John made a deposit of 1000 into a fund at the beginning of each year for 20 years. At the end of 20 years, he began making semiannual withdrawals of 3000 at the beginning of each six months, with a smaller final withdrawal to exhaust the fund. The fund earned an annual effective interest rate of 8.16%.

Calculate the amount of the final withdrawal.

• 561
• 1226
• 1430
• 1488
• 2240

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

Solution: C

The accumulated value is

[[math]]1000 \ddot{s}_{\overline{20}|0.0816}=50,382.16.[[/math]]

This must provide a semi-annual annuity-due of 3000. Let n be the number of payments. Then solve

[[math]]3000 \ddot a_{\overline{n}|0.04} = 50,382.16[[/math]]

for [math]n = 26.47[/math]. Therefore, there will be 26 full payments plus one final, smaller, payment. The equation is

[[math]]50,382.16 = 3000 \ddot a_{\overline{n}|0.04} + X (1.04)^{-26}[[/math]]

with solution [math]X = 1430[/math]. Note that the while the final payment is the 27^th payment, because this is an annuity-due, it takes place 26 periods after the annuity begins.

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