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

Suppose $X_1, X_2, \\ldots$ are independent subexponential random variables with partial sums $S_n$. We show that if the pairwise sums of the $X_i$’s are subexponential, then $S_n$ is subexponential and $(S_n > x) ∼ \\sum_{i}^n P(X_i > x)(x \\rightarrow \\infty)$. The result is applied to give conditions under which $P(\\sum_{1}^\\infty c_iX_i > x)$ as $x \\rightarrow \\infty$, where $c_1, c_2, \\ldots$ are constants such that $\\sum_{1}^\\infty c_iX_i$ 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.

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