MAT886 mean excess function (and reinsurance)

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Tomorrow, in the course on extreme value, we will focus on applications. We will discuss reinsurance pricing. Consider a random variable http://freakonometrics.blog.free.fr/public/perso5/mef01.gif, a threshold http://freakonometrics.blog.free.fr/public/perso5/mef.gif and define

http://freakonometrics.blog.free.fr/public/perso5/mef02.gif


the mean excess function. This function is known in life insurance as the average remaining life time of someone alive at age http://freakonometrics.blog.free.fr/public/perso5/mef.gif. This function can be written

http://freakonometrics.blog.free.fr/public/perso5/mef03.gif

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

http://freakonometrics.blog.free.fr/public/perso5/mef08.gif

the mean excess function is linear in http://freakonometrics.blog.free.fr/public/perso5/mef.gif,

http://freakonometrics.blog.free.fr/public/perso5/mef10.gif

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

http://freakonometrics.blog.free.fr/public/perso5/mef13.gif

If http://freakonometrics.blog.free.fr/public/perso5/mef11.gif denotes an order statistics, it is possible to calculate that quantity in those specific values. Set

http://freakonometrics.blog.free.fr/public/perso5/mef06.gif

It is possible to plot http://freakonometrics.blog.free.fr/public/perso5/mef04.gif. 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“.

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