A Closed Form Equation for the Price of a Lognormal Payoff Under Power Utility

Abstract

We imagine investors taking shares in an exogenous lognormal cash payoff with known parameters. Using a power utility (CRRA) replica of Lintner's static payoffs-based CARA-normal CAPM, we examine how an investor's expected utility is affected by the payoff parameters and surrounding market conditions. The market clearing asset price falls as a proportion of wealth when market wealth is higher, implying that CRRA investors hold a lower (rather than fixed) proportion of wealth in the risky asset when they are wealthier. Investors prefer conditions where they can obtain more risk, either because the risky payoff is exogenously riskier or because there is less risk sharing. The equilibrium asset price is "disproportionately" lower when the asset is riskier or when there are fewer willing buyers, making investment more attractive under those conditions.

Keywords: CRRA CAPM, lognormal payoff, Lambert function, attraction of risk, risk sharing