PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS Performance Aribuion for Equiy Porfolios Yang Lu and David Kane Inroducion Many porfolio managers measure performance wih reference o a benchmark. The difference in reurn beween a porfolio and is benchmark is he acive reurn of he porfolio. Porfolio managers and heir cliens wan o know wha caused his acive reurn. Performance aribuion decomposes he acive reurn. The wo mos common approaches are he Brinson-Hood-Beebower (hereafer referred o as he Brinson model) and a regression-based analysis. 1 Porfolio managers use differen variaions of he wo models o assess he performance of heir porfolios. Managers of fixed income porfolios include yield-curve movemens in he model Lord (1997) while equiy managers who focus on he effec of currency movemens use variaions of he Brinson model o incorporae local risk premium Singer and Karnosky (1995). In conras, in his paper we focus on aribuion models for equiy porfolios wihou considering any currency effec. The pa package provides ools for conducing boh mehods for equiy porfolios. The Brinson model akes an AOVA-ype approach and decomposes he acive reurn of any porfolio ino asse allocaion, sock selecion, and ineracion effecs. The regression-based analysis uilizes esimaed coefficiens from a linear model o aribue acive reurn o differen facors. Afer describing he Brinson and regression approaches and demonsraing heir use via he pa package, we show ha he Brinson model is jus a special case of he regression approach. Daa We demonsrae he use of he pa package wih a series of examples based on real-world daa ses from MSCI Barra s Global Equiy Model II(GEM2). 2 MSCI Barra is a leading provider of invesmen decision suppor ools o invesmen insiuions worldwide. According o he company: GEM2 is he laes Barra global muli-facor equiy model. I provides a foundaion for invesmen decision suppor ools via a broad range of insighful analyics for developed and emerging marke porfolios. The laes model version provides: 1 See Brinson e al. (1986) and Grinold (2006) for more informaion. 2 See www.msci.com and Menchero e al. (2008) for more informaion. 3 Global Indusry Classificaion Sandard Improved accuracy of risk forecass and increased explanaory power. An inuiive srucure ha accommodaes differen invesmen processes in developed vs. emerging markes. Greaer responsiveness o marke dynamics. Comprehensive marke coverage. GEM2 leverages he decades of experience ha MSCI Barra has in developing and mainaining global equiy muli-facor models and indices, and offers imporan enhancemens over GEM, which is uilized by hundreds of insiuional fund managers worldwide. The original daa se conains seleced aribues such as indusry, size, counry, and various syle facors for a universe of approximaely 48,000 securiies on a monhly basis. For illusraive purposes, his aricle uses hree modified versions of he original daa se, conaining 3000 securiies, namely year, quarer, and jan. The daa frame, quarer, is a subse of year, conaining he daa of he firs quarer. The daa frame, jan, is a subse of quarer wih he daa from January, 2010. > daa(year) > names(year) [1] "barrid" "name" "reurn" [4] "dae" "secor" "momenum" [7] "value" "size" "growh" [10] "cap.usd" "yield" "counry" [13] "currency" "porfolio" "benchmark" barrid: securiy idenifier by Barra. name: name of a securiy. monhly oal reurn in rading cur- reurn: rency. dae: he saring dae of he monh o which he daa belong. secor: consolidaed secor caegories based on he GICS. 3 momenum: capure susained relaive performance. value: capure he exen o which a sock is priced inexpensively in he marke. size: differeniae beween large and small cap companies. 1
THE BRISO MODEL PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS growh: capure sock s growh prospecs. cap.usd: capializaion in model base currency USD. yield: dividend of a securiy. counry: he counry in which he company is raded. currency: currency of exposure. porfolio: op 200 securiies based on value scores in January are seleced as porfolio holdings and are held hrough December 2010. This is o avoid he complexiy of rading in he analyses. benchmark: op 1000 securiies based on size each monh. The benchmark is cap-weighed. Here is a sample of rows and columns from he daa frame year: name 44557 BLUE STAR OPPORTUITIES CORP 25345 SEADRILL 264017 BUXLY PAITS (PKR10) 380927 CD IMPERIAL BK OF COMMERCE 388340 CD IMPERIAL BK OF COMMERCE reurn dae secor size 44557 0.00000 2010-01-01 Energy 0.00 25345-0.07905 2010-01-01 Energy -0.27 264017-0.01754 2010-05-01 Maerials 0.00 380927 0.02613 2010-08-01 Financials 0.52 388340-0.00079 2010-11-01 Financials 0.55 counry porfolio benchmark 44557 USA 0.000 0.000000 25345 OR 0.000 0.000427 264017 PAK 0.005 0.000000 380927 CA 0.005 0.000012 388340 CA 0.005 0.000012 The porfolio has 200 equal-weighed holdings. The row for Canadian Imperial Bank of Commerce indicaes ha i is one of he 200 porfolio holdings wih a weigh of 0.5% in 2010. Is reurn was 2.61% in Augus, and almos fla in ovember. The Brinson Model Single-Period Brinson Model Consider an equiy porfolio manager who uses he S&P 500 as he benchmark. In a given monh, she ouperformed he S&P by 3%. Par of ha performance was due o he fac ha she allocaed more weigh of he porfolio o cerain secors ha performed well. Call his he allocaion effec. Par of her ouperformance was due o he fac ha some of he socks she seleced did beer han heir secor as a whole. Call his he selecion effec. The residual can hen be aribued o an ineracion beween allocaion and selecion he ineracion effec. The Brinson model provides mahemaical definiions for hese erms and mehods for calculaing hem. The example above uses secor as he classificaion scheme when calculaing he allocaion effec. Bu he same approach can work wih any oher variable which places each securiy ino one, and only one, discree caegory: counry, indusry, and so on. In fac, a similar approach can work wih coninuous variables ha are spli ino discree ranges: he highes quinile of marke cap, he second highes quinile and so forh. For generaliy, we will use he erm caegory o describe any classificaion scheme which places each securiy in one, and only one, caegory. oaions: w B i w P i is he weigh of securiy i in he benchmark. is he weigh of securiy i in he porfolio. Wj B is he weigh of caegory j in he benchmark. Wj B = wi B, i j. Wj P is he weigh of a caegory j in he porfolio. Wj P = wi P, i j. The sum of he weigh wi B, wp i, WB j, and WP j 1, respecively. r i is he reurn of securiy i. R B j is he reurn of a caegory j in he benchmark. R B j = wi Br i, i j. R P j is he reurn of a caegory j in he porfolio. R P j = wi Pr i, i j. The reurn of a porfolio, R P, can be calculaed in wo ways: On an individual securiy level by summing over n socks: R P = n wi Pr i. i=1 On a caegory level by summing over caegories: R P = Wj P R P j. Similar definiions apply o he reurn of he benchmark, R B, R B = n wi Br i. i=1 R B = Wj B R B j. is 2
THE BRISO MODEL PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS Acive reurn of a porfolio, R acive, is a performance measure of a porfolio relaive o is benchmark. The wo convenional measures of acive reurn are arihmeic and geomeric. The pa package implemens he arihmeic measure of he acive reurn for a single-period Brinson model because an arihmeic difference is more inuiive han a raio over a single period. The arihmeic acive reurn of a porfolio, R acive, is he porfolio reurn R P less he benchmark reurn R B : R acive = R P R B. Since he caegory allocaion of he porfolio is generally differen from ha of he benchmark, allocaion plays a role in he acive reurn, R acive. The same applies o sock selecion where assuming ha he porfolio has he exac same caegorical exposures as he benchmark does, equiies wihin each caegory are differen. This conribues o R acive as well. Allocaion effec R allocaion and selecion effec R selecion over caegories are defined as: R allocaion = Wj P R B j Wj B R B j, R selecion = Wj B R P j Wj B R B j. The inuiion behind he allocaion effec is ha a porfolio would produce differen reurns wih differen allocaion schemes (Wj P vs. Wj B ) while having he same sock selecion and hus he same reurn (R B j ) for each caegory. The difference beween he wo reurns, caused by he allocaion scheme, is called he allocaion effec (R allocaion ). Similarly, wo differen reurns can be produced when wo porfolios have he same allocaion (Wj B ) ye dissimilar reurns due o differences in sock selecion wihin each caegory (R p j vs. R B j ). This difference is he selecion effec (R selecion). Ineracion effec, R ineracion, is he resul of subracing reurn due o allocaion R allocaion and reurn due o selecion R selecion from he acive reurn R acive : R ineracion = R acive R allocaion R selecion. Weakness of he Brinson Model The Brinson model allows porfolio managers o analyze he relaive reurn of a porfolio using any aribue of a securiy, such as counry or secor. One weakness of he model is o expand he analysis beyond wo caegories. 4 As he number of caegories increases, his procedure is subjec o he curse of dimensionaliy. Suppose an equiy porfolio manager wans o find ou he conribuions of any wo caegories (for insance, counry and secor) o her porfolio based on he Brinson model. She can decompose he acive reurn ino hree broad erms R allocaion, R selecion, and R ineracion. The allocaion effec can be furher spli ino counry allocaion effec, secor allocaion effec and he produc of counry and secor allocaion effecs: R allocaion = R counry allocaion + R secor allocaion + R counry allocaion R secor allocaion. Specifically, he counry allocaion effec is he reurn caused by he difference beween he acual counry allocaion and he benchmark counry allocaion while assuming he same benchmark reurn wihin each level of he caegory counry, ha is, where R counry allocaion = C W P j CR B j C W B j CR B j, C W P j and C W B j refer o he weigh of each counry j ( C counries in oal) in he porfolio and ha in he benchmark, respecively. C R B j refers o he benchmark reurn of any counry j. Similarly, he secor allocaion effec is he difference in reurn beween a porfolio s secor allocaion and he benchmark s secor allocaion while having he same benchmark reurns: R secor allocaion = S W P j SR B j S W B j SR B j, SWj P and S Wj B refer o he weigh of he secor j in he porfolio and he weigh of he secor j in he benchmark, respecively. SR B j is he benchmark reurn of any given secor j of all S secors. In he same vein, he reurn as a resul of he selecion effec R selecion is he sum of counry selecion effec, secor selecion effec, and he produc of counry and secor selecion effecs: R selecion = R counry selecion + R secor selecion +R counry selecion R secor selecion = + + ( ( 4 Brinson e al. (1991) proposed a framework o include wo variables in he Brinson analysis. CW B j CR P j SW B j SR P j CW B j CR P j SW B j SR P j CW B j CR B j SW B j SR B j CW B j CR B j ) SW B j SR B j ). 3
THE BRISO MODEL PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS The ineracion effec, R ineracion, includes he ineracion beween counry allocaion and secor selecion and ha beween counry selecion and secor allocaion. Therefore, in he case of Q caegories where Q > 1, he Brinson model becomes very complex (assume Q 3): R allocaion = R selecion = + + Q R allocaionj + Q Q Q k=1 p=1 =..., Q R selecionj + Q Q Q k=1 p=1 =..., Q Q k=1 R allocaionj R allocaionk R allocaionp Q Q k=1 R selecionj R selecionk R selecionj R selecionk R selecionp where R allocaionj is he allocaion effec of any given caegory j, j Q and R selecionj is he selecion effec of any given caegory j, j Q. i, j, k have differen values. As he number of caegories grows, he numbers of erms for he allocaion and he selecion effecs grow exponenially. Q caegories will resul in 2 Q 1 erms for each of he allocaion and selecion effec. Due o he ineracion beween allocaion and selecion of each of he Q caegories (i could be ineracion beween 2, 3 or even all Q caegories), he number of erms included in he ineracion effec grows exponenially o ake ino all he ineracion effecs among all caegories. R ineracion = + Q Q k=1 Q Q Q k=1 p=1 +.... R allocaionj R selecionk R allocaionj R selecionk R allocaionp Q caegories has 2 2n 2 n+1 + 1 erms of ineracion effecs. For insance, when here are 3 caegories, he allocaion effec and he selecion effec each have 2 3 1 = 7 erms. The ineracion effec has 2 6 2 4 + 1 = 49 erms. When here are 4 caegories, 2 4 1 = 15 erms have o be esimaed for he allocaion effec as well as he selecion effec, respecively. 2 8 2 5 + 1 = 225 erms have o be calculaed for he ineracion effec of 4 caegories. This poses a significan compuaional challenge when a porfolio manager performs a mulivariae Brinson analysis. To some exen, he regression-based model deailed laer solves he problem of mulivariae aribuion. Single-Period Brinson Tools Brinson analysis is run by calling he funcion brinson o produce an objec of class brinson. Below we show he ools provided in he pa package o analyze a single period porfolio based on he Brinson model. > daa(jan) > br.single <- brinson(x = jan, dae.var = "dae", R allocaionj R allocaionk + ca.var = "secor", + bench.weigh = "benchmark", + porfolio.weigh = "porfolio", + re.var = "reurn") > The daa frame, jan, conains all he informaion necessary o conduc a single-period Brinson analysis. dae.var, ca.var, and reurn idenify he columns conaining he dae, he facor o be analyzed, and he reurn variable, respecively. bench.weigh and porfolio.weigh specify he name of he benchmark weigh column and ha of he porfolio weigh column in he daa frame. Calling summary on he resuling objec br.single of class brinson repors essenial informaion abou he inpu porfolio (including he number of securiies in he porfolio and he benchmark as well as secor exposures) and he resuls of he Brinson analysis. > summary(br.single) Period: 2010-01-01 Mehodology: Brinson Securiies in he porfolio: 200 Securiies in he benchmark: 1000 Exposures Porfolio Benchmark Diff Energy 0.085 0.2782-0.19319 Maerials 0.070 0.0277 0.04230 Indusrials 0.045 0.0330 0.01201 ConDiscre 0.050 0.0188 0.03124 ConSaples 0.030 0.0148 0.01518 HealhCare 0.015 0.0608-0.04576 Financials 0.370 0.2979 0.07215 InfoTech 0.005 0.0129-0.00787 TeleSvcs 0.300 0.1921 0.10792 Uiliies 0.030 0.0640-0.03399 Reurns $`Aribuion by caegory in bps` Allocaion Selecion Ineracion Energy 110.934-37.52 26.059 Maerials -41.534 0.48 0.734 Indusrials 0.361 1.30 0.473 ConDiscre -28.688-4.23-7.044 ConSaples 5.467-3.59-3.673 HealhCare -6.692-4.07 3.063 Financials -43.998 70.13 16.988 InfoTech -3.255-5.32 3.255 TeleSvcs -23.106 41.55 23.348 4
THE BRISO MODEL PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS Uiliies 16.544 83.03-44.108 Toal -13.966 141.77 19.095 $Aggregae 2010-01-01 Allocaion Effec -0.00140 Selecion Effec 0.01418 Ineracion Effec 0.00191 Acive Reurn 0.01469 The br.single summary shows ha he acive reurn of he porfolio, in January, 2010 was 1.47%. This reurn can be decomposed ino allocaion effec (-0.14%), selecion effec (1.42%), and ineracion effec (0.19%). > plo(br.single, var = "secor", ype = "reurn") To obain Brinson aribuion on a muli-period daa se, one calculaes allocaion, selecion and ineracion wihin each period and aggregaes hem across ime. There are five mehods for his arihmeic, geomeric, opimized linking by Menchero (2004), linking by Davies and Laker (2001), and linking by Frongello (2002). We focus on he firs hree mehods in his paper. Arihmeic measure calculaes relaive performance of a porfolio and is benchmark by a difference; geomeric measure does so by a raio. Arihmeic measure is more inuiive bu a well-known challenge in arihmeic aribuion is ha acive reurns do no add up over muliple periods due o geomeric compounding. 5 Geomeric is able o circumven he adding-up problem. Menchero (2004) discussed various linking algorihms o connec arihmeic reurn wih geomeric reurn and argued ha he opimized linking algorihm is he bes way o link aribuion over ime. Arihmeic Aribuion. The arihmeic aribuion model calculaes acive reurn and conribuions due o allocaion, selecion, and ineracion in each period and sums hem over muliple periods. The arihmeic acive reurn over T periods R arihmeic is expressed as: Secor Energy Maerials Indusrials ConDiscre ConSaples HealhCare Financials InfoTech TeleSvcs Uiliies Reurn Porfolio vs. Benchmark 0.10 0.05 0.00 0.05 Reurn Figure 1: Secor Reurn. Type Benchmark Porfolio Figure 1 is a visual represenaion of he reurn of boh he porfolio and he benchmark secor by secor in January, 2010. This plo shows ha in absolue erms, Uiliies performed he bes wih a gain of more han 5% and Consumer Discreionary, he wors performing secor, los more han 10%. Uiliies was also he secor wih he highes acive reurn in he porfolio. Muli-Period Brinson Model R arihmeic = T R acive, =1 and R acive is he acive reurn in a single period. Geomeric Aribuion. The geomeric aribuion is o compound various reurns over T periods where, 1 + R P = T (1 + R P), =1 1 + R B = T (1 + R B), =1 and R P and R B are porfolio and benchmark reurns in a single period, respecively. Geomeric reurn R geomeric is hus he difference beween R p and R B : R geomeric = R p R B. Opimized Linking Algorihm. The wellknown challenge faced in arihmeic aribuion is ha he acual acive reurn over ime is no equal o he arihmeic summaion of single-period acive reurns, i.e., 5 See Bacon (2008) for a complee discussion of he complexiy involved. R geomeric = R arihmeic, R P R B = T R acive. =1 Menchero (2004) proposed an opimized linking coefficien b op o link arihmeic reurns of individual periods wih geomeric reurns over ime, R p R B = T b op R acive, =1 where b op is he opimized linking coefficien in a single period. The opimized linking coefficien b op is he summaion of a naural scaling A and an adjusmen a specific o a ime period, b op = A + a, 5
THE BRISO MODEL PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS where A is an coefficien for linking from he singleperiod o he muli-period reurn and a is an adjusmen o eliminae residuals 6. Since acive reurn over ime R P R B is a summaion of acive reurn in each period adjused o he opimized linking algorihm, he following is rue: R P R B = T =1 b op (R allocaion + R selecion + R ineracion ), where R allocaion, R selecion, and R ineracion represen allocaion, selecion and ineracion in each period, respecively. Wihin each period, he adjused aribuion is hus expressed as and ˆR allocaion ˆR selecion ˆR ineracion = b op R allocaion, = b op R selecion, = b op R ineracion. Therefore, across T periods, acive reurn R acive, he difference beween porfolio reurn R P and benchmark reurn R B, can be wrien as R acive = T ( ˆR allocaion + ˆR selecion + ˆR ineracion ), =1 where R acive = R P R B. $Benchmark Low 0.0681 0.0568 0.0628 2 0.0122 0.0225 0.0170 3 0.1260 0.1375 0.1140 4 0.2520 0.2457 0.2506 High 0.5417 0.5374 0.5557 $Diff Low 0.0719 0.083157 0.0922 2 0.0378 0.047456 0.0280 3 0.0490 0.007490 0.0410 4-0.0170-0.000719-0.0106 High -0.1417-0.137385-0.1507 The exposure mehod on he class br.muli objec shows he exposure of he porfolio and he benchmark based on a user-defined caegory. Here, i shows he exposure on size. We can see ha he porfolio overweighs he benchmark in he lowes quinile in size and underweighs in he highes quinile. > reurns(br.muli, ype = "linking") Muli-Period Brinson Tools In pracice, analyzing a single-period porfolio is meaningless as porfolio managers and heir cliens are more ineresed in he performance of a porfolio over muliple periods. To apply he Brinson model over ime, we can use he funcion brinson and inpu a muli-period daa se (for insance, quarer) as shown below. > daa(quarer) > br.muli <- brinson(quarer, dae.var = "dae", + ca.var = "secor", + bench.weigh = "benchmark", + porfolio.weigh = "porfolio", + re.var = "reurn") The objec br.muli of class brinsonmuli is an example of a muli-period Brinson analysis. > exposure(br.muli, var = "size") $Porfolio Low 0.140 0.140 0.155 2 0.050 0.070 0.045 3 0.175 0.145 0.155 4 0.235 0.245 0.240 High 0.400 0.400 0.405 6 See Menchero (2000) for more informaion on he opimized linking coefficiens. $Raw Allocaion -0.0014 0.0064 0.0046 Selecion 0.0146 0.0178-0.0152 Ineracion 0.0020-0.0074-0.0087 Acive Reurn 0.0151 0.0168-0.0193 $Aggregae 2010-01-01, 2010-03-01 Allocaion 0.0095 Selecion 0.0173 Ineracion -0.0142 Acive Reurn 0.0127 The reurns mehod shows he resuls of he Brinson analysis applied o he daa from January, 2010 hrough March, 2010. The opimized linking algorihm is applied here by seing he ype o linking. The firs porion of he reurns oupu shows he Brinson aribuion in individual periods. The second porion shows he aggregae aribuion resuls. The porfolio formed by op 200 value securiies in January had an acive reurn of 12.7% over he firs quarer of 2010. The allocaion and he selecion effecs conribued 0.95% and 1.73% respecively; he ineracion effec made a loss of 1.42%. 6
REGRESSIO PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS > plo(br.muli, ype = "reurn") Reurn across Periods 2010 01 01 2010 02 01 2010 03 01 Energy Maerials f k is a column vecor of lengh k. The elemens are he esimaed coefficiens from he regression. Each elemen represens he facor reurn of an aribue. u n is a column vecor of lengh n wih residuals from he regression. Secor Indusrials ConDiscre ConSaples HealhCare Financials InfoTech TeleSvcs Uiliies 0.10 0.050.00.050.10 0.10 0.050.00.050.10 0.10 0.050.00.050.10 Reurn Figure 2: Type Benchmark Porfolio Secor Reurn Across Time. Figure 2 depics he reurns of boh he porfolio and he benchmark of he allocaion effec from January, 2010 hrough March. 2010. This plo shows ha for he porfolio, Uiliies performed he bes wih a gain of more han 5% in January and February, 2010 bu anked in March, 2010. Regression Single-Period Regression Model One advanage of a regression-based approach is ha such analysis allows one o define heir own aribuion model by easily incorporaing muliple variables in he regression formula. These variables can be eiher discree or coninuous. Suppose a porfolio manager wans o find ou how much each of he value, growh, and momenum scores of her holdings conribues o he overall performance of he porfolio. Consider he following linear regression wihou he inercep erm based on a single-period porfolio of n securiies wih k differen variables: where r n = X n,k f k + u n r n is a column vecor of lengh n. Each elemen in r n represens he reurn of a securiy in he porfolio. X n,k is an n by k marix. Each row represens k aribues of a securiy. There are n securiies in he porfolio. In he case of his porfolio manager, suppose ha she only has hree holdings in her porfolio. r 3 is hus a 3 by 1 marix wih reurns of all her hree holdings. The marix X 3,3 records he score for each of he hree facors (value, growh, and momenum) in each row. f 3 conains he esimaed coefficiens of a regression r 3 on X 3,3. The acive exposure of each of he k variables, X i, i k, is expressed as X i = w acive x n,i, where X i is he value represening he acive exposure of he aribue i in he porfolio, w acive is a column vecor of lengh n conaining he acive weigh of every securiy in he porfolio, and x n,i is a column vecor of lengh n wih aribue i for all securiies in he porfolio. Acive weigh of a securiy is defined as he difference beween he porfolio weigh of he securiy and is benchmark weigh. Using he example menioned above, he acive exposure of he aribue value, X value is he produc of w acive (conaining acive weigh of each of he hree holdings) and x 3 (conaining value scores of he hree holdings). The conribuion of a variable i, R i, is hus he produc of he facor reurns for he variable i, f i and he acive exposure of he variable i, X i. Tha is, R i = f i X i. Coninuing he example, he conribuion of value is he produc of f value (he esimaed coefficien for value from he linear regression) and X value (he acive exposure of value as shown above). Therefore, he acive reurn of he porfolio R acive is he sum of conribuions of all k variables and he residual u (a.k.a. he ineracion effec), R acive = k R i + u. i=1 For insance, a hypoheical porfolio has hree holdings (A, B, and C), each of which has wo aribues size and value. Reurn ame Size Value Acive_Weigh 1 0.3 A 1.2 3.0 0.5 2 0.4 B 2.0 2.0 0.1 3 0.5 C 0.8 1.5-0.6 Following he procedure as menioned, he facor reurns for size and value are -0.0313 and -0.1250. The acive exposure of size is 0.32 and ha of value is 0.80. The acive reurn of he porfolio is -11% which can be 7
REGRESSIO PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS decomposed ino he conribuion of size and ha of value based on he regression model. Size conribues 1% of he negaive acive reurn of he porfolio and value causes he porfolio o lose he oher 10.0%. Single-Period Regression Tools Anoher convenional aribuion mehodolody is he regression-based analysis. As menioned, he pa package provides ools o analyze boh single-period and muli-period daa frames. > rb.single <- regress(jan, dae.var = "dae", + re.var = "reurn", + reg.var = c("secor", "growh", + "size"), + benchmark.weigh = "benchmark", + porfolio.weigh = "porfolio") > exposure(rb.single, var = "growh") Porfolio Benchmark Diff Low 0.305 0.2032 0.1018 2 0.395 0.4225-0.0275 3 0.095 0.1297-0.0347 4 0.075 0.1664-0.0914 High 0.130 0.0783 0.0517 reg.var specifies he columns conaining variables whose conribuions are o be analyzed. Calling exposure wih a specified var yields informaion on he exposure of boh he porfolio and he benchmark by ha variable. If var is a coninuous variable, for insance, growh, he exposure will be shown in 5 quaniles. Majoriy of he high value securiies in he porfolio in January have relaively low growh scores. > summary(rb.single) Period: 2010-01-01 Mehodology: Regression Securiies in he porfolio: 200 Securiies in he benchmark: 1000 Reurns 2010-01-01 secor 0.003189 growh 0.000504 size 0.002905 Residual 0.008092 Porfolio Reurn -0.029064 Benchmark Reurn -0.043753 Acive Reurn 0.014689 The summary mehod shows he number of securiies in he porfolio and he benchmark, and he conribuion of each inpu variable according o he regression-based analysis. In his case, he porfolio made a loss of 2.91% and he benchmark los 4.38%. Therefore, he porfolio ouperformed he benchmark by 1.47%. Secor, growh, and size conribued 0.32%, 0.05%, and 0.29%, respecively. Muli-Period Regression Model The same challenge of linking arihmeic and geomeric reurns is presen in muli-period regression model. We apply he opimized linking algorihm proposed by Menchero (2000) in he regression aribuion as well. Wihin each period, R acive = k R i, + u, i=1 where R i, represens he conribuion of a variable i of he ime period and u is he residual in ha period. Across T periods, he acive reurn can be expressed by a produc of he opimized linking coefficien b op and he individual conribuion of each of he k aribues. The adjused conribuion of each of he k variables i, ˆR i,, is expressed by ˆR i, = b op R i,. Thus, he overall acive reurn R acive can be decomposed ino R acive = T k =1 i=1 ˆR i, + U, where U is he residual across T periods. Muli-Period Regression Tools > rb.muli <- regress(quarer, dae.var = "dae", + re.var = "reurn", + reg.var = c("secor", "growh", + "size"), + benchmark.weigh = "benchmark", + porfolio.weigh = "porfolio") > rb.muli Period sars: 2010-01-01 Period ends: 2010-03-01 Mehodology: Regression Securiies in he porfolio: 200 Securiies in he benchmark: 1000 Regression-based analysis can be applied o a muli-period daa frame by calling he same mehod regress. By yping he name of he class objec rb.muli direcly, a shor summary of he analysis is provided, showing he saring and ending period of he analysis, he mehodology, and he average number of securiies in boh he porfolio and he benchmark. > summary(rb.muli) Period sars: 2010-01-01 Period ends: 2010-03-01 Mehodology: Regression Avg securiies in he porfolio: 200 Avg securiies in he benchmark: 1000 Reurns $Raw 8
BRISO AS REGRESSIO PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS secor 0.0032 0.0031 0.0002 growh 0.0005 0.0009-0.0001 size 0.0029 0.0295 0.0105 Residual 0.0081-0.0172-0.0302 Porfolio Reurn -0.0291 0.0192 0.0298 Benchmark Reurn -0.0438 0.0029 0.0494 Acive Reurn 0.0147 0.0163-0.0196 $Aggregae 2010-01-01, 2010-03-01 secor 0.0065 growh 0.0013 size 0.0433 Residual -0.0392 Porfolio Reurn 0.0190 Benchmark Reurn 0.0064 Acive Reurn 0.0127 The regression-based summary shows ha he conribuion of each inpu variable in addiion o he basic informaion on he porfolio. The summary suggess ha he acive reurn of he porfolio in year 2010 is 1.27%. The Residual number indicaes he conribuion of he ineracion among various variables including secor, growh, and growh. Visual represenaion of relaive performance of a porfolio agains is benchmark is bes viewed across a longer ime span. Here, we use he daa frame year for illusraive purposes. > rb.muli2 <- regress(year, dae.var = "dae", + re.var = "reurn", + reg.var = c("secor", "growh", + "size"), + benchmark.weigh = "benchmark", + porfolio.weigh = "porfolio") > reurns(rb.muli2, ype = "linking") $Raw secor 0.0035 0.0034 0.0002 growh 0.0005 0.0010-0.0001 size 0.0031 0.0320 0.0109 Residual 0.0088-0.0187-0.0312 Acive Reurn 0.0159 0.0177-0.0203 2010-04-01 2010-05-01 2010-06-01 secor 0.0017 0.0044 0.0077 growh 0.0001 0.0002 0.0004 size 0.0145 0.0041 0.0020 Residual -0.0043 0.0346 0.0201 Acive Reurn 0.0122 0.0433 0.0304 2010-07-01 2010-08-01 2010-09-01 secor 0.0016 0.0051-0.0023 growh -0.0005 0.0005-0.0006 size 0.0066 0.0000 0.0100 Residual -0.0333 0.0189-0.0229 Acive Reurn -0.0256 0.0246-0.0158 2010-10-01 2010-11-01 2010-12-01 secor 0.0016-0.0048-0.0084 growh -0.0011-0.0004 0.0010 size 0.0024 0.0143 0.0057 Residual 0.0149 0.0192-0.0253 Acive Reurn 0.0179 0.0282-0.0270 $Aggregae 2010-01-01, 2010-12-01 secor 0.0137 growh 0.0011 size 0.1056 Residual -0.0193 Acive Reurn 0.1014 We obained an objec rb.muli2 of class regress- Muli based on he daa se from January, 2010 hrough December, 2010. The porfolio bea he benchmark by 10.1% over his period. Based on he regression model, size conribued o he lion share of he acive reurn. > plo(rb.muli2, var = "secor", ype = "reurn") Reurn 0.10 0.05 0.00 0.05 0.10 2010 01 01 2010 02 01 2010 03 01 2010 04 01 Figure 3: Porfolio Performance 2010 05 01 2010 06 01 Dae 2010 07 01 2010 08 01 2010 09 01 2010 10 01 2010 11 01 2010 12 01 Type Benchmark Porfolio Performance Aribuion. Figure 3 displays boh he cumulaive porfolio and benchmark reurns from January, 2010 hrough December, 2010. I suggess ha he porfolio, consised of high value securiies in January, consisenly ouperformed he benchmark in 2010. Ouperformance in May and June helped he overall posiive acive reurn in 2010 o a large exen. Brinson as Regression Anoher way o hink abou he analysis as Brinson e al. (1986) have done is o consider i in he conex of a regression model. Conducing a Brinson aribuion is similar o running a linear regression wihou he inercep erm. Esimaed coefficiens will hen be he mean reurn of each caegory of he aribued specified in he universe, a.k.a. he facor reurn of each caegory. The mean reurn of each caegory also appears in he Brinson analysis. The equivalen o he allocaion effec for he universe in he Brinson model is he sum of he produc of he esimaed coefficien and he acive weigh of each caegory. 9
BRISO AS REGRESSIO PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS Using he same regression model as before, R allocaion = W P j R B j = (W P W B ) f, W B j R B j where W P is a column vecor indicaing he porfolio weigh of each caegory wihin he aribued specified by he manager; W B, a column vecor indicaing he benchmark weigh of each caegory, and f is he column vecor which has benchmark reurn of all he caegories. Assuming ha in his case, he benchmark is he universe and he porfolio holdings are all from he benchmark, R B can be esimaed by regressing reurns on he aribue specified by he porfolio manager: where r n = X n,p f + U, r n is a column vecor of lengh n. Each elemen in r n represens he reurn of a securiy in he porfolio. X n,p is an n by p marix where n refers o he number of securiies in he porfolio and p refers o he number of levels wihin he aribue specified. f is he esimaed coefficiens on he regression wihou he inercep erm. The esimaed coefficien of each aribue is he mean reurn for each of he aribue. U is he column vecor wih all he residual erms. Since R B is he same as f, he allocaion effec in he Brinson model is a special case of he regression approach. In order o esimae he selecion effec in he Brinson model, one can calculae he mean reurn of each caegory wihin he aribue in boh he porfolio and he benchmark under a regression framework and use he benchmark weighs o calculae he selecion effec. R selecion = W B j R P j = W B (f P f B ), W B j R B j where W B is he column vecor wih he benchmark weigh of each caegory wihin he aribue specified; f P and f B are he column vecors indicaing he mean reurn of he porfolio and ha of he benchmark, respecively. As menioned above, f P and f B can be esimaed by running a linear regression wihou he inercep erm wih respec o socks in he porfolio and benchmark separaely. Hence, he selecion effec in he Brinson model can be calculaed by using linear regression. Ineracion effec is he difference beween a porfolio s acual reurn and he sum of he allocaion and selecion effecs. An numerical example of showing ha he Brinson model is a special case of he regression approach is as follows. Suppose ha an equiy porfolio manager has a porfolio named es wih he universe as he benchmark. > daa(es) > es.br <- brinson(x = es, dae.var = "dae", + ca.var = "secor", + bench.weigh = "benchmark", + porfolio.weigh = "porfolio", + re.var = "reurn") > reurns(es.br) $`Aribuion by caegory in bps` Allocaion Selecion Ineracion Energy -10.4405 6.01 1.7761 Maerials 4.6486-1.59 0.1544 Indusrials 1.7606-19.03-1.5726 ConDiscre -1.0970-13.47 2.2158 ConSaples 0.1907-16.79 2.1560 HealhCare 0.0861 19.69 0.6350 Financials 0.0908 8.35-0.0116 InfoTech 0.5057-32.40-1.9313 TeleSvcs -1.7611 15.52 3.0745 Uiliies 2.6190-8.81 3.5853 Toal -3.3971-42.54 10.0816 $Aggregae 2010-01-01 Allocaion Effec -0.00034 Selecion Effec -0.00425 Ineracion Effec 0.00101 Acive Reurn -0.00359 When we apply he sandard single-period Brinson anaysis, we obain an acive reurn of -35.9 bps which can be furher decomposed ino allocaion (-3.4 bps), selecion (-42.5 bps), and ineracion (10.1 bps). We can also show he allocaion effec by running a regression model based on secor only. > es.reg <- regress(x =es, + dae.var = "dae", + re.var = "reurn", + reg.var = "secor", + benchmark.weigh = "benchmark", + porfolio.weigh = "porfolio") > reurns(es.reg) 2010-01-01 secor -0.00034 Residual -0.00325 Porfolio Reurn -0.01621 Benchmark Reurn -0.01263 Acive Reurn -0.00359 10
COCLUSIO BIBLIOGRAPHY The conribuion from secor based on he regression approach (-3.4 bps) maches he allocaion effec from he Brinson model as shown above. However, in order o calculae he selecion effec from he regression approach, we need o apply anoher regression model o a universe limied o he securiies held in he porfolio. Using he facor reurns from he regress class objec, es.reg, and hose from he linear regression, we can obain he selecion effec (-42.5 bps) via he regression approach. > lm.es <- lm(reurn ~ secor - 1, + daa = es[es$porfolio!= 0, ]) > lm.es$coefficiens secorenergy secormaerials -0.03561-0.05146 secorindusrials secorcondiscre 0.00194-0.00533 secorconsaples secorhealhcare -0.02514 0.04327 secorfinancials secorinfotech -0.02376-0.02376 secortelesvcs secoruiliies 0.00916-0.03878 > exposure(br.single, var = "secor")[,2] %*% + (lm.es$coefficiens - es.reg@coefficiens) [,1] [1,] 0.00653 Conclusion In his paper, we describe wo widely-used mehods for performance aribuion he Brinson model and he regression-based approach, and provide a simple collecion of ools o implemen hese wo mehods in R wih he pa package. We also show ha he Brinson model is a special case of he regression mehod. A comprehensive package, porfolio Enos and Kane (2006), provides faciliies o calculae exposures and reurns for equiy porfolios. I is possible o use he pa package based on he oupu from he porfolio package. Furher, he flexibiliy of R iself allows users o exend and modify hese packages o sui heir own needs and/or execue heir preferred aribuion mehodology. Before reaching ha level of complexiy, however, pa provides a good saring poin for basic performance aribuion. Yang Lu and David Kane yang.lu@williams.edu and dave.kane@gmail.com Bibliography C. Bacon. Pracical Porfolio Performance Measuremen and Aribuion. John Wiley & Sons, Ld., 2 ediion, 2008. G. Brinson, R. Hood, and G. Beebower. Deerminans of Porfolio Performance. Financial Analyss Journal, 42(4):39 44, Jul. Aug. 1986. URL hp://www.jsor.org/sable/4478947. G. Brinson, B. Singer, and G. Beebower. Deerminans of Porfolio Performance II: An Updae. Financial Analyss Journal, 47(3):40 48, May Jun. 1991. URL hp://www.jsor.org/sable/ 4479432. O. Davies and D. Laker. Muli-Period Performance Aribuion Using he Brinson Model. Journal of Performance Measuremen, Fall, 2001. J. Enos and D. Kane. Analysing Equiy Porfolios in R. R ews, 6(2):13 19, MAY 2006. URL hp://cra.r-projec.org/doc/rnews. A. Frongello. Linking Single Period Aribuion Resuls. Journal of Performance Measuremen, Spring, 2002. R. Grinold. Aribuion, Modeling asse characerisics as porfolios. The Journal of Porfolio Managemen, page 19, Winer 2006. T. Lord. The Aribuion of Porfolio and Index Reurns in Fixed Income. Journal of Performance Measuremen, Fall:45 57, 1997. J. Menchero. An Opimized Approach o Linking Aribuion Effecs over Time. Journal of Performance Measuremen, 5(1):36 42, 2000. J. Menchero. Muliperiod Arihmeic Aribuion. Financial Analyss Journal, 60(4):76 91, 2004. J. Menchero, A. Morozov, and P. Shepard. The Barra Global Equiy Model (GEM2). MSCI Barra Research oes, Sep. 2008. B. Singer and D. Karnosky. The General Framework for Global Invesmen Managemen and Performance Aribuion. The Journal of Porfolio Managemen, Winer:84 92, 1995. 11