CVaR Constraints Research Articles

A non-zero-sum investment and reinsurance game between two mean–variance insurers with dynamic CVaR constraints

This paper is devoted to investigating a non-zero-sum game between two competing insurers. The insurers can diversify their insurance risks by purchasing proportional reinsurance and investing their collected premiums into a financial market composed of one risk-free asset and one stock. The reinsurance premiums charged by the reinsurer follow the generalized mean–variance premium principle. Moreover, the dynamic CVaR constraints are incorporated in the game problem to control risks. With the dynamic mean–variance objective, we introduce two forward deterministic auxiliary processes to represent the expectations of the insurers’ wealth processes and transform the original time inconsistent game problem into a standard time consistent game problem with two state variables for each insurer. By adopting the dynamic programming principle and the Lagrange duality method, we derive the Nash equilibrium investment–reinsurance strategies for the two insurers. Finally, the effects of several important model parameters on the optimal policies are analyzed by numerical examples.

A distributionally robust approximate framework consider CVaR constraints for energy management of microgrid

As the penetration of intermittent renewable energy increases in microgrid systems, flexible power generation resources cause the power imbalance problem. To improve the absorption of renewable energy, this paper proposes a distributionally robust (DR) approximate framework considering conditional value-at-risk measures for energy management of microgrids based on cooperative scheduling energy storage and direct load control operations. First, combined with CVaR and DR theory, DR CVaR constraints that can quantitatively the power balance risk of microgrid is constructed. Then, to improve the efficiency of solving DR CVaR constraints, it is tractably approximated using Jensen’s​ inequality, and the approximate error of DR CVaR constraint is analyzed by comparing two different Jensen’s inequality gap expressions so that the decision maker can select the appropriate approximate DR CVaR constraint according to the actual needs. Finally, the appropriate DR CVaR framework of the microgrid is transformed into mixed-integer linear programming that can be directly solved by CPLEX. The approximate framework has the characteristics of risk aversion measurement, linear efficient solving, and flexible decision-making. And it is also verified on the IEEE 33-bus distribution system through a wide range of different tests.

Development of a new asset liability Management model with liquidity and inflation risks based on the Lower Partial Moment

Inflation and liquidity risks are two significant risks with which pension fund managers must contend. This study develops a new multi-period Asset Liability Management model to address these challenges. The model objective function minimizes both risks simultaneously. The Lower Partial Moment measure is used to minimize investment portfolio risk and a CVaR constraint is employed to avoid loss. A hybrid ARIMA-GARCH model is developed to generate scenarios regarding a case study from Iran. To do so, four different investment cases out of seven stocks from the stock market, one bond, three foreign currencies (Dollar, Euro, and Yen) and gold are considered. Findings affirm model efficacy and the model suggests that a portfolio of all four assets is the best investment strategy.

Optimal Allocation of Retirement Portfolios

A retiree with a savings account balance, but without a pension, is confronted with an important investment decision that has to satisfy two conflicting objectives. Without a pension, the function of the savings is to provide post-employment income to the retiree. At the same time, most retirees want to leave an estate to their heirs. Guaranteed income can be acquired by investing in an annuity. However, that decision takes funds away from investment alternatives that might grow the estate. The decision is made even more complicated because one does not know how long one will live. A long life expectancy may require more annuities, and a short life expectancy could promote more risky investments. However there are very mixed opinions about both strategies. A framework has been developed to assess consequences and the trade-offs of alternative investment strategies. We propose a stochastic programming model to frame this complicated problem. The objective is to maximize expected estate value, subject to cash outflow constraints. The model is motivated by the Markowitz mean-variance approach, but with risk measured by CVaR and additional sophisticated constraints. The cash outflow shortages are penalized in the objective function of the problem. We use the kernel method to build position adjustment functions that control how much is invested in each asset. These adjustments nonlinearly depend upon asset returns in previous years. A case study was conducted using two variations of the model. The parameters used in this case study correspond to a typical retirement situation. The case study shows that if the market forecasts are pessimistic, it is optimal to invest in an annuity. The case study results, codes, and data are posted on our website.

A column generation heuristic for VMAT planning with adaptive CVaR constraints

In this study we develop an efficient computational procedure that generates medically acceptable treatment plans for volumetric modulated arc therapy with constant gantry speed. Our proposed method is a column generation heuristic based on a mixed integer linear programming model, where the objective function contains minimization of total monitor unit of the treatment plan and dose-volume requirements are included as conditional value-at-risk constraints. Our heuristic generates a full treatment arc for the restricted master problem and calibrates the right hand side parameters of the conditional value-at-risk constraints in the first phase. In the second phase, this initial solution is improved by performing column generation. This is a fully automated procedure and produces treatment plans in a single call without any human intervention. We evaluate its performance on real prostate cancer data by comparing the quality of the generated plans with those obtained by a widely used commercial treatment planning system. Our analysis shows that the results are promising, and the generated plans satisfy the prescription restrictions and require fewer monitor units on average compared to the ones obtained using Eclipse.

A differential evolution copula-based approach for a multi-period cryptocurrency portfolio optimization

Recent years have seen a growing interest among investors in the new technology of blockchain and cryptocurrencies and some early investors in this new type of digital assets have made significant gains. The heuristic algorithm, differential evolution, has been advocated as a powerful tool in portfolio optimization. We propose in this study two new approaches derived from the traditional differential evolution (DE) method: the GARCH-differential evolution (GARCH-DE) and the GARCH-differential evolution t-copula (GARCH-DE-t-copula). We then contrast these two models with DE (benchmark) in single and multi-period optimizations on a portfolio consisting of five cryptoassets under the coherent risk measure CVaR constraint. Our analysis shows that the GARCH-DE-t-copula outperforms the DE and GARCH-DE approaches in both single- and multi-period frameworks. For these notoriously volatile assets, the GARCH-DE-t-copula has shown risk-control ability, hereby confirming the ability of t-copula to capture the dependence structure in the fat tail.

CVaR distance between univariate probability distributions and approximation problems

The paper defines new distances between univariate probability distributions, based on the concept of the CVaR norm. We consider the problem of approximation of a discrete distribution by some other discrete distribution. The approximating distribution has a smaller number of atoms than the original one. Such problems, for instance, must be solved for generation of scenarios in stochastic programming. The quality of the approximation is evaluated with new distances suggested in this paper. We use CVaR constraints to assure that the approximating distribution has tail characteristics similar to the target distribution. The numerical algorithm is based on two main steps: (i) optimal placement of positions of atoms of the approximating distribution with fixed probabilities; (ii) optimization of probabilities with fixed positions of atoms. These two steps are iterated to find both optimal atom positions and probabilities. Numerical experiments show high efficiency of the proposed algorithms, solved with convex and linear programming.

Downside risk control and optimal investment turnover around financial crises

This paper investigates tactical investment strategies for investors to survive financial crises. Compared with the buy-and-hold strategy, the buy-and-sell strategy is much more effective in mitigating downside risk before, during, and after a crisis by restricting the left-tail volatility of portfolio returns through CVaR constraints. The paper also studies investors' optimal turnovers around a crisis under the buy-and-hold strategy. Considering investors' heterogeneous behaviours, we find the wealth-weighted average optimal turnover across all investors during a crisis is much higher than that before or after the crisis. This indicates investors who enter the market before a crisis may be better off by leaving their portfolios untouched during the market downturn. In addition, the downside risk control model can detect a market downturn earlier than the mean-variance model therefore it helps to 'spread out' the required asset adjustments over a longer horizon than the crisis period itself.

Downside risk control and optimal investment turnover around financial crises

This paper investigates tactical investment strategies for investors to survive financial crises. Compared with the buy-and-hold strategy, the buy-and-sell strategy is much more effective in mitigating downside risk before, during, and after a crisis by restricting the left-tail volatility of portfolio returns through CVaR constraints. The paper also studies investors’ optimal turnovers around a crisis under the buy-and-hold strategy. Considering investors’ heterogeneous behaviors, we find the wealth-weighted average optimal turnover across all investors during a crisis is much higher than that before or after the crisis. This indicates investors who enter the market before a crisis may be better off by leaving their portfolios untouched during the market downturn. In addition, the downside risk control model can detect a market downturn earlier than the mean-variance model, therefore it helps to “spread out” the required asset adjustments over a longer horizon than the crisis period itself.

The deterministic ERM and CVaR reformulation for the stochastic generalized complementarity problem

In this paper, we consider the stochastic generalized complementarity problem (SGCP), which has a variety of important applications and has the stochastic complementarity problem as a special case. In order to give reasonable solutions of SGCP, we propose a deterministic formulation that minimizes the expected value of the residual function as the objective function and uses CVaR constraints to ensure the probabilistic feasibility of its solutions. The resulting model containsnon-smooth constraints and expected value functions both in the objective function and constraints. We present approximation problems for solving this model using smoothing and Monte-Carlo techniques. We prove two convergence results of the proposed approximation problems. One is to increase the sample size only, the other is to let smoothing parameter tend to zero as sample increase together. The effectiveness of the proposed approach was demonstrated using numerical examples.

Pension Fund Asset Allocation: A Mean-Variance Model with CVaR Constraints

In this paper, we first review some important aspects of asset allocation for some typical large Social Security Reserve Funds (SSRFs) in the world. Then we present the mean-variance model with CVaR constraints as asset allocation methodology. Concerning the real circumstance in China, we apply the model to pension fund asset allocation. The empirical results show that to maintain purchase power of pension fund, certain proportion should be invested in stocks as well as direct equity investments. We also find that time horizon significantly influence asset allocation of pension fund. If time horizon is longer, more allocations to stocks and equity investments help the pension fund to achieve better performance.

Integrated operational and financial hedging with capacity reshoring

We consider a multinational corporation that employs capacity reshoring, production switching, and financial hedging to manage supply–demand mismatches and currency risk. We optimize mean-conditional value-at-risk (CVaR) by decomposing operations and finance: operational flexibility maximizes expected profit subject to a CVaR constraint, whereas financial hedging minimizes CVaR subject to a minimum expected profit. We report three main findings. First,operational flexibility and financial hedging can be complements: operational flexibility enhances profitability and reduces downside risk, while financial hedging minimizes downside risk and can affect the feasible set of capacity portfolios (albeit indirectly) by relaxing a CVaR constraint. Second,operational flexibility and financial hedging are substitutes in risk reduction (though the latter has greater risk reduction effects in CVaR when used alone). Third,coordinating operations and finance is crucial for minimizing substitution effects. Efficient financial hedging depends on rigorous estimation of cash flow distribution as shaped by operational flexibility, and a capacity portfolio's feasibility relies on financial hedging because of the CVaR constraint.

Optimal consumption—portfolio problem with CVaR constraints

The optimal portfolio selection is a fundamental issue in finance, and its two most important ingredients are risk and return. Merton's pioneering work in dynamic portfolio selection emphasized only the expected utility of the consumption and the terminal wealth. To make the optimal portfolio strategy be achievable, risk control over bankruptcy during the investment horizon is an indispensable ingredient. So, in this paper, we consider the consumption-portfolio problem coupled with a dynamic risk control. More specifically, different from the existing literature, we impose a dynamic relative CVaR constraint on it. By the stochastic dynamic programming techniques, we derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation. Moreover, by the Lagrange multiplier method, the closed form solution is provided when the utility function is a logarithmic one. At last, an illustrative empirical study is given. The results show the distinct difference of the portfolio strategies with and without the CVaR constraints: the proportion invested in the risky assets is reduced over time with CVaR constraint instead of being constant without CVaR constraints. This can provide a good decision-making reference for the investors.

Optimal insurance contract under VaR and CVaR constraints

This study endogenously develops an optimal insurance contractual form for maximizing insured expected utility under VaR and CVaR constraints. We find that CVaR constraint does not affect the contractual form, but may increase minimum insurance premium requirement. Additionally, when the VaR constraint is binding, the optimal contract is a double deductible insurance. However, if the contract is restricted to a regular form (both indemnity schedule and retained loss schedule are continuously nondecreasing) for avoiding moral hazard problem, the optimal contract is a piecewise linear deductible insurance. Finally, we provide intuitive comparison between this study result and relevant studies.

Distributionally Robust Joint Chance Constrained Problem under Moment Uncertainty

We discuss and develop the convex approximation for robust joint chance constraints under uncertainty of first- and second-order moments. Robust chance constraints are approximated by Worst-Case CVaR constraints which can be reformulated by a semidefinite programming. Then the chance constrained problem can be presented as semidefinite programming. We also find that the approximation for robust joint chance constraints has an equivalent individual quadratic approximation form.

CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems

We reformulate a stochastic nonlinear complementarity problem as a stochastic programming problem which minimizes an expected residual defined by a restricted NCP function with nonnegative constraints and CVaR constraints which guarantee the stochastic nonlinear function being nonnegative with a high probability. By applying smoothing technique and penalty method, we propose a penalized smoothing sample average approximation algorithm to solve the CVaR-constrained stochastic programming. We show that the optimal solution of the penalized smoothing sample average approximation problem converges to the solution of the corresponding nonsmooth CVaR-constrained stochastic programming problem almost surely. Finally, we report some preliminary numerical test results.

Optimal bidding strategy for generation companies under CVaR constraint

SUMMARY This paper investigates bidding strategy problems of generation companies in electricity markets. A new bidding model is set up, and a solution approach is presented. Compared with existing models, the model proposed in this paper has some different characteristics. First, the model adopts a bilevel optimization framework where the upper-level optimization is to maximize the profit of generation companies, and the aim of the lower-level optimization is the maximal social welfare and the security of the operation system considered by the Independent System Operator. This model combines practical situations faced by market agents in real-life operations and the requirement of both the system security and the social welfare. Second, the popular risk measure, called conditional value at risk, is used in the bidding model, which can quantify the market risk and describe the competitive mechanism effectively. For solving the complicated bilevel optimization model, a hybrid particle swarm optimization algorithm is presented. Numerical examples are used to validate the model and the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

Evaluating risk-constrained bidding strategies in adjustment spot markets for wind power producers

Participation of wind energy in the day-ahead electricity market implies large deviations from the initial schedule, which leads to costs for the wind farm owner. By means of short-term wind power prediction programs, the contracted energy can be updated in adjustment spot markets, reducing the power deviations and increasing the total revenue for wind power producers. Taking into account the different uncertainties involved in the problem, an optimal bidding strategy can be used to maximize the wind power producer revenues. As the strategy could be very risky due to all these uncertainties, a CVaR constraint for the bid that maximizes the expected revenue is proposed as a way of reducing the risk. A test-case using the Spanish market rules during a 10-month period has been used to check the potential benefits of the aforementioned strategies.

A mixed 0–1 LP for index tracking problem with CVaR risk constraints

Index tracking problems are concerned in this paper. A CVaR risk constraint is introduced into general index tracking model to control the downside risk of tracking portfolios that consist of a subset of component stocks in given index. Resulting problem is a mixed 0–1 and non-differentiable linear programming problem, and can be converted into a mixed 0–1 linear program so that some existing optimization software such as CPLEX can be used to solve the problem. It is shown that adding the CVaR constraint will have no impact on the optimal tracking portfolio when the index has good (return increasing) performance, but can limit the downside risk of the optimal tracking portfolio when index has bad (return decreasing) performance. Numerical tests on Hang Seng index tracking and FTSE 100 index tracking show that the proposed index tracking model is effective in controlling the downside risk of the optimal tracking portfolio.

The value of rolling-horizon policies for risk-averse hydro-thermal planning

We consider the optimal management of a hydro-thermal power system in the mid and long terms. From the optimization point of view, this amounts to a large-scale multistage stochastic linear program, often solved by combining sampling with decomposition algorithms, like stochastic dual dynamic programming. Such methodologies, however, may entail prohibitive computational time, especially when applied to a risk-averse formulation of the problem. We propose instead a risk-averse rolling-horizon policy that is nonanticipative, feasible, and time consistent. The policy is obtained by solving a sequence of risk-averse problems with deterministic constraints for the current time step and future chance and CVaR constraints. The considered hydro-thermal model takes into account losses resulting from run-of-river plants efficiencies as well as uncertain demand and streamflows. Constraints aim at satisfying demand while keeping reservoir levels above minzones almost surely. We show that if the problem uncertainty is represented by a periodic autoregressive stochastic process with lag one, then the probabilistic constraints can be computed explicitly. As a result, each one of the aforementioned risk-averse problems is a medium-size linear program, easy to solve. For a real-life power system we compare our approach with three alternative policies. Namely, a robust nonrolling-horizon policy and two risk-neutral policies obtained by stochastic dual dynamic programming, implemented in nonrolling- and rolling-horizon modes, respectively. Our numerical assessment confirms the superiority of the risk-averse rolling-horizon policy that yields comparable average indicators, but with reduced volatility and with substantially less computational effort.