(Broverman exercise 4.1.2) A 6% bond maturing in 8 years with semiannual coupons to yield 5% convertible semiannually is to be replaced by a 5.5% bond yielding the same return. In how many years should the new bond mature? (Both bonds have the same price, yield rate and face amount.)

• 18.5
• 19
• 20
• 21.5
• 22

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.

Solution: D

Let [math]P=[/math] price of bond [math]1(=[/math] price of bond 2[math]) .8[/math] years means 16 periods. Bond 1 gives [math]P=F v^{16}+F r a_{\overline{16} \mid}=F(1.025)^{-16}+F(.03) a_{\overline{16} \mid .025}[/math].

For bond 2, let [math]n[/math] be the number of years. There are [math]2 n[/math] periods. Bond 2 gives [math]P=F(1.025)^{-2 n}+F(.0275) a_{\overline{2 n \mid} .025}[/math]. Equating the two expressions and cancelling [math]F[/math] gives

[[math]] \begin{aligned} & 1.025^{-16}+.03\left(1-1.025^{-16}\right) / .025=1.025^{-2 n}+.0275\left(1-1.025^{-2 n}\right) / .025 . \text { Thus } \\ & 1.025^{-2 n} / 10=1.1-\left(1.025^{-16}+.03\left(1-1.025^{-16}\right) / .025\right)=0.03472499 . \text { Thus } \\ & -2 n(\ln (1.025))=\ln (.3472499) \text { so } \\ & 2 n=-.5 * \ln (.3472499) / \ln (1.025) \text { and } n=21.41755 \end{aligned} [[/math]]

We round this to [math]2 n=-.5 * \ln (.3472499) / \ln (1.025)[/math] and [math]n=21.41755[/math] years. We round this to 21.5 years to get the final coupon.

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.