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This study addresses market concentration among major corporations, highlighting the utility of relative entropy for understanding diversification strategies. It introduces entropic value at risk (EVaR) as a coherent risk measure, which is an upper bound to the conditional value at risk (CVaR), and explores its generalization, relativistic value at risk (RLVaR), rooted in Kaniadakis entropy. Through extensive empirical analysis on both developed (i.e., S&P 500 and Euro Stoxx 50) and developing markets (i.e., BIST 100 and Bovespa), the study evaluates entropy-based criteria in portfolio selection, investigates model behavior across different market types, and assesses the impact of cryptocurrency introduction on portfolio performance and diversification. The key finding indicates that entropy measures effectively identify optimal portfolios, particularly in scenarios of heightened risk and increased concentration, crucial for mitigating negative net performances during low returns or high turnover. Bitcoin is primarily used for diversification and performance enhancement in the BIST 100 index, while its allocation in other markets remains minimal or non-existent, confirming the extreme concentration observed in stock markets dominated by a few leading stocks. [ABSTRACT FROM AUTHOR]

This study addresses market concentration among major corporations, highlighting the utility of relative entropy for understanding diversification strategies. It introduces entropic value at risk (EVaR) as a coherent risk measure, which is an upper bound to the conditional value at risk (CVaR), and explores its generalization, relativistic value at risk (RLVaR), rooted in Kaniadakis entropy. Through extensive empirical analysis on both developed (i.e., S&P 500 and Euro Stoxx 50) and developing markets (i.e., BIST 100 and Bovespa), the study evaluates entropy-based criteria in portfolio selection, investigates model behavior across different market types, and assesses the impact of cryptocurrency introduction on portfolio performance and diversification. The key finding indicates that entropy measures effectively identify optimal portfolios, particularly in scenarios of heightened risk and increased concentration, crucial for mitigating negative net performances during low returns or high turnover. Bitcoin is primarily used for diversification and performance enhancement in the BIST 100 index, while its allocation in other markets remains minimal or non-existent, confirming the extreme concentration observed in stock markets dominated by a few leading stocks.

As the cyber insurance market is expanding and cyber insurance policies continue to mature, the potential of including pre-incident and post-incident services into cyber policies is being recognised by insurers and insurance buyers. This work addresses the question of how such services should be priced from the insurer's viewpoint, i.e. under which conditions it is rational for a profit-maximising, risk-neutral or risk-averse insurer to share the costs of providing risk mitigation services. The interaction between insurance buyer and seller is modelled as a Stackelberg game, where both parties use distortion risk measures to model their individual risk aversion. After linking the notions of pre-incident and post-incident services to the concepts of self-protection and self-insurance, we show that when pricing a single contract, the insurer would always shift the full cost of self-protection services to the insured; however, this does not generally hold for the pricing of self-insurance services or when taking a portfolio viewpoint. We illustrate the latter statement using toy examples of risks with dependence mechanisms representative in the cyber context. [ABSTRACT FROM AUTHOR]

The concept of Risk Parity (or Equal Risk Contribution), which has been widely used in financial portfolio management, aims at explicitly enforcing diversification in a portfolio by ensuring equal contribution from each asset to the total volatility. While the Risk Parity condition has a straightforward use case in finance, several other application areas can be found in engineering and operations research. In these settings, the Risk Parity condition can be interpreted as enforcing the fairness of a decision or as a way to balance between a number of candidate solutions. In this paper, we consider Risk Parity in conjunction with modern risk-averse stochastic optimization (namely coherent measures of risk), study a generalized Risk Parity model, and propose a combined two-stage diversification-risk framework. We also introduce a bi-level formulation in a case when hierarchical decision-making is enforced. An approach to reformulate the Risk Parity problem as second-order cone programming is also proposed. We assess the performance of the proposed models based on a case study in hazardous materials transportation. The results show their effectiveness in terms of fairness and risk equity for decision-making under uncertainty with heavy-tailed distribution of losses. [ABSTRACT FROM AUTHOR]

Problem definition: Bargaining situations are ubiquitous in economics and management. We consider the problem of bargaining for a fair ex ante distribution of random profits arising from a cooperative effort of a fixed set of risk-averse agents. Our approach integrates optimal managerial decision making into bargaining situations with random outcomes and explicitly models the impact of risk aversion. The proposed solution rests on a firm axiomatic foundation and yet allows to compute concrete bargaining solutions for a wide range of practically relevant problems. Methodology/results: We model risk preferences using coherent acceptability functionals and base our bargaining solution on a set of axioms that can be considered a natural extension of Nash bargaining to our setting. We show that the proposed axioms fully characterize a bargaining solution, which can be efficiently computed by solving a stochastic optimization problem. We characterize special cases where random payoffs of players are simple functions of overall project profit. In particular, we show that, for players with distortion risk functionals, the optimal bargaining solution can be represented by an exchange of standard options contracts with the project profit as the underlying asset. We illustrate the concepts in the paper with a detailed example of risk-averse households that jointly invest into a solar plant. Managerial implications: We demonstrate that there is no conflict of interest between players about management decisions and that risk aversion facilitates cooperation. Furthermore, our results on the structure of optimal contracts as a basket of option contracts provides valuable guidance for negotiators. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.1076. [ABSTRACT FROM AUTHOR]

We study Kusuoka representations of law-invariant coherent risk measures on the space of bounded random variables, which says that any law-invariant coherent risk measure is the supremum of integrals of Average-Value-at-Risk measures. We refine this representation by showing that the supremum in Kusuoka representation is attained for some probability measure in the unit interval. Namely, we prove that any law-invariant coherent risk measure on the space of bounded random variables can be written as an integral of the Average-Value-at-Risk measures on the unit interval with respect to some probability measure. This representation gives a numerically constructive way to bound any law-invariant coherent risk measure on the space of essentially bounded random variables from above and below. The results are illustrated on specific law-invariant coherent risk measures along with numerical simulations. [ABSTRACT FROM AUTHOR]

We study Kusuoka representations of law-invariant coherent risk measures on the space of bounded random variables, which says that any law-invariant coherent risk measure is the supremum of integrals of Average-Value-at-Risk measures. We refine this representation by showing that the supremum in Kusuoka representation is attained for some probability measure in the unit interval. Namely, we prove that any law-invariant coherent risk measure on the space of bounded random variables can be written as an integral of the Average-Value-at-Risk measures on the unit interval with respect to some probability measure. This representation gives a numerically constructive way to bound any law-invariant coherent risk measure on the space of essentially bounded random variables from above and below. The results are illustrated on specific law-invariant coherent risk measures along with numerical simulations. [ABSTRACT FROM AUTHOR]

Decision making under uncertainty includes reassessing and reevaluating risk after initial decisions. To this end, it is essential to consider a governing value process and to track its evolution over time. The paper, "Risk-Averse Stochastic Programming: Time Consistency and Optimal Stopping," by Pichler, Liu, and Shapiro, develops a consistent framework by consolidating risk and optimal stopping. The paper develops the notion of time consistency for the stochastic multistage optimization problem. Supermartingales and envelopes characterizing optimal decisions are given explicitly. With that, dynamic equations are derived, which gradually reveal the optimal policy. Taking risk into account requires updating optimal policies, as an explicit example on American options demonstrates. This paper addresses time consistency of risk-averse optimal stopping in stochastic optimization. It is demonstrated that time-consistent optimal stopping entails a specific structure of the functionals describing the transition between consecutive stages. The stopping risk measures capture this structural behavior and allow natural dynamic equations for risk-averse decision making over time. Consequently, associated optimal policies satisfy Bellman's principle of optimality, which characterizes optimal policies for optimization by stating that a decision maker should not reconsider previous decisions retrospectively. We also discuss numerical approaches to solving such problems. [ABSTRACT FROM AUTHOR]

In this paper, we develop an algorithm to efficiently solve risk-averse optimization problems posed in reflexive Banach space. Such problems often arise in many practical applications as, e.g., optimization problems constrained by partial differential equations with uncertain inputs. Unfortunately, for many popular risk models including the coherent risk measures, the resulting risk-averse objective function is nonsmooth. This lack of differentiability complicates the numerical approximation of the objective function as well as the numerical solution of the optimization problem. To address these challenges, we propose a primal–dual algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the classical method of multipliers and by epigraphical regularization of risk measures. As a result, the algorithm solves a sequence of smooth optimization problems using derivative-based methods. We prove convergence of the algorithm even when the subproblems are solved inexactly and conclude with numerical examples demonstrating the efficiency of our method. [ABSTRACT FROM AUTHOR]

We use the theory of cooperative games for the design of fair insurance contracts. An insurance contract needs to specify the premium to be paid and a possible participation in the benefit (or surplus) of the company. We suppose that a convex commonotonic premium functional is used to value the aggregated liability of the insurance company. It results from the analysis that when a contract is exposed to the default risk of the insurance company, ex‐ante equilibrium considerations require a certain participation in the benefit of the company to be specified in the contracts. The fair benefit participation of agents appears as an outcome of a game involving the residual risks induced by the default possibility and using fuzzy coalitions. [ABSTRACT FROM AUTHOR]

Distortion risk measures have been popular in financial and insurance applications in recent years due to their attractive properties. The aim of the article is to investigate whether risk measures “VaR in the power of t”, introduced by the author, belong to the class of distortion risk measures, as well as to describe the corresponding distortion functions. The author introduces a new class of risk measures “ES to the power of t” and investigates whether it belongs to distortion risk measures, and also describes the corresponding distortion functions. The author used the composite method to design new distortion functions and corresponding distortion risk measures, to prove that risk measures “VaR to the power of t” and “ES to the power of t” belong to the class of distortion risk measures. The paper presents examples to illustrate the relevant concepts and results that show the importance of risk measures “VaR to the power of t” and “ES to the power of t” as subsets of distortion risk measures that allow identifying various financial catastrophic risks. The author concludes that risk measures “VaR to the power of t” and “ES to the power of t” can be used in risk management of companies when assessing remote, highly catastrophic risks.

We consider a group consisting of N business units. We suppose there are regulatory constraints for each unit; more precisely, the net worth of each business unit is required to belong to a set of acceptable risks, assumed to be a convex cone. Because of these requirements, there are less incentives to operate under a group structure, as creating one single business unit, or altering the liability repartition among units, may allow to reduce the required capital. We analyse the possibilities for the group to benefit from a diversification effect and economise on the cost of capital. We define and study the risk measures that allow for any group to achieve the minimal capital, as if it were a single unit, without altering the liability of business units, and despite the individual admissibility constraints. We call these risk measures cohesive risk measures. In the commonotonic case, we show that they are tail expectations but calculated under a different probability. [ABSTRACT FROM AUTHOR]

We revisit mean‐risk portfolio selection in a one‐period financial market where risk is quantified by a positively homogeneous risk measure ρ$\rho$. We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean‐variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation ρ$\rho$‐arbitrage, and prove that it cannot be excluded—unless ρ$\rho$ is as conservative as the worst‐case risk measure. After providing a primal characterization of ρ$\rho$‐arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterization of ρ$\rho$‐arbitrage. We show that the absence of ρ$\rho$‐arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of ρ$\rho$. A special case of our result shows that the market does not admit ρ$\rho$‐arbitrage for Expected Shortfall at level α$\alpha$ if and only if there exists an EMM Q≈P$\mathbb {Q} \approx \mathbb {P}$ such that ∥dQdP∥∞<1α$\Vert \frac{\textnormal {d}\mathbb {Q}}{\textnormal {d}\mathbb {P}} \Vert _\infty < \frac{1}{\alpha }$. [ABSTRACT FROM AUTHOR]

In Quantitative Finance 2016, Chen, Hu and Lin (CHL) claimed the following: ' ... there is yet no coherent risk measure related to investment performance.' (p. 682). Our paper suggests and analyzes four coherence axioms that portfolio performance ratios should satisfy. Our Portfolio Riskless Translation Invariance axiom must be satisfied to assure separation of the objective decision to optimize a portfolio's risky composition from the subjective decision to optimize the weight of the portfolio's level of risk-free asset. Performance ratios with fixed thresholds other than the risk-free rate do not satisfy this axiom, allowing portfolio managers to affect an ex-ante performance ratio merely by changing the proportion of the risk-free asset in the portfolio rather than by improving the composition of the portfolio's risky components. The magnitude of this potential drawback is examined using S&P-500 stock index data. Replacing the fixed threshold, T, with a threshold T (γ , α) that equals γ times the portfolio's risk premium plus (1-γ) times the risk-free rate, eliminates the above shortcoming for any selected γ. In addition, using performance ratios with threshold T (γ , α) rather than fixed T, assures consistency of performance ratios of effective stochastic dominance and risk-free asset rules. [ABSTRACT FROM AUTHOR]

We investigate how the spectral risk measure associated with holding stocks rather than a risk-free deposit, depends on the holding period. Previous papers have shown that within a limited class of spectral risk measures, and when the stock price follows specific processes, spectral risk becomes negative at long periods. We generalize this result for arbitrary exponential Lévy processes. We also prove the same behavior for all spectral risk measures (including the important special case of Expected Shortfall) when the stock price grows realistically fast and when it follows a geometric Brownian motion or a finite moment log stable process. This result would suggest that holding stocks for long periods has a vanishing downside risk. However, using realistic models, we find numerically that spectral risk initially increases for a significant amount of time and reaches zero level only after several decades. Therefore, we conclude that holding stocks has spectral risk for all practically relevant periods. [ABSTRACT FROM AUTHOR]

Uncertainty pervades virtually every branch of science and engineering, and in many disciplines, the underlying phenomena can be modeled by partial differential equations (PDEs) with uncertain or random inputs. This work is motivated by risk-averse stochastic programming problems constrained by PDEs. These problems are posed in infinite dimensions, which leads to a significant increase in the scale of the (discretized) problem. In order to handle the inherent nonsmoothness of, for example, coherent risk measures and to exploit existing solution techniques for smooth, PDE-constrained optimization problems, we propose a variational smoothing technique called epigraphical (epi-)regularization. We investigate the effects of epi-regularization on the axioms of coherency and prove differentiability of the smoothed risk measures. In addition, we demonstrate variational convergence of the epi-regularized risk measures and prove the consistency of minimizers and first-order stationary points for the approximate risk-averse optimization problem. We conclude with numerical experiments confirming our theoretical results. [ABSTRACT FROM AUTHOR]

The intuition of risk is based on two main concepts: loss and variability. In this paper, we present a composition of risk and deviation measures, which contemplate these two concepts. Based on the proposed Limitedness axiom, we prove that this resulting composition, based on properties of the two components, is a coherent risk measure. Similar results for the cases of convex and co-monotone risk measures are exposed. We also provide examples of known and new risk measures constructed under this framework in order to highlight the importance of our approach, especially the role of the Limitedness axiom. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investment strategies that combine asset pricing models and coherent risk measures. In particular, we utilize the theoretical framework of Balbas et al. (J Risk 18(4):25–52, 2016), which suggests that simply by managing a portfolio of assets, an investor can achieve risk that converges to −∞ and returns that converge to + ∞. We contribute on that framework by providing evidence that arise from the CAPM model, in regard to the efficient market hypothesis. In addition, our results suggest that an investor can exhibit returns that outperform the market index by managing a portfolio less volatile than the market. [ABSTRACT FROM AUTHOR]

This paper studies the problem of risk-averse receding horizon motion planning for agents with uncertain dynamics, in the presence of stochastic, dynamic obstacles. We propose a model predictive control (MPC) scheme that formulates the obstacle avoidance constraint using coherent risk measures. To handle disturbances, or process noise, in the state dynamics, the state constraints are tightened in a risk-aware manner to provide a disturbance feedback policy. We also propose a waypoint following algorithm that uses the proposed MPC scheme for discrete distributions and prove its risk-sensitive recursive feasibility while guaranteeing finite-time task completion. We further investigate some commonly used coherent risk metrics, namely, conditional value-at-risk (CVaR), entropic value-at-risk (EVaR), and g-entropic risk measures, and propose a tractable incorporation within MPC. We illustrate our framework via simulation studies. [ABSTRACT FROM AUTHOR]

Worst-case risk measures provide a means of calculating the largest value of risk when only partial information of the underlying distribution is available. For popular risk measures such as value-at-risk (VaR) and conditional value-at-risk (CVaR) it is now known that their worst-case counterparts can be evaluated in closed form when only the first two moments are known. We show in this paper that closed-form solutions exist for a general class of law invariant coherent risk measures, which consist of spectral risk measures (and thus CVaR also) as special cases. Moreover, we provide worst-case distributions characterized in terms of risk spectrums, which can take any form of distribution bounded from below. As applications of the closed-form results, new formulas are derived for calculating the worst-case values of higher order risk measures and higher order semideviation, and new robust portfolio optimization models are provided. The online appendix is available at https://doi.org/10.1287/opre.2018.1736. [ABSTRACT FROM AUTHOR]

We study statistical properties of the optimal value and optimal solutions of the sample average approximation of risk-averse stochastic problems. Central limit theorem-type results are derived for the optimal value when the stochastic program is expressed in terms of a law invariant coherent risk measure having a discrete Kusuoka representation. The obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk-averse convex stochastic programs on the basis of samples of the underlying random vectors. We also consider nonasymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the stochastic mirror descent algorithm. Numerical simulations show how to use our developments to choose among different distributions and on the considered class of risk-averse stochastic programs the asymptotic tests show better results. [ABSTRACT FROM AUTHOR]

A classical result in risk measure theory states that every coherent risk measure has a dual representation as the supremum of certain expected value over a risk envelope. We study this topic in more detail. The related issues include: (1) Set operations of risk envelopes and how they change the risk measures, (2) The structure of risk envelopes of popular risk measures, (3) Aversity of risk measures and its impact to risk envelopes, and (4) A connection between risk measures in stochastic optimization and uncertainty sets in robust optimization. [ABSTRACT FROM AUTHOR]

In this work, we study the problem of detecting risk-averse low-diameter clusters in graphs. It is assumed that the clusters represent k-clubs and that uncertain information manifests itself in the form of stochastic vertex weights whose joint distribution is known. The goal is to find a k-club of minimum risk contained in the graph. A stochastic programming framework that is based on the formalism of coherent risk measures is used to quantify the risk of a cluster. We show that the selected representation of risk guarantees that the optimal subgraphs are maximal clusters. A combinatorial branch-and-bound algorithm is proposed and its computational performance is compared with an equivalent mathematical programming approach for instances with k=2,3, and 4. [ABSTRACT FROM AUTHOR]

It is of crucial importance to develop risk-averse models for multicriteria decision making under uncertainty. A major stream of the related literature studies optimization problems that feature multivariate stochastic benchmarking constraints. These problems typically involve a univariate stochastic preference relation, often based on stochastic dominance or a coherent risk measure such as conditional value-at-risk, which is then extended to allow the comparison of random vectors by the use of a family of scalarization functions: All scalarized versions of the vector of the uncertain outcomes of a decision are required to be preferable to the corresponding scalarizations of the benchmark outcomes. While this line of research has been dedicated almost entirely to linear scalarizations, the corresponding deterministic literature uses a wide variety of scalarization functions that, among other advantages, offer a high degree of modeling flexibility. In this paper we aim to incorporate these scalarizations into a stochastic context by introducing the general class of min-biaffine functions. We study optimization problems in finite probability spaces with multivariate stochastic benchmarking constraints based on min-biaffine scalarizations. We develop duality results, optimality conditions, and a cut generation method to solve these problems. We also introduce a new characterization of the risk envelope of a coherent risk measure in terms of its Kusuoka representation as a tool toward proving the finite convergence of our solution method. The main computational challenge lies in solving cut generation subproblems; we develop several mixed-integer programming formulations by exploiting the min-affine structure and leveraging recent advances for solving similar problems with linear scalarizations. We conduct a computational study on a well-known homeland security budget allocation problem to examine the impact of the proposed scalarizations on optimal solutions, and illustrate the computational performance of our solution methods. [ABSTRACT FROM AUTHOR]

We solve the optimization problem of minimum portfolio risk for seven measures using linear programming under cardinality constraints. The risk measures used are Expected Loss, Expected Loss Deviation, Expected Shortfall, Shortfall Deviation Risk, Expectile Value at Risk, Deviation Expectile Value at Risk, and Maximum Loss. We assess the out-of-sample performance of seven risk-optimized portfolios with a maximum size of 20 assets for S&P 100 components. After subtracting transaction costs, the Expected Loss Deviation portfolios have shown superior performance in terms of diversification and risk, the Maximum Loss portfolios have presented a higher Sharpe ratio (1.098 against 0.990 for the benchmark), the Expected Loss portfolios have higher absolute returns (660%), Sortino and STARR ratios. Expected Shortfall portfolios have presented the lowest Beta coefficients (0.616), and all portfolios returned lower Betas than the benchmark. All portfolios present significant annual alpha of 25% after adjusting for several risk factors. Our results show that superior performance can be achieved with simple linearized optimization models with lower market exposure measures to the CAPM beta. • We provide linear programming solutions for the minimum risk portfolio. • Risk-based portfolios were constructed from a sample of S&P 100 constituents. • Maximum Loss out-of-sample portfolios have presented higher Sharpe ratios. • All risk-based portfolios present lower market exposure than the benchmark. • Usual risk factors do not fully explain the returns for risk-based portfolios. [ABSTRACT FROM AUTHOR]

This paper provides a critical analysis of the subadditivity axiom, which is the key condition for coherent risk measures. Contrary to the subadditivity assumption, bank mergers can create extra risk. We begin with an analysis how a merger affects depositors, junior or senior bank creditors, and bank owners. Next it is shown that bank mergers can result in higher payouts having to be made by the deposit insurance scheme. Finally, we demonstrate that if banks are interconnected via interbank loans, a bank merger could lead to additional contagion risks. We conclude that the subadditivity assumption should be rejected, since a subadditive risk measure, by definition, cannot account for such increased risks.

In this paper we study distributionally robust stochastic programming in a setting where there is a specified reference probability measure and the uncertainty set of probability measures consists of measures in some sense close to the reference measure. We discuss law invariance of the associated worst case functional and consider two basic constructions of such uncertainty sets. Finally we illustrate some implications of the property of law invariance. [ABSTRACT FROM AUTHOR]

We study the exploitation of a one species, multiple stand forest plantation when timber price is governed by a stochastic process. Our model is a stochastic dynamic program with a weighted mean-risk objective function, and our main risk measure is the Conditional Value-at-Risk. We consider two stochastic processes, geometric Brownian motion and Ornstein-Uhlenbeck: in the first case, we completely characterize the optimal policy for all possible choices of the parameters while in the second, we provide sufficient conditions assuring that harvesting everything available is optimal. In both cases we solve the problem theoretically for every initial condition. We compare our results with the risk neutral framework and generalize our findings to any coherent risk measure that is affine on the current price. [ABSTRACT FROM AUTHOR]

This paper addresses the Entropic Value-at-Risk ( $${{\mathrm{\mathsf {EV@R}}}}$$ ), a recently introduced coherent risk measure. It is demonstrated that the norms defined by $${{\mathrm{\mathsf {EV@R}}}}$$ induce the same Banach spaces, irrespective of the confidence level. Three vector spaces, called the primal, dual, and bidual entropic spaces, corresponding with $${{\mathrm{\mathsf {EV@R}}}}$$ are fully studied. It is shown that these spaces equipped with the norms induced by $${{\mathrm{\mathsf {EV@R}}}}$$ are Banach spaces. The entropic spaces are then related to the $$L^p$$ spaces, as well as specific Orlicz hearts and Orlicz spaces. This analysis indicates that the primal and bidual entropic spaces can be used as very flexible model spaces, larger than $$L^\infty $$ , over which all $$L^p$$ -based risk measures are well-defined. The dual $${{\mathrm{\mathsf {EV@R}}}}$$ norm and corresponding Hahn-Banach functionals are presented in closed form, which are not explicitly known for the Orlicz and Luxemburg norms that are equivalent to the $${{\mathrm{\mathsf {EV@R}}}}$$ norm. The duality relationships among the entropic spaces are investigated. The duality results are also used to develop an extended Donsker-Varadhan variational formula, and to explicitly provide the dual and Kusuoka representations of $${{\mathrm{\mathsf {EV@R}}}}$$ , as well as the corresponding maximizing densities in both representations. Our results indicate that financial concepts can be successfully used to develop insightful tools for not only the theory of modern risk measurement but also other fields of stochastic analysis and modeling. [ABSTRACT FROM AUTHOR]

In this paper we study optimal stopping problems with respect to distorted expectations with concave distortion functions. Our starting point is a seminal work of Xu and Zhou in 2013, who gave an explicit solution of such a stopping problem under a rather large class of distortion functionals. In this paper, we continue this line of research and prove a novel representation, which relates the solution of an optimal stopping problem under distorted expectation to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming-based approaches possible. Furthermore, by means of the well-known Kusuoka representation, we extend our results to optimal stopping under general law invariant coherent risk measures. Finally, based on our representations, we develop several Monte Carlo approximation algorithms and illustrate their power for optimal stopping under absolute semideviation risk measures. [ABSTRACT FROM AUTHOR]

A prospective hospital payment system induces a substantial financial risk for the provider that not only increases incentives for cost efficiency. If financial risk is not bounded at an equal level for all DRGs and patient cases, hospitals have incentives to minimize their risk level as well. This paper investigates its consequences and proposes a DRG-System redesign encompassing a fair (or risk-adjusted) compensation. We adjust the price of the provider such that the risk of a financial loss is bounded similar to the Value-at-Risk (VaR). By focusing on the risk-transfer relation we are able to understand the induced basic strategies of risk mitigation for providers: The induced incentives reinforces the objectives in the DRG system to streamline processes in order to control its costs and hence stimulates specialized health clinics, while general providers offering basic services are likely to be more exposed to make losses. [ABSTRACT FROM AUTHOR]

The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile-based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law-invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law-invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles. [ABSTRACT FROM AUTHOR]

Stochastic models used for pricing, reserving, or capital modelling in insurance companies are often very complex, which is why resulting distributions are typically approximated by Monte Carlo simulations. Both the market and regulators exert increasing pressure not to discard the resulting sample distributions, but rather to store them for future review, audit, or validation, as well as to transfer them between actuarial systems. The present work introduces a compression algorithm which approximates an empirical univariate distribution function through a piecewise linear distribution. In contrast to keeping the full sample, such an approximation facilitates the storage and data transfer of the results by drastically reducing memory requirements. The approximation algorithm preserves the mean and imposes a uniformly bounded relative error over a space of coherent risk measures (TVaR). An efficient, open source implementation is provided. [ABSTRACT FROM AUTHOR]

In this paper we consider the notion of rectangularity of a set of probability measures from a somewhat different point of view. We define rectangularity as a property of dynamic decomposition of a distributionally robust stochastic optimization problem and show how it relates to the modern theory of coherent risk measures. Consequently, we discuss robust formulations of multistage stochastic optimization problems in frameworks of stochastic programming, stochastic optimal control, and Markov decision processes. [ABSTRACT FROM AUTHOR]

Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage, however, using coherent risk measures within a standard single-level optimization framework becomes problematic. To avoid severe time-consistency difficulties, the current state of the art is to employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. This paper summarizes the potential drawbacks of nested-form risk measure issues and then presents an alternative multilevel optimization modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler's choice of objective function. This technique leads to models that are time consistent even while using time-inconsistent risk measures and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should be the starting point for all multistage optimization modeling. When used with time-consistent objective functions, we show its multilevel optimization constraints become redundant, and the associated models thus simplify to a more familiar single-objective form. Unfortunately, we also show that our proposed approach leads to NP-hard models, even in the simplest imaginable setting in which it would be needed: three-stage linear problems on a finite probability space, using the standard average value-at-risk and first-order mean-semideviation risk measures. Finally, we show that for a simple but reasonably realistic test application, the kind of models we propose, although drawn from an NP-hard family and certainly more time consuming to solve than those obtained from the nested-objective approach, are readily solvable to global optimality using a standard commercial mixed-integer linear programming solver. Therefore, there seems some promise of our proposed modeling approach being useful despite its computational complexity properties. [ABSTRACT FROM AUTHOR]

We seek to characterize the trading behavior of an agent, in the context of a continuous-time portfolio choice model, if she measures the risk by a so called weighted value-at-risk (VaR), which is a generalization of both VaR and conditional VaR. We show that when bankruptcy is allowed the agent displays extreme risk-taking behaviors, unless the downside risk is significantly penalized, in which case an asymptotically optimal strategy is to invest a very small amount of money in an extremely risky but highly rewarding lottery, and save the rest in the risk-free asset. When bankruptcy is prohibited, extreme risk-taking behaviors are prevented in most cases in which the asymptotically optimal strategy is to spend a very small amount of money in an extremely risky but highly rewarding lottery and put the rest in an asset with moderate risk. Finally, we show that the trading behaviors remain qualitatively the same if the weighted VaR is replaced by a law-invariant coherent risk measure. [ABSTRACT FROM AUTHOR]

This paper compares two frameworks for measuring risk in a multiperiod setting. The first corresponds to applying a single coherent risk measure to the cumulative future costs, and the second involves applying a composition of one-step coherent risk mappings. We characterize several necessary and sufficient conditions under which one measurement always dominates the other and introduce a metric to quantify how close the two measures are. Using this notion, we address the question of how tightly a given coherent measure can be approximated by lower or upper bounding compositional measures. We exhibit an interesting asymmetry between the two cases: the tightest upper bound can be exactly characterized and corresponds to a popular construction in the literature, whereas the tightest lower bound is not readily available. We show that testing domination and computing the approximation factors are generally NP-hard, even when the risk measures are comonotonic and law-invariant. However, we characterize conditions and discuss examples where polynomial-time algorithms are possible. One such case is the well-known conditional value-at-risk measure, which we explore in more detail. Our theoretical and algorithmic constructions exploit interesting connections between the study of risk measures and the theory of submodularity and combinatorial optimization, which may be of independent interest. [ABSTRACT FROM AUTHOR]