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

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## Abstract

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## INTRODUCTION AND SUMMARY

^{st}quarter of 1984. The rationale for concentrating on large banks is motivated in large part by the intense policy debate surrounding the New Basel Capital Accord (BCBS, 2004). The most recent incarnation of Basel incorporates operational risk, a new risk type to the regulatory calculation, which differs substantially in distributional characteristics from market and credit risk. The importance of this is highlighted by the conclusions of the BCBS Joint Forum (2001, 2003), which highlights quite clearly how banks and insurers are actively wrestling with this. While the focus herein is upon the banking sector, our methodology could just as easily extend any other kind of financial conglomerate such as an insurer.

^{th}percentile Value-at-Risk (VaR) is increasing in size of institution, but expressed as a proportion of book value it appears to be decreasing in size of the entity. Across different risk aggregation methodologies and banks we observe that the empirical copula simulation (ECS) and Archimadean-Gumbel copula simulations (AGCS) to produce the highest absolute magnitudes of VaR as compared to the Gaussian copula simulation (GCS), Student-T copula simulation (STCS) or any of the other Archimadean copulas. The variance-covariance approximation (VCA) produces the lowest VaR. The proportional diversification benefits, as measured by the relative VaR reduction vis a vis the assumption of perfect correlation, exhibit radical variation across banks and aggregation techniques. The ECS generally yields the highest values than the other methodologies (127% to 243%), the GCS “benchmark” (41-58%) and VCA (31-40%) toward the middle to lower end of the range, while the AGCS is the lowest (10-21%). We conclude that while ECS (VCA) may over-state (under-state) absolute (relative) risk, on the order of about 20% to 30% across all banks, proportional diversification benefits are generally understated (overstated) by the VCA (ECS) relative to standard copula formulations on the order of about 15% to 30% (3 to 6) across all banks and frameworks, respectively. Through differences observed across the five largest banks, we fail to find business mix10 to exert a directionally consistent an impact on total integrated risk or proportional diversification benefits above and beyond exposure to, and correlation amongst, underlying risk factors. In an application of the goodness-of-fit tests for copula models, developed by Genest et al (2009), we find mixed results and in many cases that commonly utilized parametric copula models fail to fit the data. In a bootstrapping experiment, we are able to measure the variability in the VaR integrated risk and proportional diversification benefit measures, which can be interpreted as a sensitivity analysis (Gourieroux et al, 2000.) In this experiment we find the variability of the VaR to be significantly lower for the EC, and significantly greater for the VCA, as compared to other standard copula formulations. However, amongst copula models we 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. Taken as a whole, our results constitute 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.

## 1. REVIEW OF THE LITERATURE

## 2. ESTIMATION METHODOLOGY: ALTERNATIVE RISK AGGREGATION FRAMEWORKS

#### 2.1 Value-at-Risk (VaR)

^{th}confidence level between times t and t + Δ, denoted as $Va{R}_{t,t+\text{\Delta}}^{\text{\Delta}\left(X\right)}\left(\alpha \right)$, is related to the α

^{th}quantile of ${F}_{\text{\Pi}\left(X\right)}\left(\text{\Pi}\left(X\right)\right)$ by18:

^{nd}20:

^{nd}moments to remind ourselves of the dynamic nature of this problem in a general context. To illustrate, suppose that we have 3 risk factors ${X}_{t}=\left({X}_{1t},{X}_{2t},{X}_{3t}\right)$ 21. In this case we get the familiar expression:

#### 3.2 The Method of Copulas

^{th}risk factor. We see that the copula is a relation between the quantiles of a set of random variables, rather than the original variables, and as such is invariant under monotonically increasing transformations of the raw data. As summarized by Nelson (1999), there are four technical conditions that are sufficient for a copula to exist. First, $\exists {u}_{i}\text{\hspace{0.17em}}=0\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}C\left(u\right)\text{\hspace{0.17em}}=0$. Second, we require that $\left({u}_{1},\mathrm{..},{u}_{i-1},{u}_{i+1},\mathrm{..},{u}_{K}\right)={i}_{K-1},{u}_{i}\text{\hspace{0.17em}}<1\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}C\left(u\right)\text{\hspace{0.17em}}={u}_{i}\text{\hspace{0.17em}}$. Third, $C\left(u\right)$ must be k-increasing on the sub-space $B={\times}_{i=1}^{K}\left[{x}_{i},{y}_{i}\right]\subseteq {\left[0,1\right]}^{K}$. Finally, the so-called C-volume of B should be non-negative, ${V}_{C}\left(B\right)\triangleq {\displaystyle \sum _{z\in {\times}_{i=1}^{K}\left\{{x}_{i},{y}_{i}\right\}}{\left(-1\right)}^{N(z)}C}(z)\ge 0$, where $N\left(z\right)=\mathrm{card}\left\{K|{z}_{K}={x}_{K}\right\}$. It has also been proven (Nelson, 1999) that there exist theoretical bounds to any given copula, which are important in that they represent generalizations to the conventional concepts of perfect inverse and perfect positive correlation. These are called the Frechet-Hoeffding boundaries for copulas. The minimum copula, the case of perfect inverse dependence amongst random variables, is given by:

^{th}hyper-unit interval by $u=\left({u}_{1},\mathrm{..},{u}_{K}\right)\in {\left[0,1\right]}^{K}$, then we may write the copula as a function as this as follows:

**Q**we have the degrees-of-freedom parameter $\nu $, which controls the thickness of the tails. We use separate notation for

**Q**for the reason that it may not coincide with

**P**23.

^{th}order statistic of ${x}_{j}$. An interesting computational property of (3.2.9) is that this corresponds to the historical simulation method of computing VaR, which involves simply resampling the observed history of joint losses with replacement (or bootstrapping). Historically, this was one of the standard methods for computing VaR for trading positions amongst market risk department practitioners.

## 4. DATA ANALYSIS: SUMMARY STATISTICS AND MARGINAL DISTRIBUTIONS

^{th}quarter of 2008 for the 200 largest banks (the “Top 200") in aggregate that represents a hypothetical “super-bank” (“AT200”) and individually for the top 5 banks in BVA or the “Top 5". The five largest banks by BVA as of 4Q08, in descending order, are as follows: JP Morgan Chase – “JPMC” (BVA = $1.85T), Bank of America – “BofA” (BVA = $1.70T), Citigroup – “CITI” (BVA = $1.32T), Wells Fargo –WELLS (BVA = $1.24T) and Pittsburg National Corporation – “PNC” (BVA = $290B). As of 4Q08 the AT200 represented $10.8T in BVA, and of this the Top 5 banks represents $6.4T, or 59.4% of the total. The skew in this data is extreme, as the average (median) banks amongst the Top 200 has $53.8B ($7.04B) in BVA, reflected in a skewness coefficient of 6.8 that indicates an very elongated right tail relative to a normal distribution. Indeed, our Top 5 banks reside well into the upper 5th percentile of the distribution of book value of assets (BVA = $162.9B). This distribution is shown graphically in Figure 1.1.1.

^{th}percentile) to 93.8% (95

^{th}percentile).

^{th}percentile of NIM = 1.85% for the Top 200).

## 5. ESTIMATION RESULTS: INTEGRATED RISK THROUGH ALTERNATIVE AGGREGATION METHODOLOGIES

^{th}percentile VaR (Equation 3.1.2) for alternative risk aggregation methodologies for each AT200 and the Top 5 in row-wise panels, and in Table 2.2 we replicate this for the Expected Shortfall (ES) at the 99

^{th}percentile (Equation 3.1.3). The different techniques are arrayed by column as “Gaussian Copula Simulation” (Equations 3.2.6-3.2.8; henceforth "GCS") 33, “Gaussian (Variance-Covariance) Approximation” (Equations 3.2.6-3.2.8; henceforth "VCA"), “Historical Bootstrap (Empirical Copula) Simulation” (Equation 3.2.9; henceforth "ECS"), “T - Copula Simulation” (Equation 3.2.8 henceforth "TCS"), “Archimadean Copula (Gumbel) Simulation” (Equation 3.2.13; henceforth "AGCS"), “Archimadean Copula (Clayton) Simulation” (Equation 3.2.12; henceforth "ACCS") and “Archimadean Copula (Frank) Simulation” (Equation 3.2.14; henceforth "AFCS"). The 1

^{st}row in each panel labeled “Magnitude of Risk – Fully Diversified” represents the 99.97

^{th}percentile (ES at the 99

^{th}percentile) of the loss distribution, either simulated in the case of the copula methods or analytic in normal approximation, in Table 2.1 (Table 2.2). The second rows of each panel labeled “Magnitude of Risk – Perfect Correlation” represents the simple sum of the 99.97

^{th}percentiles (ES at the 99

^{th}percentile) of the simulated loss distributions for each risk type in the case of the copula methods, or the sum of the standard deviations of the loss in the analytic normal approximation (in either case, “simple summation” of risks), in Table 2.1 (Table 2.2). In the corresponding 3

^{rd}rows we show the “Proportional Diversification Benefit” (henceforth PDB), which is defined as the difference in the risk measure between the perfect correlation and fully diversified cases, expressed as a proportion of fully diversified VaR or ES for the respective tables:

## 6. SUMMARY OF MAJOR CONCLUSIONS AND DIRECTIONS FOR FUTURE RESEARCH

^{th}percentile VaR is increasing in size of institution, VaR as a fraction of book value does not appear to be a monotonically increasing in size (however, it appears to decrease overall). Second, we saw that across different risk aggregation methodologies and banks that consistently the ECS and AGCS produce the highest absolute magnitudes of VaR as compared to either GCS “benchmark”, STCS or any of the other Archimadean copulas. Furthermore, ECS – a variant of the well-established “historical simulation” methodology in market risk practice – was in many cases found to be most conservative, a surprise in that according to asymptotic theory it should be the lower bound across copula models. On the other hand, the VCA consistently produced the lowest VaR number, which is disturbing in that several bank practitioners are (for the lack of theoretical or supervisory guidance) adopting this computational shortcut. Third, we also noted that the PDB tended to be largest for the ECS than the other methodologies, including the GCS “benchmark” or the VCA, while the AGCS produced the lowest. Therefore, if we regard ECS as a reasonable benchmark with much to recommend it, we caution that banks choosing either the VCA or other copula models may possibly understate diversifications benefits. Fourth, through differences observed across the two (three) of five largest banks having proportionately more trading (lending) assets, we failed to find business mix to exert a directionally consistent an impact on total integrated risk. Fifth, in an application of a blanket goodness-of-fit tests for copula models (Genest et al, 2009), we found mixed results: while in about half the cases commonly utilized parametric copula models fail to fit the data, confidence levels tended to be modest, so clearly this is an area that warrants further investigation and experimentation. Finally, the bootstrapping experiment revealed the variability of the VaR itself to be significantly lowest (highest) for the ECS and GCS (VCA) relative to other risk aggregation models. Furthermore, we found that the contribution of the sampling error in the parameters of the marginal distributions to be an order of magnitude greater than that of the dependency measures. Overall, our 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. This is because 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.

## TABLES AND FIGURES

**Table 1.1.**Summary Statistics on Characteristics of Top 200 and 5 Largest Banks by Asset Size (Call Report Data As of 2008

^{1})

**Table 1.2.**Summary Statistics on Market Value Characteristics of Banks by Asset Size (Call Report and CRSP Data As of 2008

^{1})

**Table 1.3.**Summary Statistics on Risk Measures for Top 200 and 5 Largest Banks by Asset Size (Call Report Data 1984-2008

^{1})

**Table 1.4.**Pairwise Correlations for Top 200 and 5 Largest Banks Risk Proxies (Call Report Data 1984-2008)

**Table 1.5.**Genest et al (2004) Mulivariate Groupwise Independence Test P-Values: Top 200 and 5 Largest Banks Risk Proxies (Call Report Data 1984-2008)

**Table 2.1.**99.97% Confidence Level Value-at-Risk for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate (200 Largest Banks: Call Report Data 1984-2008)

**Table 2.2.**99.9% Confidence Level Expected Shortfall for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate (200 Largest Banks: Call Report Data 1984-2008)

**Table 3.1.**Bootstrap Analysis of 99.97% Confidence Level Value-at-Risk for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate (200 Largest Banks: Call Report Data 1984-2008)

**Table 3.2.**Bootstrap Analysis 99.97% Confidence Percent Level Diversification Benefit for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate (200 Largest Banks: Call Report Data 1984-2008)

**Figures 1.1.1-1.1.6.**Distributions of Key Call Report Variables as of 4Q08 for Top 200 Banks by Book Value of Assets

**Figures 1.2.1-1.2.3.**Distributions of Key Call Report Variables as of 4Q08 for Top 200 Banks by Market Value of Assets

**Figures 6.1-6.4.**Dollar & Relative Value-at-Risk, Proportional Diversification Benefits and, Genest et al (2009) G.O.F. Test P-Values

**Figures 7.1-7.4.**Coefficients of Variation of Bootstrapped Value-at-Risk and Percent Diversification Benefit across Methodologies and Banks

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^{1}This is not unique to enterprise risk measurement for financial conglomerates, as it appears in several areas of finance, including corporate finance (e.g., financial management), investments (e.g., portfolio choice) as well as option pricing (i.e., hedging).^{2}However, in the U.S. supervisors are not requiring all institutions to model EC, only the largest and most systemically important (BCBS, 2009).^{3}This is one form of a form of stress testing, the other is scenario analysis (BCBS, 2009).^{4}Another name for this is hypothetical portfolio analysis (BCBS, 2009).^{5}Even in this context, there are anomalies such as the stock market crash of 1987, which is an event which should never have occurred under the normality of equity returns.^{6}However, this does not cover catastrophic losses, e.g., the terrorist attacks of 9/11.^{7}We may extend this framework to a more recently developed more general setting, where different subsets of risk types can have different dependency structures, through nested and pair copula constructions (Aas et al 2004 a,b, 2007).^{8}For a study in this area very much in this spirit of our research program, see Rosenberg and Schuermann (2006).^{9}These are, in order of decreasing size: JP Morgan Chase, Bank of America, Citibank, Wells Fargo and PNC.^{10}As measured by the relative proportion of trading to lending assets.^{11}Patton (2002) uses copulas to model exchange rate dependence. Rosenberg (2003) accomplishes multivariate contingent claims pricing through application of copulas. Fermanian and Scaillet (2003) analyze copula estimation and testing methods.^{12}Ward and Lee (2002) also apply a risk-adjusted return on capital (RAROC) framework to analyze financial performance of the institution.^{13}It follows that this description of risk is not of the Knightian variety, wherein we do not have knowledge nor means of inferring this mathematical description, also known as “uncertainty”.^{14}If a random variable X has a distribution function F, this is defined as $F\left(x\right)\equiv \mathrm{Pr}\left[X\le x\right]$.^{15}In fact, this extends more broadly to the class of elliptical distributions, of which the normal is a member.^{16}A classic example is J.P. Morgan’s RiskMetrics™ (Phelan, 1995).^{17}In the case of market risk, we consider daily changes in P&L on the positions,, whereas for credit or operational risk the horizon is conventionally 1 year. The latter is also the supervisory horizon under Basel II for credit and the Advanced Measurement Approach (AMA) for operational risk.^{18}Note the conventions: we have oriented**X**such that losses are in the positive direction and $1-\alpha $ is the tail probability.^{19}It may be the case that the distribution function is not everywhere differentiable, in which case we have to deal with the theory of generalized inverses.^{20}The later case would hold under rather restrictive assumptions, such as a quadratic utility function (Markowitz, 1959). Still, it has been considered a useful approximation in many situations.^{21}These are perhaps credit, market and operational losses; but we would abstracting from the fact that credit and operational losses – which are non-negative and highly skewed – would not be modelled as Gaussian (but perhaps log-normal.)^{22}In the case of $1-\alpha =0.9997$ , the Basel Pillar II capital calculation, we get ${\text{\Theta}}^{-1}\left(1-\alpha \right)=3.06$ .^{23}We may estimate its components jointly by maximum likelihood.^{24}This is related to the gamma frailty models of survival analysis (Clayton, 1978).^{25}Note that some authors change notation on the Archimadean parameter to $\alpha $ , as in that context has the interpretation as the tail parameter.^{26}In this study we use mainly the library “Copula” in R and the Statistics Toolbox in Matlab.^{27}In probability theory and statistics the GEV is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families. By the extreme value theorem the GEV distribution is the limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Therefore, the GEV distribution is used as an approximation to model the maxima of long (finite) sequences of random variables.^{28}See De Fountnouvelle et al (2003) for an alternative way to model operational risk.^{29}In order to illustrate, if a bank in 2008 is the result of a merger in 2008, pre-2008 data is merged on a pro-forma basis (i.e., the other non-surviving bank’s data will be represented as part of the surviving bank going back in time.)^{30}A relatively straightforward choice would be to use these fitted kernel density estimates, at a modest but material increase in computational burden. A more computationally expensive approach would be to model the body and the tails separately, say through a “conventional” distribution (e.g., lognormal or students-t) and something like a Generalized Pareto Distribution, respectively.^{31}Note that these are the ordinary Pearson correlation amongst the rank transformed variables.^{32}As outlined by Genest and Remillard (2004), this test is composed of two steps. First, for all sub-sets of the variables, the distributions of the test statistics arte simulated, under the null hypothesis of mutual independence and for the given sample size. In the second step, the approximate p-values are computed, based upon the distribution in step one.^{33}Results for H-VaR were nearly identical to the N-VaR, which is the output of the VCA methodology.^{34}Genest and Remillard (2008) note that if the parametric bootstrap is used, then the vector of dependence parameters for the copula family in question can be estimated by maximizing the pseudo-maximum likelihood (PML), inverting Spearman’s rho or by inverting Kendal’s tau. On the other hand, if the multiplier method is used, any of these may be used in the bivariate case, but in higher dimensional problems only PML may be used.

## Share and Cite

**MDPI and ACS Style**

Inanoglu, H.; Jacobs, M., Jr.
Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital. *J. Risk Financial Manag.* **2009**, *2*, 118-189.
https://doi.org/10.3390/jrfm2010118

**AMA Style**

Inanoglu H, Jacobs M Jr.
Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital. *Journal of Risk and Financial Management*. 2009; 2(1):118-189.
https://doi.org/10.3390/jrfm2010118

**Chicago/Turabian Style**

Inanoglu, Hulusi, and Michael Jacobs, Jr.
2009. "Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital" *Journal of Risk and Financial Management* 2, no. 1: 118-189.
https://doi.org/10.3390/jrfm2010118