EXAMPLE IV.-What annuity, falling due in annual instalments, would have amounted to £300 in 10 years, at 5 per cent.? If the instalments were 1 each, the amount would be £ 1·051o—1—£12′578 (nearly). The required annuity, there

*05

300

fore, is £12578 £23.85=£23 178.

EXAMPLE V. What sum must a person save every half-year in order that, the savings being invested at 3 per cent.-compound interest, he may be worth £2,000 in 16

6 years? If the sum saved half-yearly were £1, the amount realized would be £1015—1 40688 (nearly); saving I every

32

*015

half-year, and investing the money at compound interest, being obviously the same as allowing an annuity which falls due in half-yearly instalments of £1 each to remain out at compound interest. The required sum, therefore, is £2,000 = £49'154 =£49 3s. Id.

40·688

EXAMPLE VI.-What sum must a person save quarterly in order that, the savings being invested at 8 per cent.-compound interest, he may be worth £1,000 in 6 years?

If the sum saved quarterly were £1, the amount realized would be £1022-1-30-422. The required sum, therefore,

'02

I

==£32·87=£32 17s. 5d. (nearly.)

311. To determine the present value of £1 which is to become due at a future time: Find what £1, if invested as principal, would have amounted to, at compound interest, in the given time; and take the reciprocal of this amount.

At 4 per cent., for instance, £1, ready-money, is equivalent to 104 payable a year hence; so that the ready-money equivalent to 1 payable a year hence is the fourth term of the proportion—

(Payable

in a year)

(Ready-
money)

(Payable in a year)

(Readymoney)

£1.04

: £I

:

£104

I

ΙΟ

Again: 1, ready-money, is equivalent to £1042 payable in

2 years; so that the ready-money equivalent to 1 payable in 2 years is the fourth term of the proportion—

In like manner, £1, ready-money, is equivalent to £1.043 payable in 3 years; so that the ready-money equivalent to £1 payable in 3 years, is the fourth term of the proportion—

(Readymoney) £104

I

It could be shown, in the same way, that the present value

of £1 payable in

If, however, interest on principal fell due half-yearly, the present value of £1 payable in

And if interest on principal fell due quarterly, the present

value of £1 payable in

312. To determine the present value of an annuity falling due in instalments of £1 each, and continuing for a certain number of "periods "years, half-years, or quarters, as the case may be: Find the present value of £1 payable at the expiration of the given time (§ 311); subtract this present value from unity; and divide the remainder by the interest of I for one period.

Let us suppose that an annuitant, entitled to £1 a year for 12 years, sells his claim for ready-money when the rate of interest is 4 per cent. The purchaser of this claim is to receive I at the end of a year, another £1 at the end of 2 years, another I at the end of 3 years, and so on-the last I becoming due at the end of 12 years. The present value of the first

I

pound is £104

I

I

; of the second pound, 1.04; of the third £;

I'04

pound, £43; and so on-the present value of the last pound

104°

These fractions form a geometrical progression, which has

I

I

I

1'04

for common ratio; and their sum is {1-04-1-041-04)

I

}

If the instalments of £1 each were half-yearly ones, the present value of the annuity would be-

And if the instalments of 1 each were quarterly ones, the present value of the annuity would be

Here we observe that the present value of £1 payable in 12

I

I

I

years is £129 or 2 or 18-according as interest

10412

on principal falls due yearly, half-yearly, or quarterly; also, that the interest of £1 for one period is

313. To determine the present value of an annuity falling due in instalments of P pounds each, and continuing for a certain time: Find what the present value would be if the instalments were £1 each; and multiply the result by P.

It is obvious that when-other circumstances being the same -the present value of an annuity of £1 is multiplied by

2

£2

3 we obtain the present value of an annuity of £3

4

£4

&c.

&c. EXAMPLE VII.—What is the present value of an annuity falling due in annual instalments of £24 each, and continuing for 7 years-money being worth 5 per cent.?

If the instalments were £1 each, the present value would be

I

I

1057

I

I

1057

The required present value, therefore, is £*05 X24 £138.874 (nearly)=£138 17s. 6d. (nearly.)

*05

EXAMPLE VIII.-What is the present value of an annuity falling due in half-yearly instalments of £72 each, and continuing for 13 years-money being worth 6 per cent.?

If the instalments were I each, the present value would be

The required present value, therefore, is £

X72 £1,287 13=£1,287 28. 7d.

EXAMPLE IX.-What is the present value of an annuity falling due in quarterly instalments of £120 17s. 6d. each, and continuing for II years-money being worth 3 per cent?

If the instalments were £1 each, the present value would be

I

I

1'007511.
*0075

The required present value, therefore, is

£

£

I

I

1'007544

*0075

· X120·875=£4,515.89=£4,515 17s. 9d.

314. The present value of a "perpetual" annuity of 1 is the reciprocal of the interest of £1 for one period.

At 4 per cent., for instance, the present value of a perpetual annuity falling due in annual instalments of £1 each is

I

I

I

+&c., ad infinitum. 104 1042 1*043

The sum of the terms of this progression (§ 266, Note 1) is

I I'04. I

X

'04
104 104 104 104

*04 '04

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

And if the instalments of £1 each were quarterly ones, the present value would be

315. To determine the present value of a perpetual annuity falling due in instalments of P pounds each: Find what the present value would be if the instalments were 1 each, and multiply the result by P.

In other words: Divide P by the interest of £1 for one period.