Determining Tomato Crop Agricultural Insurance Premium for COVID-19 Pandemic

One type of insurance known as parametric insurance has an agreement for predetermined events made at the beginning of the contract between the insurer (insurance firm) and the insured (farmer). When the causative event occurs, the provision applies that insurer must pay insured with some amount of money (damage compensation). Ozaki has formulated parametric method of premium rates for agricultural insurance build upon yields in specific area. Indonesian Ministry of Agriculture uses this method to ensure that farmers can re-plant crops in following planting season if a crop failure occurs. However, the COVID-19 pandemic's losses were not covered by this method. Given this, we would like to develop agricultural insurance models for tomato crops which figure out COVID-19 pandemic. For make it easier to see the price of tomato commodity due to impact of COVID-19 pandemic, in this research we will take a case study on agriculture managed by PT Mitra Tani Parahyangan. This company is engaged in the horeca business, so it has been greatly affected by the quarantine policy. The results of this study are suggestions for policy makers in anticipation if a pandemic occurs again, it help farmers and Indonesia’s food availability will be maintained.


INTRODUCTION
Corona virus disease 2019 (COVID-19) suddenly spread rapidly throughout the world and hit various aspect of life such as health, economic, social and others [1], [2].All sectors affected by the pandemic are struggling to survive, but some have been hit harder than others.Agriculture has become a vulnerable sectors affected by this pandemic [3].For a nation like Indonesia, where many people still rely on the agricultural sector, this is a complicated situation [4].Research have examined the pandemic's effects on agriculture from a various aaspects of life [5], [6].Some studies examined the impact of the pandemic on food security [7], while others concentrated on the effects of the pandemic on agricultural supply chains [8] and others examined the impact of the pandemic on food safety [9].Concerning the impact of pandemics on agriculture, two problems must be solved: the first is to guarantee the production and supply of agricultural goods, and the second is to maintain farmers' income, which in turn affects

METHODS
The Data This research uses study cases method which is secondary data.Secondary data in this study contain tomato yields and sales of tomato plants per month at PT Mitra Tani Parahyangan starting from the beginning of 2015 to the end of 2021 in kilograms.Several things to note is that sometimes PT Mitra Tani Parahyangan got market demand that exceeds the tomato harvest from their farmers.To solve this, the marketing manager takes tomato crops from farmers in the surrounding area which is not their partner.

Parametric Method in Insurance
The parametric method makes the assumption that crop yields fit with specific distributions.When the yield falls below the guaranteed yield, the probability of yield loss in agriculture is equal to the area under the density function's curve.Let  denote the level of coverage, where 0 <  < 1 and   represents the expected yield.The loss probability

Anderson Darling Test
Anderson Darling test is used to test the normality of a data (normality test).The data needs to follow a normal distribution because the test statistic used in parametric analysis for parameter estimation is derived from a normal distribution.The decision making process on the Anderson Darling test is to use the critical value.If the statistical value < critical value then reject  0 , whereas if the statistical value > critical value then accept  0 with a value of (significant level) = 0.01, 0.05, 0.10, …  [20].

Kolmogorov-Smirnov Test
The Kolmogorov-Smirnov test is one type goodness of fit test.In this case what is considered is the degree of correspondence between the distribution of sample values (observed scores) and certain theoretical distributions.The advantage of this test is that it is simple and does not cause differences in perception between one researcher and another, which often occurs in normality tests using graphs.The basic concept of decision making in the Kolmogorov-Smirnov test uses use the critical value.If the statistical value < critical value then reject  0 , whereas if the statistical value > critical value then accept  0 with a value of (significant level) = 0.01, 0.05, 0.10, …  [20].

Akaike's Information Criterion (AIC)
AIC is a measure for selecting the best distribution introduced by Hirotugu Akaike in 1973 with the equation: Where: log (| ̂) : Log-likelihood function  : Empirical data  : Number of parameters The first part −2(| ̂) is the fit measure for the selected distribution and the second part 2 is the rule requirement for the complexity of the distribution.According to the AIC method, the best distribution is the distribution that has the smallest AIC value [22].

RESULTS AND DISCUSSION
The data in Table 1 will be divided into three time periods: 1.Before COVID-19 pandemic occurs, from January 2015 to December 2019 notated as

Tomato Yields Description
Let random variables of tomato yields in time periods  = 0,  = 1 and  = 0 ∪  = 1 are  0 ,  1 and  descriptive statistic value is shown by Table 2. Table 2 shows that  0 ,  1 and  have positive skewness value that means they have right skewed curve.Meanwhile the average of their kurtosis are around 3, that means they have high peak in some value of data.The following figures show the histogram of tomato crop yields data Furthermore, the expected value of tomato crop yields will denoted as  0  for conditions  = 0,  1  for conditions  = 1 and   for conditions ( = 0 ∪  = 1).The expected value of tomato crop yields is shown by Table 3.The expected value in Table 3 is used to find limit value of the maximum guaranteed yield using the percentage of yields which is denoted by symbol .In general, the value of  has range 0 <  < 1.If the value is near to 1, then a higher premium must be paid, conversely if it is close to 0 then the premium to be paid will be smaller.In this study the  value will be focused on the range 0.6 <  < 0.9 as presented in Table 4.

Sales Result Description
Let random variables of tomato yields in time periods ( = 0), ( = 1) and ( = 0 ∪  = 1) are  0 ,  1 and  descriptive statistic value is presented in Table 5.From Table 5 show that the average sales result while COVID-19 pandemic was significantly decreased.The sharp decline in sales results was due to the hotel, restaurant and cafe (horeca business) sector, which has been PT Mitra Tani Parahyangan's main sales target did not operate smoothly.This was caused by the COVID-19 pandemic which required large-scale social restrictions policy or usually called as PSBB in Indonesia.Before COVID-19 pandemic occurs, tomato sales result reached 16092 kg but in pandemic it reached the lowest point at 1385 kg.The difference between tomato yields and sales results will be shown in figure 2.

Tomato Crop Agricultural Insurance Model
The area-yield insurance model introduced by Ozaki [20] does not recognize extreme conditions such as the occurrence of a pandemic.Therefore, this study offers a model of yield-based agricultural insurance in an area (area-yield insurance) which calculated pandemic as a risk factor by assuming several conditions.The conditions to be considered are divided into three conditions, first when there is no pandemic( = 0), when there is a pandemic( = 1) and the combined condition of the two ( = 0 ∪  = 1) with the events of a pandemic is indicated by the random variable .

The Coverage Value
The coverage value of agricultural insurance is the amount of money that will be paid by the insurer to the insured if the conditions are fulfilled.In this case, coverage value is calculated based on PT Mitra Tani Parahyangan's production cost for tomato commodity.These costs include the price of seeds, fertilizers, pesticides and other needed as much as IDR 16,000,000 per hectare in one planting season.In 2016, the Ministry of Agriculture of the Republic of Indonesia proposed a yield percentage () of 75%.According to the Table 6, the recommended coverage value is IDR 12,000,000.

Tomato Sales Results Distribution and Parameter Estimation
The mathematical approach to determine the best distribution and estimating the parameters in this study by using the Kolmogorov-Smirnov test and the Anderson-Darling test to show the level of data fit with a distribution based on the statistical value and pvalue assisted by EasyFit software.The appropriate distribution in the Kolmogorov-Smirnov test is chosen based on the lowest statistical value.While in the Anderson-Darling test, the appropriate distribution is determined by comparing the extreme values in the data.Furthermore, the suitability of the two tests is compared using Akaike's Information Criterion (AIC) based on the lowest value.Hypothesis that is used as the basis for drawing conclusions from the Kolmogorov-Smirnov and Anderson-Darling tests as follows:  0 : Data fitted with  distribution  1 : Data did not fit with  distribution with  represents random variable that expresses a certain theoretical distribution to be determined for its fit with the distribution of random variable  0 &  1

 Tomato Sales Results Distribution and Parameter Estimation when 𝑻 = 𝟎
Table 7 will presented the result of Anderson-Darling & Kolmogorov-Smirnov tests for  0 with  = 0.05 Based on the results of Kolmogorov-Smirnov and Anderson-Darling tests, there are still 5 distributions that match the random variable  0 , therefore a comparison of the Akaike's Information Criterion (AIC) value will be made and the distribution with the smallest AIC value is selected which is presented in Table 8.Table 8 imply that the appropriate distribution for the random variable  0 with the smallest AIC value is the Generalized Pareto distribution.The parameter values for the Gen. Pareto distribution obtained using EasyFit software are  ̂= −0.01295,  ̂= 3438.1 and ̂= 3916.3.The assumption of the selected Generalized Pareto distribution is reinforced by the QQ-Plot graph which shows most of the data (represented by the + sign) is on the reference line and there are only a small number of outliers as presented in Figure 3.  9. Based on the results of Kolmogorov-Smirnov and Anderson-Darling tests for, there are still 5 distributions that match the random variable  1 , therefore a comparison of the Akaike's Information Criterion (AIC) value will be made and the distribution with the smallest AIC value is selected which is presented in Table 10.The assumption of the selected Log Logistic distribution is reinforced by the QQ-Plot graph which shows most of the data (represented by the + sign) is on the reference line and there are only a small number of outliers as presented in Figure 4.

Premium Rate and Premium for Sales Result  Premium Rate and Premium for Sales Result when 𝑻 = 𝟎
A series of test in the previous section implies that the Generalized Pareto distribution is the best fitted distribution for  0 .The probability density function and cumulative distribution function of the Generalized Pareto distribution with shape parameter ( ̂), scale parameter ( ̂) and location parameter () are as follows [23] Probability density function: Cumulative distribution function: Expected value The following illustrates the calculation of the cumulative distribution function value and the expected value of the Generalized Pareto distribution by substituting the parameter values  ̂= −0.01295,  ̂= 3438.1 ̂= 3916.3 &  0  = 5336.9into equations ( 12), ( 13) and ( 14).Next, the expected value of the loss will be calculated according to equation (3) based on value that has been obtained previously (| = 0) =  0 (5336.9)[5336.9− [ 0 | 0 < 5336.9]]= 0.339207[5336.9− 4768.4]= 257.2832 So we can get premium rate and premium as follows : Using the same steps, the Generalized Pareto premium rate and premium distribution values for the full yield percentage () with range 0.6 <  < 0.9 are shown in Table 11. Premium Rate and Premium for Sales Result when  =  A series of test in the previous section implies that the Log Logistic distribution is the best fitted distribution for  1 .The probability density function and cumulative distribution function of the Log Logistic distribution with shape parameter ( ̂), scale parameter ( ̂) and location parameter ( ̂) are as follows [24] Probability density function: Cumulative distribution function: Expected value: The following illustrates the calculation of the cumulative distribution function value and the expected value of the Log Logistic distribution by substituting the parameter values  ̂= 3.1296,  ̂= 2183.7  ̂= 743.68 and the value of  1  = 5890.5into equations ( 15), ( 16) and (17) Next, the expected value of the loss will be calculated according to equation (4) based on value that has been obtained previously (| = 1) =  1 (5890.Using the same steps, the Log Logistic premium rate and premium distribution values for the full yield percentage () with range 0.6 <  < 0.9 are shown in Table 12.The premium rate and the premium for sales result when  = 0 ∪  = 1 can be determined using equation (10).However, in equation (10), it is necessary to determine the value of the probability of a pandemic appearance first.Marani stated that the occurrence of major pandemics such as COVID-19 and Spanish flu is very likely to happen again due to the worsening condition of the earth [25].Furthermore, Marani estimates the chance of a pandemic similar to COVID-19 in the future is 38%.Thus ( = 1) = 0.38 and ( = 0) = 1 − ( = 1) = 0.62.Illustration of the calculation of the premium rate and the sold crop premium when  = 0 ∪  = 1 using a value of  = 0.75 as follows The full calculation of premium rates and premiums for sales result with values of 0.6 <  < 0.9 is shown in Table 13.Table 13 shows that the level of tomato crop insurance premiums when  = 0 ∪  = 1 has increased by 12%-16% from pre-pandemic conditions.In the case study of PT Mitra Tani Parahyangan in this research, the author also interviewed the marketing team of PT Mitra Tani Parahyangan who said that the decline in sales turnover of PT Mitra Tani Parahyangan during the pandemic reached 60%.So that an increasing of 12%-16% in the premium rate when  = 0 ∪  = 1 is expected to cover the losses incurred during the pandemic.

CONCLUSIONS
The research has been formulated an area yield-based tomato agricultural insurance model by considering COVID-19 pandemic as the triggering event.Based on this model, we got the premium rate is 20.46% and the premium is IDR 2,455,694 with the coverage value is IDR 12,000,000.The relative high premium rate is expected to cover the 60% decline in sales turnover when a pandemic occurs.The tomato agricultural insurance model developed in this study is expected to be a consideration for government or private insurance companies in taking agricultural insurance policies for farmers.

Table 1 .
Tomato Yields and Sales Result of PT Mitra Tani Parahyangan [20]rmining Tomato Crop Agricultural Insurance Premium for COVID-19 Pandemic PandemicBinar Aulia Setyawan 112 can be calculated over the area under the density function by using trapezoidal rule to estimate it numerically.The premium in the form of rate denote as[20]:

Table 2 .
Descriptive Statistic of Tomato Yields

Table 4 .
Limit Value of the Guaranteed Tomato Crop Yields

Table 5 .
Descriptive Statistic of Sales Results

Table 6 .
Coverage Value

Table 8 .
AIC value for random variable  0

Table 10 .
AIC Value for random variable  1 Table 10 imply that the appropriate distribution for the random variable  1 with the smallest AIC value is the Log Logistic distribution.The parameter values for the Gen. Pareto distribution obtained using EasyFit software are  ̂= 3.1296,  ̂= 2183.7 and  ̂= 743.68.

Table 11 .
Premium rate and premium Generalized Pareto distribution . Cumulative distribution function for Log Logistic:

Table 12 .
Premium rate and premium Log Logistic distribution  Premium Rate and Premium for Sales Result when  =  ∪  =