Chapter 6, Problem 6.1STP

Learning Goals 5, 6

ST6- 1 Bond valuation Lahey Industries has outstanding a $1,000 par-value bond with an 8% coupon rate. The bond has 12 years remaining to its maturity date.

1. a. If interest is paid annually, find the value of the bond when the required return is (1) 7%, (2) 8%, and (3) 10%.
2. b. Indicate for each case in part a whether the bond is selling at a discount, at a premium, or at its par value.
3. c. Using the 10% required return, find the bond’s value when interest is paid semiannually.

Summary Introduction

To calculate: Valuation of bond.

Introduction:

Bond valuation: Determining the theoretical fair value of a particular bond is known as bond valuation.

Answer to Problem 6.1STP

At 7% required return = $1,079.43

At 8% required return = $1.000

At 10% required return = $863.80

Explanation of Solution

Given:

(1)

I (interest) = 0.8%

n (years to maturity) = 12

M (dollar par value) = $1000

r_d (required return on the bond) = 7%

(2)

I (interest) = 0.8%

n (years to maturity) = 12

M (dollar par value) = $1000

r_d (required return on the bond) = 8%

(3)

I (interest) = 0.8%

n (years to maturity) = 12

M (dollar par value) = $1000

r_d (required return on the bond) = 10%

Calculation

The general formula for calculating bond valuation in annually is shown below.

${\text{B}}_{\text{0}}\text{=}\frac{\text{1}}{{\text{r}}_{\text{d}}}\left[1-\frac{\text{1}}{{(1+{\text{r}}_{\text{d}})}^{\text{n}}}\right]\text{+M}\left[\frac{\text{1}}{{(1+{\text{r}}_{\text{d}})}^{\text{n}}}\right]$ (1)

Substitute the values in equation (1) to calculate the valuation of bond. Here the interest rate is $80 of interest (0.08 multiplied by $1000).

$\begin{array}{c}{\text{B}}_{\text{0}}\text{=}\frac{\$80}{0.07}\left[1-\frac{\text{1}}{{(1+0.07)}^{12}}\right]\text{+\$1000}\left[\frac{\text{1}}{{(1+0.07)}^{12}}\right]\\ =\left(\$1,142.86\times 0.556\right)+\left(\$1000\times 0.444\right)\\ =\left(\$635.43+\$444.00\right)\\ =\$1,079.43\end{array}$

Valuation of bond is $1,079.43.

By using the same equation (1), the valuation of bond in different interest rate is shown below.

Table 1 shows the valuation of bond.

Table 1

	I	M	N	rd	Bond valuation
1	$80	$1000	12	0.07	$1,079.43
2	$80	$1000	12	0.08	$1,000.00
3	$80	$1000	12	0.10	$863.80

Summary Introduction

To discuss: Classify the bond as, sale in discount, premium or at its par value.  

Explanation of Solution

On the basis of required return and valuation, the bonds are classified as sale in discount, sale in premium or sale in its par value. The bond which has 7% return and $1, 079, 43 bond value is sells at premium. The second bond (8% return and $1,000.00 bond value) is sells at its par value and third bond (10% return and $863.80 bond value) is sells at discount.

Summary Introduction

To calculate: Valuation of bond at semiannual interest payment.

Introduction:

Bond valuation: Determining the theoretical fair value of a particular bond is known as bond valuation.

Answer to Problem 6.1STP

$862.00

Explanation of Solution

Given:

r_d (required return on the bond) = 10%

M (dollar par value) = $1000

n (years to maturity) = 12

I (interest) = 0.8%

Calculation

For calculating the value of bond in semiannual interest rate, convert the annual interest rate in to semiannual interest rate by dividing by two, likewise the required return and number of years to maturity must be divided and multiplied by two respectively.

By using the equation (1), in part (a), the calculation of valuation in semiannual interest payment is shown below.

$\begin{array}{c}{\text{B}}_{\text{0}}\text{=}\frac{\frac{\text{i}}{2}}{\frac{{\text{r}}_{\text{d}}}{2}}\left[1-\frac{\text{1}}{{(1+\frac{{\text{r}}_{\text{d}}}{2})}^{\text{n}\times \text{2}}}\right]\text{+M}\left[\frac{\text{1}}{{(1+\frac{{\text{r}}_{\text{d}}}{2})}^{\text{n}\times \text{2}}}\right]\\ \text{=}\left(\frac{\$80}{2}\times \frac{0.10}{2}\right)\left(1-\frac{1}{{\left(\frac{1+0.10}{2}\right)}^{24}}\right)\times 1000\left(\frac{1}{{\left(\frac{1+0.10}{2}\right)}^{24}}\right)\\ =\$1000\times \left(\frac{1}{{\left(\frac{0.10}{2}\right)}^{24}}\right)\\ =\left(800\times 0.690\right)+\left(1000\times 0.310\right)\\ =552.00+310.00\\ =862.00\end{array}$

Valuation of bond in semiannual interest payment is $862.00.