Chapter 8, Problem 2QP

Valuing Bonds Microhard has issued a bond with the following characteristics:

Par: $1,000

Time to maturity: 20 years

Coupon rate: 7 percent

Semiannual payments

Calculate the price of this bond if the YTM is:

1. a. 7 percent
2. b. 9 percent
3. c. 5 percent

Summary Introduction

To determine: The price of the bond.

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 the maturity and would not be called.

Explanation of Solution

Given,

The maturity period is 20 years.

The bond is a 7% coupon bond.

The par value of the bond is $1,000.

The bond is making semi-annual payments.

The yield to maturity is 7%.

The yield to maturity for semiannual payments will be 3.5% $\left(\frac{7\%}{2}\right)$

Calculation of the price of the bond:

The formula to calculate the price of the bond is,

$\text{Price}\text{of}\text{bond}=\left\{\begin{array}{l}\left[\left(\text{Par}\text{value}\times \text{Coupon}\text{rate}\right)\times {\text{PVIFA}}_{3.5\%,40}\right]\\ +\left(\text{Par}\text{value}\times {\text{PVIF}}_{3.5\%,40}\right)\end{array}\right\}$

Substitute $1,000 for the par value of the bond, 40 $\left(20\times 2\right)$ years for the time period, 21.357 for the ${\text{PVIFA}}_{3.5\%,40}$, and 0.25257 for the ${\text{PVIF}}_{3.5\%,40}$ in the given formula.

$\begin{array}{c}\text{Price}\text{of}\text{bond}=\left\{\begin{array}{c}\left[\left(\$1,000\times 3.5\%\right)\times 21.357\right]\\ +\left(\$1,000\times 0.25252\right)\end{array}\right\}\\ =\left[\left(\$35\times 21.357\right)+\left(\$252.52\right)\right]\\ =\$747.495+\$252.52\\ =\$1,000.015\end{array}$

The price of the bond is $1,000.

Working note:

Calculation of the ${\text{PVIFA}}_{3.5\%,40}$:

$\begin{array}{c}{\text{PVIFA}}_{3.5\%,40}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.035}\right)}^{40}\right\}}{0.035}\right]\\ =\frac{1-0.25257}{0.035}\\ =21.357\end{array}$

Calculation of the ${\text{PVIF}}_{3.5\%,40}$:

$\begin{array}{c}{\text{PVIF}}_{3.5\%,40}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.035\right)}^{40}}\right]\\ =0.25252\end{array}$

Conclusion

Thus, the price of the bond is $1,000.

Summary Introduction

To determine: The price of the bond.

Explanation of Solution

Given,

The maturity period is 20 years.

The bond is a 7% coupon bond.

The par value of the bond is $1,000.

The bond is making semi-annual payments.

The yield to maturity is 9%.

The yield to maturity will be 4.5% $\left(\frac{9\%}{2}\right)$

Calculation of the price of the bond:

The formula to calculate the price of the bond is,

$\text{Price}\text{of}\text{bond}=\left\{\begin{array}{c}\left[\left(\text{Par}\text{value}\times \text{Coupon}\text{rate}\right)\times {\text{PVIFA}}_{4.5\%,40}\right]\\ +\left(\text{Par}\text{value}\times {\text{PVIF}}_{4.5\%,40}\right)\end{array}\right\}$

Substitute $1,000 for the par value of the bond, 40 $\left(20\times 2\right)$ years for the time period, 18.402 for the ${\text{PVIFA}}_{4.5\%,40}$, and 0.1719 for the ${\text{PVIF}}_{4.5\%,40}$ in the given formula.

$\begin{array}{c}\text{Price}\text{of}\text{bond}=\left\{\begin{array}{c}\left[\left(\$1,000\times 3.5\%\right)\times 18.402\right]\\ +\left(\$1,000\times 0.1719\right)\end{array}\right\}\\ =\left[\left(\$35\times 18.402\right)+\left(\$171.9\right)\right]\\ =\$644.07+\$171.9\\ =\$815.97\end{array}$

Thus, the price of the bond is $815.97.

Working note:

Calculation of the ${\text{PVIFA}}_{4.5\%,40}$:

$\begin{array}{c}{\text{PVIFA}}_{4.5\%,40}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.045}\right)}^{40}\right\}}{0.045}\right]\\ =\frac{1-0.1719}{0.045}\\ =18.402\end{array}$

Calculation of the ${\text{PVIF}}_{4.5\%,40}$:

$\begin{array}{c}{\text{PVIF}}_{4.5\%,40}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.045\right)}^{40}}\right]\\ =0.1719\end{array}$

Conclusion

Conclusion:

Thus, the price of the bond is $815.97.

Summary Introduction

To determine: The price of the bond.

Explanation of Solution

Given,

The maturity period is 20 years.

The bond is a 7% coupon bond.

The par value of the bond is $1,000.

The bond is making semi-annual payments.

The yield to maturity is 5%.

The yield to maturity will be 2.5% $\left(\frac{5\%}{2}\right)$

Calculation of the price of the bond:

The formula to calculate the price of the bond is,

$\text{Price}\text{of}\text{bond}=\left\{\begin{array}{c}\left[\left(\text{Par}\text{value}\times \text{Coupon}\text{rate}\right)\times {\text{PVIFA}}_{2.5\%,40}\right]\\ +\left(\text{Par}\text{value}\times {\text{PVIF}}_{2.5\%,40}\right)\end{array}\right\}$

Substitute $1,000 for the par value of the bond, 40 $\left(20\times 2\right)$ years for the time period, 25.104 for the ${\text{PVIFA}}_{2.5\%,40}$, and 0.3724 for the ${\text{PVIF}}_{2.5\%,40}$ in the given formula.

$\begin{array}{c}\text{Price}\text{of}\text{bond}=\left\{\begin{array}{c}\left[\left(\$1,000\times 3.5\%\right)\times 25.104\right]\\ +\left(\$1,000\times 0.3724\right)\end{array}\right\}\\ =\left[\left(\$35\times 25.104\right)+\left(\$372.4\right)\right]\\ =\$878.64+\$372.4\\ =\$1,251.04\end{array}$

Thus, the price of the bond is $1,251.04.

Working note:

Calculation of the ${\text{PVIFA}}_{2.5\%,40}$:

$\begin{array}{c}{\text{PVIFA}}_{2.5\%,40}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.025}\right)}^{40}\right\}}{0.025}\right]\\ =\frac{1-0.3724}{0.025}\\ =25.104\end{array}$

Calculation of the ${\text{PVIF}}_{2.5\%,40}$:

$\begin{array}{c}{\text{PVIF}}_{2.5\%,40}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.025\right)}^{40}}\right]\\ =0.3724\end{array}$

Conclusion

Thus, the price of the bond is $1,251.04.