Bounds for Sums of Random Variables When the Marginals and the Variance of the Sum are Given

Abstract

In this paper we establish several relations between convex order, variance order, and comonotonicity.

In the first part, we extend Cheung (2008b) to show that when the marginal distributions are fixed, a sum with maximal variance is in fact a comonotonic sum. Thus the convex upper bound is achieved if and only if the marginal variables are comonotonic.

Next, we study the situation where besides the marginal distributions, the variance of the sum is also fixed. Intuitively one expects that adding this information may lead to a bound that is sharper than the comonotonic upper bound. However, we show that such upper bound does not even exist. Nevertheless, we can still identify a special dependence structure known as upper comonotonicity, in which case the sum behaves like a convex largest sum in the upper tail.

Finally, we investigate when the convex order is equivalent to the weaker variance order. Throughout this paper, interpretations and significance of the results in terms of portfolio risks will be emphasized.

Keywords: comonotonicity, copula, dependence, solvency, Basel II, Solvency II, Value-at-Risk, Tail Value-at-Risk