m«-A) 0-e-Ar )/2A]-» It can be seen from (7.10) that B,/B = [1-H(T)e-^r ]=q(f ), ie. the ratio (B, /B) is independent of r and strictly a function of time to maturity. This implies that the choice of as indicated by the relationship in equation (7.8) leads to an expression for the liquidity/term premium as (7.11) As pointed out by Ingersoll [39], for ( k, • k^ r) > 0, the term premium is a positive, increasing concave function, and for (k( r)<0, is negative, decreasing and convex. These are the usual properties associated with the liquidity premium. Further, for a qiven maturity, the relation between A- and r as qiven by equation (7.11), is consistent with some of the popular 131 assumptions about term/liquidity premia, ie. , a) a constant term premium independent of interest 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 interest rate. b) term premiums proportional to the interest rate (set k, -0). This would specify that the return on a qiven maturity of bonds be a constant ratio to the instantaneous interest rate. c) term premia that are positive as lonq as interest rates are below a threshold level, 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 in equations (7.8) and (7.9) is that it permits a simple method of estimating the parameters k( and kr because we have a closed form solution for the pure discount bond under this 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 is as qiven by equation (7.10). Given a sample of market prices on straight coupon bonds, one method of estimating k( and k2 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 interest rate process, the current interest rate, time to maturity, and coupon rate), the model price of any straight coupon bond can be computed using eguation (7. 12) 57. The simplest model that was considered was P.' = PL + €i, 17.13) where P1 and P^ are respectively market and model prices, and 2 e. r-J N(0, (T ); Cov(ec ,e ) =0 for i#j. It may be noted that P is a non -linear function of the parameters k, and kt Thus, estimating k, and k^ in the present scenario is the standard problem of coefficient estimation in a non-linear regression frameworkss. Throughout, we adopt maximum likelihood (ML) methods for parameter estimation. In this situation least squares estimation = ML(asymptotically). However since P^» and are strictly positive, it was considered more appropriate to assume a model of the form s7 P(r,^,c) can be evaluated very easily 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 results. 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 negligible. It may be noted that in any approach to estimating k, and kj_ , model prices of the total bond sample would have to be evaluated several hundred times. Even with the present assumptions, estimating k, and kL is computationally quite expensive. However, if were not , (or zero) and if 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 finite difference methods., That would mean a computation expense more than just prohibitive! 58 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 kx were estimated by both models above, and the parameter estimates were hardly different5*: (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 linear models it is well known that the estimated coefficients are unbiased, even where the residual covariance is -Q-^cr"J : the covariance matrix of the estimated parameters is biased. In a non-linear setting, whether the estimated parameters are unbiased in small samples is not known when Si£ O-5- I. TO test for heteroscedasticity, the residual vector e^ was retrieved and the following regression was performed: (The hypothesis was that var(e• ) is a function of time to matruity of P60.) 59 The standard errors of the estimates, based on asymptotic theory (ie., by inverting the Fisher Information matrix) are not reported, as their values was very different across the two models. This was investigated further and found to be due to numerical inaccuracy in evaluating the second derivative of the joint likelihood function near the optimal point, 60 P is a function of r and ? . Heteroscedasticity as a function only of X was considered. Understandably, it could have also been a function of r. However, this was not considered, as the variability of T over the sample was much larger than that of r. It was therefore felt that most of the heteroscedasticity could be explained by T alone. 134 loo^{^1) a -t- b Lft(Vi) + IA)O (7.15) where Tt=time to maturity of the i™ data point. If b=0, then we cannot reject the hypothesis that the residuals exhibit homoscedasticity61. This was done for the residuals from equation (7.14) and b was estimated at 2.091, and its t static was 1.06. This seems to indicate that there is no compellinq reason to suspect heteroscedasticity to be present. Testinq for autocorrelation among the residuals is a more complicated matter.. There are two types of error correlations to consider. 1) Serial correlation within each bond across time. , 2) Contemporaneous correlation across bonds, at any instant of time. It must be remembered that the ordinary coupon bond sample consists of time series on 18 different bonds. Serial correlation of residuals refers to the correlation between consecutive residuals of each bond. It is, however, also reasonable to expect the errors across all bonds, at a The more "correct" method of testing for heteroscedasticity would be to do a "constrained" and "unconstrained" estimation, and then perform a likelihood ratio test. Under the constrained estimation JL is assumed = cr21 and in the unconstrained JL- is diaqonal with elements o^^a?^ . The rest of the approach is 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 this would have been computationally expensive. Thus the more ad hoc approach was taken. This method of hypothesis testinq on b, is also dependent upon w,- being i.i.d and normally distributed. 135 particular point in time to be correlated. Since each bond data series starts and ends at a different point in time (and each is of different length) , accounting for contemporaneous correlation would be a horrendous task. Considering the difficulties involved, it was decided to leave the problem of contemporaneous correlation in abeyance, but tackle the serial correlation problem. When we consider serial correlation only, the covariance matrix SL of the residual vector is block diagonal in structure, with the representative matrix having the usual form as when we have first-order autocorrelation, ie, Jl = ( Si-c) where Jlj, is the matrix along the diagonal for bond i, and is of the form I f f' r f f -Hi f f \ (7.16) (Tc x- "ft ) 7-i We could further assume that "f is egual across all bonds. This simplified structure makes it computationally much easier to set up the likelihood function of the residuals and thereby estimate the parameters. What was actually done was that, along with serial correlation (using the model of equation 7. 14) , heteroscedasticity of the form discussed earlier was assumed, and ML methods were employed to estimate jointly (k, ,kt ,-f,a,b). It was computationally very expensive and so no constrained estimation was performed, (to do likelihood ratio 136 tests for testing hypotheses on any of the parameters). The log of the joint likelihood function was L = -L^\SL\ - i«'Jl e (7.17) where e is the column vector of residuals, e* is its transpose, 2 2 and e^/v/ N( 0 ( cr- ), with Q~c = aT^ and Corr (et ^ ( ) = f and is constant across all bonds. The result of the estimation was that convergence was not attained in 60 iterations using a quasi-Newton algorithm for maximizing L. The intermediate parameter values were62 k, = 0.3916 x 10-s 2 kr = -0.2144 x 10~ f = 0.0097 a * 0.1394 x 10-ft b = 1.586 The gradients on k, and kL indicated that the optimum would require both values to move towards zero. The broad conclusions that can be arrived at, based on the results, are: a) The estimates of k, and kj_ based on the model of equation (7.14) are probably not very different from the model assuminq autocorrelation and heteroscedasticity of the error vector. b) The serial correlation coefficient (f) between the »2 apparently, the converqence rate is very slow. The CPU time used for this partial converqence run was 5000 seconds on an IBH 370/168. , Since the computational cost was extremely hiqh and no additional insiqhts appeared to be likely by restarting the search for the optimum andqoinq on until converqence was attained, the matter was not pursued further. 137 residuals appears to be close to zero. c) A statistically significant level of heteroscedasticity does not seem to exist. The final question that was considered under the estimation of the parameters kv and kx , was the validity of the assumption of normality of the residuals - after all, the HL approach here is based on this assumption. The approach that we adopt in testing for normality (or departures therefrom) is probability graphing. Fama [22] uses this approach in examining the behaviour of stock prices. If u is a Gaussian random variable with mean /A and variance cr2, the standardized variable Z = (u-/^)/r will be unit normal., Since Z is just a linear transformation of u, the graph of Z against u is just a straight line. The relationship between Z and u can be used to detect departures from normality in the distribution of u. If u^(i-=1.,H are N sample values of the variable u arranged in ascending order, then a particular uL is an estimate of the f fractile of the distribution of u, where the value of f is given by 63 63 As pointed out in Fama [22], this particular convention for estimating f is only one of many that are available. Other popular conventions are i/(N + 1), (i-3/8)/(N«- ) and (i- )/N. All four techniques give reasonable estimates of the fractiles and, for the large sample that we have, it makes little difference which specific convention is chosen. 138 Now the exact value of Z for the f fractile of the unit normal distribution can easily be obtained by inverting the unit cumulative normal. Computer routines are available for this. If u is a Gaussian random variable, then a graph of the sample values of u against the values of Z derived from the theoretical unit normal cumulative distribution function should be a straight line. There may, of course, be some departure from linearity due to sampling error. If the departures from linearity are extreme, however, the Gaussian hypothesis for the distribution of u should be questioned. The normal probability plot of the residuals from the model of equation (7.14) is presented in Figure 4. Inspection of the plot indicates that the distribution of the residuals is thinner than the normal at the tails, and also more peaked at the mode. In fact, it could be that we have a mixture of normal distributions with identical means but differing variances - one (or more) corresponding to the tails; and another (or others) corresponding to the peak at the mean. This could be the result of heteroscedasticity of the form we considered earlier (but did not find statistically significant). Possibly if we had adopted the more "correct" method of testing for heteroscedasticity (see footnote 54) we might have observed it at a statistically significant level. Thus, whether heteroscedasticity exists or not is at present an unresolved issue. However, we did find that even taking it 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 alter materially the point estimates of k( and kz . He may therefore assume that our estimates of k, and kt based on the model of equation (7.14) are satisfactory. To get a better feel for the numerical values of k, and k% , the liquidity/term premium function A was plotted against time to maturity, for 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. If we represent the term structure by R(r,t ), then we have RC*,*) - -1 ^[ec-r,r)j where B(r,t ) is the pure discount bond value. Figure 6 shows the shape of the term structure6* at values of the current value of the short term interest rate varying from '/^JX. , to 2JUL . It may be of interest to note that when r= , the term/liquidity premium is a positive and increasing function of time to maturity.. When r=-k, /k2 , A =0 for all maturities. Obviously this does not imply a flat term structure - only that at this value of r, the term structure curves for the pure expectations and the liquidity/term premium hypotheses models coincide. As can be seen from Figure 6, when r=-k, /k% , the term structure is downward sloping. Ingersoll[ 39 ] has pointed out that the term structure corresponding to this interest rate process, and the assumed form of (as in equation (7.8)), could have a *•* The value of HISF in the figure corresponds to the limiting value of R(r,T ) as T -><* . From the term structure equation, this is given by (2m1 ' ,/ LIQUIDITY PREMIfl V5 TIME TO MATURITY ON DISCOUNT BONDS Kl = 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 Kl - 0.309 X 10 XX -5 K2 =-0.154 X 10 XX -2 Kl t K2 BRSEQ ON BOND DATR JRN 59 - NOV 77 15.oa.. 2.73 _ d $ iO J5 20 25 i0 i5 4) 45 40 TIME TO MATURITY IN YEARS 143 FIGURE 7 YIELD TO MATURITY VS TIME TO MATURITY ON DISCOUNT BONDS Kl - 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 is borne out in Figure 7. Before comparing Figures 6 and 7, care must be taken to note the large difference in 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 their liguidity/term premia requirements, we can proceed to value our sample of retractable/extendible bonds, with this assumption incorporated. The p.d.e. governing the bond price is only slightly altered (cf. equation 7.1) we now have l(rV<5M -i ^V"^)^' -^^^^-^ = ° (7.18) where i' and JJL' are as defined in equation (7.10). The boundary conditions remain exactly the same as for the PEXP case. Model prices were computed for all 20 bonds, and the results of regressing the market prices on model prices are presented in column 2 of Tables XXI through XL. &s expected, the mean error (defined earlier) which was consistently negative under the PEXP model, is now more often positive (except for bonds E7 to E10). For purposes of quick comparison across bonds, Table XLI presents the mean error for all 20 bonds using the different models, and Table XLII presents similar summary results on P , the slope coefficient from regressing the market price on the model prices as well as the correlation between the model and market prices., Comparing the results 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 - - El -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 Ell -5.79 0.47 -2.06 -2.61 -2.24 -1.73 -5.16 -2.50 E12 -5.67 2.18 -1.93 -2.56 -2.01 -1.28 -6.11 0.05 E13 -7.00 2.60 -3.43 -3.80 -3.03 -2.01 -6.74 0.31 EU -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 OOOOOOOOOOOOOOO OOO 1 OOOQOQOOOOOOOOOOOOO -* S ^- >>"» < -* ciOeo'-'ior-twi^S s coc^ooaD < i < OOOOOOOOOOOOOOO I 1 1 g ! 3 I I 1 1 I I I I i I I i I 3 1 o'dooodoooo'oddododddd i s i 3 5 I i I I I 1 2 s-l i ! 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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 -7.570 0.778 -0.905 -1.621 -1.258 -0.751 • -2.075 -0.841 ESTIMATED SLOPE 0.301 0.546 0.678 0.479 0.478 0.477 0.469 0.444 (S.E. OF SLOPE) 0.006 0.009 0.017 0.012 0.011 0.011 0.015 0.010 68.183 46.170 31.876 51.725 52.057 52.360 52.520 55.716 EST.INTERCEPT 0.718 0.978 1.825 1.285 1.216 1.144 1.546 1.042 (S.E. OF INTR) FRACTION OF ERROR 0.545 0.038 0.057 0.123 0.077 0.029 0.174 0.037 DUE TO BIAS ^ 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 RES. VARIANCE 94.386 6.573 2.035 9.096 8.576 8.114' 10.483 9.445 MISSPEC ERROR 10.748 8.986 12.264 12.170 11.797 11.360 14.168 9.448 RESID.ERROR -1^ 148 i model with the PEXP results we could infer that: a) Whereas the PEXP model consistently overvalues the bonds, the LIQP model tends (more often than not) to undervalue them. This is indicated by the greater number of positive mean error figures in Table XLI. b) The slope coefficient of the regression (7.7) is a measure of relative responsiveness. If f <1, then the model is 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 it contains more information on the term structure of interest rates. To enable one to compare the different models across all bonds, a qlobal measure that aqqreqates the results of Tables XXI to XL is desireable. For this purpose, the regression of equation (7.7) was performed by pooling data of all the bonds, and the results are presented in Table XLIX., Be now investigate the impact on the model of incorporating taxes. 7,6 Bond Valuation With Revenue Taxes In this section, we look at the effect of including in the model taxes on coupons and interest, but not on capital gains. In Chapter 2, we developed the p.d.e. governing the bond valuation under specific assumptions about the way taxes are 149 applicable {see equation 2,11). The assumptions did appear to be a gross over-simplification of reality. The question, however, remains; are we better off without incorporating taxes into the model? Inclusion of revenue taxes in the bond valuation equation has two opposing influences. First, the coupon yield is reduced from cdt/G to c(1-R)dt/G, where c is the coupon and R the revenue tax rate. Thus, the net gain (or benefit) from owning the bond is reduced, and so its value is lowered. On the other hand, the rate of return on the instantaneously riskless asset is also reduced from rdt to r(1-R)dt, where r is the instantaneous riskless rate of interest. This has the opposite effect on the bond value - it pushes up the bond price. Whether the net effect of these two forces pushes the model price up or down is not a priori apparent. It was not clear what value of R to use in the model. Ideally, it should represent the marginal tax rate of the representative investor,. Since no one figure was available, it was decided to try both 8=25% and 50%. The value of the tax rate was kept constant over the whole period., The results of comparing market and model prices for these two cases are also reported in the same tables as the results of the previous two models, ie.. Tables XXI to XL. (See also Table XLIX). Comparing with the results of the LIQP model, we note that the mean error value (which was almost consistently positive due to under-valuation of model price) is equally positive and negative over the 20 bonds. This seems to imply that introduction of revenue taxes has pushed up model prices - at 150 least for this sample of bonds. Comparing the , we find 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 in 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 is between the two figures., So far, 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" is supposed to measure the joint movement of the two prices - the market and the model price. However, we should recognize that joint movement has two aspects: direction and magnitude. To clarify, if market price drops from one week to the next, and so does model price, there is perfect harmony between the two with respect to direction of movement. But if market price drops by 500, whereas model price by $1, then the model is over-reacting (which would show up in a low ^ value). We know that ^ can be expressed in terms of the correlation between the independent (market price) and dependent (model price) variables of the regression as f, - f • J^L (7-19' where Smfet and s ^ represent the standard deviation of the market and model prices respectively, and -f5 represents the correlation between the two. Now we can see that f is a measure of directional co-movement, whereas the ratio of the 151 standard deviations is a measure of the magnitude. This breakdown of ^ enables us to see which aspect has led to a change in the value of p - . , From Table XLII we see that increasing the tax rate (or even including it in the first place) does not improve the correlation between market and model prices - it is the magnitude factor that is affected. Thus, introducing revenue taxes helps in fine tuning the relative volatility of model and market price movements. 7 • 7 Bond Valuation- Incorpprating Capital Gains Tax Having introduced revenue taxes into the model in the last section, we proceed to see the effect on model price behaviour, vis-a-vis market prices, when we incorporate capital gains (CG) tax into the valuation model. Here again, the approach makes assumptions that appear simplistic (as pointed out in Chapter 2) but what we want to investigate is whether there is any improvement in the predictive power of the model. The effect on model prices of introducing CG taxes is unambiguous. The benefits to owning the bond are reduced, and so the model prices will decrease with its introduction. The effect on the mean error is clear (it is expected to increase) , but the effect on ^ , is 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 far). The results are presented in 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 is reflected in the value of the 152 mean error - the positive values have increased in 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 all the bonds, introducing CG taxes marginally improves the correlation between model and market prices but, in all cases, the ^ values go down. This implies that (S^ /S^^) goes down by more than ^ goes up (see eguation (7.19)), resulting in lower ^ values., The volatility of the model prices thus consistently increases (ie. Smo^ increases) with CG taxes. By appropriately choosing revenue and CG tax values, we can achieve both an improvement in •f and the slope. 7•8 The ffHovinq Average" Hodel From our analysis in the last two sections, we find that incorporating taxes into the model leads mainly to a "fine tuning" effect in 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 correlation as a measure of joint directional movement b) the p> coefficient as a measure of equal amplitude of movement 153 c) the mean error as a measure of bias65., We noticed that use of the liquidity premium hypothesis.-led to substantial improvements in all three measures. Incorporating taxes led to improvement on the first 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 "tool" to tackle the bias. In our case, tools are created by relaxing our prior assumptions to match reality. In the analysis in Chapter 5, we found that the interest rate process parameter ^ ' had the most significant impact on bond values. Ceteris paribus , increasing (decreasing) y. would lead to an across-the-board decrease (increase) in bond values. It was felt,therefore, that the assumption of time homogeneity of the interest rate process parameters (particularly^- ) was the principal source of bias. In this section, we adopt an approximate method of relaxing that assumption. Probably the most elegant approach to the problem to date is that of Brennan & Schwartz [12], who set up the bond price as a function of both the short term interest rate and a long term interest rate, where these two rates follow correlated diffusion " it may be argued that root mean square error (RMSE) is a better measure of overall error. From the results presented in Tables XXI to XL, it may be seen that the ranking of each bond across models using either mean error or BASE is virtually identical. 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 interest rate as the value of ^ for the short term interest rate process. However, there are several problems associated with the estimation of parameters of such joint process, as well as with the solution of the p.d.e. for bond valuation, which are beyond the scope of this study. Instead, what we do is to take as the value of jx for each bond, (R1 to E19) the average value of the short term interest rate in the two years immediately prior to the date of issue.^ This value of ^ is maintained constant for the life of that bond. The results of this approach are presented in Tables XXVI through XL for bonds E5 to E19. There does not appear to be any significant improvement in the fit between market and model prices from this approach. The ) and correlations move a little, but not in any particular direction; 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 results. So far, we have not looked at the sensitivity of bond values to the liquidity premium parameters. The parameters k, and k<3_ directly affect m and jx (as shown in equation 7. 10) , alterinq them as follows: ra» = (m-k^. ) JX' - {mjx +k, ) /m' Tables XLIII and XLIV present the price sensitivity of pure discount bonds to errors in k( and kx respectively. It may be noted that bond values do not appear to be very sensitive to chanqes in these parameters. However, variations across time in TABLE XLIII THEORETICAL SENSITIVITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN Kj^ ERROR IN Kx -25% -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 96.96 0.0896 96.89 0.0179 96.87 96.85 -0.0179 96.78 -0.0895 3 89.05 0.6138 88.62 0.1225 88.51 88.40 -0.1223 87.97 -0.6101 Vi/2 5 80.63 1.3539 79.77 0.2693 79.55 79.34 -0.2686 78.49 -1.3358 7 72.61 2.1805 71.37 0.4323 71.07 70.76 -0.4305 69.55 -2.1339 -3.3670 10 61.85 3.4843 60.18 0.6873 59.77 59.36 -0.6827 57.76 1 95.07 0.0896 95.01 0.0179 94.99 94.97 -0.0179 94.90 -0.0895- 3 85.64 0.6138 85.22 0.1225 85.12 85.01 -0.1223 84.60 -0.6101 5 76.97 1.3539 76.15 0.2693 75.94 75.74 -0.2686 74.93 -1.3358 7 69.13 2.1805 67.95 0.4323 67.65 67.36 -0.4305 .66.21 -2.1339 10 58.81 3.4843 57.22 0.6873 56.83 56.44 -0.6827 54.91 -3.3670 1 91.42 0.0896 91.35 0.0179 91.34 91.32 -0.0179 91.25 -0.0895 3 79.20 0.6138 78.81 0.1225 78.71 78.62 -0.1223 78.23 -0.6101 -1.3358 r=2y 5 70.14 1.3539 69.39 0.2693 69.21 69.02 -0.2686 68.28 7 62.65 2.1805 61.58 0.4323 61.31 61.05 -0.4305 60.00 -2.1339 10 53.16 3.4843 51.72 0.6873 51.37 51.02 -0.6827 49.64 -3.3670 cn cn TABLE XLIV THEORETICAL SENSITIVITY OF PURE DISCOUNT BOND PRICES TO ERRORS IN K, ERROR IN K. •25% -5% 0% +5% +25% BOND CURRENT TIME TO BOND % BOND % BOND BOND % % INTEREST MATURITY PRICE ERROR PRICE ERROR PRICE PRICE ERROR PRICE ERROR IN YEARS 1 96.85 -0.0258 96.87 -0.0051 96.87 96.88 0.0051 96.90 0.0255 3 88.31 -0.2238 88.47 -0.0443 88.51 88.55 0.0440 88.70 0.2178 r=y 12 5 79.10 -0.5643 79.47 -0.1112 79.55 79.64 0.1104 79.99 0.5441 7 70.37 -0.9843 70.93 -0.1936 71.07 71.20 0.1920 71.74 0.9445 10 58.76 -1.6802 59.57 -0.3302 59.77 59.96 0.3273 60.73 1.6086 1 94.95 -0.0440 94.98 -0.0088 94.99 95.00 0.0087 95.03 0.0435 3 84.85 -0.3154 85.06 -0.0624 85.12 85.17 0.0620 85.38 0.3067 r=y 5 75.40 -0.7141 75.84 -0.1407 75.94 76.05 0.1397 76.47 0.6881 7 66.86 -1.1675 67.50 -0.2296 67.65 67.81 0.2277 68.41 1.1200 10 . 55.75 -1.8845 56.61 -0.3704 56.83 57.03 0.3672 57.85 1.8051 1 91.26 -0.0805 91.32 -0.0160 91.34 91.35 0.0160 91.41 0.0796 3 78.32 -0.4983 78.64 -0.0986 78.71 78.79 0.0980 79.10 0.4847 69.07 -0.1997 69.21 69.34 0.1982 69.88 0.9769 r=2U 5 68.51 -1.0131 7 60.37 -1.5329 61.13 -0.3016 61.31 61.49 0.2992 62.21 1.4721 10 50.19 -2.2919 51.13 -0.4508 51.37 51.60 0.4471 52.50 2.1990 \ 157 these parameters could account for a reasonable amount of the bias between existing model and market prices, as the extent of bias in percentage terms is also guite small. 7.9 Tests of Market Efficiency We proceed to test the efficiency of the market for retractable/extendible bonds to information contained in the models. In deriving the basic bond valuation eguation in Chapter 2, we used a hedging argument, wherein a zero net investment portfolio was formed by going long on the generic bond, short on any other bond, and finally making up the difference by borrowing or investing in the short term riskless asset.. The dollar amounts to be invested in each asset were given as: where x, = dollar investment in generic bond x = dollar investment in any other bond and G represents the generic bond price (with G( its partial derivative with respect to the interest rate) and B the price of any other bond (with B( its partial derivative with respect to the interest rate). The investment in the riskless asset is - (x^ + x2 •).-... 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 partial 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 valid. In our first test of market efficiency, we assume that at the beqinninq of each period (which is a week in 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 in a staight bond, and the balance is made up by an investment in the riskless asset. At the end of the period, we assume that we liquidate this portfolio at the then-existing market prices, and compute the return to the portfolio over the one period. Be then form a new portfolio, and proceed on until 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 indication is that the returns to the zero-investment hedge portfolios are insignificantly different from zero6*. It appears that we cannot reject the hypothesis that the market is efficient to information contained in the models. An alternative strategy was also adopted for testing market efficiency. It was observed that the hedge portfolio returns on the above test were highly serially correlated. The second strategy tested was to assume a long position in the generic bond only if the portfolio return in the previous period (based on a constant long position in the generic bond) was positive - 66 Hypothesis testing was based on the t-statistic, which assumes that the returns to the hedge portfolio are normally distributed. Thorpe [68] has shown that in the option pricing framework the hedge portfolio returns are not normally distributed., This need not be cause for concern, as the t-test is quite robust to reasonable departures from normality. The distribution of the hedqe portfolio returns is very briefly' investiqated toward the end of this section. TABLE XLV RETURN ON ZERO INVESTMENT PORTFOLIO BASED ON CONSTANT LONG POSITION IN BOND (Results for all models) BOND PURE EXP. LIQ.PREM. REV. TAX(50a:) REV.TAX(25%) C.G.TAX(IOX) C.G.TAX (20Z) MOV. AVG. R 0.0286 0.0210 -0.0012 0.0102 0.0133 0.0171 - 1 (0.2784) (0.2478) (0.1890) (0.2108) (0.2203) (0.2331) E -0.0676 0.0368 0.0440 0.0404 l 0.0392 0.0378 - (0.4558) (0.1852) (0.1800) (0.1804) (0.1980) (0.1833) E2 -0.0059 0.0506 0.0668 0.0586 0.0562 0.0533 - (0.3804) (0.280) (0.2961) (0.2837) (0.2816) (0.2801) E 0.0288 3 0.0695 0.0968 0.0829 0.0079 0.0751 (0.3399) (0.2982) (0.3462) (0.3145) (0.3087) (0.3032) _ E -0.0093 0.0515 4 0.0720 0.0671 0.0587 0.0551 - (0.3395) (0.2291) (0.2675) (0.2436) (0.2383) (0.2331) E -0.0022 0.0054 5 -0.0007 0.0024 0.0032 0.0042 0.0037 (0.3748) (0.1722) (0.1728) (0.1652) (0.1659) (0.1683) (0.1663) E -0.1011 0.0068 6 0.0169 0.0119 0.0103 0.0084 0.0069 (0.4254) (0.5235) (0.3560) (0.4254) (0.4522) (0.4869) (0.5419) F -0.0453 -0.0067 7 0.0089 -0.0052 -0.0072 -0.0091 -0.0045 (0.6514 (0.3868) (0.3119) (0.3654) (0.3790) (0.3938) (0.3710) E -0.0453 0.0020 0.0282 0.0147 0.0131 0.0094 0.0096 8 (0.7945) (0.4821) (0.3760) (0.4127) (0.4173) (0.4189) (0.4337) E -0.0207 0.0112 0.0143 9 0.0055 0.0058 0.0066 0.0144 (0.7338) (0.3547) (0.3138) (0.3681) (0.3783) (0.3885) (0.3087) E 0.0058 -0.0019 10 0.0068 0.0057 0.0045 0.0030 0.0028 (0.5242) (0.2968) (0.2455) (0.2747) (0.2835) (0.2962) (0.2732) E 0.0110 0.0034 0.0043 0.0061 0.0062 0.0060 0.0037 11 (0.3628) (0.2759) (0.2970) (0.2828) (.02825) (0.2828) (0.3069) E 0.0042 0.0047 12 -0.0019 -0.0007 0.0000 0.0009 -0.0035 (0.6296) (0.4063 (0.3577) (0.4106) (0.4204) (0.4307) (0.4626) E 0.0048 13 0.0016 -0.0067 -0.0041 -0.0036 -0.0031 -0.0039 (0.4381) (0.3558) (0.3975) (0.3600) (0.3579) (0.3568) (0.3572) E 0.0014 14 0.0024 0.0010 0.0006 -0.0007 0.0009 0.0000 (0.4031) (0.2784) (0.2753) (0.2750) (0.2792) (0.2848) (0.2846) E 0.0088 15 0.0231 0.0318 0.0271 0.0266 0.0260 0.0234 (0.4816) (0.4037) (0.4351) (0.4056) (0.4034) (0.4022) (0.4003) E 0.0164 0.0227 16 0.0242 0.0233 0.0233 0.0232 0.0196 (0.5054) (0.4195) (0.4412) (0.1463) (0.4155) (0.4158) (0.4144 E17 -0.0222 -0.0148 -0.0124 -0.0163 -0.0169 -0.0176 -0.0121 (0.4739 (0.3905) (0.3810) (0.3861) (0.3901) (0.3953) (0.3785) E 0.0008 0.0120 0.0220 18 0.0169 0.0158 0.0146 0.0173 (0.2369) (0.2209) (0.2459) (0.2279) (0.2252) (0.2229) (0.2278) E -0.0319 19 -0.0056 0.0178 0.0016 -0.0015 -0.0049 0.0066 (0.4985) (0.4221 (0.4102) (0.4116) (0.4152) (0.4203) (0.4083) 160 if negative, a short position was assumed in G, and the hedge position formed accordingly. This strategy was tested for all models, but only the results for the pure expectation hypothesis model are presented in Table XLVI, because the results are very similar for all the other models. There is no reason to alter our previous conclusion. The third test was to see if the model was able to identify over- and underpriced bonds. This test (based on a test in Galai[29]) is quite similar to the previous ones, only that each period we take a long (short) position in the generic bond if its model price is lower (higher) than the market price at that point.. If the return on the hedge portfolio based on this strategy resulted in a statistically significant increase in the mean return, over the strategy of a constant long position in the generic bond, we could say that the model is able to identify overpriced/underpriced bonds. The results of this test for all models and bonds is presented in Table XLVII. Here again, the mean return appears to be insignificantly different from zero, based on a t-test. The results of the previous three tests were based on the returns to hedge portfolios, using one bond at a time. It was felt that if the hedge returns over all bonds outstanding in each period was considered (along the lines of Brennan S Schwartz [ 11 ]), the aggregation might lead to a reduction in the variance of the returns to the hedge portfolio and thereby improve the statistical significance of the returns. To overcome the problem of heteroscedasticity caused by the different numbers of hedge portfolios in each period, the dollar 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 for PEXP model only) BOND Mean Std. Dev. of t- Stat Return ($)Return Rl -0.0412 2.777 -0.149 El -0.0808 0.454 -0.178 E2 0.0208 0.380 0.055 E3 0.0491 0.338 0.145 E4 0.0206 0.339 0.061 E5 0.0531 0.372 0.143 E6 0.0556 0.153 0.036 E7 0.0612 0.650 0.094 E8 -0.0639. 0.793 -0.081 E9 0.0961 0.728 0.132 E10 0.0159 0.524 0.032 EH -0.0099 0.363 -0.028 El 2 -0.0693 0.626 -0.111 E13 -0.0840 0.430 -0.195 E14 -0.0281 0.402 -0.070 E15 -0.0078 0.482 -0.016 E16 0.0042 0.506 0.008 E17 -0.1990 . 0.430 -0.462 E18 -0.0415 0.233 -0.178 E19 -0.1470 0.477 -0.307 TABLE XLVII RETURN ON ZERO INVESTMENT PORTFOLIO BASED ON VARYING POSITION IN BOND (Results for all models) BOND PURE EXP. LIQ. PREM. REV.TAX(50Z) REV.TAX(25Z) C.G.TAX(10%) C.G.TAX(20%) MOV.AVG. R -0.0636 -0.0210 0.0020 -0.0102 -0.0103 -0.0173 ** 1 (0.2724) (0.2478) (0.1890 (0.2108) (0.2203) (0.2331) E -0.0848 -0.0368 -0.0440 -0.0404 -0.0392 -0.0378 1 (0.4558) (0.1852) (0.1800) (0.1804) (0.1814) (0.1833) - E -0.0095 -0.0449 -0.0668 -0.0473 -0.0453 -0.04 70 2 (0.3804) (0.2814) (0.2961) (0.2858) (0.2836) (0.2812) - 0.0288 -0.0790 -0.0882 -0.0822 -0.0788 -0.0751 E3 (0.3399) (0.2988) (0.3485) (0.3147) (0.3088) (0.3032) - -0.0093 -0.0429 -0.0625 -0.0502 -0.0477 -0.0457 E4 - (0.3395) (0.2309) (0.2700) (0.2463) (0.2408) (0.2352) E -0.0223 -0.0060 0.0024 -0.0016 -0.0029 -0.0042 0.0029 5 (0.3748 (0.1722) (0.1728) (0.1652) (0.1659) (0.1683) (.16632) E -0.0433 0.0027 -0.0095 -0.0106 -0.0092 -0.0051 0.0080 6 (0.4823) (0.5236) (0.3563) (0.4254) (0.4522) (0.4870) (.54194) -0.0042 0.0110 -0.0052 -0.0094 -0.0133 -0.0172 E -0.0453 7 (0.6514) (0.3868) (0.3119) (0.3654) (0.3789) (0.3937) (0.3706) 0.0104 0.0040 -0.0094 E -0.0435 -0.0185 -0.0263 0.0055 8 (0.7549) (0.4817) (0.3761) (0.4129 (0.4173) (0.4190) (0.4337) E -0.0207 0.0063 0.0129 0.0055 0.0058 0.0229 0.0115 9 (0.7338) (0.3549) (0.3139) (0.3681) (0.3783) (0.3879) (0.3088) E 0.0045 -0.0039 -0.0143 -0.0023 -0.0055 -0.0106 -0.0091 10 (0.5242) (0.2968) (0.2452) (0.2747) (0.2835) (0.2960) (0.2730) E 0.0142 -0.0030 -0.0098 0.0096 0.0219 0.0102 -0.0037 11 (0.3627) (0.2759) (0.2968) (0.2827) (0.0281) (0.2827) (0.3069 E 0.0042 -0.0170 -0.0024 -0.0072 0.0003 0.0321 -0.0058 12 (0.6296) (0.4060 (0.3577) (0.4106) (0.4204) (0.4295) (0.4626) E 0.0048 -0.0219 -0.0175 -0.0041 T0.0093 -0.0182 -0.0039 13 (0.4381) (0.3552) (0.3972) (0.3600) (0.3577) (0.3564) (0.3572) 0.0006 -0.0056 0.0076 0.0004 E 0.0014 -0.0047 -0.0033 14 (0.4031) (0.2784) <0.2753) (0.2750 (0.2792) (0.2847) (0.2846) E 0.0088 -0.0167 -0.0335 -0.0303 -0.0221 -0.0322 -0.0338 15 (0.4816) (0.4040) 40.4349) (0.4054) (0.4037) (0.4017) (0.3996) E 0.0164 0.0086 -0.0313 -0.0177 -0.0286 -0.0240 0.0142 16 (0.5055) (0.4201) (0.4408) (0.4166) (0.4151) (0.4158) (0.4146) E -0.0222 -O.0310 -0.0124 -0.0163 -0.0169 -0.0204 0.0121 17 (0.4739) (0.3895) (0.3810) (0.3861) (0.3901) (0.3952) (0.3785) E -0.0171 -0.0120 -0.0220 -0.0169 -0.0158 -0.0146 -0.0173 18 (0.2363) (0.2209) (0.2539) (0.2279 (0.2252) (0.2229) (0.2278) E -0.1421 -0.0948 -0.0076 -0.0744 -0.0781 -0.0938 -0.0540 19 (0.4764) 10.4061) (0.4025 (0.3986) (0.4020) (0.4042) (0.3981) 163 return in each period was weighted by 1/JIT, where N represents the number of retractable/extendible bonds outstanding (which therefore represent the number of hedge portfolios formed) in each period. The results are presented in Table LI. The mean dollar return per period, as well as its standard deviation, remained of the same order of magnitude as in the case of the results in Tables XL? to XLVII - aggregation has not led to any statistically significant increased profit opportunity. This result was not unexpected. The movement of bond prices exhibits high contemporaneous correlation,so that the returns to the zero investment hedge portfolios would also be likewise correlated., Thus, aggregating across bonds at any instant in time would not lead to any significant reduction in the dispersion of returns to the hedge portfolio. In the case of options on common stock, however, the contemporaneous correlation across different stocks is not so high, which could lead to variance reduction due to aggregation on a similar test. In forming the hedge portfolios, for an investment of x, dollars in the generic bond, the strategy was to invest x^ dollars in another bond, where xL was given by In the tests performed so far, the value of B used in the above expression was the market price of the straight bond. It could be argued that model prices should be used for B. The reasoning is that we want to observe whether the retractable/extendible bond offers arbitrage profit opportunities, after controlling for other factors. When we use market price for B, due to the TABLE LI RETURN ON ZERO NET INVESTMENT PORTFOLIO (BASED ON A CONSTANT LONG POSITION IN THE GENERIC BOND) BY AGGREGATING OVER ALL BONDS (Results for all models) Mean Std. Dev. Model Return ($) of Return t-Stat PEXP -0.0197 0.955 -0.021 LIQP 0.0228 0.476 0.048 REV.TAX (50%) 0.0352 0.486 0.072 REV.TAX (25%) 0.0270 0.451 0.060 CG.TAX (10%) 0.0258 0.457 0.056 CG.TAX (20%) 0.0242 0.466 0.052 MOV.AVG. 0.0113 0.484 0.023 Notes: 1) The above test, by aggregating over all kinds outstanding in every period, was also performed on the other two market efficiency tests. The results are not reported as they are very similar to the ones above. 2) The models have been listed in the table using the abbreviations used in the text and in earlier tables. 165 valuation error in B, the correct hedge proportions are not maintained which increases the variance of the returns to the zero investment hedge portfolio. By using model values of 8, there is no other source of error - it is a pure test of the retractable/extendible bond. all three market efficiency related tests reported above were repeated, (for each individual bond and aggregated over all bonds) for each of the models used for valuing the retactabie/extendible bonds. The results were hardly any different from those obtained by using market price of B to evaluate x% , as well as for evaluating the hedge portfolio returns. To indicate the degree of similarity of results from using market and model prices of B in the tests of market efficiency, the mean and standard deviation of the zero investment hedge portfolio return for the Capital Gains Tax 20% model(using a strategy of a constant long position in the generic bond) are presented in Table L.., It was felt that no further information would be conveyed by presenting the complete results across all models for all three hedging strategies. Finally, the portfolio returns (on the zero investment hedge position) were tested for normality using the probability graphing approach outlined earlier.. This was not strictly necessary, as the t-statistic of the mean return (ie. mean/standard deviation) was almost always of the order of 0.1 and that should be statistically insignificant in most situations even with resonable departures from normality. In Figures 8 and 9, we present two sample cases., In general, the distributions appear to have more mass at the mean than a normal distribution 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 USING MARKET PRICES FOR STRAIGHT BOND STRAIGHT BOND BOND MEAN STD.DEV - t-STAT MEAN STD.DEV t-STAT Rl 0.0135 0.287 0.047 0.0171 0.233 0.073 El 0.0627 0.349 0.180 0.379 0.183 0.207 E2 0.0622 0.517 0.120 0.0534 0.280 0.191 E3 0.0831 0.664 0.125 0.0752 0.303 0.248 E4 0.0631 0.474 0.133 0.0551 0.233 0.237 E5 -0.0019 0.220 -0.009 0.0042 0.168 0.025 E6 0.0010 0.334 0.003 0.0084 0.487 0.017 E7 0.0125 0.425 0.029 -0.0091 0.394 -0.023 E8 0.0193 0.452 0.043 0.0095 0.419 0.023 E9 0.0106 0.567 0.019 0.0067 0.389 0.017 ElO 0.0058 0.505 0.011 0.0030 0.296 0.010 Ell 0.0089 0.686 0.013 0.0061 0.283 0.021 E12 -0.0034 0.343 -0.010 0.0010 0.431 0.002 E13 -0.0075 0.482 -0.016 -0.0031 0.357 -0.009 E14 -0.0032 0.375 -0.009 0.0009 0.285 0.003 E15 0.0200 0.509 0.039 0.0260 0.402 0.065 E16 0.0281 0.511 0.055 0.0232 0.416 0.056 E17 -0.0257 0.484 -0.053 -0.0176 0.395 -0.045 E18 0.0149 0.289 0.051 0.0147 0.223 0.066 El 9 -0.0152 0.588 -0.026 -0.0049 0.420 -0.012 Aggregate 0.0265 0.786 0.034 0.0242 0.466 0.052 NOTES: 1) The above returns correspond to using the constant long position in the generic bond strategy. 2) The model used for the valuation of the generic bond was the Capital 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 OF fCDEE PORTFOLIO RETURNS •-NORMRL PROBABILITY PLOT OF HEDGE PORTFOLIO RETURNS "MHEDGE BRSED ON VRRm'G POSITION IN BOND "HEDGE BRSED ON VRRTING POSITION IN BOND :IKDD£L RDJU5TING FOR tRPITRL GRINS TRX) (MODEL RD JUST ING FOR CRPITRL GRINS TRX) KWDB\-.5.5DJ «P» 4 1963 501 1 96 BOHDi f4-*- I*™ l ^ Jtlli -0.45 « 10 » -I •SmntVi 0.335 -O.OflS HU3GE K1URH IK " VPLUt tr HECSE !« FIGURE 9 168 CQMPRRISDN OF MARKET I MODEL PRICES (MODEL ADJUSTING FOR CRP1TRL GRINS TRX) BDNDi E7s 7.25X RPR 19 1974 MRRKET PRlCEl HOOtL PRItti DO00DD0 DISTRIBUTION OF HEDGE PORTFOLIO RETURNS NORHRL PROBRBILITY PLOT OF HEDGE PORTFOLIO RETURNS HEDGE BR5ED ON VRRY1NG POSITION IN BOND HEDGE BRSEO ON VRRY1NG POSITION IN BOND (MODEL BDJUSTING FOR CRPITRL GRINS TAX) IHODEL ADJUSTING FOR CRPITRL GRIN5 TRX! BONOftt T.75I RPR ]9 1974 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 in valuing bonds. To this end, we develop an ad hoc valuation model - which we shall refer to as the "naive" model. It is based on an approach suggested in Dipchand S Banrahan [9]. Based on a regression equation for the yield curve developed by Bell Canada*s Bond Research Division, we compute the yield to maturity on each extendible*7 in our sample. For each bond, at each point in time, we estimate two yields to maturity - one correspondinq to each of the alternative maturities. Usinq each yield, we discount the future coupons (assuming continuous coupon payments) and the principal, and thus find the values of the long and short bonds. The price of the extendible is then set to the higher of the long and short bonds, at every point in time. Bell Canada's yield curve regression model was where Yt represents the yield to maturity at time t on a bond having Xt months to maturity. For our study, we modified the model slightly to include in the regression equation the current value of the short term interest rate. It was felt that this inclusion should improve the fit of the model. Thus, the 67 The retractable B1 was not priced according to the naive model because it had several retraction dates. This makes it complicated to price, and it was felt that dropping one case should not affect the comparison. 170 regression model used to determine yields was -va^ -V Yt at + a3X<. c^Xt + <*5 \ + 4 CL^Xf. (7.20) The next problem was to determine the coefficients. For this purpose, the straight coupon bond sample was used. The market price of a bond at any instant is the present value of its future payoffs. Thus we can write \ = c*~* dtt + loo er where y is the yield to maturity at time t, c is the continuous coupon, X is the time to maturity, and the face value of the bond is $100. This gives (7.21) 68 Using eguation (7.21) above, we can solve for y , given B% and the other parameters. This was done for all 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 results of the regression are reported in Table XLVIII. Consistent with the experience of Dipchand S Hanrahan [19], the R2 from most of the equations was over 0.80 (except for F9 and the whole sample). The regression coefficients based on the total 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 starting value supplied in the search for a root was the current value of r. It can easily be shown that equation (7.21) has only one root. 171 TABLE XLVIII RESULTS OF YIELD EQUATION COEFFICIENT ESTDIATION FOR "NAIVE" MODEL - 2 3 (Yeild = ax + a2rt + a^T + a^/i + ajT + a6T + a7logT ) 2 3 X 2 5 9 a^lO a 2 BOND a3«10 «4 1.0 R2 2 a5*10 a6xl0 a7*10 Fl 0.6510 0.4857 0.8323 -0.0434 -0.3163 7.8739 0.8427 0.9418 (16.42) (14.96) (5.13) (-4.13) (-6.72) (7.89) (3.01) F2 0.0776 0.4636 -0.0241 0.0652 0.00151 -0.0046 -0.0940 0.9295 (13.53) (36.70) (-11.61) (12.71) (7.76) (-4.03) (-11.85) F3 0.0z82 0.5863 -0.0057 0.0210 -0.0001 - 0.0032 -0.0347 0.8977 (15.42) (52.52) (-3.10) (4.57) (-0.69) (3.90) (4.97) F4 0.0531 0.8941 0.1026 -0.1273 -0.0414 1.0565 0.1129 0.9342 (6.41) (43.28) (3.90) (-3.14) (5.69) (7.62) (2.61) F5 -0.0658 0.9979 -0.3540 0.4993 0.1211 -2.7779 -0.5007 0.9270 (-1.31) (31.38) (-2.87) (2.44) (3.89) (-4.84) (2.09) F6 0.0882 0.59S2 0.0098 -0.0105 -0.0018 0.0140 0.0004 0.8916 (20.78) (47.67) (5.31) (-2.63) (-9.12) (10.76) (0.07) F7 0.3142 0.8066 0.0918 -0.0486 -0.0546 1.4785 -0.0668 0.9513 (20.24) (20.71) (1.33) (-0.45) (-2.94) (4.28) (-0.58) F8 -2704.9 0.3566 251.09 -172.87 -0.3991 0.5026 870.01 0.9004 (-3.38) (21.87) (3.20) (-3.26) (-3.06) (2.92) . (3.33) F9 -0.0803 0.9587 0.2523 -0.4556 -0.0460 0.5171 0.5555 0.4990 (-1.17) (6.86) (3.24) (-3.23) (-3.11) (2.94) (3.15) F10 0.0738 0.5713 0.1367 -0.2001 -0.0483 1.2322 0 .2116 0.9635 (3.09) (15.66) (2.33) (-2.54) (-2.33) (2.49) (2.89) .0289 Fll 0 0.6947 -0.0081 0.0097 0.0022 -0.0338 -0.0027 0.9015 (3.50) (23.19) (-1.27) (1.01) (1.46) (-1.54) (-0.31) F12 371.99 0.4623 -8.6736 42.659 0.2689 -0.6572 -153.04 0.8614 (4.45) (25.37) (-4.83) (4.69) (5.13) (-5.47) (-4.56) F13 0.0896 0.8474 0.0677 -0.0877 -0.0189 0.2772 0.0695 0.8695 (9.19) (15.63) (4.13) (-3.76) (-4.37) (4.36) (3.21) F14 4195.5 -0.2159 80.383 -213.34 -3.1702 7.0379 -437.32 0.8404 (0.70) (-2.62) (0.79) (-0.62) (-0.81) (0.79) (-0.35) F15 2031.73 0.3441 -30.575 175.98 0.6934 -1.2435 -739.53 0.8772 (5.35) (19.14) (-5.59 (5.50) (5.76) (-5.93) (-5.42) F16 -0.2189 0.4323 0.0579 -0.1699 -0.0038 0.0148 0.3186 0.9306 (-0.25) (43.25) (1.90) (-1.28) (-3.33) (4.68) (0.77) F17 .0081 -0 0.7408 -0.0447 0.0672 0.0109 -0.1579 -0.0598 0.8818 (-1.13) (25.08) (-5.25) (4.95) (5.50) (-5.46) (-4.32) El 8 0.0154 0.4883 -0.0371 0.0599 0.0080 -0.1020 -0.0580 0.8750 (4.22) (30.39) (-11.96) (12.20) (12.41) (-12.74) (-12.18) TOTAL 0.0509 0.7049 -0.0067 0.0184 0.0005 -0.0019 -0.0271 0.7679 (16.81) (138.12) (-12.39) (11.78) (14.20) (-15.34) (-9.80) - Figures in parenthesis are the t statistic for the estimated coefficient J 172 in our sample. The results of regressing the market prices on these model prices are reported in Tables XXII to XL. The results from the summary run by aggregating over all the 19 bonds is in Table XLIX. A cursory examination of the results indicates that the naive model performs reasonably well, in comparison to the other models. Closer scrutiny however reveals the superiority of the ,-, more rigorous models of retractable/extendible bond valuation developed in this study. The three criteria used to evaluate the performance of each model were: 1) correlation 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 results of the Capital Gains Tax 20% (CG Tax 20%) model (column 6 in Tables XXI to XL and Table XLIX) with that of the naive model, it is seen that the CG Tax 2.0% model outperforms the naive model on the first two counts almost consistently. Looking at the summary results from pooling all 20 bonds (Table XLIX), we see that above observation is borne out with respect to the slope coefficient. The correlation between market and model prices (the square root of the R- squared is the simple correlation coefficient) is marginally superior in the naive model. However, as pointed out in the earlier sections; altering the Revenue Tax rate and the Capital Gains Tax rate, provides a "fine tuning" mechanism to improve the correlation and the slope coefficient. Since the objective of the present study is more one of description, rather than of "fitting" the best model, no further attempt was made to find a 173 set of tax rates that actually provided consistently improved correlations over that of the naive model. Finally, looking at the bias measures, we see from the summary results in Table XLTX that the mean error is lower for the CG Tax 20% model, whereas the P.MSE is lower for the naive model. Comparison of the mean error over individual bonds (Table XLI), we see an almost even split - the naive model performs better just as many times as the CG Tax 20% model. However, we note that if we were to increase the CG Tax rate used in the model, this would lead not only to a reduction in the mean error but also to an improvement in the correlation. Thus, it would be fair to say that, even in their present state, the partial equilibrium models developed in Chapter 2 are superior in several respects in predicting retractable/extendible bond price movements, when compared with a naive model of a reasonable level 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 specification to model the instantaneous riskless rate of interest. 2) identifying methods to estimate the parameters of such a model, given a discrete time realization of the interest rate process, and comparing the relative efficiencies of the different 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 riskless interest rate process. In the absence of any formal guidelines, economic reasoning and mathematical tractability were the only criteria. A mean-reverting diffusion process was suggested, having a drift term of the same form as that adopted by others in the existing literature, (see Vasicek [72], Cox,Ingersoll & Boss [16]) but with a more general variance element. Thus, the diffusion eguations adopted by Vasicek [72] and Cox, Ingersol 6 Ross [16], are both special cases of the more general form used in this study. The behaviour of the assumed form of the interest rate process at its singular boundaries is investigated, to ensure that its behaviour at these points is 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 interest rate process, and their relative merits and weaknesses are pointed out. All of them are maximum likelihood methods. The Transition Probability density method is exact, but the transition probability density is not known for all parameter values of the proposed process. Its use would require curtailing the generality of the interest rate process model. The other two methods (the Steady State density approach and the Simple Linearization method) are both based on approximations. Ho analytical method could be developed to compare the estimators of the parameters - Monte Carlo methods had to be employed. Chapter 5 presents the results of the Monte Carlo simulations to arrive at the distribution of the estimators, using the three alternate methods of parameter estimation. The criteria 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 all three methods produce estimators with rather similar properties, and so are quite comparable. Partial equilibrium valuation models based on the option pricing approach were developed in Chapter 2, for very general stochastic specifications of the interest rate process. The valuation models draw heavily from the earlier works of Cox, Ingersol & Boss [16], Brennan 6 Schwartz [10,12], and Vasicek [72]. The performance of models developed in Chapter 2, when the interest rate process of the chosen form is 176 incorporated, in pricing a sample of retractable/extendible bonds was tested in 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 in January 1959 and 19 extendibles issued between October 1959 and October 1975. weekly data on market prices for this set was collected from the Bank of Canada Review . Model prices based on the pure expectations hypothesis about the term structure of interest rates on the part of investors were consistently higher than actual market prices. When a provision was made for a term/liquidity premium in the term structure of interest rates, model prices were more in line with market prices. Incorporating revenue taxes (taxes on interest payments and on coupon receipts) and then capital gains taxes, improved the performance of the model in predicting market price 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. It was found that the partial equilibrium models performed atleast as well as the ad hoc model - with further refinements the equilibrium models could dominate the ad hoc model. Finally, the efficiency of the bond market to information contained in the models was tested. The approach was to set up a zero net investment hedge portfolio by investing in the retractable/extendible bond, the short term interest rate, and any other bond, and observing whether any arbitrage opportunities were available. The results indicated that the 177 market was consistently efficient to information contained in these models. 8*2 Conclusions And Directions For Further Research The interest rate process proposed in Chapter 3 is of the form The processes used in earlier studies were special 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 results in Chapter 5 indicate that increasing the generality of the model by including an extra free parameter (o() in the variance element does not materially enrich the family of processes. It was found that cr2 and c\ were very highly correlated and their influence on the process dynamics was almost totally substitutable. It appears that bond values resulting from the above interest rate process are most sensitive to the parameter jx the overall mean of the process. What is more interesting is the fact that the other parameters ( mn , impact on bond valuation. This is an indication that, even though the above model of interest rates may be quite satisfactory to portray the interest rate dynamice jper se , as far as bond valuation is concerned we have only a one-parameter process. This clearly indicates the need to look for alternative stochastic specifications for the interest rate process, where more than one parameter has a significant impact 178 on bond valuation.. The assumption of homogeneity over time of the interest rate process parameters appears restrictive. The constraint is, however, to afford mathematical tractability, both for the estimation of the parameters of the process as well as in bond valuation. The approach of Brennan S Schwartz [12] appears to be one elegant solution to the problem. In the framework of the interest rate model of this thesis, their model for the short term interest rate is eguivalent to setting = 1 and making stochastic (they set yjL as the long term interest rate), where r and jx follow correlated joint diffusion processes. As pointed out in the text, such processes pose additional problems in estimation of the parameters, and even more in solving the resultant valuation equation. However, the additional effort might well be worthwhile. We have seen that ^ is the critical parameter of the interest rate process in bond valuation; allowing it to be stochastic should lead to improved congruence between model and market prices. The term structure of interest rates plays a pivotal role in 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 interest rate. This is rather ambitious. The approach of Brennan & Schwartz [12] is an attempt to predict the term structure, at any instant in time, knowing the two extreme points - the instantaneous and the very long term yeilds. Thus, it 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 significantly better results. It is evident from the brief survey of the existing literature presented in Chapter 1 that a fair amount of work needs to be done in the area of empirical testing of bond valuation models developed in the option pricing framework. The present thesis is one step in that direction. However, we have addressed only the valuation of default-free bonds. The whole area of corporate bonds {where a positive probability of default exists) has not been tackled. The valuation theory has been developed in the literature, but empirical testing poses the problem of choosing some observable proxy for the value of the firm, as this is a required input to the bond valuation model. This would be a fruitful direction for future research. Finally, there is considerable interest at present in arrivinq at closed form or analytical solutions to the term structure equation. Vasicek [72] and Cox, Ingersoll 6 Ross [16] have two different stochastic specifications to model the course of the instantaneously riskless rate of return., It can, in qeneral, be shown that the resultinq pure discount bond valuation equation closely resembles the Kolmogorov backward eguation qoverninq the diffusion equation chosen to model the interest rate process. It is also well known that, in general, by a suitable redefinition of variables, the backward equation may be transformed into a similar forward equation, as pointed out in 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. 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Ltd., Amsterdam. 58 Richard S.F., (1976), "Analytical Model of the Term Structure Of Interest Rates", Carnegie Mellon University, WP.No. 19-76-77. 59 Roussas G, (1965), "Extension to Markov Processes of a Result By A. Wald about the Consistency of the Maximum Likelihood Estimate", Z. wahrscheinlichkeitshteorie Verw. Geb., Vol. 4. 60 Sargan J.D. (1974), "Some Discrete Approximations to Continuous Time Stochastic Models", J. of Royal Stat. Sco. B. 36. 61 Scholes M. (1976), "Taxes and the Pricing of Options", Journal Of Finance Vol. 31 (May 1976). 62 Schroedinger E. (1926), "Quantisierung als Eigenivertproblem", Ann. der Physik, Vol 80. 63 Schwartz E.S. (1975), "Generalized Option Pricing Models: Numerical Solutions and the Pricing of a New Life Insurance Contract", Ph.D. Thesis, U.B.C. 64 Sirjaev A.N. (1972), "Statistics of Diffusion Processes", European Meeting of Statisticians, Budapest, (Hungary). ^ 65 Smith CH. (1976), "Option Pricing: A Review", JFE, Vol. 3, No. 1/2. 66 Theil H. (1965), "Economic Forecasts And Policy",north Holland Publishing Co. Ltd., Amsterdam. North Holland Publishing Co. Ltd., Amsterdam. 67 Theil H. (1970), "Principles of Econometrics", John Wiley S Sons, New York. 68 Thorpe E. 0. (1976), "Common Stock Volatilities In Option Formulas", Working Paper, Centre For Research In Security Prices, Graduate School Of Business, University Of Chicago. 69 Titchmarsh B.C., (1962), "Eigenf unction Expansions associated with Second-order Differential Equations", Oxford University Press., 70 Trischka J. S Salwen H. (1959), "Dipole Moment Function of a Diatomic Molecule", J. of Chem. Physics. Vol. 31 No. 1, pp 218. , 71 VanHorne J.C. (1970), "Function and analysis of Capital Market Rates", Prentice Hall. 72 Vasicek 0. (1977), "An Equilibrium Characterization of the Term Structure", Forthcoming JFE. 73 Wolfowitz J. (1953), "Estimation by the Minimum Distance Method", Ann. Inst. Stat. Math., Vol 5. 74 Wolfowitz J., (1957), "The Minimum Distance Method", Ann. Math. Statist., Vol 28. 75 Wymer C.E. ,(1972), "Econometric Estimation of Stochastic Differential Equation Systems", Econometrica, Vol. , 40, No. 3, pp. 565. 76 Zellner A. (1971), "An Introduction to Bayesian Inference in Econometrics", John Wilny & Sons Inc. 187 APPENDIX - 1 Classification of singular boundary behaviour for the cases a=%,l. We have as our diffusion equation dr = b(r)dt + a(r) dZ (Al.l) Al 2 2a where b (r) = m(y-r) and a (r) = a r (AOi.la) The type of behaviour at the singular boundary is determined by the integrability of the following two functions fr 1 hx(r) = 7T(r) [a (s) IT (s) ] ds (A1.2a) ro fr -1 h2(r) = [a(r) TT (r) ] TT (s) dS (A1.2b) Over the interval I E [r #r ], where rQ is any interior point of the state space of the process, and r is the boundary (r might be infinite in the case of no "built in" finite boundary) The function 'k(r) in equations (A1.2) is defined by rr TT (r) = exp {-2 [b (s)/a(s)]ds} (A1.2c) When both h^ and h^ are integrable over I, the boundary is called a regular boundary and by imposing suitable boundary conditions, the behaviour can be either reflecting or absorbing. When h-^ is integrable over I, but h2 is not, the boundary is called an exit boundary and it acts as an absorbing boundary. When h-^ is not integrable over I, but h2 is, the boundary is called an entrance boundary. An extrance boundary is inaccessible from inside the open interval (rQ,f), but any 188 probability assigned to it initially, flows into the open interval. When both h-^ and h.^ar e not integrable over I, the boundary is called a natural boundary. This boundary is inaccessible from inside the open interval, and any probability assigned to it initially is trapped there forever. It can be shown that (see Keilson [41]) rr h-j_(s)ds = M^(r|rQ) = average time to reach r starting r 0 from rQ (rQ is a reflecting boundary) r h2(s)ds = M^(rQ|r) = average time to reach rQ starting ro from r (r is a reflecting boundary) This provides the intuition behind the singular boundary classi• fication. For the caseoof a=l, we have by substitution from (Al.la) into (A1.2c), and performing the required integration TT (r) = r^ exp(3u/r) 3 = 2m/a2 and further from (A1.2) rBexp(3u/r) h, (r) = x" exp(-3y/x) d-x.;6=(2+3) >0 1 2 r o rr 6 h,(r) = r" =exp(-3y/r) x$ exp(3y/x)dx ...(A1.2d) ro Performing the integration for h^(r) gives 189 ey/r. a^re h^r) =•;[ — 1 » a^ 'Constant of a2 (By)2 3ya integralton (Al.3) Clearly hj(s)ds' approaches infinity as r tends to 7ro infinity due to the second and third terras in (Al.. 3)' above. We need only evaluate the integral as r tends to zero. Here, the last two terms of (A1.3) are clearly finite. Thus we need to look at the first term only. [We may conveniently drop multiplicative constants] a ra (3y/s h-^sjds ^ se d s (A1.3a) If we now make the substitution 1/y = s, we can integrate and get (all integrals were obtained from Gradshteyn and Ryzhik[35]) y3y 2 +8yEi(eyy) (A1.3b) h1(s)ds •« y- 1/a where Ei (.) is the exponential integral. Now (A1.3b) is clearly unbounded as Y approaches infinity. Thus h^(r) is unbounded at both boundaries, which clearly implies that, either both boundaries are inaccessible, or entrance boundaries, rr depending upon h„(s)ds, as r tends to zero and infinity. Making the substitution 1/z = x in the expression for h2(r) in equation (A1.2b) gives r~ exp(-i.S/r) < '±z 6 exp (ygz)dz dr (A1.4) h2(s)ds = !; a2 1/a 1.90 Since we are only interested in the behaviour of the integral at the boundaries, we can without any loss of generality evaluate the integrals for 8=1 ie 6=3. This gives 1 2 a2.exp(-Bu/s) + • + .0;u. , (B-u.) exp (-Bu/s) h2 (s)d£ 2 aV " 2a ^s loT^l ^3 Ei(3y/s) ds . . (A1.4a) Due to the second and third terms in the expression above, the integral approaches infinity as r approaches zero. Further, as r approaches infinity, the integral is unbounded, due to the second term alone. Thus the integral of h2(r) is unbounded at both boundaries. Thus both r=0 and r=°° are natural boundaries. For the case u=h, the behaviour at the singular boundaries has been studied by Feller[18] . In brief, his results are: 1) T=y°° is a natural boundary 2) at r=0, the boundary behaviour depends upon the parameter values. a) if m is negative (or rather my were negative), the boundary is an exit boundary 2 - b) if 0< 2my a , we have an entrance boundary. APPENDIX 2 Details of the Estimation Procedure for the Linearized Model The SDE governing the diffusion process is dx = m(y-x)dt + axadz (A2.1) We can replace dx = (xt+l~ xt^ and x E xf If we further choose our unit of time equal to the discretization in• terval we have xt+l = m1X> + ^1-m^ xt + axta nt(A2.2) where nt ^ N(0,1). Equation (A2.2) implies 2 2a P(xt+1 xt,8) * N[{my+(l-m)xt},a xt ] . . . (A2.2a) We can therefore set up the likelihood function (logs taken) 2 , 0 , [x.^v--my-(1-m) x. J] 1„, 2a n , 2ol^r t+1 t - -1 as L = 2 E log x - j log c - ^ 2z{ ^ j- —} xt (A2.3) From the form of equation (A2.3), it can clearly be seen the m and y enter only in the last summation term, which is exactly the residual variance term. Thus m and y are just the least squares estimates given a. Further differentiating L w.r.t. a2 and setting to zero gives 2 •n , 1 „ f[*t+l- my-(l-m)xt] _ 2 2 2 1 { 2a (2a )a ^5a or x my (1 m)x 3 1 rT t+l" ~ " t (A2.4) " n E { 2a x, The structure of a2, m and y suggest a simple iterative procedure for estimating all the parameters. a) Pick a starting value of a = b) Using OLS (ordinary Least squares) to equation A2.2, after dividing through by x0^. , we can estimate m and y. Equation (A2.4) then gives an estimate of a2. It is well known that our estimates of m and y are inefficient c) Evaluate 8L/9a for the present parameter values and pick the next a to attempt setting 9L/3a = 0 where 2 2 [x - my-(l-m)xt] logx ^ = - Zlogxt + 2-a2 Z{ — } Xt The next question is how to pick a reasonable starting value of a? This can be done by breaking down the problem using an approximation. Squaring (A2.1) we have (dx)2 = a2x2a(dz)2, dz * N (0,dt) If we now replace differentials by differences, and choosing the unit of time as before, we have 2 o ? ot ?2 2 2 yt ~- a x x 1} 193 where yfc = Axt = (xt+J__- xfc) 2 2 2a and so we Let z E y /(a xt ) <\, X(^) have f(z) = -i- z"1/2 exp (-z/2) dz /2TT Suppose we now set up the joint likelihood of the data in terms of the y's we have T n 2a _i /n 1 Y+. (data) = n — (y2.a2.x. ) x/ exp — ) t=l /2T 2 a2x2a Taking logs and dropping additive constants .-gives V 2 2 a y L = - iT lo,__g a_9 - i1 E„ log (y,2\ „x.^ 2a), - =1• 2 2^ (——, t ) (A2.5) z z r r 2a Xt 3L T 1 ^t2 972 = "la2 + 27^" E( 7^"«)= 0 Xt 1 Yt2 which gives a2 = — E ( —— ) (A2.6) *T . 2a Xt and ^= I EC( ^2 x^T " X) lo^xt2] •••• (A2'7) The approach is to iterate between (A2.6) and (A2.7), so as to reduce 9L/9a =0. It is- found that convergance is very fast. Finally, going back to the original problem, we can get an esti• mate of the asymptotic variance r- covariance matrix by inverting the Fisher information matrix at the chosen optimal point. The elements of the hessian matrix are given by: 194 32L _ -T-" 3(a2)2 " 2 (a2)2 2 {xt+1-my-(1-m)xt} log xt , 2 3ZL 3a2 la2 * t •v 2a Xt (rxt)2 32L a2 x ToT 3m2 3_fL 3y2 a2 1 x 2a 32 (1J-x-t) 199 Xt2 L u -2 E {a(x) • ^ 2~- • >; a(x) xt+1-my-(1-m)xfc 3 a 3m log xfc2 32L m 3a3y - ^2 E { a(x) log xfc2 3Z L T^fyy EHa(x)}2 3 a 30 " xt2a Z x 3 L 1 yr 1 a(x) m(H- t) 2 El 2 3m3y. a xt * a2T i a(x)(y-x ) J 3m3a^, (a2) 2 I 2a t 3 2L "•• _ _ m , a(x) , 3y3c2'' " (a2)22 "x 2aJ 195 APPENDIX - 3 Solution to the forward equation for a = 1 The SDE for the diffusion process is dx = m(y-x)dt + axdz (A3.1) and the transitional probability density follows^ the FP eqn ff=~4 [m(u-x)P] + \ -Ii- [«r2x2p]_. (A3>2) 8x2 with the initial condition P(x| Xq,0) = 6(X-XQ) ... (A3.2a) We can transform eqn (A3.2) to the form H - -St" [* rx: z(x) = (trs) -"-ds 2 2 2 2 a(z) =[m(y-x) - j |x (a x ) ] (a x ) ... (A3.4a) . g(z,|'z0,t)= (ax)P(x| xQ,t) | x = x(z) (A3.4b) with initial condition g (Jz { 2Q> q ) = <5(Z-ZQ) (A3.4c) We shall therefore concentrate on a solution to the transformed equation (A3.3) and once we solve for g, we can retrieve P using (A3.4b) . Using the standard separation of variables we get Et/2 g(z |-'z ft) = Q(z)e~ (A3.5) Equation (A3.„3)now reduces to the .eigen value problem 2 - | [a(z)Q] + EQ = 0 (A3.6) dz where the boundary conditions on Q are given by the conditions on g through eqn (A3.5) We can further transform eqn (A3.6) by substituting Q(z) = i|.(z) [TT(Z)]1/2 (A3.7a) where •rz TT (z) = exp {- 2 a U) de.} (A3.7b) and then we have eqn (A3.6) as ,2. 5-1 + [E - U(z)] ijj = 0 ((A3.7c) dz2 where U(z) = |§ + a2 (A3.7d) The boundary conditions on are got from the boundary con• ditions on P through (A3.4B), A3.5), (A3.7a) and (A3.7b) For our process, we have the two singular boundaries as in• accessible i.e. p (x 0, ») + 0 (A3.8) We now have by our earlier definition of Z(x) z(x) = 1 to x (A3.9) Thus for 0 £ x < <*> we have - » < Z < 00 Further using (A3.4b) and assuming that P(x) -> 0 faster than x -* °° (.and using eqn (A. 3.8) we have g(z 4- ± 00) = 0 (A3. 8a) Equation (A3.5) so gives us Q (z -> ± 00) = 0 (A3.8b) We now proceed to get the functional form of eqn (A3.7c)as it is in the form of the time homogeneous Schroedinger equation of wave mechanics. a(z). = [m(y-x) - \ 2a2x] — 4 ax = [my - (m + §—) x ] -^x- my + ax < ? 7 > a z From (A3.9) we have x = eaz which gives -az a^ (z) ( £ + § ) a 2 da •az dz my e From (A3.7d we have TT/ \ /— -az -az , my e .(..'ii + °) U(z) ',- - my e + a ~ a 2 -az , , m-y, 2 my e + ( —r-) a my my e + ( —) (e_aZ- 1)2+ 2e-aztl-- - :f- ) y 2my + ^ y + 2my^ ^ . mu.2 , —az n v 2 , - a z 1) + e 2 (my) 2(1- J- - ^) - my 2 + ( HLbL ) { (-l + |1 ,2 _ ! } a y 2my which gives 2 2 CTZ U (z) = ax (e~° - l) + a2 e + a3 my , 2 where a. a 2\\W- ) 2 <1 -^'f) y 1 my - 1 ^ a ' { y 2my; Substituting into equation (A3.7c) we have dj> { 1 ,, ..- -a-2, 2 -az I-i . _ „ 0 (E>-a3) - a2 1 — (1- e -) - e I ^ (A3 dz a. *- Now by a suitable change of variable we want to transform 199 eqn (A3.10) to the following form 2 + { E1*- c(e C- 2e ?) } i|i = 0 (A3.11a) d?2 i where IJJ( 5 -> ± ») = 0 (A3, lib) and where 3, E and c are to be chosen in terms of the parameters of equation (A3.10) Let £ = a (z-z*) (A3.11c) 2 2 2 Then d_j, .a = d_J_ (A3, lid) 2 d'S, dz2 Further taking the second term in the. square bracket of equation 1(A3 .10) we have ,, „ -az. , -2az. -az a^ (l-2e + e ) - e * and substitute -az = -(£+az ) which gives ane * _ a a„ = a. + -±—* - 2e M + — zr ) (A3.11) 1 2az* az* 0 az* e e 2e Comparing with the corresponding part of equation,(A3.11) we want to choose z* so as to satisfy ^- -2az* -az* , 2 e ,C= ax e = axe + 5" • , , a2 . -az* = (ax + 2- ) e and dropping the trivial solution e =; 6 we have 2ax or z * = iogQ ( (A3.12) Thus we note that we can transform eqn (A3.10) to the form eqn (9) by the following substitutions „ _ , *x (A3.11c) £ a(z - z*) where z* is given by eqn (A3.12) 3 = a2 (A3.12a) E' = (E-a3+a1) (A3.12b) 2oz and = a± e~ * (A3.12c) The point of all this effort is that eqn (A3.11) is just the Schrfidinger equation for a diatomic molecule with a Morse potential - an equation which has been studied in the quantum physics literature by Trischka & Salwen [ ].(See also Morse [ '.. ] , Schroedinger [ ] , Dunham [ ]) . At the boundary TJ(£) = c (e~2?-2e~?)->• 0 i.e. at infinite boundaries, U(5) [by comparing eqn (A3.11a) with eqn (A.3 .7c)which is the basic time homogeneous Schrodinger equation] is not always infinite. This implies (see Titch- marsh [ J) that the set of eigenvalues of equation (A3.11a) are not strictly discrete, and there is a continuous interval 201 of eigenvalues as well. The discrete, region of the eigenvalues of eqn (A3.1la) are given in Trischka & Salwan [ ] as 2 /c E- = _c[i_ £ (n+|)] 0< n < [ r - | ] (A3.13) / c where [x] is the integral part of the number x i.e., the largest integral less than or equal to x. The correspond• ing normalized eigen functions are given by 1 (q-2n-l) 2 u/2 * U) = M u e~ Fn (u) (A3.14) rn n n where q = 7 (A3.14a) ^ exp(-az*) u = q exp UQ- O (A3.14b) and £Q is got from the initial condition (A3.2a) suitably transformed. and Pn(u) = Ji(J)Tl^r ; (J) = ITT^iri....(A3.14c) .2 1 (q-2n)i Mn = n! r(q-2n-l) ' .*(A3.:14d). where r(.) is the gamma function and (x)n is defined as (x) = {1 if n = 0 (A3.14e) x(x+l) (x+n-1) if n > 1 The solution as in eqn (A3.5) is thus given by (for the 202 discrete portion). E t/2 g(z zQ,t) = E an Qn e n (A3.15) n where Qn is got from ipn using equations (A3.7a) (A3.7b) , and the constants a are got frc the constants a n 'are got from the initial condition (A3.2a), and setting t=0, which gives us °n= Qn(z0) 7r (z0} :-• (A3-15a) the general solution is now given by g(zfzo,t) = u(z0) £ Qn(z) Qm(zQ) exp (-E^ t/2) (A3.16) + continuous spectra contribution The contribution of the continuous part (see Goel et al [,. ) has not been solved in closed form, but is known to be of the form as under F (E' , x)exp { (z-ez) ] - j E*t > °X (A3.16a) 0 where the relation between E1 and E is given by (A3.12b) and the function F depends upon confluent hypergeometrie functions. Without pursuring this line of analysis further the follow• ing comments may be made: a) It appears that an important characteristic of ex• pression (A3.16a) is that it decays very rapidly with time (t), so that by an appropriate choice of t, it may be negligibly small, and conveniently dropped. The transitional density is rather cumbersome and may not be meaningfully tractable from the point of view of parameter estimation by ML methods. APPENDIX 4 Solution of the Fokker-Plancy Equation for g=0 With No Restriction at Origin. The SDE for the diffusion process with a=0 is dx = m(y-x) dt + adz (A4.1) If we now make the substitution -y = u-x, we get dy = - my dt + adz (A4.2) which is the straight Ornstein-Uhlenbeck process and has a transitional probability density given by 2 1/2 _rnt 2 P(y yo,0) = [2rrV ] exp {-|[{y-y0e }/V] } (A4.3) where V = g- (l-e-2^) 2m The solution to (A4.1 is therefore simply got by substituting y = x-u ; yQ= xQ-y. APPENDIX 5 Derivation of the Stationary (or Steady State) densities We have our diffusion process defined by dx = m(vi-x)dt + ax 'dz" (A5.1) which has the form dx = b (x) dt + /a\(x) dz (A5.1a) where b;(x) = m(y-x) ; a.(x) = a2x201 The FP equation corresponding to A5.1a) is - |x |b(x)F] + \ Jl [ a(x)F] = (A5.1b) where F = F(x-1 XQ,t,6) is the transitional probability density. The steady state density is the solution to (A5.1b) got by setting 8F/3t = 0, and is of the form -1 [a/(x) TT (x) ] •(A5.1c) P (x 8) = -l [a,!(s)rr(s)] ds (r) where TT(S) = exp [-2 b dr] (AS.ld)- a.(r) and Q indicates integration over the total state space of x 206 For our process (A5.1) we have 2m TT (x) = x exp(Bx) ; n2 o=l/2 TT (X) x exp'(By/x) a=l TT (x) l+X a-y 1/2 ,1 = exp [ ex eyx -] ; X=l-2a i+x (A5.2) For a = 1/2 we have •1 _ 1 By-1 [b(x) TT(X)] x exp(-ex) ey ey-i 3 1 and \2 x - exp(-ex) = Ifi- (e) exp(-ex) (edx) _ (e) ey r(3y) P(x) (A5.2a) a'=l/2 For a = 1 we have -6 [b(x)TT(x)]~-L = ^ exp(-ey/x) ; 6 = (2+e) 6 (3+1) and K x" exp(-ey/x)dx = \2 (By) r(B+l) 0 U U which gives >3P)3+1 -(2 P(x) +6) exp(-6y/x) (A5.2b) l a=l r(e+l) x Finally for the general case of 1/2, 1 we have •l+A B:yx [b(x) TT (x) ] = x1_X exp[ 3x -] ; A=l-2a l + A which cannot be readily integrated, and so we have for the steady state density l+X A-l rByx x ex exp [M*\ •-- x l+X' P(x) ~-i r eyy^ eyi+xn , a^l/2,1 exp[ -p- - ] dy (A5.2c) Finally, it would be of interest for us to verify that the steady state probability density-(A5. 2c) , reduces to the functional forms (A5.2a) and (A5.2b) as a approaches 1/2 and 1 respectively (i.e. X -»- 0 ; -1) Now dropping the denominator (which is a constant) from (A5.2c) we can write the density function as l+A T-, A-l r eyx ex P «i x exp [ 1L^— ] (A5.3) A A l+A and multiplying and dividing by exp (gy/A) gives 208 Mow x - 1 _ exp(Alogx)-1 A X 12 2 X log x + 2 A"(log x) -i 00 -i -+ ± E £ (A log x)n A n=3 nij; Now as X •+ 0 (i.e. a -> 1/2) clearly xX-l (A5.3a) Lt r = log x A+0 A * x^y 1 exp(-$x) P(x) a -> 1/2 A 0 which has the same kernel as (A5.2a). To show the same sort of continuity for the a=l case we can write (A5.3) as 1+X B {x - l}i A-l PA°= exp [ 1+A J X and as in (A5.3a) above as a -> 1 ; (1+A) 0 and xl+A_ 1 Lt —T-T^ = log x X-y-1 1+A Thus taking limits as A -> - 1 we get (2+B) P, « x" exP [- ] A X which has the same form as (A5.2b) APPENDIX ~ 6 Details of the Phillip's Approach to Estimation The stochastic differential equation (s.d.e.) governing the interest rate process is dx = m(yU-x) dt + ax" dz ... (A 6.1) It is necessary to transform the above s.d.e. so as to eliminate x from the variance element. This can be done by a transformation of variables. Let the transformation be y = f(x) where the functional form of f(.) is unknown.By: Ito1s Lemma we have 2 2a a dy = Lf mf x (y-x) + %f xx a x J] dt + fx ax dz we now choose f(x) so that a f x = 1 (or any constant) which on integration gives 1-a y = -=r_—— for a ^ 1 1-a- = log x for aC^j\ Proceeding with the a^l case (as it is the more general form), we get by substitution -a "*"-a 2-1 dy = [myx - m(l-a) -= Jgaa x ] dt + adz 1-a ~ J If we now set u = x ; v = x we get the equivalent form of (A 6.1), in a form where the Phillip's approach may be applied. Thus 2 dy = m(a-l)y(t) + my u(t) - haa vr(t) + K (t) Since" 0U.T objective at present is purely expositional, - - let us proceed ahead further assuming. a=h. This gives- dy = [-(m/2)y + (my-a2)2 ] dt + adz ...(A6.2) 4 y If now we set 2/y = u, we can treat u as equivalent to Phillips [ ] exogenous variable. Then an approximate discrete time equivalent to (A6.2) is Yt= El^t-1+ E2Ut + E3Ut-l + E4Ut-2 + V(A6"3) where = exp (-m/2) 2 2 3 2 E2= (my-a /4)[(2/m ) exp(-m/2)(1-4/m)+(2/m )(m -3m+4)] 2 3 2 3 E3= (my-a /4) [ (2/m ) exp (-m/2)(8-m ) + (8/m )(m-2)] 2 3 3 E4= (my-a /4)[-(2/m ) exp (-m/2)(m+4) - (2/m )(m-4)] (A6.4) 211 and 2 nt ^ N[0, (a /m)(l-exp(-m))] Thus the log likelyhood function is L = - | log w2 - E (A6.5) 2 T~2 where w = (a2/m)(l-exp(-m)) and E = E n.• It may be t=l t- noted that the degrees of freedom have reduced by 2 as we require lagged values in (A6.3). If the time between observations is very small, m is also small in magnitude. We can then expand exp(-m), and drop terms of second order and higher. Then w2 - a2. However, we find that using a direct regression approach -uniquely determines m, and the residual 2 variance is a . Thus we find y is overdetermined. The direct regression approach fails. Constrained regression also fails as a2 enters the E's. Thus the only approach is to maximize L directly. On sample data sets, it was found that using standard non linear optimization routines, convergence was not obtained. It was therefore decided to drop further investigation. The difference between the approaches of Sargan [ ] and Phillips[ ] is very minor. We have the solution to the Stochastic differential equation D y(t) = Ay(t) + B z(t) + E(t) as shown in Sargan [ ] h y(t) = e^ytt-h) + e B z(t-s)ds + e E(t-s) ds (A6.6) 0 Both approaches approximate the integrand in the second term on the r.h.s. of (A6.6) by a polynomial in s by a Taylor series expansion of z(t-s) about s=0, and dropping terms of third and higher order. (Clearly the approximation hinges on the differentiability of z(t)). They differ only in the way they approximate the derivatives of z(t). Sargan adopts the more direct approach and sets 2Z + Z 2t~ Zt-h t-h t-2h z'(t) = z"(t) = 2h2 whereas Phillips uses the more involved Lagrange three points interpolation formula (see Conte de Boor [ J). 213 APPENDIX - 7 Details of Estimating Procedure for a = 1/2 (Known) Case This appendix outlines the method adopted to estimate the parameters (m,y,a), for the case where a = 1/2 is assumed known. The diffusion equation is given by dx = m(y-x)dt + cr/x dz (A7.1) a) Simple linearization method: Approximating the differential equation (A7.1) by a difference expression gives a x (xt~ ^t_1) = m(y - xt-1) + ^ t_i et (A7.2) v/x where et ^ N(0,1). Dividing through by t_^ and rearranging.-> terms gives Yt = % Xlt+ (1"m) X2t + nt (A7*3) where yt = xt//x^_1 ; xlt = l//x^_1 ; x2t = /x^ and 2 nt=aet^N(0,a ) ^ \^ Now in equation (A7.1), the dz1s are intertemporal • independent, which implies that E(nt nt,) =0 for all t'^ t. Thus (A7.3) is the standard regression equation and ML estimation of the parameters is equivalent to least squares estimates. Thus the log of the joint likelihood of T observations is given by T 4 2 2 2 2 2 2 2 ^ -L EtIl n = I y + m y Ex t + (1-m) Ex t- 2*^ Ix^ Ex x 2 1-m Ex Y v^ _l. 2my (1-m) it 2t~ ( ) 2t t '' (A7.4) 214 Zx2 = Ex X M = Ex Setting = j_t- '< Mi_2 lt 2t ' 22 2t ' M, = .Y. ; JVL = Ex„,y. ; M = Ey2 ly It t ^y 2t-rt yy Jt We have the first order conditions as 2 |^ = 2my M11 - 2(l-m) M^- 2 p JA^ + 2PM12- 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) yM^- M^ Substituting the above units (A7.5a) and setting 3L/9m = 0 gives v. \«12-*K% (fl7.6b) 2 - M. _M_ + M..M- - NL-M-i ) (M12 " \2% + ^1% " Wi The Fisher information matrix corresponding to the present sample is I and its elements are (by invoking asymptotic results) where 6^ = m and 02 = y, and are the M.L. estimates For the present case we have 82L 2 4VML2 - 2y Mn - 2M22 3m2 3fL Y = - 2m2 M^ 3y 9£L_ = 4m M12 + 2Mly - 2^2 - toM^ 9m9y 215 This enabless us to estimate off thee variance-covariance matrix off ththe estimates based on asymptotic theory. b) Steady State Density method: The steady state density corresponding to a = 1/2 is (see Appendix 5) F(x) = (^ .xey-i. (_BX) (A7.7) r.(By) The joint likelihood of the data for this approach is T a =.n, F(x.) i=l i Taking logs and setting lo,g (I) = L we have L = Tgu log B + (B'y- 1) E log x, - 6 Z xi~ T log [r(By)] (A7.8) where r(.) is the Gamma function. The first 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 + ya-b - TyiJ; (By) = 0 (A7.9b) dp where a = E log x^ and b = Ex^ and is the psi function i.e. the first derivative of log [F(.)]. Equating (A7.9a) and (A7.9b), and observing that is 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) in either of equations (A7.9) iJi(By) =logB+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 of the asymptotic variance-covariance matrix of the parameters, we need to evaluate the Hessian of L at the neximum. Thus |^ = - T32 r(8v) (A7.ll) = TP2 [ i - (Bu)] (A7.11b) 32L jrgp = T log 3 + T + a - Tip (3y) - T3y^' ,(3y) (A7.11c) where ip' (.) is the digamma function. Clearly the optimum is 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 in the UBCrNLE routines) which evaluate the zeros- -vof nonlinear equations. The psi function has been coded and is available in the UBC programme library. For the digaimia function, first a series expansion was used. However this was not satisfactory, as truncation (even after a large number of terms) resulted in sizeable errors in the function value, which was detected as the diagonal elements of the hessian matrix some• times became positive. An asymtotic expansion (for large arguments) was very satisfactory for the parameter values of our problem. c) Transition., . Probability Density Method: The transition probability density corresponding to a = 1/2 is given by 217 2 2 F(x |x ,8,t) = {2m/a (w-l)} . exp [-{2m (x+wxQ)}/{a (w-1)}J. •x • ( — )'"' i ° I' [4m^5wx /a2 (w-1)] wx 1 o o ^ 2mp _ ^ a2 (A7.12a) where w = exp (mt) and is the modified Bessel function of order k. The likelihood function is therefore T-l . ' * = ±£ F(xi+1|xi,0) . Pgs(x1) (A7.12b) p where ss(«) is the steady state density (A7.7) . In general, when Ti is large, the contribution of Pce(.) may be considered very small compared to the other terms, and so may be dropped from the likelihood function. The log livelihood function is not further tractable analytically as expressions for (9L/96^) require derivatives of the Bessel function with respect to its order, (for arbitrary positive orders) which are problematic. The approach towards parameter estimation has to be direct iraximization of the log of (A7.12b). For this purpose, the Fletcher algorithm using a quasi- Newton method was used. In general, it was found that convergence was obtained to a reasonable degree of accuracy within 15 iterations, given starting values for the parameters as the results of the simple linearization model. In small sample trials, to ascertain whether convergence is to a local or a "global" maximum, very different starting values were given. Without fail in all cases, the convergence took longer, but the final maximum value parameters were unchanged. The term "global" has been set within quotes, as there is no rigorous guarantee that the maximum obtained is truly 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 in 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 this, the expression was split up as F(xt|t0,9) •= f(xt,xo,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 lines of Giver [66] was used, whenever 6 was greater than 20. This was very efficient. The relative 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 Xq ranging from near 0% to 30%, by the two methods and computing its first 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, is a central moment. The asymptotic expansion performed very well, as may be seen from the tabulation in Figure 1 Just to show the shape of the transitional probability density function in equation (A7.12a), Figure 1 was prepared. What is interesting to note is that, for the parameter set used, y - 5% per annum, and when the current interest is at or above y, the transitional density function FIGURE 1 Plot of Transition Probability Density Function (& Cumulative Probability) for a = 1/2 at Different rn Values 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 r 9 r Mean of t+^/ t« Std. dev of t+1At'6 rt 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: - All 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 linearization of the diffusion equation, (which assumes Gaussian transition probabilities) may not perform too badly. Finally the second derivatives of the log likelihood 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 Analysis Of Effect Of Measurement Errors On Data : The analysis here assumes that the observed data is the combined effect of the true process and a superimposed error process. The formulation of the problem runs as follows: We may believe that the true interest rate (i) follows the process di = m(y-i) dt + Vafi dz, (A8,l) where we observe i with error. (For this analysis, I have used the square root process, as the purpose of this section at the present moment is expositional). Let us observe r as i with an error n i.e. r = i + n (A8.2) where n is white noise. To be able to proceed further, we have to impose some additional structure on the problem. Let us look at a particular form of the error structure dn = a2/T dz2 ; E(n) =0 ..(A8:3) The rationale behind" this form is that it ensures that theTerror goes to zero as i + 0. Dif ferenciating (A8.2.) and substituting (A8.1) and (A8.3) we get dr = di + dn = m(p-i) dt + oj/I'dx, + o2 ^ dz2 where E (dr) = m(y-r) dt 2 2 2.. E(dr ) = (o1+a2)r dt + 2a i a2 pr dt [since Cov (dz1 2 2 - (a + a + 20la2p) r dt = a2 r dt where a2 == (a2 + a2 + 2a^p) Thus we can represent the process r as :A8 dr =: m(y-r) dt + a3/r dz ( -4) which is exactly of the same form as equation (A8V1);-,the tr;'u'e' interest rate process. Clearly, we cannot identify a , a2 of p . Further, if we ignore the error in measurement (when an error does exist), then a| as an estimate of a2, is either over or under estimated according as 2 (a + 2ol a2 p) | 0 2a1 This implies that even when p = 0, (the error is uncorrelated with the true interest value) a2 is over estimated by o2. In this error structure, as long as we assume that both the error and true interest process have the same a exponent in the variance term, the present analysis holds in toto. This is easily verified by carrying the algebra through. 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 Theil [ ] t Goldfeld and Quandt [ ]). This result uses the property of ML estimators whereby 3^ , 92L E (— ) = (A9.1) 30i90j 90i90, 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, if we know 0 (the true value), then we can compute the assymptotic covariance matrix of 0 as T1 Cov(0) = -E (A9.2) ' 90^0^ Further if we represent by L^n\ the joint likelihood of n data points, we "can approximate a2.W ,2 (1) ID - i 3 " Equation (A9.3) is valid strictly only for independent random variables. We hope that the "bias" due to dependence of the sequence does not alter the basic nature of the analysis to follow very much. The point to be noted here is that when we compute the asymptotic correlation matrix from the covariance matrix, it is obviously immaterial whether we use the expectation over n data points or even 1 data point. Let I represent the Fisher Information matrix. Then 2h I = E r * ) 86.80. J and further P^fr,y . r ,0 ) . P (r 6J) dr dr 1 J (r r e) 30.30. fetaeT t' 0l - l t o' ss^ o t 0 0 IA9.4) where Prp(-) represents the transitional probability density and Pss(0 the stationary probability density. Since we want to evaluate the correlation matrix over all parameters (including ot we could assume PT(.) to be normal - which is the case in the SL approximation. This now gives {myr + (l-m)J r }, a r (A9.5) PT(rt rffie):a N o o a r 2r N.[ ( 0)> ° 0] i _ By By-i e Pss(^ol ) - ; . .exp (-Pro) (A9.6) r(By) /° Where B = 2m/a2. Expressions for have all been set out in 96,30, Appendix,2. Substituting (A9.5) and (A9.6) into (A9.4); noting that one of the integrals is now from -°° to +00 due to the normal density approxi• mation; and further that r PT (*t|r0,e)drt= 1 rt pT(rtlro.8)drt= a(ro> {rt - a(rQ)} Pt(rt |rQ,6) drt' = 2 {rt - a(r0)} PT(rt|r0,6) dr^- gxves 9*L E ( -) = 0 9a9ta. E ( ) = 0 9a9y E = 0 9m9cT 92L E ( -) = 0 9y9a" 226 (The definite integrals are from Gradshtein $ Rzyhik [35[) By-2 m y.rQ exp(-3ro) drQ ; y 8p m2 3 C (3u-l) E(4^ = r )2 r 3y 2 ex B dr 9m ^ " o o " P <~ ro) o By - l By-2 my r } r 6u 2 dr E(9m9y } " ^- o o " «p(-e^0> o m Q: (Bu-D 2 2 (9a/)/ 2 (a ) f00 92L 2 3y_1 (-3ro) E ( 9 ) = - 2y (log rQ) . rQ . exp drQ 9a = ^2 { i^(gy) - log er + 1(2, BU-D} Where §(z,q) is the Riemann's Zeta function = In" ( 7) and is the q+n psi function. 227 roo log r r ex r r ) dr 9a9a a o ' o * P( "^ o o 0 --V12n+ ii+ (1 -sir*- s where n = { (3y) - log3) The hessian matrix of the log likelihood function has a block diagonal form, with the two off diagonal (2x2) matricies being zero (the order of the 2 parameters is assumed {m,y,a ,a}). This means that the inverse of the hessian matrix is the matrix with the individual blocks inverted. This tells us that we can infer the sign of the correlation between m and y, and , 2 a and a. These are exactly the same as the corresponding cross derivatives of L (with the sign reversed). Thus we expect Cor (m,u) < 0 & Cor (a2,a) > 0 The correlation matrix is presented in the main text. APPENDIX <- 10 Maximum Likelihood Estimation of the Parameters {m,u,a,a} Using the Steady State Probability Density Approach The steady state probability density function corresponding to general a values (ie a ^ 1/2, 1) is A-l Bux Bx x exp 1+A P(x) (A10.1) Q X 0 1+A Buy By A-l dy y exp A 1+A x^ '''exp [ a(x) ] ^ ^ & a(x) an& D suitably B = 2m/a defined D The joint log likelihood function of n observations is n n (A10.2) L = (A-l) I log xi + E a(x±) + n log D i=l 1=1 The log likelihood function is not tractable analytically for purposes of estimating its maximum with respect to the parameter, and so only nonlinear optimization methods must be employed. However, it was found that methods that used numerical derivatives (like any modification of the Newton method) led to problems due to the complicated way in which the parameters enter the likelihood function. Expressions for the first and second derivative of L with reference to the parameters were 229 derived as under 3L n 3D/3y = 2 D 9u i=l _3D 3z where 3y. a(z) dz A 3L n a(x-i) 3D/3m 3m i i=l L 3 D 3D { }exp where 33 x a(z) dz l+A l+A A N AA D&Ax . 3L _ ? 3D/3A 3yxi m 3yxj - i , , . 3A . . log x± + T 2 i=l X log xi " * l+A" 0 ' g x-* ~ (l+A)' 3D 1 l -3yz 3z6 logz 3z6" where 3A b(z) dz . exp A Az l+A Az a+A)J 3yz 3z^ where 6=1+ 1/A ; b(z) = A l+A 2 2 32D/3y (3D/3y) ' 32L ly? = I 3^ 3a 2 z 2 where exp b(z) dz 3y2 32D/ 333y (9D/33) (9D/3y) 333y = I + D2 230 where exp b(z) . U+b(z)} bz my o A' A x 3XJ 3 i log x± = Z _ . — 3A3u -D— ~ ,00 1/A 1/A 2 2 3 D _y + z z where 3z exp b(z) + 0* 3A3u AJ X • X2 (l+X)2 (1+A) — log z) dz XZ r 3D/gg2 (3D/ae) = I + ? —5— T2 33 3^ where exp b(z) dz X 1 3 ^32 l+X l+X 32D/ x. Ux-j l 3A33 = I log X-L + lo x 3X33 (l+X) 2 Ti+I)" s ± D (3D/33QD/3A) D2 231 where 2 9 D 2Uz (1+2A) b(z) jexp b(z) + ' A L + 3A83 (1+A): A(1+A) ' Pi12 4. 3z^ gz^ log Z \ dz <: A2 (l+A)2 (1+A) A2 2 A gux^log x Bux^Clog'jCj) 23yx± Bux^^ log x± 32L = Z •' • + + 3A< A' A A3 A' o 1+A 1+A 1+A px. log X. px. 2 2px_L px^ 1 ° 1 — (log x ) ^— + —-—=— log x (1+A)2 1+A (1+A)J (1+A)' 9D/3A2 (3D/3A2) D + D2 where exp tb(z)] ( 2+ 33jz ^ l°g z . (3X2+4x3) _ g8|lo8 z) 2 3 4 8A I > > Ab(l+A) A"(1+AA (1+A-) 3z6 2 3z6 log (1+4A+3A ) z c- K 3yz A2(1+A)4 A3(1+A)2 6 + 3z log z ,+,_3zj_ ) y ( 3yz + 3z 3z log z dz 3 2 ..A (1+A) A(l+X) (1+A)' .A2 (1+A) 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, it was felt 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 its 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 first 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 00 z 0 z->°° 3D/3y +0 +0 B/X2 - 0 3D/33 +00 - 0 yM2 +0 3D/3A 1/A2 + 0 Not investigated 32D/3y2 +0 +0 +0 -0 32D/333y +0 - 0 lM2 + 0 32D/3X3y -0 3(3y-2)M3 +o -0 , A< -1 32D/332 +0 +0 +0 -0 233 A few clarifying comments are in order: a) The expression -0 and +0 indicate that the function approaches zero from the negative and positive directions respectively. b) the limits indicated are valid only, given the parameter values ie they do not represent the limits as, say, A -*• 0. It is anyway shown that A •*• 0 and -1, represent special cases (see Appendix 5) . c) the behaviour of 9D/9A was not analytically examined with 2 2 reference to its slope at the limits, 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 lie entirely either in the first or fourth quadrant, but partly in both in some cases. To investigate further the shape of each of these functions, and also to see what proportion of the total area lies in either quadrant for a broad range of parameter values, the functions were numerically evaluated and plotted. The conclusions were that for all practical purposes all the area was in either the first or fourth quadrant. All 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 trivial matter. To see the special problems that we face, let us address the problem of evaluating the seemingly innocuous integral D. We have l+A A-1 Buy By D = dy y exp l+X With a change of variables we can transform D as under A-1 Let z = y dz = Xy dy l+X -r Buz Bz dz D = X exP X l+X j o f(z) dz [In passing it may be noted thatithe limits of integration have to be interchanged for X JHo identify the mode of f(z), we set its first derivative to zero, which gives 1/X l+X -t Byz - Bz f (z) = j ( j ^-) exp = 0 X " l+X 1/X = f(z). B( (A10.3) Ruling out the alternative that f(z) = 0 at the mode gives the mode a z = y The integrand in D is clearly unimodal. Looking at f(z) we therefore see that from o to p\ the first 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) is very large (particularly when 3u is moderately large and X is near zero ie. 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 f (z) = j ^XP •.Buz 3z - P (A10.4) •IT " l+X and everytime D has to be evaluated p may be chosen such that f(z) at the mode is a reasonable number. The approach poses no problem even when we evaluate the derivatives of L; as we always have D in denominator with a derivative of D with reference to (B,u,X} in the numerator, and the same adjustment works there. The next point is that the mode jump all over the half real line as X goes from positive to negative. In our problem u is 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 itself should not cause any concern, but when coupled with the fact f(z) happens to be a very spiked function, (ie its total mass is concentrated over a very small range) poses some problems. All 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), its value is very close to zero over all 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 likely finds the value of f(z) at all those points very close to zero, and returns the value of the integral as 0. This is because the total area may lie 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 zm be the mode and and z2 the two inflection points of %/ f(z); zi < zm < z£. If now we represent the limits of integration by z and z" ', (z'' < z" ) then we can choose k^ and k2 such that z' = zm " kl (zm ~ zl) n z = zm + k2 (z2 - zm) where z'' and z"+ are required to satisfy some criteria like (say) 40 f(z*)/f(zm) and f(z" )/f(zm) < 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 1+1/X -, Buz Bz f, (z) = — exp X - X2 ~x i/x-i .1/Xv , 3 v ^ z " + (u^-" ). £ . (u^"lAv):l X Setting the above to zero, noting that f(z)^0 at the inflection points gives 1/X, - z + 3".(y - z ) = 0 Multiplying through by z and substituting y = 1+1/X gives Y^2 = zY -3 (uz - z')" = 0 (The functional form clearly suggests that the above equation has two roots) An iterative method to solve for the roots of the above equation isngot from a first order Taylor series expansion. g(z) M z? - 3 (uz - ^z'2) (A10.5) Then = z - (A10.6) z 1 Where zn+^ is the solution to (A10.5) at the (n+l)*"* iteration. In general, the scheme above should converge quite rapidly. However, it was found that for some parameter values, the scheme tended to converge always towards the same root (ie. the second solution was not obtainable) It was therefore necessary to find an approximate solution to (A10.5), and using them, and (A10.6) arrive at more accurate values of the 238 inflection points. For this we expand TX using a Binomial series, about •u . This gives T J = (z-uA) + uA = y ' (l AY 1 + (A10.7) - ¥ A •y Y Plugging equation (A10.7) for z' into (A10.5) and setting y = (z - yA)/ yA we get -i 2 AY A+1 Ay y (l - Yy) - 6 y (y+l) - y (I^YY) = 0 which can be reduced to l - Yy - y2 3U-Y)2 = 0 1/2 2 2 = - Y ±' { Y +43 (1-Y) } or '26 (1 -y)2 and that gives us the approximate solution. 239 APPJNDXX II Effect on bond valuation of using the yield to maturity on a 91-day pure discount bond instead of the instantaneously risk- free rate of interest* The basic assumption of the bond valuation model is that it is a function of the instantaneously risk free interest rate and time to maturity, By definition, the instantaneous risk free interest rate is the yield to maturity on a riskless pure discount bond due to mature the next instant in time. Thus, using the yield to maturity on a riskless bond which has a longer time to maturity, as a proxy for theinstantaneously risk free rate, would bias the bond valuation. This bias can be broken down into two parts: 1} The estimated parameters of the interest rate process (m,^A , (p2, d ) are biased because we have estimated them from a process which is 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 interest rate a proxy is used, and this biases the bond value. To analyse the nature of these biases, let us assume that the true model of the interest rate process is given by IT 7n(>--0dt + cr-f^f (A11.1) Then Ingersoll [39] has a solution for the yield to maturity on a pare discount bond having time to maturity t , and 240 current value of instantaneous interest rate r, as RC^rt") = -^J^^£^^H 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 in a fixed value of f =91 days, the coefficients in equation (A11.3) may be treated as parameters. Thus if we represent by R, the yield 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 first thing to be noted is that the assumption that R is a process of the form where jU^ * {bfkJr estimates of jk^ and based on equation (A11.6) instead of (All. 5) are incorrect to start with. However,the error due to this is complicated to investigate analytically!. Let us, therefore, only consider the relatively simpler question: what is the error from using jx^ and 0^ , as estimates of ^ and (T respectively? To quantify^ let us use numerical values for (m,yt, a-2), so that we may compute a and b. Since we are interested in the errors in the neighbourhood of the parameter range we have estimated for the interest rate process, we may use those values themselves to compute a and b. Thus, we use m = 0.002522 jj- — 0.001293 1 The extent of the error can be easily investigated by Monte Carlo methods. We could expect the error to be quite small due to the nearness of (R-a) to R. . This is because a R (and since R~]x ,in relative terms, a a 0) and thus assuming that the diffusion equation governing B has a singular boundary at R = 0, (as in equation Alt. 6) instead of at R = a, (as in eguation A11.5) should have only a marginal effect on the parameters. Further, it appears that the principal effect of the approximation, is on the variance element, ie., (A11.6) would be less than which gives us values for a and b a = 0. 001637|A, b = 0.998343 This implies that JUfc = 0.99 9981 (r2 = 0. 998343 The errors in assuming that JA^ is approximately jX and 2 is approximately It 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 yield to maturity. That equation is valid only under the pure expectations hypothesis about the term structure of interest rates. If we assume liguidity/-term premium of the form which is what we have used in 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 k2 as (see Chapter 7, section 7.3) k, = 0.3093 x 10-s k2 = -0. 1548 x 10-2 Osing these values give for a and b the values a = 0.01705/v 243 b = 0.98837 Comparison with the earlier values of a and b shows that 8 now is a poorer proxy for r ~ which is as expected. These values now give jU^ * 1.00542/^ 0^2 = 0. 98837 cr * We see that JUR is an overestimate of jx , • and The direction of the bias on both jx and Hext, we consider the effect on bond value by using H instead of r, in the valuation equation. The proportional error in r, by substituting fi instead of r may be represented as 0- + {b-\) (A11.7) T The error is clearly dependent on the current value of r. Since, on an average, the interest rate is expected to remain around jx , let 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 in the value of a pure discount bond due to the above error in r may be represented as Where is the bond value elasticity with respect to r. If 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 in Ingersoll [39], Thus Percent error in bond value = 0.542 x <-brf ) which for r = works out to 0.009% - a truly negligible error. It seems reasonable to expect that at other values of r around ^U., the error is also of similar orders of magnitude. It may therefore be concluded that the error due to the use of the yield to maturity on a 91-day discount bond as a proxy for the instantane free interest rate, is minimal..