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We make an extensive empirical study of the market impact of large orders(metaorders) executed in the U.S. equity market between 2007 and 2009. We showthat the square root market impact formula, which is widely used in theindustry and supported by previous published research, provides a good fit onlyacross about two orders of magnitude in order size. A logarithmic functionalform fits the data better, providing a good fit across almost five orders ofmagnitude. We introduce the concept of an "impact surface" to model the impactas a function of both the duration and the participation rate of the metaorder,finding again a logarithmic dependence. We show that during the execution theprice trajectory deviates from the market impact, a clear indication ofnon-VWAP executions. Surprisingly, we find that sometimes the price startsreverting well before the end of the execution. Finally we show that, althoughon average the impact relaxes to approximately 2/3 of the peak impact, theprecise asymptotic value of the price depends on the participation rate and onthe duration of the metaorder. We present evidence that this might be due to aherding phenomenon among metaorders.

Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate