We have not yet seen any fruits of this application of the calculus which appear to us of much value, nor are we at all sanguine in expecting any.
Although Lambert and Duvillard had made some efforts in this way before, Laplace (in his *Théor. Anal. des Probabilités*, No. 40) was the principal writer who thus treated the subject, and that very shortly, merely touching upon the elements. He arrived in the usual manner at the same formulæ that are given in the elementary algebraic method, and are here demonstrated by common arithmetic; only expressed in the manner of the higher calculus, in terms of the absciss and ordinate of the curve of mortality, both considered as variable quantities.
He judiciously observed that the integral might be obtained in every case by calculating all its terms from a table of mortality, and taking their sum; and that in this manner tables of the values of annuities on single and joint lives might be calculated; which is only reverting to the usual method.
But he also observed that the same thing might be effected by describing a parabolic curve through the vertices of the two extreme and several intermediate ordinates of the curve of mortality, and even that a few of these would be sufficient, since the differences between the different tables of mortality would justify us in considering that method to be equally exact with those tables themselves. And in this we should entirely concur with that profound mathematician, provided we could admit that those tables, or most of them, had equal titles to our confidence, which he appears tacitly to assume.
But here it is that M. Laplace appears to us to have fallen into the same error as most others respecting those tables of mortality, from not having paid sufficient attention to the data they were constructed from, and the manner of their construction.
After what has been advanced in this article, and in that on the *Law of Mortality* in this work, we think it quite unnecessary to say more here, than that we consider it an established truth, that tables of mortality well constructed from proper data, for determining the values of annuities and reversions, do not differ materially from each other.
If imperfect data for constructing a table of mortality be obtained, and any one already constructed, or the mean of several of them, be taken as a pattern or standard, to which it is desired that the new table should approach, it will not be difficult, by the known methods of approximation and interpolation, so to construct such new table that it shall not differ much from the standard; but such new table, being in a certain degree hypothetical, can be of little or no value.
According to the usual methods of treating these subjects, and constructing accurate tables, we never depart from the observations, but are supported by them at every step; our clear and simple methods of reasoning and calculation are much superior to the data we can obtain: proper data are alone wanting to further the science at present; government only can effectually supply them, and all who take any interest in these subjects must be grieved to find that there is little or no hope of assistance from that quarter. Even if a wiser course be adopted in future, 20 years more must elapse before we can reap the benefit of it.
This is not the proper place to enter further into that part of the subject; but to those who take an interest in it, we would recommend the perusal of the minutes of evidence taken before the committee on the population bill, ordered by the House of Commons to be printed 11th May 1830; and the minutes of the committee on the re-committed bill, printed 26th May 1830; especially, in the latter, Mr Milne's letter to Mr Davies Gilbert, the chairman, in answer to an application made to him for his opinion, with Mr Rickman's marginal notes on that letter, and his observations on it in his letter to the chairman, which Mr Milne knew nothing of till the bill was passed.
NOTE REFERRED TO IN TWO PLACES ABOVE.
According to Mr Morgan's statements in the places here referred to, the number of members, or of assurances, or of policies, found to be in the Equitable Assurance Society, was—
| At the end of the year | Reference to Mr Morgan's Statement | |------------------------|----------------------------------| | 1768 | Policies..........................View of Rise and Progress, &c. p. 10. | | 1770 | Policies..........................Ditto........................................ditto. | | 1772 | Members...........................p. 27. | | 1773 | Members...........................Address of 7th March 1793......p. 118.¹ | | 1776 | Policies..........................Ditto of 24th April 1800.......p. 140. | | 1783 | Members...........................Ditto of 7th March 1793........p. 118. | | 1786 | Members...........................Ditto of 24th April 1800.......p. 140. | | 1792 | Assurances.......................View of Rise and Progress, &c. p. 24. | | 1799 | Members...........................Ditto........................................p. 26. |
From which we infer, that if, at the end of each year, beginning with 1770, and ending with 1799, the number had been taken, the sum of all the 30 would have been 75,664, and the mean number during these 30 years 2522.
In a note on p. 443, vol. ii. of Dr Price's *Obs. on Rev. Paym.* (7th edit.), Mr Morgan states, that during 33 years, from January 1768 to January 1801, the number of assurances on single lives had been 83,201; but this great number can only be the sum of the 33 annual numbers as above mentioned, and the mean of these will be 2521.
What we wish to know is, the mean number of lives insured on which policies were in force during the observations; but that Mr Morgan never mentions. As more policies than one are not unfrequently granted for so many distinct assurances on the same life, neither the number of policies nor of assurances will answer our purpose; neither will the number of members, for if a policy be granted to A for insurance on the life of B, A is the member of the society, and not B, who is only the life assured, and several other members besides A may insure the life of B, while A may also hold more policies than one insuring B's life.
As has been already observed, Mr Morgan has repeatedly stated that the rate of mortality in the Equitable Society has always continued the same. And by tables a and b it appears, that out of 91,512 living persons in a similar society above 20 years of age, 1489 would die annually; also, in the table at the end of Mr Morgan's annuities (2d ed.), it is stated, that during the first 20 years of the 19th century, 1923 of the lives assured in the society died above 20 years of age; but 1489 : 1923 :: 91,512 : 115,185, so that this last is the number of lives in a similar society, out of which these 1923 deaths would happen in one year.
¹ These addresses are printed at the end of the deed of settlement of the society, for the use of the members; the copy quoted from was printed in 1801. Then supposing what is probably near the truth, that under 10 years of age the number of policies was the same as that of the lives insured, we shall have, by the statement last mentioned,
| Number of Lives | Number of Annual Deaths | |-----------------|------------------------| | Between 10 and 20 years of age | 1,494 | 7 | | Above 20 | 118,185 | 1923 | | Above 10 years of age | 119,679 | 1930 |
And \( \frac{119,679}{20} = 5984 \) is the mean number above 10.
That is, a population falling short of 6000, instead of exceeding 150,000, as stated by Mr Morgan.
But by the table above mentioned in Mr Morgan's work on annuities, and the explanations of it given above, it appears that by 151,754 policies in force in the society,
\( 119,679 \) lives were insured;
and \( 119,679 : 151,754 :: 7 : 9 \) nearly (more nearly :: 11:14).
So that the number of lives was to that of the policies as 7 to 9 nearly, during the twenty years ending with 1820.
In treating of Annuities, we think that it may be useful in a work of this kind to address ourselves as well to those readers who have not, as to those who have, an acquaintance with Algebra; and we shall accordingly divide what follows into two Parts, corresponding to these two views of the subject.
**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, while they have overlooked some circumstance or event, or some possible combination of events or circumstances, which materially affects 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 (ceteris paribus) be more liable to them.
**I.—On Annuities-Certain.**
No. 1. When the rate is 5 per cent., L1 improved at simple interest during one year will amount to L1·05; 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 \( 1·05 \times 1·05 \), or \( (1·05)^2 = 1·1025 \).
In the same manner it appears that this last amount, improved at interest during the third year, will be increased to \( (1·05)^3 = 1·157625 \); at the end of the fourth year it will be \( (1·05)^4 \); at the end of the fifth \( (1·05)^5 \), 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., L1 improved at compound interest will in seven years amount to \( (1·05)^7 \), and in 21 years to \( (1·05)^{21} \).
But if the rate of interest were only 3 per cent., these amounts would only be \( (1·03)^7 \) and \( (1·03)^{21} \) respectively.
2. The present value of L1 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 L1, in the same ratio as L1 is less than its amount in that time; but in three years at 5 per cent. L1 will amount to \( L(1·05)^3 \). And \( (1·05)^3 : 1 : : 1 : \frac{1}{(1·05)^3} \);
so that \( \frac{1}{(1·05)^3} = \frac{1}{1·157625} = 0·863838 \) is the present value of L1 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 L1 to be received at the end of a year is \( \frac{1}{1·04} = 0·961538 \); and if it were not to be received until the expiration of 21 years, its present value would be \( \frac{1}{(1·04)^{21}} = (0·961538)^{21} = 0·438834 \).
Hence it appears, that if unit be divided by the amount of L1 improved at compound interest during any number of years, the quotient will be the present value of L1 to be received at the expiration of the term; which may also be obtained by raising the number which expresses the present value of L1 receivable at the expiration of a year, to the power of which the exponent is 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 L1, L10, or L100, it is called an annuity of L1, of L10, or of L100.
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 L1, 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 L1.
8. The sum of which the simple interest for one year is L1, is evidently that which, being put out at interest, will just suffice for the payment of L1 at the end of every year, without any augmentation or diminution of the principal, and, being equivalent to the title to L1 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
\[ \frac{5}{1} : \frac{1}{100} = \frac{100}{5} = 20, \]
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
\[ \frac{100}{3} = 33\frac{1}{3}, \quad \text{or} \quad \frac{100}{6} = 16\frac{2}{3} \text{ years' purchase}; \]
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
| Table IV | Table III | Table I | Table II | |----------|-----------|---------|---------| | Amount of L1 per annum improved at Interest until | Amount of L1 | Present value of L1 to be received at | Present value of L1 per annum to be received until | | Term | the Expiration of the Term | | 1 yr. | 1-000000 | 1-050000 | 0-952381 | 0-952381 | | 2 yrs. | 2-050000 | 1-102500 | 0-907029 | 1-839410 | | 3 | 3-152500 | 1-157625 | 0-863888 | 2-723248 | | 4 | 4-310125 | 1-215506 | 0-822702 | 3-545950 | | 5 | 5-525631 | 1-276282 | 0-783526 | 4-329476 | | 6 | 6-801913 | 1-340096 | 0-746215 | 5-075691 | | 7 | 8-142009 | 1-407100 | 0-70681 | 5-786372 |
10. The calculation must begin with Table III, the first number in which should evidently be 1-05, the amount of L1 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 1-057625, 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 L1 in a year. (1.)
11. The number against any year in Table I. is found by dividing unit by the number against the same year in Table III. (2); thus, the number against the term of six years in Table I. is
\[ \frac{1}{1-340096} = 0-746215. \]
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 four years in Table II. being
\[ \text{the sum of } 0-822702 \] \[ \text{and } 2-723248 \]
is 3-545950.
13. If each payment of an annuity of L1 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 L1, having received no improvement at interest. That received one year before will be augmented to the amount of L1 in a year; that received two years before will be augmented to the amount of L1 in two years; that received three years before to the amount of L1 in three years; and so on until the first payment, which will be augmented to the amount of L1 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 unit 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
\[ \text{the sum of } 1-340096 \] \[ \text{and } 6-801913 \]
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 L1.
So that to find the amount or the present value of any sum or annuity for a given term and rate of interest, we have only to multiply the corresponding tabular value by
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.2191123; therefore L.100 would amount to 100 times as much, that is, to L.2191123, 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.0505068: L.400 to be received at the same time will therefore be worth, in present money, 400 times as much, or L.2020272, that is, L.202. 0s. 2½d.
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.128212; 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.56084938, as appears by Table IV.; the amount required will therefore be 10.5 times this, or L.58889185, 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 is 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 33½ 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. 3½d., 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 II. it appears, that the present value of an annuity for 21 years, is 12·8212 for 7 years, 5·7864 years' purchase.
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. 18s. 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 granter 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 long, 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 nine 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 five years of age, and the remaining 6797 completed their fifth year. Also, that out of 6797 children who attained to five years of age, 6460 survived their tenth year.
But the mortality under ten 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 to be at Carlisle during the period of these observations, except among children under ten 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 \(\frac{6047}{6460}\) 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 eleven years; and for each life the \( \frac{6047}{6460} \)th part of that, is, the \( \frac{6047}{6460} \)th 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 \( \frac{6047 \times 0.584679}{6460} = L.0.5473 \).
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 \times 5075 to 4727 \times 4073.
So that L.1 to be paid at the expiration of 15 years for each of these 5642 \times 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 \( \frac{L.4727 \times 4073}{5642 \times 5075} \) 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 \( \frac{4727 \times 4073 \times L.0.481017}{5642 \times 5075} \). 37. Hence it is sufficiently evident how the present value of L1 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 L1 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 15 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 L1 to be received upon these three jointly surviving the term of 15 years from this time, will be \( \frac{4727 \times 4073 \times 1675}{181} \times L_0 = 481017 \); 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 L1 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 L1 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, reckon interest at 5 per cent., the present value of L1 to be received at the expiration of a year, provided that a life, now 89 years of age, survived till then, is \( \frac{142 \times 0.952381}{181} \). But the age of that life will then be 90 years, and the proprietor of an annuity of L1 now depending upon it will, in that event, receive his annual payment of L1 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 L3-339, to be received then, if the life be still subsisting, or \( \frac{142 \times 0.952381}{181} \times L_3 = 2-495 \); which, therefore, will be the present value of an annuity of L1 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 unit 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 L1 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 L1 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 L1 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.
| Age of Life | Value of an Annuity on that Life increased by Unit | 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 | \( \frac{1}{2} \) | 0-317 | 103 | | 103 | 1-317 | \( \frac{2}{3} \) | 0-753 | 102 | | 102 | 1-753 | \( \frac{4}{5} \) | 1-192 | 101 | | 101 | 2-192 | \( \frac{7}{8} \) | 1-624 | 100 | | 100 | 2-624 | \( \frac{11}{12} \) | 2-045 | 99 | | 99 | 3-045 | \( \frac{14}{15} \) | 2-478 | 98 | | 98 | 3-278 | \( \frac{17}{18} \) | 2-928 | 97 | | 97 | 3-428 | \( \frac{20}{21} \) | 3-378 | 96 | | 96 | 3-555 | \( \frac{23}{24} \) | 3-826 | 95 | | 95 | 3-596 | \( \frac{25}{26} \) | 4-274 | 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 L1 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 \( \frac{142 \times 9 \times L_0}{181 \times 11} \); therefore, if the present value of an annuity of L1 on the joint continuance of two lives, now aged 90 and 100 years respectively, be L0-950, by reasoning as in No. 41, it will be found that the present value of an annuity on the joint continuance of two lives,
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 Unit, | 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 | 1×40 | 0·235 | 93 & 103 | | | | 3×54 | | | | 93 & 103 | 1·235 | 3×54 | 0·508 | 92 & 102 | | | | 5×75 | | | | 92 & 102 | 1·508 | 5×75 | 0·733 | 91 & 101 | | | | 7×105 | | | | 91 & 101 | 1·733 | 7×105 | 0·950 | 90 & 100 | | | | 9×142 | | | | 90 & 100 | 1·950 | 9×142 | 1·192 | 89 & 99 | | | | 11×181 | | | | 89 & 99 | 2·192 | 11×181 | 1·280 | 88 & 98 | | | | 14×232 | | |
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. Dem. 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. Dem. 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 Popular annuities are those which are paid during the life or lives of the insured, and the present value of such annuities can be calculated using the following formula:
\[ \text{Present Value} = \frac{\text{Annual Payment}}{\text{Interest Rate}} \times \left(1 - \frac{1}{(1 + \text{Interest Rate})^n}\right) \]
where \( n \) is the number of years.
56. Dem. 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 L1 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, is 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 wherein 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. Dem. 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 the reversion after the failure of such life or lives; and the present value of that reversion would manifestly be as stated above.
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. Dem. 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 22nd 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 from which the Carlisle table of mortality was constructed, 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:
\[ \begin{align*} \text{Value of a single life of } & (40|13-390) \\ & (50|11-660) \quad (\text{by Table VI.}) \\ \sum & 25-050 \\ \text{Subtract the value of their joint lives,} & 9-984 \quad (\text{Table VIII.}) \\ \text{remains} & 15-066 \text{ years' purchase, the required value.} \end{align*} \]
And if the annuity be L200, its present value will be L3013-2, or L3013. 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 | | 55 & 60 | 7-106 | | 50 & 60 | 7-601 |
\[ \begin{align*} \sum & 23-235 \\ \text{Subtract } 2 \times 6-289 & = 12-578 \\ \text{remains} & 10-657 \text{ years' purchase, the required value.} \end{align*} \]
Therefore, if the annuity were L300, it would be worth L3197. 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 | | 55 | 10-347 | | 60 | 8-940 | | 50, 55, & 60 | 6-289 |
\[ \begin{align*} \sum & 37-236 \\ \text{Subtract the sum of the values of annuities on the three combinations of two lives,} & 23-235 \quad (\text{No. 67.}) \\ \text{remains} & 14-001 \text{ years' purchase, the required value.} \end{align*} \]
And if the annuity were L300, it would now be worth L4200. 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 time, 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
\[ \frac{4073}{4727} \times 0.613918 = 5.473 \text{ years' purchase; so that if the annuity were L.200, its present value would be L.1094.12s.} \]
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,
\[ \frac{5473}{7175} = 0.76279. \]
And since the annual payment for the deferred annuity of L.1 per annum would be L.076279, 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 life of 40,
\[ \frac{13390}{9984} = 1.3441146; \]
for an annuity of L.1 should 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 unit; 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
\[ \frac{3406}{9984} = L.0310087; \]
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,
\[ \frac{11660}{834} = 13.984; \]
remains 8.34 years' purchase.
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 bound 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 unit 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 unit, 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
\[ \frac{20-11.66}{20+1} = \frac{8.34}{21} = L.0397143. \]
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 commence- ment 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 unit, the quotient will be the equivalent annual premium for the same assurance.
85. Ex. 2. Required the annual premium for the assurance of L1 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 \( \frac{0·397143}{12·66} = 0·313699 \) for the assurance of L1; and for the assurance of L1000 it will be L31. 7s. 5d.
86. Ex. 3. Let both the single payment in present money, and the equivalent annual premium, be required for the assurance of L1, on the joint continuance of two lives of the respective ages of 45 and 50 years.
The value of an annuity of L1 on the joint continuance of these two lives appears by Table VII. to be L9·737, therefore \( \frac{20 - 9·737}{20 + 1} = \frac{10·263}{21} = L0·488714 \) is the single premium, and \( \frac{0·488714}{10·737} = L0·0455168 \) the equivalent annual one, for the assurance of L1 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 L500, the total present value of the assurance would be L244. 7s. 2d., and the equivalent annual premium L22. 15s. 2d.
87. Ex. 4. Let both the single and the equivalent annual premium be required for the assurance of L1 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 \( \frac{20 - 15·066}{20 + 1} = \frac{4·934}{21} = L0·234952 \); and, by No. 84, the annual one will be \( \frac{L0·234952}{16·066} = L0·0146242 \).
That is, for the assurance of L1 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 L5000, the single premium will be L1174. 15s. 2d., the equivalent annual one L73. 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 \( \frac{20 - 14·001}{20 + 1} = \frac{5·999}{21} = L0·285666 \), and the annual \( \frac{L0·285666}{15·001} = L0·0190431 \), for the assurance of L1 to be received at the end of the year in which the last surviving life of the three may fail.
For the assurance of L2000, the single premium would therefore be L571 6s. 8d., the annual one L38. 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 unit 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 unit; 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 L1 to the present value of L1 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 unit 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 L1 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 happen before the expiration of ten years.
Here the present value of L.1, to be received on 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 life of 45 enters upon, is worth 7-646 years' purchase; therefore, \( \frac{7-646}{0-10693} = L.0-013985 \) 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 whole 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 unit 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 16-066 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 which 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 11-66 × L.20 = L.233.4s.; 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 have 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 \( \frac{(L.0-397143)}{2} = L.0-199 \); 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 unit 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 L1 on a life of 25 is L15308; and in No. 82 it has been shown, that the present value of the assurance of L1 on a life of 50 is L0397143. Hence, and by the last number, it appears that if the annual produce of the living were but L1, the present value of the next presentation would be L16303 × 0397143 = L647467. The required value, therefore, is L3237. 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 6475 + 1166 = 18135 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 unit, 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 15803 × 0397143 = 627605 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.
PART II.
109. We now proceed to treat the subject of annuities Algebraically.
I.—On Annuities-Certain.
Let \( r \) denote the simple interest of L1 for one year.
\( s \), any sum put out at interest.
\( n \), the number of years for which it is lent.
\( m \), its amount in that time.
\( a \), an annuity for the same time. (3 and 4.)
\( m \), the amount to which that annuity will increase when each payment is laid up as it becomes due, and improved at compound interest until the end of the term.
\( v \), the present value of the same annuity. (6.)
110. Then, reasoning as in the first number of this article, it will be found that \( m = s(1+r)^n \). And by No. 2 it appears that the present value of \( s \) pounds to be received certainly at the expiration of \( n \) years, is \( \frac{1}{(1+r)^n} \) or \( s(1+r)^{-n} \).
111. The amount of L1 in \( n \) years being \( (1+r)^n - 1 \); and when it is considered that this increase arises entirely from the simple interest (\( r \)) of L1 being laid up at the end of each year, and improved at compound interest during the remainder of the term, it must be obvious that \( (1+r)^n - 1 \) is the amount of an annuity of \( r \) pounds in that time; but \( r : a :: (1+r)^n - 1 : \frac{a}{r}[(1+r)^n - 1] \), which, therefore, is equal to \( m \), the amount of an annuity of \( a \) pounds in \( n \) years.
112. Reasoning as in No. 8, it will be found, that since \( r : 1 :: a : \frac{a}{r} \), the present value of a perpetual annuity of \( a \) pounds is \( \frac{a}{r} \).
113. If two persons, \( A \) and \( B \), purchase a perpetuity of \( a \) pounds between them, which \( A \) and his heirs or assigns are to enjoy during the first \( n \) years, and \( B \) and his heirs or assigns for ever after. Since the value of the perpetuity to be entered upon immediately has just been shown to be \( \frac{a}{r} \), the present value of \( B \)'s share, that is, the present value of the same perpetuity when the entrance thereon is deferred until the expiration \( n \) years, will be \( \frac{a}{r}(r+1)^{-n} \), (110); and the value of the share of \( A \) will be thus much less than that of the whole perpetuity (21), therefore equal to \( \frac{a}{r}[1-(1+r)^{-n}] = v \), the present value of an annuity of \( a \) pounds for the term of \( n \) years certain.
114. If the annuity is not to be entered upon until the expiration of \( d \) years, but is then to continue \( n \) years, its value at the time of entering upon it will be \( \frac{a}{r}[1-(1+r)^{-n}] \), as has just been shown; therefore its present value will be \( \frac{a}{r}(1+r)^{-d}-(1+r)^{-(d+n)} = v \). (110.)
115. In the same manner it appears that, when the entrance on a perpetuity of \( a \) pounds is deferred \( d \) years, its present value will be \( \frac{a}{r}(1+r)^{-d} \). (110 and 112.)
116. \( q \) being any number whatever, whole, fractional, or mixed, let \( \lambda q \) denote its logarithm, and \( xq \) the arithmetical complement of that logarithm; so that these equations may obtain, \( \frac{1}{q} = -\lambda q = xq \). Then, for the resolution of the principal questions of this kind by logarithms, we shall have these formulae.
1. Amount of a sum improved at interest.
\[ \lambda m = \lambda \lambda (1+r) + \lambda s. \] (110.)
2. Amount of an annuity when each payment is laid up as it becomes due, and improved at interest until the expiration of the term. 3. Value of a lease or an annuity.
\[ \lambda v = \lambda \left[ 1 - (1 + r)^{-n} \right] + \lambda x + xr. \quad (113.) \]
4. Value of a deferred annuity, or the renewal of any number of years lapsed in the term of a lease.
\[ \lambda v = \lambda \left[ (1 + r)^{-d} - (1 + r)^{-(d+n)} \right] + \lambda x + xr. \quad (114.) \]
5. Value of a deferred perpetuity, or the reversion of an estate in fee-simple, after an assigned term.
\[ \lambda v = \lambda x + dx(1 + r). \quad (115.) \]
By means of each of these equations, it is manifest that any one of the quantities involved in it may be found when the rest are given.
117. If the interest be convertible into principal \( s \) times in the year, at \( s \) equal intervals; since the interest of L.1 for one of these intervals will be \( \frac{r}{s} \) (109), and the number of conversions of interest into principal in \( n \) years \( m \); to adapt the formula in No. 110 to this case, we have only to substitute \( \frac{r}{s} \) for \( r \), and \( m \) for \( n \), in the equation
\[ m = s(1 + r)^m \]
there given, whereby it will be transformed to this,
\[ m = s \left( 1 + \frac{r}{s} \right)^m. \]
118. According as \( s \) is equal to 1, 2, 4, or is infinite, that is, according as the interest is convertible into principal yearly, half-yearly, quarterly, or continually, let \( m \) be equal to \( y, n, q, \) or \( c; \) so shall
\[ y = s(1 + r)^n, \] \[ h = s \left( 1 + \frac{r}{2} \right)^n, \] \[ q = s \left( 1 + \frac{r}{4} \right)^n, \] and \( c = s \cdot n; \)
\( n \) being the number of which \( nr \) is the hyperbolic logarithm, and \( nr \times 0.43429448 \) its logarithm in Briggs' System, and the common tables.
119. From No. 117 and 110, it follows that the present value of \( s \) pounds to be received at the end of \( n \) years, when the interest is convertible into principal at \( s \) equal intervals in each year, is \( s \left( 1 + \frac{r}{s} \right)^{-n}. \)
120. When the present values and the amounts of annuities are desired, let the interest be convertible into principal at \( s \) equal intervals in the year, while the annuity is payable at \( s \) intervals therein, the amount of each payment being \( \frac{a}{s}. \)
121. Case 1. \( \mu \) being any whole number not greater than \( s \), let \( \frac{1}{\pi} = \frac{\mu}{s} \), so that the interest may be convertible into principal \( \mu \) times in each of the intervals between the payments of the annuity.
Then will the amount of L.1 at the expiration of the period \( \frac{1}{\pi} \) be \( \left( 1 + \frac{r}{s} \right)^{\mu} \) (117), and the interest of L.1 for the same time will be \( \left( 1 + \frac{r}{s} \right)^{\mu} - 1 \); whence the present value of the perpetuity will be \( \frac{a}{\pi} \left( 1 + \frac{r}{s} \right)^{\mu} - 1 \) (8), and the value of the same deferred \( n \) years, will be \( \frac{a}{\pi} \left( 1 + \frac{r}{s} \right)^{\mu} - 1 \) (119), therefore the present value of the annuity to be entered upon immediately, and to continue \( n \) years, will be
\[ \frac{a}{\pi} \left( 1 + \frac{r}{s} \right)^{\mu} - 1 = v. \]
122. Case 2. \( \mu \) being any whole number greater than \( s \), let \( \frac{1}{\pi} = \frac{\mu}{s} \) so that the annuity may be payable \( \mu \) times in each of the intervals between the payments of interest, or the conversion thereof into principal.
Then, at the expiration of the \( \frac{1}{\pi} \)th of a year, when the interest on the purchase-money is first payable or convertible, the interest on all the \( \mu - 1 \) payments of the annuity previously made will be
\[ \frac{a}{\pi} \left[ (\mu - 1) + (\mu - 2) + (\mu - 3) + \ldots + 3 + 2 + 1 \right] = \frac{a}{\pi} \left[ \frac{\mu(\mu - 1)}{2} \right]; \]
to which adding the \( \mu \) payments of \( \frac{a}{\pi} \) each (including the one only then due), the sum
\[ \frac{a}{\pi} \left[ \frac{\mu(\mu - 1)}{2} \right], \]
is the simple interest which the value of the perpetuity should yield at the expiration of each \( \frac{1}{\pi} \)th part of a year, in order to supply the deficiency (both of principal and interest) that would be occasioned during each of those periods, in any fund out of which the several payments of the annuity might be taken, as they respectively became due; and since
\[ \frac{a}{\pi} \left[ \frac{\mu(\mu - 1)}{2} \right] = a \left( 1 + \frac{\mu - 1}{2\pi} \right), \]
this last expression will be the value of such perpetuity with immediate possession (8); the value of the same deferred \( n \) years will therefore be \( a \left( 1 + \frac{\mu - 1}{2\pi} \right) \times \left( 1 + \frac{r}{s} \right)^{-n} \) (119).
Whence it appears that the present value of the annuity to be entered upon immediately, and to continue \( n \) years, will be \( a \left( 1 + \frac{\mu - 1}{2\pi} \right) \left[ 1 - \left( 1 + \frac{r}{s} \right)^{-n} \right] = v. \)
123. Case 3. When, in consequence of the annuity being always payable at the same time that the interest is convertible, \( s = \pi. \)
Since the interest of L.1 at the expiration of the period \( \frac{1}{\pi} \) will be \( \frac{r}{s} \) the value of the perpetuity will be
\[ \frac{1}{\pi} \left( 1 + \frac{r}{s} \right)^{\mu} = a \left( 1 + \frac{r}{s} \right)^{\mu} - 1 \] (8),
whence, proceeding as before, we obtain the present value of the annuity,
\[ \frac{a}{r} \left[ 1 - \left( 1 + \frac{r}{s} \right)^{-n} \right] = v. \]
Whence \( s = \pi \), and consequently \( \mu = 1 \), the values of \( v \) given in the two preceding cases, will be found to coincide with this.
124. According as \( s \) and \( \pi \) are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let \( v \) be equal to \( y, h, q, \) or \( c, \) then will
\[ y = \frac{a}{r} \left[ 1 - \left( 1 + \frac{r}{s} \right)^{-n} \right]. \] 125. The amount of an annuity is equal to the sum to which the purchase-money would amount if it were put out and improved at interest during the whole term.
For, from the time of the purchase of the annuity, whatever part of the money that was paid for it may be in the hands of the granter, he must improve thus to provide for the payments thereof; and if the purchaser also improve in the same manner all he receives, the original purchase-money must evidently receive the same improvement during the term, as if it had been laid up at interest at its commencement.
126. The periods of conversion of interest into principal, and of the payment of the annuity being still designated as in No. 120; since in \( n \) years the number of periods of conversion will be \( m \), in the
1st Case, where the interest is convertible \( p \) times in each of the intervals between the payments of the annuity, we have
\[ (1 + r)^m v = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{(1 + r)^p - 1} \right] = M. \]
(117, 121, and 125.)
In the 2d Case, when the annuity is payable \( p \) times, in each interval between the conversions of interest,
\[ (1 + r)^m v = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{(1 + r)^p - 1} \right] = M. \]
(117, 122, and 125.)
And, in the 3d Case, when the annuity is always payable at the same time that the interest is convertible,
\[ (1 + r)^m v = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{(1 + r)^p - 1} \right] = M. \]
(117, 123, and 125.)
127. According as \( r \) and \( s \) are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let \( m \) be denoted by \( y', h', q' \), or \( c' \);
then will
\[ y' = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{(1 + r)^p - 1} \right], \]
\[ h' = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{(1 + r)^p - 1} \right], \]
\[ q' = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{(1 + r)^p - 1} \right], \]
and \( c = \frac{a}{r} (n - 1); n \) being as in No. 118.
128. Example 1. What will L320 amount to, when improved at compound interest during 40 years, the rate of interest being 4 per cent. per annum?
By the first formula in No. 116, the operation will be as follows:
\[ 1 \times r = 1.04 \times 0.01703334 + n = 40 \]
\[ (1 + r)^n = 0.6813336 \]
\[ s = 320 \times 2.5051500 \]
\[ m = 1536327 \times 3.1864836 \]
And the answer is L1536, 6s. 6½d.
129. Ex. 2. If the interest were convertible into principal every half-year, the operation, according to No. 117, would be thus:
\[ 1 + r = 1.02 \times 0.00660017 \]
\[ \times m = 80 \]
\[ 0.6880136 \]
\[ s = 320 \times 2.5051500 \]
\[ m = 1560-14 \times 3.1931636 \]
So that in this case the amount would be L1560, 2s. 9½d.
130. Ex. 3. Required the present value of an annuity of L250 for 30 years, reckoning interest at 5 per cent.
By the third formula in No. 116, the operation will be thus:
\[ \lambda (1 + r)^{-1} = x \times 1.05 = 1.9788107 \]
\[ \times n = 30 \]
\[ (1 + r)^{-n} = 2.3137704 \times 1.9788107 \]
\[ 1 - (1 + r)^{-n} = 7.6862296 \times 1.9788107 \]
\[ a = 250 \times 2.3979400 \]
\[ r = .05 \times 1.3010300 \]
\[ v = 3843.114 \times 3.5846833 \]
And the required value is L3843, 2s. 3½d.
131. Ex. 4. The rest being still the same, if the annuity in the last example be payable half-yearly, in the formula of No. 122 \( r \) will be equal to 1, \( s = 2 \), and \( p = 2 \); that formula will therefore become \( a \left( \frac{1}{r} + \frac{1}{4} \right) \left[ \frac{1 - (1 + r)^{-n}}{1 - (1 + r)^{-p}} \right] = v \); and the operation will be thus:
\[ 1 - (1 + r)^{-n} = 1.8857133 \]
\[ a = 250 \times 2.3979400 \]
\[ \frac{1}{r} + \frac{1}{4} = 20.25 \times 1.3064250 \]
\[ v = 3891.15 \times 3.5900783 \]
The value of the annuity will therefore in this case be L3891, 3s.
132. Ex. 5. To what sum will an annuity of L120 for 20 years amount, when each payment is improved at compound interest from the time of its becoming due until the expiration of the term, the rate of interest being 6 per cent.?
The operation by the second formula in No. 116 is thus:
\[ 1 + r = 1.06 \times 0.025305865 \]
\[ \times n = 20 \]
\[ (1 + r)^n = 3.207135 \times 0.5061173 \]
\[ (1 + r)^n - 1 = 2.207135 \times 0.3438289 \]
\[ a = 120 \times 2.0791812 \]
\[ r = .06 \times 1.2218487 \]
\[ m = 4414.27 \times 3.6448588 \]
And the amount required is L4414, 5s. 5½d.
133. Ex. 6. The rest being the same as in the last example; if both the interest and the annuity be payable half-yearly, the amount will be determined by the second of the formulae given in No. 127, which in this case will become \( \frac{120}{.06} \left[ (1.03)^{40} - 1 \right] \), and the operation will be as follows:
\[ 1.03 \times 0.01283723 \times 40 \]
\[ (1.03)^{40} = 3.26204 \times 0.5134892 \]
\[ (1.03)^{40} - 1 = 2.26204 \times 0.3545003 \]
\[ a = 2.0791812 \]
\[ r = .06 \times 1.2218487 \]
\[ m = 4524.08 \times 3.6555302 \]
So that the amount in this case would be L4524, 1s. 7½d. II.—On the Probabilities of Life.
134. Any persons, A, B, C, &c., being proposed, let the numbers which tables of mortality (32) adapted to them represent to attain to their respective ages be denoted by the symbols \(a\), \(b\), \(c\), &c.; while lives \(n\) years older than those respectively are denoted thus, \(A\), \(B\), \(C\), &c., and the numbers that attain to their ages, by the symbols \(a\), \(b\), \(c\), &c.; also let lives \(n\) years younger than \(A\), \(B\), \(C\), &c., be denoted thus, \(A_n\), \(B_n\), \(C_n\), &c., while the numbers which the tables show to attain to those younger ages are designated by the symbols \(a_n\), \(b_n\), \(c_n\), &c.
Then, if \(A\) be 21 years of age, and we use the Carlisle table, we shall have \(a = 6047\), and \(1a = 5362\), the number that attain to the age of 35, or that live to be 14 years older than \(A\).
Hence the number that are represented by the table to die in \(n\) years from the age of \(A\) will be \(a - na\); that is, in 14 years, \(a - 1a\); and, by the Carlisle table, in 14 years from the age of 21, that is, between 21 and 35, it will be \(6047 - 5362 = 685\).
135. Problem. To determine the probability that a proposed life \(A\) will survive \(n\) years.
Solution. \(a\) being the number of lives in the table of mortality attaining the age of that which is proposed, conceive that number of lives to be so selected (of which \(A\) must be one), that they may each have the same prospect with regard to longevity as the proposed life and those in the table, or the average of those from which it was constructed; then will the number of them that survive the term be \(na\).
So that the number of ways all equally probable, or the number of equal chances for the happening of the event in question, is \(na\); and the whole number for its either happening or failing is \(a\); therefore, according to the first principles of the doctrine of probabilities, the probability of the event happening, that is, of \(A\) surviving the term, is \(\frac{na}{a}\).
If the age of \(A\) be 14, the probability of that life surviving 7 years, or the age of 21, will, according to the Carlisle table, be \(\frac{7a}{a} = \frac{6047}{6335} = 0.95454\).
136. Since the number that die in \(n\) years from the age of \(A\) is \(a - na\) (134), it appears, in the same manner, that the probability of that life failing in \(n\) years will be \(\frac{a - na}{a} = 1 - \frac{na}{a}\); which probability, when the life, term, and table of mortality are the same as in the last No., will be 0.04546.
137. If two lives \(A\) and \(B\) be proposed, since the probability of \(A\) surviving \(n\) years will be \(\frac{na}{a}\), and that of \(B\) surviving the same term will be \(\frac{nb}{b}\), it appears from the doctrine of probabilities that \(\frac{na}{a} \cdot \frac{nb}{b}\) or \(\frac{n(ab)}{ab}\) will be the measure of the probability that these lives will both survive \(n\) years.
In the same manner it may be shown, that the probability of the three lives \(A\), \(B\), and \(C\) all surviving \(n\) years, will be measured by \(\frac{na}{a} \cdot \frac{nb}{b} \cdot \frac{nc}{c}\) or \(\frac{n(abc)}{abc}\). And, universally, that any number of lives \(A\), \(B\), \(C\), &c., will jointly survive \(n\) years, the probability is \(\frac{n(abc)}{abc}\), &c.
138. Let \(\frac{na}{a} = a\), \(\frac{nb}{b} = b\), \(\frac{nc}{c} = c\), &c.; also let \(\frac{n(abc)}{abc} = n(abc)\), &c.; so that the probabilities of \(A\), \(B\), \(C\), &c., surviving \(n\) years may be denoted by \(a\), \(b\), \(c\), &c. respectively; and that of all these lives jointly surviving that term by \(n(abc)\), &c.
Then will the probability that none of these lives will survive \(n\) years be \((1 - a) \cdot (1 - b) \cdot (1 - c)\), &c.
139. But the probability that some one or more of these lives will survive \(n\) years will be just what the probability last mentioned is deficient of certainty: its measure, therefore, being just what the measure of that probability is deficient of unit, will be
\[1 - (1 - a) \cdot (1 - b) \cdot (1 - c) \cdot \ldots\]
140. Corol. 1. When there is only one life \(A\), this will be \(na\).
141. Corol. 2. When there are two lives \(A\) and \(B\), it becomes \(na + nb - n(ab)\).
142. Corol. 3. When there are three lives \(A\), \(B\), and \(C\), it becomes \(na + nb + nc - n(ab) - n(ac) - n(bc) + n(abc)\).
143. When three lives \(A\), \(B\), and \(C\) are proposed, that at the expiration of \(n\) years there will be living dead the probability is
| Lives | Probability | |-------|-------------| | ABC | \(n(abc)\) | | AB | \(n(ab)\) | | AC | \(n(ac)\) | | BC | \(n(bc)\) | | A | \(n(a)\) |
And the sum of these four \(n(ab)\) + \(n(ac)\) + \(n(bc)\) - \(2n(abc)\), is the measure of the probability that some two at the least out of these three lives will survive the term.
III.—Of Annuities on Lives.
144. Let the number of years' purchase that an annuity on the life of \(A\) is worth, that is, the present value of L.1 to be received at the end of every year during the continuance of that life, be denoted by \(A\); while the present value of an annuity on any number of joint lives \(A\), \(B\), \(C\), &c., that is, of an annuity which is to continue during the joint existence of all the lives, but to cease with the first that fails, is denoted by \(ABC\), &c.
Then will the value of an annuity on the joint continuance of the three lives \(A\), \(B\), and \(C\), be denoted by \(ABC\).
And on the joint continuance of the two \(A\) and \(B\), by \(AB\).
145. Also let \(tA\) and \(tA\) denote the values of annuities on lives respectively older and younger than \(A\), by \(t\) years; while \(t(ABC)\), &c., designates the value of an annuity on the joint continuance of lives \(t\) years older than \(A\), \(B\), \(C\), &c., respectively; and \((ABC)\), &c., that of an annuity on the same number of joint lives, as many years younger than these respectively.
146. Let \(\frac{1}{1+r}\), the present value of L.1 to be received certainly at the expiration of a year, be denoted by \(v\).
Then will \(v^n\) be the present value of that sum certain to be received at the expiration of \(n\) years.
But if its receipt at the end of that time be dependent upon an assigned life \(A\) surviving the term, its present value will, by that condition, be reduced in the ratio of certainty to the probability of \(A\) surviving the term, that is, in the ratio of unit to \(na\), and will therefore be \(ava^n\).
In the same manner it appears, that if the receipt of the money at the expiration of the term be dependent upon any assigned lives, as \(A\), \(B\), \(C\), &c., jointly surviving that period, its present value will be \(n(abc)\), &c.,\(v^n\). 147. Let us denote the sum of any series, as \( (abc)v + (abc)^2v^2 + \ldots + (abc)^nv^n + \ldots \), by prefixing the capital \( S \) to the general term thereof. Then, from what has just been advanced, it will be evident that \( ABC, \ldots = S_n(abc, \ldots)c^n \).
When there are but three lives, \( A, B, \) and \( C \), this becomes \( ABC = S_n(abc)c^n \).
When there are but two, \( A \) and \( B \), it becomes \( AB = S_n(ab)c^n \).
And in the same manner it appears, that for a single life \( A \), \( A = S_nac^n \).
148. \( n(abc, \ldots)c^n = \frac{n(abc, \ldots)c^n}{abc, \ldots} \) (138), where the denominator \( (abc, \ldots) \) is constant, while the numerator varies with the variable exponent \( n \). And the most obvious method of finding the value of an annuity on any assigned single or joint lives, is to calculate the numerical value of the term \( n(abc, \ldots)c^n \) for each value of \( n \), and then to divide the sum of all these values by \( abc, \ldots \); for \( \frac{S_n(abc, \ldots)c^n}{abc, \ldots} = ABC, \ldots \).
In calculating a table of the values of annuities on lives in that manner, for every combination of joint lives it would be necessary to calculate the term \( n(abc, \ldots)c^n \) for as many years as there might be between the age of the oldest life involved and the oldest in the table; and the same number of the terms \( ac^n \) for any single life of the same age.
But this labour may be greatly abridged as follows:
**Prob. I.**
149. Given \( (ABC, \ldots) \), the value of an annuity on any number of joint lives, to determine \( ABC, \ldots \) that of an annuity on the same number of joint lives respectively one year younger than they.
**Solution.**
If it were certain that the lives \( ABC, \ldots \) would all survive one year, the proprietor of an annuity of \( Ll \) dependent upon their joint continuance would, at the expiration of a year, be in possession of \( Ll \) (the first year's rent), and an annuity on the same number of lives one year older respectively than \( ABC, \ldots \); therefore, in that case, the required present value of the annuity would be \( v[1 + (ABC, \ldots)] \) (146).
But the probability of the lives \( A, B, C, \ldots \) jointly surviving one year is less than certainty in the ratio of \( (abc, \ldots) \) to unit; therefore \( ABC, \ldots = (abc, \ldots)c^n[1 + (abc, \ldots)] \).
150. **Corol. 1.** When there are but three lives, \( A, B, \) and \( C \), this becomes \( ABC = (abc)v[1 + (abc)] \).
151. **Corol. 2.** When there are only two, \( A \) and \( B \), \( AB = (ab)v[1 + (ab)] \).
152. **Corol. 3.** And for a single life \( A \), it appears, in the same manner, that \( A = av[1 + a] \).
153. Hence, in logarithms, we have these equations,
\[ \begin{align*} \lambda_A &= \lambda_v + \lambda_a + \lambda[1 + \lambda'] \\ \lambda_{AB} &= \lambda_v + \lambda_a + \lambda_b + \lambda[1 + \lambda'(ab)] \\ \lambda_{ABC} &= \lambda_v + \lambda_a + \lambda_b + \lambda_c + \lambda[1 + \lambda'(abc)] \\ &\vdots \end{align*} \]
Upon which it may be observed, that \( \lambda_v + \lambda_a \), the sum of the first two logarithms that are employed in determining \( \lambda \) from \( \lambda' \), also enters the operation whereby \( AB \) is determined from \( (ab) \). And that \( \lambda_v + \lambda_a + \lambda_b \), the sum of the first three logarithms that serve to determine \( AB \) from \( (ab) \), is also required to determine \( ABC \) from \( (abc) \); which observation may be extended in a similar manner to any greater number of joint lives.
154. By these means it is easy to complete a table of the values of annuities on single lives of all ages, beginning with the oldest in the table, and proceeding regularly by age to the youngest.
Also a table of the values of any number of joint lives, the lives in each succeeding combination, in any one series of operations (according to the retrograde order of the ages in which they are computed), being one year younger respectively than those in the preceding combination.
And, if a table of single lives be computed first, then of two joint lives, next of three joint lives, and so on, the calculations made for the preceding tables will be of great use for the succeeding.
155. Having shown how to compute tables of the values of annuities on single and joint lives, we shall, in what follows, always suppose those values to be given.
156. Let the value of an annuity on the joint continuance of any number of lives, \( A, B, C, \ldots \), that is not to be entered upon until the expiration of \( t \) years, be denoted by \( -t(ABC, \ldots) \).
Then, if it were certain that all the lives would survive the term, since the value of the annuity at the expiration of the term would be \( t(ABC, \ldots) \) (145), its present value would be \( v^t(ABC, \ldots) \) (146).
But the measure of the probability that all the lives will survive the time is \( (abc, \ldots) \), therefore \( -t(ABC, \ldots) = (abc, \ldots)v^t \cdot t(ABC, \ldots) \).
In the same manner, it appears, that for a single life \( A \),
\[ -tA = av^t \cdot tA. \]
157. Let an annuity for the term of \( t \) years only, dependent upon the joint continuance of any number of lives, \( A, B, C, \ldots \) be denoted by \( -t(ABC, \ldots) \); and, since this temporary annuity, together with an annuity on the joint continuance of the same lives deferred for the same term, will evidently be of the same value as an annuity to be entered upon immediately, and enjoyed during their whole joint continuance, we have \( -t(ABC, \ldots) + -t(ABC, \ldots) = ABC, \ldots \); whence, \( -t(ABC, \ldots) = ABC, \ldots - -t(ABC, \ldots) \).
And for a single life \( A \),
\[ -tA = A - -tA. \]
**Prob. II.**
158. To determine the present value of an annuity on the survivor of the two lives \( A \) and \( B \) (155); which we designate thus, \( AB \).
**Solution.**
The probability that the survivor of these two lives will outlive the term of \( n \) years was shown in No. 141 to be \( na + nb - n(ab) \); therefore, reasoning as in No. 146, it will be found that the present value of the \( n \)th year's rent of this annuity is \( [na + nb - n(ab)]c^n \), and the value of all the rents thereof will be \( S_n[na + nb - n(ab)]c^n \), or \( S_nac^n + S_nbc^n - S_n(ab)c^n \); so that \( AB = a + b - AB \) (147), agreeably to No. 48.
**Prob. III.**
159. To determine the present value of an annuity on the last survivor of three lives, \( A, B, \) and \( C \) (155); which we denote thus, \( ABC \).
**Solution.**
The present value of the \( n \)th year's rent is \( [na + nb + nc - n(ab) - n(ac) - n(bc) + n(abc)]c^n \) (142 and 146); whence, it appears, as in the preceding number, that \( ABC = a + b + c - AB - AC - BC + ABC \), agreeably to No. 52. PROB. IV.
160. To determine the present value of an annuity on the joint existence of the last two survivors out of three lives, \(A\), \(B\), \(C\) (155); which we denote thus, \(\frac{2}{ABC}\).
Solution.
The present value of the \(n\)th year's rent is
\[ \left[ \frac{n(ab)}{a} + \frac{n(ac)}{a} + \frac{n(bc)}{a} - \frac{2n(abc)}{a} \right] t^n \]
whence, reasoning as in the two preceding numbers, we infer, that \(\frac{2}{ABC} = AB + AC + BC - 2ABC\), as was demonstrated otherwise in No. 51.
161. Since the solutions of the last three problems were all obtained by showing each year's rent (as for instance the \(n\)th) of the annuity in question to be of the same value with the aggregate of the rents for the same year of all the annuities (taken with their proper signs) on the single and joint lives exhibited in the resulting formula; if any term of years be assigned, it is manifest that the value of such annuity for the term must be the same as that of the aggregate of the annuities above mentioned, each for the same term.
PROB. V.
162. \(A\) and \(B\) being any two proposed lives now both existing, to determine the present value of an annuity receivable only while \(A\) survives \(B\).
Solution.
A rent of this annuity will only be payable at the end of the \(n\)th year, provided that \(B\) be then dead and \(A\) living; but the probability of \(B\) being then dead is \(1 - n_b\), and that of \(A\) being then living \(n_a\), and these two events are independent; therefore, the probability of their both happening, or that of the rent being received, is \((1 - n_b)n_a = n_a - n(ab)\); the present value of that rent is, therefore,
\[ \left[ n_a - n(ab) \right] t^n \]
whence it follows, that the required value of the annuity on the life of \(A\) after that of \(B\) is \(A - AB\), agreeably to No. 60.
163. If the payment for the annuity which was the subject of the last problem is not to be made in present money, but by a constant annual premium \(p\) at the end of each year, while both the lives survive; since \(AB\) is the number of years' purchase (6) that an annuity on the joint continuance of those lives is worth, the value of \(p\) will be determined by this equation, \(p \cdot AB = A - AB\), whence we have \(p = \frac{A - AB}{AB} = 1\).
164. But if one premium \(p\) is to be paid down now, and an equal premium at the end of each year while both the lives survive, we shall have \(p \cdot (1 + AB) = A - AB\), and \(p = \frac{A - AB}{1 + AB}\).
165. For numerical examples illustrative of the formulae given from No. 158 to the present, see Nos. 66-74.
PROB. VI.
166. \(A\) and \(B\) are in possession of an annuity on the life of the survivor of them, which, if either of them die before a third person \(C\) is then to be divided equally between \(C\) and the survivor during their joint lives; to determine the value of \(C\)’s interest.
Solution.
That at the end of the \(n\)th year there will be
| dead | living | |------|--------| | \(A\) | \(BC\) | | \(B\) | \(AC\) |
the probability, multiplied by \(C\)’s proportion of the annuity in that circumstance, is
\[ \frac{1}{2} \left[ \frac{n(ab)}{a} - \frac{n(abc)}{a} \right] + \frac{1}{2} \left[ \frac{n(ac)}{a} - \frac{n(abc)}{a} \right] \]
and the sum of these being \(\frac{1}{2}n(ac) + \frac{1}{2}n(bc) - n(abc)\), the value of \(C\)’s interest is \(\frac{1}{2}AC + \frac{1}{2}BC - ABC\).
PROB. VII.
167. An annuity, after the decease of \(A\), is to be equally divided between \(B\) and \(C\) during their joint lives, and is then to go entirely to the last survivor for his life; it is proposed to find the value of \(B\)’s interest therein.
Solution.
That at the end of the \(n\)th year there will be
| dead | living | |------|--------| | \(A\) | \(BC\) | | \(AC\) | \(B\) |
the probability, multiplied by \(B\)’s proportion of the annuity in that circumstance, is
\[ \frac{1}{2}n(ab) - \frac{1}{2}n(bc) + \frac{1}{2}n(abc) \]
and the sum of these being \(\frac{1}{2}n(ab) - \frac{1}{2}n(bc) + \frac{1}{2}n(abc)\), the value of \(B\)’s interest is \(B - \frac{1}{2}BC + \frac{1}{2}ABC\).
PROB. VIII.
168. \(A\), \(B\), and \(C\) purchase an annuity on the life of the last survivor of them, which is to be divided equally at the end of every year among such of them as may then be living; what should \(A\) contribute towards the purchase of this annuity?
Solution.
That at the end of \(n\) years there will be
| dead | living | |------|--------| | none | \(ABC\) | | \(C\) | \(AB\) | | \(B\) | \(AC\) | | \(BC\) | \(A\) |
the probability, multiplied by \(A\)’s proportion of the annuity in that circumstance, is
\[ \frac{1}{3}n(abc) \]
and the sum of these being \(\frac{1}{3}n(ab) - \frac{1}{3}n(ac) + \frac{1}{3}n(abc)\), the required value of \(A\)’s interest is \(A - \frac{1}{3}AB - \frac{1}{3}AC + \frac{1}{3}ABC\).
PROB. IX.
169. As soon as any two of the three lives, \(A\), \(B\), and \(C\), are extinct, \(D\) or his heirs are to enter upon an annuity, which they are to enjoy during the remainder of the survivor’s life; to determine the value of \(D\)’s interest therein.
Solution.
That at the end of \(n\) years there will be
| dead | living | |------|--------| | \(AB\) | \(C\) | | \(AC\) | \(B\) | | \(BC\) | \(A\) |
The probability is
\[ \frac{1}{3}n(abc) - \frac{1}{3}n(ac) + \frac{1}{3}n(abc) \]
and the sum of all these being \(\frac{1}{3}n(ab) - \frac{1}{3}n(ac) + \frac{1}{3}n(abc)\), the value of \(D\)’s interest is
\[ A + B + C - 2AB - 2AC - 2BC + 3ABC \]
170. The last four may be sufficient to show the method of proceeding with any similar problems.
171. Let \((abc, \&c.)\) denote the probability that the last \(m\) survivors out of \(m + \mu\) lives \(A\), \(B\), \(C\), \&c., will jointly survive the term of \(t\) years. And when \(m = 0\), the expression will become \((abc, \&c.)\), the probability that the lives will all survive the term. (138.)
When \(m = 1\) it will become \((abc, \&c.)\), the measure of Also let \( \text{abc}, \&c. \) denote the value of an annuity on the joint continuance of the same number of last survivors out of the same lives. Then, if \( \mu \) be equal to 0, it will be \( \text{abc}, \&c. \) the value of an annuity on the joint continuance of all the lives; when \( m = 1 \), it will be \( \text{abc}, \&c. \) the value of an annuity on the last survivor of them. The values of annuities on the last survivor of two and of three lives will be denoted as in Nos. 158 and 159 respectively; and that of an annuity on the joint continuance of the last two survivors out of three lives, as in No. 160.
The value of an annuity on the last \( m \) survivors out of these \( m + \mu \) lives, according as it is limited to the term of \( t \) years, or deferred during that term, will also be denoted by \( \frac{m}{t} \text{abc}, \&c. \) or \( \frac{m}{t} \text{abc}, \&c. \) (156 and 157.)
**Prob. X.**
172. An annuity certain for the term of \( t + r \) years is to be enjoyed by \( P \) and his heirs during the joint existence of the last \( m \) survivors out of \( m + \mu \) lives, \( A, B, C, \&c.; \) and if that joint existence fail before the expiration of \( t \) years, the annuity is to go to \( Q \) and his heirs for the remainder of the term; to determine the value of \( Q \)'s interest in that annuity.
**Solution.**
\( Q \)'s expectation may be distinguished into two parts:
1st. That of enjoying the annuity during the term of \( t \) years.
2nd. That of enjoying it after the expiration of that term.
The sum of the present values of the interests of \( P \) and \( Q \) together in the annuity for the term of \( t \) years, is manifestly equal to the whole present value of the annuity certain for that term; that is, equal to \( \frac{1 - v^t}{r} \) (113 and 146); and the value of \( P \)'s interest for the term of \( t \) years is \( \frac{m}{t} \text{abc}, \&c. \) (171); therefore the value of \( Q \)'s interest for the same term is \( \frac{1 - v^t}{r} - \frac{m}{t} \text{abc}, \&c. \)
The present value of the annuity certain for \( r \) years after \( t \) years is \( \frac{v^t(1 - v^r)}{r} \) (114 and 146); and \( Q \) and his heirs will receive this annuity if the joint continuance of the last \( m \) survivors above mentioned fail before the expiration of \( t \) years; but the probability of their joint continuance failing in the term is \( 1 - \left( \frac{m}{t} \text{abc}, \&c. \right) \); therefore the value of \( Q \)'s interest in the annuity to be received after \( t \) years, is \( \left[ 1 - \left( \frac{m}{t} \text{abc}, \&c. \right) \right] \frac{v^t(1 - v^r)}{r} \); and the whole value of \( Q \)'s interest is \( \frac{1}{r} \left[ 1 - v^{t+r} - v^t(1 - v^r) \cdot \left( \frac{m}{t} \text{abc}, \&c. \right) \right] - \frac{m}{t} \text{abc}, \&c. \)
173. Corol. 1. When the whole annuity certain is a perpetuity, \( v^{t+r} = 0 \), and the value of \( Q \)'s interest is \( \frac{1}{r} \left[ 1 - \left( \frac{m}{t} \text{abc}, \&c. \right) v^t \right] - \frac{m}{t} \text{abc}, \&c. \)
174. Corol. 2. When the term \( t \) is not less than the greatest joint continuance of any \( m \) of the proposed lives, according to the tables of mortality adapted to them,
\[ \left( \frac{m}{t} \text{abc}, \&c. \right) = 0, \quad \text{and} \quad \frac{m}{t} \text{abc}, \&c. = \text{abc}, \&c.; \]
therefore, in that case, the general formula of No. 172 becomes
\[ \frac{1 - v^{t+r}}{r} - \frac{m}{t} \text{abc}, \&c.; \quad \text{that is, the excess of the value of an annuity certain for the whole term } t+r, \text{ above that of an annuity on the whole duration of joint continuance of the last } m \text{ surviving lives.} \]
175. And if, in the case proposed in the last No., the annuity certain be a perpetuity, as in No. 173, the formula will become \( \frac{m}{t} \text{abc}, \&c. \) the excess of the value of the perpetuity above the value of an annuity on the joint lives of the last \( m \) survivors; agreeably to No. 63.
176. **Example 1.** Required the present value of the absolute reversion of an estate in fee simple, after the extinction of the last survivor of three lives, \( A, B, C \), now aged 50, 55, and 60 years respectively; reckoning interest at 5 per cent.
The general algebraical expression of this value has just been shown to be \( \frac{1}{r} \text{abc}. \)
But \( \frac{1}{r} = \frac{1}{0.05} = 20-000 \)
And \( \text{abc}, \quad = 14-001 \) (68.)
Therefore \( 5-999 \) years' purchase is the value required. And if the annual produce of the estate, clear of all deductions, were L.100, the title to the reversion would now be worth L.599.18s. —, agreeably to No. 76.
177. **Ex. 2.** An annuity for the term of 70 years certain (from this time) is to revert to \( Q \) and his heirs at the failure of a life \( A \), now 45 years of age; what is the present value of \( Q \)'s interest therein, reckoning the interest of money at 5 per cent.?
In No. 174 the algebraical expression of the required value is shown to be \( \frac{v^{70}}{r} - A. \)
But \( v = x 1-05 = 1-9788107 \times 70 \)
\[ v^{70} = 0.32866 \times 2.5167490 \]
\[ 1 - v^{70} = 0.967134 \]
\[ \frac{1 - v^{70}}{r} = 0.967134 = 19-34268 \]
Subtract \( A = 12-64800 \) (Tab. VI.)
remains \( 6-69468 \) years' purchase;
so that if the annuity were L.1000, the value of the reversion would be L.6694.13s.7d.
178. **Ex. 3.** An annuity for the term of 70 years certain from this time is to revert to \( Q \) and his heirs at the extinction of the survivor of two lives, \( A \) and \( B \), now aged 40 and 50 years respectively, the interest of money being 5 per cent.; it is required to determine the value of \( Q \)'s interest in this annuity.
The algebraical expression of the value is,
\[ \frac{1 - v^{70}}{r} - AB \quad (174 \text{ and } 171). \]
But by the last example \( \frac{1 - v^{70}}{r} = 19-34268 \)
and by No. 66, \( AB = 15-06600 \)
so that the required value is \( 4-27668 \) years' IV.—Of Assurances on Lives.
179. Let the present value of the assurance (77 and 78) of L1 on the life of A be denoted by the old English capital A, and that of an assurance on the joint continuance of any number of lives A, B, C, &c. by ABC, &c. Also, let the value of an assurance on the joint continuance of any m of them out of the whole number m + p be denoted by ABC, &c.
180. And, in every case, let us designate the annual premium (83) for an assurance, by prefixing the character ⊙ to the symbol for the single premium; so that ⊙A may denote the annual premium for an assurance on the life of A; ⊙ABC, &c., the same for an assurance on the joint continuance of all the lives, A, B, C, &c.; and ⊙ABC, &c., the annual premium for an assurance on the joint continuance of the last m survivors out of the whole number m + p of those lives.
181. Then will ⊙A and ⊙A, ⊙ABC, &c., and ⊙ABC, &c., ⊙ABC, &c., and ⊙ABC, &c., designate the single and annual premiums for assurances on the same life or lives for the term of t years only.
Prob. XI.
182. To determine (ABC, &c.) the present value of an assurance on the last m survivors out of m + p lives A, B, C, &c. for the term of t years only; that is, the present value of L1 to be received upon the joint continuance of these last m survivors failing in the term.
Solution.
By reasoning as in No. 79, it will be found that a perpetuity, the first payment of which is to be made at the end of the year in which the last m survivors out of these m + p lives may fail in the term, will be of the same present value as \( \frac{1}{1-v} \left[ 1 - (abc, &c.)v^t \right] - \frac{m}{t}ABC, &c. \)
whence it is manifest that \( \frac{m}{t}ABC, &c. = v \left[ 1 - (abc, &c.)v^t \right] - (1-v) \frac{m}{t}ABC, &c. \)
183. Since the annual premium for this assurance must be paid at the commencement of every year in the term, while the last m surviving lives all subsist (83); besides the premium paid down now, one must be paid at the expiration of every year in the term except the last, provided that these last m survivors all outlive it; but the present value of L1 to be received upon their surviving that last year is \( (abc, &c.)v^t \), therefore all the future premiums are now worth \( \frac{m}{t}ABC, &c. - (abc, &c.)v^t \) years' purchase, and the present value of all the premiums, or the total present value of the assurance, is
\[ \frac{m}{t}ABC, &c. + \frac{m}{t}ABC, &c. = \frac{m}{t}ABC, &c. \]
\[ \frac{m}{t}ABC, &c. = \frac{1}{1-(abc, &c.)v^t} - (1-v) \frac{m}{t}ABC, &c. = 1-(abc, &c.)v^t \]
\[ (1-v) \left[ 1 - (abc, &c.)v^t + \frac{m}{t}ABC, &c. \right], \text{ whence we have} \]
\[ \frac{m}{t}ABC, &c. = \frac{1}{1-(abc, &c.)v^t} + \frac{m}{t}ABC, &c. \]
\[ v = 1. \]
184. Corol. 1. When (t), the term of the assurance, is not less than the greatest possible joint duration of any m of the proposed lives, \( (abc, &c.) = 0, \frac{m}{t}ABC, &c. = \frac{m}{t}ABC, &c., \) and the general formulae of the two preceding numbers become respectively
\[ \frac{m}{t}ABC, &c. = v - (1-v)ABC, &c. \]
and \( \frac{m}{t}ABC, &c. = \frac{1}{1+ABC, &c.} + v-1. \)
185. Corol. 2. In the same manner it appears, that, for a single life, \( A = v - (1-v)A, \)
and \( \frac{m}{t}A = \frac{1}{1+A} + v-1. \)
186. Corol. 3. Also that \( \frac{m}{t}A = v(1-a,v^t)-(1-v) \frac{m}{t}A \)
or \( \frac{m}{t}A = v(1-\frac{ta}{a}v^t)-(1-v) \left( A-\frac{ta}{a}v^t \cdot tA \right). \)
And \( \frac{m}{t}A = \frac{1}{1-\frac{ta}{a}v^t} + v-1; \)
that is, \( \frac{m}{t}A = \frac{1}{1+\frac{ta}{a}v^t} + v-1. \)
187. Corol. 4. When the assurance is on the joint continuance of all the lives, the formulae of No. 184 become respectively
\[ \frac{m}{t}ABC, &c. = v - (1-v)ABC, &c. \]
and \( \frac{m}{t}ABC, &c. = \frac{1}{1+ABC, &c.} + v-1. \)
And those of numbers 182 and 183,
\[ \frac{m}{t}ABC, &c. = v \left( 1 - \frac{(abc, &c.)v^t}{abc, &c.} \right) - (1-v) \times \left[ \frac{m}{t}ABC, &c. - \frac{(abc, &c.)v^t}{abc, &c.} \cdot \frac{m}{t}(ABC, &c.) \right] \]
and \( \frac{m}{t}ABC, &c. = \frac{1}{1+\frac{(abc, &c.)v^t}{abc, &c.}} + v-1. \)
188. Corol. 5. According as the assurance is in the last survivor of two or of three lives, the formulae of No. 184 become respectively \[ \overline{AB} = v - (1 - v) \overline{AB}, \] and \( \overline{ABC} = \frac{1}{1 + \overline{AB}} + v - 1; \) or \( \overline{ABC} = v - (1 - v) \overline{ABC}, \) and \( \overline{ABC} = \frac{1}{1 + \overline{ABC}} + v - 1. \)
And those of numbers 182 and 183 become
\[ \overline{AB} = v \left[ 1 - \frac{(ab)c^t}{(abc)^t} \right] - (1 - v) \overline{AB}, \] and \( \overline{ABC} = \frac{1}{1 - \frac{(ab)c^t}{(abc)^t}} + v - 1; \) or \( \overline{ABC} = v \left[ 1 - \frac{(abc)c^t}{(abc)^t} \right] - (1 - v) \overline{ABC}, \) and \( \overline{ABC} = \frac{1}{1 - \frac{(abc)c^t}{(abc)^t}} + v - 1 \) respectively.
Where \( c(ab) = a + b - (ab) \) (141), and \( c(abc) = a + b + c - \left[ c(ab) + c(ac) + c(bc) \right] + c(abc). \) (142.)
For the values of \( \overline{AB}, \overline{ABC}, \overline{AB}, \) and \( \overline{ABC}, \) see numbers 157-159, and 161.
189. Corol. 6. When the assurance is on the joint continuance of the two last survivors out of the three lives \( A, B, C, \) the formulae of No. 184 become respectively
\[ \overline{ABC} = v - (1 - v) \overline{ABC}, \] and \( \overline{ABC} = \frac{1}{1 + \overline{ABC}} + v - 1. \)
Those of numbers 182 and 183,
\[ \overline{ABC} = v \left[ 1 - \frac{m}{(abc, &c.)c^t} \right] - (1 - v) \overline{ABC}, \] and \( \overline{ABC} = \frac{1}{1 - \frac{m}{(abc, &c.)c^t}} + v - 1. \)
Where \( c(abc) = c(ab) + c(ac) + c(bc) - 2c(abc). \) (143.)
For the values of \( \overline{ABC} \) and \( \overline{ABC} \) see numbers 157, 160, and 161.
190. \( v \left[ 1 - \frac{m}{(abc, &c.)c^t} \right] - (1 - v) \overline{ABC}, \) &c. the value of an assurance on any life or lives for the term of \( t \) years, which was given in No. 182, may also be expressed thus:
\[ \left[ 1 + \frac{m}{(abc, &c.)c^t} \right] v - \overline{ABC}, \) &c.
And this, in words at length, is the rule given in No. 93.
191. When \( t \) is not less than the greatest possible joint duration of any \( m \) of the proposed lives, the last expression becomes
\[ \left( 1 + \frac{m}{\overline{ABC}, &c.} \right) v - \overline{ABC}, \) &c.
which is also equivalent to the first in No. 184; and, in words at length, is the rule given in No. 97 for determining the value of an assurance on any life or lives for their whole duration.
192. By substituting \( \frac{1}{1 + r} \) for \( v \) (146) in the last expression, it becomes \( \frac{1 + \overline{ABC}, &c.}{1 + r} = \frac{m}{\overline{ABC}, &c.} \)
\[ = \frac{1 - \frac{m}{\overline{ABC}, &c.}}{1 + r}, \text{ or } \frac{1}{r} \frac{\overline{ABC}, &c.}{1 + r}. \]
And
\[ \frac{1}{r} \frac{\overline{ABC}, &c.}{1 + r} = \overline{ABC}, \) &c. is the proposition enunciated in No. 81; \( \frac{1}{r} \) being the value of the perpetuity. (112.)
193. Examples of the determination of the single premiums for assurances, and of the derivation of the annual premiums from them, have been given in No. 82-88, also in 95 and 96; but by the algebraical formulae given here, the annual premiums may be determined directly, without first finding the total present values of the assurances.
194. Example 1. Required the annual premium for an assurance on the life \( A, \) now 50 years of age, interest 5 per cent.
According to No. 185, the operation is thus,
\[ 1 + A = \frac{12660}{28975663} \]
Adding \( v = 0.9523809, \) and subtracting unit, we have \( \overline{ABC} = 0.0313699, \) agreeably to No. 85.
195. Ex. 2. What should the annual premium be for an assurance on the last survivor of three lives \( A, B, \) and \( C, \) now aged 50, 55, and 60 years respectively, rate of interest 5 per cent.?
Operation by No. 188.
\[ (68) \quad 1 + \overline{ABC} = \frac{15001}{28238798} \]
\[ \frac{1}{1 + \overline{ABC}} = \frac{0.666622}{v} = 0.9523809 \]
\[ \overline{ABC} = 0.0190431, \text{ agreeably to No. 88.} \]
196. Ex. 3. Required the annual premium for an assurance for 10 years only, on a life now 45 years of age, interest 5 per cent.
Operation according to No. 186.
\[ v^{10} = 613913 \times 17881068 \]
\[ 10a = 4073 \times 3.6099144 \]
\[ a = 4727 \times 4.3254144 \]
\[ 1 + 10A = 528976 \times 17224356 \]
\[ 11.347 \times 1.0548811 \]
Subtract \( 6.002 \times 0.7783167 \)
from \( 1 + A = 13.648 \)
remains \( 7.646 \times 1.1165657 \)
\( 1 - 10av^{10} = 471024 \times 1.6730430 \)
\[ 0.061604 \times 2.7896087 \]
\[ v = 0.952381 \]
\[ \overline{ABC} = 0.013985, \text{ agreeably to No. 96.} \]
What has been advanced from number 99 to 109 needs no algebraical illustration. **TABLE I.**
Showing the present Value of One Pound to be received at the End of any Number of Years not exceeding 50.
*(See No. 9 — 12 of the preceding Article.)*
| Years | 2 per Cent. | 2½ per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | 8 per Cent. | 9 per Cent. | Years. | |-------|-----------|---------------|--------------|-------------|-------------|-------------|-------------|-------------|-------------|-------| | 1 | -980392 | -975610 | -970874 | -965138 | -952381 | -943396 | -945759 | -925926 | -917431 | 1 | | 2 | -961169 | -951814 | -945296 | -942556 | -907029 | -889996 | -873439 | -857339 | -841680 | 2 | | 3 | -942322 | -928599 | -915142 | -888996 | -B63535 | -883968 | -816298 | -B93832 | -772183 | 3 | | 4 | -923845 | -905351 | -884487 | -854804 | -822702 | -790249 | -762995 | -7*3030* | -708425 | 4 | | 5 | -905721 | -883854 | -862600 | -821927 | -783526 | -747258 | -712958 | -685058 | -649931 | 5 | | 6 | -887921 | -81299 | -737484 | -7903 Mon | -746215 | -704961 | -666342 | -630170` | -596267 | 6 | | 7 | -870560 | -841206 | -713092 | -759918 | -710681 | -665057 | -622750 | -583490 | -547084 | 7 | | 8 | -855450 | -920747 | -775709 | -759796 | -678539 | -647142 | -618209 | -560669 | -510068 | 8 | | 9 | -838755 | -810761 | -702587 | -702587 | -646099 | -591898 | -548592 | -500249 | -460488 | 9 | | 10 | -820848 | -781198 | -744094 | -637556 | -613913 | -558457 | -463193 | -429211 | -387538 | lo | | 11 | -794503 | 662403 | -646679 | -592967 | -546947 | -513511 | -478288 | -452342 | -420288 | 11 | | 12 | -752540 | -633306 | -570199 | -511971 | -443077 | -670417 | -421348 | -391339 | -359088 | 12 | | 13 | 730183 | -615379 | -499822 | -628327 | -534467 | -632553 | -362401 | -330202 | -302669 | 13 | | 14 | 692592 | -595522 | -475516 | -601433 | -500012 | -613500 | -331504 | -304270 | -279150 | 14 | | 15 | 659914 | -568659 | -446182 | -562726 | -504260 | -592369 | -317987 | -284538 | -261052 | 15 | | 16 | 628446 | -553400 | -420628 | -573439 | -557386 | -595666 | -303529 | -254572 | -240010 | 16 | | 17 | 594231 | 564246 | -444973 | -573928 | -562588 | -575663 | -295010 | -230654 | -212392 | 17 | | 18 | z5d515 | -507593 | -455691 | -500772 | -512911 | -574661 | -26o42s | -213750 | -1ce66t | 18 | | 19 | 734613 | -657951 | -610551 | -5e8264 | -510765 | -5c751t | -253675 | -250624 | -25038l | 19 | | 20 | 665240 | -660495 | -657833 | -5991* | -558037 | -558296 | -245729 | -247841 | -230845 | 20 | | 21 | 655597 | -535346 | -562047 | -697953 | -592707 | -590423 | -236664 | -250060 | -235070 | 21 | | 22 | 6z9811 | -502213 | -476243 | -526765 | -544053 | -543567 | -229861 | -250z61 | -213872 | 22 | | 23 | 6f6022 | -6fB3.5"*-opt-| 4360 | -532t,70 | -513,.< peut}: caust .& workers l.oz w<br TABLE II.
Showings the present Value of an Annuity of One Pound for any Number of Years not exceeding 50.
(No. 9 — 12.)
| Years | 2 per Cent. | 2\(\frac{1}{3}\) per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | 8 per Cent. | 9 per Cent. | 10 Perp. | |-------|-------------|--------------------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------|---------| | 2 | -9804 | -9756 | -9709 | -9615 | -9524 | -9434 | -9346 | -9259 | -9174 | Perp. | | 5 | 1-9416 | 1-9274 | 1-9135 | 1-8861 | 1-8594 | 1-8334 | 1-8080 | 1-7833 | 1-7591 | Perp. | | 6 | 6-8839 | 2-8560 | 2-8286 | 2-7751 | 2-7323 | 2-6730 | 2-6243 | 2-5771 | 2-5313 | Perp. | | 7 | 4-8707 | 3-7650 | 3-7171 | 3-6299 | 3-5460 | 3-4651 | 3-3872 | 3-3121 | 3-2397 | Perp. | | 8 | 4-7135 | 4-6458 | 4-5797 | 4-4514 | 4-3295 | 4-2124 | 4-1002 | 3-9927 | 3-8897 | Perp. | | 9 | 6-6014 | 5-5061 | 5-4172 | 5-3241 | 5-2075 | 5-1973 | 5-0765 | 4-9629 | 4-8489 | Perp. | | 10 | 6-4720 | 6-3494 | 6-2603 | 6-1021 | 5-7864 | 5-5924 | 5-3893 | 5-2064 | 5-0330 | Perp. | | 12 | 7-3535 | 7-1701 | 7-0197 | 6-7327 | 6-4632 | 6-2098 | 5-9713 | 5-7466 | 5-5348 | Perp. | | 14 | 8-1622 | 7-9709 | 7-7661 | 7-4578 | 7-0178 | 6-5017 | 6-0152 | 5-4649 | 5-9952 | Perp. | | 16 | 9-8141 | 8-5751 | 8-3302 | 7-9109 | 7-2717 | 7-3601 | 7-0236 | 6-7101 | 6-4177 | Perp. | | 18 | 10-5753 | 10-2078 | 9-9405 | 9-3852 | 8-8633 | 8-3838 | 7-9427 | 7-5361 | 7-1607 | Perp. | | 20 | 11-3484 | 10-8893 | 10-6545 | 9-9856 | 9-3909 | 8-8357 | 8-3377 | 7-9033 | 7-4869 | Perp. | | 22 | 12-1083 | 11-6999 | 11-2695 | 10-5651 | 9-8996 | 9-2955 | 8-7455 | 8-2442 | 7-7852 | Perp. | | 24 | 12-8493 | 12-3814 | 11-9389 | 11-1192 | 10-3787 | 9-7192 | 9-1079 | 8-5595 | 8-0607 | Perp. | | 26 | 13-5777 | 13-0570 | 12-6619 | 11-6352 | 10-8578 | 10-1089 | 9-4465 | 8-8512 | 8-3169 | Perp. | | 28 | 14-2919 | 13-7122 | 13-1661 | 12-1673 | 11-2744 | 10-4773 | 9-7632 | 9-1216 | 8-5495 | Perp. | | 30 | 16-9139 | 14-3581 | 14-7355 | 12-6557 | 11-6879 | 10-8217 | 10-0589 | 9-3719 | 8-7556 | Perp. | | 32 | 17-6580 | 14-9267 | 15-4750 | 13-2017 | 12-1059 | 11-3202 | 10-5551 | 9-5697 | 8-9586 | Perp. | | 34 | 18-3800 | 15-5162 | 15-9800 | 13-7214 | 12-5650 | 11-8645 | 11-2372 | 10-7516 | 9-1503 | Perp. | | 36 | 19-4501 | 16-0886 | 16-4750 | 14-1245 | 13-0132 | 12-2802 | 11-9436 | 10-9386 | 9-3346 | Perp. | | 38 | 20-2611 | 16-5972 | 16-9359 | 14-4471 | 13-4643 | 12-7509 | 12-6167 | 11-1078 | 9-5131 | Perp. | | 40 | 21-0222 | 17-0264 | 17-3659 | 14-7611 | 13-9076 | 13-2444 | 13-2984 | 12-4406 | 9-6818 | Perp. |
... (continues with more entries) **TABLE III.**
showing the sum to which One Pound will increase when improved at Compound Interest during any Number of Years not exceeding 50.
(No. 9—12.)
| Years | 2 per Cent. | 3/2 per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | 8 per Cent. | |-------|-------------|--------------|------------|-------------|-------------|-------------|-------------|-------------| | | | | | | | | | | | 1 | 1 | 1-02000 | 1.02500 | 1.03000 | 1.04000 | 1.05000 | 1.06000 | 1.07000 | 1.080000 | | 2 | 1 | 1-04040 | 1-05053 | 1.06090 | 1.08160 & & 1.10250 & & 1.12506 | & & 1.14900 | & & 1.16640 | | 3 | 1 | 1.06121 | 1-07659 | 1.09272 | 1.124864 | 1.157652 | 1-191016 | 1-225043 | 1-259712 | | 4 | 1 | 1-08243 | 1-11311 | 1-12959 | 1-159859 | 1-202576 | 1.252738 | 1.308459 | 3 | 5 | 1 | 1.10468 | 1.13141 | 1.159274 | 1.196553 | 1-267682 | 1-338226 | 1.402532 | 1-469328 | | 6 | 1 | 1.12616 | 1.15969 | 1-18402 | 1.235319 | 1.310969 | 1.418519 | 1-500730 | 588674 | | 7 | 1 | 1-14669 | 1-18669 | 1-229874 | 1-276139 | 1.407100 | 1.506530 | 1-605781 | 71-8824 | | 8 | 1 | 1-17166 | 1.12184 | 1.260717 | 1.368659 | 1.597455 | 1-593848 | 1-718156 | 1850930 | | 9 | 1 | 1.19509 | 1-24886 | 1.304778 | 1.428312 | 1-551328 | 1.689479 | 1-884590 | 1-199905 | | 10 | 1 | 1.21899 | 1.29008 | 1-334185 | 1.482044 | 1.582895 | 1.790848 | 1-987151 | 8158925 | | 11 | 1 | 1.24355 | 1.31209 | 1.348434 | 1.538494 | 1-710339 | 1-898299 | 2-014852 | 3311639 | | 12 | 1 | 1.26824 | 1.34489 | 1.342576 | 1.610122 | 1-795856 | 2-012196 | 2-225292 | 2-518170 | | 13 | 1 | 1.29361 | 1-37851 | 1.390284 | 1-665074 | 2-028769 | 1-232998 | 2-909497 | 217624 | | 14 | 1 | 1.31948 | 1-41297 | 1.387453 | 1-681910 | 2-029123 | 2-207105 | 2-940719 | | | 15 | 1 | 134587 | 1-44830 | 1-452575 | 1-660606 | 2-071735 | 2-395464 | 2-945810 | | | 16 | 1 | 1-37279 | 1.48451 | 1.425319 | 1-695976 | 2-162203 | 2-403552 | 3-292298 | | | 17 | 1 | 1.40024 | 1.45162 | 1.488848 | 1.749701 | 2-209289 | 2-483877 | 3-229792 | | ### TABLE IV.
Showing the Amount to which One Pound per Annum will increase at Compound Interest in any Number of Years not exceeding 50.
(No. 9—12.)
| Years | 2 per Cent. | 2½ per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | |-------|-------------|--------------|-------------|-------------|-------------|-------------|-------------| | 1 | 1-0000 | 1-0000 | 1-000000 | 1-000000 | 1-000000 | 1-000000 | 1-000000 | | 2 | 2-0200 | 2-0250 | 2-030000 | 2-040000 | 2-050000 | 2-060000 | 2-070000 | | 3 | 3-0604 | 3-0756 | 3-090900 | 3-121600 | 3-152500 | 3-183600 | 3-214900 | | 4 | 4-1216 | 4-1525 | 4-183627 | 4-246464 | 4-310125 | 4-374616 | 4-439943 | | 5 | 5-2040 | 5-2563 | 5-309136 | 5-416323 | 5-525631 | 5-637093 | 5-750739 | | 6 | 6-3081 | 6-3877 | 6-468410 | 6-632975 | 6-801913 | 6-975319 | 7-153291 | | 7 | 7-4343 | 7-5474 | 7-662462 | 7-898294 | 8-142008 | 8-393838 | 8-654021 | | 8 | 8-5830 | 8-7361 | 8-892386 | 9-214226 | 9-549109 | 9-897468 | 10-259803 | | 9 | 9-7546 | 9-9545 | 10-159106 | 10-582795 | 11-026564 | 11-491316 | 11-977989 | | 10 | 10-9497 | 11-2034 | 11-463879 | 12-006107 | 12-577693 | 13-180795 | 13-816484 | | 11 | 12-1687 | 12-4835 | 12-807796 | 13-486351 | 14-206787 | 14-971643 | 15-783599 | | 12 | 13-4121 | 13-7956 | 14-192030 | 15-025805 | 15-917127 | 16-869941 | 17-888451 | | 13 | 14-6803 | 15-1404 | 15-617790 | 16-626838 | 17-712983 | 18-882138 | 20-140643 | | 14 | 15-9739 | 16-5190 | 17-086324 | 18-291911 | 19-598632 | 21-015066 | 22-550488 | | 15 | 17-2934 | 17-9319 | 18-598914 | 20-023588 | 21-578564 | 23-275970 | 25-129022 | | 16 | 18-6393 | 19-3802 | 20-156881 | 21-845311 | 23-657492 | 25-672528 | 27-888054 | | 17 | 20-0121 | 20-8647 | 21-761588 | 23-697512 | 25-840366 | 28-212880 | 30-840217 | | 18 | 21-4123 | 22-3863 | 23-414435 | 25-645413 | 28-132385 | 30-905653 | 33-999033 | | 19 | 22-8406 | 23-9460 | 25-116868 | 27-671229 | 30-539004 | 33-759992 | 37-378965 | | 20 | 24-2974 | 25-5447 | 26-870374 | 29-778079 | 33-065954 | 36-785591 | 40-995492 | | 21 | 25-7833 | 27-1833 | 28-676486 | 31-969202 | 35-719252 | 39-992727 | 44-665177 | | 22 | 27-2990 | 28-8629 | 30-536780 | 34-247970 | 38-505214 | 43-892990 | 49-005739 | | 23 | 28-8450 | 30-5844 | 32-452884 | 36-617889 | 41-430475 | 46-995828 | 53-436141 | | 24 | 30-4219 | 32-3490 | 34-426470 | 39-082604 | 44-501999 | 50-815577 | 58-176671 | | 25 | 32-0303 | 34-1578 | 36-459264 | 41-645908 | 47-727099 | 54-864512 | 63-249038 | | 26 | 33-6709 | 36-0117 | 38-553042 | 44-311745 | 51-113454 | 59-156383 | 68-676470 | | 27 | 35-3443 | 37-9120 | 40-709634 | 47-084214 | 54-669126 | 63-705766 | 74-483823 | | 28 | 37-0512 | 39-8598 | 42-930923 | 49-967583 | 58-402583 | 68-528112 | 80-697691 | | 29 | 38-7922 | 41-8563 | 45-218850 | 52-966286 | 62-922712 | 73-639798 | 87-346529 | | 30 | 40-5681 | 43-9027 | 47-575416 | 56-084938 | 66-438848 | 79-058186 | 94-460786 | | 31 | 42-3794 | 46-0003 | 50-002678 | 59-382355 | 70-706790 | 84-801677 | 102-073041 | | 32 | 44-2270 | 48-1503 | 52-502759 | 62-701469 | 75-298829 | 90-889778 | 110-218154 | | 33 | 46-1116 | 50-3540 | 55-077841 | 66-209527 | 80-063771 | 97-343165 | 118-933425 | | 34 | 48-0383 | 52-6129 | 57-730177 | 69-857909 | 85-066959 | 104-183755 | 128-258765 | | 35 | 49-9945 | 54-9282 | 60-462082 | 73-652225 | 90-320307 | 111-434780 | 138-236878 | | 36 | 51-9944 | 57-3014 | 63-275944 | 77-598314 | 95-636323 | 119-120867 | 148-913460 | | 37 | 54-0343 | 59-7339 | 66-174223 | 81-702246 | 101-628139 | 127-268119 | 160-337402 | | 38 | 56-1149 | 62-2273 | 69-159449 | 85-970386 | 107-705416 | 135-904206 | 172-561020 | | 39 | 58-2372 | 64-7830 | 72-234233 | 90-409150 | 114-095023 | 145-058458 | 185-640292 | | 40 | 60-4020 | 67-4026 | 75-401260 | 95-025516 | 120-799774 | 154-761966 | 199-635112 | | 41 | 62-6100 | 70-0876 | 78-663298 | 99-826536 | 127-839763 | 165-047684 | 214-609570 | | 42 | 64-8622 | 72-8398 | 82-023196 | 104-819598 | 135-231751 | 175-950545 | 230-632240 | | 43 | 67-1595 | 75-6608 | 85-483892 | 110-012882 | 142-993339 | 187-507577 | 247-776496 | | 44 | 69-5027 | 78-5523 | 89-048409 | 115-412877 | 151-143006 | 199-758032 | 266-120851 | | 45 | 71-9927 | 81-5161 | 92-719861 | 121-029392 | 159-700156 | 212-743514 | 285-749311 | | 46 | 74-3306 | 84-5540 | 96-501457 | 126-870568 | 168-685164 | 226-508125 | 306-751763 | | 47 | 76-8172 | 87-6679 | 100-396501 | 132-945390 | 178-119422 | 241-098612 | 329-224386 | | 48 | 79-3535 | 90-8596 | 104-408396 | 139-263206 | 188-025393 | 256-564529 | 353-270093 | | 49 | 81-9406 | 94-1311 | 108-540648 | 145-833784 | 198-426663 | 272-958401 | 378-999000 | | 50 | 84-5794 | 97-4843 | 112-798667 | 152-667084 | 209-347996 | 290-335905 | 406-528929 | ### TABLE V.
Exhibiting the Law of Mortality at Carlisle. (No. 32.)
| Age | Number who complete that Age. | Number who die in the next interval. | Age | Number who complete that Year. | Number who die in their next Year. | Age | Number who complete that Age. | Number who die in the next Year. | |-----|-------------------------------|--------------------------------------|-----|-------------------------------|------------------------------------|-----|-------------------------------|----------------------------------| | 0 | 10000 | 533 | 32 | 5528 | 56 | 69 | 2525 | 124 | | 1 Month | 9467 | 154 | 33 | 5472 | 55 | 70 | 2401 | 124 | | 2 Months | 9313 | 87 | 34 | 5417 | 55 | 71 | 2277 | 134 | | 3 | 9226 | 256 | 35 | 5362 | 55 | 72 | 2143 | 146 | | 6 | 8970 | 255 | 36 | 5307 | 56 | 73 | 1997 | 156 | | 9 | 8715 | 254 | 37 | 5251 | 57 | 74 | 1841 | 166 | | 1 Year | 8461 | 682 | 38 | 5194 | 58 | 75 | 1675 | 160 | | 2 Years | 7779 | 505 | 39 | 5136 | 61 | 76 | 1515 | 156 | | 3 | 7274 | 276 | 40 | 5075 | 66 | 77 | 1359 | 146 | | 4 | 6998 | 201 | 41 | 5009 | 69 | 78 | 1213 | 132 | | 5 | 6797 | 121 | 42 | 4940 | 71 | 79 | 1081 | 128 | | 6 | 6676 | 82 | 43 | 4869 | 71 | 80 | 953 | 116 | | 7 | 6594 | 58 | 44 | 4798 | 71 | 81 | 837 | 112 | | 8 | 6536 | 43 | 45 | 4727 | 70 | 82 | 725 | 102 | | 9 | 6493 | 33 | 46 | 4657 | 69 | 83 | 623 | 94 | | 10 | 6460 | 29 | 47 | 4588 | 67 | 84 | 529 | 84 | | 11 | 6431 | 31 | 48 | 4521 | 63 | 85 | 445 | 78 | | 12 | 6400 | 32 | 49 | 4458 | 61 | 86 | 367 | 71 | | 13 | 6368 | 33 | 50 | 4397 | 59 | 87 | 296 | 64 | | 14 | 6335 | 35 | 51 | 4338 | 62 | 88 | 232 | 51 | | 15 | 6300 | 39 | 52 | 4276 | 65 | 89 | 181 | 39 | | 16 | 6261 | 42 | 53 | 4211 | 68 | 90 | 142 | 37 | | 17 | 6219 | 43 | 54 | 4143 | 70 | 91 | 105 | 30 | | 18 | 6176 | 43 | 55 | 4073 | 73 | 92 | 75 | 21 | | 19 | 6133 | 43 | 56 | 4000 | 76 | 93 | 54 | 14 | | 20 | 6090 | 43 | 57 | 3924 | 82 | 94 | 40 | 10 | | 21 | 6047 | 42 | 58 | 3842 | 93 | 95 | 30 | 7 | | 22 | 6005 | 42 | 59 | 3749 | 106 | 96 | 23 | 5 | | 23 | 5963 | 42 | 60 | 3643 | 122 | 97 | 18 | 4 | | 24 | 5921 | 42 | 61 | 3521 | 126 | 98 | 14 | 3 | | 25 | 5879 | 43 | 62 | 3395 | 127 | 99 | 11 | 2 | | 26 | 5836 | 43 | 63 | 3268 | 125 | 100 | 9 | 2 | | 27 | 5793 | 45 | 64 | 3143 | 125 | 101 | 7 | 2 | | 28 | 5748 | 50 | 65 | 3018 | 124 | 102 | 5 | 2 | | 29 | 5698 | 56 | 66 | 2894 | 123 | 103 | 3 | 2 | | 30 | 5642 | 57 | 67 | 2771 | 123 | 104 | 1 | 1 | | 31 | 5585 | 57 | 68 | 2648 | 123 | | | |
### TABLE VI.
Showing the Value of an Annuity on a Single Life at every Age, according to the Carlisle Table of Mortality, when the Rate of Interest is Five per Cent. (No. 65.)
| Age | Value. | Age | Value. | Age | Value. | Age | Value. | Age | Value. | Age | Value. | Age | Value. | |-----|--------|-----|--------|-----|--------|-----|--------|-----|--------|-----|--------|-----|--------| | 0 | 12-083 | 18 | 15-987 | 35 | 14-127 | 52 | 11-154 | 69 | 6-643 | 86 | 2-830 | | | | 1 | 13-995 | 19 | 15-904 | 36 | 13-987 | 53 | 10-892 | 70 | 6-336 | 87 | 2-685 | | | | 2 | 14-983 | 20 | 15-817 | 37 | 13-843 | 54 | 10-624 | 71 | 6-015 | 88 | 2-597 | | | | 3 | 15-824 | 21 | 15-726 | 38 | 13-695 | 55 | 10-347 | 72 | 5-711 | 89 | 2-495 | | | | 4 | 16-271 | 22 | 15-628 | 39 | 13-542 | 56 | 10-063 | 73 | 5-435 | 90 | 2-339 | | | | 5 | 16-590 | 23 | 15-525 | 40 | 13-390 | 57 | 9-771 | 74 | 5-190 | 91 | 2-231 | | | | 6 | 16-735 | 24 | 15-417 | 41 | 13-245 | 58 | 9-478 | 75 | 4-989 | 92 | 2-142 | | | | 7 | 16-790 | 25 | 15-303 | 42 | 13-101 | 59 | 9-199 | 76 | 4-792 | 93 | 2-518 | | | | 8 | 16-786 | 26 | 15-187 | 43 | 12-957 | 60 | 8-940 | 77 | 4-609 | 94 | 2-569 | | | | 9 | 16-742 | 27 | 15-065 | 44 | 12-806 | 61 | 8-712 | 78 | 4-422 | 95 | 2-596 | | | | 10 | 16-669 | 28 | 14-942 | 45 | 12-648 | 62 | 8-487 | 79 | 4-210 | 96 | 2-555 | | | | 11 | 16-581 | 29 | 14-837 | 46 | 12-480 | 63 | 8-258 | 80 | 4-015 | 97 | 2-428 | | | | 12 | 16-494 | 30 | 14-723 | 47 | 12-301 | 64 | 8-016 | 81 | 3-799 | 98 | 2-278 | | | | 13 | 16-406 | 31 | 14-617 | 48 | 12-107 | 65 | 7-765 | 82 | 3-606 | 99 | 2-045 | | | | 14 | 16-316 | 32 | 14-506 | 49 | 11-892 | 66 | 7-503 | 83 | 3-406 | 100 | 1-624 | | | | 15 | 16-227 | 33 | 14-387 | 50 | 11-660 | 67 | 7-227 | 84 | 3-211 | 101 | 1-192 | | | | 16 | 16-144 | 34 | 14-260 | 51 | 11-410 | 68 | 6-941 | 85 | 3-009 | 102 | 0-753 | | | | 17 | 16-066 | 35 | 14-120 | 52 | 11-154 | 69 | 6-643 | 86 | 2-830 | | | ### TABLE VII.
Showing the Value of an Annuity on the Joint Continuance of Two Lives according to the Carlisle Table of Mortality, when the difference of their ages is five years, and the Rate of Interest Five per Cent. (No. 65.)
| Ages | Value | Ages | Value | Ages | Value | Ages | Value | Ages | Value | |------|-------|------|-------|------|-------|------|-------|------|-------| | 0 & 5 | 10-551 | 20 & 25 | 13-398 | 40 & 45 | 10-598 | 59 & 64 | 6-127 | 79 & 84 | 2-045 | | 1 & 6 | 12-331 | 21 & 26 | 13-272 | 41 & 46 | 10-444 | 60 & 65 | 5-895 | 80 & 85 | 1-995 | | 2 & 7 | 13-258 | 22 & 27 | 13-137 | 42 & 47 | 10-287 | 61 & 66 | 5-678 | 81 & 86 | 1-747 | | 3 & 8 | 14-019 | 23 & 28 | 13-000 | 43 & 48 | 10-121 | 62 & 67 | 5-458 | 82 & 87 | 1-626 | | 4 & 9 | 14-402 | 24 & 29 | 12-867 | 44 & 49 | 9-937 | 63 & 68 | 5-230 | 83 & 88 | 1-535 | | 5 & 10 | 14-649 | 25 & 30 | 12-742 | 45 & 50 | 9-737 | 64 & 69 | 4-988 | 84 & 89 | 1-433 | | 6 & 11 | 14-731 | 26 & 31 | 12-615 | 46 & 51 | 9-519 | 65 & 70 | 4-737 | 85 & 90 | 1-279 | | 7 & 12 | 14-736 | 27 & 32 | 12-482 | 47 & 52 | 9-292 | 66 & 71 | 4-469 | 86 & 91 | 1-203 | | 8 & 13 | 14-689 | 28 & 33 | 12-344 | 48 & 53 | 9-054 | 67 & 72 | 4-207 | 87 & 92 | 1-192 | | 9 & 14 | 14-606 | 29 & 34 | 12-208 | 49 & 54 | 8-799 | 68 & 73 | 3-961 | 88 & 93 | 1-219 | | 10 & 15 | 14-500 | 30 & 35 | 12-078 | 50 & 55 | 8-528 | 69 & 74 | 3-731 | 89 & 94 | 1-214 | | 11 & 16 | 14-389 | 31 & 36 | 11-944 | 51 & 56 | 8-242 | 70 & 75 | 3-528 | 90 & 95 | 1-167 | | 12 & 17 | 14-284 | 32 & 37 | 11-806 | 52 & 57 | 7-950 | 71 & 76 | 3-319 | 91 & 96 | 1-161 | | 13 & 18 | 14-178 | 33 & 38 | 11-661 | 53 & 58 | 7-657 | 72 & 77 | 3-127 | 92 & 97 | 1-181 | | 14 & 19 | 14-069 | 34 & 39 | 11-508 | 54 & 59 | 7-375 | 73 & 78 | 2-948 | 93 & 98 | 1-215 | | 15 & 20 | 13-959 | 35 & 40 | 11-354 | 55 & 60 | 7-106 | 74 & 79 | 2-767 | 94 & 99 | 1-191 | | 16 & 21 | 13-853 | 36 & 41 | 11-204 | 56 & 61 | 6-860 | 75 & 80 | 2-623 | 95 & 100 | 1-038 | | 17 & 22 | 13-746 | 37 & 42 | 11-056 | 57 & 62 | 6-615 | 76 & 81 | 2-467 | 96 & 101 | 0-828 | | 18 & 23 | 13-636 | 38 & 43 | 10-907 | 58 & 63 | 6-370 | 77 & 82 | 2-333 | 97 & 102 | 0-555 | | 19 & 24 | 13-520 | 39 & 44 | 10-753 | | | 78 & 83 | 2-194 | 98 & 103 | 0-249 |
### TABLE VIII.
Showing the Value of an Annuity on the Joint Continuance of Two Lives according to the Carlisle Table of Mortality, when the difference of their Ages is ten years, and the Rate of Interest Five per Cent. (No. 65.)
| Ages | Value | Ages | Value | Ages | Value | Ages | Value | Ages | Value | |------|-------|------|-------|------|-------|------|-------|------|-------| | 0 & 10 | 10-649 | 19 & 29 | 13-117 | 38 & 48 | 10-396 | 56 & 66 | 6-156 | 75 & 85 | 2-100 | | 1 & 11 | 12-275 | 20 & 30 | 13-008 | 39 & 49 | 10-195 | 57 & 67 | 5-881 | 76 & 86 | 1-958 | | 2 & 12 | 13-087 | 21 & 31 | 12-896 | 40 & 50 | 9-984 | 58 & 68 | 5-600 | 77 & 87 | 1-888 | | 3 & 13 | 13-769 | 22 & 32 | 12-776 | 41 & 51 | 9-766 | 59 & 69 | 5-319 | 78 & 88 | 1-759 | | 4 & 14 | 14-106 | 23 & 33 | 12-648 | 42 & 52 | 9-548 | 60 & 70 | 5-044 | 79 & 89 | 1-657 | | 5 & 15 | 14-334 | 24 & 34 | 12-510 | 43 & 53 | 9-329 | 61 & 71 | 4-779 | 80 & 90 | 1-515 | | 6 & 16 | 14-419 | 25 & 35 | 12-365 | 44 & 54 | 9-104 | 62 & 72 | 4-529 | 81 & 91 | 1-450 | | 7 & 17 | 14-432 | 26 & 36 | 12-214 | 45 & 55 | 8-870 | 63 & 73 | 4-302 | 82 & 92 | 1-460 | | 8 & 18 | 14-395 | 27 & 37 | 12-058 | 46 & 56 | 8-626 | 64 & 74 | 4-094 | 83 & 93 | 1-479 | | 9 & 19 | 14-321 | 28 & 38 | 11-900 | 47 & 57 | 8-372 | 65 & 75 | 3-921 | 84 & 94 | 1-468 | | 10 & 20 | 14-221 | 29 & 39 | 11-747 | 48 & 58 | 8-111 | 66 & 76 | 3-746 | 85 & 95 | 1-443 | | 11 & 21 | 14-106 | 30 & 40 | 11-607 | 49 & 59 | 7-851 | 67 & 77 | 3-580 | 86 & 96 | 1-397 | | 12 & 22 | 13-987 | 31 & 41 | 11-474 | 50 & 60 | 7-601 | 68 & 78 | 3-407 | 87 & 97 | 1-324 | | 13 & 23 | 13-864 | 32 & 42 | 11-342 | 51 & 61 | 7-370 | 69 & 79 | 3-210 | 88 & 98 | 1-280 | | 14 & 24 | 13-737 | 33 & 43 | 11-207 | 52 & 62 | 7-142 | 70 & 80 | 3-020 | 89 & 99 | 1-192 | | 15 & 25 | 13-608 | 34 & 44 | 11-063 | 53 & 63 | 6-911 | 71 & 81 | 2-807 | 90 & 100 | 0-950 | | 16 & 26 | 13-483 | 35 & 45 | 10-912 | 54 & 64 | 6-669 | 72 & 82 | 2-616 | 91 & 101 | 0-733 | | 17 & 27 | 13-359 | 36 & 46 | 10-750 | 55 & 65 | 6-418 | 73 & 83 | 2-430 | 92 & 102 | 0-508 | | 18 & 28 | 13-235 | 37 & 47 | 10-579 | | | 74 & 84 | 2-260 | 93 & 103 | 0-235 |
(J.M.)