00Thaler_FM i-xxvi.qxd

temporarily let all signals have the same precision (i.e., q=r=p), with pC
replacing pas the perceived precision of the first signal if overconfidence
occurs. Direct calculation of the covariances then shows that

(A41)

(A42)

(A43)

By direct comparison, E[∆P 1 ∆P 2 ]r=q=pand E[∆P 2 ∆P 3 ′]r=q=pare related by:

(A44)

so the covariance between the date 2 and 3 price changes is negatively pro-
portional to the covariance of the date 1 and 2 price changes. Consider the
numerator N of the proportionality factor. The first three components,
2 p(1−p), are maximized when p=1/2 while the next two components,
pC(1−pC), are maximized when pC=1/2. Since the last two components
satisfy (2p−1)(2pC−1)<1, the N≤1/8. In the denominator D, the ex-
pression pC(2p−1)+(1−p) is minimized when pC=p=1/2, resulting in a
minimum of 1/2. The second component of Dis similarly minimized when
pC=p=1/2, resulting in a minimum of 1/4. So D≥1/8. Since N≤1/8, the
ratio N/D≤1. Therefore, the negative covariance between date two and
date three price changes must be, in absolute value, less than or equal to the
positive covariance between period one and period two price changes, re-
sulting in an overall one-period covariance that is positive.
When q=rdiffers from p, direct calculation of covariances shows:
E[∆P 1 ∆P 2 ]r=q>0; (A45)
E[∆P 1 ∆P 3 ′]r=q<0; (A46)
E[∆P 2 ∆P 3 ′]r=q<0. (A47)

Now let the signal s 3 ′have a precision of rthat differs from both preci-
sions of pand q. Proceeding as above, the covariance satisfy

E[∆P 1 ∆P 2 ]>0; (A48)

E[∆P 1 ∆P 3 ′]<0; (A49)

EPP

ppp pp p

pp ppp p

rqp CC C EPP

CC

[] rqp

()( )( )( )

[( )( )][ ( ) ( ) ]

∆∆ 23 2 [],∆∆ 12

21 1 2 12 1

′ 211 211

===− ==

−− −−

−+− −+−

EPP

pp p p p p p p

rqp pp ppp p

CCC C

CC

[]

()( )( )( )( )

[( ) ][( )( )]

∆∆ 23.

22 2

22

41 12121

211 211

′ ===^0

−−−− −

−+− −+−

<

EPP

pp p p ppp

rqp pp ppp p

CC C

CC

[]

()( )( )( )

[( ) ][( )( )]

∆∆ 13 ;

3

2

21 121

211 211

′ ===^0

−−− −

−+− −+−

<

EPP

ppp pp

rqp pp p

C

C

[]

()( )( )

()()

∆∆ 12 ;

21 2 1

211

===^0

−−−

−+−

>

496 DANIEL, HIRSHLEIFER, SUBRAHMANYAM