Aggregation Functions and Personal Utility Functions in General Insurance

Abstract

The modeling of a utility function’s forms is a very interesting part of moderndecision making theory. We apply a basic concept of the personal utility theory ondetermination of minimal net and maximal gross annual premium in general insurance. Weintroduce specific values of gross annual premium on the basis of a personal utilityfunction, which is determined empirically by a short personal interview. Moreover, weintroduce a new approach to the creation of a personal utility function by a fictive gameand an aggregation of specific values by mixture operators.

Full text

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Acta Polytechnica Hungarica Vol. 7, No. 4, 2010

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Aggregation Functions and Personal Utility
Functions in General Insurance
Jana Špirková
Department of Quantitative Methods and Information Systems,
Faculty of Economics, Matej Bel University
Tajovského 10, 975 90 Banská Bystrica, Slovakia, jana.spirkova@umb.sk
Pavol Kráľ
Department of Quantitative Methods and Information Systems,
Faculty of Economics, Matej Bel University
Tajovského 10, 975 90 Banská Bystrica, Slovakia
Institute of Mathematics and Computer Science, UMB and MÚ SAV
Ďumbierska 1, 974 11 Banská Bystrica, Slovakia, pavol.kral@umb.sk
Abstract: The modeling of a utility function’s forms is a very interesting part of modern
decision making theory. We apply a basic concept of the personal utility theory on
determination of minimal net and maximal gross annual premium in general insurance. We
introduce specific values of gross annual premium on the basis of a personal utility
function, which is determined empirically by a short personal interview. Moreover, we
introduce a new approach to the creation of a personal utility function by a fictive game
and an aggregation of specific values by mixture operators.
Keywords: Utility function; Expected utility; Mixture operator; General Insurance
1 Introduction
This paper was mainly inspired by the books Modern Actuarial Risk Theory [3]
and Actuarial models – The Mathematics of Insurance [13]. The authors of the
above-mentioned books assume utility functions as linear utility ( )
quadratic utility ( )()2
wawu
−−=
describes and explains an application of the utility function in decision making in
a really interesting way. In this book also the generation of the utility function
wwu
=
,
, power utility ( )
c
wwu
=
, etc. Lapin in [5]

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J. Špirková et al. Aggregation Functions and Personal Utility Functions in General Insurance

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using information extracted from a personal interview is explained. A modern
theoretical approach to the utility function is also described by Norstad in [8].
We can find a very interesting discussion about utility functions in [4]. An
alternative approach to the determination of a utility function on the basis of the
aggregation of specific utility values can be found in [18].
However, in real life people do not behave according to the theoretical utility
functions. It is a psychological problem rather than a mathematical one. The
seriousness and also the uncertainty of a respondent's answers depend on the
situation, on the form of the asked questions, on the time which the respondents
have, and on a lot of other psychological and social factors. In our paper we
introduce the possibility of determining a personal utility function on the basis of a
personal interview with virtual money.
Moreover, we recall and apply one type of aggregation operators [2], the so-called
mixture operators –
g
M , the generalized mixture operators –
g
M , and the
specially ordered generalized mixture operators –
′
g
M
on the aggregation of so-
called risk neutral points, see [6-7], [9-11], [14-16].
This paper is organized as follows: in Section 2 we recall the basic properties of
utility functions and their applications in general insurance. In Section 3 we also
recall mixture operators and their properties, namely the sufficient conditions of
their non-decreasing-ness. In Section 4 we describe the personal utility function of
our respondent who took part in our short interview. Using this function we
calculate the maximal gross premium for a general insurance policy. In this
section we also describe an alternative approach, where a personal utility function
is determined through the result of a fictive game and theoretical utility functions.
The resulting utility function is then used for the computation of the maximal
gross premium. Moreover, we evaluate the minimal net annual premium by means
of the theoretical utility function for the insurer. Finally, some conclusions and
indications as to our next investigation about the mentioned topic are included.
2 Utility Functions
Individuals can have very different approaches to risk. A personal utility function
can be used as a basis for describing them. In general, we can identify three basic
personalities with respect to risk. The risk-averse individual, who accepts
favorable gambles only, a risk seeker, or in other words risk-loving individual,
who pays a premium for the privilege of participation in a gamble, and the risk-
neutral individual, who considers the face value of money to be its true worth.
Throughout most of their life people are typically risk averse. Only gambles with
high expected payoff will be attractive to them. The risk-averse individual’s

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marginal utility diminishes as the benefits increase, so that the risk-averse
individual’s utility function exhibits a decreasing positive slope as the level of
monetary payoff becomes higher. Such a function is concave, see Figure 1.
The behavior of a risk-loving individual is opposite. The risk-loving individual
prefers some gambles with negative expected monetary payoffs. Their marginal
utility increases. Each additional euro provides a disproportionately greater sense
of well-being. Thus, the slope of the risk-loving individual’s utility function
increases as the monetary change improves. This function is convex (see Fig. 2).
The utility function for a risk-neutral individual is a straight line. The utility is
equal to the utility of expected value. Risk-neutral individuals buy no casually
insurance since the premium charge is greater than the expected loss. Risk-neutral
behavior is typical for persons who are enormously wealthy.
Of course, a lot of people may be risk averse and risk loving at the same time,
depending on the range of monetary values being considered, which can be
illustrated using the behavior of the personal utility function of our respondent.
2.1 The Personal Utility Function
The fundamental proposition of the modern approach to utility is the possibility to
obtain a numerical expression for individual preferences. As people usually have
different approaches to risk, two persons faced with an identical decision may
actually prefer different courses of action. In this section we will discuss utility as
an alternative expression of payoff that reflects personal approaches.
Suppose that our respondent owns capitalw , and that he values wealth by the
utility functionu . The next Theorem 1, or in other words Jensen`s inequality,
describes the properties of the utility function and its expected value [3], (see also
Figure 2). It can be written as follows.
Theorem 1 [3] (Jensen's inequality)
If ( ) xu
is a convex function and X is a random variable, then the expected utility
is greater or equals to a utility value
( )
[] [ ]
()
XEuXuE
≥

with an equality if and only if ( ) xu
is linear with respect to X or
(1)
( )
X
0 var
=
.
From Jensen`s inequality and Figure 1 it follows that for a concave utility function
it holds
()
[][]
()
[ ]
()
XEwuXwEuXwuE
−=−≤−
. (2)
In this case the decision maker is called risk averse. He prefers to pay a fixed
amount [ ]
XE
instead of a risk amount X .

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J. Špirková et al. Aggregation Functions and Personal Utility Functions in General Insurance

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Figure 1
Concave utility function - risk averse approach


Figure 2
Convex utility function - risk loving approach
In the next part we illustrate whether to buy insurance or not by evaluating an
individual's decision. Now suppose that our respondent has two alternatives, to
buy insurance or not. Assume he is insured against a loss X for a premium P .
If he is insured, this means a certain alternative. This decision gives us the utility
value ()
Pwu
−
.
If he is not insured, this means an uncertain alternative. In this case the expected
utility is
()
[]
XwuE
−
.
From Jensen's inequality (2) we get
()
[]
(
EuXwuE
≤−
[]
)
[ ]
X
()()
PwuEwuXw
−=−=−
. (3)
Since a utility function u is a non-decreasing continuous function, this is
equivalent to
PP ≤
, where
P
denotes the maximum premium to be paid.
maxmax

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This so-called zero utility premium is a solution of the following utility
equilibrium equation
()
[]
PwuXwuE
−=−
.
The difference (
Pw−
certainty equivalent is defined as follows.
()
max
(4)
)
max
is also called the certainty equivalent -CE . In [ ] 3 the
Definition 1 The certainty equivalent is that payoff amount that the decision
maker would be willing to receive in exchange for undergoing the actual
uncertainty, taking into account its benefits and risks.
Remark 1 We recall that the expected utility is calculated by means of the well-
known formula
( )
X
[]
( )
x
i
n
i
i
puuE
⋅=∑
=1
, (5)
where
i
2 , 1
=
()
n xxxX
,...,,21
=
is a vector of the possible alternatives and
, are respective probabilities.
ip , for
n
,...,
Expected utilities can be calculated as function values of a linear function, which
is assigned uniquely by points A and B , where point A represents the worst
outcome and B the best outcome.
Remark 2 [ ] 5 When possible monetary outcomes fall into the decision maker's
range of risk averse, the following properties hold (see Figure 1):
[]
XwE EP
−=
are greater than their counterpart certainty
max
PwCE
−=
.
()
[]
XwuE
−
will be less than the utility of the respective
expected monetary payoff (
Pwu
−
.
1) Expected payoffs
equivalent
2) Expected utilities
)
max
3) Risk premiums
CE EPRP
−=
are positive.
If possible monetary outcomes fall into the decision maker's range of risk loving,
the following properties hold (see Figure 2):
[]
XwE EP
−=
are less than their counterpart certainty
max
Pw CE
−=
.
()
[]
XwuE
−
will be greater than the utility of the respective
expected monetary payoff (
Pwu
−
.
1) Expected payoffs
equivalent
2) Expected utilities
)
max
3) Risk premiums
CEEP RP
−=
are negative.

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The insurer with a utility function U and capital W , with insurance of loss X for
a premium P must satisfy the inequality
()
[]
( )
WUXPWUE
≥−+
, (6)
and hence for the minimal accepted premium
( )
PWUEWU
+=
min
P

()
[]
X
−
min
. (7)
2.2 The Risk Aversion Coefficient
On the basis of equation (3) we can evaluate a risk aversion coefficient. Let μ
and
expansion of the utility function u in
2
σ be the mean and variance of loss X . Using the first terms in the Taylor
μ−
w
, we obtain
()()( ) (
⋅
)( ) (
⋅
)2
2
1
XwuXwuwuXwu
−−
′ ′
+−−
′
+−≈−
μμμμμ
.
The expected utility from (
[
uE
)
Xwu
−
is given by
(
⎡
)
]
( ) (
+
μ
)()()()⎥⎦
⎤
⎢⎣
−
′ ′
⋅−+−
′
⋅−−≈
≈−
μμμμ
wuXwuXwuE
Xw
2
2
1

After some processing we get
()
[]
()()
μσμ−
′ ′
⋅+−≈−
wuwuXwuE
2
2
1
. (8)
The Taylor expansion of the function on the right side of equation (3) is given by
(
μμ
⋅−+−≈−
uPwuPwu
)
() ()
()
μ
−
′
w
maxmax
. (9)
From the equality of equations (8) and (9) we have
()()() (
+
)
()
μμμμσμ−
′
⋅−−≈−
′ ′
⋅+−
wuPwuwuwu
max2
2
1
. (10)
After some processing we get
(
(
)
)
μ
μ
σμ
−
′
−
′ ′
u
−≈
w
wu
P
2max
2
1
, (11)
where a risk aversion coefficient ( )
is given by
wr
of the utility function u at a wealth
μ
−
w

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Acta Polytechnica Hungarica Vol. 7, No. 4, 2010

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( )
w
(
(
)
)
μ
μ
−
′
−
′ ′
u
−=
w
wu
r
. (12)
()
2 max
2
1
σ
⋅
μμ
−+≈
wrP
. (13)
From (13) you can see that, if the insured has greater risk aversion coefficient,
then he is willing to pay greater premium.
3 Mixture Operators
In this part we review some mixture operators introduced in [6], [7], [9-11].
Suppose that each alternative x is characterized by a score vector x=(
[ ]n
1 , 0
, where
{} 1
−∈Nn
is the number of applied criteria. A mixture operator can
be defined as follows:
)∈
n
xx ,...
1

Definition 2 A mixture operator
[ ]
1 , 0
[ ] 1 , 0
[ ]
1 , 0
:
→
n
g
M
is the arithmetic mean
][
∞→
, 0
given by
weighted by a continuous weighting function
( )
( )
∑
=
i
1
where()
n xx ,...,
1
is an input vector.
:
g
()
⋅
=
n
i
ii
ng
xg
xxg
xxM
1,...,
,
(14)
Observe that due to the continuity of weighting function g , each mixture operator
M is continuous. Evidently,
g
M is an idempotent operator, [2], [6], [9-10].
Note that sometimes different continuous weighting functions are applied for
different criteria score, which leads to a generalized mixture operator, see [6], [9-
10].
Definition 3 A generalized mixture operator Mg: [ ]
( )
( )
∑
=
i
1
where()
n
xx ,...,
1
weighting functions.
g
[ ] 1 , 01 , 0
→
n
is given by
()
⋅
=
n
ii
iii
ng
xg
xxg
xxM
1,...,
,
(15)
is an input vector and g ()
n
gg ,...,
1
=
is a vector of continuous

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Obviously, generalized mixture operators are continuous and idempotent. A
generalized mixture operator based on the ordinal approach can be defined as
follows.
Definition 4 An ordered generalized mixture operator
g
M ′ : [ ][ ] 1 , 01 , 0
→
n
is
given by
()
( )
i
()
( )
)
( )
i
(
∑
=
i
⋅
=
′
g
n
i
ii
n
xg
xxg
xxM
1
1
,...,
,
(16)
where g (
( )
x
,...,
1
)
n
gg ,...,
1
)
is a non-decreasing permutation of an input vector.
=
( )
n
is a vector of continuous weighting functions and
(
x
An ordered generalized mixture operator is a generalization of an OWA operator
[19], corresponding to constant weighting functions
ii
wg =
,
[ ] 1 , 0
∈
i
w
,
1
1
=
∑
=
i
n
i
w
.
However, a mixture operator need not be non-decreasing. Marques-Pereira and
Pasi [6] stated the the first sufficient condition for a weighting function g in order
to a mixture operator (8) is to be non-decreasing. It can be written as follows:
[ ]][
∞→
, 01 , 0:
g
be a non-decreasing smooth weighting
function which satisfies the next condition:
( )( ) xgxg
≤
′
≤
0

Proposition 1 Let
(17)
for all
n∈
[ ] 1 , 0
∈
x
. Then
[ ]
1 , 0
[ ] 1 , 0 :
→
n
g
M
is an aggregation operator for each
N
, n >1.
We have generalized sufficient condition (17) in our previous work. In the next
part we recall more general sufficient conditions mentioned in [7], [14-16].
From (14) we see that
( )
1
x
( )
1
′
x
()( )
1
x
( )
1
x
( )
1
′
x
( )
1
x
0
2
1
1
1
1
1
1
≥
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅−⋅⋅+
=
∂
∂
∑
=
i
∑
=
i
∑
=
i
n
nn
g
g
gxggxgg
x
M
(18)
if and only if
( )
1
xg
( )
1
x
( ) (
1
′
x
)()
, 0
1
2
≥−⋅++
βα
xgg
(19)

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where
( )
∑
=
i
(
[
=
n
ix
)
g
2
α
and
( )
x
∑
=
i
]
.
⋅=⋅
n
ii
xg
2
βα
, and thus necessarily
[ ] 1 , 0
∈
β
and
( ) (
0
)( )
11,1
gngn
⋅−⋅−∈
α
Now it is easy to see that (17) implies (19). However, (19) is satisfied also
whenever
( ) ( ) ()
0
111
≥−⋅
′
+
β
xxgxg

[ ] 1 , 0
x
and each
[ ] 1 , 0
β
.
Because ( ) 0
′ xg
, (20) is fulfiled whenever
( ) () ( ) xgxxg
≤−⋅
′
≤
10
(20)
for each
1∈
1∈
1≥
for all
[ ] 1 , 0
∈
x
.
We have just shown a sufficient condition more general than (17).
[ ]]
→
, 0 1 , 0:
g
function which satisfies the condition:
( ) () ( ) xgxxg
≤−⋅
′
≤
10

Proposition 2 Let
[
∞
be a non-decreasing smooth weighting
(21)
for all
n∈
[ ] 1 , 0
∈
x
. Then
[ ]
1 , 0
[ ] 1 , 0:
→
n
g
M
is an aggregation operator for each
N
, n >1.
Moreover, we have improved sufficient condition (21), but constrained by n .
Proposition 3 For a fixed
smooth weighting function satisfying the condition:
Nn∈
, n >1, let
[ ]
1 , 0
]
, 0
[
∞→
:
g
be a non-decreasing
( )
) ( )
g
⋅
1
(
( )
x
( ) (
x
′
) xgg
n
xg
−
−⋅≥+
1
1
2
(22)
for all
[ ] 1 , 0
∈
x
.Then
[ ]
1 , 0
[ ] 1 , 0 :
→
n
g
M
is an aggregation operator. In the next
proposition we introduce a sufficient condition for the non-decreasing-ness of
generalized mixture operators.
Proof. Minimal value of ( ) ( ) (
⋅
′
+
11
xxgxg
i. e., it is ( )( ) () 1
111
−⋅
′
+
xxgxg
. Therefore, (19) is surely satisfied whenever
)
β
−
1
for
[ ] 1 , 0
∈
β
is attained for
1
=
β
,
( )
1
x
α
( )
1
x
( ) (
1
x
′
)
1
2
1 x
−
gg
g
⋅≥+
.
Suppose that (22) holds. Then

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– 16 –
( )
1
x
α
( )
1
x
( )
) ( )
1
g
⋅
(
( ) (
1
x
′
)
1
1
2
2
1
1
xg
n
xg
−
g
g
−⋅≥≥+
,
i. e., (19) is satisfied and thus g is a fitting weighting function.
□
In the next proposition we introduce a sufficient condition for the non-decreasing-
ness of generalized mixture operators.
Proposition 4 For a fixed
non-decreasing smooth weighting functions, such that
( )
( )
g
ij
≠
Nn∈
, n >1,
ni
,...,2 , 1
=
, let
[ ]
1 , 0
]
, 0
[
∞→
:
i g
be a
( )
x
( ) (
x
) xgg
xg
i
′
i
j
i
−⋅≥+
∑
1
1
2
(23)
for all
[ ] 1 , 0
∈
x
. Then
[ ]
1 , 0
[ ] 1 , 0 :
g
→
′
n
M
, where g ()
n
gg ,...,
1
=
, is an
aggregation operator.
4 Maximal Premium Determined by a Personal
Utility Function
In practice, the utility function can be determined empirically by a personal
interview made by a decision maker. In our opinion, there are at least two suitable
ways to do this. The first one is based on an interview which provides us with
probabilities estimated by an interviewed subject; the second one on a game with
known probabilities where the interviewed subject gives us only information
about a personal breaking point. The personal breaking point is the amount of
wealth at which our individual is changed from risk averse to a risk seeker, or
vice versa. An appropriate curve for a risk averse and risk loving part is then
selected from the theoretical utility functions.
4.1 A Personal Utility Function – a Probability-oriented
Approach
Following this approach a personal utility function can easily be constructed from
the information gleaned from a short interview using the classical regression
analysis. The decision maker can use this function in any personal decision
analysis in which the payoff falls between 0 and 30000 €. Now we recall the
interview, which is compiled as follows [4].

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– 17 –
Let us suppose you are owner of an investment which brings you zero payoff now
or a loss of 30000 €. However, you have a possibility to step aside from this
investment under the penalty in the amount of a sequence: A: 1000 €, B: 5000 €,
C: 10000 €, D: 15000 €, E: 25000 €. Your portfolio manager can provide you
with information expressing the probability loosing the 30000 €. Think. What
would be the biggest probability of the loss, so that you retain the above
mentioned investment? Only a few well-proportioned graphic points are required.
From our interview we took the respective person's data points(
()
75 . 0 ,5000
−
)
60. 0 ,10000
−
, 0 ,15000
−
and created the appropriate utility function of our respondent as shown in Fig. 3.
This curve has an interesting shape that reflects our respondent’s approach to
risk. The different personal utility functions for our respondent were created
using the IBM SPSS 18.0 system for the purpose of comparison. The maximal
premium
P
was calculated by
u
inverse function to the utility equilibrium
equation (4)
()
[]
()
XwuEuwP
−−=

) 1 , 0
)
, (
−
) 8 . 0 ,
00. 0 ,
1000
−
,
, , (
, ()
60
, (
40. 0 ,25000
−
, ()
30000
max1
−
−1max
(24)
with system Mathematica 5.

Figure 3
A utility function and the expected utility of our respondent (the function 2 from Table 1)
Utility functions are used to compare investments mutually. For this reason, we
can scale a utility function by multiplying it by any positive constant and (or)
transfer it by adding any other constant (positive or negative). This kind of
transformation is called a positive affine transformation. All our results are the
same with respect to such a transformation. Quadratic and cubic utility functions
are written in Table 1. On the basis of statistical parameters (adjusted R square, p-
values) we can assume that the cubic function is the best fitting function.
Moreover, Table 1 also consists of appropriate expected utilities expressed by
linear functions.

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Remark 3 Expected utilities (for the utility functions from Table 1) can be
calculated by means of a linear function which is assigned uniquely by points
()()
30000,30000
−−
u
and
( )()
0 , 0 u
, or by the formula (8), alternatively. In both
cases we get the same values for the expected utilities.
In Figure 3 you can see the personal utility function of our respondent, as well as
three interesting points that are highlighted (also in Table 3). Maximal premium
max
a
P
represents the area where our respondent is risk averse, and
he is risk seeking (loving).
max
s P
, where
Table 1
A utility function and the expected utility

1
A utility function and the expected utility
( )
( )
[]
1002. 27
⋅=
xuE
( )
( )
[]
10 23. 3
⋅=
xuE
904 . 0
904 . 010856. 1
+
10 820. 2
−
6
5210
+
+⋅⋅=
−
−−
x
xxxu

2
971. 0
971. 010 588. 710 383. 510 310. 1
5
5293 13
+
+⋅+⋅+⋅=
−
−−−
x
xxxxu

From Table 2 you can see that the insured person is willing to pay more than the
expected loss to achieve his peace of mind''.
Table 2
The expected utility and maximal premium with respect to a quadratic function

p
Probability


[ ] uE

with respect to
quadratic function
0.904000
0.895894
0.863470
0.822940
0.741880
0.660820
0.579760
0.498700
0.417640
0.336580
0.255520
0.174460
0.093400

max
P

(€)

X
[ ]
(€)
E

min
P

(€)

0.00
0.01
0.05
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.0 0.0
0
433.9
2115.7
4110.7
7808.5
11197.3
14336.3
17293.4
20079.0
22725.4
25251.4
27672.4
30000.0
300.0
1500.0
3000.0
6000.0
9000.0
12000.0
15000.0
18000.0
21000.0
24000.0
27000.0
30000.0
301.69
1508.10
3015.34
6027.24
9035.69
12040.70
15042.40
18040.60
21035.50
24027.00
27015.20
30000.00

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Acta Polytechnica Hungarica Vol. 7, No. 4, 2010

– 19 –
Table 3
The expected utility and maximal premium with respect to a cubic function
[ ] uE

with respect to
cubic function
0.00 0.971252
0.01 0.961575
0.05 0.922868
0.10 0.874485
0.20 0.777717
0.30 0.680950
0.37 0.613140
0.40 0.584182
0.50 0.487415
0.60 0.390647
0.70 0.293880
0.80 0.197112
0.90 0.100345
1.00 0.003577 30000.00
p
Probability
max
P

(€)
[ ]
X
(€)
E

min
P

(€)

0.0 0.0
0,00
128.7
669.1
1411.9
3234.4
6021.0
11102.2
18165.0
22797.1
25033.0
26643.3
27938.0
29036.7
300.0
1500.0
3000.0
6000.0
9000.0
11102.2
12000.0
15000.0
18000.0
21000.0
24000.0
27000.0
30000.0
301.69
1508.10
3015.34
6027.24
9035.69
10986.70
12040.70
15042.40
18040.60
21035.50
24027.00
27015.20
30000.00
We determine the minimal premium by means of (7) with respect to the utility
function for insurer ( )
xxU
ln
=
with his basic capital
30000
=
X
€.
51 .2655513
=
W
€ and loss
The equation can be rewritten as follows:
( )
PWUpWU
+⋅=
() (
+
) (
U
⋅
)
min min
1
PWpX
+−−
(25)
and hence
(
W
=
) ()()
pp
PWXPW
−
+⋅−+
1
min min
. (26)
We determined individual minimal premiums with corresponding probability
with the system Mathematica 5.
4.2 A Personal Utility Function– a Game-based Approach
Our expectation that our subject can appropriately estimate probabilities is the
main drawback of the previous approach. In fact, we can doubt whether
somebody without appropriate knowledge about probabilities can provide us with
reliable answers. In order to avoid this problem we can assume a game with
probabilities which are easy to understand, e.g. games based on coin tossing. Let
us assume the following game. You have two possibilities: either to toss a coin

Page 14

J. Špirková et al. Aggregation Functions and Personal Utility Functions in General Insurance

– 20 –
with two possible results, head means you will get 10 €, tail means you will get
nothing; or to choose 5 € without playing. What is amount of money for which
you will start (stop) playing? It is easy to see that the expected value is the same
in both cases and we make our decision about playing with respect to our
personal utility function. The point at which we give up (stop) playing is the
above mentioned breaking point. For simplicity we will assume the quadratic
utility function. This approach allows us to combine easily personal utility
functions to a group utility function using aggregation operators. The group
utility function can represent a specific group of customers of our insurance
company. Let us assume three utility functions based on different breaking, a
utility function for
29900
=
x

( )
x
()
()
⎪⎩
⎪⎨
⎧
≤≤+−⋅⋅
≤≤
30000
+−
2
⋅⋅−
=
−
−
29900 for 5 . 029900 105
29900 0 for 5 . 0 2990010
x
592778604. 5
5
2
10
1
x
xx
u
, (27)
a utility function for
29800
=
x

( )
x
()

()
⎪⎩
⎪⎨
⎧
≤≤+−⋅⋅
≤≤+
for
−⋅⋅−
=
−
−
3000029800 5 . 0 298001025. 1
29800 0
x
for 5 . 0 29800
2
10
x
63037701. 5
5
2
10
1
xx
u
, (28)
A utility function for
29650
=
x

( )
x
()
()
⎪⎩
⎪⎨
⎧
≤≤+−⋅⋅
≤≤
x
+
for
−⋅⋅−
=
−
−
30000 296505 . 0 29650104081 . 0
29650 0 for 5 . 029650
2
10687489514. 5
5
2
10
1
x
xx
u
. (29)
To construct the combined utility function we can use for example an ordered
generalized mixture operator
g
where
( )
8 . 02 . 0
1
+=
xxg
,
5 . 05 . 0
2
+=
xxg
that the selected weighting functions satisfy the conditions required for of non-
decreasing aggregation operators.
M ′ with weighting vector g (
( )
3
g
)
321
,,
ggg
=
,
( )
and
25. 0 75 . 0
+=
xx
. Let us note
Values 29900, 29800, 29650 we transform to the unit interval and aggregate them
by means of
g
M ′ . We obtain an aggregated value
transformation we have point of division
neutral to risk. On the basis of this division point we can create a new combined
utility function
(
()
⎪⎩
+−⋅⋅
41, 2982710 6785 . 1
x
0.00575309
g=
′
41.
M
, and after
29827
=
x
, where the insured is
( )
x
)
⎪⎨
⎧
≤≤
≤≤+
for
−⋅⋅−
=
−
−
3000029827,41 5 . 0
41, 29827 0 for 5 . 041, 2982710 620033657. 5
2
5
2
10
x
xx
u
(30)
and appropriate expected utility
( )
x
⎪⎩
⎪⎨
⎧
≤≤⋅
≤≤⋅
=
−
−
3000029827,41 for 10 897039. 2
41,29827
x
0 for
x
106763 . 1
3
5
x
E
(31)

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Acta Polytechnica Hungarica Vol. 7, No. 4, 2010

– 21 –
Table 4
The expected utility and maximal premium with respect to a function (28)
()
[]
XwuE
−

X
0
50
100
172,59
200
300
1500
3000
6000
9000
12000
15000
18000
21000
24000
27000
30000
Xw−
30000,00
29950,00
29900,00
29827,41
29800,00
29700,00
28500,00
27000,00
24000,00
21000,00
18000,00
15000,00
12000,00
9000,00
6000,00
3000,00
0,00

max
Pw−
30000,00
29972,90
29939,30
29827,40
28923,70
27877,90
23535,00
20644,10
16643,50
13600,90
11044,90
8797,37
6767,79
4902,97
3168,28
1539,73
0,00

max
P
0,00
27,10
60,70
172,60
1076,30
2122,10
6465,00
9355,90
13356,50
16399,10
18955,10
21202,63
23232,21
25097,03
26831,72
28460,27
30000,00

min
P

1,000000
0,855148
0,710296
0,500000
0,499541
0,497864
0,477748
0,452604
0,402315
0,352025
0,301736
0,251447
0,201157
0,150868
0,100579
0,050289
0,000000
0
50
100
172,59
200
300
1500
3000
6000
9000
12000
15000
18000
21000
24000
27000
30000
The minimal premium we evaluated on the basis of formula (7) with the utility
function for insurer ( )() 1ln
+=
xxU
with the system Mathematica. From Table 4
and also from the formula (7) you can see that the minimal premium is given by
the size of the expected loss. A newly-gained utility function would be required
for evaluating a decision with more extreme payoffs or if our respondent's
attitudes change because of a new job or lifestyle change. Moreover, the utility
function must be revised from the viewpoint of time.
Conclusions
We have shown two approaches to creating a personal utility function and we
have calculated the maximum premium against the loss of 30000 € with respect
to it. We think that the personal utility function of an insured person would be
very important for an insurer. On the basis of the personal utility function the
insurer would know what approaches to risk the customers have and thus, how
they will behave towards their own wealth. Creating a utility function for the
insurer is very difficult. Moreover, in our next work we want to investigate the
insurer's utility function and we want to determine the minimal premium against
the loss of 30000 € with respect to a concrete real insurer's utility function.
Acknowledgement
This work was supported by grant VEGA 1/0539/08.