بهینه سازی سبد سرمایه گذاری در چارچوب خطر بالقوه و ریسک نامطلوب

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.

The mean-lower partial moment (μ,LPM) model has been attractive to decision makers because it does not require any distributional assumptions and it is a necessary and sufficient condition for investors with various classes of Von Neumann and Morgenstern (1944) (hereafter, vNM) utility functions which is equivalent to expected utility-maximization under risk aversion. 1 Because it does not make any distributional assumption, it has been particularly useful in the management of derivative portfolios ( Merriken, 1994 and Huang et al., 2001 ; Pedersen, 2001; and Jarrow & Zhao, 2006). However, LPM has traditionally been challenged by academic researchers because of the computational complexity of the asymmetric Co-LPM matrix used in μ-LPM portfolio analysis and the persistent belief that it is an ad-hoc method that is not grounded in capital market equilibrium theory and in expected utility maximization theory. 2 A major challenge to the use of any portfolio theory formulation that does not use mean-variance analysis is by Markowitz (2010). His position is even with non-normal security distributions, the mean-variance criterion is still a useful approximation of the expected utility of the investor. In other words, any alternative to mean-variance portfolio theory has to rest on a solid foundation of utility theory. It is not sufficient for the portfolio framework to simply be a nonparametric approach. The UPM/LPM framework is powerful because it is a nonparametric approach and it implements the full richness of economic utility theory be it Friedman and Savage (1948), Markowitz (1952), Von Neumann and Morgenstern (1944), or the prospect theory of Kahneman and Tversky (1979). Markowitz (2010) does end up supporting the geometric mean-semivariance portfolio theory model in his paper because of its utility theory foundation. Semivariance is the only risk measure other than variance that is accorded any support by Markowitz, 1959 and Markowitz, 2010. While our focus is not on LPM and capital market theory, the discussion in Hogan and Warren (1974), Bawa and Lindenberg (1977), Harlow and Rao (1989), Leland (1999), and Pedersen and Satchell (2002) makes it pretty clear that LPM is not an ad-hoc model that is ungrounded in capital market theory. We are interested in solving the computational complexities of the μ-LPM and its well-known utility maximization limitation of assuming a linear utility function above the target return.3 By solving the μ-LPM computational problem, we are able to introduce the upper partial moment-lower partial moment (UPM/LPM) portfolio selection model which extends the expected utility maximization capabilities of the LPM model. The paper continues with a discussion of mean-LPM and UPM/LPM portfolio analysis and their place in expected utility theory. Next, we offer a formulation for testing UPM/LPM portfolio optimization problems and discuss the historic issue of exogenous and endogenous LPM matrices. Next, four empirical problems are discussed which include: (1) Lack of positive semi-definite UPM and LPM matrices; (2) Rank of matrix errors; (3) Estimation errors; and (4) Endogenous and Exogenous UPM and LPM matrices. The paper offers discussions and solutions for each of these problems. Finally, the paper will present four utility theory case problems to illustrate the ability of the UPM/LPM model to generate a concave efficient frontier in the appropriate UPM/LPM space.

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 was suggested by Holthausen (1981), Kang et al. (1996), and Sortino et al. (1999) as a method of dealing with investor utility above the target return.27 Unfortunately, they only provide dominance rules rather than a portfolio selection methodology that implements the intercorrelations between securities. This paper proposes a formulation of the UPM/LPM portfolio selection model which includes all intercorrelations between securities and presents four utility case studies to illustrate its ability to generate a concave efficient frontier in the appropriate UPM/LPM space. 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. • Problems solving non-linear objective function subject to nonlinear constraints. • Rank of matrix errors. • Lack of closed form solutions in the optimization problem when using endogenous matrices. There is still an issue with estimation error, but there is evidence in the literature that the LPM measure is a ‘good’ estimator of risk when there are asymmetric distributions present in the market and there is a sufficiently large sample size.28 The two-fold advantage of the general UPM/LPM model is that it encompasses a vast spectrum of utility theory as well as a large number of symmetric and asymmetric return distributions. Thus, it meets Markowitz, 1959 and Markowitz, 2010 admonishment that portfolio theory has to be built on a solid expected utility theory foundation. The UPM/LPM analysis includes the reverse S-shaped utility functions of Friedman and Savage (1948) and Markowitz (1952) as well as the family of utility functions represented by stochastic dominance that are presented in Swalm (1966), Porter (1974), Bawa (1975), and Fishburn (1977). Finally, the UPM/LPM model is consistent with the prospect theory S-shaped utility functions proposed by Kahneman and Tversky (1979) as well as being congruent with Von Neumann and Morgenstern (1944) expected utility theory. This study was focused on the expected utility and computational issues of UPM/LPM analysis and not on general equilibrium theory. If there exists an aggregate utility function as suggested by Kroll, Levy, and Markowitz (1984), Levy and Levy (2004), and Post and Levy (2005), Post and van Vliet (2002), Baltussen, Post, and van Vliet (2012), then a global optimum consistent with equilibrium asset pricing theory may possibly be obtainable using UPM/LPM analysis.29