Shapes of Yield Curve: Principal Component Analysis & Vector Auto Regressive approach

Transcription

1 Shapes of Yield Curve: Principal Componen Analysis & Vecor Auo Regressive approach By Subhash Chandra Absrac Mos economiss agree ha wo major facors affec he shape of he yield curve: invesors expecaions for fuure ineres raes and cerain risk premiums ha invesors require holding long-erm bonds. Because he yield curve can reflec boh invesors expecaions for ineres raes and he impac of risk premiums for longer-erm bonds, inerpreing he yield curve can be complicaed. Economiss and fixed-income porfolio managers pu grea effor ino rying o undersand exacly wha forces are driving yields a any given ime and a any given poin on he yield curve. The way in which hese forces simulaneously work o shape he yield curve can be undersood. The main objecive of his paper is o hrow some ligh on he shape and cause of shapes of yield using Principal Componen Analysis. Three facors have been idenified, which are almos 99% responsible for he change and shif in he shape of yield curve. A Vecor Auo Regressive approach has been applied o hose facors, which explains and esimaes he shape of yield curve *. 1. Inroducion Managing porfolios of financial insrumens is in essence managing he radeoff beween risk and reurn. Opimizaion is a well suied and frequenly used ool o manage his radeoff. Financial risks arise due o he sochasic naure of some underlying marke parameers such as ineres raes. So, i is necessary o include sochasic parameers in opimizaion for porfolio managing, urning porfolio opimizaion in o sochasic opimizaion or sochasic programming. A vial par of sochasic programming in porfolio managemen is scenario generaion. Moneary policy makers and observers pay special aenion o he shape of he yield curve as an indicaor of he impac of curren and fuure moneary policy on he economy. However, drawing inferences from he yield curve is much like reading ea leaves if one does no have he proper ools for yield-curve analysis. The objecive is herefore o consruc a model capable of capuring he ineres raes in order o generae ineres rae scenarios. 1.1 Wha is yield? Yield refers o he annual reurn on an invesmen. The yield on a bond is based on boh he purchase price of he bond and he ineres, or coupon, paymens received. There are wo ways of looking a bond yields: curren yield and yield o mauriy. Curren yield is he annual reurn earned on he price paid for a bond. I is calculaed by dividing he bond's annual coupon ineres paymens by is purchase price. For example, if an invesor bough a bond wih a coupon rae of 6% a par, and full face value of Rs.1,000, he ineres paymen over a year would be Rs.60. Tha would produce a curren yield of 6%. When a bond is purchased a full face value, he curren yield is he same as he coupon rae. However, if he same bond were purchased a less han face value, or a a discoun price, of Rs.900, he curren yield would be higher a 6.6%). Yield o mauriy reflecs he oal reurn an invesor receives by holding he bond unil i maures. A bond s yield o mauriy reflecs all of he ineres paymens from he ime of * I is imporan o emphasise ha he purpose of he model is no o produce superior yield curve predicions, i.e. predicions ha in any sense are assumed o ou-perform he marke and hereby may serve as a basis for acical invesmen decisions aimed a ouperforming a given benchmark sraegy. Raher i is a ool, which suppors he invesmen process relaed o sraegic asse allocaion decisions. 270

2 Yield (%) purchase unil mauriy, including ineres on ineres. Equally imporan, i also includes any appreciaion or depreciaion in he price of he bond. Yield o call is calculaed he same way as yield o mauriy, bu assumes ha a bond will be called, or repurchased by he issuer before is mauriy dae, and ha he invesor will be paid face value on he call dae. Because yield o mauriy (or yield o call) reflecs he oal reurn on a bond from purchase o mauriy (or he call dae), i is generally more meaningful for invesors han curren yield. By examining yields o mauriy, invesors can compare bonds wih varying characerisics, such as differen mauriies, coupon raes or credi qualiy. 1.2 Wha is yield curve? The simples Fig.1 Yield Curve kind of bond is called a zero- coupon 9.0% bond. A zero-coupon bond (also known as a discoun bond) makes a 8.5% single paymen on is mauriy dae. By conras, a coupon bond makes 8.0% periodic ineres paymens (called coupon paymens) prior o is mauriy 7.5% 30-Mar-2007 when i also makes a final paymen ha 28-Sep-2007 represens repaymen of 7.0% principal. A coupon bond may be hough of as a porfolio of zero-coupon bonds. The yield Mauriy (Yr.) curve is a line graph ha plos he relaionship beween yields o mauriy and ime o mauriy for bonds of he same asse class and credi qualiy. The line begins wih he spo ineres rae, which is he rae for he shores mauriy, and exends ou in ime, say, o 30 years. Invesors use he yield curve as a reference poin for forecasing ineres raes, pricing bonds and creaing sraegies for boosing oal reurns. The yield curve has also become a reliable leading indicaor of economic aciviy. Fig.1 shows wo yield curves for wo differen daes. Fig.2: Various shapes of Yield Curve Invered 7.5 Humped Normal Fla Mauriy (Yr.) Yield 1.3 Various shapes of yield curve? Yield curves can have various characerisics depending on economic circumsances a a given poin in ime. An upward sloping curve wih increasing bu marginally diminishing increases in he level of raes, for increasing mauriies, is commonly referred o as a normal shaped yield curve. The reason for his naming is due o he fac ha his is he shape of a yield curve considered o be normal for economically balanced condiions. Oher ypes of yield curves include a fla yield curve where he yields are consan for all mauriies. A humped shaped yield curve has shor and long erm yields of almos equal magniude, differen from he medium erm yields which are consequenly eiher higher or lower. An invered yield curve is convered inver normal shaped curve, i.e., a downward sloping yield curve wih decreasing bu marginally diminishing decreases in yields. In Fig.2 all four ypes of yield curve have been shown as an example. 1.4 Wha deermines he shape of yield curve? Mos economiss agree ha wo major facors affec he slope of he yield curve: invesors expecaions for fuure ineres raes and cerain risk premiums ha invesors require holding long-erm bonds. 271

3 Three widely followed heories have evolved ha aemp o explain hese facors in deail: The Pure Expecaions Theory holds ha he slope of he yield curve reflecs only invesors expecaions for fuure shor-erm ineres raes. Much of he ime, invesors expec ineres raes o rise in he fuure, which accouns for he usual upward slope of he yield curve. The Liquidiy Preference Theory, an offshoo of he Pure Expecaions Theory, assers ha long-erm ineres raes no only reflec invesors assumpions abou fuure ineres raes bu also include a premium for holding long-erm bonds, called he erm premium or he liquidiy premium. This premium compensaes invesors for he added risk of having heir money ied up for a longer period, including he greaer price uncerainy. Because of he erm premium, long-erm bond yields end o be higher han shor-erm yields, and he yield curve slopes upward. Anoher variaion on he Pure Expecaions Theory, he Preferred Habia Theory saes ha in addiion o ineres rae expecaions, invesors have disinc invesmen horizons and require a meaningful premium o buy bonds wih mauriies ouside heir preferred mauriy, or habia. Proponens of his heory believe ha shor-erm invesors are more prevalen in he fixed-income marke and herefore, longer-erm raes end o be higher han shor-erm raes. Because he yield curve can reflec boh invesors expecaions for ineres raes and he impac of risk premiums for longer-erm bonds, inerpreing he yield curve can be complicaed. Economiss and fixed-income porfolio managers pu grea effor ino rying o undersand exacly wha forces are driving yields a any given ime and a any given poin on he yield curve. 2. Available Daa Hisorical daa for Indian G-securiy reurns has been aken for all he analysis. Time period chosen is from January 2001 o December Though daily daa was available bu here weekly observaion has been aken because of non availabiliy of raes for some of he mauriy years. Also, daa was no available for all he mauriy years from January Some adjusmen has been Yield (%) Fig.2.1: Shor, Medium & Long erm Yield Curve 15y Long Term 4y Medium Term 1y Shor Term Jan 01 Apr 01 Jul 01 Oc 01 Jan 02 Apr 02 Jul 02 Oc 02 Jan 03 Apr 03 Jul 03 Oc 03 Jan 04 Apr 04 Jul 04 Oc 04 Jan 05 Apr 05 Jul 05 Oc 05 Jan 06 Apr 06 Jul 06 Oc 06 Jan 07 Apr 07 Jul 07 Oc 07 done in ha respec. Some proxy Daes observaions have been aken. As for example, say, for 3 year mauriy observaion for one 21 s February 2001 is no available. So, as a proxy, observaion from neares mauriy year rae has been aken. The daa se covers 363 daes wih 1 year o 15 year mauriy. Because of 272

4 unavailabiliy of observaions for more han 15 year mauriy raes, he highes mauriy year aken is 15 year. However, he same analysis can be done for higher mauriy years oo provided daa is available. Fig.2.1 displays he yield o mauriy for hree of he mauriy years, viz. 1 year, 4 year and 15 year. In erms of mauriies, 1 year mauriy rae has been considered as Shor-erm rae, 4 year mauriy as Medium-erm and 15 year as Long-erm rae. I can be observed ha movemens of he raes on various daes are mos same for he hree ypes of yield o mauriy wih some differences. 3. Facors, which decides he Shape of Yield Curve 3.1 Level, Slope & Curvaure- Researchers in finance have sudied he yield curve saisically and have found ha shifs or changes in he shape of he yield curve are aribuable o a few Yield (%) Fig.3: Effec of Level Facor Mauriy (Yr.) Yield (%) Normal 7.5 unobservable facors and Slop Facor hereby Fig.5: Effec of Curvaure Facor 7.0 Normal Level Facor unobservable. Specifically, empirical sudies reveal ha more han 95% of he movemens of various bond yields are capured by hree facors, which are ofen called "level," "slope," and "curvaure". The names describe how he yield curve shifs or changes shape in response o a shock. As an example, Fig.3 illusraes he influence of a shock o he "level" facor on he yield curve. The solid line is he original yield curve, and he dashed line is he yield curve afer he shock. A "level" shock changes he ineres raes of all mauriies by almos idenical amouns, inducing a parallel shif ha changes he level of he whole yield curve. Fig.4 shows he influence of he "slope" facor on yield curve. The shock o he "slope" facor increases shor-erm ineres raes by much larger amouns han he long-erm ineres raes, so ha he yield curve becomes less seep and is slope decreases. Fig.5 shows he response of he yield curve o a shock o he "curvaure" facor. The main Fig.4: Effec of Slop Facor effecs of he shock focus on mediumerm ineres raes, and 9.0 consequenly he yield curve becomes 8.5 more "hump-shaped" han before. Various models have been developed 8.0 and esimaed o characerize he movemen of hese financial asse- Mauriy (Yr.) models, abou wha hese facors are, abou he idenificaion of he underlying forces ha drive heir movemens, or abou heir responses o macroeconomic variables. ha of he yield curve by economiss and bond raders in pricing exercises. Few of hese however, provide any insigh Yield (%) Normal Curvaure Facor Mauriy (Yr.) 273

5 3.2 Idenifying he facors- The aim of facor analysis is, as said before, o accoun for he variance of observed daa in erms of much smaller number of variables or facors. To perform he facor analysis i.e. o recognize he facors we apply a relaed mehod called principal componen analysis (PCA). The PCA is simply a way o re-express a se of variables, possibly resuling in more convenien represenaion. PCA is essenially an orhogonal linear ransformaion of n individuals ses of p observed variables; x ij, i = 1, 2,..., n and j = 1, 2,..., p, ino an equal number of new ses of variables; y ij = y 1, y 2,..., y p along wih coefficiens a ij, where i and j are indexes for n and p respecively. In his paper he hisorical yield curves are he n individual ses, conaining p variables of differen mauriies each. Noe he following relaionships: Each y is a linear combinaion of he x s i.e. y i = a i1 x 1 +a i2 x 2 + +a ip x p The sum of he squares of he coefficiens a ij is uniy. Of all possible linear combinaions uncorrelaed wih y 1, y 2 has he greaes variance. Similarly y 3 has he greaes variance of all linear combinaions of x i uncorrelaed wih y 1 and y 2, ec. any hypohesis abou he oucome. The new combinaions y i express he variances in a decreasing order so consequenly he PCA can be used o recognize he mos significan facors i.e. he facors describing he highes raios of he variance. The mehod is perfecly general and he only assumpion necessary o make is ha he variables which he PCA is applied on are relevan o he analysis being conduced. Furhermore i should be noiced ha he PCA use no underlying model and henceforh i is no possible o es Principal Componen Mehod has been implemened on daa se of various ime periods (as menioned in secion 2) in order o recognize he key facors for Indian Yield curve. In Table1, eigen values (when ime period seleced was ) along wih Table1: PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 PC 7 PC 8 Eigenvalue % of Var Cum. % percenage of explanaion are shown. Cumulaive percenage is also included in Table1. I can be noiced ha PC1 (facor1) alone is able o explain more han 98% variaion and all he hree facors ogeher are explaining 99.6% variaions. Annexure A1.1 illusraes Table2: Coefficiens ( ) PC 1 PC 2 PC 3 1yr yr yr yr yr yr yr

6 such eigen values, individual 8yr variaion explanaion and cumulaive explanaion 9yr percenage for various ime periods seleced. I has been 10yr noed ha excep for 2001 ime period case and yr ime period case, firs facor alone is able o explain almos 98% variaion due o a 12yr shock in all res of he mauriy year raes. However, 13yr excep ime period case, in all oher ime period cases, hree facors ogeher 14yr are explaining more han 99% variaion. 15yr In Table 2, coefficien of firs hree mos significan facors, when ime period chosen was have been shown. Column heading in he able are he hree mos significan facors, whereas row heading are various mauriy year. Values inside Table 2 are coefficien of facors, which explain variaion in he respecive row heading s yield. As for example, in case when here is a shock on raes hen for 10 h row (10yr), PC1 (loading facor1) makes 0.271% variaions in 10year bond rae due o he shock; PC2 (loading facor2) explains % variaions in 10year bond rae due o he shock; & PC3 (loading facor3) explains 0.123% variaions in 10year bond rae due o he shock. Fig.6 is an example for one paricular se of ime period ( ). Annexure A1.2 illusraes he plos of firs hree loading facors for various ime periods. I has been observed ha firs facor (PC1) is almos consan for all he mauriy years. However, 2 nd facor and 3 rd facor are varying wih various shapes. Fig.6 shows he hree facor loadings corresponding o he hree larges principal componens in Table 2. The loadings we recognize as he shif, seepness and convexiy facors idenified by Lierman & Scheinkman. From looking a Fig.6 i can be observed ha he firs facor forms almos a horizonal line over he whole ime period, excluding approximaely he firs wo years. This corresponds o a change of slope for he firs wo years and a parallel shif for he res of he mauriy horizon. The horizonal line is dominan for he res of he erm srucure and hence he facor is recognized as he level facor. Facor Loading yr Fig. 6: T hree Mos S ig nific an P rinc ipal C omponen Facors (Weekly: J an2001-dec2005) 2yr 3yr 4yr 5yr 6yr 7yr 8yr Ma uriy 9yr 10yr 11yr 12yr 13yr Facor1 Facor2 Fcaor3 14yr 15yr The second facor can be inerpreed as he curvaure facor since posiive changes in i cause a decrease in yield for bonds wih shor and long mauriies bu cause an increase in yield for medium lengh mauriies. The hird facor is he slope, which corresponds o a change of he slope for he whole erm srucure accouns for 0.673% of he oal variaion. I can be seen from he plo ha he slope is decreasing as a funcion of mauriy which fis he descripion of a normal yield curve. This is in accordance o he fac ha he yield curve he period invesigaed was for mos pars a normal yield cure wih marginally diminishing yields. 3.3 Effec of Facors on Raes- A uni change of he i h facor causes a change a j for each mauriy -year rae. Since he facors are independen of each oher we may herefore express he oal change of he random variable, r, by k Δr = a jδf j j=1 Where f j is j h facor, k is he number of facors; a j is he coefficien, idenified by he eigenvecor analysis, used o approximae he variance of he porfolio. As an example les see wha effec a uni change (_f 1 = 1) of he level facor (j = 1) has on he en year rae ( = 10). From Table2, we have a 1,10 = so a uni change in facor 1 causes

7 change in he en year rae, which means ha if he en year rae is 5% a uni change in he level facor causes i o become 5.271%. In he same manner a uni changes of hree mos significance facors ( Δ f1 = 1) for j = (1, 2, 3), again for en years means: 3 Δr10 = a j 10Δf = = j= 1 j Meaning ha a 5% en year raes would become 5.126% if a uni change occurred for all he facors. 4. Choosing he Facors for VAR model The main resul from he facor analysis (Principal Componen Analysis) was ha hree facors were o be used o consruc he model. Bu how are he facors recognized in he VAR model? There are wo mehods for selecing he facors. The 1 s mehod is a naive approach and he 2 nd is buerfly mehod suggesed by Chrisiansen & Lund (2007). The former mehod is based on aking hree posiions of he yield curve, a shor, medium and long erm mauriy. The shor erm rae can be chosen as a proxy for he level facor, he curvaure can be chosen as he difference beween wo yields, a medium mauriy yield minus he sor mauriy yield. And finally he slope is chosen as wo imes he medium rae minus he long and shor rae. If we noe shor, medium and long mauriy as y s, y m and y l, respecively hen he facors can be denoed in he following way level = y s curvaure = y l y s slope = 2 y m (y s y l ) where we choose he shor rae o be he 1 year rae, he medium o be he 4 year and he long o be he 15 year rae (can vary wih respec o available daa). The 2 nd mehod, buerfly mehod, is a bi differen from naïve mehod. The main difference is ha in buerfly approach slope of he yield curve can be chosen differenly, namely by using he mechanism of he so called buerfly spread. A buerfly spread is a porfolio which consiss of a long posiion in an inermediae mauriy bond (he body of he buerfly) and wo shor posiions of bonds whose mauriies sraddle he firs bond (he wings of he buerfly). Figure below shows a digram of how a buerfly spread looks for a concave (normal) yield curve and he spread s, is given as s = y m (w 1* y s + (1 w 2 ) * y l ) where he weighs w 1 and w 2 are chosen such ha w 1 y s = w 2 y l. An example of how he weighs are chosen if he mauriies are 1, 4 and 15 years would be w 1 = (4 1)/(15 1) = 3/14 and weigh 2 would become w 2 = (15 4)/(15 1) = 11/14. The spread shown in he figure is posiive and he more concave he yield curve becomes he more posiive he spread ges and vice versa. This applies for boh normal and invered yield curves. Equivalenly, a negaive buerfly spread indicaes a convex yield curve. 276

8 By he laer mehod he level is chosen in he same way as before, by aking he shor rae as a proxy, bu he curvaure is deermined differenly compared o he former mehod. The curvaure in he laer mehod is chosen o be he difference beween he long and shor rae in sead of he difference beween he medium and shor rae before. Tha is done in order o keep he correlaion beween he curvaure and he approximaion of he slope a a reasonable level, according o Chrisiansen & Lund (2007). Using he same noaion as for he former mehod level = y s curvaure = y l y s slope = y m (w 1* y s + (w s ) * y l ). 5. VAR Model 5.1 Defining- A p h order vecor auo regression VAR(p) process can be expressed as Y = C + A1 Y 1 + A2Y 2 + A3Y ApY p + ε Aj Y where is an (n 1) vecor of ime series of random variables, C is an (n 1) vecor of consans, is an (n n) marix of auoregressive coefficiens for j = (1, 2,..., p) and ε is a vecor generalizaion of Gaussian whie noise. Since we inend o formulae a hree facor VAR process, we give an example of such a process. y 1, = c1 + a11 y1, 1 + a12 y1, 2 + a13 y1, 3 + ε1, y 2, = c2 + a21 y2, 1 + a22 y2, 2 + a23 y2, 3 + ε 2, y 3, = c3 + a31 y3, 1 + a32 y a33 y3, 3 + ε 3, In his paper, 2 nd mehod has been applied o selec proxy of hree facors as described in secion 4. Proxy for Facor 1 (level): y s = 1-yr mauriy rae Proxy for Facor 2 (curvaure): y m = 15-yr mauriy rae minus 1-yr mauriy rae Proxy for Facor 3 (slope): y l = 4-yr mauriy rae minus [ *(1-yr mauriy rae) + * (15-yr mauriy rae)] Saionary Check For all he hree proxies, i has been esed wheher hey are saionary. I has been found ha none of he hree proxies are saionary. However, all he hree proxies are saionary a heir respecive 1 s difference. A 1 s difference for a series, say y is defined as Δy = y y 1. EViews package ahs been used o es he saionariy of he series. Augmened Dicky Fuller (ADF) es has been applied o individual series o es saionariy. Saionary has been decided on he basis of Schwarz Crieria. In Annexure A2, all he saionariy es resuls have been shown. In Annexure A2.1, i can be seen ha neiher Level, nor Curvaure nor Slope is saionary. For saionary series ADF Tes Saisic should be less han criical value. Only proxy for Facor 1 is saionary a 10% level of significance. Annexure A2.2 displays ADF es for 1 s order difference series. All he 1 s difference series are saionary a all level of significance. Esimaion of parameers of VAR model will be done on he basis of hese 1 s order difference series. 277

9 5.3 Lag Selecion and Crieria For VAR modeling, how many lags are appropriae needs o be idenified. EViews package provides faciliy o idenify he lag selecion for VAR modeling. Selecion of lag has been performed using Schwarz Crieria. 5.4 Esimaion of parameers Using hree series idenified above viz. Δ (Level), Δ (Curvaure), Δ(Slope) a VAR model has been se up in EViews and he parameer values have been esimaed. The resuls can be seen in Annexure A3. The final equaion of he esimaion comes ou o be as below VAR MODEL: Subsiued Coefficien D_CURV = *D_CURV(-1) *D_LEVEL(-1) *D_SLOPE(-1) D_LEVEL = *D_CURV(-1) *D_LEVEL(-1) *D_SLOPE(-1) D_SLOPE = *D_CURV(-1) *D_LEVEL(-1) *D_SLOPE(-1) In he above able shown, D_LEVEL represens Δ (Level), D_CURV represens Δ(Curvaure) & D_SLOPE represens Δ(Slope). From hese hree equaions one can idenify he equaions for acual series very easily. Since above VAR is of order 1, for acual series of facors VAR is of order 2. The acual series, LEVEL (which was represening Facor 1 of our PCA), CURVATURE (which was represening Facor 2 of our PCA) and SLOPE (which was represening Facor 3 of our PCA) can be found from which one can idenify he change in raes using equaion given secion Conclusion Using PCA, one can idenify he facors which are responsible for changes in yield curve. Modeling VAR is one of he ways o projec he fuure values so ha yield o mauriy raes can be undersood beer. Analyzing he VAR process reveled ha a process wih lag 2 was suiable for modeling he raes, based on he resuls of informaion crieria. Invesigaing he sabiliy of he VAR (2) process reviled ha i was sable for he ime frame of ineres, bu using all he daa was no necessarily beer. Finally, one can apply Vecor Error Correcion Model (VECM) o ake care he shorfall of VAR model. However, along wih his oher modeling process can also be applied, like ARCH, GARCH or Regime-Swiching models. In hese ways, invesors can prepare heir ools, which suppor he invesmen process relaed o sraegic asse allocaion decisions Bibliography 1. Blaskowiz, O. & Herwarz, H. (2005), Modeling he FIBOR/EURIBOR Swap Term Srucure: An Empirical Approach 2. Chrisiansen, C. & Lund, J. (2007), Revisiing he shape of he yield curve: The effec of ineres rae volailiy; Working paper 3. Einarsson, A. (2007) Sochasic Scenario Generaion for he Term Srucure of Ineres Raes 278

10 4. Lierman, R. & Scheinkman, J. (1991), Common facors affecing bond reurns, The Journal of Fixed income Annexure A1 A1.1 Explained Variance in erms of Eigen Values and Cumulaive Probabiliies 2001 PC 1 PC 2 PC 3 Eigen value % of Var Cum. % PC 1 PC 2 PC 3 Eigen value % of Var Cum. % PC 1 PC 2 PC 3 Eigen value % of Var Cum. % PC 1 PC 2 PC 3 Eigen value % of Var Cum. % PC 1 PC 2 PC 3 Eigen value % of Var Cum. % PC 1 PC 2 PC 3 Eigen value % of Var Cum. % PC 1 PC 2 PC 3 Eigen value % of Var Cum. %

11 A1.2 Chars of Facor Loadings Facor Loading yr T hree Mos S ig nific an P rinc ipal C omponen F ac ors (Weekly: J an2001-dec2001) Facor1 Facor2 Facor3 2yr 3yr 4yr 5yr 6yr 7yr 8yr 9yr 10yr 11yr 12yr 13yr 14yr 15yr Facor Loading yr T hree Mos S ig nific an P rinc ipal C omponen F ac ors (Weekly: J an2001-dec2002) 2yr 3yr 4yr 5yr 6yr 7yr 8yr 9yr 10yr 11yr 12yr 13yr Facor1 Facor2 Facor3 14yr 15yr Ma uriy Ma uriy Facor Loading yr T hree Mos S ig nific an P rinc ipal C omponen F ac ors (Weekly: J an2001-dec2003) 2yr 3yr 4yr 5yr 6yr 7yr 8yr Ma uriy 9yr 10yr 11yr 12yr Facor1 Facor2 Facor3 13yr 14yr 15yr Facor Loading T hree Mos S ig nific an P rinc ipal C omponen F ac ors (Weekly: J an2001-dec2004) 1yr 2yr 3yr 4yr 5yr 6yr 7yr 8yr Ma uriy 9yr 10yr 11yr 12yr 13yr 14yr Facor1 Facor2 Facor3 15yr Facor Loading 1yr T hree Mos S ig nific an P rinc ipal C omponen F ac ors (Weekly: J an2001-dec2005) 1. 0 Facor Facor Fcaor yr 3yr 4yr 5yr 6yr 7yr 8yr Ma uriy 9yr 10yr 11yr 12yr 13yr 14yr 15yr Facor Loading yr T hree Mos S ig nific an P rinc ipal C omponen F ac ors (Weekly: J an2001-dec2006) 2yr 3yr 4yr 5yr 6yr 7yr 8yr Ma uriy 9yr 10yr 11yr 12yr 13yr Fcaor1 Facor2 Facor3 14yr 15yr Facor Loading 1yr T hree Mos S ig nific an P rinc ipal C omponen F ac ors (Weekly: J an2001-dec2007) 1. 0 Facor Facor2 Fcaor yr 3yr 4yr 5yr 6yr 7yr 8yr Ma uriy 9yr 10yr 11yr 12yr 13yr 14yr 15yr 280

12 Annexure A2. A2.1 ADF Tes for Level ADF Tes Saisic % Criical Value* % Criical Value % Criical Value *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: D(LEVEL) Mehod: Leas Squares Sample(adjused): 2/05/ /10/2007 Included observaions: 358 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. LEVEL(-1) D(LEVEL(-1)) D(LEVEL(-2)) D(LEVEL(-3)) D(LEVEL(-4)) C R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) ADF Tes for Curvaure ADF Tes Saisic % Criical Value* % Criical Value % Criical Value *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: D(CURVATURE) Mehod: Leas Squares Sample(adjused): 2/05/ /10/2007 Included observaions: 340 Excluded observaions: 18 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. CURVATURE(-1) D(CURVATURE(-1)) D(CURVATURE(-2)) D(CURVATURE(-3)) D(CURVATURE(-4)) C R-squared Mean dependen var - 281

13 Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) ADF Tes for Slope ADF Tes Saisic % Criical Value* % Criical Value % Criical Value *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: D(SLOPE) Mehod: Leas Squares Sample(adjused): 2/05/ /10/2007 Included observaions: 358 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. SLOPE(-1) D(SLOPE(-1)) D(SLOPE(-2)) D(SLOPE(-3)) D(SLOPE(-4)) C R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) A2.2 ADF Tes of 1 s Order Difference of Level ADF Tes Saisic % Criical Value* % Criical Value % Criical Value *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: D(D_LEVEL) Mehod: Leas Squares Sample(adjused): 2/12/ /10/2007 Included observaions: 357 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. D_LEVEL(-1)

14 D(D_LEVEL(-1)) D(D_LEVEL(-2)) D(D_LEVEL(-3)) D(D_LEVEL(-4)) C R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) ADF Tes of 1 s Order Difference of Curvaure ADF Tes Saisic % Criical Value* % Criical Value % Criical Value *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: D(D_CURV) Mehod: Leas Squares Sample(adjused): 2/12/ /10/2007 Included observaions: 336 Excluded observaions: 21 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. D_CURV(-1) D(D_CURV(-1)) D(D_CURV(-2)) D(D_CURV(-3)) D(D_CURV(-4)) C R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) ADF Tes of 1 s Order Difference of Slope ADF Tes Saisic % Criical Value* % Criical Value % Criical Value *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: D(D_SLOPE) 283

15 Mehod: Leas Squares Sample(adjused): 2/12/ /10/2007 Included observaions: 357 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. D_SLOPE(-1) D(D_SLOPE(-1)) D(D_SLOPE(-2)) D(D_SLOPE(-3)) D(D_SLOPE(-4)) C R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) Annexure A3 Parameer esimaion under VAR Sample(adjused): 1/15/ /10/2007 Included observaions: 352 Excluded observaions: 9 afer adjusing endpoins Sandard errors & -saisics in parenheses D_CURV D_LEVEL D_SLOPE D_CURV(-1) ( ) ( ) ( ) ( ) ( ) ( ) D_LEVEL(-1) ( ) ( ) ( ) ( ) ( ) ( ) D_SLOPE(-1) ( ) ( ) ( ) ( ) ( ) ( ) C ( ) ( ) ( ) ( ) ( ) ( ) R-squared Adj. R-squared Sum sq. resids S.E. equaion F-saisic Log likelihood Akaike AIC Schwarz SC Mean

16 dependen S.D. dependen Deerminan Residual 4.10E-06 Covariance Log Likelihood Akaike Informaion Crieria Schwarz Crieria Abou he Auhor: Subhash Chandra Subhash Chandra is Senior Manager, Acuarial in Koak Mahindra Old Muual Life Insurance, Mumbai. The views presened in his paper are hose of he auhors and are no necessarily shared by he organizaion. 285