Before revision:  To solve the initial constrained non-self-financing optimization problem , we transform it to an unconstrained self-financing problem by replicating the future contributions and the minimum guarantee . By applying dynamic programming method, analytical solutions to the equivalent optimization problem are derived and optimal investment strategies to the original problem are obtained by simple calculations. The numerical applications reveal that the longevity risk has an important impact on the investment strategies and show evidence that mortality-linked securities could provide an efficient way to hedge the longevity risk.
After revision:  To solve the initial constrained non-self-financing optimization problem , we transform it to an unconstrained self-financing problem by replicating the future contributions and the minimum guarantee provided by the scheme. We solve the resulting optimisation problem using the dynamic programming principle and through a series of numerical studies reveal that the longevity risk has an important impact on the investment strategies and show evidence that mortality-linked securities could provide an efficient way to hedge the longevity risk.