Estimating Total Loss from a Compound Process

Authors: Saifullah Amer, Douglas Johnston, Kalee Tolentino, Mark Corrado Jr., Paul D'Amour

SUNY Campus: Farmingdale State College

Presentation Type: Poster

Location: Old Union Hall

Presentation #: 35

Timeslot: Session C 1:45-2:45 PM

Abstract: For insurance companies, a critical risk factor is that total losses exceed their capital reserves leading to insolvency. Total loss is a compound process where the number of losses and their sizes are random. We model the number of claims as a Poisson process and the size of claims as an Exponential distribution. We use mathematical tools, such as convolution, the Laplace transform, and statistical methods to solve for the probability distribution of total loss at some time in the future. This method requires numerical techniques, and we illustrate its accuracy by benchmarking against a known case. One of the main issues in applying this to real world data, however, is the unknown parameters of the underlying distributions must be determined from historical data. We derive the maximum likelihood estimate for the model’s parameters which are used to compute the cumulative distribution of total loss in the future. From this cumulative distribution, quantile estimates are computed which can be used to determine adequate capital reserves. We find, via simulations, that relying on only the single estimate of the parameters leads to a significant understatement in risk with too many exceedances above our estimated quantiles. Interestingly, this occurs even though the maximum likelihood estimates are unbiased. To improve performance, we develop a Bayesian approach where we integrate over the posterior distribution of the model parameters which results in an improved total loss risk-assessment. We illustrate our results using real-world insurance-claim data.