An 18-year bond, with a price 61% higher than its face value, offers annual coupons with the coupon rate equal to 2.25 times the annual effective yield rate. An n-year bond, with the same face value, coupon rate, and yield rate, sells for a price that is 45% higher than its face value.

• 10
• 12
• 14
• 17
• 20

Calculate n.

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

Solution: B

Let face amount equal 1.

[[math]] \begin{aligned} & 1.61=2.25 i\left(\frac{1-v^{18}}{i}\right)+v^{18} \\ & 1.61=2.25\left(1-v^{18}\right)+v^{18} \\ & 1.25 v^{18}=0.64 \\ & v=0.96342 \\ & 1.45=2.25 i\left(\frac{1-v^n}{i}\right)+v^n \\ & 1.45=2.25\left(1-v^n\right)+v^n \\ & 1.25 v^n=0.8 \\ & v^n=0.64 \\ & n \ln 0.963492=\ln 0.64 \\ & n=12\end{aligned} [[/math]]

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