Systemic Risk in Banking Networks Without Monte Carlo Simulation

Abstract

An analytical approach to calculating the expected size of contagion events in models of banking networks is presented. The method is applicable to networks with arbitrary degree distributions, permits cascades to be initiated by the default of one or more banks, and includes liquidity risk effects. Theoretical results are validated by comparison with Monte Carlo simulations, and may be used to assess the stability of a given banking network topology.

Appendices

Appendix A: Generalized Eisenberg-Noe Clearing Vector Cascades

This Appendix provides a summary of the financial cascade framework of Eisenberg and Noe [8], placed in a slightly more general context. Extending their work somewhat ([8] combine the quantities Y _i and D _i into a single quantity ${e}_{i} =\mathrm{ {Y}}_{i} -\mathrm{ {D}}_{i}$), we identify the following stylized elements of a financial system consisting of N “banks” (which may include non-regulated leveraged institutions such as hedge funds). The assets A _i of bank i at a specific time consists of external assets Y _i (typically a portfolio of loans to external debtors) plus internal assets Z _i (typically in the form of interbank overnight loans). The liabilities of the bank includes external debts D _i (largely in the form of bank deposits, but also including long term debt) and internal debt X _i . The bank’s equity is defined by $\mathrm{{E}}_{i} =\mathrm{ {Y}}_{i} +\mathrm{ {Z}}_{i} -\mathrm{ {D}}_{i} -\mathrm{ {X}}_{i}$ and is constrained to be non-negative.

The amounts Y, Z, D, X refer to the notional value, or face value, of the loans, and are used to determine the relative claims by creditors in the event a debtor defaults. Internal debt and assets refer to contracts between the N banks in the system. Banks and institutions that are not part of the system are deemed to be part of the exterior, and their exposures are included as part of the external debts and assets. Let $\bar{{L}}_{ij}$ denote the notional exposure of bank j to bank i, that is to say, the amount i owes j. Note the constraints that hold for all i

$$\mathrm{{Z}}_{i} =\sum \limits _{j}\bar{{L}}_{ji},\quad \mathrm{{X}}_{i} =\sum \limits _{j}\bar{{L}}_{ij},\quad \sum \limits _{i}\mathrm{{Z}}_{i} =\sum \limits _{i}\mathrm{{X}}_{i},\quad \bar{{L}}_{ii} = 0,$$

and that the matrix of exposures $\bar{L}$ is not symmetric.

1.1 A.1 Default Cascades

A healthy bank manages its books to maintain mark-to-market values with sufficient “economic capital” to provide an “equity buffer” against shocks to its balance sheet. This means that the bank maintains its asset-to-equity ratio A _i  ∕ e _i above a fixed threshold Λ _i (a typical value imposed by regulators might be 12. 5).

Following a bank-specific catastrophic event, such as the discovery of a major fraud, or a system wide event, the assets of some banks may suddenly contract by more than the equity buffer. Assets are then insufficient to cover the debts, and these banks are deemed insolvent. The assets of an insolvent bank must be quickly liquidated, and any proceeds go to pay off that bank’s creditors, in order of seniority. We now discuss three simple settlement mechanisms for how an insolvent bank i is removed from the system.

• Version A, the original mechanism of [8], supposes that external debt is always senior to internal debt. We define fractions ${\pi }_{ij} =\bar{ {L}}_{ij}/\mathrm{{X}}_{i}$. If p _i denotes the amount available to pay i’s internal debt, this amount is split amongst creditor banks in proportion to π _ij , that is bank j receives π _ij p _i . Given $\mathbf{p} = [{p}_{1},\ldots ,{p}_{N}]$, the clearing conditions are p _i  = 0 if $\mathrm{{Y}}_{i} -\mathrm{ {D}}_{i} +\sum \limits _{j}{\pi }_{ji}{p}_{j} < 0$ and ${p}_{i} =\min (\mathrm{{Y}}_{i} -\mathrm{ {D}}_{i} +\sum \limits _{j}{\pi }_{ji}{p}_{j},\mathrm{{X}}_{i})$ if $\mathrm{{Y}}_{i} -\mathrm{ {D}}_{i} +\sum \limits _{j}{\pi }_{ji}{p}_{j} \geq 0$. We can write this as

  $${p}_{i} = {F}_{i}^{(A)}(\mathbf{p}) :=\min (\mathrm{{X}}_{ i},\max (\mathrm{{Y}}_{i} +\sum \limits _{j}{\pi }_{ji}{p}_{j} -\mathrm{ {D}}_{i},0)),\quad i = 1,\ldots ,N$$

  (2.34)

• Version B supposes that external and internal debt have equal seniority. We define fractions $\tilde{{\pi }}_{ij} =\bar{ {L}}_{ij}/(\mathrm{{D}}_{i} +\mathrm{ {X}}_{i})$. If $\tilde{{p}}_{i}$ denotes the amount available to pay i’s total debt, creditor bank j receives $\tilde{{\pi }}_{ji}\tilde{{p}}_{i}$ while the external creditors receive $\mathrm{{D}}_{i}\tilde{{p}}_{i}/(\mathrm{{D}}_{i} +\mathrm{ {X}}_{i})$. The clearing conditions are:

  $$\tilde{{p}}_{i} = {F}_{i}^{(B)}(\tilde{\mathbf{p}}) :=\min (\mathrm{{D}}_{ i} +\mathrm{ {X}}_{i},\mathrm{{Y}}_{i} +\sum \limits _{j}\tilde{{\pi }}_{ji}\tilde{{p}}_{j}),\quad i = 1,\ldots ,N.$$

• Most simply, Version C supposes as in the GK model that the recovery from any insolvent bank is zero. That means the amount p _i available to pay i’s internal debt is simply

  $${p}_{i} = {F}_{i}^{(C)}(\mathbf{p}) :=\mathrm{ {X}}_{ i}\Theta (\mathrm{{Y}}_{i} -\mathrm{ {D}}_{i} +\sum \limits _{j}{\pi }_{ji}{p}_{j})$$

  where Θ denotes the Heaviside function.

Under each of these settlement mechanisms, any solution $\mathbf{p} = ({p}_{1},\ldots ,{p}_{N}) \in {\mathbb{R}}_{+}^{N}$ of the clearing conditions is called a “clearing vector”. In the subsequent discussion we consider only version A. The existence result extends easily to versions B and C by considering fixed points of the monotonic mappings ${F}^{(B)},{F}^{(C)} : {\mathbb{R}}_{+}^{N} \rightarrow {\mathbb{R}}_{+}^{N}$.

Proposition 2.1.

Consider a financial system with $\mathrm{Y} = [\mathrm{{Y}}_{1},\ldots ,\mathrm{{Y}}_{N}],\mathrm{D} = [\mathrm{{D}}_{1},\ldots ,\mathrm{{D}}_{N}]$ and matrix $\bar{L} = {(\bar{{L}}_{ij})}_{i,j=1\ldots ,N}$ . Then the mapping ${F}^{(A)} : {\mathbb{R}}_{+}^{N} \rightarrow {\mathbb{R}}_{+}^{N}$ defined by (2.34) has at least one clearing vector or fixed point p ^∗. If in addition the system is “regular” (a natural economic constraint on the system), the clearing vector is unique.

Proof.

Existence is a straightforward application of the Tarski Fixed Point Theorem to the mapping F acting on the complete lattice

$$[\mathbf{0},\bar{X}] :=\{ \mathbf{x} = [{x}_{1},\ldots ,{x}_{N}] \in {\mathbb{R}}_{+}^{N} : 0 \leq {x}_{ i} \leq \bar{ {X}}_{i},i = 1,\ldots ,N\}.$$

One simply verifies the easy monotonicity results that for any vectors mathbf0 ≤ p ≤ p ′ ≤ X one has

$$\mathbf{0} \leq {F}^{(A)}(\mathbf{0}) \leq {F}^{(A)}(\mathbf{p}) \leq {F}^{(A)}(\mathbf{p}\prime) \leq {F}^{(A)}(\mathrm{X}) \leq \mathrm{ X}$$

(where a ≤ b for vectors means a _i  ≤ b _i for all $i = 1,\ldots ,N$). For the definition of “regular” and the uniqueness result, please see [8].

1.2 A.2 Clearing Algorithm

Cascades of defaults arise when primary defaults trigger further losses to the remaining banks. The above proposition proves the existence of a unique “equilibrium” clearing vector that characterizes the end result of any such cascade. The following algorithm for version A of the settlement mechanism resolves the cascade to the fixed point p ^∗ in at most 2N iterations by constructing an increasing sequence of defaulted banks ${A}^{k} \cup {B}^{k},k = 0,1,\ldots$. Analogous (but simpler) algorithms are available for settlement mechanisms B and C.

• 1. Step 0 Determine the primary defaults by writing a disjoint union $\{1,\ldots ,N\} = {A}^{0} \cup {B}^{0} \cup {C}^{0}$ where

  $$\begin{array}{rcl} {A}^{0}& =& \{i\vert \mathrm{{Y}}_{ i} +\mathrm{ {Z}}_{i} -\mathrm{ {D}}_{i} < 0\} \\ {B}^{0}& =& \{i\vert \mathrm{{Y}}_{ i} +\mathrm{ {Z}}_{i} -\mathrm{ {D}}_{i} -\mathrm{ {X}}_{i} < 0\} \setminus {A}^{0} \\ {C}^{0}& =& \{1,\ldots ,N\} \setminus ({A}^{0} \cup {B}^{0})\end{array}$$

• 2. Step $k,\ k = 1,2,\ldots$  Solve the | B ^k − 1 | dimensional system of equations:

  $${p}_{i} =\mathrm{ {Y}}_{i} -\mathrm{ {D}}_{i} +\sum \limits _{j\in {C}^{k-1}}{\pi }_{ji}\mathrm{{X}}_{j} +\sum \limits _{j\in {B}^{k-1}}{\pi }_{ji}{p}_{j},\ i \in {B}^{k-1}$$

  and define result to be p ^k ∗. Define a new decomposition

  $$\begin{array}{rcl} {A}^{k}& =& {A}^{k-1} \cup \{ i \in {B}^{k-1}\vert {p}_{ i}^{k{_\ast}}\leq 0\} \\ {B}^{k}& =& ({B}^{k-1} \setminus {A}^{k}) \cup \{ i \in {C}^{k-1}\vert \mathrm{{Y}}_{ i} -\mathrm{ {D}}_{i} +\sum \limits _{j\in {C}^{k-1}}{\pi }_{ji}\mathrm{{X}}_{j} +\sum \limits _{j\in {B}^{k-1}}{\pi }_{ji}{p}_{j}^{k{_\ast}}\leq \mathrm{ {X}}_{ i}\} \\ {C}^{k}& =& \{1,\ldots ,N\} \setminus ({A}^{k} \cup {B}^{k}) \\ \end{array}$$

  and correspondingly

  $${ p}_{i}^{k} = \left \{\begin{array}{ll } 0 &i \in {A}^{k} \\ \mathrm{{Y}}_{i} +\sum \limits _{j\in {C}^{k}}{\pi }_{ji}\mathrm{{X}}_{j} +\sum \limits _{j\in {B}^{k}}{\pi }_{ji}{p}_{j}^{k{_\ast}}-\mathrm{ {D}}_{i}&i \in {B}^{k} \\ \mathrm{{X}}_{i} &i \in {C}^{k}.\end{array} \right.$$

  (2.35)

  If ${A}^{k} = {A}^{k-1}$ and ${B}^{k} = {B}^{k-1}$, then halt the algorithm and set ${A}^{{_\ast}} = {A}^{k},{B}^{{_\ast}} = {B}^{k},{\mathbf{p}}^{{_\ast}} ={ \mathbf{p}}^{k{_\ast}}$. Otherwise perform step k + 1.

Appendix B: Updating of Average Shock Strength for NYYA Model

Assuming a delta function distribution approximating S ⁿ(σ) as in Sect. 2.3.1, we need to count the number of loans (edges in the directed network) which link defaulted banks to solvent banks. In the notation of Sect. 2.3.2, the number of such “d-s” (for “defaulted-to-solvent”) edges in the network at timestep n is

$$N\sum \limits _{j,k}{p}_{jk}\sum \limits _{m=0}^{j}m\,{u}_{ jk}^{n}(m),$$

(2.36)

since each solvent bank with m defaulted debtors contributes m d-s edges to the total. We assume that all these d-s edges at timestep n carry an equal shock s ⁿ.

Now consider the situation at timestep n + 1. Some of the d-s edges from timestep n are still d-s edges, although others will have become d-d (“defaulted-to-defaulted”) edges. We count the number of d-s edges which remained as d-s from timestep n to timestep n + 1 as

$${A}^{\text{ old}} = N\sum \limits _{j,k}{p}_{jk}\sum \limits _{m=0}^{{M}_{jk}^{n} }m\,{u}_{jk}^{n}(m).$$

(2.37)

Note the upper limit of M _jk ⁿ for the sum over m (cf. Eq. (2.36)); this arises because the creditor banks in question remain solvent at timestep n + 1.

The other mechanism generating d-s edges at timestep n + 1 is the default of the debtor end of a timestep-n s-s (solvent-to-solvent) edge. Similar to (2.36), we can count the number of s-s edges at timestep n as

$$N\sum \limits _{j,k}{p}_{jk}\sum \limits _{m=0}^{j}(j - m){u}_{ jk}^{n}(m),$$

(2.38)

since each (solvent) (j, k)-class bank with m defaulted debtors must also have j − m solvent debtors. Each of the s-s edges at timestep n becomes an d-s edge at timestep n + 1 if (1) the debtor bank defaults during the timestep, and (2) the creditor bank remains solvent to at least timestep n + 1. Noting that (1) occurs with probability f ^n + 1 (see Eq. (??) of the main text), and that (2) requires m ≤ M _jk ⁿ, we obtain the number of new d-s edges at timestep n + 1 as

$${A}^{\text{ new}} = {f}^{n+1}N\sum \limits _{j,k}{p}_{jk}\sum \limits _{m=0}^{{M}_{jk}^{n} }(j - m){u}_{jk}^{n}(m).$$

(2.39)

The total number of d-s edges at timestep n + 1 is then A ^{{ old}} + A ^{{ new}}, while the cumulative total of the shock sizes transmitted by these edges is

$${s}^{n}{A}^{\text{ old}} +\widetilde{ s}{A}^{\text{ new}},$$

(2.40)

where $\widetilde{s}$ is the average shock on each newly-distressed loan (using (??) of the main text):

$$\widetilde{s} = \frac{\sum \limits _{j,k}k{p}_{jk}\sum \limits _{m={M}_{jk}^{n}+1}^{j}{u}_{jk}^{n}(m)\text{ min}\left (\frac{m{s}^{n}-{c}_{ jk}+{e}_{jk}\left [1-\exp (-\alpha {\rho }^{n})\right ]} {k} ,w\right )} {\sum \limits _{j,k}k{p}_{jk}\sum \limits _{m={M}_{jk}^{n}+1}^{j}{u}_{jk}^{n}(m)}.$$

(2.41)

Thus, under the simplifying assumption on the shock size distribution (S ⁿ(σ)↦δσ − s ⁿ) , we model the shocks on d-s edges at timestep n + 1 to each be of equal size s ^n + 1, where

$${s}^{n+1} = \frac{{s}^{n}{A}^{\text{ old}} +\widetilde{ s}{A}^{\text{ new}}} {{A}^{\text{ old}} + {A}^{\text{ new}}} ,$$

(2.42)

with A ^{{ old}}, A ^{{ new}}, and $\widetilde{s}$ given in terms of u _jk ⁿ by Eqs. (2.37), (2.39), and (2.41), respectively. This gives an update equation for s ⁿ in terms of known quantities from timestep n.

Appendix C: Frequency of Cascades for Single-Seed Initiation in GK Model

In this Appendix we consider the frequency of cascades in the GK model when initiated by a single seed node. Mathematically, our theory applies to the limiting case N → ∞ of a sequence of networks of size N, with ⌊ρ⁰ N⌋ seed nodes. In Monte Carlo simulations of real banking networks, the size N of the system is fixed, and the case of a single seed corresponds to a fraction ${\rho }^{0} = 1/N$ of initial defaults. The mechanism of cascade initiation in the infinite-N network may be understood as follows. As in [27], we call bank nodes vulnerable if they default due to a single defaulting loan. When the cascade condition (2.26) is satisfied, a giant connected cluster of vulnerable nodes exists in the network. The fractional size of this vulnerable cluster is denoted S _v , and it may be calculated by solving a site percolation problem for the directed network (see [20]) in a similar fashion to the calculation for undirected networks in [27]:

$${S}_{v} =\sum \limits _{jk}{p}_{jk}\left [1 - {(1 - \phi )}^{j}\right ]\Theta \left [\frac{0.2} {j} - {c}_{jk}\right ],$$

(2.43)

where ϕ is the non-zero solution of the equation

$$\phi =\sum \limits _{jk}\frac{k} {z}{p}_{jk}\left [1 - {(1 - \phi )}^{j}\right ]\Theta \left [\frac{0.2} {j} - {c}_{jk}\right ].$$

(2.44)

Here, as in [27], the Θ term plays the role of a degree-dependent site occupation probability: sites (nodes) are deemed occupied if they are vulnerable in the sense defined above, and this happens if the shock due to a single defaulting loan (0. 2 ∕ j) exceeds their net worth c _jk . In Fig. 2.7 we directly calculate the size of the largest vulnerable cluster in a single realization of an Erdös-Rényi network with N = 10⁴ nodes and mean degree z (cf. Fig. 2.4) and show that it closely matches to the analytical result (2.43).

Fig. 2.7

Sizes of vulnerable cluster (S _v ) and of extended vulnerable cluster (S _e ) as calculated directly from (for each value of mean degree z) a single Erdös-Rényi network with N = 10⁴ nodes. The vulnerable cluster size is compared with the analytical result of Eq. (2.43), while the extended vulnerable cluster is shown to closely match the frequency of global cascades in the single-seed GK model (cf. Fig. 2.4)

The extended vulnerable cluster [27], which takes up a fraction S _e of the network, consists of nodes which are debtors of at least one bank in the vulnerable cluster. If a seed node is part of the extended vulnerable cluster, it immediately causes the default of its creditor in the vulnerable cluster, which in turn leads to default of other nodes in the vulnerable cluster, and so on until the entire vulnerable cluster is in default. Nodes outside the vulnerable cluster (i.e. banks which can withstand the default of a single asset loan) may also be defaulted later on in this cascade as the percentage of defaulted banks increases; the result is a global cascade of expected size ρ^∞, given by Eq. (2.15). On the other hand, if no seed node is part of the extended vulnerable cluster, then no further defaults will occur and the cascade immediately terminates. Thus, if only a single seed node is used in each realization, we expect cascades of size ρ^∞ to occur in a fraction S _e of realizations (corresponding to cases where the seed node lies in the extended vulnerable cluster), and no cascades to occur in the remaining fraction 1 − S _e of realizations. The size S _e of the extended vulnerable cluster thus determines the frequency of global cascades among the set of single-seed realizations. The size of S _e was calculated analytically in [13] for the undirected networks case, but the corresponding derivation for directed networks is non-trivial. Instead, we directly calculate the size of the largest extended vulnerable cluster in the network, and show in Fig. 2.7 that it corresponds very closely to the frequency of global cascades in the large ensemble of Monte Carlo simulations of Fig. 2.4 in the main text.

As argued in [13], the frequency of cascades increases with the number ⌊ρ⁰ N⌋ of seed nodes used as

$$1 -{\left (1 - {S}_{e}\right )}^{\lfloor {\rho }^{0}N\rfloor },$$

(2.45)

which reduces to S _e for the single-seed case (${\rho }^{0} = 1/N$) and to 1 for the case where ρ⁰ remains a finite fraction as N → ∞. The frequency of cascades (of size ρ^∞) in the GK model initiated by a single default is thus S _e , whereas if multiple seeds (say, 10 initial defaults among 1,000 banks) are used we find that almost all cascades are of size ρ^∞.