Transcript A factor portfolio

Topic 5 (Ch. 10)
Arbitrage Pricing Theory (APT)
and Multifactor Models of Risk and Return
• Multifactor models
• Arbitrage opportunities and profits
• The APT: A single factor model
– Well-diversified portfolios
– Betas and expected returns
– The security market line
– Individual assets and the APT
– The APT and the CAPM
• A multifactor APT
1
Multifactor Models
 The index model gave us a way of decomposing
stock variability into market or systematic risk,
due largely to macroeconomic events, versus firmspecific effects that can be diversified in large
portfolios.
In the index model, the return on the market
portfolio summarized the broad impact of macro
factors.
2
However, sometimes, rather than using a market
proxy, it is more useful to focus directly on the
ultimate sources of risk.
This can be useful in risk assessment when measuring
one’s exposures to particular sources of uncertainty.
Factor models are tools that allow us to describe and
quantify the different factors that affect the rate of
return on a security during any time period.
3
Factor models of security returns
A single-factor model
 Under a single-factor model, uncertainty in asset
returns has two sources: a common or macroeconomic
factor, and firm-specific events.
The common factor is constructed to have zero expected
value, since we use it to measure new information
concerning the macro-economy which, by definition,
has zero expected value.
4
Let E(ri): expected return on stock i
F: deviation of the common factor from its expected
value
βi: sensitivity of firm i to the common factor
ei: firm-specific disturbance
A single-factor model:
ri  E ( ri )   i F  ei
The actual return on firm i will equal its initially
expected return plus a (zero expected value) random
amount attributable to unanticipated economywide
events, plus another (zero expected value) random
amount attributable to firm-specific events.
5
The nonsystematic components of returns (the eis) are
assumed to be uncorrelated among themselves and
uncorrelated with the factor F.
Example:
Suppose that the macro factor, F, is taken to be news
about the state of the business cycle, measured by the
unexpected percentage change in gross domestic
product (GDP), and that the consensus is that GDP
will increase by 4% this year.
Suppose also that a stock’s  value is 1.2.
6
If GDP increases by only 3%, then the value of F
would be -1%, representing a 1% disappointment in
actual growth versus expected growth.
Given the stock’s beta value, this disappointment
would translate into a return on the stock that is
1.2% lower than previously expected.
This macro surprise, together with the firm-specific
disturbance (ei) determine the total departure of the
stock’s return from its originally expected value.
7
A two-factor model
 The systematic or macro factor summarized by the
market return arises from a number of sources (e.g.,
uncertainty about the business cycle, interest rates,
inflation, etc.)
When we estimate a single-index regression, we
implicitly impose an incorrect assumption that each
stock has the same relative sensitivity to each risk factor.
If stocks actually differ in their betas relative to the
various macroeconomic factors, then lumping all
systematic sources of risk into one variable such as the
return on the market index will ignore the nuances that
better explain individual-stock returns.
8
Example:

Suppose the 2 most important macroeconomic
sources of risk are uncertainties surrounding the
state of the business cycle, news of which we will
again measure by unanticipated growth in GDP and
changes in interest rates (IR).
The return on any stock will respond both to sources
of macro risk as well as to its own firm-specific
influences.
 A two-factor model describing the rate of return on
stock i in some time period:
ri  E ( ri )   GDP GDP   IR IR  ei
9
The 2 macro factors on the right-hand side of the
equation comprise the systematic factors in the
economy.
Both of these macro factors have zero expectation:
they represent changes in these variables that have not
already been anticipated.
The coefficients of each factor measure the sensitivity
of share returns to that factor and are called factor
sensitivities, factor loadings, or factor betas.
ei reflects firm-specific influences.
10
• Now, consider 2 stocks, A and B.
Stock A has a low sensitivity to GDP risk (i.e. a low
GDP beta) and has a relatively high sensitivity to
interest rates (i.e. a high interest rate beta) .
Conversely, stock B is very sensitive to economic
activity, but it is not very sensitive to interest rates (i.e.
has a high GDP beta and a small interest rate beta).
When GDP grows, A’s and B’s stock prices will rise.
When interest rates rise, A’s and B’s stock prices will
fall.
11
Suppose that on a particular day, a news item suggests
that the economy will expand.
GDP is expected to increase, but so are interest rates.
Is the “macro news” on this day good or bad?
For stock A, this is bad news, since its dominant
sensitivity is to interest rates.
But, for stock B, which responds more to GDP, this is
good news.
12
Clearly, a one-factor or single-index model cannot
capture such differential responses to varying sources
of macroeconomic uncertainty.
Of course, once a two-factor model can better explain
stock returns, it is easy to see that models with even
more factors—multifactor models—can provide even
better descriptions of returns.
However, there are many possible sets of macro
factors that might be considered.
13
• Two principles guide us when we specify a reasonable
list of factors:
 We want to limit ourselves to macro factors with
considerable ability to explain security returns.
If our model calls for hundreds of explanatory
variables, it does little to simplify our description of
security returns.
 We wish to choose factors that seem likely to be
important risk factors (i.e. factors that concern
investors sufficiently that they will demand
meaningful risk premiums to bear exposure to those
sources of risk).
14
A multifactor security market line
 The Security Market Line of the CAPM:
Securities will be priced to give investors an expected
return comprised of 2 components: the risk-free rate,
which is compensation for the time value of money,
and a risk premium, determined by multiplying a
benchmark risk premium (i.e., the risk premium
offered by the market portfolio, RPM) times the
relative measure of risk (i.e., beta):
E ( r )  r f   [ E ( rM )  r f ]  r f    RPM
15
We can think of beta as measuring the exposure of a
stock or portfolio to marketwide or macroeconomic
risk factors.
Thus, one interpretation of the SML is that investors
are rewarded with a higher expected return for their
exposure to macro risk, based on both the sensitivity
to that risk (beta) as well as the compensation for
bearing each unit of that source of risk (i.e., the risk
premium, RPM). but are not rewarded for exposure to
firm-specific uncertainty (the residual term ei).
How might this single-factor view of the world
generalize once we recognize the presence of multiple
sources of systematic risk?
16
A multifactor index model gives rise to a multifactor
security market line in which the risk premium is
determined by the exposure to each systematic risk
factor, and by a risk premium associated with each of
those factors.
Example (two-factor economy):
E ( r )  r f   GDP RPGDP   IR RPIR
The expected rate of return on a security is the sum of:
(1) the risk-free rate of return; (2) the sensitivity to
GDP risk (the GDP beta) times the risk premium for
GDP risk; and (3) the sensitivity to interest rate risk
(the interest rate beta) times the risk premium for
interest rate risk.
17
One difference between a single and multiple factor
economy is that a factor risk premium can be
negative.
For example, a security with a positive interest rate
beta performs better when rates increase, and thus
would hedge the value of a portfolio against interest
rate risk.
Investors might well accept a lower rate of return,
that is, a negative risk premium, as the cost of this
hedging attribute.
18
Arbitrage Opportunities and Profits
 An arbitrage opportunity arises when an investor
can construct a zero investment portfolio that will
yield a sure profit.
To construct a zero investment portfolio, one has to
be able to sell short at least one asset and use the
proceeds to purchase (go long on) one or more
assets.
Clearly, any investor would like to take as large a
position as possible in an arbitrage portfolio.
19
 An obvious case of an arbitrage opportunity arises
when the law of one price is violated.
When an asset is trading at different prices in two
markets (and the price differential exceeds
transaction costs), a simultaneous trade in the two
markets can produce a sure profit (the net price
differential) without any investment.
One simply sells short the asset in the high-priced
market and buys it in the low-priced market.
The net proceeds are positive, and there is no risk
because the long and short positions offset each
other.
20
 Another example:
Imagine that 4 stocks are traded in an economy
with only 4 distinct, possible scenarios.
The rates of return of the 4 stocks for each inflationinterest rate scenario:
Probability:
Stock:
A
B
C
D
High Real Interest Rates Low Real Interest Rates
High
Low
High
Low
Inflation
Inflation
Inflation
Inflation
0.25
0.25
0.25
0.25
-20
0
90
15
20
70
-20
23
40
30
-10
15
60
-20
70
36
21
The current prices of the 4 stocks and rate of return
statistics:
Correlation Matrix
Stock
A
B
C
D
Expected Standard
Current Return Deviation
Price
(%)
(%)
A
$10
25.00
29.58
1.00
10
20.00
33.91
-0.15
10
32.50
48.15 -0.29
10
22.25
8.58
0.68
B
-0.15
1.00
-0.87
-0.38
C
-0.29
-0.87
1.00
0.22
D
0.68
-0.38
0.22
1.00
22
Consider an equally weighted portfolio of the first
three stocks (A, B, and C), and contrast its possible
future rates of return with those of D:
High Real Interest Rates
High
Low
Inflation
Inflation
Equally weighted
portfolio (A, B, and C)
D
23.33
15.00
23.33
23.00
Low Real Interest Rates
High
Low
Inflation
Inflation
20.00
15.00
36.67
36.00
 The equally weighted portfolio will outperform D in
all scenarios.
23
The rate of return statistics of the 2 alternatives are:
3-stock portfolio
D
Mean
25.83
22.25
Standard
Deviation
6.40
8.58
Correlation
0.94
Since the equally weighted portfolio will fare better
under any circumstances, investors will take a short
position in D and use the proceeds to purchase the
equally weighted portfolio.
Suppose we sell short 300,000 shares of D and use
the $3 million proceeds to buy 100,000 shares each
of A, B, and C.
24
The dollar profits in each of the 4 scenarios will be:
Stock
A
B
C
D
Portfolio
Dollar
Investment
$1,000,000
1,000,000
1,000,000
-3,000,000
0
High Real Interest Rates
Low Real Interest Rates
High Inflation Low Inflation High Inflation Low Inflation
$-200,000
$200,000
$400,000
$600,000
0
700,000
300,000
-200,000
900,000
-200,000
-100,000
700,000
-450,000
-690,000
-450,000
-1,080,000
$250,000
$10,000
$150,000
$20,000
 The net investment is zero.
Yet, our portfolio yields a positive profit in any
scenario.
25
Investors will want to take an infinite position in such
a portfolio because larger positions entail no risk of
losses, yet yield evergrowing profits.
In principle, even a single investor would take such
large positions that the market would react to the
buying and selling pressure: The price of D has to
come down and/or the prices of A, B, and C have to go
up.
The arbitrage opportunity will then be
eliminated.
That is, market prices will move to rule out arbitrage
opportunities. Violation of this restriction would
indicate the grossest form of market irrationality.
26
The APT: A Single Factor Model
 Recall: A single factor model
ri  E ( ri )   i F  ei
where
E(ri): expected return on stock i
F: deviation of the common factor from its expected
value
βi: sensitivity of firm i to the common factor
ei: firm-specific disturbance
27
Risk of a portfolio of securities
n
 Construct an n-asset portfolio with weights wi (  wi  1 ).
i 1
 The rate of return on this portfolio:
rP  E ( rP )   P F  e P
where  P   wi  i is the weighted average of the i of the
n securities.
 The portfolio nonsystematic component (which is
uncorrelated with F):
e P   wi ei
which is a weighted average of the ei of the n securities.28
 We can divide the variance of this portfolio into
systematic and nonsystematic sources.
The portfolio variance is:
2
2 2
2
 p   p F   ( e p )
where  F2 : variance of the factor F
 2 (eP ) : nonsystematic risk of the portfolio.
Note:
 2 ( e P )  Variance(  wi ei )   wi2 2 ( ei )
29
Well-diversified portfolios
 If the portfolio were equally weighted (wi = 1/n), then
the nonsystematic variance would be:
2
1
1
1

( ei ) 1 2
2
2 2
 ( e P , wi  )   ( )  ( ei )  
  ( ei )
n
n
n
n
n
where  2 ( ei ) : average nonsystematic variance.
When the portfolio gets large in the sense that n is
large and the portfolio remains equally weighted
across all n securities, the nonsystematic variance
approaches 0.
30
The set of portfolios for which the nonsystematic
variance approaches 0 as n gets large consists of
more portfolios than just the equally weighted
portfolio.
Any portfolio for which each wi approaches 0 as n
gets large will satisfy the condition that the portfolio
nonsystematic risk will approach 0 as n gets large.
Define a well-diversified portfolio as one that is
diversified over a large enough number of securities
with proportions wi, each small enough that for
practical purposes the nonsystematic variance  2 (eP ) is
negligible.
31
Because the expected value of eP is 0, if its variance
also is 0, we can conclude that any realized value of
eP will be virtually 0.
 For a well-diversified portfolio:
rp  E ( rp )   P F
2
 P2   P2 F
  P   P F
Note: Large (mostly institutional) investors can hold
portfolios of hundreds and even thousands of
securities; thus the concept of well-diversified
portfolios clearly is operational in contemporary
financial markets. Well-diversified portfolios,
however, are not necessarily equally weighted.
32
Betas and expected returns
 Consider a well-diversified portfolio A with E(rA) =
10% and A = 1.
 The return on this portfolio: E ( rA )   AF  10%  1.0  F
The well-diversified portfolio’s return is determined
completely by the systematic factor.
33
 Now consider another well-diversified portfolio B, with
an expected return of 8% and B also equal to 1.0.
Could portfolios A and B coexist with the return
pattern depicted?
34
Clearly not: No matter what the systematic factor
turns out to be, portfolio A outperforms portfolio B,
leading to an arbitrage opportunity.
If you sell short $1 million of B and buy $1 million of A,
a 0 net investment strategy, your riskless payoff would
be $20,000, as follows:
35
You should pursue it on an infinitely large scale until
the return discrepancy between the two portfolios
disappears.
 Well-diversified portfolios with equal betas must have
equal expected returns in market equilibrium, or
arbitrage opportunities exist.
36
 What about portfolios with different betas?
 Their risk premiums must be proportional to beta.
e.g. Suppose that the risk-free rate is 4% and that welldiversified portfolio, C, with a beta of 0.5, has an
expected return of 6%.
Portfolio C plots below the line from the risk-free asset
to portfolio A.
37
38
Consider a new portfolio, D, composed of half of
portfolio A and half of the risk-free asset.
Portfolio D’s beta will be 0.5  1 + 0.5  0 = 0.5, and its
expected return will be 0.5  10 + 0.5  4 = 7%.
Now, portfolio D has an equal beta but a greater
expected return than portfolio C.
From our analysis, we know that this constitutes an
arbitrage opportunity.
39
Conclusion: To preclude arbitrage opportunities, the
expected return on all well-diversified portfolios
must lie on the straight line from the risk-free asset.
The equation of this line will dictate the expected
return on all well-diversified portfolios.
Note that risk premiums are indeed proportional to
portfolio betas.
40
 Formally:
Suppose that 2 well-diversified portfolios (U & V)
are combined into a zero-beta portfolio, Z, by
choosing the weights shown below:
Expected
Portfolio
Portfolio
Return
Beta
Weight
U
E(rU)
U
V
V  U
V
Notes:
Z
E(rV)
wU  wV  1
V
 U
V  U
V
 U
 wU  U  wV V 
U 
V  0
V   U
V   U
41
Portfolio Z is riskless: It has no diversifiable risk
because it is well diversified, and no exposure to the
systematic factor because its beta is zero.
To rule out arbitrage, then, it must earn only the
risk-free rate.
 E rZ   wU E ( rU )  wV E ( rV )
V
 U

E ( rU ) 
E ( rV )  r f
V   U
V   U
 E ( rU )  r f
U

E ( rV )  r f
V
(i.e. risk premiums are proportional to betas)
42
The security market line
 Now, consider the market portfolio as a welldiversified portfolio, and measure the systematic
factor as the unexpected return on the market
portfolio.
Because the market portfolio must be on the straight
line from the risk-free asset and the beta of the
market portfolio is 1, we can determine the equation
describing that line:
E ( rP )  r f  [ E ( rM )  r f ] P
(i.e. the SML relation of the CAPM)
43
The security
market line
44
 We have used the no-arbitrage condition to obtain an
expected return-beta relationship identical to that of
the CAPM, without the restrictive assumptions of the
CAPM.
This suggests that despite its restrictive assumptions
the main conclusion of the CAPM (i.e. the SML
expected return-beta relationship) should be at least
approximately valid.
45
 In contrast to the CAPM, the APT does not require
that the benchmark portfolio in the SML relationship
be the true market portfolio.
Any well-diversified portfolio lying on the SML may
serve as the benchmark portfolio.
Accordingly, the APT has more flexibility than does
the CAPM.
46
 The APT provides further justification for use of the
index model in the practical implementation of the
SML relationship.
Even if the index portfolio is not a precise proxy for
the true market portfolio, we now know that if the
index portfolio is sufficiently well diversified, the
SML relationship should still hold true according to
the APT.
47
Individual assets and the APT
 If arbitrage opportunities are to be ruled out, each
well-diversified portfolio’s expected excess return must
be proportional to its beta.
That is, for any two well-diversified portfolios P & Q:
E ( rP )  r f
P

E ( rQ )  r f
Q
If this relationship is to be satisfied by all welldiversified portfolios, it must be satisfied by almost all
individual securities.
Thus, the expected return-beta relationship holds for
all but possibly a small number of individual securities.48
 Recall that to qualify as well diversified, a portfolio
must have very small positions in all securities.
If, for example, only one security violates the
expected return-beta relationship, then the effect of
this violation on a well-diversified portfolio will be
too small to be of importance for any practical
purpose, and meaningful arbitrage opportunities will
not arise.
But if many securities violate the expected returnbeta relationship, the relationship will no longer hold
for well-diversified portfolios, and arbitrage
opportunities will be available.
49
 Conclusion:
Imposing the no-arbitrage condition on a singlefactor security market implies maintenance of the
expected return-beta relationship for all welldiversified portfolios and for all but possibly a small
number of individual securities.
50
The APT and the CAPM
 Similarities:
The APT serves many of the same functions as the
CAPM:
 The APT gives us a benchmark for rates of return that
can be used in capital budgeting, security evaluation,
or investment performance evaluation.
 The APT highlights the crucial distinction between
nondiversifiable risk (factor risk) that requires a
reward in the form of a risk premium and diversifiable
risk that does not.
51
 Dissimilarities:
 Advantages of the APT:
 The APT depends on the assumption that a rational
equilibrium in capital markets precludes arbitrage
opportunities. A violation of the APT’s pricing
relationships will cause extremely strong pressure to
restore them even if only a limited number of investors
become aware of the disequilibrium.
The CAPM relies on a number of restrictive
assumptions.
52
 The APT yields an expected return-beta relationship
using a well-diversified portfolio that practically can
be constructed from a large number of securities.
In contrast, the CAPM is derived assuming an
inherently unobservable “market” portfolio.
 Disadvantage of the APT:
The CAPM provides an unequivocal statement on the
expected return-beta relationship for all assets,
whereas the APT implies that this relationship holds
for all but perhaps a small number of securities.
53
A Multifactor APT
 We have assumed so far that there is only one
systematic factor affecting security returns.
This simplifying assumption is in fact too simplistic.
It is easy to think of several factors driven by the
business cycle that might affect security returns:
interest rate fluctuations, inflation rates, oil prices, etc.
Presumably, exposure to any of these factors will affect
a security’s risk and hence its expected return.
We can derive a multifactor version of the APT to
accommodate these multiple sources of risk.
54
 Suppose that we generalize the single-factor factor
model to a two-factor model:
ri  E ( ri )   i1F1   i 2 F2  ei
Factor 1 might be, for example, departures of GDP
growth from expectations, and factor 2 might be
unanticipated inflation.
Each factor has a zero expected value because each
measures the surprise in the systematic variable
rather than the level of the variable.
Similarly, the firm-specific component of unexpected
return ei also has zero expected value.
55
 A factor portfolio:
A well-diversified portfolio constructed to have a
beta of 1 on one of the factors and a beta of 0 on any
other factor.
This is an easy restriction to satisfy, because we have
a large number of securities to choose from, and a
relatively small number of factors.
Factor portfolios will serve as the benchmark
portfolios for a multifactor security market line.
56
 Suppose that the two factor portfolios (portfolios 1 &
2) have expected returns E(r1) = 10% & E(r2) = 12%.
Suppose further that the risk-free rate rf is 4%.
The risk premium on the first factor portfolio:
E(r1) - rf = 10% - 4% = 6%.
The risk premium on the second factor portfolio:
E(r2) - rf = 12% - 4% = 8%.
57
 Now consider an arbitrary well-diversified portfolio,
portfolio A, with beta on the first factor, A1 = 0.5, and
beta on the second factor, A2 = 0.75.
The multifactor APT states that the overall risk
premium on this portfolio A must equal the sum of
the risk premiums required as compensation to
investors for each source of systematic risk.
The risk premium attributable to risk factor 1:
= (A’s exposure to factor 1)  (risk premium earned on
the first factor portfolio)
= A1  [E(r1) - rf] = 0.5  6% = 3%.
58
The risk premium attributable to risk factor 2:
= (A’s exposure to factor 2)  (risk premium earned on
the second factor portfolio)
= A2  [E(r2) - rf] = 0.75  8% = 6%.
 The total expected return on the portfolio A:
59
 Why the expected return on A must be 13%?
Suppose that the expected return on A were 12%.
This return would give rise to an arbitrage
opportunity.
Form a portfolio B with the same betas as A:
weight on the first factor portfolio: 0.5
weight on the second factor portfolio: 0.75
weight on the risk-free asset: -0.25
 The sum of B’s weights:
0.5 + 0.75 + (-0.25) = 1.
60
B’s beta on the first factor
= 0.5  1 + 0.75  0 + (-0.25)  0
= 0.5 (same as A1)
B’s beta on the second factor
= 0.5  0 + 0.75  1 + (-0.25)  0
= 0.75 (same as A2)
 B’s expected return
= 0.5  E(r1) + 0.75  E(r2) – 0.25  rf
= 0.5  10% + 0.75  12% - 0.25  4% = 13%
61
 A long position in B and a short position in A would
yield an arbitrage profit.
The total return per dollar long or short in each
position would be:
rB  rA
 [ E ( rB )   B1F1   B 2F2 ]  [ E ( rA )   A1F1   A2F2 ]
 [13%  0.5F1  0.75F2 ]  [12%  0.5F1  0.75F2 ]
 1%
(i.e. a positive, risk-free return on a zero net
investment position).
62
 Generalization:
The factor exposure of any portfolio, P, is given by its
betas, P1 and P2.
Form a competing portfolio:
weight in the first factor portfolio: P1
weight in the second factor portfolio: P2
weight in T-bills: 1 - P1 - P2
This competing portfolio will have betas equal to
those of Portfolio P:
beta on the first factor
= P1  1 + P2  0 + (1 - P1 - P2 )  0 = P1
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beta on the second factor
= P1  0 + P2  1 + (1 - P1 - P2 )  0 = P2
 The expected return on this competing portfolio:
E ( rP )   P 1 E ( r1 )   P 2 E ( r2 )  (1   P 1   p 2 )r f
 r f   P1[ E ( r1 )  r f ]   P 2[ E ( r2 )  r f ]
Any well-diversified portfolio with betas P1 and P2
must have return given in the above equation if
arbitrage opportunities are to be precluded.
This establishes a multifactor version of the APT.
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Note:
The extension of the multifactor SML to individual
assets is precisely the same as for the one-factor APT.
If this relationship is to be satisfied by all welldiversified portfolios, it must be satisfied by almost
all individual securities.
Thus, the multifactor SML holds for all but possibly
a small number of individual securities.
Hence, the fair rate of return on any security with 1
= 0.5 and 2 = 0.75 is 13% [= 4% + 0.5  (10% - 4%)
+ 0.75  (12% - 4%)].
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 We discuss two examples of the multifactor approach
that are more well-known in the literature:
• Example 1: 5-factor model
(Chen, Roll, and Ross, 1986):
IP = % change in industrial production
EI = % change in expected inflation
UI = % change in unanticipated inflation
CG = excess return of long-term corporate bonds over
long-term government bonds
GB = excess return of long-term government bonds
over T-bills
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 5-factor model of excess security returns during
holding period t as a function of the macro indicators:
Rit   i   iIP IPt   iEI EI t   iUI UI t   iCGCGt   iGBGBt  eit
 A multidimensional security characteristic line with 5
factors.
As before, to estimate the betas of a given security we
can use regression analysis. Here, however, because
there is more than one factor, we estimate a multiple
regression of the excess returns of the security in each
period on the 5 macro factors.
The residual variance of the regression estimates the
firm-specific risk.
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• Example 2: 3-factor model (Fama & French, 1996):
An alternative approach to specifying macro factors as
candidates for relevant sources of systematic risk uses
firm characteristics that seem on empirical grounds to
represent exposure to systematic risk.
Rit   i   iM RMt   iSMB SMBt   iHML HMLt  eit
SMB (= small minus big): the return of a portfolio of small
stocks in excess of the return on a portfolio of large stocks
HML (= high minus low): the return of a portfolio of
stocks with high ratios of book value to market value in
excess of the return on a portfolio of stocks with low
book-to-market ratios.
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Notes:
 In this model, the market index does play a role and is
expected to capture systematic risk originating from
macro factors.
 These two firm-characteristic variables (SMB &
HML) are chosen because of longstanding
observations that corporate capitalization (firm size)
and book-to-market ratio seem to be predictive of
average stock returns, and therefore risk premiums.
Small firms are more sensitive to changes in business
conditions, and firms with high ratios of book-tomarket value are more likely to be in financial
distress.
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