Asset quality, debt maturity, and market liquidity

Accepted Manuscript

Asset Quality, Debt Maturity, and Market Liquidity Yaxian Gong, Xu Wei PII: DOI: Reference:

S1544-6123(18)30122-3 https://doi.org/10.1016/j.frl.2018.11.015 FRL 1045

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Finance Research Letters

Received date: Revised date: Accepted date:

24 February 2018 19 November 2018 21 November 2018

Please cite this article as: Yaxian Gong, Xu Wei, Asset Quality, Debt Maturity, and Market Liquidity, Finance Research Letters (2018), doi: https://doi.org/10.1016/j.frl.2018.11.015

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Highlights • We construct a model that endogenizes both the debt maturity choices of financial institutions and the liquidity of the asset market. • The decrease in asset quality can cause over-reliance on short-term debt and also lower market

• The result provides a new mechanism of market freeze.

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liquidity at the same time.

• The result sheds light on unifiedy understanding the maturity mismatch and liquidity dry-ups

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of the asset market during the recent global financial crisis.

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Yaxian Gong

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Asset Quality, Debt Maturity, and Market Liquidity

School of Economics, Central University of Finance and Economics Xu Wei

School of Finance, Central University of Finance and Economics

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[email protected]

Abstract

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We construct a model that endogenizes both the debt maturity choices of financial institutions and the liquidity of the asset market in a Rational Expectations Equilibrium. And

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we find that the decrease in asset quality can cause over-reliance on short-term debt and also lower market liquidity at the same time. Our result provides a new mechanism of market

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freeze and sheds light on unifiedy understanding the maturity mismatch and liquidity dry-ups of the asset market during the recent global financial crisis.

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JEL Classification Numbers: D08; G01; G02

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Key Words: Debt Maturity; Market Discipline; Asset Quality; Market Liquidity

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Introduction

The liquidity freezes of financial asset markets, e.g., the ABS market, played important roles in the 2007-2009 global financial crisis (Bond and Leitner, 2015; Chiu and Koeppl, 2016). Most previous works attribute the liquidity dry-ups to the adverse selection problem (Plantin, 2009; Bolton et al., 2011; Malherbe, 2014). However, in this paper, we find a novel mechanism: The liquidity freeze is

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related to the over-reliance on short-term financing (also important in amplifying the crisis), which is induced by the incentive of alleviating the moral hazard problem.

We present a model endogenizing both the asset market liquidity and debt maturity choices of financial institutions in a Rational Expectations Equilibrium. Financial institutions (we call them banks for simplicity), expecting the liquidity of the asset market in the future, optimally choose

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the debt maturity, facing the tradeoff between the market discipline effect and the induced asset liquidation of the short-term debt. And the market liquidity in the future is in turn determined by the maturity choice of banks, since the amount of sold assets depends on how many banks fail to roll over the short-term debt and have to liquidate their assets. This constructs an equilibrium with rational expectations.

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Within this framework, we find that as the quality of banks’ assets decreases, the required debt repayment by creditors is raised, which worsens the moral hazard problem and thus short-term debt

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will be more relied on to exploit its discipline effect. Consequently, more assets will be liquidated, causing the drop of asset market liquidity.

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Our result is consistent with the pattern of U.S. financial market around the 2007-2009 financial crisis. Before the crisis started in 2007, the asset quality, especially that of the mortgages decreased

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year by year (Demyanyk and Hemert, 2011; Mayer et al., 2009), which encouraged the financial institutions to rely more on short-term debt (Brunnermeier, 2009). Thus, as the crisis began,

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more and more financial institutions had to liquidate their assets, which led to large fire sales and extremely low market liquidity. This paper also integrates the literature of debt maturity and market liquidity freezes. Un-

like the studies pointing out that debt maturity may affect asset market liquidity (Archarya and Viswanathan, 2011; Acharya et al., 2011), and the works emphasizing that the market liquidity may impact the debt maturity choice (Huberman and Repullo, 2014; Bruche and Segura, 2017), our model investigates the interactions between them. Eisenbach (2017) also builds a model en1

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dogenizing debt maturity and market liquidity. However, liquidation of assets in his model needs not to be inefficient, and his focus is that the short-term debt may lead to excessive inefficient liquidation and insufficient efficient liquidation. In our model, liquidation is always inefficient, and realizing this, financial institutions are still willing to use the short-term debt due to the moral hazard problem.

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Model Setup

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Figure 1: Timeline

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As the timeline in Figure 1 shows, we consider a model of three dates t = 0, 1, 2. There are a continuum of homogeneous financial institutions (we call them banks), the total measure of which is M . Banks are risk neutral. At t = 0, each bank, owned by a banker, can invest in a risky

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asset, which requires a starting cost normalized to 1. The asset matures at t = 2 and pays θ˜ with probability p, or 0 with probability 1 − p. Note that θ˜ is also random, obeying a uniform prior ¯ (the pdf function is f (θ) = distribution over [θ, θ]

1 ¯ ), θ−θ

in which θ < 1 and θ¯ > 1. And we assume


that the asset has positive net present value 12 p θ¯ + θ ≥ 1.

At t = 0, the bank has no capital and therefore it needs to finance the starting cost of the asset

from outside investors via issuing debts. We assume there are also a continuum of risk neutral

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investors with sufficiently large total measure, and each of them endows 1 unit of cash. Banks can offer two kinds of debt contracts to investors: long-term debt and short-term debt. Each unit of debt costs 1 to creditors. The long-term debt matures and promises to pay D0,2 at t = 2. The shortterm debt matures and promises to pay D0,1 at t = 1. The short-term debt induces the maturity mismatch problem: At t = 1, the maturing short-term debt needs to be refinanced, exposing the bank to the rollover risk. We denote the proportion of short-term debt by α. It represents the debt

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maturity structure of the bank and will be endogenously determined in our model.

At t = 1, the possible high payoff of the asset θ˜ realizes as θ. It can be redeemed as some public information about the asset quality. After that, the short-term debt issued at t = 0 matures, and the bank needs to issue some new debts for rollover, and we call them rollover debts. If creditors are willing to buy the rollover debt, the bank can obtain enough funds and repay the maturing

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short-term debt. Then we denote the gross interest rate of the rollover debt by D1,2 (θ). Otherwise, the bank will be forced to liquidate some fraction of its asset to repay the maturing short-term debt. The liquidation value is L per unit of the asset, which can be used to indicate the market liquidity. We assume that L = χ(n) ≥ 0 is a function of n, the total amount of the assets liquidated at t = 1. 0

00

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Further, χ (n) < 0 and χ (n) < 0 hold to capture that the liquidity decreases with asset supply at an accelerating speed. Denote the proportion of liquidated assets of each bank by ν ∈ [0, 1], and

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n = M ν since banks are homogeneous. We assume Lmax ≡ χ(0) ≤ θ so that liquidation is always inefficient. Besides, Lmax is sufficiently close to θ and Lmin ≡ χ(M ) is sufficiently small, so that the amount of liquidated assets n can affect the asset liquidity significantly.

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At t = 1, given that the bank still survives, we follow Calomiris and Kahn (1991) to assume that there may be the moral hazard problem. That is, after observing the signal, the banker can

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choose not to behave and abscond a private benefit b > 0 per unit of the remaining asset, which leaves creditors nothing. We assume that b ≤ pθ, which means the misbehavior of the banker is

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always inefficient. In other words, it causes value loss pθ − b per unit of asset in expectation.1 We call this the “moral hazard” action throughout the paper.

At t = 2, if the banker has not taken the moral hazard action, each unit of asset pays θ with

probability p. Conditional on that, creditors get the debt repayment and the bank receives the remaining value as the profit. 1

This moral hazard problem can also be interpreted as that the banker may give up managing the asset, leading to 0 asset value. And this behavior saves the manager the cost of effort b.

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3

Optimal Debt Maturity Choice

In this section, we solve the optimal debt maturity choice of a representative bank, expecting the asset market value at t = 1 to be L. In order to do this, we need to calculate the profit of the bank as a function of the debt maturity α (the proportion of short-term debt), fixing the liquidation

3.1

Rollover condition at t = 1 and D1,2 (θ)

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value L. And before that, we need to determine all the debt interest rates D0,1 , D0,2 , D1,2 (θ).

We solve all the debt interest rates backwards, i.e., starting from D1,2 (θ) at t = 1, which is closely related to the rollover condition. At t = 1, after θ is realized and supposing that the bank succeeds in rolling over the short-term debt by issuing new debts with interest rate D1,2 (θ), the banker will

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not choose the moral hazard action if and only if

b p (θ − αD1,2 (θ) − (1 − α)D0,2 ) ≥ b ⇔ θ ≥ αD1,2 (θ) + (1 − α)D0,2 + . p

(1)

It means that only if the total debt payment is small enough compared with the asset value, the

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banker will find it unprofitable to take the moral hazard action.

If θ is high enough such that inequality in (1) holds, then the bank will behave and all creditors

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can expect that they can get their debts fully repaid with probability p. In this case, creditors of the rollover debt will require pD1,2 (θ) = D0,1 to break even. Thus, we can write the condition of

D0,1 b θ ≥ θˆ ≡ α + (1 − α)D0,2 + . p p

(2)

Break-even conditions at t = 0 and D0,1 , D0,2

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3.2

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successful rollover without moral hazard in (1) as

ˆ according to above analyses, each short-term debt holder from t = 0 can get D0,1 fully If θ ≥ θ, ˆ creditors will refuse to refinance the paid at t = 1. However, if θ is too small so that θ < θ, maturing short-term debt. Given the per unit liquidation value of the asset L, the bank needs to sell ν = min{ αDL0,1 , 1} fraction of its asset to repay αD0,1 . Thus, each short-term debt holder from ˆ t = 0 can get min{D0,1 , Lα } paid at t = 1. Expecting the payoffs in both cases (θ ≥ θˆ and θ < θ),

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the break-even condition for the short-term debt holders at t = 0 is ˆ

θˆ

θ



L min D0,1 , α



f (θ)dθ +

ˆ

θˆ

θ¯

D0,1 f (θ)dθ = 1.

(3)

For each long-term debt holder from t = 0, she can get the promised value D0,2 fully paid with ˆ at t = 1. However, if θ < θ, ˆ her expected probability p if the realization is high enough (θ ≥ θ)

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payoff will be much less due to the asset liquidation (ν units of assets are liquidated with 1 − ν units left) and the possible moral hazard action by the banker. If αD0,1 ≥ L, then ν = 1, which means that the whole asset of the bank is liquidated and long-term debt holders get nothing. If αD0,1 < L, then ν < 1 and ν =

αD0,1 . L

In this case, the banker will not take the moral hazard action

to the remaining 1 − ν units of assets if and only if

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     αD0,1 αD0,1 (1 − α)D0,2 b p 1− θ − (1 − α)D0,2 ≥ b 1 − ⇔θ≥ + ≡ θˇ L L p 1 − αDL0,1

(4)

We can easily derive that θˇ ≥ θˆ ⇔ L ≤ αD0,1 + p(1 − α)D0,2 , which apparently holds because

ˆ then θ < θˇ also holds, and the banker will always take the Lmax ≤ θ < 1. As a result, if θ < θ,

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moral hazard action, leaving creditors nothing. Considering the two cases, we can write down the

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break-even condition for the long-term debt holders at t = 0: ˆ

θ¯

pD0,2 f (θ)dθ = 1.

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θˆ

(5)

Combining Equation (3) and (5), we have θˆ

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ˆ

θ

min {αD0,1 , L} f (θ)dθ +

ˆ

θ¯

θˆ

  b ˆ p θ− f (θ)dθ = 1. p

(6)

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If we treat all the holders of short-term debt and long-term debt as a group, then this equation can ˆ be regarded as the break-even condition of this group. The following lemma solves θ. Assumption 1. pθ¯ ≥ 2 + b. ¯ can be solved as θˆ = φ(min {αD0,1 , L}), where Lemma 1. Under Assumption 1, θˆ ∈ (θ, θ) φ(x) ≡

(x + pθ¯ + b) −

q   (x + pθ¯ + b)2 − 4p xθ + bθ¯ + (θ¯ − θ) 2p

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.

(7)

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Furthermore, θˆ decreases with min {αD0,1 , L} and decreases with p.

3.3

Banker’s optimal debt maturity

The banker’s payoff comes from two sources. The first is the bank’s profit conditional on successful rollover without moral hazard action. If θ ≥ θˆ at t = 1, then the bank can earn profit θ − αD1,2 (θ) − (1 − α)D0,2 . Since in this case D1,2 (θ) =

D0,1 , p

the profit can be written as θ − θˆ + pb . The second is

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the banker’s private benefit after asset liquidation. If θ < θˆ at t = 1, the bank needs to liquidate

ν proportion of the asset. According to the analyses in Section 3.2, the banker will take the moral hazard action and abscond private benefit (1 − ν)b in total. Summing up these two parts, we obtain the banker’s expected payoff:

θ

θˆ

ˆ

θ¯



b p θ − θˆ + p



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π(α) =

ˆ

(1 − ν) bf (θ)dθ +

θˆ

f (θ)dθ.

(8)

In order to see the trade-off of using short-term debt more clearly, we can rewrite (8) as θˆ

νbf (θ)dθ +

θ

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π(α) = b −

ˆ

ˆ

θˆ

θ¯



 ˆ p θ − θ f (θ)dθ.

(9)

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From this expression, we can observe that the proportion of short-term debt α affects the banker’s expected payoff from two channels. On the one hand, increasing short-term debt reduces ˆ the expected payoff of the banker if liquidation happens. If the bank fails in rollover (θ < θ),

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ν = min{ αDL0,1 } fraction of its asset will be liquidated, and the banker will take the moral hazard action to the rest of the asset and obtain private benefit (1 − ν)b. Apparently, more short-term

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debt leads to more liquidation (a higher ν), which reduces the banker’s private benefit. On the other hand, increasing short-term debt α can also lower θˆ (see Lemma 1) and raise

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the probability of successful rollover. To see the intuition, recall that θˆ =

αD0,1 p

+ (1 − α)D0,2 +

b p

depends on the total debt repayment. More short-term debts alleviate the moral hazard problem and raise the payoff of creditors especially when there is liquidation. Thus, creditors will require lower interest rates and reduce the cutoff of successful rollover. With benefits and costs of the short-term debt described above, there is a trade-off for the banker. And we derive the optimal debt maturity αo in the following proposition.

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Proposition 1. There exist L1 , L2 > 0 satisfying 0 < L1 < L2 < b, and (1) if L ≤ L1 , the banker will optimally choose αo = 0;

ˆ ≡ (2) if L1 ≤ L ≤ L2 , the banker will optimally αo = Λ(θˆ0 ) ∈ [0, L], where Λ(θ) and θˆ0 =

 2  θ¯ pb θ¯ +b−L 2b−L

. Besides,

∂αo ∂L

¯ ˆ θ+( ¯ θ−θ) ¯ pθˆ2 −(pθ+b) θ+b ˆ θ−θ

> 0;

(3) if L ≥ L2 , the banker will optimally choose αo ∈ [L, 1].

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Proposition 1 shows that when the asset liquidity L is high, the banker tends to use more short-term debts. This result is quite intuitive. As L increases, creditors can receive more upon liquidation, which decreases their required debt repayment and thus reduces rollover risk measured ˆ Besides, the banker has to liquidate a less fraction of its asset if θ < θ, ˆ which increases her by θ.

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private benefit. Both effects encourage the banker to use more short-term debts.

Endogenizing the Market Liquidity The Rational Expectation Equilibrium

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In this section, we solve the full model by endogenizing the market liquidity measure L. As we have demonstrated, the market liquidity L ≡ χ(n) at t = 1 is determined by how many assets are

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liquidated, and this is affected by the debt maturity choice of banks at t = 0, i.e., n = M Lα . If

more short-term debts are issued, then more assets will be liquidated and thus n tends to be large.

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However, as we can see in Proposition 1, the debt maturity choice of banks also depends on the expectation value of L. In this sense, we need to solve a fixed point represented by a Rational Expectations Equilibrium, which is characterized in Definition 1. And we solve this equilibrium in

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Proposition 2.

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Definition 1. A Rational Expectations Equilibrium consists of each bank’s debt maturity choice α∗ , the amount of liquidated assets n∗ and the market liquidity L∗ , satisfying: (1) Expecting the market liquidity L∗ at t = 1, each banker’s maturity choice α∗ at t = 0 maximizes its expected payoff; ∗

(2) The amount of liquidated assets is n∗ = M Lα∗ ; (3) The market liquidity is determined by L∗ = χ(n∗ ).

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Proposition 2. There exists a unique Rational Expectations Equilibrium (α∗ ,n∗ ,L∗ ). In this equilibrium, the bank’s debt maturity is an interior solution, i.e., α∗ ∈ (0, L∗ ) and L∗ ∈ (L1 , L2 ) ⊂ (0, b).

4.2

The effect of asset quality

In equilibrium, we can discuss how the asset quality of banks (p) affects banks’ debt maturity

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choices and the market liquidity. Intuitively, a decrease in asset quality raises the debt repayments of banks, which induces higher rollover risk and more liquidation. To alleviates this problem, the bank needs to increase the use of short-term debt to exploit more of its discipline effect. Assumption 2. M χ0 (M ) + χ (M ) > 0.

Proposition 3. Under Assumption 2, as banks’ asset quality p decreases, more short-term debts ∂α∗ ∂p

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will be issued and the asset market liquidity will be lower, i.e.

∗

< 0, ∂L > 0. ∂p

The result in Proposition 3 can be consistent with the pattern of financial market around the 2007-2009 financial crisis. Before the crisis started in 2007, the asset quality, especially that of the sub-prime mortgage decreased year by year (Demyanyk and Hemert, 2011; Mayer et al., 2009),

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which encouraged the financial institutions to rely more and more on short-term debt. And as the crisis began, more and more financial institutions have to liquidate their assets for refinancing,

Policy implications

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4.3

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which led to large fire sales and extremely low market liquidity (market freeze).

In this section, we discuss the effect of capital requirement within our framework. Under this

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requirement, the banks’ equity/asset ratio should be no smaller than a critical value. We denote it by k. Imposing this requirement in our model means that banks have to use k > 0 units of equity,

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and issues 1 − k units of debt to finance the asset’s cost. We show in Proposition 4 that the Capital Requirement is effective: It reduces banks’ use of short-term debt and enhances market liquidity. Assumption 3. k < min {1 − (pθ − b), 1 − χ(0)}. Proposition 4. Under Assumption 3, strengthening the capital requirement can reduce banks’ use ∗

(1−k)] of short-term debt and raise the market liquidity of the asset, ie., ∂[α ∂k < 0,

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∂L∗ ∂k

> 0.

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Proposition 4 demonstrates the desirability of Capital Requirement. If banks’ assets have lower asset quality, higher capital levels are required to stabilize the asset market by reducing asset liquidation and raising the asset’s market liquidity.

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Conclusion

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In this paper, we present a model to provide a novel mechanism leading to the asset market freeze: the moral hazard channel. In our model, we endogenize both the debt maturity choices of banks and the market liquidity of assets in a Rational Expectations Equilibrium, taking into account the discipline effect and induced asset liquidation from the short-term debt. We show that the discipline effect is more significant as the quality of the assets decreases, which generates a positive

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link between the asset quality and market liquidity through the endogenously determined debt maturities. Furthermore, we show that strengthening the Capital Requirement helps to raise the

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market liquidity of assets by making issuing short-term debts less favorable.

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Appendix A: Proofs of Lemmas and Propositions A1. Proof of Lemma 1. Rearranging Equation (6), φ(α) is the solution to the following equation ˆ ≡ pθˆ2 − (min {αD0,1 , L} + pθ¯ + b)θˆ + min {αD0,1 , L} θ + bθ¯ + (θ¯ − θ) = 0. Ω(θ)

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For the existence of a solution for any α ∈ [0, L], we need   H(α) ≡ (α + pθ¯ + b)2 − 4p αθ + bθ¯ + (θ¯ − θ) ≥ 0.

Since pθ¯ ≥ 2 + b > 2 > 2pθ, H 0 (α) = 2(α + pθ¯ + b − 2pθ) > 0 and thus to make above inequality

We can calculate that

∂H(0) ¯ ∂(pθ)

  H(0) = (pθ¯ + b)2 − 4p bθ¯ + (θ¯ − θ) ≥ 0.

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hold, we only need

= 2(pθ¯ − b − 2) > 0. Thus, H(0) ≥ 4(1 + b)2 − 4p(1 + b)2 + 4pθ > 0.

ˆ = 0. We pick the smaller one θˆ = θˆ1 , Hence, there are two solutions θˆ1 and θˆ2 (θˆ1 < θˆ2 ) to Ω(θ) which is expressed in Equation (7). Thus,

∂Ω | ˆ θˆ1 ∂ θˆ θ=

< 0. Besides, we have θ <

¯ α+pθ+b 2p

¯ Ω(θ) = < θ,

∂Ω ∂min{αD0,1 ,L}

= θ − θˆ < 0, and we have

∂ θˆ1 ∂min{αD0,1 ,L}

< 0,

∂Ω ∂p

= θˆ2 − θˆθ¯ < 0, and

∂ θˆ1 ∂p

< 0.

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Thus,

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¯ = (1 − min{αD0,1 , L})(θ¯ − θ) > 0. As a result, θ < θˆ1 (α) < θ. ¯ (1 + b − pθ)(θ¯ − θ) > 0, and Ω(θ)

A2. Proof of Proposition 1. If α ≥ L, then αD0,1 ≥ L, and apparently ν = 1. Besides, from

Equation (7), we can see that θˆ is not related to α since min{αD0,1 , L} = L in this case. Hence,

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the expected profit is a constant when α ≥ L and we only need to discuss the case of α ≤ L. If αD0,1 ≤ L, then ν ≤ 1, and from Equation (3) we can obtain that D0,1 = 1. As a result,

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αD0,1 ≤ L is equivalent to α ≤ L and ν = Lα . Further, since min{αD0,1 , L} = α, we know θˆ = φ(α)

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ˆ from Equation (6). Since φ0 < 0 from Lemma 1, we can also write α as a decreasing function of θ: ˆ2 ¯ ˆ ¯ ¯ ˆ ≡ Λ(θ) ˆ ≡ pθ − (pθ + b)θ + bθ + (θ − θ) , α = φ−1 (θ) θˆ − θ

and Λ0 < 0. Replace α in (8) by above expression, and we can easily calculate 1 π= ¯ θ−θ



  b p(θ¯2 − θˆ2 ) ˆ ¯ ˆ ¯ ¯ b(θ − θ) + pθ − (pθ + b)θ + bθ + (θ − θ) 1 − + L 2 

ˆ2

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!

− 1.

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Thus, we only need to choose θˆ ∈ [φ(L), φ(0)] to maximize the profit. Write the profit function as ˆ and π(θ),

ˆ =p π 0 (θ) b2 pθ¯



b2 pθ¯

 + b − L − (2b − L) θˆ ˆ ∝ θ¯ (L4 − L) − (L5 − L) θ, L(θ¯ − θ)

+ b, L5 ≡ 2b, and L4 < L5 .

(1) If L ≥ L5 , define θˆ0 ≡ ¯

θ¯



b2 pθ¯

 +b−L

2b − L

,

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where L4 ≡

θ¯

ˆ < 0 for θˆ < θˆ0 . Since θˆ0 = θ¯ + b(θ− p ) > θ, ¯ π 0 (θ) ˆ < 0 for any θˆ < θ. ¯ Thus, in this case, the and π 0 (θ) L−2b b

banker will optimally choose θˆo = φ(L) and αo = L.

ˆ < 0 and the banker will optimally choose θˆo = φ(L) (2) If L4 ≤ L ≤ L5 , then apparently π 0 (θ)

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and αo = L.

¯ π 0 (θ) ˆ > 0 for θˆ < θˆ0 and π 0 (θ) ˆ < 0 for θˆ > θˆ0 . Define If L ≤ L4 , then θˆ0 ≤ θ,

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ˆ ≡b L = Γ(θ)

θ¯ + pb − 2θˆ . θ¯ − θˆ

ˆ and L4 = Γ(0). Note that for L ≥ 0, we need θˆ ≤ Then θˆ0 is the solution to L = Γ(θ)

b ¯ +θ p

2

¯ < θ.

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Besides, it is easy to see that Γ0 < 0 and Γ00 < 0. Thus, L4 = Γ(0) ≥ Γ(θ) ≡ L3 . Note that

L3 < b < pθ < 1.

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ˆ < 0 for θˆ ≥ θ. Thus, in this case, the banker will (3) If L3 ≤ L ≤ L4 , then θˆ0 ≤ θ. Hence, π 0 (θ)

optimally choose θˆo = φ(L) and αo = L.

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If L ≤ L3 , then θˆ0 ≥ θ. If the banker can choose θˆ = θˆ0 , then θˆ0 is optimal. However, the banker

may not be able choose it. Actually, the available set of θˆ is [φ(L), φ(0)] ⊂ [θ,

b ¯ +θ p

2

]. Hence, we need

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to compare this range with θˆ0 .

ˆ ≥ 0 for θˆ ∈ [φ(L), φ(0)]. In this (4) If L ≤ L1 ≡ Γ(φ(0)) < L3 , then θˆ0 ≥ φ(0) and thus π 0 (θ)

case, the banker will optimally choose θˆo = φ(0) and αo = 0.

If L1 ≤ L ≤ L3 , then θˆ0 ≤ φ(0) and we only need to compare θˆ0 and φ(L). From the proof of

ˆ = 0. Then Ω(θ) ˆ and Ω0 (θ) ˆ < 0 for θˆ < φ(α). Lemma 1, θˆ = φ(α) is the smaller solution to Ω(θ) ˆ = Since Λ(θ)

ˆ Ω(θ) ˆ θ−θ

ˆ Thus, α < Λ(θ) ˆ and Λ0 (θ) ˆ < 0 for + α, φ(α) is the smaller solution to α = Λ(θ). 11

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ˆ = θˆ ∈ [0, φ(α)). Besides, Λ00 (θ)

2(1+b−pθ) 3 ˆ (θ−θ)

> 0.

It can be calculated that Λ(θ) → +∞ > Γ(θ) and Λ(φ(0)) = 0 < Γ(φ(0)). Since Λ(θ) is convexly

decreasing and Γ(θ) is concavely decreasing in θˆ for θˆ ∈ (θ, φ(0)), there exists a unique solution

ˆ = Γ(θ) ˆ and Λ(θ) ˆ > Γ(θ) ˆ for θ ∈ (θ, θˆ3 ), Λ(θ) ˆ < Γ(θ) ˆ for θ ∈ (θˆ3 , φ(0)). Define θˆ3 ∈ (θ, φ(0)) to Λ(θ)

L2 ≡ Λ(θˆ3 ) = Γ(θˆ3 ), and then L2 ∈ (L1 , L3 ).

(5) If L1 ≤ L ≤ L2 , then Λ(φ(L)) = L ≤ L2 = Λ(θˆ3 ) ⇒ φ(L) ≥ θˆ3 and Γ(θˆ0 ) = L ≤ L2 =

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ˆ and Γ(θ) ˆ are both decreasing and Λ(θ) ˆ < Γ(θ) ˆ for θ ∈ (θˆ3 , φ(0)), we Γ(θˆ3 ) ⇒ θˆ0 ≥ θˆ3 . Since Λ(θ)

have φ(L) ≤ θˆ0 . In this case, the banker will optimally choose θˆo = θˆ0 and αo is determined by

θˆo = φ(αo ). Further, we can easily see

αo ∂L

> 0 from

∂ θˆ0 ∂L

< 0 and θˆ0 (α) < 0.

(6) If L2 ≤ L ≤ L3 , then Λ(φ(L)) = L ≥ L2 = Λ(θˆ3 ) ⇒ φ(L) ≤ θˆ3 and Γ(θˆ0 ) = L ≥ L2 =

ˆ and Γ(θ) ˆ are both decreasing and Λ(θ) ˆ > Γ(θ) ˆ for θ ∈ (θ, θˆ3 ), we Γ(θˆ3 ) ⇒ θˆ0 ≤ θˆ3 . Since Λ(θ)

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ˆ ≤ 0 for θˆ ∈ [φ(L), φ(0)]. In this case, the banker will optimally choose have φ(L) ≥ θˆ0 . Thus π 0 (θ) θˆo = φ(L) and αo = L.

A2. Proof of Proposition 2. (1) If L ≤ L1 , each bank chooses αo = 0 and thus n = 0 according to Proposition 1. The resulting liquidity should be χ(0) = Lmax > L1 . Thus, equilibrium can not

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exist in this case.

(2) If L ≥ L2 , each bank chooses αo ∈ [L, 1] and thus n = M according to Proposition 1. The

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resulting liquidity should be χ(0) = Lmin < L1 . Thus, equilibrium can not exist in this case. (3) If L1 ≤ L ≤ L2 , then each bank chooses αo ∈ [0, L]. If there exists an equilibrium in this

case, then χ(M αL ) = L. Note that αo = Λ(θˆ0 (L)) is an increasing function of the liquidity L

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o

according to Proposition 1, and αo = 0 when L = L1 , αo = 1 when L = L2 . As a result, when

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L = L1 , χ(0) = Lmax > L1 and when L = L2 , χ(M ) = Lmin < L2 . Thus, there must exist an L∗ ˆ

satisfying χ(M Λ(θ0L(L)) ) = L. ˆ

ˆ

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θ0 (L)) ˆ ≡ To prove the uniqueness of the equilibrium, let Ξ(L) ≡ M Λ(θ0L(L)) = M Λ( . Define Φ(θ) Γ(θˆ (L))

¯ ˆ θ+ ¯ θ−θ ¯ θ+b 1 pθˆ2 −(pθ+b) ¯ b −2θˆ b θ+

and thus

p

0

Ξ (L) =

ˆ Λ(θ) ˆ Γ(θ)

(

0

=

¯ θˆ θ− ˆ ˆ Φ(θ) θ−θ

, and

−(θ¯ − θ) Φ(θˆ0 (L)) + [θˆ0 (L) − θ]2

12

) θ¯ − θˆ0 (L) 0 ˆ Φ (θ0 (L)) θˆ00 (L). ˆ θ0 (L) − θ

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ˆ < 0, then Ξ0 (L) > 0. Further, we can calculate that Since θˆ00 (L) < 0, we can see that if Φ0 (θ) ˆ =1 Φ (θ) b 0

−2pθˆ2 + 2p(θ¯ + pb )θˆ − p(θ¯ + pb )2 + 2bθ¯ + 2(θ¯ − θ) ˆ2 (θ¯ + b − 2θ) p

!

,

and

  2  b b ˆ ∝ Θ(θ) ˆ ≡ −2pθ + 2p θ¯ + θˆ − p θ¯ + + 2bθ¯ + 2(θ¯ − θ). Φ (θ) p p h  i b ˆ = −4p θˆ − 1 θ¯ + b , and thus Θ0 (θ) > 0 for θˆ < φ(0). Besides, Apparently, φ(0) ≤ 21 θ¯ + 2p 2 2p

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ˆ2

0

ˆ is decreasing over [θˆ3 , φ(0)]. Θ(φ(0)) = 4bθ¯ + 4(θ¯ − θ) − p(θ¯ + pb )2 ≤ 0 by Assumption 1. Thus, Φ(θ) As a result, Ξ(L) is increasing over [L1 = Γ(φ(0)), L2 = Γ(θˆ3 )]. Finally χ(Ξ(L)) is decreasing over ˆ

[L1 , L2 ], leading to a unique solution to χ(M Λ(θ0L(L)) ) = L.

∂J(L,p) ∂L

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o
A3. Proof of Proposition 3. The equilibrium liquidity L∗ is determined by the functionχ M αL =  2      θ¯ pb θ¯ +b−L Λ(θˆ0 (L,p),p) Λ(θˆ0 (L,p),p) ˆ χ M = L, where θ (L, p) = − L. Then . Define J(L, p) = χ M 0 L 2b−L L

< 0 according to Appendix A2. Now we only need to discuss the sign of !

×

M

M 0 Λ(θˆ0 (L, p), p) ∂J(L, p) = χ M ∂p L L ∂J(L,p) ∂p

>0⇔Υ ≡

∂Λ(θˆ0 (L,p),p) p

+

∂Λ(θˆ0 (L,p),p) ∂ θˆ0 p ∂ θˆ0

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If

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θˆ0 (θ¯ − θˆ0 ) + Υ (θˆ0 ) = − θˆ0 − θ

∂Λ(θˆ0 (L, p), p) ∂Λ(θˆ0 (L, p), p) ∂ θˆ0 + p p ∂ θˆ0

< 0, then

∂L∗ ∂p

!

.

> 0. We can calculate that

(1 + b − pθ)(θ¯ − θ) −p (θˆ0 − θ)2

!

1 b(θ¯ − θˆ0 ) . p2 θ¯ − pb

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Rearrange the terms:

Υ (θˆ0 ) ∝ Ψ (θˆ0 ) ≡ −pθˆ02 pθ¯ + pθˆ0 θ(pθ¯ + b) + (1 + b − pθ)(θ¯ − θ)b − pbθ2 .

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 ˆ = −2pθp ˆ θ¯ + pθ(pθ¯ + b) = −2p2 θ¯ θˆ − Apparently, Ψ 0 (θ)

¯ θ(pθ+b) 2pθ¯



< 0 and

¯ pθ(pθ+b) 2p2 θ¯

<

θ 2

< θ. Thus,

ˆ < 0 for θˆ0 ∈ (θˆ3 , φ(0)). As a result, Ψ (θˆ0 ) < 0 holds if Ψ (θˆ3 ) < 0. Define θˆ4 as the larger Ψ 0 (θ) ˆ = 0, then Ψ (θˆ3 ) < 0 is equivalent to θˆ4 < θˆ3 . We next prove that θˆ4 < solution to Ψ (θ) Firstly, in order to prove θˆ4 < 1+b p

<

¯ pθ+b 2

1+b , p

1+b p

< θˆ3 .

¯ + b − pθ) < 0 since we only need to check Ψ ( 1+b ) = −θ(1 p

by Assumption 1. Secondly, in order to prove

13

1+b p

< θˆ3 , we only have to show that

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Λ( 1+b ) > Γ( 1+b ). It is easy to calculate that Λ( 1+b ) = 1 and Γ( 1+b )=b p p p p
∗ ∗ Finally, we prove that ∂α < 0. Note that χ M Lα∗ = L∗ and ∂p

¯ 1 − 1+b θ− p p ¯ 1+b θ−

< b < 1.

p

      α∗ M ∂α∗ α∗ α∗ ∂L∗ 0 M χ M ∗ − χ M + 1 = 0. L L∗ ∂p L∗ ∂p (L∗ )2 0

∗
∗ ∗ Y + 1 and we only need Define Y ≡ M Lα∗ ∈ [0, M ]. Then χ0 M Lα∗ M (Lα∗ )2 + 1 = Ψ(Y ) ≡ χ0 (Y ) χ(Y )

(χ00 (Y )Y +χ0 (Y ))χ(Y )−(χ0 (Y ))2 Y χ2 (Y )

< 0. As a result,

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Ψ(Y ) to be positive. We can calculate that Ψ0 (Y ) =

M + 1 > 0, which is confirmed by Assumption 2. we only need Ψ(M ) = χ0 (M ) χ(M )

A4. Proof of Proposition 4. In this proof, there are two steps. The first step is to derive the optimal debt maturity structure given the liquidity L, similar to Proposition 1. The second step is

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to show the existence and uniqueness of the Rational Expectations Equilibrium and the relationship between Capital Requirement and liquidity in equilibrium. Step 1. Optimal debt maturity structure.

With equity k, the rollover condition at t = 1 is

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      αD0,1 αD0,1 b p θ− + (1 − α)D0,2 (1 − k) ≥ b ⇔ θ ≥ + (1 − α)D0,2 (1 − k) + ≡ θˆk . p p p If θˆk ≤ θ, the bank can successfully roll over the short-term debts for any realization θ, and thus D0,1 = 1, D0,2 = p1 , θˆk =

b+1−k . p

As a result, θˆk ≤ θ is equivalent to k ≥ 1 + b − pθ. Assumption 3

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prevents this case from taking place and liquidation is always possible.

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The break-even condition for short-term debt holders is ˆ

 min D0,1 ,

L α(1 − k)



f (θ)dθ +

ˆ

θ¯

θˆk

D0,1 f (θ)dθ = 1.

AC

θ

θˆk

If L ≤ α(1 − k), then there is nothing left for long-term holders. If L > α(1 − k), the banker

will not take the moral hazard action after asset liquidation if and only if:

     αD0,1 (1 − k) αD0,1 (1 − k) (1 − α)D0,2 (1 − k) b p 1− θ − (1 − α)D0,2 (1 − k) ≥ b 1 − ⇔θ≥ + ≡ θˇk , αD0,1 (1−k) L L p 1− L

and θˆk ≤ θˇk is equivalent to

L 1−k

≤ αD0,1 + p(1 − α)D0,2 , which is ensured by Assumption 3. Thus, 14

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the break-even condition for long-term debt holders is ˆ

θ¯

θˆk

pD0,2 f (θ)dθ = 1.

Similar to Appendix A1, we can calculate that θˆk is the smaller solution to the following equation

and θˆk = φk (xk ), where xk = min{αD0,1 (1 − k), L} and φ (x) =

¯ − [x + b + pθ]

p ¯ 2 − 4p[(1 − k)(θ¯ − θ) + bθ¯ + (1 − k)xθ] [x + b + pθ] . 2p

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k

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pθˆ2 − ((1 − k)xk + pθ¯ + b)θˆ + bθ¯ + (1 − k)xk θ + (1 − k)(θ¯ − θ) = 0,

We can write the expected profit of the banker as k

π =

ˆ

θ

θˆk



1 − min



  ˆ θ¯  αD0,1 b k ˆ ,1 bf (θ)dθ + p θ−θ + f (θ)dθ. L p θˆk

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Apparently, the payoff is irrelevant from α if L ≤ αD0,1 (1 − k). And for L > αD0,1 (1 − k), we write

ED

ˆ2 ¯ ˆ ¯ ¯ ˆ ≡ pθ − (pθ + b)θ + bθ + (1 − k)(θ − θ) . α = Λk (θ) (1 − k)(θˆ − θ)

0

ˆ ∝ θ(L ¯ 4 − L) − (L5 − L)θ. ˆ π (θ)

CE

PT

Inserting this expression into the banker’s payoff and take first derivative. We can have

This expression is exactly the same with that in Appendix A2. Thus, by similar process, we can obtain the optimal debt maturity structure for the banker, given the liquidity L fixed. Let

AC

ˆ = Γ(θ). ˆ Lk1 = Γ(φk (0)) and Lk2 = Λk (θˆ3k ) = Γ(θˆ3k ) > Lk1 , in which θˆ3k is the unique solution to Λk (θ) Then (1) If L ≤ Lk1 , the optimal choice is α∗ = 0; (2) If Lk1 ≤ L ≤ Lk2 , the optimal choice is given

L , 1]. by φk (θˆ0 ) = α∗ ; (3) If L ≥ Lk2 , the optimal choice is α∗ ∈ [ 1−k

Step 2. The Rational Expectations Equilibrium and the effect of Capital Requirement k.

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  ∗ = L∗ and By similar process with Appendix A3, we can obtain that χ M α (1−k) L∗ χ

0



α∗ (1 − k) M L∗

Define Y k ≡ M α

∗ (1−k)

L∗



    α∗ (1 − k) M ∂ [α∗ (1 − k)] α∗ (1 − k) ∂L∗ 0 M − χ M + 1 = 0. L∗ ∂k L∗ ∂k (L∗ )2

  ∗ ∗ M α(L(1−k) ∈ [0, M ], then χ0 M α (1−k) + 1 = Ψ(Y k ). We know Ψ0 < 0 and ∗ )2 L∗

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Ψ(M ) > 0. Hence Ψ(Y k ) > 0, and α∗ (1 − k) decreases with k.

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[9] Demyanyk, Yuliya and Otto Van Hemert (2011) “Understanding the subprime mortgage crisis,” Review of Financial Studies, Vol. 24, No. 6, pp. 1848–1880. [10] Eisenbach, Thomas M (2017) “Rollover risk as market Ddiscipline: A two-sided inefficiency,” Journal of Financial Economics, Vol. 126, No. 2, pp. 252–269. [11] Huberman, Gur and Rafael Repullo (2014) “Moral hazard and debt maturity,” Working

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