The Macrotheme Review A multidisciplinary journal of global macro trends

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1 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 The Macroheme Revew A muldscplnary journal of global macro rends A mehodology o esmae he neres raes yeld curve n Illqud Marke: he Tunsan case Fama Chakroun* and Fah Abd Faculy of Managemen and Economcs, Sfax, Tunsa fama_chakroun@une.n* Absrac The am of hs paper s o develop a mehodology o esmae he neres raes yeld curve and s dynamcs n he Tunsan bond marke, whch s consdered as an llqud marke wh a low radng volume. To acheve hs, frs, we apply he cubc splne nerpolaon mehod o deal wh he mssng observaon problem. Second, we focus on he work of Vascek (1977) and Cox-Ingersoll and Ross (CIR) (1985), we esmae each model and dscuss s performance n predcng he dynamcs of he neres raes yeld curve usng Ordnary Leas Squares and Maxmum Lkelhood Esmaon (OLS and MLE) mehods. The daa sample consss of Treasury bond prces for he perod from 14 July 004 o 10 Sepember 01 colleced from over he couner marke. Ths assessmen s done by Malab sofware usng he specfed algorhms. The resuls sugges ha he cubc splne mehod s accurae for esmaon he average yeld curve of neres raes. Then, he esmaon of Vascek (1977) and CIR (1985) models generae an upward slopng yeld curves and seems suable o replcae he sylzed facs of he neres raes n he Tunsan bond marke. Fnally, he forecasng of he yeld curve predcs an economc growh n he fuure characerzed by a hgher nflaon Keywords: Illqud Marke, Yeld Curve of Ineres Raes, Cubc Splne, Vascek (1977) model, CIR (1985) model, OLS, MLE 1. Inroducon Ths paper s an aemp o develop a mehodology o esmae a yeld curve of he neres raes and s dynamcs n he Tunsan bond marke. Durng he las decade, he effcency of he fnancal marke has been mproved hrough varous reforms. The objecve of he Tunsan auhores s o promoe he developmen of he bond marke by generang new producs such as he long erm Treasury bonds (BTA), he shor erm Treasury bonds (BTC) and zero coupon Treasury bonds, as well as he esablshmen of a group of prmary dealers (SVT) responsble for he anmang of he marke. More han 90% of he ssued bonds are governmen secures and he number of corporae ssuers s sll lmed alhough fscal ncenves have been offered o nduce lsng. The bonds are usually lsed on he Sock Exchange and herefore regulaed and supervsed by he Fnancal Markes Councl (CMF). However, he bond marke was marked by a 18

2 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 sgnfcanly lower volume of radng, abou 1,678 MTD on he Sock Exchange offcal quoaon. A declne by 37.9% compared o 010 s affeced parcularly by he suspenson of radng on wo occasons over he frs quarer of 011 and shorer radng sessons a he Sock Exchange. Therefore, he Tunsan bond marke s smaller and llqud compared o oher bond markes of he developng counres where only abou a handful of lqud bonds ges raded a day. In fac, he sparseness or nfrequency 1 of daly Treasury bonds ransacons explans n consequence he naccuracy of he neres raes yeld curve on he bond marke. The drawback s ha here s no specfc erm srucure model of neres rae and he marke operaors devse a proxy of he yeld curve based only on he lqud bonds. Thus, he unevenly dsrbued maures of he dfferen bonds make he esmaon very dffcul and he marke less lkely o form an enre and smooh yeld curve. However, mos researches have been successfully appled o he developed bond markes, parcularly he US bond marke whereas lle aenon has been pad o he emergng markes (see for nsance, Jan-Hsn Chou e al. (009), Gonzalo Corazar e al. (007), Ch Xe e al. (006), Dua e al (005), Alper e al (004)). The exsence of bond prces for a wde range of dfferen maures n he developed bond markes makes easy o exrac a yeld curve of neres raes ha explans observed prces. The developed bond marke s generally well esablshed and composed of relavely lqud secures wh shor and long maures, n some counres such as he Uned Saes, zero-coupon bonds (Srps) of dfferen maures are ndvdually raded. The purpose of hs paper s wofold: frs, we esmae he neres raes yeld curve usng he Tunsan Treasury bond prce daa. We apply he cubc splne mehod o address he problem of mssng observaons n order o consruc a smooh and connuous yeld curve. The choce of hs approach can be explaned by s flexbly and easness o mplemen as opposed o he oher sascal mehods. In addon, can be used boh for nerpolaon and smoohng. Second, we concenrae on he work of Vascek (1977) and CIR (1985). We calbrae each model and dscuss s performance n predcng he yeld curve of he neres raes usng he OLS and MLE mehods. Therefore, one of he key pons n hs paper s o compare hese models n erms of her ably o fnd a smooh yeld curve whch replcaes he sylzed facs of he neres raes on he Tunsan bond marke. Ths paper s organzed as follows. The nex secon s devoed o a leraure survey. Secon 3 descrbes he mplemenaon of he cubc splne mehod. Secon 4 presens he esmaon of he Vascek and CIR erm srucure models. The emprcal resuls are dscussed n secon 5. Secon 6 concludes he paper.. Leraure Revew The yeld curve or he erm srucure of he neres raes descrbes he relaonshp beween he yeld of a bond and s maury. I s he mos mporan concep n prcng all he fxed ncome secures and neres rae dervaves such as bond opons, caps, swaps ec, whose payoffs are srongly dependen on he neres raes. In addon, he yeld curve reflecs he moneary sance and nflaon expecaon. I s also a key ndcaor monored by fnancal nsuons o manage he neres rae rsk and make nvesmen decsons. Therefore, he research of how o esmae a fed yeld curve has become a very mporan ssue and capures boh he academc and praccal 1 Tradng frequency s defned as he number of days for whch we have a leas one ransacon of a Treasury bond of a specfc maury over all avalable radng days. 19

3 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 neress. Praconers prefer an approach ha s accessble, sraghforward o mplemen and as accurae as possble. In general here are wo dsnc approaches o esmae he yeld curve of he neres raes. The frs approach ncludes he models of fng he yeld curves o he marke daa usng sascal mehods. The man purpose of hese models s o fnd a smooh funcon beween yelds of bond prces and me o maury. One of he yeld curve esmaon mehods s he Boosrap naed by Blss and Fama (1987) from dscree spo raes o f a smooh and connuous yeld curve o he marke daa. However, varous curve fng splne mehods have been nroduced. The mos popular example of hese procedures s he semnal work of MacCulloch (1971, 1975) whch focused on esmang zero-coupon yelds and dscoun facors usng he polynomal splnes. He found ha he dscoun funcon could be fed very well by cubc or hgher order splnes and he esmaed forward raes are a smooh funcon. Vascek and Fong (198) ry o use a hrd order exponenal splne o calbrae he dscoun funcon and show ha hese models have a beer fng performance han he polynomal splnes models. Then, Nelson and Segel (1987) and Svensson (1994, 1996) suggesed paramerc curves ha are flexble enough o descrbe a whole famly of he observed yeld curve shapes. However, hs approach akes a sac vew argeng solely he shape of he yeld curve of he neres raes. In he second approach, he me dmenson s concerned. An enormous leraure focused on consrucng and esmang he erm srucure models by specfyng parcular funconal forms for he dynamcs of he neres raes. In fac, hese models posulae explc assumpons abou he evoluon of he facors drvng neres raes and deduce characerzaons of shapes and movemens of he yeld curve n frconless markes havng no arbrage. One-facor models are he frs sep n modellng he erm srucure of neres raes. These models are grounded on he esmaon of bond yelds as funcons of he shor erm neres raes. Vascek (1977) s one of he frs erm srucure models whch was used exensvely n valung bond opons, fuures and oher fxed ncome dervaves. I remans he benchmark core model for s smplcy n esmaon and possbly o easly calculae bond prces and yelds. Thereafer, a famous exenson of he Vascek model s he CIR (1985) model, whch uses an neremporal general equlbrum asse prcng model o sudy he erm srucure of he neres raes. The model provdes soluons for bond prces and a complee characerzaon of he erm srucure whch ncorporaes rsk premums and expecaons for fuure neres raes. Subsequenly, he model s exended o nclude mulple facors and o value nomnal bonds and nomnal clams. Longsaff and Schwarz (199) develop a wo facor model usng he neres rae and s nsananeous varance as sae varables o derve a closed form of expressons for he prce of he dscoun bonds and dscoun bond opons. The major benef of hese models s o provde a lnk beween neremporal asse prcng heory and he erm srucure of he neres raes ha produces a frequenly convenen closed form of soluon for asse prces. However, all hese models generally mply a erm srucure of he neres raes conflcng wh he marke yeld curve. In fac, he drf and volaly of he neres raes and he marke prce of rsk are consdered as he sngle source of uncerany o deermne he dynamcs of he erm srucure. Ths problem s solved by Ho and Lee (1986) and Hull and Whe (1990) who used dfferen nformaon o characerze he yeld curve dynamcs. The nformaon se ncludes he spo neres rae, volaly and he funconal form of he yeld curve. The model of Heah, Jarrow and Moron (199) also allows capurng he full dynamcs of he enre yeld curve n an arbrage free framework. Ths s an exenson of one facor model developed by Ho and Lee (1986) a a mul facor model by consderng forward raes raher han bond prces. 0

4 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT The mplemenaon of cubc splne mehod 3.1 Daa consrucon and analyss The daa are used on a daly bass. They, generaed used he prcng of 31Tunsan reasury bonds 3 (10 long erm Treasury bonds BTA and 1 shor erm Treasury bonds BTC ). The shor erm maury reasury bonds are he governmen reasury nsrumens for he maures of less han one year as ssued wh 13, 6 and 5 weeks maury. Currenly, only he BTC of 5 weeks are lsed on he marke. However, he BTA are he long erm publc deb suppor. Ther maury vares beween 6, 7, 10, 1 and 15 years. The daa run from 14 July 004 o10 Sepember 01 hus n oal 1050 observaons. The yeld o maury 4 for all he Treasury bonds s no drecly observable on he marke, whch means ha hey need o be compued 5 from he marke prces of he Treasury bonds usng he formula of excel spreadshee. Emprcal analyss covers he perod from 1 Aprl 010 o 10 Sepember 01 provdng 137 observaons n oal. I s a shor sample due o he sparse Treasury bond prces. Then, he daa descrbe he hsorcal changes of he redempon yeld of a Treasury bond wh maures: 3 monhs, 6 monhs, 9 monhs, year. However, he 5-week Treasury bond yelds are consdered a beer approxmaon of he shor neres rae. A lo of sudes on he yeld curve of he neres raes were conduced n he developed counres lke he USA, Japan, Germany and ohers. In hese sudes, he hree-monh Treasury bonds are consdered a beer approxmaon of he shor neres rae. For nsance, Bernnan and Schwarz (198) appled a model for he prcng of he US Treasury bonds from 1948 o Chan Karoly, Longsaff and Sanders (199) compared dfferen models usng he Treasury bll yelds as a proxy o selec he bes fng model of he shor erm neres raes. Mao (005) analyzed he mmunzaon porfolo agans he neres rae rsk from he md 1990 and early 000 on US Treasury bond marke. Munnck and Schoman (1994) also developed a model for he shor erm neres rae on he Duch bond marke. Fgure 1 llusraes he evoluon of he 5-week Treasury bond yelds whch are used o esmae and smulae he shor neres rae processes. The daabase s colleced from Inernaonal Arab Tunsan Bank (BIAT). 3 These bonds are ssued by aucon on he capal marke for a dnars value and negoable a all banks nervenng on he money marke. 4 The day coun bass convenon used for BTA s acual / 365 and acual/360 for BTC. 5 To undersand he yeld o maury, one mus have ha he marke prce of a bond s equal o he presen value of s fuure cash flows, as showng n he followng formula: n P c 1 r r n 1 1 F Where; P : Marke prce of he Treasury bond, n : Number of perods, c : Coupon paymen, r : yeld o maury, F : Maury value, and : Tme perod when paymen s o be receved 1

5 Yeld Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT Evoluon of he 5-week Treasury Bond yeld Orgnal Daa Smoohed Daa Usng 'movng' Fg.1: Evoluon of he 5-week Treasury bond yelds The descrpve sascs show ha he Jarque-bera es for normaly rejecs he null hypohess wh p-value 1e-003 (0.1%). Indeed, he daa of 5-week Treasury bond yelds have respecvely coeffcens of skewness and kuross of and whch are dfferen from a normal dsrbuon. Moreover, he Augmened Dukey-Fuller (ADF) es appled o he daa rejecs he null hypohess a he 1% rsk level, whch means ha hs seres s saonary. The resuls are repored n able 1. Table1: Resul of ADF es for 5-week Treasury bond yelds Varable ADF sasc Crcal value 1% 5% 10% 5-weeks Treasury bond yelds *** Symbol (***) denoes 1% level of sgnfcance. The goodness of f of he daa s measured by he movng average fler. A movng average smoohs ou he random flucuaons n he daa o show a paern or rend more clearly. Thus, he average values from a specfc number of daa pons (se by he perod opon) are used as a pon n he rendlne. If he perod s se o, for nsance, he average of he frs wo daa pons s used as he frs pon n he movng average rendlne. The average of he second and hrd daa pons s used as he second pon n he rendlne, and so on. 3. Cubc splne mehod In mahemacs, a cubc splne s a pecewse polynomal funcon of order hree whch s wce dfferenable a each pon known as kno pons or nodes. To consruc a se of cubc splnes, le he funcon () ; R denoe he cubc polynomal assocaed wh he segmen, 1 3 b c r 1,... R ( ) a.where n s he number of marke observaons, r Tme represens kno pon, and represens he me o maury of marke

6 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 observaon, n 1splnes, and hree coeffcens per splne. The coeffcens of he cubc splne funcon defned over he nerval can be obaned by mposng he followng consrans: a ( 3a 6a 1 6a n1 ) ( ( b 0 ( n b ( ) b ) b n1 1 1 ) b ) 1 b n1 ( 0 c ( ) c, 1 ) r 1 c 1 0 r The frs se of n 1consrans requres ha he splne funcon jon he kno pons perfecly. The second and hrd se of n consrans requre ha he frs and second dervave consrans mach he adjacen splnes. Fnally, he las wo consrans are end pon consrans ha se he dervave equal o zero a boh ends. Thus, he lnear algebrac sysem consss of 3n 3equaons and 3n 3unknowns coeffcens can be solved o produce an opmal pecewse cubc splne. Therefore, he use of a pecewse cubc splne echnque ha passes hrough he observed marke daa pons creaes a fed smooh yeld curve and avods knks. MacCulloch (1971) nroduced he splne mehod o accommodae he daa srucure by dvdng he maury range no subnervals so ha he number of he Treasury bonds n each subnerval s roughly he same and he splne funcons are conneced o ensure connuy and dfferenably a he nodes. However, he lnear nerpolaon s smple o mplemen and closely racks observed he marke neres raes. All he observed marke daa pons are conneced by a sragh lne o form a complee yeld curve. As a consequence, lnear nerpolaon can be presened n a closed form, whch smplfes he nerpolaon process. R( ) R( ) R 1 R( ) 1. Here, s he marke observaon ndex wh me o maury of, and R() represens he yeld correspondng o maury, where 1. Lnear nerpolaon s napproprae for modelng he yeld curves ha change he slope frequenly and end o produce knks around ranson areas. In fac, he pon beween wo nodes canno be accuraely esmaed usng a sragh lne. Moreover, Treasury bonds rade nfrequenly so ha for every parcular day here are Treasury bond prces for only a few maures. Ths mssng observaons problem makes dffcul, and somemes mpossble, o esmae he yeld curve usng only curren daa. For hs reason we deermne he average yelds of he Treasury bonds for each maury n order o consruc a connuous curve jonng marke observed daa as smoohly as possble. We ry o apply he lnear and cubc splne nerpolaon mehods o complee he mssng daa. Therefore, he performance of he cubc splne mehod n esmang a smooh curve s beer han he lnear mehod, as depced n fgure. Indeed, he esmaon of daa usng cubc splne has he advanage ha a large number of daa pons can be conneced. Ths s he mos sraghforward means ha gves meanngful daa o consruc a correc neres rae yeld curve n he Tunsan bond marke. 3

7 Yeld Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT Cubc Tme o maury Fg.: Inerpolaon mehods o plo yeld curve Fgure 3 plos he average yeld curves for nne years. We can also nfer ha he cubc splne nerpolaon seems able of generang a general rend of neres raes on he Tunsan bond marke. All n all, we oban upward shaped yeld curves wh oscllaons whch s dfferen from he observed average yeld curve 6. The CMF has underaken, wh he marke parcpans, a smple reflecon o consruc a yeld curve. Ths curve based on he aucon prce of BTA and BTC s arbrary and mprecse. The average yeld curves from 004 o 010 are all an upward rend and he poson of he curves changes n he whole sample perod. The plos of he dos show low dfferences. Neverheless, he yeld curve of year 011 revealed ha he changes of he yelds of all he maures follow frs a fla and aler a downward rend. The downward sloped s due o he dos esmaed whch s ofen a harbnger of an economc slowdown. Therefore, he shape of he splned curve affecs s smoohness. daa CubcSplne Lnear 6 The yeld curve s dffused va he Tunsan Deposory Learnng (STICODEVAM) and he Fnancal Marke Councl (CMF). 4

8 Yeld Yeld Yeld Yeld Yeld Yeld Yeld Yeld Yeld Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT Average yeld curve Average yeld curve Average yeld curve Tme o maury Tme o maury Tme o maury 8 Average yeld curve Average yeld curve Average yeld curve Yeld curve Daa pon Cubc nerpolaed pon Tme o maury Tme o maury Tme o maury 7 Average yeld curve Average yeld curve Average yeld curve Tme o maury Tme o maury Tme o maury Fg.3: Plos of he average yeld curves of neres raes Moreover, when esmang he curve by an nerpolaon beween he nodes, he user mus consder he conflcng ssues. There s a need o balance beween smplcy and correcness, and hence a radeoff beween he ease of he use and he accuracy of he resul. In hs case, we can see ha he cubc splne does no resul n a smooh curve. We accep a lower degree of accuracy a he nodes, n favour of smoohness across he curve. Therefore, we recommend usng a cubc splne mehod whch ensures ha he curve passes hrough all he dscree daa pons. Ths enables praconers o f a reasonable and accurae yeld curve o he observed marke neres raes, bu may oscllae wldly. However, we need o use paramerc models n order o esmae a connuous yeld curve from a se of dscree observaons and elmnae he oscllaons. 4. Esmaon of he Vascek and CIR models The Vascek (1977) and CIR (1985) models assume ha he neres rae s a dffuson process, defned by he followng sochasc dfferenal equaons: dr ( r ) d dw, (1) dr r ) d r dw, () ( Where W() s a wener process whch models he random marke rsk facor 7 and parameer deermnes he volaly of he neres rae. Boh models specfy ha he neres rae ends o rever o s long erm mean, and can hen be nerpreed as he speed of hs reverson. One 7 We assume ha he marke prce of rsk 0. 5

9 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 of he drawbacks of he Vascek model s ha can adm negave values of he neres rae. The CIR model resolves hs unforunae feaure by nroducng he square roo erm r. The Vascek yeld curve of he neres raes n he rsk neural world s exraced from he followng equaon: T 1 e T R(, T) r 1 e 3 T 4 ( T ) (3) The yeld curve of he neres raes n he rsk neural world dervng from he CIR model s equal o: 1 R(, T) B, T. r loga, T T (4) Where; 1 exp A (, T).exp T 1 B (, T) exp T T exp T 1 And A common pracce s o calbrae and predc he neres raes erm srucure models. Several advanced economerc mehods have been developed. For example, he resuls of Brown and Dybvg (1986) n examnng he CIR (1985) model wh cross seconal daa are declned by Gbbons and Ramswamy (1993) usng he generalzed mehod of momens (GMM) on shor erm Treasury blls daa, bu hey are acceped by Pearson and Sun (1994) usng he MLE. Chan, Karoly, Longsaff and Sandres (199) used he GMM mehod o esmae and compare a varey of erm srucure models on he US marke. Ther resuls demonsrae ha Vascek (1977) and CIR (1985) models perform poorly compared o Dohan (1978) and CIR (1980) models. Jensen (000) has demonsraed, hrough a new mplemenaon of he Effcen mehod of momen (EMM) esmaon prncple, ha he Longsaff and Schwarz (199) model s nadequae for he descrpon of he neres raes. The EMM proposed by Gallan and Tauchen (1996) s he bes alernave wdely used n he several well known models when he MLE mehod s mpossble, for example, he sochasc volaly models (Andersen and Lund (1997)), affne erm srucure models (Da and Sngleon, 000) and quadrac erm srucure models (Ahn e al., 00). However, he non-paramerc procedure s proposed by A Sahala (1996) o esmae he dffuson coeffcen usng 7 days Eurodollar depos raes from he Bank of Amerca durng he perod from June 1973 o February The man resuls are ha he f s mperfec and he ess rejec every paramerc model of he spo rae. Sanon (1997) fnds nonlneares n he drf coeffcen of one facor models usng daly daa rae of 3 monhs of he U.S. Treasury blls from 1965 o 1995.Oher approaches have been appled o he emprcal esmaon of neres rae erm srucure models such as Kalman fler and Markov Chan Mone Carlo (MCMC) mehods. In recen years, Bayesan esmaon has also pad aenon o he developmen of MCMC 1 / 6

10 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 smulaon for calbrang he neres rae models. Ths Bayesan approach was nflexble n he pas decade due o he dffculy of s mplemenaon for large scale problems. My assgnmen s o esmae he parameers of Vascek (1977) and CIR (1985) neres erm srucure models n rsk neural world usng he OLS and MLE echnques. 4.1 Ordnary leas squares Vascek (1977) model s equvalen o a frs order auoregressve AR (1) model. The dscrezed verson wh me sep 1 r c br 1 (5) c 1 e Where he coeffcens are; b e 1 e / And s a Gaussan whe nose. Indeed, he followng expressons for, and hold, ln( b) ˆ (6) c ˆ 1 b (7) ˆ / (8) b 1 / ln( b) Smlarly for he CIR model descrbed by equaon (), he dscresed equaon s gven by: r r ( r r, ) (9) wh s a Gaussan whe nose process. For performng OLS we ransform (9) no: r r r. Then, he drf nal esmaes are found by mnmzng he r r OLS objecve funcon: ˆ N 1 1, ˆ arg mn r r r, 1 r r (10) The dffuson parameer nal esmaeˆ s found as a sandard devaon of resduals. Thus, he MATLAB code mplemenaon can solve he problem que easly. Snce our daa se consss of daly observaon, so can be aken as 1/ Maxmum lkelhood One of he alernave mehods s o esmae he parameers whch maxmse he lkelhood funcon. 7

11 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 Gven N observaons of 5-week Treasury bond yelds N, The lkelhood funcon s N 1 as follow: L pr r ; r 1 1 (11) wh me sep;,, a parameer vecor o be esmaed and p ( r ) defned as he ranson funcon of he Vascek and CIR process respecvely. Then, he log-lkelhood funcon N 1 s, ln L ln pr r ; (1) Therefore, he maxmum lkelhood esmaor ˆ of parameer vecor s: ˆ ˆ, ˆ, ˆ arg max ln L( ) (13) Moreover, he applcaon of he maxmum lkelhood requres he specfcaon of he ranson funcon of each process. Hence, he condonal densy funcon for Vascek model s gven by: 1 r r 1 1 e e pr r ;, exp ˆ ˆ (14) 1 e Wh, ˆ. The correspondng log-lkelhood funcon s: ln L (15) 1 N N 1 N 1 1 ln( ) ln( ˆ) r r e e ˆ Now, le us focus on he MLE of he CIR model. The ranson densy funcon for he CIR process has a closed form expresson. u uv r r ;, ce Iq uv p v, (16) Where; c 1 e, And I q uv u cr e v cr, q 1,, q 1 1 ; ; s modfed Bessel funcon of he frs knd of order q. Feller (1951) shows ha he dsrbuon funcon of he neres raes s a non cenral ch-squared wh q degrees of freedom and non cenral parameer u. Hence, he log-lkelhood funcon can be derved from 8

12 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 he condonal densy funcon as: N 1 v ln L N 1 ln c u v 0.5q ln lniq uv (17) 1 u Therefore, a numercal soluon based on he funcon fmnsearch 8 n MATLAB s execued o solve he opmsaon problem (13) and (17). 5. Inference - Esmaon resul analyss s carred ou usng he OLS regresson and MLE echnques for he 5-week Treasury bond yelds. Table shows he parameer esmaes of vascek (1977) and CIR models, respecvely. The esmaon of parameer vecor,, Table : Parameer esmaon of Vascek and CIR models Model Vascek CIR Parameer ˆ ˆ ˆ ˆ ˆ ˆ OLS e MLE e Jackknfe e The OLS and he MLE produce he same values of parameer esmaes wh low sandard devaon. In fac, he mos relevan problem n he presen conex s he esmaon bas n he praccal use of economerc esmaors for he connuous me dscresed models. In general, dscresaon nroduces an esmaon bas snce he nernal dynamcs beween he samplng pons are gnored. Thus, he msspecfcaon bas resuls n an nconssen esmaon of he parameers of he connuous me model wh consequen bas effecs on he shape of he yeld curve. Lke he OLS usng a nave regresson, he maxmum lkelhood esmaor can have a subsanal bas n he esmaon of connuous me models, such as dffuson models of shor he neres raes. For example, s well known ha he Black Sholes sock opon prce esmaes are based. The bas s manly found n he drf parameer whch happens when he mean reverson parameer s small. As menoned by Yu and Phllps (005), he MLE and he OLS produce a bas of more han 00% n he bond prce. In spe of s generally good asympoc properes, he approxmaon of he maxmum lkelhood esmaor s generally nconssen. Consequenly, s exremely mporan o esmae hese parameers whou any bas. Furhermore, A Sahala (008) and Tang and Chen (009) used boosrappng o reduce MLE bas. Bekaer, Hodrck, and Marshall (1997) used Kendall s mehod o correc for bas n esng he expecaons hypohess of he neres raes yeld curve. In hs conex, he jackknfe echnque s appled o assess he varably of he MLE for he Vascek and CIR parameer esmaes. bas n 1 ˆ ˆ jack.. Where: ˆ S( x) an esmaor for The bas of MLE s deermned as: daa se X x, x,..., x ) ; S( x ) for 1,,..., n be he h jackknfe replcaon of ˆ and ( 1 n ˆ ( ) 8 Ths funcon s based on he Nelder-Mead smplex mehod. 9

13 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 ˆ. s ˆe n ˆ ( ) 1 jack n. The jackknfe esmae of sandard error wll hen be defned by: n 1 ˆ ˆ n. 1/ Table 3: Jackknfe bas correced of esmaed parameers Model Vascek CIR Parameer Bas jack Se jack ê jack Bas jack Se jack ê jack ˆ ˆ e e e e ˆ e e e e e Table 3 shows ha he sandard error of he parameers s very small, so he maxmum lkelhood esmaor s unbased and he parameers of he model are esmaed very accuraely. On he oher hand, he jackknfe produces he same values of he MLE parameer esmaes. Conversely, he sandard devaon of he sample remans low. Indeed, our fndngs ndcae ha he jackknfe allows bas reducon n he parameer esmaes under he model msspecfcaon whou ncreasng he varance. However, f we plo a hsogram for each esmaon mehod based on smulaon, no preference beween OLS and jackknfe s observed (fgure 4 and 5). 70 OLS 70 MLE Jackknfe Fg.4: Dsrbuon f of hree mehods of he Vascek esmaon model based on smulaon. 30

14 Yeld Yeld Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT OLS 60 MLE Jackknfe Fg.5: Dsrbuon f of hree mehods of he CIR esmaon model based on smulaon. As a resul, he MLE has lle asymmery han he OLS, so s he bes mehod of esmang he erm srucure model parameers. Fgure 6 shows a seep upward slopng yeld curve, whch means ha he Treasury bond yelds rse as maury exends. Vascek model CIR model Tme o Maury Tme o Maury Fg.6: Esmaes of Vascek and CIR yeld curves Indeed, he resulng yeld curve of CIR (1985) model s very close o ha of Vascek (1977). A sharply upward slopng has ofen preceded an economc upurn and he neres raes wll rse sgnfcanly n he fuure. Under hs suaon, f nvesors hold nvesmen, hey may receve a hgher rae n he fuure. However, he emprcal fndngs specfed by hese models are n lne wh he sylzed fac of he neres rae yeld curve esmaed usng he cubc splne mehod. 31

15 Raes Yeld Yeld Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT Forecasng analyss The smulaon of he erm srucure models s based on he dscresaon scheme as descrbed above. In fac, we use a forecasng perod of hry years wh a me sep of one year. We run he smulaon usng he maxmum lkelhood parameer esmaes for Vascek and CIR models respecvely. We sar he smulaon from he same pon ( r % ). Vascek model CIR model Tme o Maury Tme o Maury Fg.7: Smulaon of Vascek and CIR yeld curves The smulaon of he yeld curve wh Vascek and CIR models generaes an upward slopng curve (Fgure 7). Ths explans he sylzed fac ha he shor erm yelds are lower han he long erm yelds, as shown n fgure 8. Shor rae and long rae mouvemens wh Vascek model Shor rae and long rae mouvemens wh CIR model Tme Shor rae Long rae Tme Fg.8: Spread beween shor and long erm neres raes The above fgure compares he smulaed long and shor erm Treasury yelds. The spread s relavely small whch resuls n yeld curve seep. In fac, wh Vascek and CIR models he boh raes move up and down somewha ogeher over me. Therefore, he parallel shfs are common, n lne wh he movemens of he moneary marke rae (TMM). The Cenral Bank of Tunsa (BCT) decded o reduce he money marke rae o 3.76 percen n Augus 011 whch was sll 3

16 r() r() Raes (%) Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 fallng unl 01. Fgure 9 llusraes he evoluons of he TMM rae from January 010 o December TMM Augus 010 Augus 011 Augus 01 Tme Fg.10: Evoluon of moneary marke rae However, he posve slope reflecs nvesors expecaons for an economc expanson and hgher neres raes n he fuure. Ths ancpaon leads o expecaons ha he polcy makers wll ghen moneary polcy by rasng he shor erm neres raes n he fuure o fgh nflaon. I also creaes a need for a rsk premum assocaed wh he uncerany abou he fuure rae of nflaon and he rsk whch poses for he fuure value of cash flows. The nvesors assess hese rsks no he yeld curve by demandng hgher yelds for maures furher no he fuure. In consequence, he long erm endency reflecs he expeced fuure of shor erm yelds over he holdng horzon. The smulaed fuure shor erm neres raes based on he esmaed Vascek and CIR model parameers are exhbed n fgure Vascek smulaon CIR smulaon Tme Tme Fg.11: Smulaon of he shor erm neres raes 33

17 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 A good model of he neres raes yeld curve s val for he smooh funconng of he marke. Thus, he yeld curves predced by Vascek (1977) or CIR (1985) model seem suable o forecas he neres rae movemens n he Tunsan marke. 7. Concluson The Tunsan bond marke s llqud wh a noceably smaller radng volume compared o oher bond markes of he developng counres. In hs knd of marke, he esmaon of an accurae neres raes yeld curve s very dffcul. Therefore, hs paper developed a mehodology o esmae he neres raes yeld curve and s dynamcs n he Tunsan bond marke. Our emprcal analyss was carred ou no wo sequences: Frs, we consruced he yeld curve of neres raes usng a smple nerpolaon mehod. The yeld curve s only known wh cerany for a few specfc maury daes, whle he oher maures are esmaed usng a cubc splne mehod. Second, we calbraed he parameers of Vascek and CIR models hrough OLS and MLE mehods usng 5-week Treasury bond yelds as a proxy of he shor erm neres raes. We red o ake no accoun he varous parculares of he Tunsan bond marke daa. The emprcal resuls ndcaed ha he cubc splne mehod s a racable and reasonably correc esmaon mehod ha we recommend n any marke wh nfrequen radng. We were able o consruc an accurae average yeld curve bu no smooh. Then, he esmaon of he Vascek (1977) and CIR (1985) erm srucure models generae he same shape of he neres raes yeld curve and show an ably o replcae he sylzed facs of he neres raes n he Tunsan bond marke. Fnally, he smulaon of hese models predcs ha he spread beween he long erm and he shor erm yelds s very small whch s ofen a harbnger of fuure economc growh wh hgher nflaon. Ths work s a good experence o esmae and predc he yeld curve of neres raes n he Tunsan bond marke where no yeld curve exss. Therefore, hs mehodology can be used by he polcy makers o draf s moneary polcy and by a fnancal nsuon n s daly radng acves o predc possble losses assocaed wh unfavourable movemens n he neres raes. References [1] Alper, C.E., A. Akdemr and K. Kazmov, (004), Esmang he Term Srucure of Governmen Secures n Turkey, Bogazac Unversy Research Papers: ISS/EC Alper [] Ahn, A.H., Dmar, R.F. and Gallan, A.R., (00), Quadrac erm srucure models: heory and evdence, Revew of Fnancal Sudes, 15, pp ADG [3] A-Sahala, Y., (1996b), Tesng connuous-me models of spo neres rae dervave secures, Revew of Fnancal Sudes, 9, pp Sahala96 [4] A-Sahala Y., (008), Closed-form lkelhood expansons for mulvarae dffusons. Annals of Sascs, 36, pp Sahala08 [5] Andersen, T.G. and Lund, J., (1997), Esmang connuous-me sochasc volaly models of he shor-erm neres rae, Journal of Economercs, 77, pp adlund 34

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20 Fama Chakroun and Fah Abd, The Macroheme Revew (6), SP-IMT 013 [38] Vascek, O.A., and Fong, H.G., (198), Term srucure modelng usng exponenal splnes, Journal of Fnance, 37, pp Vas8 37