You are given the following information about two bonds, Bond A and Bond B:

• Each bond is a 10-year bond with semiannual coupons redeemable at its par value of 10,000, and is bought to yield an annual nominal interest rate of i, convertible semiannually.
• Bond A has an annual coupon rate of (i + 0.04), paid semiannually.
• Bond B has an annual coupon rate of (i – 0.04), paid semiannually
• The price of Bond A is 5,341.12 greater than the price of Bond B.

Calculate i

• 0.042
• 0.043
• 0.081
• 0.084
• 0.086

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

Solution: D

Throughout the solution, let [math]j=i/2[/math].

For bond A, the coupon rate is [math](i+0.04)/2=j+0.02.[/math]

For bond B, the coupon rate is [math](i-0.04)/2=j-0.02. [/math]

The price of bond A is [math]P_{A}=10,000(j+0.02)a_{\overline{30}|j}+10,000(1+j)^{-20}. [/math]

The price of bond B is [math]P_{B}=10,000(j-0.02)a_{\overline{20}|i}+10,000(1+j)^{-20} [/math]

Thus,

[[math]] \begin{align*} P_{A}-P_{B}=5,341.12=[200-(-200)]a_{\overline{20}|j}],=400a_{\overline{{{20}}}|j} \\ a_{\overline{{{20}}}|j} = 5,341.12 / 400 =13.3528 \end{align*} [[/math]]

Using the financial calculator, [math]j = 0.042[/math] and [math]i=2(0.042)=0.084.[/math]

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