Stochastic Processes for Interest Rates and Equilibrium Bond Prices

DEFAULT FREE BONDS REPRESENT, in essence, money sold forward: the investor gives dollars (say) to the Government now in exchange for a contract which promises to pay, with probability one, a given number of dollars at maturity, and possibly a stream of intermediate coupon payments prior to maturity. Only recently have financial economists begun to examine the pricing of those bonds in a portfolio context. In that framework, nominal "liquidity premiums" or risk premiums on long-term default-free bonds are attributable to uncertainty about taste, technological, and price level changes which may occur prior to maturity of the bonds (and prior to each coupon payment if any). To the extent that the investor is unable to diversify away that uncertainty, there is a systematic risk attached to the marginal utility which he or she can derive from the use in consumption or investment of the dollar payoffs on the bonds. In some term structure models, the instantaneous or "short-term" rate of interest, denoted here as a function of time t by R(t) in nominal terms and r(t) in real terms, can be taken as a sufficient statistic1 for the source of systematic uncertainty about the marginal utility of the proceeds to be derived from defaultfree bonds maturing beyond the next instant (e.g., Merton [1974], Vasicek [1977], Cox, Ingersoll, and Ross [1977]). Specific assumptions about tastes, technology, and utility functions lead to specific equilibrium stochastic processes for the instantaneous (short-term) riskless rate, and longer term bond prices which are modelled as functionally dependent upon maturity and that short term riskless rate. If a closed form solution exists for the long-term bond price as a function of maturity and the short term interest rate, it will be conditional, inter alia, on the parameters of the stochastic process for that short term rate. Marsh [1981] proposed that the closed form solutions could be tested in terms of the restrictions implied cross-sectionally on the parameters of the time series process for the returns (or prices) of longer term bonds, each with a given term to maturity. For an analogy outside the term structure literature, think of the closed form solution given by Black and Scholes [1972] for the price of an option on a nondividend paying stock with lognormal price dynamics. Given the model, the cross-section of time series for the changes in prices for each of the several options written on the same stock are not unrestricted: their dispersion is functionally linked to the same volatility parameter for the stock price. Thus, the option model, including