Risk Aggregation: A General Approach via the Class of Generalized Gamma Convolutions

By Edward Furman, Alexey Kuznetsov, Justin Miles

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Abstract

Risk aggregation is virtually everywhere in insurance applications. Indeed, in the vast majority of situations, insurers are interested in the properties of the sums of the risks they are exposed to, rather than in the stand-alone risks per se. Unfortunately, the problem of formulating the probability distributions of the aforementioned sums is rather involved, and as a rule does not have an explicit solution. As a result, numerous methods to approximate the distributions of the sums have been proposed, with the moment-matching approximations (MMAs) being arguably the most popular. The arsenal of existing MMAs is quite impressive and contains such very simple methods as the normal and shifted-gamma approxima-tions that, respectively, match the first two and three moments only, as well as such much more intricate methods as the one based on the mixed Erlang distributions (Cossette et al. 2016). Note, however, that in practice the sums of insurance risks can have numerous and just a few summands; in the latter case the normal approximation is very questionable. Also, in practice the distributions of the stand alone risks can be light-tailed or heavy-tailed; in the latter case moments of higher orders (e.g., ?2) may not exist, and so the approximation based on mixed Erlang distributions is of limited usefulness.In this paper we put forward a refined MMA method for approximating the distributions of the sums of insurance risks. Our method approximates the distributions of interest to any desired precision, works equally well for light and heavy-tailed distributions, and is reasonably fast irrespective of the number of the involved summands.

Keywords Generalized gamma convolution, Padé approximation, Gaver-Stehfest algorithm, individual risk model, collective risk model, policy modification

Citation

Furman, Edward, Alexey Kuznetsov, and Justin Miles, "Risk Aggregation: A General Approach via the Class of Generalized Gamma Convolutions," Variance 13:2, 2021, pp. 233-249.

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Variance (ISSN 1940-6452) is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submissions by non-CAS members is encouraged.