Integrated Credit and Equity Risk Modeling

1. Integrated Credit and Equity Risk Modeling Enterprise Risk Management Symposium Bill Pauling, CFA

2. Overview • Rationale for integrating credit and equity risk • Possible approaches to integrating credit and equity risk • Cholesky decomposition • Transfer functions • Conclusions

3. Rationale for Integrating Credit and Equity Risk • Merton (1974) viewed corporate debt as a risk-free bond plus a short put option on the firm’s equity • Hence, the value of a firm’s debt and equity are fundamentally linked • Merton’s model forms the basis of many models commonly used today, including Moody’s KMV

4. Credit and Equity Markets are Connected

5. 36-Month Rolling Correlations to Equity

6. 120-Month Rolling Correlations to Equity

7. Historical Correlations • Over the long-term, the correlation between risky debt and equity is higher than the correlation between government debt and equity • The correlation between risky bonds and equity appears to be more stable that the correlation between government bonds and equity

8. Cholesky Decomposion • Method for transforming uncorrelated normal random variables into correlated normal random variables • Formula for correlating 2 random variables (i.e. 2x2 matrix) • where, • εa and εb are normally distributed random numbers • ρ is the correlation between variables a and b • errorb is a correlated random variable with unit variance • Can be used to correlate larger matrices • Can also be used with a covariance matrix

9. Cholesky Decomposition - Example • Given: • Random term used to produce the equity market return, Δy~N(0,1) • Credit spread model: • Correlate Δz in credit spread model to Δy in equity model • Revised credit spread model:

10. Transfer Functions • Useful when two series are believed to be correlated or co-integrated • Transfer functions are often used in structured economic models to link the economic factors • Transfer functions can also be used when random terms may not normally distributed (e.g. jump diffusion models)

11. Transfer Functions - Example • Given • The periodic equity return rt, and its long-term average return rbar from jump diffusion equity model • Credit spread model:

12. Transfer Functions - Example • Incorporate the difference between r and rbar in credit spread model to reflect the correlation between the series • Revised credit spread model: • Revised credit spread model now is correlated to equity model • The jump diffusion process in the equity model is ‘transferred’ to the credit spread model • Equity market crashes will be associated with rather large increases in credit spreads • Negative skewness in equity returns will also be transferred to the credit spread model as positive skewness due to the negative sign of the beta term

13. Conclusions • Credit and equity risk should be modeled in an integrated fashion • Cholesky decomposition can be used to reflect the correlation between the random terms in credit and equity models • Transfer functions can be used to integrate credit and equity models

14. References • Bevan, Andrew and Garzarelli, Franco, “Corporate Bond Spreads and the Business Cycle: Introducing the GS-SPREAD”, The Journal of Fixed Income, March 2000. • Dynkin, L., Lindener, P., Phelps, B. and Wu, W., “Equity Market Impact on Corporate Bond Excess Returns”, Lehman Brothers Portfolio Strategies, May 7, 2001. • Kealhofer, Stephen, “Quantifying Credit Risk I: Default Prediction”, Financial Analysts Journal, January/February 2003. • Kealhofer, Stephen, “Quantifying Credit Risk II: Debt Valuation”, Financial Analysts Journal, May/June 2003. • Merton, Robert, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates”, Journal of Finance, Vol. 29, no. 2, May 1974.