Operational Risk Management & Modelling ก ˘ˇˆก ˙˝˛ก ˘˚ ˝˜ ˙ ... · 2019-05-08 ·...
Transcript of Operational Risk Management & Modelling ก ˘ˇˆก ˙˝˛ก ˘˚ ˝˜ ˙ ... · 2019-05-08 ·...
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Operational Risk Management & Modelling ก�����ก�����ก������� �!!"#�$%&'() �*�+,-.!�/.ก��
� The Basel II AMA Framework
� Poomjai Nacaskul, Ph.D. Financial Institutions Policy Group, the Bank of Thailand
� FIPG Internal Seminar
� 28 November ����15 December 2006
� ก��! Basel II AMA
� ��.JK$L� +�"&กM�&�)+N)!�)&O�!�+ก��% .+ P+�"���QR ,��%STUS)
� ก��&�$$+�V�)L+ &+&.
� WX JYT�.ก�)+ ����Z P�+#�"$ W[\]
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Presentation Outline
� I. Introduction� Slide 3 _ Slide 18
� II. Modelling Methodology� Slide 19 _ Slide 40
� III. Capital Adequacy Regulation� Slide 41 _ Slide 60
� Bibliography� Slide 61 _ Slide 62
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Part One
INTRODUCTION
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Risk & Risk Management
� Risk ≡≡≡≡ {POSSIBILITY; PROBABILITY; UTILITY}� Ask: which of these would be subject to adjustment/alteration/manipulation?
� Financial Risks : � Proxy for (monotonic w) utility measured/quoted in monies/monetary units,
� arisen from/rooted in financial market variables/institution factors,
� managed/mitigated by means of financial techniques/tools, and/or
� seen as/deemed to be intrinsic/integral to financial markets/institutions.
� Risk Management Processes : IDENTIFY/define ���� MEASURE/assess ���� MITIGATE/manage ���� REVIEW/report
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Utility Theory of Risk
� Risk has upside as well as downside, so why is it undesirable?
� Utility Theory [Neumann & Morgenstern (1947)]� Certainty Equivalence Principle/Probability/Prior :
� 100%Pr(1-./.) + 0%Pr(0-./.) ~ 1-./.⇒⇒⇒⇒ U(1-./.) = 1
� 90%Pr(1-./.) + 10%Pr(0-./.) ~ 0.5-./.⇒⇒⇒⇒ U(0.5-./.) = 0.9
� 1%Pr(1-./.) + 99%Pr(0-./.) ~ 10,000/.⇒⇒⇒⇒ U(10,000./.) = 0.01
� .02%Pr(1-./.) + 99.98%Pr(0-./.) ~ 400/.⇒⇒⇒⇒ U(400/.) = 0.0002
� 0%Pr(1-./.) + 100%Pr(0-./.) ~ 0-./.⇒⇒⇒⇒ U(0-./.) = 0
� Risk Appetite/Attitude/Aversion :� U(0.5-./.) = 0.9 > 0.5 = E[50%Pr(1-./.) + 50%Pr(0-./.)] ⇒⇒⇒⇒ risk averse
� U(10,000/.) = 0.01 = 0.01 = E[1%Pr(1-./.) + 99%Pr(0-./.)] ⇒⇒⇒⇒ risk neutral
� U(400/.) = 0.0002 < 0.0004 = E[.04%Pr(1-./.) + 99.96%Pr(0-./.)] ⇒⇒⇒⇒ risk seeking
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Defining Market & Credit Risks
� Market Risk ≡≡≡≡ opportunity/possibility & probability of financially relevant gains/losses due to movements in the financial-market and monetary-economic variables, namely interest/exchange rates, equity/commodity prices, etc.
� Credit Risk ≡≡≡≡ opportunity/possibility & probability of financially relevant losses (but occasionally gains) due to Credit Events : � For bank loans : obligor defaults (incl. counterparty/settlement),
recovery/collateral, drawdown risks, respectively ~ PD, LGD, EAD.� For defaultable bonds : default + (credit rating) downgrade risks.� For credit derivatives : single-obligor events (i.e. CDS & CLN pricing),
multi-obligor events (i.e. multi-name CDS & CDO pricing), etc.
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Defining Operational Risk
� Operational Risk ≡≡≡≡ opportunity/possibility & probability of financially relevant losses due to {failures, frauds, errors} as well as {random accidents, natural catastrophes, manmade disasters}, whence leading to {damages, disruptions, incursions}, thereby negatively impacting {financial conditions, business conduct, institutional integrity} overall.� According to the Basel Committee on Banking Supervision (BCBS)xs
New Capital Accord (Basel II) : risk of loss resulting from inadequate/failed internal processes, people and systems or from external events. This definition includes legal risk, but excludes strategic risk & reputational risk.
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Operational Risk Management (ORM)
1. ก��!ก�����ก��"#�$%&'() �*�+,-.!�/.ก�� (Operational Risk Management Framework)
2. ��!!#������!"#�$%&'() �*�+,-.!�/.ก�� (Operational Risk Measurement System)
3. ��%!')!#.P'����� �!!"#�$%&'() �*�+,-.!�/.ก�� (Operational Risk Modelling Methodology)
4. ,y���)z�!�!!����� "#�$%&'() �*�+,-.!�/.ก�� (Operational Risk Model Drivers)
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BCBS (2003), Sound Practices
� Introduction� PRINCIPLES ⇒⇒⇒⇒ FRAMEWORK for effective Management & Supervision� to be used by Banks & Supervisory Authorities when evaluating
Operational Risk Management (ORM) POLICIES & PRACTICES.
� Background� Deregulation/globalisation of financial services + growing sophistication of
financial technology ⇒⇒⇒⇒ more COMPLEX bank ACTIVITIES & RISKprofiles, e.g. automation, e-commerce, post-M&A integration, large-volume service provider, risk mitigation techniques, outsourcing
� Event TYPES: i. Internal Fraud, ii. External Fraud, iii. Employment/Safety,iv. Clients/Products/Business Practices, v. Physical-Asset Damages, vi. Disruption/Failure, vii. Execution/Delivery/Process Management
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BCBS (2003), Sound Practices (2)
� In the past, Banks relied on Internal Control/Audit Function; now there emerge STRUCTURES & PROCESSESto address OR as a distinct class.
� Sound Practices� OR : not directly taken in return for an expected reward� ORM : Identification ���� Assessment ���� Monitoring ���� Control/Mitigation� BCBS structured the sound practice paper around 10 PRINCIPLES.
� Developing an Appropriate Risk Management Environment� Board & Senior Management to create ORGANISATIONAL CULTURE
that emphasises high standard of ETHICAL behaviour and establishes expectation of INTEGRITY for all.
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BCBS (2003), Sound Practices (3)
� Principle 1 : Board of Directors aware of "a distinct risk category"� Board to approve firm-wide Operational Risk Management Framework (ORMF).
� Appropriate DEFINITION of OR
� Board to establish management STRUCTURE.
� Board to REVIEW.
� Principle 2 : Board of Directors ensures ORMF subjected to Internal Audit� Board should have in place ADEQUATE Internal Audit coverage.
� Board to ensure INDEPENDENCE of the Audit Function.
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BCBS (2003), Sound Practices (4)
� Principle 3 : Senior Management responsible for IMPLEMENTING ORMF� Translate ORMF into specific POLICIES/PROCESSES/PROCEDURES
� Ensure QUALIFIED Staff & INDEPENDENT Compliance Units.
� Ensure OR Managers COMMUNICATE w Credit/Market Risk Managers.
� Ensure REMUNERATION policies consistent w bank's appetite for risk.
� Pay particular attention to the QUALITY of DOCUMATATION Controls& TRANSACTION-Handling Practices.
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BCBS (2003), Sound Practices (5)
� ORM: Identification, Assessment, Monitoring, and Control/Mitigation� Principle 4 : IDENTIFY & ASSESS
� Paramount step: consider both INTERNAL & EXTERNAL FACTORS.
� Identify not just most potentially adverse risks, but also VULNERABILITY to them.
� TOOLS for identifying & assessing OR: (1) Self Risk Assessment, (2) Risk Mapping, (3) Risk Indicators, (4) Measurement (e.g. Loss Data), etc.
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BCBS (2003), Sound Practices (6)
� Principle 5 : MONITOR� Essential: PROMPT detecting & addressing deficiencies
can reduce potential frequency and/or severity.
� Bank to develop forward-looking Key Risk/Early Warning Indicators.
� Monitoring FREQUENCY to reflect risks, frequency & nature of changes in the OPERATING ENVIRONMENT.
� Senior Management to receive regular REPORTS.
� Board to receive higher-level INFORMATION.
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BCBS (2003), Sound Practices (7)
� Principle 6 : CONTROL and/or MITIGATE� To control and/or mitigate, or to BEAR the risks?
Or to reduce or withdraw involvement in the activity?
� Board & Senior Management to establish strong Internal Control CULTURE.
� Requires segregation of duties and avoid CONFLICT OF INTEREST.
� Bank to ensure appropriate INTERNAL PRACTICE to control OR.
� OR pronounced where banks engage in NEW, UNFAMILIAR ACTIVITIES, especially outside their CORE BUSINESS STRATEGIES.
� There exists uncontrollable LOW-PROB HIGH-IMPACT OR, e.g. natural disasters.
� Risk Mitigation Tools = COMPLEMENT, not replacement for control.
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BCBS (2003), Sound Practices (8)
� Increased AUTOMATION could transform high-frequency, low-severity into low-frequency, high-severity.
� Bank to establish POLICIES for managing OUTSOURCING risks.
� Bank to understand potential impacts from VENDORS and other THIRD-PARTY/INTRA-GROUP service providers.
� Decision to self-insure risk to be TRANSPARENT within the organisation and CONSISTENT with overall business strategy & risk appetite.
� Principle 7 : CONTINGENCY & Business CONTINUITY plans� Bank to establish DISASTER RECOVERY & BUSINESS CONTINUITY plans.
� Bank to identify CRITICAL BUSINESS PROCESSES.
� Bank to conduct REVIEW.
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BCBS (2003), Sound Practices (9)
� Role of Supervisors� Principle 8 : REQUIRE banks to have in place�
� Supervisors to require banks to develop ORMF & encourage banks to develop better TECHNIQUES.
� Principle 9 : Conduct EVALUATION� Examples of independent evaluation: �� For FINANCIAL GROUPS, Supervisors to seek to ensure
appropriate & integrated ORM throughout.� Supervisors to address DEFICIENCIES IDENTIFITED
through a range of ACTIONS.� Supervisors to take ACTIVE ROLE in encouraging
ongoing internal development efforts �
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BCBS (2003), Sound Practices (10)
� Role of Disclosure� Principle 10 : "� sufficient public disclosure to allow market participants
to assess � [banks'] approach �"� BCBS recognises: timely & frequent PUBLIC DISCLOSURES
lead to enhanced market discipline, hence more effective ORM.
� BCBS believes: bank should disclose its ORMF, even if still undergoing development.
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Part Two
MODELLING METHODOLOGY
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Mathematical & Probabilistic Models
� A Mathematical Model : � Output = Function(Input; Parameter)� Specification vs. Estimation/Calibration tasks
� w.r.t. estimation, think historical volatility� w.r.t. calibration, think implied volatility� training vs. validating vs. testing data
� A Probabilistic Model : � Random Variable : X in {x}, X ~ Distribution(θθθθ), i.e. θθθθ is the parameter.
� Discrete vs. Continuous vs. Discrete-Continuous (Mixed) distributions
� Cumulative Distribution Function : F(x) = F(x;θθθθ) = Pr(X < x)� Probability Mass/Density Function (p.m.f./p.d.f.) : p(x) = Pr(X = x) & f(x) = dF(x)/dx
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Distributional Information
� Distributional Information extracted from/used to construct F(x) : � << � statistic x group>> : means µµµµ = 1st moment, mode, median, variance
σσσσ2 = 2nd central moment), std. deviation σσσσ, skewness (3rd), kurtosis (4th)� << � metric x group>> : inter-quartile range, m-Sigma distance from means,
the 100(1-αααα)% confidence interval (CI) ���� Value-at-Risk (VaR), the conditional mean exceedance (CME) ���� Expected Shortfall (ES)
� << � functional x group>> : modality (i.e.whether f(x) is unimodal, bimodal/multimodal), quantile function, limiting tail distribution
� The levels/types of information depends on application requirements.� This, in a sense, is how increasing model sophistication pays for itself.
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Descriptive Statistics
DESCRIPTIVE STATISTICS � MEANINGFUL DATA SUMMARY
Reporting sampled data via �an expensive long distance telephone call�
� MEASURES OF CENTRAL TENDENCY
(SAMPLE) MEAN ∑=
≡n
iix
nx
1
1
(SAMPLE) MEDIAN
Ι∈+=
Ι∈=
+
=+
+
mmnx
mmnxxx
n
nn
ileth
,12
,22
1
2
1
122%50
� MEASURES OF VARIATION or DISPERSION
MEAN ABSOLUTE ∑=
−n
ii xx
n 1
21 & MEAN SQUARED ( )∑
=
−n
ii xx
n 1
21
DEVIATIONS
VARIANCE ( )∑=
−−
≡n
ii xx
nS
1
22
1
1 & STANDARD DEVIATION
2SS =
RANGE minmax xx − & INTERQUARTILE RANGE ileile thth xx%25%75
−
� MEASURES OF ASYMMETRY
SKEWNESS
( )3
1
3
S
nxx
Skew
n
ii∑
=
−≡
For a positively (negatively) skewed distribution, the mean is greater (less) than the
median.
� MEASURES OF FAT-TAILEDNESS
KURTOSIS
( )4
1
4
S
nxxKurt
n
ii∑
=
−≡
A distribution with kurtosis less than/equal to/greater than the /benchmark1 value 3
is said to be a PLATYKURTIC/MESOKURTIC/LEPTOKURTIC distribution.
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Probability DistributionPROBABILITY � CONDITIONAL PROBABILITY & BAYES5 THEOREM
� CONDITIONAL PROBABILITY & BAYES� THEOREM:
)|Pr()|Pr(
)Pr(
)Pr(
)Pr()|Pr(
cABAB
BA
B
BABA
+
∩=
∩= (1)
� INDEPENDENT random variables:
)Pr()Pr(
)Pr()Pr()|Pr()Pr()Pr()Pr( A
B
BABABABA =
⋅=⇒⋅=∩ (2)
PROBABILITY � DISCRETE, CONTINUOUS, AND CUMULATIVE DISTRIBUTION
� CUMULATIVE DISTRIBUTION FUNCTION (c.d.f.):
1)()Pr()( =∞∋≤= FxXxF (3)
Note:
)()()Pr()Pr()Pr(
)(1)Pr()Pr()Pr(
aFbFaXbXbXa
xFxXSxX
−=≤−≤=≤<
−=≤−=≥ (4)
� PROBABILITY MASS FUNCTION (p.m.f.):
1)Pr()Pr()()Pr()( ==≤==∋== ∑∑≤ axa
aXaXxFxXxp (5)
� PROBABILITY DENSITY FUNCTION (p.d.f.):
1)()()()( =≤=∋ ∫∫∞
∞−∞−
dxxfdttfxFxfx
(6)
� Hence the FUNDAMENTAL THEOREM OF CALCULUS:
)()()()()(
)( aFbFxFdxxfdx
xdFxf
b
a
b
a
−==⇒= ∫ (7)
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Distributional MomentsPROBABILITY � EXPECTATION & MOMENTS OF DISTRIBUTION
� POPULATION (TRUE) MEAN, VARIANCE, STANDARD DEVIATION,
COVARIANCE, CORRELATION:
[ ]( )[ ]
[ ]( )
[ ]( )
[ ]( ) [ ]( )[ ]]1,1[
)()(
),(),(
),(
)()(
)(
)()(
)(
)(][)(
,
,
2
2
22
+−∈⋅
≡=
Ε−Ε−Ε≡=
≡=
Ε−
Ε−=Ε−Ε≡=
=Ε≡=
∫
∑∫∑
∞
∞−
∀
∞
∞−
∀
YStdDevXStdDev
YXCovYXCorrel
YYXXYXCov
XVarXStdDev
dxxfXx
xpXxXXXVar
continuousXdxxfx
discreteXxpxXXMean
YX
YX
X
j jj
X
j jj
X
ρ
σσ
σ
µ
(1)
� In practice, it5s actually more computationally convenient to use an equivalent variance
formula:
[ ]( )[ ] [ ] [ ]222 XXXX Ε−Ε=Ε−Ε (2)
� Expected Value & Variance of a LINEARLY TRANSFORMED random variable:
)()(
][][2 XVabaXV
bXabaX
=+
+Ε=+Ε (3)
� Expected Value & Variance of a SUM of random variables:
),(2)()()(
][][][
YXCovYVXVYXV
YXYX
++=+
Ε+Ε=+Ε (4)
� For INDEPENDENT r.v.5s:
)()()(0),( YVXVYXVYXCov +=+⇒= (5)
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Distributions begets Distributions
� Additional distributions derived from F(x) : � Bayesian/Mixture Model : when θθθθ in F(x;θθθθ) is itself a random variate.
� e.g. Negative Binomial = λλλλ-Gamma mixture of Poisson(λλλλ)
� Convolution : F(x) = Gn*(x) = distribution of sum of n independent, generally identically distributed random variables (i.i.d.r.v.)� i.e. n too small to be using the Central Limits Theorem (C.L.T)
� Compound Distribution : distribution of sum of N independent, invariably identically distributed random variables, where N is itself a random variate.� e.g. sum of Gamma severity events occurring with Poisson frequency
� Copula : a functional creating a joint distribution function out of �hitherto marginal� distribution functions, whereupon FXY(x,y) = C(FX(x),FY(y))
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Unifying Structure for Financial Risk Models
� Toward a UNIFYING STRUCTURE for modelling financial risks : � S = S1 + N + SN� Si ~ Individual_Distributioni(θθθθi), i = 1,N,N
� where possibly N ~ Natural_Number_Distribution(ΘΘΘΘ)� where possibly θθθθi ~ Meta/Prior_Distribution(ΘΘΘΘi)
� S ~ Aggregate_Distribution(ΘΘΘΘ), ΘΘΘΘ = {θθθθ1,N,θθθθN} ∪∪∪∪ θθθθCopula
� To wit, respectively for market, credit, and operational risks : � Multivariate Normal Equity Portfolio� Bernoulli Mixture Loan Portfolio� Compound Poisson Subexponential Losses from failures/frauds/errors
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Market/Credit vs. OR Financial Risk Type
Market Risk
Credit Risk
Operational Risk
Probabilistic Model of Individual Event Arrivals
N/A N/A for periodic credit-rating upgrade-downgrade; Poisson, Negative Binomial for sporadic default arrivals.
Poisson, Negative Binomial.
Probabilistic Model of Individual Event Severities
(Symmetric) Profit/Loss Distribution: Univariate Normal/Logistic for single-asset return.
(Skewed) Loss Distribution: Bernoulli for single-obligor default, Beta/Gamma for single-obligor loss.
(Heavy-tailed, skewed) Loss Distribution: Subexponential, Extreme Value.
Probabilistic Model of Aggregate Event Severities
Multivariate Normal for portfolio return.
Default Correlation via Bernoulli Mixture, Factor Model and/or Copula.
Independence assumption, hence convolution of probability mass/density functions.
Comments on total VaR calculation
Straightforward low-probability quantile of Multivariate Normal distributions.
Difficult, reliant on multiple-obligor loss distribution.
Difficult if reliant on convolved loss distribution; relatively easy if tail quantile approximation method is used.
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Modelling OR Probabilistically
� [Lewis (2004)] :� OR Indicators ≡≡≡≡ �random variables that are used to provide insight into
future OR events. For example, a rising number of trades that fail to settle � Loss due to the failure of a vendor to perform outsourced processing correctly and unauthorized transfers of money by employees � are examples of OR events for which we seek to find suitable OR indicators.�
� OR VaR ≡≡≡≡ �the operational risk capital sufficient, in most instances, to cover operational risk losses over a fixed time period at a given confidence level.�
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Modelling OR Frequency, Severity & Loss
1. Modelling individual OR event arrivals� Frequency generally has a Poisson or Negative Binomial distribution.
2. Modelling individual OR event severities� Loss probability measure has support [0,∞∞∞∞)� Distribution positively skewed, heavy-tailed, most likely subexponential� The use of Extreme Value Theory (EVT)
3. Modelling aggregate OR loss� independent and identically distributed (i.i.d.) random variables (r.v.)
4. Calculating OR VaR
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Empirical & Binomial DistributionsPROBABILITY � FAMILIAR DISCRETE DISTRIBUTIONS
� EMPIRICAL DISTRIBUTION (discrete r.v.): general term denoting p.m.f. not expressible
as a mathematical formulae, i.e. 1.0)5.5(,5.0)2(,4.0)1( ===− ppp , and zero
probability otw.
� BINOMIAL DISTRIBUTION (discrete r.v.):
∈−=
Ν∈=Ε
=
=−=
⇒ ∑=
−
]1,0[,)1()(
,][
)()(
,,0,)1()(
),(~ 0
ρρρρ
ρρ
ρ
nXV
nnX
ipxF
nxCxp
nBinX
x
i
xnxnx K
(1)
BERNOULLI DISTRIBUTION as a special case:
),1()( ρρ =⇔ nBinBernoulli .
As an example of the application of moment-generating function, consider the
derivation of the means of the binomial distribution:
[ ]( )( )( ) ρρρρ
ρρρρρ
ρρρρ
nnM
eentM
ThmBinomialbyeie
eCCeetM
n
tnt
nt
n
x
n
x
xnxtnx
xnxnx
txtX
=−+=
⋅−+⋅=
−+⋅=
−⋅=−=Ε=
−
−
= =
−−∑ ∑
1
1
0 0
1)0('
)(1)('
.)..()1()(
)1()()1()(
(2)
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Negative Binomial & Poisson Distributions
� NEGATIVE BINOMIAL DISTRIBUTION (discrete r.v.):
∈−
=
Ν∈=Ε
=
≥−=
⇒
∑=
−−−
]1,0[,)1(
)(
,][
)()(
,)1()(
),(~
2
11
ρρ
ρρ
ρρ
ρ
rXV
rr
X
ipxF
rxCxp
rNegBinX
x
ri
rxrxr
(1)
GEOMETRIC DISTRIBUTION as a special case:
),1()( ρρ =⇔ rNegBinGeometric .
� POISSON DISTRIBUTION (discrete r.v.):
=
>=Ε
=
==
⇒ ∑=
−
λλλ
λ
λ
λ
)(
0,][
)()(
,2,1,0,!
)(
)(~0
XV
X
ipxF
xx
exp
PoissonXx
i
x
K
(2)
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Exponential & Beta Distributions
PROBABILITY � FAMILIAR CONTINUOUS DISTRIBUTIONS
� EXPONENTIAL DISTRIBUTION (continuous r.v.):
=
>=Ε
−=
≥=
⇒−
−
2/1)(
0,/1][
1)(
0,)(
)(~
λλλ
λ
λλ
λ
XV
X
exF
xexf
ExpXx
x
(1)
� BETA DISTRIBUTION (continuous r.v.):
∫
∫
−−
−−
−=Β
+++=
+=Ε
∈=
∈Β
−=
⇒
1
0
11
2
0
11
)1(),(
)1()()(
][
]1,0[,)()(
]1,0[,),(
)1()(
),(~
dxxx
XV
X
xdttfxF
xxx
xf
BetaX
x
βα
βα
βα
βαβααβ
βαα
βα
βα
(2)
UNIFORM DISTRIBUTION as a special case: )1,1()( ==⇔ βαBetaUnif .
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Beta Distributions
00.
10.
2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Beta(
1,1)Beta(
4,4)Beta(
3,7)
-1
0
1
2
3
4
5
6
7
8
Beta (a,b) distribution
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Gamma Distributions� GAMMA DISTRIBUTION (continuous r.v.):
)!1()(
)(
0,/)(
0,/][
)()(
0,)(
)()(
),(~
0
1
2
0
1
−=Γ⇒Ν∈
=Γ
>=
>=Ε
=
≥Γ
=
⇒
∫
∫
∞−−
−−
ααα
α
λλααλα
αλλ
λα
α
αλ
duue
XV
X
dttfxF
xxe
xf
GammaX
u
x
x
(1)
Where α is the SHAPE PARAMETER, and λβ /1≡ is the SCALE
PARAMETER.
ERLANG DISTRIBUTION as a special case when the shape parameter is a whole
number.
Exponential distribution as a special case when the shape parameter is equal to one.
� GAMMA FUNCTION, denoted ℜ∈Γ αα ,)( according to the LEGENDRE
NOTATION, is a generalisation of the FACTORIAL FUNCTION beyond non-negative
whole numbers to the real number line (more generally to the complex plane):
}
{
( )
)!1()1()!1())1(())1(()1()(,
)3()3)(2)(1()2()2)(1(
)1()1()1(0lim
)1()()()(,
0
21
0
2
0
1
0
1
0
1
−=Γ−=−−Γ−−−=ΓΝ∈
−Γ−−−=−Γ−−=
−Γ−=−⋅+−−=
−⋅−−−===Γℜ∈
∫
∫∫∫∞
−−−−
∞→
∞−−
∞−−
∞−−
∞−−
αααααααααααααααα
ααα
ααα
αα
αααα
L
L
dttete
dtteetdtetdtte
tt
t
tt
dv
t
u
t
(2)
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Extreme Value Theory
EXTENSION � EXTREME VALUE THEORY
� FISHER-TIPPET-GNEDENKO THEOREM:
Let { }n
iiX 1= be a sequence of i.i.d.r.v.5s, and define SAMPLE MAXIMA as the
maximum { }nn MMM ,,max 1 K≡ of such a sequence.
As the sample size gets large, ∞→n ,
( ) ( )xXxXxMxF nn ≤≤=≤≡ ,,PrPr)( 1 K , the c.d.f. for this sample maxima,
approaches the following three-parameter GENERALIZED EXTREME VALUE
(GEV) distribution:
[ ]
<
−+−
>
−+−
=
−−−
=
−
−
)(01exp
)(01exp
)(0expexp
)(
1
1
,,
WeibullTypeIIIx
FrechetTypeIIx
GumbelTypeIx
xF
ξσµ
ξ
ξσµ
ξ
ξσµ
ξ
ξξσµ (1)
Where the three parameters ξσµ ,, are known, respectively, as LOCATION
PARAMETER, SCALE PARAMETER, and TAIL INDEX PARAMETER.
� Method of BLOCK MAXIMA (BM):
Exploits the Fisher-Tippet-Gnedenko theorem to describe the upper percentile
description:
( )( )
=−−
>−−−=
−
0lnlnˆˆ
0)ln(1ˆˆ
ˆ)(
ˆ
ξασµ
ξαξ
σµ
αξ
OpVaR (2)
36
Convolved Probability Functions
ANALYSIS OF SMALL AGGREGATE � CONVOLUTION, NOT LIMIT THEOREMS
� Distribution of a sum of two independent (though not required to be identically distributed)
random variables is given by the convolution of their probability mass/density functions:
Discrete case:
( )
( )
∗≡−=
∗≡−=
⇒
+=
==
==
∑
∑)()()(
)()()(
)()()Pr(
)()Pr(
zppyzpyp
zppxzpxp
zp
YXZ
ypyY
xpxX
XYy
XY
YXx
YX
ZY
X
(1)
Continuous case:
( )
( )
∗≡−=
∗≡−=⇒
+=
=
=
∫∫
)()()(
)()()()()(
)(
)()(
zffdyyzfyf
zffdxxzfxfzf
YXZ
yfdy
ydF
xfdx
xdF
YXXY
YXYX
zYY
XX
(2)
37
Convolution IntegralCONVOLUTION � DEFINITION & KEY THEOREM
� Given ℜ→Dgf :, , define CONVOLUTION thus:
( ) ( ) )()()()()()( xfgdttxftgdttxgtfxgf ∗≡−=−≡∗ ∫∫ ℜℜ (1)
Not only is convolution commutative, as seen above, it is also associative and
distributive:
( ) ( )( ) ( ) ( )gfgfhgf
hgfhgf
gfgf
∗+∗=+∗
∗∗=∗∗
∗=∗
(2)
Moreover:
( )
dz
zdgfg
dz
zdf
dz
zgfd )()()(∗=∗=
∗ (3)
For ℜ→Dfff :,, 321 , observe that:
( )( ) ( )( )
∫ ∫∫ ∫∫ ∫∫
ℜ ℜ
ℜ ℜ
ℜ ℜ
ℜ
−−=
−−=
−−=
−∗=∗∗
121232211
122312211
223112211
223221321
)()()(
)()()(
)()()(
)()()(
dydyyyxfyfyf
dydyyxfyyfyf
dyyxfdyyyfyf
dyyxfyffxfff
(4)
Where the change in the ORDER OF INTEGRATION is justified, i.e. provided the
mapping )()()(),( 231221121 yxfyyfyfyy −−a is INTEGRABLE on 2ℜ ,
that is, Dx∈∀ .
Moreover, along the way there was a CHANGE OF VARIABLE 122 yyy −a ,
i.e. so that )()()()()()( 12322112312211 yyxfyfyfyxfyyfyf −−−− a .
� Given ℜ→Dff n :,,0 K , one can therefore write n-FOLD CONVOLUTION in terms of:
( ) ( )∫ ∫ ∑ℜ ℜ −
−
=−− −=∗∗ 01
1
011000 )()()( dydyyxfyfyfxff n
n
i innnn LLLL (5)
38
Computational Techniques
� PanjerRs Algorithm� Works if severity distribution is discrete
and frequency distribution is of a certain form, i.e. Binomial, Negative Binomial, and Poisson distributions.
� Monte Carlo Simulation� Works almost universally, but computationally intensive,
convergence heuristically determined �
� OP VaR Approximation� Works if severity is subexponential �
39
Compound Poisson Process Model
� Stochastic Loss Arrival/Convolution Approaches : � Compound Poisson Process/Cramer-Lundberg Model :
� Counting Register : Arrival is a Poisson Process (i.i.d. exponentially-distributed interarrival times), so that the number of arrivals is given by N ~ Poisson(λλλλ)
� Individual Risks : i.i.d.r.v. credit-event losses with skewed g(x), itself NOT subexponential or �tail-heavy enough� to employ limiting tail distributions.
� Aggregating Functional : N-fold convolution integral F(x) = GN*(x)
� Renewal Model : � Generalise arrival to a general Renewal Process (interarrival times still i.i.d.r.v. with finite means E[N], but no longer required to be exponentially-distributed, hence allowing for under/over-dispersion vs. the Poisson arrival frequency.
40
Compound Poisson Process Model (2)
฿0฿250,000
฿500,000฿750,000
฿1,000,000
04
8
12
16
20
0%
2%
4%
6%
8%
10%
12%
Probability
Total Loss for each n (the n -fold Convolution of Bernoulli yields a
Binomial Distr.)
Event Count (Number of Defaults)
Toward a Cramer-Lundberg Loss Distribution with Bernoulli Severity
41
Part Three
CAPITAL ADEQUACY REGULATION
42
On regulation & supervision N
� Evolving modality of regulation & supervision : � 19th century : keeping the guilds in check.� 20th century : issuing a body of regulatory codes.� 21st century : stabilising dynamical, networked systems.
� Contemporary view of regulation & supervision : � Supervisory regulation : mandate-oriented
� Prudential/Stability vs. Efficiency/Competition vs. Fairness/Conduct (of Business)
� Regulatory supervision : risk-based� Financial Institutions = quanta & nodes of risks� Supervisory organisation & resource planning implications
43
On capital adequacy framework N
� Foundation of capital adequacy framework : � Then Economic Capital : that which achieves an optimum balance
between maximising leverage and minimising cost of capitals.
� Then Regulatory Capital : w.r.t. deposit taking financial institutions, i.e. banks, economic welfare is on the whole improved if depositors donxt have to closely monitor the �credit risk� of their �loan portfolio� all by themselves.
� Now EC : that which duly covers, in a distributional sense, downside risks (losses), whence becomes a VaR-like risk measure of the bankxs assets.
� Now RC : regulation-based calculation of a stable, conservative, and transparent EC-like VaR-proxy established as a minimum capital margin.
44
On capital adequacy framework (2) N
� Evolution of capital adequacy framework : � Basel 0 : Capital > 8% of Sum of Assets
� Basel I : Capital > 8% of Sum of Class-Weighted Assets
� Basel II : Capital > Market-risk RC + Credit-risk RC + Operational-risk RC� Market-risk RC : as per 1996-amended Basel 1
� Credit-risk RC : esp. Foundation/Advanced Internal Ratings-Based (FIRB/AIRB)
� Operational-risk RC : first time being introduced as a separate item altogether
� Basel III : Capital > Coupled Market-Credit Portfolio & Operational VaR
� Basel IV : Capital > Enterprise VaR + per-entity Systemic Risk contribution
45
On capital adequacy framework (3) N
� Basel II Capital Measurement and Capital Standards : � The First Pillar _ Minimum Capital Requirements :
� Credit Risk : � The Standardised Approach (SA)� The Foundation/Advanced Internal Ratings-Based (F/AIRB) Approach� Securitisation Framework
� Operational Risk� The Basic Indicator Approach (BIA)� The Standardised Approach (SA-OR)
� The Advanced Measurement Approach (AMA)
� Market Risk
� The Second Pillar _ Supervisory Review Process� The Third Pillar _ Market Discipline
46
Toward the BOTRs AMA Guidelines
� Other EMEAP countries (ex-Japan) : � %ก�Q�': S� Bank of Korea (BOK) $'ก��Q+���+�� AMA ��ก,��ก�TL�* � &.�+,� 2550 S�� +'�/�$ก��
&���#� EMEAP "��� Q�� &M� ($.". 49)� &. "N,��: S� Monetary Authority of Singapore (MAS) U�*ก�R�#L+ Consultation Paper �Proposals for the
Implementation of Basel II in Singapore _ Phase 1� (&.". 48) #R� �MAS will consult on the proposed rules and guidelines relating to this topic at a later date� �/Rก�$.U�*ก�R�#O� AMA /R�Q���z)�)"#�$%J.($%/.$L+�!�! �Phase 2� ($'.". 49) Q���L+�!�! �Phase 3� ($..). 49) S'(/�$$�
� +.#�'��+��: S� Reserve Bank of New Zealand (RBNZ) U�*��ก%ก��� �Basel II: Application requirements for New Zealand banks seeking accreditation to implement the Basel II internal model approaches from January 2008� ($'.". 49) �/R%+���Q�"R�+z*� ���ก�� %+�(� ��ก P+�"��S'(LQ�RS'(&M� 4 �QR z� ,��%ST (Westpac, ANZ National Bank, BNZ ��� ASB) /R� %,�+P+�"����&%/�%�') (U$Rก�%,�+&R#+Q+�( z� )
� ��&%/�%�'): S� Australian Prudential Regulation Authority (ARPA) U�*��ก�R� APRA Prudential Standard (APS) 115 �Capital Adequacy: Advanced Measurement Approaches to Operational Risk� (/.".49) (9 Q+*�) %,�+S'(%�')!�*�)��*# N�)�R� APS 115 +'� �*� ��)��%�')�/�$ APRA Guidance Note (AGN) �'กS�� Q$�&'(�!�!�*#)ก�+ (AGN 115.1 _ 115.4) ��( ��กก�R�#O� �General Requirements� (AGN 115.1 _ 6 Q+*�) �Quantitative Standards� (AGN 115.2 _ 19 Q+*�) �Allocating Group Operational Risk Capital� (AGN 115.3 _ )� U$R��ก%�)�J�R) ��� �Cross Border Guidance� (AGN 115.4 _ )� U$R��ก%�)�J�R) /�$�����!
47
Toward the BOTRs AMA Guidelines (2)
� When it comes to introducing international standards locally, should we� adopt/adjust, adapt/compromise, map/reconcile, form union, or perform intersection?
� LetRs keep/expound : � Top-level Parameters, i.e. the 99.9% confidence level.� Formulating Guidelines as a set of Qualifying Criteria.� Formulating Qualifying Criteria as a set of Standards.� Four Key Elements as Operational Risk Model Drivers.
� LetRs augment/supplement : � Vague (phenomenological) definition of operational risk, i.e. in favour of a precise (causal) definition.� Basic areas of expertise vis-à-vis Operational Risk Modelling Methodology, i.e. extreme value theory.� Scenario Analysis of Legal Risk, i.e. to include rent or mis-selling from TUVWVUXYZ[\Y]^ WX_`Wa-bcกecf_
� LetRs discard/discourage : � Group-wide Diversification Benefits & Insurance-based Mitigation Recognition� Distinction between initial vs. ongoing & qualitative vs. quantitative standards, but recognises, instead,
the distinction between system-process integrity versus theoretical-empirical soundness standards
48
Toward the BOTRs AMA Guidelines (3)
� Where system-process integrity standards address various concerns in terms of : � consistency & achievability of objectives, coherence & completeness of design,
� quality & integration of components, independence & efforts of agents,
� transparency & relevancy of documentation, continuity & reliability of maintenance,
� availability & utilisation of resources, progress & interoperability of innovations, and so on.
� Where theoretical-empirical soundness standards address various concerns in terms of : � reasonableness & applicability of assumptions, testability & falsifiability of hypotheses,
� measurability & observability of parameters, feasibility & accuracy of calibrations,
� nonlinearity & stochasticity of dynamics, stationarity & identifiability of regimes,
� tractability & computability of calculations, validity & verifiability of outcomes, and so on.
49
Toward the BOTRs AMA Guidelines (4)
� Structure of the Document : � Section 1 : Overview
� Section 1.1 : Scope
� Section 1.2 : Definition
� Section 2 : Qualifying Criteria� Section 2.1 : Operational Risk Management Framework
� Section 2.2 : Operational Risk Measurement System
� Section 2.3 : Operational Risk Modelling Methodology
� Section 2.4 : Operational Risk Model Drivers
� Section 3 : Enforcement Issues� Section 3.1 : AMA and the BIA/SA-OR frameworks
� Section 3.2 : Foreign Bank Branches and the Trans-national Banking Groups
50
BOTRs AMA Guidelines h Re: ORMF
iThe whole of Operational Risk ManagementR
� Risk ≡≡≡≡ {POSSIBILITY; PROBABILITY; UTILITY}� Ask: which of these would be subject to adjustment/alteration/manipulation?
� Financial Risks : � Proxy for (monotonic w) utility measured/quoted in monies/monetary units,
� arisen from/rooted in financial market variables/institution factors,
� managed/mitigated by means of financial techniques/tools, and/or
� seen as/deemed to be intrinsic/integral to financial markets/institutions.
� An Operational Risk Management (ORM) Framework must address : � Actors : Board of Directors, Senior Management, Risk Management Function, Business Lines,
Internal/External Audit Functions, and Regulator/Supervisor.
� Processes : IDENTIFY/define ���� MEASURE/assess ���� MITIGATE/manage ���� REVIEW/report
51
BOTRs AMA Guidelines h Re: ORMF (2)
� The Board of Directors : � must be fully responsible/accountable for the overall operational risk policy,� must have approved/signed off on the Operational Risk Management Framework in a fully informed
manner, and� must ensure independence/integrity of the Internal/External Audit Functions.
� The Senior Management : � must be charged with developing/implementing the overall operational risk policy,� must be fully responsible/accountable for the Operational Risk Management Framework,� must have approved/signed off on the Operational Risk Measurement System in a fully informed
manner,� must ensure independence/integrity of the Operational Risk Management Function, and� should be conscious of, as well as conscientious in minimising, various areas of conflicts of
interests, monetary or otherwise, that adds to operational risk or subtracts from the efficacy of the Operational Risk Management Framework, especially in terms of the effectiveness of the Operational Risk Management Function.
52
BOTRs AMA Guidelines h Re: ORMF (3)
� The Risk Management Function : � must be charged with developing/implementing the Operational Risk Management Framework,
� must be fully responsible/accountable for the Operational Risk Measurement System,
� must have approved/signed off on the Operational Risk Modelling Methodology in a fully informed manner, and
� should be conscious of, as well as conscientious in minimising, unnecessary compliance costs, monetary or otherwise.
� The Business Line : � must be charged with developing/implementing their part in the Operational Risk Management
Framework, and
� must be fully responsible/accountable for their input to the Operational Risk Measurement System.
53
BOTRs AMA Guidelines h Re: ORMF (4)
� The Internal/External Audit Functions : � should enjoy effective cooperation and efficient coordination from the Business Lines, the
Operational Risk Management Function, as well as the Senior Management of the AMA bank, and
� should be conscious of, as well as conscientious in minimising, unnecessary compliance costs, monetary or otherwise.
� The Regulator/Supervisor : � should enjoy effective cooperation and efficient coordination from the Business Lines, the
Operational Risk Management Function, the Senior Management, the Internal/External Audit Functions, as well as the Board of Directors of the AMA bank, and
� Should be conscious of, as well as conscientious in minimising, unnecessary compliance costs, monetary or otherwise.
54
BOTRs AMA Guidelines h Re: ORMF (5)
� W.r.t. Identification, an AMA bank must have in place a sound process of : � identifying various sources of operational risk exposures,
� locating operational risk events by 7 event types, namely : i. internal fraud, ii. external fraud, iii. employment practices/workplace safety, iv. clients, products/business practices, v. damage to physical assets, vi. business disruption/system failures, vii. execution, delivery, and process management, as well as
� attributing operational risk losses to 8 business lines, namely : i. corporate finance, ii. Trading/sales, iii. retail banking, iv. commercial banking, v. payment/settlement, vi. agency services, vii. asset management, viii. retail brokerage.
55
BOTRs AMA Guidelines h Re: ORMF (6)
� W.r.t. Measurement, an AMA bank must : � stipulate minimum reporting thresholds for operational risk losses, which may or may not vary
according to SA-OR-defined/compliant event types/business lines,
� relate its minimum operational risk capital charge calculation to a statistical level of confidence of 99.9%, as well as
� have developed and continue to improve the Operational Risk Measurement System, whose detailed standards are set out in Section 2.2 of the BOTxs AMA guidelines.
56
BOTRs AMA Guidelines h Re: ORMF (7)
� W.r.t. Mitigation thru third-party insurance, an AMA bankxs recognition : � is limited to no more than 20% of the calculated pre-insured minimum operational risk capital charge,
and
� may be subjected to case-by-case approvals/consents by the BOT.
� [�]
� W.r.t. Mitigation thru diversification effect, an AMA bankxs recognition : � is subjected to case-by-case approvals/consents by the BOT.
� W.r.t. Mitigation thru improved control, an AMA bankxs recognition : � is subjected to case-by-case approvals/consents by the BOT.
57
BOTRs AMA Guidelines h Re: ORMF (8)
� W.r.t. Review, an AMA bankxs Operational Risk Management Framework is to be subjected, at a minimum, to reviews/examinations: � by the Senior Management on an annual basis,� by the internal/external audit functions on an annual basis,� by the BOT on a bi-annual basis, and� by any/all of the above on a contingency basis, i.e. as circumstances warrant.
� W.r.t. Review, an AMA bankxs Operational Risk Measurement System is to be subjected, at a minimum, to reviews/examinations: � by the Operational Risk Management Function on a semi-annual basis,� by the Senior Management on an annual basis,� by the Internal/External Audit Functions on an annual basis,� by the BOT on a bi-annual basis, and� by any/all of the above on a contingency basis, i.e. as circumstances warrant.
58
BOTRs AMA Guidelines h Re: ORMS
iThe head of the Operational Risk Management FrameworkR� Target measure :
� 1-Year Operational Risk Value-at-Risk (Op-VaR) at the 99.9% Confident Level, gross of Operational Risk Expected Loss (Op-Risk-EL).
� Advancement toward Operational Risk Conditional Mean Exceedance (Op-Risk-CME) encouraged.
� Minimum features : � A mix of quantitative judgements, qualitative techniques, computational heuristics.� Operational Risk Loss Database instituted and maintained� Threshold Reporting Loss determined and enforced� Extreme Value Losses captured and modelled� Ability to delineate : Operational Risk Exposures under partial control vs. wholly uncontrollable� Ability to delineate : Operational Risk Events that could be efficiently/effectively insurable vs. not;
those that are regular/infrequent/rare in occurrence, modest/high/extreme in impact.� Ability to delineate : Operational Risk Losses to be allocated to business lines vs. to be shared
59
BOTRs AMA Guidelines h Re: ORMM
iThe heart of an Operational Risk Measurement SystemR� Minimum features :
� Modelling Operational Risk Event Arrivals� Poisson, Negative Binomial, Empirical distributions
� Modelling Operational Risk Event Severity� Gamma, Gumbel, Log-Normal, Pareto, Weibull distributions �� Fisher-Tippet-Gnedenko theorem // Generalised Extreme Value (GEV) distribution // sample maxima //
method of Block Maxima (BM)
� Picklands-Dalkema-de Hann theorem // Generalised Pareto Distribution (GPD) // sample exceedance over large threshold // Peak Over Threshold Modelling (POTM)
� Determining Aggregate Operational Risk Loss� Compound Poisson Process : Sum of N i.i.d. random Losses, where N is itself a Poisson r.v.
� PanjerRs Algorithm (only works if severity distribution is discrete and frequency distribution is of a certain form)� Monte Carlo Simulation (works almost universally, but computationally intensive, convergence not guaranteed)
� Böcker-Klüppelberg closed-form Op-VaR approximation (only works if severity subexponential)
60
BOTRs AMA Guidelines h Re: ORMD
iThe soul of an Operational Risk Modelling MethodologyR
� Minimum features (corresponding to the so-called AMA Key Elements) : � Data Environment (Internal Data)
� Calibration Benchmark (External Data)
� Applicability/application of Scenario Analysis
� (Business Environment/Internal) Control Factors
61
Bibliography
� BÖcker, K. & KlÜppelberg, C. (2005), /Operational VaR: A Closed-Form Approximation1,
available online via http://www.gloriamundi.org/picsresources/kbck.pdf.
� Brandimarte, Paolo (2001), Numerical Methods in Finance:
A MATLAB-Based Introduction, New York, NY: Wiley-Interscience.
� Embrechts, Klüppelberg, and Mikosch (1997), Modelling Extremal Events
for Insurance and Finance, Berlin: Springer.
� Glasserman, Paul (2004), Monte Carlo Methods in Financial Engineering
(Applications of Mathematics No. 53), New York, NY: Springer.
� Johnson, Kotz, Balakrishnan (1994), Continuous Univariate Distributions
Volume 1 (2nd ed.), New York: Wiley-Interscience Publication.
� Karatzas, I. & Shreve S.E. (1991), Brownian Motion and Stochastic Calculus
(2nd ed.), New York, NY: Springer.
� Lewis, Nigel Da Costa (2004), Operational Risk with Excel and VBA, Hoboken, NJ: John Wiley & Sons.
62
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� McFadden, Daniel L. (1974), Applied Logistic Regression (2nd ed.), New York: Wiley.
� Meintanis, S.G. & Koutrouvelis, I.A. (1999). /Chi-Squared Tests of Fit
for Generalized Poisson Distributions Based on the Moment Generating Function1,
InterStat, May Issue, 1-19.
� Mikosch, Thomas (2004), Non-Life Insurance Mathematics:
An Introduction with Stochastic Processes (Universitext), Berlin: Springer.
� von Neumann, J. & Morgenstern, O. (1947), Theory of Games and Economic Behavior,
Princeton, NJ: Princeton University Press.
� Nelson, Roger B. (1999), An Introduction to Copulas
(Lecture Notes in Statistics No. 139), New York, NY: Springer.
� Panjer, H.H. (1981), /Recursive Evaluation of a Family of Compound Distribution1,
ASTIN Bulletin, 12 : 22-26.
� Pao, Yoh-Han (1989), Adaptive Pattern Recognition and Neural Networks,
New York, NY: Addison-Wesley.