By rule IV. The present value of an annuity of £1 for 12 years is found by rule II. to be 9.385073; and by rule III. of Compound Interest, the present value of this for 8 years is found to be 6.857579, as before.

By rule V. The amount of £1 at compound interest for 12 years, diminished by a unit, is 601032; and the amount of £1 for 20 years is 2.191123. Divide the former by the latter, and the result by 04, and there will finally result 6.857579, the same present value as before.

Exercises. Required the present values of the following annuities in reversion, at the given rates per cent. per annum :

Exercises.

£ s. d.

25. Ann. 135 10

Answers.

£ S. d.

9, after 6 yrs. & for 8 yrs. at 5.. 622 13 6

6,

www.

4

10, 3

79 12 58

54 12

28.

wwwwww

29. A Perpetuity of £84

30.

7 6, after 8 years, at 7 701 10 7

136 17 9,

8

www.

5 ..1853 0 41

31. What fine must be paid, to change into a perpetuity, a lease for 16 years, which brings a profit rent of £71 13 3 per annum, payable yearly, compound interest being allowed at cent. per annum? Answ. £866 6 83.

per

32. What fine must be paid to add 25 years to a lease which brings a profit rent of £112 10, and of which 14 years are unexpired, compound interest being allowed at 5 cent. annum? Answ. £800 16 42.

33. What is the present value of the reversion of a perpetuity of £60 annum, payable yearly, but not to come into possession till the expiration of 100 years, compound interest being allowed at 6 cent. annum ?* Answ. £2 18 114.

ANNUITIES CONTINGENT, OR LIFE ANNUITIES. LIFE ANNUITIES are those whose commencement or termination, or both, depend on the extinction of one or more lives.

When life annuities are in possession, they are often called simply ANNUITIES ON LIVES; but when they are in reversion, they are generally called ANNUITIES ON SURVIVOR

SHIPS.

* This question may tend to correct a mistake that pretty generally prevails respecting the comparative values of long leases and perpetuities, the latter being supposed to exceed the former in value in a far greater degree than they really do. In the case on which this exercise is founded, the difference of present values is no more than £2 18 113.

The VALUE OF A LIFE is the present value of an annuity of 1 to continue during that life.

The EXPECTATION OF A LIFE of a given age, is the mean period during which persons of that age live.

The COMPLEMENT OF A LIFE is double the expectation of the same life.

The calculation of life annuities depends on the joint application of the rules of compound interest, and of the doctrine of chances, to tables deduced from observations on the duration of human life. In what follows on this subject, a selection of the rules most generally useful will be given., For the theory of these rules, which is of a nature too complicated to be given in a work like the present, the reader who wishes to become thoroughly acquainted with this interesting and difficult subject, may have recourse to the writings of Simpson, De Moivre, and more particularly of Dr. Price and Mr. Morgan, where the subject will be found treated at great length, both in theory and practice.

The duration of life being different in different countries, calculations have been founded on the registers of births and deaths kept in London, Breslaw, Northampton, and various other places. Tables IV. and V at the end of the book, which are employed in what follows, are founded on the Northampton register, which is thought to serve, for the generality of places, better than any other.

RULEI. To find the present value of an annuity to continue during the life of a person whose age is given: Take from table IV. the value of £1 for the given age and rate, and multiply it by the given annuity.

Exam. 1. What should be given at 6 cent. annum, for a farm worth £36 annum, held on a lease of one life aged 58 years? Here, by the table, the value of an annuity of £1 on the life of a person aged 58 years, is, at 6 cent., 8173, the product of which by £36 is £294-228, or £294 4 63, the value required.

Ex. 1. If a person aged 38 years, have a salary of £138 10 year for life, what is its present value at 5 cent. annum? Answ. £1678 1 32.

2. What should a person aged 32 years, pay, at 4 cent. V annum, to have for life a yearly salary of £180? Answ. £2609 2. 3. If a farm of 28 acres be held at £1 14 6 acre, on a life aged 41 years, what fine must be paid, at 5 per cent. per annum. to reduce the rent to 10 shillings per acre? Answ. £401 2 94.

RULE II. To find the present value of an annuity which is to continue during the joint lives of two persons, and to cease when either of them dies: In table V. find the age of the younger, or of either if they be equal, in the first column; and in the same division of the table, in the second column, find the age of the other; opposite to the latter is the value of an annuity of £1, which multiply by the given annuity.

Exam. 2. What is the value at 4 per cent., of an annuity of £90 per annum, to continue during the joint lives of two persons, whose ages are 15 and 50 years respectively?

Here, by table V., the value of an annuity of £1 is £9-872, the product of which by £90 is £888-480, or £888 9 74, the value required.

Exam. 3. Required the present value, at 6 per cent., of an annuity of £120 per annum, which is to cease, when either of two persons, aged 14 and 57 years respectively, shall die.

Neither of these ages being in the table, recourse must be had to the method of proportional parts. (See page 191.) Thus, the table gives, for 10 and 55, 7·951, and for 15 and 55, 7-812. The difference of these is 139, four fifths of which being subtracted from 7.951, the remainder, 7.840, is the value of £1 for the ages 14 and 55. In like manner, the table gives for 10 and 60, 7-250, and for 15 and 60, 7·135; four fifths of the difference of which being taken from 7.250, the remainder, 7-158, is the value of £1 for the ages 14 and 60. Hence we have, for 14 and 55, 7.840, and for 14 and 60, 7·158; two fifths of the difference of which being subtracted from 7.840, the remainder, 7-567, is very nearly the present value of £1 for the ages 14 and 57: and this being multiplied by £120, the product, £908'04, or £908 0 91, is the value required.

Ex: 4. Required the present value, at 6 per cent., of an annuity of £39 10, on the joint continuance of two lives of 25 and 73 years. Answ. £182 8 104.

5. Required, at 5 per cent., the present value of an annuity of £43 12 6 on the joint lives of two persons, whose ages are 44 and 51 years respectively. Answ. 342 9 54.

RULE III. To find the present value of an annuity to continue during the longer of two lives: From the sum of the values of the single lives (found in table IV.) subtract the value of the joint lives (found by rule II. :) the remainder is the present value of an annuity of 1 on the longer of the two lives.

Exam. 4. For how much should a house be sold which brings a profit rent of £41 5 per annum, and is held by a lease on the longer of two lives aged 20 and 35 years, compound interest being allowed at 6 per cent. per annum ?

By table IV. the values of the lives are 12.398 and 11-236, the sum of which is 23-634. Also, by table V., the value of them iointly is 9-451, which taken from the preceding sum, leaves 14-183, the value of an annuity of £1 on the life of the longest liver; which being multiplied by £41 5, the product, £585-049, or £585 0 11 is the present value required.

Ex. 6. What should be paid, at 5 per cent., for the purchase of

a profit rent of £68 5 per annum, to continue during the longer of two lives, aged 38 and 42 years? Answ. £1001 18 44.

7. What should a man, aged 44, pay, at 5 per cent., to secure, during his own life, and that of his wife, aged 39, an annuity of £200 a year? Answ. £2892 9 31.

RULE IV. To find the value of an annuity during the joint continuance of three lives; or which is to terminate on the e3tinction of any one of them: Find by rule II., the value of the two eldest jointly; and, by table IV., find what single life would have this same value. Then find, by rule II., the value of the joint continuance of the single life thus found, and of the youngest, and this will be the value of the three proposed lives jointly.

Exam. 5. Required the present value at 4 per cent., of an annuity of £140 on the joint continuance of three lives aged 15, 30, and 35 years.

Here, at 4 per cent., the value of the joint continuance of two lives aged 30 and 35, is 10-948; which in table IV. is found to he the value of a single age of 51 years nearly. Then, by rule II., the value of the joint continuance of two lives, of 15 and 511, is 9-6335, the value of an annuity of £1 on the joint continuance of the three given lives. The product of this by £140 is £1348-69, or £1348 13 92, the value of the given annuity.

RULE V. To find the value of an annuity on the longest of three lives; or which is to continue, till they are all extinct: From the sum of all the values of the single lives, found in table IV. and of the value of the three jointly, found by rule IV.; subtract the sum of the joint values of the lives combined two and two, by rule II.; and multiply the remainder by the given annuity.

Exam. 6. Required the present value, at 4 and farm which yield a profit rent of £32 10 a lease of three lives, aged 35, 30, and 15 years.

cent., of a house annum, held by

Here, by the last example, the value of the three lives jointly is 96335, and their values singly are 14.039, 14-781, and 16-791. Also, by rule II., the value of the first and second jointly is 10-948; of the first and third 11-787; and of the second and third 12.246. From the sum of the first four of these numbers take the sum of the others, and the remainder, 20-2635, is the value of an annuity

* The duration of the lives of females is found to be somewhat greater than that of males. It is inconsistent with the nature of this work, however, to take into calculation this ciroumstance, which in general produces but a slight difference in the result.

of £1 on the life of the longest liver of the three. This multiplied by £32 10, gives for the value required, £658.56375, or £658 11 34.

Ex. 8. How much should a man pay at 5 cent. annum, to purchase an annuity of £320 a year to continue till himself aged 55, his wife aged 49, and his son aged 20, shall all be extinct? Answ. £5103 11 0.

RULE VI. To find the value of the reversion of an annuity after the death of the present possessor: From the present value of an annuity of £1 for the entire continuance of the annuity, subtract the value of an annuity of 1 during the life of the possessor: the remainder will be the value of a reversion of £1 in the given circumstances, which multiply by the given annuity.

If there be two or three lives, the same rule will serve, their values being used instead of the value of the single life.

Exam. 7. If a person aged 47 years, possess a perpetuity of £500 per annum; what is its present value at 5 cent. to his son, who is to possess it at his death?

Here, the value of £1 in perpetuity is £20, and, by table IV, the value of an annuity of £I on a life of 47 years is 10-784. Then 2010-7849.216; which multiplied by £500 produces £4608, the value required.

Exam. 8. A man leaves an annuity of £165 annum, of which 30 years are yet to come, to his nephew, aged 38 years, during life, or till its termination; and the reversion of it to a charitable institution, in case the nephew die before the termination of the annuity. How much is the present value of the reversion at 4 cent.?

In this case, the present value of an annuity of £1 for 30 years, is by rule II. of Annuities Certain, 17-292; and, by table IV. the present value of £1 on a life of 38 years, is 13.548. Then, 17.292 13·5483·744; and 3-744 × 165 = £617·76, or £617 15 2}, the value required.

Ex. 9. What is the present value at 5 cent., of the reversion of a perpetuity of £400 annum, not to come into possession till the death of the present incumbent aged 70, and of his intended successor aged 30? Answ. £2538 16 0.

RULE VII. To find the present value of a proposed_sum payable at the decease of a person whose age is given: From the value of el in perpetuity, take the value of an annuity of £1 on the life; divide the remainder by the value of the perpetuity increased by a unit; and multiply the result by the given sum.