TESTING THE EXPECTATIONS THEORY FOR THE PORTUGUESE YIELD CURVE

Transcription

1 TESTING THE EXPECTATIONS THEORY FOR THE PORTUGUESE YIELD CURVE * Bernardino Adão** Jorge Barros Luís ** 1. INTRODUCTION * The opinions of his paper represen hose of he auhors and are no necessarily hose of he Banco de Porugal. The auhors would like o hank he coens of Nuno Cassola and José Ferreira Machado. All reainder erros are of he auhors responsibiliy. ** Research and Saisics Deparen. (1) As indicaed in Anderson e al. (1996) he preiu is due o he invesors risk aversion and o he Jensen s inequaliy. According o he expecaions heory, long-er ineres raes are deerined by he presen level and by he expeced pah of shor-er ineres raes. Thus hey can be used o esiae fuure shor-er ineres raes. The pure version of he expecaions heory susains ha such esiaes are unbiased, i.e., ha he expeced fuure shorer ineres raes iplici in he spo long-er ineres raes are on average equal o he forward shor-er ineres raes. However, long-er ineres raes end o overesiae he fuure shorer ineres raes. This bias can be due o a er preiu, i.e., he addiional reurn required by an invesor in order for hi o choose invesens in larger auriies. If his preiu is consan, he expecaions heory will sill hold in a non-pure version, while if he forer is variable he expecaions heory will no be valid (1). If he expecaions heory is valid, inforaion on he fuure pah of shor-er ineres raes can be drawn fro he yield curve. Since forward shor-er ineres raes are in fac advanced indicaors of iporan acroeconoics variables, heir knowledge is cenral for econoic agens specially for financial asse porfolio anagers. Furherore, he validiy of he expecaions heory allows Cenral Banks o easure he ipac of heir oneary policies on he er srucure of ineres raes, and o exclude he possibiliy of auriy segenaion in he financial arkes. This paper ais a esing he expecaions heory for he er srucure of Poruguese ineres raes. Though no siilar research has been developed for Porugal, an exensive se of sudies invesigae his issue as regards oher counries. These researches are alos unanious in wo findings: firs, he expecaions heory wih consan er preiu is rejeced for he USA; secondly, he expecaions heory is no rejeced for he reaining counries (2). This paper uilises he ehodology defined by Capbell and Shiller (1991) o analyse if he Poruguese yield curve is in agreeen wih he expecaions heory. A boosrap-based es was buil as o deerine he saisical significance of findings. The paper proceeds as follows: secion 2 describes he odel; secion 3 presens he epirical resuls; resuls of he saisical significance ess are shown in he fourh secion and secion 5 draws he ain conclusions. 2. THE MODEL To analyse he validiy of he expecaions heory for Porugal, we sar by assuing a siple resul ha relaes pairs of ineres raes wih differen auriies; he greaes auriy of boh is called he long-er, and he salles is naed he shor-er. R n sands for he long-er ineres rae observed in oen (n being he longes (2) Anderson e al. (1996) suarises he findings of he os recen invesigaions. Banco de Porugal / Econoic bullein / March

2 auriy) and R is he shor-er ineres rae observed in oen ( being he shores auriy). Mauriies and n are such ha <n and n/ is an ineger. The non-pure linearised version of he expecaions heory for he er srucure of ineres raes says ha he heoreical long-er ineres rae in oen (R n* ) equals a siple average of he presen and fuure values for he shorer ineres raes ha econoic agens in oen expec for he period running beween and +n ( ER + i, i = 0,...,( n ) / ), plus a consan c n : (3). er preiu ( ) R n* ( n ) / = ER + i + c n i = 0 n, (1) where E denoes aheaical expecaion. The er preiu is defined as he addiional reurn (i.e., he yield addiional o he expeced rae of reurn of a succession of n/ shor-er -auriy invesens) deanded by an invesor, during he period running fro o +n, which akes he invesor indifferen beween his invesen and a long-er invesen wih auriy in period +n. Equaion (1) does no specify he process by which expecaions are fored. Soe ess on he expecaions heory are based on adapive expecaions. However, in os ess he expressions ER + i are replaced by heir observed values R + i. Underlying his procedure is he assupion ha people do no coi syseaic predicion errors, and hence R + i are aken as unbiased esiaes for ER + i. Since our daa base refers o a very shor period (a lile longer han 3 years), R + i canno be used as an esiae for ER + i, since he nuber of esiaes would be sall or even nil (e.g., if he auriy is greaer han 4 years, no esiae is obained hrough such procedure). Furherore, he assupion of a null predicion error does no iply ha hose errors are nil a each and every oen. Capbell and Schiller (1987) proposes a es ha overcoes his obsacle. According o his es, expecaions regarding he fuure pah of ineres raes are assessed by using a vecor auoregression (VAR) (4). n n n* n* Le S R R and S R R sand respecively for he observed and he heoreical spread beween he long-er and shor-er ineres raes, and R = R R 1 sand for he firs difference of shor-er ineres raes. Rearranging, he following expression for he heoreical spread is obained: S n* = ( n ) / n i n i = 1 j 1 E R i j + c (2) n Equaion (2) indicaes ha he heoreical spread us predic fuure changes in he shorer ineres rae whenever he expecaions heory is verified. As enioned before, he predicion of he fuure change in -auriy raes is obained wih a VAR including S n, R and lagged values of hose variables; hese variable are assued o consiue all he necessary inforaion o forecas fuure ineres rae changes. The firsorder VAR can be wrien as follows: z = Az 1 + u (3) where z [ R R S n n ' = p+, S p+ ],..., 1,..., 1, A is he arix of he VAR coefficiens and u is a vecor of residuals in period. The es for each pair of auriies (, n) consiss of: (i) esiaing he VAR odel given by (3); (ii) calculaing he heoreical spread, given by (2); and (iii) esing if he esiaed series are siilar enough o hose observed, by calculaing he correlaion coefficien beween he observed and he heoreical spreads, as well as he raio beween he sandard deviaions of hose spreads. For he null hypohesis no o be rejeced, boh saisics us be close o one. Noe ha he es perfored is in fac a copound es of wo hypohesis: firs, he hypohesis according o which he expecaions heory wih consan er preiu explains he oveens exhibied by long-er ineres raes; and second, (3) Capbell (1986) shows ha equaion (1) is a linear approxiaion o any one of he non-linear forulas of he expecaions heory of he er srucure. (4) However, a es coparing R n* and R n direcly as o rejec (or no) he expecaions heory ay becoe roublesoe if he ineres rae series are non-saionary. We overcoe his difficuly by using ineres raes spreads and firs differences. 34 Banco de Porugal / Econoic bullein / March 1997

3 he hypohesis ha he bes predicion of he fuure pah of he firs differences of shor-er ineres raes is given by a VAR including he pas values of he spreads and of he firs differences of shor-er raes. Consequenly, rejecing he null hypohesis could eiher be an indicaion ha he er preiu is no consan over ie (i.e., ha he expecaions heory is no valid) or ha econoic agens are no using he inforaion encopassed in he VAR (lagged firs differences of shor-er ineres raes and/or lagged ineres rae spreads) in heir predicion of he fuure pah of he shor-er ineres raes in firs differences. 3. EMPIRICAL RESULTS Our epirical analysis used daily daa. The saple period runs fro Augus 1993 up o Ocober The daabase was buil by Cassola and Barros Luís (1996) by eploying he ehods proposed by Svensson (1994) and Nelson and Siegel (1987) o a se of inerbank oney arke and Treasury bond ineres raes. The daabase auriies run fro overnigh up o en years. The saionariy of he involved series was sudied (5) and i was observed ha beween Augus 1993 and Augus 1994, series S n exhibi a ean which is slighly differen fro ha of he period beginning in Sepeber 1994 up o Augus 1995, being saionary in boh sub-periods. In addiion, he shor-er ineres raes in firs differences are saionary in he period as a whole, bu are clearly less volaile fro Sepeber 1995 onwards (6). Given his evidence, our saple was (5) The variables were cenred as o eploy a siplificaion of (2). Following Capbell and Shiller (1987) equaion (2) can be wrien as follows: S ha I ( )( ) ( ) n I A I A 1 = I A C z 1 + * n n where h sands for he vecor seleced fro he VAR and I is he ideniy arix. The VAR does no include a consan so as o reduce he diension of arix A; a he sae ie, his choice does no exclude arix A fro being non-singular. Therefore variables R and S n were cenred o he ean, so ha he esiaes for he VAR coefficiens were unbiased. (6) As sressed in Capbell and Shiller (1987), he saionariy of he shor-er ineres raes in firs differences is a sufficien condiion o guaranee ha he heoreical spread and he observed spread are saionary. n, broken-down ino hree sub-periods wih siilar sizes: he firs fro Augus 1993 o Augus 1994, he second fro Sepeber 1994 o Augus 1995 and he hird fro Sepeber 1995 o Ocober We also ensured ha series S n and R are saionary in all hree sub-periods considered. One can associae he breakdown of he saple period ino hree sub-periods o specific doesic arke evens. Hence, he firs sub-period exhibied a negaively sloped yield curve, having occurred disurbances in he oney and exchange rae arkes. In he second sub-period, he yield curve exhibied a significanly posiive slope, and ineres raes becae less volaile. In he hird sub-period, boh he yield curve s slope and he rae's volailiy lessened progressively. The chosen breakdown of he saple period is surely no he only choice possible, bu he resuls do no depend on he size of he sub-periods considered. For exaple, if any sub-period was ade soe onhs longer or soe onhs shorer, our findings would no be alered (7). In addiion, our resuls proved lile sensiive o he frequency of observaions and o he nuber of lags considered. Tables 1, 2 and 3 show he correlaions beween spreads and he spreads sandard deviaions (in parenheses) for our hree sub-periods. These figures correspond o a VAR wih 3 lags. The shorer raes considered were he overnigh rae and he hree-onh inerbank oney arke (IMM) raes; he longer-er raes were he 10-year and 5-year Treasury bond spo raes, and he 3-onh IMM raes. The ables show ha boh he correlaion coefficiens and he sandard deviaion raios are close o one. Furherore, boh saisics are in general closer o one he greaes he difference beween shor and long auriy. Though he correlaion coefficien and he sandard deviaion raios heoreical disribuions are no known, he values we obain sugges ha he null hypohesis is no o be rejeced. To validae his conclusion, we searched (7) Engsed and Tangaard (1995) sudied he predicion abiliies of he yield spreads in he Danish bond arke. Here, oo, was he saple period broken-down ino sub-periods, hough his procedure was relaed o he guidance of he Danish oneary policy. Banco de Porugal / Econoic bullein / March

4 Table 1 CORRELATION COEFFICIENTS AND STANDARD DEVIATION RATIOS BETWEEN S AND S* FOR PORTUGAL Augus 93-Augus 94 n 3-onh 5-year 10-year 1-day (0.986) (0.951) 1,025 3-onh (0.785) (0.947) Noe: The firs value in each cell is he correlaion coefficien beween S and S*, he raio beween he respecive sandard deviaions is in parenheses. Table 2 CORRELATION COEFFICIENT AND STANDARD DEVIATION RATIOS BETWEEN S AND S* PORTUGAL Sepeber 94 - Augus 95 n 3-onh 5-year 10-year 1-day (0.680) (0.955) (0.920) 3-onh (0.713) (0.710) Noe: The firs value in each cell is he correlaion coefficien beween S and S*; he raio beween he respecive sandard deviaions is in parenheses. Table 3 CORRELATION COEFFICIENTS AND STANDARD DEVIATION RATIOS BETWEEN S AND S* FOR PORTUGAL Sepeber 95 - Ocober 96 n 3-onh 5-year 10-year 1-day (0.738) (0.836) (0.925) 3-onh (0.606) (0.785) Noe: The firs value in each cell is he correlaion coefficien beween S e S*; he raio beween he respecive sandard deviaions is in parenheses. for an epirical disribuion by using boosrap procedures. The findings are presened in he following secion. 4. STATISTICAL SIGNIFICANCE OF THE ESTIMATES In his secion he boosrap echnique (see Efron (1982)) and he VAR were uilised o generae arificial daa so as o obain a populaion in line wih he expecaions heory. The boosrap seeed ore adequae han he Mone Carlo experiens (despie he frequen usage of he laer in lieraure), since he forer allows o generae residuals wihou specifying any disribuion in paricular: only he hisogra of he residuals obained iniially fro he observaions for he VAR variables is reproduced. So, his echnique allows o deerinae he epirical disribuions of he correlaion coefficien and of he sandard deviaion raio beween he ineres rae spread and he heoreical spread when he null hypohesis is rue. The boosrap echnique uses he residuals obained in he iniial VAR esiaion in (3) for each pair of auriies analysed (, n). This echnique is hen developed as follows: (i) (ii) (iii) (iv) a saple of observaions is generaed so ha he corresponding hisogra us replicae ha of he original residuals; using he saple generaed in (i) as well n p as he iniial values of series{ R, S } =1 and he iniially esiaed arix A, new series are generaed for variables R and S n, wih =P+1,..., ; wih he generaed series for R and S n in sep (ii), arix A is esiaed; using (2), he heoreical spread S n* is obained. Noe ha he laer was calculaed by assuing he es's null hypohesis; we hen calculae he correlaion coefficiens and he raio beween he sandard deviaion of he spreads of he ineres raes obeying he expecaions heory in he populaion and he sandard deviaion of he heoreical spread; 36 Banco de Porugal / Econoic bullein / March 1997

5 (v) he procedure reurns o sep (i) and is ran 1000 ies. Proceeding in his way 1000 esiaes for he correlaion coefficien and for he sandard deviaion raio are obained. Hence, we can esiae he respecive epirical disribuions, and herefore es he hypohesis in sake. Table 4, 5 and 6 exhibi he boosrap resuls. The earlier conclusions ha he correlaion coefficiens and he sandard deviaion raios calculaed wih he original series are saisically close o one is confired, and so he null hypohesis Table 4 STATISTICAL SIGNIFICANCE OF THE CORRELATION COEFFICIENT AND STANDARD DEVIATION RATIOS BETWEEN S E S* FOR PORTUGAL Augus 93 - Augus 94 Probabiliy in 1000 rando experiences of a value ore disan fro one han he esiaed value n 3-onh 5-year 10-year 1-day (0.785) (0.569) (0.510) 3-onh (0.977) (0.996) Noe: The probabiliy of he raio beween he respecive sandard deviaions is in parenheses. Table 5 STATISTICAL SIGNIFICANCE OF THE CORRELATION COEFFICIENTS AND STANDARD DEVIATION RATIOS S E S* FOR PORTUGAL Sepeber 94 - Augus 95 Probabiliy in 1000 rando experiences of a value ore disan fro one han he esiaed value is no o be rejeced for he esed auriies. In fac, considering all he periods and all he pairs of sudied auriies, he salles nuber of rando experiences wih a correlaion coefficien beween he observed (generaed) spread and he heoreical spread below ha iniially esiaed is 39%, which occurred for he period beween Augus 1993 and Augus 1994, for 1-day and 5- year ineres raes. As regards he raio beween he sandard deviaions of hose series, ha figure sood a 51%, for he sae period and for 1-day and 10-year ineres raes. Therefore he es's null hypohesis is no rejeced a he usual significance levels (5 per cen and 10 per cen) (8). 5. CONCLUSION Table 6 STATISTICAL SIGNIFICANCE OF THE CORRELATION COEFFICIENTS AND STANDARD DEVIATION RATIOS BETWEEN S E S* FOR PORTUGAL Sepeber 95 - Ocober 96 Probabiliy in 1000 rando experiences of a value ore disan fro one han he esiaed value n 3-onh 5-year 10-year 1-day (0.728) (0.733) (0.871) 3-onh (0.633) (0.747) Noe: The probabiliy of he raio beween he respecive sandard deviaions is in parenheses. The validiy of he expecaions heory has been analysed by a wide range of inernaional epirical sudies. No siilar research had been developed as regards he Poruguese case. This paper presens a es o he er srucure of Poruguese n 3-onh 5-year 10-year 1-day (0.750) (0.999) (0.999) 3-onh (0.679) (0.837) Noe: The probabiliy of he raio beween he respecive sandard deviaions is in parenheses (8) These are one-sided lef-handed ess, i. e., he null hypohesis is rejeced whenever he percenage of experiences scoring a value lower han ha esiaed sands below he significance level of he es. As regards he sandard deviaion raio, he null hypohesis is rejeced whenever he percenage of experiences scoring a value which is ore disan fro one ha he esiaed value sands below he significan level of he es. Banco de Porugal / Econoic bullein / March

6 ineres raes, as o deerine if he expecaions heory is applicable. The resuls obained by using he ehod proposed by Capbell and Schiller (1987) poin owards he non-rejecion of his hypohesis; he boosrap echnique also suppored his finding. Hence, he hypohesis according o which he er srucure for Poruguese ineres raes exhibis a consan risk preiu was no rejeced for each sub-period considered. This finding allows us o use forward ineres raes as he cenral esiae for he fuure spo raes (afer deducing he respecive risk preiu) in he saple period. An analysis of he level of he risk preiu lies ouside he scope of his paper. The available daa on he er srucure of ineres raes sugges inerdependence in he deerinaion of shor-er and long-er ineres raes ha is o say, arkes are no segened. REFERENCES Anderson, Nicola, Francis Breedon, Mark Deacon, Andrew Derry e Gareh Murphy (1996), Esiaing and Inerpreing he Yield Curve, John Wiley & Sons, Series in Financial Econoics and Quaniaive Analysis. Capbell, John (1986), ''A Defense of Tradiional Hypoheses abou he Ter Srucure of Ineres Raes'', Journal of Finance, 41, pp Capbell, John Y. and Rober J. Shiller (1991), ''Yield Spreads and Ineres Rae Moveens: A Bird's Eye View'', Review of Econoic Sudies, 58, pp Capbell, John Y. And Rober J. Shiller (1987), ''- Coinegraion and Tess of Presen Value Models'', Journal of Poliical Econoy, 95, pp Cassola, Nuno and Jorge Barros Luís (1996), ''The Ter Srucure of Ineres Raes: A Coparison of Alernaive Esiaion Mehods wih an Applicaion o Porugal'', Banco de Porugal, Working Paper No Efron (1982), The Jacknife, he Boosrap and oher Resapling Plans, SIAM, onograph no. 38, CBMS-NSF. Engsed, To and Carsen Tangaard (1995), ''The Predicive Power of Yield Spreads for Fuure Ineres Raes: Evidence fro he Danish Ter Srucure'', Scandinavian Journal of Econoics, 97(1), pp Nelson, Charles R. and Andrew F. Siegel (1987), ''Parsionious Modelling of Yield Curves'', Journal of Business, Vol. 60, No.4. Svensson, Lars E.O. (1994), ''Esiaing and Inerpreing he Forward Ineres Raes: Sweden '', CEPR Discussion Paper Series No Banco de Porugal / Econoic bullein / March 1997