This study quantifies the material distortions in empirical return distributions caused by finite samples and overlapping returns, raising caution about the reliability of raw historical simulation for risk management. We aim to provide actionable guidance on when historical data can be informative and when it may be misleading.
Using controlled simulations of a Geometric Brownian Motion – a best-case scenario of stationarity and ergodicity – we isolate the distorting effects of sample size (1–100 years) and overlapping windows on distributional accuracy, employing Kolmogorov-Smirnov, Anderson-Darling and other distance metrics.
Even a century of daily data leaves the mean estimated with substantial relative uncertainty and tail estimates highly variable. Overlapping returns improve nominal sample size and estimates of central tendency but introduce dependencies that materially distort tail risk, inflating Anderson-Darling statistics by a factor of 14 in our simulations. The trade-off is purpose-dependent: overlapping windows help describe the distribution body but hinder tail-risk inference.
The analysis relies on ideal conditions rarely met in real markets. This limitation is intentional: documenting distortions under these favorable conditions establishes a conservative lower bound. In practice, with non-stationary, path-dependent data, distortions are likely larger.
Risk measures calibrated from raw historical return distributions should generally avoid overlapping data for tail-risk applications. Overlapping windows may be acceptable for descriptive visualization or central-shape estimation, provided that effective sample size is reported. Even the finite-sample results alone call for a move from naive historical extrapolation to prudence, combining historical estimates with scenario analysis, robustness checks and forward-looking distributions.
This controlled lower-bound study shows that even under ideal conditions, finite samples and overlapping windows materially affect distributional estimation, especially in the tails, with clear implications for risk measurement. To our knowledge, it is the first to systematically quantify these distributional biases.
