Single-index model

TODO: DELETE

Recall that CAPM assumes securities have exactly two kinds of risk: systematic and idiosyncratic. This assumption and the insights that followed form the foundation of the single-index model (SIM), a remarkably simple model widely used in the industry.

SIM models stock $i$'s returns ($R_i$) as a random variable linearly related to the market's return ($R_m$) via three stock-specific parameters ($\alpha_i$, $\beta_i$, and $\epsilon_i$):

I must take pains to point out that many of the assumptions made to get to this point are implausible, especially the assumption that there are only those two types of risk. Nevertheless, this family of models are useful simplifications.

Still, the SIM model is useful because

• It provides a simple hard-to-game measure of risk ($\beta$).
• It prevents money managers from claiming skill merely because they lived during good years and used leverage. Instead they have to beat the market some other way ($\alpha$).
• It reduces the confidence intervals when estimate a fund's overperformance - i.e. the two parameters ($\alpha$, $\beta$) typically have far less error than just an "average return" parameter.
• It is far more computationally feasible and easier to think about than the optimization implied by the Markowitz model. In this model $n$ stocks are represented by $3n$ parameters; in the Markowitz model, it takes $n(n+3)/2$.

Single-index models are often supplemented with multi-factors models, whereby a stock's return are predicted by multiple other variables - not just one. We'll discuss some specific examples later, but for now I'll just note the end result is lots of factors and this general idea underlies arbitrage pricing theory Arbitrage pricing theory.

TODO: Discuss why arbitrage implies these multi-factor models should be linear ("Given arbitrage, this has to be a linear function" - pg 36)