An Empirical Comparison of Alternative Models of the Short‐Term Interest Rate

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 provides 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 Ingerso11(1984), Schaefer and Schwartz (1984), Sundaresan (1984), Feldman (19891, 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 Warren Bailey, Emilio Barone, Fischer Black, Tim Bollerslev, Stephen Buser, John Campbell, Jennifer Conrad, George Constantinides, Ken Dunn, Margaret Forster, Campbell Harvey, Patric Hendershott, David Mayers, Huston McCulloch, Daniel Nelson, David Shimko, Ren6 Stulz, Stuart Turnbull, Curt Wells, and Finance Workshop participants a t the Commodity Futures Trading Commission, the University of Iowa, the Kansallis Research Foundation, The Ohio State University, Purdue University, and participants a t 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. 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.' The issue of how these models compare with each other is particularly important, however, 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 performance 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: 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 implications 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 'Because of this problem, empirical work in this area has tended to focus on specific models instead of comparisons across models. See, for example, Brennan and Schwartz (19821, Brown and Dybvig (1986), Gibbons and Ramaswamy (19861, Pearson and Sun (1989) and Barone, Cuoco, and Zautzik (1991). 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 I1 discusses the econometric approach. Section I11 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 , P, a , 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. Merton d r = ad t + a d z 2. Vasicek d r = ( a + pr )d t + a d z 3. CIR SR d r = ( a + pr )d t + ar l / 'dZ 4. Dothan d r = a r d Z 5. GBM dr = Prdt + a r d Z 6. Brennan-Schwartz d r = ( a + pr )d t + a r d Z 7. CIRVR d r = ar3/ 'dZ 8. CEV d r = Prdt + a r Y d Z Model 1is 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 Gaussian 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 p = 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 mortgagebacked security valuation model in Dunn and McConnell(l981), 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 The Journal o f Finance Table I 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