## 2. Asset Volatility vs. the Difference Between the Distance-to-Default and the Distance-to-Capital

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### Introduction
- Financial stability monitoring increasingly uses market-based risk measures because securities prices are forward-looking and available at high frequency.
- The distance-to-default measure (Moody’s KMV, based on Merton (1974)) is widely used for nonfinancial corporations and has predictive power for bank rating downgrades.
- Limitations of distance-to-default for banks:
  - Bank liabilities and business model (high leverage) differ from nonfinancial corporates.
  - Regulators intervene before equity is exhausted; prompt-corrective-action (PCA) frameworks mean supervisory triggers occur prior to default.
- Objective: incorporate capital-regulation/PCA insights into market-based risk measures by proposing alternative distance measures (distance-to-capital) that include PCA “triggers.”

### What Happens Before a Bank Default?
- Key features of bank failures vs. corporate failures:
  - Supervisory and regulatory actions typically precede bank defaults; interventions are often statutory rather than voluntary.
  - Basel Committee guidance: banks typically required to maintain a capital adequacy ratio in excess of a regulatory minimum of 8 percent of the risky assets.
- Prompt-Corrective-Action (PCA) frameworks:
  - Rules-based frameworks using capital thresholds to trigger statutory supervisory actions before insolvency/default.
  - U.S. FDICIA (1991) introduced: (i) early closure policy allowing closure even at positive capital levels; (ii) mandatory PCA with five capital thresholds (Section 131).
  - U.S. PCA capital thresholds (Table 1):
    - Well Capitalized: Total Risk-Based Capital ≥10% ; Tier 1 Risk-Based Ratio ≥6% ; Tier 1 Leverage Ratio ≥5%
    - Adequately capitalized: ≥8% ; ≥4% ; ≥4%
    - Undercapitalized: <8% ; <4% ; <4%
    - Significantly undercapitalized: <6% ; <3% ; <3%
    - Critically undercapitalized: Tangible equity ≤ 2%
  - Japan: PCA thresholds depend on international vs. domestic operations:
    - International operations: 8 percent, 4 percent, 2 percent, 0 percent
    - Domestic-only operations: 4 percent, 2 percent, 1 percent, 0 percent
    - CAR equal or below 0 percent interpreted as insolvency/default barrier.
  - United Kingdom: Financial Services Authority applies Individual Capital Ratios Framework (ICRF) to set bank-specific “trigger” ratios above the 8 percent base; supervisors expect banks to maintain a higher “target” ratio.
  - Some countries (e.g., India) allow supervisory discretion in actions once thresholds breached.

### A Unified Framework for Distance Measures: Distance-to-Capital
- Rationale:
  - Distance-to-default treats equity as the final buffer where default occurs when asset value falls below liabilities.
  - For banks, regulatory intervention typically occurs earlier; hence the relevant barrier is not necessarily the face value of liabilities.
  - Distance-to-capital uses PCA capital thresholds as alternative barriers to capture supervisory intervention risk.
- Derivation (structural, Merton/Black-Scholes framework):
  - Basic accounting identity: V = D + E, and E = max(0, V − D).
  - Merton distance-to-default T periods ahead, DD_T:
    - DD_T = [ ln(V_T / D) + (μ − 1/2 σ^2) T ] / (σ sqrt(T))  (Equation (2) as presented)
  - Generic distance-risk measure T periods ahead, DR_T (Merton formula):
    - DR_T = [ ln(V_T / L) + (μ − 1/2 σ^2) T ] / (σ sqrt(T)), where L is standard default barrier and λ is a correction factor (Equation (3)).
  - Z-score alternative (focus on current asset value V_t):
    - DR_T = ( V_t − λ L ) / ( V_t σ )  (Equation (4) as presented)
- Specification of λ:
  - Distance-to-Default (DD_T): λ = 1 (Equation (5))
  - Distances-to-Capital (DC_T): λ = 1 / (1 − PCAR_i) (Equation (6)), where PCAR_i are PCA capital adequacy thresholds.
  - Under conservative assumption that bank capital = equity and assets weighted at 100 percent:
    - Bank capital = Bank equity = V − L > V × PCAR_i  -> yields equation (6) implication.
- Difference between distance-to-default and distance-to-capital:
  - Merton formula difference (Equation (7)):
    - DD_T − DC_T = ln(1 / (1 − PCAR_i)) / σ_T  (as presented)
  - Z-score difference (Equation (8)):
    - DD_T − DC_T = [ 1 / PCAR_i − V_t / PCAR_i L ] / σ_T  (as presented)
  - Key comparative insights:
    - The difference increases with higher PCAR thresholds.
    - In the Merton framework, the difference depends only on PCAR and asset volatility σ.
    - In the Z-score framework, difference also depends on the asset-liability ratio (leverage).
    - When assets are very volatile, the difference between measures is negligible.
    - As leverage increases (asset-liability ratio declines), the Z-score difference declines; equity volatility and leverage usually rise when a bank faces difficulties, causing convergence of DD and DC.
- Relationship to other measures:
  - Framework accommodates distance-to-insolvency (Danmarks Nationalbank (2004)) and distance-to-closure (Liu, Papakirykos, and Yuan (2004)).
  - Model assumes simple balance sheet; extensions to account for off-balance-sheet items (as in Moody’s KMV enhancements) are straightforward.

### Case Study: The Resona and Ashikaga Banks (Japan, 2001–03)
- Background:
  - Japanese banking problems in 2001–03: high NPLs after the technology bubble; many banks’ risk-adjusted capital fell close to regulatory minima of 8 percent (international) and 4 percent (domestic).
- Ashikaga Bank:
  - Regional bank; nationalized on November 29, 2003 after found insolvent in September 2003 (debts exceeded assets; assets hit the default barrier).
  - Prior trouble: bank run in 1997; received public capital injections three times between 1999 and 2000.
  - Capital actions: issued 60 million yen of preferred shares in November 1999, and 30 billion yen of shares in January 2002.
  - Under Deposit Insurance Law, Article 102–3, FSA intervened to protect depositors and replace management; shareholders lost stake as share values reached zero.
- Resona Bank:
  - Part of Resona Holding Financial Group (then fifth largest banking group).
  - On May 17, 2003 received government capital injection after capital adequacy ratio fell below the 4 percent PCA threshold; Article 102–1 of the Deposit Insurance Law invoked citing systemic importance.
  - All deposits protected; management replaced; shares tripled in months after injection; shares exchanged for parent company shares.
  - Created in 2001 via merger of banks with balance-sheet problems; auditing firm reported undercapitalization as of end March 2003.
- Empirical observations from case study:
  - Distance-to-capital measures indicated undercapitalization problems as early as 2001 for both banks.
  - Distance-to-capital for Ashikaga dropped sharply in August–September 2001 prior to January 2002 capital injection; rose afterwards.
  - Distance-to-capital for Resona dropped sharply in Q1 2001, indicating serious undercapitalization.
  - Accounting/treatment differences mattered: deferred tax assets treatment led to divergent capital adequacy outcomes for Ashikaga and Resona.
    - Ashikaga: auditors denied deferred tax assets -> became insolvent.
    - Resona: allowed deferral of three years of tax assets (amount exceeded capital) -> not insolvent but fell below 4 percent PCA threshold.
  - Quantitative difference between distance measures:
    - Distance-to-capital was about 10–15 percent lower than distance-to-default for Ashikaga Bank and about 5–10 percent lower for Resona Bank.
    - Both measures exhibited high correlation.
    - The dynamics of DD − DC mirrored asset volatility dynamics: lower asset volatility -> higher difference between the two measures.
    - Implication: distance-to-capital likely lower than distance-to-default in “tranquil” periods.

### Conclusions
- Distance-to-capital improves upon distance-to-default for measuring individual bank risk by capturing PCA framework complexities and supervisory triggers that occur prior to default.
- Methods used to calculate distance-to-default can be extended to compute distance-to-capital.
- Potential system-wide extensions and challenges:
  - A system-wide distance-to-capital could be useful for policymakers to monitor financial stability but faces challenges:
    - Numerous capital thresholds reduce relevance; a uniform threshold (e.g., minimum capital adequacy ratio or default barrier) could be used.
    - Aggregation may mask idiosyncratic risks; one alternative is focusing on worst-case individual measures (e.g., 10th percentile of individual distances), as suggested by European Central Bank (2005) for distance-to-default measures.

*Source: _wp06215 - 2. Asset Volatility vs. the Difference Between the Distance-to-Default and the Distance-to-Capital*

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

### _wp06215 - References..............................................................................................................

### Tables
- 1. Capital Thresholds and Bank Classification Under the U.S. Prompt-Corrective-Action Framework .........................................................................................................................6

### Figures
- 1. Ashikaga Bank and Resona Bank: Distance-to-Capital.......................................................14

*Source: _wp06215 - References..............................................................................................................*

### 2. Asset Volatility vs. the Difference Between the Distance-to-Default

### 2. Asset Volatility vs. the Difference Between the Distance-to-Default and the Distance-to-Capital

### Introduction
- Financial stability monitoring increasingly uses market-based risk measures to supplement accounting-based analyses because securities prices are forward-looking and available at high frequency.
- The distance-to-default measure (Moody’s KMV, based on Merton (1974)) is widely used for nonfinancial corporations and has predictive power for bank rating downgrades.
- Limitations of distance-to-default for banks:
  - Bank liabilities and business model (high leverage) differ from nonfinancial corporates.
  - Regulators intervene before equity is exhausted; prompt-corrective-action (PCA) frameworks mean supervisory triggers occur prior to default.
- Objective: incorporate capital-regulation/PCA insights into market-based risk measures by proposing alternative distance measures (distance-to-capital) that include PCA “triggers.”

### What Happens Before a Bank Default?
- Key features of bank failures vs. corporate failures:
  - Supervisory and regulatory actions typically precede bank defaults; interventions are often statutory rather than voluntary.
  - Basel Committee guidance: banks typically required to maintain a capital adequacy ratio in excess of a regulatory minimum of 8 percent of the risky assets.
- Prompt-Corrective-Action (PCA) frameworks:
  - Rules-based frameworks using capital thresholds to trigger statutory supervisory actions before insolvency/default.
  - U.S. FDICIA (1991) introduced: (i) early closure policy allowing closure even at positive capital levels; (ii) mandatory PCA with five capital thresholds (Section 131).
  - U.S. PCA capital thresholds (Table 1):
    - Well Capitalized: Total Risk-Based Capital ≥10% ; Tier 1 Risk-Based Ratio ≥6% ; Tier 1 Leverage Ratio ≥5%
    - Adequately capitalized: ≥8% ; ≥4% ; ≥4%
    - Undercapitalized: <8% ; <4% ; <4%
    - Significantly undercapitalized: <6% ; <3% ; <3%
    - Critically undercapitalized: Tangible equity ≤ 2%
  - Japan: PCA thresholds depend on international vs. domestic operations:
    - International operations: 8 percent, 4 percent, 2 percent, 0 percent
    - Domestic-only operations: 4 percent, 2 percent, 1 percent, 0 percent
    - CAR equal or below 0 percent interpreted as insolvency/default barrier.
  - United Kingdom: Financial Services Authority applies Individual Capital Ratios Framework (ICRF) to set bank-specific “trigger” ratios above the 8 percent base; supervisors expect banks to maintain a higher “target” ratio.
  - Some countries (e.g., India) allow supervisory discretion in actions once thresholds breached.

### A Unified Framework for Distance Measures: Distance-to-Capital
- Rationale:
  - Distance-to-default treats equity as the final buffer where default occurs when asset value falls below liabilities.
  - For banks, regulatory intervention typically occurs earlier; hence the relevant barrier is not necessarily the face value of liabilities.
  - Distance-to-capital uses PCA capital thresholds as alternative barriers to capture supervisory intervention risk.
- Derivation (structural, Merton/Black-Scholes framework):
  - Basic accounting identity: V = D + E, and E = max(0, V − D).
  - Merton distance-to-default T periods ahead, DD_T:
    - DD_T = [ ln(V_T / D) + (μ − 1/2 σ^2) T ] / (σ sqrt(T))  (Equation (2) as presented)
  - Generic distance-risk measure T periods ahead, DR_T (Merton formula):
    - DR_T = [ ln(V_T / L) + (μ − 1/2 σ^2) T ] / (σ sqrt(T)), where L is standard default barrier and λ is a correction factor (Equation (3)).
  - Z-score alternative (focus on current asset value V_t):
    - DR_T = ( V_t − λ L ) / ( V_t σ )  (Equation (4) as presented)
- Specification of λ:
  - Distance-to-Default (DD_T): λ = 1 (Equation (5))
  - Distances-to-Capital (DC_T): λ = 1 / (1 − PCAR_i) (Equation (6)), where PCAR_i are PCA capital adequacy thresholds.
  - Under conservative assumption that bank capital = equity and assets weighted at 100 percent:
    - Bank capital = Bank equity = V − L > V × PCAR_i  -> yields equation (6) implication.
- Difference between distance-to-default and distance-to-capital:
  - Merton formula difference (Equation (7)):
    - DD_T − DC_T = ln(1 / (1 − PCAR_i)) / σ_T  (as presented)
  - Z-score difference (Equation (8)):
    - DD_T − DC_T = [ 1 / PCAR_i − V_t / PCAR_i L ] / σ_T  (as presented)
  - Key comparative insights:
    - The difference increases with higher PCAR thresholds.
    - In the Merton framework, the difference depends only on PCAR and asset volatility σ.
    - In the Z-score framework, difference also depends on the asset-liability ratio (leverage).
    - When assets are very volatile, the difference between measures is negligible.
    - As leverage increases (asset-liability ratio declines), the Z-score difference declines; equity volatility and leverage usually rise when a bank faces difficulties, causing convergence of DD and DC.
- Relationship to other measures:
  - Framework accommodates distance-to-insolvency (Danmarks Nationalbank (2004)) and distance-to-closure (Liu, Papakirykos, and Yuan (2004)).
  - Model assumes simple balance sheet; extensions to account for off-balance-sheet items (as in Moody’s KMV enhancements) are straightforward.

### Case Study: The Resona and Ashikaga Banks (Japan, 2001–03)
- Background:
  - Japanese banking problems in 2001–03: high NPLs after the technology bubble; many banks’ risk-adjusted capital fell close to regulatory minima of 8 percent (international) and 4 percent (domestic).
- Ashikaga Bank:
  - Regional bank; nationalized on November 29, 2003 after found insolvent in September 2003 (debts exceeded assets; assets hit the default barrier).
  - Prior trouble: bank run in 1997; received public capital injections three times between 1999 and 2000.
  - Capital actions: issued 60 million yen of preferred shares in November 1999, and 30 billion yen of shares in January 2002.
  - Under Deposit Insurance Law, Article 102–3, FSA intervened to protect depositors and replace management; shareholders lost stake as share values reached zero.
- Resona Bank:
  - Part of Resona Holding Financial Group (then fifth largest banking group).
  - On May 17, 2003 received government capital injection after capital adequacy ratio fell below the 4 percent PCA threshold; Article 102–1 of the Deposit Insurance Law invoked citing systemic importance.
  - All deposits protected; management replaced; shares tripled in months after injection; shares exchanged for parent company shares.
  - Created in 2001 via merger of banks with balance-sheet problems; auditing firm reported undercapitalization as of end March 2003.
- Empirical observations from case study:
  - Distance-to-capital measures indicated undercapitalization problems as early as 2001 for both banks (Figure 1 behavior summary).
  - Distance-to-capital for Ashikaga dropped sharply in August–September 2001 prior to January 2002 capital injection; rose afterwards.
  - Distance-to-capital for Resona dropped sharply in Q1 2001, indicating serious undercapitalization.
  - Accounting/treatment differences mattered: deferred tax assets treatment led to divergent capital adequacy outcomes for Ashikaga and Resona.
    - Ashikaga: auditors denied deferred tax assets -> became insolvent.
    - Resona: allowed deferral of three years of tax assets (amount exceeded capital) -> not insolvent but fell below 4 percent PCA threshold.
  - Quantitative difference between distance measures:
    - Distance-to-capital was about 10–15 percent lower than distance-to-default for Ashikaga Bank and about 5–10 percent lower for Resona Bank.
    - Both measures exhibited high correlation.
    - The dynamics of DD − DC mirrored asset volatility dynamics: lower asset volatility -> higher difference between the two measures.
    - Implication: distance-to-capital likely lower than distance-to-default in “tranquil” periods.

### Conclusions
- Distance-to-capital improves upon distance-to-default for measuring individual bank risk by capturing PCA framework complexities and supervisory triggers that occur prior to default.
- Methods used to calculate distance-to-default can be extended to compute distance-to-capital.
- Potential system-wide extensions:
  - A system-wide distance-to-capital could be useful for policymakers to monitor financial stability but faces challenges:
    - Numerous capital thresholds reduce relevance; a uniform threshold (e.g., minimum capital adequacy ratio or default barrier) could be used.
    - Aggregation may mask idiosyncratic risks; one alternative is focusing on worst-case individual measures (e.g., 10th percentile of individual distances), as suggested by European Central Bank (2005) for distance-to-default measures.

*Source: _wp06215 - 2. Asset Volatility vs. the Difference Between the Distance-to-Default and the Distance-to-Capital*

### REFERENCES

### REFERENCES

### Prompt Corrective Action and Bank Regulation
- Aggarwal Raj, and K. T. Jacques, 1998, “Assessing the Impact of Prompt Corrective Action on Bank Capital and Risk,” Economic Policy Review, October, pp. 23–32 (New York; Federal Reserve Bank of New York).
- Basel Committee on Banking Supervision, 2002, “Supervisory Guidance on Dealing with Weak Banks,” (Basel, Bank for International Settlements).
- Kaufman, George G., 2002, “Prompt Corrective Action in Banking: 10 Years Later,” Research in Financial Services: Private and Public Policy, Vol. 14.
- Nieto, María J., and Larry D. Wall. 2006, “Preconditions for a Successful Implementation of Supervisors’ Prompt Corrective Action: Is There a Case for a Banking Standard in the EU?” LSE Financial Markets Group Special Paper Series, No. 165, March.
- Reserve Bank of India (RBI), 2000, “Discussion Paper on Prompt Corrective Action,” Department of Banking Supervision, Central Office, Policy Planning Division, July (Mumbai).

### Option-Based, Market-Based, and Structural Default Models
- Black, Fischer, and Myron S. Scholes, 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy , Vol. 3, pp. 637–654.
- Merton, Robert C., 1974, “On the Pricing of Corporate Debt: the Risk Structure of Interest Rates,” Journal of Finance Vol. 29, pp. 449–470.
- Chan-Lau, Jorge A., 2006, “Market-Based Estimation of Default Probabilities and Its Application to Financial Surveillance,” IMF Working Paper 06/104 (Washington: international Monetary Fund).
- Chan-Lau, Jorge A., Arnaud Jobert, and Janet Q. Kong, 2004, “An Option-Based Approach to Bank Vulnerabilities in Emerging Markets,” (Washington: International Monetary Fund). Available via the internet at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=520182.
- Crosbie, Peter, and Jeff Bohn, 2003, “Modeling Default Risk,” Moody’s KMV White Paper, (San Francisco: Moody’s Investor Service) December 18.
- Hillegeist, Stephen, Elizabeth K. Keating, Donald P. Cram, and Kyle G. Lundstedt, 2004, “Assessing the Probability of Bankruptcy,” Review of Accounting Studies, Vol. 9, pp. 5–24.
- Sellers, Martha, and Navneet Arora, 2004, “Financial EDF Measures: A New Model of Dual Business Lines,” Moody’s KMV White Paper, August (San Francisco: Moody’s Investor Service).
- Tudela, Merxe, and Garry Young, 2003, “Predicting Default Among U.K. Companies: A Merton Approach,” Bank of England Financial Stability Review, June, pp. 104–114.

### Bank Fragility, Insolvency Regimes, and Crisis Management
- Bliss, Robert, and George Kaufman, 2006, “U.S. Corporate and Bank Insolvency Regimes: An Economic Comparison and Evaluation,” Working Paper 2006-01 (Chicago; Federal Reserve Bank of Chicago).
- Gropp, Reint, Jukka Vesala, and Giuseppe Vulpes, 2002, “Equity and Bond Market Signals as Leading Indicators of Bank Fragility,” forthcoming Journal of Money, Credit and Banking.
- Hoelscher, David, and Mark Quintyn, 2003, Managing Systemic Banking Crises, Occasional Paper No. 224 (Washington, International Monetary Fund).
- Morrison, Alan D., and Lucy White, 2005, “Crises and Capital Requirements in Banking,” American Economic Review, Vol. 95, No. 5, pp.1548–72.
- Ito, Takatoshi, 2003, “Banking Sector Problems in Japan,” World Bank/IMF/SEACEN Regional Seminar on Comparative Experiences in Confronting Banking Sector Problems in Asia and Pacific Region, Kuala Lumpur, December 2.

### Financial Stability Reports and Central Bank Publications
- Danmarks National Bank, 2004, Financial Stability Report (Copenhagen). Available via the internet at http://www.nationalbanken.dk
- European Central Bank, 2005, Financial Stability Review, June (Frankfurt).
- Cihak, Martin, 2005, “How Do Central Banks Write on Financial Stability?” (unpublished; Washington: International Monetary Fund).
- Liu, Ying, Eli Papakirykos, and Mingwei Yuan, 2004, “Market Valuation and Risk Assessment of Canadian Banks,” Working Paper 2004-34 (Ottawa, Bank of Canada).

*Source: _wp06215 - REFERENCES*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06215.pdf_