To find the present worth of an Annuity at Simple Interest.

RULE.*

Find the present worth of each year by itself, discounting from the time it becomes due, and the sum of all these will be the present worth required.

The reason of this rule is manifest from the nature of discount, for all the annuities may be considered separately, as so many single and independent debts, due after 1, 2, 3, &c. years; so that the present worth of each being found, their sum must be the present worth of the whole.

The estimation, however, of annuities at simple interest is highly unreasonable and absurd. One instance only will be sufficient to show the truth of this assertion. The price of an an、 nuity of 501. to continue 40 years, discounting at 5 per cent. will, by either of the rules, amount to a sum, of which one year's interest only exceeds the annuity. Would it not therefore be highly ridiculous to give, for an annuity to continue only 40 years, a sum, which would yield a greater yearly interest for

ever.

It is most equitable to allow compound interst.

Let

r

present worth, and the other letters as before,

The other two theorems for the time and rate cannot be giv

en in general terms.

EXAMPLES.

1. What is the present worth of an annuity of 1001. to continue 5 years, at 6 per cent. per annum, simple interest?

106 100 100 94 3396=present worth the first year. 112 100 :: 100; 89.2857 = 118: 100 :: 100: 84-7457 =

124 100 :: 100: 80·6451 =

130: 100 :: 100: 76.9230=

2d year.

3d year.

4th year.

5th year.

425.9391 4251. 18s. 94d. = present

worth of the annuity required.

2. What is the present worth of an annuity or pension of 500l. to continue 4 years, at 5 per cent. per annum, simple interest? Ans. 17821. Ss. 81d.

To find the Amount of an Annuity at Compound Interest.

RULE.*

Make 1 the first term of a geometrical progression, and the amount of 11. for 1 year, at the given rate per cent. the ratio.

* DEMONSTRATION. It is plain, that upon the first year's annuity, there may be due as many year's compound interest as the given number of years less one, and gradually one year's interest less upon every succeeding year to that preceding the last, which has but one year's interest, and the last bears no in

terest.

Let therefore rate, or amount of 11. for 1 year; then the series of amounts of 11. annuity, for several year's, from the

2. Carry the series to as many terms as the number of and find its sum.

years,

3. Multiply the sum thus found by the given annuity, and the product will be the amount sought.

EXAMPLES.

1. What is the amount of an annuity of 401. to continue 5 years, allowing 5 per cent. compound interest?

first to the last, is 1, r, r2, r3, &c. to rt-1. And the sum of this

according to the rule in Geometrical Progression, will be T

amount of 11. annuity for t years.

tional to their amounts, therefore 1:

1

And all annuities are propor

amount of any given annuity n. Q. E. D.

Let r rate, or amount of 11. for one year, and the other

rt-1

letters as before, then- Xn=a, and

ara

=n.

And from these equations all the cases relating to annuities, or pensions in arrears, may be conveniently exhibited in logarithmic terms, thus:

· I. Log. n+Log. 7—1—Log. r—1—Log. d.

II. Log. a.-Log. r—1+ Log. r—1=Log. n.

3

1+1 05+1.05 +1.05] +1.05]= 5.52563125

2. If 501. yearly rent, or annuity, be forborn 7 years, what will it amount to, at 4 per cent. per annum, compound interest? Ans. 3941. 18s. 31d.

To find the present value of Annuities at Compound Interest.

RULE.*

1. Divide the annuity by the ratio, or the amount of 11. for one year, and the quotient will be the present worth of the first year's annuity.

*The reason of this rule is evident from the nature of the question, and what was said on the same subject in the purchasing of annuities at Simple Interest.

Let

present worth of the annuity, and the other letters as

before, then as the amount = -Xn, and as the present worth

or principal of this, according to the principles of Compound In terest, is the amount divided by rt, therefore

And from these theorems all the cases, where the purchase

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

of annuities is concerned, may be exhibited in logarithmic terms, as follows.

I. Log. n+Log.

II. Log.p+Log.r—1—Log. 1—=Log.n.

-Log. r—1=Log.p.

Lett express the number of half years or quarters, n the half year's or quarter's payment, and r the sum of one pound and or year's interest, then all the preceding rules are applicable to half-yearly and quarterly payments, the same as to whole years.

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 amount, 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.