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.

If fundamentalists make returns above the return that they deserve, for the level of systematic risk that they take on, then they earn positive abnormal returns and semi-strong form market efficiency must be broken. Therefore markets must be semi-strong form inefficient.

Fundamentalists benefit from semi-strong form market inefficiency. Chartists, another name for 'technical traders', benefit from weak form market inefficiency.

According to Eugene Fama, who constructed this theory, the levels of market efficiency are built on one another, so if markets are weak form inefficient then they are also semi-strong and strong-form inefficient which means that all forms of efficiency are broken.

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:

The abnormal return is then the actual historical return less the required return:

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.

When reading The Economist magazine, your friend is reading publicly available information. If the views and opinions of the magazine are helpful for selecting stocks with positive expected abnormal returns, then as soon as the magazine is published and made public, stock prices should adjust to reflect the new information revealed in the magazine. This would occur if markets are informationally efficient, which is semi-strong form efficiency. Therefore there would be no future positive expected abnormal returns from reading the magazine since stock prices would instantly incorporate the information, so the under-priced share prices would rise and the over-priced share prices would fall, meaning that there are no more 'free lunches' to be had, all assets would be fairly priced.

But if your friend can use the magazine's information to make positive expected abnormal returns, then markets must be semi-strong form inefficient, so semi-strong form efficiency is broken. Alternatively, the model used to measure the abnormal returns could be broken. Or both the model is broken and semi-strong form efficiency is broken. Note that if semi-strong form efficiency is broken then strong form efficiency must also be broken.

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.

The WACC after-tax includes the benefit of the tax shield by reducing the cost of debt. Answer (e) is incorrect since it is simply the before-tax WACC without any adjustment for the benefit of the tax shield. Read on for a more detailed look at each type of WACC equation.

The WACC before tax, also known as the opportunity cost of capital or the required return on assets (##r_{VL}##), takes the time value of money and the systematic risk into account. This is apparent when you consider the two different ways to calculate the WACC before tax.

There is the familiar formula which is the weighted average cost of the equity and debt used to finance the firm's assets:

Since ##V=D+E##, this should be equal to the required return on the firm's assets using the CAPM with the firm's asset beta:

This CAPM version of the WACC before tax equation breaks the required return into the time value of money (##r_f##) and the systematic risk premium (##\beta_{VL}.(r_m - r_f)##).

The WACC after-tax is just the same as the WACC before-tax, but it also includes the benefit of the tax shield. The WACC after-tax can also be represented by two formulas, the first and last written here:

###\begin{aligned} r_\text{WACC after tax} &= r_D.(1-t_c).\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} \\ &= {r_D.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L}} - r_D.t_c.\frac{D}{V_L}\\ &= r_\text{WACC before tax} - r_D.t_c.\frac{D}{V_L}\\ &= {r_f} + \beta_{VL}.(r_m - r_f) - r_D.t_c.\frac{D}{V_L}\\ \end{aligned}###

Answer (e) double counts the interest tax shields and thus over-estimates the value of the levered firm's assets.

The cash flow from assets is big since it's levered and therefore adds the benefit of the interest tax shields from the debt ##(IntExp.t_c)## since:

The discount rate is small since it subtracts the benefit of the proportional interest tax shield since

All other answers give the correct valuation of the levered firm's assets ##(V_L)##. They are all equivalent. They count the benefit of interest tax shields only once.

Thanks to Shahzada for correcting an error in the solutions.