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 reverification, 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 disfused.

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 1 the first term of a geometrical series, and the amount of 1l. 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 1l. annuity for the given years; which, multiplied by the given annuity, will produce the amount sought.

Example. An annuity of 40l. 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?

1 : 1.05 Annuity.

\[ \frac{1}{5} : \frac{5}{5} : \frac{4}{5} : \frac{5}{5} : \frac{1}{1.05} : \frac{1.157625}{1.21550625} : \frac{1.21550625}{5.25563125} : \frac{5.25563125}{40} = 221.02525 = 221l. os. 6d. the amount sought.

The amount may also be found thus: Multiply the given annuity by the amount of \( \frac{1}{1} \) for a year; to the product add the given annuity, and the sum is the amount in two years; which multiply by the amount of \( \frac{1}{1} \) 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.

| 40 am. in 1 year. | 126.1 am. in 3 years. | |-------------------|----------------------| | 1.05 | 1.05 | | 42.00 | 132.405 | | 40 | 40 | | 82 am. in 2 years. | 172.405 am. in 4 years. | | 1.05 | 1.05 | | 86.10 | 181.02525 | | 40 | 40 | | 126.1 am in 3 years. | 221.02525 am. in 5 years. |

If the given time be years and quarters, find the amount for the whole years, as above; then find the amount of \( \frac{1}{1} \) 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 \( \frac{1}{1} \) 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.** Annuity, rate, and time given, to find the present worth, or sum of money that will purchase the annuity.

**Rule.** Find the amount of the given annuity by the former problem; and then, by compound interest, find the present worth of this amount, as a sum due at the end of the given time.

**Examp.** What is the present worth of an annuity of 40l. to continue 5 years, discounting at 5 per cent. compound interest?

By the former problem, the amount of the given annuity for 5 years, at 5 per cent., is 221.02525; and by compound interest, the amount of \( \frac{1}{1} \) for 5 years, at 5 per cent., is 1.2762815625.

And, \( \frac{1.2762815625}{221.02525} = 173l. 3s. 7d. \) the present worth sought.

The present worth may be also found thus: By compound interest, find the present worth of each year by itself; and the sum of these is the present worth sought.

The former example done in this way follows.

\[ \begin{align*} 1.2762815625 \times 40 &= 51.3410 \\ 1.21550625 \times 40 &= 32.9080 \\ 1.157625 \times 40 &= 34.5535 \\ 1.1025 \times 40 &= 36.2811 \\ 1.05 \times 40 &= 38.0952 \\ \end{align*} \]

Present worth, 173.1788

If the annuity to be purchased be in reversion, find first the present worth of the annuity, as commencing immediately, by any of the methods taught above; and then, by compound interest, find the present worth of that present worth, rebating for the time in reversion; and this last present worth is the answer.

**Examp.** What is the present worth of a yearly pension or rent of 75l. to continue 4 years, but not to commence till 3 years hence, discounting at 5 per cent.?

\[ \begin{align*} 0.05 : 1 :: 75 : 1500 \\ 1.05 \times 1.05 \times 1.05 \times 1.05 = 1.21550625 \\ 1.21550625 \times 1500 = 1234.05371 \\ 1500 \\ 1234.05371 \\ 265.94629, \text{ present worth of the annuity, if it was to commence immediately.} \\ 1.05 \times 1.05 \times 1.05 = 1.157625 \\ L. s. d. \\ 1.157625 \times 265.94629 = 229.7344 = 229l. 14s. 8½d. \end{align*} \]

**Prob. 3.** Present worth, rate and time given, to find the annuity.

**Rule.** By the preceding problem, find the present worth of \( \frac{1}{1} \) annuity for the rate and time given; and then say, As the present worth thus found to \( \frac{1}{1} \) annuity, so the present worth given to its annuity; that is, divide the given present worth by that of \( \frac{1}{1} \) annuity.

**Examp.** What annuity, to continue 5 years, will \( \frac{1}{1} \) annuity, allowing compound interest at 5 per cent.?

\[ \begin{align*} 0.05 : 1 :: 1 : 20l. \\ 1.05 \times 1.05 \times 1.05 \times 1.05 \times 1.05 = 1.2762815625 \\ 1.2762815625 \times 20 = 15.6705 \\ 20 \\ 15.6705 \\ 4.3295 \text{ present worth of } \frac{1}{1} \text{ annuity.} \\ 4.329(173.179(40l. annuity. Ans: \end{align*} \]

**II. Annuities for ever, or freehold Estates.**

In freehold estates, commonly called annuities in fee-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 occur on this head will fall under one or other of the following problems.

**Prob. 1.** Annuity and rate of interest given, to find the price.

As the rate of \( \frac{1}{1} \) to \( \frac{1}{1} \), 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.?

As \( 0.035 : 1 :: 40 : 1142.857142 = L. 1142 17 1½ \).

**Prob. 2.** Price and rate of interest given, to find the rent or annuity.

As \( \frac{1}{1} \) 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 \( 1 : 0.045 :: 4000 : 1 : 180l. \) rent sought.

**Prob. 3.** Price and rent given, to find the rate of interest.

As the price to the rent, so \( \frac{1}{1} \) 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 \( 4000 : 180 :: 1 : 0.045 \) rate sought. Annuity.

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 estate, provided he can have 2½ per cent. for his money: How many years purchase may he offer?

\[ \frac{1}{0.025} = 40 \text{ years purchase.} \]

Ans.

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?

\[ \frac{1}{40} = 0.025 \text{ rate sought.} \]

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, recourse 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 commencement; 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 £130 l. per annum, to continue 20 years; B has the reversion of the same estate from that time for ever: What is the value of the estate, what the value of the 20 years possession, and what the value of the reversion, reckoning compound interest at 6 per cent.?

By Prob. 1. \( \frac{1}{0.06} \times 130.00 = 2166.666 \) value of the estate.

By Tables \( 1491.0896 \) val. of the possession.

\[ 675.5770 \text{ val. of the reversion.} \]

Prob. 7. The price or value of a reversion, the time prior to the commencement, and rate of interest, given, to find the annuity or rent.

By the Tables, find the amount of the price of the reversion for the years prior to the commencement; and then by Prob. 3. find the annuity which that amount will purchase.

Examp. The reversion of a freehold estate, to commence 20 years hence, is bought for £675.577 l. compound interest being allowed at 6 per cent.: Required the annuity or rent?

By the Tables the amount of \( 675.577l. \times 2166.6 \)

for 20 years, at 6 per cent. is

By Prob. 2. \( 2166.6 \times 0.06 = 130.0 \) rent sought.

III. 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 Breslaw, 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.

Dr Halley's Table on the bills of mortality at Breslaw.

| Age | Perf. liv. | Perf. liv. | Perf. liv. | Perf. liv. | |-----|------------|------------|------------|------------| | 1 | 1000 | 24 | 573 | 47 | 377 | 70 | 142 | | 2 | 855 | 25 | 567 | 48 | 367 | 71 | 131 | | 3 | 798 | 26 | 560 | 49 | 357 | 72 | 120 | | 4 | 760 | 27 | 553 | 50 | 346 | 73 | 109 | | 5 | 732 | 28 | 546 | 51 | 335 | 74 | 98 | | 6 | 710 | 29 | 539 | 52 | 324 | 75 | 88 | | 7 | 692 | 30 | 531 | 53 | 313 | 76 | 78 | | 8 | 680 | 31 | 523 | 54 | 302 | 77 | 68 | | 9 | 670 | 32 | 515 | 55 | 292 | 78 | 58 | | 10 | 661 | 33 | 507 | 56 | 282 | 79 | 49 | | 11 | 653 | 34 | 499 | 57 | 272 | 80 | 41 | | 12 | 646 | 35 | 490 | 58 | 262 | 81 | 34 | | 13 | 640 | 36 | 481 | 59 | 252 | 82 | 28 | | 14 | 634 | 37 | 472 | 60 | 242 | 83 | 23 | | 15 | 628 | 38 | 463 | 61 | 232 | 84 | 20 | | 16 | 622 | 39 | 454 | 62 | 222 | 85 | 15 | | 17 | 616 | 40 | 445 | 63 | 212 | 86 | 11 | | 18 | 610 | 41 | 436 | 64 | 202 | 87 | 8 | | 19 | 604 | 42 | 427 | 65 | 192 | 88 | 5 | | 20 | 598 | 43 | 417 | 66 | 182 | 89 | 3 | | 21 | 592 | 44 | 407 | 67 | 172 | 90 | 1 | | 22 | 586 | 45 | 397 | 68 | 162 | 91 | 0 | | 23 | 579 | 46 | 387 | 69 | 152 | | |

Mr 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 57 and 58 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 1l. for the life of a single person of any given age.

Mons. 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 the way of

**Rule.** Find the complement of life; and, by the tables, find the value of 1l. annuity for the years denoted by the said complement; multiply this value by the amount of 1l. 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 1l. 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.

**Exampl.** What is the value of an annuity of 1l. for an age of 50 years, interest at 5 per cent?

| Age | Perf. liv. | A. Perf. liv. | A. Perf. liv. | A. Perf. liv. | |-----|------------|---------------|---------------|---------------| | 0 | 1280 | 24 | 434 | 48 | | 1 | 870 | 25 | 426 | 49 | | 2 | 700 | 26 | 418 | 50 | | 3 | 635 | 27 | 410 | 51 | | 4 | 600 | 28 | 402 | 52 | | 5 | 580 | 29 | 394 | 53 | | 6 | 564 | 30 | 385 | 54 | | 7 | 551 | 31 | 376 | 55 | | 8 | 541 | 32 | 367 | 56 | | 9 | 532 | 33 | 358 | 57 | | 10 | 524 | 34 | 349 | 58 | | 11 | 517 | 35 | 340 | 59 | | 12 | 510 | 36 | 331 | 60 | | 13 | 504 | 37 | 322 | 61 | | 14 | 498 | 38 | 313 | 62 | | 15 | 492 | 39 | 304 | 63 | | 16 | 486 | 40 | 294 | 64 | | 17 | 480 | 41 | 284 | 65 | | 18 | 474 | 42 | 274 | 66 | | 19 | 468 | 43 | 264 | 67 | | 20 | 462 | 44 | 255 | 68 | | 21 | 455 | 45 | 246 | 69 | | 22 | 448 | 46 | 237 | 70 | | 23 | 441 | 47 | 228 | 71 |

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 and the number of persons living at the given age.

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.

The 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 12.51; that is, 12 l. 10 s. 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 By the table the value of 1l. is 13.98

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 foreaid value by the remainder; and the quotient will be the value of 1l. annuity, or the number of years purchase sought.

Exampl. 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 - 2.0000

Remains - 1.4128

And 1.4128 x 14.68 = 10.39 value of 1l. annuity. And 10.39 x 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 quotient will be the value of 1l. annuity, or the number of years purchase sought.

Exampl. 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 x 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.

Exampl. What is the value of an annuity of 1l. np.

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.

Exampl. 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.

Exampl. A, aged 50, enjoys an estate or living of 100l. per annum; B, aged 30, is intitled to his 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 Annuity.

Prob. 6. To find the value of the joint continuance 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.

Exam. 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.

\[16.1 \times 14.12 \times 10.01 = 2275.6\] product of the three lives,

\[91.024\]

\[182.048\] double product.

Product of A and B, 16.1 \times 14.12 \times 10.01 = 161.16

A and C, 16.1 \times 10.01 = 161.16

B and C, 14.12 \times 10.01 = 141.34

Sum of all, two and two, 529.83

Double product subtract, 182.048

Remainder, 347.782

And \(347.782 \times 2275.6 = 6.54\) 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.

Exam. 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 adduced are sufficient for most purposes. The reader probably may wish that the reason of the rules, which, if well known, 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.

Annuities (Borrowing upon): one of the methods Annuity employed by government for raising supplies.

Of this there are two methods; that of borrowing upon annuities for terms of years, and that of borrowing upon annuities for lives.

During the reigns of king William and queen Anne, large sums were frequently borrowed upon annuities for terms of years, which were sometimes longer and sometimes shorter. In 1693, an act was passed for borrowing one million upon an annuity of 14 per cent., or of 140,000 l. a-year for 16 years. In 1691, an act was passed for borrowing a million upon annuities for lives, upon terms which in the present times would appear very advantageous. But the subscription was not filled up. In the following year the deficiency was made good by borrowing upon annuities for lives at 14 per cent., or at little more than seven years purchase. In 1695, the persons who had purchased those annuities were allowed to exchange them for others of 96 years, upon paying into the exchequer 63 pounds in the hundred; that is, the difference between 14 per cent. for life, and 14 per cent. for 96 years, was sold for 63 pounds, or for four and a half years purchase. Such was the supposed instability of government, that even these terms procured few purchasers. In the reign of queen Anne, money was upon different occasions borrowed both upon annuities for lives and upon annuities for terms of 32, of 89, of 98, and of 99 years. In 1719, the proprietors of the annuities for 32 years were induced to accept in lieu of them South Sea stock to the amount of eleven and a half years purchase of the annuities, together with an additional quantity of stock equal to the arrears which happened then to be due upon them. In 1720, the greater part of the other annuities for terms of years both long and short were subscribed into the same fund. The long annuities at that time amounted to 666,821 l. 8 s. 3½ d. a-year. On the 5th of January, 1775, the remainder of them, or what was not subscribed at that time, amounted only to 136,453 l. 12 s. 8 d.

During the two wars which began in 1739 and in 1755, little money was borrowed either upon annuities for terms of years, or upon those for lives. An annuity for 98 or 99 years, however, is worth nearly as much money as a perpetuity, and should, therefore, one might think, be a fund for borrowing nearly as much. But those who, in order to make family-settlements, and to provide for remote futurity, buy into the public stocks, would not care to purchase into one of which the value was continually diminishing; and such people make a very considerable proportion both of the proprietors and purchasers of stock. An annuity for a long term of years, therefore, though its intrinsic value may be very nearly the same with that of a perpetual annuity, will not find nearly the same number of purchasers. The subscribers to a new loan, who mean generally to sell their subscription as soon as possible, prefer greatly a perpetual annuity redeemable by parliament, to an irredeemable annuity for a long term of years of only equal amount. The value of the former may be supposed always the same, or very nearly the same; and it makes, therefore, a more convenient transferable stock than the latter.

During the two last-mentioned wars, annuities, either for terms of years or for lives, were seldom granted but Annuity, as premiums to the subscribers to a new loan, over and above the redeemable annuity or interest upon the credit of which the loan was supposed to be made. They were granted, not as the proper fund upon which the money was borrowed; but as an additional encouragement to the lender.

Annuities for lives have occasionally been granted in two different ways; either upon separate lives, or upon lots of lives, which in French are called Tontines, from the name of their inventor. When annuities are granted upon separate lives, the death of every individual annuitant disturbs the public revenue so far as it was affected by his annuity. When annuities are granted upon tontines, the liberation of the public revenue does not commence till the death of all the annuitants comprehended in one lot, which may sometimes consist of twenty or thirty persons, of whom the survivors succeed to the annuities of all those who die before them; the last survivor succeeding to the annuities of the whole lot. Upon the same revenue more money can always be raised by tontines than by annuities for separate lives. An annuity, with a right of survivorship, is really worth more than an equal annuity for a separate life, and from the confidence which every man naturally has in his own good fortune, the principle upon which is founded the success of all lotteries, such an annuity generally sells for something more than it is worth. In countries where it is usual for government to raise money by granting annuities, tontines are upon this account generally preferred to annuities for separate lives. The expedient which will raise most money, is almost always preferred to that which is likely to bring about in the speediest manner the liberation of the public revenue.

In France a much greater proportion of the public debts consists in annuities for lives than in England. According to a memoir presented by the parliament of Bordeaux to the king in 1764, the whole public debt of France is estimated at twenty-four hundred millions of livres; of which the capital for which annuities for lives had been granted, is supposed to amount to three hundred millions, the eighth-part of the whole public debt. The annuities themselves are computed to amount to thirty millions a-year, the fourth part of one hundred and twenty millions, the supposed interest of that whole debt. It is not the different degrees of anxiety in the two governments of France and England for the liberation of the public revenue, which occasions this difference in their respective modes of borrowing; it arises altogether from the different views and interests of the lenders.

In Britain, the seat of government being in the greatest mercantile city in the world, the merchants are generally the people who advance money to government. By advancing it they do not mean to diminish, but, on the contrary, to increase their mercantile capitals; and unless they expected to sell with some profit their share in the subscription for a new loan, they never would subscribe. But if by advancing their money they were to purchase, instead of perpetual annuities, annuities for lives only, whether their own or those of other people, they would not always be so likely to sell them with a profit. Annuities upon their own lives they would always sell with loss; because no man will give for an annuity upon the life of another whose age and state are nearly the same with his own, the same price which he would give for one upon his own. An annuity upon the life of a third person, indeed, is, no doubt, of equal value to the buyer and the seller; but its real value begins to diminish from the moment it is granted, and continues to do so more and more as long as it subsists. It can never, therefore, make so convenient a transferable stock as a perpetual annuity, of which the real value may be supposed always the same, or very nearly the same.

In France, the seat of government not being in a great mercantile city, merchants do not make so great a proportion of the people who advance money to government. The people concerned in the finances, the farmers-general, the receivers of the taxes which are not in farm, the court-bankers, &c. make the greater part of those who advance their money in all public exigences. Such people are commonly men of mean birth, but of great wealth, and frequently of great pride. They are too proud to marry their equals, and women of quality disdain to marry them. They frequently resolve, therefore, to live bachelors; and having neither any families of their own, nor much regard for those of their relations, whom they are not always very fond of acknowledging, they desire only to live in splendor during their own time, and are not unwilling that their fortune should end with themselves. The number of rich people, besides, who are either averse to marry, or whose condition of life renders it either improper or inconvenient for them to do so, is much greater in France than in England. To such people, who have little or no care for posterity, nothing can be more convenient than to exchange their capital for a revenue, which is to last just as long, and no longer than they wish it to do.