This article investigates the natural hedging strategy to deal with longevity risks for life insurance companies. We propose an immunization model that incorporates a stochastic mortality dynamic to calculate the optimal life insurance-annuity product mix ratio to hedge against longevity risks. We model the dynamic of the changes in future mortality using the well-known Lee-Carter model and discuss the model risk issue by comparing the results between the Lee-Carter and Cairns-Blake-Dowd models. On the basis of the mortality experience and insurance products in the United States, we demonstrate that the proposed model can lead to an optimal product mix and effectively reduce longevity risks for life insurance companies. Copyright (c) The Journal of Risk and Insurance, 2009.

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In this note we prove the following result: Let X be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry and with no essential incompressible space form. Then X is diffeomorphic to S 4 , or RP 4 , or S 3 × S 1 , or S 3 ×S 1 , or a possibly infinite connected sum of them. This extends work of Hamilton and Chen-Zhu to the noncompact case. The proof uses Ricci flow with surgery on complete 4-manifolds, and is inspired by recent work of Bessières, Besson and Maillot.

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