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Portfolio ManagmentPortfolio Managment3-228-073-228-07
Albert Lee ChunAlbert Lee Chun
Construction of Portfolios:Construction of Portfolios: Introduction to Modern Introduction to Modern
Portfolio Theory Portfolio Theory
Lecture 3Lecture 3
16 Sept 2008
2
Course Outline Course Outline
Sessions 1 and 2 : The Institutional Environment Sessions 1 and 2 : The Institutional Environment Sessions Sessions 33, 4 and 5: , 4 and 5: Construction of PortfoliosConstruction of Portfolios Sessions 6 and 7: Capital Asset Pricing ModelSessions 6 and 7: Capital Asset Pricing Model Session 8: Market EfficiencySession 8: Market Efficiency Session 9: Active Portfolio ManagementSession 9: Active Portfolio Management Session 10: Management of Bond PortfoliosSession 10: Management of Bond Portfolios Session 11: Performance Measurement of Managed Session 11: Performance Measurement of Managed PortfoliosPortfolios
3Albert Lee Chun Portfolio Management
Portfolio Risk as a Function of the Number Portfolio Risk as a Function of the Number of Stocks in the Portfolioof Stocks in the Portfolio
7-37-3
4Albert Lee Chun Portfolio Management
Portfolio DiversificationPortfolio Diversification
7-47-4
5Albert Lee Chun Portfolio Management
w1 = proportion of funds in Security 1w2 = proportion of funds in Security 2r1 = expected return on Security 1r2 = expected return on Security 2
1wn
1ii
Two-Security Portfolio: ReturnTwo-Security Portfolio: Return
2211P rwrwr
7-57-5
6Albert Lee Chun Portfolio Management
12 = variance of Security 1
22 = variance of Security 2
Cov(r1,r2) = covariance of returns for Security 1 and Security 2
Two-Security Portfolio: RiskTwo-Security Portfolio: Risk
)r,r(Covww2ww 21212
22
22
12
12
p
7-67-6
7Albert Lee Chun Portfolio Management
1,2 = Correlation coefficient of returns
1 = Standard deviation of returns for Security 12 = Standard deviation of returns for
Security 2
CovarianceCovariance
212,121 )r,r(Cov
7-77-7
8Albert Lee Chun Portfolio Management
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
Correlation Coefficients: Correlation Coefficients: Possible ValuesPossible Values
7-87-8
9Albert Lee Chun Portfolio Management
Three-Security PortfolioThree-Security Portfolio
332211p rwrwrwr
),(2
),(2
),(2
3232
3131
2121
23
23
22
22
21
21
2
rrCovww
rrCovww
rrCovww
wwwp
7-97-9
10Albert Lee Chun Portfolio Management
Generally, for an Generally, for an n-Security Portfolio:n-Security Portfolio:
n
1i
iip rwr
n
kj1k,j
kjkj
n
1i
2i
2i
2p )r,r(Covww2w
7-10
7-10
11Albert Lee Chun Portfolio Management
Review of Portfolio StatisticsReview of Portfolio Statistics
N
iiip REwRE
1
)()(
ji )r ,r( Cov ww + w = jiji
N
j=1
N
=1i
2i
2i
N
=1i
2p
jijiji rrCov ,),( ji
jij,i
)r,r(Cov
12Albert Lee Chun Portfolio Management
Today’s LectureToday’s Lecture
Utility Functions, Indifference CurvesUtility Functions, Indifference Curves Capital Allocation LineCapital Allocation Line Minimum Variance PortfoliosMinimum Variance Portfolios Optimal Portfolios in a Optimal Portfolios in a
2 security world (1 risk-free and 1 risky)2 security world (1 risk-free and 1 risky)
2 security world (2 risky)2 security world (2 risky)
3 security world (2 risky and 1 risk-free)3 security world (2 risky and 1 risk-free)
N security world (with and without risk-free asset)N security world (with and without risk-free asset)
13Albert Lee Chun Portfolio Management
Utility FunctionsUtility Functions
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Risk AversionRisk Aversion
Given a choice between two assets with equal Given a choice between two assets with equal rates of return, risk-averse investors will select rates of return, risk-averse investors will select the asset with the lower level of risk.the asset with the lower level of risk.
Risk-averse investors need to be compensated Risk-averse investors need to be compensated for holding risk. for holding risk.
The higher rate of return on a risky asset The higher rate of return on a risky asset ii is is determined by the determined by the risk-premiumrisk-premium: :
E(Ri) – Rf.E(Ri) – Rf.
15Albert Lee Chun Portfolio Management
Example: Risk PremiumExample: Risk Premium
W2 = $80 Profit = -
$20
W1 = $150 Profit = $50p = .6
100
Risky
Investment
T-bills Profit = $5
Expected return: (50%)(.6) + (-20%)(.4)
= 22%
Risk Premium = E(Ri) – RfE(Ri) – Rf = 22% - 5% = = 17%
1-p = .4
16Albert Lee Chun Portfolio Management
Measure of Investor Preferences Measure of Investor Preferences
A utility function captures the level of satisfaction or A utility function captures the level of satisfaction or happiness of an investor.happiness of an investor.
The higher the utility, the happier the investors.The higher the utility, the happier the investors. For example, if investor utility depends only of the For example, if investor utility depends only of the
mean (let mean (let µ= E(R)) and variance (= E(R)) and variance (2) of returns then ) of returns then it can be represented as a function:it can be represented as a function:
The locus of portfolios that provide the same level of The locus of portfolios that provide the same level of utility for an investor defines an utility for an investor defines an indifference curveindifference curve..
U = f ( µ, )
17Albert Lee Chun Portfolio Management
Example: An Indifference CurveExample: An Indifference Curve
U = 5
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).
(Rp)
18Albert Lee Chun Portfolio Management
Direction of Increasing UtilityDirection of Increasing Utility
Expected Return
Standard Deviation
Direction of Increasing Utility
U1
U2
U3 U3 > U2 > U1
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Two Different InvestorsTwo Different Investors
U3
U2
U1
U3’
U2’ U1’
Expected Return
Standard Deviation
Which investor is more risk averse?
20Albert Lee Chun Portfolio Management
Quadratic UtilityQuadratic Utility
The utility of the investor is The utility of the investor is quadraticquadratic if only the mean and if only the mean and variance of returns is important for the investor.variance of returns is important for the investor.
AA is a constant that determines the degree of risk aversion: it is a constant that determines the degree of risk aversion: it increases with the risk-aversion of the investor. (Note that the increases with the risk-aversion of the investor. (Note that the 1/21/2 is just a normalizing constant.)is just a normalizing constant.)
Note that A > 0, implies that investors dislike risk. The higher Note that A > 0, implies that investors dislike risk. The higher the variance the lower the utility.the variance the lower the utility.
2
2
2
1
))((2
1)(
A
RERAEREU
21Albert Lee Chun Portfolio Management
Indifference CurvesIndifference Curves
E(Rp) (Rp) Utility = E(Rp) – ½ A×VAR(Rp) 0.10 0.200 0.10 – ½ 4 0.2002 = 0.02 0.15 0.255 0.15 – ½ 4 0.2552 = 0.02 0.20 0.300 0.20 – ½ 4 0.3002 = 0.02 0.25 0.339 0.25 – ½ 4 0.3992 = 0.02
Quadratic utility function with A = 4.
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.
22Albert Lee Chun Portfolio Management
Certain EquivalentCertain Equivalent
The The certain equivalentcertain equivalent is the risk-free (certain) rate of return is the risk-free (certain) rate of return that offers investors the same level of utility as the risky rate that offers investors the same level of utility as the risky rate of return. of return.
The investor is indifferent between a risky return and it’s The investor is indifferent between a risky return and it’s certain equivalent.certain equivalent.
Example: Suppose an investor has quadratic utility with A = Example: Suppose an investor has quadratic utility with A = 2. A risky portfolio offers an 2. A risky portfolio offers an E(R) equal to 22% and E(R) equal to 22% and standard deviation 34%. The utility of this portfolio is: standard deviation 34%. The utility of this portfolio is:
U = 22% - ½U = 22% - ½××22××(34%)(34%)² = 10.44%² = 10.44% The The certain equivalentcertain equivalent is equal to is equal to 10.44%10.44% because the because the
utility of obtaining a certain rate of return of 10.44% isutility of obtaining a certain rate of return of 10.44% is U = 10.44% - U = 10.44% - ½ ½ ×× 2 2××(0%)(0%)²² = 10.44% = 10.44%
23Albert Lee Chun Portfolio Management
Risk-Neutral Indifference CurvesRisk-Neutral Indifference CurvesE(RP)E(RP)
PP
U4U4
U3U3
U2U2
U1U1
Neutral attitude toward risk. Investor is indifferent between Neutral attitude toward risk. Investor is indifferent between different levels of standard deviation.different levels of standard deviation.Neutral attitude toward risk. Investor is indifferent between Neutral attitude toward risk. Investor is indifferent between different levels of standard deviation.different levels of standard deviation.
U3 > U2 > U1
24Albert Lee Chun Portfolio Management
Slope of the Indifference CurveSlope of the Indifference Curve
A steep indifference curve coincides with strong risk-A steep indifference curve coincides with strong risk-aversion.aversion.
The slope of the indifference curve captures the The slope of the indifference curve captures the required compensation for each unit of additional required compensation for each unit of additional risk.risk.
This compensation is measured in units of expected This compensation is measured in units of expected return for each unit of standard deviation. return for each unit of standard deviation.
High risk-aversion implies a high degree of High risk-aversion implies a high degree of compensation for taking on an additional unit of risk compensation for taking on an additional unit of risk and is represented by a steep slope. and is represented by a steep slope.
25Albert Lee Chun Portfolio Management
Risk-Averse Indifference CurvesRisk-Averse Indifference Curves
E(RP)E(RP)
PP
U4U4
U3U3
U2U2
U1U1Expected Return
Standard Deviation
U3 > U2 > U1
26Albert Lee Chun Portfolio Management
Two Different InvestorsTwo Different Investors
U3
U2
U1
U3’
U2’ U1’
Expected Return
Standard Deviation
Which investor is more risk averse?
More risk averse
Less risk averse
27Albert Lee Chun Portfolio Management
Stochastic DominanceStochastic DominancePrefers any portfolio in
Z1 to X.
Prefers X to any portfolio in Z4.
The rankings between portfolios in Z2 or Z3 and
X, depends on the preferences of the
investor!
28Albert Lee Chun Portfolio Management
Imagine a world with Imagine a world with 1 risk-free security and 1 risky security1 risk-free security and 1 risky security
29Albert Lee Chun Portfolio Management
1 Risk-Free Asset and 1 Risky Asset1 Risk-Free Asset and 1 Risky Asset
)()( AAffp rEwrwrE
00 w = 2A
2A
2p
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.
Suppose we construct a portfolio P consisting of 1 risk-free asset f and 1 risky asset A:
w = AAp
30Albert Lee Chun Portfolio Management
1 Risk Free Asset and 1 Risky Asset1 Risk Free Asset and 1 Risky Asset
E(rA) = 15%
rf = 7%
0
A
f
P =16.5%
E(rP) = 13%P
E(rP) = .25*.07+.75*15=13% p = .75*.22 = 16.5%
Suppose WA = .75
A =22%
31Albert Lee Chun Portfolio Management
A
fA r - )rE(
fPA
fAP r
rrErE
*
)()(
E(rA)
rf
0
A
f
p
E(rp) P
A
Capital Allocation Line (CAL)Capital Allocation Line (CAL)
Slope of CAL
Equation of CAL Line
Intercept
32Albert Lee Chun Portfolio Management
Maximize Investor Utility Maximize Investor Utility
In our world with 1 risk free asset and 1 risky asset, In our world with 1 risk free asset and 1 risky asset, if an investor has quadratic utility, what is the optimal if an investor has quadratic utility, what is the optimal portfolio allocation?portfolio allocation?
PAP ArEU 2
1)(
,)1()()(222
AP
fAP
w
rwrwErE
Utility:
Expected return and variance:
Goal is to Maximize utility. How?
33Albert Lee Chun Portfolio Management
Normally a Bear Lives in a Cave, that is Normally a Bear Lives in a Cave, that is Concave,Concave,
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:
A concave A concave function has a function has a
negative negative second second
derivative.derivative.
34Albert Lee Chun Portfolio Management
However, if the Bear is Swimming in a Bowl, However, if the Bear is Swimming in a Bowl, that is Convex,that is Convex,
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:
A convex A convex function has a function has a
positive positive second second
derivative.derivative.
35Albert Lee Chun Portfolio Management
Maximize Investor UtilityMaximize Investor Utility
2A
fA*
A
r - )rE( = w
- )1()(
)(22
21
22
1
AfA
PP
AwrwrwE
ArEU
Take derivative of U with respect to w and set equal to 0:
0)()( 2 AfA AwrrE
dw
wdU
w* is the optimal weight on risky
asset A
36Albert Lee Chun Portfolio Management
Example 1Example 1
Supppose E(rSupppose E(rAA) = 15%; ) = 15%; (r(rAA) = 22% and r) = 22% and rff = 7% and we = 7% and we
have a Quadratic investor with A = 4, thenhave a Quadratic investor with A = 4, then
w* = (0.15-0.07)/[4*(0.22)w* = (0.15-0.07)/[4*(0.22)22] ]
= 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%
2A
fA*
A
r - )rE( = w
w = Ap *
)(**)1()( Afp rEwrwrE
37Albert Lee Chun Portfolio Management
Example 2Example 2
Supppose E(rSupppose E(rAA) = 15%; ) = 15%; (r(rAA) = 22% and r) = 22% and rff = 7% and = 7% and we have a less risk-averse Quadratic investor with A we have a less risk-averse Quadratic investor with A = 1, then= 1, then
w* = (0.15-0.07)/[1*(0.22)w* = (0.15-0.07)/[1*(0.22)22] ] == 1.65 1.65 > 1 > 1 This investor should place 165% of his capital in This investor should place 165% of his capital in AA. . He needs He needs
to borrow 65% of his capital at the risk free rate of 7%.to borrow 65% of his capital at the risk free rate of 7%.
E(RE(Rpp) = 1.65(0.15) + -0.65(0.07)= 20.2% ) = 1.65(0.15) + -0.65(0.07)= 20.2%
(r(rpp) ) = 1.65*0.22= 0.363 = 36.3%= 1.65*0.22= 0.363 = 36.3%
His utility is: U = 0.202 – 0.5*1*(0.3632) = 0.1361
38Albert Lee Chun Portfolio Management
Graphical ViewGraphical View
A
E(r)
7%Ex1: Lender
Ex2: Borrower
A = 22%
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
The optimal allocation is the point of tangency between the CAL and the investor’s utility function.
39Albert Lee Chun Portfolio Management
Different Borrowing RateDifferent Borrowing Rate
What if the borrowing rate is higher than the lending What if the borrowing rate is higher than the lending rate?rate? E(r)
9%
7%
A
A = 22%
w* = (0.15-0.09)/[1*(0.22)w* = (0.15-0.09)/[1*(0.22)22] = ] = 1.241.241.241.24 < 1.65 < 1.65
40Albert Lee Chun Portfolio Management
Different Borrowing Rate Different Borrowing Rate
Supppose E(rSupppose E(rAA) = 15%; ) = 15%; (r(rAA) = 22% and r) = 22% and rff = 7% lending rate, = 7% lending rate, and a and a 9%9% borrowing rate, Quadratic investor with A = 1, then borrowing rate, Quadratic investor with A = 1, then
w* = (0.15-0.09)/[1*(0.22)2] = w* = (0.15-0.09)/[1*(0.22)2] = 1.241.241.241.24 < 1.65 < 1.65
This investor should place 124% of his capital in This investor should place 124% of his capital in AA. . He needs to borrow He needs to borrow 24% of his capital at the risk free rate of 24% of his capital at the risk free rate of 99%. This is less than what he %. This is less than what he would borrow at a 7% borrowing rate.would borrow at a 7% borrowing rate.
E(RE(Rpp) = 1.24(0.15) + -0.24(0.09)= 16.44% ) = 1.24(0.15) + -0.24(0.09)= 16.44%
(r(rpp) ) = 1.24*0.22= 27.28%= 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
41Albert Lee Chun Portfolio Management
Imagine a world with Imagine a world with 2 risky securities2 risky securities
42Albert Lee Chun Portfolio Management
Expected Return and Standard Deviation with Expected Return and Standard Deviation with Various Correlation CoefficientsVarious Correlation Coefficients
7-427-42
43Albert Lee Chun Portfolio Management
Portfolio Expected Return as a Function of Investment Portfolio Expected Return as a Function of Investment ProportionsProportions
7-437-43
44Albert Lee Chun Portfolio Management
Portfolio Standard Deviation as a Function of Investment Portfolio Standard Deviation as a Function of Investment ProportionsProportions
7-447-44
45Albert Lee Chun Portfolio Management
Returning to the Returning to the Two-Security PortfolioTwo-Security Portfolio
2211p rwrw)r(E
)r,r(Covww2ww 21212
22
22
12
12
p
and
, or
)r,r(Covww2ww 21212
22
22
12
1p
Question: What happens if we use various securities’ combinations, i.e. if we vary ?
7-457-45
46Albert Lee Chun Portfolio Management
Portfolio Expected Return as a function of Portfolio Expected Return as a function of Standard DeviationStandard Deviation
7-467-46
47Albert Lee Chun Portfolio Management
Perfect CorrelationPerfect Correlation
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = +1.00
D
EWith two perfectly correlated securities, all portfolios will lie on a straight line between the two assets.
With short selling
48Albert Lee Chun Portfolio Management
Perfect CorrelationPerfect Correlation
= +1= +1
)()()( EEDDP REwREwRE
) w + w ( =
w w 2 + w + w = 2
EEDD
EDED2E
2E
2D
2D
2p
1
w w= EEDDp
DEDE rrCov ),(
thatRecall
49Albert Lee Chun Portfolio Management
Zero CorrelationZero Correlation
f
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
f
gh
ij
k1
2With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either asset!
50Albert Lee Chun Portfolio Management
Zero CorrelationZero Correlation
= 0= 0
DEDE rrCov ),(
thatRecall
w + w =
w w 2 + w + w = 2E
2E
2D
2D
EDED2E
2E
2D
2D
2p
0
w + w= 2E
2E
2D
2Dp
51Albert Lee Chun Portfolio Management
Positive CorrelationPositive Correlation
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
Rij = +0.50
f
gh
ij
k1
2With positively correlated assets a two asset portfolio lies between the first two curves
EDDEED2E
2E
2D
2D
2p ww2 + w + w =
52Albert Lee Chun Portfolio Management
Negative CorrelationNegative Correlation
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
Rij = -0.50
Rij = +0.50
f
gh
ij
k1
2
With negatively correlated assets it is possible to create a portfolio with much lower risk.
EDDEED2E
2E
2D
2D
2p ww2 + w + w =
Negative
53Albert Lee Chun Portfolio Management
Perfect Negative CorrelationPerfect Negative Correlation
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
Rij = -1.00
Rij = +0.50
f
gh
ij
k1
2
With perfectly negatively correlated assets it is possible to create a two asset portfolio with NO RISK. How?
54Albert Lee Chun Portfolio Management
Perfect Negative Correlation Perfect Negative Correlation
) w - w ( =
ww 2 - w + w = 2
EEDD
EDED2E
2E
2D
2D
2p
| w - w=| EEDDp
0 =
+
= w and +
= w
P
ED
DE
ED
ED
obtain then we
= -1= -1
To get a zero-variance portfolio we
need to set:
DEDE rrCov 1),(
thatNote
55Albert Lee Chun Portfolio Management
Minimum Variance PortfolioMinimum Variance Portfolio
56Albert Lee Chun Portfolio Management
Minimum Variance PortfolioMinimum Variance Portfolio
DEDD2E
2D
2D
2D
2p )w-(1w2 + )w-(1 + w = Min
0 = ) 2 - (2 - ) 4 - 2 + (2 w =
)w 4-(2 + )w-2(1 - w2 = w
DE2EDE
2E
2DD
DED2ED
2DD
D
2p
w - 1 = w
2 - +
- =
2 - +
- = w
DE
EDDE2E
2D
EDDE2E
DE2E
2D
DE2E
D
minmin
min
57Albert Lee Chun Portfolio Management
Minimum Variance PortfolioMinimum Variance Portfolio
ED
E2
ED
DEE
ED2E
2D
ED2E
D +
= ) + (
) + ( =
2 + +
+ = wmin
+
0 - +
0 - = w 2
E2D
2E
2E
2D
2E
D
min
2 - +
- = w
DE2E
2D
DE2E
D
min1>1> > -1 > -1
= -1= -1
= 0= 0
= 1= 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.)
*With short *With short sales can obtain sales can obtain
0 variance.0 variance.
58Albert Lee Chun Portfolio Management
• Relationship depends on correlation coefficientRelationship depends on correlation coefficient• -1.0 -1.0 << << +1.0 +1.0• The more negative the correlation, the greater the The more negative the correlation, the greater the
risk reduction potentialrisk reduction potential• IfIf= +1.0, no risk reduction is possible (absent = +1.0, no risk reduction is possible (absent
short sales).short sales).
Portfolio of Two Securities: Correlation EffectsPortfolio of Two Securities: Correlation Effects
7-587-58
59Albert Lee Chun Portfolio Management
Example: MVPExample: MVP
Example:Example: Suppose there are 2 securities A an B::
AA BB A,BA,B
E(r)E(r) 10%10% 14%14%
0.20.2 15%15% 20%20%
Find the minimum variance portfolio?Find the minimum variance portfolio?
w - 1 = w 2 - +
- = w AB
BA2B
2A
BA2B
Aminminmin
60Albert Lee Chun Portfolio Management
Example: MVPExample: MVP
61Albert Lee Chun Portfolio Management
Similar Example from the BookSimilar Example from the Book
• Suppose our investment universe comprises the two securities of Table 7.1:
DD EE D,ED,E
E(r)E(r) 8%8% 13%13%0.30.3
12%12% 20%20%
• What are the weights of each security in the minimum-variance portfolio?
7-617-61
62Albert Lee Chun Portfolio Management
Book Example: MVP Book Example: MVP
2
2 2
( , )
2 ( , )E D E
DD E D E
Cov r rw
Cov r r
• Solving the minimization problem we get:
• Numerically:2
2 2
(20) 720.82
(20) (12) 272Dw
1 0.18E Dw w
7-627-62
63Albert Lee Chun Portfolio Management
Investor’s Utility Investor’s Utility
E(r)E(r)
More Risk AverseInvestors
More Risk AverseInvestors
U’U’
U’’U’’
U’’’U’’’Less Risk AverseInvestors Less Risk AverseInvestors
64Albert Lee Chun Portfolio Management
Investor’s Utility MaximizationInvestor’s Utility Maximization
22
1)( ArEU
)()()( EEDDP rEwrEwrE
) 2 - + ( A
) - A( + )rE( - )rE( = w
DE2E
2D
DE2EED*
D
DEDD2E
2D
2D
2D
2p )w-(1w2 + )w-(1 + w =
Homework, you should be able to show that the optimal solution is:
65Albert Lee Chun Portfolio Management
ExampleExample
Example:Example: Suppose there are only 2 portfolios::
AA BB A,BA,B
E(r)E(r) 10%10% 14%14%
0.20.2 15%15% 20%20%
Find the optimal portfolio for a investor with quadratic utility ( A Find the optimal portfolio for a investor with quadratic utility ( A = 3)?= 3)?
w w
2 - +A
- A+rE - rE = w *
A*B
BA2B
2A
BA2BBA*
A 1 ,
66Albert Lee Chun Portfolio Management
Example Example
59.01
,41.015.0*2.0*2.0*15.02.03
15.0*2.0*2.02.0314.010.022
2
ww
2 - +
- + - = w
*A
*B
*A
67Albert Lee Chun Portfolio Management
Imagine a world with Imagine a world with 2 risky securities and 1 risk-free security2 risky securities and 1 risk-free security
68Albert Lee Chun Portfolio Management
Two Feasible CALsTwo Feasible CALs
7-687-68
69Albert Lee Chun Portfolio Management
Optimal CAL and the Optimal Risky PortfolioOptimal CAL and the Optimal Risky Portfolio
7-697-69
70Albert Lee Chun Portfolio Management
With a Risk Free AssetWith a Risk Free Asset
E(r)E(r)
CAL 1CAL 1
CAL 2CAL 2
CAL 3CAL 3
I’m the optimal risky portfolio, the tangent portfolio!!I’m the optimal risky portfolio, the tangent portfolio!!
I’m the one that maximizes the slope of the Capital Allocation Line !
I’m the one that maximizes the slope of the Capital Allocation Line !
EE
DDrrff
71Albert Lee Chun Portfolio Management
Optimal Portfolio Weights Optimal Portfolio Weights
p
fpp
r - )RE( = S
)()()( EEDDP REwREwRE
DEDD2E
2D
2D
2D
2p )w-(1w2 + )w-(1 + w =
*D
*E
DEfEfD2DfE
2EfD
DEfE2EfD
D
ww
rrEr rE rrE+ r rE
rrE-rrE = w
1
*
Homework: If you are ambitious, try to show that the optimal solution is:
72Albert Lee Chun Portfolio Management
The Optimal Overall PortfolioThe Optimal Overall Portfolio
7-727-72
73Albert Lee Chun Portfolio Management
The Proportions of the Optimal Overall The Proportions of the Optimal Overall PortfolioPortfolio
7-737-73
74Albert Lee Chun Portfolio Management
Optimal Overall Portfolio: Optimal Overall Portfolio: 2 Investors i and j2 Investors i and j
PT
E(r)
rf
i
jCAL
2P
fP*
T
T
A
r - )RE( = w
Optimal weight for each Optimal weight for each investor depends on risk investor depends on risk
aversion parameter Aaversion parameter A
75Albert Lee Chun Portfolio Management
Different Lending and BorrowingDifferent Lending and Borrowing
E(r)E(r)
rfrf
P1P1
P2P2
Bfr
76Albert Lee Chun Portfolio Management
Now imagine a world Now imagine a world with many risky assetswith many risky assets
77Albert Lee Chun Portfolio Management
The Markowitz ProblemThe Markowitz Problem
1i
iiw
p REwREMaxi
N
i
N
jpijjiww
1 1
*
N
iiw
1
1
Subject to the
constraint
78Albert Lee Chun Portfolio Management
Efficient FrontierEfficient Frontier
Efficient FrontierE(R)
79Albert Lee Chun Portfolio Management
• The optimal combinations result in lowest level of The optimal combinations result in lowest level of risk for a given returnrisk for a given return
• The optimal trade-off is described as the efficient The optimal trade-off is described as the efficient frontierfrontier
• These portfolios are dominantThese portfolios are dominant
Extending Concepts to All SecuritiesExtending Concepts to All Securities
7-797-79
80Albert Lee Chun Portfolio Management
Minimum Variance Frontier of Risky AssetsMinimum Variance Frontier of Risky Assets
E(r)
Minimum variancefrontier
Globalminimumvarianceportfolio
Individualassets
7-807-80
81Albert Lee Chun Portfolio Management
Efficient Variance Frontier of Risky AssetsEfficient Variance Frontier of Risky Assets
E(r)
Efficientfrontier
Globalminimumvarianceportfolio
Individualassets
7-817-81
82Albert Lee Chun Portfolio Management
The Efficient Portfolio SetThe Efficient Portfolio Set
7-827-82
83Albert Lee Chun Portfolio Management
Now imagine a world Now imagine a world with many risky assets andwith many risky assets and
1 risk-free asset 1 risk-free asset
84Albert Lee Chun Portfolio Management
Capital Allocation LinesCapital Allocation Lines
7-847-84
85Albert Lee Chun Portfolio Management
The Market PortfolioThe Market Portfolio
Capital Market Line
E(R)
MM
86Albert Lee Chun Portfolio Management
ReadingsReadings
Readings for Today’s lecture. Readings for Today’s lecture.
1. Chapter 7.1. Chapter 7.
2. If you have not taken Investments, you may want to 2. If you have not taken Investments, you may want to review Chapter 6 as well.review Chapter 6 as well.
Readings for next week:Readings for next week:
Finish reading Chapter 7, including appedicies.Finish reading Chapter 7, including appedicies.
Readings for the week after next:Readings for the week after next:
(Course Reader) Other Portfolio Selection Models(Course Reader) Other Portfolio Selection Models