Why Larger Risks Have Smaller Insurance Charges

The insurance-charge function is defined as the excess ratio (the ratio of expected loss excess of an attachment point to the expected total loss) and is expressed as a function of the entry ratio (the ratio of the attachment to the total loss expectation). Actuaries use insurance-charge algorithms to price retrospective rating maximums and excess of aggregate coverages. Many of these algorithms are based on models that can be viewed as particular applications of the Collective Risk Model (CRM) developed by Heckman and Meyers. If we examine the insurance-charge functions for risks of different sizes produced by these models, we will find invariably that the insurance charge for a large risk is less than or equal to the charge for a small risk at every entry ratio. The specific purpose of this paper is to prove that this must be so. In other words, we will show the assumptions of the CRM force charge functions to decline by size of risk. We will take a fairly general approach to the problem, develop some theory, and prove several results along the way that apply beyond the CRM. We will first prove that the charge for a sum of two non-negative random variables is less than or equal to the weighted average of their charges. We will extend that result to show that under certain conditions, the charge for a sum of identically distributed, but not necessarily independent, samples declines with the sample size. The extension is not entirely straightforward, as the desired result cannot be directly derived using simple induction or straightforward analysis of the coefficient of variation (CV). To explore size and charge in some generality, we will define the construct of a risk-size model. A risksize model may initially be viewed as a collection of non-negative random variables whose sizes are defined by their expectation values. Given an appropriate measure on the risks of a particular size, we will be able to regard the cumulative distribution and the charge as well-defined functions of risk size. In a complete and continuous model, there are risks of every size and the cumulative distribution is a continuous function of risk size. We will first show that the charge declines with size if any risk can be decomposed into the independent sum of smaller risks in the model. Then we will employ the usual Bayesian construction to introduce parameter risk and extend the result to models that are not decomposable. This is an important extension, because actuaries have long known from study of Table M that a large risk is not the independent sum of smaller ones. In particular, our result implies that charges decrease with size in the standard contagion model of the Negative Binomial used in the CRM. Finally, we will introduce severity, prove our result assuming a fixed severity distribution, and then extend it to cover the type of parameter uncertainty in risk severity modeled in the CRM. Thus we will arrive at the conclusion that the assumptions of the CRM force charges to decline by size of risk.