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ModellingModelling Operational RiskOperational Risk
CSIRO Division of Mathematical and Information Sciences, SydneyCSIRO Division of Mathematical and Information Sciences, SydneyQuantitative Risk Management groupQuantitative Risk Management group
Sydney, AustraliaSydney, AustraliaEE--mail: mail: [email protected]@csiro.au
18 October 200618 October 2006
Pavel ShevchenkoPavel Shevchenko
www.cmis.csiro.au
� Basel II recognises the importance of the potential impact of losses due to Operational Risk and requires that banks hold adequate capital to protect against these losses.
� In Australia, the national regulator (APRA) is now applying the same detailed scrutiny to Operational Risk as previously to credit risk and market risk.
� The BCBS (Basel Committee on Banking Supervision) defined Operational Risk as:
the risk of loss resulting from inadequate or failed internal processes, peopleand systems, or from external events includes legal, excludes strategic and reputationalrisks
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Under the Basel II framework, banks canestimate operational risk using one of three approaches:
� The Basic Indicator Approach (15% of Gross income averaged over three year)
� The Standardised Approach(12-18% on the business line level)
� The Advanced Measurement Approaches (AMA)Internal model for 56 risk cells (7 event types x 8 business line)
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BCBS has identified the following 7 risk event types:� Internal fraud: e.g. intentional misreporting, employee theft, insider trading
� External fraud: e.g. robbery, cheque forgery, damage from computer hacking
� Employment practices and workplace safety:e.g. workers’compensation claims, violation of employee OH&S rules, union activities, discrimination claims
� Clients, products and business practices:e.g. misuse of confidential customer information, improper trading activities, money laundering, sale of unauthorisedproducts.
� Damage to physical assets:e.g. terrorism, vandalism, earthquakes, fires, floods.
� Business disruption and system failures:e.g. hardware and software failures, telecommunication problems, utility outages, computer viruses.
� Execution, delivery and process management:e.g. data entry errors, management failures, incomplete legal documentation, unapproved access to client accounts
� Corporate finance(18%)� Payment&Settlement(18%)� Trading&Sales(18%)� Agency Services(15%)
� Retail banking(12%)� Asset management(12%)� Commercial banking(15%)� Retail brokerage(12%)
8 Business Lines
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Business line
Event type
Bank, top level
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Internal Loss Data
(scaling, bias)
External Loss Data
(scaling, bias)
Expert Opinions
(business judgments on
#losses and $ranges)
Risk Frequency and Severity distributions
Total annual loss distribution (over all risks)
Annual Capital Charge (Expected and Unexpected losses)
Capital Allocation, what if scenarios
Control/Risk IndicatorInsurance Dependence Factors
Risk annual loss distributions
Advanced Measurement Approach: bottom-up Loss Distribution Approach
Monte Carlo simulations
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Annual Capital Chargeunexpected loss=VaR0.999-Expected Loss; Pr [Loss≤VaR0.999]=0.999
loss distribution
0
0.01
0.02
0.03
0.04
0$ loss
Expected Loss
Unexpected Loss
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Challenges/Tools
� Definition, identification, measurement, monitoring, indicators/controls
� Data Truncation: known threshold, stochastic threshold, unknown threshold
� Limited Data: mixing internal, external data and expert opinions via credibility theory, Bayesian techniques
� Data sufficiency: capital charge accuracy� Correlation between risks and its estimation: copula, common
shock processes� Control indicators: regression/factorial analysis� OR insurance: point processes� Non-Gaussian distributions, Fat tails: EVT, mixed distributions,
splices� VaR pitfalls: coherent risk measures, expected shortfall� Capital allocation
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severity distribution, pdf
0
0.1
0.2
0.3
0.4
0 3 6 9 X
frequency distribution
0
0.03
0.06
0.09
0.12
0.15
0 5 10 15 20 N
aggregate distribution, pdf
0
0.01
0.02
0.03
0.04
0 30 60 Z
Xeventtheofloss ,
Neventsofnumberannual ,
∑=
=N
kkXZlossannual
1
,
Loss Distribution Approach: single risk cell (business line/event type)
)|(.~,...,
),|(.~
1 ξ
θ
fiidareXX
PN
N
tindependenareXXandN N,...,1
e.g. Poisson
e.g. LogNormal
Monte Carlo, semi-analytic
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Insurance for Operational Risks� Insurance: probability of coverage, insurer default,
cover limit, excess, regulatory cap� Modelling of loss event times is required instead of
event frequency to address OR insurance.� point process:
e.g. homogeneous Poisson process
non-homogeneous Poisson: doubly stochastic Poisson:
2t time
1 yeartodayNt
...1...0 11 <<≤<<< +NN ttt
)Poisson(~,)l(Exponentia~1 λλδ Nttt ii −= +
1t
)(tλ(.)nomialNegativeBi(.)Gamma~ =>=> Nλ
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Data Truncation modelsData Truncation models
,....2,1, => iLX ii
,....2,1, => iLX i
(.)~ gL
Known constant truncation levelKnown constant truncation level
Known variable truncation levelKnown variable truncation level
Unknown truncation level Unknown truncation level
Stochastic truncation levelStochastic truncation level
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Known thresholdKnown threshold
0),( ≥XXf
,....2,1, =≥ iLX i� Known constant truncated level, i.e. loss data
Untruncated severity distribution
Truncated severity distribution
� Severity pdf fit via e.g. Maximum Likelihood Method
∫∞
=>≥>
=>L
dXXfLXLXLX
XfLXXf )(]Pr[;;
]Pr[
)()|(
∏= >
=ΨK
i
iN LX
Xf
11 ]Pr[
)(),...,( αα
� Frequency adjustment ]Pr[/)( LXNN obstrue >≈
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Unknown truncation levelUnknown truncation level
� L is an extra parameter in likelihood function
� L is unknown:
mu vs level, L
6
6.5
7
7.5
8
8.5
9
9.5
10
1000 4000 7000 10000 13000
sigma vs level, L
1.31.41.51.61.71.81.9
22.12.2
1000 5000 9000 13000
≥+×<+×
=γβγαγβα
µL
LLL
;
;)( ∑ −
iii
obs LL 2,, )]()([min µµγβα
Example: X~LogNormal(mu=8, sigma=2), L=4000estimates: mu≈8.05, sigma≈2.01, L≈4001
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Stochastic truncation levelStochastic truncation level
(.)~ gL
0),( ≥XXfKiLX i ,...,1, =>reported lossesreported losses wherewhere
Severity distribution of lossesSeverity distribution of losses
∫∞
> =>≥>
==a
aX dyyfaXaXaX
XfaLXf )(]Pr[;;
]Pr[
)(1)|(
marginal distribution of reported lossesmarginal distribution of reported losses
∫∫ >===
∞ X
daaX
agXfdaagaLXfXf
00]Pr[
)()()()|()(
~
∫ ∫∞ ∞
=>>=0
)()(]Pr[];Pr[/L
obstrue dXXfdLLgLXLXNN
∏=Ψi
iXf )(~
conditional conditional pdfpdf
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Dependence between risks
� Diversification: C(Z=R1+…+Rn)<C(R1)+…+C(Rn)
� VaR:
� Conditional VaR (CVaR):
� Dependence between frequencies
� Dependence between event point processes
� Dependence between severities
� Dependence between annual losses
� Dependence between risk profiles (parameters)
})(,min{)()( 1 ααα ≥== − zFzFZVaR ZZ
)](|[)( ZVaRZZEZCVaR αα >=
∑∑∑∑====
+++==KN
i
Ki
N
ii
N
ii
K
kk XXXZZlossannualtotal
1
)(2
1
)2(1
1
)1(
1
...:
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Basel Committee statement:“Risk measures for different operational risk estimates must be added for purposes of calculating the regulatory minimum capital requirement. However, the bank may be permitted to use internally determined correlations in operational risk losses across individual operational riskestimates, provided it can demonstrate to a high degree of confidence and to the satisfaction of the national supervisor that its systems for determining correlations are sound, implemented with integrity, and take into account the uncertainty surrounding any such correlation estimates(particularly in periods of stress). The bank must validate its correlation assumptions.”
Adding capitals=>perfect dependence between risks (too conservative)
∑∑∑∑====
+++==KN
i
Ki
N
ii
N
ii
K
kk XXXZZlossannualtotal
1
)(2
1
)2(1
1
)1(
1
...:
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Dependence between frequencies via copula)1,0(~)),,...,,( 21 UniformUUUUC iK
e.g. Gaussian copula
][],....,[ 11
111 KKK UFNUFN −− ==
-1
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PoissonN
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Copula correlation ρ
ρvs),( 21 NNcorr
Example
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∑=
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Dependence between frequencies=>Dependence between annual losses
Example
),(vs),( 2121 NNZZS ρρ
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ρ=
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Copula correlation ρ
10,5 21 == λλ)(~
)(~
22
11
λλ
PoissonN
PoissonN
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Dependence via common Poisson process(Johnson, Kotz and Balakrishnan)
1 yeartoday
t
1st risk events
2nd risk events
))((/),();(~)();(~)(
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1probwith)(ˆprobwith)()(ˆ
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-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
X
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7.0),(
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=YXcorr
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t2 Copula
Gumbel Copula
Gaussian Copula
Dependence via copula
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Gaussiancopula
))(1),...,2(1),1(1()(),...,2,1( duNFuNFuNFNFduuuGaC −−−= ρρ
t-copula
))(1),...,2(1),1(1()(),...,2,1( dutututtFduuutC −−−Σ=Σ ννννν
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Upper tail dependence
αααα
αααΡ
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+−−→
=−>−>−→
=1
),(211
lim)](11|)(1
2[1
lim CFXFY
βλ 22−=
0=λ
0≥λ
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0.00
0.10
0.20
0 4 8 12
Severity Distribution Tail: Extreme value Theory and splicing
=−−≠+−=
−
0];/exp[1
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/1
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x
xxH
∞<≤++= XXifXKfKwXfwXf 0),(),(...)(11
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EVT Generalized Pareto Distribution (GPD)
aX
0
f1(X)=Truncated LogNormal
f2(X)=GPD
f(x)=qf1(X)+(1-q) f2(X)
Mixture distributions:
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Structural Structural ModellingModelling of of OperatinalOperatinalRisk via Bayesian Inference: Risk via Bayesian Inference:
combining loss data with expert opinionscombining loss data with expert opinions
Pavel Shevchenko (CSIRO) and Mario Wüthrich (ETH) Structural Modelling of Operational Risk using Bayesian Inference: combining loss data with expert opinions.June 2006. Submitted to The Journal of Operational Risk
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Bayesian inference to combine expert opinions with loss data
� Expert opinions for prior distribution
� Loss data
� Posterior distribution
)(απ r
)()|(ˆ)()|(),( XhXXhXhrrrrrrrr
απαπαα ==
)(/)()|()|(ˆ XhXhXrrrrr
απααπ =
)|(};,...,{ 1 αrrrXhXXX n=
www.cmis.csiro.au
),...,( 1 nNN=N 0,!
)|( ≥= − λλλ λ
NeNf
N
PoissonPoisson--GammaGamma
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1
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βλβαλπα
)ˆ/exp(!
)/exp()(
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n
Nn
ii
×+=→
+=→ ∑=
ββββ
ααα
are conditionally iid from
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estimate of the arrival rate vs year
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 3 6 9 12 15year
arriv
al rat
e j
Bayesian estimatemaximum likelihood estimate
15.0,41.3 ≈≈ βα
6.0=λ
the Bayesian estimator with Gamma prior
Annual counts N=(0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 0) from Poisson
kkk βαλ ˆˆˆ ×=
∑ == k
i ikk N1
1~λ the Maximum Likelihood estimator
expert opinions 15.0,41.33/2]75.025.0Pr[,5.0][ ≈≈→=≤≤= βαλλE
Example
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““ ToyToy”” Model for Operational Risk Model for Operational Risk using Credibility Theoryusing Credibility Theory
Hans Bühlmann (ETH), Pavel Shevchenko (CSIRO) and Mario Wüthrich (ETH)A “Toy” Model for Operational Risk Quantification using Credibility Theory. June 2006. Submitted to Risk Magazine
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0.00
0.10
0.20
0 4 8 12
Low Frequency High Impact Losses: PoissonLow Frequency High Impact Losses: Poisson--ParetoPareto
0,...,1,0,!
)|(;0,,1
)|( ≥=−=>≥−−
=
θθθθξξξξ ne
n
nnPLx
L
x
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Severity distribution
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αατσ
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Maximum Likelihood Estimator for tail parameter:Maximum Likelihood Estimator for tail parameter:
jjj
j
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2
1
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j,
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=
−==
−=
=
=≥=
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Improved credibility estimator (using all data in the bank):Improved credibility estimator (using all data in the bank):
www.cmis.csiro.au
Jja
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jjj
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αϑαϑ
ϑϑατ
τϑαϑαϑαϑ
ϑτϑϑϑϑ
Improved credibility estimator (using all data in the bank):Improved credibility estimator (using all data in the bank):
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Improved credibility estimator (using industry data):Improved credibility estimator (using industry data):
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Improved credibility estimator (using industry data):Improved credibility estimator (using industry data):
∑∑
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==
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βϑβϑ
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1
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Maximum Likelihood Estimator for arrival rate:Maximum Likelihood Estimator for arrival rate:
,~/]|ˆ[Var,]|ˆ[
~,~1ˆ
estimator likelihood maximum Then the
profilesrisk are and constantsknown are
)Poisson( from iid ally)(condition are
,...,1,sfrequencie losswith ,...,1 cellsrisk Consider
1,
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νλλλλλλ
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Improved credibility estimator for arrival rate (us ing bank dataImproved credibility estimator for arrival rate (us ing bank data))
.~
1~
1;ˆ1
;~;)ˆ(~
1;~
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then/][Var,][,/Consider
bank. theof profilerisk a is
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νν
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νν
ννλγγ
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λνθωλν
νγλγλγλ
νλλν
λωλλλλ
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Improved credibility estimator of arrival rate (usi ng industry dImproved credibility estimator of arrival rate (usi ng industry data)ata)
∑∑
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Then industry. theof profilerisk a is
][,]E[with iidare,...,1, that Assume
1,...,, profilesrisk with banks Consider
ρλρλ
γ
ωω
ρλγλ
ωλνν
γν
λ
λωλλλλ
λ
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)1()1()1(
)()1(0
)1()1()1()1(
)1()1(0
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ˆ̂ˆ̂
,...,1,ˆ̂
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down- top:rate arrivalfor estimatorsy credibilit Corrected
jjj
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Jj
λνθ
λγλγλ
λρλρλ
=
⇓
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−+=
www.cmis.csiro.au
)ˆ̂ˆ̂
(Poisson~),ˆ̂ˆ̂
Pareto(~ j,1
, jjjjjjnj
jN
nnjj NaXXZ λνθϑξ ==⇐=∑
=
www.cmis.csiro.au
Topics for further research
� Evolutionary models (stochastic risk profiles)� Dependence between risks via dependence
between risk profiles� Full Bayesian approach to get not just credibility
estimates for risk profiles but their posterior distributions
� Modelling high frequency low impact losses (with lower threshold)
� Allocation of capital into Business Units� Combining expert opinions and external data
