Value on white balls in a bag, and the probability of drawing a 15. To find the value of an annuity granted upon any number of lives; that is, for as long as they shall all continue in being together. Let A, B, C, &c. be the lives upon which the anjoint lives. nuity is granted; and let the probability of each life continuing 1, 2, 3, &c. years be, as denoted above, a" a" b b" b" C" - &c. &c. then it follows, from what has been stated above, that , &c. which, being multi 1 plied by 1+' the value of 11. certain, at the end of the year (see art. 6.), will produce a'b'c' AN NUITE for the present value of the first year's rent. a' b' c', &c. a" b" c", &c. a"b" c", &c. &c. a b с ,&c. = present value of 1st payment; 2d payment; 3d payment; &c. nth payment. This series will continue till such time as that a) a' b'c', &c. a" b" c", &e. As the numbers a', b', c'; a", b", c", &c. are subject to no determinate law, it is obvious that there can be no rule given for summing these series; they must be computed by actually substituting the numbers proper to the case in question, and then collecting the sum of all the terms; which will be the value of an annuity of 17. on the lives proposed; and this, therefore, multiplied by any given annuity, will give its present value. It is on this principle that Tables IX. and X. have been computed, deduced from the observations made subsequent page. at Northampton, and which we shall again refer to in a VII. When the annuity is deferred. 17. If the annuity is not to commence till after a foran e 1 Sab'e abc, &c. l 1+r (1+r)2 a" b" c" + + a"" "" "" +, &c. (n).(n) (n) a' B'y b + (1+r) (1+r) where a', a", a"; ß', ß", ß"' ; y', y", y", &c. represent the number living after n, n + 1, n + 2, &c. For it is obvious that the second part of the series will be the value of such an annuity; and the first part, continued to n terms, the value of an annuity on the same lives for the first n years; the two parts together, or the whole series, being the value of an annuity to commence immediately. According to this solution, however, we cannot avail ourselves of the tables above referred to, at least only for the whole series; so that we should have still to calculate the value of the first terms, or those which correspond to the assumed temporary annuity: it is best, therefore, to proceed according to the following rule. Find the value of an annuity on the same number of lives, each as many years older than the given lives, as are equal to the number of years during which the annuity is deferred. Find also the expectation of the given lives surviving to the end of the time during which the annuity is deferred, and the product of these two quantities will be the value required. The method of determining the value of a temporary contingent annuity, which is represented by the leading terms, or first part, of the preceding series, will immediately suggest itself to the reader, without any particular remark. It is the difference between the value of the whole series, and that of the deferred annuity, determined by the above rule. PROBLEM II. 18. To find the value of an annuity granted upon the longest of any number of lives; that is, for as long stofa as any one of them is in existence. num f lives. Let A, B, C be the lives upon which the annuity is granted; let the probability of each life continuing 1, 2, 3, &c. years be denoted as in Problem I; then the probability that some one or other of these will live to the end of the first is year and the sum of all these series will be the expectation Now the reader will readily observe, by comparing combined, plus the value of an annuity on all three ANNUITIES. 19. Supposing (A), (B), (C) to denote the value of the Particular annuities on A, B, C respectively; (AB), (AC), (BC), notation the values of the annuities on A and B together, A and explained. C together, and B and C together; also (ABC), the value of the annuity on A, B, and C together; then the value of the annuity on the longest of the three lives is equal to (A) + (B) + (C) — (AB) — (AC) — (BC) + (ABC); whence the same tables, which exhibit the values of annuities on the joint lives of two or three persons, may also be employed in computing the value of an -; and, therefore, annuity on the longest of those lives. a-a' b-b' c-c' a b and that they will not all die, or, which is the same, that one at least will be living, is the difference between unity, or certainty, and the above product; that is, Some examples illustrating these formula will be found immediately preceding the following tables. Of reversionary life annuities. Particular cases. We shall, for the sake of abridging the work, confine Our investigation to two lives only, A, B; and two others, P, Q; but it will be obvious that the same process will apply to any number. p' q &c. Let now the probabilities of three joint lives, attaining to 1, 2, 3, &c. years, be denoted, as above, by a' b' a" b" a" b &c. ab a b a b P q Pq Pq Then it is obvious that the chance which the joint lives A, B have of receiving the annuity after one year, will depend upon their living to the end of that year, and on the joint lives P, Q becoming extinct before the end of that period. The former, from what we have seen, is denoted by and the latter by a' b' a b (see the last proposition); consequently, ; and the value of the first payment becomes &c. The sum of these is, therefore, the value sought, which is obviously equal to the differences between the value of an annuity on the joint lives of A, B together, minus that on A, B, P, Q together. We shall illustrate the use of these results in the AN solution of the subsequent practical questions. NUITIES Of survivorships. 22. The doctrine of survivorships is one of a mixed Survivornature, and admits of a great variety of combinations; ships. we must, of course, confine ourselves to only a few of those cases which most commonly occur. In the cases we have hitherto examined, we have only considered the value of annuities, as depending upon the continuance of certain lives, or of a certain number out of any proposed lives; we now intend to compute their value, as depending upon any specified survivorship between them; and consequently the questions become so much the more embarrassing, and admit, as we have said above, of greater variety. Those of most common application are as follow: PROBLEM IV. 23. An annuity is granted upon the longest of three Probes given lives A, B, C, to be equally divided amongst them while they are all living; equally between the two survivors, when one life fails, and the whole by the longest liver, during his life. Required the value of their respective expectations; their ages being given. Let the prospect of the given lives continuing 1, 2, 3, &c. years be still denoted as in the foregoing problems; and let us first determine A's expectation. The expectation of A, as to what he may happen ta receive at the end of any one year, may be considered in four parts. First, A, B, C may be all living; the a' b'c' abc probability of which is ; in which case he will receive 4d of the annuity; and, consequently, We may, therefore, in these computations, still avait receive one half of the annuity, or 2. On a single life A, after the longest of two lives P, Q, by (A) — (AP) — (AQ) + (APQ). 3. On the longest of two lives A, B, after a single life P, by (A) + (B) — (AB) — (AP) — (BP) + (ABP). 4. On a single life A, after two joint lives P, Q, by (A) — (APQ). 5. On two joint lives A, B, after a single life P, by pectation. (AB) (ABP). By adding these several values together, we find ems, expectation first year. In the same manner, we find the value of the expect whence, using the preceding notation, A's expectation ation of the second payment, viz. 1 Sa" (1+r)2l a a" c" 2ac + a"b"c" a b c S = whole a" b" 2ab expectation second year; and so on for the 3d, 4th, &c. years. By observing that the vertical column of a series of terms, such as the above, denotes the value on single and joint lives, as explained in the foregoing problems, and using the same symbols to express the values of these lives, we shall find the expectation in the case in question, equal to (A) (AB)-(AC) + (ABC). In the same manner, B's expectation is (B) — † (BA) — 1 (BC) + 1 (ABC); and that of C is 24. An annuity is granted on three lives, as follows: A and B are to enjoy it equally while they are both living; and on the death of either, A and C, or B and C are to have it equally shared between them; and, finally, on the death of either of these, the survivor is to enjoy the whole. Required the value of their respective expectations. Here the value of A's expectation may be considered in three parts. 1. A and B may be both living, the probability of a' b' which in which case A receives the annuity; and a b a' c' X 1+r 2 ac (1-응). 25. Various other cases of survivorships might be Particular proposed, and investigated; we must, however, be cases. contented to mention only the following, with the corresponding results, leaving the investigations to be supplied by the reader; or we may refer him to the several treatises on this subject, particularly to Baily's Doctrine of Life Annuities and Assurances, the most scientific work that has yet appeared on those subjects. A, B, and C agree to purchase an annuity on the longest of their lives, to be divided amongst them in the following manner: A and B are to enjoy it equally during their joint lives; if A die first, then B and C are to enjoy it equally during their joint lives, and the survivor of them to have the whole; but if B die first, then A is to enjoy the whole during his life; and after his decease, it is to devolve wholly to C. The value of the several expectations, according to these conditions, are C's (C) (AC) - (BC) + (ABC). Illustration of the preceding deductions, solution of various problems, &c. 26. Such of our readers as are familiar with the use of algebraical formulæ, will find no difficulty in submitting those we have deduced from our investigations to the solution of any problem which falls within their range others, however, will doubtless prefer seeing Lastly, B and C may be both dead, and A living; those deductions in words at length; which we propose the probability of this is AN NUITIES. and a Illustrations by examples. Expectation of life. Value of a contin gency. (n) the number existing any given number of years (n) after that period. In the case of joint lives, it is the product of the probabilities that each of the single lives shall continue in being to the end of the given term. Ex. I. Let it be proposed to find the probability that two persons, one aged 20, and the other 40, shall individually and jointly live 30 years. Using De Parcieux's result, Table IV. We find here that the probability of A living 30 years 581 is -; that B will live the same time is 814 310 657 ; and that they will both continue in existence to the end of the proposed term, is In the same manner, the probability of any other number of lives continuing in existence for a given term, may be determined. If the probability were required that either one or both the lives, will be in existence at the end of the proposed period, we may find the probability of their both dying within the given time; and subtract that result from unity, viz. Divide the number of persons which die within the given number of years, by the number living at the proposed ages, and the product of the fractions is the probability of their both dying; the difference between which and unity will be the probability sought. 1 n a a n mula (1 + r) Ex. 2. A person, aged 20, is entitled to 1,000l. thirty years hence, providing he is then in existence, what is the value of his expectation in a present sum? Reckoning interest at 4 per cent. and using De Parcieux's Table of Expectations, viz. Table IV. By our Table II. it appears that the present worth of 1,000l. certain, at the end of 30 years, is 2671.; and the probability of a person, 20, living thirty years, is ; therefore 581 814 28. To find the value of an annuity on a single life. We have already shown the principle upon which V is made the computation of the value of annuities a on single lives; it remains, therefore, here merely to explain the use of our Table IX. for rendering the operation more easy, or indeed for determining the required value from simple inspection. Various tables have been computed for this purpose, founded on different observations on the mortality of mankind; as we could not, from the nature of our work, introduce all such tables, we have preferred those of Dr. Price, which are formed upon observations made at Northampton, being, as we have elsewhere observed, those most commonly had recourse to in this country. Rule. Multiply the tabular value by the given annuity, for the present worth sought. Ex. 4. Required the value of an annuity of 100% per annum, on a life aged 40, allowing 5 per cent. in terest. By Table IX, the value of an annuity of 11. per annum, under the proposed circumstances, is £11-837; wherefore, the value of the proposed annuity is £11837 x 100 11831. 14s. Ex. 5. Required the value of an annuity of 100%. per annum, on a life aged 30, interest being allowed at 4 cent. per By Table IX, the value of an annuity of 17. is £14-781; whence £14-781 x 100 14781. 2s. PROBLEM IV. 29. To find the value of an annuity on two joint lives, the difference of the proposed ages falling within the limits of the differences in the tables. By inspection in Table X, find the tabular value answering to the given case, and multiply that value by the proposed annuity. Ex. 6. What is the value of an annuity of 1001. per annum, depending on the joint lives of two persons, one aged 40 and the other 50; interest at 5 per cent. Here the difference of age is 10 years, and the tabolar value is £8.177; wherefore annum £8.177 x 100 8177. 14s. Ex. 7. What is the value of an annuity of 100% per on two joint lives, each being 30; interest 4 per cent. By Table X, the value of an annuity of 1s £11.313; whence £11.313 x 100 11311. 6s. 30. To find the value of an annuity on two joint ise. when the difference of age is not found in the table. |