An Annuity is a sum of money, payable yearly, half yearly, or quarterly, to continue a certain number of years, for ever, or for life.
An annuity is said to be an arrear, when it continues unpaid after it falls due. And an annuity is said to be in reversion, when the purchaser, upon paying the price, does not immediately enter upon possession; the annuity not commencing till some time after.
Interest on annuities may be computed either in the way of simple or compound interest. But compound interest, being found most equitable, both for buyer and seller, the computation by simple interest is universally disused.
I. Annuities for a certain time.
Problem 1. Annuity, rate, and time, given, to find the amount, or sum of yearly payments, and interest.
Rule. Make the first term of a geometrical series and the amount of 1 l. for a year the common ratio; continue this series to as many terms as there are years in the question; and the sum of this series is the amount of 1 l. annuity for the given years; which, multiplied by the given annuity, will produce the amount sought.
Example. An annuity of 40 l. payable yearly, is forborne and unpaid till the end of 5 years; what will then be due, reckoning compound interest at 5 per cent. on all the payments then in arrear?
\[ \begin{align*} 1 & : 1.05 : 1.1025 : 1.157625 : 1.21550625 : \text{ whose sum is } 5.52563125 \text{ l.; and } 5.52563125 \times 40 = \\ & 221.02525 = 221. l. o s. 6 d. \text{ the amount sought.} \end{align*} \]
The amount may also be found thus: Multiply the given annuity by the amount of 1 l. for a year; to the product add the given annuity, and the sum is the amount in 2 years; which multiply by the amount of 1 l. for a year; to the product add the given annuity and the sum is the amount in 3 years, &c. The former question wrought in this manner follows.
\[ \begin{align*} 40 \text{ am. in 1 year.} & \quad 126.1 \text{ am. in 3 years.} \\ 1.05 & \quad 1.05 \\ 42.00 & \quad 132.405 \\ 40 & \quad 40 \\ 82 \text{ am. in 2 years,} & \quad 172.405 \text{ am. in 4 years.} \\ 1.05 & \quad 1.05 \\ 86.10 & \quad 181.02525 \\ 40 & \quad 40 \\ 126.1 \text{ am. in 3 years.} & \quad 221.02525 \text{ am. in 5 years.} \end{align*} \]
If the given time be years and quarters, find the amount for the whole years, as above; then find the amount of 1 l. for the given quarters; by which multiply the amount for the whole years; and to the product add such a part of the annuity as the given quarters are of a year.
If the given annuity be payable half yearly, or quarterly, find the amount of 1 l. for half a year or a quarter; by which find the amount for the several half-years or quarters, in the same manner as the amount for the several years is found above.
Prob. 2. II. Annuities for ever, or freehold Estates.
In freehold estates, commonly called annuities in free- simple, the things chiefly to be considered are, 1. The annuity or yearly rent. 2. The price or present worth. 3. The rate of interest. The questions that usually oc- cur on this head will fall under one or other of the fol- lowing problems.
Prob. 1. Annuity and rate of interest given, to find the price.
As the rate of l. to l. so the rent to the price.
Examp. The yearly rent of a small estate is 40l.: What is it worth in ready money, computing interest at 3½ per cent.?
L. s. d.
As .035 : 1 :: 40 : 1142.857142 = 1142 17 1/2
Prob. 2. Price and rate of interest given, to find the rent or annuity.
As l. to its rate, so the price to the rent.
Examp. A gentleman purchases an estate for 4000l. and has 4½ per cent. for his money: Required the rent.
As 1 : .045 :: 4000 : 180l. rent sought.
Prob. 3. Price and rent given, to find the rate of interest.
As the price to the rent, so 1 to the rate.
Examp. An estate of 180l. yearly rent is bought for 4000l.: What rate of interest has the purchaser for his money?
As 4000 : 180 :: 1 : .045 rate sought.
Prob. 4. The rate of interest given, to find how many years purchase an estate is worth.
Divide 1 by the rate, and the quotient is the number of years purchase the estate is worth.
Examp. A gentleman is willing to purchase an e- state; provided he can have 3½ per cent. for his money: How many years purchase may he offer?
.025 × 1000 (40 years purchase)
Prob. 5. The number of years purchase, at which an estate is bought or sold, given, to find the rate of interest.
Divide 1 by the number of years purchase, and the quotient is the rate of interest.
Examp. A gentleman gives 40 years purchase for an estate: What interest has he for his money?
40)1000 (.025 rate sought.
The computations hitherto are all performed by a single division or multiplication, and it will scarcely be perceived that the operations are conducted by the rules of compound interest; but when a reversion occurs, re- course must be had to tables of annuities on compound interest.
Prob. 6. The rate of interest, and the rent of a freehold estate in reversion, given, to find the present worth or value of the reversion.
By Prob. 1, find the price or present worth of the estate, as if possession was to commence presently; and then, by the Tables, find the present value of the given annuity, or rent, for the years prior to the commence- ment; subtract this value from the former value, and the remainder is the value of the reversion.
Examp. A has the possession of an estate of 130l. per annum, to continue 20 years; B has the reversion of the same estate from that time for ever: What is the value
Life Annuities.
The value of annuities for life is determined from observations made on the bills of mortality. Dr Halley, Mr Simpson, and Mons. de Moivre, are gentlemen of distinguished merit in calculations of this kind.
Dr Halley had recourse to the bills of mortality at Breslau, the capital of Silesia, as a proper standard for the other parts of Europe, being a place pretty central, at a distance from the sea, and not much crowded with traffickers or foreigners. He pitches upon 1000 persons all born in one year, and observes how many of these were alive every year, from their birth to the extinction of the last, and consequently how many died each year, as in the first of the following tables; which is well adapted to Europe in general. But in the city of London, there is observed to be a greater disparity in the births and burials than in any other place, owing probably to the vast resort of people thither, in the way of commerce, from all parts of the known world. Mr Simpson, therefore, in order to have a table particularly suited to this populous city, pitches upon 1280 persons all born in the same year, and records the number remaining alive each year, till none were in life.
It may not be improper, however, to observe, that however perfect tables of this sort may be in themselves, and however well adapted to any particular climate, yet the conclusions deduced from them must always be uncertain, being nothing more than probabilities, or conjectures drawn from the usual period of human life. And the practice of buying and selling annuities on lives, by rules founded on such principles, may be justly considered as a sort of lottery or chance-work, in which the parties concerned must often be deceived. But as estimates and computations of this kind are now become fashionable, we shall subjoin some brief account of such as appear most material.
From the preceding tables the probability of the continuance or extinction of human life is estimated as follows.
1. The probability that a person of a given age shall live a certain number of years, is measured by the proportion which the number of persons living at the proposed age has to the difference between the said number... Annuites.
Thus, if it be demanded, what chance a person of 40 years has to live seven years longer? from 445, the number of persons living at 40 years of age in Dr. Halley's table, subtract 377, the number of persons living at 47 years of age, and the remainder 68, is the number of persons that died during these 7 years; and the probability or chance that the person in the question shall live these 7 years is as 377 to 68, or nearly as 5½ to 1. But, by Mr. Simpson's table, the chance is something less than that of 4 to 1.
2. If the year to which a person of a given age has an equal chance of arriving before he dies, be required, it may be found thus: Find half the number of persons living at the given age in the tables, and in the column of age you have the year required.
Thus, if the question be put with respect to a person of 30 years of age, the number of that age in Dr. Halley's table is 531, the half whereof is 265, which is found in the table between 27 and 28 years; so that a person of 30 years has an equal chance of living between 27 and 28 years longer.
3. By the tables, the premium of insurance upon lives may in some measure be regulated.
Thus, the chance that a person of 25 years has to live another year, is, by Dr. Halley's table, as 80 to 1; but the chance that a person of 50 years has to live a year longer is only 30 to 1. And, consequently, the premium for insuring the former ought to be to the premium for insuring the latter for one year, as 30 to 80, or as 3 to 8.
Prob. I. To find the value of an annuity of £1. for the life of a single person of any given age.
Mont. de Moivre, by observing the decrease of the probabilities of life, as exhibited in the table, composed an algebraic theorem or canon, for computing the value of an annuity for life; which canon we here lay down by way of
Rule. Find the complement of life; and, by the tables, find the value of £1. annuity for the years denoted by the said complement; multiply this value by the amount of £1. for a year, and divide the product by the complement of life; then subtract the quotient from 1; divide the remainder by the interest of £1. for a year; and this last quotient will be the value of the annuity sought, or, in other words, the number of years purchased the annuity is worth.
Exam. What is the value of an annuity of £1. for an age of 50 years, interest at 5 per cent?
| Age | 3 per c. | 3½ per c. | 4 per c. | 4½ per c. | 5 per c. | 6 per c. | |-----|---------|-----------|---------|-----------|---------|---------| | 9 | 19.87 | 18.27 | 16.88 | 15.67 | 14.60 | 12.80 | | 10 | 19.74 | 18.16 | 16.79 | 15.59 | 14.53 | 12.75 | | 11 | 19.60 | 18.05 | 16.64 | 15.51 | 14.47 | 12.70 | | 12 | 19.47 | 17.94 | 16.60 | 15.43 | 14.41 | 12.65 | | 13 | 19.33 | 17.82 | 16.50 | 15.35 | 14.34 | 12.60 | | 14 | 19.19 | 17.71 | 16.41 | 15.27 | 14.27 | 12.55 | | 15 | 19.05 | 17.59 | 16.31 | 15.19 | 14.20 | 12.50 | | 16 | 18.90 | 17.46 | 16.21 | 15.10 | 14.12 | 12.45 | | 17 | 18.76 | 17.33 | 16.10 | 15.01 | 14.05 | 12.40 | | 18 | 18.61 | 17.21 | 15.99 | 14.92 | 13.97 | 12.35 | | 19 | 18.46 | 17.09 | 15.89 | 14.83 | 13.89 | 12.30 | | 20 | 18.30 | 16.96 | 15.78 | 14.73 | 13.81 | 12.20 | | 21 | 18.15 | 16.83 | 15.67 | 14.64 | 13.72 | 12.15 | | 22 | 17.99 | 16.69 | 15.55 | 14.54 | 13.64 | 12.10 | | 23 | 17.83 | 16.56 | 15.43 | 14.44 | 13.55 | 12.00 | | 24 | 17.66 | 16.42 | 15.31 | 14.34 | 13.46 | 11.95 | | 25 | 17.50 | 16.28 | 15.19 | 14.23 | 13.37 | 11.90 | | 26 | 17.33 | 16.13 | 15.04 | 14.12 | 13.28 | 11.80 | | 27 | 17.16 | 15.98 | 14.94 | 14.02 | 13.18 | 11.75 | | 28 | 16.98 | 15.83 | 14.81 | 13.90 | 13.09 | 11.65 | | 29 | 16.80 | 15.68 | 14.68 | 13.79 | 12.99 | 11.60 | | 30 | 16.62 | 15.53 | 14.54 | 13.67 | 12.88 | 11.50 | | 31 | 16.44 | 15.37 | 14.41 | 13.55 | 12.78 | 11.40 | | 32 | 16.25 | 15.21 | 14.27 | 13.43 | 12.67 | 11.35 | | 33 | 16.06 | 15.05 | 14.12 | 13.30 | 12.56 | 11.25 | | 34 | 15.86 | 14.89 | 13.98 | 13.17 | 12.45 | 11.15 | | 35 | 15.67 | 14.71 | 13.82 | 13.04 | 12.33 | 11.05 | | 36 | 15.46 | 14.52 | 13.67 | 12.90 | 12.21 | 11.00 | | 37 | 15.29 | 14.34 | 13.52 | 12.77 | 12.09 | 10.90 | | 38 | 15.09 | 14.16 | 13.36 | 12.63 | 11.96 | 10.80 | | 39 | 14.84 | 13.98 | 13.20 | 12.48 | 11.83 | 10.70 | | 40 | 14.63 | 13.79 | 13.02 | 12.33 | 11.70 | 10.55 | | 41 | 14.41 | 13.59 | 12.85 | 12.18 | 11.57 | 10.45 | | 42 | 14.19 | 13.40 | 12.68 | 12.02 | 11.43 | 10.35 | | 43 | 13.96 | 13.20 | 12.50 | 11.87 | 11.29 | 10.25 | | 44 | 13.73 | 12.99 | 12.32 | 11.70 | 11.14 | 10.10 | | 45 | 13.49 | 12.78 | 12.13 | 11.54 | 10.99 | 10.00 | | 46 | 13.25 | 12.56 | 11.94 | 11.37 | 10.84 | 9.85 | | 47 | 13.01 | 12.36 | 11.74 | 11.19 | 10.68 | 9.75 | | 48 | 12.76 | 12.14 | 11.54 | 11.00 | 10.51 | 9.60 | | 49 | 12.51 | 11.92 | 11.34 | 10.82 | 10.35 | 9.45 | | 50 | 12.26 | 11.69 | 11.13 | 10.64 | 10.17 | 9.30 | | 51 | 12.00 | 11.45 | 10.92 | 10.44 | 9.99 | 9.20 | | 52 | 11.73 | 11.20 | 10.70 | 10.24 | 9.82 | 9.00 | | 53 | 11.46 | 10.95 | 10.47 | 10.04 | 9.63 | 8.85 | | 54 | 11.18 | 10.69 | 10.24 | 9.82 | 9.44 | 8.70 | | 55 | 10.90 | 10.44 | 10.01 | 9.61 | 9.24 | 8.55 | | 56 | 10.61 | 10.18 | 9.77 | 9.39 | 9.04 | 8.35 | | 57 | 10.32 | 9.91 | 9.52 | 9.16 | 8.83 | 8.20 | | 58 | 10.03 | 9.64 | 9.27 | 8.93 | 8.61 | 8.00 | | 59 | 9.73 | 9.36 | 9.01 | 8.60 | 8.39 | 7.80 |
Complement of life, 36(17.374140).482615
From unity, viz. 1.000000
Subtract .482615
Interest of £1. .05 = 517385 (10.3477 value sought.
By the preceding problem is constructed the follow- The above table shows the value of an annuity of one pound for a single life, at all the current rates of interest; and is esteemed the best table of this kind extant, and preferable to any other of a different construction. But yet those who sell annuities have generally one and a half or two years more value, than specified in the table, from purchasers whose age is 20 years or upwards.
Annuities of this sort are commonly bought or sold at so many years purchase; and the value assigned in the table may be so reckoned. Thus the value of an annuity of one pound for an age of 50 years, at 3 per cent. interest, is 12.61; that is, 12l. 10s. or twelve and a half years purchase. The marginal figures on the left of the column of age serve to shorten the table, and signify, that the value of an annuity for the age denoted by them, is the same with the value of an annuity for the age denoted by the numbers before which they stand. Thus the value of an annuity for the age of 9 and 10 years is the same; and the value of an annuity for the age of 6 and 14, for the age of 3 and 24, &c. is the same. The further use of the table will appear in the questions and problems following.
QUEST. 1. A person of 50 years would purchase an annuity for life of 200l.: What ready money ought he to pay, reckoning interest at 4½ per cent.?
By the table the value of 1l. is 10.8
Multiply by 200
Value to be paid in ready money 2164.00 Ans.
QUEST. 2. A young merchant marries a widow lady of 40 years of age, with a jointure of 300l. a-year, and wants to dispose of the jointure for ready money: What sum ought he to receive, reckoning interest at 3½ per cent.?
By the table the value of 1l. is 13.98
Multiply by 300
Value to be received in ready money 4194.00 Ans.
PROB. 2. To find the value of an annuity for the joint continuance of two lives, one life failing, the annuity to cease.
Here there are two cases, according as the ages of the two persons are equal or unequal.
1. If the two persons be of the same age, work by the following
RULE. Take the value of any one of the lives from the table; multiply this value by the interest of 1l. for a year; subtract the product from 2; divide the foresaid value by the remainder; and the quot will be the value of 1l. annuity, or the number of years purchase sought.
EXAMP. What is the value of 100l. annuity for the joint lives of two persons, of the age of 30 years each, reckoning interest at 4 per cent.?
By the table, one life of 30 years is 14.68
Multiply by .04
Subtract the product 5872
From 20000
Remains 14128
And 14128)14.68(10.39 value of 1l. annuity.
And 10.39×100=1039 the value sought.
2. If the two persons are of different ages, work as directed in the following
RULE. Take the values of the two lives from the table; multiply them into one another, calling the result the first product; then multiply the said first product by the interest of 1l. for a year, calling the result the second product; add the values of the two lives, and from their sum subtract the second product; divide the first product by the remainder, and the quot will be the value of 1l. annuity, or the number of years purchase sought.
EXAMP. What is the value of 70l. annuity for the joint lives of two persons, whereof one is 40 and the other 50 years of age, reckoning interest at 5 per cent.?
By the table the value of 40 years is 11.83
And the value of 50 years is 10.35
First product, 122.4405
Multiply by .05
Second product, 6.122025
Sum of the two lives, 22.180000
Second product deduct, 6.122025
Remainder, 16.057975
And 16.057975)122.4405(7.62 value of 1l. annuity.
533.40 value sought.
PROB. 3. To find the value of an annuity upon the longest of two lives; that is, to continue so long as either of the persons is in life.
RULE. From the sum of the values of the single lives subtract the value of the joint lives, and the remainder will be the value sought.
EXAMP. What is the value of an annuity of 1l. up- **ANNUITIES**
On the longest of two lives, the one person being 30, and the other 40 years of age, interest at 4 per cent.
By the table, 30 years is - 14.68 40 years is - 13.20
Value of their joint lives, by Prob. 2. 27.88 Cafe 2. is 9.62
Value sought, 18.26 If the annuity be any other than 1l., multiply the answer found as above by the given annuity.
If the two persons be of equal age, find the value of their joint lives by Cafe 1. of Prob. 2.
**Prob. 4. To find the value of the next presentation to a living.**
**Rule.** From the value of the successor's life subtract the joint value of his and the incumbent's life, and the remainder will be the value of 1l. annuity; which multiplied by the yearly income, will give the sum to be paid for the next presentation.
**Examp.** A enjoys a living of 100l. per annum, and B would purchase the said living for his life after A's death: The question is, What he ought to pay for it, reckoning interest at 5 per cent. A being 60, and B 25 years of age?
By the table, B's life is 13.46 Joint value of both lives, by Prob. 2. is 6.97
The value of 1l. annuity, 6.49 Multiply by 100
Value of next presentation, 649.00
The value of a direct presentation is the same as that of any other annuity for life, and is found for 1l. by the table: which being multiplied by the yearly income, gives the value sought.
**Prob. 5. To find the value of a reversion for ever, after two successive lives; or to find the value of a living after the death of the present incumbent and his successor.**
**Rule.** By Prob. 3, find the value of the longest of the two lives, and subtract that value from the value of the perpetuity, and the remainder will be the value sought.
**Examp.** A, aged 50, enjoys an estate or living of 100l. per annum; B, aged 30, is intitled to its lifetime of the same estate after A's death; and it is proposed to sell the estate just now with the burden of A and B's lives on it: What is the reversion worth, reckoning interest at 4 per cent.?
By the table, A's life of 50 is, 11.34 B's life of 30 is, 14.68
Sum, 26.02 Value of their joint lives, found by Prob. 2. Cafe 2. is, 8.60 Value of the longest life, 17.42 sub. From the value of the perpetuity, 25.00
Remains the value of 1l. reversion, 7.58 Multiply by 100
Value of the reversion, 758.00
**Prob. 6. To find the value of the joint continuance Life Annuities of three lives, one life failing, the annuity to cease.**
**Rule.** Find the single values of the three lives from the table; multiply these single values continually, calling the result the product of the three lives; multiply that product by the interest of 1l., and that product again by 2, calling the result the double product; then, from the sum of the several products of the lives, taken two and two, subtract the double product; divide the product of the three lives by the remainder, and the quotient will be the value of the three joint lives.
**Examp.** A is 18 years of age, B 34, and C 56: What is the value of their joint lives, reckoning interest at 4 per cent.?
By the table, the value of A's life is 16.1, of B's 14.12, and of C's 10.01.
16.1 × 14.12 × 10.01 = 2275.6, product of the three lives,
91.024
182.048, double product.
Product of A and B, 16.1 × 14.12 = 227.33 A and C, 16.1 × 10.01 = 161.16 B and C, 14.12 × 10.00 = 141.34
Sum of all, two and two, 529.83 Double product subtract 182.048
Remainder 347.782
And 347.782 × 2275.600 = 6.54 value sought.
**Prob. 7. To find the value of an annuity upon the longest of three lives.**
**Rule.** From the sum of the values of the three single lives, taken from the table, subtract the sum of all the joint lives, taken two and two, as found by Prob. 2, and to the remainder add the value of the three joint lives, as found by Prob. 6, and that sum will be the value of the longest life sought.
**Examp.** A is 18 years of age, B 34, and C 56: What is the value of the longest of these three lives, interest at 4 per cent.?
By the table, the single value of A's life is 16.1 single value of B's life is 14.12 single value of C's life is 10.01
Sum of the single values, 40.23
By Prob. 2, the joint value of A and B is 10.76 joint value of A and C is 8.19 joint value of B and C is 7.65
Sum of the joint lives, 26.60
Remainder, 13.63
By Prob. 6, the value of the 3 joint lives is 6.54
Value of the longest of the 3 lives, 20.17
Other problems might be added, but these aduced are sufficient for most purposes. The reader probably may wish that the reason of the rules, which, it must be owned, are intricate, had been assigned; but this could not be done without entering deeper into the subject than was practicable in this place. See Chances.
**ANNUITY**