The Ajax Co. just decided to save $1,500 a month for the next five years as a safety net for recessionary periods. The money will be set aside in a separate savings account which pays 3.25% interest compounded monthly. They deposit the first $1,500 today. If the company had wanted to deposit an equivalent lump sum today, how much would they have had to deposit?

Question

Quizplus · Accepted Answer

To find the equivalent lump sum that needs to be deposited today, we use the formula for the present value of an annuity due (since the first payment is made immediately). The formula is $PV = P \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] \times (1 + r)$, where $P$ is the payment amount, $r$ is the monthly interest rate, and $n$ is the total number of payments. Given that the monthly interest rate is $3.25\% / 12 = 0.00270833$ and there are $5 \times 12 = 60$ payments of $1,500, the calculation is as follows:$$PV = 1500 \times \left[\frac{1 - (1 + 0.00270833)^{-60}}{0.00270833}\right] \times (1 + 0.00270833)$$Solving this gives a present value of approximately $83,189.29, which matches choice B.