Optimum Consumption and Portfolio Allocations under Incomplete Information

We solve in closed form the optimal consumption / portfolio choice problem for the class of isoelastic utility functions under incomplete information about the mean return of the stock price. Our approach consists in converting the original investor's problem into an equivalent program where the agent's utility function is state dependent and for which standard martingale techniques can be easily implemented. Upon observing the realizations of the stock and possibly outside market information, the investor can revise her beliefs about the true value of the mean return, which induces optimal allocations that can be significantly different from those of a myopic agent. We find that the fraction of wealth invested into the risky asset is always increasing in the investor's assessment of the conditional mean return whereas the consumption-wealth is increasing (decreasing) in the latter if the intertemporal elasticity of substitution (I.E.S.) is below (above) unity. When the investor has access to outside market information, the more informative the signal, the higher (lower) the fraction of wealth invested into the risky asset and the lower (higher) the consumption-wealth ratio exactly when the I.E.S. is greater (smaller) than one. The reason is that the agent understands that when she receives better quality information, she can update her beliefs more quickly which in turn can lead to larger changes in her optimal consumption and portfolio allocations. She devotes more of her wealth into the risky asset only if she is willing to tolerate changes in her consumption pattern. Finally, as far as the hedging demand for the risky security is concerned, it is positive (negative) and rises (falls) with more accurate information and the investor horizon, exactly when the I.E.S. if above (below) unity. Hence, the conventional advice according to which long horizon investors should allocate aggressively their wealth to equity is founded only for agents whose intertemporal elasticity of substitution is above unity