At the beginning of the period, t = 0, the insurance company's initial equity is denoted by E₀, and policyholders pay the up-front premium (denoted by P₀) to the insurer which grants the coverage of their claims on the position of the insured parties. Insurers' liabilities come from policyholders' claims and their value is denoted by L₀. The initially available asset (denoted by A₀) is the sum of E₀ and P₀, and can be invested in the capital market.

At the end of the period, at time t, the policyholders make the claims (denoted by L_t) on the catastrophe insurance. If the available assets A_t exceed the liabilities L_t, the policyholders get the indemnity L_t. If not, the policyholders receive only the indemnity A_t which is the market value of the insurer assets. Assets A_t and liabilities L_t are assumed to vary stochastically. Therefore, the indemnity payment P_t from the insurers to the policyholders is given by

where (·)₊ = max(·, 0). The equity position (denoted by E_t) of shareholders is determined by the difference between the asset A_t and the liability L_t

For studying fairness condition, we first denote the present values of the policyholder position, the liabilities, the equity position, and insurer' default by H0P = PV[P_t], H0L = PV[L_t], H0E = PV[(A_t–L_t)₊] and H0DPO = PV[(L_t–A_t)₊], respectively, where PV stands for the valuation operator. H0E is like an equity call option with maturity t and strike price L_t. H0DPO means the present values of the insurer's default loss and is like a default put option with maturity t and strike price L_t. According to Equation (1), H0P can be expressed by

The initial premium P₀ is fair or competitive for the policyholders if the net present value of the insured position is zero, that is to say,

The initial contribution is equal to the latter receiving a risk-adequate return. To be fair or arbitrage-free for the shareholders, the present value of the expected equity position must equal to the initial contribution E₀, that is to say,

Equation (4) is equivalent to Equation (5) because of the assumption that the available asset A_t equals the sum of the equity position and policyholder position, i.e., A_t = E_t + P_t. We have P₀ = H0P if and only if E₀ = H0E (Appendix A). Under such conditions of fairness for policyholder and shareholder, E₀-P₀ have infinite possible numbers because there are different values of H0DPO. From a shareholders' point of view, they can choose a specific equity level or solvency level to benefit themselves most as long as the equity endowment E₀ is not less than minimum equity capital E^min required by solvency regulation.

We first make some assumptions that are used in this study hereafter. There are three interested parties including shareholders, policyholders, and a public catastrophe insurance scheme (PCIS). The PCIS is a non-profit organization but needs to maintain a self-financing situation. Therefore, it charges ex-ante contributions (a kind of levy) from policyholders or shareholders or both to balance its financial revenue and expenditure. On the contract, both shareholders and policyholders seek to maximize their advantage by paying less and getting more under the same insurance protection or protection. We make some following assumptions during one-period t, and then analyze the effects of different origins of contribution on the changes in the net present value of three interested parties' positions. There are N policyholders whose average underwriting cost is ε₁, and n insurers whose average underwriting cost is ε₂. It is quite obvious that N is very larger than n. The government taxes insurers' profit by a rate τ. At time t = 0, let E0* and P0* denote the initial contributions of shareholders and policyholders, respectively. The available asset is A₀ = E₀ + P₀-Nε₁. In three settings of contribution origins, let C0* denote the contribution to PCIS, and be calculated as a fraction α of the premium volume P0*, 0 ≦ α <1. The growth rate of asset value is assumed to be γ. At time t, the available asset is A_t = eγtA0.

The insurers pay the PCIS the ex-ante contribution C0* directly to get the full protection of liability caused by issuing catastrophe policies but do not pass on it to their policyholders. That is to say, the insurers lose some profit to pay the levy in exchange for the limited capacity of the claim for the unlimited one. So, at time t = 0, the value of the insurer's asset decreases to

Policyholders are not affected by the introduction of the PCIS. That is to say, P0* = P₀. In practice, policyholders are not aware of the existing default risk and the limited compensation of the insurers. This case is feasible when the higher premiums for full insurance offered by the insurers cannot be implemented in the market.

At time t, the insurers have the available asset At* = A_t–e^γtC0*, where A_t = e^γt (E0* + P0* –Nε₁). The policyholders with claims L_t can get the indemnity Pt*,I from the insurers and the indemnity Pt*,S from PCIS if the insurers' asset had exhausted. The indemnity payment from the insurers to the policyholders is given by

The PCIS pays the remainder of claim L_t. The indemnity payment is

The equity position is Et* = At* – Pt*,I = (At*–L_t)₊. The present value of the equity position is expressed as

From the perspective of shareholders, when the PCIS is introduced, the net present value of the change in their equity position is:

The net present value is zero or negative. The zero value means that the initial investment E0* is fair for shareholders or the situation with PCIS remains fair for the insurers. According to Equation (10), two extreme situations can lead to zero value. One is that the company becomes insolvent, i.e., (A_t–L_t)₊= 0. Another is that the insurers pay no premium to PCIS, i.e., C0*= 0. Such two situations do not make any sense in Case 1. The shareholders actually face the negative value, and then promote the insurers to take any action against adverse conditions. They will lower their initial equity investment E0* because that they need to decrease the negative value in Equation (10) and that the premium income P0* is reduced by C0* to (1–α) P₀. That is to say, the equity position is adjusted to E0* which is less than E₀ without contribution C0*.

When we further consider the minimum equity level E^min required by solvency regulation, the lower E0* may be larger or less than E^min. If the latter one holds, the shareholders are not willing to support the issue of such catastrophe policies by sacrificing their equities. That is, the insurer will quit the catastrophe market. On the contrary, the former condition means that the insurers can offer the shareholders a fair return by adjusting the equity position. The catastrophe insurance will operate with lower equity E0*. The insolvency risk thus increases.

From the perspective of policyholders, they obtain the indemnity payments which consists of Pt*,I in Equation (7) and Pt*,S in Equation (8). Then, the present value of the policyholder premium, H0*,P, can be derived by

According to Equations (3) and (4), the policyholder net present value is derived by

This means that the introduction of PCIS increases the policy value which equals the default put option of the insurer. In the setting, a rational policyholder in a transparent market requires lower premiums and does not care about the insolvency risk which reflects the value of the default put option. The competitive premium P0* is adjusted decreasingly by reducing equity position and then contributes to a lower charge by the insurance scheme. However, policyholders' rights and claims are not affected whether the insurers go into bankruptcy or not.

From the perspective of PCIS, the present value of the PCIS payment Pt*,S to policyholders is denoted by H0*,S = PV[Pt*,S]. So, the net present value of PCIS position is given by

A non-profit PCIS achieves a self-financing goal if and only if the net present value H0*,DPO + nε₂-C0* is zero. If the value is negative, it means that the PCIS gets some profit. The PCIS can accumulate a reserve for the next period and then will lower the contribution C0* due to the non-profit aim. On the contrary, the positive value will force the PCIS to raise the contribution C0* in order to achieve its self-financed. The premium C0* charged at time t = 0 should be H0*,DPO, which reflects the risk premium of insurer default. Therefore, the PCIS can adjust the value of C0* by setting the parameter α based on the insolvency risk level of PV [(L_t–A^* t)₊].

In Case 1, when the insurers pay the contribution, the policyholders will lower their equity investment to reduce their negative net present value or to reflect their less premium income. The low equity will lead to high default risk. The policyholders do not care about such risk because the PCIS assures their full claim. When the PCIS affords more insurers' default risk and pays more value of H0*,DPO, PCIS will charge more contribution C0* from the insurer by raising higher parameter α next period, and then the policyholders will lower their equity position again. This phenomenon will bring in a vicious cycle until either the equity reaches the minimum equity level E^min or the PCIS gets additional funding for supplementing the default (H0*,DPO–C0*)₊.

In this case, the policyholders pay the contributions to the scheme directly or indirectly through the insurers. Both insurers and shareholders are thus not affected by the contribution. At time t = 0, the initial equity E0* equals to E₀, and the up-front premium P0* equals to P₀. This implies that the insurers' available asset is

At time t, A0* equals A₀ implies that A^* t equals A_t. The policyholders can get two indemnities Pt*,I and contingent Pt*,S. The indemnity payment Pt*,I from the insurer is

Additionally, the indemnity payment Pt*,S from the scheme compensates for the default caused by policyholders' claims, which is the amount L_t over A_t. That is to say,

From the perspective of shareholders, their equity position is Et* = (A_t–L_t)₊, and then the present value of the equity position is H0*,E = PV[Et*] = PV[(A_t–L_t)₊]. The change in equity position becomes

The above equation also reflects that shareholders are not directly affected by the introduction of the scheme and that they have a fair value of equity position.

From the perspective of policyholders, their initial position in t = 0 is P0* = P₀ + C0* + Nε₁. Because C0* = αP0*, P0* = P₀ + Nε₁ + αP0* implies that the up-premium is:

The policyholders can get full claims Pt* = L_t because of PCIS's assurance, and the present value of their position is the same as Equation (11), i.e.,

Then, using Equation (19) and Equations (3) and (4), the policyholder net present value is expressed by

The fair condition for policyholders is that the net present value is zero if and only if the fraction α of their initial premium position is the ratio of (H0*,DPO–Nε₁) to H0L. We denote the ratio by the notation αpolicyholderfair:

where α^fair denotes the ratio. If α is larger than α^fair, the situation is unfair for policyholders. They will require the PCIS to re-estimate insurers' default risk H0DPO in order to lower the next-period contribution or look for low-premium policies, which are provided by the insurers with low equity, i.e., high H0*,DPO.

From the perspective of PCIS, it receives the contribution C0* at time t = 0 and pays the claim Pt* at time t. The net present value of PCIS is expressed by

where H0*,S = PV [(L_t–A_t)₊]. The PCIS is self-financing if and only if the contribution C0* equals to the value of insurer's default put option H0*,DPO+ Nε₁. As C0* = αP0*, the self-financing condition C0* = H0*DPO, implies α = (H0*,DPO + Nε₁)/H0L. We denote the ratio by the notation αPCISfair:

If α is less than αPCISfair, the PCIS will raise the parameter α next period to maintain self-finance, and then the action will force policyholders to seek a lower premium to reduce their expenses in insurance. Compared with the one in Equation (21), αPCISfair is larger than αpolicyholderfair because that the PCIS needs to pay the underwriting cost. The PCIS has a stronger pricing power than policyholders. The ratio αPCISfair will be possibly adopted. In practice, H0DPO is quite larger than Nε₁. αpolicyholderfair is thus approximately equal to αPCISfair.

In this case, although the equity capital is fairly valued, policyholders pay the contribution based on the premium P₀, and then have more motivation to find a low-premium policy. As a result of competitive pressure, the shareholders reduce their equity position to the minimum equity level E^min in order to offer a low-premium policy. This phenomenon will lead to a similar vicious cycle as that in Case 1 until either the equity reaches the minimum equity level E^min or the PCIS gets additional funding for supplementing the default (H0*,DPO–C0*)₊.

In this case, the policyholders pay their insurers a premium P0* calculated based on the full insurance, and then the insurers pay the PCIS the contribution C0*. The PCIS ensures full protection of the insurers' liability paid to the policyholders. In other words, P0* is a risk-free premium H0L, and a fraction α, 0≦α < 1, of such premium, is transferred to the contribution C0*. The PCIS assures policyholders' full claim if the insurers run out of their assets. Because the condition is fair to the policyholders, the equality H0L = P₀ + H0DPO holds according to Equations (3) and (4). At time t = 0, the insurer' available assets are given by

where the initial investments E0* are assumed to be E₀. A0* is assumed to be invested in the same assets as A₀. At time t, the indemnity payment Pt*,I from the insurer is equal to L_t-(L_t-e^γtA0*)₊. The available asset based on Equation (24) is At* = e^γtA0*+(L_t-e^γt A0*)₊. The shareholders' position at time t is Et* = e^γtA0* +(L_t-e^γtA0*)₊–L_t. Additionally, the indemnity payment Pt*,S from the scheme compensates for the shortfall between insurers' payment and policyholders' claims. That is to say, Pt*,S = (L_t-e^γtA0*)_+.

From the perspective of shareholders, by using similar transformations in Equations (9, 10), the net present value for shareholders is

where A_t = e^γt (E0* + P0*–Nε₁), The derivation is listed in Appendix B. According to Equation (25), the condition for shareholders is fair if and only if the net present value is zero. Ct* thus equals (L_t-A_t)₊. It implies that C0* = PV[(L_t–A_t)₊] = H0*,DPO, and shows that the fair contribution C0fairfor shareholders is equal to the present value of the insurer's default put option. If C0* is larger than H0*,DPO, the situation is unfair to shareholders. The insurers will either require the PCIS to reduce the contribution next period or decrease the equity capital to raise the default put value H0*,DPO. If the adjusted equity is still over the minimum equity level E^min, the business will continue. Otherwise, the business disappears.

From the perspective of policyholders, because the PCIS grants full claims, the policyholders get the indemnity payment Pt* = L_t. The net present value for policyholder equals zero because of the following equality

It is a fair situation for the policyholders in any α. The result is due that the policyholders pay the default-risk premium corresponding exactly to the full protection.

From the perspective of PCIS, the PCIS receives contributions via n insurers. It only face n insurers and spends underwriting costs nε₂, which are quite less than Nε₁ because N >> n. the net present value for PCIS can be written as H0*, S + nε₂ – C0* = PV[(L_t–A_t)]₊ + nε₂ – C0* = H0*,DPO + nε₂ – C0*. The PCIS is self-financing if and only if the contribution C0* is equal to the value of the insurers' default put option H0*,DPO. The result is similar to the discussion in Case 1. However, both the assets A^* t and the contribution C0*, in this case, are larger than those in Case 1 because that the policyholders pay their insurers a premium supplement H0DPO and that the contribution is charged by a fraction α of the default-free premium.

In this case, it is fair to the policyholders because they pay an ex-ante risk-adequate premium and get the full protection of the claim. Every policyholder pays the same premium and then the competitive premium will not be lowered under such that situation. This situation will avoid a vicious cycle of competitive pressure results in a race to the bottom, and then force the equity capital to the regulatory minimum equity level E^min. On the contrary, whether this setting is fair to shareholders or PCIS depends on how the contribution C0* is set on the parameter α. The high value of α favors PCIS but is unfavorable to policyholders. Conversely, if contrary. The fair condition for both shareholders and PCIS is C0* = H0*,DPO + nε₂. Compared to policyholders, the PCIS has enough abilities and professional knowledge of assessing insurers' default risk and find the breakeven point

If the equity position is larger than the regulatory minimum equity level E^min under such a fair situation, the business continues. Otherwise, the insurers will quit the catastrophe market.

This study makes assumptions of stakeholders: the PCIS assures insurers' liability or policyholders' claim, rational policyholders in a transparent market prefer the lowest premiums, and the shareholders tend to reduce their equity capital to gain policyholders. The shareholders in Case 1 pay the contribution and bear the negative net present value. They have more incentive to gain shareholders through high leverage (low equity capital). Simultaneously, the PCIS bears more insurers' default risk. That is to say, more value H0*,DPO which benefits the policyholders. In Case 2, policyholders pay the contribution based on the premium P₀, and then have more incentive to find a policy with a low premium. Competitive pressure results in a race to the bottom. Both Cases 1 and 2 easily lead to bring in a vicious cycle until either the equity position drops to the minimum equity level E^min or the business is discontinued. However, in Case 3, all policyholders pay the insurers the same premium for full insurance. The way avoids a premium race to the bottom. The PCIS charges the contribution C0* from the insurers. The insurers maybe want to gain the shareholders by reducing equity capital, i.e., raising financial leverage. But, the PCIS can find the breakeven point C0fair= H0*,DPO + nε₂ by exactly assessing insurers' default risk.

Setting a unique α for market participants means to assume that all insurers are homogeneous. This condition is unfair to insurers with high equity positions (good solvency or low default risk) and then leads to adverse selection in the catastrophe insurance industry. If the PCIS assigns different parameters α to insurers according to their default risk level, the insurers charged high contribution will have more incentive to improve their default risk. They can require PCIS to lower the contribution by showing their high solvency. This way may lead to a virtuous cycle of reducing default risk.

This section focus on the values of equity holders H0*,E and policyholders H0*,P by using the same framework and assumptions introduced in this section The Contingent Claim Model for Stakeholder Positions. Since the insurer contributes C0* to the PCIS without passing on charges to the policyholders, an unfavorable situation with negative net present value happens to equity holders [see Equation (10)]. We focus on their incentives and reactions. The asset value decreases to A0* = A₀–C0*. The equity value H0*,E is derived from Equation (32):

where K^* = A0*e^γ(T−t). Finally, since the policyholder can get full claim due to the introduction of PCIS, the values of the policyholder claims are obtained from Equation (11):

For purpose of the numerical experiment, the CIR model is calibrated via the global insured losses from 1970 to 2016. Table 4 shows the descriptive statistics of global insured losses. These 47 insured losses vary from a minimum of 5.5678 to a maximum of 147.3440 (billion USD). The standard deviation of 28.2100 is less than the mean of 33.0018. The data distribution has more right skewness of 2.3548 and a much higher kurtosis of 6.8158 than the normal distribution. The augmented Dickey-Fuller (ADF) test statistic rejects the null hypothesis of a unit root. It also reflects that the time series loss data are stationary. The maximum likelihood estimation is used to estimate three parameters in the CIR model, which are listed in order: the adjustment speed a = 0.9354, the long-term growth rate b = 33.6811, and the instantaneous volatility rate σ_X = 5.7062. The riskless rate r is assumed to be 0.03. Because the insurance industry covered close to USD 54 billion in 2016 (31), we take 54 as the initial liabilities L₀. The yield rate γ of fixed income securities is set to be 0.05. The PCIS charges 1 percent of the policyholder premiums P₀. That is to say, the contribution rate α equals 0.01. The value of such a rate is chosen from the one used in practice (25).

We take an example to explain the process of numerical experiments. For simplifying the model, we ignore the term Nε₁ and nε₂ because they are quite less than P0*, E0* and H0*,DPO in the real world. First, the initial default put option H0*,DPO is set to be 1. That is to say, the initial insurer's safety is fixed. According to Equation (3), the initial premium P₀ needs to be 53. The initial asset A₀ is thus estimated when H0*,DPO = 1 is substituted into Equation (35). Then, we can get the equity value H0*,E = 64.4662 by instituting the value of asset A₀ = 117.4662 into Equation (35). If an insurer joints the PCIS, the scheme charges the insurer the contribution C0* = 0.99 if the contribution rate α = 0.01. The total assets after paying the fund contribution are A0* = A₀-C0* = 116.4762. The insurer's solvency position thus decreases when its liability is not unchanged. The present value of the default put option H0*,DPO will increase to 1.5655 by instituting the initial asset A0* = 116.4762 into Equation (34). It means that the PCIS bear more insolvency risk and then needs to charge the insurer more contribution. From equity holders' point of view, their position changes from 64.4662 to 63.0161 when the asset value 117.4662 drops to 116.4762, and is not fair because the net present value drops to −1.4501. They will be incentivized to reduce the next-year equity capital from 64.4662 to 55.3022 in order to result in a fair situation. The reasons are that the asset value drops to 116.4762 and H0*,DPO value increases to 1.5655. The PCIS will adjust the contribution rate α to maintain its finance. Such a situation brings in a vicious cycle of forcing the equity capital to the regulatory minimum equity level E^min.

Table 5 illustrates multiple premium P₀-equity holder capital H0*,E–the default put option H0*,DPO combinations at equilibrium. Less premium P₀ increases the value of the default put option H0*,DPO, and reduces the equity holder capital H0*,E. The PCIS incentivizes the combination. Lager contribution rate α under the same premium P₀ increases the value of the default put option H0*,DPO, and quickly drops the equity holder capital H0*,E to the regulatory minimum equity level E^min.

Using the same framework and assumptions introduced early in section Numerical Experiment of the Impact on Stakeholder Positions, we focus on the values of policyholders H0*,P under different initial premiums P₀ and contributions rates α because the situation of the equity holders is not altered due to policyholders' paying PCIS contribution. The total initial premium P0* paid by policyholders is expressed as a function of contribution rate α (for PCIS) and the initial premium P₀ (for insurer): P0* = P₀/(1–α) [see Equation (19)]. The fair present value of the final payoff is given by P0* = H0*,P = H0L [see Equation (20)] when the policyholder claims are guaranteed fully. Without loss of generality, we assume a present value H0L of claims L₀ of 54. Then, the fair initial total premium P0* paid by policyholders is equal to 54.

In Table 6, we illustrate the policyholder total premiums for three initial premiums of P₀= 53, 43, and 33. The case of initial premium P₀ = 53 with α = 1.9% corresponds to a fair situation. However, α > 1.9% is unfair for policyholders due to the more total premium P0*. On the contrary, α < 1.9% is unfair for PCIS due to less contribution C0*. Similarly, the case of initial premium P₀ = 43 (32) with α = 2.3% (2.7%) corresponds to a fair situation for both policyholders and PCIS. The low initial premium means the PCIS paying more claims from the insurer's default. Then, it is necessary for PCIS to charge more contributions.

Although the uniform contribution rate α for different initial premiums P₀ is always unfair for either policyholders or PCIS, the mandatory PCIS, in practice, charges some fixed contribution rate α to policyholders (as shown in Table 2) due to easy implementation and political consideration. Policyholders do not care about the insurer's safety due to PCIS's guaranty and tend to choose the policy with offering the lowest premium. The insurer thus lowers the equity capital to maintain its profit and then leaves more insolvency risk to PCIS.

In this subsection, we focus on equity holder capital E0* in different contribution rates α because policyholders pay the insurer uniform premium P0* = H0L for full coverage L₁. The insurer contributes the PCIS according to different contribution rates α based on the insurer's insolvency level C0* = H0*,DPO [see in Equation (27)]. We assume a present value H0L of claims L₀ of 54 again. That is to say, P0* = 54. The initial asset A₀ is estimated when H0*,DPO = C0* is substituted into Equation (35). E0* can be obtained by the initial asset A₀ substituted into Equation (37). On the contrary, if equity holders raise their capital, the less value of H0*,DPO is obtained by the more asset A₀ substituted into Equation (35). The insurer can negotiate the fair contribution rate with the PCIS by showing its lower solvency risk (smaller value of H0*,DPO). Comparing with adjusting the contribution rate to different policyholders with different premiums, the adjustment of the contribution rate based insurers' insolvency risk is quick feasible because that the number of the insurer is far less than the number of policyholders and that both PCIS and insurers have enough professional knowledge to make a fair contribution rate.

Table 7 illustrates the importance of the choice of α with regard to an insurer's target equity capital. It lists that a variation of α from 0.0001 to 0.05 makes the fair equity capital vary from 151.1501 to 138.5996. As the contribution rate rises, the fair equity capital drops. On the contrary, the increase of equity holder capital may convince the PCIS of decreasing the contribution. The illustration in Table 7 shows that policyholders and insurers paying contributions may lead to a virtuous cycle of reducing default risk.

In order to further detail the direct influence of PCIS on the solvency level, this subsection gives a parameterization example of policyholders and insurers' paying contribution with initial liability L₀ = 33 and initial premiums P₀ = 33 because that the mean of global insured loss from 1970 to 2016 is 33.0018 and that the insured get full coverage in this case. The objective is to calculate changes in the insurer's default probability and the expected policyholder deficit when equity holders adjust their capital to get back to even.

Table 8 shows that the extreme variations arise before and after the PCIS intervenes in the catastrophe market. The fair equity capital endowment reduced by 0.35% from E₀ = 88.1200 to E0* = 57.3657 when the insurer is charged by a contribution rate α = 0.01. Simultaneously, the available assets decrease by a quarter from A₀ = 121.0800 to A0* = 90.0357. The default probability increases more than eight times from 0.17 to 1.4207%. The expected policyholder deficit increases more than seven times from 0.03 to 0.2213. The result shows the significant immediate impact of the introduction of the PCIS on capital structure.

From the perspective of policyholders, it seems disadvantageous to introduce PCIS due to the increase of the expected deficit. However, the PCIS does not default and guarantees policyholders full coverage. Although the expected deficit with PCIS is higher than that without PCIS, the peak risk of heavy losses has been smoothened. Without PCIS, policyholders may suffer heavy losses and cannot get sufficient claims because of the insurer's default once a large Catastrophe occurs. The infrequent peak risk may force the insured into financial distress.

Furthermore, from the perspective of the business cycle, whether pro-cyclical or reverse-cyclical overall economic factors will affect the insurance coverage rate; take the United States as an example. From 2004 to 2007, when it is strongly pro-cyclical, the economic growth and income the average increased, and the insurance coverage rate increased slightly, but the insurance coverage through employers decreased; during 2007 to 2009, when the poor macro-economy or reverse cycle, the company's default rate increased. In order to reduce company costs, the company reduced employee insurance, especially when it encountered recession, the insurance coverage rate dropped significantly. For example, in 2001, the technology bubble and the 2008 subprime mortgage crisis caused an economic recession. The decline in income and the sharp rise in unemployment were accompanied by the loss of health insurance. Because they lost the insurance sponsored by their employers, there was a significant decline in insurance coverage through employers at this stage. There is less medical care than insurance (33), so the resulting unemployment will lead to a higher chance of death (34). A well-designed PCIS will stabilize the insurance industry and then benefit the social stability under a bad year.

The property-liability insurance business is obviously linked to the overall economic performance of the national economy (32, 35), especially when it comes to interest rates that are related to insurance pricing theory (36), insurance prices should reflect investment returns by discounting expected losses, so insurance prices are the result of discounting future losses. Any change in the interest rate causes a change in the premium because the insurance company invests the premium from the time the premiums receives to the time they pay the loss. However, interest rates are closely linked to the economic cycle, then the premiums are also affected by business cycles.