Question 100 market efficiency, technical analysis, joint hypothesis problem
A company selling charting and technical analysis software claims that independent academic studies have shown that its software makes significantly positive abnormal returns. Assuming the claim is true, which statement(s) are correct?
(I) Weak form market efficiency is broken.
(II) Semi-strong form market efficiency is broken.
(III) Strong form market efficiency is broken.
(IV) The asset pricing model used to measure the abnormal returns (such as the CAPM) had mis-specification error so the returns may not be abnormal but rather fair for the level of risk.
Select the most correct response:
Charting and technical analysis uses past prices, or past returns which are based on past prices, to try to predict future prices and returns. According to the theory of weak form market efficiency, prices follow a random walk with a (small upward) drift and can not be predicted. The best estimate of tomorrow's price is the future value of today's price. The history of prices before today's price is irrelevant. For example, just because prices increased strongly in the past does not mean that they are expected to keep rising strongly in the future.
Positive abnormal returns are returns above the required return that investors deserve for the asset's level of systematic risk. The required return is generally found by using the CAPM or some other asset pricing model:
###r_\text{capm, i} = r_f + \beta_i (r_m - r_f)###The abnormal return is then the actual historical return less the required return:
###r_\text{abnormal, i} = r_\text{actual, i} - r_\text{capm, i}###If the charting software can consistently pick stocks with positive future abnormal returns then either weak for market efficiency is broken, or the market is weak form efficient but the CAPM is broken, or both!
In most markets, studies have shown that weak form efficiency holds. So prices follow a random walk and it is not possible to earn positive abnormal returns using past prices alone.
Note that according to the original theory by Eugene Fama, if weak-form market efficiency is broken, then all of the higher forms of market efficiency are also broken.
Select the most correct statement from the following.
'Chartists', also known as 'technical traders', believe that:
Chartists make charts of past prices or returns and try to use them to extrapolate future prices or returns.
If a chartist can make consistent returns above what they deserve according to the systematic risk they take on, then they are breaking weak-form market efficiency, they are proving the random walk hypothesis wrong.
Most finance practitioners do not believe that chartists can make consistent positive abnormal returns. On the contrary, many expect that compared to a buy-and-hold strategy, most chartists would do worse since they simply rack up transaction costs with each trade where they sell a fairly priced stock and buy another fairly priced stock.
The idea of market efficiency in finance is very similar to competitive markets in economics. In the long run, firms operating in competitive markets with low barriers to entry will make zero economic profits. Note that economic profits include opportunity costs such as the cost of capital which accounting profit ignores.
Similarly, in the highly competitive financial markets it's very hard to make positive abnormal returns. If it was easy, someone would have already done it and bid the under-priced assets up and sold the over-priced assets down.
Question 48 IRR, NPV, bond pricing, premium par and discount bonds, market efficiency
The theory of fixed interest bond pricing is an application of the theory of Net Present Value (NPV). Also, a 'fairly priced' asset is not over- or under-priced. Buying or selling a fairly priced asset has an NPV of zero.
Considering this, which of the following statements is NOT correct?
The NPV of buying any fairly priced asset is zero. Therefore the NPV of buying a fairly priced bond is also zero. Whether the bond is a premium or discount bond is irrelevant, it's unrelated to the NPV of buying it.
The fair price of a bond is the present value (PV) of its expected future cash flows, which is the present value of coupons and face value:
###\begin{aligned} P_\text{0, bond} &= PV(\text{coupons}) + PV(\text{face value}) \\ &= \frac{C_1}{r} \left(1 - \frac{1}{(1+r)^T} \right) + \frac{F_T}{(1+r)^T} \\ \end{aligned}###The net present value (NPV) of buying an asset is the present value of costs less gains.
###\begin{aligned} NPV &= -PV(\text{costs}) + PV(\text{gains}) \\ \end{aligned}###The cost of a bond is its price, and the gains from a bond are the coupons and face value. Since the price of a fairly priced bond equals the present value of the coupons and face value, then the net present value of buying a fairly priced bond must be zero.
Mathematically, we can re-arrange the bond price formula to be in the same form as the NPV formula, which shows that the NPV must be zero:
###P_\text{0, bond} = PV(\text{coupons}) + PV(\text{face value}) ### ###\underbrace{0}_{\text{NPV}} = -\underbrace{P_\text{0, bond}}_{PV(\text{costs})} + \underbrace{PV(\text{coupons}) + PV(\text{face value})}_{PV(\text{gains})} ###Note that premium bonds can also be fairly priced. The NPV of buying a fairly priced premium bond is zero. The term 'premium' does not indicate that the bond's price is above (or below) the fair price, it indicates that the bond's price is above its face value which is usually the $100 or $1,000 that's paid at maturity. Premium bonds have a higher price than their face value because the coupon rate is more than the total required return (the yield). Therefore investors are willing to pay a high price for the bond, higher than the face value, making the bond a premium bond. The highest price investors will pay for the bond will be the price that makes the NPV zero.
An economy has only two investable assets: stocks and cash.
Stocks had a historical nominal average total return of negative two percent per annum (-2% pa) over the last 20 years. Stocks are liquid and actively traded. Stock returns are variable, they have risk.
Cash is riskless and has a nominal constant return of zero percent per annum (0% pa), which it had in the past and will have in the future. Cash can be kept safely at zero cost. Cash can be converted into shares and vice versa at zero cost.
The nominal total return of the shares over the next year is expected to be:
The opportunity cost of investing in shares is investing in cash. Because shares are more risky than cash, shares deserve to earn at least as much as cash ##(r_\text{cash} = 0)##, so their expected return must be zero or greater ##(r_\text{shares} \geq 0)##.
The shares lost 2% pa in the past due to bad news and this was unlucky. The historical return was -2% pa. But investors wouldn't buy the shares now unless the expected future return was at least as much as cash. If the return of the shares was truly expected to be negative then the share price would be zero and the firm would be worthless since nobody would pay to lose money.
Statisticians often regard the historical average as a good guide to the expected future average when the standard error of the historical sample average estimate is small. But the standard error of stock returns is notoriously wide, meaning that the historical average return is not a good estimate of the expected future return. See Merton, R. C. On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 1980, vol. 8, issue 4, pages 323-361.
Another way to think about this question is to use the idea of market efficiency that most assets are fairly priced, not under- or over-priced.
If you assume that stock markets are weak form efficient, then future returns can't be predicted based on patterns in past returns. So just because the return was negative in the past it doesn't mean that it will be negative in the future.
If you assume that stock markets are semi-strong form efficient, then future returns can't be predicted based on past publicly available information such as news announcements. Most investors agree that news drives stock returns, with good news driving prices higher and bad news leading to share price falls. This particular stock has had more bad news than good news in the past which is why it has a negative historical return. But unless someone can predict the news then the chance of good or bad future news announcements might be equal so the share price would not necessarily be expected to fall in the future, regardless of the fact that it's fallen in the past. Another way to say this is that news is unpredictable and random so past bad news does not necessarily mean future bad news.
Thanks to Shahzada for his improvements to this question's answer.
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##).
The shares are expected to be worth $110 in one year, and the loan will be worth $107. So there is a positive expected cash flow of $3 in one year.
###\begin{aligned} V_1 &= V_\text{1, shares} - V_\text{1, loan} \\ &= V_\text{0, shares}(1+r_\text{shares})^1 - V_\text{0, loan}(1+r_\text{loan})^1 \\ &= 100(1+0.1)^1 - 100(1+0.07)^1 \\ &= 110 - 107 \\ &= 3 \\ \end{aligned} ###
Most people then discount the future value of $3 to get a present value of either $2.8037 or $2.7273 depending on whether they use a discount rate of 7 or 10% respectively. But this approach is wrong. The problem becomes apparent when trying to justify the use of one discount rate over another to find the present value of the $3. Should it be 10% or 7% or an average? Unfortunately this way of thinking was flawed from the beginning when the share's and loan's cash flows were added together because they have different risks and should be discounted by different required returns.
The way to analyse this question is to consider buying the shares and selling the loan separately. Note that 'borrowing' is the same thing as 'selling' a loan.
Since the shares are fairly priced, the NPV of buying them is zero. Similarly for the fairly priced loan, the NPV of selling it must be zero. So the NPV of the two transactions is zero plus zero which equals zero.
Alternatively, a more mathematical way of looking at it is that the expected returns of the fairly priced shares and loan are exactly equal to their respective discount rates. So they cancel out as follows:
###\begin{aligned} V_1 &= V_\text{1, shares} - V_\text{1, loan} \\ V_0 &= \frac{V_\text{1, shares}}{(1+r_\text{shares})^1} - \frac{V_\text{1, loan}}{(1+r_\text{loan})^1} \\ &= \frac{V_\text{0, shares}(1+r_\text{shares})^1}{(1+r_\text{shares})^1} - \frac{V_\text{0, loan}(1+r_\text{loan})^1}{(1+r_\text{loan})^1} \\ &= \frac{100(1+0.1)^1}{(1+0.1)^1} - \frac{100(1+0.07)^1}{(1+0.07)^1} \\ &= \frac{110}{(1+0.1)^1} - \frac{107}{(1+0.07)^1} \\ &= 100 - 100 \\ &= 0 \\ \end{aligned} ###
It seems nonsensical that there is a positive expected cash flow of $3 in one year, yet the NPV is zero. The reason why this scenario occurs in theory and in real life is that the expected value of the shares is $110 in one year but it could be a lot less. The loan, on the other hand, will definitely have $107 owing. In the worst case, after one year the shares become worthless (price = 0) and $107 is owed on the loan.
The expected gain of $3 is deserved for taking on the stock's higher level of systematic risk compared with the loan. Investors who suffer higher systematic risk deserve a higher return.
Other interesting view points about this scenario:
- In a risk-neutral world, all assets earn the risk-free rate thus there would be no positive expected future cash flow of $3. But in a risk-averse world, the $3 is compensation for taking on systematic risk.
- The principal of no-arbitrage says that in an efficient market it should be impossible to make unlimited risk-free gains. The portfolio of shares funded by the loan requires no capital so its payoff is unlimited, but the $3 expected gain is not risk-free. Thus the principal of no-(risk-free)-arbitrage holds.
- Banks prefer to lend with some form of security which has a value of more than the loan. The shares have the same value as the loan so they are unlikely to provide sufficient security. In the real world, margin loans on shares generally have a maximum debt-to-assets ratio of 0.7. Residential real estate lenders prefer borrowers to contribute a deposit of 20% of the house price, which equates to a debt-to-assets ratio of 0.8.
- An interesting line of research is the 'Kelly Criterion' and the 'Growth Optimal Portfolio'. The Kelly Criterion is widely known in the gambling literature and is used to calculate the optimal proportion of wealth to wager on a risky bet when the odds are in your favour. The Kelly criterion maximises the growth rate of wealth. It can also be applied to financial decisions such as this if the investor prefers to maximise her expected growth rate of wealth rather than her utility function which takes return and volatility into account.
A managed fund charges fees based on the amount of money that you keep with them. The fee is 2% of the start-of-year amount, but it is paid at the end of every year.
This fee is charged regardless of whether the fund makes gains or losses on your money.
The fund offers to invest your money in shares which have an expected return of 10% pa before fees.
You are thinking of investing $100,000 in the fund and keeping it there for 40 years when you plan to retire.
What is the Net Present Value (NPV) of investing your money in the fund? Note that the question is not asking how much money you will have in 40 years, it is asking: what is the NPV of investing in the fund? Assume that:
- The fund has no private information.
- Markets are weak and semi-strong form efficient.
- The fund's transaction costs are negligible.
- The cost and trouble of investing your money in shares by yourself, without the managed fund, is negligible.
Since the fund doesn't actually have any inside information and markets are efficient, the fund contributes nothing so its fees are value-destructive to investors. Therefore the NPV must be negative.
There are two ways of thinking about this question. The simplest way is to find how much money will be in the fund in 40 years, which will grow by 8% pa which is the expected return (10%) less fees (2%). Then discount this amount by the required return which is 10%, since the fund has the same level of systematic risk as the market portfolio which returns 10%. This present value less the original $100,000 investment will equal the NPV of investing in the fund: ###\begin{aligned} V_0 &= -C_{0} + \frac{C_{40}}{(1+r)^{40}} \\ &= -C_{0} + \frac{C_{0}(1+g)^{40}}{(1+r)^{40}} \\ &= -100,000 + \frac{100,000(1+0.08)^{40}}{(1+0.1)^{40}} \\ &= -100,000 + 48,000.17 \\ &= -51,999.83 \\ \end{aligned}###Another way to find the NPV of the decision to invest is to calculate the NPV of the fees. Since investing in an efficient market with no inside information, transaction costs or economies of scale is a zero-NPV-sum game, any gain to the fund must be a loss to the investor. Therefore the positive NPV of the fees to the fund must be the negative NPV of paying the fees for investors. The NPV of the growing fees can be found using the annuity with growth equation (see question 65 for a derivation):
###\begin{aligned} V_0 &= \frac{-C_{1\text{,fee}}}{r-g} \left( 1 - \left( \frac{1+g}{1+r} \right)^T \right) \\ &= \frac{-100,000 \times 0.02}{0.1-0.08} \left( 1 - \left( \frac{1+0.08}{1+0.1} \right)^{40} \right) \\ &= -51,999.83 \\ \end{aligned}###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:
All statements are true except the last if the forecasts and assumptions are correct.
-
Statement a is true. Since the price ##(P)## is increasing by more than the net rent ##(C)## then the net rental yield ##(C_{t+1}/P_{t})## must fall and approach zero over time.
-
Statement b is true. The total required return on real estate ##(r_\text{real estate, total})## is the sum of the rental and capital yields.
###\begin{aligned} r_\text{real estate, total} &= r_\text{real estate, rent} + r_\text{real estate, capital} \\ &= \dfrac{C_1}{P_0} + \dfrac{P_1 - P_0}{P_0} \\ \end{aligned}###Since the price is and always will be fairly priced, and the rental yield approaches zero, then the total return must fall. The total yield will approach the capital yield.
-
Statement c is true. The total required return is based on the (systematic) risk of the investment, which is determined by the capital asset pricing model (CAPM). Since real estate is and always will be fairly priced, then real estate must plot on the CAPM's security market line (SML):
###\begin{aligned} r_\text{real estate, total} &= r_f + \beta_\text{real estate}.(r_m-r_f) \\ &= r_f + \beta_\text{real estate}.(r_\text{market risk premium}) \\ \end{aligned}###So for the required total return on real estate to fall ##(r_\text{real estate, total})##, either one or more of the risk free rate ##(r_f)##, market risk premium ##(r_\text{market risk premium} = r_m-r_f)##, or systematic risk as measured by beta ##(\beta_\text{real estate})## must fall.
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Statement d is true. Real estate comprises part of a country's wealth. If real estate prices grow by more than the country's wealth forever, then eventually real estate will become the only significant asset in the economy.
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Statement e is false . Rent comprises part of a country's gross domestic product (GDP). If rents grow by less than the country's GDP forever, then eventually rent will become insignificant compared to rest of the country's production. Rent will approach zero percent of the economy's production, not 100%.
A company advertises an investment costing $1,000 which they say is underpriced. They say that it has an expected total return of 15% pa, but a required return of only 10% pa. Of the 15% pa total expected return, the dividend yield is expected to always be 7% pa and rest is the capital yield.
Assuming that the company's statements are correct, what is the NPV of buying the investment if the 15% total return lasts for the next 100 years (t=0 to 100), then reverts to 10% after that time? Also, what is the NPV of the investment if the 15% return lasts forever?
In both cases, assume that the required return of 10% remains constant, the dividends can only be re-invested at 10% pa and all returns are given as effective annual rates.
The answer choices below are given in the same order (15% for 100 years, and 15% forever):
One hundred year case
To find the NPV of the investment when the 15% pa total return lasts for 100 years, subtract the price now and add the present value of the growing dividends and the capital price in 100 years.
Note that the fair total required return is 10% so ##r_\text{total} = 0.1## and the dividend yield is 7% so ##r_\text{div} = 0.07##. The growth rate in the dividend for the first 100 years must be 8% which is the 15% expected total return less the 7% dividend yield, so ##g_\text{div growth} = 0.08##.
###\begin{aligned} NPV &= -\text{Cost} + \text{Benefit} \\ &= -P_{\text{0, actual}} + P_{\text{0, fair}} \\ &= -P_{\text{0, actual}} + \text{PresentValueOfDividendsFor100Years} + \text{PresentValueOfFairPriceIn100Years} \\ &= -P_{\text{0, actual}} + \dfrac{C_1}{r_\text{total} - g_\text{div growth}} \left( 1 - \left( \dfrac{1+g_\text{div growth}}{1+r_\text{total}} \right)^{100} \right) + \dfrac{P_\text{100, fair}}{(1+r_\text{total})^{100}} \\ &= -P_{\text{0, actual}} + \dfrac{P_{\text{0, actual}} . r_\text{div}}{r_\text{total} - g_\text{div growth}} \left( 1 - \left( \dfrac{1+g_\text{div growth}}{1+r_\text{total}} \right)^{100} \right) + \dfrac{P_{\text{0, actual}}.(1+g_\text{div growth})^{100}}{(1+r_\text{total})^{100}} \\ &= -1,000 + \dfrac{1,000 \times 0.07}{0.1 - 0.08} \left( 1 - \left( \dfrac{1+0.08}{1+0.1} \right)^{100} \right) + \dfrac{1,000(1+0.08)^{100}}{(1+0.1)^{100}} \\ &=-1,000 + 3,100.93 \\ &=2,100.93 \\ \end{aligned}###Note that there's another way to calculate the fair price at time 100 ##P_\text{100, fair} = 1000(1+0.08)^{100} = 2,199,761.25634##, which is to value the stock as a perpetuity of the dividends from year 101 onwards:
###\begin{aligned} P_\text{100, fair} &= \dfrac{C_{101}}{r_\text{total} - g_\text{div growth low}} \\ &= \dfrac{C_{1}(1+g_\text{div growth high})^{100}}{r_\text{total} - g_\text{div growth low}} \\ &= \dfrac{P_\text{0, actual}.r_\text{div}(1+g_\text{div growth high})^{100}}{r_\text{total} - g_\text{div growth low}} \\ &= \dfrac{1,000 \times 0.07 \times (1+0.08)^{100}}{0.1 - 0.02} \\ &= \dfrac{70 \times (1+0.08)^{100}}{0.1 - 0.02} \\ &= 2,199,761.25634 \\ \end{aligned}###Perpetual case
To find the NPV of the investment when the 15% pa total return lasts forever, subtract the actual price now and add the present value of the fair price which is perpetuity of growing dividends using the DDM.
###\begin{aligned} NPV &= -\text{Cost} + \text{Benefit} \\ &= -P_{\text{0, actual}} + P_{\text{0, fair}} \\ &= -P_{\text{0, actual}} + \dfrac{C_\text{1}}{r_\text{total} - g_\text{div growth}} \\ &= -P_{\text{0, actual}} + \dfrac{P_{\text{0, actual}} . r_\text{div, actual}}{r_\text{total} - g_\text{div growth}} \\ &= -1,000 + \dfrac{1,000 \times 0.07}{0.1 - 0.08} \\ &= -1,000 + 3,500 \\ &=2,500 \\ \end{aligned}###| Actual and Fair Values and Returns | ||
| Actual | Fair Perpetual | |
| Total return pa | 0.15 | 0.10 |
| Capital return and dividend growth rate pa |
0.08 | 0.08 |
| Dividend return pa | 0.07 | 0.02 |
| Price ##(P_0)## | 1,000 | 3,500 |