single index model.ppt

8/10/2019 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

single index model.ppt

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