Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period.$118,000; monthly payments for 8 years; interest rate 9%.
Question
Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period.$118,000; monthly payments for 8 years; interest rate 9%.
Solution
To solve this problem, we will use the formula for the future value of an annuity due. The formula is:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n) * (1 + r/n)
Where: FV = future value of the annuity = $118,000 P = payment per period (what we're trying to find) r = annual interest rate = 9% = 0.09 n = number of compounding periods per year = 12 (monthly payments) t = number of years = 8
We can rearrange the formula to solve for P:
P = FV / {[(1 + r/n)^(nt) - 1] / (r/n) * (1 + r/n)}
Substituting the given values:
P = $118,000 / {[(1 + 0.09/12)^(12*8) - 1] / (0.09/12) * (1 + 0.09/12)}
Now, we can calculate the value of P using a calculator.
P = $118,000 / {[(1 + 0.0075)^(96) - 1] / 0.0075 * (1.0075)}
P = $118,000 / {[(1.0075)^96 - 1] / 0.0075 * 1.0075}
P = $118,000 / {10.935 - 1} / 0.0075 * 1.0075
P = $118,000 / 9.935 / 0.0075 * 1.0075
P = $118,000 / 74.5125
P = $1,584.67
Therefore, the monthly payment that should be used for the annuity due is approximately $1,584.67.
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