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International Journal of Quantitative Research and
Modeling

e-ISSN 2721-477X
p-ISSN 2722-5046

Vol. 3, No. 3, pp. 118-123, 2022

Determining the Price of Fisherman Micro Insurance Premiums Using the
Aggregate Risk Model Approach in Cirebon Regency
Ratih Kusumadewi1*, Riaman2, Sukono3
1,*

Mathematics Undergraduate Study Program, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran,
Jatinangor, Indonesia
23,
Departemen of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jatinangor, Indonesia
*Corresponding author email: ratih18001@mail.unpad.ac.id

Abstract
Catastrophe such as hurricanes, heavy rains, and similar occurrence pose serious threats and risks to fishermen's livelihoods as
well as losses from damage to their assets. Therefore, it is necessary to have special insurance to protect the fishermen's assets
from financial losses due to the risks that can occur, namely Fisherman Micro Insurance. Micro-insurance is an insurance product
that is intended for low-income people with features and administration that are simple, easy to obtain, economical prices and
immediately in the completion of the provision of compensation. Fisherman's micro insurance guarantees assets in the form of
fishing equipment in the occurrence of a risk of an accident causing damage, this insurance product protects against worries
without a large premium burden. This study aims to calculate the premium price with an aggregate risk model approach. The data
used is data on fisherman’s losses if they did not go to sea which obtained by surveys. The occurrence data follows the Poisson
distribution, and the loss data follows the Exponential distribution. Parameter Estimation was carried out using the Maximum
Likelihood Estimation. The estimation results from numbers of occurrence and the amount of losses are used to estimate the
collective risk model. Estimators of the average and variance of the aggregate risk are used to determine the premium. The results
of the premium selection in this study amounted to IDR 153.861.958.00. The premium amount is a collective premium which is
the result of a calculation based on the standard deviation principle.
Keywords: Non-life Insurance, premium determination, aggregate risk model, exponential distribution, Poisson distribution

1. Introduction
Potentially, Indonesia's fisheries are the largest in the world, many Indonesians live as fishermen. One of them is in
the coastal areas of West Java, especially Cirebon Regency. It is noted that there are 17,192 people in Cirebon
Regency who make a living as fishermen (Central Statistics Agency, 2018). Fishermen's activities at sea have a high
risk of causing fishermen's failure to go to sea. Therefore, work as a fisherman deserves special attention to increase
the protection of fishermen from the risk of failure, namely through the fisherman's insurance program.
This research on micro insurance was previously conducted by Tietze and Anrooy (2018) with the title Assessment
of Insurance Needs and Opportunities in The Caribbean Fisheries Sector which explained the needs and opportunities
for insurance for fishermen in the Caribbean and fishermen's assets that can be guaranteed, as well as Djuric (2013)
with the title Collective Risk Model in Non-Life Insurance explains the application of the Aggregate Risk Model in
non-life insurance using the Poisson distribution and the Exponential distribution.
Based on the explanation above, the purpose of this research is to determine the estimated number of incidents of
fishermen failing to go to sea and the losses caused by not fishing, and to determine the amount of micro insurance
premiums for fishermen to be paid using the aggregate risk model approach. The results of this study can be used by
insurance companies as consideration for determining the amount of fisherman insurance premiums in Cirebon
Regency.

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119

2. Literature Review
2.1 Micro Insurance
Micro insurance is an insurance product designed to provide protection against financial risks faced by low-income
people.
2.2 Pricing Micro insurance
Pricing is an activity to determine the premium rate for an insurance product. In general, the premium rate must be
set so that the present value of all premiums collected during the coverage period will be sufficient to fund claims
costs in the future.
2.3 Risk and Aggregate Risk
Risk is a form of uncertainty about a situation that will occur later (future) with decisions taken based on
considerations. Estimated aggregate loss is the average aggregate loss that occurs in a certain period.
2.4 The Poisson Distribution
1)

The random variable

has a Poisson distribution with
(

2)

4)

(1)

)

For the moment generating function:
( )

3)

as a parameter, the probability function:

* (

)+

The expectation is the first derivative of the moment generating function with
( )

(2)
(3)

For the variance:
( )

(4)

2.5 The Exponential Distribution
1)

2)
3)
4)

The random variable

has a Poisson distribution with
( )

as a parameter, the probability function:
(5)

The moment generating function:
( )

(6)

( )

(7)

For the expectation equation:
For the variance equation
( )

(8)

2.6 Kolmogorov-Smirnov Distribution Conformity Test
The Kolmogorov-Smirnov test is the largest absolute difference between ( ) which is the cumulative
distribution function of the population and ( ) which is the empirical distribution function of the sample
( )|+
*| ( )
The hypothesis:
: data has the same distribution as theoretical distribution
: the data is not distributed the same as the theoretical distribution

(9)

The decision criterion is if
then
is accepted, meaning that the observed sample distribution has the
same distribution as the theoretical distribution (Putri, 2020).

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2.7 Chi-Square Distribution Conformity Test
The decision making of this test uses the parameter

with the equation:
(

∑

)

(10)

Explanation:
: calculated chi-square parameter
G : number of sub groups
: the number of observation values in subgroup-i
: the number of theoretical values (expected frequency of group-i) in the sub group.
2.8 Maximum Likelihood Estimation (MLE)
Each distribution uses the MLE method to estimate its parameters. From this process, the results obtained are:
1)

Poisson distribution parameters
∑

̂
2)

(11)

Exponential distribution parameters
̂

(12)
̅

2.9 Collective Risk Model
The sum of individual claims up to time is defined ( )
( )

∑

( )

(13)

where is a random variable that states the number of events that cause losses, and which states the amount of
losses received. Expectations and variances of aggregate losses can be calculated by the following equation:
( ( ))
( ( ))

( ( ))

( ( ))

( ( )) ( ( ))
( ( ))( ( ( )))

(14)
(15)

2.10 Premium Calculation Model
1) Value Expectation Principle
Premium calculation using the principle of expected value is based on the average collective risk or pure
premium multiplied by the security loading (loading factor)
( )

(

) ( ( ))

(16)

with
2)

Standard Deviation Principle
The premium calculation with this principle is based on the pure premium which is added up by the product of
the loading factor and the standard deviation of the collective risk
( )

( ( ))

√

( ( ))

(17)

with

3. Materials and Methods
3.1. Materials
The object of research used in this study is to determine the micro insurance premiums for fishermen in Cirebon
Regency using an aggregate risk model approach. The data used is primary data in the form of the loss of fishermen's

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121

income if they do not go to sea in Cirebon Regency in December 2020 to November 2021. Data collection is obtained
through a survey, namely by distributing questionnaires to fishermen in the Cirebon Regency area.)
3.2. Methods
The initial step taken in this study was to collect data in the form of many incidents of fishermen not going to sea
and the large loss of fishermen's income if they did not go to sea that occurred in Cirebon Regency.
Based on the data obtained from the survey results that have been carried out, further data analysis is carried out in
the form of the Aggregate Risk Model which is the basis for the method used to calculate the micro insurance
premiums for fishermen in Cirebon Regency. Starting with testing the suitability of the existing distribution using the
Chi-Square test for continuously distributed data and the Kolmogorov-Smirnov test for discrete distributed data, then
proceeding with estimating the data distribution parameters with the Maximum Likelihood function, to get a large
premium. Easy fit software and Microsoft Excel are used to speed up and simplify the calculation process. The
following is the data on the average number of incidents and the amount of fisherman losses due to not going to sea
per month used in this study:
Table 1: Average Number of Incidents and Loss of Fisherman if They Don't go to The Sea Per Month
No
1
2
3
4
5
6
7
8
9
10
11
12

Month
December’20
January’21
February’21
March’21
April’21
May’21
June’21
July’21
August’21
September’21
October’21
November’21

Average Number of Events
6
5
5
5
5
6
6
7
7
8
8
7

Average Loss Amount
3.614.583
3.725.500
3.390.875
3.736.833
3.576.833
3.428.041
3.374.291
3.528.875
3.433.541
3.319.583
3.503.83
3.410.166

4. Results and Discussion
For data the number of events is assumed to have a Poisson distribution, and the data for the magnitude of losses is
assumed to have an exponential distribution. This assumption is proven by modeling the data with histogram graphs
using Easy fit software as follows:

Figure 1: Data on The Number of Events

Figure 2: Data Loss Amount

The next step is to estimate the parameters for each data distribution. Parameter estimation was carried out using
the Maximum Likelihood Estimation (MLE) method. Parameter for estimating many events that cause losses, and
parameter for estimating the amount of losses received.
∑
̂
(18)

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122

̂

(19)

̅

Based on the data that given in Table 1, the values of each parameter that calculated by using equations (18) and
(19) is:

After that, the calculation of the amount of individual losses is carried out by estimating the expected value and
variance of the data on the amount of loss. For data contained many event that cause losses, ( )
and Var( )
. For data contained the amount of losses received, ( )
and
( )
( ) – ( ( )) . The results are:
( )
( )
( )
( )

(

)

So, the values of the collective risk model that calculated by using equations (14) and (15) are:
( ( ))
( ( ))
After getting the estimated value of the collective loss risk from the loss data, then it is used to determine the
amount of premiums that must be paid by fishermen. The calculation is using two principles, value expectation
principle and standard deviation principle based on equation (16) and (17), the results are:
 Value expectation principle:
( )
 Standard deviation principle:
( )

5. Conclussion
The estimation of parameters used for determining the expectation and variance from an individual risk model.
After that, the results of that individual risk model is used for find the values of a collective risk model and calculating
the premium. The results of the premium calculation are used as a reference for insurance companies in determining
the amount of individual micro insurance premiums that must be paid by fishermen. The premium value of IDR
153,861,958,00 was chosen because the value is lower than other calculations, so it can result in lower individual
premium prices by insurance companies.

References
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