Analysts’ sale and distribution of non fundamental information

Abstract

We examine an analyst’s sale and distribution of information related to short-term price movements but unrelated to underlying firm value. By selling non fundamental information, the analyst increases competition on the signal, but prices become more sensitive to net order flow, creating an offsetting increase in the non fundamental signal’s value. More precise non fundamental information is more widely distributed. In the limit, a perfect non fundamental signal will be publicly disclosed for an arbitrarily small fee, and the analyst earns profits as if he possessed fundamental information. Consistent with empirical findings, analysts’ recommendations can be profitable, even when widely distributed or seemingly inconsistent with detailed forecasts. Analysis based on non fundamental information does not contribute to greater price efficiency but reduces liquidity costs. In a multi-period setting, traders with non fundamental information do not front-run, preferring to transact only in the period in which uninformed demand is executed.

Appendix

Proof of Lemma 1

Suppose for constants β _m , γ _m and λ _m , X, Z, and P are given by

$$ X_{m}=\beta_{m} s \,\,\,Z_{m}=\gamma_{m} \tau \,\,\, P_{m} =\lambda_{m} y. $$

Given the linear rules, the proprietary trader and a (representative) demand-based trader’s profits can be written as

$$ \begin{aligned} E\{(v-P_{m})x_{m}|s,m, x_{m}\} &= E(x_{m} (v-\lambda_{m}(x_{m}+m\gamma_{m} \tau +u))|s,m,x_{m})\\ &= x_{m}(\Uppsi s-\lambda_{m}x_{m})\\ E\{(v -P_{m})z_{m}|\tau,m, z_{m}\} &= E(z_{m}(v-\lambda_{m}(\beta_{m} s + z_{m} + (m-1)\bar{\gamma}_{m}\tau+u))|\tau,m,z_{m})\\ &= -z_{m}\lambda_{m}(z_{m}+(m-1)\bar{\gamma}_{m}\tau+\theta \tau), \end{aligned} $$

where $\bar{\gamma}$ is the expected trading strategy of each of the other m − 1 demand-based traders. Taking first order conditions and replacing $\bar{\gamma}$ with γ (since all demand-based traders are identical) gives

$$\beta_{m}=\frac{\Uppsi}{2\lambda_{m}} \,\,\, \hbox{and}\,\,\,\gamma_{m}=-\frac{\theta}{m+1}. $$

(7)

The market efficiency condition P _m  = λ _m y = E{v|y} implies

$$ \lambda_{m} =\frac{\beta_{m} \Upsigma}{\beta_{m}^2 \Upsigma/\Uppsi+\sigma_u^2\left(\frac{m^2 \gamma_{m} ^2}{\theta }+2 m \gamma_{m}+1\right)}. $$

(8)

Solving (7) and (8) subject to the second order condition λ _m  > 0 gives the equilibrium. $\square$

Proof of Proposition 1

• (i) The expected profit of an individual demand-based trader in a market with a total of m demand-based traders, $\Uppi^{D}_{i,m}$ is

  $$\Uppi^{D}_{i,m}=\frac{\gamma_{m} \left(-\gamma_{m} \lambda-\left(\frac{(m-1) \gamma_{m} }{\theta }+1\right) \theta \lambda\right) \sigma_{u}^2}{\theta}. $$

  (9)

  Replacing the expression of γ _m and simplifying yields

  $$\Uppi^{D}_{i,m}=\frac{\theta \lambda \sigma_{u}^2}{(m+1)^2}, $$

  (10)

  making a demand-based trader indifferent between buying and not buying information at

  $$ c=\frac{\theta \lambda\sigma_{u}^2}{(m+1)^2}. $$

  (11)

  Replacing λ with λ _m , we get

  $$ c_{m}=\frac{\theta}{2(m+1)} \left(\frac{\Upsigma \Uppsi \sigma_u^2}{m (m+2) (1-\theta)+1}\right)^{\frac{1}{2}}. $$

  (12)

  The analyst’s profits are equal to $\Uppi^{A}=m c_{m}$. The analyst chooses the optimal number of demand-based traders to maximize profits. Taking the FOC of $\Uppi^{A}$ in m:

  $$ \theta \sqrt{\Upsigma\Uppsi \sigma_{u}^{2}} \left(\frac{\left(m \left(1-m-m^2\right)(1-\theta )+1\right) }{2 (m+1)^2 (m (m+2) (1-\theta)+1)^{3/2}}\right)=0. $$

  (13)

  Therefore m* and c* are implicitly defined by the following equations

  $$ 0 = m^{*}\left(1-m^{*}-m^{*2}\right) (1-\theta )+1 $$

  (14)
  $$ c^{*} = \frac{\theta}{2(m^{*}+1)^2} \left(\frac{\Upsigma \Uppsi\sigma_u^2}{r_{m^{*}}}\right)^{\frac{1}{2}}. $$

  (15)

• (ii) Write (14) as $\,\Upgamma(\theta,m^{*})=0\,$ where

  $$ \Upgamma(\theta, m)=m\left(1-m-m^{2}\right) (1-\theta )+1. $$

  (16)

  $\forall \, m \geq 0,\Upgamma(\theta,m)$ is a third degree polynomial, increasing in $m\in[0,1/3]$, taking values in [1, 1 + (5/27) (1 − θ)] and decreasing in $m\in(1/3,+\infty)$, taking values in $(-\infty,1+(5/27)(1-\theta)]$. Therefore, there exists a unique real solution to $\Upgamma(\theta,m^{*})=0$, which can be written as $m^{*}= \Updelta(\theta)$. Using the implicit function theorem,

  $$ \begin{aligned}\Updelta'(\theta) &=-\frac{\partial\Upgamma}{\partial\theta}(\theta,m)/\frac{\partial\Upgamma}{\partial{m}}(\theta, m)\\ &= \frac{m\left(1-m-m^2\right)}{\left(1-2m-3 m^2\right) (1-\theta)}.\end{aligned} $$

  (17)

  By (14), $\left(1-m-m^2\right)<0$ and $\left(1-2m-3m^2\right)<0$. Thus $\Updelta'(\theta) >0$, and m* is increasing in θ. As $\theta\rightarrow0,\,m^{*}\rightarrow{\left(1-m-m^2\right)}/{\left(1-2m-3m^2\right)}$ and therefore $m^{*} > 1 \,\forall\,\theta \in (0,1)$.

Taking the second order condition, solving (14) for θ, and substituting gives

$$ -\frac{\sqrt{\Upsigma\Uppsi \sigma_{u}^{2}}(1-m^{*})(1-3m^{*})}{2 m^{*2} (1+m^{*})^{2} \sqrt{-(1-m^{*}-m^{*2})}} \leq 0 $$

for all m* ≥ 1. Since m* is unique, greater than 1, and the second order condition (SOC) is negative at m*, m* is a global maximum. $\square$

Proof of Corollary 1

• (i) $\lim_{\theta\rightarrow 1}\Uppi^{A}=\frac{m}{2(m+1)}\left(\Upsigma \Uppsi\sigma_{u}^{2}\right)^{\frac{1}{2}}$ which is increasing in m and thus maximized at $m^*\rightarrow \infty$. As m* is increasing in θ, m* is increasing in $1/\sigma^2_{\epsilon}$ and σ ² _u .

• (ii) $\Uppsi$ is increasing in 1/σ². Proprietary trading profits are

  $$ \Uppi^{P*}=\sqrt{\Upsigma\Uppsi \sigma_{u}^{2}} \left(\frac{ \sqrt{(1+m^{*} (m^{*}+2)(1-\theta)) }}{2 (m^{*}+1)}\right), $$

  (18)

  and the analyst’s profits, at m* are

  $$ \Uppi^{A*}=\sqrt{\Upsigma \Uppsi \sigma^2_u }\left(\frac{m^{*} \theta }{2(m^{*}+1)\sqrt{1+ m^{*} (m^{*}+2) (1-\theta))}}\right). $$

  (19)

  Since m* is independent of $\Uppsi$ and $\Upsigma$, it follows immediately that $\Uppi^{P*}$ and $\Uppi^{A*}$ are increasing in $\Upsigma$ and 1/σ².

• (iii) Differentiating (18) with respect to θ gives

  $$ -\sqrt{\Upsigma \Uppsi\sigma_{u}^{2}}\left(\frac{ 2 \theta \Updelta^{'}(\theta )+m^{*}(m^{*}+1) (m^{*}+2)}{4 (m^{*}+1)^2 \sqrt{1+m^{*}(m^{*}+2)(1-\theta)}}\right)\leq 0, $$

  (20)

  and since $\Updelta^{'}(\theta )>0$, and thus $\Uppi^{P*}$ is decreasing in $1/\sigma^{2}_{\epsilon}$.

  First, replace $\theta =(\sigma^{2}_{u}/(\sigma_{u}^{2}+\sigma^{2}_{\epsilon}))$ and differentiate (18) with respect to σ ² _u . Then, replace m*^'(θ) with $\Updelta'(\sigma_{u}^{2}/(\sigma_{u}^{2}+\sigma^{2}_{\epsilon}))$, and $\sigma^{2}_{u}=(m^*-1)(1+m^*)^2\sigma^2_{\epsilon}$ (14) to yield

  $$ \frac{m^{*}\left(m^{*2}+m^*-2\right)+2}{4 (m^*+1) (3 m^*-1)\left(m^{*2}+m^*-1\right)^{3/2} \sqrt{(m^*-1)\sigma^2_\epsilon}}\geq0 $$

  (21)

  and thus $\Uppi^{P*}$ is increasing in $\sigma^{2}_{u}$.

  Differentiating (19) with respect to θ, substituting for $\Updelta^{'}(\theta)$ and θ yields

  $$ \sqrt{ \Upsigma \Uppsi\sigma^{2}_{u}}\left(\frac{m^{*2}\sqrt{-\left(1-m^*-m^{*2}\right)}}{4(1+m^*)}\right)\geq0. $$

  (22)

  Thus $\Uppi^{A*}$ is increasing in non fundamental precision, $1/\sigma^{2}_{\epsilon}$.

  First, replace $\theta =(\sigma^{2}_{u}/(\sigma_{u}^{2}+\sigma^{2}_{\epsilon}))$ and differentiate (19) with respect to σ ² _u . Then, replace m*^'(θ) with $\Updelta'(\sigma_{u}^{2}/(\sigma_{u}^{2}+\sigma^{2}_{\epsilon}))$ and $\sigma^{2}_{u}=(m^*-1)(1+m^*)^2\sigma^2_{\epsilon}$, which yields for m ≥ 1

  $$ \sqrt{\frac{\Upsigma\Uppsi}{\sigma^2_\epsilon} }\left(\frac{m (m+2) \sqrt{\left(m^3-2m+1\right)}}{4 (m+1) \left(m^2+m-1\right)^2}\right)\geq 0. $$

  Thus $\Uppi^{A*}$ is increasing in $\sigma^{2}_{u}$.

$\square$

Proof of Corollary 2

Price efficiency is computed as:

$$V(v|P_{m})=V(v)-cov(v,P_{m})^{2}/V(P_{m})=\Upsigma -\frac{\beta_{m}^2 \Upsigma ^2}{m^2 \gamma_{m} ^2 \sigma_u^2/\theta +2 m \gamma_{m}\sigma_u^2+\beta_{m} ^2 \Upsigma/\Uppsi +\sigma_u^2}. $$

Using the coefficients given in Lemma 1, we obtain

$$ V(v|P^{*})=\Upsigma-\Upsigma \Uppsi/2. $$

$\square$

Proof of Corollary 3

Uninformed traders’ losses are the sum of the profits of the proprietary trader and the analyst, or

$$L(\theta)=\Uppi^{A*}(\theta)+\Uppi^{P*}(\theta). $$

Losses are minimized by setting the first order condition to zero, or

$$ 0=\frac{\sqrt{\Upsigma\Uppsi \sigma_u^2 }}{4}\left(m^{*}\left(m^{*} (2 \theta-1-(1-\theta)m^{*}) -2(1-\theta) \theta m^{*'}(\theta)\right)\left((1+(1-\theta)m^{*}(m^{*}+2)\right)^{\frac{3}{2}}\right). $$

Replacing m*^' with (17) and simplifying,

$$\theta^*=\frac{(m^*+1)^2(3m^{*}-1)}{(3 m^{*} (m^{*}+2)+1)m^{*}}.$$

Using (14) and $\theta=\sigma_u^2/(\sigma^2_{u}+\sigma^2_{\epsilon})$, we have $\sigma_u^{2*}=9\sigma^2_{\epsilon}$ or θ* = 9/10 and m* = 2. $\square$

Proof of Proposition 2

Let $K=\left(\frac{\sigma_u^2}{(1+ m)^2 \Upsigma + \sigma^2(1+ m (3 + m))}\right)^{\frac{1}{2}}$. Solving for FME following Lemma 1 gives

$$ \beta_\eta = (1+m) K,\,\, \gamma_\eta = -K, \,\, \hbox{and}\,\,\lambda_\eta=\frac{\Upsigma}{(2 (1 + m) \Upsigma + (2 + m)\sigma^2)}\frac{1}{K}. $$

The difference between expressions (3) and (4) yields

$$ \left(\Upsigma\sigma_u^2\right)^{\frac{1}{2}}\left(\frac{1}{2}-\frac{\Upsigma^\frac{1}{2}((m+1)^2 \Upsigma +(m (m+3)+1) \sigma^2)^{\frac{1}{2}}}{2 (m+1)\Upsigma +(m+2) \sigma^2}\right). $$

(23)

Differentiating with respect to m, gives

$$-\frac{\Upsigma^\frac{1}{2} \sigma^2 \left((m+4) \sigma^2+4 \Upsigma\right)\left(\Upsigma \sigma_u^2\right)^{\frac{1}{2}}}{2 \left(2(m+1) \Upsigma +(m+2) \sigma^2\right)^2 ((m+1)^2 \Upsigma+(m(m+3)+1) \sigma^2)^\frac{1}{2}}\leq0. $$

Further,

• at m = 0, (23) is equal to $\frac{1}{2}\left(\Upsigma\sigma_u^2\right)^{\frac{1}{2}}\left(1-\left(\frac{\Upsigma}{\Upsigma+\sigma^2}\right)^\frac{1}{2}\right)>0$,

• as m ⇒ +∞, (23) converges to $\left(\Upsigma\sigma_u^2\right)^{\frac{1}{2}}\left(\frac{1}{2}-\frac{(\Upsigma(\Upsigma+\sigma^2))^\frac{1}{2}}{2 \Upsigma +\sigma^2}\right)>0$.

Thus ∀ m ≥ 0, (23) is positive. $\square$

Lemma 2

Let r _m  = (m(m + 2)(1 − θ) + 1)/(m + 1)² and define constants β _m,n , γ _m,n and λ _m,n by $\beta_{m,n} =( \Uppsi r_{m}\sigma_{u}^2/ (n \Upsigma) )^{1/2}, \gamma_{m,n} =-\theta/(m+1),$ and $\lambda_{m,n} =(n\Upsigma\Uppsi/((n+1)^2 r_{m}\sigma_{u}^2))^{1/2}.$ Equilibrium demand and price schedules are

$$ X^{*}_{m,n} = \beta_{m,n}s \,\,\,\,\,\, Z^{*}_{m,n} = \gamma_{m,n} \tau \,\,\,\,\,\,P^*_{m,n} = \lambda_{m,n} (x+mz+u). $$

Proof of Lemma 2

Given the linear rules, the proprietary trader (or equivalently a representative proprietary trader) and a (representative) demand-based trader choose X _m,n and Z _m,n to maximize profits, that can be written as

$$ \begin{aligned} E\{(v-P_{m})x_{m}|s,m, x_{m}\} &= x_{m,n} (\Uppsi s-\lambda_{m,n} (x_{m,n}+(n-1)\bar{\beta}_{m,n}s )))\\ E\{(v -P_{m})x_{m}|\tau,m,z_{m}\} &= -z_{m,n} \lambda_{m,n} ( z_{m,n}+ (m-1)\bar{\gamma}_{m,n} \tau + \theta \tau) \end{aligned} $$

where $\bar{\beta}_{m,n}$ and $\bar{\gamma}_{m,n}$ are the expected trading strategies of the other n − 1 proprietary traders, and m − 1 demand-based traders, respectively. Taking first order conditions and substituting $\bar{\gamma}_{m,n}$ with γ _m,n and $\bar{\beta}_{m,n}$ with β _m,n gives

$$\beta_{m,n}=\frac{\Uppsi}{(n+1)\lambda_{m,n}} \,\,\,\hbox{and}\,\,\, \gamma_{m,n}=-\frac{\theta}{m+1}. $$

(24)

The market clearing condition, P = λ _m,n y = E{v|y} implies

$$ \lambda_{m,n} = \frac{ n\beta_{m,n} \Upsigma }{(n^2\beta_{m,n}^2 \Upsigma)/\Uppsi+\sigma_u^2\left(\frac{m^2 \gamma_{m,n} ^2}{\theta}+ 2 m\gamma_{m,n} +1\right)}. $$

(25)

Solving (24) and (25) subject to the second order condition λ _m,n  > 0 gives the equilibrium. $\square$

Proof of Proposition 3

• (i) From Lemma 2, the expected profit of an individual demand-based trader in a market with a total of m demand-based traders and n proprietary traders, $\Uppi^{D}_{i,m,n}$ is

  $$\Uppi^{D}_{i,m,n}=\frac{\theta \lambda_{m,n} \sigma_{u}^2}{(m+1)^2},$$

  (26)

  leaving a demand-based trader indifferent between buying and not buying information at $c=\frac{\theta\lambda_{m,n} \sigma_{u}^2}{(m+1)^2}$. Using Lemma 2, substitute for λ _m,n , yielding

  $$c_{m,n}=\frac{\theta}{(m+1)(n+1)} \left(\frac{n \Upsigma \Uppsi\sigma_u^2}{m (m+2) (1-\theta )+1}\right)^{\frac{1}{2}}. $$

  (27)

  The expected profit of an individual proprietary trader in a market with a total of m demand-based traders and n proprietary traders, $\Uppi^{P}_{i,m,n}$ is

  $$\Uppi^{P}_{i,m,n}=\frac{\Upsigma \Uppsi }{(n+1)^2 \lambda_{m,n} },$$

  (28)

  making a trader indifferent between buying and not buying fundamental information at $\xi=\frac{\Upsigma\Uppsi}{(n+1)^2 \lambda_{m,n} }$. Substituting for λ _m,n , we get

  $$\xi_{m,n}=\left(\frac{\Upsigma \Uppsi\sigma_u^2}{n}\right)^{\frac{1}{2}}\frac{((1+m (m+2)(1-\theta))^\frac{1}{2}}{(n+1) (m+1)}. $$

  (29)

  The analyst’s profits are equal to the sum of the proprietary and demand based sales, or $\Uppi^{A}_{m,n}=(n-1)\xi_{m,n}+mc_{m,n}$. The analyst maximizes profits by choosing m and n optimally. Taking the first order condition of $\Uppi^{A}_{m,n}$ in m and n and simplifying yields the implicit expressions for the optimal number of proprietary and demand-based traders ($\hat{n}$ and $\hat{m}$, respectively)

  $$ \hat{m} \theta (-\hat{n}(\hat{m} (\hat{n}-4)+\hat{n}-7)+\hat{m}+2) +(\hat{m}+1)^2((\hat{n}-4) \hat{n}-1)= 0 $$

  (30)
  $$ \hat{m} (1-\theta)\left(2+\hat{m}-(\hat{m}+1)^2 \hat{n}\right)+1=0. $$

  (31)

  Solving (31) for $\hat{n}$ gives

  $$ \hat{n}=\frac{\hat{m}(\hat{m}+2) (1-\theta )+1}{\hat{m} (\hat{m}+1)^2 (1-\theta )}.$$

  (32)

  Applying (32) into (30) and simplifying gives

  $$ \frac{(\hat{m}(\hat{m}+2) (1-\theta)+1)^2 (1-\hat{m} (\hat{m} (\hat{m}+5)+3)(1-\theta))}{\hat{m}^2 (\hat{m}+1)^3 (1-\theta)^2}=0. $$

  (33)

  Denote $\tilde{\Upgamma}(\theta,\hat{m})=0$ where

  $$\tilde{\Upgamma}(\theta,m)=(m (m+2) (1-\theta)+1)^2 (1-m (m (m+5)+3)(1-\theta )). $$

  (34)

  Differentiating $\tilde{\Upgamma}(\theta,m)$ in m gives

  $$ (1-\theta ) (m (m+2)(1-\theta )+1) \times (m (m (55 \theta-m (7m+40) (1-\theta ) -58)-6(4-3 \theta))+1). $$

  (35)

  Solving $\tilde{\Upgamma}(\theta,\hat{m})=0$ for θ gives the two (unique) roots below, with only (36) feasible.

  $$ \begin{aligned} \theta&= \frac{(\hat{m}+1)^2}{\hat{m} (\hat{m}+2)}>1\,\, \hbox{and} \\ \theta&= \frac{(\hat{m}+1) (\hat{m} (\hat{m}+4)-1)}{\hat{m} (\hat{m}(\hat{m}+5)+3)}. \end{aligned} $$

  (36)

  Substituting (36) into (35) yields

  $$ -\frac{(\hat{m}+1)^2(\hat{m}+3) (\hat{m}+5)^2 (3 \hat{m}+1)}{\hat{m} (\hat{m}(\hat{m}+5)+3)^3}<0. $$

  (37)

  Thus $\hat{m}$ and $\hat{n}$ are unique.

• (ii) Rewriting (30) and (31) in matrix form:

  $$ \phi = \left[{\begin{array}{cc} \phi_{1}(m,n,\theta) &\phi_{2}(m,n,\theta) \\\end{array} } \right]=\left[ {\begin{array}{cc} 0 & 0 \\\end{array} } \right] $$

  The Jacobian matrix of ϕ in m and n is

  $$ J_{\phi}(m,n) = \left[{\begin{array}{cc} \frac{\partial\phi_{1}}{\partial m}(m,n,\theta)& \frac{\partial\phi_{1}}{\partial n}(m, n,\theta) \\\frac{\partial\phi_{2}}{\partial m}(m, n,\theta) &\frac{\partial\phi_{2}}{\partial n}( m, n,\theta) \\ \end{array} }\right] $$

  where

  $$ \begin{aligned}\frac{\partial\phi_{1}}{\partial m}( m, n,\theta) &= 2 (m ((n - 4) n- 1) - 1) (1 - \theta) + n (7 \theta-8 + n (2 - \theta))\\\frac{\partial\phi_{1}}{\partial n}( m, n,\theta) &= 2 (n-2)(m+1)^2+m \theta (4 m-2n (m+1) +7) \\\frac{\partial\phi_{2}}{\partial m}( m, n,\theta) &= (m+1) (2-3 mn-n) (1-\theta) \\ \frac{\partial\phi_{2}}{\partial n}( m,n,\theta) &= -m (m+1)^2 (1-\theta). \end{aligned} $$

  The Jacobian matrix of ϕ in θ is

  $$ J_{\phi}(\theta) = \left[\begin{array}{c} \frac{\partial\phi_{1}}{\partial\theta}(m,n,\theta) \\ \frac{\partial\phi_{2}}{\partial\theta}(m, n,\theta)\end{array} \right] = \left[ \begin{array}{c} m (m-n (m(n-4)+n-7)+2) \\ m \left(n (m+1)^2-m-2\right) (2 \theta -1)+1\end{array} \right]. $$

  The determinant of $J_{\phi}(\hat{m},\hat{n})$, is

  $$\frac{\partial\phi_{1}}{\partial m}( m, n,\theta)\times\frac{\partial\phi_{2}}{\partial n}( m,n,\theta)-\frac{\partial\phi_{1}}{\partial n}( m, n,\theta)\times\frac{\partial\phi_{2}}{\partial m}( m, n,\theta). $$

  (38)

  Substituting (32) and (36) into (38) and simplifying yields

  $$ \frac{(m+1) (m+3) (m+5)^2(3 m+1)}{m (m (m+5)+3)^2}. $$

  (39)

  Expression (36) positive (θ > 0) implies $m>\sqrt{5}-2$, and the determinant is always different from zero. Applying the implicit function theorem,

  $$ \eta(\theta) = \left[{\begin{array}{c} \frac{\partial\hat{m}}{\partial\theta}(\theta) \\\frac{\partial\hat{n}}{\partial\theta}(\theta) \\ \end{array} }\right]=-[J_{\phi}(m,n)]^{-1} [J_{\phi}(\theta)]. $$

  Using the expressions for $\hat{n}$ and θ ((32) and (36), respectively) and simplifying in the expressions of $\frac{\partial\hat{m}}{\partial\theta}(\theta)$ and $\frac{\partial\hat{n}}{\partial\theta}(\theta)$ gives

  $$ \begin{aligned}\frac{\partial\hat{m}}{\partial\theta}(\theta) &= \frac{m^2 (m(m+5)+3)^2}{(m+3) (3 m+1)}> 0 \\\frac{\partial\hat{n}}{\partial\theta}(\theta) &= -\frac{4 m^2 (m(m+5)+3)^2}{(m+1)^2 (m+3) (3 m+1)}< 0. \end{aligned} $$

  Thus $\hat{m}$ is increasing in θ and $\hat{n}$ is decreasing in θ.

  Further:

  • at θ = 0, from (36) and (30), $\hat{m}=\sqrt{5}-2\approx0.236$, and $\hat{n}=2+\sqrt{5}\approx4.236$

  • $\lim_{\theta\rightarrow 1}\pi^{A}_{m,n}=\frac{m \sigma_u \sqrt{n \Upsigma \Uppsi }}{mn+m+n+1}+\frac{(n-1)\sigma_u \sqrt{\Upsigma\Uppsi}}{(m+1) (n+1)\sqrt{n}}$, which is increasing in m.

  Thus profits are maximized at $\hat{m} \rightarrow\infty$ and $\hat{n}=1$.

• (iii) Substituting m = 1 into (31) and using (30), we solve for θ and find $\bar{\theta}=8/9$.

  To prove $\hat{m}$ and $\hat{n}$ characterize a global maximum, define the Hessian matrix

  $$ H(m,n) = \left[{\begin{array}{cc} \frac{\partial^2\Uppi^{A}}{\partial m^2}\left(m,n,\theta\right)& \frac{\partial^2\Uppi^{A}}{\partial m\partial n}(m, n,\theta) \\ \frac{\partial^2{\Uppi^{A}}}{\partial n\partial m}(m, n,\theta) & \frac{\partial^{2}\Uppi^{A}}{\partial n^2}( m,n,\theta) \\ \end{array} } \right]. $$

  Substituting for $\hat{n}$ and θ and simplifying yields

  $$ H(\hat m)=\left(\frac{\Upsigma \sigma^2_u\Uppsi}{\hat{m}(\hat{m}+5)+3}\right)^{\frac{1}{2}}\frac{(\hat {m}+1)}{(\hat {m}+3) (\hat {m}+5)^2} \left[ \begin{array}{cc}-\frac{(\hat m (\hat m+4)-1) (\hat m (3 \hat m+14)+3) }{2 \hat m^2(\hat m+1)^2} &-\frac{(\hat m (\hat m+4)-1) }{2 \hat m} \\-\frac{(\hat m (\hat m+4)-1) }{2 \hat m} &-\frac{(\hat m+1)^2(\hat m+7) }{8} \end{array} \right]. $$

  Define

  $$G(\hat m)=\left[\begin{array}{cc}-\frac{(\hat m (\hat m+4)-1) (\hat m (3 \hat m+14)+3) }{2 \hat m^2(\hat m+1)^2} & -\frac{(\hat m(\hat m+4)-1) }{2 \hat m} \\-\frac{(\hat m (\hat m+4)-1) }{2 \hat m} & -\frac{(\hat {m}+1)^2 (\hat {m}+7) }{8}\end{array}\right].$$

  $\hat{m}> \sqrt{5}-2\Rightarrow -\frac{(\hat m (\hat m+4)-1) (\hat m (3 \hat m+14)+3)}{2 \hat m^2 (\hat m+1)^2}<0$ and the determinant of $G(\hat m)$ is $\frac{(m+5)^2 (3 m+1) (m(m+4)-1)}{16 m^2}>0$. Thus the Hessian is negative definite. Given $\hat{m}$ and $\hat{n}$ are unique, $\Uppi^{A}$ attains a global maximum at $\hat{m}$ and $\hat{n}$. $\square$

Proof of Corollary 4

• (i) Replacing $\hat{n}$ using (32) and θ using (36) into the expression of the uninformed traders’ losses, $L_{\hat{m},\hat{n}}(\theta)$ (that is, total profits of demand-based and the proprietary traders) yields

  $$ \sqrt{\Upsigma \Uppsi\sigma_u^2}\left(\frac{(\hat{m}(\theta)+1) (\hat{m}(\theta)+4)}{2(\hat{m}(\theta)+3) \sqrt{\hat{m}(\theta)(\hat{m}(\theta)+5)+3}}\right). $$

  (40)

  Differentiating the above expression in θ yields

  $$ \frac{(\hat{m} (\hat{m}(\hat{m}+10)+27)+6) \hat{m}'(\theta )}{4 (\hat{m}+3)^2 (\hat{m}(\hat{m}+5)+3)^{3/2}}>0. $$

  (41)

  Thus uninformed traders’ losses are minimized at θ = 0.

• (ii) Price efficiency, the inverse of $Var(v|P_{m,n} )=\Upsigma-\frac{n \Upsigma \Uppsi }{n+1}$, is increasing in n, n is decreasing in θ, and thus price efficiency is maximized at θ = 0.³¹ From expression (41) the uninformed traders’ losses are increasing in θ, when there are $\hat{n}$ fundamental traders and $\hat{m}$ demand-based traders. From Corollary 3, the uninformed traders’ losses are minimized at θ = 9/10 when the analyst does not have fundamental information.

  From (30) and (36),

  • At $\theta=0,\hat{m}=\sqrt{5}-2\approx 0.236$, and $\hat{n}=2+\sqrt{5}\approx4.236$. The uninformed traders’ losses are $L_{\hat{m},\hat{n}}(\theta)=\frac{\left(\sqrt{5}-1\right)\sqrt{2+\sqrt{5}}}{3+\sqrt{5}\sqrt{1+\sqrt{5}\left(\sqrt{5}-2\right)}}\approx 0.393$.

  • At $\theta=9/10, \hat{n}\approx2.941$ and $\hat{m}\approx 1.061$. The uninformed traders’ losses are $L_{\hat{m},\hat{n}}(\theta)\approx0.418$

  • At $\theta\rightarrow 1,\hat{m} \rightarrow \infty$ and $\hat{n}=1$. The uninformed traders’ losses are $L_{\hat{m},\hat{n}}(\theta)=1/2$.

  When the analyst has only non fundamental information, the uninformed traders’ losses are

  • L(θ) = 1/2 at $\theta \in \{0,1\}$ and

  • $L(\theta)=1/\sqrt{5}\approx 0.447$ at θ = 9/10, the level at which losses are minimized.

  Therefore the uninformed traders’ losses are always lower when the analyst also has fundamental information. $\square$

Proof of Proposition 4

From Lemma 2, the price sensitivity with two proprietary traders and m demand-based traders is equal to

$$\lambda_{m}=\frac{\sqrt{2}}{3}\left(\frac{\Upsigma\Uppsi}{\sigma_u^2}\right)^{\frac{1}{2}} \left(\frac{(m+1)^2}{(m(m+2) (1-\theta )+1)}\right)^{\frac{1}{2}}. $$

(42)

Using (11), and replacing λ with (42)

$$ c_m =\frac{\sqrt{2}\theta}{3 (m+1)}\left(\frac{\Upsigma \Uppsi\sigma_u^2}{m (m+2) (1-\theta )+1}\right)^{\frac{1}{2}}. $$

Taking the FOC of the analyst’s (proprietary and demand-based) profits with respect to m

$$ \frac{\sqrt{\Upsigma\sigma^2_u \Uppsi }}{3 \sqrt{2}}\left(\frac{\theta \sqrt{1+m (m+2)(1-\theta )} \left(1-m^2 (2 m+3) (1-\theta )\right) }{\left((m+1)^3-m (m+1) (m+2) \theta \right)^2}\right)=0. $$

(43)

Therefore the optimal number of demand-based traders and fees, when there are two proprietary traders are implicitly defined by the following equations

$$ 0 = 1-(m^{**})^2 (2m^{**}+3) (1-\theta )\\ $$

(44)

$$ c^{**} =\frac{\sqrt{2}\theta}{3 (m^{**}+1)} \left(\frac{\Upsigma \Uppsi\sigma_u^2}{m^{**} (m^{**}+2) (1-\theta )+1}\right)^{\frac{1}{2}}.$$

(45)

Taking n as exogenous, we differentiate the uninformed traders’ losses in θ. It yields after simplifying:

$$ \frac{m^2\left(-\frac{n}{m (m+2) (\theta -1)-1}\right)^{3/2} \left(-\frac{2\theta \left((m+1)^2 n-m-2\right)}{(m+1) (3 m n+n-2)}+m (\theta-1)+2 \theta -1\right)}{2 n (n+1)}=0. $$

(46)

Solving (31) and (46), we obtain:

$$ \begin{aligned} \theta&= \frac{4 n^2+4 n+1}{(n+1) (4 n+1)}\\ m &= \frac{n+1}{n}.\end{aligned} $$

• When n = 1, θ* = 9/10 and m* = 2. Uninformed traders’ losses are equal to $1/\sqrt{5}\approx0.447$. For θ = 0 and θ = 1, L(θ) = 1/2.

• When n = 2, θ** = 25/27≈ 0.926 and m** = 3/2. Uninformed traders’ losses are equal to 4/9 ≈0.444. For θ = 9/10, uninformed traders’ losses are approximately equal to 0.445 and for θ = 0 and θ = 1, uninformed traders’ losses are $\sqrt2/3\approx 0.471$.

Thus $\Uppi^{D}(m^{*})+\Uppi^P(m^{*})\geq \Uppi^{D}(m^{**})+2\Uppi^{P}(m^{**})$. Given that the maximum fee the proprietary trader can extract from the analyst is

$$ \bar{f}=\Uppi^{D}(m^{**})+\Uppi^{P}(m^{**})-\Uppi^{D}(m^{*}) $$

it follows that $\Uppi^{P}(m^{*})\geq \Uppi^P(m^{**})+\bar{f}$, and thus the proprietary trader does not sell fundamental information. $\square$

Proof of Proposition 5

Since $\Uppi^{D}(m^{*})+\Uppi^P(m^{*})\geq\Uppi^{D}(m^{**})+2\Uppi^{P}(m^{**})$, interdivisional sharing cannot increase overall profits. $\square$

Proof of Proposition 6

Let the linear demand functions of the proprietary trader and demand-based traders be denoted x ₂ = β₂ v + b ₂ p ₁ and z ₂ = γ₂ τ₂ + g ₂ p ₁ with conditional expectations

$$ \begin{array}{lll}E^P(\tau|v,y_1) =\kappa v + \omega p_1; & E^P(u_2|v,y_1) = k v + wp_1= \theta \kappa v + \theta \omega p_1\\ E^D(v|\tau,y_1) = \phi\tau+ \zeta p_1; & E^M(v|y_1,y_2) = p_2 = d_2 p_1 + \lambda_2 y_2.\end{array} $$

(47)

Then, writing the proprietary trader’s objective function, replacing for the expressions of p ₂, y ₂, u ₂ and taking expectations, we have

$$ \begin{aligned}E((v-p_2)x_2|v,p_1) &= E(v-\lambda_2 y_2 - d_2 p_1) x_2\\ &=(v-\lambda_2(x_2 + m(\gamma_2 (\kappa v + \omega p_1 )+ g_2 p_1) +k v + w p_1)-d_2 p_1)x_2. \end{aligned} $$

Differentiating, setting equal to zero, and solving for β₂ and b ₂ gives

$$ \begin{aligned} \beta_2&= \frac{1 - \lambda_2 \kappa (\theta + \gamma_2 m)}{2 \lambda_2}\\b_2 &= -\frac{d_2 + \lambda_2(\theta\omega + m (g_2+ \gamma_2\omega))}{2 \lambda_2}. \end{aligned} $$

The SOC confirms λ₂ > 0.

Doing the same for an individual demand-based trader, we have

$$ \begin{aligned} E((v-p_2) z_2| \tau, p_1) =& (v- (\lambda_2(x_2+z_2' (m-1)+z_2+u_2)+d_2 p_1) )z_2\\ =& \left(\zeta p_1 + \phi \tau - (\lambda_2(\beta_2 (\zeta p_1 + \phi \tau) + b_2 p_1 +\right.\\&\left.(\gamma_{2i} \tau + g_{2i} p_1) (m - 1) + z_2 + \theta \tau)+ d_2 p_1)\right) z_2. \end{aligned} $$

Differentiating, substituting symmetric demand functions for all demand-based traders, setting equal to zero, and solving for γ₂ and g ₂ gives

$$ \begin{aligned} \gamma_2&= { \frac{2 \theta \lambda_2 - \phi (1 +\theta\kappa\lambda_2)}{\lambda_2 (m \kappa \phi - 2 (1 + m))}}\\g_2 &= \frac{-d_2+\zeta + \lambda_2(\theta\omega + \gamma_2 m \omega+ \zeta \kappa(\theta + \gamma_2 m ))}{(2+m)\lambda_2}\end{aligned} $$

The SOC confirms λ₂ > 0.

Turning to the first period problem, the proprietary trader maximizes total expected profits, or E{(v − p ₁)x ₁ + (v − p ₂)x ₂}. Using the expressions above and appropriate substitutions leads to the following optimization program

$$ \begin{aligned}&(v-\lambda_1 x_1)x_1 + (v-(\lambda_2(\beta_2 v + b_2 \lambda_1x_1)+ m(\gamma_2(\kappa v+\omega\lambda_1 x_1)+\\ & \quad g_2\lambda_1 x_1)+ (\theta\kappa v + \theta\omega\lambda_1 x_1))+d_2\lambda_1 x_1)(\beta_2 v + b_2\lambda_1 x_1). \end{aligned}$$

the FOC to zero yields

$$ \beta_1 = -{ \frac{(-1+b_2 \lambda_1 (-1 + \lambda_2 (2 \beta_2 + \kappa (\theta+\gamma_2 m)) + \beta_2 \lambda_1 (d_2 + \lambda_2 (\omega(\theta +\gamma_2 m)+ g_2 m )))}{2\lambda_1 (1 + b_2 \lambda_1 (d_2 +\lambda_2 (b_2 + \theta\omega + m(g_2 + \gamma_2 \omega))))}}.$$

The SOC confirms 2 λ₁ (− b ₂ λ₁ (λ₂ (b ₂ + γ₂ m ω + g ₂ m + θ ω) + d ₂) − 1) < 0.

Similarly an individual demand based trader optimizes total expected profits, which can be written as

$$ \begin{aligned}&-\lambda_1 ((m - 1) z_1^{\prime} + z_1) z_1 +((\zeta (\lambda_1 ((m - 1)z_1^{\prime} + z_1)) + \phi \tau) - (\lambda_2 ((\beta_2 (\zeta (\lambda_1((m -1) z_1^{\prime} \\ & \quad + z_1)) + \phi \tau) + b_2 (\lambda_1 ((m -1) z_1^{\prime} + z_1)) ) + (\gamma_2 \tau + g_2 (\lambda_1 ((m - 1) z_1^{\prime} +z_1))) m + \theta \tau) \\ & \quad +d_2 (\lambda_1 ((m - 1) z_1^{\prime} +z_1)))) (\gamma_2 \tau + g_2 (\lambda_1 ((m - 1) z_1^{\prime} + z_1))).\end{aligned} $$

Differentiating, substituting, setting equal to zero, and recognizing the symmetry in demand-based strategies, yields

$$ \gamma_1 = {\frac{-\gamma_2 (d_2 - \zeta + \lambda_2 (b_2 + 2 g_2 m + \beta_2\zeta)) + g_2 (\phi - \lambda_2 (\theta + \beta_2 \phi))}{1 + m (1+ 2 g_2 \lambda_1 (d_2 - \zeta + \lambda_2 (b_2 + g_2 m +\beta_2\zeta)))}}. $$

(48)

The SOC confirms $2 \lambda_1 (-C_2 \lambda_1(\lambda_2 (\beta_2 \zeta+b_2+g_2 m)+d_2-\zeta)-1)<0$.

Using the projection theorem to calculate the conditional expectations in (47) yields

$$ \begin{aligned} \phi =& -\frac{\gamma_1 m}{\lambda_1(\beta_1^2 \Upsigma + \sigma_u^2)} \, \zeta = \frac{\beta_1 \Upsigma}{\lambda_1(\beta_1^2 \Upsigma + \sigma_u^2)} \quad\quad \kappa =-\frac{ \beta_1\gamma_1 m }{\gamma^2_1 m^2 + \theta} \\ \omega =& \frac{\gamma_1 m}{\lambda_1(\gamma^2_1 m^2 + \theta)} \, \lambda_1= \frac{\theta \beta_1 \Upsigma}{\theta(\beta_1^2 \Upsigma + \sigma_u^2)+ \gamma_1^2 m^2 \sigma_u^2}. \\ \lambda_2 =& \frac{(\gamma_1 m^2(-\beta_2 \gamma_1 + \beta_1 \gamma_2) - \theta(\beta_2 - \beta_1 \gamma_1 m) )\Upsigma} {2\beta_1\beta_2 \gamma_1 m (\gamma_2 m + \theta)\Upsigma - \gamma_2 m(\gamma_2 m + 2 \theta) (\beta^2_1 \Upsigma + \sigma_u^2)- \gamma_1^2 m^2 (\sigma_u^2(1-\theta)+\beta_2^2 \Upsigma) - \theta(\beta_1^2+\beta_2^2)\Upsigma+\sigma_u^2)} \\ d_2 =& \Upsigma (-\beta_1 \gamma_2 m (\gamma_1 m \lambda_1 (b_2+g_2 m)+2 \theta )-\beta_1 \theta (\gamma_1 m \lambda_1 (b2+g_2 m)+1)\\ &-\beta_1 \gamma_2^2 m^2+\beta_2 \gamma_1^2 b_2 m^2 \lambda_1+\beta_2 \gamma_1^2 g_2 m^3 \lambda_1+\beta_2 \gamma_1 m (\gamma_2 m+\theta )+\beta_2 b_2 \theta \lambda_1+\beta_2 g_2 m \theta \lambda_1)/ \\ & (\lambda_1 \left(-\theta \left(\left(\beta_1^2+\beta_2^2\right)\Upsigma +\sigma_u^2\right)-\gamma_2^2 m^2 \left(\beta_1^2 \Upsigma +\sigma_u^2\right)-2 \gamma_2 m \theta \left(\beta_1^2 \Upsigma +\sigma_u^2\right)\right.\\ &+2 \beta_1 \beta_2 \gamma_1 m \Upsigma (\gamma_2 m+\theta )-\gamma_1^2 m^2 \left(\sigma_u^2 (1-\theta)+\beta_2^2 \Upsigma \right)). \end{aligned} $$

Using conjectures of γ₁ = 0 and d ₂ = 1, substitute, simplify, and solve for equilibrium. Without first period trading by the demand-based traders, the proprietary trader does not update his priors on u ₂, and demand-based traders do not update beyond p ₁, and

$$ \begin{array}{ll} E^P(\tau|v,y_1)=E(\tau)=0 \rightarrow \kappa=0; \omega=0; k=0; w=0\\ E^D(v|\tau,y_1)=E(v|y_1)=p_1 \quad \rightarrow \phi=0; \zeta=1.\\ \end{array} $$

Substituting, solve for the remaining parameters.

$$ \begin{array}{llllll} \beta_1 &= \frac{\lambda_1-2\lambda_2}{\lambda_1^2 - 4 \lambda_1\lambda_2}\,\,\,\,\,\,\,& \lambda_1 & = \frac{\beta_1 \Upsigma}{\beta_1^2 \Upsigma + \sigma_u^2}\,\,\,\,\,\, & \gamma_2 &= -\frac{\theta}{(1+m)}\\ g_2 &= 0 & \beta_2 &= \frac{1}{2\lambda_2} & b_2 &= -\frac{1}{2\lambda_2}\\ \end{array} $$

(49)

$$ \lambda_2 = \frac{(1+m)}{2}\left(\frac{\Upsigma}{(1+m(2+m)(1-\theta))(\beta_1^2 \Upsigma +\sigma_u^2)}\right)^{\frac{1}{2}}. $$

The inequalities to verify are

$$ \begin{aligned} \frac{\lambda_1 (\lambda_1-4 \lambda_2)}{2 \lambda_2}<&0 \\ \lambda_1>&0 \\ \lambda_2>&0. \end{aligned} $$

Replacing the above parameters in expressions (49) and (48) yields d ₂ = 1 and γ₁ = 0. Conjecture $\beta_1=q (\sigma_{u}^2/\Upsigma)^{1/2}$, where q ≥ 0 (as β₁ > 0 given λ₁ > 0). Substituting all the above parameters from (49) in the expression of β₁ and simplifying yields:

$$ q=-\frac{\left(q^2+1\right) \left(q \sqrt{\left(q^2+1\right) (m (m+2) (1-\theta )+1)}-(m+1) \left(q^2+1\right)\right)}{q \left(2 (m+1) \left(q^2+1\right)-q \sqrt{\left(q^2+1\right) (m (m+2) (1-\theta )+1)}\right)}. $$

Rearranging the terms implies

$$ (m+1) q^4+q \sqrt{\left(-q^2-1\right) (m (m+2) (\theta -1)-1)}-m-1=0. $$

(50)

Differentiating the left hand side (LHS) yields

$$ 4 (m+1) q^3+\frac{\left(2 q^2+1\right)\sqrt{\left(q^2+1\right)(m(m+2) (1-\theta ))}}{q^2+1}. $$

(51)

The derivative (51) is positive for q ≥ 0. For q = 0, the LHS of (50) is equal to −(1 + m) and for $q\rightarrow +\infty$, the LHS of (50) goes to m + 1. Thus there exists a unique q satisfying (50). $\square$