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I am trying to understand how factor loadings in a general factor model are computed. For simplicity sake, lets assume a simple model:

$$ R = B \times F + \epsilon $$

$$ R = N \times 1 $$ $$ B = N \times K $$ $$ F = K \times 1 $$

where $B$ are the factor loadings and $F$ is the factor return and $N$ is the number of securities.

The way this equation is setup, a factor that is common to all $N$ assets, like GDP, inflation, or market index, can easily be plugged in as a factor return ($F$) into the above equation. But how about financial ratios? Let's say for example the PE ratio. The PE ratio is different for each security, which would in turn give us a different factor loading for each security. But how can we plugin a factor return for PE ratio into $F$; its different for each security.

Or is my understanding of using fundamental ratios in a factor model totally off?

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Your looking at the dependent variable "price" against independent "factors". Its about finding statistical correlation in the factors to price.. – cdcaveman Feb 20 '13 at 8:48
You seem to be confusing alpha models and risk models. See this previous question. – chrisaycock Feb 20 '13 at 12:10

If you are doing something cross-sectional (like Fama-Macbeth regressions) you can just use the ratios where you would put the factor loadings (i.e. betas from the time series regs). You probably want to do some kind of transformation on the ratio to make it well-behaved first though. If you want an actual factor based on the ratio, you can use "factor mimicking portfolios", which are basically portfolios that are long stocks with high values of the ratio and short stocks with low values, with some attempt to orthogonalize them to other variables. See Ken French's website. The return time series from this portfolio is your factor.

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