Motivation: Two Basic Facts

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2 Motivation: Two Basic Facts 1 Primary objective of macroprudential policy: aligning financial system resilience with systemic risk to promote the real economy Systemic risk event Financial system resilience Market failures Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

3 Motivation: Two Basic Facts 1 Primary objective of macroprudential policy: aligning financial system resilience with systemic risk to promote the real economy Systemic risk event Financial system resilience Market failures 2 Bank runs as a typical symptom of financial crises in history Gorton (2012) Reinhart and Rogoff (2009) Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

4 What I Did Developed a two-period general equilibrium model that features 1 Bank runs (systemic event) in a global game framework 2 Endogenous probability of bank runs (banking system resilience) 3 Market failures? Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

5 What I Did Developed a two-period general equilibrium model that features 1 Bank runs (systemic event) in a global game framework 2 Endogenous probability of bank runs (banking system resilience) 3 Market failures? Conducted welfare analyses and studied macroprudential instruments: Leverage restriction (capital requirement) Liquidity requirement Sectoral requirement Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

6 Main Results 1 Excessive bank leverage 2 Insufficient bank liquidity 3 Too high crisis (system-wide bank run) probability 4 Need for policy coordination; risk migration Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

7 Main Results 1 Excessive bank leverage 2 Insufficient bank liquidity 3 Too high crisis (system-wide bank run) probability 4 Need for policy coordination; risk migration 5 Sources of inefficiencies: limited liability + externality specific to a global game; pecuniary externaility Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

8 Main Results 1 Excessive bank leverage 2 Insufficient bank liquidity 3 Too high crisis (system-wide bank run) probability 4 Need for policy coordination; risk migration 5 Sources of inefficiencies: limited liability + externality specific to a global game; pecuniary externaility 6 Applications Sectoral capital requirements and risk weights Risk taking Deposit insurance Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

9 Literature and Road Map Literature Rochet and Vives (2004) Diamond and Dybvig (1983) Carlsson and Van Damme (1993); Morris and Shin (1998) Christiano and Ikeda (2013, 2016) Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

10 Literature and Road Map Literature Rochet and Vives (2004) Diamond and Dybvig (1983) Carlsson and Van Damme (1993); Morris and Shin (1998) Christiano and Ikeda (2013, 2016) Road map Benchmark model with a bank leverage choice only Role of leverage restrictions (capital requirements) Extended model to incorporate a bank liquidity choice Extension to study sectoral capital requirements Example of risk-taking Preliminary result on the dynamic model Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

11 Two-period General Equilibrium Model: Environment Two periods, t = 1, 2 Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

12 Two-period General Equilibrium Model: Environment Two periods, t = 1, 2 Three types of agents Households Banks Fund managers Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

13 Two-period General Equilibrium Model: Environment Two periods, t = 1, 2 Three types of agents Households Banks Fund managers Endowment in t = 1 only Households: y > 0 (income) Banks: n > 0 (bank capital) Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

14 Two-period General Equilibrium Model: Environment Two periods, t = 1, 2 Three types of agents Households Banks Fund managers Endowment in t = 1 only Households: y > 0 (income) Banks: n > 0 (bank capital) All agents are competitive Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

15 Two-period General Equilibrium Model: Environment Two periods, t = 1, 2 Three types of agents Households Banks Fund managers Endowment in t = 1 only Households: y > 0 (income) Banks: n > 0 (bank capital) All agents are competitive Ownership: banks are owned by households Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

16 Timing of Events 1 (t = 1) Households consume c 1 and make deposits d Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

17 Timing of Events 1 (t = 1) Households consume c 1 and make deposits d 2 (t = 1) Deposit management is delegated to fund managers Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

18 Timing of Events 1 (t = 1) Households consume c 1 and make deposits d 2 (t = 1) Deposit management is delegated to fund managers 3 (t = 1) Banks make loans with the return given by R k N( R k, σ 2 R ) k Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

19 Timing of Events 1 (t = 1) Households consume c 1 and make deposits d 2 (t = 1) Deposit management is delegated to fund managers 3 (t = 1) Banks make loans with the return given by R k N( R k, σ 2 R ) k 4 (t = 2) Bank asset return R k is realized, but yet to be known Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

20 Timing of Events 1 (t = 1) Households consume c 1 and make deposits d 2 (t = 1) Deposit management is delegated to fund managers 3 (t = 1) Banks make loans with the return given by R k N( R k, σ 2 R ) k 4 (t = 2) Bank asset return R k is realized, but yet to be known 5 (t = 2) Fund manager receives signal s i = R k + ɛ i, where ɛ i N(0, σɛ 2 ), Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

21 Timing of Events 1 (t = 1) Households consume c 1 and make deposits d 2 (t = 1) Deposit management is delegated to fund managers 3 (t = 1) Banks make loans with the return given by R k N( R k, σ 2 R ) k 4 (t = 2) Bank asset return R k is realized, but yet to be known 5 (t = 2) Fund manager receives signal s i = R k + ɛ i, where ɛ i N(0, σɛ 2 ), 6 (t = 2) Fund managers decide whether to withdraw deposits or not Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

22 Timing of Events 1 (t = 1) Households consume c 1 and make deposits d 2 (t = 1) Deposit management is delegated to fund managers 3 (t = 1) Banks make loans with the return given by R k N( R k, σ 2 R ) k 4 (t = 2) Bank asset return R k is realized, but yet to be known 5 (t = 2) Fund manager receives signal s i = R k + ɛ i, where ɛ i N(0, σɛ 2 ), 6 (t = 2) Fund managers decide whether to withdraw deposits or not 7 (t = 2) Households receive interest and profits and consume c 2 Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

23 Households s.t. max u(c 1) + E(c 2 ), {c 1,c 2,d} c 1 + d y, c 2 vrd + π, where v = { 1 with prob.1 P (no bank default) < 1 with prob.p (bank default) Solution: supply curve of funds: R = u (y d) 1 P + E(v default)p Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

24 Fund Managers: Action and Payoff Fund managers are risk neutral Payoff structure for fund manager i (0, 1): Net benefit of Withdraw over Not withdraw { Γ 0 if bank defaults = Γ 1 if bank survives Fund manager i withdraws iff γ is exogenously given P }{{} i > Γ 1 γ, Γ 0 + Γ 1 Prob. of bank default perceived by i Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

25 Fund Managers: Threshold for R k Costly liquidation: early liquidation of one unit bank asset generates only a faction 1/(1 + λ) of R k, where λ > 0 Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

26 Fund Managers: Threshold for R k Costly liquidation: early liquidation of one unit bank asset generates only a faction 1/(1 + λ) of R k, where λ > 0 x = number of fund managers who withdraw deposits Let L (n + d)/n. In period t = 2, bank defaults iff or R k ( n + d) (1 + λ)xrd < (1 x)rd, ( R k < R 1 1 ) (1 + λx) L Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

27 Fund Managers: Threshold for R k Costly liquidation: early liquidation of one unit bank asset generates only a faction 1/(1 + λ) of R k, where λ > 0 x = number of fund managers who withdraw deposits Let L (n + d)/n. In period t = 2, bank defaults iff or R k ( n + d) (1 + λ)xrd < (1 x)rd, ( R k < R 1 1 ) (1 + λx) L Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

28 Fund Managers: Signal and Threshold Strategy Threshold strategy: withdraw if s i < s Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

29 Fund Managers: Signal and Threshold Strategy Threshold strategy: withdraw if s i < s Solution: Pr ( ( R k < R 1 L) 1 [ ] ) 1 + λx(r k, s ) s = γ, x(r k, s ) = Pr(R k + ɛ i < s ) Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

30 Fund Managers: Signal and Threshold Strategy Threshold strategy: withdraw if s i < s Solution: Pr ( ( R k < R 1 L) 1 [ ] ) 1 + λx(r k, s ) s = γ, x(r k, s ) = Pr(R k + ɛ i < s ) Bank goes bankrupt iff R k < R k ( R k R k ) P = Φ F (R k ) σ R k Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

31 Fund Managers: Signal and Threshold Strategy Threshold strategy: withdraw if s i < s Solution: Pr ( ( R k < R 1 L) 1 [ ] ) 1 + λx(r k, s ) s = γ, x(r k, s ) = Pr(R k + ɛ i < s ) Bank goes bankrupt iff R k < R k ( R k R k ) P = Φ F (R k ) Limit solution σ ɛ 0: s = R k = R σ R k ( 1 1 ) [1 + λ(1 γ)] L Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

32 Bank s Problem Bank defaults iff R k < R k Bank s problem: E(π) = max {L} R k (L) subject to L L max { [ ( )] } R k L R 1 + λx R k, s (L) (L 1) ndf (R k ) Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

33 Bank s Problem Bank defaults iff R k < R k Bank s problem: E(π) = max {L} R k (L) subject to L L max Optimality condition: { [ ( )] } R k L R 1 + λx R k, s (L) (L 1) ndf (R k ) 0 = R k df (R k ) (1 P)R Rλ R k ( x R k, s ) Rλ (L 1) R k s R k x s (L) L df ( R k) ( ) ( R k, s (L) df R k), Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

34 Bank s Problem, cont d Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

35 Competitive Equilibrium Bank optimality condition (limit case σ ɛ 0): [ ( R k df (R k ) = 1 F R k )] R R k ( +λ (1 γ) f R k ) [1 + λ (1 γ)] R 2 L 1 L 2 Household optimality condition: R = u (y (L 1)n) 1 P + E(v default)p Recovery rate } v = min {1, Rk L R L 1 λx(rk, s ) Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

36 Welfare Analysis Social planner problem: [ ] max SW = u(ȳ (L 1)n) + E(R k )L λe(x)r(l 1) n, L s.t. E(x) E[x(R k, s (L))], R = u (y d)/[1 P + E(v default)p] Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

37 Welfare Analysis Social planner problem: [ ] max SW = u(ȳ (L 1)n) + E(R k )L λe(x)r(l 1) n, L s.t. E(x) E[x(R k, s (L))], R = u (y d)/[1 P + E(v default)p] In the limit equilibrium σ ɛ 0: SW = u(y (L 1)n) + E(R k )Ln P λr(l 1)n }{{}}{{} Benefit of financial intermediation Cost of crisis Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

38 Excessive Bank Leverage Proposition (Excessive leverage) Suppose that the supply curve is upward sloping. Then, the bank leverage is excessive. Restricting bank leverage can improve social welfare. Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

39 Excessive Bank Leverage Proposition (Excessive leverage) Suppose that the supply curve is upward sloping. Then, the bank leverage is excessive. Restricting bank leverage can improve social welfare. Intuition: Average cost of bank default is endogenised, but, Marginal effect (cost) of leverage is underestimated due to limited liability and the global game setup Default cost pecuniary externality Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

40 Effects of Leverage Restrictions Competitive equilibrium: L = 10, P = 5% Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

41 Extended Model with Leverage and Liquidity Banks have an access to safe asset technology with gross return 1 Banks use safe assets in response to early withdrawals Trade-off: less return vs lower probability of bank runs Liquidity-deposit ratio m M/d Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

42 Extended Model with Leverage and Liquidity Banks have an access to safe asset technology with gross return 1 Banks use safe assets in response to early withdrawals Trade-off: less return vs lower probability of bank runs Liquidity-deposit ratio m M/d Proposition (Excessive leverage and insufficient liquidity) Given bank liquidity, bank leverage is excessive Given bank leverage, bank liquidity is insufficient Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

43 Leverage or Liquidity Requirements Only: Risk Migration Competitive equilibrium: L = 11.3, m = 0.12, P = 2.2% Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

44 Joint Effects of Leverage and Liquidity Requirements Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

45 Application 1: Sectoral Capital Requirements Two sectors and two types of banks Type-j bank specializes in lending to sector j {1, 2} Sector 2 is risker than sector 1 Competitive equilibrium: L 1 = 10.8, P 1 = 6.5%, L 2 = 7.7, P 2 = 9.6% Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

46 Effects of Sectoral Leverage Restrictions Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

47 Effects of Leverage Restrictions in One Sector Only Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

48 Application 2: Risk Taking One type of bank but two types of loans For simplicity, R k j N( R k, σ 2 R k ) for j {1, 2}. Loan portfolio [θ, 1 θ] on loans 1 and 2 Portfolio θ = 1/2 minimizes the risk (volatility) of bank loans Social optimum: θ = 1/2. Do banks choose θ = 1/2? Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

49 Banks prefer a higher risk than the socially optimal level Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

50 Bank Runs in an Infinite Horizon Model (work in progress) Figure: Impulse responses to a severe negative TFP shock Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27

51 Conclusion and Future Research Agenda This model provides a unified framework for analysing banking crises, banks behaviour and macroprudential policy Further research 1 Ex-ante and ex-post policy coordination 2 Dynamic model; dynamic properties of macroprudential policy 3 Macroprudential policy and monetary policy Ikeda (BoE) Bank Runs and Macroprudential Instruments 3 November / 27