similarly if st=2, we are in the second regime and the earnings shock is
generated by Model 2. The parameters λ 1 and λ 2 determine the probabili-
ties of transition from one state to another. We focus particularly on small
λ 1 and λ 2 , which means that transitions from one state to another occur
rarely. In particular, we assume that λ 1 +λ 2 <1. We also think of λ 1 as
being smaller than λ 2. Since the unconditional probability of being in state
1 is λ 2 /(λ 1 +λ 2 ), this implies that the investor thinks of Model 1 as being
more likely than Model 2, on average. Our results do not depend, however,
on λ 1 being smaller than λ 2. The effects that we document can also obtain
if λ 1 >λ 2.
In order to value the security, the investor needs to forecast earnings into
the future. Since the model he is using dictates that earnings at any time are
generated by one of two regimes, the investor sees his task as trying to un-
derstand which of the two regimes is currently governing earnings. He ob-
serves earnings each period and uses that information to make as good a
guess as possible about which regime he is in. In particular, at time t, having
observed the earnings shock yt, he calculates qt, the probability that ytwas
generated by Model 1, using the new data to update his estimate from the
previous period, qt− 1. Formally, qt=Pr(st= 1
yt, yt− 1 , qt− 1 ). We suppose
that the updating follows Bayes Rule, so that

In particular, if the shock to earnings in period t+1, yt+ 1 , is the same as the
shock in period t, yt, the investor updates qt+ 1 from qtusing

and we show in the appendix that in this case, qt+ 1 <qt. In other words, the
investor puts more weight on Model 2 if he sees two consecutive shocks of
the same sign. Similarly, if the shock in period t+1 has the opposite sign to
that in period t,

and in this case, qt+ 1 >qtand the weight on Model 1 increases.
To aid intuition about how the model works, we present a simple exam-
ple shown in table 12.1. Suppose that in period 0, the shock to earnings y 0
is positive and the probability assigned to Model 1 by the investor, that is,
q 0 , is 0.5. For a randomly generated earnings stream over the next 20 peri-
ods, the table below presents the investor’s belief qtthat the time tshock to
earnings is generated by Model 1. The particular parameter values chosen

q

qq

t qq q q

tt

tt t t

+ =

−+− −

(^1) −+− −++−− −
12
12 1
111
111 111
(( ) ( ))( )
(( ) ( ))( ) ( ( )( ))( )
L ,
L2H
λλ π
λλ πλ λ π
q
qq
t qq q q
tt
tt t t

• =
  −+−
  (^1) −+− ++−−
  12
  12 1
  11
  11 11
  (( ) ( ))
  (( ) ( )) ( ( )( ))
  L ,
  L2H
  λλ π
  λλ πλ λ π
  q
  qqsy
  qqsyq qsy
  t
  tttt
  tttt ttt


• 
  


• 
  

  ++

  
  

  −+− =
  −+− =++−− =
  1
  12 1
  12 1 11 1
  11 1
  11 1 11 2
  (( ) ( )) ( , )
  (( ) ( )) ( , ) ( ( )( )) ( , )
  (^1).
  121
  λλ
  λλ λ λ
  Pr y
  Pr y Pr y
  t+
  t+ t+
  438 BARBERIS, SHLEIFER, VISHNY