The Maturity Rat Race - Princeton
The Maturity Rat Race Markus Brunnermeier, Princeton University Martin Oehmke, Columbia University August 4, 2010
Is There Too Much Maturity Mismatch? I Households have long-term saving needs I Banks have long-term borrowing needs Why is intermediary borrowing so short-term? Rationale for ‘beneficial’ maturity mismatch: I Diamond and Dybvig (1983) I Calomiris and Kahn (1991), Diamond and Rajan (2001) There may be excessive maturity mismatch in the financial system
This Paper A financial institution can borrow I from multiple creditors I at different maturities Negative externality can cause excessively short-term financing: I shorter maturity claims dilute value of longer maturity claims I depending on type of interim information received at rollover dates Externality arises I for any maturity structure I particularly during times of high volatility (crises) Successively unravels all long-term financing: A Maturity Rat Race
Outline Model Setup One Rollover Date I Two Simple Examples I The General Case Multi-period Maturity Rat Race Discussion Related Literature
Model Setup: Long-term Project Long-term project: I investment at t 0: 1 I payoff at t T : θ F (·) on [0, θ̄] Over time, more information is learned: I st observed at t 1, . . . , T 1 I St is sufficient statistic for all signals up to t: θ F (· St ) I St orders F (·) according to FOSD Premature liquidation is costly: I early liquidation only generates λE [θ St ], λ 1
Model Setup: Credit Markets Risk-neutral, competitive lenders All promised interest rates I are endogenous I depend on aggregate maturity structure Debt contracts specifies maturity and face value: I can match project maturity: D0,T I or shorter maturity D0,t , then rollover Dt,t τ etc. I lenders make uncoordinated rollover decisions All debt has equal priority in default: I proportional to face value
Model Setup: Credit Markets (2) Main Friction: Financial institution has opaque maturity structure I simultaneously offers debt contracts to creditors I cannot commit to aggregate maturity structure I can commit to aggregate amount raised An equilibrium maturity structure must satisfy two conditions: 1. Break even: all creditors must break even 2. No deviation: no incentive to change one creditor’s maturity
Outline Model Setup One Rollover Date I Two Simple Examples I The General Case Multi-period Maturity Rat Race Discussion Related Literature
Analysis with One Rollover Date For now: focus on only one possible rollover date, t T Outline of thought experiment: I Conjecture an equilibrium in which all debt has maturity T I Calculate break-even face values I At break-even interest rate, is there an incentive do deviate? Denote fraction of short-term debt by α
A Simple Example: News about Default Probability θ only takes two values: I θH with probability p I θL with probability 1 p p random, revealed at date t If all financing has maturity T : (1 p0 ) θL p0 D0,T 1, D0,T 1 (1 p0 ) θL p0 Break-even condition for first t-rollover creditor: (1 pt ) Dt,T L θ pt Dt,T 1, D0,T Dt,T 1 (1 p0 ) θL θL p0 (1 θL ) pt
Illustration: News about Default Probability Deviation payoff: Π α E [pt D0,T ] E [pt Dt,T ] 0? α 0 Product of two quantities matters: I Promised face value under ST and LT debt (left) I Probability that face value is repaid (right) Repayment Probability Face Value 1.0 3.0 2.5 Dt,T HSt L 0.8 pt 2.0 0.6 D0,T 1.5 0.4 1.0 0.2 0.5 p p 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
Illustration: News about Default Probability Multiplying promised face value and repayment probability: Marginal Cost 1.5 Long-term financing A 1.0 Rollover financing 0.5 B p 0.0 0.2 0.4 0.6 0.8 Note: A B implies rolling over cheaper in expectation 1.0
A Simple Example: News about Recovery Value θ only takes two values: I θH with probability p 1/2 I θL with probability 1 p Low cash flow θL random, revealed at date t If all financing has maturity T : 1 1 [ ] D0,T E θL 1, 2 2 [ ] D0,T 2 E θL Break-even condition for first t-rollover creditor: 1 1 Dt,T L Dt,T θ 1, 2 2 D0,T ( ) Dt,T θL 2 [ ] 2 E θL 2 E [θL ] θL
Illustration: News about Recovery Value Deviation payoff: Π α α 0 1 1 D0,T E [Dt,T (θL )] 0? 2 2 Product of two quantities matters: I Promised face value under ST and LT debt (left) I Probability that face value is repaid (right) Repayment Probability Face Value 1.0 2.0 0.8 1.8 1.6 Dt,T HSt L 0.6 D0,T 0.4 1.4 0.2 1.2 0.0 0.2 0.4 0.6 0.8 1.0 Θ L 0.2 0.4 0.6 0.8 1.0 ΘL
Illustration: News about Recovery Value Multiplying promised face value and repayment probability: Marginal Cost 1.1 1.0 Rollover financing 0.9 B' 0.8 Long-term financing 0.7 A' 0.6 0.0 0.2 0.4 0.6 0.8 1.0 ΘL Note: A′ B ′ implies rolling over more expensive in expectation
What is going on? Interim Information Matters! Rollover face value Dt,T (promised interest rate) I is endogenous I adjusts to interim information Interim Signal Dt,T default no default Negative Positive high low LT creditors lose LT creditors gain no effect no effect If default sufficiently more likely after negative signals LT creditors lose on average
General One-Step Deviation Extend to: I general payoff distribution I start from any conjectured equilibrium that involves some amount of LT debt Assumption 1: Dt,T (St ) dF (θ St ) is weakly increasing in St D̄T (St ) {z } repayment probability I Guarantees signal has sufficient effect on default probability Proposition: One-step Deviation. Under Assumption 1, the unique equilibrium is all short-term financing (α 1).
Outline Model Setup One Rollover Date I Two Simple Examples I The General Case Multi-period Maturity Rat Race Discussion Related Literature
Many Rollover Dates: The Maturity Rat Race Up to now: focus on one potential rollover date I Assumed everyone has maturity of length T I Showed that there is a deviation to shorten maturity to t This extends to multiple rollover dates I Assume all creditors roll over for the first time at some time τ T I By same argument as before, there is an incentive to deviate I In proof: For τ T replace final payoff by continuation value Successive unraveling of maturity structure
The Maturity Rat Race: Successive Unraveling
The Maturity Rat Race: Successive Unraveling
The Maturity Rat Race: Successive Unraveling
The Maturity Rat Race: Successive Unraveling
The Maturity Rat Race: Successive Unraveling
The Maturity Rat Race: Successive Unraveling Assumption 2: Dt 1,t (St 1 ) S̃t dG (St St 1 ) is increasing in St 1 t {z } prob of rollover at t I Guarantees signal has sufficient effect on rollover probability at next rollover date Proposition: Sequential Unraveling. Under Assumption 2, successive application of the one-step deviation principle results in unraveling of the maturity structure to the minimum rollover interval.
Outline Model Setup One Rollover Date I Two Simple Examples I The General Case Multi-period Maturity Rat Race Discussion Related Literature
Rat Race Causes Inefficiencies Excessive Rollover Risk I Project could be financed without any rollover risk I Rat race leads to positive rollover risk in equilibrium Underinvestment I Creditors rationally anticipate rat race I NPV of project must outweigh eqm liquidation costs I some positive NPV projects don’t get financed
Rat Race Strongest During Crises Rat race stronger when more information about default probability is released at interim dates I ability to adjust financing terms becomes more valuable Volatile environments, such as crises, facilitate rat race Explains drastic shortening of unsecured credit markets in crisis I e.g. commercial paper during fall of 2008
Commercial Paper Issuance 2008
Seniority, Covenants Priority for LT debt and covenants may limit rat race Can reduce externality of ST debt on LT debt I Seniority for LT debt I Restrictions on raising face value of ST debt at t T But: I by pulling out early, ST creditors may still have de facto seniority I Particularly for financial institutions, covenants are hard to write/enforce
Outline Model Setup One Rollover Date I Two Simple Examples I The General Case Multi-period Maturity Rat Race Discussion Related Literature
Related Literature ‘Beneficial’ Maturity Mismatch I Diamond and Dybvig (1983) I Calomiris and Kahn (1991), Diamond and Rajan (2001) Papers on ‘Rollover Risk’ I Acharya, Gale and Yorulmazer (2009) I He and Xiong (2009) I Brunnermeier and Yogo (2009) Signaling Models of Short-term Debt I Flannery (1986) I Diamond (1991) I Stein (2005)
Conclusion Equilibrium maturity structure may be efficiently short-term I Contractual externality between ST and LT creditors I Maturity Rat Race successively unravels long-term financing This leads to I too much maturity mismatch I excessive rollover risk I underinvestment Not easily fixed through covenants or seniority for LT debt
Extra Slides
A Simple Example: News about Default Probability θ only takes two values: I θH 1.5 with probability p 0.8 I θL 0.6 with probability 1 p 0.2 p updated at date t to pt 0.8 0.1 If all financing has maturity T : (1 p0 ) θL p0 D0,T 1, D0,T 1.1 Break-even condition for first t-rollover creditor: 1.047 if p 0.9 Dt,T L t (1 pt ) θ pt Dt,T 1, Dt,T 1.158 if pt 0.7 D0,T
Illustration: News about Default Probability Deviation payoff: Π α p0 D0,T E [pt Dt,T (pt )] 0? Product of two quantities matters: I Promised face value under ST and LT debt I Probability that face value is repaid Π α 0.8 1.1 0.5 (0.9 1.047) 0.5 (0.7 1.158) 0.0033 0 Deviation profitable
A Simple Example: News about Recovery Value θ only takes two values: I θH 1.5 with probability p 0.8 I θL 0.6 with probability 1 p 0.2 Low cash flow θL random, updated at date t: 0.6 0.1 If all financing has maturity T : [ ] (1 p)E θL pD0,T 1, D0,T 1.1 Break-even condition for first t-rollover creditor: 1.078 Dt,T L (1 p) θ pDt,T 1, Dt,T 1.112 D0,T if θL 0.7 if θL 0.5
Illustration: News about Recovery Value Deviation payoff: Π α pD0,T pE [Dt,T (θL )] 0? Product of two quantities matters: I Promised face value under ST and LT debt I Probability that face value is repaid) Π α 0.8 1.1 0.5 (0.8 1.078) 0.5 (0.8 1.122) 0.0003 0 Deviation not profitable
Inefficiency 1: Excessive Rollover Risk I Project could be financed without any rollover risk I Rat race leads to positive rollover risk in equilibrium Clearly inefficient Corollary: Excessive Rollover Risk. The equilibrium maturity structure (α 1) exhibits excessive rollover risk when conditional on the worst interim signal the expected cash flow of the project is less than the initial ) ( θ̄ investment 1, i.e. 0 θdF θ StL 1.
Inefficiency 2: Underinvestment Creditors rationally anticipate rat race: I NPV of project must outweigh eqm liquidation costs I some positive NPV projects don’t get financed Corollary: Some positive NPV projects will not get financed. As a result of the maturity rat race, some positive NPV projects will not get financed. To be financed in equilibrium, a project’s NPV must exceed (1 λ) S̃t (1) E [θ St ] dGt (St ) . StL
Many Rollover Dates: The Maturity Rat Race Up to now: focus on one potential rollover date I Assumed everyone has maturity of length T I Showed that there is a deviation to shorten maturity to t This extends to multiple rollover dates I Assume all creditors roll over for the first time at some time T I By same argument as before, there is an incentive to deviate
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