Week 10
GD31403
Multifactor Models of
Risk and Return
Arbitrage Pricing Theory
• CAPM is criticized because of:
– The many unrealistic assumptions
– The difficulties in selecting a proxy for the market
portfolio as a benchmark
– The growing empirical evidence suggesting the
need to consider additional risk variables such as
size, P/E, BV/MV.
10-2
• An alternative pricing theory with fewer
assumptions was developed by Stephen Ross
(Ross, 1976, 1977): Arbitrage Pricing
Theory (APT)
Arbitrage Pricing Theory
• Major assumptions of APT:
– Capital markets are perfectly competitive
– Investors always prefer more wealth to less wealth
with certainty
– The stochastic process generating asset returns
can be expressed as a linear function of a set of K
factors
• In contrast to CAPM, APT doesn’t assume:
– Normally distributed security returns
– Quadratic utility function
– A mean-variance efficient market portfolio
10-3
Arbitrage Pricing Theory
The APT Model
E  Ri   0  1bi1  2bi 2  ...  K biK
where:
λ0 = the expected return on an asset with zero
systematic risk
λj= the risk premium related to the j-th common
risk factor
bij= the responsiveness of asset i’s returns to
movements in the j-th common risk factor
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Arbitrage Pricing Theory
CAPM versus APT
10-5
CAPM
E ( Ri ) = RFR + bi éë E ( RM ) - RFR ùû
APT
E  Ri   0  1bi1  2bi 2  ...  K biK
CAPM
APT
Form of Equation
Linear
Linear
Number of Risk Factors
Factor Risk Premium
Factor Risk Sensitivity
“Zero-Beta” Return
1
[E(RM) – RFR]
βi
RFR
K (≥ 1)
{ λj }
{ bij }
λ0
Arbitrage Pricing Theory
• Unlike CAPM that is a one-factor model, APT is a
multifactor pricing model
• However, unlike CAPM that identifies the market
portfolio return as the factor, APT model does not
specifically identify these risk factors in terms of
quantity (how many there are) or their identity (what
they are)
• These multiple factors include:
10-6
−
−
−
−
Inflation
Growth in GNP
Major political upheavals
Changes in interest rates
Arbitrage Pricing Theory
• Instead of SML, APT implies a security market
plane- K risk factors and one dimension for the
security ’ s expected return (see Exhibit 9.2 of
Textbook, pp.240)
• Similar to CAPM, the APT assumes that the
unsystematic risk will be diversified away in a large
portfolio.
• In equilibrium when the unsystematic risk are fully
diversified, the APT requires that the return on a
zero-systematic-risk portfolio is zero (λ0 =0).
10-7
APT: Illustration 1
APT
E  Ri   0  1bi1  2bi 2  ...  K biK
(1) Selecting Risk Factors (How many and
what are these factors?)
•
As discussed earlier, the primary challenge with
using the APT is identifying the risk factors
•
For this illustration, assume that there are 2
common factors:
− Unanticipated changes in the rate of inflation
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− Unexpected changes in the growth rate of real GDP
APT: Illustration 1
(2) Determining the Risk Premium
10-9
•
λ1: The risk premium related to the first risk factor is 2
percent for every 1 percent change in the rate
(λ1=0.02)
•
λ2: The average risk premium related to the second
risk factor is 3 percent for every 1 percent change in
the rate of growth (λ2=0.03)
•
λ0: The rate of return on a zero-systematic risk asset
(i.e., zero beta) is 4 percent (λ0=0.04)
APT: Illustration 1
(3) Determining the Sensitivities for Asset X
and Asset Y
bX1 = The response of asset X to changes in the
inflation factor is 0.50 (bX1=0.50)
bX2 = The response of asset X to changes in the
GDP factor is 1.50 (bX2=1.50)
bY1 = The response of asset Y to changes in the
inflation factor is 2.00 (bY1=2.00)
bY2 = The response of asset Y to changes in the
GDP factor is 1.75 (bY2=1.75)
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APT: Illustration 1
(4) Estimating the Expected Return, E(Ri)
APT
Asset X
E  Ri   0  1bi1  2bi 2
= 0.04 + 0.02  bi1  0.03  bi 2
E  RX   0.04 +  0.02  0.50    0.031.50 
= 0.095 @ 9.5%
Asset Y
E  RY   0.04 +  0.02  2.00    0.031.75 
= 0.1352 @ 13.25%
10-11
Is it possible to
overvalued assets?
identify
undervalued
or
APT: Illustration 2
• Suppose that 3 stocks (A, B, C) and 2
common systematic risk factors have the
following relationship (assume λ0=0 ):
E(RA) = (0.8) λ1 + (0.9) λ2
E(RB)=(-0.2) λ1 + (1.3) λ2
E(RC)=(1.8) λ1 + (0.5) λ2
APT
10-12
E  Ri   1bi1  2bi 2
APT: Illustration 2
• Further assume that the risk premium for the 2
common factors are:
λ1= 4%
λ2= 5%
E(RA) = (0.8)(0.04) + (0.9)(0.05) = 0.077 @ 7.7%
E(RB) = (-0.2)(0.04) + (1.3)(0.05) = 0.057 @ 5.7%
E(RC) = (1.8)(0.04) + (0.5)(0.05) = 0.097 @ 9.7%
10-13
APT: Illustration 2
• Now, assume that all three stocks are
currently priced at $35 and do not pay a
dividend
• With the expected returns of E(RA)=7.7%,
E(RB)=5.7% and E(RC)=9.7%, we can estimate
the stock price (consistent with investor return
expectations) one year later:
E(PA) = $35(1+7.7%) = $37.70
E(PB) = $35(1+5.7%) = $37.00
10-14
APT: Illustration 2
• If one “ knows ” actual future prices (from
fundamental or technical analysis) for these
stocks are different from those previously
estimated, then these stocks are either
undervalued or overvalued
• Assume
are :
Stock A
Stock B
Stock C
10-15
that the forecasted future prices
$37.20
$37.80
$38.50
(Est. Return = 6.29%)
(Est. Return = 8.00%)
(Est. Return = 10.00%)
APT: Illustration 2
Expected
Price
A $37.70
B $37.00
C $38.40
Forecasted
Price
Evaluation Recommend
$37.20 Overvalued
SELL
$37.80 Undervalued
BUY
$38.50 Undervalued
BUY
You will get the same conclusion if using
returns!
10-16
APT: Illustration 2
Computed from CAPM
Required
Return
A
7.7%
B
5.7%
C
9.7%
Derived from fundamental
or technical analysis
Estimated
Return
Evaluation Recommend
6.29%
Overvalued
SELL
8%
Undervalued
BUY
10%
Undervalued
BUY
Undervalued: Estimated Return > Required Return
Overvalued:
10-17
Estimated Return < Required Return
Empirical Tests of the APT
Roll-Ross Study (1980)
• The methodology used in their study is as
follows:
10-18
-
Estimate the expected returns and the factor
coefficients from time-series data on individual
asset returns
-
Use these estimates to test the basic crosssectional pricing conclusion implied by the APT
• The authors concluded that the evidence
generally supported the APT, but acknowledged
that their tests were not conclusive
Empirical Tests of the APT
Extensions of the Roll-Ross Study
10-19
•
Cho, Elton, and Gruber (1984) examined the number
of factors in the return-generating process that were
priced
•
Dhrymes, Friend, and Gultekin (1984) reexamined
techniques and their limitations and found the number
of factors varies with the size of the portfolio
•
Connor and Korajczyk (1993) developed a test that
identifies the number of factors in a model that does
allow the unsystematic components of risk to be
correlated across assets
Empirical Tests of the APT
Shanken’s Challenge to Testability of the APT
10-20
•
APT has no advantage because the factors need not
be observable, so equivalent sets may conform to
different factor structures
•
Empirical formulation of the APT may yield different
implications regarding the expected returns for a given
set of securities
•
Thus, the theory cannot explain differential returns
between securities because it cannot identify the
relevant factor structure that explains the differential
returns [similar to the Roll (1977) critique of CAPM]
Multifactor Models
• The empirical challenge with CAPM is to
accurately estimate the market portfolio.
• The main practical problem with the APT is
that neither the identity nor the exact number
of the risk factors are developed by theory.
10-21
• Multifactor models of risk and return attempt
to bridge this gap by specifying the exact
number and identity of risk factors
(macroeconomic variables or microeconomic
variables).
Multifactor Models In Practice
1) Macroeconomic-Based
Risk
Factor
Models: Risk factors are viewed as
macroeconomic in nature
1) Microeconomic-Based
Risk
Factor
Models: Risk factors are viewed at a
microeconomic level by focusing on relevant
characteristics of the securities themselves
1) Extensions of Characteristic-Based Risk
Factor Models
10-22
Macroeconomic-Based Risk Factor Models
Security return are governed by a set of broad
economic influences in the following fashion by
Chen, Roll, and Ross in 1986:
Rit  ai  [bi1 Rmt  bi 2 MPt  bi 3 DEIt  bi 4UI t  bi 5UPRt  bi 6UTSt ]  eit
where:
Rm= the return on a value weighted index of NYSE-listed stocks
MP=the monthly growth rate in US industrial production
DEI=the change in inflation, measured by the US consumer price index
UI=the difference between actual and expected levels of inflation
UPR=the unanticipated change in the bond credit spread
UTS= the unanticipated term structure shift (long term less short term RFR)
10-23
Macroeconomic-Based Risk Factor Models
10-24
•
The economic significance of the risk factors changed
dramatically over time
•
The coefficient for stock market proxy is never
significant
Macroeconomic-Based Risk Factor Models
Burmeister, Roll, and Ross (1994) analyzed the
predictive ability of a model based on the
following set of macroeconomic factors:
•
•
•
•
•
10-25
Confidence risk
Time horizon risk
Inflation risk
Business cycle risk
Market timing risk
Microeconomic-Based Risk Factor Models
Fama and French (1993) developed a threefactor model specifying the risk factors in
microeconomic terms using the characteristics of
the underlying securities:
( Rit - RFRt )  ai  bi1 ( Rmt - RFRt )  bi 2 SMBt  bi 3 HMLt  eit
– SMB (i.e. small minus big) is the return to a portfolio of small
capitalization stocks less the return to a portfolio of large
capitalization stocks
– HML (i.e. high minus low) is the return to a portfolio of stocks
with high ratios of book-to-market values less the return to a
portfolio of low book-to-market value stocks
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Microeconomic-Based Risk Factor Models
Carhart (1997), based on the Fama-French three
factor model, developed a four-factor model by
including a risk factor that accounts for the
tendency for firms with positive past return to
produce positive future return
( Rit - RFRt )  ai  bi1 ( Rmt - RFRt )  bi 2 SMBt  bi 3 HMLt  bi 4 MOM t  eit
where, MOMt = the price momentum factor, the return
difference between stocks with positive
and negative excess returns in the
recent past
10-27
Extensions of Characteristic-Based Risk
Factor Models
One type of security characteristic-based method
for defining systematic risk exposures involves
the use of index portfolios (e.g. S&P 500,
Wilshire 5000) as common risk factors such as
the one by Elton, Gruber, and Blake (1996), who
rely on four indexes:
10-28
• The S&P 500
• The Lehman Brothers aggregate bond index
• The Prudential Bache index of the difference
between large- and small-cap stocks
• The Prudential Bache index of the difference
between value and growth stocks
Extensions of Characteristic-Based Risk
Factor Models
The BARRA Model: Develop a model using the
following Characteristic-based the risk factors
10-29
•
Volatility (VOL)
•
•
•
•
•
•
•
•
•
•
•
•
Momentum (MOM)
Size (SIZ)
Size Nonlinearity (SNL)
Trading Activity (TRA)
Growth (GRO)
Earnings Yield (EYL)
Value (VAL)
Earnings Variability (EVR)
Leverage (LEV)
Currency Sensitivity (CUR)
Dividend Yield (YLD)
Nonestimation Indicator (NEU)
Conclusion
• The CAPM and APT will continue to be used
to value assets.
• Future work on this area will continue to seek
to identify the set of factors that best captures
the relevant dimension of investment risk.
• Another area is to explore the intertemporal
dynamics of the models (e.g., factor betas and
risk premia that change over time).
10-30