Multivariate Stochastic Dominance Research Articles

Multistage stochastic dominance: an application to pension fund management

AbstractA pension fund manager typically decides the allocation of the pension fund assets taking into account a long-term sustainability goal. Many asset and liability management models, in the form of multistage stochastic programming problem, have been proposed to help the pension fund manager to define the optimal allocation given a multi-objective function. The recent literature proposes univariate stochastic dominance constraints to guarantee that the optimal strategy is able to stochastically dominate a benchmark portfolio. In this work we extend previous results (i) considering alternative types of multivariate stochastic dominance that appear more suitable in a multistage framework, (ii) proposing a way to measure the economic cost of introducing stochastic dominance constraints, (iii) proposing a sort of augmented stochastic dominance through a safety margin. Numerical results show the difference between the alternative ways to interpret and apply the multivariate stochastic dominance. These results are evaluated thanks to the proposed economic cost of the stochastic dominance constraints and either in presence or not of a safety margin.

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.

Using multivariate stochastic dominance to enhance portfolio selection and warn of financial crises

In this paper, we develop a portfolio optimization methodology that significantly improves upon conventional portfolio selection problems. In particular, we propose a two-step optimization problem where we first select the efficient assets based on alternative parametric assumptions and then maximize portfolio wealth using well-known performance measures. In the first optimization step, we employ multivariate stochastic dominance conditions to determine efficient assets for a class of non-satiable risk-averse investors. In the second optimization step, we maximize either reward-risk or drawdown-based performance measures on the selected assets. With this two-step optimization, we propose an early-warning system for stock market crises based on the information contained in the joint-losses of financial assets. We compare the performance of the proposed strategies with their conventional counterparts when the second optimization step is directly adopted on all assets. Empirical analyses of the US market validate the suggested approaches and highlight the implications of financial crises for portfolio selection problems. The results confirm that the proposed methodologies substantially improve upon the conventional approach for out-of-sample portfolios, from which managerial insights are drawn.

Multivariate stochastic dominance applied to sector-based portfolio selection

AbstractIn this study, we investigate whether sector-weighted portfolios based on alternative parametric assumptions are consistent with multivariate stochastic dominance (MSD) conditions for a class of non-satiable risk-averse investors. Focusing specifically on stable symmetric and Student’s t distributions, we propose and motivate an MSD rule to determine a partial order among sectors, based on a comparison between (i) location, (ii) dispersion parameters and (iii) either stability indices (for stable symmetric distributions) or degrees of freedom (for Student’s t distributions). The proposed MSD rule is applied to the US equity market to evaluate whether and how the derived stochastic dominance conditions are relevant to investors’ decisions. The empirical study confirms that the proposed MSD rule is effective and that the tail behaviour of returns is relevant to the optimization of portfolios for non-satiable investors.

A REVIEW OF STOCHASTIC DOMINANCE METHODS FOR POVERTY ANALYSIS

AbstractStochastic dominance techniques have been mainly employed in poverty analyses to overcome what it is called the multiplicity of poverty indices problem. Moreover, in the multidimensional context, stochastic dominance techniques capture the possible relationships between the dimensions of poverty as they rely upon their joint distribution, unlike most multidimensional poverty indices, which are only based on marginal distributions. In this paper, we first review the general definition of unidimensional stochastic dominance and its relationship with poverty orderings. Then we focus on the conditions of multivariate stochastic dominance and their relationship with multidimensional poverty orderings, highlighting the additional difficulties that the multivariate setting involves. In both cases, we focus our discussion on first‐ and second‐order dominance, though some guidelines on higher order dominance are also mentioned. We also present an overview of some relevant empirical applications of these methods that can be found in the literature in both univariate and multivariate contexts.

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.

Multivariate stochastic dominance for multivariate normal distribution

Multivariate stochastic dominance for multivariate normal distribution

Multistage portfolio optimization with multivariate dominance constraints

In this article we focus on multistage portfolio optimization problem with usage of multivariate stochastic dominance constraints. The first part of the work is devoted to the theoretical background needed for establishing the optimization problem. We provide a general approach to multivariate stochastic dominance, we mainly recall basic definitions and several statements that are relevant for portfolio optimization, and describe an algorithm for detecting multivariate stochastic dominance of two random vectors with discrete distributions. The main part of the work introduces multivariate stochastic dominance constraints in multistage portfolio optimization problem. We compare obtained results with traditional model which employs univariate first order stochastic dominance constraints at each stage.

Strong and Weak Multivariate First-Order Stochastic Dominance

In this article we deal with three types of multivariate first-order stochastic dominance which serve for comparing random vectors. The first one is the strongest and it is generated by all non-decreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance is defined by comparison of cumulative distribution functions of considered random vectors. The last one applies the univariate stochastic dominance notion to linear combinations of marginals. We compare these multivariate stochastic dominance relations among each other. Then we present important properties of strong and weak multivariate first-order stochastic dominance, in particular we describe their generators in the sense of von Neumann-Morgenstern utility functions and we explain their relation to joint and marginal distribution functions. Moreover, we discuss formulations of strong multivariate first-order stochastic dominance between random vectors with discrete distributions. Finally, we apply this stochastic dominance relation as a constraint to multivariate and multiperiod portfolio optimization problems.

Augmented Lagrangian Methods for Solving Optimization Problems with Stochastic-Order Constraints

We investigate risk-averse stochastic optimization problems with a risk-shaping constraint in the form of a stochastic-order relation. Both univariate and multivariate orders are considered. We extend the notion of a linear multivariate order, adding flexibility with respect to the controlled portion of the distributions. We propose several methods for the numerical solution of these problems based on augmented Lagrangian framework and analyze their convergence. The methods construct finite-dimensional risk-neutral approximations of the optimization problem whose solutions converge to the solution of the original problem. In the case of univariate stochastic dominance, we explore augmented Lagrangian functionals based on inverse formulations of the stochastic-order constraint. The performance of the methods is compared to other extant numerical methods and shows the numerical advantage of the augmented Lagrangian framework. The proposed numerical approach is particularly successful when applied to problems with multivariate stochastic dominance constraints.

Multivariate stochastic dominance for risk averters and risk seekers

This paper first extends some well-known univariate stochastic dominance results to multivariate stochastic dominances (MSD) for both risk averters and risk seekers, respectively, to n order for any n ≥ 1 when the attributes are assumed to be independent and the utility is assumed to be additively and separable. Under these assumptions, we develop some properties for MSD for both risk averters and risk seekers. For example, we prove that MSD are equivalent to the expected-utility maximization for both risk averters and risk seekers, espectively. We show that the hierarchical relationship exists for MSD. We establish some dual relationships between the MSD for risk averters and risk seekers. We develop some properties for non-negative combinations and convex combinations random variables of MSD and develop the theory of MSD for the preferences of both risk averters and risk seekers on diversification. At last, we discuss some MSD relationships when attributes are dependent and discuss the importance and the use of the results developed in this paper.

Asymptotic Multivariate Dominance: A Financial Application

We propose a multivariate stochastic dominance relation aimed at ranking different financial markets/sectors from the point of view of a non-satiable risk averse investor. In particular, we assume that the vector of returns of a given market is in the domain of attraction of a symmetric stable Paretian law in order to take into account the asymptotic behaviour of the financial returns. We determine the stochastic dominance rule for stable symmetric distributions, where the stability parameter plays a crucial role. Consequently, the multivariate rule for ordering markets is based on a comparison between i) location parameters, ii) dispersion parameters, and iii) stability indices. Finally, we apply the method to the equity markets of the four countries with the highest gross domestic product in 2013, namely, the US, China, Japan and Germany. In this empirical comparison we examine the ex ante and ex post dominance between stock markets, either assuming that the returns are jointly (or conditionally, for a robust approach) Gaussian distributed, or in the domain of attraction of a stable sub-Gaussian law.

A solution to aggregation and an application to multidimensional ‘well-being’ frontiers

We propose a new technique for identification and estimation of aggregation functions in multidimensional evaluations and multiple indicator settings. These functions may represent “latent” objects. They occur in many different contexts, for instance in propensity scores, multivariate measures of well-being and the related analysis of inequality and poverty, and in equivalence scales. Technical advances allow nonparametric inference on the joint distribution of continuous and discrete indicators of well-being, such as income and health, conditional on joint values of other continuous and discrete attributes, such as education and geographical groupings. In a multi-attribute setting, “quantiles” are “frontiers” that define equivalent sets of covariate values. We identify these frontiers nonparametrically at first. Then we suggest “parametrically equivalent” characterizations of these frontiers that reveal likely weights for, and substitutions between different attributes for different groups, and at different quantiles. These estimated parametric functionals are “ideal” aggregators in a certain sense which we make clear. They correspond directly to measures of aggregate well-being popularized in the earliest multidimensional inequality measures in Maasoumi (1986). This new approach resolves a classic problem of assigning weights to multiple indicators such as dimensions of well-being, as well as empirically incorporating the key component in multidimensional analysis, the relationship between the indicators. It introduces a new way for robust estimation of “quantile frontiers”, allowing “complete” assessments, such as multidimensional poverty measurements. In our substantive application, we discover extensive heterogeneity in individual evaluation functions. This leads us to perform robust, weak uniform rankings as afforded by tests for multivariate stochastic dominance. A demonstration is provided based on the Indonesian data analyzed for multidimensional poverty in Maasoumi and Lugo (2008).

Bi-Objective Multi-Mode Project Scheduling Under Risk Aversion

The paper proposes a model for stochastic multi-mode resource-constrained project scheduling under risk aversion with the two objectives makespan and cost. Activity durations and costs are assumed as uncertain and modeled as random variables. For the scheduling part of the decision problem, the class of early-start policies is considered. In addition to the schedule, the assignment of execution modes to activities has to be selected. To take risk aversion into account, the approach of optimization under multivariate stochastic dominance constraints, recently developed in other fields, is adopted. For the resulting bi-objective stochastic integer programming problem, the Pareto frontier is determined by means of an exact solution method, incorporating a branch-and-bound technique based on the forbidden set branching scheme from stochastic project scheduling. Randomly generated test instances, partially derived from a test case from the PSPLIB, are used to show the computational feasibility of the approach.

Optimization with Multivariate Stochastic Dominance Constraints

We consider risk-averse stochastic optimization problems with a risk-shaping constraint in the form of a multivariate stochastic order constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector in the sense of the linear stochastic dominance of second order. We refine the optimality conditions for problems with this type of constraint by using atomic measures. Additionally, we propose a primal and a dual numerical method for solving the problem and formulate sufficient conditions for their convergence. Numerical experience and comparisons to other approaches are provided.

Models and formulations for multivariate dominance-constrained stochastic programs

The use of a stochastic dominance constraint to specify risk preferences in a stochastic program has been recently proposed in the literature. Such a constraint requires the random outcome resulting from one’s decision to stochastically dominate a given random comparator. These ideas have been extended to problems with multiple random outcomes, using the notion of positive linear stochastic dominance. This article proposes a constraint using a different version of multivariate stochastic dominance. This version is natural due to its connection to expected utility maximization theory and relatively tractable. In particular, it is shown that such a constraint can be formulated with linear constraints for the second-order dominance relation and with mixed-integer constraints for the first-order relation. This is in contrast with a constraint on second-order positive linear dominance, for which no efficient algorithms are known. The proposed formulations are tested in the context of two applications: budget allocation in a setting with multiple objectives and finding radiation treatment plans in the presence of organ motion.

Conditions for the most robust multidimensional poverty comparisons using counting measures and ordinal variables

A natural concern with multivariate poverty measures, as well as with other composite indices, is the robustness of their ordinal comparisons to changes in the indices’ parameter values. Applying multivariate stochastic dominance techniques, this paper derives the distributional conditions under which a multidimensional poverty comparison based on the popular counting measures, and ordinal variables, is fully robust to any values of the indices’ parameters. As the paper shows, the conditions are relevant to most of the multidimensional poverty indices in the literature, including the Alkire–Foster family, upon which the UNDP’s “Multidimensional Poverty Index” (MPI) is based. The conditions are illustrated with an example from the EU-SILC data set.

Stochastic Programming with Multivariate Second Order Stochastic Dominance Constraints with Applications in Portfolio Optimization

In this paper we study optimization problems with multivariate stochastic dominance constraints where the underlying functions are not necessarily linear. These problems are important in multicriterion decision making, since each component of vectors can be interpreted as the uncertain outcome of a given criterion. We propose a penalization scheme for the multivariate second order stochastic dominance constraints. We solve the penalized problem by the level function methods, and a modified cutting plane method and compare them to the cutting surface method proposed in the literature. The proposed numerical schemes are applied to a generic budget allocation problem and a real world portfolio optimization problem.

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.

Multidimensional Poverty Analysis: Looking for a Middle Ground

Widespread agreement that poverty is a multifaceted phenomenon encompassing deprivations in multiple dimensions clashes with the vociferous disagreement about how best to measure these deprivations. Drawing on the recent literature, this short paper reviews three methodological alternatives to the false dichotomy between scalar indices of multidimensional poverty, on the one hand, and a dashboard approach that considers only marginal distributions, on the other. These alternatives include simple Venn diagrams of the overlap of deprivations across dimensions, multivariate stochastic dominance analysis, and the analysis of copula functions, which capture the extent of interdependency across dimensions. Examples are provided from the literature on both developing and developed countries.