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Chapter 13 Alternative Models of
Systematic Risk
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
13-2
Learning Objectives
1.
Describe the empirical findings about firm size, book-tomarket, 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-FrenchCarhart 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.
13-3
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.
13-4
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.
13-5
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.
13-6
Figure 13.1 Excess Return of Size
Portfolios, 1926–2005
13-7
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.
13-8
Figure 13.2 Excess Return
of Book-to-Market Portfolios, 1926–2005
13-9
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.
13-10
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
13-11
Example 13.1
13-12
Example 13.1 (cont'd)
13-13
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?
13-14
Alternative Example 13.1A

Solution
Market ValueABC 
Market Value XYZ
$2,200,000
 $18,333,333
.12
$2,200,000

 $13, 750, 000
.16

ABC has an expected return of 12%.

XYZ has an expected return of 16%.
13-15
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?
13-16
Alternative Example 13.1B

Solution

αABC = 12% - 15% = -3%

αXYZ = 16% - 15% = 1%

The firm with the lower market value has the
higher alpha.
13-17
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.
13-18
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.
13-19
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:

Investors are systematically ignoring positiveNPV investment opportunities.
1.
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.
13-20
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.
13-21
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.
13-22
Non-tradeable Wealth

The most important example of a nontradeable wealth is human capital.

If investors have a significant amount of nontradeable 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.
13-23
13.3 Multifactor Models of Risk

The expected return of any marketable security is:
E[Rs ]  rf  seff  (E[Reff ]  rf )

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.
13-24
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:
Reff  x1 RF 1  x2 RF 2
13-25
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:
Rs  rf  s  sF1 (RF1  rf )  sF 2 (RF 2  rf )   s


This statistical technique is known as a
multiple regression.
13-26
Using Factor Portfolios (cont'd)
A portfolio, P, consisting of the two factor
portfolios has a return of:
RP  Rs   sF 1RF 1   sF 2 RF 2  ( sF 1   sF 2 )rf

 Rs   sF 1 (RF 1  rf )   sF 2 (RF 2  rf )

which simplifies to:
RP  rf   s   s
13-27
Using Factor Portfolios (cont'd)

Since εi is uncorrelated with each factor, it
must be uncorrelated with the efficient
portfolio:
Cov (Reff , s )  Cov(x1RF 1  x2 RF 2 , s )
 x1Cov (RF 1 , s )  x2Cov (RF 2 , s )
 0
13-28
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:
13-29
Using Factor Portfolios (cont'd)
E[Rs ]  rf  sF1 (E[RF1 ]  rf )  sF 2 (E[RF 2 ]  rf )

Factor Beta

The sensitivity of the stock’s excess returns to the
excess return of a factor portfolio.
13-30
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.
13-31
Building a Multifactor Model

Given N factor portfolios with returns RF1, . . . ,
RFN, the expected return of asset s is defined
as:
E[Rs ]  rf   sF 1 (E[RF 1 ]  rf )   sF 2 (E[RF 2 ]  rf ) 
 rf 

N

n 1
FN
s
  sFN (E[RFN ]  rf )
(E[RFN ]  rf )
β1…. βN are the factor betas.
13-32
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.
13-33
Building a Multifactor Model (cont'd)

If all factor portfolios are self-financing then:
E[Rs ]  rf   sF 1E[RF 1 ]   sF 2 E[RF 2 ] 
 rf 
  sFN E[RFN ]
N
FN

 s (E[RFN ])
n 1
13-34
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 riskadjusted returns.

This self-financing portfolio is widely known as the
small-minus-big (SMB) portfolio.
13-35
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 riskadjusted returns.

This self-financing portfolio is widely known as the
high-minus-low (HML) portfolio.
13-36
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 riskadjusted 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.
13-37
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
E[Rs ]  rf   sMkt (E[RMkt ]  rf )   sSMB E[RSMB ]
  sHML E[RHML ]   sPR1YR E[RPR1YR ]
13-38
Calculating the Cost of Capital Using the FamaFrench-Carhart Factor Specification
13-39
Example 13.2
13-40
Example 13.2 (cont'd)
13-41
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.
13-42
Alternative Example 13.2

Problem (continued)

Assume you have calculated the following factor betas
for Intel stock:
Mkt
 INTC
 0.171
SMB
 INTC
 0.432
HML
 INTC
 0.419
PR1YR
 INTC
 0.121

Determine the cost of capital by using the FFC
factor specification if the monthly risk-free rate
is 0.5%.
13-43
Alternative Example 13.2

Solution
E[Rs ]  rf   sMkt (E[RMkt ]  rf )   sSMB E[RSMB ]
  sHML E[RHML ]   sPR1YR E[RPR1YR ]
E[Rs ]  0.5%  (0.171)(.64%)  (0.432)(0.17%)
(0.419)(0.53%)  (0.121)(0.76%)
E[Rs ]  .005  .0010944  .0007344  .0022207  .0009196
E[Rs ]  .0099691

The annual cost of capital is .0099691 × 12 = 11.96%
13-44
Calculating the Cost of Capital Using the FamaFrench-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
13-45
positive alphas.
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.
13-46
Figure 13.3
Variation of CAPM Beta in Time
13-47
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.
13-48
Firm Characteristics Used by MSCI
Barra
13-49
13.4 Characteristic Variable Models
of Expected Returns (cont'd)
Rs  w1s Re1  ws2 Re 2 

 wsN ReN   s
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.
13-50
Table 13.3
13-51
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.
13-52
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.
E[ Rs ]  w1s E[ Re1 ]  ws2 E[ Re2 ] 
 wsN E[ ReN ] 
13-53

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.
13-54
Figure 13.4 Returns of Portfolios Ranked by the
Characteristic Variable Model
13-55
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.
13-56
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:
Cov(Ri ,R j ) 
N

n =1

N
n
i
m
j
w w Cov(Rcn ,Rcm )
m =1
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.
13-57
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.
13-58
Figure 13.5 How Firms Calculate
the Cost of Capital
13-59