An elementary derivation of Hattendorff’s theorem

Author:

Shiu Elias S. W.ORCID,Xiong Xiaoyi

Abstract

AbstractFor 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.

Publisher

Springer Science and Business Media LLC

Subject

Statistics, Probability and Uncertainty,Economics and Econometrics,Statistics and Probability

Reference8 articles.

1. Bowers NL, Gerber HU, Hickman JC, Jones DA, Nesbitt CJ (1986) Actuarial mathematics. Society of Actuaries, Itasca

2. Dickson DCM, Hardy MR, Waters HR (2020) Actuarial mathematics for life contingent risks, 3rd edn. Cambridge University Press, Cambridge

3. Gerber HU (1979) An introduction to mathematical risk theory. Huebner Foundation Monograph 8, distributed by Irwin, Homewood

4. Gerber HU (1997) Life insurance mathematics, 3rd edn. Springer, Berlin

5. Gerber HU, Leung BP, Shiu ESW (2003) Indicator function and Hattendorff theorem. N Am Actuar J 7(1):38–47

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