ANNUITIES.

302. Property of any description, when given as an equivalent for a fixed annual income, the income being payable in yearly, half-yearly, or quarterly instalments, according to agreement,--is said to be converted into an ANNUITY.

Although, however, usually employed in this restricted sense, the termannuity" may be applied to any fixed annual income -such as a salary, a pension, &c.

CONTINGENT.

303. Annuities are of two kinds-CERTAIN and An annuity is "certain" when it is to continue for a definite number of years (10, 20, 50, &c.—as the case may be), or for ever; but when the length of time depends upon the life of some particular person, or upon the life of the survivor of two or more persons, an annuity is a "contingent”—or a LIFE-annuity.

304. A" certain" annuity which is to continue for ever is called a PERPETUAL annuity, or a PERPETUITY.

305. When an annuity-either certain or contingent becomes payable at once, it is said to be immediate, or in possession: when, on the other hand, it is not to be available until a certain period of time shall have elapsed, or until a future event (somebody's death, for instance) shall have occurred, an annuity is known as a deferred—or a reversionary-annuity.

Annuities "in possession" are sometimes spoken of as annuities on lives; and "reversionary" annuities, as annuities on survivorship.

306. When an annuity is allowed to remain unpaid for a certain length of time, the sum to which, for arrears and compound interest chargeable upon them, the annuitant becomes entitled is called the AMOUNT of the annuity.

307. By the PRESENT VALUE of an annuity is
meant the ready-money which, if improved at
compound interest, would exactly pay the annui-

tant's claim in full.

If an annuity, falling due in annual instalments, remained.
unpaid for (say) 10 years, the annuitant would be entitled, not
only to the arrears, but also to compound interest for

and to a year's (simple) interest on the 9th instalment.*

If, therefore, the instalments were £1 each, and the rate of
interest 3 per cent., the amount of the annuitant's claim for
the 10 years would be (§ 297)—

39

1+103+1032+103+103'+1'03+103°+1°03'+1'03 +1.039

£

*Upon the 10th instalment, due at the end of the 10th year, no

interest would be chargeable.

It will be seen that the numbers 1, 103, 1032, 1·033, &c. form a geometrical progression, which has 103 for common 103 X 1.03-1. ratio; so that (§ 266) the sum of the terms is

103-I

If the instalments were half-yearly ones of £1 each, the amount would be

would be

66

66

18th

17th

23

66

61+

+3

The sum of the terms of this progression, which has 1015

for common ratio, is

1*015”×1*015 — I __I°015 — I ̧

1015-1

=

*015

If the instalments were quarterly ones of £1 each, the amount

Ist

The sum of the terms of this progression, which has 1.0075 for common ratio, is 1·0075X1·0075—1__ I '007540 — I

39

1'0075-1

*0075

308. To determine the amount of an unpaid annuity falling due in instalments of £1 each: Divide the interest of £1 for one "period"-i.e., for a year, a half-year, or a quarter, as the case may be-into the compound interest of £1 for the given time.

40

I

I

, and 100750-1

*0075

year

half-year

quarter

Also, that the compound interest of £1 for 10 years is 1031o — I, 1015201, or 100750-1, according as interest on principal is supposed to fall due in yearly, half-yearly, or quarterly instal

ments.

Ist

+39

66

309. To determine the amount of an unpaid annuity falling due in instalments of P pounds each: Find what the amount would be if the instalments were 1 each (§ 308), and multiply this amount by P.

All other circumstances being the same, it is obvious_that, whatever the amount may be for instalments of £1 each, the amount would be

EXAMPLE I.-An annuity, falling due in annual instalments of £36 158. each, remained unpaid for 17 years; find the amount, at 3 per cent.

17

If the instalments were 1 each, the amount would be 1035-1. The required amount, therefore, is 1035”—1

*035

X 36.75 £834'408 £834 8s. 2d.

*035

EXAMPLE II.-An annuity, falling due in half-yearly instalments of £123 7s. 6d. each, remained unpaid for 13 years; find the amount, at 4 per cent.

£

If the instalments were I each, the amount would be -I'0226. -I

10226-1. The required amount, therefore, is £

'02

× 123'375=£4154°16=£4154 38. 24d.

'02

EXAMPLE III.--An annuity, falling due in quarterly instalments of £360 each, remained unpaid for 7 years; find the amount, at 5 per cent.

If the instalments were 1 each, the amount would be £10125-1. The required amount, therefore, is £101252 — I

*0125

I

x 360=11,980'44 £11,980 8s. 94d.

28

0125

310. The "amount" of an annuity for a certain time, and at a certain rate, being known, to find the annuity itself: Divide the given amount by the amount of an annuity of £1 for the same time, and at the same rate.

This follows from § 309.