Are "market neutral" hedge funds really market neutral?

One can consider the concept of market neutrality for hedge funds as having breadth and depth: breadth reflects the number of market risks to which a fund is neutral, while depth reflects the completeness of the neutrality of the fund to market risks. We focus on market neutrality depth, and propose five different neutrality concepts. Mean neutrality nests the standard correlation-based definition of neutrality. Variance neutrality, Value-at-Risk neutrality and tail neutrality all relate to the neutrality of the risk of the hedge fund to market risks. Finally, complete neutrality. corresponds to independence of the fund to market risks. We suggest statistical tests for each neutrality concept, and apply the tests to a combined database of monthly market neutral hedge fund returns from the HFR and TASS hedge fund databases. We find that around one-quarter of these funds exhibit some significant exposure to market risk.