Conditional Value-at-risk Minimization Research Articles

Supply option purchasing decisions via mismatch cost minimization

In this paper, the optimal purchasing decisions of the option procurement of newsvendor-type commodities are studied. Based on the mismatch cost minimization criterion, the present study analyzes the option purchasing decisions of a risk-averse newsvendor under the measure of Conditional Value-at-Risk (CVaR). For the purpose of controlling the procurement risk, it is found that a newsvendor’s optimal option purchasing decision under the CVaR measure increases at the quantile level under certain conditions. The result provides alternative explanations for a newsvendor’s over-purchasing behaviors in reality. To mitigate the conservativeness of the CVaR measure, a combined objective defined as the mean-CVaR on mismatch cost is introduced, and the optimal option purchasing decision of a newsvendor minimizing the combined objective is derived. The combined objective can strike a balance between the minimization of CVaR of mismatch cost as well as the minimization of expected mismatch cost. In addition, a sensitivity analysis is performed to exhibit the results and managerial insights are presented to hedge against the potential risks for newsvendors in the option purchasing decision-making.

Bayesian portfolio selection using VaR and CVaR

We study the optimal portfolio allocation problem from a Bayesian perspective using value at risk (VaR) and conditional value at risk (CVaR) as risk measures. By applying the posterior predictive distribution for the future portfolio return, we derive relevant quantities needed in the computations of VaR and CVaR, and express the optimal portfolio weights in terms of observed data only. This is in contrast to the conventional method where the optimal solution is based on unobserved quantities which are estimated. We also obtain the expressions for the weights of the global minimum VaR (GMVaR) and global minimum CVaR (GMCVaR) portfolios, and specify conditions for their existence. It is shown that these portfolios may not exist if the level used for the VaR or CVaR computation are too low. By using simulation and real market data, we compare the new Bayesian approach to the conventional plug-in method by studying the accuracy of the GMVaR portfolio and by analysing the estimated efficient frontiers. It is concluded that the Bayesian approach outperforms the conventional one, in particular at predicting the out-of-sample VaR.

Shortest path network problems with stochastic arc weights

This paper presents an approach to shortest path minimization for graphs with random weights of arcs. To deal with uncertainty we use the following risk measures: Probability of Exceedance (POE), Buffered Probability of Exceedance (bPOE), Value-at-Risk (VaR), and Conditional Value-at-Risk (CVaR). Minimization problems with POE and VaR objectives result in mixed integer linear problems (MILP) with two types of binary variables. The first type models path, and the second type calculates POE and VaR functions. Formulations with bPOE and CVaR objectives have only the first type binary variables. The bPOE and CVaR minimization problems have a smaller number of binary variables and therefore can be solved faster than problems with POE or VaR objectives. The paper suggested a heuristic algorithm for minimizing bPOE by solving several CVaR minimization problems. Case study (posted at web) numerically compares optimization times with considered risk functions.

Optimal position of supply chain delivery window with risk-averse suppliers: A CVaR optimization approach

The study introduces the concept of risk-averse optimal position of the delivery window (RA-OPDW) into a cost-based delivery performance model. The penalty costs for early and late deliveries have attracted much attention in the background of supply delivery performance. However, minimizing the expected penalty has a critical disadvantage that it ignores the risk-averse attitude of decision-makers. Thereby, a Conditional Value-at-Risk (CVaR) measure of the penalty on untimely delivery is proposed to incorporate the risk-aversion degree of a decision-maker. Then the closed-form expression of optimal position minimizing the CVaR is derived under the condition that the risk-aversion degree is high. The influence of the risk-aversion degree, the delivery window width, the ratio of the unit penalty on late delivery to the unit penalty on early delivery are explored. Besides, a hybrid model that trades off the minimization of CVaR and the expected penalty is investigated. The conditions for the optimal position in the hybrid model is also deduced. Moreover, the minimization of CVaR and the expected penalty are just two special cases of the hybrid model. Finally, a numerical case is executed to compare the optimal position with minimum CVaR and the one minimizing the expected penalty and to illustrate the influence of several parameters on the optimal position.

A risk-averse and buyer-led supply chain under option contract: CVaR minimization and channel coordination

Option contracts are used commonly in business to tackle difficulties in working capital shortage and hedge market risks. In this paper, we consider option contract application in a buyer-led supply chain, where both the buyer and supplier are risk-averse. The effects of option price and option exercise price are investigated via conditional value-at-risk (CVaR) minimization. A Stackelberg game model is established to examine the influences of adjusting both prices on the benefits and risks at the buyer and supplier sides. We find that the increase of both prices, especially the increase of option price, benefits the supplier but causes loss to the buyer. The supply chain's total risk is not affected by either price when the buyer and supplier have the same risk preference. However, when the supplier raises the option price, the buyer who is more risk-averse will bear extra risk more than the supplier's reduced risk. A numerical study shows that the option exercise price has an opposite effect on the supply chain. We theoretically prove that when the supplier's risk preference is the same as or larger than that of the buyer, the supply chain can be coordinated under option contract; otherwise, the supply chain cannot be coordinated.

A Multi-Objective Risk-Based Game Theoretic Approach to Reservoir Operation Policy in Potential Future Drought Condition

In this paper, by using the concept of Conditional Value at Risk (CVaR), a Leader-Follower game (LFG) based multi-objective optimization model is developed to determine the optimum 12-month operation policy of a reservoir in potential future dry periods. The minimization of CVaRs of storage loss and agricultural and environmental deficits along with maximization of planned allocation to agricultural sector are considered as leader’s objectives, while the followers try to maximize their share of water rights using Nash bargaining (NB) method. This framework is then used to model the operation policy of Dorudzan basin in Fars province, southwestern Iran. Water demand and daily climate data in the period of 2003 to 2015 for this basin, as well as future projections from fifteen IPCC-AR4 global circulation models (GCMs) for 2018–2030 under A2, B1 and A1B emission scenarios are considered to evaluate future dam operation policies. Future projections are downscaled using the LARS-WG model, which then feeds the HMETS watershed model to simulate the corresponding reservoir inflow time-series. Thereafter, three-hundred 12-month rainfall, evaporation and inflow time series with least inflow volume are used as input for the optimization model, which is solved using NSGA-II and GA algorithms. The results show while the model can determine the operation policy that keeps the associated risks in the acceptable range, it can satisfy the followers demands with respect to the available resources. The results also show that the agricultural sector of the study area can be hugely affected by potential future droughts.

Risk-Aware Proactive Scheduling via Conditional Value-at-Risk

In this paper, we consider the challenging problem of riskaware proactive scheduling with the objective of minimizing robust makespan. State-of-the-art approaches based on probabilistic constrained optimization lead to Mixed Integer Linear Programs that must be heuristically approximated. We optimize the robust makespan via a coherent risk measure, Conditional Value-at-Risk (CVaR). Since traditional CVaR optimization approaches assuming linear spaces does not suit our problem, we propose a general branch-and-bound framework for combinatorial CVaR minimization. We then design an approximate complete algorithm, and employ resource reasoning to enable constraint propagation for multiple samples. Empirical results show that our algorithm outperforms state-of-the-art approaches with higher solution quality.

An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution

This paper proposes a unified framework to solve distributionally robust mean-risk optimization problem that simultaneously uses variance, value-at-risk (VaR) and conditional value-at-risk (CVaR) as a triple-risk measure. It provides investors with more flexibility to find portfolios in the sense that it allows investors to optimize a return-risk profile in the presence of estimation error. We derive a closed-form expression for the optimal portfolio strategy to the robust mean-multiple risk portfolio selection model under distribution and mean return ambiguity (RMP). Specially, the robust mean-variance, robust maximum return, robust minimum VaR and robust minimum CVaR efficient portfolios are all special instances of RMP portfolios. We analytically and numerically show that the resulting portfolio weight converges to the minimum variance portfolio when the level of ambiguity aversion is in a high value. Using numerical experiment with simulated data, we demonstrate that our robust portfolios under ambiguity are more stable over time than the non-robust portfolios.

Application of CVaR risk aversion approach in the dynamical scheduling optimization model for virtual power plant connected with wind-photovoltaic-energy storage system with uncertainties and demand response

Conditional value at risk (CVaR) and confidence degree theory are introduced to build scheduling model for VPP connecting with wind power plant (WPP), photovoltaic generators (PV), convention gas turbine (CGT), energy storage systems (ESSs) and incentive-based demand response (IBDR). Latin hypercube sampling method and Kantorovich distance are introduced to construct uncertainties analysis method. A risk aversion scheduling model is proposed with minimum CVaR objective considering maximum operation revenue. The IEEE30 bus system is used as simulation system. Results show: (1) Price-based demand response could realize peak load shifting, ESSs and IBDR could increase operation revenue. (2) Threshold α reflects risk attitude of decision maker, which has strong risk tolerant to gain the excess income with low α. (3) In peak period, decision maker would reduce WPP and PV for avoiding power shortage loss. Otherwise, WPP and PV would be called in priority since system reserve capacity is sufficient. (4) When 0.85≤β < 0.95, the decreasing slope of CVaR value is big, decision maker is sensitive on risk. When β≥0.95, VPP scheduling scheme reach the most conservative, net revenue and CVaR value are ¥8995.34 and ¥18834. Therefore, the proposed model could describe VPP risk and provide decision support tool for decision maker.

A framework for multi-stakeholder decision-making and conflict resolution

We propose a decision-making framework to compute compromise solutions that balance conflicting priorities of multiple stakeholders on multiple objectives. In our setting, we shape the stakeholder dissatisfaction distribution by solving a conditional-value-at-risk (CVaR) minimization problem. The CVaR problem is parameterized by a probability level that shapes the tail of the dissatisfaction distribution. The proposed approach allows us to compute a family of compromise solutions and generalizes multi-stakeholder settings previously proposed in the literature that minimize average and worst-case dissatisfactions. We use the concept of the CVaR norm to give a geometric interpretation to this problem and use the properties of this norm to prove that the CVaR minimization problem yields Pareto optimal solutions for any choice of the probability level. We discuss a broad range of potential applications of the framework that involve complex decision-making processes. We demonstrate the developments using a biowaste facility location case study in which we seek to balance stakeholder priorities on transportation, safety, water quality, and capital costs.

Minimizing CVaR in global dynamic hedging with transaction costs

This study develops a global derivatives hedging methodology which takes into account the presence of transaction costs. It extends the Hodges and Neuberger [Rev. Futures Markets, 1989, 8, 222–239] framework in two ways. First, to reduce the occurrence of extreme losses, the expected utility is replaced by the conditional Value-at-Risk (CVaR) coherent risk measure as the objective function. Second, the normality assumption for the underlying asset returns is relaxed: general distributions are considered to improve the realism of the model and to be consistent with fat tails observed empirically. Dynamic programming is used to solve the hedging problem. The CVaR minimization objective is shown to be part of a time-consistent framework. Simulations with parameters estimated from the S&P 500 financial time series show the superiority of the proposed hedging method over multiple benchmarks from the literature in terms of tail risk reduction.

Dynamic conditional value-at-risk model for routing and scheduling of hazardous material transportation networks

This paper illustrates a dynamic model of conditional value-at-risk (CVaR) measure for risk assessment and mitigation of hazardous material transportation in supply chain networks. The well-established market risk measure, CVaR, which is commonly used by financial institutions for portfolio optimizations, is investigated. In contrast to previous works, we consider CVaR as the main objective in the optimization of hazardous material (hazmat) transportation network. In addition to CVaR minimization and route planning of a supply chain network, the time scheduling of hazmat shipments is imposed and considered in the present study. Pertaining to the general dynamic risk model, we analyzed several scenarios involving a variety of hazmats and time schedules with respect to optimal route selection and CVaR minimization. A solution algorithm is then proposed for solving the model, with verifications made using numerical examples and sensitivity analysis.

Cutting plane algorithms for mean-CVaR portfolio optimization with nonconvex transaction costs

This paper studies the mean-risk portfolio optimization problem with nonconvex transaction costs. We employ the conditional value-at-risk (CVaR) as a risk measure. There are a number of studies that aim at efficiently solving large-scale CVaR minimization problems. None of these studies, however, take into account nonconvex transaction costs, which are present in practical situations. To make a piecewise linear approximation of the transaction cost function, we utilized special ordered set type two constraints. Moreover, we devised a subgradient-based cutting plane algorithm to handle a large number of scenarios. This cutting plane algorithm needs to solve a mixed integer linear programming problem in each iteration, and this requires a substantial computation time. Thus, we also devised a two-phase cutting plane algorithm that is even more efficient. Numerical experiments demonstrated that our algorithms can attain near-optimal solutions to large-scale problems in a reasonable amount of time. Especially when rebalancing a current portfolio that is close to an optimal one, our algorithms considerably outperform other solution methods.

Robust portfolio techniques for mitigating the fragility of CVaR minimization and generalization to coherent risk measures

The conditional value-at-risk (CVaR) has gained growing popularity in financial risk management due to the coherence property and tractability in its optimization. However, optimal solutions to the CVaR minimization are highly susceptible to estimation error of the risk measure because the estimate depends only on a small portion of sampled scenarios. The same is equally true of the other coherent measures. In this paper, by employing robust optimization modelling for minimizing coherent risk measures, we present a simple and practical way for making the solution robust over a certain range of estimation errors. More specifically, we show that a worst-case coherent risk minimization leads to a penalized minimization of the empirical risk estimate. The worst-case risk measure developed in this paper is different from the distributionally worst-case CVaR in Zhu and Fukushima's work of 2009, but these two worst-case risk measures can be simultaneously minimized. Additionally, inspired by Konno, Waki and Yuuki's work of 2002, we examine the use of factor models in coherent risk minimization. We see that, in general, factor model-based coherent risk minimization along the lines of that pursued by Goldfarb and Iyengar in 2003 becomes computationally intractable. Therefore, we apply a simplified version to the factor model-based CVaR minimization, and see that it improves on the performance, achieving better CVaR, turnover, standard deviation and Sharpe ratio than the empirical CVaR minimization and market benchmarks.

Asset allocation with conditional value-at-risk budgets

Risk budgets are frequently used to allocate the risk of a portfolio by decomposing the total portfolio risk into the risk contribution of each component position. Many approaches to portfolio allocation use ex post methods for constructing risk budgets and take the variance as a risk measure. In this paper, however, we use ex ante methods to evaluate the component contribution to conditional value-at-risk (CVaR) and allocate risk. The proposed minimum CVaR concentration portfolio draws a balance between the investor's return objectives and the diversification of risk across the portfolio. For a portfolio invested in bonds, commodities, equities and real estate, we find that over the period from January 1984 to June 2010, the minimum CVaR concentration portfolio offers an attractive compromise between the good risk-adjusted return properties of the minimum CVaR portfolio and the positive return potential and low portfolio turnover of an equally weighted portfolio.

Minimum VaR and minimum CVaR optimal portfolios: Estimators, confidence regions, and tests

Abstract In this paper, we consider the sample estimators for the expected return, the variance, the value-at-risk (VaR), and the conditional VaR (CVaR) of the minimum VaR and the minimum CVaR portfolio. Their exact distributions are derived. These expressions are used for studying the distributional properties of the estimated characteristics. We prove that the expectation does not exist for the estimated variance, while the second moment does not exist for the estimated expected return. Moreover, expressions for the joint densities and the corresponding dependence measures between the estimators for the expected return and the variance as well as between the estimated expected return and the estimated VaR (CVaR) are derived. Finally, we present a confidence region for the minimum VaR portfolio and the minimum CVaR portfolio in the mean-variance space as well as in the mean-VaR (mean-CVaR) space. The obtained results are illustrated in an empirical study throughout the paper.

Multi-objective multi-stage transmission network expansion planning considering life cycle cost and risk value under uncertainties

Compared with traditional transmission network expansion planning, the exact life cycle cost is worth further studying. A three-dimensional life cycle cost (LCC) model for entire transmission network which consists of time dimension, component dimension and cost dimension is proposed. The device layer, the system layer and the cost of externalities of cost dimension are described and formulated in detail. The conditional value at risk (CVaR) of social welfare is established through optimal power flow calculation so as to take social responsibility into consideration. The multi-objective multi-stage transmission network expansion planning model is constituted with two objectives which are minimum LCC and minimum CVaR of social welfare. Four uncertain factors including wind power are considered and simulated through Monte Carlo method. An effective normal boundary intersection algorithm integrated with improved niche genetic method is presented to solve the proposed model. The case studies carried out on the 18-bus system and 77-bus system not only generate even distributed pareto set but also recommend the optimal planning scheme. The comparisons between the proposed scheme and existing method verify the feasibility and validity. Copyright © 2012 John Wiley & Sons, Ltd.

Dynamic hedging of conditional value-at-risk

In this paper, the problem of partial hedging is studied by constructing hedging strategies that minimize conditional value-at-risk (CVaR) of the portfolio. Two dual versions of the problem are considered: minimization of CVaR with the initial wealth bounded from above, and minimization of hedging costs subject to a CVaR constraint. The Neyman–Pearson lemma approach is used to deduce semi-explicit solutions. Our results are illustrated by constructing CVaR-efficient hedging strategies for a call option in the Black–Scholes model and also for an embedded call option in an equity-linked life insurance contract.

Asset Allocation with Conditional Value-at-Risk Budgets

Risk budgets are frequently used to estimate and allocate the risk of a portfolio by decomposing the total portfolio risk into the risk contribution of each component position. Many approaches to portfolio allocation use ex post methods for constructing risk budgets and take the variance as a risk measure. In this paper, however, we use ex ante methods to evaluate the component contribution to Conditional Value at Risk (CVaR) and to allocate risk. The proposed minimum CVaR concentration portfolio draws a balance between the investor's return objectives and the diversification of risk across the portfolio. For a portfolio invested in bonds, commodities, equities, and real estate, we show that the minimum CVaR concentration portfolio offers an attractive compromise between the good risk-adjusted return properties of the minimum CVaR portfolio and the positive return potential and low portfolio turnover of an equal-weighted portfolio.

Conditional value-at-risk in portfolio optimization: Coherent but fragile

We evaluate conditional value-at-risk (CVaR) as a risk measure in data-driven portfolio optimization. We show that portfolios obtained by solving mean-CVaR and global minimum CVaR problems are unreliable due to estimation errors of CVaR and/or the mean, which are magnified by optimization. This problem is exacerbated when the tail of the return distribution is made heavier. We conclude that CVaR, a coherent risk measure, is fragile in portfolio optimization due to estimation errors.