Question 1172928: Mr Retanan was given a loan at 10% compounded Monthly. When should he Pay it so that it will just earn only 10% of the amount borrowed
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! when you use financial calculators, you normally use the rate percent.
when you use formulas, you usually use the rate.
the rate is the rate percent / 100.
the rate percent is the rate * 100.
i'll be using formulas, so your annual rate percent is equal to .10.
your growth factor is equal to 1 plus the rate.
the formula to use for this problem 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.
first we'll take a look at what happens at the end of the year.
assume your present value is 100 dollars.
your rate per year is .10
your number of years is 1.
f = p * (1 + .10) ^ 1 becomes:
f = 100 * (1 + .10) ^ 1 which becomes:
f = 110.
the interest is f - p = 10
that earning rate is 10 / 100
the percent earning rate is 10 / 100 * 100 = 10%.
that's with 1 compounding period per year (annual compounding).
with monthly compounding, you do the following:
p = 100
r = .10 / 12 = .0083333333....
n = 1 * 12 = 12
you solve for future value to get:
f = 100 * (1 + .008333333.....) ^ 12 = 110.4713067.
the interest is equal to f - p = 110.4713067 - 100 = 10.4713067.
that earning rate is 10.4713067 / 100 * 100 = 10.4713067% per year.
the effective interest rate percent is 10.4713067% per year; this is not what you wanted.
what you wanted is interest rate percent of 10% with monthly compounding.
that will require the investment to be something less than a year.
in other words, you are solving for n.
your formula beformes 110 = 100 * (1 + .00833333...) ^ n
you know what the future value needs to be, because 10/100 * 100 = 10%.
this means the future value needs to be 110.
to solve for n, you need to do the following:
divide both sides of the equation by 100 to get:
1.10 = (1 + .00833333.....) ^ n
take the log of both sides of the equation to get:
log(1.10) = log((1 + .0083333....) ^ n)
since log (x^n) = n * log(x), this becomes:
log(1.10) = n * log(1 + .00833333....)
divide both sides of this equation by log(1 + .00833333....) to get:
log(1.10) / log(1 + .00833333...) = n
solve for n to get:
n = 11.48481075.
he should pay the loan in 11.48481075 months so that the interest earned is equal to 10% of the amount borrowed.
suppose the loan is 100.
the equation becomes f = 100 * (1 + .00833333....) ^ 11.48481075.
solve for f to get:
f = 110.
the interest earned is 10 / 100 * 100 = 10%.
if you used a financial calculator, you would use 10% / 12 = .833333.....% per time period.
the calculator i used can be found at https://arachnoid.com/finance/index.html
you would be solving for the number of time periods.
your inputs would be:
pv (present value) = -100
fv (future value) = 110
pmt (payment amount) = 0]
ir (interest rate % per time period) = 10/12 = .833333.....%
payment at (beginning/end) doesn't matter because pmt = 0.
you would click on np and the calculated will tell you the number of months required.
the results of using that calculator are shown below.
this is an excellent calculator, but it unfortunately rounds all answers to two decimal places.
this might be ok with present value and future value, but it is not always ok when you solve for number of time periods
the number of months shown should have been 11.48481075, as displayed by my TI-BA-II business analyst financial calculator, and as displayed by the use of the formula.
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