Design of efficient investment portfolios with a shortfall probability as a measure of risk

The paper presents a constructive description of the set of all efficient (Pareto-optimal) investment portfolios in a new setting, where the risk measure named “shortfall probability” (SP) is understood as the probability of a shortfall of investor's capital below a prescribed level. Under a normality assumption, it is shown that SP has a generalized convexity property, the set efficient portfolios is constructed. Relations between the set of mean-SP and the set of mean-variance efficient portfolios as well as between mean-SP and mean-Value-at-Risk (VaR) sets of efficient portfolios are studied. It turns out that mean-SP efficient set is a proper subset of the mean-variance efficient set; interrelation with the mean-VaR efficient set is more complicated, however, mean-SP efficient set is proved to be a proper subset of mean-VaR efficient set under a sufficiently high confidence level. Besides a normal distribution, elliptic distributions are considered as an alternative for modeling the investor's total return distribution. The obtained results provides the investor with a risk measure, that is more vivid than the variance and Value-at-Risk, and with determination of the corresponding set of effective portfolios.