Decomposing the Money-Weighted Rate of Return an Update. Working Paper - Nummer: 21. by: Dr. Stefan J. Illmer. in Journal of Performance Measurement;

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1 Decomposig he Moey-Weighed Rae of Reur a Updae Workig aper - Nummer: by: Dr. Sefa J. Illmer i Joural of erformace Measureme; Fall 2009; Volume 14; Number 1; page Ivesme erformace IIIllmer osulig AG Koakadresse Illmer Ivesme erformace osulig AG Weibergsrasse 28 H 8200 Schaffhause sefa.illmer@iipc-ag.com

2 Decomposig he Moey-Weighed Rae of Reur a Updae Versio as of y Dr. Sefa J. Illmer INTRODUTION I 2003 Wolfgag Mary ad I published a aricle o decomposig he moey-weighed rae of reur. 1 This aricle was addiioal ipu o he coroversy discussio o he wo mai coceps of reur calculaio i he ivesme maageme idusry - ime-weighed rae of reur (TWR) versus moeyweighed rae of reur (MWR). Uforuaely his discussio is baled quie emoioally ad herefore a idusry-wide echical discussio focusig o facs could o really happe ye. Therefore, currely he idusry is i he same siuaio as some years ago where he idusry mosly egleced he value of he moey-weighed rae of reur ad of is decomposiio. 2 Afer he publicaio of he above meioed aricle I focused my research for a while o he criical discussio of he currely used TWR-aribuio ad based o my fidigs I developed a mehodology for a MWR-aribuio. The prese aricle illusraes his mehodology ad applies i i addiio o he absolue profi ad loss calculaio o also come up a mehodology for a profi ad loss aribuio. 3 ALULATION OF MWR The sarig poi for he MWR-aribuio is he calculaio of he MWR for he porfolio ad is bechmark as well as of. To calculae he MWR, i he idusry differe mehodologies are used where all bu oe are I he followig he ieral rae of reur mehodology () is used where he is he reur ha solves he followig equaio. 4 T 1 MV 0 T (1 ) 1(1 ) MV T Ieralrae of reur, egiigmarke value, Edigmarke valuea T, ash flow a, Timeof he edof hemeasureme, Timeof cash flow. The of he porfolio is herefore defied by he respecive marke values a he begiig ad he ed of he measureme ad by he respecive cash flows durig he measureme. (1 MV, ) T T 1, MV 1(1 ) of porfolio, orfoliocash flow a. 0 orfoliobegiigmarke value, orfolioedigmarke valuea T, Similarly he for he bechmark is defied by he respecive marke values a he begiig ad he ed of he measureme ad by he respecive cash flows durig he measureme. 5 (1) (2) Here i is impora ha he cash iflows (ouflows) are ivesed (de-ivesed) accordig o he acual bechmark asse allocaio a he ime of he cash flow ad ha he reurs of he moey ivesed equal he respecive reurs of he uderlyig bechmark ivesmes. (1 ) T T 1, 1(1 MV ) 0 (3) Decomposig he Moey-Weighed Rae of Reur a Updae age 2 of 9

3 MV, of bechmark, echmarkbegiigmarke value, echmarkedigmarke valuea T, echmarkcash flow a. Give he of he porfolio ad he bechmark, he excess ca be calculaed usig he formula below. E E (4) Excess. ONTRIUTION TO MWR The secod sep owards he MWR-aribuio is he calculaio of he coribuios o MWR for he porfolio ad is bechmark as well as he coribuios o he R. To calculae he coribuios o i is impora o udersad he cocep of he ieral rae of reur ad is uderlyig assumpios. The coribuios o are o based o he weighs of he asse classes a he begiig of he measureme bu isead based o he average ivesed capial which is especially releva if here are ierim cash flows i he measureme. 6 osiderig ha he is he reur ha deermies he absolue profi ad loss (&L), he average ivesed capial ca be derived usig he followig formulas. T 1 L MV (5) L L 1 & L of porfolio. L i1,i i1,i MV,i, T 1 1,i, (6) L MV,i,i,,i,i & L of asse Edigmarke valueof asse egiigmarke valueof asse ash flow of asse class i a, Number of asse classes. L (7) Averageivesedcapialof porfolio. L,i,i (8),i,i,i Averageivesedcapialof asse of asse I is impora o oe ha he average ivesed capial of he oal porfolio does o have o be equal o he sum of he average ivesed capials of all asse classes. 7 As saed below, depedig o he uderlyig daa he average ivesed capial of he oal porfolio ca be bigger, equal or less ha he sum of he average ivesed capials of all asse classes. or or (9),i i1 Usig he above meioed formulas he of he porfolio ca be expressed as he sum of he reur coribuios of each asse class. R,i L L,i i1,i i1 R i1,i,i Reur coribui oof asse (10) Formulas (5) o (10) cover he calculaios for he porfolio bu hese formulas ca also be applied for he bechmark by jus subsiuig he idex by like i formula (11). Decomposig he Moey-Weighed Rae of Reur a Updae age 3 of 9

4 R L L,i,i,i L L,i i1 i1 R i1,i,i & L of asse,i Reur coribui oof asse & L of bechmark, Averageivesedcapialof bechmark, Averageivesedcapialof asse (11) calculaed usig he followig formula. E R i1 i1 L i1,i,i,i MWR-ATTRIUTION R i1 L i1,i,i,i i1,i,i (12) The las sep owards MWR-aribuio is he dec i a way ha i reflecs he differe seps of he decisio makig process. Here he excess is decomposed accordig o he reur aribuio mehodology published by riso, Hood ad eebower i 1986 ad herefore spli up io he hree maageme effecs: asse allocaio effec, sock pickig effec ad ieracio effec. 8 E SE SE,i,i i1 i1 i1,i (13) SE SE,i,i,i Asse allocaioeffec of porfolio, Sock pickigeffec of porfolio, Ieracio effec of porfolio, Asse allocaioeffec of asse Sock pickigeffec of asse Ieracio effec of asse To calculae he differe maageme effecs he framework for aalyzig porfolio reurs by riso, e al. (1986) is used. 9 Table 1 illusraes his framework applied o he MWR-aribuio. Table 1 Asse Allocaio Acual assive Simple framework for MWR-aribuio Acual Quadra IV of acual porfolio Quadra III of oioal porfolio 2 => acive sock pickig porfolio Selecio assive Quadra II of oioal porfolio 1 => acive asse allocaio porfolio Quadra I of bechmark Quadra I represes he of he bechmark where for he calculaio all (de-)ivesmes made i each asse class based o he passive weighs of he asse class ad expressed as cash flows - ad he passive idex reurs for each asse class are eeded. Quadra II represes he of he oioal porfolio 1 which reflecs he acive asse allocaio of he porfolio assumig o sock pickig. To calculae he of he oioal porfolio 1 all (de-)ivesmes made i each asse class based o he acual weighs of he asse class ad expressed as cash flows - ad he passive idex reurs for each asse class are eeded. Quadra III represes he of he oioal porfolio 2 which reflecs he acive sock pickig of he porfolio assumig o acive asse allocaio. To calculae he Decomposig he Moey-Weighed Rae of Reur a Updae age 4 of 9

5 of he oioal porfolio 2 all (de-)ivesmes made i each asse class based o he passive weighs of he asse class ad expressed as cash flows - ad he acual reurs for each asse class are eeded. Quadra IV represes he of he porfolio where for he calculaio all (de-)ivesmes made i each asse class based o he acual weighs of he asse class ad expressed as cash flows - ad he acual reurs for each asse class are eeded. ased o he aforemeioed framework for MWRaribuio he differe maageme effecs ca be calculaed as follows. R SE R N1 N1,i N2 N2,i QuadraII QuadraI R i1 N1 N1,i R i1,i of oioalporfolio1, Reur coribui oof asse QuadraIII QuadraI R i1 N2 N2,i R i1,i of oioalporfolio2, Reur coribui oof asse QuadraIV QuadraIII QuadraII QuadraI R i1 R i1 N1,i R N1,i i1 N2 R i1 N2,i,i (14) (15) (16) The formulas (14) o (16) ca be used as well o calculae he differe maageme effecs o a asse class level. SE,i,i,i R R (17) N1,i N2,i,i R R (18),i N2,i,i R R R R (19) N1,i,i A propery of he is ha he (perceage) reurs ca be rasformed io he observable absolue profi ad loss figures per ui of currecy. Jus as for he, he profi ad loss of he porfolio ad of he asse classes are deermied by he respecive marke values a he begiig ad he ed of he measureme ad by he respecive cash flows durig he measureme ad ca be calculaed usig formulas (5) ad (6). Formulas (5) o (6) ca also be applied for he bechmark by jus subsiuig he idex by. T 1 L MV (20) L MV,i,,i,i L i1,i i1,i 1 MV,i, T 1 1,i, Edigmarke valueof asse egiigmarke valueof asse ash flow of asse class i a. (21) Give he profi ad loss of he porfolio ad he profi ad loss ad is coribuios ca be calculaed usig he below formula. EL EL L L i1 L,i L i1,i Excess profiadloss. (22) Table 2 Simple framework for profi ad loss aribuio Acual Acual Quadra IV &L of acual porfolio Selecio assive Quadra II &L of oioal porfolio 1 => acive asse allocaio porfolio ROFIT AND LOSS ATTRIUTION Decomposig he Moey-Weighed Rae of Reur a Updae age 5 of 9

6 Asse Allocaio assive Quadra III &L of oioal porfolio 2 => acive sock pickig porfolio Quadra I &L of bechmark As show i able 2 he aforemeioed framework for MWR-aribuio ca also be applied o he profi ad loss aribuio o decompose he excess profi ad loss io he hree above meioed maageme effecs o oal porfolio ad asse class level. EL AAL SL IAL AAL SL IAL,i AAL L L N1 N1,i SL L L N2,i,i N2,i IAL AAL SL,i,i,i L AAL L SL IAL AAL SL IAL,i,i i1 i1 i1 & L dueo sock pickigof porfolio, & L dueo ieracio of asse,i & L dueo asse allocaioof asse & L dueo sock pickigof asse (23) & L dueo asse allocaioof porfolio, & L dueo ieracio effecof porfolio, L L i1 N1 L N1,i & L of asse L i1,i & L of oioalporfolio1, L L i1 N2 L N2,i L i1 & L of asse,i & L of oioalporfolio2, L L L i1 L L i1 N1,i N1,i L L N1,i N2 i1 L i1,i N2,i,i (24) (25) (26) L L (27) L L (28),i N2,i N2,i,i L L L L (29) N1,i,i AN EXAMLE FOR A MWR-ATTRIUTION AND A ROFIT AND LOSS ATTRIUTION I his secio a example for a MWR-Aribuio ad a profi ad loss aribuio is illusraed by cosiderig a muli-asse class porfolio for he wo year uil which is ivesed i wo asse classes (A ad ) ad which is rebalaced o a yearly basis a he begiig of he year. This porfolio is maaged agais a bechmark which is also ivesed i hese wo asse classes ad rebalaced o a yearly basis a he begiig of he year. A he begiig of 2007 EUR 150 ad a he begiig of 2008 addiioal EUR 100 are ivesed io he porfolio accordig o he he curre acive asse allocaio ad sock pickigs. The (passive ad acual) reurs ad he (passive ad acual) weighs of he porfolio ad he bechmark are saed i able To be able o calculae he differe reurs ad reur coribuios, he weighs have o rasferred io cash flows like for example for he acual porfolio where he cash iflow io asse class A of EUR 75 is 50% of he iiial cash iflows of EUR 150 io he porfolio or he cash ouflow a he begiig of 2008 of EUR 47.6 which is he differece bewee he marke value ivesed i asse class A a he ed of 2007 of EUR 86.3 mius 15% of he marke value a he ed of 2007 of EUR (ew weigh of asse class A a he begiig of 2008). 11 Table 3 shows ha he of he porfolio is 13.8% ad he of he bechmark is 0.1% wha causes a excess of 13.7%. 12 omparig he marke values a he ed of 2008 of he porfolio ad of he bechmark leads o he fac ha he porfolio maager geeraed a excess profi of EUR Table 4 summarizes he resuls of he -aribuio ad he profi ad loss aribuio which are based o he reur ad he profi ad loss daa of he porfolio, he bechmark, he oioal porfolio 1 ad he oioal porfolio 2. A he boom, able 4 coais he deailed Decomposig he Moey-Weighed Rae of Reur a Updae age 6 of 9

7 resuls for he -aribuio showig ha he oal excess is 13.7% a oal asse allocaio effec of -7.2%, a oal sock pickig effec of 7.8% ad a oal ieracio effec of 13.3%. 13 Exemplary, aroud 2/3 of he asse allocaio effec is due o he ivesmes i asse class A (-4.9%) bu aroud 2/3 of he sock pickig is due o he ivesmes made i asse class (5.4%). Furhermore able 4 coais he resul of he profi ad loss aribuio showig ha he oal excess profi ad loss is EUR 27.2 a oal asse allocaio Table 3 Example for MWR-aribuio ad profi ad loss aribuio uderlyig daa eriod 1 eriod 2 eriod 1 eriod 2 Daes Daes ash flow a begiig ash flow a begiig Marke value a he ed ash flow a begiig ash flow a begiig Marke value a he ed Asse A Asse A Asse Asse orfolio orfolio Acual weighs a begiig Acual orfolio () Acual weighs a begiig Weighs a he ed of Acual weighs a begiig Noioal orfolio 1 () Acual weighs a begiig Weighs a he ed of Asse A 50.0% 15.0% 13.2% Asse A 50.0% 15.0% 17.0% Asse 50.0% 85.0% 86.8% Asse 50.0% 85.0% 83.0% orfolio 100.0% 100.0% 100.0% orfolio 100.0% 100.0% 100.0% Acual reur Acual reur ummulaive reur assive reur assive reur ummulaive reur Asse A 15.0% -5.0% 17.9% Asse A -20.0% 10.0% -18.2% Asse -5.0% 10.0% 12.4% Asse 10.0% -5.0% -2.0% orfolio 5.0% 7.8% 13.8% orfolio -5.0% -2.8% -7.0% eriod 1 eriod 2 eriod 1 eriod 2 Daes Daes ash flow a begiig ash flow a begiig Ivesme a he ed of ash flow a begiig ash flow a begiig Marke value a he ed Asse A Asse A Asse Asse orfolio orfolio assive weighs a begiig Noioal orfolio 2 () assive weighs a begiig Weighs a he ed of assive weighs a begiig echmark () assive weighs a begiig Weighs a he ed of Asse A 30.0% 30.0% 27.0% Asse A 30.0% 30.0% 33.2% Asse 70.0% 70.0% 73.0% Asse 70.0% 70.0% 66.8% orfolio 100.0% 100.0% 100.0% orfolio 100.0% 100.0% 100.0% Acual reur Acual reur ummulaive reur assive reur assive reur ummulaive reur Asse A 15.0% -5.0% 5.2% Asse A -20.0% 10.0% -2.2% Asse -5.0% 10.0% 8.7% Asse 10.0% -5.0% 1.3% orfolio 1.0% 5.5% 7.7% orfolio 1.0% -0.5% 0.1% Table 4 Example for MWR-aribuio ad profi ad loss aribuio aribuio deails Decomposig he Moey-Weighed Rae of Reur a Updae age 7 of 9

8 Acual orfolio () Noioal orfolio 1 () &L R &L R Asse A % 4.7% Asse A % -5.7% Asse % 9.1% Asse % -1.4% orfolio % 13.8% orfolio % -7.0% Noioal orfolio 2 () echmark () &L R &L R Asse A % 1.5% Asse A % -0.7% Asse % 6.2% Asse % 0.8% orfolio % 7.7% orfolio % 0.1% -Aribuio rofi ad Loss Aribuio SE Toal AAL SL IAL Toal Asse A -4.9% 2.2% 8.1% 5.4% Asse A Asse -2.2% 5.4% 5.2% 8.3% Asse Excess -7.2% 7.6% 13.3% 13.7% Excess effec of EUR -14.4, a oal sock pickig effec of EUR 15.1 ad a oal ieracio effec of EUR Agai aroud 2/3 of he asse allocaio effec is due o he ivesmes i asse class A (EUR -9.9) bu aroud 2/3 of he sock pickig is due o he ivesmes made i asse class (EUR 10.7). ONLUSION This aricle illusraes wo mehodologies which help o beer udersad where he hisorical reur (expressed as a ) ad where he hisorical profi ad loss is comig from. The MWR-aribuio has a lo of advaages as is resuls are i lie he profi ad loss aribuio ad as i correcly coribues he imig effec o he asse allocaio ad sock pickig effec. This is addiioal evidece ha he MWR cocep sill adds value ad should be cosidered if aalyzig he performace of ivesme maageme aciviies. ONTAT INFORMATION If you have ay queries o his aricle, please coac: Dr. Sefa J. Illmer: hoe: sefa.illmer@credi-suisse.com ENDNOTES Weighed Rae of : Joural of erformace Measureme; Summer 2003, pages 42-50; by Dr. Sefa J. Illmer ad Wolfgag Mary. For a deailed discussio o he wo mai coceps of reur calculaio as well as o he cocep of performace aribuio please be referred o exbooks o performace measureme ad performace aribuio from auhors like arl aco (racical orfolio erformace Measureme ad Aribuio), ruce Feibel (Ivesme erformace Measureme), David Spauldig (The Hadbook of Ivesme This aricle is based o he preseaio - by Dr. Sefa J. Illmer a he MAR V coferece i hiladelphia The preseaio is available from he auhor (see coac iformaio). For he differeces bewee performace aribuio ad reur as well risk aribuio, please see reseaio: A Sep Towards Sadardizaio of erformace Decomposig he Moey-Weighed Rae of Reur a Updae age 8 of 9

9 erformace Measureme; Summer 2004; pages 46-62; by Dr. Sefa J. Illmer ad Dimiri Seik. The auhor is aware of he uderlyig assumpios of he ad he shorcomigs if calculaig he. To focus more o he mehodology i should be up o oher aricles o discuss hese criics ad o provide respecive soluios. Furhermore i should also be up o oher aricles o compare he MWR-aribuio he TWR-aribuio ad o measureme ad risk aribuio as well as o he whole performace moiorig process. Ieral rae of reur () is a MWR ad is called he "rue" MWR as i is o oly he mos precise mehod for calculaig a MWR bu he oe mehodology ha solves he full calculaio problem. The is he reur / ieres rae ha causes he edig marke value ad iermediae cash flows o be discoued o he begiig marke value. I he followig MWR ad as well as MWRaribuio ad -aribuio are herefore used ierchageable. The marke value of he bechmark a he begiig of he measureme equals he respecive marke value of he porfolio a he begiig of he measureme ad is ficiiously ivesed i he differe bechmark idices accordig o he bechmark asse allocaio weighs. The marke value of he bechmark a he ed of he measureme is he deermied by he bechmark weighs, he rebalacig aciviies, he (de-) ivesme of he ierim cash flows ad he respecive idex reurs over ime. I he f used This propery is difficul o follow bu is correc as i is based o he uderlyig reivesme assumpio of he mehodology. Fiacial Aalyss Joural; July/Augus 1986; pages 39-44; by G. riso, R. Hood ad G. eebower. The MWR-aribuio ad profi ad loss aribuio ca be adjused o reflec oher coceps for decomposig porfolio reurs. The auhor is already workig o his ad is willig o share his isighs ieresed readers (see coac iformaio). The spreadshee he calculaios is available from he auhor (see coac iformaio). The marke value ivesed i asse class A a he ed of 2007 is defied by he moey ivesed a he begiig of 2007 ad is reur: %. I able 3 a egaive sig i fro of he cash flows idicae cash iflows (buy) ad a posiive sig i fro of he cash flows idicae a cash ouflows (sell). This helps he calculaio of he where cash iflows have always a egaive sig ad vice versa. I able 3 he cumulaive reurs ad reur coribuios are o aualized. If eeded he calculaio ca be rasferred o aualized reurs by aualizig he s where i is impora o choose a cosise dae coveio. This assumes ha he porfolio maager had discreio over he cash iflow a he begiig of Oherwise he calculaio has o be adjused i a way ha he imig effec of he cash iflow has o be separaed followig he mehodology Moey- edoe 1. The ieracio effec of 13.3% is quie big bu due o he exreme example. Decomposig he Moey-Weighed Rae of Reur a Updae age 9 of 9