## Conductors and CAPM

Recently I watched this video that YouTube recommended to me about why orchestras have conductors.

The basic idea is that an orchestra  needs a whole lot of coordination, in terms of when to begin and end, when to slow down or speed up, when to move to the next line and so on. And in case there is no conductor, the members of the orchestra need to coordinate among themselves.

This is easy enough when there is a small number of members, so you don’t see bands (for example) needing conductors. However, notice that if the orchestra has to coordinate among themselves, coordination is an $O(n^2)$ problem. By appointing an external conductor whose only job is to conduct and not play, this $O(n^2)$ problem is reduced to an $O(n)$ problem.

When I saw this, this took me back to my Investments course in IIMB, where the professor one day introduced what he called the “Sharpe single index model“, which is sort of similar to the CAPM.

Just before learning the Sharpe Single Index Model, we had been learning about Markowitz’s portfolio theory. And then, as he introduced the Sharpe Single Index Model, Vaidya said something to the effect that “instead of knowing so many correlation terms” (which is an $O(n^2)$ problem), “we only need to know the correlation of each stock to the market index” (makes it an $O(n)$ problem).

As someone who has studied computer science formally, converting $O(n^2)$ problems to $O(n)$ problems is a massive fascination. It is interesting how I connected two such reductions from completely different fields.

In other words, conductors are the “market of the orchestra”.
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