Worst-case Conditional Value-at-risk Research Articles

Optimal Investment Strategy for DC Pension Plan with Stochastic Salary and Value at Risk Constraint in Stochastic Volatility Model

This paper studies the optimal asset allocation problem of a defined contribution (DC) pension plan with a stochastic salary and value under a constraint within a stochastic volatility model. It is assumed that the financial market contains a risk-free asset and a risky asset whose price process satisfies the Stein–Stein stochastic volatility model. To comply with regulatory standards and offer a risk management tool, we integrate the dynamic versions of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and worst-case CVaR (wcCVaR) constraints into the DC pension fund management model. The salary is assumed to be stochastic and characterized by geometric Brownian motion. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. By using the Lagrange multiplier method and the dynamic programming method to maximize the constant absolute risk aversion (CARA) utility of terminal wealth, we obtain closed-form expressions of optimal investment strategies with and without a VaR constraint. Several numerical examples are provided to illustrate the impact of a dynamic VaR/CVaR/wcCVaR constraint and other parameters on the optimal strategy.

Worst-case Conditional Value at Risk for asset liability management: A framework for general loss functions

Asset–liability management (ALM) is a challenging task faced by pension funds due to the uncertain nature of future asset returns, employees’ wages, and interest rates. To address this challenge, this paper presents a new mathematical model that uses a Worst-case Conditional Value-at-Risk (WCVaR) constraint to ensure that, with high probability, the funding ratio remains above a regulator-mandated threshold under the worst-case density function that plausibly explains historical sample data. A tractable reformulation of this WCVaR constraint is developed based on the definition of the Worst-case Lower Partial Moment (WLPM) for a general loss function. Additionally, a data-driven moment-based ambiguity set is constructed to capture uncertainty in the moments of the density functions of random variables in the ALM problem. The proposed approach is evaluated using real-world data from the Canada Pension Plan (CPP) and is shown to outperform classical ALM models, based on either CVaR or WCVaR with fixed moments, on out-of-sample data. The proposed framework for handling correlated uncertainty using WCVaR with nonlinear loss functions can be used in other application areas.

Profit- and risk-driven credit scoring under parameter uncertainty: A multiobjective approach

Profit-driven artificial intelligence (AI) systems and profit-based performance measures are widely used in credit scoring. When assessing the performance of an AI system for credit scoring, previous research typically assumes that the cost and benefit parameters and their distributional information are available. In reality, however, these parameters and their distributions are often not precisely known. This study considers parameter uncertainty in the development of credit-scoring models and the estimation of profits and risks generated by those models. We propose a novel profit-based metric—the worst-case expected minimum cost (WEMC)—to estimate the profit of credit-scoring models with uncertain parameters. Furthermore, we introduce the worst-case conditional value-at-risk (WCVaR) metric to measure the loss incurred from employing a classification model in credit scoring under the deterioration of cost parameters. A multiobjective feature-selection framework based on WEMC (or minimum cost) and WCVaR is then presented for model development. Using a comprehensive bankruptcy database, we compare the proposed methods with wrapper methods that use traditional metrics as selection criteria, as well as filter and embedding methods. We conduct extensive experiments to evaluate the economic benefits of the proposed methods under different scenarios that simulate dynamic changes in macroeconomic conditions. The results suggest that the proposed methods outperform other feature-selection methods in the aspects of profit and risk performance metrics in most cases.

Robust enhanced indexation optimization with sparse industry layout constraint

In this paper, we investigate an enhanced indexation methodology using robust Conditional Value-at-Risk (CVaR) and group-sparse optimization. A featured difference from the existing literatures is to describe the tail risk using the worst-case CVaR of excess returns, and the process of industry selection using a weighted ℓ∞,1-norm constraint. We develop an accelerated alternating minimization algorithm for solving this problem. At each iteration, this method usually alternately solves a convex cone program, which admits a closed-form solution via convex duality theory, and a projection problem onto a weighted infinity-to-one-ball, where a fixed-point iteration projection method is developed, terminating in finite number of iterations. The global convergence rates in terms of the primal and dual residuals are also provided. Empirical tests on actual data sets are presented to demonstrate the superior out-of-sample performance of our proposed strategy.

Integrated optimization of assortment, inventory and pricing considering omnichannel retailer’s risk aversion and customer’s time preference

This paper mainly discusses the joint optimization of product assortment, inventory and pricing for an omnichannel retailer under uncertain demand, seeking to maximize the retailer’s Worst-case Conditional Value-at-Risk (WCVaR) based profit and customer’s expected utility. The risk aversion of decision-maker and the time preference of customers are depicted by the WCVaR and quasi-hyperbolic discounting function, respectively. A bi-objective stochastic optimization model is developed. To alleviate the computational burden, a linearization method is applied to convert the problem into a mixed integer linear programming that proves to be solvable within reasonable CPU times using the augmented ε-constraint method. Numerical studies are conducted to investigate the applicability of the proposed model and the efficiency of the solution approach. Our experimental results show that offering customers with high expected utility typically requires low prices and small assortments. We also make other counterintuitive observation that the price of products sometimes increases when the expected utility increases. Furthermore, WCVaR has superiority compared with the risk-neutral model in terms of robustness and stability. The time preference of customers has non-negligible impacts on decision-making results. In addition, the further sensitivity analyses investigate the impacts of various key parameters on the performance of omnichannel retailing.

Distributionally robust joint chance-constrained programming with Wasserstein metric

In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings with a general class of convex uncertain constraints under data-driven Wasserstein ambiguity sets. It is known that robust chance constraints can be conservatively approximated by worst-case conditional value-at-risk (CVaR) constraints. It is shown that the proposed worst-case CVaR approximation model can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. This approximation is essentially exact under certain conditions. We derive a convex relaxation of this approximation model by constructing new decision variables which allows us to eliminate biconvex terms. Specifically, when the constraint function is affine in both the decision variable and the uncertainty, the resulting approximation model is equivalent to a tractable mixed-integer convex reformulation for joint binary DRCCP. Numerical results illustrate the computational effectiveness and superiority of the proposed formulations.

Integrated fire/flight control coupler design for armed helicopters based on improved non-monotone adaptive trust region algorithm with distributionally robust optimization

Aiming at enhancing the extent of military automatization and the effectiveness of armed helicopters in complex battlefield environment, integrated fire/flight control (IFFC) coupler connects the fire control system and the flight control system which is used to generate the optimal flight command signals. However, it is difficult to obtain satisfactory flight command signals within a certain firing accuracy due to the unavoidable uncertain information in battlefield environment. In this paper, a new IFFC scheme is proposed for armed helicopters against complex combat by using the idea of distributionally robust optimization combined with the framework of trust region. Initially, a vector equation and coordinate transformation are employed to acquire the relative motion model of the IFFC coupler problem in complex battlefield environment. Moreover, an improved performance index is designed according to the constraints, and the original non-convex problem is handled by the Taylor formula combined with an improved BFGS (Broyden–Fletcher–Goldfard–Shanno) algorithm. Considering the uncertain parameters during optimization, the non-linear distributionally robust optimization is transformed into a linear matrix inequality optimization problem based on the worst-case conditional value-at-risk (WC-CVaR) approximation. Then, an algorithmic framework with a non-monotone adaptive trust region algorithm is given to obtain the optimal flight command signals. Finally, a simulation example is given to illustrate the effectiveness of the proposed approach.

Distortion risk measure under parametric ambiguity

This study develops closed-form solutions for distortion risk measures (DRM) in extreme cases by utilizing the first two moments and the symmetry of underlying distributions. The resultant extreme-case distributions, encompassing the worst- and best-case distributions, are identified by the envelopes of the distortion functions. The findings of this study extend previous research on worst-case risk measures such as worst-case VaR, worst-case CVaR, worst-case RVaR, and worst-case spectral risk measure, by presenting a unified framework. Furthermore, the compact solutions enhance tractability in optimization problems involving these risk measures, particularly when the true underlying distribution is unknown, and the first two moments are uncertain. The application of the extreme-case DRMs is illustrated with real data sets through numerical examples.

Risk-Aware Linear Quadratic Control Using Conditional Value-at-Risk

Stochastic linear quadratic control problems are considered from the viewpoint of risks. In particular, a worst-case conditional value-at-risk (CVaR) of quadratic objective function is minimized subject to additive disturbances whose first two moments of the distribution are known. The study focuses on three problems of finding the optimal feedback gain that minimizes the quadratic cost of: stationary distribution, one-step, and infinite time horizon. For the stationary distribution problem, it is proved that the optimal control gain that minimizes the worst-case CVaR of the quadratic cost is equivalent to that of the standard (stochastic) linear quadratic regulator. For the one-step problem, an approach to an optimal solution as well as analytical suboptimal solutions are presented. For the infinite time horizon problem, two suboptimal solutions that bound the optimal solution and an approach to an optimal solution for a special case are discussed. The presented theorems are illustrated with numerical examples.

Risk-Based Constraints for the Optimal Operation of an Energy Community

This paper formulates an energy community’s centralized optimal bidding and scheduling problem as a time-series scenario-driven stochastic optimization model, building on real-life measurement data. In the presented model, a surrogate battery storage system with uncertain state-of-charge (SoC) bounds approximates the portfolio’s aggregated flexibility. First, it is emphasized in a stylized analysis that risk-based energy constraints are highly beneficial (compared to chance-constraints) in coordinating distributed assets with unknown costs of constraint violation, as they limit both violation magnitude and probability. The presented research extends state-of-the-art models by implementing a worst-case conditional value at risk (WCVaR) based constraint for the storage SoC bounds. Then, an extensive numerical comparison is conducted to analyze the trade-off between out-of-sample violations and expected objective values, revealing that the proposed WCVaR based constraint shields significantly better against extreme out-of-sample outcomes than the conditional value at risk based equivalent. To bypass the non-trivial task of capturing the underlying time and asset-dependent uncertain processes, real-life measurement data is directly leveraged for both imbalance market uncertainty and load forecast errors. For this purpose, a shape-based clustering method is implemented to capture the input scenarios’ temporal characteristics.

The Worst Case GARCH-Copula CVaR Approach for Portfolio Optimisation: Evidence from Financial Markets

Portfolio optimisation aims to efficiently find optimal proportions of portfolio assets, given certain constraints, and has been well-studied. While portfolio optimisation ascertains asset combinations most suited to investor requirements, numerous real-world problems impact its simplicity, e.g., investor preferences. Trading restrictions are also commonly faced and must be met. However, in adding constraints to Markowitz’s basic mean-variance model, problem complexity increases, causing difficulties for exact optimisation approaches to find large problem solutions inside reasonable timeframes. This paper addresses portfolio optimisation complexities by applying the Worst Case GARCH-Copula Conditional Value at Risk (CVaR) approach. In particular, the GARCH-copula methodology is used to model the portfolio dependence structure, and the Worst Case CVaR (WCVaR) is considered as an alternative risk measure that is able to provide a more accurate evaluation of financial risk compared to traditional approaches. Copulas model the marginal of each asset separately (which may be any distribution) and also the interdependencies between assets This allows an accurate risk to investment assessment to be applied in order to compare it with traditional methods. In this paper, we present two case studies to evaluate the performance of the WCVaR and compare it against the VaR measure. The first case study focuses on the time series of the closing prices of six major market indexes, while the second case study considers a large dataset of share prices of the Gulf Cooperation Council’s (GCC) oil-based companies. Results show that the values of WCVaR are always higher than those of VaR, demonstrating that the WCVaR approach provides a more accurate assessment of financial risk.

Distributionally robust ramp metering under traffic demand uncertainty

This study presents a distributionally robust optimization model to address the ramp metering problem with uncertain traffic demand flows. The aim of this model is to minimize the total travel delay of the system based on the macroscopic cell transmission model (CTM) of traffic flow. In our model, the only required data is the partial distributional information of stochastic demand flows. Using the Worst-Case Conditional Value-at-Risk (WCVaR) constraints to approximate the distributionally robust chance constraints, the proposed problem can be conservatively approximated as a semidefinite programming (SDP), which is computationally efficient. The performances of our proposed model are illustrated by practical applications. Experimental results show that the distributionally robust control strategy can achieve reliable performances over a range of uncertain scenarios.

Risk management for hydrogen networks across refineries

Stricter environmental regulations and policies are leading to increasing requirements for the quality of the oil product. Hydrogen fluctuation in the hydrofining process of refineries remains a considerable problem that may affect the quality of refinery products and increase operating costs. As hydrogen supply across multiple refineries is common in the process industry, the optimal design of interplant hydrogen networks (multi-refinery hydrogen networks) is inevitable in improving the stability of hydrogen supply and reducing the utility consumption of the entire system. This paper introduces the Worst-Case Conditional Value at Risk (WCVaR) concept for the synthesis of the interplant hydrogen network system with hydrogen source flowrate fluctuations. WCVaR is adopted to indicate the expected value of the pure hydrogen content in hydrogen networks in the worst case. Considering the adjustability of hydrogen utility in the realistic production process, this paper studies the stability of interplant hydrogen networks under constant and adjustable hydrogen utilities. The Monte Carlo simulation verifies that the multi-refinery hydrogen network structures obtained by this method are more robust.

Taming Tail Risk: Regularized Multiple β Worst-Case CVaR Portfolio

The importance of proper tail risk management is a crucial component of the investment process and conditional Value at Risk (CVaR) is often used as a tail risk measure. CVaR is the asymmetric risk measure that controls and manages the downside risk of a portfolio while symmetric risk measures such as variance consider both upside and downside risk. In fact, minimum CVaR portfolio is a promising alternative to traditional mean-variance optimization. However, there are three major challenges in the minimum CVaR portfolio. Firstly, when using CVaR as a risk measure, we need to determine the distribution of asset returns, but it is difficult to actually grasp the distribution; therefore, we need to invest in a situation where the distribution is uncertain. Secondly, the minimum CVaR portfolio is formulated with a single β and may output significantly different portfolios depending on the β. Finally, most portfolio allocation strategies do not account for transaction costs incurred by each rebalancing of the portfolio. In order to improve these challenges, we propose a Regularized Multiple β Worst-case CVaR (RM-WCVaR) portfolio. The characteristics of this portfolio are as follows: it makes CVaR robust with worst-case CVaR which is still an asymmetric risk measure, it is stable among multiple β, and against changes in weights over time. We perform experiments on well-known benchmarks to evaluate the proposed portfolio.RM-WCVaR demonstrates superior performance of having both higher risk-adjusted returns and lower maximum drawdown.

Is Being “Robust” Beneficial? A Perspective from the Indian Market

Motivated by the progress made towards incorporating robust optimization in the framework of risk minimization, this work focuses on assessing the practical usefulness of the robust optimization approaches for the minimization of downside risk measures, such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). Accordingly, we perform empirical analysis of the performance of VaR and CVaR models with respect to their robust counterparts, namely, Worst-Case VaR and Worst-Case CVaR, using both simulated data and market data involving the Indian indices of S&P BSE 30 and S&P BSE 100. Additionally, we provide relevant insights regarding the viability of these robust models over their classical formulations from the perspective of an investment practitioner. We conclude by noting the superior performance of Worst-Case VaR and Worst-Case CVaR with respect to their classical versions in the cases involving higher number of stocks and simulated setup respectively.

Robust portfolio rebalancing with cardinality and diversification constraints

In this paper, we develop a robust conditional value at risk (CVaR) optimal portfolio rebalancing model under various financial constraints to construct sparse and diversified rebalancing portfolios. Our model includes transaction costs and double cardinality constraints in order to capture the trade-off between the limit of investment scale and the diversified industry coverage requirement. We first derive a closed-form solution for the robust CVaR portfolio rebalancing model with only transaction costs. This allows us to conduct an industry risk analysis for sparse portfolio rebalancing in the absence of diversification constraints. Then, we attempt to remedy the hidden industry risk by establishing a new robust portfolio rebalancing model with both sparse and diversified constraints. This is followed by the development of a distributed-version of the Alternating Direction Method of Multipliers (ADMM) algorithm, where each subproblem admits a closed-form solution. Finally, we conduct empirical tests to compare our proposed strategy with the standard sparse rebalancing and no-rebalancing strategies. The computational results demonstrate that our rebalancing approach produces sparse and diversified portfolios with better industry coverage. Additionally, to measure out-of-sample performance, two superiority indices are created based on worst-case CVaR and annualized return, respectively. Our ADMM strategy outperforms the sparse rebalancing and no-rebalancing strategies in terms of these indices.

Robust portfolio selection with regime switching and asymmetric dependence

This paper solves the portfolio selection problem with regime switching and asymmetric dependence in financial markets. Investors sustain substantial loss in times of crisis and expect to reduce their losses. Thus, we consider the uncertainty in hidden states of the economy and define worst-case conditional value-at-risk (WCVaR) to capture extreme portfolio loss during financial crisis. Then, we formulate the portfolio selection problem with WCVaR as the measure of risk. We conduct an empirical study using 13 global equity indices. The results show that for dynamic investments, or during financial crisis, our model outperforms other models that only consider a fixed dependence structure between assets. This is because our model can significantly reduce extreme portfolio loss in times of crisis by selecting the assets with small lower tail dependence. This new portfolio strategy can help risk-averse investors cope with financial crisis.

Robust Portfolio Rebalancing with Cardinality and Diversification Constraints

In this paper, we develop a robust conditional value at risk (CVaR) optimal portfolio rebalancing model under various financial constraints to construct sparse and diversified rebalancing portfolios. Our model includes transaction costs and double cardinality constraints in order to capture the trade-off between the limit of investment scale and the diversified industry coverage requirement. We first derive a closed-form solution for the robust CVaR portfolio rebalancing model with only transaction costs. It allows us to conduct industry risk analysis for sparse portfolio rebalancing in the absence of diversification constraints. Then, we attempt to remedy the hidden industry risk by establishing a new robust portfolio rebalancing model with both sparse and diversified constraints. This is followed by the development of a distributed-version of the Alternating Direction Method of Multipliers (ADMM) algorithm, where each subproblem admits a closed-form solution. Finally, we conduct empirical tests to compare our proposed strategy with the standard sparse rebalancing and no-rebalancing strategies. The computational results demonstrate that our rebalancing approach produces sparse and diversified portfolios with higher industry coverage. Additionally, to measure out-of-sample performance, two superiority indices are created based on the worst-case CVaR and annualized return, respectively. Our ADMM strategy also outperforms the sparse rebalancing and no-rebalancing strategies in terms of the two indices.

Multistage Model Predictive Control Co-Risk Dispatch for Coupled Electricity and Heat System

The energy transition can facilitate the rapid development of the coupled electricity and heat system (CEHS), accommodating the higher share of renewable energy. However, uncertain renewable energy sources (RESs) and overlimit risk can bring significant challenges for the safe operation of the CEHS. Considering uncertain RESs integration and overlimit risk within the upper and lower bounds jointly with high probability, the multistage model predictive control (MPC) co-risk dispatch model for the CEHS can be proposed in this paper. The model can hedge against the operational risk for the multistage rolling dispatch. In the context of the multistage real-time risk dispatch, the MPC framework is presented, which contains three processes, autoregressive moving average (ARMA) model prediction process, risk-based optimization process, and penalty-based feedback correction process. In the prediction process, the ARMA method can be employed to accurately predict the RESs power, which is characterized as the merit of the short-term forecast. In the optimization process, the violation risk of overlimit chance constraints caused by the output power fluctuation can be mitigated using the distributionally robust chance-constrained (DRCC) under the Wasserstein ambiguity set. The tractable approximation reformulation can be obtained by adopting the Worst-Case Conditional Value-at-Risk (WC-CVaR) approximation method. In the feedback correction process, penalty-based load shedding and RESs spillage can lower the risk level by minimizing the risk penalty cost. In the numerical case, the risk performance analysis based on the modified Barry Island system is implemented. The analysis demonstrates that the daily average violation probability $V_{p}$ can be mainly affected by the Wasserstein radius $\rho $ while the total cost $C_{obj}$ can be affected by the violation probability $\varepsilon $ . The dispatch energy for the CEHS is obtained using the proposed approach, which outperforms the existing robust optimization (RO) approach.

Robust risk‐averse unit commitment with solar PV systems

A challenge for energy market operators (EMOs) is to render a cost-effective generation schedule, known as unit commitment (UC), for the power system with intermittent renewable power generations. Consolidating the risk management concept and robustness into the UC not only helps EMO to maintain the secure operation of power systems but also informs the EMO about the tail-risk cost. Based on this motivation, the authors have proposed a model for UC with worst-case conditional value-at-risk (UC-WCVaR) with intermittent solar generations and loads. Unlike intractable risk-averse stochastic UC models, this model minimises the worst-case CVaR of the dispatch cost over an ambiguity set of the probability distributions. This robust tractable model relies on the historical data that can partially characterise the underlying probability distributions of solar photovoltaic (PV) systems and loads. They solved this model by its equivalent tractable robust counterpart using linear decision rules. The tests conducted on the IEEE 118-bus test system demonstrate that: (i) UC-WCVaR solution matches the results derived from normally distributed solar samples; (ii) the UC-WCVaR is less conservative than the classic robust unit commitment. Also, the computation time is directly proportional to the number of uncertain resources and inversely proportional to the number of stages.