Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.

Transcription

1 Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since we observe he dividends and reurns ex-pos, we can es he resricions he dynamic Gordon model impose on he ime-series of observed dividends and reurns. 1

2 Campbell and Shiller ask he quesion: Do exising models of expeced reurns (discoun raes) explain he variaion in he dividend-o-price raio afer subracing off dividend growh? Quick answer: NO Long answer: 1) The ex-pos log dividend-o-price raio is significanly differen han he esimaed log dividend-o-price raio. The esimae of he log D/P is based on he raional expecaions esimaes of fuure discoun raes and dividend growh raes based from he VAR. 2) The sandard deviaion of he presen value of fuure expeced discoun raes is much smaller han he variaion in log D/P no explained by expeced log dividend growh. 2

3 Log D/P model R P = +1 R + D log( R ) h log( P D ) log( P ) Campbell and Shiller wan a linear relaionship, so hey approximae h wih a firs-order Taylor series approximaion (some error induced): h ξ k + ρ log( P + 1) + (1 ρ)log( D ) log( P ) = k + ρp (1 ρ) d p 3

4 δ d p Don forge hese are all logs! 1 Based on he Taylor approximaion we ge: h k + δ ρδ +1 + d Solving his equaion, wih he condiion: i lim i ρ δ + i = 0, hey ge he relaionship hey wan : δ = j ρ ( h d ) j + j + j 0 1 k ρ. The log D/P is a funcion of he discouned value of fuure reurns and dividend growh. All ex-pos NO ECONOMIC CONTENT! 4

5 Add some heory abou he behavior of ex-pos reurns and we can es he economic model of he D/P raio. E h = Err + c, where E denoes a raional expecaion formed by using informaion se I Now ake he expecaion of D/P raio equaion from he las slide from he based on he condiional informaion se I, which is available o agens a he beginning of : δ E j= ρ ( h d ) + j + j + j 0 1 c k ρ 5

6 Daa and Uni Roo Tess Two daa ses are used: 1) S&P 500 index from , 2) Value- Weighed NYSE index from T-bill raes and CPI inflaion raes from Ibboson Ass. Table 2. Need o check for uni roos. Why? 1) Sandard errors are no good if non-saionary regressors. 2) Resuls are sensiive o saionariy assumpions when esimaing. Table 3. 6

7 Campbell and Shiller assume ha he log D/P raio, growh raes of real dividends and prices are saionary, so ha log dividends and prices are coinegraed processes. Par of coinegraion (1,1) means he variables are saionary in firs differences. This urns ou o be beer han ransforming he variables by removing a deerminisic linear rend, because i induces biases. Campbell and Shiller (1987) go ino he deails of coinegraion and wha i means for he esimaion of presen value models. The VAR s I m abou o discuss are based on he 1987 paper. Campbell and Shiller (1988 JF) also use he VAR seup bu include earnings. 7

8 VAR s Campbell and Shiller wan o forecas fuure discoun raes and dividend growh raes and compue an implied D/P raio, hen compare he implied movemens in he D/P raio o he observed movemens in he D/P raio. They use a VAR o esimae he expeced fuure discoun raes and expeced fuure dividend growh. They include he log D/P raio as a variable in he VAR o generae forecass. The log D/P summarizes all relevan info in he marke so hey aren missing anyhing, and he forecass exacly equal he D/P raio. 8

9 Campbell and Shiller assume marke observes y, a vecor of sae variables and y follows a linear sochasic process. They choose a x, so ha i is he smalles ha allows hem o es he resricions of he D/P model. x = [ ] δ, r 1 d 1 They also assume x can be wrien as a VAR wih p lags. They make a ransformaion so hey can rewrie he VAR in firs order auoregression. The resuling VAR sysem is: z = Az 1 + υ 9

10 Condiioning on he economerician s informaion se (H ) hey ge: δ E ρ h d H δ = j ( + j + j ) j 0 Rewriing his equaion in erms of he VAR forecass: δ = e z = 1 ' ρ e2' A z δ = j j+1 j o which imposes he following resricion which can be esed wih a Wald es, 10

11 e1' j j+ 1 = ρ e2' A = e2' A( I ρa j= o ) 1 A es which is algebraically equivalen is: e1 '( I ρ A) e2' A = 0 The previous x does no separae discoun raes and dividend growh. If x has hree elemens we can es he separae effecs of each variable. The new resricion is: e z = 1 ' ρ ( e3' e2') A z δ, and = j j+1 j o e1 '( I ρ A) ( e3' e2') A = 0 11

12 Key relaionship is: δ = δ δ + δ r d Finally, if we observe consumpion growh or variance in sead of he discoun rae iself, consumpion growh or variance will be in he x vecor and he following resricions mus hold. δ = δ δ + δ αe3' A( I ρa) z e2' A( I ρa) 1 1 r d z, and e1 '( I ρ A) ( αe3' e2') A = 0 where rr = α c or rr = αv, and α is he relaive risk aversion coefficien and is esimaed from he resricions. 12

13 Empirical Resuls Tables

14 Main Resuls: Conclusion, Problems, and Exensions Log D/P raio does move raionally wih expeced fuure growh in dividends. Measures of discoun raes in he paper do no explain sock-price movemens very much. (Granger causaliy ess are no sig.) Variaion in dividend growh and discoun raes do no fully explain movemens in dividends. - Dividends are oo variable Possible Problems: Taylor series approximaion induces bias. Campbell and Shiller (1988) show ha he bias is no big enough o explain he resuls. 14

15 Use a differen measure of discoun raes. I is possible ha he processes are no sable, linear sochasic processes. Recen Exensions: Bekaer and Grenadier (2000), Vuoleenaho (2000), Young (2000) 15

16 Reurn o FIN 533 hp://schwer.ssb.rocheser.edu/f533/f533.hm 16