Category: Actuarial fundamentals · Reviewed by Apex Insurance Brokers · Last reviewed
A frequency-severity model decomposes aggregate losses into the number of claims (frequency) and the size of each claim (severity), modelling each separately and combining them to obtain the aggregate distribution.
Mathematical form
Aggregate loss S = X₁ + X₂ + … + X_N
where N is the random number of claims (frequency) and X_i are the i.i.d. claim sizes (severity).
The aggregate is a compound distribution:
E[S] = E[N] × E[X]
Var(S) = E[N] × Var(X) + Var(N) × E[X]²
Common distributions
Frequency:
Poisson (mean = variance) — simple and analytically tractable.
Negative binomial (mean < variance) — for over-dispersed counts.
Zero-inflated Poisson — for portfolios with many zero-claim risks.
Severity:
Lognormal — moderate tail, often used for motor and property.
Gamma — generalised linear model staple.
Pareto — heavy tail, common for excess layers and catastrophe.
Generalised Pareto Distribution (GPD) — extreme value theory for tail.
Mixed (e.g. lognormal body + Pareto tail) — for high-severity reinsurance.
Use
Reinsurance pricing (especially per-risk and aggregate excess-of-loss).
Capital modelling (internal models under Solvency II).
Large account pricing for casualty.
Cat modelling vendor outputs.
References
Klugman, S.A., Panjer, H.H., Willmot, G.E. (2019). Loss Models: From Data to Decisions.
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