A stock has a beta of 0.5. Its next dividend is expected to be $3, paid one year from now. Dividends are expected to be paid annually and grow by 2% pa forever. Treasury bonds yield 5% pa and the market portfolio's expected return is 10% pa. All returns are effective annual rates.
What is the price of the stock now?
Starting with the CAPM's SML equation we can find the required return from the stock's beta, the market return and the risk free rate:
###\begin{aligned} \mu_E &= r_f + \beta_E(\mu_m-r_f) \\ &= 0.05 + 0.5(0.1-0.05) \\ &= 0.075 \\ \end{aligned} ###
Now that we have the required return on equity we can discount the dividends on equity using the perpetuity with growth formula, also known as the Gordon growth model.
###\begin{aligned} P_0 =& \frac{C_1}{\mu_{E} - g} \\ =& \frac{3}{0.075-0.02} \\ =& 54.5454545 \end{aligned}###A stock has a beta of 1.2. Its next dividend is expected to be $20, paid one year from now.
Dividends are expected to be paid annually and grow by 1.5% pa forever.
Treasury bonds yield 3% pa and the market portfolio's expected return is 7% pa. All returns are effective annual rates.
What is the price of the stock now?
Starting with the CAPM's SML equation we can find the required return from the stock's beta, the market return and the risk free rate:
###\begin{aligned} \mu_E &= r_f + \beta_E(\mu_m-r_f) \\ &= 0.03 + 1.2(0.07-0.03) \\ &= 0.078 \\ \end{aligned} ###
Now that we have the required return on equity we can discount the dividends on equity using the perpetuity with growth formula, also known as the Gordon growth model.
###\begin{aligned} P_0 =& \frac{C_1}{\mu_{E} - g} \\ =& \frac{20}{0.078-0.015} \\ =& 317.46031746 \end{aligned}###A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.
What do you think will be the stock's expected return over the next year, given as an effective annual rate?
A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.
In the last 5 minutes, bad economic news was released showing a higher chance of recession. Over this time the share market fell by 1%. The risk free rate was unchanged.
What do you think was the stock's historical return over the last 5 minutes, given as an effective 5 minute rate?
Over the last 5 minutes, the return on the risk free rate would be close to zero ##(r_{f\text{ 5 min}} \approx 0)## since it's only 5% per year. The historical market return ##(r_{m\text{ 5 min}})## over the last 5 minutes was -1%. Note that in the CAPM equation below all returns are effective 5 minute historical returns. Substituting this into the CAPM equation: ###\begin{aligned} r_{i\text{ 5 min}} &= r_{f\text{ 5 min}} + \beta_i(r_{m\text{ 5 min}}-r_{f\text{ 5 min}}) \\ &= 0 + 1.5 \times (-0.01-0) \\ &= -0.015 \\ \end{aligned} ###
Discussion of why the 5 minute risk free rate is close to zero
To find the exact 5 minute risk free rate and show that it is truly close to zero, let's convert this 5% effective annual risk free rate into an effective 5 minute risk free rate. Assume that there are 365 days per year, 24 hours per day, 60 minutes per hour and therefore 12 (=60/5) five minute periods per hour.
###(1+r_{f\text{ 5 min}})^\text{number of 5 min periods in a year} = (1+r_{f\text{ annual}})^1### ###(1+r_{f\text{ 5 min}})^{365 \times 24 \times 60 / 5} = (1+r_{f\text{ annual}})### ###\begin{aligned} r_{f\text{ 5 min}} &= (1+r_{f\text{ annual}})^{1/(365 \times 24 \times 60 / 5)}-1 \\ &= (1+0.05)^{1/(365 \times 24 \times 60 / 5)}-1 \\ &= 0.000000464137895 \text{ pa} \\ &= 0.0000464137895 \text{% pa} \\ &\approx 0\text{% pa} \end{aligned}###A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.
Over the last year, bad economic news was released showing a higher chance of recession. Over this time the share market fell by 1%. So ##r_{m} = (P_{0} - P_{-1})/P_{-1} = -0.01##, where the current time is zero and one year ago is time -1. The risk free rate was unchanged.
What do you think was the stock's historical return over the last year, given as an effective annual rate?
Over the last year, the historical effective return on the risk free rate was 5% ##(r_f = 0.05)##. The historical market return ##(r_m)## over the last year was -1% ##(r_m = -0.01)##. Note that in the CAPM equation below all returns are effective annual historical returns. Substituting this into the CAPM equation: ###\begin{aligned} r_i &= r_f + \beta_i(r_m-r_f) \\ &= 0.05 + 1.5 \times (-0.01-0.05) \\ &= -0.04 \\ \end{aligned} ###
The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.
A stock has a beta of 0.7.
What do you think will be the stock's expected return over the next year, given as an effective annual rate?
Using the capital asset pricing model's (CAPM) security market line (SML) formula:
###\begin{aligned} r_i &= r_f + \beta_i (r_m - r_f) \\ &= 0.05 + 0.7 (0.1 - 0.05) \\ &= 0.085 \\ &= 8.5 \text{% pa} \\ \end{aligned}###The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.
A stock has a beta of 0.7.
In the last 5 minutes, bad economic news was released showing a higher chance of recession. Over this time the share market fell by 2%. The risk free rate was unchanged. What do you think was the stock's historical return over the last 5 minutes, given as an effective 5 minute rate?
While the risk free rate is 5% per annum, over the last 5 minutes it would be close to zero since that's such a short time.
Using the capital asset pricing model's (CAPM) security market line (SML) formula based on these returns over the last 5 minutes, the stock's return (some stock ##i##) over the last 5 minutes is expected to be: ###\begin{aligned} r_{i \text{ 5min}} &= r_{f \text{ 5min}} + \beta_i (r_{m \text{ 5min}} - r_{f \text{ 5min}}) \\ &= 0 + 0.7 (-0.02 - 0) \\ &= -0.014 \\ &= -1.4 \text{% pa} \\ \end{aligned}###A firm changes its capital structure by issuing a large amount of equity and using the funds to repay debt. Its assets are unchanged. Ignore interest tax shields.
According to the Capital Asset Pricing Model (CAPM), which statement is correct?
Beta (##\beta##) is a measure of systematic risk, along with variance (##\sigma^2##) and standard deviation (##\sigma##) .
Since the firm's assets (V) are funded by debt (D) and equity (E), the systematic risk of the firm's assets equals the weighted average beta of the debt and equity, so: ###\beta_V = \frac{D}{V}\beta_D + \frac{E}{V}\beta_E###
In this question, there is no change in the firm's assets. Therefore, all things remaining equal, there shouldn't be any change in the beta of the firm's assets (##\beta_V##).
Since the firm is issuing more equity (using a rights issue or private placement for example) and using the funds to repay debt (paying back the bond or loan-holders), the amount of equity will increase (↑ E) and the amount of debt will decrease (↓ D).
Equity holders have a residual claim on the firm's assets, which means that they get paid last if the firm goes bankrupt. So shareholders get paid after debt holders. Therefore the decrease in the amount of debt means that the equity holders are more likely to receive some payment if the firm goes bankrupt. It also means that there will be a smaller amount of interest payments that the firm must meet so there is a lower chance of going bankrupt. This means that equity must have less systematic risk, so it's beta will fall (↓##\beta_E##). This is the answer.
Also note that since there are less debt-holders, the smaller amount of debt also has less systematic risk (↓##\beta_D##). This may appear impossible since how can the beta on debt and equity fall, while the beta on assets remain constant? But this is possible since the beta on debt is always less than the beta on equity (##\beta_D < \beta_E##), and while both betas fall, there is a lower weight on debt (↓##\frac{D}{V}##), and a higher weight on equity (↑##\frac{E}{V}##), so the beta on assets stays the same.
To summarise: ###\overbrace{\beta_V}^{\cdot} = \overbrace{\frac{D}{V}}^{\downarrow} \overbrace{\beta_D}^{\downarrow} + \overbrace{\frac{E}{V}}^{\uparrow} \overbrace{\beta_E}^{\downarrow} ###
Question 244 CAPM, SML, NPV, risk
Examine the following graph which shows stocks' betas ##(\beta)## and expected returns ##(\mu)##:
Assume that the CAPM holds and that future expectations of stocks' returns and betas are correctly measured. Which statement is NOT correct?
This question is related to the security market line (SML) in the capital asset pricing model (CAPM). Stocks that plot:
- Above the SML have a positive alpha (or positive abnormal return), are under-priced, and buying them is a positive NPV investment.
Assets A and C are in this category. - On the SML have a zero alpha (or zero abnormal return), are fairly-priced, and buying them is a zero NPV investment.
Assets E, M and ##r_f## are in this category. - Below the SML have a negative alpha (or negative abnormal return), are over-priced, and buying them is a negative NPV investment.
Assets B and D are in this category.
Therefore, answer choices (a), (b), (c) and (e) are all correct.
But answer (d) is not correct since stock D has a higher beta (##\beta##) than the market portfolio (M) since it plots further to the right. Because beta is a measure of systematic risk, stock D must have more systematic risk than the market portfolio, not less.
Assets A, B, M and ##r_f## are shown on the graphs above. Asset M is the market portfolio and ##r_f## is the risk free yield on government bonds. Assume that investors can borrow and lend at the risk free rate. Which of the below statements is NOT correct?
If risk-averse investors were forced to invest all of their wealth in a single risky asset A or B (not M since it's a portfolio) then they cannot diversify so total risk is important to them, not just systematic risk. Total risk is shown on the left graph's x-axis. Total variance equals systematic variance plus diversifiable variance:
###\text{TotalVariance} = \text{SystematicVariance} + \text{IdiosyncraticVariance}### ###\begin{aligned} \sigma_\text{total i}^2 &= \sigma_\text{systematic i}^2 + \sigma_{\text{idiosyncratic i}}^2 \\ &= \beta_i^2\sigma_\text{m}^2 + \sigma_{\epsilon\text{ i}}^2 \\ \end{aligned}\\###People who prefer low risk will choose asset B instead of A since ##\sigma_\text{B total} = 0.1## is less than ##\sigma_\text{B total}= 0.4##.
They're the sort of people who might carry an umbrella in their bag even when it's sunny, just in case it might rain. They suffer carrying it around but occasionally it helps them avoid getting drenched and sick. Or perhaps they purchase comprehensive car insurance. They're afraid of risk and are prepared to suffer low expected (average) returns to avoid large losses.
People who prefer high returns will choose asset A instead of B since ##\mu_A = 0.12## is greater than ##\mu_B = 0.075##.
They're the sort of people who may not purchase comprehensive car insurance because they're comfortable with the possibility of crashing their car and losing a large sum to replace it, if it means that they will have more money otherwise. They're comfortable with the possibility of suffering large losses if it means that on average they could gain more.
Choosing asset A or B is a personal choice, there's no correct answer. It depends on your return versus risk preferences. Of course in reality, you're not restricted to choose between A or B, you can choose a bit of both by making a portfolio which is the best idea.
Question 657 systematic and idiosyncratic risk, CAPM, no explanation
A stock's required total return will decrease when its:
No explanation provided.
Question 778 CML, systematic and idiosyncratic risk, portfolio risk, CAPM
The capital market line (CML) is shown in the graph below. The total standard deviation is denoted by σ and the expected return is μ. Assume that markets are efficient so all assets are fairly priced.
Which of the below statements is NOT correct?
There's only one way a portfolio can plot on the CML (red line): it must be composed of the risk free rate (rf) and the market portfolio (M, which itself is composed of all the risky assets within the Markowitz bullet).
Since the risk free asset has no risk at all (in theory), and the market portfolio has only systematic risk since it's fully diversified, any portfolio combination of these two assets rf and M will therefore have no idiosyncratic risk (also called diversifiable risk). Therefore all assets plotting on the CML have no diversifiable risk, they only have systematic risk.
Question 807 market efficiency, expected and historical returns, CAPM, beta, systematic risk, no explanation
You work in Asia and just woke up. It looked like a nice day but then you read the news and found out that last night the American share market fell by 10% while you were asleep due to surprisingly poor macro-economic world news. You own a portfolio of liquid stocks listed in Asia with a beta of 1.6. When the Asian equity markets open, what do you expect to happen to your share portfolio? Assume that the capital asset pricing model (CAPM) is correct and that the market portfolio contains all shares in the world, of which American shares are a big part. Your portfolio beta is measured against this world market portfolio.
When the Asian equity market opens for trade, you would expect your portfolio value to:
No explanation provided.
Question 809 Markowitz portfolio theory, CAPM, Jensens alpha, CML, systematic and idiosyncratic risk
A graph of assets’ expected returns ##(\mu)## versus standard deviations ##(\sigma)## is given in the graph below. The CML is the capital market line.
Which of the following statements about this graph, Markowitz portfolio theory and the Capital Asset Pricing Model (CAPM) theory is NOT correct?
Individual assets and portfolios with returns less than the risk free rate can still be fairly priced so long as they have betas less than zero, consistent with the CAPM. Note that stocks with negative betas will rise when the market falls, so they act like insurance contracts. Holding negative betas assets means you expect to earn less than the risk free rate in the future, which is bad, but when there's a crisis and the market crashes, your assets will be worth more, which is good.
However, individual assets and portfolios with returns less than the risk free rate and positive betas are over-priced, have a negative Jensen’s alpha and should be sold.
Question 810 CAPM, systematic and idiosyncratic risk, market efficiency
Examine the graphs below. Assume that asset A is a single stock. Which of the following statements is NOT correct? Asset A:
Stock A's diversifiable standard deviation is 38.7298335%, not 20%. It can be found using the total variance formula:
###\text{TotalVariance} = \text{SystematicVariance} + \text{DiversifiableVariance}### ###\begin{aligned} \sigma_\text{total A}^2 &= \sigma_\text{systematic A}^2 + \sigma_{\text{diversifiable A}}^2 \\ &= \beta_A^2\sigma_\text{m}^2 + \sigma_{\epsilon\text{ A}}^2 \\ \end{aligned}\\### ###0.4^2 = 0.5^2 \times 0.2^2 + \sigma_{\epsilon\text{ A}}^2 ### ###\begin{aligned} \sigma_{\epsilon\text{ A}}^2 &= 0.4^2 - 0.5^2 \times 0.2^2 \\ &= 0.16 - 0.01 \\ &= 0.15 \\ \end{aligned}\\###Convert the variance to a standard deviation by taking the square root:
###\begin{aligned} \sigma_{\epsilon\text{ A}} &= \sigma_{\text{diversifiable A}} = \sqrt{\sigma_{\epsilon\text{ A}}^2} \\ &= \sqrt{0.15} \\ &= 0.387298335 = 38.7298335\text{% pa} \\ \end{aligned}\\###A common phrase heard in financial markets is that ‘high risk investments deserve high returns’. To make this statement consistent with the Capital Asset Pricing Model (CAPM), a high amount of what specific type of risk deserves a high return?
Investors deserve high returns when they buy assets with high:
According to the CAPM, high systematic risk investments deserve high returns. This is because systematic risk is the only risk that can't be diversified away, it affects all systematically risky assets at the same time. Therefore it's the more frightening risk for investors and those who are willing to put up with it deserve higher total returns.
Note that systematic risk is also known as market risk or un-diversifiable risk.
Beta ##(\beta)## measures systematic risk. The higher the beta, the higher the required total return. This is reflected in the CAPM's security market line (SML) formula:
###r_i = r_f + \beta_i (r_m - r_f) ###Question 988 variance, covariance, beta, CAPM, risk, no explanation
| Price Data Time Series | |||||||||||
| Sourced from Yahoo Finance Historical Price Data | |||||||||||
| Date | S&P500 Index (^GSPC) | Apple (AAPL) | |||||||||
| Open | High | Low | Close | Adj close | Open | High | Low | Close | Adj close | ||
| 2007, Wed 3 Jan | 1418 | 1429 | 1408 | 1417 | 1417 | 12.33 | 12.37 | 11.7 | 11.97 | 10.42 | |
| 2008, Wed 2 Jan | 1468 | 1472 | 1442 | 1447 | 1447 | 28.47 | 28.61 | 27.51 | 27.83 | 24.22 | |
| 2009, Fri 2 Jan | 903 | 935 | 899 | 932 | 932 | 12.27 | 13.01 | 12.17 | 12.96 | 11.28 | |
| 2010, Mon 4 Jan | 1117 | 1134 | 1117 | 1133 | 1133 | 30.49 | 30.64 | 30.34 | 30.57 | 26.6 | |
| Source: Yahoo Finance. | |||||||||||
Which of the following statements about the above table which is used to calculate Apple's equity beta is NOT correct?
The sample covariance of the effective total annual returns between the S&P500 and Apple is actually 0.298181591. This makes Apple's equity beta equal 3.530767 (=0.298181591/0.084452) which is quite a lot higher than the market's.
Four retail business people compete in the same city. They are all exactly the same except that they have different ways of funding or leasing the shop real estate needed to run their retail business.
The two main assets that retail stores need are:
- Inventory typically worth $1 million which has a beta of 2, and;
- Shopfront real estate worth $1 million which has a beta of 1. Shops can be bought or leased.
Lease contract prices are fixed for the term of the lease and based on expectations of the future state of the economy. When leases end, a new lease contract is negotiated and the lease cost may be higher or lower depending on the state of the economy and demand and supply if the economy is:
- Booming, shop real estate is worth more and lease costs are higher.
- In recession, shop real estate is worth less and lease costs are low.
Which retail business person will have the LOWEST beta of equity (or net wealth)?
The short answer is that Anne has no leverage unlike the others, therefore her equity beta equals her asset beta, while others' equity betas are likely to be much higher due to the way leverage amplifies the required returns on equity and equity betas.
Note that the overal asset beta is a weighted average of each of its component assets:
###\begin{aligned} \beta_V &= \beta_\text{inventory}.\dfrac{V_\text{inventory}}{V} + \beta_\text{shop}.\dfrac{V_\text{shop}}{V} \\ &= 2.\dfrac{1m}{2m} + 1.\dfrac{1m}{2m} \\ &= 1.5 \end{aligned}###Therefore Anne, Bob, Carla and Dan arguably have the same asset betas since they own the shop real estate (Carla and Dan's leases are discussed more below). Bob has leverage in the form of the bank loan and so do Carla and Dan in the form of a lease. This is because leasing (or renting or hiring) an asset is equivalent to borrowing money then buying the asset at the start, then selling the asset and paying off the loan at the end. The lease expenses can be seen as interest expenses. Therefore Bob, Carla and Dan's leverage increases their required returns and betas of equity. Remember that the asset beta is a weighted average of the debt and equity betas:
###\beta_V = \beta_D.\dfrac{D}{V} + \beta_E.\dfrac{E}{V}###Re-arrange to make the equity beta the subject:
###\beta_E = \left( \beta_V - \beta_D.\dfrac{D}{V} \right).\dfrac{V}{E}###So the equity beta increases with leverage (higher D/V and V/E).
Arguably, Carla and Dan's leases may reduce the beta of the shop asset compared to owning it outright. This is because the lease decreases the variability of business asset returns and decreases the correlation between the business assets' returns and market's returns, decreasing the firm's asset beta, as shown by this formula for the beta of assets (V) from the capital asset pricing model (CAPM):
###\beta_V = \dfrac{cov(r_V, r_M)}{var(r_M)} = correl(r_V, r_M).\dfrac{sd(r_V)}{sd(r_M)}###Carla's business with 1 year leases are likely to have a lower asset beta than Dan's with 5 year leases, since Carla's lease costs and revenues will both rise in a boom, and both fall in a recession, making profit and cash flows relatively stable. This decreases the risk of her (leased) assets and the correlation of their returns with the market, resulting in a lower asset beta.
Since Dan's 5 year lease costs are constant, in a boom Dan will be making high revenues with constant lease costs, resulting in higher profit and cash flows. But in a recession, Dan will be making low revenues and constant lease costs, resulting in lower profits (maybe losses) and cash flows. This increases the risk of his (leased) assets and the correlation of their returns with the market, resulting in a higher asset beta.
The way short-term leases reduce asset betas by smoothing profits and cash flows is unlikely to be significant enough to offset the leverage effect that the lease has on the equity beta. So Anne's 100% equity capital structure where she owns her assets outright without leverage from loans or leases is still likely to have the lowest equity beta.
CommentaryIn real life, most business people prefer to lease like Carla rather than buy their shop real estate like Anne since it gives:
- Greater flexibility to shutdown within one year if the business flops (abandonment option); or
- Expand into a larger shop if the business is a big success (expansion option), without the need to buy and sell real estate with its explicit costs including stamp duty (a transactional tax) and real estate agent fees as well as the implicit costs such as the lemons problem and winners curse;
- Lets the managers focus on the 'core business' which is efficiently running a shop, not investing in real estate; and
- May be more attractive to shareholders desiring an exposure to retail business only, not retail real estate which is a different (lower risk) asset class. A retail business which leases rather than owns the real estate would be seen as a 'pure-play' investment in the retail sector; and
- Capital constraints. It's often very difficult to raise debt from the bank, or equity from investors, to buy a big real estate asset like a shop. Leasing provides a convenient financing alternative that requires less capital up-front, making it easier to start a business.
A levered firm has only 2 assets on its balance sheet with the below market values and CAPM betas. The risk free rate is 3% pa and the market risk premium is 5% pa. Assume that the CAPM is correct and all assets are fairly priced.
| Balance Sheet Market Values and Betas | ||
| Balance sheet item | Market value ($m) | Beta |
| Cash asset | 0.5 | 0 |
| Truck assets | 0.5 | 2 |
| Loan liabilities | 0.25 | 0.1 |
| Equity funding | ? | ? |
Which of the following statements is NOT correct?
The total value of the firm's assets (V) in millions of dollars is:
###\begin{aligned} V &= V_\text{cash} + V_\text{trucks} \\ &= 0.5+0.5 \\ &= 1 \\ \end{aligned}###The value of the firm's equity (E) in millions of dollars is:
###V = D + E ### ###1 = 0.25+E ### ###\begin{aligned} E &= 1 - 0.25 \\ &= 0.75 \\ \end{aligned}###Use the 'portfolio beta' formula to find the firm's overall asset beta (##\beta_V##):
###\beta_\text{portfolio} = w_1.\beta_1 + w_2.\beta_2 + w_3.\beta_3 + ...### ###\begin{aligned} \beta_V &= w_\text{cash}.\beta_\text{cash} + w_\text{trucks}.\beta_\text{trucks} \\ &= \dfrac{V_\text{cash}}{V}.\beta_\text{cash} + \dfrac{V_\text{trucks}}{V}.\beta_\text{trucks} \\ &= \dfrac{0.5}{1} \times 0 + \dfrac{0.5}{1} \times 2 \\ &= 1 \\ \end{aligned}###Find the equity beta by again using the portfolio beta formula applied to a levered firm's assets (V), remembering that the risk of owning all of the firm's assets is equal to the risk of owning all of the firm's debt (D) and equity (E): ###\beta_\text{portfolio} = w_1.\beta_1 + w_2.\beta_2 + w_3.\beta_3 + ...### ###\begin{aligned} \beta_V &= w_\text{debt}.\beta_\text{debt} + w_\text{equity}.\beta_\text{equity} \\ &= \dfrac{D}{V}.\beta_D + \dfrac{E}{V}.\beta_E \\ \end{aligned}### ###1 = \dfrac{0.25}{1} \times 0.1 + \dfrac{0.75}{1} \times \beta_E ### ###1 = 0.025 + 0.75 \times \beta_E ### ###\begin{aligned} \beta_E &= \dfrac{1 - 0.025}{0.75} \\ &= 1.3 \\ \end{aligned}###
The required return on equity is 9.5% pa:
###\begin{aligned} r_E &= r_f + \beta_E.MRP \\ &= 0.03 + 0.05 \times 1.3 \\ &= 0.095 \\ \end{aligned}###Question 1045 payout policy, leverage, capital structure, beta
A levered firm has only 2 assets on its balance sheet with the below market values and CAPM betas. The risk free rate is 3% pa and the market risk premium is 5% pa. Assume that the CAPM is correct and all assets are fairly priced.
| Balance Sheet Market Values and Betas | ||
| Balance sheet item | Market value ($m) | Beta |
| Cash asset | 0.5 | 0 |
| Truck assets | 0.5 | 2 |
| Loan liabilities | 0.25 | 0.1 |
| Equity funding | ? | ? |
The firm then pays out all of its cash as a dividend. Assume that the beta and yield on the loan liability remain unchanged. Ignore taxes, transaction costs, signalling, information asymmetries and other frictions.
Which of the following statements is NOT correct? This event led to a:
Since the only asset left will be the trucks, the asset beta will be 2 and the WACC before tax or required return on assets (##r_V##) will be:
###\begin{aligned} r_V &= r_f + \beta_V.MRP \\ &= 0.03 + 2 \times 0.05 \\ &= 0.13 \\ \end{aligned}###After the $0.5m cash is all payed out as a dividend, the only asset left is the $0.5m of trucks which is the new total value of the firm's assets (V) in millions of dollars:
###\begin{aligned} V &= V_\text{cash} + V_\text{trucks} \\ &= 0+0.5 \\ &= 0.5 \\ \end{aligned}###The value of the firm's equity (E) in millions of dollars is:
###V = D + E ### ###0.5 = 0.25+E ### ###\begin{aligned} E &= 0.5 - 0.25 \\ &= 0.25 \\ \end{aligned}###As stated above, the only asset left is the trucks so clearly the beta will be 2, same as ther trucks. But here's the maths showing this using the 'portfolio beta' formula to find the firm's overall asset beta (##\beta_V##):
###\beta_\text{portfolio} = w_1.\beta_1 + w_2.\beta_2 + w_3.\beta_3 + ...### ###\begin{aligned} \beta_V &= w_\text{cash}.\beta_\text{cash} + w_\text{trucks}.\beta_\text{trucks} \\ &= \dfrac{V_\text{cash}}{V}.\beta_\text{cash} + \dfrac{V_\text{trucks}}{V}.\beta_\text{trucks} \\ &= \dfrac{0}{0.5} \times 0 + \dfrac{0.5}{0.5} \times 2 \\ &= 2 \\ \end{aligned}###Find the equity beta by again using the portfolio beta formula applied to a levered firm's assets (V), remembering that the risk of owning all of the firm's assets is equal to the risk of owning all of the firm's debt (D) and equity (E):
###\beta_\text{portfolio} = w_1.\beta_1 + w_2.\beta_2 + w_3.\beta_3 + ...### ###\begin{aligned} \beta_V &= w_\text{debt}.\beta_\text{debt} + w_\text{equity}.\beta_\text{equity} \\ &= \dfrac{D}{V}.\beta_D + \dfrac{E}{V}.\beta_E \\ \end{aligned}### ###2 = \dfrac{0.25}{0.5} \times 0.1 + \dfrac{0.25}{0.5} \times \beta_E ### ###2 = 0.05 + 0.5 \times \beta_E ### ###\begin{aligned} \beta_E &= \dfrac{2 - 0.05}{0.5} \\ &= 3.9 \\ \end{aligned}###The required return on equity is 22.5% pa:
###\begin{aligned} r_E &= r_f + \beta_E.MRP \\ &= 0.03 + 0.05 \times 3.9 \\ &= 0.225 \\ \end{aligned}###A levered firm has only 2 assets on its balance sheet with the below market values and CAPM betas. The risk free rate is 3% pa and the market risk premium is 5% pa. Assume that the CAPM is correct and all assets are fairly priced.
| Balance Sheet Market Values and Betas | ||
| Balance sheet item | Market value ($m) | Beta |
| Cash asset | 0.5 | 0 |
| Truck assets | 0.5 | 2 |
| Loan liabilities | 0.25 | 0.1 |
| Equity funding | ? | ? |
The firm then pays off (retires) all of its loan liabilities using its cash. Ignore interest tax shields.
Which of the following statements is NOT correct? All answers are given to 6 decimal places. This event led to a:
Once the loan liabilities are paid off, there are no longer any debt holders to share the assets with, so owning all of the equity is equivalent to owning all of the assets. Therefore the equity beta and required return will equal the asset beta and required return respectively.