Risk Estimation when the Zero Probability of Financial Return is Time-Varying

The probability of an observed financial return being equal to zero is not necessarily zero. This can be due to liquidity issues (e.g. low trading volume), market closures, data issues (e.g. data imputation due to missing values), price discreteness or rounding error, characteristics specific to the market, and so on. Moreover, the zero probability may change and depend on market conditions. In ordinary models of risk (e.g. volatility, Value-at-Risk, Expected Shortfall), however, the zero probability is zero, constant or both. We propose a new class of models that allows for a time-varying zero probability, and which nests ordinary models as special cases. The properties (e.g. volatility, skewness, kurtosis, Value-at-Risk, Expected Shortfall) of the new class are obtained as functions of the underlying volatility and zero probability models. For a given volatility level, our results imply that risk estimates can be severely biased if zeros are not accommodated: For rare loss events (i.e. 5% or less) we find that Conditional Value-at-Risk is biased downwards and that Conditional Expected Shortfall is biased upwards. An empirical application illustrates our results, and shows that zero-adjusted risk estimates can differ substantially from risk estimates that are not adjusted for the zero probability.