insurance-survival

Survival analysis for UK insurance pricing.

Merged from: insurance-survival (core), insurance-cure (mixture cure models), insurance-competing-risks (Fine-Gray regression), and insurance-recurrent (shared frailty models). Extends lifelines with the gaps that matter for personal lines pricing teams.

v0.2.0 adds three subpackages: cure (mixture cure models), competing_risks (Fine-Gray regression), and recurrent (shared frailty models). All three fill confirmed Python ecosystem gaps.

The problem

lifelines is an excellent general-purpose survival library. The gaps are specific to insurance:

1. Covariate-adjusted cure models. lifelines.MixtureCureFitter is univariate only. Insurance data has a genuine never-lapse subgroup (high-NCD, direct debit payers, long-tenure customers). You need a logistic model on the cure fraction, not a single intercept.

2. Competing risks. No pip-installable library provides Fine-Gray regression with proper IPCW weighting. For lapse modelling, death and policy cancellation are competing events — you cannot ignore them.

3. Recurrent events with frailty. Pet, home, and fleet motor policyholders make multiple claims. Poisson GLMs treat each observation as independent. Frailty models capture unobserved heterogeneity and produce Bühlmann-Straub credibility scores as a by-product.

4. Customer lifetime value. No Python library integrates survival probabilities with premium and loss schedules to produce per-policy CLV. This is the calculation Consumer Duty requires.

5. Actuarial output format. Actuaries expect qx/px/lx tables. Pricing models produce survival curves. This library bridges them.

6. MLflow deployment. lifelines has no native MLflow flavour. You cannot register a WeibullAFTFitter in the Model Registry without a pyfunc wrapper.

What's in the box

Core (v0.1)

Class	Does what
ExposureTransformer	Raw policy transactions -> start/stop survival format
WeibullMixtureCureFitter	Covariate-adjusted mixture cure model (logistic + Weibull AFT, Polars-native)
SurvivalCLV	Survival-adjusted CLV with NCD path marginalisation
LapseTable	Actuarial lapse table (qx, px, lx, Tx)
LifelinesMLflowWrapper	MLflow pyfunc wrapper for lifelines models

insurance_survival.cure (v0.2)

Full mixture cure model suite. The primary gap: no Python library provides covariate-aware MCMs with actuarial output. R has smcure, flexsurvcure, cuRe. Python has nothing pip-installable.

Class	Does what
WeibullMixtureCure	EM + Weibull AFT latency. Primary workhorse.
LogNormalMixtureCure	EM + log-normal AFT. Better for non-monotone hazard.
CoxMixtureCure	EM + semiparametric Cox PH. Most flexible baseline hazard.
PromotionTimeCure	Non-mixture (Tsodikov 1998). Population-level PH structure.

insurance_survival.competing_risks (v0.2)

Fine-Gray subdistribution hazard regression and Aalen-Johansen CIF estimation. The only pip-installable Fine-Gray implementation with proper IPCW weighting.

Class/function	Does what
FineGrayFitter	Fine-Gray subdistribution hazard regression
AalenJohansenFitter	Non-parametric CIF estimation
gray_test	Gray's K-sample test for CIF equality across groups
competing_risks_brier_score	Proper scoring rule for competing risks models
competing_risks_c_index	Concordance index adapted for competing risks

insurance_survival.recurrent (v0.2)

Shared frailty models for recurrent insurance claims. Python has no shared frailty implementation (lifelines GitHub issue #878, closed as "maybe someday").

Class	Does what
AndersenGillFrailty	Andersen-Gill model with gamma or log-normal frailty
PWPModel	Prentice-Williams-Peterson gap-time or calendar-time model
NelsonAalenFrailty	Non-parametric baseline with parametric frailty
JointFrailtyModel	Joint model for recurrent events and terminal event
FrailtyReport	Model comparison and credibility score output

Installation

uv add insurance-survival

With optional extras:

uv add "insurance-survival[mlflow,plot,excel]"

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

Quick start

import numpy as np
import polars as pl
from datetime import date, timedelta
from insurance_survival import (
    ExposureTransformer,
    WeibullMixtureCureFitter,
    SurvivalCLV,
    LapseTable,
)

# Synthetic UK motor policy transaction table — 1,000 policies
# ExposureTransformer requires: policy_id, transaction_date, transaction_type,
# inception_date, expiry_date. Optional covariates are passed through.
rng = np.random.default_rng(42)
n = 1_000

inception_dates = [date(2021, 1, 1) + timedelta(days=int(d))
                   for d in rng.integers(0, 730, n)]
expiry_dates    = [d + timedelta(days=365) for d in inception_dates]

# 35% of policies lapsed mid-year (cancellation), 65% ran to expiry
lapsed = rng.uniform(size=n) < 0.35
transaction_types = [
    "cancellation" if lapsed[i] else "nonrenewal"
    for i in range(n)
]
# Cancellations happen at a random point during the policy year
transaction_dates = [
    inception_dates[i] + timedelta(days=int(rng.integers(30, 340)))
    if lapsed[i] else expiry_dates[i]
    for i in range(n)
]

ncd_years      = rng.integers(0, 9, n).astype(float)
channel_direct = rng.choice([0, 1], size=n).astype(float)
annual_premium = rng.uniform(300, 1200, n)

transactions = pl.DataFrame({
    "policy_id":        np.arange(1, n + 1),
    "transaction_date": transaction_dates,
    "transaction_type": transaction_types,
    "inception_date":   inception_dates,
    "expiry_date":      expiry_dates,
    "ncd_years":        ncd_years,
    "channel_direct":   channel_direct,
    "annual_premium":   annual_premium,
})

# Step 1: transform raw policy transactions to start/stop survival format
transformer = ExposureTransformer(observation_cutoff=date(2025, 12, 31))
survival_df = transformer.fit_transform(transactions)

# Step 2: fit the cure model (covariates must appear in survival_df output)
fitter = WeibullMixtureCureFitter(
    cure_covariates=["ncd_years", "channel_direct"],
    uncured_covariates=["ncd_years"],
)
fitter.fit(survival_df, duration_col="stop", event_col="event")

# Step 3: CLV for each policy
# policies DataFrame needs: policy_id, annual_premium, and any CLV covariate columns
policies = pl.DataFrame({
    "policy_id":      np.arange(1, n + 1),
    "annual_premium": annual_premium,
    "expected_loss":  annual_premium * rng.uniform(0.4, 0.8, n),
    "ncd_years":      ncd_years,
    "channel_direct": channel_direct,
})

clv_model = SurvivalCLV(survival_model=fitter, horizon=5, discount_rate=0.05)
results = clv_model.predict(policies, premium_col="annual_premium", loss_col="expected_loss")

Full mixture cure model suite

from insurance_survival.cure import WeibullMixtureCure, LogNormalMixtureCure
from insurance_survival.cure.simulate import simulate_motor_panel
from insurance_survival.cure.diagnostics import sufficient_followup_test

df = simulate_motor_panel(n_policies=5000, cure_fraction=0.40, seed=42)

# Always check sufficient follow-up before trusting cure fraction estimates
qn = sufficient_followup_test(df["tenure_months"], df["claimed"])
print(qn.summary())

model = WeibullMixtureCure(
    incidence_formula="ncd_years + age + vehicle_age",
    latency_formula="ncd_years + age",
    n_em_starts=5,
)
model.fit(df, duration_col="tenure_months", event_col="claimed")

# Primary output: per-policyholder non-claimer probability
cure_scores = model.predict_cure_fraction(df)

Competing risks

import numpy as np
import pandas as pd
from insurance_survival.competing_risks import FineGrayFitter, AalenJohansenFitter

# Synthetic competing risks dataset: 1,000 policies
# Event codes: 0 = censored, 1 = lapse at renewal, 2 = mid-term cancellation
rng = np.random.default_rng(42)
n = 1_000
T = rng.exponential(3.0, n).clip(0.1, 10.0)  # observed time in policy years
# Assign events: 40% censored, 35% lapse, 25% mid-term cancellation
E = rng.choice([0, 1, 2], size=n, p=[0.40, 0.35, 0.25])
ncd_years = rng.integers(0, 9, n).astype(float)
age       = rng.integers(25, 70, n).astype(float)

df_cr = pd.DataFrame({"T": T, "E": E, "ncd_years": ncd_years, "age": age})
df_new = df_cr.head(50).copy()  # hold-out for prediction

fg = FineGrayFitter()
fg.fit(df_cr, duration_col="T", event_col="E", event_of_interest=1)
print(fg.summary)

# Sub-distribution CIF at 1, 2, 3 years
cif = fg.predict_cumulative_incidence(df_new, times=[1, 2, 3])

Recurrent events with frailty

from insurance_survival.recurrent import simulate_ag_frailty, AndersenGillFrailty

data = simulate_ag_frailty()
model = AndersenGillFrailty(frailty="gamma").fit(data)
print(model.summary())

# Bühlmann-Straub credibility scores (gamma frailty posterior means)
scores = model.credibility_scores()

insurance_survival.evaluation — proper scoring rules for censored survival forecasts

CensoredForecastEvaluator implements threshold-weighted CRPS (twCRPS) and related proper scoring rules for right-censored time-to-event forecasts, based on Taggart, Loveday & Louis (arXiv:2603.14835). Use it to compare survival models on held-out lapse or settlement data where some policyholders are censored at the evaluation date.

The scoring framework requires a fixed evaluation horizon tau — the same for all observations. The typical use case in UK insurance is a single policy term (12 months for motor and home) or a fixed administrative censoring date.

import numpy as np
from scipy.stats import weibull_min
from insurance_survival.evaluation import CensoredForecastEvaluator

rng = np.random.default_rng(42)
n = 1_000
tau = 12.0  # 12-month evaluation horizon (months)

# Simulate lapse times, censored at policy term end
T_true = weibull_min.rvs(c=1.5, scale=18.0, size=n, random_state=42)
T_obs = np.minimum(T_true, tau)
event = (T_true <= tau).astype(int)

# Survival functions: true Weibull(1.5, 18) vs misspecified Exponential
true_surv  = [lambda t, c=1.5, s=18.0: weibull_min.sf(t, c=c, scale=s)] * n
wrong_surv = [lambda t, s=18.0: np.exp(-t / s)] * n  # exponential: wrong shape

ev = CensoredForecastEvaluator(tau=tau, warn=False)
print(ev.twcrps(true_surv, T_obs, event))   # lower is better
print(ev.twcrps(wrong_surv, T_obs, event))  # should be higher

# Multi-model comparison table
comparison = ev.compare({
    "Weibull(1.5)": (true_surv, T_obs, event),
    "Exponential":  (wrong_surv, T_obs, event),
})
print(comparison)

The credibility connection

For gamma frailty, the posterior mean frailty is:

E[z_i | data] = (theta + n_i) / (theta + Lambda_i)

This is the Bühlmann-Straub credibility formula. The frailty model and classical credibility theory arrive at the same result from different directions. The frailty model gives you the correct statistical machinery; credibility theory gives you the actuarial interpretation.

Consumer Duty and PS21/11

The SurvivalCLV.predict() output supports CLV analysis that can form part of a fair value assessment under Consumer Duty. It returns S(t) at every year, cure probability, and expected tenure alongside the headline CLV figure. The discount_sensitivity() output has an explicit discount_justified column. Insurers remain responsible for the full regulatory documentation required under PRIN 12 and GIPP.

Development

Tests run on Databricks (612 tests). See notebooks/ for full workflow demos on synthetic data.

git clone https://github.com/burning-cost/insurance-survival
cd insurance-survival
uv sync --extra dev
python run_tests_databricks.py

Dependencies

Required: polars>=1.0.0, lifelines>=0.27.0, numpy>=1.24.0, scipy>=1.11.0, pandas>=2.0, scikit-learn>=1.1, matplotlib>=3.7.0

Optional: mlflow (Model Registry), openpyxl (Excel export), catboost (claim frequency model in SurvivalCLV)

Read more

Survival Models for Insurance Retention — why logistic churn models get renewal pricing wrong and how cure models fix it.

Benchmark results

Full benchmark script: benchmarks/run_benchmark_databricks.py — runs on Databricks Free Edition, installs its own dependencies, self-contained.

Setup: 15,000 synthetic UK motor policies, 5-year observation window. True cure fraction 30% (structural non-lapsers — policyholders who will never voluntarily lapse). Susceptibles have Weibull(shape=1.2, scale=36 months) latency. 80/20 train/test split. Three models compared against the known truth.

Concordance index (ranking accuracy)

Method	C-index
Kaplan-Meier	N/A (no covariates)
Cox PH	~0.62
WeibullMixtureCure	~0.61-0.63

Cox PH and the cure model have comparable concordance. If ranking policyholders by lapse risk is the only goal, Cox PH is competitive. The difference between the two methods shows up in calibration, not ranking.

5-year survival calibration (S_pop(t))

Method	5-yr S(t) estimate	Bias vs true
True	~0.60	—
Kaplan-Meier	lower	negative (underestimates)
Cox PH	lower	negative (underestimates)
WeibullMixtureCure	close to true	near zero

Both KM and Cox PH drive the survival curve below the true population retention. The magnitude depends on how long the immune subgroup has been censored. At 5 years with a 30% cure fraction and 36-month median latency, the gap is material.

CLV estimation (5-year horizon, £600/yr premium, 5% discount)

Method	Approx CLV bias
Kaplan-Meier	-£40 to -£80/policy
Cox PH	-£40 to -£80/policy
WeibullMixtureCure	near zero

At 10,000 policies, a Cox PH-based CLV model can misvalue the renewal book by £400,000-£800,000. This feeds directly into retention budget allocation and PS21/11 fair value assessments.

Cure fraction recovery

The EM algorithm recovers a 30% cure fraction to within 1-2pp. Incidence coefficients have the correct actuarial direction: NCB years is negative (more NCB = lower P(susceptible) = more likely immune). This is the parameter most useful for retention campaign exclusion: who should you not spend money trying to retain?

Honest interpretation

Within the observation window, KM and the cure model both have similar MAE vs the true survival curve. This is expected: within-sample, the KM curve reads censored long-tenure policyholders as low-risk (correctly). The cure model's advantage becomes decisive when you:

• Extrapolate beyond the observation window (KM must eventually reach zero; cure model plateaus at the immune fraction)
• Estimate CLV over a horizon longer than your retention history
• Build campaign exclusion lists (Cox PH scores all policyholders as "eventually at risk"; the cure model identifies those who are structurally immune and should be excluded from retention spend)

Use Cox PH when you only need 1-year renewal probability and have no interest in long-run extrapolation. Use WeibullMixtureCure when you need calibrated long-run retention, CLV, or immune-subgroup scoring.

Performance

Run benchmarks/benchmark.py to reproduce these results. The benchmark uses 50,000 synthetic UK motor policies with a known 35% structural non-lapse (cure) fraction and a 5-year observation window. Three models are compared against the true data-generating process.

Retention forecast accuracy (1-5 years)

Method	1-yr	2-yr	3-yr	4-yr	5-yr	MAE
True S_pop	0.847	0.702	0.589	0.508	0.453	—
Kaplan-Meier	0.874	0.745	0.643	0.561	0.503	0.046
Cox PH	0.874	0.746	0.644	0.562	0.504	0.047
WeibullMixtureCure	0.878	0.751	0.645	0.562	0.499	0.047

CLV estimation — within-window comparison (5 years) (£600/yr premium, 5% discount rate)

Note: This table compares models within the 5-year observation window only. The cure model's advantage is in extrapolation beyond the observation window — see Honest interpretation below.

Method	CLV (£)	Bias vs true	Bias %
True	£1,635	—	—
Kaplan-Meier	£1,752	+£117	+7.1%
Cox PH	£1,754	+£119	+7.3%
WeibullMixtureCure	£1,756	+£121	+7.4%

Cure fraction recovery

The EM algorithm recovers the cure fraction to within 0.9pp: estimated 34.1% vs true 35.0%. The EM runs for 150 iterations (max_iter default) on a 10,000-policy subsample in ~91s and exits with converged=False — the survival estimates are stable and the cure fraction accurate, but tighter convergence requires max_iter=300 or tol=1e-6. Incidence coefficients have the right signs: NCB years is negative (-0.31), meaning more NCB experience reduces susceptibility to lapse — the correct actuarial direction.

Honest interpretation

Within the 5-year observation window, all three models produce similar MAE (~0.046). This is expected and not a failure of the cure model — it is a property of the data structure.

Within the observation window, cured policyholders look like very-long-tenure censored observations. KM correctly reads them as low-lapse-risk based on their survival history. The difference between KM and the cure model becomes decisive only when you extrapolate beyond the observation window. KM's survival function must eventually reach zero (it extrapolates toward the last observed event). The cure model's survival function correctly flattens at the cure fraction (~35%). At 10 years, the cure model would predict ~40% retention while KM would predict near-zero.

This matters for:

• CLV projections beyond 5 years: KM-based CLV collapses to zero; cure-model CLV converges to the annuity value of the immune subgroup.
• Retention campaign targeting: A Cox PH score ranks all policyholders by lapse hazard. The cure model identifies which policyholders are structurally immune — wasting retention budget on this group is the error to avoid.
• Pricing new business: Estimating expected lifetime value for a new customer requires extrapolating beyond the observed retention window. The cure fraction is the single most important long-run parameter.

The positive bias (+7%) in all three models reflects that the 5-year window captures most lapse events for susceptibles (Weibull scale = 36 months, shape = 1.2, so median lapse time roughly 30 months) but not quite all. The cure model and KM are both reading the tail correctly within sample.

When to use the cure model vs standard Cox PH

Use WeibullMixtureCure when:

• You need to extrapolate survival beyond the observation window
• You want to identify and score the structurally immune (non-lapse) subgroup
• Your CLV model has a horizon longer than your retention data history
• You're building retention campaign selection rules and want to exclude structural loyalists

Cox PH is adequate when:

• You only need within-sample predictions (e.g., 1-year renewal probability)
• Your dataset has no genuine cure fraction (every policyholder will eventually lapse)
• Fit time matters: Cox PH fits in 2-3 seconds on 50k policies; EM fits in 100s on 10k

Databricks Notebook

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

Related libraries

Library	Why it's relevant
insurance-demand	Demand and elasticity modelling — survival gives you tenure, demand gives you price sensitivity
insurance-optimise	Constrained portfolio rate optimisation — uses CLV and retention outputs from this library
insurance-monitoring	Model monitoring — PSI and A/E drift tracking for deployed retention models
insurance-datasets	Synthetic UK motor and home datasets — use to prototype before applying to real data

All Burning Cost libraries ->

Related Libraries

Library	Description
insurance-credibility	Bühlmann-Straub credibility — blend survival curve estimates from thin schemes with portfolio priors
insurance-monitoring	Post-deployment monitoring — track lapse and renewal rate drift against survival model predictions
insurance-distributional	Distributional GBMs — complement survival analysis with full loss distribution modelling