BM410: Investments
Capital Asset Pricing
Theory and APT
or
How do you value stocks?
Objectives
A. Review and solve problems using the CAL,
MPT, and the Single Index model
B. Understand the implications of capital asset
pricing theory and the CAPM to compute
security risk premiums
C. Understand the arbitrage pricing theory and
how it works
A. Solve problems using the CAL, CML,
MPT and Single Index Models
Capital Market Line Review
•
You estimate that a passive portfolio invested to
mimic the S&P 500 (an index fund) has an expected
return of 13% with a standard deviation of 25%.
Your portfolio has an expected return of 17% with a
standard deviation of 27%. With the risk-free rate at
7%, draw the CML and your fund’s CAL on an
expected return-standard deviation diagram.
A. What is the slope of the CML? Your CAL?
B. Characterize in one short paragraph the
advantage of your fund over the passive fund.
Answer
Slope of the CML = (13-7)/25 = .24
Slope of your CAL = (17-7)/27 = .37
Your Fund
17%
13%
Index Fund
7%
10%
20%
30%
b. Your fund allows an investor a higher mean for any
given standard deviation than the passive strategy.
MPT Review

Suppose that for some reason you are required to
invest 50% of your portfolio in bonds (sb = 12%,
E(rb) = 10%) and 50% in stocks (ss = 25%, E(rs) =
17%).
A. If the standard deviation of your portfolio is
15%, what must be the correlation coefficient
between stock and bond returns?
B. What is the expected rate of return on your
portfolio?
C. Now suppose that the correlation between stock
and bond returns is 0.22 but that you are free to
choose whatever portfolio proportions you desire.
Are you likely to be better or worse off that you
were in part a?
Answer

•
•
A.
sp2 = w12s12 + w22s22 + 2W1W2 (r1,2s1s2 )
(.15) 2 =[(.512.1212) +(.522.2522) + 2(.51.52)*(.121.252 )] * r1,2
r1,2 = .2183 or 21.8% (take my word for this)
B. E(rp) = (.5 * .10) + (.5 * .17) = 13.5%
C. While the current correlation is slightly lower than 22%,
this implies slightly greater benefits from diversification.
However, the 50% bond constraint represents a cost since
you cannot choose your optimal risk-return tradeoff for
your risk level. Unless you would choose to have 50%
bonds anyway, you are better off with the slightly higher
correlation and the ability to choose your own portfolio
weights.
Factor Review
 Investors expect the market rate of return to be
10%. The expected rate of return on the stock
with a beta of 1.2 is currently 12%.
If the market return this year turns out to be
8%, how would you review/change your
expectations of the rate of return on the
stock?
Answer
 The expected return on the stock would be
your beta (1.2) times the market return or:
1.2 * 8% = 9.6%
 Likewise, you could also determine how much
the return would decrease by multiplying the
beta times the change in the market return or:
1.2 * (8%-10%) = -2.4% + 12% =
9.6%
Questions
Any questions of Capital Allocation Lines,
Modern Portfolio Theory, or Single Index
Models?
B. Implications of Capital Market
Theory and CAPM
 What have we done this far?
• We have been concerned with how an individual or
institution would select an optimum portfolio.
• If investors act as we think, we should be able to
determine how investors will behave, and how
prices at which markets will clear are set
• This market clearing of prices and returns has
resulted in the development of so-called general
equilibrium models
• These models allow us to determine the risk for
any asset and the relationship between expected
return and risk for any asset when the markets
are in equilibrium, i.e. balance or constant state
Capital Asset Pricing Theory
 What is capital asset pricing theory?
• It is the theory behind the pricing of assets which
takes into account the risk and return characteristics
of the asset and the market
 What is the Capital Asset Pricing Model?
• It is an equilibrium model (i.e., a constant state
model) that underlies all modern financial theory
• It provides a precise prediction between the
relationship between the risk of an asset and its
expected return when the market is in
equilibrium
• With this model, we can identify mis-pricing of
securities (in the long-run)
CAPM (continued)
 Why is it important?
• It provides a benchmark rate of return for
evaluating possible investments, and identifying
potential mis-pricing of investments
• For example, an analyst might want to know
whether the expected return she forecast is more
or less than its “fair” market return.
• It helps us make an “educated” guess as to the
expected return on assets that have not yet been
traded in the marketplace
• For example, how do we price an initial public
offering?
CAPM (continued)
 How was it derived?
• Derived using principles of diversification with
very simplified (i.e. somewhat unrealistic)
assumptions
 Does it work, i.e. withstand empirical tests in real life?
• Not totally
• But it does offer insights that are important and
its accuracy may be sufficient for some
applications
 Do we use it?
• Yes, but with knowledge of its limitations
CAPM Assumptions
 What does the model assume (some are unrealistic)?
• Individual investors are price takers (cannot affect
prices)
•
•
•
•
•
•
Single-period investment horizon (an its identical for all)
Investments are limited to traded financial assets
No taxes, and no transaction costs (costless trading)
Information is costless and available to all investors
Investors are rational mean-variance optimizers
Investors analyze information in the same way, and
have the same view, i.e., homogeneous expectations
Resulting Equilibrium Conditions
 Based on the previous assumptions:
• All investors will hold the same portfolio for risky
assets – the market portfolio (M)
• The market portfolio (M) contains all securities and
the proportion of each security is its market value as
a percentage of total market value
• The risk premium on the market depends on the
average risk aversion of all market participants
• The risk premium on an individual security is a
function of its covariance (correlation and ss sm)
with the market
Capital Market Line
E(r)
M = Market portfolio rf = Risk free rate
E(rM) - rf= Market risk premium
[E(rM) - rf]/sM= Market price of risk
E(rM)
M
CML
The efficient frontier without
lending or borrowing
rf
sm
s
Expected Return and Risk
of Individual Securities
 What does this imply?
• The risk premium on individual securities is
a function of the individual security’s
contribution to the risk of the market
portfolio
• Individual security’s risk premium is a
function of the covariance of returns with
the assets that make up the market portfolio
CAPM Key Thoughts
 Key statements:
• Portfolio risk is what matters to investors, and
portfolio risk is what governs the risk premiums
they demand
• Non-systematic, or diversifiable risk can be reduced
through diversification.
• Investors need to be compensated for bearing only
non-systematic risk (risk that cannot be diversified
away)
• The contribution of a security to the risk of a
portfolio depends only on its systematic risk, as
measured by beta. So the risk premium of the asset
is proportional to its beta. (ß = [COV(ri,rm)] / sm2)
Expected Return – Beta Relationship
Expected return - beta relationship of CAPM:
E(rM) - rf
1.0
=
E(rs) - rf
bs
In other words, the expected rate of return of an asset
exceeds the risk-free rate by a risk premium equal to the
asset’s systematic risk (its beta) times the risk premium
of the market portfolio. This leads to the familiar rearrangement of terms to give (memorize this):
E(rs) = rf + bs [E(rM) - rf ]
The Security Market Line
E(r)
 Notice that instead of using standard
deviation, the Security Market Line uses Beta
 SML Relationships
ß = [COV(ri,rm)] / sm2
SML
Slope SML = E(rm) – rf = market risk
premium
E(rM)
rf
SML = rf + ß[E(rm) - rf]
ß M = 1.0
ß
Differences Between the SML and CML
 What are the differences?
• The CML graphs risk premiums of efficient
portfolios , i.e. complete portfolios made up of the
risk portfolio and risk-free asset, as a function of
standard deviation
• The SML graphs individual asset risk premiums as
a function of asset risk.
• The relevant measure of risk for individual
assets is not standard deviation; rather, it is beta
• The SML is also valid for portfolios
Example: SML Calculations
Put the following data on the SML. Are
they in equilibrium?
Market data: E(rm) - rf = .08
rf = .03
Asset data:
bx = 1.25
by = .60
• Calculations:
bx = 1.25 so E(r) on x =
E(rx) = .03 + 1.25(.08) = .13 or 13%
by = .60 so E(r) on y =
E(ry) = .03 + .6(.08) = .078 or 7.8%
Graph of Sample Calculations
E(r)
SML
Rx=13%
Rm=11%
Ry=7.8%
3%
.08
They are in equilibrium
.6 1.0 1.25
ßy ßm ßx
ß
Disequilibrium Example
 Suppose a security with a beta of 1.25 is
offering expected return of 15%
• According to SML, it should be 13%
• Under priced: offering too high of a rate of
return for its level of risk. Investors
therefore would:
• Buy the security, which would increase
demand, which would increase the price,
which would decrease the return until it
came back into line.
Disequilibrium Example
E(r)
The return is above the
SML, so you would buy it
15%
Rm=11%
SML
As more people bought
the security, it would
push the price up,
which would bring the
return down to the line.
rf=3%
1.0 1.25
ß
CAPM and Index Models
 CAPM Problems
• It relies on a theoretical market portfolio which
includes all assets
• It deals with expected returns
 To get away from these problems and make it testable,
we change it and use an Index model which:
• Uses an actual index, i.e. the S&P 500 for
measurement
• Uses realized, not expected returns
 Now the Index model is testable
The Index Model
 With the Index model, we can:
• Specify a way to measure the factor that affects
returns (the return of the Index)
• Separate the rate of return on a security into its
macro (systematic) and micro (firm-specific)
components
 Components
ά = excess return if market factor is zero
ßiRm = component of returns due to movements in the
overall market
ei = component attributable to company specific
events
Ri = a i + ßiRm + ei
 (Notice the similarity to the Single Index model discussed earlier)
Security Characteristic Line
Excess Returns (i)
SCL
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Excess returns
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on market index
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.. . . .
Plot of a company’s excess return as a
function of the excess return of the market
Ri = a i + ßiRm + ei
Does the CAPM hold?
 There is much evidence that supports the
CAPM
• There is also evidence that does not support the
CAPM
 Is the CAPM useful?
• Yes. Return and risk are linearly related for
securities and portfolios over long periods of time
• Yes. Investors are compensated for taking on added
market risk, but not diversifiable risk
 Perhaps instead of determining whether the CAPM is
true or not, we might ask: Are there better models?
Questions
Any questions on capital asset pricing
and the Capital Asset Pricing Model?
CAPM Problem
 Suppose the risk premium on the market portfolio is
9%, and we estimate the beta of Dell as bs = 1.3. The
risk premium predicted for the stock is therefore 1.3
times the market risk premium of 9% or 11.7%. The
expected return on Dell is the risk-free rate plus the
risk premium. For example, if the T-bill rate were 5%,
the expected return of Dell would be 5% +(1.3 * 9%)
= 16.7%.
a. If the estimate of the beta of Dell were only 1.2,
what would be Dells required risk premium?
b. If the market risk premium were only 8% and Dell’s
beta was 1.3, what would be Dell’s risk premium?
Answer

a. If Dell’s beta was 1.2 the required risk premium
would be (remember the risk premium is the
expected return less the risk-free rate):
E(rs) = rf + bs [E(rM) - rf ] or the expected return on
Dell = 5% + 1.2 (9%) = 15.8%
Dell’s risk premium (over the risk free rate) =
15.8% - 5% = 10.8%
 b. If the market risk premium was 8%:
E(rs) = rf + bs [E(rM) - rf ]
E(r) of Dell = 5% + 1.3 (8%) = 15.4%
Dell’s new risk premium is 15.4 – 5% = 10.4%
C. Understand Arbitrage Pricing Theory
(APT) and How it Works
What is arbitrage?
• The exploitation of security mis-pricing to earn
risk-free economic profits
• It rises if an investor can construct a zero
investment portfolio (with a zero net investment
position netting out buys and sells) with a sure
profit
 Should arbitrage exist?
• In efficient markets (and in CAPM theory),
profitable arbitrage opportunities will quickly
disappear as more investors try to take advantage of
them
Arbitrage Pricing Theory (APT) (continued)
 What is APT based on?
• It is a variant of the CAPM, and is an attempt to
move away from the mean-variance efficient
portfolios (the calculation problem)
• Ross instead calculated relationships among
expected returns that would rule out riskless profits
by any investor in a well-functioning capital market
 What is it?
• It is a another theory of risk and return similar to the
CAPM.
• It is based on the law of one price: two items that
are the same can’t sell at different prices
APT (continued)
 In its simplest form, it is:
Ri = a i + ßiRm + ei the same as CAPM
The only value for a which rules out arbitrage
opportunities is zero. So set a to zero and you get:
Ri = ßiRm Subtract the risk-free rate and you get the
well-known equation:
E(rs) = rf + bs [E(rM) - rf ] from CAPM
APT and CAPM Compared
 Differences:
• APT applies to well diversified portfolios and not
necessarily to individual stocks
• With APT it is possible for some individual stocks
to be mispriced – to not lie on the SML
• APT is more general in that it gets to an expected
return and beta relationship without the assumption
of the market portfolio
• APT can be extended to multifactor models, such
as:
Ri = a i + ß1R1 + ß2R2 + ß3R3 + ßnRn + ei
APT and Investment Decisions
 Roll and Ross argue that APT offers an approach to
strategic portfolio planning
• Investors need to recognize that a few systematic
factors affect long-term average returns
• Investors should understand those factors and set
up their portfolios to take those factors into
account
• Key Factors:
• Changes in expected inflation
• Unanticipated changes in inflation
• Unanticipated changes in industrial production
• Unanticipated changes in default-risk premium
• Unanticipated changes in the term structure of
interest rates
Questions
 Any questions on Arbitrage Pricing Theory
and how it differs from CAPM?
Problem
 Suppose two factors are identified for the U.S.
economy: the growth rate of industrial
production (IP) and the inflation rate (IR). IP
is expected to be 4% and IR 6% this year. A
stock with a beta of 1.0 on IP and 0.4 on IR
currently is expected to provide a rate of return
of 14%. If industrial production actually
grows by 5% while the inflation rate turns out
to be 7%, what is your best guess on the rate
of return on the stock?
Answer
 The revised estimate on the rate of return on
the stock would be:
• Before
• 14% = a +[4%*1] + [6%*.4]
a = 7.6%
• With the changes:
• 7.6% + [5%*1] + [7%*.4]
The new rate of return is 15.4%
Review of Objectives
 A. Can you solve problems using the CAL,
MPT, and the Single Index model?
 B. Do you understand the implications of
capital asset pricing theory and the CAPM to
compute security risk premiums?
 C. Do you understand arbitrage pricing theory
and how it works?