Find the monthly payment, needed to have a sinking fund accumulate the future value, $16,000. The yearly interest rate is 6.7% and the number of payments is 20.
Interest is compounded monthly. Round your answer to the nearest cent.
I do not understand how to find the monthly payment. Please help me.
Thank you.
To find the monthly payment required to accumulate a future value, we can use the formula for the present value of an ordinary annuity:
PV = PMT × (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
PMT = Monthly Payment
r = Interest rate per period
n = Number of periods
In this case, the future value (FV) is given as $16,000, and we need to find the monthly payment (PMT). The interest rate is 6.7% per year, which we can convert to a monthly interest rate by dividing by 12. The number of payments (n) is 20.
Let's substitute the given values into the formula:
PV = PMT × (1 - (1 + r)^(-n)) / r
16000 = PMT × (1 - (1 + (0.067/12))^(-20)) / (0.067/12)
Now we can solve this equation for PMT.
To find the monthly payment needed to have a sinking fund accumulate a future value, you can use the formula for calculating the future value of an ordinary annuity.
The formula for the future value of an ordinary annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value
P = Monthly payment
r = Interest rate per period
n = Number of periods
In this case, we are given:
FV = $16,000 (the desired future value)
r = 6.7% per year (which needs to be converted to a monthly rate)
n = 20 (the number of payments)
First, we need to convert the yearly interest rate to a monthly interest rate. Since the interest is compounded monthly, we divide the yearly interest rate by 12.
Monthly interest rate = 6.7% / 12 = 0.067 / 12 = 0.00558
Now we can substitute the values into the formula:
$16,000 = P * [(1 + 0.00558)^20 - 1] / 0.00558
To solve for P, we need to isolate it in the equation. Let's go through the steps:
1. Multiply both sides of the equation by 0.00558:
0.00558 * $16,000 = P * [(1 + 0.00558)^20 - 1]
2. Simplify the right side of the equation:
0.0558 * $16,000 = P * [(1.00558)^20 - 1]
3. Calculate (1.00558)^20 - 1 using a calculator:
(1.00558)^20 - 1 ≈ 0.1196
4. Substitute the value back into the equation:
0.0558 * $16,000 = P * 0.1196
5. Divide both sides of the equation by 0.1196:
(0.0558 * $16,000) / 0.1196 = P
Now you can calculate the monthly payment needed:
P ≈ $742.85
Rounding to the nearest cent, the monthly payment needed to have a sinking fund accumulate a future value of $16,000 is approximately $742.85.