Question 8.3: Rockwell Industries has a three-year bond outstanding that p...
Rockwell Industries has a three-year bond outstanding that pays a 7.25 percent coupon and is currently priced at $913.88. What is the yield to maturity of this bond? Assume annual coupon payments.
We start with a time line for Rockwell’s bond:
Use trial and error to solve for the yield to maturity (YTM). Since the bond is selling at a discount, we know that the yield to maturity is higher than the coupon rate. Try YTM = 10%.
P_B=\frac{C_1}{1+i}+\frac{C_2}{(1+i)^2}+\frac{C_3+F_3}{(1+i)^3}\$913.88=\frac{\$72.50}{1.10}+\frac{\$72.50}{(1.10)^2}+\frac{\$72.50+\$1,000}{(1.10)^3}
=\$65.91+\$59.92+\$805.79
\neq \$931.61
Try a higher rate, say YTM = 11%.
P_B=\frac{C_1}{1+i}+\frac{C_2}{(1+i)^2}+\frac{C_3+F_3}{(1+i)^3}\$913.88=\frac{\$72.50}{1.11}+\frac{\$72.50}{(1.11)^2}+\frac{\$72.50+\$1,000}{(1.11)^3}
= \$65.32+\$58.84+\$784.20
\neq \$908.36
Since this is less than the price of the bond, we know that the YTM is between 10 and 11 percent and closer to 11 percent.
Try YTM = 10.75%.
\$913.88=\frac{\$72.50}{1.1075}+\frac{\$72.50}{(1.1075)^2}+\frac{\$72.50+\$1,000}{(1.1075)^3}
=\$65.46+\$59.11+\$789.53
\cong \$914.09
Alternatively, we can use Equation 6.1 and the present value equation from Chapter 5 to solve for the price of the bond.
PVA_n=\frac{CF}{i}\times\left[1-\frac{1}{(1+i)^n}\right]=CF\times\frac{1-1/(1+i)^n}{i^2} 6.1
P_B=C\times\left[\frac{1-\frac{1}{(1+i)^{n}}}{i}\right]+\frac{F_n}{(1+i)^{n}}
\$913.88=\$72.50\times\left[\frac{1-\frac{1}{(1+0.1075)^3}}{0.1075}\right]+\frac{\$1,000}{(1.1075)^3}
=\$177.94+\$736.15
\cong \$914.09
Thus, the YTM is approximately 10.75 percent. Using a financial calculator provides an exact YTM of 10.7594%.