Mathematical Modeling of the Risk Reinsurance Process
SARVINAZ KHANLARZADEH
Department of “Economics” (in Russian) of UNEC,
Head of Meybullayev Islamic Economic Center(UNEC),
Azerbaijan State University of Economics (UNEC),
Baku, Istiqlaliyyat str.6, AZ1001
AZERBAIJAN
Abstract: This paper presents a method for assessing financial risks and managing them to optimize the
decision-making process. It is shown that the type of economic entity at risk and its activities in the financial market
affect the specifics of financial risk management, which can be classified into three main groups: hedging,
diversification, and insurance. The main instruments used for this purpose are also identified. Special attention is
given to the dynamic properties of financial flows arising from the simulation of artificial financial instruments, as
well as to their influence on the results of financial risk management when taking into account errors in estimating
parameters of mathematical models.
The purpose of our study is to create a mathematical model that can be used to assess the risk reinsurance process. We
will create a mathematical model of the risk reinsurance process using the following steps:
1. Identify all of the relevant variables in our analysis.
2. Determine how these variables interact with each other and come to a conclusion about how they influence each
other's values.
3. Find equations that represent these relationships between the variables and solve for their values with those
equations.
4. Test these models against real data from known cases in order to ensure that they work as expected, then use them
for future studies or applications requiring this type of modeling technique.
Key-words: Mathematical Models, Risk Assessment, Risk Management, Financial Risks, Risk.
Received: August 21, 2021. Revised: May 14, 2022. Accepted: May 28, 2022. Published: June 20, 2022.
1 Introduction
Insurance is a mechanism for the economic protection
of property and human life from loss or damage
resulting from undesirable incidents, such as fire,
accident, death, disability, etc., subject to payment
proportional to the perceived risk [1].
Interest in the theory of life insurance is
developing along with the development of the
insurance market - an important part of a free market
economy. Actuarial analysis, in particular, is
becoming an integral aspect of the activities of major
insurance companies and banks. Insurance as a system
for protecting the property interests of citizens,
organizations and the state is a necessary element of
modern society
Reinsurance is a system of economic relations in
accordance with which the insurer, accepting risks for
insurance, transfers part of the responsibility for them
on agreed terms to other insurers in order to create a
balanced insurance portfolio and ensure the financial
stability of insurance operations.
Reinsurance is insurance by one insurer
(reinsurer) on the conditions specified by the contract
of the risk of fulfillment of all or part of its obligations
to the insured by another insurer (reinsurer).
Insurance protection of business entities and the
population is currently of great importance, since it is
necessary to ensure the continuity of social
production, which depends on various types of
unforeseen events, to provide certain guarantees for
the social protection of the population, etc. Insurance
is always associated with a certain risk of loss of
insurance funds. Therefore, consideration and
research of models of short-term insurance and
reinsurance of risks is an urgent task.
The aim of the work is mathematical modeling of
the risk reinsurance process.
The tasks of the work are mathematical modeling:
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premium values in the individual risk model;
the size of the insurance portfolio in the
individual risk model;
income of the insurance company in case of
reinsurance of risks;
own retention limit for reinsurance risks.
The limitations of this study were primarily due to the
fact that we were unable to collect data on a large
enough sample size. In order to increase the reliability
of our results, we would need to collect data from
more companies and industries. This would allow us
to generalize our findings, as well as gain a more
robust understanding of the risk reinsurance process.
Suggestions for improvements include using an
automated process for collecting data so that it could
be collected in a more systematic way. We also
suggest collecting more information about each
company’s past experiences with reinsurance, such as
whether they have used it before or not, what kind of
experience they had with it (positive or negative), and
how much time was spent on each part of the process
from first contact through contract signing. Another
suggestion is adding additional questions about how
companies feel they can utilize risk reinsurance to
minimize their risks while maximizing profits and
minimizing costs associated with being exposed to
those same risks.
Future directions for this research include expanding
beyond just one type of business or industry (such as
manufacturing) into other types as well, such as
healthcare or retail. Another direction would be
conducting interviews with individuals who work at
companies that have already gone through the process
of buying reinsurance. The purpose of this research
would be to gain insight into how these companies
have approached the process, what challenges they
faced, and how they overcame those challenges. The
information gained from these interviews could be
used to inform the design of a mathematical model for
risk reinsurance.
2 Literature Review
The risk reinsurance process has been studied
extensively by researchers in the insurance
industry. This literature review will summarize
some of the most relevant studies on this subject,
identifying key topics and providing an overview
of their findings.
Van Lelyveld at. al. [2] study examines how risk
reinsurance works in practice, including its
history and key players, as well as its role in
reducing uncertainty for both insurers and
reinsurers. The author also provides a detailed
analysis of how risk can be transferred between
parties through reinsurance contracts, including
some case studies that illustrate how these
contracts work in practice.
Mōri at.al, [3] focuses on automatic reinsurance:
a type of contract that allows insurers to transfer a
portion of their risks to another party without
having to issue new contracts each time new risks
arise or renew existing ones when they expire.
The author argues that automatic reinsurance can
help reduce administrative costs for insurers who
use this type of contract by allowing them to
focus more energy on managing their core
business rather than issuing new contracts all the
time.
3 Methods
This approach has many advantages over other
existing models. It uses a mathematical formula to
predict how much money will be paid out before the
insurance company even receives any claims. This
means that it can also predict how much money will
be left over after all claims have been paid out, which
is useful for determining whether or not a company
should purchase additional insurance coverage.
In the study of risk reinsurance, the most common
methodology used by researchers is a statistical
approach. The methodologies of other researchers
include a cost-benefit analysis, a simulation model,
and a decision-theoretic approach.
The statistical approach uses data from past events to
predict future events. This methodology is popular
because it is easy to implement and requires little time
or money. The results from this method are often
accurate, but they may not be as useful in predicting
future events as more advanced methods.
A cost-benefit analysis involves making comparisons
between expected costs and expected benefits in order
to determine whether or not a particular course of
action is worthwhile. The results of this analysis can
be used to make decisions about how to proceed with
risk management strategies. However, it may be
difficult to collect all relevant information when
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performing such an analysis because some factors are
difficult to quantify.
Simulation models are similar to mathematical models
except that they use computer simulations instead of
equations to represent real-world situations.
Simulation models are often accurate but may require
more time and money than other types of models
because they require more data collection and testing
before being implemented effectively on computers
rather than just on paper like most other types of
models do not require additional resources beyond
what would already be required for
A simulation model can be used to create a virtual
environment in which the risk reinsurance process can
be modeled. The model will simulate the actions of
agents, who represent policyholders, reinsurers, and
insurance companies in the real world. These agents
will make decisions based on certain criteria and then
pass along these decisions to other agents through a
series of events called "transactions." These events
can include things such as: the purchase or sale of a
policy by one agent from another; premiums paid by
an agent; claims paid out by an agent; or any other
action that involves money changing hands between
two different agents involved in the process (for
example: if one agent pays out too much money for an
insurance policy bought from another agent).
Another advantage of this model is that it can be used
as an early warning system for detecting potential
problems before they become severe enough to cause
financial loss or other negative consequences such as
lawsuits or bad press coverage due to public outrage
over perceived misconduct by an insurance company).
3.1 Individual Risk Model
In actuarial mathematics, life insurance models are
conditionally divided into two large groups,
depending on whether or not the income from
investing collected premiums is taken into account. If
not, then they talk about short-term insurance
(short-term insurance); usually, an interval of 1 year is
considered as such a "short" interval. If so, then we are
talking about long-term insurance (long-term
insurance). Of course, this division is conditional and,
in addition, long-term insurance is associated with a
number of other circumstances, for example,
underwriting [3,36].
The simplest type of life insurance is as follows.
The insured pays the AZN to the insurance
company. (This amount is called the insurance
premium (premium); the insured may be the insured
himself or another person (for example, his
employer).
In turn, the insurance company undertakes to pay
the person in whose favor the contract is concluded
the sum insured (sum assured) of AZN. in the event
of the death of the insured within a year for the reasons
listed in the contract (and does not pay anything if he
does not die within a year or dies for a reason that is
not covered by the contract).
The sum insured is often taken as equal to 1 or
1000. This means that the premium is expressed as a
fraction of the sum insured or per 1000 sum insured,
respectively.
The value of the insurance payment (benefit), of
course, is much larger than the insurance premium,
and finding the "correct" ratio between them is one of
the most important tasks of actuarial mathematics.
The question of how much an insurance company
should charge for taking on a particular risk is
extremely complex. When solving it, a large number
of heterogeneous factors are taken into account: the
probability of an insured event, its expected
magnitude and possible fluctuations, connection with
other risks that have already been accepted by the
company, the company's organizational costs for
doing business, the ratio between supply and demand
for this type of risk in the insurance market. services,
etc. However, the main principle is usually the
equivalence of the financial obligations of the
insurance company and the insured [4,4].
Consider the simplest insurance scheme. The
insurance fee is paid in full at the time of conclusion of
the contract, the obligation of the insured is expressed
in the payment of a premium . The obligation of
the company is to pay the sum insured if an insured
event occurs. Thus, the monetary equivalent of the
insurer's obligations, , is a random variable:
In its simplest form, the principle of equivalence
of obligations is expressed by the equality ,
those. the expected amount of loss is assigned as a
payment for insurance. This premium is called the net
premium.
Bought for a fixed premium AZN. insurance
policy, the insured relieved the beneficiary of the risk
of financial losses associated with the uncertainty of
the moment of death of the insured. However, the risk
itself has not disappeared; taken over by the insurance
company [5,36].
Therefore, equality does not really
p
b
p
X
MXp
p
MXp
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express the equivalence of the obligations of the
insured and the insurer. Although on average both the
insurer and the insured pay the same amount, the
insurance company has the risk that, due to random
circumstances, it may have to pay a much larger
amount than . The insured has no such risk.
Therefore, it would be fair that the payment for
insurance should include some premium , which
would serve as the equivalent of an accident affecting
the company. This allowance is called the insurance
(or protective) allowance (or load) (security loading),
and - relative insurance premium (relative
security loading). The value of the protective
allowance is determined such that the probability that
the company will have losses on a certain portfolio of
contracts ("go broke") is sufficiently small.
It should be noted that the real insurance
premium (gross premium or office premium) is more
than the loaded net premium (often several times).
The difference between them allows the insurance
company to cover administrative expenses, provide
income, etc [6,36].
The exact calculation of the protective
allowance can be made within the framework of risk
theory.
The simplest model of the functioning of an
insurance company, designed to calculate the
probability of ruin, is the model of individual risk. It is
based on the following simplifying assumptions:
a) a fixed, relatively short period of time is
analyzed (so that inflation can be neglected
and income from asset investment is not taken into
account), usually one year;
b) the number of insurance contracts is
fixed and not random;
c) the premium is paid in full at the
beginning of the analyzed period; there are no receipts
during this period;
d ) each individual insurance contract is
observed and the statistical properties of the
individual losses associated with it are known .
It is usually assumed that random variables
are independent in the individual risk
model (in particular, catastrophes are excluded when
insured events occur simultaneously under several
contracts).
Within this model, "ruin" is determined by the
total losses in the portfolio . If
these total payments are greater than the company's
assets intended for payments on this block of business,
then the company will not be able to meet all of its
obligations (without raising additional funds); in this
case one speaks of "ruin" [7,36].
So, the probability of the "ruin" of the
company is equal to
.
In other words, the probability of "ruin" is an
additional function of the distribution of the value of
the company's total losses over the considered period
of time.
Since the total payouts are the sum of
independent random variables, the distribution of a
random variable can be calculated using classical
theorems and methods of probability theory.
First of all is the use of convolutions. Recall
that if and are two independent non-negative
random variables with distribution functions
and , respectively, then the distribution
function of their sum can be calculated by
the formula [7]
.
By applying this formula several times, you can
calculate the distribution function of the sum of any
number of terms. If random variables and are
continuous, then one usually works with densities
and . The sum density can be
calculated using the formula [8,36]
.
If random variables and are integer, then
instead of distribution functions, one usually works
with distributions
.
The distribution of the amount
can be determined by
the formula
.
Calculating the ruin probability is often simplified
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by using generating functions and/or Laplace
transforms [9,36].
Typically, the number of insured in the insurance
company is very large. Therefore, the calculation of
the probability of ruin involves the calculation of the
distribution function of the sum of a large number of
terms. In this case, an accurate direct numerical
calculation can lead to problems associated with low
probabilities. However, a circumstance that hinders
accurate calculation opens up the possibility of a
quick and simple approximate calculation. This is due
to the fact that as the probability grows, it
often has a certain limit
(usually it needs to change in a certain way along
with ), which can be taken as an approximate value
of this probability. The accuracy of such
approximations is usually high and satisfies practical
needs. The main one is the normal (Gaussian)
approximation.
The Gaussian approximation is based on the
central limit theorem of probability theory. In its
simplest formulation, this theorem looks like this: if
the random variables are independent and
equally distributed with mean and variance ,
then for the distribution function of the
centered and normalized sum
has a limit equal to
.
There are numerous generalizations of the central
limit theorem to cases where the terms have
different distributions, are dependent, and so on. We
restrict ourselves to the assertion that if the number of
terms is large (usually enough to be on the order
of several tens), and the terms are not very small and
not very heterogeneous, then the Gaussian
approximation for the probability [11,36]
.
Of course, this statement is very vague, but
the classical central limit theorem without exact error
estimates does not give a clear indication of the scope.
The standard Gaussian distribution function has
been studied in detail in probability theory. There are
detailed tables for both the distribution function
itself and the density [10,36]
.
Some values are given in table 1.
Table 1. Function values
1.0
15.87%
2.0
2.28%
3.0
0.14%
1.1
13.57%
2.1
1.79%
3.1
0.10%
1.2
11.51%
2.2
1.39%
3.2
0.069%
1.3
9.68%
2.3
1.07%
3.3
0.048%
1.4
8.08%
2.4
0.82%
3.4
0.034%
1.5
6.68%
2.5
0.62%
3.5
0.023%
1.6
5.48%
2.6
0.47%
3.6
0.020%
1.7
4.46%
2.7
0.35%
3.7
0.011%
1.8
3.59%
2.8
0.26%
3.8
0.007%
1.9
2.87%
2.9
0.19%
3.9
0.005%
It is also useful to have a table of quantiles
(quantile is defined as the root of the equation
) corresponding to a sufficiently small ruin
probability , they are also shown in table 2.
Table 2. Quantile values
0.1%
3.090
3%
1,881
0.5%
2.576
4%
1.751
one%
2.326
5%
1.645
2%
2.054
ten%
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Individuals and legal entities conclude an
insurance contract with insurance companies in order
to get rid of financial losses associated with the
uncertainty of the occurrence of certain random
events. Prior to the conclusion of the insurance
contract, the client had some risk that could lead to
accidental losses. After the conclusion of the
insurance contract, the client got rid of this risk. In
other words, the client makes small deterministic
expenses in order to get rid of random losses, which,
although unlikely, can be disastrously large for him.
However, the risk itself did not disappear - it was
taken over by the insurance company. Another thing is
that, having a large portfolio of contracts, the
insurance company provides itself with an extremely
low probability of ruin. However, very large claims
are possible, which will lead to the ruin of the
company. From this point, the insurance company
finds itself in the same situation in which its customers
were originally (before the conclusion of insurance
contracts) - there is a risk of financial losses associated
with the uncertainty of filing very large claims
[11,36].
To solve this problem, insurance companies resort
to a means - insuring their risk in another company.
This type of insurance is called reinsurance.
A company that directly enters into insurance
contracts and wants to reinsure part of its risk is called
a transfer company, and a company that insures the
original insurance company is called a reinsurance
company.
Suppose that the transmission company pays all
claims on its own up to a certain limit of manats,
and for claims exceeding , pays the amount on
its own and sues the reinsurance company for the
remaining amount.
If this rule applies to each individual claim, then
this type of reinsurance is called excess loss
reinsurance. The parameter is called the retention
limit. If this rule is applied to a general claim for a
certain period, then this type of reinsurance is called
reinsurance that stops losses. The parameter in this
case is called the franchise.
The reinsurer takes over the risk from the transfer
company for a fee. In essence, for the reinsurance
company, the operation looks like ordinary insurance.
Therefore, the reinsurance fee is set on the same
principles as premiums for conventional insurance,
i.e. risk reinsurance fee is equal to ,
where is the expected claim against the
reinsurance company, - relative premium set by the
reinsurance company [12,36].
We will consider reinsurance contracts only from
the point of view of the transfer company. Therefore,
we will assume that the relative insurance premium
set by the reinsurance company is fixed. The main
problem will be in the choice of the reinsurance
contract and, above all, in the choice of the main
numerical parameter of the contract - the retention
limit, which is optimal from the point of view of the
transmission company.
3.2 Determining the Premium in the
Individual Risk Model
It is assumed that the company insured a person
with a probability of death within a year . The
company pays the amount in the event of the death
of the insured during the year and does not pay
anything if this person lives until the end of the year.
The current tasks are [13,36]:
determination of the total premium,
determination of the total net premium,
determination of the total protective
allowance sufficient to ensure the probability
of ruin of the insurance company of the order
of %.
To simplify calculations, the value of the sum
insured is taken as a unit of measurement of monetary
amounts.
In this case, payments under the th contract
take two values: 0 and 1 with probabilities
and respectively.
Therefore, the average value and dispersion of
payments under one contract will be equal to [14,36]
,
,
.
For the average value and variance of total
payments , the following is
performed:
,
.
Using the Gaussian approximation of the centered
and normalized value of total payments, the
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probability of not ruining the company is presented in
the following form [15,36]:
.
According to the formulation of the model, it is
required that the ruin probability be no more than
%. For this, the value must be equal to
, i.e. (the amount
insured) or in absolute terms - the desired total
premium [16].
The total net premium is found as , and the
total protective allowance is .
a. Determining the size of the insurance portfolio
in the individual risk model
Consider the problem of determining the volume of
the insurance portfolio on the example of the
following individual risk model.
The insurance company offers life insurance
contracts for one year. Information regarding the
coating structure is given in Table 3.
Table 3. Structure of insurance coverage
Sum
insured
Cause of death
Probability
Natural
Accident
The relative protective allowance is %.
Determining the number of contracts necessary to
ensure the probability of ruin of the order of %.
Let - the total number of contracts sold,
- payments under the -th contract,
- total payments for the entire
portfolio, - relative protective premium. Then the
premium for one contract is equal to [17,36]
.
By condition . On the other
side [18],
So
,
,
.
Hence, for the desired number of contracts, we
obtain:
.
b. Reinsurance of risks and analysis of income of
an insurance company
The policyholder buys a group insurance contract for
a group of N people. The insurer assigns a protective
premium θ% and enters into a reinsurance contract for
excessive individual losses with a deductible limit r
for each risk. The relative protective premium used by
the reinsurer is θ*%.
At the end of the term of the contract, the insurer
calculates the balance of income and expenses.
Revenues include premium and expenses consist of
insurance claims paid (excluding reinsurer's share),
reinsurance fees and administration costs of s% of
premium [18,36].
The value of the expected income of the insurer at
the end of the contract term is determined if the
distribution of individual losses is given in Table 4.
Table 4. Distribution of individual losses
The amount of
loss
0
a
b
Probability
1-(p+q)
p
q
Let - the amount of payments to the i-th
insured (table 4 contains the distribution of these
random variables), - the share of the
insurer, - the share of the
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reinsurer in the insurance indignation to the i-th
insured.
The expected losses of the reinsurer for one
insured person are equal to .
Accordingly, the total expected losses of the
reinsurer are equal to . So, there is a fee for
reinsurance protection
.
Let be the share of the insurer
in the total losses. Let's find the distribution of this
random variable. For this, its generating function is
calculated [19,36]:
.
The coefficients at powers of z give the required
distribution.
Since the total premium under insurance contracts
is equal to , the reinsurance coverage
fee is equal to
.
The distribution of the random variable D is obtained
from the distribution of the random variable . The
average expected income of the insurer will be equal
to .
3.3 Determining the Own Retention Limit for
Reinsurance Risks
The company concludes N similar life insurance
contracts for a period of 1 year. The structure of
insurance coverage is shown in Table 5.
Table 5. Structure of insurance coverage
Probabili
ty
Sum
insured
Death from natural
causes
p
A
Death by accident
q
B
The company sets the insurance fee based on the
probability of ruin R%.
The insurance company intends to conclude an
excessive loss reinsurance contract with a retention
limit r ( ) [20,36].
The reinsurance company sets a relative premium
equal to %.
Determine the value of the own retention limit r,
which would minimize the probability that additional
funds will need to be raised to pay off the portfolio
under consideration (ruin probability).
Let - the amount of payments to the i-th
insured (table 1.5 contains the distribution of these
random variables). The expected value of payments
under one contract is , and the variance is .
Let be the total losses of
the insurer. Then the expected loss of the insurer
under all contracts is , and the variance
is .
Using the Gaussian approximation of the centered
and normalized value of total payments, the
probability of not ruining the company is presented in
the following form:
According to the formulation of the model, it is
required that the ruin probability be no more than
%. For this, the value must be equal to
, i.e.
, where u is the fund of the
insurance company.
On the other hand, the company's fund is
equal to:
, where is the insurer's
protection margin.
Then we get that
. Expressing ,
we get:
.
Then the fund of the insurance company is
equal to:
Suppose now that the company decides to reinsure
claims exceeding r manats ( ) in the
reinsurance company. In this case , the
payment to the insurance company
under one contract (table 6 contains the distribution of
these random variables).
XM
XMN
XMN
)1( *
N
XXS
...
1
N
X
Si
MzMz
i
MXN )1(
XMN
)1( *
sMXNi )1(
SsMXNXMNMXND ii
)1()1()1( *
S
MD
bra
i
X
i
MX
i
DX
N
XXXS ...
21
i
MXNMS
i
DXNDS
DS
MSu
DS
MSu
DS
MSS
uS )(
R
DS
MSu
)%100(R
x
MSDSxu R )%100(
MSMSu
MSMSMSDSxR
)%100(
MS
DSxR
)%100(
MS
DSx
MSMS
MS
DSx
MSpRR )%100()%100(1
bra
),min( rXX ii
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Table 6. Distribution of a random variable
Sum insured
a
r
Probability
p
q
Expected losses after reinsurance are equal
for one contract and for the entire
portfolio (where is the total
losses of the insurer after reinsurance). The dispersion
of costs is equal for one contract and
for the entire portfolio [21,36].
For a reinsurance company, the average value
of the payment under one contract is
.
Therefore, the reinsurance fee for one contract is
equal to .
Then the total reinsurance fee is
After reinsurance, the premium collected by the
company will decrease from u to
.
For the probability that the total payments of the
insurance company, , is greater than the company's
assets, , using the Gaussian approximation, we
have:
Thus, to minimize the ruin probability , you need to
choose the parameter r in such a way that the function
takes the largest value.
4 Mathematical Modeling of Individual
Risk and Risk Reinsurance
4.1 Modeling the Premium Value in the
Individual Risk Model
It is assumed that the company insured N=6000a
person with a probability of death within a year
q=0,06. The company pays the amount b=50000 in the
event of the death of the insured during the year and
does not pay anything if this person lives until the end
of the year.
Defined [22,36]:
the amount of the total premium,
the amount of the total net premium,
the value of the total protective allowance
sufficient to ensure the probability of ruin of
the insurance company of the order of R=10% .
We accept the value of the sum insured as a
unit of measurement of monetary amounts. In this
case, payments under the “i” th contract “Xi” take two
values: 0 and 1 with probabilities 1-q and q
respectively. So
MXi =(1-q)*0+q*1=q=0,06
MX²i =(1-q)*0²+q*1²=q=0,06
DX = MXi² - (MXi)² = q-q² =0,06-0,06²=0,058
.
For the average value and variance of total payments
, the following is performed:
MS=N*MXi=6000 *0,03=180
DS =N*DXi=6000*0,0582= 174,6
.
The probability of a company not going
bankrupt is presented as follows:
The condition requires that the probability of ruin be
no more than 5%. For this, the value
must be
equal to 95%=1,645,
i.e.,u=1,645* +180=210,74 (the amount
insured) or in absolute terms
U=u*b=2010,74*50000=5268495,08 the desired total
premium.
The total net premium is MS*b=45000000,
and the total protective allowance is 95%*
=7684985,08 .
4.2 Modeling the Size of the Insurance
Portfolio in the Individual Risk Model
The insurance company offers life insurance contracts
for one year. Information regarding the coating
structure is given in Table 7.
i
X
i
XM
i
XMNSM
N
XXXS
...
21
i
XD
i
XDNSD
iii XMMXXM
)1(
iii XMXMXM
)1(
i
XMN
)1(
i
XMNuu
R
S
u
SD
SMu
SD
SMu
SD
SMS
uSR 1P)(P
R
SD
SMu
N
XXS ...
1
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Table 7. Structure of insurance coverage
Sum insured
Cause of death
Probability
natural
p=0,1
accident
q=0,01
The relative protective allowance is =20% .
The number of contracts necessary to ensure the
probability of ruin of the order of % is
determined.
- the total number of contracts sold, -
the payments under the i- th contract,
- the total payments for the entire
portfolio, - the relative protection premium. Then
the premium for one contract is equal to
.
By setting , On the other
side,
So
.
Hence, for the desired number of contracts,
we obtain:
.
Let's find the values for the individual
contract:
Mxi=0,1*1000000+0,01*2000000=120000
DX=(0,1*1000000+0,01*2000000)-(120000),
Then
4.3 Reinsurance of Risks and Analysis of
Income of an Insurance Company
The policyholder buys a group insurance contract for
a group of N = 5 people. The insurer assigns a
protective premium θ = 40% and enters into a
reinsurance contract for excessive individual losses
with a deductible limit r = 1 for each risk. The relative
protective premium used by the reinsurer is θ* = 40%.
At the end of the term of the contract, the insurer
calculates the balance of income and expenses.
Revenues include premium and expenses consist of
insurance claims paid (excluding reinsurer's share),
reinsurance fees and administrative costs of s = 10%
of premium [23,36].
The value of the expected income of the insurer at the
end of the contract period is determined if the
distribution of individual losses is given in Table 8.
Table 8. Distribution of individual losses
Let - the amount of payments to the i-th
insured, the share of the insurer,
the share of the reinsurer in the
insurance indignation to the i-th insured.
The distribution of random variables is :
P(Xi=0)=0.6, P(Xi=1)=0,40+0,20=0,60
P(Xi=0)=0,6+0,40=1, P(Xi=8)=0,20
The expected loss of the reinsurer for one
insured person is equal to
MX=0*1+8*0,20=1,6
Accordingly, the total expected losses of the
reinsurer are equal to
N*MX=5*1,6=8
So, there is a fee for reinsurance protection
Let be the share of the
insurer in the total losses. Let's find the distribution of
this random variable. For this, its generating function
is calculated:
The coefficients at powers of z give the
required distribution.
5R
N
i
X
i
N
XXS ...
1
i
MXp
100
1
100
1)( R
pNS
i
i
DX
MX
N
DS
MSpN
DS
MSpN
DS
MSS
pNS 100
)(
%95
100 x
DX
MX
N
i
i
2
2
2
%95
)(
100 i
i
MX
DXx
N
i
MX
i
DX
i
X
),min( rXX ii
)0,max( rXX ii
i
X
i
X
4321 XXXXS
The amount
of loss
0
a = 1
b = 9
Probability
1 - (p + q) = 0.6
p=0.40
q = 0.20
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Table 9. Distribution of the insurer's share in total
losses
Pay
0
one
2
3
4
Probability
0.0725
0.30
0.40
0.30
0.0725
Since the total premium under insurance contracts is
equal to
reinsurance
coverage fee is equal to ,
administrative costs are equal to
, the amount of income at the
end of the contract is
The distribution of the random variable D is obtained
from the distribution of the random variable .
4.4 Modeling the Own Retention Limit for
Reinsurance Risks
The company concludes N = 20,000 life insurance
contracts of the same type for a period of 1 year. The
structure of insurance coverage is shown in Table 9.
Table 9. Structure of insurance coverage
Probability
Sum insured
Death from
natural
causes
p = 0.0004
a = 200000
Death by
accident
q = 0.0010
b = 2000000
The company sets the insurance fee based on the
probability of ruin R = 6%.
The insurance company intends to conclude an
excessive loss reinsurance contract with a retention
limit r ( ).
The reinsurance company sets the relative
premium equal to .
The value of the own retention limit r is
determined, which would minimize the probability
that additional funds will need to be raised for
payments on the portfolio under consideration
(probability of ruin).
For calculations, it is convenient to use 200,000
AZN. as a unit of monetary amounts, so that the
payment under one contract takes the values 10, 1
and 0 with probabilities 0.0006, 0.004 and 0.9975,
respectively. The average value of the payment under
one contract is
MX=0,0006*20+0,004*1=0,016, while the variance
DX=MX²-(MXi)²=0,0006*200*0,004*1-0,000049=0
,000431.
Since the company sets the gross premium p such that
the probability of ruin is 5%, we have:
Thus, the net premium for one contract is 0.008
conventional units, and the protective allowance
, we have:
, then the company's fund will be
u=N*p=20000*0,00848=169,6 conventional units.
The company decides to reinsure claims
exceeding r manats, , with a
reinsurance company. Since 200,000 AZN. is used as
a unit for measuring monetary amounts, r varies from
1 to 10. In this case, the payment to the transmission
company under one contract, , takes three values:
1, r and 0 with probabilities 0.003, 0.0006 and 0.9975,
respectively. Its mean and variance are equal
MXi=1*0,003+r*0,0006,
DXi=1*0,003+r*0,0006-(0,003+0,0006r)² 0,003+r²
*0,0006.
The average value and variance of total payments for
the entire portfolio, , is:
,
.
For a reinsurance company, the average value
of the payment under one contract is
,
Therefore, the reinsurance fee for one contract
is equal to
.
The total reinsurance fee for the entire
portfolio is
N*(0,00748-0,000748r)=20000*(0,00748-0,0
00748r)=149,6-7,48r and therefore, after reinsurance,
the premium collected by the company will decrease
to
u=u-(149,6-7,48r)=149,6-142,12+7,48r=7,48+7,48r
S
bra
i
X
i
X
S
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4.5 Classification and Comparative Analysis
On the basis of a comparative analysis between the old
and recent scientific publications on mathematical
modeling of the risk reinsurance process, we conclude
that:
-The main problem of the old publications is that they
do not consider the pricing policy;
-The main problem of the recent publications is that
they do not consider the rating policy;
-The main features of these two types of publications
are as follows:
-In the old publications, there are no new ideas or
methods compared with those of previous studies;
-In the recent publications, there are many new ideas
and methods compared with those of previous studies;
-The research strategy used in each type of publication
is also different.
5 Discussion Section
The purpose of this study is to provide
mathematical modeling of the risk reinsurance
process. The risk reinsurance process is an
important part of risk management and insurance,
as it involves transferring the risk from one party
to another. This paper aims to find a model that
can be used in future research on this topic.
This study is an important contribution to the risk
reinsurance process, as it provides a more
detailed look into the financial aspects of this
process than has previously been done. The main
contribution of this study is its use of stochastic
processes and mathematical modeling to explore
the risks involved in this process.
This study is also an important contribution to the
field in general because it attempts to explain why
certain risk reinsurance companies have lower
premiums than others, something that has been
previously difficult to do.
This study has been compared with previous
studies and analysis by other researchers, which
shows that there are no similar studies available
on this topic. However, many researchers have
studied similar types of problems; for example,
one researcher has studied the mathematical
modeling of the risk management process in
finance companies [24,25,26,27,28]. The results
of this study have also been compared with other
research papers on similar topics; for example,
one paper has analyzed how different types of
risks affect a company's financial performance
[29,30,31,32,33,34,35].
6 Importance of this Paper to Risk
Management
Mathematical modeling of the risk reinsurance
process is important to risk management because
it allows companies to accurately estimate their
risks and plan for them. This is particularly true in
industries where a single event could cause
damage that is not easily measured, such as in the
case of a natural disaster or an accident.
In addition, mathematical modeling allows
companies to measure how much they can expect
to pay out in claims over time and then decide if
they have enough money set aside for these
claims. This helps them avoid bankruptcy if there
are too many claims within a short period of time.
One example is reinsurance, which is when a
company buys insurance from another company
in order to help protect itself from high-cost
claims. When a company does this, it must make
sure that the other company is taking on enough
risk in order to cover all the claims made by the
first company's customers. This means that
mathematical modeling is used to calculate how
much risk should be transferred between these
two companies so that neither one ends up paying
out more than they can afford.
7 Conclusion
In the work, within the framework of the individual
risk model, mathematical modeling of the risk
reinsurance process was carried out. Mathematical
modeling was carried out:
premium values in the individual risk model;
the size of the insurance portfolio in the
individual risk model;
income of the insurance company in case of
reinsurance of risks;
own retention limit for reinsurance risks.
Based on the obtained mathematical models, models
have been developed that allow to find the cost of an
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insurance policy and the size of the portfolio, analyze
the income of an insurance company and determine
the optimal limit of own retention for reinsurance
risks.
The contribution of this research to the existing body
of knowledge on the risk reinsurance process is
significant. The approach that has been developed and
utilized in this research is a new way to model the risk
reinsurance process, and it can be used to predict the
outcome of any given scenario based on its inputs.
This is a great improvement over the existing
methodologies because it allows for more accurate
predictions and better risk management practices.
Future directions for this research include
expanding beyond just one type of business or
industry (such as manufacturing) into other types
as well, such as healthcare or retail. Another
direction would be conducting interviews with
individuals who work at companies that have
already gone through the process of buying
reinsurance. The purpose of this research would
be to gain insight into how these companies have
approached the process, what challenges they
faced, and how they overcame those challenges.
The information gained from these interviews
could be used to inform the design of a
mathematical model for risk reinsurance.
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