TY - JOUR

T1 - Extreme-value based estimation of the conditional tail moment with application to reinsurance rating

AU - Goegebeur, Yuri

AU - Guillou, Armelle

AU - Pedersen, Tine

AU - Qin, Jing

N1 - Funding Information:
The authors sincerely thank the editor, the associate editor and the referees for their helpful comments and suggestions that led to considerable improvements of the paper. The research of Armelle Guillou was supported by the French National Research Agency under the grant ANR-19-CE40-0013-01/ExtremReg project.
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/11

Y1 - 2022/11

N2 - We study the estimation of the conditional tail moment, defined for a non-negative random variable X as θβ,p=E(Xβ|X>U(1/p)), β>0, p∈(0,1), provided E(Xβ)<∞, 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.

AB - We study the estimation of the conditional tail moment, defined for a non-negative random variable X as θβ,p=E(Xβ|X>U(1/p)), β>0, p∈(0,1), provided E(Xβ)<∞, 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.

KW - Conditional tail moment

KW - Excess-of-loss reinsurance

KW - Order statistics

KW - Pareto-type distribution

KW - Second order condition

KW - Tail index

U2 - 10.1016/j.insmatheco.2022.08.003

DO - 10.1016/j.insmatheco.2022.08.003

M3 - Journal article

AN - SCOPUS:85136578974

SN - 0167-6687

VL - 107

SP - 102

EP - 122

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

ER -