Modeling Correlated Discrete Uncertainties in Event Trees with Copulas

Modeling the dependence between uncertainties in decision and risk analyses is an important part of the problem structuring process. We focus on situations where correlated uncertainties are discrete, and extend the concept of the copula‐based approach for modeling correlated continuous uncertainties to the representation of correlated discrete uncertainties. This approach reduces the required number of probability assessments significantly compared to approaches requiring direct estimates of conditional probabilities. It also allows the use of multiple dependence measures, including product moment correlation, rank order correlation and tail dependence, and parametric families of copulas such as normal copulas, t‐copulas, and Archimedean copulas. This approach can be extended to model the dependence between discrete and continuous uncertainties in the same event tree.     

An Empirical,Nonparametric Simulator for Multivariate Random Variables with Differing Marginal Densities and Nonlinear Dependence with Hydroclimatic Applications

Multivariate simulations of a set of random variables are often needed for risk analysis. Given a historical data set, the goal is to develop simulations that reproduce the dependence structure in that data set so that the risk of potentially correlated factors can be evaluated. A nonparametric, copula‐based simulation approach is developed and exemplified. It can be applied to multiple variables or to spatial fields with arbitrary dependence structures and marginal densities. The nonparametric simulator uses logspline density estimation in the univariate setting, together with a sampling strategy to reproduce dependence across variables or spatial instances, through a nonparametric numerical approximation of the underlying copula function. The multivariate data vectors are assumed to be independent and identically distributed. A synthetic example is provided to illustrate the method, followed by an application to the risk of livestock losses in Mongolia.     

Probabilistic,Multivariable Flood Loss Modeling on the Mesoscale with BT‐FLEMO

Flood loss modeling is an important component for risk analyses and decision support in flood risk management. Commonly, flood loss models describe complex damaging processes by simple, deterministic approaches like depth‐damage functions and are associated with large uncertainty. To improve flood loss estimation and to provide quantitative information about the uncertainty associated with loss modeling, a probabilistic, multivariable B agging decision T ree F lood L oss E stimation MO del (BT‐FLEMO) for residential buildings was developed. The application of BT‐FLEMO provides a probability distribution of estimated losses to residential buildings per municipality. BT‐FLEMO was applied and validated at the mesoscale in 19 municipalities that were affected during the 2002 flood by the River Mulde in Saxony, Germany. Validation was undertaken on the one hand via a comparison with six deterministic loss models, including both depth‐damage functions and multivariable models. On the other hand, the results were compared with official loss data. BT‐FLEMO outperforms deterministic, univariable, and multivariable models with regard to model accuracy, although the prediction uncertainty remains high. An important advantage of BT‐FLEMO is the quantification of prediction uncertainty. The probability distribution of loss estimates by BT‐FLEMO well represents the variation range of loss estimates of the other models in the case study.     

典型事实、混合Copula函数与金融市场相依结构研究

运用ARFIMA-FIAPARCH-skst模型对沪深300指数和香港恒生指数建立收益-波动模型, 然后结合估计的参数对模型进行修正以确立最终模型, 排除金融市场典型事实对相依关系的影响, 进而运用由Clayton、Frank和Gumbel组成的混合copula模型对相依结构进行建模。研究结果表明:内地市场和香港市场均未观察到显著的杠杆效应;由Clayton、Frank和Gumbel组成的混合Copula模型能够准确地描述两个市场之间的相依结构, 且两个市场下尾相依关系要强于上尾的相依关系, 通过动态混合copula也验证了这一明显的非对称关系。     

A Probabilistic Framework for Risk Analysis of Widespread Flood Events: A Proof‐of‐Concept Study

This article presents a flood risk analysis model that considers the spatially heterogeneous nature of flood events. The basic concept of this approach is to generate a large sample of flood events that can be regarded as temporal extrapolation of flood events. These are combined with cumulative flood impact indicators, such as building damages, to finally derive time series of damages for risk estimation. Therefore, a multivariate modeling procedure that is able to take into account the spatial characteristics of flooding, the regionalization method top‐kriging, and three different impact indicators are combined in a model chain. Eventually, the expected annual flood impact (e.g., expected annual damages) and the flood impact associated with a low probability of occurrence are determined for a study area. The risk model has the potential to augment the understanding of flood risk in a region and thereby contribute to enhanced risk management of, for example, risk analysts and policymakers or insurance companies. The modeling framework was successfully applied in a proof‐of‐concept exercise in Vorarlberg (Austria). The results of the case study show that risk analysis has to be based on spatially heterogeneous flood events in order to estimate flood risk adequately.     

Copulas and Temporal Dependence

An emerging literature in time series econometrics concerns the modeling of potentially nonlinear temporal dependence in stationary Markov chains using copula functions. We obtain sufficient conditions for a geometric rate of mixing in models of this kind. Geometric β‐mixing is established under a rather strong sufficient condition that rules out asymmetry and tail dependence in the copula function. Geometric ρ‐mixing is obtained under a weaker condition that permits both asymmetry and tail dependence. We verify one or both of these conditions for a range of parametric copula functions that are popular in applied work.     

国际油气价格与汇率动态相依关系研究:基于一种新的时变最优Copula模型

本文提出一个新的时变最优Copula模型,可以准确识别二元时间序列任意时点最优的相依结构。该模型构造了半旋转copula以刻画非对称的反向相依关系,并引入独立性的无分布检验证实相依关系的存在性。同时,我们对能源商品市场（原油、天然气）、外汇市场间动态相依关系进行了实证分析,实证结果表明跨市场相依结构类型确实是时变的,突发事件往往是相依结构突变的主因。另外,时变最优Copula模型的主要优势在于不仅能够捕捉相依方向和相依强度的动态性,还能有效捕捉相依结构类型的动态性。     

Implementation of Equity in Resource Allocation for Regional Earthquake Risk Mitigation Using Two‐Stage Stochastic Programming

This article presents a new methodology to implement the concept of equity in regional earthquake risk mitigation programs using an optimization framework. It presents a framework that could be used by decisionmakers (government and authorities) to structure budget allocation strategy toward different seismic risk mitigation measures, i.e., structural retrofitting for different building structural types in different locations and planning horizons. A two‐stage stochastic model is developed here to seek optimal mitigation measures based on minimizing mitigation expenditures, reconstruction expenditures, and especially large losses in highly seismically active countries. To consider fairness in the distribution of financial resources among different groups of people, the equity concept is incorporated using constraints in model formulation. These constraints limit inequity to the user‐defined level to achieve the equity‐efficiency tradeoff in the decision‐making process. To present practical application of the proposed model, it is applied to a pilot area in Tehran, the capital city of Iran. Building stocks, structural vulnerability functions, and regional seismic hazard characteristics are incorporated to compile a probabilistic seismic risk model for the pilot area. Results illustrate the variation of mitigation expenditures by location and structural type for buildings. These expenditures are sensitive to the amount of available budget and equity consideration for the constant risk aversion. Most significantly, equity is more easily achieved if the budget is unlimited. Conversely, increasing equity where the budget is limited decreases the efficiency. The risk‐return tradeoff, equity‐reconstruction expenditures tradeoff, and variation of per‐capita expected earthquake loss in different income classes are also presented.     

A Blueprint for Full Collective Flood Risk Estimation: Demonstration for European River Flooding

Floods are a natural hazard evolving in space and time according to meteorological and river basin dynamics, so that a single flood event can affect different regions over the event duration. This physical mechanism introduces spatio‐temporal relationships between flood records and losses at different locations over a given time window that should be taken into account for an effective assessment of the collective flood risk. However, since extreme floods are rare events, the limited number of historical records usually prevents a reliable frequency analysis. To overcome this limit, we move from the analysis of extreme events to the modeling of continuous stream flow records preserving spatio‐temporal correlation structures of the entire process, and making a more efficient use of the information provided by continuous flow records. The approach is based on the dynamic copula framework, which allows for splitting the modeling of spatio‐temporal properties by coupling suitable time series models accounting for temporal dynamics, and multivariate distributions describing spatial dependence. The model is applied to 490 stream flow sequences recorded across 10 of the largest river basins in central and eastern Europe (Danube, Rhine, Elbe, Oder, Waser, Meuse, Rhone, Seine, Loire, and Garonne). Using available proxy data to quantify local flood exposure and vulnerability, we show that the temporal dependence exerts a key role in reproducing interannual persistence, and thus magnitude and frequency of annual proxy flood losses aggregated at a basin‐wide scale, while copulas allow the preservation of the spatial dependence of losses at weekly and annual time scales.     

The Effects of Revealed Information on Catastrophe Loss Projection Models’ Characterization of Risk: Damage Vulnerability Evidence from Florida

We examine whether the risk characterization estimated by catastrophic loss projection models is sensitive to the revelation of new information regarding risk type. We use commercial loss projection models from two widely employed modeling firms to estimate the expected hurricane losses of Florida Atlantic University's building stock, both including and excluding secondary information regarding hurricane mitigation features that influence damage vulnerability. We then compare the results of the models without and with this revealed information and find that the revelation of additional, secondary information influences modeled losses for the windstorm‐exposed university building stock, primarily evidenced by meaningful percent differences in the loss exceedance output indicated after secondary modifiers are incorporated in the analysis. Secondary risk characteristics for the data set studied appear to have substantially greater impact on probable maximum loss estimates than on average annual loss estimates. While it may be intuitively expected for catastrophe models to indicate that secondary risk characteristics hold value for reducing modeled losses, the finding that the primary value of secondary risk characteristics is in reduction of losses in the “tail” (low probability, high severity) events is less intuitive, and therefore especially interesting. Further, we address the benefit‐cost tradeoffs that commercial entities must consider when deciding whether to undergo the data collection necessary to include secondary information in modeling. Although we assert the long‐term benefit‐cost tradeoff is positive for virtually every entity, we acknowledge short‐term disincentives to such an effort.     

Risk Finance for Catastrophe Losses with Pareto‐Calibrated Lévy‐Stable Severities

For catastrophe losses, the conventional risk finance paradigm of enterprise risk management identifies transfer, as opposed to pooling or avoidance, as the preferred solution. However, this analysis does not necessarily account for differences between light‐ and heavy‐tailed characteristics of loss portfolios. Of particular concern are the decreasing benefits of diversification (through pooling) as the tails of severity distributions become heavier. In the present article, we study a loss portfolio characterized by nonstochastic frequency and a class of Lévy‐stable severity distributions calibrated to match the parameters of the Pareto II distribution. We then propose a conservative risk finance paradigm that can be used to prepare the firm for worst‐case scenarios with regard to both (1) the firm's intrinsic sensitivity to risk and (2) the heaviness of the severity's tail.     

国际油价与中美股价的相依关系研究——基于不同行业数据的分析

石油市场和股票市场作为现代经济中两个重要的市场，在经济活动中发挥着重要的作用。二者之间的关系对研究市场间的价格波动和风险传递有着重要的意义，本文通过vine copula模型对国际油价和中美两国股价之间相依关系进行分析，并将得到的相依关系运用到风险管理中。利用国际油价和中美各十个行业股票价格指数进行相依关系建模，得到相应的相依结构和相依关系，选择出与油价相依关系较强的行业股票价格指数和油价构建投资组合，利用相依关系模拟出收益率数据，度量投资组合的风险。实证研究结果表明中美两国的行业股票价格指数与国际油价的相依关系存在着显著的不同，中国行业股票价格指数与国际油价之间的相依关系要弱于美国行业股票价格指数与国际油价的相依关系；同时利用相依关系组成投资组合，对两组投资组合进行风险度量，风险度量的结果显示vine copula-GARCH能对具有较强的相依关系的变量组成的投资组合风险有很好的估计。     

Exploring the Potential for Multivariate Fragility Representations to Alter Flood Risk Estimates

In flood risk analysis, limitations in the multivariate statistical models adopted to model the hydraulic load have restricted the probability of a defense suffering structural failure to be expressed conditionally on a single hydraulic loading variable. This is an issue at the coastal level where multiple loadings act on defenses with the exact combination of loadings dictating their failure probabilities. Recently, a methodology containing a multivariate statistical model with the flexibility to robustly capture the dependence structure between the individual loadings was used to derive extreme nearshore loading conditions. Its adoption will permit the incorporation of more precise representations of a structure's vulnerability in future analyses. In this article, a fragility representation of a shingle beach, where the failure probability is expressed over a three‐dimensional loading parameter space—water level, wave height, and period—is derived at two localities. Within the approach, a Gaussian copula is used to capture any dependencies between the simplified geometric parameters of a beach's shape. Beach profiles are simulated from the copula and the failure probability, given the hydraulic load, determined by the reformulated Bradbury barrier inertia parameter model. At one site, substantial differences in the annual failure probability distribution are observed between the new and existing approaches. At the other, the beach only becomes vulnerable after a significant reduction of the crest height with its mean annual failure probability close to that presently predicted. It is concluded that further application of multivariate approaches is likely to yield more effective flood risk management.     

Integrated Direct and Indirect Flood Risk Modeling: Development and Sensitivity Analysis

In this article, we propose an integrated direct and indirect flood risk model for small‐ and large‐scale flood events, allowing for dynamic modeling of total economic losses from a flood event to a full economic recovery. A novel approach is taken that translates direct losses of both capital and labor into production losses using the Cobb‐Douglas production function, aiming at improved consistency in loss accounting. The recovery of the economy is modeled using a hybrid input‐output model and applied to the port region of Rotterdam, using six different flood events (1/10 up to 1/10,000). This procedure allows gaining a better insight regarding the consequences of both high‐ and low‐probability floods. The results show that in terms of expected annual damage, direct losses remain more substantial relative to the indirect losses (approximately 50% larger), but for low‐probability events the indirect losses outweigh the direct losses. Furthermore, we explored parameter uncertainty using a global sensitivity analysis, and varied critical assumptions in the modeling framework related to, among others, flood duration and labor recovery, using a scenario approach. Our findings have two important implications for disaster modelers and practitioners. First, high‐probability events are qualitatively different from low‐probability events in terms of the scale of damages and full recovery period. Second, there are substantial differences in parameter influence between high‐probability and low‐probability flood modeling. These findings suggest that a detailed approach is required when assessing the flood risk for a specific region.     

重尾索赔条件下现代风险模型的破产概率估计:多险种混合情形

破产概率是非寿险保险风险理论的核心问题。与经典的Cramér-Lundberg模型相比, 由Li Zehui等建立的现代风险模型更为准确地描述了非寿险保险运营的主要特征, 对现实保险业务具有较好的解释力。本文基于现代风险模型, 考虑保险公司多个险种混合经营这一更为现实的情形, 在索赔额服从正则尾分布条件下获得了破产概率的渐近等价估计。我们发现, 在具有大额索赔特征的多个险种混合的条件下, 公司面临的极端索赔风险将由索赔额分布尾部最厚的那些险种决定, 而索赔额分布尾部相对较薄的那些险种的影响作用将被淹没。该结论的有效性可用MATLAB数值模拟得到理想的验证。本文结果是对风险模型研究的重要推广, 也为多险种混合情形下保险公司的风险控制与初始保证金界定提供了依据。     

内生性回收率与信用风险度量研究

在信用风险模型中,外生性回收率的设定会忽略回收率对损失分布尾部的影响,而且会导致潜在的模型风险。本文将因子扩散过程引入结构信用风险模型,获得了回收率和违约概率之间的内在关系,利用Monte Carlo模拟方法数值分析了预期回收率对违约概率和资产价值波动率的依赖性,结果表明预期回收率与违约率之间具有很强的负相关关系,而且这种相关关系会受到债务人资产价值波动率的正向影响。在内生性回收率下,推导了信用损失的概率分布,计算了信用风险的Credit-VaR和ETF指标。最后利用市场数据检验了内生回收率信用风险模型的有效性,结果表明该模型可以很好的描述历史违约率和回收率的变化过程。     

Modeling Insurer‐Homeowner Interactions in Managing Natural Disaster Risk

The current system for managing natural disaster risk in the United States is problematic for both homeowners and insurers. Homeowners are often uninsured or underinsured against natural disaster losses, and typically do not invest in retrofits that can reduce losses. Insurers often do not want to insure against these losses, which are some of their biggest exposures and can cause an undesirably high chance of insolvency. There is a need to design an improved system that acknowledges the different perspectives of the stakeholders. In this article, we introduce a new modeling framework to help understand and manage the insurer's role in catastrophe risk management. The framework includes a new game‐theoretic optimization model of insurer decisions that interacts with a utility‐based homeowner decision model and is integrated with a regional catastrophe loss estimation model. Reinsurer and government roles are represented as bounds on the insurer‐insured interactions. We demonstrate the model for a full‐scale case study for hurricane risk to residential buildings in eastern North Carolina; present the results from the perspectives of all stakeholders—primary insurers, homeowners (insured and uninsured), and reinsurers; and examine the effect of key parameters on the results.     

Application of a Regional Hurricane Wind Risk Forecasting Model for Wood-Frame Houses

Hurricane wind risk in a region changes over time due to changes in the number, type, locations, vulnerability, and value of buildings. A model was developed to quantitatively estimate changes over time in hurricane wind risk to wood-frame houses (defined in terms of potential for direct economic loss), and to estimate how different factors, such as building code changes and population growth, contribute to that change. The model, which is implemented in a simulation, produces a probability distribution of direct economic losses for each census tract in the study region at each time step in the specified time horizon. By changing parameter values and rerunning the analysis, the effects of different changes in the built environment on the hurricane risk trends can be estimated and the relative effectiveness of hypothetical mitigation strategies can be evaluated. Using a case study application for wood-frame houses in selected counties in North Carolina from 2000 to 2020, this article demonstrates how the hurricane wind risk forecasting model can be used: (1) to provide insight into the dynamics of regional hurricane wind risk-the total change in risk over time and the relative contribution of different factors to that change, and (2) to support mitigation planning. Insights from the case study include, for example, that the many factors contributing to hurricane wind risk for wood-frame houses interact in a way that is difficult to predict a priori, and that in the case study, the reduction in hurricane losses due to vulnerability changes (e.g., building code changes) is approximately equal to the increase in losses due to building inventory growth. The potential for the model to support risk communication is also discussed.     

The Impact of Portfolio Location Uncertainty on Probabilistic Seismic Risk Analysis

Probabilistic seismic risk analysis is a well‐established method in the insurance industry for modeling portfolio losses from earthquake events. In this context, precise exposure locations are often unknown. However, so far, location uncertainty has not been in the focus of a large amount of research. In this article, we propose a novel framework for treatment of location uncertainty. As a case study, a large number of synthetic portfolios resembling typical real‐world cases were created. We investigate the effect of portfolio characteristics such as value distribution, portfolio size, or proportion of risk items with unknown coordinates on the variability of loss frequency estimations. The results indicate that due to loss aggregation effects and spatial hazard variability, location uncertainty in isolation and in conjunction with ground motion uncertainty can induce significant variability to probabilistic loss results, especially for portfolios with a small number of risks. After quantifying its effect, we conclude that location uncertainty should not be neglected when assessing probabilistic seismic risk, but should be treated stochastically and the resulting variability should be visualized and interpreted carefully.     

MATERIALS PRICES AND PRODUCTIVITY

There is substantial within‐industry variation in the prices that plants pay for their material inputs. Using plant‐level data from the US Census Bureau, I explore the consequences and sources of this variation in materials prices. For a sample of industries with relatively homogeneous products, the standard deviation of plant‐level productivity would be 7% smaller if all plants faced the same materials prices. Moreover, plant‐level materials prices are persistent, spatially correlated, and positively associated with the probability of exit. The contribution of entry and exit to aggregate productivity growth is smaller for productivity measures that are purged of materials price variation. After documenting these patterns, I discuss three potential sources of materials price variation: geography, differences in suppliers' marginal costs, and within‐supplier markup differences. Together, these variables explain 15% of the variation of materials prices.