Aggregate loss: the average year is not the year that breaks you
Interactive aggregate loss model for a personal motor book: Poisson claim frequency with Log-Normal severity across a portfolio of policies. Compute expected loss and run a 20,000-year Monte Carlo to see the aggregate loss distribution and its 95th, 99th and 99.5th percentiles.
The collective risk model is the backbone of non-life insurance pricing: total loss for a period is the sum of a random number of claims, each of random size. Its average is simply expected claim count × mean claim size, the pure premium where every rate starts. The total itself is a random sum, though, and its spread is not a product of averages.
Two portfolios with the same expected loss can need very different capital if one has a heavier severity tail or more claims that strike together. This module starts from an illustrative personal motor book: 5,000 policies, 0.08 claims per policy per year and a $7,000 average claim, so about 400 claims, $2.8 million of expected loss and $560 of pure premium per policy. You can change the claim rate, average claim size and number of policies (claim sizes are Log-Normal, with the standard deviation held at 1.2 times the mean), then simulate 20,000 years to draw the aggregate loss histogram with 95th, 99th and 99.5th percentile markers.
Those percentiles are the bridge from pricing to capital. Solvency II calibrates its capital requirement to a 99.5% one-year value-at-risk (a 1-in-200 year), and the Swiss Solvency Test to the average loss beyond the 99th percentile (99% expected shortfall, or TVaR). At the defaults the 99.5th percentile sits only about 20% above the expected loss; cut the book to 500 policies and it is roughly 80% above. Small portfolios need proportionally more capital, the law of large numbers working in reverse. The module covers premium risk for independent claims only; a real solvency calculation also covers reserve, catastrophe, market, credit and operational risk.
Collective risk model
S = X₁ + X₂ + … + X_N, E[S] = E[N] · E[X]
N is the number of claims in the year, Poisson with mean (number of policies) × (claim rate per policy); X₁, X₂, … are independent claim sizes with mean E[X]. When N is Poisson, Var[S] = E[N] · E[X²], which is why a heavier severity tail widens the total.
Frequently asked questions
What is the pure premium in insurance pricing?
The expected loss per unit of exposure: claim frequency multiplied by average claim severity. Loadings for expenses, profit and risk are added on top to reach the gross premium.
How is value-at-risk used in an aggregate loss model?
VaR at a confidence level is the aggregate loss exceeded only rarely under the model: at 99.5%, one year in 200, the Solvency II calibration. The gap between it and the expected loss (already in the premium) is a stand-alone measure of the capital needed to absorb a bad year for that risk.
Why simulate instead of using a formula?
The mean and variance of the collective risk model have closed forms, but its full distribution does not for most severity choices. Simulation, or Panjer recursion and FFT methods on a discretised severity, is how the percentiles are obtained in practice.
Interactive explorer
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Related modules
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- Cards & hypergeometric: Interactive card-dealing simulator. Deal hands from a 3D deck and compare empirical frequencies of aces, pairs and colors against exact hypergeometric probabilities, and see why a deck's memory matters.
- Frequency: Interactive Poisson distribution explorer for insurance claim counts and hospital admissions. Move the rate λ and watch P(X=k), the mode, and the probability of zero or extreme counts update live.
- Severity: Interactive loss severity explorer comparing Log-Normal, Pareto (heavy-tail) and Gamma distributions. Adjust parameters and see the mean, median, 95th and 99th percentile claim sizes, and the share of total cost from the largest 5% of claims, update in real time.