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Published online by Cambridge University Press: 22 April 2013
Recent investigations in the theory of life contingencies have thrown new light on the treatment of problems involving the probability of survival of several lives. For the exact solution of such problems by older methods, there are required complete tables of annuities (or other integral functions) on the number of lives to be dealt with. The time and labour necessary for the computation of tables of annuities on three or more joint lives, for every combination of ages, are so great as to have hitherto effectually prevented their construction; while tables of annuities on all combinations of quinquennial ages, like the joint-life tables given by Price and Milne, or the tables of annuities on three joint lives published by Filipowski in 1850, cannot generally be applied without a troublesome and uncertain process of interpolation. When, therefore, cases involving more than two lives have occurred, it has been usual in dealing with them to employ methods of approximation, leading to more or less error in the final result. Of late years, however, it has been found possible to express, with considerable exactness, the rate of mortality during the greater period of adult life, as a function of the age, by means of an hypothesis having at the same time important a priori considerations in its favour. Corresponding expressions have been deduced, representing the values of annuities on any number of joint lives, and relations have been established which enable us to determine the values of such annuities from tables of moderate extent, with considerable certainty and accuracy.
page 39 note * It will be convenient to use the abreviation J. I. A. to denote Journal of the Institute of Actuaries.
page 46 note * These quantities are in Mr. Makeham's paper and table denoted by z and n respectively. The dashes have been added for the sake of consistency in the notation employed throughout the present paper.
page 48 note * These quantities are, in the papers referred to, denoted by v and m respectively. It has been thought better, as in the case of Mr. Makeham's constants z and n, to guard against confusion by adding dashes.
page 52 note * The D here used without a suffix is not the D of the Commutation Table, but is equivalent to 1 + Δ.
page 60 note * In calculating these values we depart from the Institute Hm Table hitherto employed, and adopt a mortality table adjusted according to Mr. Makeham's hypothesis. The constants employed are those given by MrWoolhouse, (J. I. A., xv. 408)Google Scholar, and the methods used give exact results, as illustrated by the close agreement of the latter with each other.