A characterization of optimal portfolios under the tail mean-variance criterion

The tail mean-variance model was recently introduced for use in risk management and portfolio choice; it involves a criterion that focuses on the risk of rare but large losses, which is particularly important when losses have heavy-tailed distributions. If returns or losses follow a multivariate elliptical distribution, the use of risk measures that satisfy certain well-known properties is equivalent to risk management in the classical mean-variance framework. The tail mean-variance criterion does not satisfy these properties, however, and the precise optimal solution typically requires the use of numerical methods. We use a convex optimization method and a mean-variance characterization to find an explicit and easily implementable solution for the tail mean-variance model. When a risk-free asset is available, the optimal portfolio is altered in a way that differs from the classical mean-variance setting. A complete solution to the optimal portfolio in the presence of a risk-free asset is also provided.