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APT AND MULTIFACTOR MODELS OF
RISK AND RETURN

Arbitrage
◦ Exploitation of security mispricing, risk-free
profits can be earned

No arbitrage condition, equilibrium market
prices are rational in that they rule out
arbitrage opportunities
9.1 MULTIFACTOR MODELS
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Returns on a security come from two sources
◦ Common macro-economic factor
◦ Firm specific events
Focus directly on the ultimate sources of risk, such
as risk assessment when measuring one’s
exposures to particular sources of uncertainty
Factors models are tools that allow us to
describe and quantify the different factors
that affect the rate of return on a security
ri  E (ri )  i F  ei
ri = Return for security I
i = Factor sensitivity or factor loading or factor
beta
F = Surprise in macro-economic factor
(F could be positive, negative or zero)
ei = Firm specific events
F and ei have zero expected value, uncorrelated

Example
◦ Suppose F is taken to be news about the state of
the business cycle, measured by the unexpected
percentage change in GDP, the consensus is that
GDP will increase by 4% this year.
◦ Suppose that a stock’s beta value is 1.2, if GDP
increases by only 3%, then the value of F=?
◦ F=-1%, representing a 1% disappointment in actual
growth versus expected growth, resulting in the
stock’s return 1.2% lower than previously expected

Macro factor summarized by the market return arises from a
number of sources, a more explicit representation of
systematic risk allowing for the possibility that different
stocks exhibit different sensitivities to its various components
◦ Use more than one factor in addition to market return

 Examples include gross domestic product, expected
inflation, interest rates etc.
 Estimate a beta or factor loading for each factor using
multiple regression.
Multifactor models, useful in risk management applications,
to measure exposure to various macroeconomic risks, and to
construct portfolios to hedge those risks

Two factor models
◦ GDP, Unanticipated growth in GDP, zero
expectation
◦ IR, Unanticipated decline in interest rate, zero
expectation
ri  E  ri   iGDPGDP  iIR IR  ei
Factor
betas
Multifactor model: Description of the factors that
affect the security returns

Example
◦ One regulated electric-power utility (U), one airline
(A), compare their betas on GDP and IR
 Beta on GDP: U low, A high, positive
 Beta on IR: U high, A low, negative
◦ When a good news suggesting the economy will
expand, GDP and IR will both increase, is the news
good or bad ?
 For U, dominant sensitivity is to rates, bad
 For A, dominant sensitivity is to GDP, good
◦ One-factor model cannot capture differential
responses to varying sources of macroeconomic
uncertainty
r  13.3%  1.2GDP  0.3IR  e


Expected rate of return=13.3%
1% increase in GDP beyond current
expectations, the stock’s return will increase
by 1%*1.2


Multifactor model, a description of the factors
that affect security returns, what determines E(r)
in multifactor model
Expected return on a security (CAPM)
E  r   rf    E  rM   rf   rf   RPM
Compensation for
time value of money
Compensation for
bearing the
macroeconomic risk

Multifactor Security Market Line for multifactor
index model, risk premium is determined by
exposure to each systematic risk factor and its
risk premium
E  r   rf  GDP RPGDP  IR RPIR
9.2 ARBITRAGE PRICING THEORY


Stephen Ross, 1976, APT, link expected
returns to risk
Three key propositions
◦ Security returns can be described by a factor model
◦ Sufficient securities to diversify away idiosyncratic
risk
◦ Well-functioning security markets do not allow for
the persistence of arbitrage opportunities
Arbitrage - arises if an investor can construct a zero
investment portfolio with a sure profit
 Since no investment is required, an investor can
create large positions to secure large levels of
profit
 In efficient markets, profitable arbitrage
opportunities will quickly disappear

Law of One Price
◦ If two assets are equivalent in all economically
relevant respects, then they should have the same
market price

Arbitrage activity
◦ If two portfolios are mispriced, the investor could
buy the low-priced portfolio and sell the highpriced portfolio
◦ Market price will move up to rule out arbitrage
opportunities
◦ Security prices should satisfy a no-arbitrage
condition
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
Well-diversified portfolio, the firmspecific risk negligible, only systematic
risk remain
n-stock portfolio
ri  E  ri   i F  ei
rP  E  rP    P F  eP
 P   wi i ,eP   wi ei

The portfolio variance

1
If equally-weighted portfolio wi 
, the
n
nonsystematic variance
      ep
2
P
2
P
2
F
2
 1  2 1  ei
2 2
 var iance   wi ei    wi  ei      ei  
n
n
n
2
 e2
P
1 2
  ei
n

N lager, the nonsystematic variance approaches zero,
the effect of diversification
2
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This is true for other than equally weighted one
Well-diversified portfolio is one that is diversified
over a large enough number of securities with each
weight small enough that the nonsystematic
variance is negligible, eP approaches zero
E  eP   E   wi ei    wi E  ei   0
ri  E (ri )  i F  ei
 e2  0
rP  E (rP )   P F  eP
P

For a well-diversified portfolio
rP  E  rP   P F


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Only systematic risk should command a risk
premium in market equilibrium
Well-diversified portfolios with equal betas
must have equal expected returns in market
equilibrium, or arbitrage opportunities exist
Expected return on all well-diversified
portfolio must lie on the straight line from
the risk-free asset


Only systematic risk should command a risk premium in market
equilibrium
Solid line: plot the return of A with beta=1 for various realization of
the systematic factor (Rm)
rA  E  rA    A F  10%  1 F
Expected
rate=10%,c
ompletely
determine
d by Rm
Subject to
nonsystematic risk
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B: E(r)=8%. beta=1; A:E(r)=10%. beta=1
Arbitrage opportunity exist, so A and B can’t coexist
10%  1 F  8%  1 F  2%
Long in A, Short in B
Factor risk cancels out across the long and short positions,
zero net investment get risk-free profit
infinitely large scale until return discrepancy disappears
well-diversified portfolios with equal betas must have equal
expected return in market equilibrium, or arbitrage
opportunities exist

 

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What about different betas
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A: beta=1,E(r)=10%;
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C: beta=0.5,E(r)=6%;
D: 50% A and 50% risk-free (4%) asset,
◦ beta=0.5*1+0.5*0=0.5, E(r)=7%

C and D have same beta (0.5)
◦ different expected return
◦ arbitrage opportunity
Expected Return %
A
10
7
6
Risk
premium
D
C
Risk-free rate=4
0
0.5
A/C/D, well-diversified portfolio,
D : 50% A and 50% risk-free asset,
C and D have same beta (0.5),
different expected return,
arbitrage opportunity
1
beta
0.5*0  0.5*1  0.5
0.5*4%  0.5*10%  7%

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M, market index portfolio, on the line and beta=1
no-arbitrage condition to obtain an expected returnbeta relationship identical to that of CAPM
E  rP   rf   p  E  rM   rf 

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EXAMPLE
Market index, expected return=10%;Risk-free rate=4%
Suppose any deviation from market index return can
serve as the systematic factor
E, beta=2/3, expected return=4%+2/3(10%-4%)=8%
If E’s expected return=9%, arbitrage opportunity
Construct a portfolio F with same beta as E,
◦ 2/3 in M, 1/3 in T-bill
◦ Long E, short F

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M, market index portfolio, as a well-diversified portfolio, noarbitrage condition to obtain an expected return-beta
relationship identical to that of CAPM
three assumptions: a factor model, sufficient number of
securities to form a well-diversified portfolios, absence of
arbitrage opportunities
APT does not require that the benchmark portfolio in SML be
the true market portfolio
9.3 A MULTIFACTOR APT

Use of more than a single factor
 Several factors driven by the business cycle that might
affect stock returns
 Exposure to any of these factors will affect a stock’s risk
and its expected return

Two-factor model
ri  E  ri   i1F1  i 2 F2  ei
◦ Each factor has zero expected value, surprise
 Factor 1, departure of GDP growth from expectations
 Factor 2, unanticipated change in IR
◦ e, zero expected ,firm-specific component of unexpected return

Requires formation of factor portfolios
◦ Factor portfolio:
 Well-diversified
 Beta of 1 for one factor
 Beta of 0 for any other
◦ Or Tracking portfolio: the return on such portfolio
track the evolution of particular sources of
macroeconomic risk, but are uncorrelated with
other sources of risk
◦ Factor portfolios will serve as the benchmark
portfolios for a multifactor SML

Example: Suppose two factor Portfolio 1, 2,
E  r1   10%, E  r2   12%


Risk-free rate=4%
Consider a well-diversified portfolio A ,with beta on the two factors
 A1  0.5,  A2  0.75

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Multifactor APT states that the overall risk premium on portfolio A
must equal the sum of the risk premiums required as
compensation for each source of systematic risk
Total risk premium on the portfolio A:
 A1  E  r1   rf    A2  E  r2   rf 
 0.5*(10%  4%)  0.75* 12%  4%   9%

Total return on the portfolio A: 9%+4%=13%


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Factor Portfolio 1 and 2, factor exposures of
any portfolio P are given by its  P1 and  P 2
Consider a portfolio Q formed by investing
in factor portfolios with weights
 P1 in portfolio 1
◦
 P 2 in portfolio 2
◦
1   P1   P 2 in T-bills
◦
Return of portfolio Q
E  rQ   1   P1   P 2  rf   P1 E  r1    P 2 E  r2 
 rf   P1  E  r1   rf    P 2  E  r2   rf 

Suppose return on A is 12% (not 13%), then arbitrage
opportunity
◦ Form a portfolio Q from the factor portfolios with same
betas as A, with weights:
 0.5 in factor 1 portfolio
 0.75 in factor 2 portfolio
 -0.25 in T-bill
◦ Invest $1 in Q, and sell % in A, net investment is 0, but
with positive riskless profit
1* E  rQ   1* E  rA   13%  12%  1%

Q has same exposure as A to the two sources of risk, their
expected return also ought to be equal
9.4 WHERE TO LOOK FOR FACTORS

Two principles when specify a reasonable list of
factors
◦ Limit ourselves to systematic factors with considerable
ability to explain security returns
◦ Choose factors that seem likely to be important risk factors,
demand meaningful risk premiums to bear exposure to
those sources of risk

Chen, Roll, Ross 1986
◦ Chose a set of factors based on the ability of the factors to
paint a broad picture of the macro-economy
rit  i  iIP IPt  iEI EIt  iUIUIt  iCGCGt  iGBGBt  eit
IP: % change in industrial production
EI: % change in expected inflation
UI: % change in unexpected inflation
CG: excess return of long-term corporate bonds over longterm government bonds
◦ GB: excess return of long-term government bonds over T-bill
Multidimensional SCL, multiple regression, residual variance of the
regression estimates the firm-specific risk
◦
◦
◦
◦

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Fama, French, three-factor model
rit  i  iM RMt  iSMB SMBt  iHML HMLt  eit
◦ Use firm characteristics that seem on empirical grounds to
proxy for exposure to systematic risk
◦ SMB: return of a portfolio of small stocks in excess of the
return on a portfolio of large stocks
◦ HML: return of a portfolio of stocks with high book-tomarket ratio in excess of the return on a portfolio of
stocks with low ratio
◦ Market index is expected to capture systematic risk

Fama, French, three-factor model
◦ Long-standing observations that firm size and bookto-market ratio predict deviations of average stock
returns from levels with the CAPM
◦ High ratios of book-to-market value are more likely
to be in financial distress, small stocks may be more
sensitive to changes in business conditions
◦ The variables may capture sensitivity to risk-factors in
macroeconomy
9.5 THE MULTIFACOTOR CAPM
AND THE APT

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Many of the same functions: give a benchmark for
rate of return.
APT
◦ highlight the crucial distinction between factor risk
and diversifiable risk
◦ APT assumption: rational equilibrium in capital
markets precludes arbitrage opportunities (not
necessarily to individual stocks)
◦ APT yields expected return-beta relationship using a
well-diversified portfolio (not a market portfolio)
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APT applies to well diversified portfolios
and not necessarily to individual stocks
APT is more general in that it gets to an
expected return and beta relationship
without the assumption of the market
portfolio
APT can be extended to multifactor models

A multi-index CAPM
◦ Derived from a multi-period consideration of a stream of
consumption
◦ will inherit its risk factors from sources of risk that a broad
group of investors deem important enough to hedge, from
a particular hedging motive

The APT is largely silent on where to look for
priced sources of risk