©1999 thomas a. rietz 1 diversification and the capm the relationship between risk and expected...

©1999 Thomas A. Rietz1

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


Introduction Introduction

Investors are concerned with– Risk– Returns

What determines the required compensation for risk?

It will depend on– The risks faced by investors– The tradeoff between risk and return they face


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)


Overview Overview

Investors demand compensation for risk– If investors hold “diversified” portfolios, risk

can be defined through the interaction of a single investment with the rest of the portfolios through a concept called “beta”

The CAPM gives the required relationship between “beta” and the return demanded on the investment!


VocabularyVocabulary

Expected return:– What we expect to receive

on average Standard deviation of

returns:– A measure of dispersion

of actual returns Correlation

– The tendency for two returns to fall above or below the expected return a the same or different times

Beta– A measure of risk

appropriate for diversified investors

Diversified investors– Investors who hold a

portfolio of many investments

The Capital Asset Pricing Model (CAPM)– The relationship between

risk and return for diversified investors

04/19/23©1999 Thomas A. Rietz

ii

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”).


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%!


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% The expected return E(r) is:

– 0.9x17.65% - 0.1x100% = 5.88%


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-BillsAverage Return 2.42% 3.64% 4.72% 1.75% 0.35%


$-

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

Ap

r-9

5

Ju

l-9

5

Oc

t-9

5

Ja

n-9

6

Ap

r-9

6

Ju

l-9

6

Oc

t-9

6

Ja

n-9

7

Ap

r-9

7

Ju

l-9

7

Oc

t-9

7

Ja

n-9

8

Ap

r-9

8

Ju

l-9

8

Oc

t-9

8

Ja

n-9

9

Ap

r-9

9

Ju

l-9

9

Oc

t-9

9

Month

Va

lue

of

Inv

es

tme

nt

AAPL

IBM

MSFT

SP500

T-Bill(2)

Growth of $1000 InvestmentsGrowth of $1000 Investments


2222 )()( iii

iii

i VarrErprErp

Often estimated using historical averages (excel function: “stddev”)

Measuring Risk: Standard Measuring Risk: Standard Deviation and VarianceDeviation and Variance Standard Deviation in Returns:


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


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%

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

The standard deviation is:– [0.9x(17.65%)2 + 0.1x(-100%)2 - 5.88%2]0.5 = 35.29%



Recent data from the IEM shows the following average monthly returns & standard deviations from 5/95 to 10/99:– (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)

AAPL IBM MSFT SP500 T-BillsAverage Return 2.42% 3.64% 4.72% 1.75% 0.35%Std. Dev 14.84% 10.31% 8.22% 3.82% 0.06%


$-

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

Ap

r-9

5

Ju

l-9

5

Oc

t-9

5

Ja

n-9

6

Ap

r-9

6

Ju

l-9

6

Oc

t-9

6

Ja

n-9

7

Ap

r-9

7

Ju

l-9

7

Oc

t-9

7

Ja

n-9

8

Ap

r-9

8

Ju

l-9

8

Oc

t-9

8

Ja

n-9

9

Ap

r-9

9

Ju

l-9

9

Oc

t-9

9

Month

Va

lue

of

Inv

es

tme

nt

AAPL

IBM

MSFT

SP500

T-Bill(2)

Growth of $1000 InvestmentsGrowth of $1000 Investments


Risk and Average ReturnRisk 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

Ave

rag

e R

etu

rn


Measures of AssociationMeasures of Association

Correlation shows the association across random variables

Variables with– Positive correlation: tend to move in the same

direction– Negative correlation: tend to move in opposite

directions– Zero correlation: no particular tendencies to move

in particular directions relative to each other


– AB is in the range [-1,1]

– Often estimated using historical averages (excel function: “correl”)

Covariance in returns, AB, is defined as:

)()()()( BABiAii

iBBiAAii

iAB rErErrprErrErp

BA

ABAB

Covariance and CorrelationCovariance and Correlation

The correlation, AB, is defined as:


Notation for Two Asset and Notation for Two Asset and Portfolio ReturnsPortfolio ReturnsItem Asset A Asset B Portfolio

Actual Return rAi rBi rPi

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

Variance A2 B

2 P2

Std. Dev. A B P

Correlation in Returns AB

Covariance in Returns AB = ABAB


Example: IEMExample: IEM Suppose

– You buy an MSFT090iH for $0.85 and a MSFT090iL contract 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



Recent data from the IEM shows the following monthly return correlations from 5/95 to 10/99:– (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html)

AAPL IBM MSFT SP500 T-BillsAAPL 1.000 0.262 0.102 0.046 -0.103IBM 1.000 0.240 0.362 -0.169MSFT 1.000 0.550 -0.073SP500 1.000 -0.003T-Bills 1.000


y = 0.3777x + 0.0105Correl = 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

Re

turn

Correlation of AAPL & IBMCorrelation of AAPL & IBM


Risk and Average ReturnRisk 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

Ave

rag

e R

etu

rn


The standard deviation is not a linear combination of the individual asset standard deviations

Instead, it is given by:

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

A22

Ap BA

%08.10262.01031.0.148405.5x0.2x0

1031.05.01484.05.0 22222

p

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 asset’s because of diversification.


Correlations and Correlations and DiversificationDiversification Suppose

– E(r)A = 16% and A = 30%

– E(r)B = 10% and B = 16%

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


Case 1: Perfect positive correlation Case 1: Perfect positive correlation between securities, i.e., between securities, i.e., AB AB = +1= +1

8%9%

10%11%12%13%14%15%16%17%

0% 10% 20% 30% 40%

Std. Dev.

Exp

. R

et.

(10%,16%)

(16%,30%)


Case 2: Zero correlation between Case 2: Zero correlation between securities, i.e., securities, i.e., ABAB = 0. = 0.

8%9%

10%11%12%13%14%15%16%17%

0% 10% 20% 30% 40%

Std. Dev.

Exp

. R

et.

(10%,16%)

(16%,30%)Min. Var.(11.33%,14.12%)


Case 3: Perfect negative correlation Case 3: Perfect negative correlation between securities, i.e., between securities, i.e., ABAB = -1 = -1

8%9%

10%11%12%13%14%15%16%17%

0% 10% 20% 30% 40%

Std. Dev.

Exp

. R

et.

(10%,16%)

(16%,30%)Zero Var.(11.33%,14.12%)


8%9%

10%11%12%13%14%15%16%17%

0% 10% 20% 30% 40%

Std. Dev.

Exp

. R

et.

r=1r=0r=-1

(10%,16%)

(16%,30%)

ComparisonComparison


w2w +

w2w +

w2w +

www

MSFTIBM,MSFTIBMMSFTIBM

MSFTAAPL,MSFTAAPLMSFTAAPL

IBMAAPL,IBMAAPLIBMAAPL

2MSFT

2MSFT

2IBM

2IBM

2AAPL

2AAPL

p

3 Asset Portfolios: Expected 3 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) The standard deviation is:


%59.30472.03

10364.0

3

10242.0

3

1)( PRE

%75.7

240.00822.01031.03

1

3

12 +

102.00822.01484.03

1

3

12 +

262.01031.01484.03

1

3

12 +

0822.03

11031.0

3

11484.0

3

1 22

22

22

2p

For the Naively Diversified For the Naively Diversified Portfolio, this gives:Portfolio, this gives:


For the Naively Diversified For the Naively Diversified Portfolio, this gives:Portfolio, this gives:

T-Bill

S&P500

MSFT

IBM

AAPL

Naive Portfolio

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

Ave

rag

e R

etu

rn


The Concept of Risk With N The 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 a portfolio not through its standard deviation but through its correlation with other assets in the portfolio.

This will form the basis for CAPM.


Portfolio variance consists of two parts:– 1. Non-systematic (or idiosyncratic) risk and– 2. Systematic (or covariance) risk

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

risk

Systematic

ij

risksystematicNon

ip nn

11

1 22

Variance of a naively diversified Variance of a naively diversified portfolio of N assetsportfolio of N assets


Naive DiversificationNaive Diversification

0%

20%

40%

60%

80%

100%

1 10 19 28 37 46 55 64 73 82 91 100

Number of Assets

Var

. o

f P

ort

foli

o

.5.20


Z

YES

XRXWMT

VIA

U

TS

R

QUIZPOAT

NOVL

MSFT

LEK

JNJ

IBM

HWPGE

FEK

DECATBA

AAPL

-2%

-1%

0%

1%

2%

3%

4%

5%

6%

0.00% 5.00% 10.00% 15.00% 20.00%

Standard Deviation in Return

Ex

pe

cte

d R

etu

rn26 Risky Assets Over a 10 26 Risky Assets Over a 10 Year PeriodYear Period


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

Ex

pe

cte

d P

ort

folio

Re

turn

an

d

Sta

nd

ard

De

via

tio

n

Average Monthly Return

Consider Naive Portfolios of 1 Consider Naive Portfolios of 1 through all 26 of these Assets through all 26 of these Assets (Added in Alphabetical Order)(Added in Alphabetical Order)


The Capital Asset Pricing The Capital Asset Pricing ModelModel CAPM Characteristics:

– i = imim/m2

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

CAPM is a model of what expected returns should 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


CAPM AssumptionsCAPM Assumptions

No transactions costs No taxes Infinitely divisible assets Perfect competition

– No individual can affect prices Only expected returns and variances matter

– Quadratic utility or– Normally distributed returns

Unlimited short sales and borrowing and lending at the risk free rate of return

Homogeneous expectations


Feasible portfolios withFeasible portfolios withN risky assets N risky assets

Expected

return (Ei)

Std dev (i)

Efficient

frontier

Feasible Set


Dominated and Efficient Dominated and Efficient PortfoliosPortfolios

Expected

return (Ei)

Std dev (i)

A

B

C


How would you find the How would you find the efficient frontier?efficient frontier?1. Find all asset expected returns and

standard deviations.2. Pick one expected return and minimize

portfolio risk.3. Pick another expected return and

minimize portfolio risk.4. Use these two portfolios to map out the

efficient frontier.


Expected

return (Ei)

Std dev (i)

D

Utility maximizing

risky-asset portfolio

Utility MaximizationUtility Maximization


Expected

return (Ei)

Std dev (i)

DM

E

Utility maximization withUtility maximization witha riskfree asseta riskfree asset


Three Important FundsThree Important Funds

The riskless asset has a standard deviation of zero

The minimum variance portfolio lies on the boundary of the feasible set at a point where variance is minimum

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


All risky assets

and portfoliosExpected

return (Ei)

Std dev (i)

Riskless

asset Minimum

Variance

Portfolio

Market

Portfolio

Efficient

frontier

A world with one risklessA world with one risklessasset and N risky assetsasset and N risky assets


Tobin’s Two-Fund SeparationTobin’s Two-Fund Separation

When the riskfree asset is introduced, All investors prefer a combination of 1) The riskfree asset and 2) The market portfolio Such combinations dominate all other

assets and portfolios


The Capital Market LineThe Capital Market Line

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


Equilibrium Portfolio ReturnsEquilibrium Portfolio Returns

The CML gives the expected return-risk combinations 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 of a portfolio must be offset by what it adds in terms of expected return– Equivalent increases in risk must result in equivalent

increases in returns.


How is Risk Priced?How is Risk Priced?

Consider the variance of the market portfolio:

It is the covariance with the market portfolio and not the variance of a security that matters

Therefore, the CAPM prices the covariance with the market and not variance per se


E(R R E(R R

where

i f m f i

ii m im

m

im

m

) )

2 2

The CAPM Pricing Equation!The CAPM Pricing Equation!

The expected return on any asset can be written as:

This is simply the no arbitrage condition! This is also known as the Security Market Line

(SML).


Notes on Estimating b’sNotes on Estimating b’s Let rit, rmt and rft denote historical returns for the time

period t=1,2,...,T. The are two standard ways to estimate historical ’s

using regressions:– Use the Market Model: rit-rft = i + i(rmt-rft) + eit

– Use the Characteristic Line: rit = ai + birmt + eit i = ai + (1-bi)rft and i = 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


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

AA

PL

Pre

miu

m


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

Pre

miu

m


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

MS

FT

Pre

miu

m


Notes on Estimating Notes on Estimating ’s’s

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%


Average Returns vs Average Returns vs (Adjusted) Betas (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

Av

era

ge

Re

turn


SummarySummary

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Where to get more informationWhere to get more information

Other training sessions List books, articles, electronic sources Consulting services, other sources

©1999 thomas a. rietz 1 diversification and the capm the relationship between risk and expected...

Documents

expected returns slide

risk returns

standard deviation of

concepts of risk

actual returns

market returns

measure of risk appropriate

stddev measuring risk