Some catastrophe models compensate for the inadequacies of two parameter severity distributions by including point weights at zero and total loss. These are generally undocumented but appear to be non-zero values of probability density corresponding to mean damage ratios of zero and unity. The severity distribution between zero and unity is invariably represented by a two parameter probability distribution. Therefore, at least theoretically, we have a four parameter severity distribution.
At the time of writing these parameters are not well documented but can be inferred as described below.
For a single event, the general form for the calculation of “gross” expected loss net of deductible D and limit L is:
this is true for both continuous and discrete forms of the original (or “ground up”) severity distribution P(x). Considering two different limits, L1 and L2, on the same ground up distribution, we can say that
if both cases have no deductible. As L1 → L2, we can say that P(x>L1) → P(x>L2) so that
assuming that
which should certainly be true in the case where we have a significant probability density at x = 1. If we know we have a significant point weight corresponding to total loss, we can infer that point weight by comparing ground up (GU) and gross (GR) expected losses considering a limit L close to the insured value T:
