93
Year VII, No.8/2008
RISK INSOLVABILITY MANAGEMENT THROUGH
OPTIMIZING INSURANCE PORTFOLIO –
MATHEMATICAL CALCULATIONS
Assoc. Prof. Mirela CRISTEA, PhD
Assoc. Prof. Raluca DRACEA, PhD
University of Craiova
Assoc. Prof. Murat KASIMOGLU, PhD
Onsekiz Mart University - Turkey
1. Introduction
Starting with the second half of
the year 2007, the international financial
markets indicate new essential features,
influenced by the issues existing on the
high-risk mortgage market of the USA
(known as subprime crisis) most of them
generating a wrong perception of the
investors’ risk and a liquidity diminution.
However, the insurance market,
in full expansion and finalizing the stage
of harmonization with the European
legislation, was not troubled by these
issues existing on the financial markets,
due to the insurance features and to their
development
and
integration
with
external financial markets.
The main aspects which are
essential for a careful observation are the
increase
of
the
indemnification
installment for general insurances and
the return dynamics of the personal asset
portfolio for life insurances.
It is important for the insurance
company to calculate in a very strict
manner the premiums owed by the
insured persons in order to create an
adequate insurance fund necessary to
cover the indemnifications, diminishing
thus the insolvability risk.
The structure of the subscription
portfolio should be elaborated by means
of the return analysis as well as the
financial stability analysis of the
insurance activity. Thus, the insurers
should find the adequate methods of
substantiating the premium installments,
the adequate ways of attracting
insurances in order to achieve the right
structure of the portfolio and the desired
level of financial stability within the
company.
The present paper views the
analysis of the subscription portfolio
within an insurance company considering
the number of estates comprised by the
insurance and the net premium
installment in order to obtain a certain
level of financial stability. This analysis
represents an essential instrument for
each insurer as the optimization of the
subscription portfolio generates more
important returns for the company; it also
improves the financial stability and
reduces the possibility of insolvency.
Using mathematical calculation, different
solution may be given in order to
optimize this portfolio, determining thus
its adequate structure to a certain level of
stability planned by the company. Thus,
the mathematical calculation shown
below within this paper may be applied in
practice and improved.
2. Material and method
The solvency represents an
important aspect of the operational
regulation within an insurance company,
thus, for every year of the next period,
the manager is concerned with the
knowledge of the following aspects: the
level of the indemnifications to pay; the
level of the indemnifications compared to
level of the net premium paid for each
estate category; the possibility that the
94
Finance – Challenges of the Future
amount of indemnifications exceeds the
amount of paid premiums P (D>PnT) .
Compared to the multiannual
average of the risk indicators, considered
for the net premium 1 substantiation, there
are errors reflected by the dissimilarities
between the indemnifications to pay
currently and those belonging to the
considered period.
In order to establish the value of
the difference between the indemnifications which ought to be paid currently
and the indemnifications registered
during the analyzed period, mean
squared deviation calculations are
S n q(1 q )
,
(1)
used:
where: S – insured sum of an
indemnified estate; n – number of insured
estates; q – probability of damage
occurrence; (1-q) - probability of having
no damage occurring
Pn
For an estate,
q
b
S ,
(2)
where: b = number of estates;
Pn
b - net premium for an estate.
For the total amount of estates,
q
Pnt
St
Pn
n
b
S n
,
(3)
where: Pnt – total net premium.
The interval meant for the
indemnification variation is given by the
relation: D : Pnt ; Pnt
(4)
The financial stability level is
given by the value of the K coefficient:
K
(5)
As the K coefficient is lower, the
financial stability level is higher (the
deviation is reduced).
1
Pnt .
Net insurance premium serves to the creation of
the fund needed for the indemnification or the
insurance compensation payment. For obtaining the
total premium owed by insured persons, to the net
premium is added the supplement or additional
premium, wihich covers the insurer’s purchase and
administration fees, as well as a return achievement
In order to establish the number
of years counted until a favorable year
occurs, “a” is determined as it follows:
a
100
1
K
P( D Pnt )
100
2
.
(6)
The following conditions are
necessary, in order to obtain an
improvement of the financial stability
level: a large number of insured estates;
the increase of the net premium rate; the
cession to reinsurance.
For knowing if the cession for
reinsurance is required, we calculate
the maximum insured sum for each
insured risk (X), this calculation should
be maintained by the insurer, in order to
obtain an adequate stability level:
X 2 K 2 Pnt ,
(7)
where K is the average coefficient of
financial stability, for all risks, and Pnt is
the net premium within the insurance
company.
The insurance companies resort
to reinsurance for protecting its clients,
the insured persons, whenever the
assumed risks are too important.
Through reinsurance, an insurance
company gets a higher financial stability
and, in addition to this, a higher ability of
dealing with the new risks. The
reinsurance interferes whenever the
insured sum allotted to a risk or a group
of risks exceeds the limit that an
insurance company is able to sustain
without affecting the protection of the
other insured persons. One of the
reinsurance functions is constrained by
the monitoring and control institutions of
the insurance market in each country by
imposing a minimum level of solvency.
In
order
to
apply
this
mathematical calculation, we consider an
insurance company for which we
determine the financial stability level
according to the variation of the net
premium rate and to the number of
insured estates. Thus, the insurance
company underwrites 920 estates
belonging to a certain category, being
insured for an average sum of 23,000
euros, with a premium rate of 1.3%.
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Year VII, No.8/2008
3. Results and discussions
Considering the above presented
data, the insurance company is able to
create an analysis of its financial stability
variation, for different situations that may
occur in practice.
1) When the net premium rate
increases, for example 1.6%, the
financial stability level of the company
is changed.
According to the relation (5), the
calculation of the coefficient of financial
stability indicating the financial stability
K
the insured value remain unchanged,
the financial stability level of the
insurance
company
withstands
modifications.
The initial number of insured
estates, n 0 , was 920; its rise to 50% is
expected, obtaining thus n 1 , which
becomes 1.380 estates.
The modification of the financial
stability coefficient is given by the
following formula:
IK
K1
100
K0
IK
S n1 q (1 q )
q S n1
Pnt .
level can be known:
For the initial situation,
K 0
0
Pnt0
S n q0 (1 q0 )
q0 S n
23,000 920 0.013 (1 0.013)
0.287
0.013 23,000 920
When the net premium rate
increases, the coefficient of financial
stability becomes:
K 1
1
Pnt1
S n q1(1 q1 )
q1 S n
23,000 920 0.016 (1 0.016)
0.259
0.016 23,000 920
The change index for the
coefficient of financial stability is
determined as it follows:
IK
K1
0.259
100
100 90.24%
K0
0.287
% K 90.24 100 9.76%
Therefore, the increase of the
net premium rate from 1.3% to 1.6%,
considering the fact that the number
of insured estates and the insured
value do not change, leads to a
growth of the financial stability level of
9.76%, this percentage is given by the
diminution of the financial stability
coefficient, K, from 0.287 to 0.259. Of
course, this situation is recorded only if
the market, the competition respectively,
allows it.
2) When the number of insured
estates increases with 50%, and the
net premium rate of 1.3%, as well as
IK 2
n1q (1 q ) n02
n12 n0 q (1 q )
q S n0
S n0 q (1 q )
n
920
0
0.666
n1 1,380
IK 0.6666 100 0.8165 100 81.65%
%ΔK 81.65 100 18.35%
If the competition does not allow
the increase of the net premium rate, the
insurance company will make efforts for
the growth of the number of customers,
raising the number of insured estates.
The underwriting of over 50% of the
initial number of insured estates leads
to an increase of 18.35% of the
financial stability level, creating thus a
favorable situation for the insurance
company.
3) In order to increase the
financial stability level to a certain
percentage,
for
example
40%,
compared to the initial situation, then
what is the number of estates needed
to be underwritten?
The increase of the financial
stability level with 40%, implies the
reduction of the K coefficient with 60%,
resulting I k , as it follows:
n
2
0.36 0 0.36
I K 60% I k 0.60 I K
n1
n1
n0
920
2,555 estates.
0.36 0.36
The result can be verified as it follows:
96
K 1
Finance – Challenges of the Future
1
Pnt1
S n1 q (1 q )
q S n1
23,000 2,555 0.013 (1 0.013)
0.013 23,000 2,555
% K
0.172
K1
0.172
100 100
100 100 40%
K0
0.287
For increasing the financial
stability level with 40%, the insurance
company is forced to increase the
number of insured estates about three
times, from 920 insured estates,to
2,555 estates.
4) If it is not possible to increase
the number of insured estates (the
market is saturated or other different
reasons), for increasing the financial
stability level with 40% compared to the
initial circumstances, the insurer may
resort to the increase of the net premium
rate. The question is then, which is the
level of the net premium rate in order
to provide a certain coefficient of the
financial stability?
As in the previous case, the
growth of 40% of the financial stability
level implies a reduction of 60% of the K
coefficient, given by l k , as it follows:
2
I K 60% I k 0.60 I K
0.36.
When only the net premium rate
is changed, I k 2 becomes:
n q (1 q1 ) q02 q0 (1 q1 )
2
1
IK
q1 (1 q0 )
q12 n q0 (1 q0 )
0.013 (1 q1 )
0.36
q1 (1 0.013)
0.013 (1 q1 ) 0.36 q1 (1 0.013)
terms of an important growth of the net
premium rate.
A synthesis of the four cases
leads to the following results obtained by
an insurance company, as shown in the
Table 1.
Possible cases registered within an
insurance company
Table 1
Case
Ini ială
1
2
3
4
No. of
insured
estates
920
920
1,380
2,555
920
Net
premium
rate
1.3%
1.6%
1.3%
1.3%
3.53%
Value of the
K coefficient
0.287
0.259
0.234
0.172
0.172
Analyzing these cases, for a
higher financial stability level, the insurer
has to make his option for the increase of
the net premium rate to a certain level,
taking into account the competition on
the market, and for the increase of the
number of insured estates belonging to
the same category of risk.
In practice, the portfolio of
insurance companies, for the same
category of insured estates, includes
several risk groups.
In addition to the initial case
mentioned above, belonging to a certain
category of risk (for example risk
category type I), the insurance company
disposes for the same class of three risk
categories (see Table 2), thus the
manager may adopt decisions according
to a series of eventual situations.
0.013 0.013q1 0.36q1 0.00468q1
q1 0.0353 3.53%.
So, in order to allow the growth
of the financial stability level with
40%, the insurance company has to
increase the net premium rate about
three times, from 1.3% to 3.53%, fact
that leads to a pretty high value
considering the market average. As,
there is the addition which represents the
insurer’s expenses and profit, the
insurance company is not able to
maintain the number of insured estates
unchanged on a competitive market, in
The portfolio of an insurance
company for a class of three risk
categories
Table 2
Risk
category
No. of
insured
estates
I
II
III
920
1,200
1,360
Average
insured
sum
(euro)
23,000
20,000
15,000
Net
premium
rate (%)
1.3
1.1
0.9
In order to establish the
financial stability level for the whole
insured class, respectively for the
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Year VII, No.8/2008
entire insurance company, it is
important to determine first the financial
stability coefficient for the other risk
categories (risk category type II and III).
For the risk category type II:
K II
II
Pnt II
S II nII q II (1 q II )
q II S II nII
20,000 1,200 0.011 (1 0.011)
0.011 20,000 1,200
72,263
0.274
264,000
For the risk category type III:
K III
III
Pnt III
S III nIII q III (1 q III )
q III S III nIII
15,000 1,360 0.009 (1 0.009)
0.009 15,000 1,360
52,242
0.285
183,600
The financial stability level, for
the whole class or for the entire portfolio
of the insurance company is determined
as it follows:
KT
T
,where: T
PnT
i 1
PnT Pnt i
i 1
; i – number of risk
categories.
Thus, K T becomes:
n
KT
i 1
n
Pnti
2
2
III
I2 II
Pnt I Pnt II Pnt III
i 1
100
12.12 12 years
8,25
In order to prevent losses hard to
bear, the insurance company may cede a
part of the concluded policies to
reinsurance. Thus, in order to know
whether the cession to reinsurance is
imperious or not, it is necessary to
compare the maximum insured value for
each insured risk (X) to the insured sum
meant for each risk. If the insured sum is
lower than the maximum insured sum,
the insurer is able to bear alone the
insured risks and he is not forced to cede
them
to
reinsurance
(relation
2
7): X 2 K Pnt .
X I 2 K I 2 Pnt I 2 0.287 2 275,080 45,316
X II 2 K II 2 Pnt II 2 0.274 2 264,000 39,640
X III 2 K III 2 Pnt III 2 0.285 2 183,600 29,826
.
We notice that X I >S I (45,316 >
23,000), X II >S II , X III >S III , X T >S T .
Therefore, the insurance comapny is not
forced to cede the policies to reinsurance
because it is able to undertake all the
insured risks.
Considering the crisis of the
external financial markets and the control
of financial stability, the AIG, the most
important insurer of the world, has
announced accountable reductions in
assets because of the subprime crisis, of
about 40 billion dollars, the most
considerable loss in the insurance
industry. One of the methods meant to
protect the financial stability of the
company is the reduction of premiums for
the insurance policies of the trade
estates, as an attempt to gain market
share. Prices for this type of insurances
diminished with 11% since June 2007 till
June 2008.
X T 2 K T 2 PnT 2 0.165 2 722,680 39,350
n
i2 ;
n
i2
aT
79,0232 72,2632 52,242 2
119,146
0.165
275,080 264,000 183,600
722,680
In
addition,
the
insurance
company is able to determine the
interval of years to which an
unfavorable year is recorded for each
risk category, respectively for the
entire portfolio of the insurance
company, considering the relation (7).
100
1
,
K
P( D Pnt )
100
2
100
aI
6.96 7 years
14,35
a
a II
100
7.29 7 years
13,7
a III
100
7.01 7 years
14,25
4. Conclusions
Insurances represent products
which provide protection against risks
having the financial impact for insured
persons. Thus, the consumers’ trust in
the financial stability of the insurance
98
Finance – Challenges of the Future
sector is extremely important. The
European Union proposed an indicator
meant to measure the insurance
company solvency, denominated as the
solvency margin, being available for all
the member countries.
Regarding Romania, the impact
produced by the issues existing on the
external financial markets over the
insurance sector was limited, with no
risks over its financial stability. The rate
of growth specific to the underwriting
activity in Romania was exceeded on the
whole market by the evolution of
indemnifications with 27.7% compared
with 25% registered in the case of the
underwritten premiums. This fact may
generate negative implications over the
financial stability of the insurance sector
on middle term (Report of the financial
stability, NBR, 2008). This evolution was
generated by the auto insurances,
because of the competition rise and the
high costs meant for repairs, thus the
requested premiums were inappropriate,
unable to cover the underwritten risks.
We consider that this paper, the
result of elaborated studies and analysis,
represents an useful instrument for the
insured persons, thus, they are able to
choose the right type of insurance
appropriate for their services, resting on
its
comparisons,
analysis
and
conclusions, and for the insurance
companies, being meant to improve their
subscription and investment activity, as
well as the financial stability.
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