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Outline the quantile function along with estimators that are based on quantiles.

FRM® Part I · Quantitative Analysis · Random Variables · FRM1.QA.2.e

The idea

The quantile function q(p) is the inverse of the cumulative distribution function F. It takes a probability p and returns the value x below which a fraction p of the probability lies, so F(x) = p. It converts a probability into a point on the scale of the variable, so the median is q(0.5) and a 99% value at risk is the 99th percentile of the loss distribution. For a normal variable with mean μ and standard deviation σ, the quantile is: Quantile = μ + z × σ. Here z is the standard normal quantile at p, which is 1.645 at 95% and 2.326 at 99%, one-tailed. For a discrete variable, q(p) is the smallest value x whose cumulative probability F(x) is at least p. Quantile-based estimators are robust to outliers. The interquartile range measures spread: IQR = q(0.75) − q(0.25). A common robust skewness measure divides by the interquartile range: Skewness = (q(0.75) + q(0.25) − 2 × q(0.5)) ÷ IQR. It is negative when the lower quartile is further from the median than the upper one. Watch the direction: lower-tail quantiles need the negative z.

Formulas

Quantile of a normal variable

q(p) = μ + z × σ

μ
the mean
σ
the standard deviation
z
the standard normal quantile at probability p

z is 1.645 at 95% and 2.326 at 99%, one-tailed. Lower-tail quantiles need the negative z.

Quantile-based skewness

Skewness = (q(0.75) + q(0.25) − 2 × q(0.5)) ÷ IQR

q(p)
the quantile at probability p
IQR
the interquartile range, q(0.75) − q(0.25)

Robust to outliers. Negative when the lower quartile is further from the median than the upper one.

A practice question

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

How is the quantile function q(p) of a random variable related to its cumulative distribution function F(x)?

  1. It is the inverse of F: q(p) returns the value x for which F(x) = p. - correct
  2. It is the same function as F, only plotted with probability on the horizontal axis.Why not: Treats the quantile function as a relabelled CDF; it maps probability to value, the opposite direction.
  3. It is the derivative of F with respect to x.Why not: Describes the probability density function, not the quantile function.
  4. It returns the upper-tail probability P(X > x) for each x.Why not: Describes the survival function 1 - F(x), not the inverse of F.

Given a probability p, q(p) = F^-1(p) returns the point below which a fraction p of the distribution lies, so q(0.5) is the median.

A harder one

Monthly returns are normal with a mean of 1.0 percent and a standard deviation of 4.0 percent. What is the 5th percentile of monthly returns, to the nearest hundredth of a percent?

  1. 7.58 percentWhy not: Adds 1.645 sigma to the mean, which gives the 95th percentile rather than the 5th.
  2. -5.58 percent - correct
  3. -6.58 percentWhy not: Omits the mean and uses only -1.645 sigma.
  4. -6.84 percentWhy not: Uses z = 1.96, the two-sided 95 percent value, rather than the one-sided 1.645 for the 5th percentile.

q(0.05) = mu + z sigma with z = -1.645: 1.0 - 1.645(4.0) = -5.58 percent.

The hardest tier

A risk analyst estimates quantiles of a return distribution: q(0.025) = -4.6, q(0.25) = -1.5, q(0.75) = 1.3 and q(0.975) = 5.0 percent. Using the quantile-based kurtosis (q(0.975) - q(0.025)) / (q(0.75) - q(0.25)), whose value for a normal distribution is about 2.91, what is the estimate and what does it imply?

  1. 2.91, indicating tails equal to a normal distribution.Why not: Quotes the normal benchmark rather than computing the estimate from the given quantiles.
  2. 0.29, indicating much lighter tails than a normal distribution.Why not: Inverts the ratio (IQR divided by the 2.5 to 97.5 percent range).
  3. 3.43, indicating heavier tails than a normal distribution. - correct
  4. 6.86, indicating much heavier tails than a normal distribution.Why not: Divides by half the interquartile range (the semi-interquartile range) instead of the full IQR.

Numerator = 5.0 - (-4.6) = 9.6; denominator = 1.3 - (-1.5) = 2.8. Ratio = 3.43, above the normal value of 2.91, so the tails are heavier than normal.

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