The stochastic properties of term structure movements☆

In mean–variance (M–V) analysis, an investor with a holding period [0,T] operates in a two-dimensional space—one is the mean and the other is the variance. At time 0, he/she evaluates alternative portfolios based on their means and variances, and holds a combination of the market portfolio (e.g., an index fund) and the risk-free asset to maximize his/her expected utility at time T. In our continuous-time model, we operate in a three-dimensional space—the first is the spot rate, the second is the expected return on the risky asset (e.g., an index fund), and the third is time. At various times over [0,T], we determine, for each combination of the spot rate and expected return, the optimum fractions invested in the risky and risk-free assets to maximize our expected utility at time T. Hence, unlike those static M–V models, our dynamic model allows investors to trade at any time in response to changes in the market conditions and the length of their holding period. Our results show that (1) the optimum fraction y*(t) in the risky asset increases as the expected return increases but decreases as the spot rate increases; (2) y*(t) decreases as the holding period shortens; and (3) y*(t) decreases as the risk aversion parameter-γ is larger.

Previous studies often assume a linear relation between term premiums on Treasury securities and forward interest rates even though a nonlinear relation is a theoretical and an empirical possibility. To examine the relation, this paper uses a nonparametric kernel approach that permits both linear and nonlinear associations. The linear specification yields conditional expectations of term premiums that are similar to those predicted by the kernel approach only at the mean forward premiums. Generally, kernel estimation shows that the responses of expected term premiums to changes in forward premiums are time-varying and are significantly different from the constant slope coefficients produced by linear estimations. The evidence also shows that forward premiums contain much more information content for predicting future term premiums that has been found with linear estimation procedures.

This paper provides a detailed survey of the financial economics literature pertaining to the term structure of interest rates. The analysis concentrates exclusively on the application of modern contingent claims pricing methods, with particular emphasis on the term structure as a subset of general asset pricing theory, recent models for interest rate contingent claims which are designed to replicate any current term structure, and empirical evidence.

We derive and test an alternative closed-form general equilibrium model of the term structure within the Cox, Ingersoll, and Ross theoretical framework in which yields are nonlinear functions of the risk-free rate. We show that equilibrium bond prices and the risk-free rate are not always inversely related and that bond risk need not be strictly increasing in maturity. Using Hansen's generalized method of moments to obtain parameter estimates, this nonlinear model outperforms the Cox, Ingersoll, and Ross square root model in describing actual Treasury bill yields for the 1964–1986 period.

The stochastic properties of flexible exchange rates are investigated by looking at a sample of intradaily Sfr./$ quotes for 231 trading days in 1983. Estimated moments of the distribution of returns over five-minute intervals often exhibit large changes which are mostly unpredictable even from one day to the next. Autocorrelation in returns is generally absent even inside a given day. Specific days of the week do not exhibit any special characteristics. During a single day, both relative changes in exchange rates as well as their volatility show some distinct patterns.

Term-structure models from Cox, Ingersoll, and Ross (1985) imply that conditional expected discrete-period returns on discount instruments are linear functions of forward rates. Tests reject a single-latent-variable model of expected returns on U.S. Treasury bills, but two or three latent variables appear to describe expected returns on bills of all maturities. Expected returns estimated using two-latent-variables exhibit variation with business cycles similar to what Fama (1986) observes for forward rates. Inverted term structures precede recessions and upward-sloping structures precede recoveries.