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Stability analysis of financial contagion due to overlapping portfolios
Fabio Caccioli, Munik Shrestha, Cristopher Moore, J. Doyne Farmer
TL;DR
The paper asks when overlapping portfolios and leverage amplify financial contagion. It models institutions and assets as a network, maps failures to a generalized branching process, and identifies instability across diversification, crowding, leverage, and market impact. The model finds cascade windows and robust-yet-fragile regions, with potential use in macroprudential stress testing.
Problem
The paper addresses the need to understand how contagion propagates through overlapping portfolios, a channel believed to have been central to the financial crisis.
Method
The authors develop a network model and generalized branching-process stability analysis for stress-testing contagion from localized shocks.
Results
Diversification creates a window where global cascades occur, leverage increases instability, and some regions are robust yet fragile, with rare but system-wide contagion.
Takeaways & Limitations
The framework can in principle be calibrated to real data and used for macroprudential stress tests of leveraged financial institutions.
Takeaways & Limitations
The branching-process analysis neglects failure interactions involving loops and therefore provides a sufficient, but not necessary, condition for global cascades.
Abstract
from arXiv · showhide
Common asset holdings are widely believed to have been the primary vector of contagion in the recent financial crisis. We develop a network approach to the amplification of financial contagion due to the combination of overlapping portfolios and leverage, and we show how it can be understood in terms of a generalized branching process. By studying a stylized model we estimate the circumstances under which systemic instabilities are likely to occur as a function of parameters such as leverage, market crowding, diversification, and market impact. Although diversification may be good for individual institutions, it can create dangerous systemic effects, and as a result financial contagion gets worse with too much diversification. Under our model there is a critical threshold for leverage; below it financial networks are always stable, and above it the unstable region grows as leverage increases. The financial system exhibits "robust yet fragile" behavior, with regions of the parameter space where contagion is rare but catastrophic whenever it occurs. Our model and methods of analysis can be calibrated to real data and provide simple yet powerful tools for macroprudential stress testing.
1 Introduction
The paper models contagion through overlapping portfolios and leverage, using a network and generalized branching-process framework to identify when localized shocks become global cascades.
- Motivation: Overlapping portfolios transmit contagion when fire sales depress common-asset prices, potentially triggering further failures and selling spirals.This mechanism can operate without inter-institutional lending and is intensified by leverage.
- Approach: The model represents financial institutions and assets as a bipartite network and studies shocks propagating through local portfolio overlaps.The analysis varies diversification, crowding, leverage, and market impact.
- Approach: Global cascades are failures affecting a non-zero fraction of banks in the infinite-network limit, and stability is measured through their probability.A smaller cascade probability corresponds to higher stability.
- Approach: A generalized branching-process mapping identifies parameter regions where global cascades can occur and supports calibration to real data for stress testing.The method provides an analytical stability framework alongside numerical investigation.
- Results: Diversification produces two phase transitions: insufficient connectivity prevents propagation, excessive diversification makes banks robust to a few devaluations, and an intermediate region permits global cascades.The cascade window lies between the two transitions.
- Results: Higher leverage increases overall network instability, while some parameter regions are robust to shocks yet fragile because contagion is rare but catastrophic when it occurs.These results motivate stability analysis across leverage and network properties.
2 The model
The model encodes banks’ portfolios as a bipartite bank–asset network, assigns balance-sheet variables and leverage, and simulates iterative fire-sale contagion after localized shocks.
- Banks, assets, and cascades of bankruptcies: Banks and assets form a bipartite network, with links representing investments and bank degree representing the number of assets held.In the illustrative network, N = 4, M = 3, µb = 1.5, and µa = 2.
- Banks, assets, and cascades of bankruptcies: The average diversification µb and crowding n = N/M summarize portfolio connectivity and the density of institutions investing from the same asset pool.These parameters provide a rough characterization rather than a complete description of topology.
- Balance sheets: Each solvent bank holds a fixed portfolio of asset shares, cash, and liabilities while asset prices change over time.Only assets included in a bank’s portfolio have non-zero holdings.
- Solvency and leverage: Leverage is the ratio of risky assets to equity, and a bank becomes insolvent when portfolio losses exceed its initial capital.Without leverage, the model’s maximal-loss condition cannot cause failure; leverage above one is required for insolvency.
- Fire sales: An insolvent bank fully liquidates its portfolio, and the resulting asset-price decline can transmit losses to other banks.The price update depends on the fraction of each asset liquidated.
- Shock propagation: The simulations apply either a devalued random asset or a failed random bank, then repeatedly check solvency, liquidate newly insolvent portfolios, and recompute prices.The process ends when no new bankruptcies occur, and failed banks do not re-enter.
3 Stability analysis
The paper analyzes cascading bank failures from overlapping portfolios using a generalized branching-process framework. It derives stability conditions and a matrix-based approach that can incorporate network structure, market impact, leverage, diversification, and other institutional or asset characteristics.
- Cascade mechanism: A bank failure can trigger others through leveraged fire-sale price impacts on commonly held assets, producing cascades that propagate across portfolio-overlap networks.The model treats banks as fixed-portfolios institutions until default, after which their portfolios are fully liquidated.
- Branching-process formulation: Global cascades occur in a parameter region identified analytically through the largest eigenvalue of the offspring matrix: extinction when ξ1 < 1 and positive-probability continuation when ξ1 > 1.The matrix entry Nhk is the expected number of banks of type h failing because of a bank of type k.
- Branching-process formulation: The branching-process analogy represents banks failing at successive times as offspring generated by an initially failed bank.Banks with differing degrees, leverage, or sizes can be represented as different types in a generalized process.
- Approximation and scope: The analytical process is not an exact tree model because failures can form loops and depend on the portfolios of other overlapping banks.The resulting condition is sufficient but not necessary for global cascades and provides an upper bound on stability, while remaining in rough agreement with simulations.
- Approximation and scope: The one-step approximation can be systematically improved with multiple-time-step dynamics, while the framework can generalize to non-Poisson degree distributions and heterogeneous banks or assets.The model’s market-impact treatment uses linear impact for log-prices, although cited evidence indicates large-trade impact can be concave.
- Stability matrix: For a fully specified banking system, the stability matrix B contains Bij, the probability that bank i fails when bank j fails, based on overlapping-asset market impacts exceeding bank i’s equity.The calculation focuses on direct effects at the boundary of a cascade, before shared assets have already been devalued.
4 Dependence on leverage and network properties
Simulations show that diversification and crowding create a nonmonotonic contagion window, while leverage and market impact increase instability. The system can be robust yet fragile: cascades are unlikely in some regions but affect almost all banks when they occur.
- 4.1 Effect of diversification and crowding: Two phase transitions in average diversification µb define a contagion window where global cascades occur with non-zero probability.Below µ1, poor connectivity prevents propagation; above µ2, diversified portfolios are robust to single-asset devaluations and bank failures have smaller price effects.
- 4.1 Effect of diversification and crowding: Just below µ2, global cascades have very low probability but affect almost all banks when they occur, with conditional extent nearly 1.This is the model’s “robust yet fragile” regime.
- 4.1 Effect of diversification and crowding: Crowding shifts both transition boundaries µ1 and µ2 to lower diversification, while its effect on contagion probability depends on proximity to either transition.Near µ1, increased crowding can move the network into the contagion window; near µ2, it can move the network out of the window by making assets and banks more robust.
- 4.2 Dependence on shocks: The contagion window and conditional cascade extent are the same for failed-asset and failed-bank shocks, although their contagion probabilities differ.Once a cascade begins, its dynamics determine whether it spreads or dies out, regardless of the initial shock type.
- 4.3 Leverage: For each fixed µb and n, a critical leverage λ separates a regime without global cascades from one where they occur with non-zero probability.The critical λ increases with µb, so greater diversification permits more leverage before systemic events arise.
- 4.3 Leverage: For fixed µb and λ, a critical market-impact parameter α separates regimes, and increasing α has a similar effect to increasing leverage.Larger α causes sharper price drops during fire sales, while greater diversification raises the critical α.
5 Comparison to predictions from stability analysis
The stability analysis uses the largest eigenvalue of a branching-process matrix to identify cascade conditions, then compares those predictions with finite-size simulations and phase diagrams.
- ξ1 > 1 marks the condition under which global cascades are observed in simulations.The analytic calculation provides a sufficient condition for global cascades.
- The analytic calculation underestimates the width of the contagion window relative to numerical simulations.The discrepancy is partly attributed to finite-size effects.
- As system size increases from N = 100 to N = 20000, agreement between theory and simulations improves.The theory is valid in the limit {N, M} →∞.
- The phase diagram predicts two connectivity transitions bounding a region where global cascades occur with non-zero probability.The analytic approach also predicts shifts in transition points as network parameters change.
- Monitoring ξ1 over time could warn regulators that the system is approaching a dangerous regime as ξ1 nears 1.The proposed monitoring is intended to support actions that increase system stability.
6 Conclusion
The paper frames overlapping-portfolio contagion as a stability problem in a bank–asset network, analyzed through generalized branching processes. It identifies cascade-prone parameter regions, including a leverage threshold, and highlights extensions needed for empirical calibration and more realistic dynamics.
- 6 Conclusion: A bipartite bank–asset network captures how portfolio links both diversify individual banks and transmit contagion.The system is modeled with N banks investing in M common assets, with shocks applied to a single bank or asset.
- 6 Conclusion: Global cascades occur within a cone-shaped region of leverage, diversification, and crowding, with no cascades below a critical leverage value.The analytical visualization identifies the cascade region in the model’s parameter space.
- 6 Conclusion: Contagion probability varies nonmonotonically with diversification, with two transitions defining a window in which global cascades can occur.Below the first transition, banks are insufficiently interconnected; above the second, they are robust to devaluations in a few assets.
- 6 Conclusion: Generalized branching-process analysis estimates the parameter region where global cascades occur with non-zero probability.The approach accounts for dependencies on node and neighbor degrees and can, in principle, support stress testing against real data.
- 6 Conclusion: The framework is mechanistic and assumes homogeneous balance sheets, Poisson degree distributions, fixed portfolios until default, and a specified market-impact function.The authors propose relaxing these assumptions and adding more realistic price dynamics and portfolio rebalancing.