Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions

Abstract

Abstract. Among the many possible ways to study the right tail of a real-valued random variable, a particularly general one is given by considering the family of its Wang distortion risk measures. This class of risk measures encompasses various interesting indicators, such as the widely used Value-at-Risk and Tail Value-at-Risk, which are especially popular in actuarial science, for instance. In this paper, we first build simple extreme analogues of Wang distortion risk measures and we show how this makes it possible to consider many standard measures of extreme risk, including the usual extreme Value-at-Risk or Tail-Value-at-Risk, as well as the recently introduced extreme Conditional Tail Moment, in a unified framework. We then introduce adapted estimators when the random variable of interest has a heavy-tailed distribution and we prove their asymptotic normality. The finite sample performance of our estimators is assessed on a simulation study and we showcase our techniques on two sets of real data.

Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions

“…These measures are of great value in practice, especially in actuarial science: for instance, as mentioned in Dowd and Blake (2006), the Tail-Value-at-Risk would be used if one is interested in the average loss after a catastrophic event or to estimate the cover needed for an excess-of-loss reinsurance treaty. As shown in El Methni and Stupfler (2017), the aforementioned quantities can actually be written as simple combinations of Wang distortion risk measures of a power of the variable of interest (abbreviated by Wang DRMs hereafter; see Wang, 1996). Wang DRMs are weighted averages of the quantile function, the weighting scheme being specified by the so-called distortion function; on the practical side, Wang DRMs can, among others, be useful to price insurance premiums, bonds, and tackle capital allocation problems, see e.g.…”

“…The second problem, which is theoretical, is that their asymptotic results about this class of estimators are restricted to asymptotic normality and are thus somewhat frustrating in the sense that they are stated under an integrability condition on the quantile function which is substantially stronger than the simple existence of the Wang DRM to be estimated. In particular, a consistency result under the latter condition, in the spirit of the one Jones and Zitikis (2003) obtained for the estimation of fixed-order Wang DRMs, is not provided in El Methni and Stupfler (2017).…”

“…Two weaknesses of this study can be highlighted however. The first problem, a practical one, is that it is a consequence of the results in the simulation study of El Methni and Stupfler (2017) that the finite-sample performance of the suggested class of estimators decreases sharply in terms of mean squared error as the tail of the underlying distribution gets heavier. This is due to the propensity of heavier-tailed distributions to generate highly variable top order statistics and, therefore, to increase dramatically the variability of the estimates.…”

“…To the best of our knowledge though, the only study providing estimators of extreme distortion risk measures is the recent work of El Methni and Stupfler (2017). More precisely, they show that a simple and efficient solution to estimate extreme Wang DRMs when the right tail of the underlying distribution is moderately heavy is to consider a so-called functional plug-in estimator.…”

“…This is due to the propensity of heavier-tailed distributions to generate highly variable top order statistics and, therefore, to increase dramatically the variability of the estimates. No solution is put forward in El Methni and Stupfler (2017) in order to tackle this issue. The second problem, which is theoretical, is that their asymptotic results about this class of estimators are restricted to asymptotic normality and are thus somewhat frustrating in the sense that they are stated under an integrability condition on the quantile function which is substantially stronger than the simple existence of the Wang DRM to be estimated.…”