Chapter 12, Problem 13QP

Summary Introduction

To determine: The weight average cost of capital (WACC)

Introduction:

The weighted average cost of capital (WACC) refers to the weighted average of the cost of debt after taxes and the cost of equity.

Answer to Problem 13QP

The weighted average cost of capital is 8.36 percent.

Explanation of Solution

Given information:

Company B has 5,000,000 common equity shares outstanding. The market price of the share is $73, and its book value is $9. The company declared a dividend of $2.85 per share in the current year. The growth rate of dividend is 6 percent.

It has two outstanding bond issues. The face value of Bond A is $60,000,000 and matures in 20 years. It has a coupon rate of 7 percent, and the market value is 98 percent of the face value.

The face value of Bond B is $40,000,000 and matures in 12 years. It has a coupon rate of 6.5 percent, and the market value is 97 percent of the face value. Both the bonds make semiannual coupon payments. Assume that the face value of one unit of both the bond issues is $1,000.

The tax rate applicable to Company B is 35 percent. The weights based on the market values are more relevant because they do not overstate the company’s financing through debt. The weight of equity is 0.7890, and the weight of debt is 0.2109 based on the market value of the capital structure (Refer to Questions and Problems Number 12 in Chapter 12).

The formula to calculate the cost of equity under the Dividend growth model approach:

$R_E=\frac{D_0\times \left(1+g\right)}{P_0}+g$

Here,

“R_E” refers to the return on equity or the cost of equity

“P₀” refers to the price of the equity share

“D₀” refers to the dividend paid by the company

“g” refers to the constant rate at which the dividend will grow

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:

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

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 after-tax cost of debt:

$\begin{array}{l}\text{Weighted average}\\ \text{after-tax cost of debt}\end{array}\}=\left(\begin{array}{l}\left(\frac{\text{Market value of Bond A}}{\text{Total market value}}\right)\times R_{\text{Debt A}}+\\ \left(\frac{\text{Market value of Bond B}}{\text{Total market value}}\right)\times R_{\text{Debt B}}\end{array}\right)$

Where,

“R_{Debt A}” and “R_{Debt B}” refers to the after-tax cost of debt

The formula to calculate the weighted average cost of capital:

$WACC=\left(\frac{E}{V}\right)\times R_E+\left[\left(\frac{D}{V}\right)\times R_D\times \left(1-T_C\right)\right]$

Where,

“WACC” refers to the weighted average cost of capital

“R_E” refers to the return on equity or the cost of equity

“R_D” refers to the return on debt or the cost of debt

“(E/V)” refers to the weight of common equity

“(D/V)” refers to the weight of debt

“T_C” refers to the corporate tax rate

Compute the cost of equity:

$\begin{array}{c}{R}_E=\frac{D_0\times \left(1+g\right)}{P_0}+g\\ =\frac{\$2.85\times \left(1+0.06\right)}{\$73}+0.06\\ =0.0413+0.05\\ =0.1013\text{ or 10}\text{.13}\%\end{array}$

Hence, the cost of equity is 10.13 percent.

Compute the cost of debt for Bond A:

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 7\%\\ =\$70\end{array}$

Hence, the annual coupon payment is $70.

Compute the current price of the bond:

The face value of the bond is $1,000. The bond value is 98% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{98}{100}\\ =\$980\end{array}$

Hence, the current price of the bond is $980.

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

The bond pays the coupons semiannually. The annual coupon payment is $63. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $35 $\left(\$70÷2\right)$.

The remaining 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.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$980=\$35\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{40}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{40}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above the 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 discount because the market value of the bond is lower than its face value. Hence, substitute “r” with a rate that is higher than the coupon rate until one obtains the bond value close to $980.

The coupon rate of 7 percent is an annual rate. The semiannual coupon rate is 3.5 percent $\left(\text{7 percent}÷2\right)$. The trial rate should be above 3.5 percent.

The attempt under the trial and error method using 3.599 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$35\times \frac{\left[1-\frac{1}{{\left(1+0.03599\right)}^{40}}\right]}{0.03599}+\frac{\$1,000}{{\left(1+0.03599\right)}^{40}}\\ =\$736.0835+\$243.0958\\ =\$979.17\end{array}$

The current price of the bond is $979.17 when “r” is 3.599 percent. This value is close to the bond value of $980. Hence, 3.599 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =3.599\%\times 2\\ =7.198\%\end{array}$

Hence, the yield to maturity is 7.198 percent.

Compute the after-tax cost of debt:

The pre-tax cost of debt is equal to the yield to maturity of the bond. The yield to maturity of the bond is 7.198 percent. The corporate tax rate is 35 percent.

$\begin{array}{c}\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)\\ =0.07198\times (1-0.35)\\ =0.07198\times 0.65\\ =0.04678\text{ or }4.678\%\end{array}$

Hence, the after-tax cost of debt is 4.678 percent.

Compute the cost of debt for Bond B:

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 6.5\%\\ =\$65\end{array}$

Hence, the annual coupon payment is $65.

Compute the current price of the bond:

The face value of the bond is $1,000. The bond value is 97% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{97}{100}\\ =\$970\end{array}$

Hence, the current price of the bond is $970.

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

The bond pays the coupons semiannually. The annual coupon payment is $58. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $32.5 $\left(\$65÷2\right)$.

The remaining time to maturity is 12 years. As the coupon payment is semiannual, the semiannual periods to maturity are 24 $\left(12\text{ years}\times 2\right)$. In other words, “t” equals to 24 6-month periods.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$970=\$32.5\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{24}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{24}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above the 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 discount because the market value of the bond is lower than its face value. Hence, substitute “r” with a rate that is higher than the coupon rate until one obtains the bond value close to $970.

The coupon rate of 6.5 percent is an annual rate. The semiannual coupon rate is 3.25 percent $\left(\text{6}\text{.5 percent}÷2\right)$. The trial rate should be above 3.25 percent.

The attempt under the trial and error method using 3.43 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$32.5\times \frac{\left[1-\frac{1}{{\left(1+0.0343\right)}^{24}}\right]}{0.0343}+\frac{\$1,000}{{\left(1+0.0343\right)}^{24}}\\ =\$525.7548+\$445.1264\\ =\$970.88\end{array}$

The current price of the bond is $970.88 when “r” is 3.43 percent. This value is close to the bond value of $970. Hence, 3.43 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =3.43\%\times 2\\ =6.86\%\end{array}$

Hence, the yield to maturity is 6.86 percent.

Compute the after-tax cost of debt:

The pre-tax cost of debt is equal to the yield to maturity of the bond. The yield to maturity of the bond is 6.86 percent. The corporate tax rate is 35 percent.

$\begin{array}{c}{\text{After-tax R}}_D={\text{R}}_D\times \left(1-T_c\right)\\ =0.0686\times \left(1-0.35\right)\\ =0.0686\times 0.65\\ =0.04459\text{ or 4}\text{.45\%}\end{array}$

Hence, the after-tax cost of debt is 4.45 percent.

Compute the weighted average after-tax cost of debt:

The market value of Bond A is $58,800,000 $\left(\$60,000,000\times 98\%\right)$. The market value of Bond B is $38,800,000 $\left(\$40,000,000\times 97\%\right)$. The total market value is $97,600,000 $(\$58,800,000+\$38,800,000)$. The after-tax cost of debt of Bond A (R_{Bond A}) is 4.21 percent and of Bond B (R_{Bond B}) is 4 percent.

$\begin{array}{c}\begin{array}{l}\text{Weighted average}\\ \text{after-tax cost of debt}\end{array}\}=\left(\begin{array}{l}\left(\frac{\text{Market value of Bond A}}{\text{Total market value}}\right)\times R_{\text{Bond A}}+\\ \left(\frac{\text{Market value of Bond B}}{\text{Total market value}}\right)\times R_{\text{Bond B}}\end{array}\right)\\ =\left(\frac{\$58,800,000}{\$97,600,000}\times 0.0467\right)+\left(\frac{\$38,800,000}{\$97,600,000}\times 0.0445\right)\\ =0.0281+0.176\\ =0.0457\text{ or }4.57\%\end{array}$

Hence, the weighted average after-tax cost of debt is 4.57 percent.

Compute the weighted average cost of capital of Company B:

The weight of equity is 0.7890 (E/V), and the weight of debt is 0.2109 (D/V) based on the market value of the capital structure. The cost of equity “R_E” is 10.13 percent, and the after-tax weighted average cost of debt “R_D” is 4.678 percent. As the after-tax cost of debt is already calculated, it is not necessary to multiply the cost of debt with “1 ‑‑ T_C”.

$\begin{array}{c}WACC=\left(\frac{E}{V}\right)\times R_E+\left(\frac{D}{V}\right)\times R_D\\ =\left(0.7890\times 0.1013\right)+\left(0.2109\times 0.0467\right)\\ =0.0799+0.0098\\ =0.0897\text{ or }8.97\%\end{array}$

Hence, the weighted average cost of capital is 8.97 percent.