2011年-IMF国际货币组织全球_Probabilities_of_Default_and_the_Market_Price_of_Risk_in_a_Distressed_Economy_15页_1mb

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

Core Content

This paper presents an original method to estimate the market price of risk under stress, which is essential for correcting the CDS-implied probabilities of default for risk aversion. The method is grounded in a one-factor asset pricing model, which allows for the derivation of the market price of risk from observable market data.

Main Points

• The market price of risk is defined as the conditional expectation of the stochastic discount factor, given that the market is under distress.
• The risk-neutral probability of default (from CDS spreads) overestimates the actual probability of default due to the influence of risk aversion.
• The price of risk (variance of the stochastic discount factor divided by its mean) can be estimated using various methods, including the VIX index or Principal Components (PC) of market yields.
• The threshold for distress is endogenously determined to align with the estimated probability of default, ensuring the model is consistent with market conditions.
• The adjustment factor between risk-neutral and real probabilities of distress is derived from the inverse Mills ratio and the market price of risk.
• The paper shows that during the 2007-2009 crisis, the CDS-implied probabilities of default overestimated credit risk by approximately 50%.

Key Information

• The stochastic discount factor $ m_{t+1} $ is a key component in linking risk-neutral probabilities to real probabilities of distress.

• The one-factor model assumes that the stochastic discount factor is normally distributed, and that the market price of risk is derived from the mean and variance of this factor.

• The threshold $ T $ is set such that the probability of the market price of risk exceeding it corresponds to the probability of distress.

• The adjustment factor is calculated as:

  $$
  \frac{1}{(1 + r_f) E_t[m_{t+1} \mid m_{t+1} > T]}
  $$

• The VIX index is used to estimate the price of risk, and the paper proposes normalizing it by a factor of 4 to align with the theoretical maximum Sharpe Ratio.

• The Principal Components (PC) method is also used, particularly for bond and currency markets, where the market price of risk is modeled as an affine function of the PC of returns.

• The probability of default is derived from the risk-neutral probability using the formula:

  $$
  \pi_t = \frac{\hat{\pi}_t}{(1 + r_t^f)(\mu_t + \sigma_t \cdot \lambda(\alpha_t))}
  $$

  where $ \alpha_t = (T - \mu_t)/\sigma_t $, and $ \lambda(\alpha_t) $ is the inverse Mills ratio.

• The threshold $ T $ is determined such that the probability of distress is equal to the probability that the market price of risk exceeds $ T $, and this is shown to be a non-linear equation that has a unique solution.

• The model is applied to U.S. banks during the subprime crisis, revealing that the CDS-implied default probabilities overestimated the actual risk by about 50%.

• The method is generalizable to other CDS markets, including sovereign CDS, as demonstrated by Caceres, Guizzo, and Segoviano (2010).

Methodology

• The paper uses a one-factor model (such as the Consumption CAPM) to estimate the market price of risk.
• The price of risk is derived from the variance of the stochastic discount factor and is used to adjust the risk-neutral probabilities of default.
• The threshold $ T $ is set based on the inverse Mills ratio and the estimated probability of default.
• Two methods are proposed to estimate the price of risk:
  1. VIX-based method: Normalizing the VIX index to align with the theoretical maximum Sharpe Ratio.
  2. Principal Components method: Using the first principal component of market yields to estimate the price of risk.

Conclusion

The paper provides a theoretically consistent and empirically grounded method for estimating the market price of risk under distress, which allows for the correction of risk-neutral probabilities of default. The method is free of assumptions about the utility function and is suitable for extreme event analysis. The application to U.S. banks during the crisis highlights the significant overestimation of credit risk by CDS spreads, suggesting that the method can be used to improve the accuracy of risk assessments in distressed markets.

Figures

• Figure 1: Ratio of CDS-implied Probability of Distress to Moody's KMV EDF.
• Figure 2: Uniqueness of the Threshold.
• Figure 3: Adjustment Factor and Market Price of Risk.
• Figure 4: Estimated Probabilities of Default.

References

• Adrian, T., and E. Moench, 2008, "Pricing the Term Structure with Linear Regressions," Federal Reserve Bank of New York Staff Report No. 340.
• Adrian, T., E. Etula, and H. Shin, 2010, “Risk Appetite and Exchange Rates,” Fed Reserve Bank of New York Staff Report No. 361.
• Amato, J., 2005, "Risk Aversion and Risk Premia in the CDS Market," BIS Quarterly Review, December, pp. 55-68.
• Amato, J., and M. Luisi, 2006, "Macro Factors in the Term Structure of Credit Spreads," BIS Working Paper No. 203.
• Berndt, A., D. Rohan, D. Duffie, M. Ferguson, and D. Schranzk, 2005, "Measuring Default Risk Premia from Default Swap Rates and EDFs," BIS Working Paper No. 173.
• Black, F., and M. Scholes, 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 7, pp. 637-654.
• Caceres, C., and D., Filiz Unsal, 2011, "Sovereign Spreads and Contagion Risks in Asia," IMF Working paper, forthcoming.
• Caceres, C., V. Guzzo, and M. Segoviano, 2010, “Sovereign Spreads: Global Risk Aversion, Contagion or Fundamentals?” IMF Working Paper 10/120.
• Cochrane, J., 2005, Asset Pricing, Princeton University Press.
• Coudert, V., and M. Gex, 2008, "Does Risk Aversion Drive Financial Crises? Testing the Predictive Power of Empirical Indicators," Journal of Empirical Finance, Vol. 15, pp. 167-184.
• Jackwerth, J., 2000, "Recovering Risk Aversion from Option Prices and Realized Returns," Review of Financial Studies, Vol. 13, pp. 433-451.
• Merton, R., 1974, "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, Vol. 29, pp. 449-470.