An Analytic Market Condition for Mutual Fund Separation: Demand for the Non-Sharpe Ratio Maximizing Portfolio

This study finds a necessary and sufficient condition for mutual fund separation, in which investors have the same portfolio of risky assets regardless of their utility functions. Unlike previous studies, the market condition is obtained in analytic form, using the Clark–Ocone formula of Ocone and Karatzas (Stoch Stoch Rep 34(3–4):187–220, 1991). We also find that the condition for separation among arbitrary utility functions is equivalent to the condition for separation among utility functions with constant relative risk aversion (CRRA utility functions). The condition is that a conditional expectation of an infinitesimal change in the uncertainty of an instantaneous Sharpe ratio maximizing portfolio can be hedged by the Sharpe ratio maximizer itself. In a Markovian market, such an infinitesimal change is characterized as an infinitesimal change in state variables. A closer look at the Clark–Ocone formula offers an intuition of the condition: an investor invests in (1) the Sharpe ratio maximizer; and (2) another portfolio in such a way as to reduce the uncertainty produced by an infinitesimal change in the Sharpe ratio maximizer, depending on four components: the investor’s wealth level, marginal utility and risk tolerance, at the time of consumption, and the shadow price. This decomposition is also valid in non-Markovian markets.