A loan of X is repaid with level annual payments at the end of each year for 10 years.

You are given:

• The interest paid in the first year is 3600; and
• The principal repaid in the 6th year is 4871

Calculate X.

• 44,000
• 45,250
• 46,500
• 48,000
• 50,000

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

Solution: D

Let P be the annual payment. The fifth line is obtained by solving a quadratic equation.

[[math]] \begin{array}{l c r}{P(1- v ^{10})=3600}\\ {P v ^{10-6+1}=4871}\\ {\frac{1-v^{10}}{v^5}=\frac{3600}{4871}}\\ {1- v ^{10}=0.739068 v^{5}} \\{{ v ^{5}=0.69656}}&{{}}\\ {{ v ^{10}=0.485195}}\\ i = 0.485195^{-1/10}-1=0.075 \\ X=P{\frac{1- v ^{10}}{i}}={\frac{3600}{0.075}}=48,000 \end{array} [[/math]]

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