I've been working through this blog on multi-factor models and I noticed that, when translating factors into Z scores, the author uses cap-weighted means instead of regular means. Why would he do that? Are there certain markets where this practice is unacceptable?
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1$\begingroup$ It just depends on what expected value is important for the metric. For example, see this thread for all the "not incorrect" ways there are to derive an expected value for price-to-book: quant.stackexchange.com/questions/31284/…. I personally prefer the simple aggregate method for constructing standardized scores and ranks because the math is never ambiguous this way. In certain cases, medians may be more appropriate for measures of "typical" central tendency versus "dollar-weighted" centrality. $\endgroup$– David AddisonMay 1, 2017 at 6:19
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2 Answers
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The distribution of equity market capitalizations is such that there are few large outliers (the Apples of the world), but then a lot more smaller companies. The consequence of this is that if you do a regression, such as a cross-sectional regression of the returns in period 1 against the book to price in period 0, then the coefficients will be dominated by the large number of small stocks. In other words, you would estimate a value factor return from period 0 to period 1 that's largely driven by the smaller-cap stocks. By contrast, a weighted least squares approach could be employed to put a greater emphasis on the larger market capitalization stocks.
To the extent that Z-scores in a factor model are approximating this approach, the analysis holds.
There are many other situations where models attempt to account for this. For instance, the Fama-French factors are constructed by splitting the universe into high and low market capitalization stocks. On a cross-sectional basis, one could use dummy variables to account for the different performance between the small and large stocks. More generally, a hierarchical model could be used.
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The choice affects how you construct beta among other things, and in the traditional CAPM sense beta is market cap weighted. There's also market cap squared or cubed weighting schemes. This allows the factor model to more accurately track institutional benchmarks or style of investing for risk purposes. There are also fundamental or free float weighting schemes.
