ZILLMER RESERVE ON ENDOWMENT LAST SURVIVOR LIFE INSURANCE USING LOMAX DISTRIBUTION

Article History: This article discusses Zillmer's reserves for endowment last survivor of life insurance. Zillmer reserves are a type of modification of premium reserves which are calculated using prospective reserves and the Zillmer rate. In Zillmer reserves, loading which is the difference between gross premium and net premium in the first policy year is greater than standard loading. In this article, the life insurance used is endowment last survivor of life insurance, where the reserve calculation for last survivor status is calculated for 3 cases, namely, both participants survive until the end of the policy, participant x survive but participant y died, and participant y survive but participant x died. So, the purpose of this research is to find a way to make the loading value in 3 cases on the dwiguna last survivor of life insurance Zillmer reserves smaller. To achieve this goal, this article uses the Lomax distribution with the parameters estimated using maximum likelihood estimation and then determined by a Newton-Raphson iteration method. Based on the illustration, even though in the first policy year in cases where both participants survive until the end of the policy there was still a negative loading, overall Zillmer's reserves in each case continues to increase over time


INTRODUCTION
Basically humans will face various kinds of unpredictable risks, one way to overcome these risks is to minimize the risk, namely by participating in a life insurance program [1].Endowment life insurance with last survivor status is life insurance where premium payments are made until the death of the last insurance participant has not occured until the sum insured is given by the insurance company after the death of the last insured person [2].In an insurance company, when someone becomes an insurance participant, he or she must pay a premium to the insurance company [3].From this premium, some interest income will be obtained during the premium payment period.
The income from these premiums will later be used to pay for a number of needs from the insurance company.Reserves are the amount of money the insurance company has within the coverage period [4].The reserves in the insurance company will be used to pay the insurance money to insurance participants.In calculating the reserve value, it is necessary to first know the value of the premium and life annuity which is influenced by the chance of life and the chance of death of the insurance participant [5].In determining the chance of life and the chance of death in last survivor life insurance, the Lomax distribution is used, also known as the second type of Pareto distribution introduced by K.S. Lomax in 1954 [6], [7], with its parameters need to be estimated [8], [9].
At the beginning of the policy year, large costs are usually required for various purposes, for example paying provision fees as compensation to premium collection officers and others [3].Therefore, insurance companies at the beginning of their existence needed to have a way to prevent losses.So this article aims to obtain a calculation of the reserve value that does not have a negative value in the first policy year, or if there is still a negative value then it does not indicate a number that is too large.Negative premium reserves will provide a greater chance of insurance company bankruptcy [10].
Insurance companies need to consider including operational costs in reserve management to avoid losses, one way is by modifying the premium reserve calculation, such as using the Zillmer reserve calculation.Zillmer reserves are a type of modified premium reserves whose calculation uses prospective reserves and a Zillmer rate of α [11].Research related to the formation of life insurance premium for last survivor status, premium reserves and Zillmer reserves has been carried out by many researchers.Hasriati et al. [3] explained in his research that the calculation of single premiums and annual premiums for combined life insurance with last survivor status is enough to be combined in one insurance policy so that insurance participants pay less premiums.To determine the premium, you need to know the cash value of the initial annuity which is influenced by the chances of living and the chances of dying based on the Pareto distribution, whereas in Hasriati's et al. [12] research calculation of single premium and annual premium for combined status life insurance last survivor based on the formula of Makeham's Law which is a development of Gomperzt's law.
Research Dwipayana et al. [13] determined the last survivor life insurance premium reserve formula using New Jersey method where the calculation starts in the second year, for the  years, with  = 2,3,4, … , , where  represents the term of the insurance participant's contract.In the research Hasriati's et al. [14] discussed determining prospective reserves in endowment life insurance combined last survivor and joint life for life insurance participants by formulating the chances of life and death of insurance participants based on the Gompertz distribution.

Research by Iriana et al.
[1] calculated the Zilmer reserve of whole life insurance for insurance participants aged 23 − 25 years using the Indonesian Mortality Table 1999 and 2011 as well as gender differences, which shows that the Zillmer reserve result when calculated with the 1999 TMI is greater than the 2011 TMI, if in terms of gender, the Zillmer reserve results for male insurance participants are greater than for female insurance participants.On Hasriati et al. [5] using the Zillmer method for calculating prospective reserves in modified endowment life insurance using the CIR interest rate expressed in the form of a discount factor with two parameter estimates using Indonesian interest rate data from 2010 to 2019, producing the Zillmer Reserve formula using the CIR interest rate, for insured participants who are x years, with a coverage period of m years, and payment n years with a certain Zillmer rate which is useful for insurance companies to predict the reserves they have so that they are sufficient in the event of an insurance claim.Meanwhile, in research Hasriati's et al. [10] discussed determining the loading value of the Zillmer reserve which is the difference between the gross premium and the net premium in the first policy year which is smaller than the standard loading using the survival function based on the Pareto distribution and interest rates using the CIR interest rate model.
Based on the background that has been developed and the references cited, the aim of this research is to determine how to achieve premium reserves using the Zillmer method last survivor status there were 3 cases in the first policy year each case does not have negative value, by formulating the chance of life and the chance of death of life insurance participants based on the Lomax distribution by estimating the parameters, this research is useful for insurance companies in calculating last survivor status life insurance premiums so that the premium reserve in the first year of the policy does not cause problems for calculations at a later time.

RESEARCH METHODS
The method used in this research is a qualitative method based on relevant literature studies and journals.This research does not include data so no case studies are discussed.However, in this research an example is given as an illustration of the use of last survivor status and Lomax distribution in calculating the Zillmer reserves.This section explains the statistical and actuarial theories used to analyze the problems discussed.These theories include the survival function of last survivor function, the Lomax distribution and parameter estimation on Lomax distribution.

Survival Function of Last Survivor Function
Endowment last survivor of life insurance is a combined life insurance where premium payments are made until the last death of the insurance participant.Combined life insurance in determining the amount of premium required survival function for combined status [15], which is obtained from the relationship between survival function on individual status.The random variable  is said to be a continuous random variable if there is a function () so that the cumulative distribution function can be expressed [16] The survival function is denoted by (), () = (X > ).
The relationship between the survival function and the cumulative distribution function can be obtained as follows: The function  () () is the probability that a person aged  dies within a period of  years, with  () () denoted by    .Based on Equation (1) the survival function can be stated ( The relationship between the probability of living   (3)

Lomax Distribution
The Lomax or Pareto II distribution introduced by K.S. Lomax in 1954, is a special case of the Pareto distribution whose interval starts from zero and has been widely used in survival analysis [17].The probability density function of the Lomax distribution, namely [6] (, , ) =   ( + ) −(+1) , , ,  > 0, The parameter  is a shape parameter and  is a scalar parameter.
The cumulative distribution function () in the Lomax distribution can be obtained as follows: Based on Equation (1) and Equation (4), the survival function of the Lomax distribution is obtained The cumulative distribution function of the continuous random variable in the Lomax distribution can also be expressed as follows [18]: ). (6)

Parameter Estimation on Lomax Distribution
In the Lomax distribution, there are several parameters whose values need to be known.Therefore, parameter estimation is carried out using the Maximum Likelihood Estimation (MLE) method, the way this method works is to maximize the function likelihood function [19].
The procedure for finding the maximum likelihood of a parameter is to determine the likelihood function of the probability density function in the Lomax distribution as follows: Equation ( 7) is expressed in terms of the natural logarithm (ln), the log-likelihood function The next step after obtaining the log-likelihood function that is to determine the first derivative of the parameters  and  as follows: Furthermore, the first derivative equation is equal to zero so that the a closed form equation is formed to obtain parameter estimates on the Lomax distribution, denoted by  ̂ and  ̂ as follows: Equation ( 12) is an equation that is not closed form because in the final equation there is still a parameter .One method of solving using the Newton-Raphson method [9].
The Newton-Raphson method uses the iteration approach to produce convergent values.The general equation of the Newton-Raphson method is as follows [20]: Based on Equation (11), Equation (10) which is equated to zero can be stated ) ∑ (    +   )  =1 . ( The iteration formula for  based on Equation ( 14) is obtained The function  with respect to  is based on Equation ( 15) . ( The first derivative of the function  with respect to  based on Equation ( 16) is obtained ) ) . ( The value of iteration  in Equation ( 18) will be used to determine the estimate of parameter  in Equation (11).Parameter estimation that depends on life insurance participants for age  years is denoted by  ̂ and  ̂ while for age  years is denoted by  ̂ and  ̂.

Futures Life Annuity of Last Survivor Endowment Life Insurance
Annuity is a series of payments to an insurance company in a certain amount at certain time intervals as long as the insurance participant is still alive [21].In an initial life annuity, payments are made at the beginning of the period up to the  − 1 year period.In the calculation of the annuity life is strongly influenced by the interest rate and there is also a discount factor function, namely The cash value of a term initial life annuity for insurance participants aged  years with a coverage period of  years can be stated [18] Based on Equation (5) the term initial life annuity for insurance participants aged  years with a coverage period of  years uses the Lomax distribution with parameters that depend on age  years denoted   and   , namely Initial term life annuity for insurance participants aged  years with a coverage period of  years using the Lomax distribution with age-dependent parameters  years denoted   and   based on Equation (20) can be stated, namely Term initial life annuity for insurance participants aged  years with a coverage period of ℎ years using the Lomax distribution with parameters that depend on age  years based on Equation ( 19) is stated Based on Equation ( 22) the term initial life annuity for insurance participants aged  years with a coverage period of ℎ years using the Lomax distribution with age-dependent parameters  years can be expressed (24) Based on Equation (6) the initial life annuity last survivor term in Equation (24) with a coverage period of  years using the Lomax distribution with parameters that depend on age  years and  years, namely )).
Based on Equation (25), the initial life annuity of last survivor a term with a coverage period of ℎ years using the Lomax distribution with age-dependent parameters  years and  years can be expressed )). (26)

Premium and Prospective Reserve of Last Survivor Endowment Life Insurance
In endowment life insurance, the sum insured is based on the amount of a premium.Premiums are a series of payments made by insurance participants to insurance companies over a certain period of time with a predetermined amount.The single premium for last survivor endowment life insurance is the sum of the single premium for pure last survivor ife insurance and term last survivor life insurance, is for  is a function of the discount rate expressed by  = 1 − .The single premium for the last survivor endowment life insurance with a coverage period of  years uses the Lomax distribution as follows: )), (28) with a sum insured of  paid at the end of the single premium life insurance last survivor endowment policy year to  :| .
Annual premium is a premium paid at the beginning of each year whose amount can change or be the same every year [4].The Annual premium of the endowment last survivor life insurance with a coverage period of  years is denoted by  :| and can be expressed as The annual premium for last survivor endowments life insurance with a coverage period of  years using the Lomax distribution as follows: )) − .
The sum insured if  is paid at the end of the policy year, annual premium for the endowment last survivor life insurance will be  :| .
Prospective reserves of endowment last survivor life insurance with the sum insured paid at the end of the policy year, the single premium is  = 1 − ̈, and the annual premium is  :| = )) .

Zillmer Reserve Last Survivor Endowment Life Insurance
Zillmer reserves are one type of modified premium reserves which are calculated using prospective reserves and a Zillmer level of , with an  value of 0,025.In Zillmer reserves, there are modified premiums, namely  1 and  2 , each of which is the net premium in the first year of the policy and the net premium in the second year of the policy until the ℎ-year of the policy where ℎ is Zillmer's time with  1 <  :| <  2 .The relationship of the Zillmer rate and net premium, ie [5] Premiums paid by last survivor life insurance participants with a coverage period of  years are illustrated by the following timeline: Based on the timeline in Figure 1 and that the  value is equivalent to the annual endowment life insurance premium [22].Then to obtain the cash value of the modified premiums  1 and  2 can be determined as follows: :| ̈ :ℎ| =  husband passed away and the wife was still alive husband was still alive and the wife passed away both participants live until the end of the policy year

CONCLUSIONS
Zillmer reserves are an alternative method that can be used to determine of reserves obtained from net premiums with a value of  which is the Zillmer level.In this case, calculating on Zillmer reserves using the Lomax distribution as a modification of prospective reserves which is suitable for modeling the survival function of both men and women, and endowment last survivor status, so by using a Zillmer rate  of 0.025, we obtain increasingly increasing Zillmer reserves.This research is very useful for insurance companies in predicting the reserves that must be owned by the insurance company when a claim occurs

𝜆 in Equation (13) can
be expressed in the Newton-Raphson method based on Equation

Zillmer Reserve of Last Survivor Endowment Life Insurance
1 +  2 (̈ :ℎ| − 1) =  1 −  2 +  2 ̈ :ℎ|, Calculation of Zillmer reserves last survivor endowment life insurance with  years of reserve calculation time, ℎ years of Zillmer time 1 ≤  ≤ ℎ, and coverage period of  years, denoted  :|Example: A husband and wife whose respective ages are , which is 38 years and , which is 35 years, following the last survivor endowment life insurance program with a term of 20 years.If the sum insured received by the heirs is Rp100.000.000, the current interest rate is 0,02, the Zillmer rate is 0,025 and the Zillmer time is 16 years, then determine the calculation for the following cases:(i)If both participants live until the end of the policy year (ii) At the time the husband was still alive and the wife passed away (iii) At the time the husband passed away and the wife was still alive