Portfolio Managment
3-228-07
Albert Lee Chun
Construction of Portfolios:
Introduction to Modern
Portfolio Theory
Lecture 3
16 Sept 2008
0
Course Outline
Sessions 1 and 2 : The Institutional Environment
 Sessions 3, 4 and 5: Construction of Portfolios
 Sessions 6 and 7: Capital Asset Pricing Model
 Session 8: Market Efficiency
 Session 9: Active Portfolio Management
 Session 10: Management of Bond Portfolios
 Session 11: Performance Measurement of Managed
Portfolios

1
Portfolio Risk as a Function of the Number
of Stocks in the Portfolio
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7-2
2
Portfolio Diversification
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7-3
3
Two-Security Portfolio: Return
rP  w1  r1  w2  r2
w1
w2
r1
r2
= proportion of funds in Security 1
= proportion of funds in Security 2
= expected return on Security 1
= expected return on Security 2
n
w
i
1
i 1
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4
Two-Security Portfolio: Risk
2
2
2
1
1
p  w 
2
2
 w2 2  2w1w2Cov(r1, r2)
12 = variance of Security 1
22 = variance of Security 2
Cov(r1,r2) = covariance of returns for
Security 1 and Security 2
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Covariance
Cov(r1, r2)  1,2  1  2
1,2 = Correlation coefficient of returns
1 = Standard deviation of returns for
Security 1
2 = Standard deviation of returns for
Security 2
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Correlation Coefficients:
Possible Values
Range of values for 1,2
+ 1.0 >  > -1.0
If  = 1.0, the securities would be
perfectly positively correlated
If = - 1.0, the securities would be
perfectly negatively correlated
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7
Three-Security Portfolio
rp  w1r1  w2r2  w3r3
 p  w1  1  w2  2  w3  3 
2
2
2
2
2
2
2
 2 w1w2Cov(r1 , r2 ) 
 2 w1w3Cov(r1 , r3 ) 
 2 w2 w3Cov(r2 , r3 )
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8
Generally, for an
n-Security Portfolio:
rp 
n
w
i
i 1
2
p 
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n
w
i 1
 ri
n
  2  w jwkCov(rj , rk )
2 2
i
i
j,k 1
jk
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9
Review of Portfolio Statistics
N
E ( R p )   wi E ( Ri )
i 1
N
2 2
=
  wi  i +
2
p
i=1
i , j 
Albert Lee Chun
N
N
w w
i
j
Cov ( r i , r j ) i  j
i=1 j=1
Cov ( ri , r j )
 i j
Cov(ri , rj )  i , j i j
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Today’s Lecture




Utility Functions, Indifference Curves
Capital Allocation Line
Minimum Variance Portfolios
Optimal Portfolios in a
2 security world (1 risk-free and 1 risky)
2 security world (2 risky)
3 security world (2 risky and 1 risk-free)
N security world (with and without risk-free asset)
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Utility Functions
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12
Risk Aversion



Given a choice between two assets with equal
rates of return, risk-averse investors will select
the asset with the lower level of risk.
Risk-averse investors need to be compensated
for holding risk.
The higher rate of return on a risky asset i is
determined by the risk-premium:
E(Ri) – Rf.
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Example: Risk Premium
W1 = $150 Profit = $50
Risky
Investment
1-p = .4
100
W2 = $80 Profit = -$20
Profit = $5
T-bills
Expected return: (50%)(.6) + (-20%)(.4)
= 22%
Risk Premium = E(Ri) – Rf = 22% - 5% =
= 17%
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Measure of Investor Preferences



A utility function captures the level of satisfaction or
happiness of an investor.
The higher the utility, the happier the investors.
For example, if investor utility depends only of the
mean (let µ= E(R)) and variance (2) of returns then
it can be represented as a function:
U = f ( µ, )

The locus of portfolios that provide the same level of
utility for an investor defines an indifference curve.
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Example: An Indifference Curve
U=5
(Rp)
U=5
The investor is indifferent between X and Y, as well as all
points on the curve. All points on the curve have the same
level of utility (U=5).
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Direction of Increasing Utility
Expected Return
U3
U2
U3 > U2 > U1
U1
Direction of Increasing Utility
Standard Deviation
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Two Different Investors
Expected Return
U3’
U2’
U1’
Which investor is
more risk averse?
U3
U2
U1
Standard Deviation
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Quadratic Utility

The utility of the investor is quadratic if only the mean and
variance of returns is important for the investor.
1
U  E ( R)  AE ( R  E ( R)) 2
2
1
   A 2
2


A is a constant that determines the degree of risk aversion: it
increases with the risk-aversion of the investor. (Note that the 1/2
is just a normalizing constant.)
Note that A > 0, implies that investors dislike risk. The higher the
variance the lower the utility.
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Indifference Curves
Let’s look at an example of points on an indifference curve for
an investor with a quadratic utility function. Note that higher
variance is accompanied by a higher rate of return to
compensate the risk-averse nature of the investor.
E(Rp)
0.10
0.15
0.20
0.25
(Rp)
0.200
0.255
0.300
0.339
Utility = E(Rp) – ½ A×VAR(Rp)
0.10 – ½  4  0.2002 = 0.02
0.15 – ½  4  0.2552 = 0.02
0.20 – ½  4  0.3002 = 0.02
0.25 – ½  4  0.3992 = 0.02
Quadratic utility function with A = 4.
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20
Certain Equivalent



The certain equivalent is the risk-free (certain) rate of return
that offers investors the same level of utility as the risky rate of
return.
The investor is indifferent between a risky return and it’s
certain equivalent.
Example: Suppose an investor has quadratic utility with A = 2.
A risky portfolio offers an E(R) equal to 22% and standard
deviation 34%. The utility of this portfolio is:
U = 22% - ½×2×(34%)² = 10.44%

The certain equivalent is equal to 10.44% because the
utility of obtaining a certain rate of return of 10.44% is
U = 10.44% - ½ × 2×(0%)² = 10.44%
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Risk-Neutral Indifference Curves
E(RP)
U3 > U2 > U1
U4
U3
U2
U1
P
Neutral attitude toward risk. Investor is indifferent between
different levels of standard deviation.
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Slope of the Indifference Curve




A steep indifference curve coincides with strong riskaversion.
The slope of the indifference curve captures the
required compensation for each unit of additional
risk.
This compensation is measured in units of expected
return for each unit of standard deviation.
High risk-aversion implies a high degree of
compensation for taking on an additional unit of risk
and is represented by a steep slope.
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23
Risk-Averse Indifference Curves
U4
E(RP)
U3
Expected Return
U2
U1
U3 > U2 > U1
P
Standard Deviation
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Two Different Investors
Expected Return
More risk averse
U3’
U2’
Less risk averse
U1’
Which investor is
more risk averse?
U3
U2
U1
Standard Deviation
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25
Stochastic Dominance
Prefers any portfolio in
Z1 to X.
The rankings between
portfolios in Z2 or Z3 and
X, depends on the
preferences of the
investor!
Prefers X to any portfolio
in Z4.
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26
Imagine a world with
1 risk-free security and 1 risky security
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27
1 Risk-Free Asset and 1 Risky Asset
Suppose we construct a portfolio P consisting of
1 risk-free asset f and 1 risky asset A:
E (rp )  w f r f  w A E (rA )
 =w  00
2
p
2
A
2
A
 p = wA A
Note: The variance of the risk-free asset is 0, and the covariance
between a risky asset and a risk free asset is naturally equal to 0.
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1 Risk Free Asset and 1 Risky Asset
Suppose WA = .75
E(rA) = 15%
E(rP) = 13%
P
rf = 7%
f
0
P =16.5% A =22%
E(rP) = .25*.07+.75*15=13%
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A

p = .75*.22 = 16.5%
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Capital Allocation Line (CAL)
Equation of CAL E (rP ) 
Line
E(rA)
A
A
E(rp)
P
rf
* P  rf
Slope
of CAL
E( r A ) - r f
A
f
Intercept
0
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E (rA )  r f
p
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A

30
Maximize Investor Utility

In our world with 1 risk free asset and 1 risky asset,
if an investor has quadratic utility, what is the optimal
portfolio allocation?
Utility:
U  E (rP )  1 2 A AP
Expected return E (rP )  wE (rA )  (1  w)r f ,
and variance:  2  w 2 2
P
A
Goal is to Maximize utility. How?
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Normally a Bear Lives in a Cave, that is
Concave,
A concave
function has a
negative
second
derivative.
then to find the top of the cave
(i.e. or to maximize a concave function), take
the first derivative and set it equal to 0:
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However, if the Bear is Swimming in a Bowl,
that is Convex,
A convex
function has a
positive
second
derivative.
Then to find the bottom of the bowl
(i.e. or to minimize a convex function), take
the first derivative and set equal to 0:
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Maximize Investor Utility
U  E (rP )  1 2 A P2
 wE (rA )  (1  w)r f -
1
Aw  A
2
2
2
Take derivative of U with respect to w 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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w* is the optimal
weight on risky
asset A
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Example 1
Supppose E(rA) = 15%; (rA) = 22% and rf = 7% and we
have a Quadratic investor with A = 4, then
w* =
(0.15-0.07)/[4*(0.22)2]
E( r A ) - r f
w=
A  2A
*
= 0.41
His optimal allocation is: 41% of his capital in the risky
portfolio A and 59% in the risk-free asset.
E(rp) = 0.59*7%+0.41*15%=10.28%
and
(rp) = 0.41*0.22=9.02%
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E (rp )  (1  w*)r f  w * E (rA )
 p= w*  A
35
Example 2
Supppose E(rA) = 15%; (rA) = 22% and rf = 7% and
we have a less risk-averse Quadratic investor with A
= 1, then
w* = (0.15-0.07)/[1*(0.22)2]
= 1.65 > 1
This investor should place 165% of his capital in A. He needs
to borrow 65% of his capital at the risk free rate of 7%.
E(Rp) = 1.65(0.15) + -0.65(0.07)= 20.2%
(rp) = 1.65*0.22= 0.363 = 36.3%
His utility is: U = 0.202 – 0.5*1*(0.3632) = 0.1361
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Portfolio Management
36
Graphical View
The optimal allocation along the capital allocation line
depends on the risk-aversion of the agent. Risk-seeking agents
with w* greater than 1 will borrow at the risk-free rate and
invest in security A
E(r)
A
Ex2: Borrower
7%
Ex1: Lender
A = 22%

The optimal allocation is the point of tangency between the
CAL and the investor’s utility function.
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37
Different Borrowing Rate

What if the borrowing rate is higher than the lending
rate? E(r)
A
9%
7%
A = 22%

w* = (0.15-0.09)/[1*(0.22)2] = 1.24
1.24 < 1.65
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38
Different Borrowing Rate
Supppose E(rA) = 15%; (rA) = 22% and rf = 7% lending rate,
and a 9% borrowing rate, Quadratic investor with A = 1, then
w* = (0.15-0.09)/[1*(0.22)2] = 1.24
1.24 < 1.65
This investor should place 124% of his capital in A. He needs to borrow
24% of his capital at the risk free rate of 9%. This is less than what he
would borrow at a 7% borrowing rate.
E(Rp) = 1.24(0.15) + -0.24(0.09)= 16.44%
(rp) = 1.24*0.22= 27.28%
Increasing the borrowing rate, lowers his utility from before:
U = 0.1644 – 0.5*1*(0.27282) = .1272 < .1361
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Imagine a world with
2 risky securities
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40
Expected Return and Standard Deviation with
Various Correlation Coefficients
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Portfolio Expected Return as a Function of Investment
Proportions
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Portfolio Standard Deviation as a Function of Investment
Proportions
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Returning to the
Two-Security Portfolio
E(rp )  w1r1  w2r2 and
2
2
2
1
1
2
2
p  w   w2 2  2w1w2Cov(r1, r2), or
2
1
2
1
p  w 
2
2
 w2 2  2w1w2Cov(r1, r2)
Question: What happens if we use various
securities’ combinations, i.e. if we vary ?
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7-44
44
Portfolio Expected Return as a function of
Standard Deviation
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Portfolio Management
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45
Perfect Correlation
E(R)
0.20
With two perfectly
correlated securities,
all portfolios will lie
on a straight line
between the two
assets.
0.15
0.10
0.05
E
Rij = +1.00
D
With short
selling
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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Perfect Correlation
 = +1
E( RP )  wD E( RD )  wE E( RE )
 = w  + w  + 2 wD wE 1 D  E
2
p
2
D
2
D
2
E
2
E
Recall that
Cov(rE , rD )   E  D
= ( wD  D + wE  E )
2
 p = wD  D  wE  E
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Portfolio Management
47
Zero Correlation
E(R)
f
0.20
g
2
With uncorrelated
h
assets it is possible
i
j
to create a two
Rij = +1.00
asset portfolio with
k
lower risk than
1
either asset!
Rij = 0.00
0.15
0.10
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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Zero Correlation
=0
 = w  + w  + 2 wD wE 0  D  E
2 2
2 2
= wD  D + wE  E
2
p
2
D
2
D
2
E
2
E
Recall that
Cov(rE , rD )   E  D
2
2
2
2
=
+
p
wD  D wE  E
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Portfolio Management
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Positive Correlation
E(R)
f
0.20
g
2
With positively
h
correlated assets a
i
two asset portfolio j
Rij = +1.00
lies between the
k
Rij = +0.50
first two curves
1
Rij = 0.00
0.15
0.10
0.05
2
2
2
2
=
+

wD  D wE  E + 2 wD wE  DE  D  E
2
p
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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Portfolio Management
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Negative Correlation
E(R)
0.20
0.15
0.10
With
negatively
correlated
assets it is
possible to
create a
portfolio with
much lower
risk.
Rij = -0.50
f
2
g
h
j
i
k
Rij = +1.00
Rij = +0.50
1
Rij = 0.00
Negative
0.05
2
2
2
2
=
+

wD  D wE  E + 2 wD wE  DE  D  E
2
p
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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Perfect Negative Correlation
E(R)
f
0.20
Rij = -1.00
0.15
0.10
0.05
-
2
g
h
j
i
k
Rij = +1.00
Rij = +0.50
1
Rij = 0.00
With perfectly negatively correlated
assets it is possible to create a two asset
portfolio with NO RISK. How?
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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Perfect Negative Correlation
 = -1
 = w  + w  - 2 wD wE  D  E
2
p
2
D
2
D
2
E
2
E
Note that
Cov(rE , rD )  1 E  D
= ( wD  D - wE  E )
2
 p =| wD  D - wE  E |
To get a zerovariance portfolio we
need to set:
Albert Lee Chun
E
D
and wE =
 D + E
 D + E
then we obtain
 P=0
wD =
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Minimum Variance Portfolio
Albert Lee Chun
Portfolio Management
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Minimum Variance Portfolio
Min  = w  + (1 - wD )  2E + 2 wD (1 - wD )  DE
2
p
2
D
2
D
2
  2p
= 2 wD  2D - 2(1 - wD ) 2E + (2 - 4 wD ) DE
 wD
= wD (2  2D + 2  2E - 4  DE ) - (2  2E - 2  DE ) = 0
2
 E -  DE  D  E


DE
min
= 2
wD = 2
2
2
+
2
+
D E
 DE  D  E - 2  DE  D  E
min
min
=
1
wE
wD
2
E
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Minimum Variance Portfolio
1> > -1
 = -1
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
=0
2
2
0


E
E
min
=

wD
 2D +  2E - 0  2D +  2E
=1
If no short sales, then MVP is equal to
the asset with the minimum variance.*
(Our formula doesn’t work here, why?
Think about the bear in a cave/bowl.)
Albert Lee Chun
Portfolio Management
*With short
sales can obtain
0 variance.
56
Portfolio of Two Securities: Correlation Effects
•
•
•
•
Relationship depends on correlation coefficient
-1.0 <  < +1.0
The more negative the correlation, the greater the
risk reduction potential
If= +1.0, no risk reduction is possible (absent
short sales).
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Example: MVP
 Example: Suppose there are 2 securities A an B:
E(r)

A
B
10%
14%
15%
20%
A,B
0.2
 Find the minimum variance portfolio?
2

B -   A B
min
min
=
1
w = 2
w
w
B
A
2
+
2


A B
A B
min
A
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Example: MVP
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Similar Example from the Book
• Suppose our investment universe
comprises the two securities of Table 7.1:
D
E
E(r)
8%
13%

12%
20%
D,E
0.3
• What are the weights of each security in
the minimum-variance portfolio?
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Book Example: MVP
• Solving the minimization problem we get:
 E  Cov(rD , rE )
wD 
 D 2   E 2  2Cov(rD , rE )
2
• Numerically:
(20) 2  72
wD 
 0.82
2
2
(20)  (12)  272
wE  1  wD  0.18
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Investor’s Utility
Less Risk Averse
Investors
U’’’
E(r)
U’’
U’
More Risk Averse
Investors

Albert Lee Chun
Portfolio Management
62
Investor’s Utility Maximization
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
Homework, you should be able to show that the
optimal solution is:
2
E(
)
E(
)
+
A(

r
r
D
E
E -  DE )
*
wD =
A (  2D +  2E - 2  DE )
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Portfolio Management
63
Example
 Example: Suppose there are only 2 portfolios:
E(r)

A
B
10%
14%
15%
20%
A,B
0.2
 Find the optimal portfolio for a investor with quadratic utility ( A = 3)?
2





E
E
+
A

r
r
A
B
B -  A B 
*
*
*
=
,

1

wA
wB
wA
2
2
A A + B - 2  A B 
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Portfolio Management
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Example
*
A
w =

  0.41,
- 2 * 0.2 * 0.2 * 0.15
0.10 - 0.14 + 3 0.2 - 0.2 * 0.2 * 0.15
3
 0.2
2
2
+ 0.15
2
*

1

w
w A  0.59
*
B
Albert Lee Chun
Portfolio Management
65
Imagine a world with
2 risky securities and 1 risk-free security
Albert Lee Chun
Portfolio Management
66
Two Feasible CALs
Albert Lee Chun
Portfolio Management
7-67
67
Optimal CAL and the Optimal Risky Portfolio
Albert Lee Chun
Portfolio Management
7-68
68
With a Risk Free Asset
E(r)
CAL 3
I’m the optimal risky portfolio,
the tangent portfolio!!
CAL 2
CAL 1
E
rf
I’m the one that
maximizes the slope of
the Capital Allocation
Line !
D

Albert Lee Chun
Portfolio Management
69
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
Homework: If you are ambitious, try to show that
the optimal solution is:
wD* =
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
Albert Lee Chun
Portfolio Management
70
The Optimal Overall Portfolio
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Portfolio Management
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71
The Proportions of the Optimal Overall
Portfolio
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Portfolio Management
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Optimal Overall Portfolio:
2 Investors i and j
E(r)
j
PT
Optimal weight for each
investor depends on risk
aversion parameter A
E( R PT ) - r f
w=
A 2PT
i
*
rf
Albert Lee Chun
CAL
Portfolio Management
73
Different Lending and Borrowing
E(r)
P2
P1
rfB
rf

Albert Lee Chun
Portfolio Management
74
Now imagine a world
with many risky assets
Albert Lee Chun
Portfolio Management
75
The Markowitz Problem
Max E Rp    wi E Ri 
i 1
wi
Subject to
the
constraint
N
N
 w w 
i 1 j 1
N
w
i 1
Albert Lee Chun
i
i
j
ij

*
p
1
Portfolio Management
76
Efficient Frontier
E(R)
Albert Lee Chun
Efficient
Frontier
Portfolio Management
77
Extending Concepts to All Securities
•
•
•
The optimal combinations result in lowest level of
risk for a given return
The optimal trade-off is described as the efficient
frontier
These portfolios are dominant
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Portfolio Management
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Minimum Variance Frontier of Risky Assets
E(r)
Minimum variance
frontier
Global
minimum
variance
portfolio
Individual
assets

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Portfolio Management
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Efficient Variance Frontier of Risky Assets
E(r)
Efficient
frontier
Global
minimum
variance
portfolio
Individual
assets

Albert Lee Chun
Portfolio Management
7-80
80
The Efficient Portfolio Set
Albert Lee Chun
Portfolio Management
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81
Now imagine a world
with many risky assets and
1 risk-free asset
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Portfolio Management
82
Capital Allocation Lines
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Portfolio Management
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83
The Market Portfolio
E(R)
Capital
Market Line
M
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Portfolio Management
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Readings

Readings for Today’s lecture.
1. Chapter 7.
2. If you have not taken Investments, you may want to
review Chapter 6 as well.

Readings for next week:
Finish reading Chapter 7, including appedicies.

Readings for the week after next:
(Course Reader) Other Portfolio Selection Models
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Portfolio Management
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