Insurer Capital Needs under Parameter Uncertainty

Risk theory aims at evaluating the capital needed to meet a predetermined solvency criterion set by the insurer or regulator. The results for known parameters of the underlying loss distribution are familiar. This paper extends the analysis to the realistic case in which the parameters must be estimated and third moments are still important. With uncertainty in the parameters, the safety capital must be defined in terms of a desire to meet a solvency criterion with a given reliability. With such a definition the effect of pooling risks on the financial efficiency depends on the characteristics of the uncertainty. If the variance in the estimates declines as the inverse of the number of policies on which information is available, the elasticity of capital per risk with respect to risks pooled is basically the same as with known parameters. If the uncertainty is independent of the number of risks in the data base, the elasticity declines with the size of the insurer. The elasticity of capital per risk with respect to the number of risks on which information is available declines as the ratio of relevant data on past losses to insured risks increases. If the variance in the estimates is independent of the data on losses, pooling loss information has no effect on financial efficiency. The results have important implications for the regulation of insurance and for the ways in which insurance transactions are described.