Second-order Stochastic Dominance Research Articles

Measures of stochastic non-dominance in portfolio optimization

Stochastic dominance rules are well-characterized and widely used. This work aims to describe and better understand the situations when they do not hold by developing measures of stochastic non-dominance. They quantify the error caused by assuming that one random variable dominates another one when it does not. To calculate them, we search for a hypothetical random variable that satisfies the stochastic dominance relationship and is as close to the original one as possible. The Wasserstein distance between the optimal hypothetical random variable and the original one is considered as the measure of stochastic non-dominance. Depending on the conditions imposed on the probability distribution of the hypothetical random variable, we distinguish between general and specific measures of stochastic non-dominance. We derive their exact values for random variables with uniform, normal, and exponential distributions. We present relations to almost first-order stochastic dominance and to tractable almost stochastic dominance. Using monthly returns of twelve assets captured by the German stock index DAX, we solve portfolio optimization problems with the first-order and second-order stochastic dominance constraints. The measures of stochastic non-dominance allow us to compare the optimal portfolios with respect to different orders of stochastic dominance from a new angle. We also defined the closest dominating and closest approximately dominating portfolios. They brought a better understanding of the relationship between the two types of optimal portfolios. Using moving window analysis, the relationship of the in-sample measure of stochastic non-dominance to out-of-sample performance was studied, too.

Risk sharing under heterogeneous beliefs without convexity

We consider the problem of finding (Pareto-)optimal allocations of risk among finitely many agents. The associated individual risk measures are law-invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures. Moreover, we assume that the individual risk assessments are consistent with the respective second-order stochastic dominance relations, but remain agnostic about their convexity. A simple sufficient condition for the existence of Pareto optima is provided. The proof combines local comonotonic improvement with a Dieudonné-type argument, which also establishes a link of the optimal allocation problem to the realm of “collapse to the mean” results.

A note on convex stochastic dominance and diversification with applications

. In this note, we present an overlooked result derived from Fishburn’s Convex Stochastic Dominance (CSD) (Fishburn (1974a; 1974b). We demonstrate that CSD is a sufficient condition for establishing second-order stochastic dominance in the context of linear combinations of random variables. While this result may appear straightforward, to the best of our knowledge, it has not been explicitly presented as we have done in this work. Furthermore, the implications of this finding are extensive, as it can be applied to various areas of research, as we illustrate in this study.

Stochastic optimization problems with nonlinear dependence on a probability measure via the Wasserstein metric

Nonlinear dependence on a probability measure has recently been encountered with increasing intensity in stochastic optimization. This type of dependence corresponds to many situations in applications; it can appear in problems static (one-stage), dynamic with finite (multi-stage) or infinite horizon, and single- and multi-objective ones. Moreover, the nonlinear dependence can appear not only in the objective functions but also in the constraint sets. In this paper, we will consider static one-objective problems in which the nonlinear dependence appears in the objective function and may also appear in the constraint sets. In detail, we consider “deterministic” constraint sets, whose dependence on the probability measure is nonlinear, constraint sets determined by second-order stochastic dominance, and sets given by mean-risk problems. The last mentioned instance means that the constraint set corresponds to solutions which guarantee acceptable values of both criteria. To obtain relevant assertions, we employ the stability results given by the Wasserstein metric, based on the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {{\\mathcal {L}}}_{1} $$\\end{document} norm. We mainly focus on the case in which a solution has to be obtained on the basis of the data and of investigating a relationship between the original problem and its empirical version.

Estimation of the Local Conditional Tail Average Treatment Effect

The conditional tail average treatment effect (CTATE) is defined as a difference between the conditional tail expectations of potential outcomes, which can capture heterogeneity and deliver aggregated local information on treatment effects over different quantile levels and is closely related to the notion of second-order stochastic dominance and the Lorenz curve. These properties render it a valuable tool for policy evaluation. In this article, we study estimation of the CTATE locally for a group of compliers (local CTATE or LCTATE) under the two-sided noncompliance framework. We consider a semiparametric treatment effect framework under endogeneity for the LCTATE estimation using a newly introduced class of consistent loss functions jointly for the CTE and quantile. We establish the asymptotic theory of our proposed LCTATE estimator and provide an efficient algorithm for its implementation. We then apply the method to evaluate the effects of participating in programs under the Job Training Partnership Act in the United States.

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.

Testing out-of-sample portfolio performance using second-order stochastic dominance constrained optimization approach

Second-order Stochastic Dominance (SSD) criterion can be used to support portfolio decision making under risk and uncertainty. In this paper, we develop novel robust SSD criteria to capture the strength of dominance and portfolio optimization models utilizing these criteria to identify portfolios whose in-sample SSD dominance over a given benchmark is likely to hold also out-of-sample. The developed models can incorporate incomplete probability information by allowing a set of feasible state probabilities. We also show that these portfolio optimization models can be formulated as linear programming problems. We report results from applying these SSD-based portfolio optimization models with different sets of state probabilities in an empirical application, with a focus on evaluating the out-of-sample portfolio performance of the optimized portfolios.

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.

Quasi Second-Order Stochastic Dominance Model for Balancing Wildfire Risks and Power Outages due to Proactive Public Safety De-Energizations

Quasi Second-Order Stochastic Dominance Model for Balancing Wildfire Risks and Power Outages due to Proactive Public Safety De-Energizations

Risk Sharing with Lambda Value at Risk

In this paper, we study the risk-sharing problem among multiple agents using lambda value at risk ([Formula: see text]) as their preferences via the tool of inf-convolution, where [Formula: see text] is an extension of value at risk ([Formula: see text]). We obtain explicit formulas of the inf-convolution of multiple [Formula: see text] with monotone Λ and explicit forms of the corresponding optimal allocations, extending the results of the inf-convolution of [Formula: see text]. It turns out that the inf-convolution of several [Formula: see text] is still a [Formula: see text] under some mild condition. Moreover, we investigate the inf-convolution of one [Formula: see text] and a general monotone risk measure without cash additivity, including [Formula: see text], expected utility, and rank-dependent expected utility as special cases. The expression of the inf-convolution and the explicit forms of the optimal allocation are derived, leading to some partial solution of the risk-sharing problem with multiple [Formula: see text] for general Λ functions. Finally, we discuss the risk-sharing problem with [Formula: see text], another definition of lambda value at risk. We focus on the inf-convolution of [Formula: see text] and a risk measure that is consistent with the second-order stochastic dominance, deriving very different expression of the inf-convolution and the forms of the optimal allocations.

Robust mean-to-CVaR optimization under ambiguity in distributions means and covariance

We develop a robust mean-to-CVaR portfolio optimization model under interval ambiguity in returns means and covariance. The robust model satisfies second-order stochastic dominance consistency and is formulated as a semi-definite cone program. We use two controlled experiments to document the sensitivity of the optimal allocations to the ambiguity when asset correlation varies, and to the ambiguity intervals. We find that means ambiguity has a higher impact than covariance ambiguity. We apply the model to US equities data to corroborate works showing that ambiguity in mean returns induces a home bias; it can explain the puzzle in a two-country setting but not with three countries. We further establish that covariance ambiguity also induces bias, but with lower impact that can not explain the puzzle. Our results suggest what is needed for the ambiguity channel to provide a full explanation of the puzzle. The findings are robust to alternative model specifications and outliers.

Risk-involved dominant optimization of multi-energy CCHP-P2G-based microgrids integrated with a variety of storage technologies

Environmental problems arising from fossil-based generation units in a microgrid (MG) are a pressing concern due to their contribution to air pollution and greenhouse gas emissions. This paper investigates an integrated MG with combined cooling, heating, and power (CCHP) systems, designed to efficiently supply energy demands while considering the environmental impacts of generation units associated with CO2 emissions. To mitigate these environmental impacts and promote sustainability, the proposed CCHP-based MG incorporates a power-to-gas (P2G) technology, aimed at CO2 absorbing and recycling. This integration not only optimizes system performance but enhances the environmental aspects of the microgrid. Moreover, the uncertainties related to energy demands, wholesale prices, and renewable generation poses potential challenges for the system operation. In response, the second-order stochastic dominance (SOSD) method is employed to address these uncertainties and ensure robust decision-making. Results show that the P2G system enables a 15 % reduction in carbon emissions, highlighting the eco-friendly nature of the system. However, its incorporation introduces an additional 6 % to the overall system operation cost due to its relatively high cost. In addition, it is important to note that the operator has further conservativeness in the With-SOSD operation mode leading to an approximate 30 % increase in the operation costs compared to the NO-SOSD mode.

Sortino(γ): A Modified Sortino Ratio With Adjusted Threshold

A portfolio’s Sortino ratio is strongly affected by the risk-free vs. risky assets mix, except for the case where the threshold, T is equal to the risk-free rate. Therefore, if T differs from the risk-free rate, the portfolio’s Sortino ratio could potentially be increased by merely changing the mix of the risk-free and the risky components. The widely used Sharpe ratio, on the other hand, does not share this caveat. We introduce a modified Sortino ratio, Sortino(γ), which is invariant concerning the portfolio’s risk-free vs. risky assets mix and eliminates the above deficiency. The selected threshold T(γ), mimics the portfolio composition in the sense that it equals to the risk-free rate plus γ times the portfolio’s equity risk premium. Higher selected γ reflects higher risk/loss aversion. We propose a procedure for optimizing the composition of the risky portion of the portfolio to maximize the Sortino(γ) ratio. In addition, we show that Sortino(γ) is consistent with first and second-order stochastic dominance with riskless asset rules.

Sparse spanning portfolios and under-diversification with second-order stochastic dominance

Sparse spanning portfolios and under-diversification with second-order stochastic dominance

Green Certificate-Driven Photovoltaic Promotion in Distribution Networks Hosting Hydrogen Fueling Stations for Future Sustainable Transportation: A Risk-Adjusted Dominance Analysis

The growing number of hydrogen vehicles (HVs) has necessitated the development of hydrogen fueling stations (HFSs) to meet the hydrogen demand. This development will target environmental concerns related to electricity generation as HFSs consume power to convert electricity into hydrogen. This study focuses on the optimal risk-aware scheduling problem of a distributed network highly penetrated with photovoltaic (PV) resources. The model addresses the optimal operation of HFs under time-of-use, demand response, and multi-market mechanisms with an expanded role for PV generation under the green certificate (GCT) approach. This brings further environmental and economic benefits, as there is a growing global emphasis on the shift to a low-carbon economy. However, the uncertainties arising from PV operation, HVs’ demand, electricity load, and market prices, potentially affect the decision-maker's ability under the risky conditions. Though second-order stochastic dominance (STD) is implemented for risk management. Results show that applying the GCT method increases 5% (from 0.52 to 0.61 MW) of renewable generation and reduces 23% (300 kg) of CO2 emissions. As the conservativity of decision-makers enhances, 10% of further operation costs are imposed on the system. Results indicate that next to curbing CO2 emissions, the flexibility and robustness of the system can be improved.

СТОХАСТИЧЕСКОЕ ДОМИНИРОВАНИЕ В АКЦИЯХ ИННОВАЦИОННЫХ КОМПАНИЙ НА РОССИЙСКОМ ФОНДОВОМ РЫНКЕ

During the period of significant economic changes, the role of the state in the economy is strengthening, while the need for efficient resource allocation remains, which is characteristic of the market mechanism. The criterion for determining preferences based on second-order stochastic dominance, considered in the paper, allows the planner to identify the most preferred innovative companies for conducting appropriate policies. The result of the study is a hierarchy of preferences for a set of innovative companies represented on the Russian stock market. The article presents a calculation algorithm for series of different lengths generated by a stochastic process. An overview of the organized capital market for innovative companies is presented.

On the unfairness of actuarial fair annuities

AbstractActuarial fairness pertains to the situation in which the price of an insurance contract is equal to its expected outcome. We show that actuarial fairness leads to “unfairness” in that annuitants with higher survival rates can choose a better payoff in the sense of second-order stochastic dominance than those with lower survival rates. To deal with this issue, we propose equal utility pricing, i.e., we determine prices such that all contracts have the same (nonlinear) utility from the viewpoint of a third party (e.g., a social planner). This approach is of particular relevance with respect to the design of group self-annuitization schemes.

Risk-Aware Battery Bidding With a Novel Benchmark Selection Under Second-Order Stochastic Dominance

This paper studies the risk management of a battery bidding in both day-ahead and intraday markets arising from the uncertain nature of electricity prices. To this end, a coherent risk measure, Second-order Stochastic Dominance (SSD), which is capable of expressing battery preferences in the form of a preset fixed benchmark (profit), is incorporated into the bidding model. The SSD serves the decision-maker as a risk-averse optimizer exploring for profit distribution members greater than a preset fixed benchmark. The most challenging facet of SSD-constrained methodologies is how to effectually define the preset fixed benchmark. In this regard, first, a generic approach is offered to find the feasible region for benchmark selection in SSD-constrained optimization problems. Then, a novel benchmark selection technique considering both the decision-maker's regret and out-of-sample profit, leverages the VIKOR method to get the ranking of different solutions and find the compromise benchmark in the risk-aware environment. Consequently, two decisive criteria from both ex- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ante</i> and ex- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">post</i> tests are involved in the benchmark selection procedure, making the bidding problem regret- and consequence-aware. The numerical results of the developed methodology against risk-neutral and deterministic approaches show the efficiency of the proposed model.

Representable Markov categories and comparison of statistical experiments in categorical probability

Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely used way of comparing probability distributions by their spread. Furthermore, we lay the foundation for the theory of comparing statistical experiments within Markov categories by stating and proving the classical Blackwell–Sherman–Stein Theorem. Our version not only offers new insight into the proof, but its abstract nature also makes the result more general, automatically specializing to the standard Blackwell–Sherman–Stein Theorem in measure-theoretic probability as well as a Bayesian version that involves prior-dependent garbling. Along the way, we define and characterize representable Markov categories, within which one can talk about Markov kernels to or from spaces of distributions. We do so by exploring the relation between Markov categories and Kleisli categories of probability monads.