## Abstract

This paper explores, in a multiperiod setting, the funding liquidity of a borrower that finances its operations through short-term debt. The short-term debt is provided by a continuum of creditors with heterogeneous beliefs about the prospects of the borrower. In each period, creditors observe the borrower’s fundamentals and decide on the amount they invest in its short-term debt. We formalize this problem as a coordination game, and we show that there exists a unique reasonable Nash equilibrium. We show that the borrower is able to refinance if and only if the liquid net worth is above an illiquidity barrier, and we explicitly find this barrier in terms of the distribution of capital and beliefs across creditors.

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## Acknowledgements

We are thankful to Phil Dybvig and Bob Jarrow for helpful discussions, as well as to and the anonymous referees for helpful comments that improved the presentation of the paper. This work was presented to the AMS Sectional Meeting in Rochester 2012, the INFORMS Annual Meeting 2012, the conference Research in Options 2012, the 6th Financial Risk Forum-Liquidity Risk 2013, the 30th International French Finance Association Conference 2013, the Finance and Stochastics Seminar at Imperial College, 2014, the Stochastic Methods in Finance Seminar at ENPC–INRIA–UPEMLV, 2014. We thank the participants for helpful comments. Part of this work was accomplished while the authors visited Swiss Finance Institute at EPFL, and we thank our hosts for a very good work environment.

## Author information

### Affiliations

### Corresponding author

## Appendix

### Appendix

All proofs are presented in this appendix.

Before we come to the proof of Theorem 3.2, we first provide the

### Proof of Lemma 3.3.

First, note that the function *b*
_{
t
}(⋅) is decreasing for each *t* since *β*(⋅) and hence *β*
^{−1}(⋅) are decreasing. Hence, *C*(*b*
_{
t
}(⋅)) is increasing, and hence *m*
_{
t
}(⋅) is strictly increasing by (3.3) for each *t*.

By definition, *b*
_{
T
}(⋅)=∞. Set \(m_{ T+1}^{-1}(0):= \infty \). We show by backward induction on *t*≤*T* that *b*
_{
t−1}(*x*)≤*b*
_{
t
}(*x*) and \(m_{t}^{-1}(0) \leq m_{t+1}^{-1}(0)\). The initial step *t*=*T* is clear. Now let *t*<*T* and assume that *b*
_{
t
}(*x*)≤*b*
_{
t+1}(*x*) and \(m_{t+1}^{-1}(0) \leq m_{t+2}^{-1}(0)\). Then

So \(0 = m_{t} ( m_{t}^{-1}(0) ) \geq m_{t+1} ( m_{t}^{-1}(0) )\) and thus \(m_{t+1}^{-1}(0) \geq m_{t}^{-1}(0)\) since \(m_{t+1}^{-1}(\cdot)\) is increasing. This implies *b*
_{
t
}(*x*)≥*b*
_{
t−1}(*x*) since *β*(⋅) is decreasing, finishing the induction step. □

We next show that under an allocation strategy \(\bar{\pi}\) of the structure as in Theorem 3.2, no creditor increases her investment into the short-term debt at the time of default.

### Lemma A.1

*If*
\(y_{t} \in\Delta_{t}(\bar{\pi})\), *t*>1, *then for any creditor*
*a*, *we have*

### Proof

Since *m*
_{
t
}(⋅) is strictly increasing, \(y_{t} \in\Delta_{t}(\bar{\pi})\) and (3.3) imply

Combined with Lemma 3.3, these equations imply

Therefore, again from Lemma 3.3, we obtain

which proves the statement. □

### Proof of Theorem 3.2

We prove the claim by backward induction. The induction base at *T* holds by convention. Assume now that *π*
^{∗} is an equilibrium at time *t*+1. Fix *a*∈*A* and take any allocation strategy *π* that satisfies \(\pi _{s}(a',\cdot)= \bar{\pi}_{s}(a',\cdot)\) for all creditors *a*′ and all *s*=0,…,*t*−1, and also \(\pi_{s}(a',\cdot) = \bar{\pi}_{s}(a',\cdot)\) for all creditors *a*′≠*a* and all *s*=*t*,*t*+1,…,*T*. The strategy *π* is obtained from \(\bar{\pi}\) by modifying the strategy of player *a* alone, from time *t* on.

We have to show that

for each \(Y_{t} \in\varGamma_{t-1}(\bar{\pi}) \times \mathbb {R}\). First, note that since the set {*a*} has measure zero, changing the strategy of a single player will have no impact on the default time, that is,

By (A.1), and since consumption depends only on allocations up to *t*−1 by (2.5), we obtain \(C_{t}^{\bar{\pi}, a} = C_{t}^{\pi, a}\); so it suffices to show

We now distinguish two cases.

*Case i)* Suppose that \(Y_{t} \in\Delta_{t}(\pi)=\Delta_{t}( \bar{\pi})\). By Lemma A.1 we have

and the allocations *π* and \(\bar{\pi}\) are the same until time *t*−1. It follows that

So we have (A.2).

*Case ii)* Suppose now that \(Y_{t} \in\varGamma_{t}(\pi)=\varGamma_{t}( \bar{\pi})\). On the set \(\{ T^{\bar{\pi}} = t+1 \}\), we have \(0 > X_{t+1}^{\bar{\pi}} + C(b_{t+1}(X_{t+1}^{\bar{\pi}}) ) = X_{t+1}^{\bar{\pi}} + C(b_{t+1}(X_{t+1}^{\bar{\pi}}))\). So \(b_{t+1}(X_{t+1}^{\bar{\pi}}) = \infty\) and therefore \(\bar{\pi}_{t+1}(a, Y_{t+1}) = 0\) on \(\{T^{\bar{\pi}} = t+1 \}\) by (3.6). Hence,

Since \(T^{\pi}= T^{ \bar{\pi}}\), we have that \(\mathbb {P}^{a}[T^{\pi}>t+1 \mid Y_{t}]\) and \(\mathbb {P}^{a}[T^{\bar{\pi}}>t+1 \mid Y_{t}]\) coincide; so as in (A.3), we obtain

We also have the equivalence

which yields

Since \(T^{\bar{\pi}} > t\), we have \(X_{t}^{\bar{\pi}} + C(b_{t}(X_{t}^{\bar{\pi}})) = X_{t}^{\bar{\pi}} + C(b_{t}(X_{t}^{\bar{\pi}}) ) \geq0\) and thus \(b_{t}(X_{t}^{\bar{\pi}}) = q(t,X_{t}^{\bar{\pi}})\). So it follows from (A.5) that we have the equivalence

where we used (3.1) and (3.4) in the first of the last two equalities, and (3.6) in the last equivalence. Combining (A.3) and (A.4) with (A.6) and *π*
_{
t
}(*a*,*Y*
_{
t
})∈[0,1] yields (A.2). □

### Proof of Lemma 3.6

We prove the statement by contradiction. Let *y*
_{
t
}∈Δ_{
t
}(*π*). Assume that there exists a creditor *a*∈*A* such that *π*
_{
t
}(*a*,*y*
_{
t
})>*π*
_{
t−1}(*a*,*y*
_{−1}) and consider the strategy

Then we have

in contradiction to the definition of a Nash equilibrium. □

### Proof of Proposition 3.7

On the set {*T*
^{π}>*t*}, we have

Since *π* is an equilibrium at time *t*, Lemma 3.6 yields

and therefore we obtain

If \(\mathbb {P}^{a}[T^{\pi}>t+1|Y_{t}] > \frac{1}{1+r}\), then *π*
_{
t
}(*a*,*Y*
_{
t
})=1 maximizes the right-hand side of (A.7); hence, we have *π*
_{
t
}(*a*,*Y*
_{
t
})=1. If \(\mathbb {P}^{a}[T^{\pi}>t+1|Y_{t}] < \frac{1}{1+r}\), then *π*
_{
t
}(*a*,*Y*
_{
t
})=0 maximizes the right-hand side of (A.7); hence, we have *π*
_{
t
}(*a*,*Y*
_{
t
})=0. Finally, using (2.4), the fact that *A* is nonatomic, and the continuity of the distribution of *θ*
_{
t+1} under \(\mathbb {P}^{a}\), it follows easily that the set of creditors *a*∈*A* with \(\mathbb {P}^{a}[T^{\pi }>t+1|Y_{t}] = \frac{1}{1+r}\) has measure zero. □

**For the remainder** of this Appendix, let *b*
_{
t
}(⋅), *m*
_{
t
}(⋅), and *q*(*t*,⋅) denote the functions defined by the recursion (3.2)–(3.4) and (3.8).

### Proof of Theorem 3.8

We prove the assertion by backward induction on *t*=*T*,…,1. For *t*=*T*, the assertion is trivial. So let *t*<*T* and assume that the assertion is true for *t*+1. On {*Y*
_{
t
}∈*Γ*
_{
t
}(*π*)}, we have the sequence of implications (using that \(X_{t+1}^{\pi}= x_{t+1}^{\pi}(Y_{t+1})\))

where the “⇒” in the second step follows from *Y*
_{
t+1}∈*Γ*
_{
t+1}(*π*) and the induction hypothesis. Now (A.8) yields

Combining this with the equivalence

we obtain the implications

of which the last follows from Proposition 3.7. Therefore,

Setting \(z = X_{t}^{\pi} - m_{t+1}^{-1}(0) + \varPhi^{-1} \big(\frac {r}{1+r}\big)\) and \(s = d_{t}^{\pi}(Y_{t})\), Lemma A.2 below then implies

Since *T*
^{π}>*t*, we obtain that

This implies \(\hat{b}_{t}(X_{t}^{\pi}) = \beta^{-1}( X_{t}^{\pi} - m_{t+1}^{-1}(0) + \varPhi^{-1} (\frac{r}{1+r}) )\), and hence (A.9) becomes \(d_{t}^{\pi}(Y_{t}) \leq\hat{d}_{t}(X_{t}^{\pi})\), finishing the induction step. □

### Lemma A.2

*Fix*
\(z \in \mathbb {R}\). *The function*

*is strictly increasing in*
\(s \in \mathbb {R}\), *and its zero is given by*
*s*=*C*(*β*
^{−1}(*z*)).

### Proof

Clearly, *f*(⋅) is strictly increasing, and we have

where the last equation follows from (3.1). □

In the following, we use the notation

with the function \(\hat{\pi}_{t}(a,x)\) introduced in (3.9). For the proof of Theorem 3.13, we use the following two immediate propositions.

### Proposition A.3

*Suppose that*
\(y_{t} \in\varGamma_{t}(\hat{\pi})\). *Then for all*
*a*∈*A*, *we have*

*for any*
\(\tilde{\pi}_{t}(a,y_{t}) \neq\hat{\pi}_{t}(a,y_{t})\).

The following proposition states that given that strategy \(\hat{\pi}\) is played from time *t*+1, and conditional on survival at time *t*, the strategy \(\hat{\pi}\) is the best response at time *t* for any agent.

### Proposition A.4

*Let*
\(\pi\in \mathcal {X}\)
*and*
*t*∈{1,…,*T*} *such that for all*
*s*≥*t*+1 *and*
\(y_{s} \in\varGamma_{s-1}(\pi)\times \mathbb {R}\), *we have*
\(\pi_{s}(\cdot,y_{s}) = \hat{\pi}_{s}(\cdot, x_{s}^{\pi}(y_{s}))\). *Suppose that*
*y*
_{
t
}∈*Γ*
_{
t
}(*π*). *Then for all*
*a*∈*A*, *we have*

*for any*
\(\tilde{\pi}_{t}(a,y_{t}) \neq\hat{\pi}_{t}(a,x_{t}^{\pi}(y_{s}))\). *Moreover*, *if*
*π*
*is an equilibrium at time*
*t*, *then*
\(\pi_{t}(\cdot, y_{t}) = \hat{\pi}_{t}(\cdot, x_{t}^{\pi}(y_{s}))\).

The last statement of Proposition A.4 follows by Theorem 3.8.

### Proof of Theorem 3.13

For *t*=1,…,*T*, let *Π*
_{
t
} be the set of strategies that are equilibria at time *t* and not weakly dominated for any *a* at times *s*=*t*,…,*T*. We first prove the following statement (*H*
_{
t
}) by backward induction on *t*=*T*,…,1.

(*H*
_{
t
}) We have that \(\hat{\pi}\in\varPi_{t}\). Conversely, any *π*∈*Π*
_{
t
} that is an equilibrium at time *t*−1 fulfils

if *t*≥2, and \(\pi_{1} \equiv\hat{\pi}_{1}\).

Note that (A.11) immediately implies that for any *y*
_{
t−1}∈*Γ*
_{
t−1}(*π*),

Clearly, (*H*
_{
T
}) is true since any equilibrium *π* fulfils *π*
_{
T
}≡0 by definition. For the induction step, let 1≤*t*<*T* and assume that (*H*
_{
t+1}) is true. We now prove (*H*
_{
t
}) in the following steps.

**Step 1)** Let us show that \(\hat{\pi}\in\varPi_{T}\). By the induction hypothesis, \(\hat{\pi}\) is not weakly dominated at times *t*+1,…,*T*. We now show that \(\hat{\pi}\) is not weakly dominated at time *t*. Suppose, for a contradiction, that there exists a strategy \(\tilde{\pi}\) with \(\tilde{\pi}_{s} = \hat{\pi}_{s}\) for all *s*≤*t*−1 such that \(\hat{\pi}\) is weakly dominated by \(\tilde{\pi}\) for some creditor *a* at time *t*. Then we have the following two statements:

a) For all trajectories \(y_{t} \in\varGamma_{t-1}(\hat{\pi}) \times \mathbb {R}\),

with \(\hat{\pi}\in\varPi_{t+1}^{*}\), and we have

b) There exist a trajectory \(y_{t} \in\varGamma_{t-1}(\hat{\pi}) \times \mathbb {R}\) and a strategy

with \(\pi^{*} \in\varPi_{t+1}^{*}\) such that

For *y*
_{
t
} and *π*
^{∗} as in b), if *y*
_{
t
}∈*Γ*
_{
t
}(*π*
^{∗}), then we have by Theorem 3.8 that \(y_{t} \in\varGamma_{t}(\hat{\pi})\). Now the induction hypothesis implies that \(g_{t}^{\langle\pi ^{*}|\hat{\pi}(a) \rangle, a}(y_{t})=g_{t}^{\langle\hat{\pi}|\hat{\pi}(a) \rangle, a}(y_{t})\) and \(g_{t}^{\langle\pi^{*}|\tilde{\pi}(a) \rangle, a}(y_{t}) = g_{t}^{\langle\hat{\pi}|\tilde{\pi}(a) \rangle, a}(y_{t})\), and so (A.13) yields the inequality

in contradiction to the statement of Proposition A.3. So we must have *y*
_{
t
}∈Δ_{
t
}(*π*
^{∗}), and (A.13) becomes

Since \(\tilde{\pi}_{t-1} = \hat{\pi}_{t-1}\), it follows that \(\tilde{\pi}_{t}(a,y_{t}) < \hat{\pi}_{t}(a,y_{t})\), so in particular we have \(\hat{\pi}_{t}(a,y_{t}) > 0\) and hence \(y_{t} \in\varGamma_{t}(\hat{\pi})\) by definition of \(\hat{\pi}\). Since \(\hat{\pi}\) is an equilibrium, using Proposition A.3, this is a contradiction to (A.12).

**Step 2)** Now suppose *t*>1 and fix any *π*∈*Π*
_{
t
}. Since by Theorem 3.8 we have

to show (A.10), it is sufficient to show that

**Step 2a)** We first prove that for all \(y_{t} \in\varGamma_{t-1}(\pi) \times \mathbb {R}\), we have

To this end, take any *a*∈*A* and assume, for a contradiction, that there exists a trajectory \(\tilde{y}_{t} \in\varGamma_{t-1}(\pi) \times \mathbb {R}\) such that

Since \(\hat{\pi}_{t}\) is {0,1}-valued, and *π*
_{
t−1} is {0,1}-valued for almost all creditors by Proposition 3.7, we have \(\mbox{min}(\pi_{t-1}(a, \tilde{y}_{t-1}), \hat{\pi}_{t} (a, x_{t}^{\pi }(\tilde{y}_{t})))=1\) for almost any creditor *a* that satisfies (A.16). This yields

Define the allocation strategy \(\tilde{\pi}\) by

We claim that *π* is weakly dominated by \(\tilde{\pi}\) for creditor *a* at time *t*. To prove this, let \(y_{t} \in\varGamma_{t-1}(\pi) \times \mathbb {R}\) and consider a strategy

that is an equilibrium on *A*∖{*a*} and is not weakly dominated for any creditor in *A*∖{*a*} at times *t*=*t*+1,…,*T*. Since {*a*} has measure zero, we can extend *π*
^{∗} to an equilibrium on *A*, again denoted by *π*
^{∗}, which is not weakly dominated for any creditor in *A* at times *t*=*t*+1,…,*T*. We first have to show that

If \(y \neq\tilde{y}\), then \(\tilde{\pi}= \pi\), and we have equality in (A.21). So we now assume that \(y = \tilde{y}\). First, observe that \(T^{ \langle \pi^{*}|\pi(a) \rangle} = T^{ \langle\pi^{*}|\tilde{\pi}(a) \rangle} = T^{\pi^{*}}\). Since \(\tilde{\pi}_{t-1}(a, \tilde{y}_{t-1}) = \pi_{t-1}(a, \tilde{y}_{t-1})\), we have \(C_{t}^{\langle\pi^{*} | \pi(a) \rangle, a} = C_{t}^{\langle\pi^{*} |\tilde{\pi}(a) \rangle, a}\). We now distinguish two cases.

i) Suppose that \(\tilde{y}_{t} \in\Delta_{t}(\pi^{*})\): Since \(\tilde{\pi}_{t-1}(a,\tilde{y}_{t-1}) = \pi_{t-1}(a,\tilde{y}_{t-1}) = 1\) by (A.17) and (A.19), it follows from (2.5) and (2.6) that \(g_{t}^{\langle\pi^{*} |\tilde{\pi}(a) \rangle, a}(\tilde{y}_{t}) = g_{t}^{\langle\pi^{*}|\pi(a) \rangle,a}(\tilde{y}_{t}) = 0\).

ii) Suppose that \(\tilde{y}_{t} \in\varGamma_{t}(\pi^{*})\): By the induction hypothesis, whenever *y*
_{
t
}∈*Γ*
_{
t
}(*π*
^{∗}) and \(x_{t+1}^{\pi^{*}}(y_{t+1}) \geq m_{t+1}^{-1}(0)\), it holds that \(\pi^{*}_{t +1}(a,y_{t+1}) = \hat{\pi}_{t +1}(a,x_{t+1}^{\pi^{*}}(y_{t+1})) \). Moreover, \(\pi^{*}_{t +1}(a,y_{t+1}) \leq\pi^{*}_{t}(a,y_{t})\) whenever *y*
_{
t
}∈*Γ*
_{
t
}(*π*
^{∗}) and \(x_{t+1}^{\pi^{*}}(y_{t+1}) < m_{t+1}^{-1}(0)\). By Proposition A.4, \(\hat{\pi}_{t} (a, x_{t}^{\pi^{*}}(\tilde{y}_{t}))\) is a strict maximizer over *π*(*a*) of \(g_{t}^{ \langle\pi ^{*}|\pi(a) \rangle, a }(\tilde{y}_{t})\). By (A.17)–(A.19) and the consistency condition (A.20) (see Remark 3.10), we have

Therefore,

So we have shown (A.21).

Consider now a strategy *ρ* defined recursively by *ρ*
_{
s
}(*a*,⋅)=*π*
_{
s
}(*a*,⋅) for *s*<*t* and \(\rho_{s}(a,y_{s}) = \hat{\pi}_{s}(a, x_{s}^{\rho}(y_{s}))\) for *s*≥*t* and *y*
_{
s
}∈*Γ*
_{
s
}(*ρ*). We have by Step 1 that *ρ*∈*Π*
_{
t
}. We have \(\hat{\pi}_{t} (a, x_{t}^{\rho}(\tilde{y}_{t})) = \hat{\pi}_{t} (a, x_{t}^{\pi}(\tilde{y}_{t})) = 1\). This implies that \(\tilde{y}_{t} \in\varGamma (\rho)\). So for *π*
^{∗}=*ρ*, we are in case ii), and the inequality in (A.21) is strict for \(y_{t} = \tilde{y}_{t}\). This shows that \(\tilde{\pi}\) weakly dominates *π* for creditor *a* at time *t*, in contradiction to our assumption. Hence, we have (A.15).

**Step 2b)** Let *Y*
_{
t−1}∈*Γ*
_{
t−1}(*π*). It follows immediately from Step 2a) that

Set

We then have

where the “⇐” in the third line follows from \(D_{t}^{\pi} \geq\min(\hat{d}_{t}(X_{t}^{\pi}), D_{t-1}^{\pi})\). Now (A.23), using that \(- \theta_{t} + b(a) \sim \mathcal{N}(0,1)\) under \(\mathbb {P}^{a}\), yields

Combining this with the equivalence

we obtain the implications

of which the last follows from Proposition 3.7. Therefore,

Setting \(z = X_{t-1}^{\pi} - \mu_{t-1}(Y_{t-1}) + \varPhi^{-1} (\frac {r}{1+r})\) and \(s = d_{t-1}^{\pi}(Y_{t-1})\), Lemma A.2 then implies

Since *T*
^{π}>*t*−1, we have \(X_{t-1}^{\pi} + d_{t-1}^{\pi}(Y_{t-1}) \geq0\). Suppose, for a contraction, that \(-d_{t-1}^{\pi}(Y_{t-1}) > m_{t}^{-1}(0)\). It follows that

and so we obtain, since *β*
^{−1} is decreasing,

With (A.24), we get

where the last inequality follows from

This contradicts our assumption. Hence, we have

and thus \(\mu_{t-1}(Y_{t-1}) = m_{t}^{-1}(0)\). Now (A.22) reads as (A.14), which implies (A.10).

**Step 3)** Let us now complete the induction step by showing (A.11). First, suppose that *t*>1. If \(Y_{t} \in\varGamma_{t-1}(\pi) \times \mathbb {R}\) and \(X_{t}^{\pi} < m_{t}^{-1}(0)\), then the inequality in (A.11) follows from Lemma 3.6 and Theorem 3.8. If \(Y_{t} \in\varGamma_{t-1}(\pi) \times \mathbb {R}\) and \(X_{t}^{\pi} \geq m_{t}^{-1}(0)\), then *Y*
_{
t
}∈*Γ*
_{
t
}(*π*) by (A.10), so as in (A.7), we have

where the second equation follows from the induction hypothesis. Since *π* is an equilibrium at time *t*, maximizing the above expectation with respect to *π*
_{
t
}(*a*,*Y*
_{
t
}) yields \(\pi_{t}(a, Y_{t}) = \hat{\pi}_{t}(a, X_{t}^{\pi})\) for almost all *a*, and we obtain (A.11).

Next suppose that *t*=1. Since *Y*
_{1}=0∈*Γ*
_{1}(*π*), we have (A.25) as in the case *t*>1, and again we obtain \(\pi _{1}(a,y_{1}) = \hat{\pi}_{1}(a,y_{1})\) for almost all *a*. This concludes the induction step, and so we have proved (*H*
_{
t
}) for all *t*=1,…,*T*.

To complete the proof of the theorem, first note that \(\hat{\pi}\) is a reasonable Nash equilibrium by (*H*
_{1}). Conversely, let *π* be any reasonable Nash equilibrium. Forward induction and (A.10) immediately yield \(\varGamma_{t}(\pi) = \varGamma_{t}(\hat{\pi})\). So we have condition 1 of Definition 2.2. Equation (A.11) yields condition 2 of Definition 2.2, and condition 3 of Definition 2.2 follows from (A.11) and Lemma 3.6. Hence, *π* and \(\hat{\pi}\) are identical. □

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Krishenik, A., Minca, A. & Wissel, J. When do creditors with heterogeneous beliefs agree to run?.
*Finance Stoch* **19, **233–259 (2015). https://doi.org/10.1007/s00780-015-0259-x

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### Keywords

- Liquidity risk
- Credit risk
- Nash equilibrium
- Bank run

### Mathematics Subject Classification

- 91A10
- 91A13
- 91A20
- 91A80
- 91B69
- 91B70

### JEL Classification

- C72
- C73
- D53
- D81
- G11