## FINANCIAL SECTOR ASSESSMENT PROGRAM — DENMARK: TECHNICAL NOTE STRESS TESTING (SEPTEMBER 2006)

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### Objective, scope, and approaches
- Objective: assess resilience of Denmark’s banking and insurance sectors to low probability, severe, but plausible single factor shocks and macroeconomic scenarios.
- Two complementary approaches:
  - Bottom-Up (BU): institution-led stress tests using banks’ and insurers’ internal risk models; coordinated and aggregated by the Danish Financial Supervisory Agency (DFSA). Based on December 2005 data.
  - Top-Down (TD): mission and Danmarks Nationalbank (DNB) risk‑based stress tests to estimate portfolio credit risk and validate BU results. Based on December 2004 data.
- Coverage: five largest financial groups representing roughly 80 percent of system assets on a consolidated basis:
  - Danske Bank: 46 percent of assets
  - Nordea Bank: 26 percent of assets
  - Jyske Bank: 13 percent of assets
  - FIH: 8 percent of assets
  - Sydbank: 7 percent of assets
- Risks covered: market risk (interest rates, equity prices, foreign exchange rates, real estate prices), credit risk (multi-factor or macroeconomic scenarios), liquidity, concentration, and contagion risks. Operational risk not assessed.

### Design of shocks, scenarios, and time horizons
- Shocks and scenario magnitudes based on historical information covering at least one economic cycle and episode of financial distress.
- Baseline forecast and three adverse scenarios developed using MONA macroeconomic model (DNB) to determine endogenous responses.
- Time horizons:
  - Market and credit risk stress tests: three-year horizon.
  - Liquidity, concentration, and contagion shocks: single events.

### BU implementation details and insurance sector coverage
- DFSA guidance ensured consistency; institutions used their own methodologies.
- Banks’ Expected Loss (EL) calculation: EL = Probability of Default (PoD) × Exposure (EX) × Loss Given Default (LGD).
- BU practices:
  - Most banks computed macro impacts on PoDs by loan type using EX databases and conservative LGD assumptions.
  - One bank estimated EL directly as function of macro variables.
  - Capital impact estimation varied:
    - Two banks estimated Unexpected Losses (UL) as 99.5 percent Value at Risk (VaR) of the loan portfolio loss distribution (PLD), equated to Economic Capital (EC).
    - Two banks used an “accounting approach” to estimate CAR after shocks.
- Insurance participation: five largest insurers representing slightly more than 50 percent of insurance sector assets: Danica Pension, Nordea Pension, SEB Pension, PFA Pension, Kommunernes Pensionsforsikring.

### TD data, objective, and aggregation constraints
- TD focused on portfolio credit risk and validation of BU.
- DFSA data (1991–2004): ratios of provisions (flow) to loans by sector; percentage loan exposure per sector (Group 1 banks).
- DNB data (1996–2003): PoDs by risk-rating categories (DNB failure‑rate model); proportion of EX to each rating category.
- No bank‑level or loan‑by‑loan data available; TD results reflect “average” credit risk of Group 1 banks.
- TD objective: estimate PLD to identify EL and UL and derive EC requirements (e.g., 99.9 percentile loss level). UL in internal models identified with 99.5 percent VaR of PLD.

### PMD, PLD, CoPoD and CIMDO methodologies
- Portfolio multivariate density (PMD) describes joint likelihood of changes in credit quality across loans.
- Procedure to obtain PLD:
  - Incorporate macroeconomic shocks into PoDs (CoPoD).
  - Generate PMD describing joint likelihood of PoD changes (CIMDO).
  - Simulate portfolio losses from PMD to construct PLD and determine EC.
- CoPoD:
  - Models PoDs as functions of macroeconomic and financial variables.
  - Uses MXED/minimum cross entropy reparameterization to estimate discrete distributions for coefficients and disturbances; produces smaller-variance estimators than OLS in small-sample PoD contexts.
- CIMDO:
  - Recovers PMDs from limited information—PoDs of each loan type—without requiring firm-level market data or explicit dependence structure.
  - Uses moment-consistency constraints to force posterior marginal default probabilities to match observed PoDs.

### Four-step stress-test procedure
- Step 1: Define macroeconomic scenarios.
- Step 2: Incorporate macro shocks into sector- or rating-group PoDs using CoPoD.
- Step 3: Recover PMD using CIMDO given shocked PoDs.
- Step 4: Simulate PLD from PMD and estimate Economic Capital (VaR) appropriate to portfolio risk.

### Single-factor shocks and macro scenarios (selected exact figures)
- Single-factor market risk shocks:
  - Shock 1: Parallel shifts in the yield curve: +250 and -100 basis points.
  - Shock 2: Equity prices: -30 percent.
  - Shock 3: Foreign exchange DKr/non-euro currencies: appreciation/depreciation 40 percent.
  - Shock 4: Real estate prices: -30 percent.
- Three macroeconomic scenarios (percentage deviations from baseline) — selected entries preserved exactly:
  - GDP % y/y baseline: 2006 = 2.4, 2007 = 2.2, 2008 = 2.2.
  - GDP % y/y Scenario 1: 2006 = -1.8, 2007 = -5, 2008 = -6.8.
  - GDP % y/y Scenario 3: 2006 = -2.4, 2007 = -6.7, 2008 = -9.2.
  - House price % y/y baseline: 2006 = 7, 2007 = 3, 2008 = 3.
  - House price % y/y Scenario 3: 2006 = -15.6, 2007 = -29.8, 2008 = -41.9.
  - Oil price $/barrel baseline: 2006 = 56.4, 2007 = 56.4, 2008 = 56.4; Scenario 3 oil price: 2006 = 30, 2007 = 30, 2008 = 30.
  - Discount rate % points Scenario 3: 2006 = 2.5, 2007 = 2.5, 2008 = 2.5.

### TD credit-risk modeling results — PoDs, ELs, ULs (selected figures)
- ELs as percent of Risk-Weighted Assets (Table 3a):
  - Scenario 1: 2006 = 1.45, 2007 = 1.66, 2008 = 2.32
  - Scenario 2: 2006 = 1.37, 2007 = 1.63, 2008 = 1.73
  - Scenario 3: 2006 = 1.34, 2007 = 2.00, 2008 = 3.07
- ULs as percent of Risk-Weighted Assets (Table 3b):
  - Scenario 1: 2006 = 11.94, 2007 = 12.16, 2008 = 13.76
  - Scenario 2: 2006 = 11.89, 2007 = 12.12, 2008 = 12.28
  - Scenario 3: 2006 = 11.82, 2007 = 12.39, 2008 = 14.71
- Interpretation:
  - PoDs, ELs, and ULs increase across scenarios and over time (scenario 3 most severe; scenario 1 next; scenario 2 least).
  - Sectoral concentration: credit, finance & insurance; household; manufacturing; business services together accounted for over 82 percent of exposures in 2004 and averaged over 73 percent between 1991 and 2004.
  - Key drivers: credit/GDP and house prices significant across sectors; sector sensitivities described in source (e.g., household: employment and interest rates; manufacturing: foreign exchange).

### Buffers, capital adequacy, and coverage of losses (exact historical and scenario numbers)
- Definitions:
  - EL buffer = (pre-tax income + provisions) / RWA.
  - CAR = Capital Adequacy Ratio; Economic Capital coverage evaluated using CAR and EL buffer.
- Group 1 historical data (selected exact items):
  - Total loan-loss provision: 2000 = 1,217; 2001 = 3,364; 2002 = 2,508; 2003 = 3,213; 2004 = 410
  - Total pre-tax income: 2000 = 13,582; 2001 = 15,950; 2002 = 16,549; 2003 = 20,338; 2004 = 22,075
  - Total provisions + pretax income: 2000 = 14,799; 2001 = 19,314; 2002 = 19,057; 2003 = 23,551; 2004 = 22,485
  - Total capital (capital adequacy): 2000 = 110,460; 2001 = 117,205; 2002 = 122,496; 2003 = 125,752; 2004 = 126,958
  - Total risk-weighted assets: 2000 = 1,135,287; 2001 = 1,141,915; 2002 = 1,174,840; 2003 = 1,168,680; 2004 = 1,243,096
  - EL buffer / RWA: 2000 = 1.30, 2001 = 1.69, 2002 = 1.62, 2003 = 2.02, 2004 = 1.81
  - Capital adequacy ratio: 2000 = 9.723, 2001 = 10.26, 2002 = 10.43, 2003 = 10.76, 2004 = 10.21
- Coverage of ELs by EL buffer after shocks (Table 5a: EL-buffer-S = end-2004 EL buffer 1.81 less EL ratio under scenario):
  - Scenario 1: 2006 = 0.35, 2007 = 0.14, 2008 = -0.51
  - Scenario 2: 2006 = 0.43, 2007 = 0.18, 2008 = 0.07
  - Scenario 3: 2006 = 0.46, 2007 = -0.19, 2008 = -1.26
- CAR decreases and CAR after shocks (Table 5b and 5c):
  - CAR-Decrease (percent):
    - Scenario 1: 2006 = -1.72, 2007 = -1.95, 2008 = -3.55
    - Scenario 2: 2006 = -1.68, 2007 = -1.91, 2008 = -2.07
    - Scenario 3: 2006 = -1.61, 2007 = -2.17, 2008 = -4.50
  - CAR-S (percent) [end-2004 CAR = 10.21 minus CAR-Decrease]:
    - Scenario 1: 2006 = 8.49, 2007 = 8.26, 2008 = 6.66
    - Scenario 2: 2006 = 8.54, 2007 = 8.31, 2008 = 8.14
    - Scenario 3: 2006 = 8.60, 2007 = 8.03, 2008 = 5.71
  - CAR-S-Net (CAR after shocks net of EL buffer):
    - Scenario 1: 2006 = 8.84, 2007 = 8.41, 2008 = 6.15
    - Scenario 2: 2006 = 8.97, 2007 = 8.48, 2008 = 8.22
    - Scenario 3: 2006 = 9.07, 2007 = 7.85, 2008 = 4.45
- Regulatory threshold: minimum CAR of 8 percent. Implications:
  - Minimum CAR of 8 percent breached under Scenarios 1 and 3 in 2008.
  - CAR-S-Net indicates 8 percent threshold breached in 2007 under scenario 3.
  - If uncovered ELs are absorbed by capital, CAR-S would be further reduced.

### BU results — credit and market risk (selected exact tables and figures)
- BU Expected Losses (percent of RWA) (Table 6):
  - Scenario 1 / Scenario 2 / Scenario 3
    - 2006: 0.31 / 0.28 / 0.45
    - 2007: 0.49 / 0.42 / 0.72
    - 2008: 0.78 / 0.57 / 1.10
- BU CAR-S (Table 7, percent):
  - Scenario 1 / Scenario 2 / Scenario 3
    - 2006: 10.50 / 10.08 / 9.32
    - 2007: 8.96 / 9.17 / 7.38
    - 2008: 6.65 / 7.73 / 4.50
- Conclusion from BU and TD: commercial banks can withstand moderate shocks but experience capital shortfalls under severe shocks; by year three mandatory CAR breached, substantial shortfall only in worst-case scenario.
- BU Market Risk: single-factor shocks produce modest average capital impacts; matched-duration hedging and flexible-rate loans mute market-shock transmission to capital.
- Bottom-Up single-factor CAR-S 2005 (Table 8 example values):
  - Yield Curve Up 250 bp: 10.23
  - Yield Curve Down 100 bp: 9.46
  - Property Prices Down 30 Percent: 9.89
  - Equity Prices Down 30 Percent: 9.87
  - FX Appreciation 40 Percent: 9.66; Depreciation 40 Percent: 10.16

### Insurance sector results (selected exact figures)
- Allocation principle: losses to owners absorbed by capital base and required-solvency-margin; losses to policyholders absorbed by “collective bonus component.”
- BU shows much impact absorbed by bonus component with marginal capital-base effects.
- Bottom-Up insurance sector table excerpts (Millions of DKK, 2005):
  - Capital base under shocks:
    - Yield Curve Up 250 bp: 40,987
    - Yield Curve Down 100 bp: 39,282
    - Property Prices Down 30 %: 41,437
    - Equity Prices Down 30%: 37,912
  - Required-solvency-margin (select):
    - Yield Curve Up 250 bp: 24,255
    - Yield Curve Down 100 bp: 22,189
    - Property Prices Down 30 %: 26,191
  - Solvency-Margin-Ratio (percent):
    - Yield Curve Up 250 bp: 170
    - Yield Curve Down 100 bp: 179
    - Property Prices Down 30 %: 159
    - Equity Prices Down 30%: 158
- DFSA “traffic light” monitoring:
  - Red light: decrease of 12 percent in stocks; decrease of 8 percent in real estate; interest rate change 0.7 percentage points.
  - Yellow light: decrease of 30 percent in stocks; decline of 12 percent in real estate; interest rate change 1.0 percentage point.
  - End-2005 outcome: six companies in yellow light, none in red light.

### Liquidity, funding sources, contagion and concentration
- Liquidity judged adequate; stock of liquid assets higher than regulatory minimum but has declined somewhat recently.
- Regulations: banks must have liquid assets to cover 15 percent of call deposits and liabilities due within one month; and 10 percent of all liabilities other than core capital.
- Funding shifts: loan growth outpaced deposit growth; increased reliance on bond issuance and borrowing from other financial institutions, including intragroup and foreign-bank borrowing which may be less stable.
- Interbank contagion (DNB model) findings:
  - Failure of largest debtor would cause another bank to fail on 11 out of 253 banking days in 2004 if LGD = 100 percent.
  - If LGD = 50 percent, contagion falls to one bank on one day.
  - Only first-order contagion occurred; no more than one bank would fail on any day from largest-debtor failure.
- Banking-sector concentration: Denmark shows higher concentration (CR5, Herfindahl index) than many EU states, increasing systemic vulnerability if large banks are insufficiently robust.

### Caveats, data limitations, and model constraints
- Secondary feedback effects and system-wide externalities not fully addressed; exercise not a general equilibrium model.
- TD analysis adjusted PoDs to simultaneous (single factor) shocks and ran asset-value-loss scenarios but could not model full second‑round effects.
- Data limitations:
  - Impeded modeling of adjustments to banks’ profitability and LGDs over time under stress.
  - Profitability held at 2004 levels, potentially understating shock impacts on capital.
  - LGDs not available; TD assumed LGD = 50 percent (Basel II IRB guideline).
- Use of grouped data prevented estimation of individual-bank portfolio credit risk; system‑average results may understate vulnerabilities of specific institutions and smaller, fast‑growing banks (groups 2 and 3).

### Key supervisory and policy implications (implicit and technical recommendations)
- Maintain and strengthen buffers:
  - Build EL buffer via adequate pricing and provisioning to absorb expected losses.
  - Preserve sufficient CAR to cover ULs; recognize that the 8 percent CAR minimum may be breached under severe scenarios.
- Increase supervisory focus on household-sector vulnerabilities:
  - Monitor rapid household credit growth (e.g., 2005 household lending annual growth > 21 percent), rising household leverage (homeowners’ average mortgage debt = 2.1 times gross household income; lowest income bracket average mortgage debt = 2.9 times annual household income; 10 percent of this group had mortgage debt ≥ 4.5 times annual household income), and large share of adjustable‑rate and amortization‑deferment products.
- Focus attention on smaller and faster‑growing banks (groups 2 and 3) with more concentrated, riskier portfolios.
- Strengthen stress-testing practices and data collection:
  - Perform regular stress-test exercises; standardize reporting of PoDs and require regular recovery‑rate data.
  - Model impacts of shocks on PoDs and bank profits to assess multi‑year capital dynamics.
  - Develop risk‑based methodologies for UL measurement and EC estimation to support validation of banks’ portfolio credit risk models and Basel II preparedness.
  - Perform regular interbank contagion analysis and continue close liquidity monitoring.
  - Continue close monitoring of insurance and pension sectors and attempt to assess second‑round macroeconomic shock effects.

*International Monetary Fund. FINANCIAL SECTOR ASSESSMENT PROGRAM — DENMARK: TECHNICAL NOTE STRESS TESTING, SEPTEMBER 2006.*

### 2006. The views expressed in this document are those of the staff team and do not necessarily reflect

### FINANCIAL SECTOR ASSESSMENT PROGRAM — DENMARK: TECHNICAL NOTE STRESS TESTING (SEPTEMBER 2006)

### Introduction
- Objective: assess the resilience of Denmark’s banking and insurance sectors to a variety of relevant risks using low probability, severe, but plausible single factor shocks and macroeconomic scenarios.
- Two complementary approaches used:
  - Bottom-Up (BU): institution-led stress tests using banks’ and insurers’ internal risk models; coordinated and aggregated by the Danish Financial Supervisory Agency (DFSA). Based on December 2005 data.
  - Top-Down (TD): mission and Danmarks Nationalbank (DNB) performed risk‑based stress tests to estimate portfolio credit risk and to validate BU results. Based on December 2004 data.
- Exercises developed in collaboration with DFSA and DNB as part of the Financial Sector Assessment Program (FSAP).

### Coverage
- Tests covered the five largest financial groups, representing roughly 80 percent of the system’s assets on a consolidated basis:
  - Danske Bank: 46 percent of assets
  - Nordea Bank: 26 percent of assets
  - Jyske Bank: 13 percent of assets
  - FIH: 8 percent of assets
  - Sydbank: 7 percent of assets
- Analysis performed at consolidated group level, focusing on major trading- and banking‑book risk exposures with shocks applied to domestic portfolios.
- Risks covered:
  - Market risk: interest rates, equity prices, foreign exchange rates, real estate prices
  - Credit risk: multi-factor or macroeconomic scenarios
  - Liquidity, concentration, and contagion risks assessed with a shorter horizon
  - Operational risk was not assessed

### Methodology — design of shocks and scenarios
- Shocks and scenario magnitudes based on historical information covering at least one economic cycle and episode of financial distress.
- A baseline scenario was forecast and three adverse scenarios were developed using:
  - assumptions about key variables to be shocked; and
  - the MONA macroeconomic model developed by the DNB to determine endogenous responses.
- Time horizons:
  - Market and credit risk stress tests spanned a three-year horizon because shock effects can take more than two years to fully materialize.
  - Liquidity, concentration, and contagion shocks were treated as single events.

### Bottom-Up (BU) approach — implementation details
- DFSA guidance ensured consistency; individual institutions used their own methodologies for credit and market risk.
- Expected Losses (EL) calculation used by banks:
  - EL = Probability of Default (PoD) × Exposure (EX) × Loss Given Default (LGD)
- Banks’ practices in BU:
  - Most banks computed macroeconomic impacts on PoDs by loan type using their databases on EX and conservative LGD assumptions.
  - One bank estimated EL directly as a function of macroeconomic variables.
  - Capital impact estimation varied:
    - Two banks estimated Unexpected Losses (UL) using internal models; UL equated to the 99.5 percent Value at Risk (VaR) of the loan portfolio loss distribution (PLD) and corresponds to Economic Capital (EC).
    - Two banks used an “accounting approach” to estimate CAR after shocks.
- Insurance sector:
  - Five largest insurance companies participated, representing slightly more than 50 percent of insurance sector assets: Danica Pension, Nordea Pension, SEB Pension, PFA Pension, and Kommunernes Pensionsforsikring.

### Top-Down (TD) approach — data and focus
- TD focused mainly on banks’ portfolio credit risk as the major source of system risk and to validate BU results.
- Data sources and content:
  - DFSA data (annual observations 1991–2004) contained:
    - ratios of provisions (flow) to loans grouped by economic sectors
    - percentage loan exposure to each economic sector
    - these variables corresponded to the loan portfolio of “Group 1” banks
  - DNB data (annual observations 1996–2003) contained:
    - PoDs of loans grouped by risk-rating categories (based on DNB failure-rate model)
    - proportion of total EX to each risk-rating category
- Confidentiality and aggregation constraints:
  - No bank‑level or loan‑by‑loan data were available to the mission; TD estimates reflect the “average” credit risk of Group 1 banks.
- Risk-based objective:
  - Estimate the portfolio loss distribution (PLD) to identify levels of Expected Losses (EL) and Unexpected Losses (UL) from which Economic Capital (EC) requirements (e.g., 99.9 percentile loss level) could be derived.
  - PLD summarizes possible portfolio losses and their likelihood; ULs and EC are derived from PLD percentiles.

### Key modeling and measurement points
- Expected losses should be absorbed by earnings via adequate pricing and provisioning.
- Unexpected losses should be covered by economic capital, given the variability of realized credit losses.
- UL in participating banks’ internal models was identified with the 99.5 percent VaR of the PLD.

### Data and operational notes
- BU tests used institution-specific internal models; DFSA aggregated results to ensure confidentiality and reporting uniformity.
- TD tests used aggregated supervisory (DFSA) and financial surveillance (DNB) datasets to estimate PLD and related risk measures for Group 1 banks.
- The three-year horizon was selected based on literature and modeling practices recognizing multi-year realization of shock effects.

*International Monetary Fund. FINANCIAL SECTOR ASSESSMENT PROGRAM — DENMARK: TECHNICAL NOTE STRESS TESTING, SEPTEMBER 2006.*

### 12.      In order to estimate the PLD of a loan portfolio, it is necessary to infer the

### _cr07125 - 12.      In order to estimate the PLD of a loan portfolio, it is necessary to infer the

### Portfolio multivariate density (PMD) and portfolio loss distribution (PLD)
- The PLD of a loan portfolio requires simulation of possible losses given the characteristics of loans in the portfolio.
- The PMD describes the joint likelihood of changes in the credit risk quality of the loans that make up the portfolio.
- Steps to obtain PLD from PMD:
  - Incorporate macroeconomic shocks into PoDs of loans (CoPoD).
  - Generate the PMD describing joint likelihood of PoD changes (CIMDO).
  - Simulate portfolio losses/gains from the PMD to construct the PLD.
  - Determine Economic Capital (EC) from the PLD.

### Methodologies: CoPoD and CIMDO
- CoPoD (Conditional Probability of Default) Methodology
  - Models PoDs as functions of macroeconomic and financial variables.
  - Produces efficient estimators, particularly when time series of PoDs contain few observations.
  - Produces smaller-variance estimators than OLS in small-sample PoD contexts.
  - CoPoD assigns different weights to data observations and infers their distribution from data, contrasting with ML/OLS which implicitly assigns equal weights and assumes a parametric distribution (usually normal).
  - Appendix 1: summary and key features of CoPoD (referenced in source).
- CIMDO (Consistent Information Multivariate Density Optimizing) Methodology
  - Recovers PMDs from very limited information—the PoDs of each loan type in the portfolio.
  - Does not require stock prices, market/financial variables of individual firms, or explicit loan dependence structure (correlations).
  - Produces better-specified PMDs than parametric PMDs when calibration information is unavailable.
  - Specification improvements measured under the probability integral transformation (PIT) criterion (technical proofs in Segoviano (2006b)).
  - Appendix 3: summary of CIMDO; Appendix 4: intuitive explanation of density specification improvements (referenced in source).

### Stress test procedure (four-step)
- Step 1: Definition of macroeconomic scenarios (mission and DNB defined scenarios used in both TD and BU approaches).
- Step 2: Incorporation of macroeconomic shocks into PoDs of loans grouped by sector or rating using CoPoD.
- Step 3: Modeling of the portfolio multivariate density (CIMDO) once PoDs are shocked.
- Step 4: Simulation of portfolio loss distribution (PLD) from the PMD and estimation of Economic Capital (VaR) that a bank should set aside given the portfolio risk profile.
- The framework links macroeconomic shocks explicitly to EC estimation, ensuring economic consistency and facilitating policy recommendations.

### Caveats and limitations of the exercise
- Secondary feedback effects in the financial system are not fully addressed; the exercise is not a general equilibrium model capturing financial system externalities.
- TD analysis accounted for some effects by adjusting PoDs to simultaneous (single factor) shocks and running scenarios where assets (financial and real estate) lose value.
- Data limitations:
  - Impeded modeling of adjustments to banks’ profitability and LGDs through time under stressed scenarios.
  - Profitability assumed to remain at 2004 levels, potentially understating shocks’ impact on capital.
  - LGDs were not available; an assumption was made (see Note 11: LGD equal to 50 percent as indicated in the Basel II IRB approach guidelines).
- Use of bank-grouped data prevented estimation of individual banks’ portfolio credit risk; high bank concentration in Denmark implies system-average results may understate vulnerabilities of specific institutions.

### Vulnerabilities considered beyond credit risk
- Liquidity, contagion risk in the interbank market, and concentration were incorporated into the TD approach.
  - Liquidity and concentration risks assessed by analyzing liquidity and exposures in banks’ portfolios.
  - Contagion risk tests employed a model developed by the DNB (defined in Abildgren and Arnt (2004)).

### Definition of shocks and macroeconomic scenarios
- Single factor market risk shocks:
  - Shock 1: Parallel shifts in the yield curve: +250 and -100 basis points.
  - Shock 2: Equity prices: -30 percent.
  - Shock 3: Foreign exchange DKr/non-euro currencies: appreciation/depreciation 40 percent.
  - Shock 4: Real estate prices: -30 percent.
- Three macroeconomic scenarios (percentage deviations from baseline) designed as extreme but plausible, drawing from historical information:
  - Scenario 1: Boom-bust in real estate prices and credit.
  - Scenario 2: Foreign shock due to a correction in U.S. imbalances (reduced European Net Foreign Investment, euro appreciation, loss of competitiveness).
  - Scenario 3: Boom-bust in real estate prices and credit plus an increase in European interest rates.
- Table 1 (Macroeconomic Scenarios—Percentage Deviations From Baseline Scenarios) reports exact shocks for variables across 2006, 2007, 2008. Examples preserved exactly as in source:
  - GDP % y/y baseline: 2006 = 2.4, 2007 = 2.2, 2008 = 2.2.
  - GDP % y/y Scenario 1: 2006 = -1.8, 2007 = -5, 2008 = -6.8.
  - GDP % y/y Scenario 3: 2006 = -2.4, 2007 = -6.7, 2008 = -9.2.
  - House price % y/y baseline: 2006 = 7, 2007 = 3, 2008 = 3.
  - House price % y/y Scenario 3: 2006 = -15.6, 2007 = -29.8, 2008 = -41.9.
  - Oil price $/barrel baseline: 2006 = 56.4, 2007 = 56.4, 2008 = 56.4; Scenario 3 oil price: 2006 = 30, 2007 = 30, 2008 = 30.
  - Discount rate % points Scenario 3: 2006 = 2.5, 2007 = 2.5, 2008 = 2.5.
  - Notes: shocks are percentage deviations from the baseline scenario (BLS); interpretation example provided in source (e.g., a shock to GDP in 2008 under Scenario 3 is equal to -9.2 percent from the BSL in 2008).

### Results — Top Down (TD) approach: Credit risk modelling
- PoDs modeled as functions of macroeconomic variables using CoPoD to estimate coefficients for explanatory variables significant for PoDs of loans grouped by risk-rating and economic sector.
- Equations estimated for 8 risk-rating categories and 11 economic sectors.
- Across estimated sectors, CREOVGDP (credit over GDP) and HOUP (house prices) are significant explanatory variables.
- Sector-specific significant variables and lags (Table 2, as reported exactly in source):
  - Public: CREOVGDP Sign + Lag 3, HOUP Sign + Lag 2, GDP Sign - Lag 0, R2 = 0.83.
  - Agriculture: CREOVGDP + Lag 5, HOUP + Lag 4, GDP - Lag 3, R2 = 0.81.
  - Fisheries: CREOVGDP + Lag 5, HOUP + Lag 4, GDP + Lag 2, AVGR + Lag 1, R2 = 0.78.
  - Manufacturing: CREOVGDP + Lag 4, HOUP + Lag 1, GDP - Lag 1, AVGR + Lag 0, FXDU - Lag 2, R2 = 0.74.
  - Building: CREOVGDP + Lag 5, HOUP - Lag , + 0,5 (as presented), R2 = 0.68.
  - Trade: CREOVGDP + Lag 4, HOUP - Lag 0, GDP - Lag 3, FXDU + Lag 0, R2 = 0.72.
  - Transport: CREOVGDP - Lag 1, HOUP - Lag 0, GDP - Lag 0, AVGR + Lag 0, R2 = 0.69.
  - Finance & insurance: CREOVGDP - Lag 0, HOUP + Lag 3, AVGR + Lag 0, R2 = 0.73.
  - Business: CREOVGDP + Lag 6, HOUP - Lag 0, GDP - Lag 3, R2 = 0.74.
  - Households: (formatted in source) HOUP + Lag 4, GDP - Lag 0, AVGR + Lag 1, MTGR + Lag 1, FXDU - Lag 2, R2 = 0.71.
- Illustration: Figure 5 shows estimated PoDs as functions of macroeconomic variables for selected sectors (Manufacturing, Credit/Finance/Insurance, Business Services, Household) with actual/fitted/residual plots (referenced in source).

*Source: IMF staff report content as provided in the supplied PDF excerpt.*

### 30.      The effects of macroeconomic shocks on credit risk are gauged by estimating

### _cr07125 - 30.      The effects of macroeconomic shocks on credit risk are gauged by estimating

### Methodology for gauging effects of macroeconomic shocks on credit risk
- Stressed Probability of Defaults (PoDs) are estimated for each risk-rating category and each economic sector using selected model specifications together with values of key macroeconomic variables.
- Stressed PoDs are entered as exogenous variables to recover the Portfolio Migration Density (PMD), which describes the joint likelihood of changes in credit risk quality across loans in a bank’s portfolio.
- Expected Losses (ELs) are computed as EL = PoD x exposure x LGD. Because information on LGDs was not available, LGD was set to 50 percent in accordance with Basel II guidelines.
- The PMD is used to simulate portfolio losses, constructing a Portfolio Loss Distribution (PLD). Unexpected Losses (ULs) are taken from the PLD; the Economic Capital (EC) is set equal to the loss at the 99.5 percentile of the PLD (i.e., ULs will be covered 99.5 percent of the time; a 0.5 percent probability exists of losses exceeding EC).
- The CIMDO-recovered multivariate density embeds macroeconomic shock effects on loan credit quality (methodology referenced to Segoviano and Padilla (2006) as used to recover multivariate distributions).

### Stressed PoDs, ELs, and ULs — scenario results (selected figures and paths)
- Stressed PoDs paths were produced for sectors including Manufacturing; Credit, Finance and Insurance; Business Services; Household (Figure 6 referenced).
- Expected Losses as a Percentage of Risk-Weighted Assets (Table 3a):
  - Scenario 1: 2006 = 1.45, 2007 = 1.66, 2008 = 2.32
  - Scenario 2: 2006 = 1.37, 2007 = 1.63, 2008 = 1.73
  - Scenario 3: 2006 = 1.34, 2007 = 2.00, 2008 = 3.07
- Unexpected Losses as a Percentage of Risk-Weighted Assets (Table 3b):
  - Scenario 1: 2006 = 11.94, 2007 = 12.16, 2008 = 13.76
  - Scenario 2: 2006 = 11.89, 2007 = 12.12, 2008 = 12.28
  - Scenario 3: 2006 = 11.82, 2007 = 12.39, 2008 = 14.71

### Interpretation of scenario dynamics and sectoral sensitivities
- Consistency across scenarios:
  - Increase in PoDs, ELs, and ULs is consistent across scenarios (scenario 3 most severe; scenario 1 next; scenario 2 least severe) and increases over time across the three-year horizon (2006–2008).
- Sectoral concentration and drivers:
  - Credit portfolios concentrated in loans to credit, finance, and insurance; household; manufacturing; and business services sectors — together over 82 percent in 2004 and averaging over 73 percent between 1991 and 2004.
  - Key macroeconomic variables that deteriorate fastest under the assumed scenarios: loans to the private sector, house prices, unemployment, and GDP. Deterioration is especially drastic under scenario 3.
  - Variable sensitivities: credit/GDP and house prices significant across these sectors; credit, finance, and insurance sector is highly sensitive to interest rates; household sector highly sensitive to employment levels and interest rates; manufacturing sector highly sensitive to foreign exchange; business services sector sensitive to GDP.
- Resulting mechanism:
  - Deterioration in macro variables → higher sector PoDs → higher ELs and ULs (as shown in Tables 3a and 3b and Figure 7).

### Buffers, capital adequacy, and coverage of losses
- Definitions and decomposition:
  - EL buffer = (pre-tax income + provisions) / RWA.
  - Capital Adequacy Ratio (CAR) represents buffer to cover ULs.
  - Total buffer to withstand losses = EL buffer + CAR.
- Historical and accounting effects:
  - Total buffer as a ratio of RWA has been reduced from 2004 and the trend likely to continue due to a fall in accumulated provisions and lower solvency ratios.
  - New International Accounting Standards/International Financial Reporting Standards effective January 2005 significantly reduced provisions, affecting the EL buffer.
- Group 1 historical data (selected items from Table 4a and 4b):
  - Total loan-loss provision: 2000 = 1,217; 2001 = 3,364; 2002 = 2,508; 2003 = 3,213; 2004 = 410
  - Total pre-tax income: 2000 = 13,582; 2001 = 15,950; 2002 = 16,549; 2003 = 20,338; 2004 = 22,075
  - Total provisions + pretax income: 2000 = 14,799; 2001 = 19,314; 2002 = 19,057; 2003 = 23,551; 2004 = 22,485
  - Total capital (capital adequacy): 2000 = 110,460; 2001 = 117,205; 2002 = 122,496; 2003 = 125,752; 2004 = 126,958
  - Total risk-weighted assets: 2000 = 1,135,287; 2001 = 1,141,915; 2002 = 1,174,840; 2003 = 1,168,680; 2004 = 1,243,096
  - EL buffer / RWA: 2000 = 1.30, 2001 = 1.69, 2002 = 1.62, 2003 = 2.02, 2004 = 1.81
  - Capital adequacy ratio: 2000 = 9.723, 2001 = 10.26, 2002 = 10.43, 2003 = 10.76, 2004 = 10.21
  - Noninterest income as percent of pre-tax income (total): 2000 = 57.90, 2001 = 40.26, 2002 = 4.36, 2003 = 12.86, 2004 = 22.42
- Coverage of ELs by EL buffer after shocks (EL-buffer-S) — difference between EL buffer at end-2004 (1.81 percent) and EL ratio under each scenario (Table 5a):
  - Scenario 1: 2006 = 0.35, 2007 = 0.14, 2008 = -0.51
  - Scenario 2: 2006 = 0.43, 2007 = 0.18, 2008 = 0.07
  - Scenario 3: 2006 = 0.46, 2007 = -0.19, 2008 = -1.26
  - Interpretation: Under scenario 1, ELs in 2006 and 2007 would be covered by pricing and provisioning but not in 2008; under scenario 3, ELs would not be covered in 2007 and 2008.
- CAR decreases and adjusted CAR after shocks (CAR-S) (Table 5b):
  - CAR-Decrease (percent):
    - Scenario 1: 2006 = -1.72, 2007 = -1.95, 2008 = -3.55
    - Scenario 2: 2006 = -1.68, 2007 = -1.91, 2008 = -2.07
    - Scenario 3: 2006 = -1.61, 2007 = -2.17, 2008 = -4.50
  - CAR-S (percent) [CAR at end-2004 = 10.21 minus CAR-Decrease]:
    - Scenario 1: 2006 = 8.49, 2007 = 8.26, 2008 = 6.66
    - Scenario 2: 2006 = 8.54, 2007 = 8.31, 2008 = 8.14
    - Scenario 3: 2006 = 8.60, 2007 = 8.03, 2008 = 5.71
  - CAR-S-Net (CAR after shocks net of EL buffer) (Table 5c):
    - Scenario 1: 2006 = 8.84, 2007 = 8.41, 2008 = 6.15
    - Scenario 2: 2006 = 8.97, 2007 = 8.48, 2008 = 8.22
    - Scenario 3: 2006 = 9.07, 2007 = 7.85, 2008 = 4.45
  - Regulatory threshold: Under current guidelines and Basel II proposal a minimum CAR of 8 percent should be maintained.
    - Implications: Required minimum CAR of 8 percent would be breached under Scenarios 1 and 3 in 2008; CAR-S-Net indicates the 8 percent threshold would also be breached in 2007 under scenario 3. If uncovered ELs are absorbed by capital, CAR-S would be further reduced.

### Additional observations, vulnerabilities, and limitations
- Bank grouping and heterogeneity:
  - Results represent the “average effect” for banks in group 1; individual institutions could suffer much higher losses. Individual bank portfolio credit risk was not assessed because of use of bank-grouped data.
  - Banks in groups 2 and 3 have riskier credit portfolios; impact on CAR may be higher for these banks. Smaller banks have higher exposure to agriculture and fisheries and some have grown rapidly; historically fastest-growing banks experienced largest losses in late 1980s/early 1990s.
- Household sector vulnerabilities:
  - Lending to households grew significantly; in 2005 annual growth rate exceeded 21 percent.
  - Growth driven by introduction of adjustable-rate bank mortgage loans and mortgage-credit loans with option to defer amortization.
  - As of end February 2005, homeowners’ average mortgage debt = 2.1 times gross household income; lowest income bracket average mortgage debt = 2.9 times annual household income; 10 percent of this group had mortgage debt at least 4.5 times annual household income.
  - Rising household debt relative to disposable income and higher share of adjustable-rate loans increase vulnerability to interest rate fluctuations and unemployment; low interest rates to date have temporarily neutralized budget impact.
- Consistency with Danmarks Nationalbank (DNB) analysis:
  - DNB used a simplified model regressing aggregate write-offs (proxy for PoDs) on shocked macro variables to forecast a proxy of ELs; their results are consistent in showing increasing losses from 2006 to 2008 and across scenarios, though the proxy yields lower ELs than empirically observed ELs and the DNB estimation has limitations (short sample, standard OLS with large variances).
- Caveats:
  - Macroeconomic shocks and resulting increases in PoDs, ELs, and ULs may be extreme but are characterized as plausible.
  - The stress test did not cover banks in groups 2 and 3; potential ELs and ULs may be larger among smaller banks with more concentrated, risky portfolios.

### Key policy implications and recommendations (implicit from analysis)
- Maintain and strengthen buffers:
  - Generate an EL buffer through adequate pricing and provisioning to absorb expected losses.
  - Preserve sufficient CAR to cover unexpected losses, mindful that under severe scenarios the 8 percent minimum CAR may be breached.
- Monitor household-sector vulnerabilities:
  - Pay particular attention to rapid household credit growth, rising household leverage, increasing share of adjustable-rate mortgages, and the option-to-defer-amortization products.
- Focus supervisory attention on smaller and faster-growing banks:
  - Given their more concentrated and riskier loan portfolios, banks in groups 2 and 3 warrant attention; individual institutions may face losses materially larger than group averages.
- Prepare for Basel II impacts:
  - Anticipate that future capital adequacy rules under Basel II will influence size and composition of buffers; monitor and adapt provisioning and capital strategies accordingly.

*Source: Staff estimates and author’s estimation as presented in the chapter content.*

### 43.      Liquidity in Danish banks is judged to be adequate. The stock of liquid assets is

### _cr07125 - 43.      Liquidity in Danish banks is judged to be adequate. The stock of liquid assets is

### Liquidity in Danish banks
- Liquidity is judged to be adequate.
- The stock of liquid assets is higher than the regulatory minimum although it has declined somewhat in the most recent years.
- The Danish FSA supervises individual banks and ensures they have adequate liquidity and contingency plans.
- Regulations require banks to have liquid assets (broadly defined) to cover:
  - 15 percent of call deposits and liabilities falling due within one month; and
  - 10 percent of all liabilities other than core capital.
- The DFSA publishes indicators for banks’ excess liquidity over and above the regulatory minimum. The median and the lower quartile show that most banks have far more liquidity than the regulatory minimum.
- The excess cover has declined somewhat in the most recent years, but remains plentiful for most banks.
- The stock-based liquidity requirement, including the requirement promptly to report and rectify shortfalls, appears to function as a benchmark which banks avoid getting too close to.

### Sources of funding and funding volatility risks
- Banks experienced stronger loan than deposit growth in recent years and have increasingly relied on issuance of bonds and borrowing from other financial institutions for funding.
- A portion of borrowing from other financial institutions is accounted for by intragroup lending from parents of foreign-owned banks and borrowing from foreign banks.
- Intragroup and foreign-bank borrowing may be less stable sources of funding and liquidity, especially in the event of a financial crisis.
- It is important to maintain market access and avoid potential changes in own credit spread.

### Interbank contagion and liquidity monitoring
- The DNB contagion model suggests contagion risk in the Danish interbank market currently seems to be limited.
- The methodology analyzes contagion by simulating unexpected failures of the largest debtor.
  - Failure of the largest debtor would cause another bank to fail on 11 out of 253 banking days in 2004 if the LGD was 100 percent.
  - Reducing the LGD to 50 percent greatly decreased contagion risk: only one bank on one day would fail.
  - Only first order contagion took place; no more than one bank would fail on any day by the failure of the largest debtor.
- Recommendation: perform interbank contagion risk analysis on a regular basis and continue close monitoring of banks’ liquidity.

### Banking sector concentration and implications
- Denmark shows higher concentration in the banking sector than many other EU member states.
- Market share concentration, as measured by the CR5 and the Herfindahl index, show higher concentration in Denmark than in many other EU member states.
- Implications:
  - A highly concentrated banking system may be more vulnerable to systemic crises because banks with substantial market shares present greater contagion risk via the interbank market and via macroeconomic feedback effects.
  - It is particularly important that banks with large market shares are sufficiently robust to withstand potential shocks.

### Results: Bottom-Up (BU) stress tests — Credit Risk
- Banks participating in the BU stress test on credit risk reported ELs; DFSA aggregated and presented them.
- The ELs reported by participating banks are consistent with the TD estimates across scenarios and over time, but the level of ELs is lower than the TD estimates.
  - Main reason: banks assumed lower levels of LGDs than the 50 percent assumed in the TD approach.
  - DFSA considered 50 percent LGD in the TD approach to be a conservative and appropriate assumption.
- Table 6. Bottom-Up Results: Expected Losses (As a percentage of risk-weighted assets)
  - Scenario 1 / Scenario 2 / Scenario 3
    - 2006: 0.31 / 0.28 / 0.45
    - 2007: 0.49 / 0.42 / 0.72
    - 2008: 0.78 / 0.57 / 1.10
- CAR-S obtained in the BU stress tests are consistent with mission estimates.
- BU and TD results show commercial banks can withstand moderate shocks but will experience capital shortfalls under severe shocks.
- Both approaches show that by the third year the mandatory capital adequacy ratio of the system is breached, with the shortfall substantial only in the worst case scenario.
- Table 7. Bottom-Up Results: CAR-S (In percent)
  - Scenario 1 / Scenario 2 / Scenario 3
    - 2006: 10.50 / 10.08 / 9.32
    - 2007: 8.96 / 9.17 / 7.38
    - 2008: 6.65 / 7.73 / 4.50
- Conclusion: Credit risk remains a major risk factor for the Danish banking system.

### Results: Bottom-Up — Market Risk
- On average, market risk seems relatively modest but may vary significantly among institutions.
- Impact of individual shocks (interest rates, equity prices, real estate prices, exchange rate) on bank capital is modest.
- Most loans in banks’ portfolios have flexible interest rates, so single-factor shocks transmit primarily through credit risk.
- Potential losses on banks’ holdings of financial instruments, mainly mortgage bonds, are mitigated by dynamic hedging to maintain matched duration.
- Banks’ net open position in foreign exchange (including the Euro) is modest; even a substantial exchange rate shock results in limited capital impact.
- Table 8. Bottom-Up Results of Single Factor Shocks (In percent) — CAR-S 2005
  - Yield Curve Up 250 bp: 10.23
  - Yield Curve Down 100 bp: 9.46
  - Property Prices Down 30 Percent: 9.89
  - Equity Prices Down 30 Percent: 9.87
  - Appreciation 40 Percent: 9.66
  - Depreciation 40 Percent: 10.16
  - (additional CAR-S value listed): 10.25
- Mission’s simplified checks using public indicators confirmed consistency of banks’ results and that market risk may vary significantly among institutions.
- Table 9. Danish Banking System Market Risk Indicators, 2000–2004 (In percent) — selected excerpts
  - Interest Rate Risk Over Tier 1 Capital (Loss as percent of Tier I due to increase in interest rates by 100 bps)
    - Groups 1-4: 2000: 3.30; 2001: 3.80; 2002: 2.60; 2003: 3.70; 2004: 2.00
  - Foreign Exchange Exposure Over Tier 1 Capital
    - Groups 1-4: 2000: 5.60; 2001: 7.40; 2002: 9.10; 2003: 7.90; 2004: 5.20
  - Net Open Position of Equities to Total Capital — Groups 1-4: 2000: 36.92; 2001: 31.17; 2002: 24.07; 2003: 28.11; 2004: 29.86
- Table 10. Results of Single Factor Shocks estimated by the Mission (In percent)
  - CAR-Decrease Mean — Group 1:
    - Yield Curve Up 250 bp: 0.31
    - Equity Prices Down 30 Percent: 0.91
    - FX Movement 40 Percent: 0.11
  - CAR-S Mean — Group 1:
    - Yield Curve Up 250 bp: 9.89
    - Equity Prices Down 30 Percent: 9.29
    - FX Movement 40 Percent: 10.09

### Insurance sector stress test results and monitoring
- Danish insurance companies allocate economic losses between owners and policyholders according to the Danish “contribution principle”:
  - Losses allocated to owners should be absorbed by the capital base and the required-solvency-margin.
  - Losses allocated to policyholders should be absorbed by the “collective bonus component”, which comprises excess premiums and retained earnings.
- BU stress tests show much of the impact of shocks would be absorbed by the “bonus component” with marginal effects on the capital base.
- Largest market risk effects on solvency-margin-ratios are caused by:
  - Property Prices Down 30 percent; and
  - Yield Curve Down 100 bp.
- Table 11. Bottom-Up Insurance Sector Results (In Millions of DKK) — 2005
  - Capital base:
    - Yield Curve Up 250 bp: 40,987
    - Yield Curve Down 100 bp: 39,282
    - Property Prices Down 30 %: 41,437
    - Equity Prices Down 30%: 37,912
    - Appreciation 40 %: 40,536
    - Depreciation 40 %: 40,276
    - (other column): 41,210
  - Required-solvency-margin:
    - Yield Curve Up 250 bp: 24,255
    - Yield Curve Down 100 bp: 22,189
    - Property Prices Down 30 %: 26,191
    - Equity Prices Down 30%: 24,255
    - Appreciation 40 %: 24,255
    - Depreciation 40 %: 24,255
    - (other column): 24,255
  - Solvency-Margin-Ratio (percent):
    - Yield Curve Up 250 bp: 170
    - Yield Curve Down 100 bp: 179
    - Property Prices Down 30 %: 159
    - Equity Prices Down 30%: 158
    - Appreciation 40 %: 168
    - Depreciation 40 %: 167
    - (other column): 171
- DFSA’s “traffic light” monitoring system:
  - Red light scenario: decrease of 12 percent in price of stocks; decrease of 8 percent in price of real estate; change of interest rate level of 0.7 percentage points.
  - Yellow light scenario: decrease of 30 percent in price of stocks; decline of 12 percent in price of real estate; change of interest rate level of 1.0 percentage point.
  - Results at end-2005: six companies in yellow light and no companies in red light.
  - If a company cannot meet the red scenario, DFSA may require monthly reporting and will not allow the company to increase its overall risk.

### Technical recommendations (mission’s suggestions)
- Perform stress test exercises on a regular basis to further develop off-site supervision and model-validation of banks’ internal models.
- Standardize the reporting of PoDs or empirical frequencies of default and require regular reporting of recovery rates.
- Model the impact of various shocks on the PoDs of banks’ loan exposures and on banks’ profits to improve understanding of transmission mechanisms and effects on capital adequacy over multi-year scenarios.
- Develop risk-based methodologies to measure banks’ unexpected losses and their impact on banks’ economic capital to support consistency checks of banks’ portfolio credit risk measurements and prepare for Basel II model validation.
- Perform interbank contagion risk analysis on a regular basis and continue close monitoring of banks’ liquidity.
- Continue close monitoring of the insurance and pension sector.
- Attempt to assess second-round effects of macroeconomic shocks; acknowledge the difficulty but encourage participation in efforts to develop methodologies for this purpose.

*Source: _cr07125 - 43.      Liquidity in Danish banks is judged to be adequate. The stock of liquid assets is*

### Appendix 1. The Conditional Probability of Default Methodology

### Appendix 1. The Conditional Probability of Default Methodology

### CoPoD: Rationale
- Default under the Merton (1974) model: borrower defaults at time T > t if asset value s(T) ≤ a_i_t (pre-specified barrier modeled as function of borrower’s leverage).
- PoD expression at time t for horizon T: PoD_t = Φ(a_i_t), where Φ(·) is the standard normal cumulative distribution function cdf.
- Empirical PoDs for groups (sectoral activity or risk-rating) form a T-dimensional vector PoD; each observation lies in [0,1].
- Transformation used: a = Φ^(-1)(PoD) (Φ(.) is the inverse standard normal cdf).
- Modeling objective: express PoDs as functions of identifiable macroeconomic and financial developments X via the linear relation a = Xβ + e, where:
  - a is T-dimensional vector of transformed PoDs,
  - X is known (T x K) matrix of macroeconomic and financial series,
  - β is K-dimensional vector of unknown coefficients,
  - e is disturbance vector.

### CoPoD: Formulation and Estimation Framework
- Reparameterization treats β and e as random variables with discrete compact supports:
  - Each β_k is a discrete random variable with 2 ≤ M < ∞ possible outcomes; β_k = Z_k p_k (β = Z p in matrix form) where p_k ∈ [0,1] and weights sum to 1 for each k.
  - Each e_t is a finite discrete random variable with 2 ≤ J < ∞ possible outcomes; e = V w where w weights are strictly positive and sum to 1 for each t.
- Rewritten model: a = X Z p + V w. This incorporates macro-financial effects into PoDs and accounts for noise.
- Estimation objective: recover discrete distributions p and w consistent with data-implied moment-consistency constraints by maximizing an entropy-based functional (see equation (4) in source), using Lagrange multipliers:
  - T-dimensional vector γ for moment-consistency constraints,
  - KT-dimensional θ for additive restrictions on p and w (they sum to one).
- Entropy solution expressions (from source):
  - p̂_km = exp( - x_z_{ktm}^T λ̂ ) / ∑_{k,m} exp( - x_z_{ktm}^T λ̂ )  (see equation (5))
  - ŵ_tj = exp( - v_{tj}^T λ̂ ) / ∑_{j} exp( - v_{tj}^T λ̂ )  (see equation (6))
- Point estimates:
  - β̂ = Z p̂
  - ê = V ŵ
- Conceptual rationale: choose p (and w) that are closest to uniform while satisfying the available information (constraints), i.e., the density that could have been generated in the greatest number of ways consistent with what is known.

### Appendix 2. The Conditional Probability of Default Efficiency (heuristic)
- CoPoD uses Kullback (1959) minimum cross entropy approach (MXED); objective is to minimize cross entropy distance between posterior p and prior q.
- MXED functional: C[p,q] = ∑_k p_k ln(p_k / q_k) — interpretable as expectation of a log-likelihood ratio.
- Illustration of information interpretation:
  - For two parameter vectors θ1 and θ2 with distributions f(y; θ1) and f(y; θ2), MXED information of θ1 relative to θ2 is E_{θ1}[ ln( f(y; θ1) / f(y; θ2) ) ] (equation (7) in source).
  - When comparing discrete uniform predata weights n^{-1} to postdata p(y), minimizing MXED with uniform prior is equivalent to maximizing the scaled log-likelihood ln(p_i) (equations (8) and (9)), i.e., standard maximum likelihood.
  - Reversing roles yields Shannon entropy maximization (equation (10)); CoPoD averages with respect to postdata p rather than fixed predata weights n^{-1}.
- Efficiency implication:
  - Standard ML implicitly gives equal predata weights n^{-1} to each observation Y_i.
  - CoPoD estimates probability weights p based on observed data information (moment-consistency constraints) and averages with respect to p, potentially yielding more efficient estimation of expected log-likelihood ratios when the constraints are valid.

### Appendix 3. The Consistent Information Multivariate Density Methodology (CIMDO) — overview
- CIMDO is based on MXED and recovers a posterior multivariate density p(x,y) for portfolio returns x and y by minimizing cross-entropy relative to a prior q(x,y) subject to constraints.
- Moment-consistency constraints enforce that posterior marginal probabilities of default equal empirically observed PoDs:
  - Constraints of the form ∫∫ χ_x(x) χ_y(y) p(x,y) dx dy = PoD_x_t, PoD_y_t where χ(·) are indicator functions defined by default thresholds (as in source).
- Feasibility conditions imposed: p(x,y) ≥ 0 and ∫∫ p(x,y) dx dy = 1.
- Optimization problem minimizes functional L = ∫∫ p(x,y) ln( p(x,y) / q(x,y) ) dx dy plus integrals with Lagrange multipliers for the moment-consistency constraints and the additive constraint (see equation (11) and subsequent formulation in source).
- Calculus of variations yields the optimal posterior multivariate density (see equation (12) in source).

*Source: Appendix 1. The Conditional Probability of Default Methodology*

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