## _wp1333 - References

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---

### Major empirical facts emphasized
- Fact 1: Rate of Return Dominance. Governments hold large amounts of international reserves, for which they obtain a return lower than their borrowing cost.
- Fact 2: Gross Capital Flows Dynamics. Purchases of domestic assets by non-residents and purchases of foreign assets by residents are both procyclical and collapse during crises.

### Key empirical and calibration numbers reported
- EMBI plus sovereign spread index averaged 4.5percent between 2000 and 2012.
- Average reserve holding in simulations is equivalent to 2/3 of the average short-term debt obligations that mature within a year.
- Reserve holdings in the simulations are equivalent to 1/3 of the average holdings in Mexico between 1994 and 2011.
- IMF Survey of Reserve Managers: 80 percent of respondents cite building a buffer for liquidity needs as the most frequently cited reason for reserve accumulation.

### Model setup and assumptions (concise)
- Framework: Sovereign defaultable debt a la Eaton and Gersovitz (1981) augmented with reserves.
- Government: benevolent, issues long-duration bonds (non-contingent bonds with geometrically decaying coupons), saves via a risk-free asset (reserves).
- Default consequence: output cost and temporary exclusion from debt markets; reserve holdings can be adjusted in default.
- Sudden-stop shock: during a sudden stop the government cannot issue new debt; sudden stops are independent disturbances to capital markets.
- Bond pricing: risk-neutral foreign investors in competitive markets; spreads reflect how debt and reserves affect future repayment incentives.
- Calibration reference: Mexico—targets include levels of debt and sovereign spread, spread volatility, and frequency of sudden stops.

### Main mechanisms and theoretical insights
- Reserves as insurance: Reserves provide insurance against future increases in the borrowing cost; the government may incur the financial cost of holding low-return reserves to hedge against higher future borrowing costs.
- Importance of long-duration bonds: With long-duration bonds, issuing debt and accumulating reserves allows transfer of resources to future periods in which borrowing costs are high but there is no default. With one-period debt, issuing debt to accumulate reserves only transfers resources to periods in which the government defaults, which undermines the incentive to accumulate reserves.
- Procyclicality of flows: Countercyclical default risk leads to lower sovereign spreads in good times, inducing greater borrowing and reserve accumulation in good times; sudden stops cause both inflows and outflows to collapse as borrowing is cut and reserves are used to smooth consumption.

### Three-period example (illustrative) — environment and equations
- Periods t = 0,1,2; endowments y0 = 0, y1 > 0, y2 > 0; government maximizes E[u(c1)].
- Sudden stop: occurs in period 1 with probability π ∈ [0,1]; when it occurs the government cannot borrow in period 1.
- Interest rates: reserves earn ra ≥ 0; borrowing pays rb ≥ ra.
- Bonds: pay 1 unit in period 1 and (1 − δ) units in period 2; bond price q0 = (1 + rb)−1 + (1 − δ)(1 + rb)−2. δ = 1 → one-period bonds; δ < 1 → long-duration bonds. Assumption δ > 0.
- Budget constraints:
  - a ≤ y0 + q0 b1
  - c1(0) ≤ y1 − b1 + a(1 + ra) + b2(1 + rb)−1
  - c1(1) ≤ y1 − b1 + a(1 + ra)
  - b2 ≤ y2 − (1 − δ) b1

### Proposition 1 (Optimal Reserve Holdings) — exact statements
1. If there is no rollover risk (π = 0) and ra = rb, gross asset positions are undetermined. In particular, the optimal allocation can be attained without reserves (a⋆ = 0).
2. If there is no rollover risk (π = 0) and ra < rb, optimal reserves are zero (a⋆ = 0).
3. If the government can only issue one-period debt in period 0 (δ = 1) and ra = rb, gross asset positions are undetermined. In particular, the optimal allocation can be attained without reserve accumulation (a⋆ = 0).
4. If the government can only issue one-period debt in period 0 (δ = 1) and ra < rb, optimal reserves are zero (a⋆ = 0).

### Proposition 1 — condition for reserve accumulation (point 5) and interpretation
- Government accumulates reserves in period 0 (a⋆>0) if
  π[q0(1+ra)−1]u′(y1) > (1−π)[1−δ1+rb + 1−q0(1+ra)] u′(y1 + y2(1+rb)−1).(1)
- Additional implication: if ra = rb, the government perfectly smooths consumption.
- Interpretation:
  - Left-hand side of (1): expected marginal benefit of transferring period 2 resources to period 1 via reserves when period 1 borrowing may be unavailable (probability π).
  - Right-hand side of (1): expected marginal cost of transferring resources to the state without a sudden stop using reserves instead of cheaper borrowing in period 1 (if ra < rb).
  - With rollover risk and long-duration bonds, condition (1) can hold; with one-period debt (δ = 1) the left-hand side equals zero and the right-hand side is positive, so reserves are not optimal.
  - Condition (1) not satisfied with π = 0; satisfied with π = 1 (and long-duration bonds).
- Key message: rollover risk (π>0) and long-duration bonds (δ<1) are fundamental for explaining reserve accumulation.

### Model overview — timing, default, pricing, and recursion
- Economy: dynamic small-open-economy with benevolent government issuing non-state-contingent defaultable debt and buying risk-free assets (reserves).
- Timing each period:
  1. Realization of income and sudden-stop shocks.
  2. Government decides whether to default.
  3. Government adjusts debt and reserves; can always adjust reserves and buy back debt, but can issue debt only if not in default and not in a sudden stop.
- Sudden-stop shock: Markov process; starts with probability π ∈ [0,1], ends with probability ψ_s ∈ [0,1]; during a sudden stop cannot issue new debt, income loss φ_s(y), can buy back debt and adjust reserves.
- Bonds: infinite stream of coupons decreasing at constant rate δ; debt dynamics b_{t+1} = (1−δ)b_t + i_t.
- Budget constraint (with credit access):
  c_t = y_t − b_t + a_t + i_t q_t − a_{t+1} / (1 + r).
- Default: defaults on all current and future obligations; recovery rate in default is zero; exclusion stochastic with income y − φ_d(y); reentry probability ψ_d ∈ [0,1].
- Investors: risk-neutral, discount at rate r (also return on reserves); equilibrium bond-price condition:
  q(b′,a′,y,s)(1 + r) = E_{(y′,s′)|(y,s)} [(1 − d̂(b′,a′,y′,s′))(1 + (1 − δ) q(b′′,a′′,y′,s′))].
- Recursive value functions:
  - V(b,a,y,s) = max{ V_R(b,a,y,s), V_D(a,y,s) }.
  - V_R and V_D defined with constraints including restriction b′ − (1−δ)b ≤ 0 if s = 1.

### Calibration (benchmark) — exact parameter values
- Period = quarter.
- Utility: CRRA with γ = 2.
- Risk-free rate r = 1%.
- Endowment process: log(y_t) = (1−ρ) μ + ρ log(y_{t−1}) + ε_t, ε_t ∼ N(0, σ_ε^2), ρ = 0.94, σ_ε = 1.5%.
- Debt duration δ = 0.033 (bonds have average duration of 5 years).
- Probability of reentry after default ψ_d = 0.083 (average default duration 3 years).
- Probability of entering a sudden stop π = 0.025 (one sudden stop every 10 years).
- Probability of reentry after sudden stop ψ_s = 0.25 (mean sudden-stop duration 1.12 years).
- Discount factor β = 0.9745.
- Income cost of default φ_d(y) = d_0 y + d_1 y^2 with d_0 = −1.01683, d_1 = 1.18961.
- Income cost of sudden stops φ_s(y) = λ φ_d(y) with λ = 0.5.
- Calibration targets include mean and standard deviation of sovereign spread, mean debt level, and average accumulated income cost of a sudden stop targeted at 14 percent of annual income.

### Computation — methods and grids
- Finite-horizon approximation with horizon large enough that maximum deviation between first- and second-period value and bond-price functions ≤ 10^{−6}.
- Solution method: value function iteration; portfolio allocation solved over grids then refined with nonlinear optimization.
- Function approximations: linear interpolation over y, cubic spline over debt and reserves.
- Grids: 20 points for reserves, 20 points for debt, 25 points for income realizations.
- Expectations: 50 quadrature points for income shocks.

### Quantitative results: matching moments and facts (selected outcomes)
- Simulations match calibration targets and non-targeted moments for Mexico-like business cycle properties.
- Fact 1: Model generates joint debt and reserve accumulation together with a significant default premium.
- Reserve accumulation:
  - Average reserve holdings in simulations represent 66 percent of short-term debt (i.e., debt maturing within a year).
  - The 66 percent figure is noted as close to the “Greenspan-Guidotti rule” prescribing full short-term debt coverage.
- Other model successes:
  - Generates gross capital flows dynamics consistent with data (Fact 2).
  - Long-duration bonds, sudden stops, and endogenous countercyclical default risk are important for quantitative success.
  - Allowing reserves to lower the probability of sudden stops increases reserve accumulation in the model.

### Section II. — quantitative importance and sensitivity checks (selected results)
- Average reserve holdings in the simulation are about 1/3 of average holdings in Mexico between 1994 and 2011.
- Reserve holdings in simulations are close to holdings in Mexico in the second half of the 1990s; fast growth in reserve accumulation in Mexico since the early 2000s.
- Extending the baseline model to allow reserves to lower the sudden-stop probability can account for up to 100 percent of the average reserve holdings in Mexico.
- With one-period bonds:
  - Mean debt-to-income ratio in simulations drops to 3 percent of annual income, compared with 42 percent in simulations with long-duration bonds.
  - Reserves drop to 0.01 percent compared with 2.5 percent in the benchmark with long-duration bonds.
- Simulation diagnostics (Table 2 and Table 3 excerpts preserved in source) indicate:
  - σ(c)/σ(y) around 1.3 in benchmark.
  - Mean reserves-to-GDP 2.5 in benchmark vs. 0.01 with one-period bonds.
  - Default threshold (repayment region): government repays for income shocks higher than −5.2percent and defaults otherwise.

### Role of long-duration bonds — mechanisms and three reasons
- Long-duration bonds are essential for reserves to hedge rollover risk; with one-period bonds reserves play no role in insuring against future increases in borrowing costs.
- With one-period debt, government chooses low debt levels implying negligible default risk so portfolios with positive reserves offer little expansion in consumption space (example: rolling over average data debt requires 160 percent of quarterly income).
- Long-duration bonds change how reserve accumulation affects bond prices via the Euler equation and (∂q t /∂a t+1 ) term; with long-duration debt reserve accumulation can lower expected future default probabilities and thus spreads.

### Role of sudden stops (sensitivity)
- Higher frequency of sudden stops generates higher reserve holdings and lower debt levels.
- Without sudden stops, reserve holdings decline from 2.5 percent of income in the benchmark to 0.4 percent.
- Greater sudden-stop income loss leads government to choose higher reserve holdings and lower debt levels.
- Implication: perceived increases in frequency or cost of sudden stops can significantly increase optimal reserve accumulation.

### Role of endogenous and countercyclical sovereign spread
- Endogenous, countercyclical sovereign spread is key to generating demand for reserves in the model.
- A no-default model with constant exogenous spread calibrated to match mean spread and debt levels yields simulated reserve holdings declining from 2.5 percent of income in the benchmark to 0.1 percent.
- Two driving factors:
  1. Rollover risk is lower in no-default model because borrowing opportunities are independent from income shocks.
  2. A model without default but with the spread observed in data overstates the financial cost of accumulating reserves financed by borrowing.

### Reserve accumulation for crisis prevention (reserves reduce sudden-stop probability) — functional form and results
- Sudden-stop probability specification:
  ˆπ(a, ̺(b)) = G(m − w a ̺(b)),
  where ̺(b) denotes short-term debt maturing within the next year, and G is the standard normal CDF.
- Benchmark corresponds to w = 0 with m set so probability of a sudden stop is 10 percent.
- Simulation outcomes (Table 5 summary):
  - w=0 → Mean reserves-to-GDP 2.5; Sudden stops per 100 years 10.
  - w=0.05 → Mean reserves-to-GDP 4.0.
  - w=0.10 → Mean reserves-to-GDP 5.7.
  - w=0.15 → Mean reserves-to-GDP 7.2; Sudden stops per 100 years 6.
- As reserves become more effective at reducing sudden-stop probability (higher w), optimal reserves increase; at w = 0.15 the model replicates the average reserve level in Mexico and reduces sudden-stop frequency from 10 episodes per 100 years to 6 episodes per 100 years.

### Conclusions and implications
- The model explains two salient features of international capital flows:
  1. Indebted governments hold large amounts of international reserves while the yield on their debt is significantly higher than the return on reserves.
  2. Non-resident purchases of domestic assets and resident purchases of foreign assets are procyclical and collapse during crises.
- Mechanism: government trades off insurance benefits of reserves against costs of larger gross debt positions; because default risk is countercyclical, government accumulates reserves and debt in good times and uses reserves in low-income/high-spread periods to repay debt and smooth consumption.
- Key ingredients for quantitative success: long-duration bonds, sudden stops, and countercyclical sovereign spreads.
- Suggested avenues for further research:
  - Study interaction of debt maturity structure and reserve holdings.
  - Explore relevance of mechanisms for corporate borrowers facing rollover risk.

*Source: _wp1333 - Section II. shows that the accumulation of reserves financed by debt issuances to hedge*

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

### _wp1333 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

### Major empirical facts emphasized
- Fact 1: Rate of Return Dominance. Governments hold large amounts of international reserves, for which they obtain a return lower than their borrowing cost.
- Fact 2: Gross Capital Flows Dynamics. Purchases of domestic assets by non-residents and purchases of foreign assets by residents are both procyclical and collapse during crises.

### Key empirical and calibration numbers reported
- EMBI plus sovereign spread index averaged 4.5percent between 2000 and 2012.
- Average reserve holding in simulations is equivalent to 2/3 of the average short-term debt obligations that mature within a year.
- Reserve holdings in the simulations are equivalent to 1/3 of the average holdings in Mexico between 1994 and 2011.
- IMF Survey of Reserve Managers: 80 percent of respondents cite building a buffer for liquidity needs as the most frequently cited reason for reserve accumulation.

### Model setup and assumptions (concise)
- Framework: Sovereign defaultable debt `a la Eaton and Gersovitz (1981) augmented with reserves.
- Government: benevolent, issues long-duration bonds (non-contingent bonds with geometrically decaying coupons), saves via a risk-free asset (reserves).
- Default consequence: output cost and temporary exclusion from debt markets; reserve holdings can be adjusted in default.
- Sudden-stop shock: during a sudden stop the government cannot issue new debt; sudden stops are independent disturbances to capital markets.
- Bond pricing: risk-neutral foreign investors in competitive markets; spreads reflect how debt and reserves affect future repayment incentives.
- Calibration reference: Mexico—targets include levels of debt and sovereign spread, spread volatility, and frequency of sudden stops.

### Main mechanisms and theoretical insights
- Reserves as insurance: Reserves provide insurance against future increases in the borrowing cost; the government may incur the financial cost of holding low-return reserves to hedge against higher future borrowing costs.
- Importance of long-duration bonds: With long-duration bonds, issuing debt and accumulating reserves allows transfer of resources to future periods in which borrowing costs are high but there is no default. With one-period debt, issuing debt to accumulate reserves only transfers resources to periods in which the government defaults, which undermines the incentive to accumulate reserves.
- Procyclicality of flows: Countercyclical default risk leads to lower sovereign spreads in good times, inducing greater borrowing and reserve accumulation in good times; sudden stops cause both inflows and outflows to collapse as borrowing is cut and reserves are used to smooth consumption.

### Representative quantitative outcomes from simulations
- Simulated governments hold reserves with a return lower than their borrowing cost (consistent with Fact 1).
- With one-period debt (δ = 1) and benchmark calibration, the government does not choose significant reserve holdings.
- Additional experiments in which reserves reduce the arrival probability of a sudden stop can account for the entire reserve holdings observed in Mexico.

### Relation to existing literature (high-level)
- Builds on quantitative sovereign default literature following Aguiar and Gopinath (2006) and Arellano (2008); differs by allowing joint accumulation of assets and liabilities.
- Contrasts with Alfaro and Kanczuk (2009): their result of no simultaneous reserve accumulation and debt issuance is driven by their one-period debt assumption.
- Connects to studies of long-duration debt (Arellano and Ramanarayanan (2012), Chatterjee and Eyigungor (2012), Hatchondo and Martinez (2009)).
- Related strands: models of rollover-risk hedging (Jeanne and Ranciere (2011), Caballero and Panageas (2008)), precautionary savings and liquidity models (Durdu, Mendoza, and Terrones (2009); Aizenman and Lee (2007); Hur and Kondo (2011)), and the “rate of return dominance” literature in household finance (Telyukova (2011); Telyukova and Wright (2008)).

### Three-period example (illustrative)
- Environment: periods t = 0,1,2; endowments y0 = 0, y1 > 0, y2 > 0; government values consumption only in period 1, maximizing E[u(c1)].
- Sudden stop: occurs in period 1 with probability π ∈ [0,1]; when it occurs the government cannot borrow in period 1.
- Interest rates: reserves earn ra ≥ 0; borrowing pays rb ≥ ra.
- Bonds issued in period 0 pay 1 unit in period 1 and (1 − δ) units in period 2; bond price q0 = (1 + rb)−1 + (1 − δ)(1 + rb)−2. δ = 1 → one-period bonds; δ < 1 → long-duration bonds. Assumption δ > 0.
- Budget constraints summarized:
  - a ≤ y0 + q0 b1
  - c1(0) ≤ y1 − b1 + a(1 + ra) + b2(1 + rb)−1
  - c1(1) ≤ y1 − b1 + a(1 + ra)
  - b2 ≤ y2 − (1 − δ) b1

### Proposition 1 (Optimal Reserve Holdings) — exact statements
1. If there is no rollover risk (π = 0) and ra = rb, gross asset positions are undetermined. In particular, the optimal allocation can be attained without reserves (a⋆ = 0).

2. If there is no rollover risk (π = 0) and ra < rb, optimal reserves are zero (a⋆ = 0).

3. If the government can only issue one-period debt in period 0 (δ = 1) and ra = rb, gross asset positions are undetermined. In particular, the optimal allocation can be attained without reserve accumulation (a⋆ = 0).

4. If the government can only issue one-period debt in period 0 (δ = 1) and ra < rb, optimal reserves are zero (a⋆ = 0).

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1333.pdf*

### 5. If

### 5. If

### Proposition and main condition
- Proposition 1: Government accumulates reserves in period 0 (a⋆>0) if
  π[q0(1+ra)−1]u′(y1) > (1−π)[1−δ1+rb + 1−q0(1+ra)] u′(y1 + y2(1+rb)−1).(1)
- Additional implication: if ra = rb, the government perfectly smooths consumption.
- Interpretation:
  - Left-hand side of (1): expected marginal benefit of transferring period 2 resources to period 1 via reserves when period 1 borrowing may be unavailable (probability π).
  - Right-hand side of (1): expected marginal cost of transferring resources to the state without a sudden stop using reserves instead of cheaper borrowing in period 1 (if ra < rb).
  - With rollover risk and long-duration bonds, condition (1) can hold; with one-period debt (δ = 1) the left-hand side equals zero and the right-hand side is positive, so reserves are not optimal.
  - Condition (1) not satisfied with π = 0; satisfied with π = 1 (and long-duration bonds).
- Key message: rollover risk (π>0) and long-duration bonds (δ<1) are fundamental for explaining reserve accumulation.

### Model overview
- Economy: dynamic small-open-economy with a benevolent government that issues non-state-contingent defaultable debt and buys risk-free assets (reserves).
- Endowment: y ∈ Y ⊂ R++ follows a Markov process.
- Government objective: maximize E_t ∑_{j=t}^∞ β^{j−t} u(c_j), with u strictly increasing and concave.
- Timing within each period:
  1. Realization of income and sudden-stop shocks.
  2. Government decides whether to default.
  3. Government adjusts debt and reserves; can always adjust reserves and buy back debt, but can issue debt only if not in default and not in a sudden stop.
- Sudden-stop shock:
  - Markov process: starts with probability π ∈ [0,1], ends with probability ψ_s ∈ [0,1].
  - During sudden stop: cannot issue new debt, income loss φ_s(y), but can buy back debt and adjust reserves.
  - Sudden stops capture exogenous global dislocations to international credit markets.

### Debt structure and budget constraint
- Bonds: promise infinite stream of coupons decreasing at constant rate δ; a bond issued in period t pays (1−δ)^{j−1} units in period t+j for j≥1.
- Debt dynamics: b_{t+1} = (1−δ)b_t + i_t, where b_t is number of coupons due at beginning of period t and i_t is bonds issued in period t.
- Reserves: a_t ≥ 0 held at beginning of period t.
- Budget constraint (with credit access):
  c_t = y_t − b_t + a_t + i_t q_t − a_{t+1} / (1 + r),
  where q_t is bond price, and 1 + r is per period return on reserves.
- Return on reserves per period is fixed; long-duration reserves would deliver identical results.

### Default, exclusion, and pricing
- Default: government defaults on all current and future obligations; recovery rate in default is zero.
- Default exclusion: stochastic number of periods; income during exclusion is y − φ_d(y); reentry probability ψ_d ∈ [0,1].
- Investors: risk-neutral, discount at rate r (also return on reserves); bonds priced in competitive market with zero-expected-profit condition.
- Equilibrium bond-price condition:
  q(b′,a′,y,s)(1 + r) = E_{(y′,s′)|(y,s)} [(1 − d̂(b′,a′,y′,s′))(1 + (1 − δ) q(b′′,a′′,y′,s′))],
  where b′′ = b̂(b′,a′,y′,s′) and a′′ = â_R(b′,a′,y′,s′).
- Government cannot commit to future policies; equilibrium concept: Markov Perfect Equilibrium.

### Recursive formulation
- Value function for non-defaulting government:
  V(b,a,y,s) = max{ V_R(b,a,y,s), V_D(a,y,s) }.(2)
- Repayment value V_R solves:
  V_R(b,a,y,s) = max_{a′≥0, b′, c} { u(c) + β E_{(y′,s′)|(y,s)} V(b′,a′,y′,s′) },
  subject to c = y − s φ_s(y) − b + a + q(b′,a′,y,s)[b′ − (1−δ)b] − a′/(1 + r),
  and if s = 1 then b′ − (1−δ)b ≤ 0.
- Default value V_D solves:
  V_D(a,y,s) = max_{a′≥0, c} { u(c) + β E_{(y′,s′)|(y,s)} [ (1−ψ_d) V_D(a′,y′,s′) + ψ_d V(0,a′,y′,s′) ] },
  subject to c = y − φ_d(y) + a − a′/(1 + r).
- Policy functions: default d̂(b,a,y,s), borrowing b̂(b,a,y,s), reserves â_R, â_D, consumption ĉ_R, ĉ_D.

### Calibration (benchmark)
- Period = quarter.
- Utility: CRRA with γ = 2.
- Risk-free rate r = 1%.
- Endowment process: log(y_t) = (1−ρ) μ + ρ log(y_{t−1}) + ε_t, ε_t ∼ N(0, σ_ε^2), ρ = 0.94, σ_ε = 1.5%.
- Debt duration δ = 0.033 (bonds have average duration of 5 years).
- Probability of reentry after default ψ_d = 0.083 (average default duration 3 years).
- Probability of entering a sudden stop π = 0.025 (one sudden stop every 10 years).
- Probability of reentry after sudden stop ψ_s = 0.25 (mean sudden-stop duration 1.12 years).
- Discount factor β = 0.9745.
- Income cost of default φ_d(y) = d_0 y + d_1 y^2 with d_0 = −1.01683, d_1 = 1.18961.
- Income cost of sudden stops φ_s(y) = λ φ_d(y) with λ = 0.5.
- Endowment mean μ set so that μ(−1/2)σ_ε^2 is used in the process (as specified).
- Calibration targets include: mean and standard deviation of sovereign spread, mean debt level, and average accumulated income cost of a sudden stop targeted at 14 percent of annual income.

### Computation
- Solve finite-horizon approximation with horizon large enough that maximum deviation between first- and second-period value and bond-price functions ≤ 10^{−6}.
- Solution method: value function iteration; portfolio allocation solved over grids then refined with nonlinear optimization.
- Function approximations: linear interpolation over y, cubic spline over debt and reserves.
- Grids: 20 points for reserves, 20 points for debt, 25 points for income realizations.
- Expectations: 50 quadrature points for income shocks.

### Quantitative results: matching moments and facts
- Simulations match calibration targets and non-targeted moments (e.g., consumption vs. income volatility ratio) for Mexico-like business cycle properties.
- Fact 1: Model generates joint debt and reserve accumulation together with a significant default premium.
- Reserve accumulation:
  - Average reserve holdings in simulations represent 66 percent of short-term debt (i.e., debt maturing within a year).
  - This 66 percent figure is noted as close to the “Greenspan-Guidotti rule” prescribing full short-term debt coverage.
- Other model successes:
  - Generates gross capital flows dynamics consistent with data (Fact 2).
  - Long-duration bonds, sudden stops, and endogenous countercyclical default risk are important for quantitative success.
  - Allowing reserves to lower the probability of sudden stops increases reserve accumulation in the model.

*Source: _wp1333 - 5. If*

### Section II. shows that the accumulation of reserves financed by debt issuances to hedge

### _wp1333 - Section II. shows that the accumulation of reserves financed by debt issuances to hedge

### Key findings and quantitative importance
- Section II. shows that the accumulation of reserves financed by debt issuances to hedge rollover risk is a theoretical possibility with long-duration bonds and the simulations of the model indicate that this is not only a theoretical possibility but it is also quantitatively important.
- Average reserve holdings in the simulation are about 1/3 of average holdings in Mexico between 1994 and 2011.
- Reserve holdings in the simulations are close to holdings in Mexico in the second half of the 1990s; there has been fast growth in reserve accumulation in Mexico since the early 2000s.
- Extending the baseline model to allow reserves to lower the sudden-stop probability can account for up to 100 percent of the average reserve holdings in Mexico.
- Other motives for reserve accumulation not discussed in the paper could also account for a fraction of reserve holdings.

### Simulation results (selected moments and diagnostics)
- Table 2: Simulation Results (as presented)
  - Targeted moments
    - Mean Debt-to-GDP4243
    - Meanr s 3.43.4
    - σ(r s )1.31.5
    - Mean sudden stop income cost (% annualized)    1414
  - Non-Targeted moments
    - σ(c)/σ(y)1.31.2
    - σ(tb)1.31.4
    - ρ(tb,y)-0.5    -0.7
    - ρ(c,y)0.96   0.93
    - ρ(r s ,y)-0.4    -0.5
    - ρ(r s ,tb)0.30.6
    - Mean Reserves-to-GDP2.57.0
    - ρ(∆a,y)0.40.4
    - ρ(∆b,y)0.5    0.95
    - ρ(a,r s )-0.4    -0.2
  - Notes (preserved verbatim): The standard deviation ofxis denoted byσ(x). The coefficient of correlation betweenxandzis denoted byρ(x,z).  Changes in debt and reserves levels are denoted by ∆aand ∆b, respectively. Moments are computed using detrended series. Trends are computed using theHodrick-Prescott filter with a smoothing parameter of 1,600.  Moments for the simulations correspond to the mean value of each moment in 250 simulation samples, with each sample including 120 periods (30 years) without a default episode. Default episodes are excluded to improve comparability with the data; our samples start at least five years after a default. Consumption and income are expressed in logs. Due to data availability, debt statistics are at annual frequency.
- Table 3: Simulation Results with One-Period Bonds (as presented)
  - Benchmark  One-period bonds
  - Mean debt-to-GDP423
  - Mean reserves-to-GDP2.50.01
  - Meanr s 3.40.0
  - σ(r s )1.30.0
  - σ(tb)1.32.0

### Capital flows over the cycle and during sudden stops
- Simulations show purchases of domestic assets by non-residents (changes in debt levels) and purchases of foreign assets by residents (changes in reserve levels) are both procyclical.
- Procyclicality arises because income positively affects availability of credit: when borrowing conditions are good, a fraction of debt issuances are allocated to accumulate reserves; when borrowing conditions deteriorate, the government borrows less and sells reserves.
- Policy functions (Figure 4) for changes in debt and reserves as functions of current income (starting at mean debt and reserves in simulations) show:
  - Default threshold when not in a sudden stop: government repays for income shocks higher than -5.2percent and defaults otherwise.
  - Default threshold is higher in a sudden stop (government more likely to default when facing a sudden stop).
  - In repayment region and without a sudden stop, both borrowing and reserves increase with income; government increases reserve holdings when income is above trend consistent with permanent income hypothesis.
  - In the default region the government sells reserves (debt levels equal zero).
  - During sudden stops, government reduces borrowing and reserve accumulation, sells reserves and makes coupon payments; constrained government does not repurchase debt.
- Event analysis (Figure 5) shows model predicts a collapse in both inflows and outflows during sudden stops, consistent with Broner et al. (2012) Fact 2.

### Role of long-duration bonds
- Assuming government can issue long-duration bonds is critical to simultaneously generate significant levels of debt and reserves.
- With one-period bonds:
  - Mean debt-to-income ratio in simulations drops to 3 percent of annual income, compared with 42 percent in simulations with long-duration bonds.
  - Reserves drop to 0.01 percent compared with 2.5 percent in the benchmark with long-duration bonds.
- Three reasons why long-duration bonds matter:
  1. Long-duration bonds are essential for reserves to hedge rollover risk; with one-period bonds reserves play no role in insuring against future increases in borrowing costs.
  2. With one-period debt, government chooses low debt levels implying negligible default risk so portfolios with positive reserves offer little expansion in consumption space.
     - Example: with one-period bonds, government must roll over (or pay back) 100 percent of its debt each quarter; repaying the average value of debt in the data would require 160 percent of quarterly income.
     - With long-duration bonds, reserve holdings of 2.5 percent of income represent 66 percent of the average short-term debt and provide meaningful insurance.
  3. Long-duration bonds change the link between reserve accumulation and current cost of borrowing through the Euler equation with respect to reserves:
     - u′(t)(1 + (∂q t /∂a t+1 )(b t+1 −b t (1−δ))) =RβE t u′(t+ 1) (equation preserved).
     - The term (∂q t /∂a t+1 ) reflects how reserve accumulation affects the price at which the government issues new debt in equilibrium.
- With one-period debt, a Bulow-Rogoff type argument causes spreads to increase in reserves because higher reserves reduce the cost of defaulting (autarky more attractive); next-period default probability increases when reserves are accumulated, pushing spreads up.
- With long-duration debt, current spread reflects default probabilities in multiple future periods; accumulating reserves can lower expected future debt and default probabilities in other future periods, so spread may decrease with reserve holdings (Figures 6 and 7 illustrate these mechanisms and relationships).

### Role of sudden stops (sensitivity)
- Simulations with different sudden stop frequencies and costs show:
  - Higher frequency of sudden stops generates higher reserve holdings and lower debt levels.
  - Without sudden stops, reserve holdings decline from 2.5 percent of income in the benchmark to 0.4 percent.
  - Greater sudden-stop income loss leads government to choose higher reserve holdings and lower debt levels.
- Implication: perceived increases in frequency or cost of sudden stops (e.g., crises in late 1990s) can significantly increase optimal reserve accumulation.

### Role of endogenous and countercyclical sovereign spread
- Endogenous, countercyclical sovereign spread is key to generating demand for reserves in the model.
- A no-default version with constant exogenous spread (solve recursive problem W(b,a,y,s) = max a′ ≥0,b′,c {...} subject to budget and constraints with q∗ =1/(r∗+δ) and exogenous ̄B) yields:
  - Parameters r∗ and ̄B chosen to replicate mean spread and debt levels in Mexico.
  - Table 4 results: simulated reserve holdings decline from 2.5 percent of income in the benchmark to 0.1 percent with exogenous constant spread.
- Two factors driving this result:
  1. Rollover risk is lower in no-default model because borrowing opportunities are independent from income shocks.
  2. A model without default but with the spread level observed in the data overstates the financial cost of accumulating reserves financed by borrowing, because in the default model the government does not always repay debt yet still receives returns from reserves.

### Reserve accumulation for crisis prevention (reserves reduce sudden-stop probability)
- Following Jeanne and Ranciere (2011), assume sudden-stop probability:
  - ˆπ(a ̺(b)) = G(m − w a ̺(b)) (equation preserved),
  - where ̺(b) = b ∑ t=4 t=1 (1−δ) t−1 (1+r) t denotes short-term debt maturing within the next year, and G is standard normal CDF.
- Benchmark calibration corresponds to w= 0 with m set so probability of a sudden stop is 10 percent.
- Table 5: Simulation Results when Reserves Reduce the Probability of a Sudden Stop (w values and outcomes)
  - w=0 w= 0.05 w= 0.10 w= 0.15
  - Mean debt-to-GDP42424242
  - Mean reserves-to-GDP2.54.05.77.2
  - Sudden stops per 100 years10876
- As reserves become more effective at reducing sudden-stop probability (higher w), optimal reserve holdings increase; at w= 0.15 the model replicates the average reserve level in Mexico and reduces sudden-stop frequency from 10 episodes per 100 years to 6 episodes per 100 years.

### Conclusions and implications
- The model explains two salient features of international capital flows:
  1. Indebted governments hold large amounts of international reserves while the yield on their debt is significantly higher than the return on reserves.
  2. Non-resident purchases of domestic assets and resident purchases of foreign assets are procyclical and collapse during crises.
- Mechanism: government trades off insurance benefits of reserves against costs of larger gross debt positions; because default risk is countercyclical, government accumulates reserves and debt in good times and uses reserves in low-income/high-spread periods to repay debt and smooth consumption.
- Key ingredients for quantitative success: long-duration bonds, sudden stops, and countercyclical sovereign spreads.
- Suggested avenues for further research:
  - Study interaction of debt maturity structure and reserve holdings.
  - Explore relevance of mechanisms for corporate borrowers facing rollover risk.

*Italic source: _wp1333 - Section II. shows that the accumulation of reserves financed by debt issuances to hedge*

### References

### References

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### Appendix A — Proof of Proposition 1 (selected text)
- Without rollover risk, the optimal allocation is such thatc1=y1+y2(1 +r b) −1. Ifr a=r b (point 1 of Proposition 1), any combination of debt issuances and reserve holdings such thatb0 q0 =aandb1 =y2 −(1−δ)b0 attain the optimal allocation. In particular, the optimal allocation can be attained without reserve accumulation (a=b0 = 0, andb1 =y2).
- Ifr a< r b and there is no rollover risk (point 2 of Proposition 1), the government can only attain the optimal allocation if it chooses to not accumulate reserves. Let us consider any levels of period-0 savings and borrowing ˆa= ˆ b0 q0 >0. It is easy to show that the government can do better choosinga=b0 = 0. Sincer a < r b , ˆa(1 +r a )< ˆ b0 [1 + (1−δ)(1 +r b )] −1 . Therefore, the level of period-2 consumption is higher withb0 =a= 0 than with ˆa= ˆ b0 q0 >0, and ˆa= ˆ b0 q0 >0 cannot be part of an equilibrium.
- Suppose the government can only issue one-period debt andr a =r b (point 3 of Proposition 1). Sinceq0 = (1 +r a ) −1 ,c1 =y+b1 (1 +r b ) −1 for all possible equilibrium borrowing and saving choices satisfyingb0 q0 =a. Then, gross asset positions are undetermined and the optimal allocation can be attained without reserve accumulation (a=b0 = 0).
- Suppose now the government can only issue one-period debt andr a < r b (point 4 of Proposition 1). Let us consider any levels of period-0 savings and borrowing ˆa= ˆ b0 q0 >0. Then, period-1 consumption is given by c1 =y1 +b1 (1 +r b ) −1 + ˆ b(1 +r b ) −1 (1 +r a )− ˆ b < y+b1 (1 +r b ) −1 . Therefore, the level of period-1 consumption would be higher if the government choosesa=b0 = 0, and ˆa= ˆ b0 q0 >0 cannot be part of an equilibrium.
- Next, we show that condition (1) is sufficient for reserve accumulation (point 5 of Proposition 1). Sinceb0 q0 =a, the government’s well defined maximization problem can be written as: max b0 { πu(y1 +b0 q0 (1 +r a )−b0 ) + (1−π)u( y1 +b0 q0 (1 +r a )−b0 + y2 −(1−δ)b0 1 +r b ) } .
- The first-order condition of the government’s problem is given by: π[q0 (1 +r a )−1]u′(y1 +b0 q0 (1 +r a )−b0 )≤(8) (1−π) [ 1−δ 1 +r b + 1−q0 (1 +r a ) ] u′( y1 +b0 q0 (1 +r a )−b0 + y2 −(1−δ)b0 1 +r b ) .
- Condition (1) states that the left-hand side of condition (8) is higher than the right-hand side of condition (8) when evaluated atb0 = 0. Therefore, if condition (1) holds,a=b0 = 0 cannot be part of an equilibrium.

### Table 6 — Sudden-Stop Episodes
- Note: Sudden-stop episodes correspond to years in which the ratio of net capital inflows to GDP falls by more than 5 percentage points. Source: IMF’s International Financial Statistics annual data from 1970 to2011
- Argentina 1989, 2001
- Bolivia 1980, 1982, 1994
- Botswana 1977, 1987, 1991, 1993, 2001, 2003, 2010
- Brazil 1983
- Bulgaria 1990, 1994, 1996, 1998, 2003, 2008
- Chile 1982, 1985, 1991, 1995, 1998, 2009
- China, P.R.
- Colombia
- Costa Rica 2009
- Czech Republic 1996, 2003
- Dominican Republic 1993, 2003
- Ecuador 1979, 1986, 1988, 1992, 1999, 2006
- Egypt 1990, 1993
- El Salvador 1979, 1986, 2005, 2009
- Guatemala
- Honduras 2008
- Hungary 1994, 1996, 2009
- Jamaica 1983, 1985, 1988, 2002, 2009
- Jordan 1976, 1979, 1984, 1989, 1992, 1998, 2001, 2008, 2010
- Korea, Republic of 1986, 1997, 2008
- Malaysia 1984, 1987, 1994, 1999, 2005, 2008
- Mexico 1982, 1988, 1995
- Morocco 1978, 1995
- Paraguay 1988, 1995, 2002
- Peru 1983, 1998, 2009
- Philippines 1983, 1997, 2000
- Poland 1981, 1988, 1990
- Romania 1981, 1988, 2008
- South Africa 1985
- Sri Lanka
- Thailand 1982, 1997, 2009
- Tunisia
- Turkey 2001
- Uruguay 1982, 2002, 2004, 2007, 2009

*Source: _wp1333 - References (IMF PDF).*

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