LONG-TERM BONDS: The Demand for Indexed Bonds 4

Equations (26) and (28) show that the log consumption-wealth ratio is linear in the short-term real interest rate (since xt is linearly related to n^+i)- The response of consumption to the interest rate depends on the investor’s elasticity of intertemporal substitution, but does not depend directly on her relative risk aversion. The risk aversion coefficient affects the dynamic behavior of consumption only indirectly through its effect on the log-linearization parameter p. Below we show that this indirect effect is quantitatively negligible.

The log consumption-wealth ratio is constant only when ф — 1 .In this case ct — wt equals log(l — 5). For this reason investors with unit elasticity of intertemporal substitution are called “myopic consumers.” Since 0 < p < 1 and \ф\ < 1, the consumption-wealth ratio increases with the interest rate if ф < 1 and falls with the interest rate otherwise. An increase in the short-term real interest rate is equivalent to an improvement in the investment opportunity set, and it has both income and substitution effects. An investor with low ф is reluctant to substitute intertemporally, and for her the income effect dominates, leading her to increase her consumption relative to her wealth. This increase in consumption is larger, the more persistent is the improvement in investment opportunities—the closer is ф to one. Conversely, the substitution effect dominates for an investor with high ф > 1. This investor will reduce present consumption when the interest rate increases, and will do so more aggressively when the interest rate process is persistent.

Equation (29) shows that the optimal portfolio allocation to the long-term bond is constant over time and independent of the level of the short-term interest rate. The portfolio allocation depends on the bond maturity, on the persistence of the short-term interest rate, and on the investor’s relative risk aversion, but does not depend directly on her elasticity of intertemporal substitution. The elasticity of intertemporal substitution affects portfolio choice only indirectly through its effect on the log-linearization parameter p, and we show below that this indirect effect is quantitatively negligible.

The first term inside the brackets in (29) represents the myopic demand for longterm bonds, while the second term inside the brackets represents the intertemporal hedging demand. The myopic demand depends on the parameter pmx which determines the term premium; it is zero if ftш = 0, and it shrinks as risk aversion 7 increases. The intertemporal hedging demand for bonds is zero when 7 = 1. That is, the long-term bond demand of investors with unit relative risk aversion coefficient is driven exclusively by the risk premium. For this reason they are called “myopic investors.”

Hedging demand is negative for investors with 7 < 1; these investors prefer to hold assets that pay off when investment opportunities are good, so they “reverse hedge” the risk of adverse shifts in investment opportunities. As risk aversion 7 increases, the hedging demand increases and becomes positive when 7 > 1, Meanwhile the myopic demand for bonds shrinks, so the hedging demand for bonds increases relative to the myopic demand; in section 3.7 we discuss what happens in the limit as the investor becomes infinitely risk averse. payday loan lender