A loan of amount [math]a_{\overline{4}|i}[/math] is repaid with payments of 1 at the end of each year for four years. The sum of the interest paid in the last two years is equal to [math](1+i)^2[/math] times the sum of the principal repaid in the first two years.

Calculate i.

• 0.543
• 0.567
• 0.592
• 0.618
• 0.645

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

Solution: D

[[math]] \begin{aligned} & I_3+I_4=(1+i)^2\left(P_1+P_2\right) \\ & \left(1-v^2\right)+(1-v)=(1+i)^2\left(v^4+v^3\right) \\ & \left(1-v^2+1-v\right)=(1+i)^2\left(v^4+v^3\right) \\ & v^2\left(1-v^2+1-v\right)=\left(v^4+v^3\right) \\ & 2 v^2-v^2\left(v^2+v\right)=v^2\left(v^2+v\right) \\ & 2 v^2=2 v^2\left(v^2+v\right) \\ & 1=v^2+v \\ & v^2+v-1=0 \\ & v=0.61803 \\ & 1+i=1.61803 \\ & i=0.61803\end{aligned} [[/math]]

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