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

The key thing to realise in this question is that when house prices fall by 15%, the bank will not take pity and reduce the debt owing.

In the below table, 'm' means million. Remembering that V=D+E and filling in the values for all except the equity value at t=1, we can calculate that E = V - D = 0.85m - 0.9m = -0.05m, so equity should be -0.05m which is -$50,000. Therefore the poor borrower has negative equity or negative wealth.

Asset, Debt and Equity Values
Millions of dollars
Time	V	D	E
0	1	0.9	0.1
1	0.85	0.9	-0.05

 

The fall in equity from $0.1m (=1m-0.9m) to -0.05m (=0.85m-0.9m) corresponds to a 150% fall in equity:

###\begin{aligned} r_{\text{E, }0\rightarrow1} &= \frac{p_1-p_0+c_1}{p_0} \\ &= \frac{-0.05m-0.1m+0}{0.1m} \\ &= \frac{-0.15m}{0.1m} \\ &= -1.5 = -150\% \\ \end{aligned} ###

Negative wealth is very unfortunate. Many people would declare themselves bankrupt (or for a company, insolvent) because there is no point paying off a house worth less than the value of the loan. However there are costs and limitations on people who are bankrupt for 5 years in Australia and 2 years in America, which is designed to deter bankruptcy. If the person decided to declare bankruptcy, his change in net wealth would be -100%. But in this question we must assume that he will pay his debts, therefore his change in net wealth is -150%.

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

The key thing to realise in this question is that when house prices fall by 10%, there is no fall in the debt owing. The bank will not take pity and reduce the loan!

In the below table, 'k' means thousand. Filling in the values for all except the equity value at t=1, we can calculate that E = V - D = 360k - 320k = 40k, so equity should be 40k.

Asset, Debt and Equity Values
Millions of dollars
Time	V	D	E
0	400k	320k	80k
1	360k	320k	40k

 

The fall in equity from 80k (=400k-320k) to 40k (=360k-320k) corresponds to a 50% fall in equity:

###\begin{aligned} r_{\text{E, }0\rightarrow1} &= \frac{p_1-p_0+c_1}{p_0} \\ &= \frac{40k-80k+0}{80k} \\ &= \frac{-40k}{80k} \\ &= -0.5 = -50\% \\ \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.

No explanation provided.

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

The debt-to-equity ratio can be divided by one without changing its value : ###\dfrac{D}{E} = 0.25 = \dfrac{0.25}{1}###

So debt ##(D)## could be 0.25 and equity ##(E)## could be 1. Therefore the value of assets ##(V)## could be: ###\begin{aligned} V &= D+E \\ &= 0.25+1 \\ &= 1.25 \\ \end{aligned}###

To find the debt-to-assets ratio: ###\dfrac{D}{V} = \dfrac{0.25}{1.25} = 0.2###

The more mathematically rigorous approach is to use simultaneous equations and algebra:

###\dfrac{D}{E} = 0.25### ##E = \dfrac{D}{0.25}##

Substitute this into:

###\begin{aligned} V &= D+E \\ &= D + \dfrac{D}{0.25} \\ &= \dfrac{0.25D}{0.25} + \dfrac{D}{0.25} \\ &= \dfrac{1.25D}{0.25} \\ \end{aligned}### ###D = \dfrac{0.25V}{1.25}### ###\dfrac{D}{V} = \dfrac{0.25}{1.25} = 0.2###

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

The debt-to-equity ratio can be divided by one without changing its value : ###\dfrac{D}{E} = 0.6 = \dfrac{0.6}{1}###

So debt ##(D)## could be 0.6 and equity ##(E)## could be 1. Therefore the value of assets ##(V)## could be: ###\begin{aligned} V &= D+E \\ &= 0.6+1 \\ &= 1.6 \\ \end{aligned}###

To find the debt-to-assets ratio: ###\dfrac{D}{V} = \dfrac{0.6}{1.6} = 0.375###

The more mathematically rigorous approach is to use simultaneous equations:

###\dfrac{D}{E} = 0.6### ###E = \dfrac{D}{0.6} ### ###V=D+E### ###V = D + \dfrac{D}{0.6}### ###0.6V = 0.6D + D### ###V = \dfrac{1.6D}{0.6}### ###\dfrac{D}{V} = \dfrac{0.6}{1.6} = 0.375###

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

The debt-to-assets ratio can be divided by one without changing its value : ###\dfrac{D}{V} = 0.2 = \dfrac{0.2}{1}###

So debt ##(D)## could be 0.2 and assets ##(V)## could be 1. Now the value of equity ##(E)## can be found using the market value balance sheet formula: ###V = D+E ### ###\begin{aligned} E &= V-D \\ &= 1 - 0.2 \\ &= 0.8 \\ \end{aligned}###

To find the debt-to-equity ratio: ###\dfrac{D}{E} = \dfrac{0.2}{0.8} = 0.25###

The more mathematically rigorous approach is to use simultaneous equations and algebra:

###\dfrac{D}{V} = 0.2### ##V = \dfrac{D}{0.2}##

Substitute this into the market value balance sheet formula and seek to re-arrange the terms to show D/E on the left hand side:

###V = D+E ### ###\dfrac{D}{0.2} = D + E ### ###D = 0.2 D + 0.2 E ### ###D - 0.2 D = 0.2 E ### ###0.8 D = 0.2 E ### ###\begin{aligned} \dfrac{D}{E} &= \dfrac{0.2}{0.8} \\ &= 0.25 \\ \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.

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.

Accountants use the 'effective interest method' to calculate interest expense which is the yield on the debt multiplied by its book value at the beginning of the period, with accrual adjustments if the debt matures during the year. Mathematically this is:

###IntExp_1 = r_D.D_0###

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

The interest tax shield per year is the tax saved from funding the firm using debt rather than equity. Interest payments to debt holders are tax-deductible while divided payments to equity holders are not. Thus debt has a tax-advantage over equity. The annual tax saving can be calculated by substituting Net Income (NI) into Cash Flow From Assets (CFFA):

###NI = (Rev-COGS-FC-Depr-IntExp).(1-t_c) ### ###\begin{aligned} CFFA &= NI+Depr-CapEx - \varDelta NWC+IntExp \\ &= (Rev-COGS-FC-Depr-\mathbf{IntExp}).(1-t_c)+Depr-CapEx - \varDelta NWC+\mathbf{IntExp} \\ &= (Rev-COGS-FC).(1-t_c) - CapEx - \varDelta NWC + Depr.t_c + \mathbf{IntExp.t_c} \\ \end{aligned} ###

The last term, ##IntExp.t_c##, is the annual interest tax shield. It is the tax saved from incurring interest expense.

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

The perpetuity equation can be used to find the present value of tax shields that occur forever.

###V_\text{0} = \frac{C_1}{r - g}###

Since debt is constant through time ##D=D_1 = D_2 = D_3## and so on, then the growth rate of the interest tax shields is expected to be zero ##(g=0)##.

###\begin{aligned} V_\text{0, interest tax shields} &= \frac{\text{InterestTaxShieldPerYear}_1}{r_D - 0} \\ &= \frac{\text{IntExp}_1.t_c}{r_D - 0} \\ &= \frac{D_1.r_D.t_c}{r_D} \\ &= D_1.t_c \\ &= D.t_c \\ \end{aligned}###

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

According to the 'effective interest method' which is the standard used by accountants when preparing financial reports, the interest expense for each debt contract equals the yield multiplied by its book value at the beginning of the period, with accrual adjustments if the debt matures during the year. Since the bond trades in a liquid market, the firm's accountants will 'mark to market' the bond price, so the bond's book value will be equal to its market value. Mathematically, the interest expense will be:

###IntExp_1 = r_{\text{D, 0}\rightarrow 1}.D_0###

Some good articles on the effective interest method:

• www.accountingcoach.com
• www.principlesofaccounting.com

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

Negative gearing allows investment income losses to be deducted from personal income, saving personal income tax. This is a form of interest tax shield at the personal level.

Real estate investment income is calculated as net rental income (which doesn't include unrealised capital gains) less mortgage loan interest expense. When this is negative, it is allowed to be deducted from pre-tax personal income which results in the personal income interest tax shield and is called 'negative gearing'.

Note that this is an unusual tax policy because usually, losses in one business (real estate investment) are not allowed to be offset against profits in another businesses (your personal income). Normally, losses in one business become 'carry-forward tax losses' that may be deducted from future before-tax profits earned by that business only.

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

An increase in IntExp decreases NI and increases CFFA. This is very counter-intuitive, but it's because IntExp reduces taxes since it is subtracted from pre-tax income.

But since IntExp is a 'financing cash flow' which has nothing to do with the cash flow from the assets, it is added back in CFFA, and its only lingering effect is the reduction in taxes.

This is the so-called 'interest tax shield' effect of having debt and therefore interest expense.

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

Lev’s personal tax saving due to the investment property was $4,500, compared to not having the investment property. Lev's $10,000 annual loss before tax on the investment property reduces his personal income by $10,000, meaning he pays less personal tax. Since he's taxed at a personal marginal rate of 45%, the $10,000 before-tax loss results in a $4,500 (=10,000*0.45) personal tax saving.

Negative gearing can be a successful strategy so long as the house's after-tax capital gain (house price increase) is greater than the house's after-tax income loss (rent revenue less interest and other expenses) which in this case is $5,500 (=10,000*(1-0.45)) in the first year. So if the house price increased by more than $5,500 in the first year then Lev is better off than Nolev, ignoring capital gains tax.

Notice that Lev's personal tax saving compared to Nolev is $18,000, which equals Nolev's $13,500 personal tax payable plus Lev's $4,500 personal tax saving due to the investment property. This is also equal to the benefit of the interest tax shield in that first year: ###\begin{aligned} \text{InterestTaxShield}_1 &= \text{InterestExpense}_1.t_p \\ &= D_0.r_D.t_p \\ &= 800,000 \times 0.05 \times 0.45 \\ &= 40,000 \times 0.45 \\ &= 18,000 \\ \end{aligned}###

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

The cash flows continue forever so we'll use the perpetuity formula to price the company's assets ##(V)##.

###V=\dfrac{\text{FreeCashFlow}}{r_\text{WACC}-g} ###

'Textbook method' of firm valuation with interest tax shields

The textbook method includes the interest tax shields in the discount rate by discounting the operating free cash flow (OFCF) by the weighted average cost of capital after tax:

###\begin{aligned} V_L &= \dfrac{\text{OFCF}}{\text{WACC}_\text{AfterTax} - g} \\ &= \dfrac{100m}{0.0625 - 0} \\ &= 1600m \\ \end{aligned}###

'Harder method' of firm valuation with interest tax shields

The harder method includes the interest tax shields in the cash flow by discounting the firm free cash flow (FFCF) by the weighted average cost of capital before tax:

###\begin{aligned} V_L &= \dfrac{\text{FFCF}}{\text{WACC}_\text{BeforeTax} - g} \\ &= \dfrac{112m}{0.07 - 0} \\ &= 1600m \\ \end{aligned}###

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

The cash flows continue forever so we'll use the perpetuity formula to price the company's assets ##(V)##.

###V=\dfrac{\text{FreeCashFlow}}{r_\text{WACC}-g} ###

'Textbook method' of firm valuation with interest tax shields

The textbook method includes the interest tax shields in the discount rate by discounting the operating free cash flow (OFCF) by the weighted average cost of capital after tax:

###\begin{aligned} V_L &= \dfrac{\text{OFCF}}{\text{WACC}_\text{AfterTax} - g} \\ &= \dfrac{48.5m}{0.097 - 0} \\ &= 500m \\ \end{aligned}###

'Harder method' of firm valuation with interest tax shields

The harder method includes the interest tax shields in the cash flow by discounting the firm free cash flow (FFCF) by the weighted average cost of capital before tax:

###\begin{aligned} V_L &= \dfrac{\text{FFCF}}{\text{WACC}_\text{BeforeTax} - g} \\ &= \dfrac{50m}{0.1 - 0} \\ &= 500m \\ \end{aligned}###

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

The weighted average cost of capital (WACC) before tax is:

###\begin{aligned} r_\text{WACC before tax} &= r_D.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} \\ &= 0.04 \times 0.8 + 0.163 \times (1-0.8) \\ &= 0.0646 \\ \end{aligned}### ###\begin{aligned} r_\text{WACC after tax} &= r_D.\mathbf{(1-t_c)}.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} \\ &= 0.04 \times (1 - 0.3) \times 0.8 + 0.163 \times (1-0.8) \\ &= 0.055 \\ \end{aligned}###

The cash flows continue forever so we'll use the perpetuity formula to price the company's assets ##(V)##.

###V=\dfrac{\text{FreeCashFlow}}{r_\text{WACC}-g} ###

'Textbook method' of firm valuation with interest tax shields

The textbook method includes the interest tax shields in the discount rate by discounting the operating free cash flow (OFCF) by the weighted average cost of capital after tax:

###\begin{aligned} V_L &= \dfrac{\text{OFCF}}{\text{WACC}_\text{AfterTax} - g} \\ &= \dfrac{30k}{0.055 - 0.015} \\ &= 750k \\ \end{aligned}###

The current value of debt equals the current value of assets multiplied by the debt-to-assets ratio:

###\begin{aligned} D &= V_L \times \dfrac{D}{V_L} \\ &= 750k \times 0.8 \\ &= 600k \\ \end{aligned}###

The benefit from interest tax shields in the first year is equal to the interest expense that year multiplied by the corporate tax rate:

###\begin{aligned} \text{BenefitFromInterestTaxShields}_1 &= \text{InterestExpense}_1 \times t_c \\ &= D_0 \times r_D \times t_c \\ &= 600k \times 0.04 \times 0.3\\ &= 24k \times 0.3 \\ &= 7.2k \\ \end{aligned}###

To find the market capitalisation of equity, use the market value balance sheet formula:

###V_L = D + E ### ###750k = 600k + E ### ###\begin{aligned} E &= 750k - 600k \\ &= 150k \end{aligned}###

The share price ##P## can be found based on the market capitalisation of equity formula:

###E = P \times n_\text{shares} ### ###\begin{aligned} P &= \dfrac{E}{n_\text{shares}} \\ &= \dfrac{150k}{100k} \\ &= 1.5 \\ \end{aligned}###

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

The higher interest expense does lower profit which appears bad, but this actually leads to lower tax payments and since assets are unchanged there will be higher cash flows from assets which shareholders will benefit from, leading to a higher share price.