Charta is on Google Play: practise on the go. Get the app
Charta

Describe the power law and explain how it serves for distributions that are not normal.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.e

The idea

A power law describes the tail of a distribution. The probability that the loss exceeds x is: P = k × x^(−α). Here k is a constant and α is the tail index. Taking logs, with L for ln P and s for ln x, gives: L = ln k − α × s. So a plot of log probability against log x is a straight line with slope −α. A smaller α means a fatter tail. Probabilities fall polynomially in x, far more slowly than the normal tail, which falls roughly like an exponential of minus x² over 2. A power law therefore gives much more weight to extreme losses. Only moments of order below α are finite. If α is below 2 the variance is infinite, and if α is below 4 the kurtosis is infinite. Risk managers use it to estimate extreme tail probabilities. If the threshold x is multiplied by a, the tail probability is multiplied by: Factor = a^(−α). Doubling the threshold therefore cuts the probability by a factor of 2 to the power α.

Formulas

Power-law tail probability

P(X > x) = k × x^(−α)

k
a constant
x
the loss threshold
α
the tail index; smaller means a fatter tail

Holds in the tail only. Moments of order α or above are infinite, so with α below 2 the variance does not exist.

Scaling a power-law tail probability

P(X > a × x) = a^(−α) × P(X > x)

a
the multiple applied to the threshold
α
the tail index
P(X > x)
the known tail probability at threshold x

The constant k cancels. Doubling the threshold divides the probability by 2^α, far less than the normal tail would fall.

A practice question

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

Which expression describes a power law for the tail of a loss distribution, where alpha is the tail index and k a constant?

  1. P(X > x) = k e^(-alpha x)Why not: This is exponential decay, which falls much faster than a power law.
  2. P(X > x) = k x^(-alpha) - correct
  3. P(X > x) = k x^(alpha)Why not: A positive exponent would make the tail probability grow with x.
  4. P(X > x) = k ln(x) / alphaWhy not: Has no basis as a probability tail and does not decline as a power of x.

A power law has tail probability proportional to x raised to a negative power. Taking logs gives a straight line of slope -alpha on a log-log plot.

A harder one

A trading desk's daily loss follows a power law in the tail with a tail index of 3. The probability that the loss exceeds 3% is 1.00%. The probability that it exceeds 6% is closest to:

  1. 0.500%Why not: Halves the probability because the threshold doubled, ignoring the exponent.
  2. 0.250%Why not: Uses an exponent of 2 instead of the tail index 3.
  3. 0.125% - correct
  4. 0.016%Why not: Doubles the exponent (2^-6), applying the power twice.

P(>6%) = 1.00% x (6 / 3)^(-3) = 1.00% x 1/8 = 0.125%.

The hardest tier

A loss distribution has a power-law tail with tail index 1.8. Which statement about its moments is correct?

  1. The mean is finite but the variance is infinite, so sample standard deviations will be unstable. - correct
  2. Both the mean and the variance are infinite.Why not: The mean is infinite only when the tail index is 1 or below; here it exceeds 1.
  3. The variance is finite, but the kurtosis is infinite.Why not: Variance is finite only if the tail index exceeds 2; kurtosis becomes infinite below 4.
  4. All moments are finite because probabilities still decline as x rises.Why not: Declining probabilities do not guarantee finite moments; only moments of order below the tail index exist.

A moment of order m is finite only if m < alpha. With alpha = 1.8, the first moment exists and the second does not, so variance-based risk estimates are unreliable.

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

Start free →

More in Measuring Returns, Volatility, and Correlation