I will introduce the concepts of the capital asset pricing model and beta with a analogy.
Dancing.
We all dance. Some are better dancers than others. There's a lot of trans-cultural variation to be sure. Some influencing goes on to be sure. Some don't dance at all.
Imagine your job is to build a model of how individuals dance and you come up with the following: all dance more or less like Mick Jagger. Outrageous, I know. But just imagine you watch how Mick Jagger dances. You absorb that knowledge. You have it in your head. He's the template. You now are armed with a reference point and set out to build your Moves Like Jagger (MLJ) model of human dance.
Imagine you find someone who dances incredibly like Jagger. And another who moves like Jagger, but in a less spastic way. And another who does so in a grossly caricatured way. Perhaps yet another who only has the odd echo of Jagger in the way she claps her hands like a camp pair of cymbals.
MLJ when applied to any human being ought to be able to describe two things. First, just how like Jagger this person moves, and, all the rest of his moves which don't really line up as classic Jagger moves. So the generalised MLJ model version 1 is like a pair of scissors. It analyses anyone's moves as those which correspond more or less to Jagger like moves, together with all his other moves which don't seem to fit the Jagger pattern.
V1 of your model insists that moving like Jagger is somehow a dancer's best goal. The second fraction of his analysed behaviour which isn't like Jagger is deemed by you a failure. You refer to these abortive un-Jagger moves as residual moves. A mistake. Clearly some dancers will make more mistakes than others. So, one possible initial model is to have a simplistic weighting of moves, which you assign some fraction to a dancer's Jagger moves, and the rest to mistakes. M = w_j J +(1- w_j) E. Read this as: my model treats a dancer's moves as some fraction w_j like Jagger's and the rest as an error.
In V2 you realise some people move exactly like Jagger but more or less exaggerated. Wilder flailing, more melodramatic hops, extreme chicken pecking neck moves. While others tone it down, but essentially incorporate all his moves.
So you switch to a new model, where there's a scaling factor b, measuring the degree of brio with which the dancer copies Jagger. Again, an error term E will mop up the residual error, i.e. what remains when you've catered for the brio-adjusted moves. M = bJ + E. Jagger impersonators might have a brio score of 2.0. Modest dancers might score 0.5 or lower.
The capital asset pricing model is the equivalent of the claim that the average dancer moves exactly like Jagger. So, Iggy pop, Ian Curtis, Morrisey, Tom Waits, you and , and everyone else too. When you average up our moves, we dance on average exactly like Jagger. He's kind of a Jungian dance archetype for us all.
In any case, V2 is a two-part analysis of moves. Part one compares them to some reference moves, and part two characterises the remnant.
With stock returns, the moves-like function is implemented with Cov(A,B) the co-variance between period-returns for series A and period-returns for series B. In fact, the period returns (e.g. daily returns) are adjusted to be returns in excess of a corresponding risk free rate. The dance analogy might see the equivalent of the risk free rate would be the moves linked to breathing. Co-variance is symmetric - Cov(A,B) = Cov(B,A) - but since we are interested in the Jagger reference dance moves, we normalise the co-variance by the variance of Jagger himself / the market. This achieves the goal of making Jagger get a brio (or beta) score of 1.0. He dances exactly like himself. He will also get an E score of 0.
Equity factor portfolio management adds two elements further. First, it adds multiple dance archetypes to the model. Second, it assumes that it isn't the case that the average dancer dances like Jagger (or the set of Jungian dance archetypes). That there's a possibility that some dance moves could be better than the archetypes. New moves may be possible.
There appears to be no universally accepted theory on how to pick your archetypes. Data driven selection has issues, and theory driven selection can go badly wrong too. Iggy pop after all dances quite a lot like Jagger anyway. Are you really saying much by adding him in?
A final angle on this metaphor. I always thought Jagger was doing a very poor early 60s James Brown impression anyway! With a high comedic beta. Similarly the choice of what 'the market' means is up for grabs. And certainly I can see that change over time too.
