## _wp06104

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### I. Market-based default probabilities and financial surveillance
- Estimating default probabilities for individual obligors is the first step in assessing credit exposure and potential losses for investors and financial institutions, and is foundational for evaluating systemic risk and conducting stress tests at national, regional, and global levels.
- Once default probabilities for a subset of obligors are known, the associated loss distribution can be estimated.
- Techniques for estimating default probabilities are classified into:
  - Market-based techniques: rely on security prices and ratings.
  - Fundamental-based techniques: rely on financial statement data and/or systematic market and economic factors.
- Purpose of the paper:
  - Review techniques for estimating default probabilities using market-based information.
  - Illustrate usefulness for financial surveillance.
  - Include simple, one-period cases to sharpen intuition.
  - Provide examples using real data obtained from Bloomberg LLP.
  - Emphasize ease of implementation; some examples solved using basic Excel tools.
- Advantages of simpler models:
  - Lower data requirements.
  - Applicable to a wider cross-section of countries and markets.
- Applicability of market-based techniques:
  - Require a relatively liquid secondary market for securities issued by, or credit derivatives referencing, the obligor.
  - Under market efficiency, securities and credit derivatives prices are forward-looking and capture publicly available information on default risk.
  - Market-based methods reverse-engineer asset pricing formulas to extract default probabilities.
- Note on terminology:
  - Default probabilities obtained from market prices are “risk-neutral” or “risk-adjusted” probabilities; they reflect investors’ risk aversion and can differ materially from real world probabilities.
- Structure overview:
  - Section II: default probabilities implied from credit default swaps (CDSs).
  - Section III: prices and spreads of corporate and sovereign bonds.
  - Section IV: equity price and balance sheet information to disclose firm default risk.
  - Section V: method to recover real world probabilities from risk-neutral probabilities.
  - Section VI: conclusion.

### II. Credit default swaps (CDSs)
- Description and mechanics:
  - CDSs are analogous to insurance against default.
  - Protection buyer pays a quarterly fee, or CDS spread, in exchange for protection against default of a reference obligor during the contract life.
  - If the obligor defaults, the protection buyer delivers the bond or loan of the reference obligor to the protection seller in exchange for the face value.
- Market coverage and maturities:
  - CDS contracts available for a wide universe of firms in continental Europe, Japan, United Kingdom, United States, emerging market sovereign issuers, and some selected emerging market corporations.
  - Typical contract maturity: 5 years for corporates; from 1 to 10 years for sovereign issuers.
  - CDS spreads are quoted as spreads over the swap curve rather than the Treasuries curve.
- Pricing linkage to default probability:
  - CDS spread depends heavily on the default probability of the reference obligor.
  - Chan-Lau (2003, 2005) and Neftci, Santos, and Lu (2005) exploited this dependence for predicting sovereign defaults.
- One-period illustrative equations:
  - Protection seller expected loss, L, equals:
    - (1) L = p (1 − RR) , where p is the default probability, and RR is the expected recovery rate at default; recovery rate and default assumed independent.
  - Fair pricing and risk neutrality imply the CDS spread, S, equals the present value of the expected loss:
    - (2) S = (1) p(1 − RR) / (1 + r) , where r is the risk-free rate.
  - Default probability can be recovered from (2) if recovery rate, CDS spread, and discount factor are known.
- Multi-period (constant hazard) formulation:
  - Default probability in period T(i):
    - (3) p(T(i)) = 1 − e^(−λ(T(i))) .
  - CDS spread S(T) is given by:
    - (4) S(T, λ) = (1 − RR) * BT(λ) / AT(λ) , where RR is expected recovery rate at default, and A and B are defined in the source by summations involving y(i) (risk-free yield corresponding to period T(i)) and the hazard rate λ.
  - Given the risk-free yield curve and the expected recovery rate, the hazard rate can be extracted from (4) and default probability estimated from (3).
- Empirical and calibration notes:
  - Standard CDS pricing models (Duffie (1999); Hull and White (2000)) allow recovery of a time-varying hazard rate function when CDSs with different maturities are available.
  - A variety of combinations of recovery rates and default probabilities can be consistent with observed CDS prices; higher assumed recovery rates imply higher implied default probabilities (in the context of equations (3) and (4)).
  - Chan-Lau (2003, 2005) constructed a risk measure, the maximum recovery rate, as an early warning system (EWS) of sovereign default; in Argentina the maximum recovery rate associated with a 5-year CDS experienced a structural break in mid-2001, six months ahead of the debt default.
- CDS spreads versus asset-swap spreads:
  - An asset swap package combines a fixed coupon bond with an interest-rate swap; the asset swap spread is roughly equivalent to the CDS spread if:
    - (i) the initial value of the underlying bond is at par;
    - (ii) defaults are independent from interest rate movements;
    - (iii) it is possible to short asset swaps;
    - (iv) the default-free floater trades at par at default.
  - Even when CDSs are unavailable, a liquid asset-swap market can permit extraction of default probabilities using the techniques described.
  - For corporate issuers in advanced economies, CDS markets tend to lead price discovery; for emerging-market sovereign issuers, no single market dominates price discovery.

- Example 1. GMAC (selected figures preserved exactly):
  - March 21, 2005:
    - 1-year CDS spread = 365 basis points (bps).
    - 6-month swap rate = 3.308 percent.
    - 1-year swap rate = 3.585 percent.
    - Implied 6-month and 1-year forward default probabilities = 6 percent and 12 percent, respectively.
  - By December 6, 2005:
    - 1-year CDS spread rose to an all-time high of 715 bps.
    - 6-month and 1-year swap rates standing at 4.713 percent and 4.713 percent, respectively. The corresponding 6-month and 1-year default probabilities jumped to 11 percent and 23 percent, respectively.

### Bonds: extracting default probabilities from bond prices
- One-period zero-coupon bond model (notation from the source):
  - Let default probability = p, recovery rate = RR, risk-free discount rate = r, current bond price = B.
  - Risk-neutral valuation (equation (5)): (1/(1+r)) [ (1−p) + p RR ] = B.
  - Solved for p (equation (6)): p = 1 − (1+r)B / (1 − RR).
- Multi-period generalization and flat-term-structure assumption:
  - Equations (7) and (8) express bond price B(t) as expected discounted cash flows and allow backing out a flat default probability p_t when RR, coupon C, and the risk-free yield curve are known.
  - Probability of experiencing a default in the next M coupon payments (equation (9)): P = 1 − (1 − p_t)^M.
- Example: Brazil sovereign default probability, 2001–05
  - Method applied to evaluate Brazil’s sovereign risk from January 2001 to December 2005.
  - Observed pattern: sovereign risk rose dramatically in the second half of 2002 and subsided rapidly once policy continuity was clearer.
  - Specific bond used in analysis: U.S.-dollar-denominated 8 percent coupon Brazilian C-bond with expiration date April 15, 2014.
  - Assumed recovery rate: 25 percent.
  - Cash flows discounted using the U.S. dollar swap curve available for maturities: one week, 3 and 6 months, and 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, and 30 years; cubic splines used to interpolate between maturities.
- Bond spreads versus CDS spreads:
  - No-arbitrage arguments: the CDS spread should equal the spread over LIBOR (Z-spread) of a bond trading at par (par spread) but adjustments are required when the bond is not at par, coupons differ, coupon treatment at default differs, or there are costs to unwind positions.
  - Adjusted Z-spreads can be used to estimate default probabilities using the Section II procedure, though liquidity/technical factors may cause adjusted Z-spreads to differ from CDS spreads.

### Equity prices and structural models
- Structural (contingent-claims) approach (Black–Scholes; Merton):
  - Debt and equity payoffs at expiration correspond to put/call payoffs on firm asset value V_T relative to face value of debt D (equations (10) and (11)).
  - Default probability for horizon T under Black–Scholes assumptions (equation (12)):
    - p = 1 − N( [ ln(V_t / D) + (r − (1/2) σ_A^2) T ] / (σ_A sqrt(T)) )
    - where N is the cumulative normal distribution, V_t is asset value, r is risk-free rate, σ_A is asset volatility.
- Estimation of asset value and volatility from market data (equations (13)–(16)):
  - Use market value of equity E_t and equity volatility σ_E; assume face value of liabilities D (commonly: short-term liabilities + 1/2 long-term liabilities); choose time horizon T (commonly one year).
  - Solve equations (13) and (14) for V_t and σ_A using d1 and d2 definitions (equations (15) and (16)).
- Applications and empirical illustrations:
  - Example 4 (banks): distance-to-default predicts bank distress; Dah-Sing Bank showed substantial credit-quality deterioration in second half of 1997 with recovery to precrisis levels only in mid-2001.
  - Example 5 (General Motors, GM):
    - GM 5-year default probability started to increase rapidly in March 2005 prior to downgrade to junk by Standard & Poor’s on May 5, 2005.
    - Massive stock buying temporarily supported equity prices and reduced equity-implied probabilities.
    - Bankruptcy of Delphi and missed profit expectations for Q2 2005 pushed the 5-year default probability to an all-time high of 75 percent in the first week of January 2006.
  - Example 6 (East Asia corporate vulnerability):
    - Chan-Lau and Gravelle (2005) used a Merton-type structural model and loss-distribution modeling to construct an Expected Number of Defaults (END) indicator for Korea, Malaysia, and Thailand.
    - END rose with the onset of the Asian crisis and continued to rise into 2000 for Malaysia and Thailand and into early 2001 for Korea.

### From risk-neutral probabilities to real-world probabilities
- Market-implied probabilities extracted from security prices are risk-neutral (q) rather than real-world/objective probabilities (p).
- Conceptual illustration: strong investor aversion to default implies q > p; investors demand a default risk premium.
- Transformation under Bernoulli default event and utility-based pricing (equation (17)):
  - q / (1 − q) = [ u'(100) / u'(100 q RR + 100 (1 − q)) ] × p / (1 − p)   (presented in the source as equation (17) in utility form)
  - More compact source form (equation (17)): q / (1 − q) × u'(100(1−q + q RR)) = p / (1 − p) × u'(100)
- Example 7 (GM 5-year, risk-neutral vs real-world):
  - Transforming risk-neutral probabilities for GM into real-world probabilities using investor preferences u = log(k W) with scale parameter k produces materially lower real-world probabilities.
  - Risk-neutral expected losses were 4 to 20 times higher than real-world expected losses under the calibration used.
  - Illustration included risk-neutral probability and multiple real-world curves for k=1, k=2, k=3, k=4.

### Conclusions and policy relevance
- Securities prices can be used to extract market expectations of default probabilities from credit derivatives, bonds, and equities using simple financial engineering techniques that permit real-time monitoring.
- Main caveat: estimates are risk-neutral and can be biased upward by default risk premia; the paper proposes a simple utility-based method to transform between risk-neutral and real-world probabilities.
- Estimated default probabilities are central inputs for:
  - constructing vulnerability indicators,
  - modeling credit risk and loss distributions,
  - stress-testing financial systems.
- The techniques are sufficiently simple to encourage policy institutions to adopt them for surveillance, though further development and application work remains to be done.

*Source: _wp06104 (IMF working paper content provided in the supplied PDF excerpt).*

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

### References

### I. MARKET-BASED DEFAULT PROBABILITIES AND FINANCIAL SURVEILLANCE
- Estimating default probabilities for individual obligors is the first step in assessing credit exposure and potential losses for investors and financial institutions, and is foundational for evaluating systemic risk and conducting stress tests at national, regional, and global levels.
- Once default probabilities for a subset of obligors are known, the associated loss distribution can be estimated.
- Techniques for estimating default probabilities are classified into:
  - Market-based techniques: rely on security prices and ratings.
  - Fundamental-based techniques: rely on financial statement data and/or systematic market and economic factors.
- Purpose of the paper:
  - Review techniques for estimating default probabilities using market-based information.
  - Illustrate usefulness for financial surveillance.
  - Include simple, one-period cases to sharpen intuition.
  - Provide examples using real data obtained from Bloomberg LLP.
  - Emphasize ease of implementation; some examples solved using basic Excel tools.
- Advantages of simpler models:
  - Lower data requirements.
  - Applicable to a wider cross-section of countries and markets.
- Applicability of market-based techniques:
  - Require a relatively liquid secondary market for securities issued by, or credit derivatives referencing, the obligor.
  - Under market efficiency, securities and credit derivatives prices are forward-looking and capture publicly available information on default risk.
  - Market-based methods reverse-engineer asset pricing formulas to extract default probabilities.
- Structure overview:
  - Section II: default probabilities implied from credit default swaps (CDSs).
  - Section III: prices and spreads of corporate and sovereign bonds.
  - Section IV: equity price and balance sheet information to disclose firm default risk.
  - Section V: method to recover real world probabilities from risk-neutral probabilities.
  - Section VI: conclusion.
- Note on terminology:
  - Default probabilities obtained from market prices are “risk-neutral” or “risk-adjusted” probabilities; they reflect investors’ risk aversion and can differ materially from real world probabilities.

### II. CREDIT DEFAULT SWAPS
- Description:
  - CDSs are analogous to insurance against default.
  - Protection buyer pays a quarterly fee, or CDS spread, in exchange for protection against default of a reference obligor during the contract life.
  - If the obligor defaults, the protection buyer delivers the bond or loan of the reference obligor to the protection seller in exchange for the face value.
- Availability and maturities:
  - CDS contracts available for a wide universe of firms in continental Europe, Japan, United Kingdom, United States, emerging market sovereign issuers, and some selected emerging market corporations.
  - Typical contract maturity: 5 years for corporates; from 1 to 10 years for sovereign issuers.
  - CDS spreads are quoted as spreads over the swap curve rather than the Treasuries curve.
- Pricing linkage to default probability:
  - CDS spread depends heavily on the default probability of the reference obligor.
  - Chan-Lau (2003, 2005) and Neftci, Santos, and Lu (2005) exploited this dependence for predicting sovereign defaults.
- One-period example (as presented):
  - Protection seller expected loss, L, equals:
    - (1) L = p (1 − RR) , where p is the default probability, and RR is the expected recovery rate at default; recovery rate and default assumed independent.
  - Fair pricing and risk neutrality imply the CDS spread, S, equals the present value of the expected loss:
    - (2) S = (1) p(1 − RR) / (1 + r) , where r is the risk-free rate.
  - Default probability can be recovered from (2) if recovery rate, CDS spread, and discount factor are known.
- Multi-period formulation using constant hazard model of Duffie (1999):
  - Default probability in period T(i):
    - (3) p(T(i)) = 1 − e^(−λ(T(i))) .
  - CDS spread S(T) is given by:
    - (4) S(T, λ) = (1 − RR) * BT(λ) / AT(λ) , where RR is expected recovery rate at default, and A and B are defined in the source by summations involving y(i) (risk-free yield corresponding to period T(i)) and the hazard rate λ. (Full functional forms for A and B are provided in the source.)
  - Given the risk-free yield curve and the expected recovery rate, the hazard rate can be extracted from (4) and default probability estimated from (3).
- Example 1. GMAC: 1-Year default probabilities (illustration from source):
  - Context: General Motors Assurance Company (GMAC) under pressure due to parent company downgrade to junk status in May 2005.
  - March 21, 2005:
    - 1-year CDS spread = 365 basis points (bps).
    - 6-month swap rate = 3.308 percent.
    - 1-year swap rate = 3.585 percent.
    - Implied 6-month and 1-year forward default probabilities = 6 percent and 12 percent, respectively.
  - By December 6, 2005:
    - 1-year CDS spread rose to an all-time high of 715 bps.
    - 6-month and 1-year swap rates standing at 4.676 percent and [text truncated in source].

*Source: _wp06104 - References..............................................................................................................*

### 4.713 percent, respectively. The corresponding 6-month and 1-year default probabilities

### _wp06104 - 4.713 percent, respectively. The corresponding 6-month and 1-year default probabilities

### Credit Default Swaps (CDSs) and hazard rates
- Standard CDS pricing models (Duffie (1999); Hull and White (2000)) allow recovery of a time-varying hazard rate function when CDSs with different maturities are available.
- Empirical observation: a variety of combinations of recovery rates and default probabilities can be consistent with observed CDS prices; higher assumed recovery rates imply higher implied default probabilities (in the context of equations (3) and (4)).
- Example findings:
  - The corresponding 6-month and 1-year default probabilities jumped to 11 percent and 23 percent, respectively.
  - Chan-Lau (2003, 2005) exploited the relationship between recovery rates and hazard rates to construct a risk measure, the maximum recovery rate, as an early warning system (EWS) of sovereign default.
  - In Argentina, the maximum recovery rate associated with a 5-year CDS experienced a structural break in mid-2001, six months ahead of the debt default (Figure 1).

### CDS spreads versus asset-swap spreads
- An asset swap package combines a fixed coupon bond with an interest-rate swap; the asset swap spread is roughly equivalent to the CDS spread if:
  - (i) the initial value of the underlying bond is at par;
  - (ii) defaults are independent from interest rate movements;
  - (iii) it is possible to short asset swaps;
  - (iv) the default-free floater trades at par at default (because hedging a CDS with an asset swap constructs a synthetic default-free floating bond rather than a par floating bond).
- Even when CDSs are unavailable, a liquid asset-swap market can permit extraction of default probabilities using the techniques described.

### Bonds: extracting default probabilities from bond prices
- One-period zero-coupon bond model (notation from the source):
  - Let default probability = p, recovery rate = RR, risk-free discount rate = r, current bond price = B.
  - Risk-neutral valuation (equation (5)): (1/(1+r)) [ (1−p) + p RR ] = B.
  - Solved for p (equation (6)): p = 1 − (1+r)B / (1 − RR).
- Multi-period generalization (Fons (1987)) and flat-term-structure assumption:
  - Equations (7) and (8) express bond price B(t) as expected discounted cash flows and allow backing out a flat default probability p_t when RR, coupon C, and the risk-free yield curve are known.
  - Probability of experiencing a default in the next M coupon payments (equation (9)): P = 1 − (1 − p_t)^M.
- Example: Brazil sovereign default probability, 2001–05
  - Method applied to evaluate Brazil’s sovereign risk from January 2001 to December 2005 (Figure 2).
  - Observed pattern: sovereign risk rose dramatically in the second half of 2002 and subsided rapidly once policy continuity was clearer.
  - Specific bond used in analysis: U.S.-dollar-denominated 8 percent coupon Brazilian C-bond with expiration date April 15, 2014.
  - Assumed recovery rate: 25 percent.
  - Cash flows discounted using the U.S. dollar swap curve available for maturities: one week, 3 and 6 months, and 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, and 30 years; cubic splines used to interpolate between maturities.
- Bond spreads versus CDS spreads:
  - No-arbitrage arguments: the CDS spread should equal the spread over LIBOR (Z-spread) of a bond trading at par (par spread) but adjustments are required when the bond is not at par, coupons differ, coupon treatment at default differs, or there are costs to unwind positions.
  - Adjusted Z-spreads can be used to estimate default probabilities using the Section II procedure, though liquidity/technical factors may cause adjusted Z-spreads to differ from CDS spreads (Chan-Lau, 2003).
  - Price discovery:
    - For corporate issuers in advanced economies, CDS markets tend to lead price discovery (Blanco, Brennan, and Marsh 2005).
    - For emerging-market sovereign issuers, no single market dominates price discovery (Chan-Lau and Kim, 2005).

### Equity prices and structural models
- Structural (contingent-claims) approach (Black–Scholes; Merton):
  - Debt and equity payoffs at expiration correspond to put/call payoffs on firm asset value V_T relative to face value of debt D (equations (10) and (11)).
  - Default probability for horizon T under Black–Scholes assumptions (equation (12)):
    - p = 1 − N( [ ln(V_t / D) + (r − (1/2) σ_A^2) T ] / (σ_A sqrt(T)) )
    - where N is the cumulative normal distribution, V_t is asset value, r is risk-free rate, σ_A is asset volatility.
- Estimation of asset value and volatility from market data (equations (13)–(16)):
  - Use market value of equity E_t and equity volatility σ_E; assume face value of liabilities D (commonly: short-term liabilities + 1/2 long-term liabilities); choose time horizon T (commonly one year).
  - Solve equations (13) and (14) for V_t and σ_A using d1 and d2 definitions (equations (15) and (16)).
- Examples and applications:
  - Example 4 (banks): distance-to-default predicts bank distress; Dah-Sing Bank showed substantial credit-quality deterioration in second half of 1997 with recovery to precrisis levels only in mid-2001 (Figure 3).
  - Example 5 (General Motors, GM):
    - GM 5-year default probability started to increase rapidly in March 2005 prior to downgrade to junk by Standard & Poor’s on May 5, 2005.
    - Massive stock buying temporarily supported equity prices and reduced equity-implied probabilities.
    - Bankruptcy of Delphi and missed profit expectations for Q2 2005 pushed the 5-year default probability to an all-time high of 75 percent in the first week of January 2006 (Figure 4).
  - Example 6 (East Asia corporate vulnerability):
    - Chan-Lau and Gravelle (2005) used a Merton-type structural model and loss-distribution modeling to construct an Expected Number of Defaults (END) indicator for Korea, Malaysia, and Thailand.
    - END rose with the onset of the Asian crisis and continued to rise into 2000 for Malaysia and Thailand and into early 2001 for Korea (Figure 5).

### From risk-neutral probabilities to real-world probabilities
- Market-implied probabilities extracted from security prices are risk-neutral (q) rather than real-world/objective probabilities (p).
- Conceptual illustration: strong investor aversion to default implies q > p; investors demand a default risk premium.
- Transformation under Bernoulli default event and utility-based pricing (equation (17)):
  - q / (1 − q) = [ u'(100) / u'(100 q RR + 100 (1 − q)) ] × p / (1 − p)   (presented in the source as equation (17) in utility form)
  - More compact source form (equation (17)): q / (1 − q) × u'(100(1−q + q RR)) = p / (1 − p) × u'(100)
- Example 7 (GM 5-year, risk-neutral vs real-world):
  - Transforming risk-neutral probabilities for GM into real-world probabilities using investor preferences u = log(k W) with scale parameter k produces materially lower real-world probabilities.
  - Risk-neutral expected losses were 4 to 20 times higher than real-world expected losses under the calibration used.
  - Illustration included risk-neutral probability and multiple real-world curves for k=1, k=2, k=3, k=4 (Figure 6).

### Conclusions and policy relevance
- Securities prices can be used to extract market expectations of default probabilities from credit derivatives, bonds, and equities using simple financial engineering techniques that permit real-time monitoring.
- Main caveat: estimates are risk-neutral and can be biased upward by default risk premia; the paper proposes a simple utility-based method to transform between risk-neutral and real-world probabilities.
- Estimated default probabilities are central inputs for:
  - constructing vulnerability indicators,
  - modeling credit risk and loss distributions,
  - stress-testing financial systems.
- The techniques are sufficiently simple to encourage policy institutions to adopt them for surveillance, though further development and application work remains to be done.

*Source: IMF working paper content provided in the supplied PDF excerpt.*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06104.pdf_