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Calculate and interpret ES, and set it side by side with VaR to show how the two measures are alike and where they differ.

FRM® Part I · Valuation and Risk Models · Measures of Financial Risk · FRM1.VR.1.d

The idea

Expected shortfall (ES), also called conditional VaR, is the average loss given that the loss reaches or exceeds VaR. ES = E[loss | loss ≥ VaR]. For a normal distribution with zero mean, ES = σ × φ(z) ÷ (1 − c) × portfolio value. Here φ(z) is the standard normal density at the VaR quantile z, and c is the confidence level. At 99 percent, ES is about 2.665σ against a VaR of 2.326σ. From historical data, ES is the average of the worst (1 − c) fraction of outcomes. ES is always at least VaR at the same confidence level, because it averages tail losses at or beyond the VaR. VaR ignores the size of tail losses, while ES responds to it and is subadditive. A common error is averaging only the outcomes strictly beyond the VaR point.

Formulas

Expected shortfall of a zero-mean normal portfolio

ES = σ × φ(z) ÷ (1 − c) × V

σ
the standard deviation of returns
φ(z)
the standard normal density at the VaR quantile z
c
the confidence level
V
the portfolio value

Zero mean and normal returns. At 99% it gives about 2.665σ against a VaR of 2.326σ; dividing by a two-tailed probability is the usual slip.

A practice question

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

Expected shortfall at the 99% confidence level is best described as:

  1. The loss level that is exceeded with 99% probability.Why not: Describes a VaR quantile at the other end of the distribution; it is not an average of tail losses.
  2. The largest loss that has occurred in the historical sample.Why not: Mistakes ES for the maximum loss; ES averages the tail outcomes beyond VaR rather than taking the single worst one.
  3. The expected loss given that the loss exceeds the 99% VaR. - correct
  4. The difference between the 99% VaR and the expected loss.Why not: Subtracts the mean from VaR; this measures a relative VaR, not a conditional tail average.

ES = E[loss | loss > VaR]. It averages all losses in the worst 1% of outcomes and so captures the size of tail losses, unlike VaR.

A harder one

A portfolio worth USD 10 million has normally distributed daily returns with mean zero and standard deviation 2%. Using ES = sigma x phi(z) / (1 - c) x value, where phi(z) is the standard normal density at the VaR quantile z, the one-day 99% ES is closest to:

  1. USD 465,000Why not: Reports the 99% VaR (2.326 x 2% x 10 million) instead of the ES.
  2. USD 413,000Why not: Computes the 95% ES (z = 1.645, divide by 0.05) rather than the 99% ES.
  3. USD 267,000Why not: Divides the density by a two-tailed probability of 2% instead of the one-tail 1%.
  4. USD 533,000 - correct

z = 2.326, phi(z) = 0.02665, so ES = 0.02 x 0.02665 / 0.01 x 10,000,000 = about USD 533,000. This is above the VaR of about USD 465,000.

The hardest tier

Returns are normally distributed. A regulator wants an expected shortfall calibrated to give roughly the same capital as a 99% VaR. For a normal distribution, the ratio of the 97.5% ES to the 99% VaR is closest to:

  1. 0.84Why not: Compares the 97.5% VaR with the 99% VaR (1.960 / 2.326), forgetting that the numerator is an ES.
  2. 1.15Why not: Compares the 99% ES with the 99% VaR (2.665 / 2.326), which uses the same confidence level on both sides.
  3. 1.00 - correct
  4. 1.19Why not: Compares the 97.5% ES with the 97.5% VaR (2.338 / 1.960), again using matching confidence levels.

For a standard normal, 97.5% ES = phi(1.960) / 0.025 = 2.338 and 99% VaR = 2.326, a ratio of 1.005. Under normality the two are almost equal, which is why 97.5% ES was chosen to replace 99% VaR in the Basel market risk framework; with fat tails, ES gives the larger figure.

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