## _wp16136 - References ## Source details **Canonical URL:** [_wp16136 - References](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16136.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16136.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16136.pdf.json) --- ### I. Introduction and motivation - Since 2010, financial markets have expressed recurrent concerns about risks to debt sustainability in a number of countries; symptom: observed pattern of eurozone members sovereign yields since 2010 (Figure 1). - ECB President Mario Draghi pledge and OMT: - Pledged to do “whatever it takes”; OMT introduced September 2012 to reduce country-specific distress yields via potentially unlimited purchases of short-term government bonds. - Yields declined despite purchases never taking place; Draghi: “OMT has been probably the most successful monetary policy measure undertaken in recent time”. - Legal controversy: German constitutional court hearings June 2013; European Court of Justice ruling June 16th 2015; returned to German constitutional court latest hearings February 2016. - Central controversy: whether ECB program is monetary policy or fiscal bailout, financed by reductions in seignorage revenue for other member countries or an inflation tax. - Motivation of paper: understand sovereign default dynamics and role of a large, risk-neutral investor/agency in coordinating expectations on a “good equilibrium”. ### II. Modeling approach and environment - Framework and agents: - Dynamic endogenous default model à la Eaton and Gersovitz (1981). - Agents: a single government, international lenders, a bailout agency. - Government finance and shocks: - Government finances consumption with tax receipts and non-contingent long-duration bonds. - Tax receipts y_{t} are exogenous and stochastic. - Defaults arise from negative income shocks (Arellano (2008)) or coordination failures/sunspots (Cole and Kehoe (1996,2000)). - Default consequences and costs: - Government pays exogenous one-time utility cost of default χ_{t}, temporarily excluded from debt markets, consumes tax receipts until re-entry. - Utility cost χ_{t} is time-varying and interpretable as an “embarrassment” of default; χ ∈ { χ_{L}, χ_{H} } with 0 = χ_{L} ≤ χ_{H}. - Re-entry into debt markets each period with exogenous probability 0 ≤ α < 1; re-entry starts with debt level zero. - Market structure and budget constraint: - Traders risk neutral, discount at return R; price new debt via q_{t}(B_{t+1}). - If no default: - c_{t} + (1−θ) B_{t} = y_{t} + q_{t}(B_{t+1})(B_{t+1} − θ B_{t}), with 0 < θ ≤ 1 fraction of debt that currently needs to be repaid. - State-space: s = (B,d,z) with z = (y,χ,ζ), ζ ∈ [0,1] uniform sunspot. ### III. Bailout agency: assumptions, construction, and properties - Agency description and objective: - Large, infinitely lived, risk neutral outside investor committed to rule out sunspot-driven defaults by debt purchases while earning actuarially fair return. - Agency will not prevent defaults due to fundamentals, nor impose conditionality. - Minimal guarantee (restoration-of-the-good-equilibrium): - Define minimal B′_{a}(s) the agency must commit to buy at π=0 equilibrium price q(π=0) to force markets to coordinate on π=0. - Assumes buyer’s strike lasts at most one period; continuation after no-default equals value under π=0 equilibrium. - v_{ND;a}(s) defined with constraint B′ ≤ B′_{a}(s) and pricing q(π=0)(B′;s). - B′_{a}(s) chosen so that v_{ND;a}(s=(B,0,z)) = v_{D}(z(s)) − χ(s=(B,0,z)) + ε for all 0 ≤ B ≤ ̄B(z); for B > ̄B(z) set B′_{a}(s) = 0. - Key propositions and comparative statics (iid and constant-χ simplification): - Proposition 1: If B′_{a}(s) satisfies the condition above, then B(z) = ̄B(z) (no default unless debt exceeds ̄B(z)). - Proposition 2 properties: 1. If B(s_{1}) > B(s_{2}) then B′_{a}(s_{1}) ≥ B′_{a}(s_{2}). 2. If B(s) > 0 and default set nonempty then q(π=0)(B′_{a}(s);s) ( B′_{a}(s) − θ B(s) ) < (1−θ) B(s). 3. If y(s_{1}) > y(s_{2}) then B′_{a}(s_{1}) ≤ B′_{a}(s_{2}). 4. If χ(s_{1}) > χ(s_{2}) then B′_{a}(s_{1}) ≤ B′_{a}(s_{2}). - Interpretations: - Larger current debt increases minimal guarantee needed. - Minimal guaranteed purchase must ensure net proceeds from new issuance under guaranteed pricing are less than current repayment burden when default set nonempty. - Higher current income y reduces need for guarantee; higher embarrassment cost χ reduces need for guarantee. - Implementation constraint on purchase-at-market-price commitment: - Agency need not set ex ante purchase price; commits to buy at prevailing secondary-market price once “good” equilibrium restored. - Commitment prevents bailout-by-mistake because a purchase at prevailing secondary-market price is inconsistent with a bailout at that point; practical difficulty: committing to buy at sufficiently high price contingent on that price emerging is tricky. - Nonlinear exposure: - Agency may need to be willing to potentially purchase nearly the entire amount of newly issued debt. - Small worsening in fundamentals can make agency switch from buying the entire amount to buying no debt and letting country default. ### IV. Numerical example: setup, calibration, and key parameter values - Functional forms and timing: - Utility: u(c) = (c^{1−σ} − 1)/(1−σ). - Income: log(y_{t+1}) = (1−ρ)μ + ρ log(y_t) + ε_{t+1}, E(ε) = 0, E(ε^2) = σ_ε^2. - One period = one year. - Parameter values (Table 1): - σ 1/2 - r 3.0 - ρ 0.945 - σ_ε 3.4% - μ (-1/2) σ^2_ε - α 0.2 - θ 0.8 - β 0.4 - χ_L 0 - χ_H 0.5 - π 0.05 - Income grid y_1,...,y_20 [0.73,...,1.37] - debt grid B_1,...,B_1000 - Calibration targets and achieved values (Table 2): - Target θ = 0.8 - Debt/Tax ratio target 2 .. 3 ; achieved 2.4 - Default rate target 5% .. 8% ; achieved 6.6% - Transition matrix between χ-states: - [ 0.04 0.96 ] (as presented) - Calibration interpretation: - Two parameters (χ_H and transition prob from χ_H to χ_L) chosen to hit target properties. ### V. Numerical and structural findings - Crisis zones and agency purchase policy: - Crisis zones: intervals of income and debt where government defaults only in case of buyers’ strike; left: always repays; right: always defaults. - Guaranteed purchases rise quickly over a narrow range until they reach 100%; at 100% guarantee, risk of fundamental default tomorrow becomes large so failure to sell a small fraction triggers default. - For even higher current debt, fundamental debt price collapses to zero and bailout guarantee is zero. - Bailout guarantee positive only for income-debt pairs in crisis zones. - With higher current income, agency may guarantee purchases that would lead to default at lower income levels; implication: agency should rather support the country during a boom than a recession. - Maturity variations (Table 4 highlights): - θ = 0.9: Debt/Tax ratio target 2 .. 33.3 ; Default rate 5% .. 8% achieved 6.6% - θ = 0.8 (benchmark): Debt/Tax ratio 2 .. 3 achieved 2.4 ; Default rate 6.6% - θ = 0.5: Debt/Tax ratio 1 .. 8 achieved 1.8 ; Default rate 6.2% - θ = 0: Debt/Tax ratio 1 .. 6 achieved 1.6 ; Default rate 6.2% - As maturity increases (higher θ), threat from buyers’ strike per period declines, incentive to hold higher debt rises, crisis zones shrink, default rates change modestly. - Sunspot probability (π) variations (Tables 5–6): - Debt/Tax ratio for π values: - π = 0.2: 2 .. 31.8 - π = 0.1: 2 .. 2.1 - π = 0.05: 2 .. 2.4 - π = 0: 2 .. 2.9 - Default rates: - π = 0.2 => 5% - π = 0.1 => 8% - π = 0.05 => 6.6% - π = 0 => 4% - Default composition (benchmark π = 0.05 total prob = 6.6%): - χ_L: Buyers present 38% / Buyers’ strike 2% - χ_H: Buyers present 12% / Buyers’ strike 48% - Concluding effect: reducing π modestly increases debt levels while modestly reducing overall default probability; mass of defaults shifts from buyer-strike scenarios to defaults due to fundamentals. - Debt pricing and dynamics: - Pricing shifts upward when agency assures π = 0; debt prices rise, yields decline. - Debt-to-income ratio increases after introduction of assistance: example path shows rise from about 2.2 to about 2.9 over years 0 to 10 (visual series presented). - Stationary debt distribution shifts to the right with assistance (π = 0), inducing higher occurrences of fundamental defaults. - Decision rules shift upwards under assistance; governments willing to incur more debt. - Net quantitative effect: - Guarantees induce faster debt buildup; higher guarantees and lower yields enable governments to relax fiscal discipline, increasing future likelihood of fundamental defaults. - Nonlinear response: small worsening in fundamentals can produce sharp discontinuities in agency purchase behavior (from buying nearly all new debt to buying none). ### VI. Key empirical breakdowns and statistics - Anatomy of defaults (Table 3): - Buyers present / Buyers’ strike - χ_L: 38% / 2% - χ_H: 12% / 48% - Interpretation: 12 percent of defaults occur due to fundamental problems even with “responsible” χ_H; nearly half of defaults occur due to buyers’ strike—targeted by bailout agency. - Sunspot-specific default breakdown (selected cases): - π = 0.1 total defaults 8%: - χ_L: Buyers present 27% / Buyers’ strike 3% - χ_H: Buyers present 8% / Buyers’ strike 62% - π = 0 (assistance) total defaults 4%: - χ_L: Buyers present 81% / Buyers’ strike 0% - χ_H: Buyers present 19% / Buyers’ strike 0% ### VII. Policy-relevant conclusions and implications - Three main messages: 1. An actuarially fair bailout agency can restore the “fundamentals-only” equilibrium via debt purchase guarantees without incurring losses in expectation. 2. Guarantees need to go far enough, but not too far: excessive guarantees can lead to agency losses because fundamental defaults remain possible. 3. Overall default rates may not change much: higher guarantees and lower yields enable governments to accumulate more debt, raising future fundamental-default likelihood. - Implications for OMT-style programs: - Restoration of the “fundamentals-only” equilibrium may explain yield declines after the OMT announcement. - Coordination on the “good equilibrium” need not imply transfers to the distressed country if purchases are tied to market prices. - Careful implementation and tying purchases to market prices is crucial to avoid transfers; practical commitment challenges remain. ### VIII. Scope, interpretation, limitations, and relation to literature - Analysis is explicitly “positive”, not “normative”; welfare implications are not assessed and would require additional assumptions (e.g., government impatience vs population). - Utility cost χ introduces a free parameter enabling fit to high debt-to-tax ratios and default rates; interpretable as political/embarrassment considerations. - Literature connections and distinctions (selected): - Builds on Arellano (2008), Cole and Kehoe (1996,2000), and political-economy theories for monetary unions (Beetsma and Uhlig (1999); Cooper, Kempf and Peled (2010)). - Contrasts with conditionality/subsidized or fixed-price bailout models (Boz (2011); Fink and Scholl (2014); Juessen and Schabert (2013); Kirsch and Ruhmkorf (2013)); emphasizes actuarially fair, price-taking purchases to eliminate coordination failures. - Related strands: long-term debt (Hatchondo and Martinez (2009); Chatterjee and Eyigungor (2012)), fiscal rules (Hatchondo et al. (2015)), endogenized maturity (Arellano and Ramanarayanan (2012)), renegotiation/delays (Benjamin and Wright (2009)), reserves and rollover risk (Bianchi et al. (2014)), optimal fiscal policy with distortionary taxes (Pouzo and Presno (2014)), multiple equilibria literature (Calvo (1988); Aguiar et al. (2013); Conesa and Kehoe (2013); Corsetti and Dedola (2014); Broner et al. (2014); Lorenzoni and Werning (2014); Bacchetta et al. (2015); Aguiar et al. (2015); Kriwoluzky et al. (2015); Bocola and Dovis (2015)). ### IX. Appendix summaries: alternative bailout mechanisms and no-bailout comparative statics - Appendix A (No bailouts: analysis) — key comparative statics: - Proposition 3: If z iid and functions independent of ψ, larger B implies default remains optimal (monotonicity in B). - Proposition 4: Default incentives are stronger the lower are tax receipts y. - Proposition 5: Default incentives are stronger the lower is the utility penalty χ. - Additional dynamics: pricing q either ̄q_m(B′; s) or 0; βR relation and precautionary saving implications; stationary debt distributions can arise if βR considerably smaller than unity. - Appendix B (Other bailout mechanisms) — key scenarios and implications: - One-time assisted bailout: fixed assisted price 0 < q_a < 1/R up to B′ ≤ ̄B_a; leads to temporary higher B′_a(s) and postponed default. - Permanent agency: fixed price 0 < q_a < 1/R for all future borrowing up to B′ ≤ ̄B_a; can render debt appearing safe while agency credible but may induce runaway borrowing up to agency limit and eventual default. - Probabilistic bailout (bailout sunspot): bailout probability 0 < ω < 1; crisis zone shifts right; agency postpones but does not permanently resolve fiscal crisis. - Overall insight: bailout mechanisms postpone default rather than eliminate default risk; they enable higher government consumption and debt and shift but do not remove crisis zones. *Source: _wp16136 - References.* ### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ### _wp16136 - References ### I. Introduction and motivation - Since 2010, financial markets have expressed recurrent concerns about risks to debt sustainability in a number of countries. - Symptom: observed pattern of eurozone members sovereign yields since 2010, shown in Figure 1. - ECB President Mario Draghi pledged to do “whatever it takes" to preserve the euro zone; followed by outright monetary transactions (OMT) in September 2012. - OMT: intended to reduce country-specific distress yields per potentially unlimited purchases of the short-term government bonds of that country. - Yields declined despite purchases never taking place; ECB Draghi stated that “OMT has been probably the most successful monetary policy measure undertaken in recent time”. - Legal controversy: attacked at German constitutional court hearings in June 2013 as fiscal policy and outside Maastricht treaty; favorable ruling by the European Court of Justice on June 16th 2015; returned to German constitutional court with latest hearings in February 2016. - Central controversy: whether the ECB program represents monetary policy or fiscal policy and a bailout, financed by reductions in seignorage revenue for other member countries or an inflation tax. - Motivation: understand dynamics of sovereign default crisis and the potential role of a large, risk-neutral investor or agency in coordinating expectations on a “good equilibrium”. ### II. Modeling approach and environment - Framework: dynamic endogenous default model à la Eaton and Gersovitz (1981). - Agents in model: - a single government, - international lenders, - a bailout agency. - Government finances consumption with tax receipts and non-contingent long-duration bonds. - Tax receipts are exogenous and stochastic. - Defaults can occur from: - negative income shocks (fundamental reasons, Arellano (2008)), - coordination failures among international investors (sunspot/self-fulfilling crises, Cole and Kehoe (1996,2000)). - Default consequences: - government pays an exogenous one-time utility cost of default, - temporarily excluded from debt markets, - consumes tax receipts until re-entry into debt markets. - Utility cost of default is time-varying and interpretable as an “embarrassment” of default that changes from government to government. - Re-entry into debt markets occurs with some exogenous probability. ### III. Bailout agency: assumptions and role - Bailout agency modeled as a particularly large and infinitely lived investor committed to rule out sunspot-driven defaults by debt purchases, even if other investors do not. - Agency seeks an actuarially fair return; the paper characterizes the minimal intervention. - Agency will not: - prevent defaults due to fundamental reasons (Arellano (2008)), - impose additional policy constraints such as conditionality (e.g., Fink and Scholl (2014)). - Key operational feature: agency commits to buy at prevailing market price once the “good” equilibrium is restored; does not need to set prices ex ante. - Constraint parallels legal limits on ECB sovereign bond purchases as enshrined by the Maastricht treaty. - Quantitative implication: agency may need to be willing to potentially purchase (nearly) the entire amount of newly issued debt. - Nonlinear response: a small worsening in fundamentals can make the agency switch from buying the entire amount of newly issued debt to buying no debt at all and letting the country default. - Net effect found: - policy leads to higher debt levels, - possibly rather small changes in the probability of default, - probability of default for fundamental reasons may increase. ### IV. Key numerical and structural findings (as described) - Changing maturity of debt may have little influence on default probabilities; main change may be the level of debt. - The bailout agency’s maximal involvement can leave it exposed to sharp discontinuities in purchase behavior when fundamentals worsen. - Quantitative matching: including a utility cost parameter allows the model to match high debt-to-tax ratios and default rates. ### V. Relation to the literature - Builds on and extends three strands: - Arellano (2008): defaults more likely when income is low. - Cole and Kehoe (1996,2000): self-fulfilling debt crises and characterization of crisis zone and optimal policy. - Political-economy theories for need for debt constraints in monetary unions (e.g., Beetsma and Uhlig (1999); Cooper, Kempf and Peled (2010)). - Connections and contrasts with quantitative sovereign-default literature: - Long-term debt importance: Hatchondo and Martinez (2009); Chatterjee and Eyigungor (2012). - Fiscal rules and default risk: Hatchondo et al. (2015). - Endogenized maturity: Arellano and Ramanarayanan (2012). - Debt renegotiation and delays: Benjamin and Wright (2009). - Reserves as hedge against rollover risk: Bianchi et al. (2014). - Optimal fiscal policy with distortionary taxes: Pouzo and Presno (2014). - Distinction from prior bailout literature: - Boz (2011), Fink and Scholl (2014), Juessen and Schabert (2013), Kirsch and Ruhmkorf (2013) model conditionality, subsidized loans, or fixed-price bailouts; this paper instead emphasizes actuarially fair, price-taking debt purchases to eliminate coordination failures. - Uhlig (2013) studies central bank guarantees and shifted risks in a monetary union. - Gaballo and Zetlin-Jones (2016) discuss home bias as a commitment device. - Related literature on multiple equilibria and coordination failures: Calvo (1988); Aguiar et al (2013); Conesa and Kehoe (2013); Corsetti and Dedola (2014); Broner et al (2014); Lorenzoni and Werning (2014); Bacchetta et al (2015); Aguiar et al (2015); Kriwoluzky et al. (2015); Bocola and Dovis (2015). ### VI. Scope, interpretation, and limitations - Analysis is explicitly “positive”, not “normative”; the paper refrains from welfare assessments. - The impatience of the government may differ from that of the population; efficiency and welfare implications would require additional assumptions. - Utility cost parameter: introduces a free parameter but allows interpretation of political considerations and enhances quantitative fit. ### VII. Structure of remaining material (as described) - Section 2 introduces the model without bailouts. - Appendices and tables/figures listed include: - Appendices: A. No bailouts: Analysis; B. Other Bailout Mechanisms. - Tables include parameter values, targets and numerical results, structure of defaults, variations in maturity, sunspot probabilities and default details. - Figures include: 1. 10yr yield spread to Germany; 2. Crisis zones; 3–31 various figures illustrating debt purchase assistance policy, debt and default dependence on parameters (θ, π), debt pricing functions, debt dynamics after assistance, debt distributions with/without sunspots, stationary debt dynamics and comparisons between bailout regimes. - Notation and key parameters referenced in figures/tables include θ (debt maturity parameter), π (sunspot/default probability), βR, q(B′; s), ̄q_m(B′; s), and references to debt/gdp ratios and mean income. *Source: _wp16136 - References.* ### Section 3 introduces and characterizes the bailout agency. Section 4 presents the numerical ### _wp16136 - Section 3 introduces and characterizes the bailout agency. Section 4 presents the numerical ### II. A MODEL OF SOVEREIGN DEFAULT DYNAMICS: NO BAILOUT AGENCY - Objective: single fiscal authority maximizes utility - U = ∞∑_{t=0} β^{t}(u(c_{t}) − χ_{t} δ_{t}) - β is the discount factor; u(·) is strictly increasing, strictly concave, twice differentiable felicity function. - χ_{t} is an exogenous one-time utility cost of default; δ_{t}∈{0,1} is the default decision. - Endogenous vs exogenous: - Tax receipts y_{t} exogenous; consumption c_{t}, assets B_{t} (positive = debt), and δ_{t} chosen by government. - In default c_{t} = y_{t}. - Default costs and interpretation: - χ_{t} captures non-pecuniary costs (reputation, political factors). χ∈{χ_{L},χ_{H}} with 0 = χ_{L} ≤ χ_{H}. - ζ∈[0,1] is a uniformly distributed “crisis” sunspot. - Market structure and budget constraint: - Traders risk neutral, discount repayements at return R, price new debt B_{t+1} via q_{t}(B_{t+1}). - If no default: - c_{t} + (1−θ) B_{t} = y_{t} + q_{t}(B_{t+1})(B_{t+1} − θ B_{t}) - Parameter 0 < θ ≤ 1 denotes fraction of debt that currently needs to be repaid (maturity structure). - If default: c_{t} = y_{t}; excluded from debt markets until re-entry. - Re-entry probability each period: 0 ≤ α < 1, iid; re-entry starts with debt level zero. - State-space representation: - Aggregate state s = (B,d,z) - z = (y,χ,ζ); y∈[y_{L},y_{H}], ζ∈[0,1] uniform; entries in z independent conditional on previous state. - Default status d ∈ {0,1}; δ(s) decision rule. - Recursive formulation: - Value in default (after one-time loss): - v_{D}(z) = u(y(z)) + β(1−α) E[v_{D}(z′)|z] + α E[v_{ND}(s′ = (0,0,z′))|z] - Value from not defaulting given q(B′;s): - v_{ND}(s) = max_{c,B′} { u(c) + β E[v(s′)|z] | c + (1−θ) B(s) = y(s) + q(B′;s)(B′ − θ B(s)) ; s′ = (B′, d(s), z′) } - Overall: - v(s) = max_{δ∈{0,1}} (1−δ) v_{ND}(s) + δ( v_{D}(z(s)) − χ(s) ) - Equilibrium definition: 1. Given pricing function q(B′;s), government choices c(s), δ(s), B′(s) maximize utility subject to budget and exclusion following default. 2. Market pricing q(B′;s) is consistent with risk-neutral pricing and discounting at risk-free return R. ### II.B Debt pricing and equilibrium multiplicity - Default and continuation sets (for d=0): - D(B) = { z | δ(s) = 1 for s = (B,0,z) } - A(B) = { z | δ(s) = 0 for s = (B,0,z) } - Market price in case of no current default: - ̄q(B′;s) = (1/R) ∫_{z′∈A(B)} (1−θ + θ q( B(s′=(B′,0,z′)) )) μ(dz′|z) - Probability of continuation next period: - P(B′;s) = Prob(z′ ∈ A(B′) | s) = E[ 1_{δ(s′)=0} | s ] - One-period debt maturity special case (θ = 0): - ̄q(B′;s) = (1/R) P(B′;s) - Assumptions to rule out pathological equilibria: - Assumption A.1: Given state s, either q(B′;s) = ̄q(B′;s) for all B′ or q(B′;s) = 0 for all B′. - Bounds on debt for default decisions: - ̄B(z) = inf{ B | ̄v_{ND}(s=(B,0,z)) ≤ v_{D}(z(s)) − χ(s=(B,0,z)) } - B(z) = inf{ B | v_{ND}(s=(B,0,z)) ≤ v_{D}(z(s)) − χ(s=(B,0,z)) } - For B between B(z) and ̄B(z) sunspot ζ governs default probability π. - Assumption A.2 (sunspot coordination): - For π∈[0,1], and s with B(z) ≤ B(s) ≤ ̄B(z), q(B′;s) = ̄q(B′;s) if ζ(s) ≥ π and q(B′;s) = 0 if ζ < π. - Equilibrium regimes: 1. If B > ̄B(z): government defaults now; market price for new debt = 0. 2. If B(z) ≤ B ≤ ̄B(z): - default with probability π (ζ < π) and market price = 0; - continue with probability 1−π (ζ ≥ π) and market price = ̄q(B′;s). 3. If B < B(z): government will not default; market price = ̄q(B′;s). - Crisis zone for new debt: - B′ ∈ B = [ min B(z), max ̄B(z) ] - Safe-debt price q^{*} solves: - q^{*} = (1/R) (1−θ + θ q^{*}) → q^{*} = (1−θ) / (R − θ) - Equivalent implicit safe return given price q: - R(q) = θ + (1−θ)/q ### III. BAILOUTS — introduction of a large risk-neutral outside investor - Bailout agency description: - Large, infinitely lived, risk neutral outside investor (e.g., backed by other governments). - Agency aims to ensure selection of the “good” equilibrium while earning market rate of return in expectation (actuarially fair pricing). - Actuarially fair “restoration-of-the-good-equilibrium” used as benchmark; other non-fair mechanisms discussed in appendix B (not included here). - Minimal guarantee construction: - Agency need not buy entire debt; define minimal level B′_{a}(s) the agency must commit to buy at the π=0 equilibrium price to force markets to coordinate on π=0 equilibrium. - Debt held by agency treated the same as debt held by market participants. - Assumption on length of buyer’s strike: - Optimistic assumption: buyer’s strike lasts at most one period; continuation value after a no-default today equals value function for π=0 equilibrium. - This effectively assumes a period is the maximal time for a buyers’ strike (finite upper bound). - No-default value under assistance with current buyers’ strike (except large investor): - v_{ND;a}(s) = max_{c,B′} { u(c) + β E[ v(π=0)(s′) | z ] | c + (1−θ) B(s) = y(s) + q(π=0)(B′;s)(B′ − θ B(s)), B′ ≤ B′_{a}(s), s′ = (B′, d(s), z′) } - B′ constraint encapsulates limit of assistance. - Definition of minimal guarantee B′_{a}(s): - For ε > 0 small, choose B′_{a}(s) such that - v_{ND;a}(s = (B,0,z)) = v_{D}(z(s)) − χ(s = (B,0,z)) + ε for all 0 ≤ B ≤ ̄B(z) - For B > ̄B(z), define B′_{a}(s) = 0 (market price q(B′;s) = 0 for any B′ > 0 in that region). - Proposition 1: - If B′_{a}(s) satisfies condition above, then B(z) = ̄B(z), i.e., there will not be a default unless debt exceeds ̄B(z). - IID case and constant-χ simplification: - β ̃v_{D} ≡ β E[v_{D}(z′)] is continuation value from defaulting. - β ̃v_{ND}(B′_{a}(s)) ≡ β E[v(B′_{a}(s),0,z′)] is continuation value of not defaulting when receiving full guarantee. - Criterion ((19)) becomes: - u(y(s)) − u( y(s) + q(π=0)(B′_{a}(s);s) ( B′_{a}(s) − θ B(s) ) − (1−θ) B(s) ) = β ̃v_{ND}(B′_{a}(s)) − β ̃v_{D} + χ − ε - This compares current utility gain from defaulting to continuation loss from defaulting, including embarrassment cost χ. - Proposition 2 (properties in iid and constant-χ case): 1. For s_{1}, s_{2}, if B(s_{1}) > B(s_{2}) then B′_{a}(s_{1}) ≥ B′_{a}(s_{2}). 2. If B(s) > 0 and default set nonempty then - q(π=0)(B′_{a}(s);s) ( B′_{a}(s) − θ B(s) ) < (1−θ) B(s) 3. For s_{1}, s_{2}, if y(s_{1}) > y(s_{2}) then B′_{a}(s_{1}) ≤ B′_{a}(s_{2}). 4. For s_{1}, s_{2}, if χ(s_{1}) > χ(s_{2}) then B′_{a}(s_{1}) ≤ B′_{a}(s_{2}). - Interpretation of propositions: - Larger current debt increases the minimal guarantee needed. - The minimal guaranteed purchase must be such that net proceeds to the borrower from new issuance under guaranteed pricing are less than the current repayment burden when default set is nonempty. - Higher current output y reduces the need for guarantee; higher embarrassment cost χ reduces the need for guarantee. *Source: _wp16136 - Section 3 introduces and characterizes the bailout agency. Section 4 presents the numerical* ### 4. This follows from criterion ((19)). ### 4. This follows from criterion ((19)). ### Implementation constraint on purchase-at-market-price commitment - With the restoration of the fundamental equilibrium, the agency does not need to know a priori the price at which it is prepared to buy the debt; it just needs to commit to buy at the prevailing market price, once that equilibrium is restored. - The agency only needs to know that the equilibrium has taken place and therefore must commit to buy only at secondary market prices eventually prevailing in equilibrium. - This commitment excludes a bailout-by-mistake: a purchase at secondary market prices is inconsistent with a bailout at that point in time, while buying above secondary-market prices implies either the fundamental equilibrium has not been restored or a bailout happened. - Practical difficulty: committing to purchase at a sufficiently high price so that the fundamental equilibrium is restored, while making that commitment contingent on a high market price emerging, is tricky. ### IV. A numerical example — setup, functional forms, and parametrization - Government within-period utility: u(c) = (c^(1−σ) − 1)/(1−σ). - Income process: log-normal autoregressive - log(y_{t+1}) = (1−ρ)μ + ρ log(y_t) + ε_{t+1} - E(ε) = 0, E(ε^2) = σ_ε^2 - A period in the model = one year. - Transition matrix between the two χ-states: - [ 0.04 0.96 ] (as presented) - Two parameters chosen (χ_H and transition prob from χ_H to χ_L) to hit target properties. Table 1. Parameter values for the calibration. One period is one year. - Government’s risk aversion σ 1/2 - Interest rate r 3.0 - Income autocorrelation coefficient ρ 0.945 - Standard deviation of innovations σ_ε 3.4% - Mean log income μ (-1/2) σ^2_ε - Exclusion α 0.2 - Maturity structure θ 0.8 - Discount factor β 0.4 - Cost χ_L 0 - Cost χ_H 0.5 - SFC sunspot probability π 0.05 - Income grid y_1,...,y_20 [0.73,...,1.37] - debt grid B_1,...,B_1000 ### Calibration targets and achieved values - Targets and numerical results for the debt/tax ratio and the default rate (Table 2): - Target θ = 0.8 - Debt/Tax ratio 2 .. 3 2.4 - Default rate 5% .. 8% 6.6% ### Anatomy of defaults (Table 3) - Buyers present / Buyers’ strike - χ_L: 38% / 2% - χ_H: 12% / 48% - Interpretation: - 12 percent of defaults happen due to fundamental problems even with a “responsible” χ_H government and despite buyers willing to buy bonds in principle. - Nearly half of all defaults occur due to a buyers’ strike; these are the occurrences the bailout agency aims to avoid. ### Crisis zones and bailout agency purchase policy (Figures 2–4) - Crisis zones: - Intervals of income and debt levels where government defaults only in case of a buyers’ strike. - Left of the interval: government always repays independently of buyers’ strike. - Right of the interval: government always defaults. - Debt purchase assistance policy by the bailout agency (Figure 3): - Over a fairly narrow range, guaranteed purchases quickly rise until they reach 100%. - At 100% guarantee, the risk and incentive of fundamental default tomorrow become so large that failure to sell a small fraction of new debt will trigger default. - For even higher current debt, fundamental debt price collapses to zero and the bailout guarantee is zero. - Bailout guarantee is positive only for pairs of income and debt levels in the crisis zones. - Dependence on income (Figure 4): - With higher current income, it may be worth guaranteeing debt purchases that would lead to default at lower income levels. - Policy implication: bailout agency should rather support the country during a boom than a recession. - Rationale: worsening fundamentals can move country from crisis zone into default-for-sure region where any purchase guarantee would be a subsidy and avoided by risk-neutral investors. ### Maturity variations and impact on defaults (Table 4, Figures 5–8) - Table 4 summary (θ = 0 is one-period debt; θ = 0.9 is essentially 10-period debt): - Targets: Target - θ = 0.9: Debt/Tax ratio 2 .. 33.3 ; Default rate 5% .. 8% 6.6% - θ = 0.8: Debt/Tax ratio 2 .. 3 2.4 ; Default rate 6.6% - θ = 0.5: Debt/Tax ratio 1 .. 8 1.8 ; Default rate 6.2% - θ = 0: Debt/Tax ratio 1 .. 6 1.6 ; Default rate 6.2% - Defaults breakdown: - θ = 0.9: - χ_L: Buyers present 38% / Buyers’ strike 2% - χ_H: Buyers present 16% / Buyers’ strike 44% - θ = 0: - χ_L: Buyers present 42% / Buyers’ strike 2% - χ_H: Buyers present 2% / Buyers’ strike 54% - Effects: - As maturity increases, the threat from buyers’ strike in any given period declines because smaller fraction of debt needs rollover. - Incentive to maintain higher debt levels rises; crisis zones shrink. - Default rates change modestly. ### Sunspot (π) variations and their effects (Tables 5–6, Figures 9–10) - Table 5. Sunspot probabilities and debt levels: - Target π = 0.2: Debt/Tax ratio 2 .. 31.8 - Target π = 0.1: Debt/Tax ratio 2 .. 2.1 - Target π = 0.05: Debt/Tax ratio 2 .. 2.4 - Target π = 0: Debt/Tax ratio 2 .. 2.9 - Default rate (5% .. 8%): π = 0.2 => 5% ; π = 0.1 => 8% ; π = 0.05 => 6.6% ; π = 0 => 4% - Table 6. Sunspot probabilities and default details: - Defaults for π = 0.1: total prob = 8%: - χ_L: Buyers present 27% / Buyers’ strike 3% - χ_H: Buyers present 8% / Buyers’ strike 62% - Defaults for π = 0.05 (Benchmark): total prob = 6.6%: - χ_L: Buyers present 38% / Buyers’ strike 2% - χ_H: Buyers present 12% / Buyers’ strike 48% - Defaults for π = 0: total prob = 4%: - χ_L: Buyers present 81% / Buyers’ strike 0% - χ_H: Buyers present 19% / Buyers’ strike 0% - Effects: - Reducing sunspot probability π modestly impacts overall default probability while debt level increases. - As fear of buyers’ strike declines, debt becomes more attractive and default probability mass shifts from “buyer strike” scenarios to defaults due to fundamental reasons. - Conundrum: reducing sunspot defaults (e.g., from 20% to 0%) only modestly reduces overall default rates (e.g., from 5% to 4%); problem is postponed via higher debt accumulation. ### Debt pricing, dynamics, and stationary distributions (Figures 11–17) - Debt pricing: - Pricing function shifts up when bailout agency assures π = 0 equilibrium; q if π = 0.05 vs q if π = 0 (Figures 11–12). - Debt prices rise and yields decline when the bailout agency assures the equilibrium. - Debt dynamics after assistance (Figure 13): - Debt-to-income ratio increases over years after bailout facility is introduced (starting point π = 0.05, mean income, mean debt/gdp ratio). - Example: plotted years 0 to 10 show debt-to-income ratio rising from about 2.2 to about 2.9 (visual series as presented). - Stationary debt distribution shifts (Figures 14–16): - With sunspots (π = 0.1 or π = 0.05) distributions concentrated at lower debt levels than with π = 0 (assistance). - With debt purchase assistance (π = 0), stationary debt distribution shifts to the right, inducing higher occurrences of defaults due to fundamental reasons. - Decision rules (Figure 17): - Decision rule shifts upwards under debt purchase assistance, indicating larger willingness of government to incur debt. - Net effect: - Guarantees induce faster debt buildup; higher current guarantees and lower yields enable governments to relax fiscal discipline, increasing future likelihood of fundamental defaults. ### V. Conclusions — three main messages - Message 1: An actuarially fair bailout agency may be able to restore the “fundamentals-only” equilibrium by issuing debt purchase guarantees and without incurring losses in expectation. - Message 2: Guarantees need to go far enough, but not too far. Excessive debt purchase guarantees can lead to losses for the bailout agency because fundamental defaults still remain possible. - Message 3: Overall default rates may not change much. Higher guarantees and lower yields enable current governments to accumulate more debt, raising future default likelihoods that are driven by fundamentals rather than buyers’ strikes. - Policy implication for OMT-style programs: - Restoration of the “fundamentals-only” equilibrium may explain yield declines after the OMT announcement. - Coordination on the “good equilibrium” does not imply transfers to the distressed country if purchases are tied to market prices. - Careful implementation and tying purchases to market prices is crucial to avoid transfers. *Source: _wp16136 - 4. This follows from criterion ((19)).* ### REFERENCES ### _wp16136 - REFERENCES ### References (selected list from source) - Aguiar, Mark, Manuel Amador, Emmanuel Farhi, and Gita Gopinath, 2013, “Crisis and Commitment: Inflation Credibility and the Vulnerability to Sovereign Debt Crises,” (unpublished). - Aguiar, Mark, Satyajit Chatterjee, Harold Cole, and Zachary Stangebye, 2016, “Quantitative Models of Sovereign Debt Crises,” chapter in “Handbook of Macroeconomics,” J. Taylor and H. 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Kehoe, 2000, “Self-Fulfilling Debt Crises,” Review of Economic Studies, Vol. 67, No. 1, pp. 91–116. - Cooper, Russell, Hubert Kempf, and Dan Peled, 2010, “Regional debt in monetary unions: is it inflationary?” European Economic Review, Vol. 54, No. 3, pp. 345–358. - Corsetti, Giancarlo, and Luca Dedola, 2014, “The Mystery of the Printing Press: Monetary Policy and Self-Fulfilling Debt Crises,” Unpublished. - del Negro, Marco, and Christopher A. 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Buti, Monetary and Fiscal Policies in EMU, 2003, pp. 29–56. - Uhlig, Harald, 2010, “A model of a systemic bank run,” Journal of Monetary Economics, Vol. 57, No. 1, pp. 78–96. - Uhlig, Harald, 2013, Sovereign Default Risk and Banks in a Monetary Union (German Economic Review). ### Appendix A — No bailouts: analysis (key results) - Assumptions: - q_a(B′; s) ≡ 0 (no assisted debt issuance). - The bailout sunspot ψ(s) is “irrelevant” (all functions independent of ψ). - z is iid. - Main comparative static results (statements preserved exactly): - Proposition 3: Suppose z is iid and that all functions are independent of ψ. If default is optimal for s(1) = (B(1),0,z), then default is optimal for s(2) = (B(2),0,z), whenever B(2) > B(1). - Proposition 4: Suppose z is iid and that all functions are independent of ψ. Default incentives are stronger, the lower are tax receipts. I.e., for all y(1) ≤ y(2), if z(2) = (y(2),χ,ζ,ψ) ∈ D(B), then so is z(1) = (y(1),χ,ζ,ψ) ∈ D(B). - Proposition 5: Suppose z is iid and that all functions are independent of ψ. Default incentives are stronger, the lower is the utility penalty from defaulting. I.e., for all χ(1) ≤ χ(2), if z(2) = (y,χ(2),ζ,ψ) ∈ D(B), then so is z(1) = (y,χ(1),ζ,ψ) ∈ D(B). - Additional qualitative dynamics and implications: - Pricing function q(B′; s) can take forms q = ̄q_m(B′; s) or q ≡ 0; q ≡ 0 results in a larger default set. - For βR = 1 and small income variation, countries choose to distance from the default zone via precautionary saving; asset accumulation can prevent a sovereign debt crisis. - If βR is considerably smaller than unity, countries may perch in the crisis zone and a stationary distribution for the debt level can arise. - Graphical representations referenced: figures 18–25 illustrate relationships among debt, income, pricing functions, crisis zones, and debt dynamics. ### Appendix B — Other bailout mechanisms (key scenarios and implications) - One-time assisted bailout: - Mechanism: For a single period, debt can be sold at some fixed “assisted” price 0 < q_a < 1/R to an outside agency, provided total amount B′ ≤ ̄B_a. - Effect: The available assisted price (blue line) can raise the new debt level B′_a(s) above the old debt level; government faces less pressure to cut spending; one-time bailout leads to debt dynamics shown in figure 24 (red arrow indicates path after bailout). - Permanent agency (permanent assistance): - Mechanism: All future borrowing can be done at fixed price 0 < q_a < 1/R, provided B′ ≤ ̄B_a. - Effect: Promise of permanent agency can make debt appear safe and be discounted at safe rate R while the agency is credible; borrowing increases from B′(s) to B′_a(s); country may accumulate debt up to the agency’s imposed limit and then borrow from the agency at a “penalty rate,” leading to default next period—i.e., the agency can induce a final runaway to the debt limit. Figures referenced: 27–28 illustrate pricing and stationary debt dynamics. - Note: Without a debt limit, the country could choose a Ponzi scheme (borrowing forever without repaying). - Probabilistic bailout (bailout sunspot): - Mechanism: Bailout probability 0 < ω < 1. If ψ < ω, country can borrow at 0 < q_a < 1/R from the agency up to B′ ≤ ̄B_a; if ψ ≥ ω, country must rely on private markets only. - Effect: The crisis zone shifts to the right relative to “no bailout ever” (figure 29). Debt dynamics show higher debt levels and temporary relief; the agency postpones but does not permanently resolve the fiscal crisis. Figures referenced: 29–30 (stationary debt dynamics for small income fluctuations and probabilistic bailout agency). - Overall policy-relevant insight: - Bailout mechanisms (one-time, permanent, or probabilistic) tend to postpone default rather than eliminate default risk. They provide temporary relief that can enable higher government consumption and higher debt levels, and can create incentives that shift but do not remove the crisis zone. *Source: _wp16136 - REFERENCES (PDF content provided).* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16136.pdf_