Investments 10

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Efficient Diversification II

Efficient Frontier with Risk-Free Asset

Optimal Capital Allocation Line

Single Factor Model

Eff. Frontier with Risk-Free Asset

With risky assets only

No portfolio with zero variance

GMVP has the lowest variance

With a risk-free asset

Zero variance if investing in risk-free asset only

How does it change the efficient frontier?

Investments 10 2

Optimal CAL

Mean-variance with two risky assets

 w in security 1, 1 – w in security 2

1

2

0

0

.

10

.

14

1

2

0 .

15

0 .

20

12

0 .

2

Expected return (Mean):

 p

0 .

10

 w

0 .

14

( 1

 w )

Variance

2 p

0 .

15

2 w

2 

0 .

20

2

( 1

 w )

2 

2

0 .

2

0 .

15

0 .

20

 w ( 1

 w )

What happens when we add a risk-free asset?

A riskfree asset with r f

= 5%

What is achievable now?

Investments 10 3

Eff. Frontier with Risk-Free Asset

E[r]

M

M

CAL (P)

P

P

CAL

G

F

P P&F M

Investments 10 4

Eff. Frontier with Risk-Free Asset

CAL(P) dominates other lines

Best risk and return trade-off

Steepest slope

S

P

E [ r p

] p

 r f 

E [ r

A

]

A

 r f

Portfolios along CAL(P) has the same highest

Sharpe ratio

No portfolio with higher Sharpe ratio is achievable

Dominance independent of risk preference

How to find portfolio (P)?

Investments 10 5

Optimal Portfolio

How much in each risky asset?

w

1

1

2

( E [ r

2

]

 r f

2

)

2

( E [ r

1

]

 

2

2

 r f

( E [ r

1

]

)

 r f



)

1

2

( E [ r

2



1

2

]

 r f

)

( E [ r

1

]

E [ r

2

]

2 r f

)

.4584

The expected return and standard dev.

 p

0 .

10

 w

1

0 .

14

( 1

 w

1

)

0 .

1217

 p

0 .

1394

Sharpe Ratio

S

P

E [ r p

] p

 r f 

 p

 p r f 

0 .

1217

0 .

05

0 .

1394

0 .

514

Investments 10 6

Eff. Frontier with Risk-Free Asset

What’s so special about portfolio (P)?

P is the market portfolio

Mutual fund theorem: An index mutual fund

(market portfolio) and T-bills are sufficient for investors

Investors adjust the holding of index fund and T-bills according to their risk preferences

Investments 10 7

w

Optimal Portfolio Allocation

Investment Funds y 1y

P T-Bills

1w

Bond Stock

T - Bills Bond Stock

1 y y

× w y

×(1 w )

Investments 10

Two Step Allocation

Step 1: Determine the optimal risky portfolio

Get the optimal mix of stock and bond

Optimal for all investors

(market portfolio)

Step 2: Determine the best complete portfolio

Obtain the best mix of the optimal risky portfolio and T-Bills

Different investors may have different best complete portfolios

8

Single Factor Model

Quantifies idiosyncratic versus systematic risk of a stock’s rate of return

Factor is a broad market index like S&P500

The excess return is

R

   

R

 e

 i i i i i M i

: stock’s excess return above market performance

R

M

: stock’s return attributable to market performance

 e i

: return component from firm-specific unexpected event

Example: a statistical analysis between the excess returns of DELL and market shows that

= 4.5%,

= 1.4. If expected market excess return is 17%, what is the expected excess return for DELL?

Solution: E [ R i

]

  i

  i

E [ R

M

]

4 .

5 %

1 .

4

17 %

4 .

5 %

23 .

8 %

28 .

3 %

Investments 10 9

Single Factor Model

Security Characteristic Line

Dell Excess

Returns (i)

.

.

. ..

28.3%

. .

.

.

.

.

.

.

.

. ..

.

.

.

. .

.

.

.

.

. ..

..

.

17%

.

.

Security

Characteristic

Line

23.8%

 i

4.5%

Cov [

R i

2

M

, R

M

]

Excess Returns on market index

Investments 10 10

Single Factor Model

Meaning of Beta (

)

Indicator of how sensitive a security’s return is to changes in the return of the market portfolio.

A measure of the asset’s systematic risk.

Example: market portfolio’s risk premium is

+10% during a given period, and

= 0%.

 = 1.50, the security’s risk premium will be +15%.

 = 1.00, the security’s risk premium will be +10%

 = 0.50, the security’s risk premium will be +5%

= –0.50, the security’s risk premium will be –5%

Investments 10 11

Single Factor Model

Beta coefficients for selected firms (March 2010)

Common Stock Beta

Citigroup

Bank of America

Adobe Systems

Apple

GE

Amazon.com

Google

Microsoft

McDonald

’s

Pepsi

Exxon Mobile

Wal-Mart

Question:

2.71

2.41

1.80

1.57

1.52

1.27

1.12

0.98

0.64

0.52

0.43

0.26

What are the betas of market index and T-bills?

Investments 10 12

Single Factor Model

Systematic Risk

Risk related to the macro factor or market index

Non-diversifiable/market risk

Unsystematic Risk

Risk related to company specific problems

Diversifiable/Firm-specific/Idiosyncratic risk

Total risk = Systematic + Unsystematic

 i

2

2

 i

2

2

M

2

 i

 i

2

2

M

Var [ e i

]

% of variance explained by the market

Investments 10 13

Single Factor Model

Example

Given the following data on Microsoft, analyze the systematic risk, unsystematic risk and percentage of variance explained by systematic risk. ( σ i

σ

M

= 0.15, Cov [ R i

,R

M

]=0.0315)

= 0.25,

Solution

 i

Cov [

R i

2

M

, R

M

]

0.0315

.15

2

1.4

 i

2

2

M

Var [ e i

]

  i

2

2 

1.4

2 

.15

2

2

 i

 i

2

2

M

 i

2

2

M

0 .

0441

.

25

2 

0 .

0441

0 .

0184

.0441

.7056

70 .

56 %

.25

2

Investments 10 14

Diversification in a Single Factor

Security Market

A portfolio of three equally weighted assets 1,

2, and 3.

 p

The excess return of the portfolio is

R p

  p

  p

R

M

 e p

1

 

2

3

 

3

 p

1

 

2

3

 

3 e p

 e

1

 e

2

3

 e

3

Risk of the portfolio is

Var ( R p

)

 

2 p

Var ( R

M

)

Var ( e p

)

  p

2

2

M

Var ( e p

)

Investments 10 15

Wrap-up

What does the efficient frontier look like with the presence of a risk-free asset?

What are the two steps of asset allocation?

What is a single index model?

What are the meaning of systematic and unsystematic risks?

Investments 10 16

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