Model for Determining Insurance Premiums Taking into Account the Rate of Economic Growth and Cross-Subsidies in Providing Natural Disaster Management Funds in Indonesia

Abstract

Natural disasters are increasing due to climate change, which is causing economic losses for countries affected by them. Disaster management funds need to be provided, including through purchasing insurance. Determining natural disaster insurance premiums needs to involve consideration of the geographical conditions of the country. The aim of this research was to develop a model for determining natural disaster insurance premiums using the jumping processes method and a cross-subsidy system. The model takes into account the level of economic growth and the natural disaster potential index. The data analyzed relate to cases of natural disasters and losses that occurred in each province in Indonesia. From the results of the analysis, it was found that through a cross-subsidy system, the principle of mutual cooperation can be applied in managing natural disasters. Regions with a high level of economic growth and a low natural disaster potential index need to provide subsidies to regions with a low economic growth rate and a high natural disaster potential index. It was also found that the cost of insurance premiums was influenced by the size of losses and the frequency of natural disasters in the province. The greater the potential for disasters and economic losses experienced by a province due to disasters, the greater the premium burden that must be borne, and vice versa. Based on these conditions, insurance premiums vary in each province in Indonesia. It is hoped that the results of this research can provide a reference for the government in determining policies for providing funds for natural disaster management using a cross-subsidy system. In addition, this research can provide a reference for insurance companies in determining natural disaster insurance premiums in Indonesia.

1. Introduction

Indonesia is located on the Ring of Fire and at the confluence of three major tectonic plates, namely, the Indo-Australian, Eurasian, and Pacific plates. In the Ring of Fire, there is the potential for natural disasters, especially earthquakes, volcanic eruptions, and tsunamis [1,2,3]. Natural disasters are phenomena that cannot be predicted with certainty; they can only be predicted based on their initial symptoms. Losses due to natural disasters are spread across the infrastructure and public facilities sectors [4,5], including damage to public roads [6,7,8], offices, and residential areas [9,10]; the agricultural sector, involving damage to agricultural land [11,12,13]; and socio-cultural sectors, having a direct impact on the welfare of the people around the areas in which natural disasters occur [14,15,16]. Based on economic growth, the disasters that occur have an impact on regional and national economies, both directly and indirectly [17,18,19,20]. In efforts to minimize risks due to natural disasters, it is necessary to devise natural disaster mitigation policies that are sustainable and on target. According to Mills (2007) [21], strategies that can be implemented to reduce risk vulnerability to losses due to disasters and support sustainable development can include purchasing natural disaster insurance products.

Natural disaster insurance has advantages for customers, including transferring risk to an insurance company. In this case, the risk of economic loss resulting from a disaster will be borne by the insurance company. Of course, the customer has an obligation to pay a premium as a form of risk transfer, which is charged to the insurance company. The cost of the premium and level of risk insured are agreed between the insurance company and the customer [22,23,24,25]. However, insurance companies also have the risk of bankruptcy if the insurance premiums are not calculated effectively. In addition, premium calculations must be carried out optimally, so that insurance premiums charged to customers are not too large and do not cause losses to the insurance company in the event of a claim. Therefore, insurance companies need to consider various aspects that occur in determining optimal insurance premiums. Among the aspects that must be considered are the chance of occurrence, losses incurred due to natural disasters, and socio-economic conditions in disaster-affected areas.

Every country in the world has implemented disaster mitigation efforts in facing the risk of loss. Efforts that have been made in dealing with the risk of loss due to disasters include buying insurance as a form of providing funds for disaster management in the country. Surminski et al. (2015) [26] conducted research on flood disasters affecting European Union member states. The research focused on the relationship between loss risk and flood disaster insurance. Based on the results obtained, the number of flood disasters occurring in European Union member states has increased. Therefore, efforts are needed in the form of policies and more roles from the European Union in overcoming the risk of flood losses, which will certainly increase. It is necessary to evaluate the flood disaster insurance policies that have been implemented in the European Union. The ineffectiveness of flood disaster insurance also results in ineffective loss risk management. The evaluation carried out involves European Union countries with insurance companies making new policies in the face of an increase in the occurrence of flood disasters. McAneney et al. (2019) [27] conducted research in Australia in the form of an evaluation of existing disaster insurance. In this research, it was stated that the risk of natural disasters has increased and the cost of response has also increased. The increase in the cost of dealing with natural disasters has been triggered by global warming. Global warming causes an increase in the potential for disasters. Therefore, natural disaster insurance that has been implemented needs to be adjusted to the latest risks and costs of countermeasures, so that the insurance offered can provide risk management funds in accordance with the costs incurred when a natural disaster occurs. Mol et al. (2018) [28] conducted research in the Netherlands in areas that frequently experience floods. The research was conducted by dividing the two types of home insurance due to floods into mandatory public and voluntary private. In addition, home insurance due to floods is given a premium discount that differs between the two types of insurance. Based on the results of the analysis, it was found that public investment in both types of insurance has increased. The increase in compulsory public and voluntary private insurance has been influenced by discounts on premiums. Premium discounts given to customers can help increase their interest in insuring their homes against the risk of floods. Paleari (2019) [29] conducted research on insurance schemes and mitigating the risk of natural disaster losses. The availability of funds to manage the risk of disaster loss is a major concern. Based on the results of the analysis, it was found that insurance can provide funds for dealing with the risk of losses from natural disasters. Natural disaster insurance provides more effective risk management. However, the availability of funds in risk management does not directly resolve the losses that occur. However, it can provide periodic economic recovery and can reduce the burden of greater risk.

The disaster insurance premium determination model has attracted interest from previous researchers and continues to develop. For example, Kim et al. (2015) [30] conducted research to determine the range of disaster insurance premiums based on the ability of the community using a logistics distribution model with contingent valuation methods (CVMs). Furthermore, they calculated the value of community behavior toward the desire for insurance using the travel cost method (TCM). Liang and Young (2018) [31] determined the risks that will be faced due to reinsurance against insurance premiums that have been issued and agreed upon, using the compound Poisson process (CPP) method. Furthermore, they provided optimal analysis results on reinsurance, carried out using the Black–Scholes model. Subartini et al. (2018) [32] investigated flood disaster insurance premiums in Bandung district. They determined a reasonable flood insurance premium using the fuzzy inference system (FIS) method. Peng et al. (2019) [33] conducted research by examining the level of public knowledge of the risk of natural disasters and the willingness to buy disaster insurance. Analysis using partial least-squares structural equation modelling (PLS-SEM) showed the relationship between the two variables.

There is a research gap in previous studies; in determining the premium, they have not considered jumping processes, so they cannot accommodate extreme events of natural disasters that often occur. In addition, in determining the premium, a cross-subsidy system has not been used, and the rate of economic growth in each region is considered the same. In Indonesia, which consists of 34 provinces, the economic growth rate is uneven and natural disasters occur in provinces with low economic growth. In this study, the cross-subsidy system means that regions that have high natural disaster risks and low economic growth will receive a reduction (i.e., receive subsidies) in insurance premiums. Furthermore, areas that have a low risk of natural disasters and high economic growth will be taxed (i.e., providing subsidies) for the amount of insurance premiums. This is intended to ensure that regions with a high risk of natural disasters are not overly burdened with the large premiums that must be paid due to their low economic growth. Based on the research gaps, in this study, the cost of natural disaster insurance premiums that must be paid by policyholders is calculated using the jumping processes method approach with a cross-subsidized system. The advantage of the jumping processes method is that it can capture random jumps of events. The frequency of natural disaster events has a random size between the two jumps and is based on the model developed by considering random event data in each of the data jumps.

The model developed in this study has advantages, first in determining the premium considering the random jumping of events, and this corresponds to natural disasters in Indonesia. Second, from an economic perspective, for regions with high natural disaster cases and low economic growth rates, it is necessary to obtain subsidies from other regions with high economic growth and low natural disaster risks. Determining natural disaster insurance premiums by considering the level of economic growth in each region has not been performed by previous researchers. The implications of the results of this study are expected to be a reference for insurance companies in determining premiums in accordance with the economic growth of each province in Indonesia, while for the government the study can be used as a reference in establishing natural disaster insurance policies with a cross-subsidized system in an effort to prepare funds for economic loss risk management. The model for determining natural disaster insurance premiums that have been developed considers social, economic, and justice aspects in determining premium rates. The model helps regions that have low economic growth and high potential for natural disasters, with affordable insurance premium rates. In addition, the model provides a scientific contribution to researchers in the field of actuarial mathematics. This model can be a reference for researchers to develop models for determining insurance premiums in solving problems related to natural disasters. The model can also provide researchers with an understanding of how to utilize available spatial and temporal data to estimate the level of risk of loss due to natural disasters.

2. Data and Theoretical Foundations

2.1. Methods and Data

When determining the cost of the premium that must be borne by customers, insurance companies must consider various factors, including the type of disaster and the amount of loss, economic growth in each region, and the level of disaster risk in each region. Determination of an efficient premium is needed so that the company does not go bankrupt when a claim occurs and the premium that must be paid by customers is in accordance with their economic growth. Therefore, appropriate methods and understanding are needed to obtain information and data for the process of determining efficient insurance premiums. Quantitative methods are used in this study to test theories related to the model for determining natural disaster insurance premiums. A model is developed for determining insurance premiums using the jumping processes method and a cross-subsidy system. The advantage of the jumping processes method is that it can capture random event jumps in natural disaster event data. In addition, the cross-subsidy system method means that regions with high natural disaster risk and low economic growth will receive a reduction (i.e., receive subsidies). Furthermore, areas that have a low risk of natural disasters and high economic growth will be taxed (i.e., providing subsidies). This study uses data on the frequency of occurrence of and loss due to natural disasters as well as an index of the potential for natural disasters that occur in every province in Indonesia, obtained from the National Disaster Management Agency. The data used cover the frequency of occurrence of and loss due to natural disasters for 20 years, from 2000 to 2019. This study does not include data from 2020 to 2022 because the available data are incomplete. This was influenced by the occurrence of the COVID-19 pandemic that occurred in early 2020 in Indonesia, so data on the frequency of events and losses were not recorded optimally. Therefore, in simulating the model that has been developed, available data were used, namely from the period 2000–2019. Data on the level of economic growth that occurred in each province in Indonesia were obtained from the Central Bureau of Statistics.

2.2. Statistical Analysis

The main variables in the development of a model for determining disaster insurance premiums are data on the frequency of events and losses. In supporting the developed model, it is necessary to analyze data on the frequency of occurrence of and loss due to natural disasters. This, as an effort to provide the model formulation that is being developed, can be more effective against the natural disaster data used. In research, what must be considered in data analysis is the selection of analytical methods. Interestingly, all parts of the data analysis require the use of statistics. Statistical analysis was carried out to describe the state of the data to be processed. In addition, statistical analysis can provide estimation results in the form of expectations and variances in natural disaster data. Statistical analysis of data on the frequency of occurrence of and economic losses due to natural disasters is needed as a reference in determining the price of insurance premiums. Statistical analysis for data on the frequency of natural disasters was performed using the compound Poisson method with jump processes. In previous studies, in modeling data on the frequency of disaster events, the standard Poisson process method has been used. However, the standard Poisson process itself is limited in developing a realistic price model because the jumps are of constant size. Therefore, there are several ways of considering jump processes that can have random sizes, namely, by using a compound Poisson process with jump processes. From the compound Poisson process ${\left(Y_t\right)}_{t\in {\mathbb{R}}_{+}}$ we can calculate the expectation $Y_t$ for a fixed $t$ as the product of the average number of jump times $\mathbb{E}\left[N_t\right]=\lambda t$ and the mean jump size $\mathbb{E}\left[Z\right]$, i.e., as shown in Equation (1) [34].

$$\mathbb{E}\left[Y_t\right]=\lambda t \mathbb{E}\left[Z\right]$$

(1)

Meanwhile, the variance of a compound Poisson process with jump processes is formulated as Equation (2).

$$Var \left[Y_t\right]=\lambda t \mathbb{E}\left[{\left|Z\right|}^2\right]$$

(2)

Furthermore, to model data on economic losses due to natural disasters, the Weibull distribution model can be used. The expectation and variance of the Weibull distribution model are formulated in Equations (3) and (4) [35,36].

$$E\left(X\right)=\lambda \mathsf{\Gamma }\left(1+\frac{1}{k}\right)$$

(3)

$$Var\left(X\right)={\lambda }^2\left[\mathsf{\Gamma }\left(1+\frac{2}{k}\right)-{\mathsf{\Gamma }}^2\left(1+\frac{1}{k}\right)\right]$$

(4)

The measurement of risk is divided into two parts, namely, individual risk and collective risk. In this study, in developing a model for determining natural disaster insurance premiums, a collective risk model was used. The collective risk model combines the frequency of occurrence (discrete) model and the economic loss (continuous) model from natural disaster events. Incidents and losses due to natural disasters are a factor and the main concern in developing a model for determining insurance premiums for natural disasters. This is because losses and natural disasters are an integral part of the premium calculation. The determination of the collective risk value is based on expectations and variances from natural disaster event data, which are formulated in Equations (5) and (6) [3,19,37,38,39].

$$E\left(S\right)=E\left(Y\right)E\left(X\right)$$

(5)

And

$$Var\left(S\right)=E\left(Y\right)Var\left(X\right)+{\left(E\left(X\right)\right)}^{2}Var\left(Y\right)$$

(6)

2.3. Disaster Insurance Model with Subsidies

The model for determining natural disaster insurance premiums developed by Kalfin et al. (2022) [19], which implements a cross-subsidy system, is formulated as follows:

$$P_{\left[SD\right]i}^{*}=\left\{\begin{array}{c}{\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(S\right)}+\varsigma ; 0\le {\pi }_i\le 0.5 \\ {\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(S\right)}-\varsigma ; 0.5<{\pi }_i\le 1\end{array}\right.$$

(7)

In Equation (7), the model for determining insurance premiums uses collective risk by considering the frequency of events and economic losses. If the expectation of collective risk can be expressed as $E\left(S\right)=E\left(Y\right)E\left(X\right)$, whereas the variance of collective risk can be expressed as $Var\left(S\right)=E\left(Y\right)Var\left(X\right)+{\left(E\left(X\right)\right)}^{2}Var\left(Y\right)$, then the determination of the insurance premium from Equation (7) can be formulated as follows:

$$P_{\left[SD\right]i}^{*}=\left\{\begin{array}{c}{\pi }_{i}E\left(Y\right)E\left(X\right)+\tau \sqrt{E\left(Y\right)Var\left(X\right)+{\left(E\left(X\right)\right)}^{2}Var\left(Y\right)}+\varsigma ; 0\le {\pi }_i\le 0.5 \\ {\pi }_{i}E\left(Y\right)E\left(X\right)+\tau \sqrt{E\left(Y\right)Var\left(X\right)+{\left(E\left(X\right)\right)}^{2}Var\left(Y\right)}-\varsigma ; 0.5<{\pi }_i\le 1\end{array}\right.$$

(8)

where ${\pi }_i$ denotes the index of potential natural disasters in an area and $\varsigma$ denotes the amount of subsidy.

3. Results

3.1. Insurance Premium Determination Model by Considering Economic Growth Rate

The model development carried out in this section takes into account the level of economic growth in each region. Based on Equation (7), development is carried out by adding the economic growth rate variable in each region $\left({\gamma }_i\right)$. This model is based on the model developed by Kalfin et al. (2022) [19], which did not consider the level of economic growth. Therefore, the model developed in this research divides three conditions in determining disaster insurance premiums. The model for determining insurance premiums by considering the level of economic growth in each region is formulated as Equation (9):

$$P_{\left[SD\right]i}^{*}=\left\{\begin{array}{c}{\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(S\right)}+{\gamma }_i\varsigma ; 0\le {\pi }_i\le 0.5; {\gamma }_i\ge 5\% \\ {\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(S\right)}; \begin{array}{c}0.5<{\pi }_i\le 1; {\gamma }_i\ge 5\%\\ \mathrm{o}\mathrm{r} 0\le {\pi }_i\le 0.5;{\gamma }_i<5\%\end{array} \\ {\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(S\right)}-{\gamma }_i\varsigma ; 0.5<{\pi }_i\le 1;{\gamma }_i<5\% \end{array}\right.$$

(9)

where, based on a decision from the Central Statistics Agency (BPS), for regions with an economic growth rate greater than or equal to 5% (${\gamma }_i\ge 5\%),$ areas with high economic growth are categorized. Meanwhile, regions with an economic growth rate of less than 5% $({\gamma }_i<5\%)$ are categorized as areas with low economic growth. Based on the model for determining natural disaster insurance premiums by considering the level of economic growth and cross-subsidies, it is formulated into three parts, as given in Equation (9). Regions that receive subsidies are areas with a high level of natural disaster potential and a low level of economic growth. If a region has a low potential for natural disasters and a high economic growth rate, it provides subsidies to other regions. Regions with high natural disaster potential and high economic growth, and regions with low natural disaster potential and low economic growth, are not subject to subsidies and do not provide subsidies. In addition, the development of the insurance premium model this time uses the Poisson process method with jumping. This method is used because natural disasters have a random pattern and it is not known exactly when they will occur. Therefore, the model for the determination of insurance premiums also takes into account the jump rate from the data on the frequency of natural disasters.

Theorem 1. 

If Equation (9) is assumed, in which the frequency of events is modeled using the Poisson process with jumping and economic losses following the Weibull distribution model, the calculation of disaster insurance premiums takes into account economic growth

$$P_{\left[SD\right]i}^{*}=\left\{\begin{array}{c}{\pi }_i\left[\left(\left({\lambda }_N{\lambda }_{Z}t\right)+{\gamma }_i\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right]+\tau \sigma ; 0\le {\pi }_i\le 0.5; {\gamma }_i\ge 5\% \\ {\pi }_i{\lambda }_N{\lambda }_{Z}t{\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)+\tau \sigma ; \begin{array}{c}0.5<{\pi }_i\le 1; {\gamma }_i\ge 5\%\\ \mathrm{o}\mathrm{r} 0\le {\pi }_i\le 0.5;{\gamma }_i<5\%\end{array} \\ {\pi }_i\left[\left(\left({\lambda }_N{\lambda }_{Z}t\right)-{\gamma }_i\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right]+\tau \sigma ; 0.5<{\pi }_i\le 1; {\gamma }_i<5\% \end{array}\right.$$

with $\sigma ={\left({\lambda }_N{\lambda }_x^{2}t\left[{\lambda }_z{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)+\left({\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\left(E\left[z^2\right]-{\lambda }_Z\right)\right)\right]\right)}^{\frac{1}{2}}$

Proof. 

For Theorem 1, in the proof, three conditions must be considered, namely (1) $0\le {\pi }_i\le 0.5; {\gamma }_i\ge 5\%$; (2) $0.5<{\pi }_i\le 1; {\gamma }_i<5\%$; and (3) $0.5<{\pi }_i\le 1; {\gamma }_i\ge 5\% \mathrm{or} 0\le {\pi }_i\le 0.5; {\gamma }_i<5\%$.

➢

For the first condition $0\le {\pi }_i\le 0.5; {\gamma }_i\ge 5\%$

$$\begin{array}{ll}{P}_{\left[E\right]i}^{*}& ={\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(s\right)}+{\gamma }_i\varsigma \\ & ={\pi }_{i}E\left(Y\right)E\left(X\right)+\tau \sigma +{\gamma }_i{\pi }_{i}E\left(X\right)\\ & ={\pi }_i\left({\lambda }_N{\lambda }_{Z}t\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)+{\gamma }_i{\pi }_i\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)+\tau \sigma \\ & ={\pi }_i\left(\left({\lambda }_N{\lambda }_{Z}t\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)+\left({\gamma }_i{\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right)+\tau \sigma \\ & ={\pi }_i\left[\left(\left({\lambda }_N{\lambda }_{Z}t\right)+{\gamma }_i\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right]+\tau \sigma \end{array}$$

➢

For the second condition $0.5<{\pi }_i\le 1; {\gamma }_i<5\%$

$$\begin{array}{ll}{P}_{\left[E\right]i}^{*}& ={\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(s\right)}-{\gamma }_i\varsigma \\ & ={\pi }_{i}E\left(Y\right)E\left(X\right)\tau \sqrt{Var \left(s\right)}-{\gamma }_i{\pi }_{i}E\left(X\right)\\ & ={\pi }_i\left({\lambda }_N{\lambda }_{Z}t\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)+\tau \sigma -{\gamma }_i{\pi }_i\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\\ & ={\pi }_i\left(\left({\lambda }_N{\lambda }_{Z}t\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)-\left({\gamma }_i{\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right)+\tau \sigma \\ & ={\pi }_i\left[\left(\left({\lambda }_N{\lambda }_{Z}t\right)-{\gamma }_i\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right]+\tau \sigma \end{array}$$

➢

For the third condition $0.5<{\pi }_i\le 1; {\gamma }_i\ge 5\% \mathrm{o}\mathrm{r} 0\le {\pi }_i\le 0.5; {\gamma }_i<5\%$

$$\begin{array}{ll}{P}_{\left[E\right]i}^{*}& ={\pi }_{i}E\left(S\right)+\tau \sqrt{Var \left(s\right)}\\ & ={\pi }_{i}E\left(Y\right)E\left(X\right)+\tau \sqrt{Var \left(s\right)}\\ & ={\pi }_i\left({\lambda }_N{\lambda }_{Z}t\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)+\tau \sigma \end{array}$$

Next will be shown the level of spread of risk $\sigma ={\left({\lambda }_N{\lambda }_x^{2}t\left[{\lambda }_z{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)+\left({\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\left(E\left[z^2\right]-{\lambda }_x\right)\right)\right]\right)}^{\frac{1}{2}}$

$$\begin{array}{ll}\sigma & =Var \left(s\right)=\sqrt{E\left(N\right)Var\left(X\right)+{\left(E\left(X\right)\right)}^{2}Var\left(N\right)}\\ & =\sqrt{\left({\lambda }_N{\lambda }_{Z}t\right) \left({\lambda }_X^2\left[{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)-{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right]\right)+{\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)}^2\left({\lambda }_{N}t \mathbb{E}\left[{\left|Z\right|}^2\right]\right)}\\ & =\sqrt{\left({\lambda }_N{\lambda }_{Z}t\right) \left({\lambda }_X^2\left[{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)-{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right]\right)+\left({\lambda }_X^2{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right)\left({\lambda }_{N}t \mathbb{E}\left[{\left|Z\right|}^2\right]\right)}\\ & =\sqrt{\left({\lambda }_N{\lambda }_X^{2}t\right) \left({\lambda }_Z\left[{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)-{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right]\right)+\left({\lambda }_N{\lambda }_X^{2}t \right)\left(\mathbb{E}\left[{\left|Z\right|}^2\right]{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right)}\\ & =\sqrt{\left({\lambda }_N{\lambda }_X^{2}t\right) \left({\lambda }_Z{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)-{\lambda }_Z{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right)+\left({\lambda }_N{\lambda }_X^{2}t \right)\left(\mathbb{E}\left[{\left|Z\right|}^2\right]{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right)}\\ & =\sqrt{\left({\lambda }_N{\lambda }_X^{2}t\right) \left({\lambda }_Z{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)-{\lambda }_Z{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)+\mathbb{E}\left[{\left|Z\right|}^2\right]{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\right)}\\ & =\sqrt{\left({\lambda }_N{\lambda }_X^{2}t\right) \left({\lambda }_Z{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)+\left[{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\left(\mathbb{E}\left[{\left|Z\right|}^2\right]-{\lambda }_Z\right)\right]\right)}\\ & ={\left(\left({\lambda }_N{\lambda }_X^{2}t\right) \left({\lambda }_Z{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)+\left[{\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\left(\mathbb{E}\left[{\left|Z\right|}^2\right]-{\lambda }_Z\right)\right]\right)\right)}^{\frac{1}{2}}\end{array}$$

So it is proven that the determination of disaster insurance premiums

$$P_{\left[SD\right]i}^{*}=\left\{\begin{array}{c}{\pi }_i\left[\left(\left({\lambda }_N{\lambda }_{Z}t\right)+{\gamma }_i\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right]+\tau \sigma ; 0\le {\pi }_i\le 0.5; {\gamma }_i\ge 5\% \\ {\pi }_i{\lambda }_N{\lambda }_{Z}t{\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)+\tau \sigma ; \begin{array}{c}0.5<{\pi }_i\le 1; {\gamma }_i\ge 5\%\\ \mathrm{a}\mathrm{t}\mathrm{a}\mathrm{u} 0\le {\pi }_i\le 0.5;{\gamma }_i<5\%\end{array} \\ {\pi }_i\left[\left(\left({\lambda }_N{\lambda }_{Z}t\right)-{\gamma }_i\right)\left({\lambda }_X{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{1}{k}\right)\right)\right]+\tau \sigma ; 0.5<{\pi }_i\le 1; {\gamma }_i<5\% \end{array}\right.$$

with the degree of spread of risk $\sigma ={\left({\lambda }_N{\lambda }_x^{2}t\left[{\lambda }_z{\mathsf{\Gamma }}_{\mathrm{X}}\left(1+\frac{2}{k}\right)+\left({\mathsf{\Gamma }}_{\mathrm{X}}^2\left(1+\frac{1}{k}\right)\left(E\left[z^2\right]-{\lambda }_x\right)\right)\right]\right)}^{\frac{1}{2}}$. □

Based on the model for determining insurance premiums in Theorem 1, the model parameters that influence the results of disaster insurance premiums are the average frequency of disaster events $\left({\lambda }_N\right)$, the jump in disaster events $\left({\lambda }_Z\right),$ the average disaster loss $({\lambda }_X )$, and the level of risk spread $\left(\sigma \right)$. These parameters are used as the main variables in estimating natural disaster risk values. These risk values are then used as a basis for calculating insurance premiums by considering regional zones, economic growth levels, loading factors, and the proportion of potential disasters. The greater the value of the loss and other factors, the greater the insurance premium that the customer must pay. Further, in classifying the premium determination, there are regions that provide subsidies, regions that receive subsidies, and regions that do not provide or receive subsidies. Calculations are performed based on the level of economic growth and the proportion of potential natural disasters in each region.

3.2. Determination of Insurance Premiums with Economic Growth Rates

In this section, the determination of natural disaster insurance premiums for each province in Indonesia uses Theorem 1. The measure for determining disaster insurance premium rates is borne by the provincial government. In determining insurance premiums, the area coverage insured is based on the territorial boundaries of each province in Indonesia. Therefore, the insurance premiums that have been estimated will be charged to the provincial government in Indonesia. So, in determining disaster insurance premiums, the data used in the process of determining insurance premiums include data on the frequency of natural disaster events and losses that occur in each province in Indonesia. The data are estimated to obtain the average and standard deviation, which are then used in the model that has been developed in this research. Further, the data used comprise an index of economic growth and the potential for natural disasters in each province in Indonesia. In determining natural disaster insurance premiums in Indonesia, the currency is IDR. This is because the data used in calculating natural disaster insurance premiums are presented in IDR. There is also a policy from the Indonesian government requiring the use of IDR, in an effort to avoid fluctuations in the IDR exchange rate against foreign currencies. The estimation results from the data used for 34 provinces in Indonesia are given in

Table 1. Estimated data on the frequency of incidents and losses as well as data on the rate of economic growth in 34 provinces in Indonesia.

Region	${\mathit{\gamma }}_{\mathit{i}}$	${\mathit{\pi }}_{\mathit{i}}$	${\mathit{\lambda }}_{\mathit{N}}$	${\mathit{\lambda }}_{\mathit{Z}}$	${\mathit{\lambda }}_{{\mathit{Z}}^2}$	${\mathit{\lambda }}_{\mathit{X}}$	Var (X)
Areas with low natural disaster potential and high economic growth rates
DKI Jakarta	5.46	0.2256048	21.7	0.6	47	30,106,909,810.95	$3.16537\times {10}^{20}$
Central Java	5.17	0.4686533	598.2	16.0	35222	4,530,651,960.62	$7.16825\times {10}^{18}$
Central Kalimantan	6.53	0.4676314	27.9	0.7	78	38,729,763,312.62	$5.23819\times {10}^{20}$
North Sulawesi	5.24	0.4914887	25.4	0.7	62	35,368,311,283.38	$4.36838\times {10}^{20}$
Gorontalo	7.23	0.4462761	20.4	0.5	39	28,353,108,132.67	$2.80733\times {10}^{20}$
Regions with low natural disaster potential and low economic growth rates or regions with high natural disaster potential and high economic growth rates
Jambi	4.46	0.4885638	46.6	1.2	215	64,744,471,582.52	$1.46385\times {10}^{21}$
South Sumatera	4.11	0.4906782	65.6	1.8	427	91,197,630,272.00	$2.90442\times {10}^{21}$
Riau Islang	−0.08	0.4101906	13.5	0.4	17	18,707,206,026.95	$1.22211\times {10}^{20}$
DI Yogyakarta	4.78	0.4965984	43.5	1.2	185	60.506,118,714.15	$1.27847\times {10}^{21}$
East Java	4.88	0.4735869	358.2	9.6	12633	16,222,654,836.86	$9.19043\times {10}^{19}$
Banten	5.06	0.5457579	51.9	1.4	267	72,198,125,152.80	$1.8203\times {10}^{21}$
Bali	3.90	0.456108	31.8	0.9	101	44,283,463,877.29	$6.84818\times {10}^{20}$
West Southeast Nusa	1.10	0.4512449	45.4	1.2	203	63,136,821,231.57	$1.39206\times {10}^{21}$
East Southeast Nusa	3.98	0.4964927	62.0	1.7	377	86,228,527,891.81	$2.59653\times {10}^{21}$
West Kalimantan	5.54	0.5427625	67.6	1.8	453	93,974,480.554.34	$3.08398\times {10}^{21}$
East Kalimantan	10.09	0.5413529	3.3	0.1	1	831,739,925,812.43	$2.41583\times {10}^{23}$
North Kalimantan	4.17	0.4880352	27.5	0.7	78	38,291,312,893.05	$5.12026\times {10}^{20}$
Central Sulawesi	7.70	0.5108353	25.8	0.7	66	35,806,761,702.95	$4.47736\times {10}^{20}$
South Sulawesi	6.33	0.5620386	96.2	2.6	906	133,727,296,033.42	$6.24499\times {10}^{21}$
Southeast Sulawesi	6.10	0.5558012	44.5	1.2	192	61,821,469,972.86	$1.33466\times {10}^{21}$
West Sulawesi	5.27	0.5867065	14.0	0.4	19	19,437,955,538.77	$1.31945\times {10}^{20}$
Maluku	5.72	0.5657388	14.6	0.4	22	20,314,856,377.91	$1.44118\times {10}^{20}$
North Maluku	5.99	0.5131964	11.7	0.4	14	498,079,366,701.85	$8.66341\times {10}^{22}$
Papua	−16.36	0.4330964	13.6	1.4	19	18,853,357,354.28	$1.24128\times {10}^{20}$
Areas with high natural disaster potential and low economic growth rates
Aceh	3.45	0.5412119	2.3	2.7	1031	142,496,297,300.00	$7.09086\times {10}^{21}$
North Sumatera	3.61	0.5116106	84.5	2.3	702	117,504,637,634.15	$4.82172\times {10}^{21}$
West Sumatera	3.14	0.5269398	82.1	2.2	664	114,143,189,167.33	$4.5498\times {10}^{21}$
Riau	2.51	0.5189757	33.1	0.9	107	46,037,265,555.57	$7.09086\times {10}^{21}$
Bengkulu	4.49	0.5708838	16.1	0.4	25	22,360,957,148.43	$1.74611\times {10}^{20}$
Lampung	4.18	0.5172489	43.6	1.2	190	60,652,270,041.48	$1.28465\times {10}^{21}$
Bangka Belitung Islands	3.95	0.5692628	10.0	3.0	10	13,884,254,974.09	$6.73189\times {10}^{19}$
West Java	4.30	0.5138307	391.7	10.5	15096	5,407,552,799.76	$1.02116\times {10}^{19}$
South Kalimantan	3.26	0.5108	63.9	1.7	402	88,859,230,409.23	$2.75738\times {10}^{21}$
West Papua	−0.13	0.5107965	3.9	0.1	2	544,701,230,441.91	$1.03612\times {10}^{23}$

.

Based on the results in Table 1, the level of economic growth (${\gamma }_i$) in each province in Indonesia has different levels. The determination of the level of economic growth refers to the Central Statistics Agency (BPS) in Indonesia. Based on the decision of the Central Statistics Agency (BPS), regions with an economic growth rate greater than 5.0 (${\gamma }_i\ge 5.0$) are categorized as regions with a high economic growth rate. Meanwhile, regions with an economic growth rate of less than or equal to 5.0 (${\gamma }_i<5.0$) are categorized as regions with a low economic growth rate. The proportion of potential disaster levels (${\pi }_i$) refers to the provisions made by the National Disaster Management Agency in Indonesia. Regional mapping based on an index of potential disaster risk levels that occur in each province in Indonesia is divided into two categories, namely, high and low. Areas with a disaster risk index of less than 0.5 (${\pi }_i\le 0.5$) are categorized as areas with a low level of natural disaster potential. Areas with a disaster risk index greater than or equal to 0.5 (${\pi }_i>0.5$) are categorized as areas with a low level of natural disaster potential.

Based on the level of economic growth and potential risk index in each province based on Table 1, a division is made into three conditions in determining the premium, namely, regions that provide subsidies, regions that receive subsidies, and regions that do not provide or receive subsidies. Regions that provide subsidies are provinces that have a low proportion of potential disaster levels (${\pi }_i\le 0.5$) and a high economic growth index (${\gamma }_i\ge 5.0$). Regions that receive subsidies are provinces with a high proportion of potential levels of natural disasters (${\pi }_i>0.5$) and have a low economic growth index (${\gamma }_i<5.0$). Regions that do not provide or receive subsidies are provinces with a high level of potential natural disasters (${\pi }_i>0.5$) and a high economic growth index (${\gamma }_i\ge 5.0$), or provinces with a low level of potential natural disasters (${\pi }_i\le 0.5$) and a low index of economic growth (${\gamma }_i<5.0$). The areas that provide subsidies are the provinces of DKI Jakarta to Gorontalo, which have been allocated the color green. The areas receiving subsidies are the provinces of Aceh to West Papua. The regions that do not receive subsidies and/or provide subsidies are the provinces of Jambi to Papua, which have been allocated the blue color according to Table 1. These three conditions will later serve as a benchmark in determining insurance premiums in the 34 provinces in Indonesia.

Furthermore, determining insurance premiums for natural disasters in each province in Indonesia refers to or is based on the results in Table 1. In determining insurance premiums for each province in Indonesia, the model given in Theorem 1 is used. The stages of determining the insurance premiums are based on the model that has been developed; for factor loading (unforeseen risk), the provisions of the insurance company are used. However, in determining the natural disaster insurance premium used in this study, an unexpected risk of 1% is assumed. The specified insurance premium takes into account the level of economic growth and the potential for natural disasters in each province in Indonesia. The results of the analysis of determining insurance premiums based on Theorem 1, and the obtained insurance premiums for each province in Indonesia are given in

Table 2. Disaster insurance premiums taking into account the level of economic growth in each province in Indonesia.

Province	Insurance Premium (IDR)
Determination of Insurance Premiums by Providing Subsidies
DKI Jakarta	253,834,580.539.75
Central Java	40,732,939,021,207.10
Central Kalimantan	948,484,721,449.65
North Sulawesi	805,836,911,740.21
Gorontalo	443,471,294,685.97
Determination of Insurance Premiums by Not Giving or Receiving Subsidies
Jambi	3,555,625,199,059.89
South Sumatera	10,616,274,042,384.50
Riau Islang	83,557,307,853.12
DI Yogyakarta	3,155,970,492,206.98
East Java	52,952,239,940,221.70
Banten	5,756,421,737,494.66
Bali	1,166,283,381,976.19
West Southeast Nusa	3,124,399,352,227.51
East Southeast Nusa	9,070,290,981,711.77
West Kalimantan	726,108,605,331.37
East Kalimantan	12,463,473,622,169.00
North Kalimantan	298,482,314,834.78
Central Sulawesi	664,287,106,710.12
South Sulawesi	37,717,317,612,933.40
Southeast Sulawesi	3,685,166,451,160.72
West Sulawesi	128,603,581,848.89
Maluku	135,525,788,195.30
North Maluku	2,409,094,991,428.06
Papua	311,885,835,041.86
Determination of Insurance Premiums by Receiving Subsidies
Aceh	42,296,061,111,736.00
North Sumatera	23,029,422,214,287.90
West Sumatera	21,433,896,240,152.80
Riau	1,312,950,269,006.88
Bengkulu	51,002,037,372.75
Lampung	3,040,282,937,844.76
Bangka Belitung Islands	412,425,987,629.31
West Java	22,876,927,586,639.30
South Kalimantan	9,612,571,407,490.90
West Papua	293,527,523,446.09

.

Taking into account the results in Table 2, it can be seen that insurance premiums in each province in Indonesia vary. This is because the disaster potential index and economic growth rate are considered when determining insurance premiums. With the level of disaster potential in each province varying from one province to another, as well as the varying rates of economic growth in each province, the results in calculating insurance premiums in each province in Indonesia will vary. In addition, the losses incurred due to natural disasters in each province vary in influencing the determination of insurance premiums for natural disasters. Losses experienced by each province can be influenced by the level of ability in disaster mitigation efforts before the disaster occurs. The existence of mitigation before a disaster occurs can minimize the impact of the risks posed. In the model that has been developed using Theorem 1, important aspects in determining insurance premiums are the disaster potential index, the level of economic growth, the frequency of occurrence, and losses due to natural disasters. In addition, the number of subsidies given and subsidies received is influenced by the disaster potential index, the level of economic growth, and economic losses due to natural disasters in each province. Furthermore, based on the results in Table 2, the graph for insurance premiums for each province in Indonesia based on the pure premium model can be seen in Figure 1.

In Figure 1 and Table 2, it can be seen that the size of the insurance premium for each province in Indonesia varies greatly. Based on the bar chart, the value of insurance premiums that must be paid by provincial governments in Indonesia varies. This is influenced by the large amount of data on the frequency of disaster events and losses used in determining insurance premiums based on the conditions and circumstances in each province in Indonesia. There are several provinces that have very large insurance premiums and there are several provinces with very small insurance premiums when compared to other provinces. Based on the results of the analysis, it can be seen that the province of East Java pays the largest insurance premium when compared to other provinces with premiums to be paid, i.e., IDR 52,952,239,940,221.70. Bengkulu province is the area with the smallest premium when compared to other provinces, i.e., IDR 51,002,037,372.75.

4. Discussion

A model for determining disaster insurance premiums using a cross-subsidy system and taking into account the level of economic growth was developed to overcome the problem of very large insurance premium burdens for regions that have little economic growth. A cross-subsidy system can be used to provide affordable insurance premiums for areas that have low economic growth. The development of this model is based on research by Songwathana (2018) [17], which states that areas with a high potential for natural disasters will have low economic growth. Therefore, these regions need to receive subsidies from other regions that have high economic growth in order to pay disaster insurance premiums. The model development was also based on the model developed by Kalfin et al. (2022) [19], which implemented a cross-subsidy system. However, the previously developed model did not take into account the level of economic growth of each region. Looking at the geographical conditions of Indonesia, which consists of both large and small islands, the index of potential natural disasters and economic growth that occurs in each region certainly varies. This model offers innovation in determining insurance premiums based on geographical conditions in Indonesia.

Aidi and Farida (2020) [40] stated that disaster management that is not handled quickly due to insufficient funds will hamper economic growth in areas affected by disasters. Using the model for determining natural disaster insurance premiums, we can provide disaster management funds that are appropriate to the conditions in the field. This model also takes into account the level of economic growth of each region. In addition, the insurance premium determination model developed using Theorem 1 employs a system of mutual cooperation through cross-subsidies. If a region has a low natural disaster potential index and a high level of economic growth, it will provide subsidies (subject to tax). Meanwhile, if the area has a high natural disaster potential index and a low level of economic growth, it will receive a subsidy. Therefore, the premium determination model considers the disaster potential index in each province. This is in line with research by Kalfin et al. (2022) [19], who state that the potential index in each province in Indonesia varies. Further, Klomp and Valckx (2014) [41] stated that the level of economic growth in areas that are frequently affected by natural disasters will experience a decline. Therefore, the model developed also takes into account the level of economic growth in each province, so that the premium charged does not burden the region.

Based on the results of the analysis in Table 1, the potential index for natural disasters and economic growth in each province in Indonesia varies. This will provide variations in the insurance premiums that must be paid in each province. Based on the results of the analysis in Table 2 and Figure 1, it can be seen that the premium charged by each province in Indonesia varies. The cost of insurance premiums is influenced by the frequency of natural disaster events and losses in each province. The greater the frequency of events and economic losses in a province, the greater the premium burden that must be paid. The research of Ash-shidiqqi et al. (2023) [42] states that the APBN and APBD are sources of funds for managing natural disasters in Indonesia. However, these are still insufficient in post-disaster mitigation efforts because the budgeted risk analysis does not match the conditions in the field. Therefore, the State Revenue and Expenditure Budget (APBN) and Regional Revenue and Expenditure Budget (APBD) can be allocated for the payment of natural disaster insurance premiums. With the innovation that has been developed in this model for determining insurance premiums, the model has been adjusted to the economic and geographical conditions in each province in Indonesia.

5. Conclusions

Research into natural disaster insurance is necessary, especially against the background of climate change. Various innovations are offered in risk management through natural disaster insurance. In this study, a model for determining natural disaster insurance premiums was developed to adapt to geographical conditions in Indonesia. Through the cooperation principle, a model for determining natural disaster insurance premiums was developed by implementing a cross-subsidy system. Through this cross-subsidy system, there will be mutual assistance in paying insurance premiums. Regions with high levels of economic growth and low potential for natural disasters need to help regions that have low economic growth and high potential for natural disasters. This assistance will be in the form of subsidies, which will later reduce the burden of premiums that must be paid by regions with low economic growth and a high potential for natural disasters. This model is suitable for Indonesia’s geographical conditions; in mapping the disaster potential index and regional economic growth rates, the region was divided into 34 provinces. Based on the results of data analysis using the model, the insurance premiums that must be paid for the 34 provinces in Indonesia vary. In addition, the results of the analysis show that Bengkulu province pays the lowest insurance premiums when compared to other provinces. Meanwhile, the province of East Java has to pay the largest disaster insurance premium when compared to other provinces.

The insurance premium determination model developed in this study can be used on an international scale. The model can adjust insurance premium rates to the conditions and risks that exist in several countries. It also takes into account the impact of external factors, such as government policy, potential natural disasters, and economic growth, on the premium value. The model has high accuracy, flexibility, and competitiveness in determining insurance premiums according to the geographical conditions of the country. The model adjusts insurance premium rates to existing conditions and risks. It can be used to determine global insurance premiums if the country is mapping its natural disaster potential based on each region. What is of concern is the frequency of natural disasters and losses. In addition, the proportion of potential natural disasters and the economic growth index of each region need to be considered. The model can be a reference for future researchers in developing models for determining insurance premiums that are appropriate to the geographical conditions and policies of each country. There are some similarities and differences between natural disaster insurance policies in each country. Therefore, innovation in natural disaster insurance needs to continue to develop and consider different variables. In this way, the resulting insurance premium will be more appropriate.