Non-life underwriting and pricing · RiskAtoms Academy

By the end you will be able to

1Compute frequency, severity and burning cost from exposure and claims data, and judge how much a small segment's figures can be trusted.
2Explain why one-way relativities mislead when rating factors are correlated, using the book's young urban drivers.
3Fit a multiplicative tariff by marginal totals, the Poisson GLM answer, and turn it into a technical premium.
4Explain underwriting discipline: why pricing each segment right matters even when the total is right.

Why it matters

Look at RiskAtoms Insurance SE's motor data one factor at a time and young drivers cost 2.27 times the base, urban drivers 1.57 times the rural ones. Both figures are wrong, in opposite directions, and for the same reason: young drivers live in cities and drive small cars. Price on them and a rival with a better model takes the customers you overcharge and leaves you the ones you undercharge.

Core explanation

From claims data to a price

Illustrative dataRiskAtoms Insurance SE's insurance book

The same composite European insurer as Modules 1.8 and 2.2, in € millions at 31 December 2025: four non-life lines with ten years of claims triangles, a 1.2 million vehicle-year motor pricing data set, a 17,000-event catastrophe set, its reinsurance, its life books, its assets and an outline of its Solvency II standard formula. Every policy, claim, event and figure is made up for teaching and seeded so it never changes; none is any insurer's data. Every lesson in Module 3.8 uses this book, so the figures agree.

This lesson uses the book's private-car pricing data: one country, policy years 2023 to 2025, 1,200,000 vehicle-years in 24 segments of driver age × area × vehicle group. Claim counts are Poisson and claim sizes lognormal, drawn from a stated true model that the lesson can reveal. Real pricing data has many more factors, and claims that are still open.

Frequency × severity = burning cost

Pricing starts from exposure. Over three years the book had 97,766 claims on 1,200,000 vehicle-years, costing €382.8m. Three numbers follow:

Measure	Formula	Book
Frequency	claims ÷ vehicle-years	8.15%
Severity	amounts ÷ claims	€3,916
Burning cost	amounts ÷ vehicle-years = frequency × severity	€319.03

Burning cost is the expected claims cost of one vehicle-year (M-C02: Expectation): E[S] = E[N] × E[X], the same split into how many and how large as the loss distribution approach in L3.7.07: Measuring frequency and severity. Splitting matters because the two parts move for different reasons: speed cameras and wet roads change frequency, repair prices and court awards change severity.

Each segment's frequency is an estimate. A Poisson count of $n$ claims has a standard deviation of $\sqrt n$ (M-C05: The Poisson distribution), so the largest segment (132,000 vehicle-years) pins its frequency to 5.53% ± 0.13 points, while the smallest (2,880 vehicle-years of young rural drivers in large cars) manages only 15.8% ± 1.5. Its burning cost of €922 against a true €800 is mostly noise. Pricing each cell from its own data would chase that noise; a model with a few rating factors borrows strength across cells.

One factor at a time

The simplest model: for each factor, divide each level's burning cost by the base level's. These one-way relativities are what the data say, factor by factor:

Driver age	Vehicle-years	Frequency	Severity	Burning cost	One-way
17–24	96,000	17.17%	€3,633	€623.76	2.270
25–39	360,000	8.31%	€3,959	€329.13	1.198
40–59	480,000	6.69%	€4,108	€274.81	1.000
60+	264,000	7.29%	€3,770	€274.83	1.000

Area	Vehicle-years	Frequency	Severity	Burning cost	One-way
Urban	591,600	9.99%	€3,911	€390.71	1.567
Rural	608,400	6.36%	€3,922	€249.32	1.000

Vehicle group	Vehicle-years	Frequency	Severity	Burning cost	One-way
A (small)	396,000	8.43%	€3,068	€258.55	0.865
B (medium)	562,800	7.82%	€3,818	€298.77	1.000
C (large or powerful)	241,200	8.44%	€5,516	€465.58	1.558

Now look at who sits in each level. Young drivers are 70% urban against 45% for drivers aged 40 to 59, and 60% of them drive small cars against 25%. So the urban level holds 11.4% young drivers, the rural level only 4.7%.

The exposure mix of the book: the factors are correlated.
Driver age	Vehicle-years	Urban share	Small cars (A)	Large cars (C)
17–24	96,000	70%	60%	10%
25–39	360,000	60%	35%	20%
40–59	480,000	45%	25%	25%
60+	264,000	35%	35%	15%

Two distortions follow, both visible in the tables:

Urban looks too dear. Part of the urban level's higher cost is young drivers. The one-way relativity of 1.57 charges city drivers for their age as well as their streets.
Young looks too cheap. Young drivers' average claim is only €3,633, against €4,108 for the base age, because small cars cost less to repair. The one-way relativity of 2.27 gives young drivers credit for their cars.

All factors at once

The fix is to fit all factors together, so each relativity is measured with the others held fixed. The standard tool is a generalised linear model with a log link: the expected cost is a base times one relativity per factor. On a table like this one it can be fitted without any software, by marginal totals: adjust each relativity until the tariff reproduces the actual cost of every level. That is the answer a Poisson GLM gives (the formal version shows why).

The book's data were generated from a known model, so the fit can be checked against the truth.
Relativity	One-way	Fitted	True
Young driver (17–24)	2.270	2.467	2.400
Driver 60+	1.000	1.125	1.150
Urban	1.567	1.456	1.450
Small car (A)	0.865	0.758	0.760
Large car (C)	1.558	1.588	1.595
Base cost (€ per vehicle-year)		209.85	209.00

The fit recovers the truth to within sampling noise. Across the 24 cells, weighted by exposure, the fitted tariff is 0.9% away from the true cost on average; the one-way tariff (the product of the one-way relativities, scaled to collect the right total) is 7.7% away; and each cell's own burning cost is 1.7% away. The model beats the raw data because it pools 1.2 million vehicle-years into seven parameters (a base and six relativities).

Sort itFrequency, severity or both?

Each change hits a motor book. Does it mainly move how often claims happen, how much each costs, or both?

Young, inexperienced drivers.
Large or powerful cars.
Car parts and paint cost 9% more than last year.
A city introduces a 30 km/h limit and accidents fall.
Courts start awarding more for whiplash and other injuries.
A hailstorm damages thousands of parked cars on one afternoon.

From burning cost to premium, and the discipline to charge it

The technical premium adds the costs the claims do not cover. The book loads 25% of premium for expenses and 5% for profit, so premium = burning cost ÷ (1 − 0.30), a factor of 1.4286. The base risk costs €209.85 and needs €300; a young urban driver of a large car costs €1,197 and needs €1,710.

The one-way tariff collects exactly the right total: its ratio of claims to risk premium is 100.0%. It is wrong cell by cell, and a market punishes that. If a rival prices with the fitted tariff and customers move whenever they are quoted more than 5% above it, 24% of the book walks away. The customers who stay are the ones the one-way tariff undercharges, and their claims run at 103.3% of the risk premium. That is adverse selection.

Underwriting discipline

A technical price is not the price the market pays. Brokers, competitors and growth targets all push the charged price below it. Discipline means knowing the gap for every segment, writing business below technical only on purpose and within a limit, and walking away when a market will not pay. L3.8.04: Premium risk and the underwriting cycle shows what happens to a whole market that does not.

The formal version

Why marginal totals are the Poisson GLM answer. Let segment $s$ have exposure $E_s$ , claims $N_s$ and a log-linear mean $\mu_s = E_s\,c_s$ with $\ln c_s = x_s^{\top}\beta$ , where $x_s$ holds a 1 for the intercept and 0/1 indicators for the segment's non-base levels. The Poisson log-likelihood and its derivative are

\begin{aligned} \ell(\beta) &= \sum_s \big(N_s \ln \mu_s - \mu_s - \ln N_s!\big) \\ \frac{\partial \ell}{\partial \beta_k} &= \sum_s \Big(\frac{N_s}{\mu_s} - 1\Big)\frac{\partial \mu_s}{\partial \beta_k} = \sum_s \Big(\frac{N_s}{\mu_s} - 1\Big)\mu_s\, x_{sk} = \sum_s x_{sk}\,(N_s - \mu_s) \\ 0 &= \sum_{s:\,x_{sk} = 1} N_s - \sum_{s:\,x_{sk} = 1} E_s\,\hat c_s \qquad \text{for each level } k \end{aligned}

Setting each derivative to zero says: the fitted claims of every level equal its actual claims. The intercept's equation does the same for the whole book. Replacing counts $N_s$ with amounts $A_s$ gives the same equations for burning cost, which is how the book fits it (a quasi-Poisson GLM; practitioners also use a Tweedie GLM on pure premium, or Poisson frequency and gamma severity separately). Bailey's iteration solves them directly.

The expected cost and the premium. With $N$ claims of size $X$ , independent:

\begin{aligned} E[S] &= E[N]\,E[X] = \text{frequency} \times \text{severity} = 0.08147 \times \text{€}3{,}915.80 = \text{€}319.03 \\ \text{premium} &= \frac{E[S]}{1 - \ell_{\text{exp}} - \ell_{\text{profit}}}, \qquad \text{sd}(\hat\lambda) = \frac{\sqrt{N}}{E} \end{aligned}

Try it

Step through the rounds of marginal totals and watch the young-driver and urban relativities move away from their one-way values. Then quote any of the 24 segments: where does the one-way tariff undercharge most?

Try itFit the tariff, then quote a driver

Rounds of marginal totals0

Relativity	One-way	Fit so far	True
Young driver (17–24)	2.270	1.000	2.400
Urban	1.567	1.000	1.450
Small car (A)	0.865	1.000	0.760
Large car (C)	1.558	1.000	1.595
Base cost (€ per vehicle-year)		319.03

Round 0: every relativity is 1 and the base is the book's average cost.

Driver age

Area

Vehicle group

Technical premium (fitted)€1,710

One-way tariff premium€1,590

Premium on the true cost€1,657

One-way against fitted−7.0%

Vehicle-years in the segment6,720

Claim frequency (95% interval)21.15% ± 1.10

Premiums are burning cost ÷ (1 − 25% expenses − 5% profit). The one-way tariff multiplies the three one-way relativities, scaled to collect the book's total cost. Illustrative data.

Worked example

Pricing a young urban driver of a large car

The book's fitted tariff, in euros per vehicle-year.

Technical premium for a 17–24-year-old, urban, group C

Step 1 of 4

1
The cell's own data
6,720 vehicle-years, 1,421 claims: frequency 21.15% ± 1.10 points, severity €5,190, burning cost €1,097.48. Usable, but noisy.

Regulatory lens

Pricing is mostly a firm's own business, but some factors are off limits. In the EU, the Court of Justice ruled in Test-Achats (Case C-236/09, 1 March 2011) that the exception allowing sex-based differences in premiums and benefits was invalid from 21 December 2012, so insurers can no longer use sex as a rating factor even where it predicts claims. Solvency II looks at pricing through premium risk: the chance that next year's premiums do not cover next year's claims, charged on the volume of premium in each line (L3.8.04: Premium risk and the underwriting cycle). How supervisors review pricing practices and product oversight is part of M4.6: Solvency II and insurance regulation.

Spark Atoms

Spark AtomThe noisiest cell is the dearest

The book's smallest segment, young rural drivers in large cars, has 455 claims on 2,880 vehicle-years. Its own burning cost is €922; the true cost is €800 and the fit says €822. A tariff that trusted each cell would overcharge it by 15%, and would undercharge some other small cell by as much.

Spark AtomAn old idea in new clothes

Actuaries fitted multiplicative tariffs by balancing marginal totals long before GLMs reached insurance. In 1999 Stephen Mildenhall showed that such minimum-bias rates are the same as those from a related GLM fitted by maximum likelihood, which let pricing teams use standard statistical tools, standard errors and tests on their old methods.

Key takeaways

Burning cost = frequency × severity: 8.15% × €3,916 = €319.03 a vehicle-year on the book.
Claim counts are Poisson, so a cell with n claims pins its frequency only to about ±1.96√n ÷ exposure: small cells are mostly noise.
One-way relativities mislead when factors are correlated: young drivers 2.27 against 2.47 fitted, urban 1.57 against 1.46.
A multiplicative tariff fitted by marginal totals gives the Poisson GLM answer: every level's fitted cost equals its actual cost.
Technical premium = burning cost ÷ (1 − expense − profit loadings); a right total with wrong cells invites adverse selection (103.3% on the book that stays).

Reaction

Check what stuck

Question 1 · objective 1

A van book has 50,000 vehicle-years, 4,000 claims and €14m of claims in total. What is its burning cost per vehicle-year, in €?

€

Question 2 · objective 1

A segment has 400 claims on 5,000 vehicle-years. Treating the count as Poisson, what is the half-width of a 95% interval on its claim frequency, in percentage points?

pp

Question 3 · objective 2

On the book, young drivers look 2.27 times as costly as the base one-way, but the multiplicative fit says 2.47. Why does the one-way figure understate them?

AYoung drivers mostly drive small cars, which cost less to repair, so their average claim is lower for a reason that is not their ageBYoung drivers have fewer claims than the data suggestCThe fit adds a profit loading to young driversDOne-way relativities always understate high-risk levels

Question 4 · objective 3

A segment's fitted burning cost is €350. With loadings of 25% of premium for expenses and 5% for profit, what is the technical premium, in €?

€

Question 5 · objective 4

A rival prices with the fitted tariff; RiskAtoms prices with one-way relativities. Customers quoted more than 5% above the rival leave. What happens to RiskAtoms' book?

AAbout 24% of it leaves, and the claims of the rest run at 103.3% of the risk premium chargedBNothing: the one-way tariff collects the right total, so the loss ratio stays at 100%CThe book improves, because the overpriced customers were the riskiestDEveryone leaves, because one-way prices are always higher

Glossary

Exposure: The unit of cover being priced. In motor, one vehicle insured for one year is one vehicle-year.
Claim frequency: Claims per unit of exposure: claims ÷ vehicle-years.
Claim severity: The average cost of a claim: total claim amounts ÷ number of claims.
Burning cost (pure premium): Claims cost per unit of exposure: total claim amounts ÷ vehicle-years, which is frequency × severity. Also called the pure premium or risk premium.
Rating factor: A characteristic the price depends on, such as driver age, area or vehicle group. Each value it takes is a level.
Relativity: How much dearer a level is than the base level of the same factor: 2.0 means twice the base cost.
One-way relativity: A relativity from one factor at a time: the level's burning cost over the base level's, ignoring every other factor.
Generalised linear model (GLM): Generalised linear model. In pricing, usually a log link, so the fitted cost is a base times one relativity per factor, each net of the others.
Marginal totals: The fitting rule that the tariff must reproduce the actual total cost of every level of every factor. For a multiplicative tariff it gives the Poisson GLM's answer.
Technical premium: The premium the risk needs: expected claims cost plus loadings for expenses and profit.
Adverse selection: When a price is too low for some customers and too high for others, the underpriced ones stay and buy, the overpriced ones leave for a rival who prices better.

Deep Dives & Origins

Deep Dives

M. Goldburd, A. Khare, D. Tevet and D. Guller, Generalized Linear Models for Insurance Rating, CAS Monograph No. 5, 2nd edition, Casualty Actuarial Society, 2020. Intermediate.
S. J. Mildenhall, “A systematic relationship between minimum bias and generalized linear models”, Proceedings of the Casualty Actuarial Society LXXXVI, 1999, pp. 393–487. Advanced.

Origins

M. Goldburd, A. Khare, D. Tevet and D. Guller, Generalized Linear Models for Insurance Rating, CAS Monograph No. 5, 2nd edition, 2020 (sections 2 and 2.9: Poisson frequency, gamma severity, Tweedie pure premium, log link; GLMs against univariate analysis).
S. J. Mildenhall, “A systematic relationship between minimum bias and generalized linear models”, Proceedings of the CAS, 1999 (abstract).
Court of Justice of the EU, Case C-236/09, Association belge des Consommateurs Test-Achats ASBL and Others v Conseil des ministres, judgment of 1 March 2011 (EU Agency for Fundamental Rights case summary).

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