Table 3 Out-of-sample portfolio risk—quantified in terms of variance—with and without detrending the returns by subtracting their ensemble average. \(P_{1,2}^N\) (with \(N = 20, 50\)) refer to two different portfolios made of randomly selected S&P stocks, whereas \(q = N/T\) denotes the portfolios’ “rectangularity ratio” (i.e., the ratio between the number of stocks and the length of the in-sample time window used to compute correlations and portfolio weights). The two top rows refer to portfolios whose weights are computed based on the raw returns, whereas the two bottom rows refer to portfolios whose weights are computed based on the detrended returns. In the latter case, the detrending is only performed in-sample to compute correlations and weights, and the out-of-sample risk is computed by retaining such weights on new raw returns. The numbers reported in each case refer to the average out-of-sample risk computed over a set of 30-days long non-overlapping time windows spanning the period September 2014–November 2018.

From: Maximum entropy approach to multivariate time series randomization