LONG-TERM BONDS: Introduction 3

We assume that the investor’s preferences are of the form suggested by Epstein and Zin (1989, 1991); the investor has constant relative risk aversion and constant intertemporal elasticity of substitution in consumption, but these parameters need not be related to one another. Epstein-Zin preferences nest the traditional power-utility specification in which relative risk aversion is the reciprocal of the intertemporal elasticity of substitution there.

We show that the investor’s demand for long-term bonds can be decomposed into a “myopic” demand and a “hedging” demand. Myopic demand depends positively on the term premium, and inversely on the variance of long-term bond returns and the investor’s risk aversion. As risk aversion increases, myopic demand shrinks to zero. Hedging demand, on the other hand, is proportional to one minus the reciprocal of risk aversion. It is zero when risk aversion is one but accounts for all bond demand when risk aversion is infinitely large. We show that an infinitely risk-averse investor with zero intertemporal elasticity of substitution in consumption will choose an indexed bond portfolio that is equivalent to an indexed perpetuity, that is, a portfolio that delivers a riskless stream of real consumption. In this way we are able to support the commonsense view that long-term bonds are appropriate for long-lived investors who desire stability of income.

Our analysis delivers explicit solutions for portfolio weights, consumption rules, and investor welfare. We can compare investor behavior under alternative assumptions about the available menu of assets. We find that when indexed bonds are not available, inflation risk leads investors to shorten their bond portfolios and increase their precautionary savings. This has serious welfare costs for conservative investors, who are much better off when they have the opportunity to buy indexed bonds.

We also consider optimal portfolios when equities, as well as bonds, are available. We find that the ratio of bonds to equities in the optimal portfolio increases with the coefficient of relative risk aversion. As Canner, Mankiw, and Weil (1997) have pointed out, this is consistent with conventional portfolio advice but inconsistent with static mean-variance analysis. The static mean-variance model with a riskless one-period asset (“cash”) predicts that all investors should hold a single mutual fund of risky assets; more conservative investors should increase the ratio of cash to the risky mutual fund, but should not change their relative holdings of risky assets. Our model helps to resolve the asset allocation puzzle identified by Canner, Mankiw, and Weil; more generally it underscores the dangers of using static portfolio choice theory to study the dynamic problems faced by long-term investors.

The organization of the paper is as follows. Section 2 presents the two-fact or term structure model, and shows how it can be solved for bond prices at all maturities. Section 3 sets up the investor’s intertemporal consumption and portfolio choice problem, explains our approximation to the problem, and discusses the approximate solution in the case where only indexed bonds are available. This section also explains the relation of our solution method to the approach of Cox and Huang (1989). Section 4 asks how things change when only nominal bonds, or both nominal and indexed bonds, are available. This section also shows how to impose borrowing and short-sales constraints. Section 5 considers the consumption and portfolio choice problem in the presence of equities, and section 6 concludes.