Portfolio optimization in an upside potential and downside risk framework

The lower partial moment (LPM) has been the downside risk measure that is most commonly used in portfolio analysis. Its major disadvantage is that its underlying utility functions are linear above some target return. As a result, the upper partial moment (UPM)/lower partial moment (LPM) analysis has been suggested by Holthausen (1981. American Economic Review, v71(1), 182), Kang et al. (1996. Journal of Economics and Business, v48, 47), and Sortino et al. (1999. Journal of Portfolio Management, v26(1,Fall), 50) as a method of dealing with investor utility above the target return. Unfortunately, they only provide dominance rules rather than a portfolio selection methodology. This paper proposes a formulation of the UPM/LPM portfolio selection model and presents four utility case studies to illustrate its ability to generate a concave efficient frontier in the appropriate UPM/LPM space. This framework implements the full richness of economic utility theory be it [Friedman and Savage (1948). Journal of Political Economy, 56, 279; Markowitz, H. (1952). Journal of Political Economy, 60(2), 151; Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. (3rd ed., 1953), Princeton University Press], and the prospect theory of (Kahneman and Tversky (1979). Econometrica, 47(2), 263).