0 Portfolio Managment 3-228-07 Albert Lee Chun Construction of Portfolios: Introduction to Modern...

transcript

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

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

Portfolio DiversificationPortfolio Diversification

7-47-4

w1 = proportion of funds in Security 1w2 = proportion of funds in Security 2r1 = expected return on Security 1r2 = expected return on Security 2

Two-Security Portfolio: ReturnTwo-Security Portfolio: Return

2211P rwrwr

7-57-5

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

7-67-6

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

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

Three-Security PortfolioThree-Security Portfolio

332211p rwrwrwr

rrCovww

7-97-9

Generally, for an Generally, for an n-Security Portfolio:n-Security Portfolio:

iip rwr

kj1k,j

2p )r,r(Covww2w

Review of Portfolio StatisticsReview of Portfolio Statistics

iiip REwRE

ji )r ,r( Cov ww + w = jiji

jijiji rrCov ,),( ji

)r,r(Cov

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)

Utility FunctionsUtility Functions

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.

Example: Risk PremiumExample: Risk Premium

W2 = $80 Profit = -

W1 = $150 Profit = $50p = .6

Investment

T-bills Profit = $5

Expected return: (50%)(.6) + (-20%)(.4)

Risk Premium = E(Ri) – RfE(Ri) – Rf = 22% - 5% = = 17%

1-p = .4

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, )

Example: An Indifference CurveExample: An Indifference Curve

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).

Direction of Increasing UtilityDirection of Increasing Utility

Expected Return

Standard Deviation

Direction of Increasing Utility

U3 U3 > U2 > U1

Two Different InvestorsTwo Different Investors

U2’ U1’

Expected Return

Standard Deviation

Which investor is more risk averse?

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.

RERAEREU

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.

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%

Risk-Neutral Indifference CurvesRisk-Neutral Indifference CurvesE(RP)E(RP)

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

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.

Risk-Averse Indifference CurvesRisk-Averse Indifference Curves

E(RP)E(RP)

U1U1Expected Return

Standard Deviation

U3 > U2 > U1

Two Different InvestorsTwo Different Investors

U2’ U1’

Expected Return

Standard Deviation

Which investor is more risk averse?

More risk averse

Less risk averse

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!

Imagine a world with Imagine a world with 1 risk-free security and 1 risky security1 risk-free security and 1 risky security

1 Risk-Free Asset and 1 Risky Asset1 Risk-Free Asset and 1 Risky Asset

)()( AAffp rEwrwrE

00 w = 2A

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

1 Risk Free Asset and 1 Risky Asset1 Risk Free Asset and 1 Risky Asset

E(rA) = 15%

rf = 7%

P =16.5%

E(rP) = 13%P

E(rP) = .25*.07+.75*15=13% p = .75*.22 = 16.5%

Suppose WA = .75

A =22%

fA r - )rE(

E(rp) P

Capital Allocation Line (CAL)Capital Allocation Line (CAL)

Slope of CAL

Equation of CAL Line

Intercept

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()()(222

rwrwErE

Utility:

Expected return and variance:

Goal is to Maximize utility. How?

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.

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.

Maximize Investor UtilityMaximize Investor Utility

r - )rE( = w

- )1()(

AwrwrwE

Take derivative of U with respect to w and set equal to 0:

0)()( 2 AfA AwrrE

w* is the optimal weight on risky

asset A

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%

(rp) = 0.41*0.22=9.02%

r - )rE( = w

w = Ap *

)(**)1()( Afp rEwrwrE

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

Graphical ViewGraphical View

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.

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)

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

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

Imagine a world with Imagine a world with 2 risky securities2 risky securities

Expected Return and Standard Deviation with Expected Return and Standard Deviation with Various Correlation CoefficientsVarious Correlation Coefficients

7-427-42

Portfolio Expected Return as a Function of Investment Portfolio Expected Return as a Function of Investment ProportionsProportions

7-437-43

Portfolio Standard Deviation as a Function of Investment Portfolio Standard Deviation as a Function of Investment ProportionsProportions

7-447-44

Returning to the Returning to the Two-Security PortfolioTwo-Security Portfolio

2211p rwrw)r(E

)r,r(Covww2ww 21212

Question: What happens if we use various securities’ combinations, i.e. if we vary ?

7-457-45

Portfolio Expected Return as a function of Portfolio Expected Return as a function of Standard DeviationStandard Deviation

7-467-46

Perfect CorrelationPerfect Correlation

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

Rij = +1.00

EWith two perfectly correlated securities, all portfolios will lie on a straight line between the two assets.

With short selling

Perfect CorrelationPerfect Correlation

= +1= +1

)()()( EEDDP REwREwRE

) w + w ( =

w w 2 + w + w = 2

EDED2E

w w= EEDDp

DEDE rrCov ),(

thatRecall

Zero CorrelationZero Correlation

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

Rij = 0.00

Rij = +1.00

2With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either asset!

Zero CorrelationZero Correlation

= 0= 0

DEDE rrCov ),(

thatRecall

w + w =

w w 2 + w + w = 2E

EDED2E

w + w= 2E

Positive CorrelationPositive Correlation

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

Rij = 0.00

Rij = +1.00

Rij = +0.50

2With positively correlated assets a two asset portfolio lies between the first two curves

EDDEED2E

2p ww2 + w + w =

Negative CorrelationNegative Correlation

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

Rij = 0.00

Rij = +1.00

Rij = -0.50

Rij = +0.50

With negatively correlated assets it is possible to create a portfolio with much lower risk.

EDDEED2E

2p ww2 + w + w =

Negative

Perfect Negative CorrelationPerfect Negative Correlation

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

Rij = 0.00

Rij = +1.00

Rij = -1.00

Rij = +0.50

With perfectly negatively correlated assets it is possible to create a two asset portfolio with NO RISK. How?

Perfect Negative Correlation Perfect Negative Correlation

) w - w ( =

ww 2 - w + w = 2

EDED2E

| w - w=| EEDDp

= w and +

obtain then we

= -1= -1

To get a zero-variance portfolio we

need to set:

DEDE rrCov 1),(

thatNote

Minimum Variance PortfolioMinimum Variance Portfolio

DEDD2E

2p )w-(1w2 + )w-(1 + w = Min

0 = ) 2 - (2 - ) 4 - 2 + (2 w =

)w 4-(2 + )w-2(1 - w2 = w

DE2EDE

DED2ED

w - 1 = w

EDDE2E

minmin

= ) + (

) + ( =

+ = wmin

0 - = w 2

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.

• 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

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

Aminminmin

Example: MVPExample: MVP

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

Book Example: MVP Book Example: MVP

2 ( , )E D E

DD E D E

Cov r rw

Cov r r

• Solving the minimization problem we get:

• Numerically:2

(20) 720.82

(20) (12) 272Dw

1 0.18E Dw w

7-627-62

Investor’s Utility Investor’s Utility

E(r)E(r)

More Risk AverseInvestors

U’U’

U’’U’’

U’’’U’’’Less Risk AverseInvestors Less Risk AverseInvestors

Investor’s Utility MaximizationInvestor’s Utility Maximization

1)( ArEU

)()()( EEDDP rEwrEwrE

) 2 - + ( A

) - A( + )rE( - )rE( = w

DE2EED*

DEDD2E

2p )w-(1w2 + )w-(1 + w =

Homework, you should be able to show that the optimal solution is:

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)?

2 - +A

- A+rE - rE = w *

BA2BBA*

Example Example

,41.015.0*2.0*2.0*15.02.03

15.0*2.0*2.02.0314.010.022

- + - = w

Imagine a world with Imagine a world with 2 risky securities and 1 risk-free security2 risky securities and 1 risk-free security

Two Feasible CALsTwo Feasible CALs

7-687-68

Optimal CAL and the Optimal Risky PortfolioOptimal CAL and the Optimal Risky Portfolio

7-697-69

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 !

DDrrff

Optimal Portfolio Weights Optimal Portfolio Weights

r - )RE( = S

)()()( EEDDP REwREwRE

DEDD2E

2p )w-(1w2 + )w-(1 + w =

DEfEfD2DfE

DEfE2EfD

rrEr rE rrE+ r rE

rrE-rrE = w

Homework: If you are ambitious, try to show that the optimal solution is:

The Optimal Overall PortfolioThe Optimal Overall Portfolio

7-727-72

The Proportions of the Optimal Overall The Proportions of the Optimal Overall PortfolioPortfolio

7-737-73

Optimal Overall Portfolio: Optimal Overall Portfolio: 2 Investors i and j2 Investors i and j

r - )RE( = w

Optimal weight for each Optimal weight for each investor depends on risk investor depends on risk

aversion parameter Aaversion parameter A

Different Lending and BorrowingDifferent Lending and Borrowing

E(r)E(r)

Now imagine a world Now imagine a world with many risky assetswith many risky assets

The Markowitz ProblemThe Markowitz Problem

p REwREMaxi

jpijjiww

Subject to the

constraint

Efficient FrontierEfficient Frontier

Efficient FrontierE(R)

• 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

Minimum Variance Frontier of Risky AssetsMinimum Variance Frontier of Risky Assets

Minimum variancefrontier

Globalminimumvarianceportfolio

Individualassets

7-807-80

Efficient Variance Frontier of Risky AssetsEfficient Variance Frontier of Risky Assets

Efficientfrontier

Globalminimumvarianceportfolio

Individualassets

7-817-81

The Efficient Portfolio SetThe Efficient Portfolio Set

7-827-82

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

Capital Allocation LinesCapital Allocation Lines

7-847-84

The Market PortfolioThe Market Portfolio

Capital Market Line

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

0 Portfolio Managment 3-228-07 Albert Lee Chun Construction of Portfolios: Introduction to Modern...

Documents