the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
California earthquake insurance unpopularity: the issue is the price, not the risk perception
Adrien Pothon
Philippe Gueguen
Sylvain Buisine
PierreYves Bard
Despite California being a highly seismic prone region, most homeowners are not covered against this risk. This study analyses the reasons for homeowners to purchase insurance to cover earthquake losses, with application in California. A dedicated database is built from 18 different data sources about earthquake insurance, gathering data since 1921. A new model is developed to assess the takeup rate based on the homeowners’ risk awareness and the average annual insurance premium amount. Results suggest that only two extreme situations would lead all owners to cover their home with insurance: (1) a widespread belief that a devastating earthquake is imminent, or alternatively (2) a massive decrease in the average annual premium amount by a factor exceeding 6 (from USD 980 to 160, 2015 US dollars). Considering the low likelihood of each situation, we conclude from this study that new insurance solutions are necessary to fill the protection gap.
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Since 2002, the rate of homeowners insured against earthquake risk in California has never exceeded 16 %, according to data provided by the California Department of Insurance (CDI, 2017b, California Insurance Market Share Reports). Such a low rate is surprising in a rich area prone to earthquake risk. Consequently, several studies have already investigated homeowners' behaviour regarding earthquake insurance in California, in order to identify people who might have an interest in purchasing earthquake insurance and to understand why they do not do so. Three main variables have been observed as decisive in purchasing an earthquake insurance: the premium amount, the socioeconomic background and the risk perception (Kunreuther et al., 1978; Palm and Hodgson, 1992a; Wachtendorf and Sheng, 2002). For the premium amount, both a survey conducted by Meltsner (1978) and the statistics collected from insurance data by Buffinton (1961) and Latourrette et al. (2010) show, as expected, a negative correlation between the premium amount and the insurance adoption. Nevertheless, even uninsured homeowners tend to overestimate the loss they would face in case of a major event and feel vulnerable regardless of their income level (Kunreuther et al., 1978; Palm and Hodgson, 1992a). As they are not expecting much from federal aids, they do not know how they will recover (Kunreuther et al., 1978).
Consequently the decision to purchase insurance to cover earthquake losses is not correlated to the income level (Kunreuther et al., 1978; Wachtendorf and Sheng, 2002). Several other socioeconomic factors (e.g. the duration of residence, the neighbours' behaviour, the communication strategies of mass media, real estate agencies and insurance companies) have an impact on the risk perception (Meltsner, 1978; Palm and Hodgson, 1992a; Lin, 2013).
Pricing methods used in the real estate market do not take into account seismic risk. Indeed, based on data from the World Housing Encyclopedia, Pothon et al. (2018) have shown that building construction price is not correlated to the seismic vulnerability rating. Moreover, earthquake insurance is not mandatory in California for residential mortgage (CDI, 2019, Information Guides, Earthquake Insurance). Still, lenders for commercial mortgages are used to requiring earthquake insurance only when the probable maximum loss is high (Porter et al., 2004). However, Porter et al. (2004) have shown that taking into account earthquake risk for calculating the building's net asset value can have a significant impact in earthquakeprone areas like California.
Also, people insured against earthquake risk can receive a compensation lower than the loss incurred because most earthquake insurance policies in California include a deductible amount (i.e. at the charge of the policyholder) and are calibrated based on a total reconstruction cost, declared by the policyholder. As reported by Marquis et al. (2017) after the 2010–2011 Canterbury (NZL) earthquakes, this amount can be inadequate for the actual repair costs. Both high deductible amount (Meltsner, 1978; Palm and Hodgson, 1992b; Burnett and Burnett, 2009) and underestimated total reconstruction cost (Garatt and Marshall, 2003) can make insured people feel unprotected after an earthquake, making this kind of insurance disrepute.
Since homeowners are aware of the destructive potential of a major earthquake, the risk perception reflects their personal estimate of the occurrence (Kunreuther et al., 1978; Wachtendorf and Sheng, 2002). Indeed, most of them disregard this risk because it is seldom, even if destructive earthquake experiences foster insurance underwriting during the following year (Buffinton, 1961; Kunreuther et al., 1978; Meltsner, 1978; Lin, 2015). Consequently, earthquake insurance purchasing behaviour is correlated with personal understanding of seismic risk and is not only related to scientificbased hazard level (Palm and Hodgson, 1992a; Palm, 1995; Wachtendorf and Sheng, 2002).
With limits in earthquake insurance consumption identified, the next step is to assess the contribution of each factor to the takeup rate. Kunreuther et al. (1978) first proposed a sequential model showing that the individual's purchasing behaviour depends on personal risk perception, premium amount and knowledge of insurance solutions. In the same study (Kunreuther et al., 1978), a numerical model was developed, at the same granularity, failing however to accurately reproduce the observed behaviours. According to the authors, the model failed because many surveyed people had a lack of knowledge of existing insurance solutions or were unable to quantify the risk. Models published later (Latourrette et al., 2010; Lin, 2013, 2015) assessed the takeup rate by postal code and used demographic variables to capture the disparity in insurance solution knowledge. Nevertheless because of lack of data, they assumed the premium amount or the risk perception as constant.
The main objective of this study is to introduce a new takeup rate model for homeowners at the scale of California state. Such spatial resolution allows work on most data available in financial reports, which are numerous enough to use both the premium amount and the risk perception as variables. Different from previous studies focusing on risk pricing (Yucemen, 2005; Petseti and Nektarios, 2012; Asprone et al., 2013), this study takes into account the homeowners' risk perception and behaviour. This shift in point of view approach changes the main issue from what the price of earthquake insurance should be considering the risk level to what the acceptable price for consumers to purchase an earthquake insurance cover is. Despite results being at the state level, they bring a new framework to model earthquake insurance consumption which allows us to quantify the gap between premium amount and homeowners' willingness to pay, depending on the homeowners' risk awareness. Last, this study is also innovative by separately modelling the contribution of the risk perception and the premium amount to the level of earthquake insurance consumption.
In the first section, data are collected from several sources, processed and summarized in a new database. A takeup rate model is then developed by introducing the two following explanatory variables: the average premium amount and the subjective annual occurrence probability, defined as the risk perceived by homeowners of being affected by a destructive earthquake in 1 year. The perceived annual probability of occurrence is then studied from 1926 to present, in the next section. Finally, the last section is dedicated to analysing the current low earthquake insurance takeup rate for California homeowners.
Developing such a model, even at the state level, faces a first challenge in data collection. Despite the California earthquake insurance market having been widely analysed, data volume before the 1990s remains low and extracted from scientific and mass media publications which refer to original data sources that are no longer available. As a consequence, data description is often sparse and incomplete and values can be subject to errors or bias. Consequently, data quality of each dataset has to be assessed. Here, it is done on the basis of the support (scientific publication or mass media), the data description (quality of information on the data type and the collecting process) and the number of records. The datasets with the highest quality are used and presented in Table 1 (values are available in the Supplement).
This study focuses on the following variables about earthquake insurance policies:

the total written premium (W_{N}), corresponding to the total amount paid by policyholders to insurance companies during the year N;

the takeup rate (t_{N}), defined as the ratio between the number of policies with an earthquake coverage (Nb_{N}) and those with a fire coverage (Fi_{N});

the annual average premium (P_{N}), equal to W_{N} over Nb_{N};

the average premium rate (p_{N}) equal to the ratio between the annual average premium amount P_{N} and the average value of the good insured (here a house), later referred to as the average sum insured (ASI_{N}).
All these variables can be calculated for any set of earthquake insurance policies. Insurance companies usually classify their products as follows: the residential line of business (Res) corresponds to insurance policies covering personal goods (e.g. house, condominium, mobile home, jewellery, furniture). The homeowners line of business corresponds to the insurance policies included in the residential line of business but dedicated to homeowners. As an earthquake insurance cover can be either a guarantee included in a wider policy (e.g. also covering fire or theft risks) or issued in a standalone policy, the earthquake line of business (Eq) classifies all insurance policies exclusively covering the earthquake risk. Some insurance policies within the earthquake line of business are dedicated to professional clients and do not belong to the residential line of business. For clarity, variables hold the title of the line of business when they are not related to the homeowners line of business.
Data for all the variables listed are available only since 1996 (Table 1). In order to expand the historical database, they are also estimated in an indirect way for the following periods: 1926–1930, 1956, 1971–1972, 1976–1978 and 1989–1995, leading us to consider additional data (Table 1) and to use linear regressions (Fig. 1).
By definition of the average premium rate (p_{N}), P_{N} can be approximated by the product of p_{N} and the average sum insured (ASI_{N}). The latter is estimated from the real building cost index (RBCI_{N}), which is an economic index (base: 31 December 1979=100) capturing the evolution of the cost of building construction works (Shiller, 2018). The following linear regression has been calibrated on data from Table 1 between 2006 and 2015:
where ASI_{N} is in 2015 US dollars and R^{2} is the coefficient of determination. When the average premium rate corresponds to the residential line of business (${p}_{N}^{\mathrm{Res}}$), it can be converted into p_{N} with the following linear regression built on the CDI database between 2006 and 2016 (Table 1):
Again using the CDI database between 1996 and 2016 (Table 1), the following linear regressions have also been developed between P_{N}, t_{N}, and W_{N} and the corresponding metrics for the residential line of business (${P}_{N}^{\mathrm{Res}}$, ${t}_{N}^{\mathrm{Res}}$ and ${W}_{N}^{\mathrm{Res}})$.
Before 1996, the total premium amount for earthquake insurance, mentioned in CDI (2017b) reports, was about the earthquake line of business, and in terms of total earned premium (E_{N}). While the total written premium (W_{N}) corresponds to the total amount of premium paid by policyholders to insurance companies during the year N, E_{N} is the amount of premium used to cover the risk during the year N. To illustrate these two definitions, Fig. 2 takes the example of an insurance policy for which the annual premium is paid 1 March every year. As the amount received at year N by the insurance company (W_{N}) covers the risk until 1 March, N+1, the insurance company can use only 75 % of W_{N} during the year N (9 months over 12). By adding the 25 % of W_{N−1}, this makes the total earned premium E_{N}, for the year N.
To estimate ${W}_{N}^{\mathrm{Eq}}$ from ${E}_{N}^{\mathrm{Eq}}$ for the earthquake line of business, the following relationship has been defined based on data from the CDI (2017a, b) database and over the period 1992–2016 (Table 1):
About the difference between ${W}_{N}^{\mathrm{Eq}}$ and ${W}_{N}^{\mathrm{Res}}$, despite data (Table 1) showing a significant difference (${W}_{N}^{\mathrm{Eq}}{W}_{N}^{\mathrm{Res}}$ = USD 250 m) since 1996, this study assumes that ${W}_{N}^{\mathrm{Eq}}$ is equal to ${W}_{N}^{\mathrm{Res}}$ until 1995, i.e.
This strong assumption was used by Garamendi et al. (1992) and was verified for the loss after the 1994 Northridge earthquake reported by Eguchi et al. (1998).
Lastly, when variables cannot be inferred from other variables, they are estimated based on the annual variation for ${P}_{N}^{\mathrm{Res}}$,
and the biennial variation for ${t}_{N}^{\mathrm{Res}}$,
where ${\mathrm{\Delta}}_{X}^{Y}$ is equal to the ratio Y over X. Furthermore, the variation in the total written premium for the residential line of business (${\mathrm{\Delta}}_{{W}_{N\mathrm{1}}^{\mathrm{Res}}}^{{W}_{N}^{\mathrm{Res}}}$) is linked to ${\mathrm{\Delta}}_{{P}_{N\mathrm{1}}^{\mathrm{Res}}}^{{P}_{N}^{\mathrm{Res}}}$, ${\mathrm{\Delta}}_{{t}_{N\mathrm{1}}^{\mathrm{Res}}}^{{t}_{N}^{\mathrm{Res}}}$ and ${\mathrm{\Delta}}_{{\mathrm{Fi}}_{N\mathrm{1}}^{\mathrm{Res}}}^{{\mathrm{Fi}}_{N}^{\mathrm{Res}}}$ as follows:
As we have no data regarding the annual variation in the number of fire insurance policies for the residential line of business (${\mathrm{\Delta}}_{{\mathrm{Fi}}_{N\mathrm{1}}^{\mathrm{Res}}}^{{\mathrm{Fi}}_{N}^{\mathrm{Res}}}$) before 1996 (Table 1), it is assumed to be equal to the variation in the California population Pop_{N}:
Figures 1 and 3 illustrate the goodness of fit of the regressions developed (Eqs. 1 to 6) and the variations in the population compared to the number of fire insurance policies between 1996 and 2016 (Eq. 11), respectively.
Finally, values of P_{N}, t_{N} and W_{N} collected (Table 1) and estimated in this study (Eqs. 1 to 11) are aggregated into a new database presented in Table 2.
Financial amounts are converted into 2015 US dollars using the Consumer Price Index (US Department of Labor, 2018). The data quality of each variable is also assessed at the weakest data quality used (Table 1), by as many levels as the number of equations used to assess it (listed in the column equation ID). For example, the annual average premium P_{1926} has been calculated from the premium rate for the residential line of business ${p}_{\mathrm{1926}}^{\mathrm{Res}}$ (Table 1) and Eqs. (1) and (4). The associated data quality is C (Table 2) because the data quality of ${p}_{\mathrm{1926}}^{Res}$ is A (Table 1), downgraded by two levels for the two equations used.
The model developed in the next section uses this new database to estimate the takeup rate for the California homeowners (t_{N}) from the average premium amount (P_{N}) and another variable capturing the relative earthquake risk awareness.
In order not to presume a linear trend between the consumers' behaviour and explanatory variables (e.g. the average premium amount), this study refers to the expected utility theory (Von Neumann and Morgenstern, 1944) instead of statistical linear models. The expected utility theory is a classical framework in economic science to model the economical choices of a consumer, depending on several variables like the wealth or the risk aversion (Appendix A). Here the application field is the California homeowners instead of a single consumer. Within this framework, homeowners are assumed to be rational and making decisions in order to maximize their utility. Furthermore, their relative risk aversion is considered constant whatever the wealth because the decision to purchase earthquake insurance is independent of the income level (Kunreuther et al., 1978; Wachtendorf and Sheng, 2002). One of the most used utility functions (U) is (Holt and Laury, 2002)
where g_{N} and β are the wealth function at year N and the risk profile controlling the risk aversion level (the larger β, the higher the aversion), respectively.
Then, the average household's capital K is assumed to be equal to the average sum insured (ASI_{2015}) since the earthquake insurance consumption is uncorrelated to the wealth (Kunreuther et al., 1978; Wachtendorf and Sheng, 2002) and equal to USD 604 124 (USD 2015). Consequently, the wealth of uninsured homeowners is g_{N}=K, if no damaging earthquake occurs. Regarding the loss estimation, homeowners are mostly concerned by destructive earthquakes, defined as earthquakes which can potentially damage their home, and tend to overestimate the impact (Buffinton, 1961; Kunreuther et al., 1978; Meltsner, 1978; Lin, 2015). Furthermore, according to Kunreuther et al. (1978), insured people believe that they are fully covered in case of loss. Therefore, we assume that only uninsured homeowners expect to incur a loss equal to K after a damaging earthquake. This leads us to model the fact that uninsured homeowners expect a wealth at g_{N}=0 after a damaging earthquake. About insured homeowners, we assume that they expect to have a constant wealth at ${g}_{N}=K{P}_{N}$ since they believe to be fully covered in case of losses induced by an earthquake. Table 3 summarizes these three wealth levels, based on the occurrence of a devastating earthquake and the insurance cover.
Finally, taking into account the share of insured homeowners (t_{N}) and the results in Table 3, g_{N} can be written for all homeowners as
where ℬ_{EQ}(r_{N}) is a random variable following the Bernoulli distribution (i.e. equal to 1 if a damaging earthquake occurs during the year N and 0 otherwise). The parameter r_{N}, called the subjective annual occurrence probability, controls the homeowners' risk perception through the perceived probability of being affected by a destructive earthquake during the year N (Kunreuther et al., 1978; Wachtendorf and Sheng, 2002). As homeowners want to maximize their utility, t_{N} is solution of
where argmax stands for the argument of the maxima function (i.e. which returns the value of t_{N}, which maximizes the quantity 𝔼[U(g_{N}(t_{N}))]) and 𝔼 is the expected value of U(g_{N}(t_{N})), which depends on the random variable ℬ_{EQ}(r_{N}) (Eq. 13). Assuming that homeowners are riskaverse (i.e. β>0) and noticing that $\frac{{P}_{N}}{K}$ is very small compared to 1, ${t}_{N}^{\mathrm{Estimated}}$ is shown (Appendix B) to be equal to
As a first step, the model is calibrated with data for the period 1997–2016 corresponding to the whole activity period of the California Earthquake Authority (CEA) providing highquality data (Table 2). Created on December 1996, this is a nonprofit, statemanaged organization selling residential earthquake policies (Marshall, 2017). The stability of the insurance activity and the lack of devastating earthquakes leads us to model parameters β and r_{N}, capturing the homeowners' risk perception, by constant variables. They are estimated at ${r}_{\mathrm{1997};\mathrm{\dots};\mathrm{2016}}=\mathrm{0.027}$ % and β=0.93, using the least squared method with the generalized reduced gradient algorithm. The model (Eq. 15) fits the observed data with a R^{2}=0.79 (Fig. 4).
Furthermore, the values of the two parameters are meaningful: homeowners are modelled as “very riskaverse” (β=0.93), which is consistent with a highseverity risk like an earthquake (Holt and Laury, 2002). r_{N} is also somehow consistent with the value that can be calculated from a hazard analysis, as presented later.
The model calibration between 1997 and 2016 is not appropriate for other periods corresponding to different seismic activity and insurance economic context. The next section investigates how the homeowners' risk perception has changed since 1926 in order to adapt the model to any period.
The homeowners' risk perception is controlled by both the risk materiality (what kind of earthquakes are expected to occur?) and the risk tolerance (how much are homeowners ready to lose?). The associated parameters in the model are r_{N} and β. As we cannot differentiate the one from the other, β is assumed constant in this study and all the variations in the homeowner's risk perception are passed on to r_{N}.
According to the model and the relationships developed (Eqs. 3–5, 10–11 and 15), the variations in the total written premium amount per capita (W_{N}∕Pop_{N}) depend on the variation in the average premium amount (P_{N}) and the subjective annual occurrence probability (r_{N}):
Under the assumption that ${t}_{N}^{\mathrm{Estimated}}$ has never reached 100 % in the past and with β=0.93, Eq. (16) can be simplified as follows:
The contribution of ${P}_{N}/{P}_{N\mathrm{1}}$ is clearly marginal, leading to consider the regression, built on the variations in W_{N}∕Pop_{N} and r_{N} between 1997 and 2016 (Table 2):
Equations (16) and (18) used and the goodness of fit are illustrated in Fig. 5a and b, respectively.
Considering that understanding the details of those variations is out of reach given the scarcity of data, this study focuses only on those above 15.5 % in absolute value, qualified hereafter as significant variations (V_{N}). Other variations are neglected and we assume they cancel each other out. Table 4 lists all of them, together with the most significant event that occurred the same year, for the earthquake insurance industry. Despite the fact that these events (Table 4) could have led insurance companies to restrict or enlarge the number of earthquake insurance policies (Born and KlimaszewskiBlettner, 2013), contributing in this way to V_{N}, this study focuses only on the homeowners' risk perception variations.
Then, the subjective annual occurrence probability (r_{N}) is estimated according to the following time series for N∈[1926; 2016]:
The time series of r_{N} is represented in Fig. 6 and illustrates that some earthquakes (indicated by dots) have increased r_{N} during the year as already published (Buffinton, 1961; Kunreuther et al., 1978; Meltsner, 1978; Lin, 2015), but, more surprisingly, some others had no apparent impact, like the 1952 Kern County and 1989 Loma Prieta earthquakes.
The first earthquake damaged over 400 unreinforced masonry buildings (Jones et al., 2012), but none of the buildings were designed under the latest seismic codes (Geschwind, 2001). Consequently, structural engineers claimed that new buildings were earthquakeproof and that no additional prevention measures were needed (Geschwind, 2001). This message was received well among the population despite the warnings of some earthquake researchers like Charles Richter (Geschwind, 2001).
Regarding the second earthquake, most homeowners were only partially refunded, due to large deductibles, and decided to cancel their earthquake insurance policies (Meltsner, 1978; Palm and Hodgson, 1992b; Burnett and Burnett, 2009). Moreover, the 1989 Loma Prieta earthquake struck just after a highseismicactivity period during which r_{N} increased significantly. Indeed, the number of earthquakes occurring during the preceding 5 years with a moment magnitude greater than M5 reached the highest peak since 1855 between 1984 and 1998, as illustrated in Fig. 7.
The sequence includes in particular the 1983 Coalinga and the 1987 Whittier Narrows earthquakes. This specific temporal burst in seismic activity may have contributed to homeowners' rising risk awareness (Table 4).
Some major socioeconomic events also had an impact on r_{N} (Table 4), such as the 1929 Great Depression and the post World War II economic expansion. Insurance regulation acts also have an impact. During the period 1984–1985, the California senate assembly voted on the Assembly Bill AB2865 (McAlister, 1984), which required insurance companies to offer earthquake coverage. A second bill in 1995, named AB1366 (Knowles, 1995), amended the Insurance Code Section 10089 to restrict the extent of the mandatory cover and addressed the insurance crisis caused by the 1994 Northridge earthquake. The new “minipolicy” met with limited success mainly because the price was more expensive for a lower guarantee (Reich, 1996). Furthermore, the creation of the CEA was at odds with the trend in insurance market privatization and people perceived it as an insurance industry bailout (Reich, 1996). The combined impact on r_{N} of the two assembly bills is null as (1+30 %) × (1−23 %) = 1 (Table 4). It means that the efforts to promote earthquake insurance among the population by a first assembly bill were cancelled 20 years later by the other. Last, the discredit of a mistaken prediction can lead to a decrease in r_{N}, as happened in 1928 with the occurrence of a major earthquake in Southern California (Geschwind, 1997; Yeats, 2001).
r_{N} is then compared to a scientifically based earthquake annual occurrence probability to assess the level of the homeowners' risk perception. This requires the assessment of the annual probability for a homeowner of being affected by a destructive earthquake. For that, the ShakeMap footprints, released by the US Geological Survey (Allen et al., 2009), are used. A ShakeMap footprint gives, for a historical earthquake, the modelled ground motions for several metrics, including the macroseismic intensity on the modified Mercalli intensity scale (MMI). In California, a total of 564 ShakeMap footprints are available on the USGS website (https://earthquake.usgs.gov/data/shakemap/) for the period 1952–2018 and the magnitude range M3–M7.3. According to the MMI scale, heavy damage can be observed for an intensity above or equal to VIII. Among the 564 historical earthquakes, the ShakeMap footprints report that only 21 have caused such a high intensity and are labelled as destructive earthquakes. For each of the 21 ShakeMap footprints, the area corresponding to an intensity above or equal to VIII was reported in Table 5 and the population living inside was estimated using the Global Settlement Population Grid (European Commission; Florczyk et al., 2019).
Finally, for each year since 1952, the annual rate of people was calculated by dividing the number of people having experienced an intensity above or equal to VIII by the total population of California according to the Global Human Settlement Population Grid (European Commission, Florczyk et al., 2019). The arithmetic mean of the annual rates during the period 1952–2018 is equal to r^{Hist}=0.038 %. In this study, we consider r^{Hist} as the true average probability for a California homeowner to be affected by a destructive earthquake. Therefore, the risk has always been underestimated by California homeowners because AWR_{N} has been lower than r^{Hist} since 1926 (Fig. 6). The severity underestimation is quantified using the earthquake risk awareness ratio (AWR_{N}) defined as
With AWR_{N} being estimated since 1926, the takeup rate model is generalized in the next section and used to understand the current low takeup rate.
Introducing the earthquake risk awareness ratio (AWR_{N}), we redefine the takeup rate model (Eq. 15) as follows:
From the estimated values of AWR_{N} (Eqs. 19 and 20) and P_{N} (Table 2), Fig. 8 shows the fit between ${t}_{N}^{\mathrm{Estimated}}$ (Eq. 21) and t_{N} (Table 2).
The goodness of fit is R^{2}=0.99 and the modelling errors are below ±3 %, as represented by the dashed lines. The record at 100 % corresponds to the 1992 California Residential Earthquake Recovery Fund Program (CRER), which was a mandatory public earthquake insurance, including a cover amount for households up to USD 15 000 (USD 1992), at a very low premium amount (Lagorio et al., 1992).
Figure 9 illustrates the sensitivity of ${t}_{N}^{\mathrm{Estimated}}$ to the average premium P_{N} and the earthquake risk awareness ratio AWR_{N} (Eq. 20).
The lines AWR_{N}=70 %, AWR_{N}=100 % and AWR_{N}=425 % stand for the current situation, the true risk level and the target value of AWR_{N} for ${t}_{N}^{\mathrm{Estimated}}=\mathrm{100}$ %, at the current price of USD 980 (USD 2015), respectively. According to the first value (AWR_{N}=70 %), half of homeowners (${t}_{N}^{\mathrm{Estimated}}=\mathrm{50}$ %) are not willing to pay an average premium amount exceeding P_{N} = USD 310 (USD 2015) per year for an insurance cover. This would represent a 68 % decrease in the price of earthquake insurance coverage.
Conversely, the relatively low earthquake risk awareness (AWR_{N}=70 %) contributes only marginally to the current low takeup rate (${t}_{N}^{\mathrm{Estimated}}=\mathrm{14}$ %). In fact, even with the true level of risk (AWR_{N}=100 %), only ${t}_{N}^{\mathrm{Estimated}}=\mathrm{21}$ % of homeowners are expected to buy insurance. This result is consistent with previous findings by Shenhar et al. (2015), who observed in Israel that the earthquake insurance takeup rate did not significantly increase after a large prevention campaign.
To reach a 100 % takeup rate with the current premium amount, according to the model, the earthquake risk awareness ratio has to reach 425 %, which is very unlikely. Indeed, the UCERF3 assesses the annual probability of occurrence, in California, of an earthquake with a magnitude above M6.7 at 15 % (Field et al., 2013). Under the assumption that the probability for a homeowner of experiencing a damaging earthquake is proportional to the occurrence of such an earthquake, reaching that awareness ratio would mean that homeowners consider this probability to reach the level of 425 % × 15 % = 64 %. In other words, only the belief that a destructive earthquake is imminent (i.e. a probability of occurrence somewhere in California during the year, for a 1994 Northridgelike earthquake, above 64 %) can bring all homeowners to buy an earthquake insurance plan at the current price.
The model developed in the present study shows that the low earthquake insurance takeup rate, observed until 2016 for the homeowner line of business in California, is due foremost to high premiums. Indeed, no homeowner would prefer to stay uninsured against earthquake risk if the annual average premium were to decrease from USD 980 (as observed in 2016) to USD 160 (USD 2015) or lower. Moreover, Kovacevic and Pflug (2011) have shown that a minimum capital is required to benefit from an earthquake insurance cover, otherwise the cost of the premium would be more than the potential risk. They have also found that the lower the insurance premium, the lower this minimum capital. Consequently, the results of Kovacevic and Pflug (2011) suggest that some uninsured people can be well aware of the earthquake risk but just cannot afford earthquake insurance at the current price. This corroborates the result of the present study because bringing such people to buy earthquake insurance is foremost a matter of price, not of insufficient risk perception.
Nevertheless, as the current average premium amount corresponds to the annualized risk assessment, insurance companies would not have enough reserves to pay future claims following earthquakes if they collect on average only USD 160 per policy (CEA, 2017). Hence, the current insurance mechanism cannot meet the homeowners' demand and be sustainable, as was the case for the 1992 CRER programme (Kunreuther et al., 1998).
Since this initiative, earthquake insurance has not been mandatory in California. Because compulsory insurance is not only hard to implement but also has an unpredictable impact (Chen and Chen, 2013), new insurance solutions have emerged to increase the earthquake insurance market, like parametric insurance. This insurance product stands out from traditional insurance policies by paying a fixed amount when the underlying metrics (e.g. the magnitude and the epicentre location) exceed a threshold, whatever the loss incurred by the policyholder. This new insurance claim process significantly reduces the loss uncertainty, the operating expenses and so the premium rate. In California, parametric insurance has been offered to cover homeowners against earthquake risk since 2016 (Jergler, 2017).
Nevertheless, risk awareness remains important in earthquake insurance consumption, and insurance companies also encourage local policies led by public authorities to improve prevention (Thevenin et al., 2018).
This study supports these initiatives by demonstrating that a new earthquake insurance scheme is necessary to meet the premium expected by homeowners. However, it also shows that the lower the premium amount, the more the earthquake risk awareness contributes in insurance consumption (Fig. 9). Finally, only a global effort involving all stakeholders may fill the adoption gap in earthquake insurance coverage. This also includes the financial stakeholders, involved in the mortgage loan business. A recent study (Laux et al., 2016) analysed the fact that most banks do not require earthquake insurance from their mortgage loan clients, preferring instead to keep the risk and to increase the interest rate by +0.2 % (in 2013). These additional bank fees are, on average, as expensive as the premium rate for the residential earthquake insurance (0.185 % in 2013 according to the CDI, 2017a). Nevertheless, such a market practice puts an estimated USD 50 billion–100 billion loss risk on the US mortgage system (Fuller and Kang, 2018). Consequently, expanding earthquake insurance cover, for homeowners, is also a challenge for the banking industry. In this respect, requiring earthquake insurance for mortgage loans would strongly enhance the insurance and public initiatives.
Insurance market raw data and socioeconomic indicators collected between 1921 and 2016 in California are available in the Supplement.
The expected utility theory (also called the von Neumann–Morgenstern utilities) was introduced in 1947 by von Neumann and Morgenstern. It models a decision maker's preferences from a basket of goods (materialized by the function g), according to their utility. Under this framework, the utility is defined as the satisfaction that a decision maker receives from the goods they get. The utility is measured by a utility function, U_{i}, where i is the name of the decision maker. The function U_{i} is defined by the Von Neumann and Morgenstern (1944) theorem, which is based on several axioms that could be split in six parts (Narahary, 2012). For the first two, let us consider a basket of three goods: b={g_{1}; g_{2}:g_{3}}. Then, the two first axioms state the following.

Completeness. The decision maker can always compare two goods (e.g. g_{1} and g_{2}) and decide which one they prefer or if both are equal.

Transitivity. If the decision maker prefers g_{1} to g_{2} and g_{2} to g_{3}, then they also prefer g_{1} to g_{3}.
Let us now consider that the decision maker does not select a good from a basket of goods b={g_{1}; g_{3}} but instead participates in a lottery l_{1} on b. We introduce p_{1} and 1−p_{1} as the probability for the decision maker to win the goods g_{1} and g_{3}, respectively. Similarly, l_{2} is the lottery on b with the probabilities p_{2} and 1−p_{2}. Here, the decision maker can choose between two different lotteries. On this basis, the four remaining axioms are as follows.
 3.
Substitutability. The decision maker has no preference between l_{1} and the same lottery where g_{1} is changed by g_{4} if the decision maker has no preference between g_{1} and g_{4}.
 4.
Decomposability. The decision maker has no preference on the way to carry on the lottery l_{1}, as long as long as the probabilities (p_{1} and 1−p_{1}) remain the same.
 5.
Monotonicity. Between l_{1} and l_{2}, the decision maker always prefers the one with the higher chance of winning their favourite good (i.e. if they prefer g_{1} to g_{3}, then the decision maker prefers l_{1} to l_{2} when p_{1}>p_{2}, and reciprocally).
 6.
Continuity. If the decision maker prefers g_{1} to g_{2} and g_{2} to g_{3}, there is a value of p_{1} for which the decision maker has no preference between having g_{2} and participating at l_{1}.
According to the von Neumann–Morgenstern theorem, a utility function associates a good g_{1} with a level of utility U_{i}(g_{1}) for the decision maker i. When the good g_{1} is replaced by a lottery l_{1}, this theorem states that the level of utility is the weighted average: ${U}_{i}\left({l}_{\mathrm{1}}\right)={p}_{\mathrm{1}}\times {U}_{i}\left({g}_{\mathrm{1}}\right)+(\mathrm{1}{p}_{\mathrm{1}})\times {U}_{i}\left({g}_{\mathrm{3}}\right)$.
The first property of U_{i} is to be strictly increasing (i.e. its first derivative ${U}_{i}^{\prime}$ is strictly positive). When g_{1} corresponds to the decision maker's wealth, like in this study, it means that the decision maker always wants to increase their level of wealth. Conversely, the sign of the second derivative, ${U}_{i}^{\prime \prime}$, can be positive (i.e. U_{i} is convex) or negative (i.e. U_{i} is concave). It represents the decision maker's behaviour: when ${U}_{i}^{\prime \prime}<\mathrm{0}$, the decision maker is riskaverse, meaning that the difference of utility between g_{1} and g_{1}+1 is decreasing when g_{1} increases. Still considering that g_{1} is the decision maker's wealth, it means that increasing the wealth by EUR +100 provides a higher utility for a decision maker with a wealth g_{1} = EUR 1000 than for one with a wealth g_{1} = EUR 1 000 000. In terms of insurance, a riskaverse decision maker wants to be covered. Indeed, we assume that the decision maker can be protected against losing g_{1} with a probability p_{1} when paying a premium amount ${P}_{N}={p}_{\mathrm{1}}\times {g}_{\mathrm{1}}$. Since the utility function is concave, the following equation is verified:
Conversely, when ${U}_{i}^{\prime \prime}>\mathrm{0}$, the decision maker is riskloving and behaves in the opposite way.
Depending on the choice of U_{i}, this behaviour (riskaverse or riskloving) can change depending on g_{1}. To represent it, the Arrow–Pratt measure of relative risk aversion (named after Pratt, 1964, and Arrow, 1965) is used:
When RRA, the Arrow–Pratt measure of relative risk aversion, is decreasing, increasing or constant, it means that the decision maker is less, more or indifferently riskaverse with the value of the good g_{1}.
The utility function U_{i} used in this study was developed by Holt and Laury (2002).
where β_{i} is the level of relative risk aversion and materializes the decision maker's risk behaviour from highly riskloving to highly riskaverse. Indeed, one can verify that the RRA measure for the Holt and Laury (2002) utility function is RRA(g_{1})=β_{i}. Therefore, this function belongs to the constant risk relative aversion utility function family. It means that a decision maker's behaviour is assumed to be uncorrelated to their wealth level (g_{1}). Furthermore, the parameter β_{i} controls the decision maker's behaviour (Table A1).
Let us consider the following mathematical problem in Eq. (B1):
The function f(t_{N})=𝔼[U(g_{N}(t_{N}))] can be rewritten by detailing the expression of the expected utility 𝔼:
Introducing the definition of g_{N}(t_{N}) (Eq. A3) gives
Using the definition of the function U (Eq. A2), f(t_{N}) becomes
Furthermore, the derivative of f(t_{N}) is equal to
Thus, when $\frac{\mathit{\delta}f\left(\mathrm{1}\right)}{\mathit{\delta}{t}_{N}}<\mathrm{0}$, ${t}_{N}^{\mathrm{Estimated}}$ is equal to (otherwise, ${t}_{N}^{\mathrm{Estimated}}=\mathrm{1}$)
As the expression of ${t}_{N}^{\mathrm{Estimated}}$ is complex, it can be simplified using the approximation $\frac{{P}_{N}}{K}\approx \mathrm{0}$:
The supplement related to this article is available online at: https://doi.org/10.5194/nhess1919092019supplement.
AP had a role in the following in this study: conceptualization, formal analysis, investigation, methodology and writing. PG performed the supervision, validation, and visualization and also wrote the original draft. PYB and SB were involved in the validation as well as in the writing, review and editing processes.
The authors declare that they have no conflict of interest.
The authors acknowledge MadeleineSophie Déroche, CAT Modelling R & D Leader at AXA Group Risk Management, for her contribution to this work. They are also grateful to the California Department of Insurance, the EMDAT, Robert Shiller, the US Census, the US Department of Labor, and the USGS for providing open access to their databases.
This research has been supported by the European Union's H2020 Research and Innovation programme under the Marie SkłodowskaCurie grant (grant no. 813137) and Labex OSUG@2020 (Investissements d'avenir, ANR10LABX56 grant).
This paper was edited by Sven Fuchs and reviewed by two anonymous referees.
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 Abstract
 Introduction
 Data collection and processing
 Model development for the period 1997–2016
 Evolution of the homeowners' risk perception since 1926
 Understanding the current low takeup rate
 Conclusions
 Data availability
 Appendix A: The expected utility theory
 Appendix B: Solution of the expected utility maximization equation
 Author contributions
 Competing interests
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement
 Abstract
 Introduction
 Data collection and processing
 Model development for the period 1997–2016
 Evolution of the homeowners' risk perception since 1926
 Understanding the current low takeup rate
 Conclusions
 Data availability
 Appendix A: The expected utility theory
 Appendix B: Solution of the expected utility maximization equation
 Author contributions
 Competing interests
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement