RIGA TECHNICAL UNIVERSITY
Faculty of Computer Science and Information Technology
Institute of Applied Computer Systems
Marina UHANOVA
Computer Systems Postgraduate Program Student
EVALUATION OF INSURANCE RISK BASED ON MODELLING
AND WEB TECHNOLOGIES
Summary of the Theses
Supervisor
Dr. habil. sc. ing., Professor
L. NOVICKIS
RTU Publishing House
Riga 2007
1. GENERAL REVIEW OF THE WORK
1.1. Relevance of the Research
Risk evaluation is one of the most important problems which must solve each of the
insurance companies. For risk evaluation it is necessary to know the amount of loss and loss
probability or frequency, because basic part of insurance risk is sum of insurance indemnity
[8,25].
For the evaluation of amount and probability of losses the methods of mathematical
statistics and actuarial calculations are used. For facilitating of statistical data processing in
information systems it is necessary to integrate mathematical modules, which are provided for
risk evaluation. This is especially important because Latvian insurance market continues to rise
and it is expected that this tendency will continue during the next years. Further, in connection
with Latvia's entrance to European Community and business development, the insurance sphere
is developing very quickly which leads to the fact that the number of collected premiums and
issued policies has increased rapidly starting from the year 1992 [5]. Therefore the problem of
usage of upgraded insurance information system and up-to-date information technologies in the
insurance data processing has become very actual.
Talking about the application of information technologies in insurance, it must be noted that
the transfer from local solutions of information systems of insurance companies to
corporative systems based on WEB technologies is happening right now in Baltic countries. At
present, WEB technologies in Latvia are mainly used to maintain business processes on two
stages: direct insurance (forming and accounting policies) and account of insurance cases and
losses.
It must be noted that despite the fact that great number of mathematical models is
developed which can be used successfully in insurance, usually these models are used
autonomously, independently from information systems (Excel files, that are not integrated in
insurance information systems, etc.). It hampers the statistical analysis of data and management
of business processes.
In the Promotion Work, the attention was mainly paid to the investigation of mathematical
models that provide the possibility to evaluate the total loss of insurance company according to
the collective risk model and integration of models in insurance systems, using WEB
technologies.
1.2. Goal and Tasks of the Work
The basic goal of the Work is the analysis of mathematical models of insurance risks and
developing the approach based on WEB technologies that allows to integrate these models in the
structure of insurance information system.
Tasks of the Work:
1. To perform the analysis of insurance business process to determine that processes have
influence on the financial stability of the company.
2. To perform the analysis of existing insurance information systems in order to determine their
disadvantages and advantages.
3. To investigate WEB technologies with the aim to find the most suitable technology for
improvement of insurance information system.
4. To perform analysis of different mathematical models in order to find the most suitable
models for determination of insurance reserves and total losses.
5. To perform the statistical data analysis in order to provide the adaptation of mathematical
models for conditions of concrete insurance company and to present corresponding
recommendations.
6. To develop the Web service set and demonstrate its usage for integration of mathematical
models with the insurance information system.
1.3. Scientific Innovation and Practical Value
Scientific Innovation
The main result and scientific innovation of the work is development of an approach based on
investigation results of mathematical insurance models and Web technologies with the aim to
integrate these models into insurance information systems supplying corresponding
recommendations for their implementation.
Different mathematical models for evaluation of total losses of insurance cases have been
investigated in the Work. Recommendations are given for the usage of collective risk models in
evaluation of losses and for investigation of integration of mathematical models in the insurance
information systems. Such an approach gives a possibility to increase the management effectivity
of insurance business processes, including formalization of the sum of insurance premium
determination, calculation of reserves, and prediction of stability of insurance company. This
approach allows also to combine actuary calculation models with the existing functions of
insurance information systems.
Practical value
In frames of the work, the following Web services were developed using Microsoft .NET
technology:
1) Modelling of total losses, including:
o modelling of the number of losses;
o modeling of the amount of losses;
o modelling of total losses.
2) Search of existing services of modelling of losses.
3) Convertation and receipt statistical data from data bases of insurance systems.
Practice provides an easy modification and integration of actuary models into different
insurance information systems.
The Web services, that realizes different mathematical models, can be integrated
successfully into different insurance information systems. The application of the developed Web
service set has been illustrated with the aid of DiasoftlNSURANCE / IDC system.
1.4. Structure of the Work
The work consists of introduction, three chapters and conclusion. The introduction contains
the general description of the work, the description of purposes, aims and tasks of the work.
The first chapter is devoted to the study and analysis of insurance business processes and
technological means of their support, as well as the analysis of up -to-date Web-based
technologies and existing insurance information systems.
The second chapter contains the analysis of mathematical models of estimation of the total
loss of a portfolio of risks. Models of individual and collective risk have been analyzed. In this
chapter the major attention has been paid to the selection and adaptation of model of an
estimation of the total loss to the conditions of the chosen Latvian insurance company.
The third chapter is devoted to the mathematical models realization and their integration
into information systems using Web services.
The Promotion Work is written in Latvian. There are 52 figures and 23 tables, all together
130 pages. The bibliographic list contains 82 titles.
2. THE CONTENT OF THE WORK
2.1. The analysis of Processes of Insurance Business and General
Principles of Insurance Premiums Determination
To achieve the goal of the work it is necessary to conduct the insurance business processes at
the initial stage, to carry out the analysis of existing insurance information systems and Web
technologies (Figure 1).
Analysis, investigation and formalization of business processes constitute one of the most
important stages in the development and improvement of information systems. The analysis of
insurance business processes predetermines the requirements of functionality and structure of
insurance information system [2].
Main attention in the research is paid to business processes that have influence on the
financial stability of the company, for example: calculation of the amount of premium,
determination of reserve capital and evaluation of total losses.
The final aim of actuary calculations is such tariff rate determination that would guarantee
non bankruptcy of the Insurer with the chosen high degree of probability. The tariff calculation
standard does not exist today, but the risk theory provides many premium calculation principles
[11,12,20,39].
To evaluate insurance risk, it is necessary to determine the whole sum of losses, or total
losses, as well as to predict the number of concluded insurance agreements. The obligatory
condition of the insurance company - is work without losses:
where:
N - the number of agreements ;
Xi - losses for the agreement with the number i;
Pi - net premium for the agreement with the number j;
Therefore, to calculate the amount of premiums and evaluate insurance risk, it is necessary
to evaluate the total loss for all policies. The attention in the work has been paid to the
evaluation of total losses. The evaluation of total losses is based on determination of
distribution of total losses [30,33].
If the integral function of total losses FN(x) = P(X1+ ... +XN<X) is known, then the standard
scheme is used: the satisfactory risk-degree e (probability of bankruptcy) is chosen, and the sum
of premium D is evaluated in the following way:
(2)
It means that with the probability l-  collected premiums will be enough to settle
compensation claims.
2.2. Analysis of Insurance Information System
The first stage of the development of insurance information systems (1992-1999) in Baltic
states is characterized by development and exploitation of some separate and non-compatible
with each other software products which support different insurance operations (management of
agreements, accounting, etc.). Because of the rapid increase of number and profit of issu ed sold
insurance policies, specialists in information technologies were mainly busy with the
modification of existing informative systems, so that they did not have time to develop new, more
upgraded systems [22].
The present-day situation with usage of solutions of information technologies in the
insurance business is characterized by the introduction of new platform of hardware and
software. Besides, at present moment a transition to Web technologies is taking place in the
development of insurance information systems [4,9,41].
Analyzing insurance information systems used in Latvia, it can be noted that insurance
companies demonstrate two approaches to the introduction of information systems:
• They develop continuously information systems for their own needs;
• They use information systems that software development companies work out and offer.
For example, such companies as Balta, BTA, Baltikums and others have developed their
own systems.
Other companies use existing software program platforms as the basis for the development
of their own systems. For example, the insurance company Balva, using the platform of Russian
company DIASOFT (www.diasoft.ru), together with SIA IDC IT has developed the information
system DiasoftlNSURANCE/IDC [14,15].
At the moment, the company NEXUM is trying to enter the Latvian market with the online
product that is provided for non-life insurance [13].
The analysis of insurance information systems was performed on four insurance markets:
West European, American, Latvian and Russian. In the review of information systems, the
attention was mainly paid to the Western systems because the insurance market in Western
Europe and America has already stabilized at the beginning of 1980ies.
19 insurance information systems were investigated in the Work. The investigation of
existing information systems proved that the majority part of systems does not support actuary
calculation models and data exchange with the remote data sources. This leads to the non-optimal
amount of premiums and reserves determination, as well as hampers analysis of statistical data,
upgraded Web technologies give the possibility to avoid this disadvantage.
2.3. Software Developing Technologies in Authomatization of
Actuary Calculations
The investigation of existing automation systems showed that there is necessity to integrate
the system with actuary calculations. During integration of actuary calculations, the following
problems may arise.
Firstly, the model that provides mathematical data processing must be developed in such a
way that it can be integrated with different systems which function on different platforms,
because insurance information systems are usually realized in the form of separate subsystem set,
besides, insurance companies usually have the distributed structure.
Secondly, it is very essential that the module of actuary calculations provides easy and
quick modification of mathematical models because of possible changes of mathematical
regularities in the future. Due to this reason, it is necessary to provide the possibility to add new
mathematical models to systems that were developed earlier, and as a result - search and choice
of existing models. To provide the modification and replacement of existing module of actuary
calculations, program components must be used for developing the modulus, that in their essence
are modulus of binary codes which can be made and used autonomously. In addition, taking into
consideration the structure of insurance information system, it is necessary to, provide the
possibility to receive information from different data bases and to provide the interface with
different components of the program that function on different platforms.
10
Program components can be developed:
• in the form of DLL library;
• in the form of COM / DCOM components;
• in the form of CORBA components (CCM - CORBA Component Model);
• in the form of .NET components;
• in the form of EJB components;
• in the form of Web service.
The preference for integration of actuary calculations in the structure of insurance
information system was given to Web services, because Web services are [17,34,40]:
• independent from the programming language and platform;
• specially developed to provide the interface with components that function in d ifferent
platforms;
• they offer standards for searching corresponding services;
• they offer secure protocols for data exchange;
• they offer possibility to use unified technologies for the access to data bases.
The following two technologies are used more often for the development of Web services:
Sun J2EE and Microsoft .NET [29,36], As a result of performed analysis, the preference for
integration of mathematical models into the structure of insurance information systems was given
to Microsoft .NET technology due to the following reasons:
• solutions' price;
• the chosen insurance company uses the Windows platform;
• easy Web application and Web service development.
2.4. Mathematical Models of Insurance Risk and their
Experimental Investigation
The investigation of mathematical models of insurance risk and their integration into
existing insurance information systems will give a possibility to obtain the information system of
new quality (see Figure 2) [21].
The attention in the work is paid to the investigation of different mathematical models that
are provided for calculations of total losses. Tariff calculation and determination of reserve
capital is based on determination of the amount of total losses. Calculation of total losses is one
11
of main tasks that must be solved by each insurance company because determination of insurance
tariff and reserve capital is the management basis of financial operations of insurance companies.
In addition, correctly calculated amount of tariff and reserve capital provide financial stability or
security of the insurance company. The security is one of the main indicators of insurance
company operations [1,23,31].
Figure 2. Improvement scheme of the insurance information system
For determination of the amount of total losses, most commonly are used the normal
distribution, models of individual risk and models of collective risk [3,10,33]. The description of
given models can be found on the Table 1.
In the work the attention was also paid to adaptation of models for conditions of chosen
insurance company and to analysis of statistical data. The adaptation of model was described on
the example of CASCO insurance.
Table 1
Models of Determination of Total Losses
2.5. Usage of Normal Distribution for Evaluation of Total Damages
Usually, for the determination of premiums in mass insurance, especially when the
amount of collected data is limited, the following approximating is used for evaluation of total
losses: assume that the distribution of the amount of separate losses is normal. Taking as a
basis the central limit theorem of the theory of probability, it is the possible to assume that
distribution of total losses S is normal [32,35],
The given approximation makes possible to evaluate the necessary reserves D of insurance
company in such a way that total losses S would not exceed D with the probability l-e, where
eis the probability of bankruptcy. In other words: P(S<D) = 1-.
Therefore, the necessary amount of reserves D (sum of premiums) can be evaluated in the
following way:
N
D   Pi  E ( S )  h,
i
where:
N - number of policies;
pi - premium for the policy with number i;
E(S) - the mathematical expectation of total losses;
h - loading (additional payment for the risk).
(3)
For the next evaluation of total losses and for calculation of premiums, the following
assumptions are performed: firstly, it is assumed that separate losses are independent from
each other, and, secondly, sums of losses do not differ essentially (that is true for mass
insurance kinds). From this follows that distributions of amount of losses and their numeral
characteristics (mathematical expectation and dispersion) are equal.
Assuming the approximation that for the amount of losses Y there is a normal
distribution, the mathematical expectation of total losses can be evaluated in the following
way:
E(S) = N.E(Y),
(4)
where
E(Y) — the mathematical expectation of separate loss;
N - the number of policies.
But the dispersion:
Var(S )  Var(Y ). N ,
(5)
where Var( Y) - the dispersion of separate loss.
The probability of bankruptcy of insurance company  is equal (observing formula 3):
 =P(S>D) = P(S > E(S)+h),
(6)
where h is additional payment of risk.
Using the central limit theorem of probability theory, it can be proved that
(7)
where  - is cumulative distribution function of standard normal distribution,
Var(S) - the dispersion of the sum of losses.
Therefore, if we want that the probability of non-bankruptcy of the company to be equal
to a, then the loading h, must be equal to
h  x . Var (S )
(8)
where  (x ) = ,
but reserves of the insurance company D must be equal:
D  E(S )  x . VAR(S )
(9)
Total losses evaluated in such a way provide formation of the company reserves for
payment of compensations. To calculate the insurance premium, the loading h must be divided
among all policies.
Using the total distribution of evaluation of total losses and statistical data about insurance
polices, the insurance company reserves can be evaluated in the following way.
First, the mathematical expectation and dispersion of total losses must be evaluated. To
demonstrate the evaluation of these parameters, the data of one Latvian insurance company on
CASCO insurance during the period since 2003 to 2005 have been analyzed. As a result of
calculations obtained:
E(Y) = 92.9816;
E(S) =E(Y)*N =695472;
Var(S) = 1073313.
It follows that, to provide the probability of non bankruptcy about 95% the reserve capital
must be not less than: D = 695472+1.65 1073313 = 697181.4.
The dependence between reserves of insurance company and probability of non bankruptcy is
shown on Figure 3.
Unfortunately, owing to the fact that the distribution of amounts of losses is usually very
asymmetrical, as it can be seen from the histogram, shown on Figure 4 (during the histogram
drawing, differences in the length of intervals were observed - the heights of columns are
proportional to the concentration of losses in the corresponding interval, losses that are equal with
0 are not disposed), the application of the central limit theorem of theory of the probability is
limited (all depends on the kind of distribution of separate losses). Chebishev's inequality (10)
can be used for determination of the amount of premium as an alternative to the given theorem or
risk models described below [19,42].
(10)
2.6. Individual Risk Model Application for Evaluation of Total
Losses
Individual models are provided for modelling of total losses of homogeneous risk groups.
For obtaining a distribution approximation of total losses in individual models, the approximation
of Ri distribution of losses of separate risk with uninterrupted distribution is used, that allows
direct calculation of composition. For rough approximation of distribution of separate risk loss it
is sufficiently to know that the zero loss value has the highest probability. The further inaccuracy
of approximation of the process of distribution composition levels quickly, and models of total
losses are obtained that are very close to reality.
The following distributions are used for modelling losses of separate risks R i, (i = 1,…,1
the number of risks):
• gamma distribution;
• lognormal distribution;
• reversed Gauss distribution;
The density function of gamma distribution looks as follows:
here  -is mathematical expectation, but the parameter a determines the form of the graph of
function. The dispersion of gamma distribution is equal to 2/, variation coefficient l/  , but
asymmetry coefficient 2/  .
In case of homogenous risk group, gamma distribution with parameters  = E(Ri ) and
 = (E(Ri))2 / Var(Ri ))2 can be used for loss amount approximation. Parameters of given
distribution can be found from the equality provision of theoretic and empiric moments. To reach
the goal and to find homogenous risk groups from / distribution of total losses of risks, it is
necessary to find risk distribution with parameters  and  I-the composition of that rank.
Because, as the result of gamma distribution composition again is obtained gamma
distribution, for approximation of total loss distribution, gamma distribution with paramete rs I.
and I.a can be used (follows from gamma distribution characteristics).
In some cases, for modelling of total losses, reversed Gauss distribution is used that has an
equal gamma distribution characteristics. Probability density function of reversed G auss
distribution is as follows:
Parameters of distribution are chosen equally to previous case in such a way that the
parameter coincides with the mathematical expectation, but dispersion coincides with the value
2 /.. The variation coefficient opposite to Gauss distribution is equal to l/  , asymmetry –
3/ 
Reversed Gauss distribution as a result of composition is also preserved which is
convenient for modelling of total losses.
Besides, gamma and reversed Gauss distribution preserve the independent risk R i, with
separate parameters i and i, in the case of composition, if the ratio i|i is constant for all
risks. It gives a possibility to use this distribution also in the case, when the insurance sums of
risks Ri differ.
For modelling of total losses, the lognormal distribution is also used that is equal to
gamma and reversed Gauss distribution is also used. The density function of lognormal
distribution probability is as follows:
As a result of composition the lognormal distribution is not preserved, and therefore the
total loss is immediately modelled with lognormal distribution [18,33].
Table 2 includes data on 22439 CASCO insurance policies. The data are collected, using
statistical information for the period since the 2003 to 2005.
As has been noted, for modelling of separate risks gamma and reversed Gauss distribution
can be used. For evaluation of parameters of those distributions the moment method can be
applied. Taking into account that E(Ri) = 92.98169, but Var(Ri) = 1073313, parameters of
gamma and reversed Gauss distribution  and  are equal to 92.98169 and 0.008055
respectively
Table 2
Graphs give the conception of density function of risk distribution for gamma and Gauss
models which is shown on Figure 5. To rise the visuality of models, natural logarithms from
empiric frequencies, gamma and Gauss distribution are shown on Figure 6.
For modelling of total losses, gamma, reversed Gauss and lognormal distribution can be
used. As a result of evaluation of parameters of total loss models by moment method, the
following evaluations of parameters were obtained (amount of average portfolio was used for
obtaining evaluations):  = 695472.0303,  = 60.24911427 (for gamma and Gauss models) un
2 = 0.016462, = 13.44412 for model, that is based on lognormal distribution.
Using given models, the necessary amount of reserves of insurance companies can be
evaluated. The dependence between reserves of insurance companies and non -bankruptcy
probability is shown on Figure 7 (on sample of gamma distribution).
2.7. Collective Risk Model Application for Evaluation of Total
Losses
In collective models the distribution of total losses of risk group has been searched, using
distributions of number of losses and distribution of the amount of losses in one insurance case.
Collective models give a possibility to develop "tails" of total losses, the more realistic model
and to evaluate the probability of one year loss of any amount.
The idea of collective model is to review the portfolio as the producer of losses, neglecting
the belonging of losses to concrete risks. Losses are not connected with separate risks, but are
considered as sample from the only one distribution that in its essence is the mix of different
distributions of separate risks. As practice has showed, such information loss does not diminish
the quality of the result, because the primary distributions of collective model, such as
distributions of a number of losses and distribution of amount can be determined much more
preciser than distributions of separate uniform groups of risk [7,33],
Poisson distribution is used most often for modelling of the number of losses, which appear if
the following rather general preconditions are fulfilled:
• the number of losses in two time intervals do not overlap, are independent values of the case;
• two or more losses cannot appear simultaneously;
• losses can happen at any time moments [27,37].
In practice it can be deemed that all three preconditions are implemented (except for the
insurance practice against nature accidents) both for the portfolio and separate risk. Besides,
Poisson distribution:
P(N=n) = n e-/n!, n = 0, 1,2, ...
(14)
is easy to use for modeling of the number of losses because the independent, case value sums
distributed respectively to Poisson distributions with parameters I, ... , I are also the Poisson
distribution with the parameter  = 1+... + 1 In unified case, when i = 1, parameter 
can be expressed using portfolio amount: =I.I.
Most often are used distributions for modelling of separate amount of loss which have the
asymmetric uninterrupted probability density functions. Besides, forming the distribution model
of loss amount, the following method is usually used in practice: in the basis of the model is used
the case value Y, which has asymmetric cupola shaped probability density function which is
defined for all real arguments and it is converted according to the formula:
X = exp(Y).
(15)
For modeling of the amount of losses in collective models is rarely used empiric
distribution because this empiric distribution usually inaccurately describes the right distribution
"tail" - the big area of losses.
Five most important models of the amount of losses are shown in the Table 3. The table
includes also moments of distribution.
It is possible to evaluate the chosen distribution parameters and to control the quality of the
model using standard methods, such as the method of moments, maximal likelihood method and
2 criterion.
It is not an easy task to obtain total losses S distribution from the number of losses and the
amount of distribution. Almost always, there is a lack of data for application of some distribution
model, because every year gives only one value of S observation, and in most cases the
information about previous years is not actual.
21
Table 3
Models of the amount of losses
The distribution of case value S can be expressed, using distribution of the number of losses
pn = P(N=n) and distribution F(X) = P(X≤x) of the amount of losses X :
Unfortunately, direct calculation of compositions of infinite set is possible only in rare,
unrealistic cases, when the value N has the negative binomial distribution with the parameter a
= 1, but value X has exponential distribution.
Despite the central limit theorem of probability theory, one cannot rely on the fact that with
the increase of E(N), the distribution of S value will converge into the normal distribution. The
practice shows that big portfolios are also asymmetric and the normal distribution evaluates the
total loss too low. Much better approximation of the value S can be obtained, by using
uninterrupted asymmetric models.
The risk theory proposes also other analytical and numeral approximations, the most
popular of which are Monte-Carlo and Panjer methods of numeral recursive [16,28,43].
Thus, to build the distribution of total losses under collective risk model, it is necessary:
• to evaluate parameters of distribution of number of losses;
• at least to adapt especial risk distribution model to big loss area;
• to calculate moments of total losses and the distribution itself.
In the next chapter 2.8, the modification of proposed collective risk model is described
which is provided for evaluation of total losses in CASCO insurance in conditions of the chosen
company. The model of collective risk for providing total losses was chosen due to following
reasons:
• in difference to the normal distribution, it gives more accurate results;
• it is more suitable for risk insurance than the model of individual risk;
• requires less statistic data than the model of individual risk;
• does not require distribution of risks in homogenous groups.
2.8. Adaptation of Collective Risk Model for Conditions of Chosen
Company
In this chapter, the attention is paid to the adaptation of the model of collective risk for
conditions of chosen Latvian insurance company. As it was noted before, the evaluation of
total losses is one of the main tasks that must be solved to estimate insurance risk, the
security of insurance company and the amount of the necessary reserve capital.
For illustration of modelling the amount of total losses, data on CASCO insurance polices
have been used.
Observing the necessity of automation of the model, the preference for searching
applicable distribution is given to the distributions which parameters can be evaluated
directly.
Because the insurance company saves statistical data for previous three years, it is not
possible to evaluate accurately the kind and parameters of distribution of number of losses.
As the solution of such situation, Poisson distribution can be applied because the use of this
distribution gives good results in practice [6,33].
Table 4 includes data on 2440 values of losses in CASCO insurance. Data are
summarized, using statistical information for the period from the year 2003 to 2005.
The histogram, which is shown on Figure 8, formulates the idea about the density
function. During the drawing of the histogram, differences in the lengths of intervals have
been observed (the heights of columns are proportional to the concentration of losses in the
corresponding interval).
Table 4
As it is seen from the histogram for modelling of the amount of losses, lognormal, gamma,
Laplas, logistic and others distributions with asymmetric cupola shaped probability density
functions can be applied.
Distribution parameters have been evaluated, using the method of maximal likelihood and
method of moments. Better, but not satisfactory result (after 2 criterion) from the mentioned
distributions has been demonstrated by lognormal distribution with parameters =5.32 and
=1.5 (2 is equal to 278.9, but 2kr=16.9 (9 degrees of freedom with the significance level
0.05)).
To avoid this situation, the fact can be used that losses of bigger amounts influence more
the amount of total losses, therefore it is worth considering them separately from the losses of
smaller amount. As it can be seen from Table 5, losses of bigger amount have bigger economic
meaning. Despite the fact, that more than 85% of losses are less than average, their total
amount does not exceed 20%.
Table 5
To develop the real model of total damages, the losses were examined that exceed LVL
800 and the total amount of which reaches to 85% from the all amount of losses.
Table 6 contain data on 418 compensation claims, the amount of which exceeds LVL 800.
The number of intervals was chosen according to the expression 1+3.32 lg N (where N is the
number of data) [26].
Table 6
The histogram which is shown on Figure 9, describes the loss that exceeds LVL 800 of
empiric distribution.
As it can be seen from the histogram, for modelling of big losses such asymmetric
distributions can be applied as exponential, lognormal, gamma and Pareto distribution.
One better result under 2 and 2 criteria is given by Pareto distribution with parameters: xm
= 800 and k = 0.92392 (2 criteria is equal to 8.624 that is less than 2kr=14.067 (7 degrees of
freedom with the significance level 0.05), n2 is equal to 0.147 that is less than 0.4614 (the
significance level - 0.05)).
Note:
Integral function of Pareto distribution is:
Pareto distribution parameters can be evaluated according the method of maximal likelihood
in the following way [24]:
Empiric frequencies and developed Pareto model are shown on the Figure 10. Table 7
includes actual and predicted values for Pareto model.
As it can be seen from Figure 10 and table 7, Pareto model can be applied for modelling of
losses which amount exceeds LVL 800.
The histogram shown on Figure 11, characterizes distribution of losses which are less than
LVL 800.
Table 7
As the results of the completed investigation show, for modelling of smaller amounts, the
lognormal distribution with parameters:  = 4.99 and = I . I suits the best; 2 criterion is equal
to 20.61 that is less than 2kr= 21.67, 9 degrees of freedom with the significance level 0.01.
Notes:
Cumulative function of lognormal distribution is:
ln x  
F ( x)   (
)
(20)

To evaluate the parameters of lognormal distribution according the maximal likelihood
method is possible in the following way [24]:
Empiric frequencies and discovered lognormal model are shown on the Figure 12. Table
8 contains actual and predicted values for lognormal model.
Table 8
As it can be seen from Table 8, lognormal model slightly reduces the number of losses in
the interval from 664 to 800. Considering the value of criteria x 2. it cannot be said that the
lognormal distribution gives a good result, but in fact it can be used in practice.
For modelling of smaller amount of losses the empiric distribution can be applied because
the main reason why it is not used for modelling of losses of big amount is connected with the
necessity to evaluate the probability of such losses that are bigger than all observed losses.
The distribution of total losses can be expressed, using the distribution of number of
losses N: pn = P(N = n) and distribution of the amount of losses X: F(x) = P(X ≤ x) in the
following way:
where F*n is the n-level composition of distribution (it is assumed that F*0 (x) = 0, if x<0,
and F*0 = 1, if x≥0).
As it has been mentioned above, the calculation of this endless sum is possible only in rare
unreal cases For evaluation of total losses distribution it is possible to use ri6,28,4 3]:
• asymmetric distributions;
• modelling according to Monte-Carlo method;
• Panjer recursive method.
All three methods can be simply implemented. To demonstrate realization possibilities of
evaluation of total losses, Monte-Carlo method was chosen for the development of mathematical
module for information system, but respective Web service where the given model is realized,
can be replaced by another Web service that provides evaluation using any other model.
Therefore, to evaluate the total losses for CASCO insurance policies under conditions of
chosen company the following collective risk model can be used:
• Poisson distribution can be used for modelling of the number of loss;
• Pareto and lognormal distributions for amount of losses;
• the evaluation of total losses can be provided using the Monte-Carlo method.
Comparing models, used for evaluation of total loss of insurance company, it can be noted
.that the model based on normal distribution and model of individual risk evaluate too low the
total loss. It is connected with the asymmetry of distribution of amounts of losses due to which it
is not possible to consider that the distribution of total losses converges at the normal distribution.
It is also owing to the fact that separate distributions of risk insurance cannot be deemed as
a uniform, which limits the application of models of individual risk.
Table 9 contains the total amounts of losses (from the year 2003 to 2005), as well as
evaluated reserves of insurance company with the help of all three models (the probability of
non bankruptcy 95%). For evaluation of total losses, using the model of collective risk, the set
of Web services was used that is described in Chapter 2.9.
Table 9
Observed and evaluated values of total losses
The model of collective risk for prediction of total losses was chosen due to following
reasons:
• in difference to normal distribution, it gives a more accurate result because the strongly
expressed asymmetry troubles the usage of normal distribution of amount of losses;
• it is more suitable for the risk insurance than the individual risk model, because it does not
require the division of risk in unified groups.
2.9. The Realization of Mathematical Models and their Integration in Insurance
Information Systems
As it has been mention, to increase the effectivity of application of insurance information
systems, insurance risk evaluation and analysis of statistical data, it is necessary to integrate
models of actuary calculations into the structure of information systems.
For integration of the model of total losses into existing insurance information system, Web
services were chosen due to following reasons:
• Web services provide an easy and quick modification of the system;
• Web services provide the cooperation with different systems;
• Web services provide a possibility to use secure data transferring protocol;
• Web services offer mechanisms that provide automatic searching and connection to the
developed system.
Integration of actuary calculations in the insurance information system caused the
following problems during the task completion:
• modulus that provide the processing of mathematical data should be developed in such a way
that it would be easy to integrate them in different systems that function on different
platforms;
• it is very essential that this module provides an easy and quick modification of mathematical
models in connection with possible changes of regularities in the future.
In the work process Web services for evaluation of total losses, using normal distribution
and collective risk models have been developed. Given models are used for evaluation of
insurance reserve capital and premiums as well as for determination of security of insurance
company.
Developed Web services can be divided into the following groups:
1. Services for modelling of total losses, amongst them:
• services for modelling of the number of losses;
• services for modelling of the amount of losses;
• services for modelling of total losses.
2. Services that provide the search of existing services for modelling of losses.
3. Services, that provide converting and receiving statistical data from data bases of
insurance information systems.
Therefore, the structure of developed actuary calculations module is such as it is shown on
the Figure 13. In this case, any of the developed Web services can be easily replaced by another
services that use some other methods for modeling, for searching of existing services, and for
data reading from data bases.
In the similar way, supplementing the developed Web services with new services, it is
possible to integrate in the structure of information system some other models of actuary
calculations and to provide effective processing of statistical data.
For the demonstration of realization possibilities of actuary calculations, Web services have
been developed which provide the following functions:







modelling of the number of losses, using the Poisson distribution (for the generation of
case
values the method of opposite functions was used);
modelling of the amount of losses, using empiric distribution;
modelling of the amount of losses using lognormal and Pareto distributions (for the
generation of case values the method of opposite functions was used);
the evaluation of total losses, using Monte-Carlo method;
the evaluation of total losses, using normal distribution;
searching the most suitable Web services in the determined data bases;
reading statistical data from data bases and text files.
Table 10 includes description of developed Web services.
For the development of Web services, the Microsoft .NET Framework SDK and language
C#.NET have been used. Web services of each group realize the determined interface to provide
the mutual replacement of Web services.
The client program that is realized as Windows application. This program gives a
possibility to its users to create the most suitable model of evaluation of total losses for concrete
case.
After launching the program, as the first step, the possibility to choose Web service for the
searching of existing loss modelling services is provided. (Figure 14).
As the second step, the user must choose any the of existing models of evaluation of total
losses ( 1st' group for Web services) (Figure 15).
Table 10
Conformity of developed models of actuary calculations to Web services
On the third stage, the user must define parameter values of chosen model and choose Web
services for implementation of additional works (Figure 16). Finally, if it is necessary,
parameters of subjected services must be determined. It should be noted that the view of
window,
which is provided for entrance of parameter values of model and services, depends on the
respective parameters of Web service. For example, if Web service lognormdistl .asmx
(provides modelling the amount of losses, using lognormal distribution) is chosen, then its
parameters will be necessary to determine them, using the Window, shown on Figure 17. In its
turn, for determination of parameters of Web service getdattxt. asmx, the window which is
shown on Figure 18 will be opened.
Figure 16. One of the possible views of the window that is provided for entering Web service
parameters
Figure 17. One of the possible views of the window that is provided for entering Web service
parameters
Figure 18. One of the possible views of the window that is provided for entering Web service parameters
To add the new Web service to the system, it is necessary to copy the developed service at
the Web server.
2.10. Methodology of Application and Integration of Mathematical
Models and Method of Integration in the Structure of Information
System
To provide the realization of insurance mathematical models, the following steps must be
accomplished:
1. The most applicable mathematical models for the insurance risk evaluation must be analyzed.
2. The analysis of statistical data must be performed with the goal to adapt mathematical models
for conditions of chosen insurance company.
3. Developed models must be divided in submodels with the goal to provide repeated usage of
submodels in the content of other models. For example, the modeles of collective risk can be
divided in the following submodels:
• model of the number of losses;
• model of the size of losses;
• model of total losses.
4. Each of models must be build from existing Web services, but if there are no more applicable
Web services, then each of submodels must be realized in the kind of Web services, and be
placed in the developed Web services in the corresponding Web server.
The sequence of these steps are described on Figure 19. Especial attention in the Work has
been paid to insurance risk and total losses evaluation models. The normal distribution can be
used for evaluation of total losses, as well as the individual, or collective risk models.
In cases when distribution of losses is not clearly asymmetric or small amount of collected
data is accessible, the normal distribution can be used for evaluation of total losses.
If it is possible to divide the portfolio into maximum homogenous and independent groups,
then it is possible to apply models of individual risk. In this case, it is necessary to build the
distribution of total loses for each group but next to find the composition of obtained distributions
to find the distribution of total loses of the insurance portfolio.
In situation of non-life insurance, it is advised to use models of collective risk To evaluate
total losses according to model of collective risk, the following steps must be performed:
• distributions of number of losses and their parameters must be evaluated. In cases when
there is a lack of statistical data, Poisson distribution can be used.
• At least in area of big losses, models of separate risk distribution can be applied. To
provide better results, small and big losses can be reviewed separately.
• Total losses must be evaluated, using Monte-Carlo or Panjer methods.
Figure 19. Methodology of Application and Integration of Mathematical Models and Method of Integration in
the Structure of Information System
3. RESULTS OF THE WORK
The main results of the work present an approach for integration of mathematical models into
insurance information systems based on mathematical models investigation and Web
technologies, as well as corresponding recommendation for their implementation. In the course
of the research the following results have been also obtained:
1. An investigation of insurance information system has been undertaken in the Work.
As the investigation of existing insurance information systems has shown, the most part of
systems does not support models of actuary calculations. It hampers the analysis of statistical
data. Therefore, a special attention in the Work has been paid to analysis of existing actuary
models, their adaptation to conditions of respective insurance company and general
development of recommendations.
2. Mathematical models have been analyzed in the research that are supposed to determine the
amount of total losses, and the recommendations have been given for the application of those
models.
The application of collective risk models for evaluation of total losses of the kinds of risk
insurance has been demonstrated in the Work. The usage of the model of collective risk has
been demonstrated on the example of CASCO insurance.
As investigations of statistical data has shown, to avert total losses in conditions of chosen
insurance company, the following collective risk model can be used:
• Poisson distribution can be used in modelling of number of losses;
• Pareto distribution for losses can be used for modelling of the amount of losses for
distributions which amount exceeds the mathematical expectation, and the lognormal
distribution can be used for losses of smaller amount;
• The evaluation of total losses can be provided using Monte-Carlo method.
The model of collective risk for providing total losses was chosen due to following reasons:
• in difference from the normal distribution, it gives more accurate results, because the
usage of normal distribution is hampered by strongly expressed asymmetry of distribution
of amount of losses;
• it is most suitable for risk insurance than the model of individual risk, because it does not
require distribution of risks in uniformed groups.
3. Up-to-date information technologies have been investigated in the Work with the aim to find
the most suitable technology for integration of mathematical models into the insurance
information systems.
Because of the lower price, easy application in practice and the fact that Windows platform is
the most widespread, the preference is given to Microsoft .NET technology. Mathematical
models were developed using Web services, taking into consideration, that for the successful
integration of mathematical models with information systems it is necessary to provide the
following aspects:
• simple modification;
• easy connection with different existing computer applications;
• data exchange with remote data sources.
4. Web services have been developed in the process of the research.
For the development of Web services, language C#.NET has been used. All developed Web
services can be divided into the following groups:
• services of total losses modelling;
39
• services that provide searching of existing services for modelling of losses;
• services that provide transferring statistical data from data bases.
Web services of each group realize the determined interface, which provides connection of
new services with the simple copying of Web service in the Web server. All developed
services can be replaced by each other that provides an easy and simple modification of the
mathematical model.
The developed Web services can be used for enlarging of the functionality of existing
information systems which has been demonstrated on the example of system
DiasoftlNSURANCE / IDC.
5. Methodology of application and integration of mathematical models in the structure of
information system based on Web technologies has been suggested.
At the moment, models of actuary calculations and information systems are used separately:
insurance companies use other programs of actuary calculations, but the account of policies
and insurance cases processing are performed separately in the data bases of information
system. It reduces the efficiency of application of information systems.
Work results have been used:
• In the international project BALTPORTS-IT (IST-2001-33030) of the European commission
of IT program presented in the conclusion of review monograph (analysis of mathematical
models for evaluation of total losses) [2].
• In implementation of the following themes of Latvia Science Council (Web services
development):
- 01.0862 "Development of Intellectual Financial and Insurance Program Provision
Based on Web Technologies" (2001-2004), supervisor of the theme - Prof. L.
Novickis;
- 05.1655 "Integrated Business Software Based on Web Technology" (2005-2006),
supervisor of the theme - Prof. L. Novickis.
• On the stage of development of DiasoftlNSURANCE/IDC information system with the
results practically tested in the company "Balva".
At the final stage, in the year 2005, the Work was partly financially supported with the
scholarship of the European Social Fund.
40
4.APPROBATION OF THE RESULTS OF THE WORK
4.1. Publications
1. Selected Actuarial Models in Insurance Information Systems. In: Proceedings of 5th
International Conference on Optimal Research SOBI'2006, Tallinn, May 17-20, 2006,
pp 51-56.
2. Applications of Modelling and Internet Technologies in Marine Insurance Business Processes.
In: Proceedings of COMPIT05 International Conference, Hamburg, 2005, pp 77 -87
(līdzautors: L.Novickis).
3. Information Technologies In Reinsurance Activities. // Scientific proceegings of RTU,
Computer science, volume 17, RTU, RTga-2003. pp. 185-193. (in Latvian)
4. Modelling of Marine Insurance Business Processes. In: Proceedings of The International
Workshop on Harbour, Maritime and Multimodal Logistics Modelling & Simulation
"HMS2003", RTU, Riga - 2003. pp. 345-351 (līdzautori: L.Novickis, E.Viktorova,
V.Ragozin).
5. Customization of a Marine Insurance Information System. In: Application of Similation and
IT Solutions in the Baltic Port Areas of the Associated Candidate Countries. Edited by
Eberhard Blumel, Jacques Babot, Leonid Novitsky (Results of the IST -2001-33030 Project
BALTPORTS-IT), IUMI Ltd, Riga, June 2003, pp 224-233.
6. Internet Technologies In Insurance // Scientific proceegings of RTU, Computer science,
volume 13, RTU, RTga-2002. pp.101-108. (in Latvian)
7. An Insurance Company's Indicators of Activity Analysis and Modelling. // Scientific
proceegings of RTU, Computer science, volume 8, RTU, Rīga - 2001. pp. 163-170 (coauthors:
L.Novickis, I.Jackiva) (in LAtvian).
4.2. Presentation at conferences
1. 42 International Scientifisc Conference of RTU, Riga, October 11-13, 2001. Presentation: An
Insurance Company's Indicators of Activity Analysis and Modelling
2. 43 International Scientifisc Conference of RTU, Riga, October 10-14, 2002. Presentation:
Internet Technologies In Insurance
3. 44 International Scientifisc Conference of RTU, Riga, October 9-11, 2003. Presentation:
Information Technologies In Reinsurance Activities
41
4. The International Workshop on "Harbour, Maritime and Multimodal Logistics Modelling &
Simulation" - HMS 2003, Riga, 18-20 September 2003. Presentation: Modelling of Marine
Insurance Business Processes
5. 4th International Conference on Computer and IT Applications in the Maritime Industries COMPIT05 Hamburg, 8-11 May 2005. Presentation: Applications of Modelling and Internet
Technologies in Marine Insurance Business Processes
6. 46 International Scientifisc Conference of RTU, Riga, October 13-15, 2005. Presentation:
Organization of Business Processes in Insurance Information System Based on Web
Technologies
7. 5th International Conference on Optimal Research "Simulation and Optimization in Business
and Industry", Tallinn, 17-20 May 2006. Presentation: Selected Actuarial Models in Insurance
Information Systems.
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