How to Design Rating Schemes of Risk Matrices: A Sequential Updating Approach

Risk matrices have been widely used as a risk evaluation tool in many fields due to their simplicity and intuitive nature. Designing a rating scheme, i.e., determining the number of ratings used in a risk matrix and assigning different ratings to different cells, is an essential part of risk matrix construction. However, most of the related literature has focused on applying a risk matrix to various fields, instead of researching how to design risk matrices. Based on the analysis of several current rules, we propose a new approach, namely, the sequential updating approach (SUA), to design the rating scheme of a risk matrix in a reliable way. In this article, we propose three principles and a rating algorithm based on these principles. The three principles, namely, adjusted weak consistency, consistent internality, and continuous screening, characterize a good rating scheme. The resulting rating scheme has been proven to be unique. A global rating algorithm is then proposed to create the design that satisfies the three principles. We then explore the performance of the SUA. An illustrative application is first given to explain the feasibility of our approach. The sensitivity analysis shows that our method captures a resolution‐reliability tradeoff for decisionmakers in choosing an appropriate rating scheme for a risk matrix. Finally, we compare the designs based on the SUA and Cox's axioms, highlighting the advantages of the SUA.     

期望效用视角下的风险对冲效率

与传统文献将风险下降比率作为风险对冲效率指标不同,本文引入期望效用理论来比较最小方差对冲策略、最小在险价值(VaR)对冲策略和最小条件在险价值(CVaR)对冲策略的对冲效率,从而将人们的风险态度同对冲策略选择联系起来,以实现不同风险态度的投资者选择不同风险对冲策略的目的。借用风险中性效用函数、二次效用函数和CARA效用函数,本文严格证明:在这三种对冲策略中,最小方差对冲策略过于保守,最小VaR对冲策略最为激进,风险厌恶程度大的投资者偏好最小方差对冲策略,风险中性投资者和风险厌恶程度小的投资者更偏好最小VaR对冲策略,最小CVaR对冲策略介于二者之间。     

Why Risk Is Not Variance: An Expository Note

Variance (or standard deviation) of return is widely used as a measure of risk in financial investment risk analysis applications, where mean‐variance analysis is applied to calculate efficient frontiers and undominated portfolios. Why, then, do health, safety, and environmental (HS&E) and reliability engineering risk analysts insist on defining risk more flexibly, as being determined by probabilities and consequences, rather than simply by variances? This note suggests an answer by providing a simple proof that mean‐variance decision making violates the principle that a rational decisionmaker should prefer higher to lower probabilities of receiving a fixed gain, all else being equal. Indeed, simply hypothesizing a continuous increasing indifference curve for mean‐variance combinations at the origin is enough to imply that a decisionmaker must find unacceptable some prospects that offer a positive probability of gain and zero probability of loss. Unlike some previous analyses of limitations of variance as a risk metric, this expository note uses only simple mathematics and does not require the additional framework of von Neumann Morgenstern utility theory.     

Probabilistic Risk Analysis and Terrorism Risk

Since the terrorist attacks of September 11, 2001, and the subsequent establishment of the U.S. Department of Homeland Security (DHS), considerable efforts have been made to estimate the risks of terrorism and the cost effectiveness of security policies to reduce these risks. DHS, industry, and the academic risk analysis communities have all invested heavily in the development of tools and approaches that can assist decisionmakers in effectively allocating limited resources across the vast array of potential investments that could mitigate risks from terrorism and other threats to the homeland. Decisionmakers demand models, analyses, and decision support that are useful for this task and based on the state of the art. Since terrorism risk analysis is new, no single method is likely to meet this challenge. In this article we explore a number of existing and potential approaches for terrorism risk analysis, focusing particularly on recent discussions regarding the applicability of probabilistic and decision analytic approaches to bioterrorism risks and the Bioterrorism Risk Assessment methodology used by the DHS and criticized by the National Academies and others.     

随机组合风险的风险溢价

随机组合风险在保险索赔理论、金融及经济管理等领域有广泛的应用,数学上采用随机和来刻画随机组合风险。风险溢价在金融经济学及保险经济学的理论中都是很重要的概念,它不仅与风险的大小有关,还与当事人对风险的态度有关,从理论上看就是与当事人的效用函数有关。本文研究在期望效用理论下随机组合风险的风险溢价问题,探讨了由组合数(如索赔次数)的不确定性所引起的风险溢价,给出了几种不同效用函数下随机组合风险的风险溢价的计算公式,并特别针对随机Poisson组合及随机Poisson-Geometric组合给出了其风险溢价的计算公式及性质。     

Preference Functions for Spatial Risk Analysis

When outcomes are defined over a geographic region, measures of spatial risk regarding these outcomes can be more complex than traditional measures of risk. One of the main challenges is the need for a cardinal preference function that incorporates the spatial nature of the outcomes. We explore preference conditions that will yield the existence of spatial measurable value and utility functions, and discuss their application to spatial risk analysis. We also present a simple example on household freshwater usage across regions to demonstrate how such functions can be assessed and applied.     

Decision Making Under Uncertainty—An Example for Seismic Risk Management

Decision-making techniques are used to select the "best" alternatives under multiple and often conflicting criteria. Multicriteria decision making (MCDM) necessitates to incorporate uncertainties in the decision-making process. The major thrust of this article is to extend the framework proposed by Yager^{( 1 )} for multiple decisionmakers and fuzzy utilities (payoffs). In addition, the concept of expert credibility factor is introduced. The proposed approach is demonstrated for an example of seismic risk management using a heuristic hierarchical structure. A step-by-step formulation of the proposed approach is illustrated using a hypothetical example and a three-story reinforced concrete building.     

A Multicriteria Decision Analysis Model and Risk Assessment Framework for Carbon Capture and Storage

Multicriteria decision analysis (MCDA) has been applied to various energy problems to incorporate a variety of qualitative and quantitative criteria, usually spanning environmental, social, engineering, and economic fields. MCDA and associated methods such as life‐cycle assessments and cost‐benefit analysis can also include risk analysis to address uncertainties in criteria estimates. One technology now being assessed to help mitigate climate change is carbon capture and storage (CCS). CCS is a new process that captures CO₂ emissions from fossil‐fueled power plants and injects them into geological reservoirs for storage. It presents a unique challenge to decisionmakers (DMs) due to its technical complexity, range of environmental, social, and economic impacts, variety of stakeholders, and long time spans. The authors have developed a risk assessment model using a MCDA approach for CCS decisions such as selecting between CO₂ storage locations and choosing among different mitigation actions for reducing risks. The model includes uncertainty measures for several factors, utility curve representations of all variables, Monte Carlo simulation, and sensitivity analysis. This article uses a CCS scenario example to demonstrate the development and application of the model based on data derived from published articles and publicly available sources. The model allows high‐level DMs to better understand project risks and the tradeoffs inherent in modern, complex energy decisions.     

Eliciting and Combining Decision Criteria Using a Limited Palette of Utility Functions and Uncertainty Distributions: Illustrated by Application to Pest Risk Analysis

Utility functions in the form of tables or matrices have often been used to combine discretely rated decision‐making criteria. Matrix elements are usually specified individually, so no one rule or principle can be easily stated for the utility function as a whole. A series of five matrices are presented that aggregate criteria two at a time using simple rules that express a varying degree of constraint of the lower rating over the higher. A further nine possible matrices were obtained by using a different rule either side of the main axis of the matrix to describe situations where the criteria have a differential influence on the outcome. Uncertainties in the criteria are represented by three alternative frequency distributions from which the assessors select the most appropriate. The output of the utility function is a distribution of rating frequencies that is dependent on the distributions of the input criteria. In pest risk analysis (PRA), seven of these utility functions were required to mimic the logic by which assessors for the European and Mediterranean Plant Protection Organization arrive at an overall rating of pest risk. The framework enables the development of PRAs that are consistent and easy to understand, criticize, compare, and change. When tested in workshops, PRA practitioners thought that the approach accorded with both the logic and the level of resolution that they used in the risk assessments.     

Risk Modeling of Interdependent Complex Systems of Systems: Theory and Practice

The emergence of the complexity characterizing our systems of systems (SoS) requires a reevaluation of the way we model, assess, manage, communicate, and analyze the risk thereto. Current models for risk analysis of emergent complex SoS are insufficient because too often they rely on the same risk functions and models used for single systems. These models commonly fail to incorporate the complexity derived from the networks of interdependencies and interconnectedness (I–I) characterizing SoS. There is a need to reevaluate currently practiced risk analysis to respond to this reality by examining, and thus comprehending, what makes emergent SoS complex. The key to evaluating the risk to SoS lies in understanding the genesis of characterizing I–I of systems manifested through shared states and other essential entities within and among the systems that constitute SoS. The term “essential entities” includes shared decisions, resources, functions, policies, decisionmakers, stakeholders, organizational setups, and others. This undertaking can be accomplished by building on state‐space theory, which is fundamental to systems engineering and process control. This article presents a theoretical and analytical framework for modeling the risk to SoS with two case studies performed with the MITRE Corporation and demonstrates the pivotal contributions made by shared states and other essential entities to modeling and analysis of the risk to complex SoS. A third case study highlights the multifarious representations of SoS, which require harmonizing the risk analysis process currently applied to single systems when applied to complex SoS.     

Multiattribute Risk Analysis in Nuclear Emergency Management

Radiation protection authorities have seen a potential for applying multiattribute risk analysis in nuclear emergency management and planning to deal with conflicting objectives, different parties involved, and uncertainties. This type of approach is expected to help in the following areas: to ensure that all relevant attributes are considered in decision making; to enhance communication between the concerned parties, including the public; and to provide a method for explicitly including risk analysis in the process. A multiattribute utility theory analysis was used to select a strategy for protecting the population after a simulated nuclear accident. The value-focused approach and the use of a neutral facilitator were identified as being useful.     

Implied Preference for Seismic Design Level and Earthquake Insurance

Seismic risk can be reduced by implementing newly developed seismic provisions in design codes. Furthermore, financial protection or enhanced utility and happiness for stakeholders could be gained through the purchase of earthquake insurance. If this is not so, there would be no market for such insurance. However, perceived benefit associated with insurance is not universally shared by stakeholders partly due to their diverse risk attitudes. This study investigates the implied seismic design preference with insurance options for decisionmakers of bounded rationality whose preferences could be adequately represented by the cumulative prospect theory (CPT). The investigation is focused on assessing the sensitivity of the implied seismic design preference with insurance options to model parameters of the CPT and to fair and unfair insurance arrangements. Numerical results suggest that human cognitive limitation and risk perception can affect the implied seismic design preference by the CPT significantly. The mandatory purchase of fair insurance will lead the implied seismic design preference to the optimum design level that is dictated by the minimum expected lifecycle cost rule. Unfair insurance decreases the expected gain as well as its associated variability, which is preferred by risk-averse decisionmakers. The obtained results of the implied preference for the combination of the seismic design level and insurance option suggest that property owners, financial institutions, and municipalities can take advantage of affordable insurance to establish successful seismic risk management strategies.     

A Genuine Rank‐Dependent Generalization of the Von Neumann‐Morgenstern Expected Utility Theorem

This paper uses “revealed probability trade‐offs” to provide a natural foundation for probability weighting in the famous von Neumann and Morgenstern axiomatic set‐up for expected utility. In particular, it shows that a rank‐dependent preference functional is obtained in this set‐up when the independence axiom is weakened to stochastic dominance and a probability trade‐off consistency condition. In contrast with the existing axiomatizations of rank‐dependent utility, the resulting axioms allow for complete flexibility regarding the outcome space. Consequently, a parameter‐free test/elicitation of rank‐dependent utility becomes possible. The probability‐oriented approach of this paper also provides theoretical foundations for probabilistic attitudes towards risk. It is shown that the preference conditions that characterize the shape of the probability weighting function can be derived from simple probability trade‐off conditions.     

Experimental Evidence Against the Paradigm of Mortality Risk Aversion

This article deals with the question of how societal impacts of fatal accidents can be integrated into the management of natural or man‐made hazards. Today, many governmental agencies give additional weight to the number of potential fatalities in their risk assessments to reflect society's aversion to large accidents. Although mortality risk aversion has been proposed in numerous risk management guidelines, there has been no evidence that lay people want public decisionmakers to overweight infrequent accidents of large societal consequences against more frequent ones of smaller societal consequences. Furthermore, it is not known whether public decisionmakers actually do such overweighting when they decide upon the mitigation of natural or technical hazards. In this article, we report on two experimental tasks that required participants to evaluate negative prospects involving 1–100 potential fatalities. Our results show that neither lay people nor hazard experts exhibit risk‐averse behavior in decisions on mortality risks.     

Further Thoughts on the Utility of Risk Matrices

Risk matrices are commonly encountered devices for rating hazards in numerous areas of risk management. Part of their popularity is predicated on their apparent simplicity and transparency. Recent research, however, has identified serious mathematical defects and inconsistencies. This article further examines the reliability and utility of risk matrices for ranking hazards, specifically in the context of public leisure activities including travel. We find that (1) different risk assessors may assign vastly different ratings to the same hazard, (2) even following lengthy reflection and learning scatter remains high, and (3) the underlying drivers of disparate ratings relate to fundamentally different worldviews, beliefs, and a panoply of psychosocial factors that are seldom explicitly acknowledged. It appears that risk matrices when used in this context may be creating no more than an artificial and even untrustworthy picture of the relative importance of hazards, which may be of little or no benefit to those trying to manage risk effectively and rationally.     

定常风险偏好效用函数式及其参数确定问题

按von Neumann-Morgenstern期望效用理论建立风险偏好模型,构造一种关于未知效用度量的特殊函数方程,据此导出满足特定风险偏好性质的效用函数式,由于未知参数可严格确定,效用度量因此成了一种可按已知公式进行计算的问题。     

Is Risk Analysis Scientific?

This article discusses to what extent risk analysis is scientific in view of a set of commonly used definitions and criteria. We consider scientific knowledge to be characterized by its subject matter, its success in developing the best available knowledge in its fields of study, and the epistemic norms and values that guide scientific investigations. We proceed to assess the field of risk analysis according to these criteria. For this purpose, we use a model for risk analysis in which science is used as a base for decision making on risks, which covers the five elements evidence, knowledge base, broad risk evaluation, managerial review and judgment, and the decision; and that relates these elements to the domains experts and decisionmakers, and to the domains fact‐based or value‐based. We conclude that risk analysis is a scientific field of study, when understood as consisting primarily of (i) knowledge about risk‐related phenomena, processes, events, etc., and (ii) concepts, theories, frameworks, approaches, principles, methods and models to understand, assess, characterize, communicate, and manage risk, in general and for specific applications (the instrumental part).     

Integrating Household Risk Mitigation Behavior in Flood Risk Analysis: An Agent‐Based Model Approach

Recent studies showed that climate change and socioeconomic trends are expected to increase flood risks in many regions. However, in these studies, human behavior is commonly assumed to be constant, which neglects interaction and feedback loops between human and environmental systems. This neglect of human adaptation leads to a misrepresentation of flood risk. This article presents an agent‐based model that incorporates human decision making in flood risk analysis. In particular, household investments in loss‐reducing measures are examined under three economic decision models: (1) expected utility theory, which is the traditional economic model of rational agents; (2) prospect theory, which takes account of bounded rationality; and (3) a prospect theory model, which accounts for changing risk perceptions and social interactions through a process of Bayesian updating. We show that neglecting human behavior in flood risk assessment studies can result in a considerable misestimation of future flood risk, which is in our case study an overestimation of a factor two. Furthermore, we show how behavior models can support flood risk analysis under different behavioral assumptions, illustrating the need to include the dynamic adaptive human behavior of, for instance, households, insurers, and governments. The method presented here provides a solid basis for exploring human behavior and the resulting flood risk with respect to low‐probability/high‐impact risks.     

ISO 31000:2009—Setting a New Standard for Risk Management

Last year saw the publication of IS0 31000:2009, a new globally accepted standard for risk management together with a new, associated vocabulary in ISO Guide 73:2009. These were developed through a consensus‐driven process over four years, through seven drafts, and involving the input of hundreds of risk management professionals around the world. The new standard supports a new, simple way of thinking about risk and risk management and is intended to begin the process of resolving the many inconsistencies and ambiguities that exist between many different approaches and definitions. While most decisionmakers seem to welcome the new standard and it has so far received very good reviews, it does create challenges for those who use language and approaches that are unique to their area of work but different from the new standard and guide. The need for compromise and change is the inevitable consequence of standardization