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Explain how Monte Carlo simulation builds a loss distribution out of a suitable loss frequency distribution and loss severity distribution.

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

The idea

Monte Carlo simulation builds an annual loss distribution from two separately estimated distributions. The first is frequency, how many losses occur in a year, commonly Poisson with mean λ. The second is severity, how large each loss is, commonly lognormal. In each trial, draw the number of losses N from the frequency distribution, draw N independent severities, and add them to get one year's total loss. Repeat many thousands of times, and the sorted totals form the simulated distribution. Frequency and severity are assumed independent. The average of the trials estimates expected annual loss, which should be close to EL = λ × mean severity. The 99.9th percentile of the totals estimates the one-year 99.9% loss. Unexpected loss is that percentile minus expected loss. With 10,000 trials, the 99.9th percentile is the tenth-largest outcome. A common error is to multiply the mean frequency by a single severity draw, which ignores the randomness in the count and in the individual losses.

Formulas

Expected annual operational loss

expected annual loss = λ × mean severity

λ
the mean number of losses a year
mean severity
the expected size of one loss, for a lognormal e^(μ + σ² ÷ 2)

Assumes frequency and severity independent. The tail needs simulation: the 99.9 percent loss is not the mean times anything simple.

A practice question

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

In a Monte Carlo operational risk model, which pair of distributions is the usual choice for loss frequency and loss severity?

  1. Normal for frequency and normal for severity, fitted to the sample mean and varianceWhy not: A normal distribution allows negative counts and negative losses and cannot capture the skew of severity.
  2. Lognormal for frequency and Poisson for severityWhy not: Reverses the roles: the Poisson models counts of events, not loss amounts.
  3. Poisson for frequency and uniform for severityWhy not: A uniform distribution has bounded range and no heavy right tail, unlike operational losses.
  4. Poisson for frequency and lognormal for severity - correct

Counts of events per year are typically Poisson, and loss amounts are positive and right-skewed, so lognormal is the standard severity choice.

A harder one

A bank runs 10,000 Monte Carlo trials of annual loss and sorts the results from largest to smallest. To estimate the 99.9 percent one-year loss, it should read off:

  1. The 10th largest simulated loss - correct
  2. The largest simulated lossWhy not: One in 10,000 is the 99.99th percentile, not the 99.9th.
  3. The 100th largest simulated lossWhy not: One in 100 corresponds to the 99th percentile, not 99.9.
  4. The 1,000th largest simulated lossWhy not: One in 10 corresponds to the 90th percentile.

The 99.9th percentile leaves 0.1 percent of 10,000 = 10 outcomes above it, so it is the 10th largest.

The hardest tier

In a simulation, the number of losses per year is Poisson with mean 3. The proportion of simulated years with at least 5 losses is closest to:

  1. 10.1 percentWhy not: Computes the probability of exactly 5 losses instead of 5 or more.
  2. 8.4 percentWhy not: Computes the probability of more than 5 losses, leaving out the case of exactly 5.
  3. 35.3 percentWhy not: Computes the probability of at least 4 losses, an off-by-one error in the cumulative sum.
  4. 18.5 percent - correct

P(N<=4) = e^-3 x (1 + 3 + 4.5 + 4.5 + 3.375) = 0.8153. So P(N>=5) = 0.1847.

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