Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica/Annuities/Part 1

PART I.

We shall, in this Part, demonstrate all that is most useful and important in the doctrine of Annuities and Assurances on lives, without using Algebra, or introducing the idea of probability; but the reader is, of course, supposed to understand common Arithmetic. In the first 30 numbers of this Part, Compound Interest and Annuities-certain are treated of; from the 31st to the 76th, the doctrine of Annuities on Lives is delivered; and that of Assurances on Lives, from thence to the 108th, where the popular view terminates.

What is demonstrated in this Part, will be sufficient to give the reader clear and scientific views of the subjects treated; and, with the assistance of the necessary tables, will enable him to solve the more common and simple problems respecting the values of Annuities and Assurances. He will also understand clearly the general principles on which problems of greater difficulty are resolved; but these he cannot undertake with propriety, when the object is, to make a fair valuation of any claims or interests, with a view to an equitable distribution of property, unless he has studied the subject carefully, with the assistance of Algebra; for intricate problems of this kind can hardly be solved without it; and those who are not much exercised in such inquiries, often think they have arrived at a complete solution, when they have overlooked some circumstance or event, or some possible combination of events or circumstances, which materially affect the value sought. Eminent Mathematicians have, in this way, fallen into considerable errors, and it can hardly be doubted, that those who are not mathematicians, must (cæteris paribus) be more liable to them.

I. ON ANNUITIES-CERTAIN.

No. 1. When the rate is 5 per cent., L. 1 improved at simple interest during one year, will amount to L. 1·05; which, improved in the same manner during the second year, will be augmented in the same ratio of 1 to 1·05; the amount then, will therefore be $1{\cdot }05\times 1{\cdot }05$, or $(1{\cdot }05)^{2}=1{\cdot }1025$.

In the same manner it appears, that this last amount, improved at interest during the third year, will be increased to $(1{\cdot }05)^{3}=1{\cdot }157625$; at the end of the fourth year, it will be $(1{\cdot }05)^{2}$; at the end of the fifth $(1{\cdot }05)^{5}$, and so on; the amount at the end of any number of years being always determined, by raising the number which expresses the amount at the end of the first year, to the power of which the exponent is the number of years. So that when the rate of interest is 5 per cent., L. 1 improved at compound interest, will, in seven years, amount to $(1{\cdot }05)^{7}$, and in 21 years, to $(1{\cdot }05)^{21}$.

But if the rate of interest were only 3 per cent., these amounts would only be $(1{\cdot }03)^{7}$, and $(1{\cdot }03)^{21}$ respectively.

2. The present value of L. 1 to be received certainly at the end of any assigned term, is such a less sum, as, being improved at compound interest during the term, will just amount to one pound. It must therefore be less than L. 1, in the same ratio as L. 1 is less than its amount in that time; but in three years, at 5 per cent., L. 1 will amount to L. $(1{\cdot }05)^{3}$ (1). And $(1{\cdot }05)^{3}:1::1:{\frac {1}{(1.05)^{3}}}$, so that ${\frac {1}{(1.05)^{3}}}={\frac {1}{1{\cdot }157625}}=0{\cdot }863838$ is the present value of L. 1 to be received at the expiration of three years.

In the same manner it appears that, at 4 per cent. interest, the present value of L. 1 to be received at the end of a year, is ${\frac {1}{1{\cdot }04}}=0{\cdot }961538$; and if it were not to be received until the expiration of 21 years, its present value would be ${\frac {1}{(1{\cdot }04)^{21}}}=(0{\cdot }961538)^{21}=0{\cdot }438834$.

Hence it appears, that if unity be divided by the amount of L. 1, improved at compound interest during any number of years, the quotient will be the present value of L. 1 to be received at the expiration af the term: which may also be obtained by raising the number which expresses the present value of L. 1 receivable at the expiration of a year, to the power of which the exponent ts the number of years in the term.

3. When a certain sum of money is receivable annually, it is called an Annuity, and its quantum is expressed by saying it is an annuity of so much; thus, according as the annual payment is L. 1, L. 10, or L. 100; it is called an annuity of L. 1, of L. 10, or of L. 100.

4. When the annual payment does not depend upon any contingent event, but is to be made certainly, either in perpetuity or during an assigned term, it is called an Annuity-certain.

5. In calculating the value of an annuity, the first payment is always considered to be made at the end of the first year from the time of the valuation, unless the contrary be expressly stated.

6. The whole number, and part or parts of one annual payment of an annuity, which all the future payments are worth in present money, is called the number of years purchase the annuity is worth; and, being the sum of the present values of all the future payments, is also the sum which, being put out and improved at compound interest, will just suffice for the payment of the annuity (2).

7. Hence it follows, that when the annuity is L. 1, the number of years purchase and parts of a year, is the same as the number of pounds and parts of a pound in its present value.

And throughout this article, whenever the quantum of an annuity is not mentioned, it is to be understood to be L. 1.

8. The sum of which the simple interest for one year is L. 1, is evidently that which, being put out at interest, will just suffice for the payment of L. 1 at the end of every year, without any augmentation or diminution of the principal; and, being equivalent to the title to L. 1 per annum for ever, is called the value of the perpetuity, or the number of years purchase the perpetuity is worth.

But, while the rate remains the same, the annual interests produced by any two sums, are to each other as the principals which produce them; therefore, since $5:1::100:{\frac {100}{5}}=20$, when the rate of interest is 5 per cent., the value of the perpetuity is 20 years purchase. In the same manner it appears, that according as the rate may be 3 or 6 per cent. the value of the perpetuity will be ${\frac {100}{3}}=33{\tfrac {1}{3}}$, or ${\frac {100}{6}}=16{\tfrac {2}{3}}$ years purchase; and may be found in every case, by dividing any sum by its interest for a year.

9. All the most common and useful questions in the doctrines of compound interest and annuities-certain, may be easily resolved by means of the first four tables at the end of this article. Their construction may be explained by the following specimen, rate of interest 5 per cent.

Construction of
Term.	Table IV.	Table III.	Table I.	Table II.	Term.
	Amount of L. 1 per annum.	Amount of L. 1	Present value of L. 1 to be received at	Present value of L. 1 per annum, to be received until
	improved at Interest until
the Expiration of the Term.
1 Yr.	1·000000	1·050000	·952381	0·952381	1 Yr.
2 Yrs.	2·050000	1·102500	.907029	1·859410	2 Yrs.
3	3·152500	1·157625	·863838	2·723248	3
4	4·310125	1·215506	·822702	3·545950	4
5	5·525631	1·276282	·783526	4·329476	5
6	6·801913	1·340096	·746215	5·075691	6
7	8·142009	1·407100	·710681	5·786372	7

10. The calculation must begin with Table III., the first number in which should evidently be 1·05, the amount of L. 1 improved at interest during one year; which, being multiplied by 1·05, the product is 1·025, the second number; this second number being multiplied by 1·05, the product is 1·157625, the amount at the end of three years. And so the calculation proceeds throughout the whole of the column; each number after the first, being the product of the multiplication of the preceding number, by the amount of L. 1 in a year (1).

11. The number against any year in Table I. is found by dividing unity by the number against the same year in Table III. (2); thus, the number against the term of six years in Table I. is ${\frac {1}{1{\cdot }340096}}={\cdot }746215$. All the numbers in that table after the first, may also be found, by multiplying that first number continually into itself (2).

12. The number against any year in Table II. being the sum of the numbers against that and all the preceding years in Table I.; is found by adding the number against that year in Table I. to the number against the preceding year in Table II.; thus, the number against 4 years in Table II., being

the sum of 0·822702.

and 2·723248.

---

is 3·545950.

13. If each payment of an annuity of L. 1 be put out as it becomes due, and improved at compound interest during the remainder of the term, it is evident that at the expiration of the term, the payment then due will be but L. 1, having received no improvement at interest. That received one year before will be augmented to the amount of L. 1 in a-year; that received two years before will be augmented to the amount of L. 1 in two years; that received three years before to the amount of L. 1 in three years, and so on until the first payment, which will be augmented to the amount of L. 1 in a term one year less than that of the annuity.

Hence, it is manifest, that the number against any year in Table IV. will be unity added to the sum of all those against the preceding years in Table III.

And, therefore, that the number against any year in Table IV. is the sum of those in Tables III. and IV. against the next preceding year.

Thus, the number against seven years in Table IV., being

the sum of 1·340096

and 6·801913

---

is 8·142009.

14. The method of construction is obviously the same at any other rate of interest.

15. All the amounts and values which are the objects of this inquiry, evidently depend upon the improvement of money at compound interest; it is, therefore, that the first, second, and fourth tables, all depend upon the third.

But every pound, and every part of a pound, when put out at interest, is improved in the same manner as any single pound considered separately. Whence, it is obvious, that while the term and the rate of interest remain the same, both the amount and the present value, either of any sum, or of any annuity, will be the same multiple, and part or parts of the amount or the present value found against the same term, and under the same rate of interest in these tables, as the sum or the annuity proposed is of L. 1.

So that to find the amount or the present value of any sum of annuity for a given term and rate of interest, we have only to multiply the corresponding tabular value by the sum or the annuity proposed; the product will be the amount or the value sought, according as the case may be.

16. Example 1. To what sum will L. 100 amount, when improved at compound interest during 20 years? the rate of interest being 4 per cent. per annum.

By Table III., it appears, that L. 1 so improved, would, at the expiration of the term, amount to L. 2·191123, therefore L. 100 would amount to 100 times as much, that is, to L. 219·1123, or L. 219, 2s. 3d.

17. Ex. 2. What is the present value of L. 400, which is not to be received until the expiration of 14 years, when the rate of interest is 5 per cent.?

The present value of L. 1 to be received then, will be found by Table I. to be L. 0·505068: L. 400 to be received at the same time, will therefore be worth, in present money, 400 times as much, or L. 202·0272, that is, L. 202, 0s. 6½d.

18. Ex. 3. Required the present value of an annuity of L. 50 for 21 years, when the rate of interest is 5 per cent.

Table II. shows the value of an annuity of L. 1 for the same term to be L. 12·8212; the required value must therefore be 50 times as much, or L. 641·06, that is, L. 641, 1s, 2½d.

19. Ex. 4. What will an annuity of L. 10, 10s. or L. 10·5, for thirty years, amount to, when each payment is put out as it becomes due, and improved at compound interest until the end of the term? The rate of interest being 4 per cent.

The amount of an annuity of L. 1 so improved, would be L. 56·084938, as appears by Table IV., the amount required will therefore be 10·5 times this, or L. 588·89185, that is, L. 588, 17s. 10d.

20. When the interval between the time of the purchase of an annuity and the first payment thereof, exceeds that which is interposed between each two immediately successive payments; such annuity is said to be deferred for a time equal to that excess, and to be entered upon at the expiration of that time.

21. If two persons, A and B, purchase an annuity between them, which A is to enter upon immediately, and to enjoy during a certain part of the term, and B, or his heirs, or assigns, for the remainder of it; the present value of B’s interest will evidently be, the excess of the value of the annuity for the whole of the term from this time, above the value of the interest of A.

So that when the entrance on an annuity ts deferred for a certain term, its present value will be the excess of the value of the annuity for the term of delay and continuance together, above the value of an equal annuity for the term of delay only.

22. Example 1. Required the value of a perpetual annuity of L. 120, which is not to be entered upon until the expiration of 14 years from this time, reckoning interest at 3 per cent.

The perpetuity, with immediate possession, would be worth 33⅓ years’ purchase (8); and an annuity for the term of delay is worth 11·2961 (Table II.)

From	33·3333,
subtract	11·2961,	and multiply
the remainder	22·0372,
by	120,
the product	2644·464,	= L. 2644, 9s. 3¼d.
is the required value.

23. Ex. 2. Allowing interest at 5 per cent. what sum should be paid down now for the renewal of 14 years lapsed, in a lease for 21 years of an estate producing L. 300 per annum, clear of all deductions?

This is the price of an annuity for 14 years, to be entered upon 7 years hence; the term of delay, therefore, is 7 years, and that of the delay and continuance together 21 years.

By Table Il. it appears, that the present value of an annuity

for 21 years, is 12·8212 years’ purchase. for 7 years, 5·7864

---

Value of the deferred annuity, 7·0348 Multiply by 300

---

The product L. 2110·44,  or L. 2110, 8s. 9½d. is the price required.

24. Hitherto we have proceeded upon the supposition of the annuity being payable, and the interest convertible into principal, which shall reproduce interest, only once a-year.

But annuities are generally payable half-yearly, and sometimes quarterly; and the same circumstances that render it desirable for an annuitant to receive his annual sum in equal half-yearly or quarterly portions, also give occasion to the interest of money being paid in the same manner.

But whatever has been advanced above, concerning the present value or the amount of an annuity, when both that and the interest of money were only payable once a-year, will evidently be true when applied to half the annuity, and half the interest paid twice as often, on the supposition of half-yearly payments; or to a quarter of the annuity, and a quarter of the interest, paid four times as often, when the payments are made quarterly.

25. Half-yearly payments are, however, by far the most common, and these four tables will also enable us to answer the most useful questions concerning them.

For we have only to extract the present value, or the amount, from the table, against twice the number of years in the term, at half the annual rate of interest, and, in the case of an annuity, to multiply the number so extracted, by half the annuity proposed.

26. Ex. 1. To what sum will L. 100 amount in 20 years, when the interest at the rate of 4 per cent. per annum, is convertible into principal half-yearly?

This being the amount in 40 half years at 2 per cent. interest for every half year, will be the same as the amount in 40 years at 2 per cent. per annum, which, by Table III. will be found to be 220·804, or L. 220, 16s. 1d.; and is only L. 1, 13s. 10d. more than it would amount to if the interest were not convertible more than once a-year (16).

27. Ex. 2. What is the present value of an annuity of L. 50 for 21 years, receivable in equal half-yearly payments, when money yields an interest of 2½ per cent. every half year?

By Table II. it appears, that an annuity of L. 1 for 42 years, when the interest of money is 2½ per cent. per annum, will be worth L. 25·8206 (25); 25 times this sum, or L. 645, 10s. 3½d. is therefore, the required value, and exceeds the value when the interest and the annuity are only payable once a-year by L. 4, 9s. 1d. (18).

28. The excess of an annuity-certain above the interest of the purchase-money, is the sum which, being put out at the time of each payment becoming due, and improved at compound interest until the expiration of the term, will just amount to the purchase-money originally paid.

But, while everything else remains the same, the longer the term of the annuity is, the less must its excess above the interest of the purchase-money be, because a less annuity will suffice for raising the same sum within the term. Therefore, the proportion of that excess to the annual interest of the purchase-money, continually diminishes as the term is extended; and when the annuity is a perpetuity, there is no such excess (8).

29. The reason why the value of an annuity is increased by that and the interest being both payable more than once in the year, is, that the grantor loses, and the purchaser gains, the interest produced by that part of each payment, which is in excess above the interest then due upon the purchase-money, from the time of such payment being made, until the expiration of the year.

Hence it is obvious, that the less this excess is, that is, the longer the term of the annuity is (28), the less must the increase of value be.

And when the annuity is a perpetuity, its value will be the same, whether it and the interest of money be both payable several times in the year, or once only.

30. When the annuity is not payable at the same intervals at which the interest is convertible into principal, its value will depend upon the frequencies both of payment and conversion; but its investigation without algebra, would be too tong, and of too little use, to be worth prosecuting here.

II. OF ANNUITIES ON LIVES.

31. When the payment of an annuity depends upon the existence of some life or lives, it is called a Life-annuity.

32. The values of such annuities are calculated by means of tables of mortality, which show, out of a considerable number of individuals born, how many upon an average have lived to complete each year of their age; and, consequently, what proportion of those who attained to any one age, have survived any greater age.

The fifth Table at the end of this article is one of that kind, which has been taken from Mr Milne’s Treatise on Annuities, and was constructed from accurate observations made at Carlisle by Dr Heysham, during a period of 9 years, ending with 1787.

33. By this table it appears, that during the period in which these observations were made; out of 10,000 children born, 3203 died under 5 years of age, and the remaining 6797 completed their fifth year. Also, that out of 6797 children who attained to 5 years of age, 6460 survived their 10th year.

But the mortality under 10 years of age, has been greatly reduced since then, by the practice of vaccination.

This table also shows, that of 6460 individuals who attained to 10 years of age, 6047 survived 21. And that of 5075 who attained to 40, only 3643 survived their 60th year.

34. There is good reason to believe (as has been shown in another place), that the general law of mortality, that is, the average proportion of persons attaining to any one age, who survive any greater age, remains much the same now among the entire mass of the people throughout England, as it was found te be at Carlisle during the period of these observations; except among children under 10 years of age, as was noticed above (33).

If this be so, it will follow, that of 6460 children now 10 years of age, just 6047 will attain to 21; or rather, that if any great number be taken in several instances, this $\left({\frac {6047}{6460}}\right)$ will be the average proportion of them that will survive the period.

And if 6460 children were to be taken indiscriminately from the general mass of the population at 10 years of age, and an office or company were to engage to pay L. 1, eleven years hence, for each of them that might then be living; this engagement would be equivalent to that which should bind them to pay L. 6047 certainly, at the expiration of the term. Therefore, the office, in order that it might neither gain nor lose by the engagement, should, upon entering into it, be paid for the whole, the present value of L. 6047, to be received at the expiration of 11 years; and for each life, the 1/6460th part of it; that is, the 6047/6460th part of the present value of L. 1 to be received then.

But when the rate of interest is 5 per cent. the present value of L. 1, to be received at the expiration of 11 years, is L. 0·584679; therefore, at that rate of interest, there should be paid for each life ${\frac {6047\times 0{\cdot }584679}{6460}}$ = L. 0·5473.

And the present value of L. 100, to be received upon a life now 10 years of age attaining to 21, will be L. 54·73, or L. 54, 14s. 7d.

In the same manner it will be found, that reckoning interest at 4 per cent. the value would be L. 60, 16s. 1d.

35. This is the method of calculating the present values of endowments for children of given ages; and the values of annuities on lives may be computed in the same manner.

For, from the above reasoning it is manifest, that if the present value of L. 1, to be received certainly at the expiration of a given term, be multiplied by the number in the table of mortality against the age, greater than that of any proposed life by the number of years in the term, and the product be divided by the number in the same table, against the present age of that life; the quotient will be the present value of L. 1, to be received at the expiration of the term, provided that the life survive it.

And if, in this manner, the value be determined of L. 1, to be received upon any proposed life, surviving each of the years in its greatest possible continuance, according to the table of mortality adapted to it; that is, according to the Carlisle table, upon its surviving every age greater than its present, to that of 104 years inclusive; then, the sum of all these values will evidently be the present value of an annuity on the proposed life.

36. If 5642 lives at 30 years of age be proposed, and 5075 at the age of 40; since each of the 5642 younger lives may be combined with every one of the 5075 that are 10 years older, the number of different pairs, or different combinations of two lives differing in age by 10 years, that may be formed out of the proposed lives, is 5642 times 5075.

But at the expiration of 15 years, the survivors of the lives now 30 and 40 years of age, being then of the respective ages of 45 and 55, will be reduced to the numbers of 4727 and 4073 respectively; and the number of pairs, or combinations of two, differing in age by 10 years, that can be formed out of them, will be reduced from 5642 × 5075 to 4727 × 4073.

So that L. 1 to be paid at the expiration of 15 years for each of these 5642 × 5075 pairs or combinations of two, now existing, which may survive the term, will be of the same value in present money, as 4727 times L. 4073, to be received certainly at the same time.

Now let A be any one of these lives of 30 years of age, and B any one of those aged 40; and from what has been advanced it will be evident, that the present value of L. 1 to be received upon the two lives in this particular combination jointly surviving the term, will be the same as that of the sum L. ${\frac {4727\times 4073}{5642\times 5075}}$ to be then received certainly.

But, when the rate of interest is 5 per cent. L. 1 to be received certainly at the expiration of 15 years, is equivalent to L. 0·481017 in present money (Table I).

Therefore, at that rate of interest, and according to the Carlisle table of mortality; the present value of L. 1 to be received upon A and B now aged 30 and 40 years respectively, jointly surviving the term of 15 years, will be ${\frac {4727\times 4073\times {\text{L. }}0{\cdot }481017}{5642\times 5075.}}$.

37. Hence it is sufficiently evident, how the present value of L. 1 to be received upon the same two lives jointly surviving any other year may be found. And if that value for each year from this time until the eldest life attain to the limit of the table of mortality be calculated, the sum of all these will be the present value of an annuity of L. 1 dependent upon their joint continuance.

In this manner, it is obvious that the value of an annuity on the joint continuance of any other two lives might be determined.

38. If, besides the 5642 lives at 30 years of age, and the 5075 at 40 (mentioned in No. 36), there be also proposed 3643 at 60 years of age; each of these 3643 at 60, may be combined with every one of the 5642 × 5075 different combinations of a life of 30, with one of 40 years of age; and, therefore, out of these three classes of lives 5642 × 5075 × 3643 different combinations may be formed; each containing a life of 30 years of age, another of 40, and a third of 60.

But at the expiration of 14 years, the numbers of lives in these three classes will, according to the table of mortality, be reduced to 4727, 4073, and 1675 respectively; the respective ages of the survivors in the several classes being then 45, 55, and 75 years; and the number of different combinations of three lives (each of a different class from either of the other two), that can be formed out of them, will be reduced to 4727 × 4073 × 1675.

Hence, by reasoning as in No. 36, it will be found, that if A, B, and C be three such lives, now aged 30, 40, and 60 years, the present value of L. 1 to be received upon these three jointly surviving the term of 15 years from this time, will be ${\frac {4727\times 4073\times 1675}{5642\times 5075\times 3643}}\times {\text{L. }}0{\cdot }481017$: interest being reckoned at 5 per cent.

Thus it is shown, how the present value of an annuity dependent upon the joint continuance of these three lives might be calculated, that being the sum of the present values thus determined, of the rents for all the years which, according to the table of mortality, the eldest life can survive.

39. But it is easy to see, that the same method of reasoning may be used in the case of four, five, or six lives, and so on without limit. Whence, this inference is obvious.

The present value of L. 1, to be received at the expiration of a given term, provided that any given number of lives all survive it, may be found by multiplying the present value of L. 1 to be received certainly at the end of the term, by the continual product of the numbers in the table of mortality against the ages greater respectively by the number of years in the term, than the ages of the lives proposed; and dividing the last result of these operations, by the continual product of the numbers in the table of mortality against the present ages of the proposed lives.

And by a series of similar operations, the present value of an annuity on the joint continuance of all these lives might be determined.

But it should be observed, that, in calculating the value of a life-annuity in this way, the denominator of the fractions expressing the values of the several years rents, that is, the divisor used in each of the operations, remains always the same; the division should, therefore, be left till the sum of the numerators is determined, and one operation of that kind will suffice.

40. Enough has been said to show that these methods of constructing tables of the values of annuities on lives are practicable, though excessively laborious, and, in fact, all the early tables of this kind were constructed in that manner. We proceed now to show how such tables may be calculated with much greater facility.

41. By the method of No. 34, it will be found that, reckoning interest at 5 per cent., the present value of L. 1 to be received at the expiration of a year, provided that a life, now 89 years of age, survived till then, is ${\frac {142\times 0{\cdot }95238.}{181}}$ But the age of that life will then be 90 years, and the proprietor of an annuity of L. 1 now depending upon it, will, in that event, receive his annual payment of L. 1 then due; therefore, if the value then of all the subsequent payments, that is, the value of an annuity on a life of 90 be 2·339 years’ purchase, the present value of what the title to this annuity may produce to the proprietor, at the end of the year, will be the same as that of L. 3·339, to be received then, if the life be still subsisting, or ${\frac {142\times 0{\cdot }952381}{181}}\times {\text{L. }}3{\cdot }339={\text{L. }}2{\cdot }495$; which, therefore, will be the present value of an annuity of L. 1 on a life of 89 years of age. That is to say, an annuity on that life will now be worth 2·495 years’ purchase (7).

42. In the same manner it appears generally, that, if unity be added to the number of years’ purchase that an annuity on any life is worth, and the sum be multiplied by the present value of L. 1, to be received at the end of a year, provided that a life one year younger survive till then, the product will be the number of years’ purchase an annuity on that younger life is worth in present money.

43. But according to the table of mortality, an annuity on the eldest life in it is worth nothing; therefore, the present value of L. 1 to be received at the end of a year, provided that the eldest life but one in the table survive till then, is the total present value of an annuity of L. 1 on that life. Which, being obtained, the value of an annuity on a life one year younger than it may be found by the preceding number; and so on for every younger life successively.

Exemplification.

Rate of Interest 5 per cent.

Age of Life.	Value of an Annuity on that Life, increased by Unity.	Which, being multiplied by 0·952381, and the Product by	Produces the value of an Annuity on the next younger Life	Its Age being
104	1·000	1/3	0·317	103
103	1·317	3/5	0·753	102
102	1·753	5/7	1·192	101
101	2·192	7/9	1·624	100
100	2·624	9/11	2·045	99
99	3·045	11/14	2·278	98
98	3·278	14/18	2·428	97
97	3·428	18/23	2·555	96
96	3·555	23/30	2·596	95
95	3·596	30/40	2·569	94

44. Proceeding as in No. 36, it will be found, that at 5 per cent. interest, and according to the Carlisle table of mortality, the present value of L. 1 to be received at the expiration of a year, provided that a person now 89 years of age, and another now 99, be then living, is ${\frac {142\times 9\times {\text{L. }}0{\cdot }952381}{181\times 11}}$: therefore, if the present value of an annuity of L. 1 on the joint continuance of two lives, now aged 90 and 100 years respectively, be L. 0·950; by reasoning as in No. 41, it will be found that the present value of an annuity on the joint continuance of two lives, of the respective ages of 89 and 99 years, will be worth ${\frac {142\times 9\times 0{\cdot }952381}{181\times 11}}\times 1{\cdot }950=1{\cdot }192$ years’ purchase.

45. In this manner, commencing with the two oldest lives in the table that differ in age by ten years, and proceeding according to No. 43, the values of annuities on all the other combinations of two lives of the same difference of age, may be determined.

The method is exemplified in the following specimen:

Ages of two Lives.	Value of an Annuity on their joint continuance, increased by Unity.	Which, being multiplied by 0·952381, and the Product by	Produces the value of an Annuity on the two joint Lives one year younger respectively,	their Ages being
94 & 104	1·000	${\frac {1\times 40}{3\times 54}}$	0·235	93 & 103
93 & 103	1·235	${\frac {3\times 54}{5\times 75}}$	0·508	92 & 102
92 & 102	1·508	${\frac {5\times 75}{7\times 105}}$	0·733	91 & 101
91 & 101	1·733	${\frac {7\times 105}{9\times 142}}$	0·950	90 & 100
90 & 100	1·950	${\frac {9\times 142}{11\times 181}}$	1·192	89 & 99
89 & 99	2·192	${\frac {11\times 181}{14\times 232}}$	1·280	88 & 98

46. Hence, and by what has been advanced in the 39th number of this article, it is sufficiently evident, how a table may be computed of the values of annuities on the joint continuance of the lives in every combination of three, or any greater number; the differences between the ages of the lives in each combination remaining always the same in the same series of operations, while the calculation proceeds back from the combination in which the oldest life is the oldest in the table, to that in which the youngest is a child just born.

47. But, when there are more than two lives in each combination, the calculations are so very laborious, on account, principally, of the great number of combinations, that no tables of that kind have yet been published, except three or four for three lives.

And, in the books that contain tables of the values of two joint lives, methods are given of approximating towards the values of such combinations of two and of three lives, as have not yet been calculated.

Therefore, assuming the values of annuities on single lives, and on the joint continuance of two or of three lives, to be given; we have next to show how the most useful problems respecting the values of any interests that depend upon the continuance or the failure of life, may be resolved by them.

48. Proposition 1. The value of an annuity on the survivor of two lives, A and B, is equal to the excess of the sum of the values of annuities on the two single lives, above the value of an annuity on their joint continuance.

49. Demonstration. If annuities on each of the two lives were granted to P, during their joint continuance, he would have two annuities; but if P were only to receive these upon condition that, during the joint lives of A and B, he should pay one annuity to Q; then, there would only remain one to be enjoyed by him, or his heirs or assigns, until the lives both of A and B were extinct; whence the truth of the proposition is manifest.

50. Prop. 2. The value of an annuity on the joint continuance of the two last survivors out of three lives, A, B, and C, is equal to the excess of the sum of the values of annuities on the three combinations of two lives (A with B, A with C, and B with C) that can be formed out of them, above twice the value of an annuity on the joint continuance of all the three lives.

51. Demons. If one annuity were granted to P on the joint continuance of the two lives A and B, another on the joint continuance of A and C, and a third, on the joint continuance of B and C; during the joint continuance of all the lives he would have three annuities.

But if he were only to receive these upon condition that he should pay two annuities to Q, during the joint continuance of all the three lives; then, there would only remain to himself one annuity during the joint existence of the last two survivors out of the three lives. And the truth of the proposition is manifest.

52. Prop. 3. The value of an annuity on the last survivor of three lives, A, B, and C, is equal to the excess of the sum of the values of annuities on each of the three single lives, together with the value of an annuity on the joint continuance of all the three, above the sum of the values of three other annuities; the first dependent upon the joint continuance of A and B, the second, on that of A and C, and the third, on B and C.

53. Demons. If annuities on each of the three single lives were granted to R; during the joint continuance of all the three, he would have three annuities, and from the time of the extinction of the first life that failed, till the extinction of the second, he would have two.

So that he would have two annuities during the joint existence of the two last survivors out of the three lives; and besides these, a third annuity during the joint continuance of all the three.

Therefore, if out of these, R were to pay one annuity to P during the joint continuance of the last two survivors out of the three lives, and another to Q during the joint continuance of all the three; he would only have left one annuity, which would be receivable during the life of the last survivor of the three.

But in the demonstration of the last proposition (51) it was shown, that the value of what he paid to P, would fall short of the sum of the values of annuities dependent respectively on the joint continuance of A and B, of A and C, and of B and C, by two annuities on the joint continuance of all the three lives. Whence it is evident, that the value of the annuities he paid both to P and Q, would fall short of the sum of these three values of joint lives, only by the value of one annuity on the joint continuance of all the three lives.

Wherefore, if from the sum of the values of all the three single lives, the sum of the values of the three combinations of two that can be formed out of them were taken; there would remain less than the value of an annuity on the last survivor, by that of an annuity on the joint continuance of the three lives.

But if, to the sum of the values of the three single lives A, B, and C, there be added that of an annuity on the joint continuance of the three, and from the sum of these four values, the sum of the values of the three combinations, A with B, A with C, and B with C be subtracted; then, the remainder will be the value of an annuity on the last survivor of the three lives. Which was to be demonstrated.

54. Prop. 4. Problem. The law of mortality and the values of single lives at all ages being given; to determine the present value of an annuity on any proposed life, deferred for any assigned term.

55. Solution. Find the present value of an annuity on a life older than the proposed, by the number of years during which the other annuity is deferred; multiply this by the present value of L. 1 to be received upon the proposed life surviving the term, and the product will be the value sought.

56. Demons. Upon the proposed life surviving the term, the annuity dependent upon it will be worth the same sum, that an annuity on a life so much older is now worth; therefore, it is evident, that the deferred annuity is of the same present value as that sum to be received at the expiration of the term, provided the life survive it.

57. Corollary. In the same manner it appears, that the present value of an annuity on the joint continuance of any number of lives, deferred for a given term, may be found by multiplying the present value of an annuity on the joint continuance of the same number of lives, older respectively than the proposed, by the number of years in the term; by the present value of L. 1 to be received, upon the proposed lives all surviving it.

58. A Temporary Annuity on any single life, or on the joint continuance of any number of lives, that is, an annuity which is to be paid during a certain term, provided that the single life or the other lives jointly subsist so long; together with an annuity on the same life or lives deferred for the same term, are evidently equivalent to an annuity on the whole duration of the same life or lives.

So that the value of an annuity on any life or on the joint continuance of any number of lives for an assigned term, is equal to the excess of the value of an annuity on their whole duration, with immediate possession, above the value of an annuity on them deferred for the term.

59. Whatever has been advanced from No. 48. to 53, inclusive, respecting the values of annuities for the whole duration of the lives whereon they depend, will apply equally to those which are either deferred or temporary; and, therefore, to determine the value of any deferred or temporary annuity, dependent upon the last survivor of two or of three lives; or, upon the joint continuance of the last two survivors out of three lives; we have only to substitute temporary or deferred annuities, as the case may require, for annuities on the whole duration of the lives; and the result will, accordingly, be the value of a temporary or deferred annuity on the life of the last survivor, or on the joint lives of the two last survivors.

60. Prop. 5. A and B being any two proposed lives now in existence, the present value of an annuity to be payable only while A survives B, is equal to the excess of the value of an annuity on the life of A, above that of an annuity on the joint existence of both the lives.

61. Demons. If an annuity on the life of A, and to be entered upon immediately, were now granted to P, upon condition that he should pay it to B during the joint lives of A and B; it is evident that there would only remain to P, an annuity on the life of A after the decease of B: whence the truth of the proposition is manifest.

62. When any thing is affirmed or demonstrated of any life or lives, it is to be understood as applying equally to any proposed single life, or to the joint continuance of the whole of any number of lives that may be proposed together, or to that of any assigned number of the last survivors of them, or to the last surviving life of the whole.

63. Prop. 6. The present value of the reversion of a perpetual annuity after the failure of any life or lives, is equal to the excess of the present value of the perpetuity with immediate possession, above the present value of an annuity on the same life or lives.

64. Demons. If a perpetual annuity with immediate possession were granted to P, upon condition that he should pay the annual produce to another individual, during the existence of the life or lives proposed; it is evident that there would only remain to P, the reversion after the failure of such life or lives; and the present value of that reversion would manifestly be as stated above.

65. The 6th, 7th, and 8th tables at the end of this article, which have been extracted from the 19th, 21st, and 22d, respectively, in Mr Milne’s Treatise on Annuities, will serve to illustrate the application of these propositions to the solution of questions in numbers.

In all the following examples, we suppose the lives to be such, as the general average of those the Carlisle table of mortality was constructed from, and the rate of interest to be 5 per cent.

66. Ex. 1. What is the present value of an annuity on the joint lives, and the life of the survivor of two persons now aged 40 and 50 years respectively?

According to No. 48, the process is as follows:

Value of a single life of 40 50	13·390 by Table VI. 11·660
sum	25·050
Subtract the value of their joint lives,	9·984 (Table VIII.)
remains	15·066 years’ purchase, the required value.

And if the annuity be L. 200, its present value will be L. 3013·2, or L. 3013, 4s.

67. Ex. 2. The lives A, B, and C, being now aged 50, 55, and 60 years respectively, an annuity on the joint continuance of all the three, is worth 6·289 years purchase; What is the value of an annuity on the joint existence of the last two survivors of them?

According to No. 50, the process is thus:

Ages. Values. 50 & 55 8·528 Table VII. 55 & 60 7·106 50 & 60 7·601 Table VIII. sum 23·235 Subtract 2 × 6·289  = 12·578 remains 10·657  years’ purchase, the required value.

Therefore, if the annuity were L. 300, it would be worth L. 3197, 2s. in present money.

68. Ex. 3. Required the value of an annuity on the last survivor of the three lives in the last example.

Proceeding according to No. 52, we have

Ages. Values. 50 11·660 Tab. VI. 55 10·347 60 8·940 50, 55, & 60 6·289 (No. 67). sum 37·236 Subtract the sum of the values 23·235  (No. 67.) of annuities on the three combinations of two lives, remains 14·001  years’

purchase, the required value. And if the annuity were L. 300, it would now be worth L. 4200, 6s.

69. Ex. 4. What is the present value of an annuity on a life now 45 years of age, which is not to be entered upon until the expiration of ten years; the first payment thereof being to be made at the expiration of eleven years from this tine, if the life survive till then?

Solution.

The present value of an annuity on a life of 55 is 10·347 (Table VI.), and the present value of L. 1 to be received upon the proposed life attaining to the age of 55, is ${\frac {4073}{4727}}\times 0{\cdot }613913$; therefore, by No. 55, the required value is ${\frac {4073\times 0{\cdot }613913\times 10{\cdot }347}{4727}}=5{\cdot }473$ years purchase; so that if the annuity were L. 200, its present value would be L. 1094, 12s.

70. Ex. 5. Required the present value of an annuity to be received for the next ten years, provided that a person now 45 years of age, shall so long live.

Solution.

The present value of an annuity on a life of 45, to

be entered upon immediately, is	12·648	(Table VI.)
Subtract that of an annuity on the same life deferred 10 years,	5·473	(69).
the remainder,	7·175

is the required number of years purchase. And, if the annuity were L. 200, its present value would be L. 1435.

71. Ex. 6. An annuity on a life of 45, deferred 10 years, was shown in No. 69, to be worth 5·473 years purchase in present money; let it be required to determine the equivalent annual payment for the same, to be made at the end of each of the next 10 years, but subject to failure upon the life failing in the term.

Solution.

The present value of L. 1 per annum on the proposed life for the next 10 years, has just been shown to be L. 7·175, and this, multiplied by the required annual payment, must produce L. 5·473; that payment must, therefore, be ${\frac {5{\cdot }473}{7{\cdot }175}}=0{\cdot }76279$. And, since the annual payment for the deferred annuity of L. 1 per annum would be L. 0·76279, that for an annuity of L. 200 must be L. 152, 11s. 2d.

72. Ex. 7. Let the present value be required of an annuity on a life now 40 years of age, to be payable only while that life survives another now of the age of 50 years.

From the present value of a 13·390  (Table VI.) life of 40, Subtract that of the two joint 9·984  (Table VIII.) lives, the remainder, 3·406

years purchase is the required value (60).

Therefore, if the annuity were L. 100, it would be worth L. 340, 12s. in present money.

73. If the annuity in the last example were to be paid for by a constant annual premium at the end of each year while both the lives survived; by reasoning as in No. 71, it will be found, that such annual premium for an annuity of L. 1 should be ${\frac {3{\cdot }406}{9{\cdot }984}}={\text{L. }}0{\cdot }341146$; for an annuity of L. 100 it should therefore be L. 34, 2s. 3½d.

74. But if one of the equal premiums for this annuity is to be paid down now, and another at the end of each year while both the lives survive; the number of years purchase the whole of these premiums are worth, will evidently be increased by unity, on account of the payment made now, it will, therefore, be 10·984; and each premium for an annuity of L. 1 must, in this case, be ${\frac {3{\cdot }406}{10{\cdot }984}}={\text{L. }}0{\cdot }310087$; for an annuity of L. 100 it should, therefore, be L. 31, 0s, 2d.

75. Ex. 8. Let it be required to determine the present value of the reversion of a perpetual annuity after the failure of a life now 50 years of age.

Solution.

The value of the perpetuity is 20 years purchase (8.)

Subtract that of an annuity on the life of 50,	11·660	(Table VI.)
Remains 8·34		years

purchase, the required value of the reversion (63.)

So that if the annuity were L. 300, its present value would be L. 2502.

76. In the same manner it will be found, by the 68th number and those referred to in the last example, that the reversion of a perpetuity, after the failure of the last survivor of three lives, now aged 50, 55, and 60 years respectively, is worth 5·999 years purchase in present money; therefore, if it were L. 100 per annum, its present value would be L. 599, 18s.

III. OF ASSURANCES ON LIVES.

77. An assurance upon a life, or lives, is a contract by which the Office or Underwriter, in consideration of a stipulated premium, engages to pay a certain sum upon such life or lives failing within the term for which the assurance is effected.

78. If the term of the assurance be the whole duration of the life or lives assured, the sum must necessarily be paid whenever the failure happens; and, in what follows, that payment is always supposed to be made at the end of the year in which the event assured against takes place. The anniversary of the assurance, or the day of the date of the policy, being accounted the beginning of each year.

79. At the end of the year in which any proposed life or lives may fail, the proprietor of the reversion of a perpetual annuity of L. 1 after their failure, will receive the pound then due, and will, at the same time, enter upon the perpetuity; therefore, the present value of the reversion is the same as that of L. 1 added to the money a perpetual annuity of L. 1 would cost, supposing this sum not to be receivable until the expiration of the year in which the failure of the life or lives might happen.

80. Hence we have this proportion. As the value of a perpetuity increased by unity is to L. 1, so is the present value of the reversion of a perpetual annuity of L. 1, after the failure of any life or lives, to the present value of L. 1, receivable at the end of the year in which such failure shall take place.

81. Therefore, if the value of an annuity of one pound on any life or lives, be subtracted from that of the perpetuity, and the remainder be divided by the value of the perpetuity increased by unity; the quotient will be the value, in present money, of the assurance of one pound on the same life or lives. (63)

82. Ex. 1. What is the present value of L. 1, to be received at the end of the year, in which a life now 50 years of age may fail?

The rate of interest being 5 per cent. the value of the perpetuity is 20 years purchase, and that of the life 11·66; the answer therefore is ${\frac {20-11{\cdot }66}{20+1}}={\frac {8{\cdot }34}{21}}={\text{L. }}0{\cdot }397143$. And if the sum assured were L. 1000, the present value of the assurance would be L. 397, 2s. 10d.

83. When the term of a life assurance exceeds one year, its whole value is hardly ever paid down at the time that the contract is entered into, but, in the instrument (called a Policy) whereby the assurance is effected, an equivalent annual premium is stipulated for, payable at the commencement of each year during the term, but subject to failure with the life or lives assured.

84. But by reasoning as in No. 74, it will be found, that an annual premium payable at the commencement of each year in the whole duration of the life or lives assured, will be worth one year’s purchase more, than an annuity on them payable at the end of each year; and, consequently, that if the value in present money of an assurance on any life or lives, be divided by the number of years purchase an annuity on the same life or lives is worth, increased by unity, the quotient will be the equivalent annual premium for the same assurance.

85. Ex. 2. Required the annual premium for the assurance of L. 1, on a life of 50 years of age.

In No. 82, the single premium for that assurance was shown to be 0·397143, and the value of an annuity on the life is 11·66, therefore, by the preceding number, the required annual premium will be ${\frac {0{\cdot }397143}{12{\cdot }66}}={\cdot }0313699$ for the assurance of L. 1; and for the assurance of L. 1000, it will be L. 31, 7s. 5d.

86. Ex. 3. Let both the single payment in present money, and the equivalent annual premium be required for the assurance of L. 1, on the joint continuance of two lives of the respective ages of 45 and 50 years.

The value of an annuity of L. 1 on the joint continuance of these two lives, appears by Table VII. to be L. 9·737, therefore ${\frac {20-9{\cdot }737}{20+1}}={\frac {10{\cdot }263}{21}}={\text{L. }}0{\cdot }488714$ is the single premium, and ${\frac {0{\cdot }488714}{10{\cdot }737}}={\text{L. }}0{\cdot }0455168$, the equivalent annual one for the assurance of L. 1 to be paid at the end of the year, in which that life becomes extinct which may happen to fail the first of the two.

Therefore, if the sum assured were L. 500, the total present value of the assurance would be L. 244, 7s. 2d. and the equivalent annual premium L. 22, 15s. 2d.

87. Ex. 4. Let both the single and the equivalent annual premium be required for the assurance of L. 1, on the life of the survivor of two persons now aged 40 and 50 years respectively.

The value of an annuity on the survivor of these two lives was shown in No. 66, to be 15·066, therefore, by No. 81, the single premium will be ${\frac {20-15{\cdot }066}{20+1}}={\frac {4{\cdot }934}{21}}={\text{L. }}0{\cdot }234952$; and by No. 84, the annual one will be ${\frac {{\text{L. }}0{\cdot }234952}{16{\cdot }066}}={\text{L. }}0{\cdot }0146242$.

That is, for the assurance of L. 1 to be received at the end of the year, in which the last surviving life of the two becomes extinct.

Therefore, for the assurance of L. 5000, the single premium will be L. 1174, 15s. 2d. the equivalent annual one L. 73, 2s. 5d.

88. Ex. 5. What should the single and equivalent annual premiums be for an assurance on the last survivor of three lives, of the respective ages of 50, 55, and 60 years.

The value of an annuity on the last survivor of them, was shown in No. 68, to be 14·001, the single premium should therefore be ${\frac {20-14{\cdot }001}{20+1}}={\frac {5{\cdot }999}{21}}={\text{L. }}0{\cdot }285666$, and the annual ${\frac {{\text{L. }}0{\cdot }285666}{15{\cdot }001}}={\text{L. }}0{\cdot }0190431$, for the assurance of L. 1, to be received at the end of the year, in which the last surviving life of the three may fail.

For the assurance of L. 2000, the single premium would therefore be L. 571, 6s. 8d. the annual one L. 38, 1s. 9d.

89. Lemma. If an annuity be payable at the commencement of each year, which some assigned life or lives may enter upon in a given term; the number of years purchase in its present value, will exceed by unity the number of years purchase, in the present value of an annuity on the same life or lives for one year less than the given term, but payable as annuities generally are, at the end of each year.

Demonstration. Since the proposed life or lives can only enter upon any year after the first, by surviving the year that precedes it; the receipt of each of the payments after the first, that are to be made at the commencement of the year, will take place at the same time, and upon the same conditions as that of the rent for the year then expired of the life-annuity, for a term one year less than the term proposed: this last mentioned annuity, will therefore, be worth in present money, just the same number of years’ purchase as all the payments subsequent to the first, which may be made at the commencements of the several years.

And, since the first of these is to be made immediately, the present value of the whole of them, will evidently exceed the number of years purchase last mentioned, by unity, which was to be demonstrated.

90. If, while the rest remains the same, the payment of the annuity which depends upon the assigned life or lives entering upon any year, is not to be made until the end of that year; as the condition upon which every payment is to be made, will remain the same, but each of them will be one year later; the only alteration in the value of the whole, will arise from this increase in the remoteness of payment, by which it will be reduced in the ratio of L. 1, to the present value of L. 1, receivable at the end of a year (2).

91. When the value of an annuity on any proposed life or lives for an assigned term is given, it is evident that the value of an annuity on the same life or lives for one year less may be found, by subtracting from the given value, the present value, of the rent to be received upon the proposed life or lives surviving the term assigned.

92. Proposition. The present value of an assurance on any proposed life or lives for a given term, is equal to the excess of the value of an annuity to be paid at the end of each year, which the life or lives proposed may enter upon in the term, above the value of an annuity on them for the same term, but dependent, as usual, upon their surviving each year.

Demonstration. if an annuity payable at the end of each year, which the proposed life or lives may enter upon during the given term, be granted to P, upon condition that he shall pay over what he receives to Q, at the end of each year which the same life or lives may survive; it is manifest that there will only remain to P, the rent for the year in which the proposed life or lives may fail; that is, the assurance of that sum thereon for the given term (77). Which was to be demonstrated.

93. From the last four numbers (89—92) we derive the following

Rule

for determining the present value of an assurance on any life or lives for a given term.

Add unity to the value of an annuity on the proposed life or lives for the given term, and from the sum subtract the present value of one pound, to be received upon the same life or lives surviving the term; multiply the remainder by the present value of L. 1, to be received certainly at the end of a year, and from the product subtract the present value of an annuity on the proposed life or lives for the term.

This last remainder will be the value in present money of the assurance of L. 1 during the same term, on the life or lives proposed.

94. It has been shown above (34—39), how the present value of L. 1, receivable upon any single or joint lives surviving an assigned term, may be found. And all that was demonstrated from No. 48. to 53. inclusive, respecting annuities on the last survivor of two, or of three lives, or on the joint continuance of the two last survivors out of three lives, is equally true of any particular year’s rent of those annuities. Hence it is evident, how the present value of L. 1, to be received upon the last survivor of two, or of three lives, or upon the last two survivors out of three lives, surviving any assigned term, may be found.

95. Example. Required the present value of L. 1, to be received at the end of the year, in which a life, now forty-five years of age, may fail, provided that such failure happens before the expiration of fen years.

Here the present value of L. 1, to be received upon the life surviving the term, will be found to be L. 0·528976, and the value of an annuity on the proposed life for the term, is 7·175 (70.)

From	8·175
subtract	0·528976
the remainder	7·646024
being multiplied by	0·952381
produces	7·28193
from this subtract	7·17500
remains L.	0·10693, the required

value of the assurance; and if the sum assured were L. 3000, the value of the assurance in present money would be L. 320, 15s. 7d.

96. By numbers 89, 91, and 95, it appears, that an annuity, payable at the commencement of each of the next ten years that a lite of 45 enters upon, is worth 7·646 years purchase: therefore, ${\frac {0{\cdot }10693}{7{\cdot }646}}={\text{L. }}0{\cdot }013985$ will be the annual premium for the assurance of L. 1 for ten years on that life. For the assurance of L. 3000, it will therefore be L. 41, 19s. 1d.

97. When the term of the assurance is the while duration of the life or lives assured, L. 1 to be received upon their surviving the term is worth nothing; and an annuity on the lives for the term, is also for their whole duration.

Therefore, from No. 93. we derive the following

Rule

for determining the present value of an assurance on any life or lives.

Add unity to the value of an annuity on the proposed life or lives; multiply the sum by the present value of L. 1, to be received certainly at the end of a year; and from the product, subtract the value of an annuity on the same life or lives.

The remainder will be the value of the assurance in present money.

98. Example. Required the present value of L. 1, to be received at the end of the year, in which the survivor of two lives may fail, their ages now being 40 and 50 years respectively.

The value of an annuity on these lives was shown in No. 66. to be 15·066.

Multiply 15·066 by 0·952381, from the product 15·3009, subtract 15·066, the remainder L. 0·2349 is the required value, agreeably to No. 87.

And, in all other cases, the values determined by the rule in the preceding number, will be found to agree with those obtained by the method of No. 81.

99. When an assurance on any life or lives has been effected at a constant annual premium, and kept up for some time, by the regular payment of that premium; the annual premium required for a new assurance of the same sum on the same life or lives, will, on account of the increase of age, be greater than that at which the first assurance was effected: Therefore, the present value of all these greater annual premiums, that is, the total present value of the new assurance, will exceed the present value of all the premiums that may hereafter be received under the existing policy. And the excess will evidently be the value of the policy, supposing the life or lives to be still insurable; that being the only advantage that can now be derived from the premiums already paid.

So that, if the present value of all the future annual premiums to be paid under an existing policy for the assurance of a certain sum upon any life or lives, be subtracted from the present value of the assurance of the same sum on the same life or lives; the remainder will be the value of the policy.

100. Example. L. 1000 has been assured some years, on a life now 50 years of age, for its whole duration, at the annual premium of L. 20, one of which has just now been paid: What is the value of the policy?

The present value of the assurance of L. 1000 on that life, has been shown in No. 82. to be L. 397, 2s. 10d.; and an annuity on the life, being worth 11·66 years purchase (Table VI.), the present value of all the premiums to be paid in future under the existing policy, is 11766 × L. 20 = L. 233, 4s. 0d.; the value of the policy, therefore, is L. 163, 18s. 10d.

Immediately before the payment of the premium, the value of the policy would evidently have been less by the premium then due.

101. In our investigations of the values of annuities on lives, we have hitherto assumed, that no part of the rent is to be received for the year in which the life wherewith the annuity may terminate fails.

But if a part of the annuity is to be received at the end of that year, proportional to the part of the year which may hare elapsed at the time of such failure; as, in a great number of such cases, some of the lives wherewith the annuity terminates will fail in every part of the year, and as many, or very nearly so, in any one part of it as in any other: we may assume, that, upon an average, half a year’s rent will be received at the end of the year in which such failure happens; and, therefore, that by the title to what may be received after the failure of the life or lives whereon the annuity depends, the present value of that annuity will be increased by the present value of the assurance of half a year’s rent on the same life or lives.

102. Thus, for example: the present value of the assurance of L. 1 on a life of 50 years of age, having in No. 82. been shown to be L. 0·397143, the value of an annuity of L. 1 on that life, when payable, till the last moment of its existence, will exceed L. 11·66, its value, if only payable, until the expiration of the last year it survives, by $\left({\frac {{\text{L. }}0{\cdot }397143}{2}}=\right)$ L. 0·199; it will therefore be L. 11·859.

103. If, at the end of the year in which an assigned life A may fail, Q or his heirs are to receive L. 1; and are, at the same time, to enter upon an annuity of L. 1, to be enjoyed during another life P, to be then fixed upon: the present value of Q’s interest will evidently be the same as that of the assurance on the life of A, of a number of pounds, exceeding by unity the number of years purchase in the value of an annuity on the life of P, at the time of nomination.

104. What is the present value of the next presentation to a living of the clear annual value of L. 500; A, the present incumbent, being now 50 years of age; supposing the age of the clerk presented to be 25, at the end of the year in which the present incumbent dies; also, that he takes the whole produce of the living for that year?

By Table VI. it will be found, that the value of an annuity of L. 1 on a life of 25, is L. 15·303; and in No, 82. it has been shown, that the present value of the assurance of L. 1 on a life of 50, is L. 0·397143. Hence, and by the last number, it appears, that if the annual produce of the living were but L. 1, the present value of the next presentation would be L. 16·303 × 0·397143 = L. 6·47467. The required value, therefore, is L. 3237, 6s. 9d.

105. If, to the value of the succeeding life, determined according to No. 103, the value of the present be added, the sum of these will evidently be the present value of both the lives in succession; and, in the case of the preceding number, will be 6·475 + 11·66 = 18·135 years’ purchase.

106. In No. 103, we proceeded upon the supposition that the annuity on the present life is only payable up to the expiration of the last year it survives; and, consequently, that the succeeding life takes the whole rent for the year in which the present fails.

But, if the succeeding life is only to take a part of that rent, in the same proportion to the whole, as the portion of the year which intervenes between the failure of the present life and the end of the year, is to the whole year, then, by reasoning as in No. 101, it will be found, that the portion of that rent which the succeeding life will receive, may be properly assumed to be one half. And, instead of increasing the number of years’ purchase the annuity on the succeeding life will be worth at the end of the year in which the other fails, by unity, we must only add one half to that number, in order that the present value of the assurance of the sum on the existing life, may be the number of years’ purchase, which all that may be received during the succeeding life, is now worth.

107. The value of the succeeding life, in the case of No. 104, will, upon this hypothesis, be 15·803 × 0·397143 = 6·27605 years’ purchase.

And this appears to be the most correct way of calculating the value of an annuity on a succeeding life; although that of No. 103. proceeds upon the principle on which life interests are generally valued.

108. But the value of two lives in succession, will be the same on both hypotheses. The rent for the year in which the first may fail, being, in the one case, given entirely to the successor; and, in the other, divided equally between the two.

This is also true of any greater number of successive lives.