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Actuarial note on Duhamel's Integral

Published online by Cambridge University Press:  11 August 2014

Eli A. Grossman*
Affiliation:
The Union Labor Life Insurance Company, New York
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Extract

In this note, we consider a problem and its implication. The problem is the control of the flow of new business in a portfolio in a certain type of policy. Supposing we know in advance what the investment implications of a certain trend of reserves will be so that we can decide a programme of development of the fund purely from investment consideration. If the flow of new business is ‘controllable’, e.g. by the amount of advertising, commission, and reward to the selling staff allowed from time to time, what volume of new business is required in order to build up to a predetermined reserve pattern? In solving this problem, we will use a mathematical integral employed in electricity, physics, chemistry and other fields. This is so fundamental in other fields that perhaps it will solve many more of our problems.

In this note, emphasis is given to the amount of insurance in force because that is customarily done in the United States.

Type
Research Article
Copyright
Copyright © Institute of Actuaries Students' Society 1954

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References

page 235 note * See, for example, Titchmarsh's ‘Fourier integral’.

page 235 note † See, for example, Fourier Integrals for Practical Applications, by George Campbell and Ronald Foster (Bell Telephone System Technical Publication).

page 236 note * Many other types of approximate integration formulae may be used for either equation (7) or equation (10).

page 237 note * Initial and terminal reserves in the examples are based on the 1941 Commissioners' Standard Ordinary Table at 2¼%. These reserves, as well as the other figures in the tables of the examples, are merely picked for illustrative purposes and have no significance.