Operational Risk Management: A Review

Operational Risk Management: A Review

Dean Fantazzini

Moscow

Overview of the Presentation

• Introduction

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• Introduction

• The Basic Indicator Approach

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• Introduction


• The Standardized Approach

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• Introduction



• Advanced Measurement Approaches

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• Introduction




• The Standard LDA Approach with Comonotonic Losses

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• Introduction





• The Canonical Aggregation Model via Copulas

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• Introduction






• The Poisson Shock Model

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• Introduction






• The Poisson Shock Model

• Bayesian Approaches

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Introduction

• What are operational risks?

The term “operational risks” is used to define all financial risks that are

not classified as market or credit risks. They may include all losses due to

human errors, technical or procedural problems etc.

→ To estimate the required capital for operational risks, the Basel

Committee on Banking supervision (1998-2005) allows for both a simple

“top-down” approach, which includes all the models which consider

operational risks at a central level, so that local Business Lines (BLs) are

not involved.

→ And a more complex “bottom- up” approach, which measures

operational risks at the BLs level, instead, and then they are aggregated,

thus allowing for a better control at the local level.

(LDA).

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Introduction

Particularly, following BIS (2003), banks are allowed to choose among three

different approaches:

• The Basic Indicator approach (BI),

• the Standardized Approach (SA),

• the Advanced Measurement Approach (AMA).

If the basic indicator approach is chosen, banks are required to hold a flat

percentage of positive gross income over the past three years.

If the standardized approach is chosen, banks’ activities are separated into a

number of business lines. A flat percentage is then applied to the three year

average gross income for each business line.

Instead, if the advanced measurement approach is chosen, banks are allowed to

develop more sophisticated internal models that considers the interactions

between different BL and ET and they have to push forward risks mitigation

strategies.

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The Basic Indicator Approach

Banks using the basic indicator (BI) approach are required to hold a

capital charge set equal to a fixed percentage (denoted by α) of the

positive annual gross income (GI).

If the annual gross income is negative or zero, it has to be excluded when

calculating the average. Hence, the capital charge for operational risk in

year t is given by

RCtBI =

1

Zt

3∑

i=1

α max(GIt−i

, 0) (1)

where Zt =∑3

i=1 I[GIt−i>0] and GIt−i stands for gross income in year t− i.

Note that the operational risk capital charge is calculated on a yearly basis.

the Basel Committee has suggested α = 15%.

→ This is a straightforward, volume-based, one-size-fit-all capital charge.

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The Standardized Approach

The BI is designed to be implemented by the least sophisticated banks. Moving

to the Standardized Approach requires the bank to collect gross income data by

business lines.

The model specifies eight business lines : Corporate Finance, Trading and Sales,

Retail Banking, Commercial Banking, Payment and Settlement, Agency Services

and Custody, Asset Management and Retail Broker.

For each business line, the capital charge is calculated by multiplying the gross

income by a factor denoted by β assigned to that business line.

The total capital charge is then calculated as a three-year average over positive

gross incomes, resulting in the following capital charge formula:

RCtSA =

1

3

3∑

i=1

max

8∑

j=1

βjGIt−ij , 0

(2)

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The Standardized Approach

We remark that in formula (2), in any given year t − i, negative capital charges

resulting from negative gross income in some business line j may offset positive

capital charges in other business lines (albeit at the discretion of the national

supervisor).

⇒ This kind of netting should induce banks to go from the basic indicator to the

standardized approach.

Table 1.3 gives the beta factors for each business line:

Business Line Beta factors

Corporate Finance 18%

Trading and Sales 18%

Retail Banking 12%

Commercial Banking 15%

Payment and Settlement 18%

Agency Services and Custody 15%

Asset Management 12%

Retail Broker 12%

Table 1: Beta factors for the standardized approach.

Dean Fantazzini 7

Advanced Measurement Approaches

In the 2001 version of the Basel 2 agreement, the Committee described three

specific methods within the AMA framework:

• Internal Measurement Approach (IMA): according to this method, the OR

capital charge depends on the sum of the unexpected and expected losses:

the expected losses are computed by using bank historical data, while those

unexpected are found by multiplying the expected losses by a factor γ,

derived by sector analysis.

• Loss Distribution Approach (LDA): using internal data, it is possible to

compute, for every BL/ET combination, the probability distribution for the

frequency of the loss event as well as for its impact (severity) over a specific

time horizon.

By convoluting the frequency with the severity distribution, analytically or

numerically, the probability distribution of the total loss can be retrieved.

The final capital charge will be equal to a determined percentile of that

distribution.

Dean Fantazzini 8

Advanced Measurement Approaches

• Scorecard : an expert panel has to go through a structured process of

identifying the drivers for each risk category, and then forming these into

questions that could be put on scorecards.

These questions are selected to cover drivers of both the probability and

impact of operational events, and the actions that the bank has taken to

mitigate them. In parallel with the scorecard development and piloting, the

bank’s total economic capital for operational risk is calculated and then

allocated to risk categories.

In the last version of the Basel 2 agreement, these models are not mentioned to

allow for more flexibility in the choice of internal measurement methods.

Given its increasing importance (see e.g. Cruz, 2002) and the possibility to apply

econometric methods, we will focus here only on the LDA approach.

Dean Fantazzini 9

The Standard LDA Approach with Comonotonic Losses

The actuarial approach employs two types of distributions:

• The one that describes the frequency of risky events;

• The one that describes the severity of the losses

Formally, for each type of risk i = 1, . . . , R and for a given time period,

operational losses could be defined as a sum (Si) of the random number

(ni) of the losses (Xij):

Si = Xi1 + Xi2 + . . . + Xini (3)

A widespread statistical model is the actuarial model . In this model, the

probability distribution of Si is described as follows:

Fi(Si) = Fi(ni) · Fi(Xij), where

• Fi(Si) = probability distribution of the expected loss for risk i;

• Fi(ni) = probability of event (frequency) for risk i;

• Fi(Xij) = loss given event (severity) for risk i.

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The underlying assumptions for the actuarial model are:

• the losses are random variables, independent and identically

distributed (i.i.d.);

• the distribution of ni (frequency) is independent of the distribution of

Xij (severity).

Moreover,

• The frequency can be modelled by a Poisson or a Negative Binomial

distribution.

• The severity, is modelled by a Exponential or a Pareto or a Gamma

distribution.

→ The distribution Fi of the losses Si for each intersection i among

business lines and event types, is then obtained by the convolution of the

frequency and severity distributions.

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However, the analytic representation of this distribution is computationally

difficult or impossible. For this reason, this distribution is usually

approximated by Monte Carlo simulation:

→ We generate a great number of possible losses (i.e. 100.000) with

random extractions from the theoretical distributions that describe

frequency and severity. We thus obtain a loss scenario for each loss Si.

→ A risk measure like Value at Risk (VaR) or Expected Shortfall (ES) is

then estimated to evaluate the capital requirement for the loss Si.

• The VaR at the probability level α is the α-quantile of the loss

distribution for the i − th risk: V aR(Si; α) : Pr(Si ≥ V aR) ≤ α

• The Expected Shortfall at the probability level α is defined as the

expected loss for intersection i, given the loss has exceeded the VaR

with probability level α : ES(Si; α) ≡ E [Si|Si ≥ V aR(Si; α)]

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Once the risk measures for each losses Si are estimated, the global VaR (or

ES) is usually computed as the simple sum of these individual measures:

• a perfect dependence among the different losses Si is assumed...

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• ... but this is absolutely not realistic!

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• ... but this is absolutely not realistic!

• If we used the Sklar’s theorem (1959) and the Frechet-Hoeffding

bounds, the multivariate distribution among the R losses would be

given by

H(S1t, . . . , SR,t) = min (F (S1,t), . . . , F (SR,t)) (4)

where H is the joint distribution of a vector of losses Sit, i = 1 . . . R,

and F (·) are the cumulative distribution functions of the losses’

marginals. Needless to say, such an assumption in quite unrealistic.

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The Canonical Aggregation Model via Copulas

• Brief recall to Copula theory :

→ A copula is a multivariate distribution function H of random variables

X1 . . . Xn with standard uniform marginal distributions F1, . . . , Fn,

defined on the unit n-cube [0,1]n

(Sklar’s theorem): Let H denote a n-dimensional distribution function

with margins F1. . . Fn . Then there exists a n-copula C such that for all

real (x1,. . . , xn)

H(x1, . . . , xn) = C(F (x1), . . . , F (xn)) (5)

If all the margins are continuous, then the copula is unique; otherwise C is

uniquely determined on RanF1 × RanF2 . . . RanFn, where Ran is the

range of the marginals. Conversely, if C is a copula and F1, . . . Fn are

distribution functions, then the function H defined in (2.2) is a joint

distribution function with margins F1, . . . Fn.

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By applying Sklar’s theorem and using the relation between the

distribution and the density function, we can derive the multivariate

copula density c(F1(x1),, . . . , Fn(xn)), associated to a copula function

C(F1(x1),, . . . , Fn(xn)):

f(x1, ..., xn) =∂n [C(F1(x1), . . . , Fn(xn))]

∂F1(x1), . . . , ∂Fn(xn)·

n∏

i=1

fi(xi) = c(F1(x1), . . . , Fn(xn))·

n∏

i=1

fi(xi)

where

c(F1(x1), ..., Fn(xn)) =f(x1, ..., xn)

n∏

i=1

fi(xi)· , (6)

By using this procedure, we can derive the Normal and the T-copula...

Dean Fantazzini 15


1. Normal-copula:

c(Φ(x1), ..., Φ(xn)) =fGaussian(x1, ..., xn)

n∏

i=1

fGaussiani (xi)

=

1

(2π)n/2|Σ|1/2exp

(

− 12 x′Σ−1x

)

n∏

i=1

1√2π

exp(

− 12 x2

i

)

=

=1

|Σ|1/2exp

(

−1

2ζ′(Σ

−1− I)ζ

)

where ζ = (Φ−1(u1), ..., Φ−1(un))′ is the vector of univariate Gaussian

inverse distribution functions, ui = Φ (xi), while Σ is the correlation

matrix.

2. T-copula:

c(tυ(x1), ..., tυ(xn)) =fStudent(x1, ..., xn)

n∏

i=1fStudent

i(xi)

= |Σ|−1/2Γ(

υ+n2

)

Γ(

υ2

)

Γ(

υ2

)

Γ(

υ+12

)

n

(

1 +ζ′Σ−1ζ

υ

)− υ+n2

n∏

i=1

(

1 +ζ2

i2

)− υ+12

,

where ζ = (t−1υ (u1), ..., t

−1υ (un))′ is the vector of univariate Student‘s

T inverse distribution functions, ν are the degrees of freedom,

ui = tν(xi), while Σ is the correlation matrix.

Dean Fantazzini 16


Di Clemente and Romano (2004) and Fantazzini et al. (2007,

2008) proposed to use copulas to model the dependence among

operational risk losses:

→ By using Sklar’s Theorem, the joint distribution H of a vector of losses

Si, i = 1 . . . R, is simply the copula of the cumulative distribution functions

of the losses’ marginals :

H(S1, . . . , SR) = C(F1(S1), . . . , FR(SR)) (7)

...moving to densities, we get:

h(S1, . . . , SR) = c(F1(S1), . . . , FR(SR)) · f1(S1) · . . . · fR(SR)

→ The analytic representation for the multivariate distribution of all losses

Si with copula functions is not possible, and an approximate solution with

Monte Carlo methods is necessary.

Dean Fantazzini 17


• Simulation studies: Small sample properties - Marginals

estimators [from Fantazzini et al. (2007, 2008)]

→ The simulation Data Generating Processes (DGPs) are designed to

reflect the stylized facts about real operational risks: we chose the

parameters of the DGPs among the ones estimated in the empirical section.

We consider two DGPs for the Frequency :

Fi(ni) ∼ Poisson(0.08) (8)

Fi(ni) ∼ Negative Binomial(0.33; 0.80) (9)

and three DGPs for the Severity :

Fi(Xij) ∼ Exponential(153304) (10)

Fi(Xij) ∼ Gamma(0.2; 759717) (11)

Fi(Xij) ∼ Pareto(2.51; 230817) (12)

In addition to the five DGPs, we consider four possible data situations: 1)

T = 72; 2) T = 500; 3) T = 1000; 4) T = 2000.

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→ Simulation results:

1. As for Frequency distributions, while the Poisson distribution gives

already consistent estimates with 72 observations, the Negative

Binomial shows dramatic results, instead, with 40 % of cases where we

have negative estimates, and very high MSE and Variation Coeff.

Moreover, even with a dataset of 2000 observations, the estimates are

not yet stable. Datasets of 5000 observations of higher are required.

2. As for Severity distributions, we have again mixed results.

The Exponential and Gamma distributions give already consistent

estimates with 72 observations.

The Pareto have problems in small samples instead, with 2% of cases

of negative coefficients and very high MSE and VC.

Similar to the Negative Binomial, a size of, at least, T=5000 is

required.

Dean Fantazzini 19


• Empirical Analyis

The model we described was applied to an (anonymous) banking loss

dataset, ranging from January 1999 till December 2004, for a total of 72

monthly observations.

→ The overall loss events in this dataset are 407, organized in 2 business

lines and 4 event types, so that we have 8 possible risky combinations (or

intersections) to deal with.

→ The overall average monthly loss was equal to 202.158 euro, the

minimum to 0 (for September 2001), while the maximum to 4.570.852 euro

(which took place on July 2003).

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Table 1: Pieces of the banking losses datasetFrequency 1999 1999 1999 1999 . . . 2004 2004

January February March April . . . November December

Intersection 1 2 0 0 0 . . . 5 0

Intersection 2 6 1 1 1 . . . 3 1

Intersection 3 0 2 0 0 . . . 0 0

Intersection 4 0 1 0 0 . . . 0 0

Intersection 5 0 0 0 0 . . . 0 1

Intersection 6 0 0 0 0 . . . 2 4

Intersection 7 0 0 0 0 . . . 1 0

Intersection 8 0 0 0 0 . . . 0 0

Severity 1999 1999 1999 1999 . . . 2004 2004

January February March April . . . November December

Intersection 1 35753 0 0 0 . . . 27538 0

Intersection 2 121999 1550 3457 5297 . . . 61026 6666

Intersection 3 0 33495 0 0 . . . 0 0

Intersection 4 0 6637 0 0 . . . 0 0

Intersection 5 0 0 0 0 . . . 0 11280

Intersection 6 0 0 0 0 . . . 57113 11039

Intersection 7 0 0 0 0 . . . 2336 0

Intersection 8 0 0 0 0 . . . 0 0

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Figure 1: Global Loss Distribution

(Negative Binomial - Pareto - Normal copula)

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Table 2: Correlation Matrix of the risky Intersections

(Normal Copula)Int. 1 Int. 2 Int. 3 Int. 4 Int. 5 Int. 6 Int. 7 Int. 8

Inters. 1 1 -0.050 -0.142 0.051 -0.204 0.252 0.140 -0.155

Inters. 2 -0.050 1 -0.009 0.055 0.023 0.115 0.061 0.048

Inters. 3 -0.142 -0.009 1 0.139 -0.082 -0.187 -0.193 -0.090

Inters. 4 0.051 0.055 0.139 1 -0.008 0.004 -0.073 -0.045

Inters. 5 -0.204 0.023 -0.082 -0.008 1 0.118 -0.102 -0.099

Inters. 6 0.252 0.115 -0.187 0.004 0.118 1 -0.043 0.078

Inters. 7 0.140 0.061 -0.193 -0.073 -0.102 -0.043 1 -0.035

Inters. 8 -0.155 0.048 -0.090 -0.045 -0.099 0.078 -0.035 1

Dean Fantazzini 23


Table 3: Global VaR and ES for different marginals convolutions,

dependence structures, and confidence levelsVaR 95% VaR 99% ES 95% ES 99%

Poisson Exponential Perfect Dep. 925,218 1,940,229 1,557,315 2,577,085

Normal Copula 656,068 1,086,725 920,446 1,340,626

T copula (9 d.o.f.) 673,896 1,124,606 955,371 1,414,868

Poisson Gamma Perfect Dep. 861,342 3,694,768 2,640,874 6,253,221

Normal Copula 767,074 2,246,150 1,719,463 3,522,009

T copula (9 d.o.f.) 789,160 2,366,876 1,810,302 3,798,321

Poisson Pareto Perfect Dep. 860,066 2,388,649 2,016,241 4,661,986

Normal Copula 663,600 1,506,466 1,294,654 2,785,706

T copula (9 d.o.f.) 672,942 1,591,337 1,329,130 2,814,176

Negative Bin. Exponential Perfect Dep. 965,401 2,120,145 1,676,324 2,810,394

Normal Copula 672,356 1,109,768 942,311 1,359,876

T copula (9 d.o.f.) 686,724 1,136,445 975,721 1,458,298

Negative Bin. Gamma Perfect Dep. 907,066 3,832,311 2,766,384 6,506,154

Normal Copula 784,175 2,338,642 1,769,653 3,643,691

T copula (9 d.o.f.) 805,747 2,451,994 1,848,483 3,845,292

Negative Bin. Pareto Perfect Dep. 859,507 2,486,971 2,027,962 4,540,441

Normal Copula 672,826 1,547,267 1,311,610 2,732,197

T copula (9 d.o.f.) 694,038 1,567,208 1,329,281 2,750,097

Dean Fantazzini 24


Table 4: Backtesting results with different marginals and copulas

VaR Exceedances VaR Exceedances

N / T N / T

Perfect 99.00% 1.39% Perfect 99.00% 1.39%

Dep. 95.00% 4.17% Dep. 95.00% 4.17%

Poisson Normal 99.00% 2.78% Neg. Bin. Normal 99.00% 2.78%

Exp. Copula 95.00% 6.94% Exp. Copula 95.00% 6.94%

T Copula 99.00% 2.78% T Copula 99.00% 2.78%

(9 d.o.f.) 95.00% 6.94% (9 d.o.f.) 95.00% 6.94%

Perfect 99.00% 1.39% Perfect 99.00% 1.39%

Dep. 95.00% 6.94% Dep. 95.00% 4.17%


Gamma Copula 95.00% 6.94% Gamma Copula 95.00% 6.94%

T Copula 99.00% 1.39% T Copula 99.00% 1.39%

(9 d.o.f.) 95.00% 6.94% (9 d.o.f.) 95.00% 6.94%

Perfect 99.00% 1.39% Perfect 99.00% 1.39%

Dep. 95.00% 6.94% Dep. 95.00% 6.94%


Pareto Copula 95.00% 6.94% Pareto Copula 95.00% 6.94%

T Copula 99.00% 1.39% T Copula 99.00% 1.39%

(9 d.o.f.) 95.00% 6.94% (9 d.o.f.) 95.00% 6.94%

Dean Fantazzini 25


- The empirical analysis in Di Clemente and Romano (2004) and Fantazzini et

al. (2007, 2008) showed that is not the choice of the copula, but that of the

marginals which is important.

- Among marginals distributions, particularly the ones used to model the

losses severity are fundamental.

- The best distribution for severity modelling resulted to be the Gamma

distribution, while remarkable differences between the Poisson and Negative

Binomial for frequency modelling, were not found.

- However, we have to remind that the Poisson is much more easier to

estimate, especially with small samples.

- Copula functions allow us to reduce the risk measures capital requirements.

Dean Fantazzini 26

The Poisson Shock Model

Lindskog and McNeil (2003), Embrechts and Puccetti (2008) and

Rachedi and Fantazzini (2009) proposed a different aggregation model.

In this model, the dependence is modelled among severities and among

frequencies, using Poisson processes.

Suppose there are m different types of shock or event and, for e = 1, . . . , m, let net

be a Poisson process with intensity λe recording the number of events of type e

occurring in (0, t].

Assume further that these shock counting processes are independent. Consider

losses of R different types and, for i = 1, . . . , R, let nit be a counting process that

records the frequency of losses of the ith type occurring in (0, t].

At the rth occurrence of an event of type e the Bernoulli variable Iei,r indicates

whether a loss of type i occurs. The vectors

Ier = (Ie

1,r, . . . , IeR,r)′

for r = 1, . . . , net are considered to be independent and identically distributed

with a multivariate Bernoulli distribution.

Dean Fantazzini 27


⇒ In other words, each new event represents a new independent opportunity to

incur a loss but, for a fixed event, the loss trigger variables for losses of different

types may be dependent. The form of the dependence depends on the

specification of the multivariate Bernoulli distribution and independence is a

special case.

According to the Poisson Shock Model, the loss processes nit are clearly Poisson

themselves, since they are obtained by superpositioning m independent Poisson

processes generated by the m underlying event processes.

⇒ Therefore, (n1t, . . . , nRt) can be thought of as having a multivariate Poisson

distribution. However, it follows that the total number of losses is not itself a

Poisson process, but rather a compound Poisson process:

nt =

m∑

e=1

net

∑

r=1

R∑

i=1

Iei,r

These shocks cause a certain number of losses in the i-th ET/BL, whose severity

is (Xeir), r = 1, . . . , ne

t , where (Xeir) are i.i.d. with distribution function Fit and

independent with respect to net .

Dean Fantazzini 28


⇒ As it may appear immediately from the previous discussion, the key point of

this approach is to identify the underlying m Poisson processes: unfortunately,

this field of studies is quite recent and more research has to be made with this

regard. Moreover, the paucity of data limits any precise identification.

⇒ A simple approach is to identify the m processes with the R risky intersections

(BLs or ETs or both), so that we are back to the standard framework of the LDA

approach. This is the “soft-model” proposed in Embrechts and Puccetti (2008)

and later applied to a real OP dataset by Rachedi and Fantazzini (2009)

Embrechts and Puccetti (2008) and Rachedi and Fantazzini (2009) allow for

positive/negative dependence among the shocks (nit) and also among loss

severities (Xij), but the number of shocks and loss severities are independent to

each other:

Hf (n1t, . . . , nRt) = Cf (F (n1t), . . . , F (nRt))

Hs(X1j , . . . , XRj) = Cs(F (X1,j), . . . , F (XR,j))

Hf ⊥ Hs

Dean Fantazzini 29


Equivalently, if we use the mean loss for the period, i.e. sit, we have

Hf (n1t, . . . , nRt) = Cf (F (n1t), . . . , F (nRt))

Hs(s1t, . . . , sRt) = Cs(F (s1t), . . . , F (sRt))

Hf ⊥ Hs

The operative procedure of this approach is the following one:

1. Fit the frequency and severity distributions like in the standard LDA

approach, and compute the relative cumulative distribution functions.

2. Fit a copula Cf to the frequency c.d.f.’s. (see the next subsection for an

important remark about this issue).

3. Fit a copula CS to the severity distributions c.d.f.’s.

4. Generate a random vector uf = (uf1t, ..., u

fRt

) from the copula Cf .

5. Invert each component ufit with the respective inverse distribution function

F−1(ufit), to determine a random vector (n1t, ..., nRt) describing the number

of loss observations.

Dean Fantazzini 30


6. Generate a random vector us = (us1, . . . , us

R) from the copula CS .

7. Invert each component usi with the respective inverse distribution function

F−1(usi ), to determine a random vector (X1j , ..., XRj) describing the loss

severities.

8. Convolve the frequencies’ vector (n1t, . . . , nRt) with the one of the severities

(X1j , . . . , XRj).

9. Repeat the previous steps a great number of times, i.e. 106 times.

In this way it is possible to obtain a new matrix of aggregate losses which can

then be used to compute the usual risk measures such as the VaR and ES.

Note: copula modelling for discrete marginals is an open problem, see Genest and

Neslehova (2007, “A primer on copulas for count data”, Astin Bulletin), for a

recent discussion. Therefore, some care has to be taken when considering the

estimated risk measures.

Dean Fantazzini 31


Remark 1: Estimating Copulas With Discrete Distributions

According to Sklar (1959), in the case where certain components of the joint

density are discrete (as in our case), the copula function is not uniquely defined

not on [0,1]n, but on the Cartesian product of the ranges of the n marginal

distribution functions.

Two approaches have been proposed to overcome this problem. The first method,

has been proposed by Cameron et al. (2004) and is based on finite difference

approximations of the derivatives of the copula function,

f(x1, . . . , xn) = ∆n . . . ∆1C(F (x1), . . . , F (xn))

where ∆k, for k =1, . . . , n, denotes the k-th component first order differencing

operator being defined through

∆kC[F (x1), . . . , F (xk), . . . F (xn)] = C[F (x1), . . . , F (xk), . . . F (xn)]−

− C[F (x1), . . . , F (xk − 1), . . . F (xn)]

Dean Fantazzini 32


The second method is the continuization method suggested by Stevens (1950) and

Denuit and Lambert (2005), which is based upon generating artificially continued

variables x∗

1, . . . , x∗

n by adding independent random variables u1, . . . , un (each of

them being uniformly distributed on the set [0,1]) to the discrete count variables

x1, . . . , xn and which does not change the concordance measure between the

variables.

⇒ The empirical literature clearly shows that maximization of likelihood with

discrete margins often runs into computational difficulties, reflected in the failure

of the algorithm to converge.

⇒ In such cases, it may be helpful to first apply the continuization

transformation and then estimate a model based on copulas for continuous

variables. This is why we advice to rely on the second method.

Dean Fantazzini 33


Remark 2: EVT for Modelling Severities

In short, EVT affirms that the losses exceeding a given high threshold u converge

asymptotically to the GPD, whose cumulative function is usually expresses as

follows:

GPDξβ =

1 −(

1 + ξ yβ

)

−1/ξ

1 − exp(

− yβ

)

ξ 6= 0

ξ = 0

(13)

where y = x − u, y ≥ 0 if ξ ≥ 0 and 0 ≤ y ≤ −β/ξ if ξ ≤ 0, and where y are called

excesses whereas x exceedances.

It is possible to determine the conditional distribution function of the excesses,

i.e. y , as a function of x,

Fu (y) = P (X − u ≤ y|X > u) =Fx (x) − Fx (u)

1 − Fx (u)(14)

In these representations the parameter ξ is crucial: when ξ = 0 we have an

Exponential distribution; when ξ < 0 we have a Pareto Distribution - II Type and

when ξ > 0 we have a Pareto Distribution - I Type.

Dean Fantazzini 34


Moreover this parameter has a direct connection with the existence of finite

moments of the losses distributions. We have that

E(

xk)

= ∞ if k ≥ 1/ξ

Hence, in the case of a GPD as a Pareto - I Type, when ξ ≥ 1 we have infinite

mean models.

Di Clemente and Romano (2004) and Rachedi and Fantazzini (2009), suggest to

model the mean loss severity sit using the lognormal for the body of the

distribution and EVT for the tail, in the following way:

Fi (sit) =

Φ(

ln sit−µiσi

)

1 −Nu,i

Ni

(

1 + ξisit−ui

βi

)

−1/ξ(i)

0 < x < ui

ui ≤ x

(15)

where Φ is the standardized normal cumulative distribution functions, Nu,i is the

number of losses exceeding the threshold ui, Ni is the number of the loss data

observed in the ith ET, whereas βi and ξi denote the scale and the shape

parameters of a GPD, respectively.

Dean Fantazzini 35


For example, the graphical analysis for the ET3 in Rachedi and Fantazzini

(2009) reported in Figures 1-2 clearly shows that operational risk losses are

characterized by high frequency – low severity and low frequency – high

severity losses.

⇒ Hence the behavior of losses is twofold: one process underlying small

and frequent losses and another one underlying jumbo losses.

⇒ Splitting the model in two parts allows us to estimate the impact of

such extreme losses in a more robust way.

Dean Fantazzini 36


Figure 1: Scatter plot of ET3 losses. The dotted lines represent, re-

spectively, mean, 90%, 95% and 99.9% empirical quantiles.

Dean Fantazzini 37


Figure 2: Histogram of ET3 losses

Dean Fantazzini 38


Rachedi and Fantazzini (2009) analyzed a large dataset consists of 6 years

of loss observations from 2002 to 2007, containing the data of the seven

ETs.

⇛ They compared the comonotonic approach proposed by Basel II, the

canonical aggregation model via copulas and the Poisson shock model

The resulting total operational risk capital charge for the three models is

reported below:

VaR (99.9 %) ES (99.9 %)

Comonotonic 308861 819325

Copula (Canonical aggregation) 273451 671577

Shock Model 231790 655460

Table 2: VaR and ES final estimates

Dean Fantazzini 39

Bayesian Approaches

An important limitation of the advanced measurement approaches (AMAs)

is the inaccuracy and scarcity of data, that is basically due to the relatively

recent definition and management of operational risk.

This makes the process of data recovery generally more difficult, since

financial institutions only started to collect operational loss data a few

years ago.

⇒ In this context, the employment of Bayesian and simulation methods

appears to be a natural solution to the problem.

⇒ In fact, they allow us to combine the use of quantitative information

(coming from the time series of losses collected by the bank) and

qualitative data (coming from experts’ opinions), taking the form of prior

information.

Dean Fantazzini 40

Bayesian Approaches

Besides, simulation methods represent a widely used statistical tool that

overcome computational problems. The combination of the described

methodologies leads to the Markov chain Monte Carlo (MCMC) methods,

which includes the main advantages of both Bayesian and simulation

methods.

⇒ Interesting Bayesian approaches for marginal loss distributions has been

recently proposed in Dalla Valle and Giudici (2008), while Bayesian

copulas in Dalla Valle (2008). We refer there for more details.

⇒ A word of caution: these methods work fine if there is really prior

information (like experts’ opinions).

⇒ Instead, if the prior is chosen to “close” the model, the resulting

estimates may be very poor or unrealistic (generating also numerical

errors), as clearly reported in Tables 12-14 in Dalla Valle and Giudici

(2008), where the ES estimates are higher than e+27 or e+39!

Dean Fantazzini 41

References

Basel Committee on Banking Supervision (1998). Amendment to the Capital Accord

to Incorporate Market Risks, Basel.

Basel Committee on Banking Supervision (2003). The 2002 loss data collection

exercise for operational risk : summary of the data collected, Bank for International

Settlement document.

Basel Committee on Banking Supervision (2005). Basel II: International

Convergence of Capital Measurement and Capital Standards: a Revised Framework,

Bank for International Settlement document.

Cameron, C., Li, T., Trivedi, P., and Zimmer, D. (2004). Modelling the Differences in

Counted Outcomes Using Bivariate Copula Models with Application to Mismesured

Counts, Econometrics Journal, 7, 566-584.

Cruz, M.G. (2002). Modeling, Measuring and Hedging Operational Risk. Wiley, New

York.

Dalla Valle, L., and Giudici, P. (2008). A Bayesian approach to estimate the

marginal loss distributions in operational risk management, Computational Statistics

and Data Analysis, 52, 3107-3127.

Dalla Valle, L. (2008). Bayesian Copulae Distributions, with Application to

Operational Risk Management, Methodology and Computing in Applied Probability,

11(1), 95-115.

Denuit, M. and Lambert, P. (2005). Constraints on Concordance Measures in

Bivariate Discrete Data, Journal of Multivariate Analysis, 93 , 40-57.

Dean Fantazzini 42

References

Di Clemente, A., and Romano, C. (2004). A Copula-Extreme Value Theory Approach

for Modelling Operational Risk. In: Operational Risk Modelling and Analysis:

Theory and Practice, Risk Books, London.

Embrechts, P., and Puccetti, G. (2008). Aggregating operational risk across matrix

structured loss data, Journal of Operational Risk, 3(2), 29-44.

Fantazzini, D., L. Dallavalle and P. Giudici (2007). Empirical Studies with

Operational Loss Data: DallaValle, Fantazzini and Giudici Study. In: Operational

Risk: A Guide to Basel II Capital Requirements, Models, and Analysis, Wiley, New

Jersey.

Fantazzini, D., Dallavalle, L. and P. Giudici (2008). Copulae and operational risks,

International Journal of Risk Assessment and Management, 9(3), 238-257.

Lindskog, F. and A. McNeil, A. (2003). Common Poisson shock models: applications

to insurance and credit risk modelling, ASTIN Bulletin, 33(2) , 209-238.

Rachedi, O., and Fantazzini, D. (2009). Multivariate Models for Operational Risk: A

Copula Approach using Extreme Value Theory and Poisson Shock Models, In:

Operational Risk towards Basel III: Best Practices and Issues in Modelling,

Management and Regulation, 197-216, Wiley, New York.

Stevens, W. L. (1950). Fiducial Limits of the parameter of a discontinuous

distribution, Biometrika, 37, 117-129.

⇛ ... the book I’m writing with prof. Aivazian (CEMI)... STAY TUNED!

Dean Fantazzini 43

Documents

Operational Risk Management: A Review