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©1999 Thomas A. Rietz1
Diversification and the CAPMDiversification and the CAPMThe relationship between risk and expected returns
©1999 Thomas A. Rietz2
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
©1999 Thomas A. Rietz3
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)
©1999 Thomas A. Rietz4
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!
©1999 Thomas A. Rietz5
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”).
04/19/23©1999 Thomas A. Rietz
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%!
04/19/23©1999 Thomas A. Rietz
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%
04/19/23©1999 Thomas A. Rietz
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%
04/19/23©1999 Thomas A. Rietz
$-
$2,000
$4,000
$6,000
$8,000
$10,000
$12,000
$14,000
Ap
r-9
5
Ju
l-9
5
Oc
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5
Ja
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6
Ap
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6
Ju
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Oc
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n-9
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Ap
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7
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7
Oc
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7
Ja
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Ap
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8
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8
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9
Ap
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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
04/19/23©1999 Thomas A. Rietz
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:
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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%
04/19/23©1999 Thomas A. Rietz
Example: Market ReturnsExample: Market Returns
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%
04/19/23©1999 Thomas A. Rietz
$-
$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
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Oc
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Ja
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Ap
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Ap
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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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
– 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:
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
Example: Market ReturnsExample: Market Returns
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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.
04/19/23©1999 Thomas A. Rietz
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...
04/19/23©1999 Thomas A. Rietz
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%)
04/19/23©1999 Thomas A. Rietz
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%)
04/19/23©1999 Thomas A. Rietz
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%)
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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:
04/19/23©1999 Thomas A. Rietz
%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:
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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.
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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)
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
Feasible portfolios withFeasible portfolios withN risky assets N risky assets
Expected
return (Ei)
Std dev (i)
Efficient
frontier
Feasible Set
04/19/23©1999 Thomas A. Rietz
Dominated and Efficient Dominated and Efficient PortfoliosPortfolios
Expected
return (Ei)
Std dev (i)
A
B
C
04/19/23©1999 Thomas A. Rietz
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.
04/19/23©1999 Thomas A. Rietz
Expected
return (Ei)
Std dev (i)
D
Utility maximizing
risky-asset portfolio
Utility MaximizationUtility Maximization
04/19/23©1999 Thomas A. Rietz
Expected
return (Ei)
Std dev (i)
DM
E
Utility maximization withUtility maximization witha riskfree asseta riskfree asset
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
The Capital Market LineThe Capital Market Line
All investors face the same Capital Market Line (CML) given by:
04/19/23©1999 Thomas A. Rietz
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.
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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).
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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
04/19/23©1999 Thomas A. Rietz
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%
04/19/23©1999 Thomas A. Rietz
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
©1999 Thomas A. Rietz64
SummarySummary
State what has been learned Define ways to apply training Request feedback of training session
©1999 Thomas A. Rietz65
Where to get more informationWhere to get more information
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