3. Beta distribution

Regardless of the fact that multitude distribution types have been used for the frequency and severity distribution of individual contract losses, the aggregated portfolio losses for the majority of perils can be fitted by a compound Poisson distribution with Beta distribution as the severity, somehow an attest of its prevalence. Beta distribution has min SF ¼ 1:0, so we need an in-detail study of why it cannot fit NATH.

When matching a BetaDistribution½α; β� for skewness 5.99378 and kurtosis 65.8902, we must have β < 0. When matching a Beta distribution for CV(=std/ mean, the standard deviation divided by the mean) 1.2829 and either skewness

5.99378 or kurtosis 65.8902, we must have either both α < 0 and β < 0 or at least one of α or β less than 0. Since CV, skewness, and kurtosis are scale invariant, so no scaled Beta distribution can at the same time match any two of the three statistics CV, skewness, and kurtosis.
