LONG-TERM BONDS: The Demand for Indexed Bonds 7

Table 5 summarizes the optimal consumption behavior that is associated with these portfolio rules. The left hand side of the table shows the average consumption-wealth ratio, while the right hand side shows the standard deviation of optimal consumption growth. To understand the patterns of average consumption-wealth ratios, recall that an investor with zero elasticity of intertemporal substitution consumes the annuity value of wealth each period, so the average consumption-wealth ratio for this investor is just the average expected return on the portfolio comments.

This average return declines with risk aversion, and so the average consumption-wealth ratio also declines with risk aversion as shown in the 1/5000 column. Investors with higher elasticities of consumption, shown to the left of the 1/5000 column, are willing to substitute intertemp orally in response to incentives. The direction of the substitution depends on the average return on the portfolio in relation to the time discount rate and the risk of the portfolio. Investors with low risk aversion (at the top of the panel) have high average portfolio returns so they substitute by reducing present consumption, while investors with high risk aversion (at the bottom of the panel) have low average portfolio returns so they substitute by increasing present consumption. The magnitude of these effects is such that all investors with unit elasticity of substitution have the same average consumption-wealth ratio of (1 — <5), regardless of their risk aversion.

The accuracy of our loglinear approximation to the intertemporal budget constraint depends on the volatility of the log consumption-wealth ratio. We do not report this volatility in Table 5, but it is zero for ф = 1 (the case where our approximation holds exactly) and is roughly proportional to (ф — 1). (It would be exactly proportional if the log-linearization parameter p were fixed.) The maximum standard deviation of the log consumption-wealth ratio is about 3% for ф close to zero at the far right of the table.

These numbers suggest that our approximation should be extremely accurate for a term-structure model of the sort we have estimated in 1952-96. Campbell and Koo (1997) use numerical methods to solve a model with an exogenous portfolio return that follows an AR(1) process like the endogenous portfolio return in our model; they find that approximation error is very small whenever the standard deviation of the log consumption-wealth ratio is 5% or below.

The right hand part of Table 5 illustrates some interesting patterns in the conditional volatility of consumption growth. Investors with low risk aversion hold leveraged bond portfolios that given them highly volatile consumption, regardless of their intertemporal elasticity of substitution in consumption. Conservative investors hold indexed bonds for hedging purposes. Investors who are both highly risk-averse and highly reluctant to substitute consumption intertemporally reduce the conditional volatility of their consumption growth to zero; highly risk-averse investors who are willing to substitute intertemporally, however, respond to interest rate movements by adjusting their consumption, so their conditional consumption volatility is positive. We now explore in more detail the behavior of highly risk-averse investors.