How Much Does Correlation Reduce Backtest Selection Bias?

With 252 observations and 100 zero-edge trials, the independent benchmark has a median winning annualized Sharpe of about 2.48. Under this simplified dependence model, a latent correlation of 0.50 lowers the median winner to about 1.78, while a latent correlation of 0.90 lowers it to about 0.79.

The upper tail remains important. Even at 0.90 latent correlation, the 99th-percentile winner across 100 trials is still about 3.03. Correlation reduces the breadth of the search, but selecting the best result from many related variants can still produce an unusually strong-looking backtest by chance.

The table holds the sample length at 252 observations and the search breadth at 100 candidate strategies. Only the copula dependence parameter changes.

The independent-search benchmark shows that under independent Normal zero-edge returns, a single strategy's annualized sample Sharpe can be written as a Student-t statistic scaled by the annualization frequency. This correlated-search benchmark preserves that exact marginal distribution for every candidate strategy.

Dependence is introduced with a one-factor Gaussian copula. For each simulated search, candidate latent scores share a common Normal factor with loading sqrt(rho) and retain an idiosyncratic component with loading sqrt(1-rho). Each latent score is converted through the Normal CDF and then through the Student-t quantile function used by the independent benchmark. The maximum transformed Sharpe is the reported winner.

The published grid uses 20,000 deterministic simulations per cell with seed 29,029. This construction has a useful validation property: when rho is zero, the simulation converges to the exact independent-trial order statistic from the selection-bias benchmark.

The input is a latent copula dependence parameter. It is not a direct estimate of the correlation between two live strategy return streams, and it is not a universal conversion from a parameter grid into an “effective number of independent trials.” Real research dependence can be uneven, clustered, adaptive, path-dependent, and affected by researcher choices that a one-factor model cannot represent.

The useful conclusion is narrower: treating every tested variant as independent can materially overstate search breadth, but treating correlated variants as though multiplicity disappears is also wrong. The appropriate hurdle depends on the research process that generated the winner.

See the search-adjusted Sharpe threshold for the exact independent-trial hurdle and the multiple-testing calculator for family-wise false-positive arithmetic.

The published grid crosses 20, 100, and 500 candidate strategies with latent correlation values of 0, 0.25, 0.50, 0.75, and 0.90. It reports the median, 95th percentile, and 99th percentile winning Sharpe for each cell.

Download CSV · Download JSON

This study is available for factual citation and reuse. Preserve the stated assumptions and limitation when they materially affect interpretation, and use the stable canonical URL rather than a temporary search or distribution link.