Does the CAPM use the single index model?

Question

When we derive the CAPM (i.e. find equations for the capital market line and the security market line), we nowhere assume that the individual security return is linearly dependent on the marker return (i.e. the single index model)

However, when we interpret the CAPM, we say that expected return on an individual security depends only on its non-diversifiable risk, which we denote by beta (β). But viewing β as the measure of non-diversifiable is only justified by the variance decomposition under the single index model.

So the question is, does CAPM require assuming the single index model?

(Note - For reference, I have included the derivation of the security market line in the image below, which is taken from the text Modern Portfolio Theory by Francis. As you can see, the derivation doesn't assume the single index model)

Answer 1

To determine the efficient frontier of a mean-variance framework, one needs estimates of the expected return $r_i$, the variance $\sigma_i^2$ and the co-variance $\sigma_{ij}^2$ for each stocks $i$, $j$. For $n$ stocks, you have to estimate a total of $\frac{n(n-1)}{2}$ correlation coefficients. Index models are used, to reduce this huge amount of needed estimates.

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Single Index Models

It is assumed, that the return of a stock can be written as $$r_i = a_i + \beta_i r_m + e_i$$ ,where $r_m$ denotes the market-return, $e_i$ a mean-zero error term and $\beta_i$ a stocks beta. The key assumptions are: $$\operatorname{E}[e_i(r_m-\bar{r}_m]=0$$ $$\operatorname{E}[e_ie_j]=0$$ This implies, that the only reason stocks vary together, systematically, is because of a common comovement with the market. One can show, that the covariance can be expressed as $$\sigma_{ij}^2 = \beta_i \beta_j \sigma_m^2$$ , where $\sigma_m^2$ denotes the variance of the market-return. In summary, if you assume the single-index model, you just have to estimate a total of $3n+1$ parameters for $n$ stocks.

CAPM

The CAPM is an economic theory in equilibrium with further assumptions for an investor's utility-preference function, costless diversification,...

Combing the economic theory from Markowitz portfolio-diversification, Von Neumann and Morgenstern expected utilities etc. leads to the CAPM (where $r^f_t$ denotes the risk-less rate of interest):

$$r_{i,t}-r^f_t = \alpha_i + \beta_i(r^m_t-r^f_t)+ \epsilon_{i,t}$$

, with the following (strong) assumption:

$$\alpha_i = 0$$

You may look at this excellent answer with more details.

Differences from Single Index Models and the CAPM

In fact, the single index model is just a statistical technique, because you can replace $r_m$ with any other variable you think fits best to explain a stocks return. The CAPM however is an economic model in equilibrium, where the market-portfolio return $r_m$ is a clearly determined portfolio (of all risky assets, investments, also human-capital...). See also this answer:

The $\beta_i$ for a stock in the single-index model is not the same $\beta_i$ as in the CAPM.

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Reference:

Elton/Gruber/Brown/Götzmann (2014), Modern Portfolio Theory and Investment Analysis, ed. 9, John Wiley & Sons.