Portfolio Management 3-228-07 Albert Lee Chun

1
Portfolio Management3-228-07 Albert Lee Chun
Construction of Portfolios Markowitz and the
Efficient Frontier

• Session 4

25 Sept 2008
2
Plan for Today

• A Quick Review
• Optimal Portfolios of N risky securites
• - Markowitzs Portfolio Optimization
• - Two Fund Theorem
• Optimal Portfolios of N risky securities and a
  risk-free asset
• - Capital Market Line
• - Market Portfolio
• -Different Borrowing and Lending rates

3
Une petite révision
4
We started in a simple universe of1 risky asset
and 1 risk-free asset
5
Optimal Weights Depended on Risk Aversion
Each investor chooses an optimal weight on the
risky asset, where wgt 1 corresponds to borrowing
at the risk-free rate, and investing in the risky
asset.
E(r)
Borrower
Rf
Lender
?
?A
The optimal choice is the point of tangency
between the capital allocation line and the
agents utility function.
6
Utility maximization
Take the derivative and set equal to 0
7
We then looked at a universe with 2 risky
securities
8
Correlation and Risk
E(R)
f
E
g
?DE -1.00
h
i
j
?DE 1.00
?DE 0.50
k
D
?DE 0.00
9
Minimum Variance Portfolio
1gt? gt -1
? -1
? 0
Asset with the lowest variance, in the absence of
short sales.
? 1
10
Maximize Investor Utility
The solution is
11
Then we introduced a risk-free asset
12
Optimal Portfolio is the Tangent Portfolio
D
13
Optimal Portfolio Weights
The solution is
14
Optimal Borrowing and Lending
Borrower
w gt1
The optimal weight on the optimal risky portfolio
P depends on the risk-aversion of each investor.
E
Lender
wlt1
D
15
Now imagine a universe with a multitude of risky
securities
16
Harry 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.
17
Markowitz Efficient Frontier
Efficient Frontier
E
µ
D
s
18
The Problem of Markowitz I
Subject to the constraint
Weights sum to 1
Maximize the expected return of the portfolio
conditioned on a given level of portfolio
variance.
19
The problem of Markowitz II
Subject to the constraint
Weights sum to 1
Minimize the variance of the portfolio
conditioned on a given level of expected return.
20
Does the Risk of an Individual Asset Matter?

• Does an asset which is characterized by
  relatively large risk, i.e., great variability of
  the return, require a high risk premium?
• Markowitzs theory of portfolio choice clarified
  that the crucial aspect of the risk of an asset
  is not its risk in isolation, but the
  contribution of each asset to the risk of an
  entire portfolio.
• However, Markowitzs theory takes asset returns
  as given. How are these returns determined?

21
Citation de Markowitz

• So about five minutes into my defense,
  Friedman says, well Harry Ive read this. I
  dont find any mistakes in the math, but this is
  not a dissertation in economics, and we cannot
  give you 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 began to
  sweat. At one point he says, you have a problem.
  Its not economics, its not mathematics, its
  not business administration, and Professor
  Marschak said, Its not 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 out and said congratulations Dr.
  Markowitz.

22
Two-Fund Theorem
Interesting Fact Any two efficient portfolios
will generate the entire efficient frontier!
B
Every point on the efficient frontier is a linear
combination of any two efficient portfolios A and
B.
A
23
Now imagine a risky universe with a risk-free
asset
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Capital Market Line
CML maximizes the slope.
Capital Market Line
Tangent Portfolio
E
M
D
rf
25
Tobins Separation Theorm

• James Tobin ... in a 1958 paper said if you hold
  risky securities and are able to borrow - buying
  stocks on margin - or lend - buying risk-free
  assets - and you do so at the same rate, then the
  efficient frontier is a single portfolio of risky
  securities plus borrowing and lending....
• Tobin's Separation Theorem says you can separate
  the problem into first finding that optimal
  combination of risky securities and then deciding
  whether to lend or borrow, depending on your
  attitude toward risk. He then showed that if
  there's only one portfolio plus borrowing and
  lending, it's got to be the market.

26
Market Portfolio
Capital Market Line
w gt1
Market Portfolio
E
wlt1
M
M
D
rf
27
Separation Theorem
Borrower
Capital Market Line
w gt1
w 1
Lender
M
wlt1
Separation of investment decision from the
financing decision.
rf
28
Who holds only the Market Portfolio?
w gt1
Borrower AltAM
CML
AAM
w 1
wlt1
Lender AgtAM
M
rf
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Note that we have reduce the complexity of this
universe down to simply 2 points
30
Different Borrowing and Lending Rates
Borrower
MB
Lender
rB
ML
rL
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Who are the Lenders and Borrowers
Borrower
AltAMB
MB
Lender
rB
ML
AgtAML
rL
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Who are the Lenders and Borrowers
Borrower
AltAMB
MB
Lender
rB
ML
AgtAML
rL
33
Who holds only risky assets?
Emprunteur
AltAMB
AMB ltAltAML
MB
Prêteur
rB
ML
AgtAML
rL
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Efficient Frontier
Borrower
AltAMB
AMB ltAltAML
MD
Lender
rB
ML
AgtAML
rL
35
Where is the market portfolio?
The market portfolio can be anywhere here
rB
rf
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Only Risk-free Lending
Low risk averse agents cannot borrow, so they
hold only risky assets.
Least risk-averse lender
Lender
ML
rL
37
Efficient Frontier
The market portfolio can be anywhere here
All lenders hold this portfolio of risky
securities
Lenders
rL
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For Next Week

• Next week we will
• do a few examples, both numerical and in Excel.
• - discuss Appendix A diversification.
• discuss the article from the course reader.
• wrap up Chapter 7 and pave the way for the
  Capital Asset Pricing Model.

39
The Power of Diversification
90 of the total benefit of diversification is
obtained after holding 12-18 stocks.
Standard Deviation of Return
Non systematic risk (idiosyncratic, non
diversifiable)
Total Risk
Standard Deviation of the Market (systematic
risk)
Systematic Risk
Number of Stocks in the Portfolio