Chapter 6, Problem 3P

a)

Summary Introduction

To determine: Value of bond as on April 15, 2016 with a required rate of return of 7%.

Explanation of Solution

Given information:

Cost of debt is 7%,

M is $1,000 and n is 12 (2028-2016)

Calculation of interest:

$\begin{array}{c}\text{I}={\text{k}}_{\text{d}}\times {\text{P}}_{\text{0}}\\ =\$1,000\times 0.0875\\ =\$87.50\end{array}$

Hence interest is $87.50

Calculation of value of bond:

$\begin{array}{c}{\text{P}}_{\text{0}}={\displaystyle \sum _{\text{t}=1}^{\text{n}}\frac{\text{I}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{t}}}}+\frac{\text{M}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{n}}}\\ ={\displaystyle \sum _{\text{t}=1}^{\text{12}}\frac{\text{\$87}\text{.50}}{{\left(1+\text{0}\text{.07}\right)}^{\text{12}}}}+\frac{\$1,000}{{\left(1+\text{0}\text{.07}\right)}^{\text{12}}}\\ =\text{\$87}\text{.50}\left(7.943\right)+\$1,000\left(0.444\right)\\ =\$1,139\end{array}$

Hence, value of bond is $1,139

b)

Summary Introduction

To determine: Value of bond as on April 15, 2016 with a required rate of return of 9%.

Explanation of Solution

Given information:

Cost of debt is 9%,

M is $1,000 and n is 12 (2028-2016)

Interest is $87.50

Calculation of value of bond:

$\begin{array}{c}{\text{P}}_{\text{0}}={\displaystyle \sum _{\text{t}=1}^{\text{n}}\frac{\text{I}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{t}}}}+\frac{\text{M}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{n}}}\\ ={\displaystyle \sum _{\text{t}=1}^{\text{12}}\frac{\text{\$87}\text{.50}}{{\left(1+\text{0}\text{.09}\right)}^{\text{12}}}}+\frac{\$1,000}{{\left(1+\text{0}\text{.09}\right)}^{\text{12}}}\\ =\text{\$87}\text{.50}\left(7.161\right)+\$1,000\left(0.356\right)\\ =\$983\end{array}$

Hence, value of bond is $983

c)

Summary Introduction

To determine: Value of bond as on April 15, 2016 with a required rate of return of 9%.

Explanation of Solution

Given information:

Cost of debt is 11%,

M is $1,000 and n is 12 (2028-2016)

Interest is $87.50

Calculation of value of bond:

$\begin{array}{c}{\text{P}}_{\text{0}}={\displaystyle \sum _{\text{t}=1}^{\text{n}}\frac{\text{I}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{t}}}}+\frac{\text{M}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{n}}}\\ ={\displaystyle \sum _{\text{t}=1}^{\text{12}}\frac{\text{\$87}\text{.50}}{{\left(1+\text{0}\text{.11}\right)}^{\text{12}}}}+\frac{\$1,000}{{\left(1+\text{0}\text{.11}\right)}^{\text{12}}}\\ =\text{\$87}\text{.50}\left(6.492\right)+\$1,000\left(0.286\right)\\ =\$854\end{array}$

Hence, value of bond is $854

Calculation of value of Company F bond at 8% interest rate:

Given information:

Cost of debt is 11%,

M is $1,000 and n is 24 (12x2 semi-annually)

Calculation of interest:

$\begin{array}{c}{\text{k}}_{\text{d}}=\frac{0.08}{2}\\ =0.04\end{array}$

$\begin{array}{c}\text{I}={\text{k}}_{\text{d}}\times {\text{P}}_{\text{0}}\\ =\frac{\$1,000}{2}\times 0.0875\\ =\$43.75\end{array}$

Hence, interest is $43.75

Calculation of value of bond:

$\begin{array}{c}{\text{P}}_{\text{0}}={\displaystyle \sum _{\text{t}=1}^{\text{n}}\frac{\text{I}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{t}}}}+\frac{\text{M}}{{\left(1+{\text{k}}_{\text{d}}\right)}^{\text{n}}}\\ ={\displaystyle \sum _{\text{t}=1}^{\text{24}}\frac{\text{\$43}\text{.75}}{{\left(1+\text{0}\text{.04}\right)}^{\text{24}}}}+\frac{\$1,000}{{\left(1+\text{0}\text{.04}\right)}^{\text{24}}}\\ =\text{\$43}\text{.75}\left(15.247\right)+\$1,000\left(0.390\right)\\ =\$1,057\end{array}$

Hence, value of bond is $1,057

Note: It is assumed that, this answer will require 8% nominal return, but not the effective return and the semi-annual discount rate would be 3.92%.