Failure and Rescue in an Interbank Network

Transcription

1 Failure and Rescue in an Interbank Network Luitgard A. M. Veraart London School of Economics and Political Science October 202 Joint work with L.C.G Rogers (University of Cambridge) Paris 202 Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October 202 / 30

2 Questions How can default and contagion be modelled in financial networks? Under which circumstances do banks have incentives to bail-out defaulting banks? What are resolution mechanisms of financial distress? Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

3 An Interbank Network bank bank bank bank bank bank Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

4 The Setting Market with n banks with indices in N := {,..., n}; nodes in a network. The liabilities matrix is L R n n, where the ij th entry L ij represents the nominal liability of bank i to bank j. Assumption: L ij 0 i, j and L ii = 0 i. The total nominal obligations of bank i to all other banks in the system are given by L i = n j= L ij. The relative liabilities matrix Π R n n is defined by { Lij / L π ij := i if L i > 0, 0 otherwise. We denote by e i 0 the net assets of bank i from sources outside the banking system. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

5 Default in a Financial Network Consider network with liabilities matrix L and net external assets e. Check for each bank i N whether n π ji L j + e i j= } {{ } in flow L i }{{} out flow 0. If this is < 0 for some banks, some banks are bankrupt. Assumptions in case of default: Default costs: Fraction of net external assets realized on liquidation: α (0, ], Fraction of interbank assets realized on liquidation: β (0, ]. Clearing: Clearing mechanism determines the payments between banks. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

6 The Clearing Mechanism Limited liabilities: Nodes never pay more than available cash flow. Priority of debt claims over equity: Paying off the liabilities L ij has priority, even if net assets e i have to be used. Proportionality: If default occurs the defaulting bank pays all claimant banks in proportion to the size of their nominal claims on the assets of the defaulting bank. A clearing vector for the financial system (L, e, α, β) is a vector L [0, L] such that L = Φ(L ), { Φ(L) i := L i, if L i e i + n j= L jπ ji, αe i + β n j= L jπ ji, else. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

7 Clearing Vectors - Existence Theorem (Existence of Clearing Vectors) Let α, β (0, ]. For every financial system (L, e, α, β) there exist clearing vectors L and L such that for any clearing vector L, we have L L L. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

8 Greatest Clearing Vector Algorithm (GA) constructs sequence ( Λ (ν)) : Set ν = 0, Λ (0) := L and I :=. 2 For all nodes i compute v (ν) i := n j= Λ(ν) j π ji + e i L i. 3 Insolvent banks: I ν ={i : v (ν) i < 0}, solvent banks: S ν = {i : v (ν) i 0}. 4 If I ν I ν terminate the algorithm. Otherwise set Λ (ν+) j := L j j S ν, solve { } x i = αe i + β L j π ji + x j π ji j S ν j I ν and set Λ (ν+) i := x i for i I ν. i I ν 5 Set ν ν + and go back to 2. When the algorithm has terminated, the vector Λ (ν) is a clearing vector. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

9 Example I Bank 3 Bank Bank 2 l 3 = 2 l 2 = 2 e = /2 3/2, L = ; L = 2 0 2, Π = Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

10 Example II γ 3 2 γ γ 2 Bank 3 Bank Bank 2 π 3 L3 = 2 π 2 L = 2 Net assets are reduced to γe, γ [0, ]. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

11 Example III Bank 3 Bank Bank 2 π 3Λ (0) 3 = 2 π 2Λ (0) = 2 Example: γ = 6. Define new net assets e := ( 6, 2, 4 ). ν = 0, Λ (0) = L, v (0) = ( , 2 + 2, 4 2 ). I 0 = {3}, S 0 = {, 2}. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October 202 / 30

12 Example IV Bank 3 Bank Bank 2 π 3Λ (0) 3 = 2 π 2Λ (0) = 2 I 0 = {3}, S 0 = {, 2}. Λ () = L = 2, Λ() 2 = L 2 = 0, Λ () { 3 = x 3 where x 3 = αe 3 + β L j S0 j π j3 + } j I 0 x j π j3 = α 4 + β{ } = α 4. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

13 Example V Bank 3 Bank Bank 2 π 3Λ () 3 = 8 π 2Λ () = 2 ν :=. α = 2. v () = ( , 2 + 2, 4 2 ), I = {, 3}, S = {2}. Λ (2) = x, Λ { (2) 2 = L 2 = 0, Λ () 3 = x 3, where x = αe +β L j S0 j π j + } j I 0 x j π j = 2 +β{0+x 3+x 0} = 2 + βx 3. { x 3 = αe 3 +β L j S0 j π j3 + } j I 0 x j π j3 = 8 +β{ 2 0+x 0+x 3 0} = 8. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

14 Example VI Bank 3 Bank Bank 2 π 3Λ (2) 3 = 8 π 2Λ (2) = 2 + β 8 ν := 2. v (2) = ( , I 2 = {, 3} = I. STOP! β 8, 4 2 ), L = Λ (2) = ( 2 + β 8, 0, 8 ). Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

15 Consequences The greatest clearing vector algorithm (GA) produces a sequence of vectors ( Λ (ν)) decreasing in at most n iterations to the greatest clearing payment vector. We call the set I ν the level-ν insolvency set. The level-0 insolvency set is the set of those banks which would default even if all other banks paid their obligations in full. The level-ν insolvency set is the set of all those banks which would not be able to meet their obligations if all the level-(ν ) insolvent banks were to default. The insolvency sets I ν trace the spread of default through the financial system. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

16 Bank Merger Given a financial system (L, e, α, β) with banks of indices N. After a merger of all banks in B N one obtains a new financial system ( L, ẽ, α, β) indexed by Ñ := {0} (N \ B). We assume that a merger is associated with costs. We model costs from merger in terms of a vector κ R n +. If bank i is involved in a merger, costs of size κ i occur. New liabilities matrix L: The liabilities of the merging banks to the other (non merging) banks in the network are added up. The liabilities to those banks which merge are cancelled. New net assets ẽ: The net assets of the merged bank are the sum of the net assets of the banks that merged minus the costs for merger, i B κ i. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

17 Example of Merger - Initial Network e e 2 l 2 bank bank 2 l 2 l 3 l 4 l 24 l 32 bank 4 l 43 bank 3 e 4 e 3 Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

18 Example of Merger - Change Liabilities and Net Assets e e 2 l 2 bank + 4 bank 2 l 2 l 24 l 3 l 43 l 32 bank 3 e 4 e 3 Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

19 Example of Merger - New Network ẽ 0 = e + e 4 ẽ 2 = e 2 bank 0 l0 = l 2 l20 = l 2 + l 24 bank 2 l03 = l 3 + l 43 l32 = l 32 bank 3 ẽ 3 = e 3 Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

20 Rescue Consortium Let (L, ẽ, α, β) be a financial system where the level-0 insolvency set I 0 is non-empty and let L be the greatest clearing vector. The value V of the banks is defined as V(L, ẽ) i := (Π L + ẽ L ) i I {L i L i }. The bailout costs are j I 0 δ j, with δ := max { 0, ( Π L + ẽ L )}. Let Ṽ := max{0, Π L + ẽ L}, V := Ṽ V(L, ẽ). A rescue consortium is a set A N \ I 0 such that: i A V i > j I 0 δ j + k A I 0 κ k (rescue incentive), 2 i A Ṽi > j I 0 δ j + k A I 0 κ k (rescue ability). Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

21 Rescue Incentive and Rescue Ability Theorem Every rescue consortium which has an incentive to rescue the failing banks also has the ability to rescue the failing banks. 2 Suppose that the set of banks at risk of contagious default R := ν I ν \I 0 is non-empty, and suppose, further that some subset A R is able to rescue the failing banks. Then A also has an incentive to rescue the failing banks. Definition (Rescued Financial System) Let (L, ẽ, α, β) be a financial system in which the level-0 insolvency set I 0 is non-empty, and suppose that a rescue consortium defined by a set of indices A exists. Then the rescued financial system is the financial system obtained by a merger of all banks in I 0 A. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

22 Absence of Rescue Consortium Theorem Consider a financial system (L, e, α, β) in which all banks are initially solvent. Suppose the assets e are reduced to ẽ, with ẽ i e i i with the result that at least one bank becomes level-0 insolvent. Suppose that α = β =. Then no group of banks in the network has an incentive to rescue the insolvent bank(s). Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

23 Presence of Rescue Consortium Theorem Let (L, ẽ, α, β) be a financial system with α, β [0, ). Suppose that I 0 is a proper subset of N : I 0 N. Let L be the corresponding greatest clearing vector and let κ be the vector describing the costs for merger. Suppose n n n ( α)ẽ i + ( β) L j π ji I {L i < L i } > κ k. i= j= Then there exists a rescue consortium. k= Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

24 Assessing and Controlling Contagion Risk Consider financial system (L, e, α, β) with I 0 =. Reduce net assets to γe, γ [0, ] and consider the relative losses due to default: γ λ(γ) := n i= e v(γ) n i= e The function λ measures the difference between the initial value of the system and the value of the stressed system divided by the initial value of the system. the proportion of defaulting banks: γ η(γ) := {i N L i (γ) < L i }, N where L (γ) is the greatest clearing vector if the external assets are given by ẽ = γe. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

25 An Interbank Network bank bank bank bank bank bank Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

26 Assessing and Controlling Contagion Risk III Plot of λ and η for the asymmetric network with 6 banks, β = 0.9 (first example, slide 3) λ(γ, α) η(γ, α) γ α γ α Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

27 Possible Resolution Mechanisms for Bank Rescues I Suppose that the banks in I are capable of mounting a rescue. The regulator should be empowered to compel that group of banks to rescue the banks in I 0. Main ideas for different mechanisms: Each bank in I contributes to bailout costs in proportion to losses V that they would experience if default were to occur, receives shares in rescued banks in proportion to their contribution. 2 Each bank i in I gives a sealed bid to the regulator containing the fraction α i of the bailout costs which it was willing to take: If i I α i, banks in I are allocated fractions of the defaulting banks assets and liabilities proportional to their bids α i. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

28 Possible Resolution Mechanisms for Bank Rescues II i I α i <, then each bank in I contributes to the bailout proportionally to its potential losses V as in mechanism () above, but receives a fraction of the defaulting banks proportional to its bid. 3 Allow the regulator to seize the assets of any failing bank, which would then pay out nothing to any bank to which it owed money. Assets could be used to compensate depositors, with any not used in this way being held by the government. Gives other banks a very strong incentive to mount a rescue. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

29 Conclusion Modelled contagion in financial networks. Derived conditions for existence of a rescue consortium. Default costs are necessary for the existence of a rescue consortium. Role of regulator and bail-out decisions. Model can be used for stress testing a given network. Could include random shocks etc. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

30 References I Eisenberg, L. & Noe, T. (200). Systemic risk in financial networks. Management Science 47, Rogers, L. C. G. & Veraart, L. A. M. (202+). Failure and rescue in an interbank network. Forthcoming in Management Science. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

31 Default Costs Lemma Consider a financial system (L, e, α, β) in which all banks are initially solvent. Suppose that the assets e are reduced to ẽ, with ẽ i e i i such that some banks have become insolvent: I 0. Let L be the greatest clearing vector in (L, ẽ, α, β). Then 0 = n (V( L, e) i V(L, ẽ) i ) i= n (e i ẽ i ) + n n ( α)ẽ i + ( β) L j π ji I {L i < L i }. i= i= j= Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

32 Presence of Rescue Consortium - Corollary Theorem Let (L, ẽ, α, β) be a financial system with α, β [0, ). Suppose that I 0 is a proper subset of N : I 0 N. Suppose that the costs for merger are κ 0. Let L be the corresponding greatest clearing vector and suppose that there exists a bank k such that L k < L k which satisfies at least one of the following two conditions: ẽ k > 0, 2 there exists j k such that L j π jk > 0. Then there exists a rescue consortium. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

33 Assessing and Controlling Contagion Risk Consider financial system (L, e, α, β) with I 0 =. Reduce net assets to γe, γ [0, ] and consider the relative losses due to default: γ λ(γ) := n i= e v(γ) n i= e The function λ measures the difference between the initial value of the system and the value of the stressed system divided by the initial value of the system. the proportion of defaulting banks: γ η(γ) := {i N L i (γ) < L i }, N where L (γ) is the greatest clearing vector if the external assets are given by ẽ = γe. Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

34 An Interbank Network bank bank bank bank bank bank Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30

35 Assessing and Controlling Contagion Risk III Plot of λ and η for the asymmetric network with 6 banks, β = 0.9 (first example, slide 3) λ(γ, α) η(γ, α) γ α γ α Luitgard A. M. Veraart (LSE) Failure and Rescue in an Interbank Network October / 30