the present value of that payment is 8546.88.
the formula you can use to solve this 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.
you want to know the nominal interest rate per year when the interest rate per year is compounded quarterly.
you would first solve for the quarterly interest rate and then multiply that by 4 to get the nominal interest rate per year.
the time periods will be in quarters of a year.
the number of years is multiplied by 4 to get the number of quarters.
3 * 4 = 12 quarters.
f = p * (1 + r) ^ n becomes:
11000 = 8546.88 * (1 + r) ^ 12.
f = 11000
p = 8546.88
n = 3 years * 4 quarters per year = 12
r is what you want to find.
divide both sides of the equation by 8546.88 to get:
(11000 / 8546.88) = (1 + r) ^ 12
take the 12th root of both sides of the equation to get:
(11000 / 8546.88) ^ (1/12) = 1 + r.
solve for (1 + r) to get:
1 + r = 1.021250048.
solve for r to get:
r = 1.021250048 - 1 = .021250048
that's the interest rate per quarter.
multiply that by 4 to get the nominal interest rate per year.
you will get 4 * .02149684 = .0850001915.
multiply that by 100 to get 8.50001915%.
that's your nominal interest rate per year.
your solution is that the nominal rate of interest convertible quarterly is equal to 8.50001915%.
you can also use a financial calculator to get the same results.
an online calculator that can be used is found at https://arachnoid.com/finance/index.html
here are the results of using that calculator.
inputs are everything except ir.
output is ir.
results are that ir = 2.125005%.
that's the quarterly interest rate.
multiply that by 4 to get 8.500016%.
any differences between that and what i got above are most likely due to rounding.
it showed the quarterly rate of 2.125005% while i calculated .021250048.
the difference was definitely in the rounding.
if you round your final answer to 2 decimal places, both methods would get you 8.5% nominal interest rate per year.