Analysis of Loss Development Patterns Using Infinitely Decomposable Percent of Ultimate Curves

We model loss development by starting with percent of ultimate curves which explicitly depend on the underlying pattern of exposure accumulation. We express such curves in general as a convolution of a "generating" function with an "exposure" density function. Such curves are decomposable with respect to exposures, where decomposability means, for instance, that an accident year curve is expressible as the weighted average of four appropriately shifted translates of the related accident quarter curve. This approach is theoretically attractive and leads to several useful results. The most important area of practical application is fitting, interpolating, and extrapolating age-to-age factors. An essential point to note is that it is trivial to start with a parametric percent of ultimate curve and use it to calculate age-to-age factors, but the opposite derivation is not so simple. Another area of application is in converting development patterns from one type of exposure period to another. As a natural consequence of our formulation, we are able to derive error terms for the usual "average date of loss" approximation and to generalize the approximation so that it is valid even at immature ages.