Tail Gini’s Risk Measures and Related Linear Programming Models for Portfolio Optimization

Following the seminal work by Markowitz (1952), the portfolio optimization problem is modeled as a mean-risk bicriteria optimization problem where the expected return is maximized and some (scalar) risk measure is minimized. In the original Markowitz model the risk is measured by the standard deviation or variance. Several other risk measures have been later considered thus creating the entire family of mean-risk (Markowitz-type) models. While the original Markowitz model forms a quadratic programming problem, following Sharpe (1971), many attempts have been made to linearize the portfolio optimization procedure (c.f., Speranza (1993) and references therein). The Linear Programming (LP) solvability is very important for applications to real-life financial decisions where the constructed portfolios have to meet numerous side constraints (including the minimum transaction lots, transaction costs and mutual funds characteristics). The introduction of these features leads to mixed integer LP problems. In order to guarantee that the portfolio takes advantage of diversification, no risk measure can be a linear function of the portfolio weights. Nev-ertheless, a risk measure can be LP computable in the case of discrete random variables, i.e., in the case of returns defined by their realizations under specified scenarios. The simplest LP computable risk measures are dispersion measures similar to the variance. Yitzhaki (1982) introduced the mean-risk model using Gini’s mean (absolute) difference as the risk measure. The Gini’s mean difference (GMD) turn out to be a special aggregation technique of the multiple criteria LP model (Ogryczak, 2000) based on the pointwise comparison of the absolute Lorenz curves. The latter leads to the quantile shortfall risk measures which are more commonly used and accepted. Recently, the second order quantile risk measures have been introduced in different ways by many authors. The measure, no