Example 5.

In what time will a sum of money double itself, allowing 4 per cent. pound interest?

Here s, p,r are given, and t is sought.

From the formula (7) we have

com

In like manner, if it be required to find in what time a sum will triple itself at the same rate, we have

=

log. 3. log. 1.04

PRESENT VALUE AND DISCOUNT AT COMPOUND INTEREST.

If we call p the present value of a sum s due t years hence, and d its discount, reasoning precisely in the same manner as in the case of simple interest, we

PROBLEM VIII. To find the amount of an annuity a continued for t years, compound interest being allowed at the rate r.

At the end of the first year the annuity a will become due, at the end of the second year a second payment a will become due, together with the interest of the first payment a for one year, that is, ar; the whole sum upon which interest must now be computed is thus 2 a + a r.

At the end of the third year a further payment a becomes due, together with the interest on 2 a + a r, i. e. 2 ar + a r2; the whole sum upon which interest must now be computed is 3 a + 3 ar+ar 2. The result will appear evident when exhibited under the following form:

a {1 + (1 + r) + (1 + 1)2 ++ (1 + r)'−1 }

PROBLEM IX. To find the present value of an annuity a payable for ₺ years, compound interest being allowed at the rate r.

It is manifest that the present value of this annuity must be a sum such, that if put out to interest for t years at the rate 7, its amount at the end of that period will be the same as the amount of the annuity.

Hence, if we call this present value p, we shall have, by Probs. VII. and VIII.

What is the present value of an annuity of £500, to last for 40 years, compound interest being allowed at the rate of 21 per cent. per annum.

Example 5.

In what time will a sum of money double itself, allowing 4 per cent. compound interest?

In like manner, if it be required to find in what time a sum will triple itself at the same rate, we have

=

log. 3. log. 1.04

PRESENT VALUE AND DISCOUNT AT COMPOUND INTEREST.

If we call Р the present value of a sum s due t years hence, and d its discount, reasoning precisely in the same manner as in the case of simple interest, we shall find

PROBLEM VIII. To find the amount of an annuity a continued for compound interest being allowed at the rate r.

years,

At the end of the first year the annuity a will become due, at the end of the second year a second payment a will become due, together with the interest of the first payment a for one year, that is, ar; the whole sum upon which interest must now be computed is thus 2 a + a r.

At the end of the third year a further payment a becomes due, together with the interest on 2 a + ar, i. e. 2 ar+ar2; the whole sum upon which interest must now be computed is 3 a + 3 ar+ar 2. The result will appear evident when exhibited under the following form:

Ꭶ

= a {1 + (1 + r) + (1 + r)2 + ··· + (1 + r) `-1 }

'

PROBLEM IX. To find the present value of an annuity a payable for t years, compound interest being allowed at the rate r.

It is manifest that the present value of this annuity must be a sum such, that if put out to interest for t years at the rate 7, its amount at the end of that period will be the same as the amount of the annuity.

Hence, if we call this present value p, we shall have, by Probs. VII. and VIII.

What is the present value of an annuity of £500, to last for 40 years, compound interest being allowed at the rate of 23 per cent. per annum.

PROBLEM X. To find the present value (P) of an annuity a which is to com mence after T years, and to continue for t years.

1

The present value required is manifestly the present value of a for T+1 years, minus the present value of a for T years.

By Problem IX. the present value of a for T+ t years =

Р =

a

— . {1 + r ) − T− (1 + 2) − ( + 1)} .

..(12)

PURCHASE OF ESTATES.

PROBLEM XI. To find the present value p of an estate or perpetuity, whose annual rental is a, compound interest being calculated at the rate r.

The present value of an annuity a, to continue for t years, by Prob. IX. is

but if the annuity last for ever, as in the case of an estate, then t = ¤,

and

.......(13)

Example.

What is the value of an estate, whose rental is £1000, allowing the purchaser

5 per cent. for his money?