Extreme-value based estimation of the conditional tail moment with application to reinsurance rating

We study the estimation of the conditional tail moment, defined for a non-negative random variable X as θ β , p = E ( X β | X > U ( 1 / p ) ) , β > 0 , p ∈ ( 0 , 1 ) , provided E ( X β ) < ∞ , where U denotes the tail quantile function given by U ( x ) = inf ⁡ { y : F ( y ) ⩾ 1 − 1 / x } , x > 1 , associated to the distribution function F of X. The focus will be on situations where p is small, i.e., smaller than 1 / n , where n is the number of observations on X that is available for estimation. This situation corresponds to extrapolation outside the data range, and requires extreme value arguments to construct an appropriate estimator. The asymptotic properties of the estimator, properly normalised, are established under suitable conditions. The developed methodology is applied to estimation of the expected payment and the variance of the payment under an excess-of-loss reinsurance contract. We examine the finite sample performance of the estimators by a simulation experiment and illustrate their practical use on the Secura Belgian Re automobile claim data.