# Difference between CAPM and single index model

which is the difference betwee a model like CAPM and a single index model?

Is the first a special case of the second?

Best

As you know the equation that describes them is the same.

The single index model is an empirical description of stock returns. You do some regressions using data and you come up with Alphas, Betas etc. That's all. It is useful for example in modeling risks of a bunch of stocks in a simple way.

The CAPM is an economic theory that says that Alpha in the long run has an expected value of zero, which means that the returns investors get are solely due to their exposure to the 'market factor'. This is justified by some reasoning like "other risks can be diversified away, so they will not be rewarded in equilibrium, only 'systematic risk' will be rewarded". However, as you know, this has not held up well and it seems that there are other factors that are 'rewarded' in practice. So the CAPM is seen by many as flawed in some ways.

• So could we say that Single Index is just a synonym for standard regression while capm has an equilibrium model behind it? – Klapaucius Dec 10 '16 at 11:12
• Agreed. A statistical technique vs an economic model. Note also that the "index' used in Single Index could be any index that you think has a good $R^2$, the index in CAPM is supposed to be the Market Portfolio of all risky assets, a very special portfolio. – noob2 Dec 12 '16 at 15:35

Indeed one is the special case of the other. In CAPM you are regressing stock (or portfolio) returns vs the Market (your index) . But your index could be any independent variable that you believe explains the left hand side (your returns) - it could be the returns of an industry, an ETF a different index - what not.

You write:

“which is the difference betwee a model like CAPM and a single index model? Is the first a special case of the second?”

No, the opposite is true. SIM is interpretable as a special case of the CAPM.

The CAPM can reduced to this equation:

$$E[r_i - r_f] = \beta_i E[r_m- r_f]$$

if it hold, CAPM hold as well.

Now any linear conditional expectation is also interpretable as a OLS regression. Then we can write the SIM in excess return regression form

$$re_i = \alpha_i + \beta_i re_m + \epsilon_i$$

now to make this regression well posed in econometric sense we have to make some assumption about $$\epsilon_i$$. If variance covariance matrix is diagonal we have the SIM. In SIM $$\beta$$'s describes completely also the covariance of returns not only the mean. Is not well known but this imposition about the variance structure of returns are make in SIM while CAPM make no assumption about it (apart finiteness). The CAPM is more general than SIM.