Chapter 14, Problem 22QP

Summary Introduction

To determine: The weighted average after-tax cost of debt

Introduction:

The cost of debt refers to the return that the bondholders or lenders expect on their principal. In other words, it refers to the borrowing costs of the company.

Answer to Problem 22QP

The weighted average after-tax cost of debt is 3.72 percent.

Explanation of Solution

Given information:

Company Y has four bond issues. All the bonds make semiannual coupon payments. The corporate tax rate is 34 percent. Assume that the face value of one bond is $1,000. It issued Bond 1 with a coupon rate of 6 percent. The remaining time to maturity of the bond is 5 years. The market price of the bond is 105.86 percent of the face value. The total face value of Bond 1 is $40,000,000.

The company issued Bond 2 with a coupon rate of 7.50 percent. The remaining time to maturity of the bond is 8 years. The market price of the bond is 114.52 percent of the face value. The total face value of Bond 1 is $35,000,000.

The company issued Bond 3 with a coupon rate of 7.20 percent. The remaining time to maturity of the bond is 15.5 years. The market price of the bond is 113.07 percent of the face value. The total face value of Bond 3 is $55,000,000.

The company issued Bond 4 with a coupon rate of 6.8 percent. The remaining time to maturity of the bond is 25 years. The market price of the bond is 102.31 percent of the face value. The total face value of Bond 4 is $50,000,000.

Formulae:

The formula to calculate the market value of debt:

$\text{Market value of debt}=\text{Face value of the debt}\times \text{Price quote}$

The formula to calculate the total market value of the debt:

$\text{Total market value of debt}=\text{Market value of Debt 1}+\text{Market value of Debt 2}$

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 current price or the market value of the debt:

$\text{Current price}=\text{Face value of the debt}\times \text{Last price percentage}$

The formula to calculate the yield to maturity:

$\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}$

Where,

“C” refers to the coupon paid per period

“F” refers to the face value paid at maturity

“r” refers to the yield to maturity

“t” refers to the periods to maturity

The formula to calculate the after-tax cost of debt:

$\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)$

Where,

“R_D” refers to the cost of debt

“T_C” refers to the corporate tax rate

The formula to calculate the weighted average cost of debt:

$\text{Cost of Debt}=\left(\begin{array}{l}\frac{\text{Market value of Debt 1}}{\text{Total market value of the debt}}\\ \times \text{Cost of debt of Debt 1}\end{array}\right)+\left(\begin{array}{l}\frac{\text{Market value of Debt 2}}{\text{Total market value of the debt}}\\ \times \text{Cost of debt of Debt 2}\end{array}\right)$

Compute the market value of Bond 1:

$\begin{array}{c}\text{Market value of debt}=\text{Face value of the debt}\times \text{Price quote}\\ =\$40,000,000\times \frac{105.86}{100}\\ =\$42,344,000\end{array}$

Hence, the market value of Bond 1 is $42,344,000.

Compute the market value of Bond 2:

$\begin{array}{c}\text{Market value of debt}=\text{Face value of the debt}\times \text{Price quote}\\ =\$35,000,000\times \frac{114.52}{100}\\ =\$40,082,000\end{array}$

Hence, the market value of Bond 2 is $40,082,000.

Compute the market value of Bond 3:

$\begin{array}{c}\text{Market value of debt}=\text{Face value of the debt}\times \text{Price quote}\\ =\$55,000,000\times \frac{113.07}{100}\\ =\$62,188,500\end{array}$

Hence, the market value of Bond 3 is $62,188,500.

Compute the market value of Bond 4:

$\begin{array}{c}\text{Market value of debt}=\text{Face value of the debt}\times \text{Price quote}\\ =\$50,000,000\times \frac{102.31}{100}\\ =\$51,155,000\end{array}$

Hence, the market value of Bond 4 is $51,155,000.

Compute the total market value of the debt:

$\begin{array}{c}\text{Total market value of debt}=\left(\begin{array}{l}\left(\begin{array}{l}\text{Market value}\\ \text{of Bond 1}\end{array}\right)+\left(\begin{array}{l}\text{Market value}\\ \text{of Bond 2}\end{array}\right)\\ +\left(\begin{array}{l}\text{Market value}\\ \text{of Bond 3}\end{array}\right)+\left(\begin{array}{l}\text{Market value}\\ \text{of Bond 4}\end{array}\right)\end{array}\right)\\ =\$42,344,000+\$40,082,000+\$62,188,500+\$51,155,000\\ =\$195,769,500\end{array}$

Hence, the total market value of debt is $195,769,500.

Compute the cost of debt for Bond 1:

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 6\%\\ =\$60\end{array}$

Hence, the annual coupon payment is $60.

Compute the current price of the bond:

The face value of the bond is $1,000. The bond value is 105.86% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{105.86}{100}\\ =\$1,058.6\end{array}$

Hence, the current price of the bond is $1,058.6.

Compute the semiannual yield to maturity of Bond 1 as follows:

The bond pays the coupons semiannually. The annual coupon payment is $60. However, the bondholder will receive the same in two equal installments. Hence, the semiannual coupon payment or the 6-month coupon payment is $30 $\left(\$60÷2\right)$.

The remaining time to maturity is 5 years. As the coupon payment is semiannual, the semiannual periods to maturity are 10 $\left(5\text{ years}\times 2\right)$. In other words, “t” equals to 10 6-month periods.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,058.6=\$30\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{10}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{10}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above equation. Hence, the only method to solve for “r” is the trial and error method.

The first step in trial and error method is to identify the discount rate that needs to be used. The bond sells at a premium in the market if the market rates (Yield to maturity) are lower than the coupon rate. Similarly, the bond sells at a discount if the market rate (Yield to maturity) is greater than the coupon rate.

In the given information, the bond sells at a premium because the market value of the bond is greater than its face value. Hence, substitute “r” with a rate that is lower than the coupon rate until one obtains the bond value close to $1,058.6.

The coupon rate of 6 percent is an annual rate. The semiannual coupon rate is 3 percent that is $\left(\frac{0.06}{2}\right)$. The trial rate should be below 3 percent.

The attempt under the trial and error method using 2.3361 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$30\times \frac{\left[1-\frac{1}{{\left(1+0.023361\right)}^{10}}\right]}{0.023361}+\frac{\$1,000}{{\left(1+0.023361\right)}^{10}}\\ =\$30\times \frac{\left[1-\frac{1}{1.2597}\right]}{0.023361}+\frac{\$1,000}{{\left(1+0.023361\right)}^{10}}\\ =\$30\times \frac{\left[1-0.7938\right]}{0.023361}+\frac{\$1,000}{1.2597}\end{array}$

$\begin{array}{c}=\$264.80+\$793.83\\ =\$1,058.63\end{array}$

The current price of the bond is $1,058.63 when “r” is 2.3361 percent. Hence, 2.3361 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =2.3361\%\times 2\\ =4.6722\%\end{array}$

Hence, the yield to maturity is 4.6722 percent.

Compute the after-tax cost of Bond 1:

The pre-tax cost of debt is equal to the yield to maturity of the bond. The yield to maturity of the bond is 4.6722 percent. The corporate tax rate is 34 percent.

$\begin{array}{c}\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)\\ =0.046722\times (1-0.34)\\ =0.046722\times 0.66\\ =0.0308\text{ or }3.08\%\end{array}$

Hence, the after-tax cost of Bond 1 is 3.08 percent.

Compute the cost of debt for Bond 2:

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 current price of the bond:

The face value of the bond is $1,000. The bond value is 114.52% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{114.52}{100}\\ =\$1,145.2\end{array}$

Hence, the current price of the bond is $1,145.2.

Compute the semiannual yield to maturity of Bond 2 as follows:

The bond pays the coupons semiannually. The annual coupon payment is $75. However, the bondholder will receive the same in two equal installments. Hence, the semiannual coupon payment or the 6-month coupon payment is $37.5 years that are $\left(\frac{\$75}{2}\right)$.

The remaining time to maturity is 8 years. As the coupon payment is semiannual, the semiannual periods to maturity are 16 years $\left(\text{8 years}\times 2\right)$. In other words, “t” equals to 16 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,145.2=\$37.5\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{16}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{16}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above equation. Hence, the only method to solve for “r” is the trial and error method.

The first step in trial and error method is to identify the discount rate that needs to be used. The bond sells at a premium in the market if the market rates (Yield to maturity) are lower than the coupon rate. Similarly, the bond sells at a discount if the market rate (Yield to maturity) is greater than the coupon rate.

In the given information, the bond sells at a premium because the market value of the bond is greater than its face value. Hence, substitute “r” with a rate that is lower than the coupon rate until one obtains the bond value close to $1,145.2.

The coupon rate of 7.5 percent is an annual rate. The semiannual coupon rate is 3.75 percent that is $\left(\frac{\text{0}\text{.075}}{2}\right)$. The trial rate should be below 3.75 percent.

The attempt under the trial and error method using 2.6268 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$37.5\times \frac{\left[1-\frac{1}{{\left(1+0.026268\right)}^{16}}\right]}{0.026268}+\frac{\$1,000}{{\left(1+0.026268\right)}^{16}}\\ =\$37.5\times \frac{\left[1-\frac{1}{1.5141}\right]}{0.026268}+\frac{\$1,000}{1.5141}\\ =\$37.5\times \frac{\left[1-0.6604\right]}{0.026268}+\frac{\$1,000}{1.5141}\end{array}$

$\begin{array}{c}=\$484.81+\$660.45\\ =\$1,145.26\end{array}$

The current price of the bond is $1,145.26 when “r” is 2.6268 percent. The value is close to $1,145.26. Hence, 2.6268 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =2.6268\%\times 2\\ =5.2536\%\end{array}$

Hence, the yield to maturity is 5.2536 percent.

Compute the after-tax cost of Bond 2:

The pre-tax cost of debt is equal to the yield to maturity of the bond. The yield to maturity of the bond is 5.2536 percent. The corporate tax rate is 35 percent.

$\begin{array}{c}\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)\\ =0.052536\times (1-0.34)\\ =0.052536\times 0.66\\ =0.0346\text{ or }3.46\%\end{array}$

Hence, the after-tax cost of Bond 2 is 3.46 percent.

Compute the cost of debt for Bond 3:

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 7.2\%\\ =\$72\end{array}$

Hence, the annual coupon payment is $72.

Compute the current price of the bond:

The face value of the bond is $1,000. The bond value is 113.07% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{113.07}{100}\\ =\$1,130.7\end{array}$

Hence, the current price of the bond is $1,130.7

Compute the semiannual yield to maturity of Bond 3 as follows:

The bond pays the coupons semiannually. The annual coupon payment is $72. However, the bondholder will receive the same in two equal installments. Hence, the semiannual coupon payment or the 6-month coupon payment is $36 that is $\left(\frac{\$72}{2}\right)$.

The remaining time to maturity is 15.5 years. As the coupon payment is semiannual, the semiannual periods to maturity are 31 years $\left(\text{15}\text{.5 years}\times 2\right)$. In other words, “t” equals to 31.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,130.7=\$36\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{31}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{31}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above equation. Hence, the only method to solve for “r” is the trial and error method.

The first step in trial and error method is to identify the discount rate that needs to be used. The bond sells at a premium in the market if the market rates (Yield to maturity) are lower than the coupon rate. Similarly, the bond sells at a discount if the market rate (Yield to maturity) is greater than the coupon rate.

In the given information, the bond sells at a premium because the market value of the bond is higher than its face value. Hence, substitute “r” with a rate that is lower than the coupon rate until one obtains the bond value close to $1,130.7.

The coupon rate of 7.2 percent is an annual rate. The semiannual coupon rate is 3.6 percent that is $\left(\frac{\text{0}\text{.072}}{2}\right)$. The trial rate should be below 3.6 percent.

The attempt under the trial and error method using 2.9508 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$36\times \frac{\left[1-\frac{1}{{\left(1+0.029508\right)}^{31}}\right]}{0.029508}+\frac{\$1,000}{{\left(1+0.029508\right)}^{31}}\\ =\$36\times \frac{\left[1-\frac{1}{2.4633}\right]}{0.029508}+\frac{\$1,000}{2.4633}\\ =\$36\times \frac{\left[1-0.4059\right]}{0.029508}+\frac{\$1,000}{2.4633}\end{array}$

$\begin{array}{l}=\$724.80+\$405.95\\ =1,130.75\end{array}$

The current price of the bond is $1,130.75 when “r” is 2.9508 percent. The value is close to $1,130.75. Hence, 2.9508 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =2.9508\%\times 2\\ =5.9016\%\end{array}$

Hence, the yield to maturity is 5.9016 percent.

Compute the after-tax cost of Bond 3:

The pre-tax cost of debt is equal to the yield to maturity of the bond. The yield to maturity of the bond is 5.9016 percent. The corporate tax rate is 34 percent.

$\begin{array}{c}\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)\\ =0.059016\times (1-0.34)\\ =0.059016\times 0.66\\ =0.0389\text{ or 3}\text{.89}\%\end{array}$

Hence, the after-tax cost of Bond 3 is 3.89 percent.

Compute the annual coupon payment:

Note: To computethe cost of debt for Bond 4, the annual coupon payment is calculated.

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 6.8\%\\ =\$68\end{array}$

Hence, the annual coupon payment is $68.

Compute the current price of the bond:

The face value of the bond is $1,000. The bond value is 102.31% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{102.31}{100}\\ =\$1,023.1\end{array}$

Hence, the current price of the bond is $1,023.1.

Compute the semiannual yield to maturity of Bond 4 as follows:

The bond pays the coupons semiannually. The annual coupon payment is $68. However, the bondholder will receive the same in two equal installments. Hence, the semiannual coupon payment or the 6-month coupon payment is $34 that is $\left(\frac{\$68}{2}\right)$.

The remaining time to maturity is 25 years. As the coupon payment is semiannual, the semiannual periods to maturity are 50 $\left(\text{25 years}\times 2\right)$. In other words, “t” equals to 50.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,023.1=\$34\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{50}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{50}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above equation. Hence, the only method to solve for “r” is the trial and error method.

The first step in trial and error method is to identify the discount rate that needs to be used. The bond sells at a premium in the market if the market rates (Yield to maturity) are lower than the coupon rate. Similarly, the bond sells at a discount if the market rate (Yield to maturity) is greater than the coupon rate.

In the given information, the bond sells at a premium because the market value of the bond is greater than its face value. Hence, substitute “r” with a rate that is lower than the coupon rate until one obtains the bond value close to $1,023.1.

The coupon rate of 6.8 percent is an annual rate. The semiannual coupon rate is 3.4 percent that is $\left(\frac{0.068}{2}\right)$. The trial rate should be below 3.4 percent.

The attempt under the trial and error method using 3.305 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$34\times \frac{\left[1-\frac{1}{{\left(1+0.033050\right)}^{50}}\right]}{0.033050}+\frac{\$1,000}{\left(1+0.033050\right){287}^{50}}\\ =\$34\times \frac{\left[1-\frac{1}{5.0823}\right]}{0.033050}+\frac{\$1,000}{5.0823}\\ =\$34\times \frac{\left[1-0.1967\right]}{0.033050}+\frac{\$1,000}{5.0823}\end{array}$

$\begin{array}{c}=\$826.39+\$196.76\\ =\$1,023.15\end{array}$

The current price of the bond is $1,023.15 when “r” is 3.305 percent. Hence, 3.305 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =3.305\%\times 2\\ =6.61\%\end{array}$

Hence, the yield to maturity is 6.61 percent.

Compute the after-tax cost of Bond 4:

The pre-tax cost of debt is equal to the yield to maturity of the bond. The yield to maturity of the bond is 6.61 percent. The corporate tax rate is 34 percent.

$\begin{array}{c}\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)\\ =0.0661\times (1-0.34)\\ =0.0661\times 0.66\\ =0.0436\text{or }4.36\%\end{array}$

Hence, the after-tax cost of Bond 4 is 4.36 percent.

Compute the overall after-tax cost of the debt of Company Y:

$\begin{array}{c}\begin{array}{l}\text{Weighted }\\ \text{average}\\ \text{after-tax}\\ \text{cost}\\ \text{of debt}\end{array}\}=\left[\begin{array}{l}\left(\begin{array}{l}\frac{\text{Market value of Bond 1}}{\text{Total market value of the debt}}\times \\ \text{Cost of Bond 1}\end{array}\right)+\left(\begin{array}{l}\frac{\text{Market value of Bond 2}}{\text{Total market value of the debt}}\times \\ \text{Cost of Bond 2}\end{array}\right)\\ +\left(\begin{array}{l}\frac{\text{Market value of Bond 3}}{\text{Total market value of the debt}}\times \\ \text{Cost of Bond 3}\end{array}\right)+\left(\begin{array}{l}\frac{\text{Market value of Bond 4}}{\text{Total market value of the debt}}\times \\ \text{Cost of Bond 4}\end{array}\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(\frac{\$42,344,000}{\$195,769,500}\times 0.0308\right)+\left(\frac{\$40,082,000}{\$195,769,500}\times 0.0346\right)\\ +\left(\frac{\$62,188,500}{\$195,769,500}\times 0.0389\right)+\left(\frac{\$51,155,000}{\$195,769,500}\times 0.0436\right)\end{array}\right]\\ =\left[0.2162\times 0.0308\right]+\left[0.2047\times 0.0346\right]+\left[0.3176\times 0.0389\right]+\left[0.2613\times 0.0436\right]\\ =0.0372\text{ or 3}\text{.72\%}\end{array}$

Hence, the overall cost of debt of the firm is 3.72 percent.