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.
Equation (34) does not generally have a closed form solution, so it must be solved numerically or using an analytical approximation method. We can apply the same approximation that we have already used. Taking logs on both sides of (34) and using which is linear. It is trivial to show that this equation has the same solution that we have already derived.
Empirical properties of the solution
the same approximation around the mean log consumption-wealth ratio that we use to loglinearize the budget constraint6, we can write (34) in log form as
Implications of complete markets
We have allowed the investor to form a portfolio from only two assets, a short-term indexed bond and a single long-term indexed bond. Even with only two assets, however, markets are complete with respect to real-interest-rate risk because our real term-structure model has only one factor. This fact has several interesting implications fully.
First, with complete markets the investor can combine short- and long-term bonds so that the return on her bond portfolio is independent of the maturity of the longterm bond traded in the market. That is, she can synthesize her own optimal longterm bond, with the maturity optimal for her given her risk preferences. The return on the optimal bond portfolio is given by
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) ...
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.
In order to find optimal savings and the optimal allocations to the two bonds, we adopt an approximate analytical solution method. The first step is to characterize аП)£, the optimal allocation to the n-period bond, by combining a second-order log-linear approximation to the Euler equation with a first-order approximation to the intertemporal budget constraint. We then guess a form for the optimal consumption and portfolio policies and show that policies of this form satisfy the approximate Euler equation and budget constraint. Finally we use the method of undetermined coefficients to identify the coefficients of the optimal policies from the primitive parameters of the model. By using a second-order expansion of the log Euler equation we account for second-moment effects in the model. Electronic Payday Loans Online
Following Campbell (1993, 1996), Campbell and Viceira (1996), and Restoy (1992), we first log-Iinearize the Euler equation (14) for г = n and г — 1, where asset 1 is the short-term riskless asset. Subtracting the log-linearized Euler equation for the riskless asset from the log-linearized equation for asset n, we find:
The increase in real-interest-rate persistence increases the risk premia on indexed and nominal bonds, but it also greatly increases the volatility of indexed bond returns so the Sharpe ratio for indexed bonds is lower at 0.15. In the remainder of the paper we present portfolio choice results based on our full-sample estimates for the period 1952-96, but we also discuss results for the 1983-96 period where they are importantly different.
The Demand for Indexed Bonds
Assumptions on investor preferences
The investor is infinitely-lived, lives off her financial wealth and faces the investment environment described above. We assume that her preferences are described by the recursive utility proposed by Epstein and Zin (1989) and Weil (1989):