Modern Portfolio Theory
The Factor Models and
The Arbitrage Pricing Theory
Chapter 8
By Ding zhaoyong
Return-generating Process
and Factor Models
Return-generating process
– Is a statistical model that describe how
return on a security is produced.
– The task of identifying the Markowitz
efficient set can be greatly simplified
by introducing this process.
– The market model is a kind of this
process,and there are many others.
Return-generating Process
and Factor Models
Factor models
– These models assume that the return
on a security is sensitive to the move-
ments of various factors or indices.
– In attempting to accurately estimate
expected returns,variances,and
covariances for securities,multiple-
factor models are potentially more
useful than the market model.
Return-generating Process
and Factor Models
– Implicit in the construction of a factor
model is the assumption that the returns
on two securities will be correlated only
through common reactions to one or
more of the specified in the model,Any
aspect of a security’s return unexplained
by the factor model is uncorrelated with
the unique elements of returns on other
securities.
Return-generating Process
and Factor Models
– A factor model is a powerful tool for
portfolio management.
It can supply the information needed to
calculate expected returns,variances,and
covariances for every security,which are
the necessary conditions for determining
the curved Markowitz efficient set.
It can also be used to characterize a
portfolio’s sensitivity to movement in the
factors.
Return-generating Process
and Factor Models
Factor models supply the necessary level
of abstraction in calculating covariances.
– The problem of calculating covariances
among securities rises exponentially as
the number of securities analyzed
increase.
– Practically,abstraction is an essential
step in identifying the Markowitz set,
Return-generating Process
and Factor Models
Factor models provide investment
managers with a framework to identify
important factors in the economy and the
marketplace and to assess the extent to
which different securities and portfolios
will respond to changes in these factors.
– A primary goal of security analysis is to
determine these factors and the sensitivities
of security return to movements in these
factors.
One-Factor Models
The one-factor models refer to the return-
generating process for securities involves a
single factor,These factors may be one of
the followings:
– The predicted growth rate in GDP
– The expected return on market index
– The growth rate of industrial produc-
tion,etc,
One-Factor Models
An example Page 295,Figure 11.1
G D Pf o r f a c t o r z e r o t h e
g r o w t h G D P p r e d i c t e d W i d g e t t o ofy s e n s i t i v i t
tp e r i o d inW i d g e t on r e t u r n s p e c i f i c oe u n i q u e t h e
tp e r i o d in G D P in r e t u r n of r a t e p r e d i c t e d t h e
tp e r i o d inW i d g e t on r e t u r n t h e
:w h e r e
a
b
e
G D P
r
eb G D Par
t
t
t
ttt
One-Factor Models
Generalizing the example
– Assumptions
The random error term and the factor are
uncorrelated,(Why?)
The random error terms of any two
securities are uncorrelated,(Why?)
ittiiit eFbar
One-Factor Models
– Expected return
– Variance
– Covariance
Fbar iii
2222
eiFii b
2
Fjiij bb
One-Factor Models
Two important features of one-factor model
– The tangency portfolio is easy to get.
The returns on all securities respond to a
single common factor greater simplifies the
task of identifying the tangency portfolio.
The common responsiveness of securities to
the factor eliminates the need to estimate
directly the covariances between the
securities.
The number of estimates,3N+2
One-Factor Models
– The feature of diversification is true of
any one-factor model.
Factor risk:
Nonfactor risk:
Diversification leads to an averaging of
factor risk
Diversification reduces nonfactor risk
)( 22 Fib?
2ei?
One-Factor Models
NN
N
XbXb
b
eNee
N
i
eiep
N
i
eiiep
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iip
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:w h e r e
Multiple-Factor Models
The health of the economy effects most
firms,but the economy is not a simple,
monolithic entity,Several common
influences with pervasive effects might be
identified
– The growth rate of GDP
– The level of interest rate
– The inflation rate
– The level of oil price
Multiple-Factor Models
Two-Factor Models
– Assume that the return-generating
process contains two factors.
ittitiiit eFbFbar 2211
tttt eI N FbG D Pbar 21
Multiple-Factor Models
The second equation provides a two-factor
model of a company’s stock,whose returns
are affected by expectations concerning
both the growth rate in GDP and the rate of
inflation.
Page 301,Figure 11.2
To this scatter of points is fit a two-
dimensional plane by using the statistical
technique of multiple-regression analysis.
Multiple-Factor Models
– Four parameters need to be estimated
for each security with the two-factor
model,ai,bi1,bi2,and the standard
deviation of the random error term.
– For each of the factors,two parameters
need to be estimated,These parameters
are the expected value of each factor
and the variance of each factor,Finally,
the covariance between factors.
Multiple-Factor Models
– Expected return
– Variance
– Covariance
2211 FbFbar iiii
2
2121
2
2
2
2
2
1
2
1
2 ),(2
eiiiFiFii FFC O Vbbbb
),()( 2112212 2222 111 FFC O Vbbbbbbbb jijiFjiFjiij
Multiple-Factor Models
– The tangency portfolio
The investor can proceed to use an
optimizer to derive the curve efficient set.
– Diversification
Diversification leads to an averaging of
factor risk.
Diversification can substantially reduce
nonfactor risk.
For a well-diversified portfolio,nonfactor
risk will be insignificant.
Multiple-Factor Models
pttptpp
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iit
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ii
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Multiple-Factor Models
Sector-Factor Models
– Sector-factor models are based on the
acknowledge that the prices of securities
in the same industry or economic sector
often move together in response to
changes in prospects for that sector.
– To create a sector-factor model,each
security must be assigned to a sector.
Multiple-Factor Models
A two-sector-factor model
– There are two sectors and each security
must be assigned to one of them.
– Both the number of sectors and what
each sector consists of is an open matter
that is left to the investor to decide.
– The return-generating process for
securities is of the same general form as
the two-factor model,
Multiple-Factor Models
– Differing from the two-factor model,
with two-sector-factor model,F1 and F2
now denote sector-factors 1 and 2,
respectively,Any particular security
belongs to either sector-factor 1 or
sector-factor 2 but not both.
jjjj
iiii
eFbar
eFbar
22
11
Multiple-Factor Models
– In general,whereas four parameters
need to be estimated for each security
with a two-factor model (ai1,bi1,bi2,?ei,),
only three parameters need to be
estimated with a two-sector-factor
model,(ai1,?ei,and eitherbi1 or bi2 ).
Multiple-factor models
itktiktitiiit eFbFbFbar2211
Estimating Factor Models
There are many methods of estimating
factor models,There methods can be
grouped into three major approaches:
– Time-series approaches
– Cross-sectional approaches
– Factor-analytic approaches
Factor Models and Equilibrium
A factor model is not an equilibrium model
of asset pricing.
Both equation show that the expected return
on the stock is related to a characteristic of
the stock,bi or?i,The larger the size of the
characteristic,the larger the asset’s return.
)( fMiMfi
iii
rrrr
Fbar
Factor Models and Equilibrium
– The key difference is ai and rf.
The only characteristic of the stock that
determine its expected return according
to the CAPM is?ii,as rff denotes the risk-
free rate and is the same for all securities.
With the factor model,there is a second
characteristic of the stock that needs to be
estimated to determine the stock’s
expected return,aii.
Factor Models and Equilibrium
As the size of ai differs from one stock to
another,it presents the factor model from
being an equilibrium model.
Two stocks with the same value of bi can
have dramatically different expected
returns according to a factor model.
Two stocks with the same value of?i will
have the same expected return according
to the equilibrium-based CAPM.
Factor Models and Equilibrium
The relationship between the parameters
ai and bi of the one-factor model and the
single parameter?i of the CAPM.
– If the expected returns are determined
according to the CAPM and actual
returns are generated by the one-factor
market model,then the above
equations must be true,
)( fMiMfi
iii
rrrr
Fbar
Arbitrage Pricing Theory
APT is a theory which describes how a
security is priced just like CAPM.
– Moving away from construction of
mean-variance efficient portfolio,APT
instead calculates relations among
expected rates of return that would
rule out riskless profits by any investor
in well-functioning capital markets.
Arbitrage Pricing Theory
APT makes few assumptions.
– One primary assumption is that each
investor,when given the opportunity to
increase the return of his or her portfolio
without increasing its risk,will proceed
to do so.
There exists an arbitrage opportunity and
the investor can use an arbitrage portfolios.
Arbitrage Opportunities
Arbitrage is the earning of riskless profit
by taking advantage of differential
pricing for the same physical asset or
security,
– It typically entails the sale of a security
at a relatively high price and the
simultaneous purchase of the same
security (or its functional equivalent)
at a relatively low price.
Arbitrage Opportunities
– Arbitrage activity is a critical element
of modern,efficient security markets.
– It takes relatively few of this active
investors to exploit arbitrage situations
and,by their buying and selling actions,
eliminate these profit opportunities.
– Some investors have greater resources
and inclination to engage I arbitrage
than others.
Arbitrage Opportunities
Zero-investment portfolio
– A portfolio of zero net value,established
by buying and shorting component
securities,
– A riskless arbitrage opportunity arises
when an investor can construct a zero-
investment portfolio that will yield a
sure profit.
Arbitrage Opportunities
– To construct a zero-investment portfolio,
one has to be able to sell short at least
one asset and use the proceeds to
purchase on or more assets.
– Even a small investor,using borrowed
money in this case,can take a large
position in such a portfolio.
– There are many arbitrage tactics.
Arbitrage Opportunities
An example:
– Four stocks and four possible scenarios
– the rate of return in four scenarios
– Page 180-181 in the textbook
– The expected returns,standard
deviations and correlations do not reveal
any abnormality to the naked eye.
Arbitrage Opportunities
The critical property of an arbitrage
portfolio is that any investor,regardless of
risk aversion or wealth,will want to take
an infinite position in it so that profits will
be driven to an infinite level.
– These large positions will force some
prices up and down until arbitrage
opportunities vanishes,
Factor Models and
Principle of Arbitrage
Almost arbitrage opportunities can
involve similar securities or portfolios.
– That similarity can be defined in many
ways.
– One way is the exposure to pervasive
factors that affect security prices.
– An example Page 324
Factor Models and
Principle of Arbitrage
A factor model implies that securities or
portfolios with equal-factor sensitivities
will behave in the same way except for
nonfactor risk.
– APT starts out by making the assumption
that security returns are related to an
unknown number of unknown factors.
– Securities with the same factor sensitivities
should offer the same expected returns.
Arbitrage Portfolios
An arbitrage portfolio must satisfy:
– A net market value of zero
– No sensitivity to any factor
– A positive expected return
0321 XXX
0332211 XbXbXb
0332211 rXrXrX
Arbitrage Portfolios
The arbitrage portfolio is attractive to
any investor who desires a higher return
and is not concerned with nonfactor risk.
It requires no additional dollar investment,
it has no factor risk,and it has a positive
expected return.
One-Factor Model and APT
Pricing effects on arbitrage portfolio
– The buying-and-selling activity will
continue until all arbitrage possibilities
are significant reduced or eliminated
– There will exist an approximately linear
relationship between expected returns
and sensitivities of the following sort:
ii br 10
One-Factor Model and APT
– The equation is the asset pricing equation
of the APT when returns are generated
by one factor
The linear equation means that in equili-
brium there will be a linear relationship
between expected returns and sensitivities.
The expected return on any security is,in
equilibrium,a linear function of the
security’s sensitivity to the factor,bi
One-Factor Model and APT
– Any security that has a factor
sensitivity and expected return such
that it lies off the line will be mispriced
according to the APT and will present
investors with the opportunity of
forming arbitrage portfolios.
– Page 327,Figure 12.1
One-Factor Model and APT
Interpreting the APT pricing equation
– Riskfree asset,rf
– Pure factor portfolio,p* ififi brrrr 10
11 fpfp rrrr
iffip brrrr 11,l e t
Two-Factor Model And APT
The two-factor model
Arbitrage portfolios
– A net market value of zero
– No sensitivity to any factor
– A positive expected return
iiiii eFbFbar 2211
22110 iii bbr
Two-Factor Model And APT
Pricing effects
22
11
0
2211
r a t e r i s k f r e e
)()(
f
f
f
ififf
i
r
r
r
brbrrr
Two-Factor Model And APT
–?1 is the expected return on the portfolio
which is known as a pure factor portfolio
or pure factor play,because it has:
Unit sensitivity to one factor (F1,b1=1)
No sensitivity to any other factor (F2,b2=0)
Zero nonfactor risk
This portfolio is a well-diversification
portfolio that has unit sensitivity to the first
factor and zero sensitivity to the second
factor.
Two-Factor Model And APT
– It is the same with?2,It is the well-
diversification portfolio that has zero
sensitivity to the first factor and unit
sensitivity to the second factor,meaning
that it has b1=0 and b2=1.
– Such as a portfolio that has zero sensitivity
to predicted industrial production and
unit sensitivity to predicted inflation
would have an expected return of 6%.
Multiple-factor model
The APT pricing equation
Multiple-Factor Model
And APT
ikikiiii eFbFbFbar2211
ikkiii bbbr22110
ikfkiffi brbrrr )()( 11
The APT And The CAPM
Common point
– Both require equilibrium
– Both have almost similar equation
Distinctions
– Different equilibrium mechanism
Many investors v.s,Few investors
– Different Portfolio
Market portfolio v.s,Well-diversifyed P.
Summary
The Factor Models
– One-factor models
– Multi-factor models
Factor models and equilibrium
Arbitrage opportunity and portfolio
The arbitrage pricing equation
– One-factor equation
– Multi-factor equation
Assignments For chapter 8
Readings
– Page 282 through 301
– Page 308 through 321
Exercises
– Page 304,14,15; Page 323,4,13
Q/A:
– Page 302,3
– Page 324,8
The Factor Models and
The Arbitrage Pricing Theory
Chapter 8
By Ding zhaoyong
Return-generating Process
and Factor Models
Return-generating process
– Is a statistical model that describe how
return on a security is produced.
– The task of identifying the Markowitz
efficient set can be greatly simplified
by introducing this process.
– The market model is a kind of this
process,and there are many others.
Return-generating Process
and Factor Models
Factor models
– These models assume that the return
on a security is sensitive to the move-
ments of various factors or indices.
– In attempting to accurately estimate
expected returns,variances,and
covariances for securities,multiple-
factor models are potentially more
useful than the market model.
Return-generating Process
and Factor Models
– Implicit in the construction of a factor
model is the assumption that the returns
on two securities will be correlated only
through common reactions to one or
more of the specified in the model,Any
aspect of a security’s return unexplained
by the factor model is uncorrelated with
the unique elements of returns on other
securities.
Return-generating Process
and Factor Models
– A factor model is a powerful tool for
portfolio management.
It can supply the information needed to
calculate expected returns,variances,and
covariances for every security,which are
the necessary conditions for determining
the curved Markowitz efficient set.
It can also be used to characterize a
portfolio’s sensitivity to movement in the
factors.
Return-generating Process
and Factor Models
Factor models supply the necessary level
of abstraction in calculating covariances.
– The problem of calculating covariances
among securities rises exponentially as
the number of securities analyzed
increase.
– Practically,abstraction is an essential
step in identifying the Markowitz set,
Return-generating Process
and Factor Models
Factor models provide investment
managers with a framework to identify
important factors in the economy and the
marketplace and to assess the extent to
which different securities and portfolios
will respond to changes in these factors.
– A primary goal of security analysis is to
determine these factors and the sensitivities
of security return to movements in these
factors.
One-Factor Models
The one-factor models refer to the return-
generating process for securities involves a
single factor,These factors may be one of
the followings:
– The predicted growth rate in GDP
– The expected return on market index
– The growth rate of industrial produc-
tion,etc,
One-Factor Models
An example Page 295,Figure 11.1
G D Pf o r f a c t o r z e r o t h e
g r o w t h G D P p r e d i c t e d W i d g e t t o ofy s e n s i t i v i t
tp e r i o d inW i d g e t on r e t u r n s p e c i f i c oe u n i q u e t h e
tp e r i o d in G D P in r e t u r n of r a t e p r e d i c t e d t h e
tp e r i o d inW i d g e t on r e t u r n t h e
:w h e r e
a
b
e
G D P
r
eb G D Par
t
t
t
ttt
One-Factor Models
Generalizing the example
– Assumptions
The random error term and the factor are
uncorrelated,(Why?)
The random error terms of any two
securities are uncorrelated,(Why?)
ittiiit eFbar
One-Factor Models
– Expected return
– Variance
– Covariance
Fbar iii
2222
eiFii b
2
Fjiij bb
One-Factor Models
Two important features of one-factor model
– The tangency portfolio is easy to get.
The returns on all securities respond to a
single common factor greater simplifies the
task of identifying the tangency portfolio.
The common responsiveness of securities to
the factor eliminates the need to estimate
directly the covariances between the
securities.
The number of estimates,3N+2
One-Factor Models
– The feature of diversification is true of
any one-factor model.
Factor risk:
Nonfactor risk:
Diversification leads to an averaging of
factor risk
Diversification reduces nonfactor risk
)( 22 Fib?
2ei?
One-Factor Models
NN
N
XbXb
b
eNee
N
i
eiep
N
i
eiiep
N
i
iip
epFpp
22
2
2
1
1
2
2
2
1
222
1
2222
1
1
:w h e r e
Multiple-Factor Models
The health of the economy effects most
firms,but the economy is not a simple,
monolithic entity,Several common
influences with pervasive effects might be
identified
– The growth rate of GDP
– The level of interest rate
– The inflation rate
– The level of oil price
Multiple-Factor Models
Two-Factor Models
– Assume that the return-generating
process contains two factors.
ittitiiit eFbFbar 2211
tttt eI N FbG D Pbar 21
Multiple-Factor Models
The second equation provides a two-factor
model of a company’s stock,whose returns
are affected by expectations concerning
both the growth rate in GDP and the rate of
inflation.
Page 301,Figure 11.2
To this scatter of points is fit a two-
dimensional plane by using the statistical
technique of multiple-regression analysis.
Multiple-Factor Models
– Four parameters need to be estimated
for each security with the two-factor
model,ai,bi1,bi2,and the standard
deviation of the random error term.
– For each of the factors,two parameters
need to be estimated,These parameters
are the expected value of each factor
and the variance of each factor,Finally,
the covariance between factors.
Multiple-Factor Models
– Expected return
– Variance
– Covariance
2211 FbFbar iiii
2
2121
2
2
2
2
2
1
2
1
2 ),(2
eiiiFiFii FFC O Vbbbb
),()( 2112212 2222 111 FFC O Vbbbbbbbb jijiFjiFjiij
Multiple-Factor Models
– The tangency portfolio
The investor can proceed to use an
optimizer to derive the curve efficient set.
– Diversification
Diversification leads to an averaging of
factor risk.
Diversification can substantially reduce
nonfactor risk.
For a well-diversified portfolio,nonfactor
risk will be insignificant.
Multiple-Factor Models
pttptpp
N
i
itit
N
i
iit
N
i
ii
N
i
ii
N
i
ittitiii
N
i
itipt
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eFbFbaX
rXr
2211
1
2
1
21
1
1
1
1
2211
1
)(
Multiple-Factor Models
Sector-Factor Models
– Sector-factor models are based on the
acknowledge that the prices of securities
in the same industry or economic sector
often move together in response to
changes in prospects for that sector.
– To create a sector-factor model,each
security must be assigned to a sector.
Multiple-Factor Models
A two-sector-factor model
– There are two sectors and each security
must be assigned to one of them.
– Both the number of sectors and what
each sector consists of is an open matter
that is left to the investor to decide.
– The return-generating process for
securities is of the same general form as
the two-factor model,
Multiple-Factor Models
– Differing from the two-factor model,
with two-sector-factor model,F1 and F2
now denote sector-factors 1 and 2,
respectively,Any particular security
belongs to either sector-factor 1 or
sector-factor 2 but not both.
jjjj
iiii
eFbar
eFbar
22
11
Multiple-Factor Models
– In general,whereas four parameters
need to be estimated for each security
with a two-factor model (ai1,bi1,bi2,?ei,),
only three parameters need to be
estimated with a two-sector-factor
model,(ai1,?ei,and eitherbi1 or bi2 ).
Multiple-factor models
itktiktitiiit eFbFbFbar2211
Estimating Factor Models
There are many methods of estimating
factor models,There methods can be
grouped into three major approaches:
– Time-series approaches
– Cross-sectional approaches
– Factor-analytic approaches
Factor Models and Equilibrium
A factor model is not an equilibrium model
of asset pricing.
Both equation show that the expected return
on the stock is related to a characteristic of
the stock,bi or?i,The larger the size of the
characteristic,the larger the asset’s return.
)( fMiMfi
iii
rrrr
Fbar
Factor Models and Equilibrium
– The key difference is ai and rf.
The only characteristic of the stock that
determine its expected return according
to the CAPM is?ii,as rff denotes the risk-
free rate and is the same for all securities.
With the factor model,there is a second
characteristic of the stock that needs to be
estimated to determine the stock’s
expected return,aii.
Factor Models and Equilibrium
As the size of ai differs from one stock to
another,it presents the factor model from
being an equilibrium model.
Two stocks with the same value of bi can
have dramatically different expected
returns according to a factor model.
Two stocks with the same value of?i will
have the same expected return according
to the equilibrium-based CAPM.
Factor Models and Equilibrium
The relationship between the parameters
ai and bi of the one-factor model and the
single parameter?i of the CAPM.
– If the expected returns are determined
according to the CAPM and actual
returns are generated by the one-factor
market model,then the above
equations must be true,
)( fMiMfi
iii
rrrr
Fbar
Arbitrage Pricing Theory
APT is a theory which describes how a
security is priced just like CAPM.
– Moving away from construction of
mean-variance efficient portfolio,APT
instead calculates relations among
expected rates of return that would
rule out riskless profits by any investor
in well-functioning capital markets.
Arbitrage Pricing Theory
APT makes few assumptions.
– One primary assumption is that each
investor,when given the opportunity to
increase the return of his or her portfolio
without increasing its risk,will proceed
to do so.
There exists an arbitrage opportunity and
the investor can use an arbitrage portfolios.
Arbitrage Opportunities
Arbitrage is the earning of riskless profit
by taking advantage of differential
pricing for the same physical asset or
security,
– It typically entails the sale of a security
at a relatively high price and the
simultaneous purchase of the same
security (or its functional equivalent)
at a relatively low price.
Arbitrage Opportunities
– Arbitrage activity is a critical element
of modern,efficient security markets.
– It takes relatively few of this active
investors to exploit arbitrage situations
and,by their buying and selling actions,
eliminate these profit opportunities.
– Some investors have greater resources
and inclination to engage I arbitrage
than others.
Arbitrage Opportunities
Zero-investment portfolio
– A portfolio of zero net value,established
by buying and shorting component
securities,
– A riskless arbitrage opportunity arises
when an investor can construct a zero-
investment portfolio that will yield a
sure profit.
Arbitrage Opportunities
– To construct a zero-investment portfolio,
one has to be able to sell short at least
one asset and use the proceeds to
purchase on or more assets.
– Even a small investor,using borrowed
money in this case,can take a large
position in such a portfolio.
– There are many arbitrage tactics.
Arbitrage Opportunities
An example:
– Four stocks and four possible scenarios
– the rate of return in four scenarios
– Page 180-181 in the textbook
– The expected returns,standard
deviations and correlations do not reveal
any abnormality to the naked eye.
Arbitrage Opportunities
The critical property of an arbitrage
portfolio is that any investor,regardless of
risk aversion or wealth,will want to take
an infinite position in it so that profits will
be driven to an infinite level.
– These large positions will force some
prices up and down until arbitrage
opportunities vanishes,
Factor Models and
Principle of Arbitrage
Almost arbitrage opportunities can
involve similar securities or portfolios.
– That similarity can be defined in many
ways.
– One way is the exposure to pervasive
factors that affect security prices.
– An example Page 324
Factor Models and
Principle of Arbitrage
A factor model implies that securities or
portfolios with equal-factor sensitivities
will behave in the same way except for
nonfactor risk.
– APT starts out by making the assumption
that security returns are related to an
unknown number of unknown factors.
– Securities with the same factor sensitivities
should offer the same expected returns.
Arbitrage Portfolios
An arbitrage portfolio must satisfy:
– A net market value of zero
– No sensitivity to any factor
– A positive expected return
0321 XXX
0332211 XbXbXb
0332211 rXrXrX
Arbitrage Portfolios
The arbitrage portfolio is attractive to
any investor who desires a higher return
and is not concerned with nonfactor risk.
It requires no additional dollar investment,
it has no factor risk,and it has a positive
expected return.
One-Factor Model and APT
Pricing effects on arbitrage portfolio
– The buying-and-selling activity will
continue until all arbitrage possibilities
are significant reduced or eliminated
– There will exist an approximately linear
relationship between expected returns
and sensitivities of the following sort:
ii br 10
One-Factor Model and APT
– The equation is the asset pricing equation
of the APT when returns are generated
by one factor
The linear equation means that in equili-
brium there will be a linear relationship
between expected returns and sensitivities.
The expected return on any security is,in
equilibrium,a linear function of the
security’s sensitivity to the factor,bi
One-Factor Model and APT
– Any security that has a factor
sensitivity and expected return such
that it lies off the line will be mispriced
according to the APT and will present
investors with the opportunity of
forming arbitrage portfolios.
– Page 327,Figure 12.1
One-Factor Model and APT
Interpreting the APT pricing equation
– Riskfree asset,rf
– Pure factor portfolio,p* ififi brrrr 10
11 fpfp rrrr
iffip brrrr 11,l e t
Two-Factor Model And APT
The two-factor model
Arbitrage portfolios
– A net market value of zero
– No sensitivity to any factor
– A positive expected return
iiiii eFbFbar 2211
22110 iii bbr
Two-Factor Model And APT
Pricing effects
22
11
0
2211
r a t e r i s k f r e e
)()(
f
f
f
ififf
i
r
r
r
brbrrr
Two-Factor Model And APT
–?1 is the expected return on the portfolio
which is known as a pure factor portfolio
or pure factor play,because it has:
Unit sensitivity to one factor (F1,b1=1)
No sensitivity to any other factor (F2,b2=0)
Zero nonfactor risk
This portfolio is a well-diversification
portfolio that has unit sensitivity to the first
factor and zero sensitivity to the second
factor.
Two-Factor Model And APT
– It is the same with?2,It is the well-
diversification portfolio that has zero
sensitivity to the first factor and unit
sensitivity to the second factor,meaning
that it has b1=0 and b2=1.
– Such as a portfolio that has zero sensitivity
to predicted industrial production and
unit sensitivity to predicted inflation
would have an expected return of 6%.
Multiple-factor model
The APT pricing equation
Multiple-Factor Model
And APT
ikikiiii eFbFbFbar2211
ikkiii bbbr22110
ikfkiffi brbrrr )()( 11
The APT And The CAPM
Common point
– Both require equilibrium
– Both have almost similar equation
Distinctions
– Different equilibrium mechanism
Many investors v.s,Few investors
– Different Portfolio
Market portfolio v.s,Well-diversifyed P.
Summary
The Factor Models
– One-factor models
– Multi-factor models
Factor models and equilibrium
Arbitrage opportunity and portfolio
The arbitrage pricing equation
– One-factor equation
– Multi-factor equation
Assignments For chapter 8
Readings
– Page 282 through 301
– Page 308 through 321
Exercises
– Page 304,14,15; Page 323,4,13
Q/A:
– Page 302,3
– Page 324,8
