Properties of the MBBEF D Distribution Classes

In insurance and reinsurance, policies and reinsurance contracts often involve deductibles and limits. Insurers and reinsurers thus need to assess the covered portion of the expected loss cost. This is done with exposure curves that reflect the retained portion of of the expected loss cost as a function of the deductible. These exposure curves are either confined to the support [0, 1] in the case of an upper limit for the claims amount or to the support R + in the case of unlimited claims. The curves are derived from empirical data and either tabulated or approximated with parametric functions obtained via a transformation of the underlying probability distributions. The widely used two-parametric MBBEF D distribution class introduced by the author in 1997 for reinsurance pricing [1] is quite flexible and applicable to both cases. Gatti and Wüthrich [2] have extended the class to a bigger family. This article derives expressions (involving the dilogarithm Li2(x) for the second moment, it analyzes the properties of MBBEF D distribution class in greater detail, and it provides algorithms for the implementation in practice.