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BM410: Investments
Capital Asset PricingTheory and APT
orHow do you value stocks?
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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 computesecurity risk premiums
C. Understand the arbitrage pricing theory and
how it works
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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 funds 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.
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Answer
Slope of the CML = (13-7)/25 = .24
Slope of your CAL = (17-7)/27 = .37
b. Your fund allows an investor a higher mean for any
given standard deviation than the passive strategy.
17%
13%
7%10% 20% 30%
Your Fund
Index Fund
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MPT Review
Suppose that for some reason you are required toinvest 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 is15%, what must be the correlation coefficient
between stock and bond returns?
B. What is the expected rate of return on yourportfolio?
C. Now suppose that the correlation between stockand bond returns is 0.22 but that you are free tochoose whatever portfolio proportions you desire.Are you likely to be better or worse off that you
were in part a?
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Answer
A. sp2
= w12s1
2+ w22s2
2+ 2W1W2 (r1,2s1s2 )
(.15) 2 =[(.512.121
2) +(.522.252
2) + 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 foryour 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.
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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 be8%, how would you review/change your
expectations of the rate of return on the
stock?
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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%
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Questions
Any questions of Capital Allocation Lines,
Modern Portfolio Theory, or Single Index
Models?
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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 todetermine 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 generalequilibrium models
These models allow us to determine the risk for
any asset and the relationship between expected
return and riskfor any asset when the markets
are in equilibrium, i.e. balance or constant state
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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 therelationship 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)
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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 theexpected return on assets that have not yet been
traded in the marketplace
For example, how do we price an initial public
offering?
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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 someapplications
Do we use it?
Yes, but with knowledge of its limitations
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CAPM Assumptions
What does the model assume (some are unrealistic)?
Individual investors are price takers (cannot affectprices)
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
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Resulting Equilibrium Conditions
Based on the previous assumptions:
All investors will hold the same portfolio for risky
assetsthe market portfolio (M)
The market portfolio (M) contains all securities andthe 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 sssm)
with the market
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E(r)
E(rM)
rf
MCML
sm
Capital Market Line
s
M = Market portfolio rf = Risk free rateE(rM) - rf= Market risk premium
[E(rM) - rf]/sM= Market price of risk
The efficient frontier without
lending or borrowing
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Expected Return and Risk
of Individual Securities
What does this imply?
The risk premium on individual securities is
a function of the individual securitys
contribution to the risk of the marketportfolio
Individual securitys risk premium is a
function of the covariance of returns with
the assets that make up the market portfolio
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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. ( = [COV(ri,rm)] / sm2)
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Expected ReturnBeta Relationship
Expected return - beta relationship of CAPM:
E(rM) - rf = E(rs) - rf
1.0 bsIn other words, the expected rate of return of an asset
exceeds the risk-free rate by a risk premium equal to the
assets systematic risk (its beta) times the risk premium
of the market portfolio. This leads to the familiar re-arrangement of terms to give (memorize this):
E(rs) = rf + bs [E(rM) - rf ]
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E(r)
E(rM)
rf
SML
M
= 1.0
The Security Market Line
Notice that instead of using standarddeviation, the Security Market Line uses Beta
SML Relationships
= [COV(ri,rm)] / sm2
Slope SML = E(rm)rf = market riskpremium
SML = rf + [E(rm) - rf]
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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
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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%
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E(r)
Rx=13%
SML
m
1.0
Rm=11%
Ry=7.8%
3%
x
1.25
y.6
.08
Graph of Sample Calculations
They are in equilibrium
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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 increasedemand, which would increase the price,
which would decrease the return until it
came back into line.
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E(r)
15%
SML
1.0
Rm=11%
rf=3%
1.25
Disequilibrium Example
The return is above the
SML, so you would buy it
As more people bought
the security, it would
push the price up,
which would bring the
return down to the line.
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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
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The Index Model
With the Index model, we can:
Specify a way to measure the factor that affectsreturns (the return of the Index)
Separate the rate of return on a security into its
macro (systematic) and micro (firm-specific)components
Components
= excess return if market factor is zero
iRm= component of returns due to movements in theoverall market
ei = component attributable to company specificevents
Ri
= ai
+ i
Rm
+ ei
(Notice the similarity to the Single Index model discussed earlier)
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Security Characteristic Line
Excess Returns (i)
SCL
.
.
...
.
. .
. ..
. . .. .
. ..
..
.
. .
. ..
.
..
. .
..
.
. . .. .
.
. ... .. .. .
Excess returns
on market index
Ri= a i+ iRm+ ei
Plot of a companys excess return as a
function of the excess return of the market
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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 onadded market risk, but not diversifiable risk
Perhaps instead of determining whether the CAPM is
true or not, we might ask: Are there better models?
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Questions
Any questions on capital asset pricing
and the Capital Asset Pricing Model?
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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 therisk 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
Dells beta was 1.3, what would be Dells risk
premium?
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Answer
a. If Dells 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%
Dells 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%
Dells new risk premium is 15.4 5% = 10.4%
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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 zeroinvestment 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
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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 cant sell at different prices
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APT (continued)
In its simplest form, it is:
Ri= a i+ iRm+ 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= iRmSubtract the risk-free rate and you get the
well-known equation:
E(rs) = rf + bs [E(rM) - rf ] from CAPM
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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 mispricedto 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+ 1R1+ 2R2+ 3R3+ nRn + ei
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APT and Investment Decisions
Roll and Ross argue that APT offers an approach tostrategic portfolio planning
Investors need to recognize that a few systematicfactors affect long-term average returns
Investors should understand those factors and setup their portfolios to take those factors intoaccount
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 ofinterest rates
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Questions
Any questions on Arbitrage Pricing Theory
and how it differs from CAPM?
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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. Astock 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 outto be 7%, what is your best guess on the rate
of return on the stock?
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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%
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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 tocompute security risk premiums?
C. Do you understand arbitrage pricing theory
and how it works?
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