A zero-coupon bond with a face amount of 1000 sells for a price of 640 and matures in n years, where n is a whole number. A second bond has the same price, same time until maturity, and same annual effective yield. It pays annual coupons at an annual rate equal to 50% of the annual effective yield rate.

Calculate the face value of the second bond.

• 780
• 805
• 830
• 855
• 880

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

Solution: A

[[math]] \begin{aligned} & 640(1+i)^n=1000 \\ & (1+i)^n=1.5625 \\ & v^n=0.64 \\ & 640=F \frac{i}{2} a_{\overline{n} i}+F v^n \\ & 640=F\left[\frac{i}{2} \frac{1-v^n}{i}+v^n\right] \\ & 640=F\left[\frac{1}{2}(1-0.64)+0.64\right] \\ & F=780.49\end{aligned} [[/math]]

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