Revision as of 14:24, 19 November 2023 by Admin (Created page with "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. <ul class="mw-excansopts"><li>0.543</li><li>0.567</li><li>0.592</li><li>0.618</li><li>0.645</li></ul> {{soacopyright | 2023 }}")
ABy Admin
Nov 19'23
Exercise
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.
ABy Admin
Nov 19'23
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.