July 30, 2003 Bill Pauling, CFA Integrated Credit and Equity Risk Modeling Enterprise Risk Management Symposium

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

3 Rationale for Integrating Credit and Equity Risk n Merton (1974) viewed corporate debt as a risk-free bond plus a short put option on the firms equity n Hence, the value of a firms debt and equity are fundamentally linked n Mertons model forms the basis of many models commonly used today, including Moodys 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 n Over the long-term, the correlation between risky debt and equity is higher than the correlation between government debt and equity n The correlation between risky bonds and equity appears to be more stable that the correlation between government bonds and equity

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

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

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

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

12 Transfer Functions - Example n Incorporate the difference between r and rbar in credit spread model to reflect the correlation between the series n Revised credit spread model: n Revised credit spread model now is correlated to equity model n The jump diffusion process in the equity model is transferred to the credit spread model n Equity market crashes will be associated with rather large increases in credit spreads n 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 n Credit and equity risk should be modeled in an integrated fashion n Cholesky decomposition can be used to reflect the correlation between the random terms in credit and equity models n Transfer functions can be used to integrate credit and equity models

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