capm (rohit)

8/9/2019 capm (rohit)

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1999 Thomas A. Rietz 1

Diversification and the CAPMDiversification and the CAPMThe relationship between riskand expected returns


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IntroductionIntroduction Investors are concerned with

Risk

Returns

What determines the requiredcompensation for risk?

It will depend on The risks faced by investors

The tradeoff between risk and return they face


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AgendaAgenda Concepts of risk for

A single stock

Portfolios of stocks

Risk for the diversified investor: Beta

Calculating Beta

The relationship between Beta and Return:The Capital Asset Pricing Model (CAPM)


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Overview

Overview

Investors demand compensation for risk

If investors hold diversified portfolios, risk

can be defined through the interaction of asingle investment with the rest of the

portfolios through a concept called beta

The CAPM gives the required relationshipbetween beta and the return demandedon the investment!


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Vocabulary

Vocabulary

Expected return: What we expect to receive

on average

Standard deviation ofreturns: A measure of dispersion

of actual returns

Correlation The tendency for two

returns to fall above orbelow the expected returna the same or differenttimes

Beta A measure of risk

appropriate for diversified

investors Diversified investors

Investors who hold aportfolio of manyinvestments

The Capital AssetPricing Model (CAPM) The relationship between

risk and return fordiversified investors


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i

i

irprE !)(

Measuring Expected ReturnMeasuring Expected Return We describe what we expect to receive or

the expected return:

Often estimated using historical averages

(excel function: average).


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Example: Die ThrowExample: Die Throw Suppose you pay $300 to throw a fair die.

You will be paid $100x(The Number rolled)

The probability of each outcome is 1/6. The returns are:

(100-300)/300 = -66.67%

(200-300)/300 = -33.33% etc.

The expected return E(r) is:

1/6x(-66.67%) + 1/6x(-33.33%) + 1/6x0% +

1/6x33.33% + 1/6x66.67% + 1/6x100% = 16.67%!


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Example: IEMExample: IEM Suppose

You buy and AAPLi contract on the IEM for $0.85

You think the probability of a $1 payoff is 90% The returns are:

(1-0.85)/0.85 = 17.65%

(0-0.85)/0.85 = -100%


0.9x17.65% - 0.1x100% = 5.88%


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Example: Market ReturnsExample: Market Returns Recent data from the IEM shows the following

average monthly returns from 5/95 to 10/99:

(http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)

AAPL IBM MSFT SP500 T-Bills

Average Return 2.42% 3.64% 4.72% 1.75% 0.35%


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$-

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

Apr-95

Jul-95

Oct-95

Jan-96

Apr-96

Jul-96

Oct-96

Jan-97

Apr-97

Jul-97

Oct-97

Jan-98

Apr-98

Jul-98

Oct-98

Jan-99

Apr-99

Mon

th

Value of Investment

G

rowthof$100

0Inve

stme

G

rowthof$100

0Inve

stme


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2222 )()( iii

ii

i

i Va WW !!!

Often estimated using historical averages(excel function: stddev)

Measuring Risk: StandardMeasuring Risk: Standard

Deviation and VarianceDeviation and Variance Standard Deviation in Returns:


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Example: Die ThrowExample: Die Throw Recall the dice roll example:

You pay $300 to throw a fair die.

You will be paid $100x(The Number rolled)

The probability of each outcome is 1/6.

The expected return E(r) is 16.67%.

The standard deviation is:

56.93%

%67.16%)100(6

1%)67.66(

6

1

%)33.33(6

1%)0(

6

1

%)33.33(6

1

%)67.66(6

1

222

22

22

!

vv

vv

vv


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Example: IEMExample: IEM Suppose

You buy and AAPLi contract on the IEM for $0.85

You think the probability of a $1 payoff is 90% The returns are:

(1-0.85)/0.85 = 17.65%

(0-0.85)/0.85 = -100%


0.9x17.65% - 0.1x100% = 5.88%


[0.9x(17.65%)2 + 0.1x(-100%)2 - 5.88%2]0.5 = 35.29%


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average monthly returns & standard deviations

from 5/95 to 10/99: (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)


Average Return 2.42% 3.64% 4.72% 1.75% 0.35%

Std. Dev14.84% 10.31% 8.22% 3.82% 0.06%


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$-

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

Apr-95

Jul-95

Oct-95

Jan-96

Apr-96

Jul-96

Oct-96

Jan-97

Apr-97

Jul-97

Oct-97

Jan-98

Apr-98

Jul-98

Oct-98

Jan-99

Apr-99

Mon

th

Value of Investment

G

rowthof$100

0Inve

stme

G

rowthof$100

0Inve

stme


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Risk and Average ReturnRisk and Average Return

T-Bill

S&P

MSFT

IBM

PL

. %

. %

. %

. %

. %

. %

3. %

3. %

. %

. %

. %

. % . % . % . % . % . % . % . % . %

S d rd D iation

AverageReturn


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Measures of AssociationMeasures of Association Correlation shows the association across

random variables

Variables withPositive correlation: tend to move in the

same direction

Negative correlation: tend to move inopposite directions

Zero correlation: no particular tendencies to

move in particular directions relative to each

other


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VAB is in the range [-1,1]

Often estimated using historical averages

(excel function: correl)

Covariance in returns, WAB, is defined as:

)()()()( BABiAii

iBBiAAi

i

iAB rErErrprErrErp !!

W

BA

AB

AB

WW

WV !

Covariance and CorrelationCovariance and Correlation

The correlation, VAB, is defined as:


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Notation for Two Asset andNotation for Two Asset and

Portfolio ReturnsPortfolio ReturnsItem Asset A Asset B Portfolio

Actual Return r Ai rBi rPi

Expected Return E(rA) E(rB) E(rP)Variance WA

2 WB2 WP

2

Std. Dev. WA WB WP

Correlation in Returns VABCovariance in Returns WAB = WAWBVAB


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monthly return correlations from 5/95 to 10/99: (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)


AAPL 1.000 0.262 0.102 0.046 -0.103

IBM

1.000 0.240 0.362 -0.169

M 1.000 0.550 -0.073

SP500 1.000 -0.003

T-Bill 1.000


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y = 0.3777x + 0.0105

Correl = 0.262

$(0)

$(0)

$(0)

$(0)

$-

$0

$0

$0

$0

$1

-20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

AAPL Return

IBMR

eturn

Correlation of AAPL & I MCorrelation of AAPL & I M


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Risk and Average ReturnRisk and Average Return

T-B ll

S&P500

S T

IBM

AAP

0 0

0 5

1 0

1 5

2 0

2 5

0

5

4 0

4 5

5 0

0 0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 16 0

Sta a De iati

A

e

ageRet


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The standard deviation is not a linearcombination of the individual asset standarddeviations

Instead, it is given by:

)w(12w)w(1+w ABBAAA22

A

22

Ap VWWWWW ! BA

%08.10262.01031.0.148405.5x0.2x0

1031.05.01484.05.0 22222p !

vvv

vv!W

Two Asset Portfolios: RiskTwo Asset Portfolios: Risk

The standard deviation a the 50%/50%, AAPL &

IBM portfolio is:

The portfolio risk is lower than either individual

assets because of diversification.


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Correlations andCorrelations and

DiversificationDiversification Suppose

E(r)A = 16% and WA = 30%

E(r)B = 10% and WB = 16%

Consider the E(r)P and WP of securities Aand B as wA and V vary...


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Case 1: Perfect positive correlationCase 1: Perfect positive correlation

between securities, i.e.,between securities, i.e., VVABAB = +1= +1

8%

9%

10%

11%

12%

13%

14%15%

16%

17%

0% 10% 20% 30% 40%

Std. Dev.

Exp.

Ret

(10% 16%)

(16% 30%)


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Case 2: Zero correlation betweenCase 2: Zero correlation between

securities, i.e.,securities, i.e., VVABAB = .= .

2%

13%

14%

15%

16%

17%

0% 10% 20% 30% 40%

Std. v.

p.

t

(1 %,16%)

(16%,3 %)V

(11.33%,14.1 %)


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Case 3: Perfect negative correlationCase 3: Perfect negative correlation

between securities, i.e.,between securities, i.e., VVABAB == --11

8%

9%

10%

11%

12%

13%

14%15%

16%

17%

0% 10% 20% 30% 40%

Std. Dev.

Exp.

Ret

(10%,16%)

(16%,30%).

(11.33%,14.12%)


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8%

9%

10%

11%

12%

13%

14%15%

16%

17%

0% 10% 20% 30% 40%

Std. Dev.

Exp.

Ret

1

r=0

r=-1

(10%,16%)

(16%,30%)

ComparisonComparison


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w2w+

w2w+

w2w+

www

M FTI M,M FTI MM FTI M

M FTAAPL,M FTAAPLM FTAAPL

I MAAPL,I MAAPLI MAAPL

2

M FT

2

M FT

2

I M

2

I M

2

AAPL

2

AAPL

p

V

V

V

!

3 Asset Portfolios: Expected3 Asset Portfolios: Expected

Returns and Standard DeviationsReturns and Standard Deviations Suppose the fractions of the portfolio are given

by wAAPL, wIBM and wMSFT.

The expected return is: E(rP) = wAAPLE(rAAPL) + wIBME(rIBM) + wMSFTE(rMSFT)



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%59.30 72.03

1036.0

3

102 2.0

3

1)( !vvv!

PRE

%75.7

240.00822.01031.03

1

3

12+

102.00822.01484.031

312+

262.01031.01484.03

1

3

12+

0822.0311031.0

311484.0

31 2

2

2

2

2

2

2

p

!

!W

For the Naively DiversifiedFor the Naively Diversified

Portfolio, this gives:Portfolio, this gives:


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For the Naively DiversifiedFor the Naively Diversified

Portfolio, this gives:Portfolio, this gives:

T-Bill

S&P500

SFT

IB

PL

Naive

P rtf li

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Sta dard Deviati

vera

eRet

r


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The Concept of Risk With NThe Concept of Risk With N

Risky AssetsRisky Assets As you increase the number of assets in a

portfolio:

the variance rapidly approaches a limit, the variance of the individual assets contributes less

and less to the portfolio variance, and

the interaction terms contribute more and more.

Eventually, an asset contributes to the risk of aportfolio not through its standard deviation butthrough its correlation with other assets in theportfolio.

This will form the basis forCAPM.


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Portfolio variance consists of two parts:

1. Non-systematic (or idiosyncratic) risk and

2. Systematic (or covariance) risk

The market rewards only systematic riskbecause diversification can get rid of non-systematic risk

riskSystematic

ij

risksystematicNon

ipnn

WWW

!

11

1 22

Variance of a naively diversifiedVariance of a naively diversified

portfolio ofN assetsportfolio ofN assets


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Naive DiversificationNaive Diversification

Number of Assets

V

ar.ofPortfoli

V!

V!

V!


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Sta ar

Devaiti

0

2

4

6

8

10

12

14

16

1 3 5 7 911 13 15 17 19 21 23 25

Number fSt cksi Portfolio

Expecte

Portfolio

Retur

a

S

ta

ar

Deviati

on

Avera eMont ly Return

ConsiderNaive Portfolios of 1ConsiderNaive Portfolios of 1through all 2 of these Assetsthrough all 2 of these Assets(Added in Alphabetical Order(Added in Alphabetical Order


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The Capital Asset PricingThe Capital Asset Pricing

ModelModel CAPM Characteristics:

Fi = WiWmVim/Wm2

Asset Pricing Equation:E(ri) = rf+ Fi[E(rm)-rf]

CAPM is a model of what expected returnsshould be if everyone solves the same

passive portfolio problem CAPM serves as a benchmark

Against which actual returns are compared

Against which other asset pricing models are

compared


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TheIdea Behind CAPMThe

Idea Behind CAPM

The value of an asset reflects The risk associated with that asset given

Investors own a combination of The risk free asset and

The market portfolio.

A risky asset

Has no effect on the risk free rate.Effects the portfolio through its covariancewith it.

The market price of risk is: E(Rm)-Rf

Where do these ideas come from?


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The Capital Asset PricingThe Capital Asset Pricing

ModelModel Advantages:

Simplicity

Works well on average Disadvantages:

Makes many simplifying assumptions aboutmarkets, returns and investor behavior

How do you estimate beta? Can all aspects ofrisk be summarized by beta?

What is the true market portfolio and risk freerate?


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Feasible portfolios withFeasible portfolios with

N risky assetsN risky assets

Expected

return (E i)

Std dev (Wi)

Efficientfrontier

Feasible Set


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Dominated and EfficientDominated and Efficient

PortfoliosPortfoliosExpec e

e Ei)

Std de (Wi)

AB

C


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How would you find theHow would you find the

efficient frontier?efficient frontier?1. Find all asset expected returns and

standard deviations.

2. Pick one expected return and minimizeportfolio risk.

3. Pick another expected return and minimizeportfolio risk.

4. Use these two portfolios to map out theefficient frontier.


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Exp t d

tu n (Ei)

Std d v ( Wi)

D

Utility m ximi ing

isky ss t po t olio

U

tility MaximizationU

tility Maximization


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Expected

ret r (Ei)

Std dev (Wi)

DM

E

Utility maximization withUtility maximization with

a riskfree asseta riskfree asset


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ThreeImportant

F

undsThreeImportant

F

unds The riskless asset has a standard deviation

of zero

The minimum variance portfolio lies onthe boundary of the feasible set at a pointwhere variance is minimum

The market portfolio lies on the feasibleset and on a tangent from the riskfree asset


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All isky assets

and po t oliosExpected

etu n (E i)

Std dev (Wi)

Risklessasset Minimum

Va iance

Po t olio

Ma ket

Po t olio

Efficientfrontier

A world with one risklessA world with one riskless

asset andN

risky assetsasset andN

risky assets


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50/65

e

m

fm

fe

rrErrE W

W

-

!

)()(

The Capital Market LineThe Capital Market Line

All investors face the same Capital MarketLine (CML) given by:


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Equilibrium Portfolio ReturnsEquilibrium Portfolio Returns

The CML gives the expected return-riskcombinations for efficient portfolios.

What about inefficient portfolios? Changing the expected return and/or risk of an

individual security will effect the expected return and

standard deviation of the market!

In equilibrium, what a security adds to the risk ofa portfolio must be offset by what it adds interms of expected return

Equivalent increases in risk must result in equivalent

increases in returns.


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53/65

? AE(R R E(R R

where

i f m f i

ii m im

m

im

m

) )!

! !

F

FW W V

W

W

W2 2

The CAPM Pricing Equation!The CAPM Pricing Equation!

The expected return on any asset can bewritten as:

This is simply the no arbitrage condition!

This is also known as the Security MarketLine (SML).


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? A? A%25.7)E(r

75.0035.0085.0035.0)E(rr)E(rr)E(r

IBM

IBM

ifmfi

!

!! F

Using the CAPM: FindingUsing the CAPM: Finding

E(rE(rii Suppose you have the following

information:

rf= 3.5% E(rm)=8.5% FIBM=0.75 What should E(rIBM) be?

Answer:


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? A? A? A

? A

75.0035.0085.0

035.07250.0

035.0085.0035.07250.0

)()(

IBM

D

iffi

!

!

!

!

F

F

F

Using the CAPM: FindingUsing the CAPM: Finding FFii

Suppose you have the followinginformation:

rf= 3.5% E(rm)=8.5% E(rIBM)=7.25% What should FIBMbe?

Answer:


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? A? A

%5.3

75.01

75.0085.07250.0r

75.01r75.0085.07250.075.0r75.0085.0r7250.0

75.0r085.0r7250.0

r)E(rr)E(r

f

f

ff

ff

ifmfi

!

!

!!

!

! F

Using the CAPM: Finding rUsing the CAPM: Finding rff

Suppose you have the following information:E(rm)=8.5% FDE=0.75 E(rDE)=7.25%

What should rfbe?

Answer:


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Notes on Estimating bsNotes on Estimating bs

Let rit, rmt and rft denote historical returns forthe time period t=1,2,...,T.

The are two standard ways to estimate

historical Fs using regressions: Use the Market Model: rit-rft = Ei + Fi(rmt-rft) + eit Use the Characteristic Line: rit = ai + birmt + eit

Ei = ai + (1-bi)rft and Fi = bi

Typical regression estimates: Value Line (Market Model):

5 Yrs, Weekly Data, VW NYSE as Market

Merrill Lynch (Characteristic Line): 5 Yrs, Monthly Data, S&P500 as Market


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Example Characteristic Line:Example Characteristic Line:

AAPL vs S&P500 (IEM DataAAPL vs S&P500 (IEM Data

y = 0.1844x + 0.0182

R2

= 0.0022

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

-15% -10% -5% 0% 5% 10% 15%

S&P500 Premium

AAPLPremiu


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IBM vs S&P500 (IEM DataIBM vs S&P500 (IEM Data

y = 0.9837x + 0.0191

R2 = 0.1325

-30%

-20%

-10%

0%

10%

20%

30%

40%

-15% -10% -5% 0% 5% 10% 15%

S&P500 Premium

IBMPremiu


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MSFT vs S&P500 (IEM DataMSFT vs S&P500 (IEM Datay = 1.1867x + 0.027

R2

= 0.3032

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

-15% -10% -5% 0% 5% 10% 15%

S&P500 Premium

MSFTPremiu


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Notes on EstimatingNotes on Estimating FFss

Betas for our companies

AAPL IBM MSFT SP500

Raw: 0.1844 0.9838 1.1867 1

Adjusted: 0.4563 0.9891 1.1245 1

Avg. R: 2.42% 3.64% 4.72% 1.75%


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Average Returns vsAverage Returns vs

(Adjusted Betas(Adjusted BetasMSFT

IBM

S&P500

AAPl

T-Bills

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

- 0.20 0.40 0.60 0.80 1.00 1.20

Beta

AverageRetu


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1999 Thomas A. Rietz

64

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