Answer: Good choice. You earned $10. Poor choice. You lost $10.

This question is a little tricky since the debt-to-equity ratio is given, not the debt-to-assets ratio. To transform between them, one way is:

###\dfrac{D}{E} = \dfrac{2}{1}###

So the value of the firm's assets could be:

###V = D+E=2+1=3###

Therefore, the debt to assets ratio will be:

###\dfrac{D}{V} = \dfrac{D}{D+E} = \dfrac{2}{2+1} = \dfrac{2}{3}###

For the cost of debt and the risk free rate, always use the yield since it's the total return, ignore the coupon rate which is irrelevant in this question.

Since the equity beta is one, which is the same as the market, the cost of equity must be the same as the expected market return assuming that the equity is fairly priced. The market return is its expected dividend yield plus capital return which is 10%.

For the after-tax WACC,

###\begin{aligned} r_\text{wacc after tax} &= r_\text{e, ord}.\frac{E_\text{ord}}{V} + r_\text{d}.(1 - t_c).\frac{D}{V} \\ &= 0.1 \times \left( 1- \frac{2}{3} \right) + 0.08 \times (1-0.3) \times \frac{2}{3} \\ &= 0.070666667 \\ \end{aligned} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The levered cost of equity will be the highest of all of the other rates. This is because equity has the highest systematic risk when levered since debt amplifies the risk and return of the equity in a business.

Mathematically, the systematic risk of the levered equity in a business can be found using the portfolio beta equation, since the market value of assets (V) equals debt (D) plus equity (E).

###\beta_{V_{L}} = \dfrac{D}{V_{L}}.\beta_D + \dfrac{E_{L}}{V_{L}}.\beta_{E_{L}}###

This equation says that the weighted average beta of debt and levered equity equals the beta on assets. Since the beta on debt is close to zero, the beta on levered equity might be approximately one, so the beta on assets will be the weighted average which is somewhere between the beta on debt and levered equity. So clearly levered equity has the highest beta and therefore will have the highest required return.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The WACC after-tax includes the benefit of the tax shield by reducing the cost of debt. Answer (e) is incorrect since it is simply the before-tax WACC without any adjustment for the benefit of the tax shield. Read on for a more detailed look at each type of WACC equation.

WACC before tax

The WACC before tax, also known as the opportunity cost of capital or the required return on assets (##r_{VL}##), takes the time value of money and the systematic risk into account. This is apparent when you consider the two different ways to calculate the WACC before tax.

There is the familiar formula which is the weighted average cost of the equity and debt used to finance the firm's assets:

###r_\text{WACC before tax} = r_D.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L}###

Since ##V=D+E##, this should be equal to the required return on the firm's assets using the CAPM with the firm's asset beta:

###r_\text{WACC before tax} = r_{VL} = {r_f} + \beta_{VL}.(r_m - r_f)###

This CAPM version of the WACC before tax equation breaks the required return into the time value of money (##r_f##) and the systematic risk premium (##\beta_{VL}.(r_m - r_f)##).

WACC after-tax

The WACC after-tax is just the same as the WACC before-tax, but it also includes the benefit of the tax shield. The WACC after-tax can also be represented by two formulas, the first and last written here:

###\begin{aligned} r_\text{WACC after tax} &= r_D.(1-t_c).\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} \\ &= {r_D.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L}} - r_D.t_c.\frac{D}{V_L}\\ &= r_\text{WACC before tax} - r_D.t_c.\frac{D}{V_L}\\ &= {r_f} + \beta_{VL}.(r_m - r_f) - r_D.t_c.\frac{D}{V_L}\\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The house asset (V) is financed by the home loan debt (D) and the owners wealth or equity in the house (E).

###V = D + E###

Owning all of the debt and equity is equivalent to owning the house asset. Therefore the house asset can be seen as a portfolio of debt and equity.

Method 1: Use the CAPM Portfolio beta equation to solve for the beta of equity

Applying the portfolio beta equation, the beta of the asset must equal the weighted average of the betas on debt and equity.

###\beta_\text{portfolio} = \beta_1.x_1 + \beta_2.x_2 + ... + \beta_n.x_n ### ###\begin{aligned} \beta_V &= \beta_D.x_D + \beta_E.x_E \\ &= \beta_D.\frac{D}{V} + \beta_E.\frac{E}{V} \\ 1 &= 0.2 \times \frac{800,000}{1,000,000} + \beta_E.\frac{200,000}{1,000,000} \\ \end{aligned} ### ### \beta_E = 4.2 ###

Applying the CAPM,

###\begin{aligned} r_E &= r_f + \beta_E.(r_m - r_f) \\ &= 0.05 + 4.2 \times (0.1 - 0.05) \\ &= 0.26 \\ \end{aligned} ###

It may seem surprising that the equity's beta and required total return is so high. The reason is because of leverage. The debt-to-assets ratio (D/V) is 80% and the debt-to-equity ratio (D/E) is 400%. If the value of the house asset rose by 1%, the value of equity would rise by 5%.

Method 2: Use the WACC equation to solve for the cost of equity

Find the required return on debt ##(r_D)## and assets ##(r_V)## using the CAPM:

###\begin{aligned} r_D &= r_f + \beta_D.(r_m - r_f) \\ &= 0.05 + 0.2 \times (0.1 - 0.05) \\ &= 0.06 \\ \end{aligned} ### ###\begin{aligned} r_V &= r_f + \beta_V.(r_m - r_f) \\ &= 0.05 + 1 \times (0.1 - 0.05) \\ &= 0.1 \\ \end{aligned} ###

Using the weighted average cost of capital (WACC) equation (before tax since the question says ignore taxes), the cost of equity (also known as the required return on equity or opportunity cost of equity) can be found. ###\begin{aligned} r_V &= \text{WACC}_\text{before tax} \\ &= r_D.\dfrac{D}{V} + r_E.\dfrac{E}{V} \\ 0.1 &= 0.06 \times \dfrac{800,000}{1,000,000} + r_E \times \dfrac{200,000}{1,000,000} \\ \end{aligned} ### ###\begin{aligned} r_E &= \left(0.1 - 0.06 \times \dfrac{800,000}{1,000,000} \right) \times \dfrac{1,000,000}{200,000} \\ &= 0.26 \\ \end{aligned} ###

We can use the CAPM to find the beta of equity from this required return on equity:

###r_E = r_f + \beta_E.(r_m - r_f) ### ###0.26 = 0.05 + \beta_E.(0.1 - 0.05) ### ###\begin{aligned} \beta_E &= \dfrac{0.26 - 0.05}{0.1 - 0.05} \\ &= 4.2 \\ \end{aligned} ###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The cost of debt ##(r_D)## is the yield of the bond, 10%. The cost of equity ##(r_E)## is the required return on equity which can be found using the CAPM since we have the beta on levered equity, risk free rate and expected return on the market portfolio. The cost of equity can also be found from the DDM but not in this case since we do not have the dividend or its growth rate. The risk free rate is the yield on government bonds, 6%.

###\begin{aligned} r_E &= r_f + \beta_E(r_m - r_f) \\ &= 0.06 + 2(0.1 - 0.06) \\ &= 0.14 \\ \end{aligned}###

The WACC after tax is then:

###\begin{aligned} r_\text{WACC after tax} &= {r_D.(1-t_c).\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L}} \\ &= {0.1 \times (1-0.3)\times\frac{1m}{1m+1m} + 0.14 \times \frac{1m}{1m+1m}} \\ &= 0.105 \\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

To find the market capitalisation of the ordinary shares, ###\begin{aligned} E &= P_\text{share}.n_\text{shares} \\ E_\text{ord} &= 6\times 50m = 300m\\ \end{aligned} ###

To find the market cap of the preference shares: ###\begin{aligned} E &= P_\text{share}.n_\text{shares} \\ E_\text{pref} &= 80 \times 1m = 80m \\ \end{aligned} ###

For the debentures, we do not know their coupon rate, payment frequency or maturity, but we can work out their price since it is given that they trade at 90% of their face value, so the market price is: ###D = 60m \times 0.9 = 54m###

So the value of the firm's assets must be: ###\begin{aligned} V &= E_\text{ord} + E_\text{pref} + D \\ &= 300m + 80m + 54m \\ &= 434m \\ \end{aligned} ###

The ordinary share's required return can be found using the CAPM: ###\begin{aligned} r_e &= r_f + \beta_e(r_m - r_f) \\ r_\text{e, ord} &= 0.05 + 2(0.1 - 0.05) = 0.15 \\ \end{aligned} ###

The preference shares pay a constant annual promised dividend of 10% of their $100 face value, so the dividend is $10 per year. The price of the debentures is $80 and since the promised preference dividend is paid forever, we can find the required return of the preference shares using the perpetuity formula. Note that the dividend is constant since (non-participating) preference shares only pay a promised dividend and no more (similarly to interest on debt, that's why they're often called hybrid securities), so the growth rate of dividends g is zero: ###\begin{aligned} P_\text{0, pref} &= \frac{d_\text{1, pref}}{r_\text{e, pref} - g} \\ 80 &= \frac{10}{r_\text{e, pref} - 0} \\ r_\text{e, pref} &= \frac{10}{80} = 0.125 \\ \end{aligned} ###

Note that the required return on preferred stock is less than common stock as it should be since preferred stock gets paid first in the event of bankruptcy so it has less risk and therefore a lower expected return.

The required return of the debentures is the same thing as their yield. Note that we assume that the promised yield is the same as the expected yield which is not always the case if the firm is near bankruptcy: ### r_\text{d} = 0.1 ###

Now to apply the after-tax WACC formula: ###\begin{aligned} r_\text{wacc after tax} &= r_\text{e, ord}.\frac{E_\text{ord}}{V} +r_\text{e, pref}.\frac{E_\text{pref}}{V} + r_\text{d}.(1 - t_c).\frac{D}{V} \\ &= 0.15 \times \frac{300m}{434m} + 0.125 \times \frac{80m}{434m} + 0.1 \times (1-0.3) \times \frac{54m}{434m} \\ &= 0.135437788 \\ \end{aligned} ###