Modern Portfolio Theory
Portfolio Choice
Chapter 6
By Ding zhaoyong
Main Contents
The efficient set theorem
Concavity of the efficient set
The market Model
Diversification and market risk
Allowing for riskfree lending
Allowing for risk free borrowing
Allowing for both riskfree borrowing and
lending
How to make portfolio choice
How can Markowitz’s approach be used
to select portfolio once it is recognized
that there are an infinite number of
portfolio available for investment?
What happens when the investor
considers investing in a set of securities,
one of which is riskless?
What happens if the investor is allowed
to buy securities on margin?
The efficient set theorem can be stated:
An investor will choose his or her
optimal portfolio from the set of portfolios
that
1,Offer maximum expected return for
varying levels of risks,and
2,Offer minimum risk for varying levels of
expected returns.
The Efficient Set Theorem
The Efficient Set Theorem
The feasible set
– The feasible set is also called for the
opportunity set,which represents all
portfolios that could be formed from a
group of N securities.
– All possible portfolios that could be
formed from N securities lie either on
or within the boundary of the feasible
set,which has a umbrella-type shape.
E(rP) s
H
Feasible
E set
G
The Efficient Set Theorem
Efficient
frontier
The Efficient Set Theorem
The efficient frontier
– The efficient frontier or efficient set is
a set of portfolio which meets both
conditions of the efficient set theorem.
– It is from this set of efficient portfolios
that the investor will find his or her
optimal one.
– All the other feasible portfolios are
inefficient portfolio and can be ignored
The Efficient Set Theorem
The optimal portfolio
– The optimal portfolio correspond to
the point where an indifference curve
is just tangent to the efficient set,
(WHY?)
– There will be only one tangency point
between the investor’s indifference
curves and the efficient set,(WHY?)
The Efficient Set Theorem
I3 I2
E(rP) I1 S
H
O*
Feasible
E set
G
Portfolio With Two Risky Assets
The two risky securities
Security 1,E(r1)=5%,?1=20%
Security 2,E(r2)=15%,?2=40%
The bounds on the location of portfolio
by calculating the expected returns and
standard deviations of the portfolios
under different correlation and weights.
Diversification leads to risk reduction.
Diversification
According to the market model,the total
risk of any security I,measured by its
variance,consists of two parts,Market
(or systematic) risk and unique (or unsys-
tematic) risk:
2222
iIiIi
Diversification
The portfolio total risk
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Diversification
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Diversification
Now it can be shown that increased
diversification can lead to the reduction of a
portfolio’s total risk.
Diversification leads to average of market
risk
Diversification can substantially reduce
unique risk.
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2212
Diversification
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Unique Risk
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Total Risk Market Risk
N
Allowing For Riskless Lending
We continue the seeking of efficient
frontier and expand the Markowitz’s
assumption that the assets considered for
investment are individually risk.
The investor is allowed to use financial
leverage.
– S/he is allowed to invest in not only risky assets
but also in a riskfless asset.
– S/he is also allowed to borrow money to invest.
Allowing For Riskless Lending
– It can be showed that any combination of the
riskfree asset and Able will lie some-what on the
straight line connecting them; The exact
location will depend on the relative proportions
invested in these two asset,This means that any
portfolio consisting of a combination of the
riskfree asset and a risky asset will have an
improved expected return and standard
deviation characteristic.
Allowing For Riskless Lending
Investing in both the riskfree asset and a
risky portfolio,
– Combining the riskfree asset with any
risky portfolio can be viewed as being no
different from combining the riskfree
asset with an individual risky asset,The
resulting portfolio has an expected return
and standard deviation such that it lies
somewhat on a straight line connecting
the two endpoints.
Allowing For Riskless Lending
The effect of riskfree lending on the
efficient set
– The feasible set is changed significantly
as a result of the introduction of riskfree
lending.
– Instead of the pure curve,the efficient
set now consists of a straight-line
segment and a curved segment.
Allowing For Riskless Lending
The effect of riskfree lending on portfolio
selection
– How an investor would go about
selecting an optimal efficient portfolio
when there is a risk asset available for
investment in addition to a number of
risky assets?
Allowing For Riskless Borrowing
Riskfree borrowing
– Assume the investor is no longer
restricted to his or her initial wealth
when it comes time to decided how much
money to invest in risky assets.
– Assume the rate of interest charged on
the riskfree loan is equal to the rate of
interest that could be earned from
investing in the riskfree asset.
Allowing For Riskless Borrowing
Borrowing and Investing in a risky asset
– The borrowing at the riskfree rate and
investing all the borrowed money and
the investor’s own money in a risky
asset will result in a portfolio that has an
expected return and standard deviation
such that it lies on the extension of the
straight line connecting the riskfree rate
and the risky asset,
Allowing For Riskless Borrowing
Borrowing and investing in a risky portfolio
– Borrowing to purchase a risky portfolio is
no different from borrowing to purchase
an individual risky asset.
– In both cases,the resulting portfolio lies
on an extension of the connecting the
riskfree rate with the risky investment.
Allowing For Both
Borrowing And Lending
The Effect of riskfree borrowing and
lending on the efficient set
– The line going through T is just tangent
to the Markowitsz model efficient set,
None of the portfolios,except for T,that
were on the Markowitz model efficient
set are efficient when riskfree borrowing
and lending are induced.
Allowing For Both
Borrowing And Lending
The effect of riskfree borrowing and lending
on portfolio selection
– Given the opportunity to either borrow or
lend at the riskfree rate,an investor would
proceed to identify the optimal portfolio by
plotting his or her indifference curve on
this risk-return graph and noting where
one of them is tangent to the efficient set.
Summary
Portfolio with risky assets
– Feasible set and Efficient set
Portfolio selection with all risky assets
– Optimal portfolio
The market model and Efficient diversification
– Market risk v.s,Unique risk
– Security characteristic line
Summary
Riskfree lending and borrowing
– Efficient set with riskless lending and
borrowing
– The Capital Allocation Line (CAL)
– Optimal portfolio with riskless lending
and borrowing
– The separate theorem
Portfolio Choice
Chapter 6
By Ding zhaoyong
Main Contents
The efficient set theorem
Concavity of the efficient set
The market Model
Diversification and market risk
Allowing for riskfree lending
Allowing for risk free borrowing
Allowing for both riskfree borrowing and
lending
How to make portfolio choice
How can Markowitz’s approach be used
to select portfolio once it is recognized
that there are an infinite number of
portfolio available for investment?
What happens when the investor
considers investing in a set of securities,
one of which is riskless?
What happens if the investor is allowed
to buy securities on margin?
The efficient set theorem can be stated:
An investor will choose his or her
optimal portfolio from the set of portfolios
that
1,Offer maximum expected return for
varying levels of risks,and
2,Offer minimum risk for varying levels of
expected returns.
The Efficient Set Theorem
The Efficient Set Theorem
The feasible set
– The feasible set is also called for the
opportunity set,which represents all
portfolios that could be formed from a
group of N securities.
– All possible portfolios that could be
formed from N securities lie either on
or within the boundary of the feasible
set,which has a umbrella-type shape.
E(rP) s
H
Feasible
E set
G
The Efficient Set Theorem
Efficient
frontier
The Efficient Set Theorem
The efficient frontier
– The efficient frontier or efficient set is
a set of portfolio which meets both
conditions of the efficient set theorem.
– It is from this set of efficient portfolios
that the investor will find his or her
optimal one.
– All the other feasible portfolios are
inefficient portfolio and can be ignored
The Efficient Set Theorem
The optimal portfolio
– The optimal portfolio correspond to
the point where an indifference curve
is just tangent to the efficient set,
(WHY?)
– There will be only one tangency point
between the investor’s indifference
curves and the efficient set,(WHY?)
The Efficient Set Theorem
I3 I2
E(rP) I1 S
H
O*
Feasible
E set
G
Portfolio With Two Risky Assets
The two risky securities
Security 1,E(r1)=5%,?1=20%
Security 2,E(r2)=15%,?2=40%
The bounds on the location of portfolio
by calculating the expected returns and
standard deviations of the portfolios
under different correlation and weights.
Diversification leads to risk reduction.
Diversification
According to the market model,the total
risk of any security I,measured by its
variance,consists of two parts,Market
(or systematic) risk and unique (or unsys-
tematic) risk:
2222
iIiIi
Diversification
The portfolio total risk
N
i
iIipI
N
i
iIipI
N
i
iIipI
pIIpIpI
N
i
iIIiIiIiP
X
X
X
w he r e
r
rXr
1
1
1
1
:
)(
Diversification
So:
N
i
iip
N
i
iIipI
pIpIp
X
X
w h e r e
1
222
1
22
2222
)(
:
Diversification
Now it can be shown that increased
diversification can lead to the reduction of a
portfolio’s total risk.
Diversification leads to average of market
risk
Diversification can substantially reduce
unique risk.
NNN
i
iNp
N
22
2
2
11
1
2212
Diversification
p
Unique Risk
p?I
Total Risk Market Risk
N
Allowing For Riskless Lending
We continue the seeking of efficient
frontier and expand the Markowitz’s
assumption that the assets considered for
investment are individually risk.
The investor is allowed to use financial
leverage.
– S/he is allowed to invest in not only risky assets
but also in a riskfless asset.
– S/he is also allowed to borrow money to invest.
Allowing For Riskless Lending
– It can be showed that any combination of the
riskfree asset and Able will lie some-what on the
straight line connecting them; The exact
location will depend on the relative proportions
invested in these two asset,This means that any
portfolio consisting of a combination of the
riskfree asset and a risky asset will have an
improved expected return and standard
deviation characteristic.
Allowing For Riskless Lending
Investing in both the riskfree asset and a
risky portfolio,
– Combining the riskfree asset with any
risky portfolio can be viewed as being no
different from combining the riskfree
asset with an individual risky asset,The
resulting portfolio has an expected return
and standard deviation such that it lies
somewhat on a straight line connecting
the two endpoints.
Allowing For Riskless Lending
The effect of riskfree lending on the
efficient set
– The feasible set is changed significantly
as a result of the introduction of riskfree
lending.
– Instead of the pure curve,the efficient
set now consists of a straight-line
segment and a curved segment.
Allowing For Riskless Lending
The effect of riskfree lending on portfolio
selection
– How an investor would go about
selecting an optimal efficient portfolio
when there is a risk asset available for
investment in addition to a number of
risky assets?
Allowing For Riskless Borrowing
Riskfree borrowing
– Assume the investor is no longer
restricted to his or her initial wealth
when it comes time to decided how much
money to invest in risky assets.
– Assume the rate of interest charged on
the riskfree loan is equal to the rate of
interest that could be earned from
investing in the riskfree asset.
Allowing For Riskless Borrowing
Borrowing and Investing in a risky asset
– The borrowing at the riskfree rate and
investing all the borrowed money and
the investor’s own money in a risky
asset will result in a portfolio that has an
expected return and standard deviation
such that it lies on the extension of the
straight line connecting the riskfree rate
and the risky asset,
Allowing For Riskless Borrowing
Borrowing and investing in a risky portfolio
– Borrowing to purchase a risky portfolio is
no different from borrowing to purchase
an individual risky asset.
– In both cases,the resulting portfolio lies
on an extension of the connecting the
riskfree rate with the risky investment.
Allowing For Both
Borrowing And Lending
The Effect of riskfree borrowing and
lending on the efficient set
– The line going through T is just tangent
to the Markowitsz model efficient set,
None of the portfolios,except for T,that
were on the Markowitz model efficient
set are efficient when riskfree borrowing
and lending are induced.
Allowing For Both
Borrowing And Lending
The effect of riskfree borrowing and lending
on portfolio selection
– Given the opportunity to either borrow or
lend at the riskfree rate,an investor would
proceed to identify the optimal portfolio by
plotting his or her indifference curve on
this risk-return graph and noting where
one of them is tangent to the efficient set.
Summary
Portfolio with risky assets
– Feasible set and Efficient set
Portfolio selection with all risky assets
– Optimal portfolio
The market model and Efficient diversification
– Market risk v.s,Unique risk
– Security characteristic line
Summary
Riskfree lending and borrowing
– Efficient set with riskless lending and
borrowing
– The Capital Allocation Line (CAL)
– Optimal portfolio with riskless lending
and borrowing
– The separate theorem
