Tail variance of portfolio under generalized Laplace distribution

Many popular downside risk measures such as tail conditional expectation and conditional value-at-risk only characterize the tail expectation, but pay no attention to the tail variance beyond the value-at-risk. This is a severe deficiency of risk management in finance and insurance industry, especially in measuring extreme risk with large losses. We derive the explicit formulae of the tail variance of portfolio under the assumption of generalized Laplace distribution, and mixture generalized Laplace distribution as well. Some numerical results of parameters related to the tail variance of portfolio are also provided. Finally, we present an example of application to optimization tail mean-variance portfolio. The empirical results show that the performance of optimal portfolio can be efficiently improved by controlling the tail variability of returns distribution.