The main purpose of this paper is to propose a methodology to obtain a hedge fund tail risk measure. Our measure builds on the methodologies proposed by \citet*{ag15} and \citet*{aagvg15}, which rely in solving dual minimization problems of Cressie Read discrepancy functions in spaces of probability measures. Due to the recently documented robustness of the Hellinger estimator (Kitamura et al., 2013), we adopt within the Cressie Read family, this specific discrepancy as loss function. From this choice, we derive a minimum Hellinger risk-neutral measure that correctly prices an observed panel of hedge fund returns. The estimated risk-neutral measure is used to construct our tail risk measure by pricing synthetic out-of-the-money put options on hedge fund returns of ten specific categories. We provide a detailed description of our methodology, extract a Tail risk hedge fund factor for Brazilian funds, and relate it to a commonly adopted market volatility measure.