A borrower took out a loan of 100,000 and promised to repay it with a payment at the end of each year for 30 years. The amount of each of the first ten payments equals the amount of interest due. The amount of each of the next ten payments equals 150% of the amount of interest due. The amount of each of the last ten payments is X. The lender charges interest at an annual effective rate of 10%.

Calculate X.

• 3,204
• 5,675
• 7,073
• 9,744
• 11,746

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

Solution: D

The loan balance after 10 years is still 100,000. For the next 10 payments, the interest paid is 10% of the outstanding balance and therefore the principal repaid is 5% of the outstanding balance. After 10 years the outstanding balance is 10 100, 000(0.95)⁵ = 9,874. Then,

[[math]] X = 59,874/a_{\overline{10}|0.1} = 59,874 / 6.14457 = 9, 744 [[/math]]

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