Interpreting OLS Alpha for Skewed Portfolio Returns
Summary
The document considers how to compare two optimized portfolios whose returns are positively skewed and have positive kurtosis. The author asks whether to estimate each portfolio's alpha against value-weighted and equally weighted benchmarks, or regress one portfolio's returns on the other's, and worries that non-normal returns could bias OLS alpha. The response says the right analysis depends on the investment objective and that familiar performance measures may be more useful for an investment decision than regressions against constructed benchmarks.
It also explains that regression alpha is not automatically a measure of skill: unless the model includes all relevant sources of risk as independent variables, the intercept can absorb returns associated with omitted exposures. The response does not derive a bias from skewness or kurtosis, assess OLS assumptions for a particular dataset, or recommend a specific estimator. Its guidance is therefore conceptual; a valid comparison requires a clearly defined objective, suitable metrics, and a defensible risk model.
Key ideas
- The choice between portfolio regressions and performance metrics depends on the comparison objective.
- Skewness and kurtosis alone do not establish from this discussion that OLS alpha is biased.
- An alpha intercept can include returns associated with risk exposures omitted from the regression.
- A meaningful risk-adjusted alpha requires independent variables that capture relevant sources of risk.
- The response gives conceptual guidance but no dataset-specific estimator or statistical diagnosis.
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Full text
# Highly skewed (and positive kurtosis) return distribution as a dependent variable # Highly skewed (and positive kurtosis) return distribution as a dependent variable I have two set of optimized returns over a period of time and called this portfolio 1 and 2 and two benchmark portfolio (a value-weighted and equally-weighted benchmark). I want to see the difference in performance between portfolio 1 and 2. In order to gauge the performance (alpha) of the Portfolio 1&2& I have regressed the two (EW & VW) benchmark upon the returns of portfolio 1 & 2. This results in an alpha. However, I could have also regressed (OLS) returns 1 & 2 upon eachother and examine the alpha. However, and this is the problem, the returns of both portfolio 1 and 2 are highly (positively) skewed and have positive kurtosis. How will this affect the alpha? The bottom line is: how to compare their performance using OLS? 1) Show the alpha of both portfolio (1 & 2) over the benchmark portfolio? So that at least the bencmark portfolio is somewhat normally distributed? And compare those. 2) Or regress both (skewed) portfolio upon eachother? I know that OLS does not assume normality of variables (only error terms) however I believe the alpha will somehow be biased, but not sure why. ## Answer by Chris (score 1) https://quant.stackexchange.com/a/49319 It's not entirely clear what your objective is. If you'd like to compare portfolio 1 to portfolio 2 as part of an investment decision, using some standard, or set of, performance metric(s) is probably going to be more useful than regression against manufactured benchmarks. As to the regressions, alpha here is only going to be meaningful in absolute terms if you incorporate all sources of risk as independent variables in your regression. Otherwise the alpha term, in addition to potential actual alpha, will simply incorporate return that isn't accounted for by return streams you do include. Again, depending on what you're trying to do, using OLS as you describe is somewhat unconventional.
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