American Finance Association
An Empirical Comparison of Alternative Models of the Short-Term Interest Rate Author(s): K. C. Chan, G. , . Longstaff and . : The Journal of Finance, Vol. 47, No. 3, Papers and Proceedings of the Fifty-Second Annual Meeting of the American Finance Association, , Louisiana January 3-5, 1992 (Jul., 1992), pp. 1209-1227
Published by: Wiley for the American Finance Association
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THE JOURNAL OF FINANCE * VOL. XLVII, NO. 3 * JULY 1992
An Empirical Comparison of Alternative Models of the Short-Term Interest Rate
K. C. CHAN, G. ANDREW KAROLYI, FRANCIS A. LONGSTAFF, and ANTHONY B. SANDERS*
We estimate and compare a variety of continuous-time models of the short-term riskless rate using the Generalized Method of Moments. We find that the most successful models in capturing the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the riskless rate. A number of well-known models perform poorly in the comparisons because of their implicit restrictions on term structure volatility. We show that these results have important implications for the use of different term structure models in valuing interest rate contingent claims and in hedging interest rate risk.
THE SHORT-TERM RISKLESS interest rate is one of the most fundamental and important prices determined in financial markets. More models have been put forward to explain its behavior than for any other issue in finance. Many of the more popular models currently used by academic researchers and practitioners have been developed in a continuous-time setting, which pro- vides a rich framework for specifying the dynamic behavior of the short-term riskless rate. A partial listing of these interest rate models includes those by Merton (1973), Brennan and Schwartz (1977, 1979, 1980), Vasicek (1977), Dothan (1978), Cox, Ingersoll, and Ross (1980, 1985), Constantinides and Ingersoll (1984), Schaefer and Schwartz (1984), Sundaresan (1984), Feldman (1989), Longstaff (1989a), Hull and White (1990), Black and Karasinski (1991), and Longstaff and Schwartz (1992).
Despite a bewildering array of models, relatively little is known about how these models compare in terms of their ability to capture the actual behavior
* All authors are from the College of Business, The Ohio State University. We are grateful for the comments and suggestions of , , , , , , , , , , Campbell Harvey, , , Culloch, – son, , Ren6 Stulz, , , and Finance Workshop participants at the Commodity Futures Trading Commission, the University of Iowa, the Kansallis Research Foundation, The Ohio State University, Purdue University, and participants at the 1991 Western Finance Association meetings, the 1991 European Finance Association meetings, the 1992 American Finance Association meetings, and the 1992 Federal Reserve Bank of Atlanta’s Conference on Financial Market Issues. All errors are our responsibility.
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1210 The Journal of Finance
of the short-term riskless rate. The primary reason for this has probably been the lack of a common framework in which different models could be nested and their performance benchmarked. Without a common framework, it is difficult to evaluate relative performance in a consistent way.1 The issue of how these models compare with each other is particularly important, how- ever, since each model differs fundamentally in its implications for valuing contingent claims and hedging interest rate risk.
This paper uses a simple econometric framework to compare the perfor- mance of a wide variety of well-known models in capturing the stochastic behavior of the short-term rate. Our approach exploits the fact that many term structure models-both single-factor and multifactor-imply dynamics for the short-term riskless rate r that can be nested within the following stochastic differential equation:
dr = ((a + fBr)dt + jrYdZ. (1)
These dynamics imply that the conditional mean and variance of changes in the short-term rate depend on the level of r. We estimate the parameters of this process in discrete time using the Generalized Method of Moments technique of Hansen (1982). As in Marsh and Rosenfeld (1983), we test the restrictions imposed by the alternative short-term interest rate models nested within equation (1). In addition, we compare the ability of each model to capture the volatility of the term structure. This property is of primary importance since the volatility of the riskless rate is a key variable governing the value of contingent claims such as interest rate options. In addition, optimal hedging strategies for risk-averse investors depend critically on the level of term structure volatility.
The empirical analysis provides a number of important results. Using one-month Treasury bill yields, we find that the value of y is the most important feature differentiating interest rate models. In particular, we show that models which allow y 2 1 capture the dynamics of the short-term interest rate better than those which require y < 1. This is because the volatility of the process is highly sensitive to the level of r; the unconstrained estimate of y is 1.50. We also show that the models differ significantly in their ability to capture the volatility of the short-term interest rate. We find no evidence of a structural shift in the interest rate process in October 1979 for the models that allow y 2 1.
We show that these interest rate models differ significantly in their impli- cations for valuing interest-rate-contingent securities. Using the estimated parameters for these models from the 1964 to 1989 sample period, we employ numerical procedures to value call options on long-term coupon bonds under
1 Because of this problem, empirical work in this area has tended to focus on specific mod instead of comparisons across models. See, for example, Brennan and Schwartz (1982), Brown
and Dybvig (1986), Gibbons and Ramaswamy (1986), Pearson and Sun (1989) and Barone,
Cuoco, and Zautzik (1991).
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Comparison of Models of the Short-Term Interest Rate 1211
different economic conditions. Our findings demonstrate that the range of possible call values varies significantly across the various models.
The remainder of the paper is organized as follows. Section I describes the short-term interest rate models examined in the paper. Section II discusses the econometric approach. Section III describes the data. Section IV presents the empirical results from comparing the models. Section V contrasts the models' implications for valuing options on long-term bonds. Section VI summarizes the paper and makes concluding remarks.
I. The Interest Rate Models
The stochastic differential equation given in (1) defines a broad class of interest rate processes which includes many well-known interest rate models. These models can be obtained from (1) by simply placing the appropriate restrictions on the four parameters a, /3, o-, and y. In this paper, we focus on eight different specifications of the dynamics of the short-term riskless rate that have appeared in the literature. These specifications are listed below and the corresponding parameter restrictions are summarized in Table I:
1.Mertondr=adt+odZ
2. Vasicek dr = (a + fBr)dt + odZ
3. CIR SR dr = (ax + r)dt + ? r 12dZ
4. Dothan dr = o- rdZ
5. GBM dr = fBrdt + ordZ
6. Brennan-Schwartz dr = (a + fBr)dt + ordZ
7. CIRVR dr = o-r3/2dZ
8. CEV dr = fBrdt + jrYdZ
Model 1 is used in Merton (1973), footnote 34, to derive a model of discount bond prices. This stochastic process for the riskless rate is simply a Brownian motion with drift. Model 2 is the Ornstein-Uhlenbeck process used by Vasicek (1977) in deriving an equilibrium model of discount bond prices. This Gauss- ian process has been used extensively by others in valuing bond options, futures, futures options, and other types of contingent claims. Examples include Jamshidian (1989) and Gibson and Schwartz (1990). The Merton model can be nested within the Vasicek model by the parameter restriction ,8 = 0. Both of these models imply that the conditional volatility of changes in the riskless rate is constant.
Model 3 is the square root (SR) process which appears in the Cox, Ingersoll, and Ross (CIR) (1985) single-factor general-equilibrium term structure model. This model has also been used extensively in developing valuation models for interest-rate-sensitive contingent claims. Examples include the mortgage- backed security valuation model in Dunn and McConnell (1981), the discount bond option model in CIR (1985), the futures and futures option pricing models in Ramaswamy and Sundaresan (1986), the swap pricing model in
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1212 The Journal of Finance
Parameter Restrictions Imposed by Alternative Models of
Short-Term Interest Rate Alternative models of the short-term riskless rate of interest r can be nested with appropriate parameter restrictions within the unrestricted model
dr = (a + /3r)dt + orYdZ
Model a /3 o-2 7
Merton 0 0 Vasicek 0 CIR SR 1/2 Dothan 0 0 1 GBM 0 1
Brennan-Schwartz 1
CIR VR 0 ? 3/2 CEV 0
Sundaresan (1989), and the yield option valuation model in Longstaff (1990b). The CIR SR model implies that the conditional volatility of changes in r is proportional to r.
Model 4 is used by Dothan (1978) in valuing discount bonds and has also been used by Brennan and Schwartz (1977) in developing numerical models of savings, retractable, and callable bonds. Model 5 is the familiar geometric Brownian motion (GBM) process of Black and Scholes (1973). Geometric Brownian motion is also one of the interest rate models considered by Marsh and Rosenfeld (1983). Model 6 is used by Brennan and Schwartz (1980) in deriving a numerical model for convertible bond prices. This process is also used by Courtadon (1982) in developing a model of discount bond option prices. The GBM model is nested within the Brennan-Schwartz model by the parameter restriction a = 0. In turn, the Dothan model is nested within the GBM model by the parameter restriction 8 = 0. All three of these models imply that the conditional volatility of changes in the riskless rate is propor- tional to r2.
Model 7 is introduced by CIR (1980) in their study of variable-rate (VR) securities. A similar model is also used by Constantinides and Ingersoll (1984) to value bonds in the presence of taxes. Finally, Model 8 is the constant elasticity of variance (CEV) process introduced by Cox (1975) and by Cox and Ross (1976). The application of this process to interest rates is discussed in Marsh and Rosenfeld (1983), footnote 4. Table I shows that the CEV model nests the Dothan, Brennan-Schwartz, and CIR VR models.
Although the majority of these interest rate processes were introduced in the context of a single-factor model of the term structure, it is important to
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Comparison of Models of the Short-Term Interest Rate 1213
note that our analysis is not limited to single-factor term structure models. By comparing different models of the short-term interest rate, our analysis provides insights into the properties of any economic model in which assump- tions about interest rate dynamics are made. For example, our results are applicable to any multifactor term structure model in which assumptions about the dynamic behavior of r are embedded.
Finally, we note that our framework has some features in common with Marsh and Rosenfeld (1983), Fischer and Zechner (1984), and Melino and Turnbull (1986). For example, Marsh and Rosenfeld use a general stochastic process similar to (1) in estimating the parameters of several continuous-time interest rate models. Their framework, however, nests only three interest rate processes. A comparison of their model with (1) shows that two of these three interest rate processes are nested within (1). These nested models correspond to the CIR SR and GBM models in our framework.
II. The Econometric Approach
In this section, we describe the econometric approach used in estimating the parameters of the interest rate models and in examining their explana- tory power for the dynamic behavior of short-term interest rates. To illustrate the approach clearly, we focus first on the unrestricted process given in equation (1). The same approach can then be used for the nested models after imposing the appropriate parameter restrictions.
Following Brennan and Schwartz (1982), Dietrich-Campbell and Schwartz (1986), Sanders and Unal (1988), and others, we estimate the parameters of the continuous-time model using a discrete-time econometric specification
rt+- rt a 8t + etf + (2) Et et+]= 0, E[ e2 ?] = 0J 2r 2 . (3)
This discrete-time model has the advantage of allowing the variance of interest rate changes to depend directly on the level of the interest rate in a way consistent with the continuous-time model.
It is important to acknowledge that the discretized process in (2) and (3) is only an approximation of the continuous-time specification. The reason for this is that in measuring changes in r over discrete intervals of time, integrals appear on the right side of (1). This is the temporal aggregation issue described by Grossman, Melino, and Shiller (1987), Breeden, Gibbons, and Litzenberger (1989), and Longstaff (1989b, 1990a). Given the continuity of the interest rate process, however, the amount of approximation error introduced can be shown to be of second-order importance if changes in r are measured over short periods of time.2
2 See also Campbell (1986).
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1214 The Journal of Finance
Our econometric approach is to test (2) and (3) as a set of overidentifying restrictions on a system of moment equations using the Generalized Method
of Moments (GMM) of Hansen (1982). This technique has a number of important advantages that make it an intuitive and logical choice for the estimation of the continuous-time interest rate processes. First, the GMM approach does not require that the distribution of interest rate changes be
normal; the asymptotic justification for the GMM procedure requires only
that the distribution of interest rate changes be stationary and ergodic and
that the relevant expectations exist. This is of particular importance in
testing the continuous-time term structure models since each implies a different distribution for changes in r. For example, the Vasicek and Merton
models assume that interest rate changes are normal, whereas CIR SR
assumes that they are proportional to a noncentral x2 variate. Second, the
GMM estimators and their standard errors are consistent even if the disturb-
ances, Et+ 1, are conditionally heteroskedastic. Since the temporal aggregat problem that arises from estimation of a continuous-time process with dis- crete-time data is likely to influence the distribution of the disturbances, the
GMM approach should further alleviate the impact of this approximation error on the parameter estimates. For example, even though the CIR SR continuous-time model assumes that changes in r are distributed as a random variable proportional to a noncentral x2, the discrete-time version of the model may not. Finally, the GMM technique has also been used in other empirical tests of interest rate models by Gibbons and Ramaswamy (1986), Harvey (1988), and Longstaff (1989a).
Define 0 to be the parameter vector with elements a, 3, o( 2 and y. Given Et + 1 = rt + 1 - rt - a - I rt, let the vector ft( 0 ) be
t+1 et+ lrt
ft( 0 ) = e 2 1-ff 2 r 2y . (4)
2[ 22 r2 ) rt t -o
Under the null hypothesis that the restrictions implied by (2) and (3) are
true, E[ ft(O)] = 0. The GMM procedure consists of replacing E[ ft(O)] with its sample counterpart, gT(0), using the T observations where
gT(0) = (5)t=lft()S (5)
and then choosing parameter estimates that minimize the quadratic form,
JT(0) =g(TO0)WT(0)gT(0), (6)
where WT(0) is a positive-definite symmetric weighting matrix. Matrix differ- entiation shows that minimizing JT(0) with respect to 0 is equivalent to
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Comparison of Models of the Short-Term Interest Rate 1215
solving the homogeneous system of equations (orthogonality conditions),
D (O)WT(0)9T(O) = 0, (7)
where D(O) is the Jacobian matrix of gT(O) with respect to 0.
For the unrestricted model, the parameters are just identified and JT(O) attains zero for all choices of WT(O). For the nested interest rate models, the GMM estimates of the overidentified parameter subvector of 0 do depend on the choice of WT. Hansen (1982) shows that choosing WT(O) = S1(0), where
S(O) = E[ ft(O)ft(0)], (8)
results in the GMM estimator of 0 with the smallest asymptotic covariance
matrix. Designating an estimator of this covariance matrix as SO(0), the asymptotic covariance matrix for the GMM estimate of 0 is
-(T o 0 )S (0 )D(0) (9)
where DO0W) is the Jacobian evaluated at the estimated parameters. This covariance matrix is used to test the significance of the individual parameters.
The minimized value of the quadratic form in (6) is distributed x2 under the null hypothesis that the model is true with degrees of freedom equal to the number of orthogonality conditions net of the number of parameters to be estimated. This x2 measure provides a goodness-of-fit test for the model. A high value of this statistic means that the model is misspecified.3
We also use the hypothesis-testing methods developed by Newey and West (1987) in order to evaluate the restrictions imposed by the various models on the unrestricted model. They show that for a general null hypothesis of the form, Ho: a(0) = 0, where a(0) is a vector of order k, each element repre- senting a model restriction, the test statistic,
R = T[JT(O) -JT(O)], (10)
is asymptotically distributed X2 with k degrees of freedom. This test statistic is the normalized difference of the restricted (eJT( 0)) and unrestricted (eJT( 0)) objective functions for the efficient GMM estimator (both using the same weighting matrix from the unrestricted model) and is analogous to the likelihood ratio test. We employ these tests for a number of the pairwise comparisons of performance among the various models.
In addition to these statistical tests, we also examine the economic impor- tance of di
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