Miller and Modigliani s (1961) stock price valuation formula based on constant discount rates implies ( )

Transcription

1 VARIANCE BOUNDS Mille and Modigliani s 96) sock pice valuaion fomula based on consan discoun aes implies E[ d ] ) ) If we knew he fuue dividends fo ceain we would have pice - he pefec- foesigh d 2) ) Examining ) and 2) we find E 3) Since is he condiional expeced value of we have E [ ] ε whee E [ ε ] 0 by R. E ε which implies ) Va Va ) Va ε ) Fom which we have he VARIANCE BOUND Va ) Va ) 4) he examinaion of his inequaliy is wha he vaiance bound lieaue is all abou.

2 SHILLER 98) los and fo he yeas : Shille saes ha he inequaliy 4) is violaed damaically by daa as is immediaely oblivions in looking a he figues p. 423). In he plo he va iance of uns ou o be fa in excess of he vaiance of See able 2 p 43 σ 50.2 fo S&; σ fo DJ σ 8.97 fo S&; σ 26.8 fo DJ 2

3 he quesion is -Wha do hese plos ell you? Kleidon 986) pesens anohe plo. I ceainly looks he vaiance bound is violaed i.e. i looks like Va ) > Va ) Bu his plo is based in simulaed Daa whee he Daa is geneaed such ha. Va ) > Va ) Hence nohing can be infeed fom he plos abou he validiy of equaion 4). JUS WHA IS GOING ON HERE? he elaion Va ) > Va ) given by equaion 4) applies a paicula ime. I is a coss secional elaion acoss diffeen economics. Since we only eve see ealizaions fo a single economy we can only eally examine whehe 4) holds acoss economies by simulaion. los like and 2 ae ime seies plos fo a single economy. 3

4 Can we use his daa o check he validiy of 4)? Yes, i.e. if we make addiional assumpions o ou model. we need o assume daa is saionay and egodic. Noe: Since he daa is sock pice daa which is hough o follow a Random Walk and is heefoe nonsaionay) his is a eally bad assumpion! WHA SHOULD YOU EXEC FROM A IME SERIES LO LIKE FIG.. and fuue dividends won look like each ohe a ime if hee is unceainy abou On any ealizaion will diffe fom is expeced value epeaed dawings of would equal is expeced value,. Of couse, if we had coss secionally acoss diffeen economics, on aveage How much diffeence hee exiss beween infomaion is available when pices ae se. and depends o have much 4

5 GROSSMAN AND SHILLER 98) Gossman and Shille adjus he model in ) o allow fo non consan discoun aes u' C E B D uc ) A C whee UC ), CRRA A if A 0 Risk Neualiy) use have consan discoun aes. if A > 0 we have non consan discoun aes. hey find hey ge he closes fi beween and when A 4 [ close o up unil 949] Howeve hei analysis somewha misses he poin. Since on any ealizaion will diffe fom is expeced value,. 5

6 6 2. YOU WOULD EXEC O BE SMOOH ) d ) d hese wo pices ae viually idenical and one would expec hen o be exemely highly coelaed and hence he smooh cuve of. Also noe by consucion D and he ohe pefec foesigh pices ae woked ou ecusively fom hee d ossible change in consecuive value of is limied o he capial gain equied o give he expos eun. his einfoces he noion ha he plos will be smooh.

7 FIRS -he LOS SUMMARY ) and will no have idenical plos if hee is unceainly as o fuue dividends. If we could obseve coss secionally acoss economies, on aveage would equal is expeced value. Howeve we see only one ealizaion of he vaiable fom i s expeced value. which will pobably diffe 2) By consucion will show a vey smooh plo. his is because and epesen almos exacly he same fuue dividends and hence will be vey highly coelaed. SECOND- esing he Vaiance Bound. Va ) Va ) > 4) is a Coss Secional Resicion. o es i using imes seies daa we mus assume: saionaiy and egodiciy of he daa ices and Dividends). Unfounaely pices ae hough o follow a Random Walk and ae heefoe nonsaionay. In fac in his case uncondiional vaiances do no exis µ ε µ ε ) 0 Va ) Hence Compaing ) exis. Va wih ) HIRD - CONDIIONAL VARIANCE BOUNDS ) Va / Φ Va / Φ ) k k Va is no valid since hee quaniies do no Kleidon ess his and finds ha he vaiance bound is no violaed Secion 3, p. 984 and able 3) 7