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1999 Thomas A. Rietz1

Diversification and the CAPMThe relationship between riskand expected returns

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Introduction

Investors are concerned with

Risk

Returns

What determines the requiredcompensation for risk?

It will depend onThe risks faced by investors

The tradeoff between risk and return they face

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Agenda

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

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

Expected return: What we expect to receive

on average

Standard deviation ofreturns: A measure of dispersion

of actual returns

Correlation

The tendency for tworeturns to fall above or

below 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 Return

We describe what we expect to receive orthe expected return:

Often estimated using historical averages

(excel function: average).

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Example: 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: 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%

The expected return E(r) is: 0.9x17.65% - 0.1x100% = 5.88%

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Example: Market Returns

Recent data from the IEM shows the followingaverage 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%
http://www.biz.uiowa.edu/iem/markets/compdata/compfund.htmlhttp://www.biz.uiowa.edu/iem/markets/compdata/compfund.html

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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

Jul-99

Oct-99

Month

ValueofInvestm

ent

AAPLIBM

MSFT

SP500

T-Bill(2)

Growth of $1000 Investments

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2222 )()( iii iii i VarrErprErp Often estimated using historical averages

(excel function: stddev)

Measuring Risk: Standard

Deviation and Variance Standard Deviation in Returns:

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Example: 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

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Example: 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%

The expected return E(r) is: 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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Recent data from the IEM shows the followingaverage monthly returns & standard deviationsfrom 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. Dev 14.84% 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

Jul-99

Oct-99

Month

ValueofInvestm

ent

AAPLIBM

MSFT

SP500

T-Bill(2)

Growth of $1000 Investments

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Risk and Average Return

T-Bill

S&P500

MSFT

IBM

AAPL

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%

Standard Deviation

AverageReturn

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Measures of Association

Correlation shows the association acrossrandom 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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rAB is in the range [-1,1]

Often estimated using historical averages

(excel function: correl)

Covariance in returns, AB, is defined as:

)()()()(BAB iA i

i

iBB iAA i

i

iA BrErErrprErrErp

BA

AB

AB r

Covariance and Correlation

The correlation, rAB, is defined as:

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Notation for Two Asset and

Portfolio ReturnsItem Asset A Asset B Portfolio

Actual Return rAi rBi rPi

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

2 B

2 P

2

Std. Dev. A B P

Correlation in Returns rAB

Covariance in Returns AB = ABrAB

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Example: IEM Suppose

You buy an MSFT090iH for $0.85 and a MSFT090iLcontract for $0.15.

You think the probability of $1 payoffs are 90% & 10%

The expected returns are: 0.9x17.65% + 0.1x(-100%) = 5.88%

0.1x566.67% + 0.9x (-100%) = -33.33%

The standard deviations are:

[0.9x(17.65%)2

+ 0.1x(-100%)2

- 5.88%2

]0.5

= 35.29% [0.1x(566.67%)2 + 0.9x(-100%)2 - (-33.33%)2]0.5 = 200%

The correlation is:

1-200%35.29%

(-33.33)%5.88%-(-100%)566.67%0.1(-100%)17.65%0.9

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Recent data from the IEM shows the followingmonthly 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

MSFT 1.000 0.550 -0.073

SP500 1.000 -0.003

T-Bills 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

IBM

Return

Correlation of AAPL & IBM

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Risk and Average Return

T-Bill

S&P500

MSFT

IBM

AAPL

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%

Standard Deviation

AverageReturn

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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

A22

Ap rBA

%08.10262.01031.0.148405.5x0.2x0

1031.05.01484.05.0 22222p

Two 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 and

Diversification Suppose

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

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

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

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Case 1: Perfect positive correlation

between securities, i.e., rAB = +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 between

securities, i.e., rAB = 0.

8%

9%10%

11%

12%

13%

14%15%

16%

17%

0% 10% 20% 30% 40%

Std. Dev.

Exp.Ret.

(10%,16%)

(16%,30%)Min. Var.

(11.33%,14.12%)

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Case 3: Perfect negative correlation

between securities, i.e., rAB = -1

8%9%

10%

11%

12%

13%14%

15%

16%

17%

0% 10% 20% 30% 40%

Std. Dev.

Exp.Ret.

(10%,16%)

(16%,30%)Zero Var.(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.

r=1

r=0

r=-1

(10%,16%)

(16%,30%)

Comparison

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w2w+

w2w+w2w+

www

MSFTIBM,MSFTIBMMSFTIBM

MSFTAAPL,MSFTAAPLMSFTAAPL

IBMAAPL,IBMAAPLIBMAAPL

2

MSFT

2

MSFT

2

IBM

2

IBM

2

AAPL

2

AAPL

p

rr

r

3 Asset Portfolios: Expected

Returns 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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%59.30472.03

10364.0

3

10242.0

3

1)(

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

For the Naively Diversified

Portfolio, this gives:

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For the Naively Diversified

Portfolio, this gives:

T-Bill

S&P500

MSFT

IBM

AAPL

NaivePortfolio

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%

Standard Deviation

AverageReturn

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The Concept of Risk With N

Risky 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 the

portfolio.

This will form the basis for CAPM.

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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

r i sk Systematic

i j

r i sk systematicNon

ipnn

11

1 22

Variance of a naively diversified

portfolio of N assets

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Naive Diversification

0%

20%

40%

60%

80%

100%

110

19

28

37

46

55

64

73

82

91

100

Number of Assets

Var.ofPortfo

lio

r.5

r.2

r0

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Z

YES

XRXWMT

VIA

U

TS

R

QUIZPOAT

NOVL

MSFT

LEK

JNJ

IBM

HWPGE

FEK

DE

CATBAAAPL

-2%

-1%

0%

1%

2%

3%

4%

5%

6%

0.00% 5.00% 10.00% 15.00% 20.00%

Standard Deviation in Return

ExpectedRetu

rn

26 Risky Assets Over a 10Year Period

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Standard

Devaition

0%

2%

4%

6%

8%

10%

12%

14%

16%

1 3 5 7 9 11 13 15 17 19 21 23 25

Number of Stocks in Portfolio

Expec

tedPortfolioReturnand

StandardDevia

tion

Average Monthly Return

Consider Naive Portfolios of 1through all 26 of these Assets

(Added in Alphabetical Order)

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The Capital Asset Pricing

Model CAPM Characteristics:

bi = imrim/m2

Asset Pricing Equation:E(ri) = rf+ bi[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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CAPM Assumptions

No transactions costs

No taxes

Infinitely divisible assets

Perfect competitionNo individual can affect prices

Only expected returns and variances matter Quadratic utility or

Normally distributed returns

Unlimited short sales and borrowing and lendingat the risk free rate of return

Homogeneous expectations

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Feasible portfolios with

N risky assets

Expected

return (Ei)

Std dev (i)

Efficientfrontier

Feasible Set

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Dominated and Efficient

PortfoliosExpected

return (Ei)

Std dev (

i)

A

B

C

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How would you find the

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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Expected

return (Ei)

Std dev (

i)

D

Utility maximizing

risky-asset portfolio

Utility Maximization

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Expected

return (Ei)

Std dev (i)

DM

E

Utility maximization witha riskfree asset

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Three Important Funds

The riskless asset has a standard deviationof 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 risky assets

and portfoliosExpected

return (Ei)

Std dev (i)

Risklessasset Minimum

Variance

Portfolio

MarketPortfolio

Efficientfrontier

A world with one risklessasset and N risky assets

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Tobins Two-Fund Separation

When the riskfree asset is introduced,

All investors prefer a combination of

1) The riskfree asset and2) The market portfolio

Such combinations dominate all other

assets and portfolios

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e

m

fm

fe

rrErrE

)()(

The Capital Market Line

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

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Equilibrium 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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L im X N

m i i m im

i

N

r2 1

How is Risk Priced?

Consider the variance of the marketportfolio:

It is the covariance with the marketportfolio and not the variance of a security

that matters Therefore, the CAPM prices the

covariance with the market and notvariance per se

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E(R R E(R Rwhere

i f m f i

ii m im

m

im

m

) )

b

b r

2 2

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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Notes on Estimating bs Let rit, rmt and rft denote historical returns for

the time period t=1,2,...,T.

The are two standard ways to estimatehistorical bs using regressions:

Use the Market Model: rit-rft = ai + bi(rmt-rft) + eit

Use the Characteristic Line: rit = ai + birmt + eit

ai = ai + (1-bi)rft and bi = 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:AAPL 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

AAPLPremium

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Example Characteristic Line:IBM 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

IBM

Premium

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Example Characteristic Line:MSFT vs S&P500 (IEM Data)

y = 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

MSFTPremium

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Notes on Estimating bs

Betas for our companies

AAPL IBM MSFT SP500

Raw: 0.1844 0.9838 1.1867 1Adjusted: 0.4563 0.9891 1.1245 1

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

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Average Returns vs(Adjusted) Betas

MSFT

IBM

S&P500

AAPl

T-Bills0.00%

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

AverageReturn

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Summary

State what has been learned

Define ways to apply training

Request feedback of training session

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Where to get more information

Other training sessions

List books, articles, electronic sources

Consulting services, other sources

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