You are given the following information about an n-year bond, where n > 10:

• The bond pays 8% semiannual coupons and has face amount 1000.
• The bond is redeemable at par.
• The bond is callable at par 5 years after issue or 10 years after issue.
• P is the price to guarantee a yield of 6.8% convertible semiannually and Q is the price to guarantee a yield of 8.8% convertible semiannually.
• |P – Q| = 123.36

Calculate n.

• 11
• 15
• 19
• 22
• 26

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

Solution: C

When the yield is 6.8% < 8%, the bond is sold at a premium and hence an early call is most disadvantageous. Therefore, When the yield is 8.8% > 8%, the bond is sold at discount. Hence, Q < 1000 < P. and thus Q = 1050.15 – 123.36 = 926.79. Also, because the bond is sold at a discount, the latest call is the most disadvantageous. Thus,

[[math]] \begin{array}{l l} {{926.79=40a_{\overline{2n}|0.044}+1000(1.044)^{-2n}=\frac{40}{0.044}+\left(1.044\right)^{-2n}\left(1000-\frac{40}{0.044}\right)^{2n}}}= 909.09 +90.90(1.044)^{-2n}\\ {{17.70=90.90(1.044)^{-2n}}}&{{}}\\ {{2n=-\ln(17.70/90.90)/\ln(1.044)=38}}\\ {{n=19.}}&{{}}&{{}}\end{array} [[/math]]

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