Determine the annual percentage yield, or the effective interest rate, for $900 invested at 3.29% over 18 years compounded daily. Round your answer to the nearest hundredth of a percent, if necessary.
To calculate the annual percentage yield, we will use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial deposit or loan amount)
r = annual interest rate (in decimal form)
n = number of times that interest is compounded per year
t = time the money is invested for in years
In this case:
P = $900
r = 3.29% or 0.0329
n = 365 (compounded daily)
t = 18 years
A = $900(1 + 0.0329/365)^(365*18)
A = $900(1 + 0.000090027)^(6570)
A = $900(1.000090027)^(6570)
A = $900(1.701109)
A = $1530.999
The investment will grow to $1530.999 after 18 years.
Now, we can calculate the annual percentage yield by finding the effective annual interest rate that will result in the same future value of $1530.999 as compound interest over 18 years.
APY = (1 + r/n)^n - 1
APY = (1 + r/365)^365 - 1
0.0329 = (1 + r/365)^365 - 1
1.0329 = (1 + r/365)^365
1.0329^(1/365) = 1 + r/365
r = 365(1.0329^(1/365) - 1)
r = 365(1.000078205 - 1)
r = 365(0.000078205)
r = 0.0286
Therefore, the annual percentage yield, or the effective interest rate, for $900 invested at 3.29% over 18 years compounded daily is approximately 2.86%.