research article operational risk aggregation across...

Research ArticleOperational Risk Aggregation across Business Lines Based onFrequency Dependence and Loss Dependence

Jianping Li1 Xiaoqian Zhu12 Jianming Chen1 Lijun Gao3 Jichuang Feng4

Dengsheng Wu1 and Xiaolei Sun1

1 Institution of Policy and Management Chinese Academy of Sciences Beijing 100190 China2University of Chinese Academy of Sciences Beijing 100190 China3 School of Business Administration Shandong University of Finance and Economics Jinan Shandong 250014 China4 Industrial Bank Co Ltd Fuzhou Fujian 350003 China

Correspondence should be addressed to Lijun Gao glj963217163com

Received 3 December 2013 Accepted 14 January 2014 Published 25 February 2014

Academic Editor Chuangxia Huang

Copyright copy 2014 Jianping Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In loss distribution approach (LDA) the most popular approach in operational risk modeling frequency dependence and lossdistribution dependence across business lines are two dependences which banks should consider In practice mainly for simplicitymany banks only model frequency dependence although they think that the impact of frequency dependence is insignificantIn this study two approaches respectively models frequency dependence and loss distribution dependence are introduced Bothapproaches are modeled by copula function which is capable of capturing nonlinear correlation Based on themost comprehensiveoperational risk dataset of Chinese banking as far as we know the operational risk capital charge of the overall Chinese banking iscalculated by the two approaches The results show that there is an obvious distinction between the capital calculated by modelingfrequency dependence and the capital calculated by modeling loss dependence The approach with very limited attention exactlyyields amuch larger capital result So it is advised in this paper that banks should not just rely on the approach tomodeling frequencydependence for it is natural and easy to deal with A safer and more effective way for banks is to comprehensively take the resultsof the two kinds of approach into consideration

1 Introduction

The sudden collapse of Barings Bank alerts the bankingindustry to operational risk that is why following credit riskand market risk operational risk has become the third riskcovered by Basel Accord II Operational risk is defined asthe risk of loss resulting from inadequate or failed internalprocesses people and systems or from external events [1 2]Basel Accord II proposes basic indicator approach (BIA)standardized approach (SA) and advanced measurementapproach (AMA) to calculate operational risk capital Amongthe three approaches AMA is the most sophisticated andmuch more risk sensitive than the former two approachesso the Basel Committee on Banking Supervision (hereafterBCBS) encourages banks to develop and use the AMAwithin

operational risk management [3] Many approaches such asloss distribution approaches (LDA) and extreme value theory(EVT) have been developed for operational risk modeling sofar [4 5]

Among all kinds of AMA LDA firstly developed byactuarial industry and then introduced to operational riskmodeling by Frachot et al [6] is widely accepted as themost popular approach in calculating the capital charge foroperational risk [4 7ndash9] LDA is a parametric techniquethat consists in separately estimating a frequency distributionfor the occurrence of operational losses and a severitydistribution for the economic impact of individual lossesWith these two distributions the bank then computes theprobability distribution of the aggregated operational loss

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 404208 8 pageshttpdxdoiorg1011552014404208

2 Mathematical Problems in Engineering

Basel II Accord categorizes the operational risk event intoeight business lines that is corporate finance trading andsales retail banking commercial banking payment andsettlements agency services asset management and retailbrokerage [3] however the traditional LDA does not involvethe dependence across business lines

A more reasonable way to calculate capital charge foroperational risk is to incorporate the dependence betweendifferent business lines into LDA In this kind of LDAseparate distributions for loss frequency and severity aremodeled at business line level and then combined [10] Inthis basicmodel three kinds of dependencesmight occur andso should be taken seriously that is frequency dependenceseverity dependence and loss distribution dependence [1112] Frequency dependence is the dependence between theoccurrences of loss events In practice it means that weobserve that historically the number of (say) commercialbanking events is high (resp low) when the number of (say)corporate finance events is also high (resp low) Severitydependence is the dependence between the sizes of individualloss events Loss distribution dependence is the dependencebetween the yearly losses of business lines [11]

Many approaches modeling dependence in LDA haveemerged so far As to frequency dependence there is acontradiction On one hand the approaches modeling thefrequency dependence are probably most popular in thebanking industry On the other hand many researcheshold that frequency correlation has only a very restrictedimpact on economic capital for operational risk [11 13ndash15]For example the world leading bank in risk managementDeutsche Bank only specifies the dependence of frequencydistributions in their LDA model although they think thatthe impact of frequency dependence is insignificant [11]The main reasons responsible for this contradiction are thatcorrelation between the yearly numbers of loss event withinbusiness lines can be quite easily calculated from empiricalloss data and adding correlation between frequencies ofevents is quite an easy task and does not destroy the verynature of the LDA model Therefore this type of correlationcan be taken into account at minimum cost [12] Comparedwith frequency dependence loss distribution dependencehas been paid very limited attention Chapelle et al [9] andGiacometti et al [16] directly model the dependence of lossesthrough the use of copulas in order to combine the marginaldistributions of different business lines into a single jointloss distribution Severity dependence necessarily alters in asubstantial extent the basic foundations of the standard LDAmodel and requires building an entirely new family ofmodels[12] So it is scarcely studied in the literature and will not bediscussed in this study as well

The objective of this study is to discuss whether thepopular frequency dependence modeling is enough for thebanks when deciding to allocate operational risk capitalFirstly two approaches respectively modeling frequencydependence and loss dependence in the framework of LDAare introduced When it comes to modeling dependencecopula function has attracted broad attention because itis capable of capturing the complete dependence structure

inherent in a random vector in contrast to linear correlation[17ndash19] According to the survey of BCBS [15] dependenceis introduced into the modeling process mainly by use ofcopulas So in this study copula is employed to model bothfrequency dependence and loss dependence

Then both approaches are employed to calculate theoperational risk capital charge for Chinese banking Ourlaboratory has been devoted to collecting operational riskdata of Chinese banking for about 10 years Up to nowtotally 2132 records are collected in our dataset ranging from1994 to 2012 To the best of our knowledge this operationalrisk dataset is the most comprehensive one in China In theexperiment both approaches are employed to calculate theoperational risk capital of Chinese banking based on thisdataset

The rest of this paper is organized as follows FirstlySection 2 introduces the two approaches respectively modelfrequency dependence and loss dependence Then Section 3employs the two approaches to calculate the operationalrisk capital charge of the overall Chinese banking Section 4summarizes the conclusions

2 Approaches

In this section two approaches that are capable of modelingfrequency dependence and loss dependence in the frame-work of LDA are respectively introduced Both dependencerelations in the two approaches are modeled by copulafunctions By using the two approaches aggregate opera-tional loss distribution can be simulated from Monte CarlosimulationThen VaR is used to measure the operational riskcapital charge Pioneered by JP Morgan VaR has become astandardmeasure used in financial risk [20] VaR at a specificconfidence level 120572 isin (0 1) is defined as the smallest number119897 such that the probability of loss L exceeding l is not largerthan (1 minus 120572)

VaR = inf 119897 119875 (119871 le 119897) le (1 minus 120572) (1)

It is noteworthy that frequency refers to the number ofloss events occurring a year severity refers to the size of anindividual loss event and loss refers to the total size of all theloss events a year

21 LDA with Frequency Dependence (LDA-FD) It is ob-served that historically the number of (say) commercialbanking events is high (resp low) when the number of(say) corporate finance events is also high (resp low) Thisphenomenon implies that there is underlying correlation ordependence between the frequencies of business lines

In this approach we model the dependence structure ofthe frequencies in the framework of LDA and this approachis then named LDA-FD here The whole procedure of LDA-FD approach consists of 3 stages and 9 steps in total In Stage1 the marginal frequency distribution and marginal severitydistribution are fitted and the copula used to modelingthe dependence structure should be specified Usually thegoodness-of-fit test is used to determine the best fittingdistributions and copula In Stage 2 Monte Carlo simulation

Mathematical Problems in Engineering 3

is used to derive the aggregate loss distribution and in Stage3 VaR is calculated from the aggregate loss distributionAssume the number of Monte Carlo simulation is N Thespecific steps of LDA-FD approach are given as follows

211 Stage 1 Preparation

Step 1 Choose the best fitting distribution 1198661 119866

8for the

frequency of eight business lines


8for the

severity of eight business lines

Step 3 Choose the proper copula 119862 to describe the depen-dence structure of frequencies

212 Stage 2 Simulation

Step 4 Draw a joint sample of uniform random variables(1199061 119906

8) from the specified copula 119862

Step 5 Translate the sample from copula (1199061 119906

8) into a

sample of frequency (1198991 119899

8) = (119866

minus1

1(1199061) 119866

minus1

8(1199068))

Step 6 Simulate 1198991losses from 119865

1randomly and simulate

1198998losses from 119865

8

Step 7 Sum all the simulated losses from Step 6 to deriveaggregate loss 119897

Step 8 Repeat Step 4 to Step 7 119873 times to derive119873 simulatedaggregate losses 119897

1 119897119873

213 Stage 3 Calculation

Step 9 Calculate VaR from aggregate loss 1198971 119897119873

22 LDA with Loss Dependence (LDA-LD) In this approacha higher level of dependence is taken into account Wedirectly model dependence between the losses of differentbusiness lines This kind of aggregation is just like the ldquotop-to-downrdquo aggregation of different risk types [21 22] Herewe model dependence structure of losses in the frameworkof LDA and this approach is named LDA-LDThe number ofMonte Carlo simulation is also set as119873Thewhole procedureof LDA-LD approach consisting of 3 stages and 7 steps in totalis as follows



8for

losses of eight business lines

Step 2 Choose the proper copula 119863 to describe dependencestructure of losses

7 25

1144

619

263

27 30 170

200

400

600

800

1000

1200

1400

BL1 BL2 BL3 BL4 BL5 BL6 BL7 BL8

Freq

uenc

y

Business lines

Figure 1 Summary frequency of different business lines


Step 3 Draw joint sample of uniform random variables(V1 V

8) from the specified copula119863

Step 4 Translate the sample from copula (V1 V

8) into a

sample of loss (1199041 119904

8) = (119867

minus1

1(V1) 119867

minus1

8(V8))

Step 5 Sum all the losses from Step 4 to derive simulatedaggregate loss 119897 = 119904

1+ sdot sdot sdot + 119904

8


1 119897119873

223 Stage 3 Calculation Step 7 Calculate VaR from aggre-gate loss 119897

1 119897119873

3 Application to Chinese Banking

In this section the introduced approaches LDA-FDandLDA-LD are used to calculate the operational risk capital charge ofthe overall Chinese banking Firstly the dataset is introducedThen some details of the experiment are explained At last theresults are presented and analyzed All computations in thisexperiment are performed by R 301

31 Data Description The collection of operational risk datais still in a preliminary stage in the world and this case is evenworse in Chinese banking Our laboratory has been devotedto collecting operational risk data from publicly availableinformation sources such as newspapers Internet and courtfilings for almost 10 years Up to now totally 2132 recordsare collected in our dataset ranging from 1994 to 2012 Foreach record the loss event description start time end timeexposed time business line type loss event type loss amountin CNY banks involved location key person involved and soon are collected in exact detail To the best of our knowledgethis operational risk dataset of Chinese banking is the mostcomprehensive one inChina In the experiment the end timeloss amount in CNY and business line type are used

In line with Basel II Accord the operational risk event canbe categorized in 8 business lines that is corporate finance


Table 1 Parameter estimation and goodness of fit test results of frequency distribution

Business line Frequency KS testDistribution Parameters 119863 value 119875 value

BL3 Negative binomial 119903 = 134 119901 = 098 018 059BL4 Negative binomial 119903 = 501 119901 = 087 014 087BL5 Negative binomial 119903 = 355 119901 = 080 012 093

Table 2 Parameters estimation and goodness of fit test results of severity distribution

Business line Severity KS testDistribution Parameters 119863 value 119875 value

BL3 SGED 120583 = 333120590 = 242 119896 = 167 120582 = 139 002 062BL4 SGED 120583 = 838 120590 = 215 119896 = 216 120582 = 085 002 090BL5 SGED 120583 = 647 120590 = 287 119896 = 259 120582 = 086 004 082

trading and sales retail banking commercial banking pay-ment and settlements agency services asset managementand retail brokerage which are correspondingly called BL1to BL8 for short in the following text Figure 1 presents thefrequency of these business lines in the experiment datasetIt can be seen that the occurrence of operational risk eventmostly appears in BL3 BL4 and BL5 So in the experimentwe only consider the aggregation of BL3 BL4 and BL5

32 Experiment Design Operational risk is characterizedby ldquoleptokurtosis and fat tailrdquo [15] so lognormal distribu-tion exponential distribution Pareto distribution gammadistribution Weibull distribution generalized hyperbolicdistribution (GHD) generalized error distribution (GED)skewed generalized error distribution (SGED) and gen-eralized Pareto distribution (GPD) are usually employedto fit the loss or severity distributions [4 6 8 9] GHDGED and SGED are used to fit the natural logarithm ofdata and the rest are directly employed to fit the data Asfor frequency distribution Poisson negative binomial andgeometric distributions are frequently used [1 9 15] In somestudies Poisson distribution is employed to model frequencywithout any test for its ease of use and the viewpoint of itslimited impact on capital calculation [6 11] In this studywe are ready to use Kolmogorov-Smirnov goodness-of-fittest (KS test for short) to find out which distribution canfit the frequency severity and yearly loss best Besides allthe parameters of the distributions in this experiment areestimated by maximum likelihood method

Among all types of copulas frequently used copulasinclude Gaussian copula and t copula from elliptical copulafamily and Gumbel copula Clayton copula and Frankcopula from Archimedean copula family For a detailedintroduction of these copulas please see [23 24] Com-pared with elliptical copulas some Archimedean copulasare very sensitive to asymmetric tail dependence Howevertheir highly symmetric structures prevent their use in more

than two dimensional applications [21] Therefore here inthis experiment we choose Gaussian copula and t copulato describe the dependence structure of yearly frequenciesand yearly losses The formulations of the two copulas arepresented in the appendix We use the goodness-of-fit testproposed by Kojadinovic and Yan [25] to determine whichcopula can describe the dependence of the empirical databest

As for theMonte Carlo simulation the larger the numberof simulations is the more accurate the aggregate lossdistribution is and the longer the simulation time is In orderto balance accuracy and time the times of Monte Carlosimulation are set as 100 thousand in line with most of thestudies

33 Experiment Results

331 Results from LDA-FD In this section LDA-FD theapproach modeling frequency dependence is used to calcu-late the operational risk capital charge for the overall Chinesebanking Firstly we will find the best fitting distributions forfrequency and severity by KS test Poisson negative bino-mial and geometric distributions are used to fit the yearlyfrequency The KS test results reveal that negative binomialdistribution outperforms the other two distributions on allthe three business lines For space considerations only theresults of negative binomial are presented in Table 1 For KStest the larger the P value is the better the distribution canfit the data and the confidence level is generally set as 5Table 1 shows that P value for BL3 is 059 for BL4 is 087and for BL5 is 093 which are all much larger than 5The parameters estimated by maximum likelihood methodsfor each marginal frequency distribution are also shown inTable 1

As described above lognormal distribution exponen-tial distribution Pareto distribution gamma distributionWeibull distribution GHD GED SGED and GPD are used


Table 3 Parameters estimation and goodness of fit test results of copulas in LDA-FD

Parameters Goodness of fit testGaussian copula 120588

34= 0499 120588

35= 0267 120588

45= 0649 119875 = 002

t copula 12058834= 0571 120588

35= 0287 120588

45= 0720 ] = 1 119875 = 021

Note 12058834 denotes the correlation between BL3 and BL4 12058835 denotes the correlation between BL3 and BL5 12058845 denotes the correlation between BL4 and BL5and ] denotes the degree of freedom of t copula

Table 4 Operational risk capital charge by LDA-FD

Confidence level 90 95 99 999 9997VaR 26 34 61 231 627Economic capital 11 19 46 217 613Unit billion CNY

to fit the severity data The KS results show that SGED is thebest fitting one for each business lineThe results of SGED areshown in Table 2 The P values for each business lines in KStest are 062 090 and 082 respectively which are also muchlarger than 5 The estimated parameters for each marginalseverity distribution are also shown in Table 2

Then we should choose a proper copula that is capableof accurately capturing the dependence structure of yearlyfrequencies of different business lines The goodness-of-fittest results are shown in Table 3 In this test also the largerthe P value is the better the copula can fit the empirical dataand the confidence level is generally set as 5 From Table 3we can see that test result of Gaussian copula is 119875 = 002which is smaller than 5 As for t copula t copula withdegree of freedom equal to 1 outperforms t copula with otherdegrees of freedom The test result of t copula with degree offreedom equal to 1 is 119875 = 021 which is larger than 5 Sofinally t copula is chosen to describe the yearly frequenciesdependence The parameters of the specified t copulas arealso shown in Table 3 These parameters are estimated frommaximum likelihood method

After the marginal frequency distribution marginalseverity distribution and copula are specified The capitalcharge for operational risk of the overall Chinese banking canbe calculated by using the LDA-FD approach presented inSection 21 For riskmanagement purpose confidence level isgenerally set to be larger than 90 So the VaRs at confidencelevel 90 95 99 999 and 9997 are presented inTable 4 Table 4 shows that as the confidence level increasesthe VaR and economic capital become larger The expectedloss here is 14 billion Economic capital defined as thedifference between VaR and expected loss and so referred toas the ldquounexpected lossrdquo is also presented Therefore by con-sidering the frequency dependence the overall operationalrisk of Chinese banking is 231 billion CNY at 999 Thecorresponding operational risk regulatory (economic) capitalis 217 billion CNY

332 Results from LDA-LD In this section we will useLDA-FD the approach modeling yearly loss dependenceto calculate the operational risk capital charge for Chinese

banking Firstly we should find out the best fitting distri-bution for marginal yearly loss of each business line Alsolognormal distribution exponential distribution Pareto dis-tribution gamma distribution Weibull distribution GHDGED SGED and GPD are employed to fit the empiricalyearly loss data of each business line KS test is used to selectthe distribution that can best fit the data It turns out thatlognormal distribution is the best fitting distribution for allthe marginal loss data In consideration of the length of thispaper Table 5 only presents results of lognormal distributionWe can see that the P value of KS test for each business lineis 032 046 and 059 respectively all much larger than 5Thismeans that lognormal distribution fits eachmarginal lossdata well The parameters estimated by maximum likelihoodmethod are also presented in Table 5

Secondly we should find a proper copula to describethe dependence structure of yearly losses of these businesslines The goodness-of-fit test results presented in Table 6show that both copulas have passed the goodness-of-fit test119875 value is 031 for Gaussian copula and 093 for t copulaboth larger than 5 Nevertheless because the larger the Pvalue is the better the copula fits the data we decide to uset copula to describe dependence structure of yearly loss datain the Monte Carlo simulation The P value for t copula isas high as 093 which means that t copula fits the yearlyloss dependence very well The parameters of the specifiedt copula estimated from maximum likelihood methods arealso shown in Table 6

After the marginal loss distribution and copula arespecified The aggregate loss can be simulated by the LDA-LD approach presented in Section 22The results at differentconfidence levels are presented in Table 7 The expected lossis 20 billion here Table 7 shows that as the confidencelevel increases the VaR and economic capital become largerTherefore by considering loss dependence the overall oper-ational risk of Chinese banking is 675 billion CNY at 999and the corresponding regulatory (economic) capital is 655billion CNY

333 Results Comparison and Analysis In order to furtheranalyze the results of the two approaches we decide tocompare them with the results of BIA Banks using the BIAmust hold capital for operational risk equal to the averageover the previous three years of a fixed percentage of positiveannual gross income The fixed percentage is set to 15by the BCBS For a more detailed formulation please seeBCBS [1] and BCBS [3] As shown in Table 8 the grossincome of listed banks is 1756 billion in 2010 2234 billionin 2011 and 2596 billion in 2012 So according to BIA theoperational risk capital charge of listed commercial banks is


Table 5 Parameters estimation and goodness of fit test results of loss distribution

Yearly loss KS testDistribution Parameters 119863 value 119875 value

BL3 Lognormal 120583 = 978 120590 = 230 021 032BL4 Lognormal 120583 = 1322 120590 = 144 019 046BL5 Lognormal 120583 = 1132 120590 = 131 017 059

Table 6 Parameters estimation and goodness of fit test results of copulas in LDA-LD

Parameters Goodness of fit test (119875 value)Gaussian copula 120588

34= 0616 120588

35= 0389 120588

45= 0249 119875 = 031

t copula 12058834= 0621 120588

35= 0484 120588

45= 0299 ] = 1 119875 = 093



15lowast[(1756 + 2234 + 2596)3] = 3293 billion CNY Onlythe gross income of listed commercial banks is available sowe roughly estimate the capital of the overall Chinese banksas follows The total assets of listed banks account for 67of that of overall Chinese banking in 2010 66 in 2011 and65 in 2012 so the capital charge of overall Chinese bankingis calculated by 329366 that is 4989 billion CNY

Conventionally the capital requirement is set to protectagainst losses over one year at the 999 level becauseit is roughly equivalent to the default risk of an A-ratedcorporate bond [6] BCBS also recommends 999 as aproper confidence level Therefore we aim at comparingthe VaRs from the two approaches at 999 with result ofBIA All these results are presented in Table 9 Table 9 showsthat operational risk capital of Chinese banking calculatedby BIA is 499 billion The result of LDA-FD approach is231 billion which is smaller than the result of BIA Onthe contrary the result of LDA-LD approach is 675 billionwhich is larger than the result of BIA As mentioned inSection 1 nowadays in practice most of the banks justconsider frequency dependence in their AMA In this caseof Chinese banking it turns out that this kind of approachLDA-FD here indeed gives a much smaller capital than BIAIn contrast modeling yearly loss dependence is paid lessattention However LDA-LD gives a larger capital than BIA

There is an obvious distinction between the results ofthe approach to modeling frequency dependence and theresults of the approach to modeling loss dependence Theapproach to modeling loss dependence with less attentionexactly yields amuch larger result Because of the lack of dataalthough many studies focus on operational risk modelingvery few studies give specific operational capital charge As faras we know Feng et al [8] are the only researchers who give aspecific operational risk capital charge for the overall Chinesebanking Their results show that operational risk capital ofoverall Chinese banking is about 234 billion CNY in 2008

Our results show that LDA-FD gives the capital charge of231 billion CNY and LDA-LD gives the capital charge of 675billion CNY in 2013 Chinese banks have developed very fastin recent years so generally operational risk in 2013 ought tobe larger than operational risk in 2008 However results ofLDA-FD 231 billion in 2013 are even smaller than 234 billionconcluded from Feng et al [8] in 2008 The comparisonresults also support our conclusions that results of LDA-FDmight be too small

Therefore it is advised in this paper that banks shouldnot just rely on the result of modeling frequency dependencefor it is natural and easy to deal with A safer way to allocatecapital is to comprehensively take both results into consid-eration Combination estimation a widely used method hasbeen successfully introduced to the field of operational riskmodeling by Feng et al [8] Here an easy linear combinationof the results of LDA-FD approach and LDA-LD approach isto calculate their mean which is 453 billion CNY This resultis considered safer and more effective for banks

4 Conclusion

In this paper we present two ways of modeling dependencein the framework of LDAThe approach modeling frequencydependence based on copula is named LDA-FD approachand the approach modeling yearly loss dependence is namedLDA-LD approach We then apply the two approaches tocalculate operational risk capital of the overall Chinesebanking on the most comprehensive operational risk datasetin China as far as we know

The operational risk capital of the overall Chinese bank-ing calculated by LDA-FD approach is 231 billion CNY whichis about half of the result of BIA that is is 499 billion CNYContrastively the operational risk capital calculated by LDA-LD approach is 675 billion CNY which is larger than theresult of BIA There is an obvious distinction between theresults of LDA-FD and the results of LDA-LD The approachwith less attention that is the approach to modeling lossdependence exactly yields a much larger result so it isadvised in this paper that banks should not just rely on theapproach to modeling frequency dependence for it is naturaland easy to deal with A safer andmore effectiveway for banksis to take the results of both approaches into consideration


Table 8 Gross income and total assets of Chinese banks

Gross income of listedbanks

Total assets of listedbanks

Total assets ofChinese banking

Total asset percentage(listed banksChinese banking)

2010 1756 64079 95305 672011 2234 75211 113287 662012 2596 86791 133622 65Unit billion CNY

Table 9 Operational risk capital charge of the overall Chinesebanking by different approaches

Methods BIA LDA-FD LDA-LD Combination approachVaR 499 231 675 453Unit billion CNY

Appendix

A Some Distributions and Copulas inThis Study

A1 Lognormal Distribution The lognormal distribution isone of the standard distributions used in insurance andoperational risk to model large claims and operational riskloss The probability function of lognormal is

119891 (119909 | 120583 120590) =

1

radic2120587120590119909

expminus(ln119909 minus 120583)2

21205902

119909 gt 0 119906 gt 0 120590 gt 0

(A1)

where 120583 and 120590 are the two parameters of lognormal dis-tribution denoting the mean and standard deviation oflogarithmic 119909

A2 Skewed Generalized Error Distribution (SGED) Theprobability density function of SGED is

119891 (119909 | 120583 120590 119896 120582)

=

119862

120590

exp 1

[1 + sgn (119909 minus 120583 + 120575120590) 120582]119896120579119896120590119896

times1003816100381610038161003816119909 minus 120583 + 120575120590

1003816100381610038161003816

119896

(A2)

where

119862 =

119896

2120579Γ (1119896)

120579 = Γ(

1

119896

)

05

Γ(

3

119896

)

minus05

119878(120582)minus1

120575 = 2120582119860119878(120582)minus1

119878 (120582) =radic1 + 3120582

2minus 411986021205822

119860 = Γ (

2

119896

) Γ(

1

119896

)

minus05

Γ(

3

119896

)

minus05

(A3)

120583 and 120590 are the mean and standard deviation of 119909 120582 is askewness parameter ldquosgnrdquo is the sign function and Γ is thegamma function Scaling parameters 119896 and120582 obey constraints119896 gt 0 and minus1 lt 120582 lt 1 For a more detailed introduction ofSGED please see [8]

A3 Negative Binomial Distribution Negative binomial dis-tribution is a discrete probability distribution of the numberof successes in a sequence of Bernoulli trials before a specified(nonrandom) number of failures occur The probabilityfunction of the negative binomial distribution is

119891 (119896 | 119903 119901) = 119875 (119883 = 119896) = 119862119896

119896+119903minus1(1 minus 119901)

119903

119901119896

119896 = 0 1 2

(A4)

where 119903 is a specified number of failures and 119901 is theprobability of success in each trial

A4 Gaussian Copula and t Copula Assume 1199061 1199062 119906

119899

follow univariate standard uniform distribution The Gaus-sian copula is presented as

119862Gaussian (1199061 119906119899 | 119877) = Φ119877 (Φminus1(1199061) Φ

minus1(119906119899))

(A5)

where Φ119877denotes the standard multivariate Gaussian dis-

tribution function with correlation coefficient matrix 119877 andΦminus1 denotes the inverse function of standard univariate

Gaussian distribution Gaussian copula is very popular atfirst in practice due to the common assumption of normalityin many financial modeling applications and ease of useand understanding However it does not allow for taildependence which is a major drawback and as such becomesa less favorable candidate for capital applications [26]

The t copulais presented as

119862119905(1199061 119906

119899| ] 119877) = 119905]119877 (119905

minus1

] (1199061) 119905

minus1

V (119906119899)) (A6)

where 119905V119877 denotes the 119905 distribution function with degreeof freedom ] and correlation coefficient matrix 119877 and 119905minus1Vdenotes the inverse function of 119905 distribution function Weare currently seeing growing literature on the usefulness ofthe t copula as an alternative to the Gaussian copula formodeling financial risks The main impetus for the t copularsquosrise to notoriety is associated with its ability to incorporatetail dependence [27] Theoretically the lower the degree offreedom the heavier the tail dependence for a t copulaGaussian copula is in fact a limiting case of t copula as


the degree of freedom approaches infin By contrast t copulaexhibits the heaviest tail dependence as the degree of freedomapproaches 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by Grants from theNational Natural Science Foundation of China (71071148and 71301087) Key Research Program of Institute of Policyand Management Chinese Academy of Sciences and YouthInnovation Promotion Association of the Chinese Academyof Sciences

References

[1] Basel Committee on Banking Supervision Working Paper onthe Regulatory Treatment of Operational Risk Bank for Inter-national Settlements Basel Switzerland 2001

[2] J Li J Feng H Song and C Cai ldquoIntegrating credit marketand operational risk based on risk correlationrdquo Chinese Journalof Management Science vol 18 no 1 pp 18ndash25 2010 (Chinese)

[3] Basel Committee on Banking Supervision International Con-vergence of Capital Measurement and Capital Standards ARevised Framework Bank for International Settlements BaselSwitzerland 2006

[4] J Li J Feng and J Chen ldquoA piecewise-defined severitydistribution-based loss distribution approach to estimate oper-ational risk Evidence fromchinese national commercial banksrdquoInternational Journal of Information Technology amp DecisionMaking vol 8 no 4 pp 727ndash747 2009

[5] Z Sima C Cai and J Li ldquoUsing the POT power law model toevaluate banking operational riskrdquo Chinese Journal of Manage-ment Science vol 17 no 1 pp 36ndash41 2009 (Chinese)

[6] A Frachot P Georges and T Roncalli ldquoLoss distributionapproach for operational riskrdquo Working Paper Groupe deRecherche Operationnelle Credit Lyonnais France 2001

[7] A Frachot O Moudouland and T Roncalli ldquoLoss distributionapproach in practicerdquo Working Paper Groupe de RechercheOperationnelle Credit Lyonnais France 2003

[8] J Feng J Li L Gao and Z Hua ldquoA combination model foroperational risk estimation in a Chinese banking industry caserdquoThe Journal of Operational Risk vol 7 no 2 pp 17ndash39 2012

[9] A Chapelle Y Crama G Hubner and J-P Peters ldquoPracticalmethods for measuring and managing operational risk in thefinancial sector A clinical studyrdquo Journal of Banking amp Financevol 32 no 6 pp 1049ndash1061 2008

[10] C Angela R Bisignani G Masala and M Micocci ldquoAdvancedoperational risk modelling in banks and insurance companiesrdquoInvestmentManagement and Financial Innovations vol 6 no 3pp 73ndash83 2009

[11] F Aue and M Kalkbrener ldquoLDA at work Deutsche Bankrsquosapproach to quantifying operational riskrdquoThe Journal of Oper-ational Risk vol 1 no 4 pp 49ndash95 2006

[12] A Frachot T Roncalli and E Salomon ldquoThe correlation prob-lem in operational riskrdquo Working Paper Groupe de RechercheOperationnelle Credit Lyonnais France 2004

[13] K Bocker and C Kluppelberg ldquoModeling and measuringmultivariate operational risk with Levy copulasrdquoThe Journal ofOperational Risk vol 3 no 2 pp 3ndash27 2008

[14] M Bee ldquoCopula-based multivariate models with application torisk management and insurancerdquo Working Paper University ofTrento Trento Italy 2005

[15] Basel Committee on Banking Supervision OperationalRisk-Supervisory Guidelines for the Advanced MeasurementApproaches Bank for International Settlements BaselSwitzerland 2011

[16] R Giacometti S Rachev A Chernobai and M BertocchildquoAggregation issues in operational riskrdquo The Journal of Opera-tional Risk vol 3 no 3 pp 3ndash23 2008

[17] G N F Weiszlig ldquoCopula-GARCH versus dynamic conditionalcorrelation an empirical study on VaR and ES forecastingaccuracyrdquo Review of Quantitative Finance and Accounting vol41 no 2 pp 179ndash202 2013

[18] E C Brechmann C Czado and S Paterlini ldquoModelingdependence of operational loss frequenciesrdquo The Journal ofOperational Risk vol 8 no 4 pp 105ndash126 2013

[19] S Mittnik S Paterlini and T Yener ldquoOperational risk depen-dencies and the determination of risk capitalrdquo The Journal ofOperational Risk vol 8 no 4 pp 83ndash104 2013

[20] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 3rd edition2007

[21] J Li X Zhu D Wu C-F Lee J Feng and Y Shi ldquoOn theaggregation of credit market and operational risksrdquo Review ofQuantitative Finance and Accounting 2013

[22] J Li J Feng X Sun and M Li ldquoRisk integration mechanismsand approaches in banking industryrdquo International Journal ofInformation Technology amp Decision Making vol 11 no 6 pp1183ndash1213 2012

[23] R B Nelsen An Introduction to Copulas vol 139 of LectureNotes in Statistics Springer New York NY USA 1999

[24] U Cherubini E Luciano andW Vecchiato Copula Methods inFinance Wiley Finance Series John Wiley amp Sons ChichesterUK 2004

[25] I Kojadinovic and J Yan ldquoModeling multivariate distributionswith continuous margins using the copula R packagerdquo Journalof Statistical Software vol 34 no 9 pp 1ndash20 2010

[26] Joint ForumDevelopments inModelling Risk Aggregation Bankfor International Settlements Basel Switzerland 2010

[27] A Tang and E A Valdez ldquoEconomic capital and the aggrega-tion of risks using copulasrdquo Working Paper University of NewSouth Wales Sydney Australia 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Basel II Accord categorizes the operational risk event intoeight business lines that is corporate finance trading andsales retail banking commercial banking payment andsettlements agency services asset management and retailbrokerage [3] however the traditional LDA does not involvethe dependence across business lines

A more reasonable way to calculate capital charge foroperational risk is to incorporate the dependence betweendifferent business lines into LDA In this kind of LDAseparate distributions for loss frequency and severity aremodeled at business line level and then combined [10] Inthis basicmodel three kinds of dependencesmight occur andso should be taken seriously that is frequency dependenceseverity dependence and loss distribution dependence [1112] Frequency dependence is the dependence between theoccurrences of loss events In practice it means that weobserve that historically the number of (say) commercialbanking events is high (resp low) when the number of (say)corporate finance events is also high (resp low) Severitydependence is the dependence between the sizes of individualloss events Loss distribution dependence is the dependencebetween the yearly losses of business lines [11]

Many approaches modeling dependence in LDA haveemerged so far As to frequency dependence there is acontradiction On one hand the approaches modeling thefrequency dependence are probably most popular in thebanking industry On the other hand many researcheshold that frequency correlation has only a very restrictedimpact on economic capital for operational risk [11 13ndash15]For example the world leading bank in risk managementDeutsche Bank only specifies the dependence of frequencydistributions in their LDA model although they think thatthe impact of frequency dependence is insignificant [11]The main reasons responsible for this contradiction are thatcorrelation between the yearly numbers of loss event withinbusiness lines can be quite easily calculated from empiricalloss data and adding correlation between frequencies ofevents is quite an easy task and does not destroy the verynature of the LDA model Therefore this type of correlationcan be taken into account at minimum cost [12] Comparedwith frequency dependence loss distribution dependencehas been paid very limited attention Chapelle et al [9] andGiacometti et al [16] directly model the dependence of lossesthrough the use of copulas in order to combine the marginaldistributions of different business lines into a single jointloss distribution Severity dependence necessarily alters in asubstantial extent the basic foundations of the standard LDAmodel and requires building an entirely new family ofmodels[12] So it is scarcely studied in the literature and will not bediscussed in this study as well

The objective of this study is to discuss whether thepopular frequency dependence modeling is enough for thebanks when deciding to allocate operational risk capitalFirstly two approaches respectively modeling frequencydependence and loss dependence in the framework of LDAare introduced When it comes to modeling dependencecopula function has attracted broad attention because itis capable of capturing the complete dependence structure

inherent in a random vector in contrast to linear correlation[17ndash19] According to the survey of BCBS [15] dependenceis introduced into the modeling process mainly by use ofcopulas So in this study copula is employed to model bothfrequency dependence and loss dependence

Then both approaches are employed to calculate theoperational risk capital charge for Chinese banking Ourlaboratory has been devoted to collecting operational riskdata of Chinese banking for about 10 years Up to nowtotally 2132 records are collected in our dataset ranging from1994 to 2012 To the best of our knowledge this operationalrisk dataset is the most comprehensive one in China In theexperiment both approaches are employed to calculate theoperational risk capital of Chinese banking based on thisdataset

The rest of this paper is organized as follows FirstlySection 2 introduces the two approaches respectively modelfrequency dependence and loss dependence Then Section 3employs the two approaches to calculate the operationalrisk capital charge of the overall Chinese banking Section 4summarizes the conclusions

2 Approaches

In this section two approaches that are capable of modelingfrequency dependence and loss dependence in the frame-work of LDA are respectively introduced Both dependencerelations in the two approaches are modeled by copulafunctions By using the two approaches aggregate opera-tional loss distribution can be simulated from Monte CarlosimulationThen VaR is used to measure the operational riskcapital charge Pioneered by JP Morgan VaR has become astandardmeasure used in financial risk [20] VaR at a specificconfidence level 120572 isin (0 1) is defined as the smallest number119897 such that the probability of loss L exceeding l is not largerthan (1 minus 120572)

VaR = inf 119897 119875 (119871 le 119897) le (1 minus 120572) (1)

It is noteworthy that frequency refers to the number ofloss events occurring a year severity refers to the size of anindividual loss event and loss refers to the total size of all theloss events a year

21 LDA with Frequency Dependence (LDA-FD) It is ob-served that historically the number of (say) commercialbanking events is high (resp low) when the number of(say) corporate finance events is also high (resp low) Thisphenomenon implies that there is underlying correlation ordependence between the frequencies of business lines

In this approach we model the dependence structure ofthe frequencies in the framework of LDA and this approachis then named LDA-FD here The whole procedure of LDA-FD approach consists of 3 stages and 9 steps in total In Stage1 the marginal frequency distribution and marginal severitydistribution are fitted and the copula used to modelingthe dependence structure should be specified Usually thegoodness-of-fit test is used to determine the best fittingdistributions and copula In Stage 2 Monte Carlo simulation





8for the



8for the







8) into a


8) = (119866

minus1

1(1199061) 119866

minus1

8(1199068))




8



1 119897119873






8for



7 25

1144

619

263

27 30 170

200

400

600

800

1000

1200

1400


Freq

uenc

y

Business lines






8) into a


8) = (119867

minus1

1(V1) 119867

minus1

8(V8))



8


1 119897119873


1 119897119873























34= 0499 120588

35= 0267 120588

45= 0649 119875 = 002

t copula 12058834= 0571 120588

35= 0287 120588

45= 0720 ] = 1 119875 = 021


















34= 0616 120588

35= 0389 120588

45= 0249 119875 = 031

t copula 12058834= 0621 120588

35= 0484 120588

45= 0299 ] = 1 119875 = 093








4 Conclusion









2010 1756 64079 95305 672011 2234 75211 113287 662012 2596 86791 133622 65Unit billion CNY



Appendix



119891 (119909 | 120583 120590) =

1

radic2120587120590119909


21205902

119909 gt 0 119906 gt 0 120590 gt 0

(A1)



119891 (119909 | 120583 120590 119896 120582)

=

119862

120590

exp 1

[1 + sgn (119909 minus 120583 + 120575120590) 120582]119896120579119896120590119896

times1003816100381610038161003816119909 minus 120583 + 120575120590

1003816100381610038161003816

119896

(A2)

where

119862 =

119896

2120579Γ (1119896)

120579 = Γ(

1

119896

)

05

Γ(

3

119896

)

minus05

119878(120582)minus1

120575 = 2120582119860119878(120582)minus1

119878 (120582) =radic1 + 3120582

2minus 411986021205822

119860 = Γ (

2

119896

) Γ(

1

119896

)

minus05

Γ(

3

119896

)

minus05

(A3)



119891 (119896 | 119903 119901) = 119875 (119883 = 119896) = 119862119896

119896+119903minus1(1 minus 119901)

119903

119901119896

119896 = 0 1 2

(A4)



119899



minus1(119906119899))

(A5)





119862119905(1199061 119906

119899| ] 119877) = 119905]119877 (119905

minus1

] (1199061) 119905

minus1

V (119906119899)) (A6)






Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













8for the



8for the







8) into a


8) = (119866

minus1

1(1199061) 119866

minus1

8(1199068))




8



1 119897119873






8for



7 25

1144

619

263

27 30 170

200

400

600

800

1000

1200

1400


Freq

uenc

y

Business lines






8) into a


8) = (119867

minus1

1(V1) 119867

minus1

8(V8))



8


1 119897119873


1 119897119873























34= 0499 120588

35= 0267 120588

45= 0649 119875 = 002

t copula 12058834= 0571 120588

35= 0287 120588

45= 0720 ] = 1 119875 = 021


















34= 0616 120588

35= 0389 120588

45= 0249 119875 = 031

t copula 12058834= 0621 120588

35= 0484 120588

45= 0299 ] = 1 119875 = 093








4 Conclusion









2010 1756 64079 95305 672011 2234 75211 113287 662012 2596 86791 133622 65Unit billion CNY



Appendix



119891 (119909 | 120583 120590) =

1

radic2120587120590119909


21205902

119909 gt 0 119906 gt 0 120590 gt 0

(A1)



119891 (119909 | 120583 120590 119896 120582)

=

119862

120590

exp 1

[1 + sgn (119909 minus 120583 + 120575120590) 120582]119896120579119896120590119896

times1003816100381610038161003816119909 minus 120583 + 120575120590

1003816100381610038161003816

119896

(A2)

where

119862 =

119896

2120579Γ (1119896)

120579 = Γ(

1

119896

)

05

Γ(

3

119896

)

minus05

119878(120582)minus1

120575 = 2120582119860119878(120582)minus1

119878 (120582) =radic1 + 3120582

2minus 411986021205822

119860 = Γ (

2

119896

) Γ(

1

119896

)

minus05

Γ(

3

119896

)

minus05

(A3)



119891 (119896 | 119903 119901) = 119875 (119883 = 119896) = 119862119896

119896+119903minus1(1 minus 119901)

119903

119901119896

119896 = 0 1 2

(A4)



119899



minus1(119906119899))

(A5)





119862119905(1199061 119906

119899| ] 119877) = 119905]119877 (119905

minus1

] (1199061) 119905

minus1

V (119906119899)) (A6)






Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























34= 0499 120588

35= 0267 120588

45= 0649 119875 = 002

t copula 12058834= 0571 120588

35= 0287 120588

45= 0720 ] = 1 119875 = 021


















34= 0616 120588

35= 0389 120588

45= 0249 119875 = 031

t copula 12058834= 0621 120588

35= 0484 120588

45= 0299 ] = 1 119875 = 093








4 Conclusion









2010 1756 64079 95305 672011 2234 75211 113287 662012 2596 86791 133622 65Unit billion CNY



Appendix



119891 (119909 | 120583 120590) =

1

radic2120587120590119909


21205902

119909 gt 0 119906 gt 0 120590 gt 0

(A1)



119891 (119909 | 120583 120590 119896 120582)

=

119862

120590

exp 1

[1 + sgn (119909 minus 120583 + 120575120590) 120582]119896120579119896120590119896

times1003816100381610038161003816119909 minus 120583 + 120575120590

1003816100381610038161003816

119896

(A2)

where

119862 =

119896

2120579Γ (1119896)

120579 = Γ(

1

119896

)

05

Γ(

3

119896

)

minus05

119878(120582)minus1

120575 = 2120582119860119878(120582)minus1

119878 (120582) =radic1 + 3120582

2minus 411986021205822

119860 = Γ (

2

119896

) Γ(

1

119896

)

minus05

Γ(

3

119896

)

minus05

(A3)



119891 (119896 | 119903 119901) = 119875 (119883 = 119896) = 119862119896

119896+119903minus1(1 minus 119901)

119903

119901119896

119896 = 0 1 2

(A4)



119899



minus1(119906119899))

(A5)





119862119905(1199061 119906

119899| ] 119877) = 119905]119877 (119905

minus1

] (1199061) 119905

minus1

V (119906119899)) (A6)






Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















34= 0616 120588

35= 0389 120588

45= 0249 119875 = 031

t copula 12058834= 0621 120588

35= 0484 120588

45= 0299 ] = 1 119875 = 093








4 Conclusion









2010 1756 64079 95305 672011 2234 75211 113287 662012 2596 86791 133622 65Unit billion CNY



Appendix



119891 (119909 | 120583 120590) =

1

radic2120587120590119909


21205902

119909 gt 0 119906 gt 0 120590 gt 0

(A1)



119891 (119909 | 120583 120590 119896 120582)

=

119862

120590

exp 1

[1 + sgn (119909 minus 120583 + 120575120590) 120582]119896120579119896120590119896

times1003816100381610038161003816119909 minus 120583 + 120575120590

1003816100381610038161003816

119896

(A2)

where

119862 =

119896

2120579Γ (1119896)

120579 = Γ(

1

119896

)

05

Γ(

3

119896

)

minus05

119878(120582)minus1

120575 = 2120582119860119878(120582)minus1

119878 (120582) =radic1 + 3120582

2minus 411986021205822

119860 = Γ (

2

119896

) Γ(

1

119896

)

minus05

Γ(

3

119896

)

minus05

(A3)



119891 (119896 | 119903 119901) = 119875 (119883 = 119896) = 119862119896

119896+119903minus1(1 minus 119901)

119903

119901119896

119896 = 0 1 2

(A4)



119899



minus1(119906119899))

(A5)





119862119905(1199061 119906

119899| ] 119877) = 119905]119877 (119905

minus1

] (1199061) 119905

minus1

V (119906119899)) (A6)






Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















2010 1756 64079 95305 672011 2234 75211 113287 662012 2596 86791 133622 65Unit billion CNY



Appendix



119891 (119909 | 120583 120590) =

1

radic2120587120590119909


21205902

119909 gt 0 119906 gt 0 120590 gt 0

(A1)



119891 (119909 | 120583 120590 119896 120582)

=

119862

120590

exp 1

[1 + sgn (119909 minus 120583 + 120575120590) 120582]119896120579119896120590119896

times1003816100381610038161003816119909 minus 120583 + 120575120590

1003816100381610038161003816

119896

(A2)

where

119862 =

119896

2120579Γ (1119896)

120579 = Γ(

1

119896

)

05

Γ(

3

119896

)

minus05

119878(120582)minus1

120575 = 2120582119860119878(120582)minus1

119878 (120582) =radic1 + 3120582

2minus 411986021205822

119860 = Γ (

2

119896

) Γ(

1

119896

)

minus05

Γ(

3

119896

)

minus05

(A3)



119891 (119896 | 119903 119901) = 119875 (119883 = 119896) = 119862119896

119896+119903minus1(1 minus 119901)

119903

119901119896

119896 = 0 1 2

(A4)



119899



minus1(119906119899))

(A5)





119862119905(1199061 119906

119899| ] 119877) = 119905]119877 (119905

minus1

] (1199061) 119905

minus1

V (119906119899)) (A6)






Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article operational risk aggregation across...

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