## _wp10190

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---

### Introduction
- Probability of default (PD) measures are widely used to estimate potential losses of, and contagion among, large financial institutions.
- Two common PD extraction methods examined:
  - Risk-neutral probabilities from CDS spreads (do not account for investors’ risk aversion).
  - Real-world probabilities from equity-market based methods such as Moody’s KMV (account for investor risk aversion and use balance sheet inputs).
- Key issues affecting PD-based loss and contagion analyses:
  - Use of “risk-neutral probabilities” vs. “real-world probabilities.”
  - Divergence between credit and equity markets during periods of financial stress.
  - Assumption of stochastic vs. fixed recovery for financial institutions’ assets.

### Adjusting Probabilities: The Price of Risk
- Definition: The price of risk is the ratio needed to convert risk-neutral probabilities (from CDS spreads) to real-world probabilities.
- Conversion proxy (Amato (2005) approach):
  - Pr(CDS Spread) / Equity Market Signal = Price of Risk
  - Enhanced proxy accounting for recovery R:
    - Pr(CDS Spread) / (EDFR − R) = Adjusted Price of Risk
- Numerical observations:
  - BIS Quarterly Report (March 2009) approximation: price of risk fluctuated from an average of about 4 to 12 during the 2007–08 period.
  - Example calculation: If CDS spreads imply a PD of 0.9 percent and the conversion factor is 10, the adjusted PD would be 0.09 percent.
  - At the time of Lehman bankruptcy in September 2008:
    - Price of risk for many large banks was about 5.
    - For European banks, in some cases higher than 10.
- Remarks on alternative adjustments:
  - Moody’s model adjustments to risk-neutral measures use the Sharpe Ratio and a correlation coefficient but assume similar investor treatment of financial and nonfinancial firms and update Sharpe Ratios only once a year.
  - The price of risk approach avoids these complications and may better reflect the transformation from risk-neutral to real-world PDs.

### Divergence between Credit and Equity Markets During the Recent Crisis
- Implicit conversion from risk-neutral to real-world probabilities assumes co-movement of equity and bond market signals.
- Empirical observation for large banks since 2008:
  - Equity markets were far more volatile compared with bond markets.
  - CDS spreads for large banks remained subdued due to perceptions of “too-large-to-fail.”
  - Equity variations were reflected more immediately than bond price volatility.
- Implications:
  - Asymmetric signals from debt and equity markets affect estimates of losses or guarantees implied by balance sheet components.
  - Models using distance-to-distress must account for divergence because debt and equity market volatilities critically influence outcomes.
  - Loss estimates and policy interpretation should be cautious when models do not capture these dynamics.
- Data source referenced for volatility comparison: Bloomberg’s OVCR function (see Appendix I).

### Assumption between Fixed and Variable Recovery
- Standard practice: many PD models assume fixed recovery values (commonly in the range of 20–40).
- Issues for financial institutions:
  - Financial institutions have few tangible assets; recovery during the credit crisis differed markedly from commonly assumed fixed values.
- Empirical examples of recoveries:
  - Lehman recovery roughly 8 cents on the dollar.
  - Landsbanki recovery roughly 1 cent on the dollar.
  - Fannie Mae and Freddie Mac recoveries both above 90 cents on the dollar.
- Consequence: Need for time-varying or stochastic recovery rate assumptions for distressed financial institutions.
- Cheapest-to-deliver (CTD) bonds:
  - CTD bond prices serve as a good proxy for stochastic recovery and align with ISDA physical settlement covenants.
  - CTD-based stochastic recovery can yield markedly different PDs compared with fixed recovery (example: Landsbanki PDs differ for R=40% versus R proxied by CTD).
  - The analysis uses a distressed bond price “cut off” of below 95% of par to identify distressed bonds.

### Implications for Interconnectedness and Joint Probability of Distress (JPoD)
- Using fixed recovery PDs can bias independent probabilities and thereby contaminate conditional probabilities (e.g., probability of Goldman default conditional on Citi default).
- Accurate modeling of JPoD should incorporate observed low recovery rates for large financials to avoid over- or under-estimation of connectedness.

### Conclusion and Policy Implications
- During periods of stress, PD measures for financial institutions should:
  - Be adjusted to reflect the price of risk.
  - Address potential divergence between credit and equity market dynamics.
  - Incorporate the stochastic nature of asset recovery.
- Policy cautions:
  - Loss estimates based on distance-to-distress models should be interpreted with caution for large banks until biases are better quantified and models revised.
  - Modeling interconnectedness via joint probabilities should incorporate low recovery rates observed in recent credit events involving large financials to avoid mis-estimating the degree of connectedness.

### Appendix I — Bloomberg’s OVCR function (summary)
- OVCR converts equity prices, leverage, and implied volatility to a theoretical CDS spread; this can be compared to actual CDS spread.

### Appendix II — The Moody’s KMV Model (summary)
- Uses option-theoretic framework: equity as a call option, debt as a risk-free bond short a put.
- Key variables: default point D, market value of assets V, asset volatility A σ, risk-free rate r.
- Distance-to-Default (DD) is a volatility-adjusted market-based measure of leverage.
- First-passage model approximation: probability of default ≈ 2Φ(−DD), where Φ is the cumulative normal distribution.

### Distance-to-Default (DD): definition, components, and implementation (Section 2)
- Definition and components:
  - V is the value of a firm’s assets.
  - T X is the default point to the horizon.
  - μ is the drift term.
  - A σ is the volatility of assets.
  - T is the horizon.
  - a represents cash leakages per unit time due to interest payments, coupons and dividends.
- Default point and horizon considerations:
  - The default point is computed as current liabilities plus a portion of long-term debt.
  - For longer horizons, a larger portion of long-term debt is included in the default point to reflect that long-term debt becomes more important at longer horizons.
  - The DD varies considerably with the horizon under consideration.
  - At longer horizons, the weight on volatility increases relative to the default point.
- Empirical relationship and model implementation:
  - There is a strong empirical relationship between DD and observed default rates—firms with a larger DD are less likely to default.
  - The Brownian motion assumption on asset value implies a Gaussian relationship between DD and the EDF credit measure.
    - Specifically, for a DD greater than 4, a Gaussian relationship predicts that defaults will occur 6 out of 100,000 times.
    - Taken literally, this would lead to one half of actual firms being essentially risk-free.
  - This implication is not found in the data.
  - Consequence for implementation: The model departs from the Gaussian assumption by implementing an empirical mapping.

*Prepared by Manmohan Singh and Karim Youssef; August 2010.*

### Section 1

### _wp10190 - Section 1

### Introduction
- Probability of default (PD) measures are widely used to estimate potential losses of, and contagion among, large financial institutions.
- Two common PD extraction methods examined:
  - Risk-neutral probabilities from CDS spreads (do not account for investors’ risk aversion).
  - Real-world probabilities from equity-market based methods such as Moody’s KMV (account for investor risk aversion and use balance sheet inputs).
- Key issues affecting PD-based loss and contagion analyses:
  - Use of “risk-neutral probabilities” vs. “real-world probabilities.”
  - Divergence between credit and equity markets during periods of financial stress.
  - Assumption of stochastic vs. fixed recovery for financial institutions’ assets.

### Adjusting Probabilities: The Price of Risk
- Definition: The price of risk is the ratio needed to convert risk-neutral probabilities (from CDS spreads) to real-world probabilities.
- Conversion proxy (Amato (2005) approach):
  - Pr(CDS Spread) / Equity Market Signal = Price of Risk
  - Enhanced proxy accounting for recovery R:
    - Pr(CDS Spread) / (EDFR − R) = Adjusted Price of Risk
- Numerical observations from the source:
  - BIS Quarterly Report (March 2009) approximation: price of risk fluctuated from an average of about 4 to 12 during the 2007–08 period.
  - Example: If CDS spreads imply a PD of 0.9 percent and the conversion factor is 10, the adjusted PD would be 0.09 percent.
  - At the time of Lehman bankruptcy in September 2008:
    - Price of risk for many large banks was about 5.
    - For European banks, in some cases higher than 10.
- Remarks on alternative adjustments:
  - Moody’s model adjustments to risk-neutral measures use the Sharpe Ratio and a correlation coefficient but assume similar investor treatment of financial and nonfinancial firms and update Sharpe Ratios only once a year.
  - The price of risk approach avoids these complications and may better reflect the transformation from risk-neutral to real-world PDs.

### Divergence between Credit and Equity Markets During the Recent Crisis
- Implicit conversion from risk-neutral to real-world probabilities assumes co-movement of equity and bond market signals.
- Empirical observation for large banks since 2008:
  - Equity markets were far more volatile compared with bond markets.
  - CDS spreads for large banks remained subdued due to perceptions of “too-large-to-fail.”
  - Equity variations were reflected more immediately than bond price volatility.
- Implications:
  - Asymmetric signals from debt and equity markets affect estimates of losses or guarantees implied by balance sheet components.
  - Models using distance-to-distress must account for divergence because debt and equity market volatilities critically influence outcomes.
  - Loss estimates and policy interpretation should be cautious when models do not capture these dynamics.
- Data source referenced for volatility comparison: Bloomberg’s OVCR function (see Appendix I).

### Assumption between Fixed and Variable Recovery
- Standard practice: many PD models assume fixed recovery values (commonly in the range of 20–40).
- Issues for financial institutions:
  - Financial institutions have few tangible assets; recovery during the credit crisis differed markedly from commonly assumed fixed values.
  - Empirical examples:
    - Lehman recovery roughly 8 cents on the dollar.
    - Landsbanki recovery roughly 1 cent on the dollar.
    - Fannie Mae and Freddie Mac recoveries both above 90 cents on the dollar.
  - These observations imply the need for time-varying or stochastic recovery rate assumptions for distressed financial institutions.
- Cheapest-to-deliver (CTD) bonds:
  - CTD bond prices serve as a good proxy for stochastic recovery and align with ISDA physical settlement covenants.
  - CTD-based stochastic recovery can yield markedly different PDs compared with fixed recovery (example: Landsbanki PDs differ for R=40% versus R proxied by CTD).
  - The analysis uses a distressed bond price “cut off” of below 95% of par to identify distressed bonds.
- Implications for interconnectedness and joint probability of distress (JPoD):
  - Using fixed recovery PDs can bias independent probabilities and thereby contaminate conditional probabilities (e.g., probability of Goldman default conditional on Citi default).
  - Accurate modeling of JPoD should incorporate observed low recovery rates for large financials to avoid over- or under-estimation of connectedness.

### Conclusion and Policy Implications
- During periods of stress, PD measures for financial institutions should:
  - Be adjusted to reflect the price of risk.
  - Address potential divergence between credit and equity market dynamics.
  - Incorporate the stochastic nature of asset recovery.
- Policy cautions:
  - Loss estimates based on distance-to-distress models should be interpreted with caution for large banks until biases are better quantified and models revised.
  - Modeling interconnectedness via joint probabilities should incorporate low recovery rates observed in recent credit events involving large financials to avoid mis-estimating the degree of connectedness.

### Appendices (summaries)
- Appendix I — Bloomberg’s OVCR function:
  - OVCR converts equity prices, leverage, and implied volatility to a theoretical CDS spread; this can be compared to actual CDS spread.
- Appendix II — The Moody’s KMV Model:
  - Uses option-theoretic framework: equity as a call option, debt as a risk-free bond short a put.
  - Key variables: default point D, market value of assets V, asset volatility A σ, risk-free rate r.
  - Distance-to-Default (DD) is a volatility-adjusted market-based measure of leverage.
  - First-passage model approximation: probability of default ≈ 2Φ(−DD), where Φ is the cumulative normal distribution.

*Prepared by Manmohan Singh and Karim Youssef; August 2010.*

### Section 2

### _wp10190 - Section 2

### Definition and components of Distance-to-Default (DD)
- Distance-to-Default is defined in terms of the firm’s asset value dynamics and a default point to the horizon.
- Notation and components as stated in the source:
  - V is the value of a firm’s assets.
  - T X is the default point to the horizon.
  - μ is the drift term.
  - A σ is the volatility of assets.
  - T is the horizon.
  - a represents cash leakages per unit time due to interest payments, coupons and dividends.
- The value of the firm’s assets and volatility is computed as described above in the source text.

### Default point and horizon considerations
- The default point is computed as current liabilities plus a portion of long-term debt.
- For longer horizons, a larger portion of long-term debt is included in the default point to reflect that long-term debt becomes more important at longer horizons.
- The DD varies considerably with the horizon under consideration.
- At longer horizons, the weight on volatility increases relative to the default point.

### Empirical relationship and model implementation
- Empirically, there is a strong relationship between DD and the observed default rates—firms with a larger DD are less likely to default.
- The Brownian motion assumption on asset value implies a Gaussian relationship between DD and the EDF credit measure.
  - Specifically, for a DD greater than 4, a Gaussian relationship predicts that defaults will occur 6 out of 100,000 times.
  - Taken literally, this would lead to one half of actual firms being essentially risk-free.
- This implication is not found in the data.
- Consequence for implementation:
  - The model departs from the Gaussian assumption by implementing an empirical mapping.

*Source: _wp10190 - Section 2*

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