Agricultural Insurance Premium Determination Model for Risk Mitigation Based on Rainfall Index: Systematic Literature Review

Abstract

Rainfall is significantly essential in the agricultural sector to increase productivity. However, rainfall instability serves as a potential source of risk, causing crop failure and negatively impacting the welfare of farmers. To mitigate this risk, rainfall index-based agricultural insurance offers financial protection to farmers. There is no information on how to set a reasonable premium in index-based agricultural insurance. Therefore, this research aimed to systematically explore a model for determining a rainfall index-based agricultural insurance premium, focusing on the methods used and their effectiveness in mitigating the risk of harvest failure in the agricultural sector. The Preferred Reporting Items for Systematic Review and Meta-Analysis (PRISMA) method and a bibliometric analysis were used to collect and analyze articles from Scopus, ScienceDirect, and Dimensions databases. The results showed that there were 15 articles on determining a rainfall index-based agricultural insurance premium, where 4 used the Black–Scholes method and 11 applied other main methods. Meanwhile, no articles applied the fractional Black–Scholes method in determining agricultural insurance premiums based on the rainfall index, providing new opportunities for further research. The results contributed to the development of a model for agricultural insurance premium determination that could generate more diverse and flexible premium estimates as a sustainable method to mitigate the risk of harvest failure. This research is expected to serve as a reference for developing rainfall index-based agricultural insurance in the future and contribute to the Government of the Agriculture Department’s policy formulation regarding insurance programs for farmers.

1. Introduction

The agricultural sector is a sensitive element in the economic life of society, producing raw materials for other sectors (Edame et al. 2011). Stable growth in the agricultural sector can improve community access to food, create food security, and reduce hunger (Berry et al. 2015). Growth in the sector is capable of contributing positively to regional development and improving the living standards of farmers as well as rural communities (Pingali 2007). However, farmers often face various risks that can lead to losses due to crop failures influenced by several factors, including rainfall.

Rainfall is a crucial climatic factor that significantly impacts the agricultural sector. Although rainfall is required to increase agricultural production sustainably, conditions either excessive or insufficient can damage infrastructure, cause crop failures, and reduce productivity (Lesk et al. 2016). Changes in rainfall also have the potential to trigger natural disasters such as floods and droughts, which increases the risk of losses in the agricultural sector (Atanga and Tankpa 2021). Therefore, there is a need to design effective adaptation and mitigation strategies in loss risk management in the agricultural sector, particularly those caused by rainfall instability.

According to Dick and Wang (2010), farmers can traditionally be protected against rainfall risks by the government allocating a special budget for natural disasters in the agricultural sector. This method has difficulties, including the unavailability of insurance in every country due to the economically unfeasible cost of insurance. High premium costs often make it difficult for insurance companies to offer adequate protection. Therefore, innovative new mechanisms must be developed to replace traditional farmer protection.

An alternative mechanism for protection to reduce the risk of loss due to rainfall is through agricultural insurance programs. According to Nnadi et al. (2013), agricultural insurance can be used to overcome financial risk, playing an essential role in addressing erratic rainfall changes. In Indonesia, regulations related to insurance programs in the agricultural sector are contained in Law Number 19 of 2013 concerning Protection and Empowerment of Farmers Article 30 paragraph (1). This regulation states that “The Government and Regional Governments in accordance with their authority are obliged to protect farming businesses carried out by farmers in the form of agricultural insurance” (Muin and Mubarak 2024). Agricultural insurance has also been implemented in various countries, including the United States, Japan, Thailand, Brazil, and others.

Agricultural insurance includes various types, including indemnity, index-based, and others such as livestock, fisheries, and greenhouses (Hohl 2019). In the context of index-based insurance, there are two main categories, namely yield index-based in a region and weather index-based agricultural insurance (Kajwang 2022). Weather index parameters that are used include rainfall, temperature, and humidity. To protect farmers from the risk of instability, rainfall index-based agricultural insurance can be used. This insurance allows farmers to file claims in unexpected rainfall conditions (Nagaraju et al. 2021).

In rainfall index-based agricultural insurance, farmers as insured parties must pay a premium to insurance companies acting as insurers for their participation in the program (Coble et al. 2003). This premium is often paid every planting season as a financial contribution to manage risks associated with changes in rainfall. The determination process in rainfall index-based agricultural insurance includes various crucial methods such as Black–Scholes, the loss expectation model, a pure premium, an actuarial analysis, and others. These methods are selected based on the suitability of agricultural risks’ characteristics and the accuracy of resulting premium calculation (Afshar et al. 2021).

The Black–Scholes method is the most popular for calculating rainfall index-based agricultural insurance premiums. This method was developed by Fisher Black and Myron Scholes in 1973 to determine the value of options in stock contracts. Subsequently, it was adapted to determine agricultural insurance premiums due to similar characteristics in calculating stock option prices and agricultural insurance premiums (Prabowo et al. 2020). The use of the Black–Scholes method in agricultural insurance can handle the uncertainty from rainfall variability to provide optimal protection for farmers. A previous study used the Black–Scholes method to determine rainfall index-based agricultural insurance premiums. Paramita et al. (2023) explored the calculation of pure premiums on weather-based insurance for farmers in Wonogiri, Central Java. The results showed that when rainfall was below 100 mm/month, farmers could submit claims and receive the compensation of IDR 13,600,000. Azka et al. (2021) investigated designing rainfall index-based insurance for rubber plantations in Balikpapan. The results showed that agricultural insurance premiums obtained at the lowest and highest rainfall class were IDR 521,482.73 and IDR 568,592.93, respectively.

The Black–Scholes method has been developed into fractional Black–Scholes. The development was initiated by several empirical data values showing that the distribution of stock prices often had some properties. These included long-term dependence, which did not exist in the Black–Scholes method (Meng and Wang 2010). Finally, Mandelbrot and Taylor (1967) determined the fractional character of the stock market, while Peters (1989) provided a fractional market until Duncan et al. (2000) and Hu and Oksendal (2000) improved the fractional Black–Scholes method by stochastic integration with fractional Brownian motion. The fractional Black–Scholes method was derived with fractional time using order based on Brownian motion. The selection of fractional-order parameters was essential to handle data with long-term dependence, which was relevant in a financial context. Additionally, fractional-order parameters could make stock price options more varied to determine a more flexible profit value. Several investigations have been carried out on the fractional Black–Scholes method by Meng and Wang (2010) and He and Lin (2021) to determine stock option prices.

Generally, previous research had extensively used the Black–Scholes method to determine premiums for rainfall index-based agricultural insurance. Therefore, this research aimed to explore the application of the fractional Black–Scholes method to determine premiums on rainfall index-based agricultural insurance. This method allows for innovations, as premium calculation can be adjusted to the uncertainty of rainfall that often affects crop yields to help farmers manage the risk of crop failure more effectively. The Black–Scholes method is indicated to produce more diverse and flexible premium variations, thereby facilitating farmers’ participation in agricultural insurance. It also allows for premium setting that is more adaptive to unstable rainfall patterns, given the long-term dependence on rainfall data and associated risks. The fractional Black–Scholes method is needed for development in setting rainfall index-based agricultural insurance premiums.

Previous Systematic Literature Review (SLR) research on agricultural insurance was conducted by Cogato et al. (2019), which reviewed the relationship and impact between extreme weather events such as heat waves, floods, and droughts on agriculture worldwide. The results showed that food security and economic losses from agriculture were significant concerns and needed development to overcome the impact of extreme weather on the agricultural sector. Therefore, Vyas et al. (2021) categorized the agricultural insurance literature based on insured products, research themes, geographical areas, types of insurance, and hazards, covering as well as mapping the intensity of research by country. The results showed that agricultural insurance was more prevalent in high-income countries and focused on food crops. There was a high correlation between agricultural insurance research and extreme weather events. Riaman et al. (2022) developed mathematical models for paddy agricultural insurance classified by a bibliometric analysis. The results showed that risk modeling was used for rice farming insurance, with most of the research focusing on weather index methods and extreme weather factors.

A gap in previous research exists regarding SLR on rainfall index-based agricultural insurance, particularly methods, risks, and compensation schemes used to reduce risks. Therefore, this research aimed to conduct SLR as an initial step to obtain relevant materials for understanding a model to determine rainfall index-based agricultural insurance premiums. This SLR intends to identify previous research, evaluate developments in the field, and discover potential novelty. The review also explores a potential model for determining rainfall index-based agricultural insurance premiums using the fractional Black–Scholes method. Specifically, this research will answer five Research Questions (RQs), which are as follows:

(1)

What are the main methods used to determine agricultural insurance premiums based on the rainfall index?

(2)

How is the rainfall index used in the calculation of agricultural insurance premiums?

(3)

What commodities are included in rainfall index-based agricultural insurance?

(4)

What meteorological risks are considered in the measurement of the rainfall index?

(5)

How is the compensation payment scheme for rainfall index-based agricultural insurance?

This research adopted Preferred Reporting Items for Systematic Review and Meta-Analysis (PRISMA) methodology, which is complemented by a detailed flowchart to ensure arrangement between the selected articles and the topic. Each stage was carried out independently to avoid conflicts of interest and minimize potential biases that could affect the results. The stages in PRISMA are identification, screening, eligibility assessment, and the selection of articles that meet the predefined included criteria. The article collection process followed specific criteria obtained through search engines in various databases, including Scopus, ScienceDirect, and Dimensions. Subsequently, the articles were selected based on an analysis of abstracts and titles, which were relevant to the research topic. This was followed by a full-text review of the articles that passed the abstract and title screening to ensure relevance and quality. Finally, the review stage was conducted based on the predetermined Research Questions. An in-depth analysis was also conducted regarding future research opportunities while confirming that the determination of agricultural insurance premiums based on the rainfall index using the fractional Black–Scholes model had not been previously discussed. The results provided various recommendations for the development of future research, which could serve as a reference for other investigations. A deeper understanding could be gained, serving as a reference for investigations on the determination of agricultural insurance premiums. Consequently, this research aimed to contribute to the development of effective and efficient agricultural insurance policies promoting the welfare of farmers.

2. Materials and Methods

The data used were in the form of the relevant literature regarding the determination of agricultural insurance premiums. This research was carried out to mitigate the risk of harvest failure due to the rainfall index using the fractional Black–Scholes method, which was searched for using Scopus, ScienceDirect, and Dimensions databases. The selection of these three databases was based on international reputation and ease of accessibility, including the quality of articles relevant to the research topic. The collection of the literature was based on specific criteria (Alfandari and Taylor 2022), namely

(1)

The article was written in English;

(2)

The article has reached the final publishing stage;

(3)

The publication period is twenty years, from 2005 to 2024.

The PRISMA method was used to determine the material relevant to the research topic, serving as a model for determining rainfall index-based agricultural insurance premiums using fractional Black–Scholes. The selection process was due to the application of a detailed flowchart to improve the suitability of articles obtained with the research topic (Page et al. 2021). The four stages included in the PRISMA method are identification, screening, eligibility, and inclusion.

The identification stage was carried out by collecting articles from international journal databases, namely Scopus, ScienceDirect, and Dimensions databases. This collection was performed using search engines and the selected articles met criteria (1)–(3) from the databases. Before fulfilling the criteria, an initial search was conducted using a rigorous keyword search strategy, incorporating Boolean logic operators and applying synonyms to ensure completeness.

Table 1. Keywords relevant to the research topic.

Keywords	Search Terms
A	(“Insurance”)
B	(“Crop” OR “Agriculture” OR “Agricultural”)
C	(“Premiums”)
D	(“Weather Index” OR “Rainfall Index”)
E	(“Black Scholes” OR “Black-Scholes” OR “fractional Black-Scholes” OR “fractional Black Scholes”)

shows the keywords used in the search for articles contained in the title and abstract, which are relevant to the research topic.

The screening stage was conducted based on duplicate articles, abstracts, and titles. This stage aimed to ensure that the selected articles were accurate, were relevant, and met the predetermined selection criteria, thereby allowing a further analysis. The process at the screening stage followed the procedure stated by Firdaniza et al. (2022), namely

• Duplicate articles were removed from the three databases to ensure accuracy and diversity.
• The selection of articles focused on the title and abstract, which represented the most descriptive part of the entire content. Additionally, this stage also saved time in the selection process. Articles with titles and abstracts that did not relate to the criteria and the research topic were excluded at this stage.

The eligibility stage included where articles were selected by thoroughly reading all parts. After abstract and title screening, articles were selected by reading the entire content. This stage aimed to ensure that the selected articles were relevant to the specified criteria and the research topic (Tricco et al. 2018). The articles selected at this stage entered the included stage by meeting the criteria and being relevant to the research topic.

In the included stage, an analysis was performed to find new results, identify common patterns and gaps, and evaluate the implications of the research topic on the rainfall index as a basis for modeling agricultural insurance premiums. This stage also included a bibliometric analysis, a series of descriptions, and investigations of articles that were the research material for writing the new literature. The bibliometric analysis was conducted using VosViewer 1.6.19 software to obtain information quickly, clearly, and systematically. The analysis aimed to map the words in the title, abstract, and keywords, as well as identify the most relevant to the research topic and appropriate journal publisher. This stage allowed the determination of the right collaborators and selection of the best journal publisher. Therefore, the bibliometric analysis was essential for strategic guidance in strengthening research networks and increasing the visibility of results.

3. Results

3.1. Result of Article Selection Using PRISMA Method

The collection of articles from Scopus, ScienceDirect, and Dimensions databases was structured by searching for relevant keywords presented in Table 1. This process was carried out to find articles on the main research topic, namely a model for determining agricultural insurance premiums to mitigate the risk of harvest failure due to the rainfall index using the fractional Black–Scholes method. A total of 17 articles met criteria 1 to 3 presented in Section 2. These consisted of 2, 6, and 9 articles from Scopus, ScienceDirect, and Dimensions. The articles obtained at the identification stage were collected and carried through in the screening, eligibility, and inclusion stages. In these three stages, irrelevant articles were excluded, while those that were relevant were included for a further analysis to obtain literature review material.

In this research, two manual selection processes were added. These additional processes were carried out by generalizing the keywords in Table 1. The stage aimed to ensure that this research’s literature search was thorough, from more general topics to more specific. This method provided a comprehensive understanding of the model for determining insurance premiums before focusing on the research. Specifically, the model was developed to calculate agricultural insurance premiums for mitigating the risk of harvest failure due to the rainfall index using the fractional Black–Scholes method. The first addition process was carried out by excluding the keywords (“Crop” OR “Agriculture” OR “Agricultural”) and (“Weather Index” OR “Rainfall Index”) to review a model for the determination of insurance premiums using the Black–Scholes method. The second addition process was carried out by excluding the keywords (“Black-Scholes” OR “Black-Scholes” OR “fractional Black-Scholes” OR “fractional Black Scholes”) to broadly review a model index-based agricultural insurance premium determination with various main methods. After manual full-text selection, 10 articles were obtained for determining insurance premiums in general using Black–Scholes. Meanwhile, 11 articles were relevant to determining rainfall index-based agricultural insurance premiums using various main methods. The results of all stages with the PRISMA method are shown in Figure 1.

Based on Figure 1, the results of the identification stage with the three databases, namely Scopus, ScienceDirect, and Dimensions, obtained a total of 17 articles. Each article was screened based on duplicate articles, abstracts, and titles. At the screening stage based on duplicate articles, six were obtained from the three databases used. Regarding abstracts and titles, nine articles were relevant to the research topic, and two were irrelevant. Therefore, nine relevant articles were obtained for the research focus at the screening stage.

At the eligibility stage, a more specific selection was made by reading the entire text, leading to four relevant articles and five that did not fit the research topic being excluded. For example, research conducted by Bertranda et al. (2015) discussed the impact of unseasonal weather deviations and offered derivatives to reduce the risk of loss. Ballotta et al. (2020) analyzed the effects of weather variables with a non-parametric model. Cramer et al. (2019) developed a stochastic model to price rain derivative contracts. Prabakaran et al. (2020) and Tang and Chang (2016) used stochastic models to price weather options. In addition to 4 articles being relevant to the research topic, another 10 were included, determining premium insurance using Black–Scholes. There were an additional 11 articles on determining premiums in agricultural insurance based on the rainfall index using various main methods, which passed through an overall manual selection process.

The selection results at the eligibility stage produced 25 relevant articles. These included Chicaíza and Cabedo (2009), Chang et al. (2012), Melnikov and Tong (2013), Valverde (2015), Putri et al. (2018), Pandiangan and Sukono (2020), Saputra et al. (2021), Fang et al. (2023), Rahadi et al. (2023), Purwandari et al. (2024), Heimfarth and Musshoff (2011), Bobojonov et al. (2014), Ruiz et al. (2015), Silvestre and Lansigan (2015), Poudel et al. (2016), Kath et al. (2018), Bokusheva (2018), Wen et al. (2019), Baškot and Stanić (2020), Gómez-Limón (2020), Koprivica et al. (2024), Ariyanti et al. (2020), Prabowo et al. (2023b), Marola et al. (2023), and Raharjanti et al. (2024). Therefore, 25 relevant articles could be used as literature review material and continued into the included stage for a further analysis. The data used in this research were presented in “.bib”, containing a database of 25 articles of review material that could be accessed at the following link: https://bit.ly/MyCollectionArticle (accessed on 14 October 2024).

3.2. Bibliometric Analysis of Reviewed Articles

A total of 25 articles that passed the selection stage were selected as research materials. These articles focused on research related to a premium determination model in insurance using Black–Scholes and other methods. The selected articles were from various years of publication, providing a broad perspective and covering changes in research trends over time, as shown in Figure 2.

The consideration of the journal in which the article was published has significant benefits for authors. Therefore, the presentation of the journal names of the 25 articles was necessary. Research on premium determination in various types of insurance has been published in various academic journals, providing an idea of the distribution of contributions on this topic. Journals that published the articles reviewed in this literature included Risks; Agricultural Finance Review; Innovar: Revista de Ciencias Administrativas y Sociales; Journal of Real Estate Finance and Economics; Operations Research: International Conference Series; Risk and Decision Analysis; Climate and Development; Spanish Journal of Agricultural Research; European Journal of Economics; Finance and Administrative Sciences; Asia Pacific Journal of Multidisciplinary Research; Economics of Agriculture; International Journal of Advanced and Applied Sciences; Journal of Agricultural Science and Technology; Weather and Climate Extremes; MATEMATIKA; Journal of Applied Statistics; JTAM | Jurnal Teori dan Aplikasi Matematika; Economic Annals; Agricultural Water Management; Decision Science Letters; BAREKENG: Jurnal Ilmu Matematika dan Terapan; Universal Journal of Agricultural Research; International Journal of Environmental Research and Public Health; and International Journal of Business, Economics, and Social Development. The spread of articles addressing premium determination in insurance across these journals provided a broad overview of academic contributions. It also showed the various perspectives in research related to premium determination in insurance in several academic journals.

Figure 2 shows the literature review of articles published between 2009 and 2024. From 2009 to 2014, the number of articles remained constant, with only one per year. In 2015, there was an increase to three articles, but it decreased to two in 2016. There was an increase in 2018 with three articles, followed by a decrease in 2019, which further rose to four in 2020. The number of articles decreased to one but increased in 2023 with four articles, and in 2024, there were two articles. The increase in the number of articles in 2020 showed the high research interest in premium determination in insurance. This increasing trend could be attributed to risk awareness and the need to develop more effective premium strategies. Although there was a significant decrease in 2021, the number of articles increased in 2023, showing a growing interest in this research topic. Based on this trend, the development of a model on premium determination in insurance methods remained relevant and engaging for further investigation, along with the dynamics that continue to develop in the insurance industry. The detailed information on the authors and research objectives of the 25 articles is presented in

Table 2. Information of reviewed articles based on the author, and research objectives.

Author(s)	Research Objectives
Chicaíza and Cabedo (2009)	Calculating high-cost disease insurance premiums in the Colombian health system using Black–Scholes.
Chang et al. (2012)	Modeling house price changes in the context of mortgage insurance pricing using Black–Scholes.
Melnikov and Tong (2013)	Evaluating equity-linked life insurance contracts with efficient hedging techniques based on Black–Scholes.
Valverde (2015)	Estimating insurance premiums that can be used to reduce the risk of supplier bankruptcy using Black–Scholes.
Putri et al. (2018)	Evaluating the economic value of deposit insurance using Black–Scholes.
Pandiangan and Sukono (2020)	Determining the deposit insurance premium in Indonesia using Black–Scholes.
Saputra et al. (2021)	Developing a premium model in Takaful insurance with Sharia principles using Black–Scholes.
Fang et al. (2023)	Analyzing the price of environmental insurance using Black–Scholes.
Rahadi et al. (2023)	Calculating the price of microinsurance premiums on cattle farms using Black–Scholes.
Purwandari et al. (2024)	Developing a natural disaster insurance premium model using Black–Scholes.
Heimfarth and Musshoff (2011)	Analyzing weather index-based insurance on maize and wheat farming in the North China Plain.
Bobojonov et al. (2014)	Applying index-based insurance for climate risk management using the pure premium model and rural development in Syria.
Ruiz et al. (2015)	Evaluating the feasibility of using the AquaCrop model in calculating drought insurance premiums for irrigated agriculture.
Silvestre and Lansigan (2015)	Developing a rainfall index-based crop insurance model for rice production in Pangasinan, Philippines, using Net Single Premiums.
Poudel et al. (2016)	Estimating rainfall index-based crop insurance premiums for rice and wheat in Nepal using the expected loss model.
Kath et al. (2018)	Investigating the financial benefits of excess rainfall index insurance for sugarcane farmers in Tully, Northern Australia, and calculating premiums using the expected loss model.
Bokusheva (2018)	Developing a copula-based weather index insurance design methodology that aims to cover extreme weather events.
Wen et al. (2019)	Developing a specific model for determining premium rates in index-based crop insurance.
Baškot and Stanić (2020)	Determining the premium model for parametric insurance in the agricultural sector of Bosnia and Herzegovina using the Black–Scholes method.
Gómez-Limón (2020)	Proposing an index-based hydrological drought insurance scheme for irrigated agriculture and calculated fair premiums through an actuarial analysis.
Koprivica et al. (2024)	Determining the suitability of weather index-based agricultural insurance models for each selected rice growing zone in Malaysia.
Ariyanti et al. (2020)	Determining the amount of premiums to be paid by farmers using the Black–Scholes method based on the climate index.
Prabowo et al. (2023b)	Determining premium rates in rainfall index-based agricultural insurance for shallot crops using Black–Scholes in Central Java.
Marola et al. (2023)	Calculating the value of agricultural insurance premiums based on the rainfall index in Kapuas Hulu Regency using the Black–Scholes method.
Raharjanti et al. (2024)	Determining the price of agricultural insurance premiums based on the rainfall index in Magelang City using the Black–Scholes method.

.

Based on Table 2, the Black–Scholes method has wide applications in various types of insurance, including high-cost disease, mortgage, life, corporate bankruptcy detection, deposit, Takaful, environmental, and disaster types. This shows the flexibility of the method in measuring risk and determining premiums across various insurance sectors. The Black–Scholes method also dominates rainfall index-based agricultural insurance, as shown in several cases. Previous research applied the model to calculate agricultural crop insurance premiums based on the rainfall index. Other methods, such as the expected loss model, pure premium model, and actuarial analysis, are used in determining rainfall index-based agricultural insurance premiums. This shows that alternatives can be adapted to the specific needs and context of insurance.

A further analysis of the 25 relevant articles used in the literature review was performed visually using VOSviewer software. This software produced a visual map showing the relationship between keywords in the selected literature, which was processed to map the topic. Specifically, VOSviewer was used to analyze keywords found in article titles and abstracts. According to Sukono et al. (2022), the circle size shows how often the word appears in eight articles, with a larger circle indicating the frequency of words discussed. The relationship between the words in the eight articles is represented by the lines connecting the circles. The more connecting lines that enter the circle, the greater the relationships between the words in the circle and others. Therefore, the color of the circle represents the cluster. Circles of similar colors indicate that the words belong to the same cluster. The distance between the circles shows the intensity of the relationship among the words, with a smaller distance representing a stronger correlation. The results of network visualization with VOSiewer software for the 25 articles are shown in Figure 3.

Figure 3 shows the results of network visualization with 49 keywords that appear. In the visualization results, eight clusters based on color are obtained, namely red, yellow, green, dark blue, dark purple, orange, light purple, and light blue, which are interconnected. Each cluster consists of several words that are interconnected. The yellow cluster includes terms such as “Black-Scholes”, “microinsurance”, and “family tafakul”. This cluster shows the application of the Black–Scholes model in setting insurance premiums, particularly in the context of microinsurance and tafakul, an Islamic insurance model. The Black–Scholes model, commonly used in finance for option pricing, is adopted in insurance analyses to determine premiums based on existing risks, specifically for microinsurance and Sharia-compliant tafakul products, which are more affordable for the community (see Saputra et al. 2021; Rahadi et al. 2023). Similarly, the green, purple, and light blue clusters represent other types of insurance, with terms such as “agricultural insurance”, “catastrophic illness”, and “deposit insurance”. The green cluster features various terms including “agricultural insurance”, “rice production”, “Aquacrop model”, and “actuarial analysis”, emphasizing the use of alternative methods in agricultural insurance: the Black-Scholes model, such as the AquaCrop model, and an actuarial analysis (see Ruiz et al. 2015; Gómez-Limón 2020). The dark blue cluster includes terms such as “premium”, “economic growth”, and “natural disaster insurance”, showing the relationship between premium calculations and economic growth rates for natural disaster-related insurance products (see Purwandari et al. 2024).

Red clusters with the terms “exit value” and “trigger value”, which are essential in determining when claims can be filed or indemnity payments are made in agricultural insurance. Based on rainfall data, these index values are closely related to the term “indemnity”, which refers to the concept of compensation or indemnity in the insurance context. This compensation is an essential element in insurance contracts, showing the amount of coverage received by the insured participants. In the visualization, the term “historical burn analysis” is also relevant in determining compensation, as shown by the relationships among the clusters of words. This calculates losses by analyzing historical data, such as rainfall patterns, to determine exit and trigger threshold values to support a more accurate and data-driven determination of compensation determination (see Prabowo et al. 2023b).

Figure 3 shows large circles with different clusters, comprising words such as “premium”, “Black-Scholes”, and “rainfall index”. The circle with the word “Black-Scholes” is related to “premium” and “rainfall index”, as shown by the connecting lines and the close distance between the circles. This shows that the research topic often discussed using “Black-Scholes” is determining premiums with the rainfall index. There are circles containing words related to agriculture and three-word circles containing “agricultural insurance”. This shows that research about determining premiums based on the rainfall index using Black–Scholes was conducted for insurance programs in the agricultural sector.

Density visualization was conducted to show the publication depth in the 25 articles analyzed. This visualization shows how often specific topics appear in the literature. To create this visualization, VOSviewer software was used to generate a density map based on the distribution of keywords in relevant articles. According to Riaman et al. (2022), in density visualization, the brightest color indicates that a topic is widely used in research, while the less bright color suggests rare application. This process shows the distribution of topics, providing an idea of the intensity of research. The results of density visualization for the 25 articles are presented in Figure 4.

Figure 4 shows that the term “black-scholes” has a very light color intensity. This shows the high intensity of research on the determination of premiums, particularly rainfall index-based agricultural insurance. Black–Scholes has been widely used to calculate insurance premiums through mathematical and financial methods developed in option theory. However, the report has explicitly applied the fractional Black–Scholes method in determining rainfall index-based agricultural insurance premiums. This triggers new research opportunities to explore applying the fractional Black–Scholes model to improve accuracy in estimating agricultural insurance premiums. The method allows for a more accurate adjustment of premiums by the dynamics of weather changes in a given period, potentially providing a more appropriate solution for farmers to address weather uncertainty during insurance periods.

3.3. Analysis of Reviewed Articles Based on Research Question

In this section, an analysis was conducted based on the Research Questions presented in Table 1. This analysis focused on 15 articles that discussed the determination of rainfall index-based agricultural insurance premiums. A total of 11 articles used other main methods, while 4 applied Black–Scholes. The analysis of 11 articles aimed to provide an overview of the research topic, but the other 4 were examined more specifically to gain a deeper understanding of the application of the Black–Scholes method in determining rainfall index-based agricultural insurance premiums. This method was undertaken to provide a comprehensive perspective on the factors included in premium determination and the payment scheme in rainfall index-based agricultural insurance. The 15 articles analyzed in this research included Heimfarth and Musshoff (2011), Bobojonov et al. (2014), Ruiz et al. (2015), Silvestre and Lansigan (2015), Poudel et al. (2016), Kath et al. (2018), Bokusheva (2018), Wen et al. (2019), Baškot and Stanić (2020), Gómez-Limón (2020), Koprivica et al. (2024), Ariyanti et al. (2020), Prabowo et al. (2023b), Marola et al. (2023), and Raharjanti et al. (2024).

The first part analyzed in this research was conducted to answer the problems outlined in Research Questions Number 1 and 2, which focused on the main methods and calculation models for determining rainfall index-based agricultural insurance premiums. Generally, a premium refers to the amount of money farmers must pay as the insured party to an insurance company as the insurer for their participation and in return for the protection provided in the insurance program (Markonah et al. 2023). It also shows the transfer of risk to insurance companies, taking over the responsibility for losses incurred due to weather fluctuations, particularly rainfall. The risk transfer process is the basis for insurance companies to determine the amount of premium on the risks taken (Banks 2004).

The premium amount for agricultural insurance must be paid by farmers to insurance companies each planting season, which is in line with the planting and harvesting cycles in agricultural activities. In agricultural insurance, the premium plays a significant role in supporting the sustainability of farming operations by providing compensation guarantees for potential losses. This helps reduce the financial burden on farmers, serving as an essential element in the continuity of the agricultural insurance system and supporting the economic stability of farmers during risks.

In rainfall index-based agricultural insurance, the premium calculation depends on a primary indicator, namely rainfall conditions. Rainfall is selected as a parameter because it significantly influences the success or failure of crop yields. Therefore, the rainfall index is used to determine whether farmers are at risk of crop failure or loss. When rainfall is below or above the predetermined threshold, an insurance company must pay claims to farmers per the agreed-upon scheme. In the 15 relevant articles, various main methods can be used to calculate rainfall index-based agricultural insurance premiums. Each method has its characteristics to the specific situation of the applied insurance scheme, as shown in Figure 5.

Several main methods can be used in determining rainfall index-based agricultural insurance premiums. The pure premium model applied by Heimfarth and Musshoff (2011), Bobojonov et al. (2014), and Wen et al. (2019) offered a burn analysis method to calculate fair premiums. This method depended on historical data, which could affect the model’s accuracy. The AquaCrop model used by Ruiz et al. (2015) focused on simulating crop growth based on water availability and weather conditions but depended on accurate and specific agronomic data. The Net Single Premium used by Silvestre and Lansigan (2015) considered drought risk factors and crop water requirements in determining rainfall-based agricultural insurance premiums. However, this method required a significant amount of data to produce accurate results. Koprivica et al. (2024) used a loss cost method addressing some of the existing limitations, such as adverse risks, ensuring fairer premiums that matched the risks faced by farmers.

The expected loss model was applied by Poudel et al. (2016), Kath et al. (2018), and Bokusheva (2018). This model provided a loss estimation method allowing premiums to be estimated according to potential losses, but based on inaccurate assumptions. An actuarial analysis, which was applied by Gómez-Limón (2020), focused on using historical data and financial theory to determine premiums. Baškot and Stanić (2020) used the Bernoulli and Black–Scholes methods, which calculated premiums by considering the probability of flooding in a certain period. Meanwhile, the Black–Scholes method calculated premiums by considering flood intensity.

The Black–Scholes method dominates in four articles authored by Ariyanti et al. (2020), Prabowo et al. (2023b), Marola et al. (2023), and Raharjanti et al. (2024). The dominant use was attributed to the ability to estimate the amount considered reasonable and fair to farmers. It also offered an effective method for measuring risk and setting a premium proportional to the level of risk an insurance company bears. However, there is no research that applies the fractional Black–Scholes method to determining rainfall index-based agricultural insurance premiums. Further development using the fractional Black–Scholes method can provide a more accurate estimation by considering the long-term memory effect and produce a more varied as well as adaptive calculation based on rainfall conditions.

In the next stage, the model used in the premium calculation was analyzed. This analysis was carried out to understand how premium calculations were carried out according to the methods applied. Therefore, a more accurate estimation can be obtained for the risk conditions of farmers, creating effective protection. Through this analysis, the premium calculated will better reflect the reality of farmers’ risks, regarding the potential for drought and excessive rainfall that can damage crops. Therefore, insurance systems can provide more effective protection, maintain farmers’ economic stability, and minimize losses due to crop failure. Details of the concrete form calculating agricultural insurance premiums are presented in

Table 3. Concrete form calculating agricultural insurance premiums.

Author(s)	Premium Calculation Model
Heimfarth and Musshoff (2011)	$P=K·maks\{R_T-R_A,0\}$
Bobojonov et al. (2014)	$P=e^{-rt}{\displaystyle \frac{1}{n}}{\displaystyle \sum _{t=1}^n}{I}_t$
Ruiz et al. (2015)	$P=PR(\%)\times \overline{G}$
Silvestre and Lansigan (2015)	$P=\left[{\displaystyle \frac{1}{{(1+r)}^t}}\right]\times N\left(D\right)$
Poudel et al. (2016)	$P=E\left[Loss\right]={\displaystyle \underset{0}{\overset{R_E}{\int }}}f\left(R_A\right)d{R}_A+{\displaystyle \underset{R_E}{\overset{R_T}{\int }}}\left({\displaystyle \frac{R_T-R_A}{R_T-R_E}}\right)f\left(R_A\right)d{R}_A$
Kath et al. (2018)	$P=E\left[Loss\right]={\displaystyle \sum _{t=1}^n}[I_t·P\left({RI}_t\right)]$
Bokusheva (2018)	$P={\displaystyle \sum _{t=1}^n}\left[I_t\right]$
Wen et al. (2019)	$P={\displaystyle \frac{1}{n}}{\displaystyle \sum _{t=1}^n}{I}_t$
Baškot and Stanić (2020)	$P=K{e}^{-r}E\left[X\right]$ $P=K{e}^{-rt}N(-d_2)$ $d_2={\displaystyle \frac{\ln \left(\frac{R_A}{R_T}\right)+\left(r-\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}}$
Gómez-Limón (2020)	$P=E\left[D\right]$
Koprivica et al. (2024)	$P={\displaystyle \frac{{\sum }_{s=t-1}^{t-10}Loss cost ratio}{10}}$
Ariyanti et al. (2020); Prabowo et al. (2023b); Marola et al. (2023); and Raharjanti et al. (2024)	$P=K{e}^{-rt}N(-d_2)$ $d_2={\displaystyle \frac{\ln \left(\frac{R_A}{R_T}\right)+\left(r-\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}}$

, and the description of the notation are provided in

Table 4. A notational description in relevant articles.

Variable	Description
$P$	Premium for agricultural insurance contract
$K$	Maximum amount of compensation (indemnity) received
$R_A$	Actual rainfall index
$R_T$	Trigger threshold; i.e., rainfall index at which compensation payments start to be made
$R_E$	Exit threshold; i.e., rainfall index at which maximum payment is made
$e$	Exponential constant with value 2.718281828… ≈2.718
$r$	Risk-free interest rate
$t$	Period of insurance
$n$	Sum of rainfall values from index probability
$I_t$	Amount of compensation at time $t$
$PR(\%)$	Premium rate (in percentage)
$\overline{G}$	Average income of crops
$D$	Amount of compensation (indemnity) against drought
$W$	Amount of compensation (indemnity) against flood
$N\left(D\right)$	Probability of indemnity amount against drought protection
$f\left(R_A\right)$	Gamma density function of actual rainfall
$P\left({RI}_t\right)$	Probability of each rainfall value level
$E\left[X\right]$	Expectation of Bernoulli random variable
$\sigma$	Standard deviation
$N\left({-d}_2\right)$	Standard normal cumulative distribution function
$E\left[D\right]$	Expectation of drought protection indemnity variable

.

Table 3 shows various concrete forms of calculating agricultural insurance premiums based on the rainfall index with different main methods. The research by Heimfarth and Musshoff (2011) used a pure premium model to calculate rainfall index-based agricultural insurance premiums. The analysis was carried out using the relationship between the actual rainfall and threshold. The model provided premium adjustments based on the actual level of risk for farmers. Ruiz et al. (2015) also used the AquaCrop model, emphasizing that the premium was calculated by applying two factors. These included the percentage of risk premium based on the farmer’s risk, such as drought or excessive rainfall, and the average crop revenue obtained from the estimated yield under ideal weather conditions. Therefore, the model could provide a more accurate calculation but being dependent on weather data, which varied particularly during weather changes.

Research by Silvestre and Lansigan (2015) used the Net Single Premium to calculate rainfall index-based agricultural insurance premiums. The method has an advantage in facilitating the application of discounting concepts to minimize premiums, enhancing easy understanding. However, it lacks flexibility in capturing complex risk variability, such as extreme changes or other uncertainties that affect agricultural yields. Gómez-Limón (2020) applied an actuarial analysis by calculating premiums based on expected losses without additional costs or other factors. This method provided transparency and simplicity in the calculation process to allow easy implementation but required further investigation due to the lack of flexibility.

Poudel et al. (2016), Kath et al. (2018), and Bokusheva (2018) developed a method of calculating a rainfall index-based agricultural insurance premium with a loss expectation-oriented method as a fair premium. Poudel et al. (2016) calculated the premium by integrating two components, namely losses when rainfall was below the threshold and losses experienced between the two thresholds. This method shows the relationship between losses and rainfall variation based on historical data and the appropriate selection of a distribution function. Meanwhile, Kath et al. (2018) calculated a premium by relating predicted agricultural yield losses to rainfall probability distributions. The model provided a transparent calculation and a fair premium by directly correlating the expected loss to the rainfall probability. However, historical rainfall data and distribution assumptions must be considered to produce accurate calculations. Bokusheva (2018) and Wen et al. (2019) used a more straightforward method, which provided ease of implementation but limited the variation in losses due to rainfall fluctuations.

Baškot and Stanić (2020) applied the Bernoulli and Black–Scholes methods to calculate rainfall index-based agricultural insurance premiums to distinguish between premium calculation strategies for fixed and proportional compensation products. In the Bernoulli model, the premium calculation was more straightforward, considering the probability of loss occurrence. Meanwhile, the Black–Scholes method offered a more complex analysis and was responsive to variations in rainfall risk, ensuring accuracy for dynamic situations.

The application of the Black–Scholes method dominated in calculating rainfall-based agricultural insurance premiums in four reviewed articles. These included research conducted by Ariyanti et al. (2020), Prabowo et al. (2023b), Marola et al. (2023), and Raharjanti et al. (2024). The method was commonly used in financial markets to calculate option prices, which was adopted by agricultural insurance to measure rainfall index-based premiums. Extreme rainfall conditions of either drought or flood trigger the calculations in this method. Other parameters, as outlined in Table 4 include the risk-free interest rate $\left(r\right)$ and insurance term $\left(t\right)$, which determines how much economic changes influence the premium paid. The volatility factor $\left(\sigma \right)$ in the rainfall distribution is also considered, showing the use of standard deviation. The advantage of the Black–Scholes method depends on the ability to provide a strong and transparent basis for determining the premiums that farmers must pay while considering the risk of losses from extreme rainfall conditions. However, the Black–Scholes method has limitations, mainly its flexibility in determining premium payments, which can be made at the beginning of the contract. This may hinder insurance protection for farmers who are considering joining insurance when the contract starts. Therefore, there is a need to develop the primary method using fractional Black–Scholes, capable of providing more diverse premium estimates and allowing payments to be made on time during insurance periods. This method also considers the long-term memory effect, which provides more accurate premium estimation results. In this context, fractional Black–Scholes can improve efficiency and fairness in determining rainfall index-based agricultural insurance premiums.

The commodities included in determining rainfall index-based agricultural insurance premiums need to be analyzed. This aims to answer Research Question Number 3 relating to the commodities included in agricultural insurance. The selection of commodities is essential to assess the relationship between the crop type and rainfall index used in agricultural insurance. Additionally, understanding the impact of rainfall variations on crop productivity ensures the relevance and accuracy of the primary method of determining agricultural insurance premiums. Specific details regarding the types of commodities applied in relevant articles to the research topic are shown in Figure 6.

Figure 6 shows that the articles relevant to the research topic cover a wide range of agricultural commodities with the main focus on the rainfall index. Food crops are the dominant focus, with paddy being the most discussed in five articles. Research that selected paddy as the primary commodity was carried out by Silvestre and Lansigan (2015), Wen et al. (2019), Ariyanti et al. (2020), Marola et al. (2023), and Raharjanti et al. (2024). This was followed by wheat plants comprising three articles, including research by Bobojonov et al. (2014), Bokusheva (2018), and Baškot and Stanić (2020). The results suggested that food crops, particularly paddy and wheat, depended on determining rainfall index-based agricultural insurance premiums. This was due to the strong correlation between rainfall, paddy, and wheat productivity. Kath et al. (2018) discussed sugarcane as a commodity, suggesting the further exploration of the application of rainfall index-based agricultural insurance in plantation crops. Research also focused on shallot commodities, as conducted by Prabowo et al. (2023b). As a horticultural commodity, shallots have a high sensitivity to rainfall that can affect their growth. Lacking or excess rainfall could affect the quality and quantity of shallot yields (Sholikhah and Hakim 2024).

There are articles that use several commodities in their analysis. Heimfarth and Musshoff (2011) and Koprivica et al. (2024) explored two commodities, namely corn and wheat. The results showed that both commodities had different agronomic characteristics providing insights into the context of rainfall index-based agricultural insurance. Poudel et al. (2016) investigated paddy and wheat commodities, showing the understanding of food security and yield fluctuations influenced by rainfall. Ruiz et al. (2015) expanded the scope of the research by using the commodities wheat, corn, sunflower, cotton, and olive trees. This research showed the diversity in agricultural practices and the importance of insurance that was adaptive to the specific needs of each commodity. Gómez-Limón (2020) used cotton, maize, tomatoes, sugar, and wheat as commodities, suggesting potential application to various commodities with different characteristics. Therefore, there is a need to emphasize that the analysis of the crop commodities in insurance must also be investigated further. This procedure is important in understanding and managing the diverse risks in the agricultural sector, which can support better decision making and more effective policies in facing the challenges.

In the next stage, this analysis is conducted to answer Research Question Number 4 regarding the meteorological risk used in measuring the rainfall index in determining agricultural insurance premiums. Regarding rainfall index-based agricultural insurance, mitigating the risks to farmers due to extreme weather conditions is very important. The two main risks that are the focus of protection in agricultural insurance based on the rainfall index scheme are drought and flood. These risks have great potential to cause significant losses to farmers. Drought can cause a reduction in crop yields, while flooding has the potential to physically damage crops and disrupt growth cycles (Meng and Qian 2024). These risks are identified using a rainfall index generated from modeling the determination of a rainfall index-based agricultural insurance premium that determines compensation payments.

Based on the analysis of 15 articles relevant to the research topic, 13 articles focus on drought risk protection. These include research conducted by Heimfarth and Musshoff (2011), Bobojonov et al. (2014), Ruiz et al. (2015), Silvestre and Lansigan (2015), Poudel et al. (2016), Bokusheva (2018), Wen et al. (2019), Gómez-Limón (2020), Koprivica et al. (2024), Ariyanti et al. (2020), Prabowo et al. (2023b), Marola et al. (2023), and Raharjanti et al. (2024). The intense focus on drought risk considers the significant impacts caused by drought, particularly regarding rainfall-dependent agriculture. Meanwhile, two articles discussed flood risk protection, namely the research conducted by Kath et al. (2018) and Baškot and Stanić (2020). The limited number of studies showing flood risk suggests that this topic is still understudied in rainfall index-based agricultural insurance. Flood risk also has the potential to significantly impact the agricultural sector, particularly in areas vulnerable to climate change. This indicates the need to broaden the scope of research to include flood risk protection, either primarily or in combination with drought risk, thereby increasing the mitigation framework in agricultural insurance schemes.

The compensation payment scheme that matched the risk in each relevant article in rainfall index-based agricultural insurance was analyzed to answer Research Question Number 5. This analysis is essential to identifying the advantages of each compensation payment scheme in addressing specific risks. Generally, the compensation payment scheme in rainfall index-based agricultural insurance uses exit and trigger indices adjusted to the protected risks. The threshold determination of these two rainfall indices is calculated using cumulative rainfall data over ten days (Azka et al. 2021; Ariyanti et al. 2020). Each of these indices is essential in determining the response to the protected risk such as flooding caused by excess water. When the water level reaches a specific limit, insurance companies must compensate farmers (Baškot and Stanić 2020). Similarly, insurance companies must pay compensation for drought risk caused by water shortage when the water level falls below a predetermined threshold. Further details on compensation payment schemes for drought and flood risks in each of the articles are shown in

Table 5. Compensation payment scheme for agricultural insurance in relevant articles.

Author(s)	Risk-Based Compensation Payment Scheme
Heimfarth and Musshoff (2011)	$D=\left\{\begin{array}{c}0;R_A\ge R_T\\ K;R_A<R_T\end{array}\right.$
Bobojonov et al. (2014)	$D=\left\{\begin{array}{c}0;R_A\ge R_T\\ K;R_A<R_T\end{array}\right.$
Ruiz et al. (2015)	-
Silvestre and Lansigan (2015)	$D=(h_i-C{h}_i)\times payout rate\times (total insured/1000)$
Poudel et al. (2016)	$D=\left\{\begin{array}{c}K; R_A\le R_E\\ K\left({\displaystyle \frac{R_T-R_A}{R_T-R_E}}\right); R_E<R_A<R_T\\ 0; R_A\ge R_T\end{array}\right.$
Kath et al. (2018)	-
Bokusheva (2018)	$D=\left\{\begin{array}{c}0;R_A\ge R_T\\ K;R_A<R_T\end{array}\right.$
Gómez-Limón (2020)	$D=\left\{\begin{array}{c}K\times (1-DED);\\ h\left(R_{8t}\right)-DED\times K;\\ 0;\end{array}\right.\begin{array}{c}h\left(R_{8t}\right)=K\\ DED\times K<h\left(R_{8t}\right)<K\\ 0<h\left(R_{8t}\right)<DED\times K\end{array}$
Ariyanti et al. (2020)	$D=\left\{\begin{array}{c}K; R_A\le R_E\\ K\left({\displaystyle \frac{R_T-R_A}{R_T-R_E}}\right); R_E<R_A<R_T\\ 0; R_A\ge R_T\end{array}\right.$
Prabowo et al. (2023b)	$D=\left\{\begin{array}{c}K; R_A\le R_E\\ K\left({\displaystyle \frac{R_T-R_A}{R_T-R_E}}\right); R_E<R_A<R_T\\ 0; R_A\ge R_T\end{array}\right.$
Marola et al. (2023)	-
Koprivica et al. (2024)	-
Raharjanti et al. (2024)	$D=\left\{\begin{array}{c}K; R_A\le R_E\\ K\left({\displaystyle \frac{R_T-R_A}{R_T-R_E}}\right); R_E<R_A<R_T\\ 0; R_A\ge R_T\end{array}\right.$

.

Table 5 shows that the critical variables used in the compensation payment scheme are the trigger threshold $\left(R_T\right)$, the exit threshold $\left(R_E\right)$, and the actual rainfall index $\left(R_A\right)$. The scheme is divided into two types based on the protected risks, namely drought and flood. Although each scheme provides treatment according to the threshold conditions to trigger compensation payment research on flood protection, research does not explicitly include compensation payment schemes. Kath et al. (2018) stated that compensation payment could be made to farmers enrolled in rainfall index-based agricultural insurance. Baškot and Stanić (2020) mentioned two compensation schemes for floods, namely fixed and pro-social.

Several relevant articles use the actual rainfall index with varying notations, including Poudel et al. (2016), who applied $R$. The notation was uniform to $R_A$ because it has the exact representation, facilitating the analysis and ensuring consistency between research. This stage made identifying variables and understanding compensation payment schemes more structured and accessible, particularly when comparing various articles that discussed agricultural insurance based on rainfall index schemes. Similarly, the trigger index used different notations in several relevant articles, as shown by Bokusheva (2018), who applied $q_{\alpha }\left(W\right)$.

All relevant articles include the use of trigger and exit indices in designing compensation payment schemes. This is shown in the reports by Heimfarth and Musshoff (2011), Bobojonov et al. (2014), Poudel et al. (2016), Bokusheva (2018), Wen et al. (2019), Ariyanti et al. (2020), Prabowo et al. (2023b), and Raharjanti et al. (2024). In this scheme, the trigger index is defined as the threshold at which compensation payments are made in full. Meanwhile, the exit index is the starting point at which compensation payments are made. Compensation is paid when the actual rainfall index is below or above the exit threshold based on the type of risk insured, such as drought or flood, and the value increases proportionally until it reaches the total value at the trigger threshold. The advantages of this scheme are simplicity and transparency, allowing farmers to quickly understand the timing of compensation payments based on clearly measurable changes in the rainfall index. It also allows for adjustment of payments based on the severity of risk, thereby facilitating flexibility in protecting farmers from different levels of extreme weather risk.

Silvestre and Lansigan (2015) and Gómez-Limón (2020) did not use trigger and exit indices. Specifically, Silvestre and Lansigan (2015) emphasized water requirements at different stages of crop growth as a basis for calculating compensation payments in rainfall index-based agricultural insurance. This method was characterized by using the variables rainfall day—$i {(h}_i)$—and cumulative amount of rainfall at the growth stage day—$i \left({Ch}_i\right)$. Gómez-Limón (2020) applied the distribution function of the rainfall index $\left(h\left(R_{8t}\right)\right)$ and the deductible, which referred to the percentage of the compensation amount to be borne by farmers $\left(DED\right)$. This variable reduces the risk of moral hazard and incentivizes farmers to be more proactive in taking risk management measures by considering the probability distribution of the rainfall index.

Ruiz et al. (2015) did not explicitly include a compensation payment scheme for drought protection but explained that independent experts conducted loss assessments on the ground. This was performed to ensure that the losses experienced were caused by drought, not other factors. Koprivica et al. (2024) showed that weather index-based agricultural insurance schemes would pay compensation when specific weather parameters were exceeded. However, the value of the weather parameters used needed to be explained. Marola et al. (2023) did not explain the compensation payment scheme in detail but explained that the insured party would pay a premium when suffering a rainfall-related loss.

4. Discussion

4.1. Summary of Reviewed Articles

Based on an analysis of the 25 reviewed articles, there was an increasing trend in the number of studies related to determining agricultural insurance premiums to mitigate the risk of harvest failure due to rainfall yearly. Although there was a decrease, this trend was followed by an increase. When the popularity of research on agricultural insurance based on the rainfall index continues to increase, there is a need to understand and implement agricultural insurance based on the rainfall index to overcome several challenges. Therefore, developing this research provides significant benefits for farmers who risk crop failure due to extreme weather and insurance companies offering protection for the sector. Along with the increasing relevance and need for insurance, there is a corresponding rise in research on the rainfall index as a basis for the determination of agricultural insurance premiums.

Among the 15 articles analyzed, various main methods were found in research related to determining agricultural insurance premiums to mitigate the risk of harvest failure due to rainfall. These methods include the pure premium model, loss expectation model, AquaCrop model, Net Single Premium, actuarial analysis, Bernoulli method, and Black–Scholes. Figure 5 in Section 3.3 shows that the Black–Scholes method dominates the others due to the ability to produce premium estimates considered fair and proportional to the level of risk farmers face, as explained by Raharjanti et al. (2024). This method also offers a transparent and fast calculation process, thereby lowering operational costs for insurance companies (Filiapuspa et al. 2019).

This research found that several commodities such as food crops, plantations, horticulture, and fruits were included in determining agricultural insurance premiums to mitigate the risk of harvest failure due to rainfall. However, the most dominant commodity was paddy due to high dependence on rainfall, as reported by Prabowo et al. (2023b). The review shows that drought risk is more dominant than flood risk due to the significant impact on crop productivity (Kath et al. 2018; Baškot and Stanić 2020).

Based on the description, there is an opportunity to develop a method for determining agricultural insurance premiums to mitigate the risk of harvest failure due to rainfall. This can be carried out through the application of the fractional Black–Scholes method, which is presently lacking in the existing literature. Furthermore, commodity factors and meteorological risks using the Black–Scholes method can still be explored by covering various crops, as well as drought and flood risk simulation. The development of these methods and factors has the potential to produce premium estimates that are more accurate and meet the needs of farmers. High premium accuracy is expected to provide more effective protection against the risks in the agricultural sector, particularly related to rainfall changes that significantly impact production. Therefore, this development is expected to increase the efficiency and relevance of agricultural insurance to mitigate the risk of harvest failure due to rainfall in the future.

4.2. Literature Gap Analysis and Future Research

The research gap shows that the use of the Black–Scholes method for determining agricultural insurance premiums in mitigating the risk of crop failure due to rainfall has been widely explored. This is shown in Figure 4, where the word “Black-Scholes” has a large circle, making it the main topic in the article review. There is no “fractional Black-Scholes”, suggesting a lack of research on the topic. This is also supported by the analysis of the articles in Section 3.3, which use only the Black–Scholes method. Therefore, the use of the fractional Black–Scholes method to determine agricultural insurance premiums to mitigate the risk of crop failure due to rainfall can be a novelty that has the potential to be developed in future research.

The Black–Scholes method is used to determine the price of stock options. Generally, an option is an agreement or contract that gives the holder the right, but not the obligation, to buy or sell a certain amount of an asset at a predetermined price and time (Luenberger 1998). Based on function, an option is divided into two types, namely a call option that gives the right but not the obligation to buy an asset. There is also a put option that gives the right but not the obligation to sell an asset at a predetermined price and time (Higham 2004). Regarding the use time, an option is divided into two types: American and European. The American-type option is used at any time until maturity, while the European type is applied at maturity. The assumptions used in determining option prices using the Black–Scholes method are as follows:

(1)

The reference option is a European type that can be exercised at maturity.

(2)

The stock price follows a lognormal distributed random pattern with constant stock return variance.

(3)

The risk-free interest rate is constant.

(4)

There is no dividend payment on the stock during the option’s remaining life.

(5)

No taxes and transaction costs in buying or selling options.

According to research by Hull (2002), the Black–Scholes method for determining the call option price is written in Equation (1), and the put option price is written in Equation (2).

$$C=S_{0}N\left(d_1\right)-G{e}^{-rT}N\left(d_2\right),$$

(1)

$$P=G{e}^{-rT}N\left(-d_2\right)-S_{0}N\left({-d}_1\right),$$

(2)

with

$$d_1={\displaystyle \frac{\ln \left(\frac{S_0}{G}\right)+\left(r+\frac{{\sigma }^2}{2}\right)T}{\sigma \sqrt{T}}},$$

(3)

$$d_2={\displaystyle \frac{\ln \left(\frac{S_0}{G}\right)+\left(r-\frac{{\sigma }^2}{2}\right)T}{\sigma \sqrt{T}}},$$

(4)

with

$C$: Black–Scholes European-type call option price;

$P$: Black–Scholes European-type put option price;

$S_0$: initial stock price;

$S_T$: stock price at maturity;

$G$: strike price;

$e$: exponential constant with value 2.718…;

$r$: risk-free interest rate;

$T$: time at maturity;

$\sigma$: standard deviation of stock price;

$N\left(d_1\right)$: standard normal cumulative distribution function of $\left(d_1\right)$;

$N\left(d_2\right)$: standard normal cumulative distribution function of $\left(d_2\right)$;

$N\left(-d_1\right)$: standard normal cumulative distribution function of $\left({-d}_1\right)$;

$N\left(-d_2\right)$: standard normal cumulative distribution function of $\left({-d}_2\right)$.

The Black–Scholes method has been adapted to determine rainfall index-based agricultural insurance. This adaptation is based on the similarity of characteristics between the option pricing mechanism and agricultural insurance premiums for rainfall index-based agricultural insurance to mitigate the risk of crop failure due to rainfall (Filiapuspa et al. 2019). In the concept of agricultural insurance based on the rainfall index, the initial stock price $\left(S_0\right)$ is analogous to the actual rainfall index $\left(R_A\right)$, while the strike price $\left(G\right)$ is analogous to the trigger threshold $\left(R_T\right)$. This insurance aims to protect farmers from risks associated with rainfall fluctuations, which can damage crops. Therefore, determining agricultural insurance premiums to mitigate the risk of crop failure due to rainfall can be calculated as shown in Equation (5).

$$P=K{e}^{-rt}N\left(-d_2\right),$$

(5)

with

$$d_2={\displaystyle \frac{\ln \left(\frac{R_A}{R_T}\right)+\left(r-\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}},$$

(6)

with

$K$: sum insured or maximum amount of compensation received;

$r$: risk-free interest rate;

$t$: insurance period;

$\sigma$: standard deviation of rainfall index;

$R_A$: actual rainfall index;

$R_T$: trigger threshold;

$N\left(-d_2\right)$: probability of rainfall index.

When related to the compensation payment scheme, this premium calculation applies to protection against drought risk in the agricultural sector. This is an analogous result of the Black–Scholes cash-or-nothing put option formula, as reported by Prabowo et al. (2020). In a cash-or-nothing put option, the option is valuable when the stock price is less than the strike price. Regarding rainfall index-based agricultural insurance premiums for protection against drought risk with a cash-or-nothing put option method, an insurance company provides compensation when the actual rainfall index is less than the specified trigger threshold. This applies to crops requiring less water, such as shallot, which grow better in areas with low rainfall or during the dry season, Prabowo et al. (2023a). Excessive rainfall can damage onion crops and cause their stems to rot. Therefore, the Black–Scholes method can be used to determine premiums for rainfall index-based agricultural insurance and compensation based on fluctuations that affect agricultural yields.

Determining the premium using Black–Scholes shows the potential that this method can be developed similarly into a fractional Black–Scholes method. However, the difference lies in fractionalized insurance terms. The fractional Black–Scholes method is currently still limited in option determination and has not been used to determine agricultural insurance premiums. This method has been widely used in option pricing, as conducted by Meng and Wang (2010), He and Lin (2021), and Sun et al. (2014).

The fractional Black–Scholes method is a development of the Black–Scholes method by introducing fractionalized time using the Hurst parameter. This method has the ability to represent long-term memory, particularly when interpreting financial data. Furthermore, it can be derived through a stochastic process, starting with the assumption that the stock price follows fractional Brownian motion. A key advantage of fractional Brownian motion is the ability to show memory effects, suggesting that future asset prices are influenced by current and previous prices. According to Necula (2002), the fractional Black–Scholes method for determining option prices is expressed in Equations (7) and (8).

$$C_F=S_{0}N\left(d_1\right)-G{e}^{-rT}N\left(d_2\right),$$

(7)

$$P_F=G{e}^{-rT}N\left(-d_2\right)-S_{0}N\left({-d}_1\right),$$

(8)

with

$$d_1={\displaystyle \frac{\ln \left(\frac{S_0}{G}\right)+rT+\frac{{\sigma }^2}{2}{T}^{2H}}{\sigma \sqrt{T^{2H}}}},$$

(9)

$$d_2={\displaystyle \frac{\ln \left(\frac{S_0}{G}\right)+rT-\frac{{\sigma }^2}{2}{T}^{2H}}{\sigma \sqrt{T^{2H}}}},$$

(10)

where $C_F$ is the price of the fractional Black–Scholes European call option, $P_F$ is the price of the fractional Black–Scholes European put option, and $H$ is the Hurst parameter. Equations (7) and (8) are equivalent to the standard Black–Scholes method in Equations (1) and (2) when the Hurst parameter is $H={\displaystyle \frac{1}{2}}$. Therefore, assuming that the risk-free interest rate and dividend yield are ignored, the option price based on fractional Brownian motion is determined by two parameters. These include the Hurst parameter, which measures long-term memory, and the stock price volatility measuring price uncertainty.

The application of the fractional Black–Scholes method to determine rainfall index-based agricultural insurance premiums is similar to the Black–Scholes method. However, the difference is in insurance terms fractionalized through the Hurst parameter. This is particularly observed when calculating the cumulative distribution value, which plays a role in determining the premium amount. The premium calculation to mitigate the risk of crop failure due to rainfall using the fractional Black–Scholes method can use Equation (5) but with an adjustment to the cumulative distribution using Equation (11).

$$d_2={\displaystyle \frac{\ln \left(\frac{R_A}{R_T}\right)+rt-\frac{1}{2}{\sigma }^2{t}^{2H}}{\sigma \sqrt{t^{2H}}}},$$

(11)

with $H$ showing the Hurst parameter. In the fractional Black–Scholes method, the Hurst parameter is essential in making premiums more variable and allowing flexibility in claim payments that can be made at any time during insurance periods. The value is estimated through various methods, including the rescaled range statistics method, Geweke–Porter-Hudak (GPH). It is expected that with a new research opportunity, namely the application of the fractional Black–Scholes method, the estimation of insurance premiums is more precise and flexible, showing the complex long-term risk characteristics in rainfall index-based agricultural insurance.

5. Conclusions

In conclusion, this research obtained 25 articles from Scopus, ScienceDirect, and Dimensions based on SLR results using the PRISMA method. Each article was subjected to a bibliometric analysis using VosViewer software, which provided a comprehensive overview of current trends in insurance premium determination research and identified potential opportunities for future development. Furthermore, 15 articles were analyzed, focusing on premium determination in rainfall index-based agricultural insurance using Black–Scholes and other main methods. The analysis showed that the Black–Scholes method had been widely used to determine rainfall index-based agricultural insurance premiums. The calculation was based on fluctuations in the rainfall index, leading to premium estimates that were fair, reasonable, and transparent to farmers. Despite being applied in option prices in financial markets, no research was found on the fractional Black–Scholes method to determine agricultural insurance premiums based on the rainfall index. Therefore, the fractional Black–Scholes method has the potential to be applied in determining agricultural insurance premiums based on the rainfall index. The method considered the long-term memory effect to reflect the dependency between rainfall periods and produced premiums that were more diverse as well as adaptive to the dynamics of risk in the agricultural sector. A method for a model determining agricultural insurance premiums to mitigate the risk of crop failure due to rainfall was developed and suggested. This research is expected to serve as a reference for further investigation on agricultural insurance based on the rainfall index and make a significant contribution to the Government of the Agriculture Department’s policy formulation regarding insurance programs for farmers.