Quantiles in a multi-stage nested classification credibility model
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Quantiles in a multi‑stage nested classification credibility model Georgios Pitselis1 Received: 18 July 2019 / Revised: 5 December 2019 / Accepted: 4 June 2020 © EAJ Association 2020
Abstract In insurance and finance it is often important to have a satisfactory estimate for an extreme quantile, like the one underlying capital requirements in Solvency II and Basel III. If credibility techniques on means are used for the determination of such quantiles, this can lead to quite unsatisfactory results, in particular in the presence of outliers in the data. Quantile credibility models themselves, however, cannot perform effectively when the set of data has a nested (hierarchical) structure. This paper develops multi-stage nested classification hierarchical credibility models for quantiles as an alternative to Jewell’s (G Ist Ital Attuari 38:1–16, 1975) approach, where more than one risk factor divides the portfolio into different sectors or classes. We establish hierarchical quantile credibility estimators and illustrate their performance in two numerical illustrations. Keywords Quantile · Nested classification · Hierarchical model · Credibility
1 Introduction In actuarial science, one of the fundamental problems is to predict future claims of a class of risks (contract), given the past experience of that class and other related risk classes (contracts). Credibility is a ratemaking technique predicting future premiums for a group of insurance contracts for which we have some claim experience for that group and a lot more experience for a larger group of contracts that are similar but not exactly the same. In insurance, credibility plays a special role of spreading the risk. In the calculation of premiums, there are cases where the data set has a hierarchical structure. Individual risks are classified into risk groups (classes) with common characteristics, classes into sectors and sectors into a portfolio (a line * Georgios Pitselis [email protected] 1
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou str. T. K., 18534 Piraeus, Greece
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of business). In a statistical analysis, this structure will be constructed in reverse order. Often the premium calculation follows a top-down hierarchical procedure, where the estimated expected aggregate claim amount for the whole portfolio is successively distributed over the lower levels in the form of a hierarchical tree (see [6]). For example, in worker’s compensation insurance the individual companies are grouped together based on their type of business (supermarket, pharmacies, banks etc.) and these types of businesses are grouped into danger classes (groups of types of business). Then by fixing the tariff level we calculate the premium for the various danger classes and then calculate the premium within the danger classes for the types of business. For more examples on hierarchical credibility see Goovaerts et al. [19] and Bühlmann and Gisler [6]. Jewell [22] extended the classical Bühlmann an
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