Chapter 14, Problem 15QP

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 15QP

The weighted average cost of capital is 9.12 percent.

Explanation of Solution

Given information:

Company LP has an outstanding bond issue. The bond has a coupon rate of 6.5 percent and face value of $1,000 and. The company issued 10,000 bonds. The bonds mature in 25 years and make semiannual coupon payments. The bonds sell at 106 percent of the face value in the market.

It has 310,000 common equity shares outstanding. The market price of the share is $57. The stock has a beta of 1.05. The company has also issued preferred stock. There are 15,000 outstanding preference shares. The dividend per share is 4 percent and its current market value is $72 per share. The market risk premium is 7 percent, and the risk-free rate is 4.5 percent. The tax rate is 35 percent.

Formulae:

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.

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 106% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{106}{100}\\ =\$1,060\end{array}$

Hence, the current price of the bond is $1,060.

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

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

The maturity time which is remaining 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 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,060=\$32.5\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{50}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{50}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above 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 larger 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 which must be less than the coupon rate until one obtains the bond value close to $1,060.

The coupon rate is considered as annual rate that is 6.5%. The semiannual coupon rate is 3.25 percent $\left(\frac{\text{0}\text{.065}}{2}\right)$. The trial rate should be below 3.25 percent.

The attempt under the trial and error method using 3.015 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$32.5\times \frac{\left[1-\frac{1}{{\left(1+0.03015\right)}^{50}}\right]}{0.0315}+\frac{\$1,000}{{\left(1+0.03015\right)}^{50}}\\ =\$32.5\times \frac{\left[1-\frac{1}{4.4159}\right]}{0.03015}+\frac{\$1,000}{4.4159}\\ =\$32.5\times \frac{\left[1-0.2264\right]}{0.03015}+\frac{\$1,000}{4.4159}\end{array}$

$\begin{array}{c}=\$833.89+\$226.45\\ =\$1,060.34\end{array}$

The current price of the bond is $1,060.34 when “r” is 3.015 percent. This value is close to the bond value of $1,060. Hence, 3.015 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =3.015\%\times 2\\ =6.03\%\end{array}$

Hence, the yield to maturity is 6.03 percent.

Compute the cost of equity:

$\begin{array}{c}{R}_E=R_f+\left[R_M-R_f\right]\times {\beta }_E\\ =\text{0}\text{.045}+\left[0.07\right]\times 1.05\\ =\text{0}\text{.045}+0.0735\\ =0.1185\text{ or }11.85\%\end{array}$

Hence, the cost of equity is 11.85 percent.

Compute the cost of preferred stock:

Assume that the face value of one preferred stock is $100. At 4 percent, the dividend on preferred stock is $4 $(\$100\times 4\%)$.

$\begin{array}{c}{R}_P=\frac{D}{P_0}\\ =\frac{\$4}{\$72}\\ =0.055\text{or 5}\text{.5}\%\end{array}$

Hence, the cost of preferred stock is 5.5 percent.

Compute the market value of equity:

$\begin{array}{c}\text{Market value of equity }(E)=\text{Outstanding equity shares}\times \text{Market value per share}\\ =310,000\times \$57\\ =\$17,670,000\end{array}$

Hence, the market value of equity is $17,670,000.

Compute the market value of preferred stock:

$\begin{array}{c}\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\}=\text{Outstanding preferred shares}\times \text{Market value per share}\\ =15,000\times \$72\\ =\$1,080,000\end{array}$

Hence, the market value of preferred stock is $1,080,000.

Compute the market value of debt:

$\begin{array}{c}\text{Market value of debt }(D)=\text{Face value }\times \text{Number of bonds}\times \text{Last price percentage}\\ =\$1,000\times 8,000\times \frac{106}{100}\\ =\$8,480,000\end{array}$

Hence, the market value of debt is $8,480,000.

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

$\begin{array}{c}\text{Total market value }(V)=\left(\begin{array}{l}\text{Market value}\\ \text{of equity }(E)\end{array}\right)+\left(\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\right)+\left(\begin{array}{l}\text{Market value}\\ \text{of debt }(D)\end{array}\right)\\ =\$17,670,000+\$1,080,000+\$8,480,000\\ =\$27,230,000\end{array}$

Hence, the total market value of the capital structure is $27,230,000.

Compute the weighted average cost of capital:

$\begin{array}{c}WACC=\left(\frac{E}{V}\right)\times R_E+\left(\frac{P}{V}\right)\times R_P+\left[\left(\frac{D}{V}\right)\times R_D\times \left(1-T_C\right)\right]\\ =\left(\frac{\$17,670,000}{\$27,230,000}\right)\times 0.1185+\left(\frac{\$1,080,000}{\$27,230,000}\right)\times 0.055+\left[\begin{array}{l}\left(\frac{\$8,480,000}{\$27,230,000}\right)\times \\ \text{0}\text{.0603}\times \left(1-0.35\right)\end{array}\right]\\ =\left(\text{0}\text{.64891}\times 0.1185\right)+\left(\text{0}\text{.03966}\times 0.055\right)+\left(\text{0}\text{.31142}\times \text{0}\text{.0603}\times \text{0}\text{.65}\right)\end{array}$

$\begin{array}{c}=\text{0}\text{.076895835}+\text{0}\text{.0021813}+\text{0}\text{.0122061069}\\ =\text{0}\text{.0912 or 9}\text{.12\%}\end{array}$

Hence, the weighted average cost of capital is 9.12percent.