Decomposable sums and their implications on naturally quasiconvex risk measures

When measuring risk in finance, it is natural to expect that risk decreases with diversification. For risk measures, convexity and quasiconvexity are the two properties which capture the concept of diversification. In between these two properties, there is natural quasiconvexity. Natural quasiconvexity is an old but not so well-known property which is weaker than convexity but stronger than quasiconvexity. In the literature, a lot of effort is put on the analysis of the convexity and the quasiconvexity properties of risk measures. However, a detailed discussion on naturally quasiconvex risk measures is still missing and this thesis aims to fill this gap. Natural quasiconvexity is equivalent to a property called star-quasiconvexity. By making use of this equivalence, we relate naturally quasiconvex risk measures to additively decomposable sums. A notion called convexity index, which is defined in in the literature in 1980s, plays a crucial role in the discussion of additively decomposable sums. Next, we turn our attention to naturally quasiconvex risk measures. By making use of the results on additively decomposable sums, we prove that natural quasiconvexity and convexity are exactly the same properties for conditional risk measures defined on Lp, for p≥1, under some mild conditions. Lastly, we study naturally quasiconvex risk measures on L2 as a special case.