Chapter 6, Problem 23QP

a)

Summary Introduction

To determine: The number of coupon bonds and zero coupon bonds that the company needs to issue

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 23QP

The company needs to issue 30,000 bonds to raise $30,000,000. It needs to issue 130,810.1508 zero coupon bonds to raise $30,000,000.

Explanation of Solution

Given information:

Company X intends to raise $30,000,000 through 20-year bonds. The investors expect 7.5 percent return on the bond. The bond will pay the coupons semiannually. The company has two alternatives to the bond issue. It can either issue a coupon bond at 7.5 percent coupon rate or a zero coupon bond. The bond’s par value will be $1,000. Company X falls under 35 percent tax bracket.

The formula to calculate the number of coupon bonds that the company needs to issue:

$\text{Number of coupon bonds}=\frac{\text{Total value of bonds}}{\text{The par value of one coupon bond}}$

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

The formula to calculate the number of zero coupon bonds that the company needs to issue:

$\text{Number of zero coupon bonds}=\frac{\text{Total value of bonds}}{\text{The issue price of zero coupon bond}}$

Compute the number of coupon bonds that the company needs to issue:

$\begin{array}{c}\text{Number of coupon bonds}=\frac{\text{Total value of bonds}}{\text{The par value of one coupon bond}}\\ =\frac{\$30,000,000}{\$1,000}\\ =30,000\text{}\text{ bonds}\end{array}$

Hence, the company needs to issue 30,000 bonds to raise $30,000,000.

Compute the issue price of zero coupon bonds:

The market rate required on the bond is 7.5 percent. It is an annual rate. In the given information, the company follows semiannual compounding. Hence, the semiannual or 6-month market rate is 3.75 percent $\left(\text{7}\text{.5 percent}÷2\right)$. The time to maturity is 20 years. As the coupon payment is semiannual, the semiannual periods to maturity are 40 $\left(20\text{ years}\times 2\right)$. In other words, “t” equals to 40 6-month periods.

$\begin{array}{c}\text{Bond price}=\frac{\text{Face value of the bond}}{{\left(1+r\right)}^t}\\ =\frac{\$1,000}{{\left(1+0.0375\right)}^{40}}\\ =\frac{\$1,000}{4.3603}\\ =\$229.34\end{array}$

Hence, the issue price of the zero coupon bond will be $229.34.

Compute the number of zero coupon bonds that the company needs to issue:

$\begin{array}{c}\text{Number of zero coupon bonds}=\frac{\text{Total value of bonds}}{\text{The issue price of zero coupon bond}}\\ =\frac{\$30,000,000}{\$229.34}\\ =130,810.1508\text{bonds}\end{array}$

Hence, the company needs to issue 130,810.1508 zero coupon bonds to raise $30,000,000.

b)

Summary Introduction

To determine: The repayment value if the company issues coupon bonds and zero coupon bonds after 20 years

Answer to Problem 23QP

The repayment value of coupon bonds after 20 years is $31,125,000. The repayment value of zero coupon bonds after 20 years is $130,810,150.8.

Explanation of Solution

Given information:

The company needs to issue 30,000 bonds to raise $30,000,000 (Refer Part (a) of the solution). It needs to issue 130,810.1509 zero coupon bonds to raise $30,000,000 (Refer Part (a) of the solution). The coupon bond has a 7.5 percent coupon rate. The bond’s par value will be $1,000 and the compounding is semiannual.

The formula to calculate the repayment on zero coupon bonds:

$\begin{array}{l}\text{Repayment value of}\\ \text{zero coupon bonds}\end{array}\}=\text{Face value of the bond}\times \text{Number of zero coupon bonds}$

The formula to calculate annual coupon payment:

$\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}$

The formula to calculate the semiannual coupon payment:

$\text{Semiannual coupon payment}=\frac{\text{Annual coupon payment}}{2}$

The formula to calculate the repayment on coupon bonds:

$\begin{array}{l}\text{Repayment value}\\ \text{of coupon bonds}\end{array}\}=\text{Number of coupon bonds}\times \left(\begin{array}{l}\text{Face value of the bond}+\\ \text{semiannual coupon rate}\end{array}\right)$

Compute the repayment on zero coupon bonds:

$\begin{array}{c}\begin{array}{l}\text{Repayment value of}\\ \text{zero coupon bonds}\end{array}\}=\text{Face value of the bond}\times \text{Number of zero coupon bonds}\\ =\$1,000\times 130,810.1508\text{ bonds}\\ \text{=\$130,810,150}\text{.8}\end{array}$

Hence, the repayment value of zero coupon bonds after 20 years is $130,810,150.8.

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 7.5\%\\ =\$75\end{array}$

Hence, the annual coupon payment is $75.

Compute the semiannual coupon payment:

$\begin{array}{c}\text{Semiannual coupon payment}=\frac{\text{Annual coupon payment}}{2}\\ =\frac{\$75}{2}\\ =\$37.5\end{array}$

Hence, the semiannual coupon payment is $37.5.

Compute the repayment on coupon bonds:

$\begin{array}{c}\begin{array}{l}\text{Repayment value}\\ \text{of coupon bonds}\end{array}\}=\text{Number of coupon bonds}\times \left(\begin{array}{l}\text{Face value of the bond}+\\ \text{semiannual coupon rate}\end{array}\right)\\ =30,000\text{ bonds}\times \left(\$1,000+\$37.5\right)\\ =30,000\text{ bonds}\times \$1,037.5\\ =\$31,125,000\end{array}$

Hence, the repayment value of coupon bonds after 20 years is $31,125,000.

c)

Summary Introduction

To determine: The reason why the company would choose to issue zero coupon bonds

Answer to Problem 23QP

There will be a cash outflow every year when the company issues coupon bonds. The after-tax cash outflow is $1,462,500 when the company issues coupon bonds. On the contrary, there will be cash inflow when the company issues zero coupon bonds. The interest payments on zero bonds are notional, and it is just an accounting value. Hence, there is no actual cash outflow.

Moreover, the company can claim interest deduction arising from zero coupon bonds even if there is no actual interest payment. As a result, the tax saved ($802,127,845.31) by the notional interest of zero coupon bond is the cash inflow. Hence, the company would prefer zero coupon bonds because it would generate cash inflow.

Explanation of Solution

Required information:

The interest payment or coupon payment on coupon bond for the first year is $75 (Refer to Part (b) of the solution). The issue price of the zero coupon bond is $229.34 (Refer Part (a) of the solution). The bond’s par value will be $1,000 and compounding is semiannual. The required return on zero coupon bonds is 7.5 percent.

The company needs to issue 30,000 bonds to raise $30,000,000. It needs to issue 130,810.1509 zero coupon bonds to raise $30,000,000. (Refer to Part (a) of the solution).

The formula to calculate the after-tax cash outflow of coupon bonds:

$\text{After tax cash flow}=\text{Interest payments}\times \left(1-\text{tax rate}\right)$

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 payment on zero coupon bonds:

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

The formula to calculate the cash inflow from zero coupon bonds:

$\text{After tax cash inflow}=\left(\text{Number of coupon bonds}\times \text{Interest payment}\right)\times \text{tax rate}$

Compute the after-tax cash flow of coupon bonds:

$\begin{array}{c}\text{After tax cash flow}=\text{Total interest payments}\times \left(1-\text{tax rate}\right)\\ =\left(\text{Number of coupon bonds}\times \text{Interest payment}\right)\times \left(1-\text{tax rate}\right)\\ =\left(30,000\text{ bonds}\times \text{\$75}\right)\times \left(1-0.35\right)\end{array}$

$\begin{array}{c}=\$2,250,000\times 0.65\\ =\$1,462,500\end{array}$

Hence, the after-tax cash outflow is $1,462,500 when the company issues coupon bonds.

Compute the price at the beginning of the first year for the zero coupons bond:

The price at the beginning of the first year will be the issue price. The issue price of the zero coupon bond is $229.34 (Refer Part (a) of the solution).

Compute the price at the end of the first year:

The market rate required on the bond is 7.5 percent. It is an annual rate. In the given information, the company follows semiannual compounding. Hence, the semiannual or 6-month market rate is 3.75 percent $\left(\text{7}\text{.5 percent}÷2\right)$.

The remaining time to maturity is 19 years. As the coupon payment is semiannual, the semiannual periods to maturity are 38 $\left(\text{19 years}\times 2\right)$. In other words, “t” equals to 38 6-month periods.

$\begin{array}{c}\text{Bond price}=\frac{\text{Face value of the bond}}{{\left(1+r\right)}^t}\\ =\frac{\$1,000}{{\left(1+0.0375\right)}^{38}}\\ =\frac{\$1,000}{4.0508}\\ =\$246.86\end{array}$

Hence, the price of the zero coupon bond at the end of the first year will be $246.86.

Compute the interest payment on zero coupon bonds:

$\begin{array}{c}\text{Interest payment}=\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the end of the year}\end{array}\right)-\left(\begin{array}{l}\text{Price of the bond}\\ \text{at the beginning of the year}\end{array}\right)\\ =\$246.86-\$229.34\\ =\$17.52\end{array}$

Hence, the interest payment on zero coupon bonds for the first year is $17.52.

Compute the after-tax cash inflow of zero coupon bonds:

$\begin{array}{c}\text{After tax cash inflow}=\left(\text{Number of coupon bonds}\times \text{Interest payment}\right)\times \text{tax rate}\\ =\left(130,810,150.9\text{ bonds}\times \text{\$17}\text{.52}\right)\times 0.35\\ =\$2,291,793,843.76\times 0.35\\ =\$802,127,845.31\end{array}$

Hence, the after-tax cash inflow from zero coupon bonds is $802,127,845.31.