Chapter 12, Problem 18QP

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 18QP

The weighted average cost of capital is 7.8 percent.

Explanation of Solution

Given information:

Company P has an outstanding bond issue. The bond has a face value of $1,000 and a coupon rate of 6.9 percent. The company issued 10,000 bonds. The bonds mature in 15 years and make semiannual coupon payments. The bonds sell at 104 percent of the face value in the market.

It has 275,000 common equity shares outstanding. The market price of the share is $68.50. The stock has a beta of 0.85. The company will pay $3.25 as dividend next year, and it will grow at a rate of 5 percent per year indefinitely.

The company has also issued preferred stock. There are 8,000 outstanding preference shares. The dividend per share is 4.9 percent and its current market value is $94 per share. The market return is 12 percent, and the risk-free rate is 3.5 percent. The tax rate is 35 percent.

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 cost of equity under the Security market line (SML) approach:

$R_E=R_f+\left[R_M-R_f\right]\times {\beta }_E$

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

“R_f” refers to the risk-free rate

“R_M” refers to the expected return on the market portfolio

“R_M ‑ R_f” refers to the market risk premium

“β_E” refers to the beta or risk of the equity

The formula to calculate the cost of preferred stock:

$R_P=\frac{D}{P_0}$

Where,

“R_P” refers to the return on preferred stock or cost of preferred stock

“D” refers to the dividend earned on the preferred stock

“P₀” refers to the current price of preference stock

The formula to calculate the market value of equity:

$\text{Market value of equity }(E)=\text{Outstanding equity shares}\times \text{Market value per share}$

The formula to calculate the market value of preferred stock:

$\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\}=\text{Outstanding preferred shares}\times \text{Market value per share}$

The formula to calculate the market value of debt:

$\text{Market value of debt }(D)=\text{Face value }\times \text{Number of bonds}\times \text{Last price percentage}$

The formula to calculate the total market value of the capital structure:

$\text{Total market value }(V)=\left(\begin{array}{l}\text{Market value}\\ \text{of equity }(E)\end{array}\right)+\left(\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\right)+\left(\begin{array}{l}\text{Market value}\\ \text{of debt }(D)\end{array}\right)$

The formula to calculate the weighted average cost of capital:

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

Where,

“WACC” refers to the weighted average cost of capital

“R_E” refers to the return on equity

“R_P” refers to the return on preferred equity

“R_D” refers to the return on debt

“E” refers to the amount of common equity capital

“P” refers to the amount of preferred equity

“D” refers to the amount of debt

“V” refers to the total amount of capital

“T_C” refers to the corporate tax rate

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

$R_E=\frac{D_0}{P_0}+g$

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

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 6.9\%\\ =\$69\end{array}$

Hence, the annual coupon payment is $69.

Compute the current price of the bond:

The face value of the bond is $1,000. The bond value is 104% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{104}{100}\\ =\$1,040\end{array}$

Hence, the current price of the bond is $1,040.

Compute the semiannual yield to maturity of the bond as follows:

The bond pays the coupons semiannually. The annual coupon payment is $69. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $34.5 $\left(\$69÷2\right)$.

The remaining time to maturity is 15 years. As the coupon payment is semiannual, the semiannual periods to maturity are 30 $\left(15\text{ years}\times 2\right)$. In other words, “t” equals to 30 6-month periods.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,040=\$34.5\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{30}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{30}}\end{array}$ 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 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 $1,040.

The coupon rate of 6.9 percent is an annual rate. The semiannual coupon rate is 3.45 percent $\left(\text{6}\text{.9 percent}÷2\right)$. The trial rate should be below 3.45 percent.

The attempt under the trial and error method using 3.24 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$34.5\times \frac{\left[1-\frac{1}{{\left(1+0.0324\right)}^{30}}\right]}{0.0324}+\frac{\$1,000}{{\left(1+0.0324\right)}^{30}}\\ =\$655.7103+\$384.2024\\ =\$1,039.91\end{array}$

The current price of the bond is $1,039.91 when “r” is 3.24 percent. This value is close to the bond value of $1,040. Hence, 3.24 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =3.24\%\times 2\\ =6.48\%\end{array}$

Hence, the yield to maturity is 6.48 percent.

Compute the cost of equity using “security market line” approach:

$\begin{array}{c}{R}_E=R_f+\left[R_M-R_f\right]\times {\beta }_E\\ =0.035+\left[0.12-0.035\right]\times 0.85\\ =0.035+0.07225\\ =0.1072\text{ or }10.72\%\end{array}$

Hence, the cost of equity is 10.72 percent.

Compute the cost of equity using the “dividend discount model” approach:

$\begin{array}{c}{R}_E=\frac{D_0\times \left(1+g\right)}{P_0}+g\\ =\frac{\$3.25\times \left(1+0.05\right)}{\$68.50}+0.05\\ =0.0498+0.05\\ =0.0998\text{or 9}\text{.98}\%\end{array}$

Hence, the cost of equity is 9.98 percent.

Compute the best estimate for the cost of equity:

Find the average of the cost of equity under the dividend discount model and the “security market line” approach to find the best estimate for the cost of equity.

$\begin{array}{c}{R}_E=\frac{\left(\begin{array}{l}\text{Cost of equity under the}\\ \text{dividend discount model}\end{array}\right)+\left(\begin{array}{l}\text{Cost of equity under the}\\ \text{security market line approach}\end{array}\right)}{2}\\ =\frac{0.0998+0.1072}{2}\\ =0.1035\text{or }10.35\%\end{array}$

Hence, the cost of equity is 10.35 percent.

Compute the cost of preferred stock:

Assume that the face value of one preferred stock is $100. At 4.9 percent, the dividend on preferred stock is $4.9 $(\$100\times 4.9\%)$.

$\begin{array}{c}{R}_P=\frac{D}{P_0}\\ =\frac{\$4.9}{\$94}\\ =0.0521\text{ or 5}\text{.21}\%\end{array}$

Hence, the cost of preferred stock is 5.21 percent.

Compute the market value of equity:

$\begin{array}{c}\text{Market value of equity }(E)=\text{Outstanding equity shares}\times \text{Market value per share}\\ =275,000\times \$68.50\\ =\$18,837,500\end{array}$

Hence, the market value of equity is $18,837,500.

Compute the market value of preferred stock:

$\begin{array}{c}\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\}=\text{Outstanding preferred shares}\times \text{Market value per share}\\ =8,000\times \$94\\ =\$752,000\end{array}$

Hence, the market value of preferred stock is $752,000.

Compute the market value of debt:

$\begin{array}{c}\text{Market value of debt }(D)=\text{Face value }\times \text{Number of bonds}\times \text{Last price percentage}\\ =\$1,000\times 10,000\times \frac{104}{100}\\ =\$10,400,000\end{array}$

Hence, the market value of debt is $10,400,000.

The formula to calculate the total market value of the capital structure:

$\begin{array}{c}\text{Total market value }(V)=\left(\begin{array}{l}\text{Market value}\\ \text{of equity }(E)\end{array}\right)+\left(\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\right)+\left(\begin{array}{l}\text{Market value}\\ \text{of debt }(D)\end{array}\right)\\ =\$18,837,500+\$752,000+\$10,400,000\\ =\$29,989,500\end{array}$

Hence, the total market value of the capital structure is $29,989,500.

Compute the weighted average cost of capital:

$\begin{array}{c}WACC=\left(\frac{E}{V}\right)\times R_E+\left(\frac{P}{V}\right)\times R_P+\left[\left(\frac{D}{V}\right)\times R_D\times \left(1-T_C\right)\right]\\ =\left(\frac{\$18,837,500}{\$29,989,500}\right)\times 0.1035+\left(\frac{\$752,000}{\$29,989,500}\right)\times 0.0521+\left[\begin{array}{l}\left(\frac{\$10,400,000}{\$29,989,500}\right)\times \\ 0.0521\times \left(1-0.35\right)\end{array}\right]\\ =0.0650+0.0013+0.0117\\ =0.078\text{ or }7.8\%\end{array}$

Hence, the weighted average cost of capital is 7.8 percent.