ublc Affars 974 Meze D. Ch Fall Socal Sceces 748 Uversy of Wscos-Madso Sock rces, News ad he Effce Markes Hypohess (rev d //) 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. D [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 Expecaos (such ha he marke expecao s he mahemacal codoal expecao). Hece he expecaos operaor refers o he codoal mahemacal expecaos operaor. I oe-perod: D E [8. ] Where for smplcy, he eres rae used o dscou he fuure s a cosa, ad E (Z ) E(Z formao avalable a me ). Assume a me, ha D s kow. Noe ha he prce ex perod s gve by: D E [8. ] Subsug o [8. ] yelds: D E ( E D ) E ( E ) () ( )( ) ( )( ) Noe ha by he Law of Ieraed Expecaos, vz., E ( E ( Z )) EZ Equao () becomes: D E D E ( ) ( ) [8.3 ] Oe ca coue o subsue ou for o oba he Geeralzed Dvded Valuao Model: D E D E D E... ( ) ( ) ( ) [8.4 ] Noe ha hs expresso mples, uder cera codos:
D E D E D E D E D () ( ) 3... 3 ( ) ( ) ( ) Lm E Equao () rules ou bubbles. Tha s, s assumed ha ( ) The Gordo Growh Model assumes ha dvdeds are expeced o grow deermscally a rae g, such ha D ( g) D (whch s equao [8.6]). Subsug [8.6] o () yelds, for < : D ( g) D ( g) D ( g)... [8.7] ( ) ( ) ( ) If oe allows o go o fy: D ( g) D ( g) D ( g)... (3) ( ) ( ) ( ) ( g) D ( g) ( g)... ( ) ( ) ( ) ( g) Noe he exbook re-formulaes he mg o ge a slghly dffere expresso: D (4) D ( g) [8.8] ( g) I geeral, D 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 o (4) ad [8.8] yelds: D ( rf rp g) (5) Noe he exbook re-formulaes he mg o ge a slghly dffere expresso: D ( g) [8.] ( 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., 3,6 5, 8 5 4 5 3 4 5 6 7 8 9 REALS_AND_ REALDIVIDEND Fgure : Real (CI deflaed) Sadard ad oor dex (lef scale), ad real dvdeds (rgh scale). Source: Rober Shller, hp://www.eco.yale.edu/~shller/daa/e_daa.xls. 3,,8,4,,6, 8 4 6 4 8 6 4 3 4 5 6 7 8 9 REALS_AND_ TENYEARINTRATE Fgure : Real (CI deflaed) Sadard ad oor dex (lef scale), ad e year eres rae (rgh scale). Source: Rober Shller, hp://www.eco.yale.edu/~shller/daa/e_daa.xls. 3
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 D leads o he relaoshp ha he prce/dvded rao s versely relaed o he eres rae (mus he growh rae of dvdeds). D ( g) ( g) ( g)... ( rf rp) ( rf rp) ( rf rp) ( rf rp g) (6) The posed verse relaoshp s show Fgure 3. 6 4 Te year eres rae [rgh scale] 6 4 8 6 4 rce/dvded rao [lef scale] 8 6 4 3 4 5 6 7 8 9 S_AND_/DIVIDEND TENYEARINTRATE 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. News Le s reur o a rsk eural model. Recall he prese value of a sock s gve by: D E [8. ] E D E ( E ) E D E E (7) 4
The las erm afer he secod equal sg (7) obas by he Law of Ieraed Expecaos, vz., E ( E ( Z )) EZ Now decompose he chage he prce of he asse: E ) [( E )] (8) ( 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. D E D E E ( E ) (9) ( rf rp) ( rf rp) News cludes he dvdeds aouced for perod. I s uforecasable. Ths ews may also affec people s expecaos regardg D 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 E E ( rf rp) 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 D perod could sll affec expeced asse prces he fuure, ad hece he asse prce oday. A example: The sock marke (Goldma Sachs sued by SEC, aouceme approx. :3am) 5
Wha equao () says s ha he prce wll evolve as expecaos of dvdeds o he fuure chage over me. D E D E D E D 3... 3 ( rf rp) ( ( rf rp)) ( ( rf rp)) ( ( rf rp)) E D ( ( 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: D ( rf rp ) ( ( rf E D 3 ( ( rf rp ))( ( rf rp ( ) rp ))( ( rf E D ))( ( rf rp rp ))... )) ( ( rf E D 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 Effecs 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. () s he mos well kow aalyss of he LSA s effec o log erm bod raes. Neely () aalyses he mpac of LSA aoucemes o bod raes ad exchage raes. 6
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Effce 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 E X u u _ d(, σ ) 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: E ( rf rp) Use he defo of raoal expecaos above, ad assume log-ormaly of he error erm: p E 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 (k e ), whch from oe day o aoher s quaavely very small. Refereces Joseph Gago, Mahew Rask, Jule Remache, ad Bra Sack,, Large-Scale Asse urchases by he Federal Reserve: Dd They Work? Saff Repor o. 44 (Federal Reserve Bak of New York). Chrsopher J. Neely,, The Large Scale Asse urchases Had Large Ieraoal Effecs, Workg aper -8A (Federal Reserve Bak of S. Lous). 9.. rev.. a974_emh_f 8