Conditional Value-at-risk Constraints Research Articles

CVaR Constrained Planning of Renewable Generation with Consideration of System Inertial Response, Reserve Services and Demand Participation

Integration of renewable generation can lead to both diversification of energy sources (which can improve the overall economic performance of the power sector) and cost increase due to the need for further resources to provide flexibility and thus secure operation from unpredictable, variable and asynchronous generation. In this context, we propose a cost-risk model that can properly plan generation and determine efficient technology portfolios through balancing the benefits of energy source diversification and cost of security of supply through provision of various generation frequency control and demand side services, including preservation of system inertia levels. We do so through a scenario-based cost minimization framework where the conditional value at risk (CVaR), associated with costs under extreme scenarios of fossil fuel prices combined with hydrological inflows, is constrained. The model can tackle problems with large data sets (e.g. 8760 hours and 1000 scenarios) since we use linear programming and propose a Benders-based method adapted to deal with CVaR constraints in the master problem. Through several analyses, including the Chilean main electricity system, we demonstrate the effects of renewables on hedging both fossil fuel and hydrological risks; effects of security of supply on costs, risks and renewable investment; and the importance of demand side services in limiting risk exposure of generation portfolios through encouraging risk mitigating renewable generation investment.

Penalized sample average approximation methods for stochastic programs in economic and secure dispatch of a power system

In this paper, we develop a stochastic programming model for economic dispatch of a power system with operational reliability and risk control constraints. By defining a severity-index function, we propose to use conditional value-at-risk (CVaR) for measuring the reliability and risk control of the system. The economic dispatch is subsequently formulated as a stochastic program with CVaR constraint. To solve the stochastic optimization model, we propose a penalized sample average approximation (SAA) scheme which incorporates specific features of smoothing technique and level function method. Under some moderate conditions, we demonstrate that with probability approaching to 1 at an exponential rate with the increase of sample size, the optimal solution of the smoothing SAA problem converges to its true counterpart. Numerical tests have been carried out for a standard IEEE-30 DC power system.

Portfolio Selection with Multiple Spectral Risk Constraints

We propose an iterative gradient-based algorithm to efficiently solve the portfolio selection problem with multiple spectral risk constraints. Since the conditional value-at-risk (CVaR) is a special case of the spectral risk measure, our algorithm solves portfolio selection problems with multiple CVaR constraints. In each step, the algorithm solves very simple separable convex quadratic programs; hence, we show that the spectral risk constrained portfolio selection problem can be solved using the technology developed for solving mean-variance problems. The algorithm extends to the case where the objective is a weighted sum of the mean return and either a weighted combination or the maximum of a set of spectral risk measures. We report numerical results that show that our proposed algorithm is very efficient; it is at least one order of magnitude faster than the state-of-the-art general purpose solver for all practical instances. One can leverage this efficiency to be robust against model risk by including constraints with respect to several different risk models.

Distributionally Robust Control of Constrained Stochastic Systems

We investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process, i.e., when only the first two moments of the disturbance distribution are known. We consider two types of distributionally robust constraints. In the first case, we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments. These constraints are commonly referred to as distributionally robust chance constraints. In the second case, we impose conditional value-at-risk (CVaR) constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information. Such constraints are referred to as distributionally robust CVaR constraints with second-order moment specifications. We propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems. We prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust CVaR and chance constrained problems. The proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives.

Optimization with Multivariate Conditional Value-at-Risk Constraints

For many decision-making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces, we develop a cut generation algorithm, where each cut is obtained by solving a mixed-integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible and computationally tractable way of modeling preferences in stochastic multicriteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods.

In This Issue

In This Issue

Portfolio Selection with Multiple Spectral Risk Constraints

We propose an iterative gradient-based algorithm to efficiently solve the portfolio selection problem with multiple spectral risk constraints. Since the conditional value at risk (CVaR) is a special case of the spectral risk measure, our algorithm solves portfolio selection problems with multiple CVaR constraints. In each step, the algorithm solves very simple separable convex quadratic programs; hence, we show that the spectral risk constrained portfolio selection problem can be solved using the technology developed for solving mean-variance problems. The algorithm extends to the case where the objective is a weighted sum of the mean return and either a weighted combination or the maximum of a set of spectral risk measures. We report numerical results that show that our proposed algorithm is very efficient; it is at least one order of magnitude faster than the state-of-the-art general purpose solver for all practical instances. One can leverage this efficiency to be robust against model risk by including constraints with respect to several different risk models.

Optimization of discrete broadcast under uncertainty using conditional value-at-risk

The paper considers the problem of scheduling packets in wireless broadcast systems under uncertainty with the conditional value-at-risk (CVaR) constraints. In such systems, a server periodically transmits a stream of packets over a broadcast channel. The client that needs to access the data, tunes in to the channel and waits for the next packet. This allows one to serve a large number of clients in an efficient way and also keep clients’ location secret. We formulate and solve two alternative stochastic optimization problems that minimize the transmission time subject to CVaR constraints. Our results indicate that it is possible to derive an analytical solution to the problems in certain cases of practical interest. We also propose a methodology to obtain numerical solutions for the general case.

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.

A Conditional Value-at-Risk Based Inexact Water Allocation Model

A conditional value-at-risk (CVaR) based inexact two-stage stochastic programming (CITSP) model was developed in this study for supporting water resources allocation problems under uncertainty. A CITSP model was formulated through incorporating a CVaR constraint into an inexact two-stage stochastic programming (ITSP) framework, and could be used to deal with uncertainties expressed as not only probability distributions but also discrete intervals. The measure of risks about the second-stage penalty cost was incorporated into the model, such that the trade-off between system economy and extreme expected loss could be analyzed. The developed model was applied to a water resources allocation problem involving a reservoir and three competing water users. The results indicated that the CITSP model performed better than the ITSP model in its capability of reflecting the economic loss from extreme events. Also, it could generate interval solutions within which the decision alternatives could be selected from a flexible decision space. Overall, the CITSP model was useful for reflecting the decision maker’s attitude toward risk aversion and could help seek cost-effective water resources management strategies under complex uncertainties.

An Automated Intensity-Modulated Radiation Therapy Planning System

We design and implement an intensity-modulated radiation therapy plan generation technology that effectively and efficiently optimizes beam geometry as well as beam intensities. Our approach is based on an existing linear programming-based fluence map optimization model that approximates dose-volume requirements using conditional value-at-risk (C-VaR) constraints. We show how the parameters of the C-VaR constraints can be used to control various metrics of treatment plan quality. Next, we develop an automated search strategy for parameter tuning. Finally, beam angle selection is integrated with fluence map optimization. The beam angle selection scheme employs a bicriteria scoring of beam angle geometries and a selection mechanism to choose from among the set of nondominated geometries. The overall technology is automated and generates several high-quality treatment plans satisfying dose prescription requirements in a single invocation and without human guidance. The technology has been tested on various real-patient cases with uniform success.

Constant Rebalanced Portfolio Optimization Under Nonlinear Transaction Costs

We study the constant rebalancing strategy for multi-period portfolio optimization via conditional value-at-risk (CVaR) when there are nonlinear transaction costs. This problem is difficult to solve because of its nonconvexity. The nonlinear transaction costs and CVaR constraints make things worse; state-of-the-art nonlinear programming (NLP) solvers have trouble in reaching even locally optimal solutions. As a practical solution, we develop a local search algorithm in which linear approximation problems and nonlinear equations are iteratively solved. Computational results are presented, showing that the algorithm attains a good solution in a practical time. It is better than the revised version of an existing global optimization. We also assess the performance of the constant rebalancing strategy in comparison with the buy-and-hold strategy.

Solving optimal investment problems with structured products under CVaR constraints

We consider a simple investment problem where besides stocks and bonds the investor can also include options (or structured products) into the investment portfolio. The aim of the investor is to maximize the expected return under a conditional value-at-risk (CVaR) constraint. Due to possible intermediate payments, we have to deal with a re-investment problem which turns the original one-period problem into a multi-period one. For solving this problem, an iterative scheme based on linear optimization is developed.

Simulating Sensitivities of Conditional Value at Risk

Conditional value at risk (CVaR) is both a coherent risk measure and a natural risk statistic. It is often used to measure the risk associated with large losses. In this paper, we study how to estimate the sensitivities of CVaR using Monte Carlo simulation. We first prove that the CVaR sensitivity can be written as a conditional expectation for general loss distributions. We then propose an estimator of the CVaR sensitivity and analyze its asymptotic properties. The numerical results show that the estimator works well. Furthermore, we demonstrate how to use the estimator to solve optimization problems with CVaR objective and/or constraints, and compare it to a popular linear programming-based algorithm.

Optimal ordering quantities for multi-products with stochastic demand: Return-CVaR model

Multi-product-ordering problem with stochastic demand is often optimized by the expected cost reduction or the expected profit improvement, in most of previous works. Moreover, risk preference of decision maker is often ignored. Evidently, this approach is impracticable when the corresponding volatility is larger. Based on the outstanding characteristics of conditional value at risk (CVaR) as a risk measure, the paper proposes an optimal-order model for multi-product with CVaR constraints, also the model is finally formulated as a linear programming problem. The model is simulated for the case of a newsvendor to analyze to what degree it succeeds. The solution, in fact, is bound consistent fully with the decision maker's intuition on return-risk decision making. Finally, results of return-CVaR model and the classical model are compared and shows that return-CVaR model is more flexible than the classical model.

A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model

In this paper, we analyze the portfolio selection implications arising from imposing a value-at-risk (VaR) constraint on the mean-variance model, and compare them with those arising from the imposition of a conditional value-at-risk (CVaR) constraint. We show that for a given confidence level, a CVaR constraint is tighter than a VaR constraint if the CVaR and VaR bounds coincide. Consequently, a CVaR constraint is more effective than a VaR constraint as a tool to control slightly risk-averse agents, but in the absence of a risk-free security, has a perverse effect in that it is more likely to force highly risk-averse agents to select portfolios with larger standard deviations. However, when the CVaR bound is appropriately larger than the VaR bound or when a risk-free security is present, a CVaR constraint “dominates” a VaR constraint as a risk management tool.

CVaR as a Measure of Risk: Implications for Portfolio Selection

In this paper, we analyze the implications arising from imposing a Conditional Value-at-Risk (CVaR)constraint in an agent's portfolio selection problem, and compare them with those arising from the imposition of a Value-at-Risk (VaR) constraint. For a given confidence level, a CVaR constraint is tighter than a VaR constraint is the CVaR and VaR bounds coincide. Consequently, a CVaR constraint is more effective than a VaR constraint as a tool to control slightly risk-averse agents, but has a perverse effect in that it may force highly risk-averse agents to select portfolios with larger standard deviations. While this effect may also occur with the use of a VaR constraint, it is even more perverse and more likely to occur with a CVaR constraint. However, when the CVaR bound is appropriately larger than the VaR bound, a CVaR constraint 'dominates' a VaR constraint as a risk management tool.

Asset/Liability Management for Pension Funds Using CVaR Constraints

This article studies formal optimal decision approaches for a multi‐period asset/liability management model for a pension fund. The authors use Conditional Value‐at‐Risk (CVaR) as a risk measure, the weighted average of the Value‐at‐Risk (VaR) and those losses exceeding VaR. The model is based on sample‐path simulation of the liabilities and returns of financial instruments in the portfolio. The same optimal decisions are made for groups of sample‐paths, which exhibit similar performance characteristics. Since allocation proportions are time‐dependent, these techniques are more flexible than more standard allocation procedures, e.g. “constant proportions.” Optimization is conducted using linear programming. Compared with traditional stochastic programming algorithms (for which the problem dimension increases exponentially in the number of time stages), this approach exhibits a linear growth of the dimension. Therefore, this approach allows the solution of problems with very large numbers of instruments and scenarios.

Portfolio optimization with conditional value-at-risk objective and constraints

Recently, a new approach for optimization of Conditional Value-at-Risk (CVaR) was suggested and tested with several applications. For continuous distributions, CVaR is defined as the expected loss exceeding Value-at Risk (VaR). However, generally, CVaR is the weighted average of VaR and losses exceeding VaR. Central to the approach is an optimization technique for calculating VaR and optimizing CVaR simultaneously. This paper extends this approach to the optimization problems with CVaR constraints. In particular, the approach can be used for maximizing expected returns under CVaR constraints. Multiple CVaR constraints with various confidence levels can be used to shape the profit/loss distribution. A case study for the portfolio of S&P 100 stocks is performed to demonstrate how the new optimization techniques can be implemented.