On Beta Distributions (Part 2)

Beta distributions appear everywhere in catastrophe modelling. At least, they appear more often than not when a vendor is trying to represent uncertainty in loss severity. This is odd, considering how rarely this distribution is used in other actuarial or engineering disciplines. Nonetheless, all of the major catastrophe model vendors use Beta distributions somewhere in their financial engines; and one or two even use the Beta distribution exclusively.

The most common justification for using the Beta distribution is its ability to assume many forms. It is, of course, reasonable to wonder whether a distribution capable of assuming so many different forms can represent any physical reality. There is also a more pragmatic explanation; a distribution naturally bounded by zero and unity doesn’t need artificial truncation to avoid (a) negative loss or (b) losses greater than the sum insured.

The PDF has four basic shapes, depending on the values of the (positive, real) shape parameters A and B described in Part 1. The following four figures show the basic shape of the distribution for the four cases where each parameter can take values less than or greater than unity.

(Note, in the case of either shape parameter taking a value of 1 a power law distribution results. If both shape parameters are unity, the PDF is also unity for all values of x.)

The first thing to observe is that in all cases, the probability of both zero loss and total loss is zero. In other words, wherever a Beta distribution is being used to model loss severity, neither zero loss nor total loss are considered. This becomes evident when the PDF formula in Part 1 is examined.

This is worth repeating. Whenever a Beta distribution is used to represent loss severity, total loss is not possible. Similarly, there is no possibility of zero loss. This is not unreasonable when modelling losses to aggregates. However, this is – at very best – questionable when modelling losses to individual properties (i.e. using “detailed” data).

It could be argued that cases 1 and 2 appear to make a reasonable approximation of representing zero loss; and cases 1 and 3 appear to do the same for total loss. It is therefore instructive to look at the circumstances under which the original (“ground up” or “damage”) distribution could assume each of these shapes.  This will be explored in the next post on Beta distributions.

(Note, the application of non-proportional financial terms such as excess points and first loss limits introduces more parameters and further complicates the general description made here. For this reason original loss distributions will be investigated)