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faculty: "FEB" and publication year: "2004"
| Authors||R. Kaas, M.J. Goovaerts, Q. Tang|
|Title||Some useful counterexamples regarding comonotonicity|
|Journal||Belgian Actuarial Bulletin|
|Faculty||Faculty of Economics and Business|
|Institute/dept.||FEB: Amsterdam School of Economics Research Institute (ASE-RI)|
|Abstract||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(?, 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 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.
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