Substitute Trading and the Effectiveness of Insider-Trading Regulations

Transcription

1 Substitute Trading and the Effectiveness of Insider-Trading Regulations Hui(Jane) Huang University of Western Ontario January 18, 2005 JOB MARKET PAPER Abstract US securities laws prohibit insiders from using inside information to trade their own stocks. But it is not illegal for insiders to trade related stocks, such as stocks of their firm s competitors, suppliers, and customers. Borrowing from the law literature (Ayres and Bankman, 2001), we call insiders trading in related stocks substitute trading. This paper studies the effectiveness of insider trading laws in the presence of substitute trading. We extend Kyle s (1985) model to a (static) model with two correlated firms, each with its own insider. We consider two different trading environments, an opaque one where the market maker observes only the order flow of the asset he makes the market in, and a transparent one where he also observes the order flow of the other (correlated) stock. In transparent markets, insider-trading regulations do limit insiders ability to profit from inside information. However, in opaque markets, the regulations may actually increase insiders profits. JEL Classification: G14, G18. Keywords: Substitute Trading, Stocks of Related Firms, Insider-Trading Regulations. I would like to thank Walid Busaba, Lutz-Alexander Busch, Philippe Grégoire, Srihari Govindan, and Peter Streufert for their helpful comments. All errors are my own. Department of Economics, The University of Western Ontario, London, Ontario, Canada, N6A 5C2. hhuang@uwo.ca. 1

2 1 Introduction There have been extensive discussions in finance, economics, and law about whether insider trading should be regulated, that is, whether insiders (primarily executives, board members, officers, and controlling shareholders) should be prohibited from using inside information to trade the stocks of their own firms 1. Almost exclusively, these discussions are made under the assumption that other than trading their own stocks (e.g., Kyle, 1985; Glosten, 1989 ) and/or manipulating information (e.g., Benabou and Laroque, 1992), insiders cannot benefit from their private information. However, insiders can profit by trading the stocks of related firms and it is legal for them to conduct such trades. This paper contributes to the literature in two ways. To our knowledge, this is the first paper to present a formal study of the effectiveness of insider-trading regulations when insiders are able to trade the stocks of related firms. It also adds to the microstructure literature a detailed study of the strategic trading behavior of insiders when they can trade their own stocks as well as related stocks. Section 10(b) of the 1934 Securities Exchange Act and the misappropriation doctrine clearly prohibit insiders from using private information to trade the stocks of their own firms. They do not, however, prohibit trading the stocks of related firms, such as competitors, suppliers and customers. 2 This raises the possibility that insiders of one firm who possess non-public information relevant to both their own firm as well as related firms might be able to trade on this information profitably in the related firms stocks. We call this substitute trading, borrowing from the law literature (Ayres and Bankman, 2001). How effective are insider-trading regulations in the presence of substitute trading? To address this question, we compare the strategic trading behavior and the expected profits of insiders, market depths, and the price discovery process in the presence of substitute trading. The analysis is conducted with and without insider trading regulations, in a static trading model in the spirit of Kyle (1985). The existing papers on insider trading assume one risky asset in the market. Here we consider a market with two correlated risky stocks, where the correlation could be positive or negative. There is one insider for each firm, who can legally trade the other stock. An insider has precise information about his own firm and can therefore make inferences, though 1 The term insider trading in this paper refers to the trading by registered corporate insiders. Insiders can trade the stocks of their own firms for liquidity reasons. 2 Even though some firms have internal regulations that prohibit trading in related stocks, especially those of suppliers and less often of customers and rivals, most employment contracts are silent as to such trading. Some firms allow it under certain conditions. See Ayres and Bankman (2001) for details. 2

3 imperfect, about the other firm. Liquidity trading in the two assets is exogenous, and there are competitive market makers for the two assets. The strategic behavior of insiders is studied under two different trading environments, one opaque and the other transparent. In the first, the market maker observes only the order flow of the asset he makes the market in, whereas in the second, the market maker also observes the order flow of the other (correlated) stock. There is one trading round and the trading mechanism resembles that in Kyle (1985). The distinction between the trading environments is motivated by some facts about stock exchanges. In the NYSE, each stock is handled by a single specialist. A specialist may represent one heavily traded stock and one less active one, or several stocks across a range of industries. The trading of two correlated stocks could be conducted by two specialists, or by the same specialist. Moreover, related firms are sometimes traded in different exchanges. For example, Dell is traded on NASDAQ while Hewlett-Packard (Compaq) is traded on the Big Board. The marker makers/dealers of these two stocks cannot see each other s order flows, reflecting the opaque trading environment in this paper. On the other hand, with the development of modern technology, stock exchanges are becoming more and more transparent. For example, the Toronto Stock Exchange uses Computer Assisted Trading System and Paris Bourse uses Cotation Assistée en Continu, allowing market participants to see the details of every transaction. It is reasonable to assume that market makers also can see the order flow of every stock trading on these exchanges. Our main results are as follows. Starting with an opaque environment, which turns out to be the easier one to analyze, we find that insider-trading regulations do not always reduce the aggregate profit that can be made by the two insiders. If insider trading is not regulated, insiders trade both stocks in the same direction as their information (buy on positive information and sell otherwise). This could mean that the insider will buy his own stock and sell the other if the two stocks are negatively correlated and he possesses positive information about his own stock. If insider trading is regulated, however, insiders can only do substitute trading. The regulations do limit the aggregate profit that can be made by the two insiders if the two firms are not closely correlated. However, if the two firms are closely correlated, the regulations could in fact increase the aggregate profit the two insiders make. This is because when two firms are closely correlated, the two insiders will compete against each other for almost the same piece of information, if there are no regulations. With regulations, each insider becomes an information monopolist in the other firm s stock. Additionally, without regulations, market prices are always more informative because there is more informed trading in the market. 3

4 In a transparent environment, insider-trading regulations limit the insiders ability to profit from their information. If insider trading is not regulated, insiders will definitely trade their own stocks in the same direction as their information. However, whether they trade substitute stocks will depend on how well they can hide their trade in both stocks. In general, the higher the uncertainty surrounding the value of an asset and the larger the level of noise trading, the more profit an insider can make from trading the asset. We summarize the joint effect of the two factors by their product, which we call the camouflage level. When the two firms have the same level of camouflage, insiders trade only their own stocks. When the two firms have different levels of camouflage, the insider in the smaller camouflage stock trades the other stock in the same direction as his information. However, the insider in the larger camouflage stock finds it optimal to trade the substitute stock against his information. This will muddy the water for the market makers and hence create more room for the insider to profit from trading his own stock. If, on the other hand, insider trading is regulated, only substitute trading happens. Comparing the equilibria with and without regulations, we find that insider-trading regulations are effective in the transparent environment. The aggregate profit of the insiders is lower, and price efficiency is unaffected. In summary, insider-trading regulations are more effective in a transparent environment. In such an environment, the regulations do not impact market efficiency but do limit the aggregate expected profit of the insiders. In an opaque market, the regulations do not always limit the aggregate expected profit of the insiders. Besides, regulations always lead to less efficient markets, where prices do not reflect inside information. In this paper we predict that substitute trading will be observed in both types of markets. Tookes (2003) provides empirical evidence of intra-industry substitute trading, which she calls competitor trading, in NYSE and AMEX. We also predict that substitute trading is not always in the same direction as the insiders information. If two correlated stocks are represented by the same specialist in NYSE, or traded in transparent exchanges, we will observe some informed traders trading substitutes against their information. However, if the two correlated stocks are represented by different specialists in NYSE, or traded in different exchanges, we will only observe substitute trading in the direction of the information. To our knowledge, no empirical study addresses these predictions. This paper makes a significant contribution to the literature in two ways. First, to our knowledge, this is the first paper that formally studies the effectiveness of insider-trading regulations 4

5 by taking substitute trading into account. There is a large literature on the pros and cons of insider-trading regulations. Proponents of insider-trading regulations suggest that insider trading may reduce firms value (e.g., Manove, 1989; Bebchuk and Jolls, 1999 ). They argue that insider trading also decreases price efficiency (e.g., Ausubel, 1990; Fishman and Hagerty, 1992) and market liquidity (e.g., Glosten, 1989; Leland, 1992; and many others). Opponents show that insider trading may reduce the conflict of interest between managers and shareholders(e.g., Manne, 1966; Hu and Noe, 2001). They also argue that it facilitates the process of incorporating information into prices so that prices are closer to fundamental values (e.g., Carlton and Fischel, 1983; Bernhardt, Hollifield and Hughson, 1995 ). Further, Bhattacharya and Nicodano (2001) suggest that insider trading may benefit outsiders with stochastic liquidity needs. 3 However, none of these studies consider substitute trading and its impact on the effectiveness of insider-trading regulations. This is the purpose of our paper. We show that the regulations may actually be harmful in opaque markets when the correlation between related stocks is high, but are effective in transparent markets. Second, this paper studies strategic trading by insiders when they can also trade in substitute stocks, in opaque as well as transparent markets. Caballé and Khrisnan (1994) also use a Kyletype model to study imperfect competition among risk-neutral informed traders in a multi-security transparent market where fundamentals are correlated, but this paper is very different. First, they do not address the effectiveness of insider-trading regulations. Second, they do not study informed traders strategic trading in opaque markets. And third, their analysis of strategic trading in transparent markets cannot be applied in our environment because they implicitly assume that each trader receives at least on signal on every asset. Technically, this assumption ensures that the variance-covariance matrix of informed traders best guess of the fundamentals is nonsingular. 4 In this paper, insiders naturally know private information about their own firms but make inferences about correlated firms; that is, the number of signals each insider has is less than the number of securities in the market. So, the corresponding variance-covariance matrix of informed traders best guess of the fundamentals is singular 5. Thus, we cannot use their analysis in our model. 3 Bainbridge (2000) provides a comprehensive survey of the arguments for and against insider-trading regulations. 4 Caballé and Khrisnan assume that there are N securities with a multivariate normally distributed payoff vector v = (v 1, v 2,, v N ). Each of the K informed traders observes the realization of a vector of signal s k, so the informational advantage of insider k is defined as ξ k = E(v s k ) E(v). They assume ξ k to be multivariate normally distributed with mean 0 and nonsingular variance-covariance matrix Σ ξ. The inverse of Σ ξ appears in their analytical solution. 5 Please see footnote 7 for details. 5

6 As mentioned above, most of the traditional literature on insider trading allows only one risky asset. A few papers employ multi-asset models. Huddart, Hughes and Brunnermeier (1999), and Grégoire and Huang (2004) use multi-firm models to examine the voluntary disclosure policies by insiders, but the firms are independent in these models. Easley, O Hara and Srinivas (1998) model competitive informed traders decisions whether to trade options or the underlying stocks. They assume that informed traders cannot trade options and stocks simultaneously. Tookes (2003) explores the role of product market interaction in generating correlated fundamentals and informed traders decisions whether to trade the stock of the firm for which they have private information, or that of the competing firm. But she assumes that informed traders are competitive and cannot trade both stocks at the same time. In her model, substitute trading is always restricted to being in the same direction as the informed traders information. The remainder of this paper is organized as follows. Section 2 presents the general market framework. Section 3 studies insiders strategic trading in their own stocks as well as in substitute stocks, when the market is opaque. Comparisons of insiders profits, market liquidity, and market efficiency with and without insider-trading regulations are provided. Section 4 studies the transparent market. Section 5 concludes and discusses future research. All proofs are provided in the appendix. 2 General Settings There are two firms in the economy. The values of these firms are (v 1, v 2 ), which follow a joint normal distribution with mean (0, 0) 6. v 1 N 0, σ2 1 ρσ 1 σ 2 v 2 0 ρσ 1 σ 2 σ2 2 The correlation coefficient can be positive or negative. For each firm, there is one risk-neutral insider, who knows exactly the value of his own firm. Because of the correlation of the two firms, Insider i knows something about firm j. 7 Each firm has its stock traded in the stock market. 6 We normalize the mean to be zeros. The assumption of positive means only complicates the analysis without giving us more insight. 7 The corresponding Σ ξ matrix defined in Caballé and Khrisnan (1994) (see footnote 4) for Insider 1 here, 0 1 Σ ξ1 σ2 1 ρσ 1σ 2 A, is different from that for Insider 2, Σξ2 ρ2 σ1 2 ρσ 1σ 2 A. Both matrices are singular ρσ 1σ 2 ρ 2 σ2 2 ρσ 1σ 2 σ2 2 so that we cannot simply take the inverse of them. These matrices are still singular and nonidentical when insiders 0 1 6

7 Insiders submit market orders to maximize their expected profits given their private information and legitimacy of insider trading. There are noise traders trading these two stocks in the market for exogenous reasons that are not modeled here 8. The demands from noise traders for the two assets are x 01 and x 02, which are normal with mean 0 and variance σu 2 1 and σu 2 2 respectively. x 01 and x 02 are independent of each other and of v 1 and v 2. There are competitive risk-neutral market makers for these two assets. Two types of trading environments are studied, an opaque one and a transparent one. In the first, the market maker observes only the total order flow of the stock he makes the market in, whereas in the second, the market maker observes also the total order flow of the other stock. 9 After observing the order flow information available to them, the market makers compete for order flows to maximize their expected profit from making the markets. To compare the equilibrium where insider trading is regulated with that where insider trading is not regulated, we consider two cases within each type of market structure: 1) insiders are allowed to trade their own stocks. 2) insiders are not allowed to trade their own stocks, but they can trade substitute stocks. 3 The Opaque Market In this section, we examine the opaque market. We first study the case where insider trading is regulated, since regulated market happens to be a simpler case and in practice, insider trading is regulated. Then we study the case where insider trading is not regulated. 3.1 Insider trading is regulated Suppose there are laws prohibiting insiders from using nonpublic information to trade their own stocks. However, these insiders can legally trade substitute stocks. The game tree for this case is in Figure 1. First Nature decides the values of the assets. An insider chooses his trade in the other asset to maximize his expected profit. Nature decides again the demand from noise traders for each asset. Market makers then set their prices to compete for the order flows in order to have noisy information about their own firms, as long as we keep the information structure that each insider knows something about his own firm and makes inference about the other firm from the correlation. 8 The reasons could be portfolio rebalancing or tax avoidance. 9 In Toronto and Paris, market participants can see every detailed transactions, not just the total order flows. However, Pagano and Röell (1996) show that in Kyle-type model whether the market makers see total order flows or each order does not affect the linear strategies used by insiders, nor does it affect the liquidity of the market. 7

8 maximize their profit. Formally, Insider i decides his trading strategy for asset j, X ij : v i x ij. Market maker A j decides his pricing strategy for asset j, P Aj : x ij +x 0j p Aj. Market maker B j also decides his pricing strategy for asset j, P Bj : x ij +x 0j p Bj. We can find a Nash equilibrium in this game, which is stated in the following proposition. Proposition 1 For the game in the opaque market with insider trading prohibited, there exists a Nash equilibrium (X 12, X 21, P A1, P B1, P A2, P B2 ) with X 12 (v 1 ) = ρ σ u2 v 1, ρ σ 1 P A1 (x 21 +x 01 ) = P B1 (x 21 +x 01 ) = ρ σ 1 (x 21 +x 01 ), 2σ u1 X 21 (v 2 ) = ρ σ u1 v 2, ρ σ 2 P A2 (x 12 +x 02 ) = P B2 (x 12 +x 02 ) = ρ σ 2 (x 12 +x 02 ). 2σ u2 Further, the insiders expected profits are E[π 1 ] = ρ 2 σ 2σ u2, E[π 2 ] = ρ 2 σ 1σ u1. In this equilibrium, the insiders use linear trading strategies and market makers set prices in a linear way. There exists only one Nash equilibrium where players all use linear strategies. 10 When insider trading is regulated, from the Proposition 1 we know that if the two firms are positively correlated, that is, when ρ > 0, the insiders trade substitutes in the same direction as their private information about their own firms. If the two firms are negatively correlated, that is, when ρ < 0, the insiders trade substitute stocks in the opposite direction as their private information about their own firms. Pricing rules are always positive in the total order flows. The markets work out as if there were two independent risky assets. The difference is that now the insiders do not have complete information about the assets. We have an extra term ρ in the coefficients of the linear pricing rules. So the adverse selection problem here is not as severe as in Kyle (1985). As in Kyle (1985), we define Σ 1 = V ar(v 1 X 21 (v 2 )+x 01 ) and Σ 2 = V ar(v 2 X 12 (v 1 )+x 02 ). Then Σ 1 and Σ 2 measure the information stock left in prices from the view of market makers. Simple calculations lead to the following corollary. 10 In a model with one insider and competitive market makers, Kyle (1985) shows the existence and uniqueness of the insider s trading strategy and the market maker s pricing rule in the class of linear functions. This need not be the case when non-linear pricing rules are allowed. Rochet and Vila (1994) show uniqueness of equilibrium in the version of Kyle (1985) static market order game where the insider observes liquidity trading, assuming the liquidity shock and the asset value to have compact supports. Uniqueness of equilibrium with normally distributed variables is still an open question. 8

9 N v1 (I1) (I1) N v2 I1 I1 x12 x12 (I2) (I2) (I2) (I2) N N N N x02 x02 x02 x02 A2 A2 A2 A2 pa2 pa2 pa2 pa2 B2 B2 B2 B2 B2 B2 B2 B2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 I2 I2 x21 x21 N N N N x01 x01 x01 x01 A1 A1 A1 A1 pa1 pa1 pa1 pa1 B1 B1 B1 B1 B1 B1 B1 B1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 Figure 1: The game tree for the opaque market for asset 1 when insiders are prohibited from trading their own stocks based on their private information. N stands for Nature. I1 and I2 stand for Insider 1 and Insider 2. A1 and B1 stand for market markers for asset 1. In the information set of Insider 2 he knows v2, but not v1. In the market makers information sets, y1 = x21 + x01 is constant. The corresponding game tree for asset 2 is in the background. 9

10 Corollary 1 The informativeness of prices with insider trading regulated in the opaque market is Σ 1 = σ 2 1(1 ρ2 2 ), Σ 2 = σ 2 2(1 ρ2 2 ). Results in Proposition 1 and Corollary 1 are not surprising. If ρ = 1, that is, the two assets are identical or completely opposite, but there is only one insider for each asset, we get the results exactly the same as Kyle (1985). As ρ decreases, the insiders private information becomes less valuable, so their expected profits go down. At the same time, as the insiders information becomes less accurate, prices are less efficient. If ρ = 0, there is no informed trading at all. 3.2 Insider trading is unregulated When insider trading is unregulated, the insiders can legally trade any stocks to maximize their profits. Consider Insider 1 first. Insider 2 s situation is similar. Given Insider 1 can trade his own stock, he gets positive profit if he participates in trading of stock 1. So both insiders will trade their own stocks respectively. Given Insider 2 trades stock 2 and market makers for stock 2 can not observe total demand for stock 1, Insider 1 is better off to trade stock 2 to compete with Insider 2 for the information advantage for stock 2. Thus Insider 1 will trade both stocks, so will Insider 2. The game tree for this case is in Figure 2. Nature moves fist to decide the values of the assets. An insider chooses his trades in both assets to maximize his expected profit. Nature decides again the demand from noise traders for each asset. The competitive market makers observe only the total demand of the stock he makes the market. They set their prices to compete for the order flows in order to maximize their profit. Formally, Insider i decides his trading strategy, (X i1 : v i x i1, X i2 : v 1 x i2 ). Market maker A i decides his pricing strategy for asset i, P Ai : Σ 2 k=0 x ki p Ai. Market maker B i also decides his pricing strategy for asset i, P Bi : Σ 2 k=0 x ki p Bi. Proposition 2 For the game in the opaque market with insider trading allowed, there exists a Nash equilibrium (X 11, X 12, X 21, X 22, P A1, P B1, P A2, P B2 ) with X ij (v i ) = β ij v i, i = 1, 2; j = 1, 2. (1) 10

11 and P Aj (Σ 2 k=0 x kj) = P Bj (Σ 2 k=0 x kj) = λ j (Σ 2 k=0 x kj), j = 1, 2. (2) where β 11 = σ u 1 (2 ρ 2 ) σ 1 4 3ρ 2 +ρ 4, β 12 = β 21 = σ u2 ρ σ 1 4 3ρ 2 +ρ 4, (3) σ u1 ρ σ 2 4 3ρ 2 +ρ, β 4 22 = σ u 2 (2 ρ 2 ) σ 2 4 3ρ 2 +ρ, (4) 4 Further, the insiders expected profits are λ 1 = σ 1 4 3ρ 2 +ρ 4 σ u1 (4 ρ 2, (5) ) λ 2 = σ 2 4 3ρ 2 +ρ 4 σ u2 (4 ρ 2. (6) ) E[π 1 ] = (2 ρ2 ) 2 σ 1 σ u1 + ρ 2 σ 2 σ u2 (4 ρ 2 ) 4 3ρ 2 +ρ 4, E[π 2 ] = ρ2 σ 1 σ u1 + (2 ρ 2 ) 2 σ 2 σ u2 (4 ρ 2 ) 4 3ρ 2 +ρ 4. If the two firms are independent, ρ = 0, again we get the results in Kyle (1985) one-short game. The insiders have no information advantage in substitute stocks. As the two firms become correlated, both insiders are better off to compete with each other in both assets. Because of the competition coming from Insider j, Insider i reduces his trade of stock i, but he starts to trade more stock j to compete with Insider j. We see that 2 ρ 2 decreases in ρ while ρ 4 3ρ 2 +ρ 4 4 3ρ 2 +ρ 4 increases in ρ. When the two firms become more and more correlated, the competition between the insiders become more intense. Insider i further reduces his trades of stock i, but keep increasing his trade of stock j. But he still trades more of his own stock than the other stock. When ρ = 1, the two firms and the insiders are identical, we get the special case in Holden and Subrahmanyam (1992) where the number of insiders is two. An insider expected profit comes from two sources: trading his own stock and the substitute. For example, the first part in the expected profit of Insider 1 comes from his trading in stock 1. The larger the camouflage is in stock 1, the more profit he gets. The second part comes from his trading in stock 2. Again, the larger the camouflage is in asset 2, the more profit Insider 1 gets from trading this stock. With given σ s and σ u s, relatively how much profit coming from 11

12 N x12 x12 I2 I2 I2 I2 x22 x22 x22 x22 N N N N N N N x02 x02 x02 x02 x02 x02 x02 A2 A2 A2 A2 A2 A2 pa2 pa2 pa2 pa2 pa2 pa2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 pb2 v1 I1 I1 x11 x11 I2 I2 I2 I2 N x02 x21 x21 x21 x21 N N N N N N N N x01 x01 x01 x01 x01 x01 x01 x01 A1 A1 A1 A1 A1 A1 pa1 pa1 pa1 pa1 pa1 pa1 B1 B1 B1 B1 B1 B1 B1 B1 B1 B1 B1 B1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 pb1 N v2 I1 I1 Figure 2: The game tree for the opaque market for asset 1 when insiders are allowed to trade their own stocks based on their private information. N stands for Nature. I1 and I2 stand for Insider 1 and Insider 2. A1 and B1 stand for market markers for asset 1. In the information set of Insider 1, he knows v1, but not v2. In the information set of Insider 2 he knows v2, but not v1. In the market makers information sets, y1 = x11 + x21 + x01 is constant. The corresponding game tree for asset 2 is in the background. 12

13 each trading depends on how closely the two firms are correlated, either positively or negatively. Expected profit from trading ow stock strictly decreases with ρ since competition from the other insider becomes more intense. Expected profit from substitute trading strictly increases with ρ since his information is more precise. [Figure 5] Each insider s expected profit can either increase, decrease or be non-monotonic in ρ, depending on the relative size of the camouflage the insider can use to hide informed trading. Figure 5 illustrates the situations where the expected profit of Insider 1 strictly increases, be non-monotonic, and strictly decreases with ρ. When σ 1σ u1 σ 2 σ u2 0.68, Insider 1 s expected profit increases with ρ ; when 0.68 < σ 1σ u1 σ 2 σ u2 with ρ ; when σ 1σ u1 σ 2 σ u2 1.1, Insider 1 s expected profit first decreases then increases > 1.1, Insider 1 s expected profit decreases with ρ. This is consistent with our intuition. As σ 1 σ u1 is much smaller than σ 2 σ u2, Insider 1 s expected profit mainly comes from substitute trading. So when the correlation goes up, Insider 1 expects to make more profit. If σ 1 σ u1 is much larger than σ 2 σ u2, Insider 1 s expected profit mainly comes from trading his own stock. Therefore as the two firms become more correlated, the competition from Insider 2 drives Insider 1 s expected profit down. When σ 1σ u1 σ 2 σ u2 is moderate, if ρ is small, as ρ goes up, the decrease in expected profit in asset 1 caused by more intense competition dominates the increase in expected profit in asset 2 caused by more precise information in substitute trading. After ρ reaches certain point the increase in expected profit in asset 2 dominates the decrease in expected profit in asset 1. Let Σ j = Var(v j X 1j (v 1 )+X 2j (v 2 )+x 0j ). Then Σ j measures of the informativeness of the price for asset j. The larger the Σ j is, the less of the insiders private information is released. So we have Corollary 2 as follows. Corollary 2 The informativeness of prices for the two assets in the above equilibrium is Σ 1 = 2 ρ2 4 ρ 2 σ2 1, Σ 2 = 2 ρ2 4 ρ 2 σ2 2. How much information is revealed to the market makers only depends on how closely the two firms are correlated. When the two firms are uncorrelated, half of the private information of each insider is revealed, as in Kyle (1985). When ρ increases, the competition between the insiders becomes more intense. Thus more information is released to the market by informed trading. 13

14 Therefore a smaller portion of information is left. When the two firms are identical, two-thirds of the insiders private information is revealed. 3.3 Comparison To see the effectiveness of insider-trading regulations, we compare the equilibria with and without regulations. In this subsection, for notational convenience, variables with denote those in the case where the insiders cannot trade their own stocks using nonpublic information. Variables without denote those in the case of no insider-trading regulations. First, we look at Insider 1 s expected profits in the two cases. From Propositions 1 and 2, we see that the extra expected profits from unregulated insider trading depend on ρ, that is, how closely the two firms are correlated. The extra expected profit of Insider 1 from unregulated insider trading is [ ] (2 ρ 2 ) 2 σ 1 σ u1 E[π 1 ] E[ π 1 ] = (4 ρ 2 ) 4 3 ρ 2 + ρ + ρ 4 (4 ρ 2 ) 4 3 ρ 2 + ρ 1 ρ σ 2 σ 4 u2 2 [Figure 6] The coefficient of σ 1 σ u1 is obviously always positive, while coefficient of σ 2 σ u2 is always neg- ρ ative as increases in ρ and reaches maximum of 1 (4 ρ 2 ) 4 3ρ 2 +ρ 4 3. The expected profit for 2 Insider 2 is similar except that we switch between σ 1 σ u1 and σ 2 σ u2. Whether the extra expected profit from unregulated insider trading is positive or negative depends on the relative size of camouflage, σ i σ ui, and the correlation coefficient, ρ. Figure 6 illustrates the extra expected profit of Insider 1 when σ 1σ u1 σ 2 σ u2 the correlation coefficient. When σ 1σ u1 σ 2 σ u2 equals to 0.5 and The extra expected profit always decreases with < 1.12, the expected extra profit of Insider 1 is positive if ρ is small. As ρ increases, the expected extra profit of Insider 1 could be negative. When σ 1 σ u1 σ 2 σ u2 > 1.12, the extra expected profit of Insider 1 is always positive. This means insider-trading regulations always limit an individual insider s profit when the firm s size of camouflage is about or more than 1.12 of the other firm s. However, the regulations does not always limit an individual insider s profit if the two firms have the same level of camouflage or this insider s camouflage is smaller than the other s. In this case, larger portion of the insider s expected profit comes from substitute trading if the two firms are closely correlated. The regulations establish one insider s monopoly power in the other asset. Thus the insider s profit from substitute trading in the case of regulations can be more than the profit from trading both assets in the case of no regulations. The regulations could make an individual insider better off. The regulations could also make both insiders better off simultaneously for the same reason. 14

15 The aggregate extra expected profit for the insiders from unregulated insider trading is = (E[π 1 ]+E[π 2 ]) (E[ π 1 ]+E[ π 2 ]) [ ] (2 ρ 2 ) 2 (4 ρ 2 ) 4 3 ρ 2 + ρ + ρ 2 4 (4 ρ 2 ) 4 3 ρ 2 + ρ ρ (σ 1 σ u1 +σ 2 σ u2 ) [Figure 7] In Figure 7 examples of extra expected profit for both insiders are provided when σ 1σ u1 σ 2 σ u2 equals to 0.5 and In general, when ρ < 5 17( ), the aggregate extra expected profit of the insiders is always positive. When ρ > 5 17, the aggregate extra expected profit is always negative. The aggregate extra expected profit decreases with the correlation coefficient. Next we will study the liquidity measure λ and price efficiency measure Σ. From Proposition 1 and 2, we know when ρ > 0 λ 2 i λ 2 i = σ2 i (4 3ρ2 +ρ 4 ) σ 2 u i (4 ρ 2 ) 2 ρ2 σi 2 4σu 2 = ρ6 +12ρ 4 28ρ i 4(4 ρ 2 ) 2 σ 2 i σ 2 u i When ρ < 0 the analysis is similar. So when ρ < 5 17, λ i > λ i, that is, if the two firms are not very closely correlated, markets are more liquid when insider trading is regulated. When ρ > 5 17, λ i < λ i, that is, if the two firms are closely related, markets are more liquid when insider trading is unregulated. These results are illustrated in Figure 8. From Corollaries 1 and 2 we know that [Figure 8] Σ i Σ i = (2 ρ2 ) 2 2(4 ρ 2 ) σ2 i < 0 So the prices are always more informative when the insiders can trade their own stocks, as the more informed trading goes on, the more information is revealed. The above analysis leads to the following proposition. Proposition 3 In the opaque market, (1) Insider-trading regulations decrease the insiders aggregate expected profit when ρ < 5 17( ) and increase it when ρ > (2) Whether the regulations limit an individual insider s expected profit depends on the relative size of camouflage, σ i σ ui, and the correlation coefficient, ρ. The regulations may increase an individual insider s profit. (3) The regulations cause the markets to be more liquid when ρ < 5 17 and less liquid when ρ > (4) The regulations always lead to less efficient prices. 15

16 4 The Transparent Market 4.1 Insider trading is regulated Model in this section is similar to that in Section 3 except that the market is transparent in the sense that market makers can see the total order flows for both assets. The two firms are correlated, so the market makers can infer more information about the fundamental values of the two assets from the total demands for both assets. Since the insiders are prohibited from trading their own stocks using private information, they are better off to trade substitutes. The game tree for this case is in Figure 3. Insider i decides his trading strategy for asset j, X ij : v i x ij. Market maker A decides his pricing strategy for the two assets, P1 A:(x 21+x 01, x 12 +x 02 ) p A 1, P2 A:(x 21+x 01, x 12 +x 02 ) p A 2. Market maker B also decides his pricing strategy for the two assets, P1 B:(x 21+x 01, x 12 +x 02 ) p B 1, P 2 B:(x 21+x 01, x 12 +x 02 ) p B 2. Proposition 4 For the game in the transparent market with insider trading prohibited, there exists a Nash equilibrium (X 12, X 21, P A 1, P B 1, P A 2, P B 2 ) with X 12 (v 1 ) = ρ σ u2 v 1, ρ σ 1 X 21 (v 2 ) = ρ σ u1 v 2, ρ σ 2 P k 1 (x 21 +x 01, x 12 +x 02 ) = ρ σ 1 (4 ρ 2 (x 21 +x 01 ) + ρ )σ u1 ρ (2 ρ 2 )σ 1 (4 ρ 2 (x 12 +x 02 ), k=a, B (7) )σ u2 P2 k (x 21 +x 01, x 12 +x 02 ) = ρ (2 ρ 2 )σ 2 ρ (4 ρ 2 (x 21 +x 01 ) + ρ σ 2 )σ u1 (4 ρ 2 (x 12 +x 02 ), k=a, B. (8) )σ u2 Further, the insiders expected profits are E[π 1 ] = ρ 4 ρ 2 σ 2σ u2, E[π 2 ] = ρ 4 ρ 2 σ 1σ u1. Comparing the equilibria in Propositions 1 and 4, we find that (1) the insiders use the same linear substitute trading strategies regardless of the transparency of the markets; (2) the market makers use the total order flows of both assets to set the prices. That is, other than self price pressure traditional in the literature, here informed trading has a cross-asset price pressure. When the insiders decide how much to trade in a stock, their trades affect not only the price the asset they trade, but also the price of the correlated stock. The self price pressure is always positive. The cross-asset price pressure is positive if the two firms are positively correlated and negative 16

17 v1 v 2 N I1 I1 I1 I1 x12 x12 x12 x12 I2 I2 I2 I2 I2 I2 I2 I2 x21 x21 x21 x21 x21 x21 x21 x21 N N N N N N N N N N N N N N N N x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x02 A x02 x02 A x02 x02 A x02 x02 A x02 x02 A x02 x02 A x02 x02 A x02 x02 A x02 p A 1 p A 2 B B B B B B B B B B B B B B B B p B 1 p B 2 Figure 3: The game tree for the transparent market when insiders are prohibited from trading their own stocks based on their private information. N stands for Nature. I1 and I2 stand for Insider 1 and Insider 2. A and B stand for market marker A and B. In the information set of Insider 1, he knows v1, but not v2. In the information set of Insider 2 he knows v2, but not v1. In the market makers information sets, (y1, y2) is a vector of constants, where y1 = x21 + x01 and y2 = x12 + x02. 17

18 if the two firms are negatively correlated. Since the insiders can only trade the substitute assets under regulations, the cross-asset price pressure is always larger than self price pressure. Let Σ 1 = Var(v 1 X 21 (v 2 )+x 01, X 12 (v 1 )+x 02 ) and Σ 2 = Var(v 2 X 21 (v 2 )+x 01, x 12 (v 1 )+x 02 ), then Σ 1 and Σ 2 measure the informativeness of prices. Corollary 3 In the transparent market, the informativeness of prices with insider trading regulated is Σ 1 = 2 ρ2 4 ρ 2 σ2 1, Σ 2 = 2 ρ2 4 ρ 2 σ2 2 In the opaque market, after one trading round, at most one half of private information is released to the market, while in the transparent market, at least one half of private information is released. This is because the market makers can infer more information from the demands for both assets in the transparent market than in the opaque one. Prices are more efficient in the transparent market. In the opaque market, the expected profit of substitute trading is between 0 and 1 2 σ iσ ui, while in the transparent market, the expected profit of substitute trading is between 0 and 1 3 σ iσ ui. With more efficient prices, the insiders profits are lower in the transparent market. 4.2 Insider trading is unregulated Now suppose insider trading is not regulated. The game tree for this case is in Figure 4. Insider i decides his trading strategy, X i1 : v i x i1, X i2 : v 1 x i2. Market maker A decides his pricing strategy for the two assets, P1 A:(Σ2 k=0 x k1, Σ 2 k=0 x k2) p A 1, P 2 A:(Σ2 k=0 x k1, Σ 2 k=0 x k2) p A 2. Market maker B also decides his pricing strategy for the two assets, P1 B:(Σ2 k=0 x k1, Σ 2 k=0 x k2) p B 1, P2 B:(Σ2 k=0 x k1, Σ 2 k=0 x k2) p B 2. Obviously, the insiders can earn positive profits by trading their own stocks. However, it is ambiguous whether the insiders should trade the substitutes. Since the market makers can infer much information about the fundamental values of the two assets from the total demands for both assets, substitute trading may reveal too much information about the insider s private information such that the market makers set very efficient prices. The more efficient the prices are, the less profits the insiders can make. Proposition 5 states the conditions that ensures the existence of a Nash equilibrium. 18

19 Proposition 5 For the game in the transparent market with insider trading allowed, (X 11, X 12, X 21, X 22, P A 1, P B 1, P A 2, P B 2 ) is a Nash equilibrium if X ij (v i ) = β ij v i, i = 1, 2; j = 1, 2. (9) and P A j (Σ 2 k=0 x k1, Σ 2 k=0 x k2)=p B j (Σ 2 k=0 x k1, Σ 2 k=0 x k2) = λ j1 (Σ 2 k=0 x k1) + λ j2 (Σ 2 k=0 x k2), j = 1, 2. (10) where β s and λ s satisfy the following conditions: λ 11 > 0, λ 22 > 0, 4λ 11 λ 22 > (λ 12 +λ 21 ) 2, and 2λ 11 β 11 + (λ 12 +λ 21 )β 12 = 1 λ 11 β 21 ρσ 2 σ 1 λ 12 β 22 ρσ 2 σ 1 (11) 2λ 22 β 12 + (λ 12 +λ 21 )β 11 = ρσ 2 σ 1 (1 λ 21 β 21 λ 22 β 22 ) (12) 2λ 11 β 21 + (λ 12 +λ 21 )β 22 = ρσ 1 σ 2 (1 λ 11 β 11 λ 12 β 12 ) (13) 2λ 22 β 22 + (λ 12 +λ 21 )β 21 = 1 λ 21 β 11 ρσ 1 σ 2 λ 22 β 12 ρσ 1 σ 2 (14) λ 11 = (β 11σ 2 1 +β 21ρσ 1 σ 2 )C (β 12 σ 2 1 +β 22ρσ 1 σ 2 )B AC B 2 (15) λ 12 = (β 12σ 2 1 +β 22ρσ 1 σ 2 )A (β 11 σ 2 1 +β 21ρσ 1 σ 2 )B AC B 2 (16) λ 21 = (β 11ρσ 1 σ 2 +β 21 σ 2 2 )C (β 12ρσ 1 σ 2 +β 22 σ 2 2 )B AC B 2 (17) λ 22 = (β 12ρσ 1 σ 2 +β 22 σ 2 2 )A (β 11ρσ 1 σ 2 +β 21 σ 2 2 )B AC B 2 (18) The expected profits of the insiders are as follows ( ) E[π 1 ] = σ1 [β 2 ρσ 2 ρσ λ 11 β 11 λ 11 β 21 λ 12 β 12 λ 12 β 22 σ 1 σ 1 ( )] ρσ2 ρσ 2 ρσ 2 + β 12 λ 21 β 11 λ 21 β 21 λ 22 β 12 λ 22 β 22 σ 1 σ 1 σ 1 [ ( ) E[π 2 ] = σ2 2 ρσ1 ρσ 1 ρσ 1 β 21 λ 11 β 11 λ 11 β 21 λ 12 β 12 λ 12 β 22 σ 2 σ 2 σ 2 ( )] ρσ 1 ρσ 1 + β 22 1 λ 21 β 11 λ 21 β 21 λ 22 β 12 λ 22 β 22 σ 2 σ 2 The informativeness of prices is ( Σ 1 = σ1 2 1 M ) AC B 2 ( Σ 2 = σ2 2 1 N ) AC B 2 19

20 where A, B, C, M, and N are defined as follows A = β 2 11σ β 2 21σ β 11 β 21 ρσ 1 σ 2 + σ 2 u 1 B = β 11 β 12 σ β 21 β 22 σ (β 11 β 22 +β 21 β 12 )ρσ 1 σ 2 C = β 2 12σ β 2 22σ β 12 β 22 ρσ 1 σ 2 + σ 2 u 2 M = (β 11 σ 1 +β 21 ρσ 2 ) 2 C + (β 12 σ 1 +β 22 ρσ 2 ) 2 A 2B(β 11 σ 1 +β 21 ρσ 2 )(β 12 σ 1 +β 22 ρσ 2 ) N = (β 11 ρσ 1 +β 21 σ 2 ) 2 C + (β 12 ρσ 1 +β 22 σ 2 ) 2 A 2B(β 11 ρσ 1 +β 21 σ 2 )(β 12 ρσ 1 +β 22 σ 2 ) The conditions β s and λ s must satisfy form a system of 8 equations with 8 unknowns, which cannot be solved analytically. So we first look at two special cases which we can derive analytical results: (1) the artificial case where the insiders trade only their own stocks. The insiders act as if they were constrained to trade their own stocks. We study this case in Lemma 1. (2) The two firms are symmetric in the sense that σ 1 = σ 2 and σ u1 = σ u2, which is studied in Proposition 6. We then extend Proposition 6 to a more general case where σ 1 σ u1 = σ 2 σ u2. We will discuss the asymmetric cases later in this section, using numerical methods to give us some insights Substitute trading is not allowed Lemma 1 In the transparent market, when insiders only trade their own stocks, the insiders use the following linear strategies Pricing rules are linear X 11 (v 1 ) = σ u 1 σ 1 v 1, X 22 (v 2 ) = σ u 2 σ 2 v 2. P1 k (x 11 +x 01, x 22 +x 02 ) = (2 ρ2 )σ 1 ρσ 1 (4 ρ 2 (x 11 +x 01 ) + )σ u1 (4 ρ 2 (x 22 +x 02 ), k=a, B (19) )σ u2 P k 2 (x 11 +x 01, x 22 +x 02 ) = ρσ 2 (4 ρ 2 )σ u1 (x 11 +x 01 ) + (2 ρ2 )σ 2 (4 ρ 2 )σ u2 (x 22 +x 02 ), k=a, B (20) The expected profits for the insiders are E[π 1 ] = 2 ρ2 4 ρ 2 σ 1σ u1, E[π 2 ] = 2 ρ2 4 ρ 2 σ 2σ u2. (21) Let Σ 1 = Var(v 1 X 11 (v 1 )+x 01, X 22 (v 2 )+x 02 ) and Σ 2 = Var(v 2 X 11 (v 1 )+x 01, X 22 (v 2 )+x 02 ). Σ 1 and Σ 2 measure informativeness of prices: Σ 1 = 2 ρ2 4 ρ 2 σ2 1, Σ 2 = 2 ρ2 4 ρ 2 σ

21 v1 v 2 N x11 x11 x11 x11 I2 x12 I2 I2 x12 I2 I2 x12 I2 I2 x12 I2 x21 x21 x21 x21 x21 x21 x21 x21 N x22 N N x22 N N x22 N N x22 N N x22 N N x22 N N x22 N N x22 N x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x01 x02 A x02 A x02 x02 x02 A x02 A x02 x02 x02 A x02 A x02 x02 x02 A x02 A x02 x02 p A 1 p A 2 B B B B B B B B B B B B B B B B p B 1 p B 2 Figure 4: The game tree for the transparent market when insiders are allowed to trade their own stocks based on their private information. N stands for Nature. I1 and I2 stand for Insider 1 and Insider 2. A and B stand for market marker A and B. In the information set of Insider 1, he knows v1, but not v2. In the information set of Insider 2 he knows v2, but not v1. In the market makers information sets, (y1, y2) is a vector of constants, where y1 = x11 + x21 + x01 and y2 = x12 + x22 + x02. 21

22 The trading strategies of the insiders studied in Lemma 1 are the same as in Kyle (1985) with two independent assets in the market. What makes our equilibrium different from that in Kyle is the market makers studied in Lemma 1 infer more private information by watching order flows of the two correlated assets. Thus market prices are much more efficient and the insiders make less profits. Moreover, with given amount of noise trading in each market, the self price pressure is much higher than cross-asset price pressure in the equilibrium where the insiders only trade their own stocks. When we compare the equilibrium where the insiders only trade their own stocks to that where only substitute trading happens, we find that the insiders take advantage of their private information essentially in the same way. The market makers put more weight on the order flow of the asset they make the market when the insiders only trade their own stocks, while put more weight on the order flow of the asset they do not make the market when only substitute trading happens. Prices in both equilibria are equally efficient, because of the transparency. In substitute trading, the insiders private information is noisy. Therefore, they make considerably less profit. However, it is unrealistic to assume that the insiders do not consider substitute trading. The insiders understand that substitute trading can also be profitable. Whether the insiders should trade substitute stocks depends on the cost and benefit of such trading. The benefit of substitute trading is to sneak into other markets and get more profits since the two firms are correlated and substitute trading is one form of informed trading. However, substitute trading reveals more information about the fundamental values of the assets. This is going to cost the insiders since the prices of their own stocks will be more accurate and thus insiders make less profits from trading their own stocks. The following two subsections studies the equilibrium when the firms are symmetric and asymmetric Symmetric firms (σ 1 σ u1 = σ 2 σ u2 ) We first look at the equilibrium where σ 1 = σ 2 σ and σ u1 = σ u2 σ u, which we have analytical solutions. Proposition 6 In the transparent market, when the two firms are symmetric in the sense that σ 1 = σ 2 σ and σ u1 = σ u2 σ u, the insiders act as if they are not allowed to trade substitutes, 22

23 that is X 11 (v 1 ) = σ u σ v 1, X 12 (v 1 ) = 0, X 21 (v 2 ) = 0, X 22 (v 2 ) = σ u σ v 2, and pricing rules are P1 k (x 11 +x 01, x 22 +x 02 ) = (2 ρ2 )σ ρσ (4 ρ 2 (x 11 +x 01 ) + )σ u (4 ρ 2 (x 22 +x 02 ), )σ u P k 2 (x 11 +x 01, x 22 +x 02 ) = k = A, B ρσ (4 ρ 2 (x 11 +x 01 ) + (2 ρ2 )σ )σ u (4 ρ 2 (x 22 +x 02 ), k = A, B. )σ u If σ 1 σ u1 = σ 2 σ u2, we cannot solve the equilibrium analytically due to the complication caused by losing σ 1 = σ 2. We can easily verify that in this case, β 11 = σ u 1, β 12 = β 21 = 0 and β 22 = σ u 2 σ 1 σ 2 together with corresponding λ s satisfy the equilibrium conditions we state in Proposition 5. We cannot analytically verify whether this equilibrium is the only equilibrium in this game. We resort to GAMS 11 software to check if there exists other equilibrium. We try different parameterizations and change lowerbound and upperbound in our numerical calculations. We cannot find other equilibrium. Therefore trading only own stocks is the unique linear equilibrium in the case where firms are symmetric. Claim 1 In the transparent market, when the two firms are symmetric in the sense that σ 1 σ u1 = σ 2 σ u2, there exists a Nash equilibrium where the insiders only trade their own stocks, as in Lemma 1. In the opaque market, if the firms are symmetric in the sense that σ 1 σ u1 = σ 2 σ u2, both insiders trade both assets. While in the transparent market, symmetric insiders only trade their own stocks, even though they are given the discretion to trade substitute stocks. That is, they trade the same shares of their own stocks in the case where they can trade both stocks as in the case where they are only allowed to trade their own stocks. We see here that the cost of substitute trading more efficient prices offsets the benefit of substitute trading. Therefore, the insiders are not interested in substitute trading. 11 GAMS denotes the General Algebraic Modeling System (see Brook, Kendrick and Meeraus (1997)). This software has the nice property of specifying lowerbound and upperbound which makes it convenient to check if there are other solutions. 23

24 4.2.3 Asymmetric firms In the asymmetric case, we do not have analytical solutions. We resort to numerical methods to give us some insights. In particular, we calculate the case where the two firms are positively correlated. The analysis for the case where the two firms are negatively correlated is similar. The qualitative features of the simulations were found to be robust to a wide parameter range. It is clear that the insiders should trade their own stocks since the insiders have better information about their own stocks. They then decide whether they should trade the substitutes. That may give them more profit. For example, given Insider 1 chooses to trade his own stock, Insider 2 should calculate his expected profit to decide if trading both stocks is better than trading his own stock. Table 1 is the payoff matrix of the insiders when the insiders trade only their own stocks or both stocks. The corresponding trading patterns of the insiders are in Table 2. In the examples presented in Table 1 and Table 2, we see that with given correlation of the two firms, whether an insider should do substitute trading depends only on relative size of σσ u the camouflage the insider can use to hide his trading. As we see in Table 1 that the insider with smaller camouflage (Insider 2) always wants to do substitute trading, not depending on the strategies of the other insider (Insider 1). In the examples, when two firms are not very closely correlated, ρ = 0.7, if Insider 1 only trades stock 1, Insider 2 gets 6% more of expected profit from substitute trading; if Insider 1 trade both assets, Insider 2 gets 4% more from substitute trading. When the two firms are closely correlated, ρ = 0.9, Insider 2 gets 24% and 8% more from substitute trading respectively. However, trading both assets is not a dominant strategy for the insider with larger camouflage. In the examples, when two firms are not very closely correlated, ρ = 0.7, trading both assets is a dominant strategy for Insider 1, but when the two firms are closely correlated, ρ = 0.9, substitute trading decreases Insider 1 s expected profit if Insider 2 does not trade stock 1. Given Insider 2 always trades both assets, Insider 1 also chooses to trade both assets. Therefore in equilibrium, both insiders trade both stocks. When we look at the trading patterns of the insiders in equilibrium, we see that the insiders always trade their own stocks according to their private information. The intensity of trading own stock depends on the relative size of σ u over σ. Strikingly, the insider with larger camouflage trades substitute stock against his private information, while the other insider with smaller camouflage trades substitute stock in the same direction as his own stock. The intensity of substitute trading for Insider 2 is about twice as much as Insider 1. 24