A smoothing algorithm for two-stage portfolio model with second-order stochastic dominance constraints

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.

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.

Portfolio optimization models are usually based on several distribution characteristics, such as mean, variance or Conditional Value-at-Risk (CVaR). For instance, the mean-variance approach uses mean and covariance matrix of return of instruments of a portfolio. However this conventional approach ignores tails of return distribution, which may be quite important for the portfolio evaluation. This chapter considers the portfolio optimization problems with the Stochastic Dominance constraints. As a distribution-free decision rule, Stochastic Dominance takes into account the entire distribution of return rather than some specific characteristic, such as variance. We implemented efficient numerical algorithms for solving the optimization problems with the Second-Order Stochastic Dominance (SSD) constraints and found portfolios of stocks dominating Dow Jones and DAX indices. We also compared portfolio optimization with SSD constraints with the Minimum Variance and Mean-Variance portfolio optimization.

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.

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.

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.

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

We propose new cutting plane methods for solving optimization problems with second-order stochastic dominance constraints. The methods are based on the inverse formulation of stochastic dominance constraints using Lorenz functions. Convergence of the methods is proved for general probability distributions. For general discrete distributions convergence is finite. Numerical experiments on a portfolio problem confirm efficiency of the methods.

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.

Stochastic dominance constraints allow a decision maker to manage risk in an optimization setting by requiring his or her decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first- and second-order stochastic dominance constraints, respectively. These formulations are more compact than existing formulations, and relaxing integrality in the first-order formulation yields a second-order formulation, demonstrating the tightness of this formulation. We also present a specialized branching strategy and heuristics which can be used with the new first-order formulation. Computational tests illustrate the potential benefits of the new formulations.

A neural network framework for portfolio optimization under second-order 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.

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

In this paper we present the problem faced by an electricity retailer which searches to determine its forward contracting portfolio and the selling prices for its potential clients. This problem is formulated as a two-stage stochastic program including second-order stochastic dominance constraints. The stochastic dominance theory is used in order to reduce the risk suffering from low profits. The resulting deterministic equivalent problem is a mixed-integer linear program which is solved using commercial branch-and-cut software. Numerical results for a realistic case study are reported and relevant conclusions are drawn.

This paper studies optimal path problems integrated with the concept of second order stochastic dominance. These problems arise from applications where travelers are concerned with the trade off between the risks associated with random travel time and other travel costs. Risk-averse behavior is embedded by requiring the random travel times on the optimal paths to stochastically dominate that on a benchmark path in the second order. A general linear operating cost is introduced to combine link- and path-based costs. The latter, which is the focus of the paper, is employed to address schedule costs pertinent to late and early arrival. An equivalent integer program to the problem is constructed by transforming the stochastic dominance constraint into a finite number of linear constraints. The problem is solved using both off-the-shelf solvers and specialized algorithms based on dynamic programming (DP). Although neither approach ensures satisfactory performance for general large-scale problems, the numerical experiments indicate that the DP-based approach provides a computationally feasible option to solve medium-size instances (networks with several thousand links) when correlations among random link travel times can be ignored.