Inflation Derivatives Explained

Fxed Icome Quattatve Reseach July 2005 Iflato Devatves Explaed Makets, Poducts, ad Pcg Jeoe Kekhof he flato devatves maket has acheved ctcal mass, wth a outstadg otoal volume of ove $00b Iflato devatves make t possble to solate flato sk fom teest ate sk Zeo-coupo flato swaps domate the maket ad fom the buldg blocks fo othe flato devatves Real etus, beakeve flato ad seasoalty ae explaed he mechacs, sks ad uses of flato devatves ae dscussed ISDA 2005 flato deftos ae outled

COES. Itoducto 3 2. he Makets 5 2.. Gowth ad sze... 5 2.2. Maket beadth...6 2.3. Maket patcpats... 6 2.4. Poducts ovevew... 8 3. Iflato Idces 9 3.. Euo aea... 9 3.2. Face... 0 3.3. Uted Kgdom... 3.4. Uted States... 3.5. Othe dces...2 3.6. Seasoalty flato dces... 2 4. Iflato Bascs ad Cocepts 6 4.. Iflato, omal value ad eal value... 6 4.2. Iflato-lked cashflows ad eal bods... 6 4.3. Idces, dexato lags ad aoucemets... 9 4.4. Beakeve flato... 2 4.5. Compoets of beakeve flato ate... 23 5. Zeo-coupo Iflato Swaps 25 6. Iflato Aalyss Famewok 29 6.. Costuctg a flato cuve... 29 6.2. Iflato ad teest ate sk... 34 6.3. Coutepaty sk... 36 6.4. Roudg sk... 38 6.5. Seasoalty sk... 38 6.6. Isttutoal sk...40 7. Iflato Swaps ad Futues 4 7.. Reveue flato swap... 42 7.2. OC flato bod... 43 7.3. Peod-o-peod flato swaps... 43 7.4. Iflato asset swaps... 45 7.5. Iflato futues... 49 July 2005 Please see mpotat aalyst cetfcato(s o the back page of ths epot

8. Iflato Volatlty Poducts 52 8.. Iflato caps ad floos... 52 8.2. Iflato swaptos... 53 8.3. LPI swaps...56 8.4. Iflato spead optos... 57 9. Stuctued Iflato Poducts 59 9.. otal etu swaps... 59 9.2. Iflato-lked equty... 6 9.3. Iflato-lked equty optos... 6 9.4. Iflato-lked cedt default swaps... 62 9.5. Iflato-lked CDO... 64 0. Legal, Regulatoy, ad Accoutg Issues 66 0.. ISDA flato devatves documetato... 66 0.2. Regulato fo peso fuds... 67 0.3. Accoutg stadads... 67 Appedx 69 A.. Calculatg flato asset swap speads... 69 A.2. Estmato of seasoal pattes... 70 A.3. Duato ad covexty aalyss... 7 A.4. Valuato of fowad statg zeo-coupo swaps... 72 A.5. Valuato of flato caps ad floos... 73 A.6. Valuato of spead optos ad flato-lked equty optos... 73 A.7. Valuato of flato swaptos... 74 Refeeces 75 Glossay 76 Summay of otato ad Deftos 79 July 2005 2

. IRODUCIO Jeoe Kekhof +44 (020 702 398 jkekhof@lehma.com he pupose of flato devatves s the tasfe of flato sk Iflato devatves allow fo talo-made solutos Ou focus s o explag the mechacs, sk ad valuato of flato devatves I ecet yeas the maket fo flato-lked devatve secutes has expeeced cosdeable gowth. Fom almost o-exstet ealy 200, t has gow to about 50b otoal taded though the boke maket 2004, double the otoal taded though the boke maket 2003. Rapd gowth s expected to cotue fo the comg yeas. So fa the gowth has maly bee dve by the Euopea maket, but ecetly teest the US maket has pcked up as well. he pmay pupose of flato devatves s the tasfe of flato sk. Fo example, eal estate compaes may wat to shed some of the atual exposue to flato sk, whle peso fuds may wat to cove the atual labltes to ths sk. I the smplest fom, flato devatves povde a effcet way to tasfe flato sk. But the flexblty also allows them to eplcate devatve fom the flato sks embedded othe stumets such as stadad cash stumets (that s, flatolked bods. Fo example, as we wll see late, a flato swap ca be theoetcally eplcated usg a potfolo of a zeo-coupo flato-lked bod ad a zeo-coupo omal bod. As s the case fo the omal teest ate maket, the advatage of flato devatve cotacts ove flato bods s that devatves ca be taloed to ft patcula clet demad moe pecsely tha bods. Wth the toducto of ufuded flato-lked poducts, flato devatves have fo the fst tme sepaated the ssue of fudg fom flato sk. hs has made flato makets moe accessble to pates wth hgh fudg costs ad made t cheape to leveage flato sk. Fo stace, hedge fuds ae ceasgly volved flato makets. Befoe ay patcpat etes the flato devatves maket, a sold udestadg of the mechacs, sks ad valuato of flato devatves s essetal. he am of ths epot s to povde ths udestadg. We hope that eades wll ga the ecessay comfot ad udestadg to take advatage of the ew oppotutes that the flato devatves maket pesets. hs epot s stuctued as follows. Chapte 2 toduces the eade to the flato maket by pesetg a ovevew of the maket s gowth, poducts ad patcpats. It explas how the maket developed ad helps eades udestad the lkely futue evoluto of the flato-lked bod ad flato devatves makets. Chapte 3 dscusses the ma dces used both the flato cash ad devatves maket. Futhemoe, t dscusses seasoal pattes flato tme sees. Chapte 4 dscusses the bascs of flato makets. It explas key cocepts such as eal etus, eal bods, beakeve flato ad dexato lags. Chapte 5 toduces the ma flato devatve secuty, the zeo-coupo flato swap. he zeo-coupo flato swap ca be used as a buldg block fo almost all othe flato devatves. Chapte 6 dscusses the flato sk famewok. he chapte shows how to costuct a flato cuve fom zeo-coupo flato swaps copoatg seasoalty effects. It also looks at flato, teest ates, coutepaty, oudg, seasoalty ad sttutoal sk. Chapte 7 dscusses the pla valla flato devatves poducts. he most commo ae flato swaps, the vaous foms, ad asset swaps o flato-lked bods a key lk betwee cash ad devatve secutes. Ivestos who patcpate the flato he autho thaks Albet Bodolo, Robet Campbell, Gada Ga, Bout Mklavcc, Alvao Mucda, Domc O Kae, Ya Phoa, Ram Shakach, ad Fe Zhou fo valuable commets ad dscussos. July 2005 3

cash maket stad to lose a valuable souce of fomato f they do ot udestad the devatves maket, ad vce vesa, as the two makets ae tmately elated. We also dscuss the ecet flato futues maket. Chapte 8 cosdes flato poducts wth optoalty, such as caps, floos ad swaptos. We also dscuss LPI swaps whch have a opto-elated payoff. Chapte 9 dscusses moe complex devatve stuctues, cludg some hybd stuctues. hs s elatvely uexploed tetoy the flato wold, ad s stll dve exclusvely by clet specfc eeds. Howeve, teest these moe advaced devatve secutes s gowg ad they ae lkely to be the ext aea of gowth of the flato maket. Fally, Chapte 0 dscusses the stadadsato ad deftos of the poducts by ISDA. Futhemoe, t cosdes the fluece of egulato ad accoutg stadads o the flato maket. July 2005 4

2. HE MARKES 2.. Gowth ad sze Iflato swap volumes the boke maket have doubled 2004 he flato devatves maket has gow fom a almost o-exstet ad faly exotc bach of the teest ate maket to a szeable maket wth substatal gowth potetal. he pace of ths gowth has bee swft. hs s evdeced by te-deale data o otoal volumes taded (Fgue 2.. Volume doubled 2004 to about 50b fom about 25b 2003. ote that these umbes exclude swaps betwee baks ad the clets, fo whch the volume s dffcult to estmate. he apd pace of ths gowth the past few yeas s patly attbutable to the favouable macoecoomc evomet. Hstocally, low etus o tadtoal fxed come poducts ad a eluctace o the pat of vestos to take o sk the fom of othe assets led to a apd gowth demad fo stuctued poducts 2003, as ca be see Fgue 2.2. Fgue 2.. Mothly volumes of flato swaps taded the boke maket b 7.0 6.0 5.0 4.0 3.0 2.0.0 0.0 May-0 ov-0 May-02 ov-02 May-03 ov-03 May-04 ov-04 Souce: ICAP; Lehma Bothes. he gaph also cludes o-euo swaps. Fgue 2.2. Goss ssuace of stuctued flato-lked otes (excludg soveegs b 3.0 2.5 2.0.5.0 0.5 0.0 Ma-99 Oct-99 May-00 Dec-00 Jul-0 Feb-02 Sep-02 Ap-03 ov-03 Ju-04 Ja-05 Souce: Dealogc Bodwae; Lehma Bothes. otoals the flato maket ae cuetly at about 2% of otoals the teest ates maket Howeve, eve though the maket has bee gowg apdly, t stll epesets oly a small poto of the total ates maket. Cuet volumes of flato swaps amout to about -2% of omal teest ate swap volumes. Fo the flato devatves maket to gow futhe, t wll eed to fd a balace betwee demad ad supply of flato. July 2005 5

hs s clea by lookg at the dffeet makets. Fve-yea flato swaps cuetly tade wth a 2-3bp bd-offe spead the Euopea maket as demad to eceve flato s stog ad flato ca be souced fom the Euopea flato-lked bods ssued by Face (OA ad Italy (BP. Wth these soveegs ceasg the pecetage of the debt potfolo lked to flato, t s lkely that the flato swap maket wll develop futhe. A lage bod maket does ot ecessaly mply a lage flato swap maket. Fo example, the US flato-lked bod maket (IPS s cuetly the lagest flato-lked bod maket, almost twce the sze of the Euopea flato-lked bod maket, yet 5-yea US flato swaps cuetly tade wth bd-offe speads moe tha double those the Euopea maket. Gve the lage sze of the US flato-lked bod maket ad the ceased actvty ad lqudty fo US flato devatves 2004, we expect the US flato devatves maket to gow faste tha othe makets. Wth the developmet of the boade flato maket, spued by ceased soveeg ssuace (by the UK, euo-aea coutes, the US, Japa, vestos ad boowes alke ae becomg moe awae of the beefts of flato-lked secutes. Cosdeg the beefts of flato-lked debt fo ssues, we thk t lkely that ssues wll cotue to cease the flato-lked debt potfolo elatve to the omal oe. Fo example, the Itala easuy has stated that a steady state t s seekg to ssue oughly 5% of ts goss debt the fom of flato-lked bods. If the ato of flato-lked debt to omal debt of the majo soveegs gows to about 0 to 5%, we would expect the flato devatves maket to gow a smla fasho. hs could amout to a fve- to te-fold cease flato swap otoals fom cuet volumes. he flato devatves makets maly focus o the same dces as the flato-lked bod maket Soveegs ema the ma supples of flato 2.2. Maket beadth I tems of actvely taded flato dces, the flato devatves maket focuses lagely o the same dces as the flato-lked govemet maket. he ma makets ae the Euopea maket usg the HICPx dex fom Euostat, the Fech maket usg the FRCPI dex fom ISEE, the UK maket usg the RPI dex fom atoal Statstcs, ad the US maket usg the US-CPI dex fom BLS (see Chapte 3 fo a detaled dscusso. Cuetly, the Euopea devatves maket s by fa the lagest, but the othe makets have eached a easoable lqudty as well. Clealy, the flato devatves maket s ot estcted to the dces fo whch (soveeg ssues ssue flato-lked bods. Ideed, the ma stegth of the flato devatves maket s that t ca povde aythg the cash maket ca ad moe. Fo example, t s possble to stuctue flato swaps lked to dces fo whch o flato-lked debt s outstadg. Howeve, bd-offe speads wll be qute szable o these tades uless deales ca fd a two-way maket. 2.3. Maket patcpats Iflato poducts attact a dvese goup of vestos such as baks, peso fuds, mutual fuds, suace compaes ad hedge fuds. Baks wat to eceve flato o swaps to hedge flato-lked etal poducts. Isuace compaes ad peso fuds wat to eceve flato to match the log-tem flato-lked labltes. Fgue 2.3 pesets a ovevew of the maket patcpats. We ow sepaately dscuss the atual payes ad eceves ad the cetves. July 2005 6

Fgue 2.3. Ovevew of patcpats the global flato maket Iflato payes Soveegs Utltes Ageces Poject Face (PFI Real estate Retales Asset swap fuds Othes Global flato Maket Payes / eceves Ivestmet baks Popetay desks Hedge fuds Relatve value fuds Othes Iflato eceves Peso fuds Isuace compaes Iflato mutual fuds Geeal mutual fuds Retal baks Copoates (ALM Othes Souce: Lehma Bothes. 2.3.. Iflato payes Iflato-lked secutes ca mpove the asset-lablty mx By ssug flato-lked debt, oe ca save the flato sk pemum Dvesfcato beefts fo vestos Payes of flato ae ettes that eceve flato cashflows the atual le of busess. ypcal examples ae soveegs ad utlty compaes. As the come s lked (ethe explctly o mplctly to flato, they ae deally suted to pay flato the flato maket. Below, we lst seveal easos fo ssug flato-lked debt.. Fo maket patcpats wth a come steam explctly o mplctly lked to flato, flato-lked secutes ae atual hedge stumets agast vaatos flato, makg them attactve fo asset-lablty maagemet puposes. 2. Ivestos ae teested eal etus athe tha omal etus. Iflato-lked bods guaatee a eal etu, wheeas omal bods guaatee a ceta omal, but uceta eal etu. I ode to compesate vestos fo takg the flato sk accompaed wth omal bods, omal yelds should be suffcetly hgh that the expected eal etu o omal bods s geate tha the guaateed eal etu o flato-lked bods. hs addtoal yeld o the omal bods s called the flato sk pemum. By ssug flato-lked debt the ssue ca thus save the flato sk pemum (o pat of t. 3. I peods of shotage, flato payes may pefe to pay low tal cashflows makg up fo ths wth hghe paymets late o. As flato s typcally postve, flato-lked bods ft such a cashflow stuctue. 4. Aothe easo fo ssug flato-lked debt s to attact a vesto base that s teested the potetal dvesfcato beefts offeed by these secutes. 2.3.2. Iflato eceves Iflato eceves ae typcally ettes that eed to pay flato-lked cashflows the atual le of busess. Examples clude peso fuds ad suace compaes (e.g. va sellg addtoal peso coveage polces. As the labltes ae lked (ethe explctly o mplctly to flato, they ae deally suted to eceve flato the flato maket. Fo stace, as peso fuds eed to mmse the shotfall sk 2 wth as low as possble peso pemums, they should have a atual teest 2 he sk that the assets dop below the labltes. July 2005 7

flato-lked secutes to match the flato-lked labltes. Havg flatopayg secutes o the asset sde ca substatally educe the shotfall sk as the value of the flato-lked assets ceases ad deceases wth the labltes. he same holds fo suace compaes sellg flato-lked addtoal peso coveage polces. heefoe, typcally these sttutos eceve flato the flato maket. Aothe mpotat goup of flato eceves s the etal vestmet maket. Although most people vest flato-lked cashflows va the peso schemes, may vestos pefe to addtoally vest dectly flato-lked secutes. 2.3.3. Othe playes Dvesfcato ad elatve value oppotutes make the flato makets attactve fo evey vesto Zeo-coupo flato swaps domate the flato devatves maket Zeo-coupo flato swaps ae the buldg block of the flato maket Reveue flato swaps ca be used to hedge PFI Demad fo flato optos s gowg Iflato hybds ope a ew age of possbltes Besdes vestos who ae daw to flato makets the atual le of busess, flato makets ca offe attactve featues fo flato-eutal vestos. As flato sk s ot sepaately taded o s taded as a combato of othe sks, t should povde dvesfcato beefts fo ay vesto. Futhemoe, attactve elatve value oppotutes ca ase fo flato-eutal vestos. 2.4. Poducts ovevew hee ae a umbe of stumets that ca be classfed as flato devatves, agg fom the stadad zeo-coupo flato swap to stuctued flato poducts. Cuetly the maket s domated by zeo-coupo flato swaps, but demad fo moe advaced poducts s gowg as vestos ga expeece the flato maket ad become moe awae of the beefts of these poducts. Zeo-coupo flato swaps ae the basc flato poducts. hey ae detaled Chapte 5. I Secto 6., we wll see that usg zeo-coupo flato swaps, we ca costuct a flato cuve a elatvely staghtfowad mae. Besdes zeo-coupo swaps, yea-o-yea flato swaps ae othe popula poducts. Yea-o-yea flato swaps pay flato ove oe yea, fo a peod of seveal yeas (e.g. fom Mach to Mach fo a 5-yea peod. Peod-o-peod flato swaps ae dscussed detal Secto 7.. Reveue flato swaps pay the gowth the flato dex, that s, they pay the same eal amout at all paymet dates. I the UK, eveue flato swaps ae ofte used ode to hedge pvate face tatves (PFI. Istead of accessg captal makets dectly (fo stace, by ssug a flato-lked bod, PFI poject maages boow fom baks o a floatg ate bass. As the eveues ae ofte flato-lked, they eed to hedge the flato sk whch ca be doe usg a eveue flato swap. he stuctues ae ofte moe volved ad moe specfc swaps eed to be costucted, but they ca typcally be hadled wth a appopately costucted potfolo of flato swaps. Reveue flato swaps ae teated detal Secto 7.. Demad fo opto stuctues s ceasg as well. Caps ad floos ae useful stumets to geeate patal dexato schemes. Fo example, the 995 UK Pesos Act eques schemes to dex agast the RPI, but wth a aual gowth cap of 5% ad aual gowth floo of 0%. LPI swaps ae stumets dectly lked to such a lablty steam. Othe popula poducts ae flato swaptos. Iflato swaptos gve compaes wth kow futue flato-lked cashflows the oppotuty to ete to a flato swap whe they stat havg flato-lked exposues. Futhemoe, they ae used to ceate callable ad cacellable flato swaps. Iflato optos ae dscussed Chapte 8. Aothe teestg ted s to clude flato-lked cashflows payoff schemes of othe asset classes, such as equty ad cedt. hese so-called flato hybd poducts have bee popula wth etal customes. Fo stace, aoud 200 most of the etal poducts sold Italy wee flato ad captal-potected equty-lked otes. May vaetes of flato-lked hybd stuctues ae possble. We teat some of these Chapte 9. July 2005 8

3. IFLAIO IDICES Ay flato-lked poduct eeds a efeece measue of flato. hese ae kow as flato dces. I ths secto we descbe the flato dces used the most mpotat flato makets. HICP excludg tobacco s the ma flato dex Uevsed dces ae used fo cashflow computatos 3.. Euo aea he euo aea flato swap maket s cuetly by fa the most lqud, actve ad taspaet flato swap maket. he bechmak dex fo the euo aea s the oseasoally adjusted Euo HICPx (Hamosed Idex of Cosume Pces excludg obacco publshed by Euostat. 3 Cosumpto levels of the 2 dffeet coutes ae used to weght the dex. As coutes accede to the moetay uo, they wll be cluded the dex. It s typcally publshed two weeks afte the ed of the moth. Fo stace, the Euo HICPx dex value fo Mach s aouced o about 5 Apl. he dex aouced s called the uevsed dex. Euostat mght evse the dex f afte gatheg moe data they feel the tal aoucemet was accuate. Although the value of the dex ca be evsed, the uevsed veso s used both the cash ad the devatves maket. Euostat publshes the Euo HICPx dex to the OAEI0 page o Reutes ad the CPFEMU<Idex> o Bloombeg. he majo costtuets of the dex ae gve Fgue 3.. he base yea fo HICPx s 996, meag that the aveage dex value of HICPx equaled a 00 dug 996. Fgue 3.. Costtuets of HICP excludg tobacco Restauats ad hotels Othe goods ad 9.7% sevces 8.4% Receato ad cultue 9.7% Food ad beveages 7.5% Clothg ad footw ea 7.6% Educato ad commucato 3.9% aspot 5.7% Souce: Euostat; Lehma Bothes. Health 4.2% Housg 23.2% Couty weghts ae based o GDP he Euo HICP excludg tobacco s a weghted sum of the euo-aea coutes HICP dces. he couty weghts fo the dex ae gve Fgue 3.2 fo 2005. he couty weghts ae adapted o a aual bass based o GDP. 3 See www.euopa.eu.t/comm/euostat/ fo moe fomato. July 2005 9

Fgue 3.2. Couty weghts of HICP excludg tobacco fo 2005 Potugal 2.% ethelads 5.2% Luxemboug 0.3% Spa.4% Austa 3.% Belgum 3.3% Flad.6% Face 20.7% Italy 9.2% Ielad.3% Souce: Euostat; Lehma Bothes. Geece 2.7% Gemay 29.0% he tem weghts of the Euo dex wll also vay as a cosequece of vayg couty weghts due to the fact that the dvdual HICP dces have vayg tem weghts. Fgue 3.3 shows the weghts fo the dffeet categoes fo dvdual euo-aea coutes. Fgue 3.3. Costtuets of HICP ex-tobacco dces fo dvdual coutes 00% 90% 80% 70% 60% 50% 40% 30% 20% 0% 0% Euozoe Belgum Gemay Geece Spa Face Ielad Italy Luxemboug ethelads Austa Potugal Flad Food ad beveages Clothg ad footw ea Housg Health aspot Educato ad commucato Receato ad cultue Othe goods ad sevces Restauats ad hotels Souce: Euostat; Lehma Bothes. he costtuets of the flato dex vay heavly betwee coutes Fech CPI excludg tobacco s the ma dex he weghts ca vay substatally. Fo stace, housg accouts fo about 30% of the Gema HICPx whle t accouts fo oly about 6.5% of the Spash HICPx. Ad estauats ad hotels make up about 7% of the Spash HICPx, but oly about 5.7% of the Gema HICPx. 3.2. Face Whe Face ogally decded to ssue flato-lked debt, thee was cosdeable debate about whch dex the ssue should be lked to. A atoal dex was lkely to be a bette match to the govemet s labltes, whle the Euo HICPx dex would appeal to teatoal vestos. Gve the fact that the Euo HICPx dex was elatvely ew at the tme of fst ssuace, the o-seasoally adjusted Fech CPI (Cosume Pce Idex was chose. 4 he dex fo each moth s publshed by ISEE o about the 22 d of the subsequet moth. Aga the uevsed dex s used. he 4 See www.see.f/e/dcateu/dc_cos/dc_cos.asp fo moe fomato. July 2005 0

Fech CPI s publshed by ISEE o the OAIFLAIO0 page o Reutes ad FRCPXOB<Idex> o Bloombeg. he base yea s 998. I Fgue 3.4 we peset the costtuets of the Fech CPI. Fgue 3.4. Item weghts of Fech CPI Othe sevces 24.8% aspot ad commucato 4.9% Souce: Isee; Lehma Bothes. Health sevces 5.3% ets, w ate, ad gabage collecto Eegy 7.4% 8.0% Food ad beveages 7.6% Clothg ad footw ea 5.3% Medce 5.0% Othe maufactued goods 2.9% RPI s the ma dex fo flato-lked secutes 3.3. Uted Kgdom I the UK maket, flato-lked secutes ae lked to the RPI (Retal Pce Idex. 5 hs dffes fom the RPIX whch excludes motgage teest paymets ad utl ecetly was the Moetay Polcy Commttee s (MPC flato taget. 6 he uevsed veso s used fo flato swaps. atoal statstcs publshes the RPI dex value fo each moth o about the 5 th of the subsequet moth. he Bloombeg tcke fo RPI s UKRPI<Idex>. he base efeece equals Jauay 987. he majo costtuets of the RPI ae gve Fgue 3.5. Fgue 3.5. Item weghts of UK RPI Faes ad othe tavel costs Pesoal goods 2.% ad sevces 4.2% Lesue.6% Categ 4.9% Food, beveages ad tobacco 20.8% Clothg ad footw ea 5.% Eegy 7.4% Household goods ad sevces 3.0% Souce: atoal Statstc; Lehma Bothes. Housg 20.9% he US CPI s the ma flato dex 3.4. Uted States Although the US flato maket has the lagest backg of flato-lked bod ssues, lqudty emas substatally lowe tha the euo-aea, Fech ad UK makets. It uses the same dex as the IPS maket, the o-seasoally adjusted US Cty Aveage 5 6 See www.statstcs.gov.uk/cci/ugget.asp?id=22&pos=2&colrak=&rak=60 fo moe fomato. Sce 2003 the MPC flato taget has bee 2.0% o the UK CPI dex. July 2005

All Items Cosume Pce Idex fo all Uba Cosumes (CPI-U publshed by the Bueau of Labo Statstcs (BLS. 7 he dex ca be foud o Bloombeg, CPURSA<Idex>. he base s gve by the aveage dex of 982-984. 8 he costtuets of the US CPI ae peseted Fgue 3.6. Fgue 3.6. Costtuets of the US CPI Othe Goods ad Sevces Receato 4% Educato ad 6% Commucato 6% aspot 7% Food ad Beveages 5% Clothg ad footw ea 4% Medcal cae 6% Housg 42% Souce: Bueau of Labo Statstcs; Lehma Bothes. Housg epesets a bg compoet of the US CPI Italy s the most actve devatves maket wthout atoal bod ssuace I Japa a total etu swap maket has developed Iflato sees cota stog seasoal effects Compaed wth the Euopea dces, the US CPI has a vey lage housg compoet, ad the value of the dex s theefoe lagely dve by house pces ad ets. 3.5. Othe dces By fa the most actve maket wthout a udelyg flato-lked govemet bod maket s Italy. he ma dex used s the Famgle d Opea e Impegat (FOI excludg tobacco dex. It s publshed by ISA ad ca be foud o Bloombeg ICPI <Idex>. A small Spash CPI maket has developed wthout bod ssuace suppot. he dex used s the Spash CPI, whch s publshed by IE ad avalable o Bloombeg SPCPI <Idex>. Belgum could develop to a elatvely sgfcat maket due to the fact that almost all flato labltes (e.g. ets ae lked to the same dex, the health dex. he dex s publshed by the atoal Isttute of Statstcs ad avalable o Bloombeg BECPHLH <Idex>. A small maket has developed fo Dash CPI due to a lage Dash CPI ssue to fud the taspot lk betwee Demak ad Swede. he dex used s Dash CPI publshed by Demak Statstcs ad avalable o Bloombeg DCPIEW <Idex>. Othe Euopea dces have oly aely taded so fa. Outsde of Euope, Japa lauched the fst flato-lked govemet bod JGB Mach 2004. he flato swap maket has oly developed slowly, but a maket fo total etu swaps has bee developed fo JGBs. he dex udelyg the Japaese maket s the Japaese CPI ex peshables dex, whch s publshed by the Msty of Publc Maagemet, Home Affas, Posts ad elecommucatos. It s avalable o Bloombeg JCPGEF <Idex>. 3.6. Seasoalty flato dces Fgue 3.7 shows that the Euo flato s qute volatle. Howeve, pat of the volatlty ca be attbuted to seasoal effects. Fo stace, Decembe flato teds to be above the yealy aveage (the Chstmas effect. Some of the ma easos fo seasoalty flato ae vayg food pces dug the yea ad sales pces. Seasoal effects also 7 8 See www.bls.gov/cp/home.htm fo moe fomato. he aveage US CPI value fo Ja-982 to ad cludg Dec-984 equals 00. July 2005 2

dffe sgfcatly fom oe couty to aothe, owg to dffeet weghts gve to dffeet tems the vaous atoal measues of flato as see Fgue 3.3. Moeove, thee s a lack of hamosato the teatmet of seasoal tems. I geeal, atoal sees ae moe volatle tha euo aggegate because of dffeet, ofte cotastg, atoal pattes. A close look at Fgue 3.7 eveals that thee s clealy hghe flato dug the Mach- May peod tha the Jue-August peod. We have estmated the seasoal patte fo the 200-2004 peod ad plotted seasoally adjusted sees based o these estmates besdes the HICPx flato. hs shows a much smoothe patte (the stadad devato was appoxmately half of the ogal sees. Fgue 3.7. HICPx flato hstoy 0.00% 8.00% 6.00% 4.00% 2.00% 0.00% -2.00% -4.00% -6.00% Ja-0 Jul-0 Ja-02 Jul-02 Ja-03 Jul-03 Ja-04 Jul-04 HICPx Seasoally adjusted HICPx Iflato hstoy fo the HICPx dex (evsed dex fo Jauay 200-Decembe 2004. Souce: Lehma Bothes. I Fgue 3.8 we have pefomed a egesso aalyss of flato (see Appedx A.2 fo a explaato of the seasoal dummy model estmato fo the ete HICPx flato hstoy fom Febuay 995 to Decembe 2004. I ode to vestgate whethe the seasoal pattes ae statstcally sgfcat as well as beg ecoomcally sgfcat, we also epot stadad eos. We see able 3.8 that Jauay, Febuay, Mach, Apl, July, ovembe ad Decembe ae all statstcally sgfcat dffeet fom 0 at a 5% cofdece level. 9 9 Oe ca mage moe elaboate aalyss cosdeg, fo example, mea-eveso effects, etc, but we wll ot pusue ths oute hee (see, fo example, Bockwell & Davs, 99. July 2005 3

Fgue 3.8. Seasoals fo HICPx Moth seasoal std. eos sgfcat Jauay -2.37% 0.50% Yes Febuay 2.22% 0.48% Yes Mach 2.32% 0.48% Yes Apl.30% 0.48% Yes May 0.55% 0.48% o Jue -0.80% 0.48% o July -.86% 0.48% Yes August -0.95% 0.48% o Septembe 0.30% 0.48% o Octobe -0.65% 0.48% o ovembe -.45% 0.48% Yes Decembe.39% 0.48% Yes hs table shows the estmato esults fo HICPx (evsed sees seasoal pattes, std. eos ad sgfcace at a 95% cofdece level. he data u fom Feb-95 to Dec-04, gvg us 9 obsevatos, 9 fo Jauay ad 0 fo the othe moths. Souce: Lehma Bothes. Dffeet costtuets ad computato methods lead to dffeet seasoal effects Fgue 3.9 gaphcally shows the seasoal patte peset the data. Fgue 3.9 also shows the dffeeces seasoal pattes fo dffeet dces. he seasoalty fo all dces s maly postve the fst moths (apat fom Jauay of the yea ad egatve late o. It s teestg to see that Italy has a vey low seasoal vaace ad the UK has a huge seasoal vaace. Fo Italy fesh food pces ae cluded as movg aveages smoothg out pat of the seasoalty. he lage postve Apl seasoal fo the UK s due to taxato effects fom the stat of the fscal yea (the UK fscal yea us fom Apl to Apl. Fo Euope geeal, the egatve seasoals Jauay ad July ae due to sales peods. he sales ted to be moe aggessve the UK esultg vey stog egatve seasoals. Fgue 3.9. Mothly seasoal compoets fo dffeet dces 8.00% 6.00% 4.00% 2.00% 0.00% -2.00% -4.00% -6.00% -8.00% Ja Feb Ma Ap May Ju Jul Aug Sep Oct ov Dec Itala FOI Fech CPI UK RPI US CPI Euo HICPx Seasoal pattes ae estmated usg the seasoal dummy model o data fom Febuay 995 to Decembe 2004. Souce: Lehma Bothes. Seasoalty estmates fo the Euo HICPx sees ae dffcult to calculate as the dex keeps evolvg. Fo stace, sce 200 a umbe of coutes added sales pces to the HICP umbes. Despte the lmted avalable data, we see fom Fgue 3.0 that the seasoal patte has become moe poouced ecetly. July 2005 4

Fgue 3.0. Mothly seasoal compoets HICPx fo 995-2000 ad 200-2004 5.00% 4.00% 3.00% 2.00%.00% 0.00% -.00% -2.00% -3.00% -4.00% -5.00% -6.00% Ja Feb Ma Ap May Ju Jul Aug Sep Oct ov Dec 995-2000 200-2004 995-2004 Seasoal pattes ae estmated usg the seasoal dummy model o flato data fom Febuay 995 to Decembe 2000, Jauay 200 to Decembe 2004, ad the full sample. Souce: Lehma Bothes. Most flato-lked poducts pay o a aual bass (e.g. OA bods ad most flato swaps, so the seasoal effects do ot matte fo tal pcg. Howeve, seasoal effects ae mpotat fo mak-to-maket valuato. hs mak-to-maket valuato became ceasgly mpotat whe the Euopea Uo adopted the accoutg stadads set by the Iteatoal Accoutg Stadads Boad (IASB 2005. hese stadads eque defed beeft peso fuds to eflect mak-to-maket valuatos of these schemes the facal statemets (see Secto 0.3. I Secto 6. we show how to buld a flato cuve copoatg seasoal effects ad expla how to mak-to-maket flato swaps. July 2005 5

4. IFLAIO BASICS AD COCEPS Ivestos cae about eal athe tha omal value A flato dex epesets the elatve value of a basket of goods ad sevces 4.. Iflato, omal value ad eal value Ivestos cae about goods ad sevces that moey ca buy, ot moey tself. Fo stace, people pefe a 5% cease come ad o cease pces to a doublg of come ad doublg of pces. hs smple, but mpotat ad fudametal, ecoomc axom s cucal to the udestadg of flato-lked makets. Because the (cosume maket cossts of a boad vaety of poducts, a basket of goods ad sevces s costucted that tes to epeset the basket of goods ad sevces used by a epesetatve custome. Fo stace, Fgues 3., 3.4-3.6 we peseted the majo costtuets of the Euo HICPx, Fech CPI, UK RPI, ad US CPI dces, espectvely. We assume that all vestos ae ad ema teested ths basket of goods ad sevces. he omal value of the basket of goods ad sevces s computed at egula tevals (typcally mothly. A flato dex s othg moe tha the elatve value of the basket. A base date s chose at whch the omal value of the dex s set to 00. If the omal value of the basket equals 0,000 at the base date t meas that the dex wll se pot f the basket value ceases by 00. Example 4. Let us cosde a vesto wth assets equal to 00,000. he vesto ca cuetly buy 0 baskets wth hs assets. A yea late the dex has se fom 00 to 02 (the cost of the basket has ceased to 0,200. hs meas that flato was equal to 2% (=02/00 -. Besdes the cease the dex, the omal value of the assets of the vesto has ceased to 0,000. he omal cease fo the vesto equalled: 0,000 00,000 omal chage = = +.00%. 00,000 Howeve, due to the flato of 2% the vesto ca ow oly buy 9.90 (=0,000 / 0,200 baskets of goods ad sevces. he eal come chage s theefoe equal to: eal chage 0,000 /0,200 0 = = 0.98%. 0 Of couse, thee s o eed to compute ths va the value of the efeece basket. We fd the same esult usg the omal values ad the flato dex: eal chage 0,000 / 02 = = 0.98%. 00,000 / 00 Govemets use flato dces to povde eal value cetaty to ts habtats Iflato-lked bods ca guaatee eal etus Eve though the value of the vesto s assets gew omal tems (+%, eal tems hs assets have deceased (-0.98%. Oe of the key tasks of most govemets s to cease the eal assets of ts habtats. Most govemets have legslato place that gves employees the ght to flato compesato the salay. hs s typcally acheved by a salay cease wages depedg o a flato dex at egula tevals (typcally, aually. Also popety etal pces ae ofte lked to a flato dex, meag that the popety owes may cease ets accodg to the se the flato dex at pe-specfed tevals. 4.2. Iflato-lked cashflows ad eal bods As vestos cae about eal come athe tha omal come, they pefe to vest secutes guaateeg them a eal etu athe tha a omal oe. I ths secto, we show how vestos ca get guaateed eal etus stead of omal etus by usg flato-lked bods a wold wth flato. July 2005 6

A flato-lked zeo-coupo bod s a bod that has a sgle paymet at tme, ts matuty date. We deote ts value today (that s, tme 0 as D IL (0,. he omal paymet at matuty s equal to: D IL (, = I(, the value of the dex at matuty. 0 As vestos ae teested eal etus they value all cashflows elatve to the dex, I. heefoe, the flato-lked bod has a eal value equal to I ( / I( = eal ut at matuty. I ode to get the eal value of ths flato-lked bod today, D (0,, we eed to dvde the value of the flato-lked bod by the cuet value of the flato dex, I(0. he eal value of a flato-lked zeo-coupo bod s gve by : D (0, = DIL(0,. I(0 We deote the cuet omal value of at tme by D (0,, a omal zeo-coupo bod. Smlaly, we deote the cuet eal value of eal ut at tme by D (0,, a eal zeo-coupo bod. Fgue 4. llustates the cashflows ad values of a flatolked zeo-coupo bod both omal ad eal tems. Fgue 4.. Iflato-lked paymets omal ad eal tems omal uts Real uts oday (t=0 I ( 0 D (0, D (0, = D ( 0, IL Matuty ( I ( he eal etu (that s, etu eal uts o a flato-lked zeo-coupo bod s gve by 2 : y (0, = D (0, whee y (0, deotes the aualsed eal zeo-yeld ad D (0, deotes eal value of a flato-lked bod matug at tme. hus, usg flato-lked bods oe ca get a guaateed eal etu the same way as omal bods allow oe to get a guaateed omal etu. he omal etu o a flato-lked bod s uceta ad gve by: I (0 (0, I D I( = (0 I /, / / ( / D (0,, 0 I pactce, a lag exsts betwee the dex date ad the matuty date of the bod. We dscuss flato lags detal Secto 4.3. I pactce the value of the dex s ot kow fo each date, but t s oly kow at egula tevals (typcally mothly. Futhemoe, thee s a delay betwee whe the flato takes place ad whe t s kow. I Secto 4.3, we expla how ths poblem s teated pactce ad how we ca assume that the value of the dex ca be obseved fo each date. 2 hs follows fom the geeal defto that R(0, s defed as the aualsed et etu o secuty P fo the peod [0,] usg the followg fomula: ( + R(0, =, July 2005 7 P( P(0 whee P(0 deotes the cuet pce ad P( the pce of the secuty at matuty,. We assume o temedate dvded paymets the above defto.

Example 4.2 We assume that the cuet dex equals 00, I(0=00. he maket tades a flatolked zeo-coupo bod at 98.04% ad a omal bod at 96.5%, both wth a tme-tomatuty equal to yea (=. Fom these values we ca get the aualsed omal yelds ad eal yelds the followg mae. omal yeld o omal zeo-coupo bod: y (0, = D (0, / = Real yeld o flato-lked zeo-coupo bod: y (0, = D (0, / = 0.965 0.9804 = 4.00%. = 2.00%. A vesto ca thus lock- a guaateed omal etu of 4% o a guaateed eal etu of 2%. Gve the gowth of the flato dex, we ca calculate the eal yeld o a omal bod ad the omal yeld o a flato-lked bod. Assumg the flato dex gows to 02, I(=02, these ca be computed the followg mae. Real yeld o omal zeo-coupo bod: I (0 / I ( 00 /02 = =.96%. 0.965 0.965 omal yeld o flato-lked zeo-coupo bod: I ( (0 (0, I D / 02 = 00 0.9804 = 4.04%. I Fgue 4.2 we cosde thee sceaos fo the dex afte a yea: ( the dex emas the same, that s, o flato; (2 the dex gows to 02, that s, 2% flato; the dex gows to 04, that s, 4% flato. Fgue 4.2. omal ad eal etus o omal ad flato-lked bods Iflato dex om. etu omal bod Real etu omal bod om. etu IL bod Real etu IL bod 00 4.00% 4.00% 2.00% 2.00% 02 4.00%.96% 4.04% 2.00% 04 4.00% 0.00% 6.08% 2.00% Souce: Lehma Bothes. We see Fgue 4.2 that the omal bod has a ceta omal etu but uceta eal etu, wheeas the flato-lked bod has a ceta eal etu ad a uceta omal etu. Fgue 4.2 also shows that fo 2% flato the omal bod ad the flato-lked bod have almost detcal omal ad eal etus. I Secto 4.4 we expla how oe ca compute the exact level of flato fo whch the omal ad flato-lked bods have exactly the same etus. As s the case the omal maket, ssues do ot ssue zeo-coupo bods, but typcally ssue coupo bods. I the same mae as fo omal bods we ca show that a coupo bod s othg moe tha a potfolo of zeo-coupo bods wth dffeet matutes. Cosde a flato-lked coupo bod that pays a coupo equal to c at tmes,,. We ca wte the omal value of ths bod at tme 0 as follows: July 2005 8

B (0, IL = = cd (0, + D (0, = I(0 cd (0, + D (0, = = I(0 B (0,. IL A detaled accout of flato-lked bods s gve Bodolo (2005 ad Deaco et al. (2004. Below we summase the otato toduced ths secto. IL Summay of otato used D (0, D (0, D IL (0, y (0, y (0, I( Cuet value of omal zeo-coupo bod Cuet value of a eal zeo-coupo bod Cuet omal value of a flato-lked bod Yeld o omal zeo-coupo bod Yeld o eal zeo-coupo bod Value of dex at tme Iflato lags lead to a lowe degee of eal value cetaty 4.3. Idces, dexato lags ad aoucemets As dscussed befoe, the ma pupose of flato-lked secutes s to povde eal value cetaty. I ode to acheve a hgh degee of eal value cetaty the flatolked cashflows should be lked as closely as possble to cotempoaeous flato. Howeve, pactce, the value of the dex s ot yet kow fo the cashflow date ad a lagged dex value s take. As a esult, vestos have o flato potecto ove the last peod (typcally, thee moths of the flato-potected secuty. hey ae compesated fo ths by ecevg the flato of the peod pecedg the puchase of the secuty. hs s llustated Fgue 4.3. I geeal, the flato ove the pefect dexato peod s ot equal to the flato ove the lagged flato peod leadg to a lowe degee of eal value cetaty. Fgue 4.3. Idexato lag Issue date Matuty date lag lag Pefect dexato Lagged dexato Idexato lags ae uavodable hs fgue gaphcally llustates the fluece of lagged flato o eal value cetaty. Souce: Lehma Bothes. Dffeeces ae lkely to be bgge fo loge dexato lags ad moe volatle flato evomets. Futhemoe, the fluece of the lag ceases wth deceasg tme to matuty. heefoe, a small dexato lag s pefeed fo a hgh degee of eal value cetaty. hee ae two ma easos fo dexato lags. Fst, t takes tme to pocess cosume pce data ad compute flato umbes. Due to the pocessg tme, flato s, typcally, aouced about two weeks afte the moth ude cosdeato (fo example, Jauay flato s aouced o about 5 Febuay. Secod, a lag ases due to tadg ad settlg of bods betwee coupo paymet dates. As fo omal bods, flatolked bods usually pay coupos; f the bod tades betwee coupo dates selles should be compesated fo havg held the bod fo pat of the coupo peod eve though they wll ot eceve the coupo. As fo omal bods, ths compesato s July 2005 9

he UK used a eghtmoth lag to hadle accued teest Caada toduced a ew method fo computg accued teest Refeece umbes ae computed fo each day effected va the paymet of accued teest. wo ma methods of accued teest paymet ae see pactce. he oldest s the oe employed by flato-lked glts the UK maket ssued befoe 2005, whee the ext coupo s kow at all tmes. hs s acheved by usg a eght-moth lag cosstg of a two-moth peod allowg fo publcato of the flato dex ad sx moths fo the accued teest calculato (the flato-lked glts pay sem-aual coupos. 3 A moe commo ad pefeed method these days s to base the accued teest o the cumulatve movemets the assocated flato dex. hs calculato method was tated by Caada fo flato-lked bods 4 ad has bee adopted cotetal Euope ad the US. he UK has aouced that t wll swtch the calculato method fo all flato-lked glts ssued gog fowad. he method computes (daly efeece umbes fo dates usg a lea tepolato of the dex values of, typcally, two ad thee moths ago. he efeece umbe fo the fst of ay caleda moth equals the dex value of the caleda moth thee moths eale. I(0-May-04=CPI(Feb-04, I(0-Ju-04=CPI(Ma-04, ad so o. he efeece umbes fo othe dates ca the be computed usg lea tepolato of the efeece umbes of the fst days of the caleda moths. Fo example, Fgue 4.4 we compute the efeece umbe, I(2- May-04 at 2 May 2004 fo the US CPI dex. I geeal, the daly efeece umbe ca be computed as follows: dd I( dd / mm / yy = I(0/ mm / yy + [ I (0/ mm + / yy I (0/ mm / yy ], DM whee DM deotes the umbe of days the moth fo all days betwee the fst of Jauay ad the fst of Decembe. Fo the days Decembe we have: dd I( dd /2 / yy = I (0/2 / yy + [ I(0/ 0/ yy + I (0/2 / yy ]. DM Fgue 4.4. Daly efeece umbes calculato fo US CPI (2 86.20 + 3 ( 87.40 86.20 Mach 87.40 Feb 86.20 May 2 th Feb Ma Ap May Ju Jul Feb CPI elease Ma CPI elease I ths fgue we compute the daly efeece umbe fo the US CPI dex fo 2 May 2004. Souce: Lehma Bothes. Bods ae quoted eal tems, but eed to be pad omal tems Usg the (daly efeece umbes, flato-lked bods ca be quoted the stadad mae, that s, as eal bods. Howeve, as metoed above, ode to get the value of the flato-lked bod, ths pce eal tems should be multpled by the dex ato whch s the cuet daly efeece umbe computed the mae suggested by the Caada easuy dvded by the daly efeece umbe at the stat of the bod. I Fgue 4.5 we plot the dex value fo the HICPx maket ad the lealy tepolated efeece umbes assocated wth the OA. I ode to get the elatve 3 See the documet How to calculate cash flows o flato-lked glts o http://www.dmo.gov.uk/glts/dexlk/f2l.htm fo detaled examples. 4 Caada Real Retu Bods (RRBs. July 2005 20

daly efeece umbes fo the OA s, the efeece umbes eed to be dvded by the base dex fo the patcula OA s. 5 Fgue 4.5. HICPx ad assocated daly efeece umbes 6.50 6.00 5.50 5.00 4.50 4.00 3.50 3.00 2.50 2.00.50 Feb-04 Ma-04 Ap-04 May-04 Ju-04 Jul-04 Aug-04 Sep-04 Oct-04 ov-04 Dec-04 Ja-05 Itepolated + 3M lag HICP ex-tobacco he sold gee le epesets the tepolated dex umbes fo OA s ad the dashed gold le epesets the HICPx level. ote that the sold le lags the dashed le by thee moths. Souce: Agece Face éso; Lehma Bothes. 4.4. Beakeve flato o expla the cocept of beakeve flato, we cosde two poducts avalable the maket today. he fst s a omal zeo-coupo bod wth matuty date, whose omal value today s dcated by D (0, ad pays off at matuty. he secod s a zeo-coupo flato-lked bod wth matuty date, whose omal value today we dcate by D IL (0,=I(0D (0,, whee I(0 deotes the cuet efeece umbe ad D (0, deotes the eal value of a eal bod wth matuty date. he fal payoff of ths flato-lked bod at matuty wll equal I(, the efeece umbe at matuty. We assume a vesto has 00 to vest ad eeds to choose betwee the followg two vestmets. Ivestmet s omal zeo-coupo bods, whle vestmet 2 s flato-lked zeo-coupo bods.. Ivest 00 zeo-coupo omal bods, that s, 00/D (0, uts. he omal payoff of ths vestmet at matuty s gve by: 00 = 00( + y (0,, D (0, whee y (0, s the aualsed omal yeld o the omal zeo-coupo bod. Assumg D (0,=0.965 wth =, we have a fal payoff of: 00 = 04.00. 0.965 2. Ivest 00 zeo-coupo flato-lked bods, that s, 00/( I(0D (0, uts. he omal payoff of ths vestmet at matuty s gve by: 00 I( = 00( + (0, ( + y (0,, I(0 D (0, 5 Fo the OA 25 July 202 ths s 08.9870; fo the OA 25 July 2032 t s.5484; ad fo the OA 25 July 2020 t s 2.62258. July 2005 2

whee (0, deotes the aual ealsed flato ad y (0, s the aualsed eal yeld o the eal bod. Assumg I(0=00 ad D (0,=0.9804 fo = we have as a fal payoff: 00 I ( = 02.00 ( + (0,, 98.04 whch wll deped o the flato ealsed the ext yea. he etus ae llustated Fgue 4.6. Fgue 4.6. omal vesus flato-lked vestmet oday Matuty omal 00 00 ( + y (0, Iflatolked 00 00 ( + + y (0, ( (0, hs fgue pesets the payoffs of a omal ad flato-lked zeo-coupo bod. he payoff fom the omal vestmet ca be cotacted today, whle the payoff fom the flato-lked bod depeds o ealsed flato. Fo the omal vestmet, the omal payoff at matuty s kow today as D (0,, ad theeby y (0, ae kow today. Fo the flato-lked vestmet, the omal payoff at matuty depeds o the ealsed flato fom today to matuty, (0,. If ealsed flato, (0,, tus out to equal: + y (0,.04 = =.97% + y (0,.02 the vesto would, ex-post, be dffeet betwee vestmet ad 2. We defe ths quatty as the beakeve flato ate, b(0,: + y (0, b( 0, = + y (0,. Beakeve flato gves the flato ate that makes a vesto dffeet betwee omal ad flato-lked vestmets he beakeve efeece umbe gves the dex level that makes a vesto dffeet betwee omal ad flato-lked vestmets It s easy to check that f flato equalled.97% vestos would have bee dffeet betwee vestg the flato-lked ad the omal bod. I the case of the omal bod, they would have vested 00 00/0.965=04.00 omal bods, whch etued 04.00 at matuty. I the case of the flato-lked bod, they would have vested 00 00/0.9804=02.00 flato-lked bods esultg 02.00 0.97 = 04.00 at matuty. he payoffs both omal ad eal tems thus cocde fo both the omal ad flato-lked bod f ealsed flato equals the beakeve flato. If the ealsed flato tus out to be hghe (lowe tha b(0,, vestos would have bee bette off vestg the flato-lked (omal bod. he beakeve ate gves us the dffeece pot of the ealsed flato ate betwee the flato-lked ad the omal vestmet. Aothe quatty of teest s the efeece level fo whch the vesto would be dffeet. hs efeece level s called the beakeve efeece umbe ad s deoted by I(0,. If the efeece level at matuty, I(, equals: I(0 D (0, I ( 0, = = I(0( + b(0, = 00 (.097 = 0.97 D (0, July 2005 22

the vesto would, ex post, be dffeet betwee vestmet ad 2. If the efeece dex at matuty tus out to be hghe (lowe tha I(0,, vestos would have bee bette off vestg the flato-lked (omal bod. Wth the toducto of the beakeve efeece level, we ca wte the cuet omal value of a flato-lked paymet at tme as D (0, I(0,, the dscouted omal value of the beakeve efeece umbe. hs follows fom the fact that, by defto, we have: I( 0, D (0, I(0 D (0,. = Below we summase the otato toduced ths secto. Summay of otato toduced b(0, Beakeve flato ate fo flato peod fom today to. I(0, Beakeve efeece umbe fo tme see fom today. 4.5. Compoets of beakeve flato ate It s temptg to say that the beakeve ate should equal expected flato. 6 Howeve, although expected flato typcally compses the lagest compoet of the beakeve swap ate, thee ae seveal easos why they wll ot usually be the same.. Fst, thee s the compoudg effect, whch s a mathematcal pot. If the aualsed flato fo the peod fom 0 to, that s (0,, s adom, the expected payoff of a flato-lked secuty would be hghe tha f t wee to gow at the expected aualsed flato ate. I fomulas, the compoudg effect ca be peseted as: [( + (0, ] E[ (0, ] E + (, whee E deotes expectato. he equalty oly apples f (0, s detemstc. 7 hus, the compoudg effect has upwad pessue o beakeve flato ates. Futhemoe, the dffeece betwee beakeves ad expected flato wll be hghe at tmes of hghe volatlty. Fo 5-yea flato swaps the effect s about 3 bass pots f the expected flato equals 2% wth a 0.5% stadad eo ad about 6 bass pots fo a 30-yea swap. 2. Secod, the flato covexty, meag the secod-ode pce effect case of flato chages, ceases wth the matuty of the bod. Hgh covexty s attactve fo vestos: t meas that pces se moe tha flato duato pedcts f beakeve flato ates cease, ad decease less tha flato duato pedcts f beakeve flato ates decease. As covexty s attactve fo vestos, t pushes dow the beakeve ates. 3. Fally, as flato-lked bods povde a hgh degee of eal value cetaty, vestos ae wllg to pay a flato sk pemum to eceve flato. he flato sk pemum pushes beakeve flato hghe tha expected flato. Let us expla ths moe detal. We cosde sk-avese vestos who ae teested eal come whch s pefectly matched by the daly efeece umbes, I. 8 At tme t these vestos ca vest ethe a flato-lked bod wth matuty offeg them a eal etu of y (0, o a omal bod wth matuty offeg them a omal etu of y (0,. hs gves the followg eal etus of the omal ad the flato-lked bod, espectvely. 6 ote fo the techcal eade: he beakeve swap ate s the expected flato ude the -fowad pcg measue. Howeve, ths expectato s geeally ot equal to the expectato ude the physcal (eal-wold measue. 7 he equalty s a specal case of Jese s equalty whch says that the expectato of a covex fucto s lage tha the fucto evaluated at the expectato. 8 As dscussed Secto 4.3 ths s ot stctly the case, but a easoable appoxmato. July 2005 23

I(0( + y(0, I( vesus ( + y (0, Because the eal etu o the omal bod s uceta ad the eal etu o the flato-lked bod s ceta, sk-avese vestos wll oly cosde vestg the omal bod f they ae compesated fo beag the flato sk. 9 hs wll be the case f the expected eal etu o the omal bod s hghe tha the eal etu o the flato-lked bod, o f the omal etu o the omal bod s hghe tha the expected omal etu o the flato-lked bod,.e. ( + y (0, ( + y (0, I( E. I(0 he addtoal etu that soveegs (o othe ssues eed to pay o omal ssues compaed wth flato-lked ssues s called the flato sk pemum, whch we deote by p(0,. We ca ow wte the omal ate as a Fshe equato (afte the Ameca ecoomst Ivg Fshe: 20 y (0, = ( + y (0, ( + E[ (0, ] ( + c(0, ( p(0, +. + Empcal evdece suggests that soveegs ca save substatal amouts by ssug flato-lked debt hus the omal etu equals the eal etu tmes the expected dex cease tmes the sk pemum. he sze of the flato sk pemum depeds o the volatlty of flato (hghe volatlty leads to hghe pemum ad the sk-aveseess of vestos (the moe sk-avese the hghe the pemum. It s had to put a umbe o the flato sk pemum ad the esults dffe accodg to maket ad study (see, fo example, Campbell ad Shlle (996, Gog ad Remoloa (996 fo US data ad Foes et al. (997 fo UK data amog othes. hese studes suggest that soveegs ca save substatal amouts by ssug flato-lked debt. Fgue 4.7 shows the compoets of the beakeve flato ate. Fgue 4.7. Compoets of beakeve flato Covexty Rsk pema Beakeve flato Compoudg Expected flato ote that the scalg should ot be take dcatve of the actual sze. Souce: Lehma Bothes. 9 hs s ot stctly tue as t could be that omal debt offes dvesfcato beefts that flato-lked bods do ot have wth the est of the vesto s potfolo. hs s ulkely (typcally eve the othe way ad theefoe we goe that possblty. 20 ote that E[+(0,] [+ E (0,] ad we have wtte E[+(0,] = [+ E (0,] [+ c(0,], whee c(0, 0 deotes the compoudg effect. Futhemoe, we ased both sdes of the equato to the powe of / ad the sk pema p(0, s such that the equalty sg apples. July 2005 24

5. ZERO-COUPO IFLAIO SWAPS Descpto Iflato swaps ca be used to hedge out flato sk Iflato swaps ca be spot o fowad statg he zeo-coupo flato swap has become the stadad flato devatve. Fo may, t s the basc buldg block of the flato devatves maket. Its appeal s ts smplcty ad the fact that t offes vestos ad hedges a wde age of possbltes that dd ot pevously exst the cash maket. A fxed zeo-coupo flato swap s a blateal cotact that eables a vesto o hedge to secue a flato-potected etu wth espect to a flato dex. he flato buye (also called the flato eceve pays a pedetemed fxed ate, ad etu eceves fom the flato selle (also called the flato paye flato-lked paymet(s. he mechacs ae faly smple; today a flato paye ad a flato eceve agee to exchage the chage the flato dex value fom a base moth (e.g. ovembe 2004 to a ed moth (e.g. ovembe 2009 vesus a compouded fxed ate. If the value of the dex the base moth s kow at the tme of the cepto of the cotact, we call the flato swap spot statg. If the value of the dex the base moth s ot yet kow, we speak of a fowad statg flato swap. Fgue 5. gves a example tem sheet of a spot statg flato swap fo the Euopea HICPx maket. Fgue 5.. Example of a tem sheet fo HICPx zeo-coupo flato swap otoal: 00,000,000 Idex: HICPx (o evsed Souce: Fst publcato by Euostat as show o Bloombeg CPFEMU ade date: 0 Febuay 2005 Stat date: 2 Febuay 2005 Ed date: 2 Febuay 200 Fst fxg: 5.60 (ovembe 2004 Fxed leg: 5 (+ 2.% - Iflato leg: HICPx ( ov / 09 I(0 Feb 0 = HICPx( ov / 04 I(0 Feb 05 he euo maket tades mothly dex levels We see that the flato swap stats o 2 Febuay 2005 ad eds o 2 Febuay 200 wth the exchage of cashflows. As the value of the HICPx dex the cotact moth of 2009 eeds to be kow at the paymet, the cotact moth s lagged to the cuet moth (usually by two to thee moths. I the above example, the cotact moth s ovembe, ad the HICPx dex values fo ovembe ae omally publshed md- Decembe, whch s well befoe the paymet/ed date. As the value of the HICPx dex ovembe 2004 (t equalled 5.60 s kow by 2 Febuay 2005, the flato swap ou example s spot statg. It s maket stadad to quote fxed flato swaps whose tal lfetme equals a multple of whole yeas (5 yeas ou example. ote that as the cotact moth s ovembe, the flato leg payoff tems of efeece umbes s based o Febuay ot 2.Febuay. 2 hs has the advatage that all cotacts tadg wth the same cotact moth ad matuty have the same fal payoff. hs smplfes closg out of the posto. 2 Recall that fo the HICPx dex the efeece umbes ae thee moths lagged. July 2005 25

ot all makets use the coveto to pay tems of dex levels. he maket stadad fo the Fech FR-CPI ad US-CPI s to defe the payout o the flato leg tems of efeece umbes. A tem sheet would look lke the oe show Fgue 5.2. Fgue 5.2. Example of a tem sheet fo US CPI zeo-coupo flato swap otoal: $00,000,000 Idex: US CPI-SA (o-evsed Souce: Fst publcato by BLS as show o Bloombeg CPURSA ade date: 0 Febuay 2005 Stat date: 2 Febuay 2005 Ed date: 2 Febuay 200 Fst fxg: 90.77500 Fxed leg: 5 (+ 2.75% - 20 Iflato leg: I (2 May 0 CPI ( Feb /0 + CPI ( Ma /0 3 3 = 20 I(2 May 05 CPI ( Feb / 05 + CPI ( Ma / 05 3 3 he US maket tades efeece umbes Fgue 5.3 gves a ovevew of the maket covetos used fo tadg zeo-coupo swaps the dffeet dces. he coveto ad lag ae also dcated o the LehmaLve page fo flato poducts. Fgue 5.3. Maket covetos fo zeo-coupo swaps Maket Euopea (HICPx Fech (FR CPI Uted Kgdom (UK RPI Uted States (CPI-SA Method Mothly dex level (3M lag Itepolated values Mothly dex level (2M lag Itepolated values Souce: Lehma Bothes. Less lqud makets ae typcally quoted as speads to the lqud makets he less lqud makets, such as the Belga Health dex, ae typcally quoted as a spead to the HICPx. Fo example, f the 5y beakeve ate fo HICPx equals 2.% ad the Belga Health dex s quoted at 50bp, ths meas that the 5y beakeve ate fo the Belga Health dex equals 2.6%. Valuato of zeo-coupo swaps A flato swap has a flato peod 22 statg at s ad edg at e ove whch the flato s computed ad a sgle paymet date, whch we assume to equal e fo ow, whe the flato paymet (o the flato leg s exchaged wth a fxed amout (o the fxed leg. he cashflows ae peseted Fgue 5.4. 22 he dates efe to the dates whe the efeece umbes ae obseved. Fo some makets (most otably the Euopea HICPx maket, t s customay to tade o flato moths, but oe ca always ewte ths tems of efeece dates ad umbes (see Secto 4.3 fo the elato betwee efeece umbes ad dex levels. July 2005 26

Fgue 5.4. Cash flows of zeo-coupo flato swap Iflato leg o cash flows ae exchaged at cepto I( e I( s Fxed leg s ( + b e s e he beakeve ate depeds o the flato peod he flato leg thus pays the et cease efeece umbes fom s to e, whee I( s s kow. he fxed leg pays a fxed amout whch s coveetly wtte as a accumulated ate, b. he ate b s quoted the maket ad called the beakeve swap ate. he ate b wll dffe depedg o the cuet tme ad the flato peod, ad theefoe we use the otato b = b(0; s, e fo the beakeve swap ate today fo a flato peod s to e. I geeal, s ca be dffeet fom today. Based o the tem sheet Fgue 5. we take s = 0-Feb-05 ad e = 0-Feb-0 ad assume today s gve by the 0 Febuay 2005. 23 he beakeve flato swap ate quoted the maket equals b(0;0-feb-05, 0-Feb-0 = 2.% ad the dscout facto fo 2 Febuay 200 equals 0.86. Assumg a otoal equal to,000,000 the value of the fxed leg ca the be computed as: cuet value of fxed leg = D (0, = 0.86 e [( + b(0;, s ] e, s e 5 [( + 2.% ],000,000 = 94,640. 45 he cashflow at matuty emas costat ad theefoe the fxed leg oly vaes wth the dscout facto. At cepto the beakeve swap ate s set at such a level that the maket cosdes the value of the fxed leg to equal the value of the uceta flato leg: cuet va lue of flato leg = cuet value of fxed leg = 94,640.45. he oly ukow o the flato leg s the efeece umbe at e. Usg the cocept of the flato-lked zeo-coupo bod toduced Secto 4.2 we kow that the cuet value of a payoff of I( e at e equals I(0D (0, e. hs allows us to wte the cuet value of the flato leg as follows: I (0 D (0, e cuet va lue of flato leg = D (0,,000,000 = 94,640.45 ( e I s whee the eal dscout bod, D (0, e, s the emag ukow. Usg the fact that the value of the fxed leg ad flato leg ae equal at cepto, we fd that the value of D (0, e cosstet wth the quoted beakeve swap ate equals: I( s D (0, e = D (0, e I(0 ( + b(0;, 5.60 = 0.86 ( + 2.% 5.70 s e 5 = e s 0.954, whee the value of the HICPx dex fo ovembe 2004 equals 5.60, the efeece umbe today (2 Febuay 2005 equals 5.70, ad as befoe the dscout facto fo 2 Febuay 200 equals 0.86. Besdes a beakeve swap ate fo the swap, we ca also compute a beakeve efeece umbe fo the zeo-coupo flato swap whch we deote by I(0; s, e. It s gve by: 23 ote that the efeece umbe at 0-Feb-05 equals the HICPx fo ovembe 2004. July 2005 27

I (0;, = I( s e s ( + b(0;, = 5.60 ( + 2.% s e e s 5 = 28.32. he beakeve efeece umbes oly deped o the matuty date It s also easy to show that the stat date of the peod does ot matte. Pluggg the bootstapped value fo D (0, e gves: I(0 D (0, e I( 0, e = = I( s e s e s e D (0, e e s ( + b (0;, = I(0;,. Oe ca also check that I(0D (0,=I(0,D (0,. We have 5.70 0.954 = 28.32 0.86. As a specal case of ou exteded defto of the beakeve swap ate, we have: b ( 0, = b(0;0, fo a zeo-coupo flato swap wth flato peod fom today (t=0 to. I Secto 6. we expla how oe ca mak-to-maket flato swaps afte we expla how to costuct flato cuves usg the zeo-coupo flato swaps. July 2005 28

UG G EAG : use am e= ul&plot am e= ul Lehma Bothes Iflato Devatves Explaed 6. IFLAIO AALYSIS FRAMEWORK he commodtsato ad tasfe of flato sk s oe of the majo goals of the flato devatves maket. I ode fo facal sttutos to beeft fom ths maket, a aalyss famewok s ecessay. I ths chapte we descbe such a famewok. We stat by costuctg a flato cuve gve maket quotes fo beakeve flato swap ates takg seasoalty to accout. Futhemoe, we peset ad expla a umbe of sks elated to flato-lked cashflows. he maket quotes flato swaps wth aual teos 6.. Costuctg a flato cuve I ths secto we peset a methodology to costuct flato-lked swap cuves usg quotes of beakeve ates fo zeo-coupo flato swaps. We also show how oe ca accout fo seasoalty effects. he maket povdes bd-offe beakeve flato ates fo seveal matutes. At the momet boke quotes ae typcally avalable fo,2,..,0,2,5,20,25,30 yea matutes. Lehma Bothes povdes lve quotes fo beakeve swap ates o LehmaLve as ca be see Fgue 6.. Fgue 6.. Zeo-coupo flato swap ates o LehmaLve (04-02-2005 % 3. 3. 0 2. 9 2. 8 2. 7 5.0Y, 2.743:USCPI 5.0Y, 2.690:UKRPI 2. 6 2. 5 2. 4 2. 3 2. 2 2. 5.0Y, 2.05:HICPX 5.0Y, 2.037:FRCPI 2. 0. 9 0 0Y 20Y 30Y HICPX FRCP USCP UKRP Souce: LehmaLve. he quotes gve Fgue 6. fo the HICPx dex have ovembe as the efeece moth, wheeas fo the FRCPI ad USCPI quotes ae o the efeece umbes (ov- Dec. We focus o the costucto of a flato cuve fo the HICPx dex. We toduce the followg teo stuctue 0 =0-Feb-05, =0-Ma-05,, 360 =0-Feb-35. As poted out Secto 4.3 the efeece umbes I ae lagged to the flato dex values. I the case of the HICPx maket, the flato dex value fo ovembe s equal to the efeece umbe of the fst of Febuay. heefoe, fo the quotes tems of efeece umbes we have 0 equal to 0-Feb-05. We have see Secto 4.4 that usg the beakeve swap ates the maket we ca compute the beakeve efeece umbes the followg mae: ( + b(0;,. 0 I( 0, = I( 0 0 July 2005 29

fo = 2, 24,, 20, 44, 80, 240, 300, 360. Fgue 6.2 shows the beakeve flato dex values ad beakeve efeece umbes fo the gve beakeve flato swap ates. Fgue 6.2. Computg the flato cuve o a aual bass eo Beakeve ate Matuty flato dex moth Matuty Uadjusted efeece date Beakeve efeece umbe fowads Y 2.04% ov-05 0-Feb-06 5.60 (+2.04% = 7.96 2.02% 2Y 2.04% ov-06 0-Feb-07 5.60 (+2.04% 2 = 20.36 2.02% 3Y 2.07% ov-07 0-Feb-08 5.60 (+2.07% 3 = 22.93 2.% 4Y 2.09% ov-08 0-Feb-09 5.60 (+2.09% 4 = 25.57 2.3% 5Y 2.% ov-09 0-Feb-0 5.60 (+2.% 5 = 28.32 2.7% 6Y 2.3% ov-0 0-Feb- 5.60 (+2.3% 6 = 3.8 2.2% 7Y 2.4% ov- 0-Feb-2 5.60 (+2.4% 7 = 34.07 2.8% 8Y 2.6% ov-2 0-Feb-3 5.60 (+2.6% 8 = 37.5 2.27% 9Y 2.8% ov-3 0-Feb-4 5.60 (+2.8% 9 = 40.36 2.3% 0Y 2.20% ov-4 0-Feb-5 5.60 (+2.20% 0 = 43.70 2.35% 2Y 2.24% ov-6 0-Feb-7 5.60 (+2.24% 2 = 50.80 2.4% 5Y 2.28% ov-9 0-Feb-20 5.60 (+2.28% 5 = 62. 2.4% 20Y 2.35% ov-24 0-Feb-25 5.60 (+2.35% 20 = 83.96 2.53% 25Y 2.42% ov-29 0-Feb-30 5.60 (+2.42% 25 = 20.7 2.66% 30Y 2.48% ov-34 0-Feb-35 5.60 (+2.48% 30 = 24.06 2.74% Souce: Lehma Bothes. A tepolato scheme s eeded fo o-quoted swaps he colum uadjusted fowads deotes the gowth ates the beakeve efeece umbes ad ae gve by log(7.96/5.60, log(20.36/7.96,, log(24.06/20.7/5, whee log deotes the atual logathm. Fo mak-to-maket valuato of off-maket swaps (fo stace, a flato swap wth Decembe as ts efeece moth we eed beakeve efeece umbes fo all temedate moths as well. As o othe maket fomato s avalable, these have to be detemed va a tepolato scheme. We wat to costuct a beakeve efeece cuve fo 30 yeas o a mothly bass. A (too smple appoach would be to lealy tepolate the efeece umbes gve Fgue 6.2. Although ths methodology poduces a beakeve efeece cuve cosstet wth maket data, t suffes fom two ma dawbacks. Fst, t goes ay seasoal patte. Secod, t poduces a athe bumpy beakeve ate cuve. I the ext secto we peset a alteatve methodology that allows fo easy tegato of seasoalty effects. Icopoatg seasoalty We assume that the beakeve efeece umbes ae of the followg fom: I( 0, = + I( 0 exp 0 [ f ( u s( u ] du fo =,.., 360. Usg a cotuously compouded fowad flato ate, f, allows us to sepaate the aual flato compoet fom the seasoal compoet a staghtfowad mae. We ca wte: o wods: I( 0, = I( 0 exp f ( u du exp s( u du 0 0 beakeve efeece umbe = stat efeece umbe goss beakeve flato seasoal adjustmet. July 2005 30

he beakeve efeece umbes ca be splt to a aual flato compoet ad a seasoal compoet Seveal assumptos ca be made fo the detemstc fuctos f(u, the stataeous fowad ate, ad s(u, the seasoal fucto. We assume both to be pece-wse costat. We assume the seasoals to be pece-wse costat wth a gve moth, that s: s fo u Jauay s( u = M M s2 fo u Decembe Futhemoe, the sum of the seasoal factos should equal zeo ode to esue that thee s o seasoal effect fo full-yea swaps. 24 o kowledge of the flato cuve s eeded to compute the seasoal adjustmet factos. 25 We show them Fgue 6.3. he seasoal adjustmet fo ovembe-decembe s smply the Decembe seasoal, the ovembe-jauay seasoal s the sum of the Decembe ad Jauay seasoal, ad so o. Fgue 6.3. Aualsed seasoal adjustmet factos Peod Seasoal Seasoal adjustmet Peod Seasoal Seasoal adjustmet ovembe-decembe.39%.39% ovembe-jue -0.80% 4.6% ovembe-jauay -2.37% -0.98% ovembe-july -.86% 2.75% ovembe-febuay 2.22%.24% ovembe-august -0.95%.80% ovembe-mach 2.32% 3.56% ovembe-septembe 0.30% 2.0% ovembe-apl.30% 4.86% ovembe-octobe -0.65%.45% ovembe-may 0.55% 5.4% ovembe-ovembe -.45% 0.00% Souce: Lehma Bothes. Gve the beakeve efeece umbes show Fgue 6.3, we ca compute uadjusted fowad flato ates Fgue 6.4. 26 We have assumed these fowad ates to be pecewse costat. 27 Fo stace, the uadjusted fowad ate fo the secod yea s gve by log(20.36 / 7.96 = 2.02%. We ca ow compute a mothly beakeve efeece cuve usg the followg teatve fomula: I ( 0, = I (0, exp (( ( f + s fo 3,...,360, = he fst moth s typcally aleady kow whee f deotes the aual flato (the uadjusted fowads ad s the aualsed seasoal compoet fo the peod [ -, ]. Fo example, we fd the beakeve efeece umbe fo Mach 2006 the followg mae: I (0,0 Ma 06 = I (0,0 Feb ad fo Apl 2006 we fd: = 7.96 exp I (0,0 Ap 06 = I (0,0 Feb = I (0,0 Ma = 8.29 exp 06 exp( (2.02% +.39% /2 ((2.02% +.39% /2 = 8. 30 06 exp( (2 2.02% 0.98% /2 06 exp( (2.02% 2.37% /2 ((2.02% 2.37% /2 = 8.26. he fst yea s exceptoal that, typcally, the beakeve efeece umbe of the ext moth s aleady kow. I ou example, we kow that the HICPx equals 5.90 fo Decembe 2004. heefoe, we take as the fowad ate fo the fst yea [log(7.96 / 5.90+.39%/2] 2/ = 2.05%. We subtacted the Decembe seasoal as we oly have the peod Jauay-ovembe emag ad multply by 2/ to aualse the 24 Sce s + +s 2 =0, we have that exp(f=exp(f+s + +s 2, ad so o. 25 If the maket matues ad swaps taded o seveal efeece moths ae taded, the seasoal factos ca be bootstapped out of maket quotes. Cuetly, quotes o moe tha oe efeece moth ae ae. 26 I Secto 7. ad Appedx A.4 we dscuss why these fowad ates eed to be coected to epeset the ates fo fowad statg flato swaps. 27 he pece-wse flat assumpto goes potetal depedece stuctues betwee omal ates, eal ates ad flato. July 2005 3

ate. he seasoal factos we use to adjust the ates ae Decembe-Jauay (-2.37%,, Decembe-ovembe (-.39%. Fgue 6.4. Aual fowad ates ad seasoally adjusted fowad ates 5.00% 4.00% 3.00% 2.00%.00% 0.00% -.00% Ap-05 Ap-07 Ap-09 Ap- Ap-3 Ap-5 Seasoally adjusted uadjusted Souce: Lehma Bothes. hs pocedue ca be doe ecusvely to costuct the whole cuve. Fgue 6.4 pesets the seasoally adjusted fowad ates ad uadjusted ates fo Apl 2005 to Apl 205. I Fgue 6.5 we peset the costucto of the fst yea of the flato cuve. Fgue 6.5. Computg the mothly beakeve efeece umbe cuve Beakeve dex moth Matuty efeece date y-o-y seasoal Beakeve Idex t- Beakeve Idex ov-04 0-Feb-05 5.60 Dec-04 0-Ma-05.39% 5.60 5.90 Ja-05 0-Ap-05-2.37% 5.90 exp([2.05%-2.37%]/2 = 5.87 Feb-05 0-May-05 2.22% 5.87 exp([2.05%+2.22%]/2 = 6.28 Ma-05 0-Ju-05 2.32% 6.28 exp([2.05%+2.32%]/2 = 6.7 Ap-05 0-Jul-05.30% 6.7 exp([2.05%+.30%]/2 = 7.03 May-05 0-Aug-05 0.55% 7.03 exp([2.05%+0.55%]/2 = 7.28 Ju-05 0-Sep-05-0.80% 7.28 exp([2.05%-0.80%]/2 = 7.4 Jul-05 0-Oct-05 -.86% 7.4 exp([2.05%-.86%]/2 = 7.43 Aug-05 0-ov-05-0.95% 7.43 exp([2.05%-0.95%]/2 = 7.53 Sep-05 0-Dec-05 0.30% 7.53 exp([2.05%+0.30%]/2 = 7.76 Oct-05 0-Ja-06-0.65% 7.76 exp([2.05%-0.65%]/2 = 7.90 ov-05 0-Feb-06 -.45% 7.90 exp([2.05%-.45%]/2 = 7.96 Souce: Lehma Bothes. Seasoal effects become less mpotat fo loge matutes I Fgue 6.6 the mothly beakeve ates (both seasoally adjusted ad uadjusted ae show. A clea seasoal patte ca be ecogsed, whch des out ove tme as the seasoal fluece becomes less mpotat. July 2005 32

Fgue 6.6. Beakeve flato ates 3.0% 2.90% 2.70% 2.50% 2.30% 2.0%.90%.70%.50% Ap-05 Jul-09 Oct-3 Ja-8 Ap-22 Jul-26 Oct-30 Feb-35 Seasoally adjusted BE Seasoally uadjusted BE Souce: Lehma Bothes. We ae ot estcted to usg pece-wse flat fowad ates. Fo example, oe ca apply lea o quadatc fowad ates stead of the pece-wse flat fowads. Aothe possblty s to use (cubc sples ode to get a smooth cuve. he US ad Fech makets tade zeo-coupo swaps based o efeece umbes, so we typcally have flato peods whch ae patally coveg moths. Although ths slghtly complcates the calculato of the seasoal-adjustmet facto, the cuve ca be costucted usg the same methodology. Makg to maket flato swaps Chages flato expectatos ae eflected the mak-to-maket value Eve though the payout of a flato swap s lked to ealsed flato, t s also a flato expectatos poduct. O a mak-to-maket bass, the value of the flato swap chages le wth chages expected flato as eflected the chagg beakeve flato ate. hs esults fom the fact that the mak-to-maket value of the flato swap has to eflect the cost of eteg to the offsettg tasacto. Fo a flato buye, ths eques sellg flato o a flato swap wth detcal chaactestcs. he mak-to-maket today (t=0 fo a flato swap tated at tme t B 0 wth flato peod fom s to e fo the flato buye s gve by: e s e s D ( 0, ([ + b(0;, ] [ + b( t ;, ], e s e B s e whee b(0; s, e ad b(t B ; s, e deote the beakeve flato swap ates at the cuet tme 0 ad at the cepto tme t B, espectvely. he tem o the left deotes the value of the flato leg at tme 0, whle the tem o the ght deotes the ogal value of the flato (ad fxed leg. Howeve, t s ulkely that the maket s stll quotg the ogal swap stuctue. Suppose we eteed a 5-yea flato swap (say, Febuay 2004 to Febuay 2009 ths stuctue s lkely oly quoted dug May ad Jue 2004. Gve a flato cuve cosstg of the beakeve efeece umbes I(0, fo 0 e, the mak-to-maket value of ths stuctue s gve by: I(0, (, e I tb e D (0,, ( ( e I s I s whee I(0, s ad I(t B, s deote the beakeve efeece umbes fo e (=0-May-09 fo the Febuay 2004 to Febuay 2009 flato swap today ad at the cepto tme t B. Above we have detaled the costucto of a flato cuve cosstg of the beakeve efeece umbes I(0, as a fucto of. I geeal, the beakeve efeece umbe at matuty wll deped o the seasoalty assumpto used the flato cuve costucto. Howeve, May ad Jue 2005, 4- July 2005 33

yea flato swaps ae quoted fo the peod (Febuay 2005 to Febuay 2009 ad the dex cease fom Febuay 2004 to Febuay 2005 s kow. he flato swap ca the be easly maked-to-maket by: * I( s D (0, e I( s * 4 5 [ b(0;, ] [ + b( t ;, ], + s e B s e whee s * equals May 05. hus we ca get a seasoally depedet mak-to-maket value o a aual bass. Example 6. We cosde a zeo-coupo flato swap o HICPx fo Febuay 2004 to Febuay 2009 payg May 2009. Ogally, a beakeve ate of 2.0% was pad fo the cotact o a otoal of 00,000,000. We kow that the HICPx fo Febuay 2004 equalled 3.50. Usg the beakeve efeece cuve, we fd that the beakeve efeece fo Febuay 2009 equals: 25.57 exp( 2 2.7% + 2.24% 3 = 26.38 whee 25.57 s the beakeve efeece dex level fo ovembe 2008, 2.7% the pece-wse flat fowad ate fom ovembe 2008 to ovembe 2009 (see Fgue 6.2 ad.24% the ovembe-febuay seasoal (see Fgue 6.3. he value of the flato leg at matuty s theefoe equal to: 26.38 00,000,000 =,35,405. 3.50 he cotacted paymet at matuty was gve by: 00,000,000 hs esults a mak-to-maket value of: 5 (( + 2.0% = 0,950,359. [,35,405 0,950,359] 360, 94 0.90 = whee 0.90 s the dscout facto fom today to May 2009 fom the pespectve of the flato eceve. 6.2. Iflato ad teest ate sk Smla to omal bods ad swaps, flato-lked bods ad swaps ae sestve to teest ates. Howeve, cotay to omal bods ad swaps, flato-lked bods ad swaps also have explct flato sk. We have see befoe that a flato-lked cashflow at tme e ca be wtte two maes: I(0 D (0, = I (0, D (0, ( + y (0, e e e o e ( + b(0, e = ( + y (0, Followg the fomulato o the left-had sde, we ca vestgate the sk due to chages the eal ate, y (0, e. Followg the fomulato o the ght-had sde, we ca vestgate the sk of chages the omal ate, y (0, e, ad the sk of chages the beakeve ate, b(0, e. Of couse, both appoaches should be equvalet ad the appoach take s meely a matte of taste. Below, we dscuss the sk tems of flato ad (omal teest ate sk. We pefe ths appoach because the swap maket uses the omal cuve ad flato cuve as these ae easly costucted fom e e e e July 2005 34

maket stumets. 28 I ths fomulato, the paye of flato o a zeo-coupo swap faces the followg sks:. Iflato sk, that s, chages the flato cuve. 2. Iteest ate sk, that s, chages the omal yeld cuve. he flato PV0 gves the sestvty to chages the flato cuve Iflato PV0 he flato PV0 measues the sestvty to chages the (aually compouded flato cuve, cetes pabus. 29 It s appoxmately equal to the chage value due to a bp move the flato cuve. Fo a flato swap wth flato peod s to e ad matug at tme e the flato PV0 today s gve by: I(0 D (0, e IfPV0= e. I( + b(0, s e Oe ca get the flato duato by dvdg the flato PV0 by the pce. I Appedx A.3 we gve the flato PV0 fo flato-lked coupo bods. he omal PV0 gves the sestvty to chages the omal cuve Moves the flato cuve ad omal cuve ae coelated omal PV0 he omal PV0 measues the sestvty to chages the (aually compouded omal cuve, cetes pabus. Fo a flato swap wth flato peod s to e ad matug at tme e the omal PV0 today s gve by: = e I(0 I( tb, e PV0 D (0, (0, + (0, e D. e y e I( s I( s At cepto (t B =0 the tem betwee backets equals zeo (sce I(t B D (t B, e =I(t B, e D (t B, e, mplyg that the flato swap has o teest ate sk at cepto. he stuato chages as the swap moves o out of the moey. If the flato beakeves se, cetes pabus, the swap moves to the moey fo the flato eceve (the tem betwee the backets s postve sce I(0D (0, e has ceased, ad the flato eceve has a egatve PV0 as the futue cashflow becomes less valuable wth hghe ates. It ca be deceptve to look at cetes pabus moves the flato ad omal cuves because moves these cuves ae typcally coelated. As flato ad omal ates ted to be postvely coelated, the sk fo the buye of a upwad shft the omal cuve s thus (much less tha based o the PV0. Fo stace, the case of a 50% coelato, the sk s half that of a cetes pabus up move the omal cuve. he coelato ca be made a fucto of tme wth typcally hghe coelato fo loge matutes. hs eables us to captue the empcal fact that log eal teest ates ae less volatle tha shot eal ates. he fomulas fo coupo flato-lked bods ad the devato ae detaled Appedx A.3. It also pesets fomulas fo covexty ad coss covexty fo flato ad teest ate sk. I Fgue 6.7 we gve a example of a 5-yea flato-lked coupo bod ad calculate the flato ad teest ate PV0s. 28 Of couse, gve ay two out of the thee cuves (omal, flato, ad eal the thd oe ca be costucted usg o-abtage agumets but the flato ad omal cuves ae typcally used. 29 Of couse, the PV0 ca also be defed as the sestvty of the flato cuve fo othe compoudg fequeces, e.g., cotuously compouded o sem-aually. hs has a small mpact o the fomula. July 2005 35

Fgue 6.7. Iflato PV0 ad omal PV0 example Yea Coupo Beakeve ate Beakeve efeece umbe Cashflows omal yeld 2.50% 2.00%.020 2.55 4.00% 2 2.50% 2.08%.042 2.6 4.0% 3 2.50% 2.4%.066 2.66 4.8% 4 2.50% 2.8%.090 2.73 4.24% 5 2.50% 2.20%.5 4.28 4.26% Pce 02.29 Iflato PV0 4.769 omal PV0-4.675 bp bump b s 0.04770 bp bump y s -0.04674 Souce: Lehma Bothes. I the above example we see that f the flato cuve moves up by bass pot, the value of the flato-lked bod pces moves up by 0.04770, whle t moves dow by 0.04674 f the teest ate cuve moves up by bass pot. 30 Iflato swap cashflows have a dffeet cedt qualty tha flato-lked bod cashflows 6.3. Coutepaty sk So fa, we have goed the cedt qualty whe dscussg flato-lked bods ad flato swaps to avod otatoal complexty. I ths secto, we take the cedt qualty explctly to accout ad expla the dffeet cedt qualty cuves. ypcally flato devatves deales eceve flato fom flato-lked bods (the stadadsed poducts ad pay flato o stuctued flato devatves (typcally swaps. hs toduces a ceta type of coutepaty sk. Iflato cashflows fom govemets should be dscouted fom the flato-lked govemet cuve, whle flato cashflows fom swap coutepates should be dscouted fom the Lbo cuve (plus/mus a spead depedg o the coutepaty s cedt qualty. 3 ypcally, ths meas that flato swaps should be dscouted off the Lbo cuve ad flato-lked bods should be dscouted off the govemet cuve. Let us fo ow assume that all cuves ae lqud alog the whole cuve. hs s a easoable assumpto o the omal sde, but hadly o the flato-lked cuves as we dscuss late. Fgue 6.8 gves a example of the elatos betwee the cuves. Fgue 6.8. Govemet vesus swap cuves G LBE SS L GBE G L RG RL om. govemet cuve om. Lbo cuve Real Govemet cuve Real Lbo cuve RG RSS RL SS RSS GBE LBE omal swap spead Real swap spead Govemet beakeve Lbo beakeve Schematc ovevew of the elatos betwee the omal govemet, omal Lbo, eal govemet, ad eal Lbo cuve. Souce: Lehma Bothes. 30 If we had computed flato ad teest ate PV0s o cotuously compouded ates they would be exactly offsettg. 3 Lbo s typcally take as havg AA-cedt qualty. July 2005 36

We deote the fou cuves above as 32 G D ( t,, y G D ( t,, y L D ( t,, y L D ( t,, y G G G G ( t, : eal govemet bod ( t, : omal govemet bod ( t, : eal LIBOR cuve ( t, : omal LIBOR cuve cuve cuve whee D stads fo the zeo-coupo bod ad y fo the yeld. Fgue 6.9 gves a example of the fou cuves. Fgue 6.9. Govemet ad Lbo example cuves 5.00% 4.50% 4.00% 3.50% 3.00% 2.50% 2.00%.50%.00% 0.50% 0.00% 2 3 4 5 6 7 8 9 0 Souce: Lehma Bothes. Real Gov om. Gov. Real LIBOR om LIBOR Usg ou defto of beakeve flato, we ca ow compute the beakeve flato fo both the govemet cuve ad the Lbo cuve. he beakeve efeece umbe ad beakeve ate fo the Lbo flato cuve ae gve by: L L I(0 D (0, I (0, = ad L + y + b (0, = L D (0, + y ad fo the govemet flato cuve: L L (0, (0, G G G I(0 D (0, I (0, = ad G + y (0, + b (0, =. G G D (0, + y (0, If we assume that the omal swap spead equals the eal swap spead, 33 that s: + y + y G G (0, + y = (0, + y L L (0,, (0, we get a uque beakeve efeece umbe, I G (0, = I L (0,. As metoed eale, the assumpto that both a eal govemet ad eal swap cuve exst s qute stog the cuet maket evomet. I pactce, t s ulkely that the omal ad eal swap speads ae the same. Fom a hstocal pespectve t was teestg to see what happeed whe Italy ssued the BP 2008 2003. At that tme 5y flato was maly dve by demad fom flato eceves. hs dove up flato beakeve ates the swap maket, allowg Italy to ssue at a attactve level. 32 ote that ths dstcto otato fo cedt qualty s oly made ths paagaph. I othe paagaphs we oly dcate t f t leads to ambguty. 33 he spead s based o aual compoudg ad s oly appoxmately equal to X X y ( 0, y (0, fo X=G,L. Fo cotuous compoudg ths would be exact. July 2005 37

6.4. Roudg sk A typcal sk fo flato-lked secutes s the oudg sk. Wheeas teest ates ca take ay value, dces ae set up to oe decmal. Cosde the followg example. Example 6.2 he value of the HICPx dex s 5.60 fo ovembe 2004 ad the beakeve ate fo a 2-yea zeo-coupo flato swap s gve by 2.%. hs mples a beakeve HICPx dex level fo Febuay 2006 equal to: 5.60 ( + 2.% 5 = 28.32. Iflato dces ae ouded to oe decmal Seasoal effects ae ot kow he maket povdes lttle fomato o seasoal effects Due to the oe decmal place oudg, the dex caot set at ths level. Eve f the beakeve dex ealses, the publshe ( ths case Euostat wll oud the value up o dow to the eaest decmal. I ths example, that would be 28.30 o 28.40. he oudg sk deceases quckly wth ceasg matuty ad becomes eglgble fo log matutes. Roudg sk s especally mpotat fo flato caps ad floos. Fo stace, f we have a cap wth a stke of 2.%, the oudg ca deteme whethe the cap wll ed o out of the moey. 6.5. Seasoalty sk We saw Secto 3.6 that flato sees have stog seasoal pattes. I ths secto, we aalyse the cosequeces of these seasoal pattes fo pcg ad sk maagemet. I Secto 6. we costucted a flato cuve that copoates seasoalty pattes. Fo the costucto we assumed a set of detemstc seasoal factos. I ealty, the seasoal patte s pobably ot detemstc, o at least ot kow wth cetaty as the stadad eos Fgue 3.8 dcate. heefoe, flato poducts ae sestve to seasoalty sk. Cuetly, lttle maket fomato s avalable o seasoals. Quotes ae aely gve fo off-maket swaps, although ths mght chage wth the toducto of the futues maket. hee s lkely to be a dscepacy betwee the expected seasoals ad the maket mpled seasoals due to maket postog. 34 Fo stace, Apl ad May Euo flato swaps ca be hedged by baks usg OA s, wheeas, say,. Mach flato swaps caot (yet be souced fom the flato cash maket. Example 6.3 We stat by dscussg the seasoal sk usg Example 6. based o the data Fgues 6.2 ad 6.3. he tade of teest s aga a zeo-coupo flato swap o HICPx fo Febuay 2004 to Febuay 2009 payable o May 2009. I ode to value the Febuay flato swap, we eed the beakeve efeece umbe fo Febuay 2009. We compute the beakeve efeece umbe as: I 0,0 May 09 = 25.57 exp( 3 2.7% + ( s + s + s = 26.383845, ( 2 2 2 2 whee 25.57 s the beakeve efeece dex level fo ovembe 2008, 2.7% the pece-wse flat fowad ate fom ovembe 2008 to ovembe 2009 (see Fgue 6.2 ad s 2 + s + s 2 =.24% s gve by ou seasoal assumpto Fgue 6.3. hs gves se to the questo of the sestvty of the valuato fo the seasoal fucto. We have see befoe that seasoals eed to add up to zeo o a aual bass. heefoe, f a ceta seasoal compoet moves up at least oe of the othes eeds to move dow. We deote the chages the seasoal fucto by δ,,δ 2, whch eed to sum to zeo ode to have the chaged seasoals to add up to zeo. hus, the adjusted seasoal compoets ae gve by: 34 ote fo the techcal eade. he seasoals ude the physcal measue dffe fom the seasoals ude the sk-eutal measue. July 2005 38

s * = s + δ,..., s = s + 2. * 2 2 δ Oe way to vestgate the seasoalty sk s by bumpg oe seasoal up by x bass pots ad deceasg the emag seasoals by a equal amout (x/bp such that the aual zeo costat s ot volated. I ou example, we bump the seasoal of each moth by 50bp (appoxmately std. devato ad decease the emag seasoals by 50/bp ad vestgate the mpact o the P&L of the cotact. Fo stace, f we bump the Jauay seasoal wth 50bp ad decease all othe moths by 50/bp we get: * * s = s + 0.005,..., s2 = s2 + 0.005 / hs gves s * 2 + s * + s * 2 =.65% ad ou adjusted beakeve efeece umbe s gve by I (0,0 May 09 = 25.57 exp( 3 2.7% +.65% = 26.426937. ' 2 2 he P&L mpact of such a chage the seasoal assumpto s gve by: 26.43 26.38 00,000,000 =,389,372,35,405 = 37,967. 3.50 3.50 If we would bump the Mach seasoal by 50bp ad decease the othe moths by 50/bp we get s * 2 + s * + s * 2 =.0% ad ou adjusted beakeve efeece umbe s gve by: I (0,0 May 09 = 25.57 exp( 3 2.7% +.0% = 26.369483. ' 2 2 he P&L mpact of such a chage the seasoal assumpto s gve by: 26.37 26.38 00,000,000 =,338,752,35,405 = 2,653. 3.50 3.50 We see Fgue 6.0 that bumpg the Decembe to Febuay seasoals has the same mpact, whch makes sese as oly the aggegate of the emag seasoals mpacts the value. he same holds fo the seasoals fom Mach to ovembe. As a cease oe of these moths deceases the aggegate seasoals fom Decembe to Febuay ths educes the value of the swap. Fgue 6.0. Seasoalty sk 50,000.00 40,000.00 30,000.00 20,000.00 0,000.00 - (0,000.00 (20,000.00 Ja Feb Ma Ap May Ju Jul Aug Sep Oct ov Dec hs fgue gves the mpact o the P&L of a stadad devato move the seasoal effect fo all moths. Souce: Lehma Bothes. July 2005 39

6.6. Isttutoal sk he flato dces o whch flato swaps (ad bods ae based ae typcally defed ad publshed by a statstcal bueau. hs etals seveal sks that ae dffcult to quatfy. Fo example, whe addtoal coutes ae added to the euozoe, these coutes wll be copoated the Euopea dces. As these coutes typcally have hghe flato tha the cuet coutes, ths pushes beakeve ates hghe fo loge matutes. Aothe teestg sttutoal sk s the chage of bechmak flato dex fo polcymakes. Fo stace, the UK chaged ts flato taget dex fom RPIX to UK HICP (deoted CPI by the atoal Statstcs. July 2005 40

7. IFLAIO SWAPS AD FUURES Fxed ad floatg flato swaps I Secto 5 we toduced ad descbed the (fxed zeo-coupo flato swap. Besdes zeo-coupo flato swaps a umbe of othe flato swaps ae taded, whch ae typcally potfolos of zeo-coupo swaps oe way o the othe. We descbe a eveue flato swap, a OC flato bod, ad peod-o-peod flato swaps. Iflato swaps ae blateal cotacts that eable a vesto o hedge to secue a flato-potected etu wth espect to a flato dex. he flato buye (also called the flato eceve pays a pedetemed fxed o floatg ate (usually mus a spead. I etu, the flato buye eceves fom the flato selle (also called the flato paye flato-lked paymet(s. wo ma types of (zeo-coupo flato swaps exst: fxed flato swaps (flato vesus fxed ate ad floatg flato swaps (flato vesus floatg ate, usually Lbo. he mechacs ae show Fgue 7.. Fgue 7.. Mechacs of fxed ad floatg flato swaps Iflato buye Floatg flato swap Iflato buye Fxed flato swap Iflato buye et dex cease (Iflato Iflato leg vesus LIBOR - spead Floatg leg o fxed ate Fxed leg Iflato selle Iflato selle Iflato selle Souce: Lehma Bothes. A vesto pays flato o a paye flato swap ad eceves flato o a eceve flato swap We call a flato swap a paye flato swap f you pay flato ad a eceve flato swap f you eceve flato. Usg a teest ate swap (IRS, we ca fd a o-abtage elatoshp betwee fxed ad floatg flato swaps whch we call the flato swap paty: Floatg paye flato swap = Fxed paye flato swap + paye fxed-fo-floatg swap Floatg eceve flato swap= Fxed eceve flato swap + eceve fxed-fo-floatg Applcatos May applcatos exst fo flato swaps, whch we ow summase. Hedgg / Asset Lablty Maagemet Iflato swaps ca be used to hedge cocetatos of flato sk. hs s especally useful fo peso fuds that have flato-lked labltes. July 2005 4

Iflato swaps ca be used to hedge flato exposues that ae ot taded the cash maket. It s mpactcal fo baks to hedge flato sks usg flato-lked bods. Eve f two-way lqudty ca be foud, t mposes heavy balace sheet chages. As flato swaps ae off-balace sheet tasactos, they avod the balace sheet chages. Ivestg Iflato swaps ae a ufuded way to take a vew o flato. hs allows fo leveage ad helps those wth a hgh fudg cost. As flato swaps ae customsable ove-the-coute poducts, vestos ca talo the flato exposue to match the pecse equemets tems of dex (e.g. may schemes ae lked to domestc wage dces, tmg, matuty, sze, ad so o. Iflato swaps ca be used to go ethe log o shot flato o a efeece dex. Ivestos may ot be allowed to sell a flato-lked asset due to egulatoy estctos, but may be allowed to ete a flato swap payg flato. Abtage / adg It s usually ease to pay flato o a flato swap tha to shot a flatolked asset. ades ca take advatage of pce dslocato betwee the cash ad devatves maket by buyg the cash asset ad payg flato o sellg the cash asset ad ecevg flato ad tadg the swap spead to hedge to govemet vs. Lbo cuve spead. 7.. Reveue flato swap Ulke a zeo-coupo flato swap, a eveue flato swap has multple flatolked cashflows. At the ed of each peod (usually a yea t pays the cumulatve flato fom a base moth up to the peod. Fo stace, a aual eveue flato swap o HICPx wth stat date ad ed date equal to 2 May 2004 ad 2 May 2009 has the cashflows show Fgue 7.2. Fgue 7.2. Cashflows of a typcal eveue flato swap otoal 00m Fxed ate: 2.69% Stat date 2 May 2004 Rolls 2 th Ed date 2 May 2009 HICPx Febuay 2004 3.50 Fxed leg fequecy Aual, uadjusted Fxed leg bass AC Iflato leg fequecy Aual, uadjusted Iflato leg bass BOD (30/360 Fxed ate Fxed leg Dscout facto Beakeve Idex 35 Beakeve Iflato leg 2 May 2005 (+2.69% - 2,692,509 0.977 6.25 2,420,000 2 May 2006 (+2.69% 2-5,457,53 0.948 9.34 5,44,56 4 May 2007 (+2.69% 3-8,296,966 0.92 22.69 8,099,326 2 May 2008 (+2.69% 4 -,22,87 0.874 26.36,332,008 2 May 2009 (+2.69% 5-4,207,287 0.834 30.50 4,973,878 Souce: Lehma Bothes. 35 he beakeve dex values fo Febuay 2005,, Febuay 2009 o alteatvely the beakeve efeece umbes fo 0 May 2005,, 0 May 2009. July 2005 42

he flato cashflows ca be easly costucted usg the zeo-coupo flato swaps as buldg blocks. A eveue flato swap s, theefoe, othg moe tha a potfolo of zeo-coupo swaps whee the fxed paymets ae tasfomed to a costat fxed ate pad at the cashflow dates. A OC flato bod has cashflows smla to flato-lked bods ad a cedt qualty smla to flato swaps A peod-o-peod swap pays peodc flato 7.2. OC flato bod A OC flato bod pays a eal coupo ad has a edempto paymet of the otoal at matuty. It s essetally the same as a govemet flato-lked secuty, wth the dffeece that the cedt qualty of the ssue ow equals that of the swap coutepaty. Just lke the eveue flato swap, the flato leg of a OC flato bod ca be see as a potfolo of spot statg zeo-coupo flato swaps. Usg the stuctues dscussed eale, we ca eplcate t by vestg c, the eal coupo, a eveue flato swap wth cashflows at the coupo dates ad 00 a zeo-coupo flato swap wth a matuty equal to the matuty of the bod. ote that usg ths stuctue oe ca ceate flato bods wth dffeet coupo dates/matutes as ssued the cash maket. It should be oted that these OC flato-lked bods tade at a pemum compaed wth the OC flato-lked bods wth coupo dates/matutes matchg the cash maket as they ae moe dffcult fo vestmet baks to sk maage due to seasoalty sk. OC flato bods ae aely taded as such, but ae a useful buldg block to flato asset swaps whch ae dscussed Secto 7.4. 7.3. Peod-o-peod flato swaps As a eveue flato swap, a peod-o-peod (p-o-p flato swap has multple paymets dug ts lfe. Howeve, stead of payg the cumulatve flato fom the stat date up to the coupo dates, t pays the flato ove a umbe of accual peods. he most commo stuctue s the yea-o-yea (y-o-y flato swap, whch pays aual flato at the ed of each yea. A example tem sheet s gve Fgue 7.3. Fgue 7.3. Example of a tem sheet fo HICPx yea-o-yea flato swap otoal: 00,000,000 Idex: Souce: HICPx (o evsed oday: 0 Febuay 2005 Stat date: 2 Febuay 2005 Ed date: 2 Febuay 200 Rolls: Paymet: Day cout: Fst publcato by Euostat as show o Bloombeg CPFEMU 2 th Aual, modfed followg 30/360 uadjusted Fst fxg: 5.60 (ovembe 2004 Fxed ate: 2.25% Fxed leg: day cout facto fxed ate Iflato leg: HICPx ( ov / yy I(0 Feb yy + = fo yy = 04,...,09. HICPx( ov / yy I(0 Feb yy he y-o-y flato swap Fgue 7.3 s tated o 2 Febuay 2005 ad the flato paye pays fve tmes aual flato fom ovembe to ovembe evey 2 Febuay the yeas 2006,, 200. A example of a yea-o-yea flato swap s show Fgue 7.4. July 2005 43

Fgue 7.4. Cashflows fo 5-yea y-o-y flato swaps (fxed ad floatg Fxed cashflows o fxed y-o-y flato swap pad aually to flato selle 2.25% 2.25% 2.25% 2.25% 2.25% 2-02-06 2-02-07 2-02-08 2-02-09 2-02-0 HICPx( ov / 05 HICPx( ov / 06 HICPx( ov / 07 HICPx( ov / 04 HICPx( ov / 05 HICPx( ov / 06 HICPx( ov / 08 HICPx( ov / 09 HICPx( ov / 07 HICPx( ov / 08 2-02-06 2-02-07 2-02-08 2-02-09 2-02-0 L -.47% L -.47% L -.47% L -.47% L -.47% Lbo spead cashflows pad aually to flato buye o floatg flato swap he cashflows o the flato leg ae gve the mddle. Souce: Lehma Bothes. he valuato of p-o-p swaps s model-depedet he cashflows o the flato leg ca be eplcated usg a sees of fowad statg zeo-coupo flato swaps. I the above example we ete to fowad statg zeocoupo swaps payg Febuay 2004-2005 flato,, Febuay 2008-2009 flato. heefoe, the valuato ca be doe tems of fowad statg zeo-coupo swaps. We gve futhe detals Appedx A.4. Pue peod-o-peod flato swaps ad aualsed peod-o-peod flato swaps Although the yea-o-yea flato swap s the most popula stumet, othe peodo-peod swaps tade as well. We make a dstcto betwee what we call pue peodo-peod flato swaps ad aualsed peod-o-peod flato swaps. A pue p-o-p flato swap pays the flato ove the peod o the flato leg. Fo example, a sem-aual pue p-o-p flato swap wth cotact moths Febuay ad August pays the et cease the dex fom Febuay to August ad the et cease fom August to Febuay. As the flato paymets ae ot o a aual bass, seasoalty s a mpotat ssue whe valug these swaps. Just lke a y-o-y flato swap, a aual p-o-p flato swap pays aual flato, but t pays t at a hghe fequecy ad weghted wth the appopate day cout facto. Fo example, a sem-aual aual p-o-p flato swap wth cotact moths Febuay ad August pays half the et dex cease fom last Febuay to Febuay ad half the et dex cease fom last August to August. 36 As the peod fo a yea-o-yea swap equals a yea, a y-o-y flato swap falls both categoes. Fgue 7.5 shows the cashflows fo a sem-aual pue ad a sem-aual aual flato swap. 36 Fo coveece, we assumed 30/360 as the bass such that the peod betwee Febuay ad August has a day cout facto equal to a half. July 2005 44

Fgue 7.5. Iflato leg cashflows of 2-yea sem-aual peod-o-peod swaps Iflato cashflows of a sem-aual aual p-o-p flato swap CPI( Feb / 05 2 CPI( Feb / 04 CPI( Aug / 05 2 CPI( Aug / 04 CPI( Feb / 06 2 CPI( Feb / 05 CPI( Aug 2 CPI( Aug / 06 / 05 2-05-05 2--05 2-05-06 2--06 Iflato cashflows of a sem-aual pue p-o-p flato swap CPI( Feb / 05 CPI( Aug / 04 CPI( Aug / 05 CPI( Feb / 05 CPI( Feb/ 06 CPI( Aug / 05 CPI( Aug / 06 CPI( Feb / 06 2-05-05 2--05 2-05-06 2--06 he fgue plots the flato leg cashflows fom a sem-aual p-o-p flato swap wth aual flato peods ad pue sem-aual flato swaps. he fxed / floatg leg paymets ae omtted. Souce: Lehma Bothes. As seasoalty plays a ole valug the pue peod-o-peod swaps, they would tade at a pemum as the flato selle eeds to be ewaded fo takg o the seasoal sk. Usg hgh fequecy (eg, mothly aualsed peod-o-peod flato swaps seems attactve as t speads flato paymets ove the yea stead of oe lump sum paymet each yea wthout a seasoalty pemum as seasoalty does ot affect the valuato. Makg to maket flato swaps As eveue ad peod-o-peod flato swaps ae potfolos of zeo-coupo flato swaps, mak-to-maket valuato ca be doe by makg to maket each dvdual stumet f o specfc quotes fo the whole stuctue ae avalable. I the case of flato vesus floatg flato swaps, the postos ca be maked to maket by makg to maket the fxed flato swap ad the teest ate swap dvdually. See Secto 6. fo a example of makg-to-maket a zeo-coupo flato swap. 7.4. Iflato asset swaps Descpto A flato asset swap s the combato of the puchase of a flatolked bod ad the ety to a flato swap I geeal, a asset swap s a sythetc stuctue ecomposg cashflows of exstg maket stumets. hs s usually dve by vestos eed fo cashflow pofles ot attaable the cuet maket. I ths secto, we focus o asset swaps of flatolked secutes ad, patcula, flato-lked bods. Stctly speakg, asset swaps caot be classfed as flato devatves, but they clealy epeset a key efeece the valuato of flato devatves as they povde a lk betwee the cash ad devatves maket. Whle the teest ate swap maket was bo the 980s, the asset swap maket oly came to lfe the ealy 990s. It s most wdely used by baks, whch use asset swaps to covet the log-tem fxed ate assets, typcally balace sheet loas ad bods, to floatg ate assets ode to match the shot-tem floatg ate labltes,.e. deposto accouts. I the past few yeas the asset swap has become a key stuctue the cedt makets ad s wdely used as a efeece fo cedt devatve pcg (see O Kae, 200. July 2005 45

Seveal vaatos of the flato asset swap stuctue exst. I ts smplest fom t ca be teated as cosstg of two sepaate tades. I etu fo a upfot paymet of ethe pa (pa flato asset swap o the dty pce (maket asset swap, the asset swap buye: Receves a flato-lked bod fom the asset swap selle. ypcally the bod s tadg away fom pa. Depedg o the passed lfe of the tade the bod ca tade substatally away fom pa. Etes to a sees of flato swaps (equvalet to the OC flato bod descbed Secto 7. to pay the asset swap selle flato coupos equal to that of the asset. I etu the asset swap buye eceves egula floatg ate paymets of Lbo plus (o mus a ageed fxed spead. he tasacto s show Fgue 7.6. he fxed spead to Lbo pad by the asset swap selle s kow as the asset swap spead ad t s set at a beakeve value such that the et peset value of the tasacto s zeo at cepto. Fgue 7.6. Asset swap mechacs At cepto the asset swap buye puchases a flato-lked bod woth B MK etu fo pa o a cash paymet of B MK (maket asset swap Asset swap selle IL bod woth B MK Asset swap buye pa o B MK ad etes to a flato swap, payg the bod s flato-lked cashflows etu fo Lbo plus/mus the asset swap spead tmes pa o P (fo the maket asset swap. Asset swap selle IL cash flows (LIBOR +/- spead (pa o B MK Asset swap buye IL cash flows IL bod At matuty thee s a exchage of pcpal. Asset swap selle IL edempto Asset swap buye IL edempto IL bod pa o B MK hs fgue pesets the dyamcs of a flato asset swap dug the swap ad at matuty. Souce: Lehma Bothes. Fgue 7.7 gves a example of a maket asset swap of a flato-lked bod, the OA 202 ssued by Face, whch has a matuty date of 25 July 202 ad a aual eal coupo of 3%. he fequecy of the floatg leg s take to be sem-aual. he beakeve value of the asset swap spead makes the et peset value of all cashflows equal to zeo. July 2005 46

Fgue 7.7. Maket asset swap of OA 202 (3% coupo Floatg paymets pad to asset swap buye dug swap, otoal = 2.0% L 4 bp L 4 bp L 4 bp L 4 bp L 4 bp 25-07-05 25-0-06 25-07- 25-0-2 25-07-2 I(25 Jul 05 3% I (25 Jul 0 I(25 Jul 3% I (25 Jul 0 I(25 Jul 2 3% I (25 Jul 0 Iflato paymets to the asset swap selle dug swap, otoal = 00% Cashflows at matuty Asset swap selle I(25Jul2 00% max, I(25Jul0 00 Asset swap buye Cashflows of a flato asset swap o the OA 202. he base efeece at 25 July 200 equalled 08.9870. Souce: Lehma Bothes. Accetg flato asset swaps he flato asset swaps teated befoe have the same otoal o all the floatg paymets because the asset swap selle eceves the flato-acceted edempto vs. payg the floatg leg otoal (pa fo the pa flato asset swap ad the dty pce fo the maket flato asset swap. As the flato-acceted edempto ca be substatally bgge tha the floatg leg otoal (ethe pa o the ogal dty pce, the asset swap selle us a coutepaty sk wth espect to the asset swap buye. Fo example, fo a modest aveage ealsed flato of 2%, the otoal acceto ove 20 yeas equals about 48.6% ad ove 30 yeas about 8.%. he usual soluto fo ths coutepaty exposue s to put place a collatealsato ageemet though a Cedt Suppot Aex (CSA ageemet. Howeve, ths s ot possble fo all coutepates. Oe way to mtgate ths coutepaty sk s to use a accetg flato asset swap as descbed below. Accetg the floatg leg otoal mtgates coutepaty sk Istead of havg a costat otoal o the floatg leg, a accetg flato asset swap has a accetg otoal o the floatg leg as well. he acceto ca be based o the flato dex, whch case thee s o msmatch at matuty. Aothe possblty s a acceto based o a fxed acceto ate. Uless the ealsed flato equals the fxed acceto ate, thee wll stll be a msmatch at matuty, but t s vey lkely to be much smalle tha wthout a accetg otoal. he advatage of the fxed acceto ate s that the floatg paymets ca be easly valued. I the case of the flato-acceted otoal, oe has to take to accout the depedece betwee the floatg paymets ad dex acceto. As ths depedece s faly had to estmate, a sk pemum wll be demaded by the asset swap buye. Ealy edempto flato asset swap Aothe soluto to the coutepaty sk s to pay the edempto acceto dug the lfe of the swap. he total acceto o the otoal s gve by: July 2005 47

A( I( = otoal, ( 0 I whee A( equals the edempto acceto fom 0 (the stat of flato-lked bod to whch equals matuty. hs otoal acceto ca be splt up peodcal accetos such that: whee: A ( = = [ A( A( ], Usg a ealy edempto swap mtgates coutepaty sk wthout pcg mplcatos I( A( = otoal. ( 0 I Due to the ealy edempto paymets of the patal otoal accetos at the ed of each of the peods the cedt exposue of the asset swap selle s mtgated substatally. I Appedx A. we see that ths stuctue has o pcg mplcatos whe the paymets ae dscouted usg the appopate dscout facto. We show that the ealy paymet at tme should equal: [ A A( ] D (,, ( that s, the otoal cease dug the peod fom - to dscouted fom matuty to the ealy paymet date,. Valuato Aspects he valuato of the flato asset swap stogly depeds o the type of asset swap. Fo the o-accetg swaps, valuato s elatvely staghtfowad ad le wth usual asset swaps. Calculatg the asset swap spead he asset swap spead s detemed by settg the et peset value of all cashflows equal to zeo. Whe dscoutg cashflows the swap we use the Lbo cuve, mplyg that the pates to the swap have the same cedt qualty as AA-ated bak coutepates. It s show Appedx A. that the pa asset swap spead s gve by: ad: B pa asset swap spead = B maket asset swap spead = B B PV 0 LIBOR LIBOR MK MK BMK, PV 0 whee B MK deotes the value of the flato-lked bod the maket, B LIBOR s the value of the flato-lked bod cashflows dscouted fom the Lbo cuve, ad PV0 stads fo the peset value of oe bass pot. See Appedx A. fo exact deftos. Fgue 7.8 gves a example of a maket asset swap fo the OA 202. July 2005 48

Fgue 7.8. Cashflows of asset swap o OA 202 (3% coupo Date Lbo DF Idex ato Iflato cashflow omal ate Floatg cashflow 25-Jul-05 0.994.05 3,57.92 3.5% -6,5.85 25-Ja-06 0.983-3,58.72 25-Jul-06 0.972.073 32,88.74 3.22% -4,349.96 25-Ja-07 0.959-6,8.73 25-Jul-07 0.946.095 32,862.20 3.29% -7,69.03 25-Ja-08 0.932-8,488.88 25-Jul-08 0.97.9 33,558.02 3.36% -9,256.73 25-Ja-09 0.902-20,363.20 25-Jul-09 0.886.43 34,29.9 3.43% -20,887.60 25-Ja-0 0.87-22,057.43 25-Jul-0 0.855.69 35,055.23 3.5% -22,53.37 25-Ja- 0.838-23,692.25 25-Jul- 0.822.95 35,844.87 3.58% -23,999.92 25-Ja-2 0.805-25,055.93 25-Jul-2 0.789.222,265,86.00 25.88% -,236,320.92 A Bod pce 2.0 B PV of bod cashflows 20.72 C PV of floatg leg 2.06 D PV0 6.58 00 (B-C/(A D Asset swap spead -4bp Souce: Lehma Bothes. he asset swap speads fo asset swaps wth accetg floatg otoals ae hade to compute ad ae deved Appedx A.. Applcatos Asset swaps eleve deales of balace sheet cost he ma use of flato asset swaps s to eleve deales of balace sheet cost. Iflato deales povde stuctued flato solutos usg typcally off-balace sheet stumets such as flato swaps. I ode to hedge themselves, they buy flatolked bods whch eed to be epoted o the balace sheet. o eleve them of balace sheet coveage, flato deales act as asset swap selles. By actg as asset swap selles they get the flato-lked bods off the balace sheet whle emag exposed to the flato-lked paymets. hee ae seveal vaetes of asset swaps, such as fowad asset swaps, cacellable asset swaps ad callable asset swaps. Howeve, these asset swap types seem to be moe elevat fo copoate bods cedt makets (see O Kae (200 fo a ovevew. 7.5. Iflato futues Descpto CPI futues maket has ot eally take off he CME (Chcago Mecatle Exchage stated tadg futues o the US CPI flato dex o 8 Febuay 2004. he ma advatage of CPI futues ove zeo-coupo flato swaps s mtgated coutepaty sk. he CPI futues taded o the CME ae desged to esemble the Euodolla futues cotact. Lkely due to the ll-desg of the CPI futue (the cotact taded aualsed quately flato, the maket so fa eve eally took off. he cotact specfcatos of CPI futues cuetly taded ae gve Fgue 7.9. July 2005 49

Fgue 7.9. Cotact specfcato fo CPI flato futues Refeece Idex Settlemet pce Cotact moths Cotact sze 00 - aualsed thee-moth flato based o the CPI-U o seasoally adjusted sees publshed mothly by the Bueau of Labo Statstcs (BLS. he same dex s used fo IPS. Fal settlemet amouts to 00 less the aualsed %-chage the CPI- U ove the past thee moths ad s ouded to fou decmal places. hus, 00 400[CPI( /CPI( -3 -], whee deotes the cotact moth ad -3 the base moth. Fo example, fo the Jue 2003 cotact the elevat CPI- U dex levels ae May 2003 (83.5, eleased 7 Jue 2003 ad Febuay 2003 (83., eleased 2 Mach 2003. he fal settlemet pce would have bee 00-400[83.5/83.-]=99.262. 2 cosecutve Mach quately cotact moths. $2,500 tmes the efeece dex. Mmum tck sze 0.005 dex pots whch amouts to $2.50. Expy date adg fshes 7.00am Chcago tme o the day the CPI aoucemet s made the cotact moth. I case the aoucemet s postpoed beyod the cotact moth, tadg ceases at 7.00am Chcago tme o the fst busess day followg the cotact moth. Souce: CME; Lehma Bothes. he CME stats tadg flato futues o the HICPx dex 2005. Oe of the ma advatages of the Euo cotact ove the US s that the flato s aual. he ma chaactestcs of the Euo Cosume Pce Idex (HICPx cotact ae: Fgue 7.0. Cotact specfcato fo HICPx flato futues Refeece Idex Settlemet pce Cotact moths Cotact sze 00 aual flato ate the 2 moth peod pecedg the cotact moth based o the Euozoe Hamosed Idex of Cosume Pces excludg tobacco publshed by Euostat. he same dex s used fo the Fech, Itala ad Geek Euo flato-lked bods. Fal settlemet amouts to 00 less the aual %-chage the HICPx ove the past 2 moths ad s ouded to fou decmal places. hus: HICPx ( 00 00, ( 2 HICPx whee deotes the cotact moth ad -2 the base moth. Fo example, fo the July 2004 cotact the elevat HICPx dex values ae Jue 2004 (5.0, eleased 6 July 2004 ad July 2003 (2.70, eleased 8 July 2003. he fal settlemet pce would have bee: 5.0 97.8705 = 00 00. 2.70 A pce of ove 00 dcates deflato dug the past 2-moth peod. 2 cosecutve caleda moths. 0,000 tmes efeece dex. Mmum tck sze 0.0 dex pots, whch amouts to 00.00. Expy date Souce: CME; Lehma Bothes. adg fshes 4.00pm Geewch Mea me (GM o the busess day pecedg the scheduled day the HICPx aoucemet s made the cotact moth. I case the aoucemet s postpoed beyod the cotact moth, tadg ceases at 4.00pm GM o the last busess day of the cotact moth. July 2005 50

Valuato aspects Due to daly magg, flato futues have a futues coecto Although flato futues closely esemble zeo-coupo flato swaps, the valuato s moe complcated. Aalogous to teest ate futues cotacts, flato futues have daly magg whch, due to the coelato betwee flato ad teest ates, esults a futues coecto. Futhemoe, the futues cotacts matue oce the ealsed flato s kow, whle zeo-coupo swaps usually matue wth a lag about equal to the lag the cash maket. hs ealy paymet of the futues also leads to some pcg adjustmets. Applcatos Ivestg / adg: Cosdeg the shot matuty flato futues complemet the flato-lked bod makets ad allow vestos to hedge shot-tem flato exposues. As the futues tade o 2 cosecutve moths, vestos ca take a vew o flato seasoalty. Hedgg: Usg stps of flato futues, oe ca set up a hedge agast bod cashflows. As the flato futues ae quoted fo cosecutve moths they ca be used to hedge seasoalty sk. July 2005 5

8. IFLAIO VOLAILIY PRODUCS 8.. Iflato caps ad floos Descpto Besdes swaps, optos ca also be taded o flato dces. Caps ad floos play a atual ole stuctues wth patal dexato. Fo stace, a floo o the pcpal s ofte cluded flato-lked bods ode to potect vestos agast deflato. Befoe movg to caps ad floos we fst look at the smplest of the flato optos. hese ae calls ad puts o zeo-coupo swaps. A put opto o a zeo-coupo swap pays the dffeece wth espect to a (compouded stke case flato tus out to be lowe tha ths pe-specfed stke. Combg the put wth ecevg flato o a zeocoupo flato swap esults a stuctue whch pays the maxmum of flato ad the stke, theeby floog the flato payout at the stke. Hece, puts ae usually temed floolets fo teest ates ad flato. A example tem sheet of a % floo o a zeocoupo swap s gve Fgue 8.. Fgue 8.. Example tem sheet fo HICPx zeo-coupo flato floolet otoal: 00,000,000 Idex: HICPx (o evsed Souce: Fst publcato by Euostat as show o Bloombeg CPFEMU Stat date: 2 Febuay 2004 Ed date: 2 Febuay 2009 Fst fxg: 5.60 (ovembe 2004 Buye: 5 HICPx ( ov / 09 otoal max ( +.00%, 0 HICPx ( ov / 04 May flato-lked bods have embedded zeo floos Selle: upfot pemum If the buye of the floo the above example s also ecevg flato o a zeo-coupo swap, he o she s ow potected agast a aveage flato cease lowe tha %. Floolets ae egulaly copoated flato-lked bods. Fo stace, all the US IPS ad Fech OAs have a edempto-potectg floolet guaateeg a edempto equal to pa. I the same way, a call opto s commoly efeed to as a caplet, because combg sellg the call wth ecevg flato o a flato swap esults cappg the flato cashflow o the swap. he caplets ad floolets ca be ethe spot o fowad statg, depedg o whethe the fst fxg of the dex s kow. Caplets/floolets o zeo-coupo swaps ae usually spot statg whe taded dvdually. Fowad statg caplets/floolets ae buldg blocks fo caps ad floos o peod-o-peod flato swaps as we wll see. Caps ad floos ae usually taded combato wth peod-o-peod flato swaps. Fo example, patal dexato ca be acheved by buyg a floo ad sellg a cap combed wth ecevg flato a yea-o-yea swap. A example tem sheet fo a 5% flato cap s show Fgue 8.2. July 2005 52

Fgue 8.2. Example tem sheet fo RPI yea-o-yea flato floo otoal: 00,000,000 Idex: RPI (o evsed Souce: Fst publcato by Euostat as show o Bloombeg UKRPI Stat date: 2 May 2004 Ed date: 2 May 2009 Fst fxg: 83.80 (Febuay 2004 Buye: HICPx( Feb / yy otoal max 5%, HICPx( Feb / yy 0 fo yy = 05,,09. Selle: upfot pemum Fo oday swaps, a flato cap-floo paty elato should be satsfed at ay tme: Iflato cap flato floo = paye flato swap, whee the cap ad floo have the same stke. Fally, a flato colla s the combato of a flato cap ad floo. Valuato aspects Stadad Black-type fomulas ca be used fo valuato Due to the optoalty the poduct, the valuato of caps ad floos eques a model fo the dyamcs of the dex at matuty. Fo Euopea-style optos, the smplest model s a vaato o the Black model used fo valug teest ate caps ad floos, whee stead of fowad teest ates, we model the dex cease at the opto matuty as a adom vaable whch has a logomal dstbuto. I Appedx A.5 we povde the valuato fomulae. Because selles of flato optos typcally have to hedge the optos dyamcally ad the bd-offe speads the zeo-coupo flato swap maket ae ot yet eglgble, tasactos costs wll be factoed to the pcg. Applcatos We summase a umbe of applcatos fo caps ad floos. Hedgg / Asset Lablty Maagemet Caps ad floos ae patculaly well suted fo patal dexato schemes. A paye of flato o a flato swap ca lmt the uceta flato payoff by buyg a cap. Yeld s sacfced exchage fo the guaatee that the flato paymets do ot exceed a pe-ageed stke. Ivestg / tadg Caps ad floos ca be used to leveage a vew o flato. A vesto who aleady has a posto a flato swap payg flato ca sell a floo fo yeld ehacemet. A vesto who s ecevg flato o a flato swap ca sell a cap fo yeld ehacemet. 8.2. Iflato swaptos Descpto Iflato swaptos ca be optos o both fxed o floatg flato swaps I the same way that the payoff of flato caps ad floos depeds o actual fxgs of flato, the payoff of flato swaptos depeds o flato expectatos. I the same way that thee ae two commo types of flato swaps, thee ae also two types of flato swaptos. We deote swaptos o floatg flato swaps as floatg flato July 2005 53

swaptos ad swaptos o fxed flato swaps as fxed flato swaptos. Usg swaptos we ca ceate callable ad cacellable flato swaps the same mae as fo teest ate swaps. A paye/eceve flato swapto gves the ght to ete to a paye/eceve flato swap wth teo - 0 at tme 0 at a pe-specfed coupo κ. We deote ths as a 0 ( - 0 flato swapto. he mechacs of a flato swapto ae gve Fgue 8.3 Fgue 8.3. Mechacs of a (physcally settled swapto o a flato swap At tato Swapto buye Upfot pemum Swapto selle At matuty (physcal settlemet Swapto buye Ete to the flato swap at ageed stke f flato swap has postve value fo buye Swapto selle Souce: Lehma Bothes. We stat by toducg the fxed flato swaptos. A example tem sheet s gve Fgue 8.4. Fgue 8.4. Example tem sheet fxed paye flato swapto otoal: 00,000,000 Idex: Souce: HICPx (o evsed Stat date: 2 May 2005 Opto ed date: 2 May 2006 Swap ed date: 2 May 20 ype: Fst publcato by Euostat as show o Bloombeg CPFEMU Zeo-coupo Refeece moth: Febuay Fst fxg: Buye: Selle: ot yet kow. he ght to ete to a fxed paye flato swap statg 2 May 2005 ad edg 2 May 20 wth payoff 5 HICPx( Feb / ( + b, whee b deotes the flato swap ate at HICPx( Feb / 06 2 May 2006 fo the Febuay 2006 to Febuay 20 flato swap. upfot pemum he above fxed paye flato swapto gves the buye the ght to ete at 2 May 2006 to a fxed paye zeo-coupo flato swap edg 2 May 20. hs flato swap pays the dffeece betwee the compouded maket flato ate at 2 May 2006 mus the ato of the HICPx dex values Febuay 20 ad Febuay 2006. July 2005 54

Besdes zeo-coupo swaptos, peod-o-peod flato swaps ae also possble. I the ext example tem sheet we descbe a floatg eceve yea-o-yea flato swap, also called a eal ate swapto (Fgue 8.5. Fgue 8.5. Example tem sheet floatg eceve flato swapto otoal: 00,000,000 Idex: Souce: HICPx (o evsed Stat date: 2 May 2004 Opto ed date: 2 May 2006 Swap ed date: 2 May 20 ype: Floatg ate: Fst publcato by Euostat as show o Bloombeg CPFEMU Yea-o-yea Refeece moth: Febuay Fst fxg: 3M Eubo 2.5%, quately pad (AC/360 Feb, May, Aug, ov. ot yet kow. Buye: he ght to ete to a floatg eceve flato swap statg 2- May-06 ad edg 2-May- wth cash flows Selle: HICPx ( Feb / yy otoal HICPx ( Feb / yy fo yy = 07,,, ad -otoal (EURIBOR-2.50% daycout statg 2 Aug 2006 ad edg 2 May 20. upfot pemum he above floatg eceve flato swapto gves the buye the ght to ete at 2 May 2006 to a floatg eceve yea-o-yea flato swap edg 2 May 20. hs flato swap pays the aual flato evey yea fom Febuay to Febuay etu fo quately paymets of Eubo 2.50% (the ageed spead. Valuato aspects Vaats of the Black model ca be used fo pcg Due to the optoalty the poduct, the valuato of fxed flato swaptos eques a model fo the dyamcs of the beakeve ate o the flato swap. As a floatg flato swap s a potfolo of a fxed flato swap ad a teest ate swap we eed to model the dyamcs of the beakeve ate o the fxed flato swap, the omal swap ate, ad the tedepedece fo valuato of the floatg flato swapto. Fo Euopea-style optos, the smplest models ae aga vaatos o the Black model used fo valuato, whee stead of fowad teest ates we model the fowad swap ate fo the fxed flato swapto as a logomal adom vaable. he floatg flato swapto ca be valued modellg the spead betwee the omal swap ate ad the beakeve flato ate as a logomal adom vaable. I Appedx A.7 we povde the valuato fomulae. As selles of flato optos typcally have to hedge the optos dyamcally ad the bd-offe speads the zeo-coupo flato swap maket ae ot yet eglgble, tasactos costs wll be factoed to the pcg just as fo caps ad floos. Applcatos Hedgg / Asset Lablty Maagemet Iflato swaptos gve vestos the ght to exchage futue flato-lked cash flows fo fxed o floatg paymets. Fo stace, vestos kow that they wll July 2005 55

eceve flato-lked cashflows fo fve yeas statg sx moths tme. I exchage fo a upfot paymet, they eceve the ght to ete a flato swap at a cuetly ageed stke value. Ivestg / tadg Iflato swaptos ae well suted to leveage a vew o flato. If vestos aleady have a posto a flato swap, they ca ete to a flato swapto o the opposte swap ode to have the ght to cacel the flato swap. Floatg flato swaptos allow vestos to leveage a vew o the spead betwee the omal swap ate ad the beakeve flato ate. LPI swaps ae maly taded o RPI 8.3. LPI swaps LPI (Lmted Pce Idex swaps ae useful stumets fo sttutos that have lmted dexato schemes. hey ae especally useful fo the UK maket, whch has ove 200b of LPI labltes. LPI swaps come a vaety of flavous, but what they have commo s that, oe way o aothe, the flato paymet s capped ad/o flooed. If the flato paymet s both capped ad flooed, we call t collaed. We descbe two stuctues that ae elated to the UK peso egulato, but othes exst as well. Zeo-coupo LPI swap he zeo-coupo LPI swap s patculaly useful fo peso fuds that have labltes elated to the lmted dexato of pesos defemet as toduced the 995 Peso Act. hey have labltes of the followg fom: RPI( L ( = L(0 max m,.05,. 0, RPI(0 whee L(0 deotes the lablty whe the etee etes. Fgue 8.6. Example tem sheet fo zeo-coupo LPI swap otoal: 00,000,000 Idex: RPI (o evsed Souce: Fst publcato by Euostat as show o Bloombeg UKRPI oday: 29-Ja-05 Stat date: 0-Feb-05 Ed date: 0-Feb-0 Rolls: Paymet: Day cout: st Fst fxg: LPI(0 =.0 Fxed leg: (+2.89% 5 Aual, modfed followg 30/360 uadjusted Iflato leg: I( e ( = ( max m,.05 e s LPI e LPI.0 fo s s = 0-Dec-04 I( s ad e =0-Dec-09. July 2005 56

he zeo-coupo LPI swap ca be valued usg a potfolo of a zeo-coupo flato swap, a zeo-coupo flato call, ad a zeo-coupo flato put. Of couse, othe stke values fo the cap ad floo ca be used as well. he 5% ad 0% used ae a esult of the 995 Peso Act legslato. Peodc LPI swaps A peodc LPI swap has peodc, typcally aual, paymets smla to a eveue flato swap wth the excepto that the payoff s defed tems of a atfcally ceated dex, the LPI, stead of the oday dex. he LPI ca be defed as: I( LPI( LPI ( max m,.05, fo,...,, I( = = wth LPI( 0 =I( 0, the value of a flato dex (e.g. RPI at 0. he LPI s capped at a aual gowth of 5% ad flooed at 0%. Aga, othe cap ad floo stke values ca be used. I Fgue 8.7 we peset a example tem sheet. Fgue 8.7. Example tem sheet fo aual LPI swap otoal: 00,000,000 Idex: Souce: RPI (o evsed oday: 29 Jauay 2005 Stat date: Febuay 2005 Ed date: Decembe 200 Rolls: Paymet: Day cout: Fst publcato by Euostat as show o Bloombeg UKRPI st Aual, modfed followg 30/360 uadjusted Floatg leg: (GBP Lbo + 0.20% Iflato leg: Valuato aspects b LPI( fo =,,5 wth LPI(0 =.0 ad RPI( Dec / yy LPI ( = LPI( max m,.05, fo =,...,5. ( / RPI Dec yy Futhemoe, a edempto pck-up s pad at matuty equal to (LPI(5- otoal. Ulke the zeo-coupo LPI stuctue, the peod-o-peod LPI swap caot be valued usg a appopate statc potfolo flato swaps, caps, ad floos. I ode to value a LPI swap oe eeds a pope flato tem stuctue model. A LPI swap ca the be valued by smulato of the tem stuctue model. 8.4. Iflato spead optos Descpto Iflato spead optos pay the spead betwee two flato dces, fo example, hamosed Euopea flato ad Dutch flato, f postve. hese poducts ca be useful fo peso fuds coutes whee o flato-lked bods ae ssued wth espect to the bechmak flato dex. Fo stace, let s assume a peso fud s teested beg bechmaked agast dex A (e.g. Dutch HICP, but the lqud dex (e.g. Euo HICPx s dex B. A typcal payoff to the holde s gve by: July 2005 57

{ dex cease dex A dex cease dex B,0} max he selle gets a upfot paymet etu. Combg a spead opto wth a smple flato payg stumet o the lqud dex, B, gves a best of two flato dces stumet. Valuato aspects max { dex cease dex A, dex cease dex B} A smple exteso of the Black model ca be used fo pcg he optoalty the poduct makes the valuato of spead optos model-depedet. Valug the spead optos eques modellg the spead o the jot dyamcs of the dces. Euopea-style optos ca be valued usg Magabe s (978 fomula, whch s a exteso of the Black model. We peset the valuato fomula Appedx A.6. Futhemoe, the deales take o a sk wth espect to a o-taded dex whch they ca hadly hedge. I etu they wll demad a sk pemum. Applcatos Hedgg / Asset Lablty Maagemet Iflato spead optos povde a eat hedge agast the sk that the dex used fo hedgg flato exposues (usually a lqud dex, such as HICPx dveges fom the bechmak dex fo the vesto. Ivestg / tadg Iflato spead optos allow vestos to expess a vew about spead volatlty sepaate fom a vew about the decto of the dex spead. July 2005 58

9. SRUCURED IFLAIO PRODUCS Cuetly, vestos have the opto to get flato-lked etus o bods va flatolked bods. Howeve, atual eceves of flato such as peso fuds ad suace compaes have potfolos cosstg of bods, equty, cedt-lked poducts, a so o. I addto to beg able to get eal bod-lke etus, they could be teested vestg poducts whch geeate eal equty o eal cedt-lked etus. he gowth of pue flato devatves, as descbed the pevous secto, has opeed the doo fo stuctued hybd poducts lked to flato. I ths secto, we cosde a selecto of flato-lked stuctues. 9.. otal etu swaps Descpto A total etu swap s a tool fo egulato ad balace sheet abtage athe tha vestg A total etu swap (RS s a cotact that allows vestos to eceve all the cashflow beefts of owg a asset wthout actually holdg the physcal asset o the balace sheets. As such, a total etu swap s moe a tool fo balace sheet abtage o egulato abtage tha fo vestmet puposes. Fst, we dscuss total etu swaps fo the case whee the asset s a flato-lked bod. Fgue 9. shows the mechacs. Fgue 9.. Mechacs of a total etu swap Dug the swap otal etu eceve Iflato-lked cashflows Lbo +/- spead otal etu paye At matuty otal etu eceve Souce: Lehma Bothes. Ay cease the maket value of the flato-lked bod Ay decease the maket value of the flato-lked bod otal etu paye otal etu swaps ca be lked to dvdual secutes o dces At tade cepto, the total etu eceve agees to make paymets of Lbo plus o mus a fxed spead to the total etu paye, etu fo the coupos pad by some flato-lked secuty. At the ed of the tem of the total etu swap, the total etu paye must the pay the dffeece betwee the fal maket pce of the asset ad the tal pce of the asset, f postve. If the dffeece betwee the fal maket pce of the asset ad the tal pce s egatve, the total etu eceve eeds to pay the dffeece to the total etu paye. otal etu swaps do ot ecessaly have to be lked to a patcula secuty. Fo example, oe may wsh to lk the total etu to a dex such as the Lehma Bothes Global Iflato-Lked Aggegate Idex, theeby ceatg what s kow as a dex swap. Lehma Bothes has seveal flato-elated dces, ad we peset the most mpotat oes Fgue 9.2. July 2005 59

Fgue 9.2. Lehma flato dces Lehma dces Global flato-lked Euo-zoe Pa-Euo US IPS Descpto Icludes all flato-lked bods by the US, the UK, Caada, Swede, Italy, Face, ad Geece. Icludes all flato-lked bod lked to the Euopea HICPx dex. Icludes all flato-lked bods excludg US IPS ad Caada RRBs. Icludes all US flato-lked bods, IPS I addto to the dces above, couty-specfc dces fo Caada, UK, Swede, Face (cludg both Fech ad Euo dces, Italy, ad Geece also exst. Souce: Lehma Bothes. Valuato aspects Pcg s detemed by the cost of fudg the hedge posto A total etu paye ca statcally hedge the total etu swap by buyg the flatolked secuty, fudg t o balace sheet, ad sellg t at tade matuty. Ideed, oe way the holde of a asset ca hedge agast chages the value of a asset s to become a paye o a total etu swap. hs meas that the cost of the tade,.e. the spead, wll maly deped o the fudg cost of the total etu paye. Whe the total etu swap s made wth espect to a flato dex, the total etu paye wll have to hedge by buyg the dex. he cost of ths wll be epeseted by the cost of buyg the dvdual bods. Howeve, typcally the bd-offe wll be lowe tha eplcatg the dvdual bods. hs s because the dex swap deale wll be moe wllg to take outght maket sks ad bass sks tha the vesto. Also, by havg a easoably balaced book dex swaps, the deale wll be able to aggegate sks ad so educe hedgg costs. Fo flato dces, the dffeece wll be easoably small as the dces cosst of faly few bods. As the dces gow costtuets, the eplcatg potfolo becomes moe expesve compaed wth the dex tems of bd-offe speads. Applcatos A vesto may be teested a total etu swap o a flato-lked secuty fo seveal easos. Fudg / Leveage As a ufuded tasacto, total etu swaps make t easy to leveage a vew o flato ethe dvdual assets o a potfolo of flato-lked bods. hey eable vestos to obta off-balace-sheet exposue to assets to whch they mght othewse be pecluded fo tax, poltcal o othe easos. Buyg ad sellg dex swaps may be moe lqud tha tadg the udelyg assets. Bd-offe speads wll usually be tghte. adg / Ivestg otal etu swaps allow a total etu paye to shot a asset wthout actually sellg t. hs mght be useful fom the pot of vew of tempoaly hedgg a flato exposue. Aothe easo mght be a expected udepefomace the shot u, whle matag cofdece ts log-u pefomace. Usg a total etu swap, oe ca ceate a sythetc stuctue wth the equed matuty. Clets ca use the dex swap to bechmak the potfolos to stadad flato dces, such as those summased Fgue 9.2. A potfolo maage ca eplcate a dex wthout cug a tackg eo. July 2005 60

9.2. Iflato-lked equty Descpto Iflato-lked equty s the equty wth a fxed matuty equvalet to a flatolked bod wth a fxed matuty. Fo example, cosde the stuato whee a peso fud has flato-lked labltes ad a substatal equty ad bod potfolo (we dscuss CDOs the ext secto. he peso fud ca buy flato-lked bods o ete to zeo-coupo swaps to match ts bod potfolo to the flato-lked labltes. Iflato-lked equty allows the peso fud to lk the equty to flato as well. he value of eal equty at matuty s gve by: R ( = I( S(, whee S( deotes the value of the equty potfolo at tme. he dea behd eal equty s smla to that of eal bods whch pay I(D (, =I( at tme. Valuato aspects Iflato-lked equty s sestve to the flatoequty coelato he payoff of flato-lked equty depeds o co-movemet betwee the flato dex cease ad the equty cease. hs co-movemet makes the valuato modeldepedet. Oe model s gve by assumg that the flato dex ad the equty ae both logomal dstbuted wth a detemstc coelato. he value s the gve by: I (0, S(0 R(0 = exp ( (, (,, (0, ρ IS u σ I u σ S u du D 0 whee σ I (u, deotes the stataeous volatlty of the beakeve efeece umbe I(0, today, σ S (u, deotes the stataeous volatlty of the fowad equty pce, that s, S(0 / D (0,, ad ρ IS (u deotes the stataeous coelato. Applcatos As fo flato-lked bods, the ma applcato of flato-lked equty s the ALM cotext. We summase some potetal uses. Hedgg / ALM Iflato-lked equty allows vestos to hedge the flato-lked labltes whle ejoyg equty-lke etus. Ivestg / adg Iflato-lked equty allows vestos to take a vew o the depedece betwee flato ad equty etus. 9.3. Iflato-lked equty optos Descpto A popula stuctue fo vestos omal space to hedge the dowsde s to combe a stock potfolo ad a (typcally AM put. hs stuctue esues that the value of a vesto s potfolo at matuty equals at least ts cuet value. hs payoff s gve by: s ( S(0 S(,0 = S(0 max(, + (0,, S( + max whee s (0,=S(/S(0- deotes the goss etu o the equty potfolo. I ths stuctue, the omal value of the equty potfolo emas at least equal to ts ogal omal value. Wth flato-lked labltes, the eal value of the equty potfolo deceases f the bechmak dex ses. Usg a flato-lked equty opto, we ca costuct a stuctue whch guaatees that the eal value of the equty potfolo emas at least equal to the ogal eal value. hs ca be costucted the followg mae: July 2005 6

Iflato-lked equty optos allow stuctues that guaatee the eal value of equty s s ( (0,, (0, = S( + S(0 max( (0, (0,,0. S( t + S(0 max he stuctue ca be eplcated by vestg the stock ad buyg a spead opto whch pays off the dffeece betwee flato ad the equty etu f postve. Valuato aspects As fo flato spead optos, flato-lked equty optos eque modellg the spead o the jot dyamcs of the dces. Euopea-style optos ca be valued usg Magabe s (978 fomula, whch s a exteso of the Black model. Applcatos Iflato-lked equty optos have a umbe of applcatos whch ae summased below. Hedgg / Asset Lablty Maagemet Iflato-lked equty optos allow vestos to guaatee the eal value a stock ad put costucto. Ivestg / adg Iflato-lked equty optos allow vestos to leveage a vew o equty ad flato coelato. 9.4. Iflato-lked cedt default swaps Descpto he CDS spead s flato-lked A cedt default swap (CDS s a blateal cotact that eables a vesto to buy potecto agast the default of a asset ssued by a specfed efeece etty. I the case of a defed cedt evet (typcally, default, the potecto buye eceves a default paymet teded to compesate agast the loss vestmet. I etu the flato buye pays a fee. hs fee may be up-fot o the fom of a egula sees of cashflows. I pcple, oe ca evsage seveal types of flato-lked cedt default swaps. We focus o a CDS o whch the potecto buye pays a eal spead. he default paymet s as usual. We cosde a example tade: a 00 mllo, 5-yea flato-lked CDS. he eal spead equals % aually. Fo coveece, we assume that flato wll equal 2% all subsequet yeas to compute pojected cashflows. he pemum cashflows ae the gve by.02 M fo yea =,,5 f o default has occued. If a default occus, the pemum cashflows cacel out. We peset some example cashflows to llustate default ad o-default sceaos Fgue 9.3. July 2005 62

Fgue 9.3. Example cashflows of 5-yea flato-lked CDS o default.02m.04m.06m.08m.0m 0-0-05 0-0-06 0-0-07 0-0-08 0-0-09.02M.04M.06M 0-0-05 0-0-06 0-0-07 Default Jauay 2008 Default wth 50% ecovey. Default paymet equal to 50M Souce: Lehma Bothes. Valuato aspects he value depeds o the flato ad default coelato A flato-lked CDS pays a eal spead wheeas a oday CDS pays a fxed spead. If we kew whch spead paymet would occu we could easly tasfom the fxed pemum cashflows to flato-lked pemum cashflows usg a flato swap. Howeve, due to the uceta cedt evet, ths s ot possble. Usg a cuve of suvval pobabltes (whch gves the pobablty that the asset has suvved to that pot tme, we ca compute the expected flato-lked cashflows. Whe defaults ae ucoelated wth flato, the value of a dvdual paymet s the dscouted value of the expected otoal tmes the expected dex ato at the tme of the paymet. 37 I the case of postve coelato betwee flato ad the cedt evet, the pemum wll be lowe tha whe thee s o coelato. I the case of egatve coelato, the value wll be hghe. he tuto behd ths s that peods of hgh flato the pemum s lkely to be pad loge the case of egatve coelato ad vce vesa. Applcatos Hedgg / ALM A flato-lked CDS povdes a atual way fo vestos wth flato-lked labltes to sell potecto o a CDS. Ulke wth fxed o omal floatg spead paymet, vestos eceve flato-lked spead paymets to match the labltes. Ivestg / adg A flato-lked CDS allows vestos to leveage a vew o cedt ad flato coelato. 37 he expectato s take wth espect to the appopate pcg measue. July 2005 63

9.5. Iflato-lked CDO Descpto he CDO spead s flato-lked A collatealsed debt oblgato (CDO s a stuctue of cedt sky secutes whose cashflows ae lked to the cdece of default a pool of debt stumets. hese debts ca be, fo example, loas, les of cedt, asset-backed secutes, copoate ad soveeg bods, ad othe CDOs (typcally deoted CDO 2. Fo moe detaled fomato o CDO ad othe cedt devatves, see O Kae (200 ad O Kae et al. (2004. he pefomace of a flato-lked sythetc CDO s lked to the cdece of default a potfolo of CDS ad flato swaps. he CDO edstbutes the cedt sk by allowg dffeet taches to take these default losses a specfc ode. hs ode s called the watefall. o see how the CDO edstbutes the cedt sk, we cosde the example Fgue 9.4. Fgue 9.4. Stuctue of a flato-lked CDO Refeece pool 3bp dex Seo tache 850m 00 ames CDS fomat 0m 00 assets = b total otoal Lehma Bothes 200bp dex 900bp dex Mezzae tache 00m Equty tache 50m cotget paymet Souce: Lehma Bothes. hs CDO s based o a efeece pool of 00 CDS, each wth 0m otoal. he cedt sk s edstbuted thee taches. he equty tache assumes the fst 50m of losses. As t s the skest tache, t s pad the hghest spead. ext we have the mezzae tache whch assumes the ext 00m of losses. It s less sky tha the equty tache, ad theefoe pays a lowe spead. Fally, we have the seo tache whch s potected by 50m of subodato. o get a dea of the sk of the seo tache, we ote that t would eque moe tha 25% of the assets to default wth a ecovey of 40% befoe the seo tache takes a pcpal loss. Cosequetly, t s pad a eve lowe spead. All speads ae flato-lked the same mae; the pemum eceved beg the fxed eal spead tmes the flato dex wth the pemum pad o the outstadg otoal. I Fgue 9.5 we gve a example of the paymets of a equty tache of a 5-yea flato-lked CDO, whch tates o Octobe 2004. July 2005 64

Fgue 9.5. Example cashflows of 5-yea flato-lked CDO equty tache Paymets o flato leg 4.59M 4.2M 4.30M 3.4M 3.48M 0-0-05 0-0-06 0-0-07 0-0-08 0-0-09 default 50% ecovey o 0M 2 defaults 50% ecovey o 0M each 5M 0M Paymets o potecto leg Due to flato the potecto ca become moe expesve ove the lfe ove the CDO We assume that ove the lfe of the tade flato equals 2%. he equty tache the above example s pad a flato-lked spead of 900bp. Fo the fst paymet ths equals 9% tmes 50m tmes.02 (comg fom the 2% flato. I yea 2 we see a default wth a ecovey of 50% whch educes the otoal to 45m esultg a floatg paymet of 45m tmes 9% tmes.02 2 (cumulatve flato up to yea 2 ad a default paymet fom the equty tache holde of 5m. o defaults yea 3 lets the floatg paymet gow wth flato. Due to two 50% ecovey defaults yea 4 the otoal o the floatg leg deceases to 35m (the floatg paymet ow equals.02 4 9% 35m= 3.4m ad the equty tache holde eeds to pay a 0m default paymet. I the last yea the floatg paymet gows wth flato. Valuato aspects he value depeds o the flato ad default coelato Wheeas a omal CDO pays a fxed omal spead, a flato-lked CDO pays a eal spead tmes a flato dex. If the otoals fo the paymets wee kow, the valuato could be smply accomplshed usg fxed flato swaps wth the appopate otoals. By the atue of the CDO, the otoals ae ukow ad the coelato of the defaults ad flato becomes a ssue fo valuato. Whe defaults ae ucoelated wth flato, the value of a dvdual paymet s the dscouted value of the expected otoal tmes the expected dex ato at the tme of the paymet. 38 I the case of postve coelato betwee flato ad defaults, the value wll be lowe tha f thee s o coelato. I the case of egatve coelato, the value wll be hghe. he tuto behd ths s that peods of hgh flato the expected otoals ae hgh the case of egatve coelato ad vce vesa. Applcatos Hedgg / ALM Iflato-lked CDOs povde a atual way fo vestos wth flato-lked labltes to vest CDOs. hey eceve the yeld pck-up of CDOs whle ecevg flato-lked cashflows to match the labltes. Ivestg / adg Iflato-lked CDOs allow vestos to leveage a vew o cedt ad flato coelato. 38 he expectato s take wth espect to the appopate pcg measue. July 2005 65

0. LEGAL, REGULAORY, AD ACCOUIG ISSUES 0.. ISDA flato devatves documetato he ISDA ecetly publshed documetato o flato deftos supplemetg the ISDA Maste Ageemets. 39 he ma ssues elate to delay ad dsupto the publcato of the flato dex. Futhemoe, t defes the most elevat dces. 0.. Delay of publcato A substtute dex s used f the efeece dex s ot publshed o tme If the efeece dex has ot bee publshed fve busess days po to the ext paymet date fo the tasacto elated to that dex, the calculato aget shall use a substtute dex level usg the followg methodology:. If applcable, the calculato aget takes the same acto to deteme the substtute dex level as that specfed the tems ad codtos of the elated bod. he elated bod, f ay, s specfed the cofmato of the tade. A elated bod s typcally specfed fo asset swaps, but ot fo flato swaps. 2. If. does ot esult a substtute dex level fo the affected paymet date the calculato aget detemes the substtute dex level as follows: substtute dex level = Base level (Latest level / Refeece level whee Base level meas the level of the dex 2 caleda moths po to the moth fo whch the substtute dex level (deftve o povsoal s beg detemed, fo example, Decembe 2005. Latest level meas the latest avalable level (deftve o povsoal fo the dex, fo example, ovembe 2005. Refeece level meas the level (deftve o povsoal of the dex 2 caleda moths po to the moth to whch the latest level s efeg, fo example, ovembe 2004. If a elevat level s publshed afte fve busess days po to the ext paymet date, o adjustmets wll be made to the tasacto. he detemed substtute efeece level wll be the deftve level fo that efeece moth. 0..2 Successo dex A eplacemet dex wll be used f the dex o loge gets publshed If, dug the tem of the tasacto, the dex sposo aouces that a dex wll o loge be publshed o aouced but wll be supeseded by a eplacemet dex specfed by the dex sposo, ad the calculato aget detemes that the eplacemet dex s calculated usg the same o smla methodology as the ogal dex, ths dex s deemed the successo dex. 0..3 Cessato of publcato If a dex has ot bee publshed fo two cosecutve moths o f the dex sposo (publshe has aouced that t wll o loge publsh the dex, the calculato aget shall deteme a successo dex fo the pupose of the tasacto usg the followg methodology:. If a successo dex has bee desgated by the calculato aget pusuat to the tems ad codtos of the elated bod, such successo dex shall be desgated a successo dex heeude. 2. If o elated bod exsts, the calculato aget shall ask fve leadg depedet deales to state what the eplacemet dex shall be. If thee o moe deales out of at 39 See 2005 ISDA Iflato Devatves Deftos o www.sda.og. July 2005 66

least fou esposes state the same dex, ths dex wll be deemed the successo dex. If, out of thee esposes, two o moe deales state the same dex, ths dex wll be deemed the successo dex. If o successo dex has bee decded followg esposes fom deales by the thd busess day po to the ext paymet date o by the date that s fve busess days afte the last paymet date (f o futhe paymet dates ae scheduled, the calculato aget detemes a appopate alteatve dex. hs alteatve dex wll be deemed the successo dex. If the calculato aget detemes that thee s o appopate alteatve dex, a temato evet occus ad both pates ae affected pates as defed the 2002 ISDA Maste Ageemet. 0..4 Rebasg the dex If a dex s ebased the ebased dex wll be used gog fowad If the calculato aget detemes that the dex has bee o wll be ebased at ay tme, the ebased dex wll be used fom the o. Howeve, the calculato aget shall make adjustmets pusuat to the tems ad codtos of the elated bod, f ay. If thee s o elated bod, the calculato aget shall make adjustmets to the past levels of the ebased dex so that ebased dex levels po to the ebase date eflect the same flato ate as befoe t was ebased. 0..5 Mateal modfcato po to paymet date If po to fve busess days befoe a paymet date a dex sposo aouces that t wll make a mateal chage to a dex, the the calculato aget shall make ay adjustmets to the dex cosstet wth adjustmets made to the elated bod. If thee s o elated bod, oly those adjustmets ecessay fo the modfed dex to cotue as the dex wll be made. 0..6 Mafest eo publcato If, wth 30 days of publcato, the calculato aget s otfed that the dex level has to be coected to emedy a mateal eo ts ogal publcato, the calculato aget wll otfy the pates of the coecto ad the amout payable as a esult of that coecto. Stoge dexato equemets fo peso fuds wll pobably boost the flato maket Euopea compaes eed to follow the ew Iteatoal Accoutg Stadads 0.2. Regulato fo peso fuds A mpotat ole the developmet of the flato devatves maket (especally Euope wll be played by the egulatoy famewoks fo peso fuds the espectve coutes. Fo stace, the ethelads, whch pobably has the most developed Euopea peso dusty, the socal pates (employes ad employees decded that most peso fuds eed to povde oly codtoal dexato. Codtoal dexato meas that peso fuds oly eed to povde dexato f they have suffcet fuds to allow ths. Ucodtoal dexato s a stoge equemet, whch a peso fud guaatees a flato-adjusted peso. Ucodtoal dexato schemes would lkely have a stog postve mpact o the flato maket as demad fo flato poducts by peso fuds would almost cetaly cease. So fa, o majo ettes have adopted a ucodtoal dexato scheme. 0.3. Accoutg stadads I 2005 the Euopea Uo adopted the accoutg stadads as set by the Iteatoal Accoutg Stadads Boad (IASB. he toducto of these accoutg stadads oblges all Euopea (cludg the UK publc compaes lsted o ay exchage the Euopea Uo to pepae the facal accouts followg the Iteatoal Facal Repotg Stadads (IFRS. I the devatves maket, all compaes ae affected va IAS-39, the IFRS stadad coveg devatves. he US equvalet to IAS-39, FAS-33, whch was toduced seveal yeas ago, has moe o less smla mplcatos fo flato devatves by US ettes. July 2005 67

IAS-39 eques all devatves to be valued o a mak-to-maket bass, whch essetally meas that they ae tasfeed to the balace sheet. Chages the mak-to-maket valuato wll theefoe affect poft ad loss accout of compaes, whch lkely ceases volatlty compaes eags statemets. Oly f a compay ca demostate that the devatve s used fo hedge accoutg puposes, wll t eed ot be value the devatve secuty o a mak-to-maket bass. Fo flato devatves, ths meas that compaes wth flato-lked cashflows eed to demostate that a flato swap s a effectve hedge fo ts flato-lked cashflows. If compaes have cashflows dectly lked to flato (e.g. supemakets ths ca be poblematc ad lkely costly. I the flato devatves maket all compaes wth defed beeft cotbutos wll also be affected by IAS-9, whch pescbes the accoutg ad dsclosue fo employee beefts (that s, all foms of cosdeato gve by a etepse exchage fo sevce edeed by employees. he pcple udelyg all of the detaled equemets of the Stadad s that the cost of povdg employee beefts should be ecogsed the peod whch the beeft s eaed by the employee, athe tha whe t s pad o payable. heefoe, compaes ae equed to mak-to-maket the value of dexato schemes. I the UK the atoal Accoutg Stadads Boad (ASB publshed ts Facal Repotg Stadad, whch oblges all publc compaes lsted o the Lodo Stock Exchage to apply smla stadads. Although some dffeeces exst betwee IAS-9 ad FRS-7, the bottom le emas the same; assets ad labltes should be valued o a maket stead of actuaal bass. he toducto of the above accoutg stadads s lkely to esult moe volatle facal statemets tha befoe. hs wll especally be the case fo compaes whose assets ad labltes ae ot well matched. hs makes flato devatves patculaly teestg fo peso fuds wth dexato schemes. By cludg flato devatves the vestmet potfolo, compaes ae able to match the flato-lked labltes moe closely. hs ca sgfcatly educe the volatlty of the facal statemets. July 2005 68

APPEDIX A.. Calculatg flato asset swap speads o-accetg otoal he asset swap selle sells the bod fo pa (assumed to equal o 00% plus accued teest. he et up-fot paymet theefoe has a value of -B MK fo the pa asset swap ad zeo fo the maket asset swap, whee B MK s the full pce of the bod the maket. If we assume that both pates to the swap ae AA-ated, these cashflows ae dscouted off the Lbo cuve. Fo smplcty we assume that all paymets ae aual ad ae made o the same dates. he beakeve asset swap spead s s computed by settg the et peset value of all cashflows equal to zeo. Fom the pespectve of the pa asset swap selle, the peset value s: whee BMK 4243 upfot paymet to puchase bod fo pa + BLIBOR ( + spv0 = 0 444 24443 ug cash flows B LIBOR = c = I ( 0, D (0, + I(0, D (0, s the value of the flato bod pced off the Lbo cuve wth a eal coupo equal to c, ad PV0 = δ D (0, = s the peset value of a bp auty wth the same schedule as the floatg sde of the asset swap, pced off the Lbo cuve. We used the fact that the value of a steam of Lbo paymets equals ad the spead esults a auty of s. Solvg gves us the esult: B LIBOR B s = MK, PV0 I the case of a maket asset swap, we eed to solve moe o less the same equato. he value fo the asset swap selle s gve by: B B + B B ( + spv0 0 Floatg paymets puchase paymet to asset etu 4444 24444 3 MK MK 4243 4 LIBOR MK = 44244 3 fo BMK ug cash flows whch esults a asset swap spead s equallg: Accetg otoal BLIBOR B s = PV0 B I the case of a accetg otoal, the detemato of the asset swap spead s moe tedous, but coceptually ot much hade. We susta the same assumptos as above, wth the excepto that the otoals multplcatvely cease by (+a, whee a deotes the accetg facto. he value fo the asset swap selle case of a pa accetg asset swap s gve by: MK MK. July 2005 69

BMK 4243 puchase Upfot paymet to asset etu fo pa + j L [ D (0, (0, ] + ( + (0, j D j s a D j = 0 j B LIBOR ( + a j= j = 44444444444 244444444444 3 Floatg paymets 444444444444 24444444444444 3 Rug cash flows whee B LIBOR s as befoe ad the tem betwee oud backets deotes the value of the accetg floatg leg plus a spead. Solvg fo the asset swap spead, s, gves: s B + B j = MK LIBOR j = = ( + a ( + a j [ D (0, D (0, ] j D (0, j j j Oe ca solve fo the spead of a accetg maket value asset swap a smla fasho. Ealy edempto flato asset swap If thee s o ealy edempto, the value of the otoal cease pad at matuty s gve by: I( I( j I( V (0 = D (0, E0 D (0, E0 I( 0 = j = I( 0 j = D (0, I(0,, I(0, whee the supescpt dcates that we take the expectato wth espect to the umeae assocated wth the dscout bod payg at. ow let us cosde oe of the ealy edempto paymets. he paymet that eeds to be made at tme s such that t s equvalet to ecevg (I( -I( - /I( 0 at. We kow tutvely that: V (0 = D (0, E = D (0, E 0 0 I( I( I( 0 I( I( I( 0 D (, as the ealy paymet s just the peset value at tme of the dex cease at matuty. A.2. Estmato of seasoal pattes Seasoal dummy model I ths secto we dscuss a method of computg seasoal effects. he model, whch we call the seasoal dummy model, s gve by: dext y log t dex t = 2 = β d + ε whee y t+ deotes the flato of the dex measued as the logathm dffeece of the flato dex ad t s measued moths. he d s, =,,2 deote mothly dummes ad ae defed as: d = 0 fo moth elsewhee t fo =,,2, whee Jauay epesets moth,, ad Decembe epesets moth 2. So d = fo Jauay flato, etc. ote that the β s do ot dectly epeset seasoal effects, but expected flato fo a patcula moth. Estmato of ths model s staghtfowad ad. 0 July 2005 70

July 2005 7 ca be pefomed usg oday least squaes (OLS. We subtact expected aual flato (the aveage of the βs to get the seasoal effects. amo-seats model I the seasoal dummy model teated befoe, we dd ot take to accout ay possble mea-eveso effects the flato umbes. We ca geealse ths appoach usg moe advaced tme-sees aalyss. Hee we descbe a moe advaced method, the socalled amo-seats model, whch s used by Euostat to calculate ts seasoally adjusted dces. he amo-seats model sepaates flato to fou compoets: ( explaatoy vaabes; (2 ted; (3 seasoal; (4 egula compoet. hs allows us to wte the flato y t as: t t t t t S x y ε β + + + = ' whee x t deotes the explaatoy egesso vaables kow at tme t, t deotes the ted fo peod t, S t deotes the seasoal ad ε t deotes the egula compoet. he ted, seasoal ad egula compoets ae uobseved compoets ad modelled as ARIMA pocesses. I geeal, ARIMA models ty to expla tme sees behavou usg autoegessve ad movg aveage tems, ad fo a tme sees z t ca be epeseted as: = = + Θ + = p j t j t j p j j t j z t z ξ ξ θ whee the fst sum deotes the autoegessve compoets ad the secod sum the movg aveage compoets. Fo example, we ca have: = + + + + = + = j t j t t t t t S S ω η (0, ~ (0, ~ 2 2 ω η σ ω σ η t t ote that ths specfcato, seasoals oly add up to a expected zeo o a aual bass. Assumg that ε t s Gaussa dstbuted, ths model ca be estmated by maxmum lkelhood. See Bodolo ad Ga (2005 fo a moe detaled aalyss of seasoalty flato dces. A.3. Duato ad covexty aalyss Duato ad covexty ae the bead ad butte of omal bod tades. I ths secto we exted these fst- ad secod-ode sk measues to flato secutes. We ecall the value of a flato-lked bod (fo coveece we have set I(t B = tated at t B 0 ad matug at : c s y b s y b c B I + + + + + + + + + = = (0, (0, (0, (0, (0, (0 wth =s=0. We wte the chage the value of a flato-lked bod usg a aylo expaso. c c c c c c c s s s B I d B I d s s s B I d B di s s s B di t B di B I = = + = + = + = + = + + Δ = = = = = 0 0, (0, (0 0 (0, (0 0 (0, (0 0 (0, (0 0 (0, (0 (0, (0 (0, (0 2 2 2 2 2 2 2 2 2

whee s dcates the chage y (0, ad the chage b(0,. he fst tem stads fo the oll-dow of the flato-lked bod ad s gve by the chage the flatolked bod f the matuty deceases by a day ad the omal ad flato cuves ema uchaged. he secod le gves the fst-ode chages the flato-lked bod pce f the omal ad/o flato cuve move. he last two les gve the secod-ode chages. As dscussed Secto 6.2, the omal PV0 s gve by a paallel shft the omal tem stuctue, that s, s =s., c I(0, D (0, I(0, D (0, PV0 = I(0 B (0, s= 0= c s + y (0, + y (0, = he flato PV0 of flato coupo bods s gve by a paallel shft the flato tem stuctue, that s, =, c I(0, D (0, I(0, D (0, If PV0 = I(0 B (0, 0= c +. = + b(0, + b(0, omal covexty s gve by: omal Covexty ad the flato covexty s gve by: = 2 c I(0, D (0, = I(0 B (0, 0 ( 2 s= = c 2 s ( + y (0, = + ( I(0, D (0,, 2 ( + y (0, 2 c I(0, D (0, Iflato Covexty = I(0 B (0, 0 ( 2 = = c 2 ( + b(0, = + ( I(0, D (0, 2 ( + b(0, Ulke oe-cuve poducts, we eed to take to accout the depedece betwee the moves the cuves fo two-cuve poducts. he coss covexty fo flato-lked bods s gve by: 2 c Coss covexty = I(0 B (0, s= = 0= c s = 2 2 I(0, D (0, ( + y (0, ( + b(0,. I(0, D (0, ( + y (0, ( + b(0, We see that the coss covexty s smla to the egatve of the omal ad flato covexty. 40 As the omal ad flato PV0s also have opposg sgs, we see that a equal move the omal ad flato cuve oly has a small mpact o the value of the flato bod. As dscussed Secto 6.2, moves the omal ad flato cuves ae typcally coelated. Fo stace, oe ca assume that =ρ s ad apply the above techques. A.4. Valuato of fowad statg zeo-coupo swaps he cux of valug fowad statg flato poducts s computg the expectato of the dex ato ude the appopate umeae.. whee: I(, I(0, = G(0,, I(, I(0, E0 40 It would exactly equal the egatve of the omal ad flato covexty f we had computed covexty tems of cotuously compoudg ates. July 2005 72

= G(0, exp σ I ( u, γ ( u;, 0 du deotes the fowad stat covexty tem, ad the expectato s take ude the - fowad matgale measue. he covexty coecto o peod-o-peod swaps s theefoe a fucto of the volatlty of the beakeve flato ate wth the matuty of the dex stat date, σ I (u, -, the volatlty of the eal fowad ate, γ (u; -,, ad the stataeous coelato. 4 he fowad statg zeo-coupo swap ate s ow gve by: /( I(0, b (0;, = (0,. (0, G. I Because a peod-o-peod flato swap s a potfolo of fowad statg flato swaps, ts fxed ate ca be computed by valug the udelyg flato legs of the fowad stat flato swaps ad solvg fo the beakeve ate o the fxed leg. A.5. Valuato of flato caps ad floos I ths appedx we peset a valuato fomula fo caps ad floos o the fal dex value. he fal payoff of a caplet s gve by: I( I(, max K,0 = max K,0. I( (, I Ude the assumpto that I(, /I( -, - has a logomal dstbuto: V 0 = A (0 Φ( h A ( h, ( + 2Φ whee ad h ± I(0, A (0 = G(0, I(0, A 2 = + K ( A (0 / A (0 log 2 ± = σ σ 2 2. I the Gaussa HJM model of Jaow ad Yldm(2003 σ s gve by: 2 σ = σ ( u; σ ( u; du + 0 σ ( u; July 2005 73 du I the usual mae a valuato fomula ca be deved fo floolets. Caps ad floos ca the be valued as a potfolo of caplets ad floolets, espectvely. A.6. Valuato of spead optos ad flato-lked equty optos I ths appedx we peset a valuato fomula fo spead optos, such as flato spead optos ad the flato-lked equty optos. I geeal, these ca be thought of as a opto to exchage oe asset, say X fo aothe, say Y. Ealy wok o these types of devatves s peseted Magabe (978 ad ca also be foud stadad textbooks, such as, Hull (2003. We cosde stuctues wth payoffs of the fom: { ( X (,0} max Y. Ude the assumpto that pocesses X ad Y ae logomal dstbuted, wth detemstc volatlty fuctos ad coelato, σ X (u, σ Y (u, ad ρ(u, espectvely, we have the followg pcg fomula, kow as Magabe s fomula: 4 Pecsely, γ (u; -, deotes the volatlty of log(d (t, - /D (t,,.e. the logomal volatlty of the fowad eal dscout facto. 2 / 2.

whee ad V ( 0 = Y (0 Φ( d+ X (0 Φ( d, d ± ( Y (0 / X (0 log ± = σ σ 2 2 XY 2 2 2 2 σ XY = σ X ( u + σ Y ( u 2ρ( u σ X ( u σ Y ( u du.. 0 he pocesses X ad Y ae gve by I A (0D (0, ad I B (0D (0,, that s, flato-lked zeo-coupo bods o dex A ad B, espectvely. Fo the flato-lked equty opto, X ad Y ae gve by I(0D (0, ad S(0, that s, a flato-lked zeo-coupo bod ad the udelyg equty potfolo. A.7. Valuato of flato swaptos I ths appedx we peset valuato fomulas fo fxed ad floatg flato swaptos. he fal payoff of a fxed eceve flato swapto s gve by the dffeece of the beakeve ate, b(; s, e, fo a flato swap statg at s ad matug at e ad a pe-ageed stke, K, tmes the appopate PV0 fo the swap peod,.e.: V = max{ ( b( ;, KPV0( ;,,0}. ( s e s e Ude the assumpto that b(; s, e has a logomal dstbuto ude the measue assocated wth the PV0 umeae, the value of the flato swapto today s gve by: whee [ b(0;, Φ( h K ( h ] PV0(0;,, V ( 0 = s e s e + Φ ( b(0;, / K log s e ± h± = σ σ 2 2 I the same mae, the valuato fomula fo the fxed paye flato swapto ca be deved. Fo floatg flato swaptos o eal swaptos, oe eeds to pefom a umecal tegato assumg that both the omal swap ate ad the flato swap ate ae logomally dstbuted. We wll peset the valuato fomula ude the assumpto that both the omal ad flato swap ates ae omally dstbuted ude the swap measue. he payoff at matuty s gve by: V = max( b( ;, + K S( ;, PV0( ;,, [ ]. ( s e s e s e whee S(; s, e deotes the teest ate swap ate. Assumg that both b(; s, e ad S(; s, e have a omal dstbuto ude the measue assocated wth the PV0 umeae, we kow that the dffeece s also omally dstbuted. he value today s gve by: whee ad [ b(0;, + K S(0;, ] Φ(, V ( 0 = σ φ( d + d s b( 0; s, e + K S(0; s, e d = σ e 2 2 [ σ ( u + σ ( u 2ρ( u σ ( u σ ( u ], 2 σ = b S b S du 0 whee σ b (u deotes the stataeous volatlty of the beakeve flato ate, σ S (u deotes the stataeous volatlty of the teest ate swap ate, ad ρ(u the stataeous coelato. s e July 2005 74

REFERECES Bockwell, P. ad R. Davs (99: me Sees: heoy ad Methods, Spge-Velag, Bel Hedelbeg ew Yok. Bodolo, A. (2005: Iflato Bods Explaed: Masteg Makets & Mechacs, Fxed Icome Stateges Lehma Bothes. Bodolo, A. ad G. Ga (2005: Seasoalty Pce Idces, Fxed Icome Stateges Lehma Bothes. Campbell, J.Y. ad R.J. Shlle (996: A Scoecad fo Idexed Govemet Debt Be S. Beake ad Julo Rotembeg eds, atoal Bueau of Ecoomc Reseach Macoecoomcs Aual 996, MI Pess: Cambdge, MA, pp. 55-97, 996. Deaco, M., Dey, A., ad D. Mfedeesk (2004: Iflato-dexed Secutes, 2 d edto, Wley. Foes, S., Peat, A., ad G. Peacch (997: Reducg the Cost of Govemet Debt: he Role of Iflato-lked Bods M. De Cecco, L. Pecch ad G.Pga eds, Maagg Publc Debt: Iflato-lked Bods heoy ad Pactce, Chelteham, UK, pp. 93-6. Gog F., ad E. Remoloa (996: Iflato Rsk the US Yeld Cuve: he Usefuless of Idexed Bods, Fedeal Reseve Bak of ew Yok Reseach Pape o. 9637. Hull, J. (2002: Optos, Futues ad Othe Devatves, 5 th edto, Petce-Hall. Jaow R. ad Y. Yldm (2003: Pcg easuy Iflato Potected Secutes ad Related Devatves usg a HJM model, Joual of Facal ad Quattatve Aalyss 38, pp. 337-358. O Kae, D. (200: Cedt Devatves Explaed, Stuctued Cedt Reseach Lehma Bothes. O Kae, D., ald M, Gaapat S., Bed A., Pedese C., Schlögl, L. ad R. Mashal (200: he Lehma Bothes Gude to Exotc Cedt Devatves, Stuctued Cedt Reseach Lehma Bothes. Magabe, W. (978: he Value of a Opto to Exchage Oe Asset fo Aothe, Joual of Face 33, pp. 77-86. July 2005 75

GLOSSARY BP (Buo Poleal del esoo A BP s a flato-lked bod ssued by Italy ad lked to a Euopea CPI dex, the HICPx. Bueau of Labo Statstcs (BLS US statstcal offce that s esposble fo the publcato of the US CPI dex. CPI dex Cosume pce dex. Both used as a geeal tem ad specfcally fo the US maket. Euostat Euopea statstcal offce esposble fo the publcato of the Euopea HICP ad HICPx dces. FAS 33 he US accoutg famewok fo all facal devatves, whch came to effect Jue 2000. It eques compaes to mak-to-maket the devatve postos ad to post gas o losses to eags. FRCPI he Fech atoal CPI as calculated by ISEE. hs CPI excludes tobacco. HICP / HICPx Hamozed Idex of Cosume Pces. HICPx stads fo HICP excludg tobacco. he HICP dces ae calculated by Euostat. IAS-9 Iteatoal Accoutg Stadad 9, the secto the ew Euopea accoutg stadads o employee beefts. It eques compaes to ecogse employee beefts at the tme they ae eaed, ot whe they ae payable. hs eques peso fuds wth defed beeft schemes to mak-to-maket the postos. IAS-39 Iteatoal Accoutg Stadad 39, the secto the ew Euopea accoutg stadads o devatve tasactos. It eques compaes to value devatves o a mak to maket bass. ISEE Isttut atoal de la Statstque et des Etudes Ecoomques, the Fech atoal sttute of statstcs ad ecoomc studes. ISDA Iteatoal Swaps ad Devatves Assocato. Iflato swap A flato swap s a blateal cotact volvg the exchage of flato-lked paymets fo pedetemed fxed o floatg paymets. It s typcally used to hedge flato sk. We call the flato swap a paye flato swap f the holde pays flato o the swap ad a eceve flato swap f the holde eceves flato. Iflato asset swap A combato of the puchase of a flato-lked bod ad ety a (usually offmaket floatg flato swap. July 2005 76

Iflato asset swap spead he spead wth espect to the Lbo ate eceved by the asset swap buye a flato asset swap. Iflato cap A opto that povdes a cashflow equal to the dffeece betwee flato ad a peageed stke ate f ths dffeece s postve. Iflato floo A opto that povdes a cashflow equal to the dffeece betwee a pe-ageed stke ate ad flato f ths dffeece s postve. Iflato futue Iflato futues ae futues o the flato dces (US CPI ad HICPx ad ae taded at the Chcago Mecatle Exchage (CME. Iflato paye A flato paye s a paty whch acts as a paye the flato maket. ypcally, flato payes have a come steam lked to a flato dex. he typcal example s a soveeg whose tax eveues gow wth sg flato. Payg flato the flato maket allows them to smooth the eal come. Iflato eceve A flato eceve s a paty whch acts as a eceve the flato maket. ypcally, flato eceves have a lablty steam lked to a flato dex. he typcal example s a peso fud whose pesos ae lked to a flato dex. Recevg flato the flato maket povdes them wth a atual hedge agast the flato sk. Iflato swapto A opto to ete to a flato swap (ethe zeo-coupo o peod-o-peod. he swapto s deoted a paye swapto whe t gves the ght to ete to a paye flato swap ad a eceve swapto whe t gves the ght to ete to a eceve swapto. Iteest ate swap A blateal devatve cotact volvg the exchage of fxed ate paymets fo floatg ate paymets typcally lked to a patcula Lbo ate. ypcally used to hedge teest ate sk. Lbo he Lodo Ite-Bak Offeed Rate. hs s a teest ate at whch hghly ated (typcally AA-ated baks ca boow. It s calculated by pollg 6 baks o a daly bass (though the Lodo baches to deteme the ate at whch they ca boow fo vaous tems ad cueces. Fo each tem ad cuecy the eceved ates ae aked, the top ad bottom fou ae deleted, ad a aveage of the emag eght s take. LPI Lmted Pce Idex. atoal statstcs he UK statstcal offce esposble fo the publcato of the RPI ad RPIX dces. July 2005 77

OA / (Oblgatos assmlables du éso A OA s a flato-lked bod ssued by Face ad lked to a Fech CPI dex, FRCPI. A OA s a flato-lked bod ssued by Face ad lked to a Euopea CPI dex, the HICPx. PV0 Peset value of oe bass pot. Also efeed to as PVBP ad auty. July 2005 78

SUMMARY OF OAIO AD DEFIIIOS he followg otato s used the documet. I geeal we use sub-dces of to dcate omal ad sub-dces of to dcate eal. D (0, deotes the value at tme t of a omal dscout bod wth a otoal of omal ut (eg, euo matug at tme omal uts (eg, euos. D (0, deotes the value at tme t of a eal dscout bod wth a otoal of eal ut matug at tme eal uts. I(t deotes the (daly efeece umbe at tme t. he daly efeece umbe s lked to a flato dex (fo example, the HICPx dex. It gves the umbe of omal uts pe flato dex ut, ad ca be see as a exchage ate fom the omal to the eal ecoomy (I(t uts of omal value coespod to ut of eal value. I(t s usually lagged a few moths. Fo stace, the daly efeece umbe at Apl 2004 deotes the Jauay 2004 CPI level the euo maket. 42 ote the dffeece betwee flato dex values ad daly efeece umbes. Iflato dex values ae deoted by eg, HICPx(Ja04 ad oly exst fo whole moths, wheeas daly efeece umbes exst fo each day ad ae deoted by eg, I(5/0/04. (0, deotes the aualsed flato betwee 0 ad. (0, s elated to the flato daly efeece umbes by: / I( (0, =, (0 I whee s measued yeas. y k (0,, k {,} deotes the (aual compoudg zeo-yeld o a dscout bod ad s defed by: / y k (0, = Dk (0,. B c (0, deotes the tme t value (dty pce of a coupo-beag bod the omal ecoomy wth matuty = ad face value, that s: c B (0, = cd (0, + D (0, = B c (0, deotes the tme t value of a eal coupo-beag bod (the coupo s sometmes efeed to as the eal-ate coupo the eal ecoomy wth matuty = ad face value, that s: c B (0, = cd (0, + D (0, = y c k (0,, k {,} deotes the (aual compoudg yeld to matuty o a bod ad s (mplctly defed by: c c Bk (0, = +. c c ( + y (0, ( + y (0, = k k 42 Fo otatoal coveece, we wll geeal deote the flato dex by CPI. Depedg o the maket we dscuss ths could mea HICPx, RPI, ad so o. July 2005 79

he vews expessed ths epot accuately eflect the pesoal vews of Jeoe Kekhof, the pmay aalyst(s esposble fo ths epot, about the subject secutes o ssues efeed to hee, ad o pat of such aalyst(s compesato was, s o wll be dectly o dectly elated to the specfc ecommedatos o vews expessed hee. Ay epots efeeced hee publshed afte 4 Apl 2003 have bee cetfed accodace wth Regulato AC. o obta copes of these epots ad the cetfcatos, please cotact Lay Pdyck (lpdyck@lehma.com; 22-526-6268 o Valee Moch (vmoch@lehma.com; 44-(0207-02-8035. Lehma Bothes Ic. ad ay afflate may have a posto the stumets o the Compay dscussed ths epot. he Fm s teests may coflct wth the teests of a vesto those stumets. he eseach aalysts esposble fo pepag ths epot eceve compesato based upo vaous factos, cludg, amog othe thgs, the qualty of the wok, fm eveues, cludg tadg, compettve factos ad clet feedback. hs mateal has bee pepaed ad/o ssued by Lehma Bothes Ic., membe SIPC, ad/o oe of ts afflates ( Lehma Bothes ad has bee appoved by Lehma Bothes Iteatoal (Euope, authosed ad egulated by the Facal Sevces Authoty, coecto wth ts dstbuto the Euopea Ecoomc Aea. hs mateal s dstbuted Japa by Lehma Bothes Japa Ic., ad Hog Kog by Lehma Bothes Asa Lmted. hs mateal s dstbuted Austala by Lehma Bothes Austala Pty Lmted, ad Sgapoe by Lehma Bothes Ic., Sgapoe Bach. hs mateal s dstbuted Koea by Lehma Bothes Iteatoal (Euope Seoul Bach. hs documet s fo fomato puposes oly ad t should ot be egaded as a offe to sell o as a solctato of a offe to buy the secutes o othe stumets metoed t. o pat of ths documet may be epoduced ay mae wthout the wtte pemsso of Lehma Bothes. We do ot epeset that ths fomato, cludg ay thd paty fomato, s accuate o complete ad t should ot be eled upo as such. It s povded wth the udestadg that Lehma Bothes s ot actg a fducay capacty. Opos expessed hee eflect the opo of Lehma Bothes ad ae subject to chage wthout otce. he poducts metoed ths documet may ot be elgble fo sale some states o coutes, ad they may ot be sutable fo all types of vestos. If a vesto has ay doubts about poduct sutablty, he should cosult hs Lehma Bothes epesetatve. he value of ad the come poduced by poducts may fluctuate, so that a vesto may get back less tha he vested. Value ad come may be advesely affected by exchage ates, teest ates, o othe factos. Past pefomace s ot ecessaly dcatve of futue esults. If a poduct s come poducg, pat of the captal vested may be used to pay that come. Lehma Bothes may, fom tme to tme, pefom vestmet bakg o othe sevces fo, o solct vestmet bakg o othe busess fom ay compay metoed ths documet. 2005 Lehma Bothes. All ghts eseved. Addtoal fomato s avalable o equest. Please cotact a Lehma Bothes etty you home jusdcto.