An investor owns a bond that is redeemable for 250 in 6 years from now. The investor has just received a coupon of c and each subsequent semiannual coupon will be 2% larger than the preceding coupon. The present value of this bond immediately after the payment of the coupon is 582.53 assuming an annual effective yield rate of 4%.

Calculate c.

• 32.04
• 32.68
• 40.22
• 48.48
• 49.45

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

Solution: A

The effective semi-annual yield rate is

[[math]] 1.04=\left(1+\frac{i^{(2)}}{2}\right)^{2}=\gt\frac{i^{(2)}}{2}=1.9804\% [[/math]]

Then,

[[math]] \begin{array}{l}{{582.53=c(1.02) v+c(1.02 v)^{2}+\cdots+c(1.02 v)^{12}+250 v^{12}}}\\ {{=c{\frac{1.02 v-(1.02 v)^{13}}{1-1.02 v}}+250 v^{12}=12.015c+197.579=c=32.04.}}\\ {{582.53=c{\frac{1.02 v-(1.02 v)^{13}}{1-1.02 v}}+250 v^{12}=12.015c+197.579=c=32.04.}}\end{array} [[/math]]

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