Seth borrows X for four years at an annual effective interest rate of 8%, to be repaid with equal payments at the end of each year. The outstanding loan balance at the end of the third year is 559.12.

Calculate the principal repaid in the first payment.

• 444
• 454
• 464
• 474
• 484

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

Solution: A

The outstanding balance is the present value of future payments. With only one future payment, that payment must be 559.12(1.08) = 603.85. The amount borrowed is [math]603.85 a_{\overline{4}|0.08}=2000.[/math]The first payment has 2000(0.08) = 160 in interest, thus the principal repaid is 603.85 – 160 = 443.85.

Alternatively, observe that the principal repaid in the final payment is the outstanding loan balance at the previous payment, or 559.12. Principal repayments form a geometrically decreasing sequence, so the principal repaid in the first payment is 559.12 /1.08³ = 443.85.

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