Cascades in multiplex financial networks with debts of different seniority

The seniority of debt, which determines the order in which a bankrupt institution repays its debts, is an important and sometimes contentious feature of financial crises, yet its impact on systemwide stability is not well understood. We capture seniority of debt in a multiplex network, a graph of nodes connected by multiple types of edges. Here an edge between banks denotes a debt contract of a certain level of seniority. Next we study cascading default. There exist multiple kinds of bankruptcy, indexed by the highest level of seniority at which a bank cannot repay all its debts. Self-interested banks would prefer that all their loans be made at the most senior level. However, mixing debts of different seniority levels makes the system more stable in that it shrinks the set of network densities for which bankruptcies spread widely. We compute the optimal ratio of senior to junior debts, which we call the optimal seniority ratio, for two uncorrelated Erdős-Rényi networks. If institutions erode their buffer against insolvency, then this optimal seniority ratio rises; in other words, if default thresholds fall, then more loans should be senior. We generalize the analytical results to arbitrarily many levels of seniority and to heavy-tailed degree distributions.

Figure 1

An example of a bank's balance sheet. Assets (left-hand column) consist of external assets, $l_S=4$ many senior interbank loans, and $l_J=3$ many junior interbank loans. Liabilities (right-hand column) consist of external liabilities, $b_S=2$ many senior interbank liabilities, and $b_J=3$ many junior interbank liabilities. Net worth (also called equity or, for banks, capital) is the difference between assets and liabilities.

Figure 2

An example of the balance sheet (and the corresponding multiplex network visualizations) of a bank that is (a) in junior-default: $m_J+m_S>R_J(l_J+l_S)\equiv w$ and (b) in senior-default: $m_J+m_S>R_S(l_J+l_S)\equiv w+b_J$. Circular and spiky-shaped nodes are solvent and insolvent (bankrupt, in default), respectively, in that layer. Edges point from lender to borrower. Senior loans are dashed green arrows; junior loans are solid red arrows. In panel (a), the bank labeled $A$ suffers losses of $m_J+m_S=2$ on its assets side (assets are listed on the left-hand side of the bank's balance sheet) because $m_J=1$ of its ($l_J=3$ many) junior debtors are in junior-default and because $m_S=1$ of its ($l_S=4$ many) senior debtors are in senior-default. These losses exceed bank $A$'s equity ($w$), so bank $A$ cannot pay off all its ($b_J$ many) junior liabilities. Because of the assumption that loss given default at layer $\alpha$ is $100\%$ for all creditors in that layer, all of $A$'s junior creditors do not receive repayments on their loan to $A$, and we say that $A$ goes into junior-default. However, bank $A$ does not go into senior-default because it can still pay its senior liabilities (i.e., because $m_J+m_S\le w+b_J$). In panel (b), bank $A$ suffers so many losses ($m_J+m_S=5$) that it can repay neither its junior creditors nor its senior creditors, so bank $A$ goes into senior-default.

Figure 3

(a) The multiplex cascade region [the edge-density parameters $(\langle l_J\rangle ,\langle l_S\rangle )$ satisfying Eq. (6), drawn as a blue region with a thick, solid boundary] contains the junior-only and senior-only cascade conditions given by Eqs. (7) and (8) and drawn as an orange region (with a dashed boundary) and green region (with a dotted boundary), respectively. Moreover, the multiplex cascade region is asymmetric in the densities of junior and senior loans, $(\langle l_J\rangle ,\langle l_S\rangle )$. The optimal seniority ratio, defined in Eq. (9), is drawn as a black line. This ratio $\langle l_S\rangle /\langle l_J\rangle \approx 1.79$ minimizes the cascade window, the set of mean degrees satisfying the first-order cascade condition, Eq. (6). The case of equally many junior and senior loans (on average) is shown with a dashed line, which has a slightly larger cascade window. (b) The cascade window as a function of the ratio of senior loans to junior loans, $\langle l_S\rangle /\langle l_J\rangle$, achieves a minimum size at $\approx 1.79$.

Figure 4

If banks hold less capital (equity) $w$ relative to their interbank assets $l_J+l_S$ (i.e., if the threshold for junior-default, $R_J$, decreases), then (a) the cascade region (the set of parameters satisfying the first-order cascade condition (6)] enlarges, and (b) the optimal ratio of senior to junior loans increases. The first-order cascade condition (6) considers defaults that occur due to one defaulted neighbor (for details, see Appendix pp3). Thus, when $R_J$ falls below the inverse of an integer, $1/k$, nodes with total degree $k$ and with one defaulted neighbor will junior-default, so the cascade region changes at $R_J=1/k$, and hence the optimal ratio jumps up at $R_J=1/k$.

Figure 5

Distinguishing more types of seniority reduces the size of the cascade region. In other words, with more types of seniority, the expected size of a default cascade is negligibly small for more pairs of thresholds $R_1$ and expected number of loans per bank $z={\sum }_{i=1}^M\langle l_i\rangle$. Here we consider splitting a directed Erdős-Rényi network into $M\in \{1,2,3,4\}$ many independent, identically distributed layers, each with the same mean degree $\langle l_i\rangle =z/M$.

Figure 6

For heavy-tailed degree distributions like those in the Brazilian interbank lending network [28], having $\approx 30\%$ of loans be junior (and the rest senior) reduces systemic risk, in that it minimizes the size of the set of mean total out-degrees such that bankruptcy spreads widely. The gray-scale background shows the average fraction of banks in junior-default (top panel) and in senior-default (bottom panel) in simulations on 2400 banks, begun from 10 banks initially in senior-default, averaged over 30 simulations. The green lines mark the boundary of the cascade region [${\lambda }_{max}\left(\mathcal{J}\right)=1$ with $\mathcal{J}$ computed from Eq. (C4)]. The junior-default threshold of every bank is $R_J=0.18$, as in Fig. 3. Directed graphs are generated with the static model [39, 40] with out-degree distributions having tail behavior $\propto k^{-\gamma }$ with $\gamma =2.83$, which approximates the value found in the Brazilian interbank lending network [28]. A fraction of loans (horizontal axis) chosen uniformly at random are junior, and the rest are senior. The “optimal” fraction of junior loans (that minimizes the height of the green line) is $\approx 0.30$. For the mean out-degree (vertical axis) of $\approx 8.5$ observed for Brazilian banks [28], the expected cascade size is small if and only if an intermediate fraction of loans are junior.

Figure 7

Here we compare the junior-only, senior-only, and multiplex cascade conditions [Eqs. (6, 7, 8), respectively] with the fixed points of the recursion equations (4) [in panels (a) and (b)] and with numerical simulations on random graphs with ${10}^4$ nodes [in panels (c) and (d)]. The junior-default threshold of every node is $R_J=0.18$ (as in Fig. 3). The axes are the edge densities $\langle l_J\rangle$ and $\langle l_S\rangle$ (i.e, the average numbers of junior and senior loans per bank). The solid blue, dashed orange, and dot-dashed green lines surround the multiplex, junior-only, and senior-only cascade regions [i.e., the sets of values of $(\langle l_J\rangle ,\langle l_S\rangle )$ that satisfy the inequalities (6, 7, 8), respectively]. In panels (a) and (b), the gray scale in the background shows the fixed points ${\varphi }_{\infty }^J$ and ${\varphi }_{\infty }^S$ of the recursion (4), begun from the initial condition ${\varphi }_0^J={\varphi }_0^S=5\times {10}^{-4}$. In panels (c) and (d), the gray scale in the background shows the fraction of nodes in junior- and senior-default, respectively, in numerical simulations with ${10}^4$ nodes and 5 banks initially in senior-default, averaged over 75 simulations. In the upper-left part of all four panels, where most loans are senior, the fractions of nodes $({\varphi }_{\infty }^J,{\varphi }_{\infty }^S)$ in junior- and senior-default at the end of the cascade are both $\approx$1. By contrast, in the lower-right part, where most loans are junior, the fraction ${\varphi }_{\infty }^J$ of nodes in junior-default is large [${\varphi }_{\infty }^J\approx 1$; panels (a) and (c)], whereas the fraction ${\varphi }_{\infty }^S$ of nodes in senior-default is $\approx 0.4$ [see the light-gray region surrounded by the dashed orange line in panels (b) and (d)].