A 40-year bond is purchased at a discount. The bond pays annual coupons. The amount for accumulation of discount in the 15th coupon is 194.82. The amount for accumulation of discount in the 20th coupon is 306.69.

Calculate the amount of discount in the purchase price of this bond.

• 13,635
• 13,834
• 16,098
• 19,301
• 21,135

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

Solution: E

Let F, C, r, and i have their usual interpretations. The discount is [math][/math]and the discount in the coupon at time t is . Then,

[[math]] \begin{array}{l}{{194.82=(C i-F r)\nu^{26}}}\\ {{306.69=(C i-F r)\nu^{21}}}\\ 0.63523=\nu^{5}\Longrightarrow\nu=0.91\,324\Longrightarrow i=0.095 \\ (C i-F r)=194.82(1.095)^{26}=2062.53 \\ \mathrm{Discount} = 2062.53a_{\overline{40}|0.095}=21,135. \end{array} [[/math]]

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