# MARKET MODEL vs SML-CAPM

Recently, I got a question from a student that would like to use Market Model by reference to the paper written by Brown and Warner 1985.

Talking about the Market Model, I would like to touch slighly on the difference between Market Model with the one being used by William F. Sharpe in its CAPM, particularly in estimating the beta. Well, the whole idea about CAPM is beta, right? One single factor that is very important in that model, that is market beta.

Market model:

Rj = aj + bj rm + ej

Rj = historical (realized) rate of return on Stock J

Rm = historical (realized) rate of return on the market

aj = vertical axis intercept term for Stock J

bj = slope, or beta, coefficient, for Stock J

ej  = random error = difference actual return on a certain period vs that as predicted by the regression line

So what we see here, is, we are trying to regess the stock’s realized return against the market’s realized return.

If we look at the SML of the CAPM, the realized return is a bit different from the above Market Model:

Rj = r (Rf) + (rm – r (Rf)) + ej

Rf is the historical (realized) risk free rate

Then if we want to compare the Market Model and CAPM to estimate beta, then we need to re-arrange SML to be:

(rj – r (Rf) = aj + bj x (rm – r(Rf)) + ej

In other words, to be theoretically correct, both y-variable and x-variable in regression, should be the stock’s return in excess of the risk-free rate and the market’s return in excess of the risk-free rate.

In most cases, both models will give us results that will not have big differences.

So, it is safe to use Market Model in estimating the beta without having to use SML of the CAPM.

As far as I remember, Michael C. Jensen did some initial test of the CAPM Model, which now we name ej (random error above) as Jensen’s alpha. Please see as well F.Black, M. Jensen and M. Scholes in their initial test “The Capital Asset Pricing Model” Some Empirical Tests” (1972).

Michael C. Jensen emphasized the fact that there is no theory behind the Market Model. It is simply a statement about the empirical relationship between security and market returns and tells us nothing about what causes returns to be what they are in any period. It hypothesizes only that there is a systematic linear association between the returns on individual securities and the market index, and empirical tests confirm this characteristic of the model. (M. Jensen, “Tests of Capital Market Theory and Implications of the Evidence, 1974).