57 Mea-Semivariace Optimizatio: A Heuristic Approach Javier Estrada Academics ad practitioers optimize portfolios usig the mea-variace approach far more ofte tha the measemivariace approach, despite the fact that semivariace is ofte cosidered a more plausible measure of risk tha variace. The popularity of the mea-variace approach follows i part from the fact that mea-variace problems have well-kow closed-form solutios, whereas measemivariace optimal portfolios caot be determied without resortig to obscure umerical algorithms. This follows from the fact that, ulike the exogeous covariace matrix, the semicovariace matrix is edogeous. This article proposes a heuristic approach that yields a symmetric ad exogeous semicovariace matrix, which eables the determiatio of mea-semivariace optimal portfolios by usig the well-kow closed-form solutios of mea-variace problems. The heuristic proposed is show to be both simple ad accurate. Javier Estrada is a Professor of Fiace at the IESE Busiess School i Barceloa, Spai. I would like to thak David Nawrocki ad Charles Smithso for their commets. Gabriela Giaattasio provided valuable research assistace. The views expressed below ad ay errors that may remai are etirely my ow. The heuristic proposed i this article has bee icorporated ito the dowside risk Excel add-i provided by hoadley Tradig & Ivestmet Tools. 57 As is well kow, Markowitz (1952) pioeered the issue of portfolio optimizatio with a semial article, later expaded ito a semial book (Markowitz, 1959). Also well kow is that at the heart of the portfolio-optimizatio problem, there is a ivestor whose utility depeds o the expected retur ad risk of his portfolio, the latter quatified by the variace of returs. What may be less well kow is that, from the very begiig, Markowitz favored aother measure of risk: the semivariace of returs. I fact, Markowitz (1959) allocates the etire chapter IX to discuss semivariace, where he argues that aalyses based o S [semivariace] ted to produce better portfolios tha those based o V [variace] (see Markowitz, 1991, page 194). I the revised editio of his book (Markowitz, 1991), he goes further ad claims that semivariace is the more plausible measure of risk (page 374). Later he claims that because a ivestor worries about uderperformace rather tha overperformace, semideviatio is a more appropriate measure of ivestor s risk tha variace (Markowitz, Todd, Xu, ad Yamae, 1993, page 307). Why, the, have practitioers ad academics bee optimizig portfolios for more tha 50 years usig variace as a measure of risk? Simply because, as Markowitz (1959) himself suggests, variace has a edge over semivariace with respect to cost, coveiece, ad familiarity (see Markowitz, 1991, page 193). He therefore focused his aalysis o variace, practitioers, ad academics followed his lead, ad the rest is history. Familiarity, however, has become less of a issue over time. I fact, i both practice ad academia, dowside risk has bee gaiig icreasig attetio, ad the may magitudes that
58 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 capture dowside risk are by ow well kow ad widely used. The focus of this article is o the issues of cost ad coveiece. The differece i cost, Markowitz (1959) argues, is give by the fact that efficiet sets based o semivariace took, back the, two to four times as much computig time as those based o variace. The differece i coveiece, i tur, is give by the fact that efficiet sets based o variace require as iputs oly meas, variaces, ad covariaces, whereas those based o semivariace require the etire joit distributio of returs. The ultimate goal of this article, the, is to propose a heuristic approach to the estimatio of portfolio semivariace that reders the issues of cost ad coveiece irrelevat, thus hopefully removig the last remaiig obstacles to a widespread use of mea-semivariace optimizatio. I a utshell, this article proposes to estimate the semivariace of portfolio returs by usig a expressio similar to that used to estimate the variace of portfolio returs. The advatages of this approach are twofold: 1) estimatig the semivariace of portfolio returs is just as easy as estimatig the variace of portfolio returs (ad i both cases the same umber of iputs is required); ad 2) it ca be doe with a expressio well kow by all practitioers ad academics, without havig to resort to ay black-box umerical algorithm. I additio, the heuristic proposed here yields a portfolio semivariace that is both very highly correlated ad very close i value to the exact magitude it iteds to approximate. The article is orgaized as follows. Sectio I itroduces the issue, discusses the difficulties related to the optimizatio of portfolios o the basis of meas ad semivariaces, ad shows how they are overcome by the heuristic approach proposed i this article. Sectio II, based o data o idividual stocks, markets, ad asset classes, provides empirical support for this heuristic. Sectio III cocludes with a assessmet. I. The Issue There is little doubt that practitioers rely much more o mea-variace optimizatio tha o mea-semivariace optimizatio. This is largely because, ulike the eat closedform solutios of mea-variace problems kow by most academics ad practitioers, mea-semivariace problems are usually solved with obscure umerical algorithms. This, i tur, is largely because, ulike the exogeous covariace matrix used i mea-variace problems, the semicovariace matrix of mea-semivariace problems is, as will be illustrated, edogeous. This sectio starts with some basic defiitios ad otatio, ad the itroduces the defiitio of portfolio semivariace proposed i this article. A umerical example is the used to illustrate both the edogeeity of the usual defiitio of the semicovariace matrix ad the exogeeity of the defiitio proposed here. A. The Basics Cosider a asset i with returs R it, where t idexes time. The variace of this asset s returs (σ i2 ) is give by T = 2 2 2 σ i = E[( Ri µ i) ] = (1/ T) ( Rit µ t i), (1) 1 where µ i deotes the mea retur of asset i ad T the umber of observatios; ad the covariace betwee two assets i ad j (σ ij ) is give by T = σ ij = E[( Ri µ i )( R j µ j )] = (1/ T) ( Rit µ i )( R µ t jt j ). (2) 1 The semivariace of asset i s returs with respect to a bechmark B (Σ ib2 ) is give by, (3) Σ 2 T ib = E{[Mi( R i B,0)] 2 } = (1/ T) [Mi( R t it B,0)] 2 = 1 where B is ay bechmark retur chose by the ivestor. The square root of Equatio (3) is the semideviatio of asset i with respect to bechmark B, a widely-used measure of dowside risk. Sectio A of the Appedix, 1 borrowig heavily from Estrada (2006), provides a very brief itroductio to the semideviatio ad discusses some of the advatages it has over the stadard deviatio as a measure of risk; Nawrocki (1999) provides a brief history of dowside risk ad a overview of dowside risk measures. The semicovariace betwee assets i ad j (Σ ij ) is trickier to defie. Hoga ad Warre (1974) defie it as Σ ij HW = E{( R R ) Mi( R R,0)}, (4) i f j f where the superscript HW idicates that this is defiitio proposed by Hoga ad Warre. This defiitio, however, has two drawbacks: 1) the bechmark retur is limited to the risk-free rate ad caot be tailored to ay desired bechmark, ad 2) it is usually the case that Σ ij HW Σ ji HW. This secod characteristic is particularly limitig both formally (the semicovariace matrix is usually asymmetric) ad ituitively (it is ot clear how to iterpret the cotributio of assets i ad j to the risk of a portfolio). I order to overcome these two drawbacks, Estrada (2002, 2007) defies the semicovariace betwee assets i ad j with respect to a bechmark B (Σ ijb ) as Σ ijb i j T (1/ T) R t = [Mi( it B,0) M 1 = E{Mi( R B,0) Mi( R B,0)} =. (5) i( R B,0)] This defiitio ca be tailored to ay desired B ad geerates a symmetric (Σ ijb =Σ jib ) ad, as will be show, jt
ESTRADA exogeous semicovariace matrix. Both the symmetry ad exogeeity of this matrix are critical for the implemetatio of the proposed heuristic. Fially, the expected retur (E p ) ad variace (σ p2 ) of a portfolio are give by E = x E = 1, (6) p i i i σ, (7) 2 σ p = xx i = 1 j = 1 i j ij where x i deotes the proportio of the portfolio ivested i asset i, E i the expected retur of asset i, ad the umber of assets i the portfolio. B. The Problem Portfolio-optimizatio problems ca be specified i may ways depedig o the goal ad restrictios of the ivestor. 1 The problem of miimizig the risk of a portfolio subject to a target retur (E T ) is give by 2 x, x,..., σ = xxσ x p i = 1 j = 1 i j ij Mi 1 2 MEAN-SEMIVARIANCE OPTIMIZATION: A HEURISTIC APPROACH (8) T x E = E ad x = 1, (9) i = 1 i i i = 1 where risk is measured as the variace of portfolio returs. This problem ca be solved for a specific value of E T or, alteratively, for several values of E T thus geeratig the miimum-variace set. Either way, it is importat to otice, first, that the risk of the portfolio ca be expressed as a fuctio of the risk of the idividual assets i the portfolio; secod, that all the variaces, covariaces, ad expected returs of the idividual assets are exogeous variables; ad third, that this problem has a well-kow closed-form solutio. For this reaso, although it is ot importat for the purposes of this article how the σ ij are estimated, it is importat that, oce the values of these parameters (exogeous variables) are determied, they become iputs (together with E i ad E T ) i the closed-form solutio of the problem, which i tur yields the optimal allocatio to each of the assets i the portfolio (the edogeous variables x i ). But what if, istead of defiig risk as the variace of portfolio returs, a ivestor wated to defie it as the semivariace of portfolio returs? What if, give a bechmark retur B chose by the ivestor, he wated to 2 T 2 Mi x (1/ ) [Mi(,0)] 1, x2,..., x Σ pb= T Rpt B (10) t = 1 1 There are four stadard portfolio optimizatio problems: 1) miimizig the risk of a portfolio; 2) miimizig the risk of a portfolio subject to a target retur; 3) maximizig the retur of a portfolio subject to a target level of risk; ad 4) maximizig the risk-adjusted retur of a portfolio. The heuristic proposed here applies to all four problems; oly, for cocreteess, most of the discussio is focused o the secod problem. i 59 T x E = E, x = 1, ad x i 0, (11) i = 1 i i i = 1 i where R pt deotes the returs of the portfolio ad Σ pb 2 their semivariace? The mai obstacle to the solutio of this problem is that the semicovariace matrix is edogeous; that is, a chage i weights affects the periods i which the portfolio uderperforms the bechmark, which i tur affects the elemets of the semicovariace matrix. 2 I order to overcome this obstacle, may algorithms have bee proposed to solve the problem i Equatios (10) ad (11), some of which are discussed below. More importatly, this article proposes a heuristic approach to solve this problem without havig to resort to ay black-box umerical algorithm. I fact, as will be evidet, the heuristic proposed here makes it possible to solve ot oly the problem i Equatios (10) ad (11), but also all mea-semivariace problems with the same well-kow closed-form solutios widely-used to solve mea-variace problems. More precisely, this article argues that the semivariace of a portfolio with respect to a bechmark B ca be approximated with the expressio 2 Σ pb xx i jσ i = 1 j = 1 ijb, (12) where Σ ijb is defied as i Equatio (5). This expressio yields a symmetric ad exogeous semicovariace matrix, which ca the be used i the same way the (symmetric ad exogeous) covariace matrix is used i the solutio of meavariace problems. C. A Example Table I displays the aual returs of the S&P-500 ad the Nikkei-225 betwee 1997 ad 2006, as well as the retur of two portfolios: oe ivested 80% i the S&P ad 20% i the Nikkei, ad the other ivested 10% i the S&P ad 90% i the Nikkei. Cosider for ow the 80-20 portfolio. The stadard deviatio of this portfolio ca be calculated by first estimatig its returs over the sample period, ad the calculatig the stadard deviatio of those returs. The fourth colum of Table I shows the returs of the 80-20 portfolio, ad the stadard deviatio of those returs, which ca be straightforwardly calculated usig the square root of (1), is 16.7%. Importatly, the stadard deviatio of the 80-20 portfolio ca also be calculated by usig the square root of Equatio (7). Takig ito accout that the stadard deviatios of the S&P ad the Nikkei over the 1997-2006 period are 17.8% 2 Note, also, that this formulatio of the optimizatio problem igores both the dowside risk of idividual assets ad the dowside covariace betwee idividual assets; see Sig ad Og (2000).
60 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 Table I. The Edogeous Semicovariace Matrix This table shows the returs over the 1997-2006 period of the S&P-500 (S&P), the Nikkei-225 (Nikkei), a portfolio ivested 80% i the S&P ad 20% i the Nikkei (80-20), ad a portfolio ivested 10% i the S&P ad 90% i the Nikkei (10-90). Coditioal returs are defied as 0% whe the retur of the portfolio is positive, ad the retur of the asset whe the retur of the portfolio is egative. All returs are i dollars ad accout for capital gais ad divideds. All umbers are i percetages. Year S&P Nikkei 80-20 10-90 Coditioal Returs 80-20 Portfolio 10-90 Portfolio S&P Nikkei Product S&P Nikkei Product 1997 31.0 21.2 20.6 16.0 0.0 0.0 0.0 31.0 21.2 6.6 1998 26.7 9.3 19.5 5.7 0.0 0.0 0.0 26.7 9.3 2.5 1999 19.5 36.8 23.0 35.1 0.0 0.0 0.0 0.0 0.0 0.0 2000 10.1 27.2 13.5 25.5 10.1 27.2 2.8 10.1 27.2 2.8 2001 13.0 23.5 15.1 22.5 13.0 23.5 3.1 13.0 23.5 3.1 2002 23.4 18.6 22.4 19.1 23.4 18.6 4.4 23.4 18.6 4.4 2003 26.4 24.5 26.0 24.6 0.0 0.0 0.0 0.0 0.0 0.0 2004 9.0 7.6 8.7 7.7 0.0 0.0 0.0 0.0 0.0 0.0 2005 3.0 40.2 10.4 36.5 0.0 0.0 0.0 0.0 0.0 0.0 2006 13.6 6.9 12.3 7.6 0.0 0.0 0.0 0.0 0.0 0.0 ad 24.1%, ad that the covariace betwee these two idices is 0.0163, it follows from Equatio (7) that σ p = {(0.82)(0.1782) + (0.22)(0.2412) + 2(0.8)(0.2)(0.0163)}1/2 = 16.7%, which is, of course, idetical to the umber obtaied before from the portfolio returs. So far, o mystery here. The problem arises if the proper measure of risk is ot the portfolio s variace, but its semivariace. Oe obvious way of calculatig this magitude would be by first calculatig the returs of the portfolio ad the usig Equatio (3) to calculate the semivariace of its returs. Assume a bechmark retur of 0% (B=0), ad cosider agai the 80-20 portfolio. We could first calculate the returs of this portfolio (show i the fourth colum of Table I), ad the calculate the semivariace of its returs by usig Equatio (3). That would obtai a portfolio semivariace with respect to 0% equal to 0.0092, ad a portfolio semideviatio equal to (0.0092) 1/2 =9.6%. Thus, for ay give portfolio, its semideviatio ca always be calculated as just explaied. But here is the problem: if istead of the semideviatio of oe portfolio, we wated to calculate the portfolio with the lowest semideviatio from a set of, say, 1,000 feasible portfolios, we would first eed to calculate the returs of each portfolio; the from those returs we would eed to calculate the semideviatio of each portfolio; ad fially from those semideviatios we would eed to select the oe with the lowest value. Obviously, as the umber of assets i the portfolio icreases, ad the umber of feasible portfolios icreases eve more, choosig the optimal portfolio with this procedure becomes itractable. Look at this from a differet perspective. If the elemets of the semicovariace matrix were exogeous, the we could formally solve the give optimizatio problem ad obtai a closed-form solutio. We could the iput ito this closedform solutio the values of the exogeous variables of the problem at had, ad obtai as a result the weights that satisfy the problem. This is exactly what ivestors routiely do whe solvig portfolio-optimizatio problems i the mea-variace world. But the problem i the mea-semivariace world is, precisely, that the elemets of the semicovariace matrix are ot exogeous. D. The Edogeeity of the Semicovariace Matrix Markowitz (1959) suggests estimatig the semivariace of a portfolio with the expressio, (13) 2 Σ pb = xxs i = 1 j = 1 i j ijb where K = SijB = (1/ T) ( Rit B)( R jt B) t 1, (14) where periods 1 through K are those i which the portfolio uderperforms the bechmark retur B. This defiitio of portfolio semivariace has oe advatage ad oe drawback. The advatage is that it provides a exact estimatio of the portfolio semivariace. The drawback is that the semicovariace matrix is edogeous; that is, a chage i weights affects the periods i which the portfolio
ESTRADA MEAN-SEMIVARIANCE OPTIMIZATION: A HEURISTIC APPROACH 61 uderperforms the bechmark, which i tur affects the elemets of the semicovariace matrix. To see the advatage of this defiitio of portfolio semivariace, go back to the 80-20 portfolio i Table I, ad cosider agai B=0. The sixth colum of this table shows the coditioal returs of the S&P defied, followig Equatio (14), as 0% whe the retur of the 80-20 portfolio is positive (thus outperformig the bechmark), ad the retur of the S&P whe the retur of the 80-20 portfolio is egative (thus uderperformig the bechmark). To illustrate, the coditioal retur of the S&P is 0% i 1997 because the 80-20 portfolio delivered a positive retur, ad 10.1% (the retur of the S&P) i 2000 because the 80-20 portfolio delivered a egative retur. The seveth colum shows the coditioal returs for the Nikkei, ad the eighth colum is just the product of the sixth ad the seveth colums. The four terms of the semicovariace matrix that follow from Equatio (14) ca be calculated as follows. Squarig the coditioal returs i the sixth colum ad takig their average obtais S S&P,S&P,0 =0.0082; doig the same with the coditioal returs i the seveth colum obtais S Nikkei,Nikkei,0 =0.0164; ad takig the average of the umbers i the eighth colum obtais S S&P,Nikkei,0 =0.0102. The, it follows from Equatio (13) that the semivariace of the 80-20 portfolio is The heuristic proposed is both simple ad accurate. Estimatig semicovariaces is just as easy as estimatig covariaces, ad aggregatig them ito a portfolio semivariace is, with the proposed heuristic, just as easy as aggregatig covariaces ito a portfolio variace. {(0.8 2 )(0.0082) + (0.2 2 )(0.0164) + 2(0.8)(0.2)(0.0102)}= 0.0092, ad its semideviatio is (0.0092) 1/2 =9.6%, which is exactly the same umber obtaied before. Therefore, the expressio proposed by Markowitz (1959) does ideed provide a exact estimatio of the portfolio semivariace. But the problem is that, i order to estimate this semivariace, we eed to kow whether the portfolio uderperforms the bechmark, ad we the ru ito the problem previously metioed: the semicovariace matrix is edogeous because a chage i weights affects whe the portfolio uderperforms the bechmark, which i tur affects the elemets of the semicovariace matrix. To see this more clearly, go back to Table I ad cosider ow the portfolio ivested 10% i the S&P ad 90% i the Nikkei. The returs of this portfolio are show i the fifth colum, the coditioal returs (as defied above) of the S&P ad the Nikkei i the ith ad teth colums, ad the product of these last two colums i the eleveth colum. Importatly, ote that the coditioal returs of the S&P ad the Nikkei for the 10-90 portfolio that follow from Equatio (14) are differet from those for the 80-20 portfolio that follow from the same expressio. The four terms of the semicovariace matrix that follows from Equatio (14) ca be calculated as before. Squarig the umbers i the ith colum ad the takig their average obtais S S&P,S&P,0 =0.0249; squarig the umbers i the teth colum ad the takig their average obtais S Nikkei,Nikkei,0 =0.0217; ad takig the average of the umbers i the last colum obtais S S&P,Nikkei,0 =0.0011. Ad importatly, ote that all these umbers are differet from those calculated for the 80-20 portfolio. This clearly illustrates that the semicovariace matrix is edogeous because its elemets deped o the asset weights. Fially, for the sake of completeess, with the umbers just calculated Equatio (13) ca be used to calculate the semivariace of the 10-90 portfolio, which is give by {(0.1 2 )(0.0249) + (0.9 2 )(0.0217) + 2(0.1)(0.9)(0.0011)} = 0.0181, thus implyig a semideviatio of (0.0181) 1/2 =13.4%. E. Some Possible Solutios The edogeeity of the semicovariace matrix as defied i Equatio (14) has led may authors to propose differet ways of tacklig the problem i Equatios (10) ad (11). Hoga ad Warre (1972) propose to solve this problem usig the Frak-Wolfe algorithm; they explai the two basic steps of this iterative method (the directio-fidig problem ad the step-size problem) ad illustrate its applicatio with a simple hypothetical example. Ag (1975) proposes to liearize the semivariace so that the optimizatio problem ca be solved usig liear (istead of quadratic) programmig. Nawrocki (1983) proposes a further simplificatio of the heuristic proposed by Elto, Gruber, ad Padberg (1976). The latter focus o the mea-variace problem ad impose the simplifyig assumptio that all pairwise correlatios are the same; the former further imposes a value of zero for all of these correlatios ad exteds the aalysis to other measures of risk, icludig the semivariace. I this heuristic, assets are raked accordig to the measure z i = (E i R f )/RM i,
62 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 where RM i is a risk measure for asset i, ad assets with z i >0 are icluded i the portfolio accordig to the proportios w i = z i /Σ i z i. Nawrocki ad Staples (1989) expad the scope of Nawrocki (1983) by cosiderig the lower partial momet (LPM) as a risk measure. Harlow (1991) also cosiders the problem i Equatios (10) ad (11) ad geerates mea-semivariace efficiet frotiers, which he compares to mea-variace efficiet frotiers. However, he does ot explai how these frotiers are obtaied other tha statig that the optimizatio process uses the etire distributio of returs. Similarly, Grootveld ad Hallerbach (1999) geerate mea-lpm efficiet frotiers ad state that the umerical optimizatio process they use for solvig the problem i Equatios (10) ad (11) is tedious ad demadig, but do ot provide details of such process. Markowitz et al. (1993) trasform the mea-semivariace problem ito a quadratic problem by addig fictitious securities. This modificatio eables them to apply to the modified mea-semivariace problem the critical lie algorithm origially developed to solve the mea-variace problem. More recetly, de Athayde (2001) proposes a oparametric approach to calculate the portfolio semivariace, as well as a algorithm (basically a series of stadard miimizatio problems) to optimize it ad geerate the efficiet frotier. Ballestero (2005), i tur, proposes a defiitio of portfolio semivariace (restrictig the bechmark to the mea) that, whe icorporated ito optimizatio problems, these ca be solved by applyig parametric quadratic programmig methods. F. A Heuristic Approach As previously advaced, the heuristic proposed i this article is based o estimatig the portfolio semivariace usig Equatio (12), which i tur is based o Equatio (5), which geerates a symmetric ad exogeous semicovariace matrix. Recall that with Equatio (14) kowledge of whether the portfolio uderperforms the bechmark B is eeded, which geerates the edogeeity problem discussed earlier. With Equatio (5), however, kowledge of whether the asset (ot the portfolio) uderperforms the bechmark B is eeded. Agai, a example may help. Table II reproduces the returs over the 1997-2006 period of the S&P, the Nikkei, the 80-20 portfolio, ad the 10-90 portfolio, all take from Table I. As previously illustrated, the elemets of the semicovariace matrix that follow from Equatio (14) for the 80-20 portfolio are differet from those of the semicovariace matrix that follow from Equatio (14) for the 10-90 portfolio, which cofirms the edogeeity of this defiitio of semicovariace. As will be show, the elemets of the semicovariace matrix that follow from Equatio (5) are ivariat to the portfolio cosidered ad are, therefore, exogeous. To see this, calculate the four terms of the semicovariace matrix that follow from this expressio by cosiderig oce agai a bechmark retur of 0%. First, redefie coditioal returs as 0% whe the retur of the asset is positive (thus outperformig the bechmark), ad the retur of the asset whe the retur of the asset is egative (thus uderperformig the bechmark). To illustrate, the coditioal retur of the S&P is 0% i 1997 because the S&P delivered a positive retur, ad 10.1% (the retur of the S&P) i 2000 because the S&P delivered a egative retur. These coditioal returs of the S&P ad the Nikkei are show i the sixth ad seveth colums of Table II, ad the eighth colum is the product of the previous two. Note that because these coditioal returs deped o whether the asset, ot the portfolio, uderperforms the bechmark, they are relevat ot oly to estimate the semicovariace matrix of the 80-20 portfolio, but also that of ay other portfolio. The four terms of the semicovariace matrix that follow from Equatio (5), the, ca be calculated as follows. Squarig the coditioal returs i the sixth colum ad takig their average obtais Σ S&P,S&P,0 =0.0082; doig the same with the coditioal returs of the seveth colum obtais Σ Nikkei,Nikkei,0 =0.0217; ad takig the average of the umbers i the eighth colum obtais Σ S&P,Nikkei,0 =0.0102. The, it follows from Equatio (12) that the semivariace of the 80-20 portfolio is {(0.8 2 )(0.0082) + (0.2 2 )(0.0217) + 2(0.8)(0.2)(0.0102)} = 0.0094, ad its semideviatio is (0.0094) 1/2 =9.7%, very close to the exact 9.6% umber calculated previously from the portfolio returs. Importatly, if Equatio (12) is used to calculate the semivariace of the 10-90 portfolio, the {(0.1 2 )(0.0082) + (0.9 2 )(0.0217) + 2(0.1)(0.9)(0.0102)} = 0.0195, thus implyig a semideviatio of (0.0195) 1/2 =14.0%. Note that this umber is very close to the exact 13.4% figure calculated for this portfolio i sectio I.D. More importatly, ote that the oly differece betwee this calculatio ad that for the 80-20 portfolio is i the weights; the four elemets of the semicovariace matrix are the same. I short, if semicovariaces are defied as i Equatio (14) ad portfolio semivariace as i Equatio (13), the the edogeeity problem occurs ad black-box umerical algorithms eed to be used to solve portfolio-optimizatio problems. If semicovariaces are istead defied as i Equatio (5) ad portfolio semivariace as i Equatio (12), the a symmetric ad exogeous semicovariace matrix is obtaied, ad the well-kow ad widely-used closed-form
ESTRADA MEAN-SEMIVARIANCE OPTIMIZATION: A HEURISTIC APPROACH 63 Table II. The Exogeous Semicovariace Matrix This table shows the returs over the 1997-2006 period of the S&P-500 (S&P), the Nikkei-225 (Nikkei), a portfolio ivested 80% i the S&P ad 20% i the Nikkei (80-20), ad a portfolio ivested 10% i the S&P ad 90% i the Nikkei (10-90). Coditioal returs are defied as 0% whe the retur of the asset is positive, ad the retur of the asset whe the retur of the asset is egative. All returs are i dollars ad accout for capital gais ad divideds. All umbers are i percetages. Year Assets Portfolios Coditioal Returs S&P Nikkei 80-20 10-90 S&P Nikkei Product 1997 31.0 21.2 20.6 16.0 0.0 21.2 0.0 1998 26.7 9.3 19.5 5.7 0.0 9.3 0.0 1999 19.5 36.8 23.0 35.1 0.0 0.0 0.0 2000 10.1 27.2 13.5 25.5 10.1 27.2 2.8 2001 13.0 23.5 15.1 22.5 13.0 23.5 3.1 2002 23.4 18.6 22.4 19.1 23.4 18.6 4.4 2003 26.4 24.5 26.0 24.6 0.0 0.0 0.0 2004 9.0 7.6 8.7 7.7 0.0 0.0 0.0 2005 3.0 40.2 10.4 36.5 0.0 0.0 0.0 2006 13.6 6.9 12.3 7.6 0.0 0.0 0.0 solutios of mea-variace portfolio optimizatio problems ca be applied. G. A First Look at the Accuracy of the Approximatio I order to take a prelimiary look at the accuracy of the approximatio proposed, ad to roud up the example discussed so far, Table III shows the returs of eleve portfolios over the 1997-2006 period that differ oly i the proportios ivested i the S&P ad the Nikkei. The third row from the bottom shows the exact semideviatio of each portfolio calculated from the portfolio returs ad based o Equatio (3), ad the secod row from the bottom shows the semideviatio of each portfolio based o the approximatio proposed i Equatio (12). I both cases the bechmark retur B is 0%. The last row shows the differece betwee the exact ad the approximate semideviatios. The correlatio betwee the exact semideviatios based o Equatio (3) ad the approximate semideviatios based o Equatio (12) is a whoppig 0.98. Furthermore, the differece betwee the approximate ad the exact semideviatios is uder 1% i all cases, with a average of 0.42%. Fially, the directio of the error is predictable; wheever there is a differece betwee the two, the approximate semideviatio is larger tha its exact couterpart. I other words, wheever the approximatio errs, it does so o the side of cautio, overestimatig (by a small amout) the risk of the portfolio. II. The Evidece The heuristic proposed i this article yields a symmetric ad exogeous semicovariace matrix which, as discussed earlier, makes it possible to solve mea-semivariace optimizatio problems usig the well-kow closed-form solutios widely-used for mea-variace optimizatio problems. However, as with ay heuristic, its usefuless rests o its simplicity ad accuracy. Its simplicity is hopefully evidet from the previous discussio; its accuracy is discussed ext. This sectio starts by cosiderig portfolios of stocks, markets, ad asset classes with the purpose of comparig their exact semideviatios to the approximate semideviatios based o the proposed heuristic. It the cosiders mea-variace ad mea-semivariace optimal portfolios, the latter based o the proposed heuristic, with the goal of comparig the allocatios geerated by these two approaches. A. The Accuracy of the Approximatio I order to test the accuracy of the proposed heuristic over a wide rage of assets, exact ad approximate semideviatios were calculated for over 1,100 portfolios, some cotaiig stocks, some markets, ad some asset classes. The data is described i detail i sectio A of the Appedix. Table IV summarizes the results of all the estimatios. Pael A shows the results for two-asset portfolios selected from three asset classes: 1) US stocks, 2) emergig markets stocks, ad 3) US real estate, all of which exhibit statisticallysigificat egative skewess over the sample period.
64 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 Table III. A First Look at the Accuracy of the Approximatio This table shows the retur of eleve portfolios over the 1997-2006 period, each with differet proportios ivested i the S&P- 500 (S&P) ad the Nikkei-225 (Nikkei). The returs of the S&P ad the Nikkei are those i Tables I ad II. Σ p0 Equatio (3) ad Σ p0 Equatio (12) deote the portfolio semideviatios based o Equatios (3) ad (12), both with respect to a bechmark retur of 0%. The last row shows the differeces Σ p0 Equatio (12) Σ p0 Equatio (3). All returs are i dollars ad accout for capital gais ad divideds. All umbers are i percetages. Year Proportio of the Portfolio Ivested i the S&P (%) 100 90 80 70 60 50 40 30 20 10 0 1997 31.0 25.8 20.6 15.3 10.1 4.9 0.3 5.5 10.8 16.0 21.2 1998 26.7 23.1 19.5 15.9 12.3 8.7 5.1 1.5 2.1 5.7 9.3 1999 19.5 21.3 23.0 24.7 26.4 28.2 29.9 31.6 33.3 35.1 36.8 2000 10.1 11.8 13.5 15.3 17.0 18.7 20.4 22.1 23.8 25.5 27.2 2001 13.0 14.1 15.1 16.2 17.2 18.3 19.3 20.4 21.4 22.5 23.5 2002 23.4 22.9 22.4 21.9 21.5 21.0 20.5 20.0 19.6 19.1 18.6 2003 26.4 26.2 26.0 25.8 25.6 25.4 25.2 25.0 24.8 24.6 24.5 2004 9.0 8.9 8.7 8.6 8.4 8.3 8.2 8.0 7.9 7.7 7.6 2005 3.0 6.7 10.4 14.2 17.9 21.6 25.3 29.1 32.8 36.5 40.2 2006 13.6 12.9 12.3 11.6 10.9 10.3 9.6 8.9 8.3 7.6 6.9 p0 Equatio (3) 9.05 9.29 9.57 9.88 10.23 10.60 11.00 11.56 12.36 13.44 14.75 p0equatio (12) 9.05 9.32 9.68 10.12 10.64 11.21 11.84 12.52 13.23 13.98 14.75 Differece 0.00 0.03 0.11 0.24 0.41 0.61 0.84 0.96 0.87 0.53 0.00 Table IV. Exact ad Approximate Portfolio Semideviatios This table shows the aualized semideviatios of portfolios with respect to a bechmark retur of 0% over the Jauary 1997-December 2006 period. For the assets i each lie of Pael A, 101 portfolios were geerated, with weights varyig betwee 0% ad 100% i each asset (i icremets of 1%), ad their semideviatio calculated. Avg Σ p0 Equatio (3) ad Avg Σ p0 Equatio (12) deote the average portfolio semideviatios across the 101 portfolios based o Equatios (3) ad (12). Σ p0 Equatio (3) Rage deotes the rage betwee the miimum ad the maximum values of Σ p0 Equatio (3) across the 101 portfolios. Differece is betwee Avg Σ p0 Equatio (12) ad Avg Σ p0 Equatio (3) ad Rho deotes the correlatio betweeσ p0 Equatio (3) ad Σ p0 Equatio (12), both across the 101 portfolios. For the assets i each lie of Paels B, C, ad D, 100 radom portfolios were geerated ad the process outlied for Pael A was repeated. All returs are mothly, i dollars, ad accout for capital gais ad divideds. All umbers but correlatios are i percetages. A full data descriptio is available i the appedix. p 0 Equatio (3) Rage Avg. Equatio (3) p 0 Avg. Equatio (12) p 0 Differece Pael A. Asset Classes USA-EMI 10.21 17.29 13.15 13.24 0.09 1.00 USA-NAREIT 7.37 10.21 8.16 8.48 0.32 0.99 EMI-NAREIT 8.48 17.29 11.66 11.96 0.30 1.00 Pael B. Emergig Markets Group 1 (5 markets) 14.75 19.70 16.80 18.90 2.10 0.97 Group 2 (5 markets) 15.24 17.51 16.06 16.74 0.68 0.98 Group 3 (10 markets) 13.92 16.38 15.11 16.91 1.80 0.93 Pael C. DJIA Stocks Group 1 (10 stocks) 9.99 13.16 11.26 13.33 2.07 0.90 Group 2 (10 stocks) 9.12 14.87 12.31 14.76 2.45 0.99 Group 3 (10 stocks) 9.34 12.37 10.59 12.92 2.34 0.95 Group 4 (30 stocks) 9.39 11.16 10.23 12.90 2.67 0.91 Pael D. Asset Classes 5 Asset classes 4.32 11.66 7.43 8.30 0.87 1.00 Rho
ESTRADA MEAN-SEMIVARIANCE OPTIMIZATION: A HEURISTIC APPROACH 65 Cosider first the combiatio of US stocks ad emergig markets stocks (first lie of Pael A). Portfolios were formed with weights varyig betwee 0% ad 100% i the US market (the rest beig allocated to emergig markets), i icremets of 1%. Mothly returs over the Jauary 1997-December 2006 period were the calculated for these 101 portfolios. Usig these returs, the exact semideviatio with respect to a 0% bechmark retur was calculated for all these portfolios accordig to Equatio (3) ad subsequetly aualized. The secod colum of the table shows the miimum (10.21%) ad maximum (17.29%) values across the 101 aualized semideviatios, ad the third colum shows the average (13.15%). Approximate semideviatios accordig to Equatio (12), with Semivariace is a more plausible measure of risk tha variace, as Markowitz (1991) himself suggested, ad the heuristic proposed here makes mea-semivariace optimizatio just as easy to implemet as mea-variace optimizatio. respect to a 0% bechmark retur, were the calculated ad subsequetly aualized for the 101 portfolios. The average amog all these aualized approximate semideviatios (13.24%) is reported i the fourth colum of the table. The differece betwee the average exact semideviatio ad the average approximate semideviatio is reported i the fifth colum, ad at 0.09% i aual terms is basically egligible. Furthermore, the correlatio betwee the 101 exact ad approximate semideviatios, reported i the last colum, is a perfect 1.00. These results obviously support the heuristic proposed here. The results are equally ecouragig for portfolios of the other two-asset combiatios i Pael A, geerated with the methodology already described. The average differece betwee exact ad approximate semideviatios is 0.32% for the 101 portfolios of US stocks ad real estate, ad 0.30% for the 101 portfolios of emergig markets ad real estate, both i aual terms. The correlatios betwee the 101 exact ad approximate semideviatios are 0.99 i the first case ad 1.00 i the secod. Agai, these results are clearly ecouragig. Pael B shows portfolios of emergig markets. Group 1 cosists of 5 emergig markets (Chia, Egypt, Korea, Malaysia, ad Veezuela) that over the sample period displayed statistically-sigificat positive skewess. Portfolios were formed by geeratig 100 radom weights for each of these idices, subsequetly stadardized to esure that for each portfolio their sum added to oe. As before, returs for these 100 portfolios were calculated over the Jauary 1997-December 2006 period. The, exact ad approximate semideviatios with respect to a 0% bechmark retur were calculated for all portfolios ad subsequetly aualized. The correlatio betwee the 100 exact ad approximate semideviatios is 0.97 ad the differece betwee the averages is i this case higher, 2.1% i aual terms. Group 2 cosists of five emergig markets (Chile, Hugary, Mexico, Peru, ad South Africa) that over the sample period displayed statistically-sigificat egative skewess. Portfolios combiig these five markets were geerated with the same methodology described for the markets i Group 1. As the table shows, the correlatio betwee the 100 exact ad approximate semideviatios is 0.98, ad the differece betwee the averages is substatially lower tha i group 1, 0.68% i aual terms. Combiig the five emergig markets from Group 1 ad the five from Group 2 ito a 10-market portfolio (Group 3), the methodology described was applied oce agai. The correlatio betwee the 100 exact ad approximate semideviatios is 0.93, ad the average differece betwee them is 1.80% i aual terms. For portfolios of emergig markets, the, the proposed heuristic yields almost perfect correlatios betwee exact ad approximate semideviatios, ad the average differeces betwee them are somewhat higher tha for the asset classes previously discussed. I all cases, whe the approximatio errs, it does so o the side of cautio, overestimatig the risk of the portfolio i the magitudes already discussed. Pael C cosiders portfolios of idividual stocks, i particular the 30 stocks from the Dow Joes Idustrial Average. The 30 stocks were ordered alphabetically ad split ito three groups of 10 stocks. For each of these three groups, portfolios were formed followig the same methodology described earlier: 100 radom weights were geerated for each of the 10 stocks, which were subsequetly stadardized to esure that for each portfolio their sum added to oe; returs for each portfolio were geerated over the Jauary 1997- December 2006 period; ad their exact ad approximate semideviatios were calculated ad subsequetly aualized. As the table shows, the correlatios betwee the exact ad approximate semideviatios are still very high i all three groups (0.90, 0.99, ad 0.95), ad the average differeces betwee these magitudes are 2.07%, 2.45%, ad 2.34% (i aual terms) for Groups 1, 2, ad 3, agai higher tha for asset classes. Portfolios of the 30 Dow stocks altogether, calculated with the same methodology already described, show similar results.
66 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 The correlatio betwee exact ad approximate semideviatios across the 100 portfolios remais very high (0.91); ad the average differece betwee these two magitudes is 2.67% i aual terms, agai higher tha for asset classes. As before, whe the approximatio errs it does so o the side of cautio, overestimatig the risk of the portfolio i the magitudes already discussed. Fially, Pael D cosiders portfolios of five asset classes, amely: 1) US stocks, 2) iteratioal (EAFE) stocks, 3) emergig markets stocks, 4) US bods, ad 5) US real estate. Portfolios of these five asset classes were formed with the methodology already described. The correlatio betwee the 100 exact ad approximate semideviatios is a perfect 1.00; the average differece betwee these magitudes, i tur, is a low 0.87% i aual terms. Agai, whe the approximatio errs it does so o the side of cautio; ad agai, the heuristic shows very ecouragig results for asset classes. I short, the evidece for a wide rage of portfolios shows that the heuristic proposed i this article yields portfolio semivariaces that are very highly correlated, as well as close i value, to the exact portfolio semivariaces they ited to approximate. Importatly, as argued by Nawrocki (1999), ad as is also well kow, portfolio optimizatio is owadays used much more for allocatig fuds across asset classes tha across idividual stocks. It is i the former case, precisely, where the heuristic approach proposed i this article is particularly accurate. B. Optimal Portfolios Havig show that the defiitio of portfolio semivariace proposed here is both simple ad accurate, we ca fially use it to compare the optimal portfolios that stem from meavariace ad mea-semivariace optimizatios, the latter based o the proposed heuristic approach. Give that optimizers are largely used to allocate fuds across asset classes, the assets cosidered i the optimizatios are the five asset classes i Pael D of Table IV; that is, US stocks, iteratioal (EAFE) stocks, emergig markets, US bods, ad US real estate. There are several portfolio-optimizatio problems, ad decidig which oe is more relevat simply depeds o the goals ad restrictios of differet ivestors. Some may wat to miimize risk; others may wat to miimize risk subject to a target retur; others may wat to maximize retur subject to a target level of risk; ad others may wat to maximize risk-adjusted returs. The focus of this sectio is o this last problem. More precisely, the two problems cosidered are Max 1 2 x, x,..., x Ep R f xe R = σ p i = 1 i i f 1/2 ( xx i = 1 j = 1 i jσ ij ) (15) x = i 1 ad x i = i 0, (16) 1 for the optimizatio of mea-variace (MV) portfolios, ad Max 1 2 x, x,..., x Ep R f xe R = Σ p0 i = 1 i i f 1/ ( xx i = 1 j = 1 i jσ ij0 ) 2 (17) x = i 1 ad x = i 0, (18) 1 i for the optimizatio of mea-semivariace (MS) portfolios, where Σ p0 is defied as i Equatio (12), ad the bechmark retur for the semideviatio is, as before, 0%. It is importat to otice that, with the heuristic proposed here, the problem i Equatios (17) ad (18) ca be solved with the same techiques widely used to solve the problem i Equatios (15) ad (16); these iclude professioal optimizatio packages, simple optimizatio packages available with ivestmet textbooks, ad eve Excel s solver. It is also importat to otice that i terms of the required iputs, the oly differece betwee these two problems is that Equatios (15) ad (16) require a (symmetric ad exogeous) covariace matrix ad Equatio (17) ad (18) require a (symmetric ad exogeous) semicovariace matrix, which ca be calculated usig Equatio (5). Expected returs, required as iputs i both optimizatio problems, were estimated with the (arithmetic) mea retur of each asset class over the whole Jauary 1988-December 2006 sample period. Variaces, covariaces, semivariaces, ad semicovariaces were calculated over the same sample period, the last two with respect to a 0% bechmark retur ad accordig to Equatio (5). Optimizatios were performed for combiatios of three, four, ad five asset classes. The results of all estimatios are show i Table V. Whe optimizig a three-asset portfolio cosistig of US stocks, iteratioal stocks, ad emergig markets, either the MV optimizer or the MS optimizer give a positive weight to iteratioal stocks. Perhaps usurprisigly, the MS optimizer gives a lower weight to emergig markets ad a higher weight to the US market tha does the MV optimizer. The expected mothly retur of the optimal MV ad MS portfolios is similar, 1.17% ad 1.13%. Although the riskadjusted retur of the MS optimal portfolio is higher tha that of the MV optimal portfolio, it would be deceivig to coclude that the MS optimizer outperforms the MV
ESTRADA MEAN-SEMIVARIANCE OPTIMIZATION: A HEURISTIC APPROACH 67 Table V. Optimal Portfolios Asset Classes This table shows mea-variace (MV) ad mea-semivariace (MS) optimal portfolios. Risk is defied as the stadard deviatio i MV optimizatios ad as the semideviatio i MS optimizatios. RAR deotes risk-adjusted returs defied as (Retur R f )/Risk, where R f deotes the risk-free rate. Retur ad risk are expressed i mothly terms. Mothly R f is assumed at 0.41%. All returs are mothly, over the Jauary 1988-December 2006 period, i dollars, ad accout for capital gais ad divideds. All umbers but RAR are i percetages. A full data descriptio is available i the Appedix. Weights Performace USA EAFE EMI Bods NAREIT Retur Risk RAR Pael A. Three Assets MV 67.0 0.0 33.0 1.17 4.30 0.18 MS 78.4 0.0 21.6 1.13 2.66 0.27 Pael B. Four Assets MV 17.5 0.0 18.8 63.7 0.87 2.11 0.22 MS 26.4 0.0 10.2 63.4 0.84 1.38 0.31 Pael C. Five Assets MV 10.4 0.0 13.2 49.1 27.2 0.91 2.07 0.24 MS 17.1 0.0 3.0 41.4 38.5 0.90 1.41 0.35 optimizer; this is simply a cosequece of the fact that the semideviatio is a oe-sided measure of risk. 3 The four-asset optimizatio ivolves the previous three asset classes plus US bods. Agai both optimizers assig a zero weight to iteratioal stocks, ad agai the MS optimizer allocates less to emergig markets ad more to the US market tha does the MV optimizer. Iterestigly, both optimizers allocate a substatial proportio (early two thirds) of the portfolio to bods. As was the case with three assets, the expected mothly retur of both optimal portfolios is very similar, 0.87% ad 0.84%. Fially, the five-asset optimizatio ivolves the previous four asset classes plus US real estate. Oce agai both optimizers give a zero weight to iteratioal stocks, ad oce agai the MS optimizer allocates less to emergig markets ad more to the US market tha does the MV optimizer. Both optimizers allocate more tha 40% of the portfolio to bods ad o less tha 25% of the portfolio to real estate. Ad oce agai, the expected mothly retur of both portfolios is very similar, 0.91% ad 0.90%. It is temptig to draw a coclusio regardig which optimizer performs better, but it is also largely meaigless. By defiitio, the MV optimizer will maximize the excess returs per uit of volatility, whereas the MS optimizer will maximize excess returs per uit of volatility below the 3 Comparig the risk-adjusted returs of MV ad MS optimizers is oiformative at best ad deceivig at worst. Similarly, comparig the efficiet sets geerated by these two approaches, as doe by Harlow (1991) ad others, yields little isight, if ay. By defiitio, MV efficiet sets will outperform MS efficiet sets whe plotted o a mea-variace graph, ad the opposite will be the case whe plotted o a mea-semivariace graph. chose bechmark. I the ed, it all comes dow to what ay give ivestor perceives as the more appropriate measure of risk. III. A Assessmet There is little questio that mea-variace optimizatio is far more pervasive tha mea-semivariace optimizatio. This is, at least i part, due to the fact that mea-variace problems have well-defied, well-kow closed-form solutios, which implies that users kow what the optimizatio package is doig ad what characteristics the solutio obtaied has. Whe optimizig portfolios o the basis of meas ad semivariaces, i tur, little is usually kow about the algorithms used to obtai optimal portfolios ad the characteristics of the solutio obtaied. This article proposes a heuristic approach for the calculatio of portfolio semivariace, which essetially puts mea-semivariace optimizatio withi reach of ay academic or practitioer familiar with mea-variace optimizatio. By replacig the symmetric ad exogeous covariace matrix by a symmetric ad exogeous semicovariace matrix, the well-defied, well-kow closedform solutios of mea-variace problems ca be applied to mea-semivariace problems. This takes mea-semivariace optimizatio away from the realm of black boxes ad ito the realm of stadard portfolio theory. The heuristic proposed is both simple ad accurate. Estimatig semicovariaces is just as easy as esitmatig covariaces, ad aggregatig them ito a portfolio semivariace is, with the proposed heuristic, just as easy as
68 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 aggregatig covariaces ito a portfolio variace. Similarly, fidig optimal portfolios (regardless of whether that meas miimizig risk, miimizig risk subject to a target retur, maximizig retur subject to a target level of risk, or maximizig risk-adjusted returs) whe risk is thought of as semivariace ca be doe with the same methods used as whe risk is thought of as variace. I terms of accuracy, the proposed defiitio of portfolio semivariace was evaluated usig portfolios of stocks, markets, ad asset classes. The evidece discussed shows that the portfolio semivariaces geerated by the heuristic proposed are very highly correlated, as well as close i value, to the exact portfolio semivariaces they aim to approximate. This heuristic is particularly accurate whe optimizig across asset classes, which owadays is the mai use give to optimizers. There is a growig literature o dowside risk ad a icreasig acceptace of this idea amog both academics ad practitioers. Semivariace is a more plausible measure of risk tha variace, as Markowitz (1991) himself suggested, ad the heuristic proposed here makes mea-semivariace optimizatio just as easy to implemet as mea-variace optimizatio. For this reaso, this article ot oly provides aother tool that ca be added to the fiacial toolbox, but also hopefully cotributes toward icreasig the acceptace ad use of mea-semivariace optimizatio. Appedix 1. A Brief Itroductio to the Semideviatio This sectio aims to itroduce the semideviatio to readers largely uaware of this cocept. Readers who wat to explore this issue further are referred to Estrada (2006), a article from which sectio of the appedix borrows heavily. A. Shortcomigs of the Stadard Deviatio Cosider a asset with a mea aual retur of 10%, ad assume that i the last two years the asset retured 5% ad 25%. Because both returs deviate from the mea by the same amout (15%), they both icrease the stadard deviatio of the asset by the same amout. But is a ivestor i this asset equally (u)happy i both years? Not likely, which uderscores oe of the mai problems of the stadard deviatio as a measure of risk: it treats a x% fluctuatio above ad below the mea i the same way, though ivestors obviously do ot. Should it ot, the, a proper measure of risk capture this asymmetry? The secod colum of Table A1 shows the aual returs of Oracle (R) for the years 1995 2004. As the ext-to-last row shows, the stock s mea aual retur (µ) durig this period was a healthy 41.1%. Ad as is obvious from a casual observatio of these returs without resortig to ay formal measure of risk, Oracle treated its shareholders to quite a bumpy ride. The third colum of the table shows the differece betwee each aual retur ad the mea aual retur; for example, for the year 2004, 37.4% = 3.7% 41.1%. The fourth colum shows the square of these umbers; for example, 0.1396 = ( 0.374) 2. The average of these squared deviatios from the mea is the variace (0.8418), ad the square root of the variace is the stadard deviatio (91.7%). Note that all the umbers i the fourth colum are positive, which meas that every retur, regardless of its sig, cotributes to icreasig the stadard deviatio. I fact, the largest umber i this fourth colum (the oe that cotributes to icreasig the stadard deviatio the most) is that for the year 1999 whe Oracle delivered a positive retur of almost 290%. Now, would a ivestor that held Oracle durig the year 1999 be happy or uhappy? Would he cout this performace agaist Oracle as the stadard deviatio as a measure of risk does? We will get back to this below but before we do so cosider aother shortcomig of the stadard deviatio as a measure of risk: it is largely meaigless whe the uderlyig distributio of returs is ot symmetric. Skewed distributios of returs, which are far from uusual i practice, exhibit differet volatility above ad below the mea. I these cases, variability aroud the mea is at best uiformative ad more likely misleadig as a measure of risk. B. The Semideviatio As Table A1 makes clear, oe of the mai problems of the stadard deviatio as a measure of risk is that, ulike ivestors, it treats fluctuatios above ad below the mea i the same way. However, tweakig the stadard deviatio so that it accouts oly for fluctuatios below the mea is ot difficult. The fifth colum of Table A1 shows coditioal returs with respect to the mea; that is, the lower of each retur mius the mea retur or zero. I other words, if a retur is higher tha the mea, the colum shows a 0; if a retur is lower tha the mea, the colum shows the differece betwee the two. To illustrate, i 1995 Oracle delivered a 44.0% retur, which is higher tha the mea retur of 41.1%; therefore the fifth colum shows a 0 for this year. I 2004, however, Oracle
ESTRADA MEAN-SEMIVARIANCE OPTIMIZATION: A HEURISTIC APPROACH 69 Table A1. Oracle, 1995-2004 Year R R µ (R µ) 2 Mi(R µ, 0) {Mi(R µ, 0)} 2 1995 44.0% 3.0% 0.0009 0.0% 0.0000 1996 47.8% 6.7% 0.0045 0.0% 0.0000 1997 19.8% 60.9% 0.3709 60.9% 0.3709 1998 93.3% 52.2% 0.2726 0.0% 0.0000 1999 289.8% 248.7% 6.1849 0.0% 0.0000 2000 3.7% 37.3% 0.1394 37.3% 0.1394 2001 52.5% 93.6% 0.8752 93.6% 0.8752 2002 21.8% 62.9% 0.3952 62.9% 0.3952 2003 22.5% 18.6% 0.0345 18.6% 0.0345 2004 3.7% 37.4% 0.1396 37.4% 0.1396 Average 41.1% 0.8418 0.1955 Square Root 91.7% 44.2% delivered a 3.7% retur, which is below the mea retur of 41.1%; therefore, the fifth colum shows the shortfall of 37.4% for this year. Comparig the third ad the fifth colums, it is clear that whe a retur is lower tha the mea both colums show the same umber; whe a retur is higher tha the mea, however, the third colum shows the differece betwee these two umbers ad the fifth colum shows a 0. Furthermore, it is clear that coditioal returs are either egative or 0 but ever positive. The last colum of Table A1 shows the square of the umbers i the fifth colum. As the ext-to-last row shows, the average of these umbers is 0.1955; ad as the last row shows, the square root of this umber is 44.2%. This umber, which measures volatility below the mea retur, is obviously a step i the right directio because we have isolated the dowside that ivestors associate with risk. But is there aythig special about the mea retur? Is it possible that some ivestors are iterested to assess volatility below the risk-free rate? Or volatility below 0? Or, more geerally, volatility below ay give retur they may cosider relevat? That is exactly what the dowside stadard deviatio of returs with respect to a bechmark B measures. This magitude, usually referred to as the semideviatio with respect to B (Σ B ), for short, is formally defied as Σ B T t = 1 { } = (/ 1 T ) Mi( R B, 0 ) 2 t ad measures dowside volatility; or, more precisely, volatility below the bechmark retur B. I this expressio, t idexes time ad T deotes the umber of observatios. Table A2 shows agai the returs of Oracle for the 1995-2004 period, as well as the coditioal returs with respect to three differet bechmarks: the mea retur, a risk-free rate (R f ) of 5%, ad 0. The third colum of this exhibit is the same as the last colum of Table A1 ad therefore the bechmark is the mea retur; the fourth ad fifth colums show coditioal returs with respect to the other two bechmarks (a risk-free rate of 5% ad 0). The last row shows the semideviatios with respect to all three bechmarks. (The ext-to-last row shows the semivariaces with respect to all three bechmarks, which are simply the square of the semideviatios.) How should these umbers be iterpreted? Each semideviatio measures volatility below its respective bechmark. Note that because the risk-free rate of 5% is below Oracle s mea retur of 41.1%, we would expect (ad fid) less volatility below the risk-free rate tha below the mea. Similarly, we would expect (ad agai fid) less volatility below 0 tha below the mea or the risk-free rate. It may seem that a volatility of 21.5% below a risk-free rate of 5%, or a volatility of 19% below 0, do ot covey a great deal of iformatio about Oracle s risk. I fact, the semideviatio of a asset is best used i two cotexts: oe is i relatio to the stadard deviatio of the same asset ad the other is i relatio to the semideviatio of other assets. Table A3 shows the stadard deviatio (σ) of Oracle ad Microsoft over the 1995-2004 period, as well as the semideviatios with respect to the mea of each stock (Σ µ ), with respect to a risk-free rate of 5% (Σ f ), ad with respect to 0 (Σ 0 ) over the same period. The semideviatios of Oracle are the same as those i Table A2. The mea retur of Microsoft durig this period was 35.5%. Note that although the stadard deviatios suggest that Oracle is far riskier tha Microsoft, the semideviatios tell a differet story. First, ote that although the volatility of Oracle below its mea is less tha half of its total volatility (0.442/
70 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 Table A2. Semideviatios Year R {Mi(R µ, 0)} 2 {Mi(R R f, 0)} 2 {Mi(R 0, 0)} 2 1995 44.0% 0.0000 0.0000 0.0000 1996 47.8% 0.0000 0.0000 0.0000 1997 19.8% 0.3709 0.0617 0.0393 1998 93.3% 0.0000 0.0000 0.0000 1999 289.8% 0.0000 0.0000 0.0000 2000 3.7% 0.1394 0.0002 0.0000 2001 52.5% 0.8752 0.3304 0.2754 2002 21.8% 0.3952 0.0718 0.0475 2003 22.5% 0.0345 0.0000 0.0000 2004 3.7% 0.1396 0.0002 0.0000 Average 41.1% 0.1955 0.0464 0.0362 Square Root 44.2% 21.5% 19.0% Table A3. Volatility ad Dowside Volatility Compay σ Σ µ Σ f Σ 0 Oracle 91.7% 44.2% 21.5% 19.0% M icrosoft 50.4% 38.1% 23.1% 21.1% 0.917 = 48.2%), the same ratio for Microsoft is over 75% (0.381/0.504 = 75.5%). I other words, give the volatility of each stock, much more of that volatility is below the mea i the case of Microsoft tha i the case of Oracle. (I fact, the distributio of Microsoft s returs has a slight egative skewess, ad that of Oracle a sigificat positive skewess.) Of course it is still the case that the semideviatio with respect to the mea of Oracle is larger tha that of Microsoft; but recall that the mea retur of Oracle (41.1%) is also higher tha that of Microsoft (35.5%). For this reaso, it is perhaps more tellig to compare semideviatios with respect to the same bechmark for both stocks. Comparig the semideviatios of Oracle ad Microsoft with respect to the same risk-free rate of 5%, we see that Microsoft exhibits higher dowside volatility (23.1% versus 21.5%). Ad comparig their semideviatios with respect to 0, we agai see that Microsoft exhibits higher dowside volatility (21.1% versus 19.0%). Therefore, although the stadard deviatios suggest that Oracle is riskier tha Microsoft, the semideviatios suggest the opposite. 2. The Data Table A4 describes the data used i Sectio II of the article (The Evidece). All series are mothly, i dollars, ad accout for capital gais ad divideds. Table 3 is based o data over the Jauary 1997-December 2006 period ad Table 4 o data over the Jauary 1988-December 2006 period.
ESTRADA MEAN-SEMIVARIANCE OPTIMIZATION: A HEURISTIC APPROACH 71 Table A4. The Data USA MSCI USA EMI MSCI EMI NAREIT FTSE NAREIT - All REITs Emergig Markets MSCI idices Emergig Markets Group 1: Five emergig markets (Chia, Egypt, Korea, Malaysia, Veezuela) with sigificat positive skewess Emergig Markets Group 2: Five emergig markets (Chile, Hugary, Mexico, Peru, South Africa) with sigificat egative skewess Emergig Markets Group 3: Te markets, the five from Group 1 plus the five from Group 2 DJIA Stocks Idividual stocks from the Dow Joes Idustrial Average idex DJIA Stocks Group 1: The first te stocks from a alphabetical orderig of the Dow (3M, Alcoa, Altria, Amex, AIG, AT&T, Boeig, Caterpillar, Citigroup, Coca-Cola) DJIA Stocks Group 2: The secod te stocks from a alphabetical orderig of the Dow (DuPot, ExxoMobil, GE, GM, HP, HomeDepot, Hoeywell, Itel, IBM, J&J) DJIA Stocks Group 3: The third te stocks from a alphabetical orderig of the Dow (JPM-Chase, McDoald s, Merck, Microsoft, Pfizer, P&G, Uited Tech, Verizo, WalMart, WaltDisey) DJIA Stocks Group 4: All thirty stocks i the Dow Asset Classes US stocks (MSCI USA), iteratioal stocks (MSCI EAFE), emergig markets stocks (MSCI EMI), US bods (10-year Govermet bods Global Fiacial Data), ad US real estate (FTSE NAREIT All REITs) MSCI: Morga Staley Capital Idices; EMI: Emergig Markets Idex; FTSE: Fiacial Times Stock Exchage; NAREIT: Natioal Associatio of Real Estate Ivestmet Trusts; DJIA: Dow Joes Idustrial Average; EAFE: Europe, Australia, ad the Far East. Refereces Ag, J., 1975, A Note o the E, SL Portfolio Selectio Model, Joural of Fiacial ad Quatitative Aalysis 10 (No. 5, December), 849-857. Ballestero, E., 2005, Mea-Semivariace Efficiet Frotier: A Dowside Risk Model for Portfolio Selectio, Applied Mathematical Fiace 12 (No. 1, March), 1-15. De Athayde, G., 2001, Buildig a Mea-Dowside Risk Portfolio Frotier, i F. Sortio ad S. Satchell Eds., Maagig Dowside Risk i Fiacial Markets, Oxford, UK, Butterworth-Heiema. Elto, E., M. Gruber, ad M. Padberg, 1976, Simple Criteria for Optimal Portfolio Selectio, Joural of Fiace 31 (No. 5, December), 1341-1357. Estrada, J., 2002, Systematic Risk i Emergig Markets: The D-CAPM, Emergig Markets Review 3 (No. 4, December), 365-379. Estrada, J., 2006, Dowside Risk i Practice, Joural of Applied Corporate Fiace 18 (No. 1, Witer), 117-125. Estrada, J., 2007, Mea-Semivariace Behavior: Dowside Risk ad Capital Asset Pricig, Iteratioal Review of Ecoomics ad Fiace 16 (No. 2), 169-185. Hoga, W. ad J. Warre, 1972, Computatio of the Efficiet Boudary i the E-S Portfolio Selectio Model, Joural of Fiacial ad Quatitative Aalysis 7 (No. 4, September), 1881-1896. Hoga, W. ad J. Warre, 1974, Toward the Developmet of a Equilibrium Capital-Market Model Based o Semivariace, Joural of Fiacial ad Quatitative Aalysis 9 (No. 1, Jauary), 1-11. Grootveld, H. ad W. Hallerbach, 1999, Variace vs. Dowside Risk: Is There Really that Much Differece? Europea Joural of Operatioal Research 114 (No. 2, April), 304-319. Harlow, V., 1991, Asset Allocatio i a Dowside Risk Framework, Fiacial Aalysts Joural 47 (No. 5, September/October), 28-40. Markowitz, H., 1952, Portfolio Selectio, Joural of Fiace 7 (No. 1, March), 77-91.
72 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008 Markowitz, H., 1959, Portfolio Selectio: Efficiet Diversificatio of Ivestmets, New York, NY, Joh Wiley & Sos. Markowitz, H., 1991, Portfolio Selectio: Efficiet Diversificatio of Ivestmets, 2 d ed., Cambridge, MA, Basil Blackwell. Markowitz, H., P. Todd, G. Xu, ad Y. Yamae, 1993, Computatio of Mea-Semivariace Efficiet Sets by the Critical Lie Algorithm, Aals of Operatios Research 45 (No. 1, December), 307-317. Nawrocki, D., 1999, A Brief History of Dowside Risk Measures, Joural of Ivestig (Fall), 9-25. Nawrocki, D. ad K. Staples, 1989, A Customized LPM Risk Measure for Portfolio Aalysis, Applied Ecoomics 21 (No. 2, February), 205-218. Sig, T. ad S. Og, 2000, Asset Allocatio i a Dowside Risk Framework, Joural of Real Estate Portfolio Maagemet 6 (No. 3), 213-223. Nawrocki, D., 1983, A Compariso of Risk Measures Whe Used i A Simple Portfolio Selectio Heuristic, Joural of Busiess Fiace ad Accoutig 10 (No. 2, Jue), 183-194.