The distribution of the stochastic dominance index for risk measurement

The most employed approaches about operational risk (see Cruz, 2002), considers a quantitative approach and calculate the value at risk to derive the total economic capital required to protect an institution against possible losses. Figini and Giudici (2010) show that operational risk measurement is possible also for data in ordinal scale, and suggest as measure of risk the stochastic dominance index (SDI). Operational data for risk measurement are typically summarized in a matrix of J business lines and I event types. For each event type, in a specific business line, we have two different measures: the frequency and the severity expressed in an ordinal scale. To summarize them in a tendency measure, we structure a data set which counts, for each event type-business line and for a given severity, the absolute frequency. In this paper we derive the distribution of SDI, thus allowing exact inference to be performed. Confidence intervals and testing rules are particularly useful in the context of operational risk, as they can help to prioritize and prevent operational failures in a quality control framework, so to effectively reduce the impact of risks ex ante and not ex post. We also derive the distribution of summary means of such measures for all business lines, particularly important in some applications.