Index Mathematics Methodology

Transcription

1 Index Mahemacs Mehodology S&P Dow Jones Indces: Index Mehodology November 2017

2 Table of Conens Inroducon 2 Dfferen Varees of Indces 2 The Index Dvsor 2 Capalzaon Weghed Indces 3 Defnon 3 Adjusmens o Share Couns 3 Dvsor Adjusmens 4 Necessary Dvsor Adjusmens 5 Capped Indces 7 Prce Weghed Indces 8 Defnon 8 Equal Weghed Indces 9 Defnon 9 Modfed Equal Weghed Indces 10 Corporae Acons and Index Adjusmens 10 Index Acons 10 Modfed Marke Capalzaon Weghed Indces 11 Defnon 11 Corporae Acons and Index Adjusmens 12 Index Acons 12 Capped Marke Capalzaon Indces 13 Defnon 13 Corporae Acons and Index Adjusmens 14 Dfferen Cappng Mehods 14 Pure Syle Indces 16 Weghed Reurn Indces 17 S&P Dow Jones Indces: Index Mahemacs Mehodology 1

3 Leveraged and Inverse Indces 18 Leveraged Indces for Eques 18 Leveraged Indces whou Borrowng Coss for Eques 19 Inverse Indces for Eques 19 Inverse Indces whou Borrowng Coss for Eques 20 Leveraged and Inverse Indces for Fuures 20 Daly Rebalanced Leverage or Inverse Fuures Indces 20 Perodcally Rebalanced Leverage or Inverse Fuures Indces 21 Toal Reurn Calculaons 22 Ne Toal Reurn Calculaons 23 Frankng Cred Adjused Toal Reurn Indces 24 Index Fundamenals 25 Dvdend Indces 26 Rsk Conrol Indces 27 Excess Reurn Indces 29 Exponenally-Weghed Volaly 30 Smple-Weghed Volaly 31 Fuures-Based Rsk Conrol Indces 32 Exponenally-Weghed Volaly for Fuures-Based Rsk Conrol Indces 33 Dynamc Volaly Rsk Conrol Indces 33 Varance Based Rsk Conrol Indces 33 Rsk Conrol 2.0 Indces 34 Consuen Weghng 34 Currency and Currency Hedged Indces 36 Reurn Defnons 36 The Hedge Rao 37 Calculang a Currency-Hedged Index 37 Currency Hedgng Oucomes 38 Index Compuaon 38 Domesc Currency Reurn Index Calculaon 41 Background 41 Equvalence of DCR and Dvsor Calculaons 41 DCR Calculaon 42 Essenal Adjusmens 42 Fee Indces 43 Specal Openng Quoaon 44 S&P Dow Jones Indces: Index Mahemacs Mehodology 0

4 S&P Dow Jones Indces Conac Informaon 45 Clen Servces 45 Dsclamer 46 S&P Dow Jones Indces: Index Mahemacs Mehodology 1

5 Inroducon Ths documen covers he mahemacs of equy ndex calculaons and assumes some acquanance wh mahemacal noaon and smple operaons. The calculaons are presened prncpally as equaons, whch have largely been excluded from he ndvdual ndex mehodologes, wh examples or ables of resuls o demonsrae he calculaons. Dfferen Varees of Indces A majory of S&P Dow Jones Indces 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. In addon, S&P Dow Jones Indces offers a varey ndces calculaed accordng o varous mehodologes whch are covered n hs documen: Prce weghed ndces where consuen weghs are deermned solely by he prces of he consuen socks n he ndex Equal weghed ndces where each sock s weghed equally n he ndex Indces weghed by oher facors such as maxmum wegh resrcons or ceran arbues used o choose he socks Leveraged and nverse ndces whch reurn posve or negave mulples of her respecve underlyng ndces Weghed reurn ndces commonly known as ndex of ndces, where each underlyng ndex s a componen wh an assgned wegh o calculae he overall ndex of ndces level Indces ha operae on an ndex as a whole raher han on he ndvdual socks hese nclude calculaons of varous oal reurn mehodologes and ndex fundamenals Dvdend ndces whch rack he oal dvdend paymens of ndex consuens Rsk conrol, excess reurn, currency, currency hedged, domesc currency reurn, and he specal openng quoaon calculaons 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 a fgure n he rllons of dollars. Raher han deal wh en or more dgs, he fgure s scaled o a more easly handled number (e.g. 2000). 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 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 2

6 Capalzaon Weghed Indces Many of S&P Dow Jones Indces equy ndces, ndeed he mos wdely quoed sock ndces, are capalzaon-weghed 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. 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: Index Level P Q = 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$ 20 rllon and he dvsor s US$ 10 bllon, he ndex level would be 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,1 P,0 Q, o Q, o (2) 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. 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 3

7 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, corporae acons, addon or deleon of socks o he ndex should no change he level of he ndex. If he S&P 500 closes a 2000 and one sock s replaced by anoher, afer he marke close, he ndex should open a 2000 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: ( P Q ) + Pr Qr Index Level 1 = (5) Dvsor 1 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: ( P Q ) + PsQs Index Level = Dvsor (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 4

8 ( P Q ) + Pr Qr ( P Q ) + PsQs = Index Level = Dvsor 1 Dvsor (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: MV Dvsor = (8) ( Dvsor 1 ) 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 = (10) Index Level MV CMV Dvsor New = + (11) IndexLevel IndexLevel However, he frs erm on he rgh hand sde s smply he Dvsor value before he addon or deleon of he socks. Ths yelds: CMV DvsorNew = DvsorOld + (12) IndexLevel 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 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. S&P Dow Jones Indces: Index Mahemacs Mehodology 5

9 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. 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 Indces equy ndces, IWFs and share couns updaes are appled hroughou he year based on rules defned n he mehodology. 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 less frequen 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 Company added/deleed Change n shares ousandng Sock spl Spn-off Change n IWF Specal dvdend Rghs offerng Commens Ne change n marke value deermnes he dvsor adjusmen. Any combnaon of secondary ssuance, share repurchase or buy back share couns revsed o reflec change. Share coun revsed o reflec new coun. Dvsor adjusmen s no requred snce he share coun and prce changes are offseng. The spn-off s added o he ndex on he ex-dae a a prce of zero. Each ndvdual ndex mehodology wll specfy wheher he spn-off wll be elgble o reman n he ndex beyond he ex-dae. Increasng (decreasng) he IWF ncreases (decreases) he oal 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 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 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 No Yes Yes Yes Wh corporae acons where specal dvdends such as exraordnary cash or oher corporae asses ha are dsrbued o shareholders, he prce of he sock wll gap down on he ex-dae (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. 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 6

10 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 followng secons of hs documen. S&P Dow Jones Indces: Index Mahemacs Mehodology 7

11 Prce Weghed Indces Defnon In a prce weghed ndex, such as he Dow Jones Indusral Average, consuen weghs are deermned solely by he prces of he consuen socks. Shares ousandng are se o a unform number hroughou he ndex. Indces usng hs mehodology wll adjus he ndex dvsor for any prce mpacng corporae acon on one of s member socks; hs ncludes prce adjusmens, specal dvdends, sock spls and rghs offerngs. The ndex dvsor wll also adjus n he even of an addon o or deleon from he ndex. All oher ndex calculaon deals follow he sandard dvsor based calculaon mehodology dealed n he prevous capalzaon weghed secon. For more deals on corporae acon reamen please refer o he able below. Corporae Acon Company added/deleed Change n shares ousandng Sock spl Spn-off Commens Ne change n prce deermnes he dvsor adjusmen. Share changes due o secondary ssuance, share repurchase, buy back, or any oher reason are no recognzed n he prce weghed mehodology. All shares are se o 1. Share coun s no revsed o reflec new coun. Dvsor adjusmen s requred o offse he adjusmen n prce. The spn-off s added o he ndex on he ex-dae a a prce of zero. Each ndvdual ndex mehodology wll specfy wheher he spn-off wll be elgble o reman n he ndex beyond he exdae. Dvsor Adjusmen? Yes Change n IWF No IWFs are used n prce weghed ndces. N/A Specal dvdend When a company pays a specal dvdend he share prce s assumed o drop by he amoun of he dvdend; he dvsor adjusmen reflecs hs drop n ndex marke value. Yes Rghs offerng Each shareholder receves he rgh o buy a proporonal number of addonal shares a a se (ofen dscouned) prce. The calculaon assumes ha he offerng s fully subscrbed. Dvsor adjusmen reflecs change n prce as a resul of he rghs offerng. No Yes No Yes S&P Dow Jones Indces: Index Mahemacs Mehodology 8

12 Equal Weghed Indces Defnon An equal weghed ndex s one where every sock, or company, has he same wegh n he ndex, and a porfolo ha racks he ndex wll nves an equal dollar amoun n each applcable nsrumen. 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. 2 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 Marke Value Index Level = (13) Dvsor Index Marke Value = 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 Marke Value = P Shares IWF FxRae AWF (15) where AWF (Addonal Wegh Facor) 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: AWF Z N * FloaAdjusedMarkeValue, = (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. 2 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. S&P Dow Jones Indces: Index Mahemacs Mehodology 9

13 So: (Dvsor) afer rebalancng = (Index Marke Value) afer rebalancng (Index Value) before rebalancng (16a) where: Index Marke Value = P Shares IWF FxRae AWF (16b) Modfed Equal Weghed Indces There are some equal weghed ndces ha place furher resrcons on socks ncluded n he ndex. An example resrcon mgh be a cap on he wegh allocaed o one secor or a cap on he wegh of a sngle counry or regon n he ndex. The rules could also spulae a maxmum wegh for a sock f he ndex apples addonal lqudy facors (e.g. baske lqudy) when deermnng he ndex weghs. In any of hese suaons, f a cap s appled o sasfy he resrcons, he excess wegh lefover by he cap would be dsrbued equally amongs he uncapped companes. Corporae Acons and Index Adjusmens The followng able shows he necessary adjusmens o he ndex and 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. 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 Index Acon Adjusmen Made o Index Dvsor Adjusmen? Spn-off The spn-off s added o he ndex on he ex-dae a a prce of zero. No Each ndvdual ndex mehodology wll specfy wheher he spn-off wll be elgble o reman n he ndex beyond he ex-dae. If he spn-off s dropped, he wegh of he dropped sock wll be renvesed n he paren sock. Rghs offerng The prce s adjused o he Prce of he Paren Company mnus No (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. Sock spl Shares are mulpled by and he prce s dvded by he spl facor. No Share/IWF None. The adjusmen facor s changed o keep he ndex wegh No changes he same. Specal dvdends The prce of he sock makng he specal dvdend paymen s reduced by he per share specal dvdend amoun afer he close of Yes Consuen change even number of adds and drops Consuen change deleon only radng on he day before he ex-dae. 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 specfed. 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. No, excep for he scenaro where a company s removed a a prce of zero and replacemen eners he ndex a he wegh of he deleed company pror o he zero prce. Yes S&P Dow Jones Indces: Index Mahemacs Mehodology 10

14 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. Beween ndex rebalancngs, corporae acons generally have no effec 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 Marke Value Index Level = (17) Dvsor Index Marke Value = 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 Marke Value = 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, FloaAdjusedMarkeValue, = * W (20), 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. 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. S&P Dow Jones Indces: Index Mahemacs Mehodology 11

15 So: (Dvsor) afer rebalancng = (Index Marke Value) afer rebalancng (Index Value) before rebalancng (20a) where: Index Marke Value = P Shares IWF FxRae AWF (20b) Corporae Acons and Index Adjusmens The followng able 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 Index Acon Spn-off Rghs offerng Sock spl Share/IWF changes Specal dvdends Merger or acquson Consuen change Delsng, acquson or any oher corporae acon resulng n a consuen deleon. Adjusmen Made o Index The spn-off s added o he ndex on he ex-dae a a prce of zero. Each ndvdual ndex mehodology wll specfy wheher he spn-off wll be elgble o reman n he ndex beyond he ex-dae. 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 adjusmen facor s changed o keep he ndex wegh he same. 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 exdae. 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. The company enerng he ndex goes n a he wegh of he company comng ou. The sock s dropped from he Index Dvsor Adjusmen? No No No No Yes. Yes, f here s a removal. No Yes S&P Dow Jones Indces: Index Mahemacs Mehodology 12

16 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 and he ndex mehodology for furher deals). 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 Marke Value Index Level = (17) Dvsor Index Marke Value = 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 Marke Value = 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 13

17 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: (Dvsor) afer rebalancng = (Index Marke Value) afer rebalancng (Index Value) before rebalancng (21a) where: Index Marke Value = P Shares IWF FxRae AWF (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 n he case of 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. A mes, companes may also be represened n an ndex by mulple share class lnes. In hese nsances, maxmum wegh cappng wll be based on company floa-adjused marke capalzaon, wh he wegh of mulple class companes allocaed proporonally o each share class lne based on s floa-adjused marke capalzaon as of he rebalancng reference dae. The sandard S&P Dow Jones Indces mehodologes for deermnng he weghs of capped ndces usng he mos popular cappng mehods are descrbed below. Sngle Company Cappng. In a sngle company cappng mehodology, no company 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 company 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 companes whn he ndex. 4. Afer hs redsrbuon, f he wegh of any oher company(s) hen breaches X%, he process s repeaed eravely unl no companes breach he X% wegh cap. Sngle Company and Concenraon Lm Cappng. In a sngle company and concenraon lm cappng mehodology, no company n an ndex s allowed o breach a ceran pre-deermned wegh and all companes 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 4.5%/22.5%/45% cappng (B/A/C n he followng S&P Dow Jones Indces: Index Mahemacs Mehodology 14

18 example). No sngle company s allowed o exceed 22.5% of he ndex and all companes wh a wegh greaer han 4.5% of he ndex canno exceed, as a group, 45% of he ndex. The procedure for assgnng capped weghs o each company 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 companes whn he ndex. 4. Afer hs redsrbuon, f he wegh of any oher company(s) hen breaches A%, he process s repeaed eravely unl no companes breach he A% wegh cap. 5. The sum of he companes wh wegh greaer han B% canno exceed C% of he oal wegh. 6. If he rule n sep 5 s breached, all he companes are ranked n descendng order of her weghs and he company wh he lowes wegh ha causes he C% lm o be breached s denfed. The wegh of hs company s, hen, reduced eher unl he rule n sep 5 s sasfed or reaches B%. 7. Ths excess wegh s proporonally redsrbued o all companes wh weghs below B%. Any sock ha receves wegh canno breach he B% cap. Ths process s repeaed eravely unl sep 5 s sasfed or unl all socks are greaer han or equal o B%. 8. If he rule n sep 5 s sll breached and all socks are greaer han or equal o B%, he company wh he lowes wegh ha causes he C% lm o be breached s denfed. The wegh of hs company s, hen, reduced eher unl he rule n sep 5 s sasfed or reaches B%. 9. Ths excess wegh s proporonally redsrbued o all companes wh weghs greaer han B%. Any sock ha receves wegh canno breach he A% sock cap. Ths process s repeaed eravely unl sep 5 s sasfed. For ndces ha use cappng rules across more han one arbue, S&P Dow Jones Indces wll ulze an opmzaon program o sasfy he cappng rules. The saed objecve for he opmzaon wll be o mnmze he dfference beween he pre-capped weghs of he socks n he ndex and he fnal capped weghs. S&P Dow Jones Indces: Index Mahemacs Mehodology 15

19 Pure Syle Indces For he S&P Pure Syle Indces, nroduced n 2005, a sock s wegh depends on s growh or value arbue measuremens, he same measures ha are used n he ndex sock selecon process. The dscusson here covers how hese ndces are calculaed; he selecon of socks s covered n he S&P U.S. Syle Indces mehodology. 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. 3 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. For more nformaon on he calculaon of S&P Pure Syle ndces, please refer o he S&P U.S. Syle Indces Mehodology locaed on our webse, 3 A sock canno be boh pure growh and pure value; can be neher pure growh nor pure value. S&P Dow Jones Indces: Index Mahemacs Mehodology 16

20 Weghed Reurn Indces S&P Dow Jones Indces Weghed Reurn Indces combne he reurns of wo or more underlyng ndces usng a specfed se of weghng rules o creae a new unque ndex reurn seres. An ndex ha uses he Weghed Reurn mehodology mgh also be referred o as an Index of Indces. Weghed Reurn ndces may nclude a cash componen whch for he purposes of hese ndces s reaed as an underlyng ndex. On any radng dae,, he ndex s calculaed as follows usng he componen ndces as dealed on he pror pages: Index = Index PB * (1 + IndexReurn ) (22) IndexReurn = n w R =1 (23) where: Index PB = Index value on he prevous rebalancng dae w R = Wegh of ndex = Cumulave reurn of he represenave ndex a from he prevous rebalancng dae S&P Dow Jones Indces: Index Mahemacs Mehodology 17

21 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 Re 1 Borrowng Rae Leveraged Index urn = K ( K 1) D, 1 (24) Underlyng Index In equaon (24) 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 Reurn) (25) Subsung (24) no (25) and expandng he rgh hand sde of (25) yelds: Leveraged IndexValue Leveraged IndexValue where: K (K 1) = Leverage Rao = K = 1, no leverage K = 2, Exposure = 200% K = 3, Exposure = 300% UnderlyngIndex 1+ K UnderlyngIndex Borrowng Rae 1 ( K 1) D 360 1, 1 1 Borrowng Rae = Overngh LIBOR n he U.S. or EONIA n Europe are wo common examples D, -1 = he number of calendar days beween dae and -1 In he absence of leverage (K=1), = 1 * Underlyng Index Leveraged Index Value Leveraged Index Value (27) Underlyng Index 1 The leverage poson s rebalanced daly. Ths s conssen wh he payoff from fuures based replcaon. (26) S&P Dow Jones Indces: Index Mahemacs Mehodology 18

22 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 (24) and (26) 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. 4 The general formula for he reurn o he nverse ndex s Underlyng Index Inverse Index Reurn = K 1 Underlyng Index 1 Lendng Rae + ( K + 1) D, (28) 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: InverseIndexValue Inverse IndexValue = UnderlyngIndex 1 K UnderlyngIndex LendngRae 1 ( K + 1) D 360 1, 1 1 where: K (K 1) = Leverage Rao K = 1, Exposure = -100% K = 2, Exposure = -200% K = 3, Exposure = -300% Lendng Rae = Overngh LIBOR n he U.S. or EONIA n Europe are wo common examples D, -1 = he number of calendar days beween dae and -1 (29) In he absence of leverage (K =1), InverseIndexValue Inverse IndexValue = UnderlyngIndex 1 UnderlyngIndex LendngRae 1 (2) D 360 1, 1 1 (30) The nverse poson s rebalanced daly. Ths s conssen wh he payoff from fuures based replcaon. 4 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. S&P Dow Jones Indces: Index Mahemacs Mehodology 19

23 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 (28) and (29) 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 = 1+ IndexER 1 K 1 (31) 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 ndces s calculaed, whch ncludes neres accrual on he noonal value of he ndex based on a specfed neres rae (e.g. 91-day U.S. Treasury rae), as follows: where: IndexTR IndexER = IndexTR + 1 TBR (32) IndexER 1 IndexTR -1 = The Index Toal Reurn on he precedng busness day TBR = Treasury Bll Reurn, as deermned by he followng formula: Dela Dela 91 1 TBR = 1 (33) 91 1 * TBAR = The number of calendar days beween he curren and prevous busness days S&P Dow Jones Indces: Index Mahemacs Mehodology 20

24 TBAR -1 = The mos recen weekly hgh dscoun rae for 91-day U.S. Treasury blls effecve on he precedng busness day 5 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 = 1+ IndexER _ LR K 1 (34) UnderlyngIndexER _ LR IndexER _LR = The Index Excess Reurn on he las rebalancng busness day, _LR UnderlyngIndexER _LR = The Underlyng Index Excess Reurn value on he las rebalancng busness day, _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% = The las rebalancng busness day A oal reurn verson of each of he ndces s calculaed, whch ncludes neres accrual on he noonal value of he ndex based on he 91-day U.S. Treasury rae. The formulae are he same as (32) and (33) above. 5 Generally he raes are announced by he U.S. Treasury on each Monday. On Mondays ha are bank holdays, Frday s raes apply. S&P Dow Jones Indces: Index Mahemacs Mehodology 21

25 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 (35) 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. Socks may also ssue dvdends on a monhly, sem-annual or annual bass. 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 = (36) Dvsor The nex sep s o apply he usual defnon of a oal reurn from a fnancal nsrumen o he prce ndex. Equaon (35) gves he defnon, and equaon (36) apples o he ndex: and P + D ToalReur n = ( ) 1 (37) P 1 IndexLevel + IndexDvdend DTR = ( 1) (38) IndexLevel 1 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) (1+ DTR ) (39) S&P Dow Jones Indces: Index Mahemacs Mehodology 22

26 Ne Toal Reurn Calculaons To accoun for ax whheld from dvdends, a ne oal reurn calculaon s used. The calculaon s dencal o he calculaons dealed n he prevous Toal Reurn secon, excep each dvdend s adjused o accoun for he ax aken ou of he paymen. Inserng he whholdng rae no he calculaon a he frs sep s all ha needs o be done he calculaon can follow dencally from ha pon forward: ToalDalyDvdend = Dvdend Shares ( 1 WhholdngRae ) (40) The ax raes used for S&P Dow Jones Indces global ndces are from he perspecve of a Luxembourg nvesor. However, n domesc ndex famles, ax raes from he perspecve of a domesc nvesor wll be appled. S&P Dow Jones Indces: Index Mahemacs Mehodology 23

27 Frankng Cred Adjused Toal Reurn Indces Addonal oal reurn ndces are avalable for a number of S&P/ASX Indces ha adjus for he ax effec of frankng creds aached o cash dvdends. The ndces ulze ax raes relevan o wo segmens of nvesors: one verson ncorporaes a 0% ax rae relevan for ax-exemp nvesors and a second verson uses a 15% ax rae relevan for superannuaon funds. The frankng creds aached o boh regular and specal cash dvdends are ncluded n he respecve calculaons. To calculae he gross dvdend pons renvesed n he Frankng Cred Adjused Toal Reurn Indces: Grossed-up Dvdend = [As Repored Dvdend * (1 % Franked) + (As Repored Dvdend * % Franked / (1 Company Tax Rae))] (41) The Ne Tax Effec of he frankng cred s hen calculaed based on he nvesor ax rae (.e. 0% for axexemp nvesors and 15% for superannuaon funds). Ne Tax Effec = [Grossed-up Dvdend * (1 Invesor Tax Rae)] As Repored Dvdend (42) The Ne Tax Effec of each dvdend s hen mulpled by he ndex shares of ha company o calculae he gross dvdend marke capalzaon. Gross Dvdend Marke Cap = Ne Tax Effec * Index Shares (43) These are hen summed for all dvdends gong ex on ha dae and convered o dvdend pons by dvdng by he ndex dvsor Gross Dvdend Pons = Sum of Gross Dvdend Marke Caps / Index Dvsor (44) Frankng Cred Adjused Annual Toal Reurn Indces. Ths ndex seres accrues a pool of gross dvdend pons on a daly bass and renvess hem across he ndex annually afer he end of he fnancal year. Renvesmen occurs a marke close on he frs radng day afer June 30 h. The gross dvdend pons are derved by akng he value of he gross dvdend marke capalzaon (less he as repored dvdend marke capalzaon) and dvdng by he ndex dvsor effecve on he ex-dae of he respecve dvdend. Frankng Cred Adjused Daly Toal Reurn Indces. Raher han allowng a separae accrual of gross dvdend pons, hs ndex seres renvess he gross dvdend amoun across he ndex a he close of he ex-dae on a daly bass. S&P Dow Jones Indces: Index Mahemacs Mehodology 24

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: Index EPS eps shares = (45) Dvsor 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 25

29 Dvdend Indces S&P Dow Jones Indces Dvdend Indces are desgned o rack he oal dvdend paymens from he consuens of an underlyng ndex. The level of he ndex s based on a runnng oal of dvdends of he consuens of he underlyng ndex. Some ndces rese o zero on a perodc bass, generally quarerly or annually. Thus, he ndex measures he oal dvdends pad n he underlyng ndex snce he prevous rebalancng dae, or he base dae for ndces ha do no rese on a perodc bass. For quarerly ndces, he ndex reses o zero afer he close on he hrd Frday of he las monh of he quarer, o concde wh fuures and opons expraon. For annual ndces, he ndex reses o zero afer he close on he hrd Frday of December, o concde wh fuures and opons expraon. The formula for calculang he dvdend ndex on any dae,, for a gven underlyng ndex, x, s: where: DvdendIndex = ID, x, x (47) = r + 1 ID,x = The ndex dvdend of he underlyng ndex x on day. = The curren dae. r+1 = The radng dae mmedaely followng he rese dae of he ndex (or base dae f he ndex does no rese perodcally). The ndex dvdend (ID) of he underlyng ndex s calculaed on any gven day as he oal dvdend value for all consuens of he ndex dvded by he ndex dvsor. The oal dvdend value s calculaed as he sum of dvdends per share mulpled by ndex shares ousandng for all consuens of he ndex whch have a dvdend gong ex on he dae n queson. For more deal concernng he calculaon of ndex dvdends please refer o he Toal Reurns Calculaon secon above. S&P Dow Jones Indces: Index Mahemacs Mehodology 26

30 Rsk Conrol Indces S&P Dow Jones Indces Rsk Conrol Indces are desgned o rack he reurn of a sraegy ha apples dynamc exposure o an underlyng ndex n an aemp o conrol he level of volaly. The ndex ncludes a leverage facor ha changes based on realzed hsorcal volaly. If realzed volaly exceeds he arge level of volaly, he leverage facor wll be less han one; f realzed volaly s lower han he arge level, he leverage facor may be greaer han one, assumng he ndex allows for a leverage facor of greaer han one. A gven Rsk Conrol Index may have a maxmum leverage facor ha canno be exceeded. There are no guaranees ha he ndex shall acheve s saed arges. The reurn of he ndex consss of wo componens: (1) he reurn on he poson n he underlyng ndex and (2) he neres cos or gan, dependng upon wheher he poson s leveraged or deleveraged. A leverage facor greaer han one represens a leveraged poson, a leverage facor equal o one represens an unleveraged poson, and a leverage facor less han one represens a deleveraged poson. The leverage facor may change perodcally, on a se schedule, or may change when volaly exceeds or falls below predeermned volaly hresholds. For equy ndces, he leverage facor wll no change a he close of any ndex calculaon day n whch socks represenng 15% or more of he oal wegh of he underlyng ndex are no radng due o an exchange holday. A each underlyng ndex s rebalancng, and usng each sock s wegh a ha me, a forward lookng calendar of such daes s deermned and posed on S&P Dow Jones Indces Web se a The formula for calculang he Rsk Conrol Index s as follows: Rsk Conrol Index Reurn = Underlyng Index K rb * 1 + (1 Krb ) * (1+ IneresRae 1 * D 1, / 360) 1 Underlyng Indexrb = rb+ 1 (48) The Rsk Conrol Index Value a me can, hen, be calculaed as: RskConrolIndexValue RskConrolIndex Value = ) * (1 + RskConrolIndex Reurn ) ( rb (49) Subsung equaon (48) no (49) and expandng yelds: Rsk Conrol Index Value = Rsk Conrol Index Valuerb * Underlyng Index 1+ K + + rb * 1 (1 Krb ) * (1 IneresRae D 1 * 1, / 360) 1 Underlyng Indexrb = rb+ 1 (50) S&P Dow Jones Indces: Index Mahemacs Mehodology 27