ModelRisk example · General techniques
Correlated Insurance Policies
Five lines of policies whose bad years come together: what correlation does to the total.
From the model’s saved run
The year's total: linked against independent
The same five lines, with and without the copula.
S-curve (cumulative probability) of Linked, Independent, Bad-year threshold from the saved simulation.
| Measure | Linked | Independent |
|---|---|---|
| Mean | 514.9 | 515.8 |
| Standard deviation | 69.6 | 45.2 |
| 5th percentile | 407.6 | 444.7 |
| Median | 510.7 | 514.3 |
| 95th percentile | 634.1 | 592.0 |
| 99th percentile | 692.4 | 626.4 |
| Expected shortfall (99%) | 719.4 | 643.7 |
| Chance of a bad year (above the threshold) | 11.9% | 3.6% |
| Expected total (exact) | 516.1 | 516.1 |
The business question
An insurer has five lines of policies (A to E). On each, policyholders have claims at a known rate a year, each claim's damage is a Lognormal, and the insurer pays the damage above a deductible, up to a limit. A storm or a downturn makes busy years come together.
A Clayton copula with a strong parameter (10) links the five lines' numbers of claims, through 1 − U, so that their busy years coincide; each claim's payout stays independent. The model puts the year's total next to the same five lines independent.
- Who it is for
- Insurers and reinsurers, and anyone adding up uncertain totals that tend to be high or low together.
- Main outputs
- Each line's expected claims and its annual total · the year's total with the lines correlated and independent: mean, spread, percentiles, the 99% expected shortfall and the chance of a year above a threshold · how strongly the lines move together.
How the model works
Four sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The Clayton copula's parameter θ (10: strong dependence) and the total that counts as a bad year.
Policies sheet
For each line: policy holders and claims a year per holder give a Poisson number of claims; the damage is a Lognormal; the payout is the damage above the deductible, up to the limit (VoseDeductObject). Its mean payout per claim and expected annual claims.
Annual claims sheet
VoseCopulaMultiClayton draws five linked probabilities U. Each line's number of claims is VosePoisson(λ, 1 − U), its (1 − U)-th percentile, so a low U means a busy year; VoseAggregateMC adds one payout per claim. The same five totals are also drawn independently.
Results sheet
The total correlated and independent, each line's figures, and how closely the lines move.
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The results in detail
By line
Each line keeps its own distribution: simulated (linked) against exact.
| Line | Expected (exact) | Simulated mean | 99th pct (FFT, exact) | 99th pct (simulated) |
|---|---|---|---|---|
| A | 105.7 | 105.6 | 138.6 | 138.7 |
| B | 197.1 | 196.8 | 256.8 | 254.7 |
| C | 70.7 | 70.6 | 99.5 | 99.9 |
| D | 81.3 | 80.7 | 171.5 | 168.5 |
| E | 61.3 | 61.2 | 81.3 | 81.5 |
How closely the lines move together
Rank correlation of the lines' annual totals in the simulation.
Histogram of Linked total from the saved simulation.
| Pair | Rank correlation |
|---|---|
| Lines A and B | 0.55 |
| Lines B and D | 0.50 |
| Lines C and E | 0.31 |
How to read these results
- The five lines expect about 516 ($000) of claims a year, linked or not: linking them does not change the mean, and each line keeps its own distribution (compare the exact and simulated columns by line).
- It changes the spread. With the lines' busy years together the standard deviation rises from about 46 to 70, the 99th percentile from about 631 to 694, and a year above 600 becomes three times as likely (about 12% against 4%).
- The copula links the numbers of claims almost in step (θ = 10), but each claim's size is still independent, so the lines' totals are only moderately correlated (rank correlations about 0.3 to 0.55).
- The original example linked the lines' whole annual totals instead, reading each line's FFT distribution at the copula's percentile (VoseAggregateFFT with U). In the ModelRisk build this example was made with, VoseAggregateFFT with a VoseDeductObject severity returns errors during a simulation, so the copula drives the claim numbers here. The FFT (density 18) gives each line's exact percentiles.
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.
The dependence between the lines 1 input
| Assumption | Unit | Value |
|---|---|---|
| Clayton copula parameter θ0: independent; 10: strong (Kendall's τ = θ ÷ (θ + 2) = 0.83) | 10.0 |
A bad year 1 input
| Assumption | Unit | Value |
|---|---|---|
| A bad year: total claims aboveAbout 16% above the expected total | $000 | 600 |
Units and conventions
- $000
- All amounts.
- Payout
- min(max(damage − deductible, 0), limit): a damage below the deductible is a claim that pays nothing (zeros = TRUE).
- 1 − U
- The copula's U enters each line's number of claims as 1 − U: the Clayton's dependence in the lower tail becomes dependence in the busy years.
Key assumptions and limitations
- Within a line
- Claims are independent and arrive at a constant rate (Poisson).
- Between lines
- Only through the copula: a Clayton with θ = 10, a strong and simple dependence.
- Data
- The five lines and θ = 10 are those of ModelRisk's earlier Correlated Insurance Portfolio example.
- ModelRisk build
The original read each line's FFT distribution at the copula's percentile. In the build this example was made with, VoseAggregateFFT with a VoseDeductObject severity errors during a simulation, so the copula drives the numbers of claims instead.
The original's default FFT density also overstated the lines by several per cent (line A's mean 109.5 against 105.7; even density 16 overstated line B by 2.5%); the exact percentiles here use density 18.
How to use it
- Set the five lines on the Policies sheet and the copula's strength on the Inputs sheet.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Compare the correlated and independent totals: the difference is what the correlation costs.
ModelRisk functions used
- VosePoissonObject, VoseLognormalObject
- The number of claims and the damage, as objects.
- VoseDeductObject
- The payout after a deductible and up to a limit, as an object.
- VoseAggregateFFT
- A line's annual total by Fast Fourier Transform; with U, its U-th percentile.
- VosePoisson with U
- The U-th percentile of a Poisson: the copula's way into the number of claims.
- VoseAggregateMC
- One year of a line: a number of claims, then one payout each, added up.
- VoseCopulaMultiClayton
- Five linked probabilities from a Clayton copula (array function).
- VoseMean
- The exact mean of an object: here the mean payout per claim.
- VoseSimCorrelation, VoseSimCVARp
- The lines' rank correlation and the expected shortfall, from the simulation.
Further reading
- Monte Carlo simulation explained (Vose guide)
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: copulas and aggregate distributions in insurance.
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.