Systemic Risk Index (SRI) Overview

• Systemic Risk Index (SRI) is a composite metric that quantifies financial systemic risk by integrating multi-scale network dynamics and information flows.
• It employs transfer entropy, wavelet-based decomposition, and agent-based modeling to measure network, concentration, volatility, liquidity, and contagion risks.
• Empirical validation shows robust early-warning signals with ROC AUC ≈ 0.87 and up to five trading days lead time for market instability.

The Systemic Risk Index (SRI) is a rigorously defined composite metric designed to quantify systemic risk in financial markets by explicitly incorporating multi-scale network dynamics. Unlike traditional risk indicators that rely on static or single-scale properties, the SRI operationalizes temporally and structurally heterogeneous information flows and market interactions across agent types and timescales. It is implemented through transfer entropy network analysis combined with agent-based modeling and multi-resolution decomposition, as formalized in the Model Context Protocol Financial Markets (MCPFM) framework (Bhandari, 10 Jul 2025). The SRI provides practitioners and researchers a principled, early-warning tool for crisis prediction and for macroprudential policy assessment.

1. Mathematical Formulation and Network Construction

The SRI is constructed using a multi-scale transfer entropy (TE) network. For each asset $i$ and each temporal scale $j$, discrete wavelet transforms (DWT) are applied to extract scale-specific returns $x^{(j)}_{i,t}$. TE is then estimated as

$$\mathrm{TE}^{(j)}_{i\to\ell} = \sum p\bigl(x^{(j)}_{\ell,t+1},\,x^{(j)}_{\ell,t},\,x^{(j)}_{i,t}\bigr) \log \frac{ p\bigl(x^{(j)}_{\ell,t+1}\mid x^{(j)}_{\ell,t},\,x^{(j)}_{i,t}\bigr) }{ p\bigl(x^{(j)}_{\ell,t+1}\mid x^{(j)}_{\ell,t}\bigr) }$$

and networks $G^{(j)}$ of information transfer are constructed by thresholding $\mathrm{TE}^{(j)}$ at its $85^\text{th}$ percentile.

The resultant multi-scale directed network recursively captures dependencies and spillover effects, with asset nodes and edges weighted by information flow at scale $j$.

2. Agent-Based Framework and Information Flows

Agent-based modeling within the MCP-context introduces agents $a$ of heterogenous type:

• High-frequency traders (HFTs)
• Market makers (MMs)
• Institutional investors (IIs)
• Regulators (REGs)

State variables $s_a(t)$ include portfolio weights $\mathbf{w}_a$, cash $c_a$, risk aversion $\gamma_a$, and time horizon $\theta_a$. Agent communication is governed by the Model Context Protocol, tagging each network statistic message with context (scale), protocol (intent), and payload (TE and network metrics). Agents adapt their beliefs and positions using weighted network information, propagating systemic signals across time horizons.

3. SRI Formula and Component Risks

The SRI at time $t$ is defined as a convex weighted sum: $$\mathrm{SRI}(t) = \sum_{k=1}^5 w_k\,R_k(t), \quad \sum_k w_k = 1,\,w_k \ge 0$$ where the risk components $R_k$ are:

1. Network risk: connectivity and density across scales,

$$R_{\rm network} = \frac1{|\mathcal{S}|}\sum_{j\in\mathcal{S}} \rho^{(j)}\bigl(1-\tfrac1N\sum_{i}\tfrac1{d_i^{(j)}+1}\bigr)$$

with $\rho^{(j)}$ as scale-specific density, $d_i^{(j)}$ scale-specific degree centrality.

1. Concentration risk: Herfindahl index of positions.
2. Volatility risk: multi-scale GARCH-derived volatility.
3. Liquidity risk: spreads, depth, and price impact.
4. Contagion risk: maximal spillover using TE and correlations.

Component weights $w_k$ are calibrated empirically or via regulatory policy objectives.

Decomposition Example

Empirical decomposition over 150 simulation periods (eight assets):

• Correlation risk: 50.3%
• Concentration risk: 27.7%
• Network risk: 20.4% Baseline SRI average: 0.316

4. Early Warning Performance and Comparative Analysis

The multi-scale SRI produces robust early-warning signals. Backtesting over crisis vs. non-crisis market days gives ROC AUC $\approx 0.87$, outperforming traditional single-scale metrics (AUC $\approx 0.65$). Early-warning lead times for systemic events are extended by up to five trading days.

The inclusion of transfer entropy and wavelet-based decomposition enables the index to identify cross-scale spillovers and contagion patterns that are invisible to classical covariance- or correlation-based network indexes.

5. Implementation Protocols and Practical Application

The framework is delivered as the MCPFM R package, with open-source access. Agents communicate via structured messages using the MCP protocol:

for each agent i: for each neighbor ℓ with A^{(j)}_{iℓ}=1: msg.PAYLOAD ← TE^{(j)}_{i→ℓ} msg.CONTEXT ← scale j msg.PROTOCOL ← “INFORM_FLOW” send(msg) to agent ℓ end

Upon receiving “INFORM_FLOW”, agents update belief weights and recalibrate risk expectations. This protocol supports real-time integration of multi-horizon information, facilitating both early warning and stress testing.

6. Generalization and Extensions

The SRI methodology, though developed for financial equity-credit markets, is extensible to any multi-scale, agent-based network system with heterogeneous interactions, such as epidemiological contact networks, power-grid failures, and climate teleconnection structures (Bhandari, 10 Jul 2025). The underlying approach—combining multi-resolution signal extraction, causal network inference, and structured inter-agent communication—forms a general template for systemic risk quantification in highly interconnected, multi-layer, temporally dynamic environments.

7. Application and Policy Impact

The SRI provides actionable metrics for regulators and policymakers:

• Macroprudential regulation: Rule-based interventions can directly penalize SRI excursions.
• Regulatory utility function: e.g., $$U^{\rm REG}_a = -\alpha_a\,\mathrm{SRI} - \beta_a \sum_p C_p(\pi_p)$$ for agents tasked with minimizing systemic risk and policy cost.

Empirical analysis supports its use in scenario simulation, market monitoring, and the design of interventions for enhanced market stability.

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The Systemic Risk Index as defined by (Bhandari, 10 Jul 2025) is a multi-component, multi-scale, network-based metric with demonstrable superiority in early detection of market instabilities. Its formal integration with transfer entropy, agent-based models, and protocol-driven inter-agent communication facilitates both theoretical insight and practical deployment for systemic risk assessment.