Expected Utility from a Rationally Priced Lognormal Payoff Under Power Utility

Abstract

We consider investment in a rationally priced lognormal payoff under power (CRRA) utility. The asset price is set endogenously in response to investors’ demands. By deriving a new price equation, which we call the CRRA-lognormal CAPM, we. find that CRRA investors: (i) hold a lower proportion of wealth in the risky asset when their wealth is higher, (ii) obtain higher expected utility when the risky payoff is riskier, and (iii) prefer that other investors are more risk averse and hence less willing to invest. These surprising results point to previously unknown theoretical attributes of CRRA investors that we prove analytically.