insurance-dro

Distributionally Robust Optimisation (DRO) for insurance premium setting.

The problem

Your severity model was trained on 2022-2024 motor data. In 2025, claims inflation hits 15%, a litigation funding firm enters the market, and a new risk segment (EVs) makes up 20% of new business. Your ERM-fitted model is now pricing a distribution it was never trained on.

Standard ERM (OLS, GLM) minimises expected loss on the training distribution. When the test distribution shifts, it can produce premiums that are significantly off — either underpricing tails (claims inflation) or overpricing standard risks (new market entry creates adverse selection).

DRO finds a premium that minimises the worst-case expected loss over a set of plausible distributions near the training distribution. The uncertainty set is a Wasserstein ball of radius rho — a formal way of saying "allow the test distribution to differ from training by at most this much".

The solution

WassersteinDRO solves the Esfahani-Kuhn (2018) Wasserstein DRO problem for quadratic loss. For this loss function, the DRO problem reduces to a regularised regression with an L2 penalty calibrated to rho and the operator norm of the feature matrix. No inner optimisation loop required — it's a clean, fast fit.

Also included: CVaRPremium (tail-risk-aware premium via Conditional Value at Risk) and DeterministicERM (standard OLS baseline).

Installation

pip install git+https://github.com/burning-cost/insurance-dro.git

Usage

from insurance_dro import WassersteinDRO

dro = WassersteinDRO(
    rho=0.2,             # uncertainty radius (tune to expected shift magnitude)
    loss="quadratic",    # quadratic loss (MSE)
)
dro.fit(X_train, y_train)
premiums = dro.predict(X_new)

# Evaluate worst-case loss under distributional perturbation
wc_loss = dro.worst_case_loss(X_test, y_test, perturbation_std=0.2)

Choosing rho

rho controls the size of the uncertainty set. A practical approach:

• Fit on recent years, evaluate on older years under each rho
• Choose rho where the loss ratio is stable across the historical periods
• For soft robustness, rho=0.05-0.10 is a starting point
• For explicit regulatory stress, rho should match the stated shock magnitude

Performance

Benchmarked against deterministic ERM (OLS) on synthetic motor severity data (20,000 training policies) under 5 distribution shift scenarios. See notebooks/benchmark_dro.py for full methodology.

• DRO is 15-30% more stable (lower loss ratio variance) under distribution shift scenarios (covariate shift, variance shift, tail shift, mixed) than ERM.
• ERM outperforms DRO in-distribution — this is expected. DRO trades in-sample accuracy for robustness. The tradeoff is governed by rho.
• Tail shift is the most dramatic scenario: ERM loss ratio can drift 10-20% from target under heavy-tailed perturbations; DRO stays within 3-5%.
• CVaR premium is simpler and more interpretable but assumes a fixed multiplicative loading rather than adapting to the feature distribution.
• Fit time is negligible: DRO reduces to regularised regression and fits in under 0.2s for n=20k. The main overhead is tuning rho.

References

• Esfahani, P. M., & Kuhn, D. (2018). Data-driven DRO using the Wasserstein metric. Math. Programming, 171(1).
• Blanchet, J., & Murthy, K. (2019). Quantifying distributional model risk. Math. Operations Research, 44(2).
• Rahimian, H., & Mehrotra, S. (2019). Distributionally robust optimization: A review. arXiv:1908.05659.