TY - JOUR
AU - Buss,, Adrian
AB - Abstract We study the joint portfolio and information choice problem of institutional investors who are concerned about their performance relative to a benchmark. Benchmarking influences information choices through two distinct economic mechanisms. First, benchmarking reduces the number of shares in investors’ portfolios that are sensitive to information. Hence, the value of private information declines. Second, benchmarking limits investors’ willingness to speculate. This not only reduces the value of private information but also adversely affects information aggregation. In equilibrium, investors acquire less information and informational efficiency declines. As a result, return volatility increases, and less-benchmarked institutional investors outperform more-benchmarked ones. Received May 31, 2017; editorial decision July 4, 2018 by Editor Stijn Van Nieuwerburgh. Authors have furnished supplementary code, which is available on the Oxford University Press Web site next to the link to the final published paper online. Institutional investors own a majority of U.S. equity and account for most of the transactions and trading volume in financial markets.1 Notably, the importance of relative performance concerns for institutional investors has grown steadily over the past years. That is, the performance of many institutions is nowadays evaluated relative to a benchmark portfolio (“benchmarking”), either explicitly, for example, through performance fees,2 or implicitly, for instance, because of the flow-performance relation. A growing body of literature studies the asset pricing implications of benchmarking for the case of symmetrically informed institutional investors. In contrast, the focus of this paper is on the impact of benchmarking on information acquisition and informational efficiency. In particular, our objective is to understand the economic mechanisms through which the growth of assets under management by benchmarked institutions affects informational efficiency and asset prices. For that purpose, we develop an equilibrium model of joint portfolio and information choice that explicitly accounts for relative performance concerns. The two key features of the model are that (1) all institutional investors endogenously decide on the precision of their private information and (2) a fraction of institutional investors (“benchmarked investors”) are concerned about their performance relative to a benchmark. Otherwise, the framework is kept as simple as possible to illustrate the economic mechanisms in the clearest possible way; for example, for most of the analysis we focus on the case in which investors can trade a risk-free bond and a single risky stock. Varying the fraction of benchmarked investors or, equivalently, the share of assets managed by benchmarked institutions will be our key comparative statics analysis to illustrate the implications of benchmarking. We identify two distinct economic channels through which benchmarking influences investors’ information choices: (1) information-scale effects and (2) risk-taking effects. Both mechanisms are driven by the interaction between portfolio and information choice. We illustrate these two effects by means of two economic settings, the first of which keeps, by design, investors’ risk appetite—and, hence, aggregate risk-bearing capacity—fixed. In particular, we illustrate information-scale effects arising from benchmarking in a tractable model in which institutional investors have constant absolute-risk aversion utility but a preference for the early resolution of uncertainty.3 We show that benchmarked investors’ portfolios can be decomposed into two components. First, the standard mean-variance portfolio, that is, the optimal portfolio of nonbenchmarked investors. This component is not affected by benchmarking, rather it is driven by investors’ posterior beliefs. Second, a hedging portfolio arising from benchmarking. Intuitively, because benchmarked institutional investors strive to do well when the benchmark performs well, they overweight the benchmark portfolio. This hedging portfolio is designed to track the benchmark, not to outperform it, and, consequently, it is information insensitive. Importantly, conditional on investors’ information choices, these investment decisions do not affect the sensitivity of the stock price with respect to the payoff or the noise. Hence, benchmarking does not adversely affect information aggregation. Instead, benchmarking reduces the value of information and, thus, investors’ incentives to acquire private information. Intuitively, once the stock market clears, the aggregate—information insensitive—hedging demand of the benchmarked institutions reduces the “effective supply” of the stock, that is, the number of shares in the economy that are available for speculation.4As a result, for all investors, both benchmarked and nonbenchmarked, the expected number of shares in their portfolio that are sensitive to information declines.5 Because private information is then applied to fewer shares, its marginal value is lower. With no change in information costs, investors acquire less private information. Consequently, price informativeness declines in the fraction of benchmarked investors in the economy. This, in turn, implies a higher return volatility since the price tracks the stock’s payoff less closely. The implications described thus far follow from scale effects in information acquisition. In particular, the decline in price informativeness is entirely determined by the decline in the investors’ precision choices. However, benchmarking also affects information aggregation through risk-taking effects, which we illustrate using a model with CRRA preferences.6 Benchmarked investors’ portfolios can be (approximately) decomposed into the same two components like in the mean-variance setting. The key difference is that relative performance concerns now also affect the mean-variance portfolio. In particular, benchmarking limits institutional investors’ willingness to speculate.7 Hence, benchmarked investors not only acquire less information but also trade less aggressively on a particular piece of information and, thus, benchmarking adversely affects information aggregation. This amplifies the decline in price informativeness. Risk-taking effects have important implications. For example, in contrast to the case of mean-variance preferences, the information choices of benchmarked and nonbenchmarked institutional investors differ. In particular, benchmarked investors choose a lower precision of private information. As a result, benchmarked investors are less well informed and, hence, earn lower expected portfolio returns than nonbenchmarked investors. Moreover, as the assets under management of benchmarked institutions increase, less information is revealed through the public stock price, such that the “information gap” and, hence, the return gap between investors widen. For realistic calibrations, some of the asset-pricing implications change even qualitatively. For example, the price of the stock can decline in the fraction of benchmarked investors, or, equivalently, its expected excess return can increase. Finally, we study two extensions of our basic economic framework. First, we extend our model to multiple stocks and document that informational efficiency also deteriorates for stocks that are not part of the benchmark, although to a smaller extent because the information-scale effect is absent for nonindex stocks. Second, we study benchmarking concerns that are nonlinear in the benchmark’s performance—in line with asymmetric performance fee structures and implicit in the flow-performance relationship. We demonstrate that asymmetric benchmarking concerns can mitigate the adverse effects of benchmarking on information choice and information aggregation. We also make a methodological contribution by developing a novel numerical solution approach that allows us to determine the equilibrium in noisy rational expectations models with nonlinear price functions. Our approach differs from previous attempts, such as Bernardo and Judd (2000), in that it does not rely on a parameterization of the price law and the asset demand nor on the projection method. Our solution technique is flexible and also could be used to study rational expectation equilibrium models in the presence of constraints or frictions. The two papers that are closest to our work are Admati and Pfleiderer (1997) and García and Strobl (2011), whose results and implications, however, are distinctly different. Admati and Pfleiderer (1997) study linear benchmarking concerns in the compensation of privately informed portfolio managers, but in a framework with constant absolute-risk aversion (CARA) preferences and an exogenous price process. They document that, because managers can use their own portfolio choice to “undo” the benchmarking component in their compensation, managers’ optimal information choice is not affected by relative performance concerns. In contrast, taking into account the effect of investors’ portfolio choices on the market-clearing stock price, we illustrate that benchmarking leads to a decline in information acquisition. We also document a novel effect resulting from benchmarked investors’ risk-taking. García and Strobl (2011) study how relative wealth concerns affect investors’ incentives to acquire information. They demonstrate that, when an investor’s utility depends on the consumption of the average investor, complementarities in information acquisition arise, introducing the possibility of multiple “herding” equilibria. These complementarities can lead to an increase in informed trading, thereby improving price informativeness. The key difference is that, in our framework, investors have relative performance concerns (with respect to a benchmark), and not relative wealth concerns (with respect to other investors). As a result, in our setting, no complementarities in information choice arise and price informativeness declines. Kacperczyk, Nosal, and Sundaresan (2018) develop a rational expectations model with market power and demonstrate that active investors shift their information choice in reaction to an increase in the size of passive investors. As a result, price informativeness declines. While passive investors in their model are, by definition, uninformed, the benchmarked investors in our framework endogenously choose the precision of their private information. The economic framework that we develop builds on two independent strands of research: first, the asset pricing literature on the stock market implications of benchmarking (absent information choice). Cuoco and Kaniel (2011) and Basak and Pavlova (2013) study models with CRRA preferences and relative performance concerns. They document that, in the presence of benchmarking, institutional investors optimally tilt their portfolio toward the benchmark, creating upward price pressure.8Buffa, Vayanos, and Woolley (2017) study a setup with CARA preferences in which, because of agency frictions, asset-management contracts endogenously depend on fund managers’ performance relative to a benchmark. Unlike these papers, we explicitly model investors’ joint information and portfolio choice in the presence of benchmarking. Notably, allowing for endogenous information choice can lead to qualitatively different asset pricing implications. For example, stock prices can decline in the fraction of benchmarked investors. We also provide novel predictions regarding institutional investors’ expected portfolio returns. Second, our framework builds on the literature on information acquisition in competitive markets (absent benchmarking).9 Information-scale effects are discussed in van Nieuwerburgh and Veldkamp (2009, 2010), who document a feedback effect between information and portfolio choice through the number of shares investors expect to hold, and Peress (2010), who demonstrates that better risk-sharing lowers the value of information because investors expect to hold fewer shares.10 Unlike these papers, we explicitly model institutional investors who are concerned about their relative performance, which allows us to make novel predictions about the relationship between the size of benchmarked institutions and informational efficiency. 1. The Model This section introduces our basic economic framework, which explicitly accounts for the information choices of institutional investors who are concerned about their performance relative to a benchmark. In particular, we incorporate benchmarking concerns, like in Cuoco and Kaniel (2011) and Basak and Pavlova (2013), into a competitive rational expectations equilibrium model of joint portfolio and information choice, like in Verrecchia (1982). We also discuss investors’ optimization problems and the equilibrium concept. 1.1 Economic framework 1.1.1 Information structure and timing We consider a static model, which we break up into three (sub)periods. Figure 1 illustrates the sequence of the events. In period |$1$|⁠, the information choice period, investors can spend time and resources to acquire private information about a stock. For example, they may study financial statements, gather information about consumers’ taste, hire outside financial advisers, or subscribe to proprietary databases. In particular, investor |$i$| can choose the precision |$q_i$| of the private signal |$Y_i$|⁠. Higher precision will reduce the posterior uncertainty regarding the stock’s payoff but will increase the information acquisition costs |$\kappa(q_i)$|⁠.11 Figure 1 View largeDownload slide Timing The figure illustrates the sequence of the events. Figure 1 View largeDownload slide Timing The figure illustrates the sequence of the events. In period |$2$|⁠, the portfolio choice period, investors observe their private signals (with the chosen precision) and make their investment choice. Prices are set such that markets clear. In period 3, the consumption period, investors consume the proceeds from their investments. We denote the expectation and variance conditional on prior beliefs as |$E_1[\cdot]$| and |$Var_1(\cdot)$|⁠. To denote investor |$i$|’s expectation and variance conditional on time-|$2$| information set |$\mathcal{F}_i = \{Y_i, P\}$|⁠, we use |$E_2[\, \cdot \, | \, \mathcal{F}_i]$| (or, |$E_2[\, \cdot \,]$|⁠) and |$Var_2(\, \cdot \, | \, \mathcal{F}_i)$| (or, |$Var_2(\, \cdot \,)$|⁠). 1.1.2 Investment opportunities Two financial securities are traded competitively in the market: a risk-less asset (the “bond”) and a risky asset (the “stock”). The bond pays an exogenous (gross) interest rate |$R_f$| and is available in perfectly elastic supply. It also serves as the numéraire, with its price being normalized to one. The stock is modeled as a claim to a random payoff |$X$|⁠, which is only observable in period |$3$|⁠. Its price is denoted by |$P$|⁠. The supply of the stock, denoted by |$Z$|⁠, is assumed to be random and unobservable. This prevents the price from fully revealing the information acquired by the investors and, thus, preserves the incentives to acquire private information in the first place. 1.1.3 Investors There exists a continuum of atomless investors with mass one that we separate into two groups of institutional investors: (1) a fraction |$\Gamma$| of benchmarked institutions, or, short, “benchmarked investors,” |$\mathcal{BI}$| and (2) a fraction |$1 - \Gamma$| of nonbenchmarked institutions, or, short, “nonbenchmarked investors,” |$\mathcal{NI}$|⁠. Each investor |$i \in \{\mathcal{BI}, \mathcal{NI}\}$| is endowed with the same initial wealth |$W_{0,i}$|⁠, which we normalize to |$1$|⁠. Motivated by recent theoretical contributions,12 we model the compensation of institutional investors, |$C_{i}$|⁠, as:13 \begin{equation} \label{eq:Compensation} C_i(W_i, R_B) = W_i - \gamma_i \, W_{0,i} \, R_B - \kappa(q_i).\end{equation} (1) Institutional investors’ compensation has two components; first, a standard component related to terminal wealth |$W_i$| (i.e., “assets under management”);14 and, second, a linear benchmarking component that is related to the performance of the benchmark |$R_B$|⁠.15 Consequently, |$\gamma_i$| captures the strength of investor |$i$|’s benchmarking concerns; in particular, whereas benchmarked institutional investors are concerned about their performance relative to a benchmark (⁠|$\gamma_i > 0, \forall i \in \mathcal{BI}$|⁠), nonbenchmarked investors are not (⁠|$\gamma_i = 0, \forall i \in \mathcal{NI}$|⁠). Finally, information acquisition costs |$\kappa(q_i)$| are deducted. It is important to highlight that benchmarking is the only source of heterogeneity across the two groups of institutional investors. In particular, benchmarking does not affect the investors’ financial wealth but only their utility; that is, in the portfolio choice period, both groups of institutional investors have the same capital available for investment. We consider a general form of investors’ preferences over compensation |$C_i$|⁠: \begin{equation} \label{eq:genUtility} U_{1,i} = E_1\bigl[u_1\left(\,E_2\left[u_2(C_i)\right]\,\right)\bigl], \end{equation} (2) in which the inner utility function |$u_2$| governs risk aversion and the outer utility function |$u_1$| governs the preference for the timing of the resolution of uncertainty.16 Specifically, if |$u_1$| is linear, investors are indifferent about the timing, whereas a convex (concave) function |$u_1$| implies a preference for early (late) resolution of uncertainty. In the next sections, we explore specific examples. The specification of the benchmarked investors’ utility over compensation |$C_i$| exhibits two important characteristics that let them behave differently from nonbenchmarked investors. First, benchmarked investors have an incentive to post a high return when the benchmark performs well or, formally, their marginal utility is increasing in |$R_B$|⁠. Second, benchmarked investors’ utility function is decreasing in the performance of benchmark, thus affecting information choice. Note that our definition of the investors’ compensation scheme in (1) shares many similarities with the specifications in Cuoco and Kaniel (2011) and Basak and Pavlova (2013).17 1.2 Investors’ optimization problems and equilibrium 1.2.1 Portfolio and information choice In the portfolio choice period (⁠|$t=2$|⁠), investor |$i$| chooses the number of shares of the stock, |$\theta_i$|⁠, to maximize expected utility; conditional on his posterior beliefs and price |$P$|⁠: \begin{equation} \label{eq:OptProbt2} U_{2,i} = \max_{\theta_i} \, E_2\bigl[\, u_2(C_i) \, | \, \mathcal{F}_i\bigr], \end{equation} (3) with terminal wealth, |$W_i$|⁠, being given by |$W_i = W_{0,i} \, R_f + \theta_i \, (X - P R_f)$|⁠.18 In the information choice period (⁠|$t=1$|⁠), investor |$i$| chooses the precision of his private signal, |$q_i$|⁠, in order to maximize expected utility over all possible realizations of his private signal |$Y_i$| and the public price |$P$|⁠, anticipating his optimal portfolio choice in period |$2$|⁠: \begin{equation} \label{eq:OptProbt1} \max_{q_i \geq 0} \, E_1\bigl[u_1(U_{2,i})\bigr].\end{equation} (4) 1.2.2 Equilibrium A rational expectations equilibrium is defined by portfolio choices |$\{\theta_i\}$|⁠, information choices |$\{q_i\}$|⁠, and prices |$\{P\}$| such that |$\theta_i$| and |$q_i$| solve investor |$i$|’s maximization problems in (3) and (4), taking |$P$| as given. Expectations are rational; that is, the average precision of private information implied by aggregating investors’ precision choices equals the level assumed in investors’ optimization problems (3) and (4). Aggregate demand equals aggregate supply. Note that, in equilibrium, the stock price plays a dual role: (1) it clears the security market and (2) aggregates and disseminates investors’ private information. 2. Benchmarking and Information Scale Effects In this section, we illustrate how benchmarking affects equilibrium due to information-scale effects. For this purpose, we rely on a tractable model that is designed to provide the economic intuition and allows for closed-form solutions of all quantities in the economy. In particular, we demonstrate that, in the presence of information-scale effects, price informativeness deteriorates. Importantly, this drop in price informativeness is entirely determined by a decline in the value of private information and, hence, a decline in investors’ average precision choices, whereas information aggregation is not adversely affected. 2.1 Setup We assume that the payoff of the stock, |$X$|⁠, and its supply, |$Z$|⁠, are normally distributed, with |$X \sim \mathcal{N}(\mu_X, \sigma_X^2)$| and |$Z \sim \mathcal{N}(\mu_Z, \sigma_Z^2)$|⁠. The investors’ private signals are given by |$Y_i = X + \varepsilon_i$|⁠, with |$\varepsilon_i \sim \mathcal{N}(0, 1/q_i)$|⁠. Investors have “mean-variance preferences” of the Kreps-Porteus type. Their time-|$1$| utility function, |$U_{1,i}$|⁠, is given by:19 \begin{equation} \label{eq:MVpref} U_{1,i} = E_1\left[E_2\left[C_i\right] - \frac{\rho}{2} \, Var_2\left(C_i\right)\right], \end{equation} (5) where compensation |$C_i$| is given by |$C_i = W_i - \gamma_i \, \left(X - P R_f\right) - \kappa(q_i)$|⁠, with |$W_{0,i} R_B \equiv X - P R_f$| capturing the performance of the benchmark. These preferences lead, in period |$2$|⁠, to the same portfolio that obtains under CARA expected utility, but investors have a preference for the early resolution of uncertainty.20 They can also arise in a setting with risk-neutral, profit-maximizing portfolio managers who invest on behalf of clients with CARA expected utility (see, e.g., the discussion in footnote 10 in van Nieuwerburgh and Veldkamp 2009). In this setting, by design, benchmarking does not affect the benchmarked investors’ risk appetite. As a result, changes in the fraction of benchmarked institutional investors, |$\Gamma$|⁠, do not affect aggregate risk-bearing capacity. 2.2 Portfolio choice and the equilibrium price An investor’s optimization problem must be solved in two stages, starting with the optimal portfolio choice in period |$2$|⁠, taking information choices as given. The following theorem characterizes investors’ optimal portfolio choice for arbitrary values of their posterior mean |$\hat{\mu}_{X,i} \equiv E_2[X \, | \, \mathcal{F}_i]$| and posterior precision |$h_i \equiv Var_2(X \, | \, \mathcal{F}_i)^{-1}$|⁠. Theorem 1. Conditional on an investor’s posterior beliefs, described by |$\hat{\mu}_{X,i}$| and |$h_i$|⁠, and the stock price |$P$|⁠, the optimal stock demand equals: \begin{equation} \label{eq:OptHold} \theta_i = h_i \, \frac{\hat{\mu}_{X,i} - P \, R_f}{\rho} + \, \gamma_i \, \equiv \, \theta_i^{MV} + \gamma_i. \end{equation} (6) The optimal demand for the stock has two components. First, for all investors, the standard mean-variance portfolio, |$\theta_i^{MV}$|⁠. Second, for benchmarked investors, a hedging demand |$\gamma_i>0$|⁠. Intuitively, benchmarked investors have the desire to acquire assets that do well when the benchmark does well, or, equivalently, assets that covary positively with the benchmark. In our single-asset economy, the stock, also serving as the benchmark, naturally covaries positively with the benchmark. The mean-variance portfolio, |$\theta_i^{MV}$|⁠, is independent of the strength of an investor’s benchmarking concerns, |$\gamma_i$|⁠. In contrast, the hedging component is increasing in investors benchmarking concerns, |$\gamma_i$|⁠, but is information insensitive; that is, it is not affected by their posterior beliefs. Intuitively, it is designed to closely track or, formally, covary with, the benchmark, and not meant for speculation. Aggregating the demand of both groups of institutional investors and imposing market-clearing delivers the equilibrium stock price: Theorem 2. Conditional on investors’ information choices, described by |$q_i, \, \forall i$|⁠, there exists a unique linear rational expectations equilibrium: \begin{align} P \, R_f = \frac{1}{\bar{h}} \, &\left(\frac{\mu_X}{\sigma_X^2} + \frac{\mu_Z \, \bar{q}}{\sigma_Z^2 \, \rho} + \rho \, \bar{\gamma} \right) + \frac{1}{\bar{h}} \, \left(\bar{h} - \frac{1}{\sigma_X^2}\right) \, X - \frac{1}{\bar{h}} \, \left(\frac{\bar{q}}{\rho \, \sigma_Z^2} + \rho \right) \, Z, \label{eq:EqPrc}\\ \end{align} (7) \begin{align} \text{where} \quad \, \, \, &h_0 \equiv \frac{1}{\sigma_X^2} + \frac{\bar{q}^2}{\rho^2 \, \sigma_Z^2}, \qquad \bar{h} \equiv h_0 + \bar{q}, \label{eq:Defh0hBar} \\ \end{align} (8) \begin{align} &\bar{q} \equiv \int_0^\Gamma q_i^\mathcal{BI} \, di + \int_\Gamma^1 q_i^{\mathcal{NI}} \, di, \quad \text{and} \quad \bar{\gamma} \equiv \int_0^\Gamma \gamma_i \, di. \label{eq:DefqBarGamBar} \end{align} (9) The characterization of the equilibrium price in (7) is standard for this type of economy, and the variables defined in (8) and (9) allow for intuitive interpretations. |$h_0$| equals the sum of the precisions from the prior, |$1/\sigma_X^2$|⁠, and from the price signal, |$\bar{q}^2/(\rho^2 \, \sigma_Z^2)$|⁠; hence, it characterizes the precision of public information. |$\bar{q}$| measures the precision of the private information of the average investor. Consequently, |$\bar{h}$| governs average aggregate precision, that is, the precision of public and private information of the average investor. Finally, |$\bar{\gamma}$| captures the degree of benchmarking in the economy, thereby aggregating the strength of the relative performance concerns of each benchmarked investor, |$\gamma_i$|⁠, and their size in the economy, |$\Gamma$|⁠. Note that, for ease of exposition, we assume for all graphical illustrations that the strength of the benchmarked investors’ relative performance concerns coincide: |$\gamma_i = \gamma, \forall i \in \mathcal{BI}$|⁠. As a result, |$\bar{\gamma}$| simplifies to |$\Gamma \, \gamma$| and, conditional on |$\gamma$|⁠, all variation in |$\bar{\gamma}$| is driven by the fraction of benchmarked institutions, |$\Gamma$|⁠. Conditional on information choices, the equilibrium price exceeds the price in an economy without benchmarking by |$(\rho \, \bar{\gamma})/\bar{h}$|⁠; hence, the price is increasing in the degree of benchmarking |$\bar{\gamma}$|⁠. Intuitively, conditional on a given supply |$Z$|⁠, the excess demand for the stock resulting from benchmarked investors’ hedging demand, drives up the price.21 It is important to highlight that information aggregation is not adversely affected by benchmarking in this setting. In particular, one can derive the following corollary that describes the ability of financial markets to aggregate private information in the presence of benchmarking: Corollary 1. Conditional on investors’ information choices, described by |$q_i, \, \forall i$|⁠, price informativeness, defined as the precision of the public price signal, is given by: \begin{equation} \label{eq:MVPrcInfo} h_0 - \frac{1}{\sigma_X^2} = \frac{\bar{q}^2}{\rho^2 \, \sigma_Z^2}.\end{equation} (10) Specifically, Equation (10) implies that, conditional on investors’ information choices (as aggregated by |$\bar{q}$|⁠), price informativeness is not affected by |$\bar{\gamma}$|⁠. Thus, given information choices, a shift in the degree of benchmarking in the economy has no impact on how much information is revealed by the public stock price or how much investors can learn from the price. Intuitively, because the aggregate hedging demand is fully predictable, it only affects the level of the stock price, but not its sensitivity to the payoff or the noise.22 2.3 Information choice Theorem 2 and Corollary 1 take the information environment as given, but information choices are actually an endogenous outcome of the model. At time |$t=1$|⁠, investor |$i$| chooses the precision of his private signal, |$q_i$|⁠, in order to maximize utility (5), anticipating his optimal portfolio choice in period |$2$|⁠. To impose rational expectations, we substitute the optimal stock demand (6) into wealth |$W_i$| and, hence, compensation, |$C_i$|⁠, which yields the following time-|$1$| objective function: \begin{equation} \label{eq:MVInfoObjFun} U_{1,i} = R_f \, \left(W_{0,i} - \frac{\kappa(q_i)}{R_f} \right) \,+ \frac{1}{2 \, \rho} \, E_1\left[z_i^2\right], \end{equation} (11) where |$z_i \equiv \sqrt{h_i} \, (\hat{\mu}_{X,i} - P \, R_f)$| denotes the investor’s time-|$2$| expected Sharpe ratio—a function of |$Y_i$|⁠, |$q_i$| and |$P$|⁠. Intuitively, |$E_1\left[z_i^2\right]$| governs the squared Sharpe ratio that investor |$i$| expects to achieve in the information choice period (⁠|$t=1$|⁠). The objective function captures the “information choice trade-off.” Higher precision |$q_i$| leads to higher information acquisition costs |$\kappa(q_i)$|⁠, thereby reducing time-|$1$| utility. On the other hand, higher precision |$q_i$| increases the posterior precision |$h_i$|⁠, the time-|$1$| expected squared Sharpe ratio |$E_1[z_i^2]$| and, thus, time-|$1$| utility. Notably, benchmarked investors (⁠|$i \in \mathcal{BI}$|⁠) “undo” the benchmarking component in their compensation; that is, a benchmarked investor’s time-|$1$| objective function does not depend on the strength of his individual benchmarking concerns, |$\gamma_i$| and, hence, coincides with the objective function of a nonbenchmarked investor. As a result, the two groups of institutional investors face the same information choice problem. Accordingly, in the following, we discuss the information choice problem of a generic institutional investor |$i \in \{\mathcal{BI}, \mathcal{NI}\}$|⁠, noting that |$q_i = q_i^{\mathcal{BI}} = q_i^{\mathcal{NI}}$|⁠. Computing the first-order condition of |$U_{1,i}$| with respect to |$q_i$|⁠, delivers an investor’s best information choice, given arbitrary precision choices by the other investors: Theorem 3. Conditional on the average private signal precision, |$\bar{q}$|⁠, investor |$i$|’s optimal signal precision |$q_i(\bar{q})$| is the unique solution of: \begin{align} \label{eq:MVBIR} 2 \, \kappa'(q_i) = \frac{1}{\rho} \, \underbrace{\left[\frac{1}{\bar{h}^2} \, \left(\rho^2 \left(\sigma_Z^2 + \left(\mu_Z - \bar{\gamma}\right)^2\right) + \bar{h} + \bar{q}\right)\right]}_{\equiv \, A(\bar{\gamma}, \bar{q})}.\end{align} (12) If |$\mu_Z - \bar{\gamma}>0$|⁠, an investor’s best information choice |$q_i(\bar{q})$| shifts down when the degree of benchmarking in the economy, |$\bar{\gamma}$|⁠, increases. Formally, |$\partial A/\partial \bar{\gamma} < 0$|⁠. Equation (12) characterizes that, at the optimum, the marginal cost of private information, |$\kappa'(q_i)$|⁠, equals the marginal benefit, which is governed by |$A(\bar{\gamma}, \bar{q})$| and risk aversion |$\rho$|⁠. Hence, the degree of benchmarking |$\bar{\gamma}$| affects the marginal value of private information. In particular, if the “effective supply” of the stock, that is, the supply after accounting for the aggregate hedging demand is positive (⁠|$\mu_Z - \bar{\gamma}>0$|⁠),23 the value of private information declines in the degree of benchmarking for benchmarked and nonbenchmarked investors. This effect is illustrated in Figure 2, which depicts an investor’s optimal signal precision |$q_i$| as a function of the average private signal precision |$\bar{q}$| for different degrees of benchmarking, as captured by the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Because of strategic substitutability (see, e.g., Grossman and Stiglitz 1980), the investor’s optimal signal precision is declining in the average private signal precision, regardless of the degree of benchmarking (see also the explicit expression (A12)). Intuitively, a higher average private signal precision implies that more information is revealed through the public price signal, thereby reducing investors’ incentives to acquire private information themselves. Figure 2 View largeDownload slide Information demand The figure depicts an investor’s optimal signal precision |$q_i$| as a function of the average private signal precision |$\bar{q}$| for different degrees of benchmarking, as captured by the fraction of benchmarked investors, |$\Gamma$|⁠. The graph is based on the framework described in Section 2.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = 1.0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BM}$|⁠, |$\bar{q} = 0.45$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 2 View largeDownload slide Information demand The figure depicts an investor’s optimal signal precision |$q_i$| as a function of the average private signal precision |$\bar{q}$| for different degrees of benchmarking, as captured by the fraction of benchmarked investors, |$\Gamma$|⁠. The graph is based on the framework described in Section 2.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = 1.0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BM}$|⁠, |$\bar{q} = 0.45$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. However, most important for us is that the investor’s optimal private signal precision shifts down as the degree of benchmarking increases, regardless of the average private signal precision. This result is driven by information-scale effects. Intuitively, when making information choices in period |$1$|⁠, each investor takes into account the expected number of shares in his portfolio that are sensitive to private information. In particular, one can rewrite investor |$i$|’s time-|$2$| expected squared Sharpe ratio (divided by |$\rho$|⁠) as \begin{equation*} \frac{z_i^2}{\rho} = h_i \, \frac{\hat{\mu}_{X,i} - P R_f}{\rho} \times \left(\hat{\mu}_{X,i} - P R_f\right) = \theta_i^{MV} \times \left(E_2[X] - P R_f\right).\end{equation*} Hence, the benefit of private information in (11) is governed by the product of the mean-variance portfolio (the information sensitive part of the portfolio) and the time-|$2$| expected excess stock return. Taking into account market-clearing, the aggregate hedging demand reduces the expected number of shares in the information sensitive part (⁠|$\theta_i^{MV}$|⁠).24 Because returns to information are increasing in the number of shares (i.e., one piece of information can be used for many shares), this implies a decline in the marginal value of private information for all investors. Alternatively, the aggregate hedging demand increases the stock price for all future states (see Theorem 2), thereby ambiguously reducing its expected excess return (⁠|$E_2[X] - P R_f$|⁠). This, in turn, reduces the potential rent from speculating with private information and, hence, the value of private information. It is instructive to also study the case of CARA expected utility. The optimal stock demand and the resulting price function are identical to the case of CARA utility with a preference for the early resolution of uncertainty and are given by (6) and (7). However, at the information choice stage, an investor’s best information choice differs. In particular, given arbitrary precision choices by the other investors, his optimal information choice is characterized by the following theorem: Theorem 4. Assume that investors have CARA expected utility. Conditional on the average private signal precision, |$\bar{q}$|⁠, investor |$i$| chooses a signal of precision |$q_i(\bar{q})$| such that: \begin{align*} 2 \, \kappa'(q_i) = \frac{1}{\rho} \, \frac{1}{h_0 + q_i}.\end{align*} Hence, the marginal benefit of private information is governed by the investor’s posterior precision |$h_0 + q_i$| and his risk aversion |$\rho$|⁠. Importantly, his optimal information choice |$q_i$| does not depend on the degree of benchmarking in the economy, |$\bar{\gamma}$|⁠; implying that information-scale effects are absent. Graphically, this implies that while the investor’s best information response is declining in the average private signal precision |$\bar{q}$|⁠, the three lines in Figure 2 (for different fraction of benchmarked investors) would coincide. We can now return to the case of mean-variance preferences. Recall that Theorem 3 takes the information choices of the other investors as given. However, in equilibrium, the information choice of each investor, |$q_i$|⁠, affects aggregate average precision, |$\bar{q}$|⁠, which, in turn, affects each investor’s information choice. Therefore, the equilibrium value of the average private signal precision, |$\bar{q}$|⁠, is determined by plugging |$q_i$| in (12) into its definition in (9): Theorem 5. The average private signal precision, |$\bar{q}$|⁠, is the unique solution to: \begin{equation} \label{eq:InfoEqMV} \bar{q} = \int_0^\Gamma \, q_i^\mathcal{BI}(\bar{q}) \, di + \int_\Gamma^1 \, q_i^{\mathcal{NI}}(\bar{q}) \, di = \int_0^1 q_i(\bar{q}) \, di.\end{equation} (13) If |$\mu_Z - \bar{\gamma} > 0$|⁠, the average private signal precision, |$\bar{q}$|⁠, is declining in the degree of benchmarking, |$\bar{\gamma}$|⁠. Formally, |$d \bar{q}/d \bar{\gamma} < 0$|⁠. This theorem summarizes the key result of this section: an increase in the degree of benchmarking in the economy, |$\bar{\gamma}$|⁠, leads to a decline in the precision of the private information of all investors (for the economically relevant case of |$\mu_Z - \bar{\gamma} > 0$|⁠). Graphically, this result is illustrated in Figure 2, which shows that the investors’ equilibrium precision choice, characterized by the intersection between the respective best information response function and the |$45$|-degree line, declines in the fraction of benchmarked investors, |$\Gamma$|⁠. Panel (a) of Figure 3 shows this even more explicitly by depicting the optimal precision of the investors’ private signals, |$q_i$|⁠, as a function of the fraction of benchmarked investors, |$\Gamma$|⁠. As a result of the lower precision choices, equilibrium price informativeness, given by |$\bar{q}^2/(\rho^2 \, \sigma_Z^2)$|⁠, also declines in the degree of benchmarking. This is illustrated in panel (b) of Figure 3. Figure 3 View largeDownload slide Equilibrium information choice The figure shows the optimal private signal precision, |$q_i$|⁠, in equilibrium (panel (a)) and the precision of the public price signal (“price informativeness”) |$\bar{q}^2/(\rho^2 \, \sigma_Z^2)$| (panel (b)), as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. The graphs are based on the framework described in Section 2.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = 1.0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BM}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 3 View largeDownload slide Equilibrium information choice The figure shows the optimal private signal precision, |$q_i$|⁠, in equilibrium (panel (a)) and the precision of the public price signal (“price informativeness”) |$\bar{q}^2/(\rho^2 \, \sigma_Z^2)$| (panel (b)), as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. The graphs are based on the framework described in Section 2.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = 1.0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BM}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Remark 1. It is also instructive to consider the case of a single (small) benchmarked institutional investor with constant absolute-risk aversion and a preference for the early resolution of uncertainty. Because the marginal value of private information |$A(\bar{\gamma}, \bar{q})$| does not depend on the strength of the investor’s individual benchmarking concerns, |$\gamma_i$|⁠, but only on the aggregate degree of benchmarking |$\bar{\gamma}$|⁠, such an investor’s information choice is actually not affected. Intuitively, the expected number of shares in his portfolio that are sensitive to private information does not decline in the absence of an aggregate hedging demand. 2.4 Unconditional asset prices and return moments Benchmarking also affects the unconditional stock price and return moments. The following theorem summarizes a first set of results: Theorem 6. The total derivative of the unconditional stock price |$S \equiv E_1[P R_f]$| and of the unconditional expected excess return |$M \equiv E_1[X - P R_f]$| with respect to the degree of benchmarking, |$\bar{\gamma}$|⁠, are given by: \begin{align*} \frac{d S}{d \bar{\gamma}} &= \frac{1}{\bar{h}} \, \rho + \frac{1}{\bar{h}^2} \, \rho \, (\mu_Z - \bar{\gamma}) \, \frac{d \bar{h}}{d \bar{\gamma}}, \qquad \text{and} \qquad \frac{d M}{d \bar{\gamma}} = \frac{- \, \rho}{\bar{h}} + \frac{-\rho}{\bar{h}^2} \, (\mu_Z - \bar{\gamma}) \, \frac{d \bar{h}}{d \bar{\gamma}}.\end{align*} The stock price is directly affected by the excess demand and indirectly affected by the induced change in price informativeness. The direct effect (first component) leads to an increase in the price. The indirect effect (second component) is of the opposite sign (if |$\mu_Z - \bar{\gamma}>0$|⁠) and lowers the price. Intuitively, with lower price informativeness risk-averse investors command a price discount. In equilibrium, the total impact on the stock price depends on the relative importance of the two effects. We find that for realistic parameter values the first effect always dominates; that is, the unconditional price increases, as illustrated in panel (a) of Figure 4. The impact of benchmarking on the expected excess return follows accordingly but is of opposite sign. The direct (indirect) effect reduces (increases) the expected return, because it increases (reduces) the stock price. Finally, benchmarking also affects the unconditional stock return variance: Theorem 7. The total derivative of the unconditional return variance |$V^2 \equiv Var_1(X - P R_f)$| with respect to the degree of benchmarking, |$\bar{\gamma}$|⁠, is given by: \begin{equation*} \frac{d V^2}{d \bar{\gamma}} = - \, \frac{1}{\bar{h}^3}\, \frac{d \bar{h}}{d \bar{\gamma}}.\end{equation*} In particular, an increase in the degree of benchmarking, |$\bar{\gamma}$|⁠, unambiguously leads to an increase in the unconditional return variance—due to the decline in price informativeness (⁠|$d \bar{h}/d \bar{\gamma}<0$|⁠). Panel (b) of Figure 4 illustrates this. Figure 4 View largeDownload slide Equilibrium price and return volatility The figure shows the unconditional expected equilibrium stock price |$S$| (panel (a)) and the unconditional stock return volatility |$V$| (panel (b)), as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. The graphs are based on the framework described in Section 2.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = 1.0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BM}$| and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 4 View largeDownload slide Equilibrium price and return volatility The figure shows the unconditional expected equilibrium stock price |$S$| (panel (a)) and the unconditional stock return volatility |$V$| (panel (b)), as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. The graphs are based on the framework described in Section 2.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = 1.0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BM}$| and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. 3. Benchmarking and Risk-Taking Effects In this section, we discuss a second channel through which benchmarking affects information choice. For this purpose, we rely on a model with CRRA preferences, which is considerably less tractable.25 In particular, we demonstrate that benchmarking limits an investor’s willingness to speculate. As a result, information aggregation is now also adversely affected. Hence, this “risk-taking effect” amplifies the decline in price informativeness. Moreover, benchmarked investors choose a lower precision of private information and, as a result, have lower expected portfolio returns than do nonbenchmarked investors. 3.1 Setup We assume that investors have CRRA preferences. Their time-|$1$| utility function |$U_{1,i}^{\text{CRRA}}$| is given by:26 \begin{equation} \label{eq:CRRAutil} U_{1,i}^{\text{CRRA}} = E_1\biggl[E_2\biggl[\frac{C_i^{1-\rho}}{1-\rho}\biggr]\biggr] = E_1\biggl[\frac{C_i^{1-\rho}}{1-\rho}\biggr], \end{equation} (14) where |$\rho$| denotes the curvature parameter in utility and |$C_i = W_i - \gamma_i \, W_{0,i} \, (X/P) - \kappa(q_i)$| denotes the investor’s compensation, with |$R_B \equiv (X/P)$| capturing the performance of the benchmark.27,28 For ease of exposition and numerical convenience, we assume that the payoff, |$X \geq 0$|⁠, is binomially distributed, with equally likely realizations |$X_H \equiv \mu_X + \sigma_X$| and |$X_L \equiv \mu_X - \sigma_X$|⁠. Similarly, the investors’ private signals, |$Y_i \in \{Y_H, Y_L\}$|⁠, are binomially distributed.29 Intuitively, a higher precision of the private signal, |$q_i$|⁠, increases an investor’s posterior precision by increasing the correlation between the payoff, |$X$|⁠, and his private signal, |$Y_i$|⁠; formally, |$\mathbb{P}[X_o \, | \, Y_i = Y_o] = \frac{1}{2} + \frac{1}{2} \, \sqrt{\frac{q_i}{q_i + 4}}$|⁠, |$o \in\{H,L\}$|⁠. To guarantee a positive supply of the stock and rule out negative compensation (for which CRRA preferences are undefined), we assume that the supply of the stock, |$Z$|⁠, is lognormally distributed, with |$\ln Z \sim \mathcal{N}(\mu_Z, \sigma_Z^2)$|⁠. It is important to highlight that, in this setting, the strength of an individual investor’s benchmarking concerns, |$\gamma_i$|⁠, affects his local risk aversion: Lemma 1. The local curvature of the CRRA-utility function (14) with respect to terminal wealth, |$W_i$|⁠; that is, the local coefficient of relative-risk aversion, |$\hat{\rho}_i$|⁠, is given by: \begin{equation} \label{eq:LocalRiskAversion} \hat{\rho}_{i} = \rho \, \, \frac{W_i}{W_i - \gamma_i \, W_{0,i} \, R_B - \kappa(q_i)}.\end{equation} (15) If |$\gamma_i>0$|⁠, it holds that |$\hat{\rho}_{i} \geq \rho$| and |$\partial \hat{\rho}_{i}/\partial \gamma_i \geq 0$|⁠, with strict inequalities for the case |$R_B > 0$| (or, equivalently, |$X > 0$|⁠). Lemma 1 shows that the risk aversion of a benchmarked investor, |$\hat{\rho}_{i}$|⁠, exceeds the risk aversion of a nonbenchmarked investor (equal to |$\rho$|⁠) and is increasing in the strength of his benchmarking concerns |$\gamma_{i}$|⁠.30 Intuitively, keeping |$\gamma_i \, W_{0,i} \, R_B$| fixed, a given proportional movement in wealth |$W_i$| has a stronger impact on utility (14) if the wealth in excess of the benchmarking component, |$W_i - \gamma_i \, W_{0,i} \, R_B$|⁠, is low.31 Consequently, variations in the fraction of benchmarked investors, |$\Gamma$|⁠, imply changes in the aggregate risk-bearing capacity in the economy; in particular, an increase in the fraction of benchmarked institutions leads to a decline in the aggregate risk-bearing capacity. Note that, for all graphical illustrations, we again assume that the strength of the benchmarked investors’ benchmarking concerns coincide: |$\gamma_i = \gamma > 0, \forall i \in \mathcal{BI}$|⁠, while nonbenchmarked investors have no relative performance concerns: |$\gamma_i = 0, \, \forall i \in \mathcal{NI}$|⁠. As a result, all variation in the degree of benchmarking in the economy is driven by changes in the fraction of benchmarked investors, |$\Gamma$|⁠. 3.2 Portfolio choice At time |$t=2$|⁠, investors must choose their optimal portfolio in order to maximize (3), conditional on their posterior beliefs, described by |$\mathcal{F}_i$|⁠, and taking the price |$P$| as given. For illustrative purposes, one can derive the following approximate solution for the optimal fraction of wealth invested into the stock, |$\phi_i \equiv (\theta_i \, P) / W_{0,i}$|⁠:32 \begin{equation} \label{eq:CRRAapproxPfChoice} \phi_i \approx \frac{E_2[r^e \, \bigl| \, \mathcal{F}_i\bigr]}{\frac{\rho}{1-\gamma_i} \, Var_2[r^e \, \bigl| \, \mathcal{F}_i\bigr]} \, + \, \gamma_i \equiv \phi_i^{MV} + \gamma_i, \end{equation} (16) where |$r^e$| denotes the stock’s excess return, |$r^e \equiv X/P - R_f$|⁠. The optimal demand for the stock has the same two components used in the case of constant absolute-risk aversion. First, the mean-variance portfolio, |$\phi_i^{MV}$|⁠. But, note that this is now the mean-variance portfolio of an investor with local risk aversion |$\rho/(1-\gamma_i)$|.33 As a result, it is declining in the strength of an investor’s benchmarking concerns, |$\gamma_i$|⁠. Second, the hedging portfolio, |$\gamma_i$|⁠, which is increasing in an investor’s benchmarking concerns, but is information insensitive. Naturally, for nonbenchmarked investors (⁠|$\gamma_i = 0$|⁠), the demand reduces to the standard mean-variance portfolio of an investor with risk aversion |$\rho$|⁠. Figure 5 illustrates how the optimal stock demand of a benchmarked investor varies with the strength of his benchmarking concerns, |$\gamma_i$|⁠. In particular, panel (a) shows that the overall portfolio share of the stock, |$\phi_i$|⁠, is increasing in an investor’s benchmarking concerns |$\gamma_i$|—comparable to the case of constant absolute-risk aversion. But, due to the decline in the mean-variance portfolio, |$\phi_i^{MV}$|⁠, the overall increase is substantially smaller. Figure 5 View largeDownload slide Stock demand The figure illustrates a benchmarked investor’s portfolio choice for various levels of the strength of his benchmark concerns, |$\gamma_i$| and conditional on a given signal precision |$q_i$|⁠. Panel (a) shows the expected stock demand and the mean-variance portfolio component. Panel (b) depicts the stock demand conditional on a specific signal realization |$Y_i \in \{Y_H, Y_L\}$|⁠. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$q_i = 0.1667$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 5 View largeDownload slide Stock demand The figure illustrates a benchmarked investor’s portfolio choice for various levels of the strength of his benchmark concerns, |$\gamma_i$| and conditional on a given signal precision |$q_i$|⁠. Panel (a) shows the expected stock demand and the mean-variance portfolio component. Panel (b) depicts the stock demand conditional on a specific signal realization |$Y_i \in \{Y_H, Y_L\}$|⁠. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$q_i = 0.1667$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Panel (b) further illustrates that effect. It shows an investor’s optimal stock demand, conditional on a specific signal realization |$Y_i \in \{Y_H, Y_L\}$|⁠. As expected, the investor, in general, overweights the stock following a high signal, |$Y_H$|⁠, and vice versa for a low signal, |$Y_L$|⁠. Most importantly, however, the “spread” between a benchmarked investor’s stock demand following a positive and a negative signal narrows as his benchmarking concerns, |$\gamma_i$|⁠, strengthen. Hence, in the presence of benchmarking, the investor’s willingness to speculate declines (because of his higher local risk aversion). This result is in clear contrast to the case of constant absolute-risk aversion (with or without a preference for the early resolution of uncertainty), for which the mean-variance portfolio component is independent of the strength of an investor’s benchmarking concerns. 3.3 Information choice To build the basic intuition for the incremental impact of the risk-taking effect on information choice in the clearest possible way, we first consider the case of a single (small) benchmarked investor. Hence, benchmarking does not affect the equilibrium stock price or the effective supply. Recall from Remark 1 that, in this case, the strength of the investor’s individual benchmarking concerns, |$\gamma_i$|⁠, does not affect his information choice with mean-variance preferences. This result does not hold with CRRA preferences. In particular, panel (a) of Figure 6 depicts the optimal signal precision |$q_i$| of a single (small) benchmarked investor as a function of the average private signal precision |$\bar{q}$|—for different levels of the strength of his benchmarking concerns, |$\gamma_i$|⁠. As expected, the investor’s optimal signal precision is declining in the average private signal precision. Most importantly, the optimal signal precision is now also declining in the strength of the investor’s individual benchmarking concerns, |$\gamma_i$|⁠. Intuitively, the investor anticipates that his time-|$2$| portfolio choice will be less sensitive to the realization of his private signal because he trades less aggressively, and, consequently, the value of private information declines. Panel (b) more explicitly shows this by depicting the optimal precision of the investor’s private signal, |$q_i$|⁠, as function of the strength of his benchmarking concerns, |$\gamma_i$|⁠. Figure 6 View largeDownload slide Information demand The figure depicts the optimal signal precision |$q_i$| of a single (small) benchmarked investor. Panel (a) shows the optimal signal precision as a function of the average private signal precision |$\bar{q}$|—for different levels of the strength of his benchmark concerns, |$\gamma_i$|⁠. Panel (b) shows the optimal signal precision as function of the strength of the investor’s benchmark concerns, |$\gamma_i$|⁠. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 6 View largeDownload slide Information demand The figure depicts the optimal signal precision |$q_i$| of a single (small) benchmarked investor. Panel (a) shows the optimal signal precision as a function of the average private signal precision |$\bar{q}$|—for different levels of the strength of his benchmark concerns, |$\gamma_i$|⁠. Panel (b) shows the optimal signal precision as function of the strength of the investor’s benchmark concerns, |$\gamma_i$|⁠. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Note that, because information-scale effects are also present with CRRA preferences, varying the degree of benchmarking in the economy |$\bar{\gamma}$| (e.g., the fraction of benchmarked investors) would also lead to a decline in the optimal signal precision—similar to the case of mean-variance preferences (as illustrated in Figure 2). However, the key new result is that the investor’s individual benchmarking concerns |$\gamma_i$| now also play a role. The lower value of information for benchmarked investors implies that, in equilibrium, benchmarked institutions collect less information than nonbenchmarked institutions—in stark contrast to the earlier results (in which both groups of market participants chose the same signal precision). Panel (a) of Figure 7 illustrates this. Figure 7 View largeDownload slide Equilibrium information choice The figure illustrates equilibrium signal precisions, as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) shows the optimal private signal precision of the two groups of investors, and panel (b) depicts the precision of the public price signal (“price informativeness”). Precision is measured as |$R^2$|⁠, that is, the fraction of the variance of the payoff |$X$| that is explained by the investors’ private information and the stock price, respectively. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 7 View largeDownload slide Equilibrium information choice The figure illustrates equilibrium signal precisions, as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) shows the optimal private signal precision of the two groups of investors, and panel (b) depicts the precision of the public price signal (“price informativeness”). Precision is measured as |$R^2$|⁠, that is, the fraction of the variance of the payoff |$X$| that is explained by the investors’ private information and the stock price, respectively. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. The figure also shows the novel result that both groups of investors choose a higher precision for their private signals as the share of benchmarked investors, |$\Gamma$|⁠, increases. To understand this result, recall that an increase in the fraction of benchmarked investors leads to a decline in the aggregate risk-bearing capacity. Consequently, prices reveal less information, which, in turn, increases the marginal benefit of private information and, hence, the incentives of all investors to choose a more precise signal. However, as shown in panel (b) of Figure 7, price informativeness—the amount of private information aggregated and revealed by the stock price—is declining in the fraction of benchmarked investors. This result is driven by two distinct economic forces: information acquisition and information aggregation. First, an increase in the share of benchmarked investors implies that better-informed nonbenchmarked investors are replaced by less-informed benchmarked investors (as illustrated in panel (a)). Second, it implies a shift toward a group of investors that trades less aggressively based on available private information—because of their lower risk appetite. Consequently, less information can be aggregated into the price. Both effects imply a decline in price informativeness. To (quantitatively) disentangle this effect from the information-scale effect, panel (b) of Figure 7 also shows how price informativeness varies with the fraction of benchmarked investors, if one would keep the aggregate risk-bearing capacity fixed.34 While the majority of the decline is due to the reduction in aggregate risk-bearing capacity, the impact of the information-scale effect is nonnegligible (about |$25\%$| of the decline for the parameters used for the illustration). 3.4 Unconditional asset prices and return moments Similar to the case of mean-variance preferences, the unconditional stock price is affected through two conflicting effects, such that the net effect depends on their relative importance. On the one hand, the aggregate hedging demand pushes up the price. On the other hand, the decline in price informativeness increases posterior uncertainty, and, hence, risk-averse investors command a lower price. Panel (a) of Figure 8 shows that, due to the amplification of the decline in price informativeness, the stock price can decline in the fraction of benchmarked investors. This is the case if information acquisition costs are low (⁠|$\omega_{low}$|⁠) and, hence, price informativeness declines from a high initial level. This result is distinctly different from the setting with mean-variance preferences in which—for realistic parameters values—the stock price always increases in the fraction of benchmarked investors. As one increases information acquisition costs, the stock price first becomes essentially flat (⁠|$\omega_{high}$|⁠) and in the limit approaches the case without information choice in which the price is always increasing in the fraction of benchmarked investors. The implications for the expected excess return are simply of the opposite sign. Figure 8 View largeDownload slide Equilibrium price and return volatility The figure shows the unconditional expected equilibrium stock price (panel (a)) and the unconditional stock return volatility (panel (b)), as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$| and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega_{low} = 0.015$|⁠, and |$\omega_{high} = 0.045$|⁠. Figure 8 View largeDownload slide Equilibrium price and return volatility The figure shows the unconditional expected equilibrium stock price (panel (a)) and the unconditional stock return volatility (panel (b)), as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$| and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega_{low} = 0.015$|⁠, and |$\omega_{high} = 0.045$|⁠. Similar to the case of mean-variance preferences, the stock’s return volatility is unambiguously increasing in the share of benchmarked investors because of the higher posterior uncertainty. This is illustrated in panel (b) of Figure 8. The effect is quantitatively stronger for lower information acquisition costs and unique to our setting with endogenous information choice (at least for static frameworks). 3.5 Portfolio returns In practice, institutional investors typically invest on behalf of their clients. Thus, the differences in the institutional investors’ portfolio and information choices (resulting from benchmarking) will also affect the clients’ expected returns. As shown in panel (a) of Figure 9, the difference between the expected portfolio return of nonbenchmarked and benchmarked institutional investors is always positive. Hence, nonbenchmarked investors generate higher expected returns than their benchmarked peers. Moreover, as the fraction of benchmarked investors increases, this “return gap” widens. Figure 9 View largeDownload slide Return and information gap The figure illustrates the return and information gap between the two groups of institutional investors, as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) shows the difference between the expected portfolio return of nonbenchmarked and benchmarked institutions. Panel (b) shows the precision of investors’ information sets, |$\mathcal{F}_i = \{Y_i, P\}$|⁠, measured as a fraction of the variance of the payoff |$X$| that they can explain. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 9 View largeDownload slide Return and information gap The figure illustrates the return and information gap between the two groups of institutional investors, as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) shows the difference between the expected portfolio return of nonbenchmarked and benchmarked institutions. Panel (b) shows the precision of investors’ information sets, |$\mathcal{F}_i = \{Y_i, P\}$|⁠, measured as a fraction of the variance of the payoff |$X$| that they can explain. The graphs are based on the CRRA framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Recall that benchmarked investors always choose a lower precision of private information than nonbenchmarked investors (see panel (a) of Figure 7). This has two effects for expected portfolio returns. First, nonbenchmarked investors hold, on average, more of the stock, because their superior information renders the investment less risky. Because of the stock’s positive risk premium, this increases their expected portfolio return. Second, conditional on a signal realization, nonbenchmarked investors’ trades are more profitable. Both effects unambiguously lead to a higher expected portfolio return for nonbenchmarked investors. As the fraction of benchmarked investors increases, price informativeness declines and less information is revealed through the public stock price. Thus, the importance of private information rises. Consequently, the “information gap” between investors’ information sets |$\mathcal{F}_i$| widens (panel (b) of Figure 9) and so does the “return gap.” Note that these effects are not present for mean-variance preferences because, in this case, the precision of private information of benchmarked and nonbenchmarked institutional investors coincides. 3.6 Robustness The results are robust to changes in the parameter values. Intuitively, the increase in the benchmarked investors’ risk aversion, as discussed in Lemma 1, does not rely on specific parameter values. Hence, information aggregation is always adversely affected. Moreover, their higher local risk aversion implies that benchmarked investors choose less precise private information. Consequently, the difference in the expected portfolio returns between benchmarked and nonbenchmarked investors is positive and increases in the fraction of benchmarked investors. The only quantities for which the impact of benchmarking depends on the choice of parameter values are, as discussed above, the unconditional expected stock price and excess return. In both cases, two economic forces are in conflict, with the net effect depending on their magnitudes. 4. Extensions This section introduces two extensions of our basic economic framework. First, an extension to multiple stocks and, second, an extension to asymmetric benchmarking concerns. In both cases, we rely on CRRA preferences, capturing information-scale and risk-taking effects. 4.1 Multiple stocks Our objective in this section is to study how the results generalize in an economy with multiple stocks. In particular, we consider an economy with two symmetric stocks, |$k \in \{1,2\}$|⁠, in which each investor has to decide simultaneously on the signal precision for the two stocks, |$q_{i,k}$|⁠. Similar to the one-stock setup described in Section 3.1, the stocks’ payoffs, |$X_k$|⁠, and the private signals, |$Y_{i,k}$|⁠, are assumed to be binomially distributed, and the supply |$Z_k$| is assumed to be lognormal. In addition, we assume that payoffs, signals, and supplies are independent across assets.35 The first stock (the “index stock”) also serves as the benchmark such that the compensation of the benchmarked investors is declining in its (gross) return: |$\gamma_{i,1}>0, \forall i \in \mathcal{BI}$|⁠. In contrast, the second stock (the “nonindex stock”) is not part of the benchmark: |$\gamma_{i,2}=0, \forall i$|⁠. Panel (a) of Figure 10 illustrates that price informativeness is declining for both stocks in the fraction of benchmarked investors, |$\Gamma$|⁠. However, the decline in price informativeness is more pronounced for the index stock. Intuitively, the increase in the benchmarked investors’ risk aversion limits their speculative activities in the nonindex stock as well.36 However, because of the absence of an information insensitive aggregate hedging demand, information-scale effects are absent. Hence, the expected number of shares of the nonindex stock in investors’ portfolios that are sensitive to private information does not decline. Consequently, a piece of information can be applied to more shares, and, hence, the value of private information and price informativeness is higher. Figure 10 View largeDownload slide Multiple stocks The figure shows price informativeness and the unconditional expected stock price of the index and nonindex stock, as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) depicts the precision of the public price signal (“price informativeness”), measured as the fraction of the variance (⁠|$R^2$|⁠) of the payoff |$X_k$| that is explained by the corresponding stock price. Panel (b) shows the unconditional expected equilibrium stock price. The graphs are based on the symmetric two-stock CRRA framework described in Section 4.1, with the following parameter values: |$\mu_{X,k} = 1.05$|⁠, |$\sigma_{X,k}^2 = 0.25$|⁠, |$\mu_{Z,k} = \ln(1) = 0$|⁠, |$\sigma_{Z,k}^2 = 0.2$| for |$k \in \{1,2\}$|⁠, |$\rho = 3$|⁠, |$\gamma_{i,1} = \gamma = 1/3 \, \forall i \in \mathcal{BM}$|⁠, |$\gamma_{i,2} = 0 \, \forall i$|⁠, and an information cost function |$\kappa(q_{i,k}) = \omega \, q_{i,k}^2$|⁠, |$k \in \{1,2\}$|⁠, with |$\omega = 0.015$|⁠. Figure 10 View largeDownload slide Multiple stocks The figure shows price informativeness and the unconditional expected stock price of the index and nonindex stock, as functions of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) depicts the precision of the public price signal (“price informativeness”), measured as the fraction of the variance (⁠|$R^2$|⁠) of the payoff |$X_k$| that is explained by the corresponding stock price. Panel (b) shows the unconditional expected equilibrium stock price. The graphs are based on the symmetric two-stock CRRA framework described in Section 4.1, with the following parameter values: |$\mu_{X,k} = 1.05$|⁠, |$\sigma_{X,k}^2 = 0.25$|⁠, |$\mu_{Z,k} = \ln(1) = 0$|⁠, |$\sigma_{Z,k}^2 = 0.2$| for |$k \in \{1,2\}$|⁠, |$\rho = 3$|⁠, |$\gamma_{i,1} = \gamma = 1/3 \, \forall i \in \mathcal{BM}$|⁠, |$\gamma_{i,2} = 0 \, \forall i$|⁠, and an information cost function |$\kappa(q_{i,k}) = \omega \, q_{i,k}^2$|⁠, |$k \in \{1,2\}$|⁠, with |$\omega = 0.015$|⁠. Panel (b) shows that the price of the index stock is always higher than that of the nonindex stock, with the price gap widening as the fraction of benchmarked investors increases. Intuitively, the aggregate hedging demand for the index stock increases its price relative to the nonindex stock. At the same time, the stronger decline in price informativeness for the index stock implies more of a discount. In equilibrium, the first effect is quantitatively stronger, explaining the observed price pattern. The results for the stocks’ expected returns follow accordingly. For both stocks, the unconditional return volatility is increasing because price informativeness declines. However, because of the more pronounced decline in the index stock’s price informativeness, the increase in its return volatility is stronger as well. Finally, nonbenchmarked investors’ private information in both assets is more precise, so that the results regarding their superior expected portfolio return carry over. 4.2 Asymmetric benchmarking concerns Second, we consider an extension to asymmetric benchmarking concerns. In particular, while U.S. mutual funds are limited to use linear (Fulcrum) performance fees, asymmetric performance fees for mutual funds can be used outside the United States and are also much more standard in other asset classes (such as hedge funds or private equity). Moreover, implicit benchmarking concerns at the fund manager’s level or arising from the flow-performance relation are typically also rather asymmetric.37 The easiest way to incorporate benchmarking concerns that are nonlinear in the benchmark’s return is to model a benchmarked investor’s compensation, |$C_i$|⁠, as \begin{equation*} C_i(W_i, R_B) = W_i - W_{0,i} \, R_B \, \left(\gamma_i^+ \, {\rm 1}\kern-0.24em{\rm I}_{R_B > 1} + \gamma_i^- \, {\rm 1}\kern-0.24em{\rm I}_{R_B \leq 1}\right) - \kappa(q_i), \end{equation*} where |${\rm 1}\kern-0.24em{\rm I}_{\cdot}$| denotes the indicator function. In particular, with this specification, the sensitivity of the investor’s compensation with respect to positive (net) benchmark returns, |$\gamma_i^+$|⁠, can differ from the sensitivity with respect to negative (net) benchmark returns, |$\gamma_i^-$|⁠.38 Otherwise, the setup is similar to the one described in Section 3.1. As a full analysis of the equilibrium with asymmetric benchmarking concerns is outside the scope of this paper, we concentrate on the case of a single (small) benchmarked investor with asymmetric benchmarking concerns, in which case benchmarking does not affect the equilibrium stock price. In particular, consider the case of varying the strength of the investor’s benchmarking concerns for positive benchmark returns, |$\gamma_i^+$|⁠, while setting the strength of his benchmarking concerns for negative benchmark returns, |$\gamma_i^-$|⁠, to 0.39 Panel (a) of Figure 11 shows the investor’s stock demand conditional on a specific signal realization |$Y_i \in \{Y_H, Y_L\}$|⁠. Interestingly, the investor’s willingness to speculate, measured by the “spread” between his stock demand following a high and a low signal, is declining substantially less than in the case of symmetric benchmarking concerns (see panel (b) of Figure 5). On the one hand, when the investor observes a signal indicating a low payoff and, hence, a negative benchmark return, the aggressiveness with which he trades is essentially unaffected because |$\gamma_i^- = 0$|⁠. On the other hand, when the investor observes a signal indicating a high payoff and, hence, a positive benchmark return, the aggressiveness with which he trades is only marginally affected relative to the no-benchmarking case (although |$\gamma_i^+ > 0$|⁠). Intuitively, because the investor overweights the benchmark following a high signal (i.e., the stock’s portfolio share is above |$1$|⁠), his expected final wealth, |$W_i$|⁠, relative to the benchmark component |$\gamma_i^+ \, R_B$|⁠, is high, limiting the increase in local risk aversion (15). Figure 11 View largeDownload slide Asymmetric benchmarking concerns The figure illustrates a single (small) benchmarked investor’s stock and information demand, as functions of the strength of his benchmarking concerns for positive benchmark returns |$\gamma_i^+$|⁠. Panel (a) shows the stock demand conditional on a specific signal realization |$Y_i \in \{Y_H, Y_L\}$|⁠, with exogenous signal precision |$q_i = 0.1667$|⁠. Panel (b) depicts the optimal, endogenous signal precision |$q_i$|⁠. The graphs are based on the framework described in Section 4.2, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i^- = 0$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure 11 View largeDownload slide Asymmetric benchmarking concerns The figure illustrates a single (small) benchmarked investor’s stock and information demand, as functions of the strength of his benchmarking concerns for positive benchmark returns |$\gamma_i^+$|⁠. Panel (a) shows the stock demand conditional on a specific signal realization |$Y_i \in \{Y_H, Y_L\}$|⁠, with exogenous signal precision |$q_i = 0.1667$|⁠. Panel (b) depicts the optimal, endogenous signal precision |$q_i$|⁠. The graphs are based on the framework described in Section 4.2, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i^- = 0$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Consequently, the sensitivity of the investor’s portfolio with respect to private information declines only marginally as the strength of the investor’s benchmarking concerns for positive benchmark returns, |$\gamma_i^+$|⁠, increases. Hence, as shown in panel (b) of Figure 11, the underprovision of private information seems to be considerably weaker than in the case of symmetric benchmarking concerns. That is, the investor chooses a substantially higher signal precision than in the linear case (see panel (b) of Figure 6)—although the optimal signal precision is still declining in the strength of his benchmarking concerns. 5. Key Predictions and Conclusion Relative performance concerns play a key role in the decisions of institutional investors who are in the business of acquiring information and using that information for portfolio management. In this paper, we develop an economic framework that explicitly accounts for benchmarking and investors’ joint portfolio and information choice. We identify two distinct economic channels through which benchmarking affects informational efficiency. Key to both economic mechanisms is the interaction between investors’ portfolio and information choices. First, the information insensitive aggregate hedging demand of benchmarked investors reduces, in equilibrium, the expected number of shares in investors’ portfolios that are sensitive to private information. This information-scale effect reduces the value of private information but does not affect information aggregation. Second, an investor’s individual benchmarking concerns limit his willingness to speculate, so that the sensitivity of his portfolio with respect to private information declines. This risk-taking effect leads to a decline in the value of private information but also adversely affects information aggregation. Notably, incorporating risk-taking effects leads to qualitatively different implications. For example, benchmarked and nonbenchmarked investors differ in their information choice and, hence, in their expected portfolio returns. Moreover, the equilibrium stock price can decline in the fraction of benchmarked investors. The model generates a rich set of empirically refutable predictions. For example, similar to our results, the literature on “index effects” has found a higher price and a lower Sharpe ratio for index stocks relative to nonindex stocks. While these results are not unique to our model (see, e.g., Cuoco and Kaniel 2011; Basak and Pavlova 2013), the joint analysis of portfolio and information choices also leads to novel and unique predictions. In particular, our model predicts a lower price informativeness for index stocks. In line with this prediction, Israeli, Lee, and Sridharan (2017) document that an increase in exchange traded fund (ETF) ownership is associated with less-informative security prices. Also, our model makes unique predictions about the expected portfolio returns of benchmarked and nonbenchmarked investors that would not arise in models with symmetric information and can be confronted with the data. In this paper, the benchmarking concerns of the institutional investors are exogenous. However, in the presence of agency frictions, they might be an endogenous outcome; for example, the incentives between institutional investors and their clients (see, e.g., Buffa, Vayanos, and Woolley 2017 or Sotes-Paladino and Zapatero 2016) and between fund managers within an asset management firm (see, e.g., van Binsbergen, Brandt, and Koijen 2008) might be better aligned due to benchmarking. In this regard, our paper highlights a novel tension between benchmarking as a tool to align incentives and its adverse effects on managers’ information acquisition and, hence, their portfolio returns. Consequently, a logical extension of our work would be to study optimal asset management contracts in a setting with agency frictions and information acquisition.40 In particular, allowing for nonlinear contracts might lead to many new insights regarding the optimal compensation in the asset-management industry. Also, extensions of our framework can be used to study the optimal size of benchmarked investors in the economy (Pástor and Stambaugh 2012) or the implications of passive investing. Appendix A. Proofs for Section 2 Proof of Theorem 1 Plugging wealth |$W_i$| into an investor’s compensation |$C_i = W_i - \gamma_i \, \left(X - P R_f\right) - \kappa(q_i)$| and substituting into the time-|$2$| objective function |$E_2\left[C_i\right] - \rho/2 \, Var_2\left(C_i\right)$| yields \begin{align*} E_2&\left[W_{0,i} R_f + \left(\theta_i - \gamma_i\right) (X - P R_f) - \kappa(q_i)\right] - \frac{\rho}{2} Var_2\bigl(\left(\theta_i -\gamma_i\right) (X - P R_f)\bigr) \\ &= W_{0,i} R_f - \kappa(q_i) + \left(\theta_i - \gamma_i\right) \left(\hat{\mu}_{X,i} - P R_f \right) - \frac{\rho}{2} \left(\theta_i -\gamma_i\right)^2 \frac{1}{h_i}.\end{align*} Hence, the first-order condition with respect to |$\theta_i$| is given by \begin{equation*} \left(\hat{\mu}_{X,i} - P \, R_f\right) - \rho \, (\theta_i -\gamma_i)\, \frac{1}{h_i} \triangleq 0, \end{equation*} which, after rearranging, yields Theorem 1. Proof of Theorem 2 and Corollary 1 We conjecture (and later verify) that the equilibrium stock price is a linear function of the unobservable state variables; that is, the payoff and the aggregate stock supply \begin{equation} \label{eqA:PrcConj} P R_f = a + b \, X - d\, Z.\end{equation} (A1) As a result, conditional on a specific signal |$Y_i$| and the realized price |$P R_f$|⁠, investor |$i$|’s posterior precision and mean are given by \begin{align} h_i &= \underbrace{\frac{1}{\sigma_X^2} + \frac{b^2}{d^2 \, \sigma_Z^2}}_{\equiv \, h_0} + q_i = \frac{\left(1 + q_i \, \sigma_X^2 \right) \, d^2 \, \sigma_Z^2 + b^2 \, \sigma_X^2}{\sigma_X^2 \, d^2 \, \sigma_Z^2}, \label{eqA:PostPrec}\\ \end{align} (A2) \begin{align} \hat{\mu}_{X,i} &= \mu_X + \frac{1}{h_i} \, q_i \, \left(Y_i - \mu_X\right) + \frac{1}{h_i} \, \frac{b^2}{d^2 \, \sigma_Z^2} \left(\frac{P \, R_f - a + d \, \mu_Z}{b} - \mu_X\right) \label{eqA:PostMean}\\ &= \frac{d^2 \, \sigma_Z^2 \, \left(q_i \, \sigma_X^2 \, \, Y_i + \mu_X\right) + b \, \bigl(P R_f - a + d \, \mu_Z\bigr) \, \sigma_X^2}{(1 + q_i \, \sigma_X^2) \, d^2 \, \sigma_Z^2 + b^2 \, \sigma_X^2}, \nonumber \end{align} (A3) where we substituted the posterior precision |$h_i$| with (A2) in the last step. Intuitively, investors have three pieces of information that they aggregate to form their expectation of the asset’s payoff: their prior beliefs, their private signals and the public stock price. The posterior mean is simply the weighted average of the three signals’ realizations, while the posterior precision is the weighted average of the signals’ precisions. Plugging the posterior mean and precision into the optimal stock demand in (6) yields \begin{equation} \label{eqA:PfHold3} \theta_i = \frac{1}{\rho} \, \left( \frac{\mu_x}{\sigma_X^2} + q_i \, Y_i + \frac{b \, (- a + d \mu_Z)}{d^2 \sigma_Z^2} - P \, R_f\left(\frac{1}{\sigma_X^2} + q_i + \frac{b}{d^2 \sigma_Z^2} (b-1) \right) \right) + \, \gamma_i.\end{equation} (A4) Market clearing requires that aggregate demand, that is, demand (A4) integrated over all investors, equals (random) supply: \begin{align*} &\int_0^\Gamma \left[\frac{1}{\rho} \, \left( \frac{\mu_x}{\sigma_X^2} + q_i \, Y_i + \frac{b \, (- a + d \mu_Z)}{d^2 \sigma_Z^2} - P \, R_f \, \left(\frac{1}{\sigma_X^2} + q_i + \frac{b}{d^2 \sigma_Z^2} (b-1) \right) \right) + \gamma_i \right] di \\ & \quad{} + \int_\Gamma^1 \left[\frac{1}{\rho} \, \left( \frac{\mu_x}{\sigma_X^2} + q_i \, Y_i + \frac{b \, (- a + d \mu_Z)}{d^2 \sigma_Z^2} - P \, R_f \, \left(\frac{1}{\sigma_X^2} + q_i + \frac{b}{d^2 \sigma_Z^2} (b-1) \right) \right) \right] di\triangleq Z.\end{align*} Substituting the private signals by |$Y_i = X + \varepsilon_i$| and using that investors are, on average, unbiased (⁠|$\int_0^\Lambda \varepsilon_i = 0$|⁠, |$\int_\Lambda^1 \varepsilon_i = 0$|⁠), this can be simplified to: \begin{equation} \label{eq:MC} \frac{1}{\rho} \, \left( \frac{\mu_x}{\sigma_X^2} + \bar{q} \, X + \frac{b \, (- a + d \mu_Z)}{d^2 \sigma_Z^2} - P \, R_f \, \left(\frac{1}{\sigma_X^2} + \bar{q} + \frac{b \, (b-1)}{d^2 \sigma_Z^2} \right)\right) + \bar{\gamma} \triangleq Z, \end{equation} (A5) with the precision of private information of the average agent, |$\bar{q}$|⁠, and the degree of benchmarking in the economy, |$\bar{\gamma}$|⁠, being defined in (9). Solving the market-clearing condition (A5) for the price |$P R_f$| yields \begin{equation*} P R_f = \frac{d^2 \sigma_Z^2}{\bar{h} \, d^2 \sigma_Z^2 - b} \, \left( \frac{\mu_x}{\sigma_X^2} + \rho \, \bar{\gamma} + \frac{b \, (- a + d \mu_Z)}{d^2 \sigma_Z^2} \right) + \frac{\bar{q} \, d^2 \sigma_Z^2}{\bar{h} \, d^2 \sigma_Z^2 - b} \, X - \frac{\rho \, d^2 \sigma_Z^2}{\bar{h} \, d^2 \sigma_Z^2 - b} \, Z, \end{equation*} with |$\bar{h}$| being defined in (8). Moreover, it implies a signal-to-noise ratio, defined as the ratio of the price sensitivities to the payoff and the noisy supply |$b/d$|⁠, for |$-\bar{q}/\rho$|⁠. Finally, matching the coefficients of this price function to the ones of the conjecture (A1) and solving the resulting equation system for |$a$|⁠, |$b$| and |$d$| yields the equilibrium price function (7) Finally, plugging these coefficients into |$h_0$| defined in (A2) yields its expression in (8) and Corollary 1. Proof of Theorem 3 Plugging the optimal portfolio |$\theta_i$| in (6) into compensation |$C_i$| and computing its expectation and variance, yields \begin{align} E_2[C_i] &= W_0 \, R_f + h_i \, \frac{\left(\hat{\mu}_{X,i} - P \, R_f\right)^2}{\rho} - \kappa(q_i) = R_f \, \left(W_0 - \frac{\kappa(q_i)}{R_f} \right)+ \frac{1}{\rho} \, z_i^2, \label{eqA:ExpComp}\\ \end{align} (A6) \begin{align} Var_2(C_i) &= h_i^2 \, \left(\frac{\hat{\mu}_{X,i} - P \, R_f}{\rho}\right)^2 \, \frac{1}{h_i} = h_i \, \frac{\left(\hat{\mu}_{X,i} - P \, R_f\right)^2}{\rho^2} = \frac{1}{\rho^2} \, z_i^2, \label{eqA:VarComp} \end{align} (A7) with |$z_i$| being defined as |$z_i \equiv \sqrt{h_i} \, (\hat{\mu}_{X,i} - P \, R_f)$|⁠. Substituting the expectation and variance into the investor’s utility function (5) gives the investor’s time-|$1$| objective function (11). To ease the computations of |$E_1[z_i^2]$|⁠, we introduce |$u_i = \sqrt{h_i} z_i = h_i (\hat{\mu}_{X,i} - P \, R_f)$|⁠. Moreover, note that viewed from period |$1$| both |$\hat{\mu}_{X,i}$| and |$P R_f$| are random variables. In particular, substituting |$Y_i$| by |$X + \varepsilon_i$| and |$\frac{P \, R_f - a + d\, \mu_Z}{b}$| by |$X - \frac{\rho}{\bar{q}} \, \left(Z - \mu_Z \right)$| in the expression for the posterior mean (A3) yields \begin{equation}\label{eqA:PostMeanRV} \hat{\mu}_{X,i} = \frac{1}{h_i} \, \biggl(\left(\frac{\mu_X}{\sigma_X^2} + \frac{\mu_Z \, \bar{q}}{\rho \, \sigma_Z^2} \right) + q_i \, \varepsilon_i + \biggl(q_i + \frac{\bar{q}^2}{\rho^2 \, \sigma_Z^2}\biggr) \, X - \frac{\bar{q}}{{\rho \, \sigma_Z^2}} \, Z \biggr).\end{equation} (A8) Consequently, after substituting (A8) for |$\hat{\mu}_{X,i}$| and (7) for |$P R_f$| in |$u_i$|⁠, one gets: \begin{align*} u_i &= \left(\frac{\mu_X}{\sigma_X^2} + \frac{\bar{q} \, \mu_Z}{\rho \, \sigma_Z^2} \right) \, \left(1 - \frac{h_i}{\bar{h}} \right) - \frac{h_i}{\bar{h}} \, \rho \, \bar{\gamma} + q_i \, \varepsilon_i + \left(\frac{1}{\sigma_X^2} \, \left(\frac{h_i}{\bar{h}}-1\right)\right) \, X\\ & \quad + \frac{\rho}{\bar{q}} \, \left(h_i -\frac{\bar{q}^2}{{\rho^2 \, \sigma_Z^2}} - \frac{h_i}{\bar{h} \, \sigma_X^2} \right) \, Z.\end{align*} Integrating over the distributions of |$P$| and |$Y_i$|⁠, the time-|$1$| expectation and variance are \begin{align*} E_1[u_i] &= \frac{h_i \, \rho}{\bar{h}} \, \left(\mu_Z - \bar{\gamma}\right),\\ Var_1(u_i) &= \frac{h_i^2}{\bar{h}^2} \, \left(\frac{1}{\sigma_X^2} + \frac{\rho^2}{\bar{q}^2} \, \sigma_Z^2 \, \left(\bar{h} - \frac{1}{\sigma_X^2}\right)^2 \right) - h_i = \frac{h_i^2}{\bar{h}^2} \, \left(\bar{h} + \rho^2 \sigma_Z^2 + \bar{q} \right) - h_i.\end{align*} Using that |$E_1\left[u_i^2\right] = Var_1(u_i) + E_1[u_i]^2$| (from the definition of variance), we obtain \begin{align*} E_1\left[u_i^2\right] &= h_i^2 \, \frac{1}{\bar{h}^2} \, \left(\rho^2 \left(\sigma_Z^2 + \left(\mu_Z - \bar{\gamma}\right)^2\right) + \bar{h} + \bar{q}\right) - h_i = A \, h_i^2 - h_i.\end{align*} Consequently, we get that |$E_1\left[z_i^2\right] = \frac{1}{h_i} \, E_1\left[u_i^2\right] = A \, (h_0 + q_i) - 1$|⁠, such that the time-|$1$| objective function (11) is given by \begin{equation} \label{eqA:ObjFuncInfFin} E_1\left[E_2[C_i] - \frac{\rho}{2} \, Var_2(C_i) \right] = R_f \, \left(W_0 - \frac{\kappa(q_i)}{R_f} \right) \,+ \frac{1}{2 \, \rho} \, \left(A \, (h_0 + q_i) - 1\right).\end{equation} (A9) Rearranging the first-order condition of (A9) with respect to |$q_i$| (taking |$\bar{q}$| and, hence, |$\bar{h}$| and |$A$| as given), yields Theorem 3. In particular, whereas the right-hand side of the equation is constant, the left-hand side is increasing in |$q_i$| (because |$\kappa$| is convex), so that an unique interior solution exists if |$2 \kappa'(0) < A/\rho$|⁠. Otherwise, one arrives at a corner solution: |$q_i = 0$|⁠. Finally, using the definition of |$A$| in (12), we obtain \begin{equation} \label{eqA:ADerivGamma} \frac{\partial A}{\partial \bar{\gamma}} = - \, \frac{2 \, \rho^2}{\bar{h}} \, (\mu_Z - \bar{\gamma}) < 0, \quad \text{if} \mu_Z - \bar{\gamma} > 0.\end{equation} (A10) One can also show that the value of information is declining in average aggregate precision |$\bar{h}$|⁠: \begin{equation} \label{eqA:ADerivHBar} \frac{\partial A}{\partial \bar{h}} = -\frac{2}{\bar{h}^3} \, \left(\rho^2 \left(\sigma_Z^2 + \mu_Z^2\right) + \bar{h} + \bar{q}\right) + \frac{1}{\bar{h}^2} = \underbrace{-\frac{1}{\bar{h}^3}}_{< \, 0} \, \underbrace{\left\{2 \left(\rho^2 \left(\sigma_Z^2 + \mu_Z^2\right) + \bar{q}\right) + \bar{h}\right\}}_{> \, 0} \, < 0, \end{equation} (A11) and, consequently, also declines in the average precision of private information |$\bar{q}$|⁠: \begin{equation} \label{eqA:ADerivQ} \frac{\partial A}{\partial \bar{q}} = \frac{\partial A}{\partial \bar{h}} \, \frac{\partial \bar{h}}{\partial \bar{q}} = \underbrace{\frac{\partial A}{\partial \bar{h}}}_{< \, 0} \, \left(\frac{2 \, \bar{q}}{\rho^2 \, \sigma_Z^2} + 1 \right) < 0.\end{equation} (A12) This result is reminiscent of the result on strategic substitutability in Grossman and Stiglitz (1980). Proof of Theorem 4 In the information choice period, the objective of an investor with CARA expected utility is given by \begin{align} &E_1\left[-E_2\left[\exp\left(-\rho \, C_i\right)\right]\right] = E_1\biggl[-\exp\biggl(E_2\left[-\rho \, C_i\right] + \frac{1}{2} Var_2\left(-\rho \, C_i\right)\biggr)\biggr] \nonumber\\ &\quad \quad = -\exp\left(\rho \, R_f \, \left(\frac{\kappa(q_i)}{R_f} -W_0\right)\right) \, E_1\left[\exp\left(-\frac{1}{2} \, z_i^2 \right)\right], \label{eqA:CARAObjInfo} \end{align} (A13) where we used (A6) and (A7) for the mean and variance of compensation |$C_i$|⁠. To compute the expectation of the exponential of a squared normal variable (⁠|$z_i^2$|⁠), we can use Brunnermeier (2001, p. 64): \begin{align*} E_1\left[\exp\left(-\frac{1}{2} \, z_i^2 \right)\right] &= \left|1 - 2 \, Var_1(z_i) \, \left(-\frac{1}{2}\right)\right|^{-\frac{1}{2}} \\ & \times\, \, \exp\biggl(\underbrace{\frac{1}{2} \left(-E_1[z_i]\right)^2 \, \left(1 - 2 \, Var_1(z_i) \, \left(-\frac{1}{2}\right)\right)^{-1} \, Var_1(z_i) -\frac{1}{2} \, E_1[z_i]^2}_{= \, \frac{1}{2} \, E_1[z_i]^2 \, \left(\frac{Var_1(z_i)}{1+Var_1(z_i)}-1\right) = \, \frac{1}{2} \, E_1[z_i]^2 \, \left(\frac{Var_1(z_i)-1-Var_1(z_i)}{1+Var_1(z_i)}\right)}\biggr)\\ &= \left(\left(1 + Var_1(z_i)\right) \, \exp\left(\frac{E_1[z_i]^2}{1+Var_1(z_i)} \right)\right)^{-\frac{1}{2}}.\end{align*} One can further simplify the two components within the exponential function as follows: \begin{align*} 1 + Var_1(z_i) &= 1 + \frac{Var_1(u_i)}{h_i} = h_i \, \frac{1}{\bar{h}^2} \left(\bar{h} + \rho^2 \sigma_Z^2 + \bar{q} \right) \equiv A_1 \, h_i,\\ \frac{E_1[z_i]^2}{1+Var_1(z_i)} &= \frac{h_i}{h_i + Var_1(u_i)} \, \frac{1}{h_i} \, E_1\left[u_i\right]^2 = \left(\bar{h} + \rho^2 \sigma_Z^2 + \bar{q}\right)^{-1} \, \rho^2 \, \left(\mu_Z - \bar{\gamma} \right)^2 \equiv A_2.\end{align*} Consequently, the CARA objective function (A13) can be written as \begin{align*} \max_{q_i} \, E_1\left[-\exp\left(-\rho \, C_i\right)\right] = -\exp\left(\rho \, R_f \, \left(\frac{\kappa(q_i)}{R_f} -W_0\right)\right) \, \left(A_1 \, (h_0 + q_i) \, \exp\left(A_2\right)\right)^{-\frac{1}{2}}.\end{align*} Note that neither |$A_1\left(\bar{q}\right)$| nor |$A_2\left(\bar{q},\bar{\gamma}\right)$| depends on |$q_i$|⁠. Hence, taking the first-order condition with respect to |$q_i$| and rearranging yields Theorem 4. Proof of Theorem 5 In equilibrium, precision choices are mutual best response functions. In particular, each investor’s information choice |$q_i$| affects |$\bar{q}$| and |$\bar{q}$| affects |$q_i$| through |$A$|⁠. Therefore, the equilibrium value of the precision of the private signal of the average investor, |$\bar{q}$|⁠, is determined by plugging |$q_i$| from (12) into the definition (9), which yields (13). To prove that the equilibrium is unique (within the class of linear equilibrium), it suffices to show that |$\bar{q}$| is uniquely defined. Let |$\Sigma(\bar{q}) = \frac{1}{\bar{q}} \, \int_0^1 \, q_i(\bar{q}) \, di \geq 0$|⁠. Then |$\bar{q}$| is defined as the solution of |$\Sigma(\bar{q}) = 1$|⁠. Differentiating |$\Sigma(\bar{q})$| yields \begin{equation} \label{eqA:DerivSigma} \Sigma'(\bar{q}) = \underbrace{\frac{-1}{\bar{q}^2} \, \int_0^1 \, q_i(\bar{q}) \, di}_{= \, -\frac{1}{\bar{q}} \, \Sigma(\bar{q}) \, < \, 0} + \underbrace{\frac{1}{\bar{q}}}_{> 0} \, \int_0^1 \, \frac{\partial q_i}{\partial \bar{q}} \, di.\end{equation} (A14) Differentiating (12) with respect to |$\bar{q}$| yields \begin{align*} 2 \, \kappa''(q_i) \, \frac{\partial q_i}{\partial \bar{q}} = \frac{1}{\rho} \, \frac{\partial A}{\partial \bar{q}} \quad \Leftrightarrow \quad \frac{\partial q_i}{\partial \bar{q}} = \underbrace{\frac{1}{\rho \, 2 \, \kappa''(q_i)}}_{> \, 0} \, \underbrace{\frac{\partial A}{\partial \bar{q}}}_{< \, 0 \, \, \text{(see (A12))}} < 0, \end{align*} where we used that |$\kappa$| is strictly convex. Consequently, |$\Sigma'(\bar{q})$| is negative and |$\Sigma(\bar{q})$| is decreasing over the real positive line. Together with |$\Sigma(0) = +\infty$| and |$\Sigma(\infty) = 0$|⁠, this implies that |$\Sigma(\bar{q})$| “crosses” each real point once. Hence, a unique |$\bar{q}$| satisfies |$\Sigma(\bar{q}) = 1$| and, hence, (13). Taking the derivative of (13) with respect to |$\bar{\gamma}$|⁠, yields \begin{equation} \label{eqA:DerivEqInf} \frac{d \bar{q}}{d \bar{\gamma}} = \int_0^1 \, \frac{\partial q_i}{\partial \bar{\gamma}} \, di.\end{equation} (A15) Moreover, the derivative of the best information response (12) with respect to |$\bar{\gamma}$| is \begin{equation*} 2 \, \underbrace{\frac{\partial \kappa'}{\partial q_i}}_{= \, \kappa''(q_i)} \, \frac{\partial q_i}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}} {d \bar{\gamma}} = \frac{1}{\rho} \, \biggl(\underbrace{\frac{\partial A}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}}{d \bar{\gamma}} + \frac{\partial A}{\partial \bar{q}} \, \frac{d \bar{q}}{\bar{\bar{q}} \bar{\gamma}}}_\frac{dA}{d \bar{\gamma}}\biggr) \Leftrightarrow \frac{\partial q_i}{\partial \bar{\gamma}} = \frac{1}{2 \, \rho \, \kappa''(q_i)} \, \left(\frac{\partial A}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}}{d \bar{\gamma}} + \frac{\partial A}{\partial \bar{q}} \, \frac{d \bar{q}}{\bar{q} \bar{\gamma}}\right).\end{equation*} Substituting this expression into (A15) and simplifying yields \begin{align*} \frac{d \bar{q}}{d \bar{\gamma}} &= \int_0^1 \, \frac{1}{2 \, \rho \, \kappa''(q_i)} \, \left(\frac{\partial A}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}}{d \bar{\gamma}} + \frac{\partial A}{\partial \bar{q}} \, \frac{d \bar{q}}{\bar{q} \bar{\gamma}}\right) \, di = \underbrace{\int_0^1 \, \frac{1}{2 \, \rho \, \kappa''(q_i)} \, di}_{\equiv \, H} \, \left(\frac{\partial A}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}}{d \bar{\gamma}} + \frac{\partial A}{\partial \bar{q}} \, \frac{d \bar{q}}{d \bar{\gamma}}\right)\\ \Leftrightarrow &\left(1 - \frac{\partial A}{\partial \bar{q}} \, H\right) \, \frac{d \bar{q}}{d \bar{\gamma}} = H \, \frac{\partial A}{\partial \bar{\gamma}} \quad \Leftrightarrow \quad \frac{d \bar{q}}{d \bar{\gamma}} = \underbrace{\left(1 - \frac{\partial A}{\partial \bar{q}} \, H\right)^{-1}}_{> \, 0 \, \text{ because} \frac{\partial A}{\partial \bar{q}} < 0 \, \text{(see (A12))}} \, H \, \underbrace{\frac{\partial A}{\partial \bar{\gamma}}}_{< \, 0 \, \text{(see (A10))}} < 0.\end{align*} Proof of Theorem 6 Taking the time-|$1$| expectation of the equilibrium price (7) yields \begin{equation*} S = \frac{1}{\bar{h}} \, \left(\frac{\mu_X}{\sigma_X^2} + \rho \, \bar{\gamma} + \frac{\mu_Z \, \bar{q}}{\sigma_Z^2 \, \rho}\right) + \frac{1}{\bar{h}} \, \left(\bar{h} - \frac{1}{\sigma_X^2}\right) \, \mu_X - \frac{1}{\bar{h}} \, \left(\frac{\bar{q}}{\rho \, \sigma_Z^2} + \rho \right) \, \mu_Z = \mu_X - \frac{1}{\bar{h}} \, \rho \, (\mu_Z - \bar{\gamma}).\end{equation*} Hence, the total derivative with respect to |$\bar{\gamma}$| is given by \begin{align*} \frac{d S}{d \bar{\gamma}} &= \frac{\partial S}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}}{d \bar{\gamma}} + \frac{\partial S}{\partial \bar{h}} \, \frac{d \bar{h}}{d \bar{\gamma}} = - \frac{1}{\bar{h}} \, \rho \, (-1) - \, \frac{-1}{\bar{h}^2} \, \rho \, (\mu_Z - \bar{\gamma}) \, \frac{d \bar{h}}{d \bar{\gamma}} = \frac{1}{\bar{h}} \, \rho \, + \frac{1}{\bar{h}^2} \, \rho \, (\mu_Z - \bar{\gamma}) \, \frac{d \bar{h}}{d \bar{\gamma}}.\end{align*} Accordingly, the time-|$1$| expectation of the excess return |$M \equiv E_1[X - P R_f]$| is given by \begin{align} M = E_1[X] - E_1[P R_f] = \mu_X - \left(\mu_X - \frac{1}{\bar{h}} \, \rho \, (\mu_Z - \bar{\gamma})\right) = \frac{\rho}{\bar{h}} \, \left(\mu_Z - \bar{\gamma}\right), \label{eqA:ExpRet} \end{align} (A16) and, thus, its total derivative with respect to |$\bar{\gamma}$| is \begin{align*} \frac{d M}{d \bar{\gamma}} &= \frac{\partial M}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}}{d \bar{\gamma}} + \frac{\partial M}{\partial \bar{h}} \, \frac{d \bar{h}}{d \bar{\gamma}} = \frac{- \, \rho}{\bar{h}} + \frac{-\rho}{\bar{h}^2} \, (\mu_Z - \bar{\gamma}) \, \frac{d \bar{h}}{d \bar{\gamma}}.\end{align*} Proof of Theorem 7 The unconditional variance |$V^2$| is given by \begin{equation*} V^2 \equiv Var_1(X - P R_f) = E_1\left[\left(X - P R_f\right)^2\right] - E_1\left[\left(X - P R_f\right)\right]^2 = A - M^2, \end{equation*} where we used the law of iterated expectations, to rewrite the first part as \begin{align*} E_1\left[E_2\left[\left(X - P R_f\right)^2\right]\right] = \underbrace{E_1\left[Var_2(X)\right]}_{= \, Var_2(X) = \frac{1}{h_i}} + \underbrace{E_1\left[\left(\hat{\mu}_{X,i} - P R_f)^2\right)\right]}_{= \, E_1[z_i^2] \, (1/h_i)} = \frac{1}{h_i} + \frac{1}{h_i} \, \left(A \, h_i - 1\right) = A.\end{align*} Accordingly, the unconditional variance is given by \begin{align*} V^2 &= \frac{1}{\bar{h}^2} \, \left(\rho^2 \left(\sigma_Z^2 + \mu_Z^2\right) + \bar{h} + \bar{q}\right) - \frac{\rho^2}{\bar{h}^2} \, \left(\mu_Z - \bar{\gamma} \right)^2 = \frac{1}{\bar{h}^2} \, \left(\rho^2 \sigma_Z^2 + \bar{h} + \bar{q}\right), \end{align*} which does not explicitly depend on |$\bar{\gamma}$|⁠, but implicitly depends on it (through |$\bar{q}$| in |$\bar{h}$|⁠): \begin{equation*} \frac{d V^2}{d \bar{\gamma}} = \frac{\partial V^2}{\partial \bar{\gamma}} \, \frac{d \bar{\gamma}}{d \bar{\gamma}} + \frac{\partial V^2}{\partial \bar{h}} \, \frac{d \bar{h}}{d \bar{\gamma}} = \frac{-1}{\bar{h}^3} \, \frac{d \bar{h}}{d \bar{\gamma}}.\end{equation*} B. Proofs and Derivations for Section 3 Proof of Lemma 1 The first- and second-order derivatives of the CRRA-utility function (14) with respect to wealth |$W_i$| are given by \begin{equation*} \frac{\partial C_i^{1-\rho}/(1-\rho)}{\partial W_i} = C_i^{-\rho}, \qquad \frac{\partial^2 C_i^{1-\rho}/(1-\rho)}{\partial W_i^2} = -\rho \, C_i^{-\rho-1}.\end{equation*} As a result, the local coefficient of relative-risk aversion, denoted by |$\hat{\rho}_i$|⁠, is given by \begin{equation*} \hat{\rho}_i = \frac{(-W_i) \, (-\rho) \, C_i^{-\rho-1}}{C_i^{-\rho}} = \rho \, \frac{W_i}{C_i} = \rho \, \, \frac{W_i}{W_i - \gamma_i W_{0,i} R_B - \kappa(q_i)}, \end{equation*} which yields Lemma 1. B.1 Derivations for the Approximate Portfolio Holdings Substituting wealth |$W_i = W_{0,i} \, (R_f + \phi_i \, r^e)$| into compensation |$C_i$|⁠, we get: \begin{align} C_i &= W_{0,i} (R_f + \phi_i r^e) - \gamma_i W_{0,i} R_B - \kappa(q_i) \nonumber\\ &= (1 - \gamma_i) \underbrace{\biggl(\frac{1}{1-\gamma_i} W_{0,i} \bigl(R_f + \phi_i r^e - \gamma_i R_B\bigr) - \frac{\kappa(q_i)}{1-\gamma_i}\biggr)}_{\, \equiv \tilde{C}_i}. \label{eqA:CompTilde} \end{align} (A17) Importantly, with CRRA preferences, maximizing utility over |$C_i$| is equivalent to maximizing utility over |$\tilde{C}_i$| because any multiplicative constant drops out of the optimization problem. If the mean of |$\tilde{C}_i$| is around |$1$|⁠, one can approximate |$\tilde{C}_i$| using the following lognormal distribution: \begin{equation*} \tilde{C}_i \approx \exp\left(-1 + E_1[\tilde{C}_i] - \frac{1}{2} Var_1(\tilde{C}_i) + \sqrt{Var_1(\tilde{C}_i)} \, \nu\right), \quad \nu \sim \mathcal{N}(0,1).\end{equation*} Because of lognormality of the approximated |$\tilde{C}_i$|⁠, the time-|$2$| portfolio choice problem with CRRA preferences can then be written as \begin{align*} &\max_{\phi_1} \, \frac{1}{1-\rho} \, \exp\left( \left(1-\rho\right) \, \left(-1 + E_1[\tilde{C}_i] - \frac{1}{2} \, Var_1(\tilde{C}_i) \right) + \frac{1}{2} \, (1-\rho)^2 \, Var_1(\tilde{C}_i)\right)\\ \Leftrightarrow \, &\max_{\phi_1} \, E_1[\tilde{C}_i] - \frac{\rho}{2} \, Var_1(\tilde{C}_i).\end{align*} Computing the mean and variance of |$\tilde{C}_i$|⁠, taking the first-order condition with respect to |$\phi_i$| and rearranging, yields (16). B.2 Numerical Solution Approach Before describing our novel numerical solution approach for solving rational expectations equilibrium models, we first derive the investors’ first-order conditions associated with their optimal portfolio and information choice—for the case of CRRA preferences and a single risky stock. These conditions form the basis of the numerical approach. In period |$2$|—the portfolio choice period—an investor (with arbitrary benchmarking concerns |$\gamma_i$|⁠) chooses, conditional on his posterior beliefs, the fraction of wealth to be invested into the stock market, |$\phi_i$|⁠, in order to maximize his expected time-|$2$| utility in (3) with |$u_2(x) = x^{1-\rho}/(1-\rho)$| subject to the “budget equation” |$C_i = W_{0,i} (R_f + \phi_i \, r^e) - \gamma_i W_{0,i} R_B - \kappa(q_i)$|⁠. Computing the derivative with respect to |$\phi_i$|⁠, yields the following first-order condition: \begin{equation} \label{eqA:FOCp2} E_2\bigl[C_i^{-\rho} \, r^e \, \bigl| \, Y_i, P\bigr] = 0, \end{equation} (A18) where |$r^e \equiv X/P - R_f$| denotes the stock’s excess return. In period |$1$|—the information choice period—each investor then chooses the precision of his private signal, |$q_i$|⁠, in order to maximize his expected time-|$1$| utility in (4) with |$u_1(x) = x$|⁠. Computing the derivative with respect to |$q_i$|⁠, yields the following first-order condition: \begin{equation} \label{eqA:FOCp1} E_1\left[\frac{\partial U_{2,i}}{\partial q_i}\right] = 0.\end{equation} (A19) Together with the market-clearing condition for the stock, the first-order condition (A18)—for the different investors—characterizes the equilibrium in period |$2$|⁠. Importantly, note that the stock price plays a dual role: (1) it clears the stock market, and (2) it appears in the information set used to compute each investor’s conditional expectation in (A18). The key difficulty is that, in contrast to CARA-normal frameworks, the equilibrium price function is nonlinear, and its specific functional form is unknown. As a result, one cannot explicitly compute the investors’ posterior beliefs, and, hence, no closed-form solution for the equilibrium exists. Accordingly, the model has to be solved numerically. Bernardo and Judd (2000) propose a numerical solution approach that relies on conjecturing (parameterizing) the equilibrium price function and investors’ demand functions using Hermite polynomials. Using the projection method (see also Judd (1992)), the conditional expectation in (A18) is then replaced with a finite number of integration conditions. Consequently, the first-order condition (A18) but also the market-clearing condition will only hold approximately (“|$\varepsilon$|-equilibrium”). In this paper, we take a different approach. We discretize the state space of the economy. Doing so allows us to explicitly compute the investors’ posterior beliefs and solve the investors’ first-order conditions (A18) and the market-clearing condition exactly. Moreover, the approach allows for arbitrary price and demand functions, that is, we do not parameterize (conjecture) these functions in any form. The disadvantage of the approach is that one can only compute the equilibrium for points within the discretization space (though one can make the space between grid points arbitrarily narrow or can rely on interpolation between grid points). For the setup described in Section 3.1 discretization is straightforward: Because of its binomial distribution, the payoff |$X$| can only take on two values: |$X_L$| or |$X_H$|⁠. Hence, we solely discretize the noisy supply |$Z$| by using |$N_Z$| discretization points. Conditional on investors’ information choices, the system of equations for each discretization point |$(X_o, Z_n), o \in \{L,H\}, n \in \{1,\ldots,N_Z\}$| is then composed of five equations: The first-order condition (A18) for each of the four “subgroups” of investors (that is, for benchmarked and nonbenchmarked investors and in each case for the two possible realizations of the signal |$Y_i \in \{Y_L, Y_H\}$|⁠) plus one market-clearing condition. The five unknowns are the stock price |$P(X_o, Z_n)$| and the stock demand of each group of investors and for each signal realization: |$\phi_i^{\mathcal{BI}}(X_o, Z_n, Y_i)$| and |$\phi_i^{\mathcal{NI}}(X_o, Z_n, Y_i);$||$i \in \{L,H\}$|⁠. For numerical convenience, we also treat the aggregate demand for the stock as a variable and add its definition to the equation system, thereby increasing the number of equations (and unknowns) to |$6$|⁠. If the posterior probabilities characterizing the conditional expectations in the first-order conditions (A18) were “exogenous” (e.g., a function of investors’ private signals only or given by some prior beliefs), one could easily solve this equation system numerically. However, as mentioned above, the key difficulty is that the investors’ posterior beliefs depend on the stock price. The advantage of our discretization is that it allows us to compute the corresponding posterior probabilities explicitly and exactly. In particular, note that, because each investor is small, the equilibrium stock price is a function of the stock’s payoff and its supply only: |$P(X,Z)$|⁠. Consequently, if the stock price function, conditional on a given value for the payoff |$X$|⁠, is invertible (which we numerically verify), each investor can back out the two combinations of the payoff and the noisy supply, denoted by |$\{(\hat{X}_L, \hat{Z}_L), (\hat{X}_H, \hat{Z}_H)\}$|⁠, which are consistent with a given price |$\hat{P}$| (given information choices).41 Using the distribution of the noisy supply, an investor can then compute the posterior probabilities for the payoff |$X$|⁠. Note that each “conjectured” supply |$\hat{Z}_{j}$|⁠, |$j \in \{L,H\}$| is given by the aggregate demand in the economy at price |$\hat{P}$|—conditional on the payoff being |$\hat{X}_{j}$|⁠. Hence, to arrive at |$\hat{Z}_{j}$|⁠, one needs to solve the portfolio choice problem of all four subgroups of investors at price |$\hat{P}$| and then aggregate accordingly—with the likelihood of low and high signals depending on |$\hat{X}_{j}$|⁠. Treating the aggregate demand as an unknown again, this adds |$4 + 1$| equations for each |$\hat{X}_{j}$|⁠, |$j \in \{L,H\}$| to the system—in total |$2 \times 5 = 10$| equations. Formally, given the pair |$\{(\hat{X}_L, \hat{Z}_L), (\hat{X}_H, \hat{Z}_H)\}$|⁠, investor |$i$|’s posterior probability of the unobservable payoff |$X$| being high, conditional on price |$\hat{P}$| and his private signal |$Y_i$| is:42 \begin{equation} \label{eqA:NumPosterior1} \hat{\pi}_{i} = \mathbb{P}(X_H \, | \, Y_i, \hat{P}) = \frac{f_Z(\hat{Z}_H) \, \mathbb{P}(X_H \, | \, Y_i)} {\sum_{j} f_Z(\hat{Z}_j) \, \mathbb{P}(X_j \, | \, Y_i)} , \quad j \in \{L,H\}; \end{equation} (A20) where |$f_Z(\cdot)$| denotes the density function of the stochastic supply |$Z$|⁠. The probabilities of the payoff |$X$| conditional on the private signal only, |$\mathbb{P}(X_o \, | \, Y_i)$|⁠, can be computed using the correlation between an investor’s signal and the payoff (implicit in the signal’s precision |$q_i$|⁠): \begin{equation} \label{eqA:NumPosterior2} \mathbb{P}(X_o \, | \, Y_m) = \begin{cases} \frac{1}{2} + \frac{1}{2} \, \sqrt{\frac{q_i}{q_i + 4}} & \text{if} o = m,\\ \frac{1}{2} - \frac{1}{2} \, \sqrt{\frac{q_i}{q_i + 4}} & \text{if} o \neq m; \end{cases} \qquad \text{ with } o,m \in \{L, H\}.\end{equation} (A21) Panel (a) of Figure A1 depicts the equilibrium price law for the setting discussed in Section 3 of the paper. As one can see, the equilibrium price law is highly nonlinear.43 Panel (b) of Figure A1 illustrates an investor’s corresponding posterior beliefs. In particular, it shows the posterior probability |$\hat{\pi}_{i}$|⁠, defined in (A20), for different combinations of the equilibrium stock price |$P$|⁠, and the investor’s private signal |$Y_i \in \{Y_L, Y_H\}$|⁠. Intuitively, the posterior probability of the unobservable payoff being high is increasing in the stock price and always higher for the case in which the investor receives a high private signal. Figure A1 View largeDownload slide Numerical solution approach Panel (a) shows the equilibrium price as a function of the noisy supply |$Z$| and the payoff |$X$|⁠. Panel (b) shows an investor’s posterior probability |$\hat{\pi}_{i}$| as a function of the stock price, |$P$|—for the two possible signal realizations |$Y_i \in \{Y_L, Y_H\}$|⁠. The graphs are based on the framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$| and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure A1 View largeDownload slide Numerical solution approach Panel (a) shows the equilibrium price as a function of the noisy supply |$Z$| and the payoff |$X$|⁠. Panel (b) shows an investor’s posterior probability |$\hat{\pi}_{i}$| as a function of the stock price, |$P$|—for the two possible signal realizations |$Y_i \in \{Y_L, Y_H\}$|⁠. The graphs are based on the framework described in Section 3.1, with the following parameter values: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$| and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Adding the |$10$| equations required for computing the posterior probabilities to the system of equations composed of the six equations discussed above (four “explicit” first-order conditions, one market-clearing condition and one aggregate demand definition) yields a system of |$16$| equations for each discretization point |$(X_o, Z_n)$|⁠. However, note that, at each discretization point, many of the equations (and matching variables) are redundant. For example, one solves, for each subgroup, the exact same first-order condition associated with the optimal portfolio choice (A18) three times (once for the “explicit” first-order condition, once to compute the aggregate demand |$\hat{Z}_L$|⁠, and, similarly, once to compute |$\hat{Z}_H$|⁠). Moreover, one of the aggregate demand definitions associated with computing the aggregate demand for |$\hat{X}_L$| or |$\hat{X}_H$| is equivalent to the aggregate demand definition associated with the explicit market-clearing condition for the discretization point |$X_o$|⁠, |$o \in \{L,H\}$|⁠. Taking this into account allows us to reduce the system to |$7$| equations for each discretization point—|$4$| first-order conditions (A18), |$1$| market-clearing condition and |$2$| aggregate demand definitions (⁠|$1$| associated with the explicit market-clearing condition and |$1$| associated with the “off-equilibrium” aggregate demand used to compute the posterior probabilities). In a setting with exogenous signal precisions, one would be able to solve this equation systems separately for each discretization point. However, in the presence of endogenous information choices, the problem has to be solved globally because the period-|$1$| precision choices |$q_i, \forall i$| affect the investors’ period-|$2$| posterior beliefs and portfolio choices, which, in turn, affect the investors’ information choices. That is, we arrive at a combined system for all discretization points, with |$(N_Z \times 2) \times 7$| equations. Finally, to evaluate the first-order conditions associated with the investors’ optimal information choice (A19), one needs to compute, for each discretization point, the partial derivatives of the investors’ time-|$2$| value functions |$U_{2,i}$| with respect to |$q_i$|⁠.44 The derivatives can be numerically computed by computing the investors’ time-|$2$| value functions |$U_{2,i}$| also for |$q_i+\delta$| (with |$\delta$| being small). Because investors are price takers (also in period |$1$|⁠), the equilibrium price and, hence, learning from the price are unaffected by the change of precision to |$q_i+\delta$| (technically, |$\hat{Z}_{j}, j \in \{L,H\}$| is unchanged). However, the probability of the payoff conditional on the private signal, given in (A21), changes, it is based on |$q_i + \delta$| now. In summary, for each discretization point, one needs to solve for the optimal portfolio (and, hence, value function) of each of the four subgroups using the “new” posterior probabilities (based on |$q_i + \delta$|⁠). Neither the market-clearing condition nor the equations associated with the learning from the stock price are required. This results in additional |$(N_Z \times 2) \times 4$| equations. Together with the two first-order conditions (A19), which characterize the two groups’ information choices, this yields an overall system of |$(N_Z \times 2) \times (4+7) + 2$| equations, with the same number of unknowns. We solve this large-scale fixed-point problem using Mathematica. In particular, we rely on FindRoot, which uses a damped version of the Newton-Raphson method, together with finite difference approximations to compute the Hessian. To facilitate the computations, we first solve the portfolio choice equation systems (separately for each discretization point) in the absence of private information. The solution to these systems is then used as a starting point for solving the same systems with a nonzero, but exogenous, signal precision (again, separately for each discretization point) before we use these systems’ solutions as a starting point for the full system of equations with endogenous information choice. We find that the solution of the full system is quite accurate for |$N_Z = 25$|⁠, that is, further increasing the number of discretization points hardly changes the solution.45 For that choice, the full system consists of |$25 \times 22 + 2 = 552$| equations, which can be solved on an Intel Core i7-980 workstation in about 3 minutes. The numerical solution approach is quite flexible and can be used to address a far broader range of noisy rational expectations equilibrium models. For example, it can easily handle frictions, constraints or other preferences. Such variations would simply lead to a different set of first-order conditions (A18) (plus—potentially—additional equations/constraints). Other than that, the algorithm would be unchanged. One can also extend the solution approach to a dynamic setting, as is done in Breugem (2018). C. Lognormally Distributed Payoff and Signals The binomial distributions for the asset’s payoff and the investors’ private signals, as described in Section 3.1, are chosen for ease of exposition and numerical convenience only. The results also hold for the case of a lognormally distributed payoff and lognormally distributed signals.46 Figure A2 depicts the investors’ optimal signal precisions and the resulting price informativeness for that case. Similar to the case with binomially distributed payoff and signals, the nonbenchmarked investors endogenously chose a higher signal precision than the benchmarked investors. Moreover, price informativeness declines as the fraction of benchmarked investors increases. The results for the other quantities (omitted for brevity) are directly comparable to the setting with binomial distributions as well. Figure A2 View largeDownload slide Equilibrium information choice with lognormal distributions The figure illustrates equilibrium signal precisions, as a function of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) shows the optimal signal precision of the two groups of investors, and panel (b) depicts the precision of the public price signal (“price informativeness”). Precision is measured as |$R^2$|⁠, that is, the fraction of the variance of the payoff |$X$| that is explained by the investors’ private information and the stock price, respectively. The graphs are based on the CRRA framework described in Section 3.1, but with lognormally distributed payoff and signals. The following parameter values are used: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. Figure A2 View largeDownload slide Equilibrium information choice with lognormal distributions The figure illustrates equilibrium signal precisions, as a function of the fraction of benchmarked investors in the economy, |$\Gamma$|⁠. Panel (a) shows the optimal signal precision of the two groups of investors, and panel (b) depicts the precision of the public price signal (“price informativeness”). Precision is measured as |$R^2$|⁠, that is, the fraction of the variance of the payoff |$X$| that is explained by the investors’ private information and the stock price, respectively. The graphs are based on the CRRA framework described in Section 3.1, but with lognormally distributed payoff and signals. The following parameter values are used: |$\mu_X = 1.05$|⁠, |$\sigma_X^2 = 0.25$|⁠, |$\mu_Z = \ln(1) = 0$|⁠, |$\sigma_Z^2 = 0.2$|⁠, |$\rho = 3$|⁠, |$\gamma_i = \gamma = 1/3 \, \forall i \in \mathcal{BI}$|⁠, and an information cost function |$\kappa(q_i) = \omega \, q_i^2$|⁠, with |$\omega = 0.015$|⁠. We thank the editor, Stijn van Nieuwerburgh, and two anonymous referees. For useful comments and suggestions to this paper and its previous versions, we also thank Jonathan Berk, Andrea Buffa (discussant), Will Cong (discussant), Bernard Dumas, Diego Garcia, Philipp Illeditsch (discussant), Lawrence Jin (discussant), Marcin Kacperczyk, Ron Kaniel (discussant), Antonio Miguel (discussant), Anna Pavlova (discussant), Joel Peress, Zacharias Sautner, Larissa Schaefer, Günther Strobl, Savitar Sundaresan (discussant), Raman Uppal, Dimitri Vayanos, Laura Veldkamp, Grigory Vilkov, Yajun Wang (discussant), and Jing Zeng and seminar participants at Frankfurt School of Finance and Management, INSEAD, the 2016 European Summer Symposium in Financial Markets, the 2017 Australasian Finance & Banking Conference, the 2017 Annual Meeting of the American Finance Association, the 2017 Adam Smith Workshops in Asset Pricing and Corporate Finance, the CEPR Second Annual Spring Symposium in Financial Economics, the 2017 NBER Conference on New Developments in Long-Term Asset Management Conference, the 2017 World Finance Conference, the 2018 Frontiers of Factor Investing Conference, the 2018 Annual Meeting of the Western Finance Association, the INSEAD Finance Symposium, the 2018 China International Conference in Finance, the 2018 Annual Meeting of the European Finance Association, Collegio Carlo Alberto, Geneva School of Economics and Management, BI Norwegian Business School, University of Piraeus, and Imperial Business School. This work was supported by the Europlace Institute of Finance, the Labex Louis Bachelier, and the Lyxor-Dauphine Research Academy. Supplementary data can be found on The Review of Financial Studies Web site. Footnotes 1 For evidence on institutional ownership and trading volume, see French (2008), U.S. Securities and Exchange Commission (2013), Stambaugh (2014), and Griffin, Harris, and Topaloglu (2003). 2 For example, many fund managers, including AllianceBernstein, Allianz Global Investors, Equitile, Fidelity International, and Orbis Investment Management, have recently introduced new performance-linked fee structures. Also, Japan’s Government Pension Investment Fund, the world’s largest retirement fund, introduced a system whereby it pays all active managers a fee based on their return relative to a benchmark. Moreover, individual portfolio manager’s compensation is also often performance-based (see Ma, Tang, and Gomez Forthcoming). 3 These preferences are chosen for illustration only. In the case of constant relative-risk aversion (CRRA) preferences, as will be discussed next, information-scale effects are also present. But the model is less tractable, and benchmarking affects the aggregate risk-bearing capacity as well. 4 The same effect would arise in the presence of “index investors,” that is, investors whose portfolio choice is not driven by (private) information. 5 This holds for the economically relevant case in which the effective supply is positive, which guarantees a positive risk premium for the stock. 6 Because the equilibrium price function in this model is nonlinear, it is considerably less tractable, and we rely on a novel numerical solution method to solve it. 7 Technically, benchmarking increases the local coefficient of relative-risk aversion because proportional movements in an investor’s wealth have a larger impact in the presence of relative performance concerns. 8 Related, Brennan (1993) derives a two-factor model, one of them being the benchmark, and Buffa and Hodor (2018) show how heterogeneous benchmarking can result in negative spillovers across assets. 9 Our work is also related to recent studies that have relaxed the joint CARA-normal assumption (see, e.g., Barlevy and Veronesi 2000; Peress 2004; Albagli, Hellwig, and Tsyvinski 2014; Breon-Drish 2015; Chabakauri, Yuan, and Zachariadis 2017). 10 Also related, García and Vanden (2009) show that competition between fund managers makes prices more informative. Malamud and Petrov (2014) demonstrate that convex compensation contracts lead to equilibrium mispricing. Kacperczyk, van Nieuwerburgh, and Veldkamp (2016) and Farboodi and Veldkamp (2017) discuss what data fund managers optimally choose to process. Bond and García (2018) study the consequences of investing solely via the market portfolio and find negative externalities for uninformed investors. 11 The information cost function |$\kappa$| is assumed to be continuous, increasing, and strictly convex, with |$\kappa(0) = 0$|⁠. This guarantees the existence of interior solutions and captures the idea that each new piece of information is more costly than the previous one. 12Basak and Pavlova (2013) formally demonstrate benchmarking using an agency-based argument. In Buffa, Vayanos, and Woolley (2017), because of agency frictions, investors endogenously make fund managers’ fees sensitive to the performance of a benchmark. Similarly, Sotes-Paladino and Zapatero (2016) show that a linear benchmark-adjusted component in managers’ contracts can benefit investors. 13 This compensation scheme is chosen for simplicity and analytical tractability. In earlier versions of the paper, we explicitly modeled the compensation of institutional investors as a function of their “outperformance,” that is, the difference between the investors’ return and the benchmark return. This has no qualitative effect on the results. 14 Considering instead a fraction of terminal wealth, for example, |$\beta_i \, W_i$|⁠, has qualitatively no impact on the results. It affects the total amount of information that investors acquire, but not the impact of benchmarking. 15 These types of benchmarking concerns capture linear (“Fulcrum”) performance fees. The |$1970$| Amendment of the Investment Advisers Act of |$1940$| restricts mutual fund fees in the United States to be of the Fulcrum type. An investor’s desire to perform well relative to a benchmark also may be driven by social status, instead of monetary incentives. Also, note that these benchmarking concerns do not capture relative wealth concerns, which are discussed in García and Strobl (2011). 16 Technically, |$u_1$| and |$u_2$| are assumed to be continuous, twice differentiable, and increasing functions. 17 Compensation in Cuoco and Kaniel (2011) is comprises a constant “load fee,” a fraction of terminal wealth, and a performance-related component. Basak and Pavlova (2013) study a tractable specification similar to ours. However, utility is increasing in the benchmark (which is less important in the absence of information choice because only marginal utility matters). In the case of |$\gamma_i = 1$|⁠, our specification in (1) is comparable to the alternative specification discussed in their remark 1. 18 This follows from the two budget equations |$W_i = \theta_i \, X + \theta_i^B \, R_f$| and |$W_{0,i} = \theta_i \, P + \theta_i^B$| by solving the second equation for |$\theta_i^B$| (the number of shares of the bond) and plugging the solution into the first. 19 This is a special case of the general form in (2), with |$u_1(x) = -\ln(-x)$| and |$u_2(x) = -\exp(-\rho \, x)$|⁠. 20Kreps and Porteus (1978) provide the axiomatic foundations for this class of nonexpected utility, which allows to disentangle risk aversion from the elasticity of intertemporal substitution. The generalizations of iso-elastic utility in Epstein and Zin (1989) and Weil (1990) are widely used in asset pricing. 21 Technically, |$\partial (P R_f)/\partial \bar{\gamma} = \rho / \bar{h}$|⁠. Because the magnitude of an individual investor’s hedging demand relative to his mean-variance portfolio is increasing (decreasing) in risk aversion (aggregate posterior precision), the sensitivity of the price with respect to benchmarking is positively (negatively) related to |$\rho$| (⁠|$\bar{h}$|⁠). 22 Accordingly, the signal-to-noise ratio, defined as the ratio of the price sensitivities with respect to the payoff and the noise, given by |$-\bar{q}/\rho$|⁠, is also unaffected by |$\bar{\gamma}$|⁠. 23 This is the economically interesting case, because it guarantees that the stock’s risk premium is positive (confer with (A16) in Appendix A). 24 In particular, for each investor, the time-|$1$| expectation of the number of shares in his portfolio that are sensitive to private information is lowered by |$\bar{\gamma}$| relative to the absence of benchmarking. 25 The equilibrium price function is nonlinear, and, hence, the model has to be solved numerically. Appendix B.2 provides the technical details for our novel numerical solution approach. 26 This is a special case of the general form in (2), with |$u_1(x) = x$| and |$u_2(x) = x^{1-\rho}/(1-\rho)$|⁠. 27 Note that if benchmarked investors put all their wealth into the benchmark, their compensation is given by |$C_i = (1-\gamma_i) \, W_{0,i} \, R_B - \kappa(q_i)$|⁠. Thus, we assume that |$\gamma_i <\!< 1$|⁠, such that buying the benchmark is always a feasible strategy that yields a strictly positive compensation and, hence, strictly positive marginal utility. 28 We now directly work with percentage returns—as is done in practice—rather than work with the CARA framework (in which one usually works with dollar returns because they lead to explicit expressions). 29 The results are robust to incorporating continuous distributions. In particular, the results are qualitatively unchanged if one relies on lognormal distributions for the payoff and the private signals, as discussed in Appendix C and illustrated in Figure A2 therein. 30 Risk aversion is also increasing in the benchmark portfolio’s return; formally, |$\partial \hat{\rho}_{i}/\partial R_B > 0$|⁠. 31 In that regard, benchmarking concerns are akin to a subsistence level or background risk. 32 The approximation relies on approximating (normalized) compensation, |$\tilde{C}_i$| (see (A17) in Appendix B.1), using a lognormally distributed variable. It is valid if |$\bigl(1/(1-\gamma_i)\bigr) \, \bigl(W_{0,i} \, \bigl(R_f + (\phi_i - \gamma_i) E_1[r^e] - \kappa(q_i)\bigr)$| is around one and, for example, very accurate if the risk-free rate is below |$5\%$| and the expected excess return is below |$10\%$|⁠. See also van Nieuwerburgh and Veldkamp (2010). 33 This is consistent with Lemma 1. In particular, a benchmarked investor’s risk aversion |$\hat{\rho}_{i}$| can be rewritten as |$\rho \, / \left(1 - \gamma_i \, \frac{W_{0,i} R_B - \kappa(q_i) / \gamma_i}{W_i}\right)$|⁠. In a one-stock economy and ignoring the (small) information cost, |$\kappa(q_i)$|⁠, the last part is given by |$(1 + r^e)/(R_f + \phi r^e)$|⁠, which, for realistic values of |$R_f$| (around |$1.0$|⁠) and |$\phi$| (around |$1.0$|⁠), is close to |$1$|⁠, such that |$\hat{\rho}_{i}$| reduces to |$\rho/(1-\gamma_i)$|⁠. 34 For that purpose, we reduce the curvature parameter of benchmarked investors’ utility, such that their risk aversion, after accounting for their benchmarking concerns, equals that of nonbenchmarked investors. In particular, we choose the curvature parameter such that a single benchmarked investor’s mean-variance portfolio component matches that of a nonbenchmarked investor. 35 We focus on the case of stocks with symmetric distributions of independent fundamentals and noise, so that all differences between the two stocks arise exclusively from benchmarking. 36 Technically, conditional on benchmarked investors’ posterior beliefs, their optimal demand for the index stock is characterized by (16), with |$\gamma_{i,1}>0, \forall i \in \mathcal{BI}$|⁠. Moreover, their demand for the nonindex stock is also described by (16), but with |$\gamma_{i,2} = 0$|⁠. 37 See, for example, Ibert, Kaniel, van Nieuwerburgh, and Vestman (2018) for empirical evidence on Swedish mutual fund managers and Sirri and Tufano (1998), Chevalier and Ellison (1997), and Huang, Wei, and Yan (2007) for evidence on an asymmetric flow-performance relation. 38 Technically, this introduces at kink at |$R_B = 1$|⁠, which further complicates the solution. 39 The results are qualitatively unchanged if one varies the strength of investors benchmarking concerns for negative benchmark returns, |$\gamma_i^-$|⁠, while setting the strength of their benchmarking concerns for positive benchmark returns, |$\gamma_i^+$|⁠, to 0. 40 Intuitively, because monitoring information acquisition is difficult, fund managers have an incentive to lie, which, naturally, creates an agency conflict. 41 To distinguish between the values of the payoff and the noisy supply at a given grid point |$(X_o, Z_n)$|⁠, |$o \in \{L,H\}, n \in \{1,\ldots,N_Z\}$|⁠, and the two “conjectured” payoff-noise combinations |$\{(\hat{X}_{j},\hat{Z}_{j})\}$|⁠, |$j \in \{L,H\}$| used to compute the posterior beliefs, we denote the latter ones by a “ |$\hat{}$| ”. Similarly, to distinguish between the equilibrium price |$P$| and an (arbitrary) price used to illustrate the learning, we denote the latter by |$\hat{P}$|⁠. 42 The posterior probability of the unobservable payoff |$X$| being low, |$\mathbb{P}(X_L \, | \, Y_i, \hat{P})$|⁠, is simply given by |$1-\hat{\pi}_{i}$|⁠. 43 Intuitively, the stock price is a decreasing function of the noisy supply and higher in the case of a high underlying payoff (which increases the likelihood of high signals). 44 Also, Equation (A19) requires us to compute the time-|$1$| expectation over all realizations of |$X$| and |$Z$|⁠. Numerically, this expectation is replaced by the expectation over the discretization space. This can be done either through interpolation or by “rescaling” the probabilities for the finite number of grid points (so they sum to 1.0). 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TI - Institutional Investors and Information Acquisition: Implications for Asset Prices and Informational Efficiency
JF - The Review of Financial Studies
DO - 10.1093/rfs/hhy103
DA - 2019-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/institutional-investors-and-information-acquisition-implications-for-SGvHlsvhcx
SP - 2260
VL - 32
IS - 6
DP - DeepDyve
ER -