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.

Question

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 answer with the solution

Answer 1

To determine the quarterly payment amount, we can use the formula for the future value of an ordinary annuity:

Future Value = Payment * [((1 + interest rate)^number of periods) - 1] / interest rate

In this case, we want to find the quarterly payment amount, so let's solve for "Payment." Rearranging the formula, we get:

Payment = Future Value * (interest rate / [((1 + interest rate)^number of periods) - 1])

First, let's calculate the future value of the annuity. Since we need $5,000 four years after the last payment, we can treat this as a standard time value of money problem:

Future Value = $5,000 / (1 + 0.08/4)^(4*10) = $5,000 / (1.02)^40 = $5,000 / 1.8185 ≈ $2,747.50

Now, we can substitute this value along with the interest rate (8% or 0.08) and the number of periods (24 payments over 10 years, or 40 quarters) into the payment formula:

Payment = $2,747.50 * (0.08 / [((1 + 0.08)^40) - 1])
≈ $2,747.50 * (0.08 / [1.02^40 - 1])
≈ $2,747.50 * (0.08 / 1.8185)
≈ $2,747.50 * 0.0440
≈ $120.79

Therefore, you would have to deposit approximately $120.79 (rounded to the nearest dollar) into the account every quarter to accumulate $5,000 at the end of 10 years.