Description

follow pdf as instruction, and data is the hw, the example and PPT as reference! make sure look at PPT AND EXAMPLE! !

Project 2

Due on February 3 by 5:00 pm

MGT 252 – Winter 2021

Investments and Portfolio Management

c 2021 Alexander Barinov

1

Diversification Classics (100 points)

The file Project 2.xls in the Projects folder contains the monthly return to Exxon Mobil

(XOM) and Darden Restaurants (DRI)1 in 2005-2009, as well as the market return (MKT)

and the risk-free rate (RF) in the same time period. Use these data to answer the questions

below:

i. Compute the average return, the standard deviation, and the correlation for Exxon

and Darden. (10 points)

ii. If you can only invest in the risk-free asset and either Exxon or Darden, which one

do you pick? (5 points)

iii. If you can only invest in the risk-free asset and either Exxon or Darden and your

target standard deviation is 15% per annum, what will be your best possible expected

monthly return? What weight does the risk-free asset take in your portfolio? (10

points)

iv. Compute the average return, standard deviation, and the Sharpe ratio of the portfolio

that invests 80% in Exxon and 20% in Darden (5 points)

v. Does the existence of Exxon and Darden violate the CAPM? (Hint: You will need

to use the data on the market excess returns). (5 points)

vi. Redo (iv) assuming that the correlation between Exxon and Darden is 0.6. What

does it tell you about the benefits of diversification? (5 points)

vii. What combination of Exxon and Darden creates the minimum variance portfolio

(MVP)? What are the average return, standard deviation and the Sharpe ratio of

this portfolio? (Hint: use the Solver add-in) (10 points)

1

Darden Restaurants is the holding company that owns Olive Garden, Red Lobster, and LongHorn

Steakhouse

1

viii. What combination of Exxon and Darden creates the mean-variance efficient portfolio

(MVE)? What are the average return, standard deviation and the Sharpe ratio of

this portfolio? (Hint: use the Solver add-in) (10 points)

ix. Bonus question: Estimate the market model for Exxon and the market model for

Darden. Do either Exxon or Darden violate the CAPM by a statistically significant

amount (10 points)

x. Assume that Exxon and Darden are the only two stocks in the economy. Under the

CAPM, what are the average return and standard deviation of the market portfolio?

What are the average return and standard deviation of the zero-beta portfolio? (Hint:

Use your solution to (viii) to answer this question) (10 points)

xi. Bonus question: Redo (iii) assuming that now you can invest in Exxon, Darden, and

the risk-free asset simultaneously. What are the weights of Exxon, Darden, and the

risk-free asset in the portfolio that delivers the best possible return for the target

standard deviation of 15% per annum? (10 points)

xii. Assume that Exxon and Darden are the only two stocks in the economy and there

is no risk-free asset. For each of the following three points – (0.83%, 5.44%), (1.03%,

6.53%), (0.95%, 5.31%) – answer two questions: (a) Will you want to choose it? (b)

Can you have it? The first number of the pair is always expected return. (Hint 1:

use Solver. Hint 2: do not forget comparing the points with MVP). (20 points)

xiii. How would your answer to (xii) change if the risk-free asset is available? (10 points)

2

Date

1/3/2005

2/1/2005

3/1/2005

4/1/2005

5/2/2005

6/1/2005

7/1/2005

8/1/2005

9/1/2005

10/3/2005

11/1/2005

12/1/2005

1/3/2006

2/1/2006

3/1/2006

4/3/2006

5/1/2006

6/1/2006

7/3/2006

8/1/2006

9/1/2006

10/2/2006

11/1/2006

12/1/2006

1/3/2007

2/1/2007

3/1/2007

4/2/2007

5/1/2007

6/1/2007

7/2/2007

8/1/2007

9/4/2007

10/1/2007

11/1/2007

12/3/2007

1/2/2008

2/1/2008

3/3/2008

4/1/2008

5/1/2008

6/2/2008

7/1/2008

8/1/2008

9/2/2008

10/1/2008

XOM

0,662983

23,29308

-5,85826

-4,31246

-0,96857

2,275449

2,224824

2,443681

6,093925

-11,6459

3,87674

-3,19617

11,70423

-4,88496

2,512095

3,648575

-2,94221

0,721761

10,40846

0,356969

-0,84074

6,440567

8,011642

-0,22692

-3,31201

-2,8521

5,266344

5,20414

5,23367

0,844046

1,49369

1,103781

7,968377

-0,61599

-2,70144

5,084135

-8,53254

1,975741

-2,79583

10,04163

-4,20727

-0,70608

-8,7381

-0,01321

-2,94545

-4,55906

DRI

MKT

RF

6,558045

-2,66

-9,32722

2,27

14,46037

-1,69

-2,06262

-2,52

8,236179

3,79

1,563586

1,15

5,200137

4,33

-9,49593

-0,59

-3,30579

1,06

7,469342

-2,08

10,37344

4,04

8,677945

0,35

4,554627

4,01

3,170664

-0,16

-2,19134

1,91

-3,00546

1,3

-10,5634

-3,1

11,24409

-0,04

-14,2129

-0,19

4,752475

2,51

19,9748

1,95

-0,84034

3,71

-4,15784

2,37

0,027632

1,08

-2,54144

1,94

4,705215

-1,4

0,487277

1,29

1,266164

3,99

9,869646

3,89

-3,46247

-1,48

-2,83421

-3,18

-2,29737

1,16

0,634082

4,09

3,150433

2,58

-7,45737

-4,93

-30,363

-0,43

3,00158

-6,23

8,819018

-2,2

5,567301

-1,05

9,87984

5,12

-3,73633

2,38

-6,75292

-7,86

2,639594

-1,32

-10,089

1,1

-2,23689

-9,81

-21,943

-18,47

0,16

0,16

0,21

0,21

0,24

0,23

0,24

0,3

0,29

0,27

0,31

0,32

0,35

0,34

0,37

0,36

0,43

0,4

0,4

0,42

0,41

0,41

0,42

0,4

0,44

0,38

0,43

0,44

0,41

0,4

0,4

0,42

0,32

0,32

0,34

0,27

0,21

0,13

0,17

0,17

0,17

0,17

0,15

0,12

0,15

0,08

11/3/2008

12/1/2008

1/2/2009

2/2/2009

3/2/2009

4/1/2009

5/1/2009

6/1/2009

7/1/2009

8/3/2009

9/1/2009

10/1/2009

11/2/2009

12/1/2009

8,769428

-0,4064

-4,19903

-10,7722

0,292578

-2,08813

4,641681

0,794245

0,698781

-1,1664

-0,77682

4,456489

5,347362

-9,16678

-17,4916

54,04776

-6,3138

3,510896

26,23782

8,52378

-2,16278

-8,81326

-1,02073

1,643571

3,646164

-10,5231

3,692308

11,60567

-8,52

2,15

-7,74

-10,1

8,77

11,06

6,73

-0,28

8,25

3,19

4,52

-2,84

5,74

2,92

0,02

0,09

0

0,01

0,01

0,01

0

0

0,01

0,01

0

0

0

0

Mean

StDev

Sharpe

ExpRet

Weight

Match 6.77%

Weight

StDev

FCNTX

FCPVX

RF

6.12

9.36

16.66

23.5

0.234814 0.30434

5.730209 6.773106

0.90036 0.638298

1.166949 0.638298

19.44138

15

TgtStDev

2.208

15

Date

FHKCX

FDFFX

1/31/2005

5.49

1.02

2/28/2005

-3.29

-1.72

3/31/2005

1.56

-5.01

4/30/2005

1.60

6.73

5/31/2005

1.99

0.42

6/30/2005

4.78

5.81

7/31/2005

-2.25

-0.45

8/31/2005

3.42

2.48

9/30/2005

-5.84

-3.13

10/31/2005

5.47

5.90

11/30/2005

4.35

1.23

12/31/2005

5.52

6.29

1/31/2006

-0.41

-1.69

2/28/2006

2.68

3.24

3/31/2006

5.97

1.23

4/30/2006

-3.81

-5.04

5/31/2006

-0.89

0.71

6/30/2006

0.67

-2.48

7/31/2006

3.46

2.28

8/31/2006

2.05

0.96

9/30/2006

2.59

3.62

10/31/2006

5.62

2.96

11/30/2006

3.27

0.00

12/31/2006

0.33

1.60

1/31/2007

-2.07

-1.72

2/28/2007

0.96

2.22

3/31/2007

2.10

4.52

4/30/2007

2.94

6.10

5/31/2007

9.57

0.25

6/30/2007

8.32

-1.41

7/31/2007

3.21

1.64

8/31/2007

17.22

9.08

9/30/2007

13.14

8.13

10/31/2007

-11.45

-6.25

11/30/2007

-2.81

3.04

12/31/2007

-14.69

-8.62

1/31/2008

5.48

2.79

2/29/2008

-3.64

-3.18

3/31/2008

8.49

9.76

4/30/2008

-0.34

4.70

5/31/2008

-9.86

1.81

6/30/2008

-3.57

-10.74

7/31/2008

-4.78

-3.83

8/31/2008

-14.92

-20.63

9/30/2008

-18.87

-22.78

10/31/2008

-3.21

-8.64

11/30/2008

12/31/2008

1/31/2009

2/28/2009

3/31/2009

4/30/2009

5/31/2009

6/30/2009

7/31/2009

8/31/2009

9/30/2009

10/31/2009

11/30/2009

12/31/2009

Average

StDev

Sharpe

Correl

Weight US

Average

StDev

Sharpe

6.45

1.49

-8.64

-7.50

-1.48

-9.17

11.23

8.26

13.39

11.40

20.87

9.68

-1.20

-2.59

12.58

8.61

-6.86

4.23

8.31

7.60

1.37

-5.47

4.11

7.36

0.90

4.29

1.83

2.06

FHKCX (China) FDFFX (US) RF

1.47

0.56

7.37

6.62

0.17

0.05

0.79

0.75

0.79

6.5170

0.09

Average

StDev

Corr

Weight

StDevPort

AveRetPort

Corr

Weight

StDevPort

AveRetPort

0.23

Assume

Weight US

Average

StDev

Sharpe

0.70

0.833

6.5206

0.09

MVP with Solver

Weight US

Average

StDev

Sharpe

X

0.00

0.70

0.833

5.13

0.12

MVE with Solver

Weight US

Average

StDev

Sharpe

-2.53

3.79

16.43

0.22

Y

10.00

10.00

1.00

2.00

0.00

5.00

-1.00

0.67

0.00

11.67

15.00

20.00

Weight US

Average

StDev

Sharpe

-0.17

1.626

7.76

0.18

-1.00

0.33

(a+b)^2=a^2+b^2+2*a*b

(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc

Zero-Beta Portfolio

Weight

StdDevPort

1.36

7.10

AveRetPort

0.23

SharpePort

0.00

Asset Allocation and

Optimal Portfolio

Professor Alexander Barinov

School of Business Administration

University of California Riverside

MGT 252 Investments and Portfolio Management

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

1 / 34

Outline

1

Risky Asset Plus Risk-Free Asset

Derivation

Example

No Risk-Free Borrowing

Optimal Portfolio

2

Two Risky Assets

Correlation and Covariance

Example

The Bullet

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

2 / 34

Risky Asset Plus Risk-Free Asset

Derivation

The Setup

Consider portfolio P that invests Ãâ€° in the risky

asset (stock) X and the rest (1 Ã¢Ë†â€™ Ãâ€°) in the

risk-free asset

Assume that the return of the risk-free asset is

constant at RF

The common proxy for the risk-free asset is

3-month Treasury bill, though it is not perfectly

safe

0 Ã¢â€°Â¤ Ãâ€° < 1 - we invest both in the stock and the
Treasury bill
Ãâ€° > 1 – we borrow at the risk-free rate and buy

the stock on the margin

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

3 / 34

Risky Asset Plus Risk-Free Asset

Derivation

Portfolio Characteristics

Expected return of the portfolio:

E(RP ) = E(Ãâ€°RX + (1 Ã¢Ë†â€™ Ãâ€°)RF ) = Ãâ€° Ã‚Â· E(RX ) + (1 Ã¢Ë†â€™ Ãâ€°) Ã‚Â· RF

E(RP ) = RF + Ãâ€°(E(RX ) Ã¢Ë†â€™ RF )

Variance of the portfolio return:

ÃÆ’ 2 (RP ) = ÃÆ’ 2 (Ãâ€°RX + (1 Ã¢Ë†â€™ Ãâ€°)RF ) = ÃÆ’ 2 (Ãâ€°RX ) = Ãâ€° 2 Ã‚Â· ÃÆ’ 2 (RX )

Standard deviation of the portfolio return:

ÃÆ’(RP ) = Ãâ€°ÃÆ’(RX )

E(RP ) = RF +

Alexander Barinov (SoBA, UCR)

E(RX ) Ã¢Ë†â€™ RF

Ã‚Â· ÃÆ’(RP )

ÃÆ’(RX )

Asset Allocation and Optimal Portfolio

MGT 252 Investments

4 / 34

Risky Asset Plus Risk-Free Asset

Derivation

Capital Allocation Line

E(RP ) = RF +

E(RX ) Ã¢Ë†â€™ RF

Ã‚Â· ÃÆ’(RP )

ÃÆ’(RX )

If you invest in the portfolio P, your risk can be

measured by the standard deviation ÃÆ’(RP )

The reward per unit of risk, SX =

E(RX ) Ã¢Ë†â€™ RF

,

ÃÆ’(RX )

is called the Sharpe Ratio

The portfolio expected return is the risk-free rate

plus the reward for bearing risk

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

5 / 34

Risky Asset Plus Risk-Free Asset

Derivation

CAL as the Budget Constraint

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

6 / 34

Risky Asset Plus Risk-Free Asset

Derivation

Capital Market Line

E(RP ) = RF +

E(RM ) Ã¢Ë†â€™ RF

Ã‚Â· ÃÆ’(RP )

ÃÆ’M

If the risky portfolio is the market portfolio (market

index), the CAL is called the Capital Market Line

(CML)

Notice that both CAL and CML are derived from two

equations for the portfolio variance and expected

return

We did not assume that standard deviation is the right

measure of risk, or that investors care only about

mean and variance, or that mean and variance

completely explain the behavior of returns, etc.

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

7 / 34

Risky Asset Plus Risk-Free Asset

Example

Example: Fidelity Funds

Over the past 5 years, the average risk-free rate

is 2.21% (data from Ken French)

Suppose we shoot for 15% per year standard

deviation

Our investment vehicles are the money market

account that yields precisely 2.208% and one of

the two Fidelity funds: Fidelity Small Cap Value

(FCPVX) and Fidelity Contrafund (FCNTX)

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

8 / 34

Risky Asset Plus Risk-Free Asset

Example

Which One to Pick?

Small Value: average return 9.36% per year, standard deviation

23.5% per year

9.36% Ã¢Ë†â€™ 2.21%

Sharpe ratio: SFCPVX =

= 0.304

23.5%

Expected return of Small Value fund + CD:

E(RP, FCPVX ) = 2.21% + 0.304 Ã‚Â· 15% = 6.77%

Contrafund: average return 6.12% per year, standard deviation

16.66% per year

6.12% Ã¢Ë†â€™ 2.21%

Sharpe ratio: SFCNTX =

= 0.235

16.66%

Expected return of Contrafund + CD:

E(RP, FCNTX ) = 2.21% + 0.235 Ã‚Â· 15% = 5.73%

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

9 / 34

Risky Asset Plus Risk-Free Asset

Example

How to Allocate the Money?

What fraction of your money you have to invest in the Small Value

Fund to get the return of 6.77% and standard deviation of 15%?

Ãâ€°FCPVX =

15%

ÃÆ’(RP )

=

= 0.638

ÃÆ’(RFCPVX ) 23.5%

What fraction of your money do you have to invest in Contrafund

fund to match the return of Small Value?

E(RP ) = RF + Ãâ€°(E(RFCNTX ) Ã¢Ë†â€™ RF ) Ã¢â€¡â€™

Ã¢â€¡â€™ 6.77% = 2.21% + Ãâ€°FCNTX (6.12% Ã¢Ë†â€™ 2.21%) Ã¢â€¡â€™

6.77% Ã¢Ë†â€™ 2.21%

= 1.17

6.12% Ã¢Ë†â€™ 2.21%

What is the standard deviation of your portfolio in this case?

Ã¢â€¡â€™ Ãâ€°FCNTX =

ÃÆ’(RP ) = Ãâ€°ÃÆ’(RFCPVX ) = 1.17 Ã‚Â· 16.66% = 19.44%

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

10 / 34

Risky Asset Plus Risk-Free Asset

No Risk-Free Borrowing

No Risk-Free Borrowing

Ã‚ÂµP = RF +

E(RM ) Ã¢Ë†â€™ RF

Ã‚Â· ÃÆ’P

ÃÆ’M

Suppose we can still invest in the Treasury bill,

but we have to borrow at a higher rate RB

If 0 Ã¢â€°Â¤ Ãâ€° Ã¢â€°Â¤ 1, CAL/CML does not change

If Ãâ€° > 1, CAL/CML becomes flatter

Ã‚ÂµP = RF +

Alexander Barinov (SoBA, UCR)

E(RM ) Ã¢Ë†â€™ RB

Ã‚Â· ÃÆ’P

ÃÆ’M

Asset Allocation and Optimal Portfolio

MGT 252 Investments

11 / 34

Risky Asset Plus Risk-Free Asset

No Risk-Free Borrowing

No Risk-Free Borrowing

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

12 / 34

Risky Asset Plus Risk-Free Asset

No Risk-Free Borrowing

Interpretation

If we cannot borrow at the risk-free rate, we get

less reward for a unit of risk if we buy on the

margin

Part of the reward we had when risk-free

borrowing was available, now goes to the lender

It is fair, because if “we” are investment

company, the loan is risky and we are protected

by limited liability

Limited liability pushes part of the risk (severe

downturns) onto the lender

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

13 / 34

Risky Asset Plus Risk-Free Asset

Optimal Portfolio

Portfolio Choice

CAL/CML is the budget constraint

We need a utility function to figure out where on

the constraint we want to be

If we assume U = E(RP ) Ã¢Ë†â€™ 0.5 Ã‚Â· A Ã‚Â· ÃÆ’ 2 (RP ), we

will get nice curves in the E(RP ), ÃÆ’(RP ) space

The function is called mean-variance utility

It is based on the assumption that returns are

normally distributed

In reality, the return distribution has fat tails

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

14 / 34

Risky Asset Plus Risk-Free Asset

Optimal Portfolio

Utility Maximization

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

15 / 34

Risky Asset Plus Risk-Free Asset

Optimal Portfolio

Portfolio Choice

Utility function: U = E(RP ) Ã¢Ë†â€™ 0.5 Ã‚Â· A Ã‚Â· ÃÆ’ 2 (RP )

A is the risk aversion coefficient. It measures our

unwillingness to take on risk

The value of the utility function is certainty

equivalent – the minimum certain return we would

take instead of investing in the risky portfolio

Recall our expressions for the portfolio variance and

expected return:

E(RP ) = Ãâ€° Ã‚Â· E(RX ) + (1 Ã¢Ë†â€™ Ãâ€°) Ã‚Â· RF

ÃÆ’ 2 (RP ) = Ãâ€° 2 Ã‚Â· ÃÆ’ 2 (RX )

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

16 / 34

Risky Asset Plus Risk-Free Asset

Optimal Portfolio

Portfolio Choice

Just plug in the variance and the expected

return into the utility function:

U = Ãâ€° Ã‚Â· E(RX ) + (1 Ã¢Ë†â€™ Ãâ€°) Ã‚Â· RF Ã¢Ë†â€™ 0.5 Ã‚Â· A Ã‚Â· Ãâ€° 2 Ã‚Â· ÃÆ’ 2 (RX )

Now maximize the utility w.r.t Ãâ€° and obtain the

optimal weight of the risky asset, Ãâ€° Ã¢Ë†â€”

U 0 = E(RX ) Ã¢Ë†â€™ RF Ã¢Ë†â€™ A Ã‚Â· ÃÆ’ 2 (RX ) Ã‚Â· Ãâ€° Ã¢Ë†â€” = 0

Ãâ€°Ã¢Ë†â€” =

E(RX ) Ã¢Ë†â€™ RF

E(RX ) Ã¢Ë†â€™ RF

1

=

Ã‚Â·

A Ã‚Â· ÃÆ’ 2 (RX )

ÃÆ’(RX )

A Ã‚Â· ÃÆ’(RX )

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

17 / 34

Risky Asset Plus Risk-Free Asset

Optimal Portfolio

Optimal Portfolio

The weight of the risky asset in the optimal

portfolio

Increases with the Sharpe ratio (i.e. reward for

bearing risk)

Decreases with risk aversion

Decreases with portfolio volatility (the investor cares

about ÃÆ’P2 , and the Sharpe ratio measures the

compensation only for ÃÆ’P )

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

18 / 34

Two Risky Assets

Correlation and Covariance

Covariance and Its Properties

Covariance measures comovement of two

random variables

Cov (Ri , Rj ) = E[(Ri Ã¢Ë†â€™ E(Ri )) Ã‚Â· (Rj Ã¢Ë†â€™ E(Rj )]

Excel formulas: COVAR or SUMPRODUCT if

you do “by hand”

Cov (aRi , Rj ) = a Ã‚Â· Cov (Ri , Rj )

Cov (aRi , bRj ) = a Ã‚Â· b Ã‚Â· Cov (Ri , Rj )

Cov (Ri + C, Rj ) = Cov (Ri , Rj )

Cov (Ri , Ri ) = Var (Ri )

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

MGT 252 Investments

19 / 34

Two Risky Assets

Correlation and Covariance

Correlation and Its Properties

Correlation is a scale-free measure of

comovement

Cov (Ri , Rj )

Corr (Ri , Rj ) =

ÃÆ’(Ri ) Ã‚Â· ÃÆ’(Rj )

Correlation is bounded between -1 and 1,

covariance increases with the variances of the

variables

R-square of the market model is the squared

correlation between the stock and the market

Corr (aRi , Rj ) = Corr (Ri , Rj )

Corr (Ri + C, Rj ) = Corr (Ri , Rj )

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

Correlation and Covariance

Variance of the Sum

For independent variables we had

ÃÆ’ 2 (Ri + Rj ) = ÃÆ’ 2 (Ri ) + ÃÆ’ 2 (Rj )

For dependent variables, the covariance term appears:

ÃÆ’ 2 (Ri + Rj ) = ÃÆ’ 2 (Ri ) + 2Cov (Ri , Rj ) + ÃÆ’ 2 (Rj ) =

= ÃÆ’ 2 (Ri ) + 2Corr (Ri , Rj )ÃÆ’(Ri )ÃÆ’(Rj ) + ÃÆ’ 2 (Rj )

Which explains why we had ÃÆ’2 (RMKT Ã¢Ë†â€™ RF ) = ÃÆ’2 (RMKT )covariance of the constant with anything is zero

Economic interpretation: the smaller is the correlation, the

larger is the diversification benefit

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

Correlation and Covariance

Variance of the Weighted Sum

Suppose we invest Ãâ€° and 1 Ã¢Ë†â€™ Ãâ€° in the two stocks

Recall that ÃÆ’ 2 (Ãâ€°Ri ) = Ãâ€° 2 ÃÆ’ 2 (Ri ) and

Cov (Ãâ€°Ri ; (1 Ã¢Ë†â€™ Ãâ€°)Rj ) = Ãâ€°(1 Ã¢Ë†â€™ Ãâ€°)Cov (Ri ; Rj )

ÃÆ’ 2 (Ãâ€°Ri +(1Ã¢Ë†â€™Ãâ€°)Rj ) = ÃÆ’ 2 (Ãâ€°Ri )+2Cov (Ãâ€°Ri ; (1Ã¢Ë†â€™Ãâ€°)Rj )+ÃÆ’ 2 ((1Ã¢Ë†â€™Ãâ€°)Rj ) =

= Ãâ€° 2 ÃÆ’ 2 (Ri ) + 2Ãâ€°(1 Ã¢Ë†â€™ Ãâ€°)Cov (Ri ; Rj ) + (1 Ã¢Ë†â€™ Ãâ€°)2 ÃÆ’ 2 (Rj ) =

= Ãâ€° 2 ÃÆ’ 2 (Ri )+2Corr (Ri ; Rj )Ã‚Â·(Ãâ€°ÃÆ’(Ri ))Ã‚Â·((1Ã¢Ë†â€™Ãâ€°)ÃÆ’(Rj ))+(1Ã¢Ë†â€™Ãâ€°)2 ÃÆ’ 2 (Rj )

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

Example

Benefits of Diversification

Consider two Fidelity funds:

Fidelity Independence (ticker FDFFX, US stocks,

large growth)

Fidelity China (ticker FHKCX, Chinese stocks, large

value)

In 2005-2009, ÃÆ’(RFDFFX ) = 6.62%, ÃÆ’(RFHKCX ) = 7.37%

Average returns: R FDFFX = 0.56%, R FHKCX = 1.47%

Correlation of the two returns: 0.79

What is the standard deviation of the portfolio

that is 70% FDFFX and 30% FKHCX?

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

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Two Risky Assets

Example

Benefits of Diversification

ÃÆ’ 2 (0.7RFDFFX + 0.3RFHKCX ) = 0.72 ÃÆ’ 2 (RFDFFX )+

+2Ã‚Â·0.7Ã‚Â·0.3Ã‚Â·ÃÆ’(RFDFFX )ÃÆ’(RFHKCX )Ã‚Â·Corr (RFDFFX ; RFHKCX )+0.32 ÃÆ’ 2 (RFHKCX ) =

= 0.49Ã‚Â·6.622 +2Ã‚Â·0.21Ã‚Â·6.62Ã‚Â·7.37Ã‚Â·0.79+0.09Ã‚Â·7.372 = 42.52

ÃÆ’(0.7RFDFFX + 0.3RFHKCX ) =

Ã¢Ë†Å¡

42.52 = 6.52%

The standard deviation of the portfolio is smaller than

the standard deviations of the funds

Verify that the standard deviation of the portfolio would

be 5.13% if the correlation was 0

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

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Two Risky Assets

Example

International Diversification

The example above suggests that you can get

some extra mileage diversifying your portfolio

internationally

LetÃ¢â‚¬â„¢s do the three Sharpe ratios (the average

risk-free rate is 0.23%)

R port = 0.7Ã‚Â·R FDFFX +0.3Ã‚Â·R FHKCX = 0.7Ã‚Â·0.56%+0.3Ã‚Â·1.47% = 0.83%

SFDFFX =

0.56% Ã¢Ë†â€™ 0.23%

0.9% Ã¢Ë†â€™ 0.23%

= 0.05; SFHKCX =

= 0.17

6.62%

7.37%

Sport =

Alexander Barinov (SoBA, UCR)

0.83% Ã¢Ë†â€™ 0.23%

= 0.09

6.52%

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Two Risky Assets

Example

Goal Seek

Goal Seek is a useful Excel tool that lets you tweak the

parameters to get the desired value of the formula

Data tab – What-If Analysis – Goal Seek

We want to change the weight on FDFFX so that the

Sharpe ratio of the portfolio beats the Sharpe ratio of

FHKCX

If we want Sport = 0.18, Ãâ€° = Ã¢Ë†â€™0.17 – we have to short sell

FDFFX and buy 117% of FHKCX

Note that we would never get the Sharpe ratio of 0.18

playing with FHKCX and RF

Also, bringing FHKCX into play allows us to easily beat

FDFFX on return, standard deviation and the Sharpe ratio

simultaneously

Alexander Barinov (SoBA, UCR)

Asset Allocation and Optimal Portfolio

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Two Risky Assets

Example

Problems with Diversification

Consider two more funds: Fidelity Large Cap

(FLCSX, large US stocks) and Fidelity Latin

America (FLATX, large Brazilean stocks)

In 2005-2009, what do you think was the

correlation between the returns to FDFFX and

FLCSX?

And what do you think was the correlation

between the returns to FDFFX and FLATX?

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

Example

Problems with Diversification

Country funds can have high Sharpe ratio and

you may capture some of it without really

shifting away from the US stocks

Country funds usually hold huge multinationals,

and therefore do not offer much diversification

benefit

The real diversification probably comes from

smaller foreign firms, which are hard to get

Even with those, most likely you have to short

something to beat all components on the

Sharpe ratio

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

The Bullet

Minimum Variance Frontier

The set of all possible combinations of average

returns and standard deviations one can construct

from two (or more) assets is called the Minimum

Variance Frontier (MVF)

If the two assets have perfect positive correlation,

MVF is a straight line joining the two assets

The straight line intersects with the vertical axis,

meaning that we can reach zero standard deviation if

we short the top asset

If the two assets have perfect negative correlation,

MVF is a kinked line and we can achieve zero

standard deviation with two positive weights

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Two Risky Assets

The Bullet

Numerical Example

Consider two stocks X and Y: E(RX ) = 10%,

ÃÆ’(RX ) = 10% and E(RY ) = 15%, ÃÆ’(RY ) = 20%

If the correlation between their returns is +1, buy

2 stocks X and short 1 stock Y

Verify that this portfolio has E(RP ) = 5%,

ÃÆ’(RP ) = 0%

If the correlation is -1, buy 2 stocks X and 1

stock Y

Verify that this portfolio has E(RP ) = 11.67%,

ÃÆ’(RP ) = 0%

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

The Bullet

Minimum Variance Frontier

If the correlation is between -1 and +1, the MVF

is a curved line usually called the bullet

Play with portfolio weights in the Excel file and

draw the bullet

With more than two assets, the solution is very

complicated, but we still have the bullet, as

Markowitz has shown

The more negative is the correlation, the greater

is the space inside the bullet, and we like it

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

The Bullet

Minimum Variance Frontier

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Two Risky Assets

The Bullet

Efficient Frontier

Minimum Variance Portfolio (MVP) – where

the bullet touches the vertical line

MVF is investorÃ¢â‚¬â„¢s “budget constraint”

The points inside the bullet are inefficient: you

take too much risk for too little reward

The points outside should be unattainable

The points on the frontier, but above MVP are

the efficient frontier

The points on MVF, but below MVP are also

inefficient

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Asset Allocation and Optimal Portfolio

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Two Risky Assets

The Bullet

Efficient Frontier

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Capital Asset Pricing Model

Professor Alexander Barinov

School of Business Administration

University of California Riverside

MGT 252 Investments and Portfolio Management

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Outline

1

Two-Fund Separation

2

CAPM and the Market Portfolio

3

Security Market Line

4

Zero-Beta CAPM

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Two-Fund Separation

Minimum Variance Frontier

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Two-Fund Separation

Add the Risk-Free Asset

Suppose we can invest in any point on the bullet

and the risk-free asset

Essentially, we can draw a Capital Allocation

Line through the risk-free rate and any point on

the bullet

Any point on the CAL would be a portfolio

combining the RF asset and the risky portfolio

(the point at which the CAL crosses the bullet)

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Capital Asset Pricing Model

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Two-Fund Separation

Two-Fund Separation

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Two-Fund Separation

Two-Fund Separation

Recall that the slope of CAL is the Sharpe ratio, our

reward-for-volatility ratio

Hence, we would be making the CAL as steep as

possible, until it is tangent to the bullet

The tangency point is the Mean-Variance Efficient

(MVE) portfolio

Irrespective of risk aversion, all investors will hold this

portfolio

Risk aversion will only determine how they split their

investment between the MVE portfolio and the RF asset

Essentially, we are back to CAL and optimal portfolio

given CAL

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Two-Fund Separation

Solver Add-in in Excel

Solver add-in is an extended version of Goal Seek,

which lets you minimize and maximize the output

of the formulas instead of setting it to a number

To solve for MVP, minimize the standard

deviation of the portfolio by changing the

portfolio weight

Fidelity example: if we combine Fidelity

Independence and Fidelity China, the minimum

possible standard deviation (which is the standard

deviation of the MVP) is 6.52%, attained by the

portfolio that invests 75% in Fidelity Independence

and 25% in Fidelity China

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Capital Asset Pricing Model

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Two-Fund Separation

Solver Add-in in Excel

To solve for MVE, maximize the Sharpe ratio

of the portfolio by changing the weight

Fidelity example: if we combine Fidelity

Independence and Fidelity China, the MVE has

the Sharpe ratio of 0.22, attained by the weights

of -253% on Fidelity Independence and 353%

on Fidelity China

Weights interpretation: if you have $100 to

invest, short sell Fidelity Independence for $253

and invest $353 in Fidelity China – it will give you

the best possible Sharpe ratio

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Capital Asset Pricing Model

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CAPM and the Market Portfolio

Two-Fund Separation

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CAPM and the Market Portfolio

CAPM and Two-Fund Separation

Two-Fund Separation says that everyone holds the MVE

portfolio and the RF asset

By definition, if we sum everyoneÃ¢â‚¬â„¢s holdings of risky

assets, we will get the market portfolio (i.e. everything in

the market of risky assets)

Therefore, everyone holds the market portfolio and the

market portfolio is mean-variance efficient

That is, in the long term you should not be able to beat

the market on the Sharpe ratio

If you can beat the Sharpe ratio of the market, the

CAPM is not valid

This is the first part of the CAPM (due to Sharpe, Lintner,

and Mossin)

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Capital Asset Pricing Model

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CAPM and the Market Portfolio

Market Portfolio in the CAPM

By market portfolio, the CAPM means all risky

assets in the economy, not just S&P500 or

exchange-listed stocks

Market portfolio in the CAPM includes other

securities, durable goods, human capital, etc.

Market portfolio is unobservable, therefore,

strictly speaking, the CAPM is not testable (Roll

critique)

Shanken (JF 1986, JFE 1987) argues that the

stock market portfolio can be a good proxy for

the market portfolio meant by the CAPM

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Capital Asset Pricing Model

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CAPM and the Market Portfolio

CAPM Assumptions

All changes in our wealth come from our investment

decisions (there are no other payments like wages, or

we view wages as return to our human capital)

The CAPM is one-period model – in the CAPM,

investors do not try to shift wealth around because

some periods are better than the others

We assume that investors only care about mean and

variance

Investors have the same investment opportunity set

and understand that it is the same

It is possible to diversify away all idiosyncratic risks

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Capital Asset Pricing Model

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Security Market Line

Stock Contribution to Portfolio Variance

ÃÆ’ 2 (Ãâ€°A RA + Ãâ€°B RB ) = Ãâ€°A2 ÃÆ’ 2 (RA ) + 2Ãâ€°A Ãâ€°B Cov (RA , RB ) + Ãâ€°B2 ÃÆ’ 2 (RB )

Stock A contribution to the portfolio variance

Ãâ€°A2 ÃÆ’ 2 (RA ) + Ãâ€°A Ãâ€°B Cov (RA , RB )

Stock B contribution to the portfolio variance

Ãâ€°B2 ÃÆ’ 2 (RB ) + Ãâ€°A Ãâ€°B Cov (RA , RB )

If you throw in a third stock C, it will contribute

Ãâ€°C2 ÃÆ’ 2 (RC )+Ãâ€°A Ãâ€°C Cov (RA , RC )+Ãâ€°B Ãâ€°C Cov (RB , RC )

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Capital Asset Pricing Model

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Security Market Line

Stock Contribution to Portfolio Variance

Now, recall three facts about covariance: ÃÆ’ 2 (RC ) = Cov (RC , RC ),

Cov (RC , Ãâ€°A RA ) = Ãâ€°A Cov (RC , RA ) and

Cov (RC , RA + RB ) = Cov (RC , RA ) + Cov (RC , RB )

If you add stock J to a portfolio of N stocks, its contribution to the

variance will be

Ãâ€°J

N

X

Ãâ€°i Cov (RJ , Ri ) = Ãâ€°J

i=1

If we assume that

N

X

Cov (RJ , Ãâ€°i Ri ) = Ãâ€°J Cov (RJ ,

i=1

N

X

N

X

Ãâ€° i Ri )

i=1

Ãâ€°i Ri = RMKT , then adding more of stock J to

i=1

the market portfolio increases the variance by Ãâ€°J Cov (RJ , RMKT )

The contribution of stock J to expected return is easy: it is just

Ãâ€°J E(RJ )

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Security Market Line

Two Bullets

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Security Market Line

Two Bullets

Small bullet cannot go outside of the big one,

because big one combines everything in it,

including MKT and J, in the optimal fashion

But MKT is on both: it is on the big one,

because MKT is MVE, and it is on the small

one, because if you mix MKT and J, you can

always set Ãâ€°MKT = 1 and Ãâ€°J = 0

Hence, at MKT the big bullet and the small bullet

have to be tangent, i.e. have the same slope

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Security Market Line

From CML to Beta

Because what we know is the contribution of J to MKT expected

return and variance, instead of slopes we will use

return-to-variance ratios

For the MKT, it is

E(RMKT ) Ã¢Ë†â€™ RF

ÃÆ’ 2 (RMKT )

For J, it is its contribution to the expected return over its

Ãâ€°J (E(RJ ) Ã¢Ë†â€™ RF )

contribution to variance

Ãâ€°J Cov (RJ , RMKT )

Set those two ratios equal:

E(RJ ) Ã¢Ë†â€™ RF

E(RMKT ) Ã¢Ë†â€™ RF

=

Cov (RJ , RMKT )

ÃÆ’ 2 (RMKT )

E(RJ ) Ã¢Ë†â€™ RF =

Alexander Barinov (SoBA, UCR)

Cov (RJ , RMKT )

(E(RM ) Ã¢Ë†â€™ RF )

ÃÆ’ 2 (RMKT )

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Security Market Line

Security Market Line

E(RJ ) Ã¢Ë†â€™ RF = ÃŽÂ²J (E(RM ) Ã¢Ë†â€™ RF )

Cov (RJ , RMKT )

ÃŽÂ²J =

ÃÆ’ 2 (RMKT )

ÃŽÂ²J happens to be the slope of the regression of

the stock excess return on the market excess

return

Excess return is stock return minus the risk-free rate

The equation at the top of the slide is the main

result of the CAPM

It is called the Security Market Line (SML)

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Security Market Line

Interpreting Beta

Beta as the regression coefficient: if the market

goes up by 1%, the stock goes up, on average,

by ÃŽÂ²%

Beta does not measure how closely the asset

tracks the market (we have R-squared for this)

It is possible to have high beta and low

R-squared, and vice versa

By definition, beta measures the comovement of

stock and the market

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Security Market Line

How the Notion of Risk Has Changed

We started with standard deviation as a

measure of risk (CAL)

When we mixed together two risky assets, we

realized that covariance also matters

When we looked at the portfolio of N stocks, we

concluded that the numerous covariance terms

are much more important than one single

variance term

The CAPM proclaimed that all what matters is

the covariance with the market (recall what the

CAPM means by “market portfolio”)

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Zero-Beta CAPM

Zero-Beta CAPM

Even Treasuries have some risk (inflation, small

risk of default)

Availability of borrowing at the risk-free rate is a

dubious assumption

Instead of the risk-free rate, we can take the rate of

return of any asset that has zero correlation with

the market

However, there is only one line that is tangent to

the bullet at M

Black (1972) shows that if we draw a horizontal line

from the CML intercept to the MVF, we will hit the

portfolio that has zero correlation with the market

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Zero-Beta CAPM

Zero-Beta CAPM

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Zero-Beta CAPM

Making It Work

You need to know the mean, the standard

deviation, and the Sharpe ratio of the market

portfolio

Then you take the CML and solve for “RF”, which

is now the zero-beta rate

This is the average return of the zero-beta portfolio

Now use Solver to figure out the standard deviation

Tell Solver “set average return to what I solved from

the CML for by changing the weight” and watch the

standard deviation cell to update automatically

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Zero-Beta CAPM

Zero-Beta CAPM: Example

Assume that the Fidelity China fund and the Fidelity

Independence fund are the only two assets in the

economy

We have already solved for MVE: mean 3.79%, standard

deviation 16.43%, Sharpe 0.22

CML :

E(RM ) = ZB + SM Ã‚Â· ÃÆ’(RM )

Substitute and solve for ZB – it will be 0.23%, just as the

risk-free rate, because you used 0.23 in the Sharpe ratio

when you maximized it

If you could just observe the market portfolio instead of

solving for it, you would figure out the slope of MVF at the

market portfolio and ZB will not be RF when you solve for

it from the CML

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Zero-Beta CAPM

Zero-Beta CAPM: Example

Now, what risky portfolio has the expected return of

0.23%?

Use Solver here: “set average return to 0.23% by

changing the weight”

Solver says the weight is 136% in Fidelity

Independence (-36% in Fidelity China)

Standard deviation updates to 7.1%

This risky portfolio is a substitute for the risk-free

asset, because it has zero beta and thus the CAPM

says it is as risky as the risk-free asset – adding it to

your portfolio will not increase the portfolioÃ¢â‚¬â„¢s variance

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Capital Asset Pricing Model

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Zero-Beta CAPM

Do We Want Inefficient Portfolios?

CAPM says we have to hold MVE (the market),

but it can price anything else

SML :

E(RP ) = RF + ÃŽÂ²P Ã‚Â· (RM Ã¢Ë†â€™ RF )

For example, zero-beta portfolio is from the

inefficient part of MVF, so you do not want to put

there all the money you plan to put into the stock

market

But if this portfolio has a zero alpha, it is a fair

deal on the margin: we can add a small bit of it

to our diversified portfolio, it will not hurt

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Zero-Beta CAPM

Alpha and Sharpe Ratio

Diversified portfolios with positive alpha will beat

the market portfolio on the Sharpe ratio and vice

versa

An individual stock can have a positive alpha

and lose to the market portfolio on the Sharpe

ratio, because one stock is very volatile

CAPM says you can diversify this risk away adding the positive alpha stock to your portfolio

from CML will create a portfolio that beats the

market on the Sharpe ratio

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Zero-Beta CAPM

Zero-Beta CAPM

Suppose that the zero-beta CAPM holds and the true

SML is E(Ret Ã¢Ë†â€™ RZB ) = ÃŽÂ²i Ã‚Â· E(RM Ã¢Ë†â€™ RZB )

But instead we estimate the CAPM with the risk-free rate

Deduct RF from both sides of the SML and collect the

terms:

E(Ret) Ã¢Ë†â€™ RF = E(RZB ) + ÃŽÂ²i E(RM ) Ã¢Ë†â€™ ÃŽÂ²i E(RZB ) Ã¢Ë†â€™ ÃŽÂ²i RF + ÃŽÂ²i RF Ã¢Ë†â€™ RF

E(Ret)Ã¢Ë†â€™RF = (1Ã¢Ë†â€™ÃŽÂ²i )E(RZB )+ÃŽÂ²i (E(RM )Ã¢Ë†â€™RF )Ã¢Ë†â€™(1Ã¢Ë†â€™ÃŽÂ²i )RF

E(Ret) Ã¢Ë†â€™ RF = (1 Ã¢Ë†â€™ ÃŽÂ²i )(E(RZB ) Ã¢Ë†â€™ RF ) + ÃŽÂ²i (E(RM ) Ã¢Ë†â€™ RF )

So, if you ignore the inability to borrow and lend at the

risk-free rate, you will underestimate the performance of

aggressive stocks (ÃŽÂ²i > 1) and overestimate the

performance of conservative stocks (ÃŽÂ²i < 1)
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