Markowitz Portfolio Selection

1

CHAPTER THREE: Portfolio Theory, Fund Separation and CAPM

2

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

3

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

4

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 ?

5

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

6

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

7

— 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.

8

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

9

Efficient Frontier of Risky As

sets

The Mean-Variance Frontier

E r

min 0

Indifference Curve of Utility

Optimal Portfolio of Risky Assets

10

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.

11

• Systematic risk cannot be diversified

12

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.

13

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

14

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

15

Orthogonal Characterization of the Mean-Variance Frontier

16

Orthogonal Characterization of the Mean-Variance Frontier

17

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

18

Efficient Frontier of Risky

Assets

The Portfolio Frontier: where is R*?

E r

0

R*

w1

w2

w3

in

19

Some Properties of the Orthogonal Characterization

20

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

21

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

22

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

23

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.

24

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.

25

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

26

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

27

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

28

M E rM

rf

E r

0 1.0

SML

The market becomes more aggressive

The market becomes more conservative

Risk neutral

29

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