Portfolio Management
3-228-07
Albert Lee Chun
Construction of Portfolios:
Markowitz and the Efficient
Frontier
Session 4
25 Sept 2008
0
Plan for Today
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A Quick Review
Optimal Portfolios of N risky securites
- Markowitz`s Portfolio Optimization
- Two Fund Theorem
Optimal Portfolios of N risky securities and a risk-free asset
- Capital Market Line
- Market Portfolio
-Different Borrowing and Lending rates
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Une petite révision
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We started in a simple universe of
1 risky asset and 1 risk-free asset
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Optimal Weights Depended on Risk Aversion
Each investor chooses an optimal weight on the risky asset, where
w*> 1 corresponds to borrowing at the risk-free rate, and investing in
the risky asset.
E(r)
Borrower
Rf
Lender
A

The optimal choice is the point of tangency between the capital
allocation line and the agent`s utility function.
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Utility maximization
U  E (rP )  1 2 A P2
 wE (rA )  (1  w)r f -
1
Aw  A
2
2
2
Take the derivative and set equal to 0
dU ( w)
 E (rA )  r f  Aw A2  0
dw
E( r A ) - r f
w=
A  2A
*
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We then looked at a universe with 2 risky
securities
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Correlation and Risk
E(R)
f
0,20
E
g
ρDE = -1.00
0,15
h
j
i
k
0,10
ρDE = +1.00
D
ρDE = + 0.50
ρDE = 0.00
0,05
0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,11 0,12
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Minimum Variance Portfolio
1> > -1
 = -1
=0
=1
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w
min
D
=
 2D
 2E -  DE
+  2E - 2  DE
2

 E (  E + D ) =  E
E + D  E
min
=
=
wD
2
2
2
+
+
2
D E
 D  E (  D + E )  D + E
2
2
0


E
E
min
=

wD
 2D +  2E - 0  2D +  2E
Asset with the lowest variance, in the
absence of short sales.
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Maximize Investor Utility
U  E (r ) 
1
2
A
2
E(rP )  wD E(rD )  wE E(rE )
2
=
+
(1

w 
wD )  E + 2 wD (1 - wD )  DE
2
p
2
D
2
D
2
The solution is:
2
E(
)
E(
)
+
A(

r
r
D
E
E -  DE )
*
wD =
A (  2D +  2E - 2  DE )
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Then we introduced a risk-free asset
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Optimal Portfolio is the Tangent Portfolio
E(r)
CAL 3
Every investor holds exactly
the same optimal portfolio of
CAL 2
risky assets!
CAL 1
E
D
Intuition : the optimal
solution is the CAL
with the maximum
slope!

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Optimal Portfolio Weights
S p=
E( r p ) - r f
p
E(rP )  wD E(rD )  wE E(rE )
2
 2p = w2D  2D + (1 - wD )  2E + 2 wD (1 - wD )  DE
The solution is:
*
D
w =
E r   r
D
E r   r  - E r   r 
 + E r   r   E r   r
D
f
2
E
2
E
f
E
f
E
2
D
f
D
f
DE
 E r E   r f  DE
w*E  1  w*D
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Optimal Borrowing and Lending
CAL
E(r)
The optimal weight on
the optimal risky
portfolio P depends on
the risk-aversion of each
investor.
E
P
rf
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D
Portfolio Management
E(rP ) - r f
w=
A  2P
*
13
Now imagine a universe with a multitude of
risky securities
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Harry Markowitz
1990 Nobel Prize in Economics
for having developed the theory of portfolio
choice.
The multidimensional problem of investing
under conditions of uncertainty in a large
number of assets, each with different
characteristics, may be reduced to the issue of a
trade-off between only two dimensions, namely
the expected return and the variance of the
return of the portfolio.
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Markowitz Efficient Frontier
E(R port )
Efficient
Frontier
E
µ*
D
σ*
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 port
16
The Problem of Markowitz I
Max E rp    wi E ri 
N
i 1
wi
Subject to
the
constraint:
Weights sum to 1
N
N
 w w 
i 1 j 1
N
w
i 1
i
i
j
ij

2*
p
1
Maximize the expected return of the portfolio conditioned on
a given level of portfolio variance.
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The problem of Markowitz II
N
N
Min  p2   wi w j ij
i 1 j 1
wi
Subject to
the
constraint:
Weights sum to 1
N
 w E (r )  E (r
i
i 1
N
w
i 1
i
i
p
*
)
1
Minimize the variance of the portfolio conditioned on a given
level of expected return.
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Does the Risk of an Individual Asset Matter?

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
Does an asset which is characterized by relatively
large risk, i.e., great variability of the return, require a
high risk premium?
Markowitz’s theory of portfolio choice clarified that
the crucial aspect of the risk of an asset is not its risk
in isolation, but the contribution of each asset to the
risk of an entire portfolio.
However, Markowitz’s theory takes asset returns as
given. How are these returns determined?
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Citation de Markowitz
So about five minutes into my defense, Friedman says, well
Harry I’ve read this. I don’t find any mistakes in the math, but
this is not a dissertation in economics, and we cannot give you
a PhD in economics for a dissertation that is not in economics.
He kept repeating that for the next hour and a half. My palms
began to sweat. At one point he says, you have a problem.
It’s not economics, it’s not mathematics, it’s not business
administration, and Professor Marschak said, “It’s not
literature”. So after about an hour and a half of that, they send
me out to the hall, and about five minutes later Marschak came
out and said congratulations Dr. Markowitz.
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Two-Fund Theorem
E(r port )
Interesting Fact: Any two
efficient portfolios will
generate the entire efficient
frontier!
B
A
Every point on the
efficient frontier is
a linear
combination of any
two efficient
portfolios A and B.
 port
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Now imagine a risky universe with a risk-free
asset
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Capital Market Line
E(r port )
Capital Market
Line
E
M
rf
D
*
D
w =
E r   r
D
E r   r  - E r   r 
 + E r   r   E r   r
D
f
2
E
2
E
f
E
f
E
2
D
f
D
DE
f
 E r E   r f  DE
w*E  1  w*D
 port
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Tobin’s Separation Theorm

James Tobin ... in a 1958 paper said if you hold risky
securities and are able to borrow - buying stocks on margin or lend - buying risk-free assets - and you do so at the same
rate, then the efficient frontier is a single portfolio of risky
securities plus borrowing and lending....

Tobin's Separation Theorem says you can separate the problem
into first finding that optimal combination of risky securities
and then deciding whether to lend or borrow, depending on
your attitude toward risk. He then showed that if there's only
one portfolio plus borrowing and lending, it's got to be the
market.
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Market Portfolio
E(r port )
Capital
Market
Line
E
M
E( r M ) - r f
w 
A 2M
*
rf
D
 port
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Separation Theorem
E(r port )
Capital
Market
Line
M
Separation
of investment decision
from
the financing decision.
rf
 port
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Who holds only the Market Portfolio?
E(r port )
CML
M
rf
E( r M ) - r f
w =1 
A 2M
E( r M ) - r f
M
A 
2
*
M
 port
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Note that we have reduce the complexity of
this universe down to simply 2 points
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Different Borrowing and Lending Rates
E(r port )
rB
MB
ML
rL
 port
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Who are the Lenders and Borrowers
E(r port )
A
rB
ML

E( r M L ) - r L
 2M
L
MB
ML
A
MB

E( r M B ) - r B
 2M
B
rL
 port
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Who are the Lenders and Borrowers
E(r port )
MB
rB
ML
E( r M B ) - r B
1
wB 
2
A M B
*
rL
E( r M L ) - r L
1
wL 
2
A M L
*
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Who holds only risky assets?
E(r port )
rB
rL
MB
ML
2
E(
)
E(
)
+
A(
)

r
r
M
M
M L -  M BM L
*
B
L
wM B =
A (  2M B +  2M L - 2  M B M L )
 port
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Efficient Frontier
E(r port )
rB
MD
ML
rL
 port
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Where is the market portfolio?
E(r port )
rB
rf
 port
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Only Risk-free Lending
E(r port )
Low risk averse agents
cannot borrow, so they
hold only risky assets.
Least risk-averse lender
ML
rL
AM L 
E( r M L ) - r L
 2M
L
 port
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Efficient Frontier
E(r port )
All lenders hold this
portfolio of risky securities
rL
 port
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For Next Week
Next week we will
- do a few examples, both numerical and in Excel.
- discuss Appendix A – diversification.
- discuss the article from the course reader.
- wrap up Chapter 7 and pave the way for the Capital
Asset Pricing Model.
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The Power of Diversification
90% of the total benefit of
Non systematic diversification is obtained after
risk (idiosyncratic,
holding 12-18 stocks.
Standard Deviation of Return
non diversifiable)
Total
Risk
Standard Deviation of the Market
(systematic risk)
Systematic Risk
Number of Stocks in the Portfolio
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