**DISCUSSION PAPER PI-0608**

**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.