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
Tables 3-5 explore the properties of our solution using the bond-pricing parameters estimated in Table 1 for the period 1952-96. We compute optimal portfolio and consumption rules for investors with the same time discount rate (4%) but different coefficients of relative risk aversion and elasticities of intertemporal substitution. We consider risk aversion coefficients of 0.75, 1, 2, 5, 10, and 5000 (effectively almost infinite), and elasticities of intertemporal substitution that are the reciprocals of these values.
The tables are organized so that very risk-averse investors are at the bottom, investors who are very reluctant to substitute intertemporally are at the right, and power-utility investors (for whom the elasticity of intertemporal substitution is the reciprocal of risk aversion) are along the main diagonal. The top panel of each table assumes that the bonds available to investors are one-quarter and ten-year zero-coupon indexed bonds.
The top panel of Table 3 reports the percentage portfolio share of a ten-year zero-coupon indexed bond. Since indexed bonds have attractive Sharpe ratios, we find that investors with low risk aversion have a very large myopic demand for long-term indexed bonds; they want to invest many times their total wealth in these bonds and borrow at the short-term riskless interest rate. As risk aversion increases, the demand for indexed bonds gradually declines, but it does not go to zero because highly risk averse investors have a positive intertemporal hedging demand for long-term indexed bonds.
Table 4 clarifies this point by reporting the share of intertemporal hedging demand in the total demand for long-term bonds. This share rises from zero when 7 = 1 to 99.7% when 7 = 5000. Note also that different columns of these tables, corresponding to different elasticities of intertemporal substitution, are almost identical. This confirms our theoretical claim that the elasticity of intertemporal substitution, operating only indirectly through the log-linearization parameter p, has a negligible effect on portfolio allocation. Electronic Payday Loans Online