Liquidity risk and contagion in interbank markets: a presentation of Allen and Gale Model

Transcription

1 MPRA Munih Personal RePE Arhive Liquidity risk and ontagion in interbank markets: a presentation of Allen and Gale Model El Mehdi FERROUHI and Abderrassoul LEHADIRI Mohammed V Agdal University (Rabat), Faulty of law and eonomis 1 February 013 Online at MPRA Paper No. 5985, posted 14 November :6 UTC

2 Liquidity risk and ontagion in interbank markets: a presentation of Allen and Gale Model FERROUHI El Mehdi Corresponding author, Mohammed V University (Rabat), Faulty of law and eonomis elmehdiferrouhi@gmail.om LEHADIRI Abderrassoul Mohammed V University (Rabat), Faulty of law and eonomis Abstrat The paper analyzes liquidity risk and ontagion in interbank markets. The aim of the researh is to define the different strutures of interbank markets and strutures that allow the better alloation of liquidity and thus avoid the spread of risis in the whole system. For this purpose, this paper examines Allen and Gale model. This model is the pioneer model in the management of liquidity risk in the interbank market. We will then analyze the mehanisms that explain the spread of liquidity risk in the banking system both at national and international level. Introdution: Interbank markets play a key role in the funtioning of finanial systems. Reserved to finanial institutions, the interbank market is an interbank network where banks borrow and lend in short term and thus allows hannel bank liquidity surplus ash to banks in defiit 1. Indeed, it avoids the liquidation of long term assets when the demand for liquidity exeeds liquid assets (liquidation of investments - long term assets - auses a bank assets value s derease). However, the interbank market failitates the spread of the risis from one bank to the entire banking system. Aording to FREIXAS, there are two mehanisms that an spread the failure of one bank into others: the similarity of their assets and the pure speulation 3. The first type of ontagion ours when banks invest in similar assets (whih was the ase during the subprime risis). The seond type, speulative ontagion is explained by the fat that withdrawals of some depositors affet the others depositor s behavior of the bank and of the others bank: these depositors withdraw also their deposits. Even being solvent but illiquid, and not having the opportunity to obtain liquidity, the bank is fored to liquidate long-term assets resulting bankrupty. Thus, the preipitation of depositors rush to withdraw their deposit is the element that triggers rushes withdrawals in other banks in the system. The first type of ontagion is thus onsidered ineffetive beause it leads to bankrupty illiquid and 1 COSTISOR M., «Comment appréhender le risque d illiquidité dans le système banaire?», E.R.U.D.I.T.E., otobre 006, Université PARIS FREIXAS X., «The Lender of Last Resort in Today s Finanial Environment», C.R.E.I, 1999, n 10, Universitat POMPEU FABRA, Barelone 3 Ibid. p. 4 1

3 solvent banks while the seond type is onsidered effetive beause the affeted banks are only the insolvent ones. The ontagion of liquidity risk an also be done via the interbank market. The questions that arise are: What are the different strutures of the interbank market? what struture of the interbank market allows the better alloation of liquidity? And whih one allows to avoid the spread of risis due to liquidity shoks? To answer these questions, we will examine the work done by Allen and Gale. Indeed, these authors were the first to analyze the mehanisms of propagation of the risis in the interbank market. In their model omposed of four banks, they model the demand for liquidity in the interbank omplete market and the inomplete one. This paper analyzes the Allen and Gale model. Thus, we will present this pioneer model in the management of liquidity risk in the interbank market and analyze the mehanisms that explain the spread of liquidity risk in the banking system both at national and international level. This paper is organized as follows. In a first part, we present the main works that foused on the study of interbank market strutures and the management of liquidity in these markets. In a seond part, we present Allen and Gale model that studied the effet of the interbank market struture on liquidity and ontagion risk. We will see what struture allows the better alloation of liquidity. The third and last part is reserved to the onlusions and reommendations. Related literature: The study of the interbank market has a major interest to both researhers and pratitioners in finane. In their artile, ALLEN and GALE 4 presented three types of interbank strutures: the omplete struture where eah bank is onneted reiproally to other banks in the system, the inomplete struture where banks are onneted only to their neighbors and the inomplete and disonneted struture where exists two unonneted interbank markets. If the interbank market is omplete, the effet aused by an unexpeted shok in a bank an be absorbed by other banks whih redues the intensity of the shok. However, if the interbank market is inomplete, a shok in a bank an be transmitted to its neighbors. The inomplete market seems to be more sensitive to shoks. FREIXAS, PARIGI and ROCHET 5 have defined a fourth struture of interbank markets: the money enter in whih exists a enter of money onneted to all the others banks disonneted. In ase of a bank s failure, there is no effet on the money enter. But if the failure is at the money enter, this risis is transmitted to all existing banks. ESTRADA and MORALES 6 presented these different strutures of interbank markets as follows: 4 ALLEN F and GALE D., «Finanial Contagion», Journal of Politial Eonomy, University of Chiago Press, vol. 108(1), pp. 1-33, Février FREIXAS X., PARIGI B. and ROCHET J.C., «Systemi risk, interbank relations and liquidity provision by the Central Bank», Journal of Money, Credit and Banking, n 3, pp ESTRADAY D. and MORALES P., «The Struture of the Colombian Interbank Market and Contagion Risk», Bano de la Republia de Colombia, 008

4 Complete struture Bank A Bank B Bank C Bank D Bank A Bank B Bank C Bank D Inomplete struture Bank A Bank B Bank C Bank D Bank A Bank B Bank C Bank D Disonneted struture Bank A Bank B Bank C Bank D Bank A Bank B Bank C Bank D Money enter struture Bank A Bank B Bank C Bank D Bank A Bank B Bank C Bank D Table 1 : Interbank markets strutures 7 7 Ibid,. page 1 3

5 Complete struture Inomplete struture Disonneted struture Money enter struture Figure 1: Interbank markets strutures The main works that studied the interbank market and liquidity risk were presented by BHATTAHCHARYA and GALE (1985), FLANNERY (1996), FREIXAS and JORGE (007), ACHARYA, GROMB and YORULMAZER (009), EIFELDT (004), FREIXAS and HOLTHAUSEN (001), HEIDER, HOEROVA and HOLTHAUSEN (009), BRUNNERMEIR and PEDERSEN (007) and ALLEN, CARLETTI AND GALE. The paper of BHATTAHCHARYA and GALE 8 was the first that foused on the management of liquidity in the interbank market. This work is onsidered as the pioneer theoretial study in this field. BHATTAHCHARYA and GALE examine how the interbank market an effetively manage liquidity among banks in time of risis. They analyze a market in whih individual banks fae a liquidity shok due to massive withdrawals. Sine shoks are imperfetly orrelated between different banks, they provide eah other through an interbank market. They suggest that even in the absene of liquidity shoks, banks must invest in long-term assets and under-invest in liquid assets mainly beause of the poor performane of these assets. They stipulate that the Central Bank an mitigate this problem by ensuring proper hoie of assets by banks. Reent literature interests on the effiient funtioning of the interbank market during finanial rises. In his artile, FLANNERY 9 modeled the failure of the interbank market. He presents the risis as a loss of onfidene s phase of banks on the interbank market, whih refuse to lend to other banks when 8 BHATTACHARYA S. and GALE D., «Preferene Shoks, Liquidity, and Central Bank Poliy», Caress Working, Mai 1985, n FLANNERY M.J., «Finanial rises, payment system problems and disount window lending», Journal Of Money, Credit And Banking, November 1996,Vol. 8 No. 4, pp

6 they an no longer identify insolvent banks from illiquid ones. FREIXAS and JORGE 10 analyzed the effets of market imperfetions interbank on monetary transmission. Indeed, ompanies resort to bank redit for finaning, resulting an inrease of redit s demand and a liquidity shok for banks. These banks borrow from the interbank market. Under asymmetri information, the interbank market is unable to effiiently hannel liquidity to solvent and illiquid banks. Thus, the liquidity reserves of banks must regulate bank loans. ACHARYA, GROMB and YORULMAZER 11 presented interbank markets as markets haraterized by asymmetri information and monopoly in times of risis. Thus, banks having an exess of liquidity have a bargaining power than banks having lak of liquidity. Banks in exess of liquidity offer insuffiient loans to enourage banks in need to sell their assets. The role of the Central Bank is to propose a better option for banks in lak of liquidity. In 004, EIFELDT 1 developed a model for understanding the harateristis of liquidity in asset markets. The author notes that a greater eonomi produtivity inreases liquidity. FREIXAS and HOLTHAUSEN 13 foused on the possible ross-border integration of interbank markets in the presene of asymmetri information on the solveny of banks. They model a banking setor where banks fae individual shoks. The differene of liquidity shoks between banks justifies the existene of an interbank market where banks an insure against these shoks. ALLEN, CARLETTI and GALE 14 show that the lak of opportunities to over liquidity shoks by banks makes pries exessively volatile. They analyze how the Central Bank should intervene to restore effiieny. The Central Bank an at by fixing the short-term interest rates to avoid prie volatility. HEIDER, HOEROVA and HOLTHAUSEN 15 analyzed the funtioning of interbank markets. They show the role of the interbank market in the exhange of liquidity among banks in exess of liquidity and banks in lak of liquidity. BRUNNERMEIR and PEDERSEN 16 made a distintion between market liquidity and funding liquidity. The define market liquidity as the differene between the transation prie and the fundamental value and funding liquidity as a speulator s sarity of apital. They show how the predatory behavior of some banks fores the ineffiient liquidation of other banks assets, whih may jeopardize the effiient distribution of liquidity in the interbank market. They also show that under ertain onditions, margins are destabilizing and market liquidity and funding liquidity are mutually reinforing, leading to liquidity spirals. The model: In what follows, we present a model that made a signifiant ontribution to the analysis of liquidity risk in the interbank market and the explanation of ontagion within the banking system of a ountry and / or international level 17. ALLEN and GALE model is a generalized and developed representation of Diamond and Dybvig model. This model is mainly based on the assumptions made by Diamond and Dybvid (1983). Indeed, the model has as a hypothesis the existene of two periods - 3 dates - 10 FREIXAS X. and JORGE J., (007), «The role of interbank markets in monetary poliy: a model with rationing», Eonomis Working Paper, Universitat Pompeu Fabra, no ACHARYA V.V., GROMB D. and YORULMAZER T., «Imperfet Competition in the Interbank Market for Liquidity as a rationale for Central Banking», NYU Working Paper, Otober 009, 1 EISFELDT A.L., «Endogenous Liquidity in Asset Markets», The Journal of Finane, February 004, Volume 59, Issue 1, pp. 1 30, 13 FREIXAS X. and HOLTHAUSEN C., «Interbank market integration under asymmetri information», Eonomis Working Papers, 001, Universitat POMPEU FABRA 14 ALLEN F., CARLETTI E. and GALE D., Interbank Market Liquidity and Central Bank Intervention 15 HEIDER F., HOEROVA, M. and HOLTHAUSEN, C., «Liquidity Hoarding and Interbank Market Spreads: The Role of Counterparty Risk», Disussion Paper, S, Tilburg University, Center for Eonomi Researh 16 BRUNNERMEIER M.K. and PEDERSEN L.H, «Market Liquidity and Funding Liquidity», June COSTISOR M., op.it. page 5

7 t=(0,1,), a single onsumption good whih serves as the numeraire, a ontinuum of ex ante idential agents and an investment tehnology. There are two types of agents: patients and impatients. The preferenes of the individual onsumer are given by: U 1, = u 1 with probability ω u with probability 1 ω with ω publi information First, all agents have the same hane of being impatient. At t = 1, eah agent learns his true private type. ω an be either a high value ω H, ω L is a low value with 0 <ω L <ω H <1. S 1 and S are two states of nature whose realization depends on the realization of the type of agent ω. The bank ollets funds used to finane non-risky assets in the short term and long term and offer deposit ontrats 1 for the first period and for the seond period. The short-term assets (L) are represented by the storage tehnology that produes 1 at t=1, and the long-term assets (1-L), partially illiquid and represented by the prodution tehnology. This investment is done at t = 0 and produes at t = 1 0 < r <1 and at t = R> 1. ALLEN and GALE model is haraterized by the existene of 4 idential banks (or four regions) in the interbank market, A, B, C and D where eah bank undergoes random shoks and massive withdrawals of liquidity while the demand for liquidity of the entire system as well as the overall demand for liquidity for eah region are onstant. The assumption of the onstany of the overall demand for liquidity, despite the random distribution of onsumers, allows to onsider a mehanism for interregional liquidity insurane, where banks provide exess liquidity to banks in need of liquidity. The repartition of liquidity shoks at t = 0 is shown in the following table: A B C D S 1 ω H ω L ω H ω L S ω L ω H ω L ω H Table : Regional liquidity shoks 18 Eah bank knows its different states of nature and the orrelation between these. At t = 1, depositors know if they must onsume immediately or not and the banks beome aware of the state of nature where they are atually, S 1 or S. Eah region has the same probability of being faed to a high demand for liquidity. This insurane sheme is a reassuring element for both banks and depositors. But how the mehanism of interbank deposits works? To analyze this behavior, we analyze the balane sheets of the four banks (A, B, C, D). If at t = 1, the number of impatient agents is high for banks A and C and low for B and D, it will be low for A and C and B and D at t =. The authors analyze the omplete and inomplete interbank (Figure 1). In the ase of omplete interbank market, the balane sheet of the four banks is presented as follows: 18 ALLEN F et GALE D., op.it. page 8 6

8 t=1 t= Banks with high demand for liquidity at Banks with low demand for liquidity at t=1 t=1 Assets Liabilities Assets Liabilities Short term assets Withdrawals of Short term assets Withdrawals of Deposits on the impatients agents impatients agents other banks Withdrawals of Withdrawals of banks with high banks with high demand of liquidity demand of liquidity at t=1 at t=1 Long term assets Withdrawals of patients agents Withdrawals of banks with high demand of liquidity at t= Long term assets Deposits on the other banks Withdrawals of patients agents Withdrawals of banks with high demand of liquidity at t= Tableau 3 : Struture du bilan dans le as du marhé interbanaire omplet How banks manage liquidity shoks at t = 1 and t =? At t = 1 (Table 3), the liabilities of the bank (banks A and C) with high demand of liquidity, are withdrawals of impatient agents and withdrawals of banks with high demand of liquidity at t=1. Short term assets and deposits on the other banks an satisfy all these withdrawals. The liabilities of the bank (bank B and D) with low demand of liquidity, inludes withdrawals of impatient agents and withdrawals of banks with high demand of liquidity at t=1. The assets of these banks are omposed only of short-term assets. At t =, the liabilities of the bank with low demand of liquidity (banks A and C) ontain withdrawals of patients agents, withdrawals of banks with high demand of liquidity at t= while the assets inlude sales of long term assets. For the bank that faes a high demand of liquidity (bank B and D), liabilities inludes withdrawals of patient agents and withdrawals of banks with high demand of liquidity at t= while assets inludes deposits on the other banks and sales of long term assets. 7

9 t=1 t=1 A (High demand of liquidity) B (Low demand of liquidity) Assets Liabilities Atifs Liabilities ω H 1 L ω L 1 ω H γ ω H γ 1 1 L 3 ω H γ L + 3 ω H γ L 3 ω H γ 1 ω H 1 + ω H γ ω H γ 1 L ω L 1 + (ω H γ) 1 C (High demand of liquidity) D (Low demand of liquidity) Assets Liabilities Atifs Liabilities ω H 1 L ω L 1 ω H γ ω H γ 1 1 L + 3 ω H γ 1 ω H 1 + ω H γ ω H γ 1 L ω L 1 + (ω H γ) With: Table 4: Funtioning of omplete interbank market at t=1 γ : ω H ω L : Average demand of banks; L: Short term assets; ω H : Highest value of ω; ω L : Lowest value of ω; 1 : Deposit ontrat for the first period; t= t= A (High demand of liquidity) B (Low demand of liquidity) Atifs Passifs Atifs Passifs (1 ω H ) (1 ω L ) ω H γ 3 ω H γ ω H γ ω H γ (1 ω H ) + (ω H + 3 ω γ) H γ (1 ω L ) + ω H γ C (High demand of liquidity) D (Low demand of liquidity) Atifs Passifs Atifs Passifs (1 ω H ) (1 ω L ) ω H γ 3 ω H γ ω H γ ω H γ (1 ω H ) + (ω H + 3 ω γ) H γ (1 ω L ) + ω H γ Table 5: Funtioning of omplete interbank market at t = 8

10 With: (1-L): Long term assets; : Deposit ontrat for the seond period; In what follows, we will analyze the interbank market inomplete. The balane sheets of the four banks are as follows: t=1 t= Bank High demand of liquidity at t=1 Bank Low demand of liquidity at t=1 Assets Liabilities Assets Liabilities Short term assets Withdrawals of Short term assets Withdrawals of Deposits on the impatients agents impatients agents neighboring bank Withdrawals of neighboring bank with high demand of liquidity at t=1 Long term assets Withdrawals of Long term assets Withdrawals of patients agents Deposits on the patients agents Withdrawals of neighboring bank neighboring bank with high demand of liquidity at t= Table 6: Struture of the balane sheet in the ase of inomplete interbank market At t = 1, the assets of the bank whih faes a high demand for liquidity (banks A and C) are omposed of short term assets and deposits on the neighboring bank that hedge the impatient depositor withdrawals. At t =, this bank faes a low demand of liquidity. Its assets ontains long term assets while its liabilities are withdrawals of patients agents and withdrawals of banks with high demand of liquidity at t=. For the bank (B and D) with low demand for liquidity, at t = 1, the short term assets are used to over withdrawals of impatiens agents and withdrawals of neighboring bank with high demand of liquidity at t=1. At t = 1, the bank liquidate its long term assets and deposits on the neighboring bank (faing weak demand liquidity) to over withdrawals of patients agents. t=1 t=1 A (High demand of liquidity) B (Low demand of liquidity) Assets Liabilities Assets Liabilities L ω H 1 L ω L 1 3 (ω H γ) 1 (ω H γ) 1 L + (ω H γ) 1 ω H 1 L ω L 1 + (ω H γ) 1 C (High demand of liquidity) D (Low demand of liquidity) Assets Liabilities Assets Liabilities L ω L 1 L ω L 1 (ω H γ) 1 (ω H γ) 1 L + (ω H γ) 1 ω L 1 L ω L 1 + (ω H γ) 1 Table 7: Funtioning of inomplete interbank market at t = 1 9

11 t= t= A (Low demand of liquidity) B (High demand of liquidity) Assets Liabilities Assets Liabilities (1 ω H ) (1 ω L ) (ω H γ) (ω H γ) (1 ω H ) + (ω H γ) + (ω H γ) (1 ω L ) C (Low demand of liquidity) D (High demand of liquidity) Assets Liabilities Assets Liabilities (1 ω H ) (1 ω L ) (ω H γ) (ω H γ) (1 ω H ) + (ω H γ) + (ω H γ) (1 ω L ) Table 8: Funtioning of inomplete interbank market at t = However, even if the interbank market plays the role of liquidity insurane, in some situations, it also promotes the spread of the liquidity risis in the entire banking system. In this sense, it is important to note that if there is an exess of aggregate demand for liquidity, the interdependenies between banks an ause many bankrupties. The question is if the way in whih banks are onneted (interbank market) is a key element for the spread of the liquidity risis from one region to other regions through interbank deposits. In the omplete market, eah bank is fored to pay its share as interbank onnetions are reiproal. When the total demand for liquidity exeeds the supply, and the gap is not very important, eah region absorbs some liquidity shoks through the liquidation of a small proportion of long-term investment. The impat of the risis is then muh more attenuated; it is possible to prevent the spread of the risis. Instead, in the inomplete struture, a liquidity shok in a region is strongly felt in the neighboring bank. If this shok is very strong, then there is a spread of the risis whih then transforms the illiquidity of banks into insolveny. The explanation is related to the unilateral nature of interbank onnetions. Banks in not affeted by the liquidity risis refuse to liquidate a portion of long-term assets until they are themselves exposed to the risis. Conlusions: To onlude, we an say that there are three mehanisms that an spread the failure of one bank into other. The first one is the pure speulation whih is explained by the fat that withdrawals of some depositors affet the depositor s behavior of the bank and of the others bank who withdraw their deposits. The seond mehanism is the similarity of their assets whih ours when banks invest in similar assets. The third and the last mehanism is related to the interbank market struture. In the omplete market, the risk of ontagion is almost absent. The impat of shoks are attenuated and absorbed by the interbank market s banks whih prevent the spread of the risis. In the opposite, the inomplete interbank market struture failitates the spread of the risis, and amplifies the negative effets of the interation between liquidity and solveny. The omplete interbank market is then the struture that allows the better alloation of liquidity. 10

12 The intervention of Central Bank is very important to regulate and omplete the market. Central Banks should reate trust between different banking institutions of the system so that relationships beome reiproal and thus omplete the market. They also have to distribute effiiently liquidity among banks. The effet of liquidity risk spread will be redued. Bibliography: Aharya, V., Gromb, D. and Yorulmazer. T. (01), Imperfet Competition in the Interbank Market for Liquidity as a Rationale for Central Banking. Amerian Eonomi Journal: Maroeonomis, 4(): Allen, F., Carletti,, E. and Gale, D., (009), Interbank market liquidity and entral bank intervention. Journal of Monetary Eonomis, 56: Allen, F. (011)., Liquidity and Crises, Oxford University Press, 707 pages; Allen, Franklin and Gale, D. (001), Finanial ontagion, Journal of Politial Eonomy, University of Chiago Press, vol. 108(1):1-33; Bhattaharya, S., and Gale, D. (1987). Preferene Shoks, Liquidity and Central Bank Poliy. In New Approahes to Monetary Eonomis, edited by William A. Barnett and Kenneth J. Singleton, New York: Cambridge University Press; pp Brunnermeier, M. K. and Pedersen, L.H. (009). Market Liquidity and Funding Liquidity, The Review of Finanial Studies, (6): Costisor, M. (006), «Comment appréhender le risque d illiquidité dans le système banaire?», E.R.U.D.I.T.E., University of PARIS Eisfeldt, A.L. (004), Endogenous Liquidity in Asset Markets, The Journal of Finane, Volume 59, Issue 1:1 30; Estraday, D. and Morales, P. (008), The Struture of the Colombian Interbank Market and Contagion Risk, Bano de la Republia de Colombia; 11

13 Flannery, M. J. (1996), Finanial rises, payment system problems and disount window lending, Journal Of Money, Credit And Banking, Vol. 8 No. 4: Freixas X. (1999), The Lender of Last Resort in Today s Finanial Environment, C.R.E.I, 1999, n 10, Universitat POMPEU FABRA, Barelona Freixas, X. and Holthausen, C. (001), Interbank market integration under asymmetri information, Eonomis Working, Universitat POMPEU FABRA; Freixas Xavier. and Jorge José., (007), The role of interbank markets in monetary poliy: a model with rationing, Eonomis Working Paper, Universitat Pompeu Fabra, no 107; Freixas, X.., Parigi, B. X. and Rohet J.C. (000). Systemi risk, interbank relations and liquidity provision by the Central Bank, Journal of Money, Credit and Banking, n 3, pp ; Heider, F., Hoerova, M., and Holthausen, C. (009), Liquidity Hoarding and Interbank Market Spreads: The Role of Counterparty Risk, Tilburg University, Center for Eonomi Researh., Disussion Paper, S 1