Modern Portfolio Theory
The Capital Asset Pricing Model
By Ding zhaoyong
Main Contents
The CAPM’s Assumptions
Market equilibrium
The Capital Market Line (CML)
The Security Market Line (SML)
The market model
The Security Characteristic Line (SCL)
The Capital Allocation Line (CAL)
How An Investor Makes
Investment Decision
First,identify his or her efficient frontier
– Estimate the expected returns and variances
for all securities under consideration.
– Estimate all the covariances among assets.
– Determine the riskfree rate
– Determine the tangency portfolio
– Consequently,the efficient set is linear.
How An Investor Makes
Investment Decision
Second,Identify his or her indifference
curve.
– Test his or her preference on risk-return
and risk tolerance.
Finally,identify his or her optimal portfolio
– Let the indifference curve touches but
does not intersect the efficient set.
How An Investor Makes
Investment Decision
The problem is how the investor evaluate the
expected returns and risks when they make
investment decisions.
We hereafter introduce some theories or
models about how the value of an asset is
determined or priced.
– The Capital Asset Pricing Model (CAPM)
– The Factor Models (FM)
– The Arbitrary Pricing Theory (APT)
The Assumptions of CAPM
Investors evaluate portfolios by looking at the
expected returns and standard deviations of the
portfolios over a one-period horizon.
Investors are never satiated,so when given a
choice between two otherwise identical portfolios,
they will choose the one with the higher expected
return.
Investors are risk-averse,so when given a choice
between two otherwise identical portfolios,they
will choose the one with the lower standard
deviation.
The Assumptions of CAPM
Individual assets are infinitely divisible,
meaning that an investor can by a fraction of
a share if he or she so desires.
There is a riskfree rate at which an investor
may either lend or borrow money.
Taxes and transaction costs are irrelevant.
All investors have the same one-period
horizon.
The Assumptions of CAPM
The riskfree rate is the same for all investors
Information is freely and instantly available
to all investors
Investors have homogeneous expectations,
meaning that they have the same perceptions
in regard to the expected returns,standard
deviation,and covariances of securities.
The Assumptions of CAPM
Implicit means behind the assumptions
– Efficient Market
Everyone has the same information and
agrees about the future prospects for assets
– Perfect Market
There are on frictions to impede investing
– Equilibrium Market
Risk-return trade-off
The Capital Market Line
The Separation Theorem
– Everyone would obtain in equilibrium
the same tangency.
All investors face the same efficient set.
– Each investor will choose the same
combination of risky securities
Each investor will spread his or her funds
among risky securities in the same relative
proportions.
The Capital Market Line
E(rp)
CML
M
E(rM)
M?
The Capital Market Line
The Separation Theorem
The optimal combination of risky
assets for an investor can be deter-
mined without any knowledge of
the investor’s preferences toward
risk and return.
The Capital Market Line
The determination of the optimal
combination if risky assets can be made
without determining the shape of an
investor’s indifference curve,
The reasoning behind the separation
theorem (?)
– A property of the linear efficient set.
– An important feature of CAPM.
The Capital Market Line
The Market Portfolio
– According CAPM,in equilibrium each
security must has a nonzero proportion
in the composition of the tangency
portfolio.
It results from separation theorem which
assert that the risky portion of every
investor’s portfolio is independent of the
investor;s risk-return preference.
The Capital Market Line
When the market approaches equilibrium
– Each investor want to hold a certain positive
amount of each risky security.
– The current market price of each security will
be at a level where the number of shares
demanded equals the number of shares of
shares outstanding.
– The riskfree rate will be at a level where the
total amount of money borrowed equals the
total amount of money debt.
The Capital Market Line
E(rp)
CML
M
E(rM)
M?
The Capital Market Line
The Market Portfolio
The market portfolio is a portfolio consist-
ing of all securities where the proportion
invested in each security corresponds to its
relative market value,The relative market
value of a security is simply equal to the
aggregate market value of the security
divided by the sum of the aggregate market
values of all securities,
The Capital Market Line
The efficient set in equilibrium
– The efficient set consists of an invest-
ment in the market portfolio,coupled
with a desired amount of either riskfree
borrowing or lending in the CAPM.
– All portfolios other than the market
portfolio and riskfree borrowing or
lending would be below the efficient set.
The Capital Market Line
The Capital Market Line
– The portfolio P consists of the market
portfolio and riskfree assets only.
– The linear efficient set of the CAPM is
defined as the CML,The straight line
characterizing the CML has the follow-
ing equation:
p
M
fM
fp
rrE
rrE?
)(
)(
The Capital Market Line
Equilibrium in the security market can
be characterized by two key numbers:
– The reward for waiting,the vertical
intercept of the CML.
– The reward per unit of risk borne,the
slope of the CML.
– The security market provides a place
where time and risk can be traded with
their prices determined by the forces of
supply and demand.
The Capital Market Line
E(rp)
CML
M
E(rM)
M?
The Security Market Line
We know:
– The CML represents the equilibrium
relationship between the expected
return and standard deviation for
efficient portfolios.
– Individual risky securities will always
plot below the line.
How is an individual security priced?
The Security Market Line
2
1
1 1
N
i
N
j
ijjMiMM XX
2
1
11
22
1
11?
N
j
NjjMNM
N
j
jjMM
N
j
jjMMM XXXXXX
N
j
ijjMiM X
1
212211 NMNMMMMMM XXX
The Security Market Line
The covariance of individual security
with the market portfolio
– The standard deviation of the market
portfolio is equal to the square root of
a weighted average of the covariances
of all the securities with it,where the
weights are equal to the proportions of
the respective securities in the market
portfolio.
The Security Market Line
– The contribution of each security to the
standard deviation of the market
portfolio depend on the size of its
covariance with the market portfolio.
– Accordingly each investor will note that
the relevant measure of risk for a
security is its covariance with the
market portfolio,?iM.
The Security Market Line
– The relevant measure of risk for a
security means that securities with large
values of?iM will be viewed by investors
as contributing more to the risk of the
market portfolio.
– It also means that securities with larger
standard deviation should not be viewed
as necessarily adding more risk to the
market portfolio than those securities
with smaller standard deviations.
The Security Market Line
From the above analysis it follows that
securities with larger values for?iM have
to provide proportionately larger expected
returns to interest investors in purchase
them.
The exact form of the equilibrium relation-
ship between risk and return is,
iM
M
fM
fi
rrE
rrE?
2
)(
)(
The Security Market Line
E(ri)
SML
E(rM) M
rf
2M?iM
The Security Market Line
The Security Market Line
– The above relationship between
covariance and expected return is
defined as the SML.
– A risky security with?iM=0 will have
an expected return equal to the rate on
the riskfree security,rf,(WHY?)
– It is even possible for some risky assets
to have expected return less than rf,
The Security Market Line
Beta Coefficient
– The SML can also express as follow:
– The term?iM is defined as,
iMfMfi rrErrE?])([)(
2
M
iM
iM?
The Security Market Line
E(ri)
SML
E(rM) M
rf
M=1?iM
The Security Market Line
The Beta of a portfolio
– The beta of a portfolio is simply a
weighted average of the beta of its
component securities,where the
proportions invested in the securities
are the respective weights.
N
i
iMipM X
1
The Security Market Line
The SML and CML
– Efficient portfolios plot on both the
CML and SML,but inefficient portfolios
plot on the SML but below the CML.
– The SML expresses return-beta relation-
ship and the CML expresses return-
covariance relationship.
– Both of them are based on the market
equilibrium,
The Security Market Line
The equilibrium relationship shown by the
SML comes to exist through the combined
effects of investor’s adjustments in
holdings and the resulting pressures on
security prices.
Equilibrium requires the adjustment of
each security’s price until there is contsis-
tency between the quantity desired and
the quantity available.
The Security Market Line
An example on the SML
– The market portfolio
Expected return=22.4%
Standard deviation=15.2%
– The riskfree rate=4%
– Risky assets,Able,Baker and Charlie
– Excel Sheet
The Security Market Line
The Alpha of a security
– The security’s Alpha is the difference
between the fair and actually expected
rate of return on it.
– It is a abnormal rate of return on a
security in excess of what would be
predicted by an equilibrium model
such as the CAPM or APT,
The Security Market Line
E(ri)
SML
E(rM) M
rf
M=1?iM
The Security Market Line
– Whenever the CAPM holds,all securi-
ties must lie on the SML in market
equilibrium.
– Underpriced securities plot above the
SML.
– Overpriced securities plot below the
SML.
– An example,Page 172 in textbook
The Security Market Line
The SML and stocks’ Alphas
The Security Market Line
Applications of the CAPM
– Investment management industry
The SML is taken as a benchmark to assess
the fair expected return on a risky asset and
to evaluate the investment performance.
– Capital budgeting decision
Managers can use the CAPM to obtain the
cutoff IRR or hurdle rate for a project.
– Regulation decision in utilities,
The CAPM and Market Model
The market model is a factor model,how-
ever the CAPM is an equilibrium model.
The market model utilizes a market index
(actual),whereas the CAPM involves the
market portfolio (theoretical).
The market model deals with the actual
returns,otherwise the CAPM deals with
the expected returns.
The beta iI differs from the beta im
The CAPM and Market Model
Move from a model cast in expectation to
a realized return framework
– The single-index (market) modelifMiifi rrrr )(
ifMiifi rrrr )(
iIIiIiIi rr,or
The CAPM and Market Model
– Thus,we can convert the CAPM prediction
about unobserved expectations of security
returns relative to an unobserved market
portfolio into a prediction about the intercept
in a regression of observed variables,
Realized excess returns of a security relative
to those of a specified index.
])([)( fMiifi rrErrE
The CAPM and Market Model
Estimate the Market index model
– The single-index linear regression
model.
– The parameters of riskfree rate,?,?
and Var(e) can be estimated using
standard regression techniques.
i M tMti M ti M ti M t rR
The CAPM and Market Model
The Security Characteristic Line (SCL)
– A plot of a security’s expected excess
return over the riskfree rate as a function
of the excess return on the market.
– Systematic risk,the Beta of the regression
equation.
– Nonsystematic risk,the residual variance
of the regression.
The CAPM and Market Model
RiMt
SCL
RMt
The CAPM and Market Model
Predicting Beta
– The Beta from the regression equation is
an estimate based on past history.
– The Beta can be forecasted by a weighted
average of the sample estimate with the
value 1.0
The Beta exhibit a statistical property called
regression toward the mean.
The CAPM and Market Model
– It also can be forecasted by the weight
assigned to the sample estimate of beta
on its statistical reliability.
– More precise estimates of beta could be
obtained by using more data,usually a
long time series of returns on stock.
– The other techniques include some
models,such as ARCH models.
The Conclusion Of CAPM
Market risk is related to the risk of the
market portfolio and to the Beta of the
security in question.
Securities with larger betas will have
larger amounts of market risk,In the
world of the CAPM,securities with larger
betas will have larger expected returns,So
the securities with larger market risks
should have larger expected returns.
The Conclusion Of CAPM
Non-market risk is not related to beta,
This means there is no reason why
securities with larger amounts of non-
market risks should have larger expected
returns.
Thus according to the CAPM,investors
are rewarded for bearing market risk but
not for bearing non-market risk.
Summary
The assumptions of CAPM
The CML
– The Separation Theorem
– The Market Portfolio
The SML
– The Beta
The SCL
– The Alpha
The Capital Asset Pricing Model
By Ding zhaoyong
Main Contents
The CAPM’s Assumptions
Market equilibrium
The Capital Market Line (CML)
The Security Market Line (SML)
The market model
The Security Characteristic Line (SCL)
The Capital Allocation Line (CAL)
How An Investor Makes
Investment Decision
First,identify his or her efficient frontier
– Estimate the expected returns and variances
for all securities under consideration.
– Estimate all the covariances among assets.
– Determine the riskfree rate
– Determine the tangency portfolio
– Consequently,the efficient set is linear.
How An Investor Makes
Investment Decision
Second,Identify his or her indifference
curve.
– Test his or her preference on risk-return
and risk tolerance.
Finally,identify his or her optimal portfolio
– Let the indifference curve touches but
does not intersect the efficient set.
How An Investor Makes
Investment Decision
The problem is how the investor evaluate the
expected returns and risks when they make
investment decisions.
We hereafter introduce some theories or
models about how the value of an asset is
determined or priced.
– The Capital Asset Pricing Model (CAPM)
– The Factor Models (FM)
– The Arbitrary Pricing Theory (APT)
The Assumptions of CAPM
Investors evaluate portfolios by looking at the
expected returns and standard deviations of the
portfolios over a one-period horizon.
Investors are never satiated,so when given a
choice between two otherwise identical portfolios,
they will choose the one with the higher expected
return.
Investors are risk-averse,so when given a choice
between two otherwise identical portfolios,they
will choose the one with the lower standard
deviation.
The Assumptions of CAPM
Individual assets are infinitely divisible,
meaning that an investor can by a fraction of
a share if he or she so desires.
There is a riskfree rate at which an investor
may either lend or borrow money.
Taxes and transaction costs are irrelevant.
All investors have the same one-period
horizon.
The Assumptions of CAPM
The riskfree rate is the same for all investors
Information is freely and instantly available
to all investors
Investors have homogeneous expectations,
meaning that they have the same perceptions
in regard to the expected returns,standard
deviation,and covariances of securities.
The Assumptions of CAPM
Implicit means behind the assumptions
– Efficient Market
Everyone has the same information and
agrees about the future prospects for assets
– Perfect Market
There are on frictions to impede investing
– Equilibrium Market
Risk-return trade-off
The Capital Market Line
The Separation Theorem
– Everyone would obtain in equilibrium
the same tangency.
All investors face the same efficient set.
– Each investor will choose the same
combination of risky securities
Each investor will spread his or her funds
among risky securities in the same relative
proportions.
The Capital Market Line
E(rp)
CML
M
E(rM)
M?
The Capital Market Line
The Separation Theorem
The optimal combination of risky
assets for an investor can be deter-
mined without any knowledge of
the investor’s preferences toward
risk and return.
The Capital Market Line
The determination of the optimal
combination if risky assets can be made
without determining the shape of an
investor’s indifference curve,
The reasoning behind the separation
theorem (?)
– A property of the linear efficient set.
– An important feature of CAPM.
The Capital Market Line
The Market Portfolio
– According CAPM,in equilibrium each
security must has a nonzero proportion
in the composition of the tangency
portfolio.
It results from separation theorem which
assert that the risky portion of every
investor’s portfolio is independent of the
investor;s risk-return preference.
The Capital Market Line
When the market approaches equilibrium
– Each investor want to hold a certain positive
amount of each risky security.
– The current market price of each security will
be at a level where the number of shares
demanded equals the number of shares of
shares outstanding.
– The riskfree rate will be at a level where the
total amount of money borrowed equals the
total amount of money debt.
The Capital Market Line
E(rp)
CML
M
E(rM)
M?
The Capital Market Line
The Market Portfolio
The market portfolio is a portfolio consist-
ing of all securities where the proportion
invested in each security corresponds to its
relative market value,The relative market
value of a security is simply equal to the
aggregate market value of the security
divided by the sum of the aggregate market
values of all securities,
The Capital Market Line
The efficient set in equilibrium
– The efficient set consists of an invest-
ment in the market portfolio,coupled
with a desired amount of either riskfree
borrowing or lending in the CAPM.
– All portfolios other than the market
portfolio and riskfree borrowing or
lending would be below the efficient set.
The Capital Market Line
The Capital Market Line
– The portfolio P consists of the market
portfolio and riskfree assets only.
– The linear efficient set of the CAPM is
defined as the CML,The straight line
characterizing the CML has the follow-
ing equation:
p
M
fM
fp
rrE
rrE?
)(
)(
The Capital Market Line
Equilibrium in the security market can
be characterized by two key numbers:
– The reward for waiting,the vertical
intercept of the CML.
– The reward per unit of risk borne,the
slope of the CML.
– The security market provides a place
where time and risk can be traded with
their prices determined by the forces of
supply and demand.
The Capital Market Line
E(rp)
CML
M
E(rM)
M?
The Security Market Line
We know:
– The CML represents the equilibrium
relationship between the expected
return and standard deviation for
efficient portfolios.
– Individual risky securities will always
plot below the line.
How is an individual security priced?
The Security Market Line
2
1
1 1
N
i
N
j
ijjMiMM XX
2
1
11
22
1
11?
N
j
NjjMNM
N
j
jjMM
N
j
jjMMM XXXXXX
N
j
ijjMiM X
1
212211 NMNMMMMMM XXX
The Security Market Line
The covariance of individual security
with the market portfolio
– The standard deviation of the market
portfolio is equal to the square root of
a weighted average of the covariances
of all the securities with it,where the
weights are equal to the proportions of
the respective securities in the market
portfolio.
The Security Market Line
– The contribution of each security to the
standard deviation of the market
portfolio depend on the size of its
covariance with the market portfolio.
– Accordingly each investor will note that
the relevant measure of risk for a
security is its covariance with the
market portfolio,?iM.
The Security Market Line
– The relevant measure of risk for a
security means that securities with large
values of?iM will be viewed by investors
as contributing more to the risk of the
market portfolio.
– It also means that securities with larger
standard deviation should not be viewed
as necessarily adding more risk to the
market portfolio than those securities
with smaller standard deviations.
The Security Market Line
From the above analysis it follows that
securities with larger values for?iM have
to provide proportionately larger expected
returns to interest investors in purchase
them.
The exact form of the equilibrium relation-
ship between risk and return is,
iM
M
fM
fi
rrE
rrE?
2
)(
)(
The Security Market Line
E(ri)
SML
E(rM) M
rf
2M?iM
The Security Market Line
The Security Market Line
– The above relationship between
covariance and expected return is
defined as the SML.
– A risky security with?iM=0 will have
an expected return equal to the rate on
the riskfree security,rf,(WHY?)
– It is even possible for some risky assets
to have expected return less than rf,
The Security Market Line
Beta Coefficient
– The SML can also express as follow:
– The term?iM is defined as,
iMfMfi rrErrE?])([)(
2
M
iM
iM?
The Security Market Line
E(ri)
SML
E(rM) M
rf
M=1?iM
The Security Market Line
The Beta of a portfolio
– The beta of a portfolio is simply a
weighted average of the beta of its
component securities,where the
proportions invested in the securities
are the respective weights.
N
i
iMipM X
1
The Security Market Line
The SML and CML
– Efficient portfolios plot on both the
CML and SML,but inefficient portfolios
plot on the SML but below the CML.
– The SML expresses return-beta relation-
ship and the CML expresses return-
covariance relationship.
– Both of them are based on the market
equilibrium,
The Security Market Line
The equilibrium relationship shown by the
SML comes to exist through the combined
effects of investor’s adjustments in
holdings and the resulting pressures on
security prices.
Equilibrium requires the adjustment of
each security’s price until there is contsis-
tency between the quantity desired and
the quantity available.
The Security Market Line
An example on the SML
– The market portfolio
Expected return=22.4%
Standard deviation=15.2%
– The riskfree rate=4%
– Risky assets,Able,Baker and Charlie
– Excel Sheet
The Security Market Line
The Alpha of a security
– The security’s Alpha is the difference
between the fair and actually expected
rate of return on it.
– It is a abnormal rate of return on a
security in excess of what would be
predicted by an equilibrium model
such as the CAPM or APT,
The Security Market Line
E(ri)
SML
E(rM) M
rf
M=1?iM
The Security Market Line
– Whenever the CAPM holds,all securi-
ties must lie on the SML in market
equilibrium.
– Underpriced securities plot above the
SML.
– Overpriced securities plot below the
SML.
– An example,Page 172 in textbook
The Security Market Line
The SML and stocks’ Alphas
The Security Market Line
Applications of the CAPM
– Investment management industry
The SML is taken as a benchmark to assess
the fair expected return on a risky asset and
to evaluate the investment performance.
– Capital budgeting decision
Managers can use the CAPM to obtain the
cutoff IRR or hurdle rate for a project.
– Regulation decision in utilities,
The CAPM and Market Model
The market model is a factor model,how-
ever the CAPM is an equilibrium model.
The market model utilizes a market index
(actual),whereas the CAPM involves the
market portfolio (theoretical).
The market model deals with the actual
returns,otherwise the CAPM deals with
the expected returns.
The beta iI differs from the beta im
The CAPM and Market Model
Move from a model cast in expectation to
a realized return framework
– The single-index (market) modelifMiifi rrrr )(
ifMiifi rrrr )(
iIIiIiIi rr,or
The CAPM and Market Model
– Thus,we can convert the CAPM prediction
about unobserved expectations of security
returns relative to an unobserved market
portfolio into a prediction about the intercept
in a regression of observed variables,
Realized excess returns of a security relative
to those of a specified index.
])([)( fMiifi rrErrE
The CAPM and Market Model
Estimate the Market index model
– The single-index linear regression
model.
– The parameters of riskfree rate,?,?
and Var(e) can be estimated using
standard regression techniques.
i M tMti M ti M ti M t rR
The CAPM and Market Model
The Security Characteristic Line (SCL)
– A plot of a security’s expected excess
return over the riskfree rate as a function
of the excess return on the market.
– Systematic risk,the Beta of the regression
equation.
– Nonsystematic risk,the residual variance
of the regression.
The CAPM and Market Model
RiMt
SCL
RMt
The CAPM and Market Model
Predicting Beta
– The Beta from the regression equation is
an estimate based on past history.
– The Beta can be forecasted by a weighted
average of the sample estimate with the
value 1.0
The Beta exhibit a statistical property called
regression toward the mean.
The CAPM and Market Model
– It also can be forecasted by the weight
assigned to the sample estimate of beta
on its statistical reliability.
– More precise estimates of beta could be
obtained by using more data,usually a
long time series of returns on stock.
– The other techniques include some
models,such as ARCH models.
The Conclusion Of CAPM
Market risk is related to the risk of the
market portfolio and to the Beta of the
security in question.
Securities with larger betas will have
larger amounts of market risk,In the
world of the CAPM,securities with larger
betas will have larger expected returns,So
the securities with larger market risks
should have larger expected returns.
The Conclusion Of CAPM
Non-market risk is not related to beta,
This means there is no reason why
securities with larger amounts of non-
market risks should have larger expected
returns.
Thus according to the CAPM,investors
are rewarded for bearing market risk but
not for bearing non-market risk.
Summary
The assumptions of CAPM
The CML
– The Separation Theorem
– The Market Portfolio
The SML
– The Beta
The SCL
– The Alpha
