Active and Passive Investing, Exams of Investment Management and Portfolio Theory

Download Active and Passive Investing and more Exams Investment Management and Portfolio Theory in PDF only on Docsity! Active and Passive Investing Nicolae Gârleanu and Lasse Heje Pedersen∗ October 1, 2019 Abstract We model how investors allocate between asset managers, managers choose their portfolios of multiple securities, fees are set, and security prices are determined. The optimal passive portfolio is linked to the “expected market portfolio,” while the op- timal active portfolio has elements of value and quality investing. We make precise Samuelson’s Dictum by showing that macro inefficiency is greater than micro ineffi- ciency under realistic conditions — in fact, all inefficiency arises from systematic factors when the number of assets is large. Further, we show how the costs of active and pas- sive investing affect macro and micro efficiency, fees, and assets managed by active and passive managers. Our findings help explain empirical facts about the rise of delegated asset management, the composition of passive indices, and the resulting changes in financial markets. Keywords: asset pricing, market efficiency, asset management, search, information JEL Codes: D4, D53, D8, G02, G12, G14, G23, L1 ∗Gârleanu is at the Haas School of Business, University of California, Berkeley, and NBER; e-mail: gar- leanu@berkeley.edu. Pedersen is at AQR Capital Management, Copenhagen Business School, New York Uni- versity, and CEPR; www.lhpedersen.com. We are grateful for helpful comments from Antti Ilmanen, Kelvin Lee, and Peter Norman Sørensen as well as from seminar participants at the Berkeley-Columbia Meeting in Engineering and Statistics, Federal Reserve Bank of New York, and Copenhagen Business School. Peder- sen gratefully acknowledges support from the FRIC Center for Financial Frictions (grant no. DNRF102). AQR Capital Management is a global investment management firm, which may or may not apply similar investment techniques or methods of analysis as described herein. The views expressed here are those of the authors and not necessarily those of AQR. Over the past half century, financial markets have witnessed a continual rise of delegated asset management and, especially over the past decade, a marked rise of passive manage- ment, as seen in Figure 1. This delegation has potentially profound implications for market efficiency (see, e.g., the presidential addresses to the American Finance Association of Gross- man (1995), Stein (2009), and Stambaugh (2014)), investor behavior (presidential address of Gruber (1996)), and asset management fees (e.g., the presidential address of French (2008)). The rise of delegated management raises several questions: What is the optimal portfolio of active (i.e., informed) and passive (i.e., uninformed) managers, respectively? What deter- mines the number of investors choosing active management, passive management, or direct holdings? What are the implications of delegated management on market efficiency at the micro and macro levels? How do macro and micro efficiencies depend on the costs of active and passive management? We address these questions in an asymmetric-information equilibrium model where se- curity prices, asset management fees, portfolio decisions, and investor behavior are jointly determined. Our main findings are: (1) the optimal passive portfolio is the “expected market portfolio,” tilted away from assets with the most supply uncertainty; (2) the optimal active portfolio has elements of value and quality investing; (3) macro inefficiency is greater than micro inefficiency (consistent with Samuelson’s Dictum) when there exists a strong common factor in security fundamentals or when the number of securities is large; (4) when infor- mation costs decline, the number of active managers increases, active fees decrease, market inefficiency decreases, especially macro inefficiency (counter to part of Samuelson’s Dictum); (5) when the cost of passive investing decreases, market inefficiency increases, especially macro inefficiency, the number of active mangers decreases, and active fees drop by less than passive fees; and (6) market inefficiency is linked to the economic value of information and to entropy. These findings help explain a number of empirical findings and give rise to new tests as we discuss below. To understand our results, let us briefly explain the framework. We introduce asset managers into the classic noisy-rational-expectations-equilibrium (REE) model of Grossman 1 Active investors thus exploit market inefficiency across various assets, where inefficiency is defined (following Grossman and Stiglitz (1980)) as the uncertainty about the fundamental value conditional on only knowing the price relative to the uncertainty conditional on also knowing the private information. For example, the market inefficiency is zero (fully efficient market) if the uncertainty about the fundamental value is the same whether one learns from just the price or also the signal. Further, the more information advantage one enjoys from knowing the signal, the more inefficient the market. An interesting question that can be addressed naturally in this framework is whether there are greater inefficiencies at the macro or at the micro level. Indeed, Samuelson fa- mously hypothesized that macro inefficiency is greater than micro inefficiency, a notion known as “Samuelson’s Dictum” (see quote and references in the beginning of Section 3 and the empirical evidence in Jung and Shiller (2005)). We show that Samuelson’s Dictum holds when there exists a strong common factor in security fundamentals. More precisely, we show that the factor-mimicking portfolio is the most inefficient portfolio, while the least inefficient portfolios are long-short relative-value portfolios that eliminate factor risk. Hence, this makes precise what macro and micro efficiency means, and gives precise conditions un- der which Samuelson’s Dictum applies (or does not apply). We further show that, due to diversification, when the number of securities is large, not only does Samuelson’s Dictum always hold, but in fact the combined inefficiency of all micro portfolios becomes negligible — all the inefficiency is in the pricing of systematic factors. This result is related to the Arbitrage Pricing Theory (APT) of Ross (1976). While the APT states that risk premia are driven by systematic factors when the number of assets is large, we show that inefficiencies are also driven by these factors.2 2Active managers in our model make an all-or-nothing information choice, whereas Veldkamp (2011), Van Nieuwerburgh and Veldkamp (2010), Kacperczyk et al. (2016), Glasserman and Mamaysky (2018), and others study agents’ choice of information, which can affect macro vs. micro efficiency as emphasized by the latter paper. Our assumption captures, for example, the case in which active investors decide whether or not to set up an IT system that captures the main databases, whereas the above papers capture the idea that different investors may focus on different subsets of the available information. See also Breugem and Buss (2018), who consider the effect of benchmarking considerations on information acquisition and efficiency with multiple assets, and Kacperczyk et al. (2018), who consider the effect of large investors’ market power on market efficiency. We complement the literature with regard to macro vs. micro efficiency by providing a general definition of Samuelson’s Dictum, by showing how it arises with many assets (i.e., as the number of 4 Samuelson also hypothesized that efficiency, especially micro efficiency, has improved over time (see quote and reference in Section 4). Such an improvement in efficiency may be driven by a reduction in information costs as information technology has improved. We show that reduced information costs indeed lower inefficiencies, but they actually mostly lower macro inefficiency (counter to that part of Samuelson’s hypothesis). Lower information costs also increase active management (relative to self-directed investment and passive management), consistent with the development in the 1980s and 1990s. Another trend over the past decades is the decline in the cost of passive management. We show that such a decline should lead to a rise in passive management (at the expense of self- directed investment and active management), consistent with the development in the 2000s. Further, reduced cost of passive management leads to an increase in market inefficiency, especially macro inefficiency, leading to stronger performance of active managers so that their fees decrease by less than passive fees. These predictions are consistent with the empirical findings by Cremers et al. (2016). Indeed, Cremers et al. (2016) find that lower-cost index funds leads to a larger share of passive investment, higher average alpha for active managers, and lower fees for active, but an increased fee spread between active and passive managers. Finally, we show that market inefficiency is linked to the economic value of information and to relative entropy (and the Kullback-Leibler divergence), which provides a new way to estimate efficiency. In summary, we complement the literature by studying the optimal passive and active portfolios and the nature of optimal security indices, giving general conditions for Samuel- son’s Dictum, and studying what happens when information costs and asset management- costs change. Our model provides a financial economics framework that links the CAPM, APT, and REE in a way that helps explain recent trends in financial markets and in the financial services sector. Finally, we quantify the model’s implications via a calibration. 3 assets goes to infinity), and by showing the importance of all systematic factors (not just the market, assumed exogenously by Glasserman and Mamaysky (2018)), thus linking to APT. More broadly, we complement the literature by capturing investors’ costs of active, passive, and direct investments, by studying the effects of changes in the asset-management costs, and by relating efficiency to entropy. 3Stambaugh (2014) also considers trends in the investment management sector based on a different framework where the key driving force is a reduction in the amount of noise trading. As noise trading 5 1 Model and Equilibrium This section lays out our noisy rational expectations equilibrium (REE) model and shows how to solve it. 1.1 REE Model with Multiple Assets and Asset Managers We model a two-period economy featuring a risk-free security and n risky assets. The return of the risk-free security is normalized to zero while the vector risky asset prices p is determined endogenously. The risky assets deliver final payoffs given by the vector v, which is normally distributed with mean v̄ and variance-covariance matrix Σv, which we write as v ∼ N (v̄, Σv). Agents can acquire various signals about all the assets at a cost k. We collect all the signals in a vector of dimension n that we denote s = v + ε, where ε ∼ N (0, Σε) is the noise in the signal.4 The supply of the risky assets is given by q ∼ N (q̄, Σq) and the shocks to q, ε, and v are independent. The supply is noisy for several reasons (e.g., Pedersen, 2018): New firms are listed in initial public offerings, existing firms issue new shares in seasoned equity offerings, firms repurchase shares (sometimes by buying shares in the market without telling investors), and the number of floating shares changes when control groups buy or sell shares. 5 Further, the de facto supply of publicly traded shares also implicitly changes when correlations vary between public shares and investors’ endowment (e.g., human capital, natural resources, or private equity holdings, where private firms may also issue or repurchase shares). For these declines in his model, the allocation to, and the performance of, active managers both decline. Hence, this model cannot explain the finding of Cremers et al. (2016) discussed above, namely that the size and performance of active management move in opposite directions (but the model can explain a number of other phenomena). 4If we start with a signal ŝ of any other dimension n̂, then we can focus on the conditional mean u := E(v|ŝ), which is of dimension n and can be translated into a signal s as modeled above. For example, if n̂ ≥ n, we have s := v̄ + ΣvΣ−1 u (u − v̄), where Σu = var(u). 5Many indices use a float adjustment. E.g., S&P Float Adjustment Methodology 2017 states “the share counts used in calculating the indices reflect only those shares available to investors rather than all of a company’s outstanding shares. Float adjustment excludes shares that are closely held by control groups, other publicly traded companies or government agencies.” 6 1.2 Efficiency of Assets, Portfolios, and the Market An important building block of our analysis is the notion of price efficiency. To define this concept, we build on the logic of Grossman and Stiglitz (1980), which considers the inefficiency of a single asset. We wish to define the inefficiency of any set of linearly independent portfolios {ζ1, ..., ζl} ⊂ Rn, where the number of portfolios can be anywhere from l = 1, i.e., a single asset, to l = n, that is, the entire market. We collect the portfolio weights in a matrix ζ ∈ Rn×l and define their joint inefficiency as follows. ηζ = 1 2 log ( det(var ( ζ>v|Fu ) ) det(var (ζ>v|Fi)) ) , (1) where Fi = F(p, s) is the informed information set, consisting of both the price and the signal, and Fu = F(p) is the uninformed information set, consisting only of the price. In words, this definition means that a set of portfolios is considered more inefficient if the uninformed has a larger uncertainty relative to the informed about the fundamental values of these portfolios. For example, the inefficiency of a single asset, say asset 1, is computed by considering the portfolio ζ = (1, 0, ..., 0)>, which yields ηasset 1 = 1 2 log ( var (v1|Fu) var (v1|Fi) ) = log ( var (v1|Fu) 1/2 var (v1|Fi) 1/2 ) . (2) This expression is equivalent to that of Grossman and Stiglitz (1980). The expression makes the link between our notion of inefficiency and the amount of information gleaned from signals, respectively only prices, particularly easily to see. The overall market inefficiency plays a special role in the equilibrium of the model. The overall market inefficiency is naturally the inefficiency of the set of all assets. Hence, we consider the largest possible matrix of portfolios, ζ = In, namely the identity matrix in 9 Rn×n.8 We denote the overall market inefficiency simply by η: η = ηoverall market = ηIn = 1 2 log ( det(var (v|Fu)) det(var (v|Fi)) ) . (3) This definition of overall market inefficiency is the natural extension of the one-asset definition of Grossman and Stiglitz (1980), since it retains the tight link between market inefficiency and investors’ utility of information as discussed further below and formalized in Proposition 8.9 Another natural property of our notion of market inefficiency is that it is linked to entropy, which is also stated in Proposition 8. When we analyze macro vs. micro efficiency (in Section 3), we also study the efficiency of individual securities, portfolios, and collections of portfolios; the general definition of efficiency will be very useful. 1.3 Solution: Deriving the Equilibrium To solve for an equilibrium, one proceeds backwards in time. The first step, therefore, consists of solving for an equilibrium in the asset market, taking as given the masses of investors conditioning on Fi, respectively on Fu. This is done in a standard Grossman- Stiglitz step. We conjecture and verify that prices p are linear in the information s about securities as well as the supply q: p = θ0 + θs ((s − v̄) − θq(q − q̄)) . (4) The resulting optimal demands are linear, as well. For an investor of type j ∈ {i, u}, where i means that the investor invests through an informed manager and u means that he invests uninformed (passive manager or self-directed), the optimal demand is the portfolio xj that maximizes the investor’s expected utility given the information used. 10 The resulting 8The same outcome for the overall market inefficiency obtains for any matrix ζ ∈ Rn×n of full rank. 9The link between the value of information and the ratio of determinants of the conditional variances was first derived in Admati and Pfleiderer (1987). 10When an investor has searched for a manager, confirmed that the manager is informed, and paid the fee, then the manager invests in the investor’s best interest. This lack of agency problems means that there 10 certainty equivalent utility is − 1 γ log ( E [ max xj E ( e−γ(W+x> j (v−p)) |Fj )]) =: W + uj , (5) where Fj is the information set of investor j (as defined above). The above equality defines the certainty equivalent utility of being informed, ui, respectively uninformed, uu, as well as the corresponding optimal portfolios xi and xu. With these portfolio choices by informed and uninformed investors, market clearing requires that the supply of shares q equals the total demand, q = Ixi + Uxu, (6) where the number of informed investors (I) and the number of uninformed ones (U) are defined in Section 1.1. Despite the presence of many assets, the overall asset-market equilibrium is summarized by a single number, namely the overall inefficiency η defined in equation (3). Indeed, the market inefficiency captures investors’ utility of information: γ(ui − uu) = η. (7) Thus, when the overall market is more inefficient, investors have a greater utility gain from being informed relative to being uninformed (as we show in Proposition 8). The second step consists of determining the active-management fee, taking into account the utility consequences of investing actively with an informed manager, respectively pas- sively. The active asset management fee is set through Nash bargaining, meaning that the fee maximizes the product of the manager’s and investor’s gains from trade. The investor’s gain from his investment is his certainty equivalent utility if he invests (W − c − fa + ui) over and above his outside option of going to a passive manager (W − c− fp + uu), where c is no difference between investing in a fund and sale of information as in Admati and Pfleiderer (1990). For a recent model of agency issues in asset management, see Buffa et al. (2014). 11 Number of informed active investors equilibrium market inefficiency active cost Figure 2: Equilibrium Market Inefficiency. The figure shows how an equilibrium is found by equalizing the cost and benefit of active investing. The benefit of active investing (the solid line) stems from the ability to exploit market inefficiency; expressed as certainty equivalent wealth, it equals η/γ. The net cost of active investing (the dashed line) equals the active fee plus the search cost, minus the cost of passive investing. Finally, as seen in Figure 2, the equilibrium is found as the intersection of the solid and dashed lines. Intuitively, having more investors with informed managers reduces market inefficiency, diminishing the benefit of being informed. Equalizing this benefit with the cost of active investing yields the equilibrium. This intuitive figure makes it easy to investigate the implications of varying parameters; we’ll find it useful, for instance, when we consider the effect of changing passive costs in Section 4. 1.4 Statistical Assumptions Before we discuss our main results, we note that certain properties of the equilibrium (such as investors’ portfolios and the validity of the Samuelson’s Dictum) simplify when we impose further statistical structure on the shocks. We mainly rely on Assumption 1, namely that the covariance of shocks can be represented by a factor model. 14 Assumption 1 Fundamentals have a factor structure: v = v̄ + βFv + wv (12) ε = βFε + wε (13) q = q̄ + βFq + wq, (14) where v̄, q̄ ∈ Rn are the average fundamental values, respectively supplies, β ∈ Rn is a vector of factor loadings normalized (without loss of generality) such that β>β = n, the common factors Fv, Fε, and Fq are one-dimensional random variables with zero means and variances σ2 Fv , σ2 Fε , and σ2 Fq , respectively, and the idiosyncratic shocks wv, wε, and wq ∈ Rn are i.i.d. across assets with variances σ2 wv , σ2 wε , respectively σ2 wq for each asset. We will refer to the portfolio proportional to β as the “factor portfolio,” since this port- folio is maximally correlated with the common shocks. It is natural to think of this factor as the average market portfolio — that is, under Assumption 1 it is natural to also assume that the average supply q̄ is proportional to β and we will occasionally make this additional assumption. We also make use of the following assumption, which captures the idea that may underlie Samuelson’s hypothesis, namely that the common factor-component of the risk is especially important for future security prices. Assumption 1′ Assumption 1 holds and the common factor of v is non-zero, σ2 Fv > 0, and at least as important as that of ε, i.e., σ2 Fv /σ2 wv ≥ σ2 Fε /σ2 wε . Finally, we also consider the following assumption. Assumption 2 There exist scalars zε and zq such that Σε = zεΣv and Σ−1 q = zqΣv. The first part of Assumption 2 simply says that fundamentals and signal noise have the same risk structure (which can also hold under Assumption 1). The second part, which is 15 more unusual, says that the inverse of the variance-covariance matrix of the supply noise also shares this structure.12 Assumptions 1 and 2 are both satisfied if all shocks are i.i.d. across assets, but otherwise they are different. We focus on Assumption 1, as it is the more standard and more realistic assumption. Assumption 2 is to be thought of as a generalization of the i.i.d.-shock case. In particular, the results that require narrowing Assumption 1 down to the case of i.i.d. shocks also hold under Assumption 2, and we therefore state them in this greater generality. 2 Optimal Passive and Active Portfolios We wish to understand how active and passive investors construct their portfolios. Specif- ically, we are interested in the optimal informed portfolio xi and uninformed portfolio xu, defined in Equation (5). Given that real-world uninformed investors tend to hold indices, the optimal uninformed portfolio provides a foundation for the economics of indices. A standard benchmark portfolio in financial economics is the “market portfolio,” that is, the portfolio of all assets (cf. the CAPM). However, as emphasized by Roll (1977), the market portfolio is not known in the real world. Likewise, uninformed investors do not know the market portfolio q in our noisy REE economy. While the market portfolio is central in the CAPM, it has not played a role in the REE literature (because this portfolio is public knowledge and because this literature is focused on a single asset). Nevertheless, we can bridge these literatures by introducing the concepts of the “conditional expected market portfolio,” E(q|p), and the “average market portfolio,” q̄. The conditional expected market portfolio is the uninformed investors’ best estimate of the true market portfolio, q, based on public information. We first show how the optimal 12To understand Assumption, consider what happens if any security j undergoes a two-for-onw stock split, meaning that all shareholders receive two new shares for each old share. In this case, the number of shares outstanding doubles and the value of each share drops by half. This means that, if Assumption 2 was satisfied before the stock split, then it remains satisfied after the stock split. Indeed, the split means that the volatility of the value of shares drops by half, the volatility of the information noise drops by the same ratio, and the volatility of the supply noise doubles. A less natural implication of Assumption 2 is that securities with more correlated fundamentals have less correlated supply shocks (except in the special case, which overlaps with Assumption 1, when all securities are i.i.d.). 16 securities with too low price, too low market capitalization, too low liquidity, too recent IPO, or involved in certain corporate actions.13 Finally, we note that portfolio holdings scale with the degree of risk aversion. The results therefore only depend on the actual risk aversion of any given investor through a (positive) constant of proportionality. We next turn to the portfolio holdings of the informed investors. Proposition 2 (Optimal active portfolio: value and quality) Under Assumption 1 or Assumption 2, an informed investor’s position in any asset j is more sensitive than that of an uninformed agent to (a) supply shocks for asset j, ∂E[xi,j |q] ∂qj > ∂E[xu,j |q] ∂qj (value investing); (b) the signal sj about asset j, ∂E[xi,j |s] ∂sj > 0 > ∂E[xu,j |s] ∂sj (quality investing). The first part of the proposition states that informed investors buy more when the supply increases. For example, when there is an initial public offering, informed investors likely buy a disproportional fraction of the shares during book-building process. Likewise, if the supply of an existing company increases (for a given value of the signal), this extra supply will tend to lower the price, creating buying opportunity for informed investors (who realize that the price drop is not due to bad information). Buying securities at depressed prices can be viewed as a form of “value investing.” The second part of the proposition states that, when the signal for a given security improves, informed investors tend to increase their position in this security while uninformed investors tend to lower their position. Clearly, when informed investors receive favorable information about a security, they are more inclined to buy it. This extra demand tends to increase the price, leading the uninformed to reduce their position (markets must always clear) since uninformed investors cannot know whether the price increase due to favorable information or a drop in supply. Buying securities with strong fundamentals, even if their price has increased, is called “quality investing.” 13See, e.g., and “Russell U.S. Equity Indexes 2017” and “S&P U.S. Indices Methodology 2017.” 19 The idea that informed investors should focus on value and quality goes back at least to Graham and Dodd (1934) and, following this advice, investors such as Warren Buffett have pursued these strategies (Frazzini et al. (2018)). Value and quality investment strategies have indeed been profitable on average across global markets (see, e.g., Asness et al. (2013), Asness et al. (2018), Fama and French (2017)). 3 Samuelson’s Dictum: Macro vs. Micro Efficiency We have seen that the overall level of market efficiency is linked to the level of active asset management fees in (8). It is interesting to further study the relative price efficiency of different securities and portfolios. In this connection, Paul Samuelson famously conjectured that markets would have greater micro efficiency than macro efficiency:14 “Modern markets show considerable micro efficiency (for the reason that the mi- nority who spot aberrations from micro efficiency can make money from those oc- currences and, in doing so, they tend to wipe out any persistent inefficiencies). In no contradiction to the previous sentence, I had hypothesized considerable macro inefficiency, in the sense of long waves in the time series of aggregate indexes of security prices below and above various definitions of fundamental values.” Our framework is an ideal setting to make Samuelson’s intuition precise. Indeed, we have multiple securities (so we can discuss micro vs. macro) and a precise measure of efficiency for any asset or portfolio given by (1). Inspired by Samuelson’s Dictum, we are interested in which portfolio ζ has the maximum inefficiency: max ζ∈Rn ηζ = max ζ∈Rn 1 2 log ( ζ>var(v|p)ζ ζ>var(v|s)ζ ) . 14This quote is from a private letter from Samuelson to John Campbell and Robert Shiller, as discussed by Shiller (2001). Other references to the notion of macro vs. micro efficiency appear in, e.g., Samuelson (1998). 20 We wish to determine whether the solution, say ζ∗, is micro or macro in nature. Similarly, we are interested in which portfolio has the minimum inefficiency, but since the analysis is analogous, we focus here on the maximum and state the general result in the proposition below. To solve this problem, we let G = var(v|s)−1/2var(v|p)var(v|s)−1/2, which is essentially the matrix of the informed investor’s information advantage (in terms of her reduction in uncertainty). We denote its eigenvalues by g1 ≥ g2 ≥ ... ≥ gn > 0 and the corresponding eigenvectors by w1 through wn. Using this matrix, we see that the maximum portfolio inefficiency is: max ζ∈Rn ηζ = max z∈Rn 1 2 log ( z>Gz z>z ) = 1 2 log g1, where we have used the substitution z = var (v|s)1/2 ζ. The most inefficient portfolio is the eigenvector w1 corresponding to the largest eigenvalue, translated back into portfolio coordinates (i.e., reversing the substitution), ŵ1 := var (v|s)−1/2 w1. We have almost answered Samuelson’s question namely whether the most inefficient portfolio, ζ∗ = ŵ1, is macro or micro in nature. All that is left is to determine the portfolio ŵ1 by finding the primary eigenvector of G. Before we state the answer, we note that this analysis also sheds new light on the meaning of “overall market inefficiency” in an economy with multiple assets. Specifically, we can express the overall market inefficiency in terms of the eigenvalues (gj): η = 1 2 log ( det(var(v|p)) det(var(v|s)) ) = 1 2 log (det(G)) = 1 2 n∑ j=1 log gj = n∑ j=1 ηj , (18) where ηj is the inefficiency of portfolio ŵj , defined analogously to ŵ1, as the “rotated” portfolio versions of eigenvector wj : ŵj = var (v|s)−1/2 wj. We see that the overall market inefficiency is the sum of portfolio inefficiencies for the set (ŵj) of independent15 portfolios 15While the original eigenvectors (wj) are orthogonal in the Euclidean norm, w> j wk = 0, the corresponding portfolios (ŵj) are orthogonal in the economically more interesting sense that their payoffs are conditionally uncorrelated, cov(ŵ> j v>, ŵ> k v|s) = ŵ> j var (v|s) ŵk = 0. (We also note that, under Assumption 1, the 21 the variance σ2 Fε of the common component in the signal noise is sufficiently large. In this case, the correlated signals convey little information about the factor portfolio — indeed, in the limit as σ2 Fε becomes infinite the inefficiency is zero: no information conveyed by the signals or prices. On the other hand, a portfolio with no factor exposure is predicted with finite noise (the most informative signal for such a portfolio has zero loading on the common signal noise), and only some of the information is impounded in the price; the inefficiency is strictly positive. In this case, learning about the factor portfolio is difficult because all signals contain correlated noise, but trading on the factor portfolio is relatively safe (because the common component in fundamental risks is comparatively small). Therefore, informed investors will make the factor portfolio relatively efficient in the sense that much of what can be learned about the factor portfolio is incorporated into the price. Finally, part (d) of the proposition shows that the above three cases exhaust all possible scenarios, under Assumption 1. In other words, Samuelson’s notion of macro vs. micro efficiency is a good one in the sense that the most and least efficient portfolios are always the factor portfolio (macro) and the arbitrage portfolios (micro), never anything in between. 3.1 Macro vs. Micro Efficiency with Many Assets We next consider the efficiency in a market with a large number of assets. We are interested in what happens to macro and micro efficiency as the number of assets, n, grows large. The simplest way to consider a growing number of assets is to adopt the factor structure in Assumption 1 and let βi = 1 for any asset i.18 In this case, all assets are symmetric and it is clear what happens to fundamentals v and noise ε as the number of assets increases — namely, there simply are more of the same type of securities. To keep the economy finite, we need to make the total supply of shares constant, which implies that the number of shares outstanding for each firm falls as 1/n. Specifically, we let the expected supply of each security be q̄i = q̂/n and the supply uncertainties be σFq = σ̂Fq/n and σwq = σ̂wq/n, where q̂, σ̂Fq , and σ̂wq are the corresponding (scalar) values in the economy with a single asset. 18A simple generalization can be made along the lines of the multi-factor extension below. 24 With this simple model of symmetric assets, the next proposition shows that Samuelson’s Dictum always holds when n is large — that is, without relying on Assumption 1 ′ (as in Proposition 3) that the factor structure in fundamentals is “strong.” In fact, the next proposition contains a stronger result: As the number of assets grows, the factor portfolio becomes the only inefficient portfolio. To appreciate this result, we note that, as we showed above — see equation (18) — the overall inefficiency η can be seen as the sum of the inefficiencies of n uncorrelated portfolios. More specifically, the overall inefficiency equals the inefficiency of the factor portfolio ηβ plus n − 1 other portfolio inefficiencies, and we show that the factor inefficiency dominates to a surprising extent: Proposition 4 In the symmetric single-factor model described above with σFq > 0, the most inefficient portfolio is the factor portfolio β = (1, 1, ..., 1) when the number of assets, n, is large enough. Further, the fraction of market inefficiency coming from the common factor approaches 100% as n → ∞, that is, ηβ/η → 1. The fact that micro portfolios, which load exclusively on idiosyncratic shocks, are asymp- totically efficient is not hard to intuit. The supply noise in such a portfolio is purely idiosyn- cratic, and therefore its variance goes to zero as n increases without bound. Consequently the price signal is asymptotically as informative as the fundamental signal. The reason why the combined inefficiency of all micro portfolios tends to zero is more involved. It is a combination of the facts that (i) the variance of the (unit-norm) micro port- folios decreases as 1/n2 with n; (ii) the gain in the precision of the price signal, and therefore the decrease in inefficiency, from increasing n is approximately linear in this variance; and (iii) there are n−1 independent micro portfolios. When restricted to micro portfolios, there- fore, an informed agent’s utility gain over that of an uninformed one declines as 1/n, and thus goes to zero. On the other hand, as long as the factors are not trivial, the factor portfolio offers the informed agent a utility gain that is bounded away from zero: the supply noise in the factor does not disappear as n goes to infinity. 25 Multifactor-Model: The APT of Efficiency. We can generalize this result to a multi- factor model. As we shall see, in a multi-factor economy, all systematic factors have non- zero inefficiencies even with a large number of assets, but all idiosyncratic inefficiencies are eliminated, a result that “echoes” the Arbitrage Pricing Theory (APT) of Ross (1976). To construct a model with k factors, we can reinterpret the factors F from Assumption 1 to be k-dimensional (column) vectors and, correspondingly, β is a n × k matrix of factor loadings (i.e., each column provides the loadings of that factor across assets). Further, for each vector F , we assume that the k factors are iid. Lastly, to keep the model “stationary,” we assume that the eigenvalues of β>β/n (a k× k matrix) converge to some strictly positive limits λ1, ..., λk. Of course, we let supply decrease with n in the same way as above. For example, we could have a two-factor model, β = (β1, β2), where factor 1 is the market factor (as before), β1 = (1, 1, ..., 1)>, and factor 2 is value-versus-growth, β2 = (1 2 ,−1 2 , 1 2 ,−1 2 , ..., (−1)n−1 2 )>. The specification of factor 1 means that all assets are part of the market. The alternating signs in factor 2 means that every other asset is a value stock, and the remaining ones are growth stocks. Further, the lower coefficients in factor 2 means that this factor is a weaker driver of returns. In this case, the eigenvalues of β>β/n are 1 and 1 4 for all n even. Analogously to the APT, which describes expected returns in the presence of a factor structure, our next result characterizes asset inefficiencies. Recall that equation (1) defines the overall inefficiency ηβ of the collection of factor portfolios β. Proposition 5 As the number of assets grows, n → ∞, in the multi-factor model described above with σFq > 0 it holds that [APT of Returns] market risk premia are determined by factor loadings in the limit, that is, E(vi − pi) = ∑ j=1,...,k βijλj, where λj ∈ R is the risk premium of factor j. Conse- quently, a portfolio with zero loadings on all the factors has zero expected excess return. [APT of Efficiency] the fraction of the market inefficiency coming from the systematic fac- tors approaches 100%, that is, ηβ/η → 1. Consequently, a portfolio with zero loadings on all the factors has zero inefficiency. 26 from economic history or avant garde theorizing, that MACRO MARKET INEF- FICIENCY is trending toward extinction: The future can well witness the oldest business cycle mechanism, the South Sea Bubble, and that kind of thing. We have no theory of the putative duration of a bubble. It can always go as long again as it has already gone. You cannot make money on correcting macro inefficiencies in the price level of the stock market.” [emphasis as in original] One of the ways in which markets may have improved over time is that information costs may have come down, so it is interesting to consider whether lower information costs has the implications conjectured by Samuelson’s. In particular, when Samuelson’s Dictum holds for the levels of inefficiency (Assumption 1′ as shown in Proposition 3(b)), we can look at the relative changes in macro inefficiency, ηβ, vs. micro inefficiency, η⊥. Recall ηβ is the inefficiency of the market portfolio, and we define micro inefficiency by η⊥ := ηζ for any market neutral (or micro) portfolio: ζ>β = 0. Interestingly, Samuelson’s examples of micro efficiency appear close to our definition since option arbitrage and index arbitrage are long-short portfolios that eliminate factor risk. Proposition 6 (Information cost and the evolution of macro vs. micro efficiency) When the cost of information k decreases, overall asset price inefficiency η decreases and, under Assumption 1′, the macro inefficiency (ηβ) decreases by more than the micro ineffi- ciency (η⊥) as long as γ/I is sufficiently small. Further, the number of self-directed investors remains unchanged, the numbers of informed investors I and of informed active managers M increase, the number of active investors Sa may either increase or decrease, the active management fee fa decreases, and the passive fee fp is unchanged. With the improvement in information technology, the cost of information may have de- creased over time. If so, Proposition 6 shows that overall market inefficiency should have im- proved as a result, consistent with Samuelson’s conjecture. However, Proposition 6 predicts that macro inefficiency has dropped by more than micro inefficiency, counter to Samuelson’s conjecture. The intuition behind our result is that both macro and micro inefficiency de- crease toward zero, and therefore the higher of these, macro inefficiency, must decrease by 29 more to reach zero. (We can only speculate regarding whether Samuelson would have con- sidered our model “persuasive evidence” of lower macro inefficiencies based on “avant-garde theorizing.”) Nevertheless, even if macro inefficiencies have decreased the most, the reamin the largest source of inefficiency, so “the oldest business cycle mechanism” may still be at play. Another important real-world trend is that the cost of passive investing has come down over time due to low-cost index funds and exchange traded funds (ETFs). Interestingly, the cost of passive investing varies significantly across countries, giving rise to a number of cross- sectional tests, as we discuss below. First, however, we consider the model’s implications for how the cost of passive investing affects security markets and the market for active asset management. Proposition 7 (Cost of passive investing) As the cost of passive investing kp = fp de- creases, the largest equilibrium changes as follows. The overall asset price inefficiency η increases and, under Assumption 1′, the macro inefficiency (ηβ) increases by more than the micro inefficiency (η⊥) as long as γ/I is sufficiently small. Further, the number of passive investors Sp increases, the numbers investors searching for active managers Sa, self-directed investors, informed investors I, and informed active managers M decrease, the active man- agement fee fa may decrease or increase, and active fees in excess of passive fees fa − fp increase. As seen in the proposition, we would expect that lower costs of passive investing due to index funds and ETFs should drive down the relative attractiveness of active investing and therefore reduce the amount of active investing, rendering the asset market less efficient. This effect can be visualized via Figure 2. Indeed, a reduction in the cost of passive investing implies a rise in the relative cost of active investing, corresponding to an upward shift in the dashed curve in Figure 2. In seen in the figure, such an upward shift leads to higher market inefficiency and fewer informed investors. As evidence of these predictions, Cremers et al. (2016) finds that the performance of active managers “is positively related to the market share of explicitly indexed funds [...] and negatively related to the average cost of explicit 30 indexing.” This is consistent with Proposition 7 since higher market inefficiency naturally corresponds to better performance by active managers. The proposition also makes predictions for fees, which we can compare with the empirical evidence. Cremers et al. (2016) find that a decline in the average fees of “indexed funds of 50 basis points ... is associated with 16 basis point lower fees charged by active funds. Overall, the results suggest that investors pay a higher price for active funds in markets in which explicitly indexed products exert less competitive pressure.” In other words, active fees tend to decrease when passive fees decrease, but they move less than one-for-one so that the fee difference between active and passive fact increases when passive fees decline. Proposition 7 predicts exactly such an increase in the active-minus-passive fee difference. Our model is also consistent with a reduction in the total active fee, although this need not happen in our model. To understand this feature, note that there are two effects: First, a lower cost of passive investing directly lowers the cost of active through competitive effects as seen in equation (8). Second, having fewer active investors leads to a higher market inefficiency (η), which increases the value of active management, and hence the fee. This second effect mitigates the reduction in the active fee (and can in some cases even reverse it). In summary, Propositions 6–9 show how changes in technology may have led to the observed “institutionalization” of the market, with a range of knock-on effects for security- market efficiency and the asset-management industry. Section 6 considers some quantitative implications. 5 Efficiency and Entropy In this section, we seek to shed further light on the properties of market efficiency. We show (as already discussed in Section 1.3) that market efficiency is linked to the (private) economic value of information. Hence, it is natural to further explore the connection between market efficiency and information-theoretic value of information. Indeed, the idea that the economic and information-theoretic values of information are linked goes back at least to Marschak (1959), and the following proposition further establishes a link to the degree of 31 We can also try to reconcile the finding of Cremers et al. (2016) that a decline in the average fees of “indexed funds of 50 basis points ... is associated with 16 basis point lower fees charged by active funds.” In other words, suppose that the cost kp % drops by 0.50%, leading to a drop in the passive fee fp % by the same amount and a drop in the active fee fa % of 0.16%. Then based on (19), we predict that overall market inefficiency increases by Δη = 2γ(Δfa − Δfp) + γ(Δkp − Δka) (20) = 6 × (−0.16% − (−0.50%)) + 3 × (−0.50% − 0) = 0.54%. In the model, the equilibrium fee and inefficiency are naturally determined jointly. Fig- ure 3A shows how these key variables change when the cost of passive investing fp % varies. A change in the cost of passive investing actually does not change the active management fee in the numerical example as seen in the figure. To understand why, note first that a reduction in passive fees puts competitive pressure on active managers to lower their fees (the first term in Eqn.(8)). At the same time, however, the market becomes more inefficient since some investors move to passive management, which increases the value of active invest- ing (the second term in Eqn.(8)). These two forces exactly offset under the specified search function, leaving the active management fee unchanged (but this not a general property of the model). As seen in the figure, if kp % drops by 0.50% (e.g., change of kp from 0.60% to 0.10%) then inefficiency increases by 1.5% in the numerical example. We can also consider how the 6% overall inefficiency is distributed between macro and micro inefficiencies as seen in Figure 4. As seen in the figure, when the number of assets is large, most inefficiency arises from macro sources. In fact, at the right end of the figure with 1000 assets, we see that 81% of the overall inefficiency is due to the inefficiency of the expected market portfolio β1 = (1, 1, ..., 1)>, 18% is due to the other systematic factor, namely the relative-value portfolio β2 = (.61,−.61, ...,−.61)>, and the remaining 1% is due to all the 998 micro portfolios. The low degree of inefficiency stemming from the micro portfolios may seem shocking, but it arises from investors’ ability to diversify such risk when there are as many as 1000 securities in our example. In other words, micro inefficiencies 34 Panel A: Inefficiency and active fees Panel B: Ownership structure Figure 3: Numerical Example. This figure shows properties of the model implied by dif- ferent values of the percentage cost of passive investment, f% p (listed on the x-axis). Panel A shows the inefficiency η and the active fee f% a . Panel B shows shows the fraction of ownership that is active management (informed I and uninformed N M̄−M M̄ ), passive management (Sp), and self-directed. are diminished when informed investing virtually eliminates near-arbitrage opportunities, at least in the model. Hence, most of the inefficiency arises from the non-diversifiable risk due to the two factors. Most of the inefficiency is in the expected market portfolio, but also a non-trivial part is in the second factor, which is a long-short portfolio such as the high-minus-low (HML) value factor or the small-minus-big (SMB) size factor used in much of empirical finance (see Fama and French (1993)). Hence, non-trivial mispricing can exist when many inefficient trades are correlated, consistent with the empirical evidence that most return drivers indeed are based on factor structures of such correlated trades (see Kelly et al. 2018 and references therein). In contrast, truly idiosyncratic mispricing should be minimal according to the model — quantitative predictions that may or not stand the test of data, especially before transaction costs.25 Figure 3B shows how the ownership structure depends on the passive fees in the numerical example. We see that lower passive costs imply more passive asset management, less active investment, especially informed active investment, and less self-directed investment. These findings can be viewed as the model-based counterpart to the recent trends in real- 25Gupta and Kelly (2018) find that many factors can be timed, not just the market, which likely poses a challenge to the estimate that as much as 81% of the inefficiency stems from the market portfolio. However, their setting includes many more factors than the study of Roll (1988) used for our choice of parameters (see Footnote 24) so a real test of the model should use parameters consistent with the test portfolios. 35 101 102 103 number of securities (logarithmic scale) 0 10 20 30 40 50 60 70 80 90 sh ar e of o ve ra ll in ef fic ie nc y (p er ce nt ) expected market portfolio long-short factor all micro portfolios Figure 4: Decomposing Overall Inefficiency. This figure shows the share of the over- all market inefficiency arising from the expected market portfolio (solid line), long-short portfolios such as the Fama and French (1993) factors called high-minus-low (HML) and small-minus-big (SMB) used in much of empirical finance (dashed line), and the sum of all micro portfolios. With many assets, the overall inefficiency is mostly due to the former two kinds of inefficiency, both macro in nature, consistent with Samuelson’s Dictum. world markets seen in Figure 1. Over the past two decades (moving left-to-right in Figure 1), we have seen an increase in passive management and a decrease in active management and self-directed investment, consistent with the model if we assume that the cost of passive investment have declined (moving right-to-left in Figure 3B). 7 Conclusion and Testable Implications We model how investors choose between active and passive management, how active and passive managers choose their portfolios, and how security prices are set. We provide a theoretical foundation for Samuelson’s Dictum by showing that macro inefficiency is greater than micro inefficiencies under realistic conditions. We calibrate the central economic mag- nitudes, thus providing a potential explanation for the recent trends in asset management and financial markets. Our model provides new testable implications to be explored in future empirical research. 36 Gârleanu, N. and L. H. Pedersen (2018). 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Combining this with (9)–(11) gives η = 2γ ( ka − kp 2 + (c̄kα) 1 1+α ) , (A.2) which shows how market inefficiency depends on search costs ( c̄, α), asset management costs (ka, kp), and information costs (k). A.2 Notation. In the proofs, we will use the following notation for any random variables x and y, Σx := var(x) and Σx|y := var(x|y). In addition, we define the functions gI(I,M) = c(M, I − NM/M̄) − η(I) 2γ − kp − ka 2 (A.3) gM(I,M) = M I − N M M̄ k − η(I) 2γ − kp − ka 2 . (A.4) Given (9), (11), and the definition of I, at any interior equilibrium we have gI(I,M) = 0 and gM (I,M) = 0. A.3 Deriving an equilibrium. Here we review a few of the details of the Grossman and Stiglitz (1980) logic, which de- termines the asset-market equilibrium. We explained the investors’ choices and managers’ entry decisions in the body of the paper. An agent having conditional expectation of the final value μ and variance V optimally demands a number of shares equal to x = (γV )−1 (μ − p) . (A.5) To compute the relevant expectations and variance, we conjecture the form (4) for the price and introduce a slightly simpler “auxilary” price, p̂ = v − v̄ + ε − θq(q − q̄), with the same i It is clear that the set of matrices of the above form is closed under the arithmetic operations of addition, subtraction, and multiplication. Further, for any function f such that f(x)x + 1 6= 0 ∀x ∈ R+, we have ( In + βf(β>β)β> )−1 = In + βf̂(β>β)β> (A.22) = In − β(In + β>βf(β>β))−1f(β>β)β>, (A.23) with f̂(x) = − f(x) 1 + xf(x) (A.24) also satisfying f̂(x)x + 1 6= 0. It follows that all variance-covariance matrices, their inverses, as well as any other matri- ces describing the equilibrium, have the form aIn + βf(B)β>. (Note that, since all matrices that have to be inverted are known to be invertible, it is never the case that f(x)x = −1.) It is immediately apparent that the eigenvectors of aIn + βf(B)β> are the k eigenvectors of ββ> that are not associated with zero eigenvalues (equivalently, they equal βy with y eigenvector of B), and n − k linearly independent vectors orthogonal to the columns of β. For the first type of eigenvector, given ββ>y = λy, the associated eigenvalue is 1 + λf(λ); for the second it is 1. Moreover, given two such matrices M1 = a1In+βf1(B)β> and M2 = a2In+βf2(B)β> and the bivariate function F (x, y) being either addition, subtraction, multiplication, or division, the eigenvalue of F (M1,M2) corresponding to eigenvalue λ of O equals F (a1 + λf1(λ), a2 + λf2(λ)). Consider now the matrix G = Σ − 1 2 v|s Σv|pΣ − 1 2 v|s = Σ−1 v|sΣv|p = ( Σ−1 v + Σ−1 ε ) ( Σ−1 v + (Σε + θqΣqθq) −1 )−1 . (A.25) All its eigenvectors are described above. Its eigenvalue associated with any O eigenvector is a function of the corresponding eigenvalue of O given by (A.25). Specifically, we have g(λ) = ( σ2 wv + σ2 Fv λ )−1 + ( σ2 wε + σ2 Fε λ )−1 ( σ2 wv + σ2 Fv λ )−1 + ( σ2 wε + σ2 Fε λ + γ2I−2 ( σ2 wε + σ2 Fε λ )2 ( σ2 wq + σ2 Fq λ ))−1 . (A.26) It is a simple manipulation to check that equation (A.18) holds. Further, it is also immediate that the right-hand side of that equation increases in λ as long as the positive quantities X and Y do. It is clear that Y is increasing, while X increases if and only if Assumption 1′ holds. iv Proof of Proposition 1. Part 1. (a) Start with the market clearing condition q = Uxu + Ixi = U ( γΣv|p )−1 (E[v|p] − p) + I ( γΣv|s )−1 (E[v|s, p] − p) . (A.27) Take expectations conditional on p and rewrite to get E[q|p] = ( U + IΣ−1 v|sΣv|p ) xu. (A.28) Solving for xu yields equation (15). (b) Let’s use the notation A ∼ B for two matrices that are scalar multiplies of each other. To see the implications of the sufficient condition Σv ∼ Σε ∼ Σ−1 q , we note the following. θq ∼ Σε (A.29) θs ∼ In (A.30) Σv|s = Σv (Σv + Σε) −1 Σε ∼ Σε (A.31) Σv|p = Σv (Σv + Σε + θqΣqθq) −1 (Σε + θqΣqθq) ∼ Σε (A.32) Consequently, Σ−1 v|sΣv|p is a scalar and xu is proportional to E[q|p]. (c) Under Assumption 1, Σv|s and Σv|p commute, which implies that Σ−1 v|sΣv|p is positive definite, and therefore (U + IΣ−1 v|sΣv|p) −1 is positive definite. Further, the assumptions of Lemma 1 are satisfied. It follows that H ≡ (U +IΣ−1 v|sΣv|p) −1 takes the form a0 − a1O. Equivalently, that the function h giving the eigenvalues h(λ) of H be decreasing, which is itself equivalent with the function g giving the eigenvalues of G being increasing. Assumption 1′ is sufficient for this conclusion. (b) This result follows from equation (16), taking into account the sign of A1 and the normalization of β. (c) Under the assumptions stated, the investments in each asset are as in a single-asset Grossman-Stiglitz world. Since Σv|p increases with Σq, the coefficient A — which is trivially a scalar in a one-asset world — decreases with the variance of qi. Proof of Proposition 2. (a) Consider demands conditional on realized supply: E[xi|q] = ( γΣv|s )−1 (v̄ − E[p|q]) = ( γΣv|s )−1 π + ( γΣv|s )−1 θsθq(q − q̄) (A.33) E[xu|q] = ( γΣv|p )−1 (E[E[v|p]|q] − E[p|q]) = ( γΣv|p )−1 π + ( γΣv|p )−1 ( θs − ΣvΣ −1 p̂ ) θq(q − q̄), (A.34) v where π is the risk premium π = ( U ( γΣv|p )−1 + I ( γΣv|s )−1 )−1 q̄. Each of the following inequalities holds as long as all matrices involved commute with each other, which is the case both under Assumption 1 and under Assumption 2. θs > θs − ΣvΣ −1 p̂ (A.35) θsθq > ( θs − ΣvΣ −1 p̂ ) θq (A.36) ( γΣv|s )−1 θsθq > ( γΣv|p )−1 ( θs − ΣvΣ −1 p̂ ) θq. (A.37) The desired conclusion follows, both because uninformed investors update their estimate of the value, which mitigates the direct impact of the price on their demand, and because they face more risk. (b) We have E[xi|s] = ( γΣv|s )−1 (E[v|s] − E[p|s]) = ( γΣv|s )−1 π + ( γΣv|s )−1 ( ΣvΣ −1 s − θs ) (s − v̄) (A.38) E[xu|s] = ( γΣv|p )−1 (E[E[v|p]|s] − E[p|s]) = ( γΣv|p )−1 π + ( γΣv|p )−1 ( ΣvΣ −1 p̂ − θs ) (s − v̄). (A.39) Under Assumption 1 or Assumption 2, ΣvΣ −1 p̂ < θs < ΣvΣ −1 s (see equation (A.12)). (In general, due to the market clearing, IE[xi|s] + UE[xu|s] = q̄.) The conclusion follows. Proof of Proposition 3. (a) Under Assumption 2, G is a scalar (see proof of Proposition 1, part 1a), and thus all portfolios have the same inefficiency. (b) We note that the market portfolio, β, is the only non-zero eigenvector of O. It is the maximum-inefficiency portfolio if and only if the associated eigenvalue of G is higher than for the other eigenvectors of O — the ones with O-eigenvalues zero. It is sufficient, then, that the function g in Lemma 1 be increasing, which obtains under Assumption 1 ′. (c) This can be shown via numerical example, or by fixing all other parameters and observing that g is a decreasing function when σ2 Fε is large enough. (It is perhaps easier to see that (g(λ) − 1)−1 is increasing over a fixed domain when σ2 Fε is large enough.) Proof of Proposition 4. This proposition follows from Lemma 1 by letting n be large. One point to keep in mind is that the equilibrium mass of agents investing with informed managers, I, also depends on n. It suffices for our purposes here, however, that this quantity be bounded — above and away from zero. In fact, a stronger statement holds, under our mild regularity assumptions: I has a strictly positive limit with n. This is the mass of informed agents that arises as equilibrium when the inefficiency, as a function of I, is given vi Since we already know that dI dk < 0, we are aiming to show that ∂ log(g(n)) ∂I−2 > − ∂ log(g(0)) ∂I−2 , (A.49) at least for γ I small enough. To that end, we compute ∂ log(g(λ)) ∂I−2 and check that, at γ I = 0, it increases in λ. Proof of Proposition 7. We concentrate again on interior equilibria. As in the previous proof, the dependence of I and M on kp is given as a solution to ( gM I gM M gI I gI M )( Ikp Mkp ) = 1 2 ( 1 1 ) , (A.50) and therefore by ( Ikp Mkp ) = 1 2 1 gI MgM I − gI Ig M M ( gI M − gM M gM I − gI I ) . (A.51) We note that gI M − gM M < 0 and gM I − gI I < 0, while the determinant gI MgM I − gI Ig M M is negative from (A.44). Thus, both I and M decrease as kp decreases. Consequently, the inefficiency η increases. Under Assumption 1′ the macro inefficiency is larger, and more sensitive, than the micro inefficiency. Since M decreases, while the expression M Sa k − c(M,Sa) remains equal to zero, Sa must also decrease. The lower cost fp = kp of passive investing translates into fewer self-directed investors, leaving an increased number Sp of passive investors. Proof of Proposition 9. The logic of the proof is the same as for the previous proposition. Specifically, we are solving for the derivatives Iz and Mz from ( gM I gM M gI I gI M )( Iz Mz ) = ( 1 2 ∂η ∂z )( 1 1 ) , (A.52) giving ( Iz Mz ) = 1 gI MgM I − gI Ig M M ( gI M − gM M gM I − gI I )( 1 2 ∂η ∂z ) . (A.53) We remarked in the proof of Proposition 6 that η increases with z (holding I fixed). It follows that I and M also increase with z. As in the previous proof, Sa must also increase, while Sp decreases because Sa + Sp does not change. The total effect on the inefficiency η is ambiguous. ix Proof of Proposition 8. We need to calculate the utilities uu and ui and use the formula E [ ex>Ax+b>x ] = det(In − 2ΩA)− 1 2 e 1 2 b>(In−2ΩA)−1Ωb (A.54) for x ∼ N (0, Ω). It helps to actually compute “ex-interim” utilities, conditional on p. Specifically, we compute max xi E [ e−γ(xi(v−p)) |p ] = E [ e− 1 2 (E[v|s]−p)>Σ−1 v|s(E[v|s]−p)|p ] (A.55) by letting x = E[v|s] − E[v|p], A = −1 2 Σ−1 v|s, and b> = (E[v|p] − p)> Σ−1 v|s to evaluate E [ ex>Ax+b>x− 1 2 (E[v|p]−p)>Σ−1 v|s(E[v|p]−p) ] (A.56) = det(In + ΩΣ−1 v|s) − 1 2 e 1 2 (E[v|p]−p)>Σ−1 v|s(In+ΩΣ−1 v|s) −1ΩΣ−1 v|s(E[v|p]−p)− 1 2 (E[v|p]−p)>Σ−1 v|s(E[v|p]−p) with Ω = Var (E[v|s]|p) = Σv|p −Σv|s. Simplifying this expression and the analogous one for the uninformed agent shows the equivalence with entropy. We go further by using the fact that the Kullback-Leibler divergence of a n-dimensional multi-variate normal distribution with mean μ1 and variance Σ1 from one with mean μ0 and variance Σ0 is DKL = 1 2 ( tr ( Σ−1 1 Σ0 ) − n + (μ1 − μ0) >Σ−1 1 (μ1 − μ0) + log ( det(Σ1) det(Σ0) )) . (A.57) In our case, Σ0 = Σv|s, Σ1 = Σv|p, μ0 = E[v|s], and μ1 = E[v|p]. Taking expectations, it follows that E [DKL] = η. x