Index Mathematics Methodology

Transcription

1 Index Mahemacs Mehodology S&P Dow Jones Indces: Index Mehodology Augus 2016

2 Table of Conens Inroducon 4 Dfferen Varees of Indces 4 The Index Dvsor 4 Capalzaon Weghed Indces 6 Defnon 6 Adjusmens o Share Couns 6 Dvsor Adjusmens 7 Necessary Dvsor Adjusmens 8 Capped Indces 10 Equal Weghed Indces 11 Defnon 11 Corporae Acons and Index Adjusmens 12 Index Acons 12 Corporae Acons 13 Modfed Marke Capalzaon Weghed Indces 14 Defnon 14 Corporae Acons and Index Adjusmens 15 Index Acons 15 Corporae Acons 15 Capped Marke Capalzaon Indces 16 Defnon 16 Corporae Acons and Index Adjusmens 17 Dfferen Cappng Mehods 17 Arbue Weghed Indces 19 Corporae Acons 20 S&P Dow Jones Indces: Index Mahemacs Mehodology 1

3 Leveraged and Inverse Indces 22 Leveraged Indces for Eques 22 Leveraged Indces whou Borrowng Coss for Eques 23 Inverse Indces for Eques 23 Inverse Indces whou Borrowng Coss for Eques 24 Leveraged and Inverse Indces for Fuures 24 Daly Rebalanced Leverage or Inverse Fuures Indces 24 Perodcally Rebalanced Leverage or Inverse Fuures Indces 25 Toal Reurn Calculaons 26 Index Fundamenals 27 Fund Sraegy Indces 28 Toal Reurn and Synhec Prce Indces 29 Index and Corporae Acons 30 Dvdend Indces 31 Excess Reurn Indces 32 Rsk Conrol Indces 33 Exponenally-Weghed Volaly 35 Smple-Weghed Volaly 37 Fuures-Based Rsk Conrol Indces 38 Exponenally-Weghed Volaly for Fuures-Based Rsk Conrol Indces 39 Dynamc Volaly Rsk Conrol Indces 39 Varance Based Rsk Conrol Indces 39 Rsk Conrol 2.0 Indces 40 Consuen Weghng 40 Currency Hedged Indces 42 Reurn Defnons 42 The Hedge Rao 43 Calculang a Currency-Hedged Index 43 Currency Hedgng Oucomes 44 Index Compuaon 44 S&P Dow Jones Indces: Index Mahemacs Mehodology 2

4 Domesc Currency Reurn Index Calculaon 47 Background 47 Equvalence of DCR and Dvsor Calculaons 47 DCR Calculaon 48 Essenal Adjusmens 48 Specal Openng Quoaon 49 S&P Dow Jones Indces Conac Informaon 50 Index Managemen 50 Meda Relaons 50 Clen Servces 50 Dsclamer 51 S&P Dow Jones Indces: Index Mahemacs Mehodology 3

5 Inroducon Ths documen covers he mahemacs of ndex calculaons. To undersand and successfully use ndces for nvesmen analyses, s mporan o know how hey are calculaed and how adjusmens are made when consuens change or when dfferen knds of corporae acons occur. Whle acual ndex calculaons are done almos enrely by compuer, ulzng a wde range of programs, algorhms and rounes, he underlyng mah s farly sraghforward. Ths documen assumes some acquanance wh mahemacal noaon and smple operaons, and lle more. A he same me, he calculaons are presened prncpally as equaons, wh a few examples or ables of resuls. The equaons whch have largely been excluded from he ndvdual ndex mehodologes are he mos effcen way o presen he nformaon. Dfferen Varees of Indces A majory of S&P Dow Jones equy ndces are marke cap weghed and floa-adjused, where each sock s wegh n he ndex s proporonal o s floa-adjused marke value. The frs secon covers hese ndces. A second group of ndces are equal weghed, where each sock s weghed equally n he ndex. These are dscussed n he second secon. A hrd group of ndces are weghed by oher facors, such as maxmum wegh resrcons or ceran arbues used o choose he socks. These are dscussed n secons hree, four and fve. A fourh group of ndces are leverage and nverse ndces, whch reurn posve or negave mulples of her underlyng ndces. These are dscussed n secon sx. There are also calculaons ha operae on an ndex as a whole raher han on he ndvdual socks. These nclude calculaons of oal reurns and ndex fundamenals. These are covered n secons seven and egh below. Fund sraegy ndces nclude porfolos of funds. These ndces are covered n secon nne. Dvdend ndces, whch rack he oal dvdend paymens of ndex consuens, and excess reurn ndces, whch calculae he reurn on an nvesmen n an ndex where he nvesmen was made hrough he use of borrowed funds, are covered n secons 10 and 11, respecvely. Fnally, nformaon on he mehodology for rsk conrol and currency hedged ndces are covered n he 12 h and 13 h secons of hs documen. The Index Dvsor Throughou all he calculaons here s one concep ha s crucally mporan o undersandng how ndces are calculaed he ndex dvsor. The smples capalzaon weghed ndex can be hough of as a porfolo conssng of all avalable shares of he socks n he ndex. Whle one mgh rack hs porfolo s value n dollar erms, would probably be an unweldy number for example, he S&P 500 floa-adjused marke value s roughly US$ 11.0 rllon. Raher han deal wh en or more dgs, he fgure s scaled o a more easly handled number, currenly around Dvdng he porfolo marke value by a facor, usually called he dvsor, does he scalng. An ndex s no exacly he same as a porfolo. For nsance, when a sock s added o or deleed from an ndex, he ndex level should no jump up or drop down; whle a porfolo s value would usually change as socks are swapped n and ou. To assure ha he ndex s value, or level, does no change when socks are added or deleed, he dvsor s adjused o offse he change n marke value of he ndex. Thus, he dvsor plays a crcal role n he ndex s ably o provde a connuous measure of marke valuaon when S&P Dow Jones Indces: Index Mahemacs Mehodology 4

6 faced wh changes o he socks ncluded n he ndex. In a smlar manner, some corporae acons ha cause changes n he marke value of he socks n an ndex should no be refleced n he ndex level. Adjusmens are made o he dvsor o elmnae he mpac of hese corporae acons. S&P Dow Jones Indces: Index Mahemacs Mehodology 5

7 Capalzaon Weghed Indces Mos of S&P Dow Jones ndces, ndeed he mos wdely quoed sock ndces, are capalzaonweghed ndces. Somemes hese are called value-weghed or marke cap weghed nsead of capalzaon weghed. Examples nclude he S&P 500, he S&P Global 1200 and he S&P BMI ndces. Examples from oher ndex provders (where some of he deals may vary slghly from hose descrbed here) nclude MSCI s ndces, FTSE s ndces and Russell s ndces. Whle Dow Jones does offer marke cap weghed ndces, he well-known Dow Jones Indusral Average s no cap-weghed. In he dscusson below mos of he examples refer o he S&P 500 bu apply equally o a long ls of S&P Dow Jones Indces cap-weghed ndces. Defnon The formula o calculae he S&P 500 s: P Q Index Level = Dvsor (1) The numeraor on he rgh hand sde s he prce of each sock n he ndex mulpled by he number of shares used n he ndex calculaon. Ths s summed across all he socks n he ndex. The denomnaor s he dvsor. If he sum n he numeraor s US$ 11.8 rllon and he dvsor s US$ 9.4 bllon, he ndex level would be 1250, close o he curren levels of he S&P 500. Ths ndex formula s somemes called a base-weghed aggregave mehod. 1 The formula s creaed by a modfcaon of a LasPeyres ndex, whch uses base perod quanes (share couns) o calculae he prce change. A LasPeyres ndex would be: Index = P P,1,0 Q,o Q,o In he modfcaon o (2), he quany measure n he numeraor, Q 0, s replaced by Q 1, so he numeraor becomes a measure of he curren marke value, and he produc n he denomnaor s replaced by he dvsor whch boh represens he nal marke value and ses he base value for he ndex. The resul of hese modfcaons s equaon (1) above. Adjusmens o Share Couns S&P Dow Jones Indces marke cap-weghed ndces are floa adjused he number of shares ousandng s reduced o exclude closely held shares from he ndex calculaon because such shares are no avalable o nvesors. S&P Dow Jones Indces rules for floa adjusmen are descrbed n more deal n S&P Dow Jones Indces Floa Adjusmen Mehodology or n some of he ndvdual ndex mehodology documens. As dscussed here, for each sock S&P Dow Jones Indces calculaes an Invesable Wegh Facor (IWF) whch s he percenage of oal shares ousandng ha are ncluded n he ndex calculaon. (2) 1 Ths erm s used n one of he earler and more complee descrpons of S&P Dow Jones Indces ndex calculaons n Alfred Cowles, Common Sock Indces, Prncpa Press for he Cowles Commsson of Research n Economcs, The book refers o he Sandard Sascs Company Formula; S&P was formed by he merger of Sandard Sascs Corporaon and Poor s Publshng n S&P Dow Jones Indces: Index Mahemacs Mehodology 6

8 When he ndex s calculaed usng equaon (1), he varable Q s replaced by he produc of ousandng shares and he IWF: Q = IWF Toal Shares (3a) A mes here are oher adjusmens made o he share coun o reflec foregn ownershp resrcons or o adjus he wegh of a sock n an ndex. These are combned no a sngle mulpler n place of he IWF n equaon (3a). In combnng resrcons s mporan o avod unwaned double counng. Le FA represen he fracon of shares elmnaed due o floa adjusmen, FR represen he fracon of shares excluded for foregn ownershp resrcons and IS represen he fracon of oal shares o be excluded based on he combnaon of FA and FR. If FA > FR hen IS = 1- FA, If FA < FR hen IS = 1-FR (4a) (4b) and equaon (3a) can be wren as: Q = IS Toal Shares (3b) Noe ha any me he share coun or he IWF s changed, wll be necessary o adjus he ndex dvsor o keep he level of he ndex unchanged. Dvsor Adjusmens The key o ndex manenance s he adjusmen of he dvsor. Index manenance reflecng changes n shares ousandng, capal acons, addon or deleon of socks o he ndex should no change he level of he ndex. If he S&P 500 closes a 1250 and one sock s replaced by anoher, afer he marke close, he ndex should open a 1250 he nex mornng f all of he openng prces are he same as he prevous day s closng prces. Ths s accomplshed wh an adjusmen o he dvsor. Any change o he socks n he ndex ha alers he oal marke value of he ndex whle holdng sock prces consan wll requre a dvsor adjusmen. Ths secon explans how he dvsor adjusmen s made gven he change n oal marke value. The nex secon dscusses wha ndex changes and corporae acons lead o changes n oal marke value and he dvsor. Equaon (1) s expanded o show he sock beng removed, sock r, separaely from he socks ha wll reman n he ndex: Index Level 1 = ( P Q ) + P Q Dvsor 1 r r (5) Noe ha he ndex level and he dvsor are now labeled for he me perod -1 and, o smplfy hs example, ha we are gnorng any possble IWF and adjusmens o share couns. Afer sock r s replaced wh sock s, he equaon wll read: Index Level = ( P Q ) + P Q Dvsor s s (6) In equaons (5) and (6) -1 s he momen rgh before company r s removed from and s s added o he ndex; s he momen rgh afer he even. By desgn, Index Level -1 s equal o Index Level. Combnng (5) and (6) and re-arrangng, he adjusmen o he Dvsor can be deermned from he ndex marke value before and afer he change: S&P Dow Jones Indces: Index Mahemacs Mehodology 7

9 ( P Q ) + P Q ( P Q ) + P Q Dvsor 1 r r = Index Level = Dvsor s s (7) Le he numeraor of he lef hand fracon be called MV -l, for he ndex marke value a (-1), and he numeraor of he rgh hand fracon be called MV, for he ndex marke value a me. Now, MV -1, MV and Dvsor -1 are all known quanes. Gven hese, s easy o deermne he new dvsor ha wll keep he ndex level consan when sock r s replaced by sock s: Dvsor MV = ( Dvsor 1 ) (8) MV 1 As dscussed below, varous ndex adjusmens resul n changes o he ndex marke value. When hese adjusmens occur, he dvsor s adjused as shown n equaon (8). In some mplemenaons, ncludng he compuer programs used n S&P Dow Jones Indces ndex calculaons, he dvsor adjusmen s calculaed n a slghly dfferen, bu equvalen, forma where he dvsor change s calculaed by addon raher han mulplcaon. Ths alernave forma s defned here. Rearrangng equaon (1) and usng he erm MV (marke value) o replace he summaon gves: MV Dvsor = (9) Index Level When socks are added o or deleed from an ndex here s an ncrease or decrease n he ndex s marke value. Ths ncrease or decrease s he marke value of he socks beng added less he marke value of hose socks deleed; defne CMV as he Change n Marke Value. Recallng ha he ndex level does no change, he new dvsor s defned as: or MV + CMV Dvsor New Index Level = (10) MV CMV Dvsor New + IndexLevel IndexLevel = (11) However, he frs erm on he rgh hand sde s smply he Dvsor value before he addon or deleon of he socks. Ths yelds: Dvsor Dvsor New Old + CMV IndexLevel = (12) Noe ha hs form s more versale for compuer mplemenaons. Wh hs addve form, he second erm (CMV/Index Level) can be calculaed for each sock or oher adjusmen ndependenly and hen all he adjusmens can be combned no one change o he Dvsor. Necessary Dvsor Adjusmens Dvsor adjusmens are made afer he close meanng ha afer he close of radng he closng prces are used o calculae he new dvsor based on whaever changes are beng made. I s, hen, possble o provde wo complee descrpons of he ndex one as exsed a he close of radng and one as wll S&P Dow Jones Indces: Index Mahemacs Mehodology 8

10 exs a he nex openng of radng. If he same sock prces are used o calculae he ndex level for hese wo descrpons, he ndex levels are he same. Wh prces consan, any change ha changes he oal marke value ncluded n he ndex wll requre a dvsor change. For caalogng changes, s useful o separae changes caused by he managemen of he ndex from hose semmng from corporae acons of he consuen companes. Among hose changes drven by ndex managemen are addng or deleng companes, adjusng share couns and changes o IWFs and oher facors affecng share couns or sock prces. Index Managemen Relaed Changes: When a company s added o or deleed from he ndex, he ne change n he marke value of he ndex s calculaed and hs s used o calculae he new dvsor. The marke values of socks beng added or deleed are based on he prces, shares ousandng, IWFs and any oher share coun adjusmens. Specfcally, f a company beng added has a oal marke cap of US$ 1 bllon, an IWF of 85% and, herefore, a floa adjused marke cap of US$ 850 mllon, he marke value for he added company used s US$ 850 mllon. The calculaons would be based on eher equaon (8) or equaon (12) above. For mos S&P Dow Jones ndces, here are a few daes durng he year when IWFs and share couns are updaed. (Typcally small changes n shares ousandng are refleced n ndces once a quarer o avod excessve changes o an ndex.) The revsons o he dvsor resulng from hese are calculaed and a new dvsor s deermned. Equaon (12) shows how he mpac of a seres of share coun changes can be combned o deermne he new dvsor. Corporae Acon Relaed Changes: There are a large range of dfferen corporae acons rangng from roune share ssuances or buy backs o unusual evens lke spn-offs or mergers. These are lsed on he able below wh noes abou he necessary changes and wheher he dvsor s adjused. Corporae Acon Commens Company added/deleed Ne change n marke value deermnes he dvsor adjusmen. Change n shares Any combnaon of secondary ssuance, share repurchase or buy ousandng back share couns revsed o reflec change. Share coun revsed o reflec new coun. Dvsor adjusmen s no Sock spl requred snce he share coun and prce changes are offseng. If he spun-off company s no beng added o he ndex, he dvsor Spn-off adjusmen reflecs he declne n ndex marke value (.e., he value of he spun-off un). Spun-off company added o he ndex, no company removed from he Spn-off ndex. Spun-off company added o he ndex, anoher company removed o Spn-off keep number of names fxed. Dvsor adjusmen reflecs deleon. Increasng (decreasng) he IWF ncreases (decreases) he oal Change n IWF marke value of he ndex. The dvsor change reflecs he change n marke value caused by he change o an IWF. When a company pays a specal dvdend he share prce s assumed Specal Dvdend o drop by he amoun of he dvdend; he dvsor adjusmen reflecs hs drop n ndex marke value. Each shareholder receves he rgh o buy a proporonal number of addonal shares a a se (ofen dscouned) prce. The calculaon Rghs offerng assumes ha he offerng s fully subscrbed. Dvsor adjusmen reflecs ncrease n marke cap measured as he shares ssued mulpled by he prce pad. Dvsor Adjusmen Yes Yes No Yes No Yes Yes Yes Yes Wh corporae acons where cash or oher corporae asses are dsrbued o shareholders, he prce of he sock wll gap down on he ex-dvdend day (he frs day when a new shareholder s no elgble o receve he dsrbuon.) The effec of he dvsor adjusmen s o preven hs prce drop from causng a correspondng drop n he ndex. S&P Dow Jones Indces: Index Mahemacs Mehodology 9

11 For more nformaon on he reamen of corporae acons, please refer o S&P Dow Jones Indces Equy Indces Polces & Pracces documen. Capped Indces A mes s desrable o se a maxmum wegh for some socks n an ndex. In some markes regulaons resrc he wegh of he larges sock or group of socks o be less han a ceran percenage of a porfolo. Ths s done by a furher adjusmen o he share coun, beyond he nvesable wegh facor. Snce he oal wegh of all socks n he ndex wll add up o 100%, reducng he wegh of one sock wll ncrease he wegh of he ohers. I s possble ha when he larges sock s wegh s brough down below some lm, he wegh of he nex larges or several nex larges socks wll exceed he lm. Therefore, he process mus be erave. Weghs wll change over me as sock prces move even f share couns reman consan. If a capped sock enjoys a prce run-up, may exceed he cap. In mos cases buffer zones are used e.g., f he maxmum allowable wegh s 10% he sock s shares are adjused downward unl s wegh s 9% leavng a one-percenage pon buffer before anoher adjusmen s necessary. Varous forms of capped ndces are covered n he nex hree secons of hs documen. S&P Dow Jones Indces: Index Mahemacs Mehodology 10

12 Equal Weghed Indces Defnon An equal weghed ndex s one where every sock has he same wegh n he ndex, and a porfolo ha racks he ndex wll nves an equal dollar amoun n each secury. As sock prces move, he weghs wll shf and exac equaly wll be los. Therefore, an equal weghed ndex mus be rebalanced from me o me o re-esablsh he proper weghng. (In conras, a cap-weghed ndex requres no rebalancng as long as here aren any changes o share couns, IWFs, reurns of capal, or socks added or deleed.) The overall approach o calculae equal weghed ndces s he same as n he cap-weghed ndces; however, he consuens marke values are re-defned o be values ha wll acheve equal weghng a each rebalancng. Recall wo basc formulae: and Index MarkeValue Index Level = (13) Dvsor Index MarkeValue = P Shares IWF (14) To calculae an equal weghed ndex, he marke capalzaon for each sock used n he calculaon of he ndex s redefned so ha each ndex consuen has an equal wegh n he ndex a each rebalancng dae. In addon o beng he produc of he sock prce, he sock s shares ousandng, and he sock s floa facor (IWF), as wren above and he exchange rae when applcable a new adjusmen facor s also nroduced n he marke capalzaon calculaon o esablsh equal weghng. AdjusedSock MarkeValue = P Shares IWF FxRae AWF (15) where AWF s he adjusmen facor of sock assgned a each ndex rebalancng dae,, whch makes all ndex consuens modfed marke capalzaon equal (and, herefore, equal wegh), whle mananng he oal marke value of he overall ndex. The AWF for each ndex consuen,, a rebalancng dae,, s calculaed by: Z N * FloaAdjusedMarkeValue AWF, = (16), where N s he number of socks n he ndex and Z s an ndex specfc consan se for he purpose of dervng he AWF and, herefore, each sock s share coun used n he ndex calculaon (ofen referred o as modfed ndex shares). The ndex dvsor s defned based on he ndex level and marke value from equaon (13). The ndex level s no alered by ndex rebalancngs. However, snce prces and ousandng shares wll have changed snce he las rebalancng, he dvsor wll change a he rebalancng. So: S&P Dow Jones Indces: Index Mahemacs Mehodology 11

13 where, (Dvsor) afer rebalancng = MarkeValue = (Index Marke Value) (Index Value) P Shares afer rebalancng before rebalancng IWF FxRae AWF (16a) Index (16b) Corporae Acons and Index Adjusmens The ables on he followng page show he necessary adjusmens o he ndex and he dvsor for managng an equal weghed ndex. One key ssue s how o handle evens when one sock s replaced by anoher. Gven ha sock prces move all he me, he ndex s only ruly equally weghed a he rebalancng. Therefore, when socks are added or deleed eher he new sock mus assume he acual wegh of he old sock or he enre ndex mus be rebalanced. Snce ndex rebalancng generaes radng coss for racker funds, he desgn decson s usually made o have a new sock assume he wegh of he sock beng dropped unl he nex rebalancng. However, hs s no always he case and may vary by ndex famly. For more nformaon on he reamen of corporae acons, please refer o S&P Dow Jones Indces Equy Indces Polces & Pracces documen. For more nformaon on he specfc reamen whn an ndex famly, please refer o ha ndex mehodology. Index Acons S&P Dow Jones Indces Paren Index Acon Consuen change even number of adds and drops Consuen change deleon only Adjusmen Made o he Equal Wegh Index The company enerng he ndex goes n a he wegh of he company comng ou. Ths wegh s used o compue he adjused wegh facor of he added sock, usng Equaon 15. If a company s beng removed a a prce of 0.00, he replacemen goes n a he wegh of he deleed company a he close on he day before he effecve dae. If more han one company s beng replaced n he ndex on a sngle dae, he replacemens would be n he order saed n he press release for he paren ndex change. The weghs of all socks n he ndex wll change, due o he absolue change n he number of ndex consuens. Relave weghs wll say he same. None. The adjusmen facor s changed o keep he ndex wegh he same. Dvsor Adjusmen for he S&P Dow Jones Indces EWI None Share changes beween quarerly share adjusmens None Quarerly share changes There s no drec adjusmen. None Yes S&P Dow Jones Indces: Index Mahemacs Mehodology 12

14 Corporae Acons S&P Dow Jones Indces Paren Index Acon Spn-off Rghs Offerng Sock Spl Share Issuance or Share Repurchase Specal Dvdends Adjusmen Made o he Equal Wegh Index The prce s adjused o he Prce of he Paren Company mnus (he Prce of Spn-off company/share Exchange Rao). The adjusmen facor changes accordng o Equaon 16, o manan he wegh o be he same as he company had before he spn-off. The prce s adjused o he Prce of he Paren Company mnus (he Prce of Rghs Offerng/Rghs Rao). The adjusmen facor changes accordng o Equaon 16, o manan he wegh o be he same as he company had before he rghs offerng. Shares are mulpled by and he prce s dvded by he spl facor. None. The prce of he sock makng he specal dvdend paymen s reduced by he per share specal dvdend amoun afer he close of radng on he day before he ex-dae. Dvsor Adjusmen for he S&P Dow Jones Indces EWI None None None None A dvsor adjusmen s made o ensure he ndex level remans he same. S&P Dow Jones Indces: Index Mahemacs Mehodology 13

15 Modfed Marke Capalzaon Weghed Indces Defnon A modfed marke cap weghed ndex s one where ndex consuens have a user-defned wegh n he ndex. Ths mehodology s ypcally used for ndces where some consuens are confned o a maxmum wegh, and he excess wegh s dsrbued proporonaely among he remanng ndex consuens. Beween ndex rebalancngs, corporae acons generally have no affec on ndex weghs, as hey are fxed hrough he processes defned below. As sock prces move, he weghs wll shf and he modfed weghs wll change. Therefore, as n he case of an equal-weghed ndex, a modfed marke cap weghed ndex mus be rebalanced from me o me o re-esablsh he proper weghng. The overall approach o calculae modfed marke cap weghed ndces s he same as n he capweghed ndces; however, he consuens marke values are re-defned o be values ha wll acheve he user-defned weghng a each rebalancng. Recall wo basc formulae: and Index MarkeValue Index Level = (17) Dvsor Index MarkeValue P Shares IWF FxRae (18) = To calculae a modfed marke cap weghed ndex, he marke capalzaon for each sock used n he calculaon of he ndex s redefned so ha each ndex consuen has he approprae user-defned wegh n he ndex a each rebalancng dae. In addon o beng he produc of he sock prce, he sock s shares ousandng, and he sock s floa facor (IWF), as wren above and he exchange rae when applcable a new adjusmen facor s also nroduced n he marke capalzaon calculaon o esablsh he approprae weghng. AdjusedSock MarkeValue = P Shares IWF FxRae AWF (19) where AWF s he adjusmen facor of sock assgned a each ndex rebalancng dae,, whch adjuss he marke capalzaon for all ndex consuens o acheve he user-defned wegh, whle mananng he oal marke value of he overall ndex. The AWF for each ndex consuen,, on rebalancng dae,, s calculaed by: Z AWF, = * W, (20) FloaAdjusedMarkeValue, where Z s an ndex specfc consan se for he purpose of dervng he AWF and, herefore, each sock s share coun used n he ndex calculaon (ofen referred o as modfed ndex shares). W, s he userdefned wegh of sock on rebalancng dae. S&P Dow Jones Indces: Index Mahemacs Mehodology 14

16 The ndex dvsor s defned based on he ndex level and marke value from equaon (17). The ndex level s no alered by ndex rebalancngs. However, snce prces and ousandng shares wll have changed snce he las rebalancng, he dvsor wll change a he rebalancng. So: where, (Dvsor) afer rebalancng = MarkeValue = (Index Marke Value) (Index Value) P Shares afer rebalancng before rebalancng IWF FxRae AWF (20a) Index (20b) Corporae Acons and Index Adjusmens The ables below shows he necessary adjusmens o he ndex and he dvsor for managng a modfed marke cap weghed ndex. For more nformaon on he reamen of corporae acons, please refer o S&P Dow Jones Indces Equy Indces Polces & Pracces documen. For more nformaon on he specfc reamen whn an ndex famly, please refer o ha ndex mehodology. Index Acons S&P Dow Jones Indces Paren Index Acon Consuen change Adjusmen Made o he Modfed Marke Cap Weghed Index The company enerng he ndex goes n a he wegh of he company comng ou. Dvsor Adjusmen for he Index? None Delsng, acquson or any oher corporae acon resulng n a The sock s dropped from he Index Yes consuen deleon. Share changes beween quarerly None. The adjusmen facor s changed o keep share adjusmens he ndex wegh he same. None Quarerly share changes There s no drec adjusmen. None Corporae Acons S&P Dow Jones Indces Paren Index Acon Spn-off Rghs Offerng Sock Spl Share Issuance or Share Repurchase Specal Dvdends Merger or acquson Adjusmen Made o he Modfed Marke Cap Weghed Index The prce s adjused o he Prce of he Paren Company mnus (he Prce of Spn-off company/share Exchange Rao). The adjusmen facor changes accordng o Equaon 20, o manan he wegh o be he same as he company had before he spn-off. The prce s adjused o he Prce of he Paren Company mnus (he Prce of Rghs Offerng/Rghs Rao). The adjusmen facor changes accordng o Equaon 20, o manan he wegh o be he same as he company had before he rghs offerng. Shares are mulpled by and he prce s dvded by he spl facor. None. The prce of he sock makng he specal dvdend paymen s reduced by he per share specal dvdend amoun afer he close of radng on he day before he ex-dae. If he survvng company s already an ndex member, s reaned n he ndex. If he survvng company does no mee ndex crera, s removed. Dvsor Adjusmen for he Index? None None None None A dvsor adjusmen s made o ensure he ndex level remans he same. Yes, f here s a removal. S&P Dow Jones Indces: Index Mahemacs Mehodology 15

17 Capped Marke Capalzaon Indces Defnon A capped marke cap weghed ndex s one where sngle ndex consuens or defned groups of ndex consuens are confned o a maxmum wegh and he excess wegh s dsrbued proporonaely among he remanng ndex consuens. As sock prces move, he weghs wll shf and he modfed weghs wll change. Therefore, as n he case of an equal-weghed ndex and a modfed marke cap ndex, a capped marke cap weghed ndex mus be rebalanced from me o me o re-esablsh he proper weghng. The mehodology for capped ndces proceeds smlarly o ha for modfed marke cap weghed ndces. The man dfference beween he wo mehods s he reamen of corporae acons beween rebalancng perods. For modfed marke cap weghed ndces mos corporae acons whch affec he marke capalzaon of a gven sock are counerbalanced by a correspondng change n he AWF assgned o he sock n ha ndex, hus resulng n no wegh change o he sock and no ndex dvsor change (see he pror secon for reference). On he oher hand, for capped ndces no AWF change s made due o corporae acons beween rebalancngs and, hus, he weghs of socks n he ndex as well as he ndex dvsor wll change due o corporae acons The overall approach o calculae capped marke cap weghed ndces s he same as n he pure markecap weghed ndces; however, he consuens marke values are re-defned o be values ha wll mee he parcular cappng rules of he ndex n queson. Recall equaons 17, 18 and 19 from he pror secon: and Index MarkeValue Index Level = (17) Dvsor Index MarkeValue P Shares IWF * FxRae * * = (18) To calculae a capped marke cap weghed ndex, he marke capalzaon for each sock used n he calculaon of he ndex s redefned so ha each ndex consuen has he approprae wegh n he ndex a each rebalancng dae. In addon o beng he produc of he sock prce, he sock s shares ousandng, and he sock s floa facor (IWF), as wren above and he exchange rae when applcable a new adjusmen facor s also nroduced n he marke capalzaon calculaon o esablsh he approprae weghng. AdjusedSock MarkeValue = P Shares IWF FxRae AWF (19) where AWF s he adjusmen facor of sock assgned a each ndex rebalancng dae,, whch adjuss he marke capalzaon for all ndex consuens o acheve he user-defned wegh, whle mananng he oal marke value of he overall ndex. The AWF for each ndex consuen,, on rebalancng dae,, s calculaed by: AWF CW W,, = (21), S&P Dow Jones Indces: Index Mahemacs Mehodology 16

18 where W, s he uncapped wegh of sock on rebalancng dae based on he floa marke capalzaon of all ndex consuens; and CW, s he capped wegh of sock on rebalancng dae as deermned by he cappng rule of he ndex n queson and he process for deermnng capped weghs as descrbed n Dfferen Cappng Mehods below. The ndex dvsor s defned based on he ndex level and marke value from equaon (17). The ndex level s no alered by ndex rebalancngs. However, snce prces and ousandng shares wll have changed snce he las rebalancng, he dvsor wll change a he rebalancng. So: where, (Dvsor) afer rebalancng = MarkeValue = (Index Marke Value) (Index Value) P Shares afer rebalancng before rebalancng IWF FxRae AWF (21a) Index (21b) Corporae Acons and Index Adjusmens All corporae acons for capped ndces affec he ndex n he same manner as n marke cap weghed ndces. For deals on how each corporae acon s reaed please refer o he able for Capalzaon Weghed Indces secon above. For more nformaon on he reamen of corporae acons, please refer o S&P Dow Jones Indces Equy Indces Polces & Pracces documen. Dfferen Cappng Mehods Capped ndces arse due o he need for benchmarks whch comply wh dversfcaon rules, and wh funds and lsed producs when he general desre s for a benchmark whch s hghly concenraed n one or a small number of socks. Cappng may apply o sngle sock concenraon lms or concenraon lms on a defned group of socks. The sandard S&P Dow Jones Indces mehodology for deermnng he weghs of capped ndces usng he mos popular cappng mehods are descrbed below. Sngle Sock Cappng In a sngle sock cappng mehodology no sock n an ndex s allowed o breach a ceran pre-deermned wegh as of each rebalancng perod. The procedure for assgnng capped weghs o each sock a each rebalancng s as follows: 1. Wh daa refleced on he rebalancng reference dae, each company s weghed by floaadjused marke capalzaon. 2. If any company has a wegh greaer han X% (where X% s he maxmum wegh allowed n he ndex), ha company has s wegh capped a X%. 3. All excess wegh s proporonally redsrbued o all uncapped socks whn he ndex. 4. Afer hs redsrbuon, f he wegh of any oher sock(s), hen, breaches X%, he process s repeaed eravely unl no socks breach he X% wegh cap. Sngle Sock and Concenraon Lm Cappng In a sngle sock and concenraon lm cappng no sock n an ndex s allowed o breach a ceran predeermned wegh and all socks wh a wegh greaer han a ceran amoun are no allowed, as a group, o exceed a predeermned oal wegh. One example of hs s 5/25/50 cappng (B/A/C n he example S&P Dow Jones Indces: Index Mahemacs Mehodology 17

19 below). No sngle sock s allowed o exceed 25% of he ndex and all socks wh a wegh greaer han 5% of he ndex canno exceed, as a group, 50% of he ndex. The procedure for assgnng capped weghs o each sock a each rebalancng s as follows: 1. Wh daa refleced on he rebalancng reference dae, each company s weghed by floaadjused marke capalzaon. 2. If any company has a wegh greaer han A% (where A% s he maxmum wegh allowed n he ndex), ha company has s wegh capped a A%. 3. All excess wegh s proporonally redsrbued o all uncapped socks whn he ndex. 4. Afer hs redsrbuon, f he wegh of any oher sock(s), hen, breaches A%, he process s repeaed eravely unl no socks breach he A% wegh cap. 5. The wegh of all socks whn he ndex ha have a wegh greaer han B% are added ogeher. If he oal wegh of hese socks s less han C% hen he cappng s compleed. 6. If he oal wegh s greaer han C%, hen he socks n queson are ranked n descendng order based on wegh, summed cumulavely, and he frs sock ha brngs he oal wegh of he group above C% s, hen, capped. Ths sock s capped o a wegh equal o he larger of (1) B% or (2) he dfference beween C% and he oal wegh of all he socks larger han he sock n queson. 7. All socks wh weghs greaer han B%, bu wh lower weghs han he sock capped n sep 6, are capped o a wegh of B%. 8. All excess wegh s proporonally redsrbued o all socks whn he ndex wh a wegh less han B% 9. Afer hs redsrbuon, f (1) he wegh of any sock(s) ha was orgnally less han B% hen breaches B% or (2) a he frs nsance a sock brngs he oal wegh of he group above C%, hen he process s repeaed eravely unl no socks breach he lower of (1) he B% wegh cap and (2) he wegh of he frs sock ha brough he oal wegh of he group above C%. S&P Dow Jones Indces: Index Mahemacs Mehodology 18

20 Arbue Weghed Indces In recen years varous new approaches o weghng socks n ndces have appeared o supplemen cap weghng and prce weghng. 2 S&P s Pure Syle Indces, nroduced n 2005, are arbue weghed a sock s wegh depends on he measures of s growh or value arbues, he same measure ha was used o selec socks for he ndex. The dscusson here covers how hese ndces are calculaed; he selecon of socks s covered n he S&P U.S. Syle Indces mehodology documen. There are boh Pure Growh Syle and Pure Value Syle ndces. Under he selecon process, each sock has a growh score and a value score. These scores are used o denfy pure growh socks and pure value socks. (A sock canno be boh pure growh and pure value; can be neher pure growh nor pure value.) The Pure Growh ndex ncludes only pure growh socks; a socks wegh n he ndex s deermned by s growh score; lkewse for pure value. When he ndex s rebalanced, he relave weghs of any wo socks should be proporonal o her relave syle scores. For nsance, n he pure value ndex, a he rebalancng, a sock wh a value score of 1.5 should have 50% more wegh han a sock wh a value score of 1.0 (snce 1.5 s 50% greaer han 1.0). Snce he wegh represens how much of he ndex s oal marke value s n each sock, he wegh depends on boh he prce of he sock and he number of shares held. The desgn of he ndex calculaon procedure deermnes he number of shares of each sock o be ncluded whle he marke deermnes he prce. As n he case of equal weghed ndces, he general srucure follows ha used for he cap-weghed ndces, defnng he basc equaons as: and Index MarkeValue Index Level = (22) Dvsor Index MarkeValue = P Shares (23) For smplcy, he equaons do no nclude he IWF erm even hough s used n he acual calculaons. The share coun s modfed so ha he sock s wegh reflecs s syle score raher han s marke capalzaon. Defne M_Shares as he modfed share coun: M _ Shares = Shares PWF (24) where PWF s he Pure Syle Wegh Facor for sock. The PWF depends on he rue share coun (Shares), he syle score and he sock s prce value when he ndex s rebalanced. The PWF s defned as: PWF k SV = (25) Shares, = reb Pr ce, = reb 2 Prce weghng means an unweghed or smple average of he sock prces. I s seldom used n new ndces bu connues o be used by some long-sandng ndces ncludng he Dow Jones Indusral Average and he Nkke 225 Sock Average. The Dow daes from 1896; he Nkke from S&P Dow Jones Indces: Index Mahemacs Mehodology 19

21 Shares s he rue share coun, no he modfed share coun. SV s he value score of sock. The facor k s ncluded o scale he numbers; he denomnaor s que large and whou k he PWF would be a very small number. SV s he capped value score (o preven oulers from havng hgh weghs n he ndex, he scores are capped a 2, effecvely choppng off a dsrbuon a wo sandard devaons away from he mean.) The subscrp =reb ndcaes ha he shares and prce for sock are evaluaed a he rebalancng dae. Noe also ha he IWF would be ncluded n he acual calculaon. By subsung (25) no (24) and hen (24) no (23): k SV P Shares, Index MarkeValue = (26) P Shares,, = reb, = reb Noe ha we have subsued for M_shares so ha he Shares varable n (26) s he acual shares. Now, re-arrangng erms on he rgh hand sde: P Shares, = k SV (27) P Shares,, = reb, = reb The fracon erm nsde he summaon sgn s one plus he proporonal ncrease or decrease n he value of he sock snce he rebalancng (n effec he value relave ). Therefore, he rgh hand sde of (27) represens he weghed average of he value relaves of he socks where he weghs are he value scores. Recallng (22), we can defne he ndex as: P, Shares, k SV P, = reb Shares, = reb IndexLevel = (28) Dvsor Users of S&P Dow Jones Indces arbue weghed ndces ypcally ge he modfed ndex shares and prce n daly fles from S&P Dow Jones Indces, and do no have o do he above calculaons hemselves. Havng a smple share number n erms of he modfed ndex shares helps f hese ndces seamlessly no sysems desgned for marke capalzaon weghed ndces. Corporae Acons Paren Index Acon Adjusmen Made o Pure Syle Index If he consuen beng dropped s a member of he Pure Syle Index, s removed from he Pure Syle Index. The replacemen sock can be added o eher he pure value or he pure growh ndex, or o neher. Consuen change The wegh s smply he rao of he capped syle score of he added sock dvded by he sum of syle scores of all ndex consuens. The ndcave wegh s announced hrough S&P Dow Jones Indces IDP. The wegh of socks s unchanged. (The share coun follows he paren ndex share coun. Share changes beween To keep weghs of socks unchanged followng a share quarerly share adjusmens change he PWF s adjused for he sock whose shares are beng changed.) If he annual rebalancng dae concdes wh a quarerly Quarerly share changes share updae, hs s he only me when weghs are revsed. The wegh of socks s unchanged. Spn-off The prce follows he paren ndex prce change. To keep weghs of socks unchanged followng prce change, Dvsor Adjusmen Requred? Yes No Yes f concdes, No oherwse. No S&P Dow Jones Indces: Index Mahemacs Mehodology 20

22 Rghs Offerng Sock Spl Specal Dvdends PWF s adjused for he sock whose shares are beng changed. The wegh of socks s unchanged. The prce follows he paren ndex prce change. To keep weghs of socks unchanged, PWF s adjused for he sock whose shares are beng changed. Shares are mulpled by and he prce s dvded by he spl facor. The prce of he sock makng he specal dvdend paymen s reduced by he per share specal dvdend amoun afer he close of radng on he day before he ex-dae. No No Yes For more nformaon on he reamen of corporae acons, please refer o S&P Dow Jones Indces Equy Indces Polces & Pracces documen. S&P Dow Jones Indces: Index Mahemacs Mehodology 21

23 Leveraged and Inverse Indces Leveraged Indces for Eques S&P Dow Jones Indces Leveraged Indces are desgned o generae a mulple of he reurn of he underlyng ndex n suaons where he nvesor borrows funds o generae ndex exposure beyond hs/her cash poson. The approach s o frs calculae he underlyng ndex, hen calculae he daly reurns for he leveraged ndex and, fnally, o calculae he curren value of he leveraged ndex by ncremenng he prevous value by he daly reurn. There s no change o he calculaon of he underlyng ndex. The daly reurn for he leveraged ndex consss of wo componens: (1) he reurn on he oal poson n he underlyng ndex less (2) he borrowng coss for he leverage. The formula for calculang he Leveraged Index s as follows: Underlyng Index Borrowng Rae Leveraged Index Re urn = K 1 ( K 1) D, 1 Underlyng Index (29) In equaon (29) he borrowng rae s appled o he leveraged ndex value because hs represens he funds beng borrowed. Gven hs, he Leveraged Index Value a me can be calculaed as: Leveraged IndexValue = ( Leveraged IndexValue 1 ) (1+ Leveraged Index Re urn ) (30) Subsung (29) no (30) and expandng he rgh hand sde of (30) yelds: where: Leveraged IndexValue = UnderlyngIndex Leveraged IndexValue K UnderlyngIndex K (K 1) = Leverage Rao K = 1, no leverage K = 2, Exposure = 200% K = 3, Exposure = 300% Borrowng Rae 1 ( K 1) D 360, 1 1 Borrowng Rae = Overngh LIBOR n he U.S. or EONIA n Europe D, -1 = he number of calendar days beween dae and -1 In he absence of leverage (K=1), (31) Underlyng Index Leveraged IndexValue = Leveraged Index Value * 1 (32) Underlyng Index 1 The leverage poson s rebalanced daly. Ths s conssen wh he payoff from fuures based replcaon. S&P Dow Jones Indces: Index Mahemacs Mehodology 22

24 Leveraged Indces whou Borrowng Coss for Eques In some cases, leveraged ndces ha do no accoun for coss ncurred o fnance he assocaed leverage are calculaed. For hese ndces, he borrowng rae n formulas (29) and (31) s se o zero and he calculaon follows as above. Inverse Indces for Eques S&P Dow Jones Indces Inverse ndces are desgned o provde he nverse performance of he underlyng ndex; hs represens a shor poson n he underlyng ndex. The calculaon follows he same general approach as he leveraged ndex wh ceran adjusmens: Frs, he reurn on he underlyng ndex s reversed. Second, whle he coss of borrowng he secures are no ncluded, here s an adjusmen o reflec he neres earned on boh he nal nvesmen and he proceeds from sellng shor he secures n he underlyng ndex. These assumpons reflec normal ndusry pracce. (Sraghforward adjusmens can be made o eher o nclude he coss of borrowng secures or o exclude he neres earned on he shorng proceeds and he nal nvesmen.) The general formula for he reurn o he nverse ndex s Underlyng Index Inverse Index Re urn = K 1 Underlyng Index 1 Lendng Rae + ( K + 1) D 360, 1 (33) Where he frs rgh hand sde erm represens he reurn on he underlyng ndex and he second rgh hand sde erm represens he neres earned on he nal nvesmen and he shorng proceeds. Expandng hs as done above for he leveraged ndex yelds: where: InverseIndexValue Inverse IndexValue = K (K 1) = Leverage Rao K = 1, Exposure = -100% K = 2, Exposure = -200% K = 3, Exposure = -300% UnderlyngIndex 1 K UnderlyngIndex LendngRae 1 ( K + 1) D, 360 (34) Lendng Rae = Overngh LIBOR n he U.S. or EONIA n Europe D, -1 = he number of calendar days beween dae and -1 In he absence of leverage (K =1), InverseIndexValue Inverse IndexValue = UnderlyngIndex 1 UnderlyngIndex LendngRae 1 ( 2 ) D 360 1, 1 1 (35) S&P Dow Jones Indces: Index Mahemacs Mehodology 23

25 The nverse poson s rebalanced daly. Ths s conssen wh he payoff from fuures based replcaon. Inverse Indces whou Borrowng Coss for Eques In some cases, nverse ndces ha do no accoun for any neres earned are calculaed. For hese ndces, he lendng rae n formulas (33) and (34) s se o zero and he calculaon follows as above. Leveraged and Inverse Indces for Fuures S&P Dow Jones Indces fuures-based Leveraged Indces are desgned o generae a mulple of he reurn of he underlyng fuures ndex n suaons where he nvesor borrows funds o generae ndex exposure beyond hs/her cash poson. S&P Dow Jones Indces fuures-based Inverse ndces are desgned o provde he nverse performance of he underlyng fuures ndex; hs represens a shor poson n he underlyng ndex. The approach s o frs calculae he underlyng ndex, hen calculae he daly reurns for he leveraged or nverse ndex. There s no change o he calculaon of he underlyng fuures ndex. The leveraged or nverse ndex may be rebalanced daly or perodcally. Daly Rebalanced Leverage or Inverse Fuures Indces If he S&P Dow Jones Indces fuures-based leveraged or nverse ndex s rebalanced daly, he ndex excess reurn s he mulple of he underlyng ndex s excess reurn and calculaed as follows: where: UnderlyngIndexER IndexER = IndexER 1 1+ K * 1 (36) UnderlyngIndexER 1 K (K 0) = Leverage/Inverse Rao K = 1, no leverage K = 2, leverage exposure = 200% K = 3, leverage exposure = 300% K = -1, nverse exposure = -100% A oal reurn verson of each of he Indces s calculaed, whch ncludes neres accrual on he noonal value of he ndex based on he 91-day US Treasury rae, as follows: IndexTR = IndexTR IndexER IndexER + TBR 1 (37) 1 where: IndexTR -1 TBR = The Index Toal Reurn on he precedng busness day. = Treasury Bll Reurn, as deermned by he followng formula: S&P Dow Jones Indces: Index Mahemacs Mehodology 24

26 TBR Dela TBAR -1 = * TBAR 1 Dela 91 1 = he number of calendar days beween he curren and prevous busness days. = he mos recen weekly hgh dscoun rae for 91-day US Treasury blls effecve on he precedng busness day. Generally he raes are announced by he US Treasury on each Monday. On Mondays ha are bank holdays, Frday s raes apply. Perodcally Rebalanced Leverage or Inverse Fuures Indces If he S&P Dow Jones Indces fuures-based leveraged or nverse ndex s rebalanced perodcally (e.g. weekly, monhly, or quarerly), he ndex excess reurn s he mulple of he underlyng ndex excess reurn snce las rebalancng busness day and shall be calculaed as follows: where: UnderlyngIndexER IndexER = IndexER _ LR 1+ K * 1 (39) UnderlyngIndexER _ LR IndexER _LR UnderlyngIndexER _LR _LR K (K 0) = Leverage / Inverse Rao K = 1, no leverage K = 2, leverage exposure = 200% K = 3, leverage exposure = 300% K = -1, nverse exposure = -100% (38) = The Index Excess Reurn on he las rebalancng busness day, _LR. = The Underlyng Index Excess Reurn value on he las rebalancng busness day, _LR. = he las rebalancng busness day. A oal reurn verson of each of he Indces s calculaed, whch ncludes neres accrual on he noonal value of he ndex based on he 91-day US Treasury rae. The formulae are he same as (37) and (38) above. S&P Dow Jones Indces: Index Mahemacs Mehodology 25

27 Toal Reurn Calculaons The precedng dscussons were relaed o prce ndces where changes n he ndex level reflec changes n sock prces. In a oal reurn ndex changes n he ndex level reflec boh movemens n sock prces and he renvesmen of dvdend ncome. A oal reurn ndex represens he oal reurn earned n a porfolo ha racks he underlyng prce ndex and renvess dvdend ncome n he overall ndex, no n he specfc sock payng he dvdend. The oal reurn consrucon dffers from he prce ndex and bulds he ndex from he prce ndex and daly oal dvdend reurns. The frs sep s o calculae he oal dvdend pad on a gven day and conver hs fgure no pons of he prce ndex: ToalDalyDvdend = Dvdend Shares (40) Where Dvdend s he dvdend per share pad for sock and Shares are he shares. Ths s done for each radng day. Dvdend s generally zero excep for four mes a year when goes ex-dvdend for he quarerly dvdend paymen. Some socks do no pay a dvdend and Dvdend s always zero. ToalDalyDvdend s measured n dollars. Ths s convered o ndex pons by dvdng by he dvsor for he underlyng prce ndex: ToalDalyDvdend IndexDvd end = (41) Dvsor The nex sep s o apply he usual defnon of a oal reurn from a fnancal nsrumen o he prce ndex. Equaon (40) gves he defnon, equaon (41) apples o he ndex: and P + D Toal Re urn = ( P DTR 1 ) 1 IndexLevel + IndexDvdend = ( IndexLevel 1 1) (42) (43) where he ToalReurn and he daly oal reurn for he ndex (DTR) s saed as a decmal. The DTR s used o updae he oal reurn ndex from one day o he nex: Toal Reurn Index = (Toal Reurn Index ) (1 DTR ) (44) 1 + S&P Dow Jones Indces: Index Mahemacs Mehodology 26

28 Index Fundamenals Indces are ofen used o measure marke condons or gauge valuaons among markes or beween socks and ndces hrough measures lke earnngs per share (EPS), prce-earnngs raos, dvdend yelds and so forh. These are calculaed by usng he dvsor as f represens shares for a company. The basc forma s llusraed for he EPS for an ndex: eps shares Index EPS = Dvsor (45) where IndexEPS s he EPS for he overall ndex, eps s he EPS for sock and shares are he shares used o calculae he ndex wh any adjusmens such as he IWF ncorporaed no he fgure. If he calculaon refers o an equal weghed or arbue weghed ndex, he calculaon use he shares defned for hose ndces (C_shares or M_shares, as approprae). The prce-earnngs (PE) rao for he ndex s smply he rao of he ndex level (or prce) o he ndex EPS. For a cap-weghed ndex, hs can also be calculaed drecly from he sock level daa by dvdng he oal marke cap of he ndex by oal earnngs of all companes n he ndex. In hs calculaon, he Dvsor erms n he denomnaor drop ou: P Shares IndexPE = Dvsor eps Shares (46) Dvsor The same general approach can be used for varous ndex fundamenals and raos such as book value per share, prce-o-book, dvdend-o-prce (.e. dvdend yeld) and so forh. S&P Dow Jones Indces: Index Mahemacs Mehodology 27

29 Fund Sraegy Indces A fund sraegy ndex s a porfolo of funds ha s desgned o delver excess reurns over a benchmark whn an esablshed rsk framework. A fund sraegy ndex can nclude a porfolo of funds for a sngle asse class (.e. U.S. eques) or cover mulple asse classes (.e. asse allocaon programs). Fund sraegy ndces are creaed wh U.S. muual funds for U.S. nvesors or Inernaonal (offshore) funds for non-u.s. nvesors. Each fund s wegh whn he sraegy ndex porfolo s deermned by a rules-based porfolo consrucon mehodology. The calculaon process for fund sraegy ndces are based upon he dvsor mehod commonly used for S&P Dow Jones equy ndces. However, he share coun s redefned o be a number ha wll acheve he fund s allocaon weghng a he rebalancng dae. and Index MarkeValue Index Level = (47) Dvsor Index Marke Value = NAV Shares (48) where NAV s he Ne Asse Value, a fund s share prce. To calculae a fund ndex, shares are redefned. Raher han beng he acual coun of shares mulpled by he nvesable wegh facors (IWF) or oher such adjusmen facors, he Shares number s calculaed o esablsh he allocaon weghng a he rebalancng dae. For clary, hs secon wll refer o he shares fgure as fund shares. No only are hese no he rue share coun, hey have essenally no relaon o a fund s rue share coun. Snce fund shares are beng used nsead of rue shares, he Index Marke Value defned n (44) s no he acual marke value of he ndex. In a fund sraegy ndex he Index Marke Value s an arbrary or nomnal value for he porfolo used when he fund shares fgure s esablshed. The fund shares are calculaed a each rebalancng: w,rebalancng dae fund _ shares, rebalancng dae = (49) NAV,rebalancng dae where w s he new allocaon wegh and NAV s he prce of he fund a he rebalancng dae. A each ndex rebalancng, n order o manan ndex seres connuy, s also necessary o adjus he dvsor. Therefore, (Index Level) before rebalancng = (Index Level) afer rebalancng (50) (Dvsor) afer rebalancng = (Index Marke Value) (Index Level) afer rebalancng before rebalancng (51) S&P Dow Jones Indces: Index Mahemacs Mehodology 28