Problem Chosen 选择的问题
E
MCM/ICM 管理和监测/信息和通信管理
Summary Sheet 汇总表
Team Control Number 团队控制编号
Veils of Uncertainty: Weaving Risk into the Tapestry of Preservation Under the Weather's Watch Summary 《不确定的面纱:在天气监视下将风险编织到保护工作的织锦中》摘要
As the tapestry of nature weaves its unpredictable patterns, humanity’s quest for stability becomes ever more pressing. In the shadow of uncertainty, we find resilience, crafting shields against the tempests of fate. 当大自然的织锦编织出变幻莫测的图案时,人类对稳定的追求也变得愈发迫切。在不确定性的阴影下,我们找到了韧性,打造了抵御命运风暴的盾牌。
First, we establish a Risk Analysis model to comprehensively assess the Expected Annual Loss (EAL) from extreme weather in terms of population, building, and agriculture. The assessment for each aspect is calculated from three perspectives: natural hazard exposure, Historic Loss Ratio, and the likelihood risk factor of natural hazard annualized frequency. Community Risk Factor (CRF) is calculated from social vulnerability and community resilience. EAL and CRF are used to quantify the risk levels of various regions and rank them using the K-means algorithm, resulting in a risk level map of the United States. 首先,我们建立了一个风险分析模型,从人口、建筑和农业三个方面全面评估极端天气造成的预期年损失(EAL)。各方面的评估从三个角度进行计算:自然灾害风险暴露、历史损失率和自然灾害年化频率的可能性风险系数。社区风险系数 (CRF) 是根据社会脆弱性和社区恢复力计算得出的。EAL 和 CRF 用于量化不同地区的风险水平,并使用 K-means 算法对其进行排序,从而绘制出美国的风险水平图。
Second, we develop a Risk-incorporated Capital Asset Pricing Model (CAPM) to aid insurance companies in underwriting decisions. This model combines market return rates, the risk-free rate, and bankruptcy theory with a 10%10 \% bankruptcy probability to set insurance rates. It evaluates if the region’s residents can afford these premiums, providing decision-making advice for insurance companies. 其次,我们建立了风险纳入资本资产定价模型(CAPM),以帮助保险公司做出承保决策。该模型将市场收益率、无风险利率和破产理论与 10%10 \% 破产概率相结合,以确定保险费率。它可以评估该地区的居民是否能够负担得起这些保费,从而为保险公司提供决策建议。
More specifically, we apply our Risk-incorporated Capital Asset Pricing Model in Los Angeles and Gorontalo. In Los Angeles, insurance companies see high profits and low risks. However, in Gorontalo, the required premium for $10,000\$ 10,000 coverage is $342.745\$ \mathbf{3 4 2 . 7 4 5}, beyond local affordability. We recommend insurance securitization and partnerships with local governments to reduce premiums. Consequently, Gorontalo residents could pay just $137.25\mathbf{\$ 1 3 7 . 2 5} annually, with companies projecting $245\mathbf{\$ 2 4 5} million in revenue. 更具体地说,我们将风险纳入资本资产定价模型应用于洛杉矶和戈伦塔洛。在洛杉矶,保险公司看到的是高利润和低风险。然而,在戈伦塔洛, $10,000\$ 10,000 保险所需的保费为 $342.745\$ \mathbf{3 4 2 . 7 4 5} ,超出了当地的承受能力。我们建议保险证券化,并与地方政府合作降低保费。因此,戈伦塔洛居民每年只需支付 $137.25\mathbf{\$ 1 3 7 . 2 5} ,公司预计可获得 $245\mathbf{\$ 2 4 5} 万的收入。
Third, we establish a Building Preservation Model, selecting seven secondary indicators such as the annual number of visitors and construction cost, and three primary indicators: cultural values and community influence, economy, and history. These are weighted using the Sperman-CRITIC algorithm and AHP method to calculate building value, combined with risk levels to determine the preservation level of buildings. Based on the preservation level, the community’s investment and measures for building protection can be determined. 第三,建立建筑保护模型,选取年游客量、建筑成本等七个二级指标和文化价值与社区影响、经济、历史三个一级指标。利用 Sperman-CRITIC 算法和 AHP 方法对这些指标进行加权,计算出建筑物的价值,再结合风险等级,确定建筑物的保护等级。根据保护级别,可以确定社区对建筑保护的投资和措施。
Then our models inform investment and protection strategies for Tokyo Tower, acknowledging its value and the necessity of preservation in an earthquake zone. We communicate these findings and propose protection measures to the Tokyo Tower community. 然后,我们的模型为东京塔的投资和保护战略提供信息,承认其在地震带的价值和保护的必要性。我们将向东京塔社区传达这些发现并提出保护措施。
Finally, we analyze the sensitivity and robustness of our models, the models can change the insurance rate sensitively according to the change of the market predicted return and the slight error of the risk factor calculation will not affect the models’ result, which verifies the sensitivity and robustness of our models. In addition we analyze the strengths and weaknesses of the models. 最后,我们分析了模型的灵敏性和稳健性,模型可以根据市场预测收益率的变化灵敏地改变保险费率,风险系数计算的微小误差也不会影响模型的结果,这验证了模型的灵敏性和稳健性。此外,我们还分析了模型的优缺点。
1 Introduction … 4 1 导言 ... 4
1.1 Problem Background … 4 1.1 问题背景 ... 4
1.2 Restatement of the Problem … 4
1.3 Our Work … 4 1.3 我们的工作 ... 4
2 Assumptions and Justifications … 5
3 Notations … 5 3 注释 ... 5
4 Insurance Pricing and Decision Model … 6 4 保险定价和决策模型 ... 6
4.1 Data Collection … 6 4.1 数据收集 ... 6
4.2 Risk Analysis … 6 4.2 风险分析 ... 6
4.2.1 The Expected Annual Loss(EAL) … 6
4.2.2 Community Risk Factor … 7 4.2.2 社区风险因素...... 7
4.2.3 Risk Calculation … 8 4.2.3 风险计算...... 8
4.3 Insurance Premium … 9 4.3 保险费 ... 9
4.4 Measures … 13 4.4 措施 ... 13
4.4.1 Insurance Securitization … 13 4.4.1 保险证券化 ... 13
4.4.2 Co-operation with the Government … 13
4.4.3 The Implementation of Measures … 14
4.5 Real Estate Decision Making … 14 4.5 房地产决策 ... 14
5 Building Preservation Model … 16
5.1 Building Value Quantification … 16
5.1.1 Indicators Determination … 16
5.1.2 Weight Calculation … 17 5.1.2 重量计算 ... 17
5.1.3 Quantitative Results of Building Values … 19
5.2 Determination of protection measures … 20
5.2.1 Measure Score … 20
5.2.2 Score of Protection Measures … 20
5.2.3 Mentoring for Community Leaders … 20
6 Landmark Case Analysis … 21 6 里程碑案例分析 ... 21
6.1 Insurance Pricing for Tokyo Tower … 21 6.1 东京塔的保险定价...... 21
6.2 Architectural Value of Tokyo Tower … 21 6.2 东京塔的建筑价值...... 21
7 Sensitivity and Robustness Analysis … 22
7.1 Sensitivity … 22
7.2 Robustness … 23
8 Model Evaluation … 23 8 模型评估 ... 23
8.1 Strengths … 23 8.1 优势 ... 23
8.2 Weaknesses … 23
References … 23 参考文献 ... 23
1 Introduction 1 引言
1.1 Problem Background 1.1 问题背景
Extreme weather events are becoming more frequent due to climate change. As the economic costs of disasters rise, how should the insurance industry respond to losses? Extreme weather events cause untold human suffering but and ever-growing economic costs. 由于气候变化,极端天气事件越来越频繁。随着灾害造成的经济损失不断增加,保险业应如何应对损失?极端天气事件给人类带来了无尽的苦难,但也造成了不断增长的经济损失。
Catastrophic risks consequently cause a variety of problems for insurers. First, because the losses arise from a small number of lumpy events, the insurer may not have sufficient resources to cover the losses. Less dramatically, the firm may suffer losses well in excess of the value of the premiums that it charged for the coverage. In the absence of adequate reinsurance, the firm may go bankrupt or may choose to exit a state in which there is a substantial exposure to such catastrophic risks. The unexpected catastrophes or blockbuster events maybe raise the rate that firms charge for insurance. Thus, for any given number of policies written, the total premiums will rise ^([1]){ }^{[1]}. 因此,巨灾风险给保险公司带来了各种问题。首先,由于损失是由少量的偶发事件造成的,保险公司可能没有足够的资源来弥补损失。更严重的是,公司蒙受的损失可能远远超过其收取的保险费的价值。在没有足够的再保险的情况下,公司可能会破产,或者可能会选择退出存在大量此类灾难性风险的国家。意想不到的灾难或大事件可能会提高公司的保险费率。因此,对于任何给定数量的保单,总保费都会上升 ^([1]){ }^{[1]} 。
The impact of natural disasters is equivalent to a $520\$ 520 billion loss in annual consumption, and forces some 26 million people into poverty each year. Given the gradual increase in the number of extreme-weather events in the world, how to realize the sustainability of property insurance is a great challenge we need to address. 自然灾害造成的影响相当于每年十亿 $520\$ 520 人的消费损失,每年迫使约 2600 万人陷入贫困。在全球极端天气事件逐渐增多的情况下,如何实现财产保险的可持续发展,是我们需要应对的巨大挑战。
1.2 Restatement of the Problem 1.2 问题的重述
Considering the background information and restricted conditions identified in the problem statement, we need to solve the following problems: 考虑到问题陈述中确定的背景信息和限制条件,我们需要解决以下问题: >> Problem 1: The property insurance industry faces a crisis from increased extreme weather events caused by climate change, leading to higher claims and premiums. >> 问题 1:由于气候变化导致极端天气事件增多,财产保险业面临危机,从而导致索赔和保费增加。 >> Problem 2: Insurance companies must decide on underwriting policies in weather-affected regions, balancing risk and long-term viability. >> 问题 2:保险公司必须决定在受天气影响的地区承保,平衡风险和长期生存能力。
Problem 3: Participants are tasked with creating a model to assist in underwriting decisions and a preservation model for community leaders to protect significant buildings. 问题 3:参与者的任务是创建一个模型,以协助作出承保决定,并为社区领导人创建一个保护模型,以保护重要的建筑物。 >> Problem4: The models should be applied to specific areas and a historic landmark, with recommendations for future insurance and preservation strategies. >> 问题 4:模型应适用于特定区域和历史地标,并就未来的保险和保护战略提出建议。
1.3 Our Work 1.3 我们的工作
In order to clearly illustrate our work, we draw the flowchart Figure 1. 为了清楚地说明我们的工作,我们绘制了图 1 流程图。
Figure 1: Our work 图 1:我们的工作
2 Assumptions and Justifications 2 假设和理由
Considering those practical problems always contain many complex factors, first of all, we need to make reasonable assumptions to simplify the model, and each hypothesis is closely followed by its corresponding explanation: 考虑到实际问题总是包含许多复杂的因素,我们首先需要做出合理的假设来简化模型,而每一个假设都紧跟着相应的解释:
Assumption: The data we use are accurate and valid. 假设:我们使用的数据准确有效。
Justification: Our data is collected from the World Bank and some other official web sites and research papers. it’s reasonable to assume the high quality of their data. 理由我们的数据来自世界银行和其他一些官方网站及研究论文。
Assumption: The regions under study will remain peaceful and stable, with no significant events other than natural disasters occurring in the foreseeable future. 假设:所研究的地区将保持和平与稳定,在可预见的将来不会发生除自然灾害以外的重大事件。
Justification: A stable capital market environment provides a predictable framework within which we can project our expected returns. It is important to note that this assumption does not negate the potential impact of natural disasters. 理由:稳定的资本市场环境为我们预测预期回报提供了一个可预测的框架。值得注意的是,这一假设并不否定自然灾害的潜在影响。
Assumption: The estimated figures for each region represent an average level of performance or condition for that area. 假设:每个地区的估计数字代表了该地区的平均绩效水平或状况。
Justification: For the purposes of this study, treating each region as a cohesive entity allows for a more streamlined analysis. This methodological approach simplifies the complexity inherent in regional studies by focusing on aggregate data, thereby providing a generalized view of each area’s performance or condition. 理由:就本研究而言,将每个地区视为一个整体,可以使分析更加简洁。这种方法侧重于综合数据,简化了区域研究中固有的复杂性,从而提供了对各地区绩效或状况的总体看法。
3 Notations 3 备注
The key mathematical notations used in this paper are listed in Table 1. 表 1 列出了本文使用的主要数学符号。
Table 1: Notations 表 1:符号
Symbol 符号
Description 说明
EAL
Expected Annual Loss 预期年度损失
SV
Social Vulnerability 社会脆弱性
CR
Community Resilience 社区复原力
CRF
Community Risk Factor 社区风险因素
HLR
Historic Loss Ratio 历史损失率
Symbol Description
EAL Expected Annual Loss
SV Social Vulnerability
CR Community Resilience
CRF Community Risk Factor
HLR Historic Loss Ratio| Symbol | Description |
| :--- | :--- |
| EAL | Expected Annual Loss |
| SV | Social Vulnerability |
| CR | Community Resilience |
| CRF | Community Risk Factor |
| HLR | Historic Loss Ratio |
There are some variables that are not listed here and will be discussed in detail in each section. 还有一些变量没有在此列出,将在各章节中详细讨论。
4 Insurance Pricing and Decision Model 4 保险定价和决策模型
Database Names Database Websites
Los Angeles https://geohub.lacity.org/
The National Risk Index National Risk Index | FEMA.gov
Bureau of the Census https://data.census.gov/
South Carolina https:////www.sc.edu//
World Bank https://// data.worldbank.org/| Database Names | Database Websites |
| :--- | :--- |
| Los Angeles | https://geohub.lacity.org/ |
| The National Risk Index | National Risk Index \| FEMA.gov |
| Bureau of the Census | https://data.census.gov/ |
| South Carolina | $\mathrm{https}: / / \mathrm{www} . \mathrm{sc} . \mathrm{edu} /$ |
| World Bank | $\mathrm{https}: / /$ data.worldbank.org/ |
4.2 Risk Analysis 4.2 风险分析
4.2.1 Expected Annual Loss (EAL) 4.2.1 预计年度损失(EAL)
The EAL value are in units of dollars, representing the community’s average economic loss from natural hazards each year. EAL is calculated using a multiplicative equation that considers the consequence risk factors of natural hazard exposure, HLR (Historic Loss Ratio), and the likelihood risk factor of natural hazard annualized frequency. The EAL value for each consequence type is calculated by multiplying the exposure value of an area by the estimated annualized frequency and the HLR. EAL 值以美元为单位,代表社区每年因自然灾害造成的平均经济损失。EAL 采用乘法等式计算,该等式考虑了自然灾害暴露的后果风险因子、HLR(历史损失率)和自然灾害年化频率的可能性风险因子。每个后果类型的 EAL 值都是通过将一个地区的暴露值乘以估计年化频率和 HLR 计算得出的。
EAL=" Exposure "xx" Annualized Frequency "xx HLRE A L=\text { Exposure } \times \text { Annualized Frequency } \times H L R
The total EAL value for each hazard type is the sum of three different types of consequences: population §, building (B), and agriculture (A). 每种危害类型的总 EAL 值是三种不同类型后果的总和:人口 §、建筑 (B) 和农业 (A)。
EAL_("Hazard Type ")=EAL_("Hazard Type "(P))+EAL_("Hazard Type "(B))+EAL_("Hazard Type "(A))E A L_{\text {Hazard Type }}=E A L_{\text {Hazard Type }(P)}+E A L_{\text {Hazard Type }(B)}+E A L_{\text {Hazard Type }(A)}
We add up the EAL values for 18 types of hazards to obtain the composite EAL value. 我们将 18 种危害的 EAL 值相加,得出综合 EAL 值。
EAL_("Composite ")=sum_(i=1)^(18)EAL_("Hazard Type ")E A L_{\text {Composite }}=\sum_{i=1}^{18} E A L_{\text {Hazard Type }}
In order to describe our process of calculating the value of EAL more clearly. We draw the flowchart Figure 2 below. 为了更清楚地描述我们计算 EAL 值的过程。我们绘制了下图 2 流程图。
Figure 2: Calculation procedure for EAL 图 2:EAL 的计算程序
A national ranking is computed for the composite EAL value of each community. The resulting values are then converted into national percentiles to produce an EAL score for each community. The EAL scores are very helpful for the risk calculation in the following text. 根据每个社区的 EAL 综合值计算出全国排名。然后将得出的数值转换成全国百分位数,得出每个社区的 EAL 分数。EAL 分数非常有助于下文中的风险计算。
{:[EAL Rank=Rank(EAL_("Composite "))],[EALScore=(EAL" Score "-Min(EAL" Score "))/(Max(EAL" Score ")-Min(EAL" Score "))xx100]:}\begin{gathered}
E A L \operatorname{Rank}=\operatorname{Rank}\left(E A L_{\text {Composite }}\right) \\
E A L S c o r e=\frac{E A L \text { Score }-\operatorname{Min}(E A L \text { Score })}{\operatorname{Max}(E A L \text { Score })-\operatorname{Min}(E A L \text { Score })} \times 100
\end{gathered}
4.2.2 Community Risk Factor (CRF) 4.2.2 社区风险因素(CRF)
To generate a Community Risk Factor CRF (CRF) value for a community, its Social Vulnerability (SV) value is divided by its Community Resilience (CR) value. 要生成一个社区的社区风险因子 CRF 值,需要用其社会脆弱性 (SV) 值除以社区恢复力 (CR) 值。
CRF=f((SV)/(CR))C R F=f\left(\frac{S V}{C R}\right)
where f((SV)/(CR))f\left(\frac{S V}{C R}\right) is a triangular distribution with minimum 0.5 , maximum 2 , and mode 1 . 其中 f((SV)/(CR))f\left(\frac{S V}{C R}\right) 是一个三角形分布,最小值为 0.5,最大值为 2,模式为 1。
The selection process for the CRF involved evaluating various shapes, ranges, and modes. Considering its ability to highlight communities at both ends of the distribution without attributing extreme values to a select few, we use the triangular distribution. It also takes into account the EAL as the primary driver of risk, thereby making a mode of 1 the most appropriate choice for the CRF. CRF 的选择过程包括评估各种形状、范围和模式。考虑到三角形分布能够突出分布两端的群体,而不会将极端值归因于少数人,我们采用了三角形分布。它还考虑到了作为风险主要驱动因素的 EAL,从而使 1 模式成为 CRF 的最合适选择。
The SV values correspond to SOVI values, while the CR values represent HVRI BRIC index for the community, as provided by the source data sets. SV 值对应 SOVI 值,而 CR 值代表源数据集提供的 HVRI BRIC 社区指数。
4.2.3 Risk Calculation 4.2.3 风险计算
In the most general terms, natural hazard risk is often defined as the likelihood (or probability) of a natural hazard event happening multiplied by the expected consequence if a natural hazard event occurs. 最一般而言,自然灾害风险通常被定义为自然灾害事件发生的可能性(或概率)乘以自然灾害事件发生时的预期后果。
To make the level of risk more concrete and exemplified, we estimate risk from two aspects: risk value and risk score. 为了使风险水平更加具体化和例证化,我们从两个方面来估算风险:风险值和风险分值。
{:[R_("value ")=EAL_("value ")xx CRF],[R_("score ")=EAL_("score ")xx CRF]:}\begin{aligned}
& R_{\text {value }}=E A L_{\text {value }} \times C R F \\
& R_{\text {score }}=E A L_{\text {score }} \times C R F
\end{aligned}
where RR represents Risk. 其中 RR 代表风险。
This risk equation of RR includes three components: a natural hazards risk component, a consequence enhancing component, and a consequence reduction component. EAL is the natural hazards risk component, measuring the expected loss of building value, population, and/or agriculture value each year due to natural hazards. CRF incorporates both the consequence enhancing component, denoted as SV, and the consequence reduction component, represented by CR. RR 的风险等式包括三个部分:自然灾害风险部分、后果增强部分和后果减轻部分。EAL 是自然灾害风险部分,衡量每年因自然灾害造成的建筑物价值、人口和/或农业价值的预期损失。CRF 包含增强后果部分(以 SV 表示)和减少后果部分(以 CR 表示)。
For equation (8), values for Risk and EAL are in units of dollars, representing the community’s average economic loss from natural hazards each year. For SV and CR, values are the index values for the community provided by the source data sets. 对于公式 (8),Risk 和 EAL 的值以美元为单位,代表社区每年因自然灾害造成的平均经济损失。对于 SV 和 CR,数值是源数据集提供的社区指数值。
For equation (9), each component is represented by a score that represents a community’s percentile ranking relative to all other communities at the same level. The composite risk score is calculated to measure a community’s risk to all 18 hazard types. And it is also a community’s percentile ranking in risk compared to all other communities at the same level. The risk score and EAL score are provided as both composite scores from the summation of all 18 hazard types. 对于公式 (9),每个组成部分都用一个分值来表示,该分值代表一个社区相对于同一级别的所有其他社区的百分位数排名。计算综合风险分值是为了衡量一个社区在所有 18 种危害类型中的风险。它也是一个社区与同级别所有其他社区相比的风险百分位数排名。风险分数和 EAL 分数都是由所有 18 种危害类型的总和得出的综合分数。
We used the K-algorithm to categorize the disaster levels into five classes, as shown in Figure 3. 如图 3 所示,我们使用 K 算法将灾害等级划分为五个等级。
Figure 3: Risk rating and map presentation 图 3:风险评级和地图展示
4.3 Insurance Premium 4.3 保险费
Generally, the companies adopt the principle of Level Premium to determine the price of insurance. The company calculate the pure premium by using a pre-determined claim rate and a desired return on investment. The total premium is then obtained by adding surcharges at a certain expense ratio. Then, the total premium is obtained by adding surcharges at a certain expense ratio. Insurance companies increase their profitability by increasing expense ratios and reducing expected returns on investment. In addition, they utilize the ‘Law of Large Numbers’ to set the overall payout ratio of the product so that it basically matches the statistical data, thus reducing the company’s exposure to the risk of fluctuating payout ratios. For catastrophe insurance pricing, a natural disaster can cause huge property damage when it occurs. The insurance company may have a risk of becoming insolvent. 一般来说,公司采用 "水平保费 "原则来确定保险价格。公司通过预先确定的索赔率和期望的投资回报率来计算纯保费。然后,再按一定的费用率加上附加费用,得出总保费。然后,再按一定的费用比率加上附加费用,得出总保费。保险公司通过提高费用率和降低预期投资回报率来提高盈利能力。此外,他们还利用 "大数法则 "来设定产品的总体赔付率,使其与统计数据基本吻合,从而降低公司面临的赔付率波动风险。就巨灾保险定价而言,自然灾害发生时会造成巨大的财产损失。保险公司可能有破产的风险。
In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. 在金融学中,资本资产定价模型(CAPM)是一个用于确定理论上适当的资产必要收益率的模型,以做出将资产加入分散投资组合的决策。
The reward-to-risk ratio for any individual security in the market is equal to the market reward-to-risk ratio, thus 市场上任何单个证券的收益风险比都等于市场收益风险比,因此
{:[(E(r_(i))-r_(f))/(beta_(i))=E(r_(m))-r_(f)],[beta_(i)=(Cov(R_(i),R_(m)))/(Var(R_(m)))=rho(i","m)(sigma_(i))/(sigma_(m))]:}\begin{array}{r}
\frac{E\left(r_{i}\right)-r_{f}}{\beta_{i}}=E\left(r_{m}\right)-r_{f} \\
\beta_{i}=\frac{\operatorname{Cov}\left(R_{i}, R_{m}\right)}{\operatorname{Var}\left(R_{m}\right)}=\rho(i, m) \frac{\sigma_{i}}{\sigma_{m}}
\end{array}
where 其中
E(r_(i))E\left(r_{i}\right) is the expected return on the capital asset, E(r_(i))E\left(r_{i}\right) 是资本资产的预期收益、
E(r_(m))E\left(r_{m}\right) is the expected return of the market, E(r_(m))E\left(r_{m}\right) 是市场的预期收益、
r_(f)r_{f} is the risk-free rate of interest such as interest arising from government bonds, r_(f)r_{f} 是无风险利率,如政府债券产生的利息、
beta_(i)\beta_{i} is the sensitivity of the expected excess asset returns to the expected excess mar- beta_(i)\beta_{i} 是预期超额资产回报对预期超额市场回报的敏感度。
ket returns, ket 返回、
rho(i,m)\rho(i, m) denotes the correlation coefficient between the investment ii and the market mm, rho(i,m)\rho(i, m) 表示投资 ii 与市场 mm 之间的相关系数、
sigma_(i)\sigma_{i} is the standard deviation for the investment ii, sigma_(i)\sigma_{i} 是投资 ii 的标准偏差、
sigma_(m)\sigma_{m} is the standard deviation for the investment mm. sigma_(m)\sigma_{m} 是投资 mm 的标准偏差。
Expected return on investment (ROI): 预期投资回报率 (ROI):
ROI=int(-A xx l(p)+(1+x)A xxR_("value ")-I)/(I)f(p)dpR O I=\int \frac{-A \times l(p)+(1+x) A \times R_{\text {value }}-I}{I} f(p) d p
Variance of ROI: 投资回报率的差异:
Var(ROI)=int[(-A xx l(p)+A(1+x)xxR_("value ")-l-(Ax xxR_("value ")-l))/(I)]^(2)f(p)dp\operatorname{Var}(R O I)=\int\left[\frac{-A \times l(p)+A(1+x) \times R_{\text {value }}-l-\left(A x \times R_{\text {value }}-l\right)}{I}\right]^{2} f(p) d p
where 其中
AA is the sum insured, AA 为保险金额、
II is the invested capital, II 是投资资本、
l(p)l(p) is the loss function, l(p)l(p) 是损失函数、
xx is the surcharge rate. xx 是附加费率。
The surcharge rate xx is a multiple of the average value of the loss: 附加费率 xx 是损失平均值的倍数:
where sigma\sigma is the Standard deviation of R_("value ")R_{\text {value }}. 其中, sigma\sigma 是 R_("value ")R_{\text {value }} 的标准偏差。
Pure premium per $10,000 y\$ 10,000 y : 每 $10,000 y\$ 10,000 y 纯溢价:
y=10000(1+x)xx Py=10000(1+x) \times P
where PP is the probability of a disaster causing damage. 其中, PP 是灾害造成损失的概率。
Typically, it is more reasonable to spend 3-10 percent of each person’s annual income on insurance. We assume that each person is willing to spend 5%5 \% of his or her annual income each year to purchase catastrophe insurance with a one-year term. Insurance companies can make decisions from two perspectives based on the above formula: 通常情况下,将每个人年收入的 3%-10%用于购买保险更为合理。我们假设每人每年愿意花费其年收入的 5%5 \% 来购买为期一年的巨灾保险。保险公司可以根据上述公式从两个角度做出决策: >> Introducing bankruptcy theory, after calculating the lowest order price, y , in the case where the probability of the firm’s future bankruptcy is less than 10%10 \%, and then comparing it to the local per capita annual disposable income (GNI), it is expected that people in the locality will not be able to afford to consume catastrophe insurance and will not invest in it if the ratio of premiums per $10,000\$ 10,000 to GNI is greater than 5%5 \%. >> 引入破产理论,在计算出企业未来破产概率小于 10%10 \% 的情况下的最低订单价格y后,再与当地人均年可支配收入(GNI)进行比较,如果每 $10,000\$ 10,000 的保费与GNI之比大于 5%5 \% ,预计当地居民将无力消费巨灾保险,也不会投资巨灾保险。 >> We use 5%5 \% of the local national GNI per capita as the subscription price per $10,000\$ 10,000 of premium. If this price makes the likelihood of future insolvency of the company higher than 10%10 \%, no investment is made in that location. >> 我们用当地人均国民总收入的 5%5 \% 作为每 $10,000\$ 10,000 溢价的认购价格。如果该价格使公司未来破产的可能性高于 10%10 \% ,则不在该地投资。
The price of insurance also affects people’s desire to buy to some extent, and an increase in the price of insurance may lead to a decrease in their desire to buy. 保险价格也会在一定程度上影响人们的购买欲望,保险价格的上涨可能会导致人们购买欲望的下降。
N=(1-omega y)N_(A)N=(1-\omega y) N_{A}
" Total Revenue "=y xx N\text { Total Revenue }=y \times N