Tri-Criterion Model for Constructing Low-Carbon Mutual Fund Portfolios: A Preference-Based Multi-Objective Genetic Algorithm Approach

[EN] Sustainable finance, which integrates environmental, social and governance criteria on financial decisions rests on the fact that money should be used for good purposes. Thus, the financial sector is also expected to play a more important role to decarbonise the global economy. To align financial flows with a pathway towards a low-carbon economy, investors should be able to integrate into their financial decisions additional criteria beyond return and risk to manage climate risk. We propose a tri-criterion portfolio selection model to extend the classical Markowitz's mean-variance approach to include investor's preferences on the portfolio carbon risk exposure as an additional criterion. To approximate the 3D Pareto front we apply an efficient multi-objective genetic algorithm called ev-MOGA which is based on the concept of epsilon-dominance. Furthermore, we introduce a-posteriori approach to incorporate the investor's preferences into the solution process regarding their climate-change related preferences measured by the carbon risk exposure and their loss-adverse attitude. We test the performance of the proposed algorithm in a cross-section of European socially responsible investments open-end funds to assess the extent to which climate-related risk could be embedded in the portfolio according to the investor's preferences.Hilario Caballero, A.; Garcia-Bernabeu, A.; Salcedo-Romero-De-Ávila, J.; Vercher, M. (2020). Tri-Criterion Model for Constructing Low-Carbon Mutual Fund Portfolios: A Preference-Based Multi-Objective Genetic Algorithm Approach. International Journal of Environmental research and Public Health. 17(17):1-15. https://doi.org/10.3390/ijerph17176324S1151717Morningstar Low Carbon Designationhttps://bit.ly/2SfAFUAKrueger, P., Sautner, Z., & Starks, L. T. (2020). The Importance of Climate Risks for Institutional Investors. The Review of Financial Studies, 33(3), 1067-1111. doi:10.1093/rfs/hhz137Syam, S. S. (1998). A dual ascent method for the portfolio selection problem with multiple constraints and linked proposals. European Journal of Operational Research, 108(1), 196-207. doi:10.1016/s0377-2217(97)00048-9Li, D., Sun, X., & Wang, J. (2006). OPTIMAL LOT SOLUTION TO CARDINALITY CONSTRAINED MEAN-VARIANCE FORMULATION FOR PORTFOLIO SELECTION. Mathematical Finance, 16(1), 83-101. doi:10.1111/j.1467-9965.2006.00262.xBertsimas, D., & Shioda, R. (2007). Algorithm for cardinality-constrained quadratic optimization. Computational Optimization and Applications, 43(1), 1-22. doi:10.1007/s10589-007-9126-9Bawa, V. S. (1975). Optimal rules for ordering uncertain prospects. Journal of Financial Economics, 2(1), 95-121. doi:10.1016/0304-405x(75)90025-2Konno, H., & Yamazaki, H. (1991). Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science, 37(5), 519-531. doi:10.1287/mnsc.37.5.519Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443-1471. doi:10.1016/s0378-4266(02)00271-6Mansini, R. (2003). LP solvable models for portfolio optimization: a classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220. doi:10.1093/imaman/14.3.187Hirschberger, M., Steuer, R. E., Utz, S., Wimmer, M., & Qi, Y. (2013). Computing the Nondominated Surface in Tri-Criterion Portfolio Selection. Operations Research, 61(1), 169-183. doi:10.1287/opre.1120.1140Utz, S., Wimmer, M., Hirschberger, M., & Steuer, R. E. (2014). Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds. European Journal of Operational Research, 234(2), 491-498. doi:10.1016/j.ejor.2013.07.024Utz, S., Wimmer, M., & Steuer, R. E. (2015). Tri-criterion modeling for constructing more-sustainable mutual funds. European Journal of Operational Research, 246(1), 331-338. doi:10.1016/j.ejor.2015.04.035Chang, T.-J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302. doi:10.1016/s0305-0548(99)00074-xMaringer, D., & Kellerer, H. (2003). Optimization of cardinality constrained portfolios with a hybrid local search algorithm. OR Spectrum, 25(4), 481-495. doi:10.1007/s00291-003-0139-1Shaw, D. X., Liu, S., & Kopman, L. (2008). Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optimization Methods and Software, 23(3), 411-420. doi:10.1080/10556780701722542Soleimani, H., Golmakani, H. R., & Salimi, M. H. (2009). Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm. Expert Systems with Applications, 36(3), 5058-5063. doi:10.1016/j.eswa.2008.06.007Anagnostopoulos, K. P., & Mamanis, G. (2011). The mean–variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms. Expert Systems with Applications. doi:10.1016/j.eswa.2011.04.233Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2011). Heuristic algorithms for the cardinality constrained efficient frontier. European Journal of Operational Research, 213(3), 538-550. doi:10.1016/j.ejor.2011.03.030Meghwani, S. S., & Thakur, M. (2017). Multi-criteria algorithms for portfolio optimization under practical constraints. Swarm and Evolutionary Computation, 37, 104-125. doi:10.1016/j.swevo.2017.06.005Liagkouras, K., & Metaxiotis, K. (2016). A new efficiently encoded multiobjective algorithm for the solution of the cardinality constrained portfolio optimization problem. Annals of Operations Research, 267(1-2), 281-319. doi:10.1007/s10479-016-2377-zMetaxiotis, K., & Liagkouras, K. (2012). Multiobjective Evolutionary Algorithms for Portfolio Management: A comprehensive literature review. Expert Systems with Applications, 39(14), 11685-11698. doi:10.1016/j.eswa.2012.04.053Silva, Y. L. T. V., Herthel, A. B., & Subramanian, A. (2019). A multi-objective evolutionary algorithm for a class of mean-variance portfolio selection problems. Expert Systems with Applications, 133, 225-241. doi:10.1016/j.eswa.2019.05.018Chang, T.-J., Yang, S.-C., & Chang, K.-J. (2009). Portfolio optimization problems in different risk measures using genetic algorithm. Expert Systems with Applications, 36(7), 10529-10537. doi:10.1016/j.eswa.2009.02.062Liagkouras, K. (2019). A new three-dimensional encoding multiobjective evolutionary algorithm with application to the portfolio optimization problem. Knowledge-Based Systems, 163, 186-203. doi:10.1016/j.knosys.2018.08.025Kaucic, M., Moradi, M., & Mirzazadeh, M. (2019). Portfolio optimization by improved NSGA-II and SPEA 2 based on different risk measures. Financial Innovation, 5(1). doi:10.1186/s40854-019-0140-6Babaei, S., Sepehri, M. M., & Babaei, E. (2015). Multi-objective portfolio optimization considering the dependence structure of asset returns. European Journal of Operational Research, 244(2), 525-539. doi:10.1016/j.ejor.2015.01.025Ruiz, A. B., Saborido, R., Bermúdez, J. D., Luque, M., & Vercher, E. (2019). Preference-based evolutionary multi-objective optimization for portfolio selection: a new credibilistic model under investor preferences. Journal of Global Optimization, 76(2), 295-315. doi:10.1007/s10898-019-00782-1Anagnostopoulos, K. P., & Mamanis, G. (2010). A portfolio optimization model with three objectives and discrete variables. Computers & Operations Research, 37(7), 1285-1297. doi:10.1016/j.cor.2009.09.009Hu, Y., Chen, H., He, M., Sun, L., Liu, R., & Shen, H. (2019). Multi-Swarm Multi-Objective Optimizer Based on p-Optimality Criteria for Multi-Objective Portfolio Management. Mathematical Problems in Engineering, 2019, 1-22. doi:10.1155/2019/8418369Rangel-González, J. A., Fraire, H., Solís, J. F., Cruz-Reyes, L., Gomez-Santillan, C., Rangel-Valdez, N., & Carpio-Valadez, J. M. (2020). Fuzzy Multi-objective Particle Swarm Optimization Solving the Three-Objective Portfolio Optimization Problem. International Journal of Fuzzy Systems, 22(8), 2760-2768. doi:10.1007/s40815-020-00928-4Garcia-Bernabeu, A., Salcedo, J. V., Hilario, A., Pla-Santamaria, D., & Herrero, J. M. (2019). Computing the Mean-Variance-Sustainability Nondominated Surface by ev-MOGA. Complexity, 2019, 1-12. doi:10.1155/2019/6095712Laumanns, M., Thiele, L., Deb, K., & Zitzler, E. (2002). Combining Convergence and Diversity in Evolutionary Multiobjective Optimization. Evolutionary Computation, 10(3), 263-282. doi:10.1162/106365602760234108Matlab Central: ev-MOGA Multiobjective Evolutionary Algorithmhttps://bit.ly/3f2BYQMBlasco, X., Herrero, J. M., Sanchis, J., & Martínez, M. (2008). A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization. Information Sciences, 178(20), 3908-3924. doi:10.1016/j.ins.2008.06.01

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Exact Solution Methods for the kk-item Quadratic Knapsack Problem

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Comment: 12 page