Tri-criterion model for constructing low-carbon mutual fund portfolios: a preference-based multi-objective genetic algorithm approach
A. Hilario-Caballero, A. Garcia-Bernabeu, J. V. Salcedo, M. Vercher
aa r X i v : . [ q -f i n . GN ] J un Tri-criterion model for constructing low-carbon mutual fundportfolios: a preference-based multi-objective geneticalgorithm approach
A. Hilario-Caballero , A. Garcia-Bernabeu , J. V. Salcedo , and M. Vercher Universitat Politcnica de ValnciaJune 23, 2020
Abstract
Sustainable finance, which integrates environmental, social and governance (ESG) criteria on financialdecisions rests on the fact that money should be used for good purposes. Thus, the financial sector is alsoexpected to play a more important role to decarbonise the global economy. To align financial flows witha pathway towards a low-carbon economy, investors should be able to integrate in their financial decisionsadditional criteria beyond return and risk to manage climate risk. We propose a tri-criterion portfolio selectionmodel to extend the classical Markowitz mean-variance approach in order to include investors preferences onthe portfolio carbon risk exposure as an additional criterion. To approximate the 3D Pareto front we apply ane ffi cient multi-objective genetic algorithm called ev-MOGA which is based on the concept of ǫ -dominance.Furthermore, we introduce an a posteriori approach to incorporate the investor’s preferences into the solutionprocess regarding their sustainability preferences measured by the carbon risk exposure and his / her loss-adverse attitude. We test the performance of the proposed algorithm in a cross section of European SRI open-end funds to assess the extent to which climate related risk could be embedded in the portfolio according tothe investor’s preferences. Keywords—
Genetic Algorithms; Low-Carbon Economy; Multi-objective optimization; Sustainable Finance;Investor’s preferences
Climate change will pose a challenge for the financial sector seeking a balance between purely financial goals –looking for high returns – and sustainability making a positive impact on the environment and on society. Since2015, by adopting the Paris Agreement on climate change and the UN 2030 for Sustainable Development, therehas been a clear commitment, especially in the European Union, to align financial flows with a pathway towardsa low-carbon, more resource-e ffi cient and sustainable economy. In 2018, the EU has launched and Action Planto set out a strategy for sustainable finance, that is , “the process of taking due account of environmentaland social considerations in investment decision-making, leading to increased investments in longer-term andsustainable activities” [14]. As stated in this report, to date, environmental and climate risks had not beenappropriately considered by the financial sector, which is why if the EU wants to reorient capital flows to amore sustainable economy, environmental and social goals will have to be included in the financial decision-making. To this end, the Markets in Financial Instruments Directive (MIFID II) and the
Insurance DistributionDirective (IDD) provide that investment firms and insurance distributors should ask their clients’ investmentsobjectives as regard sustainability and take their preferences into account when providing financial advice.This implies that investors should be able to integrate in their financial decisions additional criteria beyond1
Exact methods vs Multi-objective Evolutionary Algorithms for extended M-V portfolio selection: a literature review return and risk and then to extend the classical bi-criterion portfolio selection problem based on Markowitzmean-variance approach [34] by adding one more criterion.In the literature, tri-criterion portfolio selection problems have been addressed by several authors makinguse of multicriteria decision problems (MCDM). One of the first attempts to compute the variance-expectedreturn-sustainability surface was [19]. These authors, proposed an inverse portfolio optimization algorithmusing CIOS (Custom Investment Objective Solver) from the model of Markowitz and they generated a tri-criterion non dominanted surface composed of a connected collection of parabolic ”platelets”. In [42] and [43],the previous procedure was applied to construct a model that includes risk, expected return and sustainability,which is measured using ESG scores. In recent years, Multi-objective Evolutionary Algorithms (MOEAs)have been proposed to handle two or more conflicting goals subject to several constraints [3] and in particularto address complex portfolio selection problems [36]. A recent approach based on ev-MOGA [17] has beenadapted in [15] to derive the non-dominated mean-variance-sustainability surface.In line with the goals of the Paris Agreement, the financial flows should be consistent with a pathwaytowards low greenhouse gas emissions. In recent years, in response to an increasing climate-conscious financialproducts demand, Morningstar, the most important information provider in the mutual fund industry, introducedthe Low Carbon Designation eco-label [37]. This new mutual fund eco-label helps investors to easily recognizewhich mutual funds are aligned with the transition to a low-carbon economy [8]. The LCD is composed of twoindices, the Carbon risk score and the Fossil Fuel involvement. In our research we only consider the fund-levelCarbon Risk score (from ESG Sustainalytics provider) which is obtained by weighting the firm-level exposureand management of material carbon issues. As [23] highlights, institutional investors increasingly addressclimate related risk and they are also viewed as catalysing driving firms to meet the reduced emission target.Thus, the mutual fund industry, and in particular institutional investors is an ideal setting to test our proposal.Our paper makes the following contributions to the literature. First, it gives a better understanding of recentmulti-criteria decision making methodologies (MCDM) to deal with tri-criterion portfolio selection problemsby reviewing the literature on exact methods and multi-objective genetic algorithms techniques. Second, itallows us to integrate carbon risk exposure as a new objective in the portfolio optimization procedure of in-stitutional mutual funds. We then, propose a recent multi-objective genetic algorithm called ev-MOGA [17]to provide investors with the insights to make more informed decisions and to manage portfolio carbon riskexpose more e ff ectively. Third, the preferences of the decision maker are incorporated into the solution processregarding their climate or green preferences measured by the carbon risk exposure and their loss-adverse atti-tude measured by the variance of returns. Taking into account the green preferences, we define three investorprofiles: weak green investor, moderate green investor and strong green investor. Moreover, we also considertheir attitude towards risk and then, we define three types of profiles: conservative, cautious and aggressive.The rest of the paper is structured as follows. Section 2 starts with a review of the literature about thetri-criterion portfolio selection problem addressed either by exact or heuristic methodologies. In Section 3,the proposed tri-criterion genetic multi-objective evolutionary algorithm for constructing low carbon portfoliosis formulated including the a-posteriori approach to integrate sustainability preferences in financial decision-making. In Section 4, we analyze the numerical results obtained by the application of the ev-MOGA usingdi ff erent investor profiles for a data set of European Socially Responsible Investments (SRI) open-end funds.Finally, Section 5 concludes the paper. The idea of determining the Pareto e ffi cient frontier in portfolio selection from a mean-variance (M-V) opti-mization was originally conceived in [34]. The essence of the M-V model is that risk is the investor’s mainconcern and he / she tries to minimize risk for a desired level of expected returns. Over the years, the Markowitzmodel has been extended either through more complex risk measures or through additional constraints, andin recent years through the possibility to include additional objectives. In this context, two main approaches Exact methods vs Multi-objective Evolutionary Algorithms for extended M-V portfolio selection: a literature review to deal with the extended portfolio optimization problem can be found: (i) Exact methods or (ii) Heuristicmethodologies.
Since the early 1970s several authors have attempted to expand the classical bi-criteria portfolio selection modelbeyond the expected return and variance with exact methods. Three main groups of studies can be identifieddealing with this problem. A first group of authors have expanded the Markowitz model by introducing addi-tional constraints such as cardinality, round lots or buy-in threshold [41, 31, 25, 5]. Alternative risk measuressuch as down-side risk measures or CVaR have been proposed in a second group of studies [4, 22, 38]. Aliterature review on risk measures in terms of computational comparison is conducted in [30].Not until the 20th century was the idea of additional objectives was further boosted by a third group ofstudies. A tri-criterion non dominated surface can be found in [19, 42, 43] using a constrained linear program(QCLP) approach by solving a quad-lin-lin optimization problem where the third objective is linear. By definingseveral measures of liquidity in [28] a three-dimensional mean-variance-liquidity frontier is constructed. Ageneral framework for computing the non-dominated surface in tri-criterion portfolio selection that extends theMarkowitz portfolio selection approach to an additional linear criterion (dividends, liquidity or sustainability)is addressed in [19]. By solving a quad-lin-lin program, they provide an exact method for computing thenon-dominated surface that can outperform standard portfolio strategies for multicriteria decision makers. Anempirical application where the third criterion is sustainability is developed to illustrate how to compose thenon-dominated surface.In [42] sustainability is included as the third criterion to obtain the variance-expected return-sustainabilitye ffi cient frontier in order to explain how the sustainable mutual fund industry can increase its levels of sustain-ability. The tri-criterion non-dominated surface is computed through the Quadratic Constrained Linear Program(QCLP) approach, and from the experimental results it can be concluded that there was room to expand thesustainability levels without hampering the levels of risk and return.However, the existing proposals based on exact procedures to solve tri-criteria portfolio selection problemshave limited capabilities when the third objective is non-linear. In such cases, heuristic techniques have beenrecently applied to solve multi-objective problems and to provide fair approximations of the pareto front. The increasing complexity of financial decision making problems has led researchers to apply heuristic proce-dures inspired by biological processes such as Multi-objective Evolutionary Algorithms (MOEAs). Suggestedin the beginning of the 90s, MOEAs have been applied in several fields including finance, and in particularto solve the portfolio selection problem [33, 11]. These techniques provide satisfactory approximations of thee ffi cient frontier even when the problem involves non-convexity, discontinuity or non integer variables. In [3],a MOEA was proposed for the first time for optimal portfolio selection by using lower partial moments as ameasure of risk. The first attempts to propose MOEAs as an extension of the M-V model aimed at consideringadditional constrains such as, cardinality, lower and upper bounds, transaction costs, transaction round lots,non-negativity constraints or sector capitalization constraints [9, 32, 39, 5, 40, 13, 2, 44, 35, 27]. A review ofthe state of the art of MOEAs in portfolio selection can be found in [36].Another group of researchers have also tried to propose alternative risk measures to variance, the most pop-ular being: semivariance, value at risk (VaR) and conditional value at risk (CVaR), the lower partial moments(LPM), the Expected Shortfall, the Skewness, and Risk parity [16, 10, 20, 26, 21]Regarding the number of objectives, while the two-objective case is the most widely used among the au-thors, the tri-objective problem has risen in popularity in the last few years. A tri-objective optimization prob-lem is proposed in [1] to find the trade-o ff between risk, return and the number of securities in the portfolio. Inthis paper, the authors compare three evolutionary multi-objective optimization techniques for finding the besttrade-o ff between risk, return and the cardinality of the portfolio. A recent approach based on ev-MOGA [17]has been adapted in [15] to derive the non-dominated mean-variance-sustainability surface. The tri-criterion multi-objective approach by ev-MOGA to manage carbon risk exposure
During the last two decades, MOEAs for portfolio management have attracted scholars and practitioners at-tention as stated in subsection 2.2. Next, some previous notions on multi-objective optimization and geneticmulti-objective optimization techniques are provided.
Multi-objective optimization is an important subclass of multiple criteria decision making techniques involv-ing more than one objective function to be optimized simultaneously. Since the conflict degree between theobjectives makes it impossible to find a feasible solution that simultaneously optimizes all the objective func-tions, there is a set of Pareto optimal solutions denoted as Pareto front at which none of the objectives can beimproved without deteriorating at least one of the others. In general a MOP optimization problem is stated asfollows: minimize w f ( w ) = (cid:2) f ( w ) , f ( w ) , . . . , f m ( w ) (cid:3) T , subject to w ∈ S , (1)where the vector w = [ ω , ω , . . . , ω n ] T is a n -parameter set included in the decision space S , and f i ( w ) : ’ n → ’ , i = , . . . , m , are the objectives to be minimized at the same time.In recent years MOEAs have been widely accepted as useful tools for solving real world multi-objectiveproblems. Within MOEAs several powerful stochastic search techniques that mimic Darwinian principles ofnatural selection are included. In this study we focus on the ev-MOGA algorithm proposed in [17], whichcombines the concept of Pareto optimality and ǫ -dominance due to [24], thus providing an approximated ǫ -Pareto set. Definition 3.1
Dominance: Let w , w ∈ ’ n be two feasible solutions, an let f ( w ) , f ( w ) ∈ ’ m be their imagesolutions in the objective space. Then, assuming that the objective functions have to be minimized, w is saidto dominate w , denoted as f ( w ) ≺ f ( w ) , i ff : ∀ i ∈ { , . . . , m } : f i ( w ) ≤ f i ( w ) ∃ j ∈ { , . . . , m } : f j ( w ) < f j ( w ) (2) Definition 3.2
Pareto set or Pareto front: Let Ω ⊆ ’ n be a set of vectors of feasible solutions with f ( Ω ) as theirimage solutions. Then the Pareto set f ( Ω P ) of f ( Ω ) is defined as follows: f ( Ω P ) contains all vectors f ( w u ) ∈ f ( Ω ) that are not dominated by any vector f ( w v ) ∈ f ( Ω ) , i.e., f ( Ω P ) : = (cid:8) f ( w u ) ∈ f ( Ω ) | ∄ f ( w v ) : f ( w v ) ≺ f ( w u ) (cid:9) (3) Definition 3.3 ǫ –dominance: Let w , w ∈ ’ n be two feasible solutions, an let f ( w ) , f ( w ) ∈ ’ m + be theirimage solutions in the objective space. Then w is said to ǫ –dominate w for some ǫ > , denoted as f ( w ) ≺ ǫ f ( w ) , i ff : ∀ i ∈ { , . . . , m } : (1 + ǫ ) · f i ( w ) ≤ f i ( w ) (4) Definition 3.4 ǫ –approximate Pareto set: Let Ω ⊆ ’ n be a set of feasible solution vectors with f ( Ω ) as theirimage solutions. Then, f ( ˆ Ω ∗ P ) is called a ǫ –approximate Pareto set of f ( Ω ) if any vector f ( w u ) ∈ f ( Ω ) is ǫ –dominated by at least one vector f ( w v ) ∈ f ( ˆ Ω ∗ P ) , i.e., ∀ f ( w u ) ∈ f ( Ω ) : ∃ f ( w v ) ∈ f ( ˆ Ω ∗ P ) | f ( w v ) ≺ ǫ f ( w u ) (5) The set of all ǫ –approximate Pareto sets of f ( Ω ) is denoted as the ǫ –Pareto front f ( ˆ Ω P ) . The tri-criterion multi-objective approach by ev-MOGA to manage carbon risk exposure
The most outstanding feature of this algorithm is that the optimal solutions are distributed uniformly acrossthe ǫ -Pareto front. To this end, the ǫ -Pareto front is split into a fixed number of boxes forming a grid, so thatthe algorithm ensures that just one solution is stored by one box. The size of the boxes is determined by thevalue of ǫ i , which is calculated as follows: ǫ i = f i ∗ − f i ∗ n box (6)where, f ∗ i and f i ∗ correspond to the maximum and minimum value of the objective function f i , and n box is thenumber of boxes. In addition, ev-MOGA is able to adjust the width of ǫ i dynamically and prevent solutionsbelonging to the extremes of the front from being lost.For solving the ev-MOGA, the main population P ( t ) whose size is Nind p explores the searching space S defined by the multi-objective problem during a number k of iterations. In the archive population A ( t ) the ǫ i -nondominated solutions are stored, so that there are as many feasible solutions as number of boxes. Then,at the end of the iteration process, A ( t ) is an ǫ -approximate Pareto set f ( ˆ Ω ∗ P ). Furthermore, in the case thatmore than one ǫ -dominant solution is detected, thus the solution that prevails in A ( t ) will be the one that isclosest to the center of the box. Next, the new individuals obtained by crossover or mutation with probabilityof crossing / mutation P c / m are included in the auxiliary population GA ( t ).Before running the algorithm, the following parameters should be defined by the analyst: • Nind p = Size of the main population. • Nind GA = Size of the auxiliary population. • k max = Maximum algorithm iterations. • P c / m = Probability of crossing / mutation. • n box = Number of boxes.The main advantage of ev-MOGA is that they generate good approximations of a well-distributed Paretofront in a single run and within limited computational time. The original ev-MOGA algorithm is avalaible atMatlab Central [18]: ev-MOGA in Matlab Central.
In this study, beyond risk and return, we wish to consider an additional objective that minimizes the carbonrisk exposure of a portfolio. Then, by introducing a third objective into the portfolio optimization modelthe e ffi cient frontier becomes a surface in the three-dimensional space. The tri-criterion portfolio selectionproblem where the objectives are the risk of the portfolio, the returns, and the portfolio carbon risk exposurecan be mathematically formulated as follows:min f ( w ) = N X i = N X j = ω i ω j σ i j (7)max f ( w ) = N X i = ω i µ i (8)min f ( w ) = N X i = ω i c i (9)subject to N X i = ω i = N denotes the available assets, µ i is the expected return of asset i ( i = , , . . . , N ), σ i j is the covariancebetween asset i and j . In addition c i is the carbon risk score and ω i denotes the proportion of asset i in theportfolio. The tri-criterion multi-objective approach by ev-MOGA to manage carbon risk exposure
Algorithm 1
Tri-criterion ev-MOGA algorithm based on [17] Set k = Initialize the population of candidate solutions P and set A = ∅ Conduct the multi-objective evaluation of portfolios from P using Equations (7)–(10) Detect the ǫ -nondominated portfolios from P and store in the archive A while k ≤ k max do Generate the auxiliary population GA k from the main population P k and thearchive population A k following this procedure: for j ← , Nind GA / do Randomly select two portfolios X P and X A from P k and A k , respectively Generate a random number u ∈ [0 , If u > P c / m , X P and X A are crossed over by means of the extended linear recombinationtechnique, generating two new portfolios for GA k If u ≤ P c / m , X P and X A are mutated using random mutation with Gaussian distributionand then included in GA k end for Evaluate population GA k using the tri-criterion multi-objective portfolio model defined by (7)–(10). Check which portfolios in GA k must be included in A k + on the basis of their locationin the objective space. A k + will contain all the portfolios from A k that arenot ǫ -dominated by elements of GA k , and all the portfolios from GA k which arenot ǫ -dominated by elements of A k Update population P k + with portfolios from GA k . Every portfolio X GA from GA k is comparedwith a portfolio X P that is randomly selected from the portfolios in P k . X GA will replace X P in P k + if it dominates X P . Otherwise X P will not be replaced k ← k + end while With the previous multi-objective optimization design a vast region of the tri-objective whole Pareto front isgenerated. Even though it is true that the non-dominated surface allows us to better understand the trade-o ff between the three objectives, this solution doesn’t provide a useful tool from the user’s perspective. To comeup with a single solution we assume that the decision maker is available to take part in the solution process.According to [12, 7] the articulation of preferences may be done either before (a priori), during (progressive),or after (a posteriori) the optimization process. In what follows, we assume that once the investor has seenan overview of the Pareto optimal solutions, he / she takes part of the final solution. Thus, we propose ana-posteriori approach.The analyst supporting a-posteriori methodology has to inform the decision maker either providing a list ofsolutions or providing a visualization of the Pareto front [29]. In a tri-objective case, two main approaches havebeen used to visualize the Pareto frontier: (i) three-dimensional graph, and (ii) decision maps. However, a newgraphical visualization called Level Diagram is proposed in [6] to represent n-dimensional Pareto fronts. TheLevel Diagrams tool, also allows the incorporation of decision makers’ preferences and it o ff ers a good tool tohelp in the decision making process.In our proposal, information on preferences is given by the investor, who is willing to achieve a desiredaspiration level for each objective function. Let us denote the reference vector for the preferences about greeninvestments defined by the carbon risk score objective function (9) as P g and the preferences for the loss The tri-criterion multi-objective approach by ev-MOGA to manage carbon risk exposure aversion attitude defined by (7) as P r .Concerning the sustainability preferences, we consider three types of green investor profiles. They aredefined as follows:1. Weak green investor . This profile is defined by a low level of aspiration for the carbon risk score p wg .2. Moderate green investor . This profile is defined by a medium level of aspiration for the carbon risk score p mg .3. Strong green investor . This profile is defined by a high level of aspiration for the carbon risk score p sg .Thus, the reference vector for the green investor could be stated as follows: P g = h p wg , p mg , p sg i (11)Concerning the investor’s loss aversion attitude, we consider three types of investor profiles. They aredefined as follows:1. Conservative investor . This profile is characterized by investing in lower-risk securities, namely, a highloss aversion attitude p cr .2. Cautious investor . This profile is defined by a medium risk tolerance, and consequently a moderate lossaversion attitude p kr .3. Aggressive investor . It includes investors that actively seek stocks with higher riskbut a chance for higherreward, that is a low loss aversion attitude p ar .Thus, the reference vector regarding the risk aversion could be stated as follows: P r = h p cr , p kr , p ar i (12) R e t u r n s Variance
Emission Risk
Figure 1: ǫ –Pareto front Variance
Emission Risk R e t u r n s Figure 2: ǫ –Pareto front with inverstor’s preferences Empirical application
We use a set of monthly returns on 22 institutional SRI European open-end funds o ff ered in Spain for the period2009-2019. The empirical information includes the time series of 120 monthly returns and the carbon riskindices. As a previous step the expected return vector ν = ( ν , . . . , ν ) T and the covariance matrix Σ = [ σ i j ], i , j = , . . . ,
22 are computed. For the carbon risk score c i , we use the Morningstar Portfolio Carbon Risk Score,which indicates the risk that companies face from the transition to a low-carbon economy. In this set, scores c i range from 0 to 10, where lower scores are better, indicating lower carbon risk levels. All the numericalinformation to be used on this opportunity set comes from Morningstar database.Table 1 shows the parameter setting applied to the ev-MOGA algorithm. The size of the main population is Nind P = , while the population of the archive A k is Nind GA = / mutationwe select P m / c = .
2. Finally, the space of each objective function has been divided in 300 boxes.Table 1: Parameter setting of the ev-MOGAParameter ValueSize of the main population
Nind P = Size of the auxiliary population
Nind GA = k max = Probability of crossing / mutation P m / c = . • Concerning the sustainability preferences, the green investors are classified in three profiles according to(11) by using the percentile 25 for the Weak green investor p wg , 55 for the Moderate green investor p mg and 75 for the Strong green investor p sg . Thus, the reference vector for the green investor yields: P g = [25% , , = [2 . , . , . • Considering the investor loss aversion attitude, the investors are classified in three profiles according to(12) by establishing percentiles 50 for a Conservative investor p cr , 75 for a cautious investor p kr and 100for an Aggressive investor p ar . Thus, the reference vector regarding the risk aversion becomes: P r = [50% , , = [9 . , . , . ffi cient portfolios for Weak, Moderate and Stronggreen investors is made in terms of di ff erent loss aversion attitude. To this end, for each profile we display anumerical description of the portfolio composition and the objective function values attained by the portfolios.We highlight in bold optimal funds allocation when achieving the three objectives simultaneously and wealso provide the portfolio weights and the objective values for the strategy involving minimum risk, minimumcarbon risk score and maximum return.Figure 3, Figure 4 and Figure 5 show the 3D representation of the approximated ǫ -Pareto front, thus pro-viding the non-dominated mean-variance-emission surface for the three types of Green investor profile and foreach level of loss aversion. Notice that, as the level of loss aversion attitude decreases, the Green Investornon-dominated surface (coloured in blue) grows.The results for a Weak green investor profile are displayed in Table 2. Let us see, for example, the caseof an investor’s conservative attitude toward risk. If the investor wants to optimize the three objectives simul-taneously, the optimal portfolio is given by F , F , F , F , , F , F , and F . We can also view the 3D Empirical application non-dominated surface in Figure 3a in which the whole ǫ -Pareto front is coloured in grey and the investor’s re-gion of interest is coloured in blue and green. While the optimum value of the three objectives lies at the centreof the figure, the corner solutions indicate the optimum objective values involving minimum risk, minimumemissions risk and maximum return. These optimal values are marked by a red dot. Note that the region ofinterest increases as the investor’s risk aversion decreases (see Figure 3b and Figure 3c).Table 2: Weak green investor portfolio composition and objective value function Risk profile F F F F F F F F Risk Ret. EmissConservative
Variance
Emission Risk R e t u r n s (a) Conservative investor Variance
Emission Risk R e t u r n s max ret1.151.21.25 (b) Cautious investor Variance
Emission Risk R e t u r n s (c) Aggressive investor Figure 3: 3D Pareto fronts for a Weak green investor profile ( p wg ): on the left, for the conservative profile ( p cr );on the centrer for the cautious investor ( p kr ); and on the right the aggressive profile ( p ar )In Table 3 and Table 4, the optimal portfolio and the values of the objective functions are shown for thecorresponding investor profiles. As expected, the minimum region of interest correspond to an Strong greeninvestor with a conservative attitude toward risk as shown in Figure 5a. Empirical application
Table 3: Moderate green investor portfolio composition and objective value function
Risk profile F F F F F F F F Risk Ret. EmissConservative
Variance
Emission Risk R e t u r n s (a) Conservative investor Variance
Emission Risk R e t u r n s max ret1.151.21.25 (b) Cautious investor Variance
Emission Risk R e t u r n s (c) Aggressive investor Figure 4: 3D Pareto fronts for a Moderate green investor profile. ( p mg ): on the left, for the Conservative profile( p cr ); on the centrer for the Cautious investor ( p kr ); and on the right for the Aggressive profile ( p ar ) Conclusions
Table 4: Strong green investor portfolio composition and objective value function
Risk profile F F F F F F F F Risk Ret. EmissConservative
Variance
Emission Risk R e t u r n s max ret1.151.21.25 (a) Conservative investor Variance
Emission Risk min var3.41.05 8.53.2 3 2.8 min emi2.6 2.41.1 max ret R e t u r n s (b) Cautious investor Variance
Emission Risk R e t u r n s (c) Aggressive investor Figure 5: 3D Pareto fronts for a Strong Green Investor profile ( p sg ): on the left, for the conservative profile ( p cr );on the centrer for the cautious investor ( p kr ); and on the right for the aggressive profile ( p ar ) In this paper, we have proposed a new application of the ev-MOGA algorithm to handle a tri-criterion portfoliooptimization problem in which the third criterion is the carbon risk score of the portfolio. Moreover, wehave incorporated the investor’s preferences regarding the risk emissions and the loss aversion attitude into thesolution process by defining di ff erent investor profiles. This allows us to propose a solution to the investor interms of their sustainability and risk preferences.Given the urgency around climate change, investors are becoming increasingly aware of the need to makethe transition to a lower carbon economy and to address climate change related risk. It is claimed that newmethodological tools are needed to help investors to align themselves with the preservation of the planet with-out compromising returns and to more thoughtfully consider carbon risk in the investment decision makingframework. In recent years, some rating agencies have introduced eco-labels for mutual funds which allow forthe measurement of the risk that companies face in the transition to a low carbon economy. We have reviewedthe literature addressing the extended M-V portfolio optimization problem. While some scholars have devel-oped exact methodologies to derive the non-dominated surface, in recent years there is an increase number of eferences contributions applying heuristic methodologies such as Multi-objective Evolutionary Algorithms. Thus, weimplement a tri-criterion portfolio optimization problem for returns, risk and emission risks by means of anheuristic methodology based on the concept of ǫ - dominance called ev-MOGA. To the best of our knowledgethis is the first time that this methodology is used to derive the non-dominated surface in three dimensionsincluding mean, variance and carbon-related risks. The ev-MOGA allows us to obtain a 3D-Pareto front in awell-distributed manner with limited memory resources.To better understand the trade-o ff between the three objectives we have introduced an a-posteriori approachto include the investors’ preferences about green investments and risk aversion. So, by considering di ff erentinvestor profiles we can provide a more approximated solution to the investor according to their preferences.Because of the possibility to obtain an e ffi cient frontier in three dimensions while including the preferenceson risk and sustainability, we believe this is a useful tool for investors, especially for those who are willing torebalance their portfolios towards more climate-conscious firms.Finally, as the mutual fund industry is an ideal setting to test our approach, we have used a set of institutionalSRI European open-end funds for illustrative purposes. In the numerical experiments we have analyzed theportfolio generated according to the investor profiles. The results obtained show that the region of interestincreases as the investor’s risk aversion decreases, namely, aggressive investors looking for high returns areallowed to invest in funds with a lower level of carbon risk scores. Thus, we conclude that green investors havea leeway to decrease the emission risk of the portfolio at even no cost to risk and returns. References [1] K. P. Anagnostopoulos and G. Mamanis. A portfolio optimization model with three objectives and discretevariables.
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