Risk Estimation with a Time-Varying Probability of Zero Returns[On the Coherence of Expected Shortfall]

The probability of an observed financial return being equal to zero is not necessarily zero, or constant. In ordinary models of financial return, however, for example, autoregressive conditional heteroskedasticity, stochastic volatility, Generalized Autoregressive Score, and continuous-time models, the zero probability is zero, constant, or both, thus frequently resulting in biased risk estimates (volatility, value-at-risk [VaR], expected shortfall [ES], etc.). We propose a new class of models that allows for a time-varying zero probability that can either be stationary or nonstationary. The new class is the natural generalization of ordinary models of financial return, so ordinary models are nested and obtained as special cases. The main properties (e.g., volatility, skewness, kurtosis, VaR, ES) of the new model class are derived as functions of the assumed volatility and zero-probability specifications, and estimation methods are proposed and illustrated. In a comprehensive study of the stocks at New York Stock Exchange, we find extensive evidence of time-varying zero probabilities in daily returns, and an out-of-sample experiment shows that corrected risk estimates can provide significantly better forecasts in a large number of instances.