In this paper a brief overview of the structure and the possibilities of the program CRAC 2.0 is given. It will be shown how sectors can be determined in order to use the hierarchical model that is built into the software. Furthermore a general structure for defining insurance problems to be solved by CRAC 2.0 will be discussed. Mathematics Subject Classification: 62P05.the software package CRAC 2.0, Jewell’s hierarchical model, practical insurance problems, structure of CRAC – input.

Virginia ATANASIU

English

Research Papers in Economics

Revista Informatica Economică nr. 1(45)/2008 
 
28
Explicit Description of the Input Data for the Program CRAC 2.0 Used in 
the Applications of the Credibility Theory 
 
Virginia ATANASIU 
Department of Mathematics, Academy of Economic Studies 
virginia_atanasiu@yahoo.com  
 
In this paper a brief overview of the structure and the possibilities of the program 
CRAC 2.0 is given. It will be shown how sectors can be determined in order to use the hie-
rarchical model that is built into the software. Furthermore a general structure for defining 
insurance problems to be solved by CRAC 2.0 will be discussed. 
Mathematics Subject Classification: 62P05. 
Keywords: the software package CRAC 2.0, Jewell’s hierarchical model, practical insurance 
problems, structure of CRAC – input. 
 
ntroduction 
In this paper we will discuss some prac-
tical insurance problems that can be solved 
by means of credibility theory. All numerical 
results in this paper were obtained using the 
software package CRAC 2.0, which uses 
Jewell’s hierarchical model. Here we con-
sider applications of credibility theory deal-
ing with real life situations, and imple-
mented on real insurance portfolios. 
 
1.   Structure of the software 
Introduction and philosophy 
The CRAC 2.0 input is designed in such a 
way that quite a number of practical prob-
lems can be solved. We describe the input 
format for the different variables. An exam-
ple will illustrate the general guidelines. 
The CRAC – software is based on a two – 
level semi – linear hierarchical credibility 
model. It is semi – linear in the sense that 
some (fixed) transformation of the claim fig-
ures is used. It is two – level hierarchical, as 
the individual contracts on the down – level 
are grouped together in sectors on an inter-
mediate level. 
So decisions can be made on three different 
levels: 
-the top – level based on collective estima-
tors for the whole portfolio; 
-the in – between sector level based on the 
estimators for the different sectors defined 
by the user; 
-the down – level based on the individual es-
timators for all contracts. 
For every contract in the portfolio (a contract 
can also be an average of a group of con-
tracts), data for T  observation periods can 
be given. For each contract the system al-
lows for missing data. For a successful ap-
plication of the software, T should be at least 
3. 
1.2 Subdivision of the portfolio into sec-
tors 
The basic idea behind hierarchical credibility 
is described in Subsection 1.4. Instead of 
working with individual contracts at one side 
and the portfolio as a whole at the other side, 
an in – between or sectorial level is created. 
In this way we obtain three decision levels: 
global (for the portfolio as a whole) denoted 
by  m , sectorial, denoted by 
a
p N  and indi-
vidual, denoted by 
a
pj M . The portfolio is 
split into a number of sectors and each of 
these sectors contains individual contracts. It 
is evident that this subdivision of the portfo-
lio into smaller subgroups will be different 
from one application to another. The deter-
mination of the sectors could be performed 
based on classical multivariate statistical 
techniques such as discriminant analysis and 
cluster analysis. On the other hand it should 
be noted that for several reasons it is just not 
possible to construct an ideal subdivision of 
the portfolio. That should not prevent us 
from actually making the non – ideal subdi-
vision and applying credibility theory. 
The sectors can be determined by CRAC 2.0 
according to either one two or three different 
IRevista Informatica Economică nr. 1(45)/2008 
 
29
characteristics. Each individual record con-
tains up to 3 different variables ranging from 
1 to  1 S ,  2 S , and  3 S  respectively. Then the 
following subdivisions are possible: 
-on the basis of one of these variables result-
ing in  1 S ,  2 S , or  3 S  sectors; 
-on the basis of two combined criteria result-
ing in two subsequent subdivisions in cells 
according to the chosen pairs of S - va-
riables. This gives for instance a total of 
2 1 S S ⋅  cells if the first two criteria are used; 
-on the basis of all three variables, producing 
3 2 1 S S S ⋅ ⋅  sectors. 
Consider the two following examples. 
 
Example 1: Individual fire insurance 
A subdivision into sectors might be based on 
the following criteria: 
• owned house or rented house; 
• size of the house: small, medium or big 
house; 
• geographical position of the house (dis-
trict, state,…). 
The choice of this subdivision may be 
brought on by statistical differences of the 
various groups, or simply imposed by the 
management. 
 
Example 2: Health insurance 
We consider the principle of this type of in-
surance (to be discussed in Section 2) to be 
that a firm insures all its employees as a 
group for medical costs. So the client of the 
insurance company is the firm and not the 
individual employee. A subdivision of a 
portfolio of such contracts might be based on 
the following criteria: 
• kind of activities of the firm: services, 
chemical, agricultural,…; 
• geographical position of the firm: e.g. 
Northern, Central and Southern part of the 
country; 
• turnover of the firm: a number of classes 
could be defined. 
The total portfolio in these two examples can 
then be split up according to one, two or all 
three of these criteria. 
 
1.3 Definition of the problem; a standard 
procedure 
It is obvious that not only the sectors but al-
so the quantity to be calculated will be dif-
ferent from one application to another. In 
fire insurance we normally will be interested 
in calculating a permillage of the capital in-
sured, whereas in the problem of health in-
surance we want to calculate a premium (in 
some currency) per individual insured. 
For this reason a standard procedure is pro-
vided within the CRAC – software. This 
procedure allows the solution of quite a 
number of practical problems. 
The standard form for defining problems is 
as follows: 
( ) ( )
() pjr
pjr
pjr ator deno factor
deductible numerator
X
min −
−
=
+  (1), 
where  { } q q , 0 max = + ,  p  denotes the sector 
this policy belongs to,  j  denotes the con-
tract and r  denotes the observation period. 
For the numerator we usually take claim 
payments, while the denominator expresses 
some measure of exposure like premium vo-
lume or number of participants. We will illu-
strate this standard form for the two exam-
ples mentioned previously. 
 
Example 1: Individual fire insurance 
When we are interested in calculating a risk 
premium as a permillage of the capital in-
sured for each sector as defined earlier, the 
numerator is the amount paid by the insur-
ance company, the denominator is the capital 
insured. By choosing the scaling factor as 
0,001 the quantity  pjr X  represents the risk 
permillage we are interested in. 
 
Example 2: Health insurance 
In this example we calculate a risk premium 
per individual, so we have to take the nume-
rator as the amount paid by the insurer; the 
denominator represents the number of in-
sured employees in the firm. 
 
1.4 Structure of CRAC – input 
An input – file to be used in CRAC 2.0 must 
contain the following data for each record: 
1.Policy number or identification number. 
2.Specification of the sector. Revista Informatica Economică nr. 1(45)/2008 
 
30
Three fields containing the values according 
to which the subdivisions in sectors is orga-
nized. These variables range from 1 to  1 S , 
2 S  or  3 S . 
3.Number of observations. 
This value must be greater than zero and at 
most equal to T . 
4.For each observation period i 
( T i − − − = ,...., 2 , 1 ) the following fields: 
• An indicator which is set to zero in case no 
observation for the period i is available; 
otherwise it is 1; 
• A numerator for the observation period i; 
• A denominator for the observation period 
i; 
This field can only contain a value of zero 
when field 4a indicates a non – available pe-
riod; 
• A weight factor for the observation period 
i; 
This field reflects the importance in the port-
folio of each individual contract. 
1.5 A practical example 
In this subsection we will combine all ele-
ments from the previous subsections into one 
example. The goal of this is to illustrate how 
practical problems can be translated into 
problems that can be handled by the CRAC 
– software. In the following sections we will 
concentrate on the results obtained rather 
than on the practical problems arising from 
working with this software package. 
To determine the risk premium for individu-
al fire insurance, assume statistical informa-
tion sis available for the last four years. The 
risk premium must be expressed as a permil-
lage of the capital insured. The user judges a 
subdivision into the following sections ne-
cessary: 
1)  houses in the North (N) and houses in the 
South (S); 
2)  houses with an insured capital of less 
than $100.000 (SH), from $100.000 up to 
$250.000 (MH), and more than $250.000 
(BH); 
3)  houses in the city (IC) and houses out-
side the city (OC). 
 
It is not the intention to set up an individual 
tariff. The only aim of the user is to set up a 
sectorial investigation. The user should make 
the following selections: 
1. 4 = T ; 
2.no individual calculations; 
3. = 1 S 2 (N/S),  = 2 S 3 (SH/MH/BH) and 
= 3 S 2 (IC/OC); 
4.the variables defining the subdivision in 
sections; when the three criteria are com-
bined, the following sectors appear: 
 
-sector 1:  N  SH  IC  -sector 7:  S  SH  IC 
-sector 2:  N  SH  OC  -sector 8:  S  SH  OC 
-sector 3:  N  MH  IC  -sector 9:   S  MH  IC 
-sector 4:  N  MH  OC  -sector 10:  S  MH  OC 
-sector 5:  N  BH  IC  -sector 11:  S  BH  IC 
-sector 6:  N  BH  OC  -sector 12:  S  BH  OC 
 
5.names of the sectors must be given. 
In this situation a typical record for an indi-
vidual contract would have the following 
layout: 
-policy number: not required in this case 
-three numerical zones, the first one contain-
ing the value 1 (N) or 2 (S); the second one 
containing the values 1 (SH), 2 (MH) or 3 
(BH); the third one containing the values s1 
(IC) or 2 (OC). 
-the number of observation periods available 
( T ≥ ), 
-for observation period i (for 
T i − − − = , , 2 , 1 K ) 
 1 (observation available) or 0 (observation 
not available); 
 numerator, the claim size for period i; 
 denominator, the capital insured for time 
period i; 
 weight, e.g. the premium paid in period i. 
 
Because the estimate for the risk premium 
has to be expressed in terms of a permillage Revista Informatica Economică nr. 1(45)/2008 
 
31
of the capital insured SF has to be put equal 
to 0.001. A total of 75 bytes are necessary in 
this case for each record. An example of a 
record looks as follows: 
 
 
010302200000000000000000100010030000006001000000280000056000000000000000000 
             | |                                 |                                  |                                 | 
 
From this record we see that this contract be-
longs to the North (N), it is a big house (BH) 
outside the city (OC). There are data for two 
years: 
 year -2: claim size of 000100, capital in-
sured equal to 300.000 and premium of 
0600; 
 year -3: claim size of 000000, capital in-
sured equal to 280.000 and premium of 
0560. 
 
2.   An example in health insurance 
Description of health insurance  
In the example of health-insurance consi-
dered here, a firm insures all its employees 
as a group for medical costs. The clients of 
the insurance company are firms, not indi-
vidual employees. 
It has been decided that the subdivision of 
the portfolio of these contracts will be based 
on the following criteria (the values between 
brackets give the corresponding sector num-
ber in the CRAC software). 
• kind of activities of the firm: metal (1), 
chemical (2), agricultural (3), kind 4 (4), 
kind 5 (5), kind 6 (6) and food (7); 
• geographical positions of the firm: North 
(1), Central (2), South (3); 
• number of employees of the firm: we de-
fined 7 different groups from small (1) to 
immense (7). 
Notice that the portfolio could be split up ac-
cording to three criteria: the first and the 
third have values ranging from 1 to 7, the 
second has values ranging from 1 to 3. By 
combining all criteria we obtain a subdivi-
sion into 147 = 7·3·7 sectors. 
Example of the input data  
In the table below, an example is given of 
how the input data might look. The first 
record of this file contains information on 
policy Z 1024. The three following fields, 
each of length 2, have the values 030302, in-
dicating that this policy covers an agricultur-
al firm (03) in the Southern part (03) of the 
country; on the basis of the number of em-
ployees, the firm is classified as „size 2” 
(02). 
Next, we see that there is only one observa-
tion period available, namely period -3. In 
this period a amount of 00092357 has been 
paid; the number of employees insured 
equals 000045 and the weight has been cho-
sen equal to this number of employees. The 
interpretation of the other records is 
straightforward. 
 
Z1024 03 03 02 1 0 00000000 000000 000000 0 00000000 
Z1028 06 01 03 3 1 00194514 000547 000547 1 00095224 
Z1185 06 01 05 3 1 00039453 001036 001036 1 00141023 
Z1188 06 03 05 3 1 00030216 000621 000621 1 00258241 
 
000000  000000 1  00092357  000045 000045 
000547  000547 1  00109245  000547 000547 
001036  001036 1  00363791  001036 001036 
000621 000621  1 00453345  000621  000621 
 
The input data has to be checked first for 
various errors, such as denominators that are 
zero, sectors that do not exist, and so on. In 
our demonstration run 3 out of 703 policies 
had to be rejected in advance. 
 
 Revista Informatica Economică nr. 1(45)/2008 
 
32
Construction of sectors 
We could make a subdivision of this portfo-
lio in seven different ways: 
1.based on kind of activities: 7 sectors; 
2.based on geographical position: 3 sectors; 
3.based on a number of employees: 7 sec-
tors; 
4.based on a combination of (1.) + (2.):7 × 3 
= 21 sectors; 
5.based on a combination of (1.) + (3.):7 × 7 
= 49 sectors; 
6.based on a combination of (2.) + (3.):3 × 7 
= 21 sectors; 
7.based on a combination of (1.) + (2.) + 
(3.):7 × 3 × 7 = 147 sectors. 
In this application we will use subdivision 
3.. In the following section we will comment 
on the credibility results generated by the 
software for this particular case and subdivi-
sion. 
 
Conclusions 
We give a rather explicit description of the 
input data for the program CRAC 2.0 used, 
only to show that in practical situations there 
will always be enough data to apply credibil-
ity theory to a real insurance portfolio. The 
point we want to emphasize is that practical 
application of credibility theory is feasible 
nowadays using appropriate software. In this 
paper we consider applications of credibility 
theory dealing with real life situations, and 
implemented on real insurance portfolios. 
Though more examples could be given, we 
limit ourselves to the introduction of a prob-
lem of health insurance, and (briefly) one in 
fire insurance. In these examples we try to 
demonstrate what kind of data is needed to 
apply credibility theory. The examples show 
that credibility theory is really a useful tool - 
perhaps the only existing tool - for such in-
surance applications. The fact that it is based 
on complicated mathematics, involving con-
ditional expectations, needs not bother the 
user more than it does when he applies sta-
tistical tools like SAS, GLIM, discriminant 
analysis, and scoring models. These tech-
niques can be applied by anybody on his 
own field of endeavor, be it economics, 
medicine, or insurance.  
References 
[1]. De Vylder, F. & Goovaerts, M.J., Semi-
linear credibility with several approx-
imating functions, Insurance: Mathematics 
and Economics, 4, 155- 162. (Zbl. No. 
0167-6687), 1985. 
[2].Gerber, H.U. & Jones, D.A., Credibility 
formulas of the updating type, in “Credibil-
ity theory and application”, Proceedings of 
the Berkeley Actuarial Research Conference 
on credibility, Academic Press, New York, 
89-109, 1975. 
[3].  Gerber, H.U.,  Credibility for Esscher 
premiums, Mitteilungen der VSVM, 80, 
307-312, 1980. 
[4].  Goovaerts M. J.,  Kaas R.,  Van Heer-
waarden A., E., Bauwelincks T., Insurance 
Series, Volume 3, Effective Actuarial Me-
thods, Elsevier Science Publishers B.V., 
187-211, 1990. 
[5].  Pentikäinen T., Daykin C. D., Pesonen 
M.,  Practical Risk Theory for Actuaries, 
Chapman & Hall, 1993. 
[6]. Sundt B., An Introduction to Non-Life 
Insurance Mathematics,  volume of the 
“Mannheim Series”, 22-54, 1984. 

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Explicit Description of the Input Data for the Program CRAC 2.0 Used in the Applications of the Credibility Theory

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Explicit Description of the Input Data for the Program CRAC 2.0 Used in the Applications of the Credibility Theory

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Research Papers in Economics