THE BENEFITS OF FAMILY ANNUITY CALCULATION WITH VINE’S COPULA AND FUZZY INTEREST RATE

ABSTRACT


INTRODUCTION
One example of a multiple life annuity product is a reversionary annuity, which is a life annuity product for two/more annuitants whose annuity payments will begin after one of the annuitants specified in the contract dies first until the other annuitant also dies [1].Reversionary annuity products can be developed into family annuity products consisting of husband, wife, and child.The determination of the annuity greatly affects the amount of the pure premium that must be paid by the insured.The calculation of pure premiums for multiple lives requires a marginal and joint distribution of their lifetimes [2].
In more recent studies, it has alternatively captured old-age mortality rates through the proposition of multi-factor exponential models based on estimating mortality measures with Laguerre functions [3].These models cannot capture the unobserved heterogeneity of individuals.There is a Gamma-Gompertz model with a death rate function that can overcome the delay and capture the unobserved heterogeneity in individuals formed by the frailty proportional hazard model [4].However, these models cannot capture mortality in infancy, childhood, and young adolescence well, so a combined model of several mortality models such as Heligman-Pollard, Kostaki, and Kannisto-Makeham models to capture mortality at old ages.
In premium calculations assuming dependence between the insured parties, the copula model is the most widely used.The use of the Archimedean Copula family is more widely used than the Eliptical Copula family which is less able to capture asymmetric data shapes in joint life span modeling [5].Furthermore, studies on joint life annuities from survey data in Ghana [6].The dependency pattern of more than 2 random variables using one Copula cannot capture the dependency between random variables specifically.Therefore, Vine's Copula can be used as an alternative to modeling the dependency [7].
One of the factors that determine the contribution price is the actuarial margin level which is taken from the interest rate or yield.Determination of the APV can also use Fuzzy interest rates.Life insurance issues such as calculating the price of life insurance policies, life insurance portfolios, life contingencies, life actuarial obligations, and life annuities can use Fuzzy sets [8].The use of Fuzzy interest rates causes investment gains and surplus processes in the form of intervals [9].Based on this background, this study will apply the vine's copula to model the joint distribution of the insured parties which is constructed from the marginal distribution of future lifetimes.Three insured parties were chosen from each party, namely husband, wife, and child.In this study, the marginal distribution of each insured party was constructed using the 2019 Indonesian Mortality Table (TMI IV).Furthermore, the actuarial margin rate uses BI-7-day data which is estimated using the Fuzzy interest rate.Furthermore, based on the joint distribution, it can also be calculated the value of benefit from an insurance product, namely a family annuity.

Mortality Distribution
The distribution of future lifetime, (), can be calculated if the mortality rate, (), for each  >  is known.The distribution of the remaining lifetime obtained by assuming a mathematical function for the death rate is called the mortality distribution.The empirical survival function of the discrete random variable   is written as follows [2], There are several mortality distributions used to model individual mortality rates as follows, Estimation of parameters in the above models by minimizing a loss function (LF).The LF used the loglikelihood function, which is defined as follows, with  ̂ is the estimated value and  is the observed value [13].

Vine's Copula
The joint distribution of the insured parties will be modelled with the assumption of dependency using copula.This joint distribution model uses three values of Kendall's tau () as follows:  = { 0.25; weak 0.50; moderate 0.75; strong This joint distribution will describe a more complex dependency relationship.The copula model links the univariate marginal cumulative distribution function to a multivariate cumulative distribution [14].This is stated in Sklar's Theorem as follows; Theorem 1.Let H is a joint distribution function with marginal distribution  1 ,  2 , … ,   , then there is a copula  such that for every  1 ,  2 , … ,   ∈ ℝ holds, . One example of a copula family is the Archimedian copula in which there are several copulas such as Frank, Clayton Gumbel and Joe with different characteristics of the   1 , 2 ( 1 ,  2 ) function.
The concept of Vine's copula is to decompose the multivariate copula function into several bivariate copula functions [7] .Vine's Copula provides a more flexible way of constructing the joint distribution of multivariate variables used for at least three random variables for which the joint distribution function is to be known.In this study, the canonical vine copula (C-Vine) will be used with the density function, namely,

Fuzzy Interest Rate
Fuzzification is the process of converting crisp values to fuzzy values.Values in the field are expressed in the form of Fuzzy data which has two aspects, namely the Fuzzy set with its membership value.Examples of fuzzification are triangular fuzzy numbers and trapezoidal fuzzy numbers.
with   denotes an interval that contains all  that have a membership level greater than or equal to  [16].
Let ̃ represents Fuzzy interest rate with ̃= (, , ),  <  < .Fuzzy number can be defined uniquely with an interval by determining the  −  value of , namely: with () and () respectively are the lower and upper limits of α at a certain value of  [17].

Family Annuity
Life annuity is an annuity that is paid for life or for a certain period.In this study, the annuity is used for life and payments during life annuity.The annuity paid at the beginning and at the end of the period paid in 1 unit is stated by [2], with The family annuity referred to here is a lifetime annuity where payments are made if the insured is still alive, payments can be made at the beginning or end of the policy period.The terms of the APV of this annuity are as follows: The OAG benefits are paid to the insured every month according to the annuity payment date stated in the policy.This benefit will end after the insured dies.

b. Widow Guarantee (WG)
The WG benefit is paid to the widower at  1 of the OAG benefit every month, starting in the month following the last OAG payment.This benefit will end upon the death of the widower.

c. Child Guarantee (CG)
The CG benefit is paid to the child at  2 of the CG benefit every month, starting in the month following the last WG payment.This benefit ends when the child reaches the age of  or dies before reaching the age of .
Suppose a family annuity contract with the insured parties being husband ( 1 ), wife ( 2 ), and child ( 3 ), then the APV of family annuity is, In this section, we will determine the value of the annuity benefit obtained by the annuitant if pay a contribution of  with an additional cost proportion (0 <  ≤ 1).Additional cost consists of acquisition costs, general and costs, policy maintenance costs, and margin.policy, and margin.From this proportion, the pure premium is obtained, which is formulated as follows, Then, we can get the benefit of family annuity as follow,

Determination of Marginal Distribution
Based on the characteristics of the existing mortality models, the Heligman-Pollard Model will be used for age 0 ≤  ≤ 49, the Kostaki Model for age 50 ≤  ≤ 82, and the Kannisto-Makeham Model for age  > 82.This model is called HKK which will be used in male and female mortality rates.The function of   is written as follows, By minimizing Equation 5, the estimator of parameter values of the above models is obtained.

THE BENEFITS OF FAMILY ANNUITY CALCULATION WITH VINE'S COPULA…
with  ̂ is the estimate of   .Based on Table 2, the HKK model has the smallest SSE value compared to the HP and Carriere models.So, it makes this model as the best model to draw mortality rates for male and female.This is also supported by the shape of the HKK model plot which is close to the observation value in Figure 2. It is very different with HP and Carriere models where they have several gaps with the observation value especially in the age of 60 to 100 for male and female also.

Determination of Joint Distribution
Let ( 1 ), ( 2 ), ( 3 ) represent individuals aged  1 ,  2 ,  3 respectively.Furthermore, the random variables   1 ,   2 ,   3 respectively represent the time remaining ( 1 ), ( 2 ), ( 3 ) until death occurs.For 3 individuals consisting of father, mother, and son or daughter, the distribution function will be determined using Vine's copula.For example, ( 1 ,  2 ,  3 ) represents the father, mother, and child (boy or girl).Given The best copula is then selected based on the loglikehood and Akaike Information Criteria (AIC) values [19].Using Rstudio we can get the loglikehood and AIC values as follows, Table 3-Table 5 conclude that the Frank copula is the best copula for the joint distribution with two individuals.This contrasts with the case of three individuals where Table 6 shows that the best copula to choose is the Clayton copula.Furthermore, the value of the joint survival distribution or   43:40:15 is simulated with the assumption of annuitant age as above.From Figure 3, with the value of  = 0, the resulting  Since the margin rate is still in annual form, it is converted to monthly form through, (1 + ) 1 12 − 1.Thus, the actuarial margin rate of this data is written as follows,

Determination of Family Annuity Benefit
The simulation results of marginal and joint distributions are used to calculate single life and multiple life annuity values.These annuity values are the components to simulate the family annuity calculation.The family annuity calculation uses the fuzzy interest rate and benefits are paid continuously to the beneficiary with a payment rate of 1 unit each month.Let assumed,  = 125,000,000 and  = 20%.So, we can get the pure premium of family annuity  = 100,000,000.Based on Table 7, the value of annuity benefits on father, mother, and son with independence assumption ( = 0) is smaller than Case 1 and Case 2. The value of annuity benefits increases by about 4% as the value of  increases.Then, the value of annuity benefits father, mother, and daughter with independence assumption ( = 0) is smaller than Case 1 and Case 2 too.The value of annuity benefits increases by around 3% as the value of  increases.For both types of annuitants, the benefit value of Case 2 is also greater than Case 1.In contrast to the benefit value, the APV for the two types of annuitants shows that with the assumption of dependence, a smaller APV value is produced as shown in Figure 5. Overall, the value of annuity benefits on father, mother and son is greater than the annuity benefits on father, mother, and daughter.

CONCLUSIONS
Based on the analysis to calculate the annuity benefit, it can be concluded that, 1.The joint and marginal distribution of the random variables of one's survival.The Heligman-Pollard (HP) mortality model can capture the ages of infancy, childhood, and young adulthood well.Therefore, the Kostaki Model was used to capture old adults up to 83 years of age and after that, the Kannisto-Makeham Model was used up to 112 years of age.These three models are combined to get the best estimate of TMI IV.
2. The bivariate co-distribution is modeled with the Archimedian copula with the best copula being Frank's copula and the best co-conditional copula being modeled with Clayton copula.The trivariate joint distribution is modeled with the Vine's copula model which can capture dependencies more flexibly compared to multivariate copula which imposes the same correlation for all individuals.

Figure 2 .
Figure 2. Triangular Fuzzy Number Graph On Figure 2, the fuzzy triangular membership function of the Fuzzy number in the interval [, ] which is determined by three values (, , ) is defined where (, , ) are the lowest, the trusted, and the highest value respectively from the data.In triangular fuzzy numbers, the b value used is usually the average data [15].Then, given that  −  is the threshold level that changes the Fuzzy set to crisp.The process of converting Fuzzy sets to crisp is called defuzzification.The  −  of the Fuzzy set  is defined as follows,  = { ∈ :   () ≥ },  ∈ [0,1]

Figure 3 .Figure 4 .From Figure 4 ,
The value of   :: for some Cases (a) Father, Mother and Son, (b) Father, Mother and Daughter 3.3 Fuzzy Interest Rate In estimating the actuarial margin rate, BI data per year for the period April 2016 to December 2022.The descriptive statistics of this data are shown, BI Data per month (a) Scatter Plot, (b) Histogram it is known that the movement of the value of  has fluctuated significantly in the span of around 6 years with a mode value of 0.35.Then, the value α = 0.437 is obtained, which is the average of fuzzy membership values using the median as a trusted value.It is also known that  = 0.035 (minimum),  = 0.060 (maximum), and  = 0.048 (median).By using Equation (8), the actuarial margin level is obtained, 035 + ((0.048 − 0.035) × 0.437), 0.060 − ((0.060 − 0.048) × 0.437)] = [0.0404625,0.0545375]

Figure 5 .
The APV of Family annuity for some Cases (a) Father, Mother and Son, (b) Father, Mother and Daughter