Vose SoftwareModelRisk example · Insurance
Aggregate Claims: Frequency × Severity
What will a year's claims cost? Adding up a random number of random claims — and the mistake to avoid.
From the model’s saved run
What will the year's claims cost?
Total claims (US$m) by each method, against the premium.
| Method | Mean | Std dev | 95th pct | 99.5th pct 1 in 200 | Above premium |
|---|---|---|---|---|---|
| Sum of every claim (VoseAggregateMC) | 5.99 | 0.75 | 7.30 | 8.16 | 3.0% |
| Sum of every claim (VoseAggregateFFT) | 6.02 | 0.75 | 7.30 | 8.13 | 2.9% |
| Number of claims × one claim size | 5.79 | 12.40 | 21.80 | 72.36 | 19.3% |
| Expected claims × average claim (no uncertainty) | 6.00 |
| Measure | Unit | Value |
|---|---|---|
| Exact mean (VoseAggregateMoments) | US$m | 6.000 |
| Exact standard deviation | US$m | 0.750 |
| Premium | US$m | 7.50 |
| Capital above the premium, 1 in 200 (sum of every claim) | US$m | 0.66 |
S-curve (cumulative probability) of Sum of every claim, FFT, Number × one size from the saved simulation.
Histogram of Sum of every claim from the saved simulation.
The business question
A motor insurer expects about 1,200 claims next year. Claim numbers vary from year to year beyond pure chance — a hard winter or a wet summer moves them all — by about 10%. Claim sizes are very uneven: most are small repairs, a few are serious injuries, so the average of US$5,000 comes with a standard deviation of US$12,000.
The total cost of the year's claims is the sum of a random number of random amounts. A tempting spreadsheet shortcut multiplies a random number of claims by a single random claim size — as if every claim in the year were the same size. The model computes the total properly, two ways, and the shortcut, and compares them with the premium.
- Who it is for
- Pricing and reserving actuaries, underwriters, finance teams; a first insurance model.
- Main outputs
- Total claims for the year: mean, percentiles up to 99.5% and the chance of exceeding the premium, by aggregate simulation, by FFT and by the shortcut · the exact mean and standard deviation for comparison.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The expected number of claims and how much it varies from year to year; the mean and standard deviation of a claim; the premium.
Claims sheet
The frequency is a Polya object (a Poisson whose rate is Gamma), the severity a Lognormal object. VoseAggregateMC adds up a simulated claim for every claim; VoseAggregateFFT samples the same aggregate distribution computed by Fast Fourier Transform. The shortcut multiplies one claim number by one claim size.
Results sheet
Mean, 5th, 95th and 99.5th percentiles and the chance that claims exceed the premium for each method, against the exact mean and standard deviation.
ModelRisk techniques
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The results in detail
Why the shortcut is wrong
The same mean, a completely different range.
| Measure | Unit | 10th pct | Median | 90th pct | Mean |
|---|---|---|---|---|---|
| Sum of every claim (VoseAggregateMC) | US$m | 5.05 | 5.96 | 6.97 | 5.99 |
| Number of claims × one claim size | US$m | 0.39 | 2.28 | 13.11 | 5.79 |
How to read these results
- Adding up every claim gives a narrow, slightly right-skewed distribution: with many claims, the large ones and the small ones largely cancel out. Its mean and standard deviation match the exact moments.
- The FFT method computes the same distribution without simulating each claim, and agrees with the sum of every claim — but only with a fine enough grid. With ModelRisk's default density the FFT mean here would be about three times too high. Always compare the FFT mean with the exact mean (the Claims sheet does) and raise the density until they agree.
- Multiplying a random number of claims by one random claim size has the right mean but a range many times too wide: it assumes every claim in the year is the same size, so one large draw makes the whole year catastrophic. It wildly overstates the chance of exceeding the premium and the reserve needed.
- The deterministic estimate — expected claims × average claim — has the right mean but no range at all: it cannot say how likely an underwriting loss is.
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
Claim numbers 2 inputs
How many claims a year.
| Assumption | Unit | Value |
|---|---|---|
| Expected number of claimsPolicies in force × claim rate. | claims a year | 1,200 |
| Year-to-year variation in the claim rateWeather and economic conditions move all claims together. Above 0: for a plain Poisson, use 1%. | coefficient of variation | 10% |
Claim sizes 2 inputs
What one claim costs.
| Assumption | Unit | Value |
|---|---|---|
| Average claimClaims history, inflated to next year. | US$ thousand | 5.0 |
| Standard deviation of a claimMost claims are small; a few are very large. | US$ thousand | 12.0 |
Premium and method 2 inputs
| Assumption | Unit | Value |
|---|---|---|
| Premium incomeExpected loss ratio 80%. | US$m | 7.50 |
| FFT density levelMany claims a year need a fine grid: the default (12) is far too coarse here. | 2^n steps | 18 |
Units and conventions
- Money
- US$ million (US$m); claim sizes in US$ thousand.
- Frequency
- The number of claims in the year.
- Severity
- The size of one claim.
- Aggregate
- The sum of all the year's claims.
Key assumptions and limitations
- Claims
- Claim sizes are independent of each other and of the number of claims.
- Frequency
- Year-to-year variation in the claim rate makes the number of claims Polya: a Poisson whose rate is Gamma (a Negative Binomial with any s, not only whole ones).
- Data
- The portfolio and all figures are invented for this example.
How to use it
- Enter your expected claim numbers, their variation, your claim-size statistics and your premium.
- Press F9 on the Claims sheet to see other possible years.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Read the chance of an underwriting loss, and the reserve needed at 99.5%.
ModelRisk functions used
- VosePolyaObject, VoseLognormalObject
- The frequency (a Poisson whose rate is Gamma) and the severity as objects.
- VoseAggregateMC
- Adds up one simulated claim for every claim in the year.
- VoseAggregateFFT
- The same aggregate distribution by Fast Fourier Transform, then sampled.
- VoseAggregateMoments
- The exact mean, variance, skewness and kurtosis of the aggregate.
- VoseSimMean, VoseSimPercentile
- Statistics of the simulation on the Results sheet.
Further reading
- Frequency-severity modelling (Vose guide)
- Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2019). Loss Models: From Data to Decisions, 5th edition. Wiley.
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley — aggregate distributions.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
Related example models
- ModelRisk★★★★★Fitting Claim Frequency and Severity to DataWhich distributions fit the claims history — and how much does the shortness of that history add to the risk?Distributions fitted to ten years of claim counts and 250 claims, compared by SIC, used with and without parameter uncertainty.
- ModelRisk★★★★★Reinsurance Programme: Excess of Loss and Quota ShareWhat does reinsurance cost, and what does it buy?Every claim split between the insurer and two excess-of-loss layers, a quota share on top; four programmes compared on the same claims.
- ModelRisk★★★★★Catastrophe and Large-Loss TailHow large is the 1-in-200 year loss when the history covers only 25 years?Extreme value theory on 25 years of loss events: a threshold, a Generalised Pareto tail, and return periods far beyond the data.
An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.