Trading Volume and Information Distribution in a Market-clearing Framework

This paper investigates the relations between aggregate trading volume and information on nancial markets from a theoretical standpoint. Through numerical examples, it relates some statistics describing equilibrium price and volume{such as the variance of the price and its correlation with the true asset value, the volume mean, variance, skewness, and kurtosis{to the distribution of information across traders. The analysis is carried out in a static noisy rational expectations framework, with multiple informed traders, where both the precision and the correlation of the signals observed by the traders can be modi ed. Numerical examples show that the variance of the market-clearing price, and the mean and variance of the volume are increasing in the precision of the informed trader's signals and{to a lesser extent{in the liquidity shock variance. The price informativeness is increasing in the precision of the informed traders' signals and decreasing in the liquidity shock variance. Skewness and kurtosis in the trading volume distribution are not always associated with a high precision of the informed trader' signals; the relation depends on the correlation across the informed traders' signals. Introduction Trading volume has received a lot of attention from practitioners and academics alike. Technical analysts use trading volume as an indicator that \new and compelling information is owing into the market" (Saitta (1996)). Regulators have recently contemplated using trading volume as a tool to detect market manipulations. This seems to warrant further theoretical e orts to investigate the relations between volume and information. The paper's objective is to relate the patterns of aggregate volume to the underlying distribution of information across traders from a theoretical perspective. It sets out to relate some important descriptive statistics of volume, such as its mean, variance, skewness, and kurtosis{together with some price statistics{to the distribution of information among traders, i.e., the precision of their information and the degree of agreement in their valuations of the asset. The analysis is carried out in a noisy rational expectations framework with a competitive securities market and multiple informed traders. When closed-form solutions for the equilibrium price are not available, a simple numerical method is introduced to compute noisy rational expectations equilibria with a exible parameter structure. The present approach follows the seminal work of Diamond and Verrechia (1981) and the more recent papers by, among others, Grundy and McNichols (1989), and Brown and Jennings (1989). The present model features a competitive securities market with multiple informed traders and a liquidity trader. One contribution of the paper is that the signals observed by the informed traders can be both correlated and have di erent precisions. This makes it easier to study the the heterogeneity of beliefs, a key element in understanding the behavior of trading volume. In Diamond and Verrechia (1981), Brown and Jennings (1989), and He and Wang (1995), the signals observed by the informed traders are independently and identically distributed, when conditioned on the true value of the asset. 1 In Grundy and McNichols (1989), the signals are correlated across traders but have the same precision. In a di erent framework, Blume, Easley and O'Hara (1994) use signals with varying precisions. As pointed by Jain (1988), the response of trading volume to economic news may indicate the degree to which market participants disagree about the e ect of the announcement. With identically distributed signals, the heterogeneity of beliefs is ex-post, i.e., the realizations of the privately observed signals may be di erent across agents even though the signals are all independently drawn from the same distribution. In an ex-ante approach, the focus is shifted from the di erence in the realization of the signals to the di erence in their distribution. The heterogeneity of beliefs depends on the degree of disagreement among traders about the true asset value and on the di erence in the precision of the information they possessed. 1He and Wang note that \extensions to more general correlation structure, except for the correlation between the signals of di erent investors, are quite straightforward," (p. 925) [emphasis added]. 1 Analytically, the degree to which the traders agree about the value of the asset is determined by the correlation between their private signals, the quality of the traders' information is determined by the correlation of their signals with the asset value. Allowing signals to be correlated and have di erent precisions makes it easier to investigate economically relevant problems, like the e ect of a public announcement versus the e ect of private information. Public information would translate in the model as a high level for the signals' correlation. More private information could be modeled by increasing the precision of one trader's information without changing the precision of the other traders' signals. When the informed traders' signals have di erent precision, the model cannot be solved analytically and numerical solutions are used. Trade size and private information have often been related in the literature. For example, Easley and O'Hara (1987), in a market-maker setup, point out that \trade size introduces an adverse selection problem into security trading, because given that they wish to trade, informed traders prefer to trade large amounts at any given price" (p.69). In the present model, the quality of traders' information can be related to the fatness of tails, as one may conjecture that the probability of large trades would increase with the quality of the traders' information. Looking at the volume kurtosis seems a good starting point. As far as large trades would make the volume more asymmetric, the same case could be made for the skewness. Interestingly enough, the results show that the e ect of an increase in the information quality of one or more traders on the volume skewness and kurtosis is very sensitive to the the degree of agreement among the traders as measured by the coe cient of correlation between their signals. Depending on the value of this parameter, skewness and kurtosis could be low while the traders are well informed, and high when they are not. Foster and Viswanathan (1996), in a model with multiple, di erentially informed traders, also found that \the trading outcomes depend critically on the initial correlation of the informed traders' private signals" (p. 1439). However, the correlation of the informed traders' signals has little e ect on the mean and variance of the volume. The paper borrows techniques classically used by the noisy rational expectations literature, such as negative exponential utility, normally distributed random variables, and some noise trading. However, in the paper, the noise does not stem from an exogenous random asset supply, as is often the case in the literature, but from a price-sensitive liquidity demand. The rst section of the paper introduces the model. The second section presents numerical examples based on di erent assumptions about the distribution of information across traders. 1 Model The paper uses a static rational expectations framework with one liquidity trader and n informed traders, each observing a signal Gi, i = 1; : : : ; n, correlated with the true value of the asset. For each index i, the random variable Gi has mean zero and variance one, the vector (G1; : : : ; Gn; x) is 2 normally distributed. Each trader has a CARA utility function ui( i) with risk aversion coe cient i. 2 Agent i's pro t, i, is the random variable yi(x p), where yi is the demand of trader i, x the true value of the asset, and p the market price. The (n + 1)th trader is a noise trader, whose demand is yn+1 = b(p "), where b > 0, and " is uncorrelated with the true value of the asset. In the simulations, unless speci ed otherwise, var(x) equals var("), which would be the variance of the market-clearing price with no informed traders. Informed traders submit demand schedules as functions of the market-clearing price. Each informed trader chooses yi to maximize E[u( i)jGi; p]. The framework described above must be seen as a `reduced form' of a structural model, which would introduce another period where agents consume but do not trade, and a riskless asset that traders could purchase or sell to nance to desired holdings of risky securities, as in Diamond and Verrechia (1981). Let vi = E[xjGi; p] and 2 i = var[xjGi; p]. As (p;G1; : : : ; Gn) is normally distributed, informed trader i's demand is yi(p) = i(vi p), with i = 1 ivar[xjGi;p] . The agent takes into account that p is the market-clearing price, i.e., p = n1 ivi+ n+1", with = n1 i, i = i b+ , for i = 1; : : : ; n, and n+1 = b b+ (in the applications, b = 1). Note that n+1 i=1 i = 1, i is the weight of the ith trader in the determination of the market-clearing price. As a result, trader i's equilibrium valuation is vi such that vi = E[xjGi; nj=1 jvj + n+1"] (1) Equation (1) is an example of `in nite regress'. For any two agents i and j, agent i's valuation depends on agent j's valuation, in turn, agent j's valuation depends on agent i's valuation, and so forth. Besides, in (1), traders have to take into account the `noise' injected in the trading by ". Equilibrium valuations can be thought of as solutions to n simultaneous equations similar to (1), with i = 1; : : : ; n. Solutions to this system of equations may fail to exist. The limit to in nite regress is common knowledge, which, in the absence of noise, can result in an equilibrium where no trade takes place, as in Milgrom and Stockey (1982). As proposition (1) shows, if there is a normally distributed solution v to (1), then the equilibrium price is linear in the traders signals and the liquidity noise. Proposition 1 If there is a vector v satisfying (1) and v , G, x and " are jointly normally distributed, the corresponding equilibrium price p is linear in the informed traders' signals and the noise trader's liquidity shock, that is,p = a 1G1 + a 2G2 + a 3": (2) Proof of proposition 1: Suppose there is a normally distributed xed point z = (v ; G; "; x). Recall 2In the numerical examples, the risk aversion coe cient is assumed to be equal to one. 3 that p = ni=1 i v i + n+1". Consequently, z i = (Gi; p ) is a normally distributed vector, and v i = E[xjGi; p ] is linear in Gi and p . Writing v i = ciGi + di p , we get that p veri es the following equation (1 ni=1 idi)p = ni=1 i ciGi + n+1 " (3) Suppose that ni=1 i di = 1, then " and theGi would be correlated, which contradicts the assumption that the liquidity shock is uncorrelated with the signals. Hence, ni=1 i di 6= 1, and p can be written as a linear combination of the state variables. When the precision of the signal is equal across traders, the coe cient a 1, a 2, a 3 have a closed form solution. Otherwise, one has to use numerical techniques. Often in the literature (for example, Diamond and Verrechia (1981), Wang (1994)) each trader's private signal is assumed to be the sum of the true asset value and a `measurement error' term, assumed to identically and independently distributed across traders (in that sense, the signals are conditionally identically and independently distributed, when conditioned on the true asset value). In the present formulation, this boils down to having 1 = 2 = 2. 3 The `measurement error' approach doe not allow changing the precision without a ecting the correlation of the signals. 1.1 Equilibrium with two informed traders with equally precise signals 1.1.1 Rational expectation equilibrium Assume that the informed trader' signals G1 and G2 are correlated and have the same covariance with the true value of the traded asset, i.e., they have the same precision. Let = E[G1G2] and = E[xG1] = E[xG1]. Since the vector state vectors (G1; G2; x; ") and (G2; G1; x; ") have the same distribution, G1 and G2 must have the same coe cient in equation (2). We can write p = a (G1 +G2) + b " (4) The equilibrium price is also equal to an weighted average of the trader's valuation of the asset and the liquidity noise. p = (v1 + v2) + (1 2 ) " (5) where = 1 1+ 1+ 2 , and 1 = 2 = 1 var(xjG1;p) = 1 var(xjG2 ;p) , which implies that = 1 2+var(xjG1;p) = 1 3 E[v2 1 ] . The informed traders' valuations v1, v2 and the weight depend on the coe cients a and b in equation (4). The key is to nd values for a and b so that equations (4) and (5) hold 3To see this, let the private signal of trader i be gi = x+ i. The i are i.i.d with variance 2 and are uncorrelated with x. All the variables are jointly normally distributed. As E[xgi] = E[gi gj ] = 2 x, for i 6= j, and E[g2 i ] = 2 x + 2 corr(gi; x) = x p 2 x+ 2 , corr(gi; gj) = 2 x 2 x + 2 . 4 simultaneously. Trader 1's equilibrium valuation is v1 such that, v1 = E[x jG1 ; p ] = E[x jG1 ; a (G1 +G2) + b " ] = E[x jG1 ; G2 + b a " ] = E[x jG1 ; G2 G1 + b a " ] = E[x jG1 ] +E[x jG2 G1 + b a " ] = G1 + cov(x ;G2 G1 + b a ") var(G2 G1 + b a ") (G2 G1 + b a ") = G1 + (1 ) 1+ 2 +(b=a)2 (G2 G1 + b a ") (6) The fth line comes from the fact that G2 G1 and " are uncorrelated with G1 and that all the variables are normally distributed. We conclude that v1 + v2 = (G1 +G1) + (1 )2 1+ 2 +(b=a)2 (G1 +G1) + 2 (1 ) 1+ 2 +(b=a)2 " = 2 (1 ) + (b=a)2 1 2 +(b=a)2 (G1 +G2) + 2 (1 ) 1+ 2+(b=a)2 " (7) Combining equations (5) and (7), we get 1 (1 2 + (b=a)2) p = 2 (1 2) + (b=a)2 (G1 +G2) + (1 2 2(1 ) 2 + (1 2) (b=a)2 + 2 (1 ) (b=a)" (8) To compute , note that E[v2 1 ] = 2 2 (1 )+(b=a)2 1 2+(b=a)2 . Hence, b=a, the ratio of the coe cient on " to the one on G1 +G2, must be such that b=a = 2 (1 ) 2 2 (b=a)2 + 1 2 + (b=a)2 + 2 (1 ) (b=a) 2 (1 ) + (b=a)2 (9) Write = b=a, then, the equilibrium must solve the following cubic equation. '( ; ; ) = 3 (1 2) 2 + (1 ) (2 2 1) = 0 (10) For and which yield a positive de nite covariance matrix for the state vector (G1; G2; x; "), there is a unique real solution to the previous cubic equation. 4 Call a ( ; ) and b ( ; ) the resulting equilibrium coe cients in the pricing equation. When = 1, the signal of the informed traders are perfectly correlated and a and b collapse to their values in the one-informed-trader case, and 4The matrix 0B@ 1 1 1 0 1 0 0 0 0 1 1CA with 1 < < 1 and 0 < < 1 is the covariance matrix of (G1; G2; x; ") provided it is positive de nite, i.e, if and only if > 2 2 1. Once has been chosen, must be between 0 and p +1 2 . 5 p = G+(1 2) " 2 2 . 1.1.2 Naive expectation equilibrium Researchers assume sometimes, for the sake of simplicity, that the traders are \naive", that is, their valuation of the asset is E[x jGi] instead of E[x jGi; p]. 5 The equilibrium price is then p = (v1 + v2) + (1 2 ) ", with = 1 2+ var(xjG1) = 1 2+ var(xjG2) = 1 3 2 , and vi = Gi, i = 1; 2. Consequently, the equilibrium price is p = a (G1 + G2) + b ", with a = 3 2 and b = 1 2 3 2 . Those are the same coe cients as the ones obtained with two rational traders and perfectly correlated signals ( = 1). This shows that assuming that the traders are naive can lead to some serious misspeci cations. Figure (1) displays the coe cients a and b as a function of for = 0 (the solid line) and = 1 (the dotted line) obtained with rational traders, with = 1 representing also the naive traders' case. Note that when = 0, the maximum level of is 1 p2 , while when = 1, can take values up to 1. In all cases, a = 0 when = 0 and rises with . The coe cient on " is a decreasing function of in the naive case, but can be an increasing function of in the rational case, and is always greater in the rational case than in the naive case. Figure (2) displays rational equilibrium a and b as a function of , setting equal to :5, 0, :5, and 1. 1.2 Equilibrium with two informed traders when the informed traders' signals have di erent precision. This section develops an algorithm to compute a solution to the xed point problem for the informed traders' valuations (equation (1)). We checked that, when the information precision is equal across traders, the solutions obtained numerically are identical to the closed-form solutions. Equation (1) can be rewritten as an evolution equation for agent i's valuation (equation (11). v(p+1) i = E[xjGi; j (p) j v(p) j + (p) n+1"] (11) where (p) i = (p) i b+ (p) , for i = 1; : : : ; n, (p) n+1 = b b+ (p) . (p) = (p) j , (p) i = 1 i (p) i , (p) i being the conditional variance of x conditioned on agent i information in the pth run of the algorithm. (p) i = E[(x v(p) i )2] = E[x2] E[(v(p) i )2] (12) Let v(p) = (v(p) i )0i=1;:::;n, and (p) = ([ (p) i ]2)0i=1;:::;n. Given v(p), equation (11) determines v(p+1), the (p) used to compute the weights are obtained from equation (12). Moreover, if v(p) is normally distributed, so is v(p+1). Let H be the mapping de ned by v(p+1) = H(v(p)). If H has a xed 5This is done mostly in dynamic models when the rational expectation equilibrium would be intractable. 6 point, called it v , then v solves (1). To simplify computations, de ne a new state vector z(p). Let z(p) = (v(p); G; "; x), and (p) = var(z(p)). The whole state of the recursive updating of expectation is described by (p). Choose as initial value v(0) the normally distributed vector (E[xjGi])ni=1; by induction, the v(p) are normally distributed for all p's. Hence, one need only care about the (p), which evolve according to (p+1) = F ( (p)). F is a non-linear function due to the updating of the 0 is. I could not show the existence of a xed point; instead, a xed point was obtained for F using numerical techniques. However, the unicity of the equilibrium is not established. In the cases studied, the xed point matrix is nonsingular , i.e., v has a non degenerate normal distribution. One can then calculate the variance-covariance matrix of the trading volume and the true value of the asset, and compute some statistics describing the trading volume. One can verify that, without informed traders, no agent trades, which is in line with the no-trade theorems. When the informed traders' signals have the same precision, the numerical solution yields the same equilibrium covariance matrix for the state vector (and the same coe cient a and b in (4)) as the closed-form solution. To compute (11), note that, if p(k) is the market-clearing price for the kth run of the algorithm, (Gi; p(k); x) is normally distributed. Hence v(k+1) i = E[xjGi; p(k)] is the linear projection of x on (Gi; p(k)). Using the state space representation, one gets v(k+1) i = (k) i z(k). Let (k) = ( (k) i )ni=1, v(k+1) = (k)z(k). Let (k) = 0@ (k) 0 In+2 1A, 0 is a (n+ 2) n vector of zeroes and In+2 is the (n+ 2) (n+ 2) identity matrix. We get z(p+1) = (k)z(k) p+1 = (k) (k) (k)0 (13) Furthermore, at each iteration, p(k) is also linear in the state variables. The conditional variances can be recursively computed as follows. (k) i = x0 (k) x vi 0 (k) vi ; (14) where x and vi are selection matrices such that x = xz, and vi = viz. Let z0 = (G1; G2; "; x)0 and M0 = cov(z0). The elements of z0 are the fundamental parameters de ning the problem. We start the recursions with v(0) i = E[xjGi] = E[xGi]Gi, and ( (0) i ) = var(xjGi) = 2 x E[xGi]2. In the following equations, demands for the asset {as functions of the informed agents' valuations and the liquidity shock{ are derived in the case of two informed traders. Formulas can be generalized 7 to situations with more traders. y1 = 1 b+ [(b+ 2)v1 2v2 b"] y2 = 2 b+ [ 1v1 + (b+ 1)v2 b"] y3 = b b+ [ 1v1 + 2v2 "] p = b b+ [ 1v1 + 2v2 + b"] (15) Then, one can compute the variance-covariance matrix of the zero mean vector (y1; y2; y3; x; p). This matrix will be used to derive the characteristics of the trading volume. The model could be extended to study how agents with di erent private signals interpret public information di erently. Let u be a public signal observed by every trader. Agent i's equilibrium valuation vi becomes E[xjGi; u; p]. Assuming that all random variables are normally distributed, vi = i GiGi+ i uu+ i pp. The marginal e ect of the publicly released information on trader i's valuation i u depends on the correlation between x, u, Gi, and p, and may vary signi cantly across traders. Hence a rational expectation model (where by de nition agents don't disagree on the model) can generate di erential interpretation of public signals and trade. 6 2 Trading volume statistics 2.1 Preliminary studies The trading volume is de ned to be the sum of the positive parts of the traders' demands, i.e., z = n+1 i=1 y+ i , where yi is the demand (negative or positive) of agent i for the asset, and y+ i = I[yi > 0] yi, with I being the indicator function. As z is the sum of the truncated demand yi's, the statistics characterizing z depend not only on the characteristics of every individual yi, but also on the crosscorrelations, which depends in turn in very non-linear ways on the cross-correlations of the informed traders' signals, that is, on how much they agree. The traders' demands are cross-correlated even if the original signals are not for two reasons. First each individual equilibrium valuation depend on all the other valuations, this might induce correlation across them. Second, the demands depend on price, which is a weighted average of traders' valuations. Besides, as proposition (2) shows, the demand of a well-informed trader will have the same coe cient of variation, skewness, and kurtosis as the demand of a non-informed one. This is because these measures are normalized. 7 Proposition 2 The coe cient of variation, skewness and kurtosis of the each informed trader's demand are independent of the precision of the informed trader's signal and the correlation between 6A di erent approach is taken by Harris and Raviv (1993), and Kandel and Pearson (1995). 7Recall that, for a random variable !, the coe cient of variation is c(!) = ! E[!] , the skewness is s(!) = E[(! E[!])3] var(!) 32 , and the kurtosis is k(!) = E[(! E[!])4] var(!)2 . 8 these signals, and are equal to the coe cient of variation, skewness and kurtosis of a standard normal random variable truncated at zero. The proof is in the appendix. The dependence of the trading volume's normalized statistics on the correlations between the demand is made clear in the case of one liquidity trader and two informed traders, as shown in proposition (3). Proposition 3 With two informed traders and one liquidity trader the coe cient of variation, the skewness and the kurtosis of the aggregate volume depend only on the correlation between the demands of the two informed traders, and the ratio of the variances of the demands The proof is in the appendix. An important consequence of proposition (3) is that, if the informed traders have identically precise information (i.e., 1 = 2), the coe cient of variation, the skewness and the kurtosis of the aggregate trading volume depend only on the how they agree, i.e., on the correlation between the two initial signals. Proposition (3) also shows that the coe cient of variation, the skewness or the kurtosis cannot be used to detect di erences between the overall information quality of two markets when the quality of the private information is equal across traders within each market. Those statistics will be the same in a market with identically poorly informed traders and in one with identically well informed traders. With the coe cient of variation, skewness and the kurtosis, in a model with two informed traders and one liquidity trader, are plotted on Figure (3) as functions of the ratio of the variance of the informed traders and the correlation between their demands. Without loss of generality, the demand variance ratio can be constrained to be between zero and one 8 When one trader has no information, the demand variance ratio is zero, when both traders are equally well informed, the ratio is one. When the ratio decreases, the coe cient of variation tends to go up, but the skewness and the kurtosis go up or down, depending on the correlation between the demand. Besides, for any given value of the variance ratio, the e ect of increasing the correlation between the traders' demands is not monotonic. Ultimately, the non-monotonicities in the relation between demands' characteristics (variance ratio and correlation) and volume characteristics are compounded by the complexity of the relations between the characteristics of the traders' information and those of their demands. Still, the graphs reveal some characteristics of volume which are independent of the demand covariance structure. Aggregate volume exhibits positive skewness and excess kurtosis (i.e., kurtosis superior to 3, the value for a normally distributed variable). Its mean is superior 8If the ratio were superior to one, one would relabel the two agents and invert the ratio. 9 to its median and the tail of its distribution is fatter than for a normal distribution. In sum, its distribution is dominated by rare occurrence of large surges in volume. 2.2 Numerical examples The fundamental parameters are: the correlations between the signals and true asset value ( 1 and 2), the correlation between these signals ( ), and the variance of the liquidity shock ( 2 "). The nal objects of interest are the following characteristics of the equilibrium price and aggregate volume: the variance of the price, its correlation with the asset value (its `informativeness'), the mean and variance of the aggregate volume, its coe cient of variation, skewness and kurtosis. By construction, the mean price is zero. The relation between the fundamentals and the nal object of analysis are complex, and few conclusions can be derived analytically. Therefore, I resort to simulations, varying some parameters and trying to make sense of the e ect on the price and the volume. 9 The informed traders' signals interact with the liquidity shock at two stages: the formation of the equilibrium valuations, and the formation of the equilibrium volume. This complex mixing of the traders' information may explain the non-linearities in the relations between the fundamentals on one side, and the price and the aggregate volume on the other. One result that can be derived analytically is that, with a liquidity trader and a single informed trader, the equilibrium price variance, its informativeness, the mean and variance of the equilibrium volume are increasing in the precision of the informed trader's signal. As shown below, these results are still valid with multiple informed traders. I construct four experiments. As the e ect of varying one parameter might depend on the value of all the others, for each experiment, I vary two parameters, and produce a three-dimensional graph and a contour graph, where a darker shade indicates lower values for the variable. In the rst experiment, the precision of the signal is identical across traders ( 1 = 2 = ), I vary the common level of the precision and the correlation of the signals ( ). In the second experiment, the precision of the second signal is maintained constant at 0:5, while I vary the precision of the rst signal and the correlation between the two signals. In the third experiment, the correlation among signals is set equal to zero, and I vary the precision of each signal independently. In the fourth experiment, the precision of the signal is identical across traders, I vary the common level of the precision and the variance of the liquidity shock. In the following, I describe the main features of the graphs and try to give some intuitive explanations for them. Furthermore, results are summarized in table (2) For each experiment, I estimate the semi-elasticities of some observable market statistics{the price variance, the volume mean and variance{with respect to the parameters 9To better understand the simulations' results, recall that if ! is a normally distributed random variable, and !+ is the positive truncation of !, the approximate values for var(!+), c(!+), s(!+), and k(!+) are 0:34, 1:46, 1:64, and 5:41. 10 used in the experiment. The semi-elasticities are obtained by regressing the logarithms of the market statistics on a constant and the levels of the parameters used in the experiments. In the rst experiment, I vary the correlation between the signals and the common value of their precision. The graphs are displayed in Figures (4) to (7). Whatever the correlation between the signals is, increasing their common precision tends to raise the price variance, its informativeness, together with the volume mean and variance, while lowering the coe cient of variation. The e ect on the skewness and kurtosis is less clear, as it depends on the correlation between the signals. To shed some light on the patterns for price, recall that the market-clearing price is a weighted sum of the valuations and the noise: p = 1v1 + 2v2 + (1 1 2)", with i = i 1+ 2+b , and i = 1 var(xjGi;p) , for i = 1; 2. The more precise the original signals are, the higher are the variance of the valuations, their correlation, and their weight i. Hence, more precise signals generate a more volatile and informative market price. As for the patterns for volume, with more precise information, each trader's demand becomes more volatile, resulting in a higher and more volatile aggregate volume. The precision of the price tends to go down as the correlation between signals increases when :5 (for lower values of , the price informativeness as a function of tends to have a U-shape that is more di cult to explain). That the price might be less informative when the signals are more correlated is quite intuitive since the market price is a noisy aggregate of the traders' information; a more diverse pool of private signals (a low correlation between signals) will generate a more informative equilibrium price. This intuition is even clearer for a totally revealing price, i.e., for p = E[xjG1; G2]; as the correlation coe cient between G1 and G2 gets smaller, the pair (G1; G2) spans more of the space of the possible realizations of x and makes the resulting equilibrium price p more informative. Of course, in our case, the market-clearing price is a noisy aggregate of the signals, but the intuition should hold. The variance of the volume is increasing with the correlation of the signals, while the mean volume is decreasing in this parameter. The e ect of increasing the correlation of the signals on the coe cient of variation, the skewness and the kurtosis of the volume depends on the level of the information precision. The last three statistics attain their minimum when the informed traders' signals are very precise and not correlated. This shows that informed trading does not always translate into fat tails in the distribution of volume. In the second experiment, the precision of the second signal is maintained constant at 0:5, while I vary the precision of the rst signal and the correlation between signals. Graphs are displayed in Figures (8) and (11). Whatever the correlation between signals is, increasing the precision of one signal raises the variance of the price and its correlation with the true value of the asset. The mean and variance of the volume tend also to increase. The coe cient of variation, 11 the skewness and the kurtosis linger inside a narrow band. As in case 1, they are the lowest when the precision of the signals is high and their correlation low. In the third experiment, I increase the quality of the two traders' signals independently, while maintaining at zero the correlation between signals. Graphs are displayed in Figures (12) and (15). Whatever the precision of one trader's signal is, increasing the precision of the other trader's signal increases the variance of the price, its correlation with the true value of the asset, and the volume mean and variance. The coe cient of variation, the skewness and the kurtosis display non-linear patterns while varying little. Notice that those variables attain a valley when the precision of both signals precision are the highest. In the fourth experiment, I vary the variance of the liquidity shock and the common information precision of the traders. The correlation of their signals is kept xed at zero. Graphs are displayed in Figures (16) and (19). Recall that, the way the liquidity demand has been constructed, without informed traders, the variance of the market-clearing price would be equal to the variance of the liquidity shock 2 " . From the onset, the distribution of the true value of the asset was set equal to the distribution of the market-clearing price with no informed trading. As the variance of the true value of the asset is xed to unity, 1 represents the \reference" level for the liquidity shock variance, which gives an idea of the range of 2 " used in the current experiment. Increasing the liquidity variance raises the mean and variance of the volume while increasing the variance of the price and decreasing its informativeness. Notice that the e ect of a higher liquidity shock variance on the observable characteristics of price and volume are qualitatively the same as for an increase in the traders' information precision, while the e ect on the price informativeness is ipped around. An increase in the informativeness of the private signals may be distinguished from an increase in the liquidity variance via the size of the e ect on the observable characteristics of price and volume. Making the liquidity noise more volatile increase the variance of the volume much more than its mean, whereas increasing the precision of the informed traders' signals has roughly the same e ect on both variables. However, in the current example the slope of the liquidity demand is set equal to one, results could be sensitive to this parameter. 12 3 Conclusion The paper relates patterns of the market price and the trading volume to the distribution of information across traders. Through numerical examples, it is shown that, when traders have better information, the mean and variance of the trading volume and the variance of the market price tend to be higher, and that the market price naturally becomes more informative. The e ect of an increase in the liquidity shock variance is similar (though smaller), except of course for the price informativeness. Varying the correlation between the informed traders' signals has a smaller e ect than varying the information quality, except for the coe cient of variation, skewness and kurtosis, which are very sensitive to the correlation across the informed traders' signals. In particular, fat tails for volume (i.e., high kurtosis) do not indicate that the traders' information about the asset value is highly precise. Increasing the correlation across the informed traders' signals (so that the traders agree more with each other) lowers the volume mean and the price variance, while increasing the volume variance. The di erence in the impact of varying the correlation between the informed traders' signals and the true asset value (the precision of their information), and varying the correlation across their signals (the degree of consensus) on the observable price and volume statistics could help us distinguish between public and private information, since public information would increase the degree of consensus among traders while making them more informed, whereas private information would a ect primarily the precision of one or more informed trader's signals without increasing the degree of consensus amomg traders. 13 APPENDIX A Characteristics of trading volume Proposition 4 Let x a standard normal random variable and x+ = I[x > 0]x. Let Mx(t) be the moment generating function of x+. Then, Mx(t) = 1 2 + exp(1 2 t2) (t). where is the standard normal cdf. Consequently, E[x+] = 1 p2 ; E[(x+)2] = 12 E[(x+)3] = 2 p2 ; E[(x+)4] = 32 (16) It follows that c(x+) = c = p 1, s(x+) = s = 1 p2 ( 1 + 12 ) ( 12 1 2 )3=2 , and k(x+) = k = 1 2 (3 5 3 2 2 ) ( 1 2 1 2 )2 . Proof of proposition 4: By de nition, Mx(t) = E[etx+ ] = p(x 0) +E[ I[x > 0]ext]. E[ I[x > 0]ext] = Z +1 x=0 ext (x)dx = Z +1 x=0 1 p2 exte 12x2dx = Z +1 x=0 1 p2 e 1 2 [(x t)2 t2]dx = e 1 2 t2 Z t u= 1 1 p2 e 12u2du (17) Results in equation (16) can be obtained by taking derivatives of the moment generating function, then proposition (4) follows straightforwardly. Proof of proposition 2: Write yi the demand of the ith informed trader, ~ yi = 1 i yi, where 2 i = var(yi). Proposition (2) follows from the fact that ~ yi is a standard normal, and proposition (4).Proposition (2) shows that the characteristics of the traded volume will be in fact determined by the interactions between the individual demand schedules, in particular their covariances. Indeed, a sum of independently and identically distributed random variables have the same coe cient of variation, skewness and kurtosis as each random variable in the sum. Now, we use the fact that y1 + y2 + y3 = 0. Let ij = cov(yi; yj), = 12, 2 i = var(yi), and ij = cov(yi; yj), s = 1 2 . Using the fact that y1 + y2 + y3 = 0, we get 2 3 = 2 1 + 2 2 + 2 12 = 1 2(s+ 1s + 2 ), and after some manipulations 1 3 = 1 2ps2 + 2 s+ 1 2 3 = 1 2q 1 s2 + 2 1s + 1 1 3 = q s s+ 1s+ 1 3 = r 1s s+ 1s+ 13 = s+ ps2+2 s+1 23 = 1s+ q 1 s2+2 1s+1 (18) Proof of proposition 3: We begin by the skewness and the kurtosis. Let ~ yi = 1 i yi, zi = y+ i E[y+ i ], ~ zi = ~ y+ i E[~ y+ i ], and = E[z], then, z = z1 + z2 + z3. E[(z )n] is a weighted 14 sum of E[za1 1 za2 2 za3 3 ], with a1 + a2 + a3 = n, while ai is an integer in [0; n]. E[za1 1 za2 2 za3 3 ] = a1 1 a2 2 a3 3 E[~ za1 1 ~ za2 2 ~ za3 3 ] = a1 1 a2 2 a3 3 g( ; s) = ( 1 3 )a1( 2 3 )a2 n 3 g( ; s) = [p 1 2]nf(s; ) (19) The second line comes from the fact that cross moments of normal truncated variables are functions of the covariance matrix of the underlying normal variables. More precisely ~ y3 = 1 3 ~ y1 2 3 ~ y2. 1 3 , and 2 3 depend on , and s, and (~ y1; ~ y2) is completely determined by rho. Hence E[za1 1 za2 2 za3 3 ] is a function of only two parameters: , and s. In the third line, we use the fact that a1 + a2 + a3 = n. The fourth line was obtained using (18). Consequently, E[(z )n] = [p 1 2]n (s; ). The 1 2 terms cancel when we compute skewness and kurtosis. As one can always permute the indices of the two informed traders, we need only s between zero and one, in other words, if c( ; s) is the value of the coe cient of variation, c( ; s) = c( ; 1s ). The same remark applies for the skewness and the kurtosis. B Computation of volume's statistics Table (1) gives the de nition of z in terms of y1, y2, and y3 in the six di erent cases. z = 6i=1I[vi 1 > 0; vi 2 > 0](vi 1 + vi 2), with (v1 1 ; v1 2) = (y1; y2), (v2 1 ; v2 2) = ( y2; y1 + y2), (v3 1 ; v3 2) = (y1; (y1 + y2)), (v4 1 ; v4 2) = ( y1; y1 + y2), (v5 1 ; v5 2) = (y2; (y1 + y2)), (v6 1 ; v6 2) = ( y1; y2). Hence, to compute the various moments of z, one need only calculate E[I[vi 1 > 0; vi 2 > 0](vi 1 + vi 3)], for i = 1; : : : ; 4. B.1 Computing E[I[v1 > 0; v2 > 0]. Proposition 5 E[I(v1 > 0)I(v2 > 0)(v1 + v2)] = v 1 + v 2 2p2 (1 + v12) (20) proof: Let f be the density function of the normally distributed vector (v1; v2)0 with mean zero and variance . Note = ( ij)2i;j=1, and 1 = ( ij)2i;j=1. f(v) = 1 2 det( ) 12 e 12v0 1v Let rf = (f1(v1; v2); f2(v1; v2))0 = 1vf(v), which implies the following. ( f1(v1; v2) = f 11v1 + 12v2gf(v1; v2) f2(v1; v2) = f 21v1 + 22v2gf(v1; v2) (21) For i = 1; 2, we have E[I(v1 < 0; v2 < 0) vi] = Z 0 1 Z 0 1 vi f(v1; v2)dv1 dv2. 15 Note I = I[v1 > 0; v2 > 0], 11E[I y1] + 12E[I y2] = Z 0 1 Z 0 1[ 11v1 + 12v2] f(v1; v2) dv1 dv2 = Z v2=0 v2= 1[f(v1; v2)]v1=0 v1= 1 dv2 = Z 0 1 1 2 pj je 1 2 22t2dt = 1 p2 j j 22 Z 0 1 1 p2 e 1 2u2du = 1 p2 j j 22 (0) = 1 2 1p2 = c1 (22) In the fourth line, the change of variable u = p 22 t was used. For the last but one equation, by de nition, 22 = 11 j j , and 1 = p 11. In the same fashion 11E[I y1] + 12E[I y2] = 1 2 1 2p2 = c2 (23) Equations (22), and (23) can be rewritten as 1(E[I y1]; E[I y2])0 = c, with c = (c1; c2)0. Hence, (E[I y1]; E[I y2])0 = c and E[I(v1 < 0; v2 < 0)vi] = i 2p2 [1 + 12]; (24) where 12 is the correlation coe cient between v1 and v2. Also, p(v1 < 0; v2 < 0) = 1 4 + 1 2 arcsin( 12). B.2 Computing E[I[v1 > 0; v2 > 0](v1 + v2)2] Let G1(v1; v2) = v1f(v1; v2), where f is the density function of v. Let G1i = @G1 @vi , i = 1; 2. Then G11 = v1f1 + f , and G12 = v1f2. Hence, ( 11v2 1 + 12v1v2)f = v1f1 = f G11 ( 12v2 1 + 22v1v2)f = v1f2 = G12 (25) We will write v = (v1; v2), I = I[v1 > 0; v2 > 0], and p(v > 0) = p(v1 > 0; v2 > 0). Noting that, for any value of v2, Z 1 v1=0G11(v1; v2)dv1 = [G1(v1; v2)]1v1=0 = 0, we get 11E[I v2 1 ] + 12E[I v1v2] = p(v > 0): (26) 16 Also, 12E[I v2 1 ] + 22E[I v1v2] = Z 1 v1=0 Z 1 v2=0 @ @v2G1(v1; v2)dv1dv2 = Z 1 v1=0[G1(v1; v2)]1v2=0dv1 = 1 2 pj j Z 1 v1=0 v1e 1 2 11v2 1dv1 = 1 2 11pj j Z 1 u=0 ue 12u2du = 1 2 11pj j = pj j 2 22 (27) Note that 1 11pj j = pj j 22 , as, by de nition 11 = 22 j j . Hence, E[I v2 1 ] E[I v1v2] E[I v1v2] E[I v2 2 ] ! = 0@ p(v > 0) pj j 2 11 pj j 2 22 p(v > 0) 1A (28) We can simplify notations, using j j = 2 1 2 2 2 12, which yields pj j 22 = 1 2q1 212. We conclude that E[I[v1 > 0; v2 > 0]v2 1 ] = 2 1f1 4 + 1 2 arcsin( 12) + 1 2 12q1 212g E[I[v1 > 0; v2 > 0]v1v2] = 12(1 4 + 1 2 arcsin( 12)) + 1 2 1 2q1 212 (29) B.3 Computing E[I[v1 > 0; v2 > 0](v1 + v2)3] We compute E[I v3 1], E[I v2 1v2], E[I v1v2 2 ], E[I v3 2 ]. Let G1(v1; v2) = v2 1f(v1; v2), then G11(v) = 2v1f(v) + v2 1f1(v), G12(v) = v2 1f2(v). 11v3 1 + 12v2 1v2 = v2 1f1(v1; v2) = 2v1f(v) G11(v) 12v3 1 + 22v2 1v2 = v2 1f2(v1; v2) = G12(v) (30) Noting that, for any value of v2, Z 1 v1=0G11(v1; v2)dv1 = 0, we get 11E[I v3 1 ] + 12E[I v2 1v2] = 2E[I v1]. Also, 12E[I v3 1 ] + 22E[I v2 1v2] = Z 1 v1=0 Z 1 v2=0G12(v1; v2)dv1dv2 = Z 1 v1=0[G1(v1; v2)]1v2=0dv1 = 1 2 pj j Z 1 v1=0 v2 1e 12 11v2 1dv1 = 1 2 pj j( 11) 3 2 Z 1 0 u2e 12u2dv1 = 1 2p2 j j ( 22) 32 (31) 17 We used the change of variable u = p 11v1, the identity 11 = 22 j j , and the fact that Z 1 0 u2e 12udu = p2 2 . Hence, E[I v3 1 ] E[I v1v2 2 ] E[I v2 1v2]; E[v3 2 ] ! = 0B@ 2E[I v1] j j 2p2 ( 11) 3 2 j j 2p2 ( 22) 3 2 2E[I v2] 1CA (32) B.4 Computing E[I[v1 > 0; v2 > 0](v1 + v2)4] We compute E[I v4 1], E[I v3 1v2], E[I v2 1v2 2 ], etc. Let G1(v1; v2) = v3 1f(v1; v2). G11(v) = 3v2 1f(v) + v3 1f1(v), G12(v) = v3 1f2(v). 11v4 1 + 12v3 1v2 = v3 1f1(v1; v2) = 3v2 1f(v) G11(v) 12v4 1 + 22v3 1v2 = v3 1f2(v1; v2) = G12(v) (33) It is easy to see that 11E[I v4 1 ] + 12E[I v3 1v2] = 3E[I v2 1 ]. Also, 12E[I v4 1 ] + 22E[I v3 1v2] = Z 1 v1=0 Z 1 v2=0G12(v1; v2)dv1dv2 = Z 1 v1=0[G1(v1; v2)]1v2=0dv1 = 1 2 ;pj j Z 1 v1=0 v3 1e 1 2 11v2 1dv1 = 1 2 pj j( 11)2 Z 1 0 u3e 1 2u2dv1 = 1 j j 3 2 ( 22)2 (34) Hence, E[I v4 1] E[I v1v3 2 ] E[I v3 1v2] E[I v4 2 ] ! = 0B@ 3E[I v2 1 ]; j j 3 2 ( 11)2 j j 3 2 ( 22)2 ; 3E[I v2 2 ] 1CA (35) To compute E[I v2 1v2 2 ], let G1(v1; v2) = v1v2 2f(v1; v2). Then, G11(v) = v2 2f(v) + v1v2 2f1(v), G12(v) = v1v2 2f2(v) + 2v1v2f(v). 11v2 1v2 2 + 12v1v3 2 = v1v2 2f1(v1; v2) = v2 2f(v) G11(v) 12v2 1v2 2 + 22v1v3 1 = v1v2 2f2(v1; v2) = 2v1v2f(v) G12(v) (36) For any value of v2, Z 1 v1=0G11(v1; v2)dv1 = 0, and for any v1, Z 1 v2=0G12(v1; v2)dv2 = 0, hence, 11E[I v2 1v2 2 ] + 12 E[I v1v3 2 ] = E[I v2 2] 12E[I v2 1v2 2 ] + 22E[I v1v2 2] = 2E[I v1v2] (37) Finally, E[I v2 1v2 2 ] E[I v3 1v2] E[I v1v3 2 ] E[I v2 1v2 2 ] ! = E[I v2 2 ] 2E[I v1v2] 2E[I v1v2] E[I v2 1] ! (38) 18 REFERENCES Barclay, Michael, Robert H. Litzenberger, and Jerold Warner (1990), \Private Information, Trading Volume, and Stock-Return Variances", The Review of Financial Studies, Vol 3, No 2, p. 233-253. Blume Lawrence, David Easley, and Maureen O'Hara (1994), \Market Statistics and Technical Analysis: The Role of Volume", The Journal of Finance, Vol XLIX, No 1, March 1994. Brown, David, and Robert H. 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Shalen, Catherine (1993), \Volume, Volatility, and the Dispersion of Beliefs", The Review of Financial Studies, Vol 6, no 2, p 405-434. Wang, Jiang (1992), \A Model of Intertemporal Asset Prices under Asymmetric Information", The Review of Financial Studies, p 249-281. Wang, Jiang (1994), \A Model of Competitive Asset Trading Volume", Journal of Political Economy, vol 102, No 1, p 127-168. 20 cases i sign of y1 sign of y2 sign of y1 + y2 volume z vi 1 vi 2 1 + + + y1 + y2 y1 y2 2 + + y1 y2 y1 + y2 3 + y2 y1 (y1 + y2) 4 + + y2 y1 y1 + y2 5 + y1 y2 (y1 + y2) 6 + (y1 + y2) y1 y2 Table 1: Decomposition of trading volume There are three traders; y1 is the rst trader's demand, y2 is the second trader's demand, the last trader's demand is y3 = (y1 + y2). The trading volume is z = y+ 1 + y+ 2 + y+ 3 . In each case i, i = 1; : : : ; 6, z can be written as vi 1 + vi 2. 21 volume mean volume variance price variance case 1 0.69 0.57 2.00 ( 1 = 2 = ) -0.07 0.24 -0.22 case 2 1 0.49 0.70 1.10 ( 2 = 0:5) -0.09 0.20 -0.19 case 3 1 0.46 0.37 1.30 ( = 0) 2 0.46 0.37 1.30 case 4 1.90 3.10 3.80 ( 1 = 2 = , = 0) 2 " 0.40 0.87 0.33 Table 2: Semi-elasticities of some observable price and volume statistics with respect to underlying parameters 1 is the correlation between G1, the rst trader's signal, and x, the true value of the asset, 2 is the correlation between G2, the second trader's signal, and x, is the correlation coe cient between the signals G1 and G2, 2 " is the variance of the liquidity shock. 22 0.2 0.4 0.6 0.8 1 rho 0.1 0.2 0.3 0.4 0.5 Figure 1: p = a1G1 + a2G2 + a3 ", with E[xG1] = E[xG2] and in naive traders case. 23 0.2 0.4 0.6 0.8 1 rho 0.1 0.2 0.3 0.4 0.5 Figure 2: p = a1G1 + a2G2 + a3 " for = :5 (the upper dotted lines), 0 (the solid lines), :5 (the middle dashed lines) and 1 (the lower dashed lines). 24 kurtosis 0 0.2 0.4 0.6 0.8 1-1 -0.5 0 0.5 1 3.5 3.75 4 4.25 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 kurtosis skewness 0 0.2 0.4 0.6 0.8 1-1 -0.5 0 0.5 1 0.7 0.8 0.9 1 1.1 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 skewness coeff of variation 0 0.2 0.4 0.6 0.8 1-1 -0.5 0 0.5 1 0.55 0.6 0.65 0.7 0.75 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 coeff of variation Figure 3: Statistics as functions of the traders' demands' variance ratio (from 0 to 1) and correlation (from 1 to 1). 25 price variance 0 0.7 rho 0 1 gamma 0 0.7 price informativeness 0 0.7 rho 0 1 gamma 0 0.8 volume mean 0 0.7 rho 0 1 gamma 1 volume variance 0 0.7 rho 0 1 gamma 0.2 Figure 4: 1 = 2 = , and vary. 26 volume kurtosis 0 0.7 rho 0 1 gamma 3.8 volume skewness 0 0.7 rho 0 1 gamma 0.9 volume coff var 0 0.7 rho 0 1 gamma 0.7 Figure 5: 1 = 2 = , and vary. 27 0 0.7 0 1 price variance 0 0.7 0 1 price informativeness 0 0.7 0 1 volume mean 0 0.7 0 1 volume variance Figure 6: 1 = 2 = , and vary. 28 0 0.7 0 1 volume kurtosis 0 0.7 0 1 volume skewness 0 0.7 0 1 volume coff var Figure 7: 1 = 2 = , and vary. 29