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.
2.1 Introduction
The study of contagion in financial systems is currently very topical. “Contagion” refers to the spread of defaults through a system of financial institutions, with each successive default causing increasing pressure on the remaining components of the system. The term “systemic risk” refers to the contagion-induced threat to the financial system as a whole, due to the default of one (or more) of its component institutions, and it has become a familiar term since the failure of Lehman Brothers and the rescue of AIG in the autumn of 2008.
Interbank (IB) networks in the real world are financial systems that range in size from dozens to thousands of institutions [6, 26, 28]. An IB network may be modelled as a (directed) graph; the nodes or vertices of the network are individual banks, while the links or edges of the network are the loans from one bank to another. Such systems are vulnerable to contagion effects, and the importance of studying these complex networks has been highlighted by Andrew Haldane, Executive director of Financial Stability at the Bank of England in his speech [15], in which he posed the following challenge: ‘Can network structure be altered to improve network robustness? Answering that question is a mighty task for the current generation of policymakers’.
The study of complex networks has advanced rapidly in the last decade or so, with large-scale empirical datasets becoming readily available for a variety of social, technological, and biological networks (see [19, 23, 24] for reviews). By virtue of their size and complexity, such networks are amenable to statistical descriptions of their characteristics. The degree distributionp_{k} of a network, for example, gives the probability that a randomly-chosen node of the network has degree k, i.e., that it is connected by k edges to neighbours in the network. While classical random graph models of networks [10] have Poisson degree distributions, many empirical networks have been found to possess “fat-tailed” or “scale-free” degree distributions, where the probability of finding nodes of degree k decays as a power law in k (p_{k} ∝ k^{ − β}) for large k, in contrast to the exponential decay with k of the Poisson distribution [23].
This structural (topological) aspect of real-world networks has important implications for dynamical systems which run on the nodes of the network graph, see Barrat et al. [3] for a review. For example, the rate of disease spread on networks depends crucially on whether or not they have fat-tailed degree distributions. As a consequence, there is considerable interest in the effect of network structure on a range of dynamics. Cascade-type dynamics occur whenever the switching of a node into a certain state increases the probability of its neighbours making the same switch. Examples include cascading failures in power-grid infrastructure [22], congestion failure in communications networks [21], the spread of fads on social networks [27], and bootstrap percolation problems [5], among others [17]. Building on earlier work on the random field Ising model of statistical physics [7], the expected size of cascades has recently been determined analytically for a range of cascade dynamics and (undirected) network topologies [13, 14]. Our goal in this paper is to extend and develop these methods for application to default contagion on (directed) interbank networks.
Although the importance of network topologies has been recognized for many years in the finance and economics literature (e.g., [1]), it is only following the publication of empirical studies for large-scale interbank networks [6, 9, 26, 28] that theoretical models have moved beyond small networks and simple topologies. In this paper we focus on “deliberately simplified” models for contagion on interbank networks exemplified by those of Gai and Kapadia [11] (“GK” for short) and of Nier et al. [25] (“NYYA” for short), which have attracted significant recent attention [16, 18]. We develop an analytical approach to calculating the expected size of contagion events in networks of a prescribed topology.
The calculation is “semi-”analytical because it requires the iteration of a nonlinear map to its fixed point, but it nevertheless offers significantly faster calculation than Monte Carlo simulation. This reduces the computational burden of interbank network simulations, hence making network theory more useful for practical applications. Moreover, the analytical approach gives insights into the mechanisms of contagion transmission in a given network topology, and enables formulas relating critical parameter values to be derived.
Our work extends the seminal paper of May and Arinaminpathy [18] by moving beyond their assumption that every bank in the network is identical (i.e., that all banks have the same numbers of debtors and creditors). As shown by May and Arinaminpathy, this “mean-field” assumption gives reasonably accurate results for Erdös-Rényi random networks, which have independent Poisson distributions for in- and out-degrees. This means that each bank in such a network is similar to the “average” bank. However, real-world banking networks often have fat-tailed degree distributions [6], meaning that there is a significant probability of finding a bank with in-degree (or out-degree) very different to the mean degree. To analyze contagion on such networks we need to move beyond the mean-field assumption. Moreover, unlike May and Arinaminpathy, our formalism allows us to consider how the extent of the contagion is affected by the size of the bank which initiates the cascade, and so to inform the question of which banks are ‘too big to fail’.
The remainder of this paper is structured as follows. In Sect. 2.2 we review the models of GK and NYYA. Sections 2.3 and 2.4 develop a general theoretical framework for analyzing such models, while in Sect. 2.5 we compare the results of our analytical approach with full Monte-Carlo simulations, and discuss conclusions in Sect. 2.6. Three appendices give details of several results that are not crucial to the main flow of the paper.
2.2 Models of Contagion in Banking Networks
In the second step, each node (bank) of the skeleton structure is endowed with a balance sheet and the edges between banks are weighted with loan magnitudes. This process is performed in such as way as to ensure the banking system so represented is fully in equilibrium (i.e., assets exceed liabilities for each bank) in the absence of exogenous shocks. Once the banking networks are generated, the cascade dynamics can be implemented to examine the effects of various types of shocks. In Monte Carlo implementations, each step of the process (skeleton structure/balance sheets/dynamics) is repeated many times to simulate the ensemble of possible systems. The most common output from such simulations is the expected fraction of defaulted banks in steady-state (i.e., when all cascades have run their course) for the prescribed p_{jk} network topology.
We stress that this two-step procedure is only one of many possible alternatives for generating an ensemble of random networks. However, it is easily explained and reproducible by other researchers, and proves amenable to analysis. As a “deliberately oversimplified” model of the true complexities of banking networks, it is not suitable for calibration to real network data in its current form, but may nevertheless provide a starting point for improving our understanding of the interplay between network topology and default contagion cascades.
2.2.1 Generating Model Networks
We first discuss the creation of the skeleton structure for N banks (or nodes) consistent with a prescribed p_{jk} distribution. It is usually assumed that N is large (indeed theoretical results are proven only in the N → ∞ limit), but in practice values of N as low as 25 have been successfully examined (see Results section). In each realization, N pairs of (j, k) variables are drawn from the p_{jk} distribution. For each pair (j, k), a node is created with j in-edge stubs and k out-edge stubs. Then a randomly-chosen out-stub is connected to a randomly-chosen in-stub to create a directed edge of the network. This process is continued until all stubs are connected. Note it is possible under this process for multiple edges to exist between a given pair of nodes, or for a node to be linked to itself, but both these likelihoods become negligibly small (proportional to 1 ∕ N) as N → ∞. Note also that interbank positions are not netted, so directed edges may exist in both directions between any two nodes of the banking network.
GK | NYYA | ||
---|---|---|---|
Total assets of a (j, k)-class bank | a_{jk} = 1 | \({a}_{jk} =\tilde{ e} + w\,\text{ max}(j,k)\) | |
Net worth of a (j, k)-class bank | c_{jk} = γa_{jk} | c_{jk} = γa_{jk} | |
Size of asset loans of (j, k)-class bank | \(\frac{0.2} {j}\) | w | |
External assets of (j, k)-class bank | e_{jk} = 0. 8 | \({e}_{jk} =\tilde{ e} + w\,\text{ max}(0,k - j)\) |
2.2.2 Contagion Mechanisms
Having generated the banking system via the network skeleton structure and balance sheet allocations, the dynamics of cascading defaults can then be investigated. Recall that the banks’ balance sheet have been set up so that the system is initially in equilibrium, i.e., total assets for each bank equals its total liabilities plus its net worth. The effect of an exogenous shock is simulated, typically by setting to zero the external assets of one (or more) banks. The shocked bank(s) may be chosen randomly from all banks in the simulation, or a specific (j, k)-class may be targeted—the latter case allows us to investigate the impact of the size of the initially shocked bank upon the final cascade size (see Results section). The initial exogenous shock is intended to model, for example, a sudden decrease in the market value of the external assets held by the bank. The decrease may lead to a situation where the total liabilities of the bank now exceed the total assets: in this case, the bank is deemed to be in default As a consequence, the bank will be unable to repay its creditors the full values of their loans; the loans from these creditors to the defaulted bank are termed “distressed”. The creditors (in network terminology, the out-neighbors of the original “seed” bank) experience a shock to their balance sheets at the next timestep due to writing-down the value of the distressed loans. If at any time the total of the shocks received by a bank (i.e. the total losses to date on its loan portfolio) exceeds the net worth of the bank, then its liabilities exceed its assets, and it is deemed to be in default. The defaulted bank then passes shocks to its creditors in the system, and so the cascade or contagion may spread through the banking network. Timesteps are modelled as being discrete, with possibly many banks defaulting simultaneously in each timestep, and with the shocks transmitted to their creditors taking effect in the following timestep.
The mechanism of shock transmission is treated differently by GK and by NYYA, and this is an important distinction between the models.
2.2.2.1 Shock Transmission in the GK Model
In the GK model, defaulted banks do not repay any portion of their outstanding interbank debts because the timescale for any recovery on these defaulted loans is assumed to exceed the timescale of the contagion spread in the system. Consequently, all creditors of a bank which defaulted in timestep n receive, at timestep n + 1, a shock of magnitude equal to the total size of their loan to the defaulted bank. If multiple banks defaulted at timestep n, then a bank which is a creditor of several of these will receive multiple shocks at timestep n + 1. Specifically, if the creditor bank is in the (j, k) class, then it receives a total shock of size 0. 2μ ∕ j, where μ is the number of its asset loans which defaulted at timestep n (since each loan is of size 0. 2 ∕ j, see Table 2.1). This process of shock transmission continues until there are no new defaults, at which point the cascade terminates.
2.2.2.2 Shock Transmission in the NYYA Model
2.2.3 Liquidity Risk
2.2.4 Monte Carlo Simulations
The steps needed to study the models using Monte Carlo simulation are now clear. In each realization a skeleton structure for a network of N nodes with joint in- and out-degree distribution p_{jk} is first constructed. Then balance sheets are assigned to each node, consistent with the specific model chosen (see Table 2.1). The cascade of defaults initiated by an exogenous shock to one (or more) banks proceeds on a timestep-by-timestep basis, following the dynamics of either the zero recovery (GK) or non-zero recovery (NYYA) prescription for shock transmission. When no further defaults occur, the fraction of defaulted banks (the “cascade size”) is recorded, and then another realization may begin. When a sufficiently large number of realizations are recorded, the average cascade size (and potentially further statistics, i.e., the variance, of the cascade size) may be calculated in a reproducible (up to statistical scatter) manner. Monte Carlo simulations of this type were implemented in GK and NYYA; our focus in the remainder of this paper is on analytical approaches to predicting the average size of cascades, and so avoiding the need for computationally expensive numerical simulations.
2.3 Theory
In this section we derive analytical equations which allow us to calculate the expected fraction of defaults in a banking network with a given topology (defined by p_{jk}). Like related approaches for cascades on undirected networks [13, 14], the method is only approximate for finite-sized networks because it assumes the N → ∞ limit of infinite system size. However, in practice we find it nevertheless gives reasonably accurate results for networks as small as N = 25 banks (see Sect. 2.5).
2.3.1 Thresholds for Default
We begin by defining the threshold level M_{jk}^{n} as the maximum number m of distressed loans that can be sustained by a (j, k)-class bank at timestep n without the bank defaulting at timestep n + 1. If a (j, k)-class bank has m defaulted debtors, with m > M_{jk}^{n}, then it will default in the subsequent timestep, otherwise it will remain solvent. As we show below, the GK model is easily expressed in terms of thresholds, but thresholds can be defined for the NYYA model only under an approximating assumption.
Here e_{jk} is the value of external assets for (j, k)-class banks, α is the liquidity risk parameter introduced in Sect. 2.2 and we constrain M_{jk}^{n} to be between − 1 and j. Note that this expression for M_{jk}^{n} is constant over time n if α = 0, and is decreasing in time if α is positive and ρ^{n} is increasing.
In the NYYA model the size of the write-down shock on a newly-distressed loan depends on how large the shock received by the debtor bank was compared to its net worth. This means that there will, in general, be a distribution of shocks of various sizes in the system, and this distribution will change in time. Denoting the distribution of shock sizes by S^{n}(σ)—so that at timestep n a randomly-chosen distressed loan (i.e. an out-edge of a defaulted bank node) carries a shock of size σ with probability S^{n}(σ)—we would require m-fold convolutions of S^{n}(σ) to correctly describe the shock received by a bank with m distressed asset loans (as the sum of m independent draws of shock values from S^{n}(σ)). It is clearly desirable to find a simple parametrization of S^{n}(σ) to make the model computationally tractable, even at the loss of some accuracy. With this in mind, we approximate the true value of the shock received by a bank with m distressed loans at timestep n by ms^{n}, where s^{n} is the average shock on all distressed loans in the system at that timestep. Effectively we are replacing the true distribution S(σ) of shock sizes by a delta function distribution: S^{n}(σ)↦δσ − s^{n}, where s^{n} is the average shock s^{n} = ∫σS^{n}(σ)dσ; in other words, every distressed loan at timestep n is assumed to have equal recovery value w − s^{n}. This approximation turns out to work rather well because in cases where many debtors are in default, the total shock received by a creditor is well approximated by m times the average shock. However we will also show examples (in the Results section) where the approximation of the shock distribution S^{n}(σ) by a delta function leads to less accurate results.
2.3.2 General Theory
For the NYYA model, we use the mean-shock-size approximation discussed in Sect. 2.3.1, so the thresholds M_{jk}^{n} are given by Eq. (2.8). Then the iteration equation for s^{n} (see Appendix B), along with Eqs. (2.12) and (2.13), gives us a system of equations for u_{jk}^{n + 1}(m), f^{n + 1}, and s^{n + 1} in terms of the values of these quantities at the previous timstep. Results for both models are compared with Monte Carlo simulations in Sect. 2.5.
2.4 Simplified Theory
In this section we show that the iteration of the system defined by Eqs. (2.12) and (2.13) in order to obtain the expected fraction of defaulted banks (as given by Eq. (2.14)) may be dramatically simplified in certain cases. A sufficient condition for this simplified theory to exactly match the full theory of Eqs. (2.12) and (2.13) is:
Condition 1.
For every (j, k) class with p_{jk} > 0, the threshold level M_{jk}^{n} is a non-increasing function of n.
This condition holds if the threshold levels for each (j, k) class are constant, or decreasing with time, as in the GK model. For the NYYA model, cases where the shock size decreases over time may have thresholds M_{jk}^{n} which increase with n, and so this model does not satisfy Condition 1.
2.4.1 Simplified Theory for GK
The expected size of global cascades in a given GK-model network has essentially been reduced to solving the single Eq. (2.16), since ρ^{n + 1} can be immediately determined by substituting g^{n} into (2.15). Equation (2.16) is of the form \({g}^{n+1} = J\left ({g}^{n}\right )\), and the function J( ⋅) is non-decreasing on [0, 1]. It follows that g^{n + 1} ≥ g^{n} for all n, and iteration of the map leads to the solution g^{∞} of the fixed-point equation g^{∞} = Jg^{∞}. The corresponding steady-state fraction of defaulted banks is determined by substituting g^{∞} for g^{n} in (2.15).
Equations of this sort, giving the expected size of cascades on directed networks, have been previously derived in various contexts [2, 12]. In Gleeson [12], the main focus is on percolation-type phenomena (see also the undirected networks case Gleeson [13]), while Amini et al. [2] consider more complicated dynamics but take the limit ρ^{0} → 0. The general case (2.24) and (2.25) where initial default fractions can be different for each (j, k) class has not, to our knowledge, been considered previously, even in Monte Carlo simulations.
2.4.2 Frequency of Contagion Events
The simplified Eqs. (2.15) and (2.16), and indeed the more general method of Sect. 2.3, allow the specification of a fraction ρ^{0} (or ρ_{jk}^{0} in the case of targeted (j, k) classes) of initially defaulted bank nodes. This fraction need not be small, and this feature allows us to investigate features of systemic risk due to simultaneous failure of more than one bank (see Results section). However, most work to date has focussed exclusively on the case where a single initially defaulted bank leads to a cascade of defaults through the network. Because our theory assumes an infinitely large network, some special attention must be paid to the case of a single “seed” default in the GK model. As we show in Appendix C, in this model the locality of the seed node determines whether, in a given realization, a cascade will reach global size, or remain restricted to a small neighborhood of the seed. The distribution of cascade sizes observed in single-seed GK simulations is thus typically bimodal: only a certain fraction (termed the frequency) of cascades reach a network-spanning size, the remainder remain small (typically only a few nodes). The average extent (i.e. size) of the global cascades is given by Eqs. (2.15) and (2.16), whereas the frequency of cascades which escape the neighborhood of the seed may be expressed in terms of the size of connected components for a suitable percolation problem, see Appendix C and the Results section. The NYYA model does not exhibit this sensitivity to the details of the neighborhood of the seed node(s), so its distribution of cascade sizes is quite narrowly centered on the mean cascade size given by theory; the same comment applies to the GK model with multiple seed nodes.
2.5 Results
2.5.1 GK Model
2.5.2 NYYA Benchmark Case
Figure 2.5a examines the benchmark case of NYYA; note our Monte Carlo simulation results match those presented in Chart 1 of Nier et al. [25]. The fraction of defaults (extent of contagion) is here plotted as a function of the percentage net worth parameter γ, as defined in Eq. (2.1). The network structure is again Erdös-Rényi, with p_{jk} given by (2.27), and mean degree z = 5. We also show Monte Carlo results for the default fraction resulting from the clearing vector algorithm of Eisenberg and Noe (see Appendix A). This algorithm gives results which are qualitatively similar in behavior (though not identical) to those generated by the NYYA shock transmission dynamics described in Eq. (2.6). As in the NYYA paper, our Monte Carlo simulations use N = 25 nodes (banks) in each realization, and cascades are initiated by a single randomly-chosen bank being defaulted by an exogenous shock. Despite this relatively small value of N, we find very good agreement between the theoretical prediction (which assumes the N → ∞ limit) from Eqs. (2.12) and (2.13), and the Monte Carlo simulation results. The theory also enables us to examine the case where multiple banks are defaulted to begin the cascade. We demonstrate this by also showing numerical results for a larger Erdös-Rényi network of N = 250 nodes, with the same mean degree z = 5. In order to match the seed fraction of defaults, cascades in the larger networks are initiated by simultaneously shocking ten randomly-chosen banks (each shock being calibrated to wipe out the external assets of the bank), so \({\rho }^{0} = 1/25 = 0.04\). The numerical results for this case are almost indistinguishable from the N = 25 case, and both cases match very well to the theory curve.
In Fig. 2.5b we increase the liquidity risk parameter from α = 0 (as in Fig. 2.5a) to α = 0. 05 and α = 0. 1. For clarity, the results of the Eisenberg-Noe dynamics are not shown here, but as in Fig. 2.5a, they are qualitatively similar to the simulation results using the NYYA shock transmission dynamics. The theory predicts a discontinuous transition in ρ at γ values between 2 and 3 % for the α = 0. 05 and α = 0. 1 cases, but this is not well reproduced in Monte Carlo simulations with N = 25 nodes and \({\rho }^{0} = 1/N\) (triangles). However, this is due to finite-N effects (i.e., due to having a finite-sized network whereas theory assumes the N → ∞ limit), as can be seen by the much closer agreement between the theory and the N = 250 (with ten seed defaults) case (filled circles) for α = 0. 05.
A more serious discrepancy between theory and numerics can be seen in the γ range 4–5 %. Here the theory underpredicts the cascade size, and the difference is unaffected by increasing the size of the network. Detailed analysis of this case reveals that the root of the discrepancy is in fact the simplifying assumption made for the shock size distribution S^{n}(σ) in the NYYA case (see Sect. 2.3.1). By replacing all shocks with the mean shock size we are underestimating (at timestep n > 1) the residual effects of the large shock which propagated from the first defaulted node(s) at timestep n = 1. Indeed, if we modify the Monte Carlo simulations to artificially replace all shocks at each timestep by their mean, we find excellent agreement between theory and numerics over all γ values. We conclude that the simplifying assumption S^{n}(σ) → δ(σ − s^{n}) of the shock size distribution may lead to some errors, and further work on approximating S^{n}(σ) by analytically tractable distributions is desirable. Despite this caveat, overall the theory works very well on the Erdös-Rényi random graphs studied by NYYA.
2.5.3 Networks with Fat-Tailed Degree Distributions
Figure 2.6a shows the theoretical and numerical results for the case where one of the largest banks in the network (i.e., with \({j}^{{\prime}} = {k}^{{\prime}} = 50\)) is targeted initially. Note that the theory accurately matches to the NYYA Monte Carlo simulation results; also note that the Eisenberg-Noe clearing vector case is (at low γ values) somewhat further removed from the NYYA dynamics than in previous figures.
2.6 Discussion
An arbitrary joint distribution p_{jk} of in- and out-degrees (i.e., numbers of debtors and creditors) for banks in the network. This includes fat-tailed distributions; see Eq. (2.28) and Fig. 2.6;
Arbitrary initial conditions for the cascade, including the targeting of one or more banks of a specified size (see Fig. 2.6);
In the general case, the theory gives a set of discrete-time update equations (2.12), (2.13), and (2.42) for a vector of unknowns g^{n}, which is composed of the state variables f^{n}, u_{jk}^{n}(m), and s^{n}. The update equations may be written in the form \({\mathbf{g}}^{n+1} = \mathbf{H}\left ({\mathbf{g}}^{n}\right )\) and this vector mapping is iterated to steady-state to find the fixed point solution g^{∞} = Hg^{∞}, hence giving the expected fraction of defaults ρ^{∞}, see Figs. 2.5 and 2.6 for examples. Under certain conditions it proves possible to simplify the equations to be iterated: as shown in Sect. 2.4, this reduces the vector g^{n} to a scalar g^{n}, with iteration map \({g}^{n+1} = J\left ({g}^{n}\right )\). The GK model is of this type, and the simplified Eqs. (2.15) and (2.16) were used to generate the theoretical results in Fig. 2.4. In all cases we find very good agreement between Monte Carlo simulations and theory, even on relatively small (N = 25) networks.
We expect it will prove possible to improve and extend these results in several ways. As noted in Sect. 2.5.2, the approximation of the shock size distribution in the NYYA model leads to some loss of accuracy, and this merits further attention. It is also desirable to develop analytical methods for calculating the frequency of cascades caused by single seeds in the GK model (see Appendix C). Even in its current form, however, the theory presented here is ideally suited to the study of some policy questions. For example, suppose the models are modified so that the capital reserve fraction γ is not the same for all banks in the system, instead depending on the size of the bank (i.e. γ↦γ_{jk}). This requires only a slight modification of the existing equations. The question is then: how should γ_{jk} depend on the (j, k) class in order to optimally reduce the risk of contagion-induced systemic failure? Other possible extensions, such as allowing for the existence of subgroups of banks with different levels of interbank assets or with interbank loans/liabilities drawn from a prescribed distribution, are required to begin modelling the important non-homogeneities that are seen in the real banking system, and these will be the subject of future work.
For these and similar questions, it is likely that a general cascade condition (or “instability criterion”), analogous to Eq. (2.26) for the GK model, will prove very useful. Cascade conditions for dynamics with vector mappings have been derived for undirected networks (see Gleeson [13] and references therein), so we believe that similar methods may be applied to the directed networks analyzed here.
Finally, it is hoped that the methods introduced here will prove extendable beyond the stylized models of Gai and Kapadia [11] and Nier et al. [25], and in particular that related methods will be applicable to datasets from real-world banking networks. Ideally, such datasets would include information on bank sizes, connections, and the sizes of loans [4]. Modelling the distribution of loan sizes within a semi-analytical framework will be challenging, but the understanding gained of how network topology affects systemic risk on toy models will no doubt prove important to finding the solution.
Notes
Acknowledgements
We acknowledge the work of undergraduate students Niamh Delaney and Arno Mayrhofer on an early version of the simulation codes used in this paper. Discussions with the participants at the Workshop on Financial Networks and Risk Assessment, hosted by MITACS at the Fields Institute, Toronto in May 2010 (particularly Rama Cont and Andreea Minca) are also gratefully acknowledged, as are the comments of Sébastien Lleo and Mark Davis. This work was funded by awards from Science Foundation Ireland (06/IN.1/I366, 06/MI/005 and 11/PI/1026), from an INSPIRE: IRCSET-Marie Curie International Mobility Fellowship in Science Engineering and Technology, and from the Natural Sciences and Engineering Research Council of Canada.
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