Claims Development and Reserving

Frequency–Severity Reserving

Separate claim emergence from average cost to explain why a triangle develops.

Frequency–Severity Reserving

A paid-loss triangle can rise because more claims become known, known claims cost more, or payments accelerate. A frequency–severity view separates two mechanisms before recombining them.

\widehat U_i =\widehat N_i^{\text{ultimate}} \times \widehat A_i^{\text{ultimate}},

where $N$ is ultimate claim count and $A$ is average ultimate cost per claim.

A claimant, coverage, event, and claim-file record need not be the same unit. Reopened and nil claims can change counts without changing economic loss. Freeze one operational definition across all periods.

1. Add a cumulative reported-count triangle

Harbour Mutual reports:

AY	Dev 0	Dev 1	Dev 2	Dev 3
2020	50	80	95	100
2021	55	88	104	—
2022	60	96	—	—
2023	70	—	—	—

Use the same volume-weighted calculation as chain ladder:

Count transition	Selected factor	CDF to ultimate
0 → 1	$264/165=1.6000$	1.9950
1 → 2	$199/168=1.1845$	1.2469
2 → 3	$100/95=1.0526$	1.0526

Projected ultimate counts are therefore approximately 100.0, 109.5, 119.7, and 139.6 for AY 2020–2023. Fractional values are expectations, not literal future files.

2. Examine average paid per reported claim

For diagnosis, divide cumulative paid loss by cumulative reported count in the matching cell:

A_{i,j}=\frac{C_{i,j}^{\text{paid}}}{N_{i,j}^{\text{reported}}}.

AY	Dev 0	Dev 1	Dev 2	Dev 3
2020	2.000	2.250	2.526	2.800
2021	2.182	2.500	2.885	—
2022	2.333	2.708	—	—
2023	2.286	—	—	—

Values are £m per count only because the teaching triangle is deliberately small. Real work would normally use £000 per claim.

This ratio is not yet “settled severity”: its numerator is paid to date while its denominator is reported claims to date. It is useful only when we understand the timing mismatch.

Using the arithmetic mean of observed cell-level ratios gives illustrative average-cost factors:

Average-cost transition	Selected factor	CDF to ultimate
0 → 1	1.1438	1.4431
1 → 2	1.1383	1.2616
2 → 3	1.1083	1.1083

The selection rule is intentionally visible. Alternatives include volume weighting, settled-claim severity, payments-per-settlement, or a regression model with inflation and mix terms.

3. Recombine the two projections

AY	Ultimate count	Ultimate average cost	Ultimate loss	Reserve
2020	100.0	2.800	280.0	0.0
2021	109.5	3.197	350.0	50.0
2022	119.7	3.417	409.0	149.0
2023	139.6	3.299	460.6	300.6
Total			1,499.7	499.7

The result is close to the £501.5m chain-ladder reserve because both components were derived from the same paid and count observations. The value of the decomposition is not the £1.8m difference; it is the explanation of where development comes from.

4. Read the signals

Observed pattern	Plausible mechanism	Additional evidence
count factors fall, cost factors stable	faster reporting	report-date distributions and operational changes
counts stable, average cost rises by calendar year	inflation or claim mix	price indices, injury mix, repair duration
average paid cost falls while closure rises	accelerated small-claim settlement	open/closed counts and payment type
count and cost both jump	exposure, catastrophe, coding, or coverage change	policy and event-level data

Never infer the mechanism from the triangle alone. The table generates hypotheses to test.

5. A more defensible practical implementation

For each origin year:

Model ultimate claim count using cumulative reported or settled counts, with exposure where relevant.
Model ultimate average severity on a coherent claim population.
Apply inflation and mix adjustments at the level where they arise.
Multiply expected count and severity only after addressing dependence:

E[S]=E[N]E[X]

holds when severities are identically distributed and independent of $N$ . More generally, use

E[S]=E\!\left[E[S\mid N,Z]\right],

where $Z$ can represent weather, inflation, or portfolio mix affecting both frequency and severity.

6. Compare with chain ladder

Question	Chain ladder	Frequency–severity
What is projected?	aggregate cumulative loss	counts and cost per claim
Main strength	simple, transparent, reproducible	operational explanation and scenario control
Main data need	consistent loss triangle	aligned count, loss, status, and exposure definitions
Main risk	stable pattern assumed	two models can compound misspecification

Use frequency–severity as a diagnostic even when the selected reserve remains chain-ladder based.

Practice

AY 2023 ultimate count is 140 and selected average ultimate cost is £3.5m. Find ultimate loss and reserve from £160m paid.
A claims-system migration creates duplicate claim-file IDs. Which component is biased first?
Why might average settled claim cost be biased downward at early ages?

Answers

Ultimate $=140\times3.5=£490$ m; reserve $=490-160=£330$ m.
Reported frequency; the resulting average cost may be biased downward because the denominator is inflated.
Small, simple claims often settle first. Early settled claims are therefore not representative of all claims.

Next, combine observed development with an independent prior.

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Basic Chain Ladder

Calculate a deterministic chain-ladder reserve and expose the assumptions behind every step.

Bornhuetter–Ferguson Method

Blend an expected-loss prior with observed maturity without letting early random payments dominate.