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Unformatted text preview: Solution to Final Review Problem Set Question 1 You are a 1period investor trying to choose a mutual fund in which to invest (in addition to investing in the riskfree asset). You have two possible choices. The first is equity index fund, which has returns that are equal to those of the S&P 500 index. The second is an actively managed fund, also invested 100% in equity. You calculate the following statistics using monthly returns (where .01 = 1%): S&P 500 Index Fund Actively Managed Fund Mean Return .009 .013 Return Std. Dev. .045 .081 A regression of the actively managed fund on the S&P Index yields a beta of .9. The riskfree rate has been .005 (half of 1%) per month over your sample. Treating the S&P Index return as the market return, is the actively managed fund's mean return higher or lower than that implied by the CAPM? Using the formula E[r] = rf + (E[rM] rf), we get .005 + .9 (.009 .005) = .0086 Thus, the actual average return is higher than implied by the CAPM. Based on the market model, what fraction of the actively managed fund's variance would you call "systematic"? What fraction is "idiosyncratic"? First note that Var(r) = .0812 = .006561 and Var(rM) = .0452 = .002025. Next we rearrange the formula Var(r) = 2 Var(rM) + Var() to obtain Var() = Var(r) 2 Var(rM) = .006561 .92 .002025 = .004921 The fraction of risk that is idiosyncratic is therefore .004921 / .006561 = .75. The remaining fraction, .25, is systematic. Question 2 Describe two empirical findings that have led finance researchers to dispute the validity of the CAPM. Briefly summarize how you would replicate these findings. Here are some choices: 1. Momentum: the abnormally good performance of past winners or bad performers of past losers, where performance is computed using past 3month to 1year returns. 2. Value: the abnormally good performance of stocks with high B/P, E/P, or D/P ratios and the poor performance of stocks with low ratios. 3. Size: the abnormally good performance of stocks with small market capitalizations. Each of these three can be tested by running a time series regression of the excess returns of a portfolio that buys these stocks on the excess returns on the market portfolio. A significantly nonzero alpha indicates a violation of the CAPM. Two more choices are: 4. The weak relation between betas and average returns. Stocks with high betas offer just slightly higher returns than stocks with low betas. 5. The irrelevance of betas when controlling for size and B/P. Run a crosssectional regression of average returns on betas. The slope coefficient will be too small and the intercept too large. This will hold whether or not other variables like size and B/P are in the regression, but the irrelevance of beta will be clearer when they are included. Question 3 A friend notices that every time Jim Cramer is very bullish on a stock on his TV show, the market price of the stock rises immediately by about 2% but falls by about 1% per day for the following two days. Both results are statistically significant. Is the 2% rise or the subsequent decline a violation of the efficient markets hypothesis? What version or versions? How would you exploit this finding in a trading strategy? Most obviously, the 1% declines after the show are violations of semistrong market efficiency. Cramer's show is publically available, so nothing on his show should be useful in forming profitable trading strategies. To exploit the strategy, short sell the stocks Cramer is bullish on just before the market closes that day. It is arguable that the 2% rise also represents a market inefficiency. The argument goes that Cramer is not producing information, but only relaying information to his viewers that is already publically known. Under this view, the 2% rise also represents a violation of semistrong efficiency, though it is not clear how to take advantage of this with a trading strategy. One might instead think that Cramer's shows constitute a release of "insider" information since he might have insights unavailable to the general public. While I doubt this, it would instead cause us to interpret the 2% rise as a violation of the strong form of the EMH. Question 4 Your boss wants you to come up with an estimate of your firm's cost of capital. He would like you to use the FamaFrench model. Describe how you would carry this out. First, obtain the FamaFrench factors (RM, RSMB, and RHML) and compute excess returns of your firm's equity. Next, estimate the expected values of the factors, most likely simply by computing sample means. Now regress the excess returns of your firm on the FamaFrench factors to get three betas. Compute the cost of capital as rf + M E[RM] + SMB E[RSMB] + HML E[RHML] Give an example of when the FamaFrench model would imply a cost of capital lower than that implied by the CAPM. Briefly explain the mechanism through which these returns are low in the FF model. Stocks with low B/P ratios, called growth stocks, underperform relative to the CAPM. The FamaFrench captures this because of two facts. First, the typical growth stock has a negative value of HML, which measures the performance of value stocks relative to growth stocks. Second, E[RHML] is positive since value stocks usually outperform growth stocks. The term HML E[RHML] is therefore negative, which brings down the cost of capital relative to the CAPM. Question 5 You are interested in whether firms that cut dividends exhibit abnormal performance following the announcement of the dividend cut (on event day 0). Carrying out an event study, you construct the following graph of cumulative abnormal returns plotted for event days 2 to +20. (These are measured in business days, so 20 days means four weeks.) Cumulative abnormal return 0 .10 .20 What evidence, if any, do you see that any of the three versions of the efficient markets hypothesis is violated? Strongform violation because of jump at time zero, as insiders would have known this announcement before it is made. Semistrong violation because of postannouncement drift. This drift represents a predictable price movement and therefore a trading opportunity. What trading strategy would you propose to take advantage of these findings? Short sell stocks sometime between the event day and five days afterward. Close out short position 20 days after the event or whenever the downward price drift flattens out. How do you think this strategy resembles more conventional strategies based on value, growth, momentum, and/or reversal? 0 5 10 15 20 Event day The strategy will have some momentumlike features because we are selling losing stocks, though here the stocks are only losing over a single day, while momentum defines losers as stocks that have done poorly for some number of months. We are also likely buying stocks that become less "valuelike" because dividends are now lower and the D/P ratio has probably gone down. (I say probably because prices have also gone down, and it is possible that D/P actually increases if prices drop enough.) Question 6 Based on our own regression analysis and the findings of others, a number of firm characteristics have been found to be related to future stock returns. For each pair, circle the characteristic that is associated with higher expected returns: Question 7 Consider the following results from regressions of excess returns on individual stocks (R1 and R2) on excess market returns (RM). Standard errors for each estimated coefficient are reported in parentheses. Low past 5year returns vs. High past 5year returns Low past 6month returns vs. High past 6month returns Low priceearnings ratio vs. High priceearnings ratio Low booktomarket ratio vs. High booktomarket ratio Small market capitalization vs. Large market capitalization Show that stock 1 has a higher variance than stock 2. Var(R1) = 1.22 .042 + .052 = .0048 Var(R2) = 1.52 .042 + .032 = .0045 Intuitively, why does the CAPM imply that stock 2 is riskier than stock 1 even though stock 1 has a higher variance? Idiosyncratic risk is diversifiable, so it is not priced. Systematic risk is nondiversifiable and therefore priced. Stock 2 has a higher beta, which is a measure of systematic risk. Do these regression results reveal any statistically significant violations of the CAPM? If so, explain briefly. The tstatistic on the alpha of stock 1 is .023/.008 = 2.875, which indicates that the alpha is statistically significant at the 95% level. Since CAPM implies zero alphas, we can reject the CAPM on the strength of the first regression. The tstatistic on stock 2 is .012/.016 = .75, which does not indicate any evidence against the null hypothesis that alpha is zero. Question 8 It appears to you that the market overreacts of extreme bad news. Specifically, firms that loose more than 75% of their market value in a single month seem to bounce back more than the CAPM predicts they should. You decide to test this possible market inefficiency using an event study, where the "event" is any monthly return less than 75%. You decide to form abnormal excess returns using the CAPM, i.e. abnormal excess return(t) = [R(t) Rf(t)] [RM(t) Rf(t)] , where R is the return on the firm, Rf is the riskfree rate, and RM is the return on the market index. t denotes the month in event time, where t=0 is the month of the big decline in value. is estimated using the 60 months prior to the big decline. You plot cumulative average abnormal excess returns and find the following pattern: Cumulative Average Abnormal Return 0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 Event Month Under the assumption that we have computed abnormal excess returns correctly, what form/forms of the efficient markets hypothesis is/are violated? Explain briefly. If the CAPM is true and the "normal" part of the return was computed correctly, then we see positive abnormal returns from the event day through event month 120, which is ten years later. In this case we should seek out stocks that underperform by large margins and hold them for a long time. Since the negative return that we are basing our event on is a type of price information, and since it seems to predict future returns, we have a violation of weak form market efficiency. How could these results be consistent with a rational asset pricing model (such as the CAPM) rather than a market inefficiency? Fama and French would say that when prices fall so dramatically, a typical firm would become a value firm overnight. That is, price is dropping by 75%, but book value is probably not dropping at all. So we are looking at a stock that just saw it's booktomarket ratio go up by a factor of four. Fama and French say that value firms are exposed to some nonmarket form of systematic risk. This risk requires that the owners of value stocks be compensated with higher expected returns. It is possible that the positive abnormal returns in the graph simply reflect this missing component of the "normal" expected return. Had we subtracted off the FFimplied stock return, we might have ended up with no abnormal returns except for those on the event day (which does not constitute a market inefficiency). Question 9 Explain how Carhart constructed his "momentum factor." The momentum factor is the return on winners minus the return on losers. Winners and losers are defined as stocks that did particularly well or poorly over the last twelve months. We form a portfolio of the winners, a portfolio of the losers, compute the returns on both those portfolios, and then take the difference. One year later, we repeat the entire process. Question 10 The following table contains regression results for the market model. Two stocks are considered, Yahoo and Google. All regression variables are excess returns. The standard error for each coefficient is given in parentheses. RYahoo = 0.0032 + (1.29) 1.91 (19.48) 2.12 (16.65) RS&P + Yahoo RGoogle = 0.0054 + In addition, you estimate that E[RS&P] SD(RS&P) SD(Yahoo) SD(Google) = .012 = .05 = .03 = .04 (1.85) RS&P + Google What does the factor model imply is the correlation between Yahoo and Google? The factor model implies that the correlation between stock 1 and stock 2 is 1 2 2 1 Var( RM )
2 2 Var( RM ) Var( 1 ) Var( RM ) Var( 2 ) 1.91 2.12 .052 1.912 .052 .032 2.122 .052 .042 .89 Why might you expect the model to be wrong in this case? Do you think the model understates or overstates the true correlation? Why? The factor model assumes that all correlation between Yahoo and Google comes from their dependence on a common factor none comes through the two error terms. Since Google and Yahoo have much more in common than just a common dependence on the market, it seems likely that the single factor would understate their correlation. Question 11 The American Century Balanced Fund is a mutual fund that invests in a mix of stocks and bonds. Usually the ratio is 60% stocks and 40% bonds. Using monthly data starting in 1993, you run the following regression, where the "Y" variable is the excess return on the fund and the "X" variable is the excess return on the S&P 500 Index. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error 0.926503659 0.85840903 0.857489608 0.010127855 Observations ANOVA Regression Residual Total Intercept X Variable 1 df 156 SS MS 1 0.095766679 0.095766679 154 0.015796312 0.000102573 155 0.111562991 Coefficients Standard Error t Stat 0.001233794 0.000819662 1.505246138 0.647986296 0.019897755 32.56579998 Can you reject the null hypothesis that the fund beta is equal to 0.6? Justify your answer. Can you reject the null hypothesis that the CAPM prices this stock? Justify your answer. With a tstatistic of 1.5, the alpha of the regression is insignificantly different from zero, implying that we cannot reject the null hypothesis that alpha is zero. Thus, the regression provides no strong evidence against the CAPM. If the monthly return on the riskless asset if .003 and your expectation of the future return (not excess return) on the market is .01, what does the CAPM imply is the expected rate of return (again, not excess return) on the fund? The CAPM implies that E[r] = rf + ( E[rM] rf) = .003 + .648(.01 .003) = .0075 The tstatistic of the difference between the estimated beta and .6 is (.648 .6) / .0199 = 2.4. Since this is >2, we can reject at the 95% confidence interval. Question 12 In a single graph, show the relation between the minimum variance frontier that is computed normally (with both long and short positions) to the MV frontier that is computed under a noshort sales constraint. Assume that both frontiers represent portfolios that can be formed by investing directly in stocks. You happen to be an investor who cannot sell stocks short, and you are deciding where to invest. One possibility is to buy a longonly portfolio of stocks of your choosing. The other possibility is to invest in the Acme LongShort ETF. This ETF takes both long and short equity positions. You are not allowed to combine the ETF with a direct investment in stocks you must choose one or the other. A riskless asset also exists in which you can borrow or lend freely at the same riskless rate. Show where, on your graph, this ETF might feasibly lie and simultaneously be preferable to you forming your own longonly portfolio. Label this graph so that I can follow your reasoning. By imposing a noshort sales restriction, you can only make yourself worse off because you are eliminating a lot of portfolios. Thus, the minimum variance frontier computed under no short sales must lie inside the one computed with unrestricted positions. These two frontiers might look like this: Region A Region B max Sharpe ratio for a longonly investor riskless rate long and short MV frontier longonly MV frontier If the ETF lies in Region B, it has a higher Sharpe ratio than what you could achieve as a longonly investor. Region A would be even better, but portfolios in this region are infeasible because they are even beyond the MV frontier that allows shorting. Question 13 Consider the following three stocks: Mean Excess Return Return SD Stock A .007 .040 Stock B .005 .031 Stock C .005 .040 Correlation between A and B: .6 Correlation between A and C: .1 Correlation between B and C: .1 After plugging these numbers into your optimizer, you find that the maximum Sharpe ratio is achieved by investing an equal amount (1/3) in each stock. Explain intuitively, for each asset, why holding a substantial amount of that asset is desirable from a portfolio perspective. In other words, how is each stock helping us to get the highest Sharpe ratio? Stock A contributes to a high Sharpe ratio by providing the highest mean. Stock B contributes by having the lowest standard deviation, Stock C has a low mean and a high standard deviation, but it is not highly correlated with either of the other assets, so it provides great diversification benefits. ...
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 Fall '07
 Callahan

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