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FRM Part I · FRM Exam Part I · Measures of Financial Risk

A risk manager estimates the one-day 95% VaR of a portfolio at USD 2.0 million. Which statement about the one-day 95% expected shortfall (ES) of the same portfolio, estimated from the same loss distribution, is correct?

ES is at least USD 2.0 million because it averages all losses in the tail beyond the 95% VaR cutoff. Each of those losses is at or above the VaR, so their average cannot be lower than the VaR figure.

  1. AES is at least USD 2.0 million, because it is the average loss over the tail beyond the VaR quantileCorrect
  2. BES is at most USD 2.0 million, because it is a conditional average
  3. CES equals USD 2.0 million, because both use the 95% confidence level
  4. DES is the loss exceeded with 5% probability, which is the VaR figure itself

Explanation

ES averages the losses that are at or beyond the VaR threshold. Every such loss is at least the VaR, so the average cannot be below USD 2.0 million. It equals VaR only if the tail has no loss above the threshold.

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