Suppose payments were made at the end of each quarter into an ordinary annuity earning interest at the rate of 9%/year compounded quarterly. If the future value of the annuity after 8 yr is $80,000, what was the size of each payment?
i = .09/4 = .0225
n = 4(8) = 32
payment = P
P( 1.0225^32 - 1)/.0225 = 8000
solve for P
To find the size of each payment, we can use the future value formula for an ordinary annuity:
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, the future value (FV) is given as $80,000, the interest rate (r) is 9% per year compounded quarterly. Since payments are made at the end of each quarter, there are 8 years * 4 quarters = 32 compounding periods (n).
Substituting the given values into the formula:
$80,000 = P * [(1 + 0.09/4)^32 - 1] / (0.09/4)
Now, let's solve for P.
First, simplify the equation inside the square brackets:
[(1 + 0.09/4)^32 - 1] = (1.0225^32 - 1)
Using a calculator, calculate the value inside the square brackets:
(1.0225^32 - 1) = 0.40525591
Now, substitute this value and the other given values back into the formula:
$80,000 = P * (0.40525591) / (0.09/4)
To isolate P, multiply both sides of the equation by (0.09/4):
$80,000 * (0.09/4) = P * (0.40525591)
Calculate the left side of the equation:
$80,000 * (0.09/4) = $1,800
Now divide both sides of the equation by (0.40525591) to solve for P:
P = $1,800 / (0.40525591)
Using a calculator, divide $1,800 by 0.40525591:
P ≈ $4,441.78
Therefore, the size of each payment is approximately $4,441.78.