A three-year 1000 face amount bond pays coupons of X quarterly. It is bought at a price to yield an annual nominal rate of 8% convertible quarterly. If the amount of each coupon were doubled, the purchase price would have to increase by 500 for the bond to maintain the same yield rate.

Calculate X.

• 20
• 24
• 31
• 40
• 47

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

Solution: E

[[math]] \begin{aligned} & P=X a_{\overline{12} | 0.02}+1000 v^{12} \\ & 500+P=2 X a_{\overline{12} | 0.02}+1000 v^{12} \end{aligned} [[/math]]

Subtract the first equation from the second.

[[math]] \begin{aligned} & 500=X a_{\overline{12} \mid 0.02} \\ & X=47.28 \end{aligned} [[/math]]

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