Increased limit factors

Robert Miccolis set out in 1977, in the Proceedings of the Casualty Actuarial Society, the theory of increased limits and excess of loss pricing. The problem belongs to liability. In property, insured value bounds the loss and makes an exposure curve possible. In liability there is no insured value: only the policy limit bounds the commitment, and nothing says what fraction of a maximum it represents. The increased limit factor expresses the premium for a limit L as the basic limit premium times the ratio of mean losses capped at L and at the basic limit. It is therefore entirely determined by the severity distribution, and its growth is sublinear. Moving from a 1M to a 10M EUR limit rarely multiplies premium by more than two, because the mass of claims stays below the first million. Using the factors calls for care. The distribution must be that of ground-up losses, not of already capped losses, on pain of circularity. The factors must incorporate defense costs, which do not scale like indemnity. And they age fast under judicial inflation, which moves mass into the tail and makes high factors grow faster than low ones.