To split or not to split: Capital allocation with convex risk measures

Convex risk measures were introduced by Deprez and Gerber [Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Math. Econom. 4 (3), 179-189]. Here the problem of allocating risk capital to subportfolios is addressed, when convex risk measures are used. The Aumann-Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed. It is demonstrated that using a convex risk measure for capital allocation can produce an incentive for infinite fragmentation of portfolios.