Optimizing benchmark-based utility functions∗

We consider four utility functions, each of which incorporates a benchmark to better capture the motivations of today’s portfolio managers. Assuming instrument returns are normally distributed, we establish conditions under which optimal portfolios for these utilities are mean-variance efficient and we briefly discuss computing solutions of the models via standard nonlinear programming tools. When returns are not normally distributed, we cannot, in general, solve the optimal allocation problems exactly. Instead we use an approximation procedure rooted in Monte Carlo simulation. Our approach requires mixed-integer programming, and we describe computational enhancements that significantly improve our ability to solve these models.