tailrisk 0.1.0

Risk-aware machine learning for tail risk modeling

TailRisk: Risk-Aware Machine Learning for Tail Risk Modeling

TailRisk is a Python package for building machine learning models that excel at predicting extreme outcomes in insurance claims, financial losses, and other tail-risk scenarios.

Why TailRisk?

Traditional ML models optimize for average performance (MSE, MAE), but this approach fails catastrophically when predicting rare, extreme events:

• Insurance claims: Models predict $500 for a claim that costs $50,000
• Financial losses: Underestimating tail risk leads to inadequate reserves
• Healthcare costs: Missing catastrophic cases can bankrupt risk pools

Standard MSE treats all errors equally — a $10,000 error on a $500 claim has the same weight as a $10,000 error on a $100,000 claim. This is dangerous for tail risk.

The Problem

# Traditional approach
model = LinearRegression()
model.fit(X, y)  # Optimizes MSE equally across all samples

# Result: Great average performance, terrible tail prediction
# - MSE: $2.8M ✓
# - CVaR(95%): $54M ✗
# - Tail Coverage Ratio: 0.058 (captures only 5.8% of extreme losses!)

The TailRisk Solution

from tailrisk import HybridMetaLearner
from sklearn.ensemble import RandomForestRegressor, GradientBoostingRegressor

# Build tail-aware model
model = HybridMetaLearner(
    base_estimators=[
        ('rf', RandomForestRegressor(n_estimators=100)),
        ('gb', GradientBoostingRegressor(n_estimators=100))
    ],
    quantile=0.95,       # Focus on 95th percentile
    blend_lambda=0.25    # Blend quantile + LaR models
)

model.fit(X_train, y_train)
predictions = model.predict(X_test)

# Result: Dramatically improved tail prediction
# - MSE: $3.0M (acceptable tradeoff)
# - CVaR(95%): $52M ✓ (10% improvement)
# - Tail Coverage Ratio: 0.102 ✓ (76% improvement!)

Key Features

🎯 Tail-Focused Models

• Loss-at-Risk (LaR) Regressor: Weighted regression that prioritizes large claims
• CVaR-Weighted Ensemble: Combines models based on tail risk performance
• Hybrid Meta-Learner: Two-stage architecture blending quantile regression + LaR optimization

📊 Specialized Metrics

Standard metrics hide tail risk problems. TailRisk provides:

• CVaR (Conditional Value-at-Risk): Average loss in worst α% of predictions
• Tail Coverage Ratio: What fraction of extreme tail value is captured
• Detection Rate: Percentage of extreme claims correctly identified
• Loss-at-Risk: Weighted MSE emphasizing large claims

🔧 Production-Ready

• Scikit-learn compatible (works with GridSearchCV, Pipeline, etc.)
• Comprehensive validation utilities
• Built-in visualizations for tail risk analysis
• Type-safe, well-documented API

Installation

From PyPI (coming soon)

pip install tailrisk

From source

git clone https://github.com/VishalLakshmiNarayanan/TailriskLib.git
cd TailriskLib
pip install -e .

Quick Start

Basic Example: LaR Regressor

import numpy as np
from tailrisk import LaRRegressor
from tailrisk.metrics import tail_validation_summary

# Heavy-tailed data (e.g., insurance claims)
X = np.random.randn(1000, 10)
y = np.random.exponential(scale=1000, size=1000)

# Fit LaR model
model = LaRRegressor(alpha=2.0)
model.fit(X_train, y_train)

# Evaluate tail performance
predictions = model.predict(X_test)
metrics = tail_validation_summary(y_test, predictions)

print(f"CVaR(95%): {metrics['cvar_95']:.2f}")
print(f"Tail Coverage @ 99%: {metrics['tcr_99']:.3f}")

Advanced Example: Hybrid Meta-Learner

from tailrisk import HybridMetaLearner
from tailrisk.utils import print_tail_validation, plot_tail_comparison
from sklearn.ensemble import RandomForestRegressor, GradientBoostingRegressor
from sklearn.linear_model import Ridge
import xgboost as xgb

# Define diverse base models
base_models = [
    ('rf', RandomForestRegressor(n_estimators=100, max_depth=10, random_state=42)),
    ('gb', GradientBoostingRegressor(n_estimators=100, max_depth=5, random_state=42)),
    ('xgb', xgb.XGBRegressor(n_estimators=100, max_depth=5, random_state=42)),
    ('ridge', Ridge(alpha=1.0))
]

# Build hybrid meta-learner
model = HybridMetaLearner(
    base_estimators=base_models,
    quantile=0.95,        # Target 95th percentile in quantile model
    blend_lambda=0.25,    # 25% quantile, 75% LaR
    lar_alpha=1.5,        # LaR weighting strength
    cv_folds=5            # Cross-validation folds for meta-features
)

# Fit and predict
model.fit(X_train, y_train)
y_pred = model.predict(X_test)

# Comprehensive validation
print_tail_validation(y_test, y_pred, model_name="Hybrid Meta-Learner")

# Visualize performance
plot_tail_comparison(y_test, y_pred_baseline, y_pred, quantiles=[0.90, 0.95, 0.99])

Model Comparison

from tailrisk.utils.validation import compare_models
from sklearn.linear_model import LinearRegression

# Train multiple models
baseline = LinearRegression().fit(X_train, y_train)
lar_model = LaRRegressor(alpha=2.0).fit(X_train, y_train)
hybrid_model = HybridMetaLearner(base_estimators=base_models).fit(X_train, y_train)

# Compare side-by-side
compare_models(y_test, {
    'Baseline': baseline.predict(X_test),
    'LaR': lar_model.predict(X_test),
    'Hybrid': hybrid_model.predict(X_test)
})

Output:

==========================================================================================
 MODEL COMPARISON
==========================================================================================
Metric                   Baseline              LaR           Hybrid
------------------------------------------------------------------------------------------
MSE (Overall)           2,808,591        2,815,649        2,979,450
MSE (Extreme)         260,918,453      259,101,161      252,549,889
CVaR (95%)                  3,146            3,086            2,824
Detection @ 90%              3.1%             3.2%             6.1%
Detection @ 95%              0.8%             0.9%             2.5%
TCR @ 95%                   0.165            0.166            0.306
TCR @ 99%                   0.058            0.059            0.102
==========================================================================================

Methodology

TailRisk implements a novel Hybrid Meta-Learning framework:

Stage 1: Diverse Base Models

Train multiple model types (tree-based, linear, boosting) to capture different aspects of the data.

Stage 2a: Quantile Meta-Model

Use quantile regression (α=0.95) on meta-features to focus on extreme predictions.

Stage 2b: LaR-Weighted Optimization

Optimize base model weights using LaR objective function:

$$\text{LaR} = \frac{1}{n} \sum_{i=1}^n w_i (y_i - \hat{y}_i)^2$$

where $w_i = 1 + \alpha \frac{y_i}{\max(y)}$

Stage 3: Hybrid Blend

Combine both meta-models:

$$\hat{y}{\text{final}} = \lambda \cdot \hat{y}{\text{quantile}} + (1-\lambda) \cdot \hat{y}_{\text{LaR}}$$

The quantile model excels at detecting extreme events, while the LaR model maintains accuracy. Blending both (typically λ=0.25) provides optimal balance.

Metrics Guide

CVaR (Conditional Value-at-Risk)

Average squared error in the worst (1-α)% of predictions.

• Lower is better
• Focuses on tail performance, ignoring average cases
• Industry standard in finance/insurance

Tail Coverage Ratio (TCR)

Fraction of total extreme tail value captured by predictions.

TCR = sum(predictions[tail]) / sum(actuals[tail])

• 1.0 is perfect (100% coverage)
• <1.0 = underprediction (dangerous!)

• 1.0 = overprediction (conservative)

Detection Rate

Percentage of actual extreme claims correctly predicted as extreme.

• Higher is better
• Measures early warning capability
• Critical for reserve planning

Use Cases

Insurance Claims

• Problem: Catastrophic claims (1% of cases) drive 30% of losses
• Solution: HybridMetaLearner improves TCR@99% from 0.058 → 0.102 (+76%)
• Impact: Better reserves, reduced insolvency risk

Financial Risk

• Problem: VaR models underestimate tail risk (2008 crisis)
• Solution: CVaR-focused models + tail validation metrics
• Impact: Regulatory compliance (Basel III), improved capital allocation

Healthcare Costs

• Problem: Rare catastrophic cases (ICU, surgery complications)
• Solution: LaRRegressor prioritizes expensive cases
• Impact: Accurate premium pricing, sustainable risk pools

Examples

See the examples/ directory for detailed notebooks:

• insurance_claims_example.ipynb: Full walkthrough with real insurance data
• financial_losses_example.py: Portfolio tail risk modeling
• model_comparison.ipynb: Benchmarking TailRisk vs standard models

API Reference

Models

LaRRegressor(alpha=2.0, base_estimator=None)

Weighted regression with Loss-at-Risk objective.

Parameters:

• alpha: Weight scaling factor (higher = more focus on large claims)
• base_estimator: Base model (default: LinearRegression)

HybridMetaLearner(base_estimators, quantile=0.95, blend_lambda=0.25, lar_alpha=1.5, cv_folds=5)

Two-stage meta-learning with quantile + LaR blending.

Parameters:

• base_estimators: List of (name, estimator) tuples
• quantile: Quantile target for stage 2a (0.90-0.99 recommended)
• blend_lambda: Blending weight (0.2-0.3 works well)
• lar_alpha: LaR optimization strength (1.0-2.0 recommended)
• cv_folds: Cross-validation folds for meta-features

CVaRWeightedEnsemble(estimators, alpha=0.95)

Ensemble weighted by inverse CVaR performance.

Metrics

• cvar_loss(y_true, y_pred, alpha=0.95): CVaR metric
• loss_at_risk(y_true, y_pred, alpha=0.5): LaR metric
• tail_coverage_ratio(y_true, y_pred, quantile=0.99): TCR metric
• detection_rate(y_true, y_pred, quantile=0.95): Detection metric
• tail_validation_summary(y_true, y_pred): All metrics combined

Utilities

• plot_tail_comparison(y_true, y_pred_baseline, y_pred_tailrisk): Visual comparison
• print_tail_validation(y_true, y_pred, model_name): Formatted report
• compare_models(y_true, predictions_dict): Multi-model comparison

Contributing

Contributions welcome! Please see CONTRIBUTING.md for guidelines.

Citation

If you use TailRisk in academic work, please cite:

@software{tailrisk2025,
  author = {Vishal Lakshmi Narayanan},
  title = {TailRisk: Risk-Aware Machine Learning for Tail Risk Modeling},
  year = {2025},
  url = {https://github.com/VishalLakshmiNarayanan/TailriskLib}
}

License

MIT License - see LICENSE file for details.

Acknowledgments

• Methodology developed for insurance claims prediction research
• Inspired by actuarial science and quantitative finance literature
• Built on scikit-learn's excellent API design

Contact

• Issues: GitHub Issues
• Email: [e-mail]
• GitHub: @VishalLakshmiNarayanan

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Made with ❤️ for actuaries, quants, and ML engineers fighting tail risk