Available online at https://journal.
com/index.
php/ijqrm/index International Journal of Quantitative Research and Modeling e-ISSN 2721-477X p-ISSN 2722-5046 Vol.
No.
3, pp.
307-315, 2025
Premium Sufficiency Reserve on Joint Life Insurance With Laplace Distribution Meisy Triyuni1*.
Haposan Sirait2 Bachelor of Statistics Study Program.
Department of Mathematics.
Faculty of Mathematics and Natural Sciences.
University of Riau.
Bina Widya Campus.
Pekanbaru 28293.
Indonesia Department of Mathematics.
Faculty of Mathematics and Natural Sciences.
University of Riau.
Bina Widya Campus.
Pekanbaru 28293.
Indonesia *Corresponding author email: meisy.
triyuni6769@student.
Abstract Each insurance participant pays a premium to the insurance company during the coverage period.
In paying the sum insured to insurance participants, insurance companies need to prepare reserve costs.
This reserve fee is used to pay for the needs of insurance companies and insurance participants.
This research explains the calculation of premium suEciency reser ve for joint life insurance for life insurance participants aged x years and y years with Laplace distribution.
The parameters of the Laplace distribution are estimated using the method of momen and the method of maximum likelihood.
The solution of the problem is obtained by determining the initial life annuity term, single premium, and annual premium so that the premium suEciency reserve formula based on Laplace distribution is obtained.
The results of the calculation of premium suEciency reserves of joint life insurance using Laplace distribution are more less the same as the prospective reserves of joint life insurance using Laplace distribution.
Keywords: Premium SuEciency reserve, endownment joint life insurance, method of moment, method of maximum likelihood.
Laplace distribution Introduction Nowadays, all aspects that are important to human life will not always be in a good and safe condition.
In real life, these important aspects will always be surrounded by various things that might threaten security, safety and cause Anancial losses.
One of the eAorts to overcome this is by transferring the Anancial loss to another party which then led to the existence of insurance, one type of insurance is endowment life insurance.
Endowment life insurance is a combination of pure endowment life insurance and term life insurance where the insured must pay the sum insured during or at the end of the policy coverage period, either death or survival (Futami, 1.
Multi life insurance id divided into two parts according to the payment pe- riod, namely joint life and last Joint life insurance is joint life insurance.
where the premium is paid until the Arst death of the participants.
last survivor life insurance is life insurance with premium payments until the last death of the participants (Bowers et al.
, 1.
Based on theory, the amount of money available to the company during the coverage period is called reserves.
Based on the method of colculation, reserves are divided into two types, namely retrospective reserves and prospective In prospective reserves there are several reserve modiAcations, one of which is premium suEciency.
Premium suEciency is the calculation of insurance premium reserves based on gross premium assumptions.
Gross premium is an annuity, but gross premium has a greater value than net premium (Futami, 1.
Laplace distribution is one of the continuous probability distributions named after Pierre Simon Laplace.
Laplace distribution is also called double exponential distribution which has very wide applications (Al-noor & Rasheed, 2.
In this article, premium reserves are determined for dual-use joint life insur- ance with premium suEciency reserves using the Laplace distribution function.
In this case, the author limits to include only two insurance participants aged x years and y years in one policy.
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Literature Review Survival Function And Laplace Distribution The survival function is obtained from the distribution function using a probability density function.
The probability of is denoted by The symbol represents the name of the random variable, while is the value of the random variable with the probability density function symbolized by By therefore, the probability density function and distribution function are explained Arst.
DeAnition 1 (Walpole et al.
, 2.
Cumulative distribution function of a continuous random variable with probability density function The survival function or life function denoted by is used to express the chance of a person living to the age of years where If is a continuous random variable that expresses the age of the insurance participant to survive, then the survival function of the insurance participant described (Bowers et al.
, 1.
, namely The relationship between the survival function and the cumulative distribution function is as follows:
Based on DeAnisi 1 and equation .
, the survival function of an insurance participant aged years is obtained, .
The cumulative distribution function of a continuous random variable expressed as follows (Dickson et al.
, 2.
denoted by can be .
The probability of a person aged x in the next t years is expressed as follows (Bowers et al.
, 1.
Based on equation .
and equation .
the relationship between the cumulative distribution function and the probability of death is obtained, namely .
The cumulative distribution function of a continuous random variable (Walpole et al.
, 2.
can also be expressed as follow The survival function of a continuous random variable T .
is denoted by which is expressed as .
Furthermore equation .
is substituted into equation .
so that it can be writen into .
The function expressed as follows:
expresses the chance that a person aged years can survive up to years which is .
Based on equation .
and equation .
the relationship between chance and probability of death is obtained, .
Joint life insurance is an insurance program that provides protection to two or more people who have a family relationship (Futami, 1.
In this article the marger is limited to two insurance participants aged x years and y years respectively which is then expressed as a joint life.
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The probability of life insurance participants in the joint life status is expressed in a combined form.
This is because the chances of each participant are independent of each other, so that the life chances for the combined status of joint life are described in (Bowers et al.
, 1.
, and (Promislow, 2.
as follow:
In this article, the life expectancy and death expectancy of insurance participants are determined using the Laplace distribution.
The probability density function of the Laplace distribution is expressed as follows (Ogunsanya & Job, 2.
where is a shape parameter, and is a scalar parameter.
Furthermore, from equation .
the cumulative distribution function of the Laplace distribution is obtained, .
Then by substituting equation .
into equation .
we obtain the survival function Based on equation .
, the cumulative distribution function for years of Laplace distribution, namely of Laplace distribution, .
The cumulative distribution function for an insurance participant aged years dying at an interval of obtained by substituting equation .
, equation .
, and equation .
into equation .
as follows:
years is .
Based on equation .
and equation .
, the probability that a person aged with Laplace distribution is obtained, namely years will die in the next Life chances of a person aged years living up to equation .
is substituted into equation .
, namely years then using the Laplace distribution is obtained by .
Based on the equation .
life chances of a life insurance participant aged with a Laplace distribution, namely years can survive until years later Life chances the joint life status of an insured person aged years and years with Laplace distribution is obtained by substituting equation .
and equation .
into equation .
, namely .
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Materials and Methods Parameter Estimation on Laplace Distribution In Laplace distribution there are parameters and whose values are unknown.
Therefore, the parameters and are estimated using the method of moments and the maximum likelihood method.
Determining the estimated value of the parameter using the method of moments, the population moment of the Laplace distribution is obtained, namely O Furthermore by equating the population moments and sample moments the esti- mated value of the parameter is obtained which is denoted by C , namely Next determine the estimated parameter .
by using the maximum likelihood method of the Laplace distribution.
Equation .
is expressed in natural logarithm .
transformation, namely .
Then determine the Arst derivative of to obtain:
Then equalized to zero obtained .
Furthermore by using equation .
and substituting equation .
obtained the estimated value of the parameter denoted by C , namely .
Innitial Joint Life Annuity Term Life Insurance With Laplace Distribution Life annuity is an annuity whose payment is inCuenced by the chance of life and the chance of death of insurance participants over a certain period of time.
A life annuity whose payments are made at the beginning of each year for a certain period of time is called a term early life annuity (Futami, 1.
The value of the initial life annuity is inCuenced by the interest rate denoted by The interest rate used in the study is the compound interest rate, which is a way of calculating interest where the principal amount of the next investment period is the previous principal amount plus the amount of interest earned.
In compound interest there is a discount factor denoted by v, which is the present value of a payment of 1 unit of payment made one year later (Futami, 1.
, namely Triyuni et al.
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The discount rate which is the amount of interest lost if the payment is made one year earlier denoted by d (Futami, 1.
can be expressed as follows:
The cash value of the initial joint life annuity for an insured person aged period of years, namely years and years with a payout The cash value of the initial annuity of years old and years old with a coverage period of distribution is obtained by substituting equation .
into equation .
, namely .
years with Laplace Furthermore the cash value of the initial life annuity of years and years old insurance participants and the coverage period of years based on equation .
with Laplace distribution, namely .
years old insureds with a payout period of .
The cash value of the initial joint life annuity of years old and years based on equation .
with Laplace distribution, namely Furthermore based on equation .
, the cash value of the initial joint life annuity of participants aged years and years with a payment coverage period of years with Laplace distribution, namely .
Premium Of Joint Life Insurance With Laplace Distribution Before calculating the annual premium.
Asrt calculate the single premium of joint life insurance.
The single premium of joint life insurance is denoted by .
and can be expressed as follows:
Furthermore, by substituting equation .
into equation .
the single premium of joint life insurance is obtained, namely .
Based on the equation .
the single premium for insurance participants aged with a coverage period of years with Laplace distribution, namely .
years and .
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The annual premium is the premium paid at the beginning of each year, which can be the same or change every year (Futami, 1.
Annual premium of joint life insurance with Laplace distribution paid for years where by insurance participants aged years and years denoted by .
can be expessed as follows:
Premium Sufficiency Reserve On Joint Life Insurance With Laplace Distribution Premium suEciency reserve is the calculation of insurance premium reserves based on gross premium Gross premium or gross premium is the premium paid by life insurance participants to the insurance Gross premium is also an annuity, but gross premium has a greater value than net premium.
In the gross premium, the detemination is inCuenced by management fees.
The gross premium of endowment life insurance can be expressed as follows (Futami, 1.
Premium suEciency reserve is a modiAcation of prospective reserve.
In prospec- tive reserve, the calculation of reserves is obtained from the diAerence between the present value of future payments, denoted by , and the present value of future revenues denoted by , namely (Futami, 1.
In prospective reserves, the calculation of reserve can be obtained from the diAerence between the present value of future payments denoted by A, and the present value of future receipts denoted by P a.
thus equation .
of prospective reserve can be expressed as follows:
The calculation of premium suEciency reserve is carried out on the basis of future expenditures added to the insurance companyAos management costs in the form of agent commission fees for each premium collection, premium maintenance costs during the payment period, and premium maintenance costs after the payment period until the end of the coverage period.
So that the present value of future payments based on prospective reserves changes to (Futami, 1.
( O .
) .
Meanwhile the present value of future revenue use gross premium instead of net premium.
Therefore the present value of future revenue in equation .
based on prospective reserves changes (Futami, 1.
Premium sufficiency reserve denoted by .
, .
with a premium payment period on m years and a reserve calculation time of t years is obtain by substituting equation .
and equation .
into the equation .
obtained as follows (Futami, 1.
Furthermore premium suEciency reserve using the gross premium obtained by substituting the equation .
into the equation .
is expressed as follows:
) O .
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( O
) O
Premium suEciency reserve for joint life insurance life n years and insured aged x years and y years with annual premium for m years paid in advance in t years using Laplace distribution is obtained from substituting equation .
, equation .
, equation .
, equation .
, equation .
, equation .
into the equation .
Results and Discussion A couple aged 47 years and 40 years respectively joined a joint life insurance program with a term of 20 years and a premium payment period of 18 years.
If the sum insured received by the heirs is IDR 100,000,000 and the prevailing interest rate is 6% and the new policy coverage fee (A) is 0, 5% and the insurance company management fee (A) is 5% of the sum insured.
Determine:
Prospective reserve of joint life insurance using Laplace distribution.
Premium suEciency reserve of joint life insurance using Laplace distribution.
Based on the problem, it is known that Based on equation .
, the discount factor is obtained.
Discount rate based on equation .
, namely The calculation of reserves using Laplace distribution is done with Maple 13.
Before calculating the reserves Arst determine the estimated value of the parameters of the Laplace distribution, with the help of Triyuni et al.
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Maple Furthermore the calculation at t = 1 for each case using software Maple 13 and obtained the following results:
Prospective reserve of joint life insurance using Laplace distribution.
The prospective reserve for the end of the Arst year of a married couple who are members of a joint life insurance with Laplace distribution, using the equation .
Premium suEciency reserve of joint life insurance using Laplace distribution.
The premium suEciency reserve of endowment life insurance with Laplace distribution for the couple at the end of the Arst year based on the equation .
is obtained .
The complete calculation of prospective reserves and premium suEciency of Laplace distribution joint life insurance for a couple aged 47 years and 40 years respectively in year t for the above two cases is presented in Table 1 and illustrated in Agure 1.
Table 1: Prospective reserve and premium suEciency reserve with Laplace distribution Years.
Prospective reserve (IDR) .
Premium Sufficiency reserve (IDR) .
Figure 1: Prospective 2,670,301.
4,159,123.
5,854,031.
7,783,552.
9,980,163.
12,480,833.
15,327,651.
18,568,533.
22,258,026.
26,458,227.
reserve and premium suEciency 2,670,301.
4,159,123.
5,854,031.
7,783,552.
9,980,163.
12,480,833.
15,327,651.
18,568,534.
22,258,026.
26,458,227.
reserve joint life insurance with Laplace Based on Table 1 and the illustration of Agure 1 it is found that the size of the premium suEciency reserve of joint life insurance using Laplace distribution are more less the same as the prospective reserve of joint life insurance using Laplace distribution.
Conclussion Based on the discussion, it is known that the premium amount depends on the coverage period, payment period, interest rate, annuity cash value, and the age of the insured.
This thesis uses a compound interest rate that depends on the discount factor v and the discount rate d.
the calculation premium suEciency reserve in joint life insurance is aAected by management costs namely the cost of covering new policies and maintenance costs.
Triyuni et al.
/ International Journal of Quantitative Research and Modeling.
Vol.
No.
3, pp.
307-315, 2025
Premium suEciency reserve using Laplace distribution are more less the same as the prospective reserve.
Premium suEciency reserve there are insurance main- tenance costs in the calculation of reserves.
Prospective reserve and premium suE- ciency reserve with Laplace distribution increases every year.
The single premium with Laplace distribution charged to insurance participants is getting bigger with a smaller initial life annuity.
Acknowledgments Gratitude is given to Dr.
Haposan Sirait.
Si.
who has guided and provided direction in writing this article.
References