Chapter 13 Alternative Models of Systematic Risk

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Chapter Outline

13.1 The Efficiency of the Market Portfolio

13.2 Implication of Positive Alphas

13.3 Multifactor Models of Risk

13.4 Characteristic Variable Models of Expected Return

13.5 Methods Used in Practice

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Learning Objectives

1. Describe the empirical 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 models, 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.

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13.1 The Efficiency of the Market Portfolio If the market portfolio is efficient, securities

should not have alphas that are significantly different from zero.

For 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.

However, it is not difficult to find individual stocks that, in the past, have not plotted on the SML.

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13.1 The Efficiency of the Market Portfolio (cont'd) Researchers have studied whether portfolios

of stocks plot on this line and have searched for portfolios that would be most likely to have nonzero alphas.

Researchers have identified a number of characteristics that can be used to pick portfolios that produce high average returns.

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The Size Effect

Size Effect

Stocks with lower market capitalizations have been found to have higher average returns. Portfolios based on size were formed.

Portfolios consisting of small stocks had higher average excess returns than those consisting of large stocks.

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Figure 13.1 Excess Return of Size Portfolios, 1926–2005

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The Size Effect (cont'd)

Book-to-Market Ratio

The ratio of the book value of equity to the market value of equity Portfolios based on the book-to-market ratio were

formed.

Portfolios consisting of high book-to-market stocks had higher average excess returns than those consisting of low book-to-market stocks.

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Figure 13.2 Excess Return of Book-to-Market Portfolios, 1926–2005

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The Size Effect (cont'd)

When the market portfolio is not efficient, theory predicts that stocks with low market capitalizations or high book-to-market ratios will have positive alphas.

This is evidence against the efficiency of the market portfolio.

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The Size Effect (cont'd)

Data Snooping Bias

The idea that given enough characteristics, it will always be possible to find some characteristic that by pure chance happens to be correlated with the estimation error of a regression

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Example 13.1

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Example 13.1 (cont'd)

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Alternative Example 13.1A

Problem

Suppose two firms, ABC and XYZ, are both expected to pay a dividend stream $2.2 million per year in perpetuity.

ABC’s cost of capital is 12% per year and XYZ’s cost of capital is 16%.

Which firm has the higher market value?

Which firm has the higher expected return?

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Alternative Example 13.1A

Solution

ABC has an expected return of 12%.

XYZ has an expected return of 16%.

ABC

$2,200,000Market Value $18,333,333

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XYZ

$2,200,000Market Value $13,750,000

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Alternative Example 13.1B

Problem

Now assume both stocks have the same estimated beta, either because of estimation error or because the market portfolio is not efficient.

Based on this beta, the CAPM would assign an expected return of 15% to both stocks.

Which firm has the higher alpha?

How do the market values of the firms relate to their alphas?

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Alternative Example 13.1B

Solution

αABC = 12% - 15% = -3%

αXYZ = 16% - 15% = 1%

The firm with the lower market value has the higher alpha.

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Past Returns

Momentum Strategy

Buying stocks that have had past high returns (and shorting stocks that have had past low returns) When the market portfolio is efficient, past returns

should not predict alphas. However, researchers found that the best performing

stocks had positive alphas over the next six months. This is evidence against the efficiency of the market

portfolio.

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13.2 Implications of Positive Alphas If the CAPM correctly computes the risk

premium, an investment opportunity with a positive alpha is a positive-NPV investment opportunity, and investors should flock to invest in such strategies.

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13.2 Implications of Positive Alphas (cont'd) If small stock or high book-to-market

portfolios do have positive alphas, one can draw one of two conclusions:

1. Investors are systematically ignoring positive-NPV investment opportunities.

If the CAPM correctly computes risk premiums, but investors are ignoring opportunities to earn extra returns without bearing any extra risk, it is because

They are unaware of them or, The costs to implement the strategies are larger than

the NPV of undertaking them.

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13.2 Implications of Positive Alphas (cont'd) If 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 the CAPM does not capture. This would suggest that the market portfolio is not efficient.

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Proxy Error

The true market portfolio is more than just stocks—it includes bonds, real estate, art, precious metals, and any other investment vehicles available.

However, researchers use a proxy portfolio like the S&P 500 and assume that it will be highly correlated to the true market portfolio.

If 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.

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Non-tradeable Wealth

The most important example of a non-tradeable wealth is human capital.

If 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.

Given non-tradeable wealth, the market portfolio of tradeable securities will likely not be efficient.

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13.3 Multifactor Models of Risk The expected return of any marketable security is:

When 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, it is not actually necessary to identify the efficient portfolio itself.

All that is required is to identify a collection of portfolios from which the efficient portfolio can be constructed.

[ ] ( [ ] ) effs f s eff fE R r E R r

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Using Factor Portfolios

Assume that there are two portfolios that can be combined to form an efficient portfolio. These 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:

1 1 2 2 eff F FR x R x R

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Using Factor Portfolios (cont'd) To see if these factor portfolios measure risk,

regress the excess returns of some stock s on the excess returns of both factors:

This statistical technique is known as a multiple regression.

1 21 2 ( ) ( ) F F

s f s s F f s F f sR r R r R r

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Using Factor Portfolios (cont'd) A portfolio, P, consisting of the two factor

portfolios has a return of:

which simplifies to:

1 2 1 21 2

1 21 2

( )

( ) ( )

F F F FP s s F s F s s f

F Fs s F f s F f

R R R R r

R R r R r

P f s sR r

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Using Factor Portfolios (cont'd) Since εi is uncorrelated with each factor, it

must be uncorrelated with the efficient portfolio:

1 1 2 2

1 1 2 2

( , ) ( , )

( , ) ( , )

0

eff s F F s

F s F s

Cov R Cov x R x R

x Cov R x Cov R

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Using Factor Portfolios (cont'd) Recall that risk that is uncorrelated with the

efficient portfolio is diversifiable risk that does

not command a risk premium. Therefore, the expected return of portfolio P is rf , which means αs must equal zero.

Setting αs equal to zero and taking expectations of both sides, the result is the following two-factor model of expected returns:

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Using Factor Portfolios (cont'd)

Factor Beta The sensitivity of the stock’s excess returns to the

excess return of a factor portfolio.

1 21 2[ ] ( [ ] ) ( [ ] ) F F

s f s F f s F fE R r E R r E R r

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Using Factor Portfolios (cont'd) Single-Factor versus Multi-Factor Model

A singe-factor model uses one portfolio while a multi-factor model uses more than one portfolio in the model.

The CAPM is an example of a single-factor model while the Arbitrage Pricing Theory (APT) is an example of a multifactor model.

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Building a Multifactor Model

Given N factor portfolios with returns RF1, . . . , RFN, the expected return of asset s is defined as:

β1…. βN are the factor betas.

1 21 2

1

[ ] ( [ ] ) ( [ ] ) ( [ ] )

( [ ] )

F F FNs f s F f s F f s FN f

NFN

f s FN fn

E R r E R r E R r E R r

r E R r

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Building a Multifactor Model (cont'd) A self-financing portfolio can be

constructed by going long in some stocks and going short in other stocks with equal market value.

In general, a self-financing portfolio is any portfolio with portfolio weights that sum to zero rather than one.

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Building a Multifactor Model (cont'd) If all factor portfolios are self-financing then:

1 21 2

1

[ ] [ ] [ ] [ ]

( [ ])

F F FNs f s F s F s FN

NFN

f s FNn

E R r E R E R E R

r E R

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Selecting the Portfolios

A trading strategy 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.

This self-financing portfolio is widely known as the small-minus-big (SMB) portfolio.

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Selecting the Portfolios (cont'd) A 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 NYSE stocks has historically produced positive risk-adjusted returns.

This self-financing portfolio is widely known as the high-minus-low (HML) portfolio.

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Selecting the Portfolios (cont'd) Each year, after ranking stocks by their return

over the last one year, a trading strategy that buys the top 30% of stocks and finances this position by short selling bottom 30% of stocks has historically produced positive risk-adjusted returns. This self-financing portfolio is widely known as the

prior one-year momentum (PR1YR) portfolio. This trading strategy requires holding the portfolio

for a year and the process is repeated annually.

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Selecting the Portfolios (cont'd) Currently the most popular choice for the

multifactor model uses the excess return of the market, SMB, HML, and PR1YR portfolios. Fama-French-Carhart (FFC) Factor Specifications

11

[ ] ( [ ] ) [ ]

[ ] [ ]

Mkt SMBs f s Mkt f s SMB

HML PR YRs HML s PR YR

E R r E R r E R

E R E R

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Calculating the Cost of Capital Using the Fama-French-Carhart Factor Specification

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Example 13.2

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Example 13.2 (cont'd)

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Alternative Example 13.2

Problem

You are considering making an investment in a project in the semiconductor industry.

The project has the same level of non-diversifiable risk as investing in Intel stock.

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Alternative Example 13.2

Problem (continued)

Assume you have calculated the following factor betas for Intel stock:

Determine the cost of capital by using the FFC factor specification if the monthly risk-free rate is 0.5%.

1

0.171

0.432

0.419

0.121

MktINTC

SMBINTC

HMLINTC

PR YRINTC

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Alternative Example 13.2

Solution

The annual cost of capital is .0099691 × 12 = 11.96%

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[ ] ( [ ] ) [ ]

[ ] [ ]

Mkt SMBs f s Mkt f s SMB

HML PR YRs HML s PR YR

E R r E R r E R

E R E R

[ ] 0.5% (0.171)(.64%) (0.432)(0.17%)

(0.419)(0.53%) (0.121)(0.76%)sE R

[ ] .005 .0010944 .0007344 .0022207 .0009196sE R

[ ] .0099691sE R

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Calculating the Cost of Capital Using the Fama-French-Carhart Factor Specification (cont'd) Although 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. One area where researchers have found that the FFC

factor specification does appear to do better than the CAPM is measuring the risk of actively managed mutual funds. Researchers 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 alphas, no evidence was found that mutual funds with high past returns had future positive alphas.

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13.4 Characteristic Variable Models of Expected Returns Calculating the cost of capital using the

CAPM or multifactor model relies on accurate estimates of risk premiums and betas.

Accurately estimating these quantities is difficult as both risk premiums and betas may not remain stable over time.

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Figure 13.3 Variation of CAPM Beta in Time

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13.4 Characteristic Variable Models of Expected Returns (cont'd) Characteristic Variable Model

An approach to measuring risk that views firms as a portfolio of different measurable characteristics that together determine the firm’s risk and return.

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Firm Characteristics Used by MSCI Barra

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13.4 Characteristic Variable Models of Expected Returns (cont'd)

There is an important difference between this and the multifactor models considered earlier. In the multifactor models, the returns of the factor

portfolios are observed, and the sensitivity of each stock to the different factors is estimated.

In the characteristic variable model, the weight of each stock on each characteristic is observed, and then we estimate the return Rcn associated with each characteristic.

1 21 2 N

s s e s e s eN sR w R w R w R

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Table 13.3

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13.4 Characteristic Variable Models of Expected Returns (cont'd) One way to estimate relation between the

characteristic variables and returns is to use the relation to estimate each stock’s expected return.

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13.4 Characteristic Variable Models of Expected Returns (cont'd) If you view a stock as portfolio of

characteristic variables, then the stock’s 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.

1 21 2 [ ] [ ] [ ] [ ] N

s s e s e s eNE R w E R w E R w E R

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13.4 Characteristic Variable Models of Expected Returns (cont'd) Researchers have evaluated the usefulness

of the characteristic variable approach by ranking stocks based on their characteristics model.

They put stocks into 10 ranked portfolios based on their characteristics model’s prediction of expected return. They then measured the return of each portfolio over the following month. They found that the top-ranked portfolios had the highest returns.

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Figure 13.4 Returns of Portfolios Ranked by the Characteristic Variable Model

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13.4 Characteristic Variable Models of Expected Returns (cont'd) Another approach is to use the estimated

returns of the characteristic variables to estimate the covariance between pairs of stocks, or between a stock and the market index.

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13.4 Characteristic Variable Models of Expected Returns (cont'd) By viewing each stock as a portfolio of characteristics,

one can calculate the covariance between two different stocks i and j as:

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. As the firm evolves in time, its beta will change

accordingly to reflect its new level of risk.

=1 =1

( , ) ( , ) N N

n mi j i j cn cm

n m

Cov R R w w Cov R R

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13.5 Methods Used In Practice There is no clear answer to the question of

which technique is used to measure risk in practice—it very much depends on the organization and the sector.

There is little consensus in practice in which technique to use because all the techniques covered are imprecise.

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Figure 13.5 How Firms Calculate the Cost of Capital