# 3 – M4 L1B 03 Factor Returns As Latent Variables V3

We left off with the idea that the return of any stock, i, can be decomposed into the returns of factors times the stocks exposures to those factors plus an unexplained portion. But wait, this is looking a lot like multiple regression. What’s the difference? Well, regression is one tool you might use to build a factor model. Also, in a factor model, the independent variables or factor returns are usually things we think are there, but have an influence that we can’t measure directly. These are called unobserved or latent random variables. We may need to take other steps to produce these time series before we can run a multiple regression. For example, what if we think a company’s size has something to do with the performance of its stock price? Maybe, smaller companies stock prices produced higher returns? Okay. That seems reasonable, so let’s do a regression. But hang on, what data do we have? We have time series of returns for several stocks. You’ve seen those data before. Okay. So size, how do we measure that? Lets say, we’re specifically thinking that market cap is the metric of interests. Market cap is the market value of a publicly traded companies outstanding shares. It’s the share price multiplied by the number of shares outstanding. So, we’d have market cap for every company in our universe for every day, because it changes over time. That sounds like the data could be organized in a table or a matrix. But to create a single factor return time-series, we’re looking for a single time-series of values that represents the effect of the size factor across stocks. Do you see the challenge we’re facing? How do we create a single time series that represents our idea that small-cap companies generally outperform large-cap companies. Moreover, how do we quantify this effect on the returns of a whole set of stocks? A latent variable is something like this, something nebulous that we want to represent as a time series like a normal variable, but it’s hard to directly measure. What we do in this case to create a single time series for our factor, is we create a theoretical portfolio. This portfolio long small caps and shorts large-caps every day. The time series we seek is the portfolio’s daily return. This is how we quantify our idea that small caps should outperform large-caps and generate a single time series to represent this idea.