insurance-bunching

Bunching estimators for insurance threshold gaming detection.

The problem

Insurance pricing creates sharp incentives at threshold boundaries. When premiums jump at 10,000 miles, policyholders declare 9,999. When NCD resets at 60%, insurers see mass just below the threshold. When sum-insured bands change at £50,000, you get a spike of policies at exactly £50,000.

This is bunching — excess mass in a distribution at a known threshold — and it matters because:

1. Adverse selection: The risk pool is mispriced if customers gaming the threshold have different claims rates than the segment they appear to be in.
2. Premium leakage: Mileage understaters cost more claims than their premium reflects.
3. FCA Consumer Duty: You need evidence that your pricing creates fair outcomes. Unexplained bunching at kinks is a red flag in a regulatory review.

The technique comes from public economics (Saez 2010, Kleven 2016) where it was used to detect taxpayer responses to income tax kinks. The methodology is rigorous, peer-reviewed, and now adapted here for insurance.

No Python implementation existed before this library.

What it does

Fit a polynomial counterfactual to the density away from the threshold. Compare what you observe inside the exclusion window to what you'd expect from the smooth counterfactual. The difference, normalised by the counterfactual density at z*, is the excess mass B.

B = 0: no bunching. B = 1: excess mass equal to one bin-width of counterfactual density. B >> 1: strong, systematic bunching.

The iterative correction handles the fact that bunching mass must come from somewhere — observations that moved to z* are missing from just above it. Without correcting for this, the polynomial overestimates the counterfactual above z* and underestimates B.

Install

pip install insurance-bunching

Quick start

import numpy as np
from insurance_bunching import BunchingEstimator

rng = np.random.default_rng(42)
# 9,000 policies from smooth distribution + 1,000 bunched at £50,000
z = np.concatenate([rng.normal(50_000, 8_000, 9_000), [50_000] * 1_000])

est = BunchingEstimator(z, threshold=50_000, n_boot=200, seed=42)
result = est.fit()
print(result.summary())
fig = est.plot()

Insurance-specific usage

The ExposureWeightedBunching class handles policy DataFrames directly:

import pandas as pd
from insurance_bunching import ExposureWeightedBunching

ewb = ExposureWeightedBunching(
    policies,
    running_var="annual_mileage",
    exposure_col="earned_years",   # exposure weighting — critical for motor
    threshold=10_000,
    threshold_type="kink",         # or "notch" for discrete price jumps
    round_numbers=[5_000, 15_000, 20_000],  # control for rounding at other values
    n_boot=500,
)
result = ewb.fit()
fig = ewb.plot()

Why exposure weighting? A policy file has policies, not risk-years. A policy renewing in June contributes 0.5 earned years. Without weighting, short-period policies and long-period policies are treated identically. The density you actually care about is per-unit-of-exposure, not per-policy.

Scan all thresholds

Test multiple pricing boundaries simultaneously and apply Benjamini-Hochberg FDR correction to control false discoveries:

from insurance_bunching import MultiThresholdScanner

scanner = MultiThresholdScanner(
    z,
    thresholds=[5_000, 10_000, 15_000, 20_000, 25_000],
    n_boot=300,
    fdr_level=0.05,
)
summary = scanner.scan()  # polars DataFrame, sorted by significance
print(summary)
print("Significant thresholds:", scanner.significant_thresholds())

FCA Consumer Duty report

Generate a self-contained HTML report for regulatory submission:

from insurance_bunching import BunchingReport

report = BunchingReport(
    results=[result_mileage, result_sum_insured, result_ncd],
    title="Annual Bunching Analysis — UK Private Motor 2024",
    product_line="UK Private Motor",
    fdr_level=0.05,
)
report.save("bunching_analysis_2024.html")

The report includes: embedded density plots, BH-corrected p-value table, regulatory interpretation boilerplate, methodology appendix with equations and references.

API

BunchingEstimator

BunchingEstimator(
    z,                    # running variable (array-like)
    threshold,            # z* — the kink/notch point
    binwidth=None,        # auto from Silverman/10 if None
    poly_degree=9,        # polynomial degree for counterfactual
    excl_left=2,          # bins excluded left of z*
    excl_right=2,         # bins excluded right of z*
    weights=None,         # per-observation exposure weights
    round_numbers=None,   # list of round numbers to control for
    n_boot=200,           # bootstrap replications
    notch=False,          # True = notch estimator (infer z**)
    t0=None,              # pre-threshold marginal rate (for elasticity)
    t1=None,              # post-threshold marginal rate
    seed=None,
)

BunchingResult attributes:

• B — normalised excess mass
• B_se — bootstrap standard error
• B_ci — (lower, upper) 95% CI
• p_value — two-sided p-value for H0: B=0
• elasticity — kink elasticity (if t0/t1 provided)
• marginal_buncher — z** (if notch=True)
• counterfactual — pd.DataFrame with per-bin observed/counterfactual/excess
• .summary() — text summary
• .plot() — matplotlib Figure

ExposureWeightedBunching

DataFrame-first wrapper. Takes column names instead of arrays. Same output as BunchingEstimator.

MultiThresholdScanner

Runs BunchingEstimator at each threshold, applies Benjamini-Hochberg FDR correction. Returns a polars DataFrame.

BunchingReport

Jinja2 HTML report. Embeds plots as base64 PNG. Applies BH correction across all results.

Algorithm

1. Bin z into bins of width h anchored at z*
2. Fit p-th degree polynomial (WLS, exposure-weighted) to bins outside exclusion window [z* - L·h, z* + R·h]
3. Iterative correction: estimate excess mass → redistribute missing mass above z* → refit → repeat until convergence
4. B_hat = Σ(observed - counterfactual in window) / counterfactual(z*)
5. Bootstrap SE: resample observations, re-bin, re-estimate
6. Kink elasticity: e = B / [z* × log((1-t0)/(1-t1))]
7. Notch: infer z** from cumulative missing mass above threshold

References

• Saez, E. (2010). Do taxpayers bunch at kink points? AEJ: Economic Policy, 2(3), 180-212.
• Kleven, H. J. (2016). Bunching estimators. Annual Review of Economics, 8, 435-464.
• Einav, L., Finkelstein, A., & Cullen, M. (2010). Estimating welfare in insurance markets. QJE, 125(3), 877-921.
• Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate. JRSS-B, 57(1), 289-300.

License

MIT. Built by Burning Cost.