Firm Heterogeneity and Credit Risk Diversification

1. Firm Heterogeneity and Credit Risk Diversification Conference on Financial EconometricsYork, UK, June 2-3, 2006 * Any views expressed represent those of the authors only and not necessarily those of the Federal Reserve Bank of New York or the Federal Reserve System.

2. We are primarily interested in generating (conditional) credit portfolio loss distributions = 100% Credit portfolio loss distributions

3. Obtaining credit loss distributions • Credit loss distributions tend to be highly non-normal • Skewed and fat-tailed • Even if underlying stochastic process is Gaussian • Non-normality due to nonlinearity introduced via the default process • Typical computational approach is through simulation for a variety of modeling approaches • Merton-style model • Actuarial model • Closed form solutions, desired by industry & regulators, are often obtained assuming strict homogeneity (in addition to distributional) assumptions • Basel 2 Capital Accord • What are the implications of imposing such homogeneity -- or neglecting heterogeneity -- for credit risk analysis?

4. Credit risk modeling literature • Contingent claim (options) approach (Merton 1974) • Model of firm and default process • KMV (Vasicek 1987, 2002) • CreditMetrics: Gupton, Finger and Bhatia (1997) • Vasicek’s (1987) formulation forms the basis of the New Basel Accord • It is, however, highly restrictive as it imposes a number of homogeneity assumptions • A separate and growing literature on correlated default intensities • Schönbucher (1998), Duffie and Singleton (1999), Duffie and Gârleanu (2001), Duffie, Saita and Wang (2006) • Default contagion models • Davis and Lo (2001), Giesecke and Weber (2004)

5. Preview of results • Our theoretical results suggest: • Neglecting parameter heterogeneity can lead to underestimation of expected losses (EL) • Once EL is controlled for, such neglect can lead to overestimation of unexpected losses (UL or VaR) • Empirical study confirms theoretical findings • Large, two-country (Japan, U.S.) portfolio • Credit rating information (unconditional default risk: p) very important • Return specification important (conditional independence) • Under certain simplifying assumptions on the joint parameter distribution, we can allow for heterogeneity with minimal data requirements

6. Our basic multi-factor firm return process t denotes the information available at time t • Firm default condition Firm returns and default: multi-factor • Note that the multi-factor nature of the process matters only when the factor loadings diare heterogeneous across firms

7. Parameter heterogeneity can be introduced through the standard random coefficient model where vi is independent of ft+1 and et+1 Introducing parameter heterogeneity: random • Parameter heterogeneity is a population property and prevails even in the absence of estimation uncertainty • Could be the case for middle market & small business lending where it would be very hard to get estimates of i • Use estimates from elsewhere for and vv

8. For simplicity, consider single factor model • EL for Vasicek fully homogeneous case Note: • Heterogeneity is introduced through ai Can be thought of as heterogeneity in default thresholds and/or expected returns a < 0 Introducing simple heterogeneity: random

9. Can also be obtained from Jensen’s inequality since for EL  under parameter heterogeneity • Now we can compute portfolio expected loss (recall a < 0 typically) • Neglecting this source of heterogeneity results in underestimation of EL

10. Systematic and random heterogeneity • Impact on loss variance under random heterogeneity is ambiguous • EL not constant • It helps to control for/fix EL • Can only be done by introducing some systematic heterogeneity, e.g. firm types • E.g. 2 types, H, L, such that pL < pH < ½ • Calibrate exposures to types such that EL is same as in homogeneous case (need NH, NL→ )

11. Holding EL fixed • Loss variance under homogeneity Systematic and random heterogeneity

12. Loss variance under heterogeneity Loss variance (UL)  under parameter heterogeneity, for a given EL • Theorem 1:Vhom > Vhet , assuming ELhom = ELhet • Neglecting this source of heterogeneity results in overestimation of loss variance

13. Since • Under • Concavity: Vhom > Vhet • Proof draws on concavity of F(p, p, r)

14. Loss variance (UL)  under parameter heterogeneity, for a given EL • Holding EL fixed, neglecting parameter heterogeneity results in the overestimation of risk • Intuition: parameter heterogeneity across firms increases the scope for diversification • Relies on concavity of loss distribution in its arguments • Easily extended to many types, e.g. several credit ratings

15. Empirical application • Two countries, U.S. and Japan, quarterly equity returns, about 600 U.S. and 220 Japanese firms • 10-year rolling window estimates of return specifications and average default probabilities by credit grade • First window: 1988-1997 • Last window: 1993-2002 • Then simulate loss distribution for the 11th year • Out-of-sample • 6 one-year periods: 1998-2003 • To be in a sample window, a firm needs • 40 consecutive quarters of data • A credit rating from Moody’s or S&P at end of period

16. Merton default model in practice • Approach in the literature has been to work with market and balance sheet data (e.g. KMV) • Compute default threshold using value of liabilities from balance sheet • Using book leverage and equity volatility, impute asset volatility • We use credit ratings in addition to market (equity) returns • Derive default threshold from credit ratings (and thus incorporate private information available to rating agencies) • Changes in firm characteristics (e.g. leverage) are reflected in credit ratings • We use arguably the two best information sources available • Market • Rating agency

17. Modeling conditional independence • The basic factor set-up of firm returns assumes that, conditional on the systematic risk factors, firm returns are independent • A measure of conditional independence could be the (average) pair-wise cross-sectional correlation of residuals (in-sample) • Similarly, we can measure degree of unconditional dependence in the portfolio • (average) pair-wise cross-sectional correlation of returns (in-sample) • Broadly, a model is preferred if it is “closer” to conditional independence

18. Model specifications

19. Modeling conditional independence: results

20. Impact of heterogeneity: asymptotic portfolio • Calibrate using simple 1-factor (CAPM) model • Compare Vasicek (homogeneity), Vasicek + rating (heterog. in default threshold/unconditional p)

21. Finite-sample/empirical loss distribution (2003)

22. Impact of heterogeneity: finite-sample portfolio • Include multi-factor models • Conditional independence?

23. Calibrated asymptotic loss distribution (2003)

24. Finite-sample/empirical loss distribution (2003)

25. Concluding remarks • Firm typing/grouping along unconditional probability of default (PD) seems very important • Can be achieved using credit ratings (external or internal) • Within types, further differentiation using return parameter heterogeneity can matter • Neglecting parameter heterogeneity can lead to underestimation of expected losses (EL) • Once EL is controlled for, such neglect can lead to overestimation of unexpected losses (UL or VaR) • Well-specified return regression allows one to comfortably impose conditional independence assumption required by credit models • In-sample easily measured using correlation of residuals • Measuring and evaluating out-of-sample conditional dependence requires further investigation

26. Thank You! http://www.econ.cam.ac.uk/faculty/pesaran/

27. Graveyard

28. Vasicek (1987) among first to propose portfolio solution • Loans are tied together via a single, unobserved systematic risk factor (“economic index”) f and same correlation r Portfolio loss in Vasicek model • Then, as N , the loss distribution converges to a distribution which depends on just p and r • These two parameters drive the shape of the loss distribution • With equi-correlation and same probability of default, default thresholds are also the same for all firms

29. Our contribution: conditional modeling and heterogeneity • The loss distributions discussed in the literature typically do not explicitly allow for the effects of macroeconomic variables on losses. They are unconditional models. • Exception: Wilson (1997), Duffie, Saita and Wang (2006) • In Pesaran, Schuermann, Treutler and Weiner (JMCB, forthcoming) we develop a credit risk model conditional on observable, global macroeconomic risk factors • In this paper we de-couple credit risk from business cycle variables but allow for • Different unconditional probability of default (by rating) • Different systematic risk sensitivity across firms (“beta”) • Different error variances across firms

30. Introducing heterogeneity • Allowing for firm heterogeneity is important • Firm values are subject to specific persistent effects • Firm values respond differently to changes in risk factors (“betas” differ across firms) • Note this is different from uncertainty in the parameter estimate • Default thresholds need not be the same across firms • Capital structure, industry effects, mgmt quality • But it [heterogeneity] gives rise to an identification problem • Direct observations of firm-specific default probabilities are not possible • Classification of firms into types or homogeneous groups would be needed • In our work we argue in favor of grouping of firms by their credit rating: pR

31. p-H p* p p-L DD-L DD-H DD EL is under-estimated