asset pricing models: their uses and their limitations - bahattin
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CHAPTER 9
The Capital Asset Pricing Model
It is the equilibrium model that underlies all
modern financial theory
Derived using principles of diversification with
simplified assumptions
Markowitz, Sharpe, Lintner and Mossin are
researchers credited with its development
CAPITAL ASSET PRICING MODEL (CAPM)
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Individual investors are price takers
Single-period investment horizon
Investments are limited to traded financial assets
There are homogeneous expectations
ASSUMPTIONS: INVESTORS
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Information is costless and available to all
investors
No taxes and transaction costs
Risk-free rate available to all
Investors are rational mean-variance optimizers
ASSUMPTIONS: ASSETS
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All investors will hold the same portfolio for risky
assets – market portfolio, which contains all securities
and the proportion of each security is its market value
as a percentage of total market value held by all investors
includes all traded assets
suppose not: then price… -> included
is on the efficient frontier
asset weights: for each $ in risky assets, how much is in IBM?
for stock i: market cap of stock i / market cap of all stocks
RESULTING EQUILIBRIUM CONDITIONS
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iii
iii
PN
PNw
Risk premium on the market depends on the
average risk aversion of all market participants
Risk premium on an individual security is a
function of its covariance with the market
RESULTING EQUILIBRIUM CONDITIONS
CONTINUED
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FIGURE 9.1 THE EFFICIENT FRONTIER AND
THE CAPITAL MARKET LINE
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MARKET RISK PREMIUM
The risk premium on the market portfolio will be
proportional to its risk and the degree of risk
aversion of the investor:
2
2
( )
where is the variance of the market portolio and
is the average degree of risk aversion across investors
M f M
M
E r r A
A
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The risk premium on individual securities is a
function of the individual security’s contribution
to the risk of the market portfolio
An individual security’s risk premium is a function
of the covariance of returns with the assets that
make up the market portfolio
RETURN AND RISK FOR INDIVIDUAL
SECURITIES
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USING GE TEXT EXAMPLE
Covariance of GE return with the market
portfolio:
Therefore, the reward-to-risk ratio for
investments in GE would be:
1 1
( , ) , ( , )n n
GE M GE k k k k GE
k k
Cov r r Cov r w r w Cov r r
( ) ( )GE's contribution to risk premium
GE's contribution to variance ( , ) ( , )
GE GE f GE f
GE GE M GE M
w E r r E r r
w Cov r r Cov r r
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USING GE TEXT EXAMPLE CONTINUED
Reward-to-risk ratio for investment in market
portfolio:
Reward-to-risk ratios of GE and the market
portfolio:
And the risk premium for GE:
2
( )Market risk premium
Market variance
M f
M
E r r
2
( ) ( ( )
( , )
GE f M f
GE M M
E r r E r r
Cov r r
2
( , )( ) ( )GE M
GE f M f
M
Cov r rE r r E r r
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EXPECTED RETURN-BETA RELATIONSHIP
CAPM holds for the overall portfolio because:
This also holds for the market portfolio:
P
( ) ( ) andP k k
k
k k
k
E r w E r
w
( ) ( )M f M M fE r r E r r
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FIGURE 9.2 THE SECURITY MARKET LINE
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FIGURE 9.3 THE SML AND A POSITIVE-ALPHA
STOCK
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THE INDEX MODEL AND REALIZED RETURNS
To move from expected to realized returns—use
the index model in excess return form:
The index model beta coefficient turns out to be
the same beta as that of the CAPM expected
return-beta relationship
i i i M iR R e
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FIGURE 9.4 ESTIMATES OF INDIVIDUAL
MUTUAL FUND ALPHAS, 1972-1991
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THE CAPM AND REALITY
Is the condition of zero alphas for all stocks as
implied by the CAPM met
Not perfect but one of the best available
Is the CAPM testable
Proxies must be used for the market portfolio
CAPM is still considered the best available
description of security pricing and is widely
accepted
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ECONOMETRICS AND THE EXPECTED
RETURN-BETA RELATIONSHIP
It is important to consider the econometric
technique used for the model estimated
Statistical bias is easily introduced
Miller and Scholes paper demonstrated how
econometric problems could lead one to
reject the CAPM even if it were perfectly valid
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EXTENSIONS OF THE CAPM
Zero-Beta Model
Helps to explain positive alphas on low beta
stocks and negative alphas on high beta stocks
Consideration of labor income and non-traded
assets
Merton’s Multiperiod Model and hedge portfolios
Incorporation of the effects of changes in the
real rate of interest and inflation
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EXTENSIONS OF THE CAPM CONTINUED
A consumption-based CAPM
Models by Rubinstein, Lucas, and Breeden
Investor must allocate current wealth between today’s consumption and investment for the
future
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LIQUIDITY AND THE CAPM
Liquidity
Illiquidity Premium
Research supports a premium for illiquidity.
Amihud and Mendelson
Acharya and Pedersen
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FIGURE 9.5 THE RELATIONSHIP BETWEEN
ILLIQUIDITY AND AVERAGE RETURNS
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THREE ELEMENTS OF LIQUIDITY
Sensitivity of security’s illiquidity to market
illiquidity:
Sensitivity of stock’s return to market illiquidity:
Sensitivity of the security illiquidity to the market
rate of return:
1
( , )
( )
i ML
M M
Cov C C
Var R C
3
( , )
( )
i ML
M M
Cov C R
Var R C
2
( , )
( )
i ML
M M
Cov R C
Var R C
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 1
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Question:
The market price of a stock is $40.
Its expected rate of return is 13%.
The risk-free rate is 7%
The market risk premium is 8%.
Suppose its covariance w/ the market portfolio doubles (other variables are unchanged)?
Do you have enough information to find what will be the new price of the stock?
Assume that the stock is expected to pay a constant dividend in perpetuity.
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Answer:
If the covariance of the security doubles, then so will its beta and its risk premium.
The current risk premium is 6% = 13% - 7%
the new risk premium would be twice as high: 12%
the new discount rate for the security would be 19% = 12% + 7%
If the stock pays a level perpetual dividend, then we know from the original data that:
Price = Dividend/Discount rate => $40 = D/0.13 => D = $5.20.
At the new discount rate of 19%, the stock would be worth only:
$5.20/0.19 = $27.37.
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 2
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Question:
Consider the following table, which gives a security analyst’s expected return on two
stocks for two particular market returns:
Market Return Aggressive Stock Defensive Stock
-------------------------------------------------------------------------------------------
5% 2% 3.5%
20% 32% 14%
-------------------------------------------------------------------------------------------
What hurdle rate should be used by the management of the aggressive firm for a project
with the risk characteristics of the defensive firm’s stock?
The risk-free rate is 8%.
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 3
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Answer:
The hurdle rate is determined by the project beta, 0.70, not by the firm’s beta.
The correct discount rate is 11.15%, the fair rate of return on Stock D (defensive). Why?
(a) The beta is the sensitivity of the stock return to the market return movements.
Then beta is the change in the stock return per change in the market return. Therefore:
A = (2 - 32)/(5 - 20) = 2.00
D = (3.5 - 14)/(5 - 20) = 0.70
(d) The defensive stock has a fair expected return of:
E(RD) = 8% + 0.7(12.5% - 8%) = 11.5%,
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 4
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Question:
Two investment advisors are comparing performance.
One averaged a 19% rate of return and the other a 16% rate of return.
However, the beta of the first investor was 1.5, whereas that of the second was 1.
(a) Can you tell which investor was a better predictor of individual stocks (aside from the
issue of general movements in the market)?
(b) If the T-bill rate were 6% and the market return during the period were 14%, which
investor would be the superior stock selector?
(c) What if the T-bill rate were 3% and the market return were 15%?
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 5
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Answer:
(a) We know that:
R1 = 19%, R2 = 16%, 1 = 1.5, and 1 = 1.
To tell which investor was a better predictor of individual stocks, we should look at their
abnormal return, which is the ex-post (alpha)
that is, the abnormal return is the difference between
the actual return
and the return predicted by the SML.
Without information about the parameters of this equation (risk-free rate and the market
rate of return) we cannot tell which investor is more accurate.
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 6
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Answer:
(b) If Rf = 6% and Rm = 14%, then (using the notation of alpha for the abnormal return):
1 = 19% - [6% + 1.5(14% - 6%)] = 19% - 18% = 1%
2 = 16% - [6% + 1(14% - 6%)] = 16% - 14% = 2%.
Here, the second investor has the larger abnormal return
and thus he appears to be a more accurate predictor.
By making better predictions,
the second investor appears to have tilted his portfolio toward underpriced stocks.
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 7
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Answer:
(c) If Rf = 3% and Rm = 15%, then
1 = 19% - [3% + 1.5(15% - 3%)] = 19% - 21% = -2%
2 = 16% - [3% + 1(15% - 3%)] = 16% - 15% = 1%.
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CAPM: EXAMPLES OF PRACTICAL
PROBLEMS 8
INDEX MODEL VS. CAPM
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Risk CAPM (theoretical, unobservable portfolio)
Index model (observable, “proxy” portfolio)
)(
),(
2M
Mii
R
RRCov
),(),( MiMiiMi ReRCovRRCov
)(0)(0 22MiMi RR
)(
),(
2M
Mii
R
RRCov
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INDEX MODEL VS. CAPM 2
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Beta Relationship
CAPM (no expected excess return for any security)
Index model (average realized alpha is 0)
Fig 10.3
)][(][ fMifi rrErrE
ifMiifi errrr )(
)][(][ fMiifi rrErrE
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MARKET MODEL
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Idea
use realized excess returns
Equivalence
CAPM + Market model = Index model
])[(][ MMiii rErrEr
)][(][ fMifi rrErrE
ifMiifi errrr )(
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SUMMARY
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CAPM
Factor model
Index model
Market model ])[(][ MMiii rErrEr
)][(][ fMifi rrErrE
ifMiifi errrr )(
iiii eFrEr ][
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CHAPTER 10
Arbitrage Pricing Theory and Multifactor Models of Risk and Return
SINGLE FACTOR MODEL
Returns on a security come from two sources
Common macro-economic factor
Firm specific events
Possible common macro-economic factors
Gross Domestic Product Growth
Interest Rates
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SINGLE FACTOR MODEL EQUATION
ri = Return for security I
= Factor sensitivity or factor loading or factor
beta
F = Surprise in macro-economic factor
(F could be positive, negative or zero)
ei = Firm specific events
( )i i i ir E r F e
i
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MULTIFACTOR MODELS 1
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Necessity
CAPM
not practical
Index model
practical
unique factor is unsatisfactory
example: Table 10.2 (very small R2
)
Solution
multiple factors
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MULTI-FACTOR MODELS 2
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Factors in practice
business cycles factors
examples (Chen Roll Ross)
industrial production % change
expected inflation % change
unanticipated inflation % change
LT corporate over LT gvt. bonds
LT gvt. bonds over T-bills
interpretation
residual variance = firm specific risk
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MULTI-FACTOR MODELS 3
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Factors in practice
firm characteristics (Fama and French)
firm size
difference in return
between firms with low vs. high equity market value
proxy for business cycle sensitivity?
market to book
difference in return
between firms with low vs. high BTM ratio
proxy for bankruptcy risk?
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MULTIFACTOR MODELS 4
Use more than one factor in addition to market
return
Examples include gross domestic product,
expected inflation, interest rates etc.
Estimate a beta or factor loading for each
factor using multiple regression.
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MULTIFACTOR MODEL EQUATION
ri = E(ri) + GDP GDP + IR IR + ei
ri = Return for security I
GDP= Factor sensitivity for GDP
IR = Factor sensitivity for Interest Rate
ei = Firm specific events
i
i
i
i
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MULTIFACTOR SML MODELS
E(r) = rf + GDPRPGDP + IRRPIR
GDP = Factor sensitivity for GDP
RPGDP = Risk premium for GDP
IR = Factor sensitivity for Interest Rate
RPIR = Risk premium for Interest Rate
i i
i
i
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ARBITRAGE PRICING THEORY (APT)
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Nature of arbitrage
APT
well-diversified portfolios
individual assets
APT vs. CAPM
APT vs. Index models
single factor
multi-factor
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ARBITRAGE PRICING THEORY
Arbitrage - arises if an investor can construct a
zero investment portfolio with a sure profit
Since no investment is required, an investor can
create large positions to secure large levels of
profit
In efficient markets, profitable arbitrage
opportunities will quickly disappear
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APT & WELL-DIVERSIFIED PORTFOLIOS
rP = E (rP) + PF + eP
F = some factor
For a well-diversified portfolio:
eP approaches zero
Similar to CAPM,
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FIGURE 10.1 RETURNS AS A FUNCTION OF
THE SYSTEMATIC FACTOR
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FIGURE 10.2 RETURNS AS A FUNCTION OF
THE SYSTEMATIC FACTOR: AN ARBITRAGE
OPPORTUNITY
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FIGURE 10.3 AN ARBITRAGE OPPORTUNITY
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FIGURE 10.4 THE SECURITY MARKET LINE
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APT applies to well diversified portfolios and not
necessarily to individual stocks
With APT it is possible for some individual stocks
to be mispriced - 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
APT AND CAPM COMPARED
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MULTIFACTOR APT
Use of more than a single factor
Requires formation of factor portfolios
What factors?
Factors that are important to performance of
the general economy
Fama-French Three Factor Model
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TWO-FACTOR MODEL
The multifactor APR is similar to the one-factor case
But need to think in terms of a factor portfolio
Well-diversified
Beta of 1 for one factor
Beta of 0 for any other
1 1 2 2( )i i i i ir E r F F e
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EXAMPLE OF THE MULTIFACTOR APPROACH
Work of Chen, Roll, and Ross
Chose a set of factors based on the ability of
the factors to paint a broad picture of the
macro-economy
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ANOTHER EXAMPLE:
FAMA-FRENCH THREE-FACTOR MODEL
The factors chosen are variables that on past evidence seem to predict
average returns well and may capture the risk premiums
Where:
SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess
of the return on a portfolio of large stocks
HML = High Minus Low, i.e., the return of a portfolio of stocks with a high
book to-market ratio in excess of the return on a portfolio of stocks with a
low book-to-market ratio
it i iM Mt iSMB t iHML t itr R SMB HML e
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THE MULTIFACTOR CAPM AND THE APM
A multi-index CAPM will inherit its risk factors
from sources of risk that a broad group of
investors deem important enough to hedge
The APT is largely silent on where to look for
priced sources of risk
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