A company estimates that it will need 144000 in 16 years to replace a computer. If it establishes a sinking fund by making fixed monthly payments into an account paying 3.6% compounded monthly, how much should each payment be?
To find the monthly payments needed for the sinking fund, we need to use the future value formula for an ordinary annuity.
The future value (FV) of an ordinary annuity is given by the formula:
FV = P * (((1 + r)^n) - 1) / r
Where P is the monthly payment, r is the monthly interest rate, and n is the number of payments.
In this case, the future value (FV) is given as $144,000, the monthly interest rate (r) is 3.6% or 0.036, and the number of payments (n) is 16 years * 12 months = 192 months.
Substituting these values into the formula, we have:
$144,000 = P * (((1 + 0.036)^192) - 1) / 0.036
Now we can solve for P.
$144,000 * 0.036 = P * (((1 + 0.036)^192) - 1)
$5,184 = P * (((1.036)^192) - 1)
Dividing both sides by (((1.036)^192) - 1), we get:
P = $5,184 / (((1.036)^192) - 1)
Calculating this value, we find:
P ≈ $517.60
Therefore, each monthly payment should be approximately $517.60.
To find the amount of each monthly payment, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the future value (the amount needed to replace the computer in this case) = $144,000
P is the monthly payment
r is the monthly interest rate = 3.6% = 0.036
n is the number of payments = 16 years * 12 months/year = 192 months
Plugging in the values into the formula:
144,000 = P * ((1 + 0.036)^192 - 1) / 0.036
Now we can solve for P.
To calculate the amount each monthly payment should be, we need to use the formula for the future value of a sinking fund:
Future Value = Payment * (1 + r)^n - 1 / r
In this formula:
- Future Value is the amount needed in the future, which is $144,000.
- Payment is the amount of each monthly payment we're trying to find.
- r is the interest rate per compounding period, which is 3.6% or 0.036 as a decimal.
- n is the number of compounding periods or the total number of months until the replacement, which is 16 years * 12 months = 192 months.
Now, let's substitute the given values into the formula and solve for Payment:
144,000 = Payment * (1 + 0.036)^192 - 1 / 0.036
First, let's calculate the value inside the parentheses:
(1 + 0.036)^192 = 5.057914
Next, calculate the value inside the square brackets:
(1 + 0.036)^192 - 1 = 5.057914 - 1 = 4.057914
Now, substitute this value back into the formula:
144,000 = Payment * 4.057914 / 0.036
To isolate Payment, we need to divide both sides of the equation by 4.057914 / 0.036:
Payment = 144,000 / (4.057914 / 0.036)
Now, calculate the right-hand side of the equation:
Payment = 144,000 / 112.719278 = $1,279.64 (rounded to the nearest cent)
Therefore, each monthly payment should be approximately $1,279.64.