## 1. Conditional VaR and Density

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---

### Introduction: concept and motivation
- Central banks in floating or “free” floating arrangements do not rule out FX intervention (FXI); interventions can be interpreted as public insurance against exchange rate tail risks.
- Market failures (incomplete hedging markets, uninternalized risks, high hedging costs, shallow liquidity) can increase financial stability risks and justify public FX insurance.
- The paper designs an operational, rule-based framework (VaR rule) to make FXI predictable, budget-neutral over the medium term, robust to market manipulation, and adaptable to market conditions.
- The VaR rule: central bank absorbs a predetermined quantile θ of the conditional distribution of exchange rate returns; intervention regions ℛ_A( r_t ) are based on θ-quantiles Q(r_t | X_t-1, θ) and assumed symmetric for appreciation/depreciation in the paper.

### VaR interventions and exchange rate at risk (ERaR)
- Exchange rate at risk (ERaR) = the VaR: the θ-percentile of the conditional distribution of exchange rate returns r_t+1 given information X_t.
- VaR decision rule: intervene if r_t falls within ℛ_A defined by quantile thresholds (θ_down, θ_up); ℛ_A(r_t) = { r_t ≤ Q(r_t | X_t-1, θ_down) } ∪ { r_t ≥ Q(r_t | X_t-1, θ_up) }.
- Advantages of VaR rule:
  - Directly tied to financial stability objective (controls risk transferred to central bank via θ).
  - Progressive adjustment of exchange rate to new equilibrium while limiting overshooting.
  - Financially sound: with symmetric interventions the central bank buys and sells FX with same probability and tends to transact on the “good side” (sell high, buy low).
  - Robust to speculative attacks because the conditional distribution is updated in real time and depends on forward-looking variables.

### Empirical framework: model specification
- Model family: ARCH/GARCH, specifically an Exponential GARCH (EGARCH) extended with exogenous regressors (EGARCH-X) and Tskew-distributed innovations estimated by Maximum Likelihood.
- Model equations (as specified in source):
  - r_t = Intercept + Σ φ r_{t-1} + β X_t + ε_t  (AR-X(p) for drift)
  - log σ_t^2 = ω + β_g g(r_{t-1}) with g(r_t) = α r_t + γ (|r_t| − E|r_t|)^2  (volatility as exponential)
  - ε_t = σ_t ε_tilde,  ε_tilde ∼ Tskew(ν, λ)
- Model chosen: EGARCH(1,1) with Tskew innovations; lags selected by AIC/BIC.
- Rationale for EGARCH-X: parsimonious, captures nonlinearities and asymmetries, widely used and operationally implementable by central banks.
- Exogenous regressors included:
  - The average exchange rate bid-ask spread over the day (absolute).
  - The intraday spread (max–min over the day).
  - The daily interest rate differential between local and foreign currencies (vs. LIBOR).
  - The one-month forward exchange rate first difference.
  - The first difference of the VIX (Chicago Board Options Exchange Volatility Index).
  - EUR/USD log returns.
  - Oil prices log returns.
  - FX intervention dummy lag included in one specification.

### Estimation: sample and diagnostics
- Sample: daily Mexican peso data since 2001; Mexican peso floating since 1994.
- Intervention data available from Banco de México; model backtested in- and out-of-sample.
- Out-of-sample diagnostics and stylized facts:
  - Conditional volatility spikes during crises (2008 global financial crisis, March 2020 COVID-19).
  - Out-of-sample conditional density (expanding window Jan–end-Oct 2020) shows widening volatility and varying skewness during COVID-19.
  - “Fan chart” of forecasted quantiles illustrates uncertainty captured by GARCH through higher moments including skewness.
  - Probability integral transform (PIT) test over four-month out-of-sample daily data: empirical PIT distribution within confidence bands for all quantiles, indicating satisfactory out-of-sample accuracy.

### Estimation results: key coefficient signs and model fit
- Qualitative signs and interpretation:
  - Bid-ask spread: positive correlation → increase in bid-ask spread (illiquidity) signals depreciation.
  - Forward points (first difference): contributes to predicting log returns with expected sign.
  - Interbank rate vs LIBOR (interest differential): positive differential tends to appreciate the currency (carry-trade effect).
  - VIX first difference: increases in global risk sentiment tend to depreciate emerging market currencies.
  - EUR/USD log returns: moves in EUR/USD affect local currency returns as expected.
  - Oil prices log returns: increases in oil prices associated with local currency appreciation for Mexico.
  - Model explains around 28 percent of the log-return variance in the preferred specifications.
- Table 1: Results of the GARCH Estimates (selected coefficients and statistics preserved exactly)
  - Columns: Microstructure | CIP | Dollar move | Risk appetite | Baseline
  - Regressors (coefficients):
    - Intercept: -2.33*** | -2.29 | -1.84 | -2.55 | -1.63
    - Lag FX log returns: -0.07*** | -0.08 | -0.08*** | -0.08*** | -0.08***
    - Bid-ask abs: 5.71 | 24.39 | -35.66 | -2.42 | 3.23
    - Min-max abs: 35.56*** | 34.63 | 34.32 | 34.55* | 26.21
    - Forward points first difference: 23.29*** | 17.79*** | 26.44*** | 19.8*** | 19.44***
    - Interbank rate vs Libor: (blank) | 33.61*** | 39.32*** | 34.75*** | 33.86***
    - EUR/USD log returns: (blank) | (blank) | -0.14*** | -0.17*** | -0.16***
    - VIX first diff: (blank) | (blank) | (blank) | 15.66*** | 15.37***
    - Oil prices log returns: (blank) | (blank) | (blank) | (blank) | -0.02***
    - FX intervention dummy lag: (blank) | (blank) | (blank) | (blank) | 2.23
  - GARCH Parameters (exact values):
    - Omega: 0.13*** | 0.13 | 0.12*** | 0.11*** | 0.12***
    - Alpha: 0.17*** | 0.17* | 0.16*** | 0.16*** | 0.15***
    - Gamma: 0.07*** | 0.06*** | 0.06*** | 0.05*** | 0.05***
    - Beta: 0.98*** | 0.99*** | 0.99*** | 0.99*** | 0.99***
    - Nu: 8.33*** | 8.66*** | 8.92*** | 8.71*** | 8.54***
    - Lambda: 0.08*** | 0.07 | 0.09 | 0.07* | 0.08***
  - Goodness of fit and sample:
    - R2: 0.058 | 0.067 | 0.104 | 0.273 | 0.276
    - R2 adjusted: 0.058 | 0.066 | 0.104 | 0.272 | 0.275
    - Number of observations: 5986 | 5986 | 5682 | 5682 | 5680
  - Significance thresholds: *10%, **5%, ***1%

### Operational implications and advantages of VaR-based FXI
- VaR rule provides a single, forward-looking input aggregating microstructure and macro variables and thus simplifies intervention decision-making compared with large dashboards whose sub-indicators may conflict.
- Adaptive feature: conditional distribution and quantile thresholds evolve in real time as market participants change behavior, reducing predictability exploitable by speculators.
- The VaR rule allows the central bank to determine maximum FXI amounts given:
  - the reserve adequacy constraint, and
  - the minimum amount necessary to have a credible impact on the exchange rate.
- The approach encourages private sector hedging by leaving a degree of risk in the market and by making public insurance predictable but limited (quantile θ).

### Model benchmarking and methodological findings (Appendix II summary)
- Alternative models tested: unconditional kernel density estimator, quantile regressions, and variations around GARCH (including Gaussian, Tskew, GARCH versus EGARCH).
- Performance metric used: log score performance metrics; additional diagnostic: Probability Integral Transform Test.
- Main finding:
  - The GARCH family of models dominates the alternatives tested against an unconditional kernel density estimator and conditional density estimated from the quantile regression.
  - Within the GARCH family (Gaussian, Tskew, and GARCH versus EGARCH), there is no significant improvement in performance against the baseline model.

### Operational framework — Risk-Based Triggers and example thresholds
- VaR rule operational mechanics:
  - Central bank monitors cumulated returns of the exchange rate relative to the previous day.
  - If more than one intervention per business day were allowed, the second-intervention trigger would be based on cumulated return compared with the previous same-day trigger rate. (Paper assumes no more than one intervention per day.)
  - Intervention regions evolve daily as a function of market conditions; thresholds computed every day for each quantile (fan chart).
- Example threshold (specific day in the source):
  - FXI would occur only if exchange rate had depreciated by more than 3.3 percent or appreciated by more than 3.7 percent, for a 5 percent VaR (2.5 percent on each side).
- Representation:
  - VaR rule represented by the region where the cumulative distribution function falls in the central bank’s FXI zone, for example, the 2.5th and 97.5th percentiles.

### Empirical counterfactuals and frequency of interventions (Mexico, Jan–Oct 2020)
- Had the BM followed the VaR FXI rule, it might have intervened in about 15 days over ten months, from January through October 2020.
- Fifteen days of intervention correspond to about 7.5 percent of the period, even when using an intervention region of 5 percent.
- Frequency could increase in short term if volatility is unusually high (as during COVID-19), but actual interventions would likely reduce volatility and thus future interventions.
- With a VaR of 2.5 percent both for depreciation and appreciation, the central bank will, on average and over time, intervene 5 percent of the time.
- The fixed-frequency feature allows ex-ante calibration of intervention budget.

### FX intervention calibration under risk-based interventions — practical implications
- Budget calibration inputs:
  - Fixed FXI frequency (from VaR),
  - Maximum FXI amount (single intervention or per business day),
  - FX reserve adequacy constraint.
- Advantages over fixed-volatility rules:
  - Known intervention frequency ex ante enables planning of intervention budget.
  - VaR triggers interventions when price movements are unusually large, reinforcing intervention impact.
  - Symmetric VaR (same VaR for appreciation and depreciation) leads to intervening same number of times on both sides and is budget neutral for the central bank over the medium term.
- Calibration trade-offs:
  - Maximum daily intervention should be large enough to influence exchange rate.
  - If budget constraints bind, accepting higher VaR (lower intervention frequency) can keep daily interventions credible.

### The case of Mexico — descriptive benchmarking of auctions
- Auction types used by BM:
  - Auctions with a minimum (reservation) rate: preannounced reservation rate = previous day’s close × multiplier.
  - Auctions without a minimum rate: organized opportunistically; sometimes auctions were preannounced without minimum rate and preannounced amounts later adjusted.
- Auctions with a minimum rate:
  - Triggered 62 times from October 2008 through February 2016.
  - Reservation rate multipliers:
    - × 1.02 from October 9, 2008, to December 8, 2014.
    - × 1.015 from December 9, 2014, to November 23, 2015.
    - × 1.01 from November 24, 2015, to February 17, 2016.
  - Ex-post benchmarking (using GARCH-estimated conditional CDF):
    - Most FXIs with minimum rate occurred near the top of the conditional distribution, often above the 95th percentile.
    - A few outliers occurred at the median or below, but these were rare.
- Auctions without a minimum rate:
  - 322 auctions organized from September 2009 through November 2015.
  - Ex-post benchmarking (conditional CDF):
    - Interventions without minimum rate exhibited no clear risk pattern: they occurred across the full conditional distribution, including during appreciation episodes.
    - Selling USD during appreciation episodes could have amplified currency appreciation.

### Comparative static financial performances (Appendix I)
- Exercise setup:
  - Comparative static evaluation of three intervention strategies: no minimum rate, minimum rate (rule-based), and VaR rule.
  - Intervention sample period: one-year between October 2015 and October 2016.
  - Frequency of interventions used for comparison: 18 interventions for each strategy.
  - Calibration for VaR rule: 10 percent intervention threshold; only on the selling side to match other strategies; intervention amount calibrated so total FXIs under the VaR rule exactly matched the intervention amount under the no minimum rate strategy.
- Key comparative findings (reported metrics):
  - Daily variation bps:
    - No minimum rate: -6.1
    - Minimum rate: 108.8
    - VaR rule: 164.8
  - Average exchange rate:
    - No minimum rate: 16.6
    - Minimum rate: 17.8
    - VaR rule: 18
  - FX performance against without minimum rate:
    - No minimum rate: 0%
    - Minimum rate: 6.90%
    - VaR rule: 8.30%
  - Total volume bn USD:
    - No minimum rate: 3.32
    - Minimum rate: 3.6
    - VaR rule: 3.32
  - Number of interventions:
    - No minimum rate: 18
    - Minimum rate: 18
    - VaR rule: 18
- Interpretation:
  - No minimum rate rule resulted in selling USD when local currency was slightly appreciating (daily variation bps = -6.1).
  - Minimum rate and VaR rule sold USD at more favorable times (higher daily variation bps and higher average exchange rates).
  - VaR rule selects days with particularly large depreciation (reported as 164.8 bps).
  - Minimum rate outperforms no minimum rate by 6.90%; VaR rule outperforms no minimum rate by 8.30%.

### Out-of-sample benchmarking (Appendix II): models, tests, and results
- Objective: present alternative models to estimate the conditional density of foreign exchange log returns over time.
- Baseline model: EGARCH(1,1) with a Tskew parametrization of the error terms’ distribution.
- Models benchmarked:
  - Parametric variations: EGARCH(1,1) with Gaussian errors; GARCH(1,1) with Tskew; GARCH(1,1) with Gaussian errors.
  - Density via quantile regressions with rearrangement resampling.
  - Unconditional distribution via Gaussian KDE on historical data.
- Evaluation metrics and testing procedure:
  1. PIT tests: “Pass the PIT test” if estimated invert CDF distribution lies within the uniform distribution +/- 5 percent interval confidence bands.
  2. Density log score (uncensored): report log score diff against baseline and p-value of the test statistic.
- Benchmark results (PIT and log score differences):
  - Baseline:
    - PIT: Pass
  - Unconditional:
    - PIT: Fail
    - Log score diff against baseline: -6.36
    - Diff p-value: 0
  - Quantile reg:
    - PIT: Pass
    - Log score diff against baseline: -2.09
    - Diff p-value: 0.02
  - Gaussian EGARCH:
    - PIT: Fail
    - Log score diff against baseline: 1.235
    - Diff p-value: 0.892
  - TSkew GARCH:
    - PIT: Fail
    - Log score diff against baseline: 1.537
    - Diff p-value: 0.938
  - Gaussian GARCH:
    - PIT: Fail
    - Log score diff against baseline: 1.86
    - Diff p-value: 0.969
- Interpretation:
  - The baseline model outperforms the unconditional distribution and the quantile-regression fitted density by log score.
  - Variations of GARCH show higher log scores but differences are not statistically significant given p-values.
  - Conclusion: GARCH specification is appropriate to model FX returns; variations around GARCH and error distributions are statistically equivalent in this out-of-sample comparison.

### Conclusion and policy implications
- Main argument: FXIs would benefit from a risk-based framework; VaR is commonly used by central banks for other risk measures and can be applied to FXI operational rules.
- Policy implications and advantages of VaR rule:
  - Helps central banks committed to floating exchange rates manage financial stability risks from exchange rate movements.
  - Can accompany transition to exchange rate flexibility by formalizing commitment to float and gradually increasing VaR (reducing intervention frequency).
  - Can deepen hedging markets by gradually transferring exchange rate risk to market participants.
  - Advantages relative to other rules: control of exchange rate risk exposure, budget planning based on fixed frequency and maximum amounts, medium-term budget neutrality (if symmetric), forward-looking, and ability to combine several exogenous variables into one trigger.
- Limitations and extensions:
  - VaR rule does not identify nature of shocks or guarantee optimality of each FXI.
  - Determining optimal degree of hedging provided by central bank versus left to market participants remains a policy judgment; a method to estimate vulnerabilities and combine impacts on macro-financial stability is left for future research.
  - VaR methodology can be adapted to other variables (e.g., tail UIP deviations, spread between forward rates and covered interest parity to detect hedging market impairments).

*Source: wpiea2021032-print-pdf - 1. Conditional VaR and Density (IMF).*

### 1. Conditional VaR and Density .........................................................................................

### 1. Conditional VaR and Density

### Contents
- 1. Conditional VaR and Density ................................................................................................6
- 2. Mexican Peso against U.S. Dollar .......................................................................................10
- 3. Conditional FX Volatility Over Time ..................................................................................13
- 4. Out-of-sample Conditional Density .....................................................................................14
- 5. Out-of-sample Fan Chart .....................................................................................................15
- 6. Probability Integral Transform Test.....................................................................................16
- 7. VaR FX Intervention Rule Based on a Given Information Set ...........................................17
- 8. Conditional Cumulative Distribution Function and Intervention Thresholds .....................18
- 9. Conditional VaR Exceedance, Out-of-Sample ....................................................................19
- 10. FX Interventions Log Return with Minimum Rate ............................................................21
- 11. Conditional CDF of FX Intervention with Minimum Rate ...............................................22
- 12. FX Interventions Without Minimum Rate on the Mexican Peso/U.S. Dollar ...................23
- 13. Conditional CDF of FX Intervention Without Minimum Rate .........................................24

*Source: wpiea2021032-print-pdf - 1. Conditional VaR and Density (canonical PDF listing).*

### 1. Results of the GARCH Estimates ......................................................................................

### 1. Results of the GARCH Estimates

### Introduction: concept and motivation
- Central banks in floating or “free” floating arrangements do not rule out FX intervention (FXI); interventions can be interpreted as public insurance against exchange rate tail risks.
- Market failures (incomplete hedging markets, uninternalized risks, high hedging costs, shallow liquidity) can increase financial stability risks and justify public FX insurance.
- The paper designs an operational, rule-based framework (VaR rule) to make FXI predictable, budget-neutral over the medium term, robust to market manipulation, and adaptable to market conditions.
- The VaR rule: central bank absorbs a predetermined quantile θ of the conditional distribution of exchange rate returns; intervention regions ℛ_A( r_t ) are based on θ-quantiles Q(r_t | X_t-1, θ) and assumed symmetric for appreciation/depreciation in the paper.

### VaR interventions and exchange rate at risk
- Exchange rate at risk (ERaR) = the VaR: the θ-percentile of the conditional distribution of exchange rate returns r_t+1 given information X_t.
- VaR decision rule: intervene if r_t falls within ℛ_A defined by quantile thresholds (θ_down, θ_up); ℛ_A(r_t) = { r_t ≤ Q(r_t | X_t-1, θ_down) } ∪ { r_t ≥ Q(r_t | X_t-1, θ_up) }.
- Advantages of VaR rule emphasized:
  - Directly tied to financial stability objective (controls risk transferred to central bank via θ).
  - Progressive adjustment of exchange rate to new equilibrium while limiting overshooting.
  - Financially sound: with symmetric interventions the central bank buys and sells FX with same probability and tends to transact on the “good side” (sell high, buy low).
  - Robust to speculative attacks because the conditional distribution is updated in real time and depends on forward-looking variables.

### Empirical framework: model specification
- Model family: ARCH/GARCH, specifically an Exponential GARCH (EGARCH) extended with exogenous regressors (EGARCH-X) and Tskew-distributed innovations estimated by Maximum Likelihood.
- Equations (as specified in the source):
  - r_t = Intercept + Σ φ r_{t-1} + β X_t + ε_t  (AR-X(p) for drift)
  - log σ_t^2 = ω + β_g g(r_{t-1}) with g(r_t) = α r_t + γ (|r_t| − E|r_t|)^2  (volatility as exponential)
  - ε_t = σ_t ε_tilde,  ε_tilde ∼ Tskew(ν, λ)
- Model chosen: EGARCH(1,1) with Tskew innovations; lags selected by AIC/BIC.
- Rationale for EGARCH-X: parsimonious, captures nonlinearities and asymmetries, widely used and operationally implementable by central banks.

- Exogenous regressors included in EGARCH-X specification:
  - The average exchange rate bid-ask spread over the day (absolute).
  - The intraday spread (max–min over the day).
  - The daily interest rate differential between local and foreign currencies (vs. LIBOR).
  - The one-month forward exchange rate first difference.
  - The first difference of the VIX (Chicago Board Options Exchange Volatility Index).
  - EUR/USD log returns.
  - Oil prices log returns.
  - (Additional: FX intervention dummy lag included in one specification.)

### Estimation: sample and diagnostics
- Sample: daily Mexican peso data since 2001; Mexican peso floating since 1994.
- Intervention data available from Banco de México; model backtested in- and out-of-sample.
- Out-of-sample diagnostics:
  - Conditional volatility spikes during crises (2008 global financial crisis, March 2020 COVID-19).
  - Out-of-sample conditional density (expanding window Jan–end-Oct 2020) shows widening volatility and varying skewness during COVID-19.
  - “Fan chart” of forecasted quantiles illustrates uncertainty captured by GARCH through higher moments including skewness.
  - Probability integral transform (PIT) test over four-month out-of-sample daily data: empirical PIT distribution within confidence bands for all quantiles, indicating satisfactory out-of-sample accuracy.

### Estimation results: key coefficient signs and model fit
- Qualitative signs and interpretation (consistent with theory):
  - Bid-ask spread: positive correlation → increase in bid-ask spread (illiquidity) signals depreciation.
  - Forward points (first difference): contributes to predicting log returns with expected sign.
  - Interbank rate vs LIBOR (interest differential): positive differential tends to appreciate the currency (carry-trade effect).
  - VIX first difference: increases in global risk sentiment tend to depreciate emerging market currencies.
  - EUR/USD log returns: moves in EUR/USD affect local currency returns as expected.
  - Oil prices log returns: increases in oil prices associated with local currency appreciation for Mexico.
  - Model explains around 28 percent of the log-return variance in the preferred specifications.

- Table 1: Results of the GARCH Estimates (selected coefficients and statistics preserved exactly)
  - Columns: Microstructure | CIP | Dollar move | Risk appetite | Baseline
  - Regressors (coefficients shown exactly as in source):
    - Intercept: -2.33*** | -2.29 | -1.84 | -2.55 | -1.63
    - Lag FX log returns: -0.07*** | -0.08 | -0.08*** | -0.08*** | -0.08***
    - Bid-ask abs: 5.71 | 24.39 | -35.66 | -2.42 | 3.23
    - Min-max abs: 35.56*** | 34.63 | 34.32 | 34.55* | 26.21
    - Forward points first difference: 23.29*** | 17.79*** | 26.44*** | 19.8*** | 19.44***
    - Interbank rate vs Libor: (blank) | 33.61*** | 39.32*** | 34.75*** | 33.86***
    - EUR/USD log returns: (blank) | (blank) | -0.14*** | -0.17*** | -0.16***
    - VIX first diff: (blank) | (blank) | (blank) | 15.66*** | 15.37***
    - Oil prices log returns: (blank) | (blank) | (blank) | (blank) | -0.02***
    - FX intervention dummy lag: (blank) | (blank) | (blank) | (blank) | 2.23
  - GARCH Parameters (exact values):
    - Omega: 0.13*** | 0.13 | 0.12*** | 0.11*** | 0.12***
    - Alpha: 0.17*** | 0.17* | 0.16*** | 0.16*** | 0.15***
    - Gamma: 0.07*** | 0.06*** | 0.06*** | 0.05*** | 0.05***
    - Beta: 0.98*** | 0.99*** | 0.99*** | 0.99*** | 0.99***
    - Nu: 8.33*** | 8.66*** | 8.92*** | 8.71*** | 8.54***
    - Lambda: 0.08*** | 0.07 | 0.09 | 0.07* | 0.08***
  - Goodness of fit and sample:
    - R2: 0.058 | 0.067 | 0.104 | 0.273 | 0.276
    - R2 adjusted: 0.058 | 0.066 | 0.104 | 0.272 | 0.275
    - Number of observations: 5986 | 5986 | 5682 | 5682 | 5680
  - Significance thresholds: *10%, **5%, ***1%

### Operational implications and advantages of VaR-based FXI
- VaR rule provides a single, forward-looking input aggregating microstructure and macro variables and thus simplifies intervention decision-making compared with large dashboards whose sub-indicators may conflict.
- Adaptive feature: conditional distribution and quantile thresholds evolve in real time as market participants change behavior, reducing predictability exploitable by speculators.
- The VaR rule allows the central bank to determine maximum FXI amounts given:
  - the reserve adequacy constraint, and
  - the minimum amount necessary to have a credible impact on the exchange rate.
- The approach encourages private sector hedging by leaving a degree of risk in the market and by making public insurance predictable but limited (quantile θ).

*Italic source attribution: wpiea2021032-print-pdf - 1. Results of the GARCH Estimates (IMF).*

### Appendix II presents a series of alternative models (unconditional, quantile regressions, and

### Appendix II presents a series of alternative models (unconditional, quantile regressions, and

### Model benchmarking and methodological findings
- Alternative models tested: unconditional kernel density estimator, quantile regressions, and variations around GARCH (including Gaussian, Tskew, GARCH versus EGARCH).
- Performance metric used: log score performance metrics; additional diagnostic: Probability Integral Transform Test (Figure 6 referenced).
- Main finding:
  - The GARCH family of models dominates the alternatives tested against an unconditional kernel density estimator and conditional density estimated from the quantile regression.
  - Within the GARCH family (Gaussian, Tskew, and GARCH versus EGARCH), there is no significant improvement in performance against the baseline model.

### Operational framework — Risk-Based Triggers
- Concept:
  - The VaR rule derives from a simple time series model; requires a limited amount of publicly available data and is easy to operationalize and communicate.
  - VaR triggers are transparent, part of central bank communication, and suitable for wholesale FX interventions (FXIs).
- Operational mechanics:
  - Central bank monitors cumulated returns of the exchange rate relative to the previous day.
  - If more than one intervention per business day were allowed, the second-intervention trigger would be based on cumulated return compared with the previous same-day trigger rate. (Paper assumes no more than one intervention per day.)
  - Intervention regions evolve daily as a function of market conditions; thresholds computed every day for each quantile (fan chart, Figure 5).
- Example threshold (specific day in Figure 7):
  - FXI would occur only if exchange rate had depreciated by more than 3.3 percent or appreciated by more than 3.7 percent, for a 5 percent VaR (2.5 percent on each side).
- Representation:
  - VaR rule represented by the region where the cumulative distribution function falls in the central bank’s FXI zone, for example, the 2.5th and 97.5th percentiles (Figure 8).

### Empirical counterfactuals and frequency of interventions
- Mexico, Jan–Oct 2020 counterfactual:
  - Had the BM followed the VaR FXI rule, it might have intervened in about 15 days over ten months, from January through October 2020.
  - Fifteen days of intervention correspond to about 7.5 percent of the period, even when using an intervention region of 5 percent.
  - Frequency could increase in short term if volatility is unusually high (as during COVID-19), but actual interventions would likely reduce volatility and thus future interventions.
- Properties of the VaR rule:
  - With a VaR of 2.5 percent both for depreciation and appreciation, the central bank will, on average and over time, intervene 5 percent of the time.
  - The fixed-frequency feature allows ex-ante calibration of intervention budget.

### FX intervention calibration under risk-based interventions — practical implications
- Budget calibration inputs:
  - Fixed FXI frequency (from VaR),
  - Maximum FXI amount (single intervention or per business day),
  - FX reserve adequacy constraint.
- Advantages over fixed-volatility rules:
  - Known intervention frequency ex ante enables planning of intervention budget.
  - VaR triggers interventions when price movements are unusually large, reinforcing intervention impact.
  - Symmetric VaR (same VaR for appreciation and depreciation) leads to intervening same number of times on both sides and is budget neutral for the central bank over the medium term.
- Calibration trade-offs:
  - Maximum daily intervention should be large enough to influence exchange rate.
  - If budget constraints bind, accepting higher VaR (lower intervention frequency) can keep daily interventions credible.

### The case of Mexico — descriptive benchmarking
- Auction types used by BM:
  - Auctions with a minimum (reservation) rate: preannounced reservation rate = previous day’s close × multiplier.
  - Auctions without a minimum rate: organized opportunistically; sometimes auctions were preannounced without minimum rate and preannounced amounts later adjusted.
- Auctions with a minimum rate:
  - Triggered 62 times from October 2008 through February 2016.
  - Reservation rate multipliers:
    - × 1.02 from October 9, 2008, to December 8, 2014 (example: 2 percent depreciation).
    - × 1.015 from December 9, 2014, to November 23, 2015.
    - × 1.01 from November 24, 2015, to February 17, 2016.
  - Ex-post benchmarking (using GARCH-estimated conditional CDF):
    - Most FXIs with minimum rate occurred near the top of the conditional distribution, often above the 95th percentile.
    - A few outliers occurred at the median or below, but these were rare.
    - Conclusion: interventions with a minimum rate were often—but not always—consistent with a risk framework (central bank intervened most of the time when depreciation pressure was largest).
- Auctions without a minimum rate:
  - 322 auctions organized from September 2009 through November 2015.
  - These auctions overlapped with auctions with a minimum rate; mere presence of the rule could influence volatility distribution even without intervention.
  - Ex-post benchmarking (conditional CDF):
    - Interventions without minimum rate exhibited no clear risk pattern: they occurred across the full conditional distribution, including during appreciation episodes.
    - Selling USD during appreciation episodes could have amplified currency appreciation.
    - Suggests BM rationales for auctions without a minimum rate likely included objectives beyond exchange rate volatility—e.g., preventing excessive accumulation of foreign reserves.

### Conclusion and policy implications
- Main argument:
  - FXIs would benefit from a risk-based framework; VaR is commonly used by central banks for other risk measures and can be applied to FXI operational rules.
- Policy implications and advantages of VaR rule:
  - Helps central banks committed to floating exchange rates manage financial stability risks from exchange rate movements.
  - Can accompany transition to exchange rate flexibility by formalizing commitment to float and gradually increasing VaR (reducing intervention frequency).
  - Can deepen hedging markets by gradually transferring exchange rate risk to market participants.
  - Advantages relative to other rules: control of exchange rate risk exposure, budget planning based on fixed frequency and maximum amounts, medium-term budget neutrality (if symmetric), forward-looking, and ability to combine several exogenous variables into one trigger.
- Limitations and extensions:
  - VaR rule does not identify nature of shocks or guarantee optimality of each FXI.
  - Determining optimal degree of hedging provided by central bank versus left to market participants remains a policy judgment; a method to estimate vulnerabilities and combine impacts on macro-financial stability is left for future research.
  - VaR methodology can be adapted to other variables (e.g., tail UIP deviations, spread between forward rates and covered interest parity to detect hedging market impairments).

*Source: Authors’ calculations and analysis in the provided content.*

### APPENDIX I. COMPARATIVE STATIC FINANCIAL PERFORMANCES

### APPENDIX I. COMPARATIVE STATIC FINANCIAL PERFORMANCES

### Overview of exercise
- Comparative static evaluation of three intervention strategies:
  - Strategy without a minimum rate (no minimum rate).
  - Strategy with a minimum rate (rule-based).
  - VaR rule strategy.
- Intervention sample period: one-year between October 2015 and October 2016.
- Frequency of interventions used for comparison: 18 interventions for each strategy.
- Calibration for VaR rule:
  - 10 percent intervention threshold.
  - Only on the selling side to match other strategies.
  - Intervention amount calibrated so total FXIs under the VaR rule exactly matched the intervention amount under the no minimum rate strategy.

### Financial benchmarking caveat
- Exercise is purely comparative and does not account for potential selection effects (e.g., some no-minimum-rate interventions might have complemented minimum-rate interventions and been executed at less favorable times).
- Results should not be interpreted as definitive proof of the financial performance of one strategy against another.

### Key comparative findings
- Summary performance metrics (as reported):
  - Daily variation bps:
    - No minimum rate: -6.1
    - Minimum rate: 108.8
    - VaR rule: 164.8
  - Average exchange rate:
    - No minimum rate: 16.6
    - Minimum rate: 17.8
    - VaR rule: 18
  - FX performance against without minimum rate:
    - No minimum rate: 0%
    - Minimum rate: 6.90%
    - VaR rule: 8.30%
  - Total volume bn USD:
    - No minimum rate: 3.32
    - Minimum rate: 3.6
    - VaR rule: 3.32
  - Number of interventions:
    - No minimum rate: 18
    - Minimum rate: 18
    - VaR rule: 18
- Interpretation from the benchmarking:
  - Under the no minimum rate rule, the central bank was selling USD when the local currency was slightly appreciating: daily variation bps = -6.1 (negative value denotes appreciation).
  - Under the minimum rate and VaR rule schemes, the central bank was selling USD at more favorable times (higher daily variation bps and higher average exchange rates).
  - The VaR rule selects days with particularly large depreciation (order of 160 bps; reported as 164.8 bps).
  - Although the VaR rule targets tails of the conditional distribution of log returns and does not guarantee the most favorable exchange rate level, the simulation shows it triggers interventions on average under better terms than the two other schemes.
  - Performance versus the strategy without a minimum rate (selling the same volume at different times):
    - Minimum rate intervention outperforms no minimum rate by 7 percent (reported as 6.90%).
    - VaR rule outperforms no minimum rate by 8.3 percent (reported as 8.30%).

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### APPENDIX II. OUT-OF-SAMPLE BENCHMARKING

### Objective and baseline
- Objective: present alternative models to estimate the conditional density of foreign exchange log returns over time.
- Baseline model: EGARCH(1,1) with a Tskew parametrization of the error terms’ distribution.
- All conditional benchmark models use the same sets of variables and the same training/testing set as the baseline.

### Models benchmarked
- Parametric variations of GARCH:
  - EGARCH(1,1) with Gaussian errors
  - GARCH(1,1) with Tskew
  - GARCH(1,1) with Gaussian errors
- Density estimation via quantile regressions:
  - Full-fledged density obtained via quantile interpolation and resampling, following the rearrangement procedure of Chernozhukov and others (2010).
- Unconditional distribution:
  - Estimated via Gaussian KDE on historical data.

### Evaluation metrics and testing procedure
- Out-of-sample performance assessed using:
  1. PIT tests
     - Definition of “pass the PIT test”: if the estimated invert CDF distribution lies within the uniform distribution +/- 5 percent interval confidence bands, as defined in Rossi and Sekhposyan (2019); otherwise “fail”.
  2. Density log score (uncensored version), following Diks and others (2011)
     - Reported items:
       - Log score diff against baseline (average difference of the log score across time).
       - p-value of the test statistic (null hypothesis: log scores of the two models are the same).
     - Interpretation: positive test statistic indicates the model outperforms the baseline.

### Benchmark results (PIT and log score differences)
- Baseline:
  - PIT: Pass
  - Log score diff against baseline: (baseline)
  - Diff p-value: (baseline)
- Unconditional:
  - PIT: Fail
  - Log score diff against baseline: -6.36
  - Diff p-value: 0
- Quantile reg:
  - PIT: Pass
  - Log score diff against baseline: -2.09
  - Diff p-value: 0.02
- Gaussian EGARCH:
  - PIT: Fail
  - Log score diff against baseline: 1.235
  - Diff p-value: 0.892
- TSkew GARCH:
  - PIT: Fail
  - Log score diff against baseline: 1.537
  - Diff p-value: 0.938
- Gaussian GARCH:
  - PIT: Fail
  - Log score diff against baseline: 1.86
  - Diff p-value: 0.969

### Interpretation of benchmarking findings
- The baseline model has a better log score than:
  - The unconditional distribution (log score diff = -6.36; PIT = Fail).
  - The daily distribution fitted from the quantile regressions model with resampling (log score diff = -2.09; PIT = Pass for quantile regressions, but lower log score).
- The unconditional and quantile-regression models both fail the PIT test (unconditional fails; quantile reg passes the PIT test as noted).
- Variations of the GARCH model (Gaussian EGARCH, TSkew GARCH, Gaussian GARCH):
  - These models show higher log scores than the baseline (positive log score diffs), but the differences are not statistically significant given p-values (0.892, 0.938, 0.969).
  - These GARCH variations fail the PIT test (albeit only by a very few numbers of percentiles).
- Conclusion:
  - GARCH specification is appropriate to model FX returns, consistent with the literature.
  - Variations around the type of GARCH and implied distributions are statistically equivalent in this out-of-sample comparison.

*Source: Authors’ calculations.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021032-print-pdf.pdf_