ASYMPTOTIC EQUIVALENCE OF RISK MEASURES UNDER DEPENDENCE UNCERTAINTY

In this paper, we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures is asymptotically equivalent if the ratio of the worst‐case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a noncoherent risk measure and a coherent risk measure, as the worst‐case value of a noncoherent risk measure under dependence uncertainty is typically difficult to obtain. The main contribution of this paper is to establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.     

Value‐at‐Risk bounds with two‐sided dependence information

Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.     

Sharing the value‐at‐risk under distributional ambiguity

This paper considers the problem of risk sharing, where a coalition of homogeneous agents, each bearing a random cost, aggregates their costs, and shares the value‐at‐risk of such a risky position. Due to limited distributional information in practice, the joint distribution of agents' random costs is difficult to acquire. The coalition, being aware of the distributional ambiguity, thus evaluates the worst‐case value‐at‐risk within a commonly agreed ambiguity set of the possible joint distributions. Through the lens of cooperative game theory, we show that this coalitional worst‐case value‐at‐risk is subadditive for the popular ambiguity sets in the distributionally robust optimization literature that are based on (i) convex moments or (ii) Wasserstein distance to some reference distributions. In addition, we propose easy‐to‐compute core allocation schemes to share the worst‐case value‐at‐risk. Our results can be readily extended to sharing the worst‐case conditional value‐at‐risk under distributional ambiguity.     

Computational aspects of robust optimized certainty equivalents and option pricing

Accounting for model uncertainty in risk management and option pricing leads to infinite‐dimensional optimization problems that are both analytically and numerically intractable. In this article, we study when this hurdle can be overcome for the so‐called optimized certainty equivalent (OCE) risk measure—including the average value‐at‐risk as a special case. First, we focus on the case where the uncertainty is modeled by a nonlinear expectation that penalizes distributions that are “far” in terms of optimal‐transport distance (e.g. Wasserstein distance) from a given baseline distribution. It turns out that the computation of the robust OCE reduces to a finite‐dimensional problem, which in some cases can even be solved explicitly. This principle also applies to the shortfall risk measure as well as for the pricing of European options. Further, we derive convex dual representations of the robust OCE for measurable claims without any assumptions on the set of distributions. Finally, we give conditions on the latter set under which the robust average value‐at‐risk is a tail risk measure.     

MODEL UNCERTAINTY AND SCENARIO AGGREGATION

This paper provides a coherent method for scenario aggregation addressing model uncertainty. It is based on divergence minimization from a reference probability measure subject to scenario constraints. An example from regulatory practice motivates the definition of five fundamental criteria that serve as a basis for our method. Standard risk measures, such as value‐at‐risk and expected shortfall, are shown to be robust with respect to minimum divergence scenario aggregation. Various examples illustrate the tractability of our method.     

Asymptotic subadditivity/superadditivity of Value-at-Risk under tail dependence

This paper presents a new method for discussing the asymptotic subadditivity/superadditivity of Value-at-Risk (VaR) for multiple risks. We consider the asymptotic subadditivity and superadditivity properties of VaR for multiple risks whose copula admits a stable tail dependence function (STDF). For the purpose, a marginal region is defined by the marginal distributions of the multiple risks, and a stochastic order named tail concave order is presented for comparing individual tail risks. We prove that asymptotic subadditivity of VaR holds when individual risks are smaller than regularly varying (RV) random variables with index −1 under the tail concave order. We also provide sufficient conditions for VaR being asymptotically superadditive. For two multiple risks sharing the same copula function and satisfying the tail concave order, a comparison result on the asymptotic subadditivity/superadditivity of VaR is given. Asymptotic diversification ratios for RV and log regularly varying (LRV) margins with specific copula structures are obtained. Empirical analysis on financial data is provided for highlighting our results.     

A unified approach to systemic risk measures via acceptance sets

We specify a general methodological framework for systemic risk measures via multidimensional acceptance sets and aggregation functions. Existing systemic risk measures can usually be interpreted as the minimal amount of cash needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal amount of cash that secures the aggregated system by allocating capital to the single institutions before aggregating the individual risks. An important feature of our approach is the possibility of allocating cash according to the future state of the system (scenario‐dependent allocation). We also provide conditions that ensure monotonicity, convexity, or quasi‐convexity of our systemic risk measures.     

Preference robust distortion risk measure and its application

Distortion risk measure (DRM) plays a crucial role in management science and finance particularly actuarial science. Various DRMs have been introduced but little is discussed on which DRM at hand should be chosen to address a decision maker's (DM's) risk preference. This paper aims to fill out the gap. Specifically, we consider a situation where the true distortion function is unknown either because it is difficult to identify/elicit and/or because the DM's risk preference is ambiguous. We introduce a preference robust distortion risk measure (PRDRM), which is based on the worst-case distortion function from an ambiguity set of distortion functions to mitigate the impact arising from the ambiguity. The ambiguity set is constructed under well-known general principles such as concavity and inverse S-shapedness of distortion functions (overweighting on events from impossible to possible or possible to certainty and underweighting on those from possible to more possible) as well as new user-specific information such as sensitivity to tail losses, confidence intervals to some lotteries, and preferences to certain lotteries over others. To calculate the proposed PRDRM, we use the convex and/or concave envelope of a set of points to characterize the curvature of the distortion function and derive a tractable reformulation of the PRDRM when the underlying random loss is discretely distributed. Moreover, we show that the worst-case distortion function is a nondecreasing piecewise linear function and can be determined by solving a linear programming problem. Finally, we apply the proposed PRDRM to a risk capital allocation problem and carry out some numerical tests to examine the efficiency of the PRDRM model.     

Injury count model for quantification of risk of occupational injury

Reduction of risk of occupational injuries is one of the most challenging problems faced by industry. Assessing and comparing risks involved in different jobs is one of the important steps towards reducing injury risk. In this study, a comprehensive scheme is given for assessing and comparing injury risks with the development of injury count model, injury risk model and derived statistics. The hazards present in a worksystem and the nature of the job carried out by workers are perceived as important drivers of injury potential of a worksystem. A loglinear model is used to quantify injury counts and the event-tree approach with joint, marginal and conditional probabilities is used to quantify injury risk. A case study was carried out in an underground coalmine. Finally a number of indices are proposed for the case study mine to capture risk of injury in different jobs. The findings of this study will help in designing injury intervention strategies for the mine studied. The job-wise risk profiles will be used to prioritise the jobs for redesign. The absolute indices can be applied for benchmarking job-wise risks and the relative indices can be used for comparing job-wise risks across worksystems.     

CHOQUET INSURANCE PRICING: A CAVEAT

We show that, if prices in a market are Choquet expectations, the existence of one frictionless asset may force the whole market to be frictionless. Any risky asset will cause this collapse if prices depend only on the distribution with respect to a given nonatomic probability measure; the frictionless asset has to be fully revealing if such dependence is not assumed. Similar considerations apply to law-invariant coherent risk measures.     

MEASURING DISTRIBUTION MODEL RISK

We propose to interpret distribution model risk as sensitivity of expected loss to changes in the risk factor distribution, and to measure the distribution model risk of a portfolio by the maximum expected loss over a set of plausible distributions defined in terms of some divergence from an estimated distribution. The divergence may be relative entropy or another f‐divergence or Bregman distance. We use the theory of minimizing convex integral functionals under moment constraints to give formulae for the calculation of distribution model risk and to explicitly determine the worst case distribution from the set of plausible distributions. We also evaluate related risk measures describing divergence preferences.     

MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS

Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk."     

MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

This paper deals with multidimensional dynamic risk measures induced by conditional g‐expectations. A notion of multidimensional g‐expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional g‐expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g‐risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi‐monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g‐risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for ‐tolerant g‐risk measures, and risk contribution for coherent g‐risk measures are investigated. Insurance g‐risk measure and other ways to induce g‐risk measures are also studied at the end of the paper.     

传统企业开展电子商务的风险评价与防范

随着信息技术的发展,电子商务将成为企业的发展方向,给传统企业带来巨大的发展机遇,但也存在诸多风险因素,只有很好地防范风险,才能使效益得以实现。本文通过分析传统企业开展电子商务所面临的风险,用模糊综合评价法对其进行评价并进行了实例验证,从内生风险、外生风险和内外协同风险三方面分别构建了风险防范体系。     

DISTRIBUTION-INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY

In the first part of the paper, we characterize distribution-invariant risk measures with convex acceptance and rejection sets on the level of distributions. It is shown that these risk measures are closely related to utility-based shortfall risk.
In the second part of the paper, we provide an axiomatic characterization for distribution-invariant dynamic risk measures of terminal payments. We prove a representation theorem and investigate the relation to static risk measures. A key insight of the paper is that dynamic consistency and the notion of "measure convex sets of probability measures" are intimately related. This result implies that under weak conditions dynamically consistent dynamic risk measures can be represented by static utility-based shortfall risk.     

BOUNDING WRONG‐WAY RISK IN CVA CALCULATION

A credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk to estimate a counterparty's risk of default together with the market value of exposure to the counterparty at default. Wrong‐way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong‐way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, a linear program computes the worst‐case CVA. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure by constraining the deviation of the joint model from a baseline reference model. Measuring the deviation through relative entropy leads to a tractable convex optimization problem that can be solved through the iterative proportional fitting procedure. Here, too, we prove convergence of the resulting estimate of the penalized worst‐case CVA and the joint distribution that attains it. We consider extensions with additional constraints and illustrate the method with examples.     

Dependence of stock and commodity futures markets in China: Implications for portfolio investment

We examine the recent trends in dependence structure between the fast-growing commodity markets and the stock markets in China. We address this issue by using copula functions that allow for measuring both average and tail dependence. Our results provide evidence of low and positive correlations between these markets, suggesting that commodity futures are a desirable asset class for portfolio diversification. By comparing the market risks of alternative portfolio strategies, we show that Chinese investors can take advantage of commodity futures during different times to realize risk diversification and downside risk reduction benefits.     

Multi level categorical data fusion using partially fused data

Data fusion poses challenging methodological issues for inferring the joint distribution of two random variables when the information available is mainly confined to the marginal distributions. When the variables are categorical, the challenges are even more severe. Applications of categorical data fusion are of top importance in marketing, especially in advertising. A great deal of categorical data fusion methods are confined to binary variables. In this paper we develop an innovative approach to categorical data fusion that extends previous methodologies and applies to categorical variables with any number of levels. We introduce a new concept of “evident dependence” that describes a variety of patterns of joint distributions given the marginals. Using information from partially fused data, our method smoothly accommodates a Bayesian approach based on mixtures of joint distributions constructed using evident dependence. The approach is illustrated using data from the advertising industry.     

Disruptions in Supply Networks: A Probabilistic Risk Assessment Approach

Supply disruptions are attracting growing attention. Even in geographically, politically and economically stable locations, companies are exposed to disruptions, because they depend on their suppliers and suppliers’ suppliers. The analysis of these disruptions helps mitigate risks: for example, instead of relying on local measures such as safety stock or insurance, a company can introduce new supply contracts or backup risky suppliers. In this article, we analyze risks caused by supplier disruptions by introducing concepts from probabilistic risk assessment (PRA), which is a widely employed methodology for the risk analysis of complex engineering systems. We apply PRA to examine simple networks such as triads analytically, and use simulation to analyze disruption risks in random networks of realistic size. We also illustrate how PRA can support strategic decisions such as whether or not to use single or multiple suppliers; which suppliers are more risky than others; and what impacts the complexity of the supply base has on the reliability of the supplier network.     

OPTIMAL PORTFOLIOS WITH LOWER PARTIAL MOMENT CONSTRAINTS AND LPM-RISK-OPTIMAL MARTINGALE MEASURES

We optimize the ratio     over an (arbitrage-free) linear sub-space     of attainable returns in an incomplete market model. If a solution exists for  1 < r < ∞  , then the 1st order optimality condition allows to construct an equivalent martingale measure for     , which is shown to be the solution of an appropriate dual minimization problem over the set of all equivalent martingale measures for     . The dual minimization problem admits a solution iff there exists an equivalent martingale measure for     and its optimal value     equals the lowest upper bound     of all α-ratios over     . This new type of non-concave duality also provides an indifference pricing method. The duality result can be extended to the case     and leads to a new no (approximate) arbitrage condition: "no great expectations with vanishing risk."