Abstract
This contribution focuses on the methodology applied in papers that investigate the dynamics of contagion in financial networks using numerical simulations. In these papers, a propagation of losses and defaults in a financial system is modeled as a direct balance-sheet contagion (a.k.a. counterparty contagion), that is the direct transmission of losses from financially distressed debtors to their creditors. The researchers in this field perform their simulations with three different methods: (i) basic linear threshold algorithms, (ii) the graph-theoretic approach, where contagion is modeled as a propagation process in directed and weighted graphs, (iii) the lattice-theoretic approach, where contagion is modeled as a ‘fictitious default algorithm’, that computes the vector of payments that clears a net of financial obligations. Some of the results obtained by this stream of literature raised doubts about the assumptions used in such simulations. We discuss this issue and present some methodological recommendations that may improve the realism and the generality achievable in numerical investigations of financial contagion.
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Notes
- 1.
A sharp increase in the sales of an illiquid asset (e.g., a share or a long term bond) may push its price below the fundamental (true) value of the asset. This phenomenon is known as ‘liquidity pricing’. During a crisis, financial intermediaries can be forced to sell assets in response to liquidity shortages and/or excessive leverage (the so called ‘fire sales’), facing losses due to such liquidity pricing.
- 2.
Recessions can cause losses in the value of the assets held by banks, losses capable of rendering them insolvent. If depositors foresee the recession, they will protect themselves from possible bank defaults by withdrawing their deposits and, in so doing, they create the conditions for the occurrence of a widespread crisis.
- 3.
See the review by Upper [34] and the papers cited therein.
- 4.
This network model can be easily generalised by adding different liabilities with different seniority, as it is done in [1].
- 5.
See [34].
- 6.
Some authors assume that this is done without incurring bankruptcy costs, while other consider the presence of fixed or porportional liquidation costs and, finally, several authors simplify their analysis assuming that for each failing agent \(b_{i}=1,\) i.e. assume that creditors get no refund at all.
- 7.
This terminal condition can also be expressed in terms of defaulting nodes, as it is done in the linear threshold algorithm depicted above, because the condition \(\sum _{\Omega }\upbeta _{i}^{t}e_{i}+\sum _{\Omega }b_{i}^{t}h_{i}=\sum _{A}b_{k}a_{k}\) implies that no more nodes default, and vice versa.
- 8.
- 9.
See [3], for the definition of a multisource flow network and for its properties.
- 10.
A lattice is a system \(\left\langle X,\le \right\rangle \) composed of a non-empty set \(X\) and a binary relation \(\le \), where the latter induces a partial ordering on the elements of \(X\) and, for any two elements \(x,y\in X,\) there exists a least upper bound (a.k.a. join or supremum) \(x\vee y\) and a greatest lower bound (a.k.a. meet or infimum) \(x\wedge y\).
- 11.
The following description of the lattice-theoretic approach is inspired to [10], which is one of the papers that applies the Eisenberg and Noe model.
- 12.
See [34].
- 13.
For instance [30], assume that customer deposits are senior to interbank liabilities, while there is no evidence in support of such a restriction.
- 14.
See [34].
- 15.
Most of the above cited authors perform numerical simulations of the effects of the default of a single bank. Again, see [34].
- 16.
As opposed to historical value.
- 17.
More precisely, feedbacks of losses, in a default contagion process, take place in strongly connected components of a graph \(N=(\Omega ,D)\) defined below, if all member of the component default.
- 18.
See below Sect. 13.4.3.
- 19.
Several studies agree on the observation of actual interbank lending networks formed by banks that consist of a core of highly connected banks, while the remaining peripheral banks are connect only to the core banks. [33] and [7] note this feature for the US commercial banks network. [8, 12, 26], respectively, find similar structures in banking networks of the UK, Canada, Japan, Germany and Austria.
- 20.
See [32] for a model where fire sales and ‘marking-to-market’ of assets amplify the effects of contagion.
- 21.
Or of its neighbours, if the network is undirected.
- 22.
More precisely, the factorisation of the probability of the states of a node enables the analytic treatment of probabilities in networks, as it is done with Bayesian networks. To derive closed form solutions for such probabilities, it is necessary to impose further restrictions. In most of the above cited papers, the probability of default of a node depends solely on the number of defaulted debtors. In these cases the probability of default of the parent nodes can modeled as a binomial and, with this restriction, the probability of cascades is characterised in closed form. See, inter alia [6].
- 23.
For example, if the states of node \(x\) and node \(y\) are not independent of one another because they share a common ancestor, node \(z\), then \(x\) and \(y\) are conditionally independent on one another given the state of node \(z\). If, conversely, two or more of the parent nodes in \(V_{i}\) belong to a directed cycle (or, more generally, to a strongly connected component of \(N\)), then the required independence condition cannot be satisfied, not even conditionally to the state of common ancestors.
- 24.
[25] exploit this property in their double (illiquidity and insolvency) cascade model.
- 25.
- 26.
Even the authors themselves appear surprised by their own results: “...numerical studies are in reassuring, some might say surprising, agreement with the results obtained from the analytic approximations...” [28]; “Extensive cross testing with Monte Carlo estimates shows that this approximate analysis performs surprisingly well.” [22].
- 27.
As argued by [35], numerical results in random graph models approximate analytical solutions as \(n\) gets close to 10,000.
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Eboli, M. (2015). Simulations of Financial Contagion in Interbank Networks: Some Methodological Issues. In: Król, D., Fay, D., Gabryś, B. (eds) Propagation Phenomena in Real World Networks. Intelligent Systems Reference Library, vol 85. Springer, Cham. https://doi.org/10.1007/978-3-319-15916-4_13
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