characteristics of factor portfolios

Question

In the paper Characteristics of Factor Portfolios (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1601414), when it discusses pure factor portfolios, it says that simple style factor portfolios have zero exposure to all other style, country, and industry factors. Could someone help me understand the math for why the style factor portfolios have zero exposure to all other style, country, and industry factors?

Answer 1

Factor Portfolios are by-products of the cross-sectional regression techniques used to estimate style factor returns - the estimates of the return of a set of styles for a particular period. Style Factor Returns are estimated by regression of the returns for all assets in a single period on a matrix containing the styles for each asset in that period.

So, for example, if we are interested in the return of a P/E factor and a P/B factor, we would gather the P/E and P/B for all of our stocks into a matrix of loadings $B$. $B$ would have two columns – one containing P/E and one containing P/B for all assets. We then regress $R$ (a vector containing the returns of all assets) on $B$. OLS regression gives us $f= (B’B)^{-1} B’R$ = the returns of the style factors for this particular period. The rows of $(B’B)^{-1} B’$ are considered to be the factor portfolios.

So, let’s go one step further and look at the loadings of the portfolio on the individual styles by multiplying the factor portfolios with the matrix of loadings. This gives $(B’B)^{-1} B’B = I$ - an identity matrix. Hence, the loadings of each factor portfolio are 1 against the particular style and 0 against any other style.

Intuitively, this result makes sense. OLS is trying to find the best estimate of the returns to a particular style. So, OLS tries to find a set of portfolios that have the following characteristics – (1) a unit exposure to the style of interest, (2) zero exposure to any other style, and (3) minimal error by maximally diversifying the portfolio. We then look at the returns of that portfolio and say that that is the estimate of the factor return. Characteristic (1) ensures that the returns of the portfolio are approximately the returns of the style factor. Characteristic (2) ensures that the return of the factor portfolio is uncontaminated by other factors. Characteristic (3) ensures that there is minimal error on the estimate of the style factor return.