Lecture Presentation Software
to accompany
Investment Analysis and
Portfolio Management
Eighth Edition
by
Frank K. Reilly & Keith C. Brown
Chapter 8
Chapter 8 - An Introduction to
Asset Pricing Models
Questions to be answered:
• What are the assumptions of the capital asset pricing model?
• What is a risk-free asset and what are its risk-return
characteristics?
• What is the covariance and correlation between the risk-free
asset and a risky asset or a portfolio of risky assets?
• What is the expected return when you combine the risk-free
asset and a portfolio of risky assets?
• What is the standard deviation when you combine the risk-free
asset and a portfolio of risky assets?
• When you combine the risk-free asset and a portfolio of risky
assets on the Markowitz efficient frontier, what does the set of
possible portfolios look like?
Chapter 8 - An Introduction to
Asset Pricing Models
• Given the initial set of portfolio possibilities with a riskfree asset, what happens when you add financial leverage
(borrow)?
• What is the market portfolio, what assets are included in
this portfolio, and what are the relative weights for the
alternative assets included?
• What is the capital market line (CML)?
• What do we mean by complete diversification?
• How do we measure diversification for an individual
portfolio?
• What is systematic and unsystematic risk?
Chapter 8 - An Introduction to
Asset Pricing Models
• Given the capital market line (CML), what is the separation
theorem?
• Given the CML, what is the relevant risk measure for an
individual risky asset?
• What is the security market line (SML) and how does it
differ from the CML?
• What is beta and why is it referred to as a standardized
measure of systematic risk?
• How can you use the SML to determine the expected
(required) rate of return for a risky asset?
Chapter 8 - An Introduction to
Asset Pricing Models
• Using the SML, what do we mean by an undervalued and
overvalued security, and how do we determine whether an
asset is undervalued or overvalued?
• What is an asset’s characteristic line and how do we
compute the characteristic line for an asset?
• What is the impact on the characteristic line when it is
computed using different return intervals (e.g., weekly
versus monthly) and when we employ different proxies
(i.e., benchmarks) for the market portfolio (e.g., the S&P
500 versus a global stock index)?
Chapter 8 - An Introduction to
Asset Pricing Models
• What happens to the capital market line (CML) when we
assume there are differences in the risk-free borrowing and
lending rates?
• What is a zero-beta asset and how does its use impact the
CML?
• What happens to the security line (SML) when we assume
transaction costs, heterogeneous expectations, different
planning periods, and taxes?
• What are the major questions considered when empirically
testing the CAPM?
• What are the empirical results from tests that examine the
stability of beta?
Chapter 8 - An Introduction to
Asset Pricing Models
• How do alternative published estimates of beta compare?
• What are the results of studies that examine the
relationship between systematic risk and return?
• What other variables besides beta have had a significant
impact on returns?
• What is the theory regarding the “market portfolio” and
how does this differ from the market proxy used for the
market portfolio?
• Assuming there is a benchmark problem, what variables
are affected by it?
Capital Market Theory:
An Overview
• Capital market theory extends portfolio theory and
develops a model for pricing all risky assets
• We begin with the insights of James Tobin, who added a
risk-free asset to the Markowitz efficient frontier
• We then move to the Capital Asset Pricing Model
(CAPM), developed independently by Sharpe, Lintner,
Treynor & Mossin in the early 1960s, which allows us to
determine the required rate of return for any risky asset
Assumptions of
Capital Market Theory
1. All investors are Markowitz efficient investors who want to
target points on the efficient frontier.
 The exact location on the efficient frontier and,
therefore, the specific portfolio selected, will depend on
the individual investor’s risk-return utility function.
Assumptions of
Capital Market Theory
2. Investors can borrow or lend at the risk-free rate of return
(Rf).
 It is always possible to lend money at the nominal riskfree rate by buying risk-free securities, such as
government T-bills.
 It is not always possible to borrow at this risk-free rate,
but assuming a higher borrowing rate does not change
the general results.
Assumptions of
Capital Market Theory
3. All investors have homogeneous expectations - they
estimate identical probability distributions for future rates
of return.
 This assumption can be relaxed. As long as the
differences in expectations are not large, the effects are
minor.
Assumptions of
Capital Market Theory
4. All investors have the same one-period time horizon, such
as one-month, six months, or one year.
 The model is developed for a single hypothetical
period, and its results could be affected by a different
assumption.
 A difference in the time horizon would require investors
to derive risk measures and risk-free assets that are
consistent with their time horizons.
Assumptions of
Capital Market Theory
5. All investments are infinitely divisible, which means that it
is possible to buy or sell fractional shares of any asset or
portfolio.
 This assumption allows us to discuss investment
alternatives as continuous curves. Changing it would
have little impact on the theory.
Assumptions of
Capital Market Theory
6. There are no taxes or transaction costs involved in buying
or selling assets.
 This is a reasonable assumption in many instances.
• Neither pension funds nor religious groups have to
pay taxes
• The transaction costs for most financial institutions
are less than 1 percent on most financial
instruments.
 Relaxing this assumption modifies the results, but does
not change the basic thrust.
Assumptions of
Capital Market Theory
7. There is no inflation or any change in interest rates, or
inflation is fully anticipated.
 This is a reasonable initial assumption, and it can be
modified.
8. Capital markets are in equilibrium.
 This means that we begin with all investments properly
priced according to their risk levels.
Risk-Free Asset
Characteristics of a risk-free asset:
•
•
•
•
An asset with zero standard deviation
Zero correlation with all other risky assets
Provides the risk-free rate of return (Rf)
Lies on the vertical axis of a portfolio graph
Covariance with a Risk-Free
Asset
The formula for covariance is:
n
 [R - E(R )][R
i
Covij =
i
j
- E(R j )]
i=1
N
Because the returns for the risk free asset are certain,
σ Risk Free  0
Thus Ri = E(Ri), and Ri - E(Ri) = 0
Consequently, the covariance of the risk-free asset with any
risky asset or portfolio will always equal zero. Similarly the
correlation between any risky asset and the risk-free asset
will be zero.
Combining a Risk-Free Asset
with a Risky Portfolio
• Expected return on the portfolio is equal to the weighted
average of the two returns
E(Rport )  WR f (R f )  (1-WR f )E(Ri )
Where:
WRf = proportion of the portfolio invested in the riskfree asset
Rf = the return on the risk-free asset
E(Ri) = the expected return of a risky asset or
portfolio
This is a linear relationship
Combining a Risk-Free Asset
with a Risky Portfolio
The expected risk (variance) of the portfolio is:
2
E( port
)  w1212  w 22 22  2w1w 2 1,21 2
Substituting the risk-free asset for
Security 1, and the risky asset for
Security 2, this formula becomes:
Where:
Wi = the prop in asset i
σi = Std dev of asset i
ρ1,2 = correlation
2
E( port
)  w 2R f  R2 f  (1  w R f ) 2  i2  2w R f (1 - w R f )  R f ,i R f  i
Combining a Risk-Free Asset
with a Risky Portfolio
Since we know that the variance of the risk-free
asset is zero and the correlation between the riskfree asset and any risky asset i is zero, the formula
simplifies to:
E(σ 2port )  (1  wR f )2 σ i2


E σ port   1  wR f σ i
Therefore, the standard deviation of a portfolio that
combines the risk-free asset with a risky asset is a
linear proportion of the standard deviation of the risky
asset portfolio.
Portfolio Possibilities Combining the RiskFree Asset and Risky Portfolios on the
Efficient Frontier
E(R port )
D
Market
Portfolio
C
Rf
B
A
Since both the expected
return and the standard
deviation of return for
such a portfolio are
linear combinations, a
graph of possible
portfolio returns and
risks forms a straight
line between the two
assets. It is called the
Capital Market Line.
E( port )
Capital Market Line
• The Capital Market Line (CML) forms a new efficient
frontier, since points along it dominate any point below it
• Thus all investors, irrespective of their risk preferences,
should invest along the CML
• They do that by investing a portion of their funds in the
Market Portfolio, M and a portion in the risk-free asset
• The risk-averse investor will invest more in the risk-free
asset and less in the risky Market Portfolio
• The risk-seeking investor will invest more than 100% of
their own funds in the Market Portfolio
Portfolio Possibilities Combining the RiskFree Asset and Risky Portfolios on the
Efficient Frontier
E(R port )
Market
Portfolio
Rf
E  RPort   R f 
 Portfolio
 RM  R f 
 Market
E( port )
The Market Portfolio
• Because the market is in equilibrium, all assets are
included in the Market Portfolio M in proportion to their
market value
• Therefore this portfolio must include ALL RISKY
ASSETS
• Because it contains all risky assets, it is a completely
diversified portfolio, which means that all the unique risk
of individual assets (unsystematic risk) is diversified away
Systematic Risk
• Only systematic risk remains in the market portfolio
• Systematic risk is the variability in all risky assets caused
by macroeconomic variables, such as:
 Variability in growth of money supply
 Interest rate volatility
 Aggregate variability in
• Industrial production
• Corporate earnings
• Corporate cash flow
• Systematic risk can be measured by the standard deviation
of returns of the market portfolio and can change over time
How to Measure Diversification
• Perfect positive correlation implies a linear relationship
between two assets and no benefit to diversification
between them
• Thus all portfolios on the CML are perfectly positively
correlated with each other and with the completely
diversified Market Portfolio M
Diversification and the
Elimination of Unsystematic Risk
• The purpose of diversification is to reduce the standard
deviation of the portfolio
• This assumes that less then perfect correlations exist
among securities
• As you add securities, you expect portfolio variance to
approach the average covariance between the assets in the
portfolio
• How many securities must you add to a portfolio to obtain
a completely diversified portfolio?
Number of Stocks in a Portfolio and the
Standard Deviation of Portfolio Return
Standard Deviation of
Return
Unsystematic
(diversifiable) Risk
The number of securities
required to eliminate 90%
of the unique risk has
varied from 10 to 40,
depending on the study!
Total Risk
Standard Deviation of the
Market Portfolio (systematic risk)
Systematic Risk
Number of Stocks in the Portfolio
The CML and the Separation Theorem
• Because the CML dominates all other possible portfolios, all
all investors will invest some portion of their funds in the
Market Portfolio
• Risk preferences will lead to where an investor invests along
the CML
• James Tobin referred to this as the Separation Theorem
because
 The investment decision tells us to invest on the CML
 The financing decision dictates whether one lends or
borrows
The CML and the Separation Theorem
The decision of both investors is to invest in portfolio M
along the CML (the investment decision)
E ( R port )
CML
B
M
A
PFR
Utility increases
as we move to
the NorthWest
(upper left)
The risk-averse
investor will invest
in portfolio A,
comprised of both
T Bills and the
market portfolio.
The risk-seeking
investor will invest
in portfolio B
 port
A Risk Measure for the CML
• In the Markowitz portfolio model, when adding a security
to the portfolio, the relevant risk is its average covariance
with all other assets in the portfolio
• With the addition of the risk-free asset, the only relevant
portfolio is the Market Portfolio
• Thus for an individual risky asset, one is only concerned
with its covariance with the market portfolio
• And the covariance with the market portfolio is the
relevant measure of risk for an individual risky asset
A Risk Measure for the CML
• Because all individual risky assets are part of the Market
Portfolio, an asset’s rate of return in relation to the return for
the Market Portfolio may be described using the following
linear model:
R it  a i  bi R Mt  
Where:
Rit = return for asset i during period t
ai = constant term for asset i
bi = slope coefficient for asset i
RMt = return for the Market Portfolio during period t
 = random error term
Variance of Returns for a Risky
Asset
Var(Ri,t )= Var(ai + bi RM,t +  )
= Var(ai ) + Var(bi RM,t ) + Var(  )
= 0+ Var(bi RM,t ) + Var(  )
Notes:
• The variance of a constant is always zero
• The variance of biRM,t is the variance of the
market portfolio scaled by bi
• The variance of epsilon (the error term) is the
asset’s residual or unique risk
The Capital Asset Pricing Model:
Expected Return and Risk
• It is now time to develop a model for estimating the
required rate of return for any risky asset
• The existence of a risk-free asset resulted in deriving the
capital market line (CML) that became the relevant frontier
• We know that an individual risky asset’s covariance with
the market portfolio is the relevant measure of risk
• We can use this knowledge to determine an appropriate
expected rate of return on any risky asset - the capital asset
pricing model (CAPM)
The Capital Asset Pricing Model:
Expected Return and Risk
• CAPM generates the return an investor requires (expects)
to hold a risky asset
• We can use this rate of return to value an asset, such as in
the dividend discount model of stock valuation
• Alternately, we can compare an estimated rate of return to
the required rate of return implied by CAPM
 If the estimated return exceeds that given by the
CAPM, the asset is undervalued
 If the estimated return is below that given by the
CAPM, the asset is overvalued
Graph of Security Market Line
(SML)
E(R i )
Rm
Rf
SML
Market
Portfolio
The relevant measure of
risk for an individual risky
asset is its covariance
with the market portfolio
(Covi,m)
The covariance of any
asset with itself is its
variance.
 m2
Covi,M
The Security Market Line (SML)
The equation for the SML can be derived as follows:
E(R i )  R f 
R M -R f
= Rf 
Covi,M


2
M
2
M
We then define
(Covi,M )
(R M -R f )
Cov i,M

2
M
as beta
E(R i )  R f  i (R M -R f )
( i )
Graph of SML with
Normalized Systematic Risk
E(R i )
SML
Rm
In an efficient market, all
securities should plot along
the SML.
A
Negative
Beta
Asset A – return is too high;
price is too low
B
Rf
Asset B – return is too low;
price is too high
0
1.0
 Covi,M 

Beta 
2
 M 
Determining the Expected
Rate of Return for a Risky Asset
E(R i )  R f  i (R M - R f )
Quantity of
risk
Reward for
bearing risk
• The expected rate of return for a risky asset is determined
by the risk-free rate plus a risk premium for the individual
asset
• The risk premium is determined by the systematic risk of
the asset (beta) and the prevailing market risk premium
(RM - Rf)
Determining the Expected
Rate of Return for a Risky Asset
Stock
Beta
A
B
C
D
E
0.70
1.00
1.15
1.40
-0.30
E(R i )  R f  i (R M - RFR)
Assume:
Rf = 5%
RM = 9%
Market risk premium = 4%
E(RA) = 0.05 + 0.70 (0.09-0.05) = 0.078 = 7.8%
E(RB) = 0.05 + 1.00 (0.09-0.05) = 0.090 = 9.0%
E(RC) = 0.05 + 1.15 (0.09-0.05) = 0.096 = 9.6%
E(RD) = 0.05 + 1.40 (0.09-0.05) = 0.106 = 10.6%
E(RE) = 0.05 + -0.30 (0.09-0.05) = 0.038 = 3.8%
Calculating Beta
Beta is a measure
of how asset i covaries with the
market portfolio.
Ri
OLS Regression Line
R i,t   i   i R M, t  
Where:
Ri,t = the rate of return for
asset i during period t
RM,t = the rate of return for the
market portfolio M during t
i  R i - i R m
i 
Covi, M
 M2
0
RM
Slope of the
regression line
equals Beta
The Impact of the Time Interval
• The number of observations and the time interval used in
the regression varies
 Value Line Investment Services (VL) uses weekly rates of return
over five years (260 observations)
 Merrill Lynch, Pierce, Fenner & Smith (ML) uses monthly return
over five years (60 observations)
• There is no theoretically “correct” interval for analysis
 If the interval is too long, the company may have changed
• Weak relationship between the Value Line & Merrill Lynch
betas due to difference in intervals used
• The return time interval makes a difference
 Average beta for smallest decile firms using monthly data 1.682
 Average beta for smallest decile firms using weekly data 1.080
The Effect of the Market Proxy
• According to the theory, the market portfolio should
include all risky assets, both real & financial, domestic and
international
• Standard & Poor’s 500 Composite Index is most often
used as a proxy for the market portfolio
 Captures a large proportion of the total market value of
U.S. stocks
 Value weighted series
Weaknesses of Using S&P 500
as the Market Proxy
 Includes only U.S. stocks
 The theoretical market portfolio should include U.S. and
non-U.S. stocks and bonds, real estate, coins, stamps, art,
antiques, and any other marketable risky asset from
around the world
Relaxing the Assumptions
1. Differential Borrowing and Lending Rates
E(R port )
RBorrow
RLend
Now have
two
tangent
portfolios
E( port )
Relaxing the Assumptions
2. Zero Beta Model
 Developed by Fischer Black in 1972
 Does not require a risk-free asset. Instead a “zerobeta” portfolio within the feasible set of portfolios
takes the place of the risk-free asset
 The zero-beta portfolio has zero correlation with the
market portfolio (and thus has zero beta)
 If the zero-beta portfolio had a higher return than the
risk-free asset, would expect a higher intercept and a
flatter SML
Relaxing the Assumptions
3. Transaction Costs
 In a perfectly efficient market all securities should plot
along the SML
 With transactions costs, the SML will be a band,
rather than a straight line
 The width of the band is determined by the size of the
transactions costs
• The smaller the transaction costs, the narrower the
band around the SML
Relaxing the Assumptions
4. Heterogeneous Expectations and Planning Periods
 If individual investors have different expectations
about risk & return, each investor will have a unique
CML and/or SML and the graph becomes a band
 The width of the band depends on the divergence of
expectations
 The impact of planning period is similar, since the
model is developed as a one-period model
Relaxing the Assumptions
5. Taxes
 The model assumes pre-tax returns
 Since many (but not all) investors pay tax, taxable
investors would use an after-tax rate of return, thus
changing both the CML and the SML
Empirical Tests of the CAPM
•
•
•
Two major questions when testing the CAPM

How stable is the measure of systematic risk (beta)?
• Can we estimate future betas from past betas?

Is there a positive linear relationship between beta and observed
rates of return?
• Does rate of return rise as beta rises?
Stability of Beta

Betas for individual stocks are not stable, but portfolio betas are
reasonably stable.

The larger the portfolio of stocks and the longer the period, the
more stable the beta of the portfolio
Comparability of Published Estimates of Beta

Differences exist (see Slide 42: Impact of Time Interval). Hence,
consider the return interval used and the firm’s relative size
Relationship Between
Systematic Risk and Return
• Effect of Skewness on Relationship
 Investors prefer stocks with high positive skewness
(provides an opportunity for very large positive returns)
• Effect of size, P/E, and leverage
 Firm size and P/E have an inverse impact on returns
(investors require a higher return to hold small-cap
stocks and stocks with a low P/E ratio)
 Size & P/E ratio are in addition to beta
 Bhandari also found that financial leverage helps to
explain the cross-section of returns
 This implies a multivariate CAPM with three risk
variables: firm size, P/E ratio and leverage
Relationship Between
Systematic Risk and Return
• Effect of Book-to-Market Value
 Fama and French questioned the relationship between
returns and beta in their seminal 1992 study. They
found that the relationship between beta and return
disappeared during the 1963 – 1990 period studied.
 They found that during the study period, size &
BV/MV were the two key determinants of return with
BV/MV being dominant.
• Summary of CAPM Risk-Return Empirical Results
 The relationship between beta and rate of return is a
moot point
The Market Portfolio: Theory
versus Practice
• There is no index for the theoretical market portfolio. Hence, proxies
are used
• There is no unanimity about which proxy to use, although many
researchers use either the S&P500 or a similar US index (although US
stocks make up less than 20% of the global market)
• An incorrect market proxy will affect both beta and the position and
slope of the SML that is used to evaluate portfolio performance
• If using a global market portfolio, would expect:
 Lower covariance between North American stocks and the index
 Lower variance of the index
 With lower covariance dominating
 Leading to lower betas
Summary
• The assumption of capital market theory expand on those
of the Markowitz portfolio model and include
consideration of the risk-free rate of return
• The Markowitz efficient frontier is replaced with the CML,
a straight line running from the risk-free return and tangent
to the market portfolio
 All investors should target points along this line
depending on their risk preferences
Summary
• All investors want to invest in the risky portfolio, so this
market portfolio must contain all risky assets, held in
proportion to their market value
 The investment decision and financing decision can be
separated
 Everyone wants to invest in the market portfolio
 Investors finance based on risk preferences
Summary
• The relevant risk measure for an individual risky asset is
its systematic risk or covariance with the market portfolio
 Once you have determined this Beta measure and a
security market line, you can determine the required
return on a security based on its systematic risk
• Assuming security markets are not always completely
efficient, you can identify undervalued and overvalued
securities by comparing your estimate of the rate of return
on an investment to its required rate of return
Summary
• When we relax several of the major assumptions of the
CAPM, the required modifications are relatively minor and
do not change the overall concept of the model.
• Betas of individual stocks are not stable while portfolio
betas are stable
• There is a controversy about the relationship between beta
and rate of return on stocks
• Changing the proxy for the market portfolio results in
significant differences in betas, SMLs, and expected
returns
Summary
• It is not possible to empirically derive a true market
portfolio, so it is not possible to test the CAPM model
properly or to use model to evaluate portfolio performance
What is Next?
• Alternative asset pricing models
The Internet
Investments Online
http://www.valueline.com
http://www.wsharpe.com
http://gsb.uchicago.edu/fac/eugene.fama/
http://www.moneychimp.com