0 portfolio management 3-228-07 albert lee chun construction of portfolios: markowitz and the...
TRANSCRIPT
1
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
Albert Lee Chun Portfolio Management 3
Une petite révisionUne petite révision
Albert Lee Chun Portfolio Management 4
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
Albert Lee Chun Portfolio Management 5
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.
Albert Lee Chun Portfolio Management 6
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
Albert Lee Chun Portfolio Management 7
We then looked at a universe with 2 risky We then looked at a universe with 2 risky securitiessecurities
Albert Lee Chun Portfolio Management 8
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
Albert Lee Chun Portfolio Management 9
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.
Albert Lee Chun Portfolio Management 10
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:
Albert Lee Chun Portfolio Management 11
Then we introduced a risk-free assetThen we introduced a risk-free asset
Albert Lee Chun Portfolio Management 12
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
Albert Lee Chun Portfolio Management 13
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:
Albert Lee Chun Portfolio Management 14
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
Albert Lee Chun Portfolio Management 15
Now imagine a universe with a multitude of Now imagine a universe with a multitude of risky securitiesrisky securities
Albert Lee Chun Portfolio Management 16
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.
Albert Lee Chun Portfolio Management 17
Markowitz Efficient FrontierMarkowitz Efficient Frontier
port
)E(R port
D
E
Efficient Frontier
σ*
µ*
Albert Lee Chun Portfolio Management 18
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
Albert Lee Chun Portfolio Management 19
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
Albert Lee Chun Portfolio Management 20
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?
Albert Lee Chun Portfolio Management 21
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.
Albert Lee Chun Portfolio Management 22
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.
Albert Lee Chun Portfolio Management 23
Now imagine a risky universe with a risk-free Now imagine a risky universe with a risk-free assetasset
Albert Lee Chun Portfolio Management 24
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
*
Albert Lee Chun Portfolio Management 25
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.
Albert Lee Chun Portfolio Management 26
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
Albert Lee Chun Portfolio Management 27
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
Albert Lee Chun Portfolio Management 28
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
Albert Lee Chun Portfolio Management 29
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
Albert Lee Chun Portfolio Management 30
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
Albert Lee Chun Portfolio Management
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
Albert Lee Chun Portfolio Management
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
Albert Lee Chun Portfolio Management
MD
34
Efficient FrontierEfficient Frontier
rL
rB Lender
Borrower
ML
A>AML
A<AMB
)E(rport
port
AMB <A<AML
Albert Lee Chun Portfolio Management 35
Where is the market portfolio?Where is the market portfolio?
port
)E(rport
rf
The market
portfolio can be
anywhere here
rB
Albert Lee Chun Portfolio Management 36
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
Albert Lee Chun Portfolio Management 37
Efficient FrontierEfficient Frontier
port
)E(rport
rL
The market
portfolio can be
anywhere here
Lenders
All lenders hold this portfolio of risky securities
Albert Lee Chun Portfolio Management
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
Albert Lee Chun Portfolio Management 39
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.