Your friend just bought a house for $400,000. He financed it using a $320,000 mortgage loan and a deposit of $80,000.
In the context of residential housing and mortgages, the 'equity' tied up in the value of a person's house is the value of the house less the value of the mortgage. So the initial equity your friend has in his house is $80,000. Let this amount be E, let the value of the mortgage be D and the value of the house be V. So ##V=D+E##.
If house prices suddenly fall by 10%, what would be your friend's percentage change in equity (E)? Assume that the value of the mortgage is unchanged and that no income (rent) was received from the house during the short time over which house prices fell.
Remember:
### r_{0\rightarrow1}=\frac{p_1-p_0+c_1}{p_0} ###
where ##r_{0-1}## is the return (percentage change) of an asset with price ##p_0## initially, ##p_1## one period later, and paying a cash flow of ##c_1## at time ##t=1##.
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} ###
Your friend just bought a house for $1,000,000. He financed it using a $900,000 mortgage loan and a deposit of $100,000.
In the context of residential housing and mortgages, the 'equity' or 'net wealth' tied up in a house is the value of the house less the value of the mortgage loan. Assuming that your friend's only asset is his house, his net wealth is $100,000.
If house prices suddenly fall by 15%, what would be your friend's percentage change in net wealth?
Assume that:
- No income (rent) was received from the house during the short time over which house prices fell.
- Your friend will not declare bankruptcy, he will always pay off his debts.
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%.
One year ago you bought $100,000 of shares partly funded using a margin loan. The margin loan size was $70,000 and the other $30,000 was your own wealth or 'equity' in the share assets.
The interest rate on the margin loan was 7.84% pa.
Over the year, the shares produced a dividend yield of 4% pa and a capital gain of 5% pa.
What was the total return on your wealth? Ignore taxes, assume that all cash flows (interest payments and dividends) were paid and received at the end of the year, and all rates above are effective annual rates.
Hint: Remember that wealth in this context is your equity (E) in the house asset (V = D+E) which is funded by the loan (D) and your deposit or equity (E).
There are a few ways to think about this problem. One is to think of the share assets as being financed by a portfolio of debt and equity, where the total historical return on the share assets equals the weighted average total historical return on the debt and equity. Note that the total historical return on the share assets is 9%, the sum of the 4% dividend yield plus the 5% capital yield. This equation is actually the weighted average cost of capital (WACC) before tax:
###r_V = r_D.\dfrac{D}{V} + r_E.\dfrac{E}{V} ### ###0.09 = 0.0784 \times \dfrac{70k}{100k} + r_E.\dfrac{30k}{100k} ### ###r_E.\dfrac{30k}{100k} = 0.09 - 0.0784 \times \dfrac{70k}{100k} ### ###\begin{aligned}r_E &= \left( 0.09 - 0.0784 \times \dfrac{70k}{100k} \right).\dfrac{100k}{30k} \\ &= 0.117067 \\ \end{aligned}###Alternatively, a table can be used. After filling in all of the known values, the unknown return on equity from time -1 to 0 can be calculated.
| Price and Income Values | ||||||
| Time | V | D | E | |||
| -1 | 100k | 70k | 30k | |||
| 0 | 109k | 75.488k | 33.512k | |||
The capital and income components of the equity rose from 30k to 33.512k (=109k-75.488k) which corresponds to a total return on equity of:
###\begin{aligned} r_{\text{E, }-1 \rightarrow 0} &= \frac{P_0-P_{-1}+C_0}{P_{-1}} \\ &= \frac{33.512k-30k+0}{30k} \\ &= \frac{3.512k}{30k} \\ &= 0.117067 = 11.7067\% \\ \end{aligned} ###
Question 408 leverage, portfolio beta, portfolio risk, real estate, CAPM
You just bought a house worth $1,000,000. You financed it with an $800,000 mortgage loan and a deposit of $200,000.
You estimate that:
- The house has a beta of 1;
- The mortgage loan has a beta of 0.2.
What is the beta of the equity (the $200,000 deposit) that you have in your house?
Also, if the risk free rate is 5% pa and the market portfolio's return is 10% pa, what is the expected return on equity in your house? Ignore taxes, assume that all cash flows (interest payments and rent) were paid and received at the end of the year, and all rates are effective annual rates.
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} ###Question 800 leverage, portfolio return, risk, portfolio risk, capital structure, no explanation
Which of the following assets would you expect to have the highest required rate of return? All values are current market values.
No explanation provided.
Question 801 negative gearing, leverage, capital structure, no explanation
The following steps set out the process of ‘negative gearing’ an investment property in Australia. Which of these steps or statements is NOT correct? To successfully achieve negative gearing on an investment property:
No explanation provided.
Question 802 negative gearing, leverage, capital structure, no explanation
Which of the following statements about ‘negative gearing’ is NOT correct?
No explanation provided.
Question 941 negative gearing, leverage, capital structure, interest tax shield, real estate
Last year, two friends Lev and Nolev each bought similar investment properties for $1 million. Both earned net rents of $30,000 pa over the past year. They funded their purchases in different ways:
- Lev used $200,000 of his own money and borrowed $800,000 from the bank in the form of an interest-only loan with an interest rate of 5% pa.
- Nolev used $1,000,000 of his own money, he has no mortgage loan on his property.
Both Lev and Nolev also work in high-paying jobs and are subject personal marginal tax rates of 45%.
Which of the below statements about the past year is NOT correct?
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}###
Question 959 negative gearing, leverage, capital structure, interest tax shield, real estate
Last year, two friends Gear and Nogear invested in residential apartments. Each invested $1 million of their own money (their net wealth).
Apartments cost $1,000,000 last year and they earned net rents of $30,000 pa over the last year. Net rents are calculated as rent revenues less the costs of renting such as property maintenance, land tax and council rates. However, interest expense and personal income taxes are not deducted from net rents.
Gear and Nogear funded their purchases in different ways:
- Gear used $1,000,000 of her own money and borrowed $4,000,000 from the bank in the form of an interest-only loan with an interest rate of 5% pa to buy 5 apartments.
- Nogear used $1,000,000 of his own money to buy one apartment. He has no mortgage loan on his property.
Both Gear and Nogear also work in high-paying jobs and are subject personal marginal tax rates of 45%.
Which of the below statements about the past year is NOT correct?
Nogear’s total personal tax payable due to the investment property would be $13,500. This equals Nogear's $30,000 pre-tax income multiplied by his 45% personal tax rate.
Gear's net rent revenue will be $150,000 since she has 5 rental properties each earning $30,000 net rent. Gear's interest expense will be $200,000 (=4,000,000*0.05) on her $4,000,000 worth of home loans. So the annual loss before tax on the properties would be $50,000 (=150,000 - 200,000), which reduces her personal income by $50,000, meaning she pays less personal tax. Since she's taxed at a personal marginal rate of 45%, the $50,000 before-tax loss results in a $22,500 (=50,000*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 properties' after-tax income loss (rent revenue less interest and other expenses). Gear's after-tax income loss due to the investment property is $27,500 (=50,000*(1-0.45)) in the first year. So if the 5 apartments collectively increased by more than $27,500 or $5,500 each (equivalent to 0.55% pa) in the first year then Gear is better off than Nogear, ignoring capital gains tax.