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Derive expected shortfall (ES) from a sample of profit-and-loss (P&L) or return observations.

FRM® Part II · Market Risk Measurement and Management · Estimating Market Risk Measures: An Introduction and Overview · FRM2.MR.1.c

The idea

Expected shortfall (ES) is the average loss in the tail beyond VaR, that is, the expected loss given that the loss exceeds VaR. From a sample, sort the losses from worst to best, count how many fall in the tail, and average those worst losses. Tail count = n × (1 − confidence level). With 200 observations at 95 percent confidence, the tail count is 10, so ES is the mean of the 10 largest losses. If the tail count is not a whole number, include the boundary observation fractionally: a tail count of 2.5 means the two worst losses plus half of the third, divided by 2.5. Report ES as a positive loss, and convert returns to money by multiplying by the position value. Common errors are averaging the wrong number of observations and confusing ES with VaR. ES is at least as large as VaR and responds to how bad the tail losses are. Exam point: count the tail correctly.

Formulas

Expected shortfall from a sample of losses

ES = Σ Lᵢ ÷ k

Lᵢ
the i-th largest loss, summed over the k worst
k
the tail count, n × (1 − c)

Report ES as a positive loss. If k is not whole, count the boundary loss fractionally: k = 2.5 uses the two worst plus half the third, divided by 2.5.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly defines expected shortfall at the 95 percent confidence level?

  1. The expected loss conditional on the loss exceeding the 95 percent VaR. - correct
  2. The loss that is exceeded with probability 5 percent, which is the 95 percent VaR itself.Why not: Describes VaR, a single quantile, rather than the average of losses beyond it.
  3. The average loss over all observations in the sample, including profitable days.Why not: Averages the whole distribution rather than only the tail beyond VaR.
  4. The largest loss observed in the sample, since it is the worst case beyond VaR.Why not: Uses the maximum loss, which depends on a single observation and ignores the tail average.

Expected shortfall is the average of losses that are worse than VaR, conditional on exceeding it. It is therefore always at least as large as VaR and reflects the severity of tail losses.

A harder one

A USD 80 million portfolio has 80 daily returns. The five worst are -4.2%, -3.5%, -3.1%, -2.6%, -2.4% (worst first). The 95 percent expected shortfall in USD is closest to:

  1. USD 2.53 millionWhy not: Averages five returns, but a 5 percent tail of 80 observations contains only four.
  2. USD 1.92 millionWhy not: Reports the 5th-worst return as a loss, which is not a tail average and is below the tail ES.
  3. USD 3.36 millionWhy not: Uses only the single worst return instead of averaging over the tail.
  4. USD 2.68 million - correct

The tail has 80 x 0.05 = 4 observations. Their average loss is (4.2 + 3.5 + 3.1 + 2.6)/4 = 3.35 percent, and 3.35% x 80 = USD 2.68 million.

The hardest tier

From 250 daily loss observations, the four largest losses, in USD million, are 7.8, 5.6, 4.4, 3.9. Using the convention that the 99 percent tail contains exactly 2.5 observations, so the third-worst loss counts with weight 0.5, the 99 percent expected shortfall is closest to:

  1. USD 6.24 million - correct
  2. USD 7.12 millionWhy not: Adds the full third-worst loss but divides by 2.5 rather than by the matching weight of 3.
  3. USD 6.70 millionWhy not: Ignores the fractional observation and averages only the two worst losses.
  4. USD 5.93 millionWhy not: Rounds the tail up to three whole observations instead of 2.5.

Tail weight is 250 x 0.01 = 2.5 observations. ES = (7.8 + 5.6 + 0.5 x 4.4)/2.5 = 15.6/2.5 = USD 6.24 million.

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