Tail Value-at-Risk-Based Expectiles for Extreme Risks and Their Application in Distributionally Robust Portfolio Selections

Empirical evidence suggests that financial risk has a heavy-tailed profile. Motivated by recent advances in the generalized quantile risk measure, we propose the tail value-at-risk (TVaR)-based expectile, which can capture the tail risk compared with the classic expectile. In addition to showing that the risk measure is well-defined, the properties of TVaR-based expectiles as risk measures were also studied. In particular, we give the equivalent characterization of the coherency. For extreme risks, usually modeled by a regularly varying survival function, the asymptotic expansion of a TVaR-based expectile (with respect to quantiles) was studied. In addition, motivated by recent advances in distributionally robust optimization in portfolio selections, we give the closed-form of the worst-case TVaR-based expectile based on moment information. Based on this closed form of the worst-case TVaR-based expectile, the distributionally robust portfolio selection problem is reduced to a convex quadratic program. Numerical results are also presented to illustrate the performance of the new risk measure compared with classic risk measures, such as tail value-at-risk-based expectiles.

Automated summary

Recent empirical evidence shows that financial risk exhibits a heavy-tailed distribution. Building on advances in generalized quantile risk measures, a tail value-at-risk (TVaR)-based expectile is proposed to capture tail risk compared to the classic expectile. This study not only presents the well-defined nature of the risk measure but also investigates its coherency properties. The asymptotic expansion of a TVaR-based expectile, with respect to quantiles, is studied for extreme risks commonly modeled by a regularly varying survival function. Additionally, a closed-form expression for the worst-case TVaR-based expectile is derived based on moment information, motivated by recent developments in distributionally robust optimization for portfolio selection. Numerical results demonstrate the performance of the new risk measure compared to classic risk measures, such as tail value-at-risk-based expectiles.