Tomorrow, I was invited to give a (brief) talk at our working group, at France Stratégies, on (non) diversification of extreme risks. Slides are online, and results are related to recent papers by Paul Embrechts and Ruodu Wang. More precisely, here are some references

But first, before discussing large risks, I need to get back (quickly) on the Pareto distribution,

To visualize Pareto tails, one can consider the Pareto plot. If points are on a straight line with (negative) slope \alpha, then observations are Pareto distributed, with tail index precisely \alpha. Depending on the slope (compared with -1), risks have either finite or infinite mean.

Infinite mean is not that common actually. It is hard to visualize what it means, actually because for any (finite) n, the empirical average \displaystyle{\overline{x}=\frac{1}{n}\sum_{i=1}^nx_i} always exists. To visualize what’s going on, we can plot the ratio \max\{x_i\} over the sum. That could be related to the concept of “top share” in inequality.

On the left, risks with finite variance (and of course finite mean). In the middle, infinite variance by finite mean. After a while, it is quite rare to have the maximum weighting for more than 1% in the total sum. With infinite mean, on the right (not too far from the limit, since \alpha is here 0.95 – finite mean means that \alpha exceeds one).

Now, if we get back to risks and insurance, recall basic things on stochastic dominance,

Then we have the following results (that is actually the most important slide)

I did include a slide with the mathematical proof (that is quite lovely actually, and straightforward)