The following two annuities-immediate have the same present value at an annual effective interest rate of i, i > 0.

1. A ten-year annuity with annual payments of 475.
2. A perpetuity with annual payments of 400 in years 1-5, zero in years 6-10, and 400 in years 11 and beyond.

Calculate i.

• 10.65%
• 10.75%
• 10.85%
• 10.95%
• 11.05%

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

Solution: B

[[math]] \begin{aligned} & 475 a_{\overline{10} \mid i}=400\left(a_{5 \mid i}+v^{10} a_{\infty}\right) \\ & 475 \frac{1-v^{10}}{i}=400 \frac{1-v^5+v^{10}}{i} \\ & 475\left(1-v^{10}\right)=400\left(1-v^5+v^{10}\right) \\ & 875 v^{10}-400 v^5-75=0 \\ & v^5=\frac{400 \pm \sqrt{400^2+4(875)(75)}}{2(875)}=0.6 \\ & i=(1 / 0.6)^{1 / 5}-1=0.1076=10.76 \%\end{aligned} [[/math]]

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