Estimating the Conditional Tail Expectation in the Case of Heavy‐Tailed Losses

Abstract

The conditional tail expectation CTE is an important actuarial risk measure and a useful tool in financial risk assessment. Under the classical assumption that the second moment of the loss variable is finite, the asymptotic normality of the nonparametric CTE estimator has already been established in the literature. The noted result, however, is not applicable when the loss variable follows any distribution with infinite second moment, which is a frequent situation in practice. With a help of extreme-value methodology, in this paper, we offer a solution to the problem by suggesting a new CTE estimator, which is applicable when losses have finite means but infinite variances.

Reduced-Bias Estimator of the Conditional Tail Expectation of Heavy-Tailed Distributions

“…determines the sign of the bias of C n,α [X]. Note that the asymptotic variance AV(γ) does not depend on α and that this result generalizes Theorem 3.1 in Necir et al (2010) in case λ = 0. The goal of the next section is to propose a reduced-bias estimator of C α [X].…”

“…Its asymptotic normality has been first established in the literature under the classical assumption that the second moment of the loss variable is finite, this condition being very restrictive in practical applications. Such a result has been extended by Necir et al (2010) in the case of infinite second moment. In this framework, we propose a reduced-bias estimator of the CTE.…”

“…In this spirit, Necir et al (2010) introduced the following estimator of the CTE, which takes into account different asymptotic properties of moderate and high quantiles in the case of Pareto-type distributions:…”

“…First, we state an asymptotic normality result for C n,α [X] exhibiting the bias term (Section 2) and thus generalizing the one of Necir et al (2010). Second, the precise knowledge of the first order of the bias allows us to propose a reduced-bias approach.…”

Estimation of multivariate conditional-tail-expectation using Kendall's process

“…see Brazauskas et al, 2008;Necir et al, 2010;Ahn and Shyamalkumar, 2011). The estimation of multivariate riskmeasures has received some attention however, it is much less common, due to a number of theoretical and practical reasons.…”