Filling in the Blanks: Network Structure and Systemic Risk

Transcription

1 Filling in the Blanks: Network Structure and Systemic Risk Kartik Anand, Bank of Canada Ben Craig, Federal Reserve Bank of Cleveland & Deutsche Bundesbank Goetz von Peter, Bank for International Settlements The views expressed in this presentation do not necessarily reflect those of the institutions the authors are affiliated with. Restricted

2 Motivation Interbank contagion is central, but bilateral linkages often unknown Standard: estimate counterparty exposures by maximum entropy Yet spreading exposures as evenly as possible can be misleading: Conceals true structure of linkages in network analysis Diversification assumption causes bias in systemic stress tests This short paper proposes opposite benchmark: minimum density Produces a highly concentrated sparse network that Retains some of the original network structure, and Provides a bound for systemic stress tests Restricted 2

3 Outline A Primer on the Maximum Entropy Method Introducing the Minimum Density Method An Illustrative Example Application to the German Interbank Network Stress-test Results Conclusion Restricted 3

4 The Maximum Entropy Method Some preliminaries Interbank network: X [0, ) N N Bilateral exposures: X ij 0 (X ii = 0) Interbank assets: A i = j X ij Interbank liabilities: L i = j X ji Suppose we only know A i and L i (the marginals ) for each bank Find matrix E that satisfies the marginal, given prior Q ij = A i L j : min E i,j E ij log E ij Q ij s.t. j E ij = A i and j E ji = L i Restricted 4

5 The Maximum Entropy Method Advantages: Implementable using a standard iterative algorithm (RAS) Yields a unique solution for E Disadvantages: Due to maximum risk-sharing, estimate E: - Does not capture stylized facts of real networks - Is typically robust to contagion, thereby potentially under estimating systemic risk Restricted 5

6 The Minimum Density Method Premise: Interbank lending and borrowing is based on relationships (Coco et al 2009) network linkages are costly Minimally connected network are efficient outcomes for networks (Jackson 2008) min Z N j=1 N j=1 N i=1 N j i Z ij = A i Z ij = L i c 1 [Zij >0] Z ij 0 s.t. i = 1,2,, N i = 1,2,, N i, j Analogous to transport network design problems: NP-hard Restricted 6

7 The Minimum Density Method: Value Function Soften the marginals and define a (unit-less) value function AD i Z = N j=1 V Z = N i=1 N j i Penalty terms: c~o 1, α i = Z ij A i, LD i Z = N j=1 N c 1 Zij >0 i=1 1 A i 2, and δ i = α i AD i 2 + δ i LD i 2 1 L i 2 Z ji L i We can rank networks according to V Z Two issues: (1) exhaustive search; (2) does not account for our priors Restricted 7

8 The Minimum Density Method: Beliefs Economic incentives disassortative interbank relationships Q ij max AD i LD j, LD j AD i i, j Big lender to small borrower, or small lender to big borrower Sampling distribution, P(Z), trades-off information: max P(Z) Z P Z V Z + θr(p Q) P Z = Q Z eθ V(Z) Z Q Z e θ V(Z ) Restricted 8

9 The Minimum Density Algorithm: Procedure Step 1: Pick link (i, j) according to probability Q ij Step 2: if Z ij is unallocated, then Fill Z ij λ min AD i, LD j, where λ 1 - If V Z = Z + Z ij V Z, then accept link Z ij - Else accept with probability P(Z ) P(Z) =exp θ V Z V Z Step 3: Update priors Q ij Step 4: While allocations < 1 ε 100%, return to Step 1 Restricted 9

10 Actual Data Estimated Networks True Network Maximum Entropy Solution A B C D E F G A i A B C D E F G A i A B C D E F G L i Observable Interbank Market Minimum Density Solution A B C D E F G A i A B C D E F G A i A B C D E F G L i Restricted 10

11 Actual Data Estimated Networks True Network Maximum Entropy Solution A B C D E F G A i A B C D E F G A i A B C Density 33% Density 62% D E F G L i Observable Interbank Market Minimum Density Solution A B C D E F G A i A B C D E F G A i A B ? C Density 21% D E F G L i Restricted 11

12 The German Interbank Market Data collected by BuBa ( Gross- und Millionenkreditstatistik ) All large ( Є 1.5m) or concentrated (>10% K) exposures Consolidated at the bank holding company level ( Konzern ) and excludes cross-border linkages Basic network characteristics Large (N = 1802), sparse (density=0.6%) But most banks active on both sides Maximum Entropy (ME) conceals structure (density 93%) Minimum Density (MD) solution is efficient (density 0.1%) Restricted 12

13 Trade-off Between Number and Size of Links Restricted 13

14 Summary Statistics: MD preserves some features better than ME Restricted 14

15 Degree Distributions: MD Retains Some Features Restricted 15

16 Stress-testing Run stress-tests to compare ME, MD with true network in practice Standard simulation methodology: Trigger: single bank failure + a 2% capital ratio shock for all banks Mechanism: Eisenberg-Noe clearing vector LGD is endogenous + allow for bankruptcy cost Let each bank, i = 1,, 1802, fail 1x1 and solve for clearing vector # banks in default (capital ratio < 6%) due of contagion (excluding i) Interbank liabilities in default (plus bankruptcy costs) Repeat for all bankruptcy costs, and report average over i s Run separately for the 3 input networks: true X, ME, MD, LD Restricted 16

17 Stress-testing: Eisenberg and Noe Clearing Algorithm Restricted 17

18 Conclusion The paper has a simple goal: to provide a meaningful alternative to maximum entropy (minimum density) Derived using some information theory and economic rationale The approach retains more information on network structure In stress testing it may not do better than ME but together with ME provide reasonable confidence bands The broad range shows: linkages matter for systemic risk! Thank you for your attention. Restricted 18