UNITED KINGDOM DEBT MANAGEMENT OFFICE


 Blake Jordan Hawkins
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1 United Kingdom Debt Management Office Eatcheap Cout Philpot Lane London EC3M 8UD UNITED KINGDOM DEBT MANAGEMENT OFFICE Fomulae fo Calculating Gilt Pice fom Yield t edition: 8 June 998 nd edition: 5 Januay 00 3 d edition: 6 Mach 005
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3 CONTENTS INTRODUCTION 3 SECTION ONE: PRICE / YIELD FORMULAE 5 Conventional Gilt; Doubledated and Undated Gilt with Aumed (o Actual) Redemption on a QuaiCoupon Date 5 Indexlinked Gilt (8Month Indexation Lag) 7 Indexlinked Gilt (3Month Indexation Lag) Doubledated Gilt 4 Undated Gilt 7 Stip 0 SECTION TWO: CALCULATION OF DIVIDEND PAYMENTS ON GILTS SECTION THREE: CALCULATION OF ACCRUED INTEREST 5 ANNEX A: ESTIMATION OF THE NOMINAL VALUES OF FUTURE UNKNOWN CASH FLOWS ON INDEXLINKED GILTS WITH AN 8MONTH INDEXATION LAG 30 ANNEX B: METHOD OF INDEXATION FOR INDEXLINKED GILTS WITH A 3MONTH INDEXATION LAG 3 NOTES 36
4 3 INTRODUCTION Thi pape et out the United Kingdom Debt Management Office (DMO) fomulae fo calculating gilt pice fom go edemption yield, thu allowing a fomal ettlement convention to be applied to tade conducted on a yield bai. The fomulae in thi pape have been effective ince Novembe 998. Thi i the thid edition of the pape fit publihed in June 998. On Decembe 004, the DMO announced that all new indexlinked gilt iued fom the financial yea would follow the theemonth indexation lag methodology which ha become the global tandad. In thi updated veion of the pape, the fomulae fo thi deign of new indexlinked gilt have been added, and othe mino change have been made although the fomulae themelve ae unchanged. In the event that the fomulae ae to be ued to deive yield fom pice it i not poible (in mot cae) to olve fo yield in tem of pice algebaically, and o ome fom of numeical technique mut be ued if, given a pice, a value fo the edemption yield i equied. The fit ection of the pape et out the DMO pice / yield fomulae; thee ae plit into the diffeent clae of gilt (new fomulae fo new intument will be added to the pape a and when equied). Fo the pupoe of thi pape, cah flow efe to cah flow eceivable by the buye of the gilt. Alo, neaet ounding to, ay, ix decimal place mean ound the ixth decimal place up by one if the eventh decimal place i five o above, and then tuncate at the ixth decimal place. Coupon payment on all cuent gilt outtanding ae made emiannually, with the exception of thee undated gilt which pay quately: ½% Annuitie, ¾% Annuitie and ½% Conolidated Stock. Compounding will occu on quaicoupon date. Fo dated gilt, quaicoupon date ae the date on the emiannual cycle defined by the (final) matuity date, iepective of whethe cah flow occu on thoe date (example of quaicoupon date on which cah flow would not occu include the fit quaicoupon date of a new iue having a long fit dividend peiod; the next quaicoupon date of a gilt ettling in it exdividend peiod; and mot quaicoupon date of a tip). The quaicoupon date fo undated gilt ae defined by thei egula coupon cycle. A full quaicoupon peiod i defined a the peiod between
5 4 two conecutive quaicoupon date. Fo example, a gilt ettling on it iue date (auming thi i not alo a quaicoupon date) will have a quaicoupon peiod which tat on the quaicoupon date pio to the iue date and end on the fit quaicoupon date following the iue date. If the iue date fall on a quaicoupon date, then the quaicoupon peiod tat on the iue date. Cah flow and quaicoupon date which ae due to occu on nonbuine day ae not adjuted (i.e. ae not bumped ). Thi mean that cah flow which occu on date which ae not quaicoupon date (uch a ome ealy edemption payment on doubledated o undated gilt) may have an additional factional peiod aociated with thei dicounting poce to allow fo dicounting back (i.e. towad the ettlement date) by a factional peiod to the quaicoupon date immediately pio to thei occuence, befoe being dicounted back to the ettlement date. All ettlement value deived fom thee fomulae (yield to pice) hould be ounded to the neaet penny on the tade, with no intemediate ounding. In addition, the pice / yield fomulae dicount all cah flow on the quaicoupon cycle uing the actual / actual daycount convention: thi i conitent with the ageed maket conenu fo dicounting the cah flow fom a tip. Following maket conultation, it wa ageed that the RPI inflation aumption that hould be ued in the fomulae fo indexlinked gilt with an 8month indexation lag i 3% pe annum. Thi will be eviewed by the DMO a and when a majoity of maket paticipant judge that a eview i neceay. The econd ection in thi pape et out the fomulae fo calculating dividend payment on gilt; and the thid ection povide thoe fo the calculation of accued inteet. Annex A decibe the pocedue fo etimating the nominal value of unknown futue cah flow on indexlinked gilt with an 8month indexation lag. Annex B et out the method of indexation fo indexlinked gilt fit iued fom Any quetion on thi pape hould be addeed to: Guminde Bhachu
6 5 SECTION ONE: PRICE / YIELD FORMULAE Conventional Gilt; Doubledated and Undated Gilt with Aumed (o Actual) Redemption on a QuaiCoupon Date The fomula fo calculating the pice fom the yield i given by: cv n n P = v d + d v + v + v ( ) 00 fo n f ( v ) Whee: P = Dity pice pe 00 nominal of the gilt 3. d = Cah flow due on next quaicoupon date, pe 00 nominal of the gilt (may be zeo if the gilt ha a long fit dividend peiod o if the gilt c ettle in it exdividend peiod; o may be geate o le than f duing long o hot fit dividend peiod epectively). d = Cah flow due on next but one quaicoupon date, pe 00 nominal of the gilt (may be geate than f c duing long fit dividend c y peiod). = Coupon pe 00 nominal of the gilt. = Nominal edemption yield (decimal), i.e. if the yield i 5% then y = f = Numbe of coupon payable on the gilt pe yea (f will be equal to o 4). v = n y + f = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quai coupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). = Numbe of full quaicoupon peiod fom the next quaicoupon
7 6 date afte the ettlement date to the edemption date. Fo n = 0, the equation educe to ( ) P = v d + 00 In thi pecial cae, we can olve algebaically fo yield in tem of pice: d + 00 y = f P
8 7 Indexlinked Gilt (8Month Indexation Lag) 4 () Not all the nominal value of futue cah flow ae fixed Cae : Two o moe cah flow emaining The fomula fo calculating the pice fom the yield i given by: P + n au w fo n ( w) acw n = d + d ( uw ) + ( w ) ( uw ) + 00 Whee: P = Dity pice pe 00 nominal of the gilt 3. d = Cah flow due on next quaicoupon date, pe 00 nominal of the gilt (may be zeo if the gilt ha a long fit dividend peiod o if the gilt c ettle in it exdividend peiod; o may be geate o le than time the RPI Ratio duing long o hot fit dividend peiod epectively). d = Cah flow due on next but one quaicoupon date, pe 00 nominal of the gilt (may be geate than c time the RPI Ratio duing c ρ long fit dividend peiod) 5. = Coupon pe 00 nominal. = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quaicoupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). = Semiannually compounded eal edemption yield (decimal), i.e. if w = the eal yield i.5% then ρ = ρ π = The aumed annual inflation ate (decimal) = 0.03.
9 8 u = = + π.03 n = Numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date. RPIB = The Bae RPI fo the gilt, i.e. the RPI cheduled to be publihed even month pio to the month of fit iue of the gilt and elating to the month eight month pio to the month of fit iue of the gilt (fo example, if the gilt i fit iued in Novembe then it Bae RPI i the RPI fo Mach of that yea). RPIL = The latet publihed RPI at the time of valuation. k = Numbe of month between the month of the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the month of the latet publihed RPI at the time of valuation. Fo example, if the RPI fo Januay i the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the latet publihed RPI at the time of valuation i the RPI fo Apil, then k = 3. k RPIL a = u RPIB Cae : One cah flow emaining (i.e. the final dividend and edemption payment) If the RPI detemining the edemption value i publihed afte the gilt goe exdividend fo the penultimate time, the pice / yield fomula i defined a: P c a = 00 + u ( uw ) +α Whee: P = Dity pice pe 00 nominal of the gilt 3. c = Coupon pe 00 nominal. ρ = Real edemption yield (decimal), i.e. if the yield i.5% then ρ = 0.05.
10 9 w = + ρ π = The aumed annual inflation ate (decimal) = u = = + π.03 = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quaicoupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). if the gilt i ettling in it penultimate ex  dividend peiod α = 0 if the gilt i ettling on o afte it penultimate quai  coupon date RPIB = The Bae RPI fo the gilt, i.e. the RPI cheduled to be publihed even month pio to the month of fit iue of the gilt and elating to the month eight month pio to the month of fit iue of the gilt (fo example, if the gilt i fit iued in Novembe then it Bae RPI i the RPI fo Mach of that yea). RPIL = The latet publihed RPI at the time of valuation. k = Numbe of month between the month of the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the month of the latet publihed RPI at the time of valuation. Fo example, if the RPI fo Januay i the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the latet publihed RPI at the time of valuation i the RPI fo Apil, then k = 3. k RPIL a = u RPIB
11 0 In thi pecial cae, we can olve algebaically fo yield in tem of pice: c 00 + a ρ = u up + α () Nominal value of all futue cah flow ae fixed Cae : Indexlinked gilt that have paed both thei penultimate exdividend date and the point at which the RPI detemining the final edemption payment i publihed povide a known cah flow on jut one emaining date. The pice / yield fomula in thi cae i: P = v +α ( d + R) LAST Whee: P = Dity pice pe 00 nominal of the gilt 3. d LAST = Final dividend payment pe 00 nominal of the gilt, a publihed. R y v = = Final edemption payment pe 00 nominal of the gilt, a publihed. = Semiannually compounded nominal edemption yield (decimal), i.e. if the yield i 5% then y = y = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quaicoupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). if the gilt i ettling in it penultimate ex  dividend peiod α = 0 if the gilt i ettling on o afte it penultimate quai  coupon date
12 In thi pecial cae, we can olve algebaically fo yield in tem of pice: y d + LAST R = P + α Cae : When valuing indexlinked gilt between the publication of the RPI detemining the edemption payment and the penultimate exdividend date (auming that the RPI detemining the edemption value i publihed befoe the gilt goe exdividend fo the penultimate time), the pice / yield fomula i defined a: P = ( d + ( d + R) v ) v PEN LAST Whee: P = Dity pice pe 00 nominal of the gilt 3. d PEN = Penultimate dividend payment pe 00 nominal of the gilt, a publihed. d LAST = Final dividend payment pe 00 nominal of the gilt, a publihed. R y v = = Redemption payment pe 00 nominal of the gilt, a publihed. = Semiannually compounded nominal edemption yield (decimal), i.e. if the yield i 5% then y = y = Numbe of calenda day fom the ettlement date to the next quaicoupon date. = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date).
13 Indexlinked Gilt (3Month Indexation Lag) 4 () Fo tade ettling befoe the penultimate dividend date: P = w n ( w ) cw n d + d w w fo n ( w ) () Fo tade ettling on o afte the penultimate dividend date and whee the tade occu befoe the publication of the RPI that detemine the edemption payment (ee Annex B fo ome example which illutate which RPI detemine the edemption payment): ( d + 00) fo 0 P = w n = In thi cae, it i poible to olve algebaically fo yield in tem of pice: d + 00 ρ = P (3) Whee the tade occu afte the publication of the RPI that detemine the edemption payment, the indexlinked gilt will effectively become a nominal (athe than a eal) intument and the fomula fo calculating the (eal) dity pice fom the nominal yield will be given by: P = v LAST R Index Ratio Set Date ( D + ) In thi cae, it i poible to olve algebaically fo yield in tem of pice: D + LAST R y = P Index RatioSet Date
14 3 Whee: P = Real dity pice pe 00 nominal. c = Coupon pe 00 nominal. = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quaicoupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). n = Numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date. ρ = Semiannually compounded eal edemption yield (decimal), i.e. if the eal yield i.5% then ρ = w = ρ + y = Semiannually compounded nominal edemption yield (decimal), i.e. if the nominal yield i 5% then y = v = y + D LAST = Final (fixed) coupon payment pe 00 nominal of the gilt, a publihed. R = Redemption payment (fixed) pe 00 nominal of the gilt, a publihed. and Index Ratio i defined in Annex B.
15 4 Doubledated Gilt A doubledated gilt ha a final matuity date and in addition an ealie matuity date, with HM Teauy having the ight to edeem the gilt on any day between thee two date, povided that the elevant notice i given (uually 3 month). In ode to calculate the edemption yield fo uch gilt it i neceay to make ome aumption about when the gilt will be edeemed (whee a pecific edemption date ha not yet been announced by the authoitie). Cae : The ettlement date i at leat x month befoe the fit date in the edeemable band (whee x i the peiod of notice equied to be given to call the gilt a pecified in it popectu  uually 3 month). Then the yield / coupon ule i ued: if the nominal edemption yield y i geate than o equal to the coupon, the latet edemption date in the edeemable band i aumed; othewie the ealiet edemption date in the edeemable band i aumed. Fo pice to yield calculation, the pa ule i ued: if the clean pice (i.e. excluding accued inteet) i le than o equal to pa, the latet edemption date in the edeemable band i aumed; othewie the ealiet edemption date in the edeemable band i aumed. Note that in cetain bounday cae, the two ule above may not be equivalent. Cae : The ettlement date i eithe le than x month befoe the fit date in the edeemable band (whee x i the peiod of notice equied to be given to call the gilt a pecified in it popectu  uually 3 month), o the ettlement date i in the edeemable band. Then if notice ha not yet been given by the authoitie that the gilt will be edeemed ealy, the latet edemption date in the edeemable band i aumed (iepective of whethe the nominal edemption yield y i geate than o le than the coupon, o whethe the clean pice i le than o geate than pa). Having made uch an aumption about the edemption date, if thi fall on a quaicoupon date the fomula fo conventional gilt hould be ued; if it fall on a date which i not a quaicoupon date, the following fomula hould be ued:
16 5 t ( ) cv n n u P = v d + d v + ( v ) d f v v fo n ( v) Whee: P = Dity pice pe 00 nominal of the gilt 3. d = Cah flow due on next quaicoupon date, pe 00 nominal of the gilt (may be zeo if the gilt ha a long fit dividend peiod o if the gilt c ettle in it exdividend peiod; o may be geate o le than duing long o hot fit dividend peiod epectively). d = Cah flow due on next but one quaicoupon date, pe 00 nominal of the gilt (may be geate than c duing long fit dividend peiod). d f = Patial coupon due on offquaicoupon edemption date, pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. y v = t u n = Semiannually compounded nominal edemption yield (decimal), i.e. if the yield i 5% then y = y = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quai coupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). = Numbe of calenda day fom the edemption date to the peceding quaicoupon date. = Numbe of calenda day in the full quaicoupon peiod in which the edemption date occu. = Numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date.
17 6 Fo n = 0, the equation educe to: () If the peiod between the ettlement date and the edemption date pan a quaicoupon date P v d + 00 t u = ( + d ) f v () If the peiod between the ettlement date and the edemption date doe not pan a quaicoupon date P = t* u ( 00 + d f ) v Whee: t * = Numbe of calenda day fom the ettlement date to the edemption date.
18 7 Undated Gilt All cuent undated gilt in iue have a date afte which they can be edeemed (fo example, 3½% Wa Loan i dated 95 o afte ). In ode to calculate the edemption yield fo uch gilt it i neceay to make ome aumption about when the gilt will be edeemed (whee a pecific edemption date ha not yet been announced by the authoitie). If notice ha not yet been given by the authoitie that the gilt will be edeemed ealy, it i aumed that the gilt will not be edeemed and the infinite cah flow fomula hould be ued (ee below), iepective of whethe the nominal edemption yield y i geate than o le than the coupon, o whethe the clean pice i le than o geate than pa. Fo an actual ealy edemption date, if thi fall on a quaicoupon date the fomula fo conventional gilt hould be ued; if it fall on a date which i not a quaicoupon date, the following fomula hould be ued: cv P = v d + d v + 00 fo n f ( v) t n n u ( v ) + ( + d ) f v v Whee: P = Dity pice pe 00 nominal of the gilt 3. d = Cah flow due on next quaicoupon date, pe 00 nominal of the gilt (may be zeo if the gilt ha a long fit dividend peiod o if the gilt c ettle in it exdividend peiod; o may be geate o le than f duing long o hot fit dividend peiod epectively). d = Cah flow due on next but one quaicoupon date, pe 00 nominal of the gilt (may be geate than f c duing long fit dividend peiod). d f = Patial coupon due on offquaicoupon edemption date, pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. y = Nominal edemption yield (decimal), i.e. if the yield i 5% then
19 8 y = f = Numbe of coupon payable on the gilt pe yea (f will be equal to o 4). v = t u n y + f = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quai coupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). = Numbe of calenda day fom the edemption date to the peceding quaicoupon date. = Numbe of calenda day in the full quaicoupon peiod in which the edemption date occu. = Numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date. Fo n = 0, the equation educe to: () If the peiod between the ettlement date and the edemption date pan a quaicoupon date P v d + 00 t u = ( + d ) f v () If the peiod between the ettlement date and the edemption date doe not pan a quaicoupon date P = t* u ( 00 + d f ) v Whee: t * = Numbe of calenda day fom the ettlement date to the edemption date.
20 9 Infinite cah flow method: Fo an infinite et of cah flow (i.e. whee it i aumed that the gilt will not be edeemed) we ue the fomula fo a conventional gilt and take P to be the limit of the um of the dicounted cah flow a n (the numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date) tend to infinity. Since v <, thi limit exit and i equal to P = v d + d cv v + f ( v ) Whee: P = Dity pice pe 00 nominal of the gilt 3. d = Cah flow due on next quaicoupon date, pe 00 nominal of the gilt (may be zeo if the gilt ha a long fit dividend peiod o if the gilt c ettle in it exdividend peiod; o may be geate o le than f duing long o hot fit dividend peiod epectively). d = Cah flow due on next but one quaicoupon date, pe 00 nominal of the gilt (may be geate than f c duing long fit dividend c y peiod). = Coupon pe 00 nominal of the gilt. = Nominal edemption yield (decimal), i.e. if the yield i 5% then y = f = Numbe of coupon payable on the gilt pe yea (f will be equal to o 4). v = y + f = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quai coupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date).
21 0 Stip Cetain conventional gilt ae eligible to be tipped into thei epaate cah flow, which ae called tip. In Febuay 997 the Bank of England (a the UK govenment debt manage at that time) publihed a conultative pape eeking view on what tandadied fomula fo computing maket pice fom go edemption yield hould be adopted to allow gilt tip to tade on a yield bai. The eult of the conultation wa indicated by Pe Notice on 30 May 997 and June 997. The maket conenu wa that the following method would be the mot uitable fo tip: P = 00 y + + n Whee: P = Pice pe 00 nominal of the tip. y = Stip go edemption yield (decimal), i.e. if the yield i 5% then y = = Numbe of calenda day fom the ettlement date to the next quaicoupon date ( = if the ettlement date fall on a quaicoupon date). = Numbe of calenda day in the quaicoupon peiod in which the ettlement date occu (i.e. between the pio quaicoupon date and the following quaicoupon date). n = Numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date. The ettlement poceed ae ounded to the neaet penny on the taded nominal amount (with no intemediate ounding of pice). In thi pecial cae, we can olve algebaically fo yield in tem of pice: 00 y = P + n
22 SECTION TWO: CALCULATION OF DIVIDEND PAYMENTS ON GILTS New gilt ae typically fit iued pat way though a quaicoupon peiod. The DMO et a nontandad fit dividend on new gilt which ae not iued on a quaicoupon date: eithe hot if the peiod between the iue date and the payment date i hote than a nomal coupon peiod, o long othewie. The accued inteet paid by puchae at the fit iue i conequently zeo. Thi ection povide fomulae fo the calculation of nontandad fit dividend, a well a dividend paid afte the fit payment. () Standad dividend peiod (i) Conventional gilt, doubledated and undated gilt c Dividend pe 00 nominal = f Whee: c = Coupon pe 00 nominal of the gilt. f = Numbe of coupon payable on the gilt pe yea (f will be equal to o 4). (ii) Indexlinked gilt (8month indexation lag) 4 c RPID Dividend pe 00 nominal = RPIB (See endnote 7 fo how thi hould be ounded). Whee: c = Coupon pe 00 nominal of the gilt. RPID = The RPI which fixe the next dividend payment fo the gilt, i.e. the RPI cheduled to be publihed even month pio to the month of the next dividend payment and elating to the month eight month pio to the month of the next dividend payment (fo example, if the next dividend payment on the gilt will be in Novembe then the RPI which fixe it value i the RPI fo Mach of that yea). RPIB = The Bae RPI fo the gilt, i.e. the RPI cheduled to be publihed even month pio to the month of fit iue of the gilt and elating to
23 the month eight month pio to the month of fit iue of the gilt (fo example, if the gilt i fit iued in Novembe then it Bae RPI i the RPI fo Mach of that yea). (iii) Indexlinked gilt (3month indexation lag) 4 Dividend pe 00 nominal = c Index Ratio Dividend Date Whee: c = Coupon pe 00 nominal and Index Ratio i defined in Annex B. Coupon payment ae ounded to the neaet 6 th decimal place pe 00 nominal. () Shot fit dividend peiod (i) Conventional gilt c Dividend pe 00 nominal =, ounded to neaet 6 th decimal place Whee: c = Coupon pe 00 nominal of the gilt. = Numbe of calenda day fom the iue date to the next (hot) coupon date. = Numbe of calenda day in the full quaicoupon peiod in which the iue date occu.
24 3 (ii) Indexlinked gilt (3month indexation lag) 4 c Dividend pe 00 nominal = Index Ratio Dividend Date Whee: c = Coupon pe 00 nominal = Numbe of calenda day fom the iue date to the next (hot) coupon date. = Numbe of calenda day in the full quaicoupon peiod in which the iue date occu. and Index Ratio i defined in Annex B. Coupon payment ae ounded to the neaet 6 th decimal place pe 00 nominal. (3) Long fit dividend peiod (i) Conventional gilt c Dividend pe 00 nominal = +, ounded to neaet 6 th decimal place Whee: c = Coupon pe 00 nominal of the gilt. = Numbe of calenda day fom the iue date to the next quaicoupon date. = Numbe of calenda day in the full quaicoupon peiod in which the iue date occu.
25 4 (ii) Indexlinked gilt (3month indexation lag) 4 c Dividend pe 00 nominal = + Index Ratio Dividend Date Whee: c = Coupon pe 00 nominal = Numbe of calenda day fom the iue date to the next quaicoupon date. = Numbe of calenda day in the full quaicoupon peiod in which the iue date occu. and Index Ratio i defined in Annex B. Coupon payment ae ounded to the neaet 6 th decimal place pe 00 nominal. (4) Shot final dividend peiod fo doubledated o undated gilt which have been called fo edemption on a date which i not a quaicoupon date c Dividend pe 00 nominal =, ounded to neaet 6 th decimal place f Whee: c = Coupon pe 00 nominal of the gilt. f = Numbe of coupon payable on the gilt pe yea (f will be equal to o 4). = Numbe of calenda day fom the edemption date to the peceding quaicoupon date. = Numbe of calenda day in the full quaicoupon peiod in which the edemption date occu.
26 5 SECTION THREE: CALCULATION OF ACCRUED INTEREST While coupon payment on individual gilt ae uually made only twice a yea, gilt can be taded on any buine day. Wheneve a gilt change hand on a day that i not a coupon payment date, the valuation of the gilt will eflect the poximity of the next coupon payment date. Thi i effected by the payment of accued inteet to compenate the elle fo the peiod ince the lat coupon payment date duing which the elle ha held the gilt but fo which he eceive no coupon payment. The accued inteet fo gilt i computed a follow 6 (baed on the actual / actual daycount convention effective fom Novembe 998): The accued inteet on all gilt i ounded to the neaet penny on the taded nominal amount fo calculating ettlement poceed. () Standad dividend peiod (i) All gilt excluding indexlinked gilt with a 3month indexation lag AI t d = t d if the ettlement date occu on o befoe the ex  dividend date if the ettlement date occu afte the ex  dividend date Whee: AI = Accued inteet pe 00 nominal of the gilt. d t = Next dividend pe 00 nominal of the gilt, a publihed. = Numbe of calenda day fom the peviou quaicoupon date to the ettlement date (t = 0 if the ettlement date fall on a quai coupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu.
27 6 (ii) Indexlinked gilt (3month indexation lag) 4 t c RAI = t c if the ettlement date occu on o befoe the ex  dividend date if the ettlement date occu afte the ex  dividend date Whee: RAI = Real accued inteet pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. t = Numbe of calenda day fom the peviou quaicoupon date to the ettlement date (t = 0 if the ettlement date fall on a quaicoupon date). = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu. Inflationadjuted accued inteet = calculation of Index Ratio Set Date ). RAI Index Ratio Set Date (ee Annex B fo the () Shot fit dividend peiod (i) Conventional gilt AI t * c = t * c if the ettlement date occu on o befoe the ex  dividend date if the ettlement date occu afte the ex  dividend date Whee: AI = Accued inteet pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. t * = Numbe of calenda day fom the iue date to the ettlement date. = Numbe of calenda day in the full quaicoupon peiod in which the ettlement date occu. = Numbe of calenda day fom the iue date to the next (hot) coupon date.
28 7 (ii) Indexlinked gilt (3month indexation lag) 4 RAI t c t c = if the ettlement date occu on o befoe the ex  dividend date if the ettlement date occu afte the ex  dividend date Whee: RAI = Real accued inteet pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. t * = Numbe of calenda day fom the iue date to the ettlement date. = Numbe of calenda day in the full quaicoupon peiod in which the iue date occu. = Numbe of calenda day fom the iue date to the next (hot) coupon date. Inflationadjuted accued inteet = RAI Index Ratio Set Date (ee Annex B fo the calculation of Index Ratio Set Date ). (3) Long fit dividend peiod (i) Conventional gilt * t c c AI = + c if the ettlement date occu duing the fit quai  coupon peiod if the ettlement date occu duing the econd quai  coupon peiod on o befoe the ex  dividend date if the ettlement date occu duing the econd quai  coupon peiod afte the ex  dividend date Whee: AI = Accued inteet pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. * t = Numbe of calenda day fom the iue date to the ettlement date in the fit quaicoupon peiod (thi tem only applie if the gilt ettle in the fit quaicoupon peiod). = Numbe of calenda day fom the iue date to the next quai
29 coupon date. = Numbe of calenda day fom the quaicoupon date afte the iue date to the ettlement date in the quaicoupon peiod afte the quaicoupon peiod in which the iue date occu (thi tem only applie if the gilt ettle in the econd quaicoupon peiod). = Numbe of calenda day in the full quaicoupon peiod in which the iue date occu. = Numbe of calenda day in the full quaicoupon peiod afte the quaicoupon peiod in which the iue date occu. 8 (ii) Indexlinked gilt (3month indexation lag) 4 * t c c RAI = + c if the ettlement date occu duing the fit quai  coupon peiod if the ettlement date occu duing the econd quai  coupon peiod on o befoe the ex  dividend date if the ettlement date occu duing the econd quai  coupon peiod afte the ex  dividend date Whee: RAI = Real accued inteet pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. * t = Numbe of calenda day fom the iue date to the ettlement date in the fit quaicoupon peiod (thi tem only applie if the gilt ettle in the fit quaicoupon peiod). = Numbe of calenda day fom the iue date to the next quaicoupon date. = Numbe of calenda day fom the quaicoupon date afte the iue date to the ettlement date in the quaicoupon peiod afte the quaicoupon peiod in which the iue date occu (thi tem only applie if the gilt ettle in the econd quaicoupon peiod). = Numbe of calenda day in the full quaicoupon peiod in which the iue date occu. = Numbe of calenda day in the full quaicoupon peiod afte the
30 9 quaicoupon peiod in which the iue date occu. Inflationadjuted accued inteet = calculation of Index Ratio Set Date ). RAI Index Ratio Set Date (ee Annex B fo the (4) Shot final dividend peiod fo doubledated o undated gilt which have been called fo edemption on a date which i not a quaicoupon date t * * c AI = f if the ettlement date occu on o befoe the final ex  dividend date Whee: AI = Accued inteet pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. f = Numbe of coupon payable on the gilt pe yea (f will be equal to o 4). t * * = Numbe of calenda day fom the ettlement date to the peceding quaicoupon date. = Numbe of calenda day in the full quaicoupon peiod in which the edemption date occu.
31 ANNEX A: ESTIMATION OF THE NOMINAL VALUES OF FUTURE UNKNOWN CASH FLOWS ON INDEXLINKED GILTS WITH AN 8MONTH INDEXATION LAG 30 Embedded within the pice / yield fomula fo indexlinked gilt with an 8month indexation lag, the nominal value of unknown futue dividend ae etimated a: d c a = i u i + i n Whee: d i + = Dividend due on (i+)th quaicoupon date afte the ettlement date, pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. π = The aumed annual inflation ate (decimal) = u = = + π.03 RPIB = The Bae RPI fo the gilt, i.e. the RPI cheduled to be publihed even month pio to the month of fit iue of the gilt and elating to the month eight month pio to the month of fit iue of the gilt (fo example, if the gilt i fit iued in Novembe then it Bae RPI i the RPI fo Mach of that yea). RPIL = The latet publihed RPI at the time of valuation. k = Numbe of month between the month of the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the month of the latet publihed RPI at the time of valuation. Fo example, if the RPI fo Januay i the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the latet publihed RPI at the time of valuation i the RPI fo Apil, then k = 3. k RPIL a = u RPIB n = Numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date.
32 3 In addition, in mot cae the RPI detemining the edemption payment will not have been publihed, o that the nominal value of the edemption payment will not be known at the time of ettlement. Embedded within the pice / yield fomula fo indexlinked gilt with an 8month indexation lag, the nominal value of the edemption payment i etimated a: R = 00 a n u Whee: R = Redemption payment pe 00 nominal of the gilt. c = Coupon pe 00 nominal of the gilt. π = The aumed annual inflation ate (decimal) = u = = + π.03 RPIB = The Bae RPI fo the gilt, i.e. the RPI cheduled to be publihed even month pio to the month of fit iue of the gilt and elating to the month eight month pio to the month of fit iue of the gilt (fo example, if the gilt i fit iued in Novembe then it Bae RPI i the RPI fo Mach of that yea). RPIL = The latet publihed RPI at the time of valuation. k = Numbe of month between the month of the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the month of the latet publihed RPI at the time of valuation. Fo example, if the RPI fo Januay i the RPI that define the dividend due (o would odinaily be due, in the cae of a long fit dividend o a gilt ettling in it exdividend peiod) on the next quaicoupon date and the latet publihed RPI at the time of valuation i the RPI fo Apil, then k = 3. k RPIL a = u RPIB n = Numbe of full quaicoupon peiod fom the next quaicoupon date afte the ettlement date to the edemption date. All etimated indexlinked gilt cah flow ae left unounded fo pice / yield calculation pupoe 7.
33 ANNEX B: METHOD OF INDEXATION FOR INDEXLINKED GILTS WITH A 3MONTH INDEXATION LAG 3 Indexlinked gilt fit iued fom Apil 005 employ the theemonth lag indexation technique fit ued in the Canadian Real Retun Bond (RRB) maket, athe than the eightmonth lag methodology peviouly ued. In addition to uing a hote lag, RRB indexation i applied in a ignificantly diffeent way to that fo ealie indexlinked gilt iue. An Index Ratio i applied to calculate the coupon payment, the edemption payment (i.e. the uplifted pincipal) and the accued inteet. The Index Ratio fo a given date i defined a the atio of the efeence RPI applicable to that date ( Ref ) divided by the efeence RPI applicable to the oiginal iue date of the gilt ( the neaet 5 th decimal place: RPIDate Ref RPI Fit Iue Date ), ounded to Index Ratio Date Ref RPI = Ref RPIFit Date Iue Date, ounded to the neaet 5 th decimal place. The efeence RPI fo the fit calenda day of any calenda month i the RPI fo the calenda month falling thee month ealie. Fo example, the efeence RPI fo June coepond to the RPI fo Mach, the efeence RPI fo July coepond to the RPI fo Apil, etc. The efeence RPI fo any othe day in the month i calculated by linea intepolation between the efeence RPI applicable to the fit calenda day of the month in which the day fall and the efeence RPI applicable to the fit calenda day of the month immediately following. Intepolated value fo Ref RPIDate hould be ounded to the neaet 5 th decimal place, a hould value fo Index RatioDate.
34 33 The fomula ued to calculate Ref RPIDate can be expeed a follow: Ref RPI Date = Ref RPI M t + D [ Ref RPI Ref RPI ] M+ M Whee: D = Numbe of day in the calenda month in which the given date fall. t = The calenda day coeponding to the given date. Ref RPI M = Refeence RPI fo the fit day of the calenda month in which the given date fall. Ref RPI M+ = Refeence RPI fo the fit day of the calenda month immediately following the given date. Fo example, the efeence RPI fo 0 July 00 i calculated a follow: Ref RPI 9 3 [ Ref RPI Ref ] 0 July 00 = Ref RPIJuly 00 + Augut 00 RPIJuly 00 = RPI Apil [ RPI RPI ] May 00 Apil 00 9 = 73.+ [ ] = , when ounded to the 3 neaet 5 th decimal place. The Ref fo a given bond emain contant ove it life. Howeve, diffeent RPI Fit Iue Date indexlinked gilt hould have diffeent value fo Ref (depending on when they ae fit iued). RPI Fit Iue Date Calculation of the ettlement pice Indexlinked gilt with a theemonth lag tade and ae auctioned on the bai of the eal clean pice pe 00 nominal. The Inflationadjuted clean pice pe 00 nominal i calculated fom the eal clean pice uing the following fomula:
35 34 Inflationadjuted clean pice = Real clean pice Index Ratio Set Date (thi hould be left unounded). The Inflationadjuted dity pice pe 00 nominal i calculated a: Inflationadjuted dity pice pe 00 nominal = Inflationadjuted clean pice pe 00 nominal + Inflationadjuted accued inteet pe 00 nominal (thi hould be left unounded). Whee: Inflationadjuted accued inteet = Real accued inteet Index Ratio Set Date and the Real accued inteet (RAI) i defined in Section 3. The Inflationadjuted accued inteet hould be left unounded. Calculation of the edemption payment The edemption payment pe 00 nominal i calculated a follow: Redemption Payment = 00 Index Ratio Redemption Date The edemption payment i ounded to the neaet 6 th decimal place pe 00 nominal. Note: unlike in ome oveeign indexlinked bond maket, in the UK no deflation floo will be applied when calculating the edemption payment, i.e. the edemption payment fo an indexlinked gilt could fall below 00 pe 00 nominal if Ref RPI Redemption Date wee le than Ref RPI Fit Iue Date.
36 35 When doe the edemption payment become known? To illutate when the edemption payment (and the final coupon payment) will be fixed, conide ome hypothetical cae baed on the aumption of an indexlinked gilt with a theemonth lag edeeming on diffeent date in Decembe 003. Cae : Redemption on Decembe 003 The edemption payment would have been fixed when the Septembe 003 RPI wa publihed on 4 Octobe, i.e. the edemption payment would have been known appoximately 6 week (48 day) befoe the bond edeemed. Cae : Redemption on Decembe 003 The edemption payment would have been fixed when the Octobe 003 RPI wa publihed on 8 Novembe, i.e. the edemption payment would have been known week (4 day) befoe the bond edeemed. Cae 3: Redemption on 3 Decembe 003 The edemption payment would have been fixed when the Octobe 003 RPI wa publihed on 8 Novembe, i.e. the edemption payment would have been known appoximately 5 week (43 day) befoe the bond edeemed. So, in pactice, the edemption payment and the final dividend payment on an indexlinked gilt with a theemonth lag will typically be fixed aound 6 week befoe the edemption date.
37 36 NOTES. In ode to olve ome type of equation it i neceay to obtain numeical appoximation to the oot uing an iteative poce. An iteative poce tat with an appoximation x 0 to a oot λ fom which anothe appoximation x i obtained, and then anothe appoximation x, and o on. Fo an effective poce (fo a paticula oot) the ucceive value (o iteate) x, x, x 3,... hould become pogeively cloe to the oot λ. The poce i continued until an appoximation of the equied accuacy i obtained.. The ection on doubledated and undated gilt povide moe infomation on how to wok out the aumed edemption date. 3. The dity pice of a gilt i it total ettlement pice which include accued inteet. 4. The following convention will apply in the (vey ae) event that the Retail Pice Index i evied following an initial eleae. Indexlinked gilt (8month indexation lag) Indexlinked gilt (3month indexation lag) Dividend Ue fit publication Ue fit publication Accued inteet Ue fit publication Ue fit publication Pice / yield Ue evied publication fo RPIL N/A (no RPI tem) whee RPIL i the latet publihed RPI at the time of valuation. 5. If thi ha not yet been publihed by the authoitie, ee Annex A fo how to etimate it. 6. The exdividend date fo all gilt except 3½% Wa Loan i cuently the date even buine day befoe the dividend date; fo 3½% Wa Loan it i the date ten buine day befoe the dividend date. 7. Actual (i.e. publihed) cah flow on indexlinked gilt ae ounded a follow pe 00 nominal: (a) % IL 006 and ½% IL 0: ounded down to decimal place; (b) ½% IL 009, ½% IL 03, ½% IL 06, ½% IL 00, ½% IL 04
38 37 and 4⅛% IL 030: ounded down to 4 decimal place; (c) all othe indexlinked gilt (i.e. thoe fit iued afte Januay 00): ounded to the neaet 6 th decimal place.
39
40 United Kingdom Debt Management Office Eatcheap Cout Philpot Lane London EC3M 8UD
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