Find the periodic payments necessary to accumulate $20,000 in a sinking fund paying 4% per year, with monthly payments for 5 years. (Assume end-of-period deposits and compounding at the same intervals as deposits. Round your answer to the nearest cent.)
Work it like a savings account accumulation problem in which you have to accumulate $20,000 in 5 years while earing 4% interest, compounded monthly.
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I get $301.66 per month
To find the periodic payments necessary to accumulate $20,000 in a sinking fund, you can use the formula for the future value of an annuity:
FV = PMT * [(1 + r)^n - 1] / r
Where:
FV = Future Value (in this case, $20,000)
PMT = Periodic Payment
r = Interest rate per period (4% per year, or 0.04/12 per month)
n = Number of periods (5 years, or 5 * 12 = 60 months)
Now, let's plug in the values and solve for PMT:
$20,000 = PMT * [(1 + 0.04/12)^60 - 1] / (0.04/12)
First, simplify the calculation inside the brackets:
$20,000 = PMT * [(1.003333...)^60 - 1] / 0.003333...
Next, calculate the value in the brackets:
(1.003333...)^60 ≈ 1.224548...
Now, substitute this value and the interest rate into the equation:
$20,000 = PMT * (1.224548... - 1) / 0.003333...
Simplify further:
$20,000 = PMT * 0.224548... / 0.003333...
Divide both sides of the equation by 0.224548... / 0.003333... to isolate PMT:
PMT = $20,000 / (0.224548... / 0.003333...)
Now, perform the division:
PMT ≈ $20,000 / 67.386...
PMT ≈ $296.88
Therefore, the periodic payment necessary to accumulate $20,000 in a sinking fund paying 4% per year, with monthly payments for 5 years, is approximately $296.88.