The Applications of Utility Theory in Insurance Industry

Y AN Li-hua**[1]**

WANG Yong-mao

WANG De-hua

WEN Xiao-nan

**Abstract:
** In this
paper, The Applications of Utility Theory in insurance industry are discussed
from two ways.First of all we consider the insurance pricing from both insurers
and insured , and makes the strict explanation from the value example to the
St. Petersburg paradox. .Then we discuss insurance pricing between the risk
swap agreement insurers and give the value example.

**Key
words:** Utility
Theory, Utility function, Insurance premium,expected Utility, Risk TheoryLIU, WANG & GUO. 2007

1.
Introduction

The insurance pricing is always
the core of insurance business. Although the price pattern is commonly fixed by
pure insurance premium and attachment insurance premium in insurance practice
and books , theoretically speaking, the insurance product is the same as other commodity.
Its price is essentially decidied by the market supply-demand relation. What is
particularly is that it is not to fix price for the visible product merely, but
to invisible risk. Here the risk can be understanded as the adjustment or the
loss random variable(S.M.Ross. 2005). As the matter stands, the insurance
pricing in formally is to establish one kind of price measures, which is
possible to use one kind of precise quantity (insurance premium) to weigh an
indefinite loss. So we discuss the insurance pricing question from the economic
utility theory in this paper.

2. Discussing insurance pricing separately
from insurer and isured's angle(QIN Gui-xia.2008)

First, we analyse the insurance
pricing from the insurer and insured's value structure separately. Suppose somebody
has the property valuing _{}, but this property faces some kind of latent loss, which is
expressed as a random variable_{}, _{}.The probability distribution records is _{}.Our question is how many insurance premiums he have to take
out for this insurance? According
to Utility theory (WANG, JIANG & LIU. 2003), the
fewer the insurance premium _{} is, the better for
the insured The highest insurance
premium is the solution when insurance effectiveness was equal to insurance
effectiveness not to take out.

If the insured is willing to take
out insurance, he only loses the insurance premium whether the loses occur or
not. And the insured still has_{}, supposes its effectiveness for the insured is _{};If the insured does not take out insurance, in fact its
property is the random variable_{}, we record the effectiveness of this random variable as_{}. Therefore, to the property owner, the insurance premium
should satisfy:

_{}

_{}bigger, _{}is smaller, and insurance effectiveness _{} is also smaller.
When the equal sign is established, it does not matter whether to participate or
not . The highest insurance premium _{} which can be
accepted by the insured is the solution when the equation equal sign is
established.

In another inspect, considering from
insurer's angle, if insuring, the insurer may increase an insurance premium
income _{} in original
wealth foundation _{}, but undertake the risk for the insured. Its wealth becomes
the random variable _{}. How many insurance premiums should the insurer charge to insure
the property owner's risk? Similarly,
the higher_{} is, the better is to the insurer. Suppose the insurer
records the determination quantity and random variable effectiveness for _{} and _{} separately .Then
the reasonable premiums should satisfy the following effectiveness inequality:

_{}

The smaller _{} is, the smaller _{}is, When the equal sign establishes, the insurance has not
any attraction. Therefore the insurer is willing to accept the lowest insurance
premium _{}_{} which can be
accepted by the insurer. And G is the solution when the equation equal sign
establishes.

Therefore, only the highest
insurance premium _{} which the insurer
is willing to pay is more than the lowest insurance premium _{}which the insurer is willing to accept , could a reasonable
insurance contract be situated between _{} and _{} . Figure 1 shows
the relations among critical insurance premium_{},_{}and pure insurance premium _{} as well as actual
price _{}.

By Utility Theory, most people hate
the risk. By the Jensen inequality (XIE, HAN. 2000), the loathing risk's policy
holder is willing to pay higher insurance premium to take out insurance, namely_{}. If _{}, it is unable to finalize a deal.

_{}

Figure 1_{}_{},_{}and _{}

The following is a famous gambling example using the utility function to fix the safe product price. Although it is not a direct safe policy-making question, it contains the same essence.

St. Petersburg paradox (GUO.2004)
There is a fable that one kind of gambling is popular in the St. Petersburg in
the past street corner. The rule is all participant prepaid certain number
money, for instance 100 rubles, then threw the cent, the gambling was
terminated when the person surface dynasty presented first time; If the person
surface dynasty did not present until the _{} talent, the
participant took back _{} rubles. The
question is that whether the policymaker take part in the gambling

Suppose the cent is even. The
probability that the person surface dynasty does not present until the _{} talent is _{}.The corresponding repayment value is _{},_{}.Therefore, the average repayment of participating the
gambling is_{}, but the average repayment of not participating the gambling
is obviously 0.It looks like that the policymaker can win (on average) the infinite
many rubles by spending 100 rubles. It seems that participating the gambling
is absolutely worthwhile. But the actual situation was contrary; extremely few
can take back 100 rubles above situations.

In fact, according to utility
theory, what we should consider is the Utility function _{}of policymaker , not
the amount value _{} itself, and
policymaker's wealth level ( recorded as _{}) will also affect his effectiveness. Generally, suppose the
policy-maker is willing to pay the price_{}to attend this game, recorded as _{}, by now, the probability of participating in the gambling
is still_{}, _{},_{} the expected utility value of participating in the gambling is _{}=_{}.

Generally speaking, the most policy-makers are loathe the risk, only when it could bring bigger utility than expected, the policymaker is willing to take part in the gambling. Namely:

_{}

We might select a model risk
loathing function _{} to take
policy-maker's utility function. As simplified computation, here suppose
policy-maker's wealth level is for _{} rubles, therefore
the expected utility of participating in the gambling is:

_{}=_{}

When _{} , namely _{}

policy-maker will choose participating in the gambling.

That is, although this game's
expectation repayment is infinite , the policy-maker is only willing to pay
the minimum price to attend this game. If participating in the gambling is regarded
as insurance product, policymakers with 10000 rubles is willing to pay 14.25
rubles to take out insurance at most.

3. Insurance pricing between insurers

In the reinsurance arrangement, stopping the loss reinsurance (LIU.2007). is the most superior. But in reinsurance practice, what needs to consider is not only the benefit original insurance company but the reinsurance company. In safe practice, to ensure the security, often two or more insurance companies sign one risk agreement which is advantageous for both through the negotiations , namely the two companies takes the original insurer and the reinsurance person's dual statuses appears at the same time.

Supposes Insurance company A and
Insurance company B has a chit respectively, random variable_{} and_{}stand for their loss separately. And

_{}and _{} stand for the
distribution function separately .Moreover, supposes initial reserve fund of
company A and the company B respectively for _{}and _{}.For simplifing model, supposes the insurers only charge the
insurance premium from the insured, namely _{},_{}.Insurance company A and the B utility function was standed
for _{}and_{}separately. If both the two companys services have the
indemnity with their amount respectively for _{} and _{}.According to the contract provision, the amount which
insurance company A will pay is_{}, Insurance company B pays the surplus indemnity_{}. Because these two company's benefit is opposite, therefore
they have to carry on the negotiations in the function_{}, making the
bilateral expected utility value as big as possible.In which,

_{}=_{}

_{}=_{}

Obviously, both two companies are
seeking to achieving the biggest effectiveness. According to the Pareto thought
, the necessary and sufficient condition of optimal solution _{}is: _{}in which_{}._{}

the proof for details sees (WANG Gang.2003).

Only when the expected utility is bigger than do not cooperate ,can companies choose the cooperation. Namely:

_{} _{}

From this we may obtain the value
scope of _{} which satisfies
the condition

Suppose the two insurance companies are known for the effectiveness of monetary :

_{} _{}

by the type_{}, the necessary and sufficient condition of optimal
solution become:

_{}

by the above equation, _{}

=_{}

Making _{}_{}_{}=_{}

Therefore, _{}

By the above equation, we can see that if in the company A has the amount for claim, then it only pays a corresponding round number, other parts are paid by company B.

When the Insurance company A
utility function is _{}, company's initial utility is

_{}=_{}

Reorganized this type may write

_{} in which _{}

Similarly initial utility of company B is:

_{}=_{}

in which_{}

making _{}_{}_{}_{}_{}_{}

then _{}

_{}

_{}

_{}

By _{}and _{}, we can see :

_{}

The Nash solution has gave the
maximization of _{}:

_{}=_{}

Solution: _{}_{}

The optimal solution namely: _{}

So the most superior effectiveness of two company is:

_{} _{}

References

LIU Jiao,
WANG Yong-mao, GUO Dong-lin. (2007). Interference with
the Continuous Risk Model
[J]. *Economic Mathematics*, 24(1): 27-30

S.M.Ross. (2005). *Stochastic Process* [M].Bei Jing Chinese Statistics Publishing house,1-7

QIN Gui-xia. (2008).* The Reasearch of Insurance risk Securitization* [D].Qin Huang dao:Yan
Shan University, 20-34

WANG Xiao-jun,JIANG-Xing,LIU Wen-qing. (2003) *Actuarial
Science of Insurance* [M].Bei Jin. Chinese
People's University Press, 284-298

XIE Zhi-gang,HAN Tian-xiong. (2000).* Theory and Non-life Insurance Calculation*
[M].Tian Jin:Nankai University Press, 194-197

GUO Chun-yan. (2004).* Theory of Expected Utility and Ordering of Risks* [D].Shi Jia
zhuang He Bei Normal University, 6-10

LIU Jiao. (2007).* The Further Research of Bankruptcy*[D]. Qin Huang dao Yan Shan
University:10-12

WANG Gang. (2003). *The Analysis of Reinsurance Optimization Model* [D].Chang Sha Hu
Nan University, 6-31

[1] College
of science, yanshan University,Qinhuangdao,Hebei, 066004, China.

* Received 2 March 2009; accepted 6 April 2009

DOI: http://dx.doi.org/10.3968%2Fg804

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