Portfolio optimization for wealth-dependent risk preferences

Empirical and theoretical studies of preference structures of investors have long shown that personal and corporate utility is typically multimodal, implying that the same investor can be risk-averse at certain levels of wealth while risk-seeking at others. In this paper, we consider the problem of optimizing the portfolio of an investor with an indefinite quadratic utility function. The convex and concave segments of this utility reflect the investor's attitude towards risk, which changes based on deviations from a fixed goal. Uncertainty is modeled via a finite set of scenarios for the returns of securities. A global optimization approach is developed to solve the proposed nonconvex optimization problem. We present computational results which investigate the effect of short sales and demonstrate that the proposed approach systematically produces portfolios with higher values of skewness than the classical expectation-variance approach.