## _wp0889

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---

### Introduction and objectives
- Purpose: use contingent claims analysis (CCA) tools to estimate the riskiness of banks and simulate effects of macroeconomic and financial shocks on bank risk.
- Emphases:
  - Estimation of probability of default, expected loss to debt holders, and financial fragility measured as distance (in standard deviations of asset returns) by which a bank’s assets exceed its liabilities.
  - Use of market-sourced data (equity prices, debt, interest rates) to obtain forward-looking, high-frequency risk measures.
- Empirical context: Chilean banks; results highlight high heterogeneity across banks in the correlation of risk measures with macro variables.

### Methodology: Contingent Claims Analysis (CCA)
- Conceptual foundations:
  - CCA generalizes Black‑Scholes (1973) and Merton (1973) option-pricing theory.
  - Three principles: (i) values of liabilities derived from assets; (ii) liabilities have different priority; (iii) assets follow a stochastic process.
- Asset dynamics:
  - Asset process: dA/A = μ dt + σ_A dε, where μ is asset drift, σ_A is asset return standard deviation, ε ~ N(0,1).
- Probability measures:
  - “Actual” probability of default uses asset drift μ.
  - “Risk-adjusted” or “risk-neutral” probability of default uses risk-free rate r.
  - Risk-neutral default probability is larger than actual default probability when μ > r.
- Distance-to-distress definitions:
  - d_{t,μ} = [ln(A_t / B_t) + (μ - 0.5 σ_A^2) t] / (σ_A sqrt(t)), actual default prob = N(−d_{t,μ}).
  - d_{t,r} = [ln(A_t / B_t) + (r - 0.5 σ_A^2) t] / (σ_A sqrt(t)), risk-adjusted default prob = N(−d_{t,r}).
- Risky debt and guarantees:
  - Risky debt ≡ Risk-free debt — Guarantee Against Default (guarantee modeled as a put option when debt is collateralized).

### Calculating bank risk indicators using CCA
- Inputs used:
  - Daily market capitalization (V_E) and volatility of market capitalization (σ_E).
  - Bank balance sheet liabilities; distress barrier (DB) constructed as ST Dep + f × LT Dep & Debt with f = 50 percent (in this paper).
  - Assumed constant risk-free interest rate of 5 percent.
- Implied unknowns solved from two equations: implied asset value (V_A) and implied asset volatility (σ_A).
- Risk indicators derived:
  - Distance to distress.
  - Risk-adjusted and actual probabilities of default.
  - Expected losses (put option values) to depositors and debt holders.
  - Potential public-sector guarantee size.
  - Sensitivities of risk indicators to changes in assets and asset volatility.
- Practical choices and notes:
  - Default barrier: short-term debt plus 50 percent of long-term debt (50 percent).
  - Interest rate choice: constant fixed rate of 5 percent for Chilean context.
  - Volatility estimation: GARCH(1,1) used for all banks; moving-average methods produce similar results.
  - For unlisted entities, mapping from accounting data to risk indicators for listed firms can be used (e.g., Moody’s RiskCalc).

### Empirical implementation and key findings
- Heterogeneity:
  - Banks show substantial heterogeneity; correlations of estimated risk measures with macro variables differ in magnitude and sign across banks.
- Dimensionality reduction:
  - PCA retained four significant factors that account for 70 percent of the variation of all macro variables (the four factors have eigenvalues greater than one).
- Time-series modelling:
  - Implied asset outputs from CCA and PCA-derived factors are input into a vector autoregression and impulse-response framework for scenario analysis.
- Historical assessment:
  - The riskiness of Chilean banks has declined to historically low levels.
  - Under shocks of magnitudes similar to past Chilean turmoil, banks appear unlikely to reach the high risk levels associated with the 1998 financial crisis.

### Selected quantitative points and illustrative values
- Default barrier parameter used: 50 percent.
- Assumed risk-free rate for main results: 5 percent.
- Example estimated risk-neutral default probabilities for Bank 2 under varying risk-free rates (figure labels reproduced):
  - September 10 1998: 12.4%
  - December 1 2006: 0.1%
  - October 2 2002: 4.8%
  - Additional values shown in figure: 17.2 %, 6.2 %, 0.3 %
- Volatility estimation example:
  - Figure 5: 10-day moving averages of volatility measures for changes in stock price of Bank 2; one series labeled One-Month MAGARCH (0.95).
- Volatility models used: GARCH(1,1) (primary) and moving-average (reference).

### Banking risk indicators — system and bank-level patterns
- Creating time series:
  - Use daily price data and quarterly or monthly debt to calculate distress barrier; derive time series of CCA risk indicators for individual banks and aggregate/systemic indicators.
- Distance-to-distress (DTD) for Chile:
  - Portfolio treated as one “big bank”; implied assets, implied asset volatility, and DTD estimated with GARCH(1,1) historical volatility.
  - Highest-risk period: late 1998 and early 1999 (LTCM/Russia crisis).
  - Since then risk declined gradually but leveled off in late 2005 “at a very low level of risk.”
  - Other high-risk periods: decline after internet bubble in 2000; period preceding Brazilian presidential elections in Q3 2002.
- Risk-neutral default probabilities:
  - Merton model yields risk-neutral default probability for T = one year.
  - Default probability associated with a given rating is surprisingly low under Moody’s illustrative mapping:
    - AAA ≈ 0.01 percent.
    - Ba1 ≈ 0.13 percent.
    - Empirical default probability reaching 1 percent corresponds generally to Ca range.
  - Aggregation comparisons:
    - “Big bank” aggregation versus asset-weighted average of bank default probabilities — difference provides a rough indicator of which banks are most affected (smaller banks can raise the asset-weighted average).
- Distribution of default probabilities across assets (example: September 10, 1998):
  - Around 20 percent of banking sector assets were held by banks whose default probability was below 0.12 percent.
  - 15 percent of assets were held by banks with default probabilities greater than or equal to 0.3 percent.
  - During the 1998 crisis a fairly large share of assets were held by banks with default probabilities greater than 10 percent; less than 20 percent of assets were held by banks with default probabilities exceeding 30 percent.
- Correlations with other measures:
  - CCA-derived measures significantly correlated with other banking risk metrics.
  - Example: NPLs for consumption are strongly correlated with declining DTD estimates from a few months prior.

### Relating default risk to macroeconomic variables — variable selection and PCA
- Variable selection principles:
  - Monthly frequency variables, low mutual correlation, relatively independent.
- Variables used:
  - Domestic: unemployment, CPI, IMACEC index.
  - Foreign spillovers: U.S. interest rates and CPI.
  - Financial markets: VIX, S&P 500 returns, Chilean IPSA returns.
  - Monthly log returns used (interest rates use level and change).
- PCA details:
  - PCA on 15 macro variables produced 15 orthogonal factors; first four account for 70 percent of variation.
  - Varimax rotation applied to the four factors.
- Interpretation of the four rotated factors (bold loadings > 0.5):
  - Factor 1: Financials
    - Strong loadings: S&P 500 (0.86), IPSA (0.73), VIX (-0.76), U.S. 10-Yr Rate (Chg) (0.73), CLP-USD Rate (-0.49).
  - Factor 2: Interest Rates (U.S. interest rates)
    - Strong loadings: U.S. 1-Yr Rate (Lvl) (0.95), CPI Chile (0.73).
  - Factor 3: Cyclicals
    - Strong loadings: Oil (0.87), CPI U.S.A. (0.82), U.S. 1-Yr Rate (Chg) (0.61), IMACEC (0.32).
  - Factor 4: Domestic/Regional
    - Strong loadings: CLP-BRL Rate (0.81), IMACEC (0.58), Copper (0.64).
- Factor behavior:
  - Factor 1 commoves strongly with IPSA and S&P 500; shows VIX peak in 1998 as its low point.
  - Factor 2 rises from 1997 to 2000 as short-term U.S. rates rose and yield curve flattened, then reverses in 2000.
  - Factor 3 driven by commodity prices and U.S. CPI.
  - Factor 4 reflects Chile-/Latin‑America specific patterns, large decline in 1998 and recovery beginning in 2004.
- The four factors account for more than 70 percent of the variation in all 15 macro variables.

### VAR specification and main findings
- VAR setup:
  - One VAR per bank with five variables: the bank’s implied assets and the four factors.
  - Two lags used (AIC/SIC point to two lags more frequently).
  - Asset returns used because D2D and EFDs produced similar VAR results but with fewer significant coefficients.
- Persistence and factor significance:
  - First lag of bank assets significantly positive in all regressions; second lag significant in two cases.
  - Each factor significant for at least one bank.
  - Financials and domestic/regional factors significant for four banks.
  - Interest-rate and cyclical factors significant for only one bank each.
- Bank size groups and differences:
  - Banks 1–3: three largest private banks; Banks 4–7: smaller.
  - First three banks cumulatively account for 52 percent of financial system assets (or 59.9 percent of the nonstate financial system).
  - Last four banks account for 20.1 percent of system assets (23.2 percent of the nonstate system).
  - Financials variable significant for the large banks; regional factor significant for smaller banks.
  - Loan-portfolio composition: 73 percent of larger banks’ loan portfolio to companies; 67 percent for small banks.

### Impulse response functions (IRFs): dynamics and heterogeneity
- Bank-own-asset shocks:
  - A shock to a bank’s assets results in continued increase for a few months.
  - Half the shock dissipates within four months for all banks.
  - 90 percent dissipates within eight months for all banks but one.
- Financials factor shock:
  - Strongly positive responses for four banks, negative for one bank, two close to zero.
  - Timing: maximum response two to three months after shock for most affected banks.
  - Bank 5 shows later maximum response and greater persistence.
- Interest-rate factor shock:
  - Significant response only for bank 7; counterintuitive sign (positive shock increases implied assets for bank 7).
- Cyclicals factor shock:
  - Significant for banks 2 and 5; intuitive sign: improvement raises implied assets with peak after a few months then turns negative.
- Domestic/regional factor shock:
  - Significant responses for banks 5 through 7 (small banks); persistence varies (half effect dissipates within six months for some).
- Summary:
  - Larger banks are generally more responsive to financial-market developments.
  - Smaller banks respond more strongly to domestic/regional developments.
  - Shocks are relatively short-lived in most cases but vary in persistence across banks.

### Scenario testing: simulated D2D under calibrated shocks
- Method:
  - Use impulse responses to simulate assets, then compute distance to default (D2D) for each bank.
  - Initial asset values set to observed values at end of sample in February 2007.
  - Parameters constant as in historical risk measures; asset volatility set to final observed value.
  - Note: shocks may raise asset volatility in reality; higher asset volatility lowers D2D, so scenario D2D may understate subsequent risk if volatility rises.
- Scenario 1: financials shock comparable to 1998 (Russia/LTCM)
  - VIX jumped by about two standard deviations in late 1998; average innovation to financial factor slightly less than one.
  - Results:
    - D2D falls in months immediately following the shock; a decline of one in D2D represents a statistically significant increase in default probability.
    - Four of five banks show recoveries beginning a few months after the shock.
    - In two cases recovery is rapid and within 10 months assets return above original level.
    - Bank 5 does not begin to revert until a year after the shock.
  - Regulatory implication: heterogeneous recovery speeds imply some banks require closer monitoring post-shock.
- Scenario 2: domestic shock calibrated to unemployment increase in 1999
  - Results:
    - Decline in D2D between one and two standard deviations for most banks (one outlier with idiosyncratic results).
    - Shock less persistent than financials shock; recovery begins within less than six months for all banks.
    - Heterogeneity: one bank bottoms out later but recovers more quickly.
  - Historical note: a domestic shock of this magnitude hit Chile only in 1999.

### Robustness tests
- Volatility and interest rates:
  - Using another widely used volatility measure would not much affect results.
  - Allowing variable interest rates raises selection issues; impact on results is small during low-risk periods and large during high-risk periods.
- Estimation period and 1998 observations:
  - Reestimated steps dropping all 1998 observations:
    - Stepwise regressions: surviving variables and parameter estimates not substantively changed.
    - PCA: dropping 1998 increases explanatory power of first four factors to 73 percent; a fifth factor attains eigenvalue > 1.
    - VARs with five factors: broadly similar results; financial developments factor significant for same banks; domestic factor significant for smaller banks; new factor significant for two banks.
  - Conclusion: including 1998 observations does not substantively change results; as the largest-scale crisis they may overestimate bank risk, acceptable for the paper’s purposes.

### Areas for further research and modeling recommendations
- Joint bank modeling:
  - Model banks jointly to allow feedback (e.g., higher-dimensional VAR), subject to data limitations (sample back to 1998).
- Relaxing constant interest-rate assumption:
  - Include realistic riskfree series; use STV or Merton-STV extensions to allow stochastic interest rates and correlation with asset returns.
- Address non-normality of asset returns:
  - Calibrate to mixtures of lognormals.
  - Use constant elasticity of variance (CEV) approaches where volatility varies with asset level.
- Detailed bank-level analysis:
  - Examine lending portfolios to explain puzzling results and bank-size differences.
- Stress-testing and capital adequacy:
  - CCA can assess capital adequacy under shocks to assets, asset volatility, interest rates, and other parameters without detailed obligor PD/LGD inputs.
  - Merton-STV extensions allow inclusion of interest rate volatility and asset–interest rate correlation.
  - Calibrated CCA models can imply PD and LGD for central-bank stress-testing when obligor data are lacking.
- Incorporating CCA into macro models:
  - Embed an aggregate credit risk indicator (e.g., DTD weighted by implied assets) into GDP gap equations and policy-rate reaction functions.
  - Estimate whether financial stability influenced historical policy rates and study tradeoffs among GDP, inflation, and distance-to-distress.

### Conclusions and policy implications
- CCA is a useful forward-looking, market-data-based tool to assess financial company risk; it shares market limitations (markets do not always foresee risky events).
- Chile findings over the sample:
  - CCA measures show gradual improvement in bank health over the decade and identify periods of elevated risk from international shocks.
  - VAR results indicate innovations in the four factors are strongly tied to bank risk; each factor is significant in at least one VAR.
  - Financial-market developments most significant in explaining variation in banks’ risk; positive financial-market developments relate to reduced risk for most banks.
  - U.S. interest-rate changes are less strongly tied to Chilean bank risk.
  - Banks’ risk moves procyclically with the cyclical factor for most banks; some banks are less responsive or nearly countercyclical.
  - Domestic/regional factor affects smaller banks more; financials factor ties more to larger banks.
- Regulatory guidance:
  - Focus on larger, more internationally connected banks for financial-market shocks.
  - Focus on smaller banks for regional or domestic shocks.
  - Monitor heterogeneity in persistence and timing of shocks across banks—some banks’ risk rises later and persists longer.
- Broader recommendation:
  - Incorporate CCA-derived risk measures into monetary and macroeconomic models and use CCA as a building block for integrating financial risk into macro analysis.

### Annex — key technical points and numerical example
- Basic accounting identity: Assets = Equity + Risky Debt.
- Equity and put/call relationships follow Black‑Scholes‑Merton formulas (definitions of d1, d2, N(d), delta, and distance-to-distress = d2).
- Yield and credit spread definitions: y (yield to maturity), s = y − r.
- Numerical example (values preserved exactly):
  - Assuming:
    - A = $100,
    - σ = 0.40 (40%),
    - B = $75,
    - r = 0.05(5%),
    - T = 1 (one year).
  - Results:
    - The value of the equity is $32.367
    - The value of risky debt is $67.633
    - The yield to maturity on the risky debt is 10.34%
    - The credit spread 5.34%
    - The risk adjusted probability of default is 26%
- Implied assets and asset volatility solved from:
  - Equity = A N(d1) − B e^{−rT} N(d2)
  - σ_E = (A / E) N(d1) σ_A
- Market price of risk links risk-neutral and actual default probabilities and can be expressed via CAPM relations.
- Typical model relaxations and extensions noted: first passage-time models, KMV mapping, CreditGrades extensions, stochastic interest rates (STV), and Merton-STV.

*Source: _wp0889 - References..............................................................................................................*

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

### _wp0889 - References..............................................................................................................

### Introduction and objectives
- Purpose: use contingent claims analysis (CCA) tools to estimate the riskiness of banks and simulate effects of macroeconomic and financial shocks on bank risk.
- Key emphases:
  - Estimation of probability of default, expected loss to debt holders, and financial fragility measured as distance (in standard deviations of asset returns) by which a bank’s assets exceed its liabilities.
  - Use of market-sourced data (equity prices, debt, interest rates) to obtain forward-looking, high-frequency risk measures.
- Main empirical context: Chilean banks; results highlight high heterogeneity across banks in the correlation of risk measures with macro variables.

### Methodology: Contingent Claims Analysis (CCA)
- Conceptual foundations:
  - CCA generalizes Black-Scholes (1973) and Merton (1973) option-pricing theory.
  - Three principles: (i) values of liabilities derived from assets; (ii) liabilities have different priority; (iii) assets follow a stochastic process.
- Asset dynamics (as presented):
  - Asset process: dA/A = μ dt + σ_A dε, where μ is asset drift, σ_A is asset return standard deviation, ε ~ N(0,1).
- Probability measures:
  - “Actual” probability of default uses asset drift μ.
  - “Risk-adjusted” or “risk-neutral” probability of default uses risk-free rate r (risk-neutral drift).
  - Risk-neutral default probability is larger than actual default probability when μ > r.
- Distance to distress definitions (as presented):
  - Distance to distress with drift μ: d_{t,μ} = [ln(A_t / B_t) + (μ - 0.5 σ_A^2) t] / (σ_A sqrt(t)), actual default prob = N(−d_{t,μ}).
  - Distance to distress with drift r: d_{t,r} = [ln(A_t / B_t) + (r - 0.5 σ_A^2) t] / (σ_A sqrt(t)), risk-adjusted default prob = N(−d_{t,r}).
- Link between risky debt and guarantees:
  - Risky debt ≡ Risk-free debt — Guarantee Against Default (guarantee modeled as a put option when debt is collateralized).

### Calculating bank risk indicators using CCA
- Inputs used:
  - Daily market capitalization (V_E) and volatility of market capitalization (σ_E).
  - Bank balance sheet liabilities; distress barrier (DB) constructed as ST Dep + f × LT Dep & Debt with f = 50 percent (in this paper).
  - Assumed constant risk-free interest rate of 5 percent.
- Implied unknowns solved from two equations: implied asset value (V_A) and implied asset volatility (σ_A).
- Risk indicators derived:
  - Distance to distress.
  - Risk-adjusted and actual probabilities of default.
  - Expected losses (put option values) to depositors and debt holders.
  - Potential public-sector guarantee size.
  - Sensitivities of risk indicators to changes in assets and asset volatility.
- Practical considerations and methodological choices:
  - Default barrier typically equals short-term debt plus 50 percent of long-term debt (50 percent used here).
  - Interest rate choice: constant fixed rate of 5 percent assumed for Chilean context.
  - Volatility estimation: GARCH(1,1) used for all banks; moving-average methods produce similar results.
  - For unlisted entities, mapping from accounting data to risk indicators for listed firms can be used (e.g., Moody’s RiskCalc).

### Empirical implementation and findings (summary of presented results and figures)
- Heterogeneity:
  - Banks show substantial heterogeneity; correlations of estimated risk measures with macro variables differ in magnitude and sign across banks.
  - Necessitated reduction of macro state space via principal component analysis (PCA).
- Dimensionality reduction:
  - PCA retained four significant factors that account for the largest share of the variance of all macro variables.
- Time-series modelling:
  - Implied asset outputs from CCA and PCA-derived factors are input into a vector autoregression and impulse-response framework for scenario analysis.
- Historical assessment:
  - The riskiness of Chilean banks has declined to historically low levels.
  - Under shocks of magnitudes similar to past Chilean turmoil, banks appear unlikely to reach the high risk levels associated with the 1998 financial crisis.

### Selected quantitative points and illustrative figures (exact values from source)
- Default barrier parameter used: 50 percent (share of long-term debt).
- Assumed risk-free rate for main results: 5 percent.
- Example estimated risk-neutral default probabilities for Bank 2 under varying risk-free rates (figure labels reproduced):
  - September 10 1998: 12.4%
  - December 1 2006: 0.1%
  - October 2 2002: 4.8%
  - (Associated axis labels: 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%; 3.0% 3.5% 4.0% 4.5% 5.0% 5.5%)
- Alternate reporting in the same figure (additional values shown):
  - 17.2 %
  - 6.2 %
  - 0.3 %
- Volatility estimation example (Figure 5): 10-day moving averages of volatility measures for changes in stock price of Bank 2 displayed; one series labeled One-Month MAGARCH (0.95).
- Volatility models used: GARCH(1,1) (primary) and moving-average (reference).

### Policy-relevant implications and uses
- Forward-looking monitoring:
  - CCA-derived indicators provide high-frequency, market-based barometers of bank creditworthiness and financial fragility that complement accounting-based measures (nonperforming loan ratios, provisioning).
- Stress and scenario analysis:
  - The VAR and impulse-response framework fed by CCA and PCA allows simulation of macro-financial shocks and assessment of timing and magnitude of peak impact and recovery across banks.
- Regulatory considerations:
  - Choice of distress threshold matters for supervisory focus (actual default vs. intervention threshold consistent with regulatory capital), suggesting additional work to map distance-to-distress to intervention triggers.
- Calibration sensitivity:
  - Default probabilities are sensitive to the assumed risk-free rate and asset volatility, especially in high-risk periods; volatility and interest-rate misspecification both materially affect estimates during crises.

*Source: _wp0889 - References..............................................................................................................*

### Section IV below discusses other ways in which both the interest rate and the volatility may

### _wp0889 - Section IV below discusses other ways in which both the interest rate and the volatility may

### D. Banking Risk Indicators
- Using daily price data and debt data (quarterly or monthly debt data is used to calculate the distress barrier) for major banks or financial institutions, the CCA model can be used to derive a time series of CCA risk indicators for individual banks and financial institutions and time series of more aggregate systemic risk indicators.
- Distance-to-distress (DTD)
  - A useful indicator of banking or financial sector risk over time is a graph of average distance-to-distress (DTD).
  - Figure 6 shows the estimated time pattern of DTD for the Chilean banking system from 1998 to 2007, along with a three-month moving average.
  - Calculation notes:
    - The portfolio of banks in the system was treated as one “big bank.”
    - Implied assets, implied asset volatility, and distance-to-distress were estimated.
    - This procedure used historical volatility of market capitalization calculated with GARCH(1,1).
  - Observations:
    - The highest-risk period was in late 1998 and early 1999, during the fallout from the LTCM/Russia crisis.
    - Since then the Chilean banking system has gradually reduced its risk, though this trend appears to have leveled off in late 2005.
    - Footnote: “As we see below, this leveling-off has occurred at a very low level of risk.”
  - Other discernible high-risk periods:
    - Decline in world stock markets following the collapse of the internet bubble in 2000.
    - The period preceding the Brazilian presidential elections in the third quarter of 2002.

- Risk-neutral default probabilities
  - The Merton model produces the risk-neutral default probability for each financial institution analyzed, yielding the cumulative probability over the time period T that the institution will default, where T is assumed to be one year.
  - Note: This default probability is not the same as the real-world probability of default; extensions to compute the real-world probability are discussed in Section IV (beyond the scope of this paper).
  - Figure 7 findings:
    - Default probability associated with a given rating is surprisingly low.
    - Overall risk of the Chilean banks in the sample peaked around the time of the 1998 crisis, and did not approach that level during later periods of financial volatility (e.g., preceding the 2002 Brazilian election or associated with the 2005 unrest in emerging markets).
    - Figure 7 presents default probabilities for:
      - The aggregation of the banking system into one large bank (“big bank”).
      - The asset-weighted average of the various banks’ default probabilities (shows slightly higher risks during crisis periods because relatively small banks have larger impact on a default-probability weighted average).
    - The difference between the two measures is a rough indicator of the size of the banks most affected.
  - Moody’s KMV (MKMV) methodology (described as context; extension beyond scope of paper):
    - MKMV estimates actual default probabilities using historical default information which are “bucketed” and compared to modified CCA risk indicators.
    - MKMV then estimates the risk-neutral default probability of default using the market price of risk technique (described in the Annex).
    - Illustrative magnitudes reported:
      - A rating of AAA is consistent under the Moody’s methodology with a rating of around only 0.01 percent.
      - A Ba1 rating is associated with a default risk close to 0.13 percent.
      - By the time an empirical default probability reaches 1 percent, the rating is generally in the Ca range.

- Distribution of default probabilities across bank assets (Figure 8)
  - Figure 8 shows, for four dates, the percentage of banking sector assets (held by banks in sample) with default probabilities at or below a given level.
  - Interpretation: The closer the line for a given date is to the Y-axis, the higher the share of banking-sector assets held by banks associated with a given probability of default (thus the safer the banking system).
  - Example: September 10, 1998 (LTCM crisis)
    - Around 20 percent of banking sector assets were held by banks whose default probability was below 0.12 percent.
    - 15 percent of assets were held by banks with default probabilities greater than or equal to 0.3 percent.
  - Observations:
    - During the height of the 1998 crisis, a fairly large proportion of the assets held by Chile’s main banks were held by banks with default probabilities greater than 10 percent, and only a small share (less than 20 percent) was held by banks with default probabilities exceeding 30 percent.
    - Over time risks fell as the health of the banking system improved.
    - The period just before the 2002 Brazilian presidential elections in late 2006 is associated with generally lower risks of default than the May 2006 “mini-crisis”, which appears to have affected more Chilean banks, even though no banks show the more extreme values observed for late 2002.

- Correlations with other measures of banking risk
  - CCA-derived risk measures are significantly correlated with other measures of banking risk.
  - Figure 9 (from Luna and Gómez, 2007) presents correlations between various measures of systemwide bank risk and the leads and lags of system DTD.
  - Example correlation:
    - Measures of nonperforming loans (NPLs) for consumption purposes are strongly correlated with declining distance-to-distress estimates from a few months previous (i.e., falling DTD predicts rising NPLs a few months later).

### III. RELATING DEFAULT RISK TO MACROECONOMIC VARIABLES
#### A. Motivation
- Monitoring default risk of financial institutions is important for policymakers; estimating effects of forecast changes in the macroeconomic environment and shocks provides useful insights.
- Variable selection principles:
  - Focus on variables available at a monthly frequency.
  - To keep dimensionality down, focus on variables with relatively low correlation.
  - Aim for relatively independent variables (full orthogonality cannot be assured).
- Variables used (Table 1 covers domestic developments, foreign spillovers, and financial-market developments):
  - Domestic developments: unemployment, CPI, IMACEC index of economic activity.
  - Foreign spillovers: U.S. interest rates and CPI.
  - Financial-market developments: VIX volatility measure, returns on the S&P 500 and the Chilean IPSA index.
  - In each case monthly log returns are used in the VARs below; for interest rates, the level and the change are used rather than log returns.
- Notes:
  - IMACEC is a monthly index of economic activity highly correlated to GDP growth.
  - Domestic interest rate is incorporated into CCA modeling.
- Regression diagnostics:
  - Due to banking heterogeneity, no small sample of variables explains variation in bank-asset returns across all banks.
  - Stepwise regressions were run for each bank’s asset returns against the full sample of macroeconomic variables; variables were eliminated until remaining variables were significant at a specified level.
  - At the 5 percent significance level, the variables listed in Table 2 are jointly significant.
  - Findings from stepwise regressions:
    - Some variables (S&P 500 returns, level of U.S. short-term rates, slope of the U.S. yield curve) are significant for four of seven banks.
    - IMACEC is significant in six banks.
    - Stepwise regressions produce models with r-squared statistics in the 40–60 percent range.
    - Smaller models using variables significant in at least three regressions (returns on IPSA and S&P, U.S. yield curve slope and one-year rate, and IMACEC) produce adjusted r-squared statistics in the 20–40 percent range.
    - Conclusion: a more tailored set of variables yields additional information for each bank.
  - To reduce dimensionality and simplify estimations and impulse response interpretation, principal component analysis (PCA) is applied.

#### B. Scenario Testing Using Principal Component Analysis
- Principal component analysis (PCA) application:
  - PCA applied to a system of 15 macro variables.
  - Output: 15 orthogonal factors; the first four account for 70 percent of the variation of all underlying macro variables.
  - Note: “These are the four factors with eigenvalues greater than one.”
  - Varimax rotation applied to the four factors to maximize interpretability (maximizes largest loadings, minimizes smallest, preserves orthogonality).
  - After rotation, factors can be interpreted as akin to sums of a small number of variables.
- Interpretation of the four factors (Table 3 factor loadings; bold indicates factors greater than 0.5 in magnitude):
  - Factor 1: Financials
    - Strong loadings: S&P 500 (0.86), IPSA (0.73), VIX (-0.76), U.S. 10-Yr Rate (Chg) (0.73), CLP-USD Rate (-0.49).
    - Characterization: commoves with returns on the IPSA and S&P 500; negatively correlated with VIX and peso-dollar exchange rate; linked to changes in U.S. 10-year Treasury bond rate.
  - Factor 2: Interest Rates (U.S. interest rates)
    - Strong loadings: U.S. 1-Yr Rate (Lvl) (0.95), CPI Chile (0.73), U.S. 1-Yr Rate (Chg) (0.21), U.S. 10-Yr Rate (Chg) (0.00).
    - Characterization: mirrors level of U.S. 1-year Treasury bond rate and negatively the slope of the U.S. yield curve; rises from 1997 to 2000 as short-term U.S. rates rose and yield curve flattened, then reverses in 2000 as yield curve steepens when short-term rates fell.
  - Factor 3: Cyclicals
    - Strong loadings: Oil (0.87), CPI U.S.A. (0.82), U.S. 1-Yr Rate (Chg) (0.61), IMACEC (0.32).
    - Characterization: most variation comes from oil and copper prices and U.S. CPI; interpreted as cyclical factors in the world economy.
  - Factor 4: Domestic/Regional
    - Strong loadings: CLP-BRL Rate (0.81), IMACEC (0.58), Copper (0.64).
    - Characterization: Chile- or Latin-American specific factors.
- Factor behavior:
  - Factor 1 (Financials) commoves strongly with IPSA and S&P 500 (correlation of -0.6 reported between those two series).
  - Factor 1 shows the VIX peak in 1998 as its low point.
  - Factor 2 (Interest Rates) behavior described above across 1997–2000.
  - Factor 3 (Cyclicals) driven by commodity prices and U.S. CPI; ties to business cycle.
  - Figure 10 displays the time series for the four significant PCA factors.

*Source: _wp0889 - Section IV below discusses other ways in which both the interest rate and the volatility may (PDF).*

### 4.  Factor 4: Domestic/Regional

### 4.  Factor 4: Domestic/Regional

### Factor characterization and interpretation
- Factor 4 weighs most strongly Chilean variables such as the IMACEC index and unemployment, and also the exchange rate with the Brazilian real.
- The factor’s variation stems mostly from a large decline in 1998 and a subsequent recovery beginning in 2004.
- Because the peso/real exchange rate is not strictly domestic, and the 1998 crash was regional, the factor is interpreted as a domestic/regional factor: many variables in Latin America, but very few elsewhere, exhibit this pattern.
- The four extracted factors account for more than 70 percent of the variation in all 15 of the macroeconomic variables used.

### Vector autoregressions (VARs): specification and main findings
- One VAR is estimated for each bank; each VAR includes five variables: the bank’s implied assets and the four factors.
- Ordering note: factors are orthogonal and cannot contemporaneously affect each other by construction, but can be autocorrelated and Granger-cause other factors.
- Asset returns are used rather than D2D or EFDs because VAR results using D2D and EFDs were similar but with fewer significant coefficients.
- All asset returns are stationary under the Dickey-Fuller criterion at the 5 percent level of significance; log changes in macro variables used are also stationary.
- Each VAR is specified in two lags; Schwarz and Akaike information criteria were calculated for lags 1–4 and AIC/SIC point to two lags more frequently than any other number.
- Persistence in assets:
  - The first lag of the bank’s assets is significantly positive in all regressions.
  - The second lag is significant in two cases.
- Factor significance across banks:
  - Each macroeconomic factor is significant for at least one bank.
  - The financials and domestic/regional factors are significant for four banks.
  - The interest-rate and cyclical factors are significant for only one bank each.
- Bank size groups:
  - Banks 1–3 are the three largest private banks; banks 4–7 are smaller.
  - The first three banks cumulatively account for 52 percent of the assets of the financial system (or 59.9 percent of the nonstate financial system).
  - The last four banks account for 20.1 percent of system assets (23.2 percent of the nonstate system).
- Differences by bank size:
  - The financials variable is significant for the first three (large) banks but not for the smaller ones.
  - The regional factor is significant for the smaller banks, but not for the larger ones.
- Loan-portfolio composition:
  - 73 percent of the loan portfolio of the larger banks is to companies.
  - 67 percent of the small banks’ portfolios is to companies.
- Financial conditions (VIX, IPSA, S&P) effects:
  - An improvement in financial conditions is tied to significantly higher asset returns at two of the three largest banks and one smaller bank, but tied to lower returns at one of the larger banks.
- Interest-rate and cyclical factors:
  - These are less significant in the VARs; stepwise regressions had shown cyclical variables (copper, oil) and interest-rate variables were useful, but after PCA and consistent cross-bank modeling each is significant for only one bank.

### Impulse response functions (IRFs): dynamics and heterogeneity
- IRFs are derived from the VARs to show how shocks to each factor impact implied assets of each bank; factors are orthogonal so IRFs isolate factor shocks holding others constant.
- Bank-own-asset shocks:
  - A shock to a bank’s assets results in continued increase of assets for a few months.
  - Half the shock dissipates within four months for all banks.
  - 90 percent of the shock dissipates within eight months for all banks but one.
- Financials factor shock:
  - Strongly positive impulse responses for four banks, negative for one bank, two close to zero.
  - Timing: three of the four intuitively-signed banks and the negative-response bank have maximum response two to three months after the shock.
  - Bank 5 shows a later maximum response and more persistence; VAR for bank 5 had no significant coefficient on lags of the first factor but did on the fourth factor, implying delayed feedback via factor interactions.
- Interest-rate factor shock:
  - Significant impulse response only for bank 7.
  - Counterintuitive sign: a positive shock to the interest-rate factor (increase in short-term U.S. interest rates or a flattening yield curve) results in an increase in implied assets of bank 7; other banks show the intuitive, if insignificant, opposite sign.
- Cyclicals factor shock:
  - Significant for the two banks showing significant coefficients in VARs (banks 2 and 5).
  - Sign is intuitive: an improvement in cyclical conditions (rises in oil or copper prices, increases in inflation or short-term rates) increases banks’ implied assets which peaks after a few months and then turns negative.
  - Other banks show broadly similar but insignificant patterns, except bank 4.
- Domestic/regional factor shock:
  - Provides significant responses for three banks: the small banks, numbers 5 through 7.
  - In significant and some less significant cases, impulse responses are of the expected sign (improvement in local conditions increases bank assets).
  - Persistence: for two banks shocks dissipate within six months; for bank 7 half the effect dissipates within six months.
- Summary:
  - Larger banks are generally more responsive to financial-market developments.
  - Smaller banks respond more strongly to domestic/regional developments.
  - Shocks are relatively short-lived in most cases but vary in persistence across banks.

### Scenario testing: simulated D2D under calibrated shocks
- Method:
  - Use impulse responses to simulate assets, then compute distance to default (D2D) for each bank under scenarios.
  - Initial asset values set to observed values at end of sample in February 2007.
  - Constant parameters used as in historic risk measures; asset volatility set to the final observed value.
  - Note: shocks may raise asset volatility in reality; higher asset volatility lowers D2D, so scenario D2D likely overestimates risk (i.e., reports lower D2D) — robustness alternatives are presented elsewhere.
- Scenario 1: shock to financials factor comparable to 1998 (Russia/LTCM)
  - VIX jumped by about two standard deviations in late 1998; stock markets fell dramatically.
  - Multiplying log changes by factor loadings yields an average innovation to the financial factor slightly less than one.
  - Results:
    - For each bank, D2D falls in months immediately following the shock.
    - A decline of one in D2D represents a statistically significant increase in probability of assets falling beneath the default barrier.
    - Four of five banks show recoveries beginning a few months after the shock.
    - In two cases recovery is rapid and within 10 months assets return above original level.
    - In other cases recovery is slower; bank 5 does not begin to revert until a year after the shock.
  - Regulatory implication: heterogeneous recovery speeds imply some banks require closer monitoring during aftermath.
- Scenario 2: shock to domestic factor calibrated to unemployment increase in 1999
  - Results:
    - Decline in D2D following shock between one and two standard deviations for most banks (one outlying bank has idiosyncratic results).
    - This shock is less persistent than the financials shock; recovery begins in all cases within less than six months.
    - Heterogeneity: one bank bottoms out later but recovers more quickly.
  - Overall: a domestic shock of this magnitude hit Chile only in 1999; such shocks are less persistent than the financials shock used.

### Robustness tests
- Volatility and interest rates:
  - Using another widely used volatility measure would not much affect results.
  - Allowing for variable interest rates raises choice issues of which riskfree rate to use; impact on results is small during low-risk periods and large during high-risk periods (when probabilities are high under any assumption).
- Estimation period and 1998 observations:
  - Reestimated stepwise regressions, PCA, and VARs dropping all 1998 observations.
  - Stepwise regressions: variables surviving and parameter estimates are not substantively changed.
  - PCA:
    - Dropping 1998 increases explanatory power of first four factors to 73 percent of overall variance.
    - A fifth factor attains eigenvalue greater than one and would be included by selection criterion.
    - Interpretation of first four factors largely unchanged; fifth factor most heavily weighs the U.S. exchange rate, copper prices, and changes to U.S. interest rates.
  - VARs with five factors:
    - Results broadly similar to four-factor VARs.
    - Financial developments factor significant for same banks (large ones and number 6).
    - Fourth, domestic factor significant for smaller banks.
    - New factor significant for two banks, comparable to interest-rate and cyclical factors previously.
  - Conclusion: including 1998 observations does not substantively change results; as they represent largest-scale crisis they may overestimate bank risk, which is acceptable for purposes here.

### Areas for further research and modeling recommendations
- Extensions of current analysis:
  - Model banks jointly to allow feedback between banks (e.g., from large to small) via a higher-dimensional VAR, subject to data limitations (sample extends back only to 1998).
  - Relax constant interest-rate assumption by including a realistic series of riskfree interest rates; use STV (Shimko, Tejima, Van Deventer 1993) or Merton-STV extensions to allow stochastic interest rates and correlation with asset returns.
  - Address non-normality of asset returns (skewness and fat tails) by:
    - Calibrating returns to a mixture of two or more lognormals.
    - Using constant elasticity of variance (CEV) approaches where asset volatility varies with asset level.
  - Conduct detailed bank-level analysis (lending portfolios) to explain puzzling results (e.g., one large bank adversely affected by improved financial-market conditions) and differences between small and large banks.
- Stress-testing and capital adequacy using CCA:
  - CCA can be used to assess capital adequacy under shocks to assets, asset volatility, interest rates, and other parameters without requiring detailed obligor default probabilities or loss-given-default inputs.
  - Extensions of Merton-STV allow inclusion of interest rate volatility and correlation of asset return with interest rates; calibrated CCA models can imply probability of default and loss-given-default.
  - Central-bank stress-testing workflows (estimate default probabilities, combine with exposures and LGD) can be complemented or supplanted by calibrated CCA when detailed obligor data are lacking.
- Incorporating CCA into macroeconomic models:
  - Embed an aggregate credit risk indicator (CRI) derived from CCA (e.g., distance-to-distress weighted by bank implied assets) into GDP gap equations and policy-rate reaction functions.
  - Estimate whether financial stability influenced historical policy rates and study tradeoffs among GDP, inflation, and distance-to-distress.
  - Related work at the Central Bank of Chile explores including distance-to-distress in GDP and policy-rate equations.

### Conclusions and policy implications
- CCA is a useful forward-looking, market-data-based tool to assess financial company risk, though it shares market limitations (markets do not always foresee risky events).
- In Chile over the sample period:
  - CCA measures show gradual improvement in bank health over the decade but clearly identify periods of elevated risk from international shocks.
  - VAR results indicate innovations in the four factors are strongly tied to bank risk; each factor is significant in at least one VAR.
  - Financial-market developments are most significant in explaining variation in banks’ risk; positive financial-market developments relate to reduced risk for most banks.
  - U.S. interest-rate changes are less strongly tied to Chilean bank risk.
  - Banks’ risk moves procyclically with the cyclical factor for most banks, but some banks are less responsive or even nearly countercyclical.
  - The domestic/regional factor has a stronger effect on smaller banks than on larger banks; conversely, the financials factor is more tied to changes among larger banks.
- Regulatory guidance based on results:
  - Focus on larger, more internationally connected banks when assessing the potential impact of financial-market shocks.
  - Focus on smaller banks when assessing the risk of regional or domestic shocks.
  - Monitor heterogeneity in persistence and timing of shocks across banks: some banks’ risk rises later and persists longer, so system-wide assessment should account for asynchronous dynamics.
- Broader recommendation:
  - Incorporate CCA-derived risk measures into monetary and macroeconomic models to improve realism, and use CCA as a building block for integrating financial risk assessment into macro analysis.

*Source: IMF Working Paper chapter “4.  Factor 4: Domestic/Regional”*

### REFERENCES

### _wp0889 - REFERENCES

### References cited
- Allen, M., C. Rosenberg, C. Keller, B. Setser, and N. Roubini, (2002) “A Balance Sheet Approach to Financial Crisis,” IMF Working Paper No. 02/210 (Washington, DC: International Monetary Fund).
- Aspachs, O., C. Goodhart, D. Tsomocos, L. Zicchino (2006) “Towards a Measure of Financial Fragility” Annals of Finance, Special Issue.
- Avesani, R., A. Garcia Pascual, J. Li, (2006) “A New Risk Indicator and Stress Testing Tool: A Mulitfact Nth-to-Default CDS Basket.” IMF Working Paper No. 06/105 (Washington, DC: International Monetary Fund).
- Bardsen, G., K.-G. Lidquist, and D. P. Tsomocos (2006) “Evaluation of Macroeconomic Models for Financial Stability Analysis,” Norges Bank Working Paper 2006/1 (Washington, DC: International Monetary Fund).
- Belmont, D. (2004) Value Added Risk Management in Financial Institutions. John Wiley and Sons, Singapore. Pp. 181–212.
- Chacko, G., A. Sjoman, H. Motohashi, and V. Dessain (2006) “Credit Derivatives,” Wharton Book Publishers, June.
- Chan-Lau, J.A (2006) “Fundamentals-Based Estimation of Default Probabilities: A Survey,” IMF Working Paper No. 06/149 (Washington, DC: International Monetary Fund).
- Chan-Lau, J.A., and T. Gravelle (2005) “The END: A New Indicator of Financial and Nonfinancial Corporate Sector Vulnerability.” IMF Working Paper No. 05/231 (Washington D.C: International Monetary Fund).
- Chan-Lau, J.A., A. Jobert, and J. Kong (2004) “An Option-Based Approach to Bank Vulnerabilities in Emerging Markets,” IMF Working Paper No. 04/33 (Washington D.C: International Monetary Fund).
- Cherubini, U., Luciano, E., and Vecchiato, W. (2004) Copula Methods in Finance. John Wiley & Sons, London.
- Crouhy, M., Galai, D., and Mark R. (2000) Risk Management. New York McGraw Hill.
- Echeverria, C., L. Luna, D. Gray (2006) “Una medida del riesgo de insolvencia de la banca en Chile.” Financial Stability Report of the Central Bank of Chile, Second Half 2006. Santiago, Chile.
- European Central Bank (2006) “EU Banking Sector Stability,” November.
- Gapen, M.T., D. F. Gray, C. H. Lim, and Y. Xiao (2005) “Measuring and Analyzing Sovereign Risk with Contingent Claims,” IMF Working Paper No. 05/155 (Washington, DC: International Monetary Fund).
- Gray, D. F., R. C. Merton, and Z. Bodie (2006) “A New Framework for Analyzing and Managing Macrofinancial Risks of an Economy,” NBER Working Paper 12637.
- Gray, D. F. (2007) “New Framework for Measuring and Managing Macrofinancial Risk and Financial Stability,” NBER Working Paper 13607 and Presentation at Bank of Canada Conference on Frameworks for Financial Stability November 7 and 8.
- Gropp, R., J. Vesala, and G. Vulpes (2001) “Equity and Bond Market Signals as Leading Indicators of Bank Fragility,” European Central Bank Working Paper 76, August.
- International Monetary Fund (2006) “U.S. Banking: Financial Innovation and Systemic Risk.” Selected Issues Paper, US 2006 Article IV Consultation.
- International Monetary Fund (2005) Global Financial Stability Report, Statistical Appendix.
- Kozak, M., M. Aaron, and C. Gauthier (2005) “Using the Contingent Claims Approach to Assess Credit Risk in the Canadian Business Sector,” Bank of Canada, Financial System Review, December.
- Lehar, A. (2005) “Measuring Systemic Risk: A Risk Management Approach,” Journal of Banking and Finance 29 (10): 2577–2603.
- Loffler, G., P. Posch, (2007) Credit Risk Modeling Using Excel and VBA. John Wiley and Sons, England.
- Luna, L. and G. Gomez. (2007) “Riesgo de insolvencia en la Banca: Un analisis tecnico.” Unpublished document. Central Bank of Chile. Santiago, Chile.
- Luna, L., D. Gray and C. Echeverria (2006) “Riesgo de insolvencia en la Banca” Unpublished document. Central Bank of Chile. Santiago, Chile.
- Merton, Robert C. (1974) “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” The Journal of Finance, Vol. 29, pp. 449–470.
- Persson, M., and M. Blavarg (2003) “The Use of Market Indicators in Financial Stability Analysis,” Riskbank Economic Review 2.
- Restrepo, J., L. Luna, D. Gray, “Including Banking System Risk Indicators in Monetary Policy Models: The Case of Chile.” Unpublished Analysis Central Bank of Chile (draft paper).
- Saunders, A and Allen, L. (2002) Credit Risk Measurement. New York: John Wiley and Sons.
- Segoviano, M., C. Goodhart, B. Hofmann (2006a) “Default, Credit Growth, and Asset Prices” IMF Working Paper No. 06/223 (Washington, DC: International Monetary Fund).
- Segoviano, M., C. Goodhart, B. Hofmann (2006b) “Portfolio Credit Risk and Macroeconomic Shocks: Applications to Stress Testing Under Data-Restricted Environments,” IMF Working Paper No. 06/283 (Washington, DC: International Monetary Fund).
- Tudela, M., and G. Young (2002) “A Merton-Model Approach to Assessing the Default Risk of UK Public Companies,” Bank of England Working Paper 194.
- Van den End, J. W., and M. Tabbae (2005) “Measuring Financial Stability: Applying the MfRisk Model to the Netherlands,” Nederlandsche Bank Working Paper No. 30.
- Van Deventer, D. and K. Imai (2003). Credit Risk Models and the Basel Accords. John Wiley and Sons, Singapore. pp. 215-224.
- Van Deventer, D. (1997). Financial Risk Analytics. McGraw-Hill Companies, New York.

### Annex 1 — Framework for Contingent Claims Analysis, Risk Measures, and Spreads using Black‑Scholes‑Merton formula

- Basic accounting identity:
  - Assets = Equity + Risky Debt
  - ()()()AtJ tDt=+

- Risky debt decomposition:
  - Risky Debt = Default-Free Debt − Debt Guarantee
  - ()
  - ()()
  - rT t
  - DtBePt
  - −−
  - =−
  - (we omit the time subscript at t = 0 and define rT
    BBe
    −
    = .)

- Equity as call option (Black‑Scholes‑Merton):
  - The value of the equity is computed using the Black‑Scholes‑Merton formula for the value of a call:
    - 12
      ()()JANdBNd=−
  - The value of the put option:
    - 21
      (())PANdBNd=−−−
  - Definitions:
    - 1
      2
      ln
      2
      A
      r
      B
      d
      T
      T
      σ
      σ
      ++
      =
      ⎛⎞
      ⎛⎞
      ⎜⎟
      ⎜⎟
      ⎝⎠
      ⎝⎠
    - 21
      ddTσ=−
    - r is the risk-free rate
    - σ is the asset return volatility
    - N(d) is the cumulative probability of the standard normal density function below d.
    - The “risk-neutral” or “risk-adjusted” default probability is 2
      ()Nd−.
    - The formula for the “delta” of the put option is 1
      ()1Nd−.
    - The distance-to-distress is 2
      d, the number of standard deviations of asset volatility from the promised payments.

- Yield, spread, and definitions:
  - The yield to maturity on the risky debt, y, is defined by:
    - yT
      DBe
      −
      =
    - ()
      ln/BD
      y
      T
      =
  - Credit spread: s = y - r

- Numerical example (preserve original values exactly):
  - Assuming that:
    - A = $100,
    - σ = 0.40 (40%)
    - B = $75
    - r = 0.05(5%)
    - T = 1 (one year)
  - Results:
    - The value of the equity is $32.367
    - The value of risky debt is $67.633
    - The yield to maturity on the risky debt is 10.34%
    - The credit spread 5.34%
    - The risk adjusted probability of default is 26%

- Calculating implied assets and implied asset volatility:
  - In the Merton Model with traded equity the two equations to solve for A (asset value) and A
    σ (asset volatility) are:
    - 012
      N(   )N(    )JA d  Bd=−
    - 1
      ()
      JA
      JANdσσ=
  - Market Price of Risk (how to go from actual world to RN world valuation):
    - Two types of distance-to-distress:
      - 2 d with an asset drift of the risk-free rate
      - 2, d μ with an asset drift of A μ
    - Two default probabilities:
      - 2
        ()Nd− is the risk-adjusted or risk-neutral default probability
      - 2,
        ()Nd
        μ is the “actual” default probability
    - Relation:
      - 2,2
        ()()NdNdt
        μ
        λ−=−−. Market price of risk reflects a certain degree of risk aversion and can be measured in several different ways.
    - Market price of risk expression:
      - 2,2
        Af
        A
        r
        ddt  t
        μ
        μ
        λ
        σ
        −
        −==
      - One can calculate the market price of risk from the capital asset pricing model:
        - ,AM
          SRλρ=
        - CAPM: ()
          M
          rrμβμ−=−
        - Beta relation:
          - ,AM
            r
            SR
            μ
            ρ
            σ
            −
            =
            and
            rμ
            λ
            σ
            −
            =
            , so ,AM SRλρ=

- Extensions of the Merton Model (limitations and typical relaxations):
  - Original restrictive assumptions:
    - (i) default can occur only at the maturity date of the debt;
    - (ii) there is a fixed default barrier;
    - (iii) there is a constant risk-free rate;
    - (iv) asset volatility is constant.
  - Black and Cox (1976): “first passage time” model allowing default prior to maturity if assets fall below a barrier function.
  - Real-world default considerations:
    - Default may occur at asset values higher than theoretical barrier due to covenant breaches, inadequate liquidity, or sovereign-induced default/renegotiation.
    - Define a “distress barrier” as the market value of total assets at which default occurs (present value of promised payments discounted at the risk-free rate).
    - KMV approach: barrier = sum of book value of short-term debt + promised interest payments for the next 12 months + half of long-term debt.
  - MKMV / VK model:
    - Multiple layers of liabilities; MKMV’s EDF credit measure computed via iterative procedure to solve for asset volatility and mapping KMV distance-to-distress to actual default probabilities (CEDFs) using a database.
    - MKV DD definition includes real drift A μ; Merton distance-to-distress uses r for asset drift.
    - MKMV mapping: ()
      ()
      2
      0
      ln//2
      tAA
      A
      KMV
      ABt
      DDf
      t
      μσ
      σ
      +−
      =
      ⎛⎞
      ⎜⎟
      ⎜⎟
      ⎝⎠
      and ()
      ()
      tKMV
      CEDFf   DDt=
  - CreditGrades and further developments:
    - Original CreditGrades (2002): diffusion of firm’s assets and first passage time default with stochastic default barrier; later modified to incorporate equity derivatives (Stamicar and Finger 2005).
    - Research links volatility skew from equity options to CDS spreads (Hull et. al. 2003), relating implied volatility of equity options, leverage, and asset volatility—an approach to obtain risk-neutral default probabilities from implied volatility.
  - Stochastic interest rates:
    - Shimko, Tejima, and Van Deventer (1993): include Vasicek interest rate term structure model (STV model) — two stochastic factors (asset and interest rate), allowing changing risk-free rate and correlation of asset return with interest rate.
    - Longstaff and Schwartz (1995): extend Black and Cox by adding stochastic interest rates, similar to STV.

*Source: _wp0889 - REFERENCES*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0889.pdf_