A. Single Index Model

The CAPM is a theory about expected returnsThe application of the CAPM, i.e., the empirical

version, is ex-post, or after the factThe empirical version is often referred to as the

Single Index ModelOne step removed from the theoretical CAPM and all

of its assumptions

Single Index Model A broad stock market index is assumed to be the

single, common factor for all securities

i = expected return of stock i if market’s 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 )()(

Single Index ModelTextbook notation:Ri = ri – rf and Rm = rm - rf Therefore,

imiii eRR

Early ApplicationTo simplify the Markowitz modelInputs of the Markowitz model: means, standard

deviations, and covariances (or correlation coefficients) of the assets

If you have 50 assets in the investment universe – how many covariances?

n(n-1) 21225

Simplifying the Markowitz ModelAdopting the Single Index Model is a way to reduce

this number– By simplifying the covariance

According to the model,– All asset returns derive only from the common factor,

RM

– ei is firm-specific, and hence uncorrelated across assetsTherefore,

Cov(Ri, Rj) = Cov(iRM, jRM ) = ijM

Implication for Security AnalysisThis setup allows security analysts to specialize

Provides rationale for why analysts do not have to research other industries

Model says there is no relationship, only the common factor (the market) matters

Decomposing Total Risk

Single Index Model for a portfolio of stocks:

The variance of Rp is:

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

)(2222pmpp e

If Portfolios are equally weighted...

Pink curve: total risk. Can exclude proof on pp.276-7

Estimating the Single Index ModelRegression 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 indexIntercept is i, slope is i

itmtiiit eRR

Security Characteristic Line

Interpreting the Results

alpha

statistical significancebeta

The Meaning of R2

2

2

2

222 )(

1i

i

i

mi eR

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

Industry VersionsBMO 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 ModelSome firms forecast beta as a function of past betasSome firms forecast beta as a function of firm size,

growth, leverage,… etc.

Industry VersionsMerrill 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 booksMerrill Lynch: monthlyIbbotson Associates: semi-annual

Market Neutral StrategiesAn application of the Single Index ModelConsider the following “market neutral” investment

strategySuppose the returns to the manager’s portfolio can be

characterized by (example on p.288):

You like the alpha, but do not want the market exposure

pTSXp eRR 4.104.0

Market Neutral StrategiesFirst, create the following tracking portfolio, T:

T is a leveraged portfolio: 1.4 in the TSX, and -0.4 in risk-free asset (or use

derivative securities)(Typo in text: share in risk-free asset should be -0.4,

not 20.4)Buy P and sell T. End result:

TSXR4.1

pTpc eRRR 04.0