insurance-trend 0.1.3

Loss cost trend analysis for UK personal lines insurance pricing

insurance-trend

Loss cost trend analysis for UK personal lines insurance pricing.

Blog post: Loss Cost Trend Analysis in Python

The problem

Every UK motor and household pricing actuary does loss cost trend analysis every quarter. The workflow is: take aggregate accident-period experience data, fit a log-linear trend to frequency and severity separately, project forward to the next rating period, and report a trend rate with a confidence interval.

Currently this is done in Excel, SAS, or bespoke R scripts. There is no Python library for it. chainladder-python handles reserving triangles but does nothing for pricing trend — it applies user-specified factors, it does not fit them from data.

The post-2021 inflationary environment has made this more urgent: UK motor claims inflation ran at 34% from 2019 to 2023 versus CPI of 21% — a 13 percentage point superimposed component that CPI alone does not capture. A library that cannot identify structural breaks (COVID lockdown, Ogden rate change) will produce misleading trend estimates.

Quick start

import numpy as np
from insurance_trend import LossCostTrendFitter

# 24 quarters of UK motor aggregate experience data (2019 Q1 – 2024 Q4)
# Synthetic: stable frequency pre-2021, then step-down (COVID recovery lag),
# then gradual recovery. Severity accelerates post-2021 (repair cost inflation).
rng = np.random.default_rng(42)
n = 24

periods = [
    f"{yr}Q{q}"
    for yr in range(2019, 2025)
    for q in range(1, 5)
]

# Earned vehicle-years (growing book, slight seasonality)
earned_vehicles = (
    12_000
    + np.arange(n) * 150
    + rng.normal(0, 200, n)
).clip(10_000, None)

# True frequency: stable ~0.085 pre-2021, drops to ~0.065 in 2021 (COVID),
# then recovers at +3% pa through 2024
t = np.arange(n)
freq_true = np.where(
    t < 8,
    0.085 + 0.001 * t,
    0.065 * np.exp(0.007 * (t - 8)),  # post-COVID recovery trend
)
claim_counts = rng.poisson(freq_true * earned_vehicles).astype(float)

# True severity: accelerates at +8% pa from 2022 (repair inflation)
base_severity = 3_800.0
sev_true = base_severity * np.where(
    t < 12,
    1.0 + 0.03 * t / 4,
    (1.0 + 0.03) ** 3 * np.exp(0.08 * (t - 12) / 4),  # post-2022 inflation
)
total_paid = claim_counts * rng.lognormal(np.log(sev_true), 0.15)

fitter = LossCostTrendFitter(
    periods=periods,
    claim_counts=claim_counts,
    earned_exposure=earned_vehicles,
    total_paid=total_paid,
)

result = fitter.fit(
    detect_breaks=True,   # auto-detect COVID, Ogden rate change
    seasonal=True,        # quarterly seasonal dummies
)

print(result.combined_trend_rate)  # e.g. 0.085 — 8.5% pa loss cost trend
print(result.decompose())          # freq_trend, sev_trend, superimposed
print(result.summary())

With an ONS external index for severity deflation (requires network access):

from insurance_trend import LossCostTrendFitter, ExternalIndex

# Fetch ONS motor repair index (SPPI G4520, 2015=100)
motor_repair_idx = ExternalIndex.from_ons('HPTH')

fitter = LossCostTrendFitter(
    periods=periods,
    claim_counts=claim_counts,
    earned_exposure=earned_vehicles,
    total_paid=total_paid,
    external_index=motor_repair_idx,  # deflates severity; superimposed_inflation() gives residual
)
result = fitter.fit(detect_breaks=True, seasonal=True)
print(result.superimposed_inflation)  # trend component not explained by ONS index

Classes

• FrequencyTrendFitter — log-linear OLS on log(claims/exposure). Optional WLS, quarterly seasonal dummies, structural break detection via ruptures PELT, piecewise refitting on detected breaks, bootstrap CI, local linear trend alternative.

• SeverityTrendFitter — same as frequency, plus optional external index deflation. When an index is supplied, the fit runs on deflated severity and superimposed_inflation() gives the residual trend not explained by the index.

• LossCostTrendFitter — wraps the frequency and severity fitters, combines results, provides decompose() and projected_loss_cost().

• ExternalIndex — fetches ONS time series from the public API (no auth required), with a catalogue of UK insurance-relevant codes. Also accepts user-supplied CSV for BCIS and other subscription data.

Why log-linear

The industry baseline. Fits log(y) = alpha + beta*t + seasonal + epsilon via OLS. The annual trend rate is exp(beta * periods_per_year) - 1. The model is transparent, easily explainable to a regulator, and fast enough to bootstrap 1000 replicates in under a second.

The local linear trend alternative (method='local_linear_trend') uses statsmodels UnobservedComponents with a Kalman filter — useful when the trend itself is changing, but requires longer series and is harder to explain.

Structural breaks

The ruptures PELT algorithm runs on the log-transformed series. If a break is detected, the library warns and refits piecewise. The trend rate from the final segment is what gets reported — this is the defensible choice for projection, since you are projecting from the current regime.

Pass changepoints=[8, 20] to impose known breaks (e.g. 2020 Q1, 2025 Q1) rather than using auto-detection.

ONS series catalogue

Key	ONS code	Description
motor_repair	HPTH	SPPI G4520 Maintenance and repair of motor vehicles (2015=100)
motor_insurance_cpi	L7JE	CPI 12.5.4.1 Motor vehicle insurance
vehicle_maintenance_rpi	CZEA	RPI Maintenance of motor vehicles
building_maintenance	D7DO	CPI 04.3.2 Services for maintenance and repair of dwellings
household_maintenance_weights	CJVD	CPI Weights 04.3 Maintenance and repair

For household severity, use D7DO as a free proxy. BCIS is more appropriate for reinstatement cost trend — load it via ExternalIndex.from_csv().

Inputs

Aggregate accident-period data. Minimum viable: 6 quarters. Recommended: 12–20 quarters.

Column	Description
periods	Quarter identifiers, e.g. '2020Q1'
claim_counts	Number of claims in the period
earned_exposure	Earned exposure (vehicle-years, policy-years, etc.)
total_paid	Total paid claims

Both pandas and Polars DataFrames/Series are accepted as inputs. All outputs are Polars.

Installation

uv add insurance-trend

Questions or feedback? Start a Discussion. Found it useful? A star helps others find it.

Dependencies

pandas, numpy, statsmodels, scipy, ruptures, matplotlib, requests, polars.

No scikit-learn, TensorFlow, or PyTorch.

Mix adjustment

V1 does not include mix adjustment. If your portfolio composition has shifted (more young drivers, different vehicle types), apparent trends may reflect mix change rather than genuine inflation. Pre-process to mix-adjusted frequency/severity before passing to the fitters if this matters for your use case.

Scope

This library is for pricing trend — forward projection of aggregate accident-period data. It is not a reserving tool. Use chainladder-python for triangle development to ultimate; use insurance-trend for what comes after.

Databricks Notebook

A ready-to-run Databricks notebook benchmarking this library against standard approaches is available in burning-cost-examples.

Performance

Benchmarked against a naive OLS baseline on synthetic UK motor data — 36 quarters (2019 Q1 to 2027 Q4) with a -35% frequency step-down at Q12 (COVID lockdown magnitude) and a severity acceleration at Q20. This DGP is the critical test case: when there is a large structural break mid-series, naive OLS produces a blended trend rate dominated by the step-down that is useless for projection. Results from benchmarks/benchmark.py.

DGP (true post-break rates): frequency +3.0% pa, severity +8.0% pa, loss cost +11.24% pa

Trend rate accuracy:

Component	True (DGP)	Naive OLS	insurance-trend	Naive error	Lib error
Frequency	+3.0%	~−8%	~+3%	~−11 pp	~0 pp
Severity	+8.0%	~+4%	~+8%	~−4 pp	~0 pp
Loss cost	+11.2%	~−4%	~+11%	~−15 pp	~0 pp

The naive OLS frequency estimate is approximately −8% pa because the −35% lockdown step-down at Q12 drags the entire fitted line downward. The library detects the break, discards the pre-break segment, and refits on Q12–Q36 only — recovering the true +3% pa recovery trend.

Break detection (36-quarter series, penalty=1.5):

Component	True break	Detected	Within ±2Q?
Frequency	Q12	Q12	Yes
Severity	Q20	Q20	Yes

4-quarter forward projection MAPE:

Method	Loss cost MAPE
Naive OLS	~30%
insurance-trend	~10%
Improvement	~20 pp

The projection MAPE improvement of ~20 pp reflects that naive OLS extrapolates from a downward-biased trend while the library projects from the correct post-break regime. This is the use case the library exists for.

Penalty parameter guidance:

The penalty parameter controls PELT's sensitivity. Lower values detect more (and smaller) breaks; higher values require a larger signal. For large breaks (>15pp step-change, as in COVID lockdown), penalty=1.5 reliably fires. For smaller breaks, reduce further or use changepoints= to impose known dates. The default penalty=2.0 is conservative — if you know there was a structural event (Ogden rate change, lockdown), impose it explicitly rather than relying on auto-detection.

Run benchmarks/benchmark.py on Databricks to reproduce. The benchmark numbers above are indicative; exact values depend on the random seed and the PELT detection result.

Related Libraries

Library	Description
insurance-causal-policy	SDID causal evaluation of rate changes — separates genuine market trends from the effects of pricing actions
insurance-dynamics	Loss development models — trend projections inform the development assumptions in reserve models
insurance-whittaker	Whittaker-Henderson graduation for development triangles — smooth the trends before forward projection

Licence

MIT. Part of the Burning Cost insurance pricing toolkit.