Chapter 4
Appendix: Asset
Pricing Models
Portfolio Theory
Portfolio theory begins with the premise that all
investors are risk averse— they want high
returns and guaranteed outcomes.
Diversification:
The essence of portfolio theory is that portfolios
of risky stocks can be put together in a manner
such that the entire portfolio is less risky than
any of the individual stocks.
4-2
Key Lesson from Portfolio Theory
The marginal contribution of an asset to the
riskiness of a portfolio does not depend upon
the riskiness of the asset. Rather it depends on
the covariance of the asset with the portfolio— 
The systematic risk of an asset cannot be
eliminated by holding the asset as part of a
diversified portfolio.
4-3
Capital Market Theory
Capital Market Theory extends portfolio
theory and attempts to describe the way in
which the equilibrium market price or
expected return of an individual asset is
related to the risk of the asset.
4-4
Portfolio Return
• Expected Return for a portfolio
Rp  x1R1  x 2 R2 
 x n Rn
(1)
– Where
– Rp = the return on the portfolio of n assets
– Ri = the return on asset i
– xi = the proportion of the portfolio held in
asset I
E(Rp )  E(x1R1 )  E(x 2 R2 ) 
 x1E(R1)  x2 E(R2 ) 
 E(x n Rn )
 xn E(Rn ) (2)
4-5
Expected Portfolio Risk & Return
Portfolio Risk-- standard deviation of portfolio
return can be written as
 2p  E[Rp  E(Rp )]2
 E[x1{R1  E(R1)} 
 x n {Rn  E(Rn )}]2
  x11p  x11p   x n np
2
p
(3)
where  ip = the covariance of the return on asset i
whith the portfolio' s return = E[{Ri  E(Ri )}  {Rp 
The contribution of asset i to the risk of the
portfolio is x i ip
4-6
Portfolio Risk
• The proportionate contribution of asset i to the
portfolio risk is
x i ip /
2
p
2

/

and the ratio ip p tells us how sensitive an asset’s
return is to movements in the portfolio’s return.
• The marginal contribution to the risk of a portfolio

depends
not on the risk of the asset in isolation, but

on the
sensitivity of that asset’s return to changes in
the value of the portfolio.
4-7
Distribution of Monthly S&P500 Returns
Jan 1926-May 2001
4-8
Defining Risk
• For Symmetric Distributions (like the S&P
500)
– 2/3rds of all values lie within 1 standard
deviation of the mean (expected return)
– 95% of all values lie within 2 standard
deviations of the mean
– A standard deviation is the square root of the
variance, our measure of dispersion:
4-9
Relative Risk and Returns
4-10
Reducing Risk: Quiz
State 1
(heavy
rain)
State 2
(Drought)
Probability
0.7
0.3
ReturnUnbrella Co.
30%
-25%
Return Beach Chair
Co
-15%
15%
R ei
Variance
Std. Dev.
Return
Portfolio
Calculate the expected return, variance, and standard deviation for each asset. Assume an
investor holds a portfolio with half of their assets in each asset. Calculate the Expected
return on the portfolio. Explain how this portfolio demonstrates the first key lesson from
Portfolio Theory.
4-11
Reducing Risk: Answer
State 1
(heavy
rain)
State 2
(Drought)
Probability
0.7
0.3
R ei
Variance
ReturnUmbrella Co.
30%
-25%
13.5%
6.35%
25.2%
Return Beach Chair
Co
-15%
15%
-6%
1.89%
13.75%
-5%
3.75%
0.33%
5.73%
Return
Portfolio
7.5%
Calculate the expected return, variance, and standard deviation for each asset. Assume an
investor holds a portfolio with half of their assets in each asset. Calculate the Expected
return on the portfolio. Explain how this portfolio demonstrates the first key lesson from
Portfolio Theory.
Std. Dev.
4-12
Reducing Risk: Diversification
• When there is a negative correlation between
returns as in our example, diversification reduces
risk. See http://en.wikipedia.org/wiki/Correlation
• Combining assets with a large positive
covariance (high correlation between returns of
assets) will not produce diversification.
• Modern Portfolio Theory tells investors how to
combine stocks in their portfolio to give them the
least risk possible, consistent with the return they
seek.
4-13
Reducing Risk
Correlation Coefficient
Effect of Diversification
on Risk
+1.0
No Risk Reduction
0.5
Moderate Risk Reduction
0
-0.5
-1.0
Considerable Risk
Reduction
Most risk can be
eliminated
All risk can be eliminated
4-14
US &
Developed
Foreign
Country
Stocks
US & Developed Country Correlations
US & Emerging Markets Correlations
US Stock & Bond (30 Tres) Correlations
Beta
• If we consider the entire market portfolio, then
the co-movement of asset i’s return with the
market return is the asset’s beta,
 i   im /
2
m
(4)
An asset’s beta measures the asset’s marginal
contribution to the risk of the market portfolio,
or
 Various Beta: Apple, HE
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
4-20
Risk—Return & Beta
• Recognize that the return on asset i consists of a
component that moves with the market return and
a component that is uncorrelated with the market
return
(5)
R    R 
i
i
i
m
i
An asset with a beta = 1 has the same average
variability as the market. And the expected return
on asset i depends on the market return and beta.
E(Ri )  i  i E(Rm )
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
4-21
Risk—Return & Beta
• The variance of asset i’s return can be
calculated from equation (5) as
  E[Ri  E(Ri )]  E{ i (Rm  E[Rm ])  i }
2
i
2
Or, since we know that the firm specific error is
uncorrelated with the market return E[Rmi]= 0:
     
2
i
2
i
2
m
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
2
4-22
2
Types of Risk
We define the risk for an individual stock by the standard
deviation of its return, or the sqare root of
     
2
i
2
i
2
m
2
Total Risk = Systematic Risk + Non-systematic risk
The risk of a well diversified portfolio depends only on
systematic risk. This is because events that affect
individual firms tend to cancel out over time.
Firm Specific
(Non-systematic) Risk
The risk associated with factors peculiar to a
particular firm.
• For example, Airline stocks have been badly
affected by terrorism, weather, and energy costs.
Can be diversified away—when one firm has a
setback, another may have a breakthrough.
• For example, security companies may see
increased profits due to the threat of terrorism, and
oil companies have had record profits.
Market Risk
• Market Risk: The risk associated with the comovements of the entire market.
• Market risk cannot be eliminated by
diversification. Holding the entire market does not
help if all firms are falling together! Market risk is
also called systematic risk.
Diversification Again
CAPM
The Capital Asset Pricing Model was developed
by Bill Sharpe, John Lintner, and Jan Mossing
simultaneously and independently in 1965 based
on the earlier work of Harry Markowitz.
Markowitz and Sharpe received the Nobel Prize
in Economics in 1990.
CAPM
Using equations (2) and (3), investors can select assets to
add to their portfolio to create any standard deviation and
expected return combination they choose.
E(R )  x E(R )  x E(R )   x E(R ) (2)
p
1
1
2
2
  x11p  x 2 2 p   x n np
2
p
n
n
(3)
CAPM
Consider a portfolio of n=5 assets, 1/5 of your wealth in each.
Then the portfolio return is
R p  1 R1  1 R2 
5
5
n
 1 R5  1  Ri
5
n
i1
Or, using equation (5) for the return on an individual asset:
(5)
Ri  i  i Rm  i
Rp  1
n
n
n
i1
i1
i1
1
1



R





i
i
m
i
n
n
n
Rp   Rm  
CAPM
If the portfolio is well diversified so that the firm specific risks
are uncorrelated with each other (and because systematic
and non-systematic risk are uncorrelated) the portfolio risk
depends only on systematic risk
     
2
i
2
i
2
  
2
p
2
  
2
p
2
m
2
2
m
m
2
 (1/n)( i )
2
0 as n gets large
CAPM
For each standard deviation,
(risk) there is a portfolio that
generates the highest
expected return. All such
portfolios makes up the the
efficient portfolio frontier.
These are the standard
deviation-expected return
combinations risk-averse
investors would always prefer.
http://www.moneychimp.com/articles/risk/efficient_frontier.htm
CAPM
• Assumes that
investors can borrow
and lend as much as
they want at a risk-free
rate of interest, Rf. By
lending at the risk- free
rate, the investor earns
an expected return of
Rf and his investment
has a zero standard
deviation because it is
risk-free.
CAPM
• Suppose an investor
decides to put half of
her total wealth in the
risk-free loan and the
other half in the market
portfolio on the efficient
portfolio frontier. What
is the investors
expected risk/return for
this portfolio? use
CAPM
• If an investor borrows
the total amount of her
wealth at the risk-free
rate Rf and invests the
proceeds plus her
wealth (that is, twice
her wealth) in portfolio
M, What is the
investors expected
risk/return for this
portfolio?
CAPM
• By choosing
different amounts of
borrowing and
lending, an investor
can form a portfolio
with a risk–return
combination that lies
anywhere on the
Opportunity Locus.
CAPM
• Assume that all investors have same
assessment of risk and return.
 portfolio M is the same for all investors
 portfolio M is the market portfolio
• The slope of the opportunity locus gives us the
trade-off between expected returns and increased
risk for the investor. (Sharpe Ratio)
CAPM
• If an investor increases the riskiness of
his portfolio by m then he can earn an
additional expected return of E(Rm)-Rf.
• E(Rm)-Rf is referred to as the market
price of risk (or an equity (or risk)
premium)
Quiz
• If the market risk premium is
E(Rm)-Rf = 12% - 3%; and a stock’s
beta tells us how much it contributes to
the risk of the market.
• What is your required rate of return for a
stock with a beta of 1, .5, and 1.5?
CAPM
• Recall that beta tells us the marginal
contribution of an asset to a portfolio’s
risk
• The amount an asset’s expected return
exceeds the risk-free rate should equal
the market price of risk times the asset’s
beta
Ri  R f   i [E(Rm )  R f ]
(7)
Security Market Line
Ri  R f   i [E(Rm )  R f ]
Consider security S
which is below the
SML. A portfolio
consisting of .5 in
tbills and .5 in the
market would have a
 = .5 but a higher
expected return than
security S. So no
body would buy S,
lowering its P and
raising its expected
Return.
Security Mkt Line
If the beta for Google is 1.2, the risk free rate is 5%, and
the return on the market portfolio is 10% (Wilshire 5000
TMW), what return must Google provide to compensate
investors for the risk?
5% + 1.2*(10% - 5%) = 11%
Using the CAPM
Suppose you are considering buying shares in Target- a
retail store. You study the company and decide that the
future return on the company will be 14% = Dt+1 /Pt + g.
Should you buy the stock?
Compute the required return from the CAPM assuming a
beta of 1.3:
R = 3% + 1.3*(13% - 3%) = 16%
Because the required return from the CAPM (to compensate
for measured risk) is 16% and your expected return is 14%,
you should not buy the stock.
Portfolio Building
If the investor holds a portfolio with a Beta of 1, they should
earn the market return.
• To increase return, the investor needs to take on
additional risk.
• To increase risk, either buy high beta stocks, or borrow to
buy the market.
• To lower risk, mix the Risk Free and Market Portfolios.
Notice that the slope of the SML is the risk premium (RmRf). If attitudes towards risk change, the slope of the SML
will change. Increasing risk aversion will make the SML
steeper, requiring a greater return, Ri for the same Beta.
Portfolio Building
Desired Beta
Composition of Portfolio
Expected Return from
Portfolio
0
$1 in risk free asset
10%
.5
$.50 in risk-free asset
$.50 in market portfolio
1/2(0.10) + 1/2(0.15) =
.125 = 12.5%
1
$1 in market portfolio
15%
1.5
$1.50 in market portfolio
1.5(0.15) - .5(0.10) =
borrowing $.50 at an assumed 0.175 = 17.5%
rate of 10%
Assuming expected market return is 15% and risk-free rate is 10%
Only Market Risk Matters
Group I (20 stocks)
Group II (20 stocks)
Systematic Risk (B) = 1 for each stock
Systematic Risk (B) = 1 for each stock
Specific risk is high for each security
Specific risk is low for each security
Total risk is high for each security
Total risk is low for each security
Before CAPM, it was believed that the return on
each security varied with the instability of that
security’s particular return. Then group I should
have a higher return than group II. The CAPM
says they should both have the same return.
Is Beta Bettah?
•
Is it true that high-beta portfolios will provide
larger long-term returns than lower-beta ones?
•
It appears that the relationship between beta
and returns is relatively flat.
Is Beta Bettah?
Is Beta Bettah?
Is Beta Bettah?
• Beta measures the relative volatility of an asset
and does capture some of what we call risk.
• Measuring Beta relative to the S&P 500 results in
inaccurate measurements of true risk. We need a
“total asset index”! Including human capital.
• Using improved measures of “the market”, the
CAPM does much bettah.
• Also need to use other measures of systematic
risk as in multifactor models