Correlation and Credit Risk presentation to CAS / SOA ERM Symposium July 30, 2003

1. Correlation and Credit Riskpresentation to CAS / SOA ERM Symposium July 30, 2003 John A. Dodson / American Express Financial Advisors / john.a.dodson@aexp.com

2. Portfolio Credit Risk • Generally, asset portfolio gains / losses related to credit rating migration and impairment through default over time • Focus here on distribution of cumulative aggregate default losses over a single fixed horizon (E.g. 99 % upper confidence over 1 year) • Expected Loss (EL) and Unexpected Loss (UL) • Unexpected Loss can be a basis for Economic Capital

3. Note: “Coherent” Measures of Risk • Paradox of Default Value-at-Risk (CVaR): diversification may increase CVaR. • VaR is not “coherent” (Artzner): monotonic, translation-invariant, positive-homogenous, sub-additive • Expected Shortfall (ES) over a single horizon is coherent – multi-horizon coherent measure remains an open problem…

4. Unexpected Loss Measures • Value-at-Risk • Expected Shortfall N.B.: For a normal distribution, these converge for high confidence

5. Credit Default Models • Intensity Models: default rates processes are continuous • Duffie-Singleton 1996 (including RMG CreditMetrics™) • Defaults are “inaccessible” • Correlations are based on empirical factor analysis • Structural Models: asset/liab. processes are continuous • Merton 1974 (including KMV & RMG CreditGrades™) • Default are “accessible” • Correlations come through equity prices N.B.: There has been recent work to unify these

6. Intensity Models for Default • Hazard rates for each obligor are positive, continuous stochastic processes • Default occurs at a stopping time defined by • Default correlations in this context are interpreted to mean correlations amongst intensity innovations[exercise: prove these correlations carry over to the default indicator processes.] N.B.: Note that this model is amenable in the actuarial context to multi-factor CIR

7. Default Intensity Correlations • Drift and Volatility are deterministic functions • K (imperfectly) correlated, systematic factors (contagions) • l’s are entries of Cholesky matrix L, s.t. • Elements of Rho are factor correlations

8. Special Cases and Extensions • Note that if K=0 (no correlations), it can be proven that the expectations of each default probability are sufficient statistics for the aggregate cumulative loss distribution over a single horizon; that is, UL from default is independent of UL from migration (as driven by the stochastic intensities) • Also note that the loss distribution is guaranteed to be multi-normal for deterministic volatilities. Stochastic volatility is the gateway to the general copula approach

9. Effective Diversity • Consider the case of a portfolio of N equal holdings of independent defaultable assets with equal expected default frequencies and equal default severities. • The single horizon cumulative aggregate loss is simply proportional to a Binomial(N, p) random variable • The UL in this case (by whatever measure) as a proportion of the portfolio value serves as a benchmark indicating effective diversityN

10. Example • Consider a portfolio with individual loss distributions independent but not all equal; in particular consider a geometric scheme: • In this case, in the limit as the number of obligors goes to infinity (Central Limit Thm), one can show that the effective diversity is

11. Obligor-Specific Risk • Consider the case of K=1 (one systematic factor) • The correlation between the intensities of two obligors is simply the product of the two systematic correlations • Hypothesis: systematic correlation should be an increasing function of firm size (E.g., systemic risks)

12. Effective Diversity & Specific Risk • In the event that the systematic correlations are taken to be identical for all obligors, the general effect this correlation has is to reduce the effective diversity • For the benchmark case this effect is • So for K independent systematic factors,

13. Heuristic Observations • Consider again the final heuristic result: • The effective value of K is driven to some extent by the correlations amongst the systematic factors; but the obligor-specific risk (as determined by the systematic correlation here) is the dominant factor • We know from simulation experiments that this is especially true at high confidence • Unfortunately, determination of appropriate models of obligor-specific risk is largely speculation

14. Empirical Observations • If we assume that the factors driving default intensity are stationary, we can impute effective diversity from the variability of historical default rates • Using Moody’s data (mostly U. S.) from 1970, these are • N≈ 494 ± 147 for high grade issuers • N≈ 109 ± 16 for all issuers • N≈ 43 ± 5 for high yield issuers • But how can effective diversity depend on quality?

15. Conclusions • How can effective diversity depend on credit quality? • The modeling choices are stark: We can either allow factor loadings to depend explicitly on intensity levels, rendering the problem intractable… …or we can acknowledge stochastic factor volatilities and enter the brave new world of copulas.

16. Other Topics… • Valuation and Risk Measurement of Structured Credit Securities using copulas (multi-period analysis) • Panjer’s Recursion and CSFB CreditRisk+ analytic model • Glasserman’s variance reduction techniques for Monte Carlo simulation of correlated defaults