You are going to withdraw $1,000 at the end of each year for the next three years from an account that pays interest at a rate of 8% compounded annually. The account balance will reduce to zero when the last withdrawal is made. How much interest will you earn on the account over the three year life?

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Quizplus · Accepted Answer

To solve this, we use the future value of an annuity formula to find the initial amount needed to fulfill these withdrawals. The formula is $FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)$, where $FV$ is the future value (which is 0 in this case since the account depletes to zero), $P$ is the payment amount, $r$ is the interest rate per period, and $n$ is the number of periods. However, since we're looking for the initial present value (PV) that can support these withdrawals, we rearrange the formula to solve for PV of an annuity: $PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)$. Given $P = \$1,000$, $r = 0.08$, and $n = 3$, we find the initial amount needed.$PV = \$1,000 \times \left( \frac{1 - (1 + 0.08)^{-3}}{0.08} \right) = \$1,000 \times \left( \frac{1 - (1.08)^{-3}}{0.08} \right) = \$1,000 \times 2.5771 = \$2,577.10$.This is the initial deposit required. Over three years, you withdraw \$3,000 in total (\$1,000 each year for 3 years). The interest earned is the difference between what you withdraw and your initial deposit.Interest earned = Total withdrawals - Initial deposit = \$3,000 - \$2,577.10 = \$422.90.

You Are Going to Withdraw $1,000 at the End of Each