Interbank Lending and Systemic Risk

Transcription

1 Interbank Lending and Systemic Risk Rochet Tirole April 2, 2012

2 Overview Theory of decentralized interbank lending based on peer monitoring Illustrate commitment problem for central bank in decision to bail out and discuss Too big to fail argument Study the possibility of (efficient) systemic risk

3 Model t = 0, 1, 2 n banks and a continuum of investors all with risk-neutral preferences and no discounting investors have deep pockets banks have initial endowment A i Technology: convex costs borne at t = 0, C(I ); linear returns at t = 2 RI if successful. Liquidity shock: at t = 1 random amount ρi is needed to continue, i.i.d across across banks with distribution F Moral hazard: after ρ realizes bank privately chooses probability of success p {p h, p l }; choosing p l yields private benefit BI

4 Timing

5 Optimal Allocation Under risk neutrality bank i is paid an amount R i iff its project succeeds Let x i : supp(ρ i ) {0, 1} denote a continuation rule; U i (ρ i ) = x i (ρ i )p h R i denote the interim expected utility of bank i; the interim expectded utility for investors V i (ρ i ) satisfies V i (ρ i ) + U i (ρ i ) = x i (ρ i )(p h R ρ i )I i The incentive constraints are for i = 1,..., n (p h p l )x i (ρ i )R i x i (ρ i )BI i pu i (ρ i ) x i (ρ i )(p h BI i )

6 Optimal allocation Given Pareto weights ν i and λ for bank i and depositors, Pareto optima are obtained by choosing {I i, U i (ρ i ), x i (ρ i )} n i=1 to maximize L = i E [(ν i λ)u i (ρ i ) + λx i (ρ i )I i (p h R ρ i ) + λ(a i C(I i ))] s.t. pu i (ρ i ) x i (ρ i )(p h BI i )

7 Optimal allocation We must have ν i < λ for all i Incetive constraints will be binding { BI ph i if x U i (ρ i ) = p i (ρ i ) = 1; 0 if x i (ρ i ) = 0.

8 Optimal allocation Given I.C. are binding I i and x i () are found by maximizing L λ = {[ I i E ( ν i λ 1) p ] } hb p + (p hr ρ i ) x i (ρ i ) i i [C(I i ) A i ] Optimal continuation { 1 if ρi ρ x i (ρ i ) = A i ; 0 if ρ i > ρ A i. Optimal threshold satisfies ρ A i p h (R B p ) + ν i p h B λ p ρ A i > p h (R B p ) ρ 0 Optimal investment level ρ A C (Ii A i ) = (ρ A i ρ i )f (ρ i )dρ i 0

9 Implementation Liquidity requirement: Borrow C(I i ) A i + ρ A i Ii A invest ρ A i Ii A in Treasuries and commit not to dilute existing claims Credit line: Borrow C(I i ) A i and obtain credit line.

10 Interbank monitoring (Date 0) 2 banks; bank 2 monitors bank 1 s short term management At private and unobservable cost ci 1 bank 2 ensures that ρ 1 is distributed according to F 1 If not monitored bank 1 can enjoy a private benefit SI 1 and induce a distribution F 1 The densities associated to the distributions satisfy MLRP: l(ρ 1 ) f 1(ρ 1 ) f 1 (ρ 1 ) f 1 (ρ 1 ) is decreasing in ρ 1 Parameters are such that it is always efficient to monitor

11 Optimal allocation Pareto optima are obtained by choosing {I i, U i (ρ), x i (ρ)} 2 i=1 to maximize L = 2 E [(ν i λ)u i (ρ) + λx i (ρ)i i (p h R ρ i ) + λ(a i C(I i )) ν 2 ci 1 ] i s.t. pu i (ρ) x i (ρ)(p h BI i ) E {U 2 (ρ)l(ρ 1 )} ci 1 x i (ρ) = 1forρ i < ρ 0

12 Optimal allocaation (monitoree) Bank 1 closure policy is the same as under autarky optimal ivnestment size is C (I 1 ) = x 1 (ρ) = 1 ρ 1 ρ 1 = ρ A 1 ρ 1 0 C (I 1 ) = (ρ 1 ρ 1 )f (ρ 1 )dρ 1 + c( ν 2 + µ ) λ ρ 1 0 F 1 (ρ 1 )dρ 1 + c( ν 2 + µ ) λ where µ is the multiplier on the incentive constraint of the monitor.

13 Monitoree Theorem Under optimal interbank lending: The continuation decision and the welfare of the monitoree (borrowing bank) do not depend on the liquidity shock facing the monitor (lending bank) The monitoree is closed less often than under autarky When the unit cost of monitoring c is small, the monitoree invests more than under autarky

14 Optimal allocation (monitor) Bank 2 s closure policy is x 1 (ρ) = 1 ρ 1 ρ 2(ρ 1 ) = ρ 0 + max {0, ν 2 + µl(ρ 1 )} p h B λ p optimal ivnestment size is C (I 2 ) = E ρ 2 (ρ 1 ) 0 F 2 (ρ 2 )dρ 2

15 Monitor Theorem Under optimal interbank lending: The continuation decision of the monitor (lending bank) depends on the liquidity shock facing the monitoree and in particular, when the credibility constraint is not binding x 2 (ρ) = 1 ρ 2 + ρ 21 (ρ 1 ) ρ A 2 ρ 21 (ρ 1 ) = µ λ l(ρ 1) p hb p If instead ρ 1 is so high that the credibility constraint binds the threshold is the per unit pledgeable income When the credibility constraint is not binding, the monitor invests more than under autarky

16 Implementation Qualitative feature are clear: the monitor s liquidity should depend on the monitoree s shock through some form af interbank credit Specific form depends on the likelihood ratio Example: bank 1(monitoree) has liquidity L 1 = (ρ 1 ρ 0 )I 1 and option to dilute; bank 2 has liquidity L 2 and debt issued by 2 with face value β ρ 0 p h I 1 that matures at t = 2 and option to dilute. At t = 1, if bank 1 survives, the value of the loan V 21 (ρ 1 ) = βρ 1 I 1 if L 1 ρ 1 I 1, if babnk 1 needs to dilute V 21 (ρ 1 ) = β [ρ 1 I 1 (ρ 1 I 1 L 1 )] This loan implements the optimum if and only if L 2 + V 21 (ρ 1 ) = [ρ 2(ρ 1 ) ρ 0 ] I 2

17 Soft Budget Constraint and Too big to fail Soft Budget Constraint ρ 2(ρ 1 ) < ρ 2 < ρ 0 In these cases the borrowing bank is in troubles and causes the lending bank to be in distress as well even if the latter would be solvent provided one ignores interbank activities Should one assist the borrowing bank, TBTF, or should one just let the borrowing bank fail and assist the failing bank?

18 Central Bank Bailouts Theorem The central bank s inability to commit not to rescue a bank that incurs losses on the interbank market but is otherwise solvent may lead to the prohibition of interbank lending in cases where it would be allowed if the central bank s commitment were credible The SBC differs from the TBTF as the borrowing bank s closure decision is unrelated to the fragility of the lending bank. Central bank assistance to a soolvent but failing bank operates through a direct assistance to that bank rather than through a bail out of the borrowing bank

19 Symmetric Date 1 Monitoring n banks located on a circle after ρ realizes bank i can monitor bank i 1 at a cost ci i 1 as long as the latter has not been shut down. monitornig costs c and reduces the benefit of shirking to the monitoring from BI to bi it is assumed that it is always optimal to monitor

20 Incentive constraints if x i (ρ) = 0 and x i 1 (ρ) = 1 if x i (ρ) = 1 and x i 1 (ρ) = 0 (p h p l )U i (ρ) p h ci i 1 if x i (ρ) = 1 and x i 1 (ρ) = 1 (p h p l )U i (ρ) p h bi i (p 2 h p 2 l )U i (ρ) p 2 h(bi i + ci i 1 ) Behind these we have R i 11,R i 10 and R i 01 and some timing and informational assumptions

21 Incentive Constraints economies of scope U i (ρ) = p2 h p 2 h p2 h (bi i + ci i 1 ) = γp h p (bi i + ci i 1 ) U i (ρ) = p h p [γ(bi i + ci i 1 )x i x i 1 + ci i 1 x i 1 (1 x i ) + bi i x i (1 x i 1 )] Taking expectations and rearranging E [U i ] p h p [bi i E(x i (1 γ)x i x i 1 ) + ci i 1 E(x i 1 (1 γ)x i x i 1 )] The pareto optima are obtained by maximizing the weighte sum of expected payoffs subject to the above incentive constraints for each i

22 Optimal contract The optimal closure decision is obtained by maximizing H(x, ρ) = n (ν i λ) p h p [bi i E(x i (1 γ)x i x i 1 ) + ci i 1 E(x i 1 (1 γ)x i x i 1 )] i Or more compactly + n λi i x i (p h R ρ i ) i n n H(x, ρ) = u i x i + w i x i x i 1 i=1 i=1 { u i (ρ i ) I i λ(p h R ρ i ) (λ ν i ) p hb p (λ ν i+1) p } hc p w i (1 γ)(λ ν i ) p h p (bi i + ci i 1 ) > 0

23 Local and Global dependency Bank i is shut down iff H x i = u i + w i x i 1 + w i+1 xi Function H is supermodular and satisfies the single crossing property 2 H x i x j 0 2 H x i x j 0 By a theorem of Milgrom and Shannon we have that the function mapping ρ to the set of maximizers of H is monotone (w.r.t the partial rdering on ρ and the strong set order on the maximizers)

24 Systemic Risk Theorem For all i and j, bank i is more likely to be liquidated if the liquidity shock facing bank j increases. There exists a value of ρ such that H has exactly two maxima x = (0,..., 0) and x = (1,..., 1). Thus a small increase in a bank s liquidity shock may imply the closure of the entire banking system.