Investments with Heterogenous Risk Aversion

We address the problem of a fund manager pooling moneys from risk-averse investors and running a single portfolio of riskfree and risky assets in such a way that individual investors are not penalized for participating in this mutual fund. This problem has a well-known solution when the investors maximize their terminal expected utility of wealth using a power utility function with the same risk-aversion coefficient. We solve this problem in the case of investors having different risk-aversion coefficients. It turns out that the initial contributions of investors must follow a precise formula which depends on the various risk-aversion coefficients. We also apply our approach to the case of a single investor with risk aversion depending on the assets in the portfolio.