- Email: [email protected]
Risk aversion, public disclosure, and long-lived information
Economics Letters 85 (2004) 327 – 334 www.elsevier.com/locate/econbase
Risk aversion, public disclosure, and long-lived information Wei David Zhang * W317 Thompson Hall, School of Business, State University of New York-Fredonia, Fredonia, NY 14063, USA Received 15 January 2004; accepted 7 April 2004 Available online 25 August 2004
Abstract This paper presents a methodology for characterizing the optimal behavior of a risk-averse insider under disclosure requirements. We show that there is an equilibrium in which the insider camouflages his trades with a noise component so that his private information is revealed slowly and linearly. D 2004 Elsevier B.V. All rights reserved. Keywords: Risk aversion; Public disclosure; Price efficiency JEL classification: G14
1. Introduction Using an extension of the frameworks of Kyle (1985); Huddart et al. (2001), this paper presents a methodology for characterizing the optimal behavior of a risk-averse trader who possesses long-lived private information about the fundamental value of a security under mandatory disclosure requirements. Ex-post trade disclosure raises some interesting questions that have not yet been addressed within a dynamic framework. In this paper we attempt to address the following question: what is the effect of the interaction between risk aversion and trade disclosure on the inter-temporal patterns of market depth and price efficiency? This is a first attempt to model risk aversion and trade disclosure in a dynamic setting. Kyle (1985) considers the case of a single risk-neutral insider and finds that the insider trades in such a way that his information is incorporated into prices at a slow and almost linear rate, while ‘‘liquidity’’ traders provide camouflage which conceals his trading from market makers. Huddart et al. (2001) consider the case of the same risk-neutral insider under disclosure requirements as mandated by US securities laws. The ex-post disclosure of the insider’s trades changes the equilibrium strategy of the * Tel.: +1-716-673-4603; fax: +1-716-673-3332. E-mail address: [email protected] (W.D. Zhang). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2004.04.022
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W.D. Zhang / Economics Letters 85 (2004) 327–334
insider, given that the market maker can infer information from the insider’s previous trade before the next round of trading. To garble the information conveyed by his trades, the insider employs a mixed strategy at every auction except the last one. Huddart et al. (2001) find that public disclosure of the insider’s trades nevertheless accelerates the price discovery process and lowers trading costs by comparison to the case with no disclosure requirement. Holden and Subrahmanyam (1994) consider the case of risk-averse insiders with no disclosure requirement and find that the risk-averse insider chooses to trade aggressively in the initial periods and exploits his information rapidly because of his concerns about the future price risk. The main finding of this paper is that, under disclosure requirements, the risk-averse insider camouflages his trades with a noise component. As a result, his private information is revealed slowly and the market depth is the same at every auction. Under disclosure requirements, our finding suggests, the risk-averse insider is more concerned about the risk of sub-optimally revealing his information by mandatory disclosure. In Section 2 we discuss the structure of our model and derive its linear and mixed equilibriua. Section 3 presents the properties of our model’s equilibrium. Section 4 concludes.
2. The model 2.1. Structure and notation We conform to the notation of Kyle (1985). Consider a model in which there is a security with liquidation value, v, which is normally distributed with prior mean p0 and variance R0. The security is traded at N sequential auctions in a time interval that begins at t = 0 and ends at t = 1. There are three types of agents: a risk-averse insider with long-lived information, risk-neutral market makers, and liquidity traders. The insider is endowed with initial wealth of W0. Let Wn denote the wealth at the end of nth auction. The insider has a negative exponential utility with risk-aversion coefficient A, that is UðWN Þ ¼ expðAWN Þ Prior to trading, the insider observes the liquidation value of the security. Let Dxn denote the order by the insider at the nth auction. Let Dun denote the net market order of liquidity traders at the nth auction. We assume Dun is serially uncorrelated and normally distributed with a zero mean and variance of ru2Dtn, where Dtn is the time interval between the nth auction and the previous one. We assume that auctions occur at equally spaced intervals and the subscript in Dtn is hereafter dropped. The insider determines his optimal trading strategy by a process of backward induction in order to maximize his expected utility, based on the knowledge of past price and his own information signal. Trading takes place at each auction through a risk-neutral market maker who absorbs the total order flow while earning zero expected profits. At the beginning of the first round, the market maker sets the price equal to its expected value conditional on all the net order flow submitted by traders, Dx1 + Du1, and the previous price, that is p1 ¼ E½mjp0 ; Dx1 þ Du1
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329
At the end of the first round, the insider is required to disclose his trade, Dx1. This allows the market maker to observe the order of the insider and updates his first round price as p*1 = E[mjp0, Dxn 1] before setting his price for the next round, p2 ¼ E½mjp*1 ; Dx2 þ Du2 : In general, the price at the nth auction is pn ¼ E½mjp*n1; Dxn þ Dun ; where p*n1 ¼ E½mjp*n2 ; Dxn1 and p*0 up0 : We propose that, under disclosure requirements, the insider employs a mixed strategy at every auction except the last one, i.e. Dxn ¼ bn ðv p*n1 ÞDt þ zn ; where n = 1, . . ., N 1 and the noise term zn is normally distributed with a zero mean and variance of rz2n . At the last auction, the insider no longer needs to camouflage his information with a noise term and his trade is linear in his information, i.e. DxN ¼ bN ðv pN*1ÞDt: Equilibrium, hence, is defined by a market efficiency condition that price equals the expected value of the security conditional on the information available to the market maker and by a utility-maximization condition that the insider chooses the optimal strategy conditional on his information and conjectures. Further, information is impounded into prices via the reduction of the post-trade residual variance, which is Rn = E[mjDx1. . .Dxn]. 2.2. Equilibrium The proposition that characterizes our equilibrium is stated as follows. Proposition 1. In the N-period setting with ex-post trade disclosure, given the following trading strategy of the insider and price strategy of the market maker for n = 1. . .N 1 Dxn ¼ bn ðv p*n1 ÞDt þ zn
ð1Þ
pn ¼ p*n1 þ kn ðDxn þ Dun Þ
ð2Þ
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p*n ¼ p*n1 þ cn Dxn
ð3Þ
p*0 up0 for n = N, the last round of auction DxN ¼ bN ðv pN*1ÞDt
ð4Þ
pN ¼ pN*1 þ kN ðDxN þ DuN Þ
ð5Þ
there exists a Nash equilibrium characterized by the following constants k1 ¼ . . . ¼ kN ¼ k
ð6Þ
c1 ¼ . . . ¼ cN 1 ¼ c ¼ Ak2 r2u Dt þ 2k
ð7Þ
1 ðN n þ 1Þc
ð8Þ
r2zn ¼
ðN nÞR0 N ðN n þ 1Þc2
ð9Þ
Rn ¼
N n R0 N
bn Dt ¼
ð10Þ
k is the positive root to the polynomial ðAr2u DtÞ2 k4 þ 4Ar2u Dtk3 þ 4k2
AR0 R0 k ¼0 N N r2u Dt
ð11Þ
Proof. We start the proof by backward induction. At the last round of auction, the insider’s trade is linear in his information and the problem is exactly the same as that of Holden and Subrahmanyam (1994) with a single risk-averse insider. We define the insider’s utility function at the last auction as J ðWN Þ ¼ expfA½WN 1 þ aN ðv pN*1Þ2 g Tailoring Proposition 1 of Holden and Subrahmanyam (1994) to our case, we have bN Dt ¼ kN ¼
1 ð2kN þ Ak2N r2u DtÞ
bN DtRN 1 2 bN Dt 2 RN 1 þ r2u Dt
RN ¼ RN 1
b2N Dt 2 R2N 1 b2N Dt 2 RN 1 þ r2u Dt
ð12Þ ð13Þ
ð14Þ
W.D. Zhang / Economics Letters 85 (2004) 327–334
aN ¼
1 2kN ð2 þ AkN r2u DtÞ
331
ð15Þ
with the second order condition 2kN þ Ak2N r2u Dt < 0 which implies that k must be positive. For auctions at n = 1. . .N 1 period, we propose that the insider, facing disclosure requirements, submits his order as described by Eq. (1) and the market maker updates his pricing rule as in Eq. (2). We make the inductive hypothesis that 2 J ðWnþ1 Þ ¼ fnþ1 expfA½Wn þ anþ1 ðv p*Þ n g
then 2 J ðWn Þ ¼ max E½fnþ1 expfA½Wn1 þ Dxn ðv pn Þ þ anþ1 ðv p*Þ n g Dxn
Given the trading strategy of the insider and the pricing rule of the market maker, the first order condition with respect to Dxn yields ð2kn 2anþ1 c2n þ Ak2n r2u DtÞDxn þ ð2anþ1 cn 1Þðv p0 Þ ¼ 0 If our proposed mixed trading strategy Dxn = bn(v pn* 1 )Dt + zn where zn f N(0, rz2n ) is to hold in equilibrium, then the insider must be indifferent across all values of Dxn, thus, cn ¼
1
ð16Þ
anþ1
Ar2u Dtk2n þ 2kn cn ¼ 0
ð17Þ
Application of projection theorem for normally distributed variables gives us kn ¼
bn DtRn1 2 2 bn Dt R0 þ r2u Dt
cn ¼
bn DtRn1 2 2 bn Dt R0 þ r2zn
Rn ¼ Rn1
þ r2zn
b2n Dt 2 R2n1 b2n Dt 2 R0 þ r2zn
ð18Þ
ð19Þ
ð20Þ
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Combining and simplifying Eqs. (18)–(20), we get bn ¼
r2zn ¼
Rn kn
r2u Rc n þ cn r2u Dt
ð21Þ
n
bn Rn Dt cn
ð22Þ
Substituting Eqs. (1)–(3) into the insider’s utility function and using Lemma 1 in Holden and Subrahmanyam (1994), it is straightforward to show that J ðWn Þ ¼ fn expfA½Wn þ an ðv p n1 Þ2 g ¼ E½fnþ1 expfA½Wn þ anþ1 ðv p n Þ2 g " ( fnþ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp AWn þ Hn þ 0:5ðAbn kn DtÞ2 r2u Dt 1 2Ln r2zn ðAkn Þ2 r2zn r2u Dt ) # ðIn þ ðAbn kn DtÞðAkn Þr2u DtÞ2 r2zn ðv p*n1 Þ2 þ 2ð1 2Ln r2zn ðAkn Þ2 r2zn r2u DtÞ
ð23Þ
where Hn ¼ Abn Dtð1 kn bn DtÞ Aanþ1 ð1 bn cn DtÞ2 In ¼ Að2kn bn Dt 1Þ þ 2Aanþ1 cn ð1 bn cn DtÞ Ln ¼ Akn Aanþ1 c2n : Thus, " # ðIn þ ðAbn kn DtÞðAkn Þr2u DtÞ2 r2zn 1 2 2 an ¼ Hn þ 0:5ðAbn kn DtÞ ru Dt þ A 2ð1 2Ln r2z ðAkn Þ2 r2z r2u DtÞ
ð24Þ
fnþ1 ffi fn ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2Ln r2zn ðAkn Þ2 r2zn r2u Dt
ð25Þ
n
n
From Eqs. (16), (17), (24) and (25), we have f1 ¼ . . . ¼ fN ¼ 1
ð26AÞ
an ¼ anþ1
ð26BÞ
and for n ¼ 1 . . . N 1:
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333
Thus, Eqs. (6) and (7) follow from Eqs. (16), (17), (26A) and (26B). From Eqs. (20) and (21), we have R0 RN 1 ¼ ðN 1Þ
k2 ð2 þ Akr2u DtÞ2 r2u Dt 1 þ Akr2u Dt
ð27Þ
From Eqs. (12)–(14) and (27), we have k2 ð2 þ Akr2u DtÞ2 r2u Dt R0 ¼ N 1 þ Akr2u Dt
ð28Þ
Eqs. (10) and (11) follow from Eqs. (27) and (28). Eq. (8) follows from Eq. (20) and Eq. (9) follows 5 from Eq. (22) after substituting for bn and Rn. 3. A comparative analysis of the properties of the equilibrium As in Kyle (1985); Holden and Subrahmanyam (1994); Huddart et al. (2001), the parameters Rn and kn are the measures of price efficiency and market depth. The most striking features of Proposition 1 are
Fig. 1. Residual variance and market depth for N = 20 rounds of trade. The exogenous parameters are: A = 4, R0 = 1, r2u = 1. The results for the case of a risk-averse insider without disclosure are taken from Holden and Subrahmanyam (1994).
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W.D. Zhang / Economics Letters 85 (2004) 327–334
that the risk-averse insider’s private information is incorporated into prices gradually, and Eq. (10) is identical to the same equation in the case of a risk-neutral insider considered by Huddart et al. (2001). Further, the market depth in our setting is the same at every auction as it is in the case of a risk-neutral insider. The intuition is that, with disclosure, marginal trading costs must be the same across different rounds of auction. A disparity in such costs would create an incentive to deviate from a mixed strategy in order to exploit the lower costs. Fig. 1 presents some numerical simulations using Eqs. (10) and (11). Like Holden and Subrahmanyam (1994) and Huddart et al. (2001), we assume R0 = 1, ru2Dt = 1/N and auctions occur at equally spaced intervals. The figure effectively presents the contrast between the case of a risk-averse insider under disclosure requirements and the case of the same risk-averse insider under no disclosure requirement. Holden and Subrahmanyam (1994) show that a risk-averse insider in that setting without disclosure is concerned about the future price risk, and this causes him to trade more aggressively than a risk-neutral insider. Under disclosure requirements, our results suggest, the riskaverse insider is more concerned about the risk of sub-optimally revealing his information by disclosure. The risk-averse insider in our setting camouflages his trades with a noise component so that his private information is revealed slowly and linearly.
4. Concluding remarks In this paper, we characterized the dynamically optimal trading strategies of a risk-averse insider under ex-post disclosure requirements. In contrast to the case of a risk-averse insider who exploits his private information rapidly in a setting under no disclosure requirement, we show that the risk-averse insider under disclosure requirements finds it optimal to employ a mixed strategy so that his private information is incorporated into prices at a slow and linear rate.
References Holden, C.W., Subrahmanyam, A., 1994. Risk aversion, imperfect competition, and long lived information. Economics Letters 44, 181 – 190. Huddart, S., Hughes, J.S., Levine, C.B., 2001. Public disclosure and dissimulation of insider trades. Econometrica 69, 665 – 681. Kyle, A.S., 1985. Continuous auctions and insider trading. Econometrica 53, 1315 – 1335.
Risk aversion, public disclosure, and long-lived information Wei David Zhang * W317 Thompson Hall, School of Business, State University of New York-Fredonia, Fredonia, NY 14063, USA Received 15 January 2004; accepted 7 April 2004 Available online 25 August 2004
Abstract This paper presents a methodology for characterizing the optimal behavior of a risk-averse insider under disclosure requirements. We show that there is an equilibrium in which the insider camouflages his trades with a noise component so that his private information is revealed slowly and linearly. D 2004 Elsevier B.V. All rights reserved. Keywords: Risk aversion; Public disclosure; Price efficiency JEL classification: G14
1. Introduction Using an extension of the frameworks of Kyle (1985); Huddart et al. (2001), this paper presents a methodology for characterizing the optimal behavior of a risk-averse trader who possesses long-lived private information about the fundamental value of a security under mandatory disclosure requirements. Ex-post trade disclosure raises some interesting questions that have not yet been addressed within a dynamic framework. In this paper we attempt to address the following question: what is the effect of the interaction between risk aversion and trade disclosure on the inter-temporal patterns of market depth and price efficiency? This is a first attempt to model risk aversion and trade disclosure in a dynamic setting. Kyle (1985) considers the case of a single risk-neutral insider and finds that the insider trades in such a way that his information is incorporated into prices at a slow and almost linear rate, while ‘‘liquidity’’ traders provide camouflage which conceals his trading from market makers. Huddart et al. (2001) consider the case of the same risk-neutral insider under disclosure requirements as mandated by US securities laws. The ex-post disclosure of the insider’s trades changes the equilibrium strategy of the * Tel.: +1-716-673-4603; fax: +1-716-673-3332. E-mail address: [email protected] (W.D. Zhang). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2004.04.022
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insider, given that the market maker can infer information from the insider’s previous trade before the next round of trading. To garble the information conveyed by his trades, the insider employs a mixed strategy at every auction except the last one. Huddart et al. (2001) find that public disclosure of the insider’s trades nevertheless accelerates the price discovery process and lowers trading costs by comparison to the case with no disclosure requirement. Holden and Subrahmanyam (1994) consider the case of risk-averse insiders with no disclosure requirement and find that the risk-averse insider chooses to trade aggressively in the initial periods and exploits his information rapidly because of his concerns about the future price risk. The main finding of this paper is that, under disclosure requirements, the risk-averse insider camouflages his trades with a noise component. As a result, his private information is revealed slowly and the market depth is the same at every auction. Under disclosure requirements, our finding suggests, the risk-averse insider is more concerned about the risk of sub-optimally revealing his information by mandatory disclosure. In Section 2 we discuss the structure of our model and derive its linear and mixed equilibriua. Section 3 presents the properties of our model’s equilibrium. Section 4 concludes.
2. The model 2.1. Structure and notation We conform to the notation of Kyle (1985). Consider a model in which there is a security with liquidation value, v, which is normally distributed with prior mean p0 and variance R0. The security is traded at N sequential auctions in a time interval that begins at t = 0 and ends at t = 1. There are three types of agents: a risk-averse insider with long-lived information, risk-neutral market makers, and liquidity traders. The insider is endowed with initial wealth of W0. Let Wn denote the wealth at the end of nth auction. The insider has a negative exponential utility with risk-aversion coefficient A, that is UðWN Þ ¼ expðAWN Þ Prior to trading, the insider observes the liquidation value of the security. Let Dxn denote the order by the insider at the nth auction. Let Dun denote the net market order of liquidity traders at the nth auction. We assume Dun is serially uncorrelated and normally distributed with a zero mean and variance of ru2Dtn, where Dtn is the time interval between the nth auction and the previous one. We assume that auctions occur at equally spaced intervals and the subscript in Dtn is hereafter dropped. The insider determines his optimal trading strategy by a process of backward induction in order to maximize his expected utility, based on the knowledge of past price and his own information signal. Trading takes place at each auction through a risk-neutral market maker who absorbs the total order flow while earning zero expected profits. At the beginning of the first round, the market maker sets the price equal to its expected value conditional on all the net order flow submitted by traders, Dx1 + Du1, and the previous price, that is p1 ¼ E½mjp0 ; Dx1 þ Du1
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329
At the end of the first round, the insider is required to disclose his trade, Dx1. This allows the market maker to observe the order of the insider and updates his first round price as p*1 = E[mjp0, Dxn 1] before setting his price for the next round, p2 ¼ E½mjp*1 ; Dx2 þ Du2 : In general, the price at the nth auction is pn ¼ E½mjp*n1; Dxn þ Dun ; where p*n1 ¼ E½mjp*n2 ; Dxn1 and p*0 up0 : We propose that, under disclosure requirements, the insider employs a mixed strategy at every auction except the last one, i.e. Dxn ¼ bn ðv p*n1 ÞDt þ zn ; where n = 1, . . ., N 1 and the noise term zn is normally distributed with a zero mean and variance of rz2n . At the last auction, the insider no longer needs to camouflage his information with a noise term and his trade is linear in his information, i.e. DxN ¼ bN ðv pN*1ÞDt: Equilibrium, hence, is defined by a market efficiency condition that price equals the expected value of the security conditional on the information available to the market maker and by a utility-maximization condition that the insider chooses the optimal strategy conditional on his information and conjectures. Further, information is impounded into prices via the reduction of the post-trade residual variance, which is Rn = E[mjDx1. . .Dxn]. 2.2. Equilibrium The proposition that characterizes our equilibrium is stated as follows. Proposition 1. In the N-period setting with ex-post trade disclosure, given the following trading strategy of the insider and price strategy of the market maker for n = 1. . .N 1 Dxn ¼ bn ðv p*n1 ÞDt þ zn
ð1Þ
pn ¼ p*n1 þ kn ðDxn þ Dun Þ
ð2Þ
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W.D. Zhang / Economics Letters 85 (2004) 327–334
p*n ¼ p*n1 þ cn Dxn
ð3Þ
p*0 up0 for n = N, the last round of auction DxN ¼ bN ðv pN*1ÞDt
ð4Þ
pN ¼ pN*1 þ kN ðDxN þ DuN Þ
ð5Þ
there exists a Nash equilibrium characterized by the following constants k1 ¼ . . . ¼ kN ¼ k
ð6Þ
c1 ¼ . . . ¼ cN 1 ¼ c ¼ Ak2 r2u Dt þ 2k
ð7Þ
1 ðN n þ 1Þc
ð8Þ
r2zn ¼
ðN nÞR0 N ðN n þ 1Þc2
ð9Þ
Rn ¼
N n R0 N
bn Dt ¼
ð10Þ
k is the positive root to the polynomial ðAr2u DtÞ2 k4 þ 4Ar2u Dtk3 þ 4k2
AR0 R0 k ¼0 N N r2u Dt
ð11Þ
Proof. We start the proof by backward induction. At the last round of auction, the insider’s trade is linear in his information and the problem is exactly the same as that of Holden and Subrahmanyam (1994) with a single risk-averse insider. We define the insider’s utility function at the last auction as J ðWN Þ ¼ expfA½WN 1 þ aN ðv pN*1Þ2 g Tailoring Proposition 1 of Holden and Subrahmanyam (1994) to our case, we have bN Dt ¼ kN ¼
1 ð2kN þ Ak2N r2u DtÞ
bN DtRN 1 2 bN Dt 2 RN 1 þ r2u Dt
RN ¼ RN 1
b2N Dt 2 R2N 1 b2N Dt 2 RN 1 þ r2u Dt
ð12Þ ð13Þ
ð14Þ
W.D. Zhang / Economics Letters 85 (2004) 327–334
aN ¼
1 2kN ð2 þ AkN r2u DtÞ
331
ð15Þ
with the second order condition 2kN þ Ak2N r2u Dt < 0 which implies that k must be positive. For auctions at n = 1. . .N 1 period, we propose that the insider, facing disclosure requirements, submits his order as described by Eq. (1) and the market maker updates his pricing rule as in Eq. (2). We make the inductive hypothesis that 2 J ðWnþ1 Þ ¼ fnþ1 expfA½Wn þ anþ1 ðv p*Þ n g
then 2 J ðWn Þ ¼ max E½fnþ1 expfA½Wn1 þ Dxn ðv pn Þ þ anþ1 ðv p*Þ n g Dxn
Given the trading strategy of the insider and the pricing rule of the market maker, the first order condition with respect to Dxn yields ð2kn 2anþ1 c2n þ Ak2n r2u DtÞDxn þ ð2anþ1 cn 1Þðv p0 Þ ¼ 0 If our proposed mixed trading strategy Dxn = bn(v pn* 1 )Dt + zn where zn f N(0, rz2n ) is to hold in equilibrium, then the insider must be indifferent across all values of Dxn, thus, cn ¼
1
ð16Þ
anþ1
Ar2u Dtk2n þ 2kn cn ¼ 0
ð17Þ
Application of projection theorem for normally distributed variables gives us kn ¼
bn DtRn1 2 2 bn Dt R0 þ r2u Dt
cn ¼
bn DtRn1 2 2 bn Dt R0 þ r2zn
Rn ¼ Rn1
þ r2zn
b2n Dt 2 R2n1 b2n Dt 2 R0 þ r2zn
ð18Þ
ð19Þ
ð20Þ
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Combining and simplifying Eqs. (18)–(20), we get bn ¼
r2zn ¼
Rn kn
r2u Rc n þ cn r2u Dt
ð21Þ
n
bn Rn Dt cn
ð22Þ
Substituting Eqs. (1)–(3) into the insider’s utility function and using Lemma 1 in Holden and Subrahmanyam (1994), it is straightforward to show that J ðWn Þ ¼ fn expfA½Wn þ an ðv p n1 Þ2 g ¼ E½fnþ1 expfA½Wn þ anþ1 ðv p n Þ2 g " ( fnþ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp AWn þ Hn þ 0:5ðAbn kn DtÞ2 r2u Dt 1 2Ln r2zn ðAkn Þ2 r2zn r2u Dt ) # ðIn þ ðAbn kn DtÞðAkn Þr2u DtÞ2 r2zn ðv p*n1 Þ2 þ 2ð1 2Ln r2zn ðAkn Þ2 r2zn r2u DtÞ
ð23Þ
where Hn ¼ Abn Dtð1 kn bn DtÞ Aanþ1 ð1 bn cn DtÞ2 In ¼ Að2kn bn Dt 1Þ þ 2Aanþ1 cn ð1 bn cn DtÞ Ln ¼ Akn Aanþ1 c2n : Thus, " # ðIn þ ðAbn kn DtÞðAkn Þr2u DtÞ2 r2zn 1 2 2 an ¼ Hn þ 0:5ðAbn kn DtÞ ru Dt þ A 2ð1 2Ln r2z ðAkn Þ2 r2z r2u DtÞ
ð24Þ
fnþ1 ffi fn ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2Ln r2zn ðAkn Þ2 r2zn r2u Dt
ð25Þ
n
n
From Eqs. (16), (17), (24) and (25), we have f1 ¼ . . . ¼ fN ¼ 1
ð26AÞ
an ¼ anþ1
ð26BÞ
and for n ¼ 1 . . . N 1:
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333
Thus, Eqs. (6) and (7) follow from Eqs. (16), (17), (26A) and (26B). From Eqs. (20) and (21), we have R0 RN 1 ¼ ðN 1Þ
k2 ð2 þ Akr2u DtÞ2 r2u Dt 1 þ Akr2u Dt
ð27Þ
From Eqs. (12)–(14) and (27), we have k2 ð2 þ Akr2u DtÞ2 r2u Dt R0 ¼ N 1 þ Akr2u Dt
ð28Þ
Eqs. (10) and (11) follow from Eqs. (27) and (28). Eq. (8) follows from Eq. (20) and Eq. (9) follows 5 from Eq. (22) after substituting for bn and Rn. 3. A comparative analysis of the properties of the equilibrium As in Kyle (1985); Holden and Subrahmanyam (1994); Huddart et al. (2001), the parameters Rn and kn are the measures of price efficiency and market depth. The most striking features of Proposition 1 are
Fig. 1. Residual variance and market depth for N = 20 rounds of trade. The exogenous parameters are: A = 4, R0 = 1, r2u = 1. The results for the case of a risk-averse insider without disclosure are taken from Holden and Subrahmanyam (1994).
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that the risk-averse insider’s private information is incorporated into prices gradually, and Eq. (10) is identical to the same equation in the case of a risk-neutral insider considered by Huddart et al. (2001). Further, the market depth in our setting is the same at every auction as it is in the case of a risk-neutral insider. The intuition is that, with disclosure, marginal trading costs must be the same across different rounds of auction. A disparity in such costs would create an incentive to deviate from a mixed strategy in order to exploit the lower costs. Fig. 1 presents some numerical simulations using Eqs. (10) and (11). Like Holden and Subrahmanyam (1994) and Huddart et al. (2001), we assume R0 = 1, ru2Dt = 1/N and auctions occur at equally spaced intervals. The figure effectively presents the contrast between the case of a risk-averse insider under disclosure requirements and the case of the same risk-averse insider under no disclosure requirement. Holden and Subrahmanyam (1994) show that a risk-averse insider in that setting without disclosure is concerned about the future price risk, and this causes him to trade more aggressively than a risk-neutral insider. Under disclosure requirements, our results suggest, the riskaverse insider is more concerned about the risk of sub-optimally revealing his information by disclosure. The risk-averse insider in our setting camouflages his trades with a noise component so that his private information is revealed slowly and linearly.
4. Concluding remarks In this paper, we characterized the dynamically optimal trading strategies of a risk-averse insider under ex-post disclosure requirements. In contrast to the case of a risk-averse insider who exploits his private information rapidly in a setting under no disclosure requirement, we show that the risk-averse insider under disclosure requirements finds it optimal to employ a mixed strategy so that his private information is incorporated into prices at a slow and linear rate.
References Holden, C.W., Subrahmanyam, A., 1994. Risk aversion, imperfect competition, and long lived information. Economics Letters 44, 181 – 190. Huddart, S., Hughes, J.S., Levine, C.B., 2001. Public disclosure and dissimulation of insider trades. Econometrica 69, 665 – 681. Kyle, A.S., 1985. Continuous auctions and insider trading. Econometrica 53, 1315 – 1335.