Your friend is trying to find the net present value of an investment which:
- Costs $1 million initially (t=0); and
- Pays a single positive cash flow of $1.1 million in one year (t=1).
The investment has a total required return of 10% pa due to its moderate level of undiversifiable risk.
Your friend is aware of the importance of opportunity costs and the time value of money, but he is unsure of how to find the NPV of the project.
He knows that the opportunity cost of investing the $1m in the project is the expected gain from investing the money in shares instead. Like the project, shares also have an expected return of 10% since they have moderate undiversifiable risk. This opportunity cost is $0.1m ##(=1m \times 10\%)## which occurs in one year (t=1).
He knows that the time value of money should be accounted for, and this can be done by finding the present value of the cash flows in one year.
Your friend has listed a few different ways to find the NPV which are written down below.
Method 1: ##-1m + \dfrac{1.1m}{(1+0.1)^1} ##
Method 2: ##-1m + 1.1m - 1m \times 0.1 ##
Method 3: ##-1m + \dfrac{1.1m}{(1+0.1)^1} - 1m \times 0.1 ##
Which of the above calculations give the correct NPV? Select the most correct answer.
A stock was bought for $8 and paid a dividend of $0.50 one year later (at t=1 year). Just after the dividend was paid, the stock price was $7 (at t=1 year).
What were the total, capital and dividend returns given as effective annual rates? The choices are given in the same order:
##r_\text{total}##, ##r_\text{capital}##, ##r_\text{dividend}##.
Question 49 inflation, real and nominal returns and cash flows, APR, effective rate
In Australia, nominal yields on semi-annual coupon paying Government Bonds with 2 years until maturity are currently 2.83% pa.
The inflation rate is currently 2.2% pa, given as an APR compounding per quarter. The inflation rate is not expected to change over the next 2 years.
What is the real yield on these bonds, given as an APR compounding every 6 months?
Question 498 NPV, Annuity, perpetuity with growth, multi stage growth model
A business project is expected to cost $100 now (t=0), then pay $10 at the end of the third (t=3), fourth, fifth and sixth years, and then grow by 5% pa every year forever. So the cash flow will be $10.5 at the end of the seventh year (t=7), then $11.025 at the end of the eighth year (t=8) and so on perpetually. The total required return is 10℅ pa.
Which of the following formulas will NOT give the correct net present value of the project?
A project has the following cash flows:
| Project Cash Flows | |
| Time (yrs) | Cash flow ($) |
| 0 | -400 |
| 1 | 0 |
| 2 | 500 |
What is the payback period of the project in years?
Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $500 at time 2 is actually earned smoothly from t=1 to t=2.
Question 353 income and capital returns, inflation, real and nominal returns and cash flows, real estate
A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 3% pa.
Inflation is expected to be 2% pa. All rates are given as effective annual rates.
What are the property's expected real total, capital and income returns? The answer choices below are given in the same order.
A firm can issue 5 year annual coupon bonds at a yield of 8% pa and a coupon rate of 12% pa.
The beta of its levered equity is 1. Five year government bonds yield 5% pa with a coupon rate of 6% pa. The market's expected dividend return is 4% pa and its expected capital return is 6% pa.
The firm's debt-to-equity ratio is 2:1. The corporate tax rate is 30%.
What is the firm's after-tax WACC? Assume a classical tax system.
The required return of a project is 10%, given as an effective annual rate. Assume that the cash flows shown in the table are paid all at once at the given point in time.
What is the Profitability Index (PI) of the project?
| Project Cash Flows | |
| Time (yrs) | Cash flow ($) |
| 0 | -100 |
| 1 | 0 |
| 2 | 121 |
A project's Profitability Index (PI) is less than 1. Select the most correct statement:
Over the next year, the management of an unlevered company plans to:
- Make $5m in sales, $1.9m in net income and $2m in equity free cash flow (EFCF).
- Pay dividends of $1m.
- Complete a $1.3m share buy-back.
Assume that:
- All amounts are received and paid at the end of the year so you can ignore the time value of money.
- The firm has sufficient retained profits to legally pay the dividend and complete the buy back.
- The firm plans to run a very tight ship, with no excess cash above operating requirements currently or over the next year.
How much new equity financing will the company need? In other words, what is the value of new shares that will need to be issued?
Question 207 income and capital returns, bond pricing, coupon rate, no explanation
For a bond that pays fixed semi-annual coupons, how is the annual coupon rate defined, and how is the bond's annual income yield from time 0 to 1 defined mathematically?
Let: ##P_0## be the bond price now,
##F_T## be the bond's face value,
##T## be the bond's maturity in years,
##r_\text{total}## be the bond's total yield,
##r_\text{income}## be the bond's income yield,
##r_\text{capital}## be the bond's capital yield, and
##C_t## be the bond's coupon at time t in years. So ##C_{0.5}## is the coupon in 6 months, ##C_1## is the coupon in 1 year, and so on.
A fairly valued share's current price is $4 and it has a total required return of 30%. Dividends are paid annually and next year's dividend is expected to be $1. After that, dividends are expected to grow by 5% pa in perpetuity. All rates are effective annual returns.
What is the expected dividend income paid at the end of the second year (t=2) and what is the expected capital gain from just after the first dividend (t=1) to just after the second dividend (t=2)? The answers are given in the same order, the dividend and then the capital gain.
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?
Question 282 expected and historical returns, income and capital returns
You're the boss of an investment bank's equities research team. Your five analysts are each trying to find the expected total return over the next year of shares in a mining company. The mining firm:
- Is regarded as a mature company since it's quite stable in size and was floated around 30 years ago. It is not a high-growth company;
- Share price is very sensitive to changes in the price of the market portfolio, economic growth, the exchange rate and commodities prices. Due to this, its standard deviation of total returns is much higher than that of the market index;
- Experienced tough times in the last 10 years due to unexpected falls in commodity prices.
- Shares are traded in an active liquid market.
- The analysts' source data is correct and true, but their inferences might be wrong;
- All returns and yields are given as effective annual nominal rates.
Let the variance of returns for a share per month be ##\sigma_\text{monthly}^2##.
What is the formula for the variance of the share's returns per year ##(\sigma_\text{yearly}^2)##?
Assume that returns are independently and identically distributed (iid) so they have zero auto correlation, meaning that if the return was higher than average today, it does not indicate that the return tomorrow will be higher or lower than average.
Use the below information to value a mature levered company with growing annual perpetual cash flows and a constant debt-to-assets ratio. The next cash flow will be generated in one year from now, so a perpetuity can be used to value this firm. The firm's debt funding comprises annual fixed coupon bonds that all have the same seniority and coupon rate. When these bonds mature, new bonds will be re-issued, and so on in perpetuity. The yield curve is flat.
| Data on a Levered Firm with Perpetual Cash Flows | ||
| Item abbreviation | Value | Item full name |
| ##\text{OFCF}_1## | $12.5m | Operating free cash flow at time 1 |
| ##\text{FFCF}_1 \text{ or }\text{CFFA}_1## | $14m | Firm free cash flow or cash flow from assets at time 1 |
| ##\text{EFCF}_1## | $11m | Equity free cash flow at time 1 |
| ##\text{BondCoupons}_1## | $1.2m | Bond coupons paid to debt holders at time 1 |
| ##g## | 2% pa | Growth rate of OFCF, FFCF, EFCF and Debt cash flow |
| ##\text{WACC}_\text{BeforeTax}## | 9% pa | Weighted average cost of capital before tax |
| ##\text{WACC}_\text{AfterTax}## | 8.25% pa | Weighted average cost of capital after tax |
| ##r_\text{D}## | 5% pa | Bond yield |
| ##r_\text{EL}## | 13% pa | Cost or required return of levered equity |
| ##D/V_L## | 50% pa | Debt to assets ratio, where the asset value includes tax shields |
| ##n_\text{shares}## | 1m | Number of shares |
| ##t_c## | 30% | Corporate tax rate |
Which of the following statements is NOT correct?
In mid 2009 the listed mining company Rio Tinto announced a 21-for-40 renounceable rights issue. Below is the chronology of events:
- 04/06/2009. Share price opens at $69.00 and closes at $66.90.
- 05/06/2009. 21-for-40 rights issue announced at a subscription price of $28.29.
- 16/06/2009. Last day that shares trade cum-rights. Share price opens at $76.40 and closes at $75.50.
- 17/06/2009. Shares trade ex-rights. Rights trading commences.
All things remaining equal, what would you expect Rio Tinto's stock price to open at on the first day that it trades ex-rights (17/6/2009)? Ignore the time value of money since time is negligibly short. Also ignore taxes.
Question 455 income and capital returns, payout policy, DDM, market efficiency
A fairly priced unlevered firm plans to pay a dividend of $1 next year (t=1) which is expected to grow by 3% pa every year after that. The firm's required return on equity is 8% pa.
The firm is thinking about reducing its future dividend payments by 10% so that it can use the extra cash to invest in more projects which are expected to return 8% pa, and have the same risk as the existing projects. Therefore, next year's dividend will be $0.90. No new equity or debt will be issued to fund the new projects, they'll all be funded by the cut in dividends.
What will be the stock's new annual capital return (proportional increase in price per year) if the change in payout policy goes ahead?
Assume that payout policy is irrelevant to firm value (so there's no signalling effects) and that all rates are effective annual rates.
Question 568 rights issue, capital raising, capital structure
A company conducts a 1 for 5 rights issue at a subscription price of $7 when the pre-announcement stock price was $10. What is the percentage change in the stock price and the number of shares outstanding? The answers are given in the same order. Ignore all taxes, transaction costs and signalling effects.
Question 625 dividend re-investment plan, capital raising
Which of the following statements about dividend re-investment plans (DRP's) is NOT correct?
The security market line (SML) shows the relationship between beta and expected return.
Buying investment projects that plot above the SML would lead to:
A stock's correlation with the market portfolio increases while its total risk is unchanged. What will happen to the stock's expected return and systematic risk?
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?
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?
Question 707 continuously compounding rate, continuously compounding rate conversion
Convert a 10% effective annual rate ##(r_\text{eff annual})## into a continuously compounded annual rate ##(r_\text{cc annual})##. The equivalent continuously compounded annual rate is:
If a variable, say X, is normally distributed with mean ##\mu## and variance ##\sigma^2## then mathematicians write ##X \sim \mathcal{N}(\mu, \sigma^2)##.
If a variable, say Y, is log-normally distributed and the underlying normal distribution has mean ##\mu## and variance ##\sigma^2## then mathematicians write ## Y \sim \mathbf{ln} \mathcal{N}(\mu, \sigma^2)##.
The below three graphs show probability density functions (PDF) of three different random variables Red, Green and Blue.
Select the most correct statement:
Question 418 capital budgeting, NPV, interest tax shield, WACC, CFFA, CAPM
| Project Data | ||
| Project life | 1 year | |
| Initial investment in equipment | $8m | |
| Depreciation of equipment per year | $8m | |
| Expected sale price of equipment at end of project | 0 | |
| Unit sales per year | 4m | |
| Sale price per unit | $10 | |
| Variable cost per unit | $5 | |
| Fixed costs per year, paid at the end of each year | $2m | |
| Interest expense in first year (at t=1) | $0.562m | |
| Corporate tax rate | 30% | |
| Government treasury bond yield | 5% | |
| Bank loan debt yield | 9% | |
| Market portfolio return | 10% | |
| Covariance of levered equity returns with market | 0.32 | |
| Variance of market portfolio returns | 0.16 | |
| Firm's and project's debt-to-equity ratio | 50% | |
Notes
- Due to the project, current assets will increase by $6m now (t=0) and fall by $6m at the end (t=1). Current liabilities will not be affected.
Assumptions
- The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio.
- Millions are represented by 'm'.
- All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
- All rates and cash flows are real. The inflation rate is 2% pa. All rates are given as effective annual rates.
- The project is undertaken by a firm, not an individual.
What is the net present value (NPV) of the project?
A company has:
- 50 million shares outstanding.
- The market price of one share is currently $6.
- The risk-free rate is 5% and the market return is 10%.
- Market analysts believe that the company's ordinary shares have a beta of 2.
- The company has 1 million preferred stock which have a face (or par) value of $100 and pay a constant dividend of 10% of par. They currently trade for $80 each.
- The company's debentures are publicly traded and their market price is equal to 90% of their face value.
- The debentures have a total face value of $60,000,000 and the current yield to maturity of corporate debentures is 10% per annum. The corporate tax rate is 30%.
What is the company's after-tax weighted average cost of capital (WACC)? Assume a classical tax system.
Question 104 CAPM, payout policy, capital structure, Miller and Modigliani, risk
Assume that there exists a perfect world with no transaction costs, no asymmetric information, no taxes, no agency costs, equal borrowing rates for corporations and individual investors, the ability to short the risk free asset, semi-strong form efficient markets, the CAPM holds, investors are rational and risk-averse and there are no other market frictions.
For a firm operating in this perfect world, which statement(s) are correct?
(i) When a firm changes its capital structure and/or payout policy, share holders' wealth is unaffected.
(ii) When the idiosyncratic risk of a firm's assets increases, share holders do not expect higher returns.
(iii) When the systematic risk of a firm's assets increases, share holders do not expect higher returns.
Select the most correct response:
Question 235 SML, NPV, CAPM, risk
The security market line (SML) shows the relationship between beta and expected return.
Investment projects that plot on the SML would have:
The total return of any asset can be broken down in different ways. One possible way is to use the dividend discount model (or Gordon growth model):
###p_0 = \frac{c_1}{r_\text{total}-r_\text{capital}}###
Which, since ##c_1/p_0## is the income return (##r_\text{income}##), can be expressed as:
###r_\text{total}=r_\text{income}+r_\text{capital}###
So the total return of an asset is the income component plus the capital or price growth component.
Another way to break up total return is to use the Capital Asset Pricing Model:
###r_\text{total}=r_\text{f}+β(r_\text{m}- r_\text{f})###
###r_\text{total}=r_\text{time value}+r_\text{risk premium}###
So the risk free rate is the time value of money and the term ##β(r_\text{m}- r_\text{f})## is the compensation for taking on systematic risk.
Using the above theory and your general knowledge, which of the below equations, if any, are correct?
(I) ##r_\text{income}=r_\text{time value}##
(II) ##r_\text{income}=r_\text{risk premium}##
(III) ##r_\text{capital}=r_\text{time value}##
(IV) ##r_\text{capital}=r_\text{risk premium}##
(V) ##r_\text{income}+r_\text{capital}=r_\text{time value}+r_\text{risk premium}##
Which of the equations are correct?
Fundamentalists who analyse company financial reports and news announcements (but who don't have inside information) will make positive abnormal returns if:
Question 416 real estate, market efficiency, income and capital returns, DDM, CAPM
A residential real estate investor believes that house prices will grow at a rate of 5% pa and that rents will grow by 2% pa forever.
All rates are given as nominal effective annual returns. Assume that:
- His forecast is true.
- Real estate is and always will be fairly priced and the capital asset pricing model (CAPM) is true.
- Ignore all costs such as taxes, agent fees, maintenance and so on.
- All rental income cash flow is paid out to the owner, so there is no re-investment and therefore no additions or improvements made to the property.
- The non-monetary benefits of owning real estate and renting remain constant.
Which one of the following statements is NOT correct? Over time:
Question 338 market efficiency, CAPM, opportunity cost, technical analysis
A man inherits $500,000 worth of shares.
He believes that by learning the secrets of trading, keeping up with the financial news and doing complex trend analysis with charts that he can quit his job and become a self-employed day trader in the equities markets.
What is the expected gain from doing this over the first year? Measure the net gain in wealth received at the end of this first year due to the decision to become a day trader. Assume the following:
- He earns $60,000 pa in his current job, paid in a lump sum at the end of each year.
- He enjoys examining share price graphs and day trading just as much as he enjoys his current job.
- Stock markets are weak form and semi-strong form efficient.
- He has no inside information.
- He makes 1 trade every day and there are 250 trading days in the year. Trading costs are $20 per trade. His broker invoices him for the trading costs at the end of the year.
- The shares that he currently owns and the shares that he intends to trade have the same level of systematic risk as the market portfolio.
- The market portfolio's expected return is 10% pa.
Measure the net gain over the first year as an expected wealth increase at the end of the year.
A person is thinking about borrowing $100 from the bank at 7% pa and investing it in shares with an expected return of 10% pa. One year later the person intends to sell the shares and pay back the loan in full. Both the loan and the shares are fairly priced.
What is the Net Present Value (NPV) of this one year investment? Note that you are asked to find the present value (##V_0##), not the value in one year (##V_1##).
Question 339 bond pricing, inflation, market efficiency, income and capital returns
Economic statistics released this morning were a surprise: they show a strong chance of consumer price inflation (CPI) reaching 5% pa over the next 2 years.
This is much higher than the previous forecast of 3% pa.
A vanilla fixed-coupon 2-year risk-free government bond was issued at par this morning, just before the economic news was released.
What is the expected change in bond price after the economic news this morning, and in the next 2 years? Assume that:
- Inflation remains at 5% over the next 2 years.
- Investors demand a constant real bond yield.
- The bond price falls by the (after-tax) value of the coupon the night before the ex-coupon date, as in real life.
Question 322 foreign exchange rate, monetary policy, American and European terms
The market expects the Reserve Bank of Australia (RBA) to decrease the policy rate by 25 basis points at their next meeting.
Then unexpectedly, the RBA announce that they will decrease the policy rate by 50 basis points due to fears of a recession and deflation.
What do you expect to happen to Australia's exchange rate? The Australian dollar will:
A firm conducts a two-for-one stock split. Which of the following consequences would NOT be expected?