A Stochastic Framework for Incremental Average Reserve Models

Motivation: Chain ladder forecasts are notoriously volatile for immature exposure periods. The Bornhuetter-Ferguson method is one commonly used alternative but needs a priori estimates of ultimate losses. Berquist and Sherman presented another alternative that used claim counts as an exposure base and used trended incremental severities to “square the triangle.” A significant advantage of the Berquist and Sherman method is the simultaneous estimate of underlying inflation. Though not the first to do so, this paper looks to extend the incremental severity method to a stochastic environment. Rather than using logarithmic transforms or (generalized) linear models, used in many other approaches, we use maximum likelihood estimators, bringing to bear the strength of that approach avoiding limiting assumptions necessitated when taking logarithms.

Method: Given that incremental severities can be looked at as averages over a number of claims, the law of large numbers would suggest those averages follow an approximately normal distribution. We then assume the variance of the incremental payments in a cell are proportional to a power of the mean (with the constant of proportionality and power constant over the triangle). We then use maximum likelihood estimators (MLEs) to estimate the incremental severities, along with the inherent claims inflation to “square the triangle.” We also use properties of MLEs to estimate the variance-covariance matrix of the parameters, giving not only estimates of process but also of parameter uncertainty for this method. Not only do we consider the model described by Berquist and Sherman, but we also set the presentation in a more general framework that can be applied to a wide range of potential underlying models.

Results: A reasonably common and powerful method now presented in a stochastic framework allowing for assessment of variability in the forecasts of the method.

Availability: The R script for these estimates appear on the CAS Web Site.

Keywords: Stochastic reserving, maximum likelihood, normal-p, incremental severity method, PPCI