Some useful counterexamples regarding comonotonicity

This article gives counterexamples for some conjectures about risk orders. One is that in risky situations, diversification is always beneficial. A counterexample is provided by the Cauchy distribution, for which the sample means have the same distribution as the sample elements, meaning that insuring half the sum of two iid risks of this type is precisely equivalent to insuring one of them fully. In this case, independence and comonotonicity for these two risks are equivalent. We also show that if X, Y are iid Pareto(α, 1) with α < 1, for the values-at-risk we have F−1X+Y(q) > F−12X(q) for q large enough. This proves that a sum of iid risks might be worse than a sum of corresponding comonotonic risks in the sense of having lower values-at-risk in the far-right tail. Then comonotonicity is preferable to independence, so independence is certainly not a 'worst case' scenario. Finally we show that if one risk has smaller stop-loss premiums than another, this doesn't have to mean that its cdf is above the other froma certain point on.We give an example that the sum of independent risks can have a cdf that crosses infinitely often with its comonotonic equivalent. That such a distribution exists is no surprise, but an example has never been exhibited so far.