EXAMPLES.

1. What is the present worth of an annuity of 100l. to continue 5 years, at 6 per cent. per annum, simple interest?

106: 100 :: 100 : 94°3396 = present worth for 1 year. 112: 100 :: 100: 89°2857 =

118: 100 100 84 7457=

2d year.

3d year..

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. 5s. 7d.

To find the Amount of an Annuity at Compound Interest.

RULE.*

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

2. Carry

The other two theorems for the time and rate cannot be given in general terms.

* DEMONSTRATION. It is plain, that upon the first year's annuity, there will be due as many years' compound interest, as the given number of years less one, and gradually one year's interest

less

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

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

EXAMPLES.

less upon every succeeding year to that preceding the last, which has but one year's interest, and the last bears no interest.

Letr, therefore, rate, or amount of 11. for 1 year; then the series of amounts of il. annuity, for several years, from the first to the last, is 1, r, r2, r3, &c. to -1 And the sum of this, according to

the rule in geometrical progression, will be

amount of

r-I

1. annuity for years. And all annuities are proportional to

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

letters as before, then

ara

Xnxa, and

--n.

I

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. r2'—1 —Log. r—1 =Log. a.

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

EXAMPLES.

1. What is the amount of an annuity of 40l. to con tinue 5 years, allowing 5 per cent, compound interest ?

2

3

1+1·05+1·05] +1·05)+1*05]=5*52563125

5*52563125

40

22102525

20

0'505

12

6.06

Ans, 2211. 6d.

2. If 50l. yearly rent, or annuity, be forborn 7 years, what will it amount to, at 4 per cent. per annum, compound interest?

Ans. 3951

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 1 year's annuity.

2. Divide

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

Let p present worth of the annuity, and the other letters as

=

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

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

EXAMPLES.

70%

or principal of this, according to the principles of compound interest, is the amount divided by rt, therefore

And from these theorems all the cases, where the purchase of annuities is concerned, may be exhibited in logarithmic terms, as follows:

I

I. Log.n+Log. I—

-Log.r-1=Log.p.

i

II. Log.p+Log. r—1—Log. 1——=Log. n.

Let t 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 account, and it will give the whole amount required.

The

EXAMPLES.

1. What is the present worth of an annuity of 40l. to continue 5 years, discounting at 5 per cent. per annum, compound interest?

1*05)40*00000(38.095-present worth for 1 year,

ratio =

2. What is the present worth of an annuity of 211. ros. 9 d. to continue 7 years, at 6 per cent. per annum, compound interest?

Ans. 1201. 5s.

3. What is 70l. per annum, to continue 59 years, worth in present money, at the rate of 5 per cent. per annum ? Ans. 132130211.

To

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.