Active and Passive Investing: Understanding Samuelson’s Dictum (2024)

We model how investors allocate between asset managers, managers choose portfolios of multiple securities, fees are set, and security prices are determined. Investors are indifferent between higher-cost informed managers and lower-cost uninformed managers, interpreted as passive managers as their portfolio is linked to the “expected market portfolio.” We make precise Samuelson’s dictum by showing that active investors reduce micro-inefficiencies more than they do macro-inefficiencies. In fact, all inefficiency arises from systematic factors when the number of assets is large. Further, we show how the costs of active and passive investing affect macro- and micro-efficiency, fees, and assets managed by active and passive managers. Our findings help explain the rise of delegated asset management and the resultant changes in financial markets. (JEL D8, G02, G12, G14, G23, L1).

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 management, as seen in Figure1. This delegation has potentially profound implications for market efficiency (see, e.g., the presidential addresses to the American Finance Association of Grossman (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 determines 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 security 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) active investors reduce micro-inefficiencies more than macro-inefficiencies (consistent with Samuelson’s dictum) when there exists a strong common factor in security fundamentals or when the number of securities is large; (3) in fact, all inefficiency is due to systematic factors in the limit with an infinite number of assets, a form of arbitrage pricing theory (APT) for market inefficiency; (4) when information costs decline, the number of active managers increases, active fees decrease, market inefficiency decreases, especially macro-inefficiency (counter to part of Samuelson’s dictum); and (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. 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 and Stiglitz (1980), following Gârleanu and Pedersen (2018). While these models consider a single risky asset, we consider multiple securities (in the spirit of Admati [1985], who also considers multiple securities, but not asset managers) and, as a further extension of Gârleanu and Pedersen (2018), consider the costs of self-directed investment and passive investment, in addition to the cost of active management.1 Our core contribution, though, is not the extensions per se, but, rather, the novel questions that they enable us to study. Indeed, having asset managers hold multiple securities allows us to study portfolio choice and macro- versus micro-efficiency (neither can be studied with one asset), and having costly active and passive management is essential to study the effects of changes in these costs, for example, as fintech reduces the costs of index funds and ETFs.

Investors must decide whether to invest on their own, allocate to a passive manager, or search for an active manager. Each of these alternatives is associated with a cost: self-directed trade has an individual-specific cost (time and brokerage fees), passive investing has a fee (equal to the marginal cost of passive management, in equilibrium), and active investing is associated with a search cost (the cost of finding and vetting a manager to ensure that she is informed) plus an active management fee. Active and passive managers determine which portfolios to choose, and, in addition, active managers decide whether or not to acquire information. Market clearing requires that the total demand for securities equals the supply, which is noisy (e.g., because of share issuance, share repurchases, changes in the float, private holdings, noise from hedging demand, etc.).

Passive managers seek to choose the best possible portfolio conditional on observed prices, but not conditional on the information that active managers acquire. One may wonder whether passive managers should choose the “market portfolio” (the market-capitalization weighted portfolio of all assets), which is the standard benchmark in the capital asset pricing model (CAPM). While the market portfolio is the focal point of much of financial economics, it is usually not discussed in the context of REE models because supply noise renders it unobservable (likewise, in the real world no one knows the true market portfolio as emphasized by Roll (1977)). Bridging the REE literature and the CAPM, Admati (1985) points out that the unconditional expected market portfolio is generally not the optimal portfolio for uninformed investors. Indeed, uninformed investors can do better by using the information reflected in prices, as shown theoretically and empirically by Biais, Bossaerts, and Spatt (2010). Extending these insights, we consider whether uninformed investors optimally hold the closest thing that they can get to the market portfolio, namely, the “conditional expected market portfolio” based on the distribution of the supply and what can be learned from prices (and other public signals). The answer is “yes” under certain special conditions (e.g., i.i.d. shocks across securities), which resembles what real-world passive investors do, namely, choosing an index that is rebalanced based on public information. However, indexes only hold a subset of all securities, typically large and mature firms with sufficient time since their initial public offering. In a similar spirit, we show that uninformed investors optimally overweight securities with less supply uncertainty in the more general case in which shocks are correlated across securities via a factor structure. Hence, our framework presents a step toward a theory of optimal security indexes.

Active investors 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, market inefficiency is zero (fully efficient market) if uncertainty about the fundamental value is the same whether one learns only from the price or also from the signal. Further, the larger information advantage one enjoys from knowing the signal, the more inefficient the market.

An interesting question naturally addressed in our framework is whether active investors make the market more efficient at the micro level than at the macro level, as Samuelson famously hypothesized (see quote and references in the beginning of Section 2 and the evidence in Jung and Shiller (2005) and Xiao et al. (2021)). The idea, known as “Samuelson’s dictum,” is that active investors have stronger incentives to correct (micro) inefficiencies in relative prices than to correct the overall (macro) price level. For example, active investors will ensure that the price difference between General Motors and Ford is close to efficient, but active investors may leave all stocks overvalued.

We show that Samuelson’s dictum holds when a strong common factor exists in the 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 under which Samuelson’s dictum applies (or does not apply). We further show that, because of diversification, when the number of securities is large, Samuelson’s dictum actually always holds in our setting.

In fact, any micro-inefficiency goes to zero as the number of assets grows. Perhaps surprisingly, the combined inefficiency of all micro portfolios becomes negligible with many assets. Almost all the inefficiency is in the pricing of systematic factors. This result is related to the arbitrage pricing theory (APT) of Ross (1976). While APT states that risk premiums are driven by systematic factors when the number of assets is large, we show that inefficiencies are also driven by these factors.

The key simplifying assumption that allows these striking results is that active managers in our model make an all-or-nothing information choice, which captures the idea that active investors must decide whether or not to set up an IT system that captures the main databases and a staff that can process all these data. In contrast, Veldkamp (2011), Van Nieuwerburgh and Veldkamp (2010), Kacperczyk et al. (2016), and Glasserman and Mamaysky (2018) study agents’ choice of information, which can affect macro- versus micro-efficiency as emphasized by the latter paper.2 We complement the literature with regard to macro- versus 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 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.

Samuelson also hypothesized that efficiency, especially micro-efficiency, has improved over time (see quote and reference in Section 3). 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 consistent with the empirical findings of Biais, Bossaerts, and Spatt (2010) and Dávila and Parlatore (2018), 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.3 Further, a 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 lead to a larger share of passive investment, a higher average alpha for active managers, and lower fees for active investing, but an increased fee spread between active and passive managers.

In summary, we make precise Samuelson’s dictum and show that it holds when the number of assets is large enough, illustrating that our REE model with a large number of assets is a powerful tool. 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.4

1. Model and Equilibrium

This section lays out our model and shows how to solve it.

1.1 REE model with multiple assets and asset managers

Markets

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 of risky asset prices p is determined endogenously. The risky assets deliver final payoffs given by the vector v, which is normally distributed with mean $v¯$ and variance-covariance matrix Σ_v, which we write as $v∼N(v¯,Σv)$⁠.

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.5 We do not think of the signal as (illegal) insider information, but rather as the ability to access and process large amounts of relevant data, which could be available for purchase or even freely. In other words, informed investors need to acquire databases, collect other relevant information, and set up systems to optimize their portfolio in light of these signals. This is a costly process, as securities databases can be expensive and trading continually on all this information requires an investment staff. We note that, as in Grossman and Stiglitz (1980), investors must simply decide whether or not to acquire this information, which captures the idea that information processing can be seen as an infrastructure with a fixed cost while the papers cited in the introduction consider endogenous information precision or choice.

The supply of the risky assets is given by $q∼N(q¯,Σq)$ 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.6 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 and other reasons, no one knows the true market portfolio, the underpinning of the “Roll critique” (Roll 1977).

Investors

The economy has $S¯$ investors with initial wealth W and constant absolute risk aversion (CARA) coefficient γ. These investors search for an active manager, allocate to a passive manager, or directly invest in the financial market. Specifically, I investors choose to search for an informed active manager, S_p investors choose passive management, and the remaining $S¯−I−Sp$ are self-directed.

If an investor l makes an uninformed investment (i.e., without using the signal s) directly in the financial market, then he incurs a cost d_l associated with brokerage fees and time used on portfolio construction, and these costs vary across investors. If the investor makes an informed investment, the cost is $dl+k$⁠, but we show that, for what we argue to be reasonable parameters, this behavior is dominated by using an active manager due to the effective ability to share information costs across the manager’s investment base. Therefore, the total number of uninformed investors, U, is the sum of the number of self-directed investors and investors with passive managers, that is, everyone who is not searching for an informed manager, $U=S¯−I$⁠.

If the investor uses a passive manager, he incurs the passive management fee $fp$⁠. Finally, if the investor chooses to search for an informed active manager, the investor must pay a search cost c and, in addition, pay an active asset management fee $fa$ determined via Nash bargaining. The search cost c(M, I) is a smooth function satisfying $∂c∂M≤0$ and $∂c∂I≥0$⁠, meaning finding an appropriate manager is easier to do if there are more managers and more difficult when more agents are searching for a manager.7

To understand why investors may choose to use asset managers, note first that an investor with a higher cost (d_l) of self-directed investing than the passive manager’s marginal cost (k_p) clearly benefits from outsourcing any passive portfolio construction to the manager. Further, investors with active managers can benefit from both potential low marginal costs from the manager (k_a) and, importantly, the fact that the active manager can effectively share the fixed cost of information (k) across all his investors. On the other hand, an investor with very low cost of self-directed investing may be best off investing himself.

Asset managers

The economy has $M¯$ active asset management firms and a representative passive manager. The passive manager seeks to deliver the best possible portfolio that can be achieved without acquiring the signal s. The passive manager faces a marginal cost per investor of $kp$ and, since passive investing is assumed to be a competitive industry, this manager charges a fee $fp=kp$ (where the subscript p naturally stands for “passive”).

Active managers face a marginal cost of $ka$ (subscript a for “active”) and, in addition, they must decide whether to incur the fixed cost k associated with acquiring the signal s. Specifically, M active managers endogenously decide to pay the cost k to become informed while the remaining $M¯−M$ managers seek to collect active asset management fees $fa$ even though they invest without information (e.g., these managers are using “closet indexing”).

Portfolios and utilities

Equilibrium

(b) fees are set via Nash bargaining; (c) each active manager decides optimally whether to be informed; and (d) each investor decides optimally whether to use an active manager, use a passive manager, or be self-directed.

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), who consider the inefficiency of a single asset.

We see that inefficiency is zero if the market is perfectly efficient in the sense that the price reveals all information about the signal. Otherwise, the inefficiency is a positive number. A lower inefficiency corresponds to a higher price informativeness as defined by Grossman and Stiglitz (1980). Indeed, Grossman and Stiglitz (1980) define the price as more informative if it is more correlated with the signal since, in this case, uninformed agents who observe the price can learn more about the signal. This higher informativeness further implies that informed and uninformed agents have more similar information and therefore similar uncertainty, meaning that market inefficiency is lower by our definition (5).

When we analyze macro- versus micro-efficiency (in Section 2), we also study the efficiency of individual securities, portfolios, and collections of portfolios, so we need a general definition of efficiency. The correlation-based definition of Grossman and Stiglitz (1980) does not lend itself to be generalized to many assets and portfolios of assets, but our definition is both general and equivalent to Grossman and Stiglitz (1980) in the one-asset case. Further, Grossman and Stiglitz (1980) show that informativeness is linked to the ratio of the utilities of the informed and uninformed agents, and our Proposition 1 shows that our more general definition of inefficiency retains this idea.10

This definition of overall market inefficiency is the natural extension of the one-asset definition of Grossman and Stiglitz (1980), since it maintains the tight link between market inefficiency and investors’ utility of information as discussed further below and formalized in Proposition 1.

1.3 Solution: Deriving the equilibrium

Active versus passive investing

This equilibrium condition is intuitive. It says that the benefit of informed investing (the left-hand side) must equal the net cost of being informed (the right-hand side). The benefit is the certainty equivalent utility of getting an informed portfolio rather than an uninformed one. The net cost of active investing is the active fee plus the search cost minus the passive fee. Next, we will show how to compute these costs and benefits as functions of the number of informed investors, I. Figure2 illustrates how the costs and benefits of active investing depends on the number of active investors. When these costs and benefits are computed, we can find an interior equilibrium as the intersection of the lines in Figure2.

The benefit of active investing: Market inefficiency

This equation shows explicitly how the market inefficiency depends on the number of active investors I (as well as the exogenous variance-covariance matrices). We see from (9) that market inefficiency decreases in I as is also clear from Figure2. This finding is intuitive since more active investors lead to more informative prices, and, hence, lower market inefficiency.

The cost of active investing

Next, we compute the cost of active investing relative to the cost of passive investing, that is, the right-hand side of Equation (8). We first recall that the passive management fee equals the marginal passive cost $fp=kp$⁠.

The final cost of active investing is the search cost, c(M, I). The search cost depends on the number of informed managers M and the number of active investors I, so we need to consider how many managers become informed in equilibrium.

In summary, we see that the cost of active investing depends on I because (a) the active fee depends on η as seen in (10), which in turn depends on I as seen in (9); and (b) the search cost depends on I and M, where the latter depends on I directly and via η as seen in (11). The cost of active investing is illustrated in Figure2, where it is seen to decrease in the number of active investors. The reason is that more active investors means more efficient markets, leading to lower active fees. The cost of active investment can also increase in I, though, if the search cost rises sufficiently.

Computing the equilibrium

Having explicitly computed the cost and benefit of active investing as a function of I, we can find the equilibrium value of I by equating the cost and benefit, a nonlinear algebraic equation that is easy to solve numerically for any search-cost function c(M, I).13

1.4 Quantitative implications

Equation (10) provides a simple relation between market inefficiency, on the one hand, and active and passive management fees, on the other. Here we exploit this relation to make quantitative statements about the former based on observations of the latter.

As a simple calibration, we can set the relative risk aversion to be $γR=3$⁠, consider a realistic fee difference of $fa%−fp%=1%$⁠, and assume similar marginal costs of active and passive management, $kp%=ka%$ (recall that active managers must additionally pay the information cost k), yielding an overall market inefficiency of $η=6%$⁠.

This calibration derives the endogenous inefficiency based on the observed level of fees, which is itself an endogenous variable. This is similar to standard applications of the CAPM, where the expected return is derived based on the observed beta and market risk premium, both of which are endogenous. Just as the CAPM provides useful insights on expected returns, our calculation provides an interesting implication on the magnitude of market efficiency based on the level of fees that investors are willing to pay.

The model is, of course, richer, and provides both the fees and the market inefficiency as outcomes once the asset structure is fully specified and parameterized, which we do starting in the next section.

Naturally, cheaper index funds increases passive investing and decreases active investing, leading to higher equilibrium market inefficiency as seen in (14). This increase in market inefficiency leads to higher expected returns to active management, consistent with the findings of Cremers et al. (2016). The intuition behind this finding is seen in Figure2, where the reduction in passive costs corresponds to an upward shift in the dashed line. Further, the model can help quantify the magnitude of the effect as seen in (14). We will illustrate these effects further with numerical examples in Sections 2 and 3.

1.5 Understanding passive investment

In standard theories of informed trading, investors naturally fall into two groups—informed and uninformed—while real-world investors are classified as active or passive, and we have made the natural link between these dichotomies. The connection between “informed” and “active” investors seems uncontroversial and their portfolios are described further in Appendix A.1.

The link between “uninformed” and “passive” investors deserves more consideration. What these concepts clearly have in common is that the uninformed investors of abstract models minimize their costs as do real-world passive investors. Uninformed investors seek to maximize their performance subject to a minimal cost, which is presumably also the goal of passive investors. But how can we make this connection clearer?

First, note that while passive indexes typically follow relatively simple rules, competition between index providers could lead these rules to become optimal subject to minimum costs, just like the portfolios of uninformed investors in the model.

Second, passive investors are doing something slightly more complicated than “just buying the market” (as would be the prediction of the CAPM), because, in the real world, no one knows exactly what the market portfolio is (Roll 1977) and it is constantly changing. In particular, passive investors must choose how to be passive, for example, choosing a portfolio of (global) stock and bond index funds or ETFs. These indexes are typically weighted by the total market capitalization of each constituent security’s floating shares outstanding, and they are regularly reweighted to account for changes in the index constitution, share repurchases, new issuance, and changes in the insider holdings (i.e., shares not part of the float).

Likewise, uninformed investors do not know the market portfolio q in our noisy REE economy. So, how do uninformed investors choose their portfolio, x_u, defined in Equation (1)? Clearly, x_u only depends on prices and public information, such as the distribution of shares outstanding, so could there be link to real-world indexes? In this connection, it is interesting to relate the uninformed portfolio x_u to the uninformed investors’ best estimate of the true market portfolio, q, based on public information. We call this best estimate the “conditional expected market portfolio,” $E(q|p)$⁠.

So uninformed and passive investors both minimize cost, presumably both maximize their performance, and perhaps both have portfolios based on $E(q|p)$⁠, which we explore based on the following assumptions.

We see that the optimal uninformed portfolio can indeed be linked to the conditional expected market portfolio, $E(q|p)$⁠. As seen in part 1 of Proposition 3, uninformed investors use public information to proxy for the expected market portfolio as an input to the portfolio construction, similar to real-world indexes. Further, part 1 states that, under certain conditions, the optimal passive portfolio is literally just that, namely, the conditional expected market portfolio. However, under the more realistic Assumption 1, the optimal passive portfolio is similar to the conditional expected market portfolio, but tilted away from risky securities. This tilt arises because uninformed investors face an extra risk (relative to informed investors) due to supply uncertainty.

Part 2 further shows that, even when the uninformed portfolio is not equal to the expected market portfolio state by state, portfolios may coincide on average. In the more general situation in which β and $q¯$ are not proportional, the optimal passive portfolio tends to downweight risky securities or securities with more supply uncertainty. This intuition may help explain why real-world passive indexes only include a subset of listed securities, which may be interpreted as a binary version the results of part 2. While the proposition states that passive investors will hold more of securities with less supply uncertainty, real-world passive investors may exclusively hold these securities. Indeed, indexes typically exclude securities with too low price, too low market capitalization, too low liquidity, too recent initial public offering, or involved in certain corporate actions.15

2. Samuelson’s Dictum: Macro- versus Micro-Efficiency

If inefficiencies are the “crimes” in financial markets, active investors are the “police” who correct these offenses. Therefore, the overall level of market efficiency is linked to the costs and fees of active asset management, relative to the cost of passive management, as seen in Equation (10). But which securities and portfolios are made especially efficient by active investors, and which are left relatively inefficient? In other words, what “crimes” are active investors best at fighting?

From the perspective of uninformed passive investors, how well are these investors “protected” from buying a security or portfolio at an inefficient price? Do passive investors face a greater risk of buying all stocks at an inefficient overall price level, or a greater risk of buying General Motors for too much relative to the price of Ford?

In this connection, Paul Samuelson famously conjectured that markets would have greater micro-efficiency than macro-efficiency:16

Modern markets show considerable micro efficiency (for the reason that the minority who spot aberrations from micro-efficiency can make money from those occurrences 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 (4).

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.

2.1 Principal inefficiency portfolios

We have almost answered Samuelson’s question, namely, whether the most inefficient portfolio, $ζ*=w^1$⁠, is macro or micro in nature. All that is left is to determine the portfolio $w^1$ by finding the primary eigenvector of G.

To understand the economics of Equation (20), note first that, if an informed investor can choose her “loading” on the most inefficient portfolio $w^1$ (i.e., go long or short and scale the position up or down as she wishes), then she will get an expected utility benefit (relative to an uninformed investor) of $12log g1$⁠.18 Second, if the informed investor can also choose her loading on the second most inefficient portfolio, $w^2$⁠, then she will get an additional utility benefit of $12log g2$⁠, and this utility is additive because of the independence of these portfolio returns and the CARA utility. Third, if the investor can choose her portfolio freely, which we can think of as the sum of loadings on all the “basis” portfolios $(w^j)$⁠, then her expected utility (again, relative to that of an uninformed) is $η=12∑j log gj$⁠.

2.2 Samuelson’s dictum: A finite number of assets

We are ready to state our first result on macro- versus micro-efficiency.

Part (a) of the proposition gives a precise meaning to Samuelson’s dictum (in the context of our rational, information-based model). Specifically, we see that when we maximize and minimize inefficiency, the solutions turn out to be “macro” and “micro” portfolios, as conjectured by Samuelson. The most inefficient portfolio is the factor portfolio, that is, the portfolio with the most systematic risk. The least inefficient portfolios are long-short portfolios that eliminate all factor risk. This result arises because active investors who “spot aberrations from micro-efficiency can make money from those occurrences and, in doing so, they tend to wipe out any persistent inefficiencies.” Indeed, Samuelson’s words describe well what goes on in our model. An active investor can eliminate a lot of risk by holding a long-short portfolio and, to rule out aggressive trades on such portfolios, these “arbitrage portfolios” are relatively efficiently priced. In contrast, active investors leave the factor portfolio relatively inefficient because, while they can potentially learn a lot about its fundamental value (since signal noise cancels out to some extent), trading on this information is risky because the factor risk is systematic. Since inefficiency is the ratio of what can be learned from the signal (a lot) to what is incorporated in the price (not so much), the factor portfolio is relatively inefficient.19

Further, the proposition provides conditions under which Samuelson’s dictum applies. The sufficient condition states that securities are correlated (rather than independent) via a common factor, and the factor risk is at least as important for fundamentals as it is for the noise in signals. While this condition appears empirically plausible, we believe that an even stronger argument for Samuelson’s dictum is that it always holds when the number of securities is large, which would certainly be an empirically relevant condition. We provide a formal statement in the next section.

Part (b) of the proposition states that, under certain conditions, all portfolios are equally efficient. This conclusion applies, for example, when all securities are independent (in terms of the fundamental values, signals, and supply shocks). This is perhaps not so surprising since, with independent securities, it is difficult to even define micro and macro, but this conclusion also holds in cases when micro and macro can be defined under Assumption 2 (although, as explained above, Assumption 2 is unrealistic for other reasons).

Part (c) of the proposition states that, under certain conditions, the conclusion opposite to Samuelson’s dictum obtains. This can happen, for example, if, given a fixed value of $σFv2$⁠, the variance $σFε2$ of the common component in the signal noise is sufficiently large.20

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- versus 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.

In summary, we have seen how to make Samuelson’s dictum precise as the statement that active investors make factor-neutral portfolios most efficient, while leaving the factor portfolio as the most inefficient portfolio. We have also seen that the dictum holds only under certain conditions, but we will next show that the dictum always holds when the number of securities is large enough.

2.3 Samuelson’s dictum meets arbitrage pricing theory

Next, we consider what happens to macro- and micro-efficiency when the number of assets, n, is large. Indeed, thousands of securities exist in the real world, so we may achieve a tractable approximation of the real world by considering the simplifications that arise when we let n go to infinity. Further, the tractability afforded by looking at this large-asset limit also allows us to consider a more general factor structure with multiple factors, similar to the setting of the Ross (1976) arbitrage pricing theory (APT). Hence, we can study the pricing of risk and market efficiency when active investors can diversify across idiosyncratic risks.

So far, we have relied on Assumption 1 with a single factor, which is the same for payoffs v, noise ε, and supply q. We now consider Assumption 3, which allows multiple factors that differ across v, ε, and q. In the interest of readability, we state the assumption without making explicit the dependence of variables on n (and we sacrifice some of the possible generality).

To understand Assumption 3, note that (21)–(23) let factor loadings differ across payoffs v, noise ε, and supply q, and (24) is a regularity condition on the factor loadings to ensure convergence as n grows.21 We write the supply $q=q′/n$ in terms of an auxiliary variable $q′$ for notational ease. In particular, $q′i$ is of similar size and variance across securities i, but the economy would not converge if we had a fixed group of investors who had to own more and more securities as n grows (since the risk per investor would go to infinity). Therefore, as we increase the number n of securities, we simultaneously reduce the supply per security by letting $q=q′/n$⁠.22

As another motivation of Assumption 3, the next proposition shows that it implies a version of the standard APT. The standard APT describes expected returns in the presence of a factor structure, and we show that our factor structure is scaled in a way that is consistent with an APT equilibrium with many assets. The more novel part of the proposition is a related APT result for market inefficiency. To understand the APT for efficiency, recall that the inefficiency $ηζ$ of any set of portfolios ζ is always less than or equal to overall market inefficiency, η.

The standard “APT of returns” says that risk premiums must be driven by systematic factors. The economics behind the APT is that, if certain assets delivered abnormal returns relative to their factor loadings, then investors could earn a return with a risk that can be diversified away, and such near-arbitrage profits are ruled out in equilibrium. As in Ross (1976), the APT generally holds as an approximation, but having a fully specified equilibrium provides us with an explicit condition for it to hold exactly, namely, that every asset constitutes a vanishing part of the economy in the limit.

Likewise, the “APT of efficiency” says that idiosyncratic inefficiencies are arbitraged away since idiosyncratic risk can be diversified away, thus leaving just the inefficiencies associated with the systematic factors when the number of securities is large.

To fully appreciate the result in Proposition 5, note that the overall inefficiency η can be seen as the sum of the inefficiencies of n uncorrelated portfolios as given by Equation (20). With the assumed factor structure, the overall inefficiency η equals the inefficiency of the factor portfolio $ηβv$ plus the sum of inefficiencies of $n−kv$ uncorrelated micro portfolios.23 Our result is therefore not simply that each individual micro-inefficiency goes to zero; rather, the sum of all micro-inefficiencies goes to zero, even though the number of micro portfolios goes to infinity. In other words, factor inefficiency dominates overall market inefficiency to a surprising extent.

The fact that micro portfolios, which load exclusively on idiosyncratic shocks, are asymptotically efficient is not difficult to intuit. The supply noise in such a portfolio is purely idiosyncratic, 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 the combined inefficiency of all micro portfolios tends to zero is more involved and more surprising since the number of micro portfolios goes to infinity as n increases. This reason is that a hypothetical active investor who is restricted to trading micro portfolios has a utility gain over that of an uninformed investor that declines as $1/n$ and thus goes to zero as n increases.24 On the other hand, when trading fundamental factors, an informed investor has a utility gain relative to an uninformed that is bounded away from zero since the supply noise in these factors does not vanish as n goes to infinity.

The proposition states that a number of portfolios equal to that of fundamental factors (k_v) is sufficient to exploit the entire inefficiency in the limit. There are infinitely many such sets of portfolios. With more structure than imposed by Assumption 3, statements can be made about the maximal-inefficiency portfolios for finite and large n. In particular, in some cases the maximal efficiency sets of portfolios for any finite n sufficiently large can be identified explicitly. We illustrate this claim with some simple examples.

We have $τp(0)≈τs(0)$ for large n, meaning that idiosyncratic information (corresponding to micro-portfolios, λ = 0) is reflected almost equally by the prices and the signals. This idiosyncratic information affects the efficiency of a single asset, but is diversified away for the index, that is, $τp(0)$ and $τs(0)$ feature in (25), but not in (26). Further, we see that $ηe1<ηβ$ from Equations (25) and (26) and the concavity of the logarithm. In words, asset 1 is less inefficient than the index. The economic intuition is that, while the single asset is more volatile than the index, the additional volatility is due to the idiosyncratic component, and the information about this component is (almost) entirely impounded in the price. Consequently, a larger proportion of the information available about the payoff of asset 1 is inferred from the price. This is precisely our definition of efficiency.

We illustrate these statements numerically. To that end, we consider a parametric model (described further in Appendix A) and choose the following parameters. There are $S¯=108$ investors, each with a relative risk aversion of $γR=3$ and wealth of W = $100,000. There are $M¯=4,000$ asset managers. The fundamentals have a one-factor structure, as described above. The factor volatility is $σFv=0.18WS¯$ and the idiosyncratic volatility is $σwv=0.37WS¯$⁠. The noise has the same factor structure but with double the volatility of factors and idiosyncratic shocks. The number of securities is n = 100, with average supply $q¯′i=1$ and supply uncertainty characterized by $σFq′2=0.03$ and $σwq′2=0.06$ for all $i≤n$⁠. The marginal cost of asset management is $kp%=ka%=0.10%$ and the cost of information is $k=2e7$⁠. The search cost is $c=84(I/M)0.25$⁠. The cost of self-directed investment is zero for 10% of investors and uniformly distributed on $[0,2%]$ for the rest.

With these parameters, the equilibrium overall market inefficiency is $η=6%$⁠, as the parameters are chosen to match the level of inefficiency of the calibration in Section 1.4. The inefficiency of the market portfolio, as given by Equation (26), is $ηβ=5.22%$⁠, which represents, indeed, a large part of the overall inefficiency. The inefficiency of a single stock, say stock 1, is computed using Equation (25) to be $ηe1=1.09%$⁠. This inefficiency is smaller than that of the market portfolio because an individual stock has both common and idiosyncratic components, as discussed before. The idiosyncratic component has variance more than four times larger26 than the common one, but this component is virtually entirely revealed by prices. If an investor was restricted to trading in asset 1, then the use of information would only benefit her by 1.09% in terms of wealth certainty equivalent, since she would have to take on sizable idiosyncratic risk without any information advantage. Trading the market portfolio, in contrast, would confer a 5.22% benefit.

We also emphasize that a single stock is not a true micro portfolio in the sense used in our model since micro portfolios must have zero factor exposure. A true micro portfolio ζ is market neutral, $ζ⊤β=0$⁠, and we denote the inefficiency of such a portfolio as the micro-inefficiency $η⊥:=ηζ$⁠. In this example, $η⊥=12log(τs(0))−12log(τp(0))=0.0078%$⁠, so we see that the true micro-inefficiency is much lower than the inefficiency of an individual stock. This low micro-inefficiency reflects that relative prices are more “correct” than any individual price.

Further, we can see how all these inefficiencies add up to the overall inefficiency. There are $n−1=99$ micro portfolios and one macro-inefficiency, and these add to the total inefficiency, $(n−1)η⊥+ηβ=99×0.0078%+5.22%=6%=η$⁠.

We can, of course, compute all of the equilibrium quantities in this example. For example, the proportion of informed investors is $I/S¯=0.194$ and there are M = 1, 152 active managers.

Here, factor 1 is the market factor (as before), $β1=1n$⁠, which means that all assets are equal parts of the market. Factor 2 is a long-short factor, $β2=(b,−b,b,−b,…,(−1)n−1b)⊤$⁠. For example, we can think of factor 2 as the Fama-French value-minus-growth factor, HML. The alternating signs in factor 2 means that every other asset is a value stock, and the remaining ones are growth stocks. We select b = 0.61 to match the finding of Roll (1988) that “the mean R²s were, respectively, .179 for the CAPM and .244 for the APT.”

In this example, the eigenvalues of $βv⊤βv/n$ are $σFv2$ and $b2σFv2$ for all n even, so we obtain that factor 2 is a weaker driver of returns since $|b|<1$⁠. For n large enough (and even), ${β1,β2}$ achieves maximal inefficiency among all pairs of portfolios. (Naturally, any other pair generating β₁ and β₂ also does.)

Figure3 displays the proportion of the total inefficiency that is due to each of the two factors, as well as all of the micro portfolios, for a number of stocks ranging between 4 and 1000.

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 80.8% of the overall inefficiency is due to the inefficiency of the expected market portfolio $β1=1n$⁠, 18.0% is due to the other systematic factor, namely, the relative-value portfolio $β2=(.61,−.61,…,−.61)⊤$⁠, and the remaining 1.2% 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 our example contains as many as 1,000 securities. In other words, micro-inefficiencies are diminished when informed investing virtually eliminates near-arbitrage opportunities, at least in the model. Hence, most of the inefficiency arises from the nondiversifiable risk due to the two factors. Most of the inefficiency is in the expected market portfolio, though a nontrivial 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, nontrivial mispricing can exist when many inefficient trades are correlated, consistent with the evidence that most return drivers indeed are based on factor structures of such correlated trades (see Kelly, Pruitt, and Su 2019 and the references therein). In contrast, truly idiosyncratic mispricing should be minimal according to the model. This is a quantitative prediction that may not stand the test of data, especially before transaction costs.27

Implications of investment management

Proposition 5 means that, with many assets, investors have two ways to try to make money on a large scale. First, one can “buy factors,” that is, buy a portfolio of securities with positive factor loadings to profit from factor risk premiums (the first part of the proposition). Second, one can try to exploit the inefficiency of these factors, which is sometimes called “factor timing” (the second part of the proposition). Factor timing means varying the exposure to each factor based on information on its expected return. For example, one can try to time the overall market to exploit inefficient bubbles and crashes. In contrast to buying and timing factors, some may attempt to earn near-arbitrage profits based on idiosyncratic inefficiencies, but our model predicts that such opportunities are infrequent enough that only a limited number of investors can exploit them. These predictions appear consistent with some empirical observations.28

Hence, with many assets, the most interesting trading opportunities occur in systematic factors, which may help explain why many investors increasingly focus their trading on exchange traded funds (ETFs), “smart beta products,” futures, and other forms of factor-based investing. For example, BlackRock (2018) estimates that “the factor industry is $1.9 trillion in AUM.”

In summary, we derive endogenously the meaning of macro-efficiency (by looking for the most inefficient portfolio), show that Samuelson’s dictum arrives naturally as the number of securities increases, and show the potential importance of systematic factors generally, not just the overall market portfolio. Our results complement those of Glasserman and Mamaysky (2018), who also provide conditions for higher macro-inefficiency with a single factor and an exogenous definition of macro versus micro information, but endogenous information choices. Finally, we note that our results on Samuelson’s dictum would also apply in a model without asset managers in which investors choose individually whether or not to be informed. However, asset managers play a central role in real-world information acquisition and, further, asset managers are important for the next section, where we will discuss the implications of changing asset management costs.

3. Falling Costs of Active and Passive Investing

Interestingly, Samuelson (1998) also conjectured how micro-efficiency, but not necessarily macro-efficiency, has improved over the years:

The pre-1800 pattern of commercial panics had to be a case of NON MACRO-EFFICIENCY of markets. We’ve come a long way, baby, in two hundred years toward micro efficiency of markets: Black-Scholes option pricing, indexing of portfolio diversification, and so forth. But there is no persuasive evidence, either from economic history or avant garde theorizing, that MACRO MARKET INEFFICIENCY 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 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 have the implications conjectured by Samuelson. In particular, when Samuelson’s dictum holds for the levels of inefficiency, then we can look at the relative changes in macro-inefficiency versus micro-inefficiency. Samuelson’s dictum holds for the level of inefficiency under Assumption 1′ as shown in Proposition 4(b), or when n is not too small under Assumption 3 as stated in Proposition 5. Here, we focus on the latter case and impose Assumption 3 in the rest of this section. Recall that macro-inefficiency is given by $ηβv$ as in Proposition 5, while total micro-inefficiency is the residual $η−ηβv$ to the overall efficiency.

With the improvement in information technology, the cost of information may have decreased over time. If so, Proposition 6 shows that overall market inefficiency should have improved 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 decrease toward zero, and therefore the higher of these, macro-inefficiency, must decrease by more to reach zero. (We can only speculate regarding whether Samuelson would have considered our model “persuasive evidence” of lower macro-inefficiencies based on “avant-garde theorizing.”) Nevertheless, even if macro-inefficiencies have decreased the most, they remain 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.

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 Figure2. 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 Figure2. As 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) find 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 indexing.” This is consistent with Proposition 7 since higher market inefficiency naturally corresponds to better performance by active managers.

The proposition further shows that such a rise in market inefficiency is greater for macro-inefficiency than micro-inefficiency, as macro-inefficiency tends to be more variable as discussed above. The proposition also makes predictions for fees, which we can compare with the 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 investment 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 such a reduction need not obtain. To understand this feature, note the two effects at play: First, a lower cost of passive investing directly lowers the cost of active through competitive effects as seen in Equation (10). Second, having fewer active investors leads to a higher market inefficiency (η), which increases the value of active management, and hence the fee (the second term in Equation (10)). This second effect mitigates the reduction in the active fee (and can in some cases even reverse it).

Example 1, continued. Next, we illustrate some of the effects of changing costs of passive investing with an example. We use the parameters from example 1 in Section 2, but vary the cost of passive investing to illustrate the results predicted by Proposition 7.

Figure4, panel A, shows how market inefficiency and active fees change when the cost of passive investing $fp%$ varies. Figure4, panel B, shows how the ownership structure depends on the passive fees in the numerical example. Naturally, a lower passive fee leads to less active investing as seen by moving from right to left in Figure4, panel B. When there are fewer active investors, the market becomes less efficient, as seen by moving from right to left in Figure4, panel A, and most of that change is due to the macro-inefficiency. (Remember also that the total micro-inefficiency equals $(n−1)η⊥=99η⊥$⁠, highlighting the small scale of the inefficiency of any one micro portfolio, $η⊥$⁠, and of its sensitivity to parameters.)

A change in the cost of passive investing actually does not change the active management fee in the numerical example, as seen in Figure4, panel A. This result happens because, as discussed above, a reduction in passive fees both (a) lowers the active fee via competitive pressure and (b) increases the active fee via higher market inefficiency. These two effects exactly offset each other in this example. (The fact that the offset is exact, however, is not a general property of the model.)

Figure4, panel B, also shows how lower passive costs imply more passive asset management, less active investment, and less self-directed investment. These findings can be viewed as the model-based counterpart to some of the recent trends in real-world markets seen in Figure1. Since the mid-1990s, passive management has grown as passive fees have declined, self-directed investment has fallen (although this trend changes in the end of the sample, perhaps because of reductions in the cost of self-directed investment from discount brokerages), and, in the last part of this time period, active management has also starting to reverse its prior multidecade growth.

4. 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 natural conditions. We calibrate the central economic magnitudes, 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. First, the model provides a clear link between active fees and market efficiency; in our calibration, overall market efficiency is six times the cost of active management.

Second, the model shows how to decompose this overall inefficiency into the inefficiency of the market portfolio, the inefficiency of other factor portfolios, and that of truly idiosyncratic micro bets. In particular, the fraction of variance explained by each of these types of returns should be linked to the Sharpe ratios that can achieved by trading them.

Third, the model makes predictions on the impact of the ongoing, widespread reductions in the cost of passive management on capital markets and the industrial organization of the asset management industry. In particular, we show that falling fees of passive investing will increase market inefficiency, lower active fees by less than the passive fees, lower the fraction of active investors, and lower the number of active managers.

Fourth, the model has implications for optimal trading strategies. Since passive funds and indexes have incentives to mimic the optimal uninformed strategy, these results can be seen as predictions for which types of indexes should emerge as the most successful.

2

See also Breugem and Buss (2018), who consider the effect of benchmarking considerations on information acquisition and efficiency with multiple assets, and Kacperczyk, Nosal, and Sundaresan (2018), who consider the effect of large investors’ market power on market efficiency.

3

In a similar spirit, Peress (2005) shows that a decreasing market-participation cost leads to more participation and, in particular, more passive participation.

4

Stambaugh (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 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).

5

If we start with a signal $s^$ of any other dimension $n^$⁠, then we can focus on the conditional mean $u:=E(v|s^)$⁠, which is of dimension n and can be translated into a signal s as modeled above. For example, if $n^≥n$⁠, we have $s:=v¯+ΣvΣu−1(u−v¯)$⁠, where $Σu=var(u)$⁠.

6

Many indexes use a float adjustment. For example, S&P Float Adjustment Methodology 2017 states “the share counts used in calculating the indexes 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.”

7

An analytically convenient search function that satisfies these requirements is $c(M,I)=c¯(IM)α$ for constants $c¯>0$ and $α>0$⁠. Investors’ search cost capture expenses of vetting a manager. Gârleanu and Pedersen (2018) describe in their appendix B how real-world investors search and perform due diligence. While searching ensures finding an informed manager in our model, real-world search is imperfect.

8

In our model, incurring a search cost alleviates any agency issues. For a recent model of agency issues in asset management, see Buffa et al. (2020).

9

In general, an equilibrium with nonzero I, S_p, and M need not be unique, but we concentrate throughout on the equilibrium featuring the largest value of I and assume throughout parameters for which the largest equilibrium is interior, in that the numbers of each of the three types of investors are strictly positive. In a related set-up, Gârleanu and Pedersen (2018) discuss in detail equilibrium determination and multiplicity.

10

Grossman and Stiglitz (1980) define the informativeness of the price of a single asset as the squared correlation between the price and the signal in their theorem 4.A (among other places). Combining their equations (13), (17), and (18) shows that this informativeness is a decreasing function of the ratio of the utilities of the informed and uninformed agents, which in turn is equivalent to our definition of inefficiency, as seen from our Proposition 1. The Grossman and Stiglitz (1980) informativeness is thus essentially the inverse of our inefficiency. The link between the value of information and the ratio of determinants of the conditional variances was first derived in Admati and Pfleiderer (1987).

11

The same outcome for overall market inefficiency obtains for any matrix $ζ∈Rn×n$ of full rank.

12

The investor’s outside option also can be seen as searching again for another active manager, which yields the same result in an interior equilibrium.

13

Such a solution can be guaranteed to exist by either increasing $M¯$ or scaling down the cost function sufficiently. We assume hereafter the interesting case in which an interior equilibrium exists.

14

To understand this assumption, consider what happens if any security j undergoes a two-for-one 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.).

15

See, for example, and “Russell U.S. Equity Indexes 2017” and “S&P U.S. Indices Methodology 2017.”

16

This 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- versus micro-efficiency appear in, for example, Samuelson (1998).

17

While the original eigenvectors $(wj)$ are orthogonal in the Euclidean norm, $wj⊤wk=0$⁠, the corresponding inefficiency portfolios $(w^j)$ are orthogonal in the economically more interesting sense that their payoffs are conditionally uncorrelated, $cov(w^j⊤v⊤,w^k⊤v|s)=w^j⊤var(v|s)w^k=0$⁠. (We also note that, under Assumption 1, the rotation only scales the eigenvectors w₁ through w_n and the corresponding portfolios remain orthogonal in the Euclidian norm.)

18

We also note that, in a world in which an informed investor was allowed only to choose her loading on a single portfolio fixed ex ante, then the information would be most valuable if this portfolio were the most inefficient one, $w^1$⁠, since this would result in the highest expected utility differential, $12log g1$⁠.

19

We note that the measure of inefficiency is related to the Sharpe ratios of investors trading such portfolios.

20

In this case, the correlated signals convey little information about the factor portfolio. Indeed, at the limit, as $σFε2$ becomes infinite, the inefficiency is zero: no information is 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.

21

The assumption that the columns of $βε$ linearly generate those of β_v excludes the case in which a well-diversified portfolio can be constructed with positive systematic fundamental risk, but no systematic signal noise. The conditions on the matrices B_xy exclude degenerate cases, and in particular ensure that the limit market inefficiency is not zero.

22

One special case of Assumption 3 is the following: Start with an economy with a finite number of assets with a factor structure, and then make an increasing number of “copies” of all stocks in this baseline economy. When there are m copies of each baseline stock, the supply of each stock is $1/m$ so that the total payoff in the economy is roughly unchanged with the number of copies.

23

If the factor portfolios given by the columns of β_v are orthogonal, then these portfolios are in fact the same as the first principal inefficiency portfolios described in Section 2.1, that is, $ηβv,j=ηj$⁠. In this case, we can write the APT of efficiency as $∑j=1kηβj/η→1$⁠, meaning that the sum of inefficiencies of factor portfolios is nearly the entire market inefficiency.

24

To see this, note that (a) the variance of each (unit-norm) micro portfolio decreases as $1/n2$ with n; (b) 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 (c) there are $n−kv$ independent micro portfolios.

25

Here, $τp(n)−1$ and $τs(n)−1$ are the eigenvalues of $Σv|p$ and $Σv|s$⁠, respectively, corresponding to the eigenvector β, and $τp(0)−1$ and $τs(0)−1$ are the eigenvalues corresponding to all other eigenvectors, that is, any vector orthogonal to β. We note that, under Assumption 1, matrices $Σv|p, Σv|s$⁠, and G have the same eigenvectors.

26

We have $σwv2σFv2=(0.370.18)2=4.22.$

27

Gupta 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 does Roll (1988), whose study we use for our choice of parameters, so a real test of the model should use parameters consistent with the test portfolios.

28

Concerning the first part, see Roll and Ross (1980) for an early test of the APT of returns and Kelly, Pruitt, and Su (2018) for recent evidence of factors as return drivers. Regarding the second part, tests of predictability of the market (i.e., market timing) have played a central role in the debate about market efficiency (see the literature following Campbell and Shiller (1988)). For evidence of timing of other factors, see Asness et al. (2000), Greenwood and Hanson (2012), and Gupta and Kelly (2018). Regarding near-arbitrage profits, systematic evidence is rare, but it is telling that no outside investors are allowed into the famous Medallion Fund of Renaissance Technologies, which has reportedly delivered very high returns with a remarkable consistency. This impressive performance suggests a focus on micro strategies that hedge out almost all risk, so the strategy's limited scale is consistent with the model.

29

See also Admati and Pfleiderer (1987), who links the value of information to a ratio of determinants that coincides with our definition of inefficiency, Veldkamp (2011) for an overview of research on information choice, and Cabrales et al. (2013) for a recent contribution on entropy as the economic value of information and for further references.

30

For a symmetric positive definite matrix S = U DU⁻¹ with D diagonal, f (S) ≡ U f (D) U⁻¹, with f(D) a diagonal matrix of values of f. For our purposes, we can treat f (S) merely as notation for U f (D) U⁻¹.

31

In fact, by Lemma 2 the stronger statement $||(Dv+Σε)−1Σε||∼n0$ holds, rendering the product of order n⁻¹.

A Appendix: Further Analysis and Proofs

A.1 Calibration: Parametric Example with Closed-Form Solution

A.2 Notation

Given (7), (8), (10), and (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

A.4 Further Results

A.4.1 Active portfolio holdings

The following proposition characterizes the portfolio holdings of the informed investors.

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.” 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, Kabiller, and Pedersen 2018). Value and quality investment strategies have indeed been profitable on average across global markets (see, e.g., Asness, Moskowitz, and Pedersen 2013; Asness, Frazzini, and Pedersen 2019; Fama and French 2017).

A.4.2 Change in noise

Section 3 considers comparative statics with respect to some key changes in the market, namely, the costs of active and passive investing. Another potential change resulting from the rise in delegated management is a reduction in “noise trading.” While we have emphasized that supply uncertainty arises for several rational reasons (e.g., firms issuing or repurchasing shares), it can also arise simply from investors making irrational trades. If this noise trading is primarily due to individuals, then it may have gone down over time as emphasized by Stambaugh (2014). We can also consider the implications of a change in noise in the context of our model.

We see that a reduction in noise trading should lead to a reduction in the number of active investors and informed active managers. The effect on market inefficiency is ambiguous; it depends on how many patsies (noise traders) versus sharks (informed investors) have left the “poker table.”

A.4.3 Efficiency, entropy, and the value of information

We can also shed further light on the properties of market efficiency. We show (as already discussed in Section 1.2) that market efficiency is linked to the (private) economic value of information so 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 has been studied in a number of papers in the literature, and the following proposition further establishes a link to the degree of market inefficiency.29

The informativeness can be measured using entropy and, as is known from information theory, the entropy of a multivariate normal is a half times the log-determinant of the variance-covariance matrix (plus a constant). Therefore, the market efficiency is the difference in entropy of the distributions of fundamental values given prices, respectively, given private signals. Another measure of the distance between two probability distributions is the Kullback-Leibler divergence. While the Kullback-Leibler divergence is random (since it depends on the conditioning variables p and s), the proposition establishes that the expected Kullback-Leibler divergence also equals overall market inefficiency. Hence, this result establishes a new potential way to measure market efficiency and the economic value of information, namely, using entropy-based methods also applied in other sciences.

A.5 Proofs

Some of our results follow from a more general result, Lemma 1, which we state and prove below. For clarity, we spell out formally the assumption on which this lemma relies.