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The same at 8%, payable quarterly, the tabular num. ber opposite 2% for 20 years is
a = P x Rr. The amount is equal to the principal x ratio to the power of the time.
The principal is equal to the amount = ratio to the power of the time.
The ratio to the power of the time = the amt. divided by principal.
FORMULA (1).- Multiply the tabular number opposite time and rate by principal for the amount.
FORMULA (2).— The principal is equal to the amount divided by the tabular number.
FORMULA (3).—To find the rate, divide the amount by the principal, and find the quotient arising therefrom in the table opposite the time, and the rate in which it is found will be the true rate.
To find the time, do the same, and look in the column of the given rate until you find the number opposite the 5. What principal will produce $2078.928 in 15 years at 5%?
6. At what rate will $450 produce $805.8816 in 10 years?
Ans. 6% 7. In what time will $500 produce $1091.44 at 5%?
Ans. 16 years. 8. In what time will $500 produce $1000 at 6%? That is, in what time will any sum double itself at 6% ?
In the table the amount of $1 at 6% for 11 years is 1.8982986, wanting .1017014 to be double. This is 1947 of the amount = .053574, the rate that 1.8982986 is multiplied by to produce 1017014. The rate at 6% is half the number of months divided by 100.
Hence .053574 x 200 = 10.7148 months.
21.4440 days. Ans. 11 years 10 months and 21 days.
ANNUITIES. An Annuity is a certain sum of money received annually. It may be for a certain fixed time, when it is called a Certain Annuity.
It may begin or end with the birth or death of one or more persons, when it is called a Contingent Annuity.
A Perpetual Annuity is called a Perpetuity. An Immediate Annuity begins at once.
A Deferred Annuity, or an Annuity in Reversion, begins at a future time.
A Forborne Annuity is one in arrears.
A sum of money at a given rate of interest produces the same interest every year; this interest is the annuity; hence a perpetuity is the interest of a fixed principal at a fixed rate of interest, and this principal is the present value of the perpetuity.
Let P represent principal or present value.
Then (1.) P x r =p
(2.) P =
P. The present worth is equal to the perpetuity divided by the rate per cent., and the rate is equal to the perpe tuity divided by the present worth.
(3.) r =
EX AMPLES 1. What perpetuity will a fixed sum of $6000 yield at the fixed rate of 5%?
6000 X to = $300.
To 2. What perpetuity will property yield whose fixed value is $12000 at a fixed rate of 6%.
$12000 xjo $720 REM.—The present value corresponds to the principal in simple interest, and the perpetuity to the interest of the present value.
3. What is the present value of a perpetuity of $300 a year at 5%
= 300 x 100 $6000.
2p Formula. P=
4. What is the present value of a perpetuity of $250 at 4%?
= 250 x 140
5. What is the present value of a deferred annuity of $300 at 6%, to commence twenty years hence ?
300 x 100 = $5000.
As $5000 at 6% produces $300 a year, the present value is such a sum of money as will amount to $5000 in 20 years at 6% compound interest. This sum is the quotient of $5000 divided by the amount of $1 in 20 years at 6%.
Tabular numbers for 20 years at 6% is
3.207136 ) 5000.0000000 ( $1559.02 = present value.
6. What would be the present value of an annuity of $300 at 6%, to begin immediately and continue 20 years?
7. What is the present value of an annuity of $300 at 6%, to begin in 10 years and then continue 10 years?
Present value of an annuity to begin in 10 years.
1.790848 ) 5000.00000000 ( 2791.97 Present value of an annuity to begin in 20 years
1559.02 Present value of an annuity to begin in 10 years and continue 10 years. $1232.95
8. What is the forborne value of an annuity of $300 · for 10 years at 6%?
$5000 is the principal which has been at compound interest 10 years.
The compound interest of $1 for 10 years is
And for $5000 is
9. The present value is $6000, interest 6%, payable semi-annually. What annuity will it yield ?
Interest on $180 for 6 months,
10. What is the present value of an annuity of $400, payable quarterly; interest at 5%, payable yearly? $100 x 9 mo. = $100 x 18 mo. at 5% = $7.50 100 x 6 mo.
400 100 x 3 mo.
= $8150.00 = present value.
11. What is the present value of an annuity of $800, payable quarterly; interest also payable quarterly ; rate 5%?
Present value = 16000.
REM.-When the payments of the annuity and of the interest on the present value are the same, the computation is the same as f both were annual.