For a general fully continuous life insurance model, the variance of the loss-at-issue random variable is the expectation of the square of the discounted value of the net amount at risk at the moment of death. In 1964 Jim Hickman gave an elementary and elegant derivation of this result by the method of integration by parts. One might expect that the method of summation by parts could be used to treat the fully discrete case. However, there are two difficulties. The summation-by-parts formula involves shifting an index, making it somewhat unwieldy. In the fully discrete case, the variance of the loss-at-issue random variable is more complicated; it is the expectation of the square of the discounted value of the net amount at risk at the end of the year of death times a survival probability factor. The purpose of this note is to show that one can indeed use the method of summation by parts to find the variance of the loss-at-issue random variable for a fully discrete life insurance policy.

对于一般的完全连续人寿保险模型，签发损失随机变量的方差是死亡时处于风险中的净额的折现值的平方的期望。1964年，吉姆·希克曼（Jim Hickman）通过部分积分的方法对这一结果进行了基本而优雅的推导。可能有人希望将按部分求和的方法用于处理完全离散的情况。但是，有两个困难。分部求和公式涉及到移动索引，使其有些笨拙。在完全离散的情况下，发行时损失随机变量的方差更为复杂。它是死亡年末处于风险中的净额的折现值的平方乘以生存概率因子的期望。