64 S
. Luo Economics Letters 70 2001 59 –68
4. Conclusion
In Kyle’s model of insider trading, the value of public information demonstrates quite different characteristics than those of private information. Indeed, the insider puts a negative weight on the
public information in order to evade the inference of market makers. We have shown that the private information is more valuable than the public information. The insider puts a relatively larger positive
weight on his private information, and a relatively smaller negative weight on the public information in formulating his trading strategies. The insider’s profit is decreasing when the public information
becomes more precise. The informativeness of the price is increasing with more precise public information.
Appendix A
We first recall a well-known regression result which one will use.
Lemma. Let X ad X be two normal random vectors
,
1 2
X m
S S
1 1
11 12
| N m, S with
m 5 ,
S 5 .
1 2 1 2 1
2
X m
S S
2 2
21 22
Then the random variable X conditional on X we denote this as X uX has a normal distribution.
1 2
1 2
More precisely,
21 21
X uX | Nm 1 S S X 2 m , S 2 S S S .
1 2
1 12
22 2
2 11
12 22
21
In particular ,
21
E[X uX ] 5 m 1 S S X 2 m .
1 2
1 12
22 2
2
Proof of Proposition 1. First, we prepare some calculations for latter use. By assumption, we have
2 2
2
˜v s
s s
v v
v 2
2 2
˜s | N
0 , s
1 1 t s
s .
c v
v v
1 2 1121 22
2 2
2 2
˜s s
s s 1 s
i v
v v
i
Thus, by the Lemma, we have
2 2
21
˜ 1 1 t
s s
s
v v
c 2
2
˜ ˜ ˜ E[v
us ,s ] 5 s ,s
c i
v v
S D
2 2
2
1 2
˜s s
s 1 s
i v
v i
1
2 2
˜ ˜
]]]]] 5
s s 1 ts s .
2 2
i c
v i
t s 1 1 1 ts
v i
Second, we conjecture that the linear Nash equilibrium is given by Eqs. 1 and 2 with the
S . Luo Economics Letters 70 2001 59 –68
65
parameters a, b, g and l being constants which need to be determined. We will identify the
parameters and verify the conjecture. ˜
˜ ˜
˜ ˜
˜ Since x is a measurable function of s and s , and u is independent of s and s , the insider’s
c i
c i
expected profit conditional on his information is ˜
˜ ˜ ˜ ˜
˜ ˜
˜ ˜ ˜ ˜
E[v 2 Px us ,s ] 5 E[v 2 gs 2 lx 1 u xus ,s ]
c i
c c
i
˜ ˜ ˜ ˜
˜ ˜ 5 E[v
us ,s ] 2 gs 2 lx x.
c i
c
˜ To maximize the above expression over x, the first order condition is
1 ˜
˜ ˜ ˜ ˜
] x
5 E[v
us ,s ] 2 gs
c i
c
2 l
2 2
˜ ˜
s s 1 ts s 1
i c
v i
˜ ] ]]]]]
5 2
gs
S D
2 2
c
7 2
l ts 1 1 1 ts
v i
2 2
s t
s 1
i v
˜ ˜
] ]]]]]
]]]]] 5
2 g s 1
s .
SS D
D
2 2
c 2
2 i
2 l
t s 1 1 1 ts
t s 1 1 1 ts
v i
v i
The second order condition 2 l , 0, which is satisfied as will be seen from Eq. 16.
Comparing Eq. 7 with Eq. 1, we have,
2
s 1
i
] ]]]]] a 5
2 g
8
S D
2 2
2 l ts 1 1 1 ts
v i
2
t s
1
v
] ]]]]] b 5
? .
9
2 2
2 l ts 1 1 1 ts
v i
˜ ˜ ˜ ˜
Now we compute P 5 E[v us ,x 1 u]. Since by Eq. 1,
c
˜ ˜
˜ ˜
˜ ˜
˜ ˜
˜ x 1 u 5
as 1 bs 1 u 5 a 1 b v 1 ae 1 be 1 u,
c i
c i
we have
2 2
2 2
2 2
2
˜ ˜
v ar[x 1 u ] 5
a 1 b s 1 a ts 1 b s 1 s .
v v
i u
Furthermore ˜v
˜s
c
1 2
˜ ˜
x 1 u
2 2
2
s s
a 1 b s
v v
v 2
2 2
2
| N 0 ,
s 1 1 t
s a 1 b s 1 ats
.
v v
v v
1 2
1 1 22
2 2
2
˜ ˜
a 1 b s a 1 b s 1 ats
var[x 1 u ]
v v
v
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. Luo Economics Letters 70 2001 59 –68
We have the determinant
2 2
2
1 1 t s
a 1 b s 1 ats
v v
v
D ;
2 2
˜ ˜
a 1 b s 1 ats var[x 1 u ]
10
v v
2 2
2 2
2 2
5 b ts 1 1 1 ts s 1 1 1 ts s .
v i
v u
v
By market semi-strong efficiency and the lemma, we have ˜ ˜
˜ Ps ,x 1 u
c
˜ ˜ ˜ ˜
5 E[v
us ,x 1 u]
c 2
2 2
21
˜ 1 1 t
s a 1 b s 1 ats
s
v v
v c
2 2
5 s ,a 1 b s
v v
S D
2 2
1 2
˜s ˜
˜ a 1 b s 1 ats
var[x 1 u ]
i v
v 2
s
v 2
2 2
2 2
˜ ˜
˜ ]
5 2
abts 1 b s 1 s s 1 bts x 1 u .
v i
u c
v
D Comparing the above expression with Eq. 1, we have
2
s
v 2
2 2
2
] g 5
2 abs 1 b s 1 s ,
11
v i
u
D
4
s bt
v
]] l 5
. 12
D Substituting Eq. 12 into Eq. 9, and noting Eq. 10, we obtain
2
1 1 t s
u 2
]]]]] b 5
, 13
2 2
t s 1 1 1 ts
v i
thus ]]]]]
2
1 1 t s
u
]]]]] b 5
. 14
2 2
t s 1 1 1 ts
œ
v i
Substituting Eq. 13 into Eq. 10, we obtain
2 2
D 5 21 1 t s s .
15
u v
Substituting Eqs. 14 and 15 into Eq, 12, we have
S . Luo Economics Letters 70 2001 59 –68
67
]]]]]
4 2
t s
1 1 t s
v u
]]]] ]]]]]
l 5
?
2 2
2 2
21 1 t s s
t s 1 1 1 ts
œ
u v
v i
. 16
2
t s
v
]]]]]]]]] 5
]]]]]]]]
2 2
2
2 t s 1 1 1 ts 1 1 ts
œ
v i
u
From Eqs. 8 and 9,we have
2
bts g
v 2
2 2
2 2
]] 2
abts 1 b s 5 bbs 2 ats 5 .
v i
i v
2 l
Substituting the above expression and Eq. 15 into Eq. 11, we obtain
2
bts g 1
v 2
S D
]]]] ]]] g 5
1 s
.
2 u
2 l
21 1 t s
u
Substituting Eqs. 14 and 16 into the above expression, we obtain Eq. 5. Finally, substituting Eqs. 5 and 16 into Eq. 8, we have
2
s 1
1
i
] ]]]]] ]]
a 5
2
S D
2 2
2 l
1 1 t t
s 1 1 1 ts
v i
2
2 t s
1
v
] ]]]]]]]] 5
?
2 2
2 l ts 1 1 1 ts 1 1 t
v i
]]]]]]]]
2
s
u
]]]]]]]] 5 2
.
2 2
t s 1 1 1 ts 1 1 t
œ
v i
Proof of Proposition 2. Since
˜ ˜
˜ P
5 gs 1 lx 1 u
c
˜ ˜
˜ ˜
5 g 1 la 1 b v 1 g 1 lae 1 lbe 1 lu,
c i
by the Lemma, we have
2 2
2 2
2 2
2 2
2
var[P] 5 g 1 la 1 b s 1 g 1 la ts 1 l b s 1 l s .
17
v v
i u
From Eqs. 3–6, we have
2 2
2
t s 1 21 1 ts
t s
v i
v
]]]]]]]] ]]]]]]
g 1 al 5 ,
lb 5 ,
2 2
2 2
2t s 1 1 1 ts 1 1 t
2t s 1 1 1 ts
v i
v i
and
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. Luo Economics Letters 70 2001 59 –68
2 2
tt 1 2 s 1 21 1 ts
v i
]]]]]]]] g 1 la 1 b 5
.
2 2
2t s 1 1 1 ts 1 1 t
v i
Substituting the above expressions into Eq. 17 and into
2 4
g 1 la 1 b s
v
˜ ˜
]]]]]] var[v
uP] 5 var[v] 2 ,
var[P] we obtain the desired result.
References
Admati, A.R., Pfleiderer, P., 1988. A theory of intraday patterns: volume and price variability. Review of Financial Studies 1, 3–40.
Fishman, M.J., Hagerty, K.M., 1992. Insider trading and efficiency of stock prices. RAND Journal of Economics 23, 106–122.
Jain, N., Mirman, L.J., 1999. Insider trading with correlated signals. Economic Letters 65, 105–113. Kyle, A.S., 1985. Continuous auctions and insider trading. Econometrica 53, 1315–1335.
O’Hara, M., 1995. In: Market Microstructure Theory. Blackwell, Cambridge. Rochet, J.C., Vila, J.L., 1994. Insider trading without normality. Review of Economic Studies 61, 131–152.
Yu, F., 1999. What is the value of knowing uninformed trades? Economic Letters 64, 87–98.
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