European Journal of Operational Research 216 (2012) 487–494

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Socially Responsible Investment: A multicriteria approach to portfolioselection combining ethical and financial objectives

Enrique Ballestero a, Mila Bravo a,⇑, Blanca Pérez-Gladish b,1, Mar Arenas-Parra b, David Plà-Santamaria a

a Universidad Politécnica de Valencia, Spainb Dpto. de Economía Cuantitativa, Facultad de Economía y Empresa, Avda. del Cristo s/n, 33006 Oviedo, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 June 2010Accepted 7 July 2011Available online 15 August 2011

Keywords:Investment analysisStock portfolio selectionEthical investmentOR in finance

0377-2217/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.ejor.2011.07.011

⇑ Corresponding author. Address: Escuela PolitécnFerrándiz y Carbonell s/n, 03801 Alcoy (Alicante), Spa+34 966528409.

E-mail addresses: mibrasel@epsa.upv.es (M. Bravo)Gladish).

1 Tel.: +34 985 106292; fax: +34 985 102806.

In a context of Socially Responsible Investment (SRI), this paper deals with portfolio selection for inves-tors interested in ethical policies. In the opportunity set there are ethical assets and other assets whichare not characterized as ethical. Two goals are considered, the traditional financial goal in the classicalutility theory under uncertainty and an ethical goal in the same utility framework. A new financial-eth-ical bi-criteria model is proposed with absolute risk aversion coefficients and targets depending on theinvestor’s ethical profile. This approach is relevant as an increasing number of mutual funds are becominginterested in SRI strategies. From the proposed model, an actual case on green investment is developed.Concerning this case (without generalizing to other contexts), an analysis of the numerical results showsthat efficient portfolios obtained by the traditional E-V model outperform the strong green portfolios interms of expected return and risk, but this does not significantly occur with weak green investment.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Many people make their investment decisions only on the basisof financial criteria. However, for an increasing number of institu-tional investors and others, this is not enough. Since 1960, a finan-cial behaviour known as Socially Responsible Investment (SRI) orethical investment arises from the mid 20th century political cli-mate of social awareness for the environment, civil rights protec-tion, distrust towards nuclear energy, and other concerns (Baueret al., 2005). At the beginning of the 21st century this attitudehas led to a strengthening of ethical management in some mutualfunds, which invest in companies with powerful environmentalpolicies, honest practices and social guidelines inspired by moralinstitutions. At the same time, support for ethical investing hasremarkably increased. To provide precise information, an indepen-dent research organization in UK, the Experts In ResponsibleInvestment Solutions (EIRIS), reports there are almost 100 greenand ethical funds available to UK investors (year 2010) while therewere just two dozen ethical funds ten years ago. Currently, thereare £7 billion invested in UK green and ethical retail funds – upfrom £1.5 billion 10 years ago (www.eiris.org, 2010). SociallyResponsible Mutual Funds (SRMF) choose investments according

ll rights reserved.

ica Superior de Alcoy, Plazain. Tel.: +34 966528472; fax:

, bperez@uniovi.es (B. Pérez-

not only to financial criteria but also to environmental, social andgovernance criteria, so that their investments reflect ethical values.In other words, SRMFs rely on portfolios combining attractive prof-itable/risk with appropriate return for society rather than relyingon ethical investment only (www.eiris.org, 2008).

This paper deals with a large scale problem of portfolio selec-tion from the SRI perspective with a universe of assets, which issplit into two opportunity subsets, say, (a) ethical assets and (b)other assets which are not characterized as ethical. We propose afinancial-ethical model with two goals in a framework of VonNeumann and Morgenstern (1947) classical utility theory underuncertainty: (i) to minimize the normalized deviation betweenu(ER) utility of expected return and Eu(R) expected utility fromthe traditional financial perspective; and (ii) to minimize thisdeviation from the ethical perspective (Section 3). Formulating thismodel does not require using a particular utility function ad hoc,except for the limited purpose of eliciting the investor’s riskaversion (ARA) coefficients (Subsection 3.4). Potential users ofthe paper are mutual fund’s managers, independent consultantsand OR practitioners.

In Section 4, this new theoretical approach to SRI portfolioselection is applied to a wide opportunity set of 80 assets, of which20 assets are ethical (green) funds. In this example, real worldinformation is used, namely, time series of weekly returns, whichinvolves accepting Sharpe (1994) principle that historic resultshave predictive ability.

No SRI portfolio selection model based on Eu(R) has been foundin previous papers. A proposal on managing SRI portfolios from a

488 E. Ballestero et al. / European Journal of Operational Research 216 (2012) 487–494

multicriteria standpoint is Hallerbach et al. (2004), but it is a linearapproach. Conversely, there is a range of literature on portfolioselection with classical criteria (risk-return) and other criteria,but they are unrelated to the SRI problem. Concerning papers onmulticriteria decision approaches to portfolio choice and relatedmethods (from 2000 onwards), we can cite the following issues.(i) Portfolio choice with fuzzy information (Arenas et al., 2001;Pérez-Gladish et al., 2007; Ballestero et al., 2009). (ii) Approximat-ing the optimum portfolio on the mean–variance efficient frontierby linkages between utility theory and compromise programming(Ballestero and Plà-Santamaria, 2003, 2004, 2005). (iii) Extendingthe classical (risk-return) approach to other different criteria(Steuer et al., 2007). (iv) Novel approaches from multi-objectiveprogramming (Steuer et al., 2005). (v) Constructing equity mutualfunds portfolios by goal programming (Pendaraki et al., 2004). (vi)Mean-semivariance efficient frontier (Ballestero, 2005b). (vii)Hybrid models, neural networks and algorithms (Huang et al.,2005; Huang et al., 2006; Lin et al., 2006). Some GP methods givenuncertainty rely on chance constrained programming (CCP, Char-nes and Cooper, 1959). CCP only allows the analyst to solve simpleproblems under precise conditions. Expected value models (EVM)involve minimizing the sum of a cost function and an extra costfunction (recourse), but this approach is rather inappropriate(Liu, 2002, ch. 5, pp. 62, 75). A different version is random fuzzyexpected value models (Liu and Liu, 2003). Genetic algorithms at-tempt to mitigate complexity in the above methods (Luhandjula,1989; Yazenin, 1996). Satisfaction functions are proposed tointegrate the decision maker’s preferences into GP models underuncertainty (Aouni et al., 2005). Fuzzy techniques are useful whenprobability distributions are unknown (Ben Abdelaziz and Masri,2005). Papers using (or referring to) EV-SGP are, among others,Weihua et al. (2001), Tozer and Stokes (2002), Bordley andKirkwood (2004), Sahoo and Biswal (2005), Ben Abdelaziz et al.(2007), Muñoz and Ruiz (2009), Bravo and González (2009), Abde-laziz et al. (2009), Muñoz et al. (2010).

2. Ethical characterization of assets: the state of the art

As noted, the purpose of this paper is not to discern the ethicalcharacter of an asset but to select portfolios from a moderate SRIperspective which combines assets previously classified in ethicalassets and others. This could be called moderate SRI or mixed SRIportfolio choice but this terminology is not yet established in theliterature. Previous literature on SRI portfolio analysis is adeparture from this paper concerning purpose and method (seeBarracchini, 2007; Geczy et al., 2005). Readers interested in ethicalcharacterization have available literature dealing with this matter(see below). In this section, the state of the art will be conciselyreviewed.

To implement ethical characterization, the following two typesof screening are used (Knoll, 2002). Negative screening (NS) is theoldest and most basic SRI filter. If a company is not involved inbusinesses that are detrimental to a given ethical issue EI (forexample, the company is not a tobacco manufacturer or dealer),then the respective asset gets one NS concerning this EI issue.Positive screening (PS) refers to a company that pursues policiesin favor of a given ethical issue EI (for example, policies againstair and water pollution); then the respective asset gets one PSconcerning this EI issue. To have a relatively high number of NSand PS is only a good recommendation to be included in the S⁄

opportunity subset. In other words, it is a necessary but notsufficient condition. Other conditions such as transparency andcredibility of the company should also be considered.

These additional conditions are related to an ongoing ques-tion: how socially responsible investors are able to examine the

mutual funds’ prospectus to see if the fund investment strategyand socially responsible guidelines meet precise requirements(Hollingworth, 1998). This involves information problems ratherthan OR problems. A company may or may not provide reliableinformation on its SRI strategy as required (Hoggett and Nahan,2002). Moreover, mutual funds are not always aware of ethicalmotivations by which companies are included in their portfolios(Tippet, 2001). Transparency and credibility seem to be neededconcerning the information provided by SRMF’s managers. Be-cause investors have a limited ability for handling extensive data,there is a growing demand for tools tailored to the investors’needs.

Social responsibility rating of mutual funds is used to evaluatetransparency and credibility. As far as we know, although numer-ous works have been published exploring Corporate Social Perfor-mance measures and rating, very few academic studies can befound in the literature on mutual funds’ socially responsible per-formance measurement and non-financial rating. Pérez-Gladishand M’Zali (2010) conducted a computer search in the SCOPUSand ABI/Inform Global to collect the relevant studies related to mu-tual funds’ social performance measurement. These authors founda total of 61 scientific papers but only 4 of them included specificpropositions of a measure for mutual funds’ social performance,which could help individual investors in their investment decisionmaking process. This review of the available literature allows theanalyst to define fundamental criteria on the measurement of mu-tual funds’ socially responsible performance. They are the follow-ing: Investment policy (O’Rourke, 2003), Screening approach(Michelson et al., 2004), Investment criteria (Renneboog et al.,2008), Engagement policy (Renneboog et al., 2008) and Voting pol-icy (Hutton et al., 1998). Regarding transparency and credibility,the criteria are as follows: Research process (Michelson et al.,2004), Selection process (Schrader, 2006), Control companies(Michelson et al., 2004), Experts opinions (Koellner et al., 2005),External Control (Chatterji et al., 2009), Competency of fundmanager (Schrader, 2006) and Communication with investors(O’Rourke, 2003).

3. Selecting SRI portfolios: A financial-ethical bi-criteria model

This is the OR core of this paper. We start with an opportunityset S of m assets, which is split as follows: (a) a subset S⁄ of h eth-ical assets, which have been characterized by ethical and financialcriteria; (b) a subset S⁄⁄ of the (m � h) remaining assets, character-ized by financial criteria only. Notation will be Fi (i = 1,2, . . . ,h) forsubset S⁄ and Fi(i = h + 1,h + 2, . . . ,m) for subset S⁄⁄.

3.1. Goal statement under uncertainty

As we said in Section 1, the choice of SRI portfolios in this paperrelies on classical normative EuðbRÞ utility theory under uncertainty(Von Neumann and Morgenstern, 1947; Arrow, 1965, and a hugerange of literature). As well-known, bR denotes random returns whileEuðbRÞ is expected utility of these returns. According to this classicaltheory, the higher the expected utility the better the investment.

Goal 1 is defined as follows:

Eu1ðbR1Þ ! u1ðR1Þ; R1 P g0; bR1 ¼Xm

i¼1

f ixi;Xm

i¼1

xi ¼ 1 ð1Þ

Goal 2 is defined as follows:

Eu2ðbR2Þ ! u2ðR2Þ; R2 P e0; bR2 ¼Xh

i¼1

f ixi þXm

i¼hþ1

uixi;Xm

i¼1

xi ¼ 1

ð2Þ

E. Ballestero et al. / European Journal of Operational Research 216 (2012) 487–494 489

with the non-negativity conditions xi P 0 for all i, where

u1 and u2 are investor’s utility from goals 1 and 2, respectivelyEu1 and Eu2 are expected utility for u1 and u2, respectivelybR1 and bR2 are random returns on each portfolioR1 and R2 are expected returnsg0 and e0 are investor’s targets or aspiration levelsf i is weekly random return on the ith assetui is assumed to be equal to zero for i = h + 1, h + 2, . . . ,m (seejustification below)

xi is the ith portfolio weight. They are decision variables.

Symbol ? means that expected utility (left hand side of eachequation) should approximate its respective upper limit (righthand side) as close as possible.

Let us highlight the meaning of each goal. In our context, theinvestor’s profile is quite different from the traditional profile.Traditionally, most investors are merely interested in expected re-turns and risk, namely, their primary objective is financial secu-rity and income, no matter SRI. Any ethical objective fallsoutside the scope of these traditional investors. Conversely, theethical investor looks for a compromise between two goals asfollows.

� Goal 1. It reflects the purely financial side of the question, anddoes not require special explanation. It is a classic issue in finan-cial analysis, namely, the EuðbRÞ objective of traditional inves-tors, who consider series of historical returns in the recentpast as the best guidance to invest.� Goal 2. It reflects the SRI side. It is the EuðbRÞ objective of a ‘‘qua-

ker’’ (extremely ethical) investor – as well known, the ReligiousSociety of Friends or quakers movement advised their membersto invest from social criteria, as they believed in fairness andpeaceful purposes. To mathematically formulate the fact thatthe ‘‘quaker’’ will never invest in the S⁄⁄ assets, we makeui ¼ 0ði ¼ hþ 1;hþ 2; . . . ;mÞ in Eq. (2), namely, every randomreturn f iði ¼ hþ 1;hþ 2; . . . mÞ is replaced by a fictitious returnequal to zero. This means that assets Fi(i = h + 1,h + 2, . . . ,m)have no value for the ‘‘quaker’’. Mathematically, this u-basedstatement is more convenient than the alternative statementof removing the ‘‘non-ethical’’ set (h + 1,h + 2, . . . ,m) from goal2. In fact, the u-based statement allows us to define both goalsin a similar way, which leads to more elegant and easier math-ematical developments.

Therefore, the ethical investor in this paper is neither a tradi-tional nor a ‘‘quaker’’ investor, but a decision maker who seeks asatisfying solution from two conflicting goals.

3.2. Stating the portfolio selection model from goals 1 and 2

System (1) and (2) is proven to have a deterministic equivalentgiven by the following mean variance-stochastic goal program-ming (EV-SGP; Ballestero, 2001, 2005a) parametric quadratic pro-gramming model.

v ¼min XVXT ð3Þ

where

X is the row vector (x1,x2, . . . ,xi, . . . ,xm)XT is the transposed vector of XV is a m⁄m matrix, which will be defined below (this matrixsummarizes variability of returns).

Minimization (3) is subject to the following goal equations:

R1 ¼Xm

i¼1

�f ixi P g0 ð4Þ

R2 ¼Xh

i¼1

�f ixi P e0 ð5Þ

where �f i is expected return on the ith asset. In addition, the portfo-lio weight constraint which restricts the sum of the portfolioweights to be 1 is imposed, namely:

Xm

i¼1

xi ¼ 1 ð6Þ

together with the non-negativity conditions.As proven in EV-SGP, matrix V is stated as follows:

V ¼ r1V1 þ r2V2 ð7Þ

where r1 and r2 are the ARA coefficients for each goal, while V1 andV2 are covariance matrices expressing variability of returns for goal1 and 2, respectively.

Using weights for normalization purposes is not required, as thevariables in our model are normalized.

3.3. Targets

First, we define the ethical target as follows:

e0 ¼ k�f e max ð8Þ

where �f e max ¼max �f iði ¼ 1;2; . . . hÞ, while parameter k (ranging be-tween 0 and 1) increases as the investor’s aspiration level for theethical goal increases.

As usual in E-V models, target g0 is treated as a parameter mov-ing on a feasible interval.

A discussion on this matter is as follows.Suppose first that the above maximum expected return,

namely, the mean value in Eq. (8) is positive.

Case 1. Suppose k > 1. Then, no feasible solution to model (3)–(7) can be found.

Case 2. Suppose k = 1. Then, there is only one solution, namely,xi = 1 if i = p where p is the ethical asset of maximumexpected return in Eq. (8).

xi ¼ 0 if i – p

This non-diversified solution corresponds to a ‘‘quaker’’ investorwho maximizes the expected return.

Case 3. Suppose 0 6 k < 1. Then, the higher the k value thehigher the e0 ethical target. Consider a value k = k0. Thisleads to solutions such as the following ones:P P

xi ¼ q P k0; q 6 1; where is extended to set S⁄ of eth-ical assets.P

xi ¼ 1� q; whereP

is extended to set S⁄⁄ of the otherassets.

Consider k = 0.75. From the above discussion, this k value mightyield a q value close to 0.75 so that (1 � q) might reach levels closeto 0.25. Then, k = 0.75 does not generally correspond to a ‘‘quaker’’investor, although it can correspond to a strongly ethical investor.

Case 4. Suppose k < 0. Then, target e0 given by Eq. (8) would beless than zero, which has little sense because even the‘‘quaker’’ investors do not like negative expectedreturns.

Now, suppose that maximum expected return in Eq. (8) isnegative. Then, to invest in ethical assets is not advisable, as the

490 E. Ballestero et al. / European Journal of Operational Research 216 (2012) 487–494

‘‘quaker’’ investors are not satisfied with negative expected returnseither.

3.4. Estimating ARA coefficients

As well-known, risk aversion does not mean risk at all. It is apsychological concept describing the investor’s psychological atti-tude towards risk – this attitude may or may not be influenced by arisk perception. Many investors behave as risk averse, namely, theyprefer portfolios of low volatility, other things being equal. Otherinvestors are risk neutral, while a few are risk lovers. To estimatethe ARA coefficients r1 and r2 in our context, two approaches canbe alternatively used as follows.

� First approach. Coefficients r1 and r2 are straightforwardly elic-ited by comparing the investor’s attitude towards risk in aframework unrelated to Arrow’s risk aversion theory. An advan-tage of this approach is simplicity; however, there is a majordrawback that ignores Arrow’s risk aversion equation – seebelow.� Second approach. Comparison of r1 to r2 is made in the frame-

work of Arrow’s theory. Let us consider the following twoscenarios.

– Scenario 1. Several investors with different wealth face a giveninvestment, which is the same for all of them. In this scenario,which is outside this paper, the jth ARA coefficient dependson the jth investor’s wealth Wj through Arrow’s (1965), p. 94equation:

rj ¼ ð�1Þu00j ðWjÞ=u0jðWjÞ; Wj P 0

where first derivative u0j > 0 and second derivative u00j < 0. In thecase of risk neutrality, u00j ¼ 0 so that rj = 0. Financial authors assumethat rj decreases with the increase of the investor’s wealth(Copeland and Weston, 1988, p. 89).

- Scenario 2. A single investor faces several investments or goals.This is the true scenario in this paper. Then, Arrow’s equationturns into:

rj ¼ ð�1Þu00j ðRjÞ=u0jðRjÞ; Rj P 0 ð9Þ

For ease of notation, we here write Rj instead of bRj to denote ran-dom return. In our paper, j = 1, 2 for goals 1, 2, respectively. Bothderivatives are specified by making return Rj ¼ Rj. Here, rj increaseswith the increase of the Rj expected return, other things beingequal. To justify this property, let us consider an investor whosewealth amounts to $50000, and an investment H whose returnsmight be as follows. Case (a). Random returns on H are $10000and 30000 with equal probability, so that RH ¼ 20000. If so, theinvestor would perceive high risk in comparing return variabilityto wealth. Case (b). Random returns on H are $1 and 3 with equalprobability, so that RH ¼ 2. If so, the investor would perceive norisk in comparing return variability to wealth. Therefore, if theinvestor’s risk aversion is influenced by his/her perception of risk,then ARA in case (a) is higher than ARA in case (b), other thingsbeing equal. Thus, the aforementioned property is justified. Noticethat both cases (a) and (b) lead to the same risk level ðrH=RHÞ asmeasured by the coefficient of variation.

Quadratic utility is the only usual utility form that satisfies theaforementioned property (see Kallberg and Ziemba, 1983). There-fore, for the limited purpose of eliciting the ARA coefficients andonly for this purpose, we will here use quadratic utility as a labo-ratory tool, namely:

uj ¼ 2bjRj � cjR2j ; bj; cj > 0; j ¼ 1;2 ð10Þ

Eqs. (9) and (10) yield:

rj ¼1

ðbj=cjÞ � Rj

; j ¼ 1;2 ð11Þ

By maximizing utility (10) we have:

bj � cjRj ¼ 0) R�j ¼ bj=cj ð12Þ

where R�j is the return that maximises function (10). From Eqs. (11)and (12) we get:

rj ¼1

R�j � Rj

; j ¼ 1;2 ð13Þ

Remark 1. Notice that R�1 is much greater than R1. This is becauseEq. (12) gives us the so-called satiation point of the traditionalinvestor (j = 1), namely, a return so high that more return does notincrease the investor’s utility. From Eqs. (4) and (5), we haveR1 P R2 so that R�1 is also much greater than R2. Hence, ratios R1=R�1and R2=R�1 are close to zero.

To elicit the ARA coefficients, the analyst should conduct a testthrough which the investor discloses his/her risk aversion for eachgoal. It is developed as follows.

3.4.1. Test inputConcerning goal 1, the test starts with a fictitious investment H1

from an opportunity set, which is not characterized as an ethicalset of assets. Investment H1 has zero mean value and r standarddeviation. Concerning goal 2, the test requires considering a ficti-tious investment H2 from ethical assets. Investment H2 also haszero mean value and r standard deviation of the observed returns.Therefore, H1 and H2 have equal volatility; however, the investor’srisk aversion can differ from one another. From Eq. (13) we get:

rHj ¼ 1= R�j � RHj

� �¼ 1=R�j ; j ¼ 1;2 ð14Þ

where rHj is the ARA coefficient for each Hj fictitious investmentsince mean value RHj is equal to zero.

3.4.2. Formulating the testThe analyst asks the investor: ‘‘If you really are an ethical inves-

tor, then your risk aversion for an ethical investment such as H2

will be relatively low, namely, lower than your risk aversion forH1, which has the same expected return and risk (volatility) asH2 but it is not characterized as ethical. Taking this into account,would you like to compare your risk aversion for H1 to your riskaversion for H2?’’. Examples of answers are as follows.

– ‘‘My risk aversion for H1 is significantly higher than for H2, say,twice higher’’. Then, rH2/rH1 = 1/2 on a scale of ARA ratios.

– ‘‘My risk aversion for H1 is moderately higher than for H2, say, 3/2 higher’’. Then, rH2/rH1 = 2/3.

– ‘‘My risk aversion for H1 is slightly higher than for H2, say, 4/3higher’’. Then, rH2/rH1 = 3/4.

Meaning and characterization. These answers have the follow-ing meaning. Suppose first that you are an extremely strong ethicalinvestor. Then, you are willing to invest in ethical assets, evenneglecting the undesirable consequences of risk on your utility.According to Arrow’s theory, this means risk neutrality or almostrisk neutrality, namely, you have zero or very low risk aversionfor ethical investment. More precisely, a low rH2/rH1 ratio togetherwith a high ethical target e0, characterizes strongly ethical profilesof investor. In contrast, weakly ethical profiles appear when riskaversion for the ethical goal increases (namely, when the rH2/rH1 < 1 is a ratio approaching 1) and the ethical target e0 decreases.Cases in which, for example, both the rH2/rH1 ratio and the ethical

E. Ballestero et al. / European Journal of Operational Research 216 (2012) 487–494 491

target reach high values are of doubtful characterization. Finally,suppose you are a traditional (non ethical) risk-averse investor.Then, you are willing to invest neither in ethical assets nor in otherassets without previously considering the undesirable conse-quences of risk on your own utility. In this case, the appropriateportfolio selection approach is classical Markowitz E-V model.

3.4.3. Test outputOnce ratio rH2/rH1 has been specified on the above scale, Eq. (14)

yields:

R�2 ¼ ðrH1=rH2ÞR�1 ð15Þ

3.4.4. ARA coefficients for goals 1 and 2From Eqs. (13) and (15), we have:

r1R�1 ¼1

1� ðR1=R�1Þð16Þ

r2R�1 ¼1

ðrH1=rH2Þ � ðR2=R�1Þð17Þ

From Eqs. (16) and (17) and Remark 1 we obtain:

r1=r2 ffi rH1=rH2 ð18Þ

where ratio rH1/rH2 > 1 (see Section 3.4.2 above). Thus, the ARA coef-ficients are elicited in an approximate way.

3.5. Meaning of the model in terms of risk and risk aversion

To highlight this meaning in terms of risk, notice that goals (1)and (2) lead to the following Pratt’s (1964) relationship (seeBallestero, 2001):

max½EujðbRjÞ�norm ffi ½ujðRjÞ�norm � ð1=2Þrjr2j ðbRjÞ; j ¼ 1;2

where subscript ‘‘norm’’ means that the respective expression isnormalized by the first derivative of utility uj specified at the Rj

mean value. Moreover, r2j ðbRjÞ ¼ XjVjX

Tj is the portfolio variance,

which means risk for both goal j = 1 and the ethical goal j = 2.Xj is the solution (vector of portfolio weights) to the following

model:

min XjVjXTj

subject to

Rj Pg0 if j ¼ 1e0 if j ¼ 2

� �

together with the non-negativity conditions. As the objective func-tion (portfolio variance) is a measure of risk, goals (1) and (2) in-volve constrained risk minimization for the traditional (purelyfinancial) investor and the ‘‘quaker’’ investor, respectively.

Now, consider the so-called ethical investor in this paper, whois an investor between the traditional (purely financial) investorand the ‘‘quaker’’ investor. Here, vector X (portfolio weights) isthe solution to model (3)–(7). Notice that the objective function,namely,

min Xðr1V1 þ r2V2ÞXT ¼ r1XV1XT þ r2XV2XT

is a composite index of variability instead of a portfolio variance.Then, how to measure the financial risk that the ethical (non tradi-tional-non ‘‘quaker’’) investor bears? This risk is given by:

var ¼ XV1XT

which is valid whatever the ethical characterization of the portfolio.In fact, matrix V1 includes the covariances of all assets, whether theasset is ethical or not. Therefore, matrix V1 is a financial matrix

while matrix V2 does not describe financial risk. Obviously, we gen-erally have:

XV1XT – XjVjXTj ðj ¼ 1Þ

because solution X is generally different from solution Xj(j = 1).Finally, to highlight the meaning of the model in terms of risk

aversion, consider the above Pratt’s relationship focusing on itsnegative term on the right-hand side. The greater this negativeterm (in absolute value) the smaller the expected utility on theleft-hand side, other things being equal, namely, if portfolio ex-pected returns and the utility function are kept equal. This termis the product of two factors: (a) the portfolio variance, which isan observable risk measure; and (b) the rj risk aversion coefficient,which describes the investor’s perception of risk and his/her riskassessment from utility. The investor’s expected utility suffersfrom the joint impact of both factors. For example, an individualwith zero risk aversion for road accidents and high risk aversionfor air accidents will prefer travelling by car despite being lesscomfortable and riskier than travelling by air. In the context of thispaper we can plausible assume that the strongly ethical investorshave less risk aversion for ethical assets than the weakly ethicalones, other things being equal. Indeed, if one loves ethical invest-ment, then one tends to close his eyes to the risk inherent in suchinvestment.

As noted, defining goals (1) and (2) does not require assuming aspecial utility function. In contrast, the problem of eliciting riskaversion requires using a particular type of utility. In Subsection3.4, the example for justifying elicitation by quadratic utility is va-lid whatever the ethical characterization of the investor. This is be-cause even ‘‘quaker’’ investors are concerned about potential lossesin their ethical investments.

4. Actual case

Environmentally responsible investment may be the most rele-vant SRI perspective. In the actual case to be developed hereafter,the ethical assets are green (environmental) investments. Opportu-nity set S includes m = 80 assets, of which h = 20 are green whilethe remaining (m � h) are not categorized as ethical. Therefore,we have Fi (i = 1,2, . . . ,20) for subset S⁄ and Fi (i = 21,22, . . . ,80)for subset S⁄⁄. In this opportunity set, each asset is a fund, whichis domiciled in the United Kingdom. All the numerical informationto be used on this opportunity set comes from the followingsources: (a) data kindly provided to the authors by MorningstarLtd; and (b) data on social responsibility of funds, available in EIRIS(2008).

4.1. Defining investor’s profiles

Concerning ethical (environmental) investments, we considersome types of investor (or investor’s profiles), who can be institu-tions, companies, mutual funds or individuals. According to Section3, they are defined as follows.

(i) Strong green investor. This profile is defined by the followingcriteria.

Criterion (a). High level of aspiration for the ethical (green)goal. More precisely, this profile is defined by k = 0.75 inEq. (8). Hence, ethical target (8) becomes:

e0 ¼ 0:75�f e max ¼ 0:75 � 0:00226 ¼ 0:00169 ð19Þ

where the numerical value of �f e max is given in Table 1, bottom.Criterion (b). According to Subsection 3.4, the ARA coeffi-cients are defined by the r2/r1 = 1/2 ratio, namely, r1 = 2/3and r2 = 1/3 as normalized values.

492 E. Ballestero et al. / European Journal of Operational Research 216 (2012) 487–494

(ii) Weak green investor. This profile is defined by the followingcriteria.

TabCov

1

M

�

�

M

�

�

V

�f e m

Criterion (a). Low level of aspiration for the ethical (green)goal. More precisely, this profile is defined by k = 0.25 inEq. (8). Hence, ethical target (8) becomes:

e0 ¼ 0:25�f e max ¼ 0:25 � 0:00226 ¼ 0:00056 ð20Þ

where the numerical value of �f e max is given in Table 1, bottom.Criterion (b). According to Subsection 3.4, the ARA coeffi-cients are defined by the r2/r1 = 3/4 ratio, namely, r1 = 4/7and r2 = 3/7 as normalized values.

4.2. Specifying and solving the model

As a previous step, weekly returns, mean values and covari-ances are computed. For each asset in the opportunity set, weeklyreturns from January 2001 to January 2006 are considered, so that264 random returns are observed. From the random returns,covariance matrices V1 and V2 are computed. In Table 1, thesematrices have been extracted, due to their large size. At the bottomof this table, the expected returns (or mean values) are displayed.

For profiles (i) and (ii), the model is stated as follows. From Eqs.(3) and (7), we have the objective function:

v ¼ min Xðr1V1 þ r2V2ÞXT ð21Þ

where r1 = 2/3 and r2 = 1/3 for profile (i), while r1 = 4/7 and r2 = 3/7for profile (ii), according to criterion (b) in the respective cases.

Minimization (21) is subject to the following constraints, whichcome from Eqs. (4) and (5), respectively:

E1 ¼ 0:001668x1 þ 0:002259x2 þ � � � þ 0:000431x80 P g0 ð22ÞE2 ¼ 0:001668x1 þ 0:002259x2 þ � � � þ 0:001378x20 P e0 ð23Þ

Targets e0 = 0.00169 for profile (i) and e0 = 0.00056 for profile (ii)are given by Eqs. (19) and (20), respectively. Target g0 takes para-metric values to obtain efficient frontiers.

To close the model, constraint (6) is added with the non-negativity conditions. Notice that diversification constraints arenot needed because the assets in the opportunity set are alldiversified funds.

Computation is easy by using Lingo GENPRT or by using MatLabwith a format similar to the generic Markowitz portfolio.

le 1ariance matrices and vector of expected returns (fragment).

2 � � � 20 21 � � � 80

atrix V1

0.00035 0.00026 � � � 0.00026 0.00039 � � � 0.000360.00026 0.00029 � � � 0.00027 0.00032 � � � 0.00028� � � � � � � � � � � � � � � � � � � �0.00026 0.00027 � � � 0.00028 0.00033 � � � 0.000280.00039 0.00032 � � � 0.00033 0.00054 � � � 0.00041� � � � � � � � � � � � � � � � � � � �0.00036 0.00028 � � � 0.00028 0.00041 � � � 0.00041

atrix V2

0.00035 0.00026 � � � 0.00026 0 � � � 00.00026 0.00029 � � � 0.00027 0 � � � 0� � � � � � � � � � � � � � � � � � � �0.00026 0.00027 � � � 0.00028 0 � � � 00 0 � � � 0 0 � � � 0� � � � � � � � � � � � � � � � � � � �0 0 � � � 0 0 � � � 0

ector of expected returns0.00167 0.00226 � � � 0.00138 � � � � � � 0.00043

ax ¼ 0:00226.

4.3. Results

In Table 2, the efficient portfolios obtained from model (21)–(23) for green profiles (i) and (ii) are displayed. For each greenprofile, each row in the table refers to an efficient portfolio, whichis characterized by parameter g0 and the respective v objectivevalue (21), together with expected return E2 given by equation(23). For every portfolio, the expected returns E1 and E2 turn outto be equal to targets g0 and e0, respectively, so that Eqs. (22)and (23) have zero slack. To compare results for the green profilesto results for the traditional investor, the classical Markowitz E-Vefficient frontier is recorded in the last columns. As typically occursin E-V models, some irregular portfolios appear for low levels ofexpectation, namely, the efficient frontier is a curve taking thestandard bullet shape (Haugen, 1997). These portfolios are re-moved from the table.

Information on the portfolio weights for each investor’s profileand each efficient portfolio is also provided in Table 2.

4.4. Discussion

Results will be here analyzed in a comparison framework.

4.4.1. Achievement in terms of expected return and riskFrom Table 2, a comparison of efficient portfolios for strong

green, weak green and traditional investors is made as follows.

(a) Traditional investors can reach feasible portfolios of highexpectation such as g0 = 0.00300. In contrast, strong greeninvestors can only reach portfolios with at most 0.00245expected return. This is a drawback of strong green invest-ment. In the case of weak green investors, the highest feasi-ble portfolio has g0 = 0.00290, which is close to the highestfeasible g0 for traditional investors.

(b) Concerning risk as measured in Subsection 3.5, let us com-pare the three columns ‘‘var’’ for the three investor’s profilesin Table 2. We see that the risk level for strong green inves-tors is significantly higher than for traditional investors,other things being equal (namely, when expected return g0

is kept at the same level). Therefore, traditional investmentoutperforms strong green investment from the classicalSharpe (1994) ratio, namely, from a financial performanceperspective. Weak green investment ranks between tradi-tional and strong green investment – its risk levels are closerto those of traditional investors than those of the stronggreen ones.

4.4.2. Portfolio weights: a comparison from the green perspectiveIn Table 2, we see that the sum of the portfolio weights corre-

sponding to green assets ranges as follows:

0:75 6X20

i¼1

xi 6 0:81 for strong green investors

0:25 6X20

i¼1

xi 6 0:27 for weak green investors

X20

i¼1

xi ¼ 0 for traditional investors

Therefore, the strong green portfolios invest in green assets aboutthree times as much as the weak green portfolios do. In both cases,it generally (but not always) occurs that the higher the expected re-turn g0 the lower the sum of the portfolio weights corresponding togreen assets. In the case of traditional investors, zero investment ingreen assets is obtained from the model. These results are perfectlyconsistent indeed.

Table 2Results: efficient frontiers for the investor’s profiles.

Strong green investor Weak green investor Traditional investor

g0 v var (a) P20i¼1xi

g0 v var (a) P20i¼1xi

g0 var (a) P20i¼1xi

0.00160 0.00010 0.000165 0.265700.00165 0.00010 0.000165 0.26170 0.00150 0.000152 00.00170 0.00010 0.000166 0.26154 0.00170 0.000156 00.00175 0.00010 0.000168 0.26135 0.00175 0.000158 00.00180 0.00011 0.000170 0.26117 0.00180 0.000160 0

0.00185 0.00020 0.000216 0.81071 0.00185 0.00011 0.000173 0.26096 0.00185 0.000162 00.00190 0.00020 0.000217 0.80521 0.00190 0.00011 0.000175 0.26085 0.00190 0.000165 00.00195 0.00020 0.000219 0.79970 0.00195 0.00011 0.000177 0.26092 0.00195 0.000167 00.00200 0.00020 0.000222 0.79775 0.00200 0.00011 0.000180 0.26098 0.00200 0.000170 00.00205 0.00020 0.000225 0.79806 0.00205 0.00011 0.000183 0.26104 0.00205 0.000173 00.00210 0.00021 0.000228 0.79837 0.00210 0.00011 0.000186 0.26112 0.00210 0.000176 00.00215 0.00021 0.000231 0.79849 0.00215 0.00012 0.000189 0.26115 0.00215 0.000180 00.00220 0.00021 0.000235 0.79900 0.00220 0.00012 0.000193 0.26115 0.00220 0.000183 00.00225 0.00021 0.000239 0.79887 0.00225 0.00012 0.000196 0.26109 0.00225 0.000186 00.00230 0.00021 0.000243 0.79167 0.00230 0.00012 0.000200 0.26104 0.00230 0.000190 00.00235 0.00022 0.000249 0.77733 0.00235 0.00012 0.000204 0.26095 0.00235 0.000194 00.00240 0.00022 0.000256 0.76300 0.00240 0.00013 0.000208 0.26087 0.00240 0.000198 00.00245 0.00023 0.000269 0.75472 0.00245 0.00013 0.000212 0.26077 0.00245 0.000202 0

0.00250 0.00013 0.000216 0.26068 0.00250 0.000206 00.00255 0.00013 0.000221 0.26059 0.00275 0.000229 00.00260 0.00014 0.000225 0.26051 0.00300 0.000261 00.00265 0.00014 0.000230 0.260420.00270 0.00014 0.000235 0.260340.00275 0.00014 0.000240 0.260250.00280 0.00015 0.000250 0.254220.00285 0.00016 0.000271 0.249880.00290 0.00018 0.000307 0.24791

(a) var = XV1XT (see Subsection 3.5).

E. Ballestero et al. / European Journal of Operational Research 216 (2012) 487–494 493

4.4.3. Sensitivity analysis(a) Firstly, consider the case of strong green investors. What if

target e0 increases/decreases by 10%? Then, small changes in theportfolio weights (around 0.04 in average) are obtained, exceptfor portfolios of high expected return in which changes in the port-folio weights reach between 0.12 and 0.16. What if ARA coeffi-cients r1 = 2/3 and r2 = 1/3 would change to 2.20/3 and 0.80/3,respectively? What if these ARA coefficients would change to1.80/3 and 1.20/3, respectively? In both cases, no significantchanges are observed. (b) Secondly, consider the case of weakgreen investors. What if target e0 increases/decreases by 10%?Then, changes in the portfolio weights are very small (around0.012 in average), except for portfolios of high expected return inwhich changes in the portfolio weights reach between 0.25 and0.32. What if ARA coefficients r1 = 4/7 and r2 = 3/7 would changeto 4.20/7 and 2.80/7, respectively? What if these ARA coefficientswould change to 3.80/7 and 3.20/7, respectively? Then, no signifi-cant changes are observed.

5. Concluding remarks

Both the proposed financial-ethical bi-criteria model especiallydesigned for the SRI problem and the application of this model toan actual case are new and relevant. Perhaps, this is the first OR ap-proach to SRI stringently based on Eu(R) classical utility theory un-der uncertainty. As developed in Section 3, this design and analysisinvolve much more effort and content than a simple application ofthe EV-SGP model. Relevance of the analysis is given by the grow-ing interest of mutual funds in SRI investment. In Section 1, preciseinformation on this financial relevance has been provided.

Concerning the actual case developed in Section 4, it is an inno-vative application of OR, which ‘‘can be used to convince managersof the value to be gained by applying OR to SRI portfolio selection’’.Indeed, we have undertaken a large scale problem of green portfo-lios. Numerical results given in Table 2 are consistent. Concerning

the framework of our example (without generalizing to other con-texts), these results show that strong green investment in the port-folios involves more financial risk than weak green investment,other things being equal. This would mean some drawback forstrong ecological investors. Such a question should be investigatedfurther, i.e. on data sets other than the ones used in this paper.

Acknowledgement

Blanca Pérez-Gladish and Mar Arenas-Parra wish to gratefullyacknowledge financial support from the Spanish Ministry of Educa-tion, project MTM2007-67634. Thanks are given to three anony-mous referees for their comments and suggestions.

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