This paper contains an empirical analysis that studies trade-offs among risk, return, and climate risk in asset management. Using a multi-criteria optimization approach to generate nondominated portfolios in a tri-criterion context, we document how it is possible in a portfolio to reduce climate risk substantially by allowing expected return to be reduced only slightly. The empirical tests conducted use the sample of stocks that were in the S P 500 over the period 2001–2020. In demonstrating the versatility of our approach, six different linear measures of climate risk are employed.

Clearly arranged visualizations are needed in multiobjective optimization problems with a large number of objective functions, when a large number of Pareto optimal outcome vectors (vectors of objective function values) must be compared during the decision making processes. This paper contributes to visualizing such outcome vectors independent of how they have been generated. Parallel coordinate plots are a widely used visualization technique to represent different outcome vectors. We propose a novel visualization technique called SCORE bands to be used with parallel coordinate plots to support the decision maker in simultaneously identifying patterns in outcome vectors and correlations among the objective functions in a meaningful way. To do so, amongst others, we change the ordering of objective functions and modify the distances among them in parallel coordinate plots. SCORE bands also have interactive capabilities allowing the decision maker to first study general trends among the outcome vectors as bands and then zoom-in and move about different groups of outcome vectors of interest. The novelty of our approach lies in proposing a visually appealing way to support the decision maker in dealing with large amounts of information. We demonstrate the benefits of SCORE bands with different examples.

Computing efficient sets has long been a topic in multiple-objective optimization and research has made substantial progress. However, there are still limitations in the multiple-objective portfolio selection and optimization areas. Firstly, researchers typically focus on models containing only one quadratic objective. Secondly, few researchers pursue multiple quadratic objectives, but their algorithms could be relatively elusive and it could be a pity that they do not explicitly demonstrate the efficient sets’ structure. Lastly, researchers mostly limit their scope to three objectives. Within this context, this paper makes theoretical contributions to the literature. Operating with multiple quadratic objectives, we analytically derive closed-form formulae for the computation of the properly efficient and weakly efficient sets of problems and demonstrate the efficient sets’ structure in the form of a sequence of pyramids in decision space. Although we are restricted to equality-constraint-only models, our results have implications for general-constraint models. In addition, our methods can be extended to general k -quadratic objective models.

In this paper, we demonstrate a completely new approach for computing cardinality constrained meanvariance efficient frontiers. By cardinality constrained, it is meant that if there is to be investment in a security, it is to be of at least some minimum amount (a buyin threshold), and that there is also a specification on the number of securities to be held in a portfolio (called a cardinality constraint). Whereas the usual strategy, as such problems are NP -hard, is to take the original exact problem and apply heuristics to solve, in this paper the strategy is to perturb the original problem and then apply exact procedures to solve. The advantages of the approach are that the perturbations are tiny, they are only applied to the problem's correlation matrix, and they allow for the accurate computation of cardinality constrained efficient frontiers in problems with up to at least 10 0 0 securities in remarkably little time. Moreover, the simplicity of the approach is such that it can be inserted into existing portfolio management systems without requiring any re-training beyond what a typical portfolio analyst would already know. 1 (c) 2023 Elsevier B.V. All rights reserved.

The paper focuses on investors whose strength of interest in sustainability issues (such as environmental, social, and governance) causes ESG to become a third criterion alongside risk and return in portfolio selection. This causes the efficient frontier to become an efficient surface. This means that an investor's optimal portfolio is no longer the point of most preferred risk/return tradeoff on the mean-variance (M-V) efficient frontier, but is the point of most preferred risk/return/ESG tradeoff on the investor's M-V-ESG efficient surface. However, to find such a point requires non-trivial ESG integration which is the name given to the process of integrating ESG into the portfolio construction process after screening. With the third objective transporting the problem into 3D-space, it is difficult to search the efficient surface in any kind of comprehensive fashion using M-V based or other bi-criterion techniques as this is akin to a 2 -dimensional being trying to view a 3-dimensional object. To remedy the situation, the paper proposes a tri-criterion approach that computes efficient surfaces and special non-contour curves (called NC-efficient fronts in the paper) that are stretched across the efficient surface so as to dragnet it for the points of best ESG integration within it. Using the methodology and data from the S&P500, the paper conducts computational tests on problems with up to 500 securities and under different constraint conditions so as to know what to expect from the new approach over a range of situations.(c) 2022 Elsevier B.V. All rights reserved.

Our purpose in this paper is to develop an integrated multicriteria evaluation methodology for assessing the impact of COVID-19 in the 27 countries of the European Union. Initially, a specialized and comprehensive set of normalized criteria metrics that capture several dimensions of the pandemic, such as infection, mortality, recovery and testing rates, is carefully specified. Then, by means of a well-established and intuitive weighting system, which directly takes into consideration the experts’ preferences, the gravity of each criterion is determined properly. Next, two of the most popular multicriteria ranking techniques, i.e. the TOPSIS and the PROMETHEE II, are simultaneously exploited in order to derive and integrate the obtained evaluations. Moreover, the value of the suggested decision support system is enriched through the introduction of a novel element in the field of multicriteria analysis, i.e. the ‘2-dimensions evaluation plane’, a data visualization concept which provides rich information to the decision maker, by fruitfully blending the ranking results of the utilized multicriteria methods. The validity of the proposed approach is verified through an indicative illustrative application on Coronavirus data for the European Union countries, deriving a wide spectrum of insightful conclusions. Finally, the flexibility of the suggested framework is also stressed, since it can be fully customized, both in terms of the selected criteria and its weights, and run repetitively under a specific time-step frequency, according to the evolution dynamics and implications of the underlying pandemic.

Robust portfolio optimization refers to finding an asset allocation strategy whose behavior under the worst possible realizations of the uncertain inputs, e.g., returns and covariances, is optimized. The robust approach is in contrast to the classical approach, where one estimates the inputs to a portfolio allocation problem and then treats them as certain and accurate. In this paper we provide a categorized bibliography on the application of robust mathematical programming to the portfolio selection problem. With no similar surveys available, one of the aims of this review is to provide quick access for those interested, but maybe not yet in the area, so they know what the area is about, what has been accomplished and where everything can be found. Toward this end, a total of 148 references have been compiled and classified in various ways. Additionally, the number of Scopus© citations by contribution and journal is recorded. Finally, a brief discussion of the review’s major findings is provided and some solid leads on future directions are given.

This paper describes an approach for markedly reducing the time required to obtain all efficient extreme points of a multiple objective linear program (MOLP) with three objectives. The approach is particularly useful when working with such MOLPs possessing large numbers of efficient extreme points. By subdividing the criterion cone into sub-cones, the paper shows how the task of computing all efficient extreme points can be broken down into parts so that the parts can be solved concurrently, thus allowing all efficient extreme points to be computed in much reduced elapsed time. The paper investigates several schemes for conducting this task and reports on a volume of computational experience. (C) 2019 Published by Elsevier B.V.

This special issue comprises extended versions of selected papers presented during the 11th International Conference on Multiple Objective Programming and Goal Programming (MOPGP'15), held from 13 to 15 December 2015 in Tlemcen, Algeria, as well as papers that were not presented during the conference.The MOPGP is a scientific forum within which researchers and practitioners can meet and learn from each other about recent developments in multi-objective decision making and multi-criteria decision aid and their applications.MOPGP aims to disseminate this knowledge, encouraging interest and training doctorate candidates in this field of study and promoting rigorous use of multi-objective decision making and multi-criteria decision aid tools in solving real decision-making situations.The topics discussed in this special issue include, but are not limited to, all areas of multiple objective optimization (MOP), in particular those addressing managerial problems:• theories and applications of MOP/GP • fuzzy sets, soft computing and MOP/GP • neural networks, meta-heuristics and MOP/GP • interactive MOP/GP methods • evolutionary MOP • discrete and combinatorial MOP/GP • stochastic MOP/GP • fuzzy MOP/GP • dynamic MOP/

In 1952, Markowitz published his famous paper on portfolio selection that transformed the field of finance. Although over 65 years have passed since then, the mean-variance model remains today the predominant model in portfolio selection. Having endured many criticisms over this period, the one that has perhaps been the most persistent is the fact that mainstream mean-variance theory is unable to accommodate additional criteria beyond expected return and variance. With investment decision-making having become more complex, this is a real problem as many problems with additional criteria exist and are only increasing in number and importance. In this paper, we review the papers that have been published that apply methods and procedures in an exact (as opposed to evolutionary) sense to address problems in portfolio selection with criteria beyond mean and variance. We also analyse the methodologies that allow the solution of the problem in a multiple criteria context, thus extending the features of the mean-variance approach that have caused portfolio theory to have such impact.

This paper provides results in the area of the analytical derivation of the efficient set of a mean-variance portfolio selection problem that has more than three criteria. By “analytical” we mean derived by formula as opposed to being computed by algorithm. By “more than three criteria”, we mean that beyond the mean and variance of regular portfolio selection, the problems addressed have two or more additional linear objectives. The additional objectives might include sustainability, dividend yield, liquidity, and R&D as extra objectives like these are being seen with greater frequency. While not all multiple criteria portfolio selection problems lend themselves to an analytical derivation, a certain class does and the problems in this class are covered by the mathematics of this paper.

In standard mean-variance bi-criterion portfolio selection, the efficient set is a frontier. While it is not yet standard for there to be additional criteria in portfolio selection, there has been a growing amount of discussion in the literature on the topic. However, should there be even one additional criterion, the efficient frontier becomes an efficient surface. Striving to parallel Merton’s seminal analytical derivation of the efficient frontier, in this paper we provide an analytical derivation of the efficient surface when an additional linear criterion (on top of expected return and variance) is included in the model addressed by Merton. Among the results of the paper there is, as a higher dimensional counterpart to the 2-mutual-fund theorem of traditional portfolio selection, a 3-mutual-fund theorem in tri-criterion portfolio selection. 3D graphs are employed to stress the paraboloidic/hyperboloidic structures present in tri-criterion portfolio selection.

Because of size and covariance matrix problems, computing much of anything along the nondominated frontier of a large-scale (1000–3000 securities) portfolio selection problem with semi-continuous variables is a task that has not previously been achieved. But given (a) the speed at which the nondominated frontier of a classical portfolio problem can now be computed and (b) the possibility that there might be overlaps between the nondominated frontier of the classical problem and that of the same problem but with semi-continuous variables, the paper shows how considerable amounts of the nondominated frontier of a large-scale mean-variance portfolio selection problem with semi-continuous variables can be computed in very little time.

Despite many proposed alternatives, the predominant model in portfolio selection is still mean-variance. However, the main weakness of the mean-variance model is in the specification of the expected returns of the individual securities involved. If this process is not accurate, the allocations of capital to the different securities will in almost all certainty be incorrect. If, however, this process can be made accurate, then correct allocations can be made, and the additional expected return following from this is the value of information. This paper thus proposes a methodology to calculate the value of information. A related idea of a level of disappointment is also shown. How value of information calculations can be important in helping a mutual fund settle on how much to set aside for research is discussed in reference to a Taiwan Stock Exchange illustrative application in which the value of information appears to be substantial. Heavy use is made of parametric quadratic programming to keep computation times down for the methodology. (C) 2016 Elsevier B.V. All rights reserved.

One of the most important factors shaping world outcomes is where investment dollars are placed. In this regard, there is the rapidly growing area called sustainable investing where environmental, social, and corporate governance (ESG) measures are taken into account. With people interested in this type of investing rarely able to gain exposure to the area other than through a mutual fund, we study a cross section of U.S. mutual funds to assess the extent to which ESG measures are embedded in their portfolios. Our methodology makes heavy use of points on the nondominated surfaces of many tri-criterion portfolio selection problems in which sustainability is modeled, after risk and return, as a third criterion. With the mutual funds acting as a filter, the question is: How effective is the sustainable mutual fund industry in carrying out its charge? Our findings are that the industry has substantial leeway to increase the sustainability quotients of its portfolios at even no cost to risk and return, thus implying that the funds are unnecessarily falling short on the reasons why investors are investing in these funds in the first place. (C) 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.

Over 60 years ago, Markowitz introduced the mean-variance efficient frontier to finance. While mean-variance is still the predominant model in portfolio selection, it has endured many criticisms. One serious one is that it does not allow for additional criteria. The difficulty is that the efficient frontier becomes a surface. With it now possible to compute such a surface, we provide an overview on how Markowitz’s risk-return (bi-criterion) portfolio selection can be extended to tri-criterion portfolio selection. With a focus on the geometry of the extension, many graphs are used to illustrate.

We present a framework for inverse optimization in a Markowitz portfolio model that is extended to include a third criterion. The third criterion causes the traditional nondominated frontier to become a surface. Until recently, it had not been possible to compute such a surface. But by using a new method that is able to generate the nondominated surfaces of tri-criterion portfolio selection problems, we are able to compute via inverse optimization the implied risk tolerances of given funds that pursue an additional objective beyond risk and return. In applying this capability to a broad sample of conventional and socially responsible (SR) mutual funds, we find that there appears to be no significant evidence that social responsibility issues, after the screening stage, are further taken into account in the asset allocation process, which is a result that is likely to be different from what many SR investors would expect. (C) 2013 Elsevier B.V. All rights reserved.