Chapter 07 - Optimal Risky Portfolios7. The proportion of the optimal risky portfolio invested in the stock fund is given by:wS = [E(rS ) − rf ]σ 2 − [E(rB ) − rf ]Cov(rS , rB ) B [E(rS ) − rf ]σ 2 + [E(rB ) − rf ]σ 2 − [E(rS ) − rf + E(rB ) − rf ]Cov(rS , rB ) B S = [(20 − 8) × 225] − [(12 − 8) × 45] = 0.4516 [(20 − 8) × 225] + [(12 − 8) × 900] − [(20 − 8 + 12 − 8) × 45] wB = 1 − 0.4516 = 0.5484The mean and standard deviation of the optimal risky portfolio are:E(rP) = (0.4516 × 20) + (0.5484 × 12) = 15.61%σp = [(0.45162 × 900) + (0.54842 × 225) + (2 × 0.4516 × 0.5484 × 45)]1/2 = 16.54%8. The reward-to-volatility ratio of the optimal CAL is: E(rp ) − rf = 15.61 − 8 = 0.4601 σp 16.549. a. If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is: E(rC ) = rf + E(rp ) − rf σC = 8 + 0.4601σC σPSetting E(rC) equal to 14%, we find that the standard deviation of the optimalportfolio is 13.04%.b. To find the proportion invested in the T-bill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal combination of stocks and bonds (P). Let y be the proportion invested in the portfolio P. The mean of any portfolio along the optimal CAL is: E(rC) = (l − y)rf + yE(rP) = rf + y[E(rP) − rf] = 8 + y(15.61 − 8) Setting E(rC) = 14% we find: y = 0.7884 and (1 − y) = 0.2116 (the proportion invested in the T-bill fund). To find the proportions invested in each of the funds, multiply 0.7884 times the respective proportions of stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.7884 × 0.4516 = 0.3560 Proportion of bonds in complete portfolio = 0.7884 × 0.5484 = 0.4324 7-3

Chapter 07 - Optimal Risky Portfolios10. Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the appropriate proportion in the stock fund (wS) and the appropriate proportion in the bond fund (wB = 1 − wS) as follows: 14 = 20wS + 12(1 − wS) = 12 + 8wS ⇒ wS = 0.25 So the proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be: σP = [(0.252 × 900) + (0.752 × 225) + (2 × 0.25 × 0.75 × 45)]1/2 = 14.13% This is considerably greater than the standard deviation of 13.04% achieved using T- bills and the optimal portfolio.11. a.25.00 Optimal CAL20.00 P Stocks15.0010.00 Gol d 5.000.00 10 20 30 40 0 Standard Dev iation(%)Even though it seems that gold is dominated by stocks, gold might still be anattractive asset to hold as a part of a portfolio. If the correlation between gold andstocks is sufficiently low, gold will be held as a component in a portfolio, specifically,the optimal tangency portfolio.b. If the correlation between gold and stocks equals +1, then no one would hold gold. The optimal CAL would be comprised of bills and stocks only. Since the set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope (see the following graph), these combinations would be dominated by the stock portfolio. Of course, this situation could not persist. If no one desired gold, its price would fall and its expected rate of return would increase until it became sufficiently attractive to include in a portfolio. 7-4

Chapter 07 - Optimal Risky Portfolios12. Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio, in equilibrium, will be the risk-free rate. To find the proportions of this portfolio [with the proportion wA invested in Stock A and wB = (1 – wA ) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation is: σP = Absolute value [wAσA − wBσB] 0 = 5wA − [10 × (1 – wA )] ⇒ wA = 0.6667 The expected rate of return for this risk-free portfolio is: E(r) = (0.6667 × 10) + (0.3333 × 15) = 11.667% Therefore, the risk-free rate is: 11.667%13. False. If the borrowing and lending rates are not identical, then, depending on the tastes of the individuals (that is, the shape of their indifference curves), borrowers and lenders could have different optimal risky portfolios. 7-5

Chapter 07 - Optimal Risky Portfolios14. False. The portfolio standard deviation equals the weighted average of the component- asset standard deviations only in the special case that all assets are perfectly positively correlated. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights.15. The probability distribution is:Probability Rate of Return 0.7 0.3 100% −50%Mean = [0.7 × 100] + [0.3 × (−50)] = 55%Variance = [0.7 × (100 − 55)2] + [0.3 × (−50 − 55)2] = 4725Standard deviation = 47251/2 = 68.74%16. σ P = 30 = yσ = 40y ⇒ y = 0.75 E(rP) = 12 + 0.75(30 − 12) = 25.5%17. The correct choice is c. Intuitively, we note that since all stocks have the same expected rate of return and standard deviation, we choose the stock that will result in lowest risk. This is the stock that has the lowest correlation with Stock A.More formally, we note that when all stocks have the same expected rate of return, theoptimal portfolio for any risk-averse investor is the global minimum variance portfolio(G). When the portfolio is restricted to Stock A and one additional stock, the objective isto find G for any pair that includes Stock A, and then select the combination with thelowest variance. With two stocks, I and J, the formula for the weights in G is:w Min (I) = σ 2 − Cov(rI , rJ ) J σ 2 + σ 2 − 2Cov(rI , rJ ) I Jw Min (J) = 1 − w Min (I) 7-6

Chapter 07 - Optimal Risky Portfolios Since all standard deviations are equal to 20%: Cov(rI , rJ) = ρσIσJ = 400ρ and wMin(I) = wMin(J) = 0.5 This intuitive result is an implication of a property of any efficient frontier, namely, that the covariances of the global minimum variance portfolio with all other assets on the frontier are identical and equal to its own variance. (Otherwise, additional diversification would further reduce the variance.) In this case, the standard deviation of G(I, J) reduces to: σMin(G) = [200(1 + ρI J)]1/2 This leads to the intuitive result that the desired addition would be the stock with the lowest correlation with Stock A, which is Stock D. The optimal portfolio is equally invested in Stock A and Stock D, and the standard deviation is 17.03%.18. No, the answer to Problem 17 would not change, at least as long as investors are not risk lovers. Risk neutral investors would not care which portfolio they held since all portfolios have an expected return of 8%.19. No, the answers to Problems 17 and 18 would not change. The efficient frontier of risky assets is horizontal at 8%, so the optimal CAL runs from the risk-free rate through G. The best Portfolio G is, again, the one with the lowest variance. The optimal complete portfolio depends on risk aversion.20. Rearranging the table (converting rows to columns), and computing serial correlation results in the following table:Nominal Rates 1920s Small Large Long-term Intermed-term Treasury Inflation 1930s company company government government bills 1940s -1.00 1950s stocks stocks bonds bonds 3.56 -2.04 1960s -3.72 18.36 3.98 3.77 0.30 5.36 1970s 7.28 -1.25 4.60 3.91 0.37 2.22 1980s 1.87 2.52 1990s 20.63 9.11 3.59 1.70 3.89 7.36Serial Correlation 6.29 5.10 19.01 19.41 0.25 1.11 9.00 2.93 5.02 0.23 13.72 7.84 1.14 3.41 0.63 8.75 5.90 6.63 6.11 12.46 17.60 11.50 12.01 13.84 18.20 8.60 7.74 0.46 -0.22 0.60 0.59 7-7

Chapter 07 - Optimal Risky PortfoliosFor example: to compute serial correlation in decade nominal returns for large-companystocks, we set up the following two columns in an Excel spreadsheet. Then, use theExcel function “CORREL” to calculate the correlation for the data.1930s Decade Previous1940s -1.25% 18.36%1950s 9.11% -1.25%1960s 19.41% 9.11%1970s 7.84% 19.41%1980s 5.90% 7.84%1990s 17.60% 5.90% 18.20% 17.60%Note that each correlation is based on only seven observations, so we cannot arrive atany statistically significant conclusions. Looking at the results, however, it appears that,with the exception of large-company stocks, there is persistent serial correlation. (Thisconclusion changes when we turn to real rates in the next problem.)21. The table for real rates (using the approximation of subtracting a decade’s average inflation from the decade’s average nominal return) is: Real Rates 1920s Small Large Long-term Intermed-term Treasury 1930s company company government government bills 1940s 1950s stocks stocks bonds bonds 4.56 1960s 2.34 1970s -2.72 19.36 4.98 4.77 -4.99 1980s 9.32 0.79 6.64 5.95 -0.35 1990s 15.27 3.75 -1.77 -3.66 1.37Serial Correlation 16.79 -1.97 -1.11 -1.07 11.20 17.19 -1.38 0.89 3.90 1.39 5.32 -0.73 -1.25 2.09 7.36 -1.46 6.40 6.91 0.00 10.91 5.67 4.81 12.50 0.29 15.27 0.38 0.11 -0.27While the serial correlation in decade nominal returns seems to be positive, it appearsthat real rates are serially uncorrelated. The decade time series (although again too shortfor any definitive conclusions) suggest that real rates of return are independent fromdecade to decade. 7-8

Chapter 07 - Optimal Risky PortfoliosCFA PROBLEMS1. a. Restricting the portfolio to 20 stocks, rather than 40 to 50 stocks, will increase the risk of the portfolio, but it is possible that the increase in risk will be minimal. Suppose that, for instance, the 50 stocks in a universe have the same standard deviation (σ) and the correlations between each pair are identical, with correlation coefficient ρ. Then, the covariance between each pair of stocks would be ρσ2, and the variance of an equally weighted portfolio would be:σ 2 = 1 σ2 + n −1 ρσ2 P n nThe effect of the reduction in n on the second term on the right-hand side wouldbe relatively small (since 49/50 is close to 19/20 and ρσ2 is smaller than σ2), butthe denominator of the first term would be 20 instead of 50. For example, if σ =45% and ρ = 0.2, then the standard deviation with 50 stocks would be 20.91%,and would rise to 22.05% when only 20 stocks are held. Such an increase mightbe acceptable if the expected return is increased sufficiently.b. Hennessy could contain the increase in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his portfolio. This entails maintaining a low correlation among the remaining stocks. For example, in part (a), with ρ = 0.2, the increase in portfolio risk was minimal. As a practical matter, this means that Hennessy would have to spread his portfolio among many industries; concentrating on just a few industries would result in higher correlations among the included stocks.2. Risk reduction benefits from diversification are not a linear function of the number of issues in the portfolio. Rather, the incremental benefits from additional diversification are most important when you are least diversified. Restricting Hennesey to 10 instead of 20 issues would increase the risk of his portfolio by a greater amount than would a reduction in the size of the portfolio from 30 to 20 stocks. In our example, restricting the number of stocks to 10 will increase the standard deviation to 23.81%. The 1.76% increase in standard deviation resulting from giving up 10 of 20 stocks is greater than the 1.14% increase that results from giving up 30 of 50 stocks.3. The point is well taken because the committee should be concerned with the volatility of the entire portfolio. Since Hennessy’s portfolio is only one of six well-diversified portfolios and is smaller than the average, the concentration in fewer issues might have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to do stock picking may be advantageous. 7-9

Chapter 07 - Optimal Risky Portfolios4. d. Portfolio Y cannot be efficient because it is dominated by another portfolio. For example, Portfolio X has both higher expected return and lower standard deviation.5. c.6. d.7. b.8. a.9. c.10. Since we do not have any information about expected returns, we focus exclusively on reducing variability. Stocks A and C have equal standard deviations, but the correlation of Stock B with Stock C (0.10) is less than that of Stock A with Stock B (0.90). Therefore, a portfolio comprised of Stocks B and C will have lower total risk than a portfolio comprised of Stocks A and B.11. Fund D represents the single best addition to complement Stephenson's current portfolio, given his selection criteria. First, Fund D’s expected return (14.0 percent) has the potential to increase the portfolio’s return somewhat. Second, Fund D’s relatively low correlation with his current portfolio (+0.65) indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio. The other three funds have shortcomings in terms of either expected return enhancement or volatility reduction through diversification benefits. Fund A offers the potential for increasing the portfolio’s return, but is too highly correlated to provide substantial volatility reduction benefits through diversification. Fund B provides substantial volatility reduction through diversification benefits, but is expected to generate a return well below the current portfolio’s return. Fund C has the greatest potential to increase the portfolio’s return, but is too highly correlated with the current portfolio to provide substantial volatility reduction benefits through diversification. 7-10

Chapter 07 - Optimal Risky Portfolios12. a. Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9 × 0.67) + (0.1 × 1.25) = 0.728% ii. Cov = r × σOP × σABC = 0.40 × 2.37 × 2.95 = 2.7966 ≅ 2.80 iii. σNP = [w 2 σOP2 + w ABC 2 σ ABC 2 + 2 wOP wABC (CovOP , ABC)]1/2 OP = [(0.9 2 × 2.372) + (0.12 × 2.952) + (2 × 0.9 × 0.1 × 2.80)]1/2 = 2.2673% ≅ 2.27%b. Subscript OP refers to the original portfolio, GS to government securities, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = (0.9 × 0.67) + (0.1 × 0.042) = 0.645% ii. Cov = r × σOP × σGS = 0 × 2.37 × 0 = 0 iii. σNP = [w 2 σOP2 + wGS2 σ 2 + 2 wOP wGS (CovOP , GS)]1/2 OP GS = [(0.9 2 × 2.372) + (0.12 × 0) + (2 × 0.9 × 0.1 × 0)]1/2 = 2.133% ≅ 2.13%c. Adding the risk-free government securities would result in a lower beta for the new portfolio. The new portfolio beta will be a weighted average of the individual security betas in the portfolio; the presence of the risk-free securities would lower that weighted average.d. The comment is not correct. Although the respective standard deviations and expected returns for the two securities under consideration are equal, the covariances between each security and the original portfolio are unknown, making it impossible to draw the conclusion stated. For instance, if the covariances are different, selecting one security over the other may result in a lower standard deviation for the portfolio as a whole. In such a case, that security would be the preferred investment, assuming all other factors are equal.e. i. Grace clearly expressed the sentiment that the risk of loss was more important to her than the opportunity for return. Using variance (or standard deviation) as a measure of risk in her case has a serious limitation because standard deviation does not distinguish between positive and negative price movements. 7-11

Chapter 07 - Optimal Risky Portfolios ii. Two alternative risk measures that could be used instead of variance are: Range of returns, which considers the highest and lowest expected returns in the future period, with a larger range being a sign of greater variability and therefore of greater risk. Semivariance, which can be used to measure expected deviations of returns below the mean, or some other benchmark, such as zero. Either of these measures would potentially be superior to variance for Grace. Range of returns would help to highlight the full spectrum of risk she is assuming, especially the downside portion of the range about which she is so concerned. Semivariance would also be effective, because it implicitly assumes that the investor wants to minimize the likelihood of returns falling below some target rate; in Grace’s case, the target rate would be set at zero (to protect against negative returns).13. a. Systematic risk refers to fluctuations in asset prices caused by macroeconomic b. factors that are common to all risky assets; hence systematic risk is often referred to as market risk. Examples of systematic risk factors include the business cycle, inflation, monetary policy and technological changes. Firm-specific risk refers to fluctuations in asset prices caused by factors that are independent of the market, such as industry characteristics or firm characteristics. Examples of firm-specific risk factors include litigation, patents, management, and financial leverage. Trudy should explain to the client that picking only the top five best ideas would most likely result in the client holding a much more risky portfolio. The total risk of a portfolio, or portfolio variance, is the combination of systematic risk and firm- specific risk. The systematic component depends on the sensitivity of the individual assets to market movements as measured by beta. Assuming the portfolio is well diversified, the number of assets will not affect the systematic risk component of portfolio variance. The portfolio beta depends on the individual security betas and the portfolio weights of those securities. On the other hand, the components of firm-specific risk (sometimes called nonsystematic risk) are not perfectly positively correlated with each other and, as more assets are added to the portfolio, those additional assets tend to reduce portfolio risk. Hence, increasing the number of securities in a portfolio reduces firm-specific risk. For example, a patent expiration for one company would not affect the other securities in the portfolio. An increase in oil prices might hurt an airline stock but aid an energy stock. As the number of randomly selected securities increases, the total risk (variance) of the portfolio approaches its systematic variance. 7-12

Chapter 08 - Index Models CHAPTER 8: INDEX MODELSPROBLEM SETS1. The advantage of the index model, compared to the Markowitz procedure, is the vastly reduced number of estimates required. In addition, the large number of estimates required for the Markowitz procedure can result in large aggregate estimation errors when implementing the procedure. The disadvantage of the index model arises from the model’s assumption that return residuals are uncorrelated. This assumption will be incorrect if the index used omits a significant risk factor.2. The trade-off entailed in departing from pure indexing in favor of an actively managed portfolio is between the probability (or possibility) of superior performance against the certainty of additional management fees.3. The answer to this question can be seen from the formulas for w 0 and w*. Other things held equal, w 0 is smaller the greater the residual variance of a candidate asset for inclusion in the portfolio. Further, we see that regardless of beta, when w 0 decreases, so does w*. Therefore, other things equal, the greater the residual variance of an asset, the smaller its position in the optimal risky portfolio. That is, increased firm-specific risk reduces the extent to which an active investor will be willing to depart from an indexed portfolio.4. The total risk premium equals: α + (β × market risk premium). We call alpha a “nonmarket” return premium because it is the portion of the return premium that is independent of market performance. The Sharpe ratio indicates that a higher alpha makes a security more desirable. Alpha, the numerator of the Sharpe ratio, is a fixed number that is not affected by the standard deviation of returns, the denominator of the Sharpe ratio. Hence, an increase in alpha increases the Sharpe ratio. Since the portfolio alpha is the portfolio-weighted average of the securities’ alphas, then, holding all other parameters fixed, an increase in a security’s alpha results in an increase in the portfolio Sharpe ratio. 8-1

Chapter 08 - Index Models5. a. To optimize this portfolio one would need: n = 60 estimates of means n = 60 estimates of variances n 2 − n = 1,770 estimates of covariances 2 Therefore, in total: n 2 + 3n = 1,890 estimates 2b. In a single index model: ri − rf = α i + β i (r M – rf ) + e i Equivalently, using excess returns: R i = α i + β i R M + e i The variance of the rate of return on each stock can be decomposed into the components:(l) The variance due to the common market factor: βi2 σ 2 M(2) The variance due to firm specific unanticipated events: σ2 (ei )In this model: Cov(ri , rj ) = βi β jσThe number of parameter estimates is: n = 60 estimates of the mean E(ri ) n = 60 estimates of the sensitivity coefficient βi n = 60 estimates of the firm-specific variance σ2(ei ) 1 estimate of the market mean E(rM ) 1 estimate of the market variance σ 2 MTherefore, in total, 182 estimates.Thus, the single index model reduces the total number of required parameterestimates from 1,890 to 182. In general, the number of parameter estimates isreduced from: n2 + 3n to (3n + 2) 2 8-2

Chapter 08 - Index Models6. a. The standard deviation of each individual stock is given by: σi = [βi2 σ 2 + σ2 (ei )]1/ 2 MSince βA = 0.8, βB = 1.2, σ(eA ) = 30%, σ(eB ) = 40%, and σM = 22%, we get: σA = (0.82 × 222 + 302 )1/2 = 34.78% σB = (1.22 × 222 + 402 )1/2 = 47.93%b. The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities: E(rP ) = wAE(rA ) + wBE(rB ) + wf rfwhere wA , wB , and wf are the portfolio weights for Stock A, Stock B, and T-bills, respectively.Substituting in the formula we get: E(rP ) = (0.30 × 13) + (0.45 × 18) + (0.25 × 8) = 14%The beta of a portfolio is similarly a weighted average of the betas of theindividual securities: βP = wAβA + wBβB + wf β fThe beta for T-bills (β f ) is zero. The beta for the portfolio is therefore: βP = (0.30 × 0.8) + (0.45 × 1.2) + 0 = 0.78The variance of this portfolio is: σ 2 = β 2 σ 2 + σ2 (eP ) P P Mwhere β 2 σ 2 is the systematic component and σ2 (eP ) is the nonsystematic P Mcomponent. Since the residuals (ei ) are uncorrelated, the non-systematic varianceis: σ2 (e P ) = w 2 σ 2 (e ) + w 2 σ 2 (e B ) + w 2 σ 2 (ef ) A B f A = (0.302 × 302 ) + (0.452 × 402 ) + (0.252 × 0) = 405where σ 2(eA ) and σ 2(eB ) are the firm-specific (nonsystematic) variances ofStocks A and B, and σ 2(e f ), the nonsystematic variance of T-bills, is zero. Theresidual standard deviation of the portfolio is thus: σ(e P ) = (405)1/2 = 20.12%The total variance of the portfolio is then: σ 2 = (0.782 × 222 ) + 405 = 699.47 PThe standard deviation is 26.45%. 8-3

Chapter 08 - Index Models7. a. The two figures depict the stocks’ security characteristic lines (SCL). Stock A has higher firm-specific risk because the deviations of the observations from the SCL are larger for Stock A than for Stock B. Deviations are measured by the vertical distance of each observation from the SCL.b. Beta is the slope of the SCL, which is the measure of systematic risk. The SCL for Stock B is steeper; hence Stock B’s systematic risk is greater.c. The R2 (or squared correlation coefficient) of the SCL is the ratio of the explained variance of the stock’s return to total variance, and the total variance is the sum of the explained variance plus the unexplained variance (the stock’s residual variance):R2 = β 2 σ 2 i M β 2 σ 2 + σ 2 (ei ) i MSince the explained variance for Stock B is greater than for Stock A (the explainedvariance is β 2 σ 2 , which is greater since its beta is higher), and its residual B Mvariance σ 2(eB ) is smaller, its R2 is higher than Stock A’s.d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas Stock B has a negative alpha; hence, Stock A’s alpha is larger.e. The correlation coefficient is simply the square root of R2, so Stock B’s correlation with the market is higher.8. a. Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firm-specific risk: 10.3% > 9.1% b. Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: 1.2 > 0.8 c. R2 measures the fraction of total variance of return explained by the market return. A’s R2 is larger than B’s: 0.576 > 0.436 d. Rewriting the SCL equation in terms of total return (r) rather than excess return (R): rA – rf = α + β(rM – rf ) ⇒ rA = α + rf (1 − β) + βr M The intercept is now equal to: α + rf (1 − β) = 1 + rf (l – 1.2) Since rf = 6%, the intercept would be: 1 – 1.2 = –0.2% 8-4

Chapter 08 - Index Models9. The standard deviation of each stock can be derived from the following equation for R2:R 2 = β 2 σ 2 = Explained variance i i M Total variance σ 2 iTherefore:σ 2 = β 2 σ 2 = 0.72 × 202 = 980 A A M 0.20 R 2 AσA = 31.30%For stock B:σ 2 = 1.22 × 202 = 4,800 B 0.12σB = 69.28%10. The systematic risk for A is:β 2 σ 2 = 0.702 × 202 = 196 A MThe firm-specific risk of A (the residual variance) is the difference between A’stotal risk and its systematic risk:980 – 196 = 784The systematic risk for B is:β 2 σ 2 = 1.202 × 202 = 576 B MB’s firm-specific risk (residual variance) is:4800 – 576 = 422411. The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated):Cov(rA , rB ) = β A β B σ 2 = 0.70 ×1.20 × 400 = 336 MThe correlation coefficient between the returns of A and B is:ρ AB = Cov(rA , rB ) = 336 = 0.155 σAσB 31.30 × 69.28 8-5

Chapter 08 - Index Models12. Note that the correlation is the square root of R2: ρ = R 2 Cov(rA,rM ) = ρσAσM = 0.201/2 × 31.30 × 20 = 280 Cov(rB,rM ) = ρσBσM = 0.121/2 × 69.28 × 20 = 48013. For portfolio P we can compute:σ P = [(0.62 × 980) + (0.42 × 4800) + (2 × 0.4 × 0.6 × 336]1/2 = [1282.08]1/2 = 35.81%β P = (0.6 × 0.7) + (0.4 × 1.2) = 0.90σ 2 (eP ) = σ 2 − β 2 σ 2 = 1282.08 − (0.902 × 400) = 958.08 P P MCov(rP,rM ) = β P σ 2 =0.90 × 400=360 MThis same result can also be attained using the covariances of the individual stocks withthe market:Cov(rP,rM ) = Cov(0.6rA + 0.4rB, rM ) = 0.6Cov(rA, rM ) + 0.4Cov(rB,rM ) = (0.6 × 280) + (0.4 × 480) = 36014. Note that the variance of T-bills is zero, and the covariance of T-bills with any asset is zero. Therefore, for portfolio Q: [ ]σQ 2 σ 2 2 2 1/ 2 = w P P + w M σ M + 2 × w P × w M × Cov(rP , rM ) [ ]= (0.52 ×1,282.08) + (0.32 × 400) + (2 × 0.5 × 0.3× 360) 1/ 2 = 21.55% βQ = w PβP + w MβM = (0.5 × 0.90) + (0.3×1) + 0 = 0.75 σ2 (eQ ) = σ 2 − β 2 σ 2 = 464.52 − (0.752 × 400) = 239.52 Q Q M Cov(rQ , rM ) = β Q σ 2 = 0.75 × 400 = 300 M15. a. Merrill Lynch adjusts beta by taking the sample estimate of beta and averaging it with 1.0, using the weights of 2/3 and 1/3, as follows: adjusted beta = [(2/3) × 1.24] + [(1/3) × 1.0] = 1.16b. If you use your current estimate of beta to be βt–1 = 1.24, then β t = 0.3 + (0.7 × 1.24) = 1.168 8-6

Chapter 08 - Index Models16. For Stock A: αA = rA − [rf + βA(rM −rf )] = 11 − [6 +0.8(12 − 6)] = 0.2% For stock B: αB = 14 − [6 + 1.5(12 −6)] = −1% Stock A would be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable.17. a. Alpha (α) Expected excess return α i = ri – [rf + βi(rM – rf ) ] E(ri ) – rf α = 20% – [8% + 1.3(16% – 8%)] = 1.6% 20% – 8% = 12% 18% – 8% = 10% A 17% – 8% = 9% 12% – 8% = 4% α = 18% – [8% + 1.8(16% – 8%)] = – 4.4% B α = 17% – [8% + 0.7(16% – 8%)] = 3.4% C α = 12% – [8% + 1.0(16% – 8%)] = – 4.0% D Stocks A and C have positive alphas, whereas stocks B and D have negative alphas. The residual variances are: σ2(eA ) = 582 = 3,364 σ2(eB) = 712 = 5,041 σ2(eC) = 602 = 3,600 σ2(eD) = 552 = 3,025b. To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio: α α / σ2(e) σ2(e) Σα / σ2(e) A 0.000476 –0.6142 B –0.000873 1.1265 C 0.000944 –1.2181 D –0.001322 1.7058 Total –0.000775 1.0000 Do not be concerned that the positive alpha stocks have negative weights and vice versa. We will see that the entire position in the active portfolio will be negative, returning everything to good order. 8-7

Chapter 08 - Index ModelsWith these weights, the forecast for the active portfolio is:α = [–0.6142 × 1.6] + [1.1265 × (– 4.4)] – [1.2181 × 3.4] + [1.7058 × (– 4.0)]= –16.90%β = [–0.6142 × 1.3] + [1.1265 × 1.8] – [1.2181 × 0.70] + [1.7058 × 1] = 2.08 The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks.σ2(e) = [(–0.6142)2 × 3364] + [1.12652 × 5041] + [(–1.2181)2 × 3600] + [1.70582 × 3025]= 21,809.6σ(e) = 147.68% Here, again, the levered position in stock B [with high σ2(e)] overcomes the diversification effect, and results in a high residual standard deviation. The optimal risky portfolio has a proportion w* in the active portfolio, computed as follows:w0 = α / σ2 (e) σ 2 = −16.90 / 21,809.6 = −0.05124 [E(rM ) − rf ] / M 8 / 232The negative position is justified for the reason stated earlier.The adjustment for beta is:w* = w 0 = − 0.05124 = −0.0486 1 + (1 − β)w 0 1 + (1 − 2.08)(−0.05124)Since w* is negative, the result is a positive position in stocks with positive alphasand a negative position in stocks with negative alphas. The position in the indexportfolio is:1 – (–0.0486) = 1.0486c. To calculate Sharpe’s measure for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpe’s measure for the market portfolio. The information ratio for the active portfolio is computed as follows:A = α /σ(e)= –16.90/147.68 = –0.1144A2 = 0.0131Hence, the square of Sharpe’s measure (S) of the optimized risky portfolio is:S2 = SM2 + A2 = 8 2 + 0.0131 = 0.1341 23 S = 0.3662 8-8

Chapter 08 - Index ModelsCompare this to the market’s Sharpe measure: SM = 8/23 = 0.3478The difference is: 0.0184Note that the only-moderate improvement in performance results from the fact thatonly a small position is taken in the active portfolio A because of its large residualvariance.d. To calculate the exact makeup of the complete portfolio, we first compute `the mean excess return of the optimal risky portfolio and its variance. The risky portfolio beta is given by:βP = wM + (wA × βA ) = 1.0486 + [(–0.0486) × 2.08] = 0.95E(RP) = α P + βPE(RM) = [(–0.0486) × (–16.90%)] + (0.95 × 8%) = 8.42%( )σ2 2 2P = β P σ M + σ2 (eP ) = (0.95 × 23)2 + (−0.04862 ) × 21,809.6 = 528.94σP = 23.00%Since A = 2.8, the optimal position in this portfolio is: y = 8.42 = 0.5685 0.01× 2.8 × 528.94In contrast, with a passive strategy: y = 8 = 0.5401 0.01× 2.8 × 232This is a difference of: 0.0284The final positions of the complete portfolio are:Bills 1 – 0.5685 = 43.15%M 0.5685 × l.0486 = 59.61%A 0.5685 × (–0.0486) × (–0.6142) = 1.70%B 0.5685 × (–0.0486) × 1.1265 = – 3.11%C 0.5685 × (–0.0486) × (–1.2181) = 3.37%D 0.5685 × (–0.0486) × 1.7058 = – 4.71% 100.00% [sum is subject to rounding error]Note that M may include positive proportions of stocks A through D. 8-9

Chapter 08 - Index Models18. a. If a manager is not allowed to sell short he will not include stocks with negative alphas in his portfolio, so he will consider only A and C: α σ2(e) α α / σ2(e) σ2(e) Σα / σ2(e) A 1.6 3,364 0.000476 0.3352 C 3.4 3,600 0.000944 0.6648 0.001420 1.0000The forecast for the active portfolio is:α = (0.3352 × 1.6) + (0.6648 × 3.4) = 2.80%β = (0.3352 × 1.3) + (0.6648 × 0.7) = 0.90σ2(e) = (0.33522 × 3,364) + (0.66482 × 3,600) = 1,969.03σ(e) = 44.37%The weight in the active portfolio is:w0 = α / σ2 (e) = 2.80 /1,969.03 = 0.0940 8 / 232 E(R M ) / σ 2 MAdjusting for beta:w* = w 0 = 0.094 = 0.0931 1 + (1 − β)w 0 1 + [(1 − 0.90) × 0.094]The information ratio of the active portfolio is:A = α /σ(e) =2.80/44.37 = 0.0631Hence, the square of Sharpe’s measure is: S2 = (8/23)2 + 0.06312 = 0.1250Therefore: S = 0.3535The market’s Sharpe measure is: SM = 0.3478When short sales are allowed (Problem 18), the manager’s Sharpe measure ishigher (0.3662). The reduction in the Sharpe measure is the cost of the short salerestriction. 8-10

Chapter 08 - Index ModelsThe characteristics of the optimal risky portfolio are: βP = wM + wA × βA = (1 – 0.0931) + (0.0931 × 0.9) = 0.99 E(RP ) = αP + βPE(RM ) = (0.0931 × 2.8%) + (0.99 × 8%) = 8.18% σ 2 = β 2 σ 2 + σ2 (eP ) = (0.99 × 23)2 + (0.09312 ×1,969.03) = 535.54 P P M σP = 23.14%With A = 2.8, the optimal position in this portfolio is: y = 8.18 = 0.5455 0.01× 2.8 × 535.54The final positions in each asset are:Bills 1 – 0.5455 = 45.45%M 0.5455 × (1 − 0.0931) = 49.47%A 0.5455 × 0.0931 × 0.3352 = 1.70%C 0.5455 × 0.0931 × 0.6648 = 3.38% 100.00%b. The mean and variance of the optimized complete portfolios in the unconstrained and short-sales constrained cases, and for the passive strategy are: E(RC ) σ 2 0.5685 × 8.42 = 4.79 C 0.5455 × 8.18 = 4.46Unconstrained 0.5401 × 8.00 = 4.32 0.56852 × 528.94 = 170.95ConstrainedPassive 0.54552 × 535.54 = 159.36 0.54012 × 529.00 = 154.31The utility levels below are computed using the formula: E(rC ) − 0.005Aσ 2 CUnconstrained 8 + 4.79 – (0.005 × 2.8 × 170.95) = 10.40Constrained 8 + 4.46 – (0.005 × 2.8 × 159.36) = 10.23Passive 8 + 4.32 – (0.005 × 2.8 × 154.31) = 10.16 8-11

Chapter 08 - Index Models19. All alphas are reduced to 0.3 times their values in the original case. Therefore, the relative weights of each security in the active portfolio are unchanged, but the alpha of the active portfolio is only 0.3 times its previous value: 0.3 × −16.90% = −5.07% The investor will take a smaller position in the active portfolio. The optimal risky portfolio has a proportion w* in the active portfolio as follows:w0 = α / σ2 (e) = − 5.07 / 21,809.6 = −0.01537 8 / 232 E(rM − rf ) / σ 2 MThe negative position is justified for the reason given earlier.The adjustment for beta is:w* = w 0 = − 0.01537 = −0.0151 1 + (1 − β)w 0 1 + [(1 − 2.08) × (−0.01537)]Since w* is negative, the result is a positive position in stocks with positive alphas and anegative position in stocks with negative alphas. The position in the index portfolio is:1 – (–0.0151) = 1.0151To calculate Sharpe’s measure for the optimal risky portfolio we compute the informationratio for the active portfolio and Sharpe’s measure for the market portfolio. The informationratio of the active portfolio is 0.3 times its previous value: A = α /σ(e)= –5.07/147.68 = –0.0343 and A2 =0.00118Hence, the square of Sharpe’s measure of the optimized risky portfolio is: S2 = S2M + A2 = (8/23)2 + 0.00118 = 0.1222 S = 0.3495Compare this to the market’s Sharpe measure: SM = 8/23 = 0.3478The difference is: 0.0017Note that the reduction of the forecast alphas by a factor of 0.3 reduced the squaredinformation ratio and the improvement in the squared Sharpe ratio by a factor of: 0.32 = 0.0920. If each of the alpha forecasts is doubled, then the alpha of the active portfolio will also double. Other things equal, the information ratio (IR) of the active portfolio also doubles. The square of the Sharpe ratio for the optimized portfolio (S-square) equals the square of the Sharpe ratio for the market index (SM-square) plus the square of the information ratio. Since the information ratio has doubled, its square quadruples. Therefore: S-square = SM-square + (4 × IR) Compared to the previous S-square, the difference is: 3IR Now you can embark on the calculations to verify this result. 8-12

Chapter 08 - Index ModelsCFA PROBLEMS1. The regression results provide quantitative measures of return and risk based on monthly returns over the five-year period. β for ABC was 0.60, considerably less than the average stock’s β of 1.0. This indicates that, when the S&P 500 rose or fell by 1 percentage point, ABC’s return on average rose or fell by only 0.60 percentage point. Therefore, ABC’s systematic risk (or market risk) was low relative to the typical value for stocks. ABC’s alpha (the intercept of the regression) was –3.2%, indicating that when the market return was 0%, the average return on ABC was –3.2%. ABC’s unsystematic risk (or residual risk), as measured by σ(e), was 13.02%. For ABC, R2 was 0.35, indicating closeness of fit to the linear regression greater than the value for a typical stock. β for XYZ was somewhat higher, at 0.97, indicating XYZ’s return pattern was very similar to the β for the market index. Therefore, XYZ stock had average systematic risk for the period examined. Alpha for XYZ was positive and quite large, indicating a return of almost 7.3%, on average, for XYZ independent of market return. Residual risk was 21.45%, half again as much as ABC’s, indicating a wider scatter of observations around the regression line for XYZ. Correspondingly, the fit of the regression model was considerably less than that of ABC, consistent with an R2 of only 0.17. The effects of including one or the other of these stocks in a diversified portfolio may be quite different. If it can be assumed that both stocks’ betas will remain stable over time, then there is a large difference in systematic risk level. The betas obtained from the two brokerage houses may help the analyst draw inferences for the future. The three estimates of ABC’s β are similar, regardless of the sample period of the underlying data. The range of these estimates is 0.60 to 0.71, well below the market average β of 1.0. The three estimates of XYZ’s β vary significantly among the three sources, ranging as high as 1.45 for the weekly data over the most recent two years. One could infer that XYZ’s β for the future might be well above 1.0, meaning it might have somewhat greater systematic risk than was implied by the monthly regression for the five-year period. These stocks appear to have significantly different systematic risk characteristics. If these stocks are added to a diversified portfolio, XYZ will add more to total volatility. 8-13

Chapter 08 - Index Models2. The R2 of the regression is: 0.702 = 0.49 Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk.3. 9 = 3 + β (11 − 3) ⇒ β = 0.754. d.5. b. 8-14

Chapter 09 - The Capital Asset Pricing ModelCHAPTER 9: THE CAPITAL ASSET PRICING MODELPROBLEM SETS1. E(rP) = rf + β P [E(rM ) – rf ] 18 = 6 + β P(14 – 6) ⇒ β P = 12/8 = 1.52. If the security’s correlation coefficient with the market portfolio doubles (with all other variables such as variances unchanged), then beta, and therefore the risk premium, will also double. The current risk premium is: 14 – 6 = 8% The new risk premium would be 16%, and the new discount rate for the security would be: 16 + 6 = 22% If the stock pays a constant perpetual dividend, then we know from the original data that the dividend (D) must satisfy the equation for the present value of a perpetuity: Price = Dividend/Discount rate 50 = D/0.14 ⇒ D = 50 × 0.14 = $7.00 At the new discount rate of 22%, the stock would be worth: $7/0.22 = $31.82 The increase in stock risk has lowered its value by 36.36%.3. a. False. β = 0 implies E(r) = rf , not zero. b. False. Investors require a risk premium only for bearing systematic (undiversifiable or market) risk. Total volatility includes diversifiable risk. c. False. Your portfolio should be invested 75% in the market portfolio and 25% in T-bills. Then: βP = (0.75 × 1) + (0.25 × 0) = 0.754. The appropriate discount rate for the project is: rf + β[E(rM ) – rf ] = 8 + [1.8 × (16 – 8)] = 22.4%Using this discount rate:∑NPV = −$40 +10 $15 = −$40 + [$15 × Annuity factor (22.4%, 10 years)] = $18.09 t=1 1.224tThe internal rate of return (IRR) for the project is 35.73%. Recall from your introductoryfinance class that NPV is positive if IRR > discount rate (or, equivalently, hurdle rate).The highest value that beta can take before the hurdle rate exceeds the IRR isdetermined by:35.73 = 8 + β(16 – 8) ⇒ β = 27.73/8 = 3.47 9-1

Chapter 09 - The Capital Asset Pricing Model5. a. Call the aggressive stock A and the defensive stock D. Beta is the sensitivity of the stock’s return to the market return, i.e., the change in the stock return per unit change in the market return. Therefore, we compute each stock’s beta by calculating the difference in its return across the two scenarios divided by the difference in the market return:βA = − 2 − 38 = 2.00 5 − 25βD = 6 −12 = 0.30 5 − 25b. With the two scenarios equally likely, the expected return is an average of the two possible outcomes: E(rA ) = 0.5 × (–2 + 38) = 18% E(rD ) = 0.5 × (6 + 12) = 9%c. The SML is determined by the market expected return of [0.5(25 + 5)] = 15%, with a beta of 1, and the T-bill return of 6% with a beta of zero. See the following graph.The equation for the security market line is: E(r) = 6 + β(15 – 6) 9-2

Chapter 09 - The Capital Asset Pricing Model d. Based on its risk, the aggressive stock has a required expected return of: E(rA ) = 6 + 2.0(15 – 6) = 24% The analyst’s forecast of expected return is only 18%. Thus the stock’s alpha is: α A = actually expected return – required return (given risk) = 18% – 24% = –6% Similarly, the required return for the defensive stock is: E(rD) = 6 + 0.3(15 – 6) = 8.7% The analyst’s forecast of expected return for D is 9%, and hence, the stock has a positive alpha: α D = actually expected return – required return (given risk) = 9 – 8.7 = +0.3% The points for each stock plot on the graph as indicated above. e. The hurdle rate is determined by the project beta (0.3), not the firm’s beta. The correct discount rate is 8.7%, the fair rate of return for stock D.6. Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for Portfolio A is lower than the expected return for Portfolio B. Thus, these two portfolios cannot exist in equilibrium.7. Possible. If the CAPM is valid, the expected rate of return compensates only for systematic (market) risk, represented by beta, rather than for the standard deviation, which includes nonsystematic risk. Thus, Portfolio A’s lower rate of return can be paired with a higher standard deviation, as long as A’s beta is less than B’s.8. Not possible. The reward-to-variability ratio for Portfolio A is better than that of the market. This scenario is impossible according to the CAPM because the CAPM predicts that the market is the most efficient portfolio. Using the numbers supplied:SA = 16 −10 = 0.5 12SM = 18 −10 = 0.33 24Portfolio A provides a better risk-reward tradeoff than the market portfolio.9. Not possible. Portfolio A clearly dominates the market portfolio. Portfolio A has both a lower standard deviation and a higher expected return. 9-3

Chapter 09 - The Capital Asset Pricing Model10. Not possible. The SML for this scenario is: E(r) = 10 + β(18 – 10) Portfolios with beta equal to 1.5 have an expected return equal to: E(r) = 10 + [1.5 × (18 – 10)] = 22% The expected return for Portfolio A is 16%; that is, Portfolio A plots below the SML (α A = –6%), and hence, is an overpriced portfolio. This is inconsistent with the CAPM.11. Not possible. The SML is the same as in Problem 10. Here, Portfolio A’s required return is: 10 + (0.9 × 8) = 17.2% This is greater than 16%. Portfolio A is overpriced with a negative alpha: α A = –1.2%12. Possible. The CML is the same as in Problem 8. Portfolio A plots below the CML, as any asset is expected to. This scenario is not inconsistent with the CAPM.13. Since the stock’s beta is equal to 1.2, its expected rate of return is:6 + [1.2 × (16 – 6)] = 18%E(r) = D1 + P1 − P0 P00.18 = 6 + P1 − 50 ⇒ P1 = $53 5014. The series of $1,000 payments is a perpetuity. If beta is 0.5, the cash flow should be discounted at the rate: 6 + [0.5 × (16 – 6)] = 11% PV = $1,000/0.11 = $9,090.91 If, however, beta is equal to 1, then the investment should yield 16%, and the price paid for the firm should be: PV = $1,000/0.16 = $6,250 The difference, $2,840.91, is the amount you will overpay if you erroneously assume that beta is 0.5 rather than 1.15. Using the SML: 4 = 6 + β(16 – 6) ⇒ β = –2/10 = –0.2 9-4

Chapter 09 - The Capital Asset Pricing Model16. r1 = 19%; r2 = 16%; β1 = 1.5; β2 = 1 a. To determine which investor was a better selector of individual stocks we look at abnormal return, which is the ex-post alpha; that is, the abnormal return is the difference between the actual return and that predicted by the SML. Without information about the parameters of this equation (risk-free rate and market rate of return) we cannot determine which investor was more accurate. b. If rf = 6% and rM = 14%, then (using the notation alpha for the abnormal return): α 1 = 19 – [6 + 1.5(14 – 6)] = 19 – 18 = 1% α 2 = 16 – [6 + 1(14 – 6)] =16 – 14 = 2% Here, the second investor has the larger abnormal return and thus appears to be the superior stock selector. By making better predictions, the second investor appears to have tilted his portfolio toward underpriced stocks. c. If rf = 3% and rM = 15%, then: α 1 =19 – [3 + 1.5(15 – 3)] = 19 – 21 = –2% α 2 = 16 – [3+ 1(15 – 3)] = 16 – 15 = 1% Here, not only does the second investor appear to be the superior stock selector, but the first investor’s predictions appear valueless (or worse).17. a. Since the market portfolio, by definition, has a beta of 1, its expected rate of return is 12%. b. β = 0 means no systematic risk. Hence, the stock’s expected rate of return in market equilibrium is the risk-free rate, 5%. c. Using the SML, the fair expected rate of return for a stock with β = –0.5 is: E(r) = 5 + [(–0.5)(12 – 5)] = 1.5% The actually expected rate of return, using the expected price and dividend for next year is: E(r) = [($41 + $1)/40] – 1 = 0.10 = 10% Because the actually expected return exceeds the fair return, the stock is underpriced.18. In the zero-beta CAPM the zero-beta portfolio replaces the risk-free rate, and thus: E(r) = 8 + 0.6(17 – 8) = 13.4% 9-5

Chapter 09 - The Capital Asset Pricing Model19. a. E(rP) = rf + β P [E(rM ) – rf ] = 5% + 0.8 (15% − 5%) = 13% α = 14% − 13% = 1% You should invest in this fund because alpha is positive.b. The passive portfolio with the same beta as the fund should be invested 80% in the market-index portfolio and 20% in the money market account. For this portfolio: E(rP) = (0.8 × 15%) + (0.2 × 5%) = 13% 14% − 13% = 1% = α20. a. We would incorporate liquidity into the CCAPM in a manner analogous to the way in which liquidity is incorporated into the conventional CAPM. In the latter case, in addition to the market risk premium, expected return is also dependent on the expected cost of illiquidity and three liquidity-related betas which measure the sensitivity of: (1) the security’s illiquidity to market illiquidity; (2) the security’s return to market illiquidity; and, (3) the security’s illiquidity to the market return. A similar approach can be used for the CCAPM, except that the liquidity betas would be measured relative to consumption growth rather than the usual market index.b. As in part (a), non-traded assets would be incorporated into the CCAPM in a fashion similar to that described above, and, as in part (a), we would replace the market portfolio with consumption growth. However, the issue of liquidity is more acute with non traded-assets such as privately-held businesses and labor income. While ownership of a privately-held business is analogous to ownership of an illiquid stock, we should expect a greater degree of illiquidity for the typical private business. As long as the owner of a privately-held business is satisfied with the dividends paid out from the business, then the lack of liquidity is not an issue. However, if the owner seeks to realize income greater than the business can pay out, then selling ownership, in full or in part, typically entails a substantial liquidity discount. The correction for illiquidity in this case should be treated as suggested in part (a). The same general considerations apply to labor income, although it is probable that the lack of liquidity for labor income has an even greater impact on security market equilibrium values. Labor income has a major impact on portfolio decisions. While it is possible to borrow against labor income to some degree, and some of the risk associated with labor income can be ameliorated with insurance, it is plausible that the liquidity betas of consumption streams are quite significant, as the need to borrow against labor income is likely cyclical. 9-6

Chapter 09 - The Capital Asset Pricing ModelCFA PROBLEMS1. a. Agree; Regan’s conclusion is correct. By definition, the market portfolio lies on the capital market line (CML). Under the assumptions of capital market theory, all portfolios on the CML dominate, in a risk-return sense, portfolios that lie on the Markowitz efficient frontier because, given that leverage is allowed, the CML creates a portfolio possibility line that is higher than all points on the efficient frontier except for the market portfolio, which is Rainbow’s portfolio. Because Eagle’s portfolio lies on the Markowitz efficient frontier at a point other than the market portfolio, Rainbow’s portfolio dominates Eagle’s portfolio. b. Nonsystematic risk is the unique risk of individual stocks in a portfolio that is diversified away by holding a well-diversified portfolio. Total risk is composed of systematic (market) risk and nonsystematic (firm-specific) risk. Disagree; Wilson’s remark is incorrect. Because both portfolios lie on the Markowitz efficient frontier, neither Eagle nor Rainbow has any nonsystematic risk. Therefore, nonsystematic risk does not explain the different expected returns. The determining factor is that Rainbow lies on the (straight) line (the CML) connecting the risk-free asset and the market portfolio (Rainbow), at the point of tangency to the Markowitz efficient frontier having the highest return per unit of risk. Wilson’s remark is also countered by the fact that, since nonsystematic risk can be eliminated by diversification, the expected return for bearing nonsystematic is zero. This is a result of the fact that well-diversified investors bid up the price of every asset to the point where only systematic risk earns a positive return (nonsystematic risk earns no return).2. E(r) = rf + β × [E(r M ) − rf ] Fuhrman Labs: E(r) = 5 + 1.5 × [11.5 − 5.0] = 14.75% Garten Testing: E(r) = 5 + 0.8 × [11.5 − 5.0] = 10.20% If the forecast rate of return is less than (greater than) the required rate of return, then the security is overvalued (undervalued). Fuhrman Labs: Forecast return – Required return = 13.25% − 14.75% = −1.50% Garten Testing: Forecast return – Required return = 11.25% − 10.20% = 1.05% Therefore, Fuhrman Labs is overvalued and Garten Testing is undervalued.3. a.4. d. From CAPM, the fair expected return = 8 + 1.25(15 − 8) = 16.75% Actually expected return = 17% α = 17 − 16.75 = 0.25% 9-7

Chapter 09 - The Capital Asset Pricing Model5. d.6. c.7. d.8. d. [You need to know the risk-free rate]9. d. [You need to know the risk-free rate]10. Under the CAPM, the only risk that investors are compensated for bearing is the risk that cannot be diversified away (systematic risk). Because systematic risk (measured by beta) is equal to 1.0 for both portfolios, an investor would expect the same rate of return from both portfolios A and B. Moreover, since both portfolios are well diversified, it doesn’t matter if the specific risk of the individual securities is high or low. The firm- specific risk has been diversified away for both portfolios.11. a. McKay should borrow funds and invest those funds proportionately in Murray’s existing portfolio (i.e., buy more risky assets on margin). In addition to increased expected return, the alternative portfolio on the capital market line will also have increased risk, which is caused by the higher proportion of risky assets in the total portfolio.b. McKay should substitute low beta stocks for high beta stocks in order to reduce the overall beta of York’s portfolio. By reducing the overall portfolio beta, McKay will reduce the systematic risk of the portfolio, and therefore reduce its volatility relative to the market. The security market line (SML) suggests such action (i.e., moving down the SML), even though reducing beta may result in a slight loss of portfolio efficiency unless full diversification is maintained. York’s primary objective, however, is not to maintain efficiency, but to reduce risk exposure; reducing portfolio beta meets that objective. Because York does not want to engage in borrowing or lending, McKay cannot reduce risk by selling equities and using the proceeds to buy risk-free assets (i.e., lending part of the portfolio). 9-8

Chapter 09 - The Capital Asset Pricing Model12. a. Expected Return Alpha 5% + 0.8(14% − 5%) = 12.2% 14.0% − 12.2% = 1.8% Stock X 5% + 1.5(14% − 5%) = 18.5% 17.0% − 18.5% = −1.5% Stock Yb. i. Kay should recommend Stock X because of its positive alpha, compared to Stock Y, which has a negative alpha. In graphical terms, the expected return/risk profile for Stock X plots above the security market line (SML), while the profile for Stock Y plots below the SML. Also, depending on the individual risk preferences of Kay’s clients, the lower beta for Stock X may have a beneficial effect on overall portfolio risk.ii. Kay should recommend Stock Y because it has higher forecasted return andlower standard deviation than Stock X. The respective Sharpe ratios for Stocks Xand Y and the market index are:Stock X: (14% − 5%)/36% = 0.25Stock Y: (17% − 5%)/25% = 0.48Market index: (14% − 5%)/15% = 0.60The market index has an even more attractive Sharpe ratio than either of theindividual stocks, but, given the choice between Stock X and Stock Y, Stock Y isthe superior alternative.When a stock is held as a single stock portfolio, standard deviation is the relevantrisk measure. For such a portfolio, beta as a risk measure is irrelevant. Althoughholding a single asset is not a typically recommended investment strategy, someinvestors may hold what is essentially a single-asset portfolio when they hold thestock of their employer company. For such investors, the relevance of standarddeviation versus beta is an important issue. 9-9

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND RETURNPROBLEM SETS1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times the respective sensitivity coefficient: revised estimate = 12% + [(1 × 2%) + (0.5 × 3%)] = 15.5%2. The APT factors must correlate with major sources of uncertainty, i.e., sources of uncertainty that are of concern to many investors. Researchers should investigate factors that correlate with uncertainty in consumption and investment opportunities. GDP, the inflation rate, and interest rates are among the factors that can be expected to determine risk premiums. In particular, industrial production (IP) is a good indicator of changes in the business cycle. Thus, IP is a candidate for a factor that is highly correlated with uncertainties that have to do with investment and consumption opportunities in the economy.3. Any pattern of returns can be “explained” if we are free to choose an indefinitely large number of explanatory factors. If a theory of asset pricing is to have value, it must explain returns using a reasonably limited number of explanatory variables (i.e., systematic factors).4. Equation 10.9 applies here: E(rp) = rf + βP1 [E(r1 ) − rf ] + βP2 [E(r2) – rf ] We need to find the risk premium (RP) for each of the two factors: RP1 = [E(r1) − rf ] and RP2 = [E(r2) − rf ] In order to do so, we solve the following system of two equations with two unknowns: 31 = 6 + (1.5 × RP1) + (2.0 × RP2) 27 = 6 + (2.2 × RP1) + [(–0.2) × RP2] The solution to this set of equations is: RP1 = 10% and RP2 = 5% Thus, the expected return-beta relationship is: E(rP) = 6% + (βP1 × 10%) + (βP2 × 5%) 10-1

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return5. The expected return for Portfolio F equals the risk-free rate since its beta equals 0. For Portfolio A, the ratio of risk premium to beta is: (12 − 6)/1.2 = 5 For Portfolio E, the ratio is lower at: (8 – 6)/0.6 = 3.33 This implies that an arbitrage opportunity exists. For instance, you can create a Portfolio G with beta equal to 0.6 (the same as E’s) by combining Portfolio A and Portfolio F in equal weights. The expected return and beta for Portfolio G are then: E(rG ) = (0.5 × 12%) + (0.5 × 6%) = 9% βG = (0.5 × 1.2) + (0.5 × 0) = 0.6 Comparing Portfolio G to Portfolio E, G has the same beta and higher return. Therefore, an arbitrage opportunity exists by buying Portfolio G and selling an equal amount of Portfolio E. The profit for this arbitrage will be: rG – rE =[9% + (0.6 × F)] − [8% + (0.6 × F)] = 1% That is, 1% of the funds (long or short) in each portfolio.6. Substituting the portfolio returns and betas in the expected return-beta relationship, we obtain two equations with two unknowns, the risk-free rate (rf ) and the factor risk premium (RP): 12 = rf + (1.2 × RP) 9 = rf + (0.8 × RP) Solving these equations, we obtain: rf = 3% and RP = 7.5%7. a. Shorting an equally-weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the ten positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio. Denoting the systematic market factor as RM , the expected dollar return is (noting that the expectation of non-systematic risk, e, is zero): $1,000,000 × [0.02 + (1.0 × RM )] − $1,000,000 × [(–0.02) + (1.0 × RM )] = $1,000,000 × 0.04 = $40,000 The sensitivity of the payoff of this portfolio to the market factor is zero because the exposures of the positive alpha and negative alpha stocks cancel out. (Notice that the terms involving RM sum to zero.) Thus, the systematic component of total risk is also zero. The variance of the analyst’s profit is not zero, however, since this portfolio is not well diversified. 10-2

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return For n = 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will have a $100,000 position (either long or short) in each stock. Net market exposure is zero, but firm-specific risk has not been fully diversified. The variance of dollar returns from the positions in the 20 stocks is: 20 × [(100,000 × 0.30)2 ] = 18,000,000,000 The standard deviation of dollar returns is $134,164. b. If n = 50 stocks (25 stocks long and 25 stocks short), the investor will have a $40,000 position in each stock, and the variance of dollar returns is: 50 × [(40,000 × 0.30)2 ] = 7,200,000,000 The standard deviation of dollar returns is $84,853. Similarly, if n = 100 stocks (50 stocks long and 50 stocks short), the investor will have a $20,000 position in each stock, and the variance of dollar returns is: 100 × [(20,000 × 0.30)2 ] = 3,600,000,000 The standard deviation of dollar returns is $60,000. Notice that, when the number of stocks increases by a factor of 5 (i.e., from 20 to 100), standard deviation decreases by a factor of 5 = 2.23607 (from $134,164 to $60,000).8. a. σ2 = β 2 σ 2 + σ2 (e) M σ 2 = (0.82 × 202 ) + 252 = 881 A σ 2 = (1.0 2 × 202 ) + 102 = 500 B σ 2 = (1.2 2 × 202 ) + 202 = 976 Cb. If there are an infinite number of assets with identical characteristics, then a well- diversified portfolio of each type will have only systematic risk since the non- systematic risk will approach zero with large n. The mean will equal that of the individual (identical) stocks.c. There is no arbitrage opportunity because the well-diversified portfolios all plot on the security market line (SML). Because they are fairly priced, there is no arbitrage. 10-3

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return9. a. A long position in a portfolio (P) comprised of Portfolios A and B will offer an expected return-beta tradeoff lying on a straight line between points A and B. Therefore, we can choose weights such that βP = βC but with expected return higher than that of Portfolio C. Hence, combining P with a short position in C will create an arbitrage portfolio with zero investment, zero beta, and positive rate of return. b. The argument in part (a) leads to the proposition that the coefficient of β2 must be zero in order to preclude arbitrage opportunities.10. a. E(r) = 6 + (1.2 × 6) + (0.5 × 8) + (0.3 × 3) = 18.1% b. Surprises in the macroeconomic factors will result in surprises in the return of the stock: Unexpected return from macro factors = [1.2(4 – 5)] + [0.5(6 – 3)] + [0.3(0 – 2)] = –0.3% E (r) =18.1% − 0.3% = 17.8%11. The APT required (i.e., equilibrium) rate of return on the stock based on rf and the factor betas is: required E(r) = 6 + (1 × 6) + (0.5 × 2) + (0.75 × 4) = 16% According to the equation for the return on the stock, the actually expected return on the stock is 15% (because the expected surprises on all factors are zero by definition). Because the actually expected return based on risk is less than the equilibrium return, we conclude that the stock is overpriced.12. The first two factors seem promising with respect to the likely impact on the firm’s cost of capital. Both are macro factors that would elicit hedging demands across broad sectors of investors. The third factor, while important to Pork Products, is a poor choice for a multifactor SML because the price of hogs is of minor importance to most investors and is therefore highly unlikely to be a priced risk factor. Better choices would focus on variables that investors in aggregate might find more important to their welfare. Examples include: inflation uncertainty, short-term interest-rate risk, energy price risk, or exchange rate risk. The important point here is that, in specifying a multifactor SML, we not confuse risk factors that are important to a particular investor with factors that are important to investors in general; only the latter are likely to command a risk premium in the capital markets. 10-4

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return13. The maximum residual variance is tied to the number of securities (n) in the portfolio because, as we increase the number of securities, we are more likely to encounter securities with larger residual variances. The starting point is to determine the practical limit on the portfolio residual standard deviation, σ(eP), that still qualifies as a ‘well- diversified portfolio.’ A reasonable approach is to compare σ2(eP) to the market variance, or equivalently, to compare σ(eP) to the market standard deviation. Suppose we do not allow σ(eP) to exceed pσM, where p is a small decimal fraction, for example, 0.05; then, the smaller the value we choose for p, the more stringent our criterion for defining how diversified a ‘well-diversified’ portfolio must be. Now construct a portfolio of n securities with weights w1, w2,…,wn, so that Σwi =1. The portfolio residual variance is: σ2(eP) = Σw12σ2(ei)To meet our practical definition of sufficiently diversified, we require this residualvariance to be less than (pσM)2. A sure and simple way to proceed is to assume the worst,that is, assume that the residual variance of each security is the highest possible valueallowed under the assumptions of the problem: σ2(ei) = nσ2MIn that case: σ2(eP ) = Σwi2 nσ 2 MNow apply the constraint: Σwi2 n σ 2 ≤ (pσM)2 MThis requires that: nΣwi2 ≤ p2Or, equivalently, that: Σwi2 ≤ p2/nA relatively easy way to generate a set of well-diversified portfolios is to use portfolioweights that follow a geometric progression, since the computations then become relativelystraightforward. Choose w1 and a common factor q for the geometric progression such thatq < 1. Therefore, the weight on each stock is a fraction q of the weight on the previousstock in the series. Then the sum of n terms is: Σwi = w1(1– qn)/(1– q) = 1or: w1 = (1– q)/(1– qn)The sum of the n squared weights is similarly obtained from w12 and a commongeometric progression factor of q2. Therefore: Σwi2 = w12(1– q2n)/(1– q 2)Substituting for w1 from above, we obtain: Σwi2 = [(1– q)2/(1– qn)2] × [(1– q2n)/(1– q 2)] 10-5

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return For sufficient diversification, we choose q so that: Σwi2 ≤ p2/n For example, continue to assume that p = 0.05 and n = 1,000. If we choose q = 0.9973, then we will satisfy the required condition. At this value for q: w1 = 0.0029 and wn = 0.0029 × 0.99731,000 In this case, w1 is about 15 times wn. Despite this significant departure from equal weighting, this portfolio is nevertheless well diversified. Any value of q between 0.9973 and 1.0 results in a well-diversified portfolio. As q gets closer to 1, the portfolio approaches equal weighting.14. a. Assume a single-factor economy, with a factor risk premium EM and a (large) set of well-diversified portfolios with beta βP. Suppose we create a portfolio Z by allocating the portion w to portfolio P and (1 – w) to the market portfolio M. The rate of return on portfolio Z is: RZ = (w × RP) + [(1 – w) × RM] Portfolio Z is riskless if we choose w so that βZ = 0. This requires that: βZ = (w × βP) + [(1 – w) × 1] = 0 ⇒ w = 1/(1 – βP) and (1 – w) = –βP/(1 – βP) Substitute this value for w in the expression for RZ: RZ = {[1/(1 – βP)] × RP} – {[βP/(1 – βP)] × RM} Since βZ = 0, then, in order to avoid arbitrage, RZ must be zero. This implies that: RP = βP × RM Taking expectations we have: EP = βP × EM This is the SML for well-diversified portfolios.b. The same argument can be used to show that, in a three-factor model with factor risk premiums EM, E1 and E2, in order to avoid arbitrage, we must have: EP = (βPM × EM) + (βP1 × E1) + (βP2 × E2) This is the SML for a three-factor economy. 10-6

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return15. a. The Fama-French (FF) three-factor model holds that one of the factors driving returns is firm size. An index with returns highly correlated with firm size (i.e., firm capitalization) that captures this factor is SMB (Small Minus Big), the return for a portfolio of small stocks in excess of the return for a portfolio of large stocks. The returns for a small firm will be positively correlated with SMB. Moreover, the smaller the firm, the greater its residual from the other two factors, the market portfolio and the HML portfolio, which is the return for a portfolio of high book- to-market stocks in excess of the return for a portfolio of low book-to-market stocks. Hence, the ratio of the variance of this residual to the variance of the return on SMB will be larger and, together with the higher correlation, results in a high beta on the SMB factor.b. This question appears to point to a flaw in the FF model. The model predicts that firm size affects average returns, so that, if two firms merge into a larger firm, then the FF model predicts lower average returns for the merged firm. However, there seems to be no reason for the merged firm to underperform the returns of the component companies, assuming that the component firms were unrelated and that they will now be operated independently. We might therefore expect that the performance of the merged firm would be the same as the performance of a portfolio of the originally independent firms, but the FF model predicts that the increased firm size will result in lower average returns. Therefore, the question revolves around the behavior of returns for a portfolio of small firms, compared to the return for larger firms that result from merging those small firms into larger ones. Had past mergers of small firms into larger firms resulted, on average, in no change in the resultant larger firms’ stock return characteristics (compared to the portfolio of stocks of the merged firms), the size factor in the FF model would have failed. Perhaps the reason the size factor seems to help explain stock returns is that, when small firms become large, the characteristics of their fortunes (and hence their stock returns) change in a significant way. Put differently, stocks of large firms that result from a merger of smaller firms appear empirically to behave differently from portfolios of the smaller component firms. Specifically, the FF model predicts that the large firm will have a smaller risk premium. Notice that this development is not necessarily a bad thing for the stockholders of the smaller firms that merge. The lower risk premium may be due, in part, to the increase in value of the larger firm relative to the merged firms. 10-7

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and ReturnCFA PROBLEMS1. a. This statement is incorrect. The CAPM requires a mean-variance efficient market portfolio, but APT does not. b. This statement is incorrect. The CAPM assumes normally distributed security returns, but APT does not. c. This statement is correct.2. b. Since Portfolio X has β = 1.0, then X is the market portfolio and E(RM) =16%. Using E(RM ) = 16% and rf = 8%, the expected return for portfolio Y is not consistent.3. d.4. c.5. d.6. c. Investors will take on as large a position as possible only if the mispricing opportunity is an arbitrage. Otherwise, considerations of risk and diversification will limit the position they attempt to take in the mispriced security.7. d.8. d. 10-8

Chapter 11 - The Efficient Market Hypothesis CHAPTER 11: THE EFFICIENT MARKET HYPOTHESISPROBLEM SETS1. The correlation coefficient between stock returns for two non-overlapping periods should be zero. If not, one could use returns from one period 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.3. Expected rates of return differ because of differential risk premiums.4. No. The value of dividend predictability would be already reflected in the stock price.5. Over the long haul, there is an expected upward drift in stock prices based on their fair expected rates of return. The fair expected return over any single day is very small (e.g., 12% per year is only about 0.03% per day), so that on any day the price is virtually equally likely to rise or fall. However, over longer periods, the small expected daily returns accumulate, and upward moves are indeed more likely than downward ones.6. c. This is a predictable pattern in returns which should not occur if the weak-form EMH is valid.7. c. This is a classic filter rule which should not produce superior returns in an efficient market.8. b. This is the definition of an efficient market.9. c. The P/E ratio is public information and should not be predictive of abnormal security returns. 11-1

Chapter 11 - The Efficient Market Hypothesis10. 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).11. The question regarding market efficiency is whether investors can earn abnormal risk- adjusted profits. If the stock price run-up occurs when only insiders are aware of the coming dividend increase, then it is a violation of strong-form, but not semistrong-form, efficiency. If the public already knows of the increase, then it is a violation of semistrong-form efficiency.12. While positive beta stocks respond well to favorable new information about the economy’s progress through the business cycle, they should not show abnormal returns around already anticipated events. If a recovery, for example, is already anticipated, the actual recovery is not news. The stock price should already reflect the coming recovery.13. a. Consistent. Based on pure luck, half of all managers should beat the market in any year. b. Inconsistent. This would be the basis of an “easy money” rule: simply invest with last year's best managers. c. Consistent. In contrast to predictable returns, predictable volatility does not convey a means to earn abnormal returns. d. Inconsistent. The abnormal performance ought to occur in January when earnings are announced. e. Inconsistent. Reversals offer a means to earn easy money: just buy last week’s losers.14. The return on the market is 8%. Therefore, the forecast monthly return for GM is: 0.10% + (1.1 × 8%) = 8.9% GM’s actual return was 7%, so the abnormal return was –1.9%. 11-2

Chapter 11 - The Efficient Market Hypothesis15. a. Based on broad market trends, the CAPM indicates that AmbChaser stock should have increased by: 1.0% + 2.0(1.5% – 1.0%) = 2.0% Its firm-specific (nonsystematic) return due to the lawsuit is $1 million per $100 million initial equity, or 1%. Therefore, the total return should be 3%. (It is assumed here that the outcome of the lawsuit had a zero expected value.)b. If the settlement was expected to be $2 million, then the actual settlement was a “$1 million disappointment,” and so the firm-specific return would be –1%, for a total return of 2% – 1% = 1%.16. Given market performance, predicted returns on the two stocks would be: Apex: 0.2% + (1.4 × 3%) = 4.4% Bpex: –0.1% + (0.6 × 3%) = 1.7% Apex underperformed this prediction; Bpex outperformed the prediction. We conclude that Bpex won the lawsuit.17. a. E(rM ) = 12%, rf = 4% and β = 0.5 Therefore, the expected rate of return is: 4% + 0.5(12% – 4%) = 8% If the stock is fairly priced, then E(r) = 8%.b. If rM falls short of your expectation by 2% (that is, 10% – 12%) then you would expect the return for Changing Fortunes Industries to fall short of your original expectation by: β × 2% = 1% Therefore, you would forecast a “revised” expectation for Changing Fortunes of: 8% – 1% = 7%c. Given a market return of 10%, you would forecast a return for Changing Fortunes of 7%. The actual return is 10%. Therefore, the surprise due to firm-specific factors is 10% – 7% = 3% which we attribute to the settlement. Because the firm is initially worth $100 million, the surprise amount of the settlement is 3% of $100 million, or $3 million, implying that the prior expectation for the settlement was only $2 million.18. Implicit in the dollar-cost averaging strategy is the notion that stock prices fluctuate around a “normal” level. Otherwise, there is no meaning to statements such as: “when the price is high.” How do we know, for example, whether a price of $25 today will turn out to be viewed as high or low compared to the stock price six months from now? 11-3

Chapter 11 - The Efficient Market Hypothesis19. The market responds positively to new news. If the eventual recovery is anticipated, then the recovery is already reflected in stock prices. Only a better-than-expected recovery should affect stock prices.20. Buy. In your view, the firm is not as bad as everyone else believes it to be. Therefore, you view the firm as undervalued by the market. You are less pessimistic about the firm’s prospects than the beliefs built into the stock price.21. Here we need a two-factor model relating Ford’s return to those of both the broad market and the auto industry. If we call r I the industry return, then we would first estimate parameters a, b ,c in the following regression: rFORD = a + brM + cr I + e Given these estimates we would calculate Ford’s firm-specific return as: rFORD − [a + brM + cr I + e] This estimate of firm-specific news would measure the market’s assessment of the potential profitability of Ford’s new model.22. The market may have anticipated even greater earnings. Compared to prior expectations, the announcement was a disappointment.23. The negative abnormal returns (downward drift in CAR) just prior to stock purchases suggest that insiders deferred their purchases until after bad news was released to the public. This is evidence of valuable inside information. The positive abnormal returns after purchase suggest insider purchases in anticipation of good news. The analysis is symmetric for insider sales.CFA PROBLEMS1. b. Semi-strong form efficiency implies that market prices reflect all publicly available information concerning past trading history as well as fundamental aspects of the firm.2. a. The full price adjustment should occur just as the news about the dividend becomes publicly available.3. d. If low P/E stocks tend to have positive abnormal returns, this would represent an unexploited profit opportunity that would provide evidence that investors are not using all available information to make profitable investments. 11-4

Chapter 11 - The Efficient Market Hypothesis4. c. In an efficient market, no securities are consistently overpriced or underpriced. While some securities will turn out after any investment period to have provided positive alphas (i.e., risk-adjusted abnormal returns) and some negative alphas, these past returns are not predictive of future returns.5. c. A random walk implies that stock price changes are unpredictable, using past price changes or any other data.6. d. A gradual adjustment to fundamental values would allow for the use of strategies based on past price movements in order to generate abnormal profits.7. a.8. a. Some empirical evidence that supports the EMH: (i) professional money managers do not typically earn higher returns than comparable risk, passive index strategies; (ii) event studies typically show that stocks respond immediately to the public release of relevant news; (iii) most tests of technical analysis find that it is difficult to identify price trends that can be exploited to earn superior risk-adjusted investment returns. b. Some evidence that is difficult to reconcile with the EMH concerns simple portfolio strategies that apparently would have provided high risk-adjusted returns in the past. Some examples of portfolios with attractive historical returns: (i) low P/E stocks; (ii) high book-to-market ratio stocks; (iii) small firms in January; (iv) firms with very poor stock price performance in the last few months. Other evidence concerns post-earnings-announcement stock price drift and intermediate-term price momentum. c. An investor might choose not to index even if markets are efficient because he or she may want to tailor a portfolio to specific tax considerations or to specific risk management issues, for example, the need to hedge (or at least not add to) exposure to a particular source of risk (e.g., industry exposure). 11-5

Chapter 11 - The Efficient Market Hypothesis9. a. The efficient market hypothesis (EMH) states that a market is efficient if security prices immediately and fully reflect all available relevant information. If the market fully reflects information, the knowledge of that information would not allow an investor to profit from the information because stock prices already incorporate the information. i. The weak form of the EMH asserts that stock prices reflect all the information that can be derived by examining market trading data such as the history of past prices and trading volume. A strong body of evidence supports weak-form efficiency in the major U.S. securities markets. For example, test results suggest that technical trading rules do not produce superior returns after adjusting for transaction costs and taxes. ii. The semistrong form states that a firm’s stock price reflects all publicly available information about a firm’s prospects. Examples of publicly available information are company annual reports and investment advisory data. Evidence strongly supports the notion of semistrong efficiency, but occasional studies (e.g., those identifying market anomalies such as the small-firm-in-January or book-to-market effects) and events (such as the stock market crash of October 19, 1987) are inconsistent with this form of market efficiency. However, there is a question concerning the extent to which these “anomalies” result from data mining. iii. The strong form of the EMH holds that current market prices reflect all information (whether publicly available or privately held) that can be relevant to the valuation of the firm. Empirical evidence suggests that strong-form efficiency does not hold. If this form were correct, prices would fully reflect all information. Therefore even insiders could not earn excess returns. But the evidence is that corporate officers do have access to pertinent information long enough before public release to enable them to profit from trading on this information. b. i. Technical analysis involves the search for recurrent and predictable patterns in stock prices in order to enhance returns. The EMH implies that technical analysis is without value. If past prices contain no useful information for predicting future prices, there is no point in following any technical trading rule. ii. Fundamental analysis uses earnings and dividend prospects of the firm, expectations of future interest rates, and risk evaluation of the firm to determine proper stock prices. The EMH predicts that most fundamental analysis is doomed to failure. According to semistrong-form efficiency, no investor can earn excess returns from trading rules based on publicly available information. Only analysts with unique insight achieve superior returns. 11-6

Chapter 11 - The Efficient Market Hypothesis In summary, the EMH holds that the market appears to adjust so quickly to information about both individual stocks and the economy as a whole that no technique of selecting a portfolio using either technical or fundamental analysis can consistently outperform a strategy of simply buying and holding a diversified portfolio of securities, such as those comprising the popular market indexes. c. Portfolio managers have several roles and responsibilities even in perfectly efficient markets. The most important responsibility is to identify the risk/return objectives for a portfolio given the investor’s constraints. In an efficient market, portfolio managers are responsible for tailoring the portfolio to meet the investor’s needs, rather than to beat the market, which requires identifying the client’s return requirements and risk tolerance. Rational portfolio management also requires examining the investor’s constraints, including liquidity, time horizon, laws and regulations, taxes, and unique preferences and circumstances such as age and employment.10. a. The earnings (and dividend) growth rate of growth stocks may be consistently overestimated by investors. Investors may extrapolate recent growth too far into the future and thereby downplay the inevitable slowdown. At any given time, growth stocks are likely to revert to (lower) mean returns and value stocks are likely to revert to (higher) mean returns, often over an extended future time horizon.b. In efficient markets, the current prices of stocks already reflect all known relevant information. In this situation, growth stocks and value stocks provide the same risk- adjusted expected return. 11-7

Chapter 12 - Behavioral Finance and Technical Analysis CHAPTER 12: BEHAVIORAL FINANCE AND TECHNICAL ANALYSISPROBLEM SETS1. Technical analysis can generally be viewed as a search for trends or patterns in market prices. Technical analysts tend to view these trends as momentum, or gradual adjustments to ‘correct’ prices, or, alternatively, reversals of trends. A number of the behavioral biases discussed in the chapter might contribute to such trends and patterns. For example, a conservatism bias might contribute to a trend in prices as investors gradually take new information in to account, resulting in gradual adjustment of prices towards their fundamental values. Another example derives from the concept of representativeness, which leads investors to inappropriately conclude, on the basis of a small sample of data, that a pattern has been established that will continue well in to the future. When investors subsequently become aware of the fact that prices have overreacted, corrections reverse the initial erroneous trend.2. Even if many investors exhibit behavioral biases, security prices might still be set efficiently if the actions of arbitrageurs move prices to their intrinsic values. Arbitrageurs who observe mispricing in the securities markets would buy underpriced securities (or possibly sell short overpriced securities) in order to profit from the anticipated subsequent changes as prices move to their intrinsic values. Consequently, securities prices would still exhibit the characteristics of an efficient market.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. A related factor is 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. 12-1

Chapter 12 - Behavioral Finance and Technical Analysis4. Two reasons why behavioral biases might not affect equilibrium asset prices are discussed in Quiz Problems (1) and (2) above: first, behavioral biases might contribute to the success of technical trading rules as prices gradually adjust towards their intrinsic values, and; second, the actions of arbitrageurs might move security prices towards their intrinsic values. It might be important for investors to be aware of these biases because either of these scenarios might create the potential for excess profits even if behavioral biases do not affect equilibrium prices.5. Efficient market advocates believe that publicly available information (and, for advocates of strong-form efficiency, even insider information) is, at any point in time, reflected in securities prices, and that price adjustments to new information occur very quickly. Consequently, prices are at fair levels so that active management is very unlikely to improve performance above that of a broadly diversified index portfolio. In contrast, advocates of behavioral finance identify a number of investor errors in information processing and decision making that could result in mispricing of securities. However, the behavioral finance literature generally does not provide guidance as to how these investor errors can be exploited to generate excess profits. Therefore, in the absence of any profitable alternatives, even if securities markets are not efficient, the optimal strategy might still be a passive indexing strategy.6. Trin = Volume declining / Number declining = 766,901,460 / 2,068 = 0.978 Volume advancing / Number advancing 467,560,150 /1,233 This trin ratio, which is below 1.0, would be taken as a bullish signal.7. Breadth: Declines Net Advances 2,068 Advances 1,233 -835Breadth is negative. This is a bearish signal (although no one would actually use a one-day measure as in this example).8. This exercise is left to the student; answers will vary.9. The confidence index increases from (7%/8%) = 0.875 to (8%/9%) = 0.889 This indicates slightly higher confidence. But the real reason for the increase in the index is the expectation of higher inflation, not higher confidence about the economy. 12-2

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