Expected Shortfall for Discontinuous Random Variables with Application to Accounting Values

We consider the expected shortfall of accounting values, or, mathematically speaking, of random variables that are not continuous (i. e. whose cumulative distribution function is not continuous). Acerbi and Tasche show that one has to abandon the conditional expectation in order to maintain expected shortfall as a coherent risk measure. They also provide a definition of expected shortfall of discontinuous random variables which fulfills the axioms of coherence.Provided that the market values of the assets are continuous random variables, we show that the expected shortfall of the accounting values can still be interpreted as a conditional expectation, where the condition refers to the market values