Chapter 7, Problem 28QP

a)

Summary Introduction

To determine: The bond’s issue price

Introduction:

A bond refers to the debt securities issued by the governments or corporations for raising capital. The borrower does not return the face value until maturity. However, the investor receives the coupons every year until the date of maturity.

If the bond sells at a steep discount during the issue and does not make any coupon payments during its life, then the bond is a zero coupon bond.

Answer to Problem 28QP

The company will issue the bond at $239.4591 per bond.

Explanation of Solution

Given information:

Company I wants to raise debt for its expansion plans. It wants to issue 25-year zero coupon bonds with a par value of $1,000. The market requires a return of 5.8 percent on similar bonds. The bond will follow semiannual compounding.

The formula to calculate the issue price of zero coupon bonds:

$\text{Bond price}=\frac{\text{Face value of the bond}}{{\left(1+r\right)}^t}$

Where,

“r” refers to the market rate expected by the investors

“t” refers to the periods of maturity

Compute the issue price of zero coupon bonds:

The market rate required on the bond is 5.8 percent. It is an annual rate. In the given information, the company follows semiannual compounding. Hence, the semiannual or 6-month market rate is 2.9 percent $\left(5.8\text{ percent}÷2\right)$. The time to maturity is 25 years. As the coupon payment is semiannual, the semiannual periods to maturity are 50 $\left(25\text{ years}\times 2\right)$. In other words, “t” equals to 50 6-month periods.

$\begin{array}{c}\text{Bond price}=\frac{\text{Face value of the bond}}{{\left(1+r\right)}^t}\\ =\frac{\$1,000}{{\left(1+0.029\right)}^{50}}\\ =\frac{\$1,000}{4.1761}\\ =\$239.4591\end{array}$

Hence, the issue price of the zero coupon bond will be $239.4591.

b)

Summary Introduction

To determine: The interest deduction in the first and last year based on “Internal revenue service” (IRS) amortization rule

Introduction:

The “Internal revenue service” (IRS) amortization rule requires the company to amortize the zero coupon bonds and consider the difference between the ending value of the bond and the beginning value of the bond as an interest deduction.

Answer to Problem 28QP

The interest deduction at the end of the first year is $14.09. The interest deduction at the end of the last year is $55.5711.

Explanation of Solution

Required information:

The company will issue the zero coupon bonds at $239.4591 (Refer Part (a) of the solution). The face value of the 25-year zero coupon bonds is $1,000. The bond follows semiannual compounding and the required rate is 5.8 percent.

The formula to calculate the price of zero coupon bonds:

$\text{Bond price}=\frac{\text{Face value of the bond}}{{\left(1+r\right)}^t}$

Where,

“r” refers to the market rate expected by the investors

“t” refers to the periods of maturity

The formula to calculate the interest deduction:

$\text{Interest deduction}=\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the end of the year}\end{array}\right)-\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the beginning of the year}\end{array}\right)$

Compute the price at the beginning of the first year:

The price at the beginning of the first year will be the issue price. The issue price of the zero coupon bond is $239.4591 (Refer Part (a) of the solution).

Compute the price at the end of the first year:

The market rate required on the bond is 5.8 percent. It is an annual rate. In the given information, the company follows semiannual compounding. Hence, the semiannual or 6-month market rate is 2.9 percent $\left(5.8\text{ percent}÷2\right)$.

The remaining time to maturity is 24 years $\left(25\text{ years}-1\text{year}\right)$. As the coupon payment is semiannual, the semiannual periods to maturity are 48 $\left(24\text{ years}\times 2\right)$. In other words, “t” equals to 48 6-month periods.

$\begin{array}{c}\text{Bond price}=\frac{\text{Face value of the bond}}{{\left(1+r\right)}^t}\\ =\frac{\$1,000}{{\left(1+0.029\right)}^{48}}\\ =\frac{\$1,000}{3.9440}\\ =\$253.5491\end{array}$

Hence, the price of the zero coupon bond at the end of the first year will be $253.5491.

Compute the interest deduction for the first year:

$\begin{array}{c}\text{Interest deduction}=\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the end of the year}\end{array}\right)-\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the beginning of the year}\end{array}\right)\\ =\$253.5491-\$239.4591\\ =\$14.09\end{array}$

Hence, the interest deduction for the first year is $14.09.

Compute the price at the beginning of the last year:

The market rate required on the bond is 5.8 percent. It is an annual rate. In the given information, the company follows semiannual compounding. Hence, the semiannual or 6-month market rate is 2.9 percent $\left(5.8\text{ percent}÷2\right)$.

The bond has 25 years of life. The year before 25 is the 24^th year. The price at the end of the 24^th year is equal to the price at the beginning of the 25^th year. Hence, the remaining time to maturity is 1 year $\left(25\text{ years}-24\text{years}\right)$.

As the coupon payment is semiannual, the semiannual periods to maturity are two $\left(1\text{ year}\times 2\right)$. In other words, “t” equals to two 6-month periods.

$\begin{array}{c}\text{Bond price}=\frac{\text{Face value of the bond}}{{\left(1+r\right)}^t}\\ =\frac{\$1,000}{{\left(1+0.029\right)}^2}\\ =\frac{\$1,000}{1.0588}\\ =\$944.4289\end{array}$

Hence, the price at the end of the 24^th year or the price at the beginning of the 25^th year is $944.4289.

Compute the price at the end of the last year:

The last year is the 25^th year. As the end of the last year, the investor receives the face value of the bond. Hence, the price at the end of the last year is $1,000.

Compute the interest deduction for the last year:

$\begin{array}{c}\text{Interest deduction}=\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the end of the year}\end{array}\right)-\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the beginning of the year}\end{array}\right)\\ =\$1,000-\$944.4289\\ =\$55.5711\end{array}$

Hence, the interest deduction for the last year is $55.5711.

c)

Summary Introduction

To determine: The interest deduction in the first and last year based on straight-line amortization

Introduction:

The straight-line amortization method amortizes the total interest in equal installments over the life of the bond.

Answer to Problem 28QP

Under the straight-line method, the interest deduction will be equal in all the years. Hence, the interest deduction at the end of the first year and last year is $30.4216.

Explanation of Solution

Required information:

The company will issue the zero coupon bonds at $239.4591 (Refer Part (a) of the solution). The face value of the 25-year zero coupon bonds is $1,000.

The formula to calculate the total interest paid on the bond:

$\text{Total interest}=\text{Face value}-\text{Issue price}$

The formula to calculate the interest deduction based on straight-line method:

$\text{Interest deduction per year}=\frac{\text{Total interest}}{\text{The maturity period of the bond}}$

Compute the total interest paid on the bond:

$\begin{array}{c}\text{Total interest}=\text{Face value}-\text{Issue price}\\ =\$1,000-\$239.4591\\ =\$760.5409\end{array}$

Hence, the total interest paid during the life of the bond is $760.5409.

Compute the interest deduction based on straight-line method:

$\begin{array}{c}\text{Interest deduction per year}=\frac{\text{Total interest}}{\text{The maturity period of the bond}}\\ =\frac{\$760.5409}{25\text{ years}}\\ =\$30.4216\end{array}$

Hence, the interest deduction every year is $30.4216.

d)

Summary Introduction

To determine: The interest amortization preferred by the company

Introduction:

A bond refers to the debt securities issued by the governments or corporations for raising capital. The borrower does not return the face value until maturity. However, the investor receives the coupons every year until the date of maturity.

If the bond sells at a steep discount during the issue and does not make any coupon payments during its life, then the bond is a zero coupon bond.

Explanation of Solution

Required information:

The interest deduction at the end of the first year is $14.09, and the interest deduction at the end of the last year is $55.5711 under the “Internal revenue service” (IRS) amortization rule. Under the straight-line method, the interest deduction at the end of the first year and last year is $30.4216.

The company will prefer the straight-line method of amortization to the “Internal revenue service” (IRS) amortization rule. The company can avail higher value deductions in the early years of the bond under the straight-line method. However, the interest deduction is lower in the early years under the “Internal revenue service” (IRS) amortization rule.

Hence, the company would prefer the straight-line method of amortization.