Value Of Annuity Research Articles

On a general method of approximation to the Values of Annuities and Assurances, for long terms of years, depending on One or Two Lives

The discovery of some general Law of Mortality, by means of which the values of the contingencies, depending upon human life, might be expressed in some simple formulæ, without the necessity of performing the numerous calculations which are required even for the chances of a single life, and still more in the various cases, in which two or more lives are involved, has occupied the attention and labours of many scientific men. But the result has not hitherto been satisfactory, in spite of the skill with which their investigations have been conducted, and the unwearied patience which they have brought to the task. The failure has probably arisen from the want of a sufficient number of observations, the authenticity of which could be relied upon; or for want of that accuracy in the data, without which it is in vain to attempt to find the traces of a regular law, disturbed as it must be by the peculiar circumstances under which the observations were made, or by the ignorance, and, in some cases it may be, by the wilful errors of those who have been employed to collect the facts.

L. On the rule for finding the value of an annuity on three lives

L. <i>On the rule for finding the value of an annuity on three lives</i>

On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &amp;c. By Benjamin Gompertz, Esq. F. R. S

This paper, which professes to be a continuation of former researches on the same subject printed in the Transactions of the Royal Society, is divided into two chapters. In the first the author considers the nature of the law of those numbers in tables of mortality, which express the amount of persons living at the end of ages in regular arithmetical progression. He remarks that for short intervals the law approaches nearly to a decreasing geometrical progression, and that this must be the case whatever be the strict expression for the law of mortality, provided the intervals do not exceed certain limits. But he further remarks, that this property will be found to belong to very extensive portions of tables of mortality, and instances Deparcieux’s tables, where from the age of 25 to that of 45, the numbers living at the end of each year decrease very nearly in geometrical progression. Considering however the whole extent of such a table, it will be found that the ratio of this geometrical progression is not the same in all parts of the table. But before he enters on this consideration, the author draws some consequences from the hypothesis of a geometrical progression being the strict law of nature after a certain age. One of these is the equality of value of all life annuities commencing after that age. Another is, that the want of instances in history of persons living to very enormous ages (waving those of the patriarchs,) is no proof that such may not be the law of nature, as he shows by calculation, that out of 3,000,000 persons of 92, not more than one should on this supposition reach 192. This leads him to some general considerations on the causes of death, after which he resumes the consideration of the general law of the tables.

XCII. New theorems applicable to the value of annuities

XCII. <i>New theorems applicable to the value of annuities</i>

VI. Short and easy theorems for finding, in all cases, the differences between the values of annuities payable yearly, and of the same annuities payable half-yearly, quarterly, or momently. By the Rev. Richard Price, D.D. F.R.S. In a letter to Sir John Pringle, Bart. P.R.S

The values of annuities, as given in all the common tables, suppose them paid yearly. But it is well known, that generally they are paid half-yearly, and sometimes quarterly; and that this is a circumstance which always adds to their value. The difference between the values of annuities, according as they are paid in these different ways, I have seen no where stated with accuracy; and therefore, I have thought that the following attempt to do this may be of some use. Annuities are of two sorts. They are either payable certainly or conditionally. Of the former sort are all annuities which are payable at fixed times, without depending on any contingency. Of the latter sort are all annuities on lives. I will first consider the first sort of annuities.

XLIII. A letter to James West, Esq; President of the Royal Society, containing the investigations of twenty cases of compound interest, by J. Robertson, Lib. R. S

Sir, It is well known that the consideration of compound interest is of great utility in the computations respecting the values of pensions, annuities, reversions, and other affairs relating to money concerns; and therefore, many mathematicians have bestowed some time on this useful subject, and have endeavoured to discover practical methods for solving the various cases that might occur: And in these inquiries, the use of Logarithms has been found of singular service in facilitating the operations.

CXIV. A letter to the Reverend William Brakenridge, D. D. Rector of St. Mich. Bassishaw, and F. R. S. with a table of the value of annuities on lives, by Mr. James Dodson, Master of the Royal Mathematical School in Christ’s-Hospital, and F. R. S

Reverend Sir, As I have made a great many calculations, relative to Annuities on Lives, and have otherwise contributed, as much as was in my Power, to facilitate the performance of such, I thought it, almost, a duty incumbent on me, to compute the values of them according to your curious Table of the Decrements of Life, inserted in the Philosophical Transactions; accordingly I have inclosed a table of them, and if you find it will bear examination I desire you to communicate it the Royal Society, in order (if approved of) to its being inserted, as an appendix to your letter to the President.

LIII. A letter from Mr. James Dodson to Mr. John Robertson, F. R. S. concerning an improvement of the bills of mortality

Sir, As there has lately been a scheme proposed for amending the form of the bills of mortality of London, in a pamphlet called Observations on the past growth and present state of London , by Mr. Corbyn Morris, the ingenious author of which has enumerated many excellent purposes, to which it may be applied, but has omitted to mention that of giving a greater degree of certainty to the calculations of the values of annuities on lives.

X. A letter from Mr. Abraham De Moivre, F. R. S. to William Jones, Esquire, F. R. S. concerning the easiest method for calculating the value of annuities upon lives, from tables of observations

Sir, You may remember, that some time after the printing of the second edition of my book of annuities on lives, you told me, that it seem'd strange to you, that, considering I had demonstrated the chief propositions in the book.

I. Some further considerations on the Breslaw bills of mortality

Sir, What I gave you in my former discourse on these Bills, was chiefly designed for the Computation of the Values of Annuities on Lives, wherein I believe I have performed what the short Period of my Observations would permit, in relation to exactness, but at the same time do earnestly desire, that their Learned Author Dr. Newman of Breslaw would please to continue them after the same manner for yet some years further, that so the casual Irregularities and apparent Discordance in the Table, p . 599. may by a certain number of Chances be rectified and ascertain'd.