On Beta Distributions (Part 3)

This post explores the shape of Beta distributions typically used to represent original (“ground up” or damage) loss distributions in commercial catastrophe models.

The shape parameters of the Beta distribution, A and B, must be both positive and real. This places some constraints on the values of expected loss (µ) and standard deviation (σ) which can be represented with a beta distribution. Part 1 describes how the shape parameter A can be expressed as a function of µ and σ. Constraining A > 0 it is simple to show that:

where C represents the coefficient of variation; the ratio of standard deviation to expected loss. It follows that the second shape parameter, B, also satisfies the constraint because 0 < µ < 1 is always true.

The same approach shows that A < 1 results when:

and A > 1 when:

(Note that C is constrained to be <1 whenever A>1)

Similarly, it is simple to show that B < 1 where:

and therefore where:

Finally, B > 1 where:

Plotting contours of these C(µ) relationships results in the following graph:

These contours demark the four basic shapes of the beta distribution as described in Part 2. Replotting this graph as linear-log distorts the plot but allows the four basic shapes of the distribution to be shown within these contours:

Unsurprisingly this blog does not itself hold a licence for any of the commercial vendor models, so data from the model vendors will not be shown.  Nonetheless, any reader with access to a vendor model is invited to explore location- or risk-level C(µ) relationships in their models. This is easy to do: simply report expected loss and standard deviation per event for a single, geocoded location referencing a single vulnerability function.

An empirical observation is that original damage relationships never fall in the range where the beta distribution B parameter takes a value <1.  This appears to be true for all perils, all geographies and all vendors (even those that do not model original loss severity with a beta distribution).

Part 2 established that (1) the beta distribution can only approximate total losses and (2) this only occurs when the B shape parameter is less than unity.  In other words, even at location level, those commercial models using beta distributions don’t even approximate total loss to a location.

Again, this bears repeating.  Those models which use a beta distribution to model loss severity do not consider total loss.  Moreover, they do not even approximate total loss.

(It is also worth noting that – even under perfect correlation of standard deviations – C cannot increase as location loss distributions are combined.  Therefore, it is not possible for any aggregate loss distribution approximate total loss).