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1、Chapter 13 Alternative Models of Systematic RiskChapter Outline13.1 The Efficiency of the Market Portfolio13.2 Implication of Positive Alphas13.3 Multifactor Models of Risk13.4 Characteristic Variable Models of Expected Return13.5 Methods Used in Practice2Learning Objectives1.Describe the empirical
2、findings about firm size, book-to-market, and momentum strategies that imply that the CAPM does not accurately model expected returns.2.Discuss two conditions that might cause investors to care about characteristics other than expected return and volatility of their portfolios.3.Use multi-factor mod
3、els, such as the Fama-French-Carhart model, to calculate expected returns.4.Illustrate how multifactor models can be written as the expected return on a self-financing portfolio.5.Discuss the use of characteristic models in calculating expected returns and betas.313.1 The Efficiency of the Market Po
4、rtfolionIf the market portfolio is efficient, securities should not have alphas that are significantly different from zero.qFor most stocks the standard errors of the alpha estimates are large, so it is impossible to conclude that the alphas are statistically different from zero.qHowever, it is not
5、difficult to find individual stocks that, in the past, have not plotted on the SML.413.1 The Efficiency of the Market Portfolio (contd)nResearchers have studied whether portfolios of stocks plot on this line and have searched for portfolios that would be most likely to have nonzero alphas. qResearch
6、ers have identified a number of characteristics that can be used to pick portfolios that produce high average returns.5The Size EffectnSize EffectqStocks with lower market capitalizations have been found to have higher average returns.nPortfolios based on size were formed.nPortfolios consisting of s
7、mall stocks had higher average excess returns than those consisting of large stocks.6Figure 13.1 Excess Return of Size Portfolios, 192620057The Size Effect (contd)nBook-to-Market RatioqThe ratio of the book value of equity to the market value of equitynPortfolios based on the book-to-market ratio we
8、re formed.nPortfolios consisting of high book-to-market stocks had higher average excess returns than those consisting of low book-to-market stocks.8Figure 13.2 Excess Return of Book-to-Market Portfolios, 192620059The Size Effect (contd)nWhen the market portfolio is not efficient, theory predicts th
9、at stocks with low market capitalizations or high book-to-market ratios will have positive alphas.qThis is evidence against the efficiency of the market portfolio.10The Size Effect (contd)nData Snooping BiasqThe idea that given enough characteristics, it will always be possible to find some characte
10、ristic that by pure chance happens to be correlated with the estimation error of a regression11Example 13.112Example 13.1 (contd)13Alternative Example 13.1AnProblemqSuppose two firms, ABC and XYZ, are both expected to pay a dividend stream $2.2 million per year in perpetuity. qABCs cost of capital i
11、s 12% per year and XYZs cost of capital is 16%. qWhich firm has the higher market value?qWhich firm has the higher expected return? 14Alternative Example 13.1AnSolutionqABC has an expected return of 12%.qXYZ has an expected return of 16%.15Alternative Example 13.1BnProblemqNow assume both stocks hav
12、e the same estimated beta, either because of estimation error or because the market portfolio is not efficient. qBased on this beta, the CAPM would assign an expected return of 15% to both stocks. qWhich firm has the higher alpha? qHow do the market values of the firms relate to their alphas? 16Alte
13、rnative Example 13.1BnSolutionqABC = 12% - 15% = -3%qXYZ = 16% - 15% = 1%qThe firm with the lower market value has the higher alpha.17Past ReturnsnMomentum StrategyqBuying stocks that have had past high returns (and shorting stocks that have had past low returns)nWhen the market portfolio is efficie
14、nt, past returns should not predict alphas.nHowever, researchers found that the best performing stocks had positive alphas over the next six months.qThis is evidence against the efficiency of the market portfolio.1813.2 Implications of Positive AlphasnIf the CAPM correctly computes the risk premium,
15、 an investment opportunity with a positive alpha is a positive-NPV investment opportunity, and investors should flock to invest in such strategies.1913.2 Implications of Positive Alphas (contd)nIf small stock or high book-to-market portfolios do have positive alphas, one can draw one of two conclusi
16、ons:1.Investors are systematically ignoring positive-NPV investment opportunities. nIf the CAPM correctly computes risk premiums, but investors are ignoring opportunities to earn extra returns without bearing any extra risk, it is becauseqThey are unaware of them or,qThe costs to implement the strat
17、egies are larger than the NPV of undertaking them.2013.2 Implications of Positive Alphas (contd)nIf small stock or high book-to-market portfolios do have positive alphas, one can draw one of two conclusions:2.The positive-alpha trading strategies contain risk that investors are unwilling to bear but
18、 the CAPM does not capture. This would suggest that the market portfolio is not efficient.21Proxy ErrornThe true market portfolio is more than just stocksit includes bonds, real estate, art, precious metals, and any other investment vehicles available. qHowever, researchers use a proxy portfolio lik
19、e the S&P 500 and assume that it will be highly correlated to the true market portfolio. qIf the true market portfolio is efficient but the proxy portfolio is not highly correlated with the true market, then the proxy will not be efficient and stocks will have nonzero alphas.22Non-tradeable WealthnT
20、he most important example of a non-tradeable wealth is human capital.qIf investors have a significant amount of non-tradeable wealth, this wealth will be an important part of their portfolios, but will not be part of the market portfolio of tradeable securities.nGiven non-tradeable wealth, the marke
21、t portfolio of tradeable securities will likely not be efficient.2313.3 Multifactor Models of RisknThe expected return of any marketable security is:qWhen the market portfolio is not efficient, we have to find a method to identify an efficient portfolio before we can use the above equation. However,
22、 it is not actually necessary to identify the efficient portfolio itself. qAll that is required is to identify a collection of portfolios from which the efficient portfolio can be constructed.24Using Factor PortfoliosnAssume that there are two portfolios that can be combined to form an efficient por
23、tfolio. qThese are called factor portfolios and their returns are denoted as RF1 and RF2. The efficient portfolio consists of some (unknown) combination of these two factor portfolios, represented by portfolio weights x1 and x2:25Using Factor Portfolios (contd)nTo see if these factor portfolios meas
24、ure risk, regress the excess returns of some stock s on the excess returns of both factors:qThis statistical technique is known as a multiple regression.26Using Factor Portfolios (contd)nA portfolio, P, consisting of the two factor portfolios has a return of:qwhich simplifies to:27Using Factor Portf
25、olios (contd)nSince i is uncorrelated with each factor, it must be uncorrelated with the efficient portfolio:28Using Factor Portfolios (contd)nRecall that risk that is uncorrelated with the efficient portfolio is diversifiable risk that does not command a risk premium. Therefore, the expected return
26、 of portfolio P is rf , which means s must equal zero.qSetting s equal to zero and taking expectations of both sides, the result is the following two-factor model of expected returns:29Using Factor Portfolios (contd)nFactor BetaqThe sensitivity of the stocks excess returns to the excess return of a
27、factor portfolio.30Using Factor Portfolios (contd)nSingle-Factor versus Multi-Factor ModelqA singe-factor model uses one portfolio while a multi-factor model uses more than one portfolio in the model.qThe CAPM is an example of a single-factor model while the Arbitrage Pricing Theory (APT) is an exam
28、ple of a multifactor model.31Building a Multifactor ModelnGiven N factor portfolios with returns RF1, . . . , RFN, the expected return of asset s is defined as:q1. N are the factor betas.32Building a Multifactor Model (contd)nA self-financing portfolio can be constructed by going long in some stocks
29、 and going short in other stocks with equal market value.qIn general, a self-financing portfolio is any portfolio with portfolio weights that sum to zero rather than one. 33Building a Multifactor Model (contd)nIf all factor portfolios are self-financing then:34Selecting the PortfoliosnA trading stra
30、tegy that each year buys a portfolio of small stocks and finances this position by short selling a portfolio of big stocks has historically produced positive risk-adjusted returns. qThis self-financing portfolio is widely known as the small-minus-big (SMB) portfolio.35Selecting the Portfolios (contd
31、)nA trading strategy that each year buys an equally-weighted portfolio of stocks with a book-to-market ratio less than the 30th percentile of NYSE firms and finances this position by short selling an equally-weighted portfolio of stocks with a book-to-market ratio greater than the 70th percentile of
32、 NYSE stocks has historically produced positive risk-adjusted returns. qThis self-financing portfolio is widely known as the high-minus-low (HML) portfolio.36Selecting the Portfolios (contd)nEach year, after ranking stocks by their return over the last one year, a trading strategy that buys the top
33、30% of stocks and finances this position by short selling bottom 30% of stocks has historically produced positive risk-adjusted returns. qThis self-financing portfolio is widely known as the prior one-year momentum (PR1YR) portfolio. nThis trading strategy requires holding the portfolio for a year a
34、nd the process is repeated annually.37Selecting the Portfolios (contd)nCurrently the most popular choice for the multifactor model uses the excess return of the market, SMB, HML, and PR1YR portfolios.qFama-French-Carhart (FFC) Factor Specifications38Calculating the Cost of Capital Using the Fama-Fre
35、nch-Carhart Factor Specification39Example 13.240Example 13.2 (contd)41Alternative Example 13.2nProblemqYou are considering making an investment in a project in the semiconductor industry.qThe project has the same level of non-diversifiable risk as investing in Intel stock.42Alternative Example 13.2n
36、Problem (continued)qAssume you have calculated the following factor betas for Intel stock:qDetermine the cost of capital by using the FFC factor specification if the monthly risk-free rate is 0.5%.43Alternative Example 13.2nSolutionqThe annual cost of capital is .0099691 12 = 11.96%44Calculating the
37、 Cost of Capital Using the Fama-French-Carhart Factor Specification (contd)nAlthough it is widely used in research to measure risk, there is much debate about whether the FFC factor specification is really a significant improvement over the CAPM.qOne area where researchers have found that the FFC fa
38、ctor specification does appear to do better than the CAPM is measuring the risk of actively managed mutual funds. nResearchers have found that funds with high returns in the past have positive alphas under the CAPM. When the same tests were repeated using the FFC factor specification to compute alph
39、as, no evidence was found that mutual funds with high past returns had future positive alphas.4513.4 Characteristic Variable Models of Expected ReturnsnCalculating the cost of capital using the CAPM or multifactor model relies on accurate estimates of risk premiums and betas.qAccurately estimating t
40、hese quantities is difficult as both risk premiums and betas may not remain stable over time.46Figure 13.3 Variation of CAPM Beta in Time4713.4 Characteristic Variable Models of Expected Returns (contd)nCharacteristic Variable ModelqAn approach to measuring risk that views firms as a portfolio of di
41、fferent measurable characteristics that together determine the firms risk and return.48Firm Characteristics Used by MSCI Barra4913.4 Characteristic Variable Models of Expected Returns (contd)nThere is an important difference between this and the multifactor models considered earlier. qIn the multifa
42、ctor models, the returns of the factor portfolios are observed, and the sensitivity of each stock to the different factors is estimated. qIn the characteristic variable model, the weight of each stock on each characteristic is observed, and then we estimate the return Rcn associated with each charac
43、teristic.50Table 13.35113.4 Characteristic Variable Models of Expected Returns (contd)nOne way to estimate relation between the characteristic variables and returns is to use the relation to estimate each stocks expected return.5213.4 Characteristic Variable Models of Expected Returns (contd)nIf you
44、 view a stock as portfolio of characteristic variables, then the stocks expected return is the sum over all the variables of the amount of each characteristic variable the stock contains times the expected return of that variable.5313.4 Characteristic Variable Models of Expected Returns (contd)nRese
45、archers have evaluated the usefulness of the characteristic variable approach by ranking stocks based on their characteristics model.qThey put stocks into 10 ranked portfolios based on their characteristics models prediction of expected return. They then measured the return of each portfolio over th
46、e following month. They found that the top-ranked portfolios had the highest returns.54Figure 13.4 Returns of Portfolios Ranked by the Characteristic Variable Model5513.4 Characteristic Variable Models of Expected Returns (contd)nAnother approach is to use the estimated returns of the characteristic
47、 variables to estimate the covariance between pairs of stocks, or between a stock and the market index.5613.4 Characteristic Variable Models of Expected Returns (contd)nBy viewing each stock as a portfolio of characteristics, one can calculate the covariance between two different stocks i and j as:q
48、The beta of a stock is equal to the weighted-average of the characteristic variable betas where the weights are the amounts of each characteristic variable the stock contains.nAs the firm evolves in time, its beta will change accordingly to reflect its new level of risk.5713.5 Methods Used In Practi
49、cenThere is no clear answer to the question of which technique is used to measure risk in practiceit very much depends on the organization and the sector. qThere is little consensus in practice in which technique to use because all the techniques covered are imprecise.58Figure 13.5 How Firms Calculate the Cost of Capital59
