Active share is the measure the debate was built on. Introduced by Martijn Cremers and Antti Petajisto in the Review of Financial Studies in 2009, it compares a fund's holdings with its benchmark's holdings position by position, summing the absolute differences in weight and halving the result. A fund holding the index exactly scores zero. A fund with no overlap at all scores 100 percent. The appeal of the measure is that it looks at the portfolio rather than at the returns, so a manager cannot produce a high score by accident of market movement.
Tracking error and R-squared come at the same question from the returns side. Tracking error is the volatility of the difference between the fund's return and the benchmark's. R-squared is the share of the fund's return movement explained by the benchmark's. ESMA's own description of why the three are used together is the clearest available: "low active share and low tracking error indicate that the portfolio of a fund is close to that of the respective index," and "the higher the R2, the closer the performance of the fund is correlated to that of the benchmark." The three can disagree, and the disagreement is informative. A fund can hold very different stocks and still post a low tracking error if the differences offset each other, which is a genuinely active portfolio rather than a closet indexer.
The one regulatory screen with published thresholds is European, and it is worth reading precisely. In February 2016 ESMA studied UCITS equity funds domiciled in the EU that were not categorized as index-trackers, had more than €50 million under management, had launched before 2005, and charged management fees above 0.65 percent of net asset value. From more than 2,600 such funds it obtained usable data on 1,251 for 2012 to 2014, and applied three sets of criteria. Funds with active share below 60 percent and tracking error below 4 percent came to 15 percent of the sample; below 50 percent and 3 percent, 7 percent; adding an R-squared above 0.95, 5 percent. ESMA was explicit about the limits of the exercise, calling it "a statistical model" whose results "can only be a first step," and it separately noted that it would "consider the merits of developing a general definition of active and passive management." So even the regulator that measured the practice stopped short of defining it.
Size is part of the explanation. Cremers and Petajisto reported that fund size is negatively correlated with active management, and put a shape on it: in their large-cap sample the active share of a marginal dollar held steady at roughly 70 percent from a $10 million fund up to a $1 billion one, then fell to about 60 percent at $10 billion and to about 50 percent for the largest funds. They cautioned that the dispersion around that average is wide, so it describes a population rather than any individual fund. The mechanism is not mysterious. A position large enough to matter in a small fund cannot be bought in a very large one without moving the price, and diversification rules and liquidity constraints push a big portfolio toward the shape of the market it invests in.
What the research says about performance is directional rather than precise. Petajisto's follow-up work in the Financial Analysts Journal in 2013 sorted US equity funds by active share and tracking error and reported that the most active stock pickers beat their benchmarks after fees over the sample period while the least active group underperformed theirs. That is evidence from one researcher's sample over one period, and it should be read as such rather than as a law. The mechanical point underneath it needs no study: a fee charged on an index-like portfolio comes out of an index-like return.