*

RULE II.

Or, find the amount of an annuity of 11. or $1 for the given time and rate (by Case 1;) divide the given sum by this amount; and the quotient will be the annuity required.

EXAMPLES.

1. What annuity, being forborne 4 years, will amount to £262-47696, at 61. per cent. compound interest?

262.47696 amount.

Multiply by 1.06 amount of 11. for 1 year.

Or, thus.

Amount of an annuity of 11. for 4 years at 6 per cent. per annum

4.374616 (by Case 1;) and

262 47696

4-374616

=£60 Ans.

Or, by Table III. the amount of 11. is found to be 4:374616; and the answer is found, as before.

2. What annuity, being forborne 20 years, will amount to $2207-1354, at 6 per cent. compound interest?

Amount of an annuity of $1 for 20 years at 6 num-36-78559. And,

36-78559)2207-13540($60, Ans.

2207.1354

per cent. per an

0

CASE III.

When the annuity, amount and ratio are given, to find the time.

RULE I.

Multiply the amount by the ratio, to this product add the annuity, and from the sum subtract the amount; this remainder being

divided by the annuity, the quotient will be that power of the ratio signified by the time, which being divided by the amount of 11. for 1 year, and this quotient by the same, till nothing remain, the number of those divisions will be equal to the time. Or, look for this number under the given rate in table 1, and in a line with it, you will see the time. Or,

RULE II.

Divide the amount by the annuity; from the quotient subtract 1; from the remainder subtract the ratio; from successive remainders subtract the square, cube, &c. of the ratio, till nothing remain ; and the whole number of the subtractions will be the answer. Or, find the quotient in Table III. under the rate, and in a line with it stands the answer.

EXAMPLES.

1. In what time will 601. per annum, payable yearly, amount to £262-47696, allowing 61. per cent. compound interest, for the forbearance of payment?

262.47696

amount.

The number of divisions by 1:06, being 4, gives the number of years == 4, the answer.

annnuity. Or thus: Annuity=60) 262-47696-amt.

4.374616

1. Subtract 1.

3-374616 2. Subtract 1.06 =ratio.

2.314616 3. Subtract 1.1236

1.191016

=ratio

4. Subtract 1.191016 ratio.

Ans. 4 years.

der the rate, 6, the quotient Or, looking into Table III. un4-374616, stands against 4 years, Ans. as before.

Or, in Table I, under the given rate, you will find 1.262476, and in a line under years, you will find 4.

2. In what time will an annuity of $60 payable yearly, amount to $2207-1354, allowing 6 per cent. for the forbearance of payment? Ans. 20 years.

PRESENT WORTH OF ANNUITIES, &c. AT COMPOUND INTEREST.

CASE I.

When the annuity, &c. rate and time are given to find the present

worth. RULE 1.*

1. Divide the annuity by the amount of $1 or £1 for 1 year, and the quotient will be the present worth of 1 year's annuity.

This rule depends on the rule for finding the present worth in Discount at Compound Interest. For each year the present worth is to be found by that rule. Then, the sum of the present worth for the several years, must evidently be the present worth of the whole, and is the rule.

Or, suppose the annuity to be 11. or $1 at 6 per cent. then

for two years;
I

1 1.063

1 1.06 for three years;

is the present

1

will be

four years, and so on. Then the sum, or +- + + 1.06 1.062 1:063 1.064 the whole present worth. Let any annuity be substituted for the numerator of these several fractions, and you have the rule in the text.

By Note 2, Prob. I. of Geometrical Progression, the sum of the series,

1

1

1.06

1
1
X
⚫06 ⚫06 1-064

1

Now if the

1 1 1 + + -, is 1·X· or 1.062 1-063 1.064 1.064 06 annuity were to continue forever, or the number of years were infinite, then the index of the denominator of the last expression would be infinite, and the value 1 of the fraction would be infinitely diminished or become nothing, and would .06

be the present worth of an annuity of 11. or $1 to continue forever at 6 per cent. Hence, if an annuity is a perpetuity, or is to continue forever, its present worth is found by dividing the annuity by the ratio, or the interest of 11. or $1 for a year at the given rate.

1

05

The present worth of an annuity of $1 to continue forever at 5 per cent. is 100 $20, and an annuity of $100 at 5 per cent. to continue forever, would

now be worth $2000, and at 7 per cent. $14284.

Rule II. is derived from the expression 1$1 or 11. and the rate 6 per cent. That is when the annuity is $1 or 11. divide the annuity by that power of the ratio indicated by the number of years, and subtract the quotient from the annuity; the remainder divided by the ratio of the series legs 1, will be the present worth. But the present worth of annuities varies as the annuity. Hence the rule is manifest.

Note. Another rule for obtaining the present worth may be derived from the

preceding. Thus, the sum of the series,

That is, divide the difference between unity and that power

the present worth of 11. or $1 for four

2. Divide the annuity by the square of the ratio, and the quotient will be the present worth for two years.

3. In like manner, find the present worth of each year by itself, and the sum of all these will be the present value of the annuity sought.

RULE II.

Or, divide the annuity, &c. by that power of the ratio signified by the number of years, and subtract the quotient from the annuity; this remainder being divided by the ratio less 1, the quotient will be the present worth. EXAMPLES.

1.* What ready money will purchase an annuity of 601. to continue 4 years, at 61. per cent. compound interest?

First Method.

1.06)60·00000(56-603-present worth for 1 year.

1.1236)60·00000(53.399= do.

for 2 years.

1-191016)60-00000(50-377=

do.

for 3 years.

of the ratio which is indicated by the number of years, by the difference between that power of the ratio which is one greater than the number of years and that power of the ratio which is equal to the number of years, and the quotient is the present worth of 11. or $1. Then, as annuities are as their present worth, multiply this quotient by the given annuity, and the product is its present worth. The rules for the next Case are derived directly from this rule, and need no further illustration.

* The amount of an annuity may also be found for years and parts of a year, thus:

1. Find the amount for the whole years, as before.

2. Find the interest of that amount for the given parts of a year.

3. Add this interest to the former account, and it will give the whole amount required.

The present worth of an annuity for years and parts of a year may be found

thus:

1. Find the present worth for the whole years, as before.

2. Find the present worth of this present worth, discounting for the given parts of a year, and it will be the whole present worth required.

Questions in this case may also be answered by first finding the amount of the given annuity by Case I. of annuities in arrears, page 315, and then the present worth, or principal, by Case II. of Compound Interest, page 311.

Rr

BY TABLE III.

Under 61. per cent. and opposite 4, we find

4.37461 amount of 11. annuity for 4 years.

Multiply by 60=annuity.

262.47660 amount of 601. for 4 years.

Then, opposite 4 years, and under 61. per cent. in Table 2d.
We have 792093

Multiply by 262-7466

4752558

4752558

3168372

5544651

1584186

4752558

1584186

208.1197426338= £208 25. 410.

Or, opposite 4 years, and under 61. per cent. in Table 1st, we have 1.26247=the amount of 11. for 4 years :

Then, 262-7466-1.26247=208·1209

BY TABLE IV.*

£208 2s. 5d. Ans.

Multiply the tabular number, under the rate, and opposite the time, into the annuity, and the product will be the present worth. Thus, in Example 1st. What ready money will purchase £60 annuity, to continue 4 years, at £6 per cent. compound interest? Under 61. per cent. and even with 4 years,

We have 3-4651-present worth £1 for 4 years.
Multiply by 60=annuity.

Ans.207.9060 £207 18s. 11d.

2. What is the present worth of an annuity of $60 per annum, to continue 20 years, at 6 per cent. compound interest?

CASE II.

Ans. $688-65 (nearly.)

When the present worth, time, and rate are given, to find the annuity,

rent, &c.
RULE.

1. From that power of the ratio, denoted by the number of years plus 1, subtract that power of it denoted by the number of years.

* Table 4th is thus made: Divide £1 by 106=94339 the present worth of the first year, which, divided by 1.06, is equal to 88999, which, added to the first year's present worth, is = 1.83339, the second year's present worth, then ·88999, divided by 1.06, and the quotient added to 1·83339, gives 2-6701 for the third year's present worth, &c.