Quantifying operational risk guided by kernel ... - Cass Knowledge

Quantifying operational risk guided by 

kernel smoothing and continuous 

credibility 

Jim Gustafsson 

Codan Insurance 

Jens P. Nielsen, Paul Pritchard and Dix Roberts 

Royal&SunAlliance 

Codan, Gammel Kongevej 60 

DK 1790 Copenhagen V,Denmark 

e-mail: jgu@codan.dk,e-mail: npj@codan.dk 

One Plantation Place 9th Floor 

30 Fenchurch Street, London, EC3M 3BD, UK 

e-mail: paul.pritchard@gcc.royalsun.com, e-mail: dix.roberts@gcc.royalsun.com 

Abstract: The challenge of how much capital is necessary to protect an 

organisation against exposure to operational risk losses underpins this pa- 

per (operational risk itself is defined as the risk of loss arising from inad- 

equate or failed internal processes, people and systems or from external 

events). The evolutionary nature of operational risk modelling to establish 

capital charges is recognised emphasizing the importance of capturing tail 

behaviour. Challenges surrounding the quantification of operational risk 

particularly those associated with sparse data are addressed with mod- 

ern statistical methodology including nonparametric smoothing techniques 

with a particular view to comparison with extreme value theory (EVT). The 

credibility approach employed supports analysis from pooled data across 

business lines on a dataset from an internationally active insurance com- 

pany. The approach has the potential to be applied more generally, for 

example where data might be pooled across risk types or where a combi- 

nation of internal company losses and publicly reported (external) data is 

1 

imsart ver. 2005/05/19 file: fest_gustafsson.tex date: December 20, 2005

Gustafsson, Nielsen, Pritchard and Roberts/Quantifying operational risk 2 

used. 

AMS 2000 subject classifications: Primary 65D10; secondary 91B30. 

Keywords and phrases: Operational risk, kernel smoothing, credibility 

theory, extreme value theory. 



1. Introduction 

There is increasing interest in financial services companies in identifying loss dis- 

tributions associated with operational risks, driven by both regulatory consid- 

erations and also in recognition of the greater importance placed on operational 

risk management. Building on the 1988 Basel Capital Accord, the Basel Com- 

mittee on Banking Supervision published; International Convergence of Capital 

Measurement and Capital Standards 1 , in June 2004. This document addressed 

the challenge of how much capital is necessary to protect an organization against 

unexpected losses, and established the need for an explicit charge for the expo- 

sure to operational risk losses. Operational risk itself is defined as the risk of 

loss arising from inadequate or failed internal processes, people and systems or 

from external events. The framework also recognized the evolutionary nature of 

operational risk modelling emphasizing the importance of capturing tail events. 

Challenges surrounding the quantification of operational risk are the sub- 

ject of this paper and include lack of suitable data and the need to focus on 

tail behaviour. These challenges are addressed with modern statistical method- 

ologies including nonparametric smoothing techniques. One of the commonly 

applied techniques in operational risk is Extreme Value Theory (EVT) which 

offers a broadly accepted methodology for estimating the tail of a distribution, 

see Embrecht, Klüppelberg, and Mikosch [9] for a detailed mathematical treat- 

ment, also Embrecht [10], Reiss and Thomas [23] and Coles [7]. It is our view 

that EVT shows conceptual similarities to other established density estimation 

techniques including the approach we consider in this paper. In essence these 

techniques seek to estimate a density with particular emphasis on the tail. In 

our approach we transform losses into the interval [0,1]. 

The introduction of the concept of local constant and local linear density 

estimators by Jones [16] was a very important contribution since both automat- 

1 Details at http//www.bis.org/publ/bcbs107.htm. 



ically adjust at boundaries. Jones [16] also pointed out the importance of dirac 

functions while dealing with local projections of data. When dealing with re- 

gression, local constant and local linear estimators projections can be obtained 

relatively easily, see Fan and Gijbels [11] for an overview. However, in other 

areas of mathematical statistics such as density estimation and hazard estima- 

tion, the dirac function approach becomes important. See Nielsen [18] for the 

multivariate density case and Nielsen [19] and Nielsen and Tanggaard [21] for 

the hazard case. 

The understanding of density and hazard estimation has recently been taken 

to a deeper level through the papers of Jiang and Doksum [14] and Jiang and 

Doksum [15]. These papers give an overview on how local polynomial densities 

and hazards are to be understood as projections even when complicated trun- 

cation and censoring is present. In general terms, these papers introduce local 

polynomial hazards and densities as simple plug-in estimators with the empir- 

ical cumulative densities and hazards being utilized. The asymptotic theory of 

the estimators and their derivatives follow as an immediate consequence. 

The parametric transformation approach to kernel smoothing of Wand, Mar- 

ron and Ruppert [25] has recently been considered with a particular interest in 

heavy tailed distributions by Bolance, Guillen and Nielsen [1], Clements, Hurn 

and Lindsay [6] and Buch-Larsen, Nielsen, Bolance and Guillen [2]. We use 

the methodology of Buch-Larsen et al. [2] in this paper. They utilize the three 

parameter modified Champernowne distribution which not only demonstrates 

desired tail behavior, but unlike EVT, is also informed by data from the full 

distribution. 

Application of a non-parametric local constant kernel estimator allows an im- 

proved fitting to the sample data. Consideration is given here to the issues raised 

by Diebold, Schuermann and Stroughair [8], who noted that the bias/variance 

trade-off in selection of a tail cut-off for EVT applications is analogous to that 

relating to the choice of bandwidth in non-parametric density estimation. In 



our approach the established approach to selection of bandwidth in density es- 

timation replaces the complex question of selecting the optimal cut off in EVT. 

Diebold et al. [8] highlighted the ‘unfortunate’ use of automatic bandwidth se- 

lection to support their statement that kernel smoothing performs poorly in 

the tail (away from the region where most of the data falls), Buch-Larsen et 

al. [2] point out however that the transformation approach to kernel density 

estimation effectively resolves this issue. 

In summary this paper is principally concerned with tools with potentially 

broad application to operational risk quantification. These are applied to an op- 

erational loss data set obtained from an internationally active insurer. Following 

application of non-parametric kernel smoothing we apply continuous credibility 

theory to density estimation facilitating the appropriate weighting of pooled 

(portfolio) losses as compared to data from individual business lines. This theo- 

retical approach is inspired by the continuous credibility technique that Hardy 

and Panjer [12] and Nielsen and Sandqvist [20] introduced to hazard estimation. 

The techniques considered in this paper complement and add to existing knowl- 

edge demonstrating potential utility in the significant computational challenges 

posed by operational risk quantification. Not only is this potentially useful in the 

context described here i.e. pooling data across business lines but also in con- 

sidering pooling of internal (company) data with that from other companies, 

reported publicly or shared through consortia. 

Section 2 describes the structure and general assumptions underpinning the 

proposed credibility model on densities in the interval [0, 1]. Section 3 is devoted 

to the nonparametric kernel density estimator, and section 4 presents the esti- 

mation of the credibility model. Section 5 is an application to operational risk 

data. Here we lay out the transformation process also a plug-in bandwidth is 

discussed. The final section evaluates the operational risk exposure based on a 

one year simulation using the loss data. 



2. The credibility model on densities in interval [0,1] 

We assume that we have k lines of operational risk and consider an i belonging 

to the set {1, ..., k}. For such an i assume that Xi1, ..., Xini are independent 

identically distributed stochastic variables with density fi on [0, 1]. These den- 

sities are assumed to be stochastically varying around a common base density 

function f. More precisely, we assume that 

fi(x) = θi(x)f(x), (2.1) 

where x ∈ [0, 1] and θ1(x), ..., θk(x) are identically distributed stochastic pro- 

cesses defined on the interval [0, 1]. This model assumption assumes a relation- 

ship between the k business lines. Also we assume that � θi(x)f(x)dx = 1, 

E(θi(x)) = 1 and V (θi(x)) = ν 2 i 

for every i between 1 and k. The assumption 

that E{θi(x)} = 1 for all i is crucial for the estimators developed. This assump- 

tion implies that for every single x, the business lines varies around the common 

base density function f. This assumption allows us to construct our estimators 

from a Hilbert space projection of the fi(x) ′ s down at their estimators for each 

x individually. The model proposes that each line has risk stemming from a 

common source as well as its own line of risk. The multiplicative construction 

above is known from the bias reduction literature, see Hjort and Glad [13] and 

Jones, Linton and Nielsen [17]. The difference is, however, that there the pur- 

pose is to eliminate bias while here we use the element in the structure as a 

stochastic process. The multiplicative structure of our approach is an important 

part of our model building. In the bias reduction literature the multiplicative 

structure is just used as a trick to reduce bias. However, similarities remain: 

we also estimate the multiplicative error nonparametrically and use it for our 

estimation purposes - just like one does in the bias reduction literature. 



3. The kernel density estimator 

Let K denote a probability density function symmetric about zero with support 

[−1, 1] and let Kh(x) = K(x/h)/h for any x in [0, 1] and any positive h. We 

also define the following functions for the asymptotic properties of the kernel 

density estimator around boundaries. 

akl(x, h) = 

min{1,x/h} � 

max{−1,(x−1)/h} 

u k K(u) l du, for x ∈ [0, 1]. 

Note that a01(x, h) = 1 and a11(x, h) = 0 for the interior points x in the 

interval [h, 1 − h]. In the boundary points belonging to the intervals [0, h) and 

(1−h, 1], a01(x, h) and a11(x, h) take nontrivial values. The local constant kernel 

estimator 2 of fi is 

�fi(x) = (a01(x, hi)ni) −1 

ni � 

Khi (x − Xij) . (3.1) 

The local constant kernel estimator of the entire data set can be taken as an 

estimator of the common base density f. We assume that the estimation error 

of the global kernel density estimator 

⎛ 

k� 

�f(x) = ⎝(a01(x, hi)ni) −1 

⎞ 

ni � 

Khi (x − Xij) ⎠ (3.2) 

i=1 

is of lower order of magnitude than the estimation error of the individual densi- 

ties f1, ..., fk. This is important for our continuous credibility approach defined 

in the next section. 

4. Estimation of the credibility model 

Credibility theory as known from actuarial science is simply a methodology to 

find out how much information a common model carries onto a specific line of 

2 See Jones [16] for a definition of local costant and local linear density estimators and for 

j=1 

j=1 

a precision of the automatic boundary corrections of these estimators. 



business. Credibility theory tells us that we can rely more on data from one 

specific line when it is plentiful, when data for one line is sparse more weight 

should be put on a common source. In our case we fix x ∈ [0, 1] and define a 

Hilbert space projection, � fi(x), of fi(x) defined by (2.1), onto the linear space 

{ax + bx � fi(x) | ax, bx ∈ [0, 1]}. We get 

�fi(x) = (1 − zi,x) E( � fi(x)) + zi,x � fi(x), (4.1) 

where the credibility factor zi,x = COV (fi(x), � fi(x))/V ( � fi(x)) is between zero 

and one. This projection gives us the optimal linear credibility estimator � fi(x) 

minimizing E((fi(x) − � fi(x)) 2 ), see also Hardy and Panjer [12] and Nielsen and 

Sandqvist [20]. 

The credibility factor quantifies the amount of weight the common mean 

and the individual mean should have for each of the lines of operational risk. 

For the original approach to credibility theory, see Bühlmann and Straub [3]. 

For a recent overview of the origins of credibility theory and a Hilbert space 

interpretation of the Bühlmann and Straub model, see Norberg [22]. To calculate 

the three moment quantities of (4.1), we prove in the Appendix that 

and 

V ( � fi(x)) = 

E( � fi(x)) = f(x) (1 + op(1)) , 

COV (fi(x), � fi(x)) = f(x) 2 ν 2 i (1 + op(1)) 

� 

f(x) 2 ν 2 i + a02(x, hi)f(x) 

a01(x, hi) 2 � 

(1 + op(1)) . 

hini 

From these moment expressions we can approximate the optimal credibility 

estimator of (4.1) to 

where 

zi,x = 

�fi(x) = (1 − zi,x) f(x) + zi,x � fi(x), 

f(x) 2 ν 2 i a01(x, hi) 2 hini 

f(x) 2 ν 2 i a01(x, hi) 2 hini + f(x)a02(x, hi) (1 + op(1)) . 



To estimate the local variance ν 2 i of the stochastic process we note that � fi(x) is 

an estimator of fi(x) = θi(x)f(x) and 

�θi(x) = � fi(x) 

�f(x) 

(4.2) 

is an estimator of θi(x). Therefore, for a given x, we estimate the variance of 

θi(x) by 

�ν 2 i = 1 

ni � 

ni 

j=1 

� �2 �θi(xj) − 1 = 

� 1 

0 

� �2 �θi(x) − 1 d � Fi(x), (4.3) 

where � Fi is the empirical distribution function of the i’th business line. 

We can hereby estimate zi,x by 

�zi,x = 

�f(x) 2�ν 2 i a01(x, hi) 2hini �f(x) 2�ν 2 i a01(x, hi) 2hini + � . (4.4) 

f(x)a02(x, hi) 

Combining (3.1), (3.2) and (4.4) the final expression for the optimal credibility 

estimator is approximately equal to 

� �f i(x) = (1 − �zi,x) � f(x) + �zi,x � fi(x). (4.5) 

To arrive at a credibility based estimator of θi(x) we use the above equality 

(4.5) divided by � f(x) 

� �θi(x) = (1 − �zi,x) + �zi,x 

�fi(x) 

�f(x) = 1 − �zi,x(1 − � θi(x)). (4.6) 

We propose here that each line has risk stemming from a common source as well 

as its own individual source. This property of our basic model is clearly reflected 

in the expression of our final estimators � � f i(x) and � � θi(x); both are constructed 

as a sum of one common and one global element. 

5. Application to operational risk data 

We utilize data from seven lines of operational risk with observed number of 

losses N1, ..., N7. The data periods in these seven groups measured in years 

are T1, ..., T7. We assume that we have seven independent homogenous Poisson 



processes describing the number of losses in each line and therefore obtain the 

following maximum likelihood estimator of the intensity of losses in the i th 

business line: � λi = Ni/Ti. Conditional on N1 = n1, ..., N7 = n7, we now have 

our operational risk data for each group, Yi1, ..., Yini , defined on the positive 

real axis. We now transform our data from the positive real axis to the interval 

[0, 1] using the modified Champernowne distribution defined in Buch-Larsen et 

al. [2]: 

Fp(y) = 

(y + c) α − cα (y + c) α + (M + c) α , y ∈ R+, 

− 2cα where p = {α, M, c} is a parameter vector. For c = 0 this distribution is a 

special case of the parametric distribution suggested by Champernowne [4] and 

Champernowne [5] and the cumulative density function equals the transforma- 

tion used by Clements et al. [6] in their approach to density estimation based 

on transformed data. However, an extensive simulation study by Buch-Larsen 

et al. [2] show that the flexibility of the modified Champernowne distribution 

outweigh the advantages of the stability obtained in the simple case of c = 0. 

Buch-Larsen et al. [2] estimate M, the median, from the empirical median and 

the parameters (α, c) by the maximum likelihood method. We use the modified 

Champernowne distribution for the i th line of business as follows 

Fpi(y) = 

(y + c) α − c α 

(y + c) α + (Mi + c) α − 2c α , y ∈ R+, (5.1) 

where pi = {α, Mi, c} is the extended parameter vector for the individual line 

of business. We use the same values of the variables (α, c) for the seven lines of 

operational risk while the median, a scaling parameter, is estimated individually 

for each line. This parametric model is the common source of information for 

our operational risk modelling. Firstly we transform the data from each line into 

the interval [0, 1] using this parametric model. That is, we define 

Xij = F�pi (Yij), (5.2) 



as the transformed losses obtained by the cumulative modified Champernowne 

distribution, where �pi = {�α, � Mi, �c} are the estimated parameter vector for 

i ∈ {1, . . . , k} on observations j ∈ {1, . . . , ni}. Now we are ready to apply 

the methodology outlined in sections 2 and 3 using the transformed data set 

Xij. Note that if we had only one line of business, then the credibility approach 

would of course be superfluous and we have the exact same one-dimensional 

estimation method as the one suggested in Buch-Larsen et al. [2]. When esti- 

mating the initial individual kernel estimator of the i th line of business, defined 

by (3.1), we use the Epanechnikov kernel function: 

K(u) = 3 � 2 

1 − u 

4 

� 1 {|u|


Consequently, if we differentiate this with respect to the bandwidth hi, we obtain 

the theoretical optimal choice 

� 

a02(x, hi) 

hi = 

a21(x, hi) 2 �1/5 � . 

ni f ′′ (x) 2dx Here everything is known except for the density f. Assuming the unknown f 

follows a normal distribution, we obtain Silverman’s rule of thumb (Silverman 

[24]): 

� √ �1/5 40 π 

hi = 

�σi. (5.3) 

ni 

The following section shows a summary of the operational loss data set fol- 

lowed by a visual comparison applied to the data. 

Table 1: Summary Statistics- operational loss data set. 

Line of 

Business ni Ti hi mean(Yij) median(Yij) sd(Yij) max(Yij) 

1 250 3 0.24 8981 4059.5 16490 163233 

2 46 3 0.36 13450 2829.5 58512 394969 

3 924 3 0.16 7835 2990 36164 874400 

4 34 3 0.47 13963 2277 18467 52200 

5 7 3 0.48 19633 14610 14621 38205 

6 7 3 0.58 959382 197600 2123445 5765217 

7 23 2 0.44 1127077 242220 2333035 8477700 

The second column shows the number of observations ni for each line of 

business. There is considerable variation in the number of recorded losses, line 

of business 3 has 925 losses compared to line 5 and 6 which only have been 

exposed to 7 losses each. The third column gives the time over which data 

was collected, in years. The fourth column shows the estimated bandwidths, 

obtained by (5.3). Note that the estimated bandwidths decrease with sample 

size as expected. Columns five to eight show some empirical results on each line 



of business. Note that the mean is significantly larger than the median in all 

cases, consistent with right skewed distributions. 

Figure 1 shows histograms of both the transformed global data set and the 

seven individual data sets. In each graph the respective semiparametric estima- 

tors are estimated. If we examine the global estimator � f(x) more closely (top-left 

graph), we observe that in the interval [0, 0.3) the estimator is less than one. 

This means that the kernel estimator corrects the parametric density where it 

has diverged from the operational risk data. In [0.3, 0.7) the kernel function 

corrects the parametric density where it was too low. In [0.7, 1] the estimator is 

once again below one and therefore adjustment is made, equivalent to a lighter 

tail. The other individual estimators � fi(x), i = 1, ...7, should be interpreted 

analogously. 

Insert Figure 1 about here 

In Figure 2 we show the credibility approach applied to each line. Here we 

omit the histograms to allow easier visual interpretation. We compare the stan- 

dard semiparametric estimator (3.1), show with a dotted curve, with an es- 

timator where we have applied the credibility approach (solid curve), defined 

through (4.5). Note that for lines of operational risk with sparse information the 

appearance of the credibility estimator is similar to the global source. This is be- 

cause the global source provides more weight for lines with sparse information, 

particular lines 5, 6 and 7. 


Figure 3 shows the estimated stochastic processes with and without the appli- 

cation of a credibility approach. The left-hand graph shows estimated stochastic 

processes for all lines estimated through formula (4.2). Here we can see that the 

processes take values between 0.5 and 1.5. This means that when a line of 

business deviates from the horizontal we know that there is a large difference 

between the specific line and the global source. The right-hand graph shows 



the estimated stochastic processes applying the credibility approach, calculated 

using (4.6). The most obvious difference is that the lines are much closer to the 

horizontal. This is, of course, to be expected since, to a greater or lesser degree, 

the individual lines will use information from the global source. 


6. Evaluating an operational loss distribution 

In this section we simulate an operational risk loss distribution for the company. 

This is done by using the severity distribution obtained from (4.5), together with 

a Poisson based frequency distribution. The frequency and severity distributions 

are used to create simulated one year loss distribution through Monte Carlo 

analysis. We sample from a Poisson process of event times through all lines of 

operational risk, and combine with loss sizes taken form the relevant severity 

distribution. To obtain our original scale, we transform the estimator (4.5) to 

its original axis, The relevant estimators takes the form 

ni � 

�fi(y) 

−1 

= (a01(F�pi (y), hi)ni) 

�f(y) = 

j=1 

⎛ 

k� 

ni � 

⎝(a01(F�pi 

−1 

(y), hi)ni) 

i=1 

Khi (F�pi (y) − F�pi (Yij)) f�pi (y) (6.1) 

j=1 

Khi (F�pi (y) − F�pi (Yij)) f�pi (y) 

where f�pi (y) is the modified Champernowne density defined by 

f�pi (y) = α(y + c)α−1 ((Mi + c) α − cα ) 

((y + c) α + (Mi + c) α − 2cα 2 , y ∈ R+. 

) 

The loss sizes are then taken from the severity distribution 

� ∞ 

� 

Hi(y) = �f i(ξ)dξ, 

0 

where � � f i are defined through the estimators in (6.1) with structure as (4.6). 

Summation of amounts from all lines provides a single estimate, this process is 

then repeated 20000 times to create a simulated loss distribution as shown in 

imsart ver. 2005/05/19 file: fest_gustafsson.tex date: December 20, 2005 

⎞ 

⎠


Figure 4. From this we can identify the loss amount associated with the relevant 

quantile and thus the capital to be held. 


In table 2 we present summary statistics for the individual loss distribution 

and for the global source. 

Table 2: Summary statistics for simulated loss distribution 

by business line and global source. 

Line of 

Business x0.5 OpVaR0.95 OpVaR0.99 OpVaR0.999 

1 1840605 372988 528891 747380 

2 238045 138215 212161 294414 

3 5657744 1124957 1690317 1897271 

4 156984 118278 180286 258575 

5 154260 342791 594155 843894 

6 2088010 4865225 10405343 15955968 

7 9856090 8621910 13031243 19455952 

Global 20706695 9675355 14890803 22462507 

The Operational Value-at-Risk (OpVaR) from table 2 is a measure of the 

unexpected losses to a specific quantile level. To obtain this value we subtract 

the median (presented as column 1) from three different upper quantiles with 

confidence levels 95%, 99% and 99.9%. Table 2 shows that the appearance in 

the tail region varies between the lines of business. More precisely, line 1 and 3 

generate a light tailed distribution while line of business 2, 4 and 7 are charac- 

terized by moderate tails. The remaining two lines of business 5 and 6 are heavy 

tailed distributed. This is not extraordinary since both lines are embodied with 

extremely small samples. 

Table 3 shows measures of the length of the tail. This is done by dividing 

the upper quantile (with one of the three confidence levels) by the median, i.e. 



ϕ (q) = xq/x0.5 , q = 0.95, 0.99 or 0.999. Here, as in table 2 lines of business 5 

and 6 stand out being most heavy tailed. 

Table 3: Tail length characteristics. 

Line of 

Business ϕ (0.95) ϕ (0.99) ϕ (0.999) 

1 1.203 1.287 1.406 

2 1.581 1.891 2.237 

3 1.199 1.299 1.335 

4 1.753 2.148 2.647 

5 3.222 4.852 6.471 

6 3.330 5.983 8.642 

7 1.875 2.320 2.974 

Global 1.467 1.719 2.085 

In this paper we have demonstrated that a number of statistical approaches 

(complementary to those commonly applied such as EVT) can provide signif- 

icant benefit in the quantification of operational risk for estimation of capital 

requirements in organizations. The fundamental challenge, that of sparse data, 

is addressed by transforming available data and then applying non-parametric 

kernel smoothing. This approach (even without further application of credibility 

analysis) should assist in maximizing useful information from limited data sets, 

particularly as the direct application of non parametric techniques would not 

yield useful information. The subsequent credibility element could also be ap- 

plied in isolation when considering data that is pooled (e.g. across businesses or 

risk types) or for the mixing of internal and external data (e.g. from a subscrip- 

tion database). We acknowledge the limitations inherent in this work, specifically 

that the data set used does not give full coverage across all operational risk cat- 

egories and that it is based on a limited collection period. Nevertheless for the 

purposes of illustration it does demonstrate that the techniques can be applied 



to real life company data, furthermore that considerable potential exists for its 

further application in this area. 

7. Appendix 

Through standard kernel smoothing theory we obtain 

and 

� � 

E �fi(x) | θi(x) 

� � 

E �fi(x) 

= θi(x)f(x) (1 + op(1)) 

� � �� 

= E E �fi(x) | θi(x) 

= E (θi(x)f(x)) (1 + op(1)) 

= f(x) (1 + o(1)) . 

To derive the variance expressions we first note that 

V 

This implies that 

and 

V 

� � 

�fi(x) 

� � 

�fi(x) | θi(x) 

= V 

= 

= θi(x)a02(x,hi)f(x) 

a01(x,hi) 2 hini 

� � �� 

E �fi(x) | θi(x) 

� 

f(x) 2 ν 2 i 

+ a02(x,hi)f(x) 

a01(x,hi) 2 hini 

� 

+ E 

(1 + op(1)) . 

V 

� �� 

�fi(x) | θi(x) 

� 

(1 + op(1)) . 

� 

COV fi(x), � � 

fi(x) 

� 

� �� 

= COV E (fi(x) | θi(x)) , E �fi(x) | θi(x) 

� � 

+E COV fi(x), � �� 

fi(x) | θi(x) 

� � 

= E COV fi(x), � �� 

fi(x) | θi(x) 

= � E � f(x) 2 θi(x) 2� − f(x) 2� (1 + op(1)) 

= � f(x) 2 E � θi(x) 2 − 1 �� (1 + op(1)) 

= f(x) 2 ν 2 i (1 + op(1)) . 


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0.0 0.2 0.4 0.6 0.8 1.0 1.2 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 


Global 

0.0 0.4 0.8 

Line 4 

0.0 0.4 0.8 

0.0 0.5 1.0 1.5 

0.0 0.5 1.0 1.5 2.0 2.5 

Line 1 

0.0 0.4 0.8 

Line 5 

0.0 0.4 0.8 

0.0 0.5 1.0 1.5 2.0 

0.0 0.5 1.0 1.5 2.0 2.5 

Line 2 

0.0 0.4 0.8 

Line 6 

0.0 0.4 0.8 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 

0.0 0.5 1.0 1.5 2.0 

Line 3 

0.0 0.4 0.8 

Line 7 

0.0 0.4 0.8 

Fig 1: Histogram of data,including the global semiparametric estimator (3.2) in the 

top-left graph and the individual semiparametric estimators defined by (3.1) 

in the remaining graphs. 

0.6 0.8 1.0 1.2 1.4 1.6 

0.6 0.8 1.0 1.2 1.4 1.6 

Global 

0.0 0.4 0.8 

Line 4 

0.0 0.4 0.8 

0.6 0.8 1.0 1.2 1.4 1.6 

0.6 0.8 1.0 1.2 1.4 1.6 

Line 1 

0.0 0.4 0.8 

Line 5 

0.0 0.4 0.8 

0.6 0.8 1.0 1.2 1.4 1.6 

0.6 0.8 1.0 1.2 1.4 1.6 

Line 2 

0.0 0.4 0.8 

Line 6 

0.0 0.4 0.8 

0.6 0.8 1.0 1.2 1.4 1.6 

0.6 0.8 1.0 1.2 1.4 1.6 

Line 3 

0.0 0.4 0.8 

Line 7 

0.0 0.4 0.8 

Fig 2: The severity densities for all lines of operational risk including also the global 

source (3.2) by top-left graph. Dotted curve represent standard 

semiparametric approach (3.1), while the solid curve is estimators with an 

credibility approach (4.5) 


0.5 1.0 1.5 


Line 1 

Line 2 

Line 3 

Line 4 

Line 5 

Line 6 

Line 7 

0.0 0.2 0.4 0.6 0.8 1.0 

0.5 1.0 1.5 

Line 1 

Line 2 

Line 3 

Line 4 

Line 5 

Line 6 

Line 7 

0.0 0.2 0.4 0.6 0.8 1.0 

Fig 3: The stochastic processes for all lines of operational risk. The left-hand graph 

presents processes without credibility defined by (4.2), and the right-hand 

graph with the application of the credibility approach calculated in (4.6). 

Total Number 

0 200 400 600 800 1000 

Expected Losses Unexpected Losses 

Mean Value 99% quantile 

10.000.000 20.000.000 30.000.000 40.000.000 50.000.000 

Annual Aggregated Loss Amount (DKK) 

Fig 4: The simulated loss distribution based on 20000 simulation summation over all 

lines of operational risk.

Quantifying operational risk guided by kernel ... - Cass Knowledge

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