Information Production by Intermediaries: Relative Valuation and Balanced Designs. Armando Gomes, Alan Moreira, and David Sovich

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1 Information Production by Intermediaries: Relative Valuation and Balanced Designs Armando Gomes, Alan Moreira, and David Sovich Preliminary Draft: 2nd July 2017 Abstract We study the problem of an investor optimally allocating analysts to assets to learn about future asset values in order to make an optimal investment decision. In our setting, due to capacity constraints, each individual analysts produce information about only a subset of assets and their reports contain analystspecic eects. The optimal matching of analysts to assets displays a balancedness property in which pairs of distinct assets are covered by a similar number of analysts. A balanced allocation allows the investor to eciently aggregate information using the relative value between assets, eliminating the eect of the analyst-specic component. The investor utility maximization is a monotone and concave problem of the information or precision matrix, and has a unique global optimal solution. We show that the optimal matching of analysts to assets and the optimal portfolio decision depends on the structure of the analyst coverage network - the bipartite graph where the vertices are the rms and the edges are all the pairs of distinct rms that are covered by at least one common analyst. For example, capital is only reallocated between rms that are connected in the network, and the intensity of the reallocations depends on both the value of relative asset recommendations and the strength of the connection between the assets. Gomes and Sovich are from the Olin Business School, Washington University in St. Louis, and Moreira is from Yale School of Management, Yale University. The authors can be reached at gomes@wustl.edu, dsovich@wustl.edu, and alan.moreira@yale.edu. We thank seminar participants at Olin Business of Business for comments. 1

2 1 Introduction Information is often produced by intermediaries or agents on behalf of a principal or investor. For example, employees evaluate investment projects or recruiting candidates for their employer; or an equity analysts produce stock recommendations and target prices to investors. The principal then aggregates the information produced by the intermediaries in order to make a more informed investment decisions. Optimally assigning information producting agents to projects is a complex problem due to the presence of two features commonly encountered in economic situations. First, due to capacity constraints, intermediaries are often only able to produce information for a fraction of the principal's investment coverage universei.e. the set of available investment projects or stocks the principal is interested in covering. Second, intermediaries naturally produce information that contains an agent-specic random xed-eect systematically aecting all agents' observations and generating biases. For example, an employee evaluating potential recruitment candidates may be in general more optimistic or pessimistic in her assessments, or dierent equity analysts covering an industry may be in general more or less bullshish about the industry overall prospects which aects all the stocks being coverered by the analyst in that industry. 1 We propose a theoretical model to address the problem of optimal production of information by intermediaries, and we test the main predictions of the model empirically analyzing the coverage of S&P 500 stocks by equity analysts. The two main questions of interest we address are: How should a principal exante optimally assign information producing agents to projects? How should a principal ex-post eciently aggregate all the information produced by the various agents? In our Bayesian framework, an investor can invest in a set of assets with uncertain prior values, and we jointly model the investment and information production choices. Specically, the investor can deploy agents whom can only produce information about a subset of assets; each agent produces a noisy signal of asset values containing an agent specic-component common across all assets followed by the agent. The investor chooses which assets each agent will cover and how much precision or time each agent should allocate to the assets being covered. The model has three periods: in period one, the investor chooses how to allocate agents to assets; in period two, the agents report to the investor the signals observed, and then the investor makes his optimal investment decison conditional signals reported by agents; nally, in period 1 For example, consider an agent-specic bias as is typical in the equity analyst literature Kothari et al. [2016]. If the bias is known to the principal, then it can simply be netted out. However, if the principal has an imperfect prior about the bias, then the problem of inference becomes much more dicult. This is the case we study in our paper. 2

3 three, the investor receives the payos from his investments. We begin the analysis by obtaining the optimal investment decision or portfolio allocation given the signals reported by all agents for all assets they were selected to cover. Our model allows for general normal priors over analyst biases or xed-eects. Using techniques from Bayesian statistics, we obtain closedform expressions for the posterior mean and variance of asset values or returns Lindley and Smith [1972], Rendon [2013]. The optimal portfolio Bayesian portfolio is cleanly expressed as the sum of two ingredients: how much investors would invest when receiving a zero posterior signal, and how much investors reallocate among assets in response to asset recommendations. We characterize the unique optimal information production design, and show that investor's utility function can be expressed as a monotonically increasing and concave function of the information or precision matrix. The information matrix is determined by the bilateral matching of assets to agents and plays a fundamental role in both the portfolio allocation decision, and the reallocation of capital across assets, as well as the optimal information production design. The monotonicy and concavity properties allow us to show that there is a unique global optimum design, and that the rst order condition is necessary and sucient for optimality. We show that the optimal design matches agents to assets in a balanced way, so that pairs of distinct assets are covered by a similar number of agents. Moreover, we show that agents should optimally spread their time and precision evenly across the assets they cover. Intuitively, balancing maximizes the relevant information production and allows for the most ecient use of relative valuation to extract the agent specic component or bias from reports. Empirical analysis of the matching of equity analysts to S&P 500 stocks provide strong support in favor of the existence of the balancedness property among stock analysts covering the S&P 500. This new empirical nding is consistent with the intuitive view that relative valuations are important in practice, and a balanced distribution of analysts to assets allows for the relative valuation as best as possible. Our main result, that the optimal assignment should exhibit a balancedness property, can be illustrated by means of the following simple example. A balanced allocation is one where every pair of distinct assets is covered by exactly the same set of analysts. For example, suppose that we have 6 stocks and 10 analysts that can each cover 3 stocks. Then the unique balanced allocation is {123, 124, 135, 146, 156, 236, 245, 256, 345, 346} where each triple abc denotes the stocks, labelled 1 to 6, covered by each of the 10 analysts. The structure is said to be balanced because each pair of stocks is covered by exactly 2 analysts for example, stocks 4 and 5 are covered by analysts 7 and 9. Note that each asset is covered by 5 analysts, but there are many other 3

4 allocations of 10 analysts to 6 stocks where each asset is covered by exactly 5 analysts, such as for example the unbalanced allocation {123, 123, 123, 123, 123, 456, 456, 456, 456, 456}. We show that under some parametrizations, the unique balanced allocation above yields 25% higher investor utility than the unbalanced structure. Economically, balanced allocations of analysts to assets help increase investor's utility because it improve the investor's ability to eciently explore relative valuation, which allows for the elimination of the agent-specic biases. In our model, analyst recommendations provide informative signals about future returns. These signals are both noisy and biased - a common nding amongst the empirical literature Lin and McNichols [1998], Michaely and Womack [1999], Kadan et al. [2009]. For example, an analyst-specic bias may arise because of valuation errors e.g. a common spreadsheet error or an investment banking conict that systematically aect the magnitude of all their stock recommendations. Moreover, higher levels of noise may arise if the analyst's cognitive abilities are stretched too thing from covering several assets - a situation that explicitly embedded within our model from the assumption of limited information capacities. Our model allows investors to account for noise and biases in a strategic way by comparing dierent recommendations for the same rm across analysts, and dierent recommendations of the same analyst across rms. This leads to a relative-valuation approach when forming the optimal portfolio, consistent with the empirical ndings that analyst are more useful in ranking assets than predicting absolute returns Jegadeesh et al. [2004], Boni and Womack [2006]. We develop the optimal Bayesian portfolio for dierent prior beliefs over the analyst biases. In the case of a perfectly informative prior over the biases, our model perfectly coincides with the popular Black and Litterman [1992] model of portfolio allocation. However, when knowledge about the biases is imperfect, our model outperforms the Black-Litterman model by producing higher Sharpe ratios and more ecient portfolio allocations. Our model thus generalizes the Black-Litterman model by allowing for the incorporation of a wider range of signals and uncertainty when forming investment decisions. Moreover, we show that only when the quality of analyst signals is suciently poor do we optimally recover the Markowitz [1952] unconditional mean-variance ecient portfolio. We provide empirical support for our result by considering the distribution of equity analysts across stocks. Equity analysts are an enormous source of information for investors. However, little work exists that can systematically explain the distribution of analysts across the market as a whole. For example, each year brokerage rms and other sell-side nancial institutions produce an abundance of information about 4

5 stocks. In the year 2014 alone over 3,900 sell-side analysts produced over 32,000 research reports and stock recommendations. 2 We empirically verify that our model provides a consistent explanation for the observed allocation of equity analysts to S&P 500 stocks. While not perfectly balanced as would be expected in the presence of frictions and departures from the modelling assumption, we nd strong evidence that the allocation of analysts matters for information production, conditional on a xed structure of analysts. From a statistical standpoint, the hypothesis that the pattern of analysts across stocks is the result of an information maximizing structure can be translated to a statement that the observed allocation of analysts is a typical sample drawn uniformly from the set of all tables with the observed number of rms per analyst and analysts per rm Chen et al. [2005]. For example, one idea in a symmetric case could be that the allocation of anlayst precision does not matter as long as each rm receives x amount of analysts. Hence, under the null, any structure with the same row and column sums produces the same information. We nd evidence that this indeed not the case. In 17 out of the 23 industries we are able to reject the hypothesis that the observed information strucutre is not information maximizing conditoinal on the xed row and column sums. To demystify the economics of the optimal Bayesian portfolio in our paper, we also turn to the study of the analyst coverage network - the graph where the vertices are the rms and the edges are all the pairs of distinct rms that are covered by at least one common analyst. Firms are connected within the network when they share common analysts. Moreover, groups of rms and analysts form separate, sub-networks called components through which information can be disseminated. We show that the components of the analyst coverage network play a key role in extracting information from analysts' signals and reallocating capital across rms and industries. First, we show that no wealth is reallocated across disconnected components of the analyst coverage network in response to analyst recommendations. Instead, wealth is only reallocated within connected components. The economic intuition is that investors can only make relative judgements about stocks that are connected through common analysts because of the presence of bias. Inter-industry reallocations only occur when there exists at least one analyst to act as a bridge between industries. Therefore, analysts that focus on making industry-wide recommendations provide value by connecting industries within the network. Having shown that no wealth is reallocated across components in response to asset recommendations, 2 Previous work has mostly examined the determinants for analyst coverage see Kothari et al. [2016] for a survey in an empirical setting and has not considered the formation of the network as a whole. The exception is a recent paper by Hong and Chang [2016] that partially endogenizes the analyst network in a labor market matching model. 5

6 we then turn to examining how wealth is reallocated within components in the analyst coverage network. We show that the extent to which investors make reallocations of capital across connected stocks depends on both the value of the relative asset recommendations and the strength of the connection between the stocks. Strong connections that exist within industries allow investors to make signicant intra-industry capital reallocations in response to asset recommendations. Moreover, we show that a favourable analyst signal about a single rm i can eectively cause a reallocation of capital throughout rm i's entire network component. The rms which are most closely connected to rm i receive positive reallocations of capital when rm i receives a favourable signal. Since no capital can leave the component, the capital is taken from rms whom which i is less connected. This reallocation is increasing in the intensity of the connection, as the relative return precision is shown to be larger for more connected rms. Our paper contributes to several strands of literature. First, our paper is related to the literatures on information acquisition and investment Veldkamp [2011] and endogenous analyst network formation Hong and Chang [2016]. Second, our paper highlights an intricate link between the literature on Bayesian portfolio choice Black and Litterman [1992], Zhou [2009], Goman and Manela [2012] and the literature on the use of graph theory and networks in nance Anton and Polk 2013, DeGroot 1974, Golub and Jackson 2010, Kelly et al We show how the analyst coverage network impacts information aggregation and portfolio choice in a Bayesian setting. In the optimal Bayesian investment strategy, reallocation across industries depends on the structure of the Laplacian matrix of the analyst coverage network. In addition, the strength of the connections within the network determines how to adjust the weights in the optimal portfolio in response to changes in analyst recommendations. One of the key contributions of our paper is to show that the structure of the coverage network provides the information necessary for this weighting on the information. Third, our paper develops a portfolio approach that mitigates the known bias in analyst recommendations. A large literature documents that analyst recommendations may be biased because of career concerns Hong and Kubik [2004], investment banking relationships Michaely and Womack [1999], Kadan et al. [2009], and preferences for stocks with certain quantitative characteristics Jegadeesh et al. [2004]. Our paper provides a formal method for eciently extracting information about excess future returns even when analyst recommendations display systematic biases. Third, our model helps explain some of the extant empirical ndings related to analyst stock recommendations. Boni and Womack [2006] show that analysts create value only by ranking stocks within industries. Jegadeesh et al. [2004] nd that the level of the consensus analyst recommendation contains no marginal predictive power about 6

7 returns. In other words, the extant literature nds that the value of analyst recommendations comes from their ability to rank stocks relatively rather than absolutely. Our model admits this empirical nding. When analyst recommendations are biased and investors have uninformative priors, only relative valuations matter. Moreover, when all industries belong to disconnected components of the analyst coverage network, the optimal portfolio only reallocates wealth relatively among stocks within industries. Reallocation across industries only occurs when industries are bridged by a common analyst. This supports the ideas in Kadan et al. [2012] and Boni and Womack [2006] that rm recommendations only contain information about industry level prospects when analysts use a market benchmark. The rest of the paper is organized as follows. Section 2 describes the base model and Section 3 solves the optimal Bayesian portfolio problem. Section 4 discusses the optimal allocation of analysts and Section 5 provides empirical support. Section 6 discusses analyst coverage network and the reallocation of wealth in the optimal Bayesian Portfolio. Section 7 concludes. 2 The Model Our model has an investor or principal that can invest in n risky assets or projects and one risk-free asset. The model has three periods. In period 1, the investor chooses how to match m analysts to assets in order to produce information about future asset values. In period 2, the analysts report to the investor the signals observed, and then the investor chooses what assets to invest in. In period 3, the investor receives the returns from his investment. The n tradable assets in the economy are labelled i = 1,.., n. We denote by R i the realized return from investing in asset i from period t = 2 to period t = 3. That is, R i = p i,t+1 p i,t p i,t, where p i,t is i s asset price at period t. Throughout our analysis we assume that asset prices are given, and the investor is a price taker and does not aect prices. 2.1 Information Production Choice: Matching of Analysts to Assets The investor can deploy at period 1, m analysts labelled a = 1,..., m information production agents to produce an informative signal about future, period t = 3, asset prices or equivalently returns which the investor can use to make more informed investment decisions in period 2. Specically, each agent a deployed 7

8 by the investor can produce a signal y ia about i s asset return given by y ia = r i + u a + ε ia, where r i is the learnable component of asset returns, u a is an analyst specic term, and ε ia is an asset-analyst error term. The return R i for each asset i is decomposed into a learnable return component r i and an unlearnable return component η i : R i = R i + r i + η i, 1 where ER i = R i in the unconditional expected return, the term r i capture the variation in fundamentals that can learned by analysts through research, and the component η i captures risk fundamentals that cannot be learned. Both r i and η i are assumed to be normally distributed with zero mean with independent distributions, and assume that only a fraction α [0, 1] of the unconditional return variation can be learned: r N0, ασ and η N0, 1 ασ and R N R, Σ. The unobserved term u a is an analyst specic bias, or measurement error, that is common across all assets covered by the analyst, and u a is normally distributed u a N 0, φ 1 a, with precision φ a 0. Note that we could also accomodate in the model systematic analyst biases with a specication u a N b a, φ 1 a. 3 The asset-specic error ε ia of each analyst is also normally distributed with mean zero and precision τ a θ ia. The total analyst precision is τ a > 0 and θ ia 0 is the fraction of time spent by analyst a researching asset i and thus the noise ε ia has normal distribution ε ia N 0, τ a θ ia 1, 3 For example, an analyst could have a conict of interest with the investment banking division of his rm and systematically provide rms with optimistic recommendations. An analyst could also have an unknown spreadsheet error or particular valuation approach that biases all his recommendations. Empirically, u are analyst random eects. 8

9 and note that n i=1 θ ia = 1. We assume that each analyst a can produce information for at most q a assets of particular interest is the case where q a < n. This assumption is motivated by the capacity constraint of an individual agent, so that the precision of the signal produced would sharply decay if more than q a assets were being followed by the analyst. The investor choice during period 1 is the matching of analysts to assets. We denote this matching choice or assignment concisely as the n m matrix Θ = [θ ia ] = [θ 1,, θ m ], where column a represents the choices θ a of analyst a. The feasible matrices Θ are those that are: 1. Non-negative i.e., θ ia 0, a, i; 2. All m columns add-up to one i.e., n i=1 θ ia = 1, a; 3. There can be at most q a non-zero entries in column a i.e., # {i N : θ ia > 0} q a, a. Throughout the study we use extensively matrix and vector notation. Thus, we represent by r and R the n-dimensional vector of returns R = [R 1,..., R n ] and r = [r 1,..., r n ] and by u the m-dimensional vector of analyst biases u = [u 1,..., u m ]. We can represent all the asset recommendations y in vector notation as the nm dimensional column vector where the asset recommendation of all analysts from a = 1 to m are stacked-up y a = [y 1a,..., y na ] and y = [y 1,..., y m]. The rm-specic errors are denoted by the nm dimensional vector ε. All the signals can be compactly written in matrix notation. y = Xr + Bu + ε, 2 where the nm n dimensional indicator matrices X = 1 m I n the Kronecker product of the vector of m ones and the n-dimensional identity matrix and the nm m dimensional indicator matrix B = I m 1 n. For convenience, and without any loss of generality, assume for the assets i not covered by analyst a that the signals and errors y ia and ε ia are arbitrarily chosen to satisfy the equation above, and set the precision θ ia to be negligibly close to zero so these signals are going to be disregarded by the investor as pure noise. 9

10 The noisy components ε and u are normally distributed ε N0, Σ ε and u N0, Σ u, where the variance matrices are Σ u = diag φ 1 1,..., φ 1 and Σa = τ 1 m a diag θ 1 1a,..., θ 1 na and Σε = Σ 1... Σ m, where φ a is the common factor precision of agent a, τ a is the total precision of agent a, and θ ia is the fraction of the precision allocated to asset i by agent a. The random vectors r, ε, u, and η have pairwise independent distributions i.e., any two sets of random vectors are independent. 2.2 Investment Decision During period 2, the investor observes the asset recommendations and make his investment decision. After observing the signals y, the investor update his prior beliefs about the asset returns. The posterior return distribution has normal distribution with mean E R y and variance matrix var R y. The investor allocates a fraction ω = [ω 1,..., ω n ] of wealth to the n risky assets and the remainder 1 n i=1 ω i to the risk-free asset where r f is the risk-free interest rate. The expected portfolio return is therefore E R p = ω E R y r f + r f. The investor has meanvariance utility E R p γ 2 varr p, and chooses the portfolio weights ω that maximizes his expected utility conditional on the analyst signals y. This portfolio choice yields expected utility, given y, equal to Uω y = ω E R y r f γ 2 ω var R y ω. 3 Note that the posterior distribution of returns and its learnable component are related by E R y = E r y + R and var R y = var r y + 1 ασ. 10

11 3 The Portfolio Selection Decision We solve the model starting from investment decision period given the analyst reported signals. The investor has mean-variance utility, therefore the problem of choosing portfolio weights ω = ω i n i=1 to maximize expected utility 3, max ω Uω y, conditional upon analysts' recommendations is immediately obtained from from the rst order condition ω y = 1 γ var R y 1 E R y r f. 4 This portfolio choice yields expected utility to the investor, after replacing the expression above in equation 3, equal to Uω y y = 1 2γ E R y r f var R y 1 E R y r f. 5 The investor portfolio allocation choice is solved once the mean E R y and the variance var R y of the posterior return distribution are known. We obtain below the posterior distribution, in the context of the more complex information production structure allowing for analyst-specic random eects. Let the signal be y = Xr+ζ where we dene the combined noise term ζ = Bu+ε with distribution ζ N0, Σ ε +BΣ u B, and where the learnable return prior is r N0, Σ L. Proposition 1 Suppose that asset returns are given by R = R + r + η, and investors observe a signal y = Xr+Bu+ε, where the random variables have distributions r N0, Σ L, η N0, Σ U, u N0, Σ U, and ε N0, Σ ε, and are all mutually pairwise independent, and R has unconditional expected value ER = R. Then: i The investor posterior mean and variance of return is: ER y = X QX + Σ L 1 1 X Qy + R, varr y = X QX + Σ L ΣU 6 where the the matrix Q is Q = var ζ 1 = Σ ε + BΣ u B 1. 11

12 ii The information or precision matrix P = X QX is explicitly given by P = m a=1 τ a diagθ a τ a θ a θ a. 7 τ a + φ a All proofs are in the Appendix. We remark that the result above holds for general variance matrices not necessarily diagonal and for any matrices X and B. Overall, the signal y reported to the investor is adding precision given by the information or precision matrix P. The precision matrix P plays a central role, not only in the portfolio choice decision, but also, as we shall see in the next section, during the information production stage. The information matrix can also be expressed, by rearranging the term, as the weighted sum of two parts P = m φ a [diagτ a θ a ] τ a + φ a }{{} a=1 absolute valuation + τ a [ τa diagθa θ a θ ] a. 8 τ a + φ a }{{} relative valuation The rst term, diagτ a θ a, is diagonal matrix with the precision added by analyst a about the idyosincratic value of each asset. Due to the common noise term u a with precision φ a only a fraction φ a τ a+φ a is learned. Whenever the analyst-specic part is very precise, φ a, then we are in the case studied by Stijn and Veldkamp 2010 which we refer as the absolute valuation case in which the information matrix is simply m P = diagτ a θ a : absolute valuation case when all φ a =. a=1 The second term, τ a diagθ a θ a θ a, captures the relative valuation component. In the extreme case in which analyst-specic part is very imprecise, φ a = 0, then the information matrix is simply m P = τ a diagθa θ a θ a : relative valuation case when all φa = 0 a=1 The relative valuation term has column and row adding up to zero and with the eigenvector 1 n. In the last section, we explore the properties of the second component of the precision matrix. We show that it is the Laplacian matrix of the analyst coverage network graph, and contains the relevant information about the strength of the connections among rms. It dictates the information that can be gained from relative valuations from comparing pair of rms. The stronger the connection among two rms the lower is the variance of the return dierence between the pair of rms. 12

13 4 Information Production Choice We have obtained before the posterior mean and variance of returns, conditional on the information y reported to the investor by the agents. The results in the previous section yields ex ante expected utility to the investor equal to the expected values of Uω y y given in equation 5 over all possible signal realizations y : E y [Uω y y] = 1 [ ] 2γ E E R y r f var R y 1 E R y r f. 9 The investor utility with the new information learned is higher than the investor utility without learning any new information which is equal to 1 R rf Σ 1 1 R rf = 2γ 2γ s Rs R where s R is the Sharpe ratio vector s R = Σ 1 2 R rf. 10 In the remainder of the article, we represent as term U the utility gain associated with the new information learned dened as U = E y [Uω y y] 1 R rf Σ 1 R rf. 11 2γ In order to take the expectation above to obtain the investor ex-ante expected utility, we use the following result: if X Nµ, Σ is a n-dimensional random vector with mean µ and variance Σ then E [X AX] = T raσ + µ Aµ. The next proposition establishes that the investors' expected utility depends only on the precision matrix P = X QX and not any other aspects of how the information is produced. That is P = X QX is a summary statistics for investor's welfare. This result will be used in an essential way in the remainder of the article to nd the optimal allocation of information production resources. Proposition 2 The investor expected utility gain given an allocation Θ can be expressed as a function of the information or precision matrix P = X QX, the prior distribution of returns R N R, Σ, the 13

14 learnable fraction α and γ is the risk aversion coecient as follows: U = α2 T r I + α 1 α P ω 1 P ω + s R I + α 1 α P ω 1 P ω s R 2γ 12 where the weighted information or precision matrix P ω is P ω = Σ 1 2 P Σ 1 2, 13 where P is the information matrix 7 and the Sharpe ratio vector s R 10. In the specic case where all return is learnable, i.e. α = 1, the investor utility gain can be expressed more succintly as U = α 2 T r ΣP + R rf P R rf. The evaluation of investor utility is made easier by the following corollary: Corollary 1 Consider the strictly increasing and concave function f given by fx = x 1 + α 1 α x, 14 and let P ω = W ΛW be the spectral decomposition of the weighted information matrix P ω, where the diagonal matrix Λ = diagλ 1,..., λ n have the eigenvalues λ i 0 of the weighted information matrix P ω, and W is an orthonormal matrix with the eigenvectors of P ω and W W = W W = I The investor utility gain U P is given by U P = α 2 n i=1 fλ i 1 + W 2 s r. 15 i As an application of the above proposition we can derive the investor utility in two polar cases of interest. The rst is the absolute valuation case, where there is no factor specic component, i.e. φ = ; the second is the pure relative valuation case, where the precision of the analyst-specic component is zero, i.e. φ = 0. Corollary 2 Information production without agent specic factor Suppose there is no factor specic component, i.e. φ =, and let Σ = diagσ 2 1,..., σ2 n. Then the optimal allocation of information production 14

15 resources that maximizes the investor's utility gain solves the concave problem: max s.t. n α 2 σi 2τ i1+s 2 Ri i=1 σi 2τ iα1 α+1 τ i 0, i n i=1 τ i mτ where τ i = τ m a=1 θ ia is asset's i total aggregate precision and s Ri = R i r f σ i is asset i's Sharpe ratio. The problem above is concave in τ i and strictly concave if α < 1 so there are decreasing returns from learning about an asset. If the assets have the same Sharpe ratio then the optimal is for each asset to have the same total aggregate precision τ i = mτ n. How analysts are organized do not matter. The second case with pure relative valuation follows and illustrate that how analysts are organized is extremely important for investor's welfare and is in sharp contrast with the absolute valuation case above. Corollary 3 Information production with pure relative valuation Suppose that the precision of the analyst specic estimate is zero, i.e. φ = 0, and Σ = diagσ 2 1,..., σ2 n and similar Sharpe ratio s R = R i r f σ i assets i = 1,..., n. Then the allocation of resources Θ yields investor's utility gain: for all U = n 1 i=1 α 2 λ i Θ α 1 α λ i Θ + 1 where λ i Θ are the n 1 non-zero eigenvalues of the weighted information matrix P ω Θ = τσ 1 2 diagθ1m ΘΘ Σ 1 2. The Sharpe ratio vector is an eigenvector corresponding to the zero eigenvalue of the weighted information matrix, therefore the second term in the expression of the utility cancels out. Also, note that one of the eigenvalues of P ω Θ is zero the one corresponding to the eigenvector e = 1 n so the summation above has only n 1 terms. 15

16 5 Optimal Allocation of Information Production Resources We now consider the problem of optimally allocating the information producing agents across assets. The investor problem is how to determine what assets each agent should cover and how much precision or time each analyst should allocate to the assets they are covering. We will show in this section that the maximization of investor's utility is achieved with balanced allocations of information production resources. These allocations allow for the most ecient use of relative valuation information to extract the agent specic component of the signal. We start by establishing the important direct link between investor's utility and the precision matrix, and show that the precision matrix is a sucient statistics for investor's welfare. 5.1 Properties of Informational Production Designs: Concavity and Monotonicity We develop in this subsection some essential properties of the optimal design problem. Specically, the investor utility U P function is a monotonically increasing and concave function of the information matrix P endowed with the positive semidenite partial order. We start with the properties of the key operator fp := I + α 1 α P 1 P, which appears in the determination of the investor utility, which is an operator from positive semidenite matrices to positive semidenite matrices. We show below that the operator is monotone and concave in the formal sense to be dened below. These two properties are going to be used in an essential way to study the optimal design of information production. 1 Let P and Q be two arbitrary positive semidenite matrices, and let fp be an operator from positive semidenite matrices to positive semidenite matrices, and let UP be an utility function from positive semidenite matrices to the reals. Then Matrix ordering We say that P Q if the matrix P Q is positive semidenite. Operator monotone The operator fp is mononote if P Q implies that fp fq, and the utility function UP is monotone if UP UQ. Operator concave The operator is concave if for any P Q and 0 λ 1 then fλp + 1 λ Q λfp +1 λ fq, and the utility function UP is concave if UλP +1 λ Q λup +1 λ UQ 16

17 Lemma 1 Operator monotonicity and concavity The operator fp dened by fp := I + α 1 α P 1 P is monotone and concave. Moreover, the investor utility UP U P := α 2 T r fp ω + s rfp ω s r is a monotone and concave utility function. Observe that the function fx dened by 14 is monotonically increasing and concave in the domain x 0 : f x > 0 and 2 f x 2 < 0. To gain intuition for the result in the lemma consider the special case where P 1 and P 2 are positive semidenite information matrices that are simultaneously diagonalizable, that is P 1 = W diagλ 1,..., λ n W and P 2 = W diagδ 1,..., δ n W, using the same orthonormal matrix W for both matrices. Since we have shown that fp 1 = W diagfλ 1,..., fλ n W and fp 2 = W diagfδ 1,..., fδ n W, then monotonicity and concavity of the function fx suces to prove the lemma. The result however also holds more generally in the important situation where the matrices may not be simultaneously diagonalizable. The proof is signicantly more complex in this more general case, as can be seen from the details in the appendix. Another important result we use is that the information matrix obtained from averaging two designs is more informative than the average of the information matrices of each design. Specically: Lemma 2 Let P Θ be the information matrix associated with a design Θ. Let also ˆΘ and Θ be two dierent arbitrary designs and let 0 < λ < 1. The information matrix obtained by combining the designs is 17

18 more informative than the average of the information matrices. That is, P Θ λ λp ˆΘ + 1 λ P Θ where Θ λ = λ ˆΘ + 1 λ Θ, and P Θ λ has at least one eigenvalue larger than λp ˆΘ + 1 λ P Θ. The same properties hold for the weighted information matrices P ω = Σ 1/2 P Σ 1/2 for any nonsingular variance matrix Σ. Combining the two lemmas above we derive the main result of this section. Proposition 3 The investor utility function U Θ is concave in Θ. That is, given any two designs Θ 0 Θ 1 and 0 λ 1 then U P Θ λ λu P Θ λ U P Θ 0 where Θ λ = λθ λ Θ 0. The investor utility function U Θ is strictly concave stricty inequality in Θ if α 0, 1 and λ 0, 1. Note that the main proposition above holds very generally for arbitrary variance matrix Σ not necessary diagonal and for the general information production setting presented. The same methodology can also be used to extend the results from mean-variance utility to other utility functions such as CARA. 5.2 Balanced Allocations of Information Production Resources We rst formally dene what is a balanced allocation. Denition: Balanced Designs Consider a triple n, m, q where n denotes the number of assets, m the number of agents, and q < n the maximum number of assets that an agent can cover. Let N a be the subsets of assets that agents a = 1,..., m are covering. We say that the coverage is λ-balanced or a λ balanced design if all subsets N a have exactly q assets, and every pair of distinct assets is contained or covered by exactly λ agents. There is a very extensive literature studying balanced incomplete block designs BIBD. We provide below examples of balanced designs. Below, to save space, we write subsets {a, b, c} in the form abc. Example 2: Consider n = 7, m = 7, and q = 3. Then the structure A = {123, 145, 167, 246, 257, 347, 356} 18

19 denote the subset of assets followed by each of the m = 7 analysts. The structure is a λ balanced design with λ = 3. Note that each asset is followed by exactly r = 3 analysts. Example 3: Consider n = 6, m = 10, and q = 3. Then the structure A = {123, 124, 135, 145, 156, 236, 245, 256, 345, 346} denote the subset of assets followed by each of the m = 10 analysts. The structure is a λ balanced design with λ = 2. Note that each asset is followed by exactly r = 5 analysts. Every balanced design satisfy two basic properties relating the ve key parameters n, m, q, λ, r : n is the number of assets, m is the number of agents, q is the number of assets covered by each agent, r denotes the number of agents covering assets, and λ is the number of agents covering pair of assets. Given any three of the ve n, m, q, λ, r parameters the other two are determined. The rst property is that number of agents following each asset, say r, is given by the equation r = mq n. The second property is that the parameter λ is determined by the equation λ = mq q 1 n n 1. We use in our study the following fundamental results from combinatorial design theory. 4 Result Necessary condition Given a triple n, m, q if λ balanced design exists then the two necessary conditions must hold: i λ n 1 must be divisible by q 1 and ii λn n 1 must be divisible by q 1. Sucient condition Given any triple n, m, q, there exists an integer n 0, such that for all n n 0 there exists a λ balanced design if the two necessary conditions above are satised. We establish in our next result the investor utility when using a balanced allocation of information production resources. We show in the next section that this design achieves the global maximum utility 4 In the proof of our results, we rely on the result that commuting matrices preserve each other's eigenspaces. Moreover, we use the spectral theorem for symmetric matrices, and the result that if two symmetric matrices commute then their eigenspaces coincide, and they can be simultaneously diagonalizable. 19

20 among all possible feasible allocations/designs. Proposition 4 Investors' utility under a balanced design Suppose there are n assets with prior excess return R r f 1 N µ1, Σ where Σ = σ 2 I +σf 2 J, and let fraction α [0, 1] be the learnable fraction. Suppose that all m analysts choose precision τ and φ and each can produce information about q assets and let the agents be organized according to a λ balanced design with λ = mqq 1 nn 1 analysts per pair of assets, and c = mq n analysts per asset. Then: i The precision of the signal obtained by investors is the n n matrix P = X QX equal to P = τφ r τ + φ q I + τ 2 λ ni J, τ + φ q2 where I and J are, respectively, the n n identity matrix and matrix of ones in all entries. ii The expected investor utility gain with a λ balanced design is equal to: U τ, φ = α n 2 1 f λ 1 + f λ nµ 2 σ 2 + nσ 2 f 16 where λ 1 and λ 2 are the eigenvalues of the weighted information matrix P ω given by: λ 1 τ, φ = σ 2 m n λ 2 τ, φ = σ 2 m n τφ 1 + τ τ + φ φ τφ τ + φ 1 + n σ2 f σ 2 q 1 : with multiplicity n 1, 17 n 1 q : with multiplicity 1. The optimal balanced design, for any α 0, 1, the unique mix between the rm-specic and common factor precision τ and φ that maximizes the problem max U τ, φ s.t. G τ, φ κ τ 0, φ 0 18 We now proceed to establish the main result of the paper. 20

21 5.3 Optimality of Balanced Designs Consider a symmetric economy with n assets with prior return R N R, Σ where Σ = σ 2 I + ρj and where a fraction α of the return is learnable. The investor principal can deploy m analysts agents and each agent a = 1,..., m can produce information about up a subset N a of assets with cardinality #N a q. Each analyst agent produce a signal y ia = r i + u a + ε ia for the selected assets i N a where φ is the precision of signal u a N 0, φ 1 and τθ ia is the precision of the rm specic signal, ε ia N 0, τθ ia 1, θ ia 0 is the fraction of time the agent spends on asset i, and τ is the agent's total precision. We now proceed to establish the main result of the paper. Proposition 5 Optimality of symmetric balanced designs Consider the symmetric problem above with n assets and m agents that can cover q n assets and let Σ = σ 2 I + ρj. Then the most ecient allocation, i.e., the one that maximizes investors' utility among all possible feasible designs Θ is the: i The balanced design with λ = mqq 1 nn 1, where any two pair of assets i and j are covered by exactly λ agents; ii All agents allocate their time equally θ ia = 1 q exactly r = mq n agents; to cover exactly q assets, and all assets are covered by iii All agents choose the same rm-specic and common factor precision τ and φ that solves the maximization problem 18. If there is a common noise component φ < and not all information is learnable by analysts, i.e. α < 1, then the λ balanced design is the unique maximum ecient allocation of resources. This results shows the benets of broader coverage by analysts including q assets versus specialization. The investor utility is strictly increasing in the asset coverage q by analysts. A broader coverage allows for more ecient usage of relative valuation. This result shows a key trade-o between brader coverage vs specialitions see also next section *. Even, in the presence of decreased total precision as the analysts broaden his coverage, the benets of relative valuation outweighs the gains from specialization. basically it is better to cover in a shitty way several stocks than cover more precisely just one stock. Proposition 6 Comparative Statics For any α 0, 1 there is a unique solution τ and φ satisfying 21

22 the following comparative statics properties: τ µ, σ 2, σ 2 f, q, m, n =, +,, +, 0, φ µ, σ 2, σ 2 f, q, m, n = +,, +,, 0, + In the examples provided at the end of the section we utilize the following corollary to illustrate the benets of balanced allocations. Observe that as the precision φ 0 the eigenvalue λ 2 0 and the second term in the utility function converges to zero. The next corollary extends the corollaries of section 4 to the case where the asset returns are correlated. Corollary 4 Under a balanced design the investor's utility is U τ, φ given by 16 for general covariance structure Σ = σ 2 I + σf 2 J and similar prior returns where the eigenvalues are : i In the pure relative valuation case, where φ = 0, 2 mτ q 1 λ 1 = σ n 1 q and λ 2 = 0 ii In the pure absolute valuation case, where φ =, λ 1 = σ 2 mτ n and λ 2 = σ 2 + nσf 2 mτ n We illustrate the main result above with several examples. 5.4 Illustrative Examples Example 4 : Consider a symmetric economy with n = 6 assets and m = 10 analysts and the matching of analysts to assets A = {123, 124, 135, 146, 156, 236, 245, 256, 345, 346}, where we denote the subset of assets followed by each of the m = 10 analysts by triples abc. The structure is a λ balanced design with λ = 2 as we have seen before. Note that each asset is followed by exactly r = 5 analysts. The associated Θ information production structures under structure A with each analyst covering q = 3 22

23 assets is: Θ = where ΘΘ = We compare the allocation of resources above with B = {123, 123, 123, 123, 123, 456, 456, 456, 456, 456}, where Θ = where ΘΘ = The total utility of investors with the rst structure is greater than with the second structure assuming v to be the variance of each asset, and α = 1 2 and φ = 0, the utilities are: U A = 5τv τv + 3 > U B = 4τv τv This example illustrates that when the same resources are allocated more eciently there is an utility increase of approximately 25%. Example 5 : This example illustrates that the coverage choice can be important part of the analyst contribution to investors. Consider the following information production structures with 4 analyst covering 8 assets A = {1234, 1234, 5678, 5678}. A fth analyst that can cover q = 4 assets can choose to cover assets 1234 or 5678 such as the other 4 analysts, or assets 1235, or cover assets We show below that the later choice is the optimal choice and the one that maximizes investors' welfare. This choice improves the connectivity among assets the most and leads to a more balanced coverage with more interconnection among assets. 23

24 Using the parametrization v = 2, α = 1 2, φ = 10, τ = 10 the following are the associated investors' utility with each structure: U A = U A = U A = U A = An analyst by making choices that are more balanced, can improve the connectivity of assets and the performance of relative valuation. The more balanced choice can more than double her potential contribution to investors as illustrated above. 5.5 Specialization versus Broad Coverage by Analysts Inspection of the eigenvalues λ 1 = σ 2 m τφ 1 + τ n n τ + φ φ n 1 λ 2 = σ 2 + nσf 2 m τφ n τ + φ q 1 : with multiplicity n 1, q : with multiplicity 1. lead the a tradeo between specialization versus broad coverage. Our assumption so far is that total analyst precision is constant up to the point where the analyst covers q assets. More generally we can assume that the analyst precision is a decreasing and concave function of the asset coverage. Let τ G q, τ, φ = G t q, φ where G is increasing in both arguments and convex where t q is decreasing and concave in q The more assets the analyst follows the higher is the total loss of precision 24

25 τ = t φ, q = h φ t q where h φ is decreasing and concave in φ λg q, τ, φ + 1 λ G q, τ, φ G q λ, τ λ, φ λ τ τ λg t q, φ + 1 λ G t q, τλ φ G t q λ, φ λ H q, τ = τ t q is convex G q, τ, φ = G H q, τ, φ is convex λ 1 φ, q = σ 2 m h φ t q φ n h φ t q + φ λ 2 φ, q = σ 2 + nσf 2 m n 1 + h φ t q φ h φ t q + φ h φ t q n φ n Optimal Design with Asymmetric Assets : show that is concave q 1 : show that is concave, q The concavity of the utility function provides signicant structure to the analysis of the optimal design problem. In particular, the rst order necessary conditions that characterize the optimal solution are also sucient conditions. As a rst application of the results of the previous section, we obtain the optimal design in the symmetric full learning case. Proposition 7 Consider an information production design where all analysts have the same precision τ a = τ and φ a = φ <, all analysts can cover all assets, i.e., q = n, and all information is learnable, i.e., α = 1, and let Σ = diagσ1 2,..., σ2 n + ρj, and the unconditional asset returns be µ i = E R i r f. Then unique global solution to the investor optimal design problem is: All analysts choose the same allocation of time across assets, i.e., θ a = θ for all analysts a = 1,..., m, where 25

26 the optimal analyst allocation of time to asset i is given by θ i = φ 1 1 τ σi 2 λ µ i µ i 2τ + τ + φ κ where the λ and κ are constants obtained so that n i=1 θ i = 1 and κ = n i=1 µ iθ i, and the function x + := maxx, 0. Proposition 7 has several intuitive comparative statics implications. First, as φ increases analysts concentrate more of their time into assets with higher volatility and returns. In the limit, when φ only assets with maximum σ 2 i + µ2 i receive analyst attention. Moreover, as the precision φ approaches zero, i.e. φ 0, analysts disperse their attention to a broader set of assets. Second, analysts' attention θ i are increasing in both the assets' volatility, i.e. in assets' return, i.e. θ i µ i > 0. θ i σ i > 0, and increasing Interestingly, note that the optimal design does not depend on the correlation ρ of asset returns. The following numerical example helps to illustrate the results above. Example 1: Consider n = 4, m = 4, and q = 3. Suppose that φ = 0 and α = 1 2 and all assets have volatility σ = 1 and Σ = σ 2 I. The investor utility with the structure A = {123, 124, 134, 234}, denote the subset of assets followed by each analyst, is given by U = Suppose now we add another agent that can cover all assets θ ia = 1 4. The utility would be U = Now suppose that we increase the volatility of asset 1 to σ 2 1 = 1.5 while maintaining the volatility of the other three assets at σ i = 1 for i = 2, 3, and 4 and the Sharpe ratios which implies that expected return of asset 1 goes up accordingly. The same structure above with ve analysts would generate utility U = That is, in a more volatile environment the analyst can bring about more gains in utility to the investor. But the structure above can be further slightly improved if the new analyst would spend an additional 3.35% of his time on the more volatile asset 1 and less time accordingly on the other three assets. That would improve investor utility by U = If they could all reoptimize the time spent on each asset then by increasing the time spent on asset 1 by 2.1% there would be a further slight gain in utility U =