The Effect of an Unobservable Factor on Interest Rates in a Pure Exchange Economy

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1 Th Effc of an Unobsrabl Facor on Inrs Ras in a Pur Exchang Econom Hiroshi Moria 1 Inroducion In h framwork of sandard microconomics, quilibrium inrs ras ar dcrasing in h ll of aggrga consumpion. hn h ll of currn consumpion bcoms high, h marginal ra of subsiuion dcrass and h markclaring inrs ra is drmind a a low ll. This mchanism is basicall h sam in h framwork of dnamic gnral quilibrium modls. Equilibrium inrs ras ar dcrasing in h ll of aggrga consumpion whil h ar incrasing in h xpcd ra of aggrga consumpion growh. hil hos horis prdic ngai corrlaion bwn inrs ras and consumpion, i is ofn obsrd ha shor rm inrs ras nd o b high whn h aggrga consumpion is high and ic rsa. Thus, i sms ha h abo horis ha srious drawbacks o xplain h acual momns of inrs ras. Th objci of his papr is o show ha i is no ncssaril ru and h inrs ras can b incrasing in h aggrga consumpion if w assum ha conomic ariabls ar pariall obsrabl. In his papr, a pur xchang conom whr h aggrga ndowmn follows a wofacor Gaussian procss is xplord. assum ha h local drif rm in h sochasic diffrnial quaion of log ndowmn follows a Gaussian procss, bu i is no obsrabl. This mans ha h ll of aggrga ndowmn is obsrabl, bu h xpcd insananous ra of is growh is no. This is a rasonabl assumpion, bcaus h xpcd alu of growh ra is no dircl obsrabl. Undr his assumpion, i is shown ha h sima of xpcd insananous ra of growh infrrd b conomic agns can b incrasing in h ll of ndowmn. As a rsul, shor rm inrs ras can b incrasing in h ll of aggrga consumpion in quilibrium. Th ffcs of an unobsrabl facor on h mark quilibrium ha alrad bn sudid in a numbr of paprs. Dmpl(1986, Dohan and Fldman(1986 xplor h conom whr an unobsrabl sa ariabls xis, bu hs works concnra on mhodological issus. Fldman(1989 and Ridl(2 focus on h rm srucur of inrs ras, bu h xamin h funcional rlaionship bwn simas infrrd b conomic agns and inrs ras. In his papr, w sud h funcional rlaionship bwn simas and

2 obsrabl ariabls in dail o obain h quilibrium inrs ras as a funcion of obsrabl ariabls. This papr is organizd as follows. In h nx scion, w dscrib a pur xchang conom whr agns ar idnical. In scion 3, w insiga how h ll of ndowmn affcs h sima of xpcd ra of growh. In scion 4, w calcula h quilibrium inrs ra brifl. In scion 5, w impos an assumpion on h simaion rror procss of agns and show ha h quilibrium inrs ras can b monoon incrasing in h ll of aggrga ndowmn. cion 6 sas summar and conclusion. 2 Modl Considr a pur xchang conom of a singl prishabl consumpion good. Th im span of his conom is [, x ]. ( X, F, Q b a compl probabili spac. Th conom is drin b wodimnsional inr procss { Z :! [, x]} whr Z = [ Z, Z ] 1 2. assum ha Z1 and Z 2 ar indpndn. Th conom is ndowd wih a flow of h consumpion good. Th ra of aggrga ndowmn flow is,! [, x ]. In his papr, i is assumd ha follows a sochasic diffrnial quaion, d = + n d dz, 1 whr = [ 1, 2] is a cor of consans. ihou loss of gnrali, i is assumd ha >. Th drif rm is assumd o follow an rnsin Uhlnbck procss, r, 2 dn= l ( n n d+ b dz whr nr and l ar posii consans and b = [ b1, b2] is a cor of consans 1. In ordr o insiga h ffc of corrlaion bwn changs in and n, w do no rsric h sign of b1 and b2. Throughou his papr, i is assumd ha is obsrabl bu n is no. I is also assumd ha h ru alu of ach paramr is known o all of h agns. Thus, agns infr n, gin h pas hisor of ll of ndowmn up o im. Filraion { F :! [, x]} dnos h Q augmnaion of naural filraion gnrad b. I is assumd ha h disribuion of n condiiond b F is normal. This is an imporan assumpion for opimal filring usd in his papr. Th sima of n is dnod as. B dfiniion, h quaion = E[ n F ] holds. 2 Th simaion rror is dfind b z = E[( n F ]. Indiidual agns wih idnical ndowmns ar assumd o ha prfrncs or h consumpion flows,

3 Th Effc of an Unobsrabl Facor on Inrs Ras in a Pur Exchang Econom Hiroshi Moria ds E> u( cs ds F H, x 1 c c whr flici funcion is dfind b uc ( = for c >. I is also assumd ha h mark 1 c is fricionlss and scuriis ar radd coninuousl in im. PT (, dnos im pric of pur discoun bond which promiss o pa on uni of consumpion good a im TT ( x. 3 Th procss of and z Undr h assumpion on h condiional disribuion of n, h procss of sima, { : $ }, is known o follow h sochasic diffrnial quaion 2, b + z d = l( n r d+ dz, 3 whr a ondimnsional inr procss { Zr :! [, x]} is dfind b, dz 1 d r = = d G, 4 and h simaion rror{ z :! [, x]} saisfis h following quaion, R V ( 2 b+ z dz= 2lz d. 5 T X Th iniial alus, and z ar dfind b, = E[ n F ] 6 E ( F = 2 z ; n E. 7 Th simaion rror procss (5 implis an ordinar diffrnial quaion, dz d = 2lz ( 2 b + z. 8 uppos ha h iniial alu z quaion problm is gin b, 3 is gin. Thn, h soluion of his ordinar diffrnial

4 z zr zr zf z z p z = z zr 1 f z p z 2l 2l, 9 whr l = 2 l + 2l b +, z = b + ( l l, z = b + ( l l. inc z conrgs o zr as " 3, h paramr zr is inrprd as h saionar ll of h simaion rror procss. us dno h corrlaion cofficin bwn changs in and n as. I is as o pro h following lmma. mma 1 l = if and onl if h following wo qualiis hold, l =, = 1. (Proof From h dfiniion of l, l = is quialn o h quali, 1 11 l 2 + 2l b+ =. inc = b, h lf hand sid can b rxprssd as, l l l l 2 2 b = Thus, h inquali, 2 2 ( 2 l l + l b+ 12 holds. No ha (12 holds wih quali whn = 1. Clarl, l = if and onl if h lf hand sid of (12 is zro and = 1. QED.. Thrfor, sric posiinss of l gnricall holds. En in h cas of prfc ngai corrlaion, bsgu masur in h spac {(, b b} ha hos paramrs saisf h condiion for l = is zro. In h squl, w assum ha a las on of (1 and (11 dos no hold and l is sricl posii.

5 Th Effc of an Unobsrabl Facor on Inrs Ras in a Pur Exchang Econom Hiroshi Moria 4 Equilibrium prics of pur discoun bonds In our homognous conom, h quilibrium prics of pur discoun bonds ar drmind as, R V d ( T uc( T PT (, = E F u c( TR XV ( T E c d T = o F. T X B h law of iraion, his quali is rxprssd as, R R V V PT (, E E c d ( T T = o F F T T X X ( T 13 d = E[ E[ xp ( c ( ln T ln F] F ]. inc ln T is Gaussian gin h informaion srucur { F }, h innr condiional xpcaion is calculas as, l ( T N xp 1 c ( l n nr f p P xp dcc nr ( T + Var ( ln F. 2 m 2 1 c 2 T n ubsiuing his ino (13 gis, R V l ( T N ( T PT (, Exp 1 = d c ( l n nr f p F P T X xp c nr ( T + Var ( ln F N c2 T f p P 14 No ha h condiional arianc Var ( ln T F is no random and can b pu ousid h xpcaion condiiond b { F }. B Proposiion 12.6 in ipsr and hira(1977, n is Gaussian undr { F }. Thus, w can xprss h bond pric as,

6 R VN l ( T PT (, xp ( T E 1 = + d c ( l n nr f p F T XP xp Var ( F NN l ( T f l n nr p PP xp dcc nr ( T + Var ( ln F 2 m 2 1 c 2 T n l ( T N = xp ( T 1 d c ( l nr f p P xp N l ( T c2 f l p z P xp dcc nr ( T + Var ( ln F. 2 m 2 1 c 2 T n 15 In h las quali, w us h fac ha E[ n ] = and Var [ n ] = z. B diffrniaing h ngai of log pric wih rspc o T, w obain h insananous forward ra. Dnoing insananous forward ra wih mauri T as ft, (, w ha, F F l ( T l ( T (, = d+ c + ( 1 nr ft d l ( T 1 ( T 2 l c l z b l N 1 l c l f p f l P ( T ( T c 2 mn N pb. P 16 5 Timhomognous modl Using h dfiniion of Z, h sochasic diffrnial quaion of is xprssd as, r c 2 1 m, 17 d ( d ( dln = ln p + p l + d whr p = l+ b + z. This lads o h sochasic ingral form,

7 Th Effc of an Unobsrabl Facor on Inrs Ras in a Pur Exchang Econom Hiroshi Moria = p d d + ln+ ( pu l u 2 p c r m du + ( pu l p d dln u. u Thus, dpnds on h pas hisor of local ariaion of ndowmn. In gnral, h simaion rror drminisicall changs or im. Bu (9 implis z conrgs o zr and h simaion rror is approximal qual o zr for sufficinl larg. In his spiri, w impos h following assumpion in par for simplici, and bcaus w wan o obain h funcional rlaionship bwn and wihou ambigui. Assumpion 1 Th iniial simaion rror is gin b, z = zr. 19 Clarl, z = zr for all undr his assumpion. From h dfiniion of zr and p, w can asil show h following lmma. mma 2 Undr assumpion 1, p = l for all $. (proof As w mniond, undr assumpion 1, z = zr for all $. Combining his rsul wih h dfiniions of zr ilds z = b + ( l l for all $. ubsiuing his quaion ino h dfiniion of p, b + z p = l+, w obain p = l for all $. QED.. Undr assumpion 1, saisfis a imhomognous sochasic diffrnial quaions. From lmma 2, (18 is rducd o, l = l + r b1 l+ ( l l l ( u d dln u, 2 whr r = al l n + 1 l l 2 k a k. inc l $, h ingral in h righ hand sid can b inrprd as h wigd arag of h pas local ariaions of ln whr ha wighs ar pu on h rcn ariaions. B ingral b pars, h ingral in h righ hand sid of (2 can b xprssd as, l u l dln u= ln ln u l l ln u du.

8 ubsiuing his quaion ino (2, can b xprssd as, = l l + r c1 m + ( l l l cln ln m ( l l l l ( u ln u duo. 21 From his quaion, w know ha linarl dpnds on ln. hn incrass, dos incras or dcras? Th following proposiion answrs his qusion. Proposiion 1 Undr h assumpion 1, is incrasing in if and onl if h following inquali holds, $. 2 1 l 22 (proof From h dfiniion of l, l l is posii if and onl if 2 l b + is posii. This condiion is arrangd as, 2l b. Using h dfiniion of, w can chang h xprssion of his inquali o (22. QED.. From his rsul, in h cas of =, changs in p and ar posiil corrlad. En if h changs in n and ar ngail corrlad, changs in can b posiil corrlad wih h changs in. This is h imporan ffc of unobsrabili of n on h quilibrium inrs ras. To undrsand his, l us considr h cas in which h inqualiis, 2 l b 23 hold. Th scond inquali mans ha changs in n and ar ngail corrlad. Bu changs in is incrasing in sinc h condiion in proposiion 1 is m b h firs inquali. Tha is, changs in and ar posiil corrlad n if changs in n and ar ngail corrlad. This inrsing rsul holds, bcaus n is unobsrabl and incras in, for xampl, maks agns infr ha n has bcom high n undr h ngai corrlaion bwn changs in n and. As a corollar, w can sablish a sufficin condiion for posii corrlaion bwn changs in and for an corrlaion cofficin. Corollar 1 uppos ha h following condiion is m, l

9 Th Effc of an Unobsrabl Facor on Inrs Ras in a Pur Exchang Econom Hiroshi Moria Thn, undr assumpion 1, is incrasing in for an corrlaion cofficin! [ 11, ]. (proof Th inquali (24 is quialn o h inquali 1 $ 2 1 l. Combining his inquali and $ 1 ilds (22. This concluds h proof. QED.. From (16, h insananous forward ras ar monoon incrasing funcion of. Thus, w sblish h following proposiion. Proposiion 2 Undr assumpion 1, h following wo samns hold. (Auppos h condiion (22 is m. Thn, h insananous forward ras in quilibrium ar monoon incrasing in h ll of ndowmn. (Buppos h condiion (24 is m. Thn, h insananous forward ras ar monoon incrasing in h ll of ndowmn for an corrlaion cofficin! [ 11, ]. 6 ummar and Conclusion In his papr, w xamind how h xisnc of an unobsrabl facor affcs h mark inrs ras, assuming ha h xpcd ra of ndowmn growh is unobsrabl. This assumpion is rasonabl, bcaus h xpcd alu of growh ra is gnrall no obsrabl. Th agns in his conom infr h xpcd ra of ndowmn growh from h pas hisor of ralizaion alu of ndowmn. Adding an assumpion on h saionari of simaing rror procss, w obain h rsul: If h corrlaion bwn h growh ra and h ll of ndowmn is sufficinl high, hn h insananous forward ras in quilibrium ar monoon incrasing in h ll of ndowmn. En if h corrlaion cofficn is ngai, his propr can hold, bcaus h obsraion ha h ll of ndowmn incrass, for xampl, maks agns infr ha xpcd growh ra has bcom high n undr h ngai corrlaion bwn h xpcd growh ra and h ll of ndowmn. Undr h assumpion on unobsrabili of xpcd growh ra, our modl is rducd o onfacor rm srucur modl. For mpirical sudis, a las anohr risk facor mus b inroducd. This rmains for fuur rsarchs. Endnos 1 In discr im sing, his mans ha w assum h ra of chang in ndowmn follows ARMA (1, 1 procss. 2 For dails, s ipsr and hira ( This ordinar diffrnial quaion blongs o h class of Ricai quaions.

10 Rfrncs Dmpl,., 1986, Ass Pricing in a Producion Econom wih Incoml Informaion, ournal of Financ 41, Dohan, M and D. Fldman, 1986, Equilibrium Inrs Ras and Mulipriod Bonds in a Pariall bsrabl Econom ournal of Financ 41, Fldman, D., 1989, Th Trm rucur of Inrs Ras in a Pariall bsrabl Econom, ournal of Financ 44, angig, T, C., 198, A Muliaria Modl of h Trm rucur, ournal of Financ 35, ipsr, R.., and A. N. hira, 1977, aisics of Random Procss 2, pringrvrlag, Nw York. Ridl,F., 2, Imprfc Informaion and Insor Hrogni in h Bond Mark, Phsica Vrlag. Vasick,., 1977, An Equilibrium Characrizaion of h Trm rucur, ournal of Financial Economics 5, Hiroshi Moria, Profssor, Facul of Businss Adminisraion, Yokohama Naional Unirsi