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CHAPTER THREE: Portfolio Theory, Fund Separation and CAPM
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Markowitz Portfolio Selection
There is no single portfolio that is best for everyone.
• The Life Cycle — different consumption preference
• Time Horizons — different terms preference
• Risk Tolerance — different risk aversion
• Limited Variety of Portfolio — Limited “finished products” in markets
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The Trade-Off Between Expected Return and Risk
E r1
2rE
1
2
1w
2w
Expected Return Risk Weight
Asset 1
Asset 2
E r wE r w E r
w w w w
1 2
2 212 2
22
1 2
1
1 2 1
Portfolio of two assets
1 1 is correlation coefficient:
Markowitz’s contribution 1: The measurement of return and risk
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Mini Case 1: Portfolio of the Riskless Asset and a Single Risky Asset
0, 22 frrE E r r w E r r
w
f f
1
1
wE r r
E r r
E r rE r r
f
f
f
f
1
1
1
Suppose , how to achieve a target expected return ?
%20%,14%,6 11 rErf
%11rE
%5.12%20%5.62
%5.62%6%14
%6%11
1
1
w
rrE
rrEw
f
f
Is the portfolio efficient ?
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The Diversification Principle
Mini Case 2: Portfolio of Two Risky Assets
w w w w 1 2
2 21 2
21 1
1 w w1 21
The Diversification Principle — The standard deviation of the combination is less than the combination of the standard deviations.
Asset 1 Asset 2
Expected Return 0.14 0.08
Standard Deviation 0.20 0.15
Correlation Coefficient 0.6
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R 0 100% 8% 0.15
C 10% 90% 8.6% 0.1479Minimum Variance Portfolio
17% 83% 9.02% 0.1474
D 50% 50% 11% 0.1569
Symbol Proportion in Asset 1
Proportion in Asset 2
Portfolio Expected Return
Portfolio Standard Deviation
S 100% 0 14% 0.20
Hyperbola Frontier of Two Risky Assets Combination
.2000
C
0 .1569.1500.1479
.0860
.0902
.1100
.1400 S
D
R.0800
rE
Minimum Variance Portfolio
The Optimal Combination of Two Risky Assets
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— Diversification
wn
i ni 1
1, , ,Suppose , Then
n
i
n
i
n
ijj
iji
n
i
n
j
n
i
n
ijj
ij
n
iiiij nnnnnn 1 1 1
2
2
21 1 1 1
21
22 111111
Let ,n 0
n
i
n
ijj
ijij nn 1 12
1 Let ,
1 12
1
2
21n
n
nijjj i
n
iji
n
ij
Systematic ExposureMarkowitz’s contribution 2: Diversification.
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Mini Case 3: Portfolio of Many Risky Assets
E ri i n 1, ,Expected return: :
ijCovariance: : i j n, , , 1
E r w E r
w w
i ii
n
i j ijj
n
i
n
1
2
11
2 0 ?
min
. .
wi j ij
j
n
i
n
i ii
n
ii
n
w w
s t w E r E r
w
2
11
1
1
1
Resolving the quadratic programming, get the minimum variance frontier
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Efficient Frontier of Risky As
sets
The Mean-Variance Frontier
E r
min 0
Indifference Curve of Utility
Optimal Portfolio of Risky Assets
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Proposition!
The variance of a diversified portfolio is irrelevant to the variance of individual assets. It is relevant to the covariance between them and equals the average of all the covariance.
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• Systematic risk cannot be diversified
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Proposition!Only unsystematic risks can be diversified.
Systematic risks cannot be diversified. They can be hedged and transferred only.
Markowitz’s contribution 3: Distinguishing systematic and unsystematic risks.
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Proposition!There is systematic risk premium contained in the expected return. Unsystematic risk premium cannot be got through transaction in competitive markets. iirE ,
Only systematic risk premium contained, no unsystematic risk premium contained.
Both systematic and unsystematic volatilities contained
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Two Fund Separation
The portfolio frontier can be generated by any two distinct frontier portfolios.
Theorem: Practice:
If individuals prefer frontier portfolios, they can simply hold a linear combination of two frontier portfolios or mutual funds.
E r
0
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Orthogonal Characterization of the Mean-Variance Frontier
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Orthogonal Characterization of the Mean-Variance Frontier
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P(x)=0
P(x)=1R*
1
E=0 E=1
Re*
ieii nrwrr ** ~~~
in
** ~~~eii rwrr
Proposition: Every return ri can be represented as
0
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Efficient Frontier of Risky
Assets
The Portfolio Frontier: where is R*?
E r
0
R*
w1
w2
w3
in
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Some Properties of the Orthogonal Characterization
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Capital Market Line (CML)
rf
M
E r
0
Indifference Curve 2
Indifference Curve 1
CAL 1
CAL 2
CML
P P can be the linear combination of M and rf
CAL — Capital Allocation Line
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Combination of M and Risk-free Security
wM — The weight invested in portfolio M
1 wM — The weight invested in risk-free security
E r r
E r r
w
p f
m f
Mp
p M M
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Market Portfolio• Definition:
A portfolio that holds all assets in proportion to their observed market values is called the Market Portfolio.
Security Market Value Composition
Stock A $66 billion 66%
Stock B $22 billion 22%
Treasury $12 billion 12%
Total $100 billion 100%
M is a market portfolio of risky assets
1. Two fund separation
2. Market clearing
!
Substitute: Market Index
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Capital Asset Pricing Model (CAPM)
• Assumptions: 1. Many investors, they are price – takers. The market is perfectly competitive.
2. All investors plan for one identical holding period.
3. Investments to publicly traded financial assets. Financing at a fixed risk – free rate is unlimited.
4. The market is frictionless, no tax, no transaction costs.
5. All investors are rational mean – variance optimizers.
6. No information asymmetry. All investors have their homogeneous expectations.
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Derivation of CAPM
• Portfolio of risky assets p
p i j ijj
n
i
n
w w
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12
w i ni: , , 1The weights
If (market portfolio),
Mp
n
jij
MjiM w
1
)(
21
1
)(
n
iiM
MiM w
The exposure of the market portfolio of risky assets is only related to the correlation between individual assets and the portfolio.
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M E rM
rf
E r
0 1.0
SML
Derivation of CAPM: Security Market Line
E(rM)-rF
iiMM
iMi rr
2
21~~
i
iM
M
2
)( irE
E r r
E r ri f
M f
MiM
2
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Security Market Line (SML)
E r r
E r ri f
M f
MiM
2
i
iM
M
2
E r r E r ri f i M f
p i ii
n
w
1
are additive
E r r E r rp f p M f
• Model
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Understanding Risk in CAPM• In CAPM, we can decompose an asset’s
return into three pieces:
ifMiifi rrrr ~)~(~
0)~,~(
0)~(
iM
i
rCov
rE
where
• Three characteristic of an asset:– Beta– Sigma– Aplha
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M E rM
rf
E r
0 1.0
SML
The market becomes more aggressive
The market becomes more conservative
Risk neutral
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Summary of Chapter Three1. The Key of Investments Trade-Off Between
Expected Return and Risk
2. Diversification Only Systematic Risk Can Get Premium
3. Two Fund Separation Any Trade in the Market can be Considered as a Trade Between Two Mutual Funds
4. CAPM — Individual Asset Pricing