Why is the Poisson distribution used for operational loss frequency and how do you apply it?

The Poisson distribution is the standard choice for modeling how many loss events occur in a fixed time period. It's ideal for operational risk because it models the count of rare, independent events — exactly what operational losses look like.

Why Poisson?

1. Discrete counts: Operational losses come in whole numbers (0, 1, 2, 3 events per year).
2. Rare events: Individual operational failures are low-probability.
3. Independence: Each event is roughly independent of others (one rogue trading incident doesn't directly cause a cyber breach).
4. Single parameter: Only requires lambda (the average number of events per period), making it easy to estimate from data.

The Formula

P(X = k) = (e^(-lambda) x lambda^k) / k!

Where lambda is the expected number of events and k is the specific count.

Worked Example

Summitridge Bank's operational risk team has observed an average of 3.2 fraud events per year over the past decade. What is the probability of exactly 5 fraud events next year?

P(X = 5) = (e^(-3.2) x 3.2^5) / 5!

= (0.04076 x 335.54) / 120

= 13.68 / 120

= 0.1140 or 11.4%

What about the probability of 7 or more events (tail risk)?

P(X >= 7) = 1 − P(X <= 6) = 1 − sum of P(X=0) through P(X=6)

Computing each term and summing: P(X <= 6) = 0.9554

So P(X >= 7) = 1 − 0.9554 = 4.46%

Key Properties:

• Mean = Variance = lambda. If your data shows variance much larger than the mean, the Poisson assumption may be violated (overdispersion). In that case, a Negative Binomial distribution is more appropriate.
• The sum of independent Poisson variables is also Poisson: if department A has lambda=2 and department B has lambda=1.5, the combined firm has lambda=3.5.

Exam Tip: On the FRM exam, you may be asked to identify when Poisson is appropriate vs. when to use Negative Binomial (overdispersion) or Binomial (fixed number of trials). The variance-to-mean ratio is the diagnostic.

For more practice on loss distributions, try our FRM operational risk question bank.