Rebonato-Denev Bayesian Network for Black Swan Events

Overview

This implementation demonstrates the Bayesian network methodology from "Portfolio Management under Stress: A Bayesian-Net Approach to Coherent Asset Allocation" by Riccardo Rebonato and Alexander Denev (Cambridge University Press, 2014).

The examples focuses on modelling black swan events:

• the potential breakup of the Eurozone, and its cascading effects on financial markets ~2010-2012
• Trump's tariffs affecting global trade in 2025
  • TRUMP_TARIFFS_2025_ANALYSIS.md
  • trump_tariffs_2025_blackswan.py

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Taleb's Black Swan

The term Black Swan in the context of financial risk and broader impact was popularized by writer, statistician, and former options trader Nassim Nicholas Taleb.

He extensively discussed the concept in his 2007 book, The Black Swan: The Impact of the Highly Improbable.

🦢 What a Black Swan Event Means

According to Taleb, a Black Swan event is an event with three principal attributes:

🦤 Rarity (The Outlier): It is an outlier because it lies outside the realm of regular expectations, as nothing in the past definitively points to its possibility. 💣 Extreme Impact: It carries an extreme impact when it occurs, causing widespread and severe consequences across markets, economies, or societies. 🔮 Retrospective Predictability: In spite of its outlier status, human nature leads people to invent explanations for it after the fact, making it appear predictable in hindsight (known as hindsight bias).

In finance, these events are particularly dangerous because standard risk models often fail to account for them, leading to massive, unpredictable losses. Examples often cited include the 2008 Global Financial Crisis and the COVID-19 pandemic.

If you'd like to know more about Taleb's other concepts, 🕵️‍♀️ search for Antifragility.

Key Innovation: Causal Modeling for Stress Testing

Unlike traditional portfolio management approaches that rely on historical correlations, Rebonato and Denev advocate for:

1. Causal Structure: Explicitly modeling how crises propagate through markets
2. Forward-Looking Scenarios: Incorporating extreme events that may not be present in historical data
3. Expert Judgment: Encoding subjective probabilities based on domain knowledge
4. Scenario Analysis: Portfolio optimization under stress conditions, not just "normal times"

The Eurozone Crisis Network

Network Structure

The Bayesian network models the following causal relationships:

Political_Instability ──┐
                        ├──> Eurozone_Breakup ──> Credit_Spreads ──> Corporate_Bonds
Economic_Weakness ─────┘                     │                   │
                                             │                   └──> Equities
                                             │                         ↑
                                             └──> Flight_to_Quality ───┤
                                                         │              │
                                                         └──> Government_Bonds

Variables and States

Variable	States	Description
Political_Instability	Low, High	Political tensions within Eurozone
Economic_Weakness	Low, High	Economic indicators (GDP, unemployment)
Eurozone_Breakup	No, Yes	The Black Swan Event
Credit_Spreads	Normal, Widening	Corporate credit spreads
Flight_to_Quality	No, Yes	Flight to safe-haven assets
Corporate_Bonds	Stable, Falling	Corporate bond performance
Government_Bonds	Stable, Rally	Government bond performance
Equities	Stable, Falling	Equity market performance

Implementation

Framework Used

The implementation uses a custom Bayesian network framework built with:

• NumPy: For numerical computations
• NetworkX: For graph structure
• Matplotlib: For visualization

This avoids heavyweight dependencies like PyTorch while maintaining full functionality.

Key Components

1. BayesianNetwork Class: Core implementation with:

   • Conditional Probability Tables (CPDs)
   • Variable elimination for inference
   • Forward sampling for Monte Carlo simulation

2. Conditional Probability Distributions: Expert-elicited probabilities reflecting:

   • Base rates for political and economic conditions
   • Conditional probabilities linking causes to effects
   • Severity of market reactions under stress

Example Results

Scenario Analysis

Scenario 1: Normal Times

• Political Instability: Low
• Economic Weakness: Low
• → Eurozone Breakup probability: 6.7%
• → Equity decline probability: 22.3%

Scenario 2: Black Swan (High Political + Economic Crisis)

• Political Instability: High
• Economic Weakness: High
• → Eurozone Breakup probability: 76.1%
• → Equity decline probability: 62.0%

Scenario 3: Given Eurozone Breakup

• Eurozone Breakup: Yes (observed)
• → Credit Spreads Widening: 78.6%
• → Flight to Quality: 88.0%
• → Corporate Bonds Falling: 73.8%
• → Government Bonds Rally: 73.3%
• → Equities Falling: 75.7%

Scenario 4: Market Stress Signals

• Credit Spreads: Widening
• Flight to Quality: Yes
• → Inferred Eurozone Breakup probability: 95.7%
• → Equity decline probability: 90.0%

Monte Carlo Results

Under the black swan scenario (High Political Instability + High Economic Weakness), 10,000 simulations show:

• Eurozone Breakup occurs: 70.5% of scenarios
• When breakup occurs:
  • Corporate Bonds fall: 76.7%
  • Government Bonds rally: 71.9%
  • Equities fall: 77.3%

Why This Matters: The Black Swan Connection

Traditional vs. Rebonato-Denev Approach

Traditional Approach (Correlation-Based):

• Uses historical correlations
• Assumes stable relationships
• Fails during regime changes
• Cannot model unprecedented events

Rebonato-Denev Approach (Causal Bayesian):

• Models causal mechanisms
• Incorporates expert judgment about novel scenarios
• Explicitly represents extreme events
• Updates beliefs as new information arrives

Black Swan Events

Nassim Taleb defined "black swan" events as:

1. Rare: Outside the realm of regular expectations
2. High Impact: Extreme consequences
3. Retrospectively Predictable: After the fact, we rationalize them

The Eurozone breakup scenario exemplifies this:

• Rare: No precedent in modern financial markets (unlike previous currency union dissolutions)
• High Impact: Would affect ~20% of global GDP and €20 trillion in contracts
• Causal Structure: We can reason about how it would unfold, even without historical data

Practical Applications

1. Stress Testing: Banks and asset managers can model scenarios beyond historical experience
2. Portfolio Construction: Allocate assets based on causal understanding, not just correlations
3. Risk Management: Quantify tail risks that don't appear in VaR models
4. Regulatory Compliance: Meet stress testing requirements with forward-looking scenarios

Key Insights from the Methodology

1. Causality > Correlation: Understanding causal mechanisms helps predict regime changes

2. Expert Judgment is Necessary: For unprecedented events, quantitative models need qualitative input

3. Scenario Diversity: Multiple scenarios (normal, stress, extreme) provide robust portfolios

4. Ambiguity Aversion: Uncertainty about probabilities themselves should affect portfolio choice

5. Dynamic Updates: Bayesian inference allows updating beliefs as new evidence arrives

Running the Code

python rebonato_denev_eurozone_crisis.py

The script will:

1. Build the Bayesian network
2. Visualize the causal structure
3. Analyze four stress scenarios
4. Show portfolio implications
5. Run Monte Carlo simulations

Further Reading

Primary Source

• Rebonato, R., & Denev, A. (2014). Portfolio Management under Stress: A Bayesian-Net Approach to Coherent Asset Allocation. Cambridge University Press.

Related Papers

• Rebonato, R. (2012). "A Bayesian Approach to Stress Testing and Scenario Analysis." Journal of Investment Management, 10(4), 19-53.

Bayesian Networks in Finance

• Pearl, J. (2009). Causality: Models, Reasoning and Inference. Cambridge University Press.
• Koller, D., & Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. MIT Press.

Black Swan Theory

• Taleb, N. N. (2007). The Black Swan: The Impact of the Highly Improbable. Random House.

Extensions

Possible extensions to this implementation:

1. More Asset Classes: Add commodities, real estate, cryptocurrencies
2. Quantitative Returns: Map states to actual return distributions
3. Multi-Period Dynamics: Model time evolution of crises
4. Learning from Data: Estimate CPDs from historical data where available
5. Portfolio Optimization: Implement utility maximization under scenarios
6. Ambiguity Aversion: Model uncertainty about probabilities (Ellsberg paradox)

Author Notes

This implementation is educational and demonstrates the core concepts. For production use:

• Consider using established libraries like pgmpy or PyMC
• Validate CPDs with domain experts
• Incorporate real market data
• Add sensitivity analysis
• Consider computational efficiency for large networks

License

This code is provided for educational purposes. The methodology is described in detail in Rebonato & Denev's book, which should be consulted for rigorous application.

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