You plan to make 24 equal quarterly payments (payments are at the end of each period) into an account to pays 8% (per year compounded quarterly). If you need $5,000 at the end of 10 years (i.e. 4 years after the last payment is made into the account), how much would you have to deposit into the account every quarter? Round your answer to the nearest 10 dollars.
Please show me the steps
To find out how much you would have to deposit into the account every quarter, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value
P = Payment per period
r = Interest rate per period
n = Number of periods
Given:
FV = $5,000
r = 8% per year compounded quarterly
n = 10 years (40 quarters)
First, we need to convert the interest rate and the number of periods to match the compounding frequency (quarterly in this case).
Interest rate per quarter (i):
i = 8% / 4 quarters = 2% per quarter = 0.02
Number of quarters (n):
n = 10 years * 4 quarters per year = 40 quarters
Now, we can rearrange the formula to solve for the payment per period (P):
P = FV * (r / ((1 + r)^n - 1))
Substituting in the given values, we get:
P = $5,000 * (0.02 / ((1 + 0.02)^40 - 1))
P ≈ $5,000 * (0.02 / (1.02^40 - 1))
Using a calculator or spreadsheet, you can calculate the value of P to the nearest dollar.
P ≈ $78.59
Finally, rounding the result to the nearest 10 dollars, you would have to deposit approximately $80 every quarter.