macrocephalopod on Twitter: "is a critical step to take before you even consider using a portfolio optimizer (or some other method of reducing noise, but a factor model is the most common)."

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

The return of stock i on day t is modeled as a linear combination of the unknown factor returns plus an unknown residual, where the dependence of stock i on factor j on day t is given by the *known* exposure X(i, j, t)pic.twitter.com/CbdvNRGura

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

If we understand that r, f and epsilon are vectors and X is a matrix then we can drop all the subscripts and write it in matrix notation which is much easier. It's important to understand that the exposures X are known in advance, and its the factor returns f that are unknownpic.twitter.com/uKK7PuGjuQ

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

Now you have a set of linear equations on each day, and you can solve the linear equations to get the vector of factor returns for each day using the normal equation -pic.twitter.com/YwVkUzSM69

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

What's the point of this? The factor models tells you how the covariance matrix of stocks is related to the covariance matrix of factors - it's the sum of the covariance due to factor exposures - Σ_f - and the covariance of the residuals Ω which is normally assumed to be diagonalpic.twitter.com/H7ruwNaEtR

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

Now instead of estimating the ~2 million parameters of a 2000 x 2000 stock covariance matrix, you just need to estimate ~800 parameters of a 40 x 40 factor covariance matrix -- your risk model just got a whole lot simpler.

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

Even more interestingly, if you are not just a risk manager but a quant equity pm, you can take expectations and get a model for the expected return (or alpha) of each stock in terms of the expected returns on each factor --pic.twitter.com/JflXE3Qj8V

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

If you're doing this you would normally treat some factors as "risk" factors which have zero expectation (i.e. you want to hedge them to minimise risk) and some factors as "alpha" factors which have positive expectation, as well as risk --

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

you want exposure to the alpha factors, but you want *more* exposure to the ones which have higher expected return given the risk, and your factor model gives you a structured way to express that.

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

The vector of alphas, and the stock covariance matrix, are the key inputs to a portfolio optimizer (along with transaction costs, financing costs, position and turnover constraints, risk constraints etc). One particularly important fact is that portfolio optimizers are

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

known to behave *really* badly (recommending nonsense portfolios) when there is noise in the covariance matrix. You get noise when you try to estimate too many parameters from too little data, so using a factor model to reduce the number of parameters to estimate

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macrocephalopod‏ @macrocephalopod 2 Feb 2021

is a critical step to take before you even consider using a portfolio optimizer (or some other method of reducing noise, but a factor model is the most common).

3:05 PM - 2 Feb 2021

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