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Read quantile-quantile (QQ) plots to infer the features of a distribution, such as its tails.

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

The idea

A quantile-quantile (QQ) plot graphs the quantiles of a sample against the quantiles of a reference distribution, usually the normal. If the sample has the reference shape, the points fall on a straight line. The intercept shows the location difference and the slope shows the scale difference. A line with intercept 0.2 and slope 1.8 against the standard normal means a normal with mean 0.2 and standard deviation 1.8. Departures reveal features. With the sample on the vertical axis, points below the line at the left end and above it at the right end indicate fatter tails than the normal. A concave curve indicates negative skew, with a long left tail of losses. A few stray points signal outliers. A typical error is reading the direction of the bend backwards. Exam point: fat tails mean normal VaR understates high-confidence risk.

Formulas

Straight line on a normal QQ plot

sample quantile = a + b × z

a
the intercept, the location (mean) of the sample
b
the slope, the scale (standard deviation) of the sample
z
the standard normal quantile on the horizontal axis

Holds only when the points lie on a line, so the sample is normal. The slope is already a standard deviation; taking its square root is a trap.

A practice question

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

Which statement describes a quantile-quantile (QQ) plot used to examine a return distribution?

  1. It plots cumulative sample frequencies against time; a rising line means that risk is increasing.Why not: Describes a cumulative frequency or time series chart; a QQ plot has quantiles on both axes.
  2. It plots sample quantiles against the quantiles of a reference distribution; a straight line means the same shape up to location and scale. - correct
  3. It plots each return against the previous return; a straight line means there is no autocorrelation.Why not: Describes a lag scatter plot for serial dependence, not a comparison of distributions.
  4. It overlays the sample histogram on the reference density; the areas must match exactly to be normal.Why not: Describes a histogram comparison, which is not the quantile against quantile comparison of a QQ plot.

A QQ plot pairs each sample quantile with the same-probability quantile of a reference distribution. If the shapes agree, the points lie on a line whose slope and intercept reflect scale and location differences.

A harder one

In a QQ plot of 250 daily P&L observations against the normal distribution, all points lie close to a straight line except the single lowest observation, which sits far below it. What is the best interpretation?

  1. The whole distribution is fat-tailed, so a Student t distribution should replace the normal everywhere.Why not: One stray point does not show a systematic fat tail; the other tail points follow the line.
  2. The distribution has lighter-than-normal tails, because most points lie close to the line.Why not: Close agreement with the line indicates normal-like tails, not lighter ones.
  3. The distribution is positively skewed, because only one tail contains an unusual observation.Why not: An unusually large loss in the left tail would point to negative skew, not positive skew.
  4. The body looks normal and the extreme loss is an outlier to investigate, as it may be an error or a rare event. - correct

Systematic fat tails show up as bending at the ends. A single extreme point with the rest on the line signals an outlier whose cause (data error or a genuine rare event) should be checked.

The hardest tier

Standardised daily returns of a USD 40 million position (daily standard deviation 1.5 percent, mean zero) are plotted against standard normal quantiles. At the normal 1 percent quantile of -2.33, the sample quantile is -3.10. Relying on the plot's tail value, by how much does the normal-based 99 percent VaR understate the VaR, in USD?

  1. USD 0.46 million - correct
  2. USD 1.86 millionWhy not: Reports the VaR implied by the plot rather than the shortfall of the normal VaR relative to it.
  3. USD 1.40 millionWhy not: Reports the normal-based VaR itself rather than the gap between the two measures.
  4. USD 0.77 millionWhy not: Leaves the gap in standardised units, without scaling by volatility and position size.

Normal VaR = 2.33 x 0.015 x 40 = USD 1.398 million; plot-implied VaR = 3.1 x 0.015 x 40 = USD 1.86 million. Gap = (3.1 - 2.33) x 0.015 x 40 = USD 0.46 million.

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