For a given mean, which distribution would you prefer for frequency modeling where operational risk events are considered dependent, or in other…

An interesting property that distinguishes the three most used distributions for modeling event

frequency is that for a given mean, their variances differ. The ratio of variance to mean (the variance-

mean ratio, calculated as variance/mean) can then be used to decide the kind of distribution to use.

Both the variance and the mean can be estimated from available data points from the internal or

external loss databases, or the scenario exercise.

The variance-mean ratio reflects how dispersed a distribution is. (In the PRMIA handbook, the

variance to mean ratio has been described as the "Q-Factor".)

The Poisson distribution has its mean equal to its variance, and therefore the variance to mean ratio

is 1. For the negative binomial distribution, this ratio is always greater than 1, which means there is

greater dispersion compared to the mean - or more intervals with low counts as well as more

intervals with high counts. For the binomial distribution, the variance to mean ratio is less than one,

which means it is less dispersed than the Poisson distribution with values closer to the mean.

In a situation where operational risk events are seen as clustering together, or dependent, the

variance will be higher and it would be more appropriate to use the negative binomial distribution.