EXPLORATORY

Insurance Catastrophe: Tail Risk Under Bimodal Losses

A study on 23 years of California wildfire losses (2003-2025). A single-distribution fit that treats catastrophe years as outliers assigns extreme losses far less weight than a bimodal (normal + catastrophe) fit does. The figures below are model outputs on this dataset, not measured client results or an industry-wide finding.

Data Period

23 years

2003-2025

99.5% VaR (model gap)

~$11.8B

Bimodal vs single-fit, this dataset

Relative gap

~39%

Between the two fits

Tail at $35B

1-in-96 yr

Bimodal fit vs ~0 single-fit

Real Data: California Wildfire Losses

Sources: California Department of Insurance, Swiss Re sigma, Moody's RMS, Insurance Information Institute

Year	Loss ($B)	Major Fires	Type
2003	2.5	Cedar Fire, Old Fire	Normal
2007	2.1	Witch Fire, Harris Fire	Normal
2015	1.8	Valley Fire, Butte Fire	Normal
2017	12.0	Tubbs Fire, Thomas Fire	CATASTROPHE
2018	12.5	Camp Fire, Woolsey Fire	CATASTROPHE
2019-2024	0.4-2.1	Various fires	Normal
2025	35.0	Palisades Fire, Eaton Fire	CATASTROPHE

Normal Years (87%)

20 of 23 years
Mean loss: $0.80B
Range: $0.2B - $2.5B
Attritional, predictable

Catastrophe Years (13%)

3 of 23 years (2017, 2018, 2025)
Mean loss: $19.83B
Range: $12.0B - $35.0B
This is the mode standard models MISS

The Problem: Outlier Removal

Common practice: Exclude "outlier" catastrophe years

When 2017, 2018, and similar years are excluded as "outliers" during model fitting, a bimodal loss process is forced into a single distribution. Mixture LDT keeps the catastrophe years in the fit as a second mode.

The Outlier Removal Problem

When insurers exclude "outlier" catastrophe years from model fitting, they create massive blind spots:

Standard Practice

Fit single distribution, exclude "outliers" like 2017, 2018. At $35B loss level, model says probability ≈ 0.

Our Method

Bimodal fit retains the catastrophe mode. At $35B, modelled probability ≈ 1% (1 in 96 years).

At $35B, the single-fit probability collapses toward zero; the bimodal fit keeps it material.

"A single-distribution fit puts LA 2025 ($35B) at roughly a 1-in-10-billion-year event. The bimodal (Mixture LDT) fit places it near 1-in-96 years on this dataset — the catastrophe mode activating, not a black swan."

The Solution: Mixture LDT

The same mathematics that captures bimodal firebrand transport applies to insurance losses. Mixture LDT recognizes two populations and models each appropriately.

Capital Requirements (99.5% VaR - Solvency II)

STANDARD MODEL

$29.95B

Single lognormal (all 23 years)

MIXTURE LDT

$41.74B

Bimodal (normal + catastrophe)

Model gap: ~$11.8B (~39%)

Difference in 99.5% VaR between the single-distribution and bimodal fits on this dataset

Validation Against Real Data

Tested against 23 years of California wildfire loss data (2003-2025):

90th percentile: Our method matches actual data. Standard model underestimates by ~50%.
99.5% VaR: ~$11.8B gap between the two fits on this dataset.
LA 2025 ($35B): our tail-risk method flagged a 1-in-96-year event that standard models effectively priced out.

Full methodology and technical documentation available under NDA.Contact us →

Why This Matters

Regulatory Capital

Solvency II requires 99.5% VaR capital
Wrong tails = wrong capital = regulatory risk
Rating agencies assess CAT exposure

Reinsurance Pricing

CAT bonds priced on tail probability
Excess-of-loss treaties need accurate PML
Wrong tails = adverse selection or losses

Where It Could Apply Commercially

Potential application areas (not priced or contracted — illustrative of where bimodal tail modelling is relevant):

Reinsurance

Tail-risk modelling

Capital and PML estimation

CAT Bonds

Tail-probability inputs

Attachment / exhaustion pricing

CAT Software

Model component

Integration with existing platforms

The Unified Insight

Standard models fail whenever transport/losses are BIMODAL

Firebrand Transport

Normal diffusion + wave-boosted "surfing"

Insurance Losses

Normal years + catastrophe years

← You are here

Market Risk

Normal volatility + crash regime

Coming soon

Grid Failures

Normal load + cascade events

Coming soon

ONE MATH FRAMEWORK. MANY APPLICATIONS.

Key Implications

1. Capital Sizing Depends on the Tail Model

• On this dataset, the two fits differ by ~39% in 99.5% VaR capital
• Solvency II sets capital at 99.5% VaR, so the tail assumption drives the number
• A case for checking CAT exposure under a mixture model, not just a single fit
• ~$11.8B difference between the fits in this California wildfire study

2. Outlier Removal and Bimodal Losses

• Common practice: exclude "outlier" catastrophe years from model fitting
• For a genuinely bimodal process, this discards the second mode that drives the tail
• The effect is a sharply thinner modelled tail at extreme loss levels
• Mixture LDT keeps both modes in the fit instead

3. Why LA 2025 Read as a "Surprise"

• The single-distribution fit puts a $35B loss far out in the tail
• The bimodal fit places it at roughly 1-in-96 years on this dataset
• Under the bimodal view it is the catastrophe mode activating, not an anomaly
• A modelling-choice question more than a black-swan one

4. Same Math as Firebrand Transport

• Both are bimodal: normal mode + boosted mode
• Standard diffusion/single-distribution fails
• Mixture rate function captures both modes correctly
• The same bimodal structure shows up in firebrand transport physics

Tail risk, done right

A research result — the same heavy-tailed methods, applied to catastrophe exposure. See the rest of the work, or get in touch.

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