0 Portfolio Management 3-228-07 Albert Lee Chun Construction of Portfolios: Markowitz and the...

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Portfolio ManagementPortfolio Management3-228-073-228-07

Albert Lee ChunAlbert Lee Chun

Construction of Portfolios:

Markowitz and the Efficient Frontier

Session 4Session 4

25 Sept 2008

Albert Lee Chun Portfolio Management 2

Plan for TodayPlan for Today

A Quick ReviewA Quick Review Optimal Portfolios of N risky securitesOptimal Portfolios of N risky securites

- Markowitz`s Portfolio Optimization - Markowitz`s Portfolio Optimization

- Two Fund Theorem- Two Fund Theorem Optimal Portfolios of N risky securities and a risk-free assetOptimal Portfolios of N risky securities and a risk-free asset

- Capital Market Line- Capital Market Line

- Market Portfolio- Market Portfolio

-Different Borrowing and Lending rates-Different Borrowing and Lending rates


Une petite révisionUne petite révision


We started in a simple universe ofWe started in a simple universe of1 risky asset and 1 risk-free asset1 risky asset and 1 risk-free asset


Optimal Weights Depended on Risk AversionOptimal Weights Depended on Risk Aversion

E(r)

RfLender

Borrower

A

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.

The optimal choice is the point of tangency between the capital allocation line and the agent`s utility function.


Utility maximizationUtility maximization

2A

fA*

A

r - )rE( = w

- )1()(

)(22

21

22

1

AfA

PP

AwrwrwE

ArEU

0)()( 2 AfA AwrrE

dw

wdU

Take the derivative and set equal to 0


We then looked at a universe with 2 risky We then looked at a universe with 2 risky securitiessecurities


Correlation and Risk Correlation and Risk

-

0,05

0,10

0,15

0,20

0,000,010,020,030,040,050,060,070,080,090,10 0,110,12

E(R)

ρDE = 0.00

ρDE = +1.00

ρDE = -1.00

ρDE = + 0.50

f

gh

ij

kD

E


Minimum Variance Portfolio Minimum 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= 1Asset with the lowest variance, in the

absence of short sales.


Maximize Investor UtilityMaximize Investor Utility

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 =

The solution is:


Then we introduced a risk-free assetThen we introduced a risk-free asset


Optimal Portfolio is the Tangent PortfolioOptimal Portfolio is the Tangent Portfolio

E(r)E(r)

CAL 1CAL 1

CAL 2CAL 2

CAL 3CAL 3Every investor holds exactly

the same optimal portfolio of

risky assets!

Every investor holds exactly

the same optimal portfolio of

risky assets!

Intuition : the optimal solution is the CAL with the maximum slope!

Intuition : the optimal solution is the CAL with the maximum slope!

EE

DD


Optimal Portfolio WeightsOptimal 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

*

The solution is:


Optimal Borrowing and Lending Optimal Borrowing and Lending

P

E(r)

rf

CAL

2P

fP*

A

r - )E(r = w

The optimal weight on the optimal risky

portfolio P depends on the risk-aversion of each

investor.

Lender

Borrower

DD

EE

w*<1

w*

>1


Now imagine a universe with a multitude of Now imagine a universe with a multitude of risky securitiesrisky securities


Harry MarkowitzHarry 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.


Markowitz Efficient FrontierMarkowitz Efficient Frontier

port

)E(R port

D

E

Efficient Frontier

σ*

µ*


The Problem of Markowitz IThe Problem of Markowitz I

ii

N

iwp rEwrEMax

i

1

N

i

N

jpijji ww

1 1

*2

N

iiw

1

1

Subject to the

constraint:

Maximize the expected return of the portfolio conditioned on a given level of portfolio variance.

Weights sum to 1


The problem of Markowitz IIThe problem of Markowitz II

N

i

N

jijjip

w

wwMini 1 1

2

N

ipii rErEw

1

*)()(

N

iiw

1

1

Subject to the

constraint:

Minimize the variance of the portfolio conditioned on a given level of expected return.

Weights sum to 1


Does the Risk of an Individual Asset Matter?Does the Risk of an Individual Asset Matter?

Does an asset which is characterized by relatively Does an asset which is characterized by relatively large risk, i.e., great variability of the return, require a large risk, i.e., great variability of the return, require a high risk premium?high risk premium?

Markowitz’s theory of portfolio choice clarified that Markowitz’s theory of portfolio choice clarified that the crucial aspect of the risk of an asset is not its risk the crucial aspect of the risk of an asset is not its risk in isolation, but the contribution of each asset to the in isolation, but the contribution of each asset to the risk of an entire portfolio. risk of an entire portfolio.

However, Markowitz’s theory takes asset returns as However, Markowitz’s theory takes asset returns as given. How are these returns determined?given. How are these returns determined?


Citation de MarkowitzCitation de Markowitz

So about five minutes into my defense, Friedman says, well 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 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 this is not a dissertation in economics, and we cannot give you a PhD in economics for a dissertation that is not in economics. 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 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. began to sweat. At one point he says, you have a problem. It’s not economics, it’s not mathematics, it’s not business It’s not economics, it’s not mathematics, it’s not business administration, and Professor Marschak said, “It’s not administration, and Professor Marschak said, “It’s not literature”. So after about an hour and a half of that, they send 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 me out to the hall, and about five minutes later Marschak came out and said congratulations Dr. Markowitz. out and said congratulations Dr. Markowitz.


Two-Fund Theorem Two-Fund Theorem

port

)E(rport

A

B

Interesting Fact: Any two efficient portfolios will

generate the entire efficient frontier!

Every point on the efficient frontier is

a linear combination of any

two efficient portfolios A and B.


Now imagine a risky universe with a risk-free Now imagine a risky universe with a risk-free assetasset


Capital Market LineCapital Market Line

port

)E(rport

rfD

E

Capital Market LineCML maximizes th

e

slope.

Tangent

Portfolio

M

*D

*E

DEfEfD2DfE

2EfD

DEfE2EfD

D

ww

rrEr rE rrE+ r rE

rrE-rrE = w

1

*


Tobin’s Separation TheormTobin’s Separation Theorm

James Tobin ... in a 1958 paper said if you hold risky James Tobin ... in a 1958 paper said if you hold risky securities and are able to borrow - buying stocks on margin - securities and are able to borrow - buying stocks on margin - or lend - buying risk-free assets - and you do so at the same or lend - buying risk-free assets - and you do so at the same rate, then the efficient frontier is a single portfolio of risky rate, then the efficient frontier is a single portfolio of risky securities plus borrowing and lending....securities plus borrowing and lending....

Tobin's Separation Theorem says you can separate the Tobin's Separation Theorem says you can separate the problem into first finding that optimal combination of risky problem into first finding that optimal combination of risky securities and then deciding whether to lend or borrow, securities and then deciding whether to lend or borrow, depending on your attitude toward risk. He then showed that if depending on your attitude toward risk. He then showed that if there's only one portfolio plus borrowing and lending, it's got there's only one portfolio plus borrowing and lending, it's got to be the market.to be the market.


Market PortfolioMarket Portfolio

port

)E(rport

M

D

EM

Capital Market

Line

rf

Market

Portfolio

w*<1

w*

>1

2M

fM*

A

r - )rE( w


Separation TheoremSeparation Theorem

port

)E(rport

M

Capital Market

Line

rf

Separation of investment decision

from the financing decision.

Lender

Borrower

w*<1

w*

>1

w*

=1


Who holds only the Market Portfolio?Who holds only the Market Portfolio?

port

)E(rport

M

CML

rf

2M

fMM

2M

fM*

r - )rE(A

A

r - )rE( = w

1Le

nder

A>AM

Borrower

A<AM

A=AM

w*<1

w*

>1

w*

=1


Note that we have reduce the complexity of Note that we have reduce the complexity of this universe down to simply 2 pointsthis universe down to simply 2 points


Different Borrowing and Lending RatesDifferent Borrowing and Lending Rates

port

)E(rport

rL

rB Lender

Borrower

ML

MB

Albert Lee Chun Portfolio Management

MB

31

Who are the Lenders and BorrowersWho are the Lenders and Borrowers

rL

rB Lender

Borrower

2M

LMM

L

LL r - )rE(

A

2M

BMM

B

BB r - )rE(

A ML

A>AML

A<AMB

)E(rport

port


MB

32

Who are the Lenders and BorrowersWho are the Lenders and Borrowers

rL

rB Lender

Borrower

ML

A>AML

A<AMB

)E(rport

port1

2M

LML

*

L

L

A

r - )rE( w

1 2

M

BMB

*

B

B

A

r - )rE( w


MB

33

Who holds only risky assets?Who holds only risky assets?

rL

rB Prêteur

Emprunteur

ML

A>AML

A<AMB

)E(rport

port

AMB <A<AML

) 2 - + ( A

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

LBLB

LBLLB

B

MM2M

2M

MM2MMM*

M


MD

34

Efficient FrontierEfficient Frontier

rL

rB Lender

Borrower

ML

A>AML

A<AMB

)E(rport

port

AMB <A<AML


Where is the market portfolio?Where is the market portfolio?

port

)E(rport

rf

The market

portfolio can be

anywhere here

rB


Only Risk-free LendingOnly Risk-free Lending

port

)E(rport

rL

Lender

ML

2M

LMM

L

LL r - )rE(

A

Low risk averse agents cannot borrow, so they hold only risky assets.

Least risk-averse lender


Efficient FrontierEfficient Frontier

port

)E(rport

rL

The market

portfolio can be

anywhere here

Lenders

All lenders hold this portfolio of risky securities


For Next WeekFor Next Week

Next week we will Next week we will - do a few examples, both numerical and in Excel.do a few examples, both numerical and in Excel.

- discuss Appendix A – diversification.- discuss Appendix A – diversification.- discuss the article from the course reader.discuss the article from the course reader.- wrap up Chapter 7 and pave the way for the Capital wrap up Chapter 7 and pave the way for the Capital

Asset Pricing Model.Asset Pricing Model.

38


The Power of DiversificationThe Power of Diversification

Standard Deviation of Return

Number of Stocks in the Portfolio

Standard Deviation of the Market (systematic risk)

Systematic Risk

Total Risk

Non systematic risk (idiosyncratic, non diversifiable)

90% of the total benefit of diversification is obtained after

holding 12-18 stocks.

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