04/03/2015 Pr. Didier Folus 1

Topic 6 – Introduction to Portfolio Theory

1. Practitioners fundamental issues

2. Portfolio optimization using Markowitz efficient frontier

3. Portfolio diversification & beta coefficient

4. Capital asset pricing model

1. Practitioners fundamental issues

• Traditional assets classes:

o Equities

o Bonds & rates

• Alternative asset classes:

o Currencies

o Commodities

o Real estate

• Does an optimal combination of asset classes exist?

• What does diversification mean?

• What about the risk-expected return relationship?

04/03/2015 Pr. Didier Folus 2

2. Portfolio optimization : Markowitz approach

2.1. Two risky assets combination

• Security X : EX = 8 % & sX = 12 %

• Security Y : EY = 10 % & sY = 18 %

04/03/2015 Pr. Didier Folus 3

Portfolio % 1 % EP sP

A’

A

B

C

D

D’

- 25

0

25

50

100

125

125

100

75

50

0

- 25

10.5 %

10.0 %

9.5 %

9.0 %

8.0 %

7.5 %

22.7 %

18.0 %

13.8 %

10.8 %

12.0 %

15.7 %

......

. 1

... .....

..

....

.. ...

..

0

...

....

..

.

...

....

......

..

- 1

.

.

.

.

.

Feasible portfolios hyperbola (case = 0)

• P return……

• P volatility…

04/03/2015 Pr. Didier Folus 4

5 % 10 % 15 % 20 %Standard deviation

Expected return

Y

X

10 %

MVP

A’

Short in X

Short in Y

8 %

A

B

D’

D

C9 %

YXP EEE 1

YXYXP sssss 121 2222

Computing the Minimum Variance Portfolio

(case = 0)

• Program :

• First order condition :

• Solution :

• MVP :

– investing 69.23 % in X & 30.77 % in Y

– expected return EMVP = 8.7 %

– expec ted volatility sMVP = 10.1 %

04/03/2015 Pr. Didier Folus 5

2min Ps

0

s

d

d P

6923.0cov2

cov22

2*

XYYX

XYY

ss

s

04/03/2015 Pr. Didier Folus 6

2.2. Optimal risky assets combination with r.f.a.

• Risk free asset F : r = 5 % & sF = 0 %

• Combinations between (X, Y) hyperbola and F

• Optimal risky combinations : (F, T, Y) curve

0.05 0.10 0.15 0.20Standard deviation

Expected return

0.05

Y

X

T

F

Risk aversion

04/03/2015 Pr. Didier Folus 7

2.3. H. Markowitz (1952, 1959) efficient frontier

• Line [F,T) if risk free asset short sales are allowed

• Segment [F, T] if not

P

T

TP

rErE s

s

0.05 0.10 0.15 0.20Standard deviation

Expected return

0.05

T

F

Best risk/reward ratio

3. Portfolio diversification

3.1. Two risky assets combination

• Security X : EX = 8 % & sX = 12 %

• Security Y : EY = 10 % & sY = 18 %

04/03/2015 Pr. Didier Folus 8

Portfolio 1 EP sP

% % = 1 = 0 = 1

A’

A

B

C

D

D’

- 25

0

25

50

100

125

125

100

75

50

0

- 25

10.5 %

10.0 %

9.5 %

9.0 %

8.0 %

7.5 %

25.5 %

18.0 %

10.0 %

3.0 %

12.0 %

19.5 %

22.7 %

18.0 %

13.8 %

10.8 %

12.0 %

15.7 %

19.5 %

18.0 %

16.5 %

15.0 %

12.0 %

10.5 %

Portfolio return equals the average assets return

Portfolio risk is less than

average assets risk

3.2. Several risky assets combination

04/03/2015 Pr. Didier Folus 9

Number of stocks

10 %

20 %

30 %

40 %

50 %

100 %

5 10 15 20 25 30 35 40

Portfolio variance

reduction

Specific risk or diversifiable risk :

- patent

- productivity

- strike…

Systematic risk :

- geopolitical tensions

- GDP growth

- IR fluctuations…

common factor(s)

Generally

R2 # 0.40

3.3. Measuring a stock sensitivity to market: beta

• Sharpe (1964) :

• Single security risk :

04/03/2015 Pr. Didier Folus 10

Linear

regression

quality

0 < R2 < 1

m

mi

ir

rr

var

,covitmtiiit

rr

Common factor

imii rr varvarvar 2

Stock sensitivity to market

Data Risk level Model

Stock return follows closely and

amplifies index return

Systematic risk high

Specific risk low

bi > 1

R2 1

Stock return follows closely and eases

index return

Systematic risk low

Specific risk low

bi < 1

R2 1

Stock return amplifies index return, but

doesn’t follow it closely

Systematic risk high

Specific risk high

bi > 1

R2 0

Stock return eases index return, but

doesn’t follow it closely

Systematic risk low

Specific risk high

bi < 1

R2 0

Systematic risk Specific risk

Industrial sectors beta coefficients

04/03/2015 Pr. Didier Folus 11

Sector Number of

companies

Average market beta

Jan 2014 Jan 2012 Jan 2010

Oil/gas utilities

Banks

Food processing

Medical services

Biotechnology

Basic chemical

Auto. parts

22

7

97

126

349

47

75

0.82

0.75

0.85

0.83

1.12

1.01

1.46

0.66

0.77

0.91

0.91

1.03

1.36

1.59

0.68

0.75

0.86

0.97

1.10

1.27

1.72

From Value Line database

7 766 firms

5 years weekly data

against Nyse Composite

By A. Damodaran

3.4. Risk reduction through diversification

• Sharpe model……………

• Single security risk…..

• Portfolio risk………...

• Given a portfolio, a i-security

contributes to the portfolio risk

through

• Diversified portfolio risk :

04/03/2015 Pr. Didier Folus 12

imii rr varvarvar 2

mPPnrr varvarlim 2

Pi

rr ,cov

PmPP rr varvarvar 2

itmtiiitrr

Systematic risk

Diversifiable

risk

04/03/2015 Pr. Didier Folus 13

4. The Capital Asset Pricing Model

4.1. Objective & assumptions

• Implementing a risk-expected return relation requires

observation of the tangency portfolio (see Markowitz)

• CAPM has been developped by Sharpe (1964), Lintner

(1965), Mossin (1966) :

A1. Investors care only about mean/variance of portfolio returns

A2. Markets are frictionless

A3. There exists one risk-free asset

A4. Investors have homogeneous beliefs (they reach the same

conclusion about all feasible portfolios returns distribution)

04/03/2015 Pr. Didier Folus 14

4.2. CAPM major results

4.2.1. Implications for optimal investment

• (A4) : the tangency portfolio is the market portfolio M

• (A3) : Markowitz efficient frontier reduces to CML :

0.05 0.10 0.15 0.20

Standard

deviation

Expected return

0.05

M

F

P

m

mP r

r

rrErrE s

s

Optimization example :

Computing the weights of the

market portfolio

AXA LVMH Peugeot

Shares 2,000 M 500 M 200 M

Quote EUR 10 EUR 50 EUR 25

Capi. EUR 20 bn EUR 25 bn EUR 5 bn

Weight 40 % 50 % 10 %

Choosing an optimal combination investing

EUR 100,000

Mister

Haterisk

Miss

Loverisk

Risk free 90 % 20 %

Tbills EUR 90,000 EUR 20,000

Risky assets 10 % 80 %

AXA

LVMH

Peugeot

EUR 4,000

EUR 5,000

EUR 1,000

EUR 32,000

EUR 40,000

EUR 8,000

Portfolio EUR 100,000 EUR 100,000

04/03/2015 Pr. Didier Folus 15

4.2.2. Implications for the risk premium

• Only the security systematic risk is profitable

• Security i risk premium :

04/03/2015 Pr. Didier Folus 16

rrErrEmii

Market risk premiumRisk free rate

or safe rate

Average market risk aversion

Market volatility

04/03/2015 Pr. Didier Folus 17

Source : Crédit Suisse Global Investment Returns Yearbook 2011.