Introduction
The next two chapters (together with Chs. 2 – 5 of
Haugen) will briefly examine the following
aspects of quantitative investment management:
• Modeling risk and return – CAPM & APT –
theory, testing, and extensions
• Estimating risk and return – the Single-Index
Model (SIM) and multiple-factor models for
risk and expected return
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Investment Analysis and
Portfolio Management
Sixth Edition
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Chapters 9 & 10
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Modeling Risk & Return
Part One:
• The Risk-Free Asset,
• Portfolio Separation, and
• The Capital Asset Pricing Model (CAPM)
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THE RISK-FREE ASSET
• WHAT IS A RISK FREE-ASSET?
– DEFINITION:
an asset whose terminal value
is certain
• variance of returns = 0,
• covariance with other assets = 0
i  0
If
then     
ij
ij i j
 0
THE RISK-FREE ASSET
• WHAT IS A RISK FREE-ASSET?
– DEFINITION:
an asset whose terminal value
is certain
•
•
•
•
•
An investment with NO risk
An asset with zero variance
Zero correlation with all other risky assets
Provides the risk-free rate of return (RFR)
Will lie on the vertical axis of a portfolio graph
THE RISK-FREE ASSET
• DOES A RISK-FREE ASSET EXIST?
– CONDITIONS FOR EXISTENCE:
• Fixed-income security
• No possibility of default
• No interest-rate risk
• no reinvestment risk
THE RISK-FREE ASSET
• DOES A RISK-FREE ASSET EXIST?
– Given the conditions, what qualifies?
• a U.S. Treasury security with a maturity matching
the investor’s horizon
Combining the Risk-Free Asset
with a Risky Portfolio
Portfolio expected return is a linear relationship
 the weighted average of the two returns
E(R port )  WRF (RFR)  (1 - WRF )E(R i )
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Combining the Risk-Free Asset
with a Risky Portfolio
Portfolio standard deviation is also a linear
relationship, equal to the weighted average of
the two standard deviations (zero for the riskfree asset and i for the risky portfolio)
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Combining a Risk-Free Asset
with a Risky Portfolio
Standard deviation
The expected variance for a two-asset portfolio is
E(
2
port
)  w   w   2w 1 w 2 r1,2 1 2
2
1
2
1
2
2
2
2
Substituting the risk-free asset for Security 1, and the risky
asset for Security 2, this formula would become
2
2
E( port
)  w 2RF RF
 (1  w RF ) 2  i2  2w RF (1 - w RF )rRF,i RF i
Since we know that the variance of the risk-free asset is
zero and the correlation between the risk-free asset and any
risky asset i is zero we can adjust the formula
2
E( port
)  (1  w RF ) 2  i2
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Combining a Risk-Free Asset
with a Risky Portfolio
Given the variance formula
the standard deviation is
2
E( port
)  (1  w RF ) 2  i2
E( port )  (1  w RF ) 2  i2
 (1  w RF )  i
Therefore, the standard deviation of a portfolio that
combines the risk-free asset with risky assets is the
linear proportion of the standard deviation of the risky
asset portfolio.
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Combining a Risk-Free Asset
with a Risky Portfolio
Example
Assume:
– E(RF) = 7%,
– E(RS&P) = 12%,
– S&P = 20%
• Expected Return on Combined Portfolio:
E RC    F RF  1   F E R P   0.27%  1  0.212%  11.0%
• Standard Deviation on Combined Portfolio:
 C  1  F  P  1  0.2 20 %   16 %
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Combining a Risk-Free Asset
with a Risky Portfolio
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 looks like a straight line between the two
assets.
Thus, the existence of a risk-free asset adds value to
investors by expanding the set of portfolios
available to them.
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Portfolio Possibilities Combining the Risk-Free Asset
and Risky Portfolios on the Efficient Frontier
E(R port )
Figure 9.1
D
P*
C
RFR
B
A
E( port )
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The Risk-Free Asset and
Portfolio Separation Theory
Assuming the investor can both lend (by buying
Treasury bonds) and borrow (by shorting the
bonds with full use of the proceeds) at the riskfree rate, this means that the investor now faces a
linear (rather than convex) efficient frontier:
 E  RP   RF 
E  RC   RF   C 

P


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The Risk-Free Asset and
Portfolio Separation Theory
This linear efficient frontier, comprising various
combinations of the risk-free asset and the risky
portfolio P*, dominates all other possible risky
portfolios within the original (Markowitz)
efficient frontier.
This fact led to the development of the Portfolio
Separation Theory (cf., James Tobin).
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Portfolio Separation Theory
Under the portfolio separation theory, the ideal risky
portfolio in which an investor should invest is the
same (P*), regardless of how aggressive or risk
averse the investor is.
– I.e., the point on the Markowitz efficient frontier at
which the investor will invest is independent of the
investor’s risk preferences.
Where risk preferences are reflected is in terms of
how much of his or her portfolio is allocated to P*
and how much is invested in the risk-free asset.
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Portfolio Separation Theory
Thus, in order to obtain his or her optimal portfolio,
there are two separate decisions for the investor
to make:
1. The investment decision
•
•
•
Which portfolio on the Markowitz efficient frontier to
choose?
This is determined by the point of tangency between
the Markowitz efficient frontier and a line extending
from the risk-free rate
This leads to the choice of portfolio P* as the optimal
risky portfolio for the investor, regardless of the
investor’s risk preferences
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Portfolio Separation Theory
2. The financing decision
•
•
•
This is where risk preferences come into the picture
If the investor is more risk averse, he or she will put
part of his or her money in P* and the rest in Treasury
bonds (this is known as a lending portfolio, because
the rest of the investor’s money is lent to the federal
government)
If the investor is more aggressive, he or she will
leverage up his or her holdings and invest in P* on
margin by borrowing at the risk-free rate (this is
known as a borrowing portfolio)
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Portfolio Possibilities Combining the Risk-Free Asset
and Risky Portfolios on the Efficient Frontier
E(R port )
P*
RFR
E( port )
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Capital Market Theory:
An Overview
• Question: What are the general implications for
security prices if investors act the way Markowitz
portfolio theory and portfolio separation theory
say they should? If such theories hold, what
would equilibrium in the capital markets entail?
• Capital market theory extends portfolio theory and
develops a model for pricing all risky assets
• The capital asset pricing model (CAPM) will
allow you to determine the required rate of return
(for use in discounting future cash flows) for any
risky asset
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Assumptions of
Capital Market Theory
1. All investors are Markowitz mean-variance
efficient investors who want to target points
on the efficient frontier.
– Also, they include all investable assets in their
estimation of the efficient frontier
– Not necessarily a realistic assumption!
• Most investors do not use Markowitz optimization
• Of those who do, they typically optimize w.r.t. alpha
and tracking error rather than mean and variance
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Assumptions of
Capital Market Theory
2. Investors can borrow or lend any amount of
money at the risk-free rate of return (RFR).
– This means that the conditions of portfolio separation
theory will hold, at least at the individual level.
– Note: it is always possible to lend money at the riskfree rate by buying securities such as T-bills, but (unless
you’re the government) it is not usually possible to
borrow at this risk-free rate.
– However, assuming a higher borrowing rate does not
change the general results.
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Assumptions of
Capital Market Theory
3. All investors have homogeneous expectations;
that is, investors have identical estimates for the
probability distributions of future rates of return.
– This implies that all investors will estimate the efficient
frontier to be in the exact same location, and the
optimal portfolio P* (i.e., the investment decision from
portfolio separation theory) will be the same for all
investors.
– This assumption can be relaxed, and as long as the
differences in expectations are not vast their effects will
be minor.
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Assumptions of
Capital Market Theory
4. All investors have the same one-period time
horizon such as one-month, six months, or one
year.
– Markowitz portfolio theory is a single-period model;
making the model dynamic requires additional
constraints, such as on portfolio turnover, in calculating
the efficient frontier.
– With regard to capital market theory, differences in
investors’ time horizons would require investors to
derive risk measures and risk-free assets that are
consistent with their time horizons.
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Assumptions of
Capital Market Theory
5. Capital markets are “frictionless,” i.e.:
– No taxes – true for many classes of investors
– No transactions costs – becoming more true over time,
but still can be an impediment
– Fixed supply of stocks – i.e., don’t have to worry about
incorporating IPO shares into the analysis
– Infinitely divisible supply of stocks – this assumption
allows us to discuss investment alternatives as
continuous curves. Changing it would have little
impact on the theory, and it is also becoming more true
over time.
– Information is costless and available to all investors
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Assumptions of
Capital Market Theory
6. Capital markets are in equilibrium.
– This means that we begin with all investments
properly priced in line with their risk levels.
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Assumptions of
Capital Market Theory
• Note that some of these assumptions are
unrealistic,
• But relaxing many of these assumptions would
have only minor influence on the model and
would not change its main implications or
conclusions;
• Moreover, a theory can be useful for helping to
explain and predict behavior, even if not all of its
assumptions hold true (e.g., many useful models in
physics assume the absence of any friction).
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Derivation of the
Capital Market Line
a) Homogeneous expectations (together with the
same investment horizon) means that investors
all face the same estimated efficient frontier.
b) Existence of a risk-free asset means that each
investor can mix the riskless asset with a risky
portfolio.
c) (a) and (b) imply that all investors choose the
same risky portfolio to hold in combination with
the risk-free asset.
•
Call this portfolio P*
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Derivation of the
Capital Market Line
d) In order to have equilibrium (supply = demand),
all risky assets must be included in P*. If this
were not the case, then some assets would not be
held at all.
e) In view of (d), the optimal portfolio P* is called
the Market Portfolio (M)
•
Value-weighted portfolio, with E(RM) and M
f) The line connecting RFR with M now represents
the market-wide opportunities for expected
return and risk. Thus, this line is called the:
•
Capital Market Line (CML)
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The Capital Market Line (CML)
E(R port )
M
Figure 9.2
RFR
 port
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The Market Portfolio
Because portfolio M lies at the point of
tangency, it has the highest portfolio
possibility line
Everybody will want to invest in Portfolio M
and borrow or lend to be somewhere on the
CML
Therefore this portfolio must include ALL
RISKY ASSETS
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The Market Portfolio
Because the market is in equilibrium, all risky
assets are included in this portfolio in
proportion to their market value
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The Market Portfolio
Because it contains all risky assets, it is a
completely diversified portfolio (once you
already own everything, you can’t diversify
any more!), which means that all the unique
risk of individual assets (unsystematic risk)
is diversified away (all the risk that’s left
over is, by definition, systematic risk)
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Systematic Risk
Only systematic risk remains in the market
portfolio
Systematic risk is the variability in all risky
assets caused by macroeconomic variables
Systematic risk can be measured by the
standard deviation of returns of the market
portfolio and can (and does) change over
time
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Examples of Macroeconomic
Factors that Affect Systematic Risk
• Variability in growth of money supply
• Interest rate volatility
• Variability in:
industrial production
corporate earnings
and cash flow
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The Market Portfolio and
How to Measure Diversification
All portfolios on the CML are perfectly positively
correlated with each other and with the completely
diversified market Portfolio M
A completely diversified portfolio would have a
correlation with the market portfolio of +1.00
Thus, can use regression R2 of portfolio’s returns
regressed on the “market” portfolio’s returns as a
measure of the extent of diversification
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The Capital Market Line (CML)
•
•
•
•
Describes the risk / return relationship for welldiversified portfolios (idiosyncratic risk has been
diversified away).
Portfolio standard deviation (Q) is the relevant
measure of risk, and the portfolio’s expected
return (E(RQ)) will be a direct linear function of
its risk:
 E  RM   R F 
E RQ   RF  
Q

M


To obtain higher expected returns, must accept
higher risk.
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The Security Market Line (SML)
•
•
Key Question: What is the relevant measure of
risk for an individual security when it is held as
part of a well diversified portfolio (i.e., the
Market portfolio, M)?
The Security Market Line describes the risk /
return relationship for an individual security.
– Also applies to non-diversified portfolios or any other
holding for which the total risk may include some
diversifiable or idiosyncratic risk.
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The Security Market Line (SML)
• The relevant risk measure for an individual risky
asset is its covariance with the market portfolio
(Covi,m)
• This is the risk measure for the SML, which
describes the relationship between risk and
expected return for all portfolios, whether welldiversified or not, as well as for all securities
• The return for the market portfolio should be
consistent with its own risk, which is the
covariance of the market with itself - or its
2
variance:

m
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Figure 9.5
Graph of Security Market Line
(SML)
E(R i )
SML
Rm
RFR

2
m
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Cov im
The Security Market Line (SML)
The equation for the risk-return line is
R M - RFR
E(R i )  RFR 
(Cov i,M )
2
M
Cov i,M
 RFR 
(R M - RFR)
2
M
Cov i,M as beta (  )
We then redefine
i
2
M
E(R i )  RFR   i (R M - RFR)
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Figure 9.6
Graph of SML with
Normalized Systematic Risk
E(R i )
SML
Rm
Negative
Beta
RFR
0
1.0
Beta( iM / )
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2
M
Determining the Expected
Rate of Return for a Risky Asset
E(R i )  RFR   i (R M - RFR)
The expected rate of return of a risk asset is determined
by the RFR plus a risk premium for the individual asset
The risk premium is determined by the systematic risk of
the asset () and the prevailing market risk premium
(RM-RFR)
In equilibrium, to obtain higher expected returns,
investors must accept higher “covariance” risk
In equilibrium, investors receive no compensation for
diversifiable (non-systematic or idiosyncratic) risk
Q: What is market is not in equilibrium?
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The Capital Asset Pricing Model:
Expected Return and Risk
• CAPM indicates what should be the expected or
required rates of return on risky assets
• This helps to value an asset by providing an
appropriate discount rate to use in dividend
valuation models
• Conversely, you can compare an estimated rate of
return to the required rate of return implied by
CAPM – over / under valued ?
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Determining the Required
Rate of Return for a Risky Asset
Stock
Beta
A
B
C
D
E
0.70
1.00
1.15
1.40
-0.30
RFR = 6% (0.06)
RM = 12% (0.12)
Implied market risk premium = 6% (0.06)
Assume:
E(R i )  RFR   i (R M - RFR)
E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%
E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%
E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%
E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%
E(RE) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%
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Determining the Required
Rate of Return for a Risky Asset
In equilibrium, all assets and all portfolios of assets
should plot on the SML
Any security with an estimated return that plots above
the SML is underpriced
Any security with an estimated return that plots below
the SML is overpriced
A superior investor must derive value estimates for
assets that are consistently superior to the consensus
market evaluation to earn better risk-adjusted rates
of return than the average investor
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Identifying Undervalued and
Overvalued Assets
Compare the required rate of return to the
expected rate of return for a specific risky
asset using the SML over a specific
investment horizon to determine if it is an
appropriate investment
Independent estimates of return for the
securities provide price and dividend
outlooks
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Price, Dividend, and
Rate of Return Estimates
Table 9.1
Current Price
Stock
A
B
C
D
E
(Pi )
25
40
33
64
50
Expected Dividend
Expected Price (Pt+1 )
27
42
39
65
54
(Dt+1 )
0.50
0.50
1.00
1.10
0.00
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Expected Future Rate
of Return (Percent)
10.0 %
6.2
21.2
3.3
8.0
Comparison of Required Rate of Return
to Estimated Rate of Return
Table 9.2
Stock
Beta
A
B
C
D
E
0.70
1.00
1.15
1.40
-0.30
Required Return
Estimated Return
E(Ri )
Minus E(R i )
Estimated Return
10.2%
12.0%
12.9%
14.4%
4.2%
10.0
6.2
21.2
3.3
8.0
-0.2
-5.8
8.3
-11.1
3.8
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Evaluation
Properly Valued
Overvalued
Undervalued
Overvalued
Undervalued
Plot of Estimated Returns
E(R i ) on SML Graph
Figure 9.7
E
-.40 -.20
Rm
.22
.20
.18
.16
.14
.12
Rm
.10
.08
.06
.04
.02
0
C
SML
A
B
D
.20
.40
.60
.80
1.0 1.20
1.40 1.60 1.80
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Beta
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Modeling Risk & Return
Part Two:
• Extensions,
• Testing, and
• The Arbitrage Pricing Theory (APT)
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Relaxing the Assumptions
of the CAPM
•
•
CAPM assumption: all investors can borrow or
lend at the risk-free rate – unrealistic
Two possible alternatives:
1. Differential borrowing and lending rates
•
•
•
Unlimited lending at risk-free rate
Borrowing at higher rate
Leads to “bent” Capital Market Line
2. Zero-Beta CAPM
•
•
Eliminates theoretical need for risk-free asset
Leads to same form for SML but with a shallower slope
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Differential Borrowing and Lending Rates
(Cost of Borrowing higher than Cost of Lending)
Figure 10.1
E(R)
G
K
F
Rb
RFR
Risk (standard deviation )
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Zero-Beta CAPM
• Zero-beta portfolio: create a portfolio that is
uncorrelated to the market (beta 0)
– The return of the zero-beta portfolio may differ from
the risk-free rate
• Any combination of portfolios on the efficient
frontier will be on the frontier
• Any efficient portfolio will have associated with it
a zero-beta portfolio
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Implications of
Black’s Zero-beta model
• The expected return of any security can be expressed as
a linear relationship of any two efficient portfolios
E(Ri) = E(Rz) + i[E(Rm) - E(Rz)]
• If original CAPM defines the relationship between risk
and return, then the return on the zero-beta portfolio
should equal RF
– Typically, in real world, RFR < E(RZ), so the zero-beta SML
would be less steep than the original SML
– Consistent with empirical results of tests of original CAPM
• To test directly - identify a market portfolio and solve
for the return of a zero-beta portfolio
– Leads to less consistent results
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Security Market Line
With A Zero-Beta Portfolio
Figure 10.2
E(R)
SML
M
E(Rm)
E(Rm) - E(Rz)
E(Rz)
0.0
1.0
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i
Relaxing the Assumptions
of the CAPM
• Another assumption of CAPM – zero transactions
costs
• Existence of transaction costs:
– affect mispricing corrections
– affect diversification
– Leads to a “security market ‘band’” in place of the
security market line
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Security Market Line
With Transaction Costs
Figure 10.3
E(R)
SML
E(Rm)
E(RFR)
or
E(Rz)
0.0
1.0
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i
Relaxing the Assumptions
of the CAPM
• Heterogenous expectations
– If all investors have different expectations about risk and
return, each investor would have a different idea about the
position and composition of the efficient frontier, hence would
have a different idea about the location and composition of the
tangency portfolio, M
– Hence, each would have a unique CML and/or SML, and the
composite graph would be a band of lines with a breadth
determined by the divergence of expectations
– Since each investor would have a different idea about where
the SML lies, each would also have unique conclusions about
which securities are under- and which are over-valued
– Also note that small differences in initial expectations can lead
to vastly different conclusions in this regard!
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Relaxing the Assumptions
of the CAPM
• Planning periods
– CAPM is a one period model, and the period
employed should be the planning period for the
individual investor, which will vary by individual,
affecting both the CML and the SML
• Taxes
– Tax rates affect returns
– Tax rates differ between individuals and institutions
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Empirical Testing of CAPM
Key questions asked:
• How stable is the measure of systematic
risk (beta)?
• Is there a positive linear relationship as
hypothesized between beta and the rate
of return on risky assets?
• How well do returns conform to the SML
equation?
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Empirical Testing of CAPM
• Beta is not stable for individual stocks over short
periods of time (52 weeks or less)
– Need to estimate over 3 or more years (5 typically used)
• Stability increases significantly for portfolios
• The larger the portfolio and the longer the period,
the more stable the beta of the portfolio
• Betas tend to regress toward the mean ( = 1.0)
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Empirical Testing of CAPM
• In general, the empirical evidence regarding
CAPM has been mixed.
• Empirically, the most serious challenge to
CAPM was provided by Fama and French
(discussed in the Introductory lecture)
• Conceptually, the most serious challenge is
provided by Roll’s Critique
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The Market Portfolio:
Theory Versus Practice
• Impossible to test full market
• Portfolio used as market proxy may be
correlated to true market portfolio
• Benchmark error – 2 possible effects:
– Beta will be wrong
– SML will be wrong
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Criticism of CAPM by Richard Roll
• Key limit on potential tests of CAPM:
– Ultimately, the only testable implication from
CAPM is whether the market portfolio is
efficient (i.e., whether it lies on the efficient
frontier)
• Range of SML’s - infinite number of
possible SML’s, each of which produces a
unique estimate of beta
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Criticism of CAPM by Richard Roll
• Market efficiency effects - substituting a
proxy, such as the S&P 500, creates two
problems
– Proxy does not represent the true market
portfolio
– Even if the proxy is not efficient, the market
portfolio might be (or vice versa)
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Criticism of CAPM by Richard Roll
• Conflicts between proxies - different substitutes
may be highly correlated even though some may
be efficient and others are not, which can lead to
different conclusions regarding beta risk/return
relationships
• So, ultimately, CAPM is not testable and cannot
be verified, so it must be used with great caution
• Stephen Ross devised an alternative way to look at
asset pricing that uses fewer assumptions – the
Arbitrage Pricing Theory, or APT
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Assumptions of
Arbitrage Pricing Theory (APT)
1. Capital markets are perfectly competitive
2. Investors always prefer more wealth to less
wealth with certainty
3. The stochastic process generating asset
returns can be presented as K factor model
(to be described)
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Assumptions of CAPM
That Were Not Required by APT
APT does not assume:
• A market portfolio that contains all risky
assets, and is mean-variance efficient
• Normally distributed security returns
• Quadratic utility function
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
Rt  Et  bi1 i  bi 2 i  ...  bik k   i
For i = 1 to N where:
Ri = return on asset i during a specified time period
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Arbitrage Pricing Theory (APT)
Rt  Et  bi1 i  bi 2 i  ...  bik k   i
For i = 1 to N where:
Ri = return on asset i during a specified time period
Ei = expected return for asset i
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
Rt  Et  bi1 i  bi 2 i  ...  bik k   i
For i = 1 to N where:
Ri = return on asset i during a specified time period
Ei = expected return for asset i
bik = reaction in asset i’s returns to movements in a common
factor
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
Rt  Et  bi1 i  bi 2 i  ...  bik k   i
For i = 1 to N where:
Ri = return on asset i during a specified time period
Ei = expected return for asset i
bik = reaction in asset i’s returns to movements in a common
factor
 k = a common factor with a zero mean that influences the
returns on all assets
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
Rt  Et  bi1 i  bi 2 i  ...  bik k   i
For i = 1 to N where:
Ri = return on asset i during a specified time period
Ei = expected return for asset i
= reaction in asset i’s returns to movements in a common
bik factor
= a common factor with a zero mean that influences the
 k returns on all assets
= a unique effect on asset i’s return that, by assumption, is
 i completely diversifiable in large portfolios and has a
mean of zero
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
Rt  Et  bi1 i  bi 2 i  ...  bik k   i
For i = 1 to N where:
Ri = return on asset i during a specified time period
Ei = expected return for asset i
= reaction in asset i’s returns to movements in a common
bik factor
 k = a common factor with a zero mean that influences the
returns on all assets
 i = a unique effect on asset i’s return that, by assumption, is
completely diversifiable in large portfolios and has a
mean of zero
N = number of assets
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
 kEconomic factors expected to have an impact on
all assets:
–
–
–
–
–
–
–
Growth rate of GDP
The level of interest rates
Inflation
Various yield spreads
Changes in oil prices
Major political upheavals
Etc.
Or, alternatively….
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
 kMarket plus Sector factors expected to have an
impact on all assets:
– General market factor, plus
– Sector factors, such as
•
•
•
•
Utilities
Transportation
Financial
Etc.
And potentially many more alternatives,
In contrast with CAPM insistence that only beta
and the market portfolio factor are relevant.
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
 kMultiple factors expected to have an impact on
all assets:
–
–
–
–
–
Inflation
Growth in GNP
Major political upheavals
Changes in interest rates
And potentially many more….
Contrast with CAPM insistence that only beta is
relevant
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
bik determine how each asset reacts to this common
factor
Each asset may be affected by growth in GNP, but
the effects will differ
In application of the theory, the factors are not
identified
Similar to the CAPM, the unique effects (the i’s) are
independent and will be diversified away in a
large portfolio
Caveat: impossible to completely diversify away
unique risk if all the relevant systematic risk
factorsCopyright
are ©not
correctly identified
2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
• APT assumes that, in equilibrium, the return
on a zero-investment, zero-systematic-risk
portfolio is zero when the unique effects are
diversified away
• The expected return on any asset i (Ei) can
be expressed as:
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Arbitrage Pricing Theory (APT)
Ei  0  1bi1  2bi 2  ...  k bik
where:
0= the expected return on an asset with zero systematic
risk where
0
0
j = the risk premium related to each of the common
factors - for example the risk premium related to
interest rate risk
j
j 0
bij = the pricing relationship between the risk premium and
asset i - that is how responsive asset i is to this common
factor j


 E
 E E
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Example of Two Stocks
and a Two-Factor Model
in the rate of inflation. The risk premium
1= changes
related to this factor is 1 percent for every 1 percent
change in the rate
(1  .01)
2= percent growth in real GNP. The average risk premium
related to this factor is 2 percent for every 1 percent
change in the rate
(  .02)
2
on a zero-systematic-risk asset (zero
3= thebeta:rateb of=0)return
is 3 percent
oj
(3  .03)
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Example of Two Stocks
and a Two-Factor Model
bx1= the response of asset X to changes in the rate of inflation
is 0.50
(bx1  .50)
by1= the response of asset Y to changes in the rate of inflation
is 2.00
(by1  .50)
bx 2 = the response of asset X to changes in the growth rate of
real GNP is 1.50
(bx 2  1.50)
b y 2= the response of asset Y to changes in the growth rate of
real GNP is 1.75
(by 2  1.75)
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Example of Two Stocks
and a Two-Factor Model
Ei  0  1bi1  2bi 2
= .03 + (.01)bi1
+ (.02)bi2
Ex = .03 + (.01)(0.50) + (.02)(1.50)
= .065 = 6.5%
Ey = .03 + (.01)(2.00) + (.02)(1.75)
= .085 = 8.5%
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Roll-Ross Study of APT
1. Estimate the expected returns and the factor
coefficients from time-series data on
individual asset returns
2. Use these estimates to test the basic crosssectional pricing conclusion implied by the
APT
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Empirical Tests of the APT
• Studies by Roll and Ross and by Chen
support APT by explaining different rates of
return with some better results than CAPM
• But, Dhrymes and Shanken question the
usefulness of APT because it was not
possible to identify the factors
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Estimating Risk & Return
• The Market Model, the Characteristic Line, and
Beta
• Estimating Multiple Factor Models
• Haugen’s comments
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Calculating Systematic Risk:
The Characteristic Line
The systematic risk input of an individual asset is derived from
a regression model, referred to as the asset’s characteristic
line with the model portfolio:
R i,t   i   i R M, t   i,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   iM / M2
i , t  the random error term
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Scatter Plot of Rates of Return
The characteristic
Ri
line is the regression
line of the best fit
through a scatter plot
of rates of return
Figure 9.8
RM
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The Impact of the Time Interval
Number of observations and time interval used in the
regression vary
Value Line Investment Services (VL) uses weekly
rates of return over five years (260 obs.)
Merrill Lynch, Pierce, Fenner & Smith (ML) uses
monthly return over five years (60 obs.)
There is no “correct” interval for analysis
Weak relationship between VL & ML betas due to
difference in intervals used
Interval effect impacts smaller firms more
Copyright © 2000 by Harcourt, Inc. All rights reserved.
The Effect of the Market Proxy
The market portfolio of all risky assets must
be represented in computing an asset’s
characteristic line
Standard & Poor’s 500 Composite Index is
most often used
– Large proportion of the total market value of
U.S. stocks
– Value weighted series
Copyright © 2000 by Harcourt, Inc. All rights reserved.
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
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Comparisons of Beta Estimates
• Different estimates of beta for a stock vary
typically in data used
• Value Line estimates use 260 weekly observations
and compare to the NYSE Composite Index
• Merrill Lynch estimates use 60 monthly
observations and compare to the S&P 500
• ML  0.127 + 0.879VL
• Securities market value affects the size and
direction of the interval affect
• Trading volume also affects the beta estimates
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Comparing Market Proxies
Calculating Beta for Coca-Cola using Morgan
Stanley (M-S) World Equity Index and S&P 500
as market proxies results in a 1.27 beta when
compared with the M-S index, but a 1.01 beta
compared to the S&P 500
The difference is exaggerated by the small sample
size (12 months) used, but selecting the market
proxy can make a significant difference
Here are the computations from page 303:
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Computation of Beta of Coca-Cola
Table 9.3
with Selected Indexes
S&P
M-S
Return
S&P M-S
500
World
500
Index
Date
Dec-97
Jan-98
Feb-98
Mar-98
Apr-98
May-98
Jun-98
Jul-98
Aug-98
Sep-98
Oct-98
Nov-98
Dec-98
World
S&P 500
CocaCola
RS&P - E(RS&P ) RM-S - E(RM-S) RKO - E(RKO )
933.60
961.50
1.02
2.99
(2.91)
1025.30
7.04
6.64
5.98
1067.40
4.99
4.11
8.47
1059.30
0.91
-0.76
1.93
1061.80
-1.88
0.24
3.29
1085.70
3.94
2.25
9.09
1082.70
-1.16
-0.28
(5.85)
937.10
-14.52 -13.45 (19.10)
952.40
6.16
1.63 (11.52)
1032.20
8.03
8.38
17.25
1097.60
5.91
6.34
3.70
1150.00
5.64
4.77
(4.37)
Average
2.17
1.90
0.50
Standard Deviation
6.18
5.63
9.87
CovKO ,S&P= 448.74/ 12 = 37.39
VarS&P = St.Dev.S&P2 = 38.19
CovKO ,M-S =
970.43
980.28
1049.34
1101.75
1111.75
1090.82
1133.84
1120.67
957.98
1017.01
1098.67
1163.63
1229.23
460.93/ 12 = 38.41
Correlation coef.KO ,S&P= 0.61
a
VarM-S = St.Dev.M-S 2 =
Column (4) is equal to column (1) multiplied by column (3)
M-S World Coca-Cola
(1)
(2)
-1.16
4.87
2.82
-1.27
-4.06
1.77
-3.34
-16.69
3.99
5.86
3.74
3.46
1.08
4.73
2.20
-2.66
-1.67
0.35
-2.18
-15.35
-0.27
6.47
4.43
2.87
(3)
-3.40
5.49
7.97
1.43
2.80
8.59
-6.35
-19.60
-12.01
16.75
3.20
-4.87
Total =
( 4 )a
( 5 )b
3.94
26.73
22.49
-1.81
-11.35
15.21
21.17
327.10
-47.91
98.07
11.97
-16.87
448.74
-3.69
25.96
17.55
-3.82
-4.67
2.98
13.84
300.87
3.27
108.43
14.19
-13.97
460.93
BetaKO,S&P= 0.98 AlphaKO ,S&P= -1.63
BetaKO,M-S = 1.21 AlphaKO ,M-S = -1.81
31.70
Correlation coef.KO ,M-S = 0.69
b
Column (5) is equal to column (2) multiplied by column (3)
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Estimating Risk & Return
• The complications and range of choices make
things difficult with a single-index model
• For multiple-factor models, there is even
greater complexity
Copyright © 2000 by Harcourt, Inc. All rights reserved.
ESTIMATING FACTOR
MODELS
• THREE METHODS
– TIME-SERIES APPROACH
– CROSS-SECTIONAL APPROACH
– FACTOR-ANALYTIC APPROACH
ESTIMATING FACTOR
MODELS
• TIME-SERIES APPROACH
– BEGINNING ASSUMPTIONS:
•
investor knows in advance of the factors that influence a
security's returns
– e.g., the return on the S&P 500 Index, or
– the growth rate in GDP
•
•
Regress the stocks’ historical returns against the historical
values of these time series
the information may be gained from an economic analysis of
the firm
ESTIMATING FACTOR
MODELS
• CROSS-SECTIONAL APPROACH
– BEGINNING ASSUMPTION
• Identify Attributes:
estimates of a securities
sensitivities to certain factors
– e.g., the firm’s size,
– beta, or
– M/B ratio
• estimate attributes in a particular period of time
• repeat over multiple time periods to estimate the
factor’s standard deviations and correlations
ESTIMATING FACTOR
MODELS
• FACTOR-ANALYTIC APPROACH
– BEGINNING ASSUMPTIONS:
• neither factor values nor securities attributes are
know
• uses factor analysis approach
• take the returns over many time periods from a
sample to identify one or more significant factors
generating covariances
The Internet
Investments Online
www.valueline.com
www.barra.com
www.stanford.edu/~wfsharpe.com
Copyright © 2000 by Harcourt, Inc. All rights reserved.
The Internet
Investments Online
www.barra.com
www.wsharpe.com
www.cob.ohio-state.edu/~fin/journal.jof.htm
www3.oup.co.uk/revfin/scope
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Future topics
• The Efficient Markets Hypotheses – basic
theory and future directions:
– The Market as a Complex Adaptive System
• “Shift Happens” - Mauboussin
– The New Finance
• “The Wrong 20-Yard Line” - Haugen
• Introduction to Behavioral Finance
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Copyright © 2000 by Harcourt, Inc. All rights reserved.