T = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are :

Transcription

1 Bullets bods Let s descrbe frst a fxed rate bod wthout amortzg a more geeral way : Let s ote : C the aual fxed rate t s a percetage N the otoal freq ( 2 4 ) the umber of coupo per year R the redempto of captal the for a exactly years bod the cash flow at dates : T /freq T 2/freq T /freq T (umber of cash flows freq ) are : 2 freq C/freq N f < C/freq N + R f Schedule Example wth 0 freq - -

2 A fxed rate bullet bod s a bod wth costat coupo whose captal s redeemed at par : R N I geeral : last cash flow last coupo + redempto of captal redempto of captal otoal of the bod f bod redeemed at par). To descrbe the schedule of a fxed rate bullet bod oe just eed the aual fxed rate ad the frequecy of coupos. I the rest of the documet we wll cosder oly bod wth a otoal of ad redempto at par : N R Ths s the market practce for probably 99% of fxed rate costat otoal bod (o amortzg). Such bods wll be called bullet bod. The defto gve ca also be used for more geeralzed bod wth exactly same formula ad very smlar coclusos

3 Yeld for a seres of cash flows Notatos The curret market date s T. We have a seres of postve cash flow F F 2 L F at creasg dates T T 2 L T wth T > T. We wll use the otato t t 2 L t for the year fractos correspodg to dates T T 2 L T : T o For example T wll be oted t ad so o 365 The year fractos are always calculated startg from curret date T ; we wll use the same otato for ay date T obvously dt / 365 dt Defto of aual yeld The aual yeld of a seres of determstc ad postve cash flow T T2 L (also called teral rate of retur) s defed by : T F F2 L F at dates F ( + r) t Market _ PV I other words r s the uque dscout rate that matches the dscouted PV at a uque rate ad the market preset value. It ca be show easly that r s uque because all the cash-flow are postve. r ca be foud a few teratos by usg a Newto-Raphso procedure (very effcet because the PV fucto s covex). We do t expla how the market calculate the PV! It s a chcke ad egg (or loop) story The market practce s to use yelds to quote bod the a zero-coupo curve ca be defed For swaps aga the practce s of course to quote valla swaps ad to derve a zero-coupo curve

4 Yeld for a bullet bod Yeld for aual bod The yeld (taux actuarel Frech) of a bullet bod wth oe coupo by year ad rate of coupo C s the defed by : C + + C t ( + r) ( + r) t Market _ PV Market_PV s by defto the market preset value of the bod. Of course oe ca say the yeld s derved from the market value or the market value s derved from the yeld. It s well kow ad easy to demostrate that f t t 2 2 L t ad C r C + + C ( + C) ( + C) I other words a (exactly) years bod wth coupo C ad yeld C s at par (prce otoal). Yeld for sem-aual quarterly bods For a bullet bod wth more tha a coupo per year the market practce s to use yeld to maturty defed by / C / freq + C / freq freq t ( + r / freq) ( + r / freq) + freq t Market _ PV - 4 -

5 If freq 2 we have : C / 2 + C / 2 + Market _ PV ; 2 r s the called a sem-aual yeld. 2 t ( + r / 2) ( + r / 2) t The aga for freq 2 f r C C / C / 2 ( + C / 2) ( + C / 2) The relato betwee the sem-aual yeld r ad the aual yeld r s : For a quarterly bullet bod same way : ( + r ) ( + r / 2) 2 ( + r ) ( + r / 4) 4 Whe comparg bod wth dfferet frequecy t s better of course to compare yeld usg the same coveto so practce compare aual yeld. Drty prce clea prce accrued terest It s easy to see that f we use the same yeld to calculate the preset value of a bod at two cosecutve dates there s a jump the preset value whe the coupo falls. Ths s why the preset value s called the drty prce. The jump s of course the coupo (C for aual bod or C/2 for sem-aual bod) I order to have cotuty (.e o jump) from oe day to the followg the yeld beg uchaged the market practce s to use clea prce to quote bods. The clea prce ( prx ped de coupo Frech) s the drty prce ( prx ple coupo Frech) mus the accrued terest : Clea prce Drty prce - cc Accrued terest ( coupo couru Frech) at date t s defed by : - 5 -

6 Accrued Iterest ( T Tprev ) ( T T ) cc C / freq ext prev Tprev beg the coupo date for the prevous coupo at date T ad T ext the coupo date for the ext coupo at date T. Roughly speakg the accrued terest s the pro rata of coupo at date T. It s easy to check that f : C + C P + t ( + r) ( + r) t P t Pl ( + r) dt P rdt P For r C P at coupo dates C dt.i other words aroud par everyday we ga t C. Ths s the ratoal for accrued terest formula. 365 Remark : For calculatg the accrued terest or the year fractos fact to use specfc rules called bass. t t 2 L t the market practce s - 6 -

7 For each market.e corporate bods USD EUR GBP JPY govermet bods there s a specfc bass for the exact calculato of accrued terest (30/360 ACT/ACT30/360E ACT/365 )ad aother specfc bass for the calculato of the year fractos t t 2 L t. The purpose of ths documet s ot to eter to these kd of detals so we assume ACT/365 bass through all the documet. It does t chage ay cocluso or terpretato. Usg the exact rules chage very margally the calculato (durato rsk ) ad s oly ecessary whe workg dealg room. Durato sestvty for a seres of cash-flows by : The durato at date t of a seres of determstc ad postve determstc cash flows F F2 L F at creasg dates T T2 L T wll be defed w ( F F LF ) D T PV w 2 PV ( T F ) ( T F ) PV PV ( t F ) > 0 L ( t F ) t w t The durato s the w weghted average of the cash-flows at maturtes The durato of a zero-coupo s ts maturty (the year fracto). The durato always have the followg property : t D ( F F LF ) t 2 t t 2 L t ad so : We do t expla here how to calculate the preset value of each cash-flow ( T ) PV. It wll be of course calculated usg dscout factors derved from a zero-coupo curve but here they are supposed to be gve. F Let s assume ow that we use the aual yeld r assocated to the cash-flows ther total preset value PV ( T ). F F F2 L F ad PV ( T F ) t ( r) + F The t s easy to check that the durato ca also be defed by : - 7 -

8 ( F F L F r) PV 2 Durato PV ( F F L F r) 2 ( + r) We wll also defe the sestvty by : PV ( F F2 L F r) Sestvty. PV F F L F r ( ) 2 Remarks : The values of the durato ad sestvty are ot very dfferet for usual values of r. If we multply the cash-flows F F2 L F by a costat λ the yeld durato ad sestvtes are obvously uchaged. If we go back to the geeral defto of durato ad use dscout factors stead of usg r : ( T ) D( T F F2 L F ) B T F B T F ( T ) tthe umercal value of the durato wll of course be dfferet. Rsk sestvty durato for a bod Rsk Durato ad Sestvty for aual bods We ow work wth bullet bods of otoal. Let s ote : C + C P + t ( + r) ( + r) t the market preset value of ths bod r beg the aual yeld assocated to P. The rsk wth respect to r wll be defed as : - 8 -

9 P Rsk The covexty wth respect to the aual yeld r wll be defed as : The durato by : The sestvty by : P Durato P 2 P covexty 2 ( + r) rsk P P Sestvt y durato / + P ( + r) ( r) P s the drty prce. It s easy to see there s ot jump for the rsk at the coupo date r beg costat. To avod jump for durato ad sestvty at the coupo date t s ecessary to use a slghtly dfferet defto usg P-cc the clea prce : P ( + r) for the durato P cc P P cc for the sestvty. Rsk Durato ad Sestvty for o aual bods Let s assume ow we have a sem-aual bullet bod. We remd that the sem-aual yeld s defed by : P C / C / 2 2t ( + r / 2) ( + r / 2) The rsk wth respect the sem-aual yeld r s : P Rsk 2 t The durato by : - 9 -

10 P Durato + P ( r / 2) The sestvty by : P Sestvt y durato /( + r / 2) P + r + r / 2 s : The rsk wth respect the aual yeld defed by ( ) ( ) 2 P /( + r / 2) Ths formula also gves the way of calculatg the sestvty wth respect the aual yeld. It s very easy to check that whatever the frequecy of coupo o the bod the rsk covexty sestvty depeds o the frequecy used for the yeld ot the durato. More geerally for a bullet bod of frequecy freq f r s the yeld cosstet wth the frequecy of the bod : P C / freq + C / freq + freq t freq t ( + r / freq) ( r / freq) + P / freq ( + r / freq) wll be the rsk wth respect the aual yeld of the bod. Covexty ( + r / freq) rsk ( / freq) 2 ( + r / freq) + 2 freq2 freq wll be the covexty wth respect the aual yeld of the bod. To gve a more cocrete dea of the dfferet dcators we gve ther values for varous aual bullet bods wth creasg maturtes ad creasg yeld two ways : Frst the coupo s at 4% Secod the coupo s equal to the yeld - 0 -

11 Coupo 4% Rsk yeld maturty % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Durato % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Sestvty % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

12 Covexty % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Remarks : For ay gve coupo rsk durato sestvty ad covexty are : decreasg fucto of the yeld for a gve maturty creasg fucto of the maturty for a gve yeld For a gve bod.e maturty ad coupo beg gve (ad also frequecy bass ) the rsk s very ofte terpreted as follows : The rsk s the umber of bp prce for bp yeld. For example we see that for a 0years bod wth coupo 4% the prce decreases of 8.bp whe yeld moves of bp. Somebody who holds 0000 bods of otoal euro wll loose 8 euros f rates creases by bp. If we forget the covexty the same perso wll loose approxmately 8 euros whe rates creases by 00bp. The covexty ca be terpreted as follows : Holdg bods we ga more whe rates decrease by 00bp tha we loose whe rates creases by 00bp. As the maturty creases the covexty creases (strogly). So as there s o free luch the facal markets t meas the yeld curve moves caot be oly parallel shft. Otherwse a strategy such as : Beg log 30 year bods Short 0 years bods so as total rsk s zero would ga whatever the yeld curve move! To hedge a portfolo the rsk s the relevat dcator ot the durato. bp bass pot 0.0%

13 But the durato s a good dcator of the average maturty of a portfolo as t s uchaged f you multply your portfolo by a costat λ

14 Coupo yeld Rsk yeld maturty % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Durato % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Sestvty % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

15 Covexty % 2% 3% 4% 5% 6% 7% 8% 9% 0% Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Remarks : It s very easy to check that as ow we work wth bods at par the sestvty s equal to the rsk (ad so the rsk s very close to the durato) All the umbers above are very smlar whe movg to sem aual bods. Lk betwee swap ad bods We wll ow remd why bods ad swaps are fact very smlar facal products. I fact terms of rsk we ca say that a 0 years swap ad a 0 years bod are exactly the same product (f we forget the use of coverage for the detals of the schedule of a swap). We wll use as example a 0Y swap valla swap to be more cocrete the demostrato ad so the cocluso wll be the same for ay maturty. We suppose the reader famlar wth valla swap defto ad prcg. Notatos The swap s a 0Y swap recever fxed rate payer EURIBOR 6M Market date 25/0/06 Frequecy 2M bass 30/360 o the fxed leg Frequecy 6M bass ACT/360 o the float leg The otoal of the swap s We ote T T2 L T0 the coupo paymet dates dates o the fxed leg We ote T T L T the coupo paymet dates o the float leg 2 20 Of course the T T2 L T0 are cluded the T T L T

16 We kow that the EURIBOR6M pad at date ote (the fxg dates) ad calculated o perod ( ) T s the EURIBOR6M fxed at T 2 busess days T T M + 6. We frst gve the schedule of the two legs : Float leg schedule Pay dates are T T L T the paymet dates 2 20 Fx dates are the fxg dates : T 2 busess days s the fxg dates for date (Lb Start LbEd) ( T T M ) EURIBOR6M 2. for date T s the perod of calculato of the + 6 ( T ) Pay Cvge s the coverage of paymet : T 360 ( T ) + 6M T Lb coverage s the coverage for the calculato of the EURIBOR6M : δ DF Pay Dates Dscout Factor at Paymet dates B ( T ) δ 0 T Lb s for Lbor as for most curreces the tradg place s Lodo - 6 -

17 Zc Pay dates s the assocated zero-coupo rate. FRA ( T ) ( T T M ) + 6 The flow at date prevous schedule). B( 0 T ) ( ) 0 T + 6M s the forward EURIBOR6M for perod δ B pad at date T ad fxed at T 2 busess days. T s FRA ( T ) δ (otherwse colum 3 colum 9 colum 0 Lookg at the schedule we see that δ. T 6M s ot always equal to + T so δ s ot always to The preset value of the float leg s : PV 20 ( floatleg) B( 0 T ) ( δ FRA T )

18 Fxed leg schedule Pay. Cvge Pay. Dates Fx. Dates DF Pay Dates flow /0/07 25/0/ % /0/08 25/0/ % 27/0/09 23/0/ % 27/0/0 23/0/ % 27/0/ 25/0/ % /0/2 25/0/ % /0/3 25/0/ % /0/4 24/0/ % 27/0/5 23/0/ % 27/0/6 23/0/ % The deftos are exactly the same of course the fxed leg s much smple to descrbe. Pay cvge are the paymet coverage : δ s the paymet coverage at date T calculated as the year fracto betwee date T ad T bass 30/360. ( fxed _ leg) B( 0 ) 0 LVL δ 8.4 s the Level assocated to the schedule of ths 0Y swap fxe leg. T The swap rate s by defto equal to PV LVL ( floatleg) ( fxed _ leg) %. Kowg the value of the swap rate we ca calculate each cash-flow at date T : δ %. Smplfcatos We start wth the float leg : Whe readg a quattatve book or workg paper you wll fd much smple way of descrbe a float leg. Quats always do the followg assumptos : T + M δ δ 6 T. I the float leg schedule show above t meas the colums Lb Ed ad Pay Dates are detcal same thg for colums Pay Cvge ad Lb Coverage. The we get : - 8 -

19 PV 20 ( float _ leg) ( B( 0 T ) B( 0 T ) B( 0 T ) B( 0 T ) 0 20 Of course quats do t take to accout the 2 busess rules they do as f fxg date for date T. They do as f T 0. 0 Of course T 20 T0 t ca be checked o the two above schedules. T s So they get the well kow formula for valug a float leg at a fxg date (but just before the fxg tme!) of maturty T : ( 0 ) B T. We see the a very mportat property : The value of the float leg the does t deped of the dex of the float leg : EURIBOR3M EURIBOR 6M EURIBOR 2M I or example dog all these smplfcatos; we get for the floatleg : ad the a swap rate of : 4.000%. As t s well kow the approxmato s very good. Whe use these approxmato? Not to value a swap portfolo : the you have to take to accout the bass swap betwee EURIBOR3M ad EURIBOR6M for example. It meas you have oe forward curve by dex oe uque dscout curve. As you wll use dfferet curves to calculate the forwards ad to calculate the dscout factors there s o value to make such smplfcatos. I addto the p & l mpact o the valuato of thousads swaps wth otoal from 0M to more 000M s ot so small. If you work o a model for a very exotc terest product (for whch the margs fortuately much more mportat tha for valla swaps) there s value to make these approxmatos at least whe you wrte the documetato of your model! I the rest of the documet we wll do theses smplfcatos. O the fxed leg order to get closer to a bod we wll assume the coverage to be detcal (so equal to )

20 We get ow a swap rate of %; stead of 4.000% after the smplfcatos of the float leg. The level s ow Overall we just mea the detals of coveto (bass busess days 2 days rule for the fxg vs the paymet ) are of course mportat for exact valuato ot for uderstad the propertes of a valla swap terms of rsk exposure to the chage terest curves. We also remd that whe the fxg s kow the valuato of the flow leg becomes : ( Eurbor6M ( T T ) ) B( 0 T ) B( 0 T ) + δ. 0 0 To get ths formula just wrte that the leg s the sum of a floatg leg startg at date T whose B 0 T B 0 T ad the preset value of the kow fxg whose valuato s valuato s ( ) ( 0 ) Eurbor6M ( T T ) δ B( 0 T ). 0 Ths formula s of course valuable f T0 0 T. 0 beg the curret market date To come to the pot of our demostrato after all these perhaps paful prelmares let s ow start from the schedules of the two legs of our 0Y swap : Fx s ow equal to %. Let s assume ow we add a cash-flow of o each leg. We do t chage aythg to the total value of the swap of course whatever the curve. The value of the float leg s ow ( B ( 0 T ) + B( 0 T )) coupo %! 0 0 the fxed leg s ow a 0Y bod of For a trader holdg a 0Y bod facg the posto at EURIBOR6M the aalyss s the same. I other words the float leg of a 0year valla recever swap s the facg leg of a log posto o a 0Y bod

21 Whe we add the otoal o the two legs of the swaps at each fxg dates (just before the fxg tme am!) we just move all the rsk of the swap o the fxed leg. Whe the fxg s kow there s a slght rsk o the float leg whch s tally aroud the dex of the float leg ( 0. 5 for our example as the float leg s payer)