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A new Method of Approximating to the Values of Last Survivor Annuities on Two or More Lives, and to the Values of Joint-Life Annuities when the Advantages of Makeham's Law are not Available

Published online by Cambridge University Press:  18 August 2016

George J. Lidstone
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Extract

The complete calculation and tabulation of annuity values depending on two or more lives present considerable practical difficulties, owing to the multiplicity of age combinations to be dealt with. When the mortality table follows Makeham’s Law the difficulty is completely avoided so far as joint life annuities are concerned by the method of equal ages and uniform seniority, but this in no way meets the case of last survivor annuities, which necessitate a special calculation increasing in complexity with the number of lives involved. When the mortality table does not follow Makeham’s Law, the difficulty extends to joint life annuities as well as last survivor annuities. In the case of a great standard table, like the OM, the labour of calculation and expense of printing a complete set of two-life annuities may be justified, but this will rarely be so with a table of less general importance, while in any case the complete tabulation of three-life and four-life annuities will be quite impracticable; so that at the present time even for OM Mortality there are no tables that enable such annuity values to be readily calculated.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 46 , Issue 1 , January 1912 , pp. 1 - 64
Copyright
Copyright © Institute and Faculty of Actuaries 1912

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References

page 3 note * It may be noted also that owing to the way in which the three sets are chosen, the last survivor annuity-values to be calculated take a simple form, involving fewer terms than in the general case.

page 18 note * If n be the extreme range, the number of 3-life combinations is ½(n+1)(n+2).

page 19 note * If n be the extreme range, the number of 4-life combinations is ⅙(n+1)(n+2)(n+3). Thus, in Table G, where the unit is 4 years and the extreme range, 20 years, is 5 units, the number of cases is ⅙×6×7×8=56, of which 55 are given and the case of equal ages is omitted.

page 55 note * [It may easily be shewn that the maximum deduction from the oldest age where n lives are involved, is ; and that in the special case where the two older of three lives are of equal age the limit is .—Ed.]