Designing Subsidy Scheme for Marine Disaster Index Insurance in China

Abstract

Designing an optimal subsidy scheme for marine disaster index insurance (MDII) for households in coastal areas of China remains a managerial challenge. The issue of subsidies for disaster insurance has received extensive research attention, but extant studies are confined to the issue of whether to subsidize, lacking focus on how and how much to subsidize. In the existing marine disaster index insurance pilots in China, there are varying levels and scales of subsidies in spite of premium subsidies. To design an optimal subsidy scheme for marine disaster index insurance in China, this paper proposes an optimal insurance model of marine disaster index insurance with government subsidy. Excluding the behaviors of the policyholders and insurance firms, the model captures the behaviors of the subsidy scheme from the government. Furthermore, employing the storm surge disasters, the optimal trigger scheme and subsidy scheme are designed and estimated. The results recommend that the optimal subsidy ratio for MDII in China needs to be at least 92.54%. Moreover, this value increases when there are more potential victims of marine disasters who choose to insure MDII, while the total subsidy decreases. Evidently, the subsidies for pilots of MDII in China are inadequate to meet the conditions for operation currently, which explains the dilemma of the MDII in China’s pilots. These findings provide theoretical evidence for the optimization of the MDII in China.

1. Introduction

Marine disasters strongly influence coastal areas [1,2]. As one of the countries suffering most from marine disasters, the caused damages present a serious threat to China’s economy and society, as well as the marine ecological environment [3,4]. As shown in the statistics of the China Marine Disaster Bulletin [5], various marine disasters caused direct economic losses of 11.703 billion yuan to China and 22 deaths (including missing). Among them, there were 16 storm surge disasters, causing direct economic losses of 11.638 billion yuan, which accounts for 99% of the total losses. As such, as the occurrences of marine disasters increase in scale and scope, the damage caused by extreme events will be particularly worrisome considering China’s heavy reliance on the fishery sector to supply seafood, offer employment, and sustain economic growth. For a developing country, the uninsured risks can put a halt on investment and productivity growth, and China is not an exception [6].

Number of scholars paid lots of attention on the evaluation and management of catastrophic risks [7]. For example, [8] discussed the assessment of the losses of climate change from social and economic perspectives. Anthoff [9] evaluated the global economic impacts of the substantial sea-level rise considering the cost of retreat and protection. Martyr-Koller [10] assessed the losses of sea-level rise on small island developing states. With the consequences of climate change growing fast, it is becoming more and more difficult to alleviate disaster losses [11]. Przybytniowski [12]’s investigation indicates that the damage that accompanies the risk of disasters increasingly affects the property and health of populations who are unable to defend themselves within their own financial resources. Thereby, insurance is recognized as a good instrument of economic and financial for the security of the property and health of these populations with disaster vulnerability [13,14]. Accordingly, various insurances for disaster risk reduction (i.e., the flood insurance, the storm surge insurance) have emerged around the world [15,16]. Nowadays, insurance is a popular approach used for managing marine disaster-related risks, particularly for low-frequency and high-impact risk events [17,18]. During the development of insurance for managing cataphoric risk, various options for insurance operations are proposed and discussed, such as the mixed public–private system, private responsibility and insurance, and public insurance fund [19]. Cummins and Mahul [20] systematically address approaches that developing countries should adopt to manage catastrophe risk. The insurance model they proposed was widely used in designing kinds of catastrophe insurance, which was the solid foundation for innovation in catastrophe insurance, including index-based insurance [21,22].

With weather index insurance being popularized as a supplement to unaffordable traditional insurance for disasters, index-based insurance has recently received growing attention in the context of marine disasters risk reduction [23,24,25]. Using objective and transparent indices to determine insurance claims payment, such as temperature and rainfall, index-based insurance overcomes the problems associated with traditional insurances, including high verification and transactions costs [26,27]. Accordingly, such an insurance type is sometimes considered as derivatives instead of insurance as the indexing formulation relies on environmental factors rather than actual losses [28]. Due to the advantage of this feature, index-based insurance is generally free of adverse selection risks and the moral hazard problem for insurance transactions [29]. Consequently, index-based insurance is particularly relevant and suitable for use in large, less heterogeneous regions [30]. Therefore, index-based insurance can transfer the disaster risks from people to insurance market and offer compensation for economic loss resulting from marine disasters in coastal areas [31,32].

In China, the weather index-based insurance pilots started in May 2012 after the policy for promoting the development of weather index-based insurance was established by the China Insurance Regulatory Commission [33]. However, due to the limited subsidy support, the pilot programs have been restricted to small, targeted areas and none of them has reached a commercially sustainable scale for continued operations. Meanwhile, despite the allocation of various resources by China government, results from the pilot marine disaster insurance programs achieved very limited success in recent years [34]. For example, in China’s first pilot of marine aquaculture index insurance, which was started in Dalian 2013, only three insurance firms were operating to provide the insurance coverage. Moreover, in the pilot of the weather index insurance for scallop farming in Hebei 2016, only 18 of the 300 farmers were covered by the insurance; and the wind index insurance launched by PICC has also been discontinued since high claims in 2016 [35,36]. Evidently, the failure of the marine disasters index insurance pilots is mainly attributable to the lack of supply and demand for this insurance type.

Government intervention is considered the most effective solution to revert failures in the index-based insurance market [37,38]. Cummins [39,40] suggested the government should make efforts to balance the supply and demand for catastrophe insurance. Subsidy, one of the most common instruments of government intervention, can help rectify the failing situation of the disaster insurance market [41]. Government-subsidized insurance is widely used in the USA to manage the risk of hurricanes in states that suffer hurricanes badly, including Florida, Texas, and Louisiana [42,43]. Currently, the types of common subsidies mainly include premium subsidy, operating and management fee subsidy, tax deduction, and reinsurance subsidy, among which premium subsidy is most commonly used in China. Babcock et al. [44] highlight that premium subsidies have the advantages of low costs, convenience, and operability. It is a demand-side incentive with greater certainty of fiscal spending and stable inventive effects, which is suitable for use by government characterized with weak fiscal discretion [45]. By alleviating the financial burden on policyholders, the premium subsidy can improve their willingness to insure index-based insurance [46]. Consequently, the premium subsidy significantly accelerates the recovery and development of the affected areas, while allowing the policyholders to concentrate on productive activities [47]. However, some scholars do not consider subsidy to be a good tool. Ye et al. [48] believe the subsidy will decrease the revenue of agriculture production by pulling up the price of agricultural products. Ricome et al. [34] suggested insurance subsidies may be not the best use of financial resources, while lower credit rates and fertilizer subsidies can improve the utilities of farmers significantly. Although the absence of premium subsidies decreases the surplus of agricultural producers and the surplus of consumers, resulting in an unnecessary loss of economic efficiency, the premium subsidy system should also be used with caution [49]. As MDII in China is still in its initial stage, insufficient supply and demand make the government subsidies essential, otherwise, MDII cannot survive in its infancy [50]. Hence, the design of the index-based insurance should take the premium subsidy into account [51].

Generally, most of the existing index-based disaster insurance in developed countries has been produced under a subsidized insurance scheme. For example, the rubber tree wind index insurance pilot project in Hainan in 2015 was launched with a 90% premium subsidy; and the wind index insurance for laver farming in Zhejiang was supported by a 50% premium subsidy in 201. The Fujian aquaculture typhoon index insurance pilot program got 40% subsidy from government in 2019; and the red tide index insurance for marine aquaculture in Fujian 2020 received a 40% premium subsidy [52]. Evidently, premium subsidies are inconsistent with marine disaster index insurance practices. Scholars have reached a consensus that an appropriate level of subsidies promotes the development of index-based insurance [53,54]. Specifically, a subsidy that is negligible will render it difficult to reach the desired goal of encouraging the development of index-based insurance, while an excessive subsidy will cause pressures for government fiscal health to compromise the index-based insurance development [55]. Therefore, any insurance subsidy policy needs to be designed carefully such that it is ex-ante cost-effective while minimizing disincentives and not generating financial burden on the government [56].

Since the pilot marine disaster index insurance in China has begun recently, few scholars have researched the design and premium subsidies for marine disaster index insurance in China. Nevertheless, some scholars have explored other index-based disaster insurances. For instance, Wang et al. [57] established a comprehensive climate risk index for evaluating climate risks based on observations from 2288 meteorological stations and meteorological disasters in China. Torabi et al. [58] estimated the weather index insurance rates for apples based on information from the Iranian Agricultural Production Organization and local weather stations on apple production and weather variables from 1987 to 2016. Raviv [59,60,61] designed a claim payment trigger mechanism for earthquake index insurance in a dual Pareto optimal utility scenario for both the affected and the underwriter under the CARA utility framework. These studies provide useful references for exploring the design and the subsidy for marine disaster index insurance schemes.

The objective of this paper is to explore an optimal subsidy scheme for marine disaster index insurance for China. Unlike the previous studies, this paper advances knowledge of the literature in three aspects. First, this study explores the value of marine disaster index insurance based on marine disaster risk loss decomposition, which provides the groundwork for modelling the subsidy scheme of MDII. Second, this paper proposes a novel model MDII subsidy scheme, based on which the optimal trigger scheme and the optimal subsidy scheme are provided. The proposed model facilitates the design and measurement of subsidy schemes for disaster index-based insurance, i.e., earthquake index insurance, flood risk index insurance, temperature index insurance, etc. Third, employing the storm surge disasters for analyses, the typical marine disasters that attacked China, this paper estimates the optimal trigger scheme and the optimal scheme for storm surge index insurance by taking wind force as the underlying risk index. The estimation proves that the model is effective in the design of a subsidy scheme for disaster index-based insurance. Moreover, the estimation results provide theoretical evidence for the optimization of the storm surge risk index insurance in China.

The rest of the paper is structured as follows. Section 2 introduces the methodologies, mainly the proposed novel model for designing index-based insurance and the subsidy scheme. Section 3 describes the assessment of the model and the results based on the case of storms in China. Section 4 presents the discussions, and the conclusions, limitations, and future research are organized in Section 5.

2. Methodology

In this section, the related model and theories of the marine disaster index insurance model are elaborated. A novel index insurance with a subsidy for marine disasters to estimate the trigger scheme and the subsidy scheme is presented.

Table 1. Main notations.

Notations	Definitions
MDII	Marine disaster index insurance.
$L$	The total economic losses caused by the marine disaster.
$L_i$	The economic losses caused by the marine disaster that suffered by individual i.
$S$	The total impact factors of individuals on the economic losses.
$X_i$	The impact factors of individual i on the economic losses.
$I$	Impact of the physical characteristics in a marine disaster on the economic losses.
$f\left(I\right)$	Claim payments of marine disaster index insurance.
$P$	Total premiums of marine disaster index insurance.
$g$/$G_P$	The proportion/total amount of subsidy in premium.
$\delta$	The proportion of victims that choose to insure marine disaster index insurance.
$h$	The number of attacked individuals of marine disasters in coastal areas.
$w_P$/$w_G$/$w_I$	The initial wealth of the victims/government/insurance firms.
$U$/$G$/$V$	The utility of the victims/government/insurance firms.
$W_G$/$W_I$	Minimum utility that government/insurance firm is willing to participate in MDII.
$\alpha$/$\beta$/$\gamma$	The absolute risk aversion coefficients of the victims/government/insurance firms.
$k\left(x\right)$	Probability density function of continuous random variable $I$ on $\left(0,T\right)$, $T\le \infty$.

lists the main notations in subsequent mathematical models.

2.1. Value of Marine Disaster Index Insurance Product

The marine disaster index is composed of two parts: risk identification parameters and loss assessment parameters, which reflect the degree of the risk and the damage of a single marine disaster respectively [62]. Taking storm surge disaster as an example, risk identification parameters mainly include three aspects of information. The first is the environment. It mainly includes the risk information regarding the natural, social, and economic environment, from which it can be described through the risk classification of marine disasters. Second, the disaster formative factors mainly include the center of the maximum wind, the scope, the duration of the damage, the maximum rise in water caused by landfall, and the amount of rainfall, which can be obtained in real-time through third-party observation agencies, e.g., monitoring stations. Third, the characteristics of the disaster affected body mainly refer to the risk information on the effectiveness of disaster prevention and mitigation projects in coastal areas, the risk resistance, and the replacement price of enterprise plants and residential housing, etc. The marine disaster loss index should capture the information in these three aspects so that it can reflect the actual economic losses caused by marine disasters. In another word, in the marine disaster loss index, identification information of these three aspects is the input variable, and the actual marine disaster economic loss is the output variable [63]. Therefore, the marine disaster loss index can be written as Equation (1).

$$Index=f\left(E,F,B,\epsilon \right)$$

(1)

where E is the information about the environment, F is the information about marine disaster formative factors, B is the information of the characteristics of the disaster affected body, and $\epsilon$ is the stochastic perturbation factor contains other information, which is a zero-mean random error.

In index-based insurance, the unique index-triggered claim payment scheme means that its claim payment depends on the disaster formative factors F or the marine disaster loss index, rather than the actual loss. To simplify the derivation process, the marine disaster loss index is replaced here by the marine disaster loss fitting index.

On the other hand, to simplify the value of MDII, the marine disaster loss index shown in Equation (1) can be decomposed into two parts: information of the disaster formative factors, mainly the physical characteristics of the marine disaster, and the other information. Accordingly, suppose that there are h individuals (residents or businesses) in a coastal area which are attacked by marine disasters frequently. Following Schlesinger [64], the losses suffered by attacked individual i in a marine disaster can be decomposed into the multiplication of two independent parts: one is the scale of loss determined by the situation of the attacked individual ($X_i$), such as the capability of fishermen’s fishing boats to withstand typhoon attacks; the other is the physical characteristics of the marine disaster ($I$), such as the wind force at the centre of a typhoon. Hence, it can be expressed in a function as Equation (2).

$$L_i=I\cdot X_i$$

(2)

where $I$ and $X_i$ are mutually independent stochastic variables. Consequently, the total economic loss caused by the particular marine disaster is

$$L={\displaystyle \sum _{i=1}^h{L}_i}=I\cdot {\displaystyle \sum _{i=1}^h{X}_i}=I\cdot S$$

(3)

where $S$ is the total impact factors of individuals on the economic losses, $S={\displaystyle \sum _{i=1}^h{X}_i}$, $I$ and $S$ are mutually independent stochastic variables.

Given the index-triggered claim payment scheme of index insurance, the claim payment of MDII depends on one or several physical characteristics of the marine disaster. Accordingly, the claim payment is only determined concerning $I$ rather than $S$. Thus, the MDII can be triggered by the economic losses index $I$ that is calculated based on the parameters of the physical characteristics of the marine disasters. Here, the respective total claim payment is

$$f\left(x\right)\ge 0$$

(4)

where $f(\cdot )$ is a monotonically increasing function that holds when $x\ge 0$.

MDII is designed and operated by insurance firms, which are its underwriters. Meanwhile, the MDII satisfies two conditions. First, the attacked individuals in coastal areas make their insurance decisions according to the benefits from insuring MDII products. Second, the insurance firm prices MDII products according to the principle of expected loss premium. For the simplicity of the model, we focus on the pure premium and omit the premium loading temporarily. The pure premium is the main part of the insurance premium, which is determined by the expected loss [65]. Thereby, given the marine disasters losses analysis, the value of MDII products for an insurance firm can be calculated by the total premiums and total claims, as shown in (5).

$${\displaystyle {\int }_0^T\left[P-f\left(x\right)\right]}k\left(x\right)dx$$

(5)

where $k\left(x\right)$ is the probability density function of a continuous random variable on $\left(0,T\right)$ with $T\le \infty$.

2.2. MDII Model with Subsidy

Inspired by the theoretical framework of [59,60,61], we develop a novel index insurance model with a subsidy for investigating the optimal solution for the dilemma of insufficient demand and supply in MDII. Among the various government subsidy tools, premium subsidy is the most common tool used by the government to intervene in the market and make the insurance scheme popular. In the premium subsidy arrangement, the government shares $g$ the premium $P$ with the policyholders who are potential marine disaster victims. The total subsidy is $G_P=gP$.

Due to the different education experience, insurance experience, and awareness of the risks in marine disasters, people may have different attitudes towards MDII [66,67,68]. Consequently, their preferences for becoming policyholders of MDII are inconsistent. Assuming that the proportion of marine disaster victims who choose to insure MDII is $\delta$, the $\delta h$ potential victims among $h$ potential victims of marine disasters in the coastal area choose to insure MDII. Meanwhile, the premium per potential victim is $P/h$, and the claim payment per potential victim is $f\left(I\right)/h$ when MDII is triggered, respectively. Accordingly, the total premiums realized are $\delta P$, and the total claim payment after disasters is $\delta f\left(I\right)$. Hence, the actual subsidy to policyholders is $G_P=g\delta P$. Therefore, the total utilities of the victims, the government, and the insurance firms after the marine disasters are

$$U\left[w_P-\delta P+G_P-I\cdot S+\delta f\left(I\right)\right]$$

(6)

$$G\left[w_G-G_P+\delta f\left(I\right)\right]$$

(7)

$$V\left[w_I+\delta P-\delta f\left(I\right)\right]$$

(8)

where the utility functions satisfy $U^{\prime }(\cdot )>0$, $U^{″}(\cdot )<0$, $G^{\prime }(\cdot )>0$, $G^{″}(\cdot )<0$, $V^{\prime }(\cdot )>0$, $V^{″}(\cdot )<0$.

It should be noted that the MDII claims costs of insurance firms are ignored here. Different from traditional insurance products, no claim investigation and actual losses assessment is required for MDII claims [26,27,28]. Specifically, under the index claim payment scheme, once the MDII is triggered, the insurance firms need to pay each policyholder following the established claim payment plan, while eliminating the claim investigation. Hence, there is no extra costs for claims. Meanwhile, MDII is designed and operated by the insurance firms, and the index-based claim payment scheme is also given by them. Moreover, claims will be performed strictly following to the claim payment scheme. Hence, there is no external cost occurred during the process of claims. Therefore, to simplify the problem, it can be assumed that the insurance firms who underwrite the MDII do not have to bear claim costs.

Drawing on the experiences of Raviv [69], the mission of MDII, as insurance of a public benefit nature, is to safeguard the property of potential attacked residents in coastal areas. The objective of designing MDII with premium subsidy is to maximize the utilities of the victims, i.e., to maximize the expected value of the victims’ utility (6). Hence, the solution of premium subsidy to MDII is equivalent to the following optimization problem.

$$\underset{g,\delta }{\max }{\displaystyle {\int }_0^{T}E\left\{U\left[w_P-\delta \left(1-g\right)P-x\cdot S+\delta f\left(x\right)\right]\right\}k\left(x\right)dx}$$

(9)

Given its constraints as

$$\{\begin{array}{l}{\displaystyle {\int }_0^{T}G\left[w_G-\delta g_{1}P+\delta f_1\left(x\right)\right]k\left(x\right)dx}\ge W_G\\ {\displaystyle {\int }_0^{T}V\left[w_I+\delta P-\delta f_1\left(x\right)\right]k\left(x\right)dx}\ge W_I\end{array}$$

(10)

where $W_G$ and $W_I$ are the minimum utilities that government and insurance firm is willing to participate in MDII, which means they will drop out of MDII if it drives their utilities less than $W_G$ or $W_I$.

From the above optimization problem and its constraints, it appears that the optimal subsidy of MDII to policyholders $g_1^{*}$ and the optimal premium $P^{*}$ depend on the claim payment function $f\left(x\right)$. Therefore, the key to solving the optimal premium subsidy scheme is to clarify the optimal trigger scheme for MDII.

2.3. Trigger Scheme

As analyzed above, the optimal trigger scheme is the optimal solution to the optimization problem (9) in Section 2.2. Accordingly, given the proportion of potential victims who choose to insure, the premium, and the subsidy to optimize the trigger scheme function, the optimal trigger scheme will also be the function of these factors. Here, the optimal trigger scheme can be summarized in Proposition 1.

Proposition 1.

Subject to the constraints (10) and the given factors, including the proportion of potential victims who choose to insure, the premium, and the subsidy, the solution to the optimization problem (9), which is the optimal trigger scheme$f_{}^{*}\left(x\right)$

$$\{\begin{array}{cc}{f}_{}^{*}\left(x\right)=0& x\le \overline{x}\\ f_{}^{*}\left(x\right)>0& x>\overline{x}\end{array}$$

(11)

Meanwhile, when $x>\overline{x}$, the optimal trigger scheme satisfies

$$f_{}^{*}{}^{\prime }\left(x\right)=\frac{1}{\delta }\frac{\frac{E\left[U^{″}\left(A_1+\delta g_{1}P-x\cdot S\right)S\right]G^{″}\left(C_1-\delta g_{1}P\right)}{E\left[U^{\prime }\left(A_1+\delta gP-x\cdot S\right)\right]G^{\prime }\left(C_1-\delta gP\right)}}{\left\{\begin{array}{c}2\frac{E\left[U^{″}\left(A_1+\delta gP-x\cdot S\right)\right]G^{″}\left(C_1-\delta gP\right)}{E\left[U^{\prime }\left(A_1+\delta gP-x\cdot S\right)\right]G^{\prime }\left(C_1-\delta gP\right)}\\ +\frac{G^{″}\left(C_1-\delta gP\right)V^{″}\left(B_1\right)}{G^{\prime }\left(C_1-\delta gP\right)V^{\prime }\left(B_1\right)}+\frac{E\left[U^{″}\left(A_1+\delta gP-x\cdot S\right)\right]V^{″}\left(B_1\right)}{E\left[U^{\prime }\left(A_1+\delta gP-x\cdot S\right)\right]V^{\prime }\left(B_1\right)}\end{array}\right\}}$$

(12)

where$A_1=w_P-\delta P+\delta f_{}^{*}\left(x\right)$,$B_1=w_I+\delta P-\delta f_{}^{*}\left(x\right)$,$C_1=w_G+\delta f_{}^{*}\left(x\right)$.

The proof of Proposition 1 is summarized in Appendix A.

Proposition 1 presents the optimal trigger scheme of MDII obtained from the optimization problem (9), which indicates the functional relationship satisfied by the claim payment scheme with respect to the given factors, including the subsidy. This is the precondition for further exploitation of the optimal subsidy scheme for MDII. Following [61], we assume that the victims of marine disasters, the government, and the insurance firms’ utility function with constant absolute risk-averse utility function form, which means that $U(\cdot )$, $G(\cdot )$, $V(\cdot )$ are CARA utility functions. According to the explanation by Arrow (1965), their constant absolute risk-averse parameters satisfy (13).

$$\alpha =-\frac{U^{″}\left(x\right)}{U^{\prime }\left(x\right)};\beta =-\frac{V^{″}\left(x\right)}{V^{\prime }\left(x\right)};\gamma =-\frac{G^{″}\left(x\right)}{G^{\prime }\left(x\right)}$$

(13)

At the same time, their first-order derivative can be written as (14).

$$U^{\prime }\left(x\right)=C_{\alpha }{e}^{-\alpha x};V^{\prime }\left(x\right)=C_{\beta }{e}^{-\beta x};G^{\prime }\left(x\right)=C_{\gamma }{e}^{-\gamma x}$$

(14)

where $C_{\alpha }$, $C_{\beta }$, and $C_{\gamma }$ are positive constants.

Consequently, the optimal trigger scheme satisfies (15).

$$f_{}^{*}{}^{\prime }\left(x\right)=\frac{1}{\delta }\frac{\alpha \gamma E\left[e^{\alpha xS}S\right]/E\left[e^{\alpha xS}\right]}{2\alpha \gamma +\beta \gamma +\alpha \beta }$$

(15)

Considering that the optimization problem and the constraints have finite expectations, the moment generating function $M_S\left(x\right)$ of the random variable S is also finite, which can be written as $M_S\left(\alpha x\right)=E\left(e^{\alpha xS}\right)$. Accordingly, the analytic solution of the optimal trigger scheme can be rewritten as (16).

$$f_{}^{*}{}^{\prime }\left(x\right)=\frac{\gamma }{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}\log \left(\frac{M_S\left(\alpha x_2\right)}{M_S\left(\alpha x_1\right)}\right)$$

(16)

Since $S={\displaystyle \sum _{i=1}^h{X}_i}$, where $X_i$ is independent to each other, and it is identically distributed with $X$, its moment generating function is $M_S\left(x\right)={\left[M_X\left(x\right)\right]}^h$. Hence, in the case of $x>\overline{x}$, the optimal trigger scheme will be (17). It shows the relationship between the claim payment that the policyholder should receive and the variable $x$ if the conditions for Pareto optimality are satisfied.

$$f_{}^{*}\left(x\right)=\frac{\gamma h}{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}\log \left(\frac{M_X\left(\alpha x\right)}{M_X\left(\overline{\alpha }\overline{x}\right)}\right)$$

(17)

Therefore, the Pareto optimal claim payment trigger scheme for MDII under CARA utility should be (18).

$$\frac{\gamma }{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}\log \left(\max \left\{\frac{M_X\left(\alpha I\right)}{M_X\left(\overline{\alpha }\overline{I}\right)},1\right\}\right)$$

(18)

The claim payment trigger scheme has a deductible, where $\overline{I}$ is the threshold value for marine disasters characteristics. For instance, when the marine disaster index is $I$ is taken as wind level in storm surges, $\overline{I}$ can be defined as the wind level threshold that causes damage. Moreover, the claim payment of MDII is related to the marine disaster index $\overline{x}$ and the proportion of the potential victims that chose to insure $\delta$, but not to the level of premiums $P$ or the subsidy $g$.

Furthermore, given $\overline{x}=0$, the trigger scheme of MDII is (19).

$$f_{}^{*}\left(x\right)=\frac{\gamma }{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}\log \left[M_S\left(\alpha x\right)\right]$$

(19)

However, the above trigger scheme does not always satisfy the constraints (10) when the $W_G$ and $W_I$ take different values. The results show that the optimization problem described in (9) alone does not yet lead to a unique Pareto-optimal trigger mechanism for MDII. To further solve the problem, we introduce a basis difference minimization constant additionally, which is a unique characteristic of index-based insurance, to explore the Pareto optimal trigger scheme. In line with the meaning of the basis risk of the index-based insurance, the total basis risk of MDII can be measured by the difference between the total claim payment $f\left(I\right)$ and the total losses of the victims $I\cdot S$. Hence, $\overline{x}$ in the optimal trigger scheme have to meet the basis risk minimization constraint, which can be illustrated as (20).

$$\underset{\overline{x}}{\min }\left\{{\left[\frac{\gamma \log \left[M_S\left(\alpha I\right)\right]}{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}-\frac{\gamma \log \left[M_S\left(\alpha \overline{x}\right)\right]}{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}-I\cdot S\right]}^2\right\}$$

(20)

Considering that the optimization problem $\underset{F}{\min }E\left\{{\left[F\left(I\right)-I\cdot S\right]}^2\right\}$ for any function $F(\cdot )$ is equivalent to $\underset{F}{\min }E\left\{{\left[F\left(I\right)-I\cdot {\mu }_S\right]}^2\right\}$, where ${\mu }_S$ is the expectation value of the random variable $S$ (${\mu }_S=E\left[S\right]$), the optimal solution to the optimization problem ${\overline{x}}^{*}$ can be obtained by (21).

$$\log \left[M_S\left(\alpha {\overline{x}}^{*}\right)\right]=E\left\{\log \left[M_S\left(\alpha I\right)\right]\right\}-\frac{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}{\gamma }{\mu }_I\cdot {\mu }_S$$

(21)

where ${\mu }_I=E\left[I\right]$, ${\overline{x}}^{*}$ is the expectation value of the random variable $I$. In addition, given the optimal solution is non-negative, we can get ${\overline{x}}^{*}=0$ with the condition of $E\left\{\log \left[M_S\left(\alpha I\right)\right]\right\}\le \frac{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}{\gamma }{\mu }_I\cdot {\mu }_S$. Since the random variables satisfy $L=I\cdot S$ and ${\mu }_L={\mu }_I{\mu }_S$, the condition can be written as $E\left\{\log \left[M_S\left(\alpha I\right)\right]\right\}\le \frac{\delta \left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)}{\gamma }{\mu }_L$. Accordingly, ${\overline{x}}^{*}$ is positive when the condition cannot hold.

In sum, there are two critical steps to design the optimal trigger scheme of MDII with subsidy. First, determine the optimal trigger index level ${\overline{x}}^{*}$. Second, find the optimal trigger scheme function $f_1^{*}\left(x\right)$ that can make the utility of the policyholder, the government, and the insurance firms reach the Pareto optimal state, as well as the constraint of minimizing the basis risk.

2.4. Subsidy Scheme

The purpose of government subsidy for MDII is to activate and promote the development so as to strengthen the role of MDII in managing the risk of marine disasters. Therefore, it is reasonable to assume that insurance firms also underwrite MDII with the primary goal of serving society rather than making profits. Accordingly, the optimal premium subsidy $g_{}^{*}$ and the optimal premium $P^{*}$ should fulfill the conditions (30) and (31).

$$E\left\{G\left[w_G-\delta g_{}^{*}{P}^{*}+\delta f_{}^{*}\left(x\right)\right]\right\}=G\left(w_G\right)$$

(22)

$$E\left\{V\left[w_I+\delta P^{*}-\delta f_{}^{*}\left(x\right)\right]\right\}=V\left(w_I\right)$$

(23)

where $f_{}^{*}\left(x\right)$ is the optimal trigger scheme of MDII.

By introducing the utility functions $G$ and $V$ into the conditions, the optimal premium subsidy $g_{}^{*}$ and the optimal premium $P^{*}$ can be obtained.

$$P^{*}=\log \left[M_{f_{}^{*}\left(x\right)}\left(\beta \delta \right)\right]/\beta \delta$$

(24)

$$g_1^{*}=\beta \log \left[M_{f_{}^{*}\left(x\right)}\left(\gamma \delta \right)\right]/\gamma \log \left[M_{f_{}^{*}\left(x\right)}\left(\beta \delta \right)\right]$$

(25)

Therefore, $\left(f_{}^{*}\left(x\right),g_{}^{*}\right)$ is the optimal subsidy scheme for marine disaster index insurance.

3. Estimation and Results

3.1. Estimation Design

As analyzed in Section 2, the key to estimating the subsidy scheme is estimating the trigger scheme, in which the moment generating function $M_S(\cdot )$ of the random variable $S$ is one of the important parts in the estimation. However, the direct observations of the random variable $S$ are not available from the statistics, which makes it difficult to estimate the moment generating function $M_S(\cdot )$ directly. Therefore, this paper takes the storm surge, which has a high frequency of occurrence, heavy losses, and sound statistics, as an example to carry out the estimation.

Accordingly, employing the actual direct economic loss data that are available from storm surge disasters, this paper decomposes the loss variable $L$ based on the relationship between the variables $L$ and $S$. Meanwhile, considering the storm surge risk index $I$ has a correlation with the characteristics of the storm surge disaster itself only, $I$ can be separated from the direct loss by using the deconvolution approach. After excluding the effect of population, the loss variable will be (26).

$$\log \overline{L}=\log I+\log \overline{X}$$

(26)

where $\overline{L}$ and $\overline{X}$ are the mean value of the $L_i$ and $X_i$, in which $\overline{X}=S/h$,$\overline{L}=I\overline{X}$, respectively.

Assuming that the storm surge risk index is built based on $m$ observed storm surge risk characteristics, the index can be written as $d\cdot d\left(x_1,x_2,\cdots x_m\right)$. Consequently, the variable $\log I$ is a function of $d$. Hence, (26) can be written as (27).

$$\log \overline{L}=q\left(d\right)+\log \overline{X}$$

(27)

Meanwhile, the index $d$ is not accurate due to the unavoidable error that occurred in the observation of storm surge risk characteristics, which can be written as (28).

$$d_P=d_T+\epsilon$$

(28)

where $d_P$ is the estimated real storm surge risk index based on the observed indicators; $d_T$ is the theoretical storm surge risk index calculated based on the observed indicators; $\epsilon$ is the observation error.

Following [70] and [61], the function $q(\cdot )$ can be obtained by the non-parametric estimation approach, as shown in (29).

$$\widehat{q}\left(x\right)={\displaystyle \sum _j\log L_{j}K\left(\frac{x-d_j}{h}\right)}/{\displaystyle \sum _j{L}_{j}K\left(\frac{x-d_j}{h}\right)}$$

(29)

where $h$ is the bandwidth; $K\left(x\right)=\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }\exp \left(-itx\right)\frac{{\phi }_K\left(t\right)}{{\phi }_{\epsilon }\left(t/h\right)}}dt$, $K$ is a random variable with some kernel functions the probability density; ${\phi }_K(\cdot )$ denotes the characteristic function of the random variable $X$.

After estimating (29), we can get a sample set of losses per capita, based on which the distributions of the variables $\overline{X}$ and $S$ can be further inferred.

$$\overline{X}=\overline{L}\cdot \exp \left(-\widehat{q}\left(x\right)\right)$$

(30)

3.2. Data and Processing

In this paper, we focus on the direct economic losses of marine disasters in China. To ensure the authenticity and reliability of the data, the official government statistics data are collected for analyses. Hence, the storm surge disaster economic loss, storm surge parameters, affected population data, and other related data used in this paper are compiled from the China Meteorological Disaster Statistical Yearbook, China Marine Disaster Bulletin, and China Statistical Yearbook. Moreover, the data on the storm surge that attacked China in 2020 are collected from the Natural Disaster Overview of China (2020) released by China’s Ministry of Emergency Management.

Since storm surges may attack more than one place at the same time, there are usually multiple data records for a single storm surge. To ensure no duplication of data, the multiple observation records for the same storm surge are combined, while the storm surge with missing records is excluded, including those records with zero attacked population, missing affected population, and no economic losses records, etc. Also, to remove the impact of economic development on the direct economic losses, the data are corrected for inflation using 2019 as the base period. Therefore, there are 125 observations for our data analyses. The descriptive statistical characteristics of the sample are summarized in

Table 2. Descriptive statistical analysis of the sample.

Variable	Number	Mean	SD	Min	Max
Max wind speed (m/s)	124	34.60161	11.3265	15	75
Max wind force (force)	125	11.8	2.514474	7	17
Direct economic loss (billion CNY)	125	8.384164	13.14854	0.9000	70.68326
Log (Direct economic loss)	125	2.923868	2.152782	−2.4079	6.5608
Direct economic loss per capita(CNY)	125	2680.371	4733.575	21.6374	43689.32
log(Direct economic loss per capita)	125	7.265928	1.126201	3.0744	10.6849
Attacked population	125	4125772	5520115	206	31940400
log(Attacked population)	125	14.07864	1.968223	5.3300	17.2800

. From 2005 to 2019, the maximum wind force of the storm surge in China ranged from 7 to 17, and the maximum wind speed ranged from 15 to 75 m/s. At the same time, China suffered economic losses ranging from 0.9 billion to 70.6832 billion CNY. Obviously, China has been suffering serious storm surge attacks, which highlights that advanced reasonable management measures are needed urgently.

3.3. Results

3.3.1. Risk Aversion Coefficient

The utility function is the key to estimating the trigger scheme and subsidy scheme of MDII. As the decisive factor of the participant’s utility function, the risk aversion coefficient needs to be estimated first. Following [71], this paper takes the typhoon from the sample, which landed in Yangjiang Guangdong on May 27 in 2016, as the case to estimate the risk aversion coefficient of the policyholders. With a maximum wind force of 7 and a maximum wind speed of 15 m/s, it attacked 145,000 of people in Guangdong and caused 60 million CNY direct economic losses, and the direct economic loss per capita is 452.13 CNY. According to the previous analysis, the absolute risk aversion coefficient of the storm surge victims (i.e., policyholders of MDII) is $\alpha$. Assuming that the risk premium of policyholders is $\theta$ ($0<\theta <1$), the utility function of the policyholder is satisfies (31).

$$\left(1-p\right)U\left(w\right)+pU\left(w-\overline{L}\right)=U\left(w-\theta \overline{L}\right)$$

(31)

where $p$($0<p<1$) is the probability of a storm surge occurring.

Based on the properties of the CARA-utility function, the risk premium $\theta$ can be obtained as (32).

$$\theta =\frac{\log \left\{1+\left(e^{\alpha \overline{L}}-1\right)p\right\}}{\alpha \overline{L}}=\frac{\left(e^{\alpha \overline{L}}-1\right)p}{\alpha \overline{L}}+o\left(p\right)$$

(32)

It is obvious from (32) that the risk premium of policyholders $\theta$ is a strictly monotonic increasing function of $\alpha$. Accordingly, the inverse function of the risk premium with respect to the absolute risk aversion coefficient of the policyholders exists and is unique, which is also a strictly monotonically increasing function.

Based on the data sample, it can be calculated that the probability of the maximum wind force 7 is 0.11476. Thus, it can be obtained that the risk premium is an inverse function with respect to the policyholders’ absolute risk aversion coefficient, which is shown in Figure 1.

As shown in Figure 1, the absolute risk aversion of the policyholders of MDII in China’s coastal areas is positively correlated with their risk premiums, which means that the higher the storm surge risk premium, the larger the absolute risk aversion coefficient of policyholders. Considering the low willingness of coastal residents to accept the high-risk premium of storm surges, this paper employs the risk premium $\theta =0.1150$ as the study of [61] to design the trigger scheme of MDII. Thus, the absolute risk aversion coefficient of policyholders in coastal areas of China can be calculated to be $\alpha =0.0000074$. Accordingly, the estimation of the trigger scheme and the subsidy scheme of MDII is proceeded with $\alpha =0.0000074$ and $h=14.5$.

3.3.2. Results of Trigger Scheme

Concerning the existing pilot practice cases of MDII, this paper explores the Pareto optimal claim payment trigger scheme of storm surge index insurance with wind force as the risk index. Thus, the storm surge risk index is taken as the observed wind forces. Hence, the non-parametrical regression shown in (29) can be estimated based on the sample. In the estimating process, the observation error of the wind forces is assumed to be a normal random variable with a mean of 0 and a standard deviation of 1. Above, the regression is estimated by inverse convolution kernel density estimation through R [72]. The results of the estimated Pareto optimal trigger scheme model are shown in the red curve in Figure 2.

Based on the regression fitting results, a sample of variables $\overline{X}$ is calculated according to (30), of which the probability density is shown in Figure 2. Observing Figure 2, the new sample distribution is fitted with a Gamma distribution, which has a shape parameter of 0.9266, a scale parameter of 2.0252, and a rate parameter of 0.4938. The probability density fitting curve for the Gamma distribution is shown as the red curve in Figure 3. Comparing the probability density histogram of the sample of $\overline{X}$ with the fitted curve of the estimated Gamma distribution, it can be found that the fitting function fits the distribution of $\overline{X}$ very well. Meanwhile, the results of the Kolmogorov–Smirnov test show that the fit of the sample of $\overline{X}$ is $H=0$ with p-value 0.0176, which indicates the sample $\overline{X}$ can be approximated as a randomly drawn subsample from the estimated Gamma distribution sequence at 1% significance. The estimation results are summarized in

Table 3. Estimation results.

Risk Aversion Coefficient		GAMMA Distribution
Parameters	Estimated Value	Parameters	Estimated Value
$q$	0.11476	shape parameter	0.9266
$\alpha$	0.0000074	scale parameter	2.0252
$\theta$	0.1150	rate parameter	0.4938
$h$	14.5	Kolmogorov-Smirnov test (p-value)	0.0176

.

Thus, the moment generating function of $S$ can be estimated as (33).

$$M_S\left(t\right)={\left(\frac{\beta }{\beta -ht}\right)}^{\alpha }={\left(\frac{0.4938}{0.4938-ht}\right)}^{0.9266}$$

(33)

where $t\in \left(-\infty ,\frac{0.4938}{h}\right)$.

Based on the above results, the constraint of minimizing the basis difference for testing the Pareto optimal claim payment trigger mechanism for MDII can be calculated, which is shown in (34).

$$E\left\{\log \left[M_S\left(\alpha I\right)\right]\right\}=-0.65383-0.9266\times E\left\{\log \left[0.4938-h\alpha \exp \left(\widehat{q}\left(d\right)\right)\right]\right\}$$

(34)

where $h\alpha =0.0001073$, $\underset{j}{\min }\left[\exp \left(\widehat{q}\left(d_j\right)\right)\right]=797.5179$. Thus, the condition $h\alpha <\underset{j}{\min }\left[\exp \left(\widehat{q}\left(d_j\right)\right)\right]$ is satisfied.

According to the estimated result of $\widehat{q}\left(d\right)$, the value of (34) is 0.36971.

$$\begin{array}{l}E\left\{\log \left[M_S\left(\alpha I\right)\right]\right\}\\ =-0.65383-0.9266\times E\left\{\log \left[0.4938-h\alpha \exp \left(\widehat{q}\left(d\right)\right)\right]\right\}\\ =-0.65383-0.9266\times \frac{1}{125}{\displaystyle \sum _{j=1}^{125}\log \left[0.4938-0.0001073\times \exp \left(\widehat{q}\left(d_j\right)\right)\right]}\\ =0.36971\end{array}$$

(35)

Considering that the risk aversion of insurance firms and the government is higher than that of policyholders, the parameters $\beta$ and $\gamma$ are considered to be positive while less than $\alpha$, i.e., 0.000006 and 0.000003. According to the basis difference minimization constraint of the optimal claim payment trigger scheme under the optimal subsidy, the minimum proportion of insured potential victims that can make the basis difference minimization constraint hold can be obtained.

$${\delta }^{*}=\frac{0.36971\gamma }{\left(2\alpha \gamma +\beta \gamma +\alpha \beta \right)h{\displaystyle \sum _{j=1}^{125}{\overline{L}}_j}}=26.72\%$$

(36)

It can be seen that the optimal trigger index for the storm surge risk index insurance is ${\overline{x}}^{*}=0$ when there are more than 26.72% of potential victims choose to insure. At this time, the Pareto optimal trigger scheme function is (37).

$$f_{}^{*}\left(d\right)=407303.3708\frac{1}{\delta }\left\{\begin{array}{l}-0.65383\\ -0.9266\log \left[0.4938-0.0001073\exp \left(\widehat{q}\left(d\right)\right)\right]\end{array}\right\}$$

(37)

On the other hand, in the case that no more than 26.72% of potential victims choose to insure, the optimal trigger index for the storm surge risk index insurance is ${\overline{x}}^{*}>0$, and the Pareto optimal trigger scheme function is (38).

$$f_{}^{*}\left(d\right)=407303.3708\frac{1}{\delta }\left\{\begin{array}{l}-0.99093\\ -0.9266\log \left[\begin{array}{l}0.4938\\ -0.0001073\exp \left(\widehat{q}\left(d_j\right)\right)\end{array}\right]\end{array}\right\}+563549.0169$$

(38)

In sum, the optimal trigger scheme function with wind force as the storm surge risk index is a segmentation function composed of (37) and (38), which are shown in Figure 4.

3.3.3. Results of Subsidy

This paper selects the five storm surges that attacked China in 2020 for the estimation of the optimal subsidy of storm surge risk index insurance, including “Hagupit”, “Mekkhala”, “Higos”, “Maysak”, and “Haishen”. Following the estimation results before, the optimal subsidy scheme can be calculated using (39). Accordingly, the estimated results of the optimal subsidy are shown in

Table 4. Results of the optimal subsidy for storm surge risk index insurance.

Proportion of Potential Victims Choose to Insure $\mathit{\delta }$	Optimal Subsidy Ratio $\left({\mathit{g}}_{}^{*}\right)$	Total Subsidy (Million CNY) $\left({\mathit{G}}_{\mathit{P}}\right)$
5.00%	92.54%	3821.3781
10.00%	96.85%	4728.4341
15.00%	98.00%	5030.7861
20.00%	98.54%	5181.9621
26.72%	98.81%	4799.0796
30.00%	98.81%	4274.3802
40.00%	98.81%	3205.7851
50.00%	98.81%	2564.6281
60.00%	98.81%	2137.1901
70.00%	98.81%	1831.8772
80.00%	98.81%	1602.8926
90.00%	98.81%	1424.7934

. The results indicate that the optimal subsidy is always at a high ratio of over 90%. Specifically, as the proportion of the potential victims who choose to insure increases, the optimal subsidy rises accordingly. At the same time, the optimal subsidy enters its stable state of 98.81% when the proportion exceeds the threshold of 26.72%.

$$g_{}^{*}=\frac{\beta \log \left[M_{f_{}^{*}\left(x\right)}\left(\gamma \delta \right)\right]}{\gamma \log \left[M_{f_{}^{*}\left(x\right)}\left(\beta \delta \right)\right]}=\frac{\beta \log \left(\frac{1}{N_{2020}}{\displaystyle \sum _{i=1}^{N_{2020}}\exp \left(\gamma \delta f_{}^{*}\left(d_i\right)\right)}\right)}{\gamma \log \left(\frac{1}{N_{2020}}{\displaystyle \sum _{i=1}^{N_{2020}}\exp \left(\beta \delta f_{}^{*}\left(d_i\right)\right)}\right)}$$

(39)

4. Discussion

Index-based insurance is an emerging tool for managing marine disasters related risks, but there is limited success achieved in the pilots of MDII recently. To improve this situation, this paper designs and evaluates the trigger scheme and subsidy scheme for improving marine disaster index insurance. Furthermore, by applying a storm case study, this paper estimates the trigger scheme and subsidy scheme of MDII with wind index. The results provide grounds for promoting the development of MDII and the risk management of marine disasters in China’s coastal areas.

The trigger scheme is the core of MDII, as well as the foundation of its subsidy scheme. Our results show that in the MDII with wind force as the trigger index, the Pareto optimal claim payment is positively related to wind force, which is consistent with the claim payment scheme of existing related disaster index insurance [61]. Meanwhile, the claim payments increase at a smaller speed when the insured proportion of marine disaster victims increases as the wind force becomes stronger. The reason for this is that the increasing proportion of insured potential marine disaster victims implies an improvement in the demand for the MDII, which significantly flattens the fluctuations in the claim payments. This feature of MDII significantly relieves insurance firms from the sudden surge in the pressures of claim payments once a marine disaster occurs. It also shows that the Pareto optimal claim payments of MDII are consistent with the law of large numbers.

The Pareto optimal subsidy scheme for MDII requires the government to fund the majority of the premiums. Our estimation results demonstrate that the government should pay more than 90% of the premiums in the MDII, and this value increases with the proportion of those insured in the case of potential marine disaster victims. Specifically, when 5% of the potential marine disaster victims insured MDII, the optimal subsidy ratio is 92.54%, which means that the government have to pay 92.54% of the premiums and the policyholders pay the 7.64% of the premium. If there are more policyholders, the government have to raise the subsidy ratio, until the proportion of the insured potential marine disaster victims reaches or surpasses 26.72%, the minimum proportion of insured potential victims that can make the basis difference minimization constraint hold, the optimal subsidy ratio reaches and stabilizes at 98.81%. Different from the optimal subsidy ratio, the total subsidy will decrease along with the increasing proportion of the insured potential marine disaster victims, and the largest total subsidy amount achieved at the minimum proportion of insured potential victims can make the basis difference minimization constraint hold.

In the existing pilots of MDII and fishery insurances in China, the subsidy ratios range from 20% to 60% [52]. Only the wind index insurance launched in Hainan since 2015 is approaching the optimal subsidy ratio with a 90% premium subsidy. Evidently, the subsidy ratio of MDII in pilots is lower than the optimal subsidy ratio universally. The finding demonstrates that the subsidy ratios in practices are too negligible to make the MDII market prosper and flourish. It also reveals that the inadequate government subsidy schemes are an essential reason for the current shortfalls in supply and demand for MDII, which is consistent with the opinions of [53,54]. On the other hand, the government needs to pay 98.81% of the premiums to support the development of MDII, which is determined by its catastrophe insurance features, but it also creates severe financial pressures on the government [56]. Although the optimal subsidy ratio has remained high, the optimal total subsidy afforded by the government continues to decrease as the proportion of those insured in the case of potential marine disasters increases. This indicates that the subsidy pressure on the government can be lessened when the MDII market is booming. However, a policy-based insurance scheme with a large amount of subsidy may be more conducive to the development of MDII in the early stages of its development [38].

5. Conclusions and Future Research

With the increasing frequency of extreme climate events, marine disasters are becoming increasingly severe in China. Marine disaster index insurance (MDII) has attracted extensive scholarly attention as an efficient and effective emerging tool for managing marine disaster risks. Although there are some pilots of MDII in China launched with subsidies, limited success has been achieved due to the limited subsidy support. This paper proposes a novel optimal insurance model for MDII with subsidy, which can be used as a tool to design and estimate the optimal subsidy for MDII. The novel model of MDII captures the behaviors of the policyholders, insurance firms, and the government. Specifically, employing storm surge index insurance as an example, the optimal trigger scheme and optimal subsidy scheme are designed and estimated. The results recommend that the optimal subsidy ratio for MDII in China needs to be at least 92.54%. Moreover, this value increases when there are more potential victims of marine disasters who choose to insure using MDII, while the total subsidy decreases. The subsidy pressure on the government can be lessened when MDII market is flourishing. Evidently, the subsidies for pilots of MDII in China are inadequate to meet the conditions for operation currently, which explains the dilemma of the MDII in China’s pilots. These findings provide theoretical evidence for the optimization of the MDII in China. In general, the optimal model of MDII with subsidy also contributes to the design and measurement of subsidy schemes for disaster index-based insurance, such as the earthquake index insurance, flood risk index insurance, and temperature index insurance.

The accurate design and estimation of the optimal subsidy scheme of MDII should be applied in reality. Actually, the governments involved in MDII market are not limited to a single government, but rather include local governments and the central government. Conflicting or synergistic interests may exist between the governments, which will affect the operations of MDII markets as well as the subsidy schemes. Future work could capture the multiple governments’ subsidies in the optimal model of MDII, based on which a multi-subsidy scheme can be designed and estimated. On the other hand, the unique index-based claim payments scheme of index-based insurance allows it to serve the policyholders as insurance, as well as financial derivatives. In future, the dual function of insurance and financial derivatives of MDII can be considered in the framework of the optimal model of MDII, to design and estimate the trigger scheme and subsidy scheme more accurately.