Chapter 8, Problem 17QP

Bond Price Movements Miller Corporation has a premium bond making semiannual payments. The bond pays a coupon of 8.5 percent has a YTM of 7 percent and has 13 years to maturity. The Modigliani Company has a discount bond making semiannual payments. This bond pays a coupon of 7 percent, has a YTM of 8.5 percent, and also has 13 years to maturity. If interest rates remain unchanged, what do you expect the price of these bonds to be 1 year from now? In 3 years? In 8 years? In 12 years? In 13 years? What’s going on here? Illustrate your answers by graphing bond prices versus time to maturity.

Summary Introduction

To determine: The price of the bonds after 1 year, 3 years, 8 years, 12 years and 13 years and prepare a graph of bond prices versus time to maturity.

Yield to Maturity:

The yield to maturity is the total yield or return, which is derived from a bond until the time of the maturity. For this, it is assumed that the bond will be held until its maturity and would not be called.

Effective Annual Rate:

The effective annual rate is the rate, which is incurred or received on various investment or loans. The effective annual rate is affected by the increase in compounding years.

Current Yield:

The current yield represents the return on the bond, which is held by the owner for a year. The current yield is calculated by dividing the annual cash inflow from the current market price.

Explanation of Solution

Solution:

Given,

For M Company,

The bond is a premium bond and makes semi-annual payments.

The coupon rate is 8.5%.

The yield to maturity is 7%.

The maturity period is 13 years.

For MG Company,

The bond is a discount bond and makes semi-annual payments.

The coupon rate is 7%.

The yield to maturity is 8.5%.

The maturity period is 13 years.

Calculation of the value of the bond after 1 year:

The formula to calculate the value of the bond is,

$\text{Value}\text{of}\text{bond}=\left[\begin{array}{l}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{r,\text{period}}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{r,\text{period}}\right)\end{array}\right]$

For M Company,

Substitute $42.5 for interest value, $1,000 for the interest value, 1.9 for ${\text{PVIFA}}_{3.5\%,2}$ and 0.9335 for ${\text{PVIF}}_{3.5\%,2}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$42.5\times 1.9\right)+\left(\$1,000\times 0.9335\right)\right]\\ =\$80.75+\$933.5\\ =\$1,014.25\end{array}$

The value of the bond is $1,014.25.

For MG Company,

Substitute $35 for interest value, $1,000 for the interest value, 1.88 for ${\text{PVIFA}}_{4.25\%,2}$ and 0.9201 for ${\text{PVIF}}_{4.25\%,2}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$35\times 1.88\right)+\left(\$1,000\times 0.9201\right)\right]\\ =\$65.8+\$920.1\\ =\$985.9\end{array}$

The value of the bond is $985.9.

Calculation of the value of bond after 3 years:

For M Company,

Substitute $42.5 for interest value, $1,000 for the interest value, 5.329 for ${\text{PVIFA}}_{3.5\%,6}$ and 0.8135 for ${\text{PVIF}}_{3.5\%,6}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$42.5\times 5.329\right)+\left(\$1,000\times 0.8135\right)\right]\\ =\$226.48+\$813.5\\ =\$1,039.98\end{array}$

The value of the bond is $1,039.98.

For MG Company,

Substitute $35 for interest value, $1,000 for the interest value, 5.2 for ${\text{PVIFA}}_{4.25\%,6}$ and 0.779 for ${\text{PVIF}}_{4.25\%,6}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$35\times 5.2\right)+\left(\$1,000\times 0.779\right)\right]\\ =\$182+\$779\\ =\$961\end{array}$

The value of the bond is $961.

Calculation of the value of bond after 8 years:

For M Company,

Substitute $42.5 for interest value, $1,000 for the interest value, 12.094 for ${\text{PVIFA}}_{3.5\%,16}$ and 0.5767 for ${\text{PVIF}}_{3.5\%,16}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$42.5\times 12.094\right)+\left(\$1,000\times 0.5767\right)\right]\\ =\$513.995+\$575.7\\ =\$1,089.695\end{array}$

The value of the bond is $1,089.695.

For MG Company,

Substitute $35 for interest value, $1,000 for the interest value, 11.44 for ${\text{PVIFA}}_{4.25\%,16}$ and 0.5137 for ${\text{PVIF}}_{4.25\%,16}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$35\times 11.44\right)+\left(\$1,000\times 0.5137\right)\right]\\ =\$400.4+\$513.7\\ =\$914.1\end{array}$

The value of the bond is $914.1.

Calculation of the value of bond after 12 years:

For M Company,

Substitute $42.5 for interest value, $1,000 for the interest value, 16.06 for ${\text{PVIFA}}_{3.5\%,24}$ and 0.4379 for ${\text{PVIF}}_{3.5\%,24}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$42.5\times 16.06\right)+\left(\$1,000\times 0.4379\right)\right]\\ =\$682.55+\$437.9\\ =\$1,120.45\end{array}$

The value of the bond is $1,120.45.

For MG Company,

Substitute $35 for interest value, $1,000 for the interest value, 14.86 for ${\text{PVIFA}}_{4.25\%,24}$ and 0.3682 for ${\text{PVIF}}_{4.25\%,24}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$35\times 14.86\right)+\left(\$1,000\times 0.3682\right)\right]\\ =\$520.1+\$368.2\\ =\$888.3\end{array}$

The value of the bond is $888.3.

Calculation of the value of the bond after 13 years:

For M Company,

Substitute $42.5 for interest value, $1,000 for the interest value, 16.89 for ${\text{PVIFA}}_{3.5\%,26}$ and 0.4088 for ${\text{PVIF}}_{3.5\%,26}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$42.5\times 16.89\right)+\left(\$1,000\times 0.4088\right)\right]\\ =\$717.825+\$408.8\\ =\$1,126.625\end{array}$

The value of the bond is $1,126.625.

For MG Company,

Substitute $35 for interest value, $1,000 for the interest value, 15.57 for ${\text{PVIFA}}_{4.25\%,26}$ and 0.3088 for ${\text{PVIF}}_{4.25\%,26}$ in the above formula.

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\left(\$35\times 15.57\right)+\left(\$1,000\times 0.3088\right)\right]\\ =\$544.95+\$308.8\\ =\$853.75\end{array}$

The value of the bond is $853.75.

The graph showing the change in the value of bonds with the maturity period is:

Fig 1

• The graph shows the change in the price of bonds with the maturity period.
• The x-axis shows the years of maturity.
• The y-axis shows the price of the bond.
• The value of the bonds of M Company increases with time.
• The value of the bonds of MG Company decreases with time.

Working note:

Calculation of the semi-annual interest for M Company:

$\begin{array}{c}\text{Semi-annual}\text{interest}=\text{Face}\text{value}\times \text{Interest}\text{rate}\times \frac{1}{2}\\ =\$1,000\times 8.5\%\times \frac{1}{2}\\ =\$42.5\end{array}$

The semi-annual interest is $42.5.

Calculation of the semi-annual interest for MG Company:

$\begin{array}{c}\text{Semi-annual}\text{interest}=\text{Face}\text{value}\times \text{Interest}\text{rate}\times \frac{1}{2}\\ =\$1,000\times 7\%\times \frac{1}{2}\\ =\$35\end{array}$

The semi-annual interest is $35.

Calculation of the ${\text{PVIFA}}_{3.5\%,2}$:

$\begin{array}{c}{\text{PVIFA}}_{3.5\%,2}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.035}\right)}^2\right\}}{0.035}\right]\\ =\frac{1-0.9335}{0.035}\\ =1.9\end{array}$

Calculation of the ${\text{PVIF}}_{3.5\%,2}$:

$\begin{array}{c}{\text{PVIF}}_{3.5\%,2}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.035\right)}^2}\right]\\ =0.9335\end{array}$

Calculation of the ${\text{PVIFA}}_{4.25\%,2}$:

$\begin{array}{c}{\text{PVIFA}}_{4.25\%,2}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.0425}\right)}^2\right\}}{0.0425}\right]\\ =\frac{1-0.9201}{0.0425}\\ =1.88\end{array}$

Calculation of the ${\text{PVIF}}_{4.25\%,2}$:

$\begin{array}{c}{\text{PVIF}}_{4.25\%,2}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.0425\right)}^2}\right]\\ =0.9201\end{array}$

Calculation of the ${\text{PVIFA}}_{3.5\%,6}$:

$\begin{array}{c}{\text{PVIFA}}_{3.5\%,6}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.035}\right)}^6\right\}}{0.035}\right]\\ =\frac{1-0.8135}{0.035}\\ =5.329\end{array}$

Calculation of the ${\text{PVIF}}_{3.5\%,6}$:

$\begin{array}{c}{\text{PVIF}}_{3.5\%,6}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.035\right)}^6}\right]\\ =0.8135\end{array}$

Calculation of the ${\text{PVIFA}}_{4.25\%,6}$:

$\begin{array}{c}{\text{PVIFA}}_{4.25\%,6}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.0425}\right)}^6\right\}}{0.0425}\right]\\ =\frac{1-0.779}{0.0425}\\ =5.2\end{array}$

Calculation of the ${\text{PVIF}}_{4.25\%,6}$:

$\begin{array}{c}{\text{PVIF}}_{4.25\%,6}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.0425\right)}^6}\right]\\ =0.779\end{array}$

Calculation of the ${\text{PVIFA}}_{3.5\%,16}$:

$\begin{array}{c}{\text{PVIFA}}_{3.5\%,16}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.035}\right)}^{16}\right\}}{0.035}\right]\\ =\frac{1-0.5767}{0.035}\\ =12.094\end{array}$

Calculation of the ${\text{PVIF}}_{3.5\%,16}$:

$\begin{array}{c}{\text{PVIF}}_{3.5\%,16}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.035\right)}^{16}}\right]\\ =0.5767\end{array}$

Calculation of the ${\text{PVIFA}}_{4.25\%,16}$:

$\begin{array}{c}{\text{PVIFA}}_{4.25\%,16}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.0425}\right)}^{16}\right\}}{0.0425}\right]\\ =\frac{1-0.5137}{0.0425}\\ =11.44\end{array}$

Calculation of the ${\text{PVIF}}_{4.25\%,16}$:

$\begin{array}{c}{\text{PVIF}}_{4.25\%,16}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.0425\right)}^{16}}\right]\\ =0.5137\end{array}$

Calculation of the ${\text{PVIFA}}_{3.5\%,24}$:

$\begin{array}{c}{\text{PVIFA}}_{3.5\%,24}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.035}\right)}^{24}\right\}}{0.035}\right]\\ =\frac{1-0.4379}{0.035}\\ =16.06\end{array}$

Calculation of the ${\text{PVIF}}_{3.5\%,24}$:

$\begin{array}{c}{\text{PVIF}}_{3.5\%,24}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.035\right)}^{24}}\right]\\ =0.4379\end{array}$

Calculation of the ${\text{PVIFA}}_{4.25\%,24}$:

$\begin{array}{c}{\text{PVIFA}}_{4.25\%,24}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.0425}\right)}^{24}\right\}}{0.0425}\right]\\ =\frac{1-0.3682}{0.0425}\\ =14.86\end{array}$

Calculation of the ${\text{PVIF}}_{4.25\%,24}$:

$\begin{array}{c}{\text{PVIF}}_{4.25\%,24}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.0425\right)}^{24}}\right]\\ =0.3682\end{array}$

Calculation of the ${\text{PVIFA}}_{3.5\%,26}$:

$\begin{array}{c}{\text{PVIFA}}_{3.5\%,26}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.035}\right)}^{26}\right\}}{0.035}\right]\\ =\frac{1-0.4088}{0.035}\\ =16.89\end{array}$

Calculation of the ${\text{PVIF}}_{3.5\%,26}$:

$\begin{array}{c}{\text{PVIF}}_{3.5\%,26}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.035\right)}^{26}}\right]\\ =0.4088\end{array}$

Calculation of the ${\text{PVIFA}}_{4.25\%,26}$:

$\begin{array}{c}{\text{PVIFA}}_{4.25\%,26}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.0425}\right)}^{26}\right\}}{0.0425}\right]\\ =\frac{1-0.3388}{0.0425}\\ =15.57\end{array}$

Calculation of the ${\text{PVIF}}_{4.25\%,26}$:

$\begin{array}{c}{\text{PVIF}}_{4.25\%,26}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.0425\right)}^{26}}\right]\\ =0.3388\end{array}$

Conclusion

Thus, the value of the bonds of M Company after 1 year is $1,014.25, after 3 years is $1,039.98, after 8 years is $1,089.695, after 12 years is $1,120.45, and after 13 years is $1,126.625 and the value of the bonds of MG Company after 1 year is $985.9, after 3 years is $961, after 8 years is $914.10, after 12 years is $888.3, and after 13 years is $853.75.