## _wp06283

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---

### Context, motivation, and key definitions
- A sound financial system supports macroeconomic stability, national savings, and efficient resource allocation.
- Economic Capital (EC) is resources a bank should hold to withstand extreme losses.
- Unexpected Losses (ULs) are defined as the 99.5 Value at Risk (VaR) of the profit and loss distribution (PLD).
- Portfolio Multivariate Distribution (PMD) describes joint likelihoods of credit-quality changes across loans.
- Stress testing exercises (STEs) are used to assess financial-system resilience and are a tool in IMF FSAPs.
- SME exposure examples preserved from source:
  - Spain: 71.4 percent of total bank exposures.
  - Mexico: 85 percent.
- Historical crisis UL increases preserved:
  - 1992 Norwegian crisis: ULs increased on average by 48 percent from pre-crisis levels.
  - 1994 Mexican crisis: ULs increased on average by 70 percent from pre-crisis levels.

### Four-step stress-testing procedure (overview)
- Steps:
  - (i) Define extreme but plausible macroeconomic scenarios.
  - (ii) Incorporate such shocks into PoDs (Conditional Probability of Default—CoPoD).
  - (iii) Recover CIMDO-PMDs using CoPoD-PoDs as inputs.
  - (iv) Simulate PLD from CIMDO-PMD and define EC (VaR at 99.5 percentile).
- Purpose: Link macroeconomic shocks to EC estimates to evaluate bank vulnerability under different shocks.

### Methodological contributions: CoPoD (Conditional Probability of Default)
- Objectives:
  - Incorporate macroeconomic and financial developments into credit-risk modeling.
  - Recover robust estimators under short PoD time series.
- Core modeling device:
  - Transform observed PoD via a_i = Φ^(-1)(PoD) and model a = Xβ + e.
  - Treat β components as discrete with compact finite support: β = Z p.
  - Represent disturbances with finite discrete support: e = V w.
  - Reparameterized model: a = XZp + Vw.
- Estimation principle:
  - Choose probability vectors p and w by maximizing an entropy-based functional subject to moment-consistency and additivity constraints.
  - Solutions for p̂ and ŵ take exponential-form expressions; β̂ = Z p̂ and ê = V ŵ.
- Data applicability:
  - Designed for short time series (often annual, quarterly, or monthly) and when common proxies like NPL ratios are the only available PoD data.
- Empirical/efficiency claims:
  - CoPoD recovers estimators that in finite samples are superior (more efficient under MSE) to OLS (Segoviano 2006a).

### CoPoD practical findings (Denmark implementation highlights)
- Data used:
  - Bank loan books aggregated by 10 economic sectors: NPLs 1991–2004 (14 observations) and sector exposures (average 1991–2004).
  - Loans by risk-rating classifications: default frequencies 1996–2003 (8 observations) and exposures.
  - Final estimations used NPLs as PoD proxies (longest series: 14 observations).
- Significant explanatory variables (Table 4 summary):
  - CREOVGDP and HOUP significant across all sectors.
  - Sector-specific significance examples: AVGR and UNEM for “Credit, Finance, and Insurance” and households; FXDU for manufacturing; GDP for business and services.
- Manufacturing example (selected coefficient estimates; exact values preserved):
  - Adjusted R-squared: 0.74.
  - C: OLS 3.2163 p-value 0.0000 OLS S.E. 0.0130 | CoPoD 3.2164 CoPoD S.E. 0.0081 Variance Difference* 155.04
  - CREOVGDP(-4): OLS 0.9010 p-value 0.0000 OLS S.E. 0.1614 | CoPoD 0.9001 CoPoD S.E. 0.1312 Variance Difference* 51.18
  - HOUP(-1): OLS 0.5215 p-value 0.0200 OLS S.E. 0.1811 | CoPoD 0.5189 CoPoD S.E. 0.1309 Variance Difference* 91.55
  - GDP(-1): OLS -1.8838 p-value 0.0000 OLS S.E. 0.3938 | CoPoD -1.8884 CoPoD S.E. 0.2920 Variance Difference* 81.82
  - AVGR(-0): OLS 0.9701 p-value 0.0200 OLS S.E. 0.3408 | CoPoD 0.9712 CoPoD S.E. 0.2513 Variance Difference* 83.87
  - FXDU(-2): OLS -0.3140 p-value 0.0600 OLS S.E. 0.1548 | CoPoD -0.3398 CoPoD S.E. 0.1116 Variance Difference* 92.31
  - *Variance Difference estimated as: Variance OLS - Variance CoPoD / Variance CoPoD.
- CoPoD–PoDs in 2007 under Scenario 3 (exact sector PoDs):
  - Public sector 0.0055
  - Agriculture 0.0490
  - Fisheries 0.1059
  - Manufacturing 0.0351
  - Building 0.0516
  - Trade 0.0495
  - Transport 0.0287
  - Finance & insurance 0.0006
  - Property administration 0.0282
  - Households 0.0582

### Methodological contributions: CIMDO (Consistent Information Multivariate Density Optimizing)
- Objective:
  - Recover posterior multivariate PMDs consistent with observed marginal PoDs without imposing parametric multivariate distributions or specifying default correlation structures.
- Core optimization:
  - Minimize cross-entropy functional C[p,q] = ∫∫ p ln(p/q) subject to moment-consistency constraints (marginal PoDs) and density validity (p ≥ 0, ∫ p = 1).
  - Optimal posterior: p̂(x,y) = q(x,y) exp{ −1 [ μ̂ + λ̂_1 χ_x + λ̂_2 χ_y ] } (as presented).
- Advantages and performance:
  - Posterior PMDs embed default dependence and vary over time with empirical PoDs.
  - CIMDO-PMDs outperform standard parametric multivariate distributions under the PIT criterion (Segoviano 2006b).
  - Implementation requires only PoDs or NPLs by loan type and can incorporate prior information if available.

### Role of CIMDO within the stress-testing procedure
- Four-step role:
  1. Define macro scenarios.
  2. Use CoPoD to obtain stressed PoDs.
  3. Use CIMDO with CoPoD-PoDs to recover stressed PMDs.
  4. Simulate PLDs from PMDs and estimate EC (VaR at 99.5 percentile).
- Repeated under each scenario to estimate EC and evaluate vulnerability.

### Denmark scenario inputs (selected preserved cells from Table 2)
- Macroeconomic context (high-level):
  - GDP growth reached 3.0 percent after 2000; unemployment fell below 4.8 percent.
  - House prices rose over 21 percent in the fourth quarter of 2005.
  - Credit growth exceeded 20 percent in 2005.
- Scenario 1 (Boom-bust in real estate prices and credit) — example entries (percentage deviations from baseline):
  - GDP: -1.8 (2006), -5 (2007), -6.8 (2008)
  - Unemployment*: 0.5 (2006), 2.2 (2007), 3.8 (2008)
  - House price index % y/y: -7.2 (2006), -17.8 (2007), -27 (2008)
  - Credit (private sector) % y/y: -1,5; -7,0; -14,2
- Scenario 2 (Foreign shock due to correction in U.S. imbalances) — example entries:
  - GDP: -1.4 (2006), -3.2 (2007), -4.1 (2008)
  - Unemployment*: 0.6 (2006), 2 (2007), 3.2 (2008)
  - House price index % y/y: -0.6 (2006), -2.9 (2007), -5.5 (2008)
  - Credit (private sector) % y/y: 0 (2006), -1 (2007), -2.7 (2008)
- Scenario 3 (Boom-bust plus increase in European interest rates) — example entries:
  - GDP: -2.4 (2006), -6.7 (2007), -9.2 (2008)
  - Unemployment*: 0.7 (2006), 2.9 (2007), 5.1 (2008)
  - House price index % y/y: -15.6 (2006), -29.8 (2007), -41.9 (2008)
  - Oil price $/barrel shocks: 30 (2006), 30 (2007), 30 (2008)
  - Money mkt interest rate % points: 2.5 (2006), 2.5 (2007), 2.5 (2008)
  - Credit (private sector) % y/y: -1.8 (2006), -9.5 (2007), -19.5 (2008)
- Note preserved: shocks are percentage deviations from the baseline scenario (BLS); interpret with caution (example partial interpretation preserved in source).

### PLD simulation results, ELs, ULs, and EC (selected exact tables)
- LGD assumption: set equal to 50 percent (Basel II guideline) due to unavailable LGDs.
- EL-ratio (EL as percent of RWA, Table 7; exact values):
  - 2006: Scenario 1 1.41, Scenario 2 1.31, Scenario 3 1.30
  - 2007: Scenario 1 1.61, Scenario 2 1.60, Scenario 3 2.00
  - 2008: Scenario 1 2.30, Scenario 2 1.70, Scenario 3 3.01
- UL-ratio (UL as percent of RWA, Table 8; exact values):
  - 2006: Scenario 1 10.94, Scenario 2 10.89, Scenario 3 10.82
  - 2007: Scenario 1 11.16, Scenario 2 11.12, Scenario 3 11.39
  - 2008: Scenario 1 12.76, Scenario 2 11.28, Scenario 3 13.71
- EC definition: UL at the 99.5 percentile of the PLD.

### Solvency, buffers, and CAR impacts (selected exact figures)
- EL-buffer and capital data (selected rows from Table 9; exact values preserved):
  - Total provisions + pre-tax income: 2000 14,799.00 ; 2001 19,314.00 ; 2002 19,057.00 ; 2003 23,551.00 ; 2004 22,485.00
  - Total capital (capital adequacy): 2000 110,460.47 ; 2001 117,205.30 ; 2002 122,495.97 ; 2003 125,752.22 ; 2004 126,958.58
  - Total risk-weighted assets: 2000 1,135,287.00 ; 2001 1,141,915.00 ; 2002 1,174,840.00 ; 2003 1,168,680.00 ; 2004 1,243,096.00
  - EL-buffer / risk-weighted assets (In percent): 2000 1.30 ; 2001 1.69 ; 2002 1.62 ; 2003 2.02 ; 2004 1.81
  - Capital adequacy ratio (In percent): 2000 9.73 ; 2001 10.26 ; 2002 10.43 ; 2003 10.76 ; 2004 10.21
- Regulatory minimum preserved:
  - Minimum CAR of 8 percent under current regulatory guidelines and the Basel II proposal.
- CAR-decrease (Table 10; In percent):
  - 2006: Scenario 1 = -0.72; Scenario 2 = -0.68; Scenario 3 = -0.61
  - 2007: Scenario 1 = -0.95; Scenario 2 = -0.91; Scenario 3 = -1.17
  - 2008: Scenario 1 = -2.55; Scenario 2 = -1.07; Scenario 3 = -3.50
- CAR adjusted after shocks (CAR-S, Table 11; In percent):
  - 2006: Scenario 1 = 9.49; Scenario 2 = 9.54; Scenario 3 = 9.60
  - 2007: Scenario 1 = 9.26; Scenario 2 = 9.31; Scenario 3 = 9.04
  - 2008: Scenario 1 = 7.66; Scenario 2 = 9.14; Scenario 3 = 6.71
- Solvency observation preserved: "The required minimum CAR of 8 percent would be bridged under scenarios 1 and 3 in 2008. In these situations, the solvency of the analyzed banks would be threatened."

### Analysis of vulnerabilities, limitations, and dynamics
- Concentration: four sectors (“credit, finance, and insurance,” “household,” “manufacturing,” and “business services”) made up over 82 percent in 2004 and averaged over 73 percent between 1991 and 2004.
- Macroeconomic drivers of PoDs: CREOVGDP, HOUP, UNEM, AVGR, FXDU, and GDP.
- Dynamics:
  - Asset-price declines → consumption decreases → GDP falls → credit growth declines → unemployment increases → higher interest rates and depressed collateral values → rising NPLs.
  - Scenario severity ranking: Scenario 3 most severe, then Scenario 1, then Scenario 2.
- Data and aggregation limitations:
  - Grouped bank data produce average effects; individual banks (and excluded smaller banks) could face higher losses.
  - Short time series and absent dependence-structure data limit calibration of standard credit-risk models.

### Policy-relevant implications and recommendations (preserved and derived from analysis)
- Maintain and strengthen EL-buffer through adequate pricing and provisioning to absorb ELs.
- Ensure capital adequacy (CAR) sufficient to cover ULs (EC defined at PLD 99.5 percentile).
- Monitor and mitigate concentrated sectoral exposures, with focus on “credit, finance, and insurance,” “household,” “manufacturing,” and “business services.”
- Monitor macrofinancial variables that drive sectoral PoDs: CREOVGDP, HOUP, UNEM, AVGR, FXDU, and GDP.
- Improve data collection (longer PoD time series, dependence-structure information, borrower market indicators) for more robust calibration and institution-level analysis.
- Recognize that new IAS/IFRS accounting standards materially reduced provisions and hence EL-buffers; assess provisioning and capital policy responses.

### Methodological conclusions and relevance for FSAPs
- Information requirements for CoPoD and CIMDO are less stringent than standard credit-risk models; feasible under typical data constraints.
- Joint CoPoD + CIMDO implementation:
  - Produces PMDs and EC estimates that embed changing economic conditions.
  - Enables quantification of macroeconomic shocks’ impact on banking-system EC and resilience.
  - Facilitates theory- and evidence-backed dialog and policy recommendations within FSAP exercises.

*Source: IMF Working Paper content unit _wp06283 (selected sections and boxes).*

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

### References

### Context and motivation
- A sound and strong financial system is critical for a nation’s macroeconomic stability, for the development of national savings, and for the efficient allocation of resources to investment opportunities.
- The strength of the financial system depends on constituent financial institutions and on each institution’s portfolio credit risk relative to its economic capital (EC).
- EC is a measure of the resources that a bank should hold to withstand extreme losses, e.g., unexpected losses (ULs), that its portfolio could experience.
- ULs are defined by the Basel 2 Accord (Basel Committee on Banking Supervision, 2001) as the 99.5 Value at Risk (VaR) of the profit and loss distribution (PLD).
- The portfolio multivariate distribution (PMD) describes the joint likelihood of changes in the credit risk quality of the loans that make up the loan portfolio.
- Stress testing exercises (STEs) are increasingly used to assess resilience of financial systems; STEs are a tool in IMF Financial Sector Assessment Programs (FSAPs).

### Methodological contributions: CoPoD and CIMDO
- Joint implementation of the conditional probability of default (CoPoD) methodology (Segoviano 2006a) and the consistent information multivariate density optimizing (CIMDO) methodology (Segoviano 2006b) within a stress testing framework.
- CoPoD:
  - Designed to incorporate the effects of macroeconomic and financial developments into credit risk.
  - Recovers robust estimators in settings of short time series.
  - In Segoviano 2006a, CoPoD recovers estimators that in finite samples are superior (more efficient) to ordinary least squares (OLS) estimators under the mean square error (MSE) criterion.
- CIMDO:
  - Relaxes reliance on potentially unrealistic statistical assumptions when modeling PMDs in data-restricted environments.
  - Does not impose any parametric multivariate distribution, nor make any assumption of the default correlation structure among the loans in a portfolio.
  - Improves specification of PMDs and, therefore, of portfolio credit risk at any point in time.
  - In Segoviano 2006b, CIMDO-recovered distributions outperform widely used parametric distributions in portfolio credit risk modeling under the probability integral transformation (PIT) criterion.

### Implementation and data requirements
- The information requirements necessary for implementing CoPoD and CIMDO (presented in Table 3) are less stringent than those required for standard credit risk models, making implementation feasible under typical data constraints.
- PoDs of loans grouped by sectors or by ratings are the only data necessary to implement these methodologies.
- Information on exposures, stock prices, market, or financial variables of individual firms is not necessary; nor is information on loans’ default correlation structure needed for recovering CIMDO-PMDs—nonetheless, such PMDs embed the default dependence among the loans making up a portfolio.
- The proposed procedure can be extended to measure portfolio credit risk of baskets, credit derivatives, or any other synthetic instrument with underlying assets facing similar data constraints to closely held and arm’s-length firms.

### Information constraints and target applications
- Information constraints arise because either:
  - (i) credit risk modelers have arm’s-length relationships with firms and lack access to market/financial information necessary for firms’ risk assessment, or
  - (ii) certain variables do not exist for the type of firms of interest (e.g., stock prices for SMEs or unlisted firms).
- Focus is on closely held or arm’s-length firms—including small-and-medium sized enterprises (SMEs) and unlisted firms—because they often represent the backbone of the economy and a significant share of bank assets.
- Empirical examples:
  - In Spain, exposures to SMEs represent, on average, 71.4 percent of total bank exposures (Saurina and Trucharte 2004).
  - In Mexico, exposures to SMEs are about 85 percent (CNBV).

### Empirical evidence and application example
- Historical evidence indicates substantial increases in portfolio ULs during crises:
  - Goodhart, Hofmann, and Segoviano (2004) show that during the 1992 Norwegian and the 1994 Mexican crises, estimates of annual and quarterly bank portfolio ULs increased on average by 48 percent and 70 percent, respectively, from pre-crisis levels.
  - Caprio and Klingebiel (1996) document numerous episodes in which bank portfolio credit losses nearly, or completely, exhausted a banking system’s capital.
- The paper illustrates the joint methodology by presenting the STE carried out for the Danish FSAP.
- The models used are consistent with economic theory and empirical evidence, facilitating dialog with relevant entities and policy recommendations by ensuring economic consistency and identifying relevant explanatory variables.

### Key statistics and definitions preserved from source
- ULs: 99.5 Value at Risk (VaR) of the PLD.
- SME exposures examples:
  - Spain: 71.4 percent of total bank exposures.
  - Mexico: 85 percent.
- Crisis UL increases:
  - 1992 Norwegian and 1994 Mexican crises: ULs increased on average by 48 percent and 70 percent, respectively.

*Source: _wp06283 - References (PDF)._

### Section IV, we describe a four-step procedure suggested for stress testing, which involves: (i)

### Section IV, we describe a four-step procedure suggested for stress testing, which involves: (i)

### Overview of the four-step stress-testing procedure
- The procedure involves four steps:
  - (i) the definition of extreme but plausible macroeconomic scenarios;
  - (ii) the incorporation of such shocks into PoDs;
  - (iii) the recovery of the CIMDO-PMD that in step (iv) is used to simulate the PLD, from which the bank’s EC is defined.
- Purpose:
  - Link macroeconomic shocks to an estimation of Economic Capital (EC) to evaluate bank vulnerability under different macroeconomic shocks.
- Implementation context in the document:
  - Section V illustrates step by step the STE procedure developed under the aegis of the Danish FSAP.
  - Section VI presents an analysis of the obtained results.
  - Section VII concludes.

### Portfolio Loss Distribution (PLD), Expected Losses (ELs), Unexpected Losses (ULs), and Economic Capital (EC)
- Key definitions and relationships:
  - The credit risk of a bank’s portfolio of loans is summarized by its portfolio loss distribution (PLD).
  - The PLD shows possible losses and their likelihoods; average losses are referred to as expected-losses (ELs) and extreme losses are referred to as unexpected-losses (ULs).
  - Example percentile cited: loss at the 99.9 percent percentile level of the distribution is used to represent extreme losses.
  - ELs and ULs are derived from the PLD; sufficient earnings/provisioning should absorb ELs while EC should be available to cover ULs.
- Construction:
  - Changes in credit risk quality are described by the portfolio multivariate distribution (PMD); the PMD is used to simulate losses and construct the PLD.

### Structural Approach (SA) to portfolio credit risk
- Core premise:
  - Borrowing firm’s underlying asset value evolves stochastically; default occurs when asset value falls below a threshold determined by the firm’s financial structure.
  - The probability of default (PoD) summarizes the likelihood of asset value falling below the default threshold.
- Modeling assumptions and alternatives:
  - Distributions behind firms’ asset-value processes are usually assumed parametric; basic version assumes firms’ logarithmic asset values are normally distributed (Merton, 1974).
  - Empirical returns show heavier tails than the normal distribution; alternative parametric distributions proposed include t-distributions, historical simulation, and mixture models.
  - Availability of variables or proxies indicating firms’ asset-value evolution is crucial for calibration.

### Information constraints and identification problems
- Data limitations:
  - For loans to arm’s-length and closely held firms, observable data are typically limited to default frequencies grouped by borrower characteristics (e.g., economic sector or credit-risk quality classes).
  - Time series of PoDs are often very short in both developed and developing countries; number of observations may barely exceed number of parameters to be estimated.
- Consequences:
  - The problem of recovering the PMD from marginal PoDs is under-identified: infinitely many solutions exist; a basis for selecting a solution is required.
  - Parametric assumptions may be inappropriate when data are insufficient to calibrate them, producing large standard errors and potentially erroneous inferences.
  - The PoD of each loan type represents partial information on the marginal asset-value distribution (the “region of default”).

### Proposal to improve portfolio credit risk measurement (CoPoD + CIMDO)
- Combined methodology:
  - Joint implementation of the CoPoD methodology and the CIMDO methodology is proposed to estimate PMDs that incorporate effects of changing economic conditions.
  - CoPoD is developed in Segoviano (2006a) and is based on the principle of Maximum Entropy (Jaynes, 1957).
  - CIMDO is developed in Segoviano (2006b) and is based on applications of the Minimum Cross Entropy approach (Kullback, 1959).
- Advantages:
  - Designed for data-constrained environments and improves portfolio credit risk measurement.
  - CoPoD models PoDs as functions of identifiable macroeconomic and financial variables and produces efficient (smaller variance) estimators with few PoD observations.
  - CIMDO infers PMDs from estimated PoDs without relying on stock prices, market variables of individual firms, or explicit dependence-structure information; it requires only the PoDs of each type of loan in the portfolio.
  - CIMDO relaxes reliance on potentially unrealistic parametric assumptions.

### The Conditional Probability of Default (CoPoD) methodology — summary
- Rationale (under Merton (1974)):
  - A borrower defaults at time T > t if asset value s(T) ≤ borrower-specific barrier a_i; PoD at time T is PoD_t = Φ(a_i), where Φ(.) is the standard normal cdf.
- Transformation and modeling:
  - For grouped PoDs (by sector or rating), each observed empirical PoD is transformed via a_i = Φ^(-1)(PoD) to map observations into (-∞, ∞).
  - The transformed PoD vector a is modeled as a function of macroeconomic and financial series X: a = Xβ + e.
  - X is a known (T x K) matrix; β is a K-dimensional vector of unknown coefficients; e is a disturbance vector.
- Estimation approach using finite-support priors:
  - Treat each β_k as a discrete random variable with compact support of M outcomes (2 ≤ M < ∞), allowing β_k to be represented as a convex combination of support points: β = Z p.
  - Represent each disturbance e_t as a finite discrete random variable with J possible outcomes (2 ≤ J < ∞) and restrict associated weights to lie in [0,1] and sum to 1 where appropriate.
- Practical benefit:
  - This finite-support, maximum-entropy formulation permits recovery of efficient estimators for β and e in settings with short PoD time series where OLS would yield large variances or be unfeasible.

*Source: _wp06283 - Section IV, Section II, Section III, Box 1 (CoPoD summary).*

### Box 1. The Conditional Probability of Default Methodology (concluded)

### Box 1. The Conditional Probability of Default Methodology (concluded)

### Entropy decision rule and recovery of probability vectors
- Model reparameterization:
  - Reparameterized unknowns: β = Zp and e = Vw.
  - Rewritten model: a = XZp + Vw (equation (3)).
- Objective: choose relative frequencies p and w consistent with moment-consistency constraints by maximizing an entropy-based functional (equation (4)).
  - Lagrange multipliers:
    - T γ ∈ R correspond to moment-consistency constraints embedding data information.
    - T θτ Κ ∈ R correspond to additivity restrictions on p and w (they must add to one).
- Entropy solutions:
  - p̂ given by an exponential-form expression (equation (5)).
  - ŵ given by an exponential-form expression (equation (6)).
- Estimates derived from optimal probability vectors:
  - Parameter point estimate: β̂ = Z p̂.
  - Disturbance point estimate: ê = V ŵ.
- Rationale: CoPoD selects the p that could have been generated in the greatest number of ways consistent with known constraints; it chooses the distribution closest to uniform while satisfying available information (no arbitrary distributional assumptions).

### Data frequency, applicability, and contributions
- Typical data characteristics:
  - Short time series of PoDs are the norm (most often recorded annually, quarterly, or monthly).
  - PoDs have been recorded from the middle/end of the 1990s in most cases.
  - Footnote: 20 — Common proxies for PoDs are nonperforming loans (NPLs) ratios; CoPoD can be applied the same way to model NPLs.
- Twofold contribution of CoPoD:
  - Econometric: recovers estimators that show greater robustness than OLS estimators in finite sample settings under the mean square error criterion.
  - Economic: incorporates a procedure to select a relevant set of (identifiable) macroeconomic explanatory variables based on theory and empirical evidence — important for stress testing and for linking economic shocks to financial risks.

### Intuition behind efficiency improvements in small sample settings (Box 2 summary)
- Problem: with few observations, OLS/ML become imprecise or unfeasible for PoDs modeled as functions of macro variables.
- Key conceptual points:
  - ML (under normality and independence) implicitly assigns equal predata weights n^-1 to each observation and assumes a parametric form; parameters are chosen to maximize the likelihood of observed Yi.
  - MXED (Kullback, 1959) minimizes cross-entropy distance between posterior p and prior q; can be interpreted as the expectation of a log-likelihood ratio.
    - Cross-entropy functional: C[p,q] = Σ p_k ln(p_k / q_k) (as presented).
    - When q is uniform (1/n), MXED reduces to maximizing ln(p_i) scaled by n^-1 (equations (8) and (9)).
    - Reversing roles yields minimization equivalent to maximizing Shannon entropy (equation (10)).
  - Difference in averaging:
    - ML: averaging performed with respect to predata weights n^-1 (uniform), implying equal weight to each observation.
    - CoPoD: averaging performed with respect to postdata distribution p, where p is inferred from observed data via moment-consistency constraints.
  - Expected advantage of CoPoD:
    - Weights observations by probability estimates inferred from data rather than fixed uniform weights.
    - Uses a distribution of probability weights inferred from data information.
    - Assuming moment constraints are valid, CoPoD estimators are anticipated to be more efficient (smaller MSE) because they calculate expected log-likelihood ratios with respect to probability weights inferred from a more efficiently used information base.
- Empirical/theoretical support:
  - Theoretical proofs and a Monte Carlo experiment demonstrating efficiency gains are presented in Segoviano (2006a).
  - Appendix 1 of the source presents the Kullback (1959) MXED approach.

### Selection of explanatory variables (economic hypothesis and initial set)
- Economic hypothesis underlying variable selection:
  - Fluctuations in key macroeconomic and financial variables can generate endogenous cycles in credit, economic activity, and asset prices, amplifying financial imbalances and systemic vulnerability.
  - During upturns, excess lending, rising collateral prices, and confidence can sow vulnerabilities and increase leverage.
  - Consistent with models: credit constraints and financial accelerator (Kiyotaki and Moore (1997)), second-generation currency crisis models, incentive structures under financial liberalization, moral hazard lending, and carry trades.
  - Empirical evidence supports that default frequencies can be partially explained by lagged values of relevant macroeconomic and financial variables (footnotes 21–24).
- Initial set of variables (Table 1: Code — Variable — Source):
  - HOUP — House price index — National sources as per detailed documentation and BIS calculations based on national data
  - SHAPRI — Share price index — IMF International Financial Statistics
  - AGGASPRI — Aggregate asset price index — National sources as per detailed documentation and BIS calculations based on national data
  - FXDU — Nominal foreign exchange — IMF International Financial Statistics
  - REER — Real foreign exchange — IMF International Financial Statistics
  - RESER — International reserves — IMF International Financial Statistics
  - AVGR — Money market interest rate — IMF International Financial Statistics
  - UNEM — Unemployment — IMF International Financial Statistics
  - OILP — Oil price — National sources
  - MTGR — Mortgage bond interest rate — National sources
  - GDP — Real GDP — IMF International Financial Statistics and OECD
  - CRE — Real aggregate credit (private sector) — IMF International Financial Statistics and OECD
  - CON — Consumption aggregate — IMF International Financial Statistics
  - CA — Current account balance — IMF International Financial Statistics
  - FDI — Foreign direct investment — IMF International Financial Statistics
  - INVE — Investment aggregate — IMF International Financial Statistics
  - CON (duplicate code listed) — Real consumption — Authors’ calculations based on national data
  - CREOVGDP — Ratio of aggregate credit in the financial system to GDP — Authors’ calculations based on national data
  - INVOVGDP — Ratio of investment to GDP — Authors’ calculations based on national data
  - CONOVGDP — Ratio of consumption to GDP — Authors’ calculations based on national data
  - RECUAOVREINV — Ratio of real current account to real investment — Authors’ calculations based on national data
  - M2OVRES — Ratio of M2 to international reserves — Authors’ calculations based on national data
  - LOMISH — Difference of long minus short interest rates — Authors’ calculations based on national data
  - INREVO — Realized volatility of money market rates — Authors’ calculations based on national data

### CIMDO: Consistent Information Multivariate Density Optimizing Methodology
- Purpose: recover multivariate distributions (CIMDO-PMD) that describe the joint likelihood of credit quality changes in loans making up a portfolio without imposing unrealistic parametric assumptions, using only partial information (realized frequencies of default of each loan type).
- Key features:
  - Multivariate distributions can be inferred even with only partial information on marginals; correlation structure is not required (but can be incorporated if available).
  - CIMDO-PMDs embed default dependence among loans.
  - Implementation feasible for arm’s-length and closely held firms since it depends only on PoDs or NPLs statistics.
- Performance:
  - Multivariate distributions recovered with CIMDO outperform standard parametric multivariate distributions (standard and conditional normal, t-distribution, mixture of normals) under the PIT criterion (Segoviano (2006b)).
  - Intuition for improvements:
    - Posterior multivariate distribution is updated each period to be consistent with empirically observed probabilities of default for each borrower (marginal restrictions).
    - The posterior shape varies with empirical frequencies of default driven by macro shocks and business-cycle variations.
    - Partial information is used more efficiently than calibrating fixed-parameter parametric distributions under data-restricted environments.
    - Detailed PIT criterion exposition and Monte Carlo studies are in Segoviano (2006b); Box 4 provides intuitive explanation.

*Source: _wp06283 - Box 1. The Conditional Probability of Default Methodology (concluded).*

### Box 3: The Consistent Information Multivariate Density Methodology

### Box 3: The Consistent Information Multivariate Density Methodology

### Methodology formulation
- CIMDO is based on the Kullback (1959) minimum cross-entropy approach (MXED).
- Illustration setup:
  - Portfolio contains loans to two borrower classes with logarithmic returns x and y, where x,y ∈ ɭ_i s.t. i = 1,..,M.
  - Objective function: C[p,q] = ∫∫ p(x,y) ln( p(x,y) / q(x,y) ) dx dy, where q(x,y) is the prior distribution and p(x,y) the posterior distribution ∈ R^2.

### Moment-consistency constraints and density validity
- Empirical PoD incorporation via moment-consistency constraints of the form:
  - ∫∫_{[, ) x_d^∞ y_d^∞} p(x,y) χ_x(, ) χ_y(, ) dx dy = x_t^PoD and = y_t^PoD
  - Notation in source: x_t^PoD and y_t^PoD are the empirically observed probabilities of default (PoDs); χ_x and χ_y are indicator functions defined with the default thresholds.
- Density validity conditions imposed:
  - p(x,y) ≥ 0
  - ∫∫ p(x,y) dx dy = 1

### Optimization functional and Lagrange multipliers
- CIMDO posterior density recovered by minimizing the functional L:
  - L[p,q] = − ∫∫ p(x,y) ln( p(x,y) / q(x,y) ) dx dy
    + [ λ_1 ( ∫∫_{[, ) x_d^∞ y_d^∞} p(x,y) χ_x dx dy − x_t^PoD ) ]
    + [ λ_2 ( ∫∫_{[, ) x_d^∞ y_d^∞} p(x,y) χ_y dx dy − y_t^PoD ) ]
    + μ ( ∫∫ p(x,y) dx dy − 1 )
  - Where λ_1, λ_2 are Lagrange multipliers of the moment-consistency constraints and μ is the Lagrange multiplier of the probability additivity constraint.

### Optimal posterior multivariate density (solution)
- By calculus of variations, the optimal posterior multivariate density is:
  - p̂(x,y) = q(x,y) exp{ −1 [ μ̂ + λ̂_1 χ_x[, ) + λ̂_2 χ_y[, ) ] }
  - (Expression in source: { } 12 [,)[, ) ˆˆ ˆˆ (, )(, )exp1() () x y d d x x pxy qxy μ λ χ λ χ ∞ ∞ = − + + + — preserved as presented.)

### Intuition, interpretation, and advantages (Box 4)
- CIMDO selects, among all distributions satisfying the moment-consistency constraints, the posterior distribution that minimizes the probabilistic divergence (“entropy distance”) from the prior.
- Relation to information theory:
  - MXED objective function extends pure-entropy maximization; multiplicity factor interpretation (Shannon, 1948) implies maximum-entropy selects frequency distributions realizable in the greatest number of ways subject to constraints (Jaynes, 1957).
  - With a prior, the cross-entropy solution minimizes divergence to the prior (Kullback, 1959), reconciling theoretical intuition with empirical PoDs.
- Practical interpretation:
  - The problem becomes one of inference via optimization: use data information to infer unknown probability density values (inverse probabilities) rather than assuming parametric probabilities.
- Implementation and model risk:
  - Implementation is simple and straightforward.
  - The methodology appears to reduce model and parameter risks of the recovered distribution, as indicated by the PIT criterion (Segoviano, 2006b).
  - Recovered posterior requires only variables directly observable for the loan types of interest and is, by construction, consistent with empirically observed PoDs.
- Conclusion on methodological contribution:
  - CIMDO offers a flexible approach to modeling multivariate densities, making efficient use of limited available information.

### Role of CIMDO within the proposed stress-testing procedure
- Four-step stress-test procedure where CIMDO is used in step (iii):
  1. Define macroeconomic scenarios (extreme but plausible).
  2. Incorporate macroeconomic shocks into PoDs using the CoPoD econometric framework to obtain CoPoD-PoDs (stressed PoDs).
  3. Model portfolio multivariate densities (PMDs) using CIMDO with CoPoD-PoDs as exogenous inputs to recover CIMDO-PMDs (stressed PMDs).
  4. Simulate portfolio loss distributions (PLDs) from CIMDO-PMDs and estimate economic capital (EC) (VaR).
- The procedure is repeated under each macroeconomic scenario to estimate EC under different shocks and to evaluate banks’ vulnerability.

### Key empirical context and scenario inputs (Denmark implementation)
- Economic context highlights (as presented):
  - GDP growth reached 3.0 percent after 2000; unemployment fell below 4.8 percent; asset inflation high.
  - House prices rose over 21 percent in the fourth quarter of 2005.
  - Credit growth exceeded 20 percent in 2005.
- Three macroeconomic scenarios relative to a baseline (percentage deviations from baseline) — selected cells from Table 2 preserved exactly:
  - Baseline variables (2006, 2007, 2008) shown for reference in Table 2.
  - Scenario 1 (Boom-bust in real estate prices and credit): example entries
    - GDP: -1.8 (2006), -5 (2007), -6.8 (2008)
    - Unemployment*: 0.5 (2006), 2.2 (2007), 3.8 (2008)
    - House price index % y/y: -7.2 (2006), -17.8 (2007), -27 (2008)
    - Credit (private sector) % y/y: -1,5; -7,0; -14,2 (note comma decimal formatting preserved as in source)
  - Scenario 2 (Foreign shock due to a correction in U.S. imbalances): example entries
    - GDP: -1.4 (2006), -3.2 (2007), -4.1 (2008)
    - Unemployment*: 0.6 (2006), 2 (2007), 3.2 (2008)
    - House price index % y/y: -0.6 (2006), -2.9 (2007), -5.5 (2008)
    - Credit (private sector) % y/y: 0 (2006), -1 (2007), -2.7 (2008)
  - Scenario 3 (Boom-bust plus increase in European interest rates): example entries
    - GDP: -2.4 (2006), -6.7 (2007), -9.2 (2008)
    - Unemployment*: 0.7 (2006), 2.9 (2007), 5.1 (2008)
    - House price index % y/y: -15.6 (2006), -29.8 (2007), -41.9 (2008)
    - Oil price $/barrel shocks: 30 (2006), 30 (2007), 30 (2008)
    - Money mkt interest rate % points: 2.5 (2006), 2.5 (2007), 2.5 (2008)
    - Credit (private sector) % y/y: -1.8 (2006), -9.5 (2007), -19.5 (2008)
- Note from source: "The shocks are reported as percentage deviations from the baseline scenario (BLS); thus, need to be interpreted with caution. For example, a shock to GDP in 2008 under Scenario 3 is equal to -9.2 percent from the BSL in"

*Italic source attribution: IMF Working Paper content unit _wp06283 - Box 3: The Consistent Information Multivariate Density Methodology*

### 2008. However, this shock corresponds to -2.4 percent from th

### _wp06283 - 2008. However, this shock corresponds to -2.4 percent from th

### Shock dynamics and macro-financial transmission
- Reassessment of asset values is reinforced by higher interest rates and higher debt servicing, turning a boom into a bust.
- Transmission chain described:
  - Asset prices start falling.
  - Consumption decreases.
  - Aggregate demand contracts.
  - GDP falls.
  - Credit growth decreases.
  - Unemployment increases.
  - Higher interest rates, depressed collateral prices, higher unemployment, and decreasing GDP negatively impact banks’ assets and increase non-performing loans.

### Data available and limitations for model implementation
- Two data sets available:
  - Loan books of the five largest banks aggregated by economic sectors (10 sectors). For each sector:
    - Annual observations of nonperforming loan ratios (NPLs) from 1991 to 2004 (14 observations).
    - Percentage loan exposure to each economic sector (average between 1991 and 2004).
  - Loans grouped by risk-rating classifications (RRs). For each RR:
    - Annual observations of empirical frequencies of default from 1996 to 2003 (8 observations).
    - Percentage loan exposures to each RR category.
- Choice of proxy for Probability of Default (PoD):
  - Both NPLs and frequencies of default used initially.
  - Final estimations reported use NPLs as proxies for PoDs because they comprised the longest time series (14 observations).
- Data constraints:
  - Time series length: 14 observations.
  - No information on dependence structure (correlations) of borrowers’ PoDs.
  - No stock prices or other market risk indicators of borrowers.
  - Insufficient for proper calibration of standard credit risk models, but sufficient to implement CoPoD and CIMDO.

### CoPoD methodology: incorporation of macroeconomic shocks into PoDs
- Approach:
  - Use CoPoD methodology with macroeconomic variables (described in Table 1) as explanatory variables.
  - Model NPLs for 10 economic sectors as functions of macroeconomic variables, producing estimating equations per sector.
  - Model specifications selected based on consistency with economic theory and empirical evidence.
- Key explanatory variables found significant across sectors (Table 4 summary):
  - CREOVGDP (credit over GDP) and HOUP (house prices) significant in all cases.
  - Sector-specific significance examples:
    - AVGR (interest rates) and UNEM (unemployment) highly significant for “Credit, Finance, and Insurance” sector and households, respectively.
    - FXDU (foreign exchange rate) highly significant for manufacturing.
    - GDP highly important for business and services.
- Forecasting:
  - Annual forecasts of NPLs (CoPoD-PoDs) computed for 2006, 2007, and 2008 under three macroeconomic scenarios.
  - For each sector, 9 NPL forecasts estimated (3 years × 3 scenarios).
- Estimation efficiency (manufacturing sector example, Table 5):
  - CoPoD estimators show same signs as OLS estimators but with greatly reduced variances.
  - Adjusted R-squared: 0.74.
  - Selected coefficient estimates and comparison (preserve exact values as reported):
    - C: OLS 3.2163 p-value 0.0000 OLS S.E. 0.0130 | CoPoD 3.2164 CoPoD S.E. 0.0081 Variance Difference* 155.04
    - CREOVGDP(-4): OLS 0.9010 p-value 0.0000 OLS S.E. 0.1614 | CoPoD 0.9001 CoPoD S.E. 0.1312 Variance Difference* 51.18
    - HOUP(-1): OLS 0.5215 p-value 0.0200 OLS S.E. 0.1811 | CoPoD 0.5189 CoPoD S.E. 0.1309 Variance Difference* 91.55
    - GDP(-1): OLS -1.8838 p-value 0.0000 OLS S.E. 0.3938 | CoPoD -1.8884 CoPoD S.E. 0.2920 Variance Difference* 81.82
    - AVGR(-0): OLS 0.9701 p-value 0.0200 OLS S.E. 0.3408 | CoPoD 0.9712 CoPoD S.E. 0.2513 Variance Difference* 83.87
    - FXDU(-2): OLS -0.3140 p-value 0.0600 OLS S.E. 0.1548 | CoPoD -0.3398 CoPoD S.E. 0.1116 Variance Difference* 92.31
  - *Variance Difference estimated as: Variance OLS - Variance CoPoD / Variance CoPoD.

### CoPoD–PoDs in 2007 under Scenario 3 (Table 6)
- CoPoD–Probability of Defaults in 2007: Scenario 3 (sector — 2007 PoD)
  - Public sector 0.0055
  - Agriculture 0.0490
  - Fisheries 0.1059
  - Manufacturing 0.0351
  - Building 0.0516
  - Trade 0.0495
  - Transport 0.0287
  - Finance & insurance 0.0006
  - Property administration 0.0282
  - Households 0.0582

### CIMDO multivariate density and portfolio multivariate density (PMD)
- CIMDO recovers PMDs period by period using PoDs of each loan type in the portfolio.
- PMDs describe the joint likelihood of credit quality changes of portfolio loans.
- Recovery approach:
  - Minimize a functional L[p,q] = ∫..∫ p(x1,..,x10) ln p(x1,..,x10) dx1,..,dx10 - ∫..∫ p(x1,..,x10) ln q(x1,..,x10) dx1,..,dx10 (equation (13)).
  - Moment-consistency constraints imposed (Lagrange multipliers 110,..,λλ and μ).
  - Optimal posterior multivariate density of the form shown (equation (14)), exponential in the Lagrange multipliers and base density q.
- Example: CIMDO multivariate density recovered for 2007 under scenario 3 using the CoPoD-PoDs of Table 6.

### Simulation of Portfolio Loss Distribution (PLD), Expected Losses (ELs), and Economic Capital (EC)
- Expected Losses (ELs) definition:
  - EL = PoD x Exposure x Loss Given Default (LGD).
- LGD treatment:
  - LGDs unavailable; Basel II guidelines followed by setting LGD equal to 50 percent.
- ELs reported as ratio of risk-weighted assets (EL-ratio) (Table 7, In percent):
  - 2006: Scenario 1 1.41, Scenario 2 1.31, Scenario 3 1.30
  - 2007: Scenario 1 1.61, Scenario 2 1.60, Scenario 3 2.00
  - 2008: Scenario 1 2.30, Scenario 2 1.70, Scenario 3 3.01
- Simulation of PLD:
  - Use PMD to simulate portfolio credit quality migrations and resulting losses.
  - ULs estimated from PLD; Economic Capital (EC) set equal to loss level at the 99.5 percentile of the PLD.
  - Interpretation: ULs covered 99.5 percent of the time by EC; with probability 0.5 percent, losses exceed EC.
- Unexpected Losses (ULs) as ratio of RWA (UL-ratio) (Table 8, In percent):
  - 2006: Scenario 1 10.94, Scenario 2 10.89, Scenario 3 10.82
  - 2007: Scenario 1 11.16, Scenario 2 11.12, Scenario 3 11.39
  - 2008: Scenario 1 12.76, Scenario 2 11.28, Scenario 3 13.71

### Analysis of stress testing results and buffers
- Portfolio concentration:
  - Credit portfolios highly concentrated in loans to “credit, finance, and insurance,” “household,” “manufacturing,” and “business services.”
  - Together, these sectors constituted over 82 percent in 2004 and averaged over 73 percent between 1991 and 2004.
- Macroeconomic variables that deteriorate most under assumed scenarios:
  - “loans to the private sector,” “house prices,” “unemployment,” and “GDP.”
  - Deterioration most drastic under scenario 3, followed by scenarios 1 and 2.
- Buffer definitions:
  - Expected loss Buffer (EL-buffer) = (pre-tax income + provisions) / RWA.
  - Capital adequacy ratio (CAR) = buffer to cover ULs.
  - Total buffer = EL-buffer + CAR.
- Observed trends and impact of accounting standards:
  - Overall total buffer as a ratio of RWA has fallen from 2004 and trend likely to continue.
  - Reduction driven by fall in accumulated provisions and lower solvency ratios, both in absolute levels and as ratio of RWA.
  - New IAS/IFRS accounting standards, effective from January 2005, had a big impact on the EL-buffer via large reductions in provisions.
  - From 2004, pre-tax income increased; however, increase in pre-tax income was lower than decrease in provisions.
- Table 9 — Expected Loss-Buffer and Capital Adequacy Ratio (preserve exact reported values)
  - Item values:
    - Total provisions + pre-tax income: 2000 14,799.00 ; 2001 19,314.00 ; 2002 19,057.00 ; 2003 23,551.00 ; 2004 22,485.00
    - Total capital (capital adequacy): 2000 110,460.47 ; 2001 117,205.30 ; 2002 122,495.97 ; 2003 125,752.22 ; 2004 126,958.58
    - Total risk-weighted assets: 2000 1,135,287.00 ; 2001 1,141,915.00 ; 2002 1,174,840.00 ; 2003 1,168,680.00 ; 2004 1,243,096.00
  - EL-buffer / risk-weighted assets (In percent): 2000 1.30 ; 2001 1.69 ; 2002 1.62 ; 2003 2.02 ; 2004 1.81
  - Capital adequacy ratio (In percent): 2000 9.73 ; 2001 10.26 ; 2002 10.43 ; 2003 10.76 ; 2004 10.21
- Regulatory minimum:
  - Under current regulatory guidelines and the Basel II proposal, a minimum CAR of 8 percent should always be kept.

### Policy-relevant implications and recommendations implied by the analysis
- Maintain and strengthen EL-buffer through adequate pricing and provisioning to absorb expected losses.
- Ensure capital adequacy (CAR) is sufficient to cover unexpected losses, recognizing EC is defined at the 99.5 percentile of PLD.
- Pay attention to concentrated sectoral exposures, particularly to “credit, finance, and insurance,” “household,” “manufacturing,” and “business services.”
- Monitor macroeconomic variables that drive sectoral PoDs: CREOVGDP, HOUP, UNEM, AVGR, FXDU, and GDP.
- Recognize data limitations (short time series, lack of dependence structure) and improve data collection for more robust calibration of credit risk models.

*Source: Authors’ calculations (content unit: _wp06283 - 2008. However, this shock corresponds to -2.4 percent from th).*

### 10.21 percent). Results are presented in Table 10.

### _wp06283 - 10.21 percent). Results are presented in Table 10.

### Stress-test results: CAR-decrease (Table 10) and CAR adjusted after shocks (CAR-S, Table 11)
- Table 10. CAR-decrease (In percent)
  - 2006: Scenario 1 = -0.72; Scenario 2 = -0.68; Scenario 3 = -0.61
  - 2007: Scenario 1 = -0.95; Scenario 2 = -0.91; Scenario 3 = -1.17
  - 2008: Scenario 1 = -2.55; Scenario 2 = -1.07; Scenario 3 = -3.50
  - Source: Authors’ calculations.
- CAR adjusted after the shocks (CAR-S) = difference between the CAR at the end of 2004 minus the CAR-decrease.
- Table 11. CAR-S (In percent)
  - 2006: Scenario 1 = 9.49; Scenario 2 = 9.54; Scenario 3 = 9.60
  - 2007: Scenario 1 = 9.26; Scenario 2 = 9.31; Scenario 3 = 9.04
  - 2008: Scenario 1 = 7.66; Scenario 2 = 9.14; Scenario 3 = 6.71
  - Source: Authors’ calculations.
- Key solvency observation:
  - "The required minimum CAR of 8 percent would be bridged under scenarios 1 and 3 in 2008. In these situations, the solvency of the analyzed banks would be threatened."

### Limitations, heterogeneity, and vulnerability channels
- Data and aggregation limitations
  - Individual institutions could suffer much higher losses than those estimated in the stress test; results represent the "average effect" for a group of banks.
  - Individual banks’ portfolio credit risk could not be assessed due to the use of bank-grouped data.
  - The stress test excluded smaller banks.
- Risk concentration and potential higher impacts
  - The credit portfolios of other bank groups seem to be riskier; impact on those banks’ CAR might be higher.
  - Smaller banks have the highest exposure of lending to agriculture and fisheries and some have grown rapidly; historically, fastest-growing banks experienced the largest losses in late 1980s and early 1990s.
  - Thus, potential losses (ELs and ULs) of the assumed shocks might be larger among smaller banks.
- Household sector vulnerability
  - Lending by banking institutions to households increased significantly; in January 2005, the annual growth rate exceeded 20 percent.
  - Growth mainly due to introduction of adjustable-rate bank mortgage loans and mortgage-credit loans with an option to defer amortization, changing loan-portfolio composition.
  - Total household debt continued to rise at a higher rate than disposable income; low interest rates temporarily neutralized the budget impact, but households are more vulnerable to fluctuations in interest rates and unemployment.
- Scenario severity and dynamics
  - Increase in PoDs, ELs and ULs is consistent across scenarios and over time: Scenario 3 is the most severe, followed by Scenario 1 and Scenario 2.
  - Cumulative macroeconomic shocks over the three-year horizon are reflected in increasing PoDs, ELs and ULs over the period.
  - Credit risk could materialize quickly if accelerated increases in house prices and leverage develop into macroeconomic imbalances leading to a boom-bust in real estate prices and credit.
  - Foreign shocks (as indicated by scenario 3) and increased household vulnerability to interest rate fluctuations could reinforce stress to the banking system.
  - The macroeconomic shocks considered, and the simulated increases in PoDs, ELs, and ULs, "may be extreme; but they are plausible." Credit risk remains a major risk factor for the Danish banking system.

### Methodological conclusions: CoPoD and CIMDO
- Information requirements
  - Information requirements necessary for CoPoD and CIMDO (presented in Table 3) are less stringent than those for standard credit risk methodologies; implementation is feasible and straightforward under typical data constraints.
- CoPoD methodology
  - Designed to recover robust estimators in settings of short time series, incorporating effects of changing macroeconomic and financial developments into PoDs.
- CIMDO methodology
  - Recovers portfolio multivariate distributions (PMDs) while avoiding reliance on possibly unrealistic statistical assumptions in settings of partial information.
  - CIMDO can be implemented using CoPoD-PoDs, recovering PMDs that reflect the impact of changing economic conditions.
- Implications for economic capital (EC) and stress testing
  - If CIMDO-PMDs are used to simulate profit and loss distributions (PLDs) and estimate economic capital (EC), these EC estimations will embed the effect of changing economic conditions.
  - Joint implementation of CoPoD and CIMDO leads to earlier recognition of banks’ risks as macroeconomic conditions change, aiding portfolio credit risk measurement through time.
  - When performed within a stress-testing framework, the joint implementation makes it possible to quantify the impact of pre-specified macroeconomic shocks on banking systems’ EC estimates and thus quantify system resilience to specific shocks.
  - This approach allows identification of explicit links between economic shocks and risks and vulnerabilities in financial systems.
- Relevance for FSAPs
  - The risk measurements produced are consistent with economic theory and empirical evidence, facilitating discussions, negotiations, and policy recommendations within FSAP exercises because model results can be backed by theory and evidence.

### Appendix 1: Entropy in a nutshell — modeling foundations and estimation via entropy
- Inverse problem setup
  - Observables: y = (y1, y2, ..., yT)'; unobservable frequencies: p = (p1, p2, ..., pK); known matrix X (T x (K>T)); finite discrete linear problem: y = Xp (equation (15)).
  - Additional constraints: ∑k pk = 1 and pk ≥ 0.
  - Problem is under-specified/ill-posed when T < K.
- Principle of Maximum Entropy (MED)
  - Shannon entropy: H(p) = -∑k pk ln pk (equation (17)); pk ln pk = 0 for ln pk = 0.
  - Jaynes (1957): maximize H(p) subject to T moment-consistency constraints and additivity to recover the probability vector p that is maximally uninformative given the constraints.
  - Lagrangian formulation (equation (18)); Maximum Entropy solution (equation (19)):
    - p̂k = exp(∑t λt f t(xk)) / ∑k exp(∑t λt f t(xk))
- Minimum cross entropy distribution (MXED)
  - Kullback (1959) extension: incorporate prior probability vector q and minimize cross entropy C[p,q] = ∑k pk ln(pk/qk) (equation (20)) subject to the same constraints.
  - Lagrangian (equation (21)); solution (equation (22)):
    - p̂k = qk exp(∑t λt f t(xk)) / ∑k qk exp(∑t λt f t(xk))
  - When q is uniform, MXED reduces to MED.
- Interpretations and properties
  - Shannon entropy E(pk) ranges from 0 (degenerate distribution) to lnK (uniform distribution).
  - MED chooses distributions as maximally uninformative as the moment constraints allow.
  - Cross-entropy C[p,q] ≥ 0 with C[p,q] = 0 iff p = q; C[p,q] is not symmetric (i.e., not a metric), but suits the objective of choosing a posterior p close to a fixed prior q.

*Source: Authors’ calculations.*

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### Risk measurement, credit risk, and value-at-risk methodologies
- Altman, E., A. Elizondo, and Miguel Segoviano, 2002, “Medicion Integral del Riesgo de Credito,” edited Limusa, Mexico.
- Butler, J., and B. Schachter, 1998, “Estimating Value-at-Risk with a Precision Measure by Combining Kernel Estimation with Historical Simulation,” Review of Derivatives Research Vol. 1, pp. 371–90.
- Crosbie, P., and J. Bohn, 1998, “Modeling Default Risk,” KMV LLC.
- Danielsson, J. and C. G. de Vries, 1997, “Tail Index and Quantile Estimation with Very High Frequency Data,” Journal of Economics and Finance, Vol. 4, pp. 241–57.
- Duffie, J. D., and K. J. Singleton, 1999, “Modelling Term Structures of Defaultable Bonds,” Review of Financial Studies Vol. 12, pp. 687–720.
- Glasserman, P., P. Heidelberger, and P. Shahabuddin, 2000, “Portfolio Value-at-Risk with Heavy-Tailed Risk Factors,” Money, Economics, and Finance Working Paper (New York: Columbia Business School, Columbia University.).
- Gupton, M., C. Finger, and M. Bhatia, 1997, “Credit Metrics Technical Document,” Morgan Guaranty Trust Company, Risk Management Research.
- Hoskin, J.R.M., G. Bonti, and D. Siegel, 2000, “Beyond the Lognormal,” Risk, 13, No. 5 (May), pp. 59–62.
- Mina, J., and Y. Xiao, 2001, “Return to RiskMetrics: The Evolution of a Standard,” RiskMetrics.
- Vassalou, M., and Y. Xing, 2002, “Default Risk in Equity Returns,” Columbia University Working Paper.
- Zangari, P., 1996, “An Improved Methodology for Measuring VaR,” RiskMetrics Monitor, 2nd Quarter, pp.7–25.
- Butler, J., and B. Schachter, 1998, “Estimating Value-at-Risk with a Precision Measure by Combining Kernel Estimation with Historical Simulation,” Review of Derivatives Research Vol. 1, pp. 371–90.

### Monetary policy, asset prices, and macro-financial linkages
- Bank for International Settlements, 1999, “The Monetary and Regulatory Implications of Changes in the Banking Industry,” Bank for International Settlements Conference Papers No.7.
- Bank for International Settlements, 2001, 71st Annual Report.
- Borio, C., and W. Fritz, 1995, “The Response of Short-Term Bank Lending Rates to Policy Rates: A Cross-Country Perspective,” Bank for International Settlements Working Paper No. 27.
- Borio, C., N. Kennedy, and S. Prowse, 1994, “Exploring Aggregate Asset Price Fluctuations Across Countries: Measurement, Determinants and Monetary Policy Implications,” Bank for International Settlements Economic Paper 40.
- Borio, C., and P. Lowe, 2002, “Asset Prices, Financial and Monetary Stability: Exploring the Nexus,” BIS Working Paper 114.
- Goodhart, Charles, and Miguel Segoviano, 2004, “Basel and Procyclicality: A Comparison of the Standardized and IRB Approaches to an Improved Credit Risk Method,” London School of Economics, Financial Markets Group, Discussion Paper 524.
- Hofmann, B., 2001, “The Determinants of Private Sector Credit in Industrialized Countries: Do Property Prices Matter?” BIS Working Paper No. 108.
- Hodrick, R., and E. Prescott, 1980, “Post-War U.S. Business Cycles: An Empirical Investigation,” Discussion Paper 451, Carnegie-Mellon University.
- Krugman, P., 1979, “A Model of Balance of Payments Crises,” Journal of Money, Credit, and Banking, Vol. II(3), pp. 311–25.
- OECD, 2006, “2006 Economic Review, Denmark,” Economic and Development Review Committee, (Paris).

### Information theory, inference, and statistical methods
- Jaynes, E., 1957, “Information Theory and Statistical Mechanics,” Physics Review Vol. 106, pp. 620–30.
- Jaynes, E., 1984, “Prior Information and Ambiguity in Inverse Problems,” Edited by D.W. McLaughlin in Inverse Problems, pp.151–66, SIAM Proceedings, American Mathematical Society, Providence, Rhode Island.
- Judge, G., and A. Golan, 1992, “Recovering Information in the Case of Ill-Posed Inverse Problems with Noise,” University of California Berkeley.
- Kullback, J., 1959, Information Theory and Statistics, (New York: John Wiley).
- Kullback, S., and R. Leibler, 1951, “On Information and Sufficiency,” Annals of Mathematical Statistics, Vol. 22, 79–86.
- Pukelsheim, F., 1994, “The Three Sigma Rule,” American Statistician, Vol. 48(4), pp. 88–91.
- Shannon, C., 1948, “A Mathematical Theory of Communication,” Bell System, Technical Journal, Vol. 27, pp. 379–423.
- McLachlan, G., and K. Basford, 1987, “Mixture Models: Inference and Applications to Clustering, Marcel Dekker,” New York and Basel.

### Corporate finance, option pricing, and term structure of credit
- Black, F., and M. Scholes, 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, 637–59.
- Merton, R., 1974, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” The Journal of Finance, Vol. 29, pp. 449–70.
- Duffie, J. D., and K. J. Singleton, 1999, “Modelling Term Structures of Defaultable Bonds,” Review of Financial Studies Vol. 12, pp. 687–720.
- Vassalou, M., and Y. Xing, 2002, “Default Risk in Equity Returns,” Columbia University Working Paper.

### Models, theory, and supplementary works
- Bernanke, B., and S. Gilchrist, 1999, “The Financial Accelerator in a Quantitative Business Cycle Framework,” in Handbook of Macroeconomics, edited by J.B. Taylor and M. Woodford,Vol. 1C, pp. 1341–93.
- Dooley, M., 1997, “A Model of Crises in Emerging Markets,” NBER, Working Paper 6300.
- Flood, R., and N. Marion, 1999, “Perspectives on the Recent Currency Crisis Literature,” International Journal of Finance and Economics, Vol. 4, 1, pp. 1–26.
- Frankel, J., and A. Rose, 1996, “Currency Crashes in Emerging Markets: An Empirical Treatment,” Journal of International Economics, Vol. 41, pp. 351–66.
- Knight, M., and J. Santaella, 1997, “Economic Determinants of IMF\ Financial Arrangements,” Journal of Development Economics, Vol. 54, pp. 405–36.
- McKinnon, R., and H. Pill, 1996, “Credible Liberalizations and International Capital Flows: The Over Borrowing Syndrome,” edited by T. Ito and A. Krueger in Financial Deregulation and Integration in East Asia, pp. 7–45 (Chicago: University of Chicago Press).
- Minsky, H., 1982, “Can It Happen Again?” in Essays on Instability and Finance, M.E. Sharpe.

*Source: _wp06283 - References*

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