Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization

Abstract

The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of $L^{p}$ optimization. These generalized $L^{p}$-quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for $1<p<2$. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate $L^{p}$-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme $L^{p}$-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme $L^{p}$-quantiles and elaborate the choice of $p$ through applications to some simulated and financial real data.