## _wp07104

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---

### Introduction: moral hazard framing and objective
- Moral hazard defined: provision of insurance increases probability of insured event due to diminished preventive incentives; necessary condition involves asymmetric information or inability of insurer to respond fully.
- Two moral hazard channels discussed:
  - Debtor moral hazard: borrowers pursue riskier policies expecting IMF bailout.
  - Creditor (investor) moral hazard (focus of paper): private creditors underprice lending risks expecting IMF bailouts.
- Key theoretical points (preserve text terminology):
  - Lane and Phillips (2000), Jeanne and Zettelmeyer (2001): IMF resources/subsidies not large enough to create serious moral hazard.
  - Mussa (1999, 2004) and the Mussa theorem (Jeanne and Zettelmeyer (2005)): if IMF lending entails no expected losses (actuarially fair), there is no expected transfer and thus no moral hazard; IMF can reduce real hazard via better enforcement/conditionality, lowering private risk premia without moral hazard.
- Empirical challenge: changes in spreads or capital flows are observationally equivalent to optimal responses to reduced real hazard, complicating empirical tests.
- Paper approach: build a calibrated model focusing on investor moral hazard with IMF lending assumed unconditional to disentangle liquidity support from conditionality.

### Model setup and scenarios
- Time structure and financing:
  - Periods: 0, 1, 2.
  - Country invests k in period 0.
  - Short-term debt finances kδ (maturing period 1), long-term finances (1)kδ− (maturing period 2).
  - Short-term debt must be rolled over in period 1; long-term is locked in.
- Production, liquidation, and productivity:
  - If investment partially liquidated at end of period 1, remaining output: y(1, k) = exp(θ) k1 where k1 ≤ k and ω=1 if k1=k, ω=ρ<1 if k1<k.
  - Productivity θ ~ Normal(mean μ, variance 2σ).
  - Liquidation is never efficient because ρ<1.
- Creditors, signals, and IMF features:
  - Continuum of private investors (mass normalized to 1); near risk neutral with maximum actuarially fair risk premium cap r.
  - Short-term investors can run in period 1 triggering liquidation; IMF may provide crisis lending.
  - Recovery values: liquidation value per unit = λ < 1; country can pledge fraction α of output to creditors.
  - IMF and private investors receive noisy signal q = θ + ε where ε ~ Normal(0, 2τσ).
  - IMF lends unconditionally in baseline; IMF credit senior to private claims; IMF lending subject to same interest rate ceiling r.
- Scenarios calibrated and compared:
  - i) Laissez-faire (no IMF, β = 0).
  - ii) IMF bailout with no investor moral hazard (IMF lends at actuarially fair rate, γ = 0).
  - iii) IMF-induced investor moral hazard (IMF lending subsidized, implicit transfer γ > 0).

### Equilibrium characterization (no IMF vs. with IMF)
- Without IMF (notation preserved):
  - Short-term (rolled) interest rate denoted 0 SNO r; long-term 0 LNO r (annualized 1/2(1+0 LNO r)−1).
  - Productivity thresholds:
    - *θ and **θ as defined in equations (2) and (2a) determine repayment regions.
  - Zero-profit rollover condition for short-term investors (equation (3)) yields SNO R(q, θ, ψ) when solution exists.
  - Rollover threshold NO q(θ, ψ) solves (1 + SNO R) = r at the minimum signal below which no fair rollover rate ≤ r exists.
  - Crisis/default probability in period 0: DNO p and CNO p defined by integration over signals up to NO q.
  - Ex ante zero-profit for short-term investors (period 0): equation (4).
  - Long-term zero-profit condition: equation (5) determines NO ψ; long-term repayments depend on whether a run occurred.
  - Equilibrium without the IMF summarized by (6): interest rates, crisis/default probabilities, and debt structure (kδ short term, (1)kδ− long term).
- With IMF (preserve notation and definitions):
  - IMF lending fraction β: β=1 full bailout (no default if IMF lends), 0<β<1 partial bailout (fraction β of short-term investors bailed out).
  - IMF lending rate F R determined by IMF zero-profit condition (equation (8)) when γ = 0; if γ>0 IMF lends at subsidized rate per (8a).
  - Productivity thresholds with IMF: F θ and *F θ defined in (7) and (7a).
  - IMF lending introduces solvency risk measure φ = /{ [1 − (1−β)ω]βδ } and φδ used to assess feasibility of partial bailout equilibrium.
  - Under full bailout (β = 1) and positive long-term debt, IMF can lend at actuarially fair rate and reduce crisis probability without inducing moral hazard (Mussa theorem).
  - Equilibrium with IMF summarized by (11): SIMF R, LIMF r, IMF q, F q, crisis/default probabilities CIMF p and DIMF p, and ex ante zero-profit conditions (9) and (10).
  - With implicit transfers γ > 0 (IMF subsidy), investor moral hazard equilibrium (superscript MH) lowers risk premia and crisis probabilities further.

### Key model findings (analytical and numerical)
- Analytical comparative statics (γ = 0):
  - Short-term premium and default probability always lower with full IMF bailout compared to without IMF because inefficient default is avoided.
  - Possibility of IMF support lowers short-term premium; long-term premium ambiguous due to subordinated private claims vs. reduced crisis likelihood.
  - Country’s average borrowing cost (weighted short- and long-term) is lower with IMF because short-term premium reduction outweighs long-term premium effects.
  - Spread between short- and long-term rates increases with IMF even if both decline in absolute terms.
  - Risk premia sensitivity statements preserved as presented: /0,/0,/0,and/0, , jjjj rrrr jSL δ ρ μ    σ∂∂> ∂∂≤ ∂∂<∂∂>  = — interpreted in text as risk premia lower with smaller short-term share and output cost, and stronger fundamentals.
  - IMF more effective in crisis prevention the larger the short-term debt and the weaker the fundamentals (conjectures with partial derivatives reported in text).
- Non-monotonic and solvency-liquidity tradeoffs:
  - Size of IMF lending β can have non-monotonic (U-shaped or Laffer-curve like) effects on short-term and long-term premiums and on crisis probability because larger β raises IMF solvency risk via seniority.
  - Equilibrium may not exist for sufficiently large partial β because IMF solvency risk becomes too large (ceiling β implied by φδ).
- Moral hazard (γ > 0) effects:
  - IMF-induced investor moral hazard lowers risk premia and crisis probability further than actuarially fair IMF lending.
  - Moral hazard component accounts for a small portion of total reduction in premia except when short-term debt share δ is high.
  - Effect of investor moral hazard on premia relatively insensitive to changes in fundamentals.

### Calibration setup and benchmark parameters (preserve sequence as given)
- Benchmark parameter values (sequence as in source): 0.8, 1, 0.5, 0, and 0.6 r α μ σ δ ρ λ τ γ ======== =
- Parameter variation exercises:
  - Exercise 1: vary δ and ρ between 0.25 and 0.75 (with μ = σ = 1, 0.5 λ τ =).
  - Exercise 2: vary μ and σ between 0.75 and 1.25.
  - Exercise 3: vary τ between 0.25 and 0.75 and σ as in exercise 2.
  - Exercise 4: quantify IMF-induced moral hazard with γ = 0.1 (10 percent implicit transfer).
- IMF bailout sizes considered: β ∈ {0.05, 0.10, 0.25, 0.50, 0.75}, plus β = 0 (no IMF) and β = 1 (full bailout).

### Major numerical results and reported table excerpts (preserve values exactly)
- 1. Short-term premium series and excerpts:
  - "1. Short-term premium 1/0.005.787.529.525.587.319.365.427.149.23"
  - Additional short-term premium entries (selected): 0.055.467.10 ; 9.11 ; 5.276.898.815.126.728.66 ; 0.105.15 ; 6.77 ; 9.514.976.48 ; 8.45 ; 4.826.318.15 ; 0.254.347.27...4.08 ; 5.56 ; 8.983.955.20 ; 7.38 ; 0.50 ; 4.16 ; 1.000.171.715.380.171.715.380.171.715.38
- Long-term and average premiums (selected excerpts):
  - "2. Long-term premium 1/0.008.339.4410.647.237.998.786.306.757.21"
  - "0.058.429.7811.507.308.249.696.356.957.92"
  - "3. Average premium 1/ 2/0.007.698.489.806.827.659.226.086.958.73"
  - "0.057.688.44 ; 9.71 ; 6.807.579.036.056.848.47 ; 1.004.744.766.484.734.656.264.724.546.04"
- Crisis and default probabilities (selected excerpts):
  - "4. Crisis Probability 3/0.0010.3613.0715.9910.0512.7615.779.7912.5015.59"
  - "0.0510.3613.0715.9910.0412.7515.749.7812.4815.53"
  - "1.009.4311.9315.009.4211.8914.919.4211.8514.82"
  - "5. Default Probability 4/0.0010.3613.0715.9910.0512.7615.779.7912.5015.59"
  - "0.050.000.164.450.000.010.530.000.000.11"
  - "0.100.031.5015.940.000.133.160.000.020.92"
  - "1.000.343.309.710.343.309.710.343.309.71"
- Solvency risk matrix ( φδ ) excerpt:
  - Row 1: "0.050.070.190.520.030.100.260.020.060.17"
  - Row 2: "0.100.130.360.920.060.180.460.040.120.31"
  - Row 3: "0.250.310.801.710.150.400.860.100.270.57"
  - Row 4: "0.500.571.332.400.290.671.200.190.440.80"
  - Row 5: "0.750.801.712.770.400.861.380.270.570.92"
- Selected panel/table excerpts:
  - Table 2 short-term premium excerpt: "1. Short-term premium 1/0.007.932.851.0015.387.313.5622.2012.437.16 ; 0.057.462.700.9514.366.893.3720.5911.656.75"
  - Table 3 short-term premium excerpt: "1. Short-term premium 2/0.0011.8619.7025.791.344.147.6010.5215.5618.20"
  - Table 3 long-term premium excerpt: "2. Long-term premium 2/0.006.9511.8015.844.528.5812.382.433.223.46"
  - Table 4 Panel A short-term premium excerpt: "1. Short-term premium 1/0.255.685.495.325.615.415.250.070.070.07"
  - Table 4 Panel A long-term premium excerpt: "2. Long-term premium 1/0.252.231.140.212.070.980.060.160.160.15"
  - Table 4 Panel B short-term premium excerpt: "1. Short-term premium 1/0.757.3313.0417.306.9011.8815.300.431.162.00"
  - Table 4 Panel B long-term premium excerpt: "2. Long-term premium 1/0.751.382.212.570.901.521.780.480.690.79"
- Notation and table footnotes preserved:
  - "1/ Annualized."
  - "2/ Weighted by the maturity share of total debt."
  - "3/ Also refers to the probability of a default  p D under a partial bailout by the IMF."
  - "4/ Probability of a default with no bailout by the IMF denoted by p DNB in the text."
  - "Difference (A-B) 1/ ; Without the IMF (A) Full Bailout by the IMF (B) Moral Hazard (B) Total Reduction (A+B) Reduced Real Hazard (A)"

### Appendix I: Equilibrium solutions with no informational uncertainty (high-level highlights)
- Perfectly informative signal assumption: 0τ=.
- Without the IMF:
  - Rollover interest rate for short-term debt equals zero risk-free interest rate: 1 11 SS Rr≡+ =.
  - Productivity threshold equation (A.I.1) and rollover/default probability expressions reduce to closed-form integrals over θ (see (A.I.1)–(A.I.5) formulations).
  - Crisis probability equals default probability without the IMF: DC pp=.
- With the IMF:
  - IMF lending rate equals zero risk-free rate: 1 11 FF Rr≡+=.
  - Thresholds reduce to expressions in (A.I.6) and default probability defined in (A.I.7).
  - Ex ante zero-profit conditions become multidimensional integral conditions ((A.I.8), (A.I.9)).
  - Equilibrium with IMF summarized in (A.I.10).

### Appendix II: Calibration and numerical solution approach (high-level highlights)
- Posterior mean and variance expressions:
  - μ() ()/(1 ) qq μτμτ=+ + and variance 22 ()/(1 ) qστστ=+.
- Substitution used: 1 /exp() S kdαθ≡−.
- Key calibration equations (as presented) preserved:
  - (A.II.1) short-term integral condition.
  - (A.II.2) crisis threshold determination: () 1 S Rqr = +.
  - (A.II.3) short-term zero-profit integral condition.
  - (A.II.4) long-term zero-profit compact form.
- With the IMF (no investor moral hazard) calibration conditions:
  - (A.II.5) IMF ex ante zero-profit condition.
  - (A.II.6) IMF lending threshold: () 1 F F Rqr = +.
  - (A.II.7) short-term investors' condition with IMF (multidimensional integral).
  - (A.II.8) long-term investors' condition with IMF.
  - With IMF-induced investor moral hazard equation (A.II.9) replacing (A.II.5).

### Policy implications and recommendations (preserve original emphasis)
- IMF can prevent liquidity crises and inefficient defaults, reduce short-term premia, and lower average borrowing costs without inducing investor moral hazard if it lends at actuarially fair rates (Mussa theorem).
- Design of IMF lending instruments without ex post conditionality (e.g., Reserve Augmentation Line-type instruments) should:
  - Restrict eligibility to members with relatively strong fundamentals and sustainable debt to minimize IMF-induced investor moral hazard.
  - Consider appropriate access limits: larger-scale partial bailouts are not necessarily more effective for crisis prevention than smaller-scale ones because of solvency-liquidity tradeoffs and potential non-monotonicity in risk premia.
- IMF seniority in lending matters: it can help coordinate creditor behavior and reduce inefficient runs, but can also raise long-term premia by subordinating private claims—policy design should account for this tradeoff.
- Transparency and informational issues:
  - The model suggests complex interactions: improving transparency (reducing τ) can, in this framework, increase coordination-failure risk and premia; policy conclusions regarding transparency should consider the specific nature of coordination failure and informational structures.

### Conclusion (preserve textual conclusions)
- Main conclusion: unconditional IMF lending can play a useful role in crisis prevention and in reducing borrowing costs without necessarily causing investor moral hazard provided IMF lending is actuarially fair and eligibility/access are prudently designed.
- IMF-induced investor moral hazard is quantitatively likely to be small relative to the insurance (welfare-enhancing) benefits of the IMF, especially for countries with strong fundamentals and limited short-term debt.
- Implication for RAL-like instruments: exceptional and upfront access may be justified if restricted to members with relatively strong fundamentals and sustainable debt.

*Source: _wp07104 - Appendix I and II excerpts, calibration results, tables, and references as provided.*

### Appendix I.  Equilibrium Solutions with No Informational Uncertainty ................................28

### Appendix I.  Equilibrium Solutions with No Informational Uncertainty ................................28

### A. Without the IMF......................................................................................................28
- Section title as listed: "Without the IMF"
- Page reference as listed: "28"

### B. With the IMF ...........................................................................................................29
- Section title as listed: "With the IMF"
- Page reference as listed: "29"

*Source: _wp07104 - Appendix I.  Equilibrium Solutions with No Informational Uncertainty ................................28*

### References..............................................................................................................

### _wp07104 - References..............................................................................................................

### Introduction: moral hazard framing and objective
- Moral hazard defined: provision of insurance increases probability of insured event due to diminished preventive incentives; necessary condition involves asymmetric information or inability of insurer to respond fully.
- Two moral hazard channels discussed:
  - Debtor moral hazard: borrowers pursue riskier policies expecting IMF bailout.
  - Creditor (investor) moral hazard (focus of paper): private creditors underprice lending risks expecting IMF bailouts.
- Key theoretical points:
  - Lane and Phillips (2000), Jeanne and Zettelmeyer (2001): IMF resources/subsidies not large enough to create serious moral hazard.
  - Mussa (1999, 2004) and the Mussa theorem (Jeanne and Zettelmeyer (2005)): if IMF lending entails no expected losses (actuarially fair), there is no expected transfer and thus no moral hazard; IMF can reduce real hazard via better enforcement/conditionality, lowering private risk premia without moral hazard.
- Empirical challenge: changes in spreads or capital flows are observationally equivalent to optimal responses to reduced real hazard, complicating empirical tests.
- Alternative approach of this paper: build a simple calibrated model focusing on investor moral hazard with IMF lending assumed unconditional to disentangle liquidity support from conditionality.

### Model setup and scenarios
- Periods: 0, 1, 2.
- Investment and financing:
  - Country invests k in period 0.
  - Short-term debt finances kδ (maturing period 1), long-term finances (1)kδ− (maturing period 2).
  - Short-term debt must be rolled over in period 1; long-term is locked in.
- Liquidation and production:
  - If investment partially liquidated at end of period 1, remaining output: y(1, k) = exp(θ) k1 where k1 ≤ k and ω=1 if k1=k, ω=ρ<1 if k1<k.
  - Productivity θ ~ Normal(mean μ, variance 2σ).
  - Liquidation is never efficient because ρ<1.
- Creditors and constraints:
  - Continuum of private investors (mass normalized to 1); near risk neutral with maximum actuarially fair risk premium cap r.
  - Short-term investors can run in period 1 triggering liquidation; IMF may provide crisis lending.
  - Recovery values: liquidation value per unit = λ < 1; country can pledge fraction α of output to creditors.
  - IMF and private investors receive noisy signal q = θ + ε where ε ~ Normal(0, 2τσ).
  - IMF lends unconditionally in baseline; IMF credit senior to private claims; IMF lending subject to same interest rate ceiling r.
- Scenarios calibrated and compared:
  - i) Laissez-faire (no IMF, β = 0).
  - ii) IMF bailout with no investor moral hazard (IMF lends at actuarially fair rate, γ = 0).
  - iii) IMF-induced investor moral hazard (IMF lending subsidized, implicit transfer γ > 0).

### Equilibrium characterization (no IMF vs. with IMF)
- Without IMF:
  - Short-term (rolled) interest rate denoted 0 SNO r; long-term 0 LNO r (annualized 1/2(1+0 LNO r)−1).
  - Productivity thresholds:
    - *θ and **θ as defined in equations (2) and (2a) determine repayment regions.
  - Zero-profit rollover condition for short-term investors (equation (3)) yields SNO R(q, θ, ψ) when solution exists.
  - Rollover threshold NO q(θ, ψ) solves (1 + SNO R) = r at the minimum signal below which no fair rollover rate ≤ r exists.
  - Crisis/default probability in period 0: DNO p and CNO p defined by integration over signals up to NO q.
  - Ex ante zero-profit for short-term investors (period 0): equation (4).
  - Long-term zero-profit condition: equation (5) determines NO ψ; long-term repayments depend on whether a run occurred.
  - Equilibrium without the IMF summarized by (6): interest rates, crisis/default probabilities, and debt structure (kδ short term, (1)kδ− long term).
- With IMF (prospect of possible IMF support; IMF lending amount 1 S Ldβ= where 01β≤≤):
  - IMF lending fraction β: β=1 full bailout (no default if IMF lends), 0<β<1 partial bailout (fraction β of short-term investors bailed out).
  - IMF lending rate F R determined by IMF zero-profit condition (equation (8)) when γ = 0; if γ>0 IMF lends at subsidized rate per (8a).
  - Productivity thresholds with IMF: F θ and *F θ defined in (7) and (7a).
  - IMF lending introduces solvency risk measure φ = /{ [1 − (1−β)ω]βδ } (defined in text) and φδ used to assess feasibility of partial bailout equilibrium.
  - Under full bailout (β = 1) and positive long-term debt, IMF can lend at actuarially fair rate and reduce crisis probability without inducing moral hazard (Mussa theorem).
  - Equilibrium with IMF summarized by (11): SIMF R, LIMF r, IMF q, F q, crisis/default probabilities CIMF p and DIMF p, and ex ante zero-profit conditions (9) and (10).
  - With implicit transfers γ > 0 (IMF subsidy), investor moral hazard equilibrium (superscript MH) lowers risk premia and crisis probabilities further.

### Key model findings (analytical and numerical/conjectural)
- Analytical / comparative statics (no investor moral hazard, γ = 0):
  - Short-term premium and default probability always lower with full IMF bailout compared to without IMF because inefficient default is avoided.
  - Possibility of IMF support lowers short-term premium; long-term premium ambiguous due to subordinated private claims vs. reduced crisis likelihood.
  - Country’s average borrowing cost (weighted short- and long-term) is lower with IMF because short-term premium reduction outweighs long-term premium effects.
  - Spreads: the spread between short- and long-term rates (yield curve steepness) increases with IMF even if both decline in absolute terms.
  - Risk premia:
    - Decrease with stronger fundamentals μ ( /0 j r μ ∂∂>).
    - Increase with higher volatility σ ( /0 j r σ ∂∂< actually text states sign pattern as /0,/0,/0,and/0,... with directions; preserve exact textual inequalities: /0,/0,/0,and/0, , jjjj rrrr jSL δ ρ μ    σ∂∂> ∂∂≤ ∂∂<∂∂>  = — interpreted as risk premia lower with smaller short-term share and output cost, and stronger fundamentals).
  - IMF role in crisis prevention:
    - IMF can reduce crisis probability even with unconditional lending (first inequality in conjecture set).
    - IMF more effective in crisis prevention the larger the short-term debt and the weaker the fundamentals (conjectures: CC p∆ δ ρ μ σ derivatives given in text).
  - Non-monotonic effects under partial bailouts:
    - Size of IMF lending β can have non-monotonic (U-shaped or Laffer-curve like) effects on short-term and long-term premiums and on crisis probability due to tradeoff between liquidity support and increased solvency risk from IMF seniority.
    - Equilibrium may not exist for sufficiently large partial β because IMF solvency risk becomes too large (ceiling β implied by φδ).
- Calibration setup and benchmark:
  - Benchmark parameter values: 0.8, 1, 0.5, 0, and 0.6 r α μ σ δ ρ λ τ γ ======== =  (parameters listed in source exactly in that sequence).
  - Parameter variation in exercises:
    - Exercise 1: vary δ and ρ between 0.25 and 0.75 (with μ = σ = 1, 0.5 λ τ =).
    - Exercise 2: vary μ and σ between 0.75 and 1.25.
    - Exercise 3: vary τ between 0.25 and 0.75 and σ as in exercise 2.
    - Exercise 4: quantify IMF-induced moral hazard with γ = 0.1 (10 percent implicit transfer).
  - IMF bailout sizes considered: β ∈ {0.05, 0.10, 0.25, 0.50, 0.75}, plus β = 0 (no IMF) and β = 1 (full bailout).
- Calibration results (summarized findings):
  - Exercise 1 (debt structure & cost of default): short-term premiums and crisis probabilities lower with IMF (β>0), lowest under full bailout β = 1; reduction in short-term premium under full bailout substantial, crisis probability reduction limited (less than two percentage points in Table 1 benchmark cases but larger reductions for weaker fundamentals).
  - Default probabilities DNB p often lower with smaller-scale partial bailouts (β ≤ 0.1) than with full bailout because IMF repayment risk relative to post-liquidation debt-servicing capacity can be smaller for small β.
  - Long-term premiums typically higher under partial bailouts and sometimes higher even under full bailout if δ and ρ large; spread between long- and short-term rates increases with IMF.
  - Non-monotonicity: short-term premium often U-shaped in β for δ ≥ 0.5; long-term premium moves in opposite direction; average borrowing cost tends to follow short-term premium and usually lower with IMF.
  - Exercise 2 (fundamentals): risk premia and crisis/default probabilities higher with weaker fundamentals; IMF dampens sensitivity of premia to fundamentals; IMF impact larger when fundamentals weak (example: for σ = 1.0, reducing μ from 0.75 to 0.5 increased full-bailout crisis probability reduction from <3 percentage points to 10 percentage points).
  - Exercise 3 (informational vs fundamental uncertainty): premia more sensitive to output volatility σ than to signal noisiness τ; counterintuitively, increased informational uncertainty (larger τ) reduced premia and crisis probabilities in this model (because perfect signal precision raises coordination failure risk under the model’s coordination failure mechanism).
  - Exercise 4 (quantifying moral hazard, γ = 0.1):
    - IMF-induced investor moral hazard (implicit transfers 10 percent) lowers risk premia and crisis probability further than actuarially fair IMF lending.
    - Moral hazard component accounts for a small portion of total reduction in premia except when short-term debt share δ is high (Panel A).
    - Effect of investor moral hazard on premia relatively insensitive to changes in fundamentals (Panel B).
    - Combined with Zettelmeyer and Joshi (2005) estimates (implicit transfers empirically far smaller than 10 percent), suggests IMF-induced investor moral hazard unlikely to be a major concern when fundamentals are strong and short-term debt small.

### Policy implications and recommendations
- IMF can prevent liquidity crises and inefficient defaults, reduce short-term premia, and lower average borrowing costs without inducing investor moral hazard if it lends at actuarially fair rates (Mussa theorem).
- Design of IMF lending instruments without ex post conditionality (e.g., Reserve Augmentation Line-type instruments) should:
  - Restrict eligibility to members with relatively strong fundamentals and sustainable debt to minimize IMF-induced investor moral hazard.
  - Consider appropriate access limits: larger-scale partial bailouts are not necessarily more effective for crisis prevention than smaller-scale ones because of solvency-liquidity tradeoffs and potential non-monotonicity in risk premia.
- IMF seniority in lending matters: it can help coordinate creditor behavior and reduce inefficient runs, but can also raise long-term premia by subordinating private claims—policy design should account for this tradeoff.
- Transparency and informational issues:
  - The model suggests complex interactions: improving transparency (reducing τ) can, in this framework, increase coordination-failure risk and premia; policy conclusions regarding transparency should consider the specific nature of coordination failure and informational structures.

### Conclusion
- Main conclusion: unconditional IMF lending can play a useful role in crisis prevention and in reducing borrowing costs without necessarily causing investor moral hazard provided IMF lending is actuarially fair and eligibility/access are prudently designed.
- IMF-induced investor moral hazard is quantitatively likely to be small relative to the insurance (welfare-enhancing) benefits of the IMF, especially for countries with strong fundamentals and limited short-term debt.
- Implication for RAL-like instruments: exceptional and upfront access may be justified if restricted to members with relatively strong fundamentals and sustainable debt.

*Source: _wp07104 - References (excerpts and calibration results) — content as provided.*

### 1. Short-term premium 1/0.005.787.529.525.587.319.365.427.149.23

### _wp07104 - 1. Short-term premium 1/0.005.787.529.525.587.319.365.427.149.23

### Major numerical results and reported tables
- Table entries (selected excerpts as presented):
  - 1. Short-term premium 1/0.005.787.529.525.587.319.365.427.149.23
  - 0.055.467.10
  - 9.11
  - 5.276.898.815.126.728.66
  - 0.105.15
  - 6.77
  - 9.514.976.48
  - 8.45
  - 4.826.318.15
  - 0.254.347.27...4.08
  - 5.56
  - 8.983.955.20
  - 7.38
  - 0.50
  - 4.16
  - 1.000.171.715.380.171.715.380.171.715.38

- Long-term and average premiums (selected excerpts):
  - 2. Long-term premium 1/0.008.339.4410.647.237.998.786.306.757.21
  - 0.058.429.7811.507.308.249.696.356.957.92
  - 3. Average premium 1/ 2/0.007.698.489.806.827.659.226.086.958.73
  - 0.057.688.44
  - 9.71
  - 6.807.579.036.056.848.47
  - 1.004.744.766.484.734.656.264.724.546.04

- Crisis and default probabilities (selected excerpts):
  - 4. Crisis Probability 3/0.0010.3613.0715.9910.0512.7615.779.7912.5015.59
  - 0.0510.3613.0715.9910.0412.7515.749.7812.4815.53
  - 1.009.4311.9315.009.4211.8914.919.4211.8514.82
  - 5. Default Probability 4/0.0010.3613.0715.9910.0512.7615.779.7912.5015.59
  - 0.050.000.164.450.000.010.530.000.000.11
  - 0.100.031.5015.940.000.133.160.000.020.92
  - 1.000.343.309.710.343.309.710.343.309.71

- Solvency risk matrix excerpt:
  - 6. Solvency risk ( φδ )
  - 0.050.070.190.520.030.100.260.020.060.17
  - 0.100.130.360.920.060.180.460.040.120.31
  - 0.250.310.801.710.150.400.860.100.270.57
  - 0.500.571.332.400.290.671.200.190.440.80
  - 0.750.801.712.770.400.861.380.270.570.92

- Panel and scenario slices (selected excerpts from Tables 2–4 and Table 4 panels):
  - Table 2: "Risk Premium and Crisis Probability: Effect of Economic Fundamentals (0.5 δρλτ====; in percent)"
  - Example short-term premium entries for β grid:
    - 1. Short-term premium 1/0.007.932.851.0015.387.313.5622.2012.437.16
    - 0.057.462.700.9514.366.893.3720.5911.656.75
  - Table 3: "Risk Premium and Crisis Probability: Effect of Informational Uncertainty (1 μβ==,                      0.5δρλ===; in percent unless otherwise indicated)"
    - 1. Short-term premium 2/0.0011.8619.7025.791.344.147.6010.5215.5618.20
    - 2. Long-term premium 2/0.006.9511.8015.844.528.5812.382.433.223.46
  - Table 4 Panel A: "Effect of Short-term Debt and Cost of Default (1 μσβ===, 0.5λτ==; in percentage points)"
    - 1. Short-term premium 1/0.255.685.495.325.615.415.250.070.070.07
    - 2. Long-term premium 1/0.252.231.140.212.070.980.060.160.160.15
  - Table 4 Panel B: "Effect of Economic Fundamentals (1 β=, 0.5δρλτ====; in percentage points)"
    - 1. Short-term premium 1/0.757.3313.0417.306.9011.8815.300.431.162.00
    - 2. Long-term premium 1/0.751.382.212.570.901.521.780.480.690.79

- Notation and footnotes present in tables:
  - 1/ Annualized.
  - 2/ Weighted by the maturity share of total debt.
  - 3/ Also refers to the probability of a default  p D under a partial bailout by the IMF.
  - 4/ Probability of a default with no bailout by the IMF denoted by p DNB in the text.
  - Difference (A-B) 1/
  - Without the IMF (A) Full Bailout by the IMF (B)
  - Moral Hazard (B)
  - Total Reduction (A+B) Reduced Real Hazard (A)

### Analytical framework and equilibrium characterizations (Appendix I highlights)
- Perfectly informative signal assumption: 0τ=.
- Without the IMF:
  - Rollover interest rate for short-term debt equals zero risk-free interest rate: 1 11 SS Rr≡+ =.
  - Productivity threshold equation (A.I.1):
    - * ln(1)θθψ=+ +
  - Probability of a rollover crisis (A.I.2):
    - *() C pgd θ θθ −∞ = ∫
  - Crisis probability equals default probability without the IMF: DC pp=.
  - Ex ante zero-profit condition for short-term investors (A.I.3a):
    - 1 (1) SD D D kdpkpδλδ=−+
  - Reformulated integral condition (A.I.3b):
    - * * exp( )( )( ) gdgd θ θ δα θ θθ λδ θθ ∞ −∞ = + ∫∫
  - Long-term debt repayment structure uses thresholds *θ, **θ and minθ:
    - min 2 2 1 if otherwise (,) L L d DS yk θθ αθ ≥ ⎧ = ⎨ ⎩
  - Ex ante zero-profit condition for long-term investors (A.I.4):
    - min min (1)exp( )( )(1)exp( ) ( ) gdgd θ θ δαψ θ θ θθ αρ δθ θθ ∞ −∞ − = + − ∫∫
  - Equilibrium without IMF characterized (A.I.5):
    - * 00 1( / ) exp(),1( /(1)) exp(), () S NONOL NONONO DNOCNO rr ppgd θ αδ θα δ θ ψ θθ −∞ += += − == ∫

- With the IMF:
  - IMF lending rate equals zero risk-free rate: 1 11 FF Rr≡+=.
  - Thresholds reduce to (A.I.6):
    - * lnandln (1/ ) FF θθφ θ θ ψβ φ=+=+ + +
  - Default probability definition with IMF (A.I.7):
    - if1 () otherwise F DNB D C pgd p p θ β θθ −∞ ⎧ = = ⎪ = ⎨ ⎪ ⎩ ∫
  - Ex ante zero-profit condition for short-term investors with IMF (A.I.8):
    - ** * exp( )( )( )(1)( )( ) F FF gdgdgdgd θθθ θθθ δα θ θθβ θθ λδ β θ θθθ ∞ −∞ ⎡⎤ ⎡⎤ = + + − + ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ∫∫∫∫
  - Long-term repayment with IMF defined piecewise using ˆF θ and minθ and expected repayment enters long-term zero-profit condition (A.I.9).
  - Equilibrium with IMF summarized (A.I.10):
    - * 00 1( / ) exp(), 1( /(1))exp(), () F S IMFIMFL IMFIMFIMF CIMFDNBIMF rr pgdp gd θθ αδ θα δψθ θθθθ −∞ −∞ +=+=− == ∫∫

### Calibration equations and numerical solution approach (Appendix II highlights)
- Posterior distribution and calibration definitions:
  - Posterior mean and variance expressions (text):
    - μ() ()/(1 ) qq μτμτ=+ + and variance 22 ()/(1 ) qστστ=+.
  - Substitution 1 /exp() S kdαθ≡−.
- Key calibration equations (as presented):
  - (A.II.1):
    - * * 1(| , , )( | )exp() ( | ) SS mRqR vqd Svqd θ θ ψθ θθθθθθθ ∞ −∞ == + − ∫∫
  - Crisis threshold determination (A.II.2):
    - () 1 S Rqr = +
  - Short-term zero-profit condition (A.II.3):
    - exp( )( )( ) q q z q dq z q dq δ α θ λ δ ∞ −∞ = + ∫∫
  - Long-term zero-profit condition compact form (A.II.4):
    - 1 exp()[/]() ()() q S q Rz q dq f q z q dq δ α θ ψ ∞ −∞ − = + ∫∫
- With the IMF (no investor moral hazard):
  - IMF ex ante zero-profit condition reduced to (A.II.5):
    - 1 1(| , , )( | )exp() ( | ) F F FF bR q R vqd vqd θ θ θ β θ θ φ θ θ θ θ ∞ − −∞ == + − ∫∫
  - IMF lending threshold (A.II.6):
    - () 1 F F Rqr = +
  - Short-term investors' condition with IMF (A.II.7):
    - exp( )()()(1)()() F FF qqq qqq z q dq z q dq z q dq z q dq δ α θ β λ δ β ∞ −∞ ⎡⎤ ⎡⎤ = + + + − + ⎢⎥ ⎢⎥ ∫∫∫∫ ⎣⎦ ⎣⎦
  - Long-term investors' condition with IMF (A.II.8):
    - 1 exp( )( ) ( ) H q z q dq δ α θ ∞ −∞ − = ∫
  - With IMF-induced investor moral hazard, equation (A.II.5) is replaced by (A.II.9):
    - 1 1( | )exp() ( | ) F F F R vqd vqd θ θ γ θ θ φ θ θ θ θ ∞ − −∞ − = + − ∫∫

### Conceptual findings and model implications (as described in text)
- Under no informational uncertainty (0τ=), rollover and IMF lending rates equal the risk-free rate; crisis thresholds and probabilities reduce to closed-form integral expressions of the productivity θ distribution.
- Without the IMF, crisis probability equals default probability (DC pp=).
- With the IMF, the IMF lends only if *θ θ< and only if F θ θ≥; full bailouts require long-term debt (ψ>0) for the IMF to act without expecting losses.
- IMF presence lowers the crisis threshold (IMF NO θ θ< for all ψ), altering crisis and default probabilities and long-term repayment regions.
- Calibration requires numerical solution of coupled integral equations (A.II.1)-(A.II.4) without the IMF, and (A.II.1)-(A.II.2), (A.II.5)-(A.II.8) with the IMF; with moral hazard the IMF equation is (A.II.9).

*Source: Excerpt from _wp07104 (tables, Appendix I and Appendix II as provided).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07104.pdf_