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Natural hedges with immunization strategies of mortality and interest rates. (English) Zbl 1431.91339

Summary: In this paper, we first derive closed-form formulas for mortality-interest durations and convexities of the prices of life insurance and annuity products with respect to an instantaneously proportional change and an instantaneously parallel movement, respectively, in \(\mu^*\) (the force of mortality-interest), the addition of \(\mu\) (the force of mortality) and \(\delta\) (the force of interest). We then build several mortality-interest duration and convexity matching strategies to determine the weights of whole life insurance and deferred whole life annuity products in a portfolio and evaluate the value at risk and the hedge effectiveness of the weighted portfolio surplus at time zero. Numerical illustrations show that using the mortality-interest duration and convexity matching strategies with respect to an instantaneously proportional change in \(\mu^*\) can more effectively hedge the longevity risk and interest rate risk embedded in the deferred whole life annuity products than using the mortality-only duration and convexity matching strategies with respect to an instantaneously proportional shift or an instantaneously constant movement in \(\mu\) only.

MSC:

91G05 Actuarial mathematics
91G30 Interest rates, asset pricing, etc. (stochastic models)
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