The discount rate is an aprioristic statement about the future. Stern makes an aprioristic statement, and so does Klaus. It's anybody's guess whose guess is correct.
Klaus chooses to concentrate on the discount rate and ignores the value at risk, which has a lot to do with the uncertainty in the estimation of the appropriate discount rate.
Underestimating the discount rate is like taking the lower end of a confidence interval, which is the proper thing to do when calculating value at risk. Can the last politician to go out the revolving door please turn the lights off?
I was at a seminar last week where large deviations theory was briefly mentioned as part of the discussion of estimating value at risk and the expected shortfall. The speaker is a senior researcher with a large international investment bank.
The problem with large deviation theory, according to this guy, is that it only works for the very tail, but that means in practice you don't have a lot of data to fit the tail distribution. Can the last politician to go out the revolving door please turn the lights off?
Migeru and Pierre, thanks!
Mandelbrot claimed that scaling laws where better than Gauss, can be calibrated, and incur manageable computing costs, but I don't think anyone has looked into it on the industry-scale. I'll dig a link Pierre
Specially impressive is the centenial chart of stock markt without 10 largest daily changes. Note they tend to be seriously down-biased. Pierre
For most insurance applications, we would expect a decreasing unit hazard functton. That is, as we move to higher and higher layers, the chance of a partial loss would decrease. For instance, if we consider a layer such as $10,000,000 xs $990,000,000 we would expect that any loss above $990,000,000 would almost certainly be a full-limit loss. This would imply h (y) ~ 0. The decreasing hazard function is not what we generally find in the exponential family. For the Normal and Poisson, the hazard function approaches 1, implying that full-limit losses become less likely on higher layers - exactly the opposite of what our understanding of insurance phenomena would suggest. The Negative Binomial, Gamma and Inverse Gaussian distributions asymptotically approach constant amounts, mimicking the behavior of the exponential distribution The table below shows the asymptotic behavior as we move to higher attachment points for a layer of width w. DistributionLimiting Form of h (y)Comments Normallim h(y) = 1No loss exhausts the limit Poissonlim h (),) = I Negative Bmomiallim h,(y) = I - ( I - p ) " Gammalim h(y) = I-e "''°'~' Inverse Gausstanlim h(y) = I-e -''a°''~ Lognormallim h(v) = 0Every loss ts a full-limit loss From this table, we see that the members of the natural exponential family have tail behavior that does not fully reflect the potential for extreme events in high casualty insurance. It would seem that the natural exponential distributions used with GLM are more appropriate for insurance lines without much potential for extreme events or natural catastrophes.
The decreasing hazard function is not what we generally find in the exponential family. For the Normal and Poisson, the hazard function approaches 1, implying that full-limit losses become less likely on higher layers - exactly the opposite of what our understanding of insurance phenomena would suggest. The Negative Binomial, Gamma and Inverse Gaussian distributions asymptotically approach constant amounts, mimicking the behavior of the exponential distribution The table below shows the asymptotic behavior as we move to higher attachment points for a layer of width w.
From this table, we see that the members of the natural exponential family have tail behavior that does not fully reflect the potential for extreme events in high casualty insurance. It would seem that the natural exponential distributions used with GLM are more appropriate for insurance lines without much potential for extreme events or natural catastrophes.
Sorry the layout is crappy, had to rework an OCR-ed pdf. But the general idea seems to be that if it's linear, then you don't have a fat tail. I don't know if there is any kind of proof of this or of something heuristically similar (I'm by no mean a statistician) Pierre
I could suggest a correction to this in public, but then I'd have to kill you ;-) Can the last politician to go out the revolving door please turn the lights off?
Do you suggest that there are places where the computing power is much greater than the average bank (that I know, my previous job had more computing power than meteo france and the french dod combined, my new place is a lot more frugal), or that your fund developed some non-gaussian models of risk that can be valued as efficiently as the gaussian with "reasonable" computing power ? Pierre
