Convexity, two-fund separation and asset ranking in a mean-LPM portfolio selection framework

This paper addresses two most important problems of mean-lower partial moment (MLPM) portfolio selection theory, the convexity of efficient frontier and the availability of target returns that permit two-fund separation (TFS). The convexity of the efficient frontier is a very crucial property as it guarantees the existence of various important results. However, in the MLPM framework, the convexity has not been analytically proved yet. In this paper, we provide an analytical proof for this convexity. On the other hand, in the MLPM framework, the separation is guaranteed for two specific targets—risk-free rate and mean return. The question of which other targets admit the separation has not been solved for the last three decades. As a result, non-separation occurs with the use of arbitrary targets and thereby several pitfalls arise in the MLPM portfolio optimization and asset ranking (Brogan and Stidham, Eur J Oper Res 184(2):701–710, 2008; Hoechner et al., Int Rev Finance 17(4):597–610, 2017). We solve this problem by showing the existence and uniqueness of a generalized family of target returns that guarantees the MLPM separation. The discovery of generalized target provides a sound theoretical foundation to use a modified version of Kappa ratio which, unlike the usual Kappa ratio, always satisfies the maximum principle and the invariance property (Pedersen and Satchell, Quant Finance 2(3):217–223, 2002; Zakamouline, Quant Finance 14(4):699–710, 2014). Finally, we conduct empirical experiments that illustrate our theoretical results and unfold some interesting facts.