Question 1115406: Sarah invests $400 into a fund that compounds monthly. If the interest over 3 years is $84, what is the interest rate per annum?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
in this problem:
f = 400 + 83 = 483
p = 400
r = what you want to find
n = 3 * 12 = 36 months
formula becomes 483 = 400 * (1 + r) ^ 36
divide both sides of this formula by 400 to get 483/400 = (1 + r) ^ 36
take the 36th root of both side of this formula to get (483/400)^(1/36) = 1 + r
solve for r to get r = (483/400) ^ (1/36) - 1.
this makes r = .0052512985
that is the interest rate per month.
the nominal interest rate per year is 12 * that to get r = .063015582.
multiply that by 100 to get nominal interest rate percent per year = 6.3015582 percent.
the effective interest rate per year will be (1 + .0052512905) ^ 12 = 1.064867845 - 1 = .064867845 interest rate per year.
multiply that by 100 to get effective interest rate percent per year = 6.4867845 percent.
i multiplied 400 by 1.064867845 ^ 3 and got 483, as i should have.
i multiplied 400 by 1.005251299 ^ 36 and got 483.0000086 which is extremely close to 483, the difference being due to rounding in the calculator display.
i also multiplied 400 by ((483/400)^(1/36)) ^ 36 and got 483, right on the button.
if you consider that (483/400) ^ (1/36) is equal to 1.005251299, which is a rounded number in the calculator display, then by calculating (483/400) ^ (1/36) and then storing it internally in the calculator, and then using the stored number, rather than the number in the display, the accuracy is much greater because you can store more decimal digits than are displayed.
that allowed me to get it right on, rather than just very close.
bottom line:
your effective interest rate is 6.4867845 percent per year.
your nominal interest rate is 6.3015582 percent per year.
the formula requires rate, not percent, so you need to divide the percent by 100 to get the rate.
any questions, send them to me and i'll answer as best i can.
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