Annual Losses for Straight Deductible Coverage: Comment

Abstract

As a risk analysis consultant, I read Professor Schott's article with great interest. While my own work in this area leads me to agree that alternative approaches to Monte Carlo simulation can prove useful, I must disagree with Schott's claim that the Gaussian model is viable. Furthermore, some questionable arguments are given to substantiate the case for the normal approximation. In particular, employing the Kolmogorov-Smirnoff (K-S) statistic as a measure of departure from normality can lead to very misleading conclusions. For example, in Table 1 on page 626, the K-S statistic at the $2,500 deductible level is less than the K-S statistic for either the $500 or $1,000 deductible. That the distribution of annual loss becomes less and less like the normal as the deductible level increases from $500 to $2,500 can be seen definitively by calculating the kurtosis and skewness of the annual loss distribution. These calculations show that the kurtosis and positive skewness both increase as the deductible level increases from $500 to $2,500. Given the parameters of the annual loss distribution, the kurtosis and skewness can be precisely calculated, and provide easily interpreted measurements of departures from normality. As a matter of fact, combining the Poisson frequency with any severity distribution results in an annual loss distribution with larger kurtosis than the normal distribution. Furthermore, if the severity distribution is a lognormal distribution with o in the range (1,2), then any deductible greater than 202 percent of the median loss results in an annual loss distribution which is positively skewed (in the severity distribution underlying Table 1, any deductible greater than $290 leads to a positively skewed annual loss distribution). Thus, any realistic application of the normal approximation to obtaining upper percentiles of the annual loss distribution is bound to lead to underestimation. The normal approximation provides no information regarding the amount of undQrestimation, which can be substantial. Finally, simulations were run using the frequency and severity distributions underlying Table 1, assuming a $2,500 deductible. The first simulation