1 Introduction

Earthquake disasters have a serious impact on national economy, people’s livelihood, and even social stability. Today in China, the main forms of economic rescue after earthquake disasters are government support subsidies and public donations. In the long run, relying on economic rescue alone will lead to a serious dependence on the government for economic assistance and post-disaster reconstruction, as well as other problems such as disputes over property rights for buildings from post-disaster reconstruction. Therefore, in response to catastrophic events such as earthquakes, it is necessary to gather the forces of society and the market to form a mechanism for sustainable and effective development. In order to transfer and disperse risks reasonably and effectively, catastrophe insurance came into being. Catastrophe insurance plays an increasingly important role in risk management, disaster emergency response, economic security, and other aspects of disaster risk reduction. In recent years, the earthquake insurance system has developed gradually in China. Especially after the “5.12” Wenchuan Earthquake, life insurance companies such as China Life Insurance (Group) Company and Heng An Standard Life Company included earthquake, typhoon, flood, and other catastrophe risks into personal accident insurance. Subsequently, insurance companies such as Sinosafe General Insurance Company Limited, Ping An Property & Casualty Insurance of China, China Continent Property & Casualty Insurance Company Limited, and China Property Reinsurance Company Limited also began to pay attention to the accumulation of property losses caused by catastrophe and increased catastrophe property insurance. However, earthquake insurance is usually bundled with other disaster insurance (for example, insurance for fire, personal accidents, life assurance and property loss) by most insurance companies in most countries (Gkimprixis et al. 2021)

The key to developing the catastrophe insurance business is the determination of the catastrophe insurance rate, and the rate includes two parts: pure rate and additional rate (Li et al. 2005). The pure premium rate is the main part of the insurance premium rate, which is the ratio of the pure premium to the insured amount calculated based on the risk of economic loss caused by the disaster being covered. The additional rate is a secondary part of the insurance rate, which is the proportion of the insurance company’s operating expenses and profits to the insured amount. The method of setting catastrophe insurance rates varies from country to country. The General Insurance Rating Organization of Japan (GIROJ 2022) divides the country into six regions based on earthquake hazard, and sets basic rates in each region. California is divided into 19 regions and provided with earthquake insurance by the California Earthquake Administration (Goda et al. 2015). Higher-risk areas are assigned higher rates, and insurance rates also vary by structure type, year of construction, and number of floors, with rate values ranging from 0.36 to 9‰. Yucemen (2005) divided Turkey into four parts according to the seismic hazard of each city, and calculated the pure rate values of reinforced concrete structures to be between 0.73 and 6.37‰ in the way of standard and nonstandard construction classifications. Building catastrophe insurance rates are based on national rates in Iceland (Goda et al. 2015). Earthquake insurance is regulated by the Comisión Nacional de Seguros y Fianzas in Mexico (CNSF), and insurance rates vary by the type of building use, number of floors, and geographic area, ranging from 0.18 to 3.56‰ for residential and from 0.28 to 7.26‰ for commercial and industrial buildings (Goda et al. 2015). Most countries assign the benchmark rate for each area according to the size of earthquake disaster risk. Then the influence coefficient is determined according to different building types. Finally, the rate values of different buildings in different areas are obtained. Building distribution is important information in earthquake insurance applications (Tyagunov et al. 2006; Erdik 2021; Skoufias et al. 2021). Many scholars extracted building distribution information from statistical data, field surveys, and use of remote sensing technology and artificial intelligence. For example, Gao et al. (2012) established a county-level housing structure database using 1% provincial population sampling survey data for 2005. Sun and Zhang (2017) investigated thousands of various buildings in more than a hundred survey points in 26 provinces in China. Hermosilla et al. (2011) used the 3D point cloud reconstruction method to extract 3D information on buildings. Su et al. (2022) combined remote sensing data and local information in an integrated method to obtain building height, function, and other information. Xu et al. (2022) obtained the height of buildings in urban areas based on street view images and deep learning.

The distribution of buildings shows a trend of local aggregation and overall dispersion, and the different site conditions where structures are located lead to differences in their exposure to seismic hazards. The validity of the information represented by building exposure varies at different spatial scales. The fineness of building exposure and information validity are inversely proportional to the size of the spatial scale. Since earthquake insurance rates are related to the earthquake hazard, structural vulnerability, and exposure, the rationalization of earthquake insurance rates is closely related to the spatial scale and exposure. From the above-mentioned rate setting in different countries or regions, it can be seen that there are studies acknowledging different rates for different regions and different building structures. There is no mention of how much the spatial scale and the building exposure distribution affect the rates.

In China, the inland area is vast, and the regional economy and earthquake risk vary greatly, such that uniform earthquake insurance rates cannot be used nationwide. We can learn from the experience of earthquake insurance in Japan, California, and Turkey, among other countries, but how to choose the appropriate spatial scale and building exposure distribution becomes the key to developing a regional earthquake insurance rate system that meets China’s national conditions. Based on the analysis of earthquake hazard, building loss vulnerability, and building exposure, this study examined the impact of spatial scale and building exposure distribution on earthquake catastrophe insurance rates. For the time being, only the calculation of pure rate was considered in the study.

2 Overview of Modules for Earthquake Catastrophe Insurance

The premium rate is the ratio of the premium to the amount insured. The premium includes pure premium and additional premium, the former is the amount of compensation charged by insurance companies to make up for the expected losses of policyholders, and the latter is the expense consumed by insurance companies to avoid uncertainties in their profits (Yucemen 2013). The pure premium (\(\mathrm{PP}\)) not only depends on the economic loss of the building after an earthquake disaster, but also is limited by the deductible (\(D\)) and limit (\(L\)) (Gurenko 2004). \(D\) and \(L\) respectively represent the minimum and maximum losses that the insurer will compensate after an earthquake disaster occurs, the calculation of pure premium can be expressed by Eq. 1 (Gurenko 2004):

$${\text{PP}} = \left\{ {\begin{array}{ll} \begin{gathered} 0, \hfill \\ Loss - D, \hfill \\ \end{gathered} & \begin{gathered} {\text{for}}\, Loss \le D \hfill \\ {\text{for}}\, D < Loss \hfill \\ \end{gathered} \\ {L - D,} & {{\text{for}}\, Loss \ge L} \\ \end{array} } \right.$$
(1)

Pure premium rate (\(\mathrm{PPR}\)) is the ratio of pure premium to insured amount (\(I\)), which is represented by Eq. 2:

$${\text{PPR}} = \frac{{{\text{PP}}}}{I}$$
(2)

When \(D\) is 0 and the \(L\) is the replacement value of the building, the \(\mathrm{PPR}\) is the ratio of the expected loss (\(Loss\)) to \(I\). Therefore, the calculation of loss is the key to the determination of the pure rate of earthquake catastrophe insurance, which is jointly determined by the three modules of the seismic hazard module, the loss vulnerability module, and the building exposure module (Tyagunov et al. 2006; Erdik 2021; Skoufias et al. 2021):

$$Loss = f\left( {{\text{Harzard}},{\text{Vulnerability}},{\text{Exposure}}} \right),$$
(3)

where \(Loss\) is the expected annual losses, \(E\left(\mathrm{AL}\right)\), as:

$$E\left( {{\text{AL}}} \right) = \int_{0}^{\infty } {E\left( {{\text{DLR}}|{\text{IM}}} \right) \cdot C \cdot {\text{d}}H\left( {{\text{IM}}} \right),}$$
(4)

where \(E\left(\mathrm{DLR}|\mathrm{IM}\right)\) is the expected value of the loss ratio of a building under a given earthquake level (\(\mathrm{IM}\)), which usually refers to the loss vulnerability of building structures. C denotes the exposed replacement value of buildings. \(H\left(\mathrm{IM}\right)\) represents the seismic hazard at the location of buildings, which can be obtained by probabilistic seismic hazard assessment (PSHA) (Cornell 1968; Hu 2006; McGuire 2008), represented by a transcendental probability curve.

2.1 Seismic Hazard Module

Seismic hazard is the consideration of the seismic threat or danger to which the location of buildings may be exposed (Hu 2006). Seismic risk is generally expressed by the probability that a certain ground motion parameter is greater than or equal to the limit value due to the occurrence of an earthquake at this location within T years (Hu 2006). The core of seismic hazard analysis includes a potential seismic source model, a seismic activity model, and a ground motion attenuation prediction model (Afsari et al. 2022). In the seismic catastrophe model, seismicity is used to simulate the occurrence of earthquake events, and the activity of earthquakes is reflected by simulating the set of earthquake events at appropriate temporal scale (Woessner et al. 2015). The seismic event set can be used to calculate the impact of a single seismic event on the site for seismic risk analysis to meet the needs of earthquake insurance (Xu et al. 2021). In this study, taking Tangshan City and its surrounding areas as an example, the Monte Carlo method was directly used to simulate the stochastic seismic event set based on the seismic activity model used in the current Seismic Ground Motion Parameter Zonation Map of China (2015 version) (CEA 2015). The event set includes the occurrence time and spatial distribution attributes of earthquakes of magnitude 5 and above, such as earthquake occurrence time (year, month, day), earthquake occurrence location (longitude, latitude), magnitude, depth, fault strike, dip angle, and so on. Figure 1 shows the distribution of potential source area and the distribution of stochastic seismic event sets in Tangshan and its surrounding areas. The ground motion prediction model used is the ground motion parameter attenuation relationship given by Yu et al. (2013), and the seismic event set is used to calculate the bedrock ground motion parameters and exceedance probability of the exposed position of buildings. Then, the ground motion parameters are adjusted according to the site type data and site amplification factor (Li et al. 2019).

Fig. 1
figure 1

Distribution of potential seismic source and 100,000-year stochastic seismic events in Tangshan City and its surrounding areas

Full size image

2.2 Loss Vulnerability Module

The loss vulnerability module is an important part of the earthquake risk assessment work. It predicts the loss of buildings under the action of a certain intensity of ground motion and provides a basis for the determination of earthquake insurance rates (Zheng et al. 2016). The loss vulnerability concept includes not only the vulnerability of building structures in the field of earthquake engineering, but also the loss ratio that may occur in various damage states of building structures, as shown in Eq. 5:

$$E\left( {{\text{DLR}}|{\text{IM}}} \right) = \mathop \sum \limits_{i = 1}^{5} E\left( {{\text{DLR}}|{\text{DS}}_{i} ,{\text{IM}}} \right) \cdot P\left( {{\text{DS}}_{i} |{\text{IM}}} \right),$$
(5)

where \(E(\mathrm{DLR}|{\mathrm{DS}}_{i},\mathrm{IM})\) is the loss ratio of a building in the \(i\)-th damage state at the \(\mathrm{IM}\) level, that is, the ratio of the economic loss of the building structure after the earthquake to the replacement value of the building. Among them, \(\mathrm{IM}\) is a ground motion parameter, that is, an index of ground motion intensity, such as peak ground acceleration (PGA). \({\mathrm{DS}}_{i}\) is the \(i\)-th damage state, and there are five damage states—basically intact, slightly damaged, moderately damaged, severely damaged, and collapsed. \(P({\mathrm{DS}}_{i}|\mathrm{IM})\) is the possibility of occurrence of the \(i\)-th damage state for a building at a given level of IM, which reflects the seismic capacity of building structures.

The vulnerability of buildings is mainly represented by the vulnerability curve or the earthquake damage matrix. There are generally two methods to obtain it. One is the empirical method (Algermissen and Steinbrugge 1984; Tyagunov et al. 2006; Yakut et al. 2006; Barbat et al. 2010; Wu et al. 2015; Azizi-Bondarabadi et al. 2016). Based on the survey data of building structure damage after previous earthquakes, the damage matrix is obtained by statistical analysis of the damage situation, so as to obtain the empirical building vulnerability curve. The other way to establish a damage matrix is the theoretical analysis method (Kwon and Elnashai 2006; Asteris et al. 2014; Gkimprixis et al. 2020), which analyzes the response of building structures under different intensities of ground motion through numerical simulation, and obtains the probability of different damage states of structures (Eq. 6), and finally obtains the building vulnerability curve.

$$P_{f} = P\left( {{\text{DS}}_{i} |{\text{IM}}} \right) = P\left( {D \ge C_{i} |{\text{IM}}} \right),$$
(6)

where \({P}_{f}\) is the probability of a certain failure state of structures; \(D\) is the effect of an action, which is the quantitative value of the response of structures due to ground motion, such as the maximum displacement angle between layers; \({C}_{i}\) is the structural resistance, which is the capacity limit of structures under different failure states. The loss ratio refers to the ratio of the value required for the repair of buildings or engineering structures with different damage levels to the replacement value. Reference to the range of damage loss ratio of buildings is given in GB/T 18208.4-2011 Post-Earthquake Field Works—Part 4: Assessment of Direct Loss (NSPRC 2011).

Different factors impact vulnerability calculations, such as the construction age of buildings, the architectural function, the design code on which they are based, and their related social environment. There are thousands of different forms of building structures in China, showing the coexistence of old and new buildings, with different functions and structures, and the design parameters and performance of the structures are also different. In order to facilitate building vulnerability calculations, post-disaster loss estimations, and earthquake insurance claims, buildings can be classified so that those with comparable structural properties, similar seismic performance, and analogous functional uses are grouped together and employ the same vulnerability model. The building taxonomy was developed with the goal of classifying buildings according to their seismic vulnerability, and this taxonomy contained 13 building attributes, including the main material of construction, lateral load-resisting system, date of construction, and number of stories (Silva et al. 2022). Combining the characteristics of China’s buildings, a classification method was proposed in this study in line with the fundamental realities of China. This system takes the building structure form, social function characteristics, and spatial location characteristics as the base layer for classification. The structure type, number of stories, date of construction, building function, and seismic precautionary intensity were used as the classification standard layer. The classification is shown in Fig. 2. Finally, each classification standard layer is coupled into a building structure. It should be noted that not all coupling results have corresponding structures. For example, because China’s Code for Seismic Design of Buildings (NSPRC 2010) stipulates that the height and number of layers of masonry structures can extend up to 21 m and 7 stories separately, the highest height of adobe, rock, and timber structures does not exceed 6 m, and general brick-timber structures are generally 1−3 layers. Under various seismic precautionary intensities, there are no high-rise masonry structures and high-rise adobe, rock and timber structures with various functions. In addition, the coupling results of buildings have regional variability, and the coupling results of each region are not the same. For example, masonry structures are more common in villages and towns than in urban areas.

Fig. 2
figure 2

Building classification for earthquake catastrophe insurance in China

Full size image

2.3 Building Exposure Module

The building exposure module describes the spatial distribution of buildings exposed to seismic hazards. This module is a database composed of elements such as building location, area, and replacement value. It takes a grid or area of a certain size as a unit, and the geographic coordinate of the unit’s center point indicates each unit’s location. In each unit, the area and cost of various types of buildings are counted according to the above classification scheme.

The acquisition of building information is crucial to construction of an exposure database. In this study, artificial intelligence methods including image recognition and street view semantic analysis are used to analyze remote sensing images and point of interest (POI) or area of interest (AOI) data. Then, the obtained building information is fused with multisource data by superimposing the geographic location on information such as structure and function information, and the fortification intensity of the building is obtained. The building information is coupled to form individual buildings and grouped into one of the categories of building classification (see Sect. 2.2). Finally, the GIS spatial analysis method is used to count the individual buildings in each unit by category, so as to obtain the area and value of various types of buildings in each local area. Examples of building exposure data are shown in Table 1, and the seismic fortification intensity is 8 for all these buildings in the Tangshan area (CEA 2015). The establishment of the building exposure database lays the foundation for subsequent earthquake prevention and disaster mitigation, earthquake risk assessment, earthquake damage prediction, and post-disaster loss statistics.

Table 1 Example of building exposure data for selected districts in Tangshan City, China
Full size table

3 Pure Rate Calculation for Earthquake Catastrophe Insurance

With a vast territory, the degree of seismic risk and the level of economic development in different regions are quite different in China, the exposure of buildings is unevenly distributed, and local administrative management and financial resources are relatively independent. Therefore, the Chinese government and insurance companies should establish an earthquake catastrophe insurance system for different provincial administrative regions (autonomous regions and municipalities) to fully mobilize the enthusiasm of the relevant provinces (Wang and Wang 2013). In the determination of the pure rate of catastrophe insurance, attention must be paid to the rationality of geographical divisions. The rate of a small area cannot be used to represent the overall rate, nor can the overall rate be used to represent the partial rate. It is necessary to reflect the differences in disaster risk between regions, and also consider the convenience and fairness of the insurance industry in implementing rates. Our study compared earthquake insurance rates based on the cases that considered different spatial scales and exposure distributions. The assumptions, principles, and calculation methods for determining the rates are as follows.

3.1 General Assumptions

The general assumptions are as follows:

  • Assumption 1: When calculating the earthquake insurance rate, the sharing mechanism of catastrophe risk is not considered, and the participants are only the policyholder and the insurance company, and there is no reinsurance company.

  • Assumption 2: The rate calculation result is a pure rate, without considering additional expenses such as management costs and operating costs of the insurance company. Research on insurance mechanisms such as implementation strength, deductibles, limits, and so on is also not considered. The net premium is used to compensate the insurer for the average annual loss of buildings due to seismic damage.

  • Assumption 3: If the building cost is proportional to the regional economy, the more developed the economy, the higher the building cost.

  • Assumption 4: Seismic hazard within each 0.01° × 0.01°grid is determined by conditions at the center point of each grid.

  • Assumption 5: For the convenience of calculation, it is assumed that building structures are evenly distributed in the grid.

3.2 Determination Principles of Pure Rate

The pure rate of catastrophe insurance is determined according to the following principles:

  1. (1)

    The principle of break-even. For nonprofit purposes, when the pure rate is determined, the insurance claim amount is the same as the loss amount.

  2. (2)

    The principle of fairness. Different regions have different risks and rates, which reasonably reflect the level of seismic risk.

  3. (3)

    The principle of stability and flexibility. Earthquake insurance rates remain stable for a relatively long period of time, and are also adjusted with changes in building structure, function, region, and so on, with a certain degree of flexibility.

  4. (4)

    The principle of overall planning and consideration. On the basis of scientific pricing, according to regional economic levels, urban and rural differences, and so on, the rate is appropriately adjusted to achieve the principle of risk sharing and mutual relief.

3.3 Rate Calculation

Divide the study area into n grids with side lengths of 0.01°. If there is a certain type of building in the \(i\)-th grid, it is represented by the value of \({a}_{i}\). The insured amount for each grid is the replacement value \({c}_{i}\) of the buildings in the grid. According to the seismic hazard and structural vulnerability, the loss rate \({r}_{i}\) in each grid is obtained, and the regional loss rate is denoted by \(R\). When calculating the pure risk rate of the area, due to the inaccuracy or uncertainty of the distribution of buildings, according to the known conditions, the distribution of buildings is simplified by dividing the area into the following three situations based on what is known about building value and distribution:

  • Situation 1: It is known that the total value of buildings in a certain area is C, and the spatial distribution of buildings is unknown. The buildings in all grids in this area are regarded as one type of structures, and the insured value of all grids is the same, that is expressed as:

    $$c_{1} = c_{2} = \ldots = c_{i} = \ldots = c_{n}$$
    (7)

    The earthquake insurance rate for regional buildings is denoted as:

    $$R = \frac{{\sum r_{i} }}{n}$$
    (8)

    In this situation, the amount of insurance is all the same.

  • Situation 2: It is known that the total value of buildings in a certain area is expressed as \(C\), and the spatial distribution of buildings can be obtained according to the remote sensing images. The distribution of various types of buildings in all grids is regarded as the same. That is, the sum insured for each grid is the same, which is:

    $$c_{i} = \left\{ {\begin{array}{ll} {\frac{C}{{\mathop \sum \nolimits_{i = 1}^{n} a_{i} }}, {\text{when}} \,{\text{there}}\, {\text{are}}\, {\text{buildings}}\, {\text{in}}\, {\text{the}}\, {\text{grid}}} \\ { 0 , {\text{when}}\, {\text{there\, are}}\, {\text{no}}\, {\text{buildings}} \,{\text{in}}\, {\text{the}}\, {\text{grid}}} \\ \end{array} } \right\}$$
    (9)

    The building rate for the area can be obtained as:

    $$R = \frac{{\sum r_{i} *a_{i} }}{{\sum a_{i} }},$$
    (10)

    where \({a}_{i}\) is:

    $$a_{i} = \left\{ {\begin{array}{ll} {1,} & {{\text{when}}\, {\text{there}}\, {\text{are}}\, {\text{buildings}}\, {\text{in}}\, {\text{the}}\, {\text{grid}}} \\ {0,} & {{\text{when}}\, {\text{there}}\, {\text{are}}\, {\text{no}}\, {\text{buildings}}\, {\text{in}}\, {\text{the}}\, {\text{grid}}} \\ \end{array} } \right\}$$
    (11)

    In this situation, the insured amount is partially the same.

  • Situation 3: The total value of buildings in an area is known. Based on the spatial distribution of buildings obtained from remote sensing images, the building area in each grid is obtained, and the insured amount \({c}_{i}\) of each grid is obtained respectively. The insurance rate for the area can be expressed as:

    $$R = \frac{{\sum r_{i} *c_{i} }}{{\sum c_{i} }}$$
    (12)

    In this situation, the insured amount is different.

4 Influence of Spatial Scale and Exposure Distribution

Usually, earthquake disaster risk analysis is based on grid data of a certain size. For example, Sun and Zhang (2017) used a 30″ × 30″ grid as the calculation unit to analyze the differences in seismic capacity in different regions. In this study, based on the grid residential building data and the grid loss rate obtained, the pure rate of earthquake catastrophe insurance for residential buildings in a certain area was calculated according to the method in Sect. 3.3, which provides a reference for insurance companies to determine the rate. Taking Tangshan City, China as an example, the influence of spatial scale and exposure distribution on the pure rate of regional earthquake insurance was studied.

4.1 Basic Information About Tangshan City and Its Setting

Tangshan City consists of seven municipal districts (Caofeidian District, Fengnan District, Fengrun District, Guye District, Kaiping District, Lubei District, and Lunan District), three county-level cities (Luanzhou, Qian’an, and Zunhua), and four counties (Laoting County, Luannan County, Qianxi County, and Yutian County). It is located in the east of Hebei Province. It borders the Bohai Sea in the south and Yanshan Mountains in the north. The terrain is high in the north and low in the south, with a piedmont plain in the middle and a coastal saline-alkali land in the south. Tangshan is located at the intersection of the North China Plain seismic belt and the Tanlu seismic belt. The main active faults are the Luanxian-Laoting fault, the Ninghe-Fengnan fault, and the Guye fault. The geological structure is complex, although the degree of complexity varies from place to place. The fault zone has obvious neotectonic activity and strong seismic activity. Figure 1 shows the distribution of potential earthquake sources and 100,000-year seismic event sets in Tangshan and its surrounding areas. It can be seen that there are potential earthquake sources of magnitude 8 and above in Tangshan, and earthquakes of magnitude 7 or above may occur in Fengnan District, Lubei District, Lunan District, Kaiping District, Guye District, Luanzhou City, and Qian’an City.Footnote 1

There have been many strong earthquakes historically in the Tangshan area. Among them, the earthquake with the greatest damage and impact was the 28 July Tangshan Earthquake of magnitude 7.8 in 1976. In that earthquake, with the destruction of 14.79 million m2 of public housing and the collapse of 5.3 million private houses, the direct loss was at RMB 5.4 billion yuan. After the Tangshan Earthquake, Tangshan was rebuilt as a whole. The adobe, rock and timber structures in this area have almost disappeared. In residential buildings, there are many multi-rise brick-concrete, multi-rise steel-concrete, and high-rise buildings. There are also low-rise and old bungalows with masonry structures in the old urban area and the urban–rural transition zone.

4.2 Influence of Spatial Scale on Earthquake Insurance Rates

Although the grid-based seismic insurance rate determination method can accurately reflect the difference in seismic hazard among grids, it increases the difficulty of management in the insurance industry. Wang et al. (2008) established comprehensive indicators of urbanization and natural hazard-related disasters to comprehensively evaluate the economic system and disaster intensity of Chinese cities. They accomplished this objective by maintaining the administrative integrity of city units while placing each unit into one of five levels: strong, high, medium, low, and weak. Therefore, this study is based on the spatial scale of administrative divisions at the city and district (or county) levels. The distribution of earthquake catastrophe insurance rates within the spatial unit is uniform, and the earthquake risk is shared and distributed evenly within the administrative region. The control variable method is used to keep the exposure of buildings in each grid the same, the residential building structures are all assumed to be constructed of reinforced concrete, and the vulnerability of all various reinforced concrete structures is unified into the average of the vulnerability of different building type. Finally, the discrepancy between the standard rate in districts and counties and the grid rate is analyzed.

The rate value of the grid of 0.01° × 0.01° is obtained through calculation, as shown in Fig. 3. Then, according to the first situation in Sect. 3.3, we calculate the regional pure rate of the two spatial scales of district (or county) and city, that is, the average of grid rates. The calculated net rate distribution for districts (or counties) is shown in Fig. 4. The regional net rate value of Tangshan is 0.253‰. From Fig. 3, it can be seen that the rates in the central part of Tangshan are higher, which is consistent with the upper magnitude limit of the potential source area and the distribution of the seismic event set. It can be seen from Fig. 4 that Lubei District, Lunan District, and Kaiping District have the highest rates, followed by Fengnan District, Guye District, and Luanzhou City.

Fig. 3
figure 3

Rate distribution within the 0.01° × 0.01° grid areas in Tangshan City

Full size image
Fig. 4
figure 4

Rate distribution by districts and counties in Tangshan City

Full size image

With a uniform distribution of exposures and using average loss vulnerability, the rate for each spatial scale is the average of all grid rates at the spatial scale. By Eq. 5, the standard deviation between the standard value of the rate of the district (or county) and the prefecture-level city and the rate value of the 0.01° × 0.01° grid is obtained, which are called the standard deviation of the district (or county) and the standard deviation of the prefecture-level city respectively. The results are shown in Table 2. The standard deviation represents the degree of dispersion of rates at each spatial scale. The larger the standard deviation value, the more obvious the dispersion, and the less reliable is the result. It can be seen from Table 2 that the standard rate for the district (or county) has a smaller dispersion than the standard rate for the prefecture-level city (Tangshan City). The larger the spatial scale for determining the earthquake insurance rate, the greater the dispersion of earthquake insurance rates. Combining with Fig. 4, it can be seen that although the pure rate with the prefecture-level city as the spatial scale can represent the average rate of the districts (or counties) in Tangshan, it cannot express the differences between districts (or counties). The reason for the difference between the district (or county) rate and the prefecture-level city rate can be attributed to the uneven distribution of potential seismic sources and their impact on ground motion intensity in the Tangshan area.

Table 2 Pure rate values and their dispersion degrees at various spatial scales
Full size table

Table 2 also presents the percentage of rates that are overestimated or underestimated for cities and counties (or counties) compared to the grid data. From these data, it can be seen that if a single rate is implemented within Tangshan City without differentiation, there is a possibility that rates are overestimated in low seismic risk areas (Laoting County, Qianxi County, and Zunhua) and underestimated in high seismic risk areas (Guye District, Kaiping District, Lubei District, and Lunan District). This poses a potential problem. In areas where rates are undervalued, policyholders choose to carry home earthquake insurance, while in areas where rates are overvalued, policyholders choose not to purchase insurance. In this way, a large amount of risk exposure in the earthquake insurance business comes from the high-risk areas of earthquake exposure. Without the low-risk areas to balance this risk, the insurance company is bound to run a serious risk of capital breakage. For example, if all the buildings in Guye District were insured, the premium would be 52.39% less. If all of Lunan District were insured, the premium would be 60.27% less. Therefore, for cities with large differences in seismic risk, such as Tangshan City, differential rates should be applied between districts and counties, and even smaller spatial units need to be further divided for differential rate determination.

4.3 Influence of Building Exposure Distribution on Earthquake Insurance Rates

The exposure of buildings determines the amount of catastrophe insurance coverage and affects the assessment of damage to buildings after disasters. The same type of buildings exposed in different locations face different seismic risks, resulting in different insurance rates. Through the statistical yearbook of Tangshan, we obtained the total value of residential buildings in the city, that is, the total insured amount in Tangshan. The total insured amount is distributed according to the three situations in Sect. 2.3, creating a mechanism by which to analyze the influence of the distribution of residential building exposure in Tangshan on the insurance rate of earthquake catastrophe. The distribution of residential buildings insurance in the three situations is shown in Fig. 5.

Fig. 5
figure 5

Distribution of the insured amount (IA) of the residential buildings in three situations: a Distribution of the IA in Situation 1; b Distribution of the IA in Situation 2; c Distribution of the IA in Situation 3

Full size image

The rates for Tangshan City are 0.253‰, 0.282‰, and 0.358‰ for the three situations obtained by the calculation method in Sect. 2.3. The rate values for the first two situations are underestimated by 41.50% and 26.95% compared to the third situation. The rates of each district and county in Tangshan City are shown in Fig. 6. The regional rates of Caofeidian District, Laoting County, Lubei District, Luannan County, Luan County, Qianxi, Yutian County, and Zunhua are almost unchanged in the three situations. This is because large areas of these districts and counties have the same potential seismic source, and the buildings are distributed evenly. The actual density of buildings in Lubei District, in contrast, and the maximum magnitude of the potential seismic source area are larger, resulting in a larger rate value. The rates in Fengnan District, Fengrun District, Guye District, and Qian’an gradually increased from the first situation to the third situation. The reason is that all four regions span multiple potential seismic sources, and buildings are more densely distributed in potential seismic source areas with higher magnitude upper bounds. The actual buildings in Lunan District and Kaiping District are unevenly distributed. In the densely distributed areas, the magnitude of the earthquake event set is larger, so the rate value of the third situation is higher. The rate value in the third situation is increased by about 10% compared to the rates in the first two situations. Overall, the rates obtained in the second situation are better than those in the first situation and closer to the results obtained in the third situation. In the area where buildings are evenly distributed and only a single potential seismic source area exists, the second situation can be used as the basis for district and county rate determination. In areas where the density of buildings varies greatly and spans multiple potential earthquake source areas, rates are set based on the true distribution of buildings; otherwise, insurance companies will face operating losses due to low rates, or residents will refuse to take out insurance due to high rates.

Fig. 6
figure 6

Histogram of comparison of rates in Tangshan City under three situations

Full size image

4.4 Classification of Districts and Counties in Tangshan City

Taking Tangshan as an example, the influence of spatial scale and building exposure distribution on the insurance rate of earthquake catastrophe was studied. According to the calculated district (or county) rate values, combined with the geometric interval classification method in ArcGIS, the districts (or counties) of Tangshan are divided into three categories as shown Fig. 7. The rate categories of each district and county in Tangshan under the three types of residential building exposure distribution are as follows:

Fig. 7
figure 7

Classification of rate value for districts and counties in Tangshan City

Full size image

  • Category 1: Caofeidian District, Laoting County, Luannan County, Qianxi County, and Zunhua.

  • Category 2: Fengrun District, Luanzhou, Yutian County.

  • Category 3: Fengnan District, Guye District, Kaiping District, Lubei District, and Lunan District.

The value ranges of categories are roughly (0, 0.25‰], (0.25‰, 0.45‰], (0.45‰, 0.68‰]. These three categories can be summarized as follows from the distribution of potential seismic source areas and residential insured amounts:

  1. (1)

    The earthquake insurance rate values in the first category area are similar and smaller, because the upper limit of the magnitude of the potential seismic source in the region is small, and the residential buildings are distributed evenly.

  2. (2)

    The earthquake insurance rate value in the second category area is at the middle level. These areas are in the potential seismic source areas with an upper magnitude limit of 6.6−7.5, and there is a high density of residential buildings in areas of high seismic risk. Among them, the rate of Luanzhou is higher than the other three areas, because Luanzhou is located in the potential earthquake source area of magnitude 7−7.5, which accounts for a larger proportion of the exposed area.

  3. (3)

    The earthquake insurance rate value in the category 3 area is relatively large, mainly because it is located in a potential source area with a large upper limit of the magnitude of a potential earthquake, and the earthquake risk is high. At the same time, this area has the highest density and number of residential buildings.

The regions can be classified according to the spatial characteristics such as the uniformity, density, and quantity of the exposure distribution, and the earthquake hazard, so as to indicate the regional earthquake rate roughly.

5 Conclusion

Taking Tangshan City as the research object, this study examined the influence of spatial scale and building exposure distribution on the determination of the earthquake catastrophe insurance rate. This is a supplement to the current research system of earthquake catastrophe insurance rate determination, which makes up for the deficiency of existing research in China to a certain extent. It can also provide the necessary reference for earthquake catastrophe insurance companies to choose the spatial scale and the detailed level of exposure distribution in rate determination. The research results reveal three points:

  1. (1)

    Uniform rates cannot simply be used at the municipal level administrative units, and differential rates are used for the districts (or counties). For districts (or counties) with large differences in seismic risk, the risk areas can be further divided to apply differential rates.

  2. (2)

    In areas with diverse distribution of potential earthquake source areas and large differences in building density, there is a risk of overestimating or underestimating the pure rate of earthquake insurance when buildings are distributed evenly or partially evenly, which violates the break-even principle of rate setting.

  3. (3)

    The area can be classified according to the distribution of earthquake risk and buildings in the spatial unit, so as to evaluate the earthquake risk level of each area and roughly estimate the value range of earthquake insurance rates within the area.

Compared with the existing research results, this article presents superficially analogous but ultimately more nuanced and different insights. What is in common is that seismic insurance rates should be implemented with regional differences and focus on building exposure distribution. However, the spatial scale used to determine seismic insurance rates in China is mostly the provincial level, while this study analyzed the impact of spatial scale on rates based on the city, district (or county), and grid levels. Previous research emphasized the extraction of building exposure distribution. This study emphasized the effect of the detail of the distribution of building exposures on earthquake insurance rates.

Limitations of this study are threefold: (1) The calculation of the pure rate of earthquake insurance is based on an assumption that ignores the actual distribution of various building structures, and this has led to certain deviations in the research results; (2) When calculating the insurance rate, the influence of parameters of the catastrophe risk sharing mechanism, such as deductibles, compensation limits, and triggering conditions, is not considered; (3) The classification of districts and counties based on rates in this study is only a qualitative analysis and lacks quantitative research support. These limitations will serve as the direction of future research efforts.