Exposure Dependent Modeling of Percent of Ultimate Loss Development Curves

This paper presents a loss development model in which exposure period dependence is fundamental to the structure of the model. The basic idea is that an exposure period, such as an accident year or policy year, gives rise to a particular distribution of accident date lags, where the accident date lag is the time elapsed from the start of the exposure period till the accident date. The paper shows how to derive the density of the accident date lag from a familiar parallelogram diagram. A fairly general theory of development is then presented and simplified under certain conditions to arrive at a total development random variable whose cumulative distribution is related to the usual percent of ultimate development curve. After presenting the theory, the paper turns to practical applications. Simulation is used to generate consistent patterns for different exposure periods. A convenient accident period development formula is derived and then used to fit and convert factors. The average date of loss approximation is generalized. To summarize, this paper will demonstrate that modeling loss development with exposure dependent percent of ultimate curves is a theoretically sound procedure with many practical uses.