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Can a coherent risk measure be too subadditive?

J. Dhaene, R.J.A. Laeven, S. Vanduffel, G. Darkiewicz, M.J. Goovaerts

We consider the problem of determining appropriate solvency capital
requirements for an insurance company or a financial institution. We
demonstrate that the subadditivity condition that is often imposed on
solvency capital principles can lead to the undesirable situation where
the shortfall risk increases by a merger. We propose to complement the
subadditivity condition by a regulator’s condition. We find that for an
explicitly specified confidence level, the Value-at-Risk satisfies the regulator’s
condition and is the “most efficient” capital requirement in the sense that it
minimizes some reasonable cost function. Within the class of concave distortion
risk measures, of which the elements, in contrast to the Value-at-Risk, exhibit
the subadditivity property, we find that, again for an explicitly specified confidence
level, the Tail-Value-at-Risk is the optimal capital requirement satisfying the
regulator’s condition.