Modeling and Calibration Errors in Measures of Portfolio Credit Risk

1. Modeling and Calibration Errors in Measures of Portfolio Credit Risk Nikola Tarashev and Haibin Zhu Bank for International Settlements April 2007 The views expressed in this paper need not represent those of the BIS.

2. Motivation • ASRF model: a well-known model of portfolio credit risk • Main reason for the model’s popularity: “portfolio invariance” of capital • Assumption 1: perfect granularity of the portfolio • Assumption 2: single common factor of credit risk • Does ASRF allow for “bottom-up” calculation of economic capital ? Not really: estimating ρirequires a global approach !!! • To take full advantage of portfolio invariance  off-the-shelf values forρi: Pillar 1 of Basel II: ρi = ρIRB( PDi) • ASRF-based capital measures are subject to: • misspecification errors: ie, violated assumptions of the model • calibration errors: eg, off-the-shelf values for ρi.

3. Related literature (BCBS WP No 15) • Misspecification of the ASRF model: • Granularity assumption: • Martin and Wilde (2002), Vasicek (2002) Emmer and Tasche (2003), Gordy and Luetkebohmert (2006) • Sector concentration and the number of common factors • Pykhtin (2004), Duellmann (2006), Garcia Cespedes et al (2006) Duellmann and Masschelein (2006) • Both assumptions: • Heitfield et al (2006), Duellman et al (2006) • Flawed calibration: • Loeffler (2003), Morinaga and Shiina (2005)

4. This paper Decomposes the wedge between target and shortcut capital measures. Exhaustive and non-overlapping components due to: • Misspecification of the ASRF model • Multifactor effect • Granularity effect • Flawed calibration the ASRF model • Correlation dispersion effect • Correlation level effect • Plausible estimation errors Takes seriously the overall distribution of risk factors • Gaussian versus fatter-tailed distributions The paper does not: (i) Discuss calibration of PDs or LGDs; (ii) address time variation in risk parameters

5. Main results • Applying ASRF model  Large deviations from target capital for realistic bank portfolios • The main drivers of these deviations are calibration errors • noise in the estimated value of ρi • wrong distributional assumptions • Misspecification of the ASRF model has a much smaller impact • exception: granularity effect in small portfolios

6. Roadmap • The ASRF model (overview) • Alternative sources of error in capital measures (intuition) • Empirical methodology • Findings

7. The ASRF model (an overview) • PDi, LGDi, ρi and weights wi are all known • Assets (driven by a single common factor) drive defaults • M ~ F (0,1), Z ~ G (0,1), V ~ H (0,1): weak restrictions • Perfectly fine granularity: Idiosyncratic risk is diversified away (lowwi & many exposures) • Pairwise correlations: ρi ρk

8. Portfolio invariance • If V, M, Z are all normal and α = 0.001: which underpins the IRB approach of Basel II .999 The ASRF model (implications) • To attain solvency with probability (1- α):

9. target capital ASRF capital Errors in calculated capital charges • The granularity effect (specification error 1) • Stylized homogeneous portfolio: PD=1%, LGD=45%, ρ2 = 10% • Granularity  ASRF capital undershoots target capital

10. Errors in calculated capital charges (cont’d) • The multi-factor effect (specification error 2) • Stylized homogeneous portfolio: PD=1%, LGD=45% • Two sectors, weight = ω. Intra-sector correlation = 0.2; inter-sector = 0 • Owing to its single-CF structure,ASRF model undershoots target

11. Errors in calculated capital charges (cont’d) • Correlation level effect (calibration error 1) • Stylized homogeneous portfolio: PD=1%, LGD=45% • Higher correlation raises the capital measure

12. Errors in calculated capital charges (cont’d) • Correlation dispersion effect (calibration error 2) • Start with a homogeneous portfolio: PD=1%, LGD=45% ρvaries within • Then, let PDs vary too, within [0.5%,1.5%]

13. Methodology. Decomposing the gap • Select a portfolio: • Realistic distribution of exposures across industrial sectors • representative portfolio of large US banks, Heitfield et al. (2006) • Size • Large: 1000 exposures (homogeneous weights) • Small: 200 exposures (homogeneous weights) • For each portfolio, quantify the difference between two extremes: • Target capital • Shortcut capital, implied by off-the-shelf calibration of ASRF model • Three additional capital calculations bring up the (i) multifactor (ii) granularity (iii) correlation-level (iv) correlation-dispersion effects

14. Target capital N firms, (σ) +Monte Carlo simulations Multi-factor effect N firms, R() (1-factor best fit)+copula Shortcut capital N = , +ASRF model Methodology: decomposing the four effects For all measures: homogeneous portfolios, the same {PDi}iN and LGD, Gaussian distributions

15. Fitting a single-factor structure • R() is the “one-factor approximation” of (σ) • Loadings (i): allowed to differ across exposures • Approximation matches well average correlation • Approximation could miss correlation dispersion

16. Target capital N firms, (σ) +Monte Carlo simulations Multi-factor effect N firms, R() (one-factor best fit)+copula Granularity effect N = ,R() +ASRF model Correlation dispersion effect N = , average  +ASRF model Correlation level effect Shortcut capital N = , +ASRF model Methodology: decomposing the four effects For all measures: homogeneous portfolios, the same {PDi}iN and LGD, Gaussian distributions

17. Data • 10,891 non-financial firms worldwide: • 40 industrial sectors • mostly unrated firms • Risk parameters: • from Moody’s KMV, • for July 2006 • two sets of mutually consistent estimates: • EDF: 1-year physical PD, at exposure level • Global Correlation (GCORR) asset return correlations Estimated based on a multi-factor loading structure • LGD = 45%

18. Simulated Portfolios • Match sectoral distribution of the representative portfolio of US wholesale banks. Heitfield et al. (2006) • Two sizes: large (1000 exposures) and small (200 exposures) • 3000 simulations (for each size): to average out sampling errors

19. Findings • Dissecting deviations from target capital:

20. Findings, continued • For small portfolios: similar results, except for the granularity effect add-on: large -3.73%; small -15.8%

21. Findings, continued • Discrepancy: shortcut minus target capital Explaining the variation of the discrepancy across simulated portfolios

22. Delving into calibration errors • Small-sample estimation errors in calibrated correlations • Conduct the following exercise • Design true model: • 2= 9.78%, PD = 1%, Gaussian variables, N equal exposures • Draw from the true model: T periods of asset returns • Adopt the point of view of a user (has data, estimates parameters) • Construct sample correlation matrix (PD, etc: known) • Fit a one-factor model and calculate ASRF-implied capital • Repeat 1000 times for each (N,T)

23. Delving into calibration errors, contd. • Small-sample estimation errors are significant • Meaningful reduction of errors requires unrealistic sample sizes

24. Delving into calibration errors, contd. • Estimation errors affect substantially capital charges • Bias • Noise

25. The importance of distributional assumptions • Stylized fact: fat tails of asset returns • Gaussian (thin tails) assumption  too low capital measures • Quantify the bias: • Student t distributions • General ASRF formula:

26. Alternative distributions • Gaussian assumption  negative bias in measured capital

27. Conclusions • Large deviations from target capital for realistic portfolios • The main drivers of the deviations: • calibration errors • notmodel misspecification • Challenges for risk managers and supervisors