Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities

Suppose X1,X2,… are independent subexponential random variables with partial sums Sn. We show that if the pairwise sums of the Xi's are subexponential, then Sn is subexponential and (Sn>x)∼∑1nP(Xi>x)(x→∞). The result is applied to give conditions under which P(∑1∞ciXi>x)∼∑1∞P(ciXi>x) as x→∞, where c1,c2,… are constants such that ∑1∞ciXi is a.s. convergent. Asymptotic tail probabilities for bivariate linear combinations of subexponential random variables are given. These results are applied to explain the joint movements of the stocks of reinsurers. Portfolio investment and retrocession practices in the reinsurance industry expose different reinsurers to the same subexponential risks on both sides of their balance sheets. This implies that reinsurer's equity returns can be asymptotically dependent, exposing the industry to systemic risk.