Question 1082851: Marina had an accident with her car and the repair bill came to $400. She didn’t have any emergency fund money and no extra money in her monthly budget, so she ended up borrowing from a pay-day loan company. As long as she can pay the loan back at the end of the 30 day period she won’t be charged any interest, technically. However, she did have to pay $18 per $100 that she borrowed. If she were to consider the processing fee to represent interest paid in her formula, what would she discover to be the relative annual interest rate the fee she was charged on her loan would represent?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the bill was 400 dollars.
she had to pay a processing fee of 18 dollars per hundred.
the processing fee was 4 * 18 = 72 dollars.
if she paid the loan back at the end of the month, then she owed 472 dollars.
the formula for interest rate 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.
the formula becomes 472 = 400 * (1+r)^1
divide both sides of the equation by 400 to get 472/400 = (1+r)^1
this becomes 472/400 = 1+r
subtract 1 from both sides to get 472/400 - = r
solve for r to get r = .18.
that'a an interest rate of 18% per month.
if you just look at 18% per month, then the nominal annual interest rate per year would be 12 * 18 = 216%.
the effective annual interest rate would be (1.18^12 - 1) * 100 = 728.76%.
that is huge.
however, it doesn't take into account what might actually happen if you kept the loan for a whole year.
if you kept the loan for a year, they might charge you the 18 dollars per hundred processing fee plus an additional interest rate charge of, let's say 24% per year compounded monthly.
24% per year compounded monthly yields a monthly interest rate of 2% per month.
you would owe the following at the end of the year:
f = p * (1+r)^n
f is the future value to be calculated.
p is 400
r is 24%/1200 = .02 per month.
n is 1 year * 12 months per year = 12
the formula becomes f = 400 * (1.02)^12
you will get f = 507.30.
add 72 to that and you owe 579.30 at the end of the 12 month period.
the effective interest rate per month is calculated as follows:
579.30 = 400 * (1+r)^12
divide both sides of this equation by 400 to get:
579.30/400 = (1+r)^12
take the 12th root of both sides of the equation to get:
(579.30/400)^(1/12) = 1+r
subtract 1 from both sides of the equation to get:
(579.30/400)^(1/12)-1 = r
solve for r to get:
r = .0313441941 per month.
the effective interest rate per year would be 1.0313441941^12 = 1.44825 - 1 = .44825 * 100 = 44.825% per year.
the nominal interest rate per year would be .0313441941*12 = .3761303295 * 100 = 37.61303295% per year.
take your pick as to which one of these answers your instructors are looking for.
i suspect the first answer might be what they want to see.
that would be the 216% nominal annual percentage rate or the 729% effective annual interest rate.
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