Conditional Value At Risk Research Articles

Evaluating Extremes in Climatology with Average Value-at-Risk Measure: Example of Precipitation Data from Upper Vistula Region in Poland

Abstract The ability to predict changes in the right tail of daily precipitation distributions over short time periods, as well as the probability of clustering extreme values, is critical for current risk management. In the present study, we apply the well-established metric average value-at-risk (AVaR) for the first time within the field of climatology. We also investigate the evolution of the return level (RL) and the extremal index (EI), which we refer to as risk measures. In the case of precipitation processes, a rise in the first two and a reduction in the third may result in increased hazards. These methods were applied to the new data on the daily sum of precipitation from the region of Upper Vistula in Poland from 1951 to 2020 to analyze the dynamics of changes across time. We found that AVaR and RL have the smallest values for the middle of the analyzed period (years 1971–2010), while EI has a maximal value around 1963–92. Both the beginning and the end of the investigated period are characterized by more extreme and clustered precipitation events. Furthermore, this paper includes some recommendations for the tools used to compute the measures.

Maximizing profitability in integrated energy markets: A novel framework for multi-faceted energy trading

The emergence of Integrated Energy Systems (IESs) has shifted energy management from traditional electricity paradigms to a more diverse approach involving electricity, thermal energy, and natural gas. This study introduces a framework for an Integrated Energy Network (IEN) that enables simultaneous trading across these interconnected markets. The objective is to maximize the profitability of integrated energy companies by optimizing energy trading strategies. The proposed business strategy incorporates integrated load response, addressing constraints within electric distribution systems, heating networks, and natural gas pipelines. A three-stage process is used: information sharing in the unified energy market, energy conversion in hubs, and flexible load planning in the retail market. Flexible loads are categorized as displaceable, reducible, or transformable. A Conditional Value-at-Risk (CVaR) mean-variance model addresses market price volatility. Simulations on a test system validate the model, demonstrating its effectiveness in improving profitability and operational efficiency for integrated energy companies.

Enhancing Portfolio Decarbonization Through SensitivityVaR and Distorted Stochastic Dominance

Recent trends in portfolio management emphasize the importance of reducing carbon footprints and aligning investments with sustainable practices. This paper introduces SensitivityVaR (SensitivityVaR), an advanced distortion risk measure that combines Value-at-Risk (VaR) and Expected Shortfall (ES) with the Cornish–Fisher expansion. SensitivityVaR provides a more robust framework for managing risk, particularly under extreme market conditions. By incorporating first- and second-order distorted stochastic dominance criteria, we enhance portfolio decarbonization strategies, aligning financial objectives with environmental targets such as the Paris Agreement’s goal of a 7% annual reduction in carbon intensity from 2019 to 2050. Our empirical analysis evaluates the impact of integrating carbon intensity data—including Scope 1, Scope 2, and Scope 3 emissions—on portfolio optimization, focusing on key sectors like technology, energy, and consumer goods. The results demonstrate the effectiveness of SensitivityVaR in managing both risk and environmental impact. The methodology led to significant reductions in carbon intensity across different portfolio configurations, while preserving competitive risk-adjusted returns. By optimizing tail risks and limiting exposure to carbon-intensive assets, this approach produced more balanced and efficient portfolios that aligned with both financial and sustainability goals. These findings offer valuable insights for institutional investors and asset managers aiming to integrate climate considerations into their investment strategies without compromising financial performance.

A new Bayesian method for estimation of value at risk and conditional value at risk

AbstractValue at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, Extreme Value Theory initially seemed to be one of the best tools for this kind of problems, because using peaks-over-threshold method, we can assume the tail data approximately follow a Generalized Pareto distribution (GPD). The main objection to its use is that it only employs observations over the threshold, which are usually scarce. With the aim of improving the inference process, we propose a new Bayesian method that computes estimates built with all the information available. Informative prior Bayesian (IPB) method employs the existing relations between the parameters of the loss distribution and the parameters of the GPD that models the tail data to define informative priors in order to perform Metropolis–Hastings algorithm. We show how to apply IPB when the distribution of the observations is Exponential, stable or Gamma, to make inference and predictions. .Afterwards, we perform a thorough simulation study to compare the accuracy and precision of the estimates computed by IPB and the most employed methods to estimate VaR and CVaR. Results show that IPB provides the most accurate, precise and least biased estimates, especially when there are very few tail data. Finally, data from two real examples are analysed to show the practical application of the method.

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

AbstractWe investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.

Bayesian Optimization Using Simulation-Based Multiple Information Sources over Combinatorial Structures

Bayesian optimization due to its flexibility and sample efficiency has become a standard approach for simulation optimization. To reduce this problem, one can resort to cheaper surrogates of the objective function. Examples are ubiquitous, from protein engineering or material science to tuning machine learning algorithms, where one could use a subset of the full training set or even a smaller related dataset. Cheap information sources in the optimization scheme have been studied in the literature as the multi-fidelity optimization problem. Of course, cheaper sources may hold some promise toward tractability, but cheaper models offer an incomplete model inducing unknown bias and epistemic uncertainty. In this manuscript, we are concerned with the discrete case, where fx,wi is the value of the performance measure associated with the environmental condition wi and p(wi) represents the relevance of the condition wi (i.e., the probability of occurrence or the fraction of time this condition occurs). The main contribution of this paper is the proposal of a Gaussian-based framework, called augmented Gaussian process (AGP), based on sparsification, originally proposed for continuous functions and its generalization in this paper to stochastic optimization using different risk profiles for combinatorial optimization. The AGP leverages sample and cost-efficient Bayesian optimization (BO) of multiple information sources and supports a new acquisition function to select the new source–location pair considering the cost of the source and the (location-dependent) model discrepancy. An extensive set of computational results supports risk-aware optimization based on CVaR (conditional value-at-risk). Computational experiments confirm the actual performance of the MISO-AGP method and the hyperparameter optimization on benchmark functions and real-world problems.

On the Gradient Method in One Portfolio Management Problem

This study refines the methodology for solving stochastic optimal control problems with quality criteria that include the sum of the quality functional of the classical formulation and an extremal measure. A two-level optimization solution of these kinds of problems is presented already for the case where the quality functional consists only of the extremal measure. Our study shows the possibility of solving the original time inconsistency problem through solving a two-level optimization problem, where the outer problem is solved by gradient methods since the value function is convex and the inner problem is solved by classical methods. Some experiments were carried out and confirmed the validity of the theory. The results of the study can be applied to the case of portfolio management with quality criteria containing the Conditional Value-at-Risk (CVaR) metric.

Forecasting Expected Shortfall and Value‐at‐Risk With Cross‐Sectional Aggregation

ABSTRACTThe combination of the conditional autoregressive value‐at‐risk (CAViaR) process with the Fissler–Ziegel (FZ) loss function generates a recently emerging framework (CAViaR‐FZ) for forecasting value‐at‐risk (VaR) and expected shortfall (ES). However, existing CAViaR‐FZ models typically overlook the presence of long‐range dependence, a stylized fact of financial time series. This paper proposes a long‐memory CAViaR‐FZ model using the cross‐sectional aggregation (CSA) method. The CSA method is well‐recognized for its ability to generate a long‐memory process by aggregating an infinite number of short‐memory processes cross‐sectionally. The proposed CSA‐CAViaR‐FZ model flexibly captures long‐memory dynamics in both VaR and ES and includes the original short‐memory CAViaR‐FZ model as a special case. Simulation and empirical results demonstrate that the proposed model outperforms various competing models.

Day-ahead multi-criteria energy management of a smart home under different electrical rationing scenarios

With the recent global energy crisis, some countries have implemented electrical rationing (ER), making it necessary for smart homes to play a pivotal role in optimizing energy consumption and contributing to sustainable practices. To effectively manage smart home consumption, a stochastic programming approach for a grid-connected smart home energy management system (SHEMS) is proposed in this paper. The system includes PV, battery, diesel, and gas-based heating/cooling systems (HCS). Additionally, a demand response program (DRP) has been employed under time-of-use tariffs in the Syrian energy market. The main objective is to minimize the day-ahead expected cost and consumer discomfort by optimizing the operation of dispatchable units and loads. To manage the risks associated with the expected cost due to potential uncertainties in PV energy generation and electrical rationing programs, the conditional value-at-risk (CVaR) approach is adopted. Two methods are proposed to model the uncertainty in PV energy generation; interval bands and interval-based scenarios. The problem is modeled as a mixed-integer non-linear programming (MINLP) model, and coded in GAMS to test different cases. Based on the results obtained, substantial reductions reached 56.2% in worst-case cost scenarios when employing concurrent DRP-risk management.

Impact of Sharia‐Based Screening on Equity Risk: International Evidence

ABSTRACTThis paper investigates the impact of mainstream Sharia‐based screening methodologies on equity risk. The study is based on the daily returns of 23 pairs of stock indices belonging to different index providers, namely Dow Jones, FTSE, and S&P. For risk measurement, the study uses five Value at Risk (VaR) methods. Moreover, the conditional VaR (CVaR) is used to analyze the tail behavior of return distributions. The results show that Sharia‐based screening is a source of significant differences in risk. There is strong evidence that Islamic stock indices (ISIs) tend to be less risky than their conventional peers. Also, the screening process allows for keeping extreme negative returns at a lower level relative to unscreened indices. The analysis during the Dotcom and Subprime crises shows that, globally, the results are time‐invariant. In sum, Islamic screening is tantamount to prudential rules that lead to low‐risk portfolios; it is, in its essence, a risk‐reduction strategy. Findings suggest that Sharia‐compliant stocks would be of great attractiveness to risk‐averse investors. They will find their claim in this asset class, as its riskiness is relatively low. For banks, a lower VaR of Islamic investing implies fewer capital requirements for market risk when holding portfolios that include Sharia‐compliant stocks. Fewer capital requirements will enhance the banks' ability to invest and make profits, which will, in turn, result in reducing the cost of capital.

Optimal reinsurance with multivariate risks and dependence uncertainty

In this paper, we study the optimal reinsurance design from the perspective of an insurer with multiple lines of business, where the reinsurance is purchased by the insurer for each line of business, respectively. For the risk vector generated by the multiple lines of business, we suppose that the marginal distributions are fixed, but the dependence structure between these risks is unknown. Due to the unknown dependence structure, the optimal strategy is investigated for the worst-case scenario. We consider two types of risk measures: Value-at-Risk (VaR) and Range-Value-at-Risk (RVaR) including Expected Shortfall (ES) as a special case, and general premium principles satisfying certain conditions. To be more practical, the minimization of the total risk is conducted under some budget constraint. For the VaR-based model with only two risks, it turns out that the limited stop-loss reinsurance treaty is optimal for each line of business. For the model with more than two risks, we obtain two types of optimal reinsurance strategies if the marginals have convex or concave distributions on their tail parts by constraining the ceded loss functions to be convex or concave. Moreover, as a special case, the optimal quota-share reinsurance with dependence uncertainty has been studied. Finally, after applying our findings to two risks, some studies have been implemented to obtain both the analytical and numerical optimal reinsurance policies.

Risk spillovers and diversification benefits between crude oil and agricultural commodity futures markets

This study examines the dependence structure and risk spillovers between crude oil and eight major agricultural futures (wheat, corn, soybean coffee, cotton, lumber, cocoa, and live cattle) markets. It also analyzes the potential conditional diversification benefits using a variety of copula functions and Conditional Value at Risk (CoVaR) measure. The results show significant crisis-sensitive and temporal dependence between oil and agricultural markets. Moreover, crude oil shows a symmetric tail dependence with both wheat, corn, soybeans, and cotton futures, whereas oil exhibits an average dependence with coffee. A strong dependence is observed between oil and cocoa (lumber) during bearish (bullish) market conditions. Oil and Live cattle have a symmetric dependence during bearish and bullish market conditions. On the other hand, we find asymmetric and bidirectional risk spillovers from oil to agricultural markets. Furthermore, the wheat futures contract appears to be the most dominating and vulnerable asset to oil price shocks, followed by lumber and corn futures, respectively, while the live cattle contracts are the least. Finally, an equally weighted portfolio offers the highest diversification benefits at a 5 % expected shortfall.

Integration of Conditional Value-at-Risk (CVaR) in Multi-Objective Optimization

Integration of Conditional Value-at-Risk (CVaR) in Multi-Objective Optimization

Multiple energy planning in the energy hub considering renewable sources, electric vehicles and management in the daily electricity market with wind multi-objective optimization algorithm

This article presents a decision-making framework leveraging randomness to manage short-term intelligent energy hub (EH) within regenerated power systems integrating EH input electricity and natural gas. EH outputs, addressing electricity and heat needs, are managed via battery storage systems (BSS) and electric vehicle (EV) fleets in regulation markets. The energy hub operator optimizes decisions concerning natural gas and energy, minimizing costs in electricity supply and thermal energy carriers across day-ahead (DA) markets and regulations. Emphasizing the benefits of demand-side management planning in cost reduction, the model incorporates a conditional value at risk (CVaR) method to assess and manage uncertainties. Moreover, leveraging electric vehicle battery storage capacity is explored to mitigate wind and solar energy production uncertainties. An energy optimization strategy based on a Multi-Objective Wind-Driven Optimization (MOWDO) is proposed, demonstrating the effectiveness of the model and method across various system conditions and scenarios. Results indicate that increasing the number of EVs from 2000 to 10,000 reduces the expected cost by 1.5 % and the CVaR by 1.5 %. Additionally, optimizing the BSS discharge cost coefficient from $45/MWh to $15/MWh decreases the expected cost by 0.5 % and the CVaR by 0.3 %. The system achieves a 20 % reduction in peak-hour electricity purchases by leveraging the BSS and EV fleet during high price conditions. These findings underscore the positive impact of demand-side management programs in enhancing renewable resource utilization amidst uncertainty.

Asymptotic properties of conditional value-at-risk estimate for asymptotic negatively associated samples

This article examines the strong consistency of the conditional value-at-risk (CVaR) estimate for asymptotic negatively associated (ANA or ρ−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho ^{-}$\\end{document}, for short) random samples under mild conditions. It is demonstrated that the optimal rate can achieve nearly O(n−1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$O (n^{-1/2})$\\end{document} under certain appropriate conditions. Furthermore, we present numerical simulations and a real data example to corroborate our theoretical results based on finite samples.

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.

Geopolitical Risk and Extreme Risk Connectedness Among Energy and Other Strategic Commodities: Fresh Sight Using the High‐Dimensional CoVaR Model

ABSTRACTExisting studies on commodity market risk spillovers recognize the pivotal role of geopolitical risk (GPR), but scarcely address how it drives tail risk spillover networks. This study adopts the Tail‐Event driven NETwork methodology to explore high‐dimensional Conditional Value at Risk (CoVaR) spillovers within energy and other strategic commodity markets. Our findings indicate that (1) In both lower and upper tail networks, metal and food commodities primarily act as net risk transmitters, whereas energy commodities are mainly net risk receivers. Additionally, these roles undergo short‐term reversals during periods of heightened market uncertainty. (2) There exists an asymmetrical pattern of CoVaR co‐movements in these commodity markets. The total connectedness (TC) in both the upper and lower tails demonstrates distinct responses to various extreme events. GPR tends to weaken the lower tail TC and strengthen the upper tail. (3) Incorporating GPR substantially improves the effectiveness of Minimum Connectedness Portfolio (MCoP) for these strategic commodities.

A novel robust method for estimating the covariance matrix of financial returns with applications to risk management

This study introduces the dynamic Gerber model (DGC) and evaluates its performance in the prediction of Value at Risk (VaR) and Expected Shortfall (ES) compared to alternative parametric, non-parametric and semi-parametric methods for estimating the covariance matrix of returns. Based on ES backtests, the DGC method produces, overall, accurate ES forecasts. Furthermore, we use the Model Confidence Set procedure to identify the superior set of models (SSM). For all the portfolios and VaR/ES confidence levels we consider, the DGC is found to belong to the SSM.

Safety assessment of cryptocurrencies as risky assets during the COVID-19 pandemic

This study examines the safe-haven properties of cryptocurrencies during the COVID-19 pandemic by exploring various cryptocurrency allocation strategies and their impact on downside risk compared to holding only the stock market index. Spanning from 2019 to 2021, the analysis includes seven cryptocurrencies—Bitcoin, Ethereum, Litecoin, Dash, Maker, Monero, and NEO—and six stock market indexes—SP500, FTSE, CAC40, IBEX, AEX, and DJIM. The research introduces a portfolio optimization approach focused on mitigating significant losses during crises, employing Value at Risk (VaR), Modified Value at Risk (MVaR), and Conditional Value at Risk (CVaR) metrics across different portfolio compositions. Findings highlight several key insights: higher cryptocurrency allocations correlate with increased downside risk, with none of the studied cryptocurrencies acting as a safe haven for international equity indexes during the COVID-19 pandemic. Relative MVaR exceeds CVaR across most combinations, and Maker demonstrates the highest downside risk, while Bitcoin and Litecoin exhibit the lowest risk levels based on CVaR and MVaR, respectively. Integration of the FTSE index is associated with peak risk levels, whereas incorporating DJIM and AEX shows reduced risk exposure. The study contributes novel insights into cryptocurrency risk dynamics during the COVID-19 pandemic, offering valuable implications for risk management strategies in investment portfolios.

Joint value-at-risk and expected shortfall regression for location-scale time series models

. This article studies the joint value-at-risk (VaR) and expected shortfall (ES) regression for a wide class of location-scale time series models including autoregressive and moving average models with generalized autoregressive conditional heteroscedasticity errors. In contrast to the quasi-maximum likelihood estimation, we estimate the model parameters with the aim of more accurate VaR and ES estimation. Then, we show consistency and asymptotic normality for parameter estimators under weak regularity conditions. Finally, a simulation study and a real data analysis are shown to illustrate our results.