Four annual tuition payments of 25,000 are to be paid at a future date. The payments will be funded by investing 1000 at the beginning of each month. The last deposit will be made six months before the first tuition payment. Interest is payable at a nominal interest rate of 6% convertible monthly.

Calculate the minimum number of monthly deposits required to fund the total tuition.

• 70
• 71
• 73
• 74
• There is not enough information to calculate the minimum number of monthly deposits.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

The effective annual rate of interest is (1.005)¹²-1 = 0.06168. The present value of the tuition payments six months before the first payment is

[[math]] 25,000^{-6}\,\ddot{a}_{\overline{4}|0.06168}=24,262.95(3.66473)=88,917.16 [[/math]]

The accumulated value of the deposits at that time is [math]1000s_{\overline{n}|0.05}[/math]. Equating the two amounts:

[[math]] \begin{aligned} 88,917.16 = 1000 \frac{1.005^n-1}{0.005} \\ 1.44459 = 1.005^n \\ n = \ln(1.44459)/\ln(1.005) = 73.75. \end{aligned} [[/math]]

Therefore, at least 74 payments will be required.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.