To start, we need to define what
operations research is. Operations research is an interdisciplinary branch of mathematics
which uses methods like mathematical modeling, statistics,
and algorithms
to arrive at optimal or good decisions in complex problems which are concerned
with optimizing the maxima (profit, faster assembly line, greater
crop yield, higher bandwidth, etc) or minima
(cost loss, lowering of risk, etc) of some objective function.[1]
As it is seen from the definition,
the related technique is mainly used for optimizing the decision making. This
will decrease the uncertainty and of course mitigate the risk. Nevertheless,
although this technique is useful for reducing the risk, since the computations
are mainly dependant on the previous data or personal experiences, this doesn’t
mean that the risk is totally avoided.
After these statements, we will
concentrate on the basic tools used under this principle. One of the most
important tools used for this purpose is decision tree. The decision tree
classifier is one of the possible approaches to multi-stage decision making.
The basic idea involved in any multi-stage approach is to break up a complex
decision into a union of several simpler decisions, hoping the final solution
obtained this way would resemble the intended desired solution.[2]
In order to use this tool, the
originator should be very careful about the sequence of the events and the
demonstration of the nodes become vital. At the end of each line, the results
should be considered. If the result of the decision is uncertain, a circle has
to be drawn. If the result is another decision that the originator needs to
make, a square must be drawn. Squares represent decisions, and circles
represent uncertain outcomes. The originator should write the decision or
factor whether it is a square or circle.[3]
This is illustrated in the following decision tree.
A Decision Tree
As it has been defined in the last
paragraph and illustrated in the above figure, the nodes; A-B-C stands for
uncertain events and 1-2 stands for certain conditions. Therefore, the decision
maker needs to take decisions in the numerical nodes but he/she must be aware
of uncertain situations in the alphabetical nodes.
For business, this technique is mainly
dealing with the calculation of expected utility or expected monetary value of
the project. The formulas are depicted
as follows.[4]
where,
i: Event number
Pi :
The probability corresponds to same event
Vi: The value or return
of the event i
Ui: The utility of the
event i
After stating the formulas, let’s
continue with the given example in the following.
Example
An investment bank is utilizing its
funds in overnight investments (A), bank loans (B), corporate loans (C) and real
estate investments (D). These funds are all put by the stake holders and the
stake holders have the opportunity to invest in O/N investments while they are
sure about a risk free gain of 5% p.a. However, what ever invested rather than
this investment has risk. If the risk becomes real, the outcome will be 0 for
the investor. This will be named as non-performing. Concerned structure will yield
us the following decision tree.
As it is seen from the decision
tree, if the stake holders invest in a bank, the bank will utilize the funds in
four different investments. Nevertheless, since the demand from the market may
vary, they expect that all investments will be distributed evenly (25%). When
we use above formula in order to calculate the expected value of “invest in
bank” alternative, the calculations can be seen as follows:
E (V)invest in bank = 25%
x (100% x 5% + 0% x 0%) + 25% x (90% x
6% + 10% x 0%) + 25% x (80% x 8% + 20% x
0%) + 25% x (70% x 11% + 30% x 0%)
E (V)invest in bank = 6.125%
Since E (V)invest in bank
= 6.125% > E (V)invest in O/N = 5% for the stake holders, they
will prefer to invest in an investment bank. This doesn’t mean that investing
in a bank will be better than investing in O/N in all circumstances. However,
since the probability of the default in “invest in bank” alternative is
highlighted, the stake holders will be more certain in making a bank
investment.
Same computation also goes for
expected utility. But, for expected utility, the originator put his subjective
opinions that may cause a big fuss in the calculation and decision making. This
calculation may be useful for calculating operational risk. An example is shown in the following.
ABCD bank is trying to find what
will be the possible burden on the bank’s operation due to operational risk. The bank has
calculated the possible utility loss per each type as depicted in the following
decision tree. The outcomes were given minus U’s (Utility) since they are
subjective measures.
Decision Tree of ABCD Bank
E (U) = 20% x (- 20 U) + 15% x (- 40
U) + 5% x (- 15 U) + 20% x (- 25 U) + 10% x (- 20 U) + 30% x (- 35 U)
E (U) = - 28.25 U
According to the following table,
the bank is under high risk and the bank should cover it by implementing
different operational risk management models.
Risk Level for Different Utilities (An Example)
Risk Level
|
Utility Score
|
None
|
Below –5 U
|
Low
|
Between –5 U and –10 U
|
Medium
|
Between –10 U and –20 U
|
High
|
Between –20 U and –30 U
|
Very high
|
Above –30 U
|
To sum up, when we come to risk
management, it is for sure that these decision trees will mitigate the risk
incurred. This is because it is helping to decrease the uncertainty. The
managers are more aware of what will happen in the future by considering the
previous information on the probability of the different and possible outcomes.
After analyzing operations research,
we will concentrate on Statistical tools. Before going into statistical tools,
we will define statistics. Statistics is the branch of scientific inquiry that
provides methods for organizing and summarizing data and for using data
information in the data to draw various conclusions.[5]
One of the main concepts of
statistics is normal distribution. Many numerical populations[6]
have distributions that can be fit very closely by an appropriate normal curve.[7]
This makes normal distribution very much popular beyond the other
distributions.
The Normal distribution describes a
special class of such distributions that are symmetric and can be described by
the distribution mean µ and the standard deviation σ. If X is normally distributed
with mean µ and the standard deviation σ, we write X~N (µ, σ2).[8]
In order to calculate Z the following formula will be used.[9]
Let’s see this formula and the
meaning of Z in the following example.
An individual wants to insure his
house against fire and robbery. He applies to an insurance company and asks for
10 years insurance scheme. The insurance company knows that (depending on the
previous data) the claims are coming with an average of 7 years with a standard
deviation of 5 years. The insurance company wants to know the probability of
realization of risk in this case.
When we apply the above formula, we
want to calculate P(X<10). First of all Z should be found. This is
calculated as follows.
Z = (10 – 7) / 5 = 0.6 à P(X<10)
= P(Z<0.6)
The correspondent normal
distribution for the value X is depicted as follows.
Normal Distribution Curve
And, the other normal distribution
regarding Z values which are the corresponding values of X values that were
depicted above is shown below.
Z Curve
P (Z < 0.6) = 0.7257 which is
depicted in the below table.
Z Table
This means that the insurance company
forecasts a claim probability of 72.5% in the coming 10 years. By factoring
this claim probability, the insurance company will do a good pricing which
covers the probable claims. Other than normal distribution, there are other
distributions such as gamma, exponential, logarithmic etc. Nevertheless, due to
the lack of place in this study we have only given the example from normal
distribution which has helped us to understand how normal distribution will be
used in the risk management concept.
[1] Wikipedia, “Operations Research”,http://en.wikipedia.org/wiki/Operations_research
[2] S. Rasoul Safavian and David Landgrebe, “A Survey of Decision Tree Classifier Methodology”, IEEE Trans. Systems, Man,& Cybernetics, V: 21, No. 3, 1991, p.661.
[3] Mind Tools, “Decision Tree”, http://www.mindtools.com/dectree.html
[4] Hamdy A.Taha,
[5] Jay L.Devore, Probability and Statistics for Engineering and the Sciences,
[6] Numerical Populations: The values of each item in a population
[7] Devore, Ibid, p. 154.
[8] Jonattan Marchini,“Normal Distribution”,http://www.stats.ox.ac.uk/~marchini/teaching/L6/L6.notes.pdf
[9] Devore, Ibid,p.158.
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