# Diversification and the CAPM The relationship between risk and expected returns 1

```Diversification and the CAPM
The relationship between risk
and expected returns
1
&copy;1999 Thomas A. Rietz
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
2
&copy;1999 Thomas A. Rietz
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)
3
&copy;1999 Thomas A. Rietz
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!
4
&copy;1999 Thomas A. Rietz
Vocabulary

Expected return:

– What we expect to receive
on average

Standard deviation of
returns:
– A measure of risk
appropriate for diversified
investors

Correlation
– The tendency for two
returns to fall above or
below the expected return
a the same or different
times
Diversified investors
– Investors who hold a
portfolio of many
investments
– A measure of dispersion
of actual returns

Beta

The Capital Asset
Pricing Model (CAPM)
– The relationship between
risk and return for
diversified investors
5
&copy;1999 Thomas A. Rietz
Measuring Expected Return

We describe what we expect to receive or
the expected return:
E (r )   pi ri
i
– Often estimated using historical averages
(excel function: “average”).
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%!
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%
Example: 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
Average Return 2.42%
IBM
MSFT SP500 T-Bills
3.64% 4.72% 1.75% 0.35%
Month
Oct-99
Jul-99
Apr-99
Jan-99
Oct-98
Jul-98
Apr-98
Jan-98
Oct-97
Jul-97
Apr-97
Jan-97
Oct-96
Jul-96
Apr-96
Jan-96
Oct-95
Jul-95
Apr-95
Value of Investment
Growth of \$1000 Investments
\$14,000
\$12,000
\$10,000
\$8,000
\$6,000
AAPL
IBM
MSFT
SP500
T-Bill(2)
\$4,000
\$2,000
\$-
Measuring Risk: Standard
Deviation and Variance

Standard Deviation in Returns:

 pi ri  E (r ) 
2
i

2
2
2
p
r

E
(
r
)
Var


 ii
i
i
Often estimated using historical averages
(excel function: “stddev”)
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:
1
1
2
 (66.67% )   (33.33% )2 
6
6
1
1
2
 (0% )   (33.33% )2 
 56.93%
6
6
1
1
2
 (66.67% )   (100% )2  16.67%2
6
6
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%

The standard deviation is:
– [0.9x(17.65%)2 + 0.1x(-100%)2 - 5.88%2]0.5 = 35.29%
Example: Market Returns

Recent data from the IEM shows the following
average monthly returns &amp; standard deviations
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%
Std. Dev
14.84% 10.31% 8.22% 3.82% 0.06%
Month
Oct-99
Jul-99
Apr-99
Jan-99
Oct-98
Jul-98
Apr-98
Jan-98
Oct-97
Jul-97
Apr-97
Jan-97
Oct-96
Jul-96
Apr-96
Jan-96
Oct-95
Jul-95
Apr-95
Value of Investment
Growth of \$1000 Investments
\$14,000
\$12,000
\$10,000
\$8,000
\$6,000
AAPL
IBM
MSFT
SP500
T-Bill(2)
\$4,000
\$2,000
\$-
Risk and Average Return
5.0%
MSFT
4.5%
Average Return
4.0%
IBM
3.5%
3.0%
AAPL
2.5%
2.0%
S&amp;P500
1.5%
1.0%
0.5%
T-Bill
0.0%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
Standard Deviation
12.0%
14.0%
16.0%
Measures 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
Covariance and Correlation

Covariance in returns, AB, is defined as:
 AB   pi rAi  E (rA )rBi  E (rB )   pi rAi rBi  E (rA )E (rB )
i

i
The correlation, rAB, is defined as:
r AB
 AB

 A B
– rAB is in the range [-1,1]
– Often estimated using historical averages
(excel function: “correl”)
Notation for Two Asset and
Portfolio Returns
Item
Actual Return
Expected Return
Variance
Std. Dev.
Asset A
rAi
E(rA)
A 2
A
Correlation in Returns
Covariance in Returns
Asset B
rBi
E(rB)
B2
B
rAB
AB = ABrAB
Portfolio
rPi
E(rP)
 P2
P
Example: 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% &amp; 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:
0.9  17.65%  (-100%)  0.1  566.67%  (-100%) - 5.88%  (-33.33)%
-1
35.29%  200%
Example: 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
AAPL 1.000 0.262
IBM
1.000
MSFT
SP500
T-Bills
MSFT
0.102
0.240
1.000
SP500
0.046
0.362
0.550
1.000
T-Bills
-0.103
-0.169
-0.073
-0.003
1.000
Correlation of AAPL &amp; IBM
\$1
y = 0.3777x + 0.0105
Correl = 0.262
\$0
\$0
IBM Return
\$0
\$0
\$-20.00%
-10.00%
0.00%
\$(0)
10.00%
\$(0)
\$(0)
\$(0)
AAPL Return
20.00%
30.00%
40.00%
Risk and Average Return
5.0%
MSFT
4.5%
Average Return
4.0%
IBM
3.5%
3.0%
AAPL
2.5%
2.0%
S&amp;P500
1.5%
1.0%
0.5%
T-Bill
0.0%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
Standard Deviation
12.0%
14.0%
16.0%
Two Asset Portfolios: Risk


The standard deviation is not a linear
combination of the individual asset standard
deviations
 p  w 2A A2 + (1  w A )2 B2  2w A (1  w A ) A B r AB

The standard deviation a the 50%/50%, AAPL &amp;
IBM portfolio is:
p

2
0.52  0.14842  0.52  0.10312

 10.08%
 2x0.5x0.5  0.1484  0.1031 0.262
The portfolio risk is lower than either individual
asset’s because of diversification.
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 A
and B as wA and r vary...
Exp. Ret.
Case 1: Perfect positive correlation
between securities, i.e., rAB = +1
17%
16%
15%
14%
13%
12%
11%
10%
9%
8%
(16%,30%)
(10%,16%)
0%
10%
20%
Std. Dev.
30%
40%
Exp. Ret.
Case 2: Zero correlation between
securities, i.e., rAB = 0.
17%
16%
15%
14%
13%
12%
11%
10%
9%
8%
(16%,30%)
Min. Var.
(11.33%,14.12%)
(10%,16%)
0%
10%
20%
Std. Dev.
30%
40%
Exp. Ret.
Case 3: Perfect negative correlation
between securities, i.e., rAB = -1
17%
16%
15%
14%
13%
12%
11%
10%
9%
8%
Zero Var.
(11.33%,14.12%)
(16%,30%)
(10%,16%)
0%
10%
20%
Std. Dev.
30%
40%
Exp. Ret.
Comparison
17%
16%
15%
14%
13%
12%
11%
10%
9%
8%
(16%,30%)
r=1
r=0
r=-1
(10%,16%)
0%
10%
20%
Std. Dev.
30%
40%
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)

The standard deviation is:
w AAPL  AAPL  w
2
p 
2

 w MSFT  MSFT
  IBM  r AAPL, IBM
2
IBM
+ 2w AAPL w IBM AAPL
2
IBM
2
+ 2w AAPL w MSFT AAPL   MSFT  r AAPL, MSFT
+ 2w IBMw MSFT IBM   MSFT  rIBM,MSFT
2
For the Naively Diversified
Portfolio, this gives:
1
1
1
E (RP )   0.0242   0.0364   0.0472  3.59%
3
3
3
2
p
2
2
2
1
1
1
0.14842  0.10312  0.08222
3
3
3
1 1
+ 2  0.1484  0.1031 0.262
3 3

1 1
+ 2  0.1484  0.0822  0.102
3 3
1 1
+ 2  0.1031 0.0822  0.240
3 3
 7.75%
For the Naively Diversified
Portfolio, this gives:
5.0%
MSFT
4.5%
Average Return
4.0%
Naive
Portfolio
3.5%
IBM
3.0%
AAPL
2.5%
2.0%
S&amp;P500
1.5%
1.0%
0.5%
T-Bill
0.0%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
Standard Deviation
12.0%
14.0%
16.0%
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 a
portfolio not through its standard deviation but
through its correlation with other assets in the
portfolio.
This will form the basis for CAPM.
Variance of a naively diversified
portfolio of N assets

Portfolio variance consists of two parts:
– 1. Non-systematic (or idiosyncratic) risk and
– 2. Systematic (or covariance) risk
p 
2
1 2
i
n

Non systematic
risk

1

  1   ij
n


Systematic
risk
The market rewards only systematic risk
because diversification can get rid of nonsystematic risk
Naive Diversification
80%
r.5
r.2
r0
60%
40%
20%
Number of Assets
100
91
82
73
64
55
46
37
28
19
10
0%
1
Var. of Portfolio
100%
26 Risky Assets Over a 10
Year Period
6%
5%
MSFT
Expected Return
4%
IBM
U
3%
2%
1%
0%
0.00%
-1%
JNJ XRX HWP
GE
WMT
VIA
AAPL
F
CAT
QUIZ
S
OAT
EK BA T
NOVL
P
K
DE
LE
YES
R
5.00%
10.00%
15.00%
Z
-2%
Standard Deviation in Return
20.00%
Consider Naive Portfolios of 1
through all 26 of these Assets
14%
12%
10%
8%
Standard
Devaition
6%
4%
2%
Number of Stocks in Portfolio
25
23
21
19
17
15
13
11
7
5
3
9
Average Monthly Return
0%
1
Expected Portfolio Return and
Standard Deviation
16%
The Capital Asset Pricing
Model

CAPM Characteristics:
– bi = imrim/m2

Asset Pricing Equation:
– E(ri) = rf + bi[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 Assumptions




No transactions costs
No taxes
Infinitely divisible assets
Perfect competition
– No individual can affect prices

Only expected returns and variances matter
– Normally distributed returns


Unlimited short sales and borrowing and lending
at the risk free rate of return
Homogeneous expectations
Feasible portfolios with
N risky assets
Feasible Set
Expected
return (Ei)
Efficient
frontier
Std dev (i)
Dominated and Efficient
Portfolios
Expected
return (Ei)
B
C
A
Std dev (i)
How would you find the
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.
Utility Maximization
Expected
return (Ei)
Utility maximizing
risky-asset portfolio
D
Std dev (i)
Utility maximization with
a riskfree asset
Expected
return (Ei)
E
M
D
Std dev (i)
Three 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
A world with one riskless
asset and N risky assets
Expected
return (Ei)
Riskless
asset
All risky assets
and portfolios
Market
Portfolio
Minimum
Variance
Portfolio
Efficient
frontier
Std dev (i)
Tobin’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 Line

All investors face the same Capital Market
Line (CML) given by:
 E (rm )  rf 
E (re )  rf  
e

 m

Equilibrium Portfolio Returns


The CML gives the expected return-risk
combinations for efficient 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?

Consider the variance of the market
portfolio:
Lim 
N


2
m

N
X 
i
i 1
i
m
r im
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
The CAPM Pricing Equation!

The expected return on any asset can be
written as:
E(Ri )  R f  E(Rm )  R f b i
 i m r im  im
where b i 

2
2
m
m


This is simply the no arbitrage condition!
This is also known as the Security Market
Line (SML).
Notes 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 b’s 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&amp;P500 as Market
Example Characteristic Line:
AAPL vs S&amp;P500 (IEM Data)
50%
40%
30%
20%
y = 0.1844x + 0.0182
R2 = 0.0022
10%
0%
-15%
-10%
-5%
0%
-10%
-20%
-30%
-40%
5%
10%
15%
Example Characteristic Line:
IBM vs S&amp;P500 (IEM Data)
40%
y = 0.9837x + 0.0191
R2 = 0.1325
30%
20%
10%
0%
-15%
-10%
-5%
0%
-10%
-20%
-30%
5%
10%
15%
Example Characteristic Line:
MSFT vs S&amp;P500 (IEM Data)
30%
y = 1.1867x + 0.027
R2 = 0.3032
25%
20%
15%
10%
5%
0%
-15%
-10%
-5%
0%
-5%
-10%
-15%
-20%
5%
10%
15%
Notes on Estimating b’s

Betas for our companies
Raw:
Avg. R:
AAPL
0.1844
0.4563
2.42%
IBM
0.9838
0.9891
3.64%
MSFT
1.1867
1.1245
4.72%
SP500
1
1
1.75%
Average Returns vs
5.00%
MSFT
4.50%
Average Return
4.00%
IBM
3.50%
3.00%
2.50%
AAPl
2.00%
S&amp;P500
1.50%
1.00%
0.50%
T-Bills
0.00%
-
0.20
0.40
0.60
Beta
0.80
1.00
1.20
Summary



State what has been learned
Define ways to apply training
Request feedback of training session
64
&copy;1999 Thomas A. Rietz