A Multivariate Model of the Term Structure*

Abstract

THE TERM STRUCTURE OF interest rates has a very important role in economic theory. On the macroeconomic level the term structure serves as a key transmission link between the monetary and real sectors. On the microeconomic level the term structure not only serves as a tool for bond pricing, but the term structure also has an important role in the valuation of most financial claims. The concept of an equilibrium term structure under uncertainty was introduced by Merton [41] in 1973. Since then, Brennan and Schwartz [4, 5], Vasicek [51], Cox, Ingersoll, and Ross [11, 12], Richard [46], Long [33], and Dothan [16] have made important contributions to the theory of the term structure and equilibrium bond pricing. At this time, closed solutions for the equilibrium term structure have been derived in several special, but important, cases where the term structure is generated from univariate or bivariate stochastic processes such as the elastic random walk, geometric random walk, and square root process. However, it is arguable that the term structure of interest rates is embedded in a large macroeconomic system. Hence, it is arguable that there are generally many economic factors which are related to the term structure. The primary objective of this paper is to develop a multivariate model of the term structure that can accommodate an arbitrary number of economic relationships. The model is of a general form and can be linked with many alternative macroeconomic systems. The major assumptions of the model to be developed are that the set of stochastic factors related to the term structure follow a joint elastic random walk, that the instantaneous riskless rate can be represented as a linear combination of the same factors, and that the market prices of risk (corresponding to the different factors) are, at most, time dependent. The resulting model of the term structure can be expressed in a form that is consistent with traditional notions of the term structure. Even with multiple factors, the term structure can always be expressed in a form characterized by expectations of future short-term rates plus a liquidity premium. We will restrict our attention to the pricing of default-free, purediscount bonds. While this special case is of interest in its own right, a model for pricing the default-free, pure-discount bond also plays an important role in pricing default-free coupon bonds, risky pure-discount bonds, and, as Cox, Ingersoll, and Ross show, the valuation of most financial claims [12].