The expected shortfall (ES), also called the conditional value-at-risk, is a tail-risk measure used to accommodate some shortcomings of VaR. The expected shortfall calculates the expected return (loss) based on the x% worst occurrences. As such, it relationship towards VaR becomes more clear. Specifically, the VaR tells you that the loss will not be greater than a certain amount over a certain period with x% probability. The expected shortfall tells you what to average loss will be over a certain period given the VaR has been breached.
Expected shortfall formula
The ES can be calculated both using empirical data or simulated data or even based on closed form solution for a given distribution. First, calculate the daily (weekly etc.) returns of a portfolio over a certain period (1 year, 5 years, etc.). Second rank the returns for smallest to largest. Third, calculate the mean of the x% (eg 5%) worst returns. Now you have calculated the x% daily, (weeklky) ES.
Expected shortfall intuition
The expected shortfall has as benefit to counter possible manipulation on the use of VaR. This is because the VaR only measures the amount at risk up to a certain percentile. Therefore risk could be intentionally or unintentionally be built up right after the VaR threshold without being observed at all. This could for example happen through an excessive amount of option-writing
The ES is a tail-risk measure developed to counter some shortcomings of VaR. It used used both for managing market and credit risk of investment portfolios. The expected shortfall tells you what the expected loss is given the x% worst return occurrences in the portfolio.
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