Critical Connectivity in Banking Networks

Transcription

1 Critical Connectivity in Banking Networks Agam Gupta, Molly King, James Magdanz, Regina Martinez, Matteo Smerlak, and Brady Stoll 2nd CIRANO-CIREQ Workshop on Networks Presented by Regina Martinez Oct. 25th, 2013 CIRANO-CIREQ Workshop Oct. 25th, / 38

2 Motivated by: Last financial crises Understand the interconnections of banks Trade-off: individual versus systemic Research question: Which is the optimal level of connectivity? CIRANO-CIREQ Workshop Oct. 25th, / 38

3 Figure: Number of failures and level of connectivity (k) 8 Number of failures k Mean degree Number of failures as a function of the degree, k. Regular network of N = 20 banks. R = 1.05, r = 1.01, f =.7, Λ = 20. Vertical line: critical degree, k = critical level of connectivity CIRANO-CIREQ Workshop Oct. 25th, / 38

4 Using a simple standard model of a banking network where: Banks are connected through interbank lending One bank is shocked exogenously We study the critical level of connectivity and find... CIRANO-CIREQ Workshop Oct. 25th, / 38

5 Summary of results: 1. Find analytical solution for the critical degree, k. Contagion occurs when k < k k = Φ (Λ, f, R, r) where Λ: leverage; f : interbank lending over total assets R: return on investment; r: interbank lending rate 2. Test validity of our result in different network structures CIRANO-CIREQ Workshop Oct. 25th, / 38

6 Figure: Number of failures in a random (ER) and scale-free network Directed networks directed ER directed static Figure: Distribution of failures for two directed networks: Erdös-Rényi random network (ER), and directed scale-free network. N = 50 banks with mean degree k = 4, and parameter values: R = 1.05, r = 1.01, f = 70% and Λ = 20 CIRANO-CIREQ Workshop Oct. 25th, / 38

7 Outline 1 Introduction 2 The Model 3 Simulation Results 4 Conclusion CIRANO-CIREQ Workshop Oct. 25th, / 38

8 Hamilton (1781), 1 st Secretary of the Treasury, said: Banks are the happiest engines that ever were invented for spurring economic growth Role of the banks? Mobilize resources from savers to investors Deal with information asymmetries Provide liquidity and manage risk CIRANO-CIREQ Workshop Oct. 25th, / 38

9 But when things go wrong... And a systemic failure can be devastating CIRANO-CIREQ Workshop Oct. 25th, / 38

10 What s special about banks? Banks are like any other firm Profit maximizers but banks are special based on Confidence Banks are highly regulated Traditionally: regulation at individual banks Problem: It doesn t capture the effect of the network CIRANO-CIREQ Workshop Oct. 25th, / 38

11 Goal of this paper What s the critical level of connectivity? (ie, level of connectivity at which the phase transition occurs) CIRANO-CIREQ Workshop Oct. 25th, / 38

12 Literature Models of individual banking crises Diamond and Dybvig (1983), Santomero (1984) Models of contagion Kaminsky and Reinhart (1999), Allen and Gale (2000) Network theory Inconclusive results Acemoglu et al. (2013) try to reconcile CIRANO-CIREQ Workshop Oct. 25th, / 38

13 Contribution Analytical solution for the critical connectivity It only depends on financial parameters Heterogenity of network structures: Critical degree applies to different structures Network structure matters for number of failures CIRANO-CIREQ Workshop Oct. 25th, / 38

14 The Model Based on Acemoglu et al. (2013) Main features: 2-period economy System of N risk neutral banks linked through interbank lending Simple balance sheet One bank receives an exogenous shock Study how that shock spreads throughout the system CIRANO-CIREQ Workshop Oct. 25th, / 38

15 At t = 0 Banks borrow from depositors and other banks Banks invest in investment opportunities and lend to other banks At t = 1 Banks get the return on the project Banks repay their liabilities Depositors first (senior liabilities) Banks after (junior liabilities) CIRANO-CIREQ Workshop Oct. 25th, / 38

16 Invesment opportunity yields a return of R Funding cost: Interbank rate is r Deposit rate is 1. where R > r > 1 One bank receives an exogenous shock on R Bank fails if total resources < total obligations Contagion may occur if failed bank is unable to repay other banks in full CIRANO-CIREQ Workshop Oct. 25th, / 38

17 CIRANO-CIREQ Workshop Oct. 25th, / 38

18 Bank Balance Sheet Total Assets Total Liabilities (A i ) (L i + K i ) Investments, λ i Interbank loans, l i Senior liabilities, σ i Interbank borrowings, b i Capital, K i where A i = λ i + l i ; L i = σ i + b i ; l i = i l ij; and b i = j b ji Financial ratios (assumed to be constant): f = l i /A i f λ = λ i /A i Leverage = Λ i = A i /K i CIRANO-CIREQ Workshop Oct. 25th, / 38

19 Repayment of i to its creditors, x ij : depends on what its own debtors repay to i, x ji. Bank i can repay its creditors in full when: R i L i σ i + j x ji rb i = x ij = rl ji but when R i L i σ i + j x ji < rb i = partial repayment of: x ij = l ( ji R i L i σ i + ) x ji b i j i Repayment equation: x ij = l { ji max b i min R i L i σ i + j i x ji, rb i }, 0 A payment equilibrium exists and it s unique (fixed point theorem) CIRANO-CIREQ Workshop Oct. 25th, / 38

20 After repayments are made, bank i has new capital: K i = R i L i σ i + j (x ji x ij ) When K i 0, bank is safe When K i < 0, bank defaults CIRANO-CIREQ Workshop Oct. 25th, / 38

21 Critical degree, k : degree below which contagion starts Assumptions Regular directed network All links have lending value of 1. Then, l i = b i = k Repayment equation becomes: { x i = min η i k + j i x } j k, rk + where x i j i x ij; CIRANO-CIREQ Workshop Oct. 25th, / 38

22 Solve for k, critical degree: ( ) r [r (Λ 2)/(Λf )] k + = Λf (R 1)Λ + (2 R) CIRANO-CIREQ Workshop Oct. 25th, / 38

23 In a Regular network 8 Number of failures k Mean degree Figure: Number of failures as a function of the degree, k. Regular network of N = 20 banks with parameter values: R = 1.05, r = 1.01, f =.7, Λ = 20). The vertical line: critical degree, k CIRANO-CIREQ Workshop Oct. 25th, / 38

24 Figure: Sample regular networks with k = 5 (left) and k = 10 (right); the shocked bank is represented in black, the failing banks in red and the safe banks in green. CIRANO-CIREQ Workshop Oct. 25th, / 38

25 Figure: Critical degree, k Figure: Critical degree. Left: as a function of interbank exposure f and leverage factor Λ (with R = 1.05, r = 1.01). Right: critical degree as a function of R and r (with f = 70% and Λ = 20) CIRANO-CIREQ Workshop Oct. 25th, / 38

26 Simulations results Compute expected number of failures F for more general networks First neighbors of the shocked bank will fail if k i < k Calculate F as a function of k for various parameter values (Λ, f, R, r). For R = 1.05, r = 1.01, f = 70%, and Λ = 20 Black node: shocked bank; red node: failed bank; green node: safe CIRANO-CIREQ Workshop Oct. 25th, / 38

27 Visualization of failure propagation From top left to bottom right: Erdös-Rényi, Barabási-Albert, directed Erdös-Rényi, directed static scale free. CIRANO-CIREQ Workshop Oct. 25th, / 38

28 1. Test the validity of our results for other network structures We compare the mean field approximation with the simulation results. Mean number of failures Directed ErdösRényi networks 5 Av. over 500 networks Meanfield approx Mean in and outdegree Mean number of failures Directed static scalefree networks Av. over 100 networks Meanfield approx Mean in and outdegree Expected number of failures in directed Erdös-Rényi (left) and directed static scale-free (right) networks with N = 50 banks, for R = 1.05, r = 1.01, f = 70% and Λ = 20 CIRANO-CIREQ Workshop Oct. 25th, / 38

29 2. Catastrophic failures F is similar in random and scale free networks. But prob. of catastrophic failure is larger in skewed networks Robust-yet-fragile property Directed networks directed ER directed static Figure: Distribution of failures for directed Erdös-Rényi and directed static scale-free networks of n = 50 banks with mean (in- and out-)degree k = 4, for R = 1.05, r = 1.01, f = 70% and Λ = 20 CIRANO-CIREQ Workshop Oct. 25th, / 38

30 Conclusion This paper studies contagion in a banking network Closed-form solution for degree at which a shock propagates Critical degree depends on banking leverage It is valid for different network structures Network structures affect the probability of catastrophic failures CIRANO-CIREQ Workshop Oct. 25th, / 38

31 Future Extensions - Several shocks - Taking R i from a probability distribution - Changing lending sizes - And more... CIRANO-CIREQ Workshop Oct. 25th, / 38

32 MANY THANKS CIRANO-CIREQ Workshop Oct. 25th, / 38

33 [Aside 1: Definitions] Degree of a node is the number of links attached to the node Critical degree is the degree under which the initial failure will extend to other banks Directed network: network in which each node has 2 degree: in-degree (number of ingoing links) & outdegree (number of outgoing links) Regular network: all nodes have the same degree Complete network: regular network in which all banks lend equally to each other CIRANO-CIREQ Workshop Oct. 25th, / 38

34 [Aside 2: Types of networks] Type of networks Undirected networks If there is a link between i and j, then a link between j and i. Directed networks Bank i may lend to j but j may not lend to i. CIRANO-CIREQ Workshop Oct. 25th, / 38

35 Undirected networks 1. Erdös-Rényi network: Prob. that a pair of nodes is connected by an undirected link is fixed, p. The network has a Poisson degree distribution with mean k = p(n 1) 15 Frequency Figure: Erdös-Rényi network with N = 50 and k = 4 (left) and the histograms of their nodes degrees (right) Degree CIRANO-CIREQ Workshop Oct. 25th, / 38

36 2. Barabási-Albert network: it s a growing network in which the prob of a link attaching to a node is proportional to the degree of the target (preferential attachment). It generates a degree distribution with a power-law tail Frequency Figure: Barabási-Albert network with N = 50 and k = 4 (left) and the histograms of their vertex degrees (right) Degree CIRANO-CIREQ Workshop Oct. 25th, / 38

37 Directed networks 1. Erdös-Rényi directed network 15 Frequency Indegreeblue and outdegreered CIRANO-CIREQ Workshop Oct. 25th, / 38

38 2. Goh-Kahng-Kim static directed network 10 8 Frequency Indegreeblue and outdegreered CIRANO-CIREQ Workshop Oct. 25th, / 38