Translation-Invariant and Positive-Homogeneous Risk Measures and Optimal Portfolio Management

The problem of risk portfolio optimization with translation-invariant and positive-homogeneous risk measures, which includes value-at-risk (VaR) and tail conditional expectation (TCE), leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. In this paper we provide an explicit closed-form solution of this minimization problem, and the condition under which this solution exists. The results are illustrated using data of 10 stocks from NASDAQ/Computers. The distance between the VaR and TCE optimal portfolios is investigated