Aggregation Issues in Operational Risk

Transcription

1 Aggregation Issues in Operational Ris Rosella Giacoetti Departent of Matheatics, Statistics, Coputer Science and Applications L. Mascheroni School of Econoics and Business Adinistration, University of Bergao Via dei Caniana, 4 Bergao, Italy E-ail: rosella.giacoetti@unibg.it Svetlozar Rachev Departent of Econoetrics, Statistics and Matheatical Finance School of Econoics and Business Engineering, University of Karlsruhe Postfach 6980, 768 Karlsruhe, Gerany and Departent of Statistics and Applied Probability, University of California, Santa Barbara CA , USA E-ail: rachev@statisti.uni-arlsruhe.de Anna Chernobai Departent of Finance, M.J. Whitan School of Manageent, Syracuse University 7 University Ave, Syracuse, NY ,USA E-ail: annac@syr.edu Marida Bertocchi Departent of Matheatics, Statistics, Coputer Science and Applications L. Mascheroni School of Econoics and Business Adinistration, University of Bergao Via dei Caniana, 4 Bergao, Italy E-ail: arida.bertocchi@unibg.it Aggregation Issues in Operational Ris

2 Abstract In this paper we study copula-based odels for aggregation of operational ris capital across business lines in a ban. A coonly used ethod of suation of the value-at-ris (VaR easures, that relies on a hypothesis of full correlation of losses, becoes inappropriate in the presence of dependence between business lines and ay lead to over-estiation of the capital charge. The proble can be further aggravated by the persistence of heavy tails in operational loss data; in soe cases, the subadditivity property of value-at-ris ay fail and the capital charge becoes underestiated. We use α-stable heavy-tailed distributions to odel the loss data and then apply the copula approach in which the arginal distributions are consolidated in the syetric and sewed Student t- copula fraewor. In our epirical study, we copare VaR and conditional VaR estiates with those obtained under the full correlation assuption. Our results deonstrate significant reduction in capital when a t-copula is eployed. However, the capital reduction is significantly saller than in cases where a oderately heavy-tailed or thin-tailed distribution is calibrated to loss data. We also show that for confidence levels below 94% VaR exhibits the super-additivity property.

3 . INTRODUCTION The financial industry s attention to operational ris has been on increase in recent years. Operational ris is defined as the ris of loss resulting fro inadequate or failed internal processes, people and systes or fro external events. Operational ris is believed to be largely a fir-specific non-systeatic ris: According to the Basel Coittee, unlie aret and perhaps credit ris, the [operational] ris factors are largely internal to the ban. In 00, the Basel II Capital Accord (hereforth, Basel II provided a detailed set of guidelines for bans on the basis of which they are required to estiate operational ris capital to serve as a buffer against potential future losses. The fir-specific nature of operational ris has propted developent of statistical odels that ae efficient use of historic operational loss data on the basis of which the capital charge can be estiated. The ost sophisticated of such approaches, and the one ost favoured by regulators 3 is the Loss Distribution Approach (LDA. LDA falls in the category of the Advanced Measureent Approaches. 4 LDA is the ost accurate fro the statistical point of view as it utilizes the exact distribution of historic losses both frequency and severity and is based on an individual ban s internal loss data. The core principle of the capital charge estiation under this approach is the value-at-ris (hereforth, VaR etric that is easured based on a principle of aggregation of the frequency and severity distributions of losses forecasted for a one-year ahead tie horizon. Two tass are central to an accurate estiation of operational VaR. The shape of the upper tail of the loss distribution largely deterines the aount of the ris capital. One approach, extree value theory (EVT, involves a separation of the ain body of the loss distribution fro the tails and odelling the tails with a Generalized Pareto Distribution (GPD; see for exaple, Chavez-Deoulin, Ebrechts, and Neslehova (006. In operational ris, EVT has been also used to odel external data that is then used to populate scarce internal data; see Baud, Frachot, and Roncalli (00 for a ore detailed discussion. 5 Aggregation of operational losses across business units or event types (or both reains as iportant issue. A siplistic approach involves estiation of VaR easures for each cell independently and the adding the up to produce the aggregate easure of ban s ris. The proble with such an approach is that it assues a perfect positive correlation between cells. As a result, the aggregate easure of ris represents an upper bound for a ban s true total level of ris. This property, often referred to as sub-additivity of ris easures, suggests that a ban can effectively reduce its ris capital by taing into account the dependence structure that exists between cells. See BCBS (00b. See BCBS ( Bans are allowed to choose an approach based on the ban s size and ris exposure and the ability to eet required criteria. 4 A detailed description of the approaches to easure the operational ris capital charge is docuented in BCBS (00a,b; Overview of EVT and its applications to ris anageent can be found in Ebrechts, Kluppelberg, and Miosch (997. Applications to operational ris are discussed in Chernobai, Rachev, and Fabozzi (007 and McNeil, Frey, and Ebrechts (005, aong others. 3

4 Soe probles with sub-additivity are often cited in literature. The sub-additivity property ay fail when ris is easured by VaR (see Artzner, Delbaen, Eber, and Heath (999. In this sense, VaR is not a coherent easure of ris. On the contrary, conditional value-at-ris (CVaR is coherent in that the sub-additivity property holds. The failure of sub-additivity (i.e., super-additivity ay be aggravated in the presence of heavy tails in the loss data. As shown in Ibragiov and Walden (007, this ay have adverse iplication for diversification. Our epirical analysis, presented later in this paper, also shows evidence of super-additivity. In this paper, we extend discussion in Giacoetti, Rachev, Chernobai, Bertocchi, and Consigli (007 that focused on various odelling techniques for operational loss data. Here, we exaine the issue of aggregation of operational losses across different business lines using a copula approach. Copulas are gaining popularity in easuring dependence in financial ris. They will be described later in this paper. In operational ris, copulas have been applied by Chavez-Deoulin, Ebrechts, and Neslehova (006, Ebrechts and Puccetti (006a,b, Dalla Valle, Fantazzini, and Giudici (007, and Chapelle, Craa, Hubner, and Peters (004, to nae a few. Using copulas tends to lead to significant reductions in total VaR. For exaple, Chapelle, Craa, Hubner, and Peters (004 showed that for a 99% VaR a ban can achieve a 35% reduction in capital estiates by using copula, echoing the results in Dalla Valle, Fantazzini, and Giudici (007 who deonstrated that using copulas can result in savings for a ban in the range of 30% to 50%. We use t-copulas to odel the dependence between losses. t-copulas are optial for odelling dependence between operational losses fro different groups in that they succeed in capturing dependence in the tails. Other copulas, such as the Gaussian copula, has no tail dependence. For the loss distribution we use (i a variation of the α-stable distribution and (ii a ixture of the α-stable distribution and Generalized Pareto distribution. Such choice of loss distributions ensures that we do not leave out any extree events and they are appropriately accounted for. We apply the ethodology in an epirical study to operational loss data of a European ban. The paper is organized as follows. Section describes a statistical odel for operational ris. Section 3 presents the data and epirical ethodology. Section 4 describes the results of applying copulas to operational loss data and its iplications for operational ris anageent. Finally, Section 5 concludes the paper and suarizes the findings.. A Modified Loss Distribution Approach for Operational Ris The concept of copound Poisson process provides an accurate analytical fraewor to address the odelling proble of operational ris and is utilized in the Loss Distribution Approach (LDA of the Advanced Measureent Approaches proposed by Basel II. The tiing of the events is captured by the intensity of the Poisson process and the losses by an appropriate state distribution. Consider a ban with K business lines/event type 4

5 cobinations, i=,,,k. Then the aggregate operational loss for i-th business line are i considered to follow a rando process L t with t 0 i N t i i i i Lt X, X F, Nt Poi(. ( 0 The operational loss distribution is thus jointly deterined by the average nuber of losses per unit of tie the intensity of the Poisson process N t, the counting process with integer variables and by the loss agnitudes X in onetary ters observed over a pre-deterined interval of tie usually taen to be one year. In our case X belong to a faily F of paraetric continuous distributions. Value-at-Ris is a ris easure that can be used as proxy for capital charge. It is coputed as a high quantile (such as 99.9% of the aggregate loss distribution. For cell i, i VaR G (0.999 ( t L Where G is the cuulative distribution of L. Then, under the LDA, the total capital charge for a ban with M cells can be estiated as the su of VaR easures 6 across cells: K M i i VaR t This LDA ethodology assues that the losses belonging to different business line/event type cells are perfectly positively correlated with each other. However, if the correlation is not perfect, under nice conditions LDA provides an upper bound for the total capital charge for a ban. As a result, LDA would generally tend to over-estiate the aount of ris capital a ban should use. (3 3. Data and Methodology The data set selected for the study covers operational losses for a large European ban fro January 00 to August 005 a total of 3 years and 8 onths. The saple consists of a total of slightly under,700 observations. The data are classified in accordance with the Basel II guidelines into business lines and event types. Due to relatively sall data saples, the business lines considered for this study are: Retail Baning (using the Basel II definitions, this refers to business line 3 or BL3, Coercial Baning (BL4, and Retail Broerage (BL8. We do not further granulate the data by event types for our estiations. BL3 accounts for 77.69% of the data, BL4 accounts for 7.8%, and BL8 accounts for 8.7% of our saple. 6 An alternative to VaR is conditional VaR (or CVaR which is estiated as the average loss given that it exceeds VaR. 5

6 Table. Descriptive statistics for internal, external, and pooled operational loss data (in Euro. Business Line Retail Baning (BL3 Coercial Baning (BL4 Retail Broerage (BL8 Internal Data External Data Pooled Data Saple Statistic Logscale original Log-scale original Log-scale original Mean 5, , , Median, , , St. Dev. 97, , , Sewness Kurtosis Mean 8, , , Median, , , St. Dev. 33, , , Sewness Kurtosis Mean 3, , , Median, , , St. Dev. 48, , , Sewness Kurtosis , , We populate our saple with additional data extracted fro an external database. The external database is a consortiu-type database provided by a European vendor. 7 The data that constitute the database are collected fro nearly 00 institutions. The size of the external data used for the analysis in this paper is approxiately six ties the saple size of internal data. In the external database, BL3 accounts for 60.63%, BL4 accounts for 6.46%, and BL8 accounts for 8.7% of the data. To cobine the internal data with external, we first standardized all data by the respective eans and standard deviations and then scaled the pooled data bac using the ean and the standard deviation of the original internal data to obtain an expanded internal dataset. Table suarizes descriptive statistics of the data. For the frequency distributions we considered a non-hoogeneous Poisson process odel; see Giacoetti et al. (007 for further details. For the loss distributions, we considered a large spectru of candidate distributions: Lognoral, Generalized Pareto, Weibull, Logweibull, and syetrised α-stable. 7 We refrain fro providing the consortiu nae in order to preserve confidentiality of the ban. 6

7 Syetrised α-stable distribution belongs to the class of stable distributions. 8 For the syetrised α-stable distribution we syetrised the data by applying the transforation Y = [X; -X] to the original data X and then fitting a -paraeter (shape and dispersion syetric α-stable distribution. The advantage of fitting a syetric α- stable distribution to syetrised data over fitting a 4-paraeter (shape, dispersion, sewness, and location α-stable distribution to original data is that syetric α-stable distribution requires estiation of only two paraeters (shape and dispersion allowing for ore efficient estiates. The shape paraeter α identifies the heaviness in the tail: α> refers to a oderately heavy tail with a finite ean, while α indicates a very heavy-tailed distribution with an infinite ean. We refer as Model to the approach in which a hypothesized loss distribution is fitted to the entire dataset. Goodness-of-fit test results showed that the syetric α-stable distribution resulted in best fit to our full-saple loss data. 9 For the three business lines, BL3, BL4, and BL8, the shape paraeters were estiated as 0.99, 0.83, and 0.99, respectively, and point to a very heavy tail. Table. Estiates of the shape paraeters. Panel A: Full-saple distribution approach (Model α of the syetrised α-stable distribution BL BL BL Panel B: Mixture distribution approach (Model Body α of the syetrised α-stable distribution Tail /ξ of the Generalized Pareto distribution MLE estiate Hill estiate BL BL BL In a separate approach (Model, we cobined Extree Value Theory with the notion of ixture distributions to odel the loss data. We first identified a level of threshold beyond which the losses were assued to follow the Generalized Pareto Distribution (GPD. We then estiated the paraeters of the GPD for the tails with the Maxiu 8 See Rachev and Mittni (000 for a thorough discussion. 9 See Giacoetti et al. (007 for a detailed description of the goodness-of-fit test results. 7

8 Lielihood estiator (MLE and the Hill estiator, and used syetric α-stable distribution to odel the body of the loss data. Finally, we cobined the two distributions into a -odel ixture distribution constructed as the weighted average of the two eber distributions. Table suarizes the estiated shape paraeters fro Models and. The estiates suggest very heavy-tailed ixture distributions especially for BL4. 4. Aggregation of Losses Using Copulas This section focuses on the discussion of two issues. The first issue is related to dependence between losses belonging to different business line/event type cobinations. 0 When the correlation is sub-perfect, i.e. Equation 3 provides an upper bound for a ban s ris capital, the nowledge on the for of the dependence structure is needed. Coputing correlation between cells iplies a linear for of dependence and therefore ay not be an optial solution. If the dependence is of a non-linear nature, then copulas provide a natural solution. A second issue we will address in this section is exaining scenarios in which subadditivity property of VaR fails. Sub-additivity property dictates that i VaR i VaR Li (4 i Sub-additivity property ay fail in cases when a loss distribution is very heavy-tailed. In essence, this eans that the VaR of aggregate losses ay actually exceed the su of the VaR easures coputed for each cell separately. 4.. Definition of Copula Definition (Copula: A copula function is a apping fro a set of univariate arginals to their full ultivariate distribution. For unifor rando variables U, U,..., U, the joint distribution function C, or copula, is C ( u, u,..., u, P U u, U u,..., U u. (5 Copula functions can be used to lin given arginal distributions with a joint distribution, since for given arginal distribution functions F ( l, F ( l,..., F ( l, we have: C( F ( l,..., F ( l, P U F ( l,..., U F ( l P P F ( U l,..., F ( U L l,..., L l F( l,..., l l (6 0 For the sae of siplicity and data consideration, in this paper we classify losses by business lines only. 8

9 Slar (959 established the fundaental converse result: he showed that any ultivariate distribution function F can be written in the for of a copula function, naely: If F l, l,..., l is a joint ultivariate distribution function with univariate ( arginal distribution functions F l, F ( l,..., F ( l ( ( l, l,..., l C( F ( l, F ( l,..., F ( l, then there exists a copula function C( u, u,..., u such that F. If each Fi is continuous then the copula is unique. Although copulas ay be difficult to wor with, the convenient aspect of copula estiation is that it can be perfored independently fro the estiation of the arginal severity distributions. In this paper, we consider syetric Student t-copula and a sewed Student t-copula because these copulas are capable of capturing the dependence in the upper tail of the distributions. 4.. Syetric Student t-copula A syetric Student t-copula c( u,..., u, R, is a function of the degrees of freedo and the correlation atrix R. For the 3-business line exaple, the syetric t-copula has joint distribution function of the for: 3 T ( 3 / ( 3 / ( / ( w R w 3 ( / ( / ( / ( w / c ( u, u, u3; R, (7 0.5 R i,,3 T where the vector w ( w, w, w, wi t ( ui has coponents directly coputed fro the inverse t distribution. The degrees of freedo of the copula can be estiated using a recursive procedure proposed by Mashal and Naldi (00. Given the initial estiate T 3 T i ~ ~, (,,, i i i R0 zt zt zt zt zt zt zt ( lt, lt F( lt (8 T t,.., T the degrees of freedo in Equation 7 are deterined as optial values of the log-copula: For increasing j j d, 0, we generate recursively for 0,,,... 3 T T j tttt 3 ~ ~ ~ 3 ~ i i R tt : ( tt, tt, tt t ( lt, lt, lt, lt F( lt (9 T t j T tt Rtt j Then, a liit correlation atrix is input into the copula function up to the point in which the log-lielihood is axiised for the current correlation atrix and degrees of freedo: T t 3 ~ ~ arg ax ln c ~ ( l, l, l, R, R li ( R. (0 t, R t t t j,,..., j j The iterative procedure will converge to a copula estiate that can be then incorporated into the siulator. i 9

10 We estiated the liit value of the degrees of freedo to be between 6 and 7. The degrees of freedo and the optial correlation atrix allow for a correct definition of the ultivariate density for a ban s aggregate losses. The epirical correlation and copula-based correlation structures can be illustrated in a 3- diensional space. Figure shows the joint losses of the three business lines internal data on the top left and the epirical joint distribution on the top right. Siulated losses are on the botto left and the syetric t-copula-based joint distribution are in the botto right corner. Coparison of the figures in the top row and in the botto row suggests that the syetric t-copula succeeds in effectively reproducing the true dependence structure between the losses. Figure. Correlation structure of the losses in the three business lines and a canonical exaple with a syetric t-copula. Top left: Weely aggregated losses, epirical data; top right: Epirical distribution of weely aggregated losses; botto left: Weely aggregated losses, siulated data; and botto right: Distribution of siulated losses, weely aggregated using syetric t-copula. 0

11 4.3. Sewed Student t-copula We use the following for of the ultivariate sewed Student t-distribution for the copula function, defined by the stochastic representation as follows: X : W Z W ( where W IG(, and Z N(0,, Z is independent of W, (,, n is an n- diensional vector accounting for the sewness, (,, n is n-diensional location paraeter vector, and ν is the degrees of freedo. We denote this distribution by X t n (,,. The notation IG(, stands for the inverse Gaa distribution with both paraeters equal to. Thus, W is a one-diensional rando variable and Z is a rando vector having a zero-ean ultivariate noral distribution with covariance atrix n n. ( n n nn The ultivariate sewed Student s t-distribution allows for closed-for expression of its density, f X ( x ak ( n ( ( ( x ( ( x ( x ( x n 4 exp(( x ( x ( ( x n n (3 where n x R, K is the odified Bessel function of the third ind and n a ( ( n. (4 The sewed Student s t copula is defined as the copula of the ultivariate distribution of X. Therefore, the copula function is C( u, u,..., un FX ( F ( u,, Fn ( un (5 For ore inforation, see Section.7 in Rachev and Mittni (000 and Dearta and McNeil (005.

12 where FX is the ultivariate distribution function of X and F ( u,,, n, is the inverse of the distribution function of the -th arginal of X. That is, F X (x has the density defined in Equation 3 and the density function f (x of each arginal is f ( x ak ( n ( ( x ( ( x (( 4 exp(( x ( x (, x R (6 where is the -th diagonal eleent in the covariance atrix. defined in Equation. For the sewed Student s t-copula estiation we assue that there are n business lines/event types cobinations of aggregate loss data saples: X X, X,, X T n ( n. We first estiate the paraeters of the sewed Student s t-ultivariate distribution on historical operational losses following the following procedure fro Rachev, Stoyanov, and Milov (007: Step. Fit one-diensional sewed Student s t-distribution over all ris variables on a stand-alone basis. The result fro that step is: ( 5, ˆ, ˆ, ˆ, i, n (7 i i i, We use MLE ethod to obtain the estiates. For the degrees of freedo we use γ=5 because at this value the copula is ost sensitive to the asyetry of paraeters; the sewed distribution reduces to the syetric case at 0. Step. Estiate the correlation atrix of the ultivariate sewed Student s t distribution by the following forula: ˆ (cov( X ˆˆ (8 ( ( 4 where 5 and ( ˆ, ˆ,, ˆ. ˆ n Step 3. Adjust the atrix ˆ so that it becoes positive definite. Having estiated the paraeters of the sewed t-distribution, we obtain the sewed t- copula using the following -step siulation algorith: Algorith Step. Draw N independent n-diensional vectors fro the ultivariate sewed Student s t distribution using the stochastic representation defined in Equation and the set of fitted paraeters ( 5, ˆ i, ˆ ˆ i, i, i,, n. The result fro that step is N n atrix S s ij with siulations. This is obtained by, first, drawing N independent n-diensional vectors fro the ultivariate Noral distribution N (0,, second, drawing N independent rando

13 nubers fro the inverse Gaa distribution with paraeters IG (,, and, third, obtaining final siulations using Equation with the estiated paraeter values. Algorith Step. Transfor siulations S to unifor siulations U using the saple distribution function of the arginals. Denote by Fˆ ( x the saple cuulative distribution function of the -th arginal, j N F ˆ ( x I S j x N, (9 where I{A} stands for the indicator function of the set A. Then U Fˆ ( S, j,, N, n. (0 j j, 4.4. Results of Copula Estiation with Operational Loss Data In this section, we apply the syetric Student t-copula and sewed Student t-copula to our data saples. We consider two odels for the arginal distributions of the losses that were suarized in Section 3. Table 3 suarizes the estiates of population descriptive statistics and ris capital easures at confidence levels 97.5, 98, 99, and 99.9 percent, by business line. These statistics refer to the arginals that will later be aggregated with copulas. It is notable that Model produces uniforly significantly larger estiates than Model. In order to estiate concordance easures between the three business lines, we aggregated losses on a weely basis. Table 3 presents results of the Spearan correlation coefficient and Kendall correlation coefficient for each pair of the business lines. The estiates suggest very low degree of dependence between the losses. To aggregate the losses belonging to the three business lines, we use the syetric Student t-copula and sewed Student t-copula, described in Sections 4. and 4.3, respectively. The degrees of freedo were estiated for the forer copula as ν=6.7 and for the latter copula as ν=5. To apply the copulas to the loss data, we first estiate the copula paraeters using historical losses aggregated weely. We then repeat the following steps a very large nuber of ties. In the first step, fro the estiated copula we saple a ultivariate rando vector with arginals distributed as unifor [0,] rando variables. In the next step, for each business line, we obtain scenarios for the cuulative loss realization by inverting the unifor [0,] variate fro the previous step and then su the up to obtain the total loss for the ban. Repeating these steps produces the distribution of aggregate losses. Finally, VaR and CVaR are coputed fro the obtained aggregate distributions. 3

14 Table 3. Population descriptive statistics and ris capital estiates by business line. Panel A: Model. Population descriptive statistics (Euro Ris capital estiates (Euro VaR CVaR Mean St.Dev. 97.5% 98% 99% 99.9% 97.5% 98% 99% 99.9% BL BL BL Panel B: Model. Population descriptive statistics (Euro Ris capital estiates (Euro VaR CVaR Mean St.Dev. 97.5% 98% 99% 99.9% 97.5% 98% 99% 99.9% BL BL BL Table 4. Estiates of concordance. BL3 Spearan Kendall BL4 Spearan Kendall BL8 Spearan Kendall BL3 BL4 BL

15 Table 5. Population descriptive statistics and ris capital estiates for aggregate loss data. The nubers in parentheses indicate percentage reduction (if - or increase (if + fro the corresponding ris easure under perfect positive correlation scenario. Panel A: Model. Perfect positive correlation Syetric t-copula Sewed t-copula Population descriptive statistics (Euro Ris capital estiates (Euro VaR CVaR EL UL 97.5% 98% 99% 99.9% 97.5% 98% 99% 99.9% (-.8% (-.8% 0.77 (-.% 0.98 (+6.4% 0.87 (-3.0% 0.8 (-5.4% (-5.5% (-.9% (-9.8% (-3.7% (-6.4% (-5.%(-38.4% (+.% (-3.7% (-5.7% (-3.% (-3.7% (-.9%(-34.% Panel B: Model. Perfect positive correlation Syetric t-copula Sewed t-copula Population descriptive statistics (Euro Ris capital estiates (Euro VaR CVaR EL UL 97.5% 98% 99% 99.9% 97.5% 98% 99% 99.9% (-.9% 0.48 (-4.5% (-.6%.004 (0.0% Panel C: Historical data. Population descriptive statistics (Euro (-0.7% (-.8%.0 (+3.%.00 (+3.0%.936 (-.4%.857 (-5.4% (-0.8% (-3.7% (-4.7% (-3.0% (-37.4% (-8.% (-33.3% (-6.8% (-4.6% (-35.% Ris capital estiates (Euro VaR CVaR EL UL 97.5% 98% 99% 99.9% 97.5% 98% 99% 99.9% Historical data Historical data excluding worst losses

16 Table 5 suarizes the estiates for expected and unexpected losses (EL and UL and the ban s cuulative ris capital, derived fro the copula-based approach and copared to the perfect correlation approach. The estiates are based on a one wee horizon. It is notable that in our results there is a negligible difference in EL and UL and the effect is ore pronounced for the VaR and CVaR estiates. In vast ajority of cases copula approach results in a substantial reduction in ris capital for the ban. For exaple, for 99.9% VaR, reduction in capital ranges fro % to 9% and for 99.9% CVaR, reduction in capital is roughly in the 34%-38% range. Another notable result is that if Model is used, syetric Student t-copula generally produces higher reduction in ris capital than the sewed Student t-copula, while the relation is reverse in ost part if Model is used. However, because sewed Student t-copula is a generalized version of the syetric Student t-copula, a ris anager would prefer the forer one. Panel C of Table 5 shows capital estiates based on historical data. Coparison of the figures with those under Model and Model reveals that for percentiles below 99.9, the forer odel uniforly under-estiates the true historic loss while the latter odel produces results fairly consistent (only slightly higher with those actually experienced. Then, because Model corresponds to a ore heavy-tailed distribution than Model, the CVaR figures under Model reveal overestiation of the historic counterparts roughly by a factor of. On the contrary, Model in ost part underestiates CVaR. A ris anager would favour Model over Model based on the above discussion. Our findings indicate lower reduction in ris capital fro those reported by Chapelle, Craa, Hubner, and Peters (004 and Dalla Valle, Fantazzini, and Giudici (007. Chapelle, Craa, Hubner, and Peters (004 showed that for a 99% VaR a ban can achieve reduction in capital estiates by 35% by using copula, and Dalla Valle, Fantazzini, and Giudici (007 showed reduction in the range of 30%-50%. One explanation is that we used a uch ore heavy-tailed loss distribution than in the two other studies. This eans that, if a thin-tailed loss distribution is used to odel loss data, copula-based aggregation would result in uch ore significant reduction in total capital. In essence, this iplies that if a ban has istaenly chosen to use a thin-tailed loss distribution when the data are in fact heavy-tailed, the resulting capital would be understated. The converse is also true. The choice of the loss distribution thus becoes of central concern and ust be estiated with high degree of accuracy. 6

17 VaR( L i - (VaR i % 8.0% 8.40% 83.60% 84.80% 86.00% 87.0% 88.40% 89.60% 90.80% 9.00% 93.0% 94.40% 95.60% 96.80% 98.00% 99.0% Euro Confidence level Figure. Illustration of super-additivity and sub-additivity of historic VaR Super-Additivity of VaR Panels A and B of Table 5 show that for the 98% confidence level, we observe superadditivity of VaR easures in 3 out of 4 cases. Super-additivity is also observed in our results for the historic estiates of VaR under 99.9% for Models and and CVaR estiates under 99.9% for Model. Super-additivity has been docuented in literature; see, for exaple, McNeil, Frey, and Ebrechts (005. Figure illustrates superadditivity and sub-additivity in historical VaR estiates. The horizontal line is the benchar representing a VaR estiate based on the aggregation of historical data for BL3, BL4, and BL8 using historical correlations. The plot represents the suation of individual VaR estiates for each of the three business lines. Super-additivity is observed for confidence levels under 94%. One ust pay extra care when choosing a confidence level to provide VaR and CVaR easures: for confidence levels chosen too low, the estiated capital charge ay be under-estiated while for confidence levels too high, it ay be over-estiated. The phenoenon of super and sub-additivity sees ore severe for ore heavy-tailed loss distributions. 7

18 5. Conclusions In this epirical paper we used a copula approach to odel the dependence between operational losses belonging to different business lines. The contributions of this paper can be suarized as follows: ( For the loss distributions that constitute the arginals of copulas we used very heavy-tailed syetric α-stable distribution (a eber of α-stable distributions and a cobination of a syetric α-stable distribution with the Generalized Pareto distribution for the tails. In operational ris, such heavy-tailed distributions are superior to thin-tailed and oderately heavy-tailed distribution due to their capacity to detect and account for the heaviness in the upper tails. ( We selected syetric Student t-copula and sewed Student t-copula due to their ability to capture tail dependence. Our epirical study has shown that using t-copulas results in substantial reduction in ris capital for the ban. Although our findings are consistent with other epirical studies that have docuented capital reduction when using copula, upon drawing coparisons with other siilar studies, we have found that the agnitude in capital reduction is saller than if thin-tailed or oderatelytailed distributions, such as Lognoral and Gaa, are used to constitute the arginals. (3 Another finding was presence of the super-additivity phenoenon in the VaR estiates. For our data saple, super-additivity was observed for confidence levels below 94%. Our findings effectively deonstrate that falsely settling on a loss distribution that is not sufficiently heavy-tailed for given data puts the capital estiates at the ris of severe underestiation for very high confidence levels or overestiation for confidence levels insufficiently high. Careful analysis and a variety of goodness-of-fit tests are thus crucial in selecting a loss distribution which is central to the accuracy of the capital charge estiates. References Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (999, Coherent easures of ris, Matheatical finance, 9(3, pp Baud, N., Frachot, A., and Roncalli, T. (00, Internal data, external data, and consortiu data for operational ris easureent: How to pool data properly, technical report, Groupe de Recherche Operationnelle, Credit Lyonnais, France. BCBS (998, Operational ris anageent, Ban of International Settleents, Basel, Switzerland, BCBS (00a, Consultative docuent: Operational ris, Ban of International Settleents, Basel, Switzerland, 8

19 BCBS (00b, Woring paper on regulatory treatent of operational ris, Ban of International Settleents, Basel, Switzerland, BCBS (006, International convergence of capital easureent and capital standards, Ban of International Settleents, Basel, Switzerland, Chernobai, A., Rachev, S. T., and Fabozzi, F. (007, Operational ris: A guide to Basel II capital requireents, odels, and analysis, Wiley Finance Series, John Wiley & Sons, Hoboen, New Jersey. Dearta, S. and McNeil, A. (005. The t copula and related copulas, International Statistical Review, 73(, -9. Ebrechts, P., Kluppelberg, C., and Miosch, T. (997, Modeling extreal events for insurance and finance, Springer-Verlag, Berlin. Ebrechts, P. and Puccetti, G. (006a, Aggregating ris capital, with an application to operational ris, The Geneva Ris and Insurance Review, 3(, pp Ebrechts, P. and Puccetti, G. (006b, Bounds for functions of dependent riss, Finance and Stochastics, 0, pp Giacoetti, R., Rachev, S. T., Chernobai, A., Bertocchi, M., and Consigli, G. (007, Heavy-tailed distributional odel for operational ris, The Journal of Operational Ris, (, pp Ibragiov, R. and Walden, J. (007, The liits of diversification when losses ay be large, Journal of Baning and Finance, 3, pp Mashal, R. and Naldi, M. (00, Pricing ulti-nae credit derivatives: Heavy tailed hybrid approach, Woring paper, Colubia Business School. McNeil, A. J., Frey, R., and Ebrechts, P. (005, Quantitative ris anageent: Concepts, techniques, and tools, Princeton Series in Finance, Springer, Boston. Rachev, S. T. and Mittni, S. (000, Stable Paretian odels in finance, John Wiley & Sons, New Yor. Rachev, S. T., Stoyanov, S., and Milov, G. (007, Sewed Student t Copula, Technical report, FinAnalytica Inc. Slar, A., (959, Fonctions de répartition à n diensions et leurs arges, Publications de l'institut de Statistique de L'Université de Paris, 8, pp