Research Paper 305 March Alternative Term Structure Models for Reviewing Expectations Puzzles

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1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 305 March 2012 Alernaive Term Srucure Models for Reviewing Expecaions Puzzles Chrisina Nikiopoulos Sklibosios and Eckhard Plaen ISSN

2 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 305 March 2012 Alernaive Term Srucure Models for reviewing Expecaions Puzzles Chrisina Nikiopoulos Sklibosios and Eckhard Plaen ISSN

3 Alernaive Term Srucure Models for reviewing Expecaions Puzzles Chrisina Nikiopoulos Sklibosios and Eckhard Plaen March 6, 2012 Absrac According o he expecaions hypohesis, he forward rae is equal o he expeced fuure shor rae, an argumen ha is no suppored by mos empirical sudies ha demonsrae he exisence of erm premiums. An alernaive arbirage-free erm srucure model for reviewing he expecaions hypohesis is presened and racable expressions for ime-varying erm premiums are obained. The model is consruced under he real-world probabiliy measure and depends on wo sochasic facors: he shor rae and he marke price of risk. The model suggess ha for shor mauriies he shor rae conribuion deermines he erm premiums, while for longer mauriies, he conribuion of he marke price of risk dominaes. Key Words: expecaions hypohesis, ime-varying erm premiums, real-world probabiliy measure, marke price of risk. JEL Classificaion: G13 1 Inroducion The Expecaions Hypohesis (EH) plays an imporan role in he common undersanding of coninuous ime erm srucure models as i relaes equilibrium condiions, Corresponding auhor: Universiy of Technology Sydney, Finance Discipline Group, PO Box 123, Broadway, NSW, 2007, Ausralia, chrisina.nikiopoulos@us.edu.au Universiy of Technology Sydney, Finance Discipline Group and School of Mahemaical Sciences, PO Box 123, Broadway, NSW, 2007, Ausralia, eckhard.plaen@us.edu.au The auhors would like o hank Mikael Elhouar and Carl Chiarella for useful suggesions on an earlier version of his work, as well as, paricipans a he II World Finance Conference (Rhodes, Greece). 1

4 marke prices of risk and associaed premiums, see Ingersoll(1987) and Cochrane(2001). The appeal of his hypohesis is unsurprising since i can provide valuable views on fuure spo ineres raes by using only he curren erm srucure of ineres raes. I provides an efforless forecas abou marke movemens o praciioners. Furhermore, i has been widely used in many rading sraegies in several markes such as currency and bond markes, see Chance and Rich (2001). The main noion of he pure EH is ha he forward rae is an unbiased predicor of he fuure spo rae. Originaed by Fisher (1896), and followed by Hicks (1939), Luz (1940) and many ohers, he EH of he erm srucure of ineres raes has been sudied exensively over he years. Cox, Ingersoll and Ross(1981) carried ou a horough heoreical analysis of he pure expecaions hypoheses demonsraing ha i is no compaible wih oher EH. Prime early works by Fama (1984), Campbell (1986), Fama and Bliss (1987), Fama (1990), Campbell and Shiller (1991) wih more recen works by Sarno, Thornon and Valene (2007), Della Core, Sarno and Thornon (2008) provide compelling evidence on he empirical failure of he pure EH. The firs aemp o reconcile empirical properies of he erm srucure of ineres raes and he EH, involved he inclusion of a consan or mauriy dependen (bu ime invarian) erm premium, see for insance Campbell (1986) and Fama and Bliss (1987). Furher, Campbell and Shiller (1991) and Harris (1998), among ohers, demonsraed ha he premium should be ime-varying. However, he EH which accouns for an addiive erm premium, is also rejeced. This version of he EH canno explain he empirically observed feaure of falling long raes when yield spreads are high. As suppored by he ARCH lieraure, he ineres rae volailiies evolve sochasically. By using he Cox, Ingersoll and Ross (1985) model for ineres raes, Fracho and Lesne (1993) demonsrae ha i is essenial o addiionally muliply by a erm premium, which arises as a resul of his sochasic volailiy requiremen. Neverheless, his specificaion fails again, in he sense ha, i does no work very well for longer mauriies. Using a CIR model for he shor rae, we encouner he issue of variances falling o zero for long mauriies, hus he shor rae is regressed wih iself. The abiliy of dynamic erm srucure models o accoun for he empirical feaures of he ineres rae erm srucure, in erms of providing an explanaory basis o he empirical failure of he EH, has been sudied by Musiela and Sondermann (1993), Fracho and Lesne (1993), Backus, Foresi, Mozumdar and Wu (2001) and Dai and Singleon (2002). More specifically, Backus e al. (2001) show ha one-facor CIR models canno mach erm premium elemens and he average upward slope of he yield curve, while 2

5 affine models wih negaive facors perform beer, especially in he long run. Dai and Singleon (2002) re-examine he EH and show ha one of he key facors o maching dynamic erm srucure models o empirically observed feaures is he marke price of risk specificaion. They consider a sae dependen marke price of risk wihin a family of Gaussian affine erm srucure models and demonsrae ha i fully maches he (sample-based linear projecion) coefficiens in yield regressions. We propose an alernaive dynamic erm srucure model and examine wheher i is able o accommodae he empirically observed feaures of ineres raes, as a basis for explaining expecaions puzzles. The proposed model for he erm srucure of ineres raes incorporaes sochasic ineres raes, sochasic ineres rae volailiies and ime-varying marke prices of risk. By using he growh opimal porfolio (GOP) as numeraire, a erm srucure model is esablished under he real-world probabiliy measure, see Plaen and Heah (2006). The dynamics of he GOP are deermined by he Markovian marke price of risk and risk-free shor rae. Consequenly, he forward raes, yields o mauriy and he corresponding erm premiums can be expressed in erms of hese wo facors. Under he proposed model, here are wo conribuions o he (addiive) erm premium: he conribuion of he sochasic ineres rae and he conribuion of he sochasic marke price of risk. The marke price of risk conribuion is minimal for shor mauriies (up o five o en years, depending on he parameer specificaions). I deermines he shape of he forward rae (and yield) for longer mauriies. The shor rae conribuion is he cenral deerminan of he shape of he forward (yield) curve for shor mauriies. By using wo well known shor rae models, he Vasicek (1977) and he Cox e al. (1985) we gauge he effec of he shor rae conribuion o he forward curve and yield curve. Thus, he model has he poenial o fi boh he shor and he long end of he forward (and yield) curve. Addiionally, as he model can also have sochasic volailiy specificaions, under he Cox e al. (1985) model assumpion, he (muliplicaive) erm premium sill remains. The key advanage of he proposed model is ha i does no require he exisence of an equivalen risk-neural probabiliy measure. Tradiional models of he erm srucure of ineres raes are usually specified under an equivalen risk neural probabiliy measure. Previous heoreical sudies of he EH separaely model he change of measure from hese equivalen probabiliy measures o he real-world (objecive) probabiliy measure, see Fracho and Lesne (1993) and Musiela and Sondermann (1993). The conclusions of hese sudies heavily depend on how his change of measure is specified. In 3

6 our approach, we sidesep hese complicaions as we model direcly he erm srucure under he real-world probabiliy measure. Furhermore, real-world pricing recognises rends in he long run, in paricular, he presence of an equiy premium, which characerises he superior expeced long erm reurn of equiies over he expeced shor rae. In he long run, he bias beween he expeced shor rae and he forward rae is dominaed by he marke price of risk. However, in he shor run, he bias depends largely on he specificaions of he shor rae model. The paper is srucured as follows. Secion 2 inroduces a dynamic erm srucure model under he real-world probabiliy measure which depends on wo facors: he marke price of risk and he shor rae. The forward rae and yield o mauriy are compleely deermined by hese wo facors. The effec of hese facors on he forward curve and yield curve is examined for a range of parameer specificaions. By employing his model, we review he EH in Secion 3, derive he relaed erm premiums and sudy heir properies. The suiabiliy of he proposed model o explain expecaions puzzles is also examined. Secion 4 concludes. 2 Alernaive Models for he Term Srucure of Ineres Raes Weassumeafileredprobabiliyspace(Ω,A T,A,P), T [0, )wiha = (A ) [0,T], saisfying he usual condiions. The coninuous uncerainy is modeled as an A-adaped Wiener process W = {W, [0,T]} under he real-world probabiliy measure P. Le P(,T) be he price a ime [0,T] of a zero-coupon bond wih mauriy T. For all [0,T] we define he following quaniies. Definiion 2.1 The yield o mauriy Y(,T) for he period [,T] is defined as Y(,T) := lnp(,t). (2.1) T The insananeous forward rae f(,t) wih mauriy T conraced a ime is defined as f(,t) := T The insananeous shor rae r a ime is hen defined as lnp(,t). (2.2) r := f(,). (2.3) 4

7 We propose an alernaive model for he erm srucure of ineres raes under he realworld probabiliy measure. The cenral elemen of his alernaive model is he growh opimal porfolio (GOP). The porfolio which maximises he expeced logarihm of erminal wealh for all imes [0,T] represens he GOP, see Karazas and Shreve (1998) and Plaen (2002) building on he early resuls by Long (1990). The GOP is he porfolio ha almos surely provides he bes oucome in he long run. I has he long erm growh rae ha is almos surely greaer han or equal o he long erm growh rae of any oher sricly posiive porfolio. Ye, he GOP is a radable porfolio, as i can be approximaed by a well diversified global accumulaion index, see Plaen and Heah (2006). Assume ha S 0 is he value of he locally riskless savings accoun a ime, which coninuously accrues he shor rae r, hus { S 0 = exp 0 } r s ds, (2.4) for [0, ), where r = {r, [0,T]} denoes he adaped shor rae process. When he oal marke price of Wiener process risk follows he predicable vecor process Θ = {θ, [0,T]}, hen he unique GOP, S, saisfies he sochasic differenial equaion (SDE) ds =S (r d+θ (θ d+dw )), (2.5) for all [0,T], wih S 0 = 1. We recall ha W = {W, [0,T]} is a sandard Wiener process under he real-world probabiliy measure P. I is noed ha he dynamics of he GOP are deermined solely by he shor rae r and he marke price of Wiener process risk θ. The benchmarked value of an asse is he value of he asse which is expressed in unis of he GOP. When a benchmarked price process forms a maringale, hen he price process is called fair. Plaen and Heah (2006) show ha any nonnegaive benchmarked porfolio forms an (A,P)-supermaringale. 1 Therefore, when benchmarked asse prices are fair, hen heir pricing is performed under he real-world probabiliy measure wih he GOP as numeraire. Thus, he price of he fair zero-coupon bond P(,T) under he 1 An A-adaped process X = {X, [0, )} is an (A,P)-supermaringale when X s E(X A s ) for 0 s. Supermaringales are imporan for financial marke modelling as i has been shown empirically ha he savings accoun expressed in unis of GOP resembles a supermaringale, see Plaen (2004). Classical risk-neural modelling implies ha he savings accoun expressed in unis of he marke index is a maringale. 5

8 real-world probabiliy measure is evaluaed by he real-world pricing formula as ( ) 1 P(,T) = S E A, (2.6) S T for [0,T]. This implies ha he benchmarked zero-coupon bond ˆP(,T) = P(,T) S, (2.7) is an (A, P)-maringale. By using he savings accoun (2.4), he zero-coupon bond price P(,T), asevaluaed in(2.6), can beexpressed inerms of he discouned GOP, S = S, S 0 as ( ) S P(,T) = E A S T ( { T } ) S = E exp r s ds A, (2.8) S T for [0,T], T [0, ). Using (2.5) and Io s lemma, he discouned GOP, S = S /S 0, is found o saisfy he SDE d S = S ( θ 2 d+θ dw ), (2.9) for all [0,T]. In he general case, he shor rae process and he discouned GOP process are correlaed which leads o pricing relaionships ha ypically have o be handled numerically. For he sake of racabiliy, we assume here independence beween he shor rae process and he discouned GOP process. Laer we will see ha he shor rae impacs mainly he shor erm dynamics of a fair zero-coupon bond, whereas he marke price of risk governs is long erm dynamics and independence is no really required. Then (2.8) yields ( { T } ) P(,T) = M (T, S )E exp r s ds A, (2.10) where M (T, S ) is he marke price of risk conribuion o he bond price, namely ( ) M (T, S S ) = E A. (2.11) S T The second facor, in (2.10), is he shor rae conribuion o he bond price. By subsiuing (2.10) ino he definiion (2.2), he forward rae is expressed as f(,t) = n f (,T)+ρ f (,T), (2.12) where n f (,T) is he marke price of risk conribuion o he forward rae n f (,T) = T ln[m (T, S )], (2.13) 6

9 and ρ f (,T) is he shor rae conribuion o he forward rae ρ f (,T) = [ ( { T } )] T ln E exp r s ds A. (2.14) By subsiuing (2.10) ino he definiion (2.1), he yield o mauriy is expressed as Y(,T) = n y (,T)+ρ y (,T), (2.15) where n y (,T) is he marke price of risk conribuion o he yield o mauriy n y (,T) = ln[m (T, S )], (2.16) T and ρ y (,T) is he shor rae conribuion o he yield o mauriy ρ y (,T) = 1 [ ( { T } )] T ln E exp r s ds A. (2.17) The yield o mauriy and he forward rae depend on wo facors: he discouned GOP (or more direcly he marke price of risk θ ) and he shor rae r. 2.1 Marke Price of Risk Conribuion To model he marke price of risk conribuion funcion, we adop a paricular model o specify he discouned GOP, he Minimal Marke Model (MMM), as discussed in Plaen (2001) and Plaen (2002). Assumpion 2.1 Assume ha he drif α of he discouned GOP, S, is of he form α := S θ 2 = α 0 exp{η}, (2.18) for [0,T], where η > 0 is he consan ne growh rae and α 0 > 0 is an iniial parameer. Empirical observaions demonsrae ha in he long erm he world economy has been growing exponenially, which suggess ha he discouned GOP should also grow in a similar manner. Under his assumpion we obain he MMM, where he marke price of risk conribuion funcion is explicily obained as follows. Proposiion 2.1 Under Assumpion 2.1, he marke price of risk conribuion funcion o he bond M (T, S ), defined in (2.11), is given by { M (T, S ) = 1 exp 2R(,T) S }, (2.19) α 7

10 where R(,T) = η exp{η(t )} 1. Furhermore, he marke price of risk conribuion funcion n f (,T) o he forward rae, see (2.13), is given by n f (,T) := T ln[ M (T, S ) ] = 2R(,T)(η+R(,T)) S, (2.20) α [exp{ 2R(,T) S } 1] and he marke price of risk conribuion funcion n y (,T) o he yield, see (2.16), will be n y (,T) = 1 [ { T ln 1 exp 2R(,T) S }]. (2.21) α Proof: The proof of he above resul is presened in Appendix A. α Noe ha assumpion (2.18) implies ha he marke price of risk conribuion funcions (2.20) and (2.21) are fully deermined by he oal marke price of risk process Θ = {θ = α / S, [0,T]}. In Figure 1, he marke price of risk conribuions, n f (,T) and n y (,T), see (2.20) and (2.21) respecively, aredisplayed as funcions of mauriy T [0,30]and ne growh rae η [0.01,0.25]. The iniial marke price of risk is here se o he value θ 0 = α0 / S 0 = 0.2, as proposed in Le and Plaen (2006) Η Η T 20 T Figure 1: Marke price of risk conribuions, n f (,T) (in he firs panel) and n y (,T) (in he second panel) 8

11 The marke price of risk conribuion n f (,T) o he forward rae is approximaely zero for shor mauriies and approaches asympoically he ne growh rae η, for very long mauriies. The conribuion n y (,T) of he marke price of risk o he yield o mauriy has similar feaures. I also approaches η asympoically, for exremely long mauriies bu wih a lower rae of convergence. 2.2 Shor Rae Conribuion To model he shor rae conribuion funcions in a paricular case, we assume ha he shor rae process r follows a process leading o affine bond prices. Then here exis unique deerminisic funcions of ime A(,T) and B(,T) such ha ( { T } ) E exp r s ds A = exp{a(,t) B(,T)r }, (2.22) where Appendix B provides he echnical deails. Noe ha he above expecaion is aken under he real-world probabiliy measure. By using he expression (2.22), he shor rae conribuions, (2.14) and (2.17), will be obained as ρ f (,T) = A(,T) T + B(,T) r, (2.23) T ρ y (,T) = A(,T) T + B(,T) T r, (2.24) respecively. For furher illusraion, we consider below wo examples of affine erm srucure models for he shor rae process. 1. The Vasicek (1977) model: The shor rae dynamics are specified by he SDE dr = κ ( r r ) d+σdw, (2.25) where κ, r, and σ are posiive consans. Then B(,T) and A(,T) are given by and respecively. B(,T) = 1 e κ(t ), (2.26) κ A(,T) = ( r )(B(,T) (T σ2 )) σ2 2κ 2 4κ B(,T)2, (2.27) 2. The Cox e al. (1985) model: he shor rae dynamics are specified by he SDE dr = κ ( r r ) d+σ r dw, (2.28) 9

12 where κ, r,σ are posiive consans wih 2κ r > σ 2, which ensures sricly posiive soluions o (2.28). Then B(,T) and A(,T) are given by and respecively, where wih π 1 = κ 2 +2σ 2. B(,T) = L 1(T ) L 2 (T ), (2.29) A(,T) = 2κ r σ 2 ln L 3(T ) L 2 (T ), (2.30) L 1 (x) = 2(e π 1x 1), L 2 (x) = π 1 (e π 1x +1)+κ(e π 1x 1), (2.31) L 3 (x) = 2π 1 e (π 1+κ)x/2, r0 0 r T 20 T Figure2: Shorraeconribuions, ρ f (,T)(inhefirspanel)andρ y (,T)(inhesecond panel) under he CIR model For illusraive purposes, Figure 2 presens he shor rae conribuion funcions ρ f (,T) and ρ y (,T), (see (2.23) and (2.24), respecively) under he CIR model, as funcions of mauriy T [0,30] and r 0 [0.005,0.15]. The parameer specificaions are σ = 0.10, κ = 1 and r = The conribuions ρ f (,T) and ρ y (,T) of he shor rae o he forward rae and o he yield o mauriy, respecively, are very pronounced for shor mauriies up o 10 years (subjec o parameer values) while hey remain minor for 10

13 longer mauriies. The shor rae conribuions under he Vasicek model are similar and no given here o save space. Figure 3 displays he forward rae (2.12) under he MMM for he marke price of risk and he CIR shor rae model. In he firs panel, he forward rae is displayed as a funcion of mauriy T [0,30] and ne growh rae η [0.01,0.25]. In he second panel, he forward rae is displayed as a funcion of mauriy T [0,30] and iniial shor rae r 0 [0.005,0.15]. The parameer specificaions are σ = 0.10, η = 0.1, θ 0 = 0.2, r = r 0 = 0.05 and κ = Η r0 T T Figure 3: Forward rae surfaces f(,t) = n f (,T)+ρ f (,T) under he CIR model, in he firs panel as a funcion of η, in he second panel as a funcion of r 0 Figure 3 illusraes ha he forward rae does no only depend on he evoluion of he shor rae. An addiional facor is required, especially for longer mauriies, which depends on he marke price of risk. For longer mauriies, he impac of he shor rae is raher limied. Furhermore, he model has he abiliy o generae a variey of forward curves. These may be, for insance, increasing, decreasing or humped. Noe ha similar paerns are also obained for he yield o mauriy. Moivaed by he above observaions arising from hese paricular models, we propose an alernaive formulaion for he EH. 3 Reviewing he Expecaions Hypohesis For compleeness, we firsly presen he classical formulaions of he EH presened by Cox e al. (1981). By using sandard arbirage pricing heory hese auhors have 11

14 classified various expecaions hypoheses for ineres raes, he so-called pure EH, and sudied heir properies. We summarise heir main resuls as follows: 1. The Unbiased Expecaions Hypohesis (U-EH): The forward rae is assumed o be an unbiased esimae of he fuure shor rae, namely f(,t) = E(r T A ). (3.1) Then from (2.2), he value of a zero-coupon bond can be expressed as { T } { T } P(,T) := exp f(,s)ds = exp E(r s A )ds. (3.2) This hypohesis holds only under he so-called T forward measure, see for insance Björk (2004) (Lemma p. 357), and no under he real-world probabiliy measure. 2. The Yield To Mauriy Expecaions Hypohesis (YTM-EH): From definiions (2.1) and (2.2), and inegraion of (3.1) he yield o mauriy is By (2.1) his is equivalen o Y(,T) = 1 T E ( T r s ds A ). (3.3) { ( T P(,T) = exp E r s ds A )}. (3.4) From (3.2) and (3.4), i is easy o conclude ha he (U-EH) and (YTM-EH) are equivalen. 3. The Local Expecaions Hypohesis (L-EH): The expeced insananeous reurn from holding a zero coupon bond is assumed o equal he shor rae. Under sandard arbirage pricing heory his hypohesis is always rue under an assumed risk neural probabiliy measure Q, ha is d d EQ ( dp(,t) P(,T) A ) = r. (3.5) Empirical work has rejeced he classificaion of he expecaions hypoheses presened in Cox e al. (1981) as i ignores he imporan impac of observed erm premiums, see Fama (1984) and Campbell (1986), Fama and Bliss (1987), Fama (1990), and Campbell and Shiller (1991) o name jus a few relevan papers in his direcion. 12

15 Reviewing he above pure EH, he U-EH assers a zero erm premium while he L- EH assers a mauriy dependen erm premium. Empirical lieraure has rejeced boh hese wo hypoheses and shows evidence of ime-dependen erm premiums, see Backus e al. (2001), Sarno e al. (2007) and lieraure referred o herein. I will be our aim o formulae an alernaive EH ha will allow us o accommodae his fac in a very general marke seing. Dynamic erm srucure models can capure empirical feaures of he ineres rae erm srucure. More specifically, a ime-dependen (addiive) erm premium seems o be needed and arises from he assumpion for he sochasic evoluion of ineres raes, while a ime-dependen (muliplicaive) erm premium seems o be relevan for he requiremen of modelling sochasic volailiy of ineres raes, as Fracho and Lesne (1993) have indicaed. I is imporan o noe ha hese erm premiums fail o capure feaures of he ineres rae erm srucure for longer mauriies. This is mainly a consequence of relying on he classical arbirage pricing heory. The curren paper allows us o go beyond he classical framework by employing an alernaive approach as discussed in Secion 2. Longsaff and Schwarz (1992) and Backus e al. (2001) provide empirical evidence ha muli-facor erm srucure models perform beer. Dai and Singleon (2002) demonsrae he imporance of inegraing he marke price of risk ino he model when fiing observed ineres rae erm srucures. These auhors consider a ime-varying erm premium ha depends on boh he shor rae and he marke price of risk. They show ha here is a large subclass of dynamic erm srucure models, such as affine and quadraic-gaussian models, which are consisen wih he key empirical findings presened in Fama and Bliss (1987) and Campbell and Shiller (1991). Moivaed by he above menioned empirical findings, we sudy an alernaive erm srucure model o review he expecaions hypoheses. By using real-world pricing for he specific racable model class described in Secion 2, we employ expecaions under he real-world probabiliy measure, and provide expressions for ime-varying erm premiums ha depend on he marke price of risk and he shor rae. The proposed real-world pricing model does no require he exisence of an equivalen risk-neural probabiliy measure, hus, our resuls do no rely on he specificaions of he probabiliy measure change, as i has been reaed by mos of he above menioned sudies. Noe also ha radiional affine erm srucure models, as well as models wih sochasic ineres rae volailiies, can also be accommodaed in our suggesed approach. Wha his paper essenially suggess is o use real-world pricing for a reasonably realisic marke model and he ypically ime-varying premiums will auomaically emerge in 13

16 a manner consisen wih wha is empirically observed. We call his approach he Alernaive Expecaion Hypohesis (AEH). 3.1 Alernaive Expecaion Hypohesis By using he ype of erm srucure model described in Secion 2, and applying real-world pricing, a paricular relaionship beween he forward rae and he expeced shor rae is obained. This relaionship can be inerpreed as being represenaive of wha he EH lieraure aims o capure. Proposiion 3.1 Under he model specificaions of Secion 2, he forward rae can be expressed as where f(,t) = c f (,T)E(r T A )+γ f (,T)+n f (,T), (3.6) c f (,T) = B(,T) T γ f (,T) = A(,T) T and n f (,T) is specified in (2.20). Proof: See Appendix C. e κ(t ), (3.7) r B(,T) (e κ(t ) 1), (3.8) T This relaionship demonsraes ha forward raes are biased predicors of fuure shor raes. There is a muliplicaive risk premium and an addiive risk premium presen. Corollary 3.2 Under he Vasicek (1977) model for he shor rae one obains c f (,T) = 1. (3.9) Proof: Subsiue B(,T) as evaluaed by (2.26) in (3.7). Corollary 3.3 Under he Cox e al. (1985) (CIR) model for he shor rae one obains 4(κ 2 +2σ 2 )e (κ+ κ 2 +2σ 2 )(T ) c f (,T) = [κ(e κ 2 +2σ 2 (T ) 1)+(e κ 2 +2σ 2 (T ) +1) (3.10) κ 2 +2σ 2 ] 2. Proof: Subsiue B(, T), as evaluaed by (2.29), in (3.7). I is easy o confirm ha, under he CIR model, c y (,T) akes non-negaive values and for very shor mauriies, c y (,T) converges o 1. One has, 0 < c y (,T) < 1, which is consisen wih empirical evidence. Nex we derive a relaionship beween he yield o mauriy and he expeced shor rae. 14

17 Proposiion 3.4 Under he model specificaions of Secion 2, he yield o mauriy can be expressed as where Y(,T) = c ( T y(,t) T E r s ds A )+γ y (,T)+δ y (,T), (3.11) c y (,T) = κb(,t) (3.12) 1 e κ(t ) [ ] γ y (,T) = A(,T) T 1 κ r (T ) 1 e κ(t ) B(,T) κ (3.13) T 1 e κ(t ) δ y (,T) = 1 [ { T ln 1 exp 2R(,T) S }] (3.14) α Proof: See Appendix D. Corollary 3.5 Under he Vasicek (1977) model for he shor rae i follows Proof: Subsiue B(, T) as evaluaed by (2.26) in (3.12). c y (,T) = 1. (3.15) A erm srucure model, such as he Vasicek model, which ignores he sochasic naure of he volailiy of he ineres raes has no muliplicaive erm premium, see also Fracho and Lesne (1993). Corollary 3.6 Under he Cox e al. (1985) model for he shor rae one has c y (,T) = where L 1 (T ) and L 2 (T ) are defined in (2.31). Proof: Subsiue B(, T) as evaluaed by (2.29) in (3.12). κl 1 (T ) L 2 (T )(1 e κ(t ) ), (3.16) Figure 4 plos c y (,T) under he CIR ineres rae model as a funcion of T. The parameer specificaions are σ = 0.10 and κ = 0.1. Noe ha c y (,T) < 1 and for long mauriies i can also ake negaive values, which is consisen wih empirical lieraure, see for insance Campbell and Shiller (1991). Campbell and Shiller (1991) provide empirical evidence ha long-erm ineres raes under reac o shor-erm ineres raes, which implies ha c y (,T) < 1. Under our model specificaions, when he shor rae volailiy is sochasic (as in he CIR model) c y (,T) is less han 1, and negaive for longer mauriies, which is again suppored by he empirical lieraure, see he yields regressions of Campbell and Shiller (1991). 15

18 Term Premiums Figure 4: c y (,T) as a funcion of T From a differen poin of view, paricular aenion has also been given o he erm premiums alone. We inroduce wo relaed noions of erm premiums ha have been exensively sudied in he lieraure. Definiion 3.1 The forward erm premium Ψ(, T) is Ψ(,T) = f(,t) E(r T A ), (3.17) and he yield erm premium Φ(,T) is Φ(,T) = Y(,T) 1 ( T ) T E r s ds A. (3.18) By employing he erm srucure model presened in Secion 2 we derive he following resuls. Corollary 3.7 The forward erm premium Ψ(, T) is expressed as Ψ(,T) = (c f (,T) 1)E(r T A )+γ f (,T)+n f (,T), (3.19) where c f (,T), γ f (,T) and n f (,T) are given in Proposiion 3.1. The yield erm premium Φ(,T) is expressed as ( 1 T Φ(,T) = (c y (,T) 1) T E r s ds A )+γ y (,T)+δ y (,T), (3.20) where c y (,T), γ y (,T) and δ y (,T) are given in Proposiion 3.4. Proof: The definiions (3.17) and (3.18) ogeher wih (3.6) and (3.11), respecively, provide he resul. 16

19 Recall ha under he Vasicek (1977) shor rae model, one has c f (,T) = c y (,T) = 1. Under Cox e al. (1985) i follows ha c f (,T) < 1 and c y (,T) < 1, which coincides wih he empirical resuls of he regressions in Campbell and Shiller (1991). Thus, he sochasic volailiy of ineres raes conribues o he erm premiums and can explain some empirical findings. Addiionally, his specificaion breaks down ino wo separae conribuions o he erm premiums; he shor rae conribuion and he marke price of risk conribuion. In paricular, noe ha under he Vasicek (1977) shor rae model, by using (2.27) and (2.26), he forward erm premium Ψ(,T), (3.19), is reduced o Ψ(,T) = σ2 2κ 2(e κ(t ) 1) 2 +n f (,T), (3.21) where n f (,T) is given by (2.20). Figure 5 displays he shor rae conribuion, he marke price of risk conribuion δ y (,T) and he combined forward erm premium as a funcion of mauriy T. The firs panel displays he shor rae conribuion for he Vasicek model and he second panel he CIR model. The parameer specificaions are σ = 0.10, κ = 0.5, r = 0.05, η = 0.1 and θ = Shor Rae Conribuion Marke Price of Risk Conribuion Forward Term Premium Shor Rae Conribuion Marke Price of Risk Conribuion Forward Term Premium Figure 5: Forward Term premium Ψ(,T) a) Vasicek model. b) CIR model The shor rae conribuion o he forward erm premium is always non posiive and for he Vasicek model does no depend on he long erm mean shor rae r, see (3.21). Figure 5 offers an ineresing profile for he forward erm premium (3.19). The marke price of risk conribuion o he erm premium is negligible a he shor end, so ha only he conribuion of he shor rae is observed. A he long end of he forward erm premium curve, here is a posiive conribuion of he marke price of risk which asympoically approaches he ne growh rae of he equiy marke, η = 0.10 in our example. In addiion, he shor rae conribuion converges o an asympoic level as 17

20 well. In general for long mauriies, he forward erm premium is dominaed by he marke price of risk conribuion. Finally, by making he simplisic assumpion ha in he long run lim E(r T A ) r T from (3.19) and for he Vasicek model, we obain lim T while for he CIR model, we have ha σ2 Ψ(,T) +η, (3.22) 2k2 lim Ψ(,T) r T κ2 +2σ 2 k κ2 +2σ 2 +k +η. (3.23) One noes ha he proposed alernaive approach recognises real-world rends in he long run. More specifically, he expeced long erm reurn over he expeced shor rae akes ino accoun he average equiy premium, which is expeced o be he ne growh rae η. 3.3 An Empirical Illusraion The esimaed coefficiens for forward regressions, as presened in he empirical sudy by Backus e al. (2001), are displayed in Table 1. We calculae he heoreical regression coefficiens implied by he proposed model in Secion 2 for he forward regression and evaluae hese for a range of parameer values. We demonsrae ha he heoreical regression coefficiens can ake values similar o he ones empirically observed, subjec o he parameer values, as presened in Table 1. We consider he forward rae regression as proposed by Backus e al. (2001), Dai and Singleon (2002) and Chrisiansen (2003) namely f(+1,n 1) r(,1) = a f n[f(,n) r(,1)]+consan+residual. (3.24) Coefficien values ha are differen from one indicae ha he erm premia are ime dependen. This regression does no involve runcaion, hus, i allows us o es he hypohesis of consan erm premium for long mauriies. The heoreical coefficiens implied by he proposed model for he forward rae regression are (see Appendix E) a f n = B(+1,n)e κ B(+1,n 1)e κ 1, (3.25) B(,n+1) B(,n) 1 18

21 Mauriy Forward Regressions n monhs ˆa f n (0.0916) (0.0570) (0.0393) (0.0172) (0.0124) (0.0102) Table1: Theesimaedcoefficiens ˆa f n ofheforwardregressions f n 1 +1 r = ˆa f n(f n r )+ consan + residual, where f n is he coninuously compounded 1 monh forward rae andr = f 0 ishe1 monhshorrae. ThesmoohedFama-Blissmehodswihmonhly daa from January 1970 o November 1995 have been used. The numbers in parenheses are he Newey-Wes sandard errors. 19

22 and addiionally a f n = lnm +1(+n 1) lnm +1 (+n), (3.26) lnm (+n) lnm (+n+1) where M (T) = M (T, S ) for noaional convenience, see (2.19). Figure 6 displays he heoreical regression coefficien (3.25) for κ [0.001, 2] and he heoreical regression coefficien (3.26) for η [0.01, 0.25] as a funcion of n [1, 20]. Empirical sudies ypically find ha his coefficien should be less han 1, which is saisfied by he values of he heoreical regression coefficien for a wide range of model parameer values. Indeed, he heoreical regression coefficien of he forward regression approaches 1 as mauriy increases Κ Η n 15 n Figure 6: Theoreical regression coefficiens a f n (3.26) respecively. for forward regressions, see (3.25) and 4 Conclusion A simple erm srucure model is presened wih flexibiliy o mach empirical feaures of he erm srucure of ineres raes and hus he poenial o explain expecaions puzzles. The model facors are he shor rae and he marke price of risk and he model accommodaes sochasic volailiy and a marke price of risk. The erm premiums implied from his formulaion are ime-varying and depend on hese wo facors. A key feaure of he model is ha he shor rae conribuion deermines he erm premia for shor mauriies. For longer mauriies, he main deerminan of he erm premia is he marke price of risk. 20

23 A subjec of furher research is he esimaion of he model parameers, and he examinaion of he exen o which he model explains empirical findings. A Appendix: Minimal Marke Model The discouned GOP S is a ime ransformed squared Bessel process of dimension four wih deerminisic ransformed ime, see Revuz and Yor (1999). Thus, he oal marke price of risk, which is inversely proporional o he square roo of he discouned GOP, see (2.18), is given by and saisfies he SDE θ = α S, d θ 2 θ 2 = ηd θ dw. (A.1) (A.2) Furhermore, by applying he explici ransiion densiy of S, he marke price of risk conribuion o he bond price, see (2.11), is obained by he formula ( ) { M (T, S S ) = E A = 1 exp 2R(,T) S }, (A.3) S T α wih R(,T) = η exp{η(t )} 1. Plaen (2002) provides all he echnical deails. The marke price of risk conribuion funcion o he forward rae (2.13) follows hen as in (2.20). In addiion, by subsiuing (2.19) ino (2.16), (2.21) is derived. B Appendix: Affine Term Srucure Consider he one-dimensional shor rae process dr = (α 1 ()+α 2 ()r )d+ β 1 ()+β 2 ()r dw. (B.1) Then by he Feynman-Kac heorem i follows ha he funcional ( { T ) r u(,x) := E exp r s ds} = x, (B.2) saisfies he parial differenial equaion u(,x) +Lu(,x) u(,x)x = 0, u(t,x) = 1, 21 (B.3)

24 for (,x) [0,T] R, wih Lu(,x) = (α 1 ()+α 2 ()x) u(,x) x + β 1()+β 2 ()x 2 u(,x). (B.4) 2 x 2 The funcions A(,T) and B(,T) solve he sysem of ordinary differenial equaions B(,T) A(,T) +α 2 ()B(,T) 1 2 β 2()B 2 (,T) = 1, B(T,T) = 0, α 1 ()B(,T)+ 1 2 β 1()B 2 (,T) = 0, A(T,T) = 0. (B.5) (B.6) I is sraighforward o show ha he funcional (2.22) saisfies (B.3), and hus generaes an affine erm srucure. We emphasize ha he expecaion in (2.22) is aken under he real-world probabiliy measure. In he radiional lieraure he expecaion is aken under a risk neural probabiliy measure. C Appendix: Proposiion 3.1 obain By subsiuing he specificaion (2.23) for he shor rae conribuion ino (2.12) we f(,t) = η f (,T) A(,T) T + B(,T) r. T (C.1) Under boh he Vasicek (1977) shor rae model (2.25), or he Cox e al. (1985) shor rae model (2.28), i follows ha E(r T A ) = r e κ(t ) + r(1 e κ(t ) ). (C.2) Thus, by rearranging (C.2), he shor rae saisfies he relaion r = E(r T A )e κ(t ) r(e κ(t ) 1). (C.3) By subsiuing (C.3) in (C.1) and by performing some basic algebraic manipulaions, he forward rae can be expressed as f(,t) = B(,T) T from which (3.6) is derived. e κ(t ) E(r T A ) r B(,T) (e κ(t ) 1) A(,T) +n f (,T), (C.4) T T 22

25 D Appendix: Proposiion 3.4 By (2.16) and (2.17), he yield o mauriy (2.15) is expressed as Y(,T) = 1 T lnm (T, S ) 1 ( { T } T lne exp r s ds A ). (D.1) For he seleced shor rae model yielding (2.22) and he marke price of risk modelled as in Secion 2.1, see (2.21), one obains [ }] δ 2R(,T) S (T )Y(,T) = ln 1 exp { [A(,T) B(,T)r ]. α (D.2) Nex we evaluae ( T ) E r s ds A. For a mean-revering shor rae, as modelled by he Vasicek (1977) model, see (2.25), or he Cox e al. (1985) model, see (2.28), we have ha E(r T A ) = r e κ(t ) + r(1 e κ(t ) ), (D.3) and hus ( T ) T E r s ds A = E(r s A )ds T = (r e κ(s ) + r(1 e κ(s ) ))ds ] 1 e κ(t ) = r + r [(T ) 1 e κ(t ). (D.4) κ κ By eliminaing r from (D.2) and (D.4), he relaionship (3.11) beween he yield o mauriy and he expeced value of he inegral of he shor rae is obained. E Appendix: Regression Coefficiens We denoe as f n he ime 1 monh forward rae commencing in n periods (or commencing a dae +n) and r = f 0 is he 1 monh shor rae. Then ( ) P(,+n) f n ln. P(,+n+1) By using he ( expression (2.10) ) for he bond price, and seing M (T) = M (T, S ) and ρ(,t) = E e T r sds A o ease he noaion, hen ( ) ( ) f n M (+n) ρ(,+n) = ln +ln. M (+n+1) ρ(,+n+1) 23

26 Nex subsiue he specificaion (2.22) o obain ( ) f n M (+n) = ln +A(,+n) A(,+n+1)+[B(,+n+1) B(,+n)]r. M (+n+1) The forward rae regression is given by f n 1 +1 r = a f n [fn r ]+consan+residual. (E.1) Noe ha, by assuming r +m = r e κm, he lef-hand side of he regression (E.1) is expressed as f+1 n 1 r =ln M +1(+n 1) +A(+1,+n 1) A(+1,+n) M +1 (+n) +[B(+1,+n)e κ B(+1,+n 1)e κ 1]r, (E.2) and he regressed erm of he righ-hand side is given by a f n[f n r ] [ ln = a f n M (+n) M (+n+1) +A(,+n) A(,+n+1)+(B(,+n+1) B(,+n) 1)r (E.3) ]. (E.4) Based on he assumpion ha shor rae and discouned GOP (marke price of risk) are independen, he heoreical regression coefficien can be expressed as a f n = B(+1,+n)e κ B(+1,+n 1)e κ 1, (E.5) B(,+n+1) B(,+n) 1 and addiionally a f n = lnm +1(+n 1) lnm +1 (+n). (E.6) lnm (+n) lnm (+n+1) 24

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