A loan of 20,000 is to be repaid with ten increasing installments payable at the end of each year. Each installment will be 10% greater than the preceding installment. The annual effective interest rate on the loan is 9%.

Calculate the amount of principal in the second installment.

• 292
• 527
• 975
• 1435
• 1774

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

Solution: B

[[math]] \begin{aligned} R_1= & \frac{20000}{\frac{1-\left(\frac{1.1}{1.09}\right)^{10}}{0.09-0.1}}=\frac{20000}{9.562} \\ = & 2091.61 \end{aligned} [[/math]]

The fourth payment is [math]2091.61(1.1)=2300.71[/math] The principal outstanding before the fourth payment is

[[math]] 2300.71 \frac{1-\left(\frac{1.1}{1.09}\right)^9}{0.09-0.1}=19708.94 [[/math]]

Interest in the fourth payment is [math]=19708.94(0.09)=1773.80[/math] [math]X=[/math] Principal repaid [math]=2300.71-1773.80=526.91[/math]

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