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studying with a factor model, I get confused more and more as I think about factor exposure and factor return

The concept (or mechanism) I get used to is evaluating a factor's Long Short Return(Q1-Q5) and see whether it gives us statistically significant return by checking t-stat. If t-stat turns out to be valid, then I'm free to use that factor. (Below picture is the one I get used to regarding to factor) enter image description here

However, in the factor equation "r=Bf+s" I'm not sure where the procedure above takes into account. Does factor return "f" mean the Q1-Q5 long short return? Moreover, there are roughly 3 types of risk factors. fundamental factor, macroeconomic factor and statistical factor. I think fundamental factor like value factor or quality factor can be used to calculate long short return because every single stock has a factor data related to value or quality. However how can I adopt such long short return evaluation mechanism in macroeconomic factor? If I have to estimate every asset's beta related to a certain macroeconomic factor through linear regression, where does the long short return procedure kicks in?

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The r = Bf+s is the general formula that explains how each ticker is affected by a specific factor. It will not hold for entire portfolios, in that case you have to use more complex approaches.

What lots of hedge funds do is to divide the factors into 2 groups: Risk Factors and Alpha Factors. Risk Factors are drivers of variance while Alpha Factors are drivers of return. Those factors are very alike and, in fact, an Alpha Factor that used to work 10 years ago, may now be a Risk Factor, since people got to learn about it and apply it to their portfolio managemnt.

Since Risk Factors are drivers of variance, and well known by market players, you can model them using PCA. No need to research for them, altho companies sell that kind of information.

For a dollar neutral portfolio:

You have to decide how many alpha factors you are going to use. You will combine the Alpha Factors and will probably make them market neutral, sector neutral, rank them and then z-score your Alpha Vectors. The final product will be an Alpha Vector for each period, with values summing up to 0.

Finaly, you will have to optmize your portfolio taking your Risk Factor and setting a series of constraints. Then you will apply an optimization algorithm to reduce the variance and rebalance the weights applied to each ticker in that period. The weights are exactly your Alpha Vector. The negative numbers represent your short positions while the positive numbers the long positions.

And here you have your Alpha Model.

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