MODELLING OPERATIONAL RISKIN FINANCIAL INSTITUTIONSUSING BAYESIAN NETWORKS
By
YEW KHUEN YOON
1 May 2003

A dissertation submitted for the award of the degree of Master of Science
In
Actuarial Management

Faculty of Actuarial Science and Statistics

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Abstract

Recent failures and losses of financial institutions have prompted the regulatory authorities such as the
Financial Services Authority in the UK and the Bank for International Settlements to highlight theimportance of adequate systems and controls to deal with operational risk (OR). An importantmeasure taken is requiring these organisations to allocate OR capital. This has given the industry animpetus to develop methods for measuring and modelling OR.

Although most firms have only begun to collect data and initiate modifications to their systems to deal
with quantification of OR, there has not been a shortage of methods proposed. Broadly, these can bedivided into (i) top-down methods that starts with enterprise-wide risk, which is then allocated to thebusiness units and (ii) bottom-up methods that analyses risk at the business unit level and then builds
up to form a firm-wide evaluation of OR. The consensus appears to be that bottom-up methods willbe the favoured approach in the long run as data is gathered and as more sophisticated methods
develop. The preferred approach is one that deals with the cause and effects of OR events.

Amongst the more popular methods are linear models such as time series models, econometric models
and empirical actuarial models. Since OR losses tend to have ‘fat-tailed’ distributions, extreme value
theory has been proposed as a solution. These models tend to rely too much on data availability, aproblem due to the non-standard and infrequent nature of operational loss events. Furthermore, thecomplex interaction between OR variables often render linear methods mathematically intractable.
Various non-linear methods have been proposed to overcome these problems, one of which is
Bayesian networks (BNs).

iModelling Operational Risk in Financial Institutions Using Bayesian Networks

BNs are a combination of graph theory and Bayesian statistical theory to produce networks ofvariables (called nodes) that are linked via directed edges. These edges imply a causal relationshipbetween nodes, hence BNs are also classified as a type of Directed Acyclic Graph. Expert input and pastdata are combined in the process of specifying the structure and the underlying probability
distributions at each node. Rules on conditional independence of the nodes allow alternative causes tobe ‘explained away’ when events occur – providing the foundation for causal analysis.

The BN can be structured in such a way to have the various OR factors cascade into an overall loss
distribution, thus allowing risk capital to be specified. The causal dependencies inherent in the
structure make BNs useful for scenario analyses in identifying and measuring the impact of operationalloss events. This would meet the main requirements of a model for OR in financial institutions.

Developments in an algorithm known as triangulation allow directed graphs to be expressed as a
network of undirected sub-graphs. This allows computations to be performed locally and therefore
more efficiently. Bayesian statistical theory is applied in a straightforward and intuitively appealing way
to update the probabilities in the model as new data arrives – this process is called learning. Modelassessment is also relatively easy to do via monitors that quantify in real-time how well the model
prediction compares to actual data.

An illustration of these aspects is provided by way of an online insurance business belonging to a
fictitious company. A BN is set up and used to model the risk of failure in the system network
supporting this business. The BN is used to calculate risk capital, scenario testing, causal analysis andsimulation of future scenarios. The learning process is also shown and monitors used to discriminate
between models with and without learning.

iiModelling Operational Risk in Financial Institutions Using Bayesian Networks

Contents

Item

Page

List of Abbreviations……………………………………………………………………

iv

Chapter One: Developments in Regulatory Capital Requirements Driver forOperational Risk Modelling………………………………… …………

1

Chapter Two: A Brief Survey of Existing Methods for Modelling Operational Risk ……

4

Chapter Three: Bayesian Networks ………………………………………………………

12

Chapter Four: Applying Bayesian Networks to Modelling Operational Risk ……………

39

Chapter Five: Discussion ………………………………………………………………

60

Post-script………………………………………………………………………………..

63

Acknowledgements………………………………………………………………………

64

References ………………………………………………………………………………

65

Appendix I:

Network Risk – Prior Distributions………………………………………

67

Appendix II:

Network Risk – Propagation Example……………………………………

69

Appendix III: Network Risk – Simulation Results ………………………………………

75

Appendix IV: Network Risk – Calculations for Parent-Child Monitor for NF Node ……

77

iii

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

List of Abbreviations Used

AMA
BN
BCBS
DAG
EVT
FI
LS
NR
OR
VaR

Advanced Measurement Approach
Bayesian network
Basel Committee for Banking Supervision
Directed acyclic graph
Extreme value theory
Financial institution
Logarithmic score
Network risk
Operational risk
Value-at-riskiv

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Chapter One
Developments in Regulatory Capital Requirements as a Driver forOperational Risk Modelling

In the past decade, the financial world has been hit by catastrophic failures and losses – the causes ofwhich are largely non-financial in nature. Names such as Barings, Daiwa Bank, Orange County,
Metallgesellschaft, and Long Term Capital Management have entered the vocabulary in the history ofoperational risk (OR). OR has been increasingly recognized as an important source of businessuncertainty, especially in financial institutions (FI), thus being given much attention by the financial
community, the public and not least of all, the industry regulators.

It is in such an environment that the new framework for capital adequacy proposed by the BaselCommittee for Banking Supervision (BCBS) of the Bank for International Settlements has highlighted
the importance of OR in determining the adequacy of regulatory capital1. OR has been defined by
BCBS as:“The risk of loss resulting from inadequate or failed internal processes,
people and systems or from external events.” (p2, BCBS (2001a))

A regulatory standard for capital adequacy calls for quantification of risks that are to be supported by
the economic capital of FI’s. Naturally, there has been increasing interest in methods of measuring andmodelling OR in response to these developments.

1 See ¶82 – 87 of BCBS (1999).

1

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

In the United Kingdom, the Financial Services Authority (FSA) has set a 2004 target date for its policy
on OR management systems and controls (scheduled to be published in 2003) to take effect, with
consultation currently well under way2. With the release of the Integrated Prudential Sourcebook by
the FSA, FIs will be required to demonstrate adequate systems and controls in managing OR.
Invariably this would involve some level of scrutiny of the company’s framework for measuring andmodelling OR.

BCBS is proposing a continuum of approaches to the allocation of capital to OR, starting from the
Basic Indicator Approach where a fixed percentage is applied across the board, to the Standardised
Approach where the percentage to be applied for each business line varies as defined by thesupervisory authority and finally, the Advanced Measurement Approach (AMA) where the bank may
use an internal mechanism. The policies proposed encourage banks to develop more sophisticatedmethods by allowing those using the AMA a possibility of lower capital charges (up to 25% lower thanthe Standardised Approach) 3. However, only banks that meet a set of qualitative and quantitative
criteria set by BCBS may use the AMA.

The interesting question would be “How prepared are banks and FIs in general to operate an internal
system of OR management and capital allocation?” Or more importantly, “What sort of OR
management system would be sufficient to meet the criteria set by BCBS?” Authors on the subject
reveal that most FIs do not have sophisticated treatments of OR. King (1999), one of the earlier
reference texts on the subject, summarized the current framework used by organizations into thefollowing broad categories:

2 See Consultation Paper 142: Operational risk systems and control and the feedback on the FSA’s website: www.fsa.gov.uk
3 For more details, refer to BCBS (2001a) and chapter 2 of van den Brink (2002).

2Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Framework

Description

Control self-

This involves questionnaires and workshops to solicit qualitative opinion of employees on areas

assessmentof risk in the organization. The subjective nature of this method restricts its suitability forcapital allocation and management performance review.

Process Analysis

Operational processes are analysed in detail at task level to identify and classify causation of loss events.
The problem of subjectivity remains and the documentation is difficult to maintain.

Loss

Losses are entered into a database and attributed to loss categories with sufficiently detailed classification

Categorisations

for risk measures. Double-counting and identification of causes are challenges to overcome.

Performance

Performance measures (such as Value Chain and Balanced Scorecard) are used to develop associated risk

Analysismeasures, ensuring consistency between performance and risk. However, the focus is more on gains andnot losses.

More recently, Hoffman (2002) sounded a more optimistic note by stating “The best firms are making
heroic strides toward risk definition, data collection, aggregation and first-level analysis” (p7). Some
promising trends were identified, including the creation of internal loss event databases, integration ofinternal and external loss data for experimental risk capital calculations, and new risk information
measures are emerging.

In the light of these developments, it would be interesting to see what solutions are currently available
for FIs to turn to in handling this important area in risk management. Hence, we proceed to discuss
some of the existing methods available for modelling OR.

3

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Chapter Two
A Brief Survey of Existing Methods for Modelling Operational Risk

2.1 Chapter introduction
This chapter is an overview of the various existing methods for measuring and modelling OR, bothin theory and what is used in practice. We will discuss briefly how useful and appropriate thesemethods are in meeting two vital business requirements of the whole OR modelling endeavour: (i)determine risk capital and (ii) identify causation of OR in a way that is quantifiable and actionable.
The considerations will be made vis-a-vis the underlying developments in the regulatoryenvironment, which was briefly discussed in the introductory chapter.

2.2 Top-down and bottom-up approaches
The whole spectrum of methods available for modelling OR can be broadly divided into two maincategories:(i)Top-down approaches start from a firm-wide determination of OR. This is then allocated tobusiness units according to how much each unit is thought to contribute to the overall
OR. Some examples include earnings volatility, scorecard approaches and CAPM-basedmodelling.

(ii)

Bottom-up approaches model the risk of loss events at the business units and build up to an
overall picture of firm-wide OR. This is where the latest modelling methods are being4Modelling Operational Risk in Financial Institutions Using Bayesian Networks

developed with increasing involvement of the actuarial profession. The main examples inthis area are loss distribution analysis and causal modelling4.

At the moment, most organizations are still in the process of classification of OR losses andgathering data. As such, top-down approaches are still the main methods of managing, measuringand modelling OR. In general, as companies accumulate more extensive internal loss databases andas modelling techniques get more sophisticated there will be a move towards bottom-up approaches.

2.3 Operational risk capital
The broad approach to setting of economic capital for banks at the moment revolves aroundobtaining a single figure known as the Value-at-Risk (VaR). This is often defined as a high quantile
of a loss distribution that is projected over a required period based on past data. Thus, for example,
BCBS may require that “the bank must be able to demonstrate that the risk measure used forregulatory capital purposes reflects a holding period of one-year and a confidence level of 99.9%” (BCBS(2001b)). A whole body of literature exists detailing the workings of VaR in setting economic capitalfor market risks and credit risks (see, for example, Jorion (2001) and Dowd (1998)).

Presently, if VaR is calculated for OR, the approach is commonly rather simplistic – ranging fromthe residual approach (firmwide VaR minus market VaR and credit VaR) to setting the VaR as amultiple of the standard error. Some of the more advanced approaches use models that essentiallytry to fit a loss distribution from which a VaR figure can be extracted5. With market VaR and credit
VaR this is more straightforward as an underlying normality is assumed to exist in the data.
However, the nature of OR losses are such that they either occur fairly often in a majority of minor
4 Ceske et al. (2000) and Marshall (2001) have fairly comprehensive tables on the advantages and disadvantages ofvarious examples of both approaches. Both authors agree that bottom-up models are the way forward.
5 Several recent texts on OR give descriptions of the methodology e.g. Cruz (2002), Hoffman (2002) and King (1999).
See also BCBS (2001), chapter 19 of Jorion (2001) and p198 of Dowd (1998).
5Modelling Operational Risk in Financial Institutions Using Bayesian Networks

cases (high frequency and low severity) or very rarely with catastrophic consequences (low frequency
and high severity) thus the assumption of normality is not appropriate. The former type of losses isless of a concern as sufficient information is available on them to calculate to a fair degree of
accuracy the standard risk charges (say, via pricing of products) or reserves to absorb the costs. It is
the latter that is more of concern in allocating economic capital. It is also the main motivation
behind most OR modelling endeavours.

The frequency of OR losses are usually modelled using counting processes. The severity of OR
losses are generally modelled using ‘fat-tailed’ distributions, or distributions with high kurtosis. The
more popular empirical distributions include Poisson and negative binomial distributions (forfrequency), and lognormal, gamma and Weibull distributions (for severity). However, due to the lackof data at the extremes, it was found that certain asymptotic distributions which had their origins in
the natural sciences (e.g. in the modelling of hurricanes and sea levels) work fairly well when applied
to the tails of OR data. This branch of statistics is called Extreme Value Theory (EVT) and consistsof two main families of distributions: the Generalised Extreme Value (which models maxima andminima) and the Generalised Pareto Distribution (which models data above a selected level, also
known as peaks-over-threshold).

2.4 Causal modelling of OR
The approach taken by BCBS in the classification of OR losses is to deal in terms of the causes ofthe loss. This can be seen from the definition of OR by BCBS given above and their proposed loss
event type classification (See Annex 2 of BCBS (2002)). The Operational Risk Working Party of theInstitute of Actuaries proposed a framework for analysis of OR based on cause and consequence in6Modelling Operational Risk in Financial Institutions Using Bayesian Networks

their recent report to GIRO 2002. Throughout the report, they stressed the importance of working
in terms of causes rather than consequence to avoid double-counting or omissions6.

Modelling of causation of OR losses is not only a neat framework for comprehensive analysis. It is avital basis for the understanding of the how the risk of OR losses arise within the structure andoperations of the organization. It also provides a basis on which management may intervene toachieve the desired alteration in risk profile.

Clearly, causal modelling is not only crucial inunderstanding and managing risks internally; it will be of necessity in the new regulatory regime.

Some existing techniques currently used for causal modelling in risk management include time-series
analysis and econometric models. A common model is the Autoregressive time-series with Conditional
Heteroscedasticy (ARCH). Factor analysis is also used to decompose uncertainty in profit and loss figures
into various causal factors of manageable sizes. Chapter 8 of Cruz (2002) provides a brief overview
of these methods. Non-linear and non-parametric models are increasingly popular for causal
modelling as they offer more flexibility and as more research is done in these areas. Some examples
include neural networks, fuzzy logic and Bayesian networks, which is the focus of this paper.

2.5 Challenges of modelling OR
The besetting problem with any attempt to model OR losses is the lack of data. Internal data of low
frequency events are rarely sufficient to model the loss distributions to the required accuracy.

The usage of parametric loss distributions requires parameter estimation based on the existing data.
As such, these methods tend to be too dependent on data quality and quantity. Even for the more
promising EVT methods, the small sample sizes result in the shape of the tails being very sensitive

6 See Institute of Actuaries (2002) especially ¶2.2.1 and ¶3.5.

7

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

to inclusions or exclusions of single events – implying a greater degree of subjectivity than may atfirst appear.

There is also the challenge of model selection. Most goodness-of-fit tests use actual data to verify thetheoretical model, which itself is obtained using the actual data. Lack of data exacerbates thisinherent cyclical problem in model selection. As regards EVT the existing tests either require visualinspection or tend to over-fit (Cruz (2002)). The peaks-over-threshold method, for example, requires
a visual inspection of the mean excess plot (the graph of threshold versus the mean of all points over
the threshold) to identify the point where the graph begins to behave linearly to determine thethreshold on which the rest of the modelling is based. King (1999) gets around this problem by
fixing the threshold as a quantile of the empirical distribution. Points below the threshold aremodelled by the empirical distribution whereas points above the threshold are modelled using EVT.

In an attempt to overcome the lack of internal data, the usage of external data has been suggested asa solution. The idea is that data from diverse locations may share certain features, thus making sense
to combine them to obtain a larger database. To achieve this, external data need to be carefully
filtered and scaled. Cruz (2002) describes a method for pooling data from different locations called‘frequency analysis’ and Frachot & Roncalli (2002) propose using linear credibility (a methodcommonly used for premium rating in general insurance) to combine internal and external data.
There have also been some efforts made at industry-wide level collection of data to provide a sharedsource of external data (e.g. British Bankers Association). Obviously, merging internal and external
data is not a straightforward exercise – especially in the case of operational loss data. There are
qualitative problems such as quality of data from external parties, the dissimilarity of OR
management practices across firms, lack of detailed breakdown of data and lack of up-to-date datathat may pose difficulties to using external data to set internal capital requirements (which areessentially prospective).

8Modelling Operational Risk in Financial Institutions Using Bayesian Networks

In any case, the modelling of OR losses would require an in-depth understanding of the complex
nature of company operations and a model that adequately represents such complexities. Linear
models may need to be mathematically intractable to sufficiently achieve this objective. As a logicalstep forward, various non-linear models have been proposed and experimented with in as yet alimited number of financial and non-financial institutions.

2.6 Expert input and Non-linear Methodologies

The lack of data and complexity of operations in FIs intuitively suggests the inclusion of expert input.
An expert in this case would be anyone whose knowledge and expertise enables him/her to makesufficiently credible conjectures about how company operations affect the company’s risk profile.
Such input can be used as a proxy for data and yield valuable information about the complexity ofcompany operations that is difficult to capture from data alone. The challenge of the modeler, then,is to incorporate such input into the overall OR modelling framework.

It has been found that qualitative information, such as management decisions, competencies andpreferences, can be better incorporated into a measurable (and hence quantitative) framework using
non-linear methods. Some of these include fuzzy logic, neural networks, system dynamics, and Bayesian
networks.

Fuzzy logic uses a multivariate logical set that recognizes that human decisions are often not binary(e.g. Yes/No, Hot/Cold) by allowing gradations in its formulations (e.g. rather hot, very hot etc.).
Hoffman (2002) has a brief case study on how this has been used in a bank in its OR management7.
Although this method has advantages in its ease-of-understanding, “…the theory of fuzzy logic

7 See pp 326-330.

9Modelling Operational Risk in Financial Institutions Using Bayesian Networks

cannot replace robust statistical methods in measuring operational risk in a capital-at-risk sense…”8Thus, its application for our present concern is limited.

Although not strictly a method that is used for incorporation of qualitative opinion, neural networks
are useful for modelling complex relationships between variables that would be difficult to do using
linear methods. The network consists of nodes with values of input, output and intermediatevariables. Data mining techniques are used to ‘train’ the model by using complex algorithms thatlearn the relationships between the variables. The model is then calibrated such that its output is as
close to the actual data output as possible. A drawback of this approach is its heavy reliance on theavailability of data. A brief example of its application can be found in Hoffman (2002), although it
involves a non-financial institution9.

System dynamics was developed by Jay Forrester of the Massachusetts Institute of Technology and has
been promoted by Tillinghast-Towers Perrin10 as an OR modelling solution. This approach involves
using expert input to map a network of cause-and-effect relationships between variables affecting
the OR of a business unit. The relationship between each cause-and-effect set of variables is thenquantified by combining data and expert input to obtain a plot on two axes (one for each of thecause and effect variables).

This dissertation discusses the usage of Bayesian networks (BNs) as a framework for modelling OR.
We shall see that it is an efficient and intuitively coherent methodology for incorporating expert
input. In addition, BNs are useful for capturing causal dependencies.

This satisfies a vital

requirement of any OR modelling framework: ability to model causation. Developments in the fieldof graphical models (of which BNs are an example of) have made BNs very user friendly and as8

Hoffman (2002) p327.

9 More specifically, the National Aeronautic and Space Administration (NASA) in the United States of America. See

pp299-300.
10 Miccolis & Shah (2000) and Shah (2001).

10

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

such, a widely accepted tool for making inference from complex networks of causal relationships.
For example, BNs have been used for almost two decades now in medical diagnoses and more
recently in criminal forensic sciences using DNA analysis. Underlying BNs is the powerful Bayesianstatistical property that allows the combination of subjective input and empirical observations. This
lends it very well to situations with a high degree of uncertainty and where data is costly or sparse.

Alexander (2000) provides a brief introduction to modelling OR using BNs. Marshall (2001), Cruz
(2002) and Hoffman (2002) give brief overviews of BNs and where they fit into the wholeframework of OR modelling. There is also an illustrative albeit high-level discussion on causal
modelling using BNs via a banking example in King (1999).

2.7 Chapter conclusion
We have seen in this chapter that although many new methods have been proposed in the past few
years, many of these methods place heavy reliance on the data used and not enough on implicit
knowledge of experts. Furthermore, most of these methods are linear and thus do not fully utilise
nor capture causal relationships inherent in the data – a step that is vital considering the regulatoryemphasis on causal identification and action. BNs have been proposed as a potential approach as it
offers intuitive yet mathematically and computationally tractable means of dealing with these two
aspects.

In the next chapter, we will proceed with a cursory recall of basic concepts of Bayesian statistics
before examining how BNs work.

11Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Chapter Three

Bayesian Networks11

3.1 Chapter Introduction
We will start off by looking at some fundamental inferential properties of Bayesian statistical theory,
specifically Bayes’ theorem and probability calculus. Then, some of these ideas will be combined with afield of information sciences known as graph theory and applied to a well-known example to illustrate
the basic building block behind BNs: the Directed Acyclic Graph. The rest of the chapter will develop
the theory behind BNs and how it can be used for probabilistic inference by incorporating expert
opinion and data.

3.2 Bayesian Statistics

3.2.1 Introduction
In classical probability theory, sample statistics are assumed to belong to a certain population with a
specified distribution, which is defined by a set of parameters that have a fixed value. The main taskof the statistician is to estimate these parameters as best as possible based on whatever data isavailable (i.e. the sample statistics). Where possible ‘experiments’ are performed repeatedly to obtaina suitably large sample to assign values to these parameters.

11

The theory and techniques discussed in this chapter follow closely the ideas presented in Cowell et al (1999). An
alternative text with some useful introductory examples is Jensen (1996).
12Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Bayesian statistics allow these parameters themselves to be random variables. Furthernore, assertions
made regarding the characteristics of a population are necessarily dependent not just on empiricalobservations or data (objective information) but also on any knowledge available to the statistician
prior to making the observations (subjective information). This knowledge may come in the form ofdata from a different location that is considered to have a certain degree of relevance to theobserved population. More importantly, it may also come in the form of information obtained fromknowledgeable parties i.e. experts whose familiarity with the subject matter makes them a credible
source.

As a consequence, where classical probability deals predominantly with assessing unconditional
probability statements such as “the probability of event A is x”, denoted P ( A) = x , Bayesianstatistics vocabulary expand into conditional probabilities with statements such as “the probability ofevent A given event B has occurred is y” (or “the probability of event A conditioned on event B
has occurred is y”) denoted P( A | B) = y .

3.2.2 Bayes’ Theorem and Probability Calculus
Manipulation of such probabilities involves treating them as functions of variables using certain wellknown rules, collectively known as probability calculus. We introduce the fundamental rule in probability
calculus:

P ( A, B ) = P ( A | B ) P ( B )(3.1)

This states that the probability of joint occurrence of events A and B is equal to the probability of A
conditional on B multiplied by probability of event B. Since the function P( A, B) is symmetrical, we
can express it in the following form:

P ( A, B ) = P ( B | A) P ( A)

(3.2)

Equating the right hand sides of (3.1) and (3.2) we obtain the definitive theorem in Bayesianstatistics, Bayes’ Theorem:
13

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

P ( B | A) =P( A | B) P( B)
P( A)

(3.3)

This can be interpreted as follows: the posterior probability P( B | A) is equal to the prior probability
P (B) multiplied by the ratio P ( A | B ) / P ( A) .

3.2.3 Inference Using Bayes’ Theorem
Applying Bayes’ theorem to random variables (e.g. one where observations x sampled from a
population with a probability distribution defined by the parameter θ ) we obtain the followingform:

P(θ | x) =P ( x | θ ) P(θ )
P( x)

(3.4)

This allows us to understand Bayes’ theorem in terms of the discussion above on objective andsubjective information. The prior is the subjective information alluded to earlier. In this case it is an
unconditional probability representing the uncertainty about the parameter θ . The function
P ( x | θ ) is commonly called the likelihood of the dataset x (also denoted L( x | θ ) ) and can beinterpreted as the probability of observing a certain dataset x given that certain characteristics of the
population (in this case the parameter θ ) are true.
Combining the subjective information and the empirical observations (i.e. the prior and thelikelihood), we get the posterior probability i.e. the probability that the parameter θ takes a certain
value given that observations x have been made. This is denoted P(θ | x) . If we sum P(θ | x) ∗ P( x)over the values of x we get the marginal probability of the parameter θ .

Treating P(x) as a constant of proportionality, (3.4) can be expressed as follows:
P (θ | x) ∝ L( x | θ ) ∗ P(θ )(3.5)

Or in words:

14

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Posterior ∝ Likelihood * Prior
Intuitively, we can think of this relationship as representing the Bayesian idea that conjectures about
a population is the combination of presuppositions (or prior knowledge) about the population and any
observations made regarding the population.

3.2.4 Epistemology
The reason why this way of looking at reality is so appealing to our present concern is because
information about the OR profile of a company is very often incomplete, derived from quantitativeand qualitative sources and is continuously updated as new data is gathered. Bayesian statistics offersa practical, yet intuitively appealing, methodology to deal with such a situation.

We will now examine how BNs apply Bayesian statistics to solve a variety of problems. It is withapologies to the reader that the rest of this chapter is rather heavy on terminology.

3.3 Introduction to Graphical Models

3.3.1 Introductory Terminology and the Direct Acyclic Graph
BNs, or variously called belief networks, causal probabilistic networks, directed graphical models or
generative models, are a type of graphical model. Graphical models are a combination of probability
theory and graph theory. It is the result of converging developments in statistical modelling,
engineering and artificial intelligence that began in the 1980’s. Initially the extensive calculations inprobability theory rendered these efforts unfeasible. However, the utilisation of conditional
independencies in graph theory and recent developments in efficient algorithms for propagation ofevidence across graphical structures has made this field much more feasible computationally.

15

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

To illustrate the usage of graphical models, consider the following diagrams:
AB(a) With directed edge

AB(b) With undirected edge

fig 3.1 Nodes in graphical models

A and B are called nodes and represent variables A and B. In fig. 3.1(a), the directed edge from A to B
implies a causal relationship between A and B. To be more precise, it states that a change in what is
known about A (usually affecting the probabilities of events in A) causes a change in what is known
about B (and thus the probabilities of events in B). This change is usually the result of new
information arriving about A (henceforth we will use such statements as “information about A” and“information about the events in A” interchangeably). This new information is sometimes called
evidence.

When variables are connected in this way, we call variable from which the edge originates the parent
and the variable to which the edge leads the child. When the edge between the nodes are not directed,
as illustrated in fig. 3.1(b), then no causation is implied, but rather that some ‘weaker’ form ofassociation (e.g. correlation) exists between A and B. Using the same sort of descriptive language, A
and B are called neighbours. It follows, finally, that if no edges exist between A and B, then A and B
are independent i.e. occurrence of events in A has no bearing on occurrence of events in B and viceversa. However, as we shall see, this may depend on whether any intervening variables exist between A
and B.

Fig 3.1(a) and 3.1(b) also represents the joint probability of A and B. However, (a) and (b) expresses
this joint probability differently. In fig. 3.1(a), the causal relationship that exists between A and B
means that the joint distribution can be expressed as a product of the probability of A and the16Modelling Operational Risk in Financial Institutions Using Bayesian Networks

probability of B conditional on A or simply written: P ( A) P( B | A) . This, as we have seen in (3.1), is
simply the fundamental rule in pictorial form. Since no such relationship is defined in fig. 3.1(b), thisgraph only expresses the joint distribution itself: P( A, B).

In general, graphical models comprise of a network of such nodes with edges to connect variables
that have some form of relationship, whether of correlation or of causation. BNs on the other handcomprise almost entirely of causal relationships and would thus involve nodes connected by directededges.

BNs belong to a subset of graphical models that are known as a directed acyclic graph (DAG). DAGs
are constructed with relationships such as those in fig. 3.1(a) as its basic building block. Thesebuilding blocks are arranged in such a way that the variables are not cyclical i.e. moving along theedges in the directions implied, it is impossible to return to a previous node. Hence, the term
“acyclic”.

Associated with each node of a DAG that has at least one parent is a set of conditional probabilities.
These describe the behaviour of the node conditioned on all its parents. This is often written asP( X | pa ( X )) where pa( X ) represents the parents of X. For example, the parent set for node Din fig. 3.2 is the set of nodes {A, B, C}.

A

B

C

D

Fig. 3.2 Parent and child nodes17Modelling Operational Risk in Financial Institutions Using Bayesian Networks

A useful property of a DAG is that it illustrates the assumption that the joint density over a set ofvariables U (comprising of variables denoted by the vector X) can be expressed as a product of theconditional densities at each node. This is expressed as follows:
P (U ) = ∏ P( X | pa( X ))

(3.6)

X

This factorization comes from the conditional independence property inherent in the structure of
DAGs. However, even this expression can make manipulations of DAGs exceedingly complex,
especially if the nodes represent variables with many states.

3.3.2 Wet grass example
We pause briefly at this point to consider a popular example to illustrate some of the concepts
presented so far. In this example, grass in a garden is observed in the morning to be either wet or
dry. If the grass is wet it could be due to either the sprinkler being on or some rain falling earlier on.
The probability of rain is 0.1 whereas the probability that the sprinkler is on is 0.2. The graph andprobabilities are shown in fig. 3.

R

Ry Sy
Ry Sn
Rn Sy
Rn Sn

S
W

Ry
0.1

Rn
0.9

Sy
0.2

Wy
1
1
0.5
0

Wn
0
0
0.5
1

Sn
0.8

The subscripts represent the state of the variable e.g. Wy = wet grass, Sn=sprinkler not on etc.
Fig. 3.3 Sprinkler example.

The probability of the grass being wet conditional on the state of the sprinkler and rain is also shownin fig 3. Thus, for example, given that neither the sprinkler was on nor was there rain, the probabilityof the grass being dry is 1. As a first step, we find the joint density of R, W and S. Using the18Modelling Operational Risk in Financial Institutions Using Bayesian Networks

fundamental rule (3.1) and assuming that R and S are independent (as is represented in the lack of anedge between the respective nodes) we know that:

P(W , R, S ) = P(W | R, S ) P( R) P( S )

Ry Sy
Ry Sn
Rn Sy
Rn Sn

Wy Wn
0.02 00.08 00.09 0.09
0 0.720.19 0.81(a) Joint distribution

Ry Sy
Ry Sn
Rn Sy
Rn Sn

Wy
0.02
0.08
0.09
0
0.19

Wn
0
0
0
0
0

Ry Sy
Ry Sn
Rn Sy
Rn Sn

Wy
0.11
0.42
0.47
0
1

(b) Incorporation of(c) Normalisation
information on wet grass
Tables 3.1 Manipulation of the joint distribution

Ry Sy
Ry Sn
Rn Sy
Rn Sn

Wy
0
0
0.9
0
0.9

(b) Incorporation of
information on no rain

Thus, multiplying the marginals of R and S into the table of conditional probabilities, we get theresulting joint distribution in Table 3.1(a). By summing up the columns, we get the marginal or
unconditional probabilities P(W ) . We call this marginalisation. We can similarly marginalise for theother variables by summing up cells containing the required variable.

Suppose that we received the information that the grass is wet, Wy. We might then be interested toknow if this was caused by rain or the sprinkler. We would want P( R y W y ) and P( S y W y ) . Using
Bayes’ theorem (3.3) we can find these as follows:P( R y Wy ) =P ( R yW y )P (W y )=0 .1= 0.53
0.19

Similarly, P( S y W y ) =0.58. Another way to arrive at this is to set the probabilities of all events
involving Wn to 0 (Table 3.1(b)). Then normalise the remaining probabilities (producing Table 3.1(c))
before extracting the required values by marginalising (e.g. P( S y W y ) =0.11 + 0.47=0.58). The
normalisation process has the effect of setting P (W y ) to 1, which is effectively making the statementthat “the grass is wet”.

19Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Thus, we see that with the evidence of wet grass the probabilities of both rain and the sprinkler
being on have increased substantially, which is what we would expect.

Next, suppose we received additional information that it did not rain earlier. We would set all events
involving Ry to 0, producing Table 3.1(d). It is trivial to see that normalization then resultsin P ( S y ) = 1 . This is intuitively obvious: if the grass is wet and it was not caused by rain, then thesprinkler caused it. It is easy to show that the converse is also true.

What we have seen here is called conditional dependence. Before anything is known about W, R and S
are independent. However, they cease to be independent once evidence is entered into W. Thus, R
and S are conditionally dependent given W. This property exists wherever there are more than one
parent nodes to a common child node. It can similarly be shown that in the cases where more than
one child nodes share a common parent node, the child nodes are conditionally independent given theparent node.

3.4 Inference in Bayesian Networks

3.4.1 The Advent of Efficient Algorithms
Inference in DAGs generally involves the incorporation of evidence entered at the nodes along thegraph. The evidence is then propagated along the network to the other nodes by reversing
conditional distributions through forming joint distributions (using Bayes’ theorem) and establishingmarginal posterior distributions from the resulting distributions. The wet grass example above was asimple illustration of this process. However, for a while this approach was still computationally
unfeasible since these joint densities can get very large. For example, the wet grass model involved 3
binary nodes and resulted in a joint density with 8 cells. If a model had 5 ternary nodes, the joint

20Modelling Operational Risk in Financial Institutions Using Bayesian Networks

density would increase to 243 cells. In reality, a BN for most applications would be much more
extensive than this.

Developments in this area have shown that DAGs can be transformed in such a way thatcomputations can be done ‘locally’ among clusters of nodes, thus involving much smaller joint
distributions. Efficient algorithms were devised to facilitate the propagation of the ‘message’ to eachcluster across the network. This drastically reduced the amount of computation required and has
made inference using BNs fairly user-friendly. In fact, most complex operations using BNs can now
be performed on laptop computers using software downloadable from the Internet.

These convenient representations of the DAG are called junction trees. These are non-directionalgraphs consisting of a collection of maximal sub-graphs known as cliques. Cliques (sometimes called
belief universes) are groups of nodes where each node in the group is connected to every other node in
the group. When a group of nodes are connected this way, we say that they are maximal. Different
cliques within a junction tree are connected via separators. These are just common nodes shared by
two adjacent cliques.

3.4.2 Triangulation
The process of transforming a DAG into a junction tree is called triangulation. Although this strictly
refers to one of the steps in the process, we will also use it to mean the whole process. Broadly, we
can look at this in three stages:(i)Moralization(ii)

Triangulation(iii)

Specifying the junction tree

We will illustrate these steps using the DAG in Fig. 3.4(a)21Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Moralization
In this step, all parent nodes of a common child node that are not connected are firstly joined withan undirected edge. For example, C, E and F are common parents of G. Thus, edges are added
between all the pairs of these three nodes. Then all remaining directed edges are rendered nondirectional. The result as shown in Fig. 3.4(b) is what is known as a ‘moral graph’ as all parents are‘married’.

Triangulation
A cycle is a sequence of nodes connected by edges that start and end at the same node. A cycle oflength n consists of a sequence of n consecutive edges. For example, the cycle C, D, H, G, C in Fig.
3.4(b) has length 4. Triangulation is the process of adding undirected edges such that any cycle that has
length over 3 posesses a chord, where a chord is defined as an undirected edge joining two nonconsecutive nodes in the cycle in question. Thus, for example, an edge has to be added between C
and H or D and G. Suppose we choose to add an edge between D and G, we get the triangulated
graph in Fig. 3.4(c).

Specifying the Junction Tree
Once a triangulated graph is obtained, a junction tree can be specified. This involves identifying thecliques within the graph and the separators that connect them. Thus, for example, the cliques(A,B,C) and (C,D,G) have separator C, whereas the cliques (C,D,G) and (D,G,H) have as a separator
the clique (D,G). A junction tree is usually shown as in Fig. 3.5.

When the graph is expressed in this form, the evidence can be incorporated locally at the cliques
where the calculations involve fewer dimensions. Information from the updated cliques is thenpropagated in step to the other cliques of the graph via the separators. Thus, there is no need to deal
with large joint distributions of the whole graph. This is possible because there is a series of22Modelling Operational Risk in Financial Institutions Using Bayesian Networks

common nodes shared by adjacent cliques running throughout the whole tree – one can picture thisto act like a series of interlocking chains. This property is called the running intersection property.

A

B

E

F

A

B

E

F

A

B

E

F

C

G

C

G

C

G

D

H

D

H

D

H

I

I

(a) Example of DAG

I(b) Moral graph of (a)(c) Triangulated graphFig 3.4 Triangulation

ABC
C

CDG

CG

CEFG

DG

DGH
DH

CEFG

CliqueCG

Separator

DHI
Fig 3.5 Junction tree

3.4.3 Inference along the junction tree
To facilitate the illustration of the steps involved in inference along the junction tree, we will use theexample in the previous section where the junction tree is as shown in fig. 3.5. For simplicity, we
assume that all the nodes are binary variables, with the states (Yes, No) denoted with subscripts (e.g.
Dy for the state D=Yes). The main steps to be taken for inference using the junction tree are:23

Modelling Operational Risk in Financial Institutions Using Bayesian Networks(i)Defining prior distributions;(ii)

Initializing the tree; and(iii)

Two-phased propagation.

(i) Defining prior distributions
The first thing that needs to be obtained would be the prior distributions. This would be theunconditional prior distribution for nodes without parents and conditional prior distributions forchild nodes – similar to the example in fig 3.3. For our example, we would need unconditional priors
P ( A) , P (B) , P (E ) and P (F ) and conditional priors P (C | A, B) , P (G | C , E , F ) , P ( D | C ) , P ( H | G )and P ( I | D, H ) . For each prior, we would need the probabilities for each configuration of the
combination of states of variables involved. Thus, for P (C | A, B) we would need figures forP(C y | A y , B y ) , P(C n | A y , B y ) , P(C y | An , B y ) etc.

These can be determined by:(a)

Subjective opinion of the expert.
Experts are interviewed in a series of questionnaires to arrive at quantified conclusions of
the probabilities. Of course, sufficient confidence in the accuracy of the expert’s advice isa prerequisite to use this method.

(b)

Maximum likelihood estimation. For conditional priors this method would entail taking aratio of frequency of the event to the frequency of the parent configuration.
For example: P(C y Ay , B y ) =n(C y , Ay , B y )n( Ay , B y ).

For unconditional priors, this would simply be the proportion of occurrence between thevarious states of the variable.

24Modelling Operational Risk in Financial Institutions Using Bayesian Networks

The method of maximum likelihood assumes that past data is relevant and complete. In practice,there will not be a clear dichotomy between these two methods as the experts will also rely on pastdata but tempered with experience and knowledge regarding its applicability for future events. §3.5
below will discuss updating the priors in the light of new data.

(ii) Initializing the tree
Next, we need to initialize the tree. For each clique and separator, a table exists with cells that
correspond to each combination of states of the variables in the clique or separator. For example,
the clique (A,B,C) will have 8 cells for combinations AyByCy, AnByCy, AnBnCy etc. The numericalvalues in these cells are called potentials. These will change as information is passed along the junction
tree and as each clique or separator is updated. We will see how this works below.

To initialize, all potentials are set to unity. Priors that factorize as per (3.6) are then multiplied into
cliques that contain the variables in the factorized set of priors. For example, the potentials in theclique (A,B,C) will be multiplied by the values in the joint distribution P( A, B, C ) =P (C | A, B ) * P ( A) * P ( B) . This process is analogous to the steps taken to arrive at table 3.1(a).

Some cliques may have more variables than any available set of factorized priors. In these cases, thepriors are still multiplied into the cliques, so long as the variables in the priors are a subset of theclique variables. Cliques for which this operation is not possible are left with all potentials as unity.
The potentials in the separators also remain as unity. Once this has been done, then the tree can beused for propagation.

25

Modelling Operational Risk in Financial Institutions Using Bayesian Networks(iii) Two-phased propagation
Propagation involves a messages originating from the leaves of the tree (i.e. the cliques at theextremities of the tree) and passed along the tree to the root clique and then back out again to theleaves (the choice of the root clique is arbitrary). Along the way the potentials in each clique will be
updated.

Message passing between two cliques takes place as follows (we call the sending and receiving
cliques source and sink respectively):
1. Marginalisation at the source clique: Sum up cells in the clique for common configurations of thevariables of interest – these are the ones in the separator. For example, (A,B,C) is connectedto (C,D,G) via separator C. Thus, we marginalize (A,B,C) for variable C by summing up all
the cells with configurations containing Cy (e.g. AyByCy, AnByCy, etc) to obtain the sum
margin Cy and similarly for Cn. This is analogous to how P(W y ) and P(Wn ) were extracted
from table 3.1(a).

2. Update separator node. The separator node is updated with the sum margins obtained bymarginalizing the source clique. This becomes the updated potential of the separator.

3. Update sink clique. The clique receiving the message is updated using update ratio. This is theratio of the potentials of the separator after the message is passed to the potentials before themessage is passed (if this is the collect phase, which takes place right after the initialization,
the separator potentials would be unity. Hence the update ratio would just be the newpotentials). The updating is performed by multiplying the update ratio into the relevant cells
in the sink clique. These are the ones with variables corresponding to the separator variables.

26

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

This step is performed sequentially: the leaves are the first to be sum-marginalized, then eachadjacent clique will be updated in turn until the root clique is updated. Cliques due to receive updates
from more than one source only send out a message once that has occurred. This flow towards theroot clique is called the collect phase.

Then the whole process is repeated but with the messages going out from the root clique to theleaves. This flow outwards from the root clique is called the distribute phase. Hence, this process issometimes called two-phased propagation. Once both phases are complete, the tree has achieved
equilibrium. At this stage, the potentials at each clique need to be normalised – i.e. scaled so that all
the potentials add up to 1. They can then be marginalised to obtain the posterior marginal
distributions for every node variable. This is the desired result.

3.4.4 Entering evidence into the junction tree
The scope of this investigation covers only evidence of certain events (e.g. “the grass is wet”).
Junction trees can incorporate evidence very efficiently. For this purpose, the two-phased
propagation method is also utilised. The one difference is that the evidence is incorporated at theinitialization stage. Evidence would involve that a node variable be known for certain to assume oneof the possible states (e.g. A=Yes or P( Ay ) = 1 ), implying certain knowledge that the variable has notassumed any of the other states. This is incorporated into the initial clique potentials by setting all
potentials involving the other states of the same variable to be equal to 0. In this example, we set thepotentials for AnByCy, AnByCn, AnBnCy and AnBnCn to 0. The other initial potentials remain as they
were.

Once this is done, the rest of the propagation then carries on as before. The marginal probabilities
obtained now will reflect the impact of the evidence entered.

27

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

3.4.5 Simulation
A useful application of the junction tree is to generate random samples of configurations for thevarious probability distributions in the DAG. There are two methods of sampling:(i)Probabilistic logic samplingThis method samples directly from the DAG itself. Samples are first taken from theunconditional nodes based on the marginal probability distributions at these nodes.
Then, the child nodes of these nodes are sampled based on the conditional probability
distributions at these nodes conditional on the outcome of the sampled configuration of
their respective parents. For example, A and B are first sampled from P( A) and P(B)respectively. Supposing the outcome is Ay and Bn. Then for node C, we sample from
P (C | Ay Bn ) . This continues until all nodes have been sampled. This will be counted as

one sample for the whole DAG. This is carried out for as many sample as required. The
frequencies at each node can then be used to express the sampled marginal probabilities.

If evidence is entered, the process is carried out as before (using the prior probabilities).
However, samples that include values that are not equal to the evidence entered (at the
specific nodes) are rejected. This results in much redundancy in the sampling process.

(ii)

Sampling from the junction tree
A better way to sample with evidence is to exploit the junction tree. The process issimilar to two-phased propagation. The junction tree is first initialized with the evidence
incorporated. Then the collect phase is carried out. When the root clique has been
updated, a configuration is sampled on the root clique. This sample is then immediately
entered into the root clique as evidence. The distribute phase is then carried out but withthe modification of this step of sampling immediately to enter evidence at the clique28

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

before passing the message on to the adjacent cliques. The algorithm continues in thisway until the whole tree has a sampled configuration at each clique.

This method is much more efficient because none of the samples are rejected ascompared to the probabilitstic logic sampling method. This is due to the fact that
evidence is already incorporated into the junction tree, thus the sampling process will not
result in samples with values that are at odds with the evidence.

3.5 Updating the BN in the light of fresh data

The method for prior specification so far has been rather simplistic, not taking into account that new
data from time to time might be useful in updating the probabilities in the BN. In this section, we
will explore how probabilities derived in §3.4.3(i) can be updated in a Bayesian fashion as new data
arrives. The methods described are applicable for complete data only. Incomplete data will require
additional methods which will not be discussed here.

What we will see here is that the expert’s opinion can be quantified in an intuitive way yetmathematically tractable in the form of a Dirichlet distribution. This distribution allows forconvenient updating using Bayesian inference as new data arrives.

3.5.1 Dirichlet distribution as conjugate prior of a multinomial likelihood
Recall from (3.5) that Bayesian inference can be expressed in the following way:
P (θ | x ) ∝ L( x | θ ) ∗ P (θ )or:Posterior ∝ Likelihood * Prior

29

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Suppose variable X in a BN that can take on the values of x1 , x2 ,..., xk with probabilitiesθ1 ,θ 2 ,...,θ k . The likelihood for a sequence of n independent samples, where n1 samples have thevalue x1 , n2 samples have the value x2 , etc, can be represented by the multinomial distribution (ofwhich the binomial distribution is a two-parameter case) :p(n1 , n2 ,..., nk θ ) =

n!

∏

k

k

∏θ
n!

i =1 i

ni
i

i =1

(3.7)

A convenient candidate to be used as the conjugate prior for the θ i ’s is the Dirichlet distribution:
D(α1 ,...,α k )with the density:p (θ ) =Γ(α + )

∏

k

i =1

k

θα
∏
(α )
i

i

i −1

(3.8)

i =1

where α + = ∑i α i , also known as the precision. The Beta distribution is actually a two-parameter
specific case of the Dirichlet distribution. Since binary nodes (e.g. Yes/No variables) are quite
common in BNs the Beta distribution is often used.

The parameters of the Dirichlet distribution, (α 1 ,..., α k ) , have a rather intuitive interpretation: they
can be seen as the implied relative sample sizes for the θ i ’s. In practice, the Dirichlet distribution
would usually not be specified precisely by the expert. Rather, the expert’s conjecture about the θ i ’s(e.g. expected value and range) can be used to work backwards to arrive at a Dirichlet distribution
since we know helpful statistics such as:
E (θ j ) =

αj
α+

and

Var (θ j ) =α j (α + − α j )α +2 (α + + 1)

(3.9)

from which the α i ’s can be derived.

The posterior analysis is fairly straightforward when using the Dirichlet prior. The posterior itself is a
Dirichlet with its functional form arrived at by multiplying (3.7) and (3.8):30

Modelling Operational Risk in Financial Institutions Using Bayesian Networksp(θ n1 , n2 ,..., nk ) ∝

k

∏θ α
i

i + ni −1(3.10)

i =1

This is a D(α 1 + n1 ,...,α k + nk ) . Thus, the posterior is a Dirichlet distribution with the newparameters, and hence the new implied relative sample sizes, equal to the prior implied relative
sample sizes adjusted by the obtained sample counts.

3.5.2 Specifying the Dirichlet parameters
Suppose we are considering the variable P(C A, B) . C is a binary variable, thus the priordistribution will be Β(α1 ,α 2 ) . Its parameters can be specified in the following way. We first gather
the expert’s opinion on the mean and standard deviation of the probabilities of the values in thevariable. Some experts who are not familiar with statistical concepts can be asked to quote a best
estimate and the range of most likely values. The best estimate can then be taken as the mean andthe range can be taken to encompass two standard deviations about the mean, from which thevariance can be easily obtained. The results can be stored as shown in Table 3.2.
ParentCy

Cn
Var (θ y )E (θ y )E (θ n )Var (θ n )Ay , B y
An , B y
A y , Bn

An , Bn

Table 3.2 Table for recording mean and variance of probability values from prior elicitation exercise

Then, for each parent configuration we have, for a two-parameter case of (3.9), the following pairsof simultaneous equations:Cy :E (θ y ) =αyα y +αnandVar (θ y ) =Cn :E (θ n ) =αnα y +αnandVar (θ n ) =α yα n(α 1 + α 2 ) 2 (α y + α n + 1)

α yα n(α 1 + α 2 ) 2 (α y + α n + 1)

31

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

These can then be solved for α y and α n . There will be two sets of parameters (α1 ,α 2 ) to choose
from. In general, it is more conservative to choose the one that gives a lower precision (and thus ahigher variance).

3.5.3 Bayesian updating with new data
This can be done in a fairly straightforward manner. As shown in the posterior analysis that resultedin (3.10), all that is required to arrive at the posterior Dirichlet distribution is to add the data counts
to the relevant prior parameter to obtain the posterior parameter. Supposing the prior distribution
for P(C Ay , B y ) is Β(α y ,α n ) and the following set of data were obtained (only the relevant variables
are shown):{ ( Ay , B y , C y ...) , ( Ay , B y , C n ...) , ( Ay , B y , C n ...) , ( Ay , B y , C n ...) , ( Ay , B y , C y ...) }It then follows that the posterior is Β(α y + 2,α n + 3) . It is worth noting that the same result is arrived
at whether the data is updated sequentially or in batches.

The updated probability estimates can be obtained by taking the mean from the Β(α y + 2,α n + 3)
distribution. Thus, the posterior for P(C y Ay , B y ) is α y (α y + α n + 5) whereas the posterior forP(C n A y , B y ) is α n (α y + α n + 5) .

This whole process of Bayesian updating is sometimes also described as learning. Thus, when a BN iscarried out without learning, then the probabilities remain unchanged with data.

32

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

3.6 Usage of monitors in model assessment

Perhaps of paramount importance, the resulting model needs to be compared against actual data toverify that it adequately corresponds to reality and to assess its usefulness as a predictive tool. Veryoften there will be two or more alternatives with regards to model structures or the sets ofprobabilities. In such cases, there will also be a need to distinguish the better model with regards
these two criteria.
3.6.1 Logarithmic score
The main indicator used is the logarithmic score (LS):S m = − log p m ( y m )Where p m (⋅) is the predictive distribution for the event, Ym , after m − 1 occurrences of events. If
learning is allowed then p m (⋅) incorporates all updates resulting from the m-1 events. The LS is thenegative log of the probability of the event in the actual outcome y m .

Actually, what the LS does is indicate the level of ‘surprise’ resulting from the actual outcome. For
example, if the event which occurred was predicted to happen with a probability of 0.1 the LS would
be –log (0.1) = 2.3. Conversely, an event expected to occur with probability of 0.9 would carry ascore of 0.1. Thus, the less likely an event is predicted to happen, the more ‘surprising’ it is if it didhappen.

For a series of M events, the total penalty is S = ∑m=1 S m . If the probabilities p m (⋅) were updatedMwith each subsequent event, then the S is invariant to the order in which the y m ’s occur. A lower
penalty is always more desirable.

33Modelling Operational Risk in Financial Institutions Using Bayesian Networks

3.6.2 Model assessment
Now we need some criteria by which to accept or reject a model. The total penalty incurred as asingle figure is quite arbitrary on its own. It needs to be compared to a standard. This can be done in
two ways:(i)

Relative standardisation
In this method the model defined by the expert is tested against a reference model,
which is a pre-defined benchmark by which the model is assessed. The total penalty
incurred by the model defined by the expert, S , is compared against the penalty incurred(using the same dataset) by a reference model, S ref . The model is rejected if S exceeds
S ref . The degree to which the expert model is preferred over the reference model isindicated by:exp( S ref − S ) =P (data | expert' s prior)
P (data | reference prior)This is also known as the Bayes’ Factor in favour of the expert’s model.

(ii)

Absolute standardisation
In this method, a test statistic is compiled using the penalties and tested against a null
hypothesis (H0) that the data fits the probabilities assumed in the BN. We define theexpectation and variance of the penalty at each update:

Em = −K∑ p (d ) log p (d )mkm

(3.11)kk =1

Varm =K∑ p (d ) log p (d ) − ( E )
m

k

2

m

k

m

2

(3.12)

k =1

where the d k ’s are all the possible states of the node(s) considered. E m could be seen as aprobability-weighted average of the logarithmic scores.

34

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

The sums of the actual penalties incurred by the data, S , together with the expectationand variance at each update are used to compute the following test statistic:

∑ E
∑ Var
M

S−

m =1

m

M

m =1

(3.13)m

Under H0, the actual penalties are from same underlying distribution as the expectedpenalties. Thus, this test statistic will have the standard Normal distribution and forvalues outside the range [-1.96,1.96], H0 will be rejected at the 5% significance level.

3.6.3 Parent-child monitors
There exists a set of monitors that can be used to diagnose the validity of BN’s in the light of data.
In general, these monitors are used to obtain the LS for each piece of data from which the total
penalties can be compiled and tested according to the two criteria laid out above. The one thing thatdifferentiates one type of monitor from another is the form of the LS. This allows various aspects of
the BN to be diagnosed.

The first of these is known as the Parent-Child Monitor. For each parent-child set of nodes, we havethe following LS:− log p m ( x k X pa ( k ) = ρ )where p m (⋅ X pa ( k ) = ρ ) is the probability distribution of the child node k for the parent
configuration X pa (k ) = ρ after m-1 cases of complete data have arrived and xk is the state of the knode on the mth state. This monitor measures how well the conditional probabilities in the BN thatlink parent and child nodes predict the outcome of the child nodes.

For example, if we are monitoring the parent-child set expressed in the conditional probability
P(C | Ay , B y ) then as the first piece of data arrives (say, Ay , B y , C y ) the LS can be determined as35Modelling Operational Risk in Financial Institutions Using Bayesian Networks

S1 = − log P1 (C y | Ay , B y ) and this is obtained from the expert prior distribution, Β(α y ,α n ) . The datais then incorporated into the conditional probability as illustrated in §3.4.3 to obtain the posterior
P2 (C | Ay , B y ) which has the distribution Β(α y + 1,α n ) . The process then continues iteratively as eachsubsequent case arrives.

The total penalty, S , can then be compared with the total penalty from various alternative reference
models, S ref . Some examples of reference models could be using the same expert prior without
learning or using a reference prior Β(0.5,0.5) with learning.
For absolute standardisation, the penalties incurred would be the same. Using (3.11) and (3.12), theexpectation and variance can be calculated for the first data set as follows:

E1 = −P1 (Cy | Ay , By ) log P1 (Cy | Ay , By ) − P1 (Cn | Ay , By ) log P1 (Cn | Ay , By )Var1 = P1 (C y | Ay , B y )[ log P1 (C y | Ay , B y )]2 + P1 (C n | Ay , B y )[ log P1 (C n | Ay , B y )]2 − E12This is similarly performed for all subsequent updates. The test statistic can then be found using(3.13). If the absolute value is under 1.96 then the child node is being correctly predicted by theconditional probability.

It is often useful to plot the cumulative values of the penalty (for relative standardisation) or
cumulative values of the test statistic (for absolute standardisation) against the data. The graph canbe used to compare the different alternative models and their paths give an indication of how
well/soon the models adapt to the data. The preferred model is the one where the cumulative
penalty increases very little with new data and the cumulative test statistic centers about zero.

36

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

3.6.4 Node monitors
Node monitors measure what is happening at the level of the individual nodes. There are two typesof node monitors: unconditional node monitors and conditional node monitors. Unconditional node monitors
detect poorly estimated marginal distributions. The LS is:− log p m ( xυ )where p m (⋅) is the marginal distribution of the states of the node X υ after m-1 cases. The scoreexpresses the ‘surprise’ at obtaining X υ = xυ on the mth case. Conditional node monitors can beused to detect poor structure, usually upon obtaining reasonable results for the unconditional nodemonitors first. The LS is:− log pm ( xυ ε m \ X υ )where p m (⋅ ε m \ X υ ) is the probability distribution of the node X υ after m-1 cases of evidence havebeen incorporated and the latest set of evidence ε m has just been propagated throughout the BN
except the evidence at node X υ . This score then measures the ‘surprise’ at obtaining X υ = xυ after allother nodes have been propagated with evidence.

The tests can then be performed in a similar manner.

3.6.5 Global monitors
The global monitor of a BN measures the LS of the total evidence entered for the mth case after m1 cases have been entered. This is:− log p m (ε m )The overall global monitor is just a sum of the LS for all the cases:G=M∑ − log p (ε )
m

m

m =1

37

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

The global monitors of two competing models, G1 and G 2 are then compared, with the preferredmodel having the lower value. The Bayes’ factor in favour of model 2 is exp (G1 − G 2 ) .

3.7 Chapter Conclusion
In this chapter, we have examined some theoretical foundations describing the construction of aBN, its usage for inference and ways to assess the validity of the resultant model. We shall see in thefollowing chapter how all this theory can be put to practice in an OR example. This example will
also serve to clarify the methods explained in this chapter.

38

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Chapter Four
Applying Bayesian Networks to the Modelling of Operational Risk

4.1 Chapter Introduction
In this chapter we examine how BNs might be applied in practice for OR modeling. This will beillustrated using a hypothetical example of the risks involved in setting up an Internet on-line
business. A fair amount of detail has been provided within the text to facilitate the clarification ofconcepts introduced in the previous chapter. More complete numerical information underlying theillustrations can be found in the Appendices. Reference to relevant sections in chapter 3 will bemade at various points to allow quick recall of the theory being illustrated.

4.2 Network Risk
The example we are considering is that of a fictitious medium sized insurance company (BayeSure
Insurance Co. or BSI) that decided to set up an on-line business (called BSNet) so that customers
may purchase via the Internet more basic products such as travel insurance, personal accident andeven some cleverly designed health insurance policies. The company has invested a significant
amount of capital to set up the on-line network infrastructure, involving fancy widgets such as highend servers and firewalls. However, as with any on-line business in a high-volume, high-value andhighly competitive industry, the management is concerned that they have the adequate financial
resources to deal with contingencies that may damage the business i.e. Operational Risks. We shallcall the aspect of OR that results from operating this on-line business Network Risk (NR).

39

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

After a year of operating BSNet, the management decided to put the I.T. managers, marketing
personnel and various BSNet user departments together with some statisticians from Risk
Management Unit to set up a BN to serve a two-pronged strategy:(i)

Identify causation of NR events such as transaction downtime, server downtime andapplication failure; and(ii)Help the management to decide how much risk capital to allocate to cover NR in all but
the most extreme of scenarios.

Throughout this chapter, we will view samples of output from a program for BN inference called
XBaies12 which was developed by Robert Cowell of City University.

4.3 Defining the Structure
After some deliberation the working committee was able to put together a structure that looks like fig
4.113. Table 4.1 is a list of the variables employed in this model. The set of values for each variable
are also given along with the abbreviations to be used throughout this chapter. We now briefly
introduce the causal dependencies in the BN and how it relates to the business environment.

The three main causes of NR loss are transaction downtime (TD), data loss (DL) and server
downtime (SD). These can have varying degrees of severity but can be grouped into half a day or afull day to get a server or network up and running, or in the case of data loss, either 50% orcomplete data loss. Based on data gathered in the first year of operation with some additional expert

12

The latest version Xbaies is still in development. However, a very similar version 2.0 is available at his website
http://www.staff.city.ac.uk/~rgc. Various computer soft wares are available to perform the operations and calculations
described in this dissertation (e.g HUGIN, Netica, XBaies). The steps involved in triangulation and updating of thecliques are usually automatically done in the program and not seen in the display. The user need not worry about theunderlying calculations but instead sees only the marginal densities of each node – and how they change as evidence isentered, as we shall see below.

13 Fig. 4.1 and all figures of the BN in this chapter are screen dumps from Xbaies 3.0.

40Modelling Operational Risk in Financial Institutions Using Bayesian Networks

input by the marketing department the total cost as a result of various combinations of such eventswere estimated. This would include, for example, the cost of repairs, lost business opportunities
during the downtime, wages paid to idle staff, costs to recover loss data and damage to reputation(perhaps quantified as loss of future business). All this is combined into one variable, Cost,
representing the bottom-line effect.

Meanwhile, TD is caused by network failure (NF) but there is a mitigating factor: whether or not ahigh-availability network (HAN) was employed in the running of BSNet. SD is caused by server
failure (SF) which in turn is due to power surges (PS) and the server quality (SQ). Again there are
mitigating factors here: the availability of uninterrupted power supply (UPS) – perhaps an internal
generator, and the usage of a high availability server (HAS).
No. Description
1.
Application Failure

2.
3.
4.
5.

Cost of losses from network risk
(Cost)
Data Loss
End User Modification
Firewall

6.
7.
8.
9.
10.
11.
12.

File Access Control
High Availability Network
High Availability Server
Hacker Attack
Network Failure
Power Surge
Server Downtime

13.
14.
15.

Server Failure
Server Hardware Quality
Transaction Downtime

16.
17.

Uninterrupted Power Supply
Virus Attack

ValuesApplication
corruption,
Lockup, OK(Nofailure)0.0m, 0.5m, 1.0m,
1.5m, 2.0m, 2.5m
0%, 50%, 100%
Yes, No
Application Proxy,
Packet Filter
High, Low
Yes, No
Yes, No
Yes, No
Yes, No
Yes, No
0 day, 0.5 day, 1
day
Yes, No
High, Low
0 day, 0.5 day, 1
day
Yes, No
Yes, No

Abbreviation
AF

Cost
DL
EUM
F
FAC
HAN
HAS
Hack
NF
PS
SD
SF
SQ
TD
UPS
V

Table 4.1 Values and abbreviations for Network Risk model

41

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

fig. 4.1 DAG of Network Risk example

DL can be caused by either SF or application failure (AF). AF has three main causes: (i)Modifications made by end users in BSI to the system, either intentionally or accidentally (EUM); (ii)
Virus attacks (V); and (iii) Malicious hacking by external parties (Hack). These three events arelargely controlled by the level of file access given to various parties (FAC). The type of firewall (F)used will also affect the ease with which malicious hackers can access the system. Malicious hacking
not only causes AF but NF as well.

This model has been defined to reflect a ‘holding period’ of one week. Thus, for example, themarginal distribution for the variable cost would be the expected probability distribution of NR
costs incurred by the company over the period of one week. The choice of holding period isarbitrary as far as the mechanics of the model is concerned. The only consideration in this case42

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

would be the availability of data. As the system has commenced for only a year, it makes more sense
to use 52 sets of weekly data than one set of annual data.

4.4 Setting up the BN

4.4.1 Prior specification
At this point the DAG will need to be populated with the prior probabilities at each node. The main
methods of prior specification have been described in §3.4.3 and we assume that BSI uses thesemethods to elicit the priors from the experts mentioned above and from past data.

For example, since attack by computer viruses on online networks are a relatively common and well
documented occurrence, sufficient data might exist for the conditional probabilities for the V|FAC
node to be obtained using maximum likelihood estimation. Thus, having combined industry
experience with one year’s worth of BSNet experience, it might have been observed that only in
10% of cases where a high level of File Access Control is implemented was there a virus attack.

For events like end user modification, the circumstances of different companies (e.g. training andrecruitment policies) might be so different as to render any external data irrelevant. In this case, theI.T. Department together with staff appraisal information from Human Resources Department
would need to decide on a figure for the probability of adverse end user modification for systems
with high/low levels of file access control. This corresponds with the EUM|FAC node.

Having performed these exercises, the company arrives at the probabilities detailed in Appendix I.
For the purpose of this illustration, the figures have been simplified and as such, might not berealistic – with apologies to readers well versed in network risks.

43Modelling Operational Risk in Financial Institutions Using Bayesian Networks

For flexibility in the model, various unconditional priors have been assigned neutral probabilities.
These will be treated as input variables to reflect the actual state of BSNet once ‘evidence’ as beenentered. Power surge, obviously, is outside of the control of the company. Thus past data has beenused to arrive at the probabilities and it will not be treated as an input variable.

Theoretically, any node can be treated as an input variable. To prevent confusion, when evidence isentered in the other nodes it will be treated as a stress or scenario test.

fig. 4.2 Unconditional priors

4.4.2 Junction Tree Specification
Before we go on to use the DAG for inference, we need to obtain a junction tree via the process oftriangulation as described in §3.4.2. We begin with the moralization of the DAG by establishing moral
links between parent nodes. The result is shown in fig. 4.3.

44

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

fig 4.3 Moral links

Then, replacing the directed edges with undirected edges, we get the moral graph for this model.
Additional edges are added between the following pairs of nodes: (Hack, TD), (Hack, EUM),(AF,TD), and (SF,TD). We now have the triangulated graph as shown in fig 4.4.

fig 4.4 Triangulated graph45Modelling Operational Risk in Financial Institutions Using Bayesian Networks

We can now specify a junction tree from the cliques identified in the triangulated graph. Junction
trees are not unique, though it is useful to avoid cliques that are too large. The one chosen for thisexample is shown in fig 4.5. The boxes indicate the cliques and give the members of the clique alongwith the clique size. The root clique is denoted by the box with a bold outline.
UPS : SD : SF : HAS ( 24 )Cost : DL : TD : SD ( 162 )DL : TD : SD : SF ( 54 )PS : SQ : SF ( 8 )AF : DL : TD : SF ( 54 )V : Hack : AF : FAC : EUM ( 48 )NF : Hack : TD ( 12 )Hack : AF : TD ( 18 )F : Hack : FAC ( 8 )HAN : TD : NF ( 12 )Fig 4.5 Junction tree

Inference
Having specified the junction tree and the prior distributions, we are now ready to make someinference about the posterior distributions. Firstly, we need to initialize the junction tree. As stated in§3.4.3, this entails setting all the clique potentials to 1. Then, combinations of priors that factorise asin (3.6) are multiplied into cliques that contain the corresponding set of variables. Thus, for example,
the potentials in the clique (V, Hack, AF, FAC, EUM) will be multiplied by the probabilities inP(AF|V, EUM, Hack) and P(V|FAC).

To illustrate, consider the potential for the cell corresponding to (V="No"; EUM="No"; AF="App
Corr"; FAC="Low"; Hack="Yes"). The initialized potential will be:
P ( AFAppCorr V No , EUM No , HackYes ) × P(V No FAC Low ) = 0.6 × 0.3 = 0.18

46Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Table 4.2 is a summary of cliques with their corresponding initial potentials:
CliqueCost : DL : TD : SD
UPS : SD : SF : HAS
PS : SQ : SF
AF : DL : TD : SF
V : Hack : AF : FAC : EUM
NF : Hack : TD
HAN : TD : NF
F : Hack : FAC

Initial Potentials
P(Cost|DL, TD, SD)
P(SD|UPS, SF, Has)*P(UPS)*P(HAS)
P(SF|PS, SQ)*P(PS)*P(SQ)
P(DL|AF, SF)
P(AF|V, EUM, Hack)*P(V|FAC)
P(NF|Hack)
P(TD|NF, HAN)*P(HAN)
P(Hack|FAC, F)*P(FAC)*P(F)Table 4.2 Initial potentials

The initialized junction tree can be found in Appendix II.

Two-Phased Propagation
To illustrate the two-phased propagation algorithm for the whole junction tree, we will focus onwhat goes on at the local level. We shall do this by zooming into the cliques (HAN, TD, NF),
denoted C1 and (NF, Hack, TD), denoted C2. The separator for these cliques consists of thevariables NF and TD. First, initialization takes place according to the initial potentials given above.
The separator potentials are left as unity. The result is seen in fig 4.6(a).

The collect phase begins from the clique C1 because it is a leaf clique. Firstly, the clique potentials arenormalized. The normalized potentials for C1 are shown in fig 4.6(b). Then, the sum-marginals
corresponding to the various combinations of states for variables NF and TD are obtained from C1
and used to update the separator.

For example, all the cells with configuration NF= “Yes” and TD = “0.5 day” are summed up:
0.2250 + 0.0250 = 0.2500. This is used to update the corresponding potential in the separator. Oncethe separator has been updated, the sink clique C2 then receives the message from the separator.

47

NF="Yes";TD="0 day";
NF="No";TD="0 day";
NF="Yes";TD="0.5 day";
NF="No";TD="0.5 day";
NF="Yes";TD="1 day";
NF="No";TD="1 day";(a) Initialization – factorized priors multiplied into
the cliques to obtain initial potentials. Please note thatzero potentials have been suppressed.

"HAN" "TD" "NF"
0.45000 HAN="Yes";TD="0.5 day";NF="Yes";
0.05000 HAN="No";TD="0.5 day";NF="Yes";
0.05000 HAN="Yes";TD="1 day";NF="Yes";
0.45000 HAN="No";TD="1 day";NF="Yes";
0.50000 HAN="Yes";TD="0 day";NF="No";0.50000 HAN="No";TD="0 day";NF="No";1.00000
1.00000
1.00000
1.00000
1.00000
1.00000"NF" "Hack" "TD"0.80000 NF="Yes";TD="0 day";Hack="Yes";
0.20000 NF="No";TD="0 day";Hack="Yes";
0.80000 NF="Yes";TD="0.5 day";Hack="Yes";
0.20000 NF="No";TD="0.5 day";Hack="Yes";
0.80000 NF="Yes";TD="1 day";Hack="Yes";
0.20000 NF="No";TD="1 day";Hack="Yes";
1.00000 NF="No";TD="0 day";Hack="No";1.00000 NF="No";TD="0.5 day";Hack="No";1.00000 NF="No";TD="1 day";Hack="No";NF="Yes";TD="0 day";
NF="No";TD="0 day";
NF="Yes";TD="0.5 day";
NF="No";TD="0.5 day";
NF="Yes";TD="1 day";
NF="No";TD="1 day";Fig. 4.6 –Illustration of Two-phased Propagation(b) Collect Phase – message is propagated to
the rest of the tree and collected at the root.

"HAN" "TD" "NF"
0.22500 HAN="Yes";TD="0.5 day";NF="Yes";
0.02500 HAN="No";TD="0.5 day";NF="Yes";
0.02500 HAN="Yes";TD="1 day";NF="Yes";
0.22500 HAN="No";TD="1 day";NF="Yes";
0.25000 HAN="Yes";TD="0 day";NF="No";0.25000 HAN="No";TD="0 day";NF="No";0.00000
0.50000
0.25000
0.00000
0.25000
0.00000"NF" "Hack" "TD"0.0000 NF="Yes";TD="0 day";Hack="Yes";
0.1000 NF="No";TD="0 day";Hack="Yes";
0.2000 NF="Yes";TD="0.5 day";Hack="Yes";
0.0000 NF="No";TD="0.5 day";Hack="Yes";
0.2000 NF="Yes";TD="1 day";Hack="Yes";
0.0000 NF="No";TD="1 day";Hack="Yes";
0.5000 NF="No";TD="0 day";Hack="No";0.0000 NF="No";TD="0.5 day";Hack="No";0.0000 NF="No";TD="1 day";Hack="No";NF="Yes";TD="0 day";
NF="No";TD="0 day";
NF="Yes";TD="0.5 day";
NF="No";TD="0.5 day";
NF="Yes";TD="1 day";
NF="No";TD="1 day";(b) Distribute Phase – message returning from the root
after being propagated across the tree.

"HAN" "TD" "NF"
0.1530 HAN="Yes";TD="0.5 day";NF="Yes";
0.0170 HAN="No";TD="0.5 day";NF="Yes";
0.0170 HAN="Yes";TD="1 day";NF="Yes";
0.1530 HAN="No";TD="1 day";NF="Yes";
0.3300 HAN="Yes";TD="0 day";NF="No";0.3300 HAN="No";TD="0 day";NF="No";0.00000
0.66000
0.17000
0.00000
0.17000
0.00000"NF" "Hack" "TD"0.0000 NF="Yes";TD="0 day";Hack="Yes";
0.0850 NF="No";TD="0 day";Hack="Yes";
0.1700 NF="Yes";TD="0.5 day";Hack="Yes";
0.0000 NF="No";TD="0.5 day";Hack="Yes";
0.1700 NF="Yes";TD="1 day";Hack="Yes";
0.0000 NF="No";TD="1 day";Hack="Yes";
0.5750 NF="No";TD="0 day";Hack="No";0.0000 NF="No";TD="0.5 day";Hack="No";0.0000 NF="No";TD="1 day";Hack="No";Modelling Operational Risk in Financial Institutions Using Bayesian Networks

48

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

The potentials in C2 are multiplied by the update ratio, defined as:
Separator potential after recent updating
Separator potential before recent updating
Thus, for example, the cell (NF="No";TD="0 day";Hack="Yes")=0.2 before updating would be
multiplied by 0.5/1 to get 0.1. When C2 is updated completely (as shown in fig 4.6(b)), it will in turnbe the sender clique to the next clique in the sequence of the propagation. This algorithm carries on
until the root clique is updated. Then the collect phase is completed. The result for the whole tree canbe seen in Appendix II.

The distribute phase then commences and the updating proceeds outwards from the root until itreaches C2 again. In fig 4.6(c) we have the updated C2 after the propagation has passed through therest of the junction tree. This time, C2 is the sender and C1 the sink. The separator potentials thentake on the new set of sum-marginals from C2. C1 is then updated according to the update ratio as
defined above. For example, the new potential for (HAN="Yes";TD="0.5 day";NF="Yes") wouldbe:Previous potential for (HAN="Yes";TD="0.5 day";NF="Yes") * Update ratio=0.225*0.17/0.25=0.153.

From the updated C2, we can sum-marginalize to obtain the posterior marginals for each of thevariables in the cliques. Thus, for TD the marginal distribution would be P(0 day, 0.5 day, 1 day) =(0.66, 0.17, 0.17).

Again, the updated cliques for the whole junction tree after the distribute phase can be seen in
Appendix II. The marginal distributions are shown in fig 4.7.

49

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

fig 4.7 Prior Marginal Distributions

At this juncture it is worth mentioning that, although this procedure appears rather tedious, theentire two-phased propagation takes place almost instantaneously at a mouse-click when using
computer software such as XBaies.

4.5 Applying the BN to OR modeling4.5.1 Risk Capital Allocation

We now come to one of the main applications of BNs in modeling of OR. Having set up the model
as above, the company would like to decide on the amount of financial capital to allocate towards
protecting the company from all but the most extreme cases of NR.

50

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

At this point the model can be configured to reflect the actual known state of certain variables toreflect the actual position of the company. For example, the I.T. Department may provide theinformation that Application Proxies are used 24 hours as the firewall for the network and that High
Availability Networks have been purchased and implemented for round-the-clock availability. Suchinformation is incorporated by inserting evidence at the relevant cliques as described in §3.4.4. For
example, to indicated the usage of Application Proxies, all cells of the clique (F, Hack, FAC)
corresponding to F= “PF” is set to 0. This is due to the fact that the prior distribution for P(FAP,FPF)
is now (1,0) since we know for certain that the type of firewall used is an Application Proxy.

Supposing the set of evidence in table 4.3 represents the status of BSI’s BSNet system. This is thenincorporated into the junction tree at the initialization stage. The tree is then propagated as before.
The new set of posterior marginals incorporating the evidence is shown in fig 4.814. The evidence
was not updated for power surge, as this remains uncertain for a future period.
Variable
F
FAC
HAN
HAS
SQ
UPS

Evidence
Application Proxy
High
Yes
Yes
High
Yes

Table 4.3 Evidence entered

It is now fairly straightforward to determine the risk capital to be allocated to cover NR in BSNet for
95% of cases over a holding period of one week. This is simply calculated by linearlyinterpolating to obtain the 95th percentile of the probability distribution for cost. In this case, theresult is 0.32 million.

14

XBaies has a user-friendly graphic interface that allows the user to incorporate evidence at the nodes by clicking at thedesired state of the variable of interest. E.g., to enter FAC= “High”, all that is needed is a click on the bar representing
the marginal probability of the state “High” at the FAC node chart. This automatically sets the bar to 1.
51

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

fig 4.8 Posterior marginals after incorporating evidence

4.5.2 Scenario testing and causal analysis
The model can be used to test various scenarios to help management in optimizing its risk profile.
For example, if BSI wishes to reduce costs by reducing the level of File Access Control to “Low”,
the effect of this action can be easily investigated by entering the evidence FAC= “Low” into thejunction tree. The tree is then propagated and the resulting marginals observed as before (this isshown in fig 4.9). We can deduce that the trade-off of the lower costs is an increased capitalrequirement of 0.97 million. We assume that the 95% confidence level is maintained.

52

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

fig 4.9 Posterior marginals of scenario testing

The company may wish to perform some stress testing by investigating the impact of adverse events.
Supposing the management is interested to examine the effects of situation of complete loss ofdata. This can be done by setting the node DL to “100%”. The effect is then propagated again andeach node is updated accordingly. The final result can be seen in fig 4.10.

53

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

fig 4.10 Cause and effect of a 100% data loss

Note the shift in the marginal probability distribution of the cost. We can now read off a few
statistics from this new distribution. The expected cost is now 0.75 million with standard deviation
0.57 million. The 95th percentile has now increased to 1.64 million.

Note also the changes in the other nodes. For example, we now have P ( SF y ) and P ( PS y ) equal to 1.
Conversely, the probability of “No” has increased significantly for Hack, V, EUM and NF. Similarly,
AF is overwhelmingly “OK”. We can read from this that server failure is the definite cause of data
loss. Consequently, we also deduced that a power surge was the definite cause of server failure. As aresult, other events that might have been the causation of data loss have been explained away, giving
the increase in “No” probabilities for these nodes.

This is an example of how BNs can be useful in causal analysis.

54Modelling Operational Risk in Financial Institutions Using Bayesian Networks

4.5.3 Simulation
We might want to use the BN to generate random sample configurations to simulate actual 52-weekperiods in which events may occur with probabilities specified in the BN. This can be useful todetermine aggregate capital requirements for one-year holding periods.

Using the algorithm for sampling from a junction tree as described in §3.4.5, Xbaies was used togenerate 52 sample configurations and the results are displayed in Appendix III. This can becompared to a sample set of size 9999 (using the same evidence) – also shown in Appendix III. We
can see that the larger sample size is much closer to the probabilities shown in fig 4.8 from which
the samples have been obtained.

4.6 Updating the BN with new data
As the model is implemented, there will be fresh data arriving every week. These can be used to
update the probabilities in the model using Dirichlet priors as discussed in §3.6. As an example weconsider the conditional probability P(NF|Hacky) underlying the node NF.

Initially, we have to specify the prior Beta distribution (since this is a binary node). Assuming thatthe process of prior elicitation is done indirectly via the I.T. Department’s opinion about therespective ranges of the probabilities specified in Appendix I, we obtain the results in Table 4.4.

NF
NF= “Yes”
Parentµ P ( NF Hack )

Range

Hack
=Yes

0.8

0.70-0.90

y

y

σ

2
P ( NFy Hack y )αyµ P ( NF Hack )

0.12

4.3

0.2

n

NF= “No”
Range

σ P2 ( NF Hack )αn0.15-0.25

0.052

1.1

y

n

y

Table 4.4 Prior elicitation to specify Beta parameters55Modelling Operational Risk in Financial Institutions Using Bayesian Networks

The statistics for the case of NF= “Yes” result in Β(12.0,3.0) whereas the statistics for NF= “No”
result in Β(4.3,1.1) . The second set of results was chosen as it results in a lower precision.

Suppose that over the course of a year, the system was penetrated by malicious hackers in 6 out of
the 52 weeks and half of those resulted in network failure. The Beta parameters then can be updated
accordingly and the latest probability estimates can be obtained from the mean of the posterior Beta.
The results are shown in Table 4.5.
NF
NF= “Yes”
Parent

Prior
mean

Data

αy

Hack
=Yes

0.8

3

4.3

NF= “No”

α y + ny

Posterior
mean

Prior
mean

Data

αn

α n + nn

Posterior
mean

7.3

0.64

0.2

3

1.1

4.1

0.36

Table 4.5 Bayesian updating with data

The results obtained are fairly intuitive: the initial prior estimate of network failure in the event of ahacker attack was reduced from 80% to 64% when the data, which indicates a 50% occurrence, wasincorporated. Similar updates can be performed simultaneously for all the nodes with the complete
dataset.

We see that Bayesian updating for BNs given new data is fairly easy to perform and would be costeffective to conduct regularly as new cases arrive.

4.7 Model Assessment
To illustrate how the model may be assessed, we shall consider the application of the Parent-Child
Monitor at the NF node. The conditional probability underlying this node is P(NF|Hack). We shallinvestigate the particular case where malicious hacking has occurred. Supposing sufficient time has
elapsed such that data is available for 16 weeks where this was the case. We shall see how well themodel fits the data.

56

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

What I have done is to simulate 50 cases of data from a slightly different population. The
distributions are given in table 4.6. This was done to investigate how well the model adapts to thedata, given that the original prior was not specified very well.

Distribution P ( NF = Yes | Hack = Yes) P ( NF = No | Hack = Yes)
Prior
0.8
0.2
Simulated
0.5
0.5
Table 4.6 Probability distribution of NF|Hack

Two alternatives were compared: one where the probabilities were updated in the same way shownin §4.6 as each new cases arrive and the other where the probabilities were left as originally specified.
At each update, the following statistics were compiled:(i)

The logarithmic score (or penalty);(ii)

The expectation of penalty; and(iii)

The variance of penalty.

The cumulative penalties and the absolute standardisation test statistics were calculated and thenplotted. The results are found in fig. 4.11 and fig. 4.12. Details of the calculations are in Appendix IV.
Parent-Child Monitor for NF|Hack=Yes
With learning
20
18

Without learning

Cumulative Penalty

16
Bayes Factor

14
12
10
8
6
4
2
0
-2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Data

fig. 4.11 Parent-child monitor for NF node: relative standardisation

57

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

We can see that just after 16 cases, the model with learning has an advantage of a Bayes factor of
4.34, which implies an odds of 76:1 in its favour.

For the absolute standardisation, the model with learning shows much better results when comparedto the model without learning. However, in both cases, the hypothesis test rejects H0 since thethreshold of 1.96 had been exceeded. Some modifications might be necessary in the structure of themodel as this parent-child structure might not be a good predictor of the NF node.

Parent-Child Monitor for NF|Hack=Yes

5

With learning
Without learningStandardised test statistic

4
3
2
1
0
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

-1
Data

fig. 4.11 Parent-child monitor for NF node: absolute standardization

Similar tests can be performed using node and global monitors in forming an opinion on theviability of the model. In each case, an alternative structure can be tested in parallel with the results
plot on the same graph for comparison. This visualisation also facilitates communication of theresults.

58

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

4.8 Chapter conclusion
We have used a fictitious company’s Internet business to illustrate how a BN can be set up using a
combination of past data and expert input. In this illustration we have also demonstrated theapplication of BNs to the business needs of financial institutions in the following areas:(i)

Setting of regulatory capital for OR;(ii)

Scenario testing for causal analysis; and(iii)

Simulation of future scenarios.

The model has been shown to be easily adaptable to incorporate new input. Techniques forassessing the suitability of the model have also been briefly demonstrated.

59Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Chapter Five
Discussion

5.1 Why Bayesian networks?

We shall now summarise the reasons for adopting BNs as an approach to modelling OR in financial
institutions.

The main advantage of using BN for modelling OR is the incorporating of expert opinion through:(i)

Choosing the variables of interest;(ii)

Defining the structure of the model via the causal dependencies; and(iii)

Specification of the prior distributions and the conditional probabilities at each node.

Bayesian probability updating ensures that the model is not static, but quickly adapts to new input
and incorporates it with prior expert opinion in a mathematically tractable manner. Monitors are alsoavailable to enable the efficacy of this process to be observed in real-time, thus facilitating informedmodel criticism and choice.

We have seen the usage of BNs to model an OR loss distribution, on which business decisions could
be based – particularly in the allocation of economic capital. Thus, there is tremendous potential
here for an internal model for the setting of regulatory risk capital for OR, such as is required by the
emerging regulatory regime for financial institutions. Stress and scenario testing is often a feature ofearly warning systems in regulatory regimes. We have seen how these can be done fairly quickly on
BNs.

60

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Finally, the graphical presentation of BNs aids the understanding of the causal structure and presents
the risk profile of the company in an intuitive way – improving management understanding of, andhence participation in, the management of OR.

5.2 Some caveats

Having sung these praises, there are various challenges to overcome in the usage of BN to model
OR. Firstly, the model can get very complex with many nodes to specify – this is especially so if thenodes have many parents. In such cases, there can be many conditional probabilities to specify – if
the maximum likelihood method of prior eliciation is used, significant volume of data might actually
be required, thus reducing one of the main advantages of using Bayesian methods.

The alternative to that would be a rigourous exercise in prior elicitation from experts through costly
methods such as the Delphi method, which involves many rounds of questionnaires. A main
challenge in this area is in dealing with experts who may not be comfortable thinking in terms offrequencies, although one would hope that this is not the case in financial institutions.

There is also the issue of the non-uniqueness of the causal structure of the model. It is easily seenthat choosing a suitable model structure can be as much an art as a science. Although a BN is cheapand easy to run, the whole process of setting up one can be costly, resource consuming andpotentially politically messy if many business units/cost centres are involved. In many cases, a fairly
complex BN is required to capture all the necessary variables. In addition, accuracy of results might
be pursued at the expense of model parsimony especially when business (or even regulatory)
decisions are at stake.

61

Modelling Operational Risk in Financial Institutions Using Bayesian Networks

The main advantage of BNs – ability to incorporate subjective knowledge – can be a disadvantage
when it comes to setting regulatory capital. Regulators who require an objective standard to approveof internal models may find it difficult to find a standard for acceptance of BNs due to its highsubjective content. Regulators might need to specify rules on the process of model specification andprior elicitation to reduce the subjectivity.

5.3 Extensions

In the Network Risk example, the distributions have been expressed as discrete probabilities for
simplicity. More advanced modelling can be used to deal with continuous distributions in BNs –these are also known as conditional Gaussian distributions. Chapter 7 of Cowell et al (1999) provides
a fairly comprehensive treatment of CG.

In the example above, we have assumed the existence of complete data. In reality, this is seldom thecase. Certain data cells will be missing or just impossible to obtain. Cowell et al (1999) suggests
various methods such as the EM algorithm and the Gibbs Sampler to deal with updating of the BNwith incomplete data. The Gibbs sampler uses a Markov Chain Monte Carlo process to recursively
update the parameters of a Bayesian network to obtain a predictive distribution.

As discussed above, specification of the structure is often subject to debate. Thus, a logicalprogression is to move towards structural learning – i.e. letting the data speak for itself not just withregards the probability distributions of the variables but even the very structure itself. This iscurrently an active area of research and chapter 11 of Cowell et al (1999) provides useful suggestionsof work being done.

62Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Post-script
Probabilistic reasoning using expert systems, of which BNs are an example, is a specific case of alarger universe of scientific modelling known as Artificial Intelligence. This exciting confluence ofvarious streams of scientific knowledge – statistics, engineering, econometrics, and information
technology – will continue to be a fertile space for collaboration to discover newer and better ways
to understand the world we live in. In researching for this dissertation, I am humbled by the amountof thought that has been put into this field yet I am also inspired to be part of its evolution throughmy pilgrimage within the actuarial profession and beyond.

63Modelling Operational Risk in Financial Institutions Using Bayesian Networks

Acknowledgements

I would like to thank the following people for their invaluable input and assistance in thisdissertation:
Professor Richard Verrall – for being as excited as I have been by this whole project and for being
such an encouraging supervisor.
Dr. Robert Cowell – for introducing me to the world of probabilistic reasoning and for the
priceless signed copy of Probabilistic Networks and Expert Systems (see Cowell et. al (1999)).
Professor Peter England – for the industry perspective and for suggesting I.T. related risks as an
example.
Professor Steven Haberman – for having spotted this as a doable dissertation in the first place.
Kevin Lai of Network Associates Incorporated, Kuala Lumpur – for helping me set up the
Network Risk example and for your friendship.
My colleagues and seniors at the Insurance Supervision Department and Cik Roslina Hitam
of Human Resources, Bank Negara Malaysia – for all the support from home.
Marilyn Parris – for running so many of my errands and for inspiration to be salt and light in themarketplace.
City University Student Union – for maintaining the photocopier.
All the staff at the Faculty of Actuarial Science, CASS Business School.

Special thanks to my Lord and Saviour, Jesus Christ, without whom none of this would be possible.
To Him be the Glory.

64Modelling Operational Risk in Financial Institutions Using Bayesian Networks

References

ALEXANDER, CAROL (2000). Bayesian methods for measuring operational risk. Discussion Papers in
Finance 2000-02. ISMA Centre, University of Reading.
BASEL COMMITTEE ON BANKING SUPERVISION (1999). A new capital adequacy framework. Bank forInternational Settlements.
BASEL COMMITTEE ON BANKING SUPERVISION (2001a). Working paper on the regulatory treatmentof operational risk. Bank for International Settlements.
BASEL COMMITTEE ON BANKING SUPERVISION (2001b). Consultative Paper 2.5. Bank forInternational Settlements.
BASEL COMMITTEE ON BANKING SUPERVISION (2002). Operational risk data collection exercise.
Bank for International Settlements.
CESKE, R., HERNANDEZ, J.V., SANCHEZ, L.M. (2000). Quantifying event risk: the next convergence.
Journal of Risk Finance. Spring 2000, 1 – 13.
COWELL, R. G., DAWID, A. P., LAURITZEN, S. L., SPIEGELHALTER, D. J., (1999). Probabilistic
networks and expert systems. Springer-Verlag.
CRUZ, MARCELO G. (2002) Modeling, measuring and hedging operational risk. Wiley.
DOWD, KEVIN (1998). Beyond value at risk: the new science of risk management. Wiley.
FRACHOT, ANTOINE & RONCALLI, THIERRY (2002). Mixing internal and external data for managing
operational risk. Groupe de Recherche Operationelle, Credit Lyonnais.
HOFFMAN, DOUGLAS (2002). Managing operational risk: 20 firmwide best practice strategies. Wiley.
INSTITUTE OF ACTUARIES (2002). Report of the operational risk working party to GIRO 2002.
Available to members at http://www.actuaries.org.uk
JENSEN, FINN V. (1996). An introduction to Bayesian networks. Springer.

65Modelling Operational Risk in Financial Institutions Using Bayesian Networks

JORION, PHILIPPE (2001). Value at risk: the new benchmark for managing financial risk. McGrawHill.
KING, JACK L. (1999). Operational Risk. Wiley.
MARSHALL, CHRISTOPHER (2001). Measuring and managing operational risks in financial institutions.
Wiley
MICCOLIS, JEREMY A. & SHAH, SAMIR (2000). Getting a handle on operational risks. Emphasis
2000/1. Tillinghast-Towers Perrin.
SHAH, SAMIR (2001). Operational risk management. Casualty Actuarial Society 2001 Seminar on
Understanding the Enterprise Risk Management Process. Powerpoint slides available from Casualty
Actuarial Society at http://www.casact.org

VAN DEN BRINK, GERRIT J. (2002). Operational risk: the new challenge for banks. Palgrave.

66

APPENDIX I
Network Risk: Prior Distributions
Unconditional Priors
FAC
High
0.5

HAN
Yes
0.5

Low
0.5

PS

HAS
Yes
0.5

No
0.5

SQ

Yes
0.25

No
0.75

F
AP
0.5

No
0.5

PF
0.5

UPS

High
0.5

Low
0.5

Yes
0.5

No
0.5

Conditional Priors
SF|PS, SQ

Hack|FAC, F

TD|HAN, NF

SF
SQ
H
L

PS
Y
N
Y
N

Hack

Y
0.2
0
0.9
0

N
0.8
1
0.1
1

F
AP
PF

FAC
High
Low
High
Low

Y
0.1
0.5
0.3
0.8

Y
Y
N
Y

AF
App Corr Lockup
0.9
0.1
0.6
0.4
0.7
0.3
0.6
0.2
0.4
0.3
0.1
0.3
0.5
0
0
0

V
Y
N
Y
N
Y
N
Y
N

V|FAC

HAS

OK
0
0
0
0.2
0.3
0.6
0.5
1

Y
0.1
0.7

SF
Y

Y
N

N

EUM|FACTD
0.5 day
0.9
0
0.1
0

1 day
0.1
0
0.9
0

UPS
Y
N
Y
N
Y
N
Y
N

0 day
0.5
0
1
1
0
0
1
1

SD
0.5 day
0.5
0.2
0
0
0.1
0
0
0

NF

EUM
N
0.9
0.3

1 day
0
0.8
0
0
0.9
1
0
0

NF|Hack

V
FAC
High
Low

HAN NF 0 day
Y
0
Y
N
1
Y
0
N
N
1

SD|UPS, SF, HAS

AF|V, EUM, Hack
Hack EUM

N
0.9
0.5
0.7
0.2

FAC
High
Low

Y
0.1
0.6

DL
50%
0
0.1
0.2
0.5
0.3
0

100%
0.9
0.8
0.6
0
0
0

N
0.9
0.4

Hack
Y
N

Y
0.8
0

N
0.2
1

DL|AF,SF
SF
Y

N

AF
App Corr
LockupOK
App Corr
Lockup
OK

0%
0.1
0.1
0.2
0.5
0.7
1

67

APPENDIX I
Cost|SD, TD, DL
Cost
DL

TD
0 day

0%

0.5 day

1 day

0 day

50%

0.5 day

1 day

0 day

100%

0.5 day

1 day

SD
0 day
0.5 day
1 day
0 day
0.5 day
1 day
0 day
0.5 day
1 day
0 day
0.5 day
1 day
0 day
0.5 day
1 day
0 day
0.5 day
1 day
0 day
0.5 day
1 day
0 day
0.5 day
1 day
0 day
0.5 day
1 day

0.0m
1
0
0
0.7
0
0
0.6
0
0
0.6
0
0
0.3
0
0
0.3
0
0
0.4
0
0
0.2
0
0
0
0
0

0.5m
0
0.6
0.1
0.3
0.5
0
0.4
0.4
0
0.4
0.5
0
0.3
0.3
0
0.2
0.3
0
0.4
0.4
0
0.2
0.2
0
0.1
0
0

1.0m
0
0.4
0.2
0
0.3
0.2
0
0.3
0.1
0
0.4
0.1
0.2
0.3
0.1
0.2
0.2
0
0.2
0.3
0
0.3
0.2
0
0.2
0.2
0

1.5m
0
0
0.3
0
0.2
0.4
0
0.2
0.2
0
0.1
0.3
0.2
0.2
0.2
0.2
0.2
0.1
0
0.2
0.3
0.2
0.3
0.2
0.3
0.3
0

2.0m
0
0
0.2
0
0
0.2
0
0.1
0.4
0
0
0.4
0
0.2
0.5
0.1
0.2
0.4
0
0.1
0.4
0.1
0.2
0.4
0.2
0.3
0.2

2.5m
0
0
0.2
0
0
0.2
0
0
0.3
0
0
0.2
0
0
0.2
0
0.1
0.5
0
0
0.3
0
0.1
0.4
0.2
0.2
0.8

68

APPENDIX II"DL" "TD" "SD" "SF"

1.00000
0.70000
0.30000
0.60000
0.40000
0.60000
0.40000
0.50000
0.30000
0.20000
0.40000
0.30000
0.20000
0.10000
0.10000
0.20000
0.30000
0.20000
0.20000
0.20000
0.40000
0.20000
0.20000
0.10000
0.20000
0.40000
0.30000
0.60000
0.40000
0.30000
0.30000
0.20000
0.20000
0.30000
0.20000
0.20000
0.20000
0.10000
0.50000
0.40000
0.10000
0.30000
0.30000
0.20000
0.20000
0.30000
0.20000
0.20000
0.20000
0.10000
0.10000
0.30000
0.40000
0.20000
0.10000
0.20000
0.50000
0.20000
0.10000
0.40000
0.50000
0.40000
0.40000
0.20000
0.20000
0.20000
0.30000
0.20000
0.10000
0.10000
0.20000
0.30000
0.20000
0.20000
0.40000
0.30000
0.20000
0.10000
0.20000
0.20000
0.30000
0.20000
0.10000
0.20000
0.30000
0.30000
0.20000
0.30000
0.40000
0.30000
0.20000
0.40000
0.40000
0.20000
0.80000

1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000

Cost="0.0m";TD="0 day";SD="0 day";DL="0%";Cost="0.0m";TD="0.5 day";SD="0 day";DL="0%";Cost="0.5m";TD="0.5 day";SD="0 day";DL="0%";Cost="0.0m";TD="1 day";SD="0 day";DL="0%";Cost="0.5m";TD="1 day";SD="0 day";DL="0%";Cost="0.5m";TD="0 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="0 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="1 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="1 day";SD="0.5 day";DL="0%";Cost="1.5m";TD="1 day";SD="0.5 day";DL="0%";Cost="2.0m";TD="1 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="0 day";SD="1 day";DL="0%";Cost="1.0m";TD="0 day";SD="1 day";DL="0%";Cost="1.5m";TD="0 day";SD="1 day";DL="0%";Cost="2.0m";TD="0 day";SD="1 day";DL="0%";Cost="2.5m";TD="0 day";SD="1 day";DL="0%";Cost="1.0m";TD="0.5 day";SD="1 day";DL="0%";Cost="1.5m";TD="0.5 day";SD="1 day";DL="0%";Cost="2.0m";TD="0.5 day";SD="1 day";DL="0%";Cost="2.5m";TD="0.5 day";SD="1 day";DL="0%";Cost="1.0m";TD="1 day";SD="1 day";DL="0%";Cost="1.5m";TD="1 day";SD="1 day";DL="0%";Cost="2.0m";TD="1 day";SD="1 day";DL="0%";Cost="2.5m";TD="1 day";SD="1 day";DL="0%";Cost="0.0m";TD="0 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0 day";SD="0 day";DL="50%";
Cost="0.0m";TD="0.5 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0.5 day";SD="0 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="0 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="0 day";DL="50%";
Cost="0.0m";TD="1 day";SD="0 day";DL="50%";
Cost="0.5m";TD="1 day";SD="0 day";DL="50%";
Cost="1.0m";TD="1 day";SD="0 day";DL="50%";
Cost="1.5m";TD="1 day";SD="0 day";DL="50%";
Cost="2.0m";TD="1 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="0 day";SD="0.5 day";DL="50%";
Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="2.0m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="0.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="2.0m";TD="1 day";SD="0.5 day";DL="50%";
Cost="2.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0 day";SD="1 day";DL="50%";
Cost="1.5m";TD="0 day";SD="1 day";DL="50%";
Cost="2.0m";TD="0 day";SD="1 day";DL="50%";
Cost="2.5m";TD="0 day";SD="1 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="1 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="1 day";DL="50%";
Cost="2.0m";TD="0.5 day";SD="1 day";DL="50%";
Cost="2.5m";TD="0.5 day";SD="1 day";DL="50%";
Cost="1.5m";TD="1 day";SD="1 day";DL="50%";
Cost="2.0m";TD="1 day";SD="1 day";DL="50%";
Cost="2.5m";TD="1 day";SD="1 day";DL="50%";
Cost="0.0m";TD="0 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0 day";SD="0 day";DL="100%";
Cost="1.0m";TD="0 day";SD="0 day";DL="100%";
Cost="0.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0.5 day";SD="0 day";DL="100%";
Cost="1.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="0 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="0.5m";TD="1 day";SD="0 day";DL="100%";
Cost="1.0m";TD="1 day";SD="0 day";DL="100%";
Cost="1.5m";TD="1 day";SD="0 day";DL="100%";
Cost="2.0m";TD="1 day";SD="0 day";DL="100%";
Cost="2.5m";TD="1 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="0 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="0 day";SD="0.5 day";DL="100%";
Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="2.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="1 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="1 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="1 day";SD="0.5 day";DL="100%";
Cost="2.5m";TD="1 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0 day";SD="1 day";DL="100%";
Cost="2.0m";TD="0 day";SD="1 day";DL="100%";
Cost="2.5m";TD="0 day";SD="1 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.5m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.0m";TD="1 day";SD="1 day";DL="100%";
Cost="2.5m";TD="1 day";SD="1 day";DL="100%";

DL="0%";TD="0 day";SD="0 day";SF="Yes";
DL="50%";TD="0 day";SD="0 day";SF="Yes";
DL="100%";TD="0 day";SD="0 day";SF="Yes";
DL="0%";TD="0.5 day";SD="0 day";SF="Yes";
DL="50%";TD="0.5 day";SD="0 day";SF="Yes";
DL="100%";TD="0.5 day";SD="0 day";SF="Yes";
DL="0%";TD="1 day";SD="0 day";SF="Yes";
DL="50%";TD="1 day";SD="0 day";SF="Yes";
DL="100%";TD="1 day";SD="0 day";SF="Yes";
DL="0%";TD="0 day";SD="0.5 day";SF="Yes";
DL="50%";TD="0 day";SD="0.5 day";SF="Yes";
DL="100%";TD="0 day";SD="0.5 day";SF="Yes";
DL="0%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="50%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="100%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="0%";TD="1 day";SD="0.5 day";SF="Yes";
DL="50%";TD="1 day";SD="0.5 day";SF="Yes";
DL="100%";TD="1 day";SD="0.5 day";SF="Yes";
DL="0%";TD="0 day";SD="1 day";SF="Yes";
DL="50%";TD="0 day";SD="1 day";SF="Yes";
DL="100%";TD="0 day";SD="1 day";SF="Yes";
DL="0%";TD="0.5 day";SD="1 day";SF="Yes";
DL="50%";TD="0.5 day";SD="1 day";SF="Yes";
DL="100%";TD="0.5 day";SD="1 day";SF="Yes";
DL="0%";TD="1 day";SD="1 day";SF="Yes";
DL="50%";TD="1 day";SD="1 day";SF="Yes";
DL="100%";TD="1 day";SD="1 day";SF="Yes";
DL="0%";TD="0 day";SD="0 day";SF="No";
DL="50%";TD="0 day";SD="0 day";SF="No";DL="100%";TD="0 day";SD="0 day";SF="No";
DL="0%";TD="0.5 day";SD="0 day";SF="No";
DL="50%";TD="0.5 day";SD="0 day";SF="No";DL="100%";TD="0.5 day";SD="0 day";SF="No";
DL="0%";TD="1 day";SD="0 day";SF="No";
DL="50%";TD="1 day";SD="0 day";SF="No";DL="100%";TD="1 day";SD="0 day";SF="No";
DL="0%";TD="0 day";SD="0.5 day";SF="No";
DL="50%";TD="0 day";SD="0.5 day";SF="No";DL="100%";TD="0 day";SD="0.5 day";SF="No";
DL="0%";TD="0.5 day";SD="0.5 day";SF="No";
DL="50%";TD="0.5 day";SD="0.5 day";SF="No";DL="100%";TD="0.5 day";SD="0.5 day";SF="No";
DL="0%";TD="1 day";SD="0.5 day";SF="No";
DL="50%";TD="1 day";SD="0.5 day";SF="No";DL="100%";TD="1 day";SD="0.5 day";SF="No";
DL="0%";TD="0 day";SD="1 day";SF="No";
DL="50%";TD="0 day";SD="1 day";SF="No";DL="100%";TD="0 day";SD="1 day";SF="No";
DL="0%";TD="0.5 day";SD="1 day";SF="No";
DL="50%";TD="0.5 day";SD="1 day";SF="No";DL="100%";TD="0.5 day";SD="1 day";SF="No";
DL="0%";TD="1 day";SD="1 day";SF="No";
DL="50%";TD="1 day";SD="1 day";SF="No";DL="100%";TD="1 day";SD="1 day";SF="No";

AL:DL:TD:SF node

"Cost" "DL" "TD" "SD"Continued on next page

NETWORK RISK: PROPAGATION EXAMPLE
INITIALIZATION"PS" "SQ" "SF"0.05000
0.22500
0.20000
0.75000
0.02500
0.75000

PS="Yes";SQ="H";SF="Yes";
PS="Yes";SQ="L";SF="Yes";
PS="Yes";SQ="H";SF="No";
PS="No";SQ="H";SF="No";PS="Yes";SQ="L";SF="No";
PS="No";SQ="L";SF="No";"UPS" "SD" "SF" "HAS"
0.25000
0.25000
0.10000
0.40000
0.05000
0.45000
0.50000
0.50000
0.50000
0.50000
0.50000

UPS="Yes";SD="0 day";HAS="Yes";SF="Yes";
UPS="Yes";SD="0.5 day";HAS="Yes";SF="Yes";
UPS="No";SD="0.5 day";HAS="Yes";SF="Yes";
UPS="No";SD="1 day";HAS="Yes";SF="Yes";
UPS="Yes";SD="0.5 day";HAS="No";SF="Yes";
UPS="Yes";SD="1 day";HAS="No";SF="Yes";
UPS="No";SD="1 day";HAS="No";SF="Yes";
UPS="Yes";SD="0 day";HAS="Yes";SF="No";
UPS="No";SD="0 day";HAS="Yes";SF="No";
UPS="Yes";SD="0 day";HAS="No";SF="No";
UPS="No";SD="0 day";HAS="No";SF="No";69

APPENDIX IINETWORK RISK: PROPAGATION EXAMPLE
INITIALIZATION (Continued)
DL:TD:SD:SF node
From previous page

"AF" "DL" "TD" "SF"0.10000
0.10000
0.20000
0.10000
0.10000
0.20000
0.10000
0.10000
0.20000
0.10000
0.20000
0.10000
0.20000
0.10000
0.20000
0.90000
0.80000
0.60000
0.90000
0.80000
0.60000
0.90000
0.80000
0.60000
0.50000
0.70000
1.00000
0.50000
0.70000
1.00000
0.50000
0.70000
1.00000
0.50000
0.30000
0.50000
0.30000
0.50000
0.30000

AF="App Corr";TD="0 day";DL="0%";SF="Yes";
AF="Lockup";TD="0 day";DL="0%";SF="Yes";
AF="OK";TD="0 day";DL="0%";SF="Yes";
AF="App Corr";TD="0.5 day";DL="0%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="0%";SF="Yes";
AF="OK";TD="0.5 day";DL="0%";SF="Yes";
AF="App Corr";TD="1 day";DL="0%";SF="Yes";
AF="Lockup";TD="1 day";DL="0%";SF="Yes";
AF="OK";TD="1 day";DL="0%";SF="Yes";
AF="Lockup";TD="0 day";DL="50%";SF="Yes";
AF="OK";TD="0 day";DL="50%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="50%";SF="Yes";
AF="OK";TD="0.5 day";DL="50%";SF="Yes";
AF="Lockup";TD="1 day";DL="50%";SF="Yes";
AF="OK";TD="1 day";DL="50%";SF="Yes";
AF="App Corr";TD="0 day";DL="100%";SF="Yes";
AF="Lockup";TD="0 day";DL="100%";SF="Yes";
AF="OK";TD="0 day";DL="100%";SF="Yes";
AF="App Corr";TD="0.5 day";DL="100%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="100%";SF="Yes";
AF="OK";TD="0.5 day";DL="100%";SF="Yes";
AF="App Corr";TD="1 day";DL="100%";SF="Yes";
AF="Lockup";TD="1 day";DL="100%";SF="Yes";
AF="OK";TD="1 day";DL="100%";SF="Yes";
AF="App Corr";TD="0 day";DL="0%";SF="No";
AF="Lockup";TD="0 day";DL="0%";SF="No";
AF="OK";TD="0 day";DL="0%";SF="No";
AF="App Corr";TD="0.5 day";DL="0%";SF="No";
AF="Lockup";TD="0.5 day";DL="0%";SF="No";
AF="OK";TD="0.5 day";DL="0%";SF="No";
AF="App Corr";TD="1 day";DL="0%";SF="No";
AF="Lockup";TD="1 day";DL="0%";SF="No";
AF="OK";TD="1 day";DL="0%";SF="No";
AF="App Corr";TD="0 day";DL="50%";SF="No";
AF="Lockup";TD="0 day";DL="50%";SF="No";
AF="App Corr";TD="0.5 day";DL="50%";SF="No";
AF="Lockup";TD="0.5 day";DL="50%";SF="No";
AF="App Corr";TD="1 day";DL="50%";SF="No";
AF="Lockup";TD="1 day";DL="50%";SF="No";"NF" "Hack" "TD"0.80000
0.20000
0.80000
0.20000
0.80000
0.20000
1.00000
1.00000
1.00000

NF="Yes";TD="0 day";Hack="Yes";NF="No";TD="0 day";Hack="Yes";NF="Yes";TD="0.5 day";Hack="Yes";NF="No";TD="0.5 day";Hack="Yes";NF="Yes";TD="1 day";Hack="Yes";NF="No";TD="1 day";Hack="Yes";NF="No";TD="0 day";Hack="No";NF="No";TD="0.5 day";Hack="No";NF="No";TD="1 day";Hack="No";"V" "Hack" "AF" "FAC" "EUM"
0.09000
0.54000
0.07000
0.54000
0.01000
0.36000
0.03000
0.18000
0.18000
0.63000
0.18000
0.49000
0.18000
0.07000
0.12000
0.21000
0.06000
0.06000
0.04000
0.09000
0.05000
0.03000
0.27000
0.03000
0.54000
0.05000
0.90000
0.28000
0.03000
0.35000
0.21000
0.09000
0.21000
0.18000
0.35000
0.30000

V="Yes";EUM="Yes";AF="App Corr";FAC="High";Hack="Yes";
V="No";EUM="Yes";AF="App Corr";FAC="High";Hack="Yes";
V="Yes";EUM="No";AF="App Corr";FAC="High";Hack="Yes";
V="No";EUM="No";AF="App Corr";FAC="High";Hack="Yes";
V="Yes";EUM="Yes";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="Yes";AF="Lockup";FAC="High";Hack="Yes";
V="Yes";EUM="No";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="No";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="No";AF="OK";FAC="High";Hack="Yes";
V="Yes";EUM="Yes";AF="App Corr";FAC="Low";Hack="Yes";
V="No";EUM="Yes";AF="App Corr";FAC="Low";Hack="Yes";
V="Yes";EUM="No";AF="App Corr";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="App Corr";FAC="Low";Hack="Yes";
V="Yes";EUM="Yes";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="Yes";AF="Lockup";FAC="Low";Hack="Yes";
V="Yes";EUM="No";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="OK";FAC="Low";Hack="Yes";
V="Yes";EUM="Yes";AF="App Corr";FAC="High";Hack="No";
V="No";EUM="Yes";AF="App Corr";FAC="High";Hack="No";
V="Yes";EUM="No";AF="App Corr";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="Lockup";FAC="High";Hack="No";
V="No";EUM="Yes";AF="Lockup";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="OK";FAC="High";Hack="No";
V="No";EUM="Yes";AF="OK";FAC="High";Hack="No";
V="Yes";EUM="No";AF="OK";FAC="High";Hack="No";
V="No";EUM="No";AF="OK";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="App Corr";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="App Corr";FAC="Low";Hack="No";
V="Yes";EUM="No";AF="App Corr";FAC="Low";Hack="No";
V="Yes";EUM="Yes";AF="Lockup";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="Lockup";FAC="Low";Hack="No";
V="Yes";EUM="Yes";AF="OK";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="OK";FAC="Low";Hack="No";
V="Yes";EUM="No";AF="OK";FAC="Low";Hack="No";
V="No";EUM="No";AF="OK";FAC="Low";Hack="No";"Hack" "AF" "TD"1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1 00000

Hack="Yes";TD="0 day";AF="App Corr";
Hack="No";TD="0 day";AF="App Corr";
Hack="Yes";TD="0.5 day";AF="App Corr";
Hack="No";TD="0.5 day";AF="App Corr";
Hack="Yes";TD="1 day";AF="App Corr";
Hack="No";TD="1 day";AF="App Corr";
Hack="Yes";TD="0 day";AF="Lockup";
Hack="No";TD="0 day";AF="Lockup";
Hack="Yes";TD="0.5 day";AF="Lockup";
Hack="No";TD="0.5 day";AF="Lockup";
Hack="Yes";TD="1 day";AF="Lockup";
Hack="No";TD="1 day";AF="Lockup";
Hack="Yes";TD="0 day";AF="OK";Hack="No";TD="0 day";AF="OK";Hack="Yes";TD="0.5 day";AF="OK";Hack="No";TD="0.5 day";AF="OK";Hack="Yes";TD="1 day";AF="OK";Hack="No";TD="1 day";AF="OK";"F" "Hack" "FAC""HAN" "TD" "NF"
0.45000
0.05000
0.05000
0.45000
0.50000
0.50000

HAN="Yes";TD="0.5 day";NF="Yes";
HAN="No";TD="0.5 day";NF="Yes";
HAN="Yes";TD="1 day";NF="Yes";
HAN="No";TD="1 day";NF="Yes";
HAN="Yes";TD="0 day";NF="No";
HAN="No";TD="0 day";NF="No";0.05000
0.15000
0.25000
0.40000
0.45000
0.35000
0.25000
0.10000

F="AP";FAC="High";Hack="Yes";
F="PF";FAC="High";Hack="Yes";
F="AP";FAC="Low";Hack="Yes";
F="PF";FAC="Low";Hack="Yes";
F="AP";FAC="High";Hack="No";F="PF";FAC="High";Hack="No";F="AP";FAC="Low";Hack="No";F="PF";FAC="Low";Hack="No";70

APPENDIX II"DL" "TD" "SD" "SF"0.03704
0.025930.011110.022220.014810.022220.014810.01852
0.011110.007410.014810.011110.007410.003700.00370
0.007410.011110.007410.007410.007410.014810.007410.007410.00370
0.007410.014810.011110.022220.014810.011110.011110.007410.007410.011110.007410.007410.007410.003700.018520.014810.003700.011110.011110.007410.007410.011110.007410.007410.007410.003700.003700.011110.014810.007410.00370
0.007410.01852
0.007410.003700.014810.018520.014810.014810.007410.007410.007410.011110.007410.003700.00370
0.007410.011110.007410.007410.014810.011110.007410.00370
0.007410.007410.011110.007410.00370
0.007410.011110.011110.007410.011110.014810.011110.007410.014810.014810.007410.029630.01860.01620.07520.00300.00100.02440.00300.00100.02440.01860.01620.07520.00300.00100.02440.00300.00100.02440.01860.01620.07520.00300.00100.02440.00300.00100.02440.09510.01490.00000.01620.01210.00000.01620.01210.00000.09510.01490.00000.01620.01210.00000.01620.01210.00000.09510.01490.00000.01620.01210.00000.01620.01210.0000

Cost="0.0m";TD="0 day";SD="0 day";DL="0%";Cost="0.0m";TD="0.5 day";SD="0 day";DL="0%";Cost="0.5m";TD="0.5 day";SD="0 day";DL="0%";Cost="0.0m";TD="1 day";SD="0 day";DL="0%";Cost="0.5m";TD="1 day";SD="0 day";DL="0%";Cost="0.5m";TD="0 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="0 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="1 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="1 day";SD="0.5 day";DL="0%";Cost="1.5m";TD="1 day";SD="0.5 day";DL="0%";Cost="2.0m";TD="1 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="0 day";SD="1 day";DL="0%";Cost="1.0m";TD="0 day";SD="1 day";DL="0%";Cost="1.5m";TD="0 day";SD="1 day";DL="0%";Cost="2.0m";TD="0 day";SD="1 day";DL="0%";Cost="2.5m";TD="0 day";SD="1 day";DL="0%";Cost="1.0m";TD="0.5 day";SD="1 day";DL="0%";Cost="1.5m";TD="0.5 day";SD="1 day";DL="0%";Cost="2.0m";TD="0.5 day";SD="1 day";DL="0%";Cost="2.5m";TD="0.5 day";SD="1 day";DL="0%";Cost="1.0m";TD="1 day";SD="1 day";DL="0%";Cost="1.5m";TD="1 day";SD="1 day";DL="0%";Cost="2.0m";TD="1 day";SD="1 day";DL="0%";Cost="2.5m";TD="1 day";SD="1 day";DL="0%";Cost="0.0m";TD="0 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0 day";SD="0 day";DL="50%";
Cost="0.0m";TD="0.5 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0.5 day";SD="0 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="0 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="0 day";DL="50%";
Cost="0.0m";TD="1 day";SD="0 day";DL="50%";
Cost="0.5m";TD="1 day";SD="0 day";DL="50%";
Cost="1.0m";TD="1 day";SD="0 day";DL="50%";
Cost="1.5m";TD="1 day";SD="0 day";DL="50%";
Cost="2.0m";TD="1 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="0 day";SD="0.5 day";DL="50%";
Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="2.0m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="0.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="2.0m";TD="1 day";SD="0.5 day";DL="50%";
Cost="2.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0 day";SD="1 day";DL="50%";
Cost="1.5m";TD="0 day";SD="1 day";DL="50%";
Cost="2.0m";TD="0 day";SD="1 day";DL="50%";
Cost="2.5m";TD="0 day";SD="1 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="1 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="1 day";DL="50%";
Cost="2.0m";TD="0.5 day";SD="1 day";DL="50%";
Cost="2.5m";TD="0.5 day";SD="1 day";DL="50%";
Cost="1.5m";TD="1 day";SD="1 day";DL="50%";
Cost="2.0m";TD="1 day";SD="1 day";DL="50%";
Cost="2.5m";TD="1 day";SD="1 day";DL="50%";
Cost="0.0m";TD="0 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0 day";SD="0 day";DL="100%";
Cost="1.0m";TD="0 day";SD="0 day";DL="100%";
Cost="0.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0.5 day";SD="0 day";DL="100%";
Cost="1.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="0 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="0.5m";TD="1 day";SD="0 day";DL="100%";
Cost="1.0m";TD="1 day";SD="0 day";DL="100%";
Cost="1.5m";TD="1 day";SD="0 day";DL="100%";
Cost="2.0m";TD="1 day";SD="0 day";DL="100%";
Cost="2.5m";TD="1 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="0 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="0 day";SD="0.5 day";DL="100%";
Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="2.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="1 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="1 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="1 day";SD="0.5 day";DL="100%";
Cost="2.5m";TD="1 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0 day";SD="1 day";DL="100%";
Cost="2.0m";TD="0 day";SD="1 day";DL="100%";
Cost="2.5m";TD="0 day";SD="1 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.5m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.0m";TD="1 day";SD="1 day";DL="100%";
Cost="2.5m";TD="1 day";SD="1 day";DL="100%";

DL="0%";TD="0 day";SD="0 day";SF="Yes";
DL="50%";TD="0 day";SD="0 day";SF="Yes";
DL="100%";TD="0 day";SD="0 day";SF="Yes";
DL="0%";TD="0.5 day";SD="0 day";SF="Yes";
DL="50%";TD="0.5 day";SD="0 day";SF="Yes";
DL="100%";TD="0.5 day";SD="0 day";SF="Yes";
DL="0%";TD="1 day";SD="0 day";SF="Yes";
DL="50%";TD="1 day";SD="0 day";SF="Yes";
DL="100%";TD="1 day";SD="0 day";SF="Yes";
DL="0%";TD="0 day";SD="0.5 day";SF="Yes";
DL="50%";TD="0 day";SD="0.5 day";SF="Yes";
DL="100%";TD="0 day";SD="0.5 day";SF="Yes";
DL="0%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="50%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="100%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="0%";TD="1 day";SD="0.5 day";SF="Yes";
DL="50%";TD="1 day";SD="0.5 day";SF="Yes";
DL="100%";TD="1 day";SD="0.5 day";SF="Yes";
DL="0%";TD="0 day";SD="1 day";SF="Yes";
DL="50%";TD="0 day";SD="1 day";SF="Yes";
DL="100%";TD="0 day";SD="1 day";SF="Yes";
DL="0%";TD="0.5 day";SD="1 day";SF="Yes";
DL="50%";TD="0.5 day";SD="1 day";SF="Yes";
DL="100%";TD="0.5 day";SD="1 day";SF="Yes";
DL="0%";TD="1 day";SD="1 day";SF="Yes";
DL="50%";TD="1 day";SD="1 day";SF="Yes";
DL="100%";TD="1 day";SD="1 day";SF="Yes";
DL="0%";TD="0 day";SD="0 day";SF="No";
DL="50%";TD="0 day";SD="0 day";SF="No";DL="100%";TD="0 day";SD="0 day";SF="No";
DL="0%";TD="0.5 day";SD="0 day";SF="No";
DL="50%";TD="0.5 day";SD="0 day";SF="No";DL="100%";TD="0.5 day";SD="0 day";SF="No";
DL="0%";TD="1 day";SD="0 day";SF="No";
DL="50%";TD="1 day";SD="0 day";SF="No";DL="100%";TD="1 day";SD="0 day";SF="No";
DL="0%";TD="0 day";SD="0.5 day";SF="No";
DL="50%";TD="0 day";SD="0.5 day";SF="No";DL="100%";TD="0 day";SD="0.5 day";SF="No";
DL="0%";TD="0.5 day";SD="0.5 day";SF="No";
DL="50%";TD="0.5 day";SD="0.5 day";SF="No";DL="100%";TD="0.5 day";SD="0.5 day";SF="No";
DL="0%";TD="1 day";SD="0.5 day";SF="No";
DL="50%";TD="1 day";SD="0.5 day";SF="No";DL="100%";TD="1 day";SD="0.5 day";SF="No";
DL="0%";TD="0 day";SD="1 day";SF="No";
DL="50%";TD="0 day";SD="1 day";SF="No";DL="100%";TD="0 day";SD="1 day";SF="No";
DL="0%";TD="0.5 day";SD="1 day";SF="No";
DL="50%";TD="0.5 day";SD="1 day";SF="No";DL="100%";TD="0.5 day";SD="1 day";SF="No";
DL="0%";TD="1 day";SD="1 day";SF="No";
DL="50%";TD="1 day";SD="1 day";SF="No";DL="100%";TD="1 day";SD="1 day";SF="No";

AL:DL:TD:SF node

"Cost" "DL" "TD" "SD""PS" "SQ" "SF"0.05000
0.22500
0.20000
0.75000
0.02500
0.75000

PS="Yes";SQ="H";SF="Yes";
PS="Yes";SQ="L";SF="Yes";
PS="Yes";SQ="H";SF="No";
PS="No";SQ="H";SF="No";PS="Yes";SQ="L";SF="No";
PS="No";SQ="L";SF="No";"UPS" "SD" "SF" "HAS"
0.06250
0.06250
0.02500
0.10000
0.01250
0.11250
0.12500
0.12500
0.12500
0.12500
0.12500

UPS="Yes";SD="0 day";HAS="Yes";SF="Yes";
UPS="Yes";SD="0.5 day";HAS="Yes";SF="Yes";
UPS="No";SD="0.5 day";HAS="Yes";SF="Yes";
UPS="No";SD="1 day";HAS="Yes";SF="Yes";
UPS="Yes";SD="0.5 day";HAS="No";SF="Yes";
UPS="Yes";SD="1 day";HAS="No";SF="Yes";
UPS="No";SD="1 day";HAS="No";SF="Yes";
UPS="Yes";SD="0 day";HAS="Yes";SF="No";
UPS="No";SD="0 day";HAS="Yes";SF="No";
UPS="Yes";SD="0 day";HAS="No";SF="No";
UPS="No";SD="0 day";HAS="No";SF="No";71

Continued on next page

NETWORK RISK: PROPAGATION EXAMPLE
COLLECT PHASEAPPENDIX IINETWORK RISK: PROPAGATION EXAMPLE
COLLECT PHASE (Continued)
DL:TD:SD:SF node
From previous page

"AF" "DL" "TD" "SF"0.007070.003120.04561
0.006120.001890.000960.006120.001890.000960.003120.045610.001890.000960.001890.000960.06367
0.02498
0.136810.055120.015160.00288
0.055120.015160.002880.03537
0.02186
0.228020.030620.01326
0.004800.030620.01326
0.00480
0.03537
0.009370.030620.005680.030620.00568

AF="App Corr";TD="0 day";DL="0%";SF="Yes";
AF="Lockup";TD="0 day";DL="0%";SF="Yes";
AF="OK";TD="0 day";DL="0%";SF="Yes";
AF="App Corr";TD="0.5 day";DL="0%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="0%";SF="Yes";
AF="OK";TD="0.5 day";DL="0%";SF="Yes";
AF="App Corr";TD="1 day";DL="0%";SF="Yes";
AF="Lockup";TD="1 day";DL="0%";SF="Yes";
AF="OK";TD="1 day";DL="0%";SF="Yes";
AF="Lockup";TD="0 day";DL="50%";SF="Yes";
AF="OK";TD="0 day";DL="50%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="50%";SF="Yes";
AF="OK";TD="0.5 day";DL="50%";SF="Yes";
AF="Lockup";TD="1 day";DL="50%";SF="Yes";
AF="OK";TD="1 day";DL="50%";SF="Yes";
AF="App Corr";TD="0 day";DL="100%";SF="Yes";
AF="Lockup";TD="0 day";DL="100%";SF="Yes";
AF="OK";TD="0 day";DL="100%";SF="Yes";
AF="App Corr";TD="0.5 day";DL="100%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="100%";SF="Yes";
AF="OK";TD="0.5 day";DL="100%";SF="Yes";
AF="App Corr";TD="1 day";DL="100%";SF="Yes";
AF="Lockup";TD="1 day";DL="100%";SF="Yes";
AF="OK";TD="1 day";DL="100%";SF="Yes";
AF="App Corr";TD="0 day";DL="0%";SF="No";
AF="Lockup";TD="0 day";DL="0%";SF="No";
AF="OK";TD="0 day";DL="0%";SF="No";
AF="App Corr";TD="0.5 day";DL="0%";SF="No";
AF="Lockup";TD="0.5 day";DL="0%";SF="No";
AF="OK";TD="0.5 day";DL="0%";SF="No";
AF="App Corr";TD="1 day";DL="0%";SF="No";
AF="Lockup";TD="1 day";DL="0%";SF="No";
AF="OK";TD="1 day";DL="0%";SF="No";
AF="App Corr";TD="0 day";DL="50%";SF="No";
AF="Lockup";TD="0 day";DL="50%";SF="No";
AF="App Corr";TD="0.5 day";DL="50%";SF="No";
AF="Lockup";TD="0.5 day";DL="50%";SF="No";
AF="App Corr";TD="1 day";DL="50%";SF="No";
AF="Lockup";TD="1 day";DL="50%";SF="No";"NF" "Hack" "TD"0.00000.10000.20000.00000.20000.00000.50000.00000.0000

NF="Yes";TD="0 day";Hack="Yes";NF="No";TD="0 day";Hack="Yes";NF="Yes";TD="0.5 day";Hack="Yes";NF="No";TD="0.5 day";Hack="Yes";NF="Yes";TD="1 day";Hack="Yes";NF="No";TD="1 day";Hack="Yes";NF="No";TD="0 day";Hack="No";NF="No";TD="0.5 day";Hack="No";NF="No";TD="1 day";Hack="No";"V" "Hack" "AF" "FAC" "EUM"
0.01125
0.06750
0.00875
0.067500.00125
0.04500
0.00375
0.02250
0.02250
0.07875
0.02250
0.06125
0.02250
0.00875
0.01500
0.02625
0.00750
0.00750
0.00500
0.01125
0.00625
0.00375
0.03375
0.00375
0.06750
0.006250.11250
0.03500
0.00375
0.04375
0.02625
0.01125
0.02625
0.02250
0.04375
0.03750

V="Yes";EUM="Yes";AF="App Corr";FAC="High";Hack="Yes";
V="No";EUM="Yes";AF="App Corr";FAC="High";Hack="Yes";
V="Yes";EUM="No";AF="App Corr";FAC="High";Hack="Yes";
V="No";EUM="No";AF="App Corr";FAC="High";Hack="Yes";
V="Yes";EUM="Yes";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="Yes";AF="Lockup";FAC="High";Hack="Yes";
V="Yes";EUM="No";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="No";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="No";AF="OK";FAC="High";Hack="Yes";
V="Yes";EUM="Yes";AF="App Corr";FAC="Low";Hack="Yes";
V="No";EUM="Yes";AF="App Corr";FAC="Low";Hack="Yes";
V="Yes";EUM="No";AF="App Corr";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="App Corr";FAC="Low";Hack="Yes";
V="Yes";EUM="Yes";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="Yes";AF="Lockup";FAC="Low";Hack="Yes";
V="Yes";EUM="No";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="OK";FAC="Low";Hack="Yes";
V="Yes";EUM="Yes";AF="App Corr";FAC="High";Hack="No";
V="No";EUM="Yes";AF="App Corr";FAC="High";Hack="No";
V="Yes";EUM="No";AF="App Corr";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="Lockup";FAC="High";Hack="No";
V="No";EUM="Yes";AF="Lockup";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="OK";FAC="High";Hack="No";
V="No";EUM="Yes";AF="OK";FAC="High";Hack="No";
V="Yes";EUM="No";AF="OK";FAC="High";Hack="No";
V="No";EUM="No";AF="OK";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="App Corr";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="App Corr";FAC="Low";Hack="No";
V="Yes";EUM="No";AF="App Corr";FAC="Low";Hack="No";
V="Yes";EUM="Yes";AF="Lockup";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="Lockup";FAC="Low";Hack="No";
V="Yes";EUM="Yes";AF="OK";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="OK";FAC="Low";Hack="No";
V="Yes";EUM="No";AF="OK";FAC="Low";Hack="No";
V="No";EUM="No";AF="OK";FAC="Low";Hack="No";"Hack" "AF" "TD"0.06130.08030.12250.00000.12250.00000.01900.04350.03790.00000.03790.00000.00480.45130.00960.00000.00960 0000Hack="Yes";TD="0 day";AF="App Corr";
Hack="No";TD="0 day";AF="App Corr";
Hack="Yes";TD="0.5 day";AF="App Corr";
Hack="No";TD="0.5 day";AF="App Corr";
Hack="Yes";TD="1 day";AF="App Corr";
Hack="No";TD="1 day";AF="App Corr";
Hack="Yes";TD="0 day";AF="Lockup";
Hack="No";TD="0 day";AF="Lockup";
Hack="Yes";TD="0.5 day";AF="Lockup";
Hack="No";TD="0.5 day";AF="Lockup";
Hack="Yes";TD="1 day";AF="Lockup";
Hack="No";TD="1 day";AF="Lockup";
Hack="Yes";TD="0 day";AF="OK";Hack="No";TD="0 day";AF="OK";Hack="Yes";TD="0.5 day";AF="OK";Hack="No";TD="0.5 day";AF="OK";Hack="Yes";TD="1 day";AF="OK";Hack="No";TD="1 day";AF="OK";"F" "Hack" "FAC""HAN" "TD" "NF"
0.22500
0.02500
0.02500
0.22500
0.25000
0.25000

HAN="Yes";TD="0.5 day";NF="Yes";
HAN="No";TD="0.5 day";NF="Yes";
HAN="Yes";TD="1 day";NF="Yes";
HAN="No";TD="1 day";NF="Yes";
HAN="Yes";TD="0 day";NF="No";
HAN="No";TD="0 day";NF="No";0.02500
0.07500
0.12500
0.20000
0.22500
0.17500
0.12500
0.05000

F="AP";FAC="High";Hack="Yes";
F="PF";FAC="High";Hack="Yes";
F="AP";FAC="Low";Hack="Yes";
F="PF";FAC="Low";Hack="Yes";
F="AP";FAC="High";Hack="No";F="PF";FAC="High";Hack="No";F="AP";FAC="Low";Hack="No";F="PF";FAC="Low";Hack="No";72

APPENDIX II"DL" "TD" "SD" "SF"

0.49399
0.05901
0.025290.05058
0.033720.001840.001230.000250.000150.000100.000200.000150.000100.00005
0.00104
0.002070.003110.002070.002070.00033
0.000670.00033
0.00033
0.000170.00033
0.000670.000500.047310.031540.01882
0.01882
0.01255
0.01255
0.01882
0.01255
0.01255
0.01255
0.006270.001340.001070.000270.00005
0.00005
0.000030.000030.00005
0.000030.000030.000030.000020.000900.00271
0.003620.001810.00005
0.000110.000260.000110.00005
0.000210.00026
0.003100.003100.00155
0.000500.000500.000750.000500.000250.000250.000500.000750.000500.00050
0.004960.00372
0.002480.001240.000800.000800.001210.000800.000400.000800.001210.001210.00080
0.01256
0.01674
0.01256
0.00272
0.005430.005430.00272
0.01087

0.001920.00168
0.007750.000310.000100.002520.000310.000100.00252
0.003070.00268
0.012400.000490.000160.004020.000490.000160.004020.010360.00905
0.041850.001670.00053
0.01358
0.001670.00053
0.01358
0.492070.077180.00000
0.08399
0.062630.00000
0.08399
0.062630.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000

Cost="0.0m";TD="0 day";SD="0 day";DL="0%";Cost="0.0m";TD="0.5 day";SD="0 day";DL="0%";Cost="0.5m";TD="0.5 day";SD="0 day";DL="0%";Cost="0.0m";TD="1 day";SD="0 day";DL="0%";Cost="0.5m";TD="1 day";SD="0 day";DL="0%";Cost="0.5m";TD="0 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="0 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="1 day";SD="0.5 day";DL="0%";Cost="1.0m";TD="1 day";SD="0.5 day";DL="0%";Cost="1.5m";TD="1 day";SD="0.5 day";DL="0%";Cost="2.0m";TD="1 day";SD="0.5 day";DL="0%";Cost="0.5m";TD="0 day";SD="1 day";DL="0%";Cost="1.0m";TD="0 day";SD="1 day";DL="0%";Cost="1.5m";TD="0 day";SD="1 day";DL="0%";Cost="2.0m";TD="0 day";SD="1 day";DL="0%";Cost="2.5m";TD="0 day";SD="1 day";DL="0%";Cost="1.0m";TD="0.5 day";SD="1 day";DL="0%";Cost="1.5m";TD="0.5 day";SD="1 day";DL="0%";Cost="2.0m";TD="0.5 day";SD="1 day";DL="0%";Cost="2.5m";TD="0.5 day";SD="1 day";DL="0%";Cost="1.0m";TD="1 day";SD="1 day";DL="0%";Cost="1.5m";TD="1 day";SD="1 day";DL="0%";Cost="2.0m";TD="1 day";SD="1 day";DL="0%";Cost="2.5m";TD="1 day";SD="1 day";DL="0%";Cost="0.0m";TD="0 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0 day";SD="0 day";DL="50%";
Cost="0.0m";TD="0.5 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0.5 day";SD="0 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="0 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="0 day";DL="50%";
Cost="0.0m";TD="1 day";SD="0 day";DL="50%";
Cost="0.5m";TD="1 day";SD="0 day";DL="50%";
Cost="1.0m";TD="1 day";SD="0 day";DL="50%";
Cost="1.5m";TD="1 day";SD="0 day";DL="50%";
Cost="2.0m";TD="1 day";SD="0 day";DL="50%";
Cost="0.5m";TD="0 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="0 day";SD="0.5 day";DL="50%";
Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="2.0m";TD="0.5 day";SD="0.5 day";DL="50%";
Cost="0.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="2.0m";TD="1 day";SD="0.5 day";DL="50%";
Cost="2.5m";TD="1 day";SD="0.5 day";DL="50%";
Cost="1.0m";TD="0 day";SD="1 day";DL="50%";
Cost="1.5m";TD="0 day";SD="1 day";DL="50%";
Cost="2.0m";TD="0 day";SD="1 day";DL="50%";
Cost="2.5m";TD="0 day";SD="1 day";DL="50%";
Cost="1.0m";TD="0.5 day";SD="1 day";DL="50%";
Cost="1.5m";TD="0.5 day";SD="1 day";DL="50%";
Cost="2.0m";TD="0.5 day";SD="1 day";DL="50%";
Cost="2.5m";TD="0.5 day";SD="1 day";DL="50%";
Cost="1.5m";TD="1 day";SD="1 day";DL="50%";
Cost="2.0m";TD="1 day";SD="1 day";DL="50%";
Cost="2.5m";TD="1 day";SD="1 day";DL="50%";
Cost="0.0m";TD="0 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0 day";SD="0 day";DL="100%";
Cost="1.0m";TD="0 day";SD="0 day";DL="100%";
Cost="0.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0.5 day";SD="0 day";DL="100%";
Cost="1.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="0 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="0 day";DL="100%";
Cost="0.5m";TD="1 day";SD="0 day";DL="100%";
Cost="1.0m";TD="1 day";SD="0 day";DL="100%";
Cost="1.5m";TD="1 day";SD="0 day";DL="100%";
Cost="2.0m";TD="1 day";SD="0 day";DL="100%";
Cost="2.5m";TD="1 day";SD="0 day";DL="100%";
Cost="0.5m";TD="0 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="0 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="0 day";SD="0.5 day";DL="100%";
Cost="0.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="2.5m";TD="0.5 day";SD="0.5 day";DL="100%";
Cost="1.0m";TD="1 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="1 day";SD="0.5 day";DL="100%";
Cost="2.0m";TD="1 day";SD="0.5 day";DL="100%";
Cost="2.5m";TD="1 day";SD="0.5 day";DL="100%";
Cost="1.5m";TD="0 day";SD="1 day";DL="100%";
Cost="2.0m";TD="0 day";SD="1 day";DL="100%";
Cost="2.5m";TD="0 day";SD="1 day";DL="100%";
Cost="1.5m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.0m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.5m";TD="0.5 day";SD="1 day";DL="100%";
Cost="2.0m";TD="1 day";SD="1 day";DL="100%";
Cost="2.5m";TD="1 day";SD="1 day";DL="100%";

DL="0%";TD="0 day";SD="0 day";SF="Yes";
DL="50%";TD="0 day";SD="0 day";SF="Yes";
DL="100%";TD="0 day";SD="0 day";SF="Yes";
DL="0%";TD="0.5 day";SD="0 day";SF="Yes";
DL="50%";TD="0.5 day";SD="0 day";SF="Yes";
DL="100%";TD="0.5 day";SD="0 day";SF="Yes";
DL="0%";TD="1 day";SD="0 day";SF="Yes";
DL="50%";TD="1 day";SD="0 day";SF="Yes";
DL="100%";TD="1 day";SD="0 day";SF="Yes";
DL="0%";TD="0 day";SD="0.5 day";SF="Yes";
DL="50%";TD="0 day";SD="0.5 day";SF="Yes";
DL="100%";TD="0 day";SD="0.5 day";SF="Yes";
DL="0%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="50%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="100%";TD="0.5 day";SD="0.5 day";SF="Yes";
DL="0%";TD="1 day";SD="0.5 day";SF="Yes";
DL="50%";TD="1 day";SD="0.5 day";SF="Yes";
DL="100%";TD="1 day";SD="0.5 day";SF="Yes";
DL="0%";TD="0 day";SD="1 day";SF="Yes";
DL="50%";TD="0 day";SD="1 day";SF="Yes";
DL="100%";TD="0 day";SD="1 day";SF="Yes";
DL="0%";TD="0.5 day";SD="1 day";SF="Yes";
DL="50%";TD="0.5 day";SD="1 day";SF="Yes";
DL="100%";TD="0.5 day";SD="1 day";SF="Yes";
DL="0%";TD="1 day";SD="1 day";SF="Yes";
DL="50%";TD="1 day";SD="1 day";SF="Yes";
DL="100%";TD="1 day";SD="1 day";SF="Yes";
DL="0%";TD="0 day";SD="0 day";SF="No";
DL="50%";TD="0 day";SD="0 day";SF="No";DL="100%";TD="0 day";SD="0 day";SF="No";
DL="0%";TD="0.5 day";SD="0 day";SF="No";
DL="50%";TD="0.5 day";SD="0 day";SF="No";DL="100%";TD="0.5 day";SD="0 day";SF="No";
DL="0%";TD="1 day";SD="0 day";SF="No";
DL="50%";TD="1 day";SD="0 day";SF="No";DL="100%";TD="1 day";SD="0 day";SF="No";
DL="0%";TD="0 day";SD="0.5 day";SF="No";
DL="50%";TD="0 day";SD="0.5 day";SF="No";DL="100%";TD="0 day";SD="0.5 day";SF="No";
DL="0%";TD="0.5 day";SD="0.5 day";SF="No";
DL="50%";TD="0.5 day";SD="0.5 day";SF="No";DL="100%";TD="0.5 day";SD="0.5 day";SF="No";
DL="0%";TD="1 day";SD="0.5 day";SF="No";
DL="50%";TD="1 day";SD="0.5 day";SF="No";DL="100%";TD="1 day";SD="0.5 day";SF="No";
DL="0%";TD="0 day";SD="1 day";SF="No";
DL="50%";TD="0 day";SD="1 day";SF="No";DL="100%";TD="0 day";SD="1 day";SF="No";
DL="0%";TD="0.5 day";SD="1 day";SF="No";
DL="50%";TD="0.5 day";SD="1 day";SF="No";DL="100%";TD="0.5 day";SD="1 day";SF="No";
DL="0%";TD="1 day";SD="1 day";SF="No";
DL="50%";TD="1 day";SD="1 day";SF="No";DL="100%";TD="1 day";SD="1 day";SF="No";

AL:DL:TD:SF node

"Cost" "DL" "TD" "SD""PS" "SQ" "SF"0.02500.11250.10000.37500.01250.3750

PS="Yes";SQ="H";SF="Yes";
PS="Yes";SQ="L";SF="Yes";
PS="Yes";SQ="H";SF="No";
PS="No";SQ="H";SF="No";PS="Yes";SQ="L";SF="No";
PS="No";SQ="L";SF="No";"UPS" "SD" "SF" "HAS"
0.01720.01720.00690.02750.00340.03090.03440.21560.21560.21560.2156

UPS="Yes";SD="0 day";HAS="Yes";SF="Yes";
UPS="Yes";SD="0.5 day";HAS="Yes";SF="Yes";
UPS="No";SD="0.5 day";HAS="Yes";SF="Yes";
UPS="No";SD="1 day";HAS="Yes";SF="Yes";
UPS="Yes";SD="0.5 day";HAS="No";SF="Yes";
UPS="Yes";SD="1 day";HAS="No";SF="Yes";
UPS="No";SD="1 day";HAS="No";SF="Yes";
UPS="Yes";SD="0 day";HAS="Yes";SF="No";
UPS="No";SD="0 day";HAS="Yes";SF="No";
UPS="Yes";SD="0 day";HAS="No";SF="No";
UPS="No";SD="0 day";HAS="No";SF="No";73

Continued on next page

NETWORK RISK: PROPAGATION EXAMPLE
DISTRIBUTE PHASEAPPENDIX IINETWORK RISK: PROPAGATION EXAMPLE
DISTRIBUTE PHASE (Continued)
DL:TD:SD:SF node
From previous page

"AF" "DL" "TD" "SF"0.00195
0.00086
0.012540.001680.000520.000260.001680.000520.000260.00086
0.012540.000520.000260.000520.00026
0.01751
0.00687
0.037620.015160.004170.00079
0.015160.004170.00079
0.061020.03770
0.39334
0.05283
0.022880.00828
0.05283
0.022880.008280.06102
0.016160.052830.009810.052830.00981

AF="App Corr";TD="0 day";DL="0%";SF="Yes";
AF="Lockup";TD="0 day";DL="0%";SF="Yes";
AF="OK";TD="0 day";DL="0%";SF="Yes";
AF="App Corr";TD="0.5 day";DL="0%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="0%";SF="Yes";
AF="OK";TD="0.5 day";DL="0%";SF="Yes";
AF="App Corr";TD="1 day";DL="0%";SF="Yes";
AF="Lockup";TD="1 day";DL="0%";SF="Yes";
AF="OK";TD="1 day";DL="0%";SF="Yes";
AF="Lockup";TD="0 day";DL="50%";SF="Yes";
AF="OK";TD="0 day";DL="50%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="50%";SF="Yes";
AF="OK";TD="0.5 day";DL="50%";SF="Yes";
AF="Lockup";TD="1 day";DL="50%";SF="Yes";
AF="OK";TD="1 day";DL="50%";SF="Yes";
AF="App Corr";TD="0 day";DL="100%";SF="Yes";
AF="Lockup";TD="0 day";DL="100%";SF="Yes";
AF="OK";TD="0 day";DL="100%";SF="Yes";
AF="App Corr";TD="0.5 day";DL="100%";SF="Yes";
AF="Lockup";TD="0.5 day";DL="100%";SF="Yes";
AF="OK";TD="0.5 day";DL="100%";SF="Yes";
AF="App Corr";TD="1 day";DL="100%";SF="Yes";
AF="Lockup";TD="1 day";DL="100%";SF="Yes";
AF="OK";TD="1 day";DL="100%";SF="Yes";
AF="App Corr";TD="0 day";DL="0%";SF="No";
AF="Lockup";TD="0 day";DL="0%";SF="No";
AF="OK";TD="0 day";DL="0%";SF="No";
AF="App Corr";TD="0.5 day";DL="0%";SF="No";
AF="Lockup";TD="0.5 day";DL="0%";SF="No";
AF="OK";TD="0.5 day";DL="0%";SF="No";
AF="App Corr";TD="1 day";DL="0%";SF="No";
AF="Lockup";TD="1 day";DL="0%";SF="No";
AF="OK";TD="1 day";DL="0%";SF="No";
AF="App Corr";TD="0 day";DL="50%";SF="No";
AF="Lockup";TD="0 day";DL="50%";SF="No";
AF="App Corr";TD="0.5 day";DL="50%";SF="No";
AF="Lockup";TD="0.5 day";DL="50%";SF="No";
AF="App Corr";TD="1 day";DL="50%";SF="No";
AF="Lockup";TD="1 day";DL="50%";SF="No";"NF" "Hack" "TD"0.00000.08500.17000.00000.17000.00000.57500.00000.0000

NF="Yes";TD="0 day";Hack="Yes";NF="No";TD="0 day";Hack="Yes";NF="Yes";TD="0.5 day";Hack="Yes";NF="No";TD="0.5 day";Hack="Yes";NF="Yes";TD="1 day";Hack="Yes";NF="No";TD="1 day";Hack="Yes";NF="No";TD="0 day";Hack="No";NF="No";TD="0.5 day";Hack="No";NF="No";TD="1 day";Hack="No";"V" "Hack" "AF" "FAC" "EUM"
0.000900.005400.00630
0.04860
0.000100.00360
0.002700.016200.016200.122850.035100.063700.023400.01365
0.023400.02730
0.00780
0.007800.00160
0.00360
0.01800
0.001200.010800.001200.02160
0.01800
0.32400
0.029400.003150.02450
0.02205
0.009450.02205
0.018900.02450
0.02100

V="Yes";EUM="Yes";AF="App corr";FAC="High";Hack="Yes";
V="No";EUM="Yes";AF="App Corr";FAC="High";Hack="Yes";
V="Yes";EUM="No";AF="App Corr";FAC="High";Hack="Yes";
V="No";EUM="No";AF="App Corr";FAC="High";Hack="Yes";
V="Yes";EUM="Yes";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="Yes";AF="Lockup";FAC="High";Hack="Yes";
V="Yes";EUM="No";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="No";AF="Lockup";FAC="High";Hack="Yes";
V="No";EUM="No";AF="OK";FAC="High";Hack="Yes";
V="Yes";EUM="Yes";AF="App Corr";FAC="Low";Hack="Yes";
V="No";EUM="Yes";AF="App Corr";FAC="Low";Hack="Yes";
V="Yes";EUM="No";AF="App Corr";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="App Corr";FAC="Low";Hack="Yes";
V="Yes";EUM="Yes";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="Yes";AF="Lockup";FAC="Low";Hack="Yes";
V="Yes";EUM="No";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="Lockup";FAC="Low";Hack="Yes";
V="No";EUM="No";AF="OK";FAC="Low";Hack="Yes";
V="Yes";EUM="Yes";AF="App Corr";FAC="High";Hack="No";
V="No";EUM="Yes";AF="App Corr";FAC="High";Hack="No";
V="Yes";EUM="No";AF="App Corr";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="Lockup";FAC="High";Hack="No";
V="No";EUM="Yes";AF="Lockup";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="OK";FAC="High";Hack="No";
V="No";EUM="Yes";AF="OK";FAC="High";Hack="No";
V="Yes";EUM="No";AF="OK";FAC="High";Hack="No";
V="No";EUM="No";AF="OK";FAC="High";Hack="No";
V="Yes";EUM="Yes";AF="App Corr";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="App Corr";FAC="Low";Hack="No";
V="Yes";EUM="No";AF="App Corr";FAC="Low";Hack="No";
V="Yes";EUM="Yes";AF="Lockup";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="Lockup";FAC="Low";Hack="No";
V="Yes";EUM="Yes";AF="OK";FAC="Low";Hack="No";
V="No";EUM="Yes";AF="OK";FAC="Low";Hack="No";
V="Yes";EUM="No";AF="OK";FAC="Low";Hack="No";
V="No";EUM="No";AF="OK";FAC="Low";Hack="No";"Hack" "AF" "TD"0.06130.08030.12250.00000.12250.00000.01900.04350.03790.00000.03790.00000.00480.45130.00960.00000.00960 0000Hack="Yes";TD="0 day";AF="App Corr";
Hack="No";TD="0 day";AF="App Corr";
Hack="Yes";TD="0.5 day";AF="App Corr";
Hack="No";TD="0.5 day";AF="App Corr";
Hack="Yes";TD="1 day";AF="App Corr";
Hack="No";TD="1 day";AF="App Corr";
Hack="Yes";TD="0 day";AF="Lockup";
Hack="No";TD="0 day";AF="Lockup";
Hack="Yes";TD="0.5 day";AF="Lockup";
Hack="No";TD="0.5 day";AF="Lockup";
Hack="Yes";TD="1 day";AF="Lockup";
Hack="No";TD="1 day";AF="Lockup";
Hack="Yes";TD="0 day";AF="OK";Hack="No";TD="0 day";AF="OK";Hack="Yes";TD="0.5 day";AF="OK";Hack="No";TD="0.5 day";AF="OK";Hack="Yes";TD="1 day";AF="OK";Hack="No";TD="1 day";AF="OK";"F" "Hack" "FAC""HAN" "TD" "NF"
0.15300.01700.01700.15300.33000.3300

HAN="Yes";TD="0.5 day";NF="Yes";
HAN="No";TD="0.5 day";NF="Yes";
HAN="Yes";TD="1 day";NF="Yes";
HAN="No";TD="1 day";NF="Yes";
HAN="Yes";TD="0 day";NF="No";
HAN="No";TD="0 day";NF="No";0.02500.07500.12500.20000.22500.17500.12500.0500F="AP";FAC="High";Hack="Yes";
F="PF";FAC="High";Hack="Yes";
F="AP";FAC="Low";Hack="Yes";
F="PF";FAC="Low";Hack="Yes";
F="AP";FAC="High";Hack="No";F="PF";FAC="High";Hack="No";F="AP";FAC="Low";Hack="No";F="PF";FAC="Low";Hack="No";74

APPENDIX III
Network Risk: Simulation Results
Sample size 52

App Corr
Lock Up
OK

Yes
No

Yes
No

Yes
No

0.0m
0.5m
1.0m
1.5m
2.0m
2.5m

AF
Counts Prob
7 13.5%2
3.8%
43 82.7%
52 100.0%

EUM
Counts Prob4
7.7%
48 92.3%
52 100.0%

NF
Counts Prob3
5.8%
49 94.2%
52 100.0%

V
Counts Prob4
7.7%
48 92.3%
52 100.0%

DL
Counts Prob
0%
45 86.5%
50%
4
7.7%
100%
3
5.8%
52 100.0%

Yes
No

HAN
CountsProb
52
100.0%
00.0%
52
100.0%TDCounts Prob
0 day
49 94.2%
0.5 day
3
5.8%
1 day
0
0.0%
52 100.0%

Yes
No

PS
Yes
No

Counts
13
39
52

AP
PF

Counts
52
0
52

Prob
25.0%
75.0%
100.0%

F
Prob
100.0%
0.0%
100.0%YesNo

HASCounts Prob
52 100.0%0
0.0%
52 100.0%

SFCounts Prob5
9.6%
47 90.4%
52 100.0%

FAC
Counts ProbHigh
52 100.0%
Low
0
0.0%
52 100.0%

SD
Counts Prob
0 day
50 96.2%
0.5 day
2
3.8%
1 day
0
0.0%
52 100.0%

Yes
No

Yes
No

Hack
Counts
Prob
6
11.5%
46
88.5%
52
100.0%UPSCountsProb
52
100.0%
00.0%
52
100.0%

SQ
CountsProb
High
52
100.0%
Low
0
0.0%
52
100.0%

Cost
CountsProb
49
94.23%
2
3.85%
0
0.00%
0
0.00%
1
1.92%
0
0.00%
52 100.00%

75

APPENDIX III

Sample size 9999

App Corr
Lock Up
OK

Yes
No

Yes
No

Yes
No

0.0m
0.5m
1.0m
1.5m
2.0m
2.5m

AF
Counts Prob
1027 10.27%
473 4.73%
8499 85.00%
9999 100.00%

EUM
Counts Prob949
9.5%
9050 90.5%
9999 100.0%

NF
Counts Prob719
7.2%
9280 92.8%
9999 100.0%

V
Counts Prob963
9.6%
9036 90.4%
9999 100.0%

DL
Counts Prob
0%8991 89.92%
50%
688 6.88%
100%
320 3.20%
9999 100.00%

Yes
No

HAN
CountsProb
9999
100.0%
00.0%
9999
100.0%TDCounts Prob
0 day
9280 92.81%
0.5 day643 6.43%
1 day
76 0.76%
9999 100.00%

Yes
No

PS
Yes
No

Counts
2403
7596
9999

Prob
24.0%
76.0%
100.0%FAPPFCounts
9999
0
9999

Prob
100.0%
0.0%
100.0%YesNo

HASCounts Prob
9999 100.0%0
0.0%9999 100.0%

SF
Counts Prob501
5.0%9498 95.0%
9999 100.0%

FAC
Counts Prob
High
9999 100.0%
Low
0
0.0%
9999 100.0%

SD
Counts Prob
0 day
9740 97.41%
0.5 day259 2.59%
1 day
0 0.00%9999 100.00%

Yes
No

Yes
No

Hack
Counts
Prob
899
9.0%
9100
91.0%
9999
100.0%UPSCountsProb
9999
100.0%
00.0%
9999
100.0%

SQ
High
Low

Counts
9999
0
9999

Prob
100.0%
0.0%
100.0%

Cost
CountsProb
9186
91.87%
529
5.29%
162
1.62%
97
0.97%
24
0.24%
1
0.01%
9999 100.00%

76

APPENDIX IV
Network Risk: Calculations for Parent-Child Monitor for NF NodeWith learning:
m

Data
NF

αy

αn

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Prior
Yes
No
No
Yes
Yes
No
No
No
No
No
Yes
No
No
Yes
Yes
No

4.32
5.32
5.32
5.32
6.32
7.32
7.32
7.32
7.32
7.32
7.32
8.32
8.32
8.32
9.32
10.32
10.32

1.08
1.08
2.08
3.08
3.08
3.08
4.08
5.08
6.08
7.08
8.08
8.08
9.08
10.08
10.08
10.08
11.08

E (θ y ) E (θ n ) - log θ y - log θ n
0.800
0.831
0.719
0.633
0.672
0.704
0.642
0.590
0.546
0.508
0.475
0.507
0.478
0.452
0.480
0.506
0.482

0.200
0.169
0.281
0.367
0.328
0.296
0.358
0.410
0.454
0.492
0.525
0.493
0.522
0.548
0.520
0.494
0.518

0.223
0.185
0.330
0.457
0.397
0.351
0.443
0.527
0.605
0.677
0.744
0.679
0.738
0.794
0.733
0.681
0.729

Sm

S

Em

0.223
1.779
1.269
0.457
0.397
1.217
1.028
0.892
0.790
0.710
0.744
0.708
0.650
0.794
0.733
0.705

0.223
2.002
3.272
3.728
4.125
5.342
6.370
7.262
8.052
8.762
9.506
10.21410.86411.65812.39113.096

0.500
0.454
0.594
0.657
0.633
0.608
0.652
0.677
0.689
0.693
0.692
0.693
0.692
0.689
0.692
0.693

1.609
1.779
1.269
1.003
1.116
1.217
1.028
0.892
0.790
0.710
0.645
0.708
0.650
0.602
0.655
0.705
0.658

Varm

0.307
0.357
0.178
0.069
0.114
0.156
0.079
0.032
0.009
0.000
0.002
0.000
0.002
0.009
0.002
0.000

Test
∑ E ∑Var Statistic
m

m

0.500
0.954
1.548
2.205
2.838
3.446
4.098
4.774
5.463
6.156
6.848
7.541
8.233
8.922
9.614
10.308

0.307
0.664
0.842
0.912
1.026
1.182
1.260
1.293
1.301
1.301
1.304
1.304
1.306
1.315
1.317
1.317

-0.500
1.286
1.878
1.595
1.271
1.745
2.024
2.188
2.270
2.284
2.328
2.341
2.302
2.386
2.420
2.430

Where θ y = P ( NF = Yes | Hack = Yes) and θ n = P ( NF = No | Hack = Yes)
Table A4.1 Penalties and test statistics for model with learning
With learning:
m

Data
NF

E (θ y )

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Prior
Yes
No
No
Yes
Yes
No
No
No
No
No
Yes
No
No
Yes
Yes
No

0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800

E (θ n ) - log θ y - log θ n
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200

0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223
0.223

1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609
1.609

Sm

S

Em

0.223
1.609
1.609
0.223
0.223
1.609
1.609
1.609
1.609
1.609
0.223
1.609
1.609
0.223
0.223
1.609

0.223
1.833
3.442
3.665
3.888
5.498
7.107
8.717
10.326
11.93512.15913.76815.37815.60115.82417.433

0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500

Varm
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307
0.307

∑ E ∑Var
m

m

0.500
1.001
1.501
2.002
2.502
3.002
3.503
4.003
4.504
5.004
5.504
6.005
6.505
7.006
7.506
8.006

0.307
0.615
0.922
1.230
1.537
1.845
2.152
2.460
2.767
3.075
3.382
3.690
3.997
4.305
4.612
4.920

Test
Statistic
-

0.500
1.061
2.021
1.500
1.118
1.837
2.457
3.005
3.500
3.953
3.618
4.041
4.438
4.143
3.873
4.250

Table A4.2 Penalties and test statistics for model without learning

77

APPENDIX IV

m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

With learning
0.223
2.002
3.272
3.728
4.125
5.342
6.370
7.262
8.052
8.762
9.506
10.21410.86411.65812.39113.096

Sm
Without learning
0.223
1.833
3.442
3.665
3.888
5.498
7.107
8.717
10.326
11.93512.15913.76815.37815.60115.82417.433

Bayes Factor
0.000-0.1700.170-0.063-0.237
0.156
0.737
1.455
2.274
3.173
2.653
3.554
4.513
3.943
3.433
4.337

Test Statistic
With learning Without learning-0.500-0.500
1.286
1.061
1.878
2.021
1.595
1.500
1.271
1.118
1.745
1.837
2.024
2.457
2.188
3.005
2.270
3.500
2.284
3.953
2.328
3.618
2.341
4.041
2.302
4.438
2.386
4.143
2.420
3.873
2.430
4.250

Table A4.3 Comparison of model alternatives

78