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Explain how operational risk can be measured with the power law.

FRM® Part I · Valuation and Risk Models · Operational Risk · FRM1.VR.7.i

The idea

The power law describes the heavy tail of operational losses. The probability that a loss v exceeds x is: Prob = K × x^(−α). Here K is a scale constant and α is the tail index. Taking logs gives a straight line on a log-log plot: Logprob = ln K − α × ln x. So α is minus the slope, and it can be estimated by regression or from two points: Alpha = ln(p1 ÷ p2) ÷ ln(x2 ÷ x1). A smaller α means a fatter tail. In the Pareto form the mean is finite only if α > 1 and the variance only if α > 2. Combined with a frequency, the power law yields tail quantiles. If λ losses exceed x0 per year, losses above x occur at this annual rate: Rate = λ × (x ÷ x0)^(−α). Setting this rate to about 0.001 and solving for x gives the 99.9% one-year loss. Typical errors are using +α instead of −α, or forgetting the frequency.

Formulas

Power law for operational losses

Prob(v > x) = K x^(−α)

v
the loss
x
the loss level
K
a scale constant
α
the tail index

A smaller α means a fatter tail. On a log-log plot α is minus the slope; using +α is the usual slip.

Tail index from two points

α = ln(p1 ÷ p2) ÷ ln(x2 ÷ x1)

p1
the probability of exceeding the smaller loss x1
p2
the probability of exceeding the larger loss x2
x1
the smaller loss level
x2
the larger loss level

Assumes the losses follow a power law between the two points. α above 1 gives a finite mean, above 2 a finite variance.

A practice question

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

When ln Prob(v > x) is plotted against ln x for loss data and the points fall roughly on a straight line, the tail index alpha is:

  1. The slope of the line as read off the graph, which is negativeWhy not: The slope itself is negative, but alpha is defined as a positive number, so alpha is the negative of the slope.
  2. The intercept of the lineWhy not: The intercept equals ln K, the scale constant.
  3. The exponential of the interceptWhy not: That would recover K, not alpha.
  4. Minus the slope of the line - correct

ln Prob = ln K - alpha ln x, so the intercept is ln K and the slope is -alpha.

A harder one

Operational losses follow a power law Prob(v > x) = K x^(-alpha) with alpha = 1.2. The probability of a loss exceeding USD 5 million is 4 percent. The probability of exceeding USD 25 million is closest to:

  1. 0.80 percentWhy not: Assumes alpha = 1, scaling the probability by 5/25.
  2. 27.6 percentWhy not: Sign error: multiplies by 5^(+1.2) instead of dividing by 5^1.2.
  3. 0.58 percent - correct
  4. 1.05 percentWhy not: Inverts the exponent, using (5/25)^(1/1.2) instead of (5/25)^1.2.

P(>25) = 0.04 x (25/5)^(-1.2) = 0.04 x 0.1450 = 0.58 percent.

The hardest tier

A bank has on average 20 losses per year above USD 1 million. For losses above that level, Prob(v > x | v > 1) = x^(-1.5), with x in USD million. The loss that is expected to be exceeded with probability about 0.1 percent in a year (set the expected number of exceedances per year equal to 0.001) is closest to:

  1. USD 737 million - correct
  2. USD 100 millionWhy not: Leaves out the frequency of 20 per year and sets x^-1.5 = 0.001.
  3. USD 159 millionWhy not: Uses a 1 percent exceedance rate, 0.01, instead of 0.1 percent.
  4. USD 13.6 millionWhy not: Multiplies by the frequency, setting x^-1.5 = 20 x 0.001, instead of dividing.

Rate above x is 20 x^-1.5 = 0.001, so x = (20/0.001)^(1/1.5) = 737.

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