SOLUTION: formula of rate in rearranging the compound interest formula

Question 1173080: formula of rate in rearranging the compound interest formula
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
compound interest formula is:
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.

normally, the time period is years.
you would be given the rate per year and the number of years.

in this formula the rate is assumed to be equal to the percent rate divided by 100.
for example:
15% / 100 = a rate of .15.

if given the rate and number of time periods in years, then you would divide by the number of compounding periods per year to get the rate per time period and multiply by the number of compounding periods per year to get the number of time periods.

with annual compounding, r = rate per year / 1 and n = number of years * 1.
with semi-annual compounding, r = rate per year / 2 and n = number of years * 2.
with quarterly compounding, r = rate per year / 4 and n = number of years * 4.
with monthly compounding, r = rate per year / 12 and n = number of years * 12.
with daily compounding, r = rate per year / 365 and n = number of years * 365.
the number of days in the year is not exactly correct, because of leap years and other anomalies, but is assumed to be 365 by convention.

the general formula, is, once again:
f = p * (1 + r) ^ n

to solve for r, do the following:
start with f = p * (1 + r) ^ n.
the time periods are assumed to be in months because of monthly compounding.
divide both sides of the equation by p to get:
f/p = (1 + r) ^ n
take the nth root of both sides of the equation to get:
(f/p) ^ (1/n) = 1 + r
subtract 1 from both sides of the equation to get:
(f/p) ^ (1/n) - 1 = r
solve for r to get:
r = (f/p) ^ (1/n) - 1

the formula assumes that you know what f is and you know what p is and you know what n is.

for example:

assume that n = 20 years and f = 7000 and p = 2000 and monthly compounding is assumed.

n will be equal to 20 * 12 which is equal to 240.

you have:
r = r
n = 240
f = 7000
p = 2000

the formula becomes:
7000 = 2000 * (1 + r) ^ 240
divide both sides of the equation by 2000 to get:
7/2 = (1 + r) ^ 240
take the 240th root of both sides of the equation to get:
(7/2) ^ (1/240) = 1 + r
subtract 1 from both sides of the equation to get:
(7/2) ^ (1/240) - 1 = r
solve for r to get:
r = (7/2) ^ (1/240) - 1 = .0052334928.

that's your interest rate per month.

confirm this is true by replacing r in the original equation with that to get:
f = 2000 * (1 + .0052334928) ^ 240 = 7000.

.0052334928 is the interest rate per month.
the nominal interest rate per year would be 12 times that = .062801914.
the effective interest rate per year would be (1 + that) ^ 12 - 1 = .0646415273.

to find the percent, you multiply these figures by 100.

.062801914 * 100 = 6.2801914%
.0646415273 * 100 = 6.46415273%

6.28.....% is the nominal interest rate per year.
6.46.....% is the effective interest rate per year.

your solution is:

f = p * ( + r) ^ n is rearranged to solve for r to get:
r = (f/p) ^ (1/n) - 1

keep in mind, that some texts will tell you that the formula is:
f = p * (1 + r/c) ^ (n*c)
f = future value
p = present value
r = interest rate per year
n = number of years
c = number of compounding periods per year.

this is really the same formula, but it assumes that you are given the rate per year and the number of years.
even though that is most often the case, it is not always the case.

to solve for r in this formula, you would do the following:
start with f = p * (1 + r/c) ^ (n*c)
divide both sides of the formula to p to get:
f/p = (1 + r/c) ^ (n*c)
take the (n*c)th root of both sides of the equation to get:
(f/p) ^ (1/(n*c)) = 1 + r/c
subtract 1 from both sides of the equation to get:
(f/p) ^ (1/(n*c)) - 1 = r/c
solve for r/c to get:
r/c = (f/p) ^ (1/(n*c)) - 1
solve for r to get:
r = ((f/p) ^ (1/(n*c)) - 1) * c

using this formula, rather than the one i showed you above, the example becomes:

7000 = 2000 * (1 + r/12) ^ (20*12)
to solve for r, you would do the following:
divide both sides of the formula by 2000 to get:
7/2 = (1 + r/12) ^ 240
take the 240th root of both sides of the equation to get:
(7/2) ^ (1/240) = 1 + r/12
subtract 1 from both sides of the equation to get:
(7/2) ^ (1/240) - 1 = r/12
solve for r/12 to get:
r/12 = (7/2) ^ (1/240) - 1 = .0052334928
solve for r to get:
r = 12 * that = .062801914.

in this case, you have solved for the nominal interest rate per year.

to find the effective interest rate you would solve for (1 + r/12) ^ 12 = (1 + .0052334928) ^ 12 - 1 = .0646415273.

i'll be available to answer any questions you might have about this.
theo

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