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Title: Multifactor models

1
Topic 5 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

2
Multifactor Models
• The index model gave us a way of decomposing
stock variability into market or systematic risk,
due largely to macroeconomic events, versus
firm-specific 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.

3
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 ones
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.
4
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.

5
Let E(ri) expected return on stock i F
deviation of the common factor from its expected
value ßi sensitivity of firm
i to the common factor ei firm-specific
disturbance A single-factor model
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.
6
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 stocks ? value is 1.2.
7
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 stocks 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 stocks return from
its originally expected value.
8
A two-factor model
• The systematic or macro factor summarized by the
market return arises from a number of sources
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.

9
• 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

10
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
ei reflects firm-specific influences.
11
• 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, As and Bs stock prices
will rise. When interest rates rise, As and Bs
stock prices will fall.

12
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.
13
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 factorsmultifactor
modelscan provide even better descriptions of
returns. However, there are many possible
sets of macro factors that might be considered.

14
• 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).

15
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
market portfolio, RPM) times the relative measure
of risk (i.e., beta)

16
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?
17
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) 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.
18
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.
19
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.

20
• 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.

21
• 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
inflation-interest rate scenario

22
The current prices of the 4 stocks and rate
of return statistics
23
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 ? The equally weighted
portfolio will outperform D in all scenarios.
24
The rate of return statistics of the 2
alternatives are ?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.
25
• ?The dollar profits in each of the 4 scenarios
will be
• The net investment is zero.
• Yet, our portfolio yields a positive profit
in any scenario.

26
?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.
27
The APT A Single Factor Model
• Recall A single factor model

where E(ri) expected return on
stock i F deviation of the common factor
from its expected value ßi
sensitivity of firm i to the common factor
ei firm-specific disturbance
28
Risk of a portfolio of securities
• Construct an n-asset portfolio with weights wi (
).
• The rate of return on this portfolio
• where is the weighted
average of the ?i of the n securities.
• ? The portfolio nonsystematic component (which is
uncorrelated with F)
• which is a weighted average of the ei of the
n securities.

29
• We can divide the variance of this portfolio into
systematic and nonsystematic sources.
• The portfolio variance is
• where variance of the factor F
• nonsystematic risk of
the portfolio.
• Note

30
Well-diversified portfolios
• If the portfolio were equally weighted (wi
1/n), then the nonsystematic variance would be
• where 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.

31
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 is negligible.
32
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 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.

33
Betas and expected returns
• Consider a well-diversified portfolio A with
E(rA) 10 and ?A 1.
• ? The return on this portfolio
• The well-diversified portfolios return is
determined completely by the systematic factor.

34
• 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?

35
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
36
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.
37
• 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 well-diversified 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.

38
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39
Consider a new portfolio, D, composed of
half of portfolio A and half of the risk-free
asset. Portfolio Ds 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.
40
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.
41
• Formally
• Suppose that 2 well-diversified portfolios
(U V) are combined into a zero-beta portfolio,
Z, by choosing the weights shown below
• Notes

42
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. ? ?
(i.e. risk premiums are proportional to betas)
43
The security market line
• Now, consider the market portfolio as a
well-diversified 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
• (i.e. the SML relation of the CAPM)

44
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45
• 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.

46
• 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.

47
• 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.

48
Individual assets and the APT
• If arbitrage opportunities are to be ruled out,
each well-diversified portfolios expected excess
return must be proportional to its beta.
• That is, for any two well-diversified
portfolios P Q
• If this relationship is to be satisfied by
all well-diversified 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.

49
• 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
return-beta relationship, the relationship will
no longer hold for well-diversified portfolios,
and arbitrage opportunities will be available.

50
• Conclusion
• Imposing the no-arbitrage condition on a
single-factor security market implies maintenance
of the expected return-beta relationship for all
well-diversified portfolios and for all but
possibly a small number of individual securities.

51
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.

52
• Dissimilarities
• The APT depends on the assumption that a rational
equilibrium in capital markets precludes
arbitrage opportunities. A violation of the
APTs 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.

53
• 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.
• 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.

54
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 securitys risk and hence its
expected return.
• We can derive a multifactor version of the
APT to accommodate these multiple sources of
risk.

55
• Suppose that we generalize the single-factor
factor model to a two-factor model
• 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.

56
• 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.

57
• 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.

58
• 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
• (As exposure to factor 1) ? (risk premium
earned on the first factor portfolio)
• ?A1 ? E(r1) - rf 0.5 ? 6 3.

59
The risk premium attributable to risk factor 2
(As 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
60
• 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 Bs weights
• 0.5 0.75 (-0.25) 1.

61
• ?Bs beta on the first factor
• 0.5 ? 1 0.75 ? 0 (-0.25) ? 0
• 0.5 (same as ?A1)
• Bs beta on the second factor
• 0.5 ? 0 0.75 ? 1 (-0.25) ? 0
• 0.75 (same as ?A2)
• Bs expected return
• 0.5 ? E(r1) 0.75 ? E(r2) 0.25 ? rf
• 0.5 ? 10 0.75 ? 12 - 0.25 ? 4 13

62
• 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
• (i.e. a positive, risk-free return on a zero
net investment position).

63
• 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

64
• beta on the second factor
• ?P1 ? 0 ?P2 ? 1 (1 - ?P1 - ?P2 ) ? 0
?P2
• The expected return on this competing portfolio
• 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.

65
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
well-diversified 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).
66
• 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

67
• 5-factor model of excess security returns during
holding period t as a function of the macro
indicators
• 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.

68
• 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.
• 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.

69
• 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