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 μ* (the force of mortality-interest), the addition of μ (the force of mortality) and δ (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 μ* 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 μ only.

中文翻译：

---

具有死亡率和利率的免疫策略的自然对冲

在本文中，我们首先导出死亡率-利息持续时间和人寿保险和年金产品的价格相对于瞬时比例变化和瞬时平行移动的凸度的封闭形式公式，单位为μ *（死亡率利息），μ（死亡率）和δ的相加（感兴趣的力量）。然后，我们建立几种死亡率-利息期限和凸度匹配策略，以确定投资组合中的整个人寿保险和递延的整个人寿年金产品的权重，并在零时评估风险价值和加权投资组合盈余的对冲有效性。数值说明表明，与仅使用死亡率的持续时间相比，针对μ *的瞬时比例变化使用死亡率-利息持续时间和凸率匹配策略可以更有效地对冲延期终生年金产品中的寿命风险和利率风险。仅针对μ中的瞬时比例移动或瞬时恒定运动的凸和凸匹配策略。