Kate buys a five-year 1000 face amount bond today with a 100 discount. The annual nominal coupon rate is 5% convertible semiannually. One year later, Wallace buys a four-year bond. It has the same face amount and coupon values as Kate’s and is priced to yield an annual nominal interest rate of 10% convertible semiannually. The discount on Wallace’s bond is D. The book value of Kate’s bond at the time Wallace buys his bond is B.

Calculate B – D.

• 724
• 738
• 756
• 838
• 917

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

Solution: C

The yield rate on Kate's bond is

[[math]] \begin{aligned} & (1000-100)=25 a_{\overline{10}| \frac{i^{(2)}}{2}}+1000 v^{10} \\ & \frac{i^{(2)}}{2}=0.0371551 \end{aligned} [[/math]]

The discount on Wallace's bond is

[[math]] \begin{aligned} & (1000-D)=25 a_{\overline{8} \mid 0.05}+1000 v^8 \\ & 1000-D=838.42, \quad D=161.58 \end{aligned} [[/math]]

The book value of Kate's bond at time 1 is

[[math]] \begin{aligned} & B=25 a_{\overline{8}|0.0371551}+1000 v^8 \\ & B=917.19 \end{aligned} [[/math]]

The difference is [math]B-D=917.19-161.58=755.61[/math]

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