Insider trading with correlated signals

Insider trading with correlated signals

Economics Letters 65 (1999) 105–113 Insider trading with correlated signals Neelam Jain a , Leonard J. Mirman b , * a Jones Graduate School of Manag...

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Economics Letters 65 (1999) 105–113

Insider trading with correlated signals Neelam Jain a , Leonard J. Mirman b , * a

Jones Graduate School of Management, Rice University, MS 531, P.O. Box 1892, 6100 Main Street, Houston, TX, USA b Department of Economics, University of Virginia, 114 Rouss Hall, Charlottesville, VA 22903, USA Received 13 April 1998; accepted 24 February 1999

Abstract We examine the effects of insider trading on the stock price determination and its informational efficiency, when the market maker observes two, correlated, signals of the value of the firm. We show that the stock price is more informative and the insider’s profits are lower than when the market maker only observes the order flow, as in Kyle [Kyle, A., 1985, Continuous auctions and insider trading, Econometrica 53].  1999 Elsevier Science S.A. All rights reserved. Keywords: Insider trading; Correlated signals; Information JEL classification: G14; D82

1. Introduction Most of the theoretical research on insider trading (see Kyle, 1985; Rochet and Vila, 1994, for example) focuses on only one signal of information for the market maker, namely the total noisy order flow. We generalize this work by adding another source of information for the market maker. It seems natural to allow the market maker to gather information from more than one source since the market maker has access to other publicly available information relevant to the value of the stock.1 The aim of this paper is to examine the effect on stock prices and the amount of insider trading when the market maker observes correlated signals of the value of the stock. As Jain and Mirman (1998), show, such a model is a useful stepping stone for studying the interaction between the decisions made in the real sector and in the financial sector. The model we present here builds on the pioneering work of Kyle (1985) in which there is an insider who knows the value of the stock and a *Corresponding author. Tel.: 11-804-924-6756; fax: 11-804-982-2904. E-mail address: [email protected] (L.J. Mirman) 1 See Jain and Mirman (1998), for an example of another signal for the market maker from the real sector. 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00121-4

 1999 Elsevier Science S.A. All rights reserved.

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market maker who only knows the distribution of the values of the stock, gets information from the total noisy, stock order flow, and sets the stock price in a way that his expected profits are zero (i.e. the price is set in a competitive manner). In this paper, in addition to the total order flow signal, we allow the market maker to observe another, ‘second’, signal of the value of the asset and examine how the stock price and the amount of the insider trade change. We then examine the comparative statics with respect to the variability in the noise trade, the value of the firm and in the additional signal. Finally, we examine the effect of the additional signal on the information revealed through the stock price and on the profits of the insider. We show that there exists a linear-normal equilibrium, in the spirit of Kyle. An effect of this exercise is to generalize the stock pricing rule set up by the market maker, since there is more than one source of information. An interesting property of the equilibrium stock price function is that a zero order flow does not imply zero information for the market maker. The stock price varies positively with both signals, given positive realizations of the signals and negatively if the realizations are negative. Interestingly, although the level of insider trading remains unchanged, the informational content of the trade changes. The model yields rich comparative statics. In contrast to Kyle, we show that a higher variance of the value of the firm does not necessarily increase the weight, placed by the market maker, on the total order flow signal. This weight increases if and only if the noise in the additional signal is sufficiently high. Further, as the noise in the additional signal increases, the weight on that signal decreases but at the same time, the weight on the total order flow signal increases. Thus, the addition of a second signal affects the weights on both signals. In contrast to the effect of the noisiness of the additional signal, the noise in the order flow only affects (lowers) the weight of the total order flow. Thus, we find a certain asymmetry in the responses of the two signals to changing parameters. Finally, we find that the stock price becomes more informative than in Kyle (where the information revealed is exactly half of the variance of the value of the firm) and that the amount of information revealed varies with some of the underlying parameters in contrast to Kyle where the amount of information revealed is independent of the parameters. Thus, the addition of a signal that is correlated with the value of the asset and cannot be manipulated by the insider leads to significant changes in the information revelation process. Finally, as is intuitive, the insider’s profits are lower compared to the profit level in Kyle. The paper is organized as follows: in Section 2, we present the model; in Section 3 we analyze the comparative statics of the model; in Section 4 we discuss the effects on information revealed and the insider’s profits and finally, we conclude in Section 4.

2. Model We assume that there is one financial asset, i.e. the stock of the firm, in the economy. The value of 2 this asset, denoted by z, is a random variable, distributed normally with mean z¯ and variance s z . There are three types of agents in the financial market for this asset (as in Kyle, 1985): an insider who knows the value of the asset before trading, the noise traders whose stock order, u, is a random variable, distributed independently of z and normally with mean 0 and variance s 2u and a market maker who sets the stock price, p. The stock market is competitive in the sense that the pricing rule is chosen by the market maker to make zero expected profits.

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2.1. Information structure The insider is assumed to know the value of the asset before trading. The market maker on the other hand, is assumed to know only the distribution of the value of the asset, z. However, he is assumed to observe two signals about the value of the asset: one is the total order flow and the other is a random variable z 1 e where e is a random variable distributed normally with mean 0 and variance s 2e and independently of z and u. It is assumed that the insider does not observe either u (as in Kyle) or e 2 .

2.2. Maximization problem of the insider The insider maximizes net profits from the financial transactions:

P 5 Eu Ee [z 2 p]x where x is the stock order of the insider. The insider chooses x taking the pricing rule set by the market maker as given.

2.3. Setting the price function The market maker sets the stock price so as to make zero expected profits. This zero-profit condition boils down to p 5 E(z /x 1 u, z 1 e ) That is, the stock price is set equal to the expected value of the asset, conditional on the observation of the total order flow, x 1 u, and the second signal, z 1 e. We shall use the following in our analysis: Fact. If the random variables z, x 1 u and z 1 e are jointly normally distributed, then there exist constants, m0 , m1 and m2 such that, E(z /x 1 u, z 1 e ) 5 m0 1 m1 (z 1 e ) 1 m2 (x 1 u)

(1)

(See Graybill, 1961.) We first assume that Eq. (1) holds and then verify that the condition of this equation is satisfied in our model.

2.4. Solving for equilibrium We now determine the equilibrium values of the stock to be traded by the insider, the stock price and the stock pricing rule, given by Eq. (1). Substituting for p from Eq. (1) into the insider’s profit function yields

P 5 (z 2 m0 2 m1 z 2 m2 x)x The first-order necessary condition for a maximum is, 2

For the analysis when this assumption is relaxed, see Rochet and Vila (1994).

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z(1 2 m1 ) 2 m0 2 2m2 x 5 0 This yields z(1 2 m1 ) 2 m0 x 5 ]]]]] 2m2

(2)

It is easy to show that the second-order condition for a maximum at this value of x is satisfied. Remark. Note that the insider’ s stock order x is a linear function of the normally distributed variable z and, thus, is normally distributed. It is easy to verify that the conditions for joint normality of the three random variables, z, z 1 e and x 1 u are also satisfied. Thus, the Fact applies. Next we determine the stock pricing function or equivalently the constants m0 , m1 and m2 . Applying Theorem 3.10 of Graybill, we have

m0 5 z¯ 2 m1 z¯ 2 m2 x¯ where x¯ is the mean stock order of the insider. Substituting for x from Eq. (2), we obtain,

S

(1 2 m1 )z¯ 2 m0 m0 5 z¯ 2 m1 z¯ 2 m2 ]]]]] 2m2

D

Simplifying and solving for m0 , we obtain

m0 5 (1 2 m1 )z¯

(3)

Next we determine the coefficients m1 and m2 . By Theorem 3.10, Graybill, the coefficients m1 and m2 are given by the following equations:

sz,z1 e s 2x1u 2 sz,x 1u sz1 e ,x 1u m1 5 ]]]]]]]]] s 2z1 e s 2x 1u 2 (sz 1 e ,x 1u )2

(4)

sz,x1u s 2z1 e 2 sz,z1 e sz1 e ,x 1u m2 5 ]]]]]]]]] 2 2 2 s z1 e s x 1u 2 (sz 1 e ,x 1u )

(5)

Substituting for x and then solving these two equations gives us the results, which are presented in the lemma below: Lemma.

s 2z (i) m1 5 ]]]] s 2z 1 2s 2e sz s 2e (ii) m2 5 ]]]]] su (s 2z 1 2s 2e )

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This yields the following proposition: Proposition 1. A linear equilibrium exists with the values of x and p given by: (1 2 m1 )(z 2 z¯ ) x 5 ]]]]] 2m2

S

D

s 2z sz s 2e z(1 2 m1 ) 2 m0 ]]]]] p 5 (1 2 m1 )z¯ 1 ]]]] 1u 2 2 (z 1 e ) 1 ]]]]] 2 2 2m2 su (s z 1 2 s e ) s z 1 2s e

Discussion of the equilibrium. Note that m0 is not equal to the unconditional mean of the value of the asset as in Kyle. Thus, even when the market maker observes a zero value for both signals, he receives information from them. In contrast, in Kyle, a zero order flow implies no information. In our model, a zero order flow implies no information if and only if the second signal z 1 e equals its ¯ Secondly, note that the weight on the signal z 1 e is positive, thus implying that the expected value, z. market maker believes the value of the asset is higher, on average, on seeing a higher realization of z 1 e. Finally, the stock pricing function is now generalized to include two sources of information.

3. Comparative statics analysis In this section, we analyze the effect of the underlying variances on the stock order of the insider and the stock pricing function. These variances are the variances of the value of the asset, s 2z , of the 2 2 noise trade, s u and of the noise in the second signal, s e . The results are summarized in the following proposition. Proposition 2. ( i) m1 varies positively with s z2 ; negatively with s 2e and is independent of s 2u . ( ii) m2 varies positively with s e2 and negatively with s 2u . It varies positively with s 2z if and only if s 2e . ]21 s 2z . ( iii) If z is positive (negative), the level of insider trading, x varies positively (negatively) with s 2u , negatively with s z2 . x is independent of s e2 . Proof. (i) follows from the expression for m1 in the Lemma. (ii) follows from the expression for m2 in the Lemma; and (iii) follows from substituting the values of m1 and m2 into the expression for x in Proposition 1. h Discussion. Proposition 2 presents results that are easy to derive but not so easy to interpret. The stock pricing coefficients m1 and m2 depend in a complicated way on the variances of the two signals as well as their covariances with the value of the firm. Another complicating factor in interpreting m1 and m2 is that some of these variances and covariances are also functions of m1 and m2 since the 2 insider’s trade, x, depends on m1 and m2 . Thus, as any given parameter (such as s u ) changes, there is a direct effect on the coefficients m1 and m2 and an indirect effect through a change in the insider’s trade. Proposition 2 gives the net outcome of these two effects. There are some interesting patterns that emerge in the comparative statics summarized in

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Proposition 2. First, note that all three underlying variances, namely, the variances of the value of the asset, s 2z , of the noise trade, s 2u and of the noise in the second signal, s 2e , affect variables m1 , m2 and the level of insider trading x differently. For example, while s e2 has no effect on x, s 2u and s 2z do. To provide intuition for the differences in the effects of changes in s u2 and s e2 , note, first, that the noise trade has no effect on the signal z 1 e. Thus, it is pure noise in the order flow and has the effect of making the signal of total order flow less informative, while leaving the information content of the signal z 1 e unchanged. The effect of a change in s 2e is more complicated than a change in s 2u since s 2e affects the signal of the total order flow through its effect on m1 while s 2u does not affect the signal z 1 e. Finally, the effect of s 2z on m1 , m2 and x is different from the effects of s e2 and s u2 since an increase in s 2z increases all variances and covariances. This is in contrast to the effects of changes in the variances of u and e. Another interesting property of the results in Proposition 2 is the asymmetry in the effects that the noise in the two signals has on m1 , m2 and x. To be specific, we have shown first, that, while the variance of noise trading, that is, s 2u does not affect m1 , the variance of the noise in z 1 e, that is, s 2e affects m2 . We refer to this as the cross-variance asymmetry. Secondly, while m1 increases when s 2z increases, m2 does so only for ‘high’ values of s e2 . At first puzzling, this lack of symmetry is a natural consequence of a more fundamental asymmetry between the two signals, z 1 e and the total order flow. Note that the signal z 1 e does not depend on the level of insider trading x, while the choice of x directly depends on m1 . An implication of this difference is that the noise in the signal z 1 e affects the level of insider trading x but the noise in the order flow does not affect the signal z 1 e. This provides an intuitive explanation for the asymmetric effects of s 2e and s 2u . Finally, consider part (iii) of Proposition 2. While the effects of s u2 and s z2 are similar to that in 2 Kyle, the effect of s e on the level of insider trading is striking. To illustrate the reasoning behind it, we focus on the case where z is positive. Note from parts (i) and (ii) of the proposition, as s 2e increases the market maker decreases m1 and increases m2 . The expression for insider trading, x, shows that these two changes have opposite effects on x. The change in m1 increases the profitability of insider trading while the change in m2 decreases the profitability of trading. The two changes cancel each other out because the market maker can correctly incorporate the change in s 2e into the stock prices. This is so because neither the market maker nor the insider observe e.

4. Information revelation and insider profits In the first subsection we examine the extent to which the stock price reveals information. In the second subsection we examine the effect of information revelation on the profits of the insider. We also examine the effects of the underlying parameters on these profits.

4.1. Informativeness of stock price Kyle (1985) and Rochet and Vila (1994) find that the stock price reveals half of the inside information, regardless of the parameter values. We show that in our model, the stock price reveals

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more than half of the information possessed by the insider. We also show that the amount of information revealed varies with some of the underlying parameters. Thus, a constant amount of information release is a special feature of Kyle’s model (and related models). Our model generalizes this result. A measure of informativeness when dealing with multivariate normal distributions is the variance of the value of the firm conditional on the information of the market maker. The lower the conditional variance, the higher the information content of the stock price. In our set-up, this can be written as: (see Graybill, 1961) Var(z /z 1 e, x 1 u) 5 s 2z 2 m1 sz,z1 e 2 m2 sz,x 1u This reduces to (1 2 m1 )s z2 ]]]] Var(z /z 1 e, x 1 u) 5 2

(6)

Note that the coefficient of s 2z is less than half. Now, substituting for m1 from the Lemma, we get, 2

se Var(z /z 1 e,x 1 u) 5 ]]]] s2 2 s z 1 2s 2e v

(7)

which is the variability in the value of the firm that still remains unexplained by the stock price. Note that the fraction of the unexplained variance varies negatively with the underlying variability s 2z and positively with the variability s 2e in contrast to Kyle. Thus, we have the following important property of stock prices. Proposition 3. The stock price reveals more than half of the information possessed by the insider. Further, the amount of information revealed varies with s 2z and s 2e . From Eq. (6), it is clear that this difference between the information revealed in our model and in Kyle’s is entirely due to the addition of signal z 1 e (as reflected in the coefficient m1 ). The amount of information revealed in Kyle remains unchanged regardless of the level of noise in the order flow or the variability of the true value of the firm because the order flow is the only signal for the market maker and the informed component of the order flow is a decision variable for the insider. The insider’s trading strategy and the market maker’s price-setting rule completely offset changes in either the variability of the value of the firm or of the noise trade. In our model, the amount of information revealed depends on various parameters, for example the level of s 2e . Thus, the observation of a second signal by the market maker adds a fundamental change to the model. Indeed, the two signals z 1 e and x 1 u are fundamentally different in that, z 1 e cannot be manipulated by the insider to affect its information content whereas x 1 u or the total order flow can be manipulated through the choice of x.

4.2. Insider’ s profits In this subsection, we examine two properties of the insider’s equilibrium profits. First we examine the effects of changes in the variability of z, e and u on the insider’s profits. We show that, as in Kyle,

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the variability in u, has a positive effect on the insider’s profits, and the variability in z has a negative effect. The noisiness of z 1 e, that is s 2e , has a positive effect on the insider’s profits. However, this positive effect is the net outcome of two opposing factors. On the one hand, a more noisy z 1 e leads to a lower m1 (see Proposition 2) and hence a lower stock price. On the other hand, a more noisy z 1 e leads to a higher m2 and, thus, a higher stock price. Secondly, we examine whether greater information revelation in our model also leads to lower profits for the insider. We show that greater information revelation leads to a lower profit level for the insider. The insider’s expected profits in equilibrium are given by: 2

((1 2 m1 )(z 2 z¯ )) P 5 ]]]]]] 4m2 Note that the insider’s profits are positive for all z. Now substituting for m1 and m2 from the Lemma, into P, we obtain 2 2

4((z 2 z¯ )s e ) su P 5 ]]]]] 4sz (s z2 1 2s 2e )2

(8)

The next proposition summarizes the comparative statics of profits with respect to the three underlying variances, as in Proposition 2. Proposition 4. In equilibrium, the following holds: dP (i) ]] . 0 dsu dP (ii) ]]2 . 0 ds e dP (iii) ] , 0 dsz Finally, we examine the effect of the greater information release through the stock price on the level of the insider’s profits. We compare the profit level given by Eq. (8) with the profits of the insider in the Kyle model. In the latter, the insider maximizes the following function by choosing x:

P 9 5 [z 2 a0 2 a1 x]x Solving for the profit maximizing level of x is straightforward. We skip the details and simply report the profit level below 3 : 2 (z 2 z¯ ) su P 9 5 ]]] 2sz

The next proposition presents the results. 3

It can be easily verified that the proportion of information revealed by the stock price is half.

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Proposition 5. P 9 , P. Thus, the addition of a correlated signal to the market maker’ s information has the effect of reducing the insider’ s profits.

References Graybill, F., 1961. An Introduction to Linear Statistical Models, Vol. I, McGraw Hill. Jain, N., Mirman, L.J., 1998. Real and financial effects of insider trading with correlated signals, Working Paper. Kyle, A., 1985. Continuous auctions and insider trading. Econometrica 53. Rochet, J., Vila, J., 1994. Insider trading without normality. Review of Economic Studies 61.