LONG-TERM BONDS: The Demand for Indexed Bonds 3

This equation was first derived by Restoy (1992). The first term is the myopic component of asset demand; it is proportional to the risk premium on the n-period bond and the reciprocal of the coefficient of relative risk aversion. The second term is Merton’s (1969, 1971, 1973) intertemporal hedging demand. It reflects the strategic behavior of the investor who wishes to hedge against future adverse changes in investment opportunities, as summarized by the consumption-wealth ratio. In our setup the investment opportunity set is time-varying because interest rates are time-varying (although expected excess returns are constant); accordingly the investor may want to hedge her consumption against adverse changes in interest rates. Intertemporal hedging demand is zero when risk aversion 7 = 1, but as 7 increases myopic demand shrinks to zero and hedging demand does not. In the limit as 7 becomes arbitrarily large, hedging demand accounts for all the demand for the risky asset this.

An important special case arises when the elasticity of intertemporal substitution is unity. As ф —> 1, the log consumption-wealth ratio becomes constant so the covariance of asset returns with this ratio approaches zero. However the covariance is divided by 1 — which also approaches zero. Giovannini and Weil (1989), by taking appropriate limits, have shown that portfolio choice is not myopic in this case even though the consumption-wealth ratio is constant. The solution presented in this paper is exact for the case ip = 1.

An explicit solution

Equation (25) is recursive in the sense that it relates current portfolio decisions to future consumption and portfolio decisions. In order to get a complete solution to the model we need to derive consumption and portfolio rules that depend only on current state variables. We do this by guessing that the consumption function takes the form
These solutions are analytical, given the log-linearization parameter p. But p itself is a nonlinear function of the coefficients 60 and b\, since p = 1 — exp{E[c£ — wt]} = 1 — exp{&o + binx}. Equations (27), (28), and the expression for p define implicitly a nonlinear mapping of p onto itself which has an analytical solution only in the case ‘0=1, when p = 6. In all other cases we solve for p numerically using a simple recursive algorithm. We set p to some initial value (typically p ~ 8) and compute the coefficients of the optimal policies; given these coefficients we compute a new value for p, from which we obtain a new set of coefficients, and so forth. We continue until the difference between two consecutive values of p is less than 10~4. This recursion converges very rapidly to a number between zero and one in cases where the value function of the model is finite; there are some cases, however, in which p is driven to zero or one because the value function is infinitely positive or negative and the infinite-horizon optimization problem is not well defined. We discuss these cases further below.