The Term Structure of Interest Rates

Transcription

1 The Term Srucure of Ieres Raes Wha is i? The relaioship amog ieres raes over differe imehorizos, as viewed from oday, = 0. A cocep closely relaed o his: The Yield Curve Plos he effecive aual yield agais he umber of periods a ivesme is held (from ime =0). Empirical evidece suggess he effecive aual yield is icreasig i, i.e. he umber of periods remaiig uil mauriy. ( 1) (2) (1) y > y >... > y > y, where () y refers o he yield a ime over periods.

2 We will cocer ourselves wih possible reasos for his: Begi by buildig simple model ha capures esseials. The iroduce complexiies. Assume he fuure is kow wih ceraiy. The iroduce uceraiy We should oe ha ime is a esseial eleme i our aalysis. A period is a porio of ime ha defied over is begiig ad ed poi.

3 Spo versus Shor Raes Spo rae: Tha rae of effecive aual growh ha equaes he prese wih he fuure value. Thus, he spo rae is he cos of moey over some ime-horizo from a cerai poi i ime. This is ideical wih he yield o mauriy, or ieral rae of reur, o a zero coupo bod. Deoe he yield of a bod a ime wih periods o mauriy by y (). Shor rae: Refers o he ieres rae ha prevails over a specific ime period. Oly kow wih ceraiy ex-pos. The shor rae refers o he (aualised) cos of moey bewee ay wo daes, hus i may provide us wih he correc rae of discou o apply over a

4 cerai ime period, e.g. he rae ha prevailed bewee year oe ad year wo. Deoe he shor rae applicable bewee ime = 1 & = 2 as r 1. We (ypically) use a combiaio (i.e. he produc) of shor raes o discou over a series of imeperiods.

5 Expecaios If we kew wih ceraiy he shor ieres raes ha will hold over he fuure periods, we could calculae he effecive aual yield ha applies for a specific imehorizo. I realiy he fuure sequece of ieres raes is ukow. Similarly, if we kow he spo-raes (he yield o mauriy of a zero coupo bod) a which moey is le/borrowed over he various ime-periods from ow (3 moh moey, six moh moey, ec.), we have a idea abou wha he bes guess is, as o he likely developme of ieres raes over he comig periods. [However, hese expecaios could chage dramaically i he ex isa.]

6 Aoher disicio we mus draw is bewee ieres raes, shor or spo, ad he yield of a ivesme. By akig he ieres raes ha prevailed over ay oe period, ad formig a average of hese (weighed by he amou of ime hey prevailed for over a give period), we ca obai he effecive aual ieres rae ha prevailed over a specific period, or, equivalely, he yield ha accrued o our ivesme. We ca plo hese over ime o obai a yield curve. (Sricly speakig he yield is simply he effecive aual rae of growh our ivesme would have o grow by i each period i order for i o grow from he price paid o he value a mauriy). The yields over -periods are give by he geomeric average of he shor raes ha prevailed i each period, i.e. i is he sigle effecive aual yield ha would have give our ivesme he same fuure value as we

7 obaied from he series of shor raes ha acually prevailed.

8 Ceraiy If we assume we kow he fuure shor raes wih ceraiy, we ca calculae he yield of ivesmes locked i a hese raes. E.g. assume r 1 = 8%, r 2 = 10%, r 3 = 10%, r 4 = 11%, where r 1 is he ieres rae ha applied i he firs year. (N.B.: The shor raes i cosecuive periods are risig!) The he yield o a 3-year ivesme should be: (1 + y (3) ) 3 = (1 + r 1 ) (1 + r 2 ) (1 + r 3 ) or y (3) = [(1 + r 1 ) (1 + r 2 ) (1 + r 3 )] 1/3-1. I his case of ceraiy, we will oe how he yield acually icreases wih he legh of ime a ivesme is locked i for. This, however, is oly because he shor raes are risig over ime. You ca calculae y (i), wih i = 1, 2, 3, 4, yourself.

9 y The Yield Curve Geerally, () (1 + y ) = (1 + ri ) i= 1 () 1/ [ (1 ri )] i= 1 y = + 1 N.B., he holdig period reur, HPR, o ivesmes of differe mauriies & wih kow (fuure) shors would have o be ideical, eve if he yields (over heir life-ime) differ.

10 Why? The HPR of ay ivesme opporuiy mus be he ideical over oe period if everyhig is kow (uder ceraiy), sice oherwise ivesmes wih higher reurs would sricly domiae he lower reur oes, hus causig he prices o adjus, ad hece he raes of reur. Cosider a roll-over sraegy uder ceraiy: ( 1 r ) 1 ( ) ( ) ( 1) (1 ) y = + y + r + = (1 + y ) ( 1) ( 1+ y ) 1

11 Forward Raes (uder ceraiy) A forward rae agreeme (FRA) is a agreeme a ime o led moey a some fuure dae, say +1, o be repaid wih ieres a some dae hereafer, say +2. Imagie, he spo raes for hree moh ad six moh moey are give by r 0,3 ad r 0,6, respecively. Wha should he forward rae from mohs four o six, f 4,6 be? Clearly, differe ivesme sraegies over he same ime horizo should have ideical reurs i a world of ceraiy. Thus, r1,6 r1,3 f4,6 1+ = Wha his says is ha he fuure value of a six-moh ivesme should be equal o wo successive 3-moh ivesme sraegies, whe ceraiy prevails. More geerally (ad omiig mohly cosideraios), he yield o a -period ivesme should equal he produc of he yield of a (-1)-period ivesme ad he rae of reur of he forward rae for he h-period, f :

12 ( 1) 1 ( ) (1 )(1 ) f (1 + + y = + y ) (1 + ( 1 + f ) = ( 1) 1 (1 + y y ) ) The forward rae is ideical o he fuure spo rae ude ceraiy. Thus, we have see, ha uder ceraiy, he oly way he yield curve ca be risig wih a isrume s imeo-mauriy is if fuure shor raes rise.

13 Aside: Syheic FRA We ca replicae a FRA wihou explicily drawig up such a agreeme by usig spo raes. Assume: M = 1000 for all bods The price of a oe-period zero, PP(1), is Tha o a wo period zero, (2), is PP By (P/M) 1/ - 1 = y (), we kow ha y (1 ) 8% ad y (2) 9%. Also, by we kow, f %. (1 + y (1 + y ) ) ( 1 + f ) = ( 1) 1 For a sfra we wa o ake ou a loa a some fuure dae, a a kow ieres rae, f, ad repay ha loa a some laer dae agai. C ash-flow Zero a = 0 Posiive whe we wa he loa, a = 1 Negaive whe we repay, = 2.

14 =0 Sraegy Buy a sigle oe-period zero a P (1) Sell (1+ f a P (2) 2 ) imes he wo period spo = 1 = 2 Noe, our cash flow is zero a = 0. [ =0]. Receive M for our oe-period zero, i.e Mus pay back he liabiliy icurred by sellig f 2 imes he wo-period spo: (1+f 2 )*M = *1000 = I is obvious ha he rae of ieres bewee ime =1 ad =2 applied would have bee quoed as 10.01%, i.e. f 2.

15 Uceraiy I a world of uceraiy, we are usure of fuure reurs, e.g. ieres raes vary, much of which is o eirely predicable. Thus, we ed o cosider expeced reurs as our bes guess, as o wha fuure shors are likely o be. We will rea he shor rae a ime, r, as a radom variable. Deoe he expecaio a ime abou he shor rae from ime +i o +i+1 as E (r +i ). Codiioal Expecaios: The expecaio, formed a ime, abou a radom variable, r, codiioal o a se of iformaio available a ime, Ω, is deoed: E (r Ω ). Furhermore, he furher away a poi i ime lies i he fuure, he less predicable he oucome is from oday s daa.

16 We equae uceraiy wih risk, which we will measure as he variace of a radom variable, Var( ). For (goverme) bods we assume defaul risk is egligible, ad hece we do o iclude a risk premium i our cosideraios for he required rae of reur. However, here is risk associaed wih he legh of ime oe is locked io a ivesme, as he reur o oher isrumes i he fuure chage. Thus, bods may have a erm premium. Prefereces Ivesors prefereces may become a impora deermia of asse valuaio i a world of uceraiy. I paricular, ages may be risk averse, which implies ha a cerai reur is preferred o a ucerai (expeced) reur. Also, hey may have prefereces abou he legh of ime hey are ivesed i a projec, e.g. shor- vs. logerm ivesors.

17 If such a ivesor had o choose bewee a logru, bu ucerai ivesme ad a shor-erm bu cerai ivesme ha offers he same reur, he would choose he shor-erm ivesme. I a world of ceraiy, his does o affec he choice over ivesmes wih differe horizos. For ivesmes ha have a loger horizo ha he ivesor desires, he ivesor ca sell his ivesme, such ha i has he same HPR as would a ivesme wih a shorer life-ime. However, i a world of uceraiy, here is (resale) price uceraiy whe ivesors wish o liquidae loger-erm ivesmes. O he oher had, if he ivesor does o wish o liquidae he ivesme prior o mauriy, he faces uceraiy over fuure reivesme raes. Thus, he ivesor could ives i a oe-period bod ad gai a cerai reur for ha oe period. O he oher had, a bod wih a loger horizo may oly offer a expeced reur, sice is fuure resale value is ucerai.

18 If fuure ieres raes icrease, he value of he bod will fall. [Acually, deviaios from he expeced ieres raes (i.e. a shock) will affec he price of loger-erm bods more ha ha of shorer-erm securiies. Why? Similar comparig he effec of he shock of you wiig he loery i your weies (disa from your expiraio), raher ha wiig i i your eighies]. Why should a risk-averse, or shor-em, ivesor hold a log-erm ivesme, if he shor-erm ivesme offers a risk-less HPR ha is ideical o he (risky) expeced HPR of a loger-erm bod? Clearly, a risk-averse (or shor-erm) ivesor mus be compesaed for assumig risk, ad his may ake he form of a erm, or liquidiy premium. Le us assume, here exiss a required rae of reur, k, for ay ivesor o hold a -period bod bewee periods +i ad +i+1, ha depeds o he oe period

19 rae of reur bewee hose daes, r +i, (ha period s shor) ad a erm premium for a -period bod, T () +i. () k + i r + i + T + i (i > 1) If his is he case, he he perie discou rae whe pricig a bod should be give by k +i, raher ha by r +i. By he same oke, hough he forward rae equals he rae of reur of ha period s shor rae uder ceraiy, uder uceraiy his is o loger he case. I fac, for a world domiaed by risk-averse ages, he, cerai, forward rae should be less ha he, ucerai, expeced shor rae, ha prevails a ha ime. The liquidiy premium would be give by he differece bewee he forward rae for period ad he expeced shor rae for period a ime : premium = f E ( r ).

20 Theories of he Term Srucure If we have a -period coupo bod ad marke deermied spo raes exis for all mauriies, he he marke price of a bod should be deermied by: (1) P C C C + M = = Vi (1) (2) (1 rs ) (1 rs ) (1 rs ) i= 1 where rs () is he -period spo rae. If his were o so, he arbirage would allow for riskless profis, i.e. we could sell he righs o he paymes. Why? Now, assume ivesors require a asse o provide a specific reur bewee ime ad +1 i order for hem o hold he asse, deoed k. E P P + C ( 1) P = k This ca be solved forward o give he curre bod price as he discoued prese value of fuure coupo

21 paymes discoued a he expeced oe-period spo reurs k +i. (2) P C + j = E + j 1 1 j= 1 (1 + k+ i) i= o i= 0 M (1 + k+ i) N.B.: The oly ukow ivesors eed o form expecaios abou are he ks. We ca spli hese io a risk-less ieres rae compoe ad a erm premium. () k + i r + i + T + i (i > 1). There exis various heories as o he reasos for he yield curve o be risig i uder uceraiy. All of he oes we will cosider make cerai hypohesis abou he erm premium, () T, whose value may be see o deped o he ime-period we are i,, ad he umber of periods a securiy has o mauriy, (). We will use he cocep of oe-period HPR o illusrae his. (1) ad (2) cao boh deermie a bods price. Uderlyig he differe equaios are differig behavioural assumpios: (1) is deermied by arbirage.

22 (2) deermies a se of expeced oe-period reurs ha yield he same bod price as (1). (1) ad (2) will provide he same price o a wo period bod if: (1) (1 + rs ) = (1 + k0) (2) 2 (1 + rs ) = (1 + k0) E (1 + k1) E + k = + rs + rs (2) 2 (1) (1 1) (1 ) (1 ) The Expecaios Hypohesis (EH) Uder he EH he assumpio made abou he erm premium is ha i is cosa i ime ad periods-omauriy. E H r = T =. + 1 T A specific versio of he EH is he Pure Expecaios Hypohesis (PEH), which saes ha he (cosa) premium is zero, T = 0, for all ime-periods ad all bods, regardless of ime-o-mauriy (simples assumpio). I ur, his implies risk-eural ivesors ha are oly cocered wih expeced reurs, ad all bods

23 expeced oe-period holdig period reur is equal o ha of a risk-less oe-period bod. I his case all marke paricipas are plugers. Loosely speakig, he forward rae will equal he marke cosesus expeced fuure shor rae, i.e. f = E ( r ). Thus, ages are risk-eural, he expeced excess HPR is zero, ad liquidiy premia are zero. (The yields o log-erm bods are give by he curre expeced fuure shor raes.) Thus, we ca use he yield curve i order o gauge expecaios o fuure ieres raes. Geerally, he EH holds ha T is cosa, bu o ecessarily zero; Thus, he expeced excess reur is idepede of ime-period ad ime-o-mauriy. Thus, i order o move from a risk-free o a risky loger erm ivesme, ivesors require some fixed premium. The Liquidiy Preferece Model (PEH) The erm-premium is ime-ivaria, bu does deped o a securiy s life-ime-o-mauriy. Hece, he

24 expeced excess reur is cosa for securiies of - period lifespa, bu depeds o : T = T. E H r = T + 1 Loosely speakig, depedig o wheher a ivesor has log- or shor-ru liquidiy prefereces, hey will require a differe erm premium: shor-ru ivesors would require a expeced fuure shor rae i excess of he forward rae, whereas he opposie would hold for log-erm ivesors. Usually, we assume he marke is domiaed by shor-erm ivesors, ad R E ( r) = T ad T > T ( 1) >... > T (1). Marke Segmeaio Theory Here, log- ad shor-erm bods are perceived o exis i eirely differe markes, wih heir very ow ad idepede equilibria. Excess expeced reurs are

25 iflueced by he ousadig sock of mauriies of similar lifeime. where () z E H + 1 r = T ( z ), is a measure of he relaive weigh of securiies wih lifespa i he porfolio of oal asses. Thus, T depeds, i par, o he ousadig sock of asses of differe mauriies. ) R E ( r ) = T ( z ( Preferred Habia Hypohesis Bods wih similar have similar T, as hey ca be regarded as subsiues. I his view, ivesors of oe ype ca be lured io he oher marke if he premium is large eough. ) Time Varyig Risk Expeced excess reurs ad he erm premium are a fucio of (vary over) ime ad. E H r = T (, z + 1, where z is some se of variables. )