Aggregate loss distributions have extensive applications in actuarial practice. Several approaches have been suggested to estimate the aggregate loss distribution, including the Heckman-Meyers method, the Panjer algorithm, and fast Fourier transformation, to name a few. All of these methods rely on separate assumptions about frequency and severity components of the aggregate losses. Quite often, however, obtaining frequency and severity expectations independently is not practical, and only aggregate information is available for analysis. In that case, the a priori assumption about the shape of the aggregate loss distribution becomes critical, especially for assessing the probability of very high aggregate loss values, in the tail.
In this work we seek to determine which statistical two-parameter distribution, out of several, serves best to approximate aggregate loss distributions for property and casualty products. We focus on ground-up losses limited by a per occurrence limit. These results are relevant for quota share agreements. In addition, we consider layer losses, the results of which are important for umbrella quota share transactions.
We simulate samples of aggregate loss, fit statistical distributions to the samples, and then use goodness-of-fit tests to determine the best-fitting distribution. In all realistic scenarios with limited losses, we find that the gamma distribution uniformly provides the most reasonable approximation to the aggregate loss.
