00Thaler_FM i-xxvi.qxd

Proposition 5:Using standard normal distribution properties,

*=E[
P 1 , θ+η] (A3)

It is straightforward to show that the ratio of the date 2 mispricing

to 
* is

(A4)

which is constant (for a given level of confidence). Thus, selective

events can alternatively be viewed as events that are linearly related

to 
*.

High values of 
* signify overpricing and low values underpricing.

The proposition follows by observing that

(A5)

and

(A6)

Since cov(P 3 −P 2 , 
*)<0, by the conditioning properties of mean-

zero normal distributions, E[P 3 −P 2 
*] can be written in the form

k
*, where k<0 is a constant. Thus, E[P 3 −P 2 
*]<0 if and only

if
*>0. Since this holds for each positive realization of 
*, E[P 3 −

P 2 
*>0]<0. By symmetric reasoning, E[P 3 −P 2 
*<0]>0. The

result for event-date price reactions uses a similar method. Since

cov(P 2 −P 1 ,
*)<0, it follows that E[P 2 −P 1 
*]<0 if and only if


*>0.

Proposition 6: We interpret the “fundamental/price” ratio or “run-up”
as For part 1,

(A7)

By our assumption that the selective event is linearly related to 
*,

the selective event is positively correlated with the mispricing mea-

sure, proving part 1.

cov( , *)

[( ) ]

()

θ.

σσσ σ σσ

σσ σ σσ

θθ

θθ

−=

++

++

P >

pp

Cp p

1

222 2 22


 22 2 22 0

θ−P 1.

cov( , *)

[( ) ]

()[()]

PP Cpp.

CCp p

21

224 2 2 2 22

−=−22222 222 0

++

+++

 

 <

σσσ σ σ σ σσ

σσσσσ σσ

θθθ

θθθ

cov( , *)

()( )

[( ) ][( ) ]

PP

pC

ppC pp

32

222 2 2 2 2

−= 22 2 22 2 2 2 220

++ −

++ ++

 <

σσσ σ σ σ σ

σσ σ σσ σ σ σ σσ

ηθ θ

θθθθ

σσσ σ σσ

σσ σ σ

θθ

θθ

22 2 2 22

22 2 2

[( ) ]

()

Cpp,

pC

++

−

=

++− +

++

σσ σ θ σσθ η

σσ σ σσ

θθ

θθ

(^222)
22
22 2 22
()()()
()
.
p
pp
INVESTOR PSYCHOLOGY 491