J. Service Science & Management, 2010, 3, 352-362
doi:10.4236/jssm.2010.33041 Published Online September 2010 (http://www.SciRP.org/journal/jssm)
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex
Ivan De Noni1, Luigi Orsi2, Luciano Pilotti1
1University of Milan, DEAS, Milan, Italy; 2University of Padua, Padua, Italy.
Received June 11th, 2010; revised July 19th, 2010; accepted August 21st, 2010.
This paper examines the relationship between operational risk management and knowledge learning process, with an
emphasis on establishing the importance of statistical and mathematical approach on organizational capability to
forecast, mitigate and control uncertain and vulnerable situations. Knowledge accumulation reduces critical situations
unpredictability and improves organizational capability to face uncertain and potentially harmful events. We retain
mathematical and statistical knowledge is organizational key factor in risk measuring and management process. Statis-
tical creativity contributes to make quicker the innovation process of organization improves exploration capacity to
forecast critical events and increases problem solving capacity, adaptation ability and learning process of organization.
We show some important features of statistical approach. First, it makes clear strategic importance of risk culture
within every level of organization; quantitative analysis support the emergence of latent troubles and make evident vul-
nerability of organization. Second, innovative tools allow to improve risk management and organizational capability to
measure total risk exposition and to define a more adequate forecasting and corrective strategy. Finally, it’s not so easy
to distinguish between measurable risk and unmeasurable uncertainty, it depends on quantity and quality of available
knowledge. Difficulty predictable extreme events can bring out crisis and vulnerable situations. Every innovative ap-
proach which increases knowledge accumulation and improves forecasting process should be considered.
Keywords: Complexity, Extreme Events, Operational Losses, Quantitative Management
In literature we find out paper explaining mathematical
techniques application to operational risk evaluation and
other concerning risk management principles and fea-
tures. What we would like to do in this paper is to merge
this too often separated concepts. Thus, in the first part,
we describe some aspects of operational risk mainly with
respect to relationship between uncertainty and corporate
learning process; in the second one, we argue general
mathematical approaches implications on operational risk
and knowledge management and then we show an inno-
vative mathematical method and exploit its advantages,
disadvantages and further extensions.
In modern economic context, knowledge management
is a more and more important resource for success of
firms. However, getting information is only the first step
for a long time sustainable development. In a dynamic
system in fact, knowledge may result just a short time
vantage because its non-excludability property makes
its transfer and competitors’ imitation easy . To ac-
quire competitive vantages, firms have to develop and
continually improve suitable capabilities and organize
routines to control, manage and use the knowledge in a
profitable way .
Thus, knowledge management becomes progressively
the success key for securing organization’s sustainability
, even if a successful learning process needs a good
adaptation capability at the same time. Only a supple firm
is able to adapt its cognitive patterns to environment and
market changes  quickly and properly enough to de-
velop an effective generative learning process for new
knowledge creation . Knowledge management is a
dynamic capability ; a successful strategy in the short
term may become less efficient in the long one if the
firm is unable to increase continually its knowledge,
improving competences and innovating competitive ad-
So we can conclude efficient knowledge management
and adaptation capability are both required to organiza-
Evaluating Enterprise Risk in a Complex Environment353
tion sustainability. When these conditions are both satis-
fied, an organization can implement a generative learning
process to preserve new knowledge creation and innova-
tion. In this perspective, statistical analysis and mathe-
matical models are some of possible tools which can help
firms to improve knowledge accumulation and decision
making processes. More and more detailed and reliable
forecasting models, in fact, permit to better predict envi-
ronment changes and manage correlated risk, reducing
uncertainty and increasing the organization’s problems-
2. Organization, Risk and Knowledge
Each enterprise can be thought as a socio-economic or-
ganization, headed by one or more persons with a pro-
pensity to risk [7,8]. So risk is an integral part of the firm.
The increasing complexity of modern society makes risk
a particularly critical factor because company manage-
ment is often unable to face it [9-11]. When we talk about
risk we refer in particular to operational risk1.
If you consider the more and more critical role of risk
in company governance, it’s easy to understand because
overlooking or thinking risks in simplistic way may lead
to inadequate exposition or unconscious acceptance by
the organization. In contrast, when risk is implemented
in the corporate culture, it develops into a production
factor and its suitable management becomes an essential
part of value creation chain . Just when the concept
of risk is integrated in corporate culture, you can iden-
tify the basis for prudent and responsible corporate gov-
However we need to distinguish between governance
and management of business risk. They are two inter-
connected but different moments of decision-making
process. Government takes care of placing the organiza-
tional bounds qualifying the logic of value creation and
the maximum tolerable risk, while management tends to
decompose the overall business risk in a variety of risks,
following the stages of assessment, treatment and reas-
sessment of all relevant risk .
In other words, risk governance relates to risk culture
sharing throughout the all organization as well as to re-
duce risk overlooking. Risk management relates to every
process or technique which allows to mitigate or remove
risk for organization. In first instance potential efficient
management depend on efficient governance.
Thus, a careful administration of risks leads to some
A corporate governance is based on a complex and
dynamic mix of risks linked inextricably to the en-
Usually a business risk cannot be fully cancelled;
Each action to reduce risk exposition involves an
organizational cost and brings out other hazards;
The sustainability of any risk depends on the amount
of enterprise knowledge and competences.
Last point focuses on the relationship between risk and
knowledge. The risk reflects the limitations of human
knowledge or bias, indicating the possible events to which
it is exposed due to the combination of their choices,
external conditions and the flow of time. Just if know-
ledge was complete and perfect, firm would operate in
conditions of certainty . So risk and knowledge are
each other mutually dependent: the risk marks the limit
of knowledge and it allows the perception of risk .
Over time, the learning process of an organization leads
to a better understanding of reality and a more awareness
of risk and thus allows to reduce uncertainty and to de-
velop greater risk management and forecasting capabili-
In literature we found three different approaches which
try to define the relationship between knowledge and risk:
scientist current, social current and critical current.
The first one is based on logical-mathematical app-
roaches and believes in primacy of knowledge on risk
[18,19]. The second one retains limits of knowledge and
social interaction processes make the risk a feature of
contemporary society, where knowledge contribution mere-
ly asserts organizational inability to eliminate risk .
The critical current based its thesis on knowledge and
risk dynamics, where risk cannot be eliminated but it can
be reduced or contained ; so the learning creative
process of a business organization needs to track the
business risk, trying to transform or mitigate it through
company skills and competences, which organizational
knowledge renews over time.
Spread risk culture to any firm level means developing
an attitude to adaptation and risk knowledge, looking for
useful concepts and approaches to address critical issues
of risk assessment and management. Neglecting risks
leads the entrepreneurship to a state of myopia that makes
it unable to predict or otherwise mitigate critical situa-
tions, increasing the vulnerability of all organization,
constantly exposed to uncertainty and possible crisis.
The information collection and the learning process,
increasing knowledge about nature and behavior of a
particular event, can help the ability to manage risks and
can play a key role in raising resources, tools or new
knowledge which, without allow a precise prediction,
will be able at least to reduce risk impact.
1According to Basel Committee operational risk is defined as the
risk of loss resulting from inadequate or failed internal processes,
eople and systems or from external events.
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment
However, the knowledge related to risk management,
if misinterpreted, could lead to controversial situations.
The larger uncertainty perception is, the stronger will be
the people’s inclination to not act; in other words, ma-
nager who is facing unknown situation usually tends to
be more cautious, ready to come back or change his
strategy just perceive discrepancy with respect of his
expectations. At opposite, when the decision maker be-
lieves to know the event distribution, his strategic be-
havior is determined by a cognitive model built on its
previous experience and thus makes him less sensitive to
environment changes and to perception of every signal
which could prevent or handle unexpected situations (ex-
treme events) and crisis.
3. Crisis Management and Vulnerability of
We need a more stable and comprehensive concept of
crisis in organizations facing complexity and uncertainty
as well as a way to reduce risk by better prevention .
Crisis management and vulnerability of the organiza-
tion are two very important concepts that have had con-
siderable attention in management literature as the basis
for defining processes of contingency planning such as
operational crisis and crisis of legitimation for disaster
recovery (bankruptcy, market breakthrough, change in
leadership, fraud, etc.). Rarely has it focused on the pro-
cesses by which the crisis has been generated, its long-
term phases and embedded sources . We need to de-
velop a better perspective to explore the generation and
nature of crisis events in organizations where, according
to Smith  “management should not be seen as oper-
ating in isolation from the generation of those crisis that
they subsequently have to manage, but rather as an inte-
gral component of the generation of such events”. But we
will see that the customer or the user becomes part of the
strategic process to prevent and respond to crisis as
“partners” of management and shareholders, sharing col-
lective knowledge and information about value creation.
The main factors to create the pre-conditions for crisis
and vulnerability of organizations are their interaction
with the market and the often huge differentiation of this
one; the growing importance of consumer expectations
and perceptions which are closely connected with the
image and reputation of the company and of the man-
agement; the technological innovations and long (or
short) wave of change that pushes companies to change
rapidly not only their products or markets, but also deal-
ers, managers and organization; the nature of leadership,
its stability and evolution, as in the case of a jump in
family control or a huge change of shareholders .
For those reasons and many others that serve to create
a complex portfolio of potential crisis scenarios, is nec-
essary to outline the nature of the crisis management
process and to know how the crisis evolve.
In the literature we can identify a focus on the devel-
opment of contingency plans to cope with a range of cri-
sis scenarios in terms of response teams, strategies for
continuity service provision, and procedures to protect
organizational assets and damage limitation. These are
activities very important in stopping dangerous conse-
quences of a crisis, but not useful to avoid it, as in the
case of an organization’s reputational harm, often irre-
versible. Effective crisis management should include a
diffuse and systematic attempt to prevent crisis by occur-
According to many authors, the notion of crisis starts
from a circularity and interaction between different pro-
cess stages not always linearly connected: crisis manage-
ment, operational crisis, crisis of legitimation, process of
organizational learning. In that view, a good preventative
measure is creating resilience [21,25,26,27, 28]. This can
be achieved within the organization by:
trying to eradicate error traps as a path to explor-
developing a culture that encourages near-miss re-
dealing with the aftermath of crisis;
learning lessons from the event;
defining an accountability to connect stakeholders
for crisis potential [26,27].
According to Smith  we need a shift in the way of
representing organizations’ dynamic change and proc-
esses of management (before, during and after crisis) as
nonlinear connections in space and time. This would al-
low us to explore pathways of vulnerability and erosion
of defences (Figure 1):
1) Crisis of management: the problems are well known;
managers believe that their organizations are safe, secure
and well run because their short-term perspective sees
the comparison between the cost of prevention and per-
ceived costs of limitation and recovery derived from an
underestimation of the second in respect to the first. The
long-term view for many managers is rare, and often
“hostages” of operational phases and controls are by-
2) Operational crisis: many characteristics of the crisis
are here often visible, but in many cases, contingent and
temporarily defined. But we face often situations of deep
incubation or latent conditions of crisis factors as if the
organization is living in a permanent present, without a
past and without a future.
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment355
Figure 1. The nature of the crisis management process and
the potential for emergence .
3) Crisis of legitimation: it starts internally but with
immediate external effect attracting media coverage,
governmental intervention or public inquiry as in the
case of the recent bank crisis in Europe or great compa-
nies in USA. It could start from a failure in a specific
product or service that causes a loss of customer confi-
dence (and/or shareholders as well). At the same time, it
will lead to erosion of the demand and probably to finan-
cial instability in the long run, with an impact on stability
of shareholders and larger stakeholders too. The organ-
izational learning will show a probable great fragility.
4) Processes of organizational learning: a crisis of le-
gitimation leads to a crisis of procedure and practices and
then to the lack of confidence of the middle management.
5) Crisis of interchange between tacit and codified
knowledge: we can underline the role of knowledge dif-
fusion as a factor of crisis in case of overestimation of
codified knowledge with respect to tacit ones, considered
as residual errors to remove by a hierarchical control
mechanism. Interdependences between tacit and codified
knowledge (or between voluntary and involuntary be-
havior) are the link with the past and future of the com-
pany. They represent both the main source of strategic
competences in connection with its identity, and the ar-
chitecture of organizational learning able to sustain more
generations of entrepreneurs and managers as well as
customers. This is key to describing one of the pathways
Critical situation can be due to a lot of internal and
external factors. Bad managed critical situation can turn
into organizational crisis. Every risk, extreme event and
uncertain situation, which cannot be forecasted, miti-
gated and controlled, is a potential factor of vulnerability
and crisis for organization. It’s important to exploit every
innovation and tool which can make management proc-
ess easier, more reliable and more responsible.
4. Certainty, Uncertainty and Risk in
In economic context there are measurable uncertainty and
unmeasurable uncertainty. If nothing you can say about
uncertain event, risk may be thought as a measure- able
uncertainty. Catastrophic event (rare event) is intermedi-
ate situation, it’s not a fully unmeasurable uncertainty but
an accurate foreseeing could be hard.
Extreme event is located in tails of distribution, it’s
featured by low probability of occurrence and high nega-
tive impact. No information on past behavior allow to
exactly understand its dynamic evolution and conse-
quently risk managers are unable to forecast it.
However mathematical models or statistical tools are
not so useless as a lot of authors believe, if you judge
they allow increasing knowledge accumulation. Better is
your knowledge about a phenomena, better you can face
it. According to Epstein  you can say a priori whe-
ther a risk is measurable or unmeasurable, it depends on
the knowledge and information you have about it. For
example, we are not yet able to predict an earthquake but
we have learned to distinguish the potentially seismic
zones according to their geological composition, to build
using materials and techniques which sure high seismic
resistance. Thus was possible on base of statistical
analyses which allowed to verify where earthquakes
happened with larger probability rate and to value the
capacity of some materials to respond efficiently to stress
actions. In other words, even if you cannot predict a
catastrophic event, you can study its features and im-
prove organization capability to adequately face it and to
act fast and efficiently to reduce its impact when the
In front to these situations, manager can choice differ-
ent strategies. He may decide on responsible manage-
ment of every risk or only more ordinary risks account-
ing. Many decision makers tend to overlook highly im-
probable events, though very dangerous; most of them
usually think coverage costs related to the extreme event
prediction are in the time higher than costs needed to
face their occurrence. Probability estimation for an out-
come based on judgment and experience may result suc-
cessful but it depends on entrepreneurship or manage-
ment capability and competence and in uncertain case on
randomness. Mathematicals and statisticals allow to get
an objective and more adequate probability estimation.
Use of statistical techniques for objective management of
risks effects on organizational culture and enhance pre-
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment
dictive capacity at every firm’s level.
However, neglect to study extreme event distribution
slows knowledge learning process and reduces organiza-
tion’ absorptive and adaptation capability.
Then overlooking behavior attends a high risk propen-
sity of management and thus reflects negatively on the
corporate culture, reducing liability and risk awareness
and exposing the company to a higher probability of
harmful events occurrence in the long time.
Knight  claimed “there is no difference for con-
duct between a measurable risk and unmeasurable un-
certainty”. Risk management doesn’t mean necessarily to
consider every potential risk, it regards awareness con-
cept. Prudent management suggests you can choose to
apply corrective measures and mitigation actions or
rather decide consciously to neglect any specific risks.
What is important a manager should make decisions
within a rational and responsible approach based on pro-
bability of occurrence, predictability and impact estima-
Then, a prudent and responsible decision maker should
consider, in its evaluation and selection of strategies to
follow, each time the reference context, differentiating
his behavior from situations of certainty, risk or uncer-
tainty. We talk about decision making under certainty
when you can be sure, without doubt the authenticity of a
case; in risk conditions, when you cannot be sure of the
authenticity of a case but this one can be estimated with a
certain probability rate; in uncertainty, when you cannot
assess the authenticity of a case and cannot understand
the probability that this is true, because you haven’t in-
formation to make a reasonable estimate .
It is essential to distinguish between conditions of cer-
tainty, uncertainty or risk, because it has power over the
chance to estimate probability of occurrence of an event
and to determine the most appropriate and effective deci-
sion-making strategies in a specific situation. Under con-
ditions of certainty, you will choose the action whose
outcome provides the greatest usefulness; in a risky
situation, you will value the greatest expected utility; in
uncertain case, no decision can be considered completely
reliable or reasonable and effects should be considered
random . In situation of uncertainty, manager doesn’t
know historical data suitability and future probability of
occurrence, and there are infinite factors which could
influence events evolution and change preliminary deci-
Normally, in order to know the risks we are exposed,
we need to choose the probability distribution form so
well as to calculate the risks and find out the probability
that a past event comes up again in the future. If you
need a probability distribution to understand the future
behavior, it’s also true past events knowledge is neces-
sary to determine the probability distribution. In other
words, we are in a vicious circle .
One of the main problems is that risk management
science has been addressed, in the last century, by an
econometric and mathematical point of view, looking for
potential models which were able to forecast loss distri-
bution. What it’s not clear is that risks of using a wrong
probability distribution aren’t obviously predictable and
can be even more dangerous than those detected by cho-
sen distribution tails .
The main problem of risk management is determined
by the fact that general properties of the generators (dis-
tribution shape)2 usually outline a uncertain situation and
not hazardous one. The worst mistake that can make the
risk manager is to confuse uncertainty with risk, failing
to define class and generator parameters.3
A generator has specific parameters that determine
certain values of the distribution, allows analysts to cal-
culate the probability that a certain event occurred. Usu-
ally the generator isn’t known and there is no independ-
ent way to determine the parameters, unless you try to
deduce from the past behavior of the generator. To esti-
mate the parameters from historical data is still necessary
to assume the generator class (Normal, Poisson, Bino-
mial and other).
These estimates are more accurate as larger is the
amount of data available. Logically, an inappropriate
choice of the type of generator immediately affect the
reliability of results (in terms of imprecise probability of
occurrence of risky events).
The risk manager can run into several situations:
the type of generator is easily identifiable and there
is an adequate supply of past data;
the generator can be considered reliable, but the lack
of historical data makes impossible to correctly de-
termine the moments;
you cannot determine either the parameters or the
general class of membership.
2The term “generator” refers to the type of distribution that best esti-
mate a well-defined set of data.
3See [30,33]. The distinction between risk and uncertainty is roughly
that risk refers to situations where the perceived likelihoods of events
of interest can be represented by probabilities measured by statistical
tools using historical data, whereas uncertainty refers to situations
where the information available to the decision-maker is too imprecise
to be summarized by a probability measure [30,33,34].
The last point is usually the dominant situation in
which the relationship between expected risk and actual
risk is still undetermined or accidental. There is a state of
uncertainty, where it cannot be sure what will happen, or
make any estimates on the probability of hypothetical
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment357
Therefore, risk managers should not confuse uncer-
tainty with risk, failing to define both the class generator
and its parameters. A fact is making decisions under risk,
where you know the distribution of a given phenomenon
and the probability that a harmful event, another fact is to
do it under conditions of uncertainty where the distribu-
tion is only conceivable and the probability of events is
not accurately defined.
Further perspective is provided by recalling the role of
differentiability in decision theory under risk, where util-
ity functions are defined on cumulative distribution func-
tions. Much as calculus is a powerful tool, Machina 
has shown that differential methods are useful in decision
theory under risk. Epstein  adds to the demonstration
in Machina that differential techniques are useful also for
analysis of decision-making under uncertainty.
Operational losses are usually forecasted using para-
metric and actuarial approaches as LDA (Loss Distribu-
tion Approach) or more cautious EVT (Extreme Value
Theory) . The main problems of this approaches
concern the choice to reduce any event distribution to
consolidated generator (normal, lognormal, GPD or other)
and the need for consistent time series to obtain values
The key attraction of EVT is that it offers a set of
ready-made approaches to a challenging problem of
quantitative operational risk analysis and try to make
risks, which are both extreme and rare, appropriately
modeled. Applying classical EVT to operational loss data
however raises some difficult issues. The obstacles are
not really due to a technical justification of EVT, but
more to the nature of the data. EVT is a natural set of
statistical techniques for estimating high quantiles of a
loss distribution, which well works with sufficient accu-
racy only when the data satisfy specific conditions .
The innovation introduced by fractal model is the
flexibility to adapt event distribution to real one without
setting up the best generator. So you don’t need to as-
sume the shape of the generator because ifs allows to
reproduce the structure of real distribution on different
scales, exploiting the properties of self-similarity4 of fra-
The fractal building is based on an innovative algo-
rithm which is iterated a theoretically infinite number of
times so that, in each iteration, the approximated distri-
bution better estimates the real one. The IFS (Iterated
Functions System) technique finds out the more appro-
priate generator without a known model application .
Moreover the properties of fractals allow to estimate the
event distribution in a reliable manner even if we have a
lack of historical data.
Therefore the advantage of this approach is to give risk
managers a tool to avoid the mistake of neglecting risks,
regardless having to fix a suitable standard generator. A
best estimate of an event risk level enhances the effi-
ciency of its management, monitoring and control and re-
duces exposure of the organization.
5. Sample and Methodology
Our analysis is based on a two year operational loss data
collection by an Italian banking group5. The dataset con-
tains operational losses broken down by eight business
lines and seven event types in accordance with the Basel
II rules. The business lines are: Corporate Finance, Tra-
ding & Sales, Retail Banking, Commercial Banking, Pay-
ment & Settlement, Agency Services, Asset Management
and Retail Brokerage. The event types are: Internal Fraud,
External Fraud, Workplace Safety, Business Practice,
Damage to Physical Assets, Systems Failures, Execution
& Process Management.
The available data consist of a collection of opera-
tional losses of an Italian banking group for a time period
of two years divided into company code, type of business
line, risk drivers, event type, amount of loss, date of oc-
currence, frequency of occurrence.
To have a significant analysis, we had to use a time
horizon of one month instead of one year for our estima-
tions (then, we have 24 observations of the aggregate
monthly loss), and to focus only on the business line (we
do not care of the event type); we analyze two different
business lines: the traditionally HFLI6 retail banking 
and the traditionally LFHI7 retail brokerage .
In Table 1 we show the descriptive statistics of retail
banking and retail brokerage. In the retail banking busi-
ness line we have 940 loss observations (high frequency
business lines), the minimum loss is 430 euros, the
maximum loss is 1066685 euros and the average loss is
13745 euros. In the retail brokerage business line we
have 110 loss observations (low frequency business line)
with a minimum loss of about 510 euros, maximum loss
of 700000 euros and an average loss of 28918 euros
(higher than retail banking business line).
We have used for our analysis three types of ap-
proaches: the traditional Loss distribution approach
(LDA), the refined Extreme Value Theory (EVT) and the
innovative fractal based approach Iterated function sys-
5For reasons of confidentiality we cannot use the group name
6High Frequency Low Impact
7Low Frequency High Impact
4The basis of self-similarity is a particular geometric transformation
called dilation that allows you to enlarge or reduce a figure leaving
unchanged the form.
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment
Table 1. Descriptive statistics of retail banking (left) and
retail brokerage (right).
Min. 438 509.6
1st Qu. 1302 1071.4
Median 3000 2124.3
Mean 13745 28918.3
3rd Qu. 8763 13375.0
Max. 1066685 700000.0
First of all, we consider the actuarial-based Loss Dis-
tribution Approach. The goal of the LDA methodology
consists in identifying the loss severity and frequency
distributions and then calculate the aggregate loss distri-
bution through a convolution between severity and fre-
LDA is built upon two different distributions, esti-
mated for every cell of the Business Line-Event Type1
matrix (ij): the distribution FXij(x) of the random variable
Xij which represent the loss amount trigged by a single
loss event. This distribution is called loss severity distri-
The distribution Pij(n) of the counting random variable
Nij, which probability function is pij(k) = P(N = k). Pij is
said loss frequency distribution and corresponds to:
These two distributions (which have to be independent
from each other) represent the core o the LDA approach,
and are used to obtain the operational loss calculated on
(mainly) a one-year horizon (in our case one-month ho-
rizon) for the ij cell:
The main approach to study extreme events is the Ex-
treme Value Theory (EVT), which is a statistical meth-
odology that allows analysts to handle separately the tail
and the body of the distribution, so that the influence of
the normal losses on the estimation of the particularly
high quantiles can be eliminated. This technique was
developed to analyze the behavior of rare events and to
prevent natural catastrophes (i.e., floods or losses due to
fires). There are two classes of distribution in particular
which prove to be useful for modeling extreme risks: the
first one is the Generalized Extreme Value (GEV) and
the second one is the Generalized Pareto Distribution
The Iterated Function Systems (IFS) represent a
mathematically complex class of innovative and non-
parametric fractal methods to create and generate fractal
objects as an approach to shift between time towards
space . The fractals are geometric figures that can be
represented at different dimensional levels, but they con-
sist in an infinite replica of the same pattern on a smaller
and smaller scale and so they are made up of multiple
copies of themselves. This fundamental property of in-
variance is called self-similarity, one of the two principal
properties that describe a fractal. The second property,
not less important, is the indefiniteness, which is the pos-
sibility to fractionate virtually till infinite every part be-
fore going to the next one. Hence, to “draw” a fractal
through a processor, the maximum number of iterations
must be specified, because a finite time is insufficient to
calculate a point of the fractal at infinite iterations.
One of the IFS possible fields of application is risk
management, in which fractal methods are used as
non-parametric estimation methods as an alternative to
the Loss Distribution Approach (LDA) and the EVT .
These innovative methods can both interpret the data of
loss in a more accurate way, and estimate and recreate
the possible population from which they could derive,
especially when there are few observations or data. The
advantage of this approach is that the elaboration proc-
esses can be reduced while improving, at the same time,
the capacity of estimation of the patrimonial require-
ments necessary to cover the expected and unexpected
operational losses8. IFS methods eliminate the necessity
to analyze the distribution of frequency9, of severity10
of the correlated convolutions between them, simulating
directly the cumulative distribution of the aggregate
losses and then applying the most efficient measure of
risk11. Consequently, they become a fundamental element
in the analysis of losses, especially in correlation with the
so called extreme losses, which are the losses that, due to
their very low probability of occurrence, tend to be ig-
nored, although their occurrence could determine catas-
In the next section, the fractal approach will be applied
8The expected loss is defined as the mean of the losses observed in the
revious periods. The unexpected loss is defined as the difference
between the VaR and the expected loss and, resulting more difficult to
e represented in a model, represent for the business management an
element of uncertainty which can be minimized only through ad
9For distribution of frequency we intend the frequency of the occurrence
of the events.
10For distribution of severity we intend the financial impacts generated
by the losses.
11In our case Value at Risk
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment359
to a specific case with the intent to demonstrate, at a prac-
tical level, what kind of information can emerge from the
use of this approach and how this information can be
used to improve the business management.
6. Estimation by IFS Approach
The non-parametric estimation we offered as an alterna-
tive way to LDA methodology and the Extreme Value
Theory is the IFS fractal based approach which should be
able to interpret loss data in the best possible way and to
simulate a population (see Figure 2) from which our
observations could come and especially in the presence
of a few observations and missing data (as in the case of
retail brokerage). The advantage of this approach is to
eliminate the analysis on frequency and severity distribu-
tion with related convolution and the previous choice of a
known distribution to directly simulate the cumulative
distribution function of the aggregate operational losses
and then apply a risk measure to calculate capital re-
quirements to cover operational losses in the business
lines we studied.
The Iterated Function Systems, designed originally
from Barnsley for the digital image transfer , depth
by Forte and Vrscay  for solving inverse problems,
and used by Iacus and La Torre for their estimation and
simulation mathematical property of probability func-
tions  and, finally, adapted to the capital require-
ments calculation of operational risk, have shown over
other methods (such as LDA and EVT) more relative
efficiency in terms of ability to reconstruct a population
We used the function arctang for transforming the
values of loss in values between 0 and 1 to allow the IFS
approach, that functions on a finite support (in our case
precisely between 0 and 1), to estimate the operational
cumulative distribution function starting from the em-
pirical distribution function (EDF).
First, we have to demonstrate that IFS approach is bet-
ter than LDA and EVT to reconstruct a population from
Figure 2. Example of IFS simulation in retail banking.
which our information could come and especially in the
presence of few data. The methodology to demonstrate
the capability of the IFS estimator to reconstruct the
original population is to calculate a statistical distance
from the original distribution of the IFS simulation and
comparing the results obtained with the classic method
used in the actuarial science (LDA) and the most innova-
tive method of Extreme Value Theory. For our analysis
we use observations of operational losses caused by fires.
The database Danish Fire Losses in literature has been
used to test both classical techniques such as LDA that
EVT as the newest techniques and lends itself very well
to the study of extreme and complex events as demon-
strated by McNeil et al.  and therefore it represents a
good test for analyzing and evaluating the relative effi-
ciency of the fractal approach versus parametric tech-
niques such as LDA and EVT. This database contains
daily observations of loss arising fire in Denmark in mil-
lions of Danish crowns from January 1980 to 1990 and is
free downloadable in the package QRMLib by R 2.10.0
project. We proceed extracting random samples of dif-
ferent sizes from the database and then simulate the real
distribution of losses through the use of IFS maps for
small sample size (n = 10, 20, 30) and medium sample (n
= 50, 100, 250). Finally we compare the results obtained
by IFS approach in terms of AMSE12 distance with the
results obtained by LDA and EVT approaches.
In Table 2 we show the behavior of the AMSE index13
(calculated over 100 simulations for each sample size).
The indices show us how IFS approach is more effi-
cient for small sample size than LDA and EVT, while for
samples of medium and large size we find an asymptotic
behavior between LDA approach and IFS one.
Secondly, we apply this new fractal methodology for
estimating enterprise capital requirements that a firm
must allocate to cover operational risks. We know the
main problem for the operational risk analysis is the lack
of information and the lack of data, so we use the power
and the capability of the fractal objects to reconstruct a
population starting from this lack of information to better
understand the real risk profile of the firm and to im-
prove the decision making process inside the company.
In Table 3 we show the result of non-parametric estima-
tion made by IFS approach for the retail banking and
retail brokerage business lines compared with the value
obtained by LDA VaR, EVT VaR and the maximum
value of the empirical distribution function.
In the two business lines in which we were able to ap-
ply all the three approaches proposed in this paper, we
12Average mean square error
13Lower is the value of the AMSE index better is IFS approach versus
LDA and EVT
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment
Table 2. Relative efficiency of IFS estimator over LDA and
n AMSE over LDA AMSE over EVT
10 65.13 24.05
20 73.52 31.26
30 75.15 25.95
50 79.33 22.27
100 111.12 21.05
250 115.67 21.11
Table 3. Value at Risk by different approaches.
line LDA VaR EVT VaR IFS
VaR Max EDF
banking 1531369 4420998 3200000 2248354
brokerage 1061966 4929569 1451000 931607
obtained, with IFS method, estimates in the middle be-
tween the Extreme Value Theory which tends to overes-
timate widely the capital requirements and the Loss Dis-
tribution Approach which tends to underestimate the re-
quirements (for example in the case of retail banking).
Another IFS advantage is the ease of use, meaning that
you do not need to simulate a distribution for the severity
and for the frequency and the related convolution, but is
enough to measure directly the aggregated operational
loss distribution (monthly in our case given the number
of observations available) and apply to this distribution a
measure of risk.
The strength of this fractal approach has demonstrated
its ability to capture the true risk profile of a company
even with lack of data.
Of course not everything is positive, but we must take
into account and know perfectly well there are inherent
disadvantages and limitations that make each model less
accurate and sensitive to errors. Firstly like all nonpara-
metric approaches, the IFS are very sensitive to the sam-
ple data used. Several samples extracted from the same
population can lead to estimates and simulations widely
different between them. It is a good idea to use a very
large number of simulations in order to obtain consistent
In recent years, there has been increasing concern among
researchers, practitioners, and regulators over how to
evaluate models of operational risk.
The actuarial methods (LDA) and the Extreme Value
Theory (EVT) constitute a basis of great value to fully
understand the nature and the mathematical-statistical
properties of the process underlying the operational losses,
as regards the severity of losses, the frequency of losses
and the relationship between them.
Several authors have commented that only by having
thousands of observations can interval forecasts be as-
sessed and traditional techniques need a large amount of
data to be precise and effective. However, currently
available data are still limited and traditional methodolo-
gies fail to grasp the correct risk profile of firms and fi-
nancial institutions, as shown the empirical analysis un-
dertaken in this article. With IFS technique you can es-
timate appropriate risk value even if you have just few
data. Increasing data availability, IFS outcomes tend to
be as suitable as traditional methods.
So, we can say that the study, the analysis and the im-
provement of innovative methods such as Iterated func-
tion systems (IFS) is a valuable support for measurement
and management the operational risk alongside actuarial
parametric techniques. Not only that, their use is not lim-
ited only in the next years, pending more complete series.
Their use should include those areas of operational risk,
which is by definition heterogeneous and complex, have
a limited numbers of events, but their impacts can, how-
ever, be devastating for firms and stakeholders. Precisely
for this reason the use and improvement of these innova-
tive tools must continue for the foreseeable future be-
cause there will always be the need in this area of inno-
vative approaches able to predict, with good approxima-
tion, situations starting from this lack of data and recon-
struct a precise and faithful population to integrate and
assess the results of other different approaches.
As such as every mathematical method, IFS could help
risk management making quicker knowledge learning
process but what is really important is a careful and pru-
dent management behavior.
Thus we can conclude, firstly, mathematical approa-
ches are effective only when they are integrated and
shared in a responsible corporate culture. Secondly, IFS
as a nonparametric method is sensitive to the composi-
tion of used sample; for an appropriate estimation of op-
erational loss with IFS you should reapply the methods
more times and then estimate an adequate average level.
More cautiously you apply mathematical techniques,
more probable is a accurate estimation of hedging value
of the total operational risk. Finally, being the first time
IFS is applied to operational risk, future improvements
In our intentions is we would extend the application of
this methods to financial and credit risk. However this
goal needs we are able to use IFS to estimate not just
probability function but density one. The main problem it
Copyright © 2010 SciRes. JSSM
Evaluating Enterprise Risk in a Complex Environment361
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