On the formation and stability of core-periphery networks in the interbank market

Transcription

1 On the formation and stability of core-periphery networks in the interbank market Marco van der Leij 1 joint with Cars Hommes 1, Daan in t Veld 1 1 Universiteit van Amsterdam - CeNDEF Lorentz Workshop Econophysics and Networks, 28 May 2013

2 Introduction The risk of the collapse of the interbank market (systemic risk) depends on the structure of the network of interbank loans. Theoretical and simulation analysis of contagion (Allen & Gale, 2000; Gai & Kapadia, 2010; and many others) Stress tests on empirically derived network structures (Upper, 2011)

3 Introduction The risk of the collapse of the interbank market (systemic risk) depends on the structure of the network of interbank loans. Theoretical and simulation analysis of contagion (Allen & Gale, 2000; Gai & Kapadia, 2010; and many others) Stress tests on empirically derived network structures (Upper, 2011) In these analyses it is assumed that the financial network is exogenously fixed. However, networks do not come out of the blue, but are consciously formed by profit-maximizing banks. What structure do financial networks have? Why do financial networks have the structure that they have?

4 Introduction Empirically, it seems that before the financial crisis financial networks were fitted well by a core-periphery structure. Craig & Von Peter (2010); Fricke & Lux (2012); Van Lelyveld & In t Veld (2013) Two tiers of banks: core and periphery Core banks are completely connected within the core Peripheral banks only have connections with core banks

5 Finding the core in the Netherlands Estimation gives core of 10 to 19 banks. Interbank market in 1998Q1 (left) and its fit on the CP model (right). Core banks represented as inner circle of small white dots.

6 Degree distribution CCDF of the degree distribution of the Dutch interbank exposure network. No power law, but two groups. CCDF of the interbank market in 1998Q1.

7 Introduction Why do financial networks have such a core-periphery structure? Answering this question will help us understand: what the benefits and costs of a particular network structure are (besides changing systemic risk) whether the current network structure is likely to be socially efficient (i.e. not over-/underconnected or over-/underdispersed) how a social planner can influence network structure (and thus mitigate systemic risk) These are important questions in the light of discussions to regulate the over-the-counter interbank loan market and introducing central counterparty clearinghouses.

8 Introduction In this paper we consider an economic theoretical approach to network formation of a (financial) trade network. a network is unilaterally stable if there is no player with a profitable opportunity to create or delete links. We focus on the role of intermediation/brokerage Trade opportunities can be realized directly or indirectly through an intermediator (broker) Traders benefit from lower transaction costs Brokers obtain intermediation benefits

9 Introduction In this paper we consider an economic theoretical approach to network formation of a (financial) trade network. a network is unilaterally stable if there is no player with a profitable opportunity to create or delete links. We focus on the role of intermediation/brokerage Trade opportunities can be realized directly or indirectly through an intermediator (broker) Traders benefit from lower transaction costs Brokers obtain intermediation benefits In this version, we do not consider the possibility of a crisis or systemic risk.

10 Trade and distribution Consider an undirected network g between n agents. g ij = g ji = 1 denotes a link (trading relationship), and g ij = g ji = 0 the absence of a link. Payoffs of agents are determined as follows: each pair of agents (i,j) creates a trade surplus of α ij as long as i and j are connected by a path of at most length 2 if i and j are at distance 3 or more, trade opportunities cannot be realized

11 Trade and distribution Consider an undirected network g between n agents. g ij = g ji = 1 denotes a link (trading relationship), and g ij = g ji = 0 the absence of a link. Payoffs of agents are determined as follows: each pair of agents (i,j) creates a trade surplus of α ij as long as i and j are connected by a path of at most length 2 if i and j are at distance 3 or more, trade opportunities cannot be realized if i and j are directly connected, then they each receive half of the surplus if i and j are indirectly connected by m ij intermediators, then i, j and the intermediators share the trade surplus i and j each receive f e (m ij,δ) each of the m ij intermediators receive f m (m ij,δ) By definition 2f e (m,δ) + mf m (m,δ) = 1

12 Trade and distribution Consider intermediated trade for some pair (i, j). We assume if there is one intermediator, then the three parties split the surplus evenly f e (1,δ) = f m (1,δ) = 1/3. i k j 1/3 1/3 1/3

13 Trade and distribution Trade between i and j: m = 3, δ = 0 k 1 1/9 i 1/3 k 2 1/9 j 1/3 k 1 1/9

14 Trade and distribution Trade between i and j: m = 3, δ = 3/5 k 1 1/15 i 2/5 k 2 1/15 j 2/5 k 1 1/15

15 Trade and distribution Trade between i and j: m = 3, δ = 1 k 1 0 i 1/2 k 2 0 j 1/2 k 1 0

16 Payoff function If g ij = 1, then both i and j pay a cost c. The payoff function, π i (g), for an agent i is: π i (g) = ( 1 α 2 ij c) + α ij f e (m ij,δ) + α kl f m (m kl,δ) j Ni 1 j Ni 2 k,l Ni 1 g kl =0 direct trade indirect trade brokerage benefits where Ni r denotes the set of nodes at distance r 1 from i in g and ni r = Ni r its size.

17 Unilateral stability We consider the economic concept of stable or equilibrium networks. In particular, we consider the concept of unilateral stability A network g is unilaterally stable if there is no agent i, such that simultaneously agent i strictly benefits from severing (some of) its existing links or proposing new links involving i none of the agents to which i proposes a new link is worse off

18 Example of a complete core-periphery network

19 Core-periphery network with homogeneous agents Consider first the case of homogeneous agents trade surplus is 1: i,j,i j : α ij = 1

20 Core-periphery network with homogeneous agents Consider first the case of homogeneous agents trade surplus is 1: i,j,i j : α ij = 1 Witregel Theorem For any c and δ, any complete core-periphery network of size n and core size k with 2 k n 2 is not unilaterally stable.

21 Intuition Proof Suppose a complete core-periphery network is unilaterally stable.

22 Intuition Proof g 3, It is better for agents 3 and 4 to trade indirectly through the 2 core agents.

23 Intuition Proof g 1, But then it should be better for agents 1 and 2 to trade indirectly through 4 agents.

24 Stable networks If the core-periphery network is not (unilaterally) stable, what kind of networks are? We consider a dynamic network formation process: Initial network is the empty network Round-robin best response dynamics (Kleinberg et al., 2008) This process always converges for any c and δ, and the resulting network is unilaterally stable There may be more unilateral stable networks to which the network does not converge to

25 Absorbing states dynamic process. n = 8. c Empty network Star Multipartite networks Complete network δ

26 Extensions Why are core-periphery networks unstable in the framework above? What additional assumption are necessary to explain the existence of core-periphery networks? Instability follows from homogeneity banks If banks in the periphery want to trade indirectly through core banks, then banks in the core want to trade indirectly through periphery banks as well!

27 Heterogenous agents Consider now a case in which banks are heterogeneous Two types of banks: k big banks and n k small banks Big banks generate more trade surplus α ij : with some α > 1. α ij L S L α 2 α S α 1

28 Heterogenous agents Theorem Suppose c > 1 2 f e(k,δ). Then there exists an α > 0, such that for all α > α the complete core-periphery network with k big banks in the core is unilaterally stable. Intuition: if α is very large, then it is more attractive for core banks to trade directly with each other.

29 Absorbing states of homogeneous dynamic process (n = 100) Star c Multipartite networks 0.2 Complete network δ

30 Contourplot of heterogeneity α > 1 required to stabilize network g CP(k=10) com for n = 100

31 Conclusions In this paper we ask ourselves: why do financial networks have a core-periphery structure? We focus on the role of intermediation benefits. We find that: A core-periphery network is not stable if agents are homogenous If there is heterogeneity in trade surplus, then a core-periphery network with big banks in the core is stable This result suggests that we cannot abstract away heterogeneity in banking size if we want to understand financial networks.

32 Conclusions In this paper we ask ourselves: why do financial networks have a core-periphery structure? We focus on the role of intermediation benefits. We find that: A core-periphery network is not stable if agents are homogenous If there is heterogeneity in trade surplus, then a core-periphery network with big banks in the core is stable This result suggests that we cannot abstract away heterogeneity in banking size if we want to understand financial networks. In future research we would like to introduce default probabilities in order to understand network formation in stress situations.

33 Related Literature Network formation theory (Jackson & Wolinsky, 1996; Bala & Goyal, 2000; see Goyal, 2007, for textbook review) Network formation, intermediation benefits and structural holes (Burt, 1994; Kleinberg et al., 2007; Goyal & Vega-Redondo, 2007; Buskens & Van de Rijt, 2008) Network formation of financial networks (Babus, 2009; Babus, 2011) Networks and trade (Kranton & Minehart, 2001; Coraminas-Bosch, 2004; Blume et al. 2009) Intermediation and bargaining on networks (Siedlarek, 2011; Manea, 2012; Gofman, 2011) Network formation and core-periphery networks (Galeotti & Goyal, 2010; Persitz, 2012)