MAT886 mean excess function (and reinsurance)

Tomorrow, in the course on extreme value, we will focus on applications. We will discuss reinsurance pricing. Consider a random variable , a threshold and define

For instance, if has a Generalized Pareto Distribution (GPD),

the mean excess function is linear in ,

A natural estimator for that function is the empirical average of observations exceeding the threshold,

If denotes an order statistics, it is possible to calculate that quantity in those specific values. Set

It is possible to plot . If the points are on a straight line, then the GPD should be an appropriate model,

> set.seed(100)
> b=1;xi=.5
> n=1000
> X=sort(b/xi*((1-runif(n))^(-xi)-1))
> e=function(u){mean(X[X>=u]-u)}
> E=Vectorize(e)
> plot(X[-n],E(X[-n]))
> abline(b/(1-xi),xi/(1-xi),col="red")

We can also use directly cumulated sums on order statistics,

> plot(rev(X),cumsum(rev(X))/1:n-rev(X),col="blue")
> abline(b/(1-xi),xi/(1-xi),col="red")

Nevertheless, that estimator are not very robust. If we generate not one, but 5,000 samples, we obtain almost everything,

with below in (dark) blue monte carlo confidence 90% confidence intervals. Nevertheless, this quantity is extremely popular in reinsurance, and is used under the name “burning cost“.