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This paper considers the asymptotic behavior for the tail probability of randomly weighted sum S 2 θ = θ 1 X 1 + θ 2 X 2 , where X 1 , X 2 , θ 1 , and θ 2 are non-negative dependent random variables with distributions F 1 , F 2 , G 1 , and G 2 , respectively. We obtain the tail-equivalence of P S 2 θ > x and P (θ 1 X 1 > x) + P (θ 2 X 2 > x) as x → ∞ and some closure properties of distribution classes in three cases: (i). θ 1 , θ 2 are bounded and F 1 , F 2 are subexponential; (ii). θ 1 , θ 2 satisfy the condition of Theorem 2.1 of Tang (Extremes 9(3):231–241 2006) and F 1 , F 2 are subexponential with positive lower Matuszewska indices; (iii). θ 1 , θ 2 satisfy the condition of Theorem 3.3 (iii) of Cline and Samorodnitsky (Stochastic Process and their Appl 49(1):75-98 1994) and F 1 , F 2 are long-tailed and dominatedly-varying-tailed. Furthermore, when F 1 and F 2 are regularly-varying-tailed, a more transparent result is established and applied to obtain asymptotic results for risk measures. Some numerical studies are conducted to check the accuracy of the obtained results. [ABSTRACT FROM AUTHOR]

Stock market returns often tend to follow a non-normal probability distribution due to extreme losses in the tails. These cause fatter tails than normal and consequently heavy-tailed probability distributions are mostly used for modeling returns. In this work, we consider the generalized T (GT) distribution which can be heavy-tailed through its parameters and propose to use it in modeling the random stock returns. The GT distribution also contains the normal, Student t and generalized Laplace distributions as special or limiting cases of the shape parameters. The closed form expressions for the important risk measures are obtained. In case of a portfolio modeling, a multivariate extension of the distribution within the class of elliptical distributions is used. The tail mean-variance portfolio model based on the multivariate GT distribution is developed and optimal portfolio problem is solved. Risk measures for the random return of an asset whose density function is a mixture of the GT densities are obtained. The computability of all expressions derived is shown, and a real data application consisting of seven real stocks from the same sector is given. [ABSTRACT FROM AUTHOR]

In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting. [ABSTRACT FROM AUTHOR]

Increasing rearrangement is a rewarding instrument in financial risk management. In practice, risks must be managed from different perspectives. A common example is the portfolio risk, which often can be seen from at least two perspectives: market value and book value. Different perspectives with different distributions can be coupled by increasing rearrangement. One distribution is regarded as underlying, and the other distribution can be expressed as an increasing rearrangement of the underlying distribution. Then, the risk measure for the latter can be expressed in terms of the underlying distribution. Our first objective is to introduce increasing rearrangement for application in financial risk management and to apply increasing rearrangement to the class of distortion risk measures. We derive formulae to compute risk measures in terms of the underlying distribution. Afterwards, we apply our results to a series of special distortion risk measures, namely the value at risk, expected shortfall, range value at risk, conditional value at risk, and Wang's risk measure. Finally, we present the connection of increasing rearrangement with inverse transform sampling, Monte Carlo simulation, and cost-efficient strategies. Butterfly options serve as an illustrative example of the method. [ABSTRACT FROM AUTHOR]

We consider the operation of a non-financial corporation and present a novel risk-management framework that supports the joint optimization of the firm's operational and financial decisions. Specifically, we study the problem of maximizing the firm's operating profits when these profits depend on movements in the financial markets, and where the decision maker—the firm's manager—is risk averse. Critically, our framework (i) imposes risk aversion through a generic class of risk constraints, and (ii) considers explicitly the differences that can exist between the shareholders' and manager's beliefs about future market risks. Together, these aspects allow us to remove speculation so that financial instruments can only be used for hedging purposes. We analyze our problem in an incomplete financial market setting and consider different informational assumptions. Within this framework, we characterize the optimal financial hedging strategy so that the problem reduces to a constrained maximization over operational policies. Our results show how financial hedging can be effectively used to expand the set of admissible operating policies, and identify conditions under which the risk constraint can be completely ignored when solving for an optimal operating policy. We illustrate our methodology in the context of an exporter firm which faces exchange rate risk in trading. [ABSTRACT FROM AUTHOR]

This paper addresses a two-machine re-entrant flow shop scheduling problem with stochastic processing times where each job is expected to require a rework phase, flowing twice within the whole system. Due to the stochastic characteristics of the addressed problem, the proposed approach aims to devise robust schedules, i.e., schedules that are less sensitive to the occurrence of uncertain events, specifically, to the variability of the processing times. Two classes of approaches are proposed: the first is a branch-and-bound algorithm capable of solving the problem optimally, although with limitations regarding the size of the scheduling instances; the second is heuristic algorithms that can be applied to medium/large instances. For both approaches, the goal is to minimise the value-at-risk associated with the makespan, to assist decision-makers in balancing expected performance and mitigating the impact of extreme scenarios. A Markovian Activity Network (MAN) model is exploited to estimate the distribution of the makespan and evaluate its value-at-risk. Phase-type distributions are used to cope with general distributions for the processing times while exploiting a Markovian approach. A set of computational experiments is conducted to demonstrate the effectiveness and performance of the proposed approaches. [ABSTRACT FROM AUTHOR]

In this paper, we provide an alternative trend (time)-dependent risk measure to Ruttiens' accrued returns variability (Ruttiens in Comput Econ 41:407–424, 2013). We propose to adjust the calculation procedure to achieve an alternative risk measure. Our modification eliminates static mean component and it is based on the deviation of squared dispersions, which reflects the trend (time factor) precisely. Moreover, we also present a new perspective on dependency measures and we apply a PCA to a new correlation matrix in order to determine a parametric and nonparametric return approximation. In addition, the two-phase portfolio selection strategy is considered, where the mean–variance portfolio selection strategies represent the first optimization. The second one is the minimization of deviations from their trend leading to identical mean and final wealth. Finally, an empirical analysis verify the property and benefit of portfolio selection strategies based on these trend-dependent measures. In particular, the ex-post results show that applying the modified measure allows us to reduce the risk with respect to the trend of several portfolio strategies. [ABSTRACT FROM AUTHOR]

We develop a distributionally robust optimisation (DRO) model based on a risk measure for the parallel machine scheduling problem (PMSP) with random job processing times. We propose an underperformance risk index (URI) to control the extent of the total weighted completion time (TWCT) that exceeds target level T. With partially characterised uncertainty set information, we transform the model with URI to its equivalent mixed-integer linear programming (MILP) counterparts. Due to the NP-hardness of PMSP with different job weights, we design a hybrid algorithm with a heuristic assignment and exact subproblem for large-scale problems. The proposed hybrid algorithm reduces the computation time significantly at the expense of solution quality. We also introduce a reformulation approach under the setting of equally weighted and identical machines. Numerical results show that our model performs better than the distributionally β-robust optimisation models. Our proposed URI accounts for both the frequency and magnitude of violation from the target. The uncertainty set we used preserves a linear structure under partially characterised distributional information. Our computational results and sensitivity analysis show the effectiveness and efficiency of our proposed DRO model under various settings, including different problem sizes, different processing time variations, and information misalignment. [ABSTRACT FROM AUTHOR]

The paper Zhao et al. (Ann Oper Res 226:727–739, 2015) shows that mean-CVaR-skewness portfolio optimization problems based on asymetric Laplace (AL) distributions can be transformed into quadratic optimization problems for which closed form solutions can be found. In this note, we show that such a result also holds for mean-risk-skewness portfolio optimization problems when the underlying distribution belongs to a larger class of normal mean–variance mixture (NMVM) models than the class of AL distributions.We then study the value at risk (VaR) and conditional value at risk (CVaR) risk measures of portfolios of returns with NMVM distributions.They have closed form expressions for portfolios of normal and more generally elliptically distributed returns, as discussed in Rockafellar and Uryasev (J Risk 2:21–42, 2000) and Landsman and Valdez (N Am Actuar J 7:55–71, 2003). When the returns have general NMVM distributions, these risk measures do not give closed form expressions. In this note, we give approximate closed form expressions for the VaR and CVaR of portfolios of returns with NMVM distributions.Numerical tests show that our closed form formulas give accurate values for VaR and CVaR and shorten the computational time for portfolio optimization problems associated with VaR and CVaR considerably. [ABSTRACT FROM AUTHOR]

Problem definition: We consider a surgery sequencing and scheduling problem with uncertain durations of surgeries in the context of an operating theater. From real data collected from a hospital, we observe the common practice, namely, "to follow," in which surgeries are conducted sequentially and immediately one after another, according to a specific schedule. Methodology/results: Based on this practice, we propose a mathematical framework to balance the risk of delay and idling using the punctuality index, which takes into consideration both the probability and intensity of delay and idle time. We develop a computationally efficient procedure based on Benders decomposition to derive exact solutions for the problem. The scheduling problem is solvable in polynomial time when the sequence is given. The framework can also accommodate a robust setting when the underlying probability distribution is not fully available. For practical use, we propose two effective heuristics for sequencing decisions by approximating the model. Using real data, we demonstrate that our framework is significantly better than the risk-neutral and probability-maximizing approaches in both performance and computational efficiency. Moreover, the robust setting can effectively lessen the risk of extremely long delay and idle time, and the heuristics are efficient with only a small sacrifice of performance. Managerial implications: With the to-follow policy, our sequencing and scheduling model describes the actual practice better. The two heuristics can be applied easily and directly to help managers of operating theaters to make decisions. Funding: J. Qi is supported by the Hong Kong Research Grants Council [Grants 26202016, 16212419, and 16204521]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0029. [ABSTRACT FROM AUTHOR]

We present a federated learning framework that is designed to robustly deliver good predictive performance across individual clients with heterogeneous data. The proposed approach hinges upon a superquantile-based learning objective that captures the tail statistics of the error distribution over heterogeneous clients. We present a stochastic training algorithm that interleaves differentially private client filtering with federated averaging steps. We prove finite time convergence guarantees for the algorithm: O (1 / T) in the nonconvex case in T communication rounds and O (exp (- T / κ 3 / 2) + κ / T) in the strongly convex case with local condition number κ . Experimental results on benchmark datasets for federated learning demonstrate that our approach is competitive with classical ones in terms of average error and outperforms them in terms of tail statistics of the error. [ABSTRACT FROM AUTHOR]

People usually create a portfolio in order to diversify the risk coming from individual investments. To get a high yield with a good level of diversification, investors usually seek professional advice from portfolio managers. However, the true performance of an optimized portfolio usually depends on the correctness of the estimates of the distribution of future returns, which is often a matter of luck rather than skill. Thus, the optimization models may not be better than randomly selected portfolios. Our aim is to find how the so-called strategy portfolios, i.e., portfolios obtained by some decision optimized for a long-run horizon, perform compared to a benchmark, namely, a random investment, under specific market conditions. For this purpose, we evaluate several portfolio strategies over two periods of crisis: the subprime mortgage crisis and the Covid-19 pandemic, as well as run a moving window analysis over a longer horizon. In each case, the results are compared with the performance of random-weight portfolios. We find that if the strategy is minimization, the portfolios perform well; however, for the maximization of the objectives, the results are rather mixed. [ABSTRACT FROM AUTHOR]

We study some properties of weighted survival extropy measure. This is a shift dependent measure and put more weights to the large values of the observed random variable. Then we introduce weighted extended survival extropy and its dynamic version. Various properties of the proposed measures are obtained. Some generalized inequalities related to weighted extended survival extropy are studied. Non-parametric estimators of these measures are proposed and their asymptotic properties are investigated. The performance of the estimators is assessed through a simulation study. Finally, it is shown that weighted survival extropy measure can be used in model discrimination and as an alternative risk measure. Analysis of real life data is considered for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, the classification of crisis states and their dynamics is carried out. The asset crisis model is approximated and formalized. Invariant stationary components and the boundaries of crisis fluctuations are determined, which allows for predictions of the course of a crisis. The hypothesis of the phase invariance of crises is statistically confirmed, and four main phases and five marker points in the development of an investment asset crisis are distinguished. A statistical study of the changes in the value of 620 investment assets over 7 years (1,584,720 values) was conducted, including 710 crisis asset states (9940 values), which allowed us to formulate a mathematical model of crisis variability that makes it possible to detect the onset of crisis phases or the passage of marker points via the values of the first- and second-order derivatives of the model. This model has not only evaluative properties, but also predictive properties relative to the risk of a crisis asset, which is also confirmed experimentally. An experiment is conducted to assess the quality of the econometric crisis model, which is based on real historical data. The results of the experiment showed the high quality of the model as a crisis detection tool. A method of early detection of the initial crisis phase is developed, in which it is seen that a statistically unpredictable state causes such a change in the indicator that it violates the valid bound constraints. [ABSTRACT FROM AUTHOR]

Risk measurements are clearly central to risk management, in particular for banks, (re)insurance companies, and investment funds. The question of the appropriateness of risk measures for evaluating the risk of financial institutions has been heavily debated, especially after the financial crisis of 2008/2009. Another concern for financial institutions is the pro-cyclicality of risk measurements. In this paper, we extend existing work on the pro-cyclicality of the Value-at-Risk to its main competitors, Expected Shortfall, and Expectile: We compare the pro-cyclicality of historical quantile-based risk estimation, taking into account the market state. To characterise the latter, we propose various estimators of the realised volatility. Considering the family of augmented GARCH(p , q) processes (containing well-known GARCH models and iid models, as special cases), we prove that the strength of pro-cyclicality depends on the three factors: the choice of risk measure and its estimators, the realised volatility estimator and the model considered, but, no matter the choices, the pro-cyclicality is always present. We complement this theoretical analysis by performing simulation studies in the iid case and developing a case study on real data. [ABSTRACT FROM AUTHOR]

The paper presents the properties of an alternative method, which measures market risk over time-horizon exceeding one day: Mark to market value at risk (MMVaR). Relying on the minimal returns during the time interval, this method not only considers the non-normality of data and information about sample size, but also meets the requirement of increasing the minimal capital ratio in Basel III, basically. The authors theoretically prove the translation invariance, monotonicity and subadditivity of MMVaR as a risk measure under some conditions, and study its finite sample properties through Monte Carlo simulations. The empirical analysis shows that MMVaR can measure multi-period risk accurately. [ABSTRACT FROM AUTHOR]

The Russian Law “On Radiation Safety of the Population” defines the effective dose as “the amount of exposure to ionizing radiation used as a measure of the risk of long-term consequences of human’s body exposure…” In turn, the Russian “Radiation Safety Standards” (RSS 99/2009) establish a procedure for assessing the health risk, associated with exposure to low doses, for two types of harmful effects (oncological diseases and hereditary effects) and two age and sex groups (“whole population” and “adults”) by multiplying corresponding linear risk coefficients per effective dose. At the same time, these documents do not impose restrictions on the risk assessment procedure, without excluding the use of any other risk indicators or risk assessment for other sex and age groups of the population. The value of radiation detriment to health, calculated by the method described in RSS 99/2009 to characterize the risk, as a unit of measurement, uses the fatal oncological disease caused by exposure to ionizing radiation, weighted by the number of years of life lost, or weighted by the severity of non-fatal cancer. Currently, mortality-based health measures are considered insufficiently informative characteristics of the impact of external factors on population health. Such indicators are poorly suited for a comparative analysis of risks, especially taking into account the different distribution of negative consequences over time. The paper presents an applied approach to expand the application of the radiation risk assessment methodology, without making significant changes to the established practice of radiation protection. The possibility of using the DALY value (disability-adjusted life years) as an measure of radiation risk is being considered; the expediency of changing approaches to calculating the effective dose (by using different values of weighting factors for tissues and organs during effective dose calculation for different age and sex groups of the population) and the possible scope of the proposed approaches to risk assessment in practice are discussed; an approach is proposed for calculating the DALY value using the effective dose and the corresponding risk factors (DALY×Sv-1).

This study investigates the properties of risk measure, value at risk (VaR) and conditional VaR (CVaR), using high-frequency Bitcoin data. These data allow us to conduct a high statistical analysis. Our findings reveal a disparity in VaR and CVaR values between the left and right tails of the return probability distributions. We refer to this disparity as "long–short asymmetry". In the high-frequency domain, the tail distribution can be accurately described by a power-law function. Moreover, the ratio of CVaR to VaR is expected to be determined solely by the power-law exponent. Through empirical analysis, we confirm that this ratio property holds true for high confidence levels. Furthermore, we investigate the relationship between risk measures (VaR and CVaR) and realized volatility. We observe that they trace a trajectory in a two-dimensional plane. This trajectory changes gradually, indicating periods of both high and low risk. [ABSTRACT FROM AUTHOR]

We develop simple versions of upper bounds of the widely used systemic risk measure of ΔCoVaR that are straightforward to calculate, and may prove useful as (conservative) benchmarks in an applied context. [ABSTRACT FROM AUTHOR]

It is not uncommon that the living situation of a unit (e.g. a component in a reliability system) plays a critical role in affecting the performance of another unit or even the whole system. The study on stochastic behaviors of conditional residual lifetimes appears to be particularly important for analyzing the performances of interested units. This paper studies stochastic comparisons on the conditional residual lifetimes of units under the dependent information when another unit has survived up to a given time. Sufficient (and necessary) conditions are established for comparing conditional residual lifetimes of different units under various dependence structures as well as different events according to some traditional stochastic orders. We provide numerical examples based on the well-known FGM copula and Archimedean copulas to verify the conditions. Three real scenarios in reliability theory and risk theory are also considered to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In the multivariate setting, estimates of extremal risk measures are important in many contexts, such as environmental planning and structural engineering. In this paper, we propose new estimation methods for extremal bivariate return curves, a risk measure that is the natural bivariate extension to a return level. Unlike several existing techniques, our estimates are based on bivariate extreme value models that can capture both key forms of extremal dependence. We devise tools for validating return curve estimates, as well as representing their uncertainty, and compare a selection of curve estimation techniques through simulation studies. We apply the methodology to two metocean data sets, with diagnostics indicating generally good performance. [ABSTRACT FROM AUTHOR]

Expectiles form a family of risk measures that have recently gained interest over the more common value-at-risk or return levels, primarily due to their capability to be determined by the probabilities of tail values and magnitudes of realizations at once. However, a prevalent and ongoing challenge of expectile inference is the problem of uncertainty quantification, which is especially critical in sensitive applications, such as in medical, environmental, or engineering tasks. We address this issue by developing a novel distribution, termed the multivariate expectile-based distribution (MED), that possesses an expectile as a closed-form parameter. Desirable properties of the distribution, such as log-concavity, make it an excellent fitting distribution in multivariate applications. Maximum likelihood estimation and Bayesian inference algorithms are described. Simulated examples and applications to expectile and mode estimation illustrate the usefulness of the MED for uncertainty quantification. • MED distribution is log-concave, unimodal, and its unique mode is an expectile. • It yields well-calibrated credible intervals for uncertainty quantification in expectile estimation. • MEDdist R package provides density, distribution, and quantile functions, and random sampling for the MED. [ABSTRACT FROM AUTHOR]

Summary: We deal with a multivariate conditional value at risk. Compared with the usual notion for the single random variable, a multivariate value at risk is concerned with several variables, and thus, the relation between each risk factor should be considered. We here introduce a new definition of copula‐based conditional value at risk, which is real valued and ready to be computed. Copulas are known to provide a flexible method for handling a possible nonlinear structure; therefore, copulas may be naturally involved in the theory of value at risk. We derive a formula of our copula‐based conditional value at risk in the case of Archimedean copulas, whose effectiveness is shown by examples. Numerical studies are also carried out with real data, which can be verified with analytical results. [ABSTRACT FROM AUTHOR]

Measuring risk effectively is crucial for managing risk in financial markets. The expected shortfall has become an increasingly popular metric for risk in recent years. How to estimate it is important in statistics and financial econometrics. Based on the single index quantile regression, we introduce a new semiparametric approach, namely, weighted single index quantile regression. We assess the performance of the proposed expected shortfall estimator with backtesting. Our simulation results indicate that the estimator has a good finite sample performance and often outperforms existing methods. By applying the new method to both a market index and individual stocks, we show that it not only exhibits the best performance but also reveals an insight about the effect of the COVID pandemic, that is, the pandemic increases the market risk. [ABSTRACT FROM AUTHOR]

In this work, we propose a theoretical framework for computing pessimistic and optimistic estimates of effective properties in the case of heterogeneous elastic materials with uncertain microscopic elastic properties. We rely on a risk-averse measure widely used in finance called the conditional-value at risk (CVaR). The CVaR computes the conditional expectation of events occurring above a given risk level, thereby characterizing the extreme tails of the probability distribution of a random variable. In the context of elastic materials, we propose to use the CVaR on the elastic free energy to compute an optimistic estimate of the global stiffness for some confidence level $\alpha $. Similarly, we also use the CVaR on the complementary elastic energy to compute a pessimistic estimate of the global stiffness. The obtained CVaR estimates benefit from a convex optimization formulation. The resulting material behavior is still elastic but not necessarily linear anymore. We discuss approximate formulations recovering a linear elastic behavior. We apply the proposed formulations to the micromechanical estimates of effective elastic properties of random heterogeneous materials.

Value-at-Risk (VaR) and Expected Shortfall (ES) are common high quantile-based risk measures adopted in financial regulations and risk management. In this paper, we propose a tail risk measure based on the most probable maximum size of risk events (MPMR) that can occur over a length of time. MPMR underscores the dependence of the tail risk on the risk management time frame. Unlike VaR and ES, MPMR does not require specifying a confidence level. We derive the risk measure analytically for several well-known distributions. In particular, for the case where the size of the risk event follows a power law or Pareto distribution, we show that MPMR also scales with the number of observations n (or equivalently the length of the time interval) by a power law, MPMR(n) ∝ nη, where η is the scaling exponent (SE). The scale invariance allows for reasonable estimations of long-term risks based on the extrapolation of more reliable estimations of short-term risks. The scaling relationship also gives rise to a robust and low-bias estimator of the tail index (TI) ξ of the size distribution, ξ = 1/η. We demonstrate the use of this risk measure for describing the tail risks in financial markets as well as the risks associated with natural hazards (earthquakes, tsunamis, and excessive rainfall).

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment will be studied under a general framework. Indeed, there is no specific type of reinsurance contract, there is no specific dynamics of the involved financial instruments and the financial market does not have to be free of frictions. On the other hand, it will be pointed out how the optimal combination above may provide us with new premium principles making the insurer global risk vanish. The risk will be managed with a coherent risk measure, and the new premium principles will seem to reflect several properties, which are desirable from both the analytical and the economic perspectives. From the analytical viewpoint, the premium principles will be continuous, homogeneous and increasing. From the economic viewpoint, the premium principles will lead to cheaper prices with respect to both the insurance market and the financial one. In other words, the premium principles will make the insurer more competitive in prices under a null risk. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Several methods preventing unbounded optimization problems will warrant special attention, and one particular case will be more thoroughly studied, namely, the combination of the Black–Scholes–Merton pricing model with the conditional value at risk. [ABSTRACT FROM AUTHOR]

In this work, we propose weighted cumulative Tsallis residual and past entropy measures and their dynamic versions. The properties of the proposed entropy measures are studied. It is shown that the residual entropy uniquely determines the survival function. Characterizations for Rayleigh and Power distributions are obtained based on the proposed entropy measures. The empirical versions of the residual and past entropies are provided and their asymptotic behaviors are discussed. It is shown that the weighted cumulative Tsallis entropy can be used as a risk measure. A dataset is analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

We show that pro‐cyclicality is inherent in risk measure estimates based on historical data. Taking the example of VaR, we show that the empirical VaR measure is mean‐reverting over a 1‐year horizon when the portfolio is held fixed. It means that a capital requirement rule based on historical measurements of VaR tends in calm times to understate future required capital and tends in volatile times to overstate it. To quantify this pro‐cyclicality, we develop a simple and efficient methodology, which we apply to major equity market indices. We make the interesting point that the pro‐cyclicality property holds true even in a world with constant volatility, though the empirical magnitude of the mean‐reversion is greater than what would be observed in that special case. [ABSTRACT FROM AUTHOR]

The emergence of bike-sharing systems has considerably improved last- and first-mile transportation systems. To ensure attractiveness to end users, operators aim to design effective service-oriented operational strategies to meet the desired service targets for users. Most existing studies focus on the service efficiency of bike-sharing systems, while service equity is overlooked. In this study, we propose a target-based stochastic distributionally robust optimization (TSDRO) model that addresses both the efficiency and equity of the service level in docked bike-sharing systems under demand uncertainty. We first employ a dissatisfaction risk measure to jointly quantify the probability and magnitude of user dissatisfaction in a zone. Then, we apply a lexicographic-order approach to define the objective function to achieve equity of service among different zones. This lexicographic approach optimizes the worst-off individual and the second-worst zone in an iterative manner. To address demand ambiguity, we use a data-driven method to explore the relationship between the demand distribution and several exogenous factors, including weather and weekends, and then construct a scenario-based distributionally robust optimization model. Based on duality theory and linear decision approximation, this model can be reformulated as a tractable equivalent deterministic model, which can be solved via a bisection-search approach to optimality. Numerical experiments based on real operational data show that compared with the benchmark models, the TSDRO model achieves (i) better out-of-sample performance in terms of service efficiency and (ii) higher service equity among users in different zones. Moreover, setting a lower target level may generate a better solution. • We propose a target-based service efficiency and equity optimization model. • We consider the spatial service equity by employing the lexicographic approach. • We address demand uncertainty by a stochastic distributionally robust approach. • Numerical experiments show that our model has good out-of-sample performance. [ABSTRACT FROM AUTHOR]

Portfolio optimization is a well-known and beneficial procedure used by shareholders to select their portfolios. An investor must seek an equilibrium between risk and profit while making investment decisions. The fundamental concern is risk because the responsibility of the risk for each investor is different. A risk profile of each investor is characterized as a risk measure. In this paper, we focus on Conditional Value-at-Risk (CVaR). We numerically consider an optimal portfolio which minimizes CVaR under CEV model. Finally, the numerical results of CVaR and the optimal portfolio are discussed. [ABSTRACT FROM AUTHOR]

A new class of skewed distributions, with a matrix skewness parameter, called extended mean mixtures of multivariate normal (EMMN) distributions, is constructed. The family of EMMN distributions includes the SN and MMN distributions as special cases. Some basic properties of this family, such as characteristic function, moment generating function, affine transformation and canonical forms of the distributions are derived. An EM-type algorithm is developed to carry out the maximum likelihood estimation of the parameters. Two special cases of this family are studied in detail. A simulation is carried out to examine the performance of the estimation method, and the flexibility is illustrated by fitting a special case of this family to a real data. Finally, the theoretical formula of the multivariate tail conditional expectation of the EMMN distribution is derived.

In this paper, we obtain a comparison theorem and an invariant representation theorem for backward stochastic differential equations (BSDEs) without any assumption on the second variable z. Using the two results, we further develop the theory of g -expectations. Filtration-consistent nonlinear expectation (F -expectation) provides an ideal characterization for the dynamical risk measures, asset pricing and utilities. We propose two new conditions: an absolutely continuous condition and a (locally Lipschitz) domination condition. Under the two conditions respectively, we prove that any F -expectation can be represented as a g -expectation. Our results contain a representation theorem for n -dimensional F -expectations in the Lipschitz case, and two representation theorems for 1-dimensional F -expectations in the locally Lipschitz case, which contain quadratic F -expectations. [ABSTRACT FROM AUTHOR]

The current study offers a sustainable supply chain model that can be used to identify opportunities and constraints for the future of the microalgae-based bio-diesel and bio-plastic industry. The network's carbon emissions are restricted by an uncertain constraint that allows for flexibility in controlling emissions as needed. The social constraints of the model provides a transparent depiction of labour needs in dynamic business landscape and foster women's empowerment through a preference coefficient. The proposed model introduces a new upside risk measures under fuzzy-stochastic setting to quantify the upward financial risk. To characterize and manage the model uncertainty, a fuzzy-stochastic distributional robust model is presented. Credibility distribution is utilized to handle fuzzy uncertainty, while box and polyhedral uncertainty sets are included to handle stochastic uncertainty. After that, tractable deterministic formulations of the uncertain model for two ambiguity sets are presented to solve the model. Finally, a number of insights are obtained by means of five test problems that are implemented in Python and resolved by one exact and five meta-heuristic algorithms. Experiments indicate that, when faced with uncertainty, two robust models have produced more stable solutions (16.23% and 19.67% higher stability, respectively) than nominal stochastic model. Further, a 4.68% decline in total cost and 3.27% increment in bio-diesel production can be achieved by increasing bio-diesel conversion rate to 10%. In addition, a 9.09% downturn in total cost and 10.44% rise in bio-plastic production can be obtained by 10% increment in bio-plastic conversion rate. Moreover, 45.38% more woman labours can be recruited by increasing the preference coefficient from − 0. 2 to + 0. 2. • Designing a microalgae-based bio-diesel and bio-plastic supply chain. • Defining the upper partial moment under fuzzy-stochastic setting to measure risk. • Controlling carbon emissions by imposing a flexible uncertain constraint. • Empowering woman labour requirement in the network. • Forming a fuzzy-stochastic distributional robust optimization approach. [ABSTRACT FROM AUTHOR]

In this paper we introduce a novel Tail Mean-Variance (TMV) model intended to optimize capital allocation in financial decision-making. The TMV model has many intriguing properties, such as considering the variability and tail risk of the loss function simultaneously. We specifically consider the multivariate extended skew-elliptical (ESE) distributions, which are widely applicable in financial and insurance data modeling. Additionally, we explore the probabilistic properties of the multivariate extended skew-elliptical (ESE) distributions and present explicit formulas for the TMV model within this distribution. Finally, a numerical example is provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, the classification of crisis states and their dynamics is carried out. The asset crisis model is approximated and formalized. Invariant stationary components and the boundaries of crisis fluctuations are determined, which allows for predictions of the course of a crisis. The hypothesis of the phase invariance of crises is statistically confirmed, and four main phases and five marker points in the development of an investment asset crisis are distinguished. A statistical study of the changes in the value of 620 investment assets over 7 years (1,584,720 values) was conducted, including 710 crisis asset states (9940 values), which allowed us to formulate a mathematical model of crisis variability that makes it possible to detect the onset of crisis phases or the passage of marker points via the values of the first- and second-order derivatives of the model. This model has not only evaluative properties, but also predictive properties relative to the risk of a crisis asset, which is also confirmed experimentally. An experiment is conducted to assess the quality of the econometric crisis model, which is based on real historical data. The results of the experiment showed the high quality of the model as a crisis detection tool. A method of early detection of the initial crisis phase is developed, in which it is seen that a statistically unpredictable state causes such a change in the indicator that it violates the valid bound constraints.

Abstract Tail risk is a classic topic in stressed portfolio optimization to treat unprecedented risks, while the traditional mean–variance approach may fail to perform well. This study proposes an innovative semiparametric method consisting of two modeling components: the nonparametric estimation and copula method for each marginal distribution of the portfolio and their joint distribution, respectively. We then focus on the optimal weights of the stressed portfolio and its optimal scale beyond the Gaussian restriction. Empirical studies include statistical estimation for the semiparametric method, risk measure minimization for optimal weights, and value measure maximization for the optimal scale to enlarge the investment. From the outputs of short-term and long-term data analysis, optimal stressed portfolios demonstrate the advantages of model flexibility to account for tail risk over the traditional mean–variance method.

Contemporary risk management is based on statistical analysis. Such an approach has a few crucial disadvantages. First of all, it has limited applicability to new technological solutions. In this paper, a new idea for risk evaluation and management is put forward. The proposed approach is based on the autonomous systems theory. The theoretical foundation of the proposed idea is described and its prospective applications are discussed. The proposed measures of risk are based on the idea of the controllability of the system—the greater the level of controllability, the lower the risk. Various aspects of controllability are analyzed—economic, technological, and industrial. For each aspect of controllability, the problem of defining adequate measures for the level of risk is discussed. The proposed approach allows the risk assessor to analyze the system deeply. As a consequence, the analyst can assess the risk based not only on a posteriori statistics but also on an analysis of the crucial properties of the system. This allows the investigator to predict a priori possibilities of critical events. The proposed methodology is applied to the power industry. [ABSTRACT FROM AUTHOR]