1. Introduction

2. Materials and Methods

2.2. Conceptual framework

There are many studies of agricultural production or yield modelling that use the weather condition index in combination with other various factors. The reason for creating numerous models is that productions changes between different areas dependent on various factors. A suitable model in one area may not be suitable for others. Thus, the procedures for this study consist of two parts. In the first part, the yield model using the weather index are formulated with various statistical tools according to the selected cultivation area. The appropriate models are adopted in the second part. This part designs and price for weather derivative in order to obtain suitable premiums, corresponding to the yield model. Consequently, the efficiency of the model is measured in scenarios in which farmers are insured or not. The following flowchart outlines the research methodology.

Figure 1. Schematic of weather index insurance design process and efficiency evaluation.

Figure 1. Schematic of weather index insurance design process and efficiency evaluation.

2.2.1. Step 1: Specification of weather variables

Table 1 pressents a review of the literature on the weather index related to crop yield. Many studies show that the weather index has a significant influence on crop yields.

From Table 1, the maximum temperature and rainfall variables are frequently used in the specification of weather variables. These two variables were therefore selected to create weather-yield models in the study. The model of the relationship between climate variables and crop yields is obtained using a statistical method to estimate the relationship between climate and crop production.

2.2.2. Step 2: Integration of data and data preparation

Climate data in the growing season, i.e., maximum temperature and rainfall were adopted to develop the weather-yield model. Price data is taken from farm gate prices from 1992 to 2022. Details on the variables and calculation methods are in Table 2.

In some casts there is missing data which could have resulted from various factors, such as due to a power cut problem. Data preprocessing was used to deal with this issue. From our historical data, only the rainfall index has missing data. As a result, we employed the work of Kanchai et al. (2022) to handle the problem.

2.2.3. Steps 3-5: Regression model linking yield to climate indices

This study was conducted using data from 1992 to 2022. Data from 1992 to 2017 were used to fit the models, while the remaining data (2018–2022) were used for model validation. Most analytical techniques are carried out with "R" and Mathematica software.

Before constructing the weather-yield model, we considered the effects that time series might have on yield productivity. Detrending the yield is one of the most popular methods for removing the effects of changes in management, technology, and production extent yields over time. One of the techniques for detrending the yield is by adjusting only the yield variable in the model (Vedenov and Barnett 2004; Johari et al. 2022). An alternative approach can be used because the annual impacts do not need to be identical, in which the yield data is implicitly adjusted for trends by incorporating the year as part of the model (Kath et al. 2018, 2019; Verón et al. 2015; Verma et al. 2020). The meteorological factors generally show upward temporal trends.

In our study, the weather-yield model was formulated using different statistic tools, including a quadratic regression model, Generalized Additive Models (GAMs), and Quantile generalized additive model (QGAM). The quadratic regression models are widely used to create yield models, especially where adjusted yield is treated as a variable (Johari et al. 2022). Since several factors influence agricultural yields, removing non-weather influences from agricultural yield data throughout the course of the observation period is the goal of detrending crop output. For example, the factors include geography, governance policies, and technological development.

Alternative approaches in the study are to use GAMs or QGAM. These models were utilized by addressing the function of year into the model. GAMs can be understood as Generalized Linear Models (GLMs) that are estimated while being subjected to smoothing penalties. The primary challenge that arises from this penalization is the requirement to choose the appropriate level of penalization, meaning that the smoothing parameters must be estimated. To analyze and predict productivity in the study, GAMs are utilized corresponding to the following equation:

$$Y_t=a+s\left({RI}_t\right)+s\left({Tmax}_t\right)+s\left(t\right)+{\epsilon }_t,$$

(1)

where ${ Y}_t$ is sugarcane yield at time $t$, $s$ is smooth function, $R{I}_t$ is the rainfall index at time $t$, ${\epsilon }_t$ is the error term, and $t$stands for year.

QGAM was developed by Fasiolo et al. (2020). This regression model is based on Quantile regression (QR) and GAM. Instead of focusing only on the mean as is the case with GAM, the rainfall index relationship with the lower quantiles of the sugar cane yield distribution is evaluated using the quantile generalized additive model. The equation of QGAM considered in this study is as same as GAM as shown in (1).

It is possible that the model could be overfitted in the model construction step. It is possible to check whether the model is overfitted. For this purpose, the data is divided into a training and a testing subset. After that, the split test technique is employed to check the overfitting. Next, we measure the quality of the fitting model. The adjusted R² value was used as a criterion (greater than 0.6 in the study) for model selection.

2.2.4. Steps 6 and 7: Design of weather derivatives and premium estimation

To evaluate the efficiency of weather derivatives, a particular contract must be designed. From the litarature in Table 1, the rainfall index is desired as the main factor affecting sugarcane production, especially in excessive rainfall events (Glaz and Lingle, 2012; Gomathi et al., 2015). Thus in the study, an elementary contract in case of excessive rainfall triggered by rainfall index is proposed (adapted from Vedenov and Barnett 2004). This contract pays an optimal indemnity conditional on the realization of an index, as depicted in Figure 3.

$$\mathrm{f}\left(i|x,i^{\left(\alpha \right)},L\right)=x\times \begin{array}{cc}\left\{\begin{array}{cc}1& ;i>L{i}^{\left(\alpha \right)},\\ \frac{(i-i^{\left(\alpha \right)})}{L{i}^{\left(\alpha \right)}-i^{\left(\alpha \right)}}& ;i^{\left(\alpha \right)}<i<L{i}^{\left(\alpha \right)},\\ 0& ;i\le i^{\left(\alpha \right)},\end{array}\right.& \end{array}$$

(2)

where $i^{\left(\alpha \right)}$ is the rainfall index with $\alpha$ percentile levels, $x$ denotes maximum indemnity. $L$ is a constant, where $L{i}^{\left(\alpha \right)}$ is the limit of rainfall to pay the maximum insurance claim.

We assume that potential insurance customers will not take any risks at all, so they will be compensated if there is a loss in sugarcane yield the following year. From Figure 3, the contract starts to pay whenever the index $i$ raises above a specific rainfall index with $\alpha$ percentile levels$i^{\left(\alpha \right)}$. The maximum indemnity (equal to $x$) is paid whenever the index raises above the limit $L{i}^{\left(\alpha \right)}$, where 0 < $L$ < 1. Therefore, an elementary contract can be identified by fixing three parameters: the rainfall index with $\alpha$ percentile levels, limit, and maximum indemnity.

To price an elementary contract, the contract parameters as well as the probability distribution of the essential index must be determined. For weather derivatives, the distribution can be derived based on historical data by using either the parametric or non-parametric approach. If $h\left(i\right)$ is a probability density function of the index, the expected payoff (equal to pure premium) of the contract can be determined by:

$$\mathsf{\pi }(\mathrm{x},\mathrm{L})=x{\int }_{L{i}^{\left(\alpha \right)}}^{\infty }h(i)di+x{\int }_{i^{(\alpha )}}^{L{i}^{(\alpha )}}h(i)\frac{(i-i^{(\alpha )})}{L{i}^{(\alpha )}-i^{(\alpha )}}di$$

(3)

Notice that for any of the rainfall indices with $\alpha$ percentile levels $i^{\left(\alpha \right)}$ and limit parameter $L{i}^{\left(\alpha \right)}$, if $x$ is the price of a contract with maximum indemnity, then $\mathsf{\pi }(\mathrm{x},\mathrm{L})$ is the premium of a contract with maximum indemnity of $x$ baht.

• Selection of contract parameters

The weather-yield model resulting from Step 5 was used to calculate payoffs in insurance contracts. At this point, a rainfall index as a trigger value of the contract must be determined. The rainfall index at strike level $i^{\left(strike\right)}$ can be selected in the sense that the predicted yields were equal to the corresponding long-time yields averages. The remaining parameters, i.e., the limit parameter and the optimal number of ฿1 contracts, were selected in order to minimize an aggregate measure of downside loss. In addition, the parameters $x$ and $L$ were determined to be solved (adapted from Vedenov and Barnett 2004):

$$\underset{x,L}{\min }\sum _{t=1}^{26}({\mathrm{m}\mathrm{a}\mathrm{x}\left\{P\right(\overline{Y}-Y_t)-\mathrm{f}\left(i|x,i^{\left(strike\right)},L\right)+\mathsf{\pi }(\mathrm{x},\mathrm{L}),0\}}^2$$

(4)

where $P$ denotes the farm gate price in the last harvest year (in the training subset, i.e., the year 2017), $Y_t$ is the yield at time $t$, $\mathrm{f}\left(i|x,i^{\left(strike\right)},L\right)$ is the insurance payout for that strike level of insurance, and $\pi (x,L)$ is the price of a contract with a maximum indemnity of $x$ baht.

Furthermore, we are interested in a rainfall index at different a percentile levels of $\alpha$, $i^{\left(\alpha \right)}$ exposed to excessive rainfall risk (i.e., $\alpha =$ 70^th, 80^th and 90^th percentiles of the rainfall index). The payoffs and premium of the contracts corresponding to these percentile levels are illustrated for the purpose of comparison to the strike level. The efficiency of these contracts for excessive rainfall risk were also examined.

2.2.5. Step 8: Efficiency Analysis

Efficiency analysis methods were adapted from Kath et al. (2018). Three methods were used, including conditional tail expectation (CTE), certainty equivalence revenue (CER), and mean root square loss (MRSL), to asses the benefit of the contract. To analyze the impact of the contract on farmer revenue, we compared revenue with and without insurance at $\alpha$ level of the rainfall index $i^{\left(\alpha \right)}$. We derive the parameters of optimal contracts and measure the in-sample performance (adapted from Vedenov and Barnett 2004). In addition, at different percentile coverage levels for the selected regression model, the revenue without a contract is given by:

$$R_t=P{Y}_t$$

(5)

and with a contract is:

$$R_t^{\left(\alpha \right)}=P{Y}_t+\mathrm{f}\left(i_t|x,i^{\left(\alpha \right)},L\right)-\mathsf{\pi }\left(\mathrm{x},i^{\left(\alpha \right)},L\right),$$

(6)

where $R_t$ denotes revenue at time t without insurance, $P$ is the farm gate price, $R_t^{\left(\alpha \right)}$ is revenue at time $t$ with $\alpha$ percentile levels of insurance (here the strike level and the 70^th, 80^th, 90^th percentiles of the excessive rainfall index), $Y_t$ is yield at time $t$, $\mathrm{f}\left(i_t|x,i^{\left(\alpha \right)},L\right)$ is the insurance payout for that level of insurance in that year (predicted from the regression models), and the yearly premium for that $\alpha$ percentile levels of insurance and is constant throughout the years in question so it is written as $\mathsf{\pi }(\mathrm{x},i^{\left(\alpha \right)},L)$.

Conditional tail expectation (CTE) is a method to assess the effectiveness of insurance to protect against financial risks. The certainty equivalence revenue with $\alpha$ percentile levels of insurance ($CT{E}^{\left(\alpha \right)}$) has the following equation:

$$CT{E}^{\left(\alpha \right)}= \frac{1}{T}{\sum }_{i=1}^T{R}_t^{\left(\alpha \right)}$$

(7)

Next, Certainty equivalence revenue (CER) was used to measure willingness to pay the farmer. Assuming that the risk aversion was taken into account and the constant relative risk aversion was assumed. Therefore,

$$CE{R}^{\left(\alpha \right)}=\frac{1}{T}{\sum }_{i=1}^{T}ln{(R}_t^{\left(\alpha \right)}),$$

(8)

where $CE{R}^{\left(\alpha \right)}$ is the certainty equivalence revenue with $\alpha$ percentile level of insurance.

Finally, Mean root square loss (MRSL) is a value that represents an insurance contract’s ability to mitigate risk. MRSLwith $\alpha$ percentile levels of insurance was calculated as

$$MRS{L}^{\left(\alpha \right)}=\sqrt{\frac{1}{T}{\sum }_{t=1}^T{\left[max\right(P\overline{Y}-R_t^{\left(\alpha \right)},0\left)\right]}^2},$$

(9)

where $P$ is the price of agricultural commodity and $\overline{Y}$ is the long-term average yield.

3. Results

4. Discussion

5. Conclusions

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