erformace Attrbuto Methodology Overvew Faba SUAREZ March 2004
erformace Attrbuto Methodology 1.1 Itroducto erformace Attrbuto s a set of techques that performace aalysts use to expla why a portfolo's performace dffered from the bechmark. Ths dfferece betwee the portfolo retur ad the bechmark retur s kow as the actve retur/excess retur/etc. The actve retur s the compoet of a portfolo's performace that arses from the fact that the portfolo s actvely maaged. For pure equty portfolos, the attrbuto aalyss dssects the value added to three compoets (or four f there s a currecy effect): asset allocato captures the pure effect of the portfolo s asset allocato betwee sectors, wthout ay stock selecto effect the sectors, stock selecto s the value added by decsos wth each sector of the portfolo, teracto captures the value added that s ot attrbutable solely to the asset allocato ad stock selecto decsos. The three attrbuto terms (asset allocato, stock selecto, ad teracto) sum exactly to the actve retur wthout the eed for ay "fudge factors". Obvously, ths approach s applcable to bod or mxed portfolos. Nevertheless the models appled for these portfolo types are, geeral more complex as vestmet decsos are dfferet for these portfolos (e.g. durato, qualty of ssuers, etc). 1.2 IS Requremets to Calculate ortfolo Returs Achevg comparablty amog vestmet maagemet frms performace presetatos requres uformty methods used to calculate returs. Ths s the reaso why the lobal Ivestmet erformace Ivestmet Stadards madate the use of certa calculato methodologes. Valug the portfolo each tme there s a exteral cash flow ought to result the most accurate method to calculate the tme-weghted rates of retur, referred to as the true Tme-Weghted Rate of Retur Method. A formula for calculatg a true tme-weghted portfolo retur wheever cash flows occur s: R TR EMV BMV =, (2.1) BMV where EMV s the market value of the portfolo at the ed of the sub-perod, before ay cash flows the perod, but cludg accrued come for the perod. BMV s the market value at the ed of the prevous sub-perod (.e., the begg of the curret sub-perod); cludg ay cash flows at the ed of the prevous sub-perod ad cludg accrued come up to the ed of the prevous perod. The sub-perod returs are the geometrcally lked accordg to the followg formula: where RTR s the total retur ad R 1, R 2,, (( 1) ( 2 ) K ( )) R = 1+ R 1+ R 1+ R 1, (2.2) TR R are the sub-perod returs for sub-perod 1 through respectvely. Sub-perod 1 exteds from the frst day of the perod up to ad cludg the date of the frst cash flow. Sub-perod 2 begs the ext day ad exteds to the date of the secod cash flow, ad so forth. The fal sub-perod exteds from the day after the fal cash flow through the last day of the perod. Ths method assumes that the cash flow s ot avalable for vestmet utl the begg of the ext day. Accordgly, whe the portfolo s revalued o the date of a cash flow, the cash flow s ot reflected the Edg Market Value, but s added to the Edg Market Value to determe the Begg Market Value for the ext day. If the cash flow s avalable for vestmet at the begg of the day the value of the cash flow should be added to the Begg Market Value.
Note that some day-weghtg methods assume the cash flow s avalable mdday ad half weght the cash flow that day. The IS stadards do ot specfy whch cash flow recogto method frms must use; however, oce a method(s) s chose ad the crtera ad assumptos are determed, they must be cosstetly appled. Begg 1 Jauary 2010, ths methodology s lkely to be requred by the IS stadards. Utl 2010, approxmato methods are permtted such as the orgal Detz method, the Modfed Detz method, the orgal Iteral Rate of Retur (IRR) method, ad the Modfed IRR method. 1.3 erformace Attrbuto Models As the cocept of performace attrbuto matures the vestmet maagemet dustry, stadard methodologes are begg to receve the same scruty as the methodologes behd the calculato of total retur. Whle the AIMR has yet to produce a hadbook o IS complat attrbuto, the drecto s clear, as clet creasgly seek attrbuto reports whose bottom-le returs match those produced by ther performace measuremet systems. Secto 1.1 has showed that the calculato of the total retur of a portfolo requres the followg formato: market values of the portfolo ad the cash flows occurred durg the perod. By aalogy, to calculate the returs for a asset held a portfolo, we eed to kow the market values ad cash flows for that asset. May portfolo maagers struggle wth the practcal problems volved collectg market values ad cash flows at the total portfolo level. Ideed, because these practcal problems are so great, IS does ot atcpate requrg the use of tme weghted returs for portfolos utl 1 Jauary 2010. However, the magtude of the data problem becomes much larger performace attrbuto. For example, f a portfolo maager wshes to attrbute a portfolo by dustry, they wll requre weghts ad returs for every dustry. If the portfolo maagers wshe to do stock-level attrbuto, they wll requre a cosstet set of weghts ad returs for every stock held the portfolo. Thus, the task of gatherg weghts ad returs for performace attrbuto may volve collectg oe or two orders of magtude more data tha s requred smply to calculate a portfolo retur. As the roles of performace measuremet ad performace attrbuto coverge, t s mportat to dstgush betwee the goals of these two cocepts ad to determe the best combato order to preserve the formato cotet that represets the heret value of each approach. Attrbuto s probably fraught wth more cotroversy tha just about ay other aspect of performace measuremet: from geometrc vs. arthmetc, to the varous lkg methods, daly vs. mothly, to securty vs. sector level. Ad oe of those areas whch has t bee addressed at legth, but s deftely cotroversal, s holdgs-based versus trasacto-based attrbuto. For some tme practtoers have bee tryg to ga greater sght to the dstctos betwee these two geeral approaches to attrbuto. There appear to be two very dfferet camps: the pro-trasacto based group, who beleve that accuracy ca oly be acheved by a trasacto-based model, ad the holdgs-based group, who feel that (a) the purported accuracy s a myth, (b) that the data requremets are such that you wll be troducg error ad/or ose to the math, ad (c) that ay margal ga accuracy wll be offset by the huge costs. Defg the holdgs-based approach s qute smple: t s a attrbuto model that reles upo the portfolos begg-perod holdgs to derve the attrbuto effects. Defg the trasacto-based method s a bt more challegg. Whle we would expect that there s some attempt to capture tra-perod actvty (ad ot rely solely o the tal holdgs), how we do t ad to what extet we capture the trasacto detals are ope to debate. Some authors suggest that there are varous degrees of trasacto-based methods ad that there s probably a pot where the dstcto betwee holdgs ad trasacto methods becomes kd of grey. Ideed, a daly holdgs-based approach ca be labelled as ether holdgs or trasacto-based: each day the portfolo s revalued, so f stocks are sold/bought, the day after we start wth the ew portfolo composto. Obvously, ths approach has to be cosdered as a rather low-level trasacto-
based model comparso to a approach whch captures 100% of trasacto actvty: capture of every trasacto (buy/sale, dvded, corporate acto, etc), each trasacto eeds to be labelled for ts effect (teral cash flow, exteral cash flow, come, etc) ad must have the rght mpact date. Ths s probably the heght of trasacto-based attrbuto. We are capturg all the detals. I the holdg-based approach to performace attrbuto, the portfolo s treated each sub-perod o a strctly buy-ad-hold bass, ad the attrbuto effects are computed usg the stadard sgle-perod equatos of the Brso model. These attrbuto effects are the lked together usg a mult-perod lkg algorthm. Trasactos are reflected through the portfolo holdgs, whch are updated at the ed of ay day whch there was a trasacto. Actual trasacto prces, however, are ot used to compute returs. Ths approach s equvalet to usg a tme-weghted method assumg that all trasactos occur at the ed of the day at the closg prce. Whle, prcple, these assumptos may lead to dscrepaces total portfolo retur f both the daly turover ad daly prce volatlty are large, practce, such effects are typcally mor. I the trasacto-based approach, sub-perods cocde wth the tmg of exteral cash flows. A example of ths method would be to assume that exteral cash flows occur at the start of each day, ad to use trasacto prces to calculate the daly performace. Sce there are o exteral cash flows durg the day, prcple ths yelds the exact soluto to the tme-weghted retur for the portfolo. May practtoers assume that ths trasacto-based approach, by exteso, also provdes the most accurate soluto to performace attrbuto. However, ths assumpto s flawed due to oe subtle pot: although there are o exteral cash flows ths method, there are varably teral cash flows as securtes are traded throughout the sub-perod (.e. traday trasactos). The sector weghts ad returs must therefore stll be computed usg a moey-weghted approach, whch tur wll geerate a source of errors. I other words, although the trasacto-based approach wll provde the most accurate actve retur, t wll oetheless cota errors the compoets of actve retur (.e. attrbuto effects). I order to aswer to the followg questo whch approach s better?, t s mportat to bear md that the objectve of performace attrbuto s to measure the sources of actve retur as accurately as possble, ad ot to measure total portfolo retur. The key advatage of the holdgs-based approach s that t uses tme-weghted returs to compute the attrbuto effects, ad hece avods the errors assocated wth the use of moey-weghted returs. Its dsadvatage s that t does ot take to accout the traday tradg effect. As suggested by Laker (2003), to solve ths problem, we ca easly add a tradg effect the attrbuto model, at the global portfolo level. Ths effect wll be equal to the dfferece betwee the offcal retur mus the sum of the retur calculated wth the attrbuto tool ad the fees appled. 1.4 Attrbuto udeles Suggested by Davd Spauldg As performace attrbuto has become a creasgly stadard part of a moey maager s performace measuremet ad reportg fucto, t s ecessary to have stadards to sure that all the relevat detals are kow ad uderstood by the recpets of the report. Attrbuto aalyss s smply too complex ad vared to assume that the recpet wll uderstad how the results were prepared. To accomplsh ths, Davd Spauldg (2002/2003) publshed reasoable stadards. The stadards are dvded to four ma sectos: termology, model selecto, dsclosures, ad supplemetal formato to a IS presetato. 1. Termology A revew of the varous terms that are typcally used wth attrbuto. A agreemet o these has to be obtaed to sure uformty. 2. Model Selecto ckg the rght attrbuto model s a crtcal step mplemetg performace attrbuto aalyss. We dscuss varous model characterstcs ad calculato ssues. 3. Dsclosures To comply wth these stadards, frms must dsclose certa formato about ther attrbuto model. 4. Supplemetal Iformato to a IS resetato We ve see a sgfcat terest attrbuto statstcs beg corporated to a IS presetato, albet as a recommedato at ths stage.
I ths documet, we preset oly the Termology ad the Model Selecto parts of the stadards. 1.4.1 Termology 1.A erformace Attrbuto A aalytcal process or techque to detfy the sources that cotrbute to a retur ad/or excess retur. 1.B Relatve erformace Attrbuto Attrbuto of a portfolo relatve to a bechmark or dex. 1.C Absolute erformace Attrbuto Attrbuto of a portfolo aloe; also kow as cotrbuto. 1.D Excess Retur The dfferece betwee a portfolo s retur ad the retur of ts bechmark. Ths value may be calculated ether arthmetcally (also called addtve ) or geometrcally (also called multplcatve ). Excess retur s also referred to as actve retur ad alpha. 1.E eometrc Excess Retur The dfferece retur betwee a portfolo ad ts bechmark, calculated as follows: ER 1+ R = 1, (2.3) 1+ R B ER = eometrc Excess Retur, R = ortfolo Retur, expressed as a decmal, ad R = Bechmark Retur, expressed as a decmal. B 1.F Arthmetc Excess Retur The dfferece retur betwee a portfolo ad ts bechmark, calculated as follows: ERA = R RB. (2.4) 1. eometrc erformace Attrbuto A attrbuto approach that reles upo the geometrc approach to derve excess retur. Also referred to as multplcatve performace attrbuto. 1.H Arthmetc erformace Attrbuto A attrbuto approach that reles upo the arthmetc approach to calculate excess retur. Also referred to as addtve performace attrbuto. 1.I Iteracto Effect A attrbuto effect that s used to accout for the teracto betwee two or more effects (e.g., betwee the stock selecto effect ad asset allocato effect for a equty portfolo). Some models may use ths effect for uaccouted-for effects, but ths should more properly be referred to as resdual. 1.J Resdual There are two ways the term resdue or resdual s used. Oe s for a sgle perod ad oe for multple perods. I both cases, t refereces a uaccouted for amout. For a sgle perod, t s a uaccouted for amout that may arse because of prcg rregulartes betwee the dex ad portfolo, effects whch are uaccouted for by the model or other factors. For multple perods, t s a amout that s uaccouted-for whch may arse whe lkg attrbuto effects over tme. Ths s typcally a problem wth arthmetc models but ot geometrc models. 1.4.2 Model Selecto Because the purpose of attrbuto aalyss s to evaluate the source(s) of a portfolo s retur, ad because dfferet models ca yeld dfferet (ad sometmes coflctg) results, t s mportat that the model that s selected coform wth the vestmet approach used for the style of vestg for the portfolo beg evaluated. Because styles of vestg may vary eve wth a frm, t s ot
cocevable that the frm wll calculate attrbuto usg dfferet models, depedg upo the partcular style, asset class, etc. Whle these stadards are ot teded to be calculato stadards, that they wll ot dctate that specfc models be utlzed, there are certa aspects to the calculato whch we requre. 2.A The attrbuto model must coform wth the vestmet approach for the portfolo beg evaluated. 2.B Whe calculatg absolute performace attrbuto (also kow as cotrbuto), the sum of the cotrbuto values must equal the total retur of the portfolo. Mathematcally: = R C. (2.5) = 1 C = calculated cotrbuto values ad = umber of sectors, securtes, etc., beg evaluated. 2.C Whe calculatg relatve performace attrbuto usg a arthmetc model, the sum of the attrbuto effects must equal the arthmetc excess retur. Mathematcally: = 1 AE = ER. (2.6) A AE = Attrbuto Effects. 2.D Whe calculatg relatve performace attrbuto usg a geometrc model, the product of the attrbuto effects must equal the multplcatve excess retur. Mathematcally: = 1 AE = ER. (2.7) If a geometrc model was employed but a adjustmet made so that the effects actually sum to the arthmetc excess retur, the ths must be stated ad the methodology that was employed to accomplsh ths must be explaed. 2.E Whe lkg arthmetc attrbuto effects over tme, the sum of the lked attrbuto effects must equal the lked arthmetc excess retur. Mathematcally: m, = 1 t = 1 AE = LR LR = LER. (2.8) t B A AE = Attrbuto Effects, = the Idvdual attrbuto effects, t = Tme perods over whch effects are beg lked, LR = Lked ortfolo Retur, LR = Lked Bechmark Retur, ad B LER = Lked Arthmetc Excess Retur. A 2.F Whe lkg geometrc attrbuto effects over tme, the product of the lked attrbuto effects must equal the lked geometrc excess retur. Mathematcally: m AE t = 1 t = 1 1+ LRB, 1+ LR = 1 = LER. (2.9)
2 Refereces Boafede, J. K.; McCarthy, M. C.: Trasacto-based vs. Holdg-based Attrbuto: the Devl s the Deftos, The Joural of erformace Measuremet, Fall 2003. Laker, D.: erspectves o Trasacto-based Attrbuto, The Joural of erformace Measuremet, Fall 2003. Mechero, J.; Hu, J.: Errors Trasacto-based erformace Attrbuto, The Joural of erformace Measuremet, Fall 2003. Spauldg, D.: A case for Attrbuto Stadards, The Joural of erformace Measuremet, Wter 2002/2003. Spauldg, D.: Holdgs vs. Trasacto-based Attrbuto, a Overvew, The Joural of erformace Measuremet, Fall 2003.