Financial Markets and Valuation MGNT808
Tutorial 6, Week 6
Homework questions.
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 a $1,000,000 house partly funded using a mortgage loan. The loan size was $800,000 and the other $200,000 was your wealth or 'equity' in the house asset.
The interest rate on the home loan was 4% pa.
Over the year, the house produced a net rental yield of 2% pa and a capital gain of 2.5% pa.
Assuming that all cash flows (interest payments and net rental payments) were paid and received at the end of the year, and all rates are given as effective annual rates, what was the total return on your wealth over the past year?
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 house asset as being financed by a portfolio of debt and equity, where the total historical return on the house asset equals the weighted average total historical return on the debt and equity. Note that the total historical return on the house is 4.5%, the sum of the 2% net rental yield plus the 2.5% capital yield.
###\begin{aligned} r_V &= r_D.\dfrac{D}{V} + r_E.\dfrac{E}{V} \\ 0.045 &= 0.04 \times \dfrac{0.8m}{1m} + r_E.\dfrac{0.2m}{1m} \\ \end{aligned}### ###\begin{aligned}r_E &= \left( 0.045 - 0.04 \times \dfrac{0.8}{1} \right) \times \dfrac{1}{0.2} \\ &= 0.065 \\ \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 | 1m | 0.8m | 0.2m | |||
| 0 | 1.045m | 0.832m | 0.213m | |||
The capital and income components of the equity rose from 0.2m to 0.213m (=1.045m-0.832m) 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{0.213m - 0.2m+0}{0.2m} \\ &= \frac{0.013m}{0.2m} \\ &= 0.065 = 6.5\% \\ \end{aligned} ###
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 337 capital structure, interest tax shield, leverage, real and nominal returns and cash flows, multi stage growth model
A fast-growing firm is suitable for valuation using a multi-stage growth model.
It's nominal unlevered cash flow from assets (##CFFA_U##) at the end of this year (t=1) is expected to be $1 million. After that it is expected to grow at a rate of:
- 12% pa for the next two years (from t=1 to 3),
- 5% over the fourth year (from t=3 to 4), and
- -1% forever after that (from t=4 onwards). Note that this is a negative one percent growth rate.
Assume that:
- The nominal WACC after tax is 9.5% pa and is not expected to change.
- The nominal WACC before tax is 10% pa and is not expected to change.
- The firm has a target debt-to-equity ratio that it plans to maintain.
- The inflation rate is 3% pa.
- All rates are given as nominal effective annual rates.
What is the levered value of this fast growing firm's assets?
Since all cash flows and discount rates are nominal, inflation can be ignored.
Since the cash flow given is the unlevered cash flow from assets, it does not include the value of the annual interest tax shield so we should discount using the WACC after tax (9.5%) which will result in a levered value of assets that includes the present value of interest tax shields. This is known as the 'textbook' method. Alternatively we could also use the 'adjust present value' technique where the tax shields are added on separately.
Using a multi-stage growth model,
###\begin{aligned} V_L &= \dfrac{CFFA_\text{U, 1}}{(1+r_\text{VwITS})^1} + \dfrac{CFFA_\text{U, 2}}{(1+r_\text{VwITS})^2} + \dfrac{CFFA_\text{U, 3}}{(1+r_\text{VwITS})^3} + \dfrac{\left( \dfrac{CFFA_\text{U, 4}}{r_\text{VwITS}-g_\text{low}} \right) }{(1+r_\text{VwITS})^3}\\ &= \dfrac{1m}{(1+0.095)^1} + \dfrac{1m(1+0.12)^1}{(1+0.095)^2} + \dfrac{1m(1+0.12)^2}{(1+0.095)^3} + \dfrac{\left( \dfrac{1m(1+0.12)^2(1+0.05)^1}{0.095-(-0.01)} \right) }{(1+0.095)^3}\\ &= 12.36m \end{aligned} ###An alternative formulation that gives the same result is:
###\begin{aligned} V_L &= \dfrac{CFFA_\text{U, 1}}{(1+r_\text{VwITS})^1} + \dfrac{CFFA_\text{U, 2}}{(1+r_\text{VwITS})^2} + \dfrac{CFFA_\text{U, 3}}{(1+r_\text{VwITS})^3} + \dfrac{CFFA_\text{U, 4}}{(1+r_\text{VwITS})^4} + \dfrac{\left( \dfrac{CFFA_\text{U, 5}}{r_\text{VwITS}-g_\text{low}} \right) }{(1+r_\text{VwITS})^4}\\ &= \dfrac{1m}{(1+0.095)^1} + \dfrac{1m(1+0.12)^1}{(1+0.095)^2} + \dfrac{1m(1+0.12)^2}{(1+0.095)^3} + \dfrac{1m(1+0.12)^2(1+0.05)^1}{(1+0.095)^4} + \dfrac{\left( \dfrac{1m(1+0.12)^3(1+0.05)^1(1-0.01)^1}{0.095-(-0.01)} \right) }{(1+0.095)^4}\\ &= 12.36m \end{aligned} ###A firm has a debt-to-equity ratio of 25%. What is its debt-to-assets ratio?
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###A firm has a debt-to-equity ratio of 60%. What is its debt-to-assets ratio?
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###A firm has a debt-to-assets ratio of 20%. What is its debt-to-equity ratio?
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}###In the home loan market, the acronym LVR stands for Loan to Valuation Ratio. If you bought a house worth one million dollars, partly funded by an $800,000 home loan, then your LVR was 80%. The LVR is equivalent to which of the following ratios?
The loan-to-valuation ratio (LVR) is a debt-to-assets ratio since it divides the loan liability value by the house asset value that secures the loan.
If the house asset value is ##V## and this is funded by the loan debt liability ##D## and the home owner's deposit or equity in the house ##E## (so ##V = D+E##), then:
###\text{LVR} = \dfrac{D}{V}###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.
Use the below information to value a levered company with annual perpetual cash flows from assets that grow. The next cash flow will be generated in one year from now. Note that ‘k’ means kilo or 1,000. So the $30k is $30,000.
| Data on a Levered Firm with Perpetual Cash Flows | ||
| Item abbreviation | Value | Item full name |
| ##\text{OFCF}## | $30k | Operating free cash flow |
| ##g## | 1.5% pa | Growth rate of OFCF |
| ##r_\text{D}## | 4% pa | Cost of debt |
| ##r_\text{EL}## | 16.3% pa | Cost of levered equity |
| ##D/V_L## | 80% pa | Debt to assets ratio, where the asset value includes tax shields |
| ##t_c## | 30% | Corporate tax rate |
| ##n_\text{shares}## | 100k | Number of shares |
Which of the following statements is NOT correct?
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}###