Modeling Operational Risk: Estimation and Effects of Dependencies

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Modeling Operational Risk:

Estimation and Effects of Dependencies

Stefan Mittnik Sandra Paterlini Tina Yener

Financial Mathematics Seminar

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Modeling Operational Risk: Estimation and Effects of Dependencies Outline

Outline of the Talk

1 Motivation

Operational Risk VaR and Subadditivity 2 The data

3 Modeling Dependence Correlation Copulas

Nonparametric Measures of Extremal Dependence 4 Effects on Risk Capital

Range of Risk–Capital Estimates Bounds on Risk–Capital Estimates 5 Conclusion

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Modeling Operational Risk: Estimation and Effects of Dependencies Introduction

Operational Risk

Operational Risk: Definition

The Basel Committee defines operational risk as

The risk of loss resulting from inadequate or failed internal processes, people and systems, or from external events.

This definition includes legal risk, but excludes strategic and reputational risk.

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Operational Risk

Operational Risk: Introduction

Operational loss events can be very heterogeneous.

Event Types Business Lines

Internal fraud Corporate Finance

External fraud Trading and Sales

Employment practices and workplace safety Retail Banking Clients, products and business practices Commercial Banking

Damage to physical assets Payment and Settlement

Business disruption and system failures Agency Services Execution, delivery and process management Asset Management

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Operational Risk

Operational Risk: Introduction

Three distinct approaches to determine minimal capital requirements:

1 Basic Indicator Approach 2 Standardised Approach

3 Advanced Measurement Approach (AMA)

Under AMA, the risk measure to be used is the Value–at–Risk (VaRα), i.e., the quantile of the loss distribution, withα = 99.9%.

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Operational Risk

Operational Risk: Heterogeneity of Events

The heterogeneity of Operational Risk leads to the regulatory requirement of a separate modeling within 56

event–type/business-line combinations.

Business Lines

Corporate Finance . . . Retail Brokerage

Event T yp es Internal Fraud L1,1 . . . L1,8 . . . ... . .. ...

Execution, Delivery & L7,1 . . . L7,8

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Operational Risk

Operational Risk: Heterogeneity of Events

The heterogeneity of Operational Risk leads to the regulatory requirement of a separate modeling within 56

event–type/business-line combinations.

Business Lines

Corporate Finance . . . Retail Brokerage

Event T yp es Internal Fraud L1,1 . . . L1,8 . . . ... . .. ...

Execution, Delivery & L7,1 . . . L7,8

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Operational Risk

Operational Risk: Total Risk Capital

The quantity of interest is

VaR.999(L) = VaR.999 56 X i =1 Li ! ; (1)

clearly, it is influenced by dependencies among cells i and j .

However, Basel II prescribes to calculate Total Risk Capital as

TRC =

56

X

i =1

VaR.999(Li) ; (2)

only under certain qualifying conditions, banks may explicitly model dependencies.

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Operational Risk

Operational Risk: Total Risk Capital

The quantity of interest is

VaR.999(L) = VaR.999 56 X i =1 Li ! ; (1)

clearly, it is influenced by dependencies among cells i and j . However, Basel II prescribes to calculate Total Risk Capital as

TRC =

56

X

i =1

VaR.999(Li) ; (2)

only under certain qualifying conditions, banks may explicitly model dependencies.

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VaR and Subadditivity

For comonotonic risks,

VaRco_α(Li+ Lj) = VaRα(Li) + VaRα(Lj). (3)

Comonotonicity translates into perfect positive correlation in the elliptical (Gaussian) world.

For elliptical distributions, the sum of the single VaRs provides an upper bound and thus a worst–case scenario for VaRα(L).

Given that we hardly have perfect positive correlation, the idea of modeling dependencies seems to be appealing in order to reduce capital estimates...

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VaR and Subadditivity

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VaR and Subadditivity

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VaR and Subadditivity

However, for non–elliptical distributions, it may happen that

VaRα(L1+ L2)> VaRα(L1) + VaRα(L2), (4)

the reason being the lack of subadditivity of the VaR measure (Artzner et al., 1999).

Does this mean that banks may not be rewarded for a more realistic dependency modeling by a decrease in risk capital, but instead be punished by an increase?

Yes! (see, e.g., Embrechts et al., 2002)

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VaR and Subadditivity

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VaR and Subadditivity

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VaR and Subadditivity

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Aim

Aim of our Analyses

Based on n = 60 observations of monthly aggregate losses from the Italian DIPO1 database, we aim at evaluating

VaR (Li + Lj) − (VaR (Li) + VaR (Lj))

| {z }

=TRC

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for different cells i and j of the event–type/business–line matrix.

This task is non–trivial, because it means analyzing the 99.9% quantile of a distribution estimated from a small sample with

extreme data under consideration of dependencies .

1

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Aim

Aim of our Analyses

Based on n = 60 observations of monthly aggregate losses from the Italian DIPO1 database, we aim at evaluating

VaR (Li + Lj) − (VaR (Li) + VaR (Lj))

| {z }

=TRC

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for different cells i and j of the event–type/business–line matrix.

This task is non–trivial, because it means analyzing the 99.9% quantile of a distribution estimated from a small sample with

extreme data under consideration of dependencies .

1

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Data

The DIPO data base

DIPO (Database Italiano delle Perdite Operative):

31 members (209 different entities) of all sizes and mainly not AMA; about 75% of the Italian banking system in terms of Gross Income and Operating Costs

Collection of operational risk data (loss threshold 5,000 EUR), starting from 2003

Data collection twice a year; common definition of gross loss, event–type decision tree and business line mapping

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Data

Distribution across business lines (in %)

Share in...

Business line Number

of events

Total loss

1 (Corporate Finance) 0.14 0.17

2 (Trading & Sales) 5.08 11.51

3 (Retail Banking) 62.46 45.25

4 (Commercial Banking) 7.61 23.64

5 (Payment & Settlement) 0.69 0.42

6 (Agency Services) 0.62 0.85

7 (Asset Management) 0.68 0.59

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Data

Distribution across event types (in %)

We aggregate losses on a monthly basis across event types.

Share in...

Event type Number

of events

Total loss

1 (Internal Fraud) 2.56 17.60

2 (External Fraud) 37.01 19.37

3 (Employment Practices & Workplace Safety) 5.93 6.87 4 (Clients, Products & Business Practices) 28.69 37.31

5 (Damage to Physical Assets) 2.65 1.34

6 (Business Disruption & System Failures) 1.16 1.12 7 (Execution, Delivery & Process Management) 21.97 16.39

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Data

Distribution across event types (in %)

We aggregate on a monthly level and obtain n = 60 observations (aggregate losses) per event type category. Due to data scarcity, we do not further subdivide on the business–line level.

0 10 20 30 ℓ1 Abs. Frequency 0 5 10 ℓ2 Abs. Frequency 0 5 10 15 ℓ3 Abs. Frequency 0 10 20 30 ℓ4 Abs. Frequency 0 5 10 15 20 ℓ5 Abs. Frequency 0 10 20 30 40 ℓ6 Abs. Frequency 0 5 10 15 ℓ7 Abs. Frequency

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Modeling Operational Risk: Estimation and Effects of Dependencies Modeling Dependencies

Measures of Dependence

For Operational Risk quantification, it is crucial to model dependencies for small tail probabilities.

In our investigation, we consider correlation,

copulas,

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Correlation

Linear (Pearson) Correlation

The well–known fact that linear correlation is prone to extremes is quickly revealed by the data.

For example, for event type combination2/5, as the two most extreme observations drop out of the sample, correlation becomes negative.

01/03–12/07 (entire sample): 01/03–04/06 (2/3 of sample):

ℓ2

ℓ5

ℓ2

ℓ5

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Correlation

Linear (Pearson) Correlation

Similarly, for event type combination3/4:

01/03–12/07: 01/03–04/06: ℓ3 ℓ4 ℓ3 ℓ4 ρ3,4= 0.5284 ρ3,4= 0.5882

If we further remove the most extreme observation between 01/03 and 04/06, the correlation decreases toρ3,4= 0.3232.

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Correlation

Correlation: Results

The well–known sensitivity of linear correlation with respect to extremes leads to substantial variations, depending on the sample size.

Its inability to capture possible nonlinear dependency structures provides another important reason for discarding linear correlation as a reliable measure of dependency. Rank correlations were also considered but not found to lead to considerably more stable results.

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Correlation

Correlation: Results

Its inability to capture possible nonlinear dependency structures provides another important reason for discarding linear correlation as a reliable measure of dependency.

Rank correlations were also considered but not found to lead to considerably more stable results.

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Correlation

Correlation: Results

Its inability to capture possible nonlinear dependency structures provides another important reason for discarding linear correlation as a reliable measure of dependency. Rank correlations were also considered but not found to lead to considerably more stable results.

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Copulas

Instead of boiling down dependency into one single number, copulas contain the dependence structure of a joint distribution.

The central theorem of copula theory can be traced back to

0 0.5 1 0 0.5 1 0 0.5 1 u1 u2 C (u 1 ,u 2 )

Sklar (1959) and summed up by

Fi ,j(`i, `j) = C (Fi(`i) | {z } ui , Fj(`j) | {z } uj ), (6)

where C denotes the copula of Li and Lj and

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Copulas

Instead of boiling down dependency into one single number, copulas contain the dependence structure of a joint distribution. The central theorem of copula theory can be traced back to

0 0.5 1 0 0.5 1 0 0.5 1 u1 u2 C (u 1 ,u 2 )

Sklar (1959) and summed up by

Fi ,j(`i, `j) = C (Fi(`i) | {z } ui , Fj(`j) | {z } uj ), (6)

where C denotes the copula of Li and Lj and

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Copulas

Copulas and Tail Dependence

Tail dependence accounts for—possibly nonlinear—dependence among extremes and thus overcomes one drawback of correlation.

In our context of loss distributions, we only consider the upper tail dependence coefficient

λU = lim

t→1−Pr[Fi(Li)> t|Fj(Lj)> t] . (7)

It can also be expressed in terms of the copula of X and Y :

λU = lim t→1−

1 − 2t + C (t, t)

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Copulas

Copulas and Tail Dependence

Tail dependence accounts for—possibly nonlinear—dependence among extremes and thus overcomes one drawback of correlation. In our context of loss distributions, we only consider the upper tail dependence coefficient

λU = lim

t→1−Pr[Fi(Li)> t|Fj(Lj)> t] . (7)

λU = lim t→1−

1 − 2t + C (t, t)

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Copulas

Copulas and Tail Dependence

Tail dependence accounts for—possibly nonlinear—dependence among extremes and thus overcomes one drawback of correlation. In our context of loss distributions, we only consider the upper tail dependence coefficient

λU = lim

t→1−Pr[Fi(Li)> t|Fj(Lj)> t] . (7)

λU = lim t→1−

1 − 2t + C (t, t)

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Copulas

Copulas and Tail Dependence

Different copulas imply different tail dependence structures.

ℓi ℓj ℓi ℓj ℓi ℓj

Gaussian copula: Gumbel copula: Clayton copula: no tail dependence upper tail dependence lower tail dependence

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Copulas

Copulas: Estimation

Upper tail dependence coefficient implied by the Gumbel copula parameter estimates: ET 2 ET 3 ET 4 ET 5 ET 6 ET 7 ET 1 0.136 0.275 0.243 0.147 0.182 0.251 ET 2 0.176 0.235 0.020 0.000 0.115 ET 3 0.574 0.318 0.272 0.297 ET 4 0.359 0.353 0.154 ET 5 0.037 0.000 ET 6 0.218

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Copulas

Copulas: Estimation

Upper tail dependence coefficient implied by the Clayton survival copula parameter estimates:

ET 2 ET 3 ET 4 ET 5 ET 6 ET 7 ET 1 0.015 0.229 0.171 0.018 0.100 0.220 ET 2 0.052 0.109 0.000 0.000 0.000 ET 3 0.609 0.277 0.213 0.304 ET 4 0.332 0.353 0.047 ET 5 0.000 0.000 ET 6 0.149

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Copulas

Copulas: Estimation

Upper tail dependence coefficient implied by the Student-t copula parameter estimates: ET 2 ET 3 ET 4 ET 5 ET 6 ET 7 ET 1 0.000 0.000 0.000 0.000 0.000 0.000 ET 2 0.137 0.203 0.095 0.097 0.149 ET 3 0.402 0.000 0.000 0.286 ET 4 0.122 0.000 0.223 ET 5 0.183 0.000 ET 6 0.000

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Copulas

Copulas: How can we find the “right” one?

1. Goodness–of–fit testing.

We consider two g–o–f tests (A2 and A4) found to perform

particularly well by Berg (2009).

Both are based on the distance between the estimated copula and the empirical copula of Deheuvels (1979).

Unfortunately, we cannot discard any of the copula families considered based on these tests.

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Copulas

Copulas: How can we find the “right” one?

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Copulas

Copulas: How can we find the “right” one?

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Copulas

Copulas: How can we find the “right” one?

2. Consideration of different sample sizes.

Contours of fitted copulas, event type combination 3/4:

u3 u4 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Empirical Gaussian Student-t Gumbel Gumbel Surv. Clayton Clayton Surv. u3 u4 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Empirical Gaussian Student-t Gumbel Gumbel Surv. Clayton Clayton Surv. 01/03–12/07 01/03–04/06

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Copulas

Copulas: Results

Copulas can be used to detect specific dependence structures, i.e., tail dependence.

It seems that there are some event type combinations which are characterized by tail dependence, while others are not. However, we do neither find an overall “best–fitting” copula, nor can we exclude any copula family considered.

Again, the availability of a small data set strongly affects the stability of estimation results.

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Copulas

Copulas: Results

It seems that there are some event type combinations which are characterized by tail dependence, while others are not.

However, we do neither find an overall “best–fitting” copula, nor can we exclude any copula family considered.

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Copulas

Copulas: Results

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Copulas

Copulas: Results

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Nonparametric Tail Dependence

Nonparametric Tail Dependence: Estimators

It is also possible to estimate tail dependence without a parametric assumption for the copula, i.e., using the empirical copula of Deheuvels (1979).

We consider two common nonparametric estimators ofλU:

bλU(t) = 2 − ln bCn(t, t) ln (t) , (Coles et al., 1999) b χ(t) = 2 −1 − bCn(t, t) 1 − t . (Joe et al., 1992)

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Nonparametric Tail Dependence: Estimators

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Nonparametric Tail Dependence: Estimators

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Nonparametric Tail Dependence: Estimators

However, as pointed out by Coles et al. (1999), asymptotic dependence implies not only that bλU > 0, but also that

¯ χ = lim t→1− 2 ln(1 − t) ln( ˜C (t, t)) − 1 = limt→1−χ(t) = 1 ,¯ (9) where ˜C (t, t) = Pr[U > t, V > t].

This quantity can be estimated via

b¯

χ(t) = 2 ln(1 − t) ln( bC (t, t))˜

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Nonparametric Tail Dependence: Estimators

However, as pointed out by Coles et al. (1999), asymptotic dependence implies not only that bλU > 0, but also that

¯ χ = lim t→1− 2 ln(1 − t) ln( ˜C (t, t)) − 1 = limt→1−χ(t) = 1 ,¯ (9) where ˜C (t, t) = Pr[U > t, V > t]. This quantity can be estimated via

b¯

χ(t) = 2 ln(1 − t) ln( bC (t, t))˜

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Asymptotic vs. Quantile Dependence

Summing up, asymptotic and quantile dependence are characterized by the following situations.

Asymptotic Independence Asymptotic Dependence

λU 0 (0,1]

¯

χ [-1,1] 1

ForλU = 0, we have asymptotic independence, but we may still

have quantile dependence at t< 1, the strength (and direction) of which is determined by ¯χ.

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Asymptotic vs. Quantile Dependence

Summing up, asymptotic and quantile dependence are characterized by the following situations.

Asymptotic Independence Asymptotic Dependence

λU 0 (0,1]

¯

χ [-1,1] 1

ForλU = 0, we have asymptotic independence, but we may still

have quantile dependence at t< 1, the strength (and direction) of which is determined by ¯χ.

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Nonparametric Tail Dependence: Estimation

Nonparametric tail dependence estimation, event type combination

2/5: 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 t λ2 ,5 (t ) bλU(t) b χ(t) b¯ χ(t) 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 t λ2 ,5 (t ) bλU(t) b χ(t) b¯ χ(t) 01/03–12/07 01/03–04/06

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Nonparametric Tail Dependence: Estimation

Nonparametric tail dependence estimation, event type combination

3/4: 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 t λ3 ,4 (t ) bλU(t) b χ(t) b¯ χ(t) 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 t λ3 ,4 (t ) bλU(t) b χ(t) b¯ χ(t) 01/03–12/07 01/03–04/06

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Nonparametric Tail Dependence: Results

We do not find clear–cut evidence of tail dependence.

A reliable estimation of tail dependence is complicated by the fact that the relevant area (t ≈ 1) is characterized by few observations.

With respect to quantile dependence, results differ among event types.

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Nonparametric Tail Dependence: Results

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Nonparametric Tail Dependence: Results

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Modeling Operational Risk: Estimation and Effects of Dependencies Effects on Risk Capital

Range of Risk–Capital Estimates

Now, we want to assess the effects of realistic dependency structures on risk–capital estimates.

To this end, we estimate 250 99.9% VaR figures per model and event type combination, using different numbers of replications. For each event type combination, we use the copula parameter values obtained from Maximum Likelihood estimation.

For the margins, a lognormal distribution was fitted and is used here to derive risk–capital estimates.

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Range of Risk–Capital Estimates

To this end, we estimate 250 99.9% VaR figures per model and event type combination, using different numbers of replications.

For each event type combination, we use the copula parameter values obtained from Maximum Likelihood estimation.

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Range of Risk–Capital Estimates

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Range of Risk–Capital Estimates

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Range of Risk–Capital Estimates

We consider

VaR.999(Li+ Lj) − (VaR.999(Li) + VaR.999(Lj))

(VaR.999(Li) + VaR.999(Lj))

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under two different assumptions:

1 the Gaussian copula for all event type combinations,

2 the “worst–case” copula, i.e., that copula yielding the highest

tail dependence coefficient for the respective event type combination.

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Range of Risk–Capital Estimates

We consider

(VaR.999(Li) + VaR.999(Lj))

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Range of Risk–Capital Estimates

We consider

(VaR.999(Li) + VaR.999(Lj))

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Range of Risk–Capital Estimates: Boxplots

Gaussian copula, lognormal margins, risk–capital estimates based on 10,000 simulated losses: −50 −25 0 25 50 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

Event Type Combination

R el . D iff . (% ) Brc= 10,000

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Range of Risk–Capital Estimates: Boxplots

“Worst–case” copula, lognormal margins, risk–capital estimates based on 10,000 simulated losses: −50 −25 0 25 50 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

R el . D iff . (% ) Brc= 10,000

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Range of Risk–Capital Estimates: Boxplots

Gaussian copula, lognormal margins, risk–capital estimates based on 50,000 simulated losses: −50 −25 0 25 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

R el . D iff . (% ) Brc= 50,000

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Range of Risk–Capital Estimates: Boxplots

“Worst–case” copula, lognormal margins, risk–capital estimates based on 50,000 simulated losses: −50 −25 0 25 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

R el . D iff . (% ) Brc= 50,000

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Range of Risk–Capital Estimates: Boxplots

Gaussian copula, lognormal margins, risk–capital estimates based on 100,000 simulated losses: −50 −40 −30 −20 −10 0 10 20 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

R el . D iff . (% ) Brc= 100,000

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Range of Risk–Capital Estimates: Boxplots

“Worst–case” copula, lognormal margins, risk–capital estimates based on 100,000 simulated losses: −50 −40 −30 −20 −10 0 10 20 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

R el . D iff . (% ) Brc= 100,000

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Bounds on Risk–Capital Estimates

Obviously, increasing the number of replications for VaR

calculations narrows the range of possible risk–capital estimates.

It is, thus, not clear which part of a change is due to the subadditivity problem, and which one is due to computational issues.

A natural question is then: What could be the worst capital estimate? Statistically, this means to study whether there are theoretical bounds on VaR.

This topic has been treated, for example, by Makarov (1981) and Frank et al. (1987), and recently by Embrechts and Puccetti (2006).

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Bounds on Risk–Capital Estimates

calculations narrows the range of possible risk–capital estimates. It is, thus, not clear which part of a change is due to the

subadditivity problem, and which one is due to computational issues.

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Bounds on Risk–Capital Estimates

calculations narrows the range of possible risk–capital estimates. It is, thus, not clear which part of a change is due to the

subadditivity problem, and which one is due to computational issues.

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Bounds on Risk–Capital Estimates: Starting Point

The Fréchet–Höffding bounds (Fréchet, 1951; Höffding, 1940) apply to any n–dimensional copula, i.e.,

max(u1+. . . + un− n + 1, 0) | {z } C`(u) ≤ C (u) ≤ min(u) | {z } Cu(u) . (12) 0 0.5 1 0 0.5 1 0 0.5 1 u1 u2 C (u 1 ,u 2 ) 0 0.5 1 0 0.5 10 0.5 1 u1 u2 C (u 1 ,u 2 ) 0 0.5 1 0 0.5 1 0 0.5 1 u1 u2 C (u 1 ,u 2 )

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Bounds on Risk–Capital Estimates: Bounds on VaR

The tightness of the bounds on VaR depends on the dependence assumption.

If we assume that the copula C of X1, . . . , Xn satisfies C ≥ C0 and

Cd ≤ Cd

1 with Cd(u, v ) = u + v − C (u, v ), we obtain the bounds

F_min−1(α) = inf C0(u1,...,un)=α F₁−1(u1) + Fn−1(un) , (13) F_max−1(α) = sup Cd 1(u1,...,un)=α F₁−1(u1) + Fn−1(un) . (14)

For the two–dimensional case, C ≥ C0 implies that Cd ≤ C0d, so

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Bounds on Risk–Capital Estimates: Bounds on VaR

Cd ≤ Cd

1 with Cd(u, v ) = u + v − C (u, v ),

we obtain the bounds

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Bounds on Risk–Capital Estimates: Bounds on VaR

Cd ≤ Cd

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Bounds on Risk–Capital Estimates: Bounds on VaR

Cd ≤ Cd

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Bounds on Risk–Capital Estimates: Assumptions

We evaluate upper and lower bounds for three scenarios.

1 C0= C1 = C`: We do not use any restriction on the

dependence structure and thus use the lower Fr´echet bound, C`.

2 C₀= C₁ = u_iu_j: We assume that C ≥ u_iu_j, that is, we have

positive quadrant dependence (PQD).

3 C0= C b θi ,j CS, bC1 = C b θi ,j

C : We take the Clayton Survival copula as

lower bound, using the parameter values estimated for the DIPO data. For the survival copula of C1, we accordingly

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Bounds on Risk–Capital Estimates: Assumptions

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Bounds on Risk–Capital Estimates: Assumptions

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Bounds on Risk–Capital Estimates: Boxplots

“Worst–case” copula, lognormal margins, risk–capital estimates based on 10,000 simulated losses: −50 −25 0 25 50 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

R el . D iff . (% ) Brc= 10,000 C0= CL C0= CI C0= CC S, bC1= CC

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Bounds on Risk–Capital Estimates: Boxplots

“Worst–case” copula, lognormal margins, risk–capital estimates based on 50,000 simulated losses: −50 −25 0 25 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

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Bounds on Risk–Capital Estimates: Boxplots

“Worst–case” copula, lognormal margins, risk–capital estimates based on 100,000 simulated losses: −50 −40 −30 −20 −10 0 10 20 1/2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7

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Bounds on Risk–Capital Estimates: Results

Risk–capital estimates may increase when departing from the comonotonicity assumption.

However,

this effect depends on the presence of extremal (tail/quantile) dependence;

such an increase may as well be caused by an insufficient number of replications in the simulation of losses.

Theoretical bounds may help to assess which part of the change in risk capital is due to computational effects.

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Bounds on Risk–Capital Estimates: Results

However,

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Bounds on Risk–Capital Estimates: Results

However,

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Bounds on Risk–Capital Estimates: Results

However,

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Modeling Operational Risk: Estimation and Effects of Dependencies Conclusion

Conclusion

Contrary to expectations, risk capital may not always decrease when modeling dependencies.

Measuring risk capital for operational risk is a challenging task due to the scarcity of data, high variability, presence of extremes and the need to compute the VaR at a very high confidence level.

Simple methods, as correlations, may not lead to a complete and/or reliable picture of dependencies in operational risk losses. More sophisticated methods, such as copulas, could provide relevant information about dependencies and their effect on risk capital estimates.

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Conclusion

The question whether risk–capital estimates may increase compared to the comonotonicity case crucially depends on the presence of extremal (tail/quantile) dependence and the ellipticity of the multivariate distribution.

Therefore, the estimation of extremal dependence is of paramount importance for an assessment of the relevance of this effect.

For small sample sizes, parametric (copula) and

nonparametric methods should both be used in order to gain a picture as comprehensive as possible.

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Conclusion

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Conclusion

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References I

Artzner, P., Delbaen, F., Eber, J., and Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9:203–228.

Berg, D. (2009). Copula goodness-of-fit testing: An overview and power comparison. European Journal of Finance, 15:675–701. Coles, S. G., Heffernan, J. E., and Tawn, J. A. (1999). Dependence

measures for extreme value analyses. Extremes, 2:339–365. Deheuvels, P. (1979). La fonction de d´ependance empirique et ses

propriétés. un test non paramétrique d’independance. Bulletin de la Classe des Sciences, 65:274–292.

Embrechts, P., McNeil, A., and Straumann, D. (2002). Correlation and dependence in risk management: properties and pitfalls. In Dempster, M., editor, Risk Management: Value at Risk and Beyond. Cambridge University Press.

(94)

References II

Embrechts, P. and Puccetti, G. (2006). Bounds for functions of multivariate risks. Journal of Multivariate Analysis,

97(2):526–547.

Frank, M. J., Nelsen, R. B., and Schweizer, B. (1987).

Best–possible bounds for the distribution of a sum—a problem of kolmogorov. Probability Theory and Related Fields,

74(2):199–211.

Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont donnés. Annales de l’Université de Lyon, 3(14):53–77. Höffding, W. (1940). Masstabinvariante korrelationstheorie.

Schriften des Mathematischen Instituts und des Instituts fur Angewandte Mathematik der Universit¨at Berlin, 5:179–233.

(95)

References III

Joe, H., Smith, R. L., and Weissman, I. (1992). Bivariate threshold models for extremes. Journal of the Royal Statistical Society, Series B, 54:171–183.

Makarov, G. (1981). Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed. Theory of Probability and its Applications, 26:803–806. Sklar, A. (1959). Fonctions de r´epartition a n dimensions et leurs

marges. Publications de l’Institut de Statistique de L’Universit´a de Paris, 8:229–231.