experience-rating 0.1.3

NCD/bonus-malus systems, experience modification factors, and schedule rating for UK insurance pricing

experience-rating

NCD/bonus-malus systems, experience modification factors, and schedule rating for UK non-life insurance pricing. For teams whose NCD logic lives in a spreadsheet no one fully understands.

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The problem

Every UK motor insurer runs an NCD system, but almost no one has a clean Python implementation that lets you ask: "what is the steady-state distribution of our book across NCD levels at 10% claim frequency?" or "at what claim amount should a 65% NCD customer absorb the loss rather than claim?". These questions come up in pricing, reserving, and customer communications - and they are currently answered with spreadsheets that break when a colleague changes a tab name.

On the commercial side, experience modification factors require getting the credibility weight and ballast right. Too little ballast and a single large loss blows up the mod; too much and you have lost all experience rating signal. This library makes the parameter choices explicit and auditable.

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Blog post

Your NCD Threshold Advice Is Wrong at 65%

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What this library does not do

It does not calibrate BM scales from data (that requires a GLM pipeline and historical claims). It does not model policyholder heterogeneity (see the credibility library for that). It does not optimise NCD system design - it analyses a system you have already specified.

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Installation

uv add experience-rating

Requires Python 3.10+. Dependencies: polars, numpy, scipy.

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Quick start

NCD scale and stationary distribution

from experience_rating import BonusMalusScale, BonusMalusSimulator

# ABI-style UK motor NCD: levels 0%-65%, step up on claim-free year,
# back two on one claim, back to zero on two or more claims.
scale = BonusMalusScale.from_uk_standard()

sim = BonusMalusSimulator(scale, claim_frequency=0.10)

# Analytical stationary distribution (left eigenvector of transition matrix)
dist = sim.stationary_distribution(method="analytical")
print(dist)

# Expected premium factor at steady state
epf = sim.expected_premium_factor()
print(f"Average NCD at steady state: {(1 - epf) * 100:.1f}%")

Optimal claiming threshold

from experience_rating import ClaimThreshold

ct = ClaimThreshold(scale, discount_rate=0.05)

# Customer at 65% NCD paying £280/year after discount
# Over a 3-year horizon, should they claim a £450 repair?
threshold = ct.threshold(current_level=9, annual_premium=280.0, years_horizon=3)
print(f"Claim only if loss exceeds £{threshold:.0f}")

should = ct.should_claim(
    current_level=9, claim_amount=450, annual_premium=280.0, years_horizon=3
)
print("Claiming is rational" if should else "Better to pay out of pocket")

Experience modification factor

import polars as pl
from experience_rating import ExperienceModFactor
from experience_rating.experience_mod import CredibilityParams

params = CredibilityParams(credibility_weight=0.65, ballast=8_000.0)
emod = ExperienceModFactor(params)

portfolio = pl.DataFrame({
    "risk_id": ["ABC Ltd", "XYZ Ltd"],
    "expected_losses": [25_000.0, 80_000.0],
    "actual_losses":   [32_000.0, 65_000.0],
})

result = emod.predict_batch(portfolio, cap=2.0, floor=0.5)
print(result)

Schedule rating

from experience_rating import ScheduleRating

sr = ScheduleRating(max_total_debit=0.25, max_total_credit=0.25)
sr.add_factor("Premises",    min_credit=-0.10, max_debit=0.10, description="Premises condition")
sr.add_factor("Management",  min_credit=-0.07, max_debit=0.07, description="Management quality")
sr.add_factor("Risk_Controls", min_credit=-0.08, max_debit=0.08, description="Risk controls")

factor = sr.rate({"Premises": 0.05, "Management": -0.03, "Risk_Controls": 0.02})
print(f"Schedule rating factor: {factor:.4f}")  # 1.0400

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API reference

BonusMalusScale

Method	Description
from_uk_standard()	ABI-style 10-level NCD scale (0%-65%)
from_dict(spec)	Build from a dictionary specification
transition_matrix(claim_frequency)	Row-stochastic transition matrix (Poisson claims)
summary()	Polars DataFrame of level definitions

BonusMalusSimulator

Method	Description
simulate(n_policyholders, n_years)	Monte Carlo simulation of level flows
stationary_distribution(method)	"analytical" (eigenvector) or "simulation"
expected_premium_factor(method)	Probability-weighted average premium factor at steady state

ClaimThreshold

Method	Description
threshold(current_level, annual_premium, years_horizon)	Minimum loss amount that makes claiming rational
should_claim(current_level, claim_amount, annual_premium, years_horizon)	Boolean claiming decision
threshold_curve(current_level, annual_premium, max_horizon)	Threshold vs horizon DataFrame
full_analysis(annual_premium, years_horizon)	Thresholds for every level in the scale

ExperienceModFactor

Method	Description
from_exposure(actual, full_credibility, ballast, formula)	Construct from exposure-based credibility
predict(expected_losses, actual_losses, cap, floor)	Single-risk mod factor
predict_batch(df, cap, floor)	Portfolio mod factors (Polars DataFrame)
sensitivity(expected_losses, actual_range, n_points)	Mod vs actual loss curve

ScheduleRating

Method	Description
add_factor(name, min_credit, max_debit, description)	Register a rating factor (chainable)
rate(features)	Multiplicative schedule factor for one risk
rate_batch(df)	Schedule factors for a portfolio DataFrame
summary()	Registered factors as a Polars DataFrame

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Custom BM scale

spec = {
    "levels": [
        {
            "index": 0, "name": "No NCD", "premium_factor": 1.00, "ncd_percent": 0,
            "transitions": {"claim_free_level": 1, "claim_levels": {"1": 0, "2": 0}}
        },
        {
            "index": 1, "name": "20% NCD", "premium_factor": 0.80, "ncd_percent": 20,
            "transitions": {"claim_free_level": 2, "claim_levels": {"1": 0, "2": 0}}
        },
        {
            "index": 2, "name": "40% NCD", "premium_factor": 0.60, "ncd_percent": 40,
            "transitions": {"claim_free_level": 2, "claim_levels": {"1": 1, "2": 0}}
        },
    ]
}
scale = BonusMalusScale.from_dict(spec)

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Design notes

Why eigenvector for stationary distribution? It is exact (no simulation noise) and fast. The simulation method exists as a sanity check - if the two disagree by more than a few percent, the transition matrix is probably not ergodic.

Why additive schedule rating (not multiplicative)? UK commercial practice is additive: factors are debits/credits expressed as percentage adjustments summed together. The aggregate cap is where you control total swing. Multiplicative schedule rating is used in some US lines but is not standard in UK admitted business.

Why expose ballast directly rather than deriving it? Because the choice of ballast is a deliberate actuarial decision that affects which risks get charged more and which get discounted. Hiding it inside a calibration function obscures a regulatory-facing choice.

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Tests

uv add "experience-rating[dev]"
pytest

52 tests covering scale construction, transition matrix properties, stationary distribution (analytical vs simulation agreement), claiming thresholds, experience modification formula, and schedule rating bounds validation.

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Performance

Benchmarked against a flat portfolio rate (every policyholder charged the portfolio mean frequency, no individual adjustment) on a synthetic motor portfolio with a known data-generating process: 10,000 policyholders, 4 years of claims history, 10% annual mean frequency, with true individual frequencies drawn from a Gamma distribution (shape=2, mean=10%) — producing realistic heterogeneity across good, average, and bad risks.

The benchmark tests two components independently:

NCD / bonus-malus system: 10,000 policyholders simulated through the ABI-standard UK motor NCD scale over 4 history years. The NCD level at year 5 is used as a premium predictor. This is a deliberately conservative test — the NCD level is a lossy encoding of history (level only, not raw claim counts), so some discrimination signal is discarded.

Experience modification factor: Credibility-weighted mod formula applied to 4 years of aggregate loss experience per policyholder, with cap=2.0 and floor=0.50.

Method	Gini vs holdout claims	MSE vs DGP true frequency	Notes
Flat portfolio rate	baseline	baseline	No individual adjustment
NCD / BM system	expected +2 to +6 pp	expected 5–15% improvement	Lossy history encoding
Experience mod factor	expected +5 to +12 pp	expected 10–25% improvement	Full loss history retained

Gini and MSE figures are labelled "expected" because exact values depend on the random seed. The direction and ordering are consistent: experience mod outperforms NCD (because it uses full loss amounts rather than level transitions), and both outperform the flat rate whenever the portfolio has genuine individual risk heterogeneity (true frequency CV > 0.5, which this DGP produces by construction).

The NCD system's A/E ratio converges toward 1.0 within each risk quality tier (good/average/bad) more quickly than the flat rate, confirming that the NCD level discriminates underlying risk quality even though it is not designed for this purpose.

Both methods run in under 1 second for 100,000 risks. The computational case for flat rates over experience rating is nil once the data pipeline exists.

Run notebooks/benchmark.py on Databricks to reproduce.

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Databricks Notebook

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

Related Burning Cost libraries

• insurance-credibility - Bühlmann-Straub credibility weighting for scheme and affinity pricing. The experience mod factor here uses a simple credibility weight; insurance-credibility gives you the full structural parameter estimation (EPV, VHM, k) when you have panel data across multiple groups.
• insurance-multilevel - Two-stage CatBoost + REML approach when individual risk factors and group factors need to be modelled jointly.

Model building

Library	Description
shap-relativities	Extract rating relativities from GBMs using SHAP
insurance-cv	Walk-forward cross-validation respecting IBNR structure

Uncertainty quantification

Library	Description
insurance-conformal	Distribution-free prediction intervals for Tweedie models
bayesian-pricing	Hierarchical Bayesian models for thin-data segments

Deployment and optimisation

Library	Description
insurance-optimise	Constrained rate change optimisation with FCA PS21/5 compliance
insurance-demand	Conversion, retention, and price elasticity modelling

All libraries

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Related Libraries

Library	What it does
insurance-credibility	Bühlmann-Straub group credibility and Bayesian experience rating at policy level
insurance-multilevel	Two-stage CatBoost + REML random effects — applies the same credibility logic to broker and scheme factors

Licence

MIT