Chapter 6, Problem 19QP

a)

Summary Introduction

To determine: The price of 4-year bond if the bond has a yield to maturity of 9%.

Yield to maturity (YTM) is the overall return anticipated on a bond throughout its maturity period and it is considered as a long-term bond yield and represented as an annual rate.

Explanation of Solution

Computation of price of the bond is as follows:

$\begin{array}{c}\text{Price }=\text{ coupon amount}\times \frac{1-{\left(1+\text{YTM}\right)}^{-\text{n}}}{\text{YTM}}+\frac{\text{Face value}}{{\left(\text{1+YTM}\right)}^{\text{n}}}\\ =\text{ \$80}\times \frac{1-{\left(1+0.09\right)}^{-4}}{0.09}+\frac{\$1,000}{{\left(\text{1+0}\text{.09}\right)}^4}\\ =\$967.60\\ \end{array}$

Therefore, the price of the bond is $967.60.

b)

Summary Introduction

To determine: The price of 8-year bond if the bond has a yield to maturity of 9%.

Explanation of Solution

Computation of price of the bond is as follows:

$\begin{array}{c}\text{Price }=\text{ coupon amount}\times \frac{1-{\left(1+\text{YTM}\right)}^{-\text{n}}}{\text{YTM}}+\frac{\text{Face value}}{{\left(\text{1+YTM}\right)}^{\text{n}}}\\ =\text{ \$80}\times \frac{1-{\left(1+0.09\right)}^{-8}}{0.09}+\frac{\$1,000}{{\left(\text{1+0}\text{.09}\right)}^8}\\ =\$944.65\\ \end{array}$

Therefore, the price of the bond is $944.65.

c)

Summary Introduction

To determine: The price of 30-year bond if the bond has a yield to maturity of 9%.

Explanation of Solution

Computation of price of the bond is as follows:

$\begin{array}{c}\text{Price }=\text{ coupon amount}\times \frac{1-{\left(1+\text{YTM}\right)}^{-\text{n}}}{\text{YTM}}+\frac{\text{Face value}}{{\left(\text{1+YTM}\right)}^{\text{n}}}\\ =\text{ \$80}\times \frac{1-{\left(1+0.09\right)}^{-30}}{0.09}+\frac{\$1,000}{{\left(\text{1+0}\text{.09}\right)}^{30}}\\ =\$897.26\\ \end{array}$

Therefore, the price of the bond is $897.26.

d)

Summary Introduction

To determine: The price of 4-year bond if the bond has a yield to maturity of 7%.

Explanation of Solution

Computation of price of the bond is as follows:

$\begin{array}{c}\text{Price }=\text{ coupon amount}\times \frac{1-{\left(1+\text{YTM}\right)}^{-\text{n}}}{\text{YTM}}+\frac{\text{Face value}}{{\left(\text{1+YTM}\right)}^{\text{n}}}\\ =\text{ \$80}\times \frac{1-{\left(1+0.07\right)}^{-4}}{0.07}+\frac{\$1,000}{{\left(\text{1+0}\text{.07}\right)}^4}\\ =\$1,033.87\\ \end{array}$

Therefore, the price of the bond is $1,033.87.

e)

Summary Introduction

To determine: The price of 8-year bond if the bond has a yield to maturity of 7%.

Explanation of Solution

Computation of price of the bond is as follows:

$\begin{array}{c}\text{Price }=\text{ coupon amount}\times \frac{1-{\left(1+\text{YTM}\right)}^{-\text{n}}}{\text{YTM}}+\frac{\text{Face value}}{{\left(\text{1+YTM}\right)}^{\text{n}}}\\ =\text{ \$80}\times \frac{1-{\left(1+0.07\right)}^{-8}}{0.07}+\frac{\$1,000}{{\left(\text{1+0}\text{.07}\right)}^8}\\ =\$1,059.71\\ \end{array}$

Therefore, the price of the bond is $1,059.71.

f)

Summary Introduction

To determine: The price of 30-year bond if the bond has a yield to maturity of 7%.

Explanation of Solution

Computation of price of the bond is as follows:

$\begin{array}{c}\text{Price }=\text{ coupon amount}\times \frac{1-{\left(1+\text{YTM}\right)}^{-\text{n}}}{\text{YTM}}+\frac{\text{Face value}}{{\left(\text{1+YTM}\right)}^{\text{n}}}\\ =\text{ \$80}\times \frac{1-{\left(1+0.07\right)}^{-30}}{0.07}+\frac{\$1,000}{{\left(\text{1+0}\text{.07}\right)}^{30}}\\ =\$1,124.09\\ \end{array}$

Therefore, the price of the bond is $1,124.09.

g)

Summary Introduction

To determine: Whether the long-term bonds more or less affected than short-term bonds by a rise in interest rates.

Explanation of Solution

From the computation of sub parts (a), (b), and (c), it is clear that the long term bonds are high sensitive with respect to changes in interest, regardless of the interest rate directions.

h)

Summary Introduction

To determine: Whether the long-term bonds more or less affected than short-term bonds by a rise in interest rates.

Explanation of Solution

From the computation of sub parts (d), (e), and (f), it is clear that the long term bonds are high sensitive with respect to changes in interest, regardless of the interest rate directions.