Dynamic Risk Measures Research Articles

On the Value of Risk-Averse Multistage Stochastic Programming in Capacity Planning

We consider a risk-averse stochastic capacity planning problem under uncertain demand in each period. Using a scenario tree representation of the uncertainty, we formulate a multistage stochastic integer program to adjust the capacity expansion plan dynamically as more information on the uncertainty is revealed. Specifically, in each stage, a decision maker optimizes capacity acquisition and resource allocation to minimize certain risk measures of maintenance and operational cost. We compare it with a two-stage approach that determines the capacity acquisition for all the periods up front. Using expected conditional risk measures, we derive a tight lower bound and an upper bound for the gaps between the optimal objective values of risk-averse multistage models and their two-stage counterparts. Based on these derived bounds, we present general guidelines on when to solve risk-averse two-stage or multistage models. Furthermore, we propose approximation algorithms to solve the two models more efficiently, which are asymptotically optimal under an expanding market assumption. We conduct numerical studies using randomly generated and real-world instances with diverse sizes, to demonstrate the tightness of the analytical bounds and efficacy of the approximation algorithms. We find that the gaps between risk-averse multistage and two-stage models increase as the variability of the uncertain parameters increases and decrease as the decision maker becomes more risk averse. Moreover, a stagewise-dependent scenario tree attains much higher gaps than a stagewise-independent counterpart, whereas the latter produces tighter analytical bounds. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work of Dr. X. Yu was partially supported by the U.S. National Science Foundation Division of Information and Intelligent Systems [Grant 2331782]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0396 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0396 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Optimizing sequential decision-making under risk: Strategic allocation with switching penalties

This paper considers the multiarmed bandit (MAB) problem augmented with a critical real-world consideration: the cost implications of switching decisions. Our work distinguishes itself by addressing the largely unexplored domain of risk-averse MAB problems compounded by switching penalties. Such scenarios are not just theoretical constructs but are reflective of numerous practical applications. Our contribution is threefold: firstly, we explore how switching costs and risk aversion influence decision-making in MAB problems. Secondly, we present novel theoretical results, including the development of the Risk-Averse Switching Index (RASI), which addresses the dual challenges of risk aversion and switching costs, demonstrating its near-optimal efficacy. This heuristic solution method is grounded in dynamic coherent risk measures, enabling a time-consistent evaluation of risk and reward. Lastly, through rigorous numerical experiments, we validate our algorithm’s effectiveness and practical applicability, providing decision-makers with valuable insights and tools for navigating the multifaceted landscape of risk-averse environments with inherent switching costs.

Risk-Averse Markov Decision Processes Through a Distributional Lens

By adopting a distributional viewpoint on law-invariant convex risk measures, we construct dynamic risk measures (DRMs) at the distributional level. We then apply these DRMs to investigate Markov decision processes, incorporating latent costs, random actions, and weakly continuous transition kernels. Furthermore, the proposed DRMs allow risk aversion to change dynamically. Under mild assumptions, we derive a dynamic programming principle and show the existence of an optimal policy in both finite and infinite time horizons. Moreover, we provide a sufficient condition for the optimality of deterministic actions. For illustration, we conclude the paper with examples from optimal liquidation with limit order books and autonomous driving. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada [Grants RGPAS-2018-522715 and RGPIN-2018-05705].

Stochastic orders and distortion risk contribution ratio measures

Relative spillover effects play a crucial role in the analysis and comparison of systemic risks. This paper introduces a novel approach, referred to as distortion risk contribution ratio measures, for quantifying such effects. Various types of contribution ratio measures are defined based on the newly proposed conditional distortion risk measures by Dhaene et al. (2022), and useful integral-based representations are provided as well. An interesting equivalent characterization result for the convex transform order is also presented, which is not only relevant to proving our main results but also has independent value in other research areas. We then establish comparison results between the distortion risk contribution ratio measures of two different bivariate random vectors with either the same or different copulas. Sufficient conditions are established in terms of stochastic orders, copula functions, distortion functions, and stress levels. Furthermore, we investigate the ordering behaviors of these measures in relation to the interaction between paired risks. Numerical examples are presented to illustrate the conditions and main findings.

Fully Dynamic Risk Measures: Horizon Risk, Time-Consistency, and Relations with BSDEs and BSVIEs

Fully Dynamic Risk Measures: Horizon Risk, Time-Consistency, and Relations with BSDEs and BSVIEs

Time consistency of dynamic risk measures and dynamic performance measures generated by distortion functions

. The aim of this work is to study risk measures generated by distortion functions in a dynamic discrete time setup and to investigate the corresponding dynamic coherent acceptability indices (DCAIs) generated by families of such risk measures. First, we show that conditional versions of Choquet integrals are indeed dynamic coherent risk measures (DCRMs) and also introduce the class of dynamic weighted value at risk measures. We prove that these two classes of risk measures coincides. In the spirit of robust representations theorem for DCAIs, we establish some relevant properties of families of DCRMs generated by distortion functions and then define and study the corresponding DCAIs. Second, we study the time consistency of DCRMs and DCAIs generated by distortion functions. In particular, we prove that such DCRMs are sub-martingale time consistent, but they are not super-martingale time consistent. We also show that DCRMs generated by distortion functions are not weakly acceptance-time consistent. We also present several widely used classes of distortion functions and derive some new representations of these distortions.

Eliciting Risk Perceptions: Does Conditional Question Wording Have a Downside?

To assess the impact of risk perceptions on prevention efforts or behavior change, best practices involve conditional risk measures, which ask people to estimate their risk contingent on a course of action (e.g., "if not vaccinated"). To determine whether the use of conditional wording-and its drawing of attention to one specific contingency-has an important downside that could lead researchers to overestimate the true relationship between perceptions of risk and intended prevention behavior. In an online experiment, US participants from Amazon's MTurk (N = 750) were presented with information about an unfamiliar fungal disease and then randomly assigned among 3 conditions. In all conditions, participants were asked to estimate their risk for the disease (i.e., subjective likelihood) and to decide whether they would get vaccinated. In 2 conditional-wording conditions (1 of which involved a delayed decision), participants were asked about their risk if they did not get vaccinated. For an unconditional/benchmark condition, this conditional was not explicitly stated but was still formally applicable because participants had not yet been informed that a vaccine was even available for this disease. When people gave risk estimates to a conditionally worded risk question after making a decision, the observed relationship between perceived risk and prevention decisions was inflated (relative to in the unconditional/benchmark condition). The use of conditionals in risk questions can lead to overestimates of the impact of perceived risk on prevention decisions but not necessarily to a degree that should call for their omission. Conditional wording, which is commonly recommended for eliciting risk perceptions, has a potential downside.It can produce overestimates of the true relationship between perceived risk and prevention behavior, as established in the current work.Though concerning, the biasing effect of conditional wording was small-relative to the measurement benefits that conditioning usually provides-and should not deter researchers from conditioning risk perceptions.More research is needed to determine when the biasing impact of conditional wording is strongest.

Well-posedness and regularity of mean-field backward doubly stochastic Volterra integral equations and applications to dynamic risk measures

In this paper, the theory of mean-field backward doubly stochastic Volterra integral equations (MF-BDSVIEs) is studied. First, we derive the well-posedness of M-solutions to MF-BDSVIEs, and prove the comparison theorem for such a type of equations. Furthermore, the regularity result of the M-solution for MF-BDSVIEs is established by virtue of Malliavin calculus. Finally, as an application of the comparison theorem, we obtain the properties of dynamic risk measures governed by MF-BDSVIEs.

Risk-Averse Decision Making Under Uncertainty

A large class of decision making under uncertainty problems can be described via Markov decision processes (MDPs) or partially observable MDPs (POMDPs), with application to artificial intelligence and operations research, among others. In this paper, we consider the problem of designing policies for MDPs and POMDPs with objectives and constraints in terms of dynamic coherent risk measures rather than the traditional total expectation, which we refer to as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constrained risk-averse problem</i> . Our contributions can be described as follows: For MDPs, under some mild assumptions, we propose an optimization-based method to synthesize Markovian policies. We then demonstrate that such policies can be found by solving difference convex programs (DCPs). We show that our formulation generalize linear programs for constrained MDPs with total discounted expected costs and constraints; For POMDPs, we show that, if the coherent risk measures can be defined as a Markov risk transition mapping, an infinite-dimensional optimization can be used to design Markovian belief-based policies. For POMDPs with stochastic finite-state controllers (FSCs), we show that the latter optimization simplifies to a (finite-dimensional) DCP. We incorporate these DCPs in a policy iteration algorithm to design risk-averse FSCs for POMDPs. We demonstrate the efficacy of the proposed method with numerical experiments involving conditional-value-at-risk (CVaR) and entropic-value-at-risk (EVaR) risk measures.

Collective dynamic risk measures

Collective dynamic risk measures

Conditionally Elicitable Dynamic Risk Measures for Deep Reinforcement Learning

Conditionally Elicitable Dynamic Risk Measures for Deep Reinforcement Learning

Duality and stable compactness in Orlicz-type modules

Orlicz-type modules are module analogues of classical Orlicz spaces. We study duality and stable compactness in Orlicz-type modules. We characterize the conditional Köthe dual of an Orlicz-type module as the space of all σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-order continuous module homomorphisms. We find an order continuity criterion for stable compactness in Orlicz-type modules. As an application, we obtain a robust representation result for conditional risk measures on Orlicz spaces.

Index policy for multiarmed bandit problem with dynamic risk measures

The multiarmed bandit problem (MAB) is a classic problem in which a finite amount of resources must be allocated among competing choices with the aim of identifying a policy that maximizes the expected total reward. MAB has a wide range of applications including clinical trials, portfolio design, tuning parameters, internet advertisement, auction mechanisms, adaptive routing in networks, and project management. The classical MAB makes the strong assumption that the decision maker is risk-neutral and indifferent to the variability of the outcome. However, in many real life applications, these assumptions are not met and decision makers are risk-averse. Motivated to resolve this, we study risk-averse control of the multiarmed bandit problem in regard to the concept of dynamic coherent risk measures to determine a policy with the best risk-adjusted total discounted return. In respect of this specific setting, we present a theoretical analysis based on Whittle’s retirement problem and propose a priority-index policy that reduces to the Gittins index when the level of risk-aversion converges to zero. We generalize the restart formulation of the Gittins index to effectively compute these risk-averse allocation indices. Numerical results exhibit the excellent performance of this heuristic approach for two well-known coherent risk measures of first-order mean-semideviation and mean-AVaR. Our experimental studies suggest that there is no guarantee that an index-based optimal policy exists for the risk-averse problem. Nonetheless, our risk-averse allocation indices can achieve optimal or near-optimal policies which in some instances are easier to interpret compared to the exact optimal policy.

Vine Copula based Portfolio Level Conditional Risk Measure Forecasting

Accurately estimating risk measures for financial portfolios and validating their robustness is critical for both financial institutions and regulators. However, many existing models operate at the aggregate portfolio level, hence they fail to capture the complex cross-dependencies between portfolio components and particularly provide no methodology to perform a sensitivity analysis on the estimates. To address both aspects, a new approach is presented that uses vine copulas in combination with univariate ARMA-GARCH models for marginal modelling to compute conditional portfolio-level risk measure estimates by simulating portfolio-level forecasts conditioned on a stress factor. A quantile-based approach is then presented to observe the behaviour of risk measures given a particular state of the conditioning asset(s). In an illustrative case study of Spanish equities with different stress factors, the results show that the portfolio is quite robust to a sharp downturn in the American market. At the same time, there is no evidence of this behaviour with respect to the European market. The novel algorithms presented are ready for use through the R package portvine, which is publicly available on CRAN.

Dynamic inflation hedging performance and downside risk: A comparison between Islamic and conventional stock indices

Inflation has attained the highest level in decades, with the Russian/Ukrainian war adding upward pressure on prices of energy and food. The recent war has exacerbated the already growth of consumer prices resulting from the COVID-19 and disrupted supply-chains. In a high-inflation environment where investors face inflationary pressures during their investment decision-making, it becomes increasingly important to diversify portfolios towards assets with potential inflation hedging abilities. This paper applies the dependence-switching copula and different dynamic downside risk measures to assess the role of Islamic stocks as a hedge for inflation, in comparison to their conventional counterparts. We use data for the period January 2000–October 2022, covering the periods prior to and post-COVID-19 and the war in Ukraine. We consider an investor simply wishing to hedge inflation (or a “safety first” investor) and an investor having a more ambitious target real return. For each of the investor targets, we identify the optimal portfolio composition depending on various scenarios. Prior to the pandemic and the Ukraine conflict, Islamic stock indices exhibit the best hedging properties under their bull states, extreme inflation episodes and for an investor willing just to protect against inflation. In considering the post-2020 concerns, Islamic stocks’ diversification potentials are going to be more relevant, reinforcing their strategic benefits as risk mitigation assets. But by increasing the target of real return, it is shown that the weight assigned to the Islamic equities in the portfolio is smaller relative to the conventional stocks. The fund managers and financial analysts can rely on these findings in order to help investors reach a well-diversified portfolio and achieve an optimal risk reduction in an inflationary environment.

Conditional Distributionally Robust Functionals

This paper addresses decision making in multiple stages, where prior information is available and where consecutive and successive decisions are made. Risk measures assess the random outcome by taking various candidate probability measures into account. To justify decisions in multiple stages, it is essential to have conditional risk measures available, which respect the information, which was already revealed in the past. The paper addresses different variants of risk measures, discusses their properties in the specific context and their implications in multistage decision making. Various examples of risk measures on simple probability spaces with finite support illustrate the content. The Wasserstein and nested distance are involved to make decision making with numerous scenarios numerically tractalbe.

Anticipated Backward Stochastic Volterra Integral Equations with Jumps and Applications to Dynamic Risk Measures

Anticipated Backward Stochastic Volterra Integral Equations with Jumps and Applications to Dynamic Risk Measures

Dynamic risk measures via backward doubly stochastic Volterra integral equations with jumps

In this article, we study dynamic risk measures by means of backward doubly stochastic Volterra integral equations (BDSVIEs, for short) with jumps. We establish the well-posedness of BDSVIEs with jumps in the sense of M-solution and prove a comparison theorem of BDSVIEs with jumps. Finally, we study properties of dynamic risk measures induced by BDSVIEs with jumps. Our results extend the well-posedness and the comparison theorem of BDSVIEs without jumps to the setting with jumps, and extend dynamic risk measures induced by BSDEs, BDSDEs, and BSVIEs to the case of BDSVIEs with jumps.

Reinforcement learning with dynamic convex risk measures

AbstractWe develop an approach for solving time‐consistent risk‐sensitive stochastic optimization problems using model‐free reinforcement learning (RL). Specifically, we assume agents assess the risk of a sequence of random variables using dynamic convex risk measures. We employ a time‐consistent dynamic programming principle to determine the value of a particular policy, and develop policy gradient update rules that aid in obtaining optimal policies. We further develop an actor–critic style algorithm using neural networks to optimize over policies. Finally, we demonstrate the performance and flexibility of our approach by applying it to three optimization problems: statistical arbitrage trading strategies, financial hedging, and obstacle avoidance robot control.

The importance of dynamic risk constraints for limited liability operators

Previous literature shows that prevalent risk measures such as value at risk or expected shortfall are ineffective to curb excessive risk-taking by a tail-risk-seeking trader with S-shaped utility function in the context of portfolio optimisation. However, these conclusions hold only when the constraints are static in the sense that the risk measure is just applied to the terminal portfolio value. In this paper, we consider a portfolio optimisation problem featuring S-shaped utility and a dynamic risk constraint which is imposed throughout the entire trading horizon. Provided that the risk control policy is sufficiently strict relative to the Sharpe ratio of the asset, the trader’s portfolio strategies and the resulting maximal expected utility can be effectively constrained by a dynamic risk measure. Finally, we argue that dynamic risk constraints might still be ineffective if the trader has access to a derivatives market.