Optimal collective investment: an analysis of individual welfare

We analyze optimal asset allocation in continuous time for a collective of tied-together investors. We rely on a specific collective utility function which dates back to Karatzas et al. (Math Oper Res 15(1):80–128, 1990), by which the fund manager maximizes the weighted average of expected individual utilities for the investors in the collective. This problem allows for a closed form solution. The payoffs allocated to the investors correspond to the individually optimal ones which can be reached with a modified initial wealth that results from redistribution. The redistribution of wealth follows from the weights of the individual investors in the collective utility function, the condition of financial fairness, the condition of maximal average utility gains, or some other condition on net certainty equivalent returns. We illustrate the resulting solutions both for a Black–Scholes economy and a model with stochastic interest rates.