Parameter Uncertainty in Loss Ratio Distributions and its Implications

This paper addresses the issue of parameter uncertainty in loss ratio distributions and its implications for primary and reinsurance ratemaking, underwriting downside risk assessment and analysis of sliding scale commission arrangements. It is in some respects a prequel to Van Kampen's 2003 CAS Forum paper [1], which described a Monte Carlo method for quantifying the effect of parameter uncertainty on expected loss ratios. He showed the effect was especially significant in pricing applications involving the right tail of the loss ratio distribution. While Van Kampen focused purely on the objective of quantification, this paper develops the functional form of the loss ratio distribution incorporating parameter uncertainty that is implicit in his approach. This paper thus both underpins Van Kampen's work and allows us to apply it more efficiently, because it is easier to work with the loss ratio distribution directly than to perform Van Kampen's simulation. Suppose we have a set of on-level loss ratios from a stable portfolio of business of substantial enough size that it is plausible that the loss ratios can be viewed as a sample arising from an approximately normal or lognormal distribution, the parameters of which are unknown. What is the distribution of the prospective loss ratio? This paper discusses the drawbacks of using the "best fit" normal or lognormal distribution to model the loss ratio, particularly for pricing or risk assessment applications that depend on the tails of the distribution. While one fit is "best", frequently a number of parameter sets provide nearly as good a fit. Choosing only the "best fit" distribution means ignoring the information contained in the sample about the other possible distributions. That information can be relected in the loss ratio distribution by weighting together all the plausible normal or lognormal distribution, given the sample, by their relative likelihoods. In the continuous case, where the weighting function is the density function of the parameters, the resulting distribution is the Student's t or log t distribution, respectively. This distribution, which incorporates the uncertainty about the parameters, is preferable to the "best fit" distribution for modeling the prospective loss ratio. The paper illustrates applications ranging from aggregate excess reinsurance pricing to measurement of underwriting downside risk to estimation of the expected cost or benefit of sliding scale commissions, in each case comparing the results arising from underlying normal and lognormal assumptions and both parameter "certainty" and parameter uncertainty.