coomcs 435 Meze. Ch Fall 07 Socal Sceces 748 Uversy of Wscos-Madso Sock rces, News ad he ffce Markes Hypohess The rese Value Model Approach o Asse rcg The exbook expresses he sock prce as he prese dscoued value of he dvded pad ad he prce of he sock ex perod. [8.] I realy, we do kow he prce ex perod. Raher we have a expecao of he prce ex perod, ad for ow we assume for ow Raoal xpecaos (such ha he marke expecao s he mahemacal codoal expecao. Hece he expecaos operaor refers o he codoal mahemacal expecaos operaor. I oe-perod: [8. ] Where for smplcy, he eres rae used o dscou he fuure s a cosa, ad (Z+ = (Z formao avalable a me +. Assume a me, ha s kow. Noe ha he prce ex perod s gve by: [8. ] Subsug o [8. ] yelds: ( ( ( ( ( ( ( Noe ha by he Law of Ieraed xpecaos, vz., ( ( Z Z quao ( becomes: ( ( [8.3 ] Oe ca coue o subsue ou for + o oba he Geeralzed vded Valuao Model: ( (... ( [8.4 ] Noe ha hs expresso mples, uder cera codos:
( ( 3... 3 ( ( ( Lm quao ( rules ou bubbles. Tha s, s assumed ha 0 ( k The Gordo Growh Model assumes ha dvdeds are expeced o grow deermscally a rae g, such ha ( g (whch s equao [8.6]. Subsug [8.6] o ( yelds, for < : ( g ( g ( g... [8.7] ( ( If oe allows o go o fy: ( g ( g ( g... (3 ( ( ( g ( g ( g... ( g ( ( ( g I geeral, wll o grow a smooh deermsc fasho, or wll be cosa. As a cosequece, he flucuaos prces wll o move oe for oe wh coemporaeous dvdeds. The prevous calculaos assume ha he eres rae used o dscou he fuure values s cosa, ad equal o a rsk free rae. I geeral, he eres rae used s he sum of he rsk free rae ad a rsk premum (rf ad rp, respecvely, he exbook. Subsug: e (4 ( g (5 ( rf rp g I he below fgures, mohly daa from Rober Shller s webse ( hp://www.eco.yale.edu/~shller/daa/e_daa.xls are used o hghlgh he relaoshps bewee sock prces, dvdeds ad eres raes. I Fgure, real (CI deflaed sock prces ad dvdeds are show; ad Fgure real prces ad e year eres raes.
5,000 80 3,500,500,500,000 500 350 50 Real S& [lef scale] Real dvded [rgh scale] 40 3 4 0 6 50 8 00 50 80 90 00 0 0 30 40 50 60 70 80 90 00 0 4 RALS_AN_ RALIVIN Fgure : Real (CI deflaed Sadard ad oor dex (lef scale, log, ad real dvdeds (rgh scale, log. Source: Rober Shller, hp://www.eco.yale.edu/~shller/daa/e_daa.xls accessed 0/5/07. 5,000 3,500,500,500 Te year rae, % [rgh scale] 6 4,000 0 500 350 50 Real S& [lef scale] 8 6 50 4 00 50 0 80 90 00 0 0 30 40 50 60 70 80 90 00 0 RALS_AN_ GS0 Fgure : Real (CI deflaed Sadard ad oor dex (lef scale, log, ad e year eres rae (rgh scale. Source: Rober Shller, hp://www.eco.yale.edu/~shller/daa/e_daa.xls, accessed 0/5/07. As predced, he sock prce dex covares posvely wh dvdeds (dvdeds are hghly serally correlaed, so a moveme up dvdeds persss, ad s egavely relaed o he eres rae. Noe ha dvdg boh sdes of equao 5 by leads o he relaoshp ha he prce/dvded rao s versely relaed o he eres rae (mus he growh rae of dvdeds. 3
( g ( g... ( g rf rp ( rf rp ( rf rp ( rf rp g (6 The posed verse relaoshp s show Fgure 3. 60 40 0 00 Te year rae, % [rgh scale] 6 4 0 80 60 40 0 0 S& rcedvded rao [lef scale] 80 90 00 0 0 30 40 50 60 70 80 90 00 0 8 6 4 0 S_AN_/IVIN GS0 Fgure 3: Sadard ad oor prce o dvded rao (lef scale, ad e year eres rae (rgh scale. Source: Rober Shller, hp://www.eco.yale.edu/~shller/daa/e_daa.xls accessed 0/5/07. News Le s reur o a rsk eural model. Recall he prese value of a sock s gve by: [8. ] ( (7 The las erm afer he secod equal sg (7 obas by he Law of Ieraed xpecaos, vz., ( ( Z Z Now decompose he chage he prce of he asse: [( ] (8 ( 4
The frs erm s he expeced poro of he prce chage. The secod erm he brackes s he uexpeced poro. Ths secod poro square brackes ca be furher broke up. ( (9 ( rp rf ( rp rf News cludes he dvdeds aouced for perod +. I s uforecasable. Ths ews may also affec people s expecaos regardg he fuure, ad hece he fuure (whch ur affecs expecaos of perod +. Hece, ew formao drecly resuls a ew prce, ad revsos expecaos. Noce he secod erm he square bracke s ( rp rf whch s he chage he expecaos regardg he asse prce perod +, based upo wha he marke kew perod + versus wha kew perod. Noe ha oher ews ha does affec perod + could sll affec expeced asse prces he fuure, ad hece he asse prce oday. A example: The sock marke (Goldma Sachs sued by SC, aouceme approx. 0:30am 5
Wha equao ( says s ha he prce wll evolve as expecaos of dvdeds o he fuure chage over me. 3... 3 ( rf rp ( ( rf rp ( ( rf rp ( ( rf rp ( ( rf rp ( Those dvded sreams deped par upo he eargs ha frms are expeced o ear he fuure. As he ecoomy looks more lkely o slow dow, expecaos of eargs (ad hece dvdeds are lkely o be revsed dowward. I addo, here s o reaso he requred reur o equy has o rema cosa. If vares over me, he ( becomes: ( rf rp ( ( rf 3 ( ( rf rp ( ( rf rp ( rp ( ( rf ( ( rf rp rp... ( ( rf rp...( ( rf rp To he exe ha he requred reur vares wh he eres rae (say o he 3 moh Treasury ad rsk averso, a addoal source of varao s roduced o he sock prce. Aouceme ffecs Oher Markes Ths framework for aalyzg asse prces has bee appled umerable cases. Mos recely, has bee appled o he Fed s mplemeao of large scale asse purchases of log erm Treasures ad morgage backed secures. Gago e al. (00 s he mos well kow aalyss of he LSA s effec o log erm bod raes. Neely (00 aalyses he mpac of LSA aoucemes o bod raes ad exchage raes. 6
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ffce Markes Hypohess I he above dscusso, I have assumed ha he subjecve marke expecao of fuure dvdeds ad prces equals he mahemacal expecaos. Ths assumpo s called he raoal expecaos hypohess. Oe mplcao of raoal expecaos s: X X u u _ d(0, u Tha s, he acual realzao of X equals he expecao of X based o me - formao, plus a uforecasable radom error, u. Cosder for sace f dvdeds were pad oly every year, ad each perod was oe day. The equao ( becomes: ( rf rp Use he defo of raoal expecaos above, ad assume log-ormaly of he error erm: p p l( rf rp p p rf rp u~ ( So sock prces follow a radom walk (wh drf. I s o que rgh o say he bes predcor of omorrow s sock prce s oday s. Raher here s a small predcable compoe (ke, whch from oe day o aoher s quaavely very small. Refereces Joseph Gago, Mahew Rask, Jule Remache, ad Bra Sack, 00, Large-Scale Asse urchases by he Federal Reserve: d They Work? Saff Repor o. 44 (Federal Reserve Bak of New York. Chrsopher J. Neely, 00, The Large Scale Asse urchases Had Large Ieraoal ffecs, Workg aper 00-08A (Federal Reserve Bak of S. Lous. 0.5.07 435_emh_f7 8