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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.

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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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  • $\begingroup$ Thanks for your reply, I fail to see why OLS would be unconventional. It is similar to finding an alpha when controlling for the Fama-French factors right? Just instead of the FF factor I use the benchmark return as a factor. $\endgroup$ – incognito Oct 23 '19 at 14:25
  • $\begingroup$ for some of the reasons I'd mentioned. calculating 'alpha' is highly dependent on the factors you include in your regression. you could conceivably deduce portfolio 1 is superior using FF factors and portfolio 2 using, say, AQR factor returns. aside from any of this, it simply isn't a standard approach for this sort of comparison in the industry. as I'd mentioned, more common is comparing a set of performance metrics (eg, sharpe, sortino, max DD, potentially skew/kurtosis). $\endgroup$ – Chris Oct 23 '19 at 17:00

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