Portfolio optimization in an upside potential and downside risk framework

• First full nonlinear unconstrained optimization formulation for UPM/LPM optimal portfolios.
• Solves many problems that occur with the implementation of two complex CLPM and CUPM matrices.
• Allows the implementation of a full array of utility functions including Von Neumann–Morgenstern, Prospect Theory's S-shaped utility curves, and Markowitz reverse S-shaped utility curves.
• Provides an explanatory foundation for future research into heuristic UPM/LPM models used for prediction purposes.

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).The methods and techniques proposed in this paper are focused on the following computational issues with UPM/LPM optimization.
• Lack of positive semi-definite UPM and LPM matrices.
• Rank of matrix errors.
• Estimation errors.
• Endogenous and exogenous UPM and LPM matrices.