## 5. Projects and gambles pay out

## Source details

**Canonical URL:** [5. Projects and gambles pay out](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14116.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14116.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14116.pdf.json)

---

### Timing of the game and funding market setup
- Banks enter funding stage after asset-side stages so banks are sorted by type (project quality and gambles) before funding, enabling adverse selection.
- Internal funds: e < 1 committed to the project; additional financing d = 1 - e from wholesale debt market (no deposit-insured retail deposits, no external equity in baseline).
- Wholesale funding assumptions:
  - Perfectly competitive, infinitely elastic supply, participants risk neutral.
  - Creditors do not observe individual bank asset structures but know model parameters and charge a fair risk premium.
  - Risk-free rate normalized to zero; gross market borrowing rate is r.
- Mixed funding extension with retail deposits considered in section 5C.

### Bank maximization (stage 1 optimization) and gamble conditions
- Define d = 1 - e as amount a sound bank needs to borrow so project yields return R.
- Parameter restriction assumed: x > R / d (equation (1)). When R, d, x satisfy that restriction, unsound banks always choose to gamble.
- Gamble characteristics (from earlier sections repeated here):
  - Gamble multiplies invested amount by x if successful; invested funds lost if unsuccessful.
  - Probability of success is ϑ, where 0 < ϑ < 1 / x.
  - Gamble has negative present value: ϑ x < 1.
- Stage 1 objective (Lemma 1):
  - If R - r d ≥ e + max{ϑ d (x - r); 0}:
    - E[Π_i] = q_i (R - r d) + (1 - q_i) ϑ d (x - r) - c q_i^2
  - Otherwise:
    - E[Π_i] = q_i (e) - c q_i^2
  - (Equation (2) and no-breakdown condition (3))
- No-breakdown condition: R - r d ≥ e + max{ϑ d (x - r); 0}; if not satisfied, creditors provide no funds (funding market freezes), and banks optimize only to prevent entrepreneur shirking.

### Breakdown threshold and market-rate solution
- Definition 2: Funding market breakdown occurs when no interest rate exists at which financiers are willing to lend.
- Solution procedure:
  1. Derive r assuming no-breakdown holds.
  2. Solve no-breakdown condition and equilibrium given that r.
  3. Verify no-breakdown condition holds.
- Fair market rate under no-breakdown (equation (4)):
  - r = 1 / [q̄ + (1 - q̄) ϑ]
- No-breakdown condition rewritten (Lemma 3, equation (5)):
  - q̄ > bq = max { ϑ (1 - ϑ x) / [x (1 - ϑ)]; d / [R - 1 + d (1 - ϑ x)] - ϑ / (1 - ϑ) } ∈ (0;1)
- Interpretation:
  - Equilibrium monitoring effort q̄ must exceed threshold bq; otherwise adverse selection leads to funding-market freeze.
  - Threshold bq falls when R increases or d decreases.
  - Effects of gambling parameters x and ϑ on bq ambiguous: higher ϑ raises charter value (reduces gambling) but raises expected gambling return relative to sound investment (opposite effect).

### Reaction function, monitoring incentives, and complementarities
- If no-breakdown holds, bank solves (equation (6)):
  - max_{q_i} q_i (R - r d) + (1 - q_i) ϑ d (x - r) - c q_i^2
- Individual best response (equation (7)):
  - q_i^* = [R - d ( r (1 - ϑ) + ϑ x )] / (2 c)
- Using r from (4) yields reaction function (equation (8)):
  - q_i^* = (1 / 2 c) [ R - d { (1 - ϑ) / [q̄ (1 - ϑ) + ϑ] + ϑ x } ]
- Comparative statics from reaction function:
  - q_i^* increases with R, decreases with d, decreases with higher c.
  - ϑ enters ambiguously due to countervailing effects.
- Complementarity and curvature (Lemma 4):
  - ∂ q_i^* / ∂ q̄ > 0 and ∂^2 q_i^* / ∂ q̄^2 < 0 (equations (9) and (10)).
  - Monitoring by other banks increases a given bank's incentive to monitor (cross-sectional complementarity).
  - Marginal effect of others' monitoring on r strongest when q̄ is low, producing concavity.

### Equilibrium constellations and stability
- Nash equilibrium: no bank has incentive to deviate. Interior equilibrium q̄ ∈ (0;1) is stable if ∂ q_i^* / ∂ q̄ < 1.
- Proposition 6: Three possible constellations of Nash equilibria:
  - Case I. One equilibrium: low-end corner (q̄ = 0)
  - Case II. Three equilibria: one interior (unstable), both corners
  - Case III. Three equilibria: two interior (one stable, one unstable), one low-end corner
- Graphical implications:
  - Reaction function increasing and concave, starts below 45-degree line at q̄ = 0.
  - No crossing with 45-degree line ⇒ Case I.
  - Single crossing ⇒ Case II (unstable interior plus corners; dynamics push to corners).
  - Two crossings ⇒ Case III (stable high interior, unstable interior, low corner).
- Parameter conditions for constellations (Proposition 7):
  - Case II iff R > 2 c + d (1 + ϑ (x - 1))
  - Case III iff all of:
    - R ≤ 2 c + d (1 + ϑ (x - 1));
    - R > 2 sqrt(2 c d + ϑ x d) - 2 c ϑ / (1 - ϑ);
    - [ q̄ d / (2 c) - ϑ / (1 - ϑ) ]^2 ∈ [0;1]
  - Case I for any other parameterization
- Key insight: Case III requires many conditions and is empirically harder to generate; model often reduces to Case II vs Case I.

### Comparative statics and qualitative behavior (“silence before the storm”)
- Vary R across high R_H, medium R_M, low R_L with R_H > R_M > 2 c + d (1 + ϑ (x - 1)) ≥ R_L:
  - Declines in R lower individual banks' optimal monitoring for any given q̄ (reaction function shifts rightward).
  - Counterintuitively, interior equilibrium monitoring can be higher at R_M than at R_H; further decline to R_L can eliminate interior equilibrium and produce the low-end corner — falling default rates may precede sudden collapse.
  - High-end corner displays non-linear response: moderate declines in R may do nothing, but further declines cause jumps between corners.
- Endogenous breakdown threshold:
  - Breakdown threshold itself shifts in response to shocks; e.g., declining R shifts threshold rightward so breakdown occurs at higher R than if threshold were constant.
  - Policy implication: treating threshold as fixed while focusing only on bank profitability may overestimate resilience.

### Numerical examples (illustrative calibrations)
- Case II example:
  - Parameters: d = 0.925, recovery rate ϑ x = 0.5, R from 1.10 to 1.08.
  - Calibrated to initial survival probability q̄ = 0.99 using c free.
  - Derived values: c = 0.048, x = 1.19, ϑ = 0.5 / x = 0.42.
  - Small drop in R (from 1.10 to 1.08) leads from interior equilibrium with very low default rates or a high-end corner to certain market gridlock (breakdown).
  - Low initial default risk implies greater systemic sensitivity (smaller shock needed for breakdown).
- Case III example:
  - Parameters: R = 1.1, d = 0.45, x = R / d, ϑ = 0.2 / x, c = 0.3.
  - Produces a stable interior equilibrium near maximum monitoring, far above breakdown threshold.
  - Increasing debt ratio from 45% to 50% eradicates interior equilibrium and induces funding market breakdown.
  - Finding: Case III requires less realistic parameterization (e.g., low debt ratio); generating Case III for realistic values is difficult.

### Extensions: informed creditors, idiosyncratic risk, and retail deposits
- Informed creditors (probability ' of portfolio revelation):
  - Banks' maximization when creditor information may be revealed (equation (11)):
    - E[Π_i] =
      - (1 - ') q_i (R - r d) + (1 - ') (1 - q_i) ϑ d (x - r) + ' q_i (R - d) - c q_i^2 if R - r d ≥ e + max{ϑ d (x - r); 0}
      - q_i (e) - c q_i^2 otherwise
  - Reaction functions (equation (12) and r-expression):
    - q̃_i = [R - r d + ' d (r - 1) - (1 - ') ϑ d (x - r)] / (2 c)
    - r = 1 / [q̃ + (1 - q̃) ϑ]
  - Comparative statics: q̃_i is monotonically increasing in ' (greater probability of information revelation increases monitoring).
- Idiosyncratic bank soundness (heterogeneous p_i = q_i + ϵ_i, ϵ_i ~ U(0; a)):
  - Monitoring rescaled q_i ∈ [0; (1 - a)].
  - Interest rate (equation (15)):
    - r = 1 / [ q̃ + a/2 + (1 - q̃ - a/2) ϑ ]
  - No-breakdown threshold (equation (16)):
    - q̃_b = max { (1 - ϑ x) / [ x (1 - ϑ) ] ; d / [ R - 1 + d (1 - ϑ x) - ϑ ] / (1 - ϑ) }
  - Comparative implications:
    - Increasing a (reducing correlation across bank exposures) lowers breakdown threshold bq.
    - Effect on q̃_i ambiguous: larger a reduces funding pressures (lower r) raising q̃_i, but larger idiosyncratic noise weakens linkage between monitoring and default probability, lowering q̃_i.
    - Net effect parameter-dependent; reducing correlation does not unambiguously reduce systemic crises.
- Retail depositors and funding composition:
  - Funding shares: equity e, wholesale w, retail d (retail fully insured, pays zero).
  - Bank maximization with retail deposits (equation (18)):
    - E[Π_i] =
      - q_i (R - r w - d) + (1 - q_i) ϑ ((w + d) x - r w - d) - c q_i^2 if R - r w - d ≥ e + max{ϑ ((w + d) x - r w - d); 0}
      - q_i (e) - c q_i^2 + ϑ d (x - 1) otherwise
  - No-breakdown condition (equation (19)):
    - q̃_b = max { (1 - ϑ x) / [ x (1 - ϑ) ] ; w [ (R - d - e) - ϑ ((w + d) x + d) ] / [ w (1 - ϑ) ((w + d) x - d) ] - ϑ / (1 - ϑ) }
  - Optimal monitoring (equation (20)):
    - q̃_i = (1 / (2 c)) * { R - (1 - ϑ) [ d + w q̃ (1 - ϑ) + ϑ ] - ϑ x (w + d) }, with constraint [ q̃ (1 - ϑ) + ϑ ] ≤ 1
  - Key findings:
    - Replacing wholesale with retail deposits (raising d and lowering w) increases bank optimal monitoring q̃_i.
    - bq declines when retail funding share increases; retail funding makes funding-market freeze less likely.
    - Intuition: retail funding reduces the externality by making banks internalize more benefits of project soundness.

### Policy-relevant conclusions and regulatory implications
- Moral hazard in monitoring combined with adverse selection in funding produces non-linear effects and multiplicity of equilibria; parameter changes can trigger sudden transitions into systemic crises.
- Three extension-based policy-relevant results:
  - Better information among bank creditors (higher ') alleviates risk of systemic freezes.
  - Reducing correlation among bank portfolios (higher a) does not necessarily make funding market more robust; monitoring responses can offset diversification benefits.
  - Increasing share of retail deposits reduces systemic vulnerability of wholesale market.
- Regulatory implications:
  - Systems that appear stable (very low default rates) can be closer to breakdown thresholds; maintain high vigilance.
  - Microprudential policies that lower individual default risk can paradoxically move system closer to systemic-freeze threshold (micro- vs macroprudential trade-offs).
  - Transparency and regulator communication matter: regulator information that is more granular than wholesalers' and public communication about system-wide distress (without bank identifiers) can produce sudden equilibrium shifts.
  - Macroprudential policy should support reducing reliance on wholesale funding; systemic-risk taxation should not focus solely on reducing common exposures.

### Appendix: analytical thresholds and equilibrium cases (explicit expressions)
- No-breakdown condition (Lemma 2; equation (21) summarized):
  - If q̃ > (1 - ϑ x) / [ x (1 - ϑ) ] then no breakdown iff q̃ ≥ d [ R - 1 + d (1 - ϑ x) - ϑ ] / (1 - ϑ).
  - Breakdown threshold:
    - bq = max { (1 - ϑ x) / [ x (1 - ϑ) ] ; d / [ R - 1 + d (1 - ϑ x) - ϑ ] / (1 - ϑ) }, bq ∈ (0;1).
- Equilibrium classification (Propositions 1 and 2):
  - Case I: only low-end corner equilibrium (q̃ = 0).
  - Case II: one interior equilibrium plus high-end corner; occurs iff R > 2 c + d (1 + ϑ (x - 1)).
  - Case III: two interior equilibria (multiplicity); occurs iff
    - R ≤ 2 c + d (1 + ϑ (x - 1)) and R > 2 sqrt{2 c d + ϑ x d - 2 c ϑ / (1 - ϑ)} and [ q d / (2 c) - ϑ / (1 - ϑ) ] ∈ [0;1].
  - Algebraic conditions and regions derived analytically in appendix.

*Source: _wp14116 - 5. Projects and gambles pay out (IMF Working Paper).*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   31

### _wp14116 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   31

### I. Introduction — key mechanism and core results
- Problem posed: how a financial system with low observed default rates and low funding costs can nonetheless be highly vulnerable to shocks.
- Conceptual distinction: risk within an equilibrium (individual bank default risk) versus risk across equilibria (likelihood that a shock leads to market breakdown).
- Main qualitative result: risk within and across equilibria can move in opposite directions; default rates can be lowest when the system as a whole is most sensitive to shocks.
- Mechanism summary:
  - Mutual reinforcement between individual bank risk taking and asymmetric information on the wholesale market for unsecured bank funding.
  - Adverse selection: financiers cannot distinguish between banks with genuine liquidity shortfalls and insolvent banks that "gamble for resurrection".
  - Prisoners' dilemma among banks: collectively they would prefer to limit risk taking to reduce adverse selection premia, but individually they under-internalize the effect of their risk choices on aggregate premia, creating externalities (moral hazard).
  - Two reinforcing channels:
    - Bank risk increases adverse selection premia on the funding market (cited modeling by Freixas, Parigi and Rochet (2004) and Heider, Hoerova and Holthausen (2009)).
    - Higher funding costs reduce banks' charter values, increasing incentives to undertake risk (Keeley (1990) literature).
  - Non-linear dynamics and multiple equilibria arise from the combination of these channels (illustrated in Figure 1).
- Equilibrium dynamics (as described with Figure 1):
  - Banks' monitoring effort q_i is a strategic complement to others' effort q.
  - Left panel: three equilibria can coexist — one interior equilibrium and two corner equilibria; a shaded "breakdown zone" exists where adverse selection leads to funding market freeze (no interest rate at which financiers lend).
  - Right panel: a parameter shock (e.g., decline in asset returns) can move the reaction function so the breakdown equilibrium becomes unique.
- Extensions analyzed (overview):
  - Providing creditors with imperfect information about individual bank quality reduces likelihood of funding market breakdown by increasing ex-ante monitoring effort.
  - Introducing idiosyncratic risk in bank portfolios and varying its size relative to common exposures; greater correlation between bank portfolios has intricate effects on systemic risk.
  - Introducing retail depositors: less reliance on wholesale funding reduces funding-market vulnerability.

### II. Relation to literatures
- Macro- vs. microprudential trade-offs:
  - Microprudential regulation can induce banks to shift risks to the system (Acharya (2009); Acharya and Yorulmazer (2007)).
  - Capital requirements can induce less diversification at the bank level, raising common exposures (Ibragimov, Jaffee and Walden (2010); Zhou (2013)) or raise tail risks (Perotti, Ratnovski and Vlahu (2011)).
  - Diversification can sometimes raise systemic risk (Wagner (2010); related extension in this paper).
  - Liquidity regulation may create incentives for opacity unless combined with transparency rules (Ratnovski (2013)).
- Wholesale funding market literature:
  - Established adverse-selection concerns and inefficiencies (Bhattacharya and Gale (1987); Rochet and Tirole (1996); Flannery (1996); Allen and Gale (2000); Freixas, Parigi and Rochet (2000); Rochet and Vives (2004); Freixas and Jorge (2008); Allen, Carletti and Gale (2009); Heider and Hoerova (2009); Freixas, Martin and Skeie (2011); Huang and Ratnovksi (2011); Fecht, Grüner and Hartmann (2012); Acharya, Gromb and Yorulmazer (2012); Anand, Gai and Marsili (2012); Ennis and Weinberg (2013); Castiglionesi and Wagner (2013)).
  - Distinction: prior models typically take bank risk taking and solvency as exogenous; in this paper credit risk premia feed back into bank decisions, producing new effects.
- Multiple equilibria in banking:
  - Multiplicity is a recurring feature since Diamond and Dybvig (1983), but triggers for shifts between equilibria are often unmodeled.
  - This paper relates equilibrium shifts to parameter changes, enabling discussion of shock sizes needed to induce shifts.
  - Related work: Goldstein and Pauzner (2005) (global games technique), Freixas and Holthausen (2005), Boissay (2011).
  - Novelty: distinction between stable and unstable equilibria emphasized as key non-linear features (Rosser, 1999).

### III. II. Model — setup and primitives
- Agent population:
  - A mass of identical, risk-neutral banks indexed by i 2 [0;1].
- Project and returns:
  - Asset side initially consists of one project yielding a total return R >1 if it succeeds.
  - Minimum funding needed for project to be of any value: 1 (fixed development costs). If funding <1 then entrepreneur can only invest in a risk-free technology yielding zero net return.
- Monitoring and project soundness:
  - Bank exerts monitoring effort q_i 2 [0;1] with associated non-pecuniary cost cq 2 i ,c0experienced by bank management.
  - q_i is the probability the bank has a "sound" project; with probability 1q_i monitoring fails and the project is unsound (certainly of no value).
  - Realization of project soundness is private information to the bank.
- Project payoff table (preserved exactly as in source):
  - Unsound quality / Sound quality
  - Funding<101(funding)
  - Funding10R(funding)=R
  - (Table heading: Table 1: Project payo§)

### IV. A. Gambles — optional post-realization decision
- After observing project soundness, banks may invest an amount between0and1 in a gamble.
- Gamble specifics:
  - If successful, the gamble multiplies the amount invested by x.
  - If unsuccessful, invested funds are lost.
  - Probability of success is , where0<  < 1/x.
  - The gamble has negative present value: x <1 (destroys value).
- Limited liability and adverse selection consequences:
  - Creditors would never lend to a known gambling bank; at r = x the expected gross return r = x <1, making loans loss-making in expectation.
  - Because of information asymmetry, sound banks may cross-subsidize gambles of unsound banks, enabling gambles to occur in equilibrium.

### V. B. Funding — timing of stages (as listed)
- Stage sequence (preserved exactly):
  1. Banks set monitoring e§ort
  2. Project soundness realizes
  3. Banks decide on gambles
  4. Bank funding

### VI. Figures and tables referenced (inventory preserved exactly as named)
- Tables:
  - Table 1.  Project payo§  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    8
  - Table 2.  Timing of the game  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    9
  - Table 3.  Timing of the game with creditor information . . . . . . . . . . . . . . .   19
- Figures:
  - Figure 1. A shift to breakdown  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    5
  - Figure 2. Low-end corner only . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   14
  - Figure 3. One interior equilibrium, two corners . . . . . . . . . . . . . . . . . .   14
  - Figure 4. Two interior equilibria . . . . . . . . . . . . . . . . . . . . . . . . . .   15
  - Figure 5. The e§ects of falling returns  . . . . . . . . . . . . . . . . . . . . . . .   16
  - Figure 6. From Case II to Case I  . . . . . . . . . . . . . . . . . . . . . . . . . . .   17
  - Figure 7. From Case III to Case I . . . . . . . . . . . . . . . . . . . . . . . . . . .   18
  - Figure 8. Breakdown threshold shift . . . . . . . . . . . . . . . . . . . . . . . . . .   19

*Source: _wp14116 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   31*

### 5. Projects and gambles pay out

### 5. Projects and gambles pay out

### Timing of the game and funding market setup
- Banks' funding stage is modeled as occurring after the asset side stages so banks are sorted by type (project quality and gambles) before turning to the funding market, enabling adverse selection in funding.
- Banks enter with internal funds worth e < 1 committed to the project; additional financing must come from a wholesale debt market (no deposit-insured retail deposits, no issuance of external equity).
- Wholesale funding is modeled as:
  - Perfectly competitive, infinitely elastic in supply, participants risk neutral.
  - Creditors do not observe individual bank asset structures but know model parameters and charge a fair risk premium.
  - Risk-free rate normalized to zero; gross market borrowing rate for banks is r and reflects default risk.
- Rationale: wholesale funding reacts to bank riskiness and was central to the recent funding market freeze; a mixed funding extension with retail and wholesale is considered in section 5C.

### Bank maximization (stage 1 optimization)
- Define d = 1 - e as the amount a sound bank needs to borrow so its project yields return R.
- Parameter restriction assumed: x > R / d (equation (1)).
- When R, d, x satisfy that restriction, unsound banks always choose to gamble.
- Stage 1 objective (Lemma 1): a bank maximizes expected payoff E[Π_i] given by
  - If R - r d ≥ e + max{ϑ d (x - r); 0}:
    - E[Π_i] = q_i (R - r d) + (1 - q_i) ϑ d (x - r) - c q_i^2
  - Otherwise:
    - E[Π_i] = q_i (e) - c q_i^2
  - (Equation (2) and no-breakdown condition (3))
- No-breakdown condition: R - r d ≥ e + max{ϑ d (x - r); 0}. If not satisfied, creditors realize they face only gambling banks and provide no funds (funding market freezes); banks then optimize only to prevent entrepreneur shirking.

### Breakdown threshold and definition
- Definition 2: Funding market breakdown occurs when no interest rate exists at which financiers are willing to lend.
- Solution procedure:
  1. Derive r assuming no-breakdown holds.
  2. Solve no-breakdown condition and equilibrium given that r.
  3. Verify no-breakdown condition holds.
- Fair market rate under no-breakdown (equation (4)):
  - r = 1 / [q̄ + (1 - q̄) ϑ]
- No-breakdown condition rewritten (Lemma 3, equation (5)):
  - q̄ > bq = max { ϑ (1 - ϑ x) / [x (1 - ϑ)]; d / [R - 1 + d (1 - ϑ x)] - ϑ / (1 - ϑ) } ∈ (0;1)
- Interpretation:
  - Equilibrium monitoring effort q̄ must exceed threshold bq; otherwise adverse selection leads to market freeze.
  - Threshold bq falls when R increases or d decreases.
  - Effects of gambling parameters x and ϑ on bq are ambiguous: higher ϑ raises charter value (reduces gambling) but raises expected gambling return relative to sound investment (works opposite).

### Reaction function and monitoring incentives
- If no-breakdown holds, bank solves (equation (6)):
  - max_{q_i} q_i (R - r d) + (1 - q_i) ϑ d (x - r) - c q_i^2
- Individual best response (equation (7)):
  - q_i^* = [R - d ( r (1 - ϑ) + ϑ x )] / (2 c)
- Replacing r from (4) yields reaction function (equation (8)):
  - q_i^* = (1 / 2 c) [ R - d { (1 - ϑ) / [q̄ (1 - ϑ) + ϑ] + ϑ x } ]
- Comparative statics from reaction function:
  - q_i^* increases with R, decreases with d, decreases with higher c.
  - ϑ enters ambiguously due to countervailing effects.
- Complementarity and curvature (Lemma 4):
  - ∂ q_i^* / ∂ q̄ > 0 and ∂^2 q_i^* / ∂ q̄^2 < 0 (equations (9) and (10)).
  - Monitoring by other banks increases a given bank's incentive to monitor (cross-sectional complementarity).
  - Marginal effect of others' monitoring on r strongest when q̄ is low, producing concavity.

### Equilibrium constellations and stability
- Nash equilibrium: no bank has incentive to deviate. Interior equilibrium q̄ ∈ (0;1) is stable if ∂ q_i^* / ∂ q̄ < 1.
- Proposition 6: Three possible constellations of Nash equilibria:
  I. One equilibrium: low-end corner (q̄ = 0)
  II. Three equilibria: one interior (unstable), both corners
  III. Three equilibria: two interior (one stable, one unstable), one low-end corner
- Graphical implications:
  - Reaction function is increasing and concave, starts below 45-degree line at q̄ = 0.
  - When no crossing with 45-degree line → Case I (only low-end corner).
  - Single crossing → Case II (unstable interior plus corners; dynamics push to corners).
  - Two crossings → Case III (stable high interior, unstable interior, low corner).
- Parameter conditions for constellations (Proposition 7):
  - Case II iff R > 2 c + d (1 + ϑ (x - 1))
  - Case III iff all of:
    - R ≤ 2 c + d (1 + ϑ (x - 1));
    - R > 2 sqrt(2 c d + ϑ x d) - 2 c ϑ / (1 - ϑ);
    - [ q̄ d / (2 c) - ϑ / (1 - ϑ) ]^2 ∈ [0;1]
  - Case I for any other parameterization
- Key insight: Case III requires many conditions and is empirically harder to generate; model often boils down to Case II vs Case I.

### Comparative statics and qualitative behavior
- Varying R across high R_H, medium R_M, low R_L with R_H > R_M > 2 c + d (1 + ϑ (x - 1)) ≥ R_L:
  - Declines in R lower individual banks' optimal monitoring for any given q̄ (reaction function shifts rightward).
  - Counterintuitively, interior equilibrium monitoring can be higher at R_M than at R_H; further decline to R_L can eliminate interior equilibrium and produce the low-end corner — i.e., falling default rates may precede a sudden collapse (“silence before the storm”).
  - High-end corner displays non-linear response: moderate declines in R may do nothing, but further declines cause jumps between corners.

### Numerical examples
- Case II example:
  - Parameters: d = 0.925 (debt ratio 92.5%, own capital 7.5%), recovery rate ϑ x = 0.5, R from 1.10 to 1.08.
  - Calibrated to initial survival probability q̄ = 0.99 using c as free parameter.
  - Derived/calculated values: c = 0.048, x = 1.19, ϑ = 0.5 / x = 0.42.
  - Small drop in R (from 1.10 to 1.08) leads from an interior equilibrium with very low default rates or a high-end corner to certain market gridlock (breakdown).
  - Low initial default risk implies greater systemic sensitivity (smaller shock needed for breakdown).
- Case III example:
  - Parameters: R = 1.1, d = 0.45, x = R / d, ϑ = 0.2 / x, c = 0.3.
  - Produces a stable interior equilibrium near maximum monitoring, far above breakdown threshold.
  - Increasing debt ratio from 45% to 50% eradicates interior equilibrium and induces funding market breakdown.
  - Finding: Case III requires less realistic parameterization (e.g., low debt ratio); generating Case III for realistic values is difficult.
- Endogenous breakdown threshold:
  - Breakdown threshold itself shifts in response to shocks. Example: declining R shifts threshold rightward so breakdown occurs at higher R than if threshold were constant.
  - Policy implication: regulators who treat threshold as fixed while focusing only on bank profitability may overestimate financial sector resilience.

### Extensions — informed creditors (brief)
- Main model baseline assumes creditors entirely uninformed about individual bank soundness.
- Extension adds a stage on creditor information occurring after realization of project soundness and banks' gambling decisions (timing updated in table 3).
- Rationale: creditor information played a central role during the recent crisis; this extension relaxes the extreme baseline assumption.

*Source: IMF Working Paper — section "5. Projects and gambles pay out" (content unit: _wp14116 - 5. Projects and gambles pay out).*

### 1. Banks set monitoring e§ort

### 1. Banks set monitoring e§ort

### Creditor information and monitoring incentives
- Model assumption: bank portfolios are revealed to the wholesale funding market with probability ' 2 (0;1).
- Bank maximization problem when creditor information may be revealed (equation (11)):
  - E[i] = 
    - (1') q_i (Rr d) + (1') (1q_i)  d (xr) + ' q_i (Rd)  c q_i^2 if Rr d  e + max{ d (xr); 0}
    - q_i (e)  c q_i^2 otherwise
  - Terms when bank quality remains private are multiplied by (1'); if bank quality is revealed only sound banks receive funding, captured by ' q_i (Rd).
- Reaction functions (equations (12) and r-expression):
  - q_i = [Rr d + ' d (r1)  (1')  d (xr)] / (2 c)
  - r = 1 / [q + (1q) ]
- Key comparative-static result:
  - q_i is monotonically increasing in '. A greater probability of information revelation at stage 4 increases banks' monitoring e§ort at stage 1.
  - Intuition: higher likelihood that asset quality affects funding costs reduces free-rider externalization of risk premia.

### Idiosyncratic bank soundness (heterogeneous bank portfolios)
- Extension: project soundness probability p_i = q_i + _i, where _i  U(0; a) with a 2 (0;1). Monitoring choice rescaled q_i 2 [0; (1a)].
- Bank expected profit (equation (14)):
  - E[i] = 
    - p_i (Rr d) + (1p_i)  d (xr)  c/(1a) q_i^2 if Rr d  e + max{ d (xr); 0}
    - p_i (e)  c/(1a) q_i^2 otherwise
- Interest rate expression (equation (15)):
  - r = 1 / (q + E[_i]) + (1(q + E[_i]))  = 1 / [q + a/2 + (1q  a/2) ]
- No-breakdown threshold (equation (16)):
  - q_b = max { 1 x / [x (1)] ; d / [R1 + d (1 x)  ] / (1) }
- Optimal monitoring (equation (17)):
  - q_i = (1a) / (2 c) * { (Rd) / [1 ( q + a/2 + (1q  a/2)  )] +  x }
- Comparative implications:
  - Increasing a (reducing correlation across bank exposures) lowers breakdown threshold bq.
  - Effect on q_i is ambiguous: larger a reduces funding pressures (lower r) which raises q_i, but larger idiosyncratic noise weakens the link between monitoring and default probability, which lowers q_i.
  - Net effect on monitoring and systemic risk is parameter-dependent; reducing correlation does not unambiguously reduce systemic crises.

### Retail depositors and funding composition
- Funding structure extension: of total funding 1, e from equity, w from wholesale funding, d from retail deposits; retail deposits are fully insured and pay the zero risk-free rate.
- Bank maximization problem with retail deposits (equation (18)):
  - E[i] =
    - q_i (Rr wd) + (1q_i)  ((w + d) x  r w  d)  c q_i^2 if Rr w  d  e + max{ ((w + d) x  r w  d); 0}
    - q_i (e)  c q_i^2 +  d (x1) otherwise
- No-breakdown condition adjusted (equation (19)):
  - q_b = max { 1 x / [x (1)] ; w [ (Rd  e)   [(w + d) x + d] ] / [ w (1) [(w + d) x  d] ]   / (1) }
- Optimal monitoring with retail funding (equation (20)):
  - q_i = (1 / (2 c)) * { R  (1) [ d + w q (1) +  ]   x (w + d) }
  - Constraint: [ q (1) +  ]  1
- Key findings:
  - Replacing wholesale with retail deposits (raising d and lowering w) increases bank optimal monitoring q_i.
  - bq declines when retail funding share increases; retail funding makes a funding-market freeze less likely.
  - Intuition: retail funding reduces the externality by making banks internalize more of the benefits of project soundness; the funding-market externality (one bank’s effort lowering all banks’ funding costs) weakens with retail funding.

### Conclusions and policy implications
- Interaction between moral hazard in monitoring and adverse selection in funding creates non-linear effects and multiplicity of equilibria; parameter changes can trigger sudden transitions into systemic crises.
- Three extension-based policy-relevant results:
  - Better information among bank creditors (higher ') alleviates risk of systemic freezes.
  - Reducing correlation among bank portfolios (higher a) does not necessarily make the funding market more robust; effects on monitoring are ambiguous.
  - Increasing the share of retail deposits in funding reduces systemic vulnerability of the wholesale market.
- Regulatory implications:
  - Regulators should maintain high awareness when the financial system appears stable; early-warning signals can be deceptively benign before a crisis.
  - Trade-off between microprudential and macroprudential objectives: lowering individual default risk may move the system closer to a systemic freeze threshold.
  - Regulatory transparency matters: if regulators have more information than wholesale financiers, public communication about system-wide distress (without bank-level identifiers) can produce sudden equilibrium shifts.
  - Macroprudential policy should support reducing reliance on wholesale funding, but systemic-risk taxation should not focus solely on reducing common exposures.

### Appendix: analytical thresholds and equilibrium cases (selected explicit expressions)
- No-breakdown condition (Lemma 2; equation (21) summarized):
  - If q > 1 x / [x (1) ] then no breakdown if and only if q  d [ R1 + d (1 x)   ] / (1).
  - Breakdown threshold bq:
    - bq = max { 1 x / [x (1)] ; d / [ R1 + d (1 x)   ] / (1) }
  - bq 2 (0;1).
- Equilibrium classification (Propositions 1 and 2):
  - Case I: only low-end corner equilibrium (q = 0).
  - Case II: one interior equilibrium plus high-end corner equilibrium; occurs if and only if
    - R > 2 c + d (1 +  (x  1))
  - Case III: two interior equilibria (multiplicity); occurs if and only if
    - R  2 c + d (1 +  (x  1)) and R > 2 sqrt{2 c d +  x d  2 c  / (1)} and [ q d / (2 c)   / (1) ] 2 [0;1]
  - Algebraic conditions derived in appendix show parameter regions for Cases I, II, III using exact expressions above.

*Source: _wp14116 - 1. Banks set monitoring e§ort*

### References

### References

### Systemic risk and prudential regulation
- Acharya, Viral V., 2009, "A Theory of Systemic Risk and Design of Prudential Bank Regulation,"Journal of Financial Stability, Vol. 5, pp. 224-255.
- Acharya, Viral V., Hamid Mehran, and Anjan V. Thakor, 2010, "Caught Between Scylla and Charybdis? Regulating Bank Leverage When There Is Rent Seeking and Risk Shifting," Federal Reserve Bank of New York Sta§Reports No. 469 (New York: Federal Reserve Bank).
- Acharya, Viral V., Lasse H. Pedersen, Thomas Philippon, and Matthew P. Richardson, 2012, "Measuring Systemic Risk," CEPR Discussion Paper No. 8824.
- Adrian, Tobias, and Markus K. Brunnermeier, 2011, "CoVaR," NBER Working Paper No. 17454 (Cambridge, MA: National Bureau of Economic Research).
- Arnold, Bruce, Claudio Borio, Luci Ellis, and Fariborz Moshirian, 2012. "Systemic Risk, Macroprudential Policy Frameworks, Monitoring Financial Systems and the Evolution of Capital Adequacy,"Journal of Banking and Finance,Vol. 36, pp. 3125-3132.
- Battiston, Stefano, Domenico D. Gatti, Mauro Gallegatti, Bruce C. Greenwald, and Joseph E. Stiglitz, 2012, "Liaisons Dangereuses: Increasing Connectivity, Risk Sharing, and Systemic Risk,"Journal of Economic Dynamics and Control, Vol. 36(8), pp. 1121ñ1141.
- Bisias, Dimitrios, Mark Flood, Andrew W. Lo and Stavros Valavanis, 2012, ìA Survey of Systemic Risk Analytics,î US Treasury, O¢ ce of Financial Research Working Paper No. 0001 (Washington, DC: US Department of the Treasury).
- Galati, Gabriele, and Richhild Moessner, 2012, ìMacroprudential Policy ñ A Literature Review,îJournal of Economic Surveys, Vol. 27(5), pp. 846-878.
- Perotti, Enrico, Lev Ratnovski, and Razvan Vlahu, 2011, "Capital Regulation and Tail Risk,"International Journal of Central Banking, Vol. 27, pp. 123-163.
- Zhou, Chen, 2013, "The Impact of Imposing Capital Requirements on Systemic Risk,"Journal of Financial Stability, Vol. 9(3), pp. 320-329.

### Interbank markets, liquidity provision, and central bank roles
- Acharya, Viral V., Denis Gromb, and Tanju Yorulmazer, 2012, "Imperfect Competition in the Interbank Market for Liquidity as a Rationale for Central Banking,"American Economic Journal: Macroeconomics, Vol. 4(2), pp. 184-217.
- Afonso, Gara, Anna Kovner, and Antoinette Schoar, 2013, "Trading Partners in the Inter-bank Lending Market," Federal Reserve Bank of New York Sta§ Report No. 620 (New York: Federal Reserve Bank).
- A¢ nito, Massimiliano, 2012, "Do Interbank Customer Relationships Exist? And How Did They Function in the Crisis? Learning from Italy,"Journal of Banking and Finance, Vol. 36(12), pp. 3163-3184.
- Aikman, David, Piergiorgio Alessandri, Bruno Eklund, Prasanna Gai, Sujit Kapadia, Elizabeth Martin, Nada Mora, Gabriel Sterne, and Matthew Willison, 2009, "Funding Liquidity Risk in a Quantitative Model of Systemic Stability," Bank of England Working Paper No. 372 (London: Bank of England).
- Borio, Claudio, 2014, "The Financial Cycle and Macroeconomics: What Have We Learnt,"Journal of Banking and Finance, in press.
- De Socio, Antonio, 2013, "The Interbank Market After the Financial Turmoil: Squeezing Liquidity in a ìLemons Marketîor Asking Liquidity ìOn Tap","Journal of Banking and Finance, Vol. 37, pp. 1340-1358.
- Diamond, Douglas, and Philip H. Dybvig, 1983, "Bank Runs, Deposit Insurance, and Liquidity,"Journal of Political Economy, Vol. 91, pp. 401-419.
- Diamond, Douglas W., and Raghuram G. Rajan, 2009, "The Credit Crisis: Conjectures about Causes and Remedies,"American Economic Review, Papers and Proceedings, pp. 606-610.
- Freixas, Xavier, Bruno M. Parigi and Jean-Charles Rochet, 2000, "Systemic Risk, Interbank Relations, and Liquidity Provision by the Central Bank,"Journal of Money, Credit and Banking, Vol. 32(3 - part 2), pp. 611-638.
- Freixas, Xavier, Bruno M. Parigi and Jean-Charles Rochet, 2004, "The Lender of Last Resort: A 21st Century Approach,"Journal of the European Economic Association, 2(6), pp. 1085-1115.
- Freixas, Xavier, and JosÈ Jorge, 2008, "The Role of Interbank Markets in Monetary Policy: a Model with Rationing,"Journal of Money, Credit and Banking, Vol.  40(6), pp. 1151-1176.
- Freixas, Xavier, Antoine Martin, and David R. Skeie, 2011, "Bank Liquidity, Interbank Markets, and Monetary Policy,"Review of Financial Studies, Vol. 24(8), pp. 2656-2692.
- Heider, Florian, and Marie Hoerova, 2009, "Interbank Lending, Credit-Risk Premia, and Collateral,"International Journal of Central Banking, Vol. 5(4), pp. 5-43.
- Heider, Florian, Marie Hoerova, and Cornelia Holthausen, 2009, "Liquidity Hoarding and Interbank Market Spreads: The Role of Counterparty Risk," ECB Working Paper No. 1126 (Frankfurt: European Central Bank).
- Rochet, Jean-Charles, and Jean Tirole, 1996, "Interbank Lending and Systemic Risk,"Journal of Money, Credit and Banking, Vol. 28(4 - Part 2), pp. 733-762.
- Rochet, Jean-Charles, and Xavier Vives, 2004, "Coordination Failures and the Lender of Last Resort: Was Bagehot Right After All?"Journal of the European Economic Association, Vol. 2(6), pp. 1116-1147.
- Upper, Christian, 2011, "Simulation Methods to Assess the Danger of Contagion in Inter-bank Markets,"Journal of Financial Stability, Vol. 7(3), pp. 111-125.

### Network structure, contagion, and simulation approaches
- Allen, Franklin, and Douglas Gale, 2000, "Financial Contagion,"Journal of Political Economy, Vol. 108, pp. 1-33.
- Allen, Franklin, Ana Babus, and Elena Carletti, 2009, "Financial Crises:  Theory and Evidence,"Annual Review of Financial Economics,Vol. 1, pp. 97-116.
- Allen, Franklin, and Ana Babus, 2010, "Networks in Finance," InThe Network Challenge, edited by Paul K. Kleindorfer and Yoram Wind (Prentice Hall: New Jersey).
- Anand, Kartik, Prasanna Gai, and Matteo Marsili, 2012, "Rollover Risk, Network Structure and Systemic Financial Crises,"Journal of Economic Dynamics and Control, Vol. 36(8), pp. 1088-1100.
- Castioglionesi, Fabio, and Wolf Wagner, 2013, "On the E¢ ciency of Bilateral Interbank Insurance,"Journal of Financial Intermediation, Vol. 22, pp. 177-200.
- Fecht, Falko, Hans Peter Gr ̧ner, and Philipp Hartmann, 2012, "Financial Integration, Specialization, and Systemic Risk,"Journal of International Economics, Vol. 88(1), pp. 150-161.
- Georg, Co-Pierre, 2013, "The E§ect of the Interbank Network Structure on Contagion and Common Shocks,"Journal of Banking and Finance, Vol. 37, pp. 2216-2228.
- Glasserman, Paul, and H. Peyton Young, 2014, "How Likely Is Contagion in Financial Networks?"Journal of Banking and Finance, In Press.
- Nier, Erlend W., Jing Yang, Tanju Yorulmazer, and Amadeo Alentorn, 2007, ìNetwork Models and Financial Stability,îJournal of Economic Dynamics and Control, Vol. 31, pp. 2033-2060.
- Garratt, Rodney J., Lavan Mahadeva, Katsiaryna Svirydzenka, 2014, "The Great Entanglement: The Contagious Capacity of the International Banking Network Just Before the 2008 Crisis,"Journal of Banking and Finance, In Press.
- FurÖne, Craig H., 2003, "Interbank Exposures: Quantifying the Risk of Contagion,"Journal of Money, Credit and Banking, Vol. 35(1), pp. 111-128.

### Liquidity, funding, market frictions, and price effects
- Brunnermeier, Markus K., 2009, "Deciphering the Liquidity and Credit Crunch 2007-2008,"Journal of Economic Perspectives,Vol. 23, pp. 77-100.
- Brunnermeier, Markus K., Andrew Crockett, Charles Goodhart, Avinash D. Persaud and Hyun Song Shin, 2009,The Fundamental Principles of Financial Regulation,Geneva Reports on the World Economy 11 (Geneva).
- Eisenschmidt, Jens, and Jens Tapking, 2009, "Liquidity Risk Premia in Unsecured Inter-bank Money Markets," ECB Working Paper No. 1025 (Frankfurt: European Central Bank).
- Ennis, Huberto M., and John A. Weinberg, 2013, "Over-the-Counter Loans, Adverse Selection, and Stigma in the Interbank Market,"Review of Economic Dynamics, Vol. 16, pp. 601-616.
- Fecht, Falko, Kjell G. Nyborg, and Jˆrg Rocholl, 2011, "The Price of Liquidity: The E§ects of Market Conditions and Bank Characteristics,"Journal of Financial Economics, Vol. 102(2), pp. 344-362.
- Flannery, Mark J., 1996, "Financial Crises, Payment Systems Problems, and Discount Window Lending,"Journal of Money, Credit and Banking, Vol. 28(4 - Part 2), pp. 804-824.
- Heider, Florian, Marie Hoerova, and Cornelia Holthausen, 2009, "Liquidity Hoarding and Interbank Market Spreads: The Role of Counterparty Risk," ECB Working Paper No. 1126 (Frankfurt: European Central Bank).
- Huang, Rocco, and Lev Ratnovski, 2011. "The Dark Side of Bank Wholesale Funding,"Journal of Financial Intermediation, Vol. 20(2), pp. 248ñ263.
- LÛpez-Espinosa, Germ·n, Antonio Moreno, Antonio Rubia, and Laura Valderrama, 2012, "Short-Term Wholesale Funding and Systemic Risk: A Global CoVaR Approach,"Journal of Banking and Finance, Vol. 36, pp. 3150-3162.
- Ratnovski, Lev, 2013, "Liquidity and Transparency in Bank Risk Management,"Journal of Financial Intermediation, Vol. 22(3), pp. 422-439.
- Shleifer, Andrei, and Robert W. Vishny, 2011, "Fire Sales in Finance and Macroeconomics,"Journal of Economic Perspectives, Vol. 25(1), pp. 29-48.

### Information frictions, incentives, and theoretical foundations
- Akerlof, George, 1970, "The Market for "Lemons": Quality Uncertainty and the Market Mechanism,"Quarterly Journal of Economics, Vol. 84(3), pp. 488-500.
- Bhattacharya, Sudipto, and Douglas Gale, 1987, "Preference Shocks, Liquidity and Central Bank Policy," InNew Approaches to Monetary Economics, edited by William A. Barnett and Kenneth J. Singleton (Cambridge: Cambridge University Press).
- Goldstein, Itay, and Ady Pauzner, 2005, "Demand-Deposit Contracts and the Probability of Bank Runs,"Journal of Finance, Vol. 60(3), pp. 1293-1327.
- Myers, Stewart C., and Nicholas S. Majluf, 1984, "Corporate Financing and Investment Decisions When Firms Have Information Investors Do Not Have,"Journal of Financial Economics, Vol. 13, pp. 187-221.
- Keeley, Michael C., 1990, "Deposit Insurance, Risk and Market Power in Banking,"American Economic Review, Vol. 80(5), pp. 1183-1200.
- Goldstein, Itay, and Ady Pauzner, 2005, "Demand-Deposit Contracts and the Probability of Bank Runs,"Journal of Finance, Vol. 60(3), pp. 1293-1327.
- Ibragimov, Rustam, Dwight Ja§ee, and Johan Walden, 2010, "DiversiÖcation Disasters,"Journal of Financial Economics, Vol. 99, pp. 333-348.
- Rochet, Jean-Charles, and Jean Tirole, 1996, "Interbank Lending and Systemic Risk,"Journal of Money, Credit and Banking, Vol. 28(4 - Part 2), pp. 733-762.
- Rosser, J. Barkley, Jr., 1999, "On the Complexities of Complex Economic Dynamics,"Journal of Economic Perspectives, Vol. 13(4), pp. 169-192.
- Stahl, Dale O., 1988, "Bertrand Competition for Inputs and Walrasian Outcomes,"American Economic Review, Vol. 78(1), pp. 189-201.
- Thakor, Anjan V., 1996, "Capital Requirements, Monetary Policy, and Aggregate Bank Lending: Theory and Empirical Evidence,"Journal of Finance, Vol. 51, pp. 279-324.
- Wagner, Wolf, 2010, "DiversiÖcation at Financial Institutions and Systemic Crises,"Journal of Financial Intermediation, Vol. 19, pp. 373-386.

*Source: _wp14116 - References*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14116.pdf_