Applications of the Python Package

Welcome to applications of the Ergodic-Insurance Python package, designed to help risk professionals make more informed decisions about insurance coverage limits and risk retention strategies.

Learn the Framework

mostlyoptimal.com/tutorial — a guide to get started applying the framework to your specific use cases.

mostlyoptimal.com/research — a research paper describing the framework in detail.

Install the Framework:

pip install ergodic-insurance

Start Here

Short on time? These three posts give you the core argument. Read them in order.

1. Insurance Limit Selection Through Ergodicity — The foundation: why time-average growth changes everything about limit selection.
2. Volatility Drag vs Premium Drag — The core insight: why insurance works even when it’s actuarially “unfair.”
3. The Objective Frontier — The practical payoff: multi-objective optimization narrows the defensible deductible range.

All Posts by Topic

Each post includes a downloadable notebook so you can adapt the analysis to your own business.

The Ergodic Foundation

Why your company’s single trajectory through time behaves nothing like the industry average, and where classical statistics breaks down entirely.

• Insurance Limit Selection Through Ergodicity — Why the 99.9th percentile isn’t enough over 50-year horizons.
• When Variance Doesn’t Exist: Infinite Moments — Where Pareto $\alpha$ drops below 2, standard deviation is an artifact of sample size, not a property of the risk.

Loss Modeling and Tail Uncertainty

Your severity model is only as good as the data behind it. These posts address estimation under scarce data and deep tail uncertainty.

• Loss Severity Estimation and the Shadow Mean — Five years of company history can’t see the catastrophic tail; the Hill estimator with industry data recovers what’s hiding.
• Stochasticizing Tail Uncertainty With Sobol Sequences — Nobody knows their tail shape; sweep 25,000 configurations across 1,000 tail scenarios to find the patterns.

Risk Measurement

Standard risk metrics miss the tail. These posts show why, and what to use instead.

• VaR vs CTE Under Catastrophic Tail Variation — Break a distribution’s variance and VaR barely notices; CTE more than doubles.
• Exploring Expectiles — The only law-invariant risk measure that is both coherent and backtestable, and 5x more responsive to tail thickening than VaR.

Deductible and Retention Optimization

From the mechanics of volatility drag to multi-objective frontiers, these posts build a complete framework for retention decisions.

• Volatility Drag vs Premium Drag — The tug-of-war between premium cost and variance destroying compounding power.
• The Insurance Cliff — Capital vs. limits vs. bankruptcy risk doesn’t form a gentle slope; it’s a cliff.
• Exploring Volatility: Optimal Deductibles Under Dual Uncertainty — Operational volatility drives optimal deductible 5x more than loss volatility.
• The Objective Frontier — Frame deductible selection as multi-objective optimization and the defensible range is far narrower than most renewal conversations suggest.
• The Decision Cloud — Stress-test the frontier under Bayesian severity uncertainty; at both extremes of loss volatility, optimal retention rises.

---

Additional applications will be added regularly as we explore new use cases and risk scenarios.