## 1. The Timeline

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---

### Introduction and purpose
- Banks use short-term debt to invest in long-term assets, creating liquidity risk when maturing debt cannot be rolled over.
- Recent bank liquidity crises in developed economies were often driven by increased uncertainty over a bank’s solvency and played out in wholesale funding markets.
- Basel III addresses liquidity risk via:
  - the Liquidity Coverage Ratio, and
  - the Net Stable Funding Ratio.
- Purpose of the paper:
  - Offer a model of bank liquidity risk driven by solvency concerns.
  - Study regulatory implications, especially the interaction between liquidity requirements, access to refinancing (linked to bank transparency), and liquidity risk.
- Core setup highlights:
  - A bank has a long-term project that with probability σ yields zero value.
  - At date 1 the bank faces an exogenous random withdrawal; refinancing can be compromised by informational frictions.
  - Most states: bank is solvent and refinancing is available; with some probability a negative signal raises the posterior insolvency probability to σ/(σ + θ) and may cause investors to refuse to lend, creating liquidity risk for potentially solvent banks.

### Key mechanisms and comparative statics
- Two hedges for liquidity risk:
  - Liquidity buffer: accumulate easily tradeable assets to cover refinancing internally.
  - Transparency: enhance ability to communicate solvency to attract external refinancing.
- Interaction:
  - Complements: both hedge the same underlying risk via different imperfections; liquidity buffers fully cover only small refinancing needs; transparency can improve access for needs of any size but only probabilistically (with probability τ).
  - Strategic substitutes: adoption of one instrument reduces the incremental value of the other.
- Costs and incentive problems:
  - Holding liquidity buffers is costly (administrative cost δ per unit of balance sheet).
  - Transparency is costly because the owner-manager sacrifices private benefits of control B.
  - Shareholders do not internalize all benefits to creditors → leveraged banks may under-invest in liquidity buffers and transparency.

### Information, withdrawals, and conditions generating liquidity risk
- Timeline and information:
  - Dates 0, 1, 2. Project requires investment 1 at date 0 and returns X > 0 at date 2 with probability 1 − σ, or 0 with probability σ.
  - At date 1 two independent events: random withdrawal and a noisy signal on solvency.
- Refinancing need:
  - With probability λ the need is low: share ω_L < 1 of initial funds.
  - With probability 1 − λ the need is high: ω_H = 1 (must refinance all initial funding).
  - If the bank cannot refinance, it fails and the long-term project is liquidated at zero.
- Signal structure at date 1:
  - Positive signal with probability 1 − (σ + θ): bank is always solvent and can refinance at risk-free rate.
  - Negative signal with probability σ + θ: received by all insolvent banks and by some solvent banks; posterior insolvency probability equals σ/(σ + θ).
- Two sufficient conditions imposed to generate liquidity risk:
  - Existence of initial funding at date 0 and positive owner-manager return:
    - X > 1/(1 − (σ + θ)) + δ/(1 − (σ + θ)).  (Equation (1))
  - Under a negative signal at date 1 a bank cannot obtain refinancing even for a low refinancing need:
    - X < (1 − ω_L) + ω_L · (σ + θ)/θ.  (Equation (2))
  - These can hold simultaneously for parameter values such as σ + θ << 1, θ << σ, and ω_L not too close to zero.

### Lemma on existence of liquidity risk
- Lemma 1:
  - There exist parameter values such that a bank can attract initial funding at date 0, but cannot refinance in case of solvency concerns (a negative signal) at date 1.
  - Result: some solvent banks are liquidated under these parameters.

---

### Appendix A — Risk management, social first best, private choices, and regulation

### Risk management instruments (model specifics)
- Liquidity buffer:
  - At date 0 the bank can attract additional funds and invest them in short-term assets that return 0.
  - Per-unit cost of effort: .
  - Size α required to cover refinancing need ωΛ is given by:
    - (1 + α)·ωΛ = α.  (Equation (3))
    - Implies α = ωΛ / (1 − ωΛ).  (Equation (4))
  - Cost to maintain buffer: α.
  - If ωH = 1, no liquidity buffer can cover a large refinancing need.
  - If liquidated at date 1, the liquidity buffer is returned to creditors in full.
- Transparency:
  - Ex-ante (date 0) decision enabling communication of asset values.
  - In case of a negative signal at date 1, a transparent solvent bank obtains funding with probability τ; with probability 1 − τ it cannot.
  - Cost of transparency: owner-manager’s loss of private benefits of control B.
- Ex-ante preference ordering for exposition: hedging preferred to no hedging, and liquidity buffers preferred to transparency on the margin.

### First best (social welfare) — payoffs and optimality conditions
- Social welfare Ω is joint surplus (bank NPV). Four strategies: N (none), L (liquidity only), T (transparency only), LT (both).
- Payoffs:
  - ΩN = (1 − (σ + θ))X −  + B.  (Equation (5))
  - ΩL = (1 − (σ + θ) + θλ)X −  − α + B.  (Equation (6))
  - ΩT = (1 − (σ + θ) + θτ)X − .  (Equation (7))
  - ΩLT = (1 − (σ + θ) + θ(λ + τ − λτ))X −  − α.  (Equation (8))
- First-best hedging inequalities:
  - Be liquid (from no hedge): α ≤ θλX.  (Equation (9))
  - Be transparent (from no hedge): B ≤ θτX.  (Equation (10))
  - Prefer liquidity to transparency: α − B ≤ θ(λ − τ)X.  (Equation (11))
  - Both hedges optimal:
    - α ≤ θλ(1 − τ)X
    - B ≤ θ(1 − λ)τX.  (Equation (12))
- Insight: marginal benefit of the second hedge is lower than the first; liquidity and transparency are strategic substitutes.

### Private owner-manager choices — payoffs and distortions
- Owner-manager payoffs Π (limited liability: creditors’ benefits not fully internalized):
  - ΠN = (1 − (σ + θ))(X − 1) −  + B.  (Equation (13))
  - ΠL = (1 − (σ + θ) + θλ)(X − 1) −  − α + B.  (Equation (14))
  - ΠT = (1 − (σ + θ) + θτ)(X − 1) − .  (Equation (15))
  - ΠLT = (1 − (σ + θ) + θ(λ + τ − λτ))(X − 1) −  − α.  (Equation (16))
- Private hedging conditions (replace X by X − 1 on right-hand sides):
  - Be liquid: α ≤ θλ(X − 1).  (Equation (17))
  - Be transparent: B ≤ θτ(X − 1).  (Equation (18))
  - Prefer liquidity to transparency: α − B ≤ θ(λ − τ)(X − 1).  (Equation (19))
  - Both hedges privately preferred:
    - α ≤ θλ(1 − τ)(X − 1)
    - B ≤ θ(1 − λ)τ(X − 1).  (Equation (20))
- Comparative outcome: private threshold lines lie below/left of first-best → potential under-insurance.
- Proposition 2:
  - Owner-manager may under-hedge; there exist parameter values where first-best is both hedges but privately only one is chosen.
- Specific parameter regions where private choice diverges from first-best:
  - Bank only liquid while socially optimal to be both:
    - θ(1 − λ)τ(X − 1) < B ≤ θ(1 − λ)τX
    - α ≤ θλ(1 − τ)X
    - α − B ≤ θ(λ − τ)(X − 1).  (Condition (21))
  - Bank only transparent while socially optimal to be both:
    - θλ(1 − τ)(X − 1) < α ≤ θλ(1 − τ)X
    - B ≤ θ(1 − λ)τX
    - α − B > θ(λ − τ)(X − 1).  (Condition (22))

### Regulation and unintended effects of liquidity requirements
- Transparency is not easily verifiable and cannot be directly regulated; liquidity buffers can be imposed via liquidity requirements.
- Because liquidity buffers and transparency are strategic substitutes, liquidity requirements may affect transparency choices.
- Consider parameters satisfying (22) where the bank privately chooses transparency only while first-best is both hedges.
- Two outcomes after imposing liquidity requirements:
  - Low cost of transparency (point A0): B ≤ θτ(1 − λ)(X − 1).  (Equation (23))
    - Bank maintains transparency; liquidity requirements induce socially optimal both-hedge outcome.
  - Higher cost of transparency (point A): B > θτ(1 − λ)(X − 1).  (Equation (24))
    - Bank abandons transparency in response to liquidity requirements; liquidity buffers substitute for transparency.
    - This shift can reduce social welfare; for τ > λ the shift increases refinancing risk.
- Proposition 3:
  - There exist parameter values where liquidity requirements cause a transparent bank to abandon transparency, reducing social welfare — an unintended effect.

### Policy responses to address transparency shortfalls
- Because transparency is endogenous and hard to verify, policy can:
  - Reduce owner-manager private benefits of control B (e.g., stronger corporate governance, less concentrated ownership).
    - Reducing B can restore condition (23) and prevent liquidity requirements from crowding out transparency.
  - Influence funding structure to make large refinancing needs less likely (increase λ).
    - Higher λ reduces the social value of transparency when liquidity buffers exist.
    - Practical instrument: maturity mismatch limits (e.g., Net Stable Funding Ratio) that promote long-term/stable funding and raise λ.
    - Caveat: longer-term funding may increase cost of funds (add ρ per-unit to cost of liquidity buffer), possibly shifting choices and making welfare effects ambiguous.
- Policy implication:
  - Liquidity requirements should be complemented by measures that improve bank transparency (corporate governance improvements) or measures that reduce likelihood of large refinancing needs (maturity mismatch limits).

### Conclusion (key takeaways)
- Liquidity buffers and transparency are both important:
  - Liquidity buffers insure small withdrawals with certainty.
  - Transparency can enable refinancing of large withdrawals probabilistically (with probability τ).
- Banks may under-insure because owner-managers do not internalize creditors’ benefits under limited liability.
- Liquidity requirements alone can unintentionally crowd out transparency and increase refinancing risk for some parameter values.
- Policy should combine liquidity requirements with measures to improve transparency and funding stability to avoid welfare-reducing outcomes.

---

### Appendix B — Quantitative example and robustness (highlights)

### Parameterization (10-year interpretation)
- X = 1:35 (corresponding to a 3:5% per annum return on bank assets)
- σ = 0:08 and θ = 0:02 (corresponding to a 1% a year risk of bank failure)
- ωL = 0:09, giving α = 0:1 (liquidity buffer of 10% of bank assets)
- λ = τ = 0:9 (most shocks are small and transparency is often effective)
- These parameter values satisfy condition (1) for  < 0:215, and satisfy condition (2).
- Substituting into (22) and applying additional restriction on  yields that the bank is only transparent while it is optimal to be both liquid and transparent for:
  - 0:063 <  < 0:215
  - B <  / 10
- Using (20), the bank will abandon transparency in response to liquidity requirements for:
  - B > 0:0063
- Example closure with  = 0:15 (administrative costs of 1:5% per year), making B < 0:015:
  - For 0:0063 < B < 0:015 the bank will abandon transparency in response to liquidity requirements.
  - For B ≤ 0:0063 the bank will maintain transparency even under liquidity requirements.
- Qualitative implication: the range of B for which liquidity requirements compromise transparency choices is wider than the range for which they do not, suggesting the model’s effects are plausible.

*Source: IMF working paper content unit "1. The Timeline" and Appendix A & B (chapter/section of provided PDF).*

### 1. The Timeline ........................................................................................................

### 1. The Timeline

### Introduction
- Banks use short-term debt to invest in long-term assets (Diamond and Dybvig, 1983), creating liquidity risk when a bank cannot roll over maturing debt and can fail despite being solvent.
- A majority of recent bank liquidity crises in developed economies were caused by increased uncertainty over a bank’s solvency and played out primarily in wholesale funding markets (Gatev and Strahan, 2006, Shin, 2008, Yorulmazer, 2008, Huang and Ratnovski, 2011).
- Basel III addresses liquidity risk through:
  - the Liquidity Coverage Ratio (a liquidity requirement), and
  - the Net Stable Funding Ratio (a restriction on maturity mismatch that limits the volume of refinancing coming due each period; see Basel Committee, 2010).
- Purpose of the paper:
  - Offer a model of bank liquidity risk driven by solvency concerns.
  - Study regulatory implications, in particular the interaction between liquidity requirements, access to refinancing (linked to bank transparency), and liquidity risk.
- Core setup:
  - A bank has a valuable long-term project that with a small probability can turn out to be of zero value.
  - At an intermediate date, the bank needs to refinance an exogenous random withdrawal; ability to refinance can be compromised by informational frictions.
  - In most states the bank is solvent and refinancing is available; with some probability the posterior probability of insolvency is high (but less than one), investors may refuse to lend, creating liquidity risk and possible failure of a potentially solvent institution.

### Key mechanisms and findings
- Two hedges for liquidity risk:
  - Liquidity buffer: accumulate easily tradeable assets to cover refinancing internally in a crisis.
  - Transparency: enhance ability to communicate solvency information to outsiders so the bank can attract external refinancing.
- Interaction between liquidity buffers and transparency:
  - Complements: they hedge the same risk through different imperfections. Liquidity buffers can cover only small refinancing needs; transparency can improve access to external refinancing for needs of any size but only probabilistically (ex-post communication can fail).
  - Strategic substitutes: adoption of one instrument reduces the incremental value of the other.
- Costs and incentive problems:
  - Holding liquidity buffers is costly due to maintenance effort (administrative cost) borne by shareholders.
  - Transparency is costly because the owner-manager sacrifices private benefits of control.
  - Some benefits of hedging accrue to creditors and are not internalized by shareholders (Jensen and Meckling, 1976), so leveraged banks may under-invest in liquidity buffers and transparency.
- Regulatory implications:
  - Liquidity buffers can be imposed via regulation; transparency is not easily verifiable and is harder to regulate.
  - Liquidity requirements may unintentionally compromise endogenous transparency choices; for some parameter values deterioration of transparency may more than offset the positive effect of larger liquidity buffers, unintentionally increasing overall refinancing risk.
  - Indirect policy mechanisms identified to address transparency:
    - Reduce the alternative cost of transparency (the owner-manager’s private benefits of control), e.g., stronger corporate governance.
    - Reduce the risk of large refinancing needs that exceed the liquidity buffer via maturity mismatch limits (e.g., Net Stable Funding Ratio of Basel III).
  - These solutions may be essential complements to liquidity requirements.

### Related literature (high level)
- Bank liquidity risk:
  - Early models (Diamond and Dybvig, 1983; Chari and Jagannathan, 1988) assumed absence of informed refinancing even for banks with valuable assets.
  - Goodfriend and King (1988): solvent banks should be able to refinance in well-functioning interbank markets; models need to show how market failures restrict refinancing of solvent institutions.
  - Informational frictions literature: Flannery (1996), Rochet and Vives (2004), Freixas et al. (2004), Huang and Ratnovski (2011).
  - Distinction from moral hazard / net liquidity need frameworks (Holmstrom and Tirole, 1998, 2011): this paper studies a gross liquidity need (substitute outflow), not additional funds for project continuation.
  - Empirical support: both stock liquidity and access to external refinancing matter for bank financial pressures (Paravisini, 2007; Kashyap and Stein, 1990; Holod and Peek, 2004). Evidence banks may be insufficiently liquid (Gatev et al., 2004; Gonzalez-Eiras, 2003) or insufficiently transparent (Morgan, 2002).
- Bank transparency:
  - Two formalizations: credible communication channels (this paper’s focus) or asset choice.
  - Transparency costs: reduces insiders’ private benefits of control (Doidge, 2003; Leuz et al., 2003).
  - Banks can conceal information via organizational complexity (Berger et al., 2000) or obfuscation (Carlin, 2009; Carlin and Manso, 2011).
  - Distinction between transparency (strategic ex-ante decision) and disclosure (ex-post action); ad hoc disclosure may not be credible in crisis (Perotti and Von Thadden, 2003; Boot and Thakor, 2001).
  - The paper abstracts from scenarios where transparency has negative consequences (e.g., revealing negative but incorrect news).

### The model — setup and parameters
- Dates: three dates (0, 1, 2) with no discounting.
- Agents: multiple competitive investors and a single owner-managed bank.
- Project:
  - Requires investment of 1 at date 0.
  - Returns at date 2 a high X > 0 with probability 1 − σ or 0 with probability σ (σ stands for solvency risk).
- Owner-manager payoff components:
  - Cost of effort δ per unit of the bank’s balance sheet at date 0.
  - Non-verifiable private benefits of control B from running the bank.
  - Owner-manager maximizes profits − costs + private benefits.
- Funding:
  - No initial capital; financed with debt.
  - Some debt is short-term and must be refinanced at date 1.
- Role of δ:
  - Interpreted as the administrative cost of running the bank and a cost on large balance sheets and holding idle assets — liquidity buffers.
  - Alternative micro-foundations (e.g., pledgeable income) yield similar implications.

### Information, withdrawals, and refinancing risk
- Two events at date 1 (independent):
  - Random withdrawal (refinancing need).
  - Noisy signal on bank solvency.
- Refinancing need:
  - With probability λ, refinancing need is low: share ω_L < 1 of initial funds.
  - With probability 1 − λ, refinancing need is high: ω_H = 1 (must refinance all initial funding).
  - If a bank cannot refinance, it fails and the long-term project is liquidated at zero residual value.
- Signal structure at date 1:
  - Bank solvency prior: solvent with probability 1 − σ, insolvent with probability σ.
  - With probability 1 − (σ + θ) there is a “positive” signal, conditional on which bank is always solvent and yields X with certainty; bank can refinance at a risk-free rate.
  - With probability σ + θ there is a “negative” signal, received by all insolvent banks and by some solvent banks, producing posterior probability of insolvency equal to σ/(σ + θ), higher than prior σ.
  - A solvent bank affected by a negative signal is pooled with insolvent banks, possibly rendering it unable to refinance due to increased solvency risk.
- Two imposed restrictions (sufficient conditions to generate liquidity risk):
  - Existence of initial funding at date 0 and positive owner-manager return:
    - X > 1/(1 − (σ + θ)) + δ/(1 − (σ + θ)).  (Equation (1))
  - Under a negative signal at date 1 a bank cannot obtain refinancing even for a low refinancing need:
    - X < (1 − ω_L) + ω_L · (σ + θ)/θ.  (Equation (2))
  - These two inequalities can hold simultaneously for parameter values such as σ + θ << 1, θ << σ, and ω_L not too close to zero.
- Assumption for exposition:
  - Date 0 initial funding attracted at risk-free rate of 0 (not priced deposit insurance).
  - Date 1 refinancing is not covered by deposit insurance and is risk-sensitive (market-based wholesale funding).
  - This assumption makes results weaker; risk-sensitive funding increases interest owed to creditors, which can make owner-manager’s decisions more distorted.

### Lemma on existence of liquidity risk
- Lemma 1:
  - There exist parameter values such that a bank can attract initial funding at date 0, but cannot refinance in case of solvency concerns (a negative signal) at date 1.
  - In that case, some solvent banks are liquidated.
- Parameter example conditions:
  - Inequalities (1) and (2) are satisfied simultaneously for example when σ + θ << 1, θ << σ, and ω_L not too close to zero.

*Source: IMF working paper content unit "1. The Timeline" (chapter/section of source PDF).*

### Appendix A, we present a version of the model that incorporates risk-sensitive debt.

### _wp1316 - Appendix A, we present a version of the model that incorporates risk-sensitive debt.

### Risk management tools
- The bank has two instruments of liquidity risk management:
  - Liquidity buffer:
    - At date 0 the bank can attract additional funds and invest them in short-term assets that return 0.
    - Holding liquidity is costly: per-unit cost of effort .
    - The size of the liquidity buffer  required to cover a refinancing need 
       is given by:
      - (1 + )·
         = , (3)
      - which implies:
        -  = 
           / (1 − 
           ). (4)
    - Cost of maintaining a liquidity buffer to cover small refinancing needs is .
    - If 
       = 1, no liquidity buffer can cover a large refinancing need (all funding must be refinanced).
    - If a bank is liquidated at date 1, or if it does not use the liquidity buffer, the bank returns the liquidity buffer to creditors in its entirety at date 1.
  - Transparency:
    - Transparency is an ex-ante (date 0) decision enabling more effective communication of asset values.
    - In case of a negative signal at date 1, a transparent solvent bank can communicate solvency and obtain funding with probability ; with probability 1 −  it cannot.
    - Cost of transparency is the owner-manager’s loss of private benefits of control .
- Ex-ante preference ordering imposed: hedging preferred to no hedging, and liquidity buffers preferred to transparency on the margin.

### First best (social welfare maximization)
- Social welfare  defined as joint surplus of owner-manager and creditors (equivalent to bank NPV).
- Four strategies: do nothing (N), liquidity buffer only (), transparency only (T), both liquidity and transparency (T).
- Payoffs (social welfare):
  - No hedge:
    - 
       = (1 − ( + )) −  + . (5)
  - Liquidity buffer only:
    - 
       = (1 − ( + ) + ) −  −  + . (6)
  - Transparency only:
    - 
       = (1 − ( + ) + ) − . (7)
  - Both liquidity and transparency:
    - 
       = (1 − ( + ) + ( +  − )) −  − . (8)
- First-best hedging conditions (inequalities):
  - Be liquid (from no hedge):  ≤ . (9)
  - Be transparent (from no hedge):  ≤ . (10)
  - Prefer liquidity to transparency:  −  ≤ ( − ). (11)
  - Both hedges optimal:
    -  ≤ (1 − )
    -  ≤ (1 − ). (12)
- Key analytical insights:
  - Marginal benefit of second hedge is lower than first: liquidity buffers and transparency are strategic substitutes (redundancy captured by term − in (8)).
- Proposition 1:
  - Both liquidity buffers and transparency can be socially desirable; when costs are low enough, combining them is optimal.
  - Liquidity buffers fully insure small refinancing needs; transparency provides partial hedge for large withdrawals.

### Private risk management choices (owner-manager payoffs)
- Owner-manager does not fully internalize benefits to creditors (limited liability).
- Payoffs Π:
  - No hedge:
    - Π
       = (1 − ( + ))( − 1) −  + . (13)
  - Liquidity buffer:
    - Π
       = (1 − ( + ) + )( − 1) −  −  + . (14)
  - Transparency:
    - Π
       = (1 − ( + ) + )( − 1) − . (15)
  - Both:
    - Π
       = (1 − ( + ) + ( +  − ))( − 1) −  − . (16)
- Private hedging conditions (replace  by ( − 1) on right-hand sides):
  - Be liquid:  ≤ ( − 1). (17)
  - Be transparent:  ≤ ( − 1). (18)
  - Prefer liquidity to transparency:  −  ≤ ( − )( − 1). (19)
  - Both hedges privately preferred:
    -  ≤ (1 − )( − 1)
    -  ≤ (1 − )( − 1). (20)
- Comparative outcome:
  - Private threshold lines are below/left of first-best lines → possible under-insurance.
- Proposition 2:
  - Owner-manager may choose insufficient hedging; there are parameter values where socially optimal choice is both hedges but privately only one is chosen.
- Conditions where bank is only liquid while socially optimal to be both (21):
  - (1 − )( − 1) <  ≤ (1 − )
  -  ≤ (1 − )
  -  −  ≤ ( − )( − 1)
- Conditions where bank is only transparent while socially optimal to be both (22):
  - (1 − )( − 1) <  ≤ (1 − )
  -  ≤ (1 − )
  -  −  > ( − )( − 1)

### Regulation: unintended effects of liquidity requirements
- Regulatory constraint: transparency is not verifiable and thus cannot be directly regulated; liquidity buffers can be regulated via liquidity requirements.
- Strategic-substitute relationship implies liquidity requirements may affect transparency choices.
- Consider parameter set satisfying (22): bank privately chooses transparency only, first-best is both hedges.
- Two possible outcomes after imposing liquidity requirements:
  - Low cost of transparency (point A0) where:
    -  ≤ (1 − )( − 1). (23)
    - Bank maintains transparency; liquidity requirements induce socially optimal both-hedge outcome.
  - Higher cost of transparency (point A) where:
    -  > (1 − )( − 1). (24)
    - Bank abandons transparency in response to liquidity requirements; liquidity buffers substitute for transparency.
    - This shift can reduce social welfare: point A lies below social indifference line (11); for  >  the shift increases refinancing risk.
- Proposition 3:
  - There exist parameter values where liquidity requirements cause a transparent bank to abandon transparency, reducing social welfare — an unintended effect.

### Addressing bank transparency (policy responses)
- Because transparency is endogenous and not directly verifiable, policy can:
  - Reduce the owner-manager’s private benefits of control  (improve corporate governance or encourage less concentrated ownership).
    - Reducing  can restore the condition (23) and prevent liquidity requirements from crowding out transparency (analogous to moving from point A to A0).
  - Influence funding structure to make large refinancing needs less likely (increase ).
    - Higher  reduces the social value of transparency in presence of liquidity buffers (second inequality in (12) may cease to bind).
    - Practical instrument: maturity mismatch limits, such as the Net Stable Funding Ratio of Basel III, which raise  by promoting more long-term/stable funding.
    - Note: longer-term funding may increase cost of funds (add  per-unit to cost of liquidity buffer), potentially shifting choices (point A → A00), and the overall welfare effect of stable funding ratios can be ambiguous (trade-offs such as deadweight costs, distortion of maturity transformation, or reduced market discipline).
- Policy implication:
  - Liquidity requirements should be complemented by measures that improve bank transparency (corporate governance reforms) or by measures that reduce the likelihood of large refinancing needs (maturity mismatch limits).

### Conclusion (key takeaways)
- Both liquidity buffers and bank transparency are important components of liquidity risk management:
  - Liquidity buffers insure small withdrawals with certainty.
  - Transparency can enable refinancing of large withdrawals, but effectiveness is probabilistic (parameter ).
- Banks may under-insure because owner-managers do not internalize benefits to creditors under limited liability.
- Liquidity requirements alone can have unintended adverse effects by crowding out transparency for some parameter values.
- Policy should combine liquidity requirements with measures to improve bank transparency and funding stability (corporate governance improvements, maturity mismatch limits) to avoid unintended welfare-reducing outcomes.

*Source: Appendix A of the provided IMF working paper content.*

### References

### _wp1316 - References

### Appendix A — A Model Without Deposit Insurance: Key Analytical Findings
- Removing the assumption of not priced deposit insurance implies creditors charge a risky bank a positive interest rate to achieve zero expected return; funding for the investment project is charged a gross interest rate >1 while funding for the liquidity buffer is risk-free.
- Under most complete hedging (creditors expect the bank to be both liquid and transparent), creditors charge the interest rate:
  - 2 = 1 / [1 − ( + ) + ( +  − )].
- Under 2 the owner-manager will choose to be liquid and transparent only when:
  -  < (1 − )( − 2)
  -  < (1 − )( − 2)
- If creditors expect the bank to be only liquid (or only transparent), the interest rate will be:
  - 1 = 1 / [1 − ( + ) + ] and = 1 / [1 − ( + ) + ], since  = .
- The bank will choose to be at least liquid if:
  -  < ( − 1)
- The bank will choose to be at least transparent if:
  -  < ( − 1)
- Because 2 > 1 and 1 > 1, abandoning the simplifying assumption of not priced deposit insurance increases the wedge between private and first-best risk management choices. This strengthens Proposition 2: private choices of liquidity buffers and transparency may be suboptimal.
- Liquidity requirements can induce the bank to abandon transparency even when socially optimal policy requires both liquidity and transparency. The condition under which the bank is transparent while it is socially optimal to be both liquid and transparent (substituting 1 and 2 into the relevant expressions) is:
  - (1 − )( − 2) <  < (1 − )
  - (1 − )( − 1) <  < 
- This confirms robustness of Proposition 3: liquidity requirements may compromise bank transparency choices when debt is risk-sensitive.

### Appendix B — Quantitative Example: Parameterization and Implications
- Example parameter values (interpreted as describing bank returns and risk over a 10-year period):
  -  = 135 (corresponding to a 35% per annum return on bank assets)
  -  = 008 and  = 002 (corresponding to a 1% a year risk of bank failure)
  - L = 009, giving  = 01 (corresponding to a liquidity buffer of 10% of bank assets)
  -  =  = 09 (so that most shocks are small and transparency is most often effective)
- These parameter values satisfy condition (1) for  < 0215, and satisfy condition (2) (as stated in the model).
- Substituting parameter values into (22) and applying the additional restriction on  yields that the bank is only transparent while it is optimal that a bank is both liquid and transparent for:
  - 0063 <  < 0215
  -  <  / 10
- Using (20), the bank will abandon transparency in response to liquidity requirements for:
  -  > 00063
- Example closure with  = 015 (corresponding to administrative costs of 15% per year), making  < 0015:
  - For 00063 <  < 0015 the bank will abandon transparency in response to liquidity requirements.
  - For  ≤ 00063 the bank will maintain transparency even under liquidity requirements.
- Qualitative implication from the numerical example:
  - The range of values of  for which liquidity requirements compromise bank transparency choices is wider than the range for which they do not, suggesting the model’s effects are plausible and reasonably likely.

### Figures and Conceptual Illustrations (as described)
- Figure 1: The Timeline — depicts the model timeline.
- Figure 2: The Information Structure — shows the probability distribution of bank fundamentals and of the intermediate signal; a bank can be solvent with probability 1 − s but receive a negative signal with probability q, making it unable to refinance.
- Figure 3: Socially Optimal Liquidity Risk Management — maps regions where optimal hedging is none (N), one hedge (L or T), or both hedges (LT); note figures drawn for t > l.
- Figure 4: Private Risk Management Choices — contrasts private choices (thick solid lines) with first best (dashed lines); identifies area where the bank privately chooses one hedge while socially two hedges are optimal.
- Figure 5: The Unintended Effects of Liquidity Requirements — shows that a bank that is privately only transparent may abandon transparency in response to liquidity requirements for higher transparency cost (point A), but may maintain it for lower cost (point A’).
- Figure 6: Maturity Mismatch Limits — illustrates effect of higher probability l that a refinancing need is small; threshold lines shift down and to the right and longer-term funding (costlier by r) can shift point A to A’’; either point can render transparency unnecessary for socially optimal liquidity risk management.

*Italic: Source — _wp1316 - References (appendixes and figures descriptions).*

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