Internet Appendix for
“Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance”
VIKAS AGARWAL, KEVIN A. MULLALLY, YUEHUA TANG, and BAOZHONG YANG
This Appendix consists of two sections. Section I provides propositions and their proofs for our
model of informed trading with different mandatory disclosure frequencies. Section II
tabulates additional results mentioned in the paper but omitted for brevity.
I.
Propositions and Proofs
A. Propositions
The following proposition characterizes the strategies and expected profits of the informed
trader, and the pricing rules of the market maker. In the proof of the proposition, we also show
that this is the unique equilibrium when strategies are constrained to be of the forms given in (1)
to (4) in the main article.
PROPOSITION 1: If k  1 , then the equilibrium strategies can be characterized as follows.
2
(i) There are constants  n ,  n , n ,  n ,  n ,  n , and  zn such that the strategies satisfy
(1) to (4), and the informed trader’s expected profits are given by
E[ n | p1* ,, pn* 1 , v]   n 1 (v  pn1 ) 2   n 1 , for 1  n  N .
(IA.1)
We define constants n   nn , for 1  n  N  1 and  N  0 to facilitate the presentation of
2
the results below. Given  0 and  u2 , the constants  n , n ,  n ,  n ,  n , and  zn solve the
following recursive equation system:
(a) If n = N, then

Citation format: Agarwal, Vikas, Kevin Andrew Mullally, Yuehua Tang, and Baozhong Yang, Internet Appendix
for “Mandatory Portfolio Disclosure, Stock Liquidity, and Mutual Fund Performance,” Journal of Finance,
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1
 N 1 
1
4N
, N 
1
2N
, N   N
N

2
u
, and  N 
1
 N 1.
2
(IA.2)
(b) If n  N 1 , or n < N is not equal to km or km  1 for some integer m > 0, then
n 1 
n
4 n (1  n 1 )
,  n 1 
1  2n
1
, n 
,
4n (1   n )
2n (1   n )

1
n  n 2 n , and  n 
 n 1.
u
2(1  n )
(IA.3)
(c) If n  N 1 is equal to km  1 for some integer m > 0, then
n 1 
n
1  2n
1
,  n 1 
, n 
4n
4n (1  n )
2n (1  n )
(IA.4)
n
1
n   n 2 , and  n 
 n 1.
u
2(1  n )
(d) If n < N is a multiple of k, then
n
1  2 n 1
1
,  n 1 
, n 
,
1  n 1
4n 1 (1  n 1 )
4n (1  n 1 )
(IA.5)
 n  n 1 (1  n 1 )  n  n
1
1
n 

, n 
 n 1 ,  n  2n , and  z2 
 u2 .
2 u2
 u2
2(1  n 1 )
2(1  n 1 )
n 1 
n
(e) In the first period, the market depth parameter is given by
1 
1  21 0
.
2(1  1 )  u
(IA.6)
(ii) The sequence of constants {n }1n N that appear in the recursive formulas (IA.2) to (IA.6)
do not depend on  0 and  u , and are uniquely determined by the following equations:
(a) If n = N, then  N  0 .
(b) If n  N 1 , or n < N is not equal to km or km  1 for some integer m > 0, then
0  n  1/ 2 and
8( n3  n2 ) 
1
(2n  1)  0 .
1  2n 1
2
(IA.7)
(c) If n  N 1 is equal to km  1 for some m > 0, then 0  n  1/ 2 and
8( n3  n2 ) 
2(1  n  2 )
(2n  1)  0.
1  2n  2
(IA.8)
(d) If n < N is a multiple of k, then n  1/ 4.
(iii) In the case of full disclosure in each period (or the case of k = 1), the equilibrium strategies
are characterized below. Denote the constants by ˆ n , ˆn , ˆn , ˆn , ˆ n , ˆn ,
2
and  zˆn .
(a) If n = N, then
ˆ N 1 
1 ˆ
1
, N 
ˆ
2 u
4N
0 ˆ
1
1
, N 
, and ˆ N  ˆ N 1.
ˆ
N
2
2N
(IA.9)
(b) If n < N, then
ˆ n 1 
1 ˆ
1
, n 
ˆ
2 u
4n
0 ˆ
1
, n 
,
N
2( N  n  1)ˆn
(IA.10)
N n ˆ
N n
N n
ˆ n 
 n 1 
 0 , ˆn  2ˆn , and  zˆ2n 
 u2 .
N  n 1
N
N  n 1
Part (i) gives the recursive formulae for the strategy parameters. Part (ii) directly
computes the series of key constants  n (used in the recursive formulae) through backward
induction. Part (iii) for the case k = 1 simply replicates the solution given in Proposition 4 in
Huddart, Hughes, and Levine (2001). In the special case k  N , the equilibrium given in the
above proposition reduces to the Kyle (1985) model.
PROPOSITION 2: (i) Assume k = 2, that is, the informed trader is required to disclose once
every two periods. Denote the average illiquidity for the case in which the informed trader is
required to disclose every two periods by  N 
1
N
N

i 1
i
and denote the average illiquidity for
the case in which the informed trader is required to disclose every period by ˆ N 
Then
3
1
N
N
 ˆ
n 1
n
.
ˆ  .

N
N
(IA.11)
That is, more frequent disclosure leads to lower average illiquidity or higher average liquidity.
ˆ increases with the extent of asymmetric information
Furthermore, the difference  N  
N
0 .
(ii) Denote the expected profits of the informed trader in the case in which the informed trader
ˆ . Then
is required to disclose every two periods by  N and every period by 
N
ˆ .
N  
N
(IA.12)
In other words, the informed trader’s profits are decreasing in the frequency of disclosure. The
ˆ increases with the extent of asymmetric information 0 .
difference  N  
N
(iii) If N '  N  2 , then
ˆ     
ˆ  .

N
N
N
N
(IA.13)
In other words, the informed trader’s profit decline from more frequent disclosure is greater
when the total number of periods is larger.
This proposition shows that market liquidity increases as a result of more frequent
disclosure. Furthermore, the liquidity improvement depends positively on the extent of
asymmetric information about the underlying security. The informed trader, however, makes
less profits due to the more frequent mandatory disclosure. His profit decline is greater when
information asymmetry is higher or when trading takes longer. Note that the cases k = 2 and k =
1 in the proposition correspond closely to the regulation where the mandatory disclosure
frequency is increased from semi-annual to quarterly.
B. Proofs
Proof of Proposition 1:
Part (i): We first prove cases (a) to (d) by induction.
Case (a): Because n = N is the last period, the disclosure requirement does not change
4
the insider’s strategy and thus the solution given in Theorem 2 of Kyle (1985) applies and (IA.2)
holds. Now assume that (a) to (d) hold for the (n+1)-th period. We will show that they also hold
for the n-th period.
Case (b): If n  N 1 , or n < N is not equal to km or km  1 for some integer m > 0,
then Theorem 2 of Kyle (1985) applies in period n and we have
 n 1 
1  2n
1
, n 
,
4n (1  n )
2n (1  n )
n
1
n   n 2 , and  n  (1  n  n ) n 1 
 n 1.
u
2(1  n )
(IA.14)
Furthermore, because n+1 is not a multiple of k, cases (a) to (c) for the (n+1)-th period imply
that
n 
1
,
n 1 (1  n 1 )
n
n
1
n 1 


.
4 n (1  n 1 ) 4(1  n 1 ) n n 4 n (1  n 1 )
(IA.15)
Equations (IA.14) and (IA.15) complete the proof of (IA.3) in case (b).
Case (c): If n  N 1 is equal to km  1 for some integer m > 0, then the insider is not
required to disclose and Theorem 2 of Kyle (1985) also applies in period n and we have
 n 1 
1  2n
1
, n 
,
4n (1  n )
2n (1  n )
n
1
n   n 2 , and  n  (1  n  n ) n 1 
 n 1.
u
2(1  n )
Since n+1 is equal to km, case (d) for the (n+1)-th period implies
n  2 
n
n
1


4 n (1  n  2 ) 4(1  n  2 ) n n 4 n (1  n  2 )
and
n  2 
Therefore,
5
n 1
.
1  n  2
(IA.16)
n 1  n  2 (1  n  2 ) 
n
.
4n
(IA.17)
Equations (IA.16) and (IA.17) complete the proof of (IA.4) in case (c).
Case (d): If n < N is a multiple of k, consider the insider’s expected profits conditional
on his information set in the (n-1)-th period:
En1[ n ( pn1 , v) | v]  En1[ xn (v  pn )   n (v  pn ) 2   n | v]
 En 1[ xn (v  pn1  n ( xn  un ))   n (v  pn1   n xn ) 2   n | v]
(IA.18)
 En 1[( n n2  n ) xn2  (1  2 n n ) xn (v  pn1 )   n (v  pn1 )2 | v]   n .
The strategy xn   n (v  pn*1 )  zn with the noise term zn ~ N (0,  z2n ) implies that the insider
is indifferent between different values of xn , and therefore
 n n2  n  0,
1  2 n n  0.
This implies that
 n  2n , n 
1
4 n

n 1
,
1  n 1
(IA.19)
where in the last step we use the equation for  n given by cases (a) to (c) for the period (n+1).
The breakeven conditions of the market maker are
pn  En1[v | xn  un ]  pn*1  n ( xn  un ),
pn*  En1[v | xn ]  pn*1   n xn ,
implying that
n 
Cov(v, xn  un )
 n  n 1
 2
,
Var ( xn  un )
 n  n 1   z2n   u2
Cov(v, xn )

n 
 2 n n 1 2 .
Var ( xn )
 n  n 1   zn
Equations (IA.19) and (IA.20) imply that
6
(IA.20)
n2n1   z2   u2
n
(IA.21)
and
n 
 n  n1
.
2 u2
(IA.22)
We also have
n  Varn 1[v | xn ]  Varn 1[v] 
Covn 1 (v, xn )2
 n 1   n n n 1.
Varn 1 ( xn )
(IA.23)
Equations (IA.22) and (IA.23) imply that
 n   n 1  4n 2 u2   n 1  4n21 (1  n 1 )2  u2 .
(IA.24)
Recall from the (n+1)-th period that
 n 1 

2
n 1
1
n ,
2(1  n 1 )
  
1  2 n 1  n 1
1  2 n 1  n
 n 1 n 21 n 1 

.
2
u
2(1  n 1 )  u
4(1  n 1 ) 2  u2
(IA.25)
Plugging into (IA.24), we obtain
n 
 n 1
.
2(1  n 1 )
(IA.26)
Equations (IA.22), (IA.25), and (IA.26) imply that
n 
2n u2 2 u2 (1  n 1 )n 1 2 u2 (1  n 1 )n21


 n 1
 n 1
n 1 n 1
(1  2n 1 ) n 1
(1  2 n 1 )
1  2 n 1



.
2
n 1 n 1
4n 1 (1  n 1 )
4n (1  n 1 )
(IA.27)
Finally, equations (IA.18) and (IA.19) imply that
 n 1   n 
1
1

.
4n 1 (1  n 1 ) 4n
(IA.28)
Equations (IA.19), (IA.22), (IA.26), (IA.27), and (IA.28) complete the proof of (IA.5) in case
7
(d).
Case (e): Since k > 1, cases (b) and (c) imply that
1  1
Therefore, 1 
1

2
u

10
1  2 1  0

.
2
2(1  1 ) u 41 (1  1 ) 2  u2
1  21 0
.
2(1  1 )  u
Part (ii): The proof of this part will need the following lemma.
LEMMA 1: Suppose K  0 , then there is a unique solution   (0,1) to the following
equation
8 3  8 2  K (2  1)  0.
(IA.29)
Furthermore, 0    1/ 2 .
Proof of Lemma 1: By taking the derivative, it is easy to show that the function
f ( ) 

8 2 (1   )
is increasing for   (0,1/ 2) . Because f (  ) approaches zero as
1  2
0 , and  as 
1

2
, there is a unique   (0,1/ 2) such that f (  )  K , that is,
(IA.29) is satisfied. Because f (  )  0 for   (1/ 2,1) , the above solution is also the unique
solution in the interval (0,1) . Q.E.D.
We proceed to prove cases (a) to (d) sequentially. Case (a) is trivial as we define
 N  0 . In case (b), n  N 1 , or n < N is not equal to km or km 1 for some integer m > 0.
Applying Part (i) cases (a) to (c) to the periods n and (n+1), we have
n   n
n

2
u
n 1   n 1
, n 
 n 1
 u2
1  2n
,
2n (1  n )
1  2n 1
1
,  n 1 
 n , and  n 1 
.
2(1  n 1 )
2n 1 (1  n 1 )
(IA.30)
From (IA.30), we obtain
n 1
 n 1
1

.
n
2(1  n 1 )  n
8
(IA.31)
Next, plugging  n 
1  2n
1  2n 1
and  n 1 
from (IA.30) into (IA.31) and
2n (1  n )
2n 1 (1  n 1 )
reorganizing, we obtain
2
 n 1 
2(1  n ) 1  2 n 1
.

 
1  2 n 4(1  n 1 ) 2
 n 
(IA.32)
n 1
1

.
n
4  n (1   n 1 )
(IA.33)
Part (i), case (b) implies that
Substituting (IA.33) into (IA.32), we obtain the recursive equation
8( n3  n2 ) 
1
(2n  1)  0.
1  2n 1
(IA.34)
By the second-order condition in Kyle (1985), 0  n  1 . It then follows from Lemma
1 that if 0  n1  1/ 2 , there is a unique root  n of (IA.34) in (0,1) such that 0  n  1/ 2 ,
which proves Case (b).
Case (c): Applying Part (i) Cases (c) and (d) to the periods n and (n+1), respectively,
we have
n   n
n

2
u
, n 
1  2n
,
2n (1  n )

1  2n  2

n 1   n 1 n2 ,  n 1 
, and n 1  n .
2 u
4n 1 (1  n  2 )
4n
(IA.35)
Using (IA.35) and similar algebra as in case (b), we obtain
8( n3  n2 ) 
2(1  n  2 )
(2n  1)  0.
1  2n  2
(IA.36)
The lemma now together with induction then implies that case (c) holds.
Case (d): By Part (i), cases (a) to (d),  n 
n   n n  1/ 4.
9
1
1
, which implies that

4n 1 (1  n 1 ) 4n
Part (iii): This is simply a replication of the solution for the case of disclosure in every period
given in Proposition 4 of Huddart, Hughes, and Levine (2001). We refer the reader to Huddart,
Hughes, and Levine (2001) for the proof.
Q.E.D.
Proof of Proposition 2: Since n is proportional to
N
function  N  n is proportional to
0
u
, the aggregate illiquidity
0  u , and so is the aggregate illiquidity in the
n 1
ˆ , or
ˆ . Therefore, if (IA.11) is true, the decrease in illiquidity   
full-disclosure case 
N
N
N
improvement in liquidity, is proportional to 0  u , and thus is increasing in  0 . We next
proceed to prove (IA.11).
To facilitate the proof, we explicitly indicate the total number of periods in our notations
below, for instance, we use n , N  n , and n, N  n . We also assume that
0
u
 1 , since it is
just a normalizing constant in (IA.11). We first show the following lemma which is useful for
our proof.
LEMMA 2: i) 2 k 1, N , k  1, 2, ,
N
, is decreasing with k.
2
ii) For N  4 ,
2 m1, N  2 m, N  2 m1, N  2 m 2, N , if m 
N
 2.
2
(IA.37)
iii) For any 0  k  N ,
N  2 k  2, N  2 ˆN  2, N  2

,
N  2 k , N
ˆN , N
(IA.38)
N  2 k 1, N  2 ˆN  2, N  2

.
N  2 k 1, N
ˆN , N
(IA.39)
Proof of Lemma 2:
Part i): Define vi  N 2i 1 . The recursive formula (IA.8) implies that
10
f (vi 1 ) 
where f ( x) 
2(1  vi )
,
1  2vi
(IA.40)
8 x 2 (1  x)
. We also have vk  (0,1/ 2) . Therefore,
1 2x
8vi21 (1  vi 1 ) 2(1  vi ) 8vi2 (1  vi )
f (vi 1 ) 


 f (vi ).
1  2vi 1
1  2vi
1  2vi
Since f (·) is increasing for x  (0,1/ 2) , vi is increasing in i and thus  2 k 1 is decreasing in
k.
Part ii): We have
1
1
2 m1  2 m
42 m1
1  42 m1

42 m1 (1  2 m1 ) 
42 m1 (1  2 m1 ).
2 m1  2 m 2 1  1
1  42 m1
42 m1
We know that
f ( y) 
f (  2 m 1 ) 
1   2 m 1
1  2  2 m 1
and the function
y
1  4 x2
16 x(1  x)
(IA.41)
satisfies
2(1  x)
, if 1/2>x  0.36 . Therefore, the increasing property of the function f implies
1 2x
that
2 m1 
1  422m1
, if 2 m1  0.36.
162 m1 (1  2 m1 )
(IA.42)
Plugging (IA.42) into (IA.41), we have
2 m 1  2 m
 1, if 2 m 1  0.36.
2 m 1  2 m  2
Since m 
(IA.43)
N
 2 , Part i) and the fact that  N 3  v2  0.387  0.36 imply that 2m1  0.36
2
and thus (IA.37) holds.
Part iii): First, note that by the recursive formulas in Proposition 1,
11
N  2 k , N 
1
4 N  2 k 1
N  2 k 1, N 
1
4 N  2 k 1
1
N 2 k 3, N  
4 N 2 k 3 (1   N 2 k 1 )
1  21  N /2 k
 N /2 k
 1

1
1
 





1, N
2
 m  2 4 2 m 1 (1  2 m 1 )  4 1
 m 1 4 2 m 1 (1  2 m 1 ) 

1  2vN / 2  N /2

1
 
.
2
4
v
(1

v
)
i

N
/
2

k
i
i 

Therefore,
N  2 k  2, N  2
1  2vN /21
1
2

N  2 k , N
2
4vN /21 (1  vN /2 1 ) 1  2vN /2




1  2vN /21
1  2v N / 2
1
4vN /21 (1  vN /21 )
4vN2 /21 (1  vN /21 )
1
1  vN / 2
4vN /21 (1  vN /21 )
1

2 (1  vN /2 )(1  vN /21 )
N ( N  2)
( N  1)( N  3)
where we used the fact that vi 

1
2 (1 
N 1
N 1
)(1 
)
2N
2( N  2)
ˆN  2, N  2
N

,
N 2
ˆN , N
i 1
, which is easily verified using the recursive formula
2i
(IA.40) and the increasing property of the function f. This proves (IA.38). (IA.39) now
follows from (IA.38) and the recursive relations N  2 k 1, N  4 N  2 k 1N  2 k , N  4vk 1N  2 k , N and
N 2 k 1, N  2  4vk 1N 2 k  2, N  2 . This completes the proof of the lemma.
Q.E.D.
Given the lemma, we will prove (IA.11) using induction on the number of periods N. In
the case N = 2, using the recursive formulas in Proposition 1 it is easy to obtain that
1,2  2,2  0.462  0.416  2ˆ2,2  0.707.
We next show that the following equations hold for any N  4 :
N 3, N  N 2, N  2ˆN , N ,
N 2, N  N 1, N  2ˆN , N .
For N  4 , we have
12
(IA.44)
1,4  2,4  0.388  0.251  2ˆ2,2  0.5,
3,4  4,4  0.345  0.312  2ˆ2,2  0.5.
Now suppose (IA.44) hold for N. Equation (IA.38) from Lemma 2 then imply that
N 1, N  2  N , N  2
ˆN  2, N  2

N 1, N  2  N  2, N  2
ˆN  2, N  2
N 3, N  N  2, N
ˆN , N

N 1, N  N , N
ˆN , N
 2,
 2.
Thus, (IA.44) also hold for N + 2. Combing (IA.44) and (IA.37), we see that
N /2
 N  (2 m1, N  2 m, N ) 
m 1
N 2
(N 3, N  N 2, N )  (N 1, N  N , N )
2
(IA.45)
N
ˆ .
  2ˆN , N  
N
2
This completes the proof of (IA.11).
(ii) We first show that the expected profits of the informed trader in period n for the cases k  1
and k  2 are given by
E[ n ]  n n2 , 1  n  N ,
E[ˆ n ]  ˆn n2 , 1  n  N .
(IA.46)
Indeed, in the case in which the informed trader is required to disclose every period (k = 1),
Proposition 4 of Huddart, Hughes, and Levine (2001) shows that the expected profit is given by
E[ˆ n ] 
u ˆ
0 / N  ˆn u2 .
2
In the case in which the informed trader is only required to disclose every two periods
(k = 2), if n = N or n is not a multiple of two, then by (1), (2), (IA.12), and (IA.15), the expected
profit is
E[ xn (v  pn )]  E[ n (v  pn*1 )(v  pn*1  n ( n (v  pn*1 )  un ))]
  n (1  n n )n 1  nn  n u2 .
If n < N is a multiple of 2, then by (1), (2), and (IA.5),
13
E[ xn (v  pn )]  E[(  n (v  pn*1 )  zn )(v  pn*1  n (  n (v  pn*1 )  zn  un ))]
  n (1  n  n ) n 1  n z2n   n

3  2  n 1
1
 n 1 
n u2
4(1   n 1 )
2(1   n 1 )
3  2 n 1
1
n u2 
n u2
2(1  n 1 )
2(1  n 1 )
 n u2 .
Therefore, (IA.46) always hold. Combining part (i) and (IA.46), we obtain the desired result on
total expected profits,
N
N
n 1
n 1
ˆ 2 
ˆ .
 N   E[ n ]   n u2   N  u2  
N u
N
As the expected profits are proportional to the aggregate illiquidity, by part (i), the difference
ˆ decreases with
N  
N
0 .
(iii) We only need to show that for any N  2 ,
ˆ
ˆ

N 2   N 2   N   N .
(IA.47)
If this is true, then (A17) follows by induction. Because of (IA.46), this is equivalent to
ˆ
ˆ

N 2   N 2   N   N .
(IA.48)
The case N  2 can be directly verified using the expressions of  calculated in (i). Next,
we show that (IA.48) holds for any N  4 . From (IA.38) and (IA.39) in Lemma 2, we have
2i 1, N  2  2i  2, N  2  (2i 1, N  2i , N )
ˆN  2, N  2
ˆN , N
, 1  i  N / 2.
(IA.49)
From (IA.37) and induction, it is easy to show that
1, N  2  2, N  2  2i 1, N  2  2i  2, N  2 , 1  i  N / 2.
Using (IA.49) and (IA.50),
14
(IA.50)
N /2
ˆ
ˆ
 N 2  
N  2  (2 i 1, N  2  2 i  2, N  2 )  ( N  2)N  2, N  2
i 0
ˆN  2, N  2
N 2

(2i1, N 2  2i, N ) ˆ  ( N  2)ˆN 2, N 2
N i 1
N ,N
N /2


ˆ
ˆ
N  2  N /2
ˆ  N  2, N  2  N  2 (  
ˆ ) N  2, N  2
(



)

N

2i, N
N ,N 
N
N
  2i 1, N  2
N  i 1
N
ˆN , N
 ˆN , N
N 2
ˆ )   
ˆ .
( N  
N
N
N
N
Q.E.D.
15
II. Additional Tables for Robustness Checks
In this section, we present results omitted from the main text of the paper for brevity.
IA.I. Analysis of the 1985 Regulation Change
This table presents results for tests on stock liquidity and fund performance conducted
using December 1985 as another event month. In 1985, the SEC changed the frequency of
disclosure required for mutual funds from a quarterly frequency to a semi-annual frequency.
We repeat our analyses in Panel A of Table III and Panel B of Table VII of the paper and present
the results in this table.
IA.II. Tests Controlling for the Change in Mutual Fund Ownership
This table presents results of regressions of the change in stock liquidity on Mutual
Fund Ownership and the change in mutual fund ownership. It is possible that mutual fund
trading changes around the regulation change. Including the change in mutual fund ownership
in the regressions helps control for this possibility. We find that our results on the impact of the
regulation change on stock liquidity are robust to the inclusion of this variable.
IA.III and IV. Tests using Abnormal Ownership
These two tables present results using abnormal mutual fund ownership as the main
independent variable in our tests. It is possible that mutual fund ownership of stocks is related
to stock characteristics. To control for this possibility, we employ a two-stage procedure. In the
first stage, we regress mutual fund ownership on both stock characteristics and a lagged
liquidity variable. We define the residual from this regression as Abnormal Mutual Fund
Ownership. In the second stage, we regress the change in stock liquidity on Abnormal Mutual
Fund Ownership and other control variables. Table IA.III shows that our results in Table II,
Panel B and Table III, Panel D in the paper are robust to this specification. Further, the results
in Table IA.IV show that our findings on NonMF Ownership and Hedge Fund Ownership are
also robust to this two-stage procedure.
16
IA.V. Fund Subsample Tests using Alternative Performance Measures
This table reports results from our fund subsample tests in which we use alternative
measures of fund performance. Specifically, we classify informed funds using
Liquidity-Adjusted DGTW (Rspread) and the impatient trading measure of Da, Gao, and
Jagannathan (2011). Liquidity-Adjusted DGTW (Rspread) is analogous to Liquidity-adjusted
DGTW in the paper (stocks sorted using Rspread instead of Amihud when forming the DGTW
benchmark portfolios). Our results using these alternative measures are qualitatively similar to
those presented in Table V in the paper.
IA.VI. Tests using Changes in Mutual Fund Characteristics
In this table, we present results using changes in mutual fund characteristics as the
independent variables in regressions estimated using equation (13) of the paper. It is possible
that top mutual funds themselves experienced changes around the SEC rule change in 2004;
such changes in fund characteristics, rather than the regulation change, may explain the
performance deterioration in top funds. Our results on fund performance in this table rule out
such a possibility.
IA.VII. Full-Period Time-Series Placebo Tests excluding Crisis Periods
This table presents results analogous to those presented in Table IX of the paper. We
exclude known crisis years (1998, 2000, and 2001) from our placebo period to ensure that our
results are not driven by these years. We continue to find that the difference in the performance
decline for top mandatory and top voluntary funds is statistically larger in May 2004 compared
to the 1994 to 2006 placebo period after excluding the crisis years.
17
REFERENCES
Huddart, Steven, John S. Hughes, and Carolyn B. Levine, 2001, Public disclosure and
dissimulation of insider trades, Econometrica 69, 665–681.
Kyle, Albert S., 1985, Continuous auctions and insider trading, Econometrica 53, 1315–1335.
Da, Zhi, Pengjie Gao, and Ravi Jagannathan, 2011, Impatient trading, liquidity provision, and
stock selection by mutual funds, Review of Financial Studies 24, 675–720.
18
Table IA.I
Impact of the 1985 Regulation Change on Stock Liquidity and Fund Performance
This table presents results related to the 1985 regulation change. Panel A presents regressions of the change in
liquidity around December 1985 on Mutual Fund Ownership, NonMF Ownership, and the lagged stock
characteristic variables we use in Table II, Panel B of the paper. The last two rows report differences between
the coefficients on Mutual Fund Ownership and NonMF Ownership and p-values from F-tests of the differences.
Panel B presents regressions of the change in mutual fund performance on an indicator variable equal to one if
the fund was in the top quartile of a given performance measure and zero otherwise, and the fund characteristics
we use in Table VII, Panel B in the paper. Standard errors are adjusted for heteroskedasticity and clustered at the
stock level, and t-statistics are reported below the coefficients. Coefficients marked with ***, **, and * are
significant at the 1%, 5%, and 10% level, respectively.
Panel A: 1985 Liquidity Analysis
(1)
ΔAmihud
(2)
ΔRspread
0.453
(0.39)
–0.258***
(–3.61)
–0.484***
(–18.95)
0.067***
(3.02)
–0.193***
(–11.15)
–0.224***
(–9.76)
–0.860***
(–6.78)
–1.004
(–1.14)
–0.165***
(–3.25)
–0.392***
(–19.80)
–0.037**
(–2.10)
–0.112***
(–10.40)
–0.246***
(–12.46)
–0.278***
(–6.01)
Observations
Adj. R2
1,386
0.524
1,386
0.496
Difference (MF – NonMF)
p-value (Difference)
0.7112
0.547
–0.8388
0.349
MF Ownership
NonMF Ownership
Momentum
Book-to-Market
Size
Lagged Liquidity
Constant
19
Panel B: 1985 Performance Analysis
(1)
(2)
Four-Factor Five-Factor
Alpha
Alpha
(3)
DGTW
–0.195
(–1.40)
Four-Factor Alpha
–0.157
(–1.39)
Five-Factor Alpha
0.156*
(2.11)
0.044
(0.35)
45.559
(1.29)
–1.283
(–1.80)
0.069
(1.26)
0.120
(1.40)
9.551
(0.38)
–0.521
(–1.02)
–0.092
(–0.96)
0.066
(1.69)
–0.012
(–0.16)
20.339
(1.12)
–0.514
(–1.46)
11
0.418
11
0.367
12
0.049
DGTW
Log(TNA)
Turnover
Expense Ratio
Constant
Observations
Adjusted R2
20
Table IA.II
Impact of Mandatory Portfolio Disclosure on Stock Liquidity: Regressions Including the Change in
Mutual Fund Ownership
This table reports regression results of changes in the stock liquidity variables around May 2004 on mutual fund
ownership and other control variables as in Table II, Panel B of the paper. We augment these regressions by
including ΔMutual Fund Ownership as an additional control variable. The independent variables are the
averages of the variables in Table II, Panel A in the year prior to May 2004. Standard errors are adjusted for
heteroskedasticity and clustered at the stock level, and t-statistics are reported below the coefficients in
parentheses. Coefficients marked with ***, **, and * are significant at the 1%, 5%, and 10% level, respectively.
(1)
(2)
ΔAmihud
ΔRspread
–1.194***
–2.088***
–2.416***
–1.966***
(–10.49)
(–13.64)
(–14.53)
(–11.73)
–4.218***
–3.460***
–3.612***
–5.461***
(–19.79)
(–12.75)
(–12.30)
(–17.59)
–0.061***
–0.102***
–0.119***
–0.105***
(–6.42)
(–10.64)
(–11.74)
(–7.42)
–0.123***
–0.049***
–0.029*
–0.136***
(–9.12)
(–3.34)
(–1.88)
(–6.47)
–0.153***
–0.119***
–0.138***
–0.051***
(–14.16)
(–16.10)
(–20.41)
(–5.75)
–0.233***
–0.227***
–0.278***
–0.118***
(–14.25)
(–12.87)
(–16.79)
(–8.85)
–1.137***
–0.361***
–0.457***
–0.351***
(–13.06)
(–8.33)
(–10.35)
(–8.23)
Observations
4,635
4,634
4,634
4,634
2
0.175
0.173
0.199
0.120
Mutual Fund Ownership
ΔMutual Fund Ownership
Momentum
Book-to-Market
Size
Lagged Liquidity
Constant
Adj. R
21
(3)
ΔSize-Weighted
Rspread
(4)
ΔEff. Spread
Table IA.III
Impact of Mandatory Portfolio Disclosure on Stock Liquidity: Base Regressions Using Abnormal
Ownership
This table reports results of a two-stage regression procedure. In the first stage, we regress aggregate mutual
fund ownership on Momentum, Size, Book-to-Market, and the corresponding lagged liquidity variable. We
define Abnormal MF Ownership as the residual of this first-stage regression. We then regress the change in
stock liquidity around May 2004 on this abnormal ownership variable and other control variables as in Table II,
Panel B of the paper. Panels A and C report results of the first-stage analysis in 2004 and the placebo period in
2006, respectively. Panels B and D report the second-stage regressions in 2004 and 2006, respectively. Panel E
reports differences between coefficients on abnormal mutual fund ownership in 2004 and 2006 and p-values
from F-tests. Standard errors are adjusted for heteroskedasticity and clustered at the stock level, and t-statistics
are reported below the coefficients in parentheses. Coefficients marked with ***, **, and * are significant at the
1%, 5%, and 10% level, respectively.
Panel A. First-Stage Analysis in 2004
Dependent Variable: MF Ownership
(1)
Momentum
Book-to-Market
Size
Liquidity (X)
Constant
Observations
Adj. R2
(2)
X = Amihud
X = Rspread
(3)
X=
Size-Weighted
Rspread
–0.002**
(–2.53)
0.004***
(3.55)
–0.020***
(–15.99)
–0.056***
(–35.08)
–0.330***
(–42.03)
0.004***
(5.28)
0.002
(1.60)
0.010***
(13.30)
–0.027***
(–16.38)
–0.101***
(–29.09)
0.005***
(5.47)
0.002*
(1.72)
0.010***
(14.52)
–0.029***
(–18.80)
–0.108***
(–31.29)
0.006***
(7.29)
–0.002*
(–1.81)
0.011***
(16.39)
–0.016***
(–18.39)
–0.070***
(–30.13)
4,635
0.547
4,634
0.449
4,634
0.458
4,634
0.45
22
(4)
X = Effective
Spread
Panel B: Second-Stage Regressions in 2004
(1)
(2)
ΔAmihud
ΔRspread
–0.815***
–1.795***
–2.100***
–1.459***
(–7.17)
(–11.96)
(–12.94)
(–8.83)
–0.081***
–0.127***
–0.146***
–0.140***
(–8.04)
(–12.80)
(–13.96)
(–9.92)
–0.133***
–0.055***
–0.036**
–0.129***
(–9.13)
(–3.70)
(–2.35)
(–6.14)
–0.139***
–0.143***
–0.165***
–0.068***
(–13.24)
(–19.24)
(–24.26)
(–7.64)
–0.177***
–0.174***
–0.211***
–0.077***
(–12.56)
(–10.46)
(–13.59)
(–5.94)
–0.795***
–0.160***
–0.206***
–0.224***
(–10.87)
(–4.02)
(–5.14)
(–5.59)
Observations
4,635
4,634
4,634
4,634
2
0.0827
0.137
0.165
0.0586
Abnormal MF Ownership
Momentum
Book-to-Market
Size
Lagged Liquidity
Constant
Adj. R
(3)
(4)
ΔSize-Weighted
Rspread
ΔEff. Spread
Panel C: First-Stage Analysis in 2006
Dependent Variable: MF Ownership
(1)
Momentum
Book-to-Market
Size
Liquidity (X)
Constant
Observations
Adj. R2
(3)
X = Amihud
X = Rspread
(4)
X=
Size-weighted
Rspread
–0.004**
(–2.22)
0.001
(0.79)
–0.022***
(–17.67)
–0.059***
(–38.26)
–0.344***
(–46.43)
0.009***
(4.70)
–0.002
(–1.59)
–0.011***
(–10.93)
–0.051***
(–36.39)
–0.134***
(–46.24)
0.008***
(4.66)
–0.003*
(–1.90)
–0.008***
(–9.31)
–0.054***
(–39.54)
–0.166***
(–50.65)
0.009***
(4.90)
–0.007***
(–4.81)
–0.011***
(–10.88)
–0.049***
(–37.10)
–0.136***
(–47.16)
4,467
0.531
4,467
0.517
4,467
0.536
4,467
0.518
23
(5)
X = Effective
Spread
Panel D: Second-Stage Regressions in 2006
Abnormal MF Ownership
Momentum
Book-to-Market
Size
Lagged Liquidity
Constant
Observations
Adj. R-squared
(1)
(2)
(3)
ΔAmihud
ΔRspread
–0.525***
(–4.99)
–0.216***
(–10.84)
–0.036**
(–2.42)
–0.072***
(–7.36)
–0.085***
(–6.50)
–0.439***
(–6.35)
–0.629***
(–6.67)
–0.271***
(–15.66)
–0.038***
(–2.73)
–0.114***
(–14.28)
–0.127***
(–10.79)
–0.395***
(–13.37)
–0.639***
(–6.80)
–0.301***
(–17.48)
–0.021
(–1.43)
–0.129***
(–19.81)
–0.156***
(–14.17)
–0.481***
(–14.45)
–0.575***
(–6.44)
–0.287***
(–16.33)
–0.017
(–1.19)
–0.090***
(–11.84)
–0.182***
(–15.89)
–0.802***
(–25.40)
4,467
0.0521
4,466
0.126
4,466
0.172
4,466
0.147
ΔSize-Weighted
Rspread
(4)
ΔEff. Spread
Panel E: Differences in the Coefficients on Abnormal Mutual Fund Ownership (Panels B and D)
Diff. of Coefficients (2004–2006)
Test of Differences (p-value)
(1)
(2)
(3)
(4)
ΔAmihud
ΔRspread
ΔSize-Weighted Rspread
ΔEff. Spread
–0.290*
0.059
–1.166***
<.0001
–1.461***
<.0001
–0.884**
<.0001
24
Table IA.IV
Impact of Mandatory Portfolio Disclosure on Stock Liquidity:
Cross-sectional Placebo Regressions Using Abnormal Ownership
This table reports the results of a two-stage regression procedure. In the first stage, we regress the aggregate
mutual fund (or nonmutual fund or hedge fund) ownership on Momentum, Size, Book-to-Market, and the
corresponding lagged liquidity variable. We define Abnormal MF Ownership (or Abnormal NonMF Ownership
or Abnormal Hedge Fund Ownership) as the residual of the first-stage regression. We then regress the change in
stock liquidity around May 2004 on this abnormal ownership variable and other control variables as in Table II,
Panel B of the paper. Panels A and C report results of the first-stage analysis in 2004 for NonMF Ownership and
Hedge Fund Ownership, respectively. Panels B and D report the second-stage regressions in which we compare
Abnormal MF Ownership with Abnormal NonMF Ownership and Abnormal Hedge Fund Ownership,
respectively. The last two rows in Panels B and D compare the coefficients on abnormal mutual fund ownership
and the corresponding abnormal institutional ownership variable and p-values from F-tests. Standard errors are
adjusted for heteroskedasticity and clustered at the stock level, and t-statistics are reported below the
coefficients in parentheses. Coefficients marked with ***, **, and * are significant at the 1%, 5%, and 10%
level, respectively.
Panel A. First-Stage Analysis for NonMF Institutions
Dependent Variable: NonMF Ownership
(1)
(2)
(3)
X=
X=
X=
Size-Weighted
Amihud
Rspread
Rspread
Momentum
Book-to-Market
Size
Liquidity (X)
Constant
Observations
Adj. R2
(4)
X=
Effective
Spread
–0.025***
(–12.44)
0.032***
(8.45)
–0.029***
(–8.75)
–0.122***
(–27.67)
–0.713***
(–32.37)
–0.010***
(–5.04)
0.024***
(6.19)
0.047***
(23.70)
–0.031***
(–7.37)
–0.172***
(–18.11)
–0.010***
(–5.03)
0.025***
(6.36)
0.045***
(24.46)
–0.037***
(–9.29)
–0.187***
(–19.61)
–0.007***
(–3.41)
0.017***
(4.37)
0.035***
(21.87)
–0.040***
(–19.20)
–0.152***
(–23.21)
4,635
0.572
4,634
0.483
4,634
0.487
4,634
0.512
25
Panel B. Second-Stage Regressions for Abnormal NonMF Ownership
Abnormal MF Ownership
Abnormal NonMF Ownership
Momentum
Book-to-Market
Size
Lagged Liquidity
Constant
Observations
Adj. R2
Diff. of Coeffs. (MF – NonMF)
Test of Difference (p-value)
(1)
(2)
(3)
ΔSize-Weighted
Rspread
(4)
ΔEff.
Spread
ΔAmihud
ΔRspread
–0.636***
(–5.29)
–0.228***
(–3.95)
–0.081***
(–8.06)
–0.133***
(–9.16)
–0.139***
(–13.25)
–0.177***
(–12.55)
–0.795***
(–10.86)
–1.302***
(–7.66)
–0.447***
(–6.64)
–0.127***
(–12.84)
–0.055***
(–3.72)
–0.143***
(–19.42)
–0.173***
(–10.49)
–0.160***
(–4.01)
–1.562***
(–8.60)
–0.494***
(–6.93)
–0.146***
(–14.00)
–0.036**
(–2.36)
–0.165***
(–24.69)
–0.211***
(–13.71)
–0.206***
(–5.13)
–1.057***
(–5.82)
–0.400***
(–4.93)
–0.140***
(–9.93)
–0.128***
(–6.16)
–0.068***
(–7.69)
–0.077***
(–5.95)
–0.224***
(–5.61)
4,635
0.0861
4,634
0.146
4,634
0.174
4,634
0.0632
–0.408***
.007
–0.855***
<.0001
–1.068***
<.0001
–0.657***
.004
26
Panel C. First-Stage Analysis for Hedge Funds
(1)
Momentum
Book-to-Market
Size
Liquidity (X)
Constant
Observations
Adj. R2
Dependent Variable: HF Ownership
(2)
(3)
(4)
X=
Amihud
X=
Rspread
X=
Size-Weighted
Rspread
X=
Effective
Spread
–0.005***
(–3.79)
0.009***
(3.69)
–0.031***
(–16.41)
–0.059***
(–23.45)
–0.279***
(–21.79)
0.002*
(1.89)
0.005**
(2.11)
0.006***
(5.25)
–0.015***
(–6.20)
–0.017***
(–2.99)
0.002*
(1.92)
0.005**
(2.23)
0.005***
(4.78)
–0.018***
(–8.05)
–0.025***
(–4.31)
0.004***
(3.08)
0.002
(0.76)
0.001
(0.95)
–0.018***
(–13.50)
–0.007
(–1.54)
4,635
0.190
4,634
0.0934
4,634
0.0985
4,634
0.120
27
Panel D. Second-Stage Regressions for Abnormal HF Ownership
Abnormal MF Ownership
Abnormal HF Ownership
Momentum
Book-to-Market
Size
Lagged Liquidity
Constant
Observations
Adj. R2
Diff. of Coeffs. (MF – HF)
Test of Difference (p-value)
(1)
(2)
(3)
ΔSize-Weighted
Rspread
(4)
ΔEff.
Spread
ΔAmihud
ΔRspread
–0.720***
(–6.19)
–0.313***
(–3.67)
–0.081***
(–8.06)
–0.133***
(–9.14)
–0.139***
(–13.24)
–0.177***
(–12.55)
–0.795***
(–10.85)
–0.108
(–0.99)
–0.118
(–1.43)
–0.083***
(–9.22)
–0.115***
(–8.35)
–0.073***
(–10.28)
–0.102***
(–9.22)
–0.249***
(–7.21)
–1.691***
(–9.91)
–0.881***
(–8.03)
–0.146***
(–14.19)
–0.036**
(–2.37)
–0.165***
(–24.77)
–0.211***
(–13.73)
–0.206***
(–5.14)
–1.205***
(–7.04)
–0.590***
(–4.79)
–0.140***
(–9.96)
–0.128***
(–6.16)
–0.068***
(–7.67)
–0.077***
(–5.94)
–0.224***
(–5.60)
4,635
0.0855
4,634
0.148
4,634
0.177
4,634
0.0629
–0.407**
0.010
–0.679***
0.002
–0.81***
0.0004
–0.615***
0.009
28
Table IA.V
Impact of Mandatory Portfolio Disclosure on Stock Liquidity: Subsamples of Mutual Funds
This table reports regression results of the changes in stock liquidity on mutual fund ownership of top- and
nontop-performing funds. The dependent variables are the changes in the liquidity variables after May 2004. All
regressions include controls for lagged stock liquidity and other stock characteristics as in Table II, Panel B in the
paper. The last two rows report differences between the coefficients on the ownership of top-quartile and
nontop-quartile funds and p-values from F-tests of the differences. Liquidity-Adjusted DGTW (Rspread) is
calculated by augmenting size, book-to-market, and momentum with stock liquidity (using Rspread) in the
characteristics used to form the DGTW benchmark portfolios. Da, Gao, and Jagannathan DGTW is the impatient
trading measure of Da, Gao and Jagannathan (2011). Panels A and B report the results when funds are separated
based on whether they are in the top quartile of these performance measures for the prior year. Standard errors are
adjusted for heteroskedasticity and clustered at the stock level, and t-statistics are reported below the coefficients
in parentheses. Coefficients marked with ***, **, and * are significant at the 1%, 5%, and 10% level, respectively.
Panel A: Liquidity-Adjusted DGTW (Rspread)
Top Fund Ownership
NonTop Fund Ownership
Difference (Top – NonTop)
p-value (diff.)
(1)
(2)
ΔAmihud
ΔRspread
–0.0019*** –0.0040***
(–5.97)
(–10.11)
–0.0006** –0.0022***
(–2.25)
(–6.34)
4,635
0.0872
4,634
0.163
(3)
(4)
ΔSize-Weighted
ΔEff. Spread
Rspread
–0.0045***
(–10.97)
–0.0027***
(–7.31)
–0.0033***
(–6.81)
–0.0017***
(–3.87)
4,634
0.195
4,634
0.0696
Panel B: Da, Gao, and Jagannathan (2011)
Top Fund Ownership
NonTop Fund Ownership
Difference (Top – NonTop)
p-value (diff.)
(1)
(2)
ΔAmihud
ΔRspread
–0.0014*** –0.0039***
(–4.14)
(–9.39)
–0.0011*** –0.0024***
(–4.49)
(–7.97)
4,635
0.0876
4,634
0.169
29
(3)
(4)
ΔSize-Weighted
ΔEff. Spread
Rspread
–0.0043***
(–9.74)
–0.0031***
(–9.48)
–0.0033***
(–6.23)
–0.0019***
(–4.86)
4,634
0.202
4,634
0.0711
Table IA.VI
Impact of Disclosure Regulation on Mutual Fund Performance: Changes on Changes Regressions
This table reports results of multivariate regressions of changes in fund performance after 2004 on lagged fund
performance and changes in fund characteristics. In all regressions, we control for changes in fund characteristics,
including ΔLog(TNA), ΔTurnover, ΔFlow, ΔExpense Ratio, and ΔLoad. All variables are defined as in Table VII of
the paper. Standard errors are adjusted for heteroskedasticity and clustered at the fund level, and t-statistics are
reported in parentheses. Coefficients marked with ***, **, and * are significant at the 1%, 5%, and 10% level,
respectively.
Top Four-Factor Alpha
(1)
(2)
Four-Factor
Alpha
Five-Factor
Alpha
(3)
(4)
DGTW
Liquidity-Adj.
DGTW
–0.103***
(–16.78)
–0.087***
(–13.53)
Top Five-Factor Alpha
–0.152***
(–22.89)
Top DGTW
0.004
(0.58)
–0.005
(–0.88)
0.125**
(2.56)
0.026
(0.01)
–0.006
(–0.29)
0.040***
(12.23)
–0.016**
(–2.10)
–0.006
(–1.15)
0.036
(0.70)
–0.866
(–0.38)
–0.007
(–0.33)
0.033***
(9.69)
0.028***
(3.69)
0.012**
(2.00)
0.048
(0.91)
1.915
(0.80)
–0.002
(–0.07)
0.027***
(6.92)
–0.067***
(–22.57)
0.005
(1.56)
–0.000
(–0.06)
–0.040*
(–1.70)
1.619
(1.53)
0.007
(0.74)
0.019***
(11.22)
1,113
0.211
1,113
0.157
1,171
0.312
1,171
0.305
Top Liquidity-Adj. DGTW
ΔLog(TNA)
ΔTurnover
ΔFlow
ΔExpense Ratio
ΔLoad
Constant
Observations
Adjusted R2
30
Table IA.VII
Impact on Mutual Fund Performance: Full Placebo Periods Excluding Crisis Periods
This table compares regression results of the changes in fund performance for matched samples of
mandatory and voluntary funds (see Table IV of the paper) in a two-year period around the SEC disclosure
change in 2004 with the same regressions conducted for placebo periods constructed using each placebo
month in the period 1994 to 2006 (excluding 2004 and the known crisis years of 1998, 2000, and 2001).
The independent variables in the placebo tests are the lagged variables. All performance variables are
annualized. In all regressions, we control for Log(TNA), Turnover, Flow, Expense Ratio, and Load. Panels
A and B report results for samples matched using Models 1 and 2 in Table IV of the paper, respectively.
Standard errors are adjusted for heteroskedasticity and clustered at the fund level, and t-statistics are
reported in parentheses. Coefficients marked with ***, **, and * are significant at the 1%, 5%, and 10%
level, respectively
Four-Factor
Five-Factor
Alpha
Alpha
Liq.-Adj.
DGTW
DGTW
Panel A. Mandatory and Voluntary Funds Matched by Model 1
Mandatory – Voluntary (May 2004)
–0.021
–0.014
–0.06
–0.047
Mandatory – Voluntary (Placebo period mean)
–0.016
–0.013
–0.012
–0.001
Quad. Diff. (May 2004 – Placebo period)
–0.005
–0.001
–0.048***
–0.046***
t-statistic
(–0.97)
(–0.24)
(–12.83)
(–6.66)
Panel B. Mandatory and Voluntary Funds Matched by Model 2
Mandatory – Voluntary (May 2004)
–0.030
–0.024
–0.041
–0.040
Mandatory – Voluntary (Placebo period mean)
–0.013
–0.010
–0.009
–0.007
–0.017***
–0.014**
–0.032***
–0.033***
(–3.53)
(–2.14)
(–9.94)
(–5.22)
Quad. Diff. (May 2004 – Placebo period)
t-statistic
31