A 20-year loan of 1000 is repaid with payments at the end of each year. Each of the first ten payments equals 150% of the amount of interest due. Each of the last ten payments is X. The lender charges interest at an annual effective rate of 10%.

Calculate X.

• 32
• 57
• 70
• 97
• 117

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

Solution: D

For the first 10 years, each payment equals 150% of interest due. The lender charges 10%, therefore 5% of the principal outstanding will be used to reduce the principal. At the end of 10 years, the amount outstanding is 1000(1-0.05)¹⁰ = 598.74.

Thus, the equation of value for the last 10 years using a comparison date of the end of year 10 is

[[math]] 598.74 = X a_{\overline{10}|10\%} = 6.1446X, X = 97.44. [[/math]]

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