Understanding saving and portfolio choices with predictable changes in assets returns

We consider an expected-utility-maximizing consumer living two periods who can invest in two assets, one of which is risk free. We do not restrict relative risk aversion to be constant. We first examine the effect that a change in the opportunity set in the second period has on the optimal saving in the first period. We show that an increase in the future risk free rate (keeping the equity premium unchanged) reduces savings if relative risk aversion is uniformly larger than unity. An increase in the equity premium or a reduction in the volatility of the risky asset raises savings if the index of cautiousness, i.e., the derivative of absolute risk tolerance, is smaller than unity. In a second stage, we use these results to determine the sign of the hedging demand for the risky asset for the following three types of predictability: predictable changes in the interest rate, mean-reversion in stock returns, and predictable changes in volatility. Depending upon the type of predictability under scrutiny, what matters to sign the hedging demand is whether relative risk aversion or cautiousness is smaller or larger than unity.