markowitz portfolio selection

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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

11

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