Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital

1. Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital Hulusi Inanoglu and Michael Jacobs, Jr. Enterprise and Credit Risk Analysis Divisions U.S. Office of the Comptroller of the Currency Presentation to PRMIA / CIRANO Luncheon, Montreal, Quebec November 2009 The views expressed herein are those of the authors and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury.

2. Outline • Background and Motivation • Introduction and Conclusions • Review of the Literature • Methodology • Data and Summary Statistics • Empirical Results • Summary of Findings

3. Background and Motivation Central challenge to enterprise risk measurement and management faced by diversified financial institutions: a coherent approach to aggregating different risk types • Impetus from rapid financial innovation, evolving supervisory standards (Basel 2) and now recent financial crises • Main risks faced (market, credit and operational) have distinct distributional properties & historically modeled differently • Extend the scope of the analysis by analyzing interest rate (income) and liquidity risk (Pillar II of IRB framework implications) • Utilize actual data representative of major banking institutions’ loss experience (Call Reports) • Explore effect of business mix & inter-risk correlations on total risk • Apply copula methods for capturing realistic distributional features of & combining different risk types • Compare different copula frameworks (including goodness-of-fit to the data) & evaluate sensitivity to sampling error

4. Background and Motivation (continued) ICAAP: Internal Capital Adequacy Assessment Process Not a model for economic capital (EC), but a bank’s overall framework and mechanism for assessing if EC is appropriate EC may be a quantitative component of ICAAP, but it is not required of all banks by supervisors (only the largest) All banks must perform Stress Testing, which includes analysis around the impact on EC from the following: Scenario Analysis: extreme broad systematic events (or high quantiles of underlying risk factors) Sensitivity Analysis: variation in key parameters due to sampling error or different specifications of the model The contribution of this work is in the latter, as we explore the variability of EC due to underlying statistical noise (sampling error) or to alternative models (specification of copula or method of aggregating risks)

5. Summary and Conclusions • Estimated loss distributions for 5 largest banks as of 4Q08 (& Top 200) using quarterly Call Report data 1984-2008 • Proxy for 5 risk types with financials: credit, operational, market, liquidity & interest income • Compare different risk aggregation methodologies in magnitude & stability of risk measures (VaR) • Empirical copula simulation-ECS, normal approximation-VCA, Gaussian copulas-GCS, Student T-TCS & Archimedean copulas (GCS,CCS&FCS) • Empirical copula simulation-ECS & AGCS (VCA) is found to be more (least) conservative & more (least) stable in bootstrap experiment vs. standard copula methods • But ECS implies significantly greater proportional diversification benefits • Result on magnitude of risk measure for ECS is contrary to asymptotics • Document significant differences across banks & aggregation methodologies in absolute risk measures & proportional diversification benefits (ranging 10% to 60%)

6. Summary and Conclusions (continued) Simple addition over-states risk relative to standard formulations (including copula methods) by about 20%-30% Goodness-of-fits tests are mixed across copula models, but in many cases show evidence of poor fit to the data Fail to find the effect of business mix to exert a directionally consistent impact on total integrated risk or proportional diversification benefits In a bootstrapping experiment, find the variability of the VaR to be on the low side (high) for the ECS & Gaussian copula than other formulations (Normal approximation) Find that the contribution of the sampling error in the parameters of the marginal distributions to be an order or magnitude greater than that of the correlations The beauty of ECS: no need for estimating marginal distributions! Results constitute a sensitivity analysis that argues for practitioners to err on the side of conservatism in considering a non-parametric EC approach to quantify integrated risk

7. Review of the Literature • Sklar (1956): mathematical foundation of copula methodology • Embrechts (1999, 2002): first application of copulas to risk management • Li (2000): credit risk management • Frey & McNeil (2001): model copulas as a generalization of dependence according to linear correlations • Poon (2001): alternative of a data intensive multivariate extension of extreme value theory (need joint tail events) • Most finance applications in portfolio risk measurement: Bouye (2001), Longin and Solnik (2001) and Glasserman et al (2002) • Dimakos and Aas (2004): model a bank with life insurance subsidiary • Schuermann & Rosenberg (2006): integrated risk measurement for typical large, internationally active financial institution

8. Methodology: Value-at-Risk • The Value-at-Risk at a certain confidence level is closely related to (the complement of or one minus) the quantile function of a distribution with respect to a set of risk factors • In layman’s terms, this measures risk as an amount that we should loose more than with only a very small probability • It is well-known that there are serious issues with VaR: • Coherence (Artzner 1997, 1999) • Loss of information vs. focusing on entire distribution (Diebold et al,1998; Christoffersen and Diebold, 2000; Berkowitz, 2001) • Possibility for unbounded concentration risk & “gaming” (Embrechts et al. 1999, 2002) • Also look the expected shortfall (ES), measuring expectation of the risk exposure conditional upon exceeding a VaR threshold • ES is indeed coherent • But issue of determining the threshold

9. Methodology: The Method of Copulas • Fundamental result (Sklar, 1956): under the appropriate & general mathematical regularity conditions any joint distribution can be expressed in terms of a copula (or dependence) function & set of marginal distributions • If we have a K-vector of risk factors, then a copula is a multivariate joint distribution defined on the K-dimensional unit cube, such that each marginal distribution is uniformly distributed on the unit interval • Nelson (1999): there are a technical conditions sufficient for a copula to even exist or be unique, but a the joint distribution is continuously differentiable to the kth degree is sufficient for both • There are Frechet-Hoeffding boundaries for copulas: minimum (maximum) copula, the case of perfect inverse (positive) dependence amongst random variables

10. Methodology: The Method of Copulas (continued) • May always compute a copula by the “method of inversion”: intuitively, “removing” the effects of the marginal distributions on dependence the relation • Substituting in the marginal quantile functions in lieu of the arguments to the original distribution function • Commonly choices for the copula are Gaussian or Student-t – these are highly tractable, being from the elliptical family of distributions • But does not exhibit the desirable feature of tail dependence • Important class of copulas: the Archimedean family, with simple forms, “nice” properties (e.g., associativity) & a variety of dependence structures • Unlike elliptical copulas, most have closed-form solutions and are not derived from the multivariate distribution functions using Sklar’s Theorem

11. Methodology: The Method of Copulas (continued) • A commonly employed copula in the Archimedean family is the Gumbel copula, having the property of positive tail dependence • Considered by Gumbel (1960) in the context of extreme value theory: • The Clayton copula family exhibits negative tail dependence • Related to the gamma frailty models of survival analysis (Clayton, 1978) • The Frank copula: neither positive nor negative tail dependence (Nelsen, 1986) • Often neglected but fundamental & interesting: empirical copula • A useful tool where there is high uncertainty on the underlying data distribution • Procedure: transform the empirical data distribution into an "empirical copula" by warping such that the marginal distributions become uniform • Computationally equivalent to historical simulation method of simply resampling the observed history of joint losses with replacement (or bootstrapping) • Historically, this was on of the standard method for computing VaR for trading positions amongst market risk department practitioners.

12. Data Description • Quarterly call report data for top 200 banks 1Q84-4Q08 • Corrected for mergers & acquisitions: legacy banks synthetically added into currently surviving banks on pro forma basis • Proxy for 5 risk types using financial statement data • Credit Risk (CR): gross charge-offs (“GCO”) • Operational Risk (OR): total other non-interest expense (“ONIE”) • Market Risk (MR): (minus of) net trading revenues deviation from moving 4 quarter moving average (“NTR-4QD”) • Liquidity Risk (LR): liquidity gap (total loans minus total deposits) deviation from 4 quarter moving average (“LR-4QD”) • Interest (Income) Risk (IR): interest rate gap (interest expense on deposits minus interest income on loans) deviation from 4 quarter moving average (“IRG-4QD”) • The latter 3 (MR, LR & IR) follow from Jorion (2001)

13. Empirical Results: Summary Statistics (Call Report Variables)

14. Empirical Results: Summary Statistics (Call Report & CRSP)

15. Historical Quarterly Risk Proxies: Loss Distributions (Top 200 Banks 1984-2008)

16. Historical Quarterly Risk Proxies: Time Series (Top 200 Banks 1984-2008)

17. Pairwise Correlations, Scatters & Histograms: 5 Risk Types (Top 200 Banks 1984-2008)

18. Dependogram of Multivariate Groupwise Independence Tests – Top 200 Banks

19. 99.97th Percentile Dollar VaR Across Banks and Methodologies

20. 99.97th Percentile VaR/BVA Across Banks and Methodologies

21. 99.97th Percentile VaR Diversification Benefit Across Banks and Methodologies

22. Genest et al (2009) Copula Goodness-of-Fit Test P-values Across Banks and Methodologies

23. Discussion of 99.97th Percentile VaR and % Diversification Benefit & GOF Tests of Model Fit • Dollar VaR (VaR/BVA) increases montonically (generally decreases) with size of the institution • VCA (ECS or the AGCS) produces consistently the lowest (highest) VaR; TCS in follows in conservativeness, while the GCS & AFCS (ACCS) is usually toward the middle (low side) • ECS yields highest PDBs (127%-252%) vs. other models (10%-50%); GCS tends to lie in the middle (41-58%), VCA to the lower end (31-41%) & GCAS is the lowest (10-21%) • No directionally consistent pattern in VaR/PDB across different business mixes (i.e., higher % trading vs. lending assets) • GOF tests highly mixed (reject null 14/30 cases), no pattern, not at very high levels of significance->models generally OK?

24. Bootstrap of Correlations: 99.97th Percentile Dollar VaR Across Banks and Methodologies

25. Bootstrap of Margins: 99.97th Percentile Dollar VaR Across Banks and Methodologies

26. Bootstrap of Correlations: 99.97th Perc. VaR % Diversification Benefit Across Banks and Methodologies

27. Bootstrap of Margins: 99.97th Perc. VaR % Diversification Benefit Across Banks and Methodologies

28. Discussion: Bootstrap of 99.97th Perc. VaR & % Diver. Benefit Across Banks and Methodologies • We fail to observe a consistent pattern in the variability of VaR or PDB across size or types of banks (i.e., business mix). • Bootstrap of correlations for VaR (PDB): GCS (ECS) is lowest, ECS (GCS) follows closely & VCA yields highest NCVs • Bootstrap of margins for VaR or PDB: ECS (CVA) universally lower (higher) than copulas • But excepting that GCS lower NCV than TCS & close to AGCS • Across models or banks NCVs are an order of magnitude higher for the resampling of margins vs. correlations • This difference is accentuated for VaR vs. PDB. • NCVs higher for the PDB vs. VaR: excluding ECS, NCVs for VaR in bootstrap of correlations (margins) range in 5.9%-32.8% (25.2%-56.1%); resp. PDB #s:17.3%-158.2% (19.9%-83.9%)

29. Summary of Contributions and Major Findings We have compared alternative risk aggregation methodologies used in practice: the VCA, well-known GCS & less-well known ECS First major exercise involved fitting the models, describing & comparing VaR & PDBs across banks & models, GOF statistics The second part involved measuring the statistical variation in VaR & PBD Across models & banks ECS & AGCS produce the highest absolute magnitudes of VaR vs. either GCS, STCS or other Archimadean copulas ECS – a variant of legacy “historical simulation” in market risk practice – in many cases the most conservative (a surprise according to asymptotic theory) VCA consistently produces the lowest VaR number (disturbing in that several practitioners for the lack of theory or supervisory guidance adopt this But PDB tended to be largest for the ECS vs.GCS or VCA, while AGCS produces the lowest, a point of caution if banks choose this route Failed to find business mix to exert a directionally consistent impact on risk

30. Summary of Contributions and Major Findings (continued) In application of a blanket GOF tests (Genest et al, 2009) found mixed results: about ½ the cases parametric copula models fail to fit the data, confidence levels tended to be modest, so clearly needs more study Bootstrapping experiment revealed the variability of the VaR or PDB itself to be significantly lowest (highest) for the ECS (VCA) relative to other copulas Contribution of the sampling error in the parameters of the marginal distributions is an order of magnitude greater than the correlations Results constituted a sensitivity analysis that argues for practitioners to err on the side of conservatism in considering a non-parametric copula alternative in order to quantify integrated risk Standard copula formulations produced a wide divergence in measured VaR, diversification benefits as well as the sampling variation in both of these across different measurement frameworks and types of institutions