Aggregated Systemic Risk Index (ASRI)

• ASRI is a quantitative risk measure that aggregates multiple systemic vulnerabilities into a unified index.
• It integrates methodologies from measure theory, network analysis, and deep learning to assess interdependent risks.
• Applications range from evaluating CCP default funds and interbank networks to stress-testing decentralized finance systems.

The Aggregated Systemic Risk Index (ASRI) is a class of systemic risk measures designed to provide a unified, quantitative assessment of the vulnerabilities of a financial system or network. The ASRI framework integrates multiple risk channels, accounting for interdependencies between components, and is deployed in both legacy financial sectors and emerging domains such as decentralized finance. ASRI methodologies share the goal of aggregating disparate risk sources into a single or vector-valued index, facilitating macroprudential monitoring, capital allocation, regulatory intervention, and empirical analysis of systemic events.

1. Formal Definitions and Mathematical Structures

The ASRI paradigm comprises several rigorously formulated approaches rooted in measure-theoretic probability, convex optimization, network theory, and aggregation operators.

1.1. Primal–Dual Formulation via Systemic Risk Measures

In deep learning-based systemic risk measures, ASRI is defined for a system of $N$ institutions with terminal loss vector $\mathbf{X} = (X^1,\ldots,X^N)$, utility functions $u_n$, and safety level $B < 0$ as follows (Feng et al., 2022):

Primal:

$$\rho(\mathbf{X}) = \inf_{\mathbf{Y} \in \mathcal{C}} \left\{ \sum_{n=1}^N Y^n : \mathbb{E} \left[ \sum_{n=1}^N u_n(X^n + Y^n) \right] \geq B \right\}$$

$\mathcal{C}$ denotes admissible capital allocations with full-system constraints.

Dual:

$$\rho(\mathbf{X}) = \max_{\mathbf{Q} \in \mathcal{D}} \left\{ \sum_{n=1}^N \mathbb{E}_{Q^n}[ -X^n ] - \alpha_B(\mathbf{Q}) \right\}$$

with

$$\alpha_B (\mathbf{Q}) = \sup_{\mathbf{Z} \in \mathcal{A}} \left\{ \sum_{n=1}^N \mathbb{E}_{Q^n}[ -Z^n ] \right\}, \quad \mathcal{A} = \left\{ \mathbf{Z} : \mathbb{E} \left[ \sum_{n=1}^N u_n(Z^n) \right] \geq B \right\}$$

1.2. Multivariate Shortfall-Based ASRI

Under the multivariate shortfall paradigm (Armenti et al., 2015), ASRI for $d$ losses $X = (X_1,\ldots,X_d)$, loss function $\ell$, and risk level $\alpha \geq 0$ is: $$R_\alpha(X) = \inf_{m \in \mathbb{R}^d} \left\{ \sum_{k=1}^d m_k : \mathbb{E}[\ell(X - m)] \leq \alpha \right\}$$ Here, $\ell$ encodes system-aggregate tail risk and component interdependence.

1.3. Aggregation Operator Frameworks

RiskRank and Choquet-integral-based methods represent ASRI as an aggregation, where nodes (countries, sectors, institutions) are combined using a monotone capacity $\mu$ and possibly pairwise or higher-order interaction indices (Mezei et al., 2014, Mezei et al., 2016): $$\operatorname{ASRI} = C_\mu(v_1,\ldots,v_n) = \sum_{i=1}^n (v_{(i)} - v_{(i-1)}) \mu(C_{(i)})$$ Where $v_{(i)}$ indicates ordered vulnerabilities and $C_{(i)}$ the corresponding set.

1.4. Multi-layer and Tensor Network ASRI

In tensorial, multi-layer networks (Sergueiva, 2013, Poledna et al., 2018), ASRI collects normalized, weighted measures over $K$ risk layers ($R_k^*$) via a scalar-valued operator: $\operatorname{ASRI} = \sum_{k=1}^K \alpha_k R_k^*$ Common $R_k$ include tensor network metrics ($\lambda_{\max}$), market stress (VaR, eigenvalues), and liquidity shortfall.

1.5. Data-Driven and Machine Learning Approaches

Recent advances employ deep neural networks, both in direct regression (primal) and adversarial dual settings (GAN-inspired) for risk allocations and scenario density estimation. These architectures are optimized under stochastic microfoundations and large-scale simulation (Feng et al., 2022). In cryptocurrency settings, empirical construction leverages compositional risk indices with normalization, channel weights, and operational thresholds (Farzulla et al., 1 Feb 2026).

2. Key Methodological Components

2.1. Risk Aggregation Principles

Core to all ASRI frameworks is aggregation over systemic risk sources:

• Componentwise normalization: Each risk factor is re-scaled (typically via z-score or quantile normalization) to a common risk scale (Sergueiva, 2013).
• Weighting: Expert-assigned, data-driven (PCA), or optimization-derived layer/channel weights allocate relative systemic importance.
• Dependence and Interconnections: Multivariate, network, and Choquet frameworks move beyond simple summation, modeling synergistic (contagion, spillover) and redundant effects.

2.2. Network and Multilayer Models

Tensorial and bipartite network models encode exposures within and across risk dimensions:

• In multi-layer adjacency $M_{ij}^{h k}$, each layer and interlayer coupling can be analyzed for per-layer and aggregate systemic propagation (Sergueiva, 2013).
• Overlapping asset portfolios yield indirect exposure matrices whose weighted DebtRank scores are normalized to generate ASRI (Poledna et al., 2018).
• Fuzzy cognitive maps (FCMs) aggregate expert-linked segment vulnerabilities, with capacities reflecting connectivity (Mezei et al., 2014).

2.3. Allocation and Dual Interpretation

Allocation mechanisms ensure capital call or risk assignment compatible with regulatory or utility-based system acceptability:

• Primal algorithms numerically optimize capital allocation to meet utility or loss-acceptance constraints.
• Dual measures interpret ASRI as the worst-case expected loss under alternative probability scenarios, penalized for disappointment risk.
• Allocation rules, derived from dual variables or Euler derivatives of the risk measure, yield componentwise contribution (Shapley indices, fair shares) (Feng et al., 2022, Armenti et al., 2015).

2.4. Implementation and Empirical Calibration

Empirical computation depends on scenario generation, expectation evaluation, and scenario-based stress testing:

• Deep learning algorithms for primal-dual estimation deploy mini-batch SGD or Adam, with scenario-based loss computation and deterministic constraint enforcement (Feng et al., 2022).
• Network-based ASRI evaluates node-level DebtRanks via layered exposure matrices, normalizing and summing as an average country-level risk index (Poledna et al., 2018).
• Hypothesis testing for ASRI change: Confidence intervals are built via ensemble simulation, and observed ASRI is compared to detect statistically significant shifts (Gangi et al., 2015).

2.5. Backtesting, Thresholds, and Regime Identification

Empirical validation strategies include:

• Event-study analysis, abnormal signal computation, and cumulative abnormal statistics.
• Operational thresholds (e.g., ASRI ≥ 50) for real-time crisis detection and alerting (Farzulla et al., 1 Feb 2026).
• Markov regime-switching analysis: Hidden Markov Models (HMMs) fitted to ASRI time series separate risk regimes and provide persistence and transition probabilities.

3. Illustrative Applications and Use Cases

3.1. CCP Default Fund and Market Infrastructure

ASRI applied to central clearing counterparties (CCPs) yields default funds sized to cover tail shortfall risk, with allocation weights shifting up to 50% when systemic (dependence-sensitive) loss functions are used (Armenti et al., 2015).

3.2. Interbank Networks and Capital Buffer Policy

Tensorial ASRI models support:

• Ranking of banks by marginal effect on systemic risk (gradient metrics).
• Stress scenario propagation under hypothetical portfolio, market, or liquidity shocks, with sensitivity analysis for capital buffers and regulatory tools (Sergueiva, 2013).

3.3. Cross-Border Banking and Macroeconomic Risk

RiskRank/Choquet-based ASRIs support early warning and crisis prediction, outperforming indicator-averaging at both system and country level, and provide relative usefulness gains in out-of-sample policy evaluation (Mezei et al., 2016).

3.4. Crypto/DeFi Systems

Composite ASRIs in cryptocurrency and DeFi record statistically significant signals for historical crisis events, with operational lead times. The index integrates DeFi-native vulnerabilities (composability, flash loans, regulatory opacity) and correlates them with cross-asset and macro linkages (Farzulla et al., 1 Feb 2026).

4. Economic and Policy Interpretation

ASRI quantifies the minimal capital required to render a system risk-acceptable, conditioning on the worst-case aggregation of losses or utilities. Key interpretive aspects include:

• Capital efficiency: ASRI ensures capital is pooled and allocated to minimize total required funding for system acceptability (Feng et al., 2022, Armenti et al., 2015).
• Systemic penalization: Dual-derived allocations penalize entities with tail co-movement or high interconnectedness.
• Dynamic monitoring: Rolling ASRI calculation and stress scenario analysis enable time-varying risk surveillance, regime identification, and targeted intervention.
• Policy levers: Regulatory choices (capital surcharges, liquidity add-ons, policy tilting) affect ASRI via weights and constraint sets, offering a direct link to macroprudential policy impact (Sergueiva, 2013).

5. Comparative and Empirical Performance

Extant ASRI studies demonstrate:

• Deep-learning-based ASRI achieves sub–5% relative errors on synthetic and empirical system loss distributions (Feng et al., 2022).
• Multilayer/nodal network ASRI detects up to 50% underestimation of system risk if indirect portfolio effects are neglected (Poledna et al., 2018).
• Aggregation via Choquet or RiskRank raises the measured ASRI relative to simple weighted sums by capturing synergistic dependencies (Mezei et al., 2014, Mezei et al., 2016).
• In cryptocurrency applications, ASRI delivers high recall in crisis regime detection and zero false positives in specificity testing (Farzulla et al., 1 Feb 2026).

6. Limitations and Modelling Considerations

ASRI construction is sensitive to:

• Specification of loss or utility functions; separability reduces systemic sensitivity, while coupled terms encode dependence.
• Choice of network/topology model (e.g., single vs multilayer exposure, quality and normalization of connectivity matrices).
• Quality and timeliness of data feeds (especially for DeFi/crypto); handling of missing data and frequency alignment is critical (Farzulla et al., 1 Feb 2026).
• Estimation approach for aggregation weights and capacities; subjectivity or ad hoc normalization may affect comparability (Mezei et al., 2014, Mezei et al., 2016).
• Scalability of numerical routines: Monte Carlo/Chebyshev interpolation may be required at high dimensions, but remains tractable up to $d \sim 30$ (Armenti et al., 2015).
• Ignoring higher-order interactions (beyond pairwise) may underestimate cascade or contagion potential in dense/interconnected networks.

7. Summary Table: Major ASRI Approaches

Framework	Risk Aggregation Principle	Computational Strategy
Primal/Dual (Utility) (Feng et al., 2022)	Minimal capital, expectation constraint	Deep learning (direct & GAN-dual nets)
Multivar. Shortfall (Armenti et al., 2015)	Expected shortfall, convex allocation	Convex program, MC/Fourier/Chebyshev integration
Tensor/Network (Sergueiva, 2013, Poledna et al., 2018)	Normalized sum over K risk layers	Network measures (λ, DebtRank), stress testing
Choquet/FCM (Mezei et al., 2014, Mezei et al., 2016)	Capacity-weighted aggregation	Expert scoring, capacity, interaction indices
Crypto/DeFi Composite (Farzulla et al., 1 Feb 2026)	Weighted normalization, channel-specific	Empirical channels, regime-HMM, dashboards

Each methodology targets the central challenge of aggregating disparate systemic vulnerabilities into a monitorable and interpretable risk index, suitable for use in empirical surveillance, capital planning, and policy analysis.