ModelRisk example · Insurance
Pricing a Policy: Premium Principles
Two policies cost the same on average — should they cost the same to insure? Four premium principles compared.
From the model’s saved run
What should each policy cost?
The risk premium (US$ thousand, before expenses) and its loading above the expected claims.
| Principle | Policy A fleet | Loading | Policy B property | Loading | B ÷ A premium |
|---|---|---|---|---|---|
| Expected claims (no loading) | 100.0 | 100.0 | 1.00 | ||
| Expected value | 115.0 | 15.0% | 115.0 | 15.0% | 1.00 |
| Standard deviation | 108.7 | 8.7% | 135.4 | 35.4% | 1.24 |
| Risk-adjusted (proportional hazards) | 106.5 | 6.5% | 128.4 | 28.4% | 1.20 |
| Cost of capital | 111.8 | 11.8% | 165.7 | 65.7% | 1.48 |
S-curve (cumulative probability) of Policy A: delivery fleet, Policy B: manufacturing property from the saved simulation.
The business question
An insurer's underwriters are quoting two commercial policies. The first covers a delivery company's 150 vans: about 20 claims a year, averaging US$5,000 — scratches, broken mirrors, the occasional collision. The second covers a manufacturing group's twelve sites against fire and damage: about one claim a year, averaging US$100,000, but a fire can cost many times that. Both are expected to cost US$100,000 a year.
A premium of expected claims plus a fixed percentage would charge them the same, yet the second is far riskier: in a bad year it can cost seven times its average, the first barely twice. The model prices both with four premium principles and shows how often each premium would fail to cover the year's claims.
- Who it is for
- Underwriters, pricing actuaries, reinsurance buyers; a first pricing model.
- Main outputs
- The risk premium and its loading for each policy under four principles — expected value, standard deviation, risk-adjusted and cost of capital · the gross premium after expenses · the chance that a year's claims exceed each premium · each policy's distribution of annual claims.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
For each policy, the expected number of claims and the average and standard deviation of a claim. The parameter of each premium principle; the expense ratio.
Pricing sheet
Claim numbers are Poisson and claim sizes Lognormal, as objects. VoseAggregateMoments gives the exact mean and standard deviation of a year's claims, and VoseAggregateFFTObject its percentiles. The expected-value, standard-deviation and cost-of-capital premiums are arithmetic on these; VosePrincipleRA gives the risk-adjusted premium; VosePrincipleEV and VosePrincipleStdev cross-check the arithmetic. Each year's claims are simulated with VoseAggregateMC.
Results sheet
The risk premium and loading of each policy under each principle; the gross premium to quote; the chance that the year's claims exceed each risk premium. Each policy's claims, simulated and calculated.
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The results in detail
What to quote: the gross premium
The risk premium plus expenses (US$ thousand): risk premium ÷ (1 − expense ratio).
| Principle | Policy A fleet | Policy B property |
|---|---|---|
| Expected value | 153.3 | 153.3 |
| Standard deviation | 145.0 | 180.5 |
| Risk-adjusted (proportional hazards) | 142.0 | 171.2 |
| Cost of capital | 149.1 | 220.9 |
How often is the premium not enough?
The chance that a year's claims exceed the risk premium.
Chart: Policy A: delivery fleet, Policy B: manufacturing property.
| Principle | Policy A | Policy B |
|---|---|---|
| Expected value | 29.2% | 31.2% |
| Standard deviation | 35.3% | 26.9% |
| Risk-adjusted (proportional hazards) | 37.5% | 28.3% |
| Cost of capital | 32.4% | 21.9% |
A year's claims
Each policy's total claims in a year (US$ thousand): simulated, and calculated on the Pricing sheet.
| Policy | Mean | Std dev | 75th pct | 95th pct | 1-in-200 99.5th pct |
|---|---|---|---|---|---|
| Policy A: delivery fleet | 99.9 | 35.1 | 120.2 | 163.1 | 214.1 |
| calculated (no simulation) | 100.0 | 34.9 | 120.1 | 163.0 | 217.9 |
| Policy B: manufacturing property | 101.3 | 142.8 | 145.6 | 381.0 | 783.6 |
| calculated (no simulation) | 100.0 | 141.4 | 145.0 | 373.7 | 756.5 |
How to read these results
- Both policies are expected to cost US$100,000 a year, but Policy B's claims vary four times as much — a standard deviation of about US$141,000 against US$35,000 — and its 1-in-200 year is about seven and a half times its average, against about twice for Policy A.
- The expected-value principle charges both the same US$115,000: its loading ignores risk. With it, Policy B's premium falls short in nearly a third of years, much as Policy A's does — but B's shortfalls are far larger.
- The standard-deviation, risk-adjusted and cost-of-capital principles all charge Policy B 20% to 50% more than Policy A, because its claims are riskier. The cost-of-capital principle is the most sensitive to the tail: it prices the capital that a 1-in-200 year requires.
- The price quoted is the gross premium: a quarter of it pays brokers and administration, so each risk premium is divided by 0.75. Under the expected-value principle that is about US$153,000 for either policy. Under the cost-of-capital principle about US$149,000 for Policy A and US$221,000 for Policy B.
- The chance of a shortfall alone is a poor guide: under the risk-sensitive principles Policy A's premium falls short more often than Policy B's, but by small amounts. A premium must reflect how large the bad years are, not only how often they come.
- ModelRisk computes the risk-adjusted premium and the percentiles of a year's claims from a Fast Fourier Transform, which can be coarse for rare, very large claims: the Pricing sheet checks that the risk-adjusted principle returns the expected claims when ρ = 1, and the table of a year's claims shows each calculated figure beside the simulated one.
- The simulated 1-in-200 year can differ from the calculated one by a few per cent: only 50 of 10,000 samples lie beyond it, so it moves from run to run. The calculated one does not, which is why the cost-of-capital premium is built on it. ModelRisk's Esscher principle (VosePrincipleEsscher) needs claims with a light tail and is not used here.
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
Policy A: delivery fleet 3 inputs
About 150 vans: many small claims.
| Assumption | Unit | Value |
|---|---|---|
| Expected number of claimsFleet size × claim rate. | claims a year | 20 |
| Average claimRepairs, glass, the occasional collision. | US$ thousand | 5.0 |
| Standard deviation of a claim | US$ thousand | 6.0 |
Policy B: manufacturing property 3 inputs
Twelve sites: about one claim a year, sometimes very large.
| Assumption | Unit | Value |
|---|---|---|
| Expected number of claimsSites × claim rate. | claims a year | 1.0 |
| Average claimFires, floods, machinery damage. | US$ thousand | 100.0 |
| Standard deviation of a claim | US$ thousand | 100.0 |
Premium principles 4 inputs
The loading parameter of each principle.
| Assumption | Unit | Value |
|---|---|---|
| Expected value principle: loading θPremium = (1 + θ) × expected claims. | of expected claims | 15% |
| Standard deviation principle: multiple αPremium = expected claims + α × standard deviation of the year's claims. | × std deviation | 0.25 |
| Risk-adjusted principle: risk aversion ρEach chance of a loss raised to the power 1 ÷ ρ; ρ = 1 gives the expected claims. | 1 or more | 1.20 |
| Cost-of-capital principle: cost of capitalPremium = expected claims + cost of capital × (1-in-200 claims − expected claims). | a year | 10% |
Expenses 1 input
| Assumption | Unit | Value |
|---|---|---|
| Expense ratioBrokers' commission and administration. | of gross premium | 25% |
Units and conventions
- Money
- US$ thousand.
- Risk premium
- Expected claims plus a loading for risk, before expenses.
- Gross premium
- The risk premium plus expenses: risk premium ÷ (1 − expense ratio).
- Loading
- How far the premium is above the expected claims, as a percentage of them.
- 1-in-200
- The 99.5th percentile of a year's claims.
Key assumptions and limitations
- Claims
- Claim numbers are Poisson; claim sizes are Lognormal and independent of each other and of their number.
- Policies
- One year of cover, no deductible or limit; the two policies are priced separately.
- Principles
- The risk-adjusted principle and the percentiles of a year's claims are computed by ModelRisk through a Fast Fourier Transform. A check confirms it reproduces the expected claims when ρ = 1, and Results shows each calculated figure beside the simulated one.
- Data
- The policies and all figures are invented for this example.
How to use it
- Enter each policy's expected claim numbers and claim sizes.
- Set the loading of each principle to your company's practice.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Compare the premiums: a loading that ignores risk undercharges the volatile policy.
ModelRisk functions used
- VosePoissonObject, VoseLognormalObject
- Claim numbers and claim sizes as distribution objects.
- VoseAggregateMoments
- The exact mean, variance, skewness and kurtosis of a year's claims (an array over four cells).
- VoseAggregateFFTObject, VosePercentile
- A year's claims as a calculated distribution, and its percentiles: the 1-in-200 year for the cost-of-capital premium.
- VosePrincipleRA
- The premium by the risk-adjusted (proportional hazards) principle.
- VosePrincipleEV, VosePrincipleStdev
- ModelRisk's expected-value and standard-deviation premiums: a cross-check of the arithmetic.
- VoseAggregateMC
- One year's total claims: a number of claims, then a size for every claim.
- VoseSimPercentile, VoseSimProbability
- Statistics of the simulation on the Results sheet.
Further reading
- Monte Carlo simulation explained (Vose guide)
- Kaas, R., Goovaerts, M., Dhaene, J. & Denuit, M. (2008). Modern Actuarial Risk Theory, 2nd edition. Springer.
- Wang, S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics 17(1), 43–54.
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
Related example models
- 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★★★★★Aggregate Claims: Frequency × SeverityWhat will a year's claims cost?A year's claims as a random number of random claim sizes, three ways — two right and one common mistake.
- 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.
An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.