Bond X and Bond Y are n-year bonds with face amount of 10,000 and semiannual coupons, each yielding an annual nominal interest rate of 7% convertible semiannually. Bond X has an annual coupon rate of 6% and redemption value c. Bond Y has an annual coupon rate of 5% and redemption value c + 50. The price of Bond X exceeds the price of Bond Y by 969.52.

Calculate n.

• 14
• 17
• 23
• 34
• 46

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

Solution: B

Let X be the price of Bond X. Then, for the two bonds:

[[math]] \begin{array}{l}{{X=10,000(0.03)a_{\overline{{{2n}}}|0.035}+c(1.035)^{-2n}}}\\ {{X-969.52=10,000(0.025)a_{\overline{{{2n}}}|0.035}+(c+50)(1.035)^{-2n}}}\end{array} [[/math]]

Subtracting the second equation from the first gives

[[math]] \begin{array}{l}{{969.52=50a_{\overline{{{2n}}}|0.035}-50(1.035)^{-2n}}}\\ {{969.52=\frac{50}{0.035}[1-(1.035)^{-2n}]-50(1.035)^{-2n}}}\\ {{1.035^{-2n}=459.05/1478.57=0.310469}}\\ {{m=-(0.5)\ln(0.310469)/\vert\ln(1.035)=17.}} \end{array} [[/math]]

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