Statistics for General Insurance

A compact, example-led course in reserving, loss models, reinsurance, aggregate risk, and ruin.

Statistics for General Insurance

An insurer receives premiums now and pays uncertain claims later. This course develops one connected answer to five questions:

Reserve: how much do claims that have already happened still cost?
Model: how often will claims occur, and how large can they be?
Transfer: which losses remain after reinsurance?
Aggregate: what can the whole portfolio lose in one period?
Survive: how much capital is needed against adverse paths through time?

The pages are concise by design. Each new idea is introduced through a calculation, followed by the assumption that makes the calculation defensible.

Before choosing a formula, write down the unit, time period, information date, loss basis, and decision. Most serious insurance-model errors begin before any arithmetic is done.

Learning outcomes

By the end, you should be able to:

validate paid, incurred, and claim-count data before constructing a run-off triangle;
calculate and critique chain-ladder, frequency–severity, and Bornhuetter–Ferguson reserves;
choose frequency and severity models using exposure, tail, censoring, and validation evidence;
translate proportional and excess-of-loss wording into retained-loss functions;
derive or simulate aggregate-loss moments and tail measures;
interpret classical ruin results without confusing them with accounting or regulatory capital;
present a recommendation with assumptions, diagnostics, uncertainty, and limitations.

These outcomes suit an upper-undergraduate course and provide a bridge to postgraduate stochastic reserving, extreme-value theory, credibility, and solvency modelling.

One portfolio, used throughout

Harbour Mutual is a fictional UK motor insurer. All figures are synthetic unless a page explicitly labels an external data source.

Item	Course convention
Valuation date	31 December 2023
Origin period	Accident year (AY)
Development age	Years since the start of the accident year
Main triangle	Cumulative paid own-damage loss, £m
Exposure	Earned vehicle-years
Severity	Ultimate cost per claim
Reinsurance	Applied to each loss unless stated otherwise
Capital horizon	One year for VaR/TVaR; multiple years for ruin

Its paid triangle is:

AY	Dev 0	Dev 1	Dev 2	Dev 3
2020	100	180	240	280
2021	120	220	300	—
2022	140	260	—	—
2023	160	—	—	—

Blank cells are unobserved future development, not zero. With volume-weighted age-to-age factors,

f_{0:1}=\frac{660}{360}=1.8333,\qquad f_{1:2}=\frac{540}{400}=1.35,\qquad f_{2:3}=\frac{280}{240}=1.1667.

This gives the following baseline—not a final professional opinion:

AY	Latest paid	Projected ultimate	Reserve
2020	280.0	280.0	0.0
2021	300.0	350.0	50.0
2022	260.0	409.5	149.5
2023	160.0	462.0	302.0
Total	1,000.0	1,501.5	501.5

The calculation is easy. The hard question is whether past development is stable enough to project the unobserved cells. The reserving module makes that question explicit.

Vocabulary that changes the model

Term	Operational meaning	Common mistake
Paid loss	Cash paid by the valuation date	Treating settlement speed as claim severity
Case reserve	Current estimate on a reported, unsettled claim	Assuming adjuster practice is stable
Incurred loss	Usually paid plus case reserve	Using the label without documenting the organisation's definition
IBNR	Future recognition of losses already incurred; usage may include development on known claims	Treating every organisation's IBNR convention as identical
Ultimate loss	Expected cost after all development	Calling the oldest observed cell “ultimate” without evidence
Exposure	Amount of insured risk, such as vehicle-years	Comparing raw counts when portfolio size changed
Deductible	Amount borne by the policyholder	Fitting the observed payment distribution as if it were ground-up loss
Policy/reinsurance limit	Maximum payment under stated terms	Using an unlimited severity model without applying the contract

Two identities organise the course:

\text{reserve}=\widehat{\text{ultimate loss}}-\text{loss observed to date},

S=\sum_{i=1}^{N}X_i,

where $N$ is claim frequency, $X_i$ is severity, and $S$ is aggregate loss. These identities do not specify a model; they tell us what must be estimated.

Course route

Rendering diagram…

Unit	Preparation	In-class question	Evidence produced
1. Data and reserving	cumulative vs incremental claims	Which cells are observed, and as of when?	validated triangle and data contract
2. Reserving methods	weighted averages; expected loss ratio	What makes a development pattern credible?	chain-ladder, frequency–severity, and BF comparison
3. Severity	survival functions and quantiles	Which model fits the centre and the tail?	fitted models and out-of-sample diagnostics
4. Frequency	Poisson and conditional expectation	Is variance larger than the mean because exposure or risk differs?	exposure-adjusted count model
5. Reinsurance	functions and layer notation	What exactly is retained for this claim or event?	contract loss transformation
6. Aggregate risk	expectation, variance, simulation	How do count, severity, and dependence combine?	simulated loss distribution and tail measures
7. Ruin and capstone	surplus processes	What does the model omit from a real capital decision?	recommendation with limitations

Suggested workload is 2 hours of preparation, 2 hours in class, and 2 hours of practice per unit. Students who have not used simulation should add one hour to Units 3 and 6. Every executable example runs in the page; no local Python or R installation is required.

A 10-minute readiness check

A portfolio has 25,000 vehicle-years and 1,000 claims. What is observed frequency per vehicle-year?
Expected claim count is 100 and expected severity is £2,000. Is aggregate loss always £200,000?
A £120,000 claim enters an unlimited £50,000 excess-of-loss layer. What is retained?
Why can paid and incurred triangles produce different reserves?
Why is a 99.5% one-year VaR not the same object as an ultimate ruin probability?

Answers

$1{,}000/25{,}000=0.04$ claims per vehicle-year.
No. Under standard independence assumptions £200,000 is $E[S]=E[N]E[X]$ , not the realised loss.
The insurer retains £50,000 and cedes £70,000.
Paid data reflect settlement timing; incurred data also reflect case-reserving practice. Either process may change.
They use different horizons, loss definitions, and dynamics. VaR is a quantile of a specified one-year distribution; ruin is a path event in a surplus process.

First executable example: from claims to retained loss

Suppose annual count is Poisson with mean 40 and individual loss is lognormal with mean about £5,000. Compare gross loss with a £20,000 per-loss retention. Change the seed or retention and rerun.

Gross and retained annual loss

The example previews three important points: an expected value does not describe tail risk; reinsurance acts on a precisely defined loss; and simulation answers only the model we programmed.

Contemporary context—and its boundary

Accounting: IFRS 17 is effective for annual periods beginning on or after 1 January 2023. It governs recognition and measurement of insurance contracts; it does not turn the deterministic chain ladder into an accounting standard.
Regulation: Solvency UK reporting applies for reporting dates from 31 December 2024. A classroom ruin probability is not the Solvency Capital Requirement.
Emerging loss: Swiss Re Institute estimated 2025 natural-catastrophe economic losses at USD 220 billion, with 49% insured, and reported that secondary perils generated 92% of insured losses. This motivates heavy-tail and dependence analysis; it does not identify a distribution by itself.

Reading-list provenance

The module reading lists were last checked on 1 August 2026. They form a selective teaching map, not a PRISMA systematic review. Sources are used in three layers:

peer-reviewed papers and established texts for mathematical foundations;
official standard-setter and regulator pages for current institutional boundaries;
recent professional research, data releases, and clearly labelled preprints for examples and open methods.

Coverage is therefore illustrative rather than exhaustive. A current statistic can motivate a stress, and a preprint can motivate an extension; neither supplies a portfolio parameter without validation.

Current industry figures are used to pose modelling questions, not to prove universal parameters. Check scope, definitions, currency, exposure, event date, publication date, and revisions before reusing any number.

How to study and report

For every method, keep a one-page model record:

Field	Minimum answer
Decision	reserve selection, treaty comparison, or capital measure
Data	unit, scope, valuation date, and transformations
Assumptions	stability, independence, distribution, and contract terms
Validation	residuals, holdout/diagonal test, sensitivity, or simulation error
Uncertainty	process, parameter, model, and operational sources
Boundary	what the result must not be interpreted as

Start with the claims-development data contract.

Sources and further reading

IFRS Foundation — IFRS 17 Insurance Contracts
Bank of England — Solvency UK insurance-sector reporting
Swiss Re Institute — natural catastrophes in 2025
Mack, T. (1993), “Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates,” ASTIN Bulletin, 23(2), 213–225.
Klugman, S. A., Panjer, H. H., and Willmot, G. E. (2019), Loss Models: From Data to Decisions, 5th ed.

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Claims Development and Reserving

Build a defensible claims-development data contract before projecting a reserve.