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On an Improved Theory of Annuities and Assurances

Published online by Cambridge University Press:  18 August 2016

W. S. B. Woolhouse
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Extract

The existing theory and practice of Annuities and Assurances, and the tables ordinarily constructed for the purpose of expediting the various computations relating to the same, are radically founded on annual payments, or upon a series of mathematical values which appertain to certain annual periods that are specifically defined. Indeed, the methods employed are not strictly applicable unless the prescribed condition of annual periodicity is realized or assumed; and this, in fact, is the reason why assurances, as well as annuities, are necessarily supposed to be payable on the exact completion of one of the recurring periods, viz., at the end of the year of age in which the death takes place.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 15 , Issue 2 , July 1869 , pp. 95 - 125
Copyright
Copyright © Institute and Faculty of Actuaries 1870

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References

page 98 note * Mr. Sprague has since gone at great length into the subject of annuities, and in confirming the accuracy of these results has been most liberal in his allusions to the decided improvement effected by my formulæ. It is well that Mr. Sprague has been induced to follow me in this particular investigation, as his methods are so essentially different from mine and afford such a scope for Ms acute mathematical talent, that his paper must be esteemed as a most valuable contribution to the Journal.

page 100 note * It has been pointed out to me by Mr. Merrifield that Legendre employs this formula in evaluating certain elliptic integrals in his valuable work on that subject. The formula is ascribed to Euler, and is also used by Laplace and others in the Theory of Probabilities.

page 105 note * Otherwise, if p denote the probability of surviving the interval x, then by (3),

And, since –dp is the proportion of deaths or probability of decease during the instent dx,

Also, by (2),

And, after deducting the advance payment

page 108 note * The value of a life annuity, when it is to be completed by the payment of a proportionate part up to the day of death, was first investigated in Mr. Sprague's paper, before referred to. It may be added that the formulæ (11) and (14) I have deduced for Complete Annuities are identical with his results.

Note.—The fundamental formulæ for assurances are comprehended in three types. The first of these is applicable generally, and may be written in an abbreviated form, by separating the symbols of operation, thus: which may be regarded as the radix of the formulæ (27), (31), (32), (40), and (41). The other two forms, which appertain exclusively to absolute assurances, are and they are respectively exhibited in (28), (33), (42) and (29), (34), (43). There is yet another form, to which special reference has not been made, viz which for calculation may be practically represented by