The Lender of Last Resort and Bank Failures Some Theoretical Considerations

Transcription

1 The Lender of Last Resort and Bank Failures Some Theoretical Considerations Philipp Johann König 5. Juni 2009

2 Outline 1 Introduction 2 Model 3 Equilibrium 4 Bank's Investment Choice 5 Conclusion and Outlook

3 Main Questions Main questions: 1 To which extent do banks trade o the probability of default (due to illiquidity) against possible higher returns due to risky, potentially illiquid investments? 2 How does the availability of LLR-loans inuences this trade-o?

4 Motivation Literature agrees on two purposes of LLR: 1 Mitigate secondary repercussions of a bank run - avert contagion; 2 Dispel uncertainty and forestall crises by committing in advance to LLR-lending and thereby generate stabilizing pattern of expectations. Here, focus on (2): Announcement has eect on creditors / depositors, but at the same time on debtors / banks. To which extent do the latter take this into account when they decide to allocate funds between risky and safe assets? * e.g. Freixas et al. (2004), Rochet and Vives (2004), Goodhart (1984), Solow (1984), Bagehot (1873), Thornton (1802)

5 Motivation How to proceed? Simple model, basic mechanisms. Need for endogenous liquidity shock. Possibility to distinguish between insolvent and illiquid nancial institutions.

6 Framework Model à la Rochet & Vives (2004), Corsetti et al. (2006) Three dates, t {0, 1, 2}; Three types of agents: Bank, Fund Managers, LLR Two assets: Very liquid asset (e.g. cash) with zero net return in periods t = 1 or t = 2, and risky asset yielding stochastic gross return R U[0, R]. R is realized in t = 1, asset pays out in t = 2. Risky asset can be sold in t = 1, but only at re sale price instead of at fundamental price R. δ > 0 renders risky asset illiquid, potential source of failure of solvent bank. R 1+δ

7 Framework Bank: The bank's balance sheet reads: assets liabilities I D 0 M E D 0 1: depostis in t = 0, against promise to return D > 1 in t = 1 or t = 2. M (0, 1): investments in liquid asset, I (0, 1): investments in illiquid asset, E : equity.

8 Framework Fund Managers: Unit mass; unit endowment, deposited with bank in t = 0; right to withdraw in t = 1 or to roll over to t = 2. Payo table reads: bank fails bank succeeds withdraw b c b c roll over c b

9 Framework LLR: Grants zero-interest loan λ in t = 1 whenever considers bank to be `sound'. Similar to fund managers, payos depend only on whether he decides `correctly': bank fails bank succeeds intervene C B do not intervene 0 0

10 Illiquidity vs. Insolvency without LLR (Early) Liquidity pressure: nd (n [0, 1] - fraction of fund managers who withdraw in t = 1) R is Common Knowledge. Normal circumstances, δ = 0: D > D 0 bank needs least return R l (I ). Fund manager rolls over if R > R l (I ). (Note: R (I ) < 0.) l Non-normal circumstances, δ > 0: Coordination / illiquidity problem critical return depends on n multiple equilibria. δ > 0 source of strategic complementarities.

11 Illiquidity vs. Insolvency without LLR Liquidity demand bounded by D: `illiquidity range' bounded above by R h (I ). For any R R h (I ) bank is `supersolvent'. Focus on R [R l, R h ): function n c (R), gives critical fraction above which failure is triggered.

12 n 1 Failure due to Illiquidity Failure due to Insolvency M D Bank never fails R l R h R

13 Illiquidity vs. Insolvency with LLR Additional liquidity injection λ does not change least return R l (I ). But changes upper bound of illiquidity region to R λ h (I ) < R h(i ). Hence, with increasing λ, size of illiquidity region R h R l shrinks. But, for λ not too large, multiplicity remains.

14 n 1 Failure due to Insolvency Failure due to Illiquidity M + λ D M D Bank never fails R l λ R h R h R

15 Global Game Selection Multiplicity eliminated by Global Game Selection. Instead of Common Knowledge, noisy signals about R: Fund managers: x k = R + υ k, LLR: y = R + η, where η, υ k U[±ε]. Fund managers and LLR cannot pool their information, each decides on his own.

16 Equilibrium Global Game Selection provides unique Bayesian Nash Equilibrium in monotone strategies. Equilibrium described by tuple {ˆx, ŷ, R λ, R 0 }, where (a) all fund managers withdraw for signals below ˆx, (b) LLR does not intervene for signals below ŷ, the bank fails (c) for returns below R λ, and (d) for returns between R λ and R 0 i LLR decides not to intervene.

17 Limit Case: ε 0 For ε 0, the fundamental uncertainty vanishes. However, strategic uncertainty does not vanish. In the limit equilibrium: ŷ = R λ = R 0 = ˆx(I, M, λ) = R l (I ) + max δ(d(1 c/b) M Θ(λ)) I, 0, where dθ(λ)/dλ > 0.

18 n 1 M + λ D M D R l λ R h R h R

19 Comparative Statics ˆx/ λ 0: LLR can shrink the region of illiquidity failures. ˆx/ I 0: Bank has more funds available in t = 2 to pay o liabilities. ˆx/ M 0: Larger liquidity buer, for given n, bank has to re-sale less assets.

20 Comparative Statics d ˆx/dI : sign ambiguous. I and M connected through balance sheet. Partial eects now move in opposite directions. Which eect will dominate depends crucially on parameters, in particular on size of re sales discount δ.

21 Comparative Statics δ low, reducing M has only small impact, asset can be sold at price near to fundamental price. First eect dominates. δ high, given liquidity demand forces bank to sell o larger proportion of assets, impacts the likelihood of survival in t = 2. Second eect dominates.

22 Bank's Investment Choice Bank decides in t = 0 about the allocation of funds between the two types of assets. Takes into account ˆx, i.e. fact that prots accrue only above threshold. In case ε 0, the maximization problem reads max {I,M} R The FOC w.r.t. I is given by: φ di R ˆx ˆx [RI + M D] df (R) s.t. I + M = D 0 + E. (R 1)dR (ˆxI E I D) d ˆx = 0.

23 Profit Case: Δxˆ > 0 L: Loss due to change in I G: Gain due to change in I G L xˆ xˆ + Δxˆ R R

24 Bank's Investment Problem ˆx (I ) > 0: Bank trades o a quantity eect (due to larger amount invested), against a 'return eect' (because it reduces the range where it can make prots). ˆx (I ) < 0: Bank does not face a trade o, quantity and return eect move into the same direction corner solution. However, numerical computations have shown so far that even if ˆx (I ) > 0, for a broad range of parameters, the quantity eect dominates and the bank always chooses I = 1.

25 Bank's Investment Problem Latter implies that bank concentrates on decreasing the insolvency range and leaves illiquidity problem to LLR. LLR-policy: Set λ = λ (I = 1), where λ (I = 1) solves ˆx(I = 1, λ) = R l (I = 1).

26 Conclusion and Outlook Model presumes unlimited availability of LLR-funds. No costs involved besides 'reputational' cost in payo functions too simplistic. Demand in secondary market too elastic. Strategic runs do not generate strong enough eect s.t. LLR can indeed inuence the investment decision.

27 Thank You Very Much!

28 Literature Bagehot, W. (1873): Lombard Street: A Description of the Money Market Humphrey, T. (1975): The Classical Concept of the LLR, Richmond FED ER, pp. 2-9 Rochet, J.-C. and Vives, X. (2004): Coordination Failures and the LLR: Was Bagehot Right?, JEEC, 6, pp Thornton, H. (1802): An Enquiry into the Nature and Eects of the Paper Credit of Great Britain