Risk Aggregation Inanoglu Jacobs 6 09 V1

1. Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital Hulusi Inanoglu and Michael Jacobs, Jr. Enterprise and Credit Risk…
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  • 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 June 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 <ul><li>Background and Motivation </li></ul><ul><li>Introduction and Conclusions </li></ul><ul><li>Review of the Literature </li></ul><ul><li>Methodology </li></ul><ul><li>Data and Summary Statistics </li></ul><ul><li>Empirical Results </li></ul><ul><li>Summary and Future Directions </li></ul>
  • 3. Background and Motivation <ul><li>Central challenge to enterprise risk measurement and management faced by diversified financial institutions: a coherent approach to aggregating different risk types. </li></ul><ul><li>Impetus from rapid financial innovation, evolving supervisory standards (Basel 2) and now recent financial crises </li></ul><ul><li>Main risks faced (market, credit and operational) have distinct distributional properties & historically modeled differently </li></ul><ul><li>Extend the scope of the analysis by analyzing A/L mismatch and liquidity risk (Pillar II of IRB framework implications) </li></ul><ul><li>Utilize actual data representative of major banking institutions’ loss experience (call reports) </li></ul><ul><ul><li>Explore effect of business mix & inter-risk correlations on total risk </li></ul></ul><ul><li>Apply copula methods for capturing realistic distributional features of & combining different risk types </li></ul><ul><ul><li>Compare different copula frameworks (including goodness-of-fit to the data) & evaluate sensitivity to sampling error </li></ul></ul>
  • 4. Background and Motivation (continued) <ul><li>ICAAP: Internal Capital Adequacy Assessment Process </li></ul><ul><li>Not a model for economic capital (EC), but a bank’s overall framework and mechanism for assessing if EC is appropriate </li></ul><ul><li>EC may be a quantitative component of ICAAP, but it is not required of all banks by supervisors (only the largest) </li></ul><ul><li>All banks must perform Stress Testing , which includes analysis around the impact on EC from the following: </li></ul><ul><ul><li>Scenario Analysis: extreme broad systematic events (or high quantiles of underlying risk factors) </li></ul></ul><ul><ul><li>Sensitivity Analysis: variation in key parameters due to sampling error or uncertainty or different specifications of the model </li></ul></ul><ul><li>The contribution of this work is in the latter, as we explore the variability of EC due to underlying statistical noise (sampling error) and to alternative models (specification of copula) </li></ul>
  • 5. Summary and Conclusions <ul><li>Estimated loss distributions for 5 largest banks as of 4Q08 (& Top 200) using quarterly Call Report data 1984-2008 </li></ul><ul><ul><li>Proxy for 5 risk types with financials: credit (GCO), operational (ONIE), market (4QDNTR), liquidity (4QDLGD) & interest income (4QDIG) </li></ul></ul><ul><ul><li>Different risk aggregation methodologies: historical bootstrap (empirical copula), Normal approx., Copulas (Gaussian,Student-t,Archimadean) </li></ul></ul><ul><li>Empirical copula (normal approx.) is found to be most (least) conservative (contrary to asymptotics) & most (least) stable in bootstrap experiment vs. standard copula methods </li></ul><ul><ul><li>But EC implies significantly greater proportional diversification benefits </li></ul></ul><ul><li>Document significant differences across banks & aggregation methodologies in absolute risk measures & diversification benefits (ranging 10% to 60%) </li></ul><ul><li>Simple addition over-states risk relative to standard copula formulations by about 30%-20% </li></ul>
  • 6. Summary and Conclusions (continued) <ul><li>Goodness-of-fits tests are mixed across copula models, but in many cases show evidence of poor fit to the data </li></ul><ul><li>Fail to find the effect of business mix to exert a directionally consistent an impact on total integrated diversification benefits </li></ul><ul><li>In a bootstrapping experiment, find the variability of the VaR to be significantly lower (higher) for the empirical & Gaussian copula than other formulations (Normal approximation) </li></ul><ul><li>Find that the contribution of the sampling error in the parameters of the marginal distributions to be an order or magnitude greater than that or the correlation matrices. </li></ul><ul><li>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 </li></ul><ul><li>. </li></ul>
  • 7. Review of the Literature <ul><li>Sklar (1956): mathematical foundation of copula methodology </li></ul><ul><ul><li>Existence of a copula to connect any set of marginal distributions </li></ul></ul><ul><li>Embrechts (1999, 2002): first applications to risk management </li></ul><ul><ul><li>Li (2000): credit risk management </li></ul></ul><ul><li>Frey & McNeil (2001): copulas as a generalization of dependence according to linear correlations </li></ul><ul><ul><li>Motivation for applying the technique to understanding tail events </li></ul></ul><ul><li>Poon (2001): alternative of a data intensive multivariate extension of extreme value theory (need joint tail events) </li></ul><ul><li>Most finance applications in portfolio risk measurement: Bouye (2001), Longin and Solnik (2001) and Glasserman et al (2002) </li></ul><ul><li>Embrechts et al (2003): reviews & extends recent results on distributional bounds for functions of dependent risks </li></ul><ul><ul><li>Main emphasis on Value-at-Risk as a risk measure </li></ul></ul><ul><li>Ward and Lee (2002): joint loss distributions (pair-wise roll-ups Gaussian copula marginal distributions) analytical & numerical </li></ul><ul><li>Kuritzkes et al. (2003): financial conglomerate & Gaussian copula for a large set of diversification results </li></ul>
  • 8. Review of the Literature (continued) <ul><li>Dimakos and Aas (2004): bank with life insurance subsidiary (risk = conditional marginal + unconditional credit risk) </li></ul><ul><ul><li>Imposing conditional independence through set of sufficient conditions such that only pair-wise dependence remains </li></ul></ul><ul><li>Schuermann & Rosenberg (2006): integrated risk management for typical large, internationally active financial institution </li></ul><ul><ul><li>Copula approach for aggregating 3 main risk types (market, credit & operational) where the distributional properties varies widely </li></ul></ul><ul><ul><li>Impact of business mix and inter-risk correlations on total risk: former found more important (“good news” for supervisors) </li></ul></ul><ul><ul><li>Compare various simplified approaches applied by practitioners (variance-covariance approach & regulatory addition approach) </li></ul></ul><ul><li>Aas (2007): incorporates ownership risk from a life insurance subsidiary & combines a base-& top-level aggregation </li></ul><ul><ul><li>Risk factors: multivariate GARCH model with Student-t errors </li></ul></ul><ul><ul><li>The model, originally developed DnB Nor is adapted to of Basel II </li></ul></ul><ul><li>Genest et al (2009): reviews literature on goodness-of-fit tests for copula models and proposes a “blanket” test with good size/power properties </li></ul>
  • 9. Methodology: Value-at-Risk <ul><li>Consider a single-valued function (simple sum of losses) of the risk factors (dollar losses: e.g., P&L, credit losses) from time t to t + Δ (Δ = horizon): </li></ul><ul><li>The Value-at-Risk at the confidence level α between times t and t + Δ (Δ is the horizon) is related to the α th quantile of F( π ( X)) as and denoted by: </li></ul><ul><li>Vector of K risk factors at time t having a joint distribution function. </li></ul><ul><li>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). </li></ul><ul><li>Therefore, we also look at the expected shortfall (ES), measuring expectation of the risk exposure conditional upon exceeding a VaR threshold: </li></ul>
  • 10. Value-at-Risk: The Variance-Covariance Approximation <ul><li>Note simply summing losses so no portfolio weights so that standard deviation of horizon losses is the root of the simple quadratic form: </li></ul><ul><li>Interesting & ubiquitous special case (motivated by Markowitz (1959) investment theory), seen in many practical EC frameworks (HSBC, 2008), where risk factors have a valid variance-covariance matrix & are multivariate Gaussian (or risk managers/investors do not care about moments > 2 nd ) : </li></ul><ul><li>Under the assumptions that minimizing the variance of the total loss is the object, NVaR (N=“normal”) is proportional to the standard deviation of the position according to the quantile of the standard normal distribution: </li></ul><ul><li>Case in which the standardized distribution of the positions is the same as that of the total loss yields “Hybrid Value-at-Risk” (HVaR) as follows </li></ul>
  • 11. Value-at-Risk: The Variance-Covariance Approximation (continued) <ul><li>The case in which we assume risk factors or losses to be perfectly correlated we call “Perfectly-correlated Value-at-Risk” (PVaR): </li></ul><ul><li>The case in which we assume risk factors or losses to be uncorrelated we call “Uncorrelated Value-at-Risk” (UVaR): </li></ul><ul><li>Obviously that in this framework and in a “mean-variance world”, PVaR (UVaR) forms an upper (lower) bound on the HVaR measure of risk: </li></ul>
  • 12. Methodology: The Method of Copulas <ul><li>If the joint distribution is continuously differentiable to the k th degree, that is sufficient for the copula to exist and be unique </li></ul><ul><li>Frechet-Hoeffding boundaries for copulas: minimum (maximum) copula, the case of perfect inverse (positive) dependence amongst random variables: </li></ul><ul><li>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. </li></ul><ul><li>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: . </li></ul><ul><li>Four technical conditions sufficient for a copula to exist (Nelson, 1999): </li></ul>
  • 13. Methodology: The Method of Copulas (continued) <ul><li>Note that P is not necessarily the correlation matrix of X , but in this context the Spearman rank-order correlations of the transformed variables (in cases of other copulas this may a different dependence measure of dependence) </li></ul><ul><li>While for a random vector having a valid joint distribution function the copula will always exist, there is no guarantee that it will be unique. </li></ul><ul><li>May always construct a copula for any multivariate distribution according to the method of inversion </li></ul><ul><ul><li>Intuitively: removing the effects of the marginal distributions on dependence relation by substituting in the marginal quantile functions in lieu of the arguments to the original distribution function </li></ul></ul><ul><li>If we have a random vector in the k th hyper-unit, them we may write the copula as a function as this as follows: </li></ul><ul><li>Consider a rather common choice of copula function, the Gaussian copula , simply a multivariate standard normal distribution with covariance matrix P : </li></ul>
  • 14. Methodology: The Method of Copulas (continued) <ul><li>Computationally equivalent to historical simulation method of simply resampling the observed history of joint losses with replacement (or bootstrapping) </li></ul><ul><ul><li>Historically, this was on of the standard method for computing VaR for trading positions amongst market risk department practitioners. </li></ul></ul><ul><li>Another commonly employed and closely related choice of copula in the elliptical family is the t-copula with degrees-of-freedom ν : </li></ul><ul><li>Often neglected but fundamental & interesting: empirical copula , a useful tool where there is high uncertainty on the underlying data distribution </li></ul><ul><ul><li>Procedure: transform the empirical data distribution into an &quot;empirical copula&quot; by warping such that the marginal distributions become uniform </li></ul></ul><ul><li>Mathematically the empirical copula frequency function has the following representation: </li></ul>
  • 15. Methodology: The Method of Copulas (continued) <ul><li>Where the generator function is indexed by a parameter θ , a whole family of copulas may be Archimedean, as in the Clayton copula : </li></ul><ul><li>An important class of copulas: Archimadean family , having simple forms with properties (e.g., associativity) & a variety of dependence structures </li></ul><ul><li>Unlike elliptical copulas, most have closed-form solutions and are not derived from the multivariate distribution functions using Sklar’s Theorem </li></ul><ul><li>One particularly simple form of k-dimensional Archimadean copula having generator function (satisfying certain conditions) : </li></ul><ul><li>Several special cases of note. In the product (independent) copula there is no dependence between variates (i.e., density function is unity everywhere): </li></ul><ul><li>Where parameter θ =0 we have the case of statistical independence </li></ul><ul><li>The Clayton copula exhibits negative tail dependence </li></ul>
  • 16. Methodology: The Method of Copulas (continued) <ul><li>Another commonly employed copulas in the Archimadean family include the Gumbel copula (having the property of positive tail dependence ): </li></ul><ul><li>Finally, we consider the Frank copula (having the property of neither positive nor negative tail dependence ): </li></ul><ul><li>We may simulate realizations from a multivariate distribution by generating independent random vectors </li></ul><ul><li>For example, in the Gaussian case, it is either standard normal and independent random variables that we generate </li></ul><ul><li>With knowledge of the marginal distributions of the risk factors (which can be estimated either parametrically or non-parametrically), we can derive a rank-order correlation matrix of the transformed marginal data </li></ul><ul><li>We can make our independent random vectors correlated (by means of a Cholesky decomposition, for instance) </li></ul>
  • 17. Data Description <ul><li>Quarterly call report data for top 200 banks 1Q84-4Q08 </li></ul><ul><ul><li>Corrected for mergers & acquisitions: legacy banks synthetically added into currently surviving banks on pro forma basis </li></ul></ul><ul><li>Proxy for 5 risk types using financial statement data </li></ul><ul><li>Credit Risk (CR): gross charge-offs (“GCO”) </li></ul><ul><li>Operational Risk (OR): total other non-interest expense (“ONIE”) </li></ul><ul><li>Market Risk (MR): (minus of) net trading revenues deviation from moving 4 quarter moving average (“NTR-4QD”) </li></ul><ul><li>Liquidity Risk (LR): liquidity gap (total loans minus total deposits) deviation from 4 quarter moving average (“LR-4QD”) </li></ul><ul><li>Interest Rate Risk (IR): interest rate gap (interest expense on deposits minus interest income on loans) deviation from 4 quarter moving average (“IRG-4QD”) </li></ul>
  • 18. Empirical Results: Summary Statistics (Call Report Data)
  • 19. Empirical Results: Summary Statistics (Call Reports & CRSP)
  • 20. Empirical Results: Summary Statistics (Call Report Variables)
  • 21. Empirical Results: Summary Statistics (Call Report & CRSP)
  • 22. Empirical Results: Summary Statistics (Risk Proxies)
  • 23. Historical Quarterly Risk Proxies: Loss Distributions (1984-2008)
  • 24. Historical Quarterly Risk Proxies: Loss Distributions (Top 200 Banks 1984-2008)
  • 25. Historical Quarterly Risk Proxies: Time Series (1984-2008)
  • 26. Historical Quarterly Risk Proxies: Time Series (Top 200 Banks 1984-2008)
  • 27. Pairwise Correlations: Pearson vs. Spearman (5 Risk Types 1984-2008)
  • 28. Pairwise Correlations, Scatters & Histograms (5 Risk Types 1984-2008)
  • 29. Pairwise Correlations, Scatters & Histograms: 5 Risk Types (Top 200 Banks 1984-2008)
  • 30. Spearman Correlations: 5 Risk Types Transformed Data
  • 31. Spearman Correlations: 5 Risk Types Transformed Data (Top 200 Banks)
  • 32. Dependograms of Multivaria
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