LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES ¨ ¨ PHILIPP J. KONIG AND TIJMEN R. DANIELS

Technische Universit¨ at Berlin Department of Macroeconomics ** Preliminary version ** please do not redistribute ** Abstract. We derive a unique equilibrium in a model of emerging market banking crises. In equilibrium, variations in short-term capital flows and holdings of international reserves can have ambiguous effects on the economy’s vulnerability, i.e. the likelihood which with a crisis occurs. We identify deadly combinations of parameters under which rising short-term capital flows and falling reserve holdings actually increase the vulnerability to a crisis. When we endogenise the maturity structure, multiple shortterm debt levels can arise. This leads to a self-fulfilling trap: foreign creditors consider an investment so risky that they prefer to hold shortterm claims, thereby vindicating their belief that the likelihood of a crisis is large. JEL Codes:G01,F32,D82

1. Deadly Combinations A large build-up of short-term capital flows is widely perceived to be a crucial factor in the outbreak of financial sector or international banking crises.1 The role of short-term capital flows became particularly apparent during the 1990s, when more than sixty percent of all international bank This version: January 2011. Views expressed are those of the authors and do not necessarily reflect official positions of De Nederlandsche Bank. Support from Deutsche Forschungsgemeinschaft through the SFB 649 project on “economic risk” is gratefully acknowledged. 1 See Montiel and Reinhart (1997) for a literature survey on the instability created by capital flows. Manasse, Roubini, and Schimmelpfennig (2003), Rodrik and Velasco (1999), Detragiache and Spilimbergo (2004), or Furman and Stiglitz (1998) provide empirical evidence for the detrimental effects of large short-term capital flows. Eichengreen and Rose (1998) or Frankel and Rose (1996) point to no relationship, or even negative effects. 1

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claims on emerging market economies had a maturity of less than a year.2 On the eve of the Asian crisis, for the involved countries, the fraction of short-term liabilities to total liabilities exceeded this number.3 But not only the crisis in 1997, also the debt crises during the 80s, the Mexican crisis in 1994, and the Russian crisis in 1998 are associated with sudden reversals of capital inflows. And during the current financial crisis, even IMF staff—usually prominent advocates of capital account liberalisation— suggested that4 “(. . . ) capital controls on certain types of inflows might usefully complement prudential regulations to limit financial fragility and can be part of the toolkit”. This essay analyses the hypothesis that large short-term capital flows to emerging market economies raise the vulnerability of these countries to banking crises and financial turmoil. Our aim is to identify the conditions under which short-term funding flows give indeed rise to a higher likelihood of financial crises. Concretely, we determine deadly combinations of parameters: configurations of the maturity composition of debt and reserves holdings such that an increase in short term flows or a reduction of reserves unequivocally raises the probability of a banking crisis. Our approach fits into the “panic view” of banking crises, put forward by Cole and Kehoe (1996), Sachs and Radelet (1998), Rodrik and Velasco (1999), Chang and Velasco (2000), or Chang and Velasco (2001), which identifies the presence of a short maturity structure as a key factor in international financial crises. A short maturity structure is a pre-condition if a panic on the side of foreign short-term investors is to be vindicated in a “self-fulfilling” crisis through a sudden stop of capital inflows. Detragiache and Spilimbergo (2004) observe that this view has led to policy recommendations that propose to minimize the exposure of debtors to a self-fulfilling panic by hoarding international reserves or by restricting inflows of foreign capital. The literature on self-fulfilling panics leaves unsettled, however, why the panic occurs. After all, being vulnerable to a sudden stop (due to a short maturity structure) is still different from actually experiencing a sudden stop. If the beliefs that give rise to the panic are not explicitly determined, it is impossible to identify those conditions which ceteris paribus induce a higher 2

See Dadush, Dasgupta, and Ratha (2000). See Corsetti, Pesenti, and Roubini (1999). 4 IMF Staff Position Note: “Capital Inflows: The Role of Controls”. 3

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likelihood of a crisis. The above mentioned policy recommendations may even be misguided. More specifically, a problem of the “panic view” is its exclusive focus on the potential illiquidity of an otherwise solvent borrower, while the demur of Goodhart (1999, p. 345) that “[. . . ] illiquidity implies at least a suspicion of insolvency” is disregarded. In reality it may be difficult, if not impossible, to disentangle default due to illiquidity from default due to insolvency. But the distinction is still theoretically instructive, since variations in the maturity structure or the ratio of reserve holdings to illiquid investments can have opposite effects on the liquidity and the solvency position of a borrower. Let us clarify the intuition behind this with a short example. Consider a borrower who has to decide on the maturity structure of her debt, which she uses to finance a long-term investment. Assume that the investment is illiquid, and suppose further that the yield curve slopes upward. If the debtor would raise the average maturity of her debt, she might be less exposed to panic runs, and thus she would have a higher chance of surviving any roll-over date. But now (given that she manages to refinance any other short term debt as planned) she needs to fetch relatively higher returns on her investments because long-term debt is more expensive than short-term debt. In sum, while long-term debt reduces the likelihood of becoming illiquid, the likelihood of becoming insolvent increases. The relevance of this trade-off is real for economies that are able to borrow long term only against a substantial term premium. This example shows that the effect of changes in the maturity structure on the total probability of default is not clear-cut. Rather, it depends on the weights that agents attach to the probabilities of insolvency and of illiquidity in deriving the total probability of default. The weights in turn must be determined as the equilibrium outcome of a model which specifies the beliefs of short-term claimants. Since these beliefs account for the fact that the likelihood of future insolvency already influences every creditor’s present decision to roll over or not, the probability of illiquidity is connected to the probability of insolvency, and hence to the fundamental return process. In what follows, we build an international bank run model based on the global game bank run models by Morris and Shin (2009), Goldstein and Pauzner (2005), and Rochet and Vives (2004) and augment it with

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heterogeneous lenders by using results of Steiner and S´akovics (2010).5 The model allows us to separate the effects of reserves and maturity structure on the equilibrium and thereby on the likelihood of a panic. We then show that, given the induced structure of equilibrium beliefs, it is generally not true that lower reserves or a shorter maturity structure of debt unequivocally raise the likelihood of a panic. However, deadly combinations of exist parameters, under which the probability of a banking crisis is raised without doubt. Finally we endogenise the supply of short-term capital in order to derive a simple condition that the term structure of interest rates has to meet in order for a positive supply of long-term capital to exist. This condition verifies the idea that emerging markets tend to borrow short-term “because it is cheaper than long term debt” (an issue also emphasised by Chang and Velasco (2000)). However, multiple equilibrium levels of short-term debt can arise. It may lead to a self-fulfilling trap: foreign creditors consider an investment so risky that they prefer to hold short-term claims, thereby vindicating their belief that the likelihood of a crisis is large. Our paper is structured as follows. Section 2 sets out the basic theoretical framework. Comparative statics results are discussed in section 3. Section 4 contains the extension to an endogenous capital structure and section 5 concludes. 2. The Model §2.1. Economic Environment Consider a small open economy with three periods indexed by t ∈ {0, 1, 2}. There exist two groups of agents, domestic depositors (indexed with subscript d) and foreign investors (indexed with subscript f ). We assume that domestic depositors are present in measure ω ∈ (0, 1), whereas foreign investors are present in measure (1 − ω). Agents in both groups are risk-neutral and they want to consume at either date 1 or date 2. Only one good exists in the economy which can be used for consumption as well as for investment purposes. We assume that at date 0 5

Takeda (2001) also considers a global game international bank run model, but he focuses on the investment and consumption decisions of domestic agents and neither on foreign creditors’ decisions to roll over debt, nor on the influence of the maturity structure of debt.

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each agent in each group receives an initial endowment of 1 unit of this good. At subsequent dates, agents receive no endowment. This creates a need for agents to invest. Neither domestic nor foreign agents have direct access to the economy’s investment and storage technologies. But they can invest at date 0 with a banking sector which then channels funds either into investment projects or into a storage technology. Storing one unit of the good at date 0 provides the bank with immediate access to one unit at either date 1 or date 2. Thus, the storage technology creates liquidity. We will denote the amount of stored funds by % and refer to it as “reserves”. In contrast, the investment technology is risky and illiquid. In line with Morris and Shin (2009), we assume that the banking sector’s investment technology follows a stochastic return process given by θ1 = θ0 + σ1 ε1 , θ2 = θ1 + σ2 ε2 , where θ0 is fixed and ε1 and ε2 are i.i.d. random variables which are drawn from a standard normal distribution. We denote the cumulative distribution function of the standard normal distribution by Φ(·) and the density function by Φ0 (·).6 The the standard deviations σ1 and σ2 may be interpreted as the degrees of interim and terminal period uncertainty respectively. Assets pay out θ2 at date 2, yet only fetch ψθ1 at date 1. The parameter ψ ∈ (0, 1) reflects the illiquidity of the assets. One can think of (1 − ψ) as the haircut that is applied in domestic repo markets or at a central bank’s window when the asset is pledged as collateral. Furthermore, as we think in terms of a banking sector rather than an individual bank, the size of ψ also reflects the liquidity and thickness of the domestic market. We shall refer to ψθ1 as the collateral value of the asset. The date 0 price of the asset is normalised to one and the amount of investment into the asset is denoted by y.

6The distributional assumptions entail that θ , conditional on θ , has mean θ and standard 2 0 0 p

deviation σ12 + σ22 . At date 1, conditional on the realization of ε1 , its mean is given by θ1 and its standard deviation by σ2 .

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§2.2. Debt Contracts The bank enters into debt contracts with the agents in order to obtain funds. At date 0, the bank issues demand deposit contracts to domestic agents. These contracts have a face value of w2,d units at date 2 and contain the option to prematurely withdraw at date 1. If the option is exercised, the depositor receives a safe payment of w1,d > 1. The assumption that the date 1 payment is safe highlights the precaution motive of depositors who withdraw. The choice between withdrawing and rolling over is almost always equivalent to trading off risky, higher returns at a later date against safer, but lower returns at an earlier date. We emphasize this trade-off by assuming the early payoff is always safe. The bank also borrows on international capital markets from foreign investors. Following Chang and Velasco (2001), we assume that foreign funds are in perfectly elastic supply and that the bank faces a credit ceiling which, without loss of generality, we set equal to (1 − ω). A fraction ϕ ∈ (0, 1) of these funds is borrowed short-term. For now ϕ is treated as exogenous, but this will be relaxed below when we endogenise the foreign supply of short-term debt. Just as demand deposit contracts, short-term debt includes the option to withdraw at date 1. In case they withdraw, foreign short-term claimants receive a safe payment of w1,f > 1. If they decide to roll over, the date 2 face value of their claims is given by w2,f > w1,f . In contrast to short term creditors, long-term creditors do not have the possibility to withdraw at date 1. The face value of their claims is wl > w2,f . The latter assumption restricts the slope of the yield curve to be positive. As we shall see in section 4, where we endogenise the debt supply, this is a necessary condition to attract a positive amount of long-term debt in equilibrium. In fact, we will show that premium that investors require for investing into long-term debt must exceeds wl − w2,f > 0. In case that the bank defaults, its investments are liquidated and the remaining claimants of group g can only get hold of their part in the liquidation value of the bank. We denote these payments by `g and assume that `g ≤ min {w1,d , w1,f } for g ∈ {d, f }.

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§2.3. Bank Default The bank’s date 0 balance sheet constraint is given by y + % = ω + (1 − ω) = 1.

(1)

Its date 2 net worth, conditional on all domestic and foreign agents having rolled over at date 1, is given by % + θ2 y − ωw2,d − (1 − ω)[ϕw2,f − (1 − ϕ)wl ]. If its date 2 net worth becomes negative, the bank is said to be insolvent. The solvency bound is given by θs =

ωw2,d + (1 − ω)[ϕw2,f + (1 − ϕ)wl ] − % , y

which, by substituting for y from equation (1), can be expressed as a function of reserves % and foreign short-term debt ϕ, ωw2,d + (1 − ω)[ϕw2,f + (1 − ϕ)wl ] − % . 1−% Hence, the bank is declared insolvent when

(2)

θs (%, ϕ) :=

θ2 < θs (%, ϕ). Besides becoming insolvent at date 2, the bank can also fail at date 1 due to (interim) illiquidity. This happens when it does not have the liquidity to meet aggregate interim withdrawals of its creditors. The liquidity pool that the bank can draw on at date 1 is % + ψyθ1 , which can be expressed as a function of reserves % using (1): (3)

L(%, θ1 ) := % + ψ(1 − %)θ1 .

We shall call L(%, θ1 ) the bank’s liquidity bound. Let λ denote the fraction of depositors who withdraw and let κ be the fraction of foreign short-term debt holders who refuse to roll over at date 1. The bank fails at date 1 due to illiquidity whenever (4)

L(%, θ1 ) < ωλw1,d + (1 − ω)κϕw1,f .

The solvency bound θs (%, ϕ) and the liquidity bound L(%, θ1 ) are important quantities for our analysis. The solvency bound is the minimal return that the bank needs to realise at date 2. It is an increasing function of % because reserves do not yield the positive returns that the bank needs to pay off its

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creditors. It is a decreasing function of ϕ, because the yield curve is increasing, so short-term funding is cheaper than long-term funding. Thus the bank’s liability burden is smaller when the average maturity structure is shorter. The liquidity bound, which measures the bank’s capacity to withstand interim withdrawals, is also a function of reserves, but the direction of the effect depends on the size θ. The liquidity bound increases in the amount of reserves held, except if the collateral value of the asset exceeds unity. In that case, the asset is more liquid than cash, which seems rather unlikely. It would imply that banks would never hold any reserves. Although we will sometimes discuss this case in what follows, we do not consider it as being particularly relevant for our equilibrium results. §2.4. The Roll-Over Problem The question about the likelihood of a liquidity crisis is tantamount to the question under which circumstances short-term creditors and depositors will exercise their option to withdraw prior to date 2. At date 1, the probability  s of  1 date 2 insolvency (conditional on the realization of θ1 ) is given by Φ θ σ−θ . 2 For an agent of group g ∈ {d, f }, who compares the expected payoffs of each action, withdrawing is a the dominant action when   s −1 w2,g − w1,g θ1 < θ − σ 2 Φ =: θg . w2,g − `g

Thus, and short-term creditors will withdraw whenever θ1 <  all depositors min θd , θf =: θ. We refer to the interval (−∞, θ] as the lower dominance region. The converse claim, that an agent of group g would roll over when θ1 > θg , is not necessarily correct. That would ignore that each agent’s payoff from rolling over depends also on the fraction of others who roll over. From equation (4) we see that illiquidity will never constitute a problem when θ1 >

ωw1,d + (1 − ω)ϕw1,f − % =: θ. ψ(1 − %)

The characterization of the roll over decisions and the resulting outcome at date 1 would be trivial if θ > θ. All claimants would roll over for θ1 > θ because they could be certain that θ1 would be so large that the bank would never face any liquidity shortage.

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The more interesting, and more relevant, case occurs when θ < θ. For any θ1 ∈ (θ, θ), agents face a situation of strategic uncertainty because the decisions of other agents affect the likelihood of the bank becoming illiquid and thus affect their own decisions. A sufficient condition for θ < θ to occur is (5)

ωw1,d + (1 − ω)ϕw1,f > L(%, θs ).

This condition basically says that the bank would be unable to fully cover its debt at date 1 when is at the brink of date 2 insolvency.7 In our opinion this condition is not restrictive, since in reality banks with such default prospects would immediately loose access to funding. In what follows we therefore assume that condition (5) holds and we refer to the interval [θ, ∞) as the upper dominance region. If θ1 ∈ (θ, θ), this gives rise to a situation reminiscent of the results of Diamond and Dybvig (1983), or what has been labeled by Sachs and Radelet (1998) or Tirole (2002) the “panic-view” of financial crises. Different sentiments about the future prospects of the bank may lead to different coordinated actions, which in turn give rise to exactly those outcomes which were initially anticipated. In order to circumvent this problem of equilibrium multiplicity we apply the global game technique (Morris and Shin 1998) to specify short-term claimants’ beliefs and to derive a unique equilibrium of the coordination game at date 1. To this end, we assume that the shock realization ε1 is not common knowledge among short-term claimants. But before making her decision to withdraw or not, each agent i ∈ [0, ω + ϕ(1 − ω)] receives some precise information about the true state θ1 . This information leads her to believe that the true value of θ1 is distributed around some xi ∈ R. This xi is henceforth called the signal of the agent. The signal errors, given by xi − θ1 , are normally and independently distributed with mean zero and standard deviation τ1 . A strategy for typical agent i is then defined as a decision rule si : xi 7→ ai , that associates a decision ai ∈ {withdraw, roll over} with each possible signal xi . A domestic agent is said to use a threshold strategy if she demands her deposit back if and only if her signal xi falls below a threshold value x∗d . Analogously, a foreign short-term creditor uses a threshold strategy if she refuses to roll over whenever her signal xi falls below some threshold value 7Note that the expectation value of θ conditional on θ s , E (θ |θ s ), equals θ s . 2 2

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x∗f . We henceforth call the agents who receive a signal exactly equal to x∗g the critical agents. Since threshold strategies are strategies where the actions of agents depend monotonically on their information about the fundamental θ1 , they are the most natural strategies to adopt. In a joint threshold strategy, all foreign short-term creditors use a threshold strategy, i.e. it is a joint strategy profile that is characterised by a pair of thresholds (x∗d , x∗f ), one for each group. A joint strategy profile constitutes a (Bayes-Nash) equilibrium point of the model if no agent can improve her expected payoff by unilaterally deviating to a different strategy. As the following proposition shows, the coordination game between the short-term creditors has a unique equilibrium point in threshold strategies if the signals are sufficiently precise. Proposition 1. For sufficiently small τ1 , there exists a unique equilibrium point in joint threshold strategies. Default at date 1 occurs for any θ1 < θ∗ . Otherwise liquidity is sufficient to continue until date 2. Since τ is small, (6)

x∗g → θ∗ ,

g ∈ {d, f }.

The equilibrium threshold θ∗ is given by the solution to   X θ1 − θ s (7) mg w1,g og = Φ (% + ψ(1 − %)θ1 ) , σ2 where md := ω, mf := ϕ(1 − ω), and og :=

w1,g −`g w2,g −`g .

Proof. See Appendix.



One can show that this is the unique equilibrium point of the model. We mentioned in subsection 2.3 that it is theoretically possible that the date 1 collateral value of the asset exceeds unity and that a unit of the asset would generate more liquidity at date 1 than one unit of cash. How does the critical agent assess the liquidity of the asset? A critical agent believes the date 1 return is close to θ∗ . Thus she believes the asset is illiquid if the collateral value is smaller than unity, which is equivalent to ψθ∗ ≤ 1. From equation (7), a necessary condition for this belief is X wg mg og < 1.

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If this condition fails to hold, the only equilibria that can exist are equilibria where agents always run unless the asset is super-liquid, which strikes us as unrealistic. For σ2 sufficiently large, we can actually prove a necessary and sufficient condition for the asset to be considered illiquid by the critical agent. P Lemma 2. For sufficiently large σ2 , ψθ∗ ≤ 1 if and only if wg mg og ≤ ∗ µ := Φ(0); moreover, θ tends to P wg mg og − µ% ∗ ˜ θ = . ψµ(1 − %) Proof. See Appendix.



The lemma reveals some interesting implications with respect to the liquidity of the asset for the case of large terminal uncertainty. Observe that the functions w1,g − `g og = , g ∈ {d, f }, w2,g − `g are measures for the relative opportunity cost of the two actions. The numerators contain the safe gain from withdrawing, i.e. the difference between the safe payoff from withdrawing and the safe payoff in case of default. As an agent foregoes this safe gain when she rolls over, the numerators measure the cost from rolling over. The denominators contain the net gain from rolling over which the agent obtains in case that the bank does not default. The lemma relies on a restriction on the weighted sum of these opportunity cost measures. This restriction is that the date 2 premium over the liquidation value is sufficiently large relative to the opportunity cost from rolling over. The lemma therefore states that for large terminal uncertainty, either the opportunity costs of rolling over must be small enough, or the equilibrium liquidity pool must become (unrealistically) large. §2.5. Critical Beliefs and Risk Tolerance For small τ , the critical domestic agent and the critical foreign agent know that the true θ1 is close to θ∗ (cf. equation (6)). Thus, they more or less agree on the probability that has to be attached to insolvency at date 2, Pr(θ2 < θs |θ∗ ). But different critical agents still hold different beliefs about the overall default probability since they do not agree on how other agents will

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behave. The probability that a critical agent of a particular group attaches to the event {θ1 < θ∗ } is called the critical belief of this group. The critical beliefs of the two groups can be characterised by the variable xd − xf δ := , τ1 which is a measure of the relative distance between the two thresholds.8 If (x∗d − x∗f )/τ1 =: δ ∗ < 0, the critical domestic agent’s signal is slightly smaller than the critical foreign agent’s signal. Compared to the critical foreign agent, the critical domestic agent attaches a greater probability to the event of illiquidity in equilibrium. Therefore the foreign agents are willing to bear relatively less risk than the domestic agents. A dual statement holds for δ ∗ > 0. What determines the sign of δ ∗ and the relative risk tolerance of the two groups? It is determined by the relation between domestic and foreign agents’ opportunity costs of each action. In order to provide some intuition for this fact, suppose for a moment that the bank is solvent at date 2 with probability one. Let qgi stand for the probability that agent i in group g attaches to the event that the bank survives a run.9 This probability is strictly increasing in xi . When, say, a foreign agent withdraws, she obtains w1,f for sure. When she rolls over, she attaches probability qfi to obtaining w2,f and converse probability to obtaining `f . An equivalent consideration holds for any domestic agent. Recall that the functions w1,g − `g og = , g ∈ {d, f } w2,g − `g are measures for the opportunity cost from rolling over. To choose to roll over is risky, because the survival of the bank until date 2 is a probabilistic event. Hence, for the critical agents to be indifferent, the probability that they attach to this event must exactly balance their respective opportunity cost measure, i.e. qg∗ = og . As the beliefs qg are increasing in the signals, it follows that whenever δ ∗ > 0, then qd∗ > qf∗ and thus od > of . By reversing this chain of thought, we can conclude that the critical agent with lower opportunity 8Note that δ is even well-defined when τ → 0. Numerator and denominator of δ shrink at 1

the same rate, thereby keeping the relative distance unchanged. 9This probability is a non-linear function of the model’s parameters; see the subsequent proposition.

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cost from withdrawing (greater og ) has a lower tolerance towards this risk and therefore she must attach a higher probability to the bank being liquid before she would decide to roll over. The equilibrium solution of δ ∗ is given in the following proposition. Its derivation, which can be found in the Appendix, generalizes the previous example, taking into account that in equilibrium the probability of being solvent is not equal to unity. Proposition 3. For sufficiently small τ , the marginal agents’ beliefs can be characterized by     of L(%, θ∗ ) od L(%, θ∗ ) ∗ −1 P −1 P (8) δ (%, ϕ) = Φ −Φ ∈ (−1, 1). mg w1,g og mg w1,g og Proof. See Appendix.



The proposition reveals a source of risk tolerance that is independent of particular (individual) preferences (recall that we assumed risk-neutrality), but is rather a function of contractual parameters. One interesting interpretation is that, times of abundant liquidity, which depresses yields on international loans, may sow the seeds for a crisis. 3. Comparative Statics §3.1. Illiquidity versus Insolvency How is the equilibrium affected by changes in the models parameters? We are particularly interested in how changes in reserves % and short-term debt ϕ affect the ex ante probability of a liquidity crisis. The latter is, conditional on θ0 , given by  ∗  θ − θ0 ∗ pIL := Pr (θ1 < θ |θ0 ) = Φ . σ1 According to this definition, the probability of a liquidity crisis includes cases where funds are withdrawn because the bank would become insolvent at the subsequent date. Morris and Shin (2009) treat such cases as insolvency, thus referring to illiquidity only in case the bank is fully solvent at date 2 yet defaults at date 1 due to a coordination failure. However, we prefer the definition above because we believe that it is impossible in reality to ex post

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determine whether a bank is unable to roll over short-term debt because it would have been virtually insolvent, or whether it would have been solvent would it not have been denied credit. Agents might deny to roll over for different reasons; either because they fear that the bank becomes insolvent, or because they fear that too many other agents withdraw, or because of a combination of both; but the default at date 1 has only one cause, namely the bank’s lack of liquidity.10 The impossibility to ex post determine whether a defaulted bank would have been solvent or insolvent does not imply that the creditors’ ex ante assessment of the bank’s solvency situation is unimportant. In contrast, everything that affects the bank’s solvency also affects the bank’s liquidity, because illiquidity is to some extent conditional on agents putting a sufficiently high weight on the bank being subsequently insolvent. Charles Goodhart’s (1999) remark that “(...) illiquidity implies at least a suspicion of insolvency” is reflected in our model because the threshold θ∗ is a function of the solvency bound θs (compare equation (7)). As already hinted at in the example presented in the introduction, parameter variations can bring about entirely opposite effects on the solvency bound and on the liquidity bound. And as the probability of illiquidity is affected by both of these bounds, it is a priori not clear how such variations translate into a change of the probability of illiquidity. To highlight this issue, we introduce the following terminology. We shall henceforth say that a parameter causes a solvency effect when a change in this parameter alters the probability (1 − pIL ) through a change in the solvency bound θs . Similarly, a parameter is said to cause a liquidity effect when its effect on (1 − pIL ) is brought about by a change in the capacity to withstand a run.11 We call effects which decrease (1 − pIL ) negative, and those that increase (1 − pIL ) positive. 10The discussion that surrounded the Asian financial crisis can be used to illustrate this

matter. It is well-documented empirical fact that the East Asian countries that were hit by the crisis all suffered from a sudden reversal of capital flows. But there is no universal agreement on the reason for this sudden stop. For example, Corsetti et al (. . . ) emphasise the “moral hazard” perspective, questioning the solvency of the affected economies. In contrast, Sachs and Radelet (1998) attribute the withdrawal to a self-fulfilling panic, i.e. to a pure mis-coordination of creditors’ beliefs. 11I.e. a liquidity effect is either caused by a change in the RHS of equation (3) for a given LHS, or by a change in the LHS, for a given RHS.

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Since ϕ and % affect pIL only through the threshold θ∗ , and since pIL is a strictly monotone function of θ∗ , it suffices for the comparative statics to examine the signs of the derivatives ∂θ∗ /∂% and ∂θ∗ /∂ϕ. After taking the derivatives, we find that the effect of the amount of short-term capital flows on the equilibrium threshold and thus on the probability of a liquidity crisis is   mf wf1 of 1 0 θ∗ − θs ∂θs ∂θ∗ ≷0 ⇔ − Φ ≶ (9) . ∂ϕ σ2 σ2 ∂ϕ L(%, θ∗ ) | {z } | {z } Positive Solvency Effect

Negative Liquidity Effect

Short-term capital flows are associated with a positive solvency effect, which is displayed on the left-hand side of the last inequality. Short-term capital is cheaper than long-term capital and the probability of insolvency therefore decreases with a shorter maturity structure. By contrast, the liquidity effect on the right-hand side is negative. An increase in short-term debt increases, for a given liquidity pool, the amount of possible date 1 claims and thus raises the risk of illiquidity. Similarly, for the effect of reserve holdings on the threshold, we find   0 θ∗ −θs Φ ∗ σ2 ∂θ 1 ∂θs (1 − ψθ∗ )   . (10) ≶0 ⇔ ≶ ∂% σ2 Φ θ∗ −θs ∂% L(%, θ∗ ) | {z } σ2 | {z } Liquidity Effect Negative Solvency Effect

In this case, the solvency effect is negative. As reserves do not yield positive net returns, an increase in reserves moves the solvency bound up, thus increasing the probability of becoming insolvent. The sign of the liquidity effect is a priori not clear. If the collateral value of the asset at the critical threshold exceeds unity—if ψθ∗ ≥ 1—an increase in reserves would reduce the available liquidity, thus rendering the liquidity effect negative. This is intuitive, since at the critical margin, the bank would be better off not to hold any reserves at all. Conversely, if the collateral value is below unity, the sign of the liquidity effect reverses because additional reserves strengthen the capacity to meet withdrawals. As explained above, we consider this to be the relevant case in our setting. In this case the liquidity effect is indeed positive. It follows that an increase in short-term capital flows and/or a reduction in reserve holdings increase the vulnerability (measured as the ex ante probability

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of a liquidity crisis) if the liquidity effects outweigh the respective solvency effects. But under what conditions will liquidity effects dominate solvency effects? A key parameter in this respect turns out to be the degree of terminal uncertainty, σ2 . From the perspective of date 1, solvency is crucially dependent on the eventual realization of the random variable ε2 . Thus, the marginal impact on the solvency bound is weighted, in essence, by σ2 . When σ2 gets large, the conditional probability of being solvent becomes less sensitive with respect to a change in the solvency bound. Agents attach a fairly constant probability to solvency, independently of any event that would potentially alter this probability. As the liquidity effect is not affected by σ2 , one can find (finite) bounds such that for a σ2 exceeding it, the liquidity effect dominates. Proposition 4. Assume θ∗ ψ < 1 holds. For sufficiently large σ2  0, the liquidity effect dominates the solvency effect. That is, ∂θ∗ ∂θ∗ > 0 and < 0. ∂ϕ ∂% Proof. Take derivatives.



Note that during financial crises, fundamental uncertainty is typically large, lending relevance to this result. By lemma 2, the condition that θ∗ ψ < 1 may P be replaced by the corresponding condition that wg mg og ≤ Φ(0). §3.2. Welfare In the previous sections we have analysed the effects of variations of shortterm capital flows and international reserves on the crisis probability. The sign of such variations depends on whether the liquidity effects outweigh the solvency effects. Increases in short-term debt can have detrimental effects whenever terminal returns are sufficiently uncertain. Consider the case where σ2 is very large. As shown in lemma 2, the equilibrium threshold then tends to P g wg og mg − µ% ∗ , θ˜ = ψµ(1 − %)

and the liquidity effect dominates the solvency effect by proposition 4. The first-best solution from the point of view of the domestic agents would be to have the banking sector being financed through foreign long-term debt.

LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES

17

autarky (θˇ∗ )

long term debt (θˆ∗ )

ϕ

θ˜∗ (ϕ)

θ

Figure 1. Thresholds as a Function of Short Term Debt But suppose that this solution is not feasible. Then the question remains whether some foreign short-term debt could improve the outcome relative to an autarky situation where no foreign financing is available at all. We will compare these situations in detail (see figure 1). The closed form threshold in case that foreign agents only hold long-term claims is given by w1,d md od − µ% θˆ∗ = , ψµ(1 − %) while the autarky level can be written as w1,d md od − µ% θˇ∗ = > θˆ∗ . ψµ(md − %)

Without foreign financing, the banking sector can at most invest ω into liquid and illiquid assets. This implies a negative solvency effect which shifts the run point upwards relative to a situation where the banking sector has some foreign liabilities. However, compared with an economy with short term debt, the absence of the negative externalities induced by foreign claimants who withdraw early induces a liquidity effect which leads to a relative downward shift of the run point. It is now obvious that when the amount of short-term debt becomes too large, the economy is worse off than it would be if it would

¨ ¨ PHILIPP J. KONIG AND TIJMEN R. DANIELS

18

be completely cut off from international capital markets. Indeed, as θ˜∗ is an increasing function of ϕ, this happens when w1,d md od − µ% . θ˜∗ ≥ θˇ∗ ⇔ ϕ ≥ ϕcrit := (md − %)w1,f of §3.3. Deadly Combinations We have shown that the liquidity effects outweigh the solvency effects for sufficiently large σ2 , and that in this case, an increase in the level of short-term debt or a reduction in reserves indeed raise the likelihood of a liquidity crisis. However, the exact size of a sufficient σ2 is dependent on the prevailing levels of short-term debt and reserves. For example, if the actual level of reserves is small and the actual level of short-term debt is quite large, the sufficient degree of uncertainty which is needed to cause additional short-term debt flows to raise the probability of a crisis is rather small, compared to a situation where, say, a high level of reserves meets a low level of short-term debt. From equation (9) follows that the liquidity effect of additional short-term debt outweighs the solvency effect if  ∗  s mf wf1 of 1 ∂θs 0 θ −θ > − · Φ . · ∗ L(%, θ ) σ2 σ2 ∂ϕ √ As Φ0 (·) has a well-defined maximum given by 1/ 2π, we have that the liquidity effect outweighs the solvency effect if s

σ2 > √

−L(%) ∂θ ∂ϕ

2π(mf wf1 of )

.

The latter term is represented by a vertical line in ϕ-%-space.12 When the equilibrium level (ϕ, %) is to the left of this line, the given σ2 is sufficiently large to cause the liquidity effect of short-term debt to dominate its solvency effect. Similarly, from equation (10), the liquidity effect of reserves dominates if  ∗   ∗  s 1 ∂θs θ − θs (1 − ψθ∗ ) 0 θ −θ ·Φ · <Φ · . σ2 σ2 ∂% σ2 L(%) 12It is vertical because the right-hand side of the inequality is independent of ϕ, since ∂ 2 θs ∂ϕ2

= 0.

LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES



∗

19

 s

Now the term Φ θ σ−θ reflects the probability of being solvent conditional 2 ∗ on θ1 = θ , which has a well-defined, non-zero lower bound, which we denote by δ. The sufficient condition becomes ∂θs ∂% L(%)

σ2 > √ . 2πδ 2 (1 − ψθ)

The term on the right-hand side is represented by an increasing curve in ϕ-%space.13 When the equilibrium level (ϕ, %) lies to the left of this line, the given σ2 is sufficiently large to cause the liquidity effect of reserves to dominate its solvency effect. The shaded area to the left of both curves is the area of deadly combinations: Whenever a country’s reserves and short-term liqbilities, (%, ϕ), come to lie in this set, the country’s likelihood of being subject to a liquidity crisis increases, when short-term debt is raised or reserves are decreased. However, the reverse implication, that these countries can easily reduce their exposure by limiting the inflows of short-term debt and increasing the level of reserves does not hold in general. Although such policies work out in a small neighborhood of the equilibrium, a too large variation drag the country into an area below the curves, where the solvency effect may dominate, and the effects reverse. This is one reason why we call these combinations deadly: the downside risk is obvious, but the seemingly obvious cure might turn out to be in fact a deadly trap. 4. Maturity Structure and Ex Ante Traps §4.1. Endogenous Maturity Structure The maturity structure which eventually prevails in equilibrium depends crucially on the ex ante decision by foreign creditors to invest into short-term or long-term debt. In this section, we endogenise the maturity structure. The option to withdraw contained in short-term contracts has a positive value since it makes debt less risky. To be in positive supply, long-term debt must fetch a sufficiently high premium over short-term funds. This reveals a potential trap: as the maturity of debt profile of debt shortens, the probability of a crisis may increase so much that, from the perspective of foreign agents, 13The curve is increasing because

∂ 2 θs ∂%∂ϕ

< 0.

¨ ¨ PHILIPP J. KONIG AND TIJMEN R. DANIELS

20

ϕ

!

Figure 2. Deadly Combinations in (%, ϕ)-space

a switch to less riskier, short-term debt becomes justified for a given term premium. Hence, there may be multiple consistent combinations of maturity structure and likelihood of a crisis. We will show below that if liquidity effects dominate solvency effects, it is indeed possible that multiple consistent maturity structures exist. Chang and Velasco (2001) determine term and maturity structure simultaneously by taking the perspective of the borrower. By contrast, here, we consider the problem from the perspective of a lender who ultimately must be indifferent between the two forms of debt. Assume foreign investors have the choice between investing in long-term or short-term debt shortly prior to date 0. Short-term debt provides them with the option to prematurely withdraw at date 1, while long-term debt does not provide this option. We assume that when making their investment, foreign agents do not know the state of the economy, summarised by θ0 . However, they receive a signal (perhaps due to some research they conduct) given by x0 = θ0 + τ0 ηi ,

LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES

21

where ηi is i.i.d. according to a standard normal distribution and τ0 is a positive scale parameter. Given their signals, agents calculate the ex ante probabilities of illiquidity and insolvency and use these to compute the expected payoffs from long- and short-term debt respectively. We use the following abbreviations: pIL (x0 ) denotes the ex ante probability of the bank becoming illiquid (conditional on signal x0 ), pL∧S (x0 ) is the ex ante probability of the bank being liquid and solvent (conditional on signal x0 ), i.e. the probability of the bank surviving up and until the end of date 2, pL∧IS (x0 ) is the ex ante probability of the bank being liquid at date 1 and becoming insolvent at date 2, and pIL∨IS (x0 ) is the probability of either being illiquid or being insolvent (conditional on signal x0 ). For notational simplicity we will omit explicit reference to the argument x0 whenever possible. Using this notation, the ex ante payoff from investing into short-term debt can be written as pIL w1,f + pL∧S w2,f + pL∧IS `f = pIL (w1,f − `f ) + pL∧S w2,f + pIL∨IS `f . The expected payoff from investing into long-term debt is given by pL∧S wl + pIL∨IS `f . Investors who receive the signal x∗0 are indifferent between the two forms of debt, whereby x∗0 solves pIL (x0 )(w1,f −`f )+pL∧S (x0 )w2,f +pIL∨IS (x0 )`f = pL∧S (x0 )wl +pIL∨IS (x0 )`f . Rewritting this gives the indifference condition (11)

pIL (x∗0 )(w1,f − `f ) = pL∧S (x∗0 )(wl − w2,f ).

The probabilities are continuous functions of x0 . And since the expected payoff from investing into short-term debt is decreasing in x0 whereas the payoff from investing long-term is increasing in x0 , there exists a unique intersection x∗0 that solves equation (11). Agents who receive a signal x0 < x∗0 will therefore invest into short-term debt, while agents with a signal x0 > x∗0 will invest into long-term debt. Given that the critical signal x∗0 is a function of the parameters of the model and thus is a function of the amount of

22

¨ ¨ PHILIPP J. KONIG AND TIJMEN R. DANIELS

short-term debt ϕ, we can now write the supply of short-term debt as  ∗  x0 (ϕ) − θ0 ∗ (12) ϕ = Pr (x0 < x0 (ϕ)|θ0 ) = Φ . τ0 Any fixed point of equation (12) constitutes an equilibrium level of short-term debt. §4.2. An Ex-Ante Trap However, it is possible that multiple such levels exist. A necessary and sufficient condition for the latter phenomenon to occur would be that the slope of Pr (x0 < x∗0 (ϕ∗ )|θ0 ), when evaluated at some θ0 = x∗ (ϕ∗ ), exceeds unity (see figure 3). The slope is given by  ∗  ∂Pr (x0 < x∗0 (ϕ)|θ0 ) 1 ∂x∗0 (ϕ) 0 x0 (ϕ) − θ0 =Φ , ∂ϕ τ0 τ0 ∂ϕ and the necessary and sufficient condition for multiple levels of short-term debt, consistent with the unique threshold θ∗ , becomes √ ∂x∗0 (ϕ) > τ0 2π. ∂ϕ |ϕ=ϕ∗ Since the right-hand side is positive, the latter condition requires that is positive as well. The following lemma shows that this is in fact true, if σ2  0. ∂x∗0 (ϕ) ∂ϕ

Lemma 5. The derivative of the signal x∗0 with respect to ϕ is given by ∂x∗0 (ϕ) ∂θ∗ ∂θs = (1 − α(σ2 )) + α(σ2 ) . ∂ϕ ∂ϕ ∂ϕ For sufficiently large σ2  0, we have α(σ2 ) ≈ 0, and thus ∂x∗0 (ϕ) ∂θ∗ ≈ > 0. ∂ϕ |ϕ=ϕ∗ ∂ϕ |ϕ=ϕ∗

Proof. See Appendix.



Hence, the effect of a marginal increase in short-term capital flows on the probability with which an investor supplies short-term debt is given by a weighted average of the marginal effects of short-term debt on the threshold θ∗ and the solvency bound θs . Whenever the terminal uncertainty becomes

LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES

23

ϕ

Pr (x0 < x∗0 (ϕ)|θ0 )

ϕ

Figure 3. Multiple fixed points in the maturity structure of debt sufficiently large, so that the probability of insolvency becomes sufficiently insensitive with respect to variations in the solvency bound, multiple shortterm debt levels are consistent with the unique equilibrium θ∗ . All these levels must lie to the right of the horizontal line in figure 2. This implies that the economy can become stuck in an equilibrium with a high level of short-term debt because they fear that the economy defaults at date 1 with a high probability. This short-term debt trap is self-fulfilling as it actually causes the high default probability. 5. Conclusion We have analysed the hypothesis that large short-term capital flows to emerging market economies raise the vulnerability of these countries from a theoretical perspective, using a global game banking model. We showed that policy recommendations that propose to minimize the exposure of debtors to a self-fulfilling panic by hoarding international reserves or by restricting inflows of foreign capital may be good advice in some circumstances, but not in all. We identified configurations of the maturity composition of debt or reserves holdings such that an increase in short term flows or a reduction

¨ ¨ PHILIPP J. KONIG AND TIJMEN R. DANIELS

24

of reserves unequivocally raises the probability of a banking crisis. A key parameter for this to be the case unequivocally is the degree of uncertainty associated with the fundamental return process. Since during financial crises, fundamental uncertainty is typically large, this degree of uncertainty also plays a crucial part in reality. We also endogenised the supply of short-term capital in order to derive a simple condition that the term structure of interest rates has to meet in order for a positive supply of long-term capital to exist. This analysis reveals a potential deadly trap when the degree of uncertainty about the return process is high. As the maturity profile of debt becomes shorter, the probability of a crisis may increase so much that, from the perspective of foreign agents, a switch to less riskier, short-term debt becomes justified for a given term premium. Hence, there may be multiple equilibrium-consistent combinations of maturity structure and likelihood of a crisis. One combination is characterised by a low amount of short term debt, and a low probability of a crisis; in another combination, the probability of a crisis is high, and consequently, most foreign creditors refuse to invest in anything else than debt with short maturity.

6. Appendix Proof of Proposition 1. We normalise payoffs so that they are identical to the payoff difference: Bank Survives Bank Fails Rollover (action 1) wg2 − wg1 `g − wg1 Withdraw (action 0) 0 0 Normalising payoffs in this way does not change the game underlying our model. Define the aggregate action, given by: Z md +mf 1 a= wg(i) ai di, ai ∈ {0, 1}, where g(i) denotes i’s group. 0

The bank survives date 1 if: (13)

a≥

X

mg wg1 − L(%, θ1 ).

LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES

25

We set u(a, θ1 ) = 1 if (13) holds and u(a, θ1 ) = 0 if not. By using the normalised payoffs, we can write the payoff from rolling over for a typical agent of group g as ( ug (a, θ1 ) := (1 − p(θ1 ))(wg2 − `g ) + `g − wg1 if u(a, θ1 ) = 1 ug (a, θ1 ) = 1 otherwise. ug (a, θ1 ) := `g − wg Note that if we set βg (θ1 ) = (1 − p(θ1 ))(wg2 − `g ) and γg = wg1 − `g we find ug (a, θ1 ) = βg (θ1 ) · u(a, θ1 ) − γg . Our model now matches the assumptions of Steiner and S´akovics (2010). Using their results we find that the equilibrium threshold is given by the solution to: Z P mg wg1 X 1 γg = u(a, θ1 ) da. m g wg βg (θ1 ) 0 Using equation (13), we rewrite this as X γg mg wg1 = L(%, θ1 ). βg (θ1 ) By substituting out βg (θ1 ), γg and L(%, θ1 ), and by multiplying both sides by (1 − p(θ1 )), we obtain   X w1 − `g θ1 − θ s 1 g (14) mg wg 2 =Φ (% + ψ(1 − %)θ1 ) .  wg − `g σ2 Proof of Lemma 2. Necessity follows from the observations in the main text. As for sufficiency, we claim that, if σ2 becomes large, Φ((θ∗ − θs )/σ2 ) tends to P µ. Hence equation (7) can hold only if µ · (% + ψ(1 − %)θ∗ ) tends to wg mg og , ∗ ∗ and thus for large σ2 , (% + ψ(1 − %)θ ) ≤ 1 if and only if ψθ ≤ 1 if and only P wg mg og ≤ µ. The expression for θ˜∗ follows readily. To prove our claim, it suffices to show (θ∗ − θs )/σ2 → 0 as σ2 → ∞. First observe that θ∗ , being a solution to equation (7), cannot be negative. So it is assured that (θ∗ − θs )/σ2 → 0 if θ∗ is bounded from above, i.e., for some constant k, we have θ∗ ≤ k for all σ2 . Of course, θ∗ may not be bounded at all. To deal with this case let us suppose that θ∗ > 0 is unbounded as σ2 → ∞ and, towards a contradiction, that (θ∗ − θs )/σ2 > c > 0 as σ2 → ∞. Since θs does not depend on σ2 , we have θ∗ /σ2 > c, viz. θ∗ > cσ2 as σ2 → ∞. But this means that for sufficiently

26

¨ ¨ PHILIPP J. KONIG AND TIJMEN R. DANIELS

large σ2 ,  ∗  X θ − θs Φ (% + ψ(1 − %)θ∗ ) > Φ(c) (% + ψ(1 − %)cσ2 ) > wg mg og , σ2 and hence equation (7) cannot hold, contradicting that θ∗ is a solution.



Proof of Proposition 3. A critical agent of group g is indifferent if and only if βg (θ∗ )qg (δ) − γg = 0,  ∗ ∗
x −θ where qg (δ) := Pr θ1 > θ∗ |x∗g = Φ gτ1 . As Φ(·) has a well-defined inverse, we have   x∗g − θ∗ γg . = Φ−1 τ1 βg (θ∗ ) By using the definition x∗f = x∗d − τ1 δ ∗ , we rewrite the equation for g = f in terms of x∗d and the distance δ ∗ ,   γf x∗d − θ∗ ∗ −1 −δ =Φ . τ1 βf (θ∗ ) Equating with the respective equation for g = d and solving for δ ∗ yields,     γf γd −1 ∗ −1 −Φ . δ =Φ βf (θ∗ ) βd (θ∗ ) The expression in equation (8) then follows by substituting for βg (θ∗ ) and γg using their definitions, and by using the fact that, from equation (7), P   mg wg1 og θ1 − θ s =Φ .  L(%, θ∗ ) σ2 Proof of Lemma 5. The indifference condition implicitly defines the function x∗ (ϕ∗ ), which involves the probabilities pIL and pL∧S . We have the following explicit expressions. For the probability of illiquidity, Z θ∗ −x0 σ ˆ pIL = Pr (θ1 ≤ θ∗ |x0 ) = Φ0 (z)dz, −∞

p where σ ˆ := σ12 + τ02 . The probability of being liquid and solvent involves a more complex expression. It can be written as pL∧S = Pr ({θ1 > θ∗ } ∩ {θ2 > θs } |x0 ) ,

LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES

27

which, when the return process and the definition of the signal are used, can be written as Pr ({σ1 ε1 − τ0 η0 > θ∗ − x0 } ∩ {σ1 ε1 + σ2 ε2 − τ0 η0 > θs − x0 }) Now we have that {σ1 ε1 − τ0 η0 > θ∗ − x0 } ⇒ {σ1 ε1 + σ2 ε2 − τ0 η0 > θs − x0 } , which is equivalent to ε2 > s

θs − θ∗ , σ2

∗

. and conversely for ε2 < θ σ−θ 2 Since all error terms are independently distributed, the joint density of σ1 ε1 − τ0 η0 and ε2 is given by Φ0 (σ1 ε1 − τ0 η0 ) × Φ0 (ε2 ).

We can thus write the probability pL∧S as Z θs −θ∗ Z ∞ Z σ2 0 0 Φ (z)Φ (t)dz dt + s −∞

θ −x0 −σ2 t σ ˆ

∞ θ s −θ ∗ σ2

Z

∞ θ ∗ −x0 σ ˆ

Φ0 (z)Φ0 (t)dz dt

The existence of a signal x∗0 which solves the indifference condition can be easily verified, by noting that the probabilities are continuous functions of x∗0 and that for x0 → ∞, we have pIL → 0 and pL∧S → 1, and for x0 → −∞, we w1,f have pIL → 1 and pL∧S → 0. Since wl −w > 0, there exists at least one x0 2,f which solves the indifference condition. Now consider the derivative of

∂x0 ∂ϕ .

G(x0 , ϕ) := pIL (x0 ) The derivative is then given by −

Define x0 (ϕ) implicitly through

w1,f − `f − pL∧S (x0 ) = 0. wl − w2,f

Gϕ (x∗0 , ϕ) . Gx0 (x∗0 , ϕ)

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¨ ¨ PHILIPP J. KONIG AND TIJMEN R. DANIELS

This involves the derivatives of pIL and pL∧S with respect to ϕ and x0 . For the derivatives of pIL , we have  ∗  ∂θ∗ ∂pIL ∂ϕ 0 θ − x0 = Φ , ∂ϕ σ ˆ σ ˆ which is positive when the liquidity effect dominates the solvency effect (and vice versa), and   ∂pIL −1 0 θ∗ − x0 = Φ , ∂x0 σ ˆ σ ˆ which is strictly negative. The derivative of pL∧S with respect to x0 is given by    s  ∗ Z θs −θ∗ Z σ2 ∂pL∧S 1 1 ∞ 0 0 θ − x0 − σ2 t 0 θ − x0 = Φ (t) dt+ Φ0 (t) dt, Φ Φ ∂x0 σ ˆ −∞ σ ˆ σ ˆ θs −θ∗ σ ˆ σ2

which is strictly positive. For the derivative of pL∧S with respect to ϕ, define Z ∞ h(ϕ, t) := s Φ0 (z)Φ0 (t) dz θ −x0 −σ2 t σ ˆ

and

Z k(ϕ, t) :=

∞ θ ∗ −x0 σ ˆ

Φ0 (z)Φ0 (t) dz.

From the definition of pL∧S , we compute, by using Leibniz’s rule, Z θs −θ∗ Z ∞ σ2 ∂pL∧S = hϕ (ϕ, t) dt + kϕ (ϕ, t) dt, θ s −θ ∗ ∂ϕ −∞ σ2



s



∂θ s ∂ϕ

where hϕ (ϕ) = −Φ0 θ −xσˆ0 −σ2 t · Φ0 (t) · σˆ which is strictly positive, and  ∗  ∂θ ∗ 0 (t) · ∂ϕ which is negative if the liquidity effect 0 kϕ (ϕ) = −Φ0 θ −x Φ · σ ˆ σ ˆ dominates the solvency effect. Combining everything, we find, after some tedious algebra, (15)

Gϕ (x∗0 , ϕ) ∂θs ∂x∗0 ∂θ∗ =− + α(σ ) , = (1 − α(σ )) 2 2 ∂ϕ Gx0 (x∗0 , ϕ) ∂ϕ ∂ϕ

LIQUIDITY AND SOLVENCY IN A MODEL OF EMERGING MARKET CRISES

29

where ∂θs ∂ϕ

α(σ2 ) := Φ0 and w ˆ :=



θ∗ −x0 σ ˆ

w1,f −`f wl −w2,f .

R

θ s −θ ∗ σ2

−∞

h R∞ w ˆ + θs −θ∗ σ2



θs −x0 −σ2 t σ ˆ



Φ0 (t) dt ,  s i R θs −θ∗  θ −x0 −σ2 t σ2 0 0 0 Φ (t) dt + −∞ Φ Φ (t) dt σ ˆ Φ0

By the dominated convergence theorem, we have that Z

lim

θ s −θ ∗ σ2

σ2 →0 −∞

0

Φ



θ s − x0 − σ2 t σ ˆ



Φ0 (t) dt → 0.

Hence, for sufficiently large, but finite σ2  0, the term  s  Z θs −θ∗ σ2 0 θ − x0 − σ2 t Φ Φ0 (t) dt σ ˆ −∞ becomes negligibly small, so that we have ∂x0 (ϕ) ∂θ∗ ≈ . ∂ϕ ∂ϕ From proposition 4 follows that this is positive for sufficiently large σ2 .



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