Factor Risk Premia Versus Long-Short Portfolio Sort Returns

Summary

The document compares two approaches to estimating factor returns: cross-sectional regression using asset returns and factor exposures, and sorting assets by a characteristic to form a long-short portfolio. The answer says these approaches do not necessarily estimate the same quantity. In its outline, a researcher first uses time-series regressions to estimate exposures to established factors and a proposed factor, then runs a cross-sectional regression of returns on those exposures to estimate the proposed factor’s risk premium.

A portfolio sort based on the characteristic instead produces a portfolio alpha, according to the response, rather than the factor risk premium. This distinction helps explain why both methods might be used: a sort describes the performance of a particular construction, while the regression framework aims to estimate compensation associated with factor exposure. The answer is brief and gives no equations, assumptions, or empirical comparison. It also does not explain how portfolio weighting, controls, or estimation choices affect the results, so its distinction is a conceptual starting point rather than a complete methodology.

Key ideas

• A cross-sectional regression on estimated factor exposures can be used to estimate a factor risk premium.
• The described regression approach includes time-series estimation of exposures before the cross-sectional step.
• A long-short sort on a characteristic measures the return or alpha of that portfolio construction.
• Portfolio-sort returns and factor risk premia should not be treated as interchangeable quantities.
• The answer outlines the distinction but does not discuss assumptions or implementation details.

Tags

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


# 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?

## Answer by phdstudent (score 2)

https://quant.stackexchange.com/a/72148

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).

Now the first thing you mention is:

- 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 $\lambda_t$.

- The second thing you mention which is a portfolio sort, does not give you a factor risk-premium. But will give you an $\alpha$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.