Second-order Stochastic Dominance Constraints Research Articles

A neural network framework for portfolio optimization under second-order stochastic dominance

Traditional portfolio optimization strategies often rely on statistical methods and linear programming tools to achieve a balance between return and risk. Despite their usefulness for portfolio optimization, they cannot efficiently capture the specific differences and complexity of real-world financial markets. Several modern approaches attempt to overcome these limitations by using nonlinear models, machine learning, and advanced risk measures. In this study, we propose a novel strategy for optimizing portfolios that incorporates second-order stochastic dominance constraints and solves them with neural networks. we show that portfolios subject to second-order stochastic dominance constraints outperform their traditional counterparts, especially in tail-risk situations.

Adjustable light robust optimization with second order stochastic dominance constraints

Robust optimization suffers from excessive conservatism and infeasibility due to stringent requirements on all constraints within the uncertainty set, rendering robust solutions impractical. The light robust approach has been proposed to mitigate these challenges. This method aims to identify a solution that minimizes the weighted sum of all constraint violations while adhering to a predetermined limit on the deterioration of expected return. Drawing inspiration from light robustness, our paper introduces the adjustable light robust optimization models with second-order stochastic dominance constraints. We empirically examine their feasibility, out-of-sample performance, and the dynamic trade-off between return and robustness.

Investments in transmission lines and storage units considering second-order stochastic dominance constraints

Increasing intermittent renewable generation to meet the climate goals entails a deep transformation of current power systems. The transmission system must adapt to ensure a rapid and flexible response to the changes in the energy flows. Flexibility can be provided by reinforcing the interconnection among all the market agents and/or by installing facilities with fast response to the changes. In this work, we study the investment problem of a central planner that seeks to expand the transmission network and install storage units considering decarbonizing measures. We propose a two-stage stochastic problem with uncertainty on the demand growth and including representative days to characterize the hourly demand and renewable power variability. To obtain better expansion strategies according to a limit on the carbon emissions, second-order stochastic dominance constraints are imposed. Numerical analyses based on the power system of the Canary island (Spain) are provided. The results show that to install storage units is key to efficiently integrate renewable generation. The investments in the transmission system are mostly in lower-voltage lines. The formulation with stochastic dominance constraints results in higher second-stage investments, allowing a better adaptation to the demand growth evolution.

Robust reward-risk performance measures with weakly second-order stochastic dominance constraints

Robust reward-risk performance measures with weakly second-order stochastic dominance constraints

Distributionally robust optimization with multivariate second-order stochastic dominance constraints with applications in portfolio optimization

In this paper, we study the stochastic optimization problem with multivariate second-order stochastic dominance (MSSD) constraints where the distribution of uncertain parameters is unknown. Instead, only some historical data are available. Using the Wasserstein metric, we construct an ambiguity set and develop a data-driven distributionally robust optimization model with multivariate second-order stochastic dominance constraints (DROMSSD). By utilizing the linear scalarization function, we transform MSSD constraints into univariate constraints. We present a stability analysis focusing on the impact of the variation of the ambiguity set on the optimal value and optimal solutions. Moreover, we carry out quantitative stability analysis for the DROMSSD problems as the sample size increases. Specially, in the context of the portfolio, we propose a convex lower reformulation of the corresponding DROMSSD models under some mild conditions. Finally, some preliminary numerical test results are reported. We compare the DROMSSD model with the sample average approximation model through out-of-sample performance, certificate and reliability. We also use real stock data to verify the effectiveness of the DROSSM model.

Analysing decarbonizing strategies in the European power system applying stochastic dominance constraints

In this paper we develop an analysis of the efficiency of the expansion strategies to be followed to attain the emissions targets established by the European Commission in the Energy Roadmap 2050. A multi-stage investment model in generating and storage capacity from the point of view of a central planner is presented, considering long-term uncertainties in the decision-making process, such as the demand growth and the investment and fuel costs, and short-term variability. To evaluate the wellness of the expansion strategies according to the CO2 emissions generated and the total cost, second-order stochastic dominance constraints are introduced in the model. This approach allows to obtain better expansion strategies enforcing acceptable distributions of CO2 emissions. The numerical study is carried out considering the case of the European power system. The predictions and suggestions made by the European Commission towards 2050 are the basis to define the benchmark solutions, whose outcomes are analysed. The results obtained from this study highlight that a renewable capacity of at least 2900 GW is needed to attain a net zero CO2 emission European power system. The strategy based on carbon capture and storage does not reduce effectively CO2 emissions while it represents an expensive alternative. Including stochastic dominance in the optimization model allows to obtain less expensive alternative expansion strategies with comparatively lower CO2 emissions in the worst scenarios.

Pension fund management with investment certificates and stochastic dominance

This paper considers an extension of the common asset universe of a pension fund to investment certificates. Investment certificates are a class of structured products particularly interesting for their special payoff structures and they are acquiring relevancy in the worldwide markets. In fact, some subclasses of certificates offer loss protection and show high liquidity and, thus, they can be very appreciated by pension fund managers. We consider the problem of a pension fund manager who has to implement an Asset and Liability Management model trying to achieve a long-term sustainability. Therefore, we formulate a multi-stage stochastic programming problem adopting a discrete scenario tree and a multi-objective function. We propose a technique to price highly structured products such as investment certificates on a discrete scenario tree. Finally, we solve the investment problem considering some investment certificate types both in term of payoff structure and protection level, and we test whether they are preferred or not to standard hedging contract such as put options. Moreover, we test the inclusion of first-order and second-order stochastic dominance constraints on multiple stages with respect to a benchmark portfolio. Numerical results show that the portfolio composition reacts to the inclusion of the stochastic dominance constraints, and that the optimal portfolio is efficiently able to reach several targets such as liquidity, returns, sponsor’s extraordinary contribution and funding gap.

Portfolio optimization with relaxation of stochastic second order dominance constraints via conditional value at risk

A portfolio optimization model with relaxed second order stochastic dominance (SSD) constraints is presented. The proposed model uses Conditional Value at Risk (CVaR) constraints at probability level \begin{document}$ \beta\in(0,1) $\end{document} to relax SSD constraints. The relaxation is justified by theoretical convergence results based on sample average approximation (SAA) method when sample size \begin{document}$ N\to\infty $\end{document} and CVaR probability level \begin{document}$ \beta $\end{document} tends to 1. SAA method is used to reduce infinite number of inequalities of SSD constraints to finite ones and also to calculate the expectation value. The proposed relaxation on the SSD constraints in portfolio optimization problem is achieved when the probability level \begin{document}$ \beta $\end{document} of CVaR takes value less than but close to 1, and the model can then be solved by cutting plane method. The performance and characteristics of the portfolios constructed by solving the proposed model are tested empirically on three sets of market data, and the experimental results are analyzed and discussed. Furthermore, it is shown that with appropriate choices of CVaR probability level \begin{document}$ \beta $\end{document} , the constructed portfolios are sparse and outperform the portfolios constructed by solving portfolio optimization problems with SSD constraints, with either index portfolios or mean-variance (MV) portfolios as benchmarks.

Second-Order Stochastic Dominance Constraints for Risk Management of a Wind Power Producer's Optimal Bidding Strategy

Risk management is critical for wind producers to participate in electricity markets. Beside market price volatility and uncertainty, wind producers are facing an additional uncertainty in the level of wind power generation. Instead of using common risk measures, such as conditional value at risk (CVaR), this paper proposes the use of the second-order stochastic dominance constraints (SOSDCs) for risk management of wind producer's bidding strategies. As benchmark selection is the major obstacle against applying SOSDCs, a novel optimization-based benchmark selection method is proposed. Case studies are carried out for an 80 MW wind producer using the SOSDCs-based bidding model with the proposed benchmark selection method and the CVaR-based bidding model. Results demonstrate the superior flexibility of the SOSDCs in risk management. Moreover, the SOSDCs can effectively manage the negative tail of the profit distribution. Compared to the SOSDCs, the CVaR is more suitable for modeling risk rather than managing risk, as it does not use a profit target value but uses the $({1 - \boldsymbol {\alpha }})$ -quantile of the profit distribution. As the negative tail is the best representative of risk in the problem under study, the SOSDCs with the proposed benchmark selection method are more suitable than the CVaR for risk management of a wind power producer's bidding strategy.

Long-term individual financial planning under stochastic dominance constraints

We analyse an optimal goal-based households’ asset-liability management problem characterised by a real estate target and a retirement goal over a long-term planning horizon. The problem is formulated as a multistage stochastic program and we evaluate the impact of second order stochastic dominance (SSD) constraints on different specifications of a family objective function and with respect to three alternative benchmark policies. We define a stochastic linear program in which the SSD constraints are based on a double stochastic matrix, whose effectiveness in determining the decision maker strategies is studied in a case study developed in the second part of the article. We show that depending on the adopted benchmark policy, SSD constraints even if binding far on the planning horizon, may influence the root node investment decision and affect both the investment and the liability optimal policies. Based on an extended computational study we analyse under which conditions and problem formulation, an SSD condition may also imply first order stochastic dominance (FSD). Finally we analyse the relationship between the specification of a minimum shortfall objective with respect to the goals and the introduced SSD constraints at the terminal horizon.

Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACTBy utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.

Risk-Tabanlı VZA ve Stokastik Baskınlık Kriteri ile OECD Üyelerinin Hisse Senedi Endekslerinin Etkinliği

A stock market index gives some illustrative information regarding the financial market. In this study, we are interested in stock indices efficiency of OECD member countries. We use Data Envelopment Analysis (DEA) methodology and Second Order Stochastic Dominance (SSD) Criteria as an efficiency metrics. DEA is a linear programming based technique for measuring the relative efficiency of homogenous decision making units by their input-output rates. In the Risk-Based DEA, traditional and modern risk measures are used as inputs of the model and the mean return as an output. We consider Conditional Value at Risk (CVaR) as a modern risk measure of financial asset returns. Another approach for the efficiency is Stochastic Dominance (SD) rule that takes into account the entire distribution of return, rather than the return distribution characteristics. There are several papers show that SSD constraints related to the CVaR constraints in an optimization model. Therefore, we compare Risk-Based DEA results with optimization problem with SSD constraints in the empirical study. We also test SSD efficiency of stock index pairs. The results are valuable for the asset managers who need to evaluate the performance of a stock index among others.

Second-order stochastic dominance constrained portfolio optimization: Theory and computational tests

Due to the definition of second-order stochastic dominance (SSD) in terms of utility theory, portfolio optimization with SSD constraints is of major practical interest. We contribute to the field in two ways: first, we present a self-contained theory with some new results and new proofs of known results; second, we perform a set of tests for computational efficiency. We provide new and simple arguments for the formulation of SSD constraints in a mathematical programming framework. For many individuals, an SSD constraint may seem too severe wherefore various relaxations (ASSD), have been proposed. We introduce yet another relaxation, directional SSD, where a candidate portfolio is admissible if a step from the benchmark in the direction of the candidate yields a dominating portfolio. Optimal step size depends on individual preferences reflected by the objective function. We compare computational efficiency of seven approaches for SD constrained portfolio problems, including SSD and ASSD constrained cases.

An interactive cooperation model for neighboring virtual power plants

Future distribution systems will accommodate an increasing share of distributed energy resources (DERs). Facing with this new reality, virtual power plants (VPPs) play a key role to aggregate DERs with the aim of facilitating their involvement in wholesale electricity markets. In this paper, the trading strategies of a VPP in cooperation with its neighboring VPPs are addressed. Toward this aim, a portfolio of inter-regional contracts is considered to model this cooperation and maximize the energy trade opportunities of the VPP within a medium-term horizon. To hedge against profit variability caused by market price uncertainties, two efficient risk management approaches are also implemented in the VPP decision-making problem based on the concepts of conditional value at risk (CVaR) and second-order stochastic dominance constraints (SSD). The resulting models are formulated as mixed-integer linear programming (MILP) problems that can be solved using off-the-shelf software packages. The efficiency of the proposed risk-hedging models is analyzed through a detailed case study, and thereby relevant conclusions are drawn.

Optimal diversification, stochastic dominance, and sampling error

PurposeThe purpose of this paper is to contribute to the literature in three ways: first, the authors investigate the impact of the sampling errors on optimal portfolio weights and on financial investment decision. Second, the authors advance a comparative analysis between various domestic and international diversification strategies to define a stochastic optimal choice. Third, the authors propose a new methodology combining the re-sampling method, stochastic optimization algorithm, and nonparametric stochastic dominance (SD) approach to analyze a stochastic optimal portfolio choice for risk-averse American investors who care about benefits of domestic diversification relative to international diversification. The authors propose a new portfolio optimization model involving SD constraints on the portfolio return rate. The authors define a portfolio with return dominating the benchmark portfolio return in the second-order stochastic dominance (SSD) and having maximum expected return. The authors combine re-sampling procedure and stochastic optimization to establish more flexibility in the investment decision rule.Design/methodology/approachThe authors apply the re-sampling procedure to consider the sampling error in the optimization process. The authors try to resolve the problem of the stochastic optimal investment strategy choice using the nonparametric SD test by Linton et al. (2005) based on sub-sampling simulated p values. The authors apply the stochastic portfolio optimization algorithm with SSD constraints to define optimal diversified portfolios beating benchmark indices.FindingsFirst, the authors find that reducing sampling error increases the dominance relationships between different portfolios, which, in turn, alters portfolio investment decisions. Though international diversification is preferred in some cases, the study’s results show that for risk-averse US investors, in general, there is no difference between the diversification strategies; this implies that there is no increase in the expected utility of international diversification for the period before and after the 2007-2008 financial crisis. Nevertheless, the authors find that stochastic diversification in domestic, global, and Europe, Australasia, and Far East markets delivers better risk returns for the US risk averters during the crisis period.Originality/valueThe originality of the idea in this paper is to introduce a new methodology combining the concept of portfolio re-sampling, stochastic portfolio optimization with SSD constraints, and the nonparametric SD test by Linton et al. (2005) based on subsampling simulated p values to analyze the impact of sampling errors on optimal portfolio returns and to investigate the problem of stochastic optimal choice between international and domestic diversification strategies. The authors try to prove more coherence in the portfolio choice with the stochastically and the uncertainty characters of the paper.

Portfolios Dominating Indices: Optimization with Second-Order Stochastic Dominance Constraints vs. Minimum and Mean Variance Portfolios

The paper compares portfolio optimization with the Second-Order Stochastic Dominance (SSD) constraints with mean-variance and minimum variance portfolio optimization. As a distribution-free decision rule, stochastic dominance takes into account the entire distribution of return rather than some specific characteristic, such as variance. The paper is focused on practical applications of the portfolio optimization and uses the Portfolio Safeguard (PSG) package, which has precoded modules for optimization with SSD constraints, mean-variance and minimum variance portfolio optimization. We have done in-sample and out-of-sample simulations for portfolios of stocks from the Dow Jones, S&P 100 and DAX indices. The considered portfolios’ SSD dominate the Dow Jones, S&P 100 and DAX indices. Simulation demonstrated a superior performance of portfolios with SD constraints, versus mean-variance and minimum variance portfolios.

Enhanced indexing for risk averse investors using relaxed second order stochastic dominance

The less efficient markets offer scope for enhanced indexing (EI), an investment strategy of portfolio selection which seeks to earn more return than the benchmark index. In this context, we examine the use of relaxed second order stochastic dominance (RSSD) by introducing underachievement and overachievement variables in the second order stochastic dominance (SSD), for EI. We propose a linear optimization model that maximizes the mean return subject to the constraints formed using RSSD. We impose bounds on the ratio of the total underachievement to the sum of total underachievement and total overachievement variables depicting the risk-return tradeoff in the model. The proposed model for EI is inspired from many applications of SSD and almost SSD (ASSD). We examine the performance of the proposed model with the SSD model, EI model of maximizing mean return and minimizing the underperformance (MM) from the benchmark index, \(\epsilon \)-almost second order stochastic dominance (\(\epsilon \)-ASSD) model, and the naive 1/N portfolio, on two Indian stock indices, CNX 100 and CNX 200, through a rolling window strategy. To widen the empirical analysis, we also compare all models on the eight publicly available real financial data sets from Beasley OR library through a single window strategy. The portfolios from the proposed model are shown to produce statistically significant mean excess return and excess Sharpe ratio (both from the benchmark indices) more often than the MM and \(\epsilon \)-ASSD models. Also, the portfolios from the proposed model always have smaller violation area in SSD constraints from benchmark indices than the MM and \(\epsilon \)-ASSD models.

On time stochastic dominance induced by mixed integer-linear recourse in multistage stochastic programs

We propose in this work a new multistage risk averse strategy based on Time Stochastic Dominance (TSD) along a given horizon. It can be considered as a mixture of the two risk averse measures based on first- and second-order stochastic dominance constraints induced by mixed integer-linear recourse, respectively. Given the dimensions of medium-sized problems augmented by the new variables and constraints required by this new risk measure, it is unrealistic to solve the problem up to optimality by plain use of MIP solvers in a reasonable computing time, at least. Instead of it, decomposition algorithms of some type should be used. We present an extension of our Branch-and-Fix Coordination algorithm, so named BFC-TSD, where a special treatment is given to cross scenario group constraints that link variables from different scenario groups. A broad computational experience is presented by comparing the risk neutral approach and the tested risk averse strategies. The performance of the new version of the BFC algorithm versus the plain use of a state-of-the-art MIP solver is also reported.

Finding Efficient and Environmentally Friendly Paths for Risk-Averse Freight Carriers

This paper aims to incorporate two important measures into a freight shortest path problem, namely reliability and sustainability. Reliability measure deals with the uncertainty of link travel time while sustainability measure tends to reduce the fuel consumption and emission along the path. Greenhouse gas (GHG) emission rates are generated from Motor Vehicle Emission Simulator (MOVES) model and approximated as a function of the average link travel speed. To model uncertainty, the link travel speed is treated as a discrete random variable with a given distribution. Freight carriers are assumed to be risk-averse; for example, given two paths with the same average cost, carriers prefer the one with less variability. The risk-averse behavior is captured by the second order stochastic dominance (SSD) relationship. Specifically, SSD constraints are introduced in our model to narrow down the feasible paths which dominate a chosen benchmark path. The reliable and sustainable routing model is formulated as an integer program that can be easily tailored to a variety of modeling preferences. The study experiments with eight variants of the base model, each corresponding to a different trade-off strategy between three objectives, namely, efficiency, reliability and sustainability. The numerical experiments illustrate the benefits of the models discussed in the paper.

Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints

Sample average approximation (SAA) method has recently been applied to solve stochastic programs with second order stochastic dominance (SSD) constraints. In particular, Hu et al. (Math Program 133:171–201, 2012) presented a detailed convergence analysis of \(\epsilon \)-optimal values and \(\epsilon \)-optimal solutions of sample average approximated stochastic programs with polyhedral SSD constraints. In this paper, we complement the existing research by presenting convergence analysis of stationary points when SAA is applied to a class of stochastic minimization problems with SSD constraints. Specifically, under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts. Moreover, by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations, we derive exponential rate of convergence of stationary points.