CAPM & APT
akson.sgh.waw.pl/~dserwa/fe.htm
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Literature
• Elton, Gruber, Brown, Goetzmann (2007)
Modern portfolio theory and investment analysis,
John Wiley and Sons. (Ch. 13-16 [, 5, 7])
• Campbell, Lo, MacKinlay (1997) The
econometrics of financial markets, Princeton
University Press. (Ch. 5, 6)
• Cuthbertson, Nitzsche (2010) Quantitative
financial economics…, John Wiley and Sons
(Ch. 5, 8)
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Financial Econometrics
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Asset pricing models
• Authors (independent) of CAPM
– Sharpe (1964)
– Lintner (1965)
– Mossin (1966)
• APT
– Ross (1976, 1977)
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Applications of CAPM
• The relevant measure of risk for each
quoted financial instrument,
• The link between asset returns and asset
risk
• Calculations of the expected rate of return
for the financial instrument (estimations of
the cost of capital, evaluation of portfolio
performance, event studies)
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Financial Econometrics
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Assumptions underlying CAPM
• No transaction costs (or other frictions)
• Assets are infinitely divisible
• The absence of personal income tax
• An individual cannot affect the price by his
buying or selling action (perfect competition)
• Investors make decisions solely in terms of
expected values and standard deviations of
asset returns
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Financial Econometrics
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Assumption underlying CAPM
• Unlimited short sales allowed
• Unlimited lending and borrowing at the riskless
rate
• Investors have the same expectations with
respect to:
– rates of return, standard deviations, return
correlations
– investment horizons
• All assets are marketable
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Financial Econometrics
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CAPM – short introduction
• Efficient frontier – frontier of efficient
investments
• Capital market line (CML) sets the CAPM:
RM − RF
σe
Re = RF +
σM
σe
= RF +
( RM − RF )
σM
• Efficient portfolio lies along the CML
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Interpretation
RM − RF
σe
Re = RF +
σM
(Expected return)=(price of time)+(market price of risk)x(amount of risk)
• All investors hold identical (most
efficient) portfolios of risky assets market portfolio
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Financial Econometrics
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CAPM – short introduction
• The following formula applies for a single
asset or a portfolio i (efficient or not):
Ri = RF + β i ( RM − RF )
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Derivation (simple version)
• We assume one-factor model is correct
Ri = α + β i RM
• Expected return from portfolio i is a linear
function of
βi
Ri = a + bβ i
• For a riskless investment β i = 0 :
RF = a + b(0)
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Financial Econometrics
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Derivation cont’d
• For investments in the market portfolio β i = 1 :
RM = a + b(1)
b = ( RM − a ) = ( RM − RF )
• Thus, the true model is:
• where:
Ri = RF + β i ( RM − RF )
βi =
σ iM
2
σM
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Financial Econometrics
 RM − RF
Ri = RF + 
 σM
 σ iM

 σM
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Interpreting „beta”
• It measures dependence of our portfolio
return on the market return
• Index of systematic risk – nondiversifiable
risk
• Investor expects an additional return for
taking the risk that cannot be divesfied
and not for the risk that can be eliminated
by diversifying the portfolio.
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Financial Econometrics
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Extensions of CAPM
•
•
Short sales disallowed – no impact
No riskless lending or borrowing:
”zero-beta CAPM” / ”two-factor model”
Ri = RZ + β i ( RM − RZ )
•
•
•
•
Personal taxes
Heterogeneous expectations
Multi-period CAPM, Multi-beta CAPM,
”Consumption-oriented CAPM”, etc.
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Financial Econometrics
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Empirical results
• Assumption: market model correct
Rit = α i + β i RMt + eit
• Expected returns from this model:
Ri = α i + β i RM
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Financial Econometrics
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Empirical results
• After subtracting two equations
above: Rit = Ri + β i ( RMt − RM ) + eit
but
Ri = RF + β i ( RM − RF )
• Final model:
Rit = RF + β i ( RMt − RF ) + eit
Rit − RF = β i ( RMt − RF ) + eit
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Financial Econometrics
Methods to estimate
parameters of the CAPM
• standard CAPM (assuming constant beta)
– Use covariance between Ri and Rm , and variance of
Rm in the sample
– Regression model estimated using the OLS method
– Regression model + GARCH
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Methods to estimate
parameters of the CAPM
• conditional CAPM
– EWMA model for covariance between Ri and Rm ,
and variance of Rm
– MGARCH (e.g. GARCH-BEKK), GARCH-M
– State-space models
– GMM
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Financial Econometrics
Methods to estimate
parameters of the CAPM
Capital Asset Pricing Model:
E ( Rit ) − RF = β i ( E ( RMt ) − RF )
can be expressed using the statistical properties of
beta:
 cov(Rit , RMt ) 
E ( Rit ) − RF = 
( E ( RMt ) − RF )

 var(RMt ) 
or:
Rit − RF = β i ( RMt − RF ) + eit
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Methods to estimate
parameters of the CAPM
• covariance between Ri and Rm , and variance of Rm
in the sample
• Regression model + Ordinary Least Squares (OLS)
method
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Sposoby szacowania CAPM (3)
• Expotential Weighted Moving Average
– λ usually set at the level of 0.94 or similar
– Starting values: sample covariance and variance
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Financial Econometrics
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Example:
simulating conditional CAPM
1,4
cov(Ri,Rm)
1,2
var(Rm)
beta
1
0,8
0,6
0,4
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Financial Econometrics
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Problem
• Compute the beta estimates using the
OLS and EWMA methods
• Use the file ewma_eng.xls to learn how to
perform computations in Excel
• Use random data or data from the popular
data sources (e.g. finance.yahoo.com)
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Financial Econometrics
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