Systemic stress test model for shared portfolio networks

We propose a dynamic model for systemic risk using a bipartite network of banks and assets in which the weight of links and node attributes vary over time. Using market data and bank asset holdings, we are able to estimate a single parameter as an indicator of the stability of the financial system. We apply the model to the European sovereign debt crisis and observe that the results closely match real-world events (e.g., the high risk of Greek sovereign bonds and the distress of Greek banks). Our model could become complementary to existing stress tests, incorporating the contribution of interconnectivity of the banks to systemic risk in time-dependent networks. Additionally, we propose an institutional systemic importance ranking, BankRank, for the financial institutions analyzed in this study to assess the contribution of individual banks to the overall systemic risk.


S1 Comparison of BankRank in the unstable regime with other centrality measures
We found that BankRank in the unstable regime (e.g. α = β = 1.5) did not correlate well with holdings, unlike the in the stable regime. We further compare BankRank in the unstable regime with other centrality measures to see if BankRank is truly capturing a different property of the system. Most centrality measures such as eigenvector, closeness and betweenness centrality are defined for monopartite networks. In order to compare BankRank of the banks with these cetrality measures, we first construct a monopartite Bank-Bank network with adjacency B = AA T where A iµ are the elements of the matrix of bank holdings. We then compute the degree, eigenvector, closeness and betweenness centrality on B and calculate the Spearman rank correlation of these centrality measures with the BankRank calculated at α = β = 1.5. The results, shown in Fig none of these centrality measures has considerable correlation with BankRank in the unstable regime. This lack of correlation is expected as BankRank in the unstable regime is sensitive to both the holdings as well as the equities of the banks, whereas all of these centrality measures only use the holdings and do not incorporate the bank equities. S2 Analytical derivation of the mean-field phase space Figure 3 shows an example of the average final prices and the time the system needs to reach its final state for various values of α and β . The system exhibits two prominent phases: one in which a new equilibrium is reached without a significant loss in asset value (upper left and lower right quadrants), and another in which the assets lose significant value (above dashed line in the upper right quadrant and lower left quadrant). At the transition in both the first and the third quadrants, the system requires a long time to reach its new equilibrium. Such a behavior in the relaxation time may signal the existence of a second order phase transition, described by γ = αβ = 1.
To obtain exact solutions for this phenomenological model, we simplify the equations (6)-(8). We assume that there is one major holder of each asset µ and reduce the system to the interactions of one bank with equity E and holdings A and one asset with price p. With this mean field assumption, we break the network apart and analytically derive the phase transition for this simplified model. As we show below, the 1-by-1 system exhibits the same phases, even if it does not have the richness and complexity of the entire system.
Analytical results from the system with one bank and one asset. In the following, we present the analytical solution of the 1-by-1 model and derive the curve along which the phase transition is happening in Figure 3. The equations for a 1-by-1 system simplify to We eliminate A and E, and to this end, we find an expression for ∂ 2 t A/A. Taking another derivative in equation (21) yields Combining this result with the equation (20) results in: where the nonlinear term is again quadratic in P and thus a generalized form of the Fisher equation. More specifically, Next, we show that in the stable regime the nonlinearity in the frequency, that is, the γAP/E = γλ term, is of the order O(∂ t A∂ t P) and thus remains small if we show that at small times the behavior of ∂ t P in the stable regime is oscillating around zero.
In this regime, the dynamics are richer, and we have a damped oscillator with a driving force coupled to P and nonlinearities of type (∂ t P) 2 . Taking the price change u ≡ ∂ t P as the fundamental variable, the nonlinearities are roughly of type u 2 + a∂ t u 2 . In short, the equations are Although ω 2 depends on A, P and E, we can use an approximate time-dependent exponential ansatz u ∼ u 0 exp [λt]. The solutions to η are: When ω 2 > 0 and 1 − 4τω 2 < 0, there will be oscillatory solutions. One such example arises when γλ < −1, which only happens for negative γ. This is consistent with the simulations which showed that the oscillatory behavior was in the αβ < 0 quadrants. For the stability, however we care about the real solutions.

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When ω 2 < 0, which happens when γλ > 1, we have two real solutions with opposite signs. The presence of the positive root signals an instability because the solution diverges. For a delta function shock of magnitude f at t = 0 we find that: Having initially scaled to E 0 = A 0 = P 0 = 1, the condition for existence of the positive root becomes: Now the question is, to which solution does the system trend when it is shocked. The price change ∂ t P is Since at t = 0 the initial conditions dictated ∂ t P(0) = 0, we have Therefore, both solutions are equally likely. It follows that whenever one of the solutions (u − in our case) is positive, the solution diverges. When f > 0 a bubble forms and grows exponentially and when f < 0, because our variables are non-negative, the price crashes to zero. This proves that the sufficient condition for stability is γ < 1. Further note that the all nonlinear terms are proportional to ∂ t P, and therefore, at t = 0, Thus, the solution is exact at t = 0 and we get three regimes: 1. When ω 2 > 1 4τ , there will be oscillatory solutions. This happens when γ < −(τ P −τ A ) 2 τ P τ A . For τ P = τ A this is just the γ < 0 condition we observed for oscillations in our simulations.
3. When ω 2 < 0, i.e., γ > 1 we will have two real solutions for with opposite signs. The presence of the positive root η + > 0 signals an instability because this solution diverges.
Thus we have proven the existence of the three phases we had observed earlier and derived the transition conditions analytically.
Validity of perturbation theory near the phase transition. For the above solution to be valid we must confirm that the corrections are small. We must find a small parameter that exists in the neglected terms which allows perturbative solutions to be viable. We have two sets of nonlinearities: (1) O (∂ t P) 2 ; (2) γAP/E.
• The nonlinearity O (∂ t P) 2 : Note that the instability occurs when the larger root η − becomes positive. Therefore, near the transition we have Thus, being close to the phase transition means η − 1/τ. As a consequence, for O (∂ t P) 2 and using the u + = −u − found above, we get:  Figure S4. Dynamics of the model for a system with one bank and one asset with E(0) = A(0) = P(0) = 1. Here f indicates the magnitude of the shock, which can be positive or negative. In the top row, γ < 0. Following a the shock, the variables of the system fluctuate for some time and eventually return to their original place or somewhere close. In the middle row, 0 < γ < 1. A shock may result in a non-negligible change to the variables of the system, and this change increases as γ → 1. In the bottom row, γ > 1. A negative shock results in a collapse, and positive shocks results in the formation of a bubble. Either way, the system is unstable.
• The nonlinearity γAP/E: We next have to examine if the assumption that ∂ t A, ∂ t P, ∂ E remain small in the stable regime is a consistent assumption, thus making perturbative expansion valid. Any term above non-linear in ∂ t A, ∂ t P, ∂ E is of higher order in this approximation. We wish to find the part of γAP/E∂ t P that is linear in the first time derivative. In the stable regime changes are slow and thus a short time after the shock we can find the Taylor expansions for the variables near t = 0. Again, we rescale the variables at t = 0 to E 0 = P 0 = A 0 = 1. Using equation (8), we get: Thus the assumption of smallness of the derivatives is consistent. We may use perturbation theory and safely discard the non-linear terms in finding the stability conditions. The stability condition is simply that ω 2 be positive. We combine equations (6)-(8) in a 1-by-1 system by taking another ∂ t derivative from (7). Thus, we eliminate most occurrences of E and A an find an equation for P, which contains non-linear terms in it. Using the price change u ≡ ∂ t P as the fundamental variable, the nonlinearities are roughly of type u 2 + a∂ t u 2 , described in the following equations: For a small shock f 0 = −ε we may safely use AP/E = 1. Thus for a time-scale where γ < 1 and is not changing much we are essentially dealing with a damped harmonic oscillator. Notice that equation (29) is almost identical to what Bouchaud proposes in 30 to explain the 1987 crash. Table S3. GIIPS debt data used in the analysis. All numbers are in million euros. Our data is based on two sources: 1) The EBA 2011 stress test data, which only includes exposure of European banks and funds (these are the ones where the "Code Name" is of the form CC123); 2) A list of top 50 global banks, insurance companies and funds with largest exposures to GIIPS debt by end of 2011 provided by S. Battiston et al. (These have a name as their "Code Name"), which was consolidated by the authors.