insurance-credibility 0.1.8

Credibility models for UK non-life insurance pricing: classical Bühlmann-Straub and individual-policy Bayesian experience rating

insurance-credibility

Bühlmann-Straub credibility and Bayesian experience rating for UK insurance pricing teams.

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The problem

Small segments have unstable loss experience. A fleet scheme with 200 vehicle-years has a loss ratio that is mostly noise, but ignoring it entirely prices a segment you have genuine data on. How much should you trust the scheme's own history versus the portfolio average?

The same question arises at individual policy level: a commercial motor policy with 5 years of no-claims history deserves a discount, but how large? Flat NCD tables assign the same maximum discount regardless of policy size or the underlying claim frequency — a 0.5-vehicle-year policy gets the same credit as a 50-vehicle-year fleet.

Blog post: Bühlmann-Straub Credibility in Python: Blending Thin Segments with Portfolio Experience

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Why this library?

Bühlmann-Straub is the actuarial standard for this problem — a statistically optimal blend of segment experience with the portfolio mean, weighted by earned exposure. Most existing implementations assume non-insurance data structures: equal group sizes, no exposure weights, no distinction between within-group and between-group variance.

This library is built for insurance: it handles unequal exposures, nested hierarchies (scheme → book, district → area), and individual policy experience rating in a consistent framework.

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Compared to alternatives

	Manual credibility weights	Random effects GLM	Hierarchical Bayes	insurance-credibility
Statistically optimal blend	No (rule-of-thumb)	Yes	Yes	Yes (B-S formula)
No prior specification needed	Yes	Yes	No	Yes
Handles unequal exposures	Manual	Yes	Yes	Yes
Nested group hierarchies	Manual	Partial	Yes	Yes (HierarchicalBuhlmannStraub)
Individual policy experience rating	No	No	Partial	Yes
Closed-form, < 1 second	Yes (simple)	No	No	Yes
Full posterior distribution	No	No	Yes	Yes (DynamicPoissonGammaModel)

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Quickstart

uv add insurance-credibility

import polars as pl
from insurance_credibility import BuhlmannStraub

df = pl.DataFrame({
    "scheme":    ["A", "A", "A", "B", "B", "B", "C", "C", "C"],
    "year":      [2022, 2023, 2024, 2022, 2023, 2024, 2022, 2023, 2024],
    "loss_rate": [0.12, 0.09, 0.11, 0.25, 0.28, 0.22, 0.08, 0.07, 0.09],
    "exposure":  [120.0, 135.0, 140.0, 45.0, 50.0, 48.0, 300.0, 310.0, 320.0],
})

bs = BuhlmannStraub()
bs.fit(df, group_col="scheme", period_col="year",
       loss_col="loss_rate", weight_col="exposure")

print(bs.z_)         # credibility factors per scheme
print(bs.k_)         # Bühlmann's k: noise-to-signal ratio
print(bs.premiums_)  # credibility-blended premium per scheme

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Group credibility: schemes and large accounts

BuhlmannStraub fits structural parameters — within-group variance (v) and between-group variance (a) — from the portfolio using method of moments. It then computes the credibility factor Z_i for each group:

Z_i = w_i / (w_i + k)    where k = v/a

Z approaches 1.0 as exposure grows — thick schemes are trusted almost entirely. Z shrinks toward 0 on thin schemes — the portfolio mean gets most of the weight. On a 30-scheme, 5-year benchmark with known true parameters:

Tier	Raw MAE	Portfolio avg MAE	Credibility MAE
Thin (< 500 exposure)	0.0074	0.0596	0.0069
Medium (500–2000)	0.0030	0.0423	0.0029
Thick (2000+)	0.0014	0.0337	0.0014 (tie)

Credibility beats raw experience on thin and medium tiers. On thick tiers, Z approaches 1.0 and the two methods converge — which is correct behaviour.

HierarchicalBuhlmannStraub extends this to nested group structures: scheme → book, sector → district → area. Following Jewell (1975).

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Individual policy experience rating

For commercial motor and fleet pricing, where you want to move individual policies away from the GLM rate based on their own claims history:

from insurance_credibility import ClaimsHistory, StaticCredibilityModel

histories = [
    ClaimsHistory("POL001", periods=[1, 2, 3], claim_counts=[0, 1, 0],
                  exposures=[1.0, 1.0, 0.8], prior_premium=400.0),
    ClaimsHistory("POL002", periods=[1, 2, 3], claim_counts=[2, 1, 2],
                  exposures=[1.0, 1.0, 1.0], prior_premium=400.0),
]

model = StaticCredibilityModel()
model.fit(histories)

cf = model.predict(histories[0])
posterior_premium = histories[0].prior_premium * cf

exposures is the key parameter that distinguishes this from flat NCD tables: a policy with 0.5 years of exposure gets far less credibility than one with 5 years, regardless of claim count.

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Model tiers

StaticCredibilityModel — Bühlmann-Straub at individual policy level. Fits kappa = sigma^2 / tau^2 from a portfolio of policy histories. Credibility weight for a policy is omega = e_total / (e_total + kappa). Closed-form, fast, suitable for production.

DynamicPoissonGammaModel — Poisson-gamma state-space model following Ahn, Jeong, Lu & Wüthrich (2023). Seniority-weighted updates: recent years count more. Produces the full posterior distribution per policy, not just a point estimate — useful when communicating uncertainty to a pricing committee or reinsurer.

SurrogateModel — IS-surrogate (Calcetero et al. 2024). For large portfolios where computing the exact posterior for every policy is expensive.

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Structural parameter recovery

On a 30-group, 5-year benchmark with known true parameters (mu=0.650, v=0.020, a=0.005, k=4.0):

• mu recovered within 1.4%
• k recovered within factor of 2 (conservative shrinkage direction)

k is over-estimated in small samples — a known property of the method-of-moments estimator. Conservative shrinkage is safe: it means you trust thin segments slightly less than the theory would dictate. On portfolios with 100+ groups over 7+ years, k converges to the true value.

Full validation: notebooks/databricks_validation.py.

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Bühlmann-Straub vs random effects GLM

The actuarial credibility approach and the random effects GLM (e.g. statsmodels MixedLM) estimate the same quantity under a Gaussian approximation. The differences are practical:

• Bühlmann-Straub is closed-form and fits in under a second on a 150-row scheme panel. No iteration, no convergence issues.
• Random effects GLM requires a correctly specified likelihood and converges slowly on unbalanced panels with many groups.
• Bühlmann-Straub exposes the structural parameters (mu, v, a, k) directly, making them easy to inspect and challenge.

For Poisson-Gamma likelihoods and non-Gaussian random effects, use DynamicPoissonGammaModel.

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Limitations

• Structural parameter estimation (v, a) requires at least 30–50 groups and 3+ years to converge reliably. On the 30-group benchmark, VHM was underestimated by 57.6%. In thin portfolios, treat credibility factors as directional and apply a floor on Z.
• StaticCredibilityModel assumes homoscedastic within-policy variance. Segment by policy size tier on portfolios with large fleets alongside small ones.
• Kappa estimation needs at least 50–100 policies with 2+ years of history. Below this, the estimate is unreliable.
• Structural parameters must be refitted as portfolio composition changes. Stale kappa from a different historical book produces miscalibrated experience adjustments.

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Part of the Burning Cost stack

Takes segment-level experience data: earned exposure, observed loss ratios, scheme panels. Feeds credibility-weighted estimates into insurance-gam (as adjusted targets for tariff fitting). See the full stack

Library	Description
insurance-whittaker	Whittaker-Henderson smoothing — smooths the raw experience rates that credibility weighting then blends
insurance-gam	Interpretable GAMs — credibility-adjusted targets as input to tariff fitting
insurance-conformal	Distribution-free prediction intervals — uncertainty quantification for credibility-blended estimates
insurance-monitoring	Model drift detection — monitors whether credibility parameters remain valid
insurance-governance	Model validation and MRM governance — sign-off pack for credibility models

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References

• Bühlmann, H. & Gisler, A. (2005). A Course in Credibility Theory and Its Applications. Springer.
• Jewell, W.S. (1975). "Multidimensional Credibility." Operations Research, 23(5), 904–920.
• Ahn, J.Y., Jeong, H., Lu, Y. & Wüthrich, M.V. (2023). "Dynamic Bayesian Credibility." arXiv:2308.16058.
• Calcetero, V., Badescu, A. & Lin, X.S. (2024). "Credibility theory for the 21st century." ASTIN Bulletin.

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Community

• Questions? Start a Discussion
• Found a bug? Open an Issue
• Blog and tutorials: burning-cost.github.io
• Training course: Insurance Pricing in Python — Module 6 covers credibility theory. £97 one-time.

Licence

MIT