# 2 methods for estimating factor return - differences between those 2 methods

I have a question for estimating factor return. I’ve found that there seems to be 2 methods for estimating factor return.

First, with return of an asset i(r_i) and factor loadings such as PER, EPS, Momentum etc(B_i), factor return can be estimated by doing regression - cross sectional regression

Second, sorting assets into deciles according to a factor value, and by taking long on the 1st decile and shorting on 10th decile, we can also get factor return.

I think ultimate goals of both methods are same - estimating the return I can expect when I expose myself into a certain factor, the factor return. But what is the difference between them? Are they catching different aspects of factor return? If I can get a factor return just by building Long-Short Portfolio, what is the need of doing a cross sectional regression?

You are confusing two different things. Let's say you have a factor that you identified, call it: $$\lambda_t$$. There are also other factors out there that are widely know. Let me call them: $$F_t$$ (potentially a vector of factors).
1. Run a time-series regressions on the factors $$F_t$$ and $$\lambda_t$$. Then run a cross-section regression of the loadings on those factors. This will get you the factor risk-premium for $$\lambda_t$$.
2. The second thing you mention which is a portfolio sort, does not give you a factor risk-premium. But will give you an $$\alpha$$.
• @geonhwa - The underlying theory is that cross-sectional average returns are driven by each asset's $\beta$ (aka covariance/correlation with) the factors in question. As such - your cross sectional regression should be factor $\beta$ on the right-hand side, and average return on the left-hand side. Commented Sep 9, 2022 at 14:14
• We don't know the values of these $\beta$s (which are so called "population parameters"), so we need to estimate them - the most common method being Time-Series regression of asset return onto the factor. Commented Sep 9, 2022 at 14:16