Network Models and Financial Stability Amadeo Alentorn Erlend Nier Jing Yang

Greenspan’s open letter… May 26

Financial System and Real Economy Savings Investment

Research questions How the generic structure of banking system affect resilience to systemic failure. How the resilience of the inter-bank network to shocks relates to the following key parameters of the system: the capacity of banks to absorb shocks the size of inter-bank exposures the degree of connectivity the degree of concentration of the banking sector.

Network approach Existing economics theory: Allen and Gale (2000) Studied two types of network: a ‘complete structure’ and an ‘incomplete structure’. Nodes in a network represent banks and links represent financial obligations between banks. Results: depends on the pattern of interconnectedness: 1. In a complete structure, the initial impact of financial stress may be attenuated 2. An incomplete structure is more prone to contagion While the study by Allen and Gale (2000) provides valuable insights into stability of inter-bank markets, their model has only four banks and both the network structures employed and the financial structure of banks are too simplistic to be sure that the intuitions generated generalise to real world financial systems.

1. power-law degree distribution Real world networks : 1. power-law degree distribution 2. clustering 3. small degree of separation: small world phenomenon Albert (2000), the highly skewed degree distributions of real world networks have substantial implications for the robustness of networks to the removal of specific nodes. Financial networks fit in this category since financial institutions (FI) typically do not play equal roles in the networks. Some banks are more important than others with respect to extending credit or channelling payments for other banks; Boss, Summer and Thurner (2004) showed that the Austrian interbank network exhibits clustering, where many banks would first link with a particular bigger bank and a few big ‘central’ banks are linked among each other. Similar patterns can be found in many payment, clearing and settlement networks as well, eg ‘tiering’ in UK payment systems. Summer and Thurner (2004) show that the Austrian interbank market can be characterised by a power–law degree distribution with a relatively small power exponent compared to other real world networks.

Empirical studies Empirical research on the importance of interbank linkages as a channel of contagion: Sheldon and Maurer (1998) for Switzerland Furfine (1999) for the US Upper and Worms (2000) for Germany Wells (2002) for the UK Boss, Elsinger, Summer and Thurner (2003) for Austria. Limitation: no generic relationship between stability of a financial system and features of the network. However, their results are invariably driven by the particulars of the banking system under study and cannot therefore provide easily generalisable insights into the drivers of systemic risk. In particular, these studies are silent as regards the influence of key parameters, such as net worth, interbank exposures, connectivity, concentration and asymmetry for the resilience of the banking system.

The Eboli (2004) Framework Network is a directional graph, where links represent exposures. Each of N nodes (banks) is connected to a source (ie source of shock/loss). Each of the N nodes is assigned a sink (representing net worth). Flow network: losses flow across a network of banks When losses reach a bank, they are absorbed by the sink, or flow further through inter-bank links.

Extending the Eboli (2004) framework Identify source with banks’ external assets Introduce depositors as the second sink deposits are senior to interbank, in turn, senior to net worth Introduce a probability law describing likelihood of interbank link between any two nodes (banks) symmetric structures (random graph a la Erdos and Reiny) or asymmetric structures (eg power law).

Constructing the inter-bank market: Fix the total external assets of the banking system (investor’s borrowing) Decide on the number of nodes (banks) Fix average size of interbank assets in percent of external assets Decide on probability that any two banks are connected (random graph, Erdos-Reiny probability, p) and simulate this. Allocate interbank exposures (equal sized) Allocate net worth as a proportion of total assets Allocate deposits and external assets in a way that preserves balance sheet identity for each bank.

Demonstration Construction of a banking system Construction of individual bank’s balance sheet Shock propagation Experiments

Simulation results

Experiments Four parameters: Number of nodes (N) Erdos-Reyni probability (p) Percentage of Interbank assets (w) Net worth (c) In each of the following experiments, we vary one parameter at a time; In each experiment, we shock one bank at a time to study the default dynamics, then take average across all banks. A first step in the experiment is thus to generate one such realisation. As a second step, for each realisation of the network, we shock one bank at a time (that is, we consider idiosyncratic shocks) and in this way successively shock each bank in the system. For each bank that is shocked, we count the total number of defaults arising until the shock has been completely absorbed. We then take the average number of defaults arising across all banks for each realisation of the system. For each set of parameters (c, w, p, N) we repeat this exercise for 100 draws of the network.

How does bank capitalisation affect contagion? weakly monotonic and negative relationship between bank capitalisation and contagion contagion does not decrease linearly in bank capitalisation for low values of net worth, a slight decrease in net worth leads to a sharp increase in number of failures Figure 5 also shows the role of the size of interbank exposure for shock transmission. In particular, it suggests that a larger proportion of interbank exposures tends to lead to a less stable banking system, as can be seen in the shift from the pink region to the blue region. For example, at 5% net worth (blue graph), a system with 30% interbank exposure renders five banks defaulting one average, compared to one default for a system with only 20% of interbank assets (pink graph). Moreover, under-capitalised banks impose a negative externality for the rest of the system by increasing the likelihood of contagious defaults further down the chain.

A multiple rounds of default scenario below illustrates this further, by showing more detail on one experiment with 1% net worth. One can see that a shock to the node on the upper left corner caused 9 banks to default although the node only has direct links with 4 banks.

How does the size of interbank exposure affect contagion? A decrease in the percentage of external assets has two opposing effects: shock propagation and shock absorption When the net worth is still sufficiently small, the shock propagation dominates the shock absorption. Nonlinear relationship: for different levels of percentage of external assets, a banking system can achieve same level of resilience. In this exercise, we examine further the effect of the size of interbank lending/borrowing on the number of failures. We hold the absolute size of external assets constant, while increasing total assets by reducing the percentage of external assets in total assets at a system level. Therefore, in this experiment, a decrease in percentage of external assets implies an increase in interbank assets and also, by construction an increase in total assets. Also, given the number of links is fixed and determined by the Erdös Rényi probability, an increase in interbank assets implies an increase in the size of each interbank lending, that is in the weight (w) of the link. As we observe, for very high levels of external assets, there is no contagion effect as the share of interbank asset is small enough to be absorbed by its creditors’ net worth. When the percentage of external asset decreases, increased interbank assets raise the likelihood of contagion. When the net worth is still sufficiently small, the shock propagation effect dominates the shock absorption effect. We observe default increases quickly from one to five. However, after a certain threshold, for example, when external asset is less than 80%, the contagion decreases when percentage of external assets decreases. The reason lies in the cushion effect provided by enlarged net worth. Since net worth is 5% of total assets, when the percentage of external asset decreases, the total assets increase and net worth increases as well. This provide banks more of a cushion when facing an external shock, which means that less of the shock is transmitted through interbank exposures. In a certain region the shock absorption effect dominates the shock propagation effect.

How does likelihood of interbank exposure affect contagion? Interbank connections have two opposing effects: shock-transmitter and shock-absorber. two mechanisms dominate over different ranges, generating an M-shaped graph. an interesting dynamic between interbank linkages and net worth In this experiment, we investigate the effect of connectivity on the resilience of the banking system. We also check how the relationship changes for different levels of net worth. The results from this experiment thus give us a first helpful insight into the interplay between connectivity and net worth. On the horizontal axis, we have the Erdös-Rényi probability p, where, as p increases, on average banks become more connected. The blue line and the blue range represent networks with net worth equal to 1% of total assets, whereas, the red and yellow ones represent networks with net worth equal to 3% and 7% of total assets, respectively. In Figure 8, we see that these two mechanisms dominate over different ranges, generating an M-shaped graph. First, for very low levels of connectivity (p close to zero), an increase in connectivity reduces system resilience, since connectivity increases the chance of shock transmission. For higher levels of connectivity, increases in connectivity may decrease of increase system resilience. But when connectivity is sufficiently high, further increases in connectivity unambiguously decrease contagion as the shock absorbtion effect comes to dominate and the initial shock is spread over more and more banks, each able to withstand the shock received. we can conclude that under-capitalised banking systems are fragile and even more so when connectivity is high; Well-capitalised banking systems are resilient to shocks, even more so when connectivity is high.

How does banking concentration affect contagion contagion risk monotonically increases in size of the shock regardless of concentration level. The more concentrated is the banking system, a higher risk of contagion for same shock. In this exercise, we investigate the effect of concentration of the banking system on its resilience to contagion resulting from interbank linkages. To do so, we vary the number of banks (N) in the banking sector from 10 to 25. As in the previous exercises we keeping constant the aggregate size of the banks’ external assets. The banking system with 10 (25) banks therefore is the most (least) concentrated, for a given size of the economy. In simulations, the shock applied wipes out a certain percentage of a bank’s external asset. However, one may ask to whether this positive relationship between concentration and the fraction of defaults is driven merely by the fact that the absolute size of the shock applied also increases when concentration increases? The answer is that while this is a factor, it is not the only one driving the result. The other is that when the system is more concentrated, interbank connections have an enhanced capacity to lead to a meltdown of the entire system.

Interdependence of the parameters Recap: up to now we have shown the

Inter-dependence of net worth and connectivity First, the figure confirms the ‘M’ shape nonlinear relationship between connectivity and contagion which embodies the two countervailing effects of shock transmission and shock diversification, described in the previous section. The experiments on connectivity show that there is a threshold value for the connectivity in a banking system under which the contagion effect can be completely eliminated. This experiment demonstrates how the threshold value changes with the level of net worth. In particular, the cut-off value for zero contagion effects is a negative and non-linear function of net worth: the higher net worth, the lower level of connectivity is required to eliminate contagion.

Heterogeneous networks Nonlinearity: as the degree of centrality increases, contagious defaults first increase, but then start to decrease, as the number of connections to the central node start to lead to greater dissipation of the shock. As we increase l, the clustering coefficient of the network also increases, and the number of defaults increases until l reaches around 42%, where the number off defaults peaks at 10. From then on, the number of defaults start to decrease, because as the large bank becomes more connected, the shock received by the large bank is distributed among more of the smaller banks, and therefore, the individual shocks that small banks receive is smaller and can be absorbed by the small banks’ net worth.

Summary Under-capitalised banks impose an externality on other banks in the system. Decreases in net worth increase the number of contagious defaults and that this effect is non-linear. Contagion risk first increases with the connectivity of the banking system, then decreases. More concentrated banking systems tend to be more prone to systemic meltdown ‘too systemic to fail’ ? This paper analysed how systemic risk depends on the structure of the banking system. We applied network theory to construct banking systems and then analysed the resilience of the system to contagious defaults, for both ex ante homogenous structures and for ex ante heterogeneous structures. We analysed the degree of system resilience with respect to each of these parameters and also checked how the parameters might interact to produce a smaller or larger degree of resilience. A number of results emerged that are interesting from the point of central banks charged with the analysis and mitigation of risks to the system. In concentrated systems, the failure of a large bank has adverse effects, not merely due to its size - that may render the bank ‘too big to fail’-, but also because the failure is more likely to affect a large fraction of the other banks in the system. In concentrated systems each bank may be thus be ‘too systemic to fail’.