Upper and lower bounds for annuities and life insurance from incomplete mortality data

ABSTRACT This study aimed to set upper and lower bounds for the expected present value of whole life annuities and whole life insurance policies from incomplete mortality data, generalizing previous results on life expectancy. Since its inception, in the 17th century, actuarial science has been devoted to the study of annuities and insurance plans. Thus, setting intervals that provide an initial idea about the cost of these products using incomplete mortality data represents a theoretical contribution to the area and this may have major applications in markets lacking historical records or those having little reliability of mortality data, as well as in new markets still poorly explored. For both the continuous and discrete cases, upper and lower bounds were constructed for the expected present value of whole life annuities and whole life insurance policies, contracted by a person currently aged x, based on information about the expected present value of these respective financial products subscribed to by a person of age x + n and the probability that an individual of age x survives to at least age x + n. Through the bounds of a continuous annuity, in an environment where the instantaneous interest rate is equal to zero, the results shown also set bounds for the complete life expectancy, which implies that the contribution of this research generalizes previous results in the literature. It was also found that, for both annuities and insurance plans, the length of constructed intervals increases as the data gap size increases and it decreases as the survival curve becomes more rectangular. Illustratively, bounds for life expectancy at 40 and 60 years of age, for the 10 municipalities showing the highest life expectancy at birth in Brazil in 2010, were constructed by using data available in the Atlas of Human Development in Brazil.

摘要 本研究旨在基于不完全死亡率数据（incomplete mortality data），确定终身年金（whole life annuities）与终身寿险保单（whole life insurance policies）的期望现值（expected present value）上下界，推广了既有研究中关于预期寿命（life expectancy）的相关结论。自17世纪创立以来，精算学（actuarial science）便致力于年金与保险产品的研究。利用不完全死亡率数据构建可初步反映此类产品成本的区间，不仅为该领域提供了理论层面的贡献，还可在缺乏历史记录、死亡率数据可靠性欠佳的市场，以及尚待深入挖掘的新兴市场中获得重要应用价值。针对连续与离散两种情形，本研究针对当前年龄为x的个体，基于年龄为x+n的个体所投保的同类金融产品的期望现值，以及x岁个体存活至至少x+n岁的概率，构建了其终身年金与终身寿险保单的期望现值上下界。在瞬时利率（instantaneous interest rate）为零的场景下，通过终身年金的上下界，本研究成果还可推导完整预期寿命的上下界，这意味着本研究的贡献推广了现有文献中的相关结论。此外研究还发现，无论针对年金还是保险产品，所构建区间的长度均随数据缺口规模的扩大而增加，随生存曲线（survival curve）趋近矩形而收窄。作为示例，本研究借助《巴西人类发展图集（Atlas of Human Development in Brazil）》中的公开数据，针对2010年巴西出生时预期寿命最高的10个城市，构建了40岁与60岁人群的预期寿命上下界。