Reative Vauation and Information Production Armando Gomes, Aan Moreira, and David Sovich Apri 26, 2018 Abstract We study the probem of an investor that aocates anaysts to assets to earn about future asset vaues. We show that when anaysts are better at reative rather than absoute asset vauations the optima matching of anaysts to assets dispays a baancedness property in which pairs of distinct assets are covered by a simiar number of anaysts. A baanced aocation aows the investor to ecienty aggregate information using the reative vaue between assets, eiminating the eect of the anayst-specic component. We show that the optima matching of anaysts to assets and the optima portfoio decision depends on the structure of the anayst coverage network - the bipartite graph where the vertices are the rms and the edges are a the pairs of distinct rms that are covered by at east one common anayst. For exampe, capita is ony reaocated between rms that are connected in the network, and the intensity of the reaocations depends on both the vaue of reative asset recommendations and the strength of the connection between the assets. Gomes and Sovich are from the Oin Business Schoo, Washington University in St. Louis, and Moreira is from the Simon Graduate Schoo of Business at the University of Rochester. The authors can be reached at gomes@wust.edu, dsovich@wust.edu, and aan.moreira@simon.rochester.edu. We thank seminar participants at Oin Business of Business, University of Rochester, University of Caifornia at Irvine, and Finance Theory Group for comments.
1. Introduction How shoud investors aggregate information produced by a wide variety of approaches? Theory typicay mode the information acquisition probem as a cost function that is increasing in the precision of the acquired signa. In this paper, we go inside the information production back box and study how an investor decentraizes information production and aggregates information of a variety of sources. The key ingredient of our anaysis is the idea that we are better at producing reative vaue signas. For exampe, a nancia anayst that covers two dierent rms can provide a more informative signa about the reative vaue across rms, i.e. how much the rst rm is overpriced reative to the second rm, than a signa about their absoute vaue, i.e how much each rm shoud worth. Whie it is we known that reative vauation is ubiquitous in nance, both among practitioners and academics, the impications are not competey understood. In this paper, we study the importance of reative vauation from the perspective of an investor that must decide how to produce and aggregate information. The investor probem starts with the probem of how to aocate anaysts to rms. Reative vauation impies that the production of information depends not ony of the tota number of hours anaysts spend researching each rm, but on the entire information production network, the bipartite graph where the vertices are the rms and the edges are a the pairs of distinct rms that are covered by at east one common anayst. The entire information production network matters because it changes the information content of a signa produced by a particuar information source. To understand the importance of the information production network an exampe is usefu. Consider three rms A,B,C and three anaysts 1,2,3 with capacity to anayze two rms each and a tota time budget of one day. When signas produced are absoute, ets say a better forecast of the rm discounted cash-ows, the foowing two arrangements are equivaent from the perspective of investors: one can concentrate one day of each anayst in each rm, or have each anayst earn about any two rms with a tota time aocation of one day per rm. As a matter of fact, in the case of absoute vauation, there is no notion of the information production unit, the anayst is this exampe. A that matters in this case is the tota aocation of anayst time per rm. When the information is reative the 1
rst arrangement woud produce no information at a, and the information produced in the second arrangement woud depend how exacty each anayst spits her time between rms. Whie our theoretica anaysis wi focus on the portfoio probem of a stock market investor, the paper insights appy whenever signas generated by mutipe information production units need to be aggregated by a principa. For exampe, Metrick and Yasuda 2010 show that Genera Partners in Private Equity and Venture Capita funds often are responsibe for no more that four rms. In the credit market, Fracassi, Petry, and Tate 2016 show that credit anaysts inside credit rating rms cover eeven rms on average. In the context of the stock market, Gomes, Gopaan, Leary, and Marcet 2016 document that nancia anayst cover on average seven dierent rms. The fact that anaysts concentrate on anayzing ony a sma subset of rms, together with the fact that anaysts do not seem to produce informative absoute signas for exampe, see Fracassi et a. 2016 impies that the information production network is essentia to the optima aggregation of information. Our modes is as foows. An investor depoys agents whom can ony produce information about a subset of assets; each agent produces a noisy signa of asset vaues containing an agent specic-component common across a assets foowed by the agent. The variance of this agent specic component contros the degree of reative vauation. A very high variance of this agent specic component captures situations when agents have modes that are so distinct that render comparison of signas produced my dierent agents uninformative. The investor chooses which assets each agent wi cover and how much time each agent aocates to the assets being covered. The mode has three periods: in period one, the investor chooses how to aocate agents to assets; in period two, the investor observes the signas produced by the agents, and then the investor makes his investment decision; nay, in period three, the investor receives the payos from his investments. We begin the anaysis by obtaining the optima investment decision given the signas produced by the agents. We obtain cosed-form expressions for the posterior mean and variance of asset returns as function of the signas and the information production network. Specicay, we show that the posterior covariance between two dierent stock returns is a function of the anayst connection strength between these rms. The posterior covariance is in turn the key determinant of how signas are interpreted. We show that a positive signa 2
about rm i produced by anayst a impies positive negative information about rm j if the connection between rm i and j i.e., the posterior covariance is stronger weaker than the average connection between rm i and a the other rms covered directy by anayst a. Because the connection strength between any two rms depend on paths connecting both rms, the mapping from signa to information requires knowedge of the entire information production network. These insights have sharp impications for how the investor responds to new information. First, we show that weath is ony reaocated within a connected component of the network. That is, the investor ony reaocate his portfoio across rms that have a chain of anaysts inking them. Second, we show that the extent to which investors make reaocations of capita across connected stocks depends on the the strength of the connection between the stocks. For exampe, a positive signa about a singe rm i causes a reaocation of capita throughout rm i's entire network component. The rms which are most cosey connected but not inked to rm i receive positive reaocations of capita when rm i receives a positive signa. Since no capita can eave the component, the capita is taken from rms whom which i is ess connected. We show further that in the knife edge case that a the information is potentiay earnabe, the increase in the posterior covariance between connected but not inked rms exacty osets the changes in expected returns due the information produced. In this case capita is ony reaocated across rm covered by the same anayst. In order to study the optima aocation of information resources, we rst characterize in cosed form the investor ex-ante utiity as function of the information production network. We show that investor's utiity function can be expressed as a monotonicay increasing and concave function of the information production network. The monotonicity and concavity properties aow us to show that there is a unique goba optimum for information production network. We nd it usefu to iustrate the impications of reative vauation for the optima arrangement of information production resources by contrasting it with the absoute vauation case. We start by showing that reative vauation is a strong force for diversication in information production even when preferences are conductive to speciaization. Reative va- 3
uation introduces compementaries in information production because the vaue of the signa produced about a singe rm increases the more other rms' signas it can be compared with. Another distinctive impication of reative vauation is that the investor aocates anaysts to pair of rms that are ex-ante expected to form protabe ong-short portfoios. Specicay, they start by covering pairs that the return dierence have the highest uncentered second moments. Our anaysis emphasizes reative vauation,the situation where comparisons of signas across anaysts is competey uninformative. But our framework is exibe to consider ess extreme cases. For exampe, our framework coincides with the popuar Van Nieuwerburgh and Vedkamp 2010 mode of information production and portfoio aocation in the extreme case where signas ony have absoute information. We show how our framework can be used to study the choice of reative versus absoute information production. Specicay, we show how an increase in the anayst capacity to anayze mutipe rms, for exampe due to a reduction in the xed costs associated with information production, eads the investor to optimay shift production towards reative vauation. Intuitivey, when the anayst can ook at more rms, reative vauation becomes more powerfu as it aows the investor to reaocate capita across more rms. This again emphasizes the important roe that the information production unit pays in a reative vauation word. Having studied the probem of how to aocate one singe information production unit, we study the probem of how to design the entire information production network. We show that reative vauation introduces a strong force for baancedness in the network. We start by showing that when the rms are symmetric, the optima network has exacty the same number of anaysts covering any pairs of stocks, i.e. the network is said to be baanced. We then extend our anaysis to show this property hods even in ess symmetric environments. Specificay, we show that in a muti-industry setting that the information production network is bock baanced, with intra-industry pairs having more anayst coverage than across-industry pairs. The resut that the optima assignment shoud exhibit a baancedness property, can be iustrated by means of the foowing simpe exampe. A baanced aocation is one where every pair of distinct assets is covered by exacty the same set of anaysts. For exam- 4
pe, suppose that we have 6 stocks and 10 anaysts that can each cover 3 stocks. Then the unique baanced aocation is {123, 124, 135, 146, 156, 236, 245, 256, 345, 346} where each tripe abc denotes the stocks, abeed 1 to 6, covered by each of the 10 anaysts. The structure is said to be baanced because each pair of stocks is covered by exacty 2 anaysts for exampe, stocks 4 and 5 are covered by anaysts 7 and 9. Note that each asset is covered by 5 anaysts, but there are many other aocations of 10 anaysts to 6 stocks where each asset is covered by exacty 5 anaysts, such as for exampe the unbaanced aocation {123, 123, 123, 123, 123, 456, 456, 456, 456, 456}. We show that under some parametrizations, the unique baanced aocation above yieds 25% higher investor utiity than the unbaanced structure. Economicay, baanced aocations of anaysts to assets hep increase investor's utiity because it improve the investor's abiity to ecienty expore reative vauation. Our paper contributes to severa strands of iterature. First, our paper is reated to the iteratures on information acquisition and investment Vedkamp 2011 and endogenous anayst network formation Hong and Chang 2016. Second, our paper highights an intricate ink between the iterature on Bayesian portfoio choice Back and Litterman 1992; Zhou 2009; Goman and Manea 2012 and the iterature on the use of graph theory and networks in nance Anton and Pok 2013, DeGroot 1974, Goub and Jackson 2010, Key et a. 2013. We show how the anayst coverage network impacts information aggregation and portfoio choice in a Bayesian setting. In the optima Bayesian investment strategy, reaocation across industries depends on the structure of the Lapacian matrix of the anayst coverage network. In addition, the strength of the connections within the network determines how to adjust the weights in the optima portfoio in response to changes in anayst 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 deveops a portfoio approach that mitigates the known bias in anayst recommendations. A arge iterature documents that anayst recommendations may be biased because of career concerns Hong and Kubik 2004, investment banking reationships Michaey and Womack 1999; Kadan, Madureira, Wang, and Zach 2009, and preferences for stocks with certain quantitative characteristics Jegadeesh, Kim, Krische, and Lee 5
2004. Our paper provides a forma method for ecienty extracting information about excess future returns even when anayst recommendations dispay systematic biases. Fourth, our mode heps expain some of the extant empirica ndings reated to anayst stock recommendations. Boni and Womack 2006 show that anaysts create vaue ony by ranking stocks within industries. Jegadeesh et a. 2004 nd that the eve of the consensus anayst recommendation contains no margina predictive power about returns. In other words, the extant iterature nds that the vaue of anayst recommendations comes from their abiity to rank stocks reativey rather than absoutey. Our mode admits this empirica nding. When anayst recommendations are biased and investors have uninformative priors, ony reative vauations matter. Moreover, when a industries beong to disconnected components of the anayst coverage network, the optima portfoio ony reaocates weath reativey among stocks within industries. Reaocation across industries ony occurs when industries are bridged by a common anayst. This supports the ideas in Kadan, Madureira, Wang, and Zach 2012 and Boni and Womack 2006 that rm recommendations ony contain information about industry eve prospects when anaysts use a market benchmark. 2. The Mode The mode has three periods. In period t = 1, an investor chooses how to aocate m information production units anaysts to n risky assets in order to produce information about period t = 3 asset vaues. In period t = 2, the investor receives the information produced by the anaysts, and then chooses how to invest her weath across n risky assets and one risk-free asset. In period t = 3, the investor receives the returns from her investment. 2.1 Assets The n tradabe assets in the economy are abeed i = 1,.., n. We denote by R i the return from investing in asset i from period t = 2 to period t = 3, and by R the vector of asset returns. Throughout the anaysis, we assume that asset prices are given and normaized to 1 in period t = 2. Furthermore, asset returns foow a norma distribution, R N R, Σ, with prior expected return R and variance Σ. 6
We decompose individua asset returns, R i, into a earnabe return component, r i, and an unearnabe return component, η i : R i = R i + r i + η i, 1 where ER i = R i is the unconditiona expected return. The term r i captures the variation in fundamentas that can be earned by anaysts through research, whie the term η i captures risk fundamentas that cannot be earned. 1 Both r i and η i are normay distributed with zero mean and independent distributions: r N0, Σ and η N0, Σ u and R N R, Σ with Σ = Σ + Σ u. 2 A case of particuar interest is when ony a fraction α [0, 1] of the unconditiona return variation is earnabe. In this case, the earnabe component of returns has variance Σ = ασ and the unearnabe component has variance Σ u = 1 α Σ. Our mode coapses to a fuy earnabe case, where Σ = Σ and Σ u = 0, whenever α = 1. This is the case that is most commony studied in the iterature e.g., Van Nieuwerburgh and Vedkamp 2010. 2.2 Information Structure In period t = 1, the investor can depoy m anaysts, abeed a = 1,..., m, to produce information signas about period t = 3 asset returns. Specicay, each anayst a can produce a signa y ia about the earnabe component of asset returns, r i : y ia = r i + u a + ε ia, 3 where u a is an anayst-specic error term, and ε ia is an asset-anayst error term which is independent of u a. The term u a can be conceptuaized as a measurement error or bias that is common across a signas produced by an anayst. 2 We assume that u a foows a norma distribution 1 The introduction of an unearnabe component impies an upper bound to the Sharpe ratio the investor can obtain by producing information. 2 For exampe, an anayst may be an optimist and consistenty report upward biased signas. 7
with mean zero and precision φ a, where the precision contros the degree of reative vauation. 3 Specicay, as the variance of u a grows, anayst signas become ess informative about the eve of asset returns, whie the information content of signa dierences, i.e. reative comparisons, remains unchanged. We refer to the case when the anayst-specic error is very precise i.e., when φ a as absoute vauation, and we refer to the case when the error is very imprecise i.e., when φ a = 0 as reative vauation. The asset-anayst error term, ε ia, foows a norma distribution with mean zero and precision τ a θ ia, where the term τ a > 0 denotes the tota precision of anayst a and θ ia 0 denotes the fraction of time spent by anayst a researching asset i s.t. n i=1 θ ia = 1. Whie τ a is given exogenousy, the investor chooses how to aocate the research eorts of the anaysts given by the set {θ ia } a i in period t = 1. Therefore, τ a θ ia 0 is the amount of anayst a's tota precision that the investor aocates to asset i. We assume that the distribution of ε ia is independent across i, and that each anayst a can produce information for at most q a < n assets. 4. To summarize, the signa structure is given by: u a N 0, φ 1 a and εia N 0, τ a θ ia 1 with n θ ia = 1. 4 i=1 The signa structure can be represented in vector notation as y a = r + u a 1 + ε a, where ε a N0, τ a diag θ a 1 and ε a and u a are independent across anaysts. 5 2.3 Portfoio Choice and Utiity The investor has weath normaized to W=1. In period t = 2, the investor observes the anayst signas y = [y 1,..., y m] and chooses portfoio weights ω = [ω 1,..., ω n ] for the n risky assets, with the remainder 1 n i=1 ω i aocated to the risk-free asset. The investor's tota 3 As is standard in the iterature, we dene precision as the inverse of variance. That is, for a random variabe x, precisionx = variancex 1. 4 Such a constraint arises naturay in a setting with rm specic xed costs in information production. 5 The important aspect of this information structure is what it impies for the information content of the signa produced by the anayst. Furthermore, the signa content is invariant to any ane transformation of the signa y a. Therefore, reative vauation does not impy that the actua voatiity of the signas reported to investor by the anayst have innite variance. In fact, because the mode is invariant to any ane transformation, the variance of the signa is not determined. We thank David Hirsheifer for this point. 8
return, R p, is equa to R p = ω y R r f +r f, where ω y is the portfoio choice conditiona on the signas and r f is the risk-free interest rate. We consider cases in which the investor has either mean-variance utiity or CARA utiity. The investor's ex-ante utiity is given by the expectation taken over the joint distribution of returns and signas: 6 U = E [ E [R p y] γ ] 2 V ar [R p y] Mean-variance preference 5 U = 1 γ og [E [exp γr p]] CARA preference 6 3. The Portfoio Decision 3.1 Optima Portfoio Weights and the Precision Matrix We sove the mode backwards starting from the period t = 2 asset aocation probem. The investor takes as given the signas produced by the anaysts, and the optima portfoio weights are obtained from the standard rst-order condition: 7 ω y = 1 γ var R y 1 E R y r f. 7 The posterior mean, E R y, and variance, var R y, of returns are obtained from Bayesian updating. Their cosed-form expressions are given by the foowing Proposition. Proposition 1. Suppose that each anayst a = 1,..., m produces a vector of signas y a = r + u a 1 + ε a about asset returns R = R + r + η. Let the distributions of the random variabes be given by: r N0, Σ, η N0, Σ u, u a N 0, φ 1 a, and ε a N0, τ a diag θ a 1, where each distribution is mutuay pairwise independent and R = ER. Then each anayst produces a quantity Θ a = τ a diagθ a τa τ a+φ a θ a θ a of information about asset returns, and 6 See appendix B.1 for this derivation. 7 The portfoio wights are the same for both preferences because we normaize prices and investor weath to one. See appendix B.1 for detais. 9
the precision matrix of signas is given by: Θ = m Θ a. 8 a=1 Moreover, the posterior mean and variance of asset returns is given by: varr y = ˆΣ Θ = Σ u + Θ + Σ 1 1 ER y = Θ + Σ 1 m 1 Θ a y a + R. a=1 9 A proofs are in the Appendix. The Proposition highights the importance of the precision matrix, Θ, in the investor's posterior beiefs and asset aocation decision. Criticay, as Θ becomes arger or more informative, the posterior variance decines and the posterior mean assigns more weight to the anayst signas reative to the prior mean of Er = 0. This, in turn, renes the investor's asset aocation decision. Later in the paper, we show that more informative Θ's stricty increase the investor's ex-ante utiity, and hence Θ pays a key roe in the period t = 1 information production decision as we. The precision matrix, Θ, is intricatey inked to the degree of reative vauation in the economy. To see this, note that Equation 8 can be re-written as foows: Θ = m φ a [τ a diagθ a ] τ a + φ a }{{} a=1 absoute vauation + τ a [τ a diagθ a θ a θ τ a + φ a]. 10 a }{{} reative vauation The rst term in the summation, τ a diagτ a θ a, is a diagona matrix with eements equa to the precision added by anayst a about each asset i. This matrix captures the amount of information produced about the eve of asset returns. Due to the anayst-specic error term u a, ony a fraction φa τ a+φ a of this information is earned by the investor. In the extreme case of absoute vauation i.e, φ a, the precision matrix is given by: Θ A = m τ a diagθ a : absoute vauation case when a φ a =. 11 a=1 10
The second term in the summation, τ a diagθ a θ a θ a, is a square matrix with coumn and row sums adding up to zero. For a xed anayst, this matrix captures the amount of information produced about the reative vaue of the assets. Moreover, if an anayst ony covers one asset, then the amount of information she produces about the reative vaue of the assets is zero. 8 In the extreme case of reative vauation i.e, φ a = 0, the precision matrix contains is given by: Θ R = m τ a diagθ a θ a θ a : reative vauation case when a φ a = 0. 12 a=1 Whenever the anaysts are symmetric i.e., τ a = τ and φ a = φ for a a, the precision matrix can be expressed as a convex combination of the absoute and reative vauation cases: Θ = φ τ Θ A + Θ R. 13 τ + φ τ + φ 3.2 Understanding the Portfoio Decision - Reative Vauation We now anayze in more detai the investor's portfoio choice for the case of reative vauation: φ a = 0 for a a. 9 We begin by estabishing a usefu connection between reative vauation and the network induced by the aocation of anaysts to assets. We then show that this network determines how the investor earns from observabe signas, in addition to how she reaocates her capita across the assets. Knowing these facts are crucia for understanding the investor's optima aocation of anaysts to assets in period t = 1. 8 Consider an aocation {θ ia } a i in which each anayst a produces information about just one asset i: for a a: θ ia {0, 1} subject to i θ ia = 1. In the case of absoute vauation, the investor earns a tota of a τ aθ ia information about each asset. In the case of reative revauation, the investor does not earn any information about any of the assets: diagθ a θ a θ a = 0 a. Economicay, this aocation is useess to the investor because she does not have a benchmark to compare each anayst's signa to. She cannot separate the information about returns from the anayst-specic errors embedded within the signas. 9 We anayze the ess interesting case of absoute vauation in Section??. 11
3.2.1 The Information Production Network The information production network is dened as the graph, G, where the vertices, V G, are the n assets and the edges, EG, are the pairs of distinct assets that are covered by at east one common anayst. 10 Two assets i and j are adjacement in the network, i j, if there is a common anayst covering both assets - i.e., i and j are joined by an edge. More generay, two assets i and j are connected in the network if there exists a set of rms i = 1,..., N = j such that k 1 k for k = 1,..., N. Connected assets that are adjacent are considered directy connected, whie connected assets that are non-adjacent are considered indirecty connected. Ony connected assets can be compared in a reative vauation setting. Connections between assets faciitate the ow of information throughout the network. To measure the strength of direct connections between pairs of assets, we dene the n n weighted adjacency matrix, AG, as foows: 0 for i = j AG ij = a τ aθ ia θ ja for i j The vaue of AG ij is increasing in 1 the number of common anaysts covering both assets i and j and 2 the amount of anayst precision aocated to covering both assets. The matrix has nu entries aong its diagona, and assets which are non-adjacent aso satisfy AG ij = 0. The weighted adjacency matrix can be used to measure the strength of indirect connections in the network as we. 11 To see this, et [ A k] ij denote the ij th eement of AG raised to the k th power. A we-known resut from graph theory e.g., Newman 2010 states that [ ] A k is equa to the weighted number of paths of ength k connecting vertices/assets i and ij j. Therefore, the sum [ ] k=1 A k measures the strength of indirect connections between ij assets i and j in the network. Simiary, [ ] k=0 A k measures the strength of a direct ij and indirect connections between i and j in the network. 10 Formay, et A ij = {a A : i N a and j N a } denote the set of a anaysts covering both rms i and j, where N a = {i : θ ia > 0} is the set of a assets covered by anayst a. The graph G consists of a set of vertices V G = N {1,..., n} and a set of edges EG = {{i, j} N N : i j and A ij }. 11 Two non-adjacent assets i and j can be indirecty connected if, for exampe, there is an asset such that i and j. More generay, two non-adjacent assets are indirecty connected if there is a sequence of assets 1,..., k, such that i 1, 1 2,..., k 1 k, k j. 12
We dene the degree of an asset i as di = j AG ij and we dene the n n degree matrix as DG = diagd1,..., dn. Finay, we dene the n n weighted Lapacian matrix, LG, as the dierence between the degree matrix and the weighted adjacency matrix: LG = DG AG. It is immediate to verify that the Lapacian matrix is equa to the precision matrix in the case of reative vauation: 12 Θ R = LG = a τ a diagθ a θ a θ a. These connections to graph theory wi aow us to better our understanding of how the investor incorporates the signas into her posterior beiefs and portfoio decision. Further detais are provided beow. 3.2.2 Impications for Learning - Reative Vauation The foowing Proposition aows us to interpret the posterior Asset Correations: The foowing proposition is hepfu in interpreting the posterior correations among assets in terms of the strength of connections between rms in the anayst coverage network AG. We show beow that reative vauation creates positive correation among assets. Moreover, we aso show that posterior correations among assets i and j are arger when the direct ink between the two assets as given by the weights A ij = a τ aθ ia θ ja are strong pus there are many indirect inks connecting the two assets. Proposition 2. Let Σ = diagσ1, 2..., σn 2 and posterior variance matrix var R y = Σ = Σ 1 + Θ 1, where Θ = a τ a diagθ a θ a θ a and et A = Θ diagθ be the adjacency matrix aso given by A ij = a τ aθ ia θ ja, for i j. Then: i The posterior correations among assets are non-negative Σ ij 0, for a i and j, and moreover, Σ ii Σ ij ; 12 Note that DG = diagθ R because di = j AG ij = a τ aθ ia j θ ja = a τ aθ ia a τ aθia 2, and that AG = diagθ R Θ R. 13
ii The posterior variance can be expressed as Σ = N 1/2 A k N 1/2, where A = N 1/2 AN 1/2 is the normaized adjacency matrix, and N = Σ 1 + diagθ is a normaizing diagona matrix. Thus the stronger weaker is the connection between two rms i and j, the arger smaer is the posterior correation among i and j. The Proposition above essentiay uses that I A 1 = k=0 Ak which hods when a eigenvaues of A have norm ess than one which we show hods in the proof i.e., ρa < 1. Moreover, we know from graph theory see for exampe, Newman 2010, that the number of paths of ength k connecting two vertices i and j is given by [ A k] i.e., the ij ijth eement of the k th power of the adjacency matrix A and that the sum [ ] k=0 A k is the tota number ij of paths connecting i and j. Thus Proposition 2 formay shows that we can interpret the posterior correations between assets i and j as arger or smaer depending on how strong or weak are the connection between i and j, as measured by the number of the paths connecting i and j in the graph with adjacency matrix A. Posterior Expected Returns: The posterior mean of returns 9 can be expressed aso as the sum of each anayst signa averaged by the precision of each signa mutipied by the posterior variance: m ER y = Σ Θ a y a + R, a=1 We now show that how investor update the mean returns after observing investors' signas depend both on direct and indirect connections. The posterior expected return of asset i is greater when either asset i receives a favorabe anayst recommendation y ia, or when an asset j, which asset i is strongy connected to, receives a favorabe anayst recommendation y ja. In other words, each signa provided by the anaysts aects the expected returns a the assets for which they cover. We state this resut beow. k=0 14
Proposition 3. The sensitivity of returns to asset recommendations satisfy E R y j y ia = τ a θ ia Σ ji k Σ jk θ ka. The foowing comparative statics resuts hod: i The asset i return increase when any anayst foowing asset i revise it upwards. That is ER y i y ia 0. ii When the strength of the connection between asset j and asset i is stronger weaker than the average strength of the connection among asset j and the other assets covered by anayst a, then asset j return increases when an anayst a revise asset i upwards. Formay, ER y j y ia 0 if and ony if Σ ji k Σ jk θ ka. 3.2.3 Impications for Portfoio Choice We now show how the information network provides insight about how capita is reaocated in response to asset recommendations. information produced as foows: Remind that capita is reaocated based on the ω y y ia = 1 γ Σ u + Σ 1 E R y. 14 y ia In the information network graph G, two rms i and j are dened as connected if and ony if there is a path connecting them. That is, there is a distinct set of rms i = i 0, i 1, i 2,...i m = j such that i k 1 and i k share a common anayst for k = 1,..., m. Note that connection we formay dene before is an equivaence reation. A graph, G, is connected if any two rms can be joined by a path, and is otherwise disconnected. A maxima connected subgraph of the graph is dened as a connected component, where a subgraph is any graph S formed from a subset of the vertices and edges of G. We wi aso refer to the components of the graph as the maxima equivaence casses of the connection reation. We show beow that no capita ows in or out of the separate connected components of the information production network. This resut directy reates to the concept of reative vauation. Since asset anayst recommendations have no eve anchor, investors can ony 15
make inferences about the reative vaue of assets that are evauated by the same anayst or through a chain of anaysts. For exampe, suppose the assets in one component receive on average more positive signas than the assets in second component Then investors are unabe to infer whether the assets in rst component have higher expected returns, or if the signas produced in second component have a dierent eve, i.e. information produced by competey disconnected information sources cannot be combined. Ony when there is a connection between the assets can inferences be made about expected returns and weath reaocated. Proposition 4. Weath Reaocations Consider the pure reative vauation setting with Σ u = 1 α Σ and Σ = ασ, for some α [0, 1], which incudes the fu-earnabe case. Then: i The assets can be partitioned into connected and disjoint components of the graph G dened above; ii There is no reaocation of weath among disconnected components. Formay, et ωy be the optima portfoio choice upon earning signa y and et ω no earn be the portfoio choice under no earning. The weath aocated to each connected component, say component G, satisfy i G ωy = i G ωno earn i, for a possibe signas y; iii In particuar, there is no reaocation of weath between risky and riskess assets, i.e. i N ωy = i N ωno earn i for a signas y. In the knife-edge case of fu earning, the weath reaocations take a particuary simpe Σ 1 form because Σ u + Σ = I when Σu = 0. In response to a signa received by anayst a for asset i, the optima investment on assets j i is ω j y ia = 1 γ τ aθ ia θ ja 0 and ω i y ia = 1 γ τ a θia θia 2 0. 15 From equation 15 there is a reaocation of weath across assets in response to signas in the presence of reative vauation. In particuar, in response to a signa received by anayst a for asset i j, the optima oading on asset j marginay decreases. Note that the decrease 16
is arger the more anayst a foows both assets i and j, and if the anayst does not foow both assets there is not reaocation between the assets. Therefore the intensity of capita reaocation depends on the strength of the anayst foowing connection between assets. Whenever, the information between the pair of assets i and j, τ a θ ia θ ja is high, good news about asset i eads the investor to pu more capita from asset j. To concude this section, no capita ows across connected components. In addition, the extent of intra-component capita reaocations depends on both the strength of connections and the vaue of asset recommendations. A favorabe signa about a rm propagates throughout the entire component, and causes capita to reaocate to the assets in the component to which the rm shocked is most cosey connected. 4. Investors Ex-ante Preferences over Aternative Information Production Networks In the this section we focus on characterizing the investor ex-ante utiity associated with aocations of information producing resources. Specicay in the Proposition 5 beow we obtain the ex-ante expected utiity and show that it depends essentiay ony on the information matrix Θ for both the CARA and mean-variance investor. Proposition 5. Let the prior excess return vector be µ = R r f 1 and variance of returns R N R, Σ with earnabe and unearnabe variances Σ and Σ u. Whenever the information matrix about the earnabe component is Θ, so that the posterior precision is ˆΣ 1 Θ = Σ u + Σ 1 + Θ 1 1, then the ex-ante investor utiity is: i In the CARA preference case, the ex-ante utiity is U Θ = 1 ˆΣ 1 og det Θ Σ + µ Σ 1 µ. 16 2γ ii In the mean-variance preference case, the ex-ante utiity is U Θ = 1 ˆΣ 1 T r Θ Σ + µ 2γ ˆΣ 1 Θ µ n. 17 17
Note that in fu earning case, i.e. Σ u = 0, the posterior precision is simpy ˆΣ 1 Θ = Σ 1 + Θ, and thus inear in the information matrix Θ. Moreover, in the mean-variance preference case with fu-earning, the ex-ante utiity is aso inear in the information matrix, U Θ = 1 T r ΘΣ + µ Θµ + µ Σ 1 µ. 18 2γ Thus, the investor optima soution in this specia case is determined without any regards for interdependencies among anaysts' choices. In a other cases, there are important interdependencies among anaysts' information production choices. We show beow the posterior precision matrix ˆΣ 1 Θ and the investor ex-ante utiity UΘ are stricty monotonic and stricty concave mappings of the information matrix Θ. We introduce beow denitions to formaize these concepts: Informativeness Matrix Θ is more informative than Θ, i.e. Θ Θ, if the matrix Θ Θ is positive semidenite. Concavity The utiity function and posterior precision mapping are concave if UΘ λ λuθ + 1 λ UΘ and ˆΣ 1 Θ λ λˆσ 1 Θ + 1 λ ˆΣ 1 Θ, for any λ [0, 1] and Θ λ = λθ + 1 λ Θ. Monotonicity The utiity function and posterior precision mapping are monotonic if UΘ UΘ and ˆΣ 1 Θ ˆΣ 1 Θ, for any Θ Θ. We provide more detais in the Appendix B, incuding the natura extensions to strict concavity and monotonicity. Lemma 1. Consider the posterior precision mapping ˆΣ 1 Θ = the ex-ante utiity UΘ. Then: Σ u + Σ 1 + Θ 1 1 and i The posterior precision mapping is stricty concave and stricty monotonic in the partia earning case with Σ u and Σ invertibe; and it is inear and stricty monotonic in the fu earning case with Σ u = 0. ii The investor ex-ante utiity function UΘ of a CARA or mean-variance investor is stricty monotonic and stricty concave in the information matrix Θ [except that in the mean variance and fu earning case Σ u = 0, it is inear in the information matrix Θ]. 18
The strict concavity for the CARA utiity arises both from the strict concavity of the og determinant mapping X og detx and the strict concavity of the posterior precision with partia earning, whie the stricty concavity for the mean-variance utiity arises ony from the atter. We wi show ater on how these genera strict monotonicity and concavity properties impy the strict optimay of baanced designs among anaysts see Section 6. Rather than use the investor utiity directy, it wi save on notation going forward to focus on the utiity gain reative to no-information earning, U Θ = U Θ 1 2γ µ Σ 1 µ, 19 subtracting away the constant utiity term 1 2γ µ Σ 1 µ due to no-earning. The evauation of investor utiity in appications is made easier by the foowing resut which provides the investor utiity gain as a function of the eigenvaues of the weighted information matrix ΘΣ. Lemma 2. Consider the partia earning setting, Σ u = 1 α Σ and Σ = ασ, for some α [0, 1]. Dene the stricty increasing and concave function fx by fx = The ex-ante utiity gain, in the CARA preference case, is U Θ = 1 2γ 1 + αx 1 + α 1 α x. 20 n og fλ i. 21 where λ i 0 are the eigenvaues of the weighted information matrix ΘΣ. i=1 The proof is in the Appendix, where we aso provide the simiar expression for the meanvariance case in terms of the eigenvaues of the weighted information matrix. 19
5. Reative versus Absoute Vauation In this section we contrast the properties of information production under reative and absoute vauation. To keep the anaysis as simpe as possibe and better provide the intuition for the resuts we consider the mean-variance preference and fu-earning in this section. 5.1 Broad Coverage versus Speciaization Decision Using the properties that the reative vauation component of the information matrix is the Lapacian of the graph we can estabish a series of identities that gives us a more intuitive understanding of the how reative vauation works. Proposition 6. Suppose an anayst produce information about q assets i = 1,..., q with attention θ and precision φ and τ, so that the precision matrix is Θ = τ diagθ. τ τ+φ θθ Then the incrementa utiity gain, U = 1 T r ΣΘ + 2γ µ Θµ, can be expressed as U = 1 2γ τ q [ θ i θ j varri r j + µ i µ j 2] τ φ + τ + τ i,j=1:i<j }{{} U R 2 θ i varri + µ 2 φ i φ + τ i=1. }{{} U A The utiity gain is the weighted average of the gain coming from absoute vauation and reative vauation reated to the respective components of the precision matrix. Intuitivey, under reative vauation the utiity gain is proportiona to the sum of variances of the ongshort portfoio comparing a pair of assets weighted by the precision θ i θ j aocated to the pair. These resuts foow directy from the fact that the precision matrix Θ R is the Lapacian matrix of the information production network. We now show that reative vauation increases the gains of diversication in the production of information. 22 Consider a symmetric setting such as an industry with variance Σ = σ 2 I + ρj and equa excess returns µ = R i r f so that varr i r j = 2σ 2 and µ i µ j 2 = 0 for a pair of assets. This impies that the summation appearing in the 20
expression for the utiity gain is q [ θ i θ j varri r j + µ i µ j 2] q q 2 = σ 2 2θ i θ j = σ 2 θ i i,j=1:i<j i,j=1:i<j i=1 q i=1 θ 2 i. Since q i=1 θ i = 1 then the optima time aocation probem is the unique soution of: max θ s.t. 1 q i=1 θ2 i q i=1 θ i = 1 and θ i 0. Therefore spreading out equay the time aocation to each asset, θ i = 1, is a unique goba q optima aocation of time by each anayst whenever reative vauation is important i.e., φ <. The maximum vaue is σ 2 1 q i=1. Note that the time aocation 1 = σ 2 q 2 above is the unique optima regardess of the choice of precision φ and τ. The proposition beow makes this point formay. Proposition 7. Consider the optima information production design of a mean-variance investor and the assets have covariance Σ = σ 2 I + ρj and equa excess returns µ = R i r f, and anayst can cover up to q assets with precision τ and φ at a convex cost c τ, φ. Then the anayst optimay spread their aocation of time equay across a q assets, i.e., θ ia = 1 q, and the optima reative and absoute vauation precision φ and τ are given by the unique goba soution of the utiity maximization probem U = τ σ 2 1 + ρ + µ 2 φ 2γ τ + φ + σ2 q 1 q q 1 q τ c τ, φ τ + φ subject to τ, φ 0. Moreover, whenever φ < this information production choice is the unique goba optimum. With absoute vauation the gain in utiity is composed of the sum of the utiity coming from µ Θ A µ, which is equa to τµ 2, and the gain in utiity coming from T rσθ A = τσ 2 1 + ρ which is the tota voatiity times the tota precision. With strict reative vauation the gain in utiity is T rσθ R = τσ 2 equa to the idiosyncratic asset voatiity q 1 q 21
q 1 times the tota precision times the fraction ost due the fact that ony reative signas q have information. For intermediate cases the utiity gain is the average of the absoute and φ τ+φ and reative vauation cases with weights respectivey τ. τ+φ The resuts above iustrate that in genera it wi be optima for an anayst to sacrice some speciaization and precision in exchange for a broader and more imprecise asset coverage. The utiity gain attributed to the reative vauation is proportiona to τ. Thus if anaysts coud cover more assets without any overa precision oss there woud be an utiity gain. However, the margina gain is decreasing with the asset coverage. In particuar, increasing coverage from 2 to 3 assets increases utiity from τ 1 2 to τ 2 3, which is an utiity gain of 33%, but the utiity gain from increasing coverage from 9 to 10 assets woud ony be 1%. More generay it is ikey that the overa precision is decreasing in the tota number of assets covered as information production is ikey to have substantia xed costs associated with eaning about a particuar rm. q 1 q Specicay, et us assume that the tota anayst precision is decreasing in the number of assets covered, τ q < 0, then the decision of how many rms to cover woud be determined by max q τ q. The rst order condition yieds that the optima quantity is a decining function of the precision easticity, q = 1 ogτq/ ogq 1. In particuar, if the tota quaity of the information decays very sowy, the anayst wi optimay choose to cover a arge number of rms. q 1 q 5.2 Asymmetric Assets: Long-Short Portfoios We now address the optima design under asymmetric assets in the mean-variance utiity case. We focus on the case that the anayst can ony choose two assets, where the investor expected utiity is as foows U = 1 2γ τθ [ iθ j varri r j + µ i µ j 2] τ }{{} φ + τ + τ U R N θ i varri + µ 2 φ i φ + τ i=1, }{{} U A In the case where there is ony reative vauation the anayst choose the assets i and j 22
to cover, and aways optimay choose to dedicate equa attention to them, i.e. θ i = θ j = 1, 2 and choose the pair of assets in order to maximize max i,j varr i r j + µ i µ j 2, i.e. the pair of assets with the ong-short portfoio with the argest uncentered second moment. In, contrast in the case of absoute vauation, the anayst shoud dedicate a attention to the asset that maximizes max i varr i + µ 2 i. In the Appendix B, we generaize this resut for a setting where the anayst covers more than two assets q > 2. 6. Optimaity of Baanced Designs Having studied in Section 5 the probem of how to aocate one singe information production unit, now we study the probem of how to design the entire information production network. We show that reative vauation introduces a strong force for baancedness in the network. We start by showing that when the rms are symmetric, the optima network has exacty the same number of anaysts covering any pairs of stocks, i.e. the network is said to be baanced. We then extend our anaysis to show this property hods even in ess symmetric environments. Specicay, we show that in a muti-industry setting that the information production network is bock baanced, with intra-industry pairs having more anayst coverage than across-industry pairs. 6.1 Baanced Designs We rst start introducing the forma denition of baanced aocation. Denition: Baanced Network Consider a tripe 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 baanced, or a baanced design, if a subsets N a have exacty q assets, and every pair of assets is covered by exacty λ agents. Consider the foowing exampe to iustrate this point. Exampe 1: Consider a symmetric economy with n = 6 assets and m = 10 anaysts which can cover q = 3 assets, and et the matching of anaysts to assets be A = {123, 124, 135, 146, 156, 236, 245, 256, 345, 346}, where we denote the subset of assets foowed 23
by each of the m = 10 anaysts by tripes abc. The structure is a baanced design with λ = 2 anaysts covering each pair of assets. Note that each asset is foowed by exacty c = 5 anaysts. There can be mutipe structures with the same aocation of anayst coverage per asset but with variation in the amount of anayst coverage per pair. For exampe consider the aocation aocation of resources B = {123, 123, 123, 123, 123, 456, 456, 456, 456, 456}. The tota utiity of investors with the rst structure, the baanced aoaction, is more than 25% higher than the second structure see detais in Appendix B. A centra resuts of our paper is that the maximization of investor's utiity is uniquey achieved with baanced aocations of information production resources. Proposition 8. Optimaity of symmetric baanced designs Consider the symmetric probem above where the investor have CARA preference with fu or partia earning or meanvariance preference with partia earning with n assets and m agents that can cover q n assets with prior return R r f 1 N µ1, Σ where Σ = σ 2 I + σf 2 J such that there exists a baanced design.then the most ecient aocation among a possibe feasibe aocations is a baanced design in which a anaysts choose the same precision τ and φ. Whenever at the optimum φ <, then the baanced design is the unique maximum ecient aocation of resources. We estabish in our next resut the cosed-form soution for the investor utiity under CARA preferences when using a baanced aocation of information production resources. Proposition 9. Investors' utiity under a baanced design Suppose the investor has CARA preference, there are n assets with prior excess return with variance Σ = σ 2 I + ρj, and fu or partia earning with Σ u = 1 α Σ and Σ = ασ, for some α [0, 1]. Suppose that a m anaysts choose precision τ and φ and each can produce information about q assets and et the agents be organized according to a baanced design with λ = mqq 1 nn 1 anaysts per pair of assets, and c = mq n anaysts per asset. Then: i The precision of the signa obtained by investors is the n n matrix Θ equa to Θ = τm n [ φ τ + φ I + τ n q 1 τ + φ n 1 q I 1n J ], 23 where I and J are, respectivey, the n n identity matrix and matrix of ones in a entries. 24
ii The expected investor utiity gain with a baanced design is equa to: U τ, φ = 1 2γ n 1 og f λ 1 + og f λ 2 24 where f is give by equation 20, and λ 1 and λ 2 are the eigenvaues of the weighted information matrix ΘΣ given by: λ 1 τ, φ = σ 2 m τ n τ + φ λ 2 τ, φ = σ 2 m n φ + τ n q 1 : with mutipicity n 1, 25 n 1 q τφ 1 + nρ : with mutipicity 1. τ + φ iii There is a unique τ and φ that maximize the utiity net of costs U τ, φ mc τ, φ given by 24 subject to τ, φ 0. This soution is such that: τ a an increase in q eads to more weight being paced on reative vauation / q 0; φ b an increase in asset correation ρ eads to ess weight being paced on reative vauation τ / ρ 0. φ The optima precision τ and φ maximize the utiity U τ, φ given by 24 subject to τ, φ 0. Any other aocation with an information matrix dierent than the one given by 23 yieds stricty ess utiity to the investor. Note that the term q 1 qn 1 mutipying the reative vauation part of the information matrix is increasing in q, the number of assets anayst can foow. Thus if anayst can spread out the same tota precision τ across mutipe assets the information matrix becomes more informative. However, the rate of utiity gain is decreasing in the number of assets covered and for a arge number of assets q, such as q = 10 assets, over 90% of the possibe gains are aready achieved. 6.2 Information Production within and across Industries: Bock Baanced Designs Whie anaysts tend to be industry speciaists, there is aso a signicant across-industry coverage by anaysts. For exampe, Boni and Womack 2006, Tabe 3 document that se- 25