A ten-year 1000 par value bond with coupons paid annually at an annual rate of r is callable at par at the end of the 6th , 7th , 8th , or 9th year. The price of the bond is 1023. If the bond is called in the worst-case scenario for the bond investor, the resulting annual effective yield rate, i, is 96% of r.

Calculate i.

• 4.41%
• 7.46%
• 8.36%
• 10.56%
• 14.32%

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

Solution: E

The bond sells at a premium, so the worst-case scenario is redemption at time six. Then,

[[math]] \begin{array}{l}{{1023=1000\frac{i}{0.96}a_{\overline{{{6}}}|i}+1000(1+i)^{-6}}}\\ {{=\frac{1000}{0.96}[1-(1+i)^{-6}]+1000(1+i)^{-6}}}\\ {{(1+i)^{-6}=0.448}}\\ {{\dot{\displaystyle i=0.448^{-1/6}-1=0.1432}}}\\ i=14.32\% \end{array} [[/math]]

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