TABLE XI. Showing the value of an annuity on three joint lives, each of the same age, deduced from the Northampton Showing the value of an annuity on three joint lives, whose difference of ages are 10 and 20 years, deduced from 1-11-21 8.627 16-26-36 9.584 31-41-51 7.420 46-56-66 4.965 61-71-81 2.224 3-13-23 4-14-24 5-15-25 6-16-26 7-17-27 8-18-28 9-19-29 10-20-30 11-21-31 12-22-32 13-23-33 *767 *548 *362 ⚫169 ANUITIES. surance lives. r ad § III. Assurances. 34. When a person purchases for himself an annuity, during his natural life, or if the purchase be made for an annuity depending upon the joint continuance of two or more given lives, or on the longest of any given lives, he may be said to have assured an annuity under the specified conditions; and thus far we have, therefore, already entered upon the subject of assurances ; but what is most commonly understood by this term, is, when a certain sum of money is to be paid, or a certain annuity to commence upon the extinction of some specified life; on condition of the assurer either paying down a certain gross sum, or making a certain yearly payment, to be continued during his life, or in any other manner that may be agreed upon between the parties. AN increase the value of the assurance, as the prospects of PROBLEM I. 35: To determine the present value of a given sum, payable at the end of the year, in which any given lives become extinct. Let us denote, as in the preceding articles, the Value of a given lives by A, B, C, and the prospect of each living sum de1, 2, 3, &c. years, by a' a" a"" α a a and let the given sum to be received be s. We have already observed, that the doctrine of annuages to ities must always be considered a subject of the first imty. portance, in a commercial state like that of Great Britain; but that of assurances, we conceive, according to the definition we have given of the term, to involve a still greater number of interest. When we consider the thousands of families in this coun ry, who are living in a state of comparative affluence, without possessing any, or very little, disposable property; whose income, in fact, depends almost entirely on the exertions of the head of the family, and with the extinction of whose life every source of income ceases; when we contemplate the poverty and distress in which many widows, with their helpless children, would be plunged by such an event, we cannot estimate too highly the advantages which are held out by those societies, who, on honourable principles, furnish the means whereby every provident father and husband may, in part, avert the consequences of a premature death; to which every one is liable, and against which event every man ought to be provided. Perhaps, no part of the civil economy of this country shows more decidedly the high moral state of the middling classes of the people, than the immense amount of life assurances effected in the different offices the third year of the metropolis, and in those of like local companies in several of the counties in England; nor, perhaps, can we have a stronger instance of the high degree of confidence that the people are disposed to place in the moral rectitude of the government; by far the greater part of the capital of the companies to which we have alluded being invested under government securities. A subject of such importance ought to be well understood, and every means ought to be taken to reduce it to accurate computation. This, however, is not a very easy task; the data must be formed from bills of mortality, such as we have alluded to in the preceding part of this article; and these bills, as we have remarked, are subject to great irregularity, the consequence of which is, that very different results will be obtained, according to the tables from which the computations are made. If the assurances are set at too high a rate, the assurer is injured instead of being benefited; and if too low, the company, after a few years, must cease to be effective, and the persons depending on its stability, are ultimately involved in all the distress they were endeavouring to avert. The most eligible plan, therefore, and that which is adopted by many assurance companies, is to assure at rather a high than at a low rate, and then from time to time to VOL. XVII. 1 = abc-a'b' c In like manner, the probability of the joint lives failing Now, by referring to art. 16, it will appear, that if we pending on the extinction of a given life. AN (ABC), as in the preceding part of this article, the being 17.; we have the following rule for determining AN NUITIES. first of the above series is expressed by the annual payment sought. Illustrated 36. Multiply the value of an annuity on the given by example. lives, whether they are joint lives, or on the longest of any given lives, or a single life, by the rate of interest on 17. and subtract the product from unity. Divide the remainder by the amount of 11. in one year, and the quotient multiplied by the given sum will be the present value sought; or the premium which must be paid to assure the sum under the proposed conditions. Ex. 1. What sum ought a person, aged 20, to pay down to ensure 100l. to his executors at his death, taking interest at 5 per cent. and according to the Northampton tables? Value de Divide the present value found, as in Problem I. by the value of an annuity on the given lives (as shown in the tables), plus 1, and the quotient will be the annual payment sought. Ex. 1. A person, aged 30, wishes to ensure 1001. payable at his death. Required the annual payments that he ought to make, allowing 3 per cent. interest, and using the Northampton tables. By Table IX. the value of an annuity of 11. for such a life, and at the proposed rate of interest, is 16:920; whence NUITIES Here, the chance of recovering the sum at the end of Va any one year, will depend on the happening of one or pe other of these two events; viz. first, that A dies in the the year, and that B lives to the end of it; secondly, oftwo that both lives fail in the year, but that A's happen lives. first. The probability of the first of these two events is (a—a′) b' a b the probability of the second isand the sum of the two (a—d') b—3'); -; 2 ab a b is the whole expectation; which, therefore, multiplied 3d ab a' b a'b + ab ab); ab a b ab a b 2 It is obvious that the present value of the sum, as pending on determined in the last problem, is that to which we the extinc- must refer in this determination; we have, therefore, tion of two only to ascertain what annual payment ought to be joint lives. made as an equivalent for the present sum found as above; and this again is nearly the same as determining what annuity, on a given life, a given sum will purchase; the only difference being, that in this case, the payment is made at the commencement instead of the end of each year, as in questions of simple annuities; that is, the number of payments will be one more in this case than in that. And as the value of annuities, as exhibited in the tables, are the same as the number of years' purchase, the tabular annuity the third is equal to S a b 4th year, &c. a corresponding signification. This result is given in words at length as follows:39. Let A' represent a life one year older than A, and A a life one year younger. Add unity to the value of an annuity on the two joint lives A'B; and multiply the sum by the number of persons living at the age of A'. Then divide th product by the amount of 17. for a year, and reserve fuotient. Multiply the value of an an uity on two joint lives A,B by the number of person living at the age of A, ; and having subtracted the product from the above reserved quotient, divide the remainder by the number of persons living at the age of A. Subtract this last quotient from the present value of 11. payable on the extinction of the two joint lives AB, and the remain multiplied by half the given sum, will be the vee required. Cor. To find the annual payments equivalent to this present sum, we must divide the latter by unity plus the value of an annuity on the joint lives of AB; for the same reason as that assigned for a similar determination in the last problem. 40. We shall not detain the reader with an illustration of this rule; it will be sufficient to observe that tables have been computed of all the most probable cases, involved in the three last problems; of which those most commonly made use of are the three followAng; indeed, till very lately, none of the assurance offices in London employed any others; but within a short period new tables have been issued by some of the principal institutions of this kind, which are computed upon more liberal principles. The tables to which we have referred, viz. Tables XII. XIII. and XIV. are all computed at 3 per cent. interest, and conformably with the observations made at Northampton. What data have been employed in the computation of the new tables we are not informed. We have before observed, that the only four principles upon which assurances can be established is that AN adopted by the Amicable, and some other offices, viz. We have given, in the conclusion of this article, Dr. Price, on Reversionary Payments, &c. first published in 1769; a fourth edition of which, very considerably augmented and improved, appeared in 1783. Simpson's Doctrine of Annuities and Reversions, 1742; with a continuation in his Select Exercises, 1752. Morgan's Doctrine of Annuities and Assurances, 1779. Masere's Principles of the Doctrine of Life Annuities, 1783. Bailey, on the Doctrine of Life Annuities and Assurances, 1813; a highly scientific and valuable performance; and lastly, to Milne's Treatise on the same subject, published in 1815. |