PORTFOLIO OPTIMIZATION WITH PERFORMANCE RATIOS

Abstract

We consider the portfolio selection problem of maximizing a performance measure in a continuous-time diffusion model. The performance measure is the ratio of the overperformance to the underperformance of a portfolio relative to a benchmark. Following a strategy from fractional programming, we analyze the problem by solving a family of related problems, where the objective functions are the numerator of the original problem minus the denominator multiplied by a penalty parameter. These auxiliary problems can be solved using the martingale method for stochastic control. The existence of solution is discussed in a general setting and explicit solutions are derived when both the reward and the penalty functions are power functions.