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AN ANNUITY is a sum of money payable every year, for a certain number of years, or for ever.
When the debtor keeps the annuity in his own hands, beyond the time of payment, it is said to be in arrears.
The sum of all the annuities for the time they have been forborn, together with the interest due upon each, is called the amount.
If an annuity be to be bought off, or paid all at once, at the beginning of the first year, the price, which ought to be given for it, is called the present worth.
To find the Amount of an Annuity at Simple Interest.
1. Find the sum of the natural series of numbers 1, 2, 3, &c. to the number of
* DEMONSTRATION. Whatever the time is, there is due upon the first year's annuity, as many years' interest as the whole number of years less one ; and gradually one less upon every succeeding year to the last but one ; upon which there is due only one year's interest, and none upon the last ; therefore in the whole there is due as many years' interest of the annuity, as the sum of the series 1, 2, 3, 4, &c. to the number of years less one. Consequently one year's interest, multiplied by this sum, must be the whole interest due ; to which if all the annuities be added, the sum is inly the amount. Q. E. D.
Let r be the ratio, n the annuity, t the time, and a the amount.
Then will the following theorenis give the solutions of all the different cases.
tarnirn I. fun=a.
2. Multiply this sum by one year's interest of the annuity, and the product will be the whole interest due upon the annuity:
3. To this product add the product of the annuity and time, and the sum will be the amount sought.
Note. When the annuity is to be paid half-yearly or quarteriy; then take, in the foriner case, the ratio, half the annuity, and twice the number of years ; and, in the Latter case, the ratio, the annuity, and 4 times the number of years, and proceed as before,
1. What is the amount of an annuity of 5ol. for 7 years, allowing simple interest at 5 per cent. ?
cannot be found equal to the amount, the problem is impossible in
Note. Some writers look upon this method of finding the amount of an annuity as a species of compound interest ; the annuity itself, they say, being properly the simple interest, and the сарital, whence it arises, the principal.
2. If a pension of 6ool. per annum be forborn 5 years, what will it amount to, allowing 4 per cent. simple interest ? 3.
What will an annuity of 250l. amount to in 7 years, to be paid by half-yearly payments, at 6 per cent. per annum, simple interest?
Ans. 209 11. 55.
To find the present Worth of an Annuity at Simple Interest.
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 sufficientto shew the truth of this assertion. The price of an annuity of gol. 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, sum, which would yield a greater yearly interest for ever?
It is most cquitable to allow compound interest.
&c. to -
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 for 1 year. 112 : 100 :: 100 : 8992857= 118 : 100 :: 100 : 84074573 124 : 100 :: '100 : 806451 =
100 : 7699230
2d year. 3d year.. 4th year. 5th year.
130 : 100 ::
425'9391 = 4251. 185. 9 d.
4251. 18s. 9 . = present worth of the annuity required.
2. What is the present worth of an annuity or pension of sool. to continue 4 years, at 5 per cent. per annum, simple interest ?
Ans. 17821. 55. 7d.
To find the Amount of an Annuity at Compound Interest.
1. Make i the first term of a geometrical progression, and the amount of il. for one year, at the given rate per cent. the ratio.
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
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.
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 1l. for 1 year; then the series of amounts of 1l. annuity, for several years, from the first to the last, is 1, r, r?gr?, &c. to p?-1. And the sum of this, according to
the rule in geometrical progression, will be
= amount of
she annuity for 4 years. And all annuities are proportional to
their amounts, therefore I :
-I amount of any given annuity n.
Q. E. D. Let r = rate, or amount of 11. for one year, and the other letters as before, then
And from these equations all the cases relating to annuities, or pensions in arrears, may be conveniently exhibited in fogarithmic terms, thus :
I. Log.n+- Log. 7-1 - Log. 1-1=Log.a.
II. Log.a-Log. I + Log.r-I=Log.n.