## Probabilities of Default and the Market Price of Risk in a Distressed Economy

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### Abstract and key objective
- Paper: WP/11/75 (© 2011 International Monetary Fund).
- Objective: provide a method to compute the market price of risk under stress and thereby correct CDS-implied probabilities of distress for risk aversion; deliver an alternative to Moody’s KMV EDF during abnormal times such as the 2007-2009 crisis.
- Main innovation: compute the market price of risk under stress as the conditional expectation of the market price of risk given it exceeds an endogenously determined threshold, within a one-factor asset pricing framework.
- Notable assumptions: pricing based on a one-factor model; that factor is normally distributed.
- Empirical claim: application to major U.S. banks during the crisis shows CDS-implied probabilities of default overestimated credit risk by roughly 50 percent.

### Introduction — motivation and empirical context
- Two main empirical measures of default probability: market-based CDS spreads and balance-sheet based Moody’s KMV EDF.
- Divergence during the 2007-2009 crisis: ratio of CDS-implied probability to EDF varied dramatically across banks and time (examples: AIG, AXP, BAC, GS, JPM, MS, WFC).
- Literature link: variation in the market price of insurance (price of risk) explains divergence; price of risk varies across sectors, correlates negatively with real activity and positively with real interest rates.
- EDFs use slowly-moving balance sheet information and can lag market signals in crises; typical methods are ill-suited when distress probabilities change abruptly.
- Empirical factor used: a transformation of the VIX (strongly correlated with first principal component of asset returns).

### Theoretical background — core framework and identification
- Pricing kernel and one-factor model:
  - Stochastic discount factor m_{t+1} defined from consumption Euler equation; linear pricing formula: E_t[m_{t+1} x_{t+1}] = P_t.
  - Equivalent returns pricing: E_t[m_{t+1} R_{t+1}] = 1.
- Risk-neutral probabilities (π̂):
  - Risk-neutral measure π̂(s) related to actual probabilities π(s) via the pricing kernel m(s).
  - CDS spreads approximate risk-neutral default probabilities: π̂(1) ≈ S_N / (1 - K), where K is the recovery rate (assumed to be at 60 percent) and 1 + r_f can be proxied by the OIS rate.
- Key missing element from market-implied default ⇢ actual default: the market price of risk m(s).

### Conditional expectation of the market price of risk under distress
- Two-state reduction: distress and no distress. Relationship:
  - π̂_d = (1 + r_f)^{-1} E_t[ m_{t+1} | distress ] · π_d  (rearranged in the paper).
- Objective: estimate E_t[ m_{t+1} | m_{t+1} > T ] from the mean and variance of m_{t+1}.
- Under normality of m_{t+1}, truncated normal identity used:
  - E[ m_{t+1} | m_{t+1} > T ] = μ + σ λ(α), with α = (T - μ)/σ, μ = E_t[m_{t+1}], σ^2 = Var_t(m_{t+1}), and λ(α) = φ(α) / (1 - Φ(α)) (inverse Mills ratio).
- Initial simplifying threshold: T = μ + σ, corresponding to P(m > T) = 84.0 percent when set once and for all; section III endogenizes T.

### Estimating the price of risk (Var and λ_m)
- Define λ_m = Var_t(m_{t+1}) / E_t[m_{t+1}] (as presented).
- CAPM rearrangement: excess expected returns driven by price of risk λ and quantity of risk β.
- Empirical estimation approaches:
  - Method 1 — VIX normalization and maximum Sharpe ratio:
    - Theoretical bound: price of risk is the maximum attainable Sharpe Ratio for an efficient portfolio.
    - Empirical heuristic: Sharpe ratios higher than 3 are very high; authors normalize the VIX by dividing by 4 to scale VIX consistent with the price-of-risk property.
  - Method 2 — Principal Components (PC) approach:
    - Decompose the market price of risk into an expected component affine in the expected PC of stock market returns and an innovation.
    - Identification via cross-sectional asset regressions linking coefficients a_i and c_i to β_i λ_0 and β_i λ_1, allowing recovery of normalization coefficients λ_0 and λ_1 for the PC series.

### Endogenous threshold — joint determination of threshold and default probability
- Goal: choose threshold T so distress definition is consistent with estimated bank-specific probability π.
- Core relation (nonlinear fixed-point equation for π):
  - π̂ = (1 + r_f)^{-1} [ μ + σ λ(α) ] · π, with α = (T - μ)/σ and T chosen so that 1 - Φ( (T - μ)/σ ) = π (i.e., π = P(m > T)).
- The paper consolidates this into equation (9) (non-linear fixed-point equation for π) and demonstrates:
  - Existence and uniqueness: there is a unique probability of default that solves equation (9).
  - Sketch proof: mapping π → f_t(π) is increasing, with limiting behavior f_t(π) → 1̂π_t as π → 0 and f_t(π) → 0 as π → 1; derivative properties imply uniqueness of fixed point.

### Empirical implementation notes and calibration choices
- Recovery rate (K) used in mapping CDS spreads to π̂: 60 percent.
- Factor used: transformation of VIX correlated with first principal component of asset returns.
- VIX normalization: divide VIX by 4 to align with historically plausible maximum Sharpe ratios.
- Static alternative threshold: choose T so that P(m > T) = 1 - Φ((T - μ)/σ) = 0.84 (i.e., T = μ + σ); model extended to make T bank-specific via equation (9).

### Application and main empirical result
- Application: major U.S. banks during the 2007-2009 crisis (details in section IV and Figure 4 of the paper).
- Main empirical result: CDS-implied probabilities of default overestimated credit risk by roughly 50 percent relative to the method correcting risk-neutral probabilities for the market price of risk under distress.
- Estimated increase in market price of risk for stress states during the turmoil: 30 percent (authors’ estimate).
- Figures reported in the paper:
  - Adjustment Factor plotted with vertical axis values from 0.3 to 0.5 over dates 12/4/2001 to 6/4/2009.
  - E(M_t / M_{t}>T) plotted with vertical axis values from 2 to 3.4 over the same date range.
- Empirical comparison findings:
  - Risk-neutral probability and associated subjective probability respond to the crisis very quickly while EDF lags.
  - Subjective probability and EDF are more in line with each other than either is with the risk-neutral default probability.

### Uniqueness of the Threshold and numerical illustration
- The function t ↦ f_t(π) is continuous and strictly increasing over ]0, π̂ ], with π̂ < π < 1.
- The derivative at 0 is greater than 1; equation t π = f_t(π) (equation 8) admits exactly one solution within ]0,1].
- The authors illustrate uniqueness by plotting t π and f_t(π) in Figure 2.

### Conclusion — practical implications, methodological remarks, and limitations
- Practical use: extracts real-world probabilities of default from CDS spreads under stress by estimating the conditional market price of risk for distress states.
- Advantages:
  - Does not require specifying the shape of the utility function or stationarity of returns.
  - Uses observable/estimable moments: mean and variance of m.
  - Produces bank-specific, self-consistent thresholds for distress via the non-linear fixed-point relation.
- Key caveats:
  - Relies critically on the one-factor pricing assumption and normality of the pricing kernel factor m_{t+1}.
  - Requires calibration/identification of the price-of-risk series (via VIX normalization or PC normalization).
- Methodological remarks:
  - Approach is free of model assumptions on utility function shape because it is based on an empirical one-factor model.
  - Does not require data on past defaults and is suitable for estimating probabilities of extreme events, but depends on CDS market assessment of risk.
- Limitations and directions for further research:
  - One-factor model limitation: a multiple factor model would better fit data and capture determinants of asset prices, but identifying which factor to filter out is challenging.
  - Normality assumption of the price of risk: relaxing it may require estimating more factors because a one-factor model identifies only one moment (variance) of the price of risk.
  - Further research needed to generalize the method to multi-factor and non-normal settings.

*Source: _wp1175 - Sections 1–2, IMF Working Paper WP/11/75 (April 2011), authors Raphael Espinoza and Miguel Segoviano.*

### Section 1

### Probabilities of Default and the Market Price of Risk in a Distressed Economy

### Abstract and key objective
- Paper: WP/11/75 (© 2011 International Monetary Fund).
- Objective: provide a method to compute the market price of risk under stress and thereby correct CDS-implied probabilities of distress for risk aversion; deliver an alternative to Moody’s KMV EDF during abnormal times such as the 2007-2009 crisis.
- Main innovation: compute the market price of risk under stress as the conditional expectation of the market price of risk given it exceeds an endogenously determined threshold, within a one-factor asset pricing framework.
- Notable assumptions: pricing based on a one-factor model; that factor is normally distributed.
- Empirical claim: application to major U.S. banks during the crisis shows CDS-implied probabilities of default overestimated credit risk by roughly 50 percent.

### Introduction — motivation and empirical context
- Two main empirical measures of default probability: market-based CDS spreads and balance-sheet based Moody’s KMV EDF.
- Divergence observed during the 2007-2009 crisis: ratio of CDS-implied probability to EDF varied dramatically across banks and time (Figure 1 for AIG, AXP, BAC, GS, JPM, MS, WFC).
- Literature links this divergence to variation in the market price of insurance (price of risk), with prior findings: price of risk varies across sectors, correlates negatively with real activity and positively with real interest rates.
- Rationale for the method: EDFs are based on slowly-moving balance sheet information and can lag market signals in crises; typical approaches (e.g., Jackwerth (2000)) are ill-suited when distress probabilities change abruptly.
- Factor used empirically: a transformation of the VIX (strongly correlated with first principal component of asset returns).

### Theoretical background — core framework and identification
- Pricing kernel and one-factor model:
  - Stochastic discount factor m(t+1) defined from consumption Euler equation; linear pricing formula: E_t[m_{t+1} x_{t+1}] = P_t.
  - Equivalent pricing in returns: E_t[m_{t+1} R_{t+1}] = 1.
- Risk-neutral probabilities (π̂):
  - Risk-neutral measure π̂(s) related to actual probabilities π(s) via the pricing kernel m(s).
  - CDS spreads provide an approximation to risk-neutral default probabilities: π̂(1) ≈ S_N / (1 - K), where K is the recovery rate (assumed to be at 60 percent) and 1 + r_f can be proxied by the OIS rate.
- Key missing element from market-implied default ⇢ actual default: the market price of risk m(s).

### Conditional expectation of the market price of risk under distress
- Reduce states to two: distress and no distress. Relationship between risk-neutral and real probability of distress:
  - π̂_d = (1 + r_f)^{-1} E_t[ m_{t+1} | distress ] · π_d  (rearranged in the paper presentation).
- Objective: estimate E_t[ m_{t+1} | m_{t+1} > T ] (market price of risk under stress) from the mean and variance of m_{t+1}, both observable/measurable.
- Under normality of m_{t+1}, truncated normal identity:
  - E[ m_{t+1} | m_{t+1} > T ] = μ + σ λ(α), with α = (T - μ)/σ, μ = E_t[m_{t+1}], σ^2 = Var_t(m_{t+1}), and λ(α) = φ(α) / (1 - Φ(α)) (inverse Mills ratio).
- Initial (simplifying) threshold choice: T = μ + σ (one unconditional standard deviation above mean), corresponding to P(m > T) = 84.0 percent when set once and for all (see section III for endogenizing).

### Estimating the price of risk (Var and λ_m)
- Define λ_m = Var_t(m_{t+1}) / E_t[m_{t+1}] (equation as presented in the paper).
- CAPM rearrangement: excess expected returns driven by price of risk λ and quantity of risk β.
- Empirical approaches to estimate the market price of risk:
  - Method 1 — VIX normalization and maximum Sharpe ratio:
    - Theoretical bound: price of risk is the maximum attainable Sharpe Ratio for an efficient portfolio; correlation ρ_{i,Rm} ≤ 1 implies λ ≤ (E[m] / Var(m)) ≤ (RVar_i / (RVar_i - ...)) in the paper’s derivation.
    - Historical heuristic used: Sharpe ratios higher than 3 are very high, so the authors normalize the VIX by a factor of 4 to scale VIX to be consistent with the price-of-risk property.
  - Method 2 — Principal Components (PC) approach (Adrian and Moench style):
    - Decompose the market price of risk into an expected component affine in the expected PC of stock market returns and an innovation.
    - Identification via cross-sectional asset regressions that link estimated coefficients a_i and c_i to β_i λ_0 and β_i λ_1, allowing recovery of normalization coefficients λ_0 and λ_1 for the PC series.

### Endogenous threshold — joint determination of threshold and default probability
- Aim: choose threshold T so that the definition of distress is consistent with the estimated probability of distress π (bank-specific).
- Core relation (nonlinear equation to solve for π):
  - π̂ = (1 + r_f)^{-1} [ μ + σ λ(α) ] · π, with α = (T - μ)/σ and T chosen so that 1 - Φ( (T - μ)/σ ) = π (i.e., π = P(m > T)).
- The paper rewrites and consolidates this into equation (9) (non-linear fixed-point equation for π):
  - Equation (9) as stated in the source: (full symbolic expression retained in the paper for precise implementation).
- Proposition and existence/uniqueness:
  - There is a unique probability of default that solves equation (9).
  - Sketch of proof: function mapping π → f_t(π) is increasing, with limiting behavior f_t(π) → 1̂π_t as π → 0 and f_t(π) → 0 as π → 1; derivative properties imply uniqueness of fixed point.

### Empirical implementation notes and calibration choices
- Recovery rate (K) assumption used in mapping CDS spreads to π̂: 60 percent.
- Factor used empirically: transformation of VIX correlated with first principal component of asset returns.
- VIX normalization: divide VIX by 4 to align with historically plausible maximum Sharpe ratios (authors’ heuristic; historically Sharpe ratios > 3 are considered very high).
- Threshold initial calibration (static alternative): choose T so that P(m > T) = 1 - Φ((T - μ)/σ) = 0.84 (i.e., T = μ + σ), but model extended to make T bank-specific via the non-linear equation (9).

### Application and main empirical result (summary)
- Application: major U.S. banks during the 2007-2009 crisis (details and figures provided in the paper’s section IV and Figure 4).
- Main empirical result summarized in the introduction: CDS-implied probabilities of default overestimated credit risk by roughly 50 percent relative to the method that corrects risk-neutral probabilities for the market price of risk under distress.

### Conclusion — practical implications
- The proposed method provides a way to extract real-world probabilities of default from CDS spreads under stress by estimating the conditional market price of risk for distress states.
- Advantages:
  - Does not require specifying utility function shape or stationarity of returns.
  - Uses observable/estimable moments: mean and variance of the stochastic discount factor (m).
  - Produces bank-specific, self-consistent thresholds for distress by solving the non-linear fixed-point relation.
- Key caveats:
  - Relies critically on the one-factor pricing assumption and normality of the pricing kernel factor m_{t+1}.
  - Requires calibration/identification of the price-of-risk series (via VIX normalization or PC normalization).

*Source: _wp1175 - Section 1, IMF Working Paper WP/11/75 (April 2011), authors Raphael Espinoza and Miguel Segoviano.*

### Section 2

### _wp1175 - Section 2

### Uniqueness of the Threshold
- The function t fπ )( is continuous and strictly increasing over ]0, πˆ ], with πˆ < π < 1.
- The derivative in 0 is greater than 1.
- Therefore, the equation t π = )(t fπ (equation 8) admits exactly one solution within ]0,1].
- The authors plot t π and )(t fπ in Figure 2 to illustrate uniqueness.

### Application to US Banks During the Crisis
- The authors compute an adjustment factor (between risk-neutral and real probabilities of distress) using the first method described earlier.
- In crisis times, real probabilities of distress tend to be much lower than risk-neutral probabilities because “insurance costs” increase as investors seek higher compensation for risk.
- The authors compare their calculated subjective default probability with Moody’s KMV Estimated Default Frequency (EDF).
  - EDF is based on modeling equity as a call option on assets; asset value is derived from market capitalization and an estimate of asset volatility is used to model the probability distribution of asset value. Given estimates of the value of debt, the value of assets and the volatility of assets, the value of equity yields the probability of default.
- Figure 4 displays risk-neutral probability of default, EDF, and the authors’ estimated default probability under method 2 for several US financial institutions.
  - The risk-neutral probability and associated subjective probability respond to the crisis very quickly while EDF seems to lag.
  - The subjective probability and EDF are more in line with each other than either is with the risk-neutral default probability.
- The authors estimate that the market price of risk (for states of nature in which banks would be under stress) increased by 30 percent in the recent turmoil (see Figure 3).
- Figure 3 (Adjustment Factor and Market Price of Risk) shows:
  - An Adjustment Factor plotted with values on the vertical axis from 0.3 to 0.5 across dates from 12/4/2001 to 6/4/2009.
  - E(Mt / Mt>T) plotted with values on the vertical axis from 2 to 3.4 across the same date range.
- Source and note included in the section:
  - Source: authors’ calculations
  - Note: the adjustment factor is ][)1( 1 11 TmmEr tttf  

### Conclusion and Methodological Remarks
- Computing the market price of risk under situations of distress is crucial when using asset prices to assess probabilities of distress.
- The paper offers a simple but theoretically consistent approach to calculate the market price of risk under distress.
- Key methodological features:
  - The approach is free of model assumptions on the shape of the utility function because it is based solely on an empirical one-factor model.
  - It does not require data on past defaults and therefore is appropriate for estimating probabilities of extreme events—although it still depends on CDS market assessment of risk.
- Applicability:
  - The method was applied to US banks during the subprime crisis but is general enough to be applied to other CDS markets.
  - Caceres, Guzzo and Segoviano (2010) used this method to analyze the contribution of risk aversion to the evolution of sovereign spreads in the eurozone.
  - Caceres and Filiz Unsal (2011) provide another application for Asia.
- Limitations and directions for further research:
  - The model is based on a one-factor model; a multiple factor model would better fit the data and capture determinants of asset prices.
    - Difficulty: matching the risk to be measured (market price of risk under distress) with the moments of the pricing factor that can be observed.
    - When several factors determine pricing, identifying which factor to filter out requires additional understanding of the meaning of the factors and the risk one wants to filter out. This issue does not arise in a one-factor model.
  - A second assumption is that the price of risk is normally distributed.
    - Relaxing this assumption may require estimating more factors because with a one-factor model only one moment (i.e., the variance) of the price of risk can be identified.
  - Further research is needed to generalize the method to these more complex settings.

*Source: _wp1175 - Section 2*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1175.pdf_