Gretta wants to retire in 13 years. At that time she wants to be able to withdraw $12,500 at the end of each 6 months for 10 years. Assume that money can be deposited at 12% per year compounded semiannually. What exact amount will Gretta need in 13 years?
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To determine the exact amount Gretta will need in 13 years, we need to calculate the future value of the annuity payments.
The formula to calculate the future value of an annuity is:
FV = P * [(1 + r)^(n) - 1] / r
Where:
FV: Future value of the annuity
P: Payment amount
r: Interest rate per compounding period
n: Number of compounding periods
In this case:
P = $12,500 (payment amount)
r = 12% per year compounded semiannually, so the interest rate per compounding period is 12% / 2 = 6%
n = 10 years * 2 compounding periods per year = 20 compounding periods
Now, plug in these values into the formula:
FV = $12,500 * [(1 + 0.06)^(20) - 1] / 0.06
Simplify the equation:
FV = $12,500 * [1.79084708 - 1] / 0.06
FV = $12,500 * 0.79084708 / 0.06
FV ≈ $164,658.45
Therefore, Gretta will need approximately $164,658.45 in 13 years to be able to withdraw $12,500 at the end of each 6 months for 10 years.