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.