# “Improving Segmentation” (using Lorenz curves, or sort of)

This afternoon, André did send me an interesting graph about the use of Lorenz curve in the context of insurance pricing (and modeling)

It is some sort of Lorenz curve, upside-down, with on the x-axis the proportion of the population, and on the y-axis the proportion of the losses. The important point is that the population is sorted according the their risk, i.e. their premium. The code to generate such a curve is actually quite simple,

```L <- function(u,varx="premium",vary="losses"){
base=base[order(base[,varx],decreasing=TRUE),]
base\$cum=(1:nrow(base))/nrow(base)
return(sum(base[base\$cum<=u,vary])/
sum(base[,vary]))}

vu=seq(0,1,by=.01)
vv=Vectorize(function(u) L(u))(vu)```

My concern was more on two labels on the figure, with on the top-left “perfect pricing” and on the first diagonal “average pricing“. What could that possibly mean? Is there even such a thing as a “perfect pricing“? In order to understand what we plot here, let us generate some dataset, and fit some model. Including things that might be seen as the “perfect model“: the price base on the parameters used to generate the data, and the model used to generate the data, fitted on the data.

# Some stylized facts about large risk covers

A couple of weeks ago, David Cummins (here) was giving a talk in Laval University. And we’ve seen a series of extremely interesting graphs and figures about catastrophe reinsurance market, as well as Cat Bonds prices. The first one was the rate one line index for catastrophe reinsurance (the rate on line is the excess of loss premium expressed as a percentage of the reinsurance cover), from Guy Carpenter (2010, page 10 here).

Following hurricane Andrew in 1992, prices went up quite high. But following hurricane Katrina (which is, so far, the most costly insured disaster following the second World War, with a cost exceeding 70 billions US\$ – 2008 \$ – while Andrew was only 24 billions – again 2008 \$), the bump is much smaller. I though cycles where much larger in the reinsurance industry.

Then there was a discussion about cat bond pricing, with a graph from Lane Financial (2010, page 13, here) with the ratio premium over expected loss

This is extremely interesting, even if it is only about cat bond, and not about reinsurance covers. Usually, when we introduce premium principles in actuarial courses, we start with the pure premium, i.e.

Then we explain that with such a price, ruin probability is certain (with an infinite time horizon), so we need to add a safety margin, and a standard idea (but that can be criticized since the expected value has – usually – nothing to do with the variability) is to add a loading proportional to the pure premium. Then the premium is

For small risks, like motor insurance, the loading is not huge. Actually, if risks have finite variance, it can be obtained simply using the central limit theorem (but I’ll get back on that point in a couple of weeks). Here, we see that loading  can be large (up to 400% in 2009).

An finally an updated graph with a comparison between BB corporate bondscoupon, and BB catastrophe bonds coupon,

(I guess the source is again Morton Lane). I found surprising the recent gap (following Katrina) between the two spreads. I guess financial market started to be scared and understood that catastrophes are not that rare…. I wonder what 2008 and 2009 prices looked like.