ModelRisk example · General techniques
Moments of an Aggregate Distribution
The mean, spread, skewness and kurtosis of a year's total claims, without simulating, and what they are good for.
From the model’s saved run
Moments of the total
Exact (VoseAggregateMoments), simulated, and those of the two matched distributions.
| Moment | Exact | Simulated | Shifted Gamma | Normal |
|---|---|---|---|---|
| Mean | 125.0 | 125.1 | 125.0 | 125.0 |
| Standard deviation | 43.0 | 43.0 | 43.0 | 43.0 |
| Skewness | 1.02 | 0.98 | 1.02 | 0.00 |
| Kurtosis | 6.07 | 5.47 | 4.56 | 3.00 |
S-curve (cumulative probability) of Simulated total, Shifted Gamma, Normal from the saved simulation.
The business question
An insurer expects about 25 claims next year (a Poisson number) and each claim's size is a Lognormal with a mean of 5 and a standard deviation of 7 ($ thousand). The total is a random sum: its moments follow exactly from those of the frequency and the severity, by formula or with VoseAggregateMoments, without any simulation.
A distribution matched to them gives a quick approximation of the total. The model matches a shifted Gamma (three moments) and a Normal (two), and compares both with the simulated total: in the upper tail, where the money is, the Gamma is close and the Normal is not.
- Who it is for
- Actuaries, risk analysts and anyone working with totals of a random number of random amounts: claims, operational losses, repair costs, demand made of many orders.
- Main outputs
- The mean, variance, skewness and kurtosis of the frequency, the severity and the total, by formula and with VoseAggregateMoments · a shifted Gamma and a Normal matched to the total's moments · their percentiles and tail probabilities against the simulated total.
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 (Poisson), the claim size's mean and standard deviation (Lognormal), and what counts as a large and a small year.
Moments sheet
The first four moments of the frequency and the severity (VoseMean, VoseVariance, VoseSkewness, VoseKurtosis). The total's moments by the formulas for a random sum, and with VoseAggregateMoments. A shifted Gamma matched to the total's mean, variance and skewness, and a Normal matched to its mean and variance. One sample of each per simulation, with the total simulated by VoseAggregateMC.
Results sheet
The moments of the three, their percentiles, and the chance of a large or a small year.
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The results in detail
Percentiles
The simulated total against the two matched distributions (exact percentiles).
| Percentile | Simulated total | Shifted Gamma | Normal |
|---|---|---|---|
| 5th | 65.6 | 68.7 | 54.3 |
| 50th (median) | 119.6 | 117.8 | 125.0 |
| 90th | 180.0 | 182.7 | 180.1 |
| 95th | 203.0 | 205.9 | 195.7 |
| 99th | 254.3 | 255.5 | 225.1 |
| 99.5th | 269.8 | 275.7 | 235.8 |
Large and small years
The chance of a year beyond the thresholds on the Inputs sheet.
Histogram of Simulated total from the saved simulation.
| Year | Simulated total | Shifted Gamma | Normal |
|---|---|---|---|
| Total above the large-year threshold | 1.13% | 1.20% | 0.18% |
| Total below the small-year threshold | 1.11% | 0.14% | 4.06% |
How to read these results
- The moments need no simulation: VoseAggregateMoments gives a mean of 125, a variance of 1,850 (standard deviation 43), a skewness of 1.02 and a kurtosis of 6.07, exactly what the formulas for a random sum give. The simulated moments agree within sampling error. Kurtosis, driven by rare large claims, is the noisiest.
- A shifted Gamma matched to the first three moments describes the upper tail well: its 99th percentile, 255.5, is within a few units of the simulated total's. A Normal matched to two moments puts it at 225 and gives a large year (above 250) a 0.2% chance instead of about 1%.
- The Gamma's weakness is the lower tail: it cannot go below its shift (40.5), so it gives a small year (below 50) about 0.1% against about 1%. Its kurtosis, 4.56 against 6.07, shows the fourth moment is not matched.
- Use the moments to check a simulation, and a matched shifted Gamma for a quick estimate of the upper tail. Simulate when the lower tail or extreme percentiles matter.
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.
Claims in a year 1 input
A Poisson number of claims.
| Assumption | Unit | Value |
|---|---|---|
| Expected number of claimsPoisson: variance = mean | claims a year | 25 |
Claim size 2 inputs
A Lognormal size for each claim.
| Assumption | Unit | Value |
|---|---|---|
| Mean claim size | $ thousand | 5.0 |
| Standard deviation of the claim sizeLarger than the mean: a long right tail | $ thousand | 7.0 |
Large and small years 2 inputs
Thresholds for the chance of an extreme year.
| Assumption | Unit | Value |
|---|---|---|
| A large year: total aboveTwice the expected total | $ thousand | 250 |
| A small year: total below | $ thousand | 50 |
Units and conventions
- Kurtosis
- Not the excess: a Normal has 3.
- $ thousand
- All amounts.
Key assumptions and limitations
- Independence
- Claim sizes are independent of each other and of the number of claims.
- Matching
- The shifted Gamma matches three moments: mean, variance and skewness; its kurtosis is whatever follows. The Normal matches two.
- Data
- The frequency and severity are those of ModelRisk's earlier Aggregate Moments example. Its formula table typed C10 for C19 in the skewness formula. The model uses the cell's formula, which is right.
How to use it
- Set the claim frequency and size on the Inputs sheet.
- Read the moments at once on the Moments sheet. Run a simulation (10,000 samples with a fixed seed are set) to compare the approximations with the simulated total.
- Use the matched Gamma for a quick upper-tail estimate; simulate when the lower tail matters.
ModelRisk functions used
- VosePoissonObject, VoseLognormalObject, VoseGammaObject, VoseNormalObject
- Distributions as objects.
- VoseMean, VoseVariance, VoseSkewness, VoseKurtosis
- An object's moments, exactly.
- VoseAggregateMoments
- The first four moments of a random sum, from its frequency and severity.
- VoseShift
- Shifts a distribution by a constant (here the Gamma's start).
- VoseAggregateMC
- One simulated total: a number of claims, then a size for each, added up.
- VosePercentile, VoseProb
- A distribution object's percentile and cumulative probability.
- VoseSimMean, VoseSimStdev, VoseSimSkewness, VoseSimKurtosis, VoseSimPercentile, VoseSimProbability
- Statistics of the simulation.
Further reading
- Frequency-severity modelling (Vose guide)
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: aggregate distributions and the method of moments.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.