Some useful counterexamples regarding comonotonicity

This article gives counterexamples for some conjecturesabout risk orders. One is that in risky situations, diversificationis always beneficial. A counterexample is providedby the Cauchy distribution, for which the sample means havethe same distribution as the sample elements, meaning that insuringhalf the sum of two iid risks of this type is preciselyequivalent to insuring one of them fully. In this case, independenceand comonotonicity for these two risks are equivalent.We also show that if X, Y are iid Pareto(?, 1) with ? < 1,for the values-at-risk we have F?1 X+Y (q) > F?1 2X(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 ofhaving lower values-at-risk in the far-right tail. Then comonotonicityis preferable to independence, so independence is certainlynot a `worst case? scenario. Finally we show that if onerisk has smaller stop-loss premiums than another, this doesn?thave to mean that its cdf is above the other froma certain pointon.We give an example that the sum of independent risks canhave a cdf that crosses infinitely often with its comonotonicequivalent. That such a distribution exists is no surprise, butan example has never been exhibited so far.