Correlation order, merging and diversification

We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses.

We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall, expressed in terms of an appropriate integral stochastic order. Furthermore, increasing the dependence between losses decreases the diversification benefit.

We also consider merging comonotonic losses and show that even in this extreme case a strictly positive diversification benefit will often arise.