You expect to receive $16,000 three years from today. If the present value of this amount is $12,701.13, what is the APY? What is the monthly return?
To determine the annual percentage yield (APY) and the monthly return, we'll need to use the present value formula and the compound interest formula.
The present value formula is:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = interest rate
n = number of periods
Given that the present value (PV) is $12,701.13 and the future value (FV) is $16,000, with a time period of three years (n), we can rearrange the formula to solve for the interest rate (r).
12,701.13 = 16,000 / (1 + r)^3
Next, we can isolate (1 + r)^3 and solve for r:
(1 + r)^3 = 16,000 / 12,701.13
Now, take the cube root of both sides:
1 + r = (16,000 / 12,701.13)^(1/3)
Subtract 1 from both sides to isolate r:
r = (16,000 / 12,701.13)^(1/3) - 1
By plugging this equation into a calculator or using a spreadsheet, we can find the solution for r, which represents the annual interest rate or APY.
To find the monthly return, we can use the compound interest formula:
FV = PV * (1 + r/12)^(n*12)
We know the future value (FV) is $16,000, the present value (PV) is $12,701.13, the interest rate per period (r) is the APY divided by 12, and the number of periods (n) is three.
16,000 = 12,701.13 * (1 + (APY/12))^(3*12)
Rearrange the formula and solve for the monthly return (APY/12):
(APY/12) = [(16,000 / 12,701.13)^(1/(3*12))] - 1
By plugging this equation into a calculator or using a spreadsheet, we can find the solution for APY/12, which represents the monthly return.