## _wp0772 - 5. Sterilized Foreign Exchange Intervention - High Volatility of Fiscal Shocks

## Source details

**Canonical URL:** [_wp0772 - 5. Sterilized Foreign Exchange Intervention - High Volatility of Fiscal Shocks](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0772.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0772.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0772.pdf.json)

---

### Major contributions and setup
- Presents a general equilibrium monetary portfolio balance model for a small open economy that:
  - Endogenizes effects of government tax and spending policies on portfolio balance.
  - Demonstrates conditions under which domestic currency denominated government bonds are imperfect substitutes for foreign currency denominated bonds.
  - Analyzes policy implications: monetary policy can affect the level of exchange rate depreciation and inflation via a nominal anchor, and can affect interest rates and the volatility of exchange rate depreciation and inflation via balance sheet operations (sterilized foreign exchange intervention).
- Identifies two factors determining imperfect asset substitutability and effectiveness of sterilized interventions:
  - The volatility of exogenous fiscal spending shocks: the more volatile such shocks, the wider the range of portfolio shares over which sterilized interventions have a large impact.
  - The government’s initial balance sheet position: sterilized intervention has the largest effects on interest rates if there are only small outstanding amounts of domestic currency denominated government debt.
- Motivation: addresses tension between practice (central bankers using sterilized interventions) and theoretical critiques that often find sterilized intervention irrelevant under certain assumptions.

### Key model elements and sources of uncertainty
- Agents and preferences:
  - Continuum of identical infinitely lived households with time-separable logarithmic preferences: E0 ∫∞0 e−βt ln(c_t − y) dt, 0 < β < 1.
  - Consumption modeled in excess of constant endowment stream y.
- Financial assets available:
  - Money M_t (zero nominal return), domestic currency bonds Q_t (nominal return i^q_t dt), international foreign-currency bonds b_t (real return dr^b_t).
  - Real stocks: m_t = M_t / E_t, q_t = Q_t / E_t; total private financial wealth a_t = m_t + q_t + b_t.
  - Portfolio shares: n^m_t = m_t / a_t, n^q_t = q_t / a_t.
- Four sources of risk:
  - Three-dimensional Brownian motion B_t = [B^M_t B^α_t B^r_t]′ affecting dlog(M_t), dlog(α_t), dr^b_t.
  - One-dimensional Brownian motion W_t representing shocks to exogenous government spending dG_t.
- Distinction between shocks:
  - B_t-shocks: fiscal response endogenous (government redistributes resulting net fiscal revenue via lump-sum transfers).
  - W_t-shocks: fiscal spending shocks exogenous; exchange rate adjusts to balance the government’s budget; money supply is endogenous with respect to such shocks.
- Key stochastic processes (as in source):
  - dM_t / M_t = μ_t dt + σ_M dB_t + σ^g_{M,t} dW_t. (Equation (1))
  - dα_t / α_t = ν dt + σ_α dB_t. (Equation (2))
  - dr^b_t = r dt + σ_r dB_t. (Equation (3))
  - dG_t a_t = σ^g_g dW_t (zero drift). (Equation (4))
  - dE_t / E_t = ε_t dt + σ_{E,t} dB_t + σ^g_{E,t} dW_t. (Equation (5))
  - Cash constraint: c_t − y = α_t m_t = α_t n^m_t a_t. (Equation (12))
  - Household lump-sum taxes: dT_t a_t = τ_t dt + σ_{T,t} dB_t. (Equation (7))

### Government behavior and budget interactions
- Policy variables:
  - Nominal anchor via target path for money growth {μ_t}.
  - Target path for stock of nominal government debt {Q_t} (or equivalently interest rate on government debt {i^q_t}).
- Consolidated government and central bank budget constraint:
  - a_t τ_t dt + a_t σ_{T,t} dB_t + h_t dr^b_t = m_t dr^m_t + q_t dr^q_t + a_t σ^g_g dW_t. (Equation (19))
  - Assumption: expected budget balance always zero; net government wealth h_t − m_t − q_t does not change; initial condition h_0− = m_0− + q_0− implies h_t = m_t + q_t ∀ t. (Equation (20))
- Derived endogenous responses:
  - τ_t = n^q_t i^q_t − (n^m_t + n^q_t) [μ_r + ε_t − (σ_{E,t})^2 − (σ^g_{E,t})^2]. (Equation (21))
  - σ_{T,t} = −(n^m_t + n^q_t) (σ_{E,t} + σ_r). (Equation (22))
  - σ^g_{E,t} = σ^g_g (n^m_t + n^q_t). (Equation (23))
    - Interpretation: fiscally induced exchange rate volatility is increasing in σ^g_g and decreasing in the amount of nominal government liabilities in household portfolios.

### Household optimization and portfolio balance
- Hamilton-Jacobi-Bellman solution with conjectured value function:
  - J(a_t, t) = x [ ln(a_t) + ln(X(t; χ_t)) ], solution yields x = β−1. (Equations (25), (26))
- First-order conditions:
  - Consumption:
    - c_t = y + β a_t (1 + (i^q_t / α_t)). (Equation (27))
  - General equilibrium portfolio balance (key equation):
    - n^q_t + n^m_t = [ μ (i^q_t − r − ε_t + (σ_{E,t})^2 + (σ^g_{E,t})^2 + (σ_r)^2 + σ_{E,t} σ_r + 2γI) ] / [ μ ( (σ_{E,t})^2 + (σ^g_{E,t})^2) + (σ_r)^2 + 2σ_{E,t} σ_r ]. (Equation (28))
  - When σ^g_g = 0, UIP adjusted for Jensen terms:
    - i^q_t = r + ε_t − (σ_{E,t})^2 − (σ_r)^2 − σ_{E,t} σ_r. (Equation (29))
  - With σ^g_g > 0, extra risk discount:
    - i^q_t = r + ε_t − (σ_{E,t})^2 − (σ_r)^2 − σ_{E,t} σ_r − (σ^g_g)^2 (1 − n^m_t − n^q_t) / (n^m_t + n^q_t)^2. (Equation (30))
    - Interpretation: the additional risk discount depends on σ^g_g and the endogenous domestic-asset portfolio share n^m_t + n^q_t; a second monetary instrument (balance sheet operations or interest rate) can affect this portfolio share.

### Equilibrium system
- Ten equilibrium equations (equations (35)–(42)) determine ε_t, σ_{E,t} = [σ^M_{E,t} σ^α_{E,t} σ^r_{E,t}], σ^g_{E,t}, σ^g_{M,t}, n_t, i^q_t, c_t and E_t given policies μ_t and Q_t.
- Notation in calibration: n_t = n^m_t + n^q_t and n_t = (M̄ + Q_t) / E_t. (Equations (41), (42))

### Calibration and estimation (Mexico example)
- Sample: quarterly data from first quarter 1996 through second quarter 2004 (34 observations).
- Benchmark calibration parameters:
  - β = 0.04, y = 1, γ = 0.0001.
  - Assumption: fraction of consumption financed by financial asset income ((c − y) / c) = 0.02.
  - Recovered values: ̄α = 0.688, ̄E = 9.35, ̄M = 0.277.
  - Set i^q = 0.095 (average over second half of sample); recover ̄a from (27), n^m from (12); baseline n̄ = 0.241; implied ̄Q = 1.0311.
- Estimation of B_t shock processes (continuous-time geometric Brownian motion framework):
  - Rewritten processes used for estimation:
    - d log M_t = [ μ_t − 1/2 (σ_M)^2 − 1/2 (σ^g_{M,t})^2 ] dt + σ_M dB_t + σ^g_{M,t} dW_t. (Equation (43))
    - d log α_t = [ ν − 1/2 (σ_α)^2 ] dt + σ_α dB_t. (Equation (44))
    - dr^b_t = r dt + σ_r dB_t. (Equation (45))
  - Identification assumptions to separate σ^g_{M,t} from σ_M:
    - (σ^g_{M,t})^2 = 3 (σ_M)^2 (assumption).
    - Real interest rate exogenous: σ^M_r = σ^α_r = 0.
    - Money supply responds instantaneously to velocity shocks but not vice versa: σ^M_α = 0.
  - Estimated parameters:
    - μ = 0.0772, ν = −0.0178, r = 0.0229,
    - σ^M_M = 0.0159, σ^α_M = −0.0009, σ^r_M = 0.0026,
    - σ^α_α = 0.0294, σ^r_α = −0.0059, σ^r_r = −0.0186, σ^g_{M,t} = 0.0276.
- Calibration of σ^g_g:
  - Baseline solved endogenously gives σ^g_g = 0.0045.
  - Sensitivity analysis considers higher and lower σ^g_g.

### Policy implications and numerical results (sterilized vs unsterilized interventions)
- Policy instruments in experiments:
  - Inflation target via μ_t (held constant at estimated Mexican value).
  - Balance sheet operations via Q_t (or equivalently i^q_t).
- Experiment decomposition:
  - Unsterilized foreign exchange purchase: foreign bonds purchased in exchange for money M (increases M).
  - Sterilization via domestic open market operation (OMO): subsequent sale of domestic currency bonds Q against money M to keep M constant.
- Key numerical findings from Mexico calibration:
  - Unsterilized foreign exchange purchase (doubling nominal money supply in the experiment) is highly inflationary: causes a doubling of exchange rate and price level.
  - Unsterilized purchase effects:
    - Reduces real value of domestic bonds and overall share of domestic assets in portfolios.
    - Interest rate drops by around 0.15%.
    - Exchange rate volatility doubles to 0.4% in annual interest-equivalent terms.
    - Required tax rate rises; consumption is barely affected because velocity is extremely high in the calibration.
  - Sterilized intervention (foreign exchange purchase followed by OMO leaving M unchanged but increasing nominal domestic liabilities Q):
    - Exchange rate depreciation is reversed, but endogenous variables do not return to baseline.
    - Real quantity of outstanding domestic liabilities increases significantly, increasing interest rates, lowering exchange rate volatility, and lowering the required tax rate to balance the government budget.
    - Sterilized intervention therefore has significant real effects (and with lower velocity would significantly affect consumption).
- Role of initial debt share n_t and σ^g_g:
  - In baseline calibration (n_t just below 25%), issuing more nominal debt (increasing Q_t) raises i^q_t by around 0.30% as n_t rises from 7% to 30%, and fiscally induced exchange rate volatility σ^g_{E,t} falls from 0.5% to near zero.
  - From around a 25% debt share onwards, further expansions of nominal debt have modest effects on interest rates and exchange rate volatility; at such levels the nominal interest rate is within about 10 basis points of uncovered interest parity.
  - If σ^g_g is reduced by factor of four to 0.0011, balance sheet operations over the same Q range have almost no effect on interest rates and exchange rate volatility.
  - If σ^g_g is increased by factor of four, balance sheet operations become more potent.
- Explicit quantitative statement from source:
  - "0.018 increases the range over which balance sheet operations have very significant effects."

### Mechanism summary
- Raising i^q_t raises n_t because higher mean return on domestic bonds attracts holdings.
- A higher n_t reduces fiscally induced exchange-rate risk on domestic bonds via σ^g_{E,t} = σ^g_g n_t, reinforcing portfolio demand for domestic bonds.
- There is a monotonically increasing relationship between i^q_t and n_t, and a monotonically decreasing relationship between i^q_t and the risk discount.
- Balance sheet operations that change Q_t (or i^q_t) can act as a second monetary policy instrument affecting interest rates and volatility when σ^g_g > 0.

### Comparative context and interpretation
- Explains why empirical studies may find little evidence for sterilized intervention effects in industrialized countries:
  - Fiscal situations are generally more robust; fiscal dominance is much less of a problem.
  - These countries can issue substantial amounts of domestic currency denominated debt, so induced exchange rate volatility would be comparatively low.
- Developing countries:
  - Face serious fiscal dominance problems and have more difficulty issuing substantial stocks of domestic currency debt.
  - In such countries, sterilized intervention can act as "a second tool of monetary policy that gives the government autonomy to set nominal interest rates independently of the inflation target."

### Main conclusions from the general equilibrium model
- Relaxing the assumption of full lump-sum redistribution of stochastic seigniorage income and allowing exogenous fiscal spending shocks makes government balance sheet operations effective:
  - Balance sheet operations in domestic and foreign currency bonds change:
    - domestic interest rates,
    - the mean and variance of exchange rate depreciation and inflation,
    - household consumption and portfolio choices.
- Uncovered interest parity (UIP) fails in this economy and is replaced by "a general equilibrium portfolio balance equation."
- Quantitative implications:
  - "Sizeable currency risk discounts are obtained when a government’s nominal liabilities are small and when the volatility of its fiscal shocks is high."
  - "Borrowing risk premia are possible when a government issues large amounts of nominal liabilities."

### Analytical results and key equations (Appendix highlights)
- Real returns (Itô derivations):
  - dr_m_t = [−ε_t + (σ_E,t)^2 + (σ_gE,t)^2] dt − σ_E,t dB_t − σ_gE,t dW_t. (A.1)
  - dr_q_t = [i_q_t − ε_t + (σ_E,t)^2 + (σ_gE,t)^2] dt − σ_E,t dB_t − σ_gE,t dW_t. (A.2)
  - dr_b_t = r dt + σ_r dB_t. (A.3)
- Conjectured value function and differential equation for X(t;χ_t):
  - V(a_t,t) = e^{−βt} J(a_t,t) = e^{−βt} x[ln(a_t) + ln(X(t;χ_t))], with x = β^{−1}. (Appendix 2)
  - ̇X(t;χ_t)/X(t;χ_t) = β ln(X(t;χ_t)) − β ln(β) + β ln(1 + (i_q_t/α_t)) + β/(1 + (i_q_t/α_t)) − r + 1/2[(σ_r)^2 + (σ_g/g)^2] + γ I(n_m_t + n_q_t − 1)^2. (B.2)
  - Saddle path stability condition: ∂(̇X/X)/∂(̇X)|_{̇X=0} = β > 0, implying X(t;χ_t) is saddle path stable and uniquely determined. (B.3)
- Verification of optimality via HJB yields:
  - For admissible controls, −DJ(a_t,t) = ln(α_t n_m_t a_t − y) and J(a_0,0) = ∫_0^∞ e^{−βs} ln(α_s n_m_s a_s − y) ds, with feedback controls (n_q^*_t, n_m^*_t) optimal. (B.7–B.8)

### Policy and research implications
- Policy:
  - In economies with fiscal dominance and limited ability to issue domestic-currency debt, sterilized intervention can be an effective additional monetary instrument to influence nominal interest rates, inflation, and exchange rate risk characteristics.
- Research directions:
  - Provides apparatus for incorporating borrowing risk premia when governments issue large amounts of nominal liabilities.
  - Suggests extending the model to a two-country framework; such extensions are "the subject of ongoing work."

*Source: _wp0772 - 5. Sterilized Foreign Exchange Intervention - High Volatility of Fiscal Shocks (IMF Working Paper content provided).*

### 1. Appendix1:ReturnsonAssets .......................  25

### Appendix1:ReturnsonAssets

### Appendix listings and page references
- 1. Appendix1:ReturnsonAssets .......................  25
- 2. Appendix2:TheValueFunction ......................  25
- References..........................................  28

### Figures (titles as listed)
- 1.   Foreign Holdings of Mexican Peso Denominated Government Securities, Percent
ofTotal;Source:BancodeMexico.........................   8
- 2.   Unsterilized Foreign Exchange Purchase and Open Market Sale . . . . . . . . . . . . . . . . . . . . . . . . .   20
- 3.  SterilizedForeignExchangeIntervention-BaselineCase ............  21
- 4.   Sterilized Foreign Exchange Intervention - Low Volatility of Fiscal Shocks . . .   22

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0772.pdf*

### 5.   Sterilized Foreign Exchange Intervention - High Volatility of Fiscal Shocks  . .   23

### 5.   Sterilized Foreign Exchange Intervention - High Volatility of Fiscal Shocks

### Major contributions and setup
- Presents a general equilibrium monetary portfolio balance model for a small open economy that:
  - Endogenizes effects of government tax and spending policies on portfolio balance.
  - Demonstrates conditions under which domestic currency denominated government bonds are imperfect substitutes for foreign currency denominated bonds.
  - Analyzes policy implications: monetary policy can affect the level of exchange rate depreciation and inflation via a nominal anchor, and can affect interest rates and the volatility of exchange rate depreciation and inflation via balance sheet operations (sterilized foreign exchange intervention).
- Identifies two factors determining imperfect asset substitutability and effectiveness of sterilized interventions:
  - The volatility of exogenous fiscal spending shocks (shocks that induce budget-balancing exchange rate movements instead of being financed by endogenous tax responses). The more volatile such shocks, the wider the range of portfolio shares over which sterilized interventions have a large impact.
  - The government’s initial balance sheet position: sterilized intervention has the largest effects on interest rates if there are only small outstanding amounts of domestic currency denominated government debt.
- Motivation: tension between practice (central bankers in developing countries routinely use sterilized interventions intending to affect interest rates and real activity without changing money supply) and theoretical critiques (e.g., Obstfeld and Rogoff (1996), Backus and Kehoe (1989)) that emphasize the government budget constraint and often find sterilized intervention irrelevant under certain assumptions.

### Key model elements and sources of uncertainty
- Agents and preferences:
  - Continuum of identical infinitely lived households with time-separable logarithmic preferences: E0 ∫∞0 e−βt ln(c_t − y) dt, 0 < β < 1.
  - Consumption modeled in excess of constant endowment stream y.
- Financial assets available:
  - Money M_t (zero nominal return), domestic currency bonds Q_t (nominal return i^q_t dt), international foreign-currency bonds b_t (real return dr^b_t).
  - Real stocks: m_t = M_t / E_t, q_t = Q_t / E_t; total private financial wealth a_t = m_t + q_t + b_t.
  - Portfolio shares: n^m_t = m_t / a_t, n^q_t = q_t / a_t.
- Four sources of risk:
  - Three-dimensional Brownian motion B_t = [B^M_t B^α_t B^r_t]′ affecting dlog(M_t), dlog(α_t), dr^b_t.
  - One-dimensional Brownian motion W_t representing shocks to exogenous government spending dG_t.
- Distinction between shocks:
  - B_t-shocks: fiscal response endogenous (government redistributes resulting net fiscal revenue via lump-sum transfers).
  - W_t-shocks: fiscal spending shocks exogenous; exchange rate adjusts to balance the government’s budget; money supply is endogenous with respect to such shocks.
- Money supply dynamics (Itô process):
  - dM_t / M_t = μ_t dt + σ_M dB_t + σ^g_{M,t} dW_t. (Equation (1))
- Velocity:
  - dα_t / α_t = ν dt + σ_α dB_t. (Equation (2))
- International bond returns:
  - dr^b_t = r dt + σ_r dB_t. (Equation (3))
- Exogenous fiscal spending process:
  - dG_t a_t = σ^g_g dW_t (zero drift). (Equation (4))
- Exchange rate (nominal) dynamics:
  - dE_t / E_t = ε_t dt + σ_{E,t} dB_t + σ^g_{E,t} dW_t. (Equation (5))
- Cash constraint linking consumption and real money balances:
  - c_t − y = α_t m_t = α_t n^m_t a_t. (Equation (12))
- Household lump-sum taxes:
  - dT_t a_t = τ_t dt + σ_{T,t} dB_t. (Equation (7))

### Government behavior and budget interactions
- Government policy variables:
  - Nominal anchor via target path for money growth {μ_t}.
  - Target path for stock of nominal government debt {Q_t} (or equivalently interest rate on government debt {i^q_t}).
- Government budget constraint (consolidated government and central bank):
  - a_t τ_t dt + a_t σ_{T,t} dB_t + h_t dr^b_t = m_t dr^m_t + q_t dr^q_t + a_t σ^g_g dW_t. (Equation (19))
  - Assumption of expected budget balance always zero; net government wealth h_t − m_t − q_t does not change; initial condition h_0− = m_0− + q_0− implies h_t = m_t + q_t ∀ t. (Equation (20))
- Endogenous responses (derived from budget constraint and requirements):
  - τ_t = n^q_t i^q_t − (n^m_t + n^q_t) [μ_r + ε_t − (σ_{E,t})^2 − (σ^g_{E,t})^2]. (Equation (21))
  - σ_{T,t} = −(n^m_t + n^q_t) (σ_{E,t} + σ_r). (Equation (22))
  - σ^g_{E,t} = σ^g_g (n^m_t + n^q_t). (Equation (23))
    - Interpretation: fiscally induced exchange rate volatility is increasing in σ^g_g and decreasing in the amount of nominal government liabilities in household portfolios.

### Household optimization and portfolio balance
- Hamilton-Jacobi-Bellman approach with conjectured value function:
  - J(a_t, t) = x [ ln(a_t) + ln(X(t; χ_t)) ], solution yields x = β−1. (Equations (25), (26))
- First-order conditions lead to:
  - Consumption:
    - c_t = y + β a_t (1 + (i^q_t / α_t)). (Equation (27))
  - General equilibrium portfolio balance (key equation):
    - n^q_t + n^m_t = [ μ (i^q_t − r − ε_t + (σ_{E,t})^2 + (σ^g_{E,t})^2 + (σ_r)^2 + σ_{E,t} σ_r + 2γI) ] / [ μ ( (σ_{E,t})^2 + (σ^g_{E,t})^2) + (σ_r)^2 + 2σ_{E,t} σ_r ]  . (Equation (28), formatted per source)
  - When σ^g_g = 0, uncovered interest parity adjusted for Jensen terms:
    - i^q_t = r + ε_t − (σ_{E,t})^2 − (σ_r)^2 − σ_{E,t} σ_r. (Equation (29))
  - With σ^g_g > 0, additional risk discount term appears:
    - i^q_t = r + ε_t − (σ_{E,t})^2 − (σ_r)^2 − σ_{E,t} σ_r − (σ^g_g)^2 (1 − n^m_t − n^q_t) / (n^m_t + n^q_t)^2. (Equation (30))
    - Interpretation: the additional risk discount depends on σ^g_g and the endogenous domestic-asset portfolio share n^m_t + n^q_t; a second monetary instrument (balance sheet operations or interest rate) can affect this portfolio share.

### Equilibrium system (compact description)
- Ten equilibrium equations (equations (35)–(42)) determine ε_t, σ_{E,t} = [σ^M_{E,t} σ^α_{E,t} σ^r_{E,t}], σ^g_{E,t}, σ^g_{M,t}, n_t, i^q_t, c_t and E_t given policies μ_t and Q_t.
- Notation: n_t = n^m_t + n^q_t and n_t = (M̄ + Q_t) / E_t in the calibration context. (Equations (41), (42))

### Calibration and estimation (Mexico example)
- Sample: quarterly data from first quarter 1996 through second quarter 2004 (34 observations).
- Benchmark calibration parameters:
  - β = 0.04, y = 1, γ = 0.0001.
  - Assumption: fraction of consumption financed by financial asset income ((c − y) / c) = 0.02.
  - Recovered values: ̄α = 0.688, ̄E = 9.35, ̄M = 0.277.
  - Set i^q = 0.095 (average over second half of sample); recover ̄a from (27), n^m from (12); baseline n̄ = 0.241; implied ̄Q = 1.0311.
- Estimation of B_t shock processes (continuous-time geometric Brownian motion framework):
  - Rewritten processes:
    - d log M_t = [ μ_t − 1/2 (σ_M)^2 − 1/2 (σ^g_{M,t})^2 ] dt + σ_M dB_t + σ^g_{M,t} dW_t. (Equation (43))
    - d log α_t = [ ν − 1/2 (σ_α)^2 ] dt + σ_α dB_t. (Equation (44))
    - dr^b_t = r dt + σ_r dB_t. (Equation (45))
  - Model-consistent variance-covariance matrix Σ given in (46).
  - Identification assumptions to separate σ^g_{M,t} from σ_M:
    - (σ^g_{M,t})^2 = 3 (σ_M)^2 (assumption).
    - Real interest rate exogenous: σ^M_r = σ^α_r = 0.
    - Money supply responds instantaneously to velocity shocks but not vice versa: σ^M_α = 0.
  - Estimated parameters:
    - μ = 0.0772, ν = −0.0178, r = 0.0229,
    - σ^M_M = 0.0159, σ^α_M = −0.0009, σ^r_M = 0.0026,
    - σ^α_α = 0.0294, σ^r_α = −0.0059, σ^r_r = −0.0186, σ^g_{M,t} = 0.0276.
- Calibration of σ^g_g (exogenous fiscal volatility):
  - Solving baseline system endogenously gives σ^g_g = 0.0045.
  - Sensitivity analysis considers higher and lower σ^g_g.

### Policy implications and numerical results (sterilized vs unsterilized interventions)
- Policy instruments focused on:
  - Inflation target via μ_t (held constant at estimated Mexican value for experiments).
  - Balance sheet operations via Q_t (or equivalently i^q_t).
- Experiment decomposition:
  - Unsterilized foreign exchange purchase: foreign bonds purchased in exchange for money M (increases M).
  - Sterilization via domestic open market operation (OMO): subsequent sale of domestic currency bonds Q against money M to keep M constant.
- Key numerical/qualitative findings from Mexico calibration:
  - Unsterilized foreign exchange purchase (doubling nominal money supply in the experiment) is highly inflationary: causes a doubling of exchange rate and price level (figure description).
  - Unsterilized purchase effects:
    - Reduces real value of domestic bonds and overall share of domestic assets in portfolios.
    - Interest rate drops by around 0.15%.
    - Exchange rate volatility doubles to 0.4% in annual interest-equivalent terms.
    - Required tax rate rises; consumption is barely affected in calibration because velocity is extremely high.
  - Sterilized intervention (foreign exchange purchase followed by OMO that leaves M unchanged but increases nominal domestic liabilities Q):
    - Exchange rate depreciation is reversed, but endogenous variables do not return to baseline.
    - Real quantity of outstanding domestic liabilities increases significantly, increasing interest rates, lowering exchange rate volatility, and lowering the required tax rate to balance the government budget.
    - Sterilized intervention therefore has significant real effects (and with lower velocity would significantly affect consumption).
  - Role of initial debt share n_t and σ^g_g:
    - In baseline calibration (n_t just below 25%), issuing more nominal debt (increasing Q_t) raises i^q_t by around 0.30% as n_t rises from 7% to 30%, and fiscally induced exchange rate volatility σ^g_{E,t} falls from 0.5% to near zero.
    - From around a 25% debt share onwards, further expansions of nominal debt have modest effects on interest rates and exchange rate volatility; at such levels the nominal interest rate is within about 10 basis points of uncovered interest parity.
    - If σ^g_g is reduced (example: reduced by factor of four to 0.0011), balance sheet operations over same Q range have almost no effect on interest rates and exchange rate volatility.
    - If σ^g_g is increased (example: raised by factor of four), balance sheet operations become more potent (figure discussion truncated in source).
- Mechanism summary:
  - Raising i^q_t raises n_t (portfolio share of domestic assets) because higher mean return on domestic bonds attracts holdings.
  - A higher n_t reduces exchange-rate-induced risk on domestic bonds (via σ^g_{E,t} = σ^g_g n_t), reinforcing portfolio demand for domestic bonds.
  - Thus there is a monotonically increasing relationship between i^q_t and n_t, and a monotonically decreasing relationship between i^q_t and the risk discount; balance sheet operations that change Q_t (or i^q_t) can therefore act as a second monetary policy instrument affecting interest rates and volatility when σ^g_g > 0.
- Additional considerations:
  - The model rationalizes why portfolio balance channels might be more significant in developing countries: higher volatility of transitory government spending and smaller outstanding stocks of domestic-currency government debt increase the scope for sterilized intervention to affect interest rates.
  - The model connects to literature on interest rate risk premia and highlights currency risk originating from fiscal shocks and portfolio considerations as a powerful source of interest differentials beyond consumption-exchange rate covariance channels.

*Source: _wp0772 - 5.   Sterilized Foreign Exchange Intervention - High Volatility of Fiscal Shocks  . .   23 (IMF Working Paper content provided).*

### 0.018 increases the range over which balance sheet operations have very significant effects.

### _wp0772 - 0.018 increases the range over which balance sheet operations have very significant effects.

### Sterilized foreign exchange intervention: comparative context and mechanism
- Empirical studies may find little evidence for sterilized intervention effects in industrialized countries because:
  - "the fiscal situation is generally much more robust, and fiscal dominance is much less of a problem."
  - These countries can issue substantial amounts of domestic currency denominated debt, so "the induced exchange rate volatility would be comparatively low."
- Developing countries face the opposite scenario:
  - They "face serious fiscal dominance problems" (Catão and Terrones (2005)).
  - They "have much more difficulty issuing substantial stocks of domestic currency debt" (Eichengreen and Hausmann (2005)).
  - In such countries, sterilized intervention can act as "a second tool of monetary policy that gives the government autonomy to set nominal interest rates independently of the inflation target."

### Main conclusions from the general equilibrium model
- Relaxing the assumption of full lump-sum redistribution of stochastic seigniorage income and allowing exogenous fiscal spending shocks makes government balance sheet operations effective:
  - Balance sheet operations in domestic and foreign currency bonds change:
    - domestic interest rates,
    - the mean and variance of exchange rate depreciation and inflation,
    - household consumption and portfolio choices.
- Uncovered interest parity (UIP) fails in this economy and is replaced by "a general equilibrium portfolio balance equation."
- Quantitative implications from the model:
  - "Sizeable currency risk discounts are obtained when a government’s nominal liabilities are small and when the volatility of its fiscal shocks is high."
  - "Borrowing risk premia are possible when a government issues large amounts of nominal liabilities," though the paper does not focus on that aspect.

### Analytical results and key equations (Appendix highlights)
- Real return on money (Itô derivation) yields (A.1):
  - dr_m_t = [−ε_t + (σ_E,t)^2 + (σ_gE,t)^2] dt − σ_E,t dB_t − σ_gE,t dW_t.
- Real return on the domestic bond (A.2):
  - dr_q_t = [i_q_t − ε_t + (σ_E,t)^2 + (σ_gE,t)^2] dt − σ_E,t dB_t − σ_gE,t dW_t.
- Real return on international bonds (A.3):
  - dr_b_t = r dt + σ_r dB_t.
- Conjectured value function and implications (Appendix 2):
  - V(a_t,t) = e^{−βt} J(a_t,t) = e^{−βt} x[ln(a_t) + ln(X(t;χ_t))].
  - Equating terms on ln(a_t) yields x = β^{−1}. (B.1)
  - Differential equation for X(t;χ_t) (B.2):
    - ̇X(t;χ_t)/X(t;χ_t) = β ln(X(t;χ_t)) − β ln(β) + β ln(1 + (i_q_t/α_t)) + β/(1 + (i_q_t/α_t)) − r + 1/2[(σ_r)^2 + (σ_g/g)^2] + γ I(n_m_t + n_q_t − 1)^2.
  - Saddle path stability condition (B.3):
    - ∂(̇X/X)/∂(̇X)|_{̇X=0} = β > 0, implying X(t;χ_t) is saddle path stable and uniquely determined for each t and χ_t.
- Verification of optimality:
  - The Hamilton-Jacobi-Bellman verification leads to:
    - For arbitrary admissible controls, −DJ(a_t,t) = ln(α_t n_m_t a_t − y) and (B.7) J(a_0,0) = ∫_0^∞ e^{−βs} ln(α_s n_m_s a_s − y) ds.
    - For the feedback controls (n_q^*_t, n_m^*_t), equality holds (B.8), showing J(a_0,0) dominates other admissible controls and (n_q^*_t, n_m^*_t) are optimal.

### Policy and research implications
- Policy:
  - In economies with fiscal dominance and limited ability to issue domestic-currency debt, sterilized intervention can be an effective additional monetary instrument to influence nominal interest rates, inflation, and exchange rate risk characteristics.
- Research directions:
  - The paper provides an analytical apparatus for incorporating borrowing risk premia when governments issue large amounts of nominal liabilities.
  - The authors suggest extending the model to a two-country framework; such extensions are "the subject of ongoing work."

*Source: _wp0772 - 0.018 increases the range over which balance sheet operations have very significant effects.*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0772.pdf_