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Uncorrected Proof PHYSA 6127 PROD. TYPE: COM ED: Rohini pp: 1-7 (col.fig.: nil) PAGN: Murthy.N -- SCAN: Shobha Physica A 000 (2001) 000–000 www.elsevier.com/locate/physa 1 A simple model of bank bankruptcies 3 Agata Aleksiejuka , Janusz A. Ho lysta;b;∗ aFaculty of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland 5 bInstitute for Economics and Tra&c, Dresden University of Technology, Andreas-Schubert-Stra+e 23, D-01062 Dresden, Germany 7 Abstract Interbank deposits (loans and credits) are quite common in bankingsystem all over the world. 9 Such interbank co-operation is proÿtable for banks but it can also lead to collective ÿnancial failures. In this paper, we introduce a new model of directed percolation as a simple represen- 11 tation for contagion process and mass bankruptcies in banking systems. Directed connections that are randomly distributed between junctions of bank lattice simulate 1ows of money in our 13 model. Critical values of a mean density of interbank connections as well as static and dynamic scalinglaws for the statistics of avalanche bankruptcies are found. Results of computer simula- 15 tions for the universal proÿle of bankruptcies spreadingare in a qualitative agreementwith the third wave of bank suspensions duringThe Great Depression in USA. c 2001 Published by 17 Elsevier Science B.V. Keywords: Random directed percolation; Interbank deposits; Mass bankruptcies 19 1. Introduction Makinga short review of latest publications on the percolation phenomenon one 21 could come to the conclusion that the percolation theory [1,2] is a universal paradigm for physics, sociology and economy. In fact, percolating systems composed of large 23 number of interactingunits can be simply adopted for simulations of complex be- haviours and environments. A number of such adoptions have been done so far in- 25 cludingmicroscopic simulations of the stock market [3–7], social percolation models [8,9]UNCORRECTED and marketingpercolation describingdi@usion of innovationsPROOF [10]. Here we pro- 27 pose a simple model based on the intuitive similarity between percolation and banking networks. ∗ Correspondingauthor. Tel.: +48-22-6607133; fax: +48-22-6282171. E-mail addresses: [email protected] (A. Aleksiejuk), [email protected] (J.A. Ho lyst). 0378-4371/01/$ - see front matter c 2001 Published by Elsevier Science B.V. PII: S 0378-4371(01)00296-5 PHYSA 6127 2 A. Aleksiejuk, J.A. Ho lyst / Physica A 000 (2001) 000–000 1 At present, economists agree that the robustness of a country is ÿnancial system is related to the strength of a domestic economy. Bank bankruptcies usually follow 3 dramatic changes in the banking capital, assets as well as liabilities and can be so- cially costly. In general, two factors may cause a bank failure: bad credits and rapid 5 withdrawingof deposits. Economic researches conÿrm that solvent and insolvent banks alike can experience withdrawals for reasons unrelated to the bank failure risk in cir- 7 cumstances of a bankingpanic [11]. The same investigationsemphasize the importance of withdrawal rates. In fact, sudden withdrawals can have dramatic e@ects on the bank 9 stability and may force a bank to bankruptcy in a short time if it does not receive assistance from other banks. On the other hand, a bankruptcy of a single bank can 11 start an avalanche of other bank failures due to the domino e9ect. 2. The model 13 In our model, banks are represented by vertices in a lattice, which for simplicity has a square or cubic symmetry. Directed connections that are randomly distributed 15 between banks simulate 1ows of money. Bankingcapital consists of assets and liabil- ities as in reality. Arrows enteringinto vertices represent liabilities (deposits of other 17 banks). Branches with opposite direction re1ect assets (investments and given cred- its). It follows that an average number of arrows entering into a vertex is equal to 19 an average number of exiting arrows. We assume that even one withdrawal or bad credit can force the bank to bankruptcy and one failingbank can cause bankruptcies 21 of other banks. Only interbank credit connections are considered, i.e., bank deposits and investments are neglected and no insurance system is assumed in our model. 23 Dynamical rules governing time evolution of the model are as follows. Initially, each bank is solvent. The ÿrst bankrupt is selected at random and we do not specify 25 the reason for this bankruptcy that can be a bad credit or sudden deposit withdrawal. Duringthe next time step, neighbouringbankslose their solvency if they gavea loan 27 to the bankrupt. This process is repeated until no bank survives that gave a bad inter- bank credit. Above mentioned rules become comprehensible after tracingFig.1. The 29 ÿgure presents a system with N = 25 banks. All possible 1ows of money (connections between vertices) are realized in this pattern. Let us choose the seventh vertex as the UNCORRECTED PROOF Fig.1. Bankruptcy spreadingin bankingnetwork based on square lattice. PHYSA 6127 A. Aleksiejuk, J.A. Ho lyst / Physica A 000 (2001) 000–000 3 1 ÿrst bankrupt. Accordingto the rules assumed earlier the collapse of this bank forces suspension of two other banks with numbers {2; 6}. Duringthe next step three other 3 banks are swept {1; 3; 11}. At the end, the avalanche originating from the bank with the number 7 includes nine banks {1; 2; 3; 4; 6; 7; 11; 12; 16}. 5 Despite the seemingsimilarity of our model to the well known directed percolation [1] it is based on a new approach to this phenomenon. In the traditional directed 7 percolation directions in space are not equal, i.e., the system is anisotropic and one direction, which is called the growth direction, is special. In our model, all directions 9 are equal. There is also another feature distinguishing the presented model from the standard percolation. In both cases of the traditional site and bond percolation, each 11 occupied site=bond belongs to only one cluster. In our model, this condition is not valid and the same bank can be included to various avalanches dependingon the ÿrst 13 bankrupt. 3. Computer simulations 15 We investigated statistics of bankrupt avalanches in systems characterized by di@er- ent mean concentrations p of existinginterbank deposits. It follows that the system 17 parameter is the same as in the percolation theory. In analogy with the traditional per- colation one can expect a critical value pc when an avalanche composed of bankrupts 19 can spread all over the bankingnetwork. This phenomenon is related to the percola- tion phase transition. We performed numerical calculations in order to estimate pc and −1= 21 based on the ÿnite size scalinglaw p(L) − pc ∼ L (where L is the linear size of the system) we found that critical values pc in our model are approximately two times 2D 3D 23 larger than in the usual bond percolation, i.e., pc ≈ 1:00±0:01 and pc ≈ 0:51±0:02 for the square and the cubic lattice, respectively. 25 We observed that the distributions of avalanche lengths have the same properties as statistics of cluster numbers in the usual percolation system. At the percolation 27 threshold, the probability that a random bank causes l-avalanche (failures of l other 1− banks) fulÿlls the power law Pl(pc) ∼ l , where is the Fisher exponent. In both 29 two- and three-dimensional systems numerically calculated Fisher exponents are con- sistent with their equivalents taken from the literature (Fig. 2). For p near pc and 31 for l →∞we found a good agreement with the scaling law describing avalanche 1− distribution Pl(p)=l f[(p − pc)l ], where f is a universal scalingfunction. Fig.3 36 33 illustrates this law for a square lattice when the scalingexponent 2D = 91 has been used. 35 Dynamical properties of our model are described by the number of banks n(l; t) sweptUNCORRECTED duringthe bankruptcy avalanche up to the moment PROOFt, where l is the total 37 avalanche length (limt→∞ n(l; t)=l). A typical plot of n(l; t) (Fig. 4) has two regions separated by a crossover time tx [12]. Initially, when t tx the number of bankruptcies 39 increases as n(l; t) ∼ tÿl. In the dynamic scalingtheory of surface growth,the anal- ogous exponent ÿ is called the growth exponent. Fortunately, for bank shareholders, PHYSA 6127 4 A. Aleksiejuk, J.A. Ho lyst / Physica A 000 (2001) 000–000 Fig. 2. Avalanche length distribution at pc in square lattice (solid squares) and cubic lattice (open squares). Fig. 3. Scaling behaviour for the renormalized avalanche statistics described by f(z)=Pl(p)=Pl(pc). 1 the power law increase is followed by the saturation regime for t tx. The satura- z tion time tx depends on the avalanche length as tx ∼ l . By analogy to the standard 3 terminology [12], we call z as the dynamic exponent. We found that the avalanche growth in our model fulÿlls the Family–Vicsek scaling relation n(l; t) ∼ lg(t=lz), where 5 g is a universal scalingfunction (Fig.4). The scalingexponents ÿ; ; z are connected by the equation + zÿ = 1. Accordingto our numerical studies for the square lattice 7 the exponents account to ÿ =1:60 ± 0:01;=0:08 ± 0:02;z=0:56 ± 0:01 and do not depend on the system parameter p. 9 Fig. 5 shows time distributions of bankruptcies belonging to avalanches presented in Fig.UNCORRECTED 4, i.e., the curves in Fig. 5 are the ÿrst derivatives PROOF of those in Fig. 4. Ob- 11 servingthe speed of avalanche spreadingwe ÿnd a clear maximum which corresponds to the highest probability of bankruptcy.
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