Return Calculation of U.S. Treasury Constant Maturity Indices


 Magdalen Briggs
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1 Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or in par wihou he prior wrien consen of Morningsar Inc is prohibied
2 Conen Inroducion 3 Indices wih One or More Years o Mauri 4 Assupions 4 Reurn Forula 5 Price Forula 7 MonhEnd Pricing 8 Calculaion Seps and Exaple 9 Indices wih Less han One Year o Mauri 0 Assupions 0 Forula Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
3 Inroducion The Federal Reserve Board publishes a group of Treasur bond ineres raes of various auriies These are naed Treasur Consan Mauriies he bes known being he TenYear Treasur Consan Mauri The ineres rae also known as he ield of he TenYear Treasur Consan Mauri is ofen used as a reference poin in valuaion of oher pes of bonds such as corporae unicipal ec The ields are inerpolaed b he US Treasur fro he dail ield curve This curve which relaes he ield on a securi o is ie o auri is based on he closing arke bid ields on acivel raded Treasur securiies in he overhecouner arke These arke ields are calculaed fro coposies of quoaions obained b he Federal Reserve Bank of New York The consan auri ield values are read fro he ield curve a fixed auriies currenl one hree and six onhs and one wo hree five seven 0 0 and 30 ears All consan auri ields are quoed on a ieldoauri basis regardless of auri and he da coun is based on acual over 365 or 366 das a ear For benchark coparison and various oher purposes knowing he ield is ofen insufficien and he reurn inforaion is required However based on he naure of he ield being inerpolaed fro he ield curve insufficien inforaion exiss o ake a precise calculaion of reurns possible bu approxiaions can be esiaed b aking assupions This ehodolog addresses he assupions and forulas used in calculaing he Toal Reurn Capial Appreciaion and Incoe Reurn This docuen is divided ino wo secions o separael address indices wih one or ore ears o auri fro hose wih less han one ear o auri Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
4 Indices wih One or More Years o Mauri Assupions Following are he assupions ha Morningsar akes in he reurn calculaion of he US Treasur Consan Mauri indices wih one or ore ears o auri: Each index consiss of a single coupon bond A he beginning of each onh a bond is purchased a he prior onhend price and dail reurns in he onh reflec he change in dail valuaion of his bond 3 Coupon is paid on he onhend da of ever six onhs fro he purchase da 4 Each bond is rading a par upon purchase 5 The ield curve is fla a he desired ie o auri Four facors drive a coupon bond s price and he are he ield he ie o auri he coupon paens and he redepion or face value The coupon raes of hese indices are no provided b he Federal Reserve Board and we assue ha he coupon rae is he sae as he ield b assuing ha he bond is rading a par B definiion a bond ha is rading a par is priced a 00 and is ield is he sae as he coupon rae These indices ields a he end of he holding period are no available For exaple a he end of a oneonh holding period a bond ha had one ear o auri a he beginning of he holding period now has onhs o auri and he Federal Reserve Board does no publish he ield of a onh bond B assuing ha he ield curve is fla a his segen he ield of a newl published oneear bond is used as he ield of he old bond ha has onhs lef o auri These assupions are refleced in forula [4] The price of a bond is has wo ajor coponens in he forula The firs coponen reflecs he discouned face value of a bond and he second coponen represens he presen value of coupon paens The ield of he bond on he purchase dae is se equal o he coupon rae in he second coponen of forula [4] so ha he bond is a par upon purchase Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
5 Indices wih One or More Years o Mauri (coninued) Reurn Forula A bond is purchased a he beginning of each onh a he prior onhend price as saed in he Assupions This bond s price is racked dail during he onh o arrive a he dail oal reurn of he bond A he end of he onh his bond is sold and a new bond is purchased for nex onh The purchase dae of he bond denoed as p is he prior onhend of he desired onh For exaple o calculae he price of he bond on an da in he onh of Januar 008 he purchase dae of he bond is Deceber The auri dae denoed as reflecs he auri of he index a purchase For exaple for he enear consan auri index he bond purchased on Deceber has a auri dae of Deceber 3 07 In he forulas below a bond is idenified b is auri dae ) [] TR ) p ) [] IR ) p [3] CA TR IR Where: TR oal reurn for he holding period fro o IR incoe reurn for he holding period fro o CA capial appreciaion also known as price reurn for he holding period fro o P ( ) price of he bond wih auri dae ield a ie See forula [4] Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
6 Indices wih One or More Years o Mauri (coninued) Reurn Forula (coninued) Noe: p < ( p + one onh) In oher words p denoes he purchase dae of he bond and boh and us be wihin he oneonh holding period of his bond For exaple for he bond ha is purchased on Deceber can be an da in he onh of Januar 008 And can be on or afer Deceber and before If is no a business da is ield is he ield of he prior business da Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
7 Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied Indices wih One or More Years o Mauri (coninued) Price Forula [4] N S D p S D N P ) ( [5] c D [6] p c S Where: ) ( P price of he bond wih auri dae ield a ie Yield also known as ineres rae expressed in decial fora N auri of he bond expressed in nuber of ears D nuber of das beween ie and he nex coupon dae of he bond S nuber of das in he coupon period in which ie falls which is he firs coupon period p coupon rae expressed in decial fora and i is he ield of he bond on is purchase da p c da of firs coupon paen which is he onhend da six onhs fro purchase da p
8 Indices wih One or More Years o Mauri (coninued) MonhEnd Pricing A onhend here are wo differen bonds because his is he da when he old bond ha has been held during he onh is sold and a new bond is purchased for nex onh s use Therefore a onhend here are wo prices one represening each bond For exaple on /3/008 he bond held during Januar 008 is sold Le us idenif his bond wih is auri of and is price is calculaed using a coupon rae ha is he ield on is purchase dae of /3/007 This price is used as he nueraor and he price of bond on /30/008 serves as he denoinaor in calculaing he dail reurn for /3/008 In addiion on /3/008 a new bond is purchased o be held during Februar 008 and le us idenif his bond wih is auri of This bond s price is 00 on /3/008 because i is a par bond This price serves as he denoinaor o he price of bond on //008 in order o produce he dail reurn for //008 Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
9 Indices wih One or More Years o Mauri (coninued) Calculaion Seps and Exaple The following exaple calculaes he reurns of he OneYear US Treasur Consan Mauri index assuing ha he holding period is one da Reurn is calculaed for Januar o illusrae an exaple in a leap ear Tie is /5/008 ie is /4/008 purchase da p is /3/007 and he nex coupon paen da c is on he onhend da ha is six onhs fro purchase da and is 6/30/008 The seps are as follow: Obain he ield of he OneYear US Treasur Consan Mauri index on /3/007 /4/008 and /5/008 Calculae ) using forula [4] b plugging in he /5/008 ield as enering N D 67 S 8 and using he /3/007 ield as p 3 Calculae ) using forula [4] b plugging in he /4/008 ield as enering N D 68 S 8 and using he /3/007 ield as p 4 Calculae he /5/008 oal reurn using equaion [] b plugging in ) and ) fro seps # and #3 above 5 Deerine p ) using forula [4] b plugging in he /3/007 ield as enering N D 67 S 8 and using he /3/007 ield as p 6 Deerine p ) using forula [4] b plugging in he /3/007 ield as enering N D 68 S 8 and using he /3/007 ield as p 7 Deerine he /5/008 incoe reurn using forula [] b plugging in p ) and p ) fro seps #5 and #6 above 6 Calculae he /5/008 capial appreciaion using forula [3] b plugging in he oal reurn and incoe reurn derived fro seps #4 and #7 above Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
10 Indices wih Less Than One Year o Mauri Assupions Following are he assupions ha Morningsar akes in he reurn calculaion of he US Treasur Consan Mauri indices wih less han one ear o auri: Each index consiss of a single bill ha does no pa coupons A he beginning of each onh a bill is purchased a he prior onhend price and dail reurns in he onh reflec he change in dail valuaion of his bill 3 The ield curve is fla a he desired ie o auri These indices ields a he end of he holding period are no available For exaple a he end of a oneonh holding period a bill ha had six onhs o auri a he beginning of he holding period now has five onhs o auri and he Federal Reserve Board does no publish he ield of a fiveonh bill B assuing ha he ield curve is fla a his segen he ield of a newl published fiveonh bill is used as he ield of he old bill ha has five onhs lef o auri Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
11 Indices wih Less Than One Year o Mauri (coninued) Forula ) [7] TR ) 00 [8] P ( ) ( ) / d + Where: TR oal reurn for he holding period fro P ( ) price of he bond wih auri dae ield a ie Yield also known as ineres rae expressed in decial fora auri dae of he bill ie d nuber of das beween he purchase dae and he onhend da six onhs fro he purchase da (ie his is he nuber of das in he ½ ear fro he bond s purchase) Noe: p < ( p + one onh) In oher words p denoes he purchase dae of he bill and boh and us be wihin he oneonh holding period of his bill For exaple for he bill ha is purchased on Deceber can be an da in he onh of Januar 008 And can be on or afer Deceber and before If is no a business da is ield is he ield of he prior business da This pricing forula is differen fro he discoun forula ofen associaed wih bills This is because he Federal Reserve Board convers he ield of bills quoed in discoun fora o he couponequivalen for prior o consrucing he US Treasur Yield Curve so ha ields on he enire curve are quoed on a ieldoauri basis Morningsar Reurn Calculaion of US Treasur Consan Mauri Indices Sepeber Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion or ranscripion b an eans in whole or par wihou he prior wrien consen of Morningsar Inc is prohibied
Morningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
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