00Thaler_FM i-xxvi.qxd

Appendix A

Proposition 1.If the investor believes that earnings are generated

by the regime-switching model described in section 4, then prices

satisfy

wherep 1 and p 2 are given by the following expressions:

where

Proof of Proposition 1: The price will simply equal the value as gauged

by the uninformed investors which we can calculate from the pres-

ent value formula:

Since

Et(Nt+ 1 )=Nt+Et(yt+ 1 ),

Et(Nt+ 2 )=Nt+Et(yt+ 1 )+Et(yt+ 2 ), and so on,

we have

PNy

yy

tttt

=+ +tt tt

+

+

+

+⋅⋅⋅

















+

(^1) ++
(^111)
23
δδE( ) δ^2
E( ) E( )
()
.
PE
NN
tt
= tt

• 
  


• 
  


• 
  

  +⋅⋅⋅
  
  
  
  
  
  
  
  
  ++ 12
  1 δ () 1 δ^2.
  Q=
  −−− −
  −− − −
  −− −−
  −−−
  () ()( ) ( )
  ()( )() ( )
  () () ()( )
  () ()(
  LLL L
  LL L L
  HHH H
  HH H
  111 1
  11 1 1
  11 11
  111
  11 2 2
  112 2
  11 2 2
  112
  λπ λ π λπ λ π
  λπ λπ λπ λπ
  λπ λ π λ π λ π
  λπ λπ λ π))( )
  .
  1 − 2 H
  
  
  
  
  
  
  
  
  
  
  
  
  
  λπ 
  ′= − −
  ′=
  ′= −
  γ
  γ
  γ
  0
  1
  2
  1111
  0010
  10 10
  (, , , ),
  (, ,, ),
  (, , , ),
  pIQQ
  pIQQ
  10
  1
  1
  20
  1
  2
  1
  11
  1
  11
  =′+ +−
  =− ′ + + −
  −
  −
  δ
  γδ δ γ
  δ
  γδ δ γ
  (( )[( ) ]),
  (( )[( ) ]),
  P
  N
  t=+ −t yp pqtt
  δ
  (), 12
  450 BARBERIS, SHLEIFER, VISHNY