Operational Risk Management: A Review

Operational Risk Management: A Review Dean Fantazzini

Moscow

Overview of the Presentation • Introduction

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Overview of the Presentation • Introduction • The Basic Indicator Approach

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Overview of the Presentation • Introduction • The Basic Indicator Approach • The Standardized Approach

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Overview of the Presentation • Introduction • The Basic Indicator Approach • The Standardized Approach • Advanced Measurement Approaches

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Overview of the Presentation • Introduction • The Basic Indicator Approach • The Standardized Approach • Advanced Measurement Approaches • The Standard LDA Approach with Comonotonic Losses

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Overview of the Presentation • Introduction • The Basic Indicator Approach • The Standardized Approach • Advanced Measurement Approaches • The Standard LDA Approach with Comonotonic Losses • The Canonical Aggregation Model via Copulas

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Overview of the Presentation • Introduction • The Basic Indicator Approach • The Standardized Approach • Advanced Measurement Approaches • The Standard LDA Approach with Comonotonic Losses • The Canonical Aggregation Model via Copulas • The Poisson Shock Model

Dean Fantazzini

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Overview of the Presentation • Introduction • The Basic Indicator Approach • The Standardized Approach • Advanced Measurement Approaches • The Standard LDA Approach with Comonotonic Losses • The Canonical Aggregation Model via Copulas • 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). Dean Fantazzini

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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. Dean Fantazzini

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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 3 X 1 t RCBI = α max(GI t−i , 0) (1) Zt i=1 P3 where Zt = i=1 I[GI t−i >0] and GI t−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. Dean Fantazzini

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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:   3 8 X 1X t  RCSA = (2) max βj GIjt−i , 0 3 i=1 j=1

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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

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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

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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.

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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. Dean Fantazzini

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The Standard LDA Approach with Comonotonic Losses 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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The Standard LDA Approach with Comonotonic Losses 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)] Dean Fantazzini

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The Standard LDA Approach with Comonotonic Losses 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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The Standard LDA Approach with Comonotonic Losses 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... • ... but this is absolutely not realistic!

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The Standard LDA Approach with Comonotonic Losses 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... • ... 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 , . . . , F n, 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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The Canonical Aggregation Model via Copulas

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 ),, . . . , F n (xn )), associated to a copula function C(F1 (x1 ),, . . . , F n (xn )): n n Y ∂ n [C(F1 (x1 ), . . . , Fn (xn ))] Y f (x1 , ..., xn ) = · fi (xi ) = c(F1 (x1 ), . . . , Fn (xn ))· fi (xi ) ∂F1 (x1 ), . . . , ∂Fn (xn ) i=1 i=1

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

f (x1 , ..., xn ) · , n Q fi (xi )

(6)

i=1

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

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The Canonical Aggregation Model via Copulas 1. Normal-copula: c(Φ(x1 ), ..., Φ(xn ))

=

f

Gaussian n Q

i=1

=

(x1 , ..., xn )

=

fiGaussian (xi )

1

|Σ|1/2

exp



1 n/2 (2π) |Σ|1/2 n Q √1 2π i=1

1 ′ −1 − I)ζ − ζ (Σ 2


exp − 12 x′ Σ−1 x

exp

− 12 x2i



=




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: f Student (x1 , ..., xn ) c(tυ (x1 ), ..., tυ (xn )) = = n Q Student f (xi ) i i=1

|Σ|

−1/2

Γ

υ+n 2   υ Γ 2 

  

Γ

Γ 

 n υ 2  υ+1 2



ζ ′ Σ−1 ζ 1+ υ n Q

i=1

ζ2 1+ i 2

! − υ+n 2

! − υ+1 2

−1 ′ where ζ = (t−1 υ (u1 ), ..., tυ (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.

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,

The Canonical Aggregation Model via Copulas 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

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The Canonical Aggregation Model via Copulas • 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 )

∼

P oisson(0.08)

(8)

Fi (ni )

∼

N egative 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 ) ∼ P areto(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. Dean Fantazzini

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The Canonical Aggregation Model via Copulas → 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.

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The Canonical Aggregation Model via Copulas

• 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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The Canonical Aggregation Model via Copulas Table 1: Pieces of the banking losses dataset Frequency

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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The Canonical Aggregation Model via Copulas

Figure 1: Global Loss Distribution (Negative Binomial - Pareto - Normal copula)

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The Canonical Aggregation Model via Copulas

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

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The Canonical Aggregation Model via Copulas Table 3: Global VaR and ES for different marginals convolutions, dependence structures, and confidence levels Poisson Exponential

Poisson Gamma

Poisson Pareto

Negative Bin. Exponential

Negative Bin. Gamma

Negative Bin. Pareto

Dean Fantazzini

VaR 95%

VaR 99%

ES 95%

ES 99%

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

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

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

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

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

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

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The Canonical Aggregation Model via Copulas Table 4: Backtesting results with different marginals and copulas VaR

Exceedances

VaR

N / T

Exceedances 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%

Poisson

Normal

99.00%

1.39%

Neg. Bin.

Normal

99.00%

1.39%

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%

Poisson

Normal

99.00%

1.39%

Neg. Bin.

Normal

99.00%

1.39%

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%

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The Canonical Aggregation Model via Copulas - 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.

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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]. e indicates At the r th occurrence of an event of type e the Bernoulli variable Ii,r whether a loss of type i occurs. The vectors e e Ier = (I1,r , . . . , IR,r )′

for r = 1, . . . , net are considered to be independent and identically distributed with a multivariate Bernoulli distribution. Dean Fantazzini

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The Poisson Shock Model ⇒ 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: e

nt =

nt R m X X X

e Ii,r

e=1 r=1 i=1

These shocks cause a certain number of losses in the i-th ET/BL, whose severity e ), r = 1, . . . , ne , where (X e ) are i.i.d. with distribution function F and is (Xir it t ir e independent with respect to nt . Dean Fantazzini

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The Poisson Shock Model ⇒ 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: H f (n1t , . . . , nRt )

=

C f (F (n1t ), . . . , F (nRt ))

H s (X1j , . . . , XRj )

=

C s (F (X1,j ), . . . , F (XR,j ))

Hf

⊥

Hs

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The Poisson Shock Model Equivalently, if we use the mean loss for the period, i.e. sit , we have H f (n1t , . . . , nRt )

=

C f (F (n1t ), . . . , F (nRt ))

H s (s1t , . . . , sRt )

=

C s (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 C f to the frequency c.d.f.’s. (see the next subsection for an important remark about this issue). 3. Fit a copula C S to the severity distributions c.d.f.’s. 4. Generate a random vector uf = (uf1t , ..., ufRt ) from the copula C f . 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.

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The Poisson Shock Model 6. Generate a random vector us = (us1 , . . . , usR ) from the copula C S . 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 Neˇslehov´ a (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.

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The Poisson Shock Model 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 . . . ∆1 C(F (x1 ), . . . , F (xn )) where ∆k , for k =1, . . . , n, denotes the k-th component first order differencing operator being defined through ∆k C[F (x1 ), . . . , F (xk ), . . . F (xn )] = C[F (x1 ), . . . , F (xk ), . . . F (xn )]− − C[F (x1 ), . . . , F (xk − 1), . . . F (xn )]

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The Poisson Shock Model

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.

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The Poisson Shock Model 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:    ξ 6= 0  1 − 1 + ξ y −1/ξ  β  GP Dξβ = (13)  1 − exp − y β ξ=0

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

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The Poisson Shock Model 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:    ln s −µ it i  Φ σi   Fi (sit ) =  1 − Nu,i 1 + ξ sit −ui −1/ξ(i) i N β i

i

0 < x < ui (15) ui ≤ x

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

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The Poisson Shock Model

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.

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The Poisson Shock Model

Figure 1: Scatter plot of ET3 losses. The dotted lines represent, respectively, mean, 90%, 95% and 99.9% empirical quantiles. Dean Fantazzini

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The Poisson Shock Model

Figure 2: Histogram of ET3 losses

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The Poisson Shock Model 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

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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

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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

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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

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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

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