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FINS2624 PROBLEM SET 7 SOLUTIONS Question 1. c) is a violation of weak form EMH, because trading on past price/return information is able to generate abnormal returns (alphas). And if prices do not fully reflect even past price information (a sub-set of all public information), prices surely do not fully reflect all public information (thus a violation of semi-strong form EMH) and all information, both public and private (thus also a violation of strong-form EMH).

e) Here fund managers are trading on private information they obtain from corporate executives and are able to generate abnormal returns (alphas). So it is a violation of strong-form EMH, but not other forms of EMH. In all other cases, it is not clear whether or not you can generate abnormal returns or alphas. For example, positive returns in a) and above than average capital gains in d) do not necessarily mean positive alphas (which means that the returns are higher than the expected returns based on some asset pricing models like CAPM).

Question 2. a) does not say anything about EMH, because we don’t know what are the alphas of these mutual funds. And even nearly half of mutual funds are able to generate positive alphas in a year, it does not necessarily violate EMH because it may be due to luck.

b) is a violation of EMH because it implies those outperforming mutual funds could consistently generate positive alphas (and thus outperformance in a year is not due to luck).

c) it does not say anything about alphas, and thus say nothing about EMH.

d) violation of semi-strong form (and hence strong-from also) EMH, because you design a trading strategy buying in those stocks with increased earnings in January and holding them in February and generate superior performance. This strategy uses public information.

e) implies a violation of weak form EMH (and thus semi-strong and strong forms also).

Question 3. a) the optimal fraction to invest in the market portfolio could be calculated from y* = 1/A*(E(rM) – rf)/σ2M With the given believes of two investors, we can easily do the calculation. y* = 1/3*(15%-3%)/(40%*40%) = 25% for rational investor and y* = 1/3*(15%-3%)/(25%*25%) = 64% for overconfident investor.

b) Based on the calculated allocations to the market portfolio in a), we can calculate expected return and volatility of the complete portfolio for both investors. Rational investors: E(rc) = 0.75*rf + 0.25* E(rM) = 6% σc = 0.25* σM = 10% U = E(rc) – 1.5 σ2c = 4.5% Overconfident investors: E(rc) = 0.36*rf + 0.64* E(rM) = 10.68% σc = 0.64* σM = 25.6% U = E(rc) – 1.5 σ2c = 0.08% We can see the following points: [1] overconfident investor’s portfolio has a higher expected return and a higher volatility, due to it invests much more in the risky assets. [2] however, the utility is lower for overconfident investors. So overconfident investors are making sub-optimal investment.

c & d) So updated E(rM) = 17% for both types of investors.

With this new expected market return, the optimal fraction of investment in the market portfolio is y* ~= 29% for rational investors, a 4% additional weight in the market portfolio. and y* ~= 75% for overconfident investors, an 11% additional weight in the market portfolio. So we can see that though the magnitude of revision in return expectation is the same for both types of investors, overconfident investors tend to trade (i.e., rebalance their portfolios) much more aggressively.

Question 4. a) Based on the given information, we can calculate the alpha of asset X as: 12.5% - 4% - 0.5*(16% - 4%) = 2.5%. So both investors will deviate from the market portfolio by holding additional asset X beyond its weight in the market portfolio. Based on lecture 7, we know the optimal additional weight in asset X is given by

So we need to calculate the idiosyncratic risk (the variance of residual return), σ2ε = σ2 - (beta* σM)^2, of asset X first. For rational investor, her estimate of σ2ε = 50%*50% - (0.5*20%)^2 = 24%, which results in W0A ~= 3.47% and W*A ~= 3.41%. For overconfident investor, her estimate of σ2ε = 30%*30% - (0.5*20%)^2 = 8%, which results in W0A ~= 10.42% and W*A ~= 9.9%.

We can see that overconfident investors deviate much further away from the market portfolio than the rational investors and hold a much more concentrated position in the active asset, although they have the same alpha estimates. So overconfident investors tend to hold underdiversified portfolios.

Selected end-of-chapter questions BKM chapter 11 1.

The correlation coefficient between stock returns for two nonoverlapping periods should be zero. If not, returns from one period could be used to predict returns in later periods and make abnormal profits.

2.

No. Microsoft’s continuing profitability does not imply that stock market investors who purchased Microsoft shares after its success was already evident would have earned an exceptionally high return on their investments. It simply means that Microsoft has made risky investments over the years that have paid off in the form of increased cash flows and profitability. Microsoft shareholders have benefited from the risk-expected return tradeoff, which is consistent with the EMH.

3.

Expected rates of return differ because of differential risk premiums across all securities.

9. c.

This is a predictable pattern in returns that should not occur if the weak-form EMH is

valid. 14.

d.

In a semistrong-form efficient market, it is not possible to earn abnormally high profits by trading on publicly available information. Information about P/E ratios and recent price changes is publicly known. On the other hand, an investor who has advance knowledge of management improvements could earn abnormally high trading profits (unless the market is also strong-form efficient).

BKM chapter 12 3.

One of the major factors limiting the ability of rational investors to take advantage of any ‘pricing errors’ that result from the actions of behavioral investors is the fact that a mispricing can get worse over time. An example of this fundamental risk is the apparent ongoing overpricing of the NASDAQ index in the late 1990s. Related factors are the inherent costs and limits related to short selling, which restrict the extent to which arbitrage can force overpriced securities (or indexes) to move towards their fair values. Rational investors must also be aware of the risk that an apparent mispricing is, in fact, a consequence of model risk; that is, the perceived mispricing may not be real because the investor has used a faulty model to value the security.

6.

a. Davis uses loss aversion as the basis for her decision making. She holds on to stocks that are down from the purchase price in the hopes that they will recover. She is reluctant to accept a loss.

9.

a. iv b. iii c. v

d. i e. ii...