Single Index Model.ppt
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Transcript of Single Index Model.ppt
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Single Index Model
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A. Single Index Model
The CAPM is a theory about expected returns The application of the CAPM, i.e., the empirical
version, is ex-post, or after the fact
The empirical version is often referred to as the
Single Index Model One step removed from the theoretical CAPM and all of
its assumptions
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Single Index Model
A broad stock market index is assumed to be thesingle, common factor for all stocks
i= expected return of stock i if markets excess return is zero
i(rmt - rft) = component of return due to market movements
eit= component of return due to unexpected firm-specific events
ifmiifi errrr )()(
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Single Index Model
Textbook notation: Ri= rirfand Rm= rm- rf
Therefore,
imiii eRR
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Early Application
To simplify the Markowitz model Inputs of the Markowitz model: means, standard
deviations, and covariances (or correlation
coefficients) of the assets
If you have 25 assets in the investment universehow many unique covariances?
n(n-1) 2
= 300
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Simplifying the Markowitz Model
Adopting the Single Index Model is a way to reducethis number
By simplifying the covariance
According to the model,
All asset returns derive only from the common factor, RM eiis firm-specific, and hence uncorrelated across assets
Therefore,
Cov(Ri, Rj) = Cov(iRM, jRM) = ijs2M
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Implication for Security Analysis
This setup allows security analysts to specialize Provides rationale for why analysts do not have to
research other sectors
Model says only the common factor (the market) matters;
there is no relationship otherwise
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Decomposing Total Risk
Single Index Model for a portfolio of stocks:
The variance of Rpis:
As the number of stocks increases, the last term
becomes less important as a result of diversification
Total risk = systematic risk + diversifiable risk
pmppp eRR
)(2222 pmpp esss
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If Portfolios are equally weighted...
Pink curve: total risk. Can exclude proof onpp.276-7
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Estimating the Single Index Model
Regression analysis
Typically, use monthly returns over the past 5 years
(i.e., 60 observations) to estimate
Y: excess return on individual security (or individual
portfolio)
X: excess return on market index
Intercept is i, slope isi
itmtiiit eRR
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Security Characteristic Line
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Interpreting the Results
alpha
statistical significancebeta
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The Meaning of R2
2
2
2
222 )(1
i
i
i
mi eRs
s
s
s
The goodness-of-fit measure, R2, from the
Single Index Model regression (the SCL) is:
In words, the R2= the percentage of total risk
of asset i that can be explained by its
systematic risk
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Industry Versions
BMO Nesbitt Burns, Merrill Lynch, Value Line These (and several other) beta estimate providers use
raw returns, not excess returns
That model is called the Market Model
Some firms forecast beta as a function of past betas Some firms forecast beta as a function of firm size,
growth, leverage, etc.
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Industry Versions
Bank of America Merrill Lynch Adjusted : 2/3 sample beta and 1/3 beta of one
Adjusted = 2/3+ 1/3
Tendency for to move toward one over time Hence, take
this into account in forecasts Beta books
Merrill Lynch: monthly
Ibbotson Associates: semi-annual
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Market Neutral Strategies
An application of the Single Index Model A long/short market neutral investment strategy:
Extract the alpha of another managers portfolio
Example on p.288:
To extract alpha, need to get rid of the exposure to
the TSX
pTSXp eRR 4.104.0
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Market Neutral Strategies
First, define the following tracking portfolio, T:
Can think of T as a leveraged portfolio: 1.4 in the TSX,
and -0.4 in risk-free asset Typo in text: share in risk-free asset should be -0.4,
not20.4
End result:
TSXR4.1
pTpc eRRR 04.0