ModelRisk example · General techniques
Counts of Random Events with Trend and Seasonality
How many car crashes will there be next year, week by week? Random events with a trend, seasons and a new law.
From the model’s saved run
Crashes in the year
The year's total against the number expected without the law.
| Measure | Unit | Value |
|---|---|---|
| Expected crashes without the law | crashes | 5,441 |
| Mean crashes in the year | crashes | 5,236 |
| Chance that the law passes (Inputs sheet) | probability | 70% |
| Mean crashes avoided by the law | crashes | 204 |
| Standard deviation of the year's total | crashes | 156 |
| Standard deviation, Poisson noise alone | crashes | 74 |
| Measure | Unit | 10th pct | Median | 90th pct | Mean |
|---|---|---|---|---|---|
| Crashes in the year | crashes | 5,053 | 5,202 | 5,468 | 5,236 |
Line chart: 90th percentile, Median, Mean, 10th percentile.
Histogram of Crashes in the year from the saved simulation.
The business question
An insurer covers motor accidents across the country, about a hundred crashes a week at the start of the year, a number that grows by about 10% a year. Crashes follow the seasons: half as many again in midwinter, a quarter fewer in summer. Parliament is debating a road-safety law, with a 70% chance of passing.
If it does, it takes effect in a week between the 23rd and the 26th and cuts crashes by 5% to 15% from then on. Within each week crashes happen at random, so the number is Poisson around its expected value. The insurer wants the year's total and its weekly pattern, for reserving and for staffing its claims centre.
- Who it is for
- Anyone forecasting counts of random events over time: claims, accidents, failures, arrivals, sales of a slow-moving product.
- Main outputs
- Crashes in the year at P10 / median / P90 against the expected number without the law · week-by-week bands · the crashes the law avoids · how much of the year's uncertainty comes from the law.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The weekly number of crashes at the start of the year and its annual growth, and the law: its chance of passing, the weeks in which it could take effect, and its effect as a PERT.
Weeks sheet
Each week's expected crashes are the trend × the seasonal index × the law factor, which is 1 until the law applies. The law passes with its probability (VoseBernoulli) in a week drawn from its window (VoseStepUniform). Each week's count is Poisson around its expected value.
Results sheet
The year's total against the expected number without the law, bands for every week, and the crashes avoided by the law.
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The results in detail
How to read these results
- Each week's crashes are Poisson around an expected number that grows with the trend, rises in winter and falls in summer, and drops by 5% to 15% once the law applies. The weekly band is widest in winter, when more crashes are expected.
- The law passes in seven years out of ten and then avoids nearly 300 crashes; on average it avoids about 200.
- The year's total varies more than Poisson randomness alone would make it: its standard deviation is about twice the square root of the expected number. The extra comes from the law, which is why the histogram has two humps: years with the law and years without it.
- The 52 weekly counts are one output range, 'Crashes by week': the Results Viewer can show them as a time series.
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.
Crashes now 2 inputs
The level at the start of the year and its growth.
| Assumption | Unit | Value |
|---|---|---|
| Crashes per week at the start of the year (an average week)Before the seasonal index. | crashes | 100 |
| Growth in crashes per yearMore cars, more driving. | per year | 10% |
The road-safety law 4 inputs
A law that may pass and cut crashes from the week it takes effect.
| Assumption | Unit | Uncertainty | Low | Most likely | High |
|---|---|---|---|---|---|
| Chance that the law passes | probability | Fixed | 70% | ||
| Earliest week it could take effect | week | Fixed | 23 | ||
| Latest week it could take effect | week | Fixed | 26 | ||
| Change in crashes once it appliesA 5% to 15% cut, most likely 10%. | change | PERT | −15% | −10% | −5% |
Units and conventions
- Weeks
- Week 1 is the first week of the year.
- Seasonal index
- The week's crashes relative to an average week: 1.5 is 50% more.
- Trend
- Annual growth, applied as a constant weekly rate: (1 + growth)^(1/52) − 1.
Key assumptions and limitations
- Poisson
- Crashes happen independently at a steady rate within a week, so a week's count is Poisson.
- Law
- Once the law applies, its effect holds for the rest of the year and is the same every week.
- Data
- The weekly seasonal pattern, starting level and growth are those of ModelRisk's earlier Poisson Series example. The law's chance, timing and effect those of its Seasonal Poisson Random Walk example.
How to use it
- Set the starting level, the growth and the law on the Inputs sheet.
- Replace the seasonal indices on the Weeks sheet with your own: they should average 1.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Open the Results Viewer's time-series view of 'Crashes by week' for more charts.
ModelRisk functions used
- VosePoisson
- Each week's number of crashes around its expected value.
- VoseBernoulli
- Whether the law passes.
- VoseStepUniform
- The week the law takes effect: each week of its window equally likely.
- VosePERT
- The law's effect on crashes.
- VoseOutput
- The 52 weekly counts as one output range ('Crashes by week') and the year's total.
- VoseSimMean, VoseSimPercentile
- Statistics of the simulation, and the weekly bands.
Further reading
- Monte Carlo simulation explained (Vose guide)
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: forecasting random events with trend and seasonality.
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.