Seasonally adjusted prices for inflation-linked bonds

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1 CUING EDGE. INFLAION Seasoally adjusted prices for iflatio-liked bods Iflatio-liked bod markets are used ever more frequetly by policy-makers, ecoomists ad commetators to assess the market s opiio about the future path of iflatio ad real yields. But importat effects such as seasoality ad carry are ofte igored, which ca make such assessmets seriously flawed. here is a lack of cosesus amog ivestors ad traders o the best way to accout for these effects. Paul Caty proposes a method for calculatig the seasoally adjusted clea price of iflatio-liked bods that ca be used to isolate ad remove seasoality ad short-term carry factors Iflatio is highly seasoal. Food ad eergy prices esure that iflatio is also volatile from moth to moth. Oe-off future iflatioary evets may be kow about moths i advace. hese three factors apply to the cosumer price idex (CPI) measures used i the calculatio of global iflatio-liked bod (ILB) markets. hey make it difficult to extract meaigful iformatio from the prices of ILBs. I particular, they hide the tred rate of iflatio, which is, arguably, the most valuable piece of iformatio. Cetral bak policy-makers ofte refer to the level of breakeve iflatio ( the differece betwee the yield of a omial bod for a give maturity date ad the real yield of a iflatioliked bod of the same maturity) implied by the bod markets to justify iterest rate policy decisios. But they ted to do so without quatifyig or clarifyig the effects of seasoality ad carry. For example, if the has icreased by 5 basis poits over the course of a moth, oe may coclude that iflatioary expectatios have rise. However, if the carry for that period was a icrease of 10bp, the iflatioary expectatios have actually falle by 5bp. I the US, the Federal Ope Market Committee teds to avoid this problem by focusig o the forward iflatio rate betwee five ad 10 years time (the 5Y 5Y forward) as a key idicator. his has the advatage of always beig a whole umber of years i legth (so avoidig seasoal cosideratios) ad it bypasses the short-term, volatile part of the curve (up to five years i this case). he approach outlied below makes it uecessary to so limit the amout of tred iformatio that ca be extracted from the market. May ecoomists ad market commetators exhibit a lack of clarity whe talkig about real yield ad breakeve iflatio levels of ILBs. It is ot really meaigful to talk about the level of or chages to that level without also quatifyig the effects of seasoality ad carry (the differece betwee the forward rate ad the spot rate). Frequetly, there is also a lack of uderstadig of the dyamics of ILB prices. For example, cocepts that are relatively simple i omial bod markets, such as the steepess of the yield curve, become much more complicated i iflatio-liked bod ad swap markets. he real yield of a bod with a maturity of five years will react to seasoality differetly from a 10-year bod. hus the spread betwee the two yields (the steepess of the curve) is seasoally depedet. Aswerig the questio How steep is the real yield curve? is ot straightforward. his article addresses these problems ad describes a method that allows the effects of seasoality ad other short-term factors to be quatified ad elimiated, thus exposig the uderlyig iflatioary tred. It is beyod the scope of the article to outlie the geeral status of seasoality modellig. For most markets, seasoally adjusted series are published by the official statistical agecies (for example, Eurostat i the case of eurozoe Harmoised Idex of Cosumer Prices (HICP) ad the Bureau of Labor Statistics for US CPI). See Belgrade & Behamou (2004) for a more complete discussio o seasoality modellig i the cotext of iflatio markets, or DeLurgio (1998) for a more geeral treatmet of time-series aalysis. Here, we focus o the applicatio of the seasoal factors rather tha their calculatio. It is orgaised as follows. he first part outlies the uderlyig causes of the issues described above. he we itroduce the cocept of seasoally adjusted prices, before extedig the approach to take accout of volatile compoets (such as eergy) ad other short-term effects i the sectio o fully adjusted prices. he methodology ca be exteded to other iflatio products, icludig derivatives such as zero-coupo swaps. A ote o carry: iflatio strategists focus much of their atte- 106 Risk Jauary 2009

2 tio o the carry of breakeve iflatio rates, sice the forward rate several moths i the future ca be rather differet from the spot level. hese cosideratios become much less importat oce the trasformatio has bee made to forward-valued, fully adjusted breakeve iflatio rates. For a discussio of the carry effect o reasury Iflatio Protected Securities (ips), see D Amico, Kim & Wei (2007). Causes of seasoality, volatility ad other factors Seasoal iflueces are defied as periodic ad recurret; the effects should be reasoably stable i terms of timig, directio ad magitude. he mai causes of seasoality i cosumer price iflatio measures are clothig, accommodatio ad, particularly i the US, motor fuels. Oe ca see from figure 1 that most of the seasoality i the eurozoe HICP comes from clothig ad accommodatio, which represet oly aroud 10 of the idex (see table A for example compoet weights of the eurozoe HICP). he mai causes of volatility, or oise, i the CPI are the food ad eergy compoets, i particular motor fuels. hese compoets ca make short-term or tactical tradig decisios difficult as their effect o breakeve iflatio ca be tricky to quatify. I additio to seasoality ad volatility, there may be kow, or at least expected, future iflatioary or deflatioary shocks that do ot sit well uder the oise (or volatility) category as they are kow about some time i advace ad with some certaity of magitude. hese items would iclude the value-added tax icrease i Germay i Jauary 2007 ad the impact of bak iterest rate chages o the mortgage iterest paymet compoet of the Retail Price Idex (RPI) i the UK. Seasoally adjusted prices For simplicity, we deal iitially with Caadia-style ILBs with aual coupos. Later, the approach is exteded to semiaual Caadia-style ILBs ad the specific case of idex-liked gilts i the UK with a eight-moth lag. he approach ca be exteded to cover bods with other paymet covetios. Wheever the maturity of a ILB is ot a whole umber of years after its settlemet date, seasoality becomes a issue. For example, if a bod settles i April ad matures some years later i tember, the idexatio period icludes a extra six moths of iflatio from Jauary to Jue (due to the threemoth lag). Iflatio i this period is typically much higher tha from July to December, which meas that the overall breakeve iflatio rate should be higher tha the same bod with a whole umber of years left to maturity. his article focuses o quatifyig this effect. he timelie for the aual coupo Caadia-style ILB is show i figure 2. he idexatio period covered by the bod is from three moths before the settlemet date to three moths before the maturity date, where each idex observatio is a liear iterpolatio betwee the idex values for two moths ad three moths prior to the moth of the settlemet date (see later for precise defiitio). he times {t i (i = 1 to )} represet the coupo paymet dates of the bod. We start with the defiitio of the dirty price (DP) of a ILB with aual real cashflows at times {t i (i = 1 to )}: DP = I i df i (1) 1 Moth-o-moth seasoal variatios for the past five years due to clothig, eergy ad the rest of the idex for theeurozoe HICP Ex-obacco idex Clothig Accommodatio Eergy Rest of idex A. Compoet weights of the eurozoe HICP: 2008 Compoet Weight () Food ad o-alcoholic beverages 15.8 Alcohol ad tobacco 3.7 Clothig ad footwear 6.8 Housig ad utilities 15.3 Household furishigs ad equipmet 7.0 Health 4.0 rasport 15.7 Commuicatios 3.3 Recreatio ad culture 9.7 Educatio 1.0 Restaurats ad hotels 9.3 Miscellaeous imelie of cashflows for Caadia-style iflatioliked bod Iflatio protectio Idexatio time t Base-lag t Settle-lag t 1-lag t ( 1)-lag t -lag t Base t Settle t 1 t ( 1) t Settlemet time where df i is the omial discout factor relatig to the cashflow at time t i. We ca apply a multiplicative decompositio to the idex I t by writig: I t = t S t (2) where t is the tred compoet ad S t is the seasoal compoet. See the appedix for a example of seasoal factors defied i this way. Substitutig (2) ito expressio (1) gives: risk.et 107

3 CUING EDGE. INFLAION 3 Breakeve iflatio history of the BPS Eurozoe HICPx breakeve curve i April Seasoally adjusted Seasoally adjusted Apr 06 Jul Oct Ja 07 Apr Jul Oct Ja 08 Apr DP = i S i df i (3) Sice the paymets o the bod are aual, all the S i are equal, ad i particular equal to the seasoal factor at maturity, which we shall call. We make a importat assumptio here that the seasoal factors remai costat over time. here are alterative approaches to seasoality i iflatio models, such as expoetially decayig seasoal factors so that, after a log time, there is o seasoality evidet. Give the defiitio of seasoality (periodic ad recurret), it is a reasoable assumptio that the factors should remai costat. he we have: DP C i i df i (4) i1 he expressio i brackets is the seasoally adjusted dirty price (SADP). It cotais oly the tred growth rate of iflatio. So we have: DP = SADP (5) or: DP SADP = (6) he clea price (CP) of the bod is defied as: CP = I Settle DP RAI (7) where RAI is the real accrued iterest (the iterest eared o the bod sice the last coupo date before adjustig for iflatio). Substitutig the DP i (4) ito (7) gives: CP = I Settle RAI (8) Applyig the decompositio a secod time to the idex at the settlemet date of the bod, I Settle, gives: I CP = Base Settle RAI (9) Simplifyig this gives: CP = Settle RAI (10) he expressio i brackets is close to the seasoally adjusted clea price (SACP). We defie SACP as: SACP i1 i Settle df i RAI (11) he, from (10) ad (11), we have the relatioship: or: CP = ( SACP + RAI ) RAI (12) SACP = CP S Settle S + RAI Settle 1 (13) I markets where the real accrued iterest is small relative to the clea price (most developed iflatio-liked bod markets have coupos lower tha 5, ad are ofte paid semiaually) ad ( / 1) is close to zero, the secod term may be igored ad the followig approximatio may be used: SACP CP (14) Hece, the seasoally adjusted clea price is approximately equal to the origial CP multiplied by the ratio of seasoal factors for the settlemet date of the bod ad its maturity date. Note that whe the maturity date of the bod is o a aiversary of the settlemet date the = ad the seasoally adjusted clea price is equal to the quoted clea price. he expressios i (13) ad (14) form the key result of this article. It is a very quick ad cocise way to strip out the seasoal oise from a idividual iflatio-liked bod to expose the uderlyig tred iflatio rate. here are alterative methods (such as bootstrappig or fittig aual, zero-coupo iflatio curves through a set of bod prices) to extract the iflatio tred rate, but this approach keeps the aalysis at the idividual, tradable istrumet level ad does ot deped o a complex model. he seasoally adjusted real yield is simply the real yield cal- 108 Risk Jauary 2009

4 culated i the usual way usig the seasoally adjusted clea price. he seasoally adjusted rate is the differece betwee the omial yield to the same maturity ad the seasoally adjusted real yield. hese cocepts are useful for a umber of reasos, icludig: Historical aalysis of real yields ad breakeve iflatio rates. Plottig the term structure of breakeve iflatio whe bods mature at differet times of the year. Relative value tradig decisios across the curve. For example, whether to buy a bod maturig i July versus oe maturig i tember, or whether to ivest i a five-year bod or a 10-year bod. Pricig ew issues of iflatio-liked bods where there is o comparable bod of the same maturity moth. Figure 3 shows the history of the BPS Italia iflatio-liked bod for both uadjusted ad seasoally adjusted series. We see that the seasoally adjusted series is much less volatile tha the origial series ad follows a arrower rage. I particular, the seasoal peak i April 2007 is much less proouced. Figure 4 shows the eurozoe HICPx breakeve curve i April 2007 for both uadjusted ad seasoally adjusted rates. here are two mai features to poit out here: first, the uadjusted curve is extremely iverted while the seasoally adjusted is very well behaved at the short ed (that is, it is smoother ad moderately upward slopig). Secod, the DBRei April 2016 poit sits icely o the seasoally adjusted curve rather tha beig a outlier o the uadjusted curve. Note that the of this bod does ot really chage uder seasoal adjustmet. his is because the settlemet date ad maturity date are both i April so the bod has almost a whole umber of years left to maturity, a property touched upo earlier. he yield calculatio of the old-style eight-moth lag ILBs i the UK covers iflatio idexatio from the latest RPI release to the moth, which is eight moths prior to the maturity date. his meas that wheever a ew RPI umber is released the quoted yield shows a jump, eve for o chage i price. But a lot of this jump is due to the seasoal variatio of the idex. Figure 5 shows the recet breakeve iflatio history of the UKI 2009, ad demostrates the fact that seasoality accouts for most of the jump i yield experieced as a result of a ew RPI release. his is a good example of the applicatio of the seasoal adjustmet of yields because historical aalysis of UK eight-moth lag bods is otoriously difficult due to the discotiuous ature of the yield. So far we have cosidered oly bods with aual coupos. May ILBs pay coupos semiaually: ips i the US, Italia BPs i Europe ad all UK govermet iflatio-liked bods. I this case, the assumptio made i (5) above (that all the S i are equal, ad i particular equal to the seasoal factor at maturity) o loger holds. here will be two seasoal factors: oe for each of the two coupo paymet moths. We ca separate the two factors as follows. Recallig the defiitio of DP i (5) above, we rewrite it i terms of the two seasoal factors. he DP the becomes a weighted average of two seasoal compoets S 1 ad S 2. We assume a eve umber of coupo paymets for simplicity: DP i1,3,5,...,3,1 i S 1 df i i2,4,6,...,2, i S 2 df i (15) 5 Breakeve iflatio history of the UKI Ja 5, 2007 Seasoally adjusted RPI Release Mar 2 Apr 27 Ju 22 Aug 17 Oct 12 Dec 7 Feb 1, Mar Give a sigle bod, we do ot kow the term structure of discout factors so we caot calculate the weights exactly, but we ca make a guess by usig the real yield of the bod to calculate a approximate value for each seasoally adjusted real discout factor, ( i / )df i, i (15). he approximatio is: i 1 df i = 1 + RY (16) where RY is the real yield of the bod ad t i is the time util paymet. Followig a similar aalysis as earlier, the seasoally adjusted clea price i the semiaual case ca be writte as: SACP CP w 1 S 1 S w Settle 2 S 2 w 1 w 2 (17) where w 1 ad w 2 are the approximate cotributios to the total clea price due to each of the two seasoal factors, as i (18) ad (19). We have substituted the real discout factors implied by the yield of the bod for the product of the idex ratio ad omial discout factor, ( i /I Settle )df i. he weights are defied as: w 1 = w 2 =,3,5,..., 3, 1 i=2,4,6,..., 2, RY (18) RY (19) Seasoally adjusted forward prices he iflatio protectio embedded i iflatio-liked bods cotais some kow, historical iflatio due to the delay mechaism (or lag) i the idexatio (for example, three moths i Caadia-style bods). If this kow period cotaied evets that meat that iflatio diverged from the seasoally adjusted tred, it is importat to accout for them. We do this by cosiderig forward-lookig breakeve rates startig from the last kow idex publicatio. Recall the timelie i figure 2. I the period betwee t Settle-lag ad t Settle (a period usually of three moths) there will be oe or two CPI releases. So we kow for certai some iformatio about iflatio i the period of idexatio of the bod. But we are iter- risk.et 109

5 CUING EDGE. INFLAION 6 imelie of cashflows for Caadia-style iflatioliked bod ad latest CPI release dates Kow iflatio period t Settle-lag Origial idexatio t Fwd Settle-lag Forward idexatio out of latest kow CPI (April) Mar 1 Apr 1 May 1 Ju 1 Jul 1 April CPI published t Settle t Fwd Settle Settlemet/publicatio time ested i the market s expectatio of future iflatio, so we eed to exclude the kow iflatio i the period betwee t Settle-lag ad the latest kow CPI release date. o do this, we calculate the forward price up to the furthest kow settlemet date give the latest CPI release. For example, whe the Frech CPI idex for April 2007 was published o May 15, 2007, the furthest forward date i the future where we could calculate the clea price with certaity (assumig a give repo rate) was July 1, O this date, the three-moth lag would refer to the April idex value; o ay later date, we would eed to iterpolate betwee the April ad May values, ad so would require kowledge of the May prit. Figure 6 illustrates this poit. he applicatio of the seasoal adjustmet the follows i exactly the same way as earlier except that, i the case of the breakeve iflatio rate, the forward omial yield to the same settlemet date should be used. Fully adjusted prices he aalysis ca be exteded further by icorporatig a extra term i the idex decompositio used i (2) earlier: I t = t S t O t (20) he term O t is called the outlier idex ad cotais iformatio about expected future oe-off shocks to the iflatio idex. For example, if we expected the Moetary Policy Committee i the UK to icrease iterest rates i three moths time, we might iclude a term that looked like this: {O t } = {1.000, 1.000, 1.002, 1.002, 1.002} i order to reflect the icrease i iflatio due to the mortgage iterest paymet compoet of the RPI. Usig a similar argumet to that above for the seasoal adjustmet, we defie the fully adjusted clea price (FACP) as: FACP CP 1 (21) O Maturity his exteded aalysis is required whe there are sigificat oseasoal items affectig the shorter ed of the real yield curve. For example, i the ips market, gasolie price volatility ca mea that eve sice the last CPI release there has bee cosiderable movemet i motor fuel prices. his is iformatio that should be icorporated ito ay breakeve iflatio adjustmet. Paul Caty is Europea head of iflatio tradig at Deutsche Bak. paul.caty@db.com Appedix A: seasoal factors We model the iflatio idex as I t = t S t where t is the tred idex ad S t is the idex of seasoal factors. For example, S t could be the list of factors show i the secod colum of table B (for completeess the equivalet set of additive seasoal factors is icluded i the third colum). A importat assumptio i the aalysis is that the complete series of seasoal factors is geerated by repeatig the set of 12 mothly factors. hat is, the seasoal factors remai costat over time. For a give base date t ad a forward date, the forward idex is give by the followig expressio: Refereces Belgrade N ad E Behamou, 2004 Impact of seasoality i iflatio derivatives pricig Workig paper QRFI 08-04/2, CDC Ixis Quatitative Research D Amico S, D Kim ad M Wei, 2007 ips from ips: the iformatioal cotet of reasury Iflatio- Protected Security prices Workig paper ( ( )) S I = I t exp r t, t S t where r t, is the cotiuously compouded tred rate of growth betwee times t ad. I most iflatio-liked bod markets, the iflatio idex is calculated usig a iterpolated three-moth lag. he iterpolated idex is defied as: I Iterp = I 1 + w( I 2 I 1 ) where I 1 is the idex three moths prior to the moth of settlemet, I 2 is the idex two moths prior to the moth of settlemet, w = 1 (d 1 1/d 2 ), d 1 is the day of the moth of the settlemet date ad d 2 is the umber of days i the settlemet moth. he seasoal adjustmet of prices i this case requires a approximatio. We use the followig approximatio: I Iterp Iterp S Iterp where Iterp is the tred idex ad S Iterp is the seasoal idex i the usual decompositio, which ca be made as the cross terms i the expasio are egligible. B. Example of multiplicative ad additive seasoal factors Moth Multiplicative Additive () Jauary February March April May Jue July August tember October November December DeLurgio S, 1998 Forecastig priciples ad applicatios McGraw-Hill 110 Risk Jauary 2009

CHAPTER 2 PRICING OF BONDS

CHAPTER 2 PRICING OF BONDS CHAPTER SUARY This chapter will focus o the time value of moey ad how to calculate the price of a bod. Whe pricig a bod it is ecessary to estimate the expected cash flows ad