Managing Default Contagion in Financial Networks

Transcription

1 Managing Default Contagion in Financial Networks Nils Detering University of California, Santa Barbara with Thilo Meyer-Brandis, Konstantinos Panagiotou, Daniel Ritter (all LMU) CFMAR 10th Anniversary Conference, May 2017 Financial support by: Institut Europlace de Finance and Frankfurt Institute for Risk Management and Regulation

2 Contents Introduction Default fraction Managing systemic risk N. Detering Default Contagion Introduction 2

3 Introduction Systemic risk: risk that in case of an adverse (local) shock substantial parts of the system default due to contagion effects. Aim: to determine capital requirements in terms of statistical network characteristics that secure financial systems against systemic risk (default contagion). Tool: asymptotic analysis of default contagion in weighted, directed, inhomogeneous random graphs. in the spirit of Gai and Kapadia (2010), Amini et al. (2013), Hurd (2016) N. Detering Default Contagion Introduction 3

4 Random financial network model Stylized features of financial networks: the graph is directed there are weights on the edges the graph is sparse: number of edges linear in network size the networks are large strong inhomogeneity in degrees (core/periphery structure) as well as exposures and capital endowments N. Detering Default Contagion Introduction 4

5 Random financial network model Let i {1,..., n}. The in-degree resp. out-degree of vertex i is given by D i := #{j edge from j to i} resp. D + i := #{j edge from i to j} The empirical distribution of the degree sequence is given by: P(k, l) := 1 n #{i D i = k, D + i = l} N. Detering Default Contagion Introduction 5

6 Random financial network model Let i {1,..., n}. The in-degree resp. out-degree of vertex i is given by D i := #{j edge from j to i} resp. D + i := #{j edge from i to j} The empirical distribution of the degree sequence is given by: P(k, l) := 1 n #{i D i = k, D + i = l} Observation: Financial networks exhibit core-periphery structures with empirical degree distributions that might not have second moments ((Boss et al., 2004; Cont et al., 2013; Craig and von Peter, 2014) ) N. Detering Default Contagion Introduction 5

7 Random graph model Gai and Kapadia (2010), Amini et al. (2013), Hurd (2016) analyze default contagion in the configuration model with second moment of the degree sequence We will see that lack of second moment has significant implications for the resilience of financial networks! N. Detering Default Contagion Introduction 6

8 Random financial network The directed inhomogeneous random graph: Financial institutions (vertices) [n] := {1,..., n} N. Detering Default Contagion Introduction 7

9 Random financial network The directed inhomogeneous random graph: Financial institutions (vertices) [n] := {1,..., n} Each bank i [n] has two (deterministic) weights (dependence on n suppressed): w i w + i R + determines tendency to have incoming edges R + determines tendency to have outgoing edges N. Detering Default Contagion Introduction 7

10 Random financial network The directed inhomogeneous random graph: Financial institutions (vertices) [n] := {1,..., n} Each bank i [n] has two (deterministic) weights (dependence on n suppressed): w i w + i R + determines tendency to have incoming edges R + determines tendency to have outgoing edges A directed edge from i to j appears independently with probability { p i,j := min 1, w } + i w j n Proposed and analyzed in Detering, Meyer-Brandis, and Panagiotou (2015) N. Detering Default Contagion Introduction 7

11 Random financial network model Capital and Exposure weights: We associate to each bank i [n] Net worth (capital) x i (possibly random) List of potential exposures L ji 0, j [n] (possibly random) Lji 0 is potential monetary liability of bank j to bank i Capital and exposures independent of random graph N. Detering Default Contagion Introduction 8

12 Random financial network model Capital and Exposure weights: We associate to each bank i [n] Net worth (capital) x i (possibly random) List of potential exposures L ji 0, j [n] (possibly random) Lji 0 is potential monetary liability of bank j to bank i Capital and exposures independent of random graph Random financial network: {L ij } i,j [n] (ω) := {1 {edge from j to i} Lji } i,j [n] (ω) N. Detering Default Contagion Introduction 8

13 Default contagion Default cascades: The set of initially defaulted banks: D 0 = {i [n] x i 0} Contagion process triggered by D 0 : Default of bank j induces loss L ji for its counterpart i. Default cascade D 0 D 1... D n 1 is given by D k = {i [n] x i L ji } j D k 1 D n 1 is the final default cluster in the network generated by the fundamental defaults D 0. N. Detering Default Contagion Introduction 9

14 Systemic risk indicator In the following we will focus on the final fraction of defaulted banks after contagion as systemic risk indicator: Λ n := D n 1 n Determine the fraction Λ n. Categorize resilient/non-resilient cases. When is Λ n big, when small? Manage the network, lets make Λ n small for most shocks! N. Detering Default Contagion Introduction 10

15 Systemic risk indicator In the following we will focus on the final fraction of defaulted banks after contagion as systemic risk indicator: Λ n := D n 1 n Determine the fraction Λ n. Categorize resilient/non-resilient cases. When is Λ n big, when small? Manage the network, lets make Λ n small for most shocks! Asymptotic analysis for n of the default fraction via generalized bootstrap percolation in weighted, directed, inhomogeneous random graph (use results from Detering, Meyer-Brandis, and Panagiotou (2015)). N. Detering Default Contagion Introduction 10

16 Contents Introduction Default fraction Managing systemic risk N. Detering Default Contagion Default fraction 11

17 Asymptotic default fraction For each bank i: { L j,i } j [n] sequence of exchangeable random variables we associate a hypothetical default threshold { 0, if x i 0, c i := inf{s [n 1] s L j=1 j,i x i }, else, N. Detering Default Contagion Default fraction 12

18 Asymptotic default fraction For each network size n, the (random) financial network model is characterized by the empirical distribution of (w i, w + i, c i ) i=1,...,n ; i.e. by a corresponding random vector (Wn, W n +, C n ) For n we assume (W n, W + n, C n ) (W, W +, C) We assume existence of first moment of (W, W + ) (equiv. of degree distribution) N. Detering Default Contagion Default fraction 13

19 Asymptotic default fraction Define [ ] f (z; (W, W +, C)) := E W + P[Poi(z W ) C] [ ] g(z; (W, C)) := E P[Poi(z W ) C] z Let ẑ be smallest, nonnegative root of f (z; (W, W +, C)) N. Detering Default Contagion Default fraction 14

20 Asymptotic default fraction Theorem: 1 For all ε > 0 with high probability: Λ n (ω) g(ẑ; (W, C)) ε 2 If f (ẑ; (W, W +, C)) < 0, then Λ n = Λ n (ω) p g(ẑ; (W, C)), as n. N. Detering Default Contagion Default fraction 15

21 Example default fraction f z g z z Figure: Determination of the theoretical final default fraction with p = 1% initial defaults and constant threshold 2. In blue: f (z) = (1 p)e[w + P(Poi(W z) 2)] + pe[w + ] z with root ẑ In red: g(z) = (1 p)e[p(poi(w z) 2)] + p with g(ẑ) N. Detering Default Contagion Default fraction 16

22 Resilient/non-resilient networks When do small infections spread? Start with P(C = 0) = 0, so no initial defaults. Then some banks default ex post: Vertices i [n] receive binary mark m i, which is either 0 (infected) or 1 (not infected), new threshold value is c i m i (W, W +, C) ex post infection ========= (W, W +, C M ) C M :=CM Assume P(C M = 0) > 0: When is Λ n lower bounded independent of M? (non-resilient case) When is Λ n small, given that P(C M = 0) is small? (resilient case) N. Detering Default Contagion Default fraction 17

23 Resilient networks (Amini et al., 2013) analyze resilience of networks in the configuration model with second moment of the degree sequence We will see that lack of second moment has significant implications for the resilience of financial networks! N. Detering Default Contagion Default fraction 18

24 Application to systemic risk Theorem (Non-Resilience): Assume f (0; (W, W +, C)) > 0. Then, for any C M with P(C M = 0) > 0 with high probability [ ] Λ n E P[Poi(ẑ W ) C] > 0 Lower bound does not depend on C M! Any small shock spreads to a substantial part (linear fraction)! N. Detering Default Contagion Default fraction 19

25 Application to systemic risk Theorem: f (0; (W, W +, C)) > 0 non-resilient system f z g z N. Detering Default Contagion Default fraction 20

26 Application to systemic risk Theorem: f (0; (W, W +, C)) > 0 non-resilient system f z g z Any small shock spreads to a substantial part (linear fraction)! N. Detering Default Contagion Default fraction 21

27 Application to systemic risk Theorem (Resilience): Assume f (0; (W, W +, C)) < 0. Then for any δ there is an ɛ such that if P(C M = 0) < ɛ then with high probability Λ n δ Small shocks remain local! N. Detering Default Contagion Default fraction 22

28 Application to systemic risk Theorem: f (0; (W, W +, C)) < 0 resilient system f z g z N. Detering Default Contagion Default fraction 23

29 Application to systemic risk Theorem: f (0; (W, W +, C)) < 0 resilient system f z g z Small shocks remain local! N. Detering Default Contagion Default fraction 24

30 Contents Introduction Default fraction Managing systemic risk N. Detering Default Contagion Managing systemic risk 25

31 Systemic risk capital requirements Question: Given a financial network {L ij } i,j [n], how do we have to set the capital levels {x i } i [n] to make the network resilient? minimal capital requirements Depends on distribution in the tails of (W +, W ) (or equivalently in/out degrees) and monetary exposures L ij. We show in the paper how to estimate these quantities. N. Detering Default Contagion Managing systemic risk 26

32 Systemic risk capital requirements Theorem: If W +, W have finite variance and x i > max j [n] L j,i a. s. for all i, then the network is resilient. No contagious links resilient network [Amini, Cont, Minca, 2013] However, many financial (and other real world) networks are ultra small W +, W power law with infinite variance N. Detering Default Contagion Managing systemic risk 27

33 Systemic risk capital requirements Let W, W + have power law with exponents β, β + (2, 3) γ := 2 β + (β 1)/(β + 1) α := (β + 1)/(β + 2) τ : R + N such that lim inf w τ(w)/w γ > α N. Detering Default Contagion Managing systemic risk 28

34 Systemic risk capital requirements Let W, W + have power law with exponents β, β + (2, 3) γ := 2 β + (β 1)/(β + 1) α := (β + 1)/(β + 2) τ : R + N such that lim inf w τ(w)/w γ > α Theorem: For all banks i [n] let x i > max j [n] L j,i and x i τ(w i )E[ L 1,i ] a.s.. Then the financial network is resilient. N. Detering Default Contagion Managing systemic risk 28

35 Systemic risk capital requirements Theorem makes statements about resilience that depend on the quantity γ Since β > 2 and β + > 2, always γ < 1 β > 3 and β + > 3: then γ < 0 and τ(w) = 2 ensures resilience β < 3 and β + > 3 or vice versa: γ < 0, γ = 0 and γ > 0 are possible. Proof based on an estimates of f (z, (W, W +, C)) = E[W + W P(Poi(zW ) = C 1)1 {C 1} ] 1 = E[W + W P(Poi(zW ) = τ(w ) 1)] 1 for z (0, z 0 ) (simlliar to Laplace method, see de Bruijn (1970)) N. Detering Default Contagion Managing systemic risk 29

36 Example τ(w i ) = max { 2, (α(1 + δ)(w i ) γ(1+δ) } p = Figure: Influence of δ on the shape of f (z) = E[W + P(Poi(W z) C)] z. N. Detering Default Contagion Managing systemic risk 30

37 Example τ(w i ) = max { 2, (α(1 + δ)(w i ) γ(1+δ) } δ = p 0.02 p p z 0.1 Figure: Influence of p on the shape of f (z) = (1 p)e[w + P(Poi(W z) C)] + pe[w + ] z for the example of δ = N. Detering Default Contagion Managing systemic risk 31

38 Simulations 1.0 final fraction number of banks Figure: Convergence of the final fraction of defaulted banks for networks of finite size. β = and β + = , threshold value 2 N. Detering Default Contagion Managing systemic risk 32

39 References I Hamed Amini, Rama Cont, and Andreea Minca. Resilience to contagion in financial networks. Mathematical Finance, pages n/a n/a, Michael Boss, Helmut Elsinger, Martin Summer, and Stefan Thurner. Network topology of the interbank market. Quantitative Finance, 4(6): , R. Cont, A. Moussa, and E.B. Santos. Network structure and systemic risk in banking systems. In J-P. Fouque and J.A. Langsam, editors, Handbook on Systemic Risk. Cambridge, Ben Craig and Goetz von Peter. Interbank tiering and money center banks. Journal of Financial Intermediation, (0):, N.G. de Bruijn. Asymptotic Methods in Analysis. Bibliotheca mathematica. Dover Publications, ISBN URL Nils Detering, Thilo Meyer-Brandis, and K. Panagiotou. Bootstrap percolation in inhomogeneous directed random graphs. submitted, P. Gai and S. Kapadia. Contagion in Financial Networks. SSRN elibrary, Tom R. Hurd. Contagion! Systemic Risk in Financial Networks. Springer, ISBN N. Detering Default Contagion Managing systemic risk 33

40 Thank you! Nils Detering Department of Statistics and Applied Probability University of California, Santa Barbara CA USA N. Detering Default Contagion 34