Fig 1.
Interbank network and balance sheet.
Sketch of a portion of the interbank network and stylised representation of the balance sheet of a bank, with interbank assets and liabilities highlighted. The difference between assets and liabilities is the equity. A negative equity is usually considered a good proxy for the default of a bank.
Fig 2.
Comparison of different algorithms.
Probability of default as a function of the relative equity loss hj(t) for different algorithms. The non-linear DebtRank interpolates between the Furfine algorithm and the non-linear DebtRank.
Fig 3.
Unraveling of shocks propagation over time.
Fraction of stressed banks S(t) (blue line), fraction of defaulted banks D(t) (red line), and H(t) (violet line), total relative equity loss experienced by the system as a function of the time t over which shocks propagate. Banks experience a shock in the external assets, which suffer a relative loss equal to xshock = 0.5%. All points are averaged over a sample of 100 reconstructed networks with connectivity p = 0.05 and compatible with 2008 balance sheets, and over 10 realisations of the shock in which each bank is shocked with probability pshock = 0.05. Error bars span three standard errors. α = 0 in panel A and the algorithm reduces to the linear DebtRank, while α = 1 in panel B, and α = 2 in panel C. We see that the dynamics unravels within a few time steps in the panels A and C, while it takes considerably more time steps in panel B.
Fig 4.
Surface plot of H∞, the total relative equity loss in the steady state, as a function of the size of the shock suffered by external assets of banks xshock and of the parameter α, which tunes the non-linearity of the algorithm. All points are averaged over a sample of 100 reconstructed networks with connectivity p = 0.05, and compatible with 2008 balance sheets, and over 10 realization of the shock in which each bank is shocked with probability pshock = 0.05 (panel A) and pshock = 1 (panel B). Note that the range of the total size of the shock pshock ⋅ xshock is the same for both panels. As α increases, the propagation of the shock is dampened, resulting in smaller losses, and in two different regimes, whose separation is especially evident for pshock = 1, i.e. when all banks are shocked.
Fig 5.
Analogous of Fig 4, but for the years 2010 and 2013. Here the connectivity of the reconstructed networks is p = 0.05 and the fraction of shocked banks on average is pshock = 0.05. Going from 2008 (see Fig 4, panel A) to 2013 H∞ is less and less sensitive to changes in α and it is generally smaller, meaning that banks are more and more robust.