Extreme Value Models

Abstract

This chapter discusses a statistical modeling strategy based on extreme value theory to describe the behavior of data far in the tails of the distributions, with a particular emphasis on large claims in property and casualty insurance and mortality at oldest ages in life insurance. Large claims generally affect liability coverages and require a separate analysis. The reason for a separate analysis of small or moderate losses (also referred to as attritional ones in Solvency 2) and large losses is that no ED distribution seems to emerge as providing an acceptable fit to both small and large claims. As a first application of extreme value theory, the “peak over threshold” strategy is presented, allowing the actuary to determine an optimal threshold separating the two types of losses. As a second application, the force of mortality at the oldest ages is studied with the help of individual ages at death above 95 for extinct cohorts born in Belgium between 1886 and 1904. The analysis supports the existence of an upper limit to human lifetime for these cohorts. Therefore, assuming that the force of mortality becomes ultimately constant, that is, that the remaining lifetime distribution tends to the Negative Exponential one as the attained age grows, appears to be a conservative strategy for the closure of life tables when dealing with life annuities.