## ppea2024067

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---

### EXECUTIVE SUMMARY — context, objective, and key findings
- Context and motivation:
  - Many central banks expanded their balance sheets substantially after the Global Financial Crisis and again in response to the COVID-19 pandemic, exposing them to balance sheet risks that materialized into sizeable central bank losses as high inflation induced policy tightening.
  - Before the Global Financial Crisis, central banks maintained much smaller balance sheets and were almost always profitable.
- Objective and scope:
  - Develops a tractable modeling framework—building on Hall and Reis (2015)—to gauge balance sheet risks arising from interest rate duration risk, foreign exchange risk, and credit risk.
  - Framework allows quantitative assessment of how central bank equity would evolve under alternative scenarios, including macroeconomic divergences from baseline and specific policy actions.
  - Forecast horizon for central bank equity: a five-year horizon.
- Key illustrative findings:
  - Larger balance sheets generate considerably more volatility in central bank equity and a higher likelihood of sizeable losses than lean balance sheets.
  - When the main liability currency is circulation (lean balance sheet), the central bank tends to be “robustly” profitable, and profits rise with higher interest rates.
  - For larger balance sheets, duration risk and foreign exchange risk importantly shape profits and equity.
  - Credit risk increases expected credit losses (ECL) and can materially affect equity.
- Use cases and policy implications:
  - Utility for central banks: assess risk to capital from policy actions, inform profit retention and building buffers, guide recapitalization decisions, improve transparency and accountability.
  - Utility for governments: simulated net income forecasts inform fiscal planning and expectations about transfers under institutional arrangements.
- Distinguishing features of central bank stress testing:
  - Central banks share risk types with commercial banks but differ: do not seek profits, are not subject to solvency ratios, do not face liquidity constraints in own currency, and hold interest-free indefinite-duration liabilities (e.g., banknotes) that can generate seigniorage.
  - Central banks can be policy insolvent if they lack income to pursue inflation objectives without persistent losses.
  - Regular, dynamic stress testing can inform board decisions on profit distribution and recapitalization and is particularly important for large, volatile balance sheets.

### FISCAL THEORY OF THE PRICE LEVEL and literature survey
- FTPL key points:
  - Price adjusts to ensure the real value of government debt equals the real present value of fiscal surplus; monetary policy passive while fiscal policy active.
  - Theoretical studies mentioned: Park (2015), Williamson (2018), Bassetto and Messer (2013), Cochrane (2023).
- Applications and empirical studies:
  - Country-specific central bank balance sheet models: Fujiki and Tomura (2017); Franta et al. (2022); Buiter and Rahbari (2012); Carpenter et al. (2015); Ize (2005); Manuelli and Vizcaino (2017).
  - Financial stress testing: Benjamin et al. (2022) — Bank of England capital framework.
  - Empirical findings: Perera et al. (2013) find a statistically significant negative relationship between central bank financial strength and inflation; Goncharov et al. (2023) dataset of more than 150 countries finds central banks report slightly positive profits more often than slightly negative profits.

### STYLIZED FACTS — balance sheet patterns, capital and dividends (numeric facts preserved)
- By level of development and time:
  - Balance sheet size (monetary base to GDP): remained between 5 and 10 percent of GDP in 2005; by 2022 distribution widened to between 15 and 45 percent.
  - CiC used to be more than 80 percent of the monetary base in AEs. With balance sheet expansion, bank reserves at the central bank are now more than two times larger than CiC.
  - Capital median: 2.1 percent when expressed as a percentage of GDP and 10.5 percent as a percentage of the monetary base.
  - Capital in terms of GDP increased from 1.3 to 2.1 percent from 2005 to 2022.
  - Capital as percentage of the monetary base declined from 14.7 to 10.5 percent from 2005 to 2022.
- By exchange rate arrangement:
  - Fixed exchange rate economies traditionally had larger balance sheets due to FX reserve needs; floaters now often have larger balance sheets with more varied distributions.
  - Erosion of seigniorage mainly impacts floating arrangements and capitalization weakened in floating arrangements.
- Dividends and transfers:
  - Most central banks distributed dividends until 2022; profits typically transferred to government in profitable years.

### INTERNATIONAL EXPERIENCE — selected central bank loss figures and communications (figures preserved exactly)
- Bank of Canada (BOC): USD 3.3 billion (nine-month period ended in September 2023). Communication: losses do not affect ability to conduct monetary policy or operations; expects to return to net income.
- Bank of England (BoE): Losses projected to be USD 50 billion in 2023 and 2024. Communication: APF indemnified by Treasury; realized profits or losses reconciled with Treasury quarterly; by May 2024, total transferred from APF to the Treasury was £63bn.
- Bank of Ghana (BoG): USD 5.3 billion (2022). Large part (USD 4.1 billion) resulted from Government’s domestic debt restructuring.
- Danmarks Nationalbank: USD 1.3 billion (2022). Communication: no transfers to central government for 2022 financial year.
- De Nederlandsche Bank (DNB): USD 3.8 billion (2023). USD 1.2 billion of 2023 losses covered by reserves.
- Deutsche Bundesbank: USD 0.00 (2023). Released risk provisions in full (USD 21.1 billion); USD 2.6 billion drawn from reserves to cover 2023 losses.
- European Central Bank (ECB): USD 1.41 billion (2023). Released USD 7.1 billion (€6.6bn) general risk provision.
- Federal Reserve (Fed): USD 114.3 billion (2023). Federal Reserve Act mandates recording a deferred asset if earnings insufficient.
- Monetary Authority of Singapore (MAS): USD 22.8 billion (financial year ending in March 2023). Doubled issued and paid-up capital to SGD 50 billion (or USD 37 billion).
- Reserve Bank of Australia (RBA): USD 3.96 billion (fiscal year 2022/23). Negative position of USD 11.6 billion by 30 June 2023.
- Reserve Bank of New Zealand (RBNZ): No surplus equity for distribution (fiscal year 2022/23). Minister agreed to provide USD 2 billion in new capital (on top of USD 806 million in July 2023); new indemnity covering losses up to USD 8 billion on bond purchase programs.
- Riksbank: USD 7.94 billion (2022). Equity negative USD 1.76 billion (SEK 18.2 billion). New Act targets equity level of SEK 60 billion and basic level of SEK 40 billion.
- Swiss National Bank (SNB): USD 3.54 billion (2023). Note: in 2022 it had a record loss of USD 152 billion (CHF 133 billion).

### MODELING FRAMEWORK — core model, satellite models, horizons, and clearing item
- Core approach:
  - Starts from accounting identities of the balance sheet and income statement (the “core model”).
  - Uses satellite models to estimate risks and forecast each core-model item (e.g., currency in circulation [CiC], operational cost) conditional on macroeconomic variables.
  - Derives baseline capital path from combined forecasts and stress-tests it using deterministic and at-risk scenarios.
  - Forecast horizon used for central bank equity: five years.
- Equity identities (consistency check):
  - (1) Equity_t = Total Assets_t − Total Liabilities_t
  - (2) Equity_t = Equity_{t−1} + [realized earnings]_t + Δ Revaluation Account_t
    - Δ Revaluation Account includes Net Gains or Losses from Foreign Currency Revaluation and other non-realized earnings.
- Clearing item — Open Market Operations (OMO):
  - OMO_t = OMO_{t−1} + [ − ΔFXInt_t − ΔDAH_t + ΔCi_C_t + ΔRR_t + IRIncome_t − Op_t ]
  - OMO outcomes can be positive or negative; generally generate income for central banks with lean balance sheets and incur costs for those with large balance sheets.
- Foreign reserve dynamics:
  - FXRes_t = FXRes_{t−1} + FIRIncome_t + Revaluation_{t}^{FXRes} + ΔFXInt_t

### ACCOUNTING, VALUATION, AND IFRS 9 ALIGNMENT
- Convert balance sheet to an internationally acceptable accounting standard for comparability.
- Valuation approaches:
  - Foreign reserves denominated in cash: term deposits valued at amortized cost.
  - Foreign currency securities (longer-term): fair valuation.
  - Domestic loans and securities: amortized cost or fair value depending on maturity.
  - Local and foreign loans acquired by central bank: typically recorded at amortized cost.
- Guidance discusses valuation in line with IFRS 9 for classifying and measuring financial assets and liabilities.

### CREDIT RISK MODELING — ECL, PD methods, LGD, and EAD
- ECL standard formula:
  - ECL_t = PD_t ⋅ LGD ⋅ EAD_t
- PD methods:
  - (i) Credit rating-based PD (preferred benchmark): forecast credit ratings using an internal credit rating model incorporating macro forecasts, fiscal forecasts, and governance/event-risk assumptions; map ratings to historical PDs; use one-year PD standard; lifetime PD for non-investment grade sovereigns with heightened risk.
  - (ii) CDS-implied PD: s_spread_t = (1 − R_CDS) ⋅ λ_t with 1 − exp(−λ_t) ≈ λ_t as 1-year default probability approximation; market convention R_CDS = 25 percent.
  - (iii) Discount-implied PD (DCF): bond premium (bond yield − policy rate) considered; LGD = 100 percent equivalence noted.
- LGD and recovery:
  - LGD derived from recovery rate R = 1 − LGD; sovereign recovery rates harder to estimate—regional LGDs may be used.
- EAD and yield curve modeling:
  - Yield curve models (e.g., Dynamic Nelson-Siegel) used to price assets and determine EAD; IFRS 9 fair value hierarchy applied: Level 1, Level 2, Level 3.

### INTEREST RATE, DURATION, AND MARK-TO-MARKET RISKS
- Interest rate risk:
  - Losses depend on size of policy rate increase, size of asset portfolio, and pace of re-pricing (average duration).
  - Policy rate modelled as reaction function consistent with central bank framework; for inflation-targeters a specification with inflation gap, output gap, lagged rate, and shocks is used.
- Duration risk:
  - CBST explicitly models domestic portfolio duration via rollover satellite model; typical simulation approach replaces maturing assets to keep average duration unchanged.
- Mark-to-market:
  - Mark-to-market losses primarily concern FX portfolios in fixed exchange regimes or EMs; FX portfolios usually marked to market.
  - Domestic securities often held to maturity in banking book to avoid mark-to-market.
- Exchange rate risk and revaluation reserve:
  - FX revaluation gains/losses largely booked to revaluation reserve (cannot be distributed); revaluation reserve functions as a buffer.
  - CBST forecasts conditional distribution of exchange rate (5th percentile) to calibrate revaluation account distribution rules.

### EMERGENCY LENDING, FX LIQUIDITY, AND COLLATERAL RISK
- Emergency lending:
  - Collateralized lending provides defense lines: counterparty solvency and collateral value; haircuts mitigate risk.
  - If collateralized capacity insufficient, government guarantee should cover shortfalls.
  - Emergency lending credit risk modelled as ECL with contingent EAD; PDs inferred from counterparty credit ratings.
- FX liquidity risk:
  - Central banks not liquidity-constrained in local currency but face constraints in foreign currencies.
  - Fully dollarized economies are subject to standard liquidity risk.
  - FX reserve depletion can lead to policy insolvency.

### FORECASTING BALANCE SHEET ITEMS — CiC, reserves, costs, and Treasury relations
- CiC forecasting:
  - Seasonal ARIMA model with nominal GDP as exogenous variable; second specification adds interest rate effects with sensitivity parameter γ_CiC.
  - Nonlinearities may appear when demand for money drops during high inflation.
- Reserve requirement balances:
  - Assumed to depend on demand for money; remunerated versus unremunerated reserve requirements have differing impacts on interest paid/received.
- Operational cost:
  - Modeled as log-linear with nominal GDP: Operational Cost_t = μ ⋅ NGDP_t^α (notation preserved in source form).
- Relation with Treasury:
  - Two projection approaches: Treasury with account but no borrowing (constant historical average), or Treasury borrows from central bank (projected jointly or per legal provisions).

### SCENARIO DESIGN, MODEL TYPES, AND HORIZON
- Risk-horizon: five years.
- Scenario model options:
  - (i) at-risk models (preferred for “extreme but plausible” shocks),
  - (ii) other IMF-used models including SVAR and DSGE,
  - (iii) deterministic models.
- At-risk models:
  - Use quantile regressions, conditional distributions, and quantile-based techniques (e.g., Growth-at-Risk, Inflation-at-Risk, Exchange-rate-at-Risk).
  - Best suited for short horizons (ideally up to one year) but can be extended using VARs and policy reaction functions.
- Deterministic models:
  - Assess impacts of fixed parameter changes and policy decisions (e.g., remunerating required reserves).
- Balance sheet projection method:
  - Recursive projection: macro scenarios → satellite models → cash flows and net profit → capital path; iterate without dividend distribution.

### STYLIZED BALANCE SHEET SIMULATIONS — lean, large domestic, large foreign, and sovereign ECL cases
- Lean balance sheet (assumptions and results):
  - Net foreign assets represent 30 percent of total assets.
  - Starting equity is 5 percent of total assets.
  - Initial operational costs are 0.5 percent.
  - Result: inflation shock is positive for equity (channels: higher interest income and larger OMO stocks from faster CiC growth). Equity increases as a percentage of GDP when the real interest rate turns positive.
- Large domestic portfolio:
  - Total assets increased 7.6 times versus lean balance sheet due entirely to domestic asset purchases.
  - CiC share drops to 9 percent of total assets.
  - Domestic policy rate increases from 1 percent to 16 percent.
  - Flat yield curve assumed.
  - Two duration cases: one year and five years. Valuation via amortization (held to maturity).
  - Result: capital declines; short duration ends at −3 percent of nominal GDP; long duration ends at −5 percent of nominal GDP.
- Large foreign portfolio:
  - Total assets increased 4.7 times versus lean balance sheet due to foreign government bonds under fixed exchange regime.
  - Domestic policy rate follows U.S. policy rate plus 25 basis points.
  - Two U.S. term-spread cases: flat and steepening then flattening after December 2025.
  - Average duration cases: 0.5 years and 2.5 years. U.S. policy rate assumed to rise from 4.8 percent to 6 percent over five years.
  - Result: capital declines; flat yield curve case: short duration ends at −0.5 percent of nominal GDP; long duration ends at −0.7 percent of nominal GDP. Equity declines up to December 2025 then recovers under long-duration steepening scenario.
- Large domestic portfolio under debt restructuring (ECL-focused scenario):
  - Total assets increased 7.6 times compared to lean balance sheet.
  - Assumes no return from domestic bond portfolio and zero remuneration rate on excess reserves to focus on ECL impact.
  - PD inferred from sovereign CDS spreads responding to real GDP growth, inflation, and public debt as percentage of GDP.
  - Two debt paths:
    - (i) debt as percentage of GDP starts from 90 percent and declines by 1 percent in each quarter;
    - (ii) debt as percentage of GDP holds at 90 percent over time.
  - Results:
    - In both paths capital declines for first two years then increases.
    - Declining debt path: capital reaches 1 percent of nominal GDP at end of simulation (ECL declines quickly).
    - Constant debt path: capital reaches 0.4 percent of nominal GDP (ECL declines more slowly).

### YIELD-CURVE, DYNAMIC NELSON-SIEGEL, AND POLICY-RATE DYNAMICS
- Dynamic Nelson-Siegel:
  - Projects yields as linear function of three factors: level, slope, curvature, with factor dynamics modeled by a VAR.
  - Short-end of yield as maturity → 0: y_t(0) = L_t + S_t interpreted as policy rate i_t.
  - Separates long-term yield dynamics (L_t) and policy rate (i_t = L_t + S_t), allowing capture of divergence affecting net interest income.

### INTERNAL CREDIT RATING MODEL and CBST inputs for PDs
- Internal credit rating model:
  - Four blocks + ESG overlay: economic strength, institutional strength, fiscal strength, susceptibility to event risks.
  - Blocks weighted based on historical calibration; ESG overlay used in exceptional circumstances.
  - Back testing against historical agency ratings for past five to ten years.
  - Conversion to PD via mapping tables to historical PDs; one-year PD standard, lifetime PD for non-investment grade with heightened risk.
- Country risk premium estimation (fixed exchange rate context):
  - Target spread: SseRPRt = It − I0t (three-month interbank rate minus international benchmark).
  - ARIMAX monthly model with benchmark factors including Government revenue; CDS 1 year; CDS 5 years; CDS 10 years; GDP; Stock market index; Stock market turnover; Unemployment (international/domestic); VIX; CPI.
  - Model selection via Akaike’s Information Criterion (AICc).

### CBST IMPLEMENTATION, CODE STRUCTURE, AND WORKFLOW
- Modular code structure:
  - Macroeconomic component separated from main CBST component for replaceability.
- Main execution codes:
  - SelectScenarioRunModel: choose scenario ID and number of Monte-Carlo paths.
  - Simulate_CBBS_MonteCarlo: generates projections across Monte-Carlo paths.
  - Core central balance sheet model exe: computes all balance sheet and income statement items using macro inputs, initial balance sheet, and parameters.
- At-risk model and synthetic regressors:
  - Synthetic variables via partial least squares (PLS) to extract common trends under thematic groups (e.g., Domestic Demand, External Demand, Domestic Financial Conditions).
  - Conditional mean estimation uses jackknife Theil-Sen estimator; asymmetry estimated with Firth penalized logistic regression.

### AT-RISK MODEL TECHNIQUES — Theil-Sen / Firth dual model and parametrization
- Dual model elements:
  - Theil-Sen regression: conditional expectation 피[ y_{t+h} | X_t ] = α̂ + β̂_TTC X_t; residual variance used as variance assumption.
  - Firth logistic model: estimates asymmetry as ℙ[ y_{t+h} > ȳ | X_t ] = α̂ + β̂_LGL X_t using penalized likelihood to reduce bias.
- Assembly of conditional moments:
  - Expectation: Theil-Sen projection.
  - Variance: assumed unconditional equal to Theil-Sen residual variance.
  - Skewness/asymmetry: inferred from Firth model via cumulative density at conditional mean.
- Parametrization:
  - Fits an asymmetrical Gaussian distribution from the three moments, enabling closed-form parameter derivation and parsimonious representation given limited sample size.
  - Rationale: balances parsimony and interpretability; avoids requiring higher moments (kurtosis) infeasible in small samples.

_Italicized source attribution: CENTRAL BANK STRESS TESTING—GUIDANCE NOTE, INTERNATIONAL MONETARY FUND_

### EXECUTIVE SUMMARY

### EXECUTIVE SUMMARY

### Context and motivation
- Many central banks expanded their balance sheets substantially in the wake of the Global Financial Crisis and again in response to the COVID-19 pandemic.
- These expansions, aimed at price and financial stability and often spurred by limited conventional policy space, exposed central banks to significant balance sheet risks that have materialized into sizeable central bank losses as high inflation induced policy tightening.
- By contrast, before the Global Financial Crisis, central banks maintained much smaller balance sheets and were almost always profitable.

### Objective and scope of the guidance note
- Develops a tractable modeling framework—building on Hall and Reis (2015)—to gauge balance sheet risks arising from:
  - interest rate duration risk,
  - foreign exchange risk,
  - credit risk.
- The framework allows quantitative assessment of how central bank equity would evolve under alternative scenarios, including:
  - macroeconomic divergences from baseline (for example, if inflation runs higher than expected),
  - specific policy actions (for example, a lower remuneration rate on excess reserves).
- Addresses a gap: few central banks currently utilize full-fledged stress testing to guide capital policy.

### Modeling approach and features
- Starts from fundamental accounting identities of the balance sheet and income statement (the “core model”).
- Uses satellite models to estimate risks and forecast each core-model item (e.g., currency in circulation [CiC], operational cost) conditional on macroeconomic variables.
- Derives a baseline capital path from combined forecasts and stress-tests it using deterministic and at-risk macroeconomic scenarios.
- Forecast horizon used for central bank equity: a five-year horizon.

### Key illustrative findings from model simulations
- Larger balance sheets generate considerably more volatility in central bank equity and a higher likelihood of sizeable losses than lean balance sheets.
- When the main liability currency is circulation (i.e., a lean balance sheet), the central bank tends to be “robustly” profitable, and profits rise with higher interest rates.
- For larger balance sheets, duration risk and foreign exchange risk importantly shape the path of profits and central bank equity.
- Credit risk (including risks from deteriorating fiscal balances in some EMs and from developmental activities such as subsidized lending during COVID-19) increases expected credit losses (ECL) and can materially affect equity.

### Use cases and policy implications
- Utility for central banks:
  - Assess risks that policy actions pose for capital positions.
  - Inform decisions on profit retention and building financial buffers.
  - Guide determinations of when recapitalization may be needed.
  - Improve transparency and accountability for balance-sheet policies.
- Utility for governments:
  - As shareholders, governments can use simulated net income forecasts from CBST to evaluate claims on central bank profits and losses, given institutional agreements.
  - Helps anticipate transfers emanating from prevailing arrangements with the central bank and informs fiscal planning.
- Caveat: balance sheet actions have a range of macroeconomic, financial, and fiscal effects beyond their effects on central bank profits and these must be considered.

### Distinguishing features of central bank stress testing
- Central banks share risk types with commercial banks (credit, interest rate, exchange rate) but differ in key dimensions:
  - Do not seek profits;
  - Are not subject to financial solvency ratios;
  - Do not face liquidity constraints;
  - Hold interest-free indefinite-duration liabilities (e.g., banknotes) that can generate seigniorage.
- Central banks can be policy insolvent if they do not have enough income to pursue their inflation objective without incurring persistent losses.
- Regular, dynamic stress testing can inform board decisions on profit distribution and potential recapitalization and is particularly important for large, volatile balance sheets.

### Structure of the guidance note (contents overview)
- Reviews academic literature (Section II) and stylized facts (Section III).
- Describes the core model (Section IV) and risk modeling approaches (Section V: credit, interest rate, exchange rate, commodity, FX liquidity).
- Discusses forecasting of balance sheet and financial account items (Section VI).
- Presents scenario design (Section VII) including at-risk and deterministic models.
- Shows model outputs under stylized balance sheets (Section VIII): lean balance sheet; large domestic portfolio; large foreign portfolio; exposure to restructured assets (expected credit loss).

*Source: ppea2024067 - EXECUTIVE SUMMARY (December 5, 2024).*

### 10.      Relevant but theoretical studies are conducted under the Fiscal Theory of Price Level

### 10.      Relevant but theoretical studies are conducted under the Fiscal Theory of Price Level

### A. Fiscal Theory of the Price Level (FTPL) — key points
- The basic FTPL emphasizes the role of fiscal policy for analyzing inflation based on the analysis of the consolidated government’s intertemporal budget constraint.
- The intertemporal budget constraint arises to rule out Ponzi scheme finance.
- The key implication of the FTPL: the price adjusts itself to ensure the real value of the government debt is equal to the real present value of fiscal surplus.
- Under FTPL: monetary policy is passive while fiscal policy is active in determining the price level.
- Theoretical studies mentioned include Park (2015), Williamson (2018), Bassetto and Messer (2013) (who develop a stylized competitive equilibrium model analyzing conditions under which a central bank can ensure positive profits and transfers to the fiscal authority), and Cochrane (2023).

### B. Applications — central bank balance sheet and stress-testing literature
- Country-specific central bank balance sheet models and studies:
  - Fujiki and Tomura (2017): model of the Bank of Japan balance sheet.
  - Franta et al. (2022): balance sheet model of the Czech National Bank.
  - Buiter and Rahbari (2012): stylized balance sheet of central bank and government from ECB perspective.
  - Carpenter et al. (2015): projections of selected balance sheet items and income/expenses of the Federal Reserve.
  - Ize (2005): simple balance sheet model for central banks in Costa Rica, Chile, Mozambique during crises.
  - Manuelli and Vizcaino (2017): simple model of budget constraints of central bank and fiscal authority; study of Argentina.
- Financial stress testing for central banks:
  - Benjamin et al. (2022) present the Bank of England (BoE) capital framework developed in 2018; assesses “potential losses in a set of severe but plausible events for activities that are backed by the BoE’s capital” using “a forward-looking, scenario-based approach.”
  - Benjamin et al. (2022) explain analytical underpinnings and institutional setup of the BoE model and explore general applicability to other central banks.
- Empirical studies:
  - Perera et al. (2013): find a statistically significant negative relationship between central bank financial strength and inflation using selected advanced and emerging country datasets.
  - Goncharov et al. (2023): dataset of more than 150 countries; central banks tend to report slightly positive profits more often than slightly negative profits, associated with political and market pressures and central bankers’ career concerns.

### C. Other relevant theoretical and modeling studies
- Quantifying macroeconomic impact of balance sheet normalization (examples):
  - Domínguez and Gomis-Porqueras (2023): dynamic equilibrium model with fiscal policy passive and monetary policy following the Taylor rule; central bank holds short-term and long-term bonds; show balance sheet changes affect long-run inflation and the real economy when short-term and long-term bond premia exist due to imperfect asset substitutability.
  - Bolt et al. (2023): game-theoretic model analyzing the limit of keeping the value of fiat money in context of the failure of the Bank of Amsterdam (early 19th century).
  - Arce et al. (2020): endogenous market for interbank loans in a new-Keynesian model to analyze transmission of balance sheet policy (do not study intertemporal insolvency of central bank).
  - Benigno and Benigno (2022): extend New-Keynesian model by incorporating reserves to study monetary policy normalization when a liquidity trap exists; solve optimal combination of reserves and interest rate policy but assume central bank is not subject to solvency condition.
  - Hayashi and Koeda (2019): regime-switching SVAR to analyze QE and QT in Japan; find higher reserves at the zero lower bound raise inflation and output; effect of QT depends on economy state; do not model entire BOJ balance sheet.
- Note on current CBST framework:
  - The current CBST framework does not allow feedback effects from reducing balance sheet size to the macroeconomy because it models all balance sheet items in a granular manner and such feedbacks would complicate analysis.
  - Capturing those feedback effects under CBST is a potentially fruitful future research and development topic.
- Balance-sheet concern and optimal monetary policy:
  - Jeanne and Svensson (2007): model where an independent central bank can commit to a higher future price level to escape a liquidity trap; optimal commitment achievable if balance sheet concern is sufficiently strong.
  - Berriel and Bhattarai (2009): central bank minimizes deviations of inflation and output and the real value of its own capital; balance sheet concern and fiscal independence can act as commitment device under cost-push inflation.
  - These studies justify CBST focus on balance sheet concern despite not adopting a general equilibrium framework.

### D. Accounting, profit distribution, and capital literature
- Reviews and surveys:
  - Archer and Moser-Boehm (2013) and Bunea et al. (2016): review central bank accounting frameworks and profit distribution practices.
  - Tanaka (2021): recent survey of the literature on central bank capital.

### E. Stylized facts — balance sheet patterns and capital
- By level of economic development:
  - Large advanced economies (AEs) with floating exchange rates traditionally had lean balance sheets with minimal risk exposures and were robustly profitable because most liabilities consisted of interest-free CiC investable in income-earning assets.
  - Emerging market central banks often faced financial issues due to relatively large FX portfolios to support fixed exchange rate commitments.
  - Post-Global Financial Crisis, AE central banks engaged in asset purchase programs that significantly increased balance sheet size and risk exposure, surpassing other countries.
- Key numeric and comparative findings (preserve original figures exactly):
  - Balance sheet size, measured as the ratio of monetary base to GDP, remained between 5 and 10 percent of GDP in 2005. However, by 2022, the distribution widened to between 15 and 45 percent.
  - CiC used to be more than 80 percent of the monetary base in AEs. With the balance sheet expansion, bank reserves at the central bank are now more than two times larger than CiC.
  - Capital median: 2.1 percent when expressed as a percentage of GDP and 10.5 percent as a percentage of the monetary base.
  - Capital in terms of GDP increased from 1.3 to 2.1 percent from 2005 to 2022.
  - Capital as percentage of the monetary base declined from 14.7 to 10.5 percent from 2005 to 2022, reflecting capital accumulation lagging the increase in central bank balance sheets.
  - Decline in capital vis-a-vis the monetary base concerns mainly AE central banks.
- By type of exchange rate arrangement:
  - Fixed exchange rate economies traditionally had larger balance sheets than floating ones due to the need for FX reserves to support an exchange rate commitment; they were more exposed to exchange rate risk and duration risk.
  - The scenario evolved: balance sheets of central banks with a floating exchange rate often now exceed those with fixed exchange rates; distribution among floaters is more varied with large-economy central banks as outliers.
  - Countries with floating rate systems are more likely to hold large portfolios of government securities; large AE expansions typically reflect quantitative easing and shift to floor operating systems.
  - Erosion of seigniorage mainly impacts central banks with floating arrangements.
  - Capitalization seems to have weakened in floating arrangements: floaters started with lower capital as percentage of base money than fixers; despite faster growth in floaters’ balance sheets, capitalization did not keep pace, leading to declined capital as percentage of base money for floating systems while it slightly increased for fixed systems.
- Dividends and transfers:
  - Most central banks distributed dividends until 2022; distribution rules vary.
  - Central banks in the sample distributed non-negligible dividends, as a percentage of GDP, to their governments through 2022.
  - In profitable years, most profits were transferred to the government, with few exceptions.

### F. International experience on central bank losses and communications — selected examples
- After 2022, central bank losses became widespread and large; communications often stated losses would not hinder policy objectives and detailed offsetting approaches such as future profits or recapitalization; some central banks stopped remunerating certain accounts to save costs.
- Selected central bank loss figures and communicated actions (figures preserved exactly as in source):
  - Bank of Canada (BOC): USD 3.3 billion (nine-month period ended in September 2023). Communication: losses do not affect ability to conduct monetary policy or operations; expects to return to net income.
  - Bank of England (BoE): Losses projected to be USD 50 billion in 2023 and 2024. Communication: APF indemnified by Treasury; realized profits or losses reconciled with Treasury quarterly; transfers occur each quarter; by May 2024, total transferred from APF to the Treasury was £63bn.
  - Bank of Ghana (BoG): USD 5.3 billion (2022). Communication: losses explained; present financial position has no immediate impact on ability to pursue policies; large part (USD 4.1 billion) resulted from Government’s domestic debt restructuring; equity and earning capacity expected to be high enough in long run to ensure financial independence.
  - Danmarks Nationalbank: USD 1.3 billion (2022). Communication: no transfers to central government for 2022 financial year.
  - De Nederlandsche Bank (DNB): USD 3.8 billion (2023). Communication: expect losses to continue until 2028; buffers expected to remain positive but capital contribution from the Dutch State may be necessary in extreme cases; USD 1.2 billion of 2023 losses covered by reserves.
  - Deutsche Bundesbank: USD 0.00 (2023). Communication: balanced financial result after releasing risk provisions and withdrawals from reserves; released risk provisions in full (USD 21.1 billion); USD 2.6 billion drawn from reserves to cover 2023 losses; expects considerable burdens in 2024 likely to exceed remaining reserves; will report a loss carryforward and offset through future profits; no profit transferred to federal budget since 2020.
  - European Central Bank (ECB): USD 1.41 billion (2023). Communication: expects further losses after fully releasing remaining USD 7.1 billion (€6.6bn) general risk provision; losses to be carried forward against future profits, avoiding need for recapitalization.
  - Federal Reserve (Fed): USD 114.3 billion (2023). Legal framework: Federal Reserve Act mandates recording a deferred asset if earnings insufficient; deferred asset represents future excess earnings regional Reserve Banks need to accumulate before resuming remittances to U.S. Treasury.
  - Monetary Authority of Singapore (MAS): USD 22.8 billion (financial year ending in March 2023). Communication: largest loss recorded; reflected monetary policy tightening and broad appreciation of SGD; losses mostly due to negative currency translation effects; loss did not affect ability to conduct monetary policy or support financial stability; unlikely to contribute to Consolidated Fund over next few years; doubled issued and paid-up capital to SGD 50 billion (or USD 37 billion).
  - Reserve Bank of Australia (RBA): USD 3.96 billion (fiscal year 2022/23). Communication: suspended transfers to government; emphasized restoring equity over time; Board expects future distributable earnings to offset accumulated losses and restore Reserve Fund balance; measure endorsed by government; negative position of USD 11.6 billion by 30 June 2023.
  - Reserve Bank of New Zealand (RBNZ): No surplus equity for distribution (fiscal year 2022/23). Communication: indemnity agreements with the Treasury; no dividend paid to Treasury in 2020, 2022, 2023; Minister of Finance agreed to provide USD 2 billion in new capital (on top of USD 806 million in July 2023); new indemnity covering losses up to USD 8 billion on bond purchase programs.
  - Riksbank: USD 7.94 billion (2022). Result: equity negative USD 1.76 billion (SEK 18.2 billion). Communication: new Act targets equity level of SEK 60 billion and basic level of SEK 40 billion; Executive Board intends to petition government in March 2024 for capital injections and proposals for additional earnings capacity.
  - Swiss National Bank (SNB): USD 3.54 billion (2023). Communication: will not make payments to Swiss central or local governments nor pay dividend to investors due to 2023 loss. Note: in 2022 it had a record loss of USD 152 billion (CHF 133 billion).
- Source of table: central banks’ communiqués.

*Italicized source attribution: CENTRAL BANK STRESS TESTING—GUIDANCE NOTE, INTERNATIONAL MONETARY FUND*

### 25.      The CBST projects the capital of central banks by combining macroeconomic forecasts

### 25.      The CBST projects the capital of central banks by combining macroeconomic forecasts

### Modeling approach and data inputs
- All balance sheet items are modeled separately, then aggregated to provide a calculation for realized income, provisioning, statutory capital, and revaluation income.
- Macroeconomic variable forecasts are usually sourced from the country’s IMF macroeconomic framework and typically include projections of real GDP, inflation, net foreign reserves, and the exchange rate.
- Policy-determined macro variables (e.g., forecast of the policy interest rate) are determined by satellite models.
- Exogenously determined macro forecasts—such as foreign interest rates or CDS spreads—are sourced from the World Economic Outlook and Bloomberg.
- Several forecasting techniques are used for balance sheet items; the choice varies by the nature of the item.

### Accounting, income distribution, and capital definitions
- The model forecasts comprehensive income, which is then distributed to different equity items:
  - Based on exchange rate projections, part of the profit is assigned to the revaluation reserve.
  - The remainder is distributed between general reserves and a transfer to the government (dividend payment).
- For simplicity in the framework, all retained earnings are assigned to a general reserve; losses reduce the general reserve.
- No distribution is assumed ex ante because the stress test informs the distribution decision.
- Definitions:
  - Statutory capital = initial “paid-up” capital + general reserves.
  - Equity = statutory capital + revaluation reserve.

### Core model equations and clearing item
- Equity identities (consistency check between two representations):
  - (1) Equity_t = Total Assets_t − Total Liabilities_t
  - (2) Equity_t = Equity_{t−1} + [realized earnings]_t + Δ Revaluation Account_t
    - Note: Δ Revaluation Account includes Net Gains or Losses from Foreign Currency Revaluation and other non-realized earnings.
- A clearing item: Open Market Operations (OMO) captures changes in assets and liabilities.
  - OMO_t = OMO_{t−1} + [ − ΔFXInt_t − ΔDAH_t + ΔCi_C_t + ΔRR_t + IRIncome_t − Op_t ]
    - Where DAH_t is domestic asset holdings, Ci_C_t is currency in circulation, IRIncome_t is net interest rate income, RR_t is required reserves, and Op_t is operational expenses.
  - OMO outcomes can be positive or negative; generally generate income for central banks with lean balance sheets and incur costs for those with large balance sheets.
- Foreign reserve dynamics:
  - FXRes_t = FXRes_{t−1} + FIRIncome_t + Revaluation_{t}^{FXRes} + ΔFXInt_t
    - Components: FIRIncome_t (foreign currency interest income), Revaluation_{t}^{FXRes} (FX revaluation gains/losses), ΔFXInt_t (net USD purchase for Net Foreign Reserves accumulation).

### Balance sheet valuation and IFRS 9 alignment
- First step: convert the balance sheet to an internationally acceptable accounting standard for comparability.
- Valuation approaches:
  - Foreign reserves denominated in cash: term deposits valued at amortized cost.
  - Foreign currency securities (longer-term): fair valuation.
  - Domestic loans and securities: amortized cost or fair value depending on maturity.
  - Local and foreign loans acquired by the central bank: typically recorded at amortized cost.
- The guidance note discusses valuation in line with IFRS 9 for classifying and measuring financial assets and liabilities.

### Modeling credit risk — ECL provisioning
- Credit risk is measured using ECL provisioning; expected losses on all assets including FX and sovereign exposures should be modeled.
- ECL provisioning for sovereign debts: forecasted by conditioning the ECL calculations on projected macroeconomic variables from the country’s macro framework (forward-looking).
- ECL provisioning for corporate debts: in the model’s benchmark, forecasted by assuming the ECL provision, as a percentage of outstanding debt, remains constant over the forecast horizon.
- ECL standard formula:
  - ECL_t = PD_t ⋅ LGD ⋅ EAD_t
    - Where EAD_t is set equal to estimated exposure at time of default t.
- LGD derived from recovery rate R, where R = 1 − LGD.
  - Recovery rates for corporate bonds can be sourced (e.g., Bloomberg); sovereign recovery rates are harder to estimate—regional LGDs may be used.

### Probability of Default (PD) methods and practices
- Three PD calculation methods:
  - (i) Credit rating-based PD (preferred benchmark)
  - (ii) CDS-implied PD
  - (iii) Discount-implied PD (DCF method)
- Credit rating-based PD:
  - Employed as benchmark; for sovereigns, forecast credit ratings over the horizon using an internal credit rating model.
  - Credit-rating forecasts incorporate:
    - (i) Macroeconomic forecasts (changes in GDP and related components: GDP growth and volatility, economic diversification, credit growth, per capita GDP).
    - (ii) Fiscal forecasts from the latest Debt Sustainability Analysis (debt level and trajectory, fiscal and primary balances, indicators for interest payments and coverage).
    - (iii) Assumptions on governance and susceptibility to event risks (political risk, climate risk, ESG considerations).
  - Mapping: sovereign credit ratings map to historical PDs from rating agencies; CBST uses a one-year PD as standard. Lifetime PD used for non-investment grade sovereigns with heightened credit risk.
- CDS-implied PD:
  - Sovereign CDS spread approximately given by: s_spread_t = (1 − R_CDS) ⋅ λ_t, where λ_t is default intensity and 1 − exp(−λ_t) ≈ λ_t is the 1-year default probability in approximation.
  - Market convention often assumes sovereign CDS recovery rate R_CDS = 25 percent.
  - CDS-implied PDs are risk-neutral; may diverge from physical PDs and may suffer from endogeneity if central bank has large market footprint.
  - CDS method can be linked to macro variables by conditioning historical CDS spreads on inflation, real GDP growth, and debt/GDP, estimating a linear relationship, and deriving scenario-based ECL provisioning.
- Discount-implied PD (DCF method):
  - Includes a discount rate reflecting sovereign default risk; difference between risk-free and default-risk-embedded discount rate (bond premium) can be considered equivalent to ECL provisioning under LGD = 100 percent.
  - Bond premium (bond yield − policy rate) can be conditioned on macroeconomic variables similar to the CDS method.

### Exposure at Default (EAD) and yield curve modeling
- Yield curve model is crucial to determine price and remuneration of domestic assets bought by the central bank and to determine EAD.
- IFRS 9 fair value hierarchy applied where possible:
  - Level 1: exact market prices (preferred).
  - Level 2: prices from market transactions of comparable securities.
  - Level 3: internal model discounting future cash flows using an estimated yield curve when no comparables exist.
- Yield curves estimated using reduced-form models such as the Nelson-Siegel functional form.
  - Dynamic Nelson-Siegel projects yields of any maturity as a linear function of three factors: level, slope, curvature.
  - The three factors can be driven by exogenous regressors (real GDP growth, inflation, exchange rate, others), enabling generation of stressed yield curves.
  - Interpretations:
    - Level = very long-term interest rate.
    - Slope = difference between short-term yield and long-term yield.
    - Curvature = degree to which the yield curve is hump-shaped.

*Source: IMF staff; CENTRAL BANK STRESS TESTING—GUIDANCE NOTE, selected excerpts.*

### 40.      Central banks are not expected to provide unsecured loans even when providing

### 40.      Central banks are not expected to provide unsecured loans even when providing

### Emergency lending and credit risk
- Collateralized lending provides two lines of defense for the central bank: (i) the solvency of the institutions that receive the loan; and (ii) the collateral itself.
- Central bank risk on counterparty default is the potential need to liquidate the collateral; an appropriate haircut can be applied to the collateral value to mitigate this risk.
- If demand for liquidity exceeds what the central bank could provide with full collateralization after the appropriate haircut, the government should provide a guarantee to the central bank to cover any shortfall.
- In crises, unsecured lending or collateralization shortfalls are not a null probability:
  - Central banks may deliberately increase risk tolerance (lower haircut) to preserve financial stability.
  - Haircuts may be inaccurately calibrated for less liquid collateral under urgent conditions.
  - The sovereign may be under stress and unable to provide a credible guarantee.
- After such episodes, the central bank’s balance sheet could be durably damaged.
- “Emergency liquidity” credit risk is modelled as an ECL with a contingent EAD.
  - Emergency lending could be construed as a contingent credit line implicitly extended to predetermined counterparties.
  - The PDs for these counterparties would be computed based on their credit ratings, including solvency ratios.
  - The EAD should reflect the likelihood of a large liquidity need from eligible counterparties.
  - The ECL could also be applied or refined ex-post.

### Interest rate risk — Inflation risk and the policy rate
- Central banks with large balance sheets are particularly exposed to an inverted yield curve; losses depend on:
  - the size of the increase in the policy rate,
  - the size of the asset portfolio held by the central bank,
  - the pace of re-pricing of this portfolio (i.e., its average duration).
- A portfolio with a shorter duration reduces exposure because it adjusts quicker to the new interest rate environment.
- The policy rate is modelled as the reaction function most consistent with the monetary policy framework of the central bank.
- For inflation-targeting central banks, the policy rate is typically modelled as a function of inflation and possibly other variables. Example specification (domestic policy rate 푅푅
푡푡
):
  - 푅푅
푡푡
=휋휋�+푒푒
∗
+ 훾훾
휋휋
(휋휋
푡푡
−휋휋�
)
+훾훾
푦푦
(푦푦
푡푡
−푦푦�
)
+훾훾
푖푖
푅푅
푡푡−1
+휀휀
푡푡
  - It is often assumed that central banks adjust their policy rates gradually toward the target rate.
- For a fixed exchange rate regime, the policy rate should reflect the anchor currency policy rate plus an additional premium for country-specific risk:
  - UIP conventional: 푅푅
푡푡
−푅푅
푡푡
∗
=(푠푠
푡푡+1
푒푒
− 푠푠
푡푡
) 푠푠
푡푡
⁄
  - With time-varying country risk premium 휑휑
푡푡
: 푅푅
푡푡
−푅푅
푡푡
∗
=(푠푠
푡푡+1
푒푒
− 푠푠
푡푡
) 푠푠
푡푡
+휑휑
⁄
  - Under fixed exchange rate (change in exchange rate = 0): 휑휑
푡푡
=푅푅
푡푡
−푅푅
푡푡
∗
- Country risk premium theoretically affected by multiple factors including debt, currency, political stability, geopolitical factors, regulatory factors, liquidity and market access, perceptions of risk and global economic conditions.
- In the CBST, a model-determined risk premium is forecasted by conditioning the risk premium time series on quantified explanatory factors; the forecasted risk premium is denoted by 휑휑
푡푡+1
푒푒
 and the interest rate reaction function becomes:
  - 푅푅
푡푡
=푅푅
푡푡
∗
+휑휑
푡푡+1
푒푒

### Duration risk
- The CBST explicitly models the duration of the domestic portfolio using a satellite model in which the central bank rolls over the domestic bond portfolio to capture average yield effects and gradual adjustment to the domestic policy rate.
- Assumptions that could be considered:
  - The maturing assets will not be replaced: Reserves shrink as portfolio matures, lowering the cost of monetary policy.
  - The maturing assets are replaced as they expire, and the average duration of the portfolio is kept unchanged: Reserve stock remains unchanged; monetary policy expense derived from forecasted policy rate path; average return on assets changes based on policy rate and yield curve forecasts.
  - The portfolio is partially replaced and the average maturity changes: combination approach used in practice under quantitative tightening.
- Simulations employ the second approach (maturing assets replaced to keep average duration unchanged) to analyze the impact of duration and yield curve changes on central bank equity.

### Mark-to-market
- Mark-to-market losses are primarily a concern for the FX portfolio of central banks with fixed exchange rate agreements or those in emerging markets.
- These central banks might need to sell FX securities at any time to meet FX intervention objectives, facing market price risk.
- Exposure is pronounced when major central banks engage in sharp and prolonged tightening cycles after long low-rate periods, during which portfolio duration often increases.
- Most or all of the FX portfolio is expected to be mark-to-market and exposed to valuation changes due to foreign policy rate changes and major currency yield curve slopes.
- FX portfolio valuation approach:
  - Net present value approach using foreign yield curves.
  - Assume equal amounts spread across maturities up to a predetermined longest maturity; when the shortest maturity bond expires, principal is reinvested in the longest maturity bond based on current yield.

### Domestic securities and banking book treatment
- Less need to mark-to-market for domestic securities: central banks are not liquidity constrained in their own currency and may place domestic securities in the banking book (held to maturity), avoiding mark-to-market.

### Exchange rate risk
- Exchange rate assumptions align with the country team macroframework or are modelled as an exogenous process in its absence.
- Exchange rate risk depends on FX reserves composition. Example measurement items for foreign assets in USD include cash deposits, FX securities held at fair value, and derivative financial instruments measured as:
  - 퐶퐶푅푅푠푠ℎ
푈푈퐶퐶퐶퐶,푡푡
=퐹퐹푋푋
푈푈퐶퐶퐶퐶,푡푡
퐹퐹푋푋
푈푈퐶퐶퐶퐶,푡푡−1
⁄
⋅�1 +푅푅
푐푐푐푐 푠푠ℎ,푈푈퐶퐶퐶퐶,푡푡
�⋅퐶퐶푅푅푠푠ℎ
푈푈퐶퐶퐶퐶,푡푡−1

  - 퐹퐹푅푅푅푅푒푒 푉푉푅푅푅푅푅푅푅푅
푈푈퐶퐶퐶퐶 ,푡푡
=퐹퐹푋푋
푈푈퐶퐶퐶퐶,푡푡
퐹퐹푋푋
푈푈퐶퐶퐶퐶,푡푡−1
⁄
⋅�1 +푅푅
푌푌푇푇푌푌,푈푈퐶퐶퐶퐶,푡푡
�⋅퐹퐹
푈푈퐶퐶퐶퐶,푡푡
(1 +푅푅
푈푈퐶퐶퐶퐶,푡푡
)
휏휏
⁄

  - 퐼퐼푒푒퐴퐴푅푅 퐼퐼푅푅 푅푅푒푒 퐷퐷푅푅푒푒푅푅푅푅푅푅푅푅푅푅푅푅푅푅푠푠
푈푈퐶퐶퐶퐶 ,푡푡
=퐹퐹푋푋
푈푈퐶퐶퐶퐶,푡푡
퐹퐹푋푋
푈푈퐶퐶퐶퐶,푡푡−1
⁄
⋅�푅푅
 푈푈퐶퐶퐶퐶,푡푡
−푅푅
 푈푈퐶퐶퐶퐶,퐶퐶푒푒 퐷퐷푖푖퐷퐷𝑐푐푡푡𝑖𝑖 퐷퐷푒푒
�.�퐷퐷
퐴퐴,푡푡
−퐷퐷
퐿퐿,푡푡
�,
  - Where definitions include 퐹퐹푋푋
푈푈퐶퐶퐶퐶,푡푡 as the local currency-to-USD exchange rate, 푅푅
푐푐푐푐 푠푠ℎ,푈푈퐶퐶퐶퐶,푡푡 as the short-term (three months) foreign interest rate, 푅푅
푌푌푇푇푌푌,푈푈퐶퐶퐶퐶,푡푡 as yield to maturity, 퐹퐹
푈푈퐶퐶퐶퐶,푡푡 as face value of USD investments, 푅푅
 푈푈퐶퐶퐶퐶,퐶퐶푒푒 퐷퐷푖푖퐷퐷𝑐𝑐푡푡𝑖𝑖 퐷퐷푒푒 as interest rate used to price derivatives at execution, and �퐷퐷
퐴퐴,푡푡
−퐷퐷
퐿퐿,푡푡
� as the net position of the value of derivatives on the balance sheet.

### FX revaluation gains and losses
- A significant part of exchange rate gains or losses does not affect statutory capital; unrealized FX revaluation gains or losses are booked in the revaluation reserve account and cannot be distributed.
- Revaluation account functions as a capital buffer specifically dedicated to exchange rate risk:
  - Revaluation gains are typically held to offset future losses, though central banks with long-standing positive revaluation gains may realize a portion that exceeds possible exchange rate losses with high probability.
  - A revaluation deficit remains unrealized if exchange rate gains are expected to cover the losses with high probability; otherwise it would be deducted from statutory capital.
- In the CBST, the conditional distribution of the exchange rate is forecasted to identify tail exchange rate risk; the revaluation account distribution rule could be calibrated on this forecast.
  - Forecast employs an at-risk functional form using the 5th percentile to simulate exchange rate appreciation (revaluation losses).
  - General specification:
    - ∆푆푆
푡푡+푘푘
=훼훼+∆푆푆
푡푡
+ 훾훾퐷퐷푅푅퐼퐼푅푅푠푠푅푅푅푅퐴퐴 푀푀푅푅푒푒퐴퐴푅푅
푡푡
+훿훿퐵퐵푅푅푅푅푅푅푒푒퐴퐴푅푅 푅푅푑푑 푃푃푅푅푦푦퐼퐼푅푅푒푒푅푅푠푠
푡푡
+휃휃퐷퐷푅푅퐼퐼푅푅푠푠푅푅푅푅퐴퐴 푃푃푒푒푅푅퐴퐴푅푅
푡푡
+휎휎퐸퐸퐸퐸 푅푅푅푅푒푒푒푒푅푅푅푅 퐹퐹푅푅퐴퐴푅푅푅푅푒푒푠푠
푡푡
+휀휀
푡푡
  - ∆푆푆
푡푡 is the log return of the exchange rate (percentage change); forecasting the distribution based on the 5 percent value at risk ensures that the top 5 percent of exchange rate appreciations are forecasted.

### Cross-currency exposure and derivatives
- Cross-currency exposure matters when a central bank has borrowed one currency to invest in another; U.S. dollar depreciations against foreign currencies lead to revaluation gains if the central bank holds net asset positions in those foreign currencies.
- Derivative exposure:
  - Central banks may provide FX derivatives (forwards, FX swaps); main risk is liquidation in event of counterparty default.
  - Example: if the central bank lends local currency for U.S. dollars and the counterparty defaults, the central bank must sell the U.S. dollars in the market, creating market risk mitigated by haircuts and regular margin calls.
  - The CBST assesses exposure to market risk after factoring mitigation measures.

### Commodity risk
- Central banks are rarely exposed to commodity price changes, but exposure arises when central banks have purchased equities or gold.
- Commodity-like items are modelled as a stochastic process.

### FX liquidity risk
- Central banks are not liquidity constrained in their local currency but face constraints in currencies they do not issue.
- Fully dollarized economies are subject to standard liquidity risk.
- In non-dollarized economies, central banks typically have long FX positions and are not normally exposed to depositor runs, but FX reserve depletion can prevent aid in a FX liquidity crisis, akin to policy insolvency.
- This risk is not covered in this guidance note as a tool has been proposed in Oura (2022).

### Forecasting balance sheet and related items

Currency in circulation and reserve requirement
- CiC forecasting:
  - Proposed seasonal ARIMA model with nominal GDP as an exogenous variable to capture inflation and real GDP impacts on CiC; conditions CiC on macro variables so CiC responds to stress testing scenarios.
  - Second specification adds interest rate effects:
    - CiC
t
=η
CiC
 NGDP
t
⋅
�
i
0
i
t
�
γ
CiC
, 
    - Where 휂휂
퐶퐶푖푖퐶퐶
 is a scaling factor and �
i
0
i
t
�
γ
CiC
 captures impact of policy rate changes to CiC, with γ
CiC
 measuring sensitivity.
  - The drop in demand for money during high inflation could introduce nonlinearities in the CiC–prices relationship that need modelling.

Reserve requirement balances
- Assuming a constant ratio and definition of the base, balances held at the central bank for the reserve requirement should depend on demand for money, similar to currency in circulation.
- If the reserve requirement is not remunerated, its growth changes interest paid or received depending on banks’ deficit or surplus with the central bank; this dynamic does not occur if requirement is remunerated at the policy rate.
- The CBST can simulate the impact on equity of remunerated versus unremunerated reserve requirements.

Operational cost
- Operational costs modelled to increase log-linearly with nominal GDP:
  - Operational Cost
푡푡
=휇휇휇휇퐿퐿퐷퐷푃푃
푡푡
훼훼
, 
  - μ is a scaling factor; α represents growth path for operational costs relative to nominal GDP.
- Specification allows dynamic response of operational costs to stress scenarios and simulation of impact on retained earnings and central bank capital via operational efficiency.

Relation with the Treasury
- Central bank often acts as the government’s banking agent; projected positions depend on two situations:
  - Treasury has an account but does not borrow from the central bank: project a constant balance based on historic average of the account as a percentage of GDP.
  - Treasury has an account and does borrow from the central bank: projections should come from government and central bank on expected borrowing and reimbursement, possibly based on legal provisions if central bank borrowing is regulated.

*Central Bank Stress Testing—Guidance Note (excerpt).*

### 67.      Asset purchase programs are designed to determine the size of the balance sheet. In

### ppea2024067 - 67.      Asset purchase programs are designed to determine the size of the balance sheet. In

### E. Foreign Reserves
- Impact of exchange reserves depends on the FX policy; three cases:
  - Freely floating exchange rate: FX reserves are not actively used. The stock, denominated in foreign currency, is assumed constant.
  - Managed float: The central bank has an active FX intervention policy. The FX reserves trajectory would be determined based on the central bank’s FX accumulation or decumulation plan.
  - Fixed exchange rate: FX reserves are unpredictable as the central bank responds on demand to market requests to buy or sell FX. However, if the interest rate rule correctly captures the risk premium, FX reserves could be assumed as growing with the demand for money because the need arising from CiC and the reserve requirement in the market would be satisfied by selling FX to the central bank. This would be equivalent to keeping FX reserves stable as percentage of nominal GDP.

### Scenario design (baseline and risk scenarios)
- Risk-horizon of the CBST model spans five years because forecast accuracy declines with the forecast horizon.
- Baseline and medium-term macroeconomic assumptions/forecasts:
  - Could be derived from official central bank forecasts developed by the research (or equivalently entitled) department and provided to the Risk Unit.
  - The Risk Unit develops macroeconomic assumptions for stress testing scenarios and obtains forecasting results from the research department.
- Sufficient variation around the baseline is necessary for risk assessment and robustness checks.
- Scenario design model options:
  - (i) at-risk models;
  - (ii) other IMF-used models, including SVAR (Structural Vector Autoregression) and DSGE (Dynamic Stochastic General Equilibrium);
  - (iii) deterministic models.
- Preference for at-risk models because they provide “extreme but plausible” shocks and are relatively easy to set up; general equilibrium approaches are ideal for endogeneity but are quantitatively difficult and resource-intensive.

### Balance sheet projection method
- Central bank balance sheet projection follows a recursive method over the stressed horizon:
  - Scenarios on real GDP growth, inflation, foreign and domestic interest rates, and the exchange rate determine the macro-financial environment for the period.
  - Satellite models map scenarios into balance sheet and income statement items.
  - Cash flows, net profit (or losses), and capital for the period are computed. The balance sheet is projected without dividend distribution.
  - The model is iterated to the next period.

### A. At-risk models (design and use)
- At-risk models:
  - Based on explanatory variables and projections of future distributions conditional on a given economic situation.
  - Useful for designing “extreme but plausible” macroeconomic scenarios.
  - Project asymmetric distributions reflecting future risks using quantile regressions and variables related to financial conditions and macro-financial vulnerabilities.
  - Capture structural changes and quantify upside and downside risks for variables such as GDP growth, inflation rates, exchange rates, using most recent observations.
  - Capture nonlinear interactions between shocks, financial conditions, and economic outcomes.
- Horizon suitability:
  - Mainly suitable for short horizons, ideally up to one year.
  - After shock estimation, various econometric techniques (e.g., VARs, frequentist or Bayesian estimation) can be employed to construct trajectories for remaining forecast horizons; should be combined with policy responses via central bank reaction functions.
- Growth-at-risk specifics:
  - Forecasts conditional distributions for changes in real GDP.
  - Four-step approach to obtain full distribution:
    - (i) dimensionality reduction on explanatory variables;
    - (ii) conditional average prediction using the Theil-Sen model;
    - (iii) skewness estimation of the distribution by the Firth model;
    - (iv) along with an assumption of Theil-Sen variance, parametrization of an asymmetrical Gaussian distribution.
  - Growth-at-risk transmits through higher PDs and other balance sheet impacts from lower output changes, such as slower growth in CiC.
- Inflation-at-risk:
  - Used to stress test equity against extreme but plausible inflation levels over a short forecast horizon.
  - Satellite models estimate the impact of inflation shocks on other macro variables and autonomously driven balance sheet items such as CiC, operational expenses, and the policy rate.
- Exchange-rate-at-risk:
  - Can determine how much of the revaluation reserve should be realized based on tail distribution assessment of the exchange rate risk.

### B. Deterministic models
- Focus on a fixed set of parameters to assess impacts of specific scenarios without random variables.
- Useful to assess robustness of equity path to costs mandated by discretionary central bank decisions (e.g., whether to remunerate required reserves).
- Can assess policy changes that increase equity (e.g., cost-efficiency gains).

### C. Other models
- Considered if data permit:
  - SVAR: assesses economy response to policy and external shocks by capturing dynamic relationships between multiple time series.
  - DSGE: uses microeconomic principles to forecast macro phenomena; useful for understanding impact of shocks and policy changes over time.

### Model output using stylized balance sheets — A. Macroeconomic environment
- Simulation purpose: assess impact of macro-economic environment on central bank capital using hypothetical balance sheet proportions.
- Macroeconomic scenario: inflation shock—high at simulation start, declines over time back to target due to central bank raising the policy rate via model reaction function; increase in policy rate leads to a short recession necessary to lower inflation.
- Exchange rate: depreciates over time, speed of depreciation slows after two years.
- Assumptions made about returns on claims on government and remuneration of reserves to ensure plausible simulations (key assumptions summarized in a table in source).

### B. Lean balance sheet simulation (assumptions and results)
- Lean balance sheet features:
  - Most central bank liabilities are CiC.
  - Banks hold what is necessary to fulfill reserve requirements at the central bank.
  - As demand for CiC increases, banks borrow from the central bank (OMO) at the policy rate.
  - Central bank operates a flexible exchange rate and keeps a modest FX portfolio.
  - Net foreign assets represent 30 percent of total assets.
  - Starting equity is 5 percent of total assets.
  - Initial operational costs are 0.5 percent.
- Result: inflation shock is positive for equity of the lean central bank, assuming no distribution:
  - Channels: (i) increased interest income due to higher policy rate; (ii) larger OMO stocks due to faster growth in CiC from higher inflation.
  - Equity also increases as a percentage of GDP when the real interest rate turns positive because realized earnings grow faster than nominal GDP.
  - In this scenario, central bank capital exceeds the minimum required to break even over the medium term. Decision to retain profits in general reserves versus distributing to the government would require stressing the macro framework.
- Note: exchange rate held fixed for this scenario; if exchange rate responds dynamically some capital gains would be marginally tempered by appreciation causing losses on FX holdings, but effect is limited due to small FX portfolio.

### C. Large domestic portfolio (assumptions and results)
- Features:
  - Central bank holds a large domestic securities portfolio; total assets increased 7.6 times compared with the lean balance sheet due entirely to domestic asset purchases.
  - CiC nominal starting point remains the same; its share drops to 9 percent of total assets.
  - Domestic policy rate increases over time from 1 percent to 16 percent.
  - Flat yield curve assumed for simplicity.
- Portfolio duration modeling:
  - Satellite model holds equal amounts across maturities up to a pre-fixed longest maturity; portions mature each quarter and principal reinvested into longest maturity.
  - Two average maturity cases considered: (i) one year; (ii) five years.
  - Valuation based on amortization method (held to maturity) so no mark-to-market loss from domestic yield increases.
- Result:
  - Central bank capital declines over time because interest income from domestic portfolio is smaller than interest expenses on excess reserves.
  - Longer durations exacerbate capital decline due to slower adjustment of bond portfolio returns:
    - Short duration case: capital ends at -3 percent of the nominal GDP.
    - Long duration case: capital ends at -5 percent of the nominal GDP.

### D. Large foreign portfolio (assumptions and results)
- Features:
  - Central bank has a large portfolio of foreign securities within a fixed exchange regime; total assets increased 4.7 times compared to the lean balance sheet due entirely to foreign government bonds.
  - Reaction function under fixed exchange rate: domestic policy rate follows the U.S. policy rate plus a spread assumed to be 25 basis points.
  - Two term-spread U.S. yield curve cases considered: (i) U.S. yield curve remains flat over time; (ii) U.S. yield curve steepens in the first two years before gradually flattening after December 2025.
- Foreign portfolio duration and valuation:
  - Satellite model rolls over foreign bond portfolio valuated under mark-to-market accounting.
  - Average duration cases: (i) 0.5 years; (ii) 2.5 years.
  - U.S. policy rate assumed to increase over time from 4.8 percent to 6 percent in the next five years.
  - Prudent FX risk management requires a short average duration; longer duration may reflect yield-seeking in low-rate environment.
- Results:
  - Central bank capital declines over time because interest income from foreign portfolio is smaller than domestic interest expenses on excess reserves.
  - The average yield of the foreign bond portfolio gradually adjusts through rollover toward the U.S. policy rate.
  - Flat yield curve case outcomes:
    - Short duration: capital ends at -0.5 percent of nominal GDP.
    - Long duration: capital ends at -0.7 percent of nominal GDP.
  - Longer duration leads to poorer capital outcomes due to slower adjustment of bond portfolio returns.
  - Mark-to-market losses impact equity path more when duration is longer:
    - Equity declines up to December of 2025 and then begins to recover under the long duration assumption because (i) long-end yield curve declines after 2025 producing mark-to-market gains; and (ii) returns for long-term bonds are higher in the steepened yield curve scenario than in the flat yield curve case.

### E. Sovereign credit risk due to possibility of debt restructuring
- (Section heading present; detailed content not included within provided excerpt.)

*Source: CENTRAL BANK STRESS TESTING—GUIDANCE NOTE (excerpt).*

### 90.      In the final hypothetical case, the central bank holds a large portfolio of domestic

### ppea2024067 - 90.      In the final hypothetical case, the central bank holds a large portfolio of domestic

### Large domestic portfolio under debt restructuring: scenario setup and key assumptions
- Total assets increased 7.6 times compared to the lean balance sheet.
- Initial balance sheet mirrors that of the large domestic bond portfolio case.
- Key distinctions in this scenario:
  - Expected Credit Losses (ECL) on the domestic government bond portfolio arise due to heightened sovereign credit risk associated with the possibility of a debt restructuring process.
  - Assumption of no return from the domestic bond portfolio and zero remuneration rate on the excess reserves (therefore no duration risk).
    - Note: The assumption of zero remuneration rate is applied only for expository reasons to remove the duration impact and focus the analysis on the ECL impact.

### ECL modeling approach and drivers
- ECL on government debt is simulated and allowed to be driven by the macroeconomic environment via a satellite model.
- Probability of default (PD) is inferred from changes in sovereign CDS spreads, which vary in response to three macroeconomic variables:
  - real GDP growth,
  - inflation,
  - public debt as a percentage of GDP.
- For public debt as percentage of GDP, two cases are considered:
  - (i) debt as percentage of GDP starts from 90 percent and declines by 1 percent in each quarter;
  - (ii) debt as a percentage of GDP holds at 90 percent over time.

### Impact on central bank capital: outcomes across scenarios
- In both debt-path scenarios, the capital of the central bank declines for the first two years, and then increases over time.
- Divergent long-run outcomes:
  - Declining debt path: capital reaches 1 percent of nominal GDP at the end of the simulation because ECL declines quickly after the first two years.
  - Constant debt path: capital reaches 0.4 percent of nominal GDP because ECL declines more slowly.
- The disparity arises from differing dynamics of ECL under the two public debt paths.

### Dynamic Nelson-Siegel model: yield-curve and policy-rate dynamics
- The Dynamic Nelson-Siegel (NS) model extends the NS model by explicitly modeling factor dynamics (Level, Slope, Curvature).
- NS functional form summarizes yield curve shapes with three factors and one parameter 휆휆.
- Short-end of the yield as maturity goes to zero: 푦푦푡(0) = 퐸퐸푡 + 푆푆푡.
  - Policy rate is interpreted as 풊풊푡 = 푦푦푡(0) (=푳푳푡 + 푺푺푡).
- Matrix form: 풚풚푡 = 훽훽 ⋅ 푿푿푡, with factor dynamics given by a VAR:
  - 푿푿푡 = 훷훷 푿푿푡−1 + 흁흁 + εt, where εt ∼ multivariate normal(0, 횺횺).
- The dynamic NS model separates dynamics of long-term yield (proxied by 퐿푡) and policy rate 풊풊푡 (=퐿푡+푆푡), enabling capture of divergence that impacts central bank net interest income (e.g., lower net interest income if policy rates increase faster than long-term yields).

### Internal credit rating model: structure and PD derivation
- Purpose: derive forecasts for PDs used to compute ECLs within CBST models.
- Model structure:
  - Four blocks plus an ESG overlay: economic strength, institutional strength, fiscal strength, susceptibility to event risks (political turmoil, banking sector crisis, natural disasters).
  - Blocks and sub-blocks are weighted based on historical calibration; ESG overlay applied only in exceptional circumstances.
  - All blocks except event risk use portions of variables as adjustment factors for the weighted sum; choices are historically calibrated.
- Back testing:
  - Model is back tested by deriving historical credit ratings over the past five to ten years and comparing to actual rating-agency ratings to assess accuracy.
- Conversion to PD:
  - Forecasted credit ratings are converted to PDs using tables mapping sovereign credit ratings to historical PDs from major rating agencies.
  - Standard approach employs a one-year PD; lifetime PD is employed for non-investment grade sovereigns with heightened credit risk.
- ECL calculation then uses the chosen PD alongside estimates for LGD and EAD.

### Estimating country risk premium (fixed exchange rate context)
- Target variable (spread): SseRPRt = It − I0t, where:
  - It is the three-month interbank offered rate in the country,
  - I0t uses the three-month LIBOR rate (replaceable by other international benchmarks).
- Model: ARIMAX (auto-regressive integrated moving average with exogenous variables) at monthly frequency.
- Benchmark list of factors (Table IV.1): includes Government revenue; CDS 1 year; CDS 5 years; CDS 10 years; GDP; Stock market index; Stock market turnover; Unemployment (international); Unemployment (domestic); VIX; CPI.
- Input variables are differenced and normalized before estimation.
- Model selection: uses Akaike’s Information Criterion (AICc) to identify best combinations of input variables and ARIMA orders.

### CBST code structure and execution workflow
- Codes are modular for readability and extensibility:
  - Macroeconomic component separated from main CBST component: replaceable by overwriting codes in “generation_modeling” or “macro_variables” folders.
- Main component contains three execution codes:
  - SelectScenarioRunModel: choose scenario ID and number of Monte-Carlo simulation paths.
  - Simulate_CBBS_MonteCarlo: generates projections of central bank balance sheet items and income statements across Monte-Carlo paths.
  - Core central balance sheet model exe: computes all balance sheet and income statement items over time using macroeconomic inputs, initial balance sheet, and fundamental parameters.

### At-Risk model and synthetic regressors
- Synthetic variables obtained through data reduction using partial least squares (PLS) regression to extract common trends under thematic groups.
- PLS projects Y and X onto a lower-dimensional subspace maximizing covariance; useful for many multicollinear variables.
- Examples of synthetic regressors and underlying variables provided for:
  - Growth at Risk model (Domestic Demand, External Demand, Domestic Financial Conditions).
  - Inflation at Risk model (Domestic Macro, External Demand, Monetary Factor).
  - Exchange Rate at Risk model (Domestic Macro, Inflation and Interest Rate, Balance of Payment).
- Conditional mean estimation uses a jackknife Theil-Sen estimator (robust to outliers), with specification analogous to OLS:
  - 푦푡+ℎ = 훼 + 훽T TC 푋푡 + 휖푡TC.

*Source: IMF staff calculations and guidance note content provided in the specified content unit.*

### 3. Where 풚풚

### 3. Where 풚풚

### Theil-Sen regression: conditional expectation and residuals
- The model specifies 풚풚_t+ h as the real GDP growth in t+h, with 푿푿_t a vector of conditional variables and 휶휶 the intercept.
- Residuals of the Theil-Sen regression are denoted 흐흐_t_TST (presented in the source as 흐흐 풕풕 푻푻푺푺).
- Theil-Sen provides the conditional expectation:
  - 피피[푦푦_t0+ℎ | 푋푋_t0] = 훼훼� + 훽훽̂_TTC 푋푋_t0
- The residual variance from the Theil-Sen estimation is used as the model’s assumption for variance (see variance discussion below).

### Firth logistic model: estimating asymmetry (probability of exceeding a threshold)
- The Firth model is a logistic regression with penalized likelihood used to estimate asymmetry around the average projection of growth, inflation, or exchange rate.
- The dependent event is coded as a binary indicator: 1 if the event occurs (푦푦_t+ℎ > 푦푦�) and 0 otherwise.
- Specification written in logistic form:
  - ℙ[푦푦_t+ℎ > 푦푦� | 푋푋_t] = 훼훼 + 훽훽_LGL 푋푋_t + 휖휖_t_LGL
- In the notation used:
  - 풚풚_t+ h is the real GDP growth in t+h.
  - 풚풚� is a given growth threshold.
  - 푿푿_t is the vector of conditional variables.
  - 휶휶 is the intercept.
  - 흐흐_t_LGL denotes the residuals of the logistic regression (presented as 흐흐 풕풕 푳푳푳푳).
- Firth’s penalized-likelihood innovation addresses bias in classical logistic estimation for small/noisy samples or weak separation by modifying the likelihood and introducing a penalizing term, eliminating estimation bias (see Firth 1993).

### Theil-Sen / Firth dual model: assembling conditional moments
- The dual model estimates two moments from the conditional distribution:
  - First moment (conditional expectation): Theil-Sen projection, 피피[푦푦_t0+ℎ | 푋푋_t0] = 훼훼� + 훽훽̂_TTC 푋푋_t0.
  - Second statistic (asymmetry): cumulative density at the conditional mean estimated by the logistic (Firth) model:
    - F(푦푦_t0 | 푋푋_t) = 훼훼� + 훽훽̂_LGL 푋푋_t0.
  - These two statistics do not identify the full distribution because the variance (second-order moment) is missing.
- To address limited-sample issues in variance estimation, the model assumes:
  - Variance is unconditional and equal to the residual variance of the Theil-Sen estimation (constant heteroskedasticity over time).
  - This avoids projection variance inflation with horizon and provides a stable variance input for parametric fitting.
- Under these assumptions, the at-risk model obtains three conditional moments:
  - Expectation (Theil-Sen projection).
  - Variance (constant heteroskedasticity, derived from Theil-Sen residual variance).
  - Skewness (inferred from the Firth logistic model).

### Parametrization: asymmetric Gaussian distribution and rationale
- The team parametrizes an asymmetrical Gaussian distribution from the three estimated moments (expectation, variance, skewness).
- Rationale for asymmetric Gaussian:
  - Encompasses both standard normal and asymmetric cases while remaining parsimonious given only three moments.
  - Enables closed-form algebraic relationships between moments, cumulative density, and parameters, allowing manual parameter derivation rather than numerical optimization.
  - Avoids using distributions requiring more moments (e.g., asymmetric Student needs four moments including kurtosis) which are infeasible with limited samples.
  - Nonparametric approaches (e.g., kernel methods) are unsuitable due to limited sample size.
- Note on inference of asymmetry:
  - The cumulative quantity F(y_t0 | X_t) is not directly a measure of asymmetry, but in the case of an asymmetrical Gaussian distribution, the asymmetry coefficient can be inferred from F(y_t0 | X_t) via a simple bijective transformation.

### Practical considerations and model strengths
- The Theil-Sen/Firth dual approach:
  - Stabilizes projections under limited data by avoiding direct conditional variance estimation.
  - Provides interpretable metrics useful for economic analysis: central tendency, interquartile range, and balance of risks.
  - Retains generality while conserving simplicity through the asymmetric Gaussian assumption and closed-form parameter relationships.

*Source: CENTRAL BANK STRESS TESTING—GUIDANCE NOTE (excerpt).*

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_Source: https://www.imf.org/-/media/files/publications/pp/2024/english/ppea2024067.pdf_