# Statistical significance of mean returns between two portfolios

Suppose I have developed two versions ($$A$$ and $$B$$) of a factor model for ranking stocks. Both versions of the model use the same scoring system: stocks are percentile ranked within a given universe of stocks from 0-100% based on a weighted sum of standardized factor scores.

The difference between model $$A$$ and model $$B$$ is that $$B$$ includes several new factors (but overall, mostly the same factors), and also uses drastically different weights for the factors.

Now, suppose I construct two portfolios (one for $$A$$, and one for $$B$$) using an identical strategy: I select the Top $$N$$ stocks based on their model ranking, equal-weighted, and rebalance monthly.

I construct the portfolio returns for each month by taking the cross-sectional mean of returns, giving me two vectors (time-series) of monthly returns, $$R_A$$ and $$R_B$$. I have a good sample size, with the time-series starting around 1995.

What I am interested in is comparing the statistics of $$R_A$$ and $$R_B$$. To start, I am wondering how I can determine whether the mean of $$R_B$$ is significantly different from that of $$R_A$$. A t-test would seem like a reasonable tool to use, but then I'm not sure if it's more appropriate to use a paired t-test vs. unpaired, given the similarity between the two models.

Furthermore, suppose I wanted to compare the standard deviations of returns $$\sigma_A$$ and $$\sigma_B$$. How could I assess whether these two statistics differ significantly?

• paired t test is for sample at two different times Commented May 1, 2020 at 21:45
• Here is a test for equality of standard deviations: atomic.phys.uni-sofia.bg/local/nist-e-handbook/e-handbook/eda/… Commented May 1, 2020 at 21:58
• @Permian so unpaired two-sample t-test would be appropriate in this case? Commented May 1, 2020 at 22:55