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Obtain risk measures by estimating the relevant quantiles of the distribution.

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

The idea

Every standard risk measure is a function of the quantiles of the P&L or loss distribution. VaR at confidence α is the (1 − α) quantile of P&L with the sign reversed, which equals the α quantile of the loss distribution. Expected shortfall is the average of VaRs over all confidence levels from α to 100 percent. It can be approximated by slicing the tail into equal probability slices and averaging the VaR at each slice. More generally, a spectral risk measure is a weighted average of quantiles. Order the quantiles from the better to the worse outcome. The measure is coherent if the weights are non-negative, sum to 1 and never fall towards the worse tail, so a more extreme loss gets at least as much weight. For a normal distribution, the quantile (Quantile) is the mean plus z standard deviations. Quantile = μ + z × σ. Use the lower tail for long positions and the upper tail of returns for short positions. Exam point: first decide which tail is relevant, then estimate that quantile.

Formulas

Quantile of a normal distribution

Quantile = μ + z × σ

μ
the mean
z
the standard normal quantile at the chosen probability, negative in the lower tail
σ
the standard deviation

For a long position VaR is the lower-tail quantile of P&L with the sign reversed; a short position uses the upper tail of returns.

Spectral risk measure as a weighted average of quantiles

M = Σ wᵢ × qᵢ

wᵢ
the weight on quantile i
qᵢ
the i-th quantile, ordered from better to worse outcome

Coherent only if the weights are non-negative, sum to 1 and never fall towards the worse tail. Equal weights over the tail give expected shortfall.

A practice question

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

A system reports the 99 percent one-day VaR of a long position. Which quantile of the P&L distribution is it estimated from?

  1. The 99th percentile of P&L, with the sign reversed to give a positive loss.Why not: Picks the wrong tail: the 99th percentile of P&L is a large profit, not a loss.
  2. The 1st percentile of P&L (the 99th percentile of losses), with the sign reversed to give a positive loss. - correct
  3. The average of all P&L outcomes below the 1st percentile.Why not: Describes expected shortfall, which averages the tail beyond the quantile instead of taking the quantile.
  4. The standard normal 99th percentile of 2.326, whatever the mean and standard deviation of P&L.Why not: Omits the scaling by the P&L standard deviation and the adjustment for the mean.

VaR at confidence alpha is a quantile: the (1 - alpha) quantile of P&L, equivalently the alpha quantile of the loss distribution. At 99 percent that is the 1st percentile of P&L, sign-reversed.

A harder one

P&L is normally distributed with mean zero and standard deviation USD 10 million. A risk manager approximates the 95 percent expected shortfall by splitting the 5 percent tail into five equal-probability slices and averaging the VaRs at the confidence levels 95.5, 96.5, 97.5, 98.5 and 99.5 percent. The approximation is closest to:

  1. USD 16.45 millionWhy not: Takes the 95 percent VaR alone, which is the start of the tail rather than its average.
  2. USD 23.26 millionWhy not: Takes the 99 percent VaR alone, which is deeper in the tail than the tail average.
  3. USD 20.43 million - correct
  4. USD 19.31 millionWhy not: Averages VaRs at 95, 96, 97, 98 and 99 percent, using slice edges that leave out the far end of the tail.

The normal quantiles are 1.695, 1.812, 1.960, 2.170, 2.576; their mean is 2.043, and times sigma of 10 gives USD 20.43 million. The exact normal ES is about 20.63.

The hardest tier

A risk measure is defined as a weighted average of the VaRs at 95, 97.5 and 99 percent confidence, with weights 0.2, 0.3 and 0.5 respectively, so that more extreme losses get more weight. P&L is normal with mean zero and standard deviation USD 10 million. The measure is closest to:

  1. USD 19.77 millionWhy not: Uses equal weights of one third, ignoring the stated weights that emphasise the extreme quantile.
  2. USD 20.80 million - correct
  3. USD 18.76 millionWhy not: Reverses the weights so the least extreme quantile gets the largest weight.
  4. USD 23.26 millionWhy not: Uses only the 99 percent VaR and ignores the weighted mixture of quantiles.

Normal quantiles are 1.645, 1.960 and 2.326. Weighted sum: 0.2 x 1.645 + 0.3 x 1.960 + 0.5 x 2.326 = 2.0801; times 10 gives USD 20.80 million. Weights rising with severity make this a valid spectral measure.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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