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Who Benefits from Innovations in Financial Technology?
Roxana Mihet∗
NYU, Stern Economics
(Latest version here)
November 10, 2019
Abstract
Financial technology affects both efficiency and equity in the stock market. The
impact is non-trivial because several key technological developments have altered mul-
tiple dimensions of investors’ opportunity sets at the same time. For example, better
and faster computing has made it cheaper for retail investors to participate and to find
funds that meet their needs. However, it has also made it cheaper for sophisticated
investors to learn about asset returns. Some experts believe these innovations will
increase financial inclusion. Others worry about possible anti-competitive effects that
can lead to more unequal rent distribution. To address this debate, I first build a the-
oretical model of intermediated trading under asymmetric information that allows me
to differentiate between the effects of each innovation. Second, I interpret US macro
data from the last 40 years through the lens of my model and find that, although the
gains from financial technology were accruing to low-wealth investors throughout the
1990s, they have been accruing to high-wealth investors since the early 2000s. The
key theoretical finding is that, even if investors have access to the equity premium
through cheap funds, improvements in financial technology disproportionately benefit
informed, sophisticated traders. This reduces the participation rate of low-wealth in-
vestors, improves price informativeness, enlarges (but, at the same time, consolidates)
the sophisticated asset management industry, and amplifies capital income inequality.
Further advances in modern computing, big data, and artificial intelligence in asset
management, in the absence of any gains redistribution, may accelerate the rate of
change.
Keywords: Financial technology; capital income inequality; stock market; asset
management; information; participation.
∗Contact: rmihet@stern.nyu.edu. Acknowledgements: I am indebted to my advisors, Laura Veldkamp,
Thomas Philippon, and Venky Venkateswaran for their unwavering support and patient advice. I have
learned immensely from them. I am grateful to Jess Benhabib, Stijn Claessens, Jerome Dugast, Mark
Gertler, Avi Goldfarb, Paymon Khorrami, Luc Laeven, Andrew Lo, Joseba Martinez, Pricila Maziero, Guido
Menzio, Emily Moschini, John Muellbauer, Cecilia Parlatore, Luigi Pistaferri, Thomas Sargent, Hyun Shin,
and Ansgar Walther for their useful suggestions, as well as to participants at the 14th Macro-Finance Society
Meeting (2019), The Future of Financial Information Conference (2019), The Young Economist Symposium
(2019), Chicago Booth Asset Pricing Conference (2018), The Macro-Financial Modeling Summer Session
(2018), and Wharton Women in Business (2018), and to seminar participants at the BIS, IMF, NYU Stern,
and NYU. I thank Chase Coleman, Clara Dolfen, Paul Dolfen, Sally Lanar, Adam Nahum, Bang Nguyen,
and Desi Volker for their helpful encouragement. Financial support by NYU Stern and the Becker Friedman
Institute through the Macro-Financial Modeling Dissertation Grant is gratefully acknowledged. Part of this
work was completed during a PhD Fellowship at the Bank of International Settlements. This paper was
awarded the Special Mention Runner-up Prize at the 2019 European Best JM Paper Competition by the
European Economic Association. All errors are my own.
1
1 Motivation
Progress in financial technology affects both efficiency and equity in the stock market.
Its impact is non-trivial because several key technological changes have altered multiple
dimensions of investors’ opportunity sets. For example, better and faster computing has
made it cheaper for retail investors to participate and to find funds that meet their needs.
However, it has also made it cheaper for sophisticated investors to learn about asset returns.1
To understand the ultimate impact of progress in financial technology, we need to understand
which one of these dimensions is altered most. The answer is not obvious, as evidenced by
the disparate opinions of several experts in this field. For example, Stiglitz (2014) believes
financial technology will increase financial inclusion, but Philippon (2019) worries about its
potential anti-competitive effects, and Piketty (2014) about more unequal rent distribution.
To address this debate, I build a theoretical model of intermediated trading under asym-
metric information that decomposes progress in financial technology into an improvement
in three distinct frictions: participation, search, and information costs. The model allows to
study the effects of each friction separately. In the model, there is a continuum of two types
of agents: investors and asset managers. Investors differ in their wealth and risk-aversion,
and are uninformed. Asset managers, on the other hand, are informed and trade in the best
interest of their investors. Investors make two decisions: whether or not to pay a fixed cost
of participation to enter the stock market, and whether or not to pay a search cost to find an
informed asset manager with access to information that can improve return performance. In
other words, the cost of participation represents the cost of investing through a cheap fund,
such as a mutual fund, or an exchange-traded fund. The cost of searching for an informed
asset manager represents the cost of investing through a sophisticated fund, such as a hedge
fund or a family-owned office. The information cost represents the cost of researching asset
returns, which facilitates a more educated portfolio decision.
While both investors and asset managers can acquire private information about asset
returns, due to economies of scale in asset management,2a natural outcome of my model
is that no investor acquires private information directly. Investors optimally either invest
directly and uninformedly (without any private signals) or indirectly and informedly, by
delegating their portfolios to informed asset managers. In other words, the equilibrium of
my model is characterized by two wealth thresholds that separate non-participating investors
from those who invest directly uninformedly, and the latter from those who invest indirectly
informedly. I also solve several extensions: an exogenous information structure with manager
free-entry; and an endogenous information structure with manager free-entry.
I then perform three comparative statics exercises of progress in financial technology.
The first exercise is a decrease in the costs of stock market participation. The second is a
1Technology has reduced transaction and information costs (Puschmann and Alt (2016)), improved inclu-
sion and transparency (Claessens et al. (2002)), facilitated risk-sharing among financial participants (Aron
and Muellbauer (2019)), and reduced search frictions in markets for asset management (Lester et al. (2018)).
2Asset managers can spread the information costs among many investors. This can improve the returns
of the less wealhy and increase market efficiency (Garleanu and Pedersen (2018)).
2
decrease in search and matching costs which allows investors to perform due dilligence and
vet the asset managers they choose to invest through. The third is a decrease in the costs
of acquiring private information about the risky assets.
In my model, improvements in participation, search and information processing costs
have opposing implications for stock market efficiency and equity. Lower participation costs
make the stock market more inclusive by improving participation, but less efficient because
of a rise in uninformed participation. Lower search and information costs make the stock
market less inclusive because they reduce uninformed participation, but they also make it
more efficient because of a rise in informed participation. Lower search costs also lead to
a consolidation of the sophisticated investment management industry, because, when the
stock market becomes almost perfectly efficient, the value of informed asset management
falls. This tradeoff between efficiency and equity is explained in more detail below.
A lower participation cost leads to a natural rise in participation. However, this increase
in participation lowers the incentives to delegate to informed asset managers because a fixed
asset supply implies that stock prices go up for all assets. So, incumbent indirectly informed
investors reduce their stock-holdings. The marginally indirectly informed investor, who
was indifferent between investing directly uninformedly and delegating to a manager is now
worse off. It becomes less attractive to pay the search cost to find an informed manager,
because the gains from trade on a smaller portfolio fall. Overall expected returns fall, the
equity premium falls, inequality decreases, and stock market prices become less informative
because there are fewer indirectly informed investors.
A lower information cost implies the opposite. Because information is cheaper, asset
managers process more private information. Delegation fees fall, which incentivizes more
investors to delegate their portfolios to informed asset managers. As the size of the asset
management industry grows, price informativeness goes up because there are more indi-
rectly informed investors, but crucially, uninformed participation falls. When the informed
investor measure grows, the marginal uninformed investor, who was indifferent between non-
participation and investing directly uninformedly, now competes for the equity premium
against more indirectly informed traders who drive asset prices up. It is no longer attractive
to pay the participation fee just to invest uninformedly, because the gains from doing so
are smaller. Therefore, participating uninformed investors exit the stock market altogether,
which amplifies inequality.
A lower search cost also allows a larger measure of investors to delegate to informed
asset managers. Consequently, relatively lower-wealth investors, who were investing directly
uninformedly before and who do not benefit from the lower search cost, exit the stock market
altogether, forgoing access to the equity premium. In a model extension with manager
free-entry, lower search costs, unlike information costs, increase the concentration of the
sophisticated asset management industry (i.e., the total revenue grows, but the number of
asset managers falls). This is because the stock market becomes so efficient, that one big
manager captures the entire market. This is an efficiently inefficient outcome.
3
My contribution comes from obtaining original general equilibrium effects from the inter-
action of these three distinct aspects of financial technology. The difficulty of writing down
this model of asymmetric information is conceptual. The model is elegant and tractable
with only the necessary ingredients to emphasize key tensions at play. Seemingly innocuous
changes to the model quickly render it intractable and opaque. We already know from pre-
vious work by Peress (2005), Bond and Garcia (2018), Kacperczyk et al. (2018), Garleanu
and Pedersen (2018), and Davila and Parlatore (2019) that the three distinct margins: par-
ticipation, search and information matter separately. What we do not know is how they
interact in equilibrium. In this paper, I emphasize that it is important to separate these
three aspects of financial technology because their effects and normative implications are
starkly different and their interaction amplifies the economic mechanism I propose.
The net result on participation/equity and price informativeness/efficiency depends on
how these aspects of financial technology interact in equilibrium. The comparative statics
exercises discussed above emphasize a trade-off between information/ efficiency in financial
markets and participation/ risk-sharing. Improvements in financial technology are pulling
this tradeoff one way or another. Knowing that their effects are different allows me to use
the model to interpret the data and assign a dominating lessening friction to different time
intervals in the last 40 years. This resolves the identification challenge discussed previously.
Viewed through the lens of my model, the data from Figures (1) and (2) indicates that the
lessening of the participation cost was being more dominant before 2001, which coincides
with the emergence of e-trading, while the lessening in information and search costs has been
dominating since 2001. Only this sequence of technological improvements can explain the
patterns in the data. I elaborate on this interpretation in Section (7).
Figure 1: The participation effect drives the increase in uninformed participation
before 2001. Search & information costs drive it down after 2001.
a) Participation Rate b) Price Informativeness
Legend: The share of stocks by top 20% capital wealth is from Saez and Zucman (2016). Stock market
participation (weighted) is from SCF and includes direct & indirect holdings. Bai et al. (2016) compute
price informativeness by running cross-sectional regressions of future earnings on current market prices.
In addition, interpreted through the lens of my model, the data suggests that the gains
from financial technology were accruing to low-wealth investors before the 2000s, but they are
4
Figure 2: The participation effect drives the number of hedge funds and their fees
up before 2001. The information effect drives the number up and the fees down,
while the search effect drives both of them down after 2005.
a) Number of Hedge Funds (US) b) HF Incentive Fees per Rate of Return
Source: Lipper TASS Hedge Fund Database.
now accruing to high-wealth investors. The reason this is happening is that even if investors
have access to the equity premium through cheap funds, improvements in financial technol-
ogy disproportionately benefit informed, big data players. This reduces the participation
rate of low-wealth investors, improves price informativeness, enlarges (but consolidates) the
sophisticated asset management industry and amplifies capital income inequality. Fagereng
et al. (2016), Di Maggio et al. (2018), Calvet et al. (2019), and Campbell et al. (2018) pro-
vide evidence that wealthy investors already achieve higher Sharpe-ratios, while low-wealth
investors lose money in the stock market. Moreover, the percentage of households delegating
their investment portfolios increases with their initial wealth. The two phenomena are clearly
related, and the idea is that wealthier investors benefit from searching for informed asset
managers, since their search cost is low relative to their capital. And while ‘dumb money’
investing has been a popular and optimal option for low-wealth investors, it is not the right
vehicle to earn high payoffs. Leon Cooperman, the famed CEO of Omega Advisors has been
saying for years that “The rich didn’t get to their net worth by buying an index.” I expect
to continue to see an amplification of capital income inequality and polarization of capital
returns due to financial technology. Further advances in modern computing, big data, and
artificial intelligence in investment management may accelerate the rate of change.
The key theoretical finding of my paper is that even if investors have access to cheap
funds, low-wealth investors are still going to exit the market in the presence of search and
information frictions. More importantly, however, lower search and information frictions do
not solve this problem. On the contrary, they amplify it. Improvements in financial technol-
ogy make information-based trading (i.e.,‘smart money’) more attractive than uninformed
trading (i.e.,‘dumb money’). Low-wealth uninformed investors end up competing with more
aggresively informed traders, who drive prices up and the returns from uninformed trading
down. Thus, some investors no longer find it attractive to pay participation fees just to
5
invest uninformedly, as the gains from doing so are smaller. Because uninformed trading
becomes a worse option than before, the stock market participation of uninformed investors
falls. This mechanism amplifies inequality by lowering stock market participation, improves
price informativeness, and leads to a larger and more concentrated asset management sector.
The paper proceeds as follows. Section (2) places the contribution in the literature.
Section (3) describes some motivating facts. Section (4) explains the model and solution.
Section (5) presents the comparative statics exercises. Section (6) comprises of model exten-
sions. Section (7) interprets the data through the lens of the theory. Section (8) discusses
policy implications and finally, section (9) concludes.
2 Contribution to Existing Literature
First, my paper relates to the stock market participation literature in household finance.
Participation costs encompass a number of different things, such as paying signup fees, time
spent understanding and filing the necessary paperwork associated with stockholdings, and
downloading e-trading apps. Participation matters not only for capital income inequality but
also for the propagation of shocks in the economy (see Allen and Gale (1994), and Morelli
(2019)). Lusardi et al. (2017) argue that investors with low financial literacy are significantly
less likely to invest in stocks. Financial education has been proposed regularly as one way
of increasing participation (van Rooij et al. (2011)). My paper contributes to literature.
I argue that lower participation costs are not enough to improve participation rates in a
world with asymmetric information and co-existence of ‘smart money’ with ‘dumb money’.
One needs to take into account the information externalities generated by technological
innovations. When improvements in financial technology lower the incentives for uninformed
participation, additional solutions are needed.
Second, my paper adds to the literature on endogenous information acquisition in fi-
nancial markets (Grossman and Stiglitz (1980), Verrecchia (1982), Kyle (1985), Admati and
Pfleiderer (1990), Veldkamp (2006), and Peress (2010)). Noisy rational expectations models
with endogenous information costs are a useful framework for thinking about the impact
of the financial information revolution. They facilitate the study of complex general equi-
librium effects while remaining highly tractable. Generally, these papers find that better
information increases price informativeness and market efficiency.
I contribute to this literature by disentangling information itself into two components:
information about hedge fund managers (modeled as a search cost) and information about
assets (modeled as an information cost). The two generate different results in the model, and
their coexistence explains the data better. Only cheaper information about asset managers
drives, in the model, the drop in the number of hedge fund managers observed in the data.
I build on the modeling framework of Peress (2005), which is successful in explaining
the rise in stock market participation and the rise in passive investing before the 2000s.
However, the focus is not on the decline in stock market participation since the 2000s. I
6
base my model on this framework because it generates a tradeoff between participation
and information, which is useful for explaining the data before and after 2001. I employ a
similar setting with heterogeneity in absolute risk aversion, to which I add a market for asset
management, and distinguish between information about asset managers and information
about assets. I use the model to study the effects of new technologies on price efficiency, the
market structure of the asset management industry, and capital income inequality. Similar to
Peress (2005), Bond and Garcia (2018) look at the impact of a fall in the cost of participation
over time. While this leads to more indexing, it does not explain the fall in participation in
the last 20 years (i.e., the retrenching of low-wealth investors). The key missing ingredient
is the lack of information technology effects. I do look at the impact of lower information
costs, both for asset managers and for investors searching for asset managers, and trace out
effects for capital income inequality, which could be extracted from this paper, but are not
the focus there. Malikov (2019) builds a related, but different model. He also sets out to
explain the growth in passive indexing over time. I set out to understand the interaction
between non-participation, ‘dumb money’ and ‘smart money’, with a focus on explaining the
fall in overall stock market participation in the last 20 years. Malikov (2019) does not have
a margin for stock market non-participation and cannot capture this.3Moreover, Malikov
(2019) does not model search frictions for ‘smart money’. Because I also have a distinction
between information about asset managers and information about assets, I can generate
predictions for the hedge fund industry, for stock market participation, and ultimately for
inequality. Inequality, in my model, hinges on the difference in ex-post returns conditional
on participation (that exists in these other models of asymmetric information), but also on
the stock market non-participation margin, which is the novel contribution of my paper.
It has been known since Arrow (1987) that endogenous information acquisition in an
asymmetric information context amplifies inequality. This property of information has been
quantified in static discrete choice models of capital income inequality (Kacperczyk et al.
(2018)) and dynamic models of capital wealth inequality (Kasa and Lei (2018), Lei (2019),
and Azarmsa (2019)) and shown to hold without loss of generality. It has also been verified
in reduced form models that use alternative sources of data as information. For example,
Katona et al. (2018) use satellite imagery of parking lot traffic across major U.S. retailers
and find that sophisticated investors, who can afford to incur the costs of processing satellite
imagery data, formulate profitable trading strategies at the expense of individual investors,
who tend to be on the other side of the trade. Kacperczyk et al. (2018) is the most closely
related paper, as it also studies capital income inequality in a static portfolio choice model
where technological change improves the information constraints. I extend this literature
by separating information frictions into information about good asset managers (modeled as
3Besides different research questions, our models also differ in their assumptions and applications. I con-
sider the impact of lower participation, search and information costs in a model of asymmetric information
with wealth effects and a distinction between ‘smart money’, ‘dumb money’, and non-participation. Malikov
(2019) considers a decrease in the cost of information and an increase in the costs of ‘smart money’ partici-
pation in a model of symmetric information, without wealth effects, and with a distinction between an index
asset and multiple risky assets.
7
search costs for ‘smart money’) and information about assets. These two types of information
have very different implications. My model also has additional amplification mechanisms for
inequality due to the participation margin and the market structure of the asset management
industry. Thus, one can go to the data to qualitatively or quantitatively distinguish between
the effects of each mechanism.
Third, my paper extends the theoretical literature on the macroeconomic implications
of technological innovations. The majority of existing work has focused on the impact of
automation on labor income inequality through skill-biased technological change (Aghion
et al. (2019), Acemoglu and Restrepo (2017), Autor et al. (2017), Martinez (2018), and
Benhabib et al. (2017b)). My paper extends this discussion by considering the impact of
technological change on capital markets through information effects.4My contribution to
this literature is theoretical. I show that lower search and information costs increase capital
income inequality through information externalities, consistent with empirical evidence by
Ellis (2016), Dyck and Pomorski (2016) and Brei et al. (2018).
Lastly, my paper contributes to the literature on investment management. The bench-
mark paper is Berk and Green (2004) which studies the implications of fully efficient asset
management markets in the context of exogenous and inefficient asset prices. I extend the
analysis to consider an imperfect market for asset management due to search and information
frictions. I use elements from Garleanu and Pedersen (2018) who find that the efficiency of
asset prices is linked to the efficiency of the asset management market. Using a model where
ex-ante identical investors can invest directly or search for an asset manager, Garleanu and
Pedersen (2018) find that when investors can find asset managers more easily, more money
is allocated to informed management, fees are lower, and asset prices become more informa-
tive. While the model generates a number of verifiable predictions, it does not say anything
about capital income inequality over time, or about the rate of stock market participation.
It does not have a margin for participation; hence, there is no tradeoff between informa-
tion and participation. My contribution is to add a margin for participation and show that
there is a tradeoff between information (i.e., efficiency) and participation (i.e., risk-sharing).
This tradeoff is important because it amplifies capital income inequality when information
becomes cheaper to acquire.
Abis (2017) is also related to my paper, but the focus is on variation in learning be-
tween quantitative and discretionary funds over the business cycle. Importantly, it does not
distinguish between non-participation and the decision to establish a fund. As mentioned
before, the non-participation margin is important for the mechanism of my model.
4As opposed to Schumpeterian theory, in which growth is driven by quality-improving innovations which
destroy the rents generated by previous ones, I lay down a theory in which the value extracted from cost-
saving innovations is different for the rich and the poor, allowing the rich to extract more surplus from their
investments. Bilias et al. (2017) emphasizes that increased access to a risky financial instrument offering an
expected return premium can reduce wealth inequality, but Favilukis (2013) shows that there is a conflict
between participation, which lowers inequality by making the equity premium available to more investors,
and stock market booms, which widen the wealth gap between stockholders and non-stockholders.
8
3 Motivating Facts
The first fact is that less-wealthy investors are withdrawing from the stock market. Ac-
cording to various surveys, the US has the lowest level of direct and indirect stock ownership
in almost 20 years, as shown in Figure (3, a). Yet, stock ownership for the wealthy is at
a new time high, as shown in Figure (3, b), and this has accounted for most of their good
fortune compared to the rest of America.
Figure 3: The Wealthy Own the Stock Market; The Less Wealthy Are Exiting
a) Stock Market Participation b) Who Owns the Equity Market?
Source: Participation from the SCF, Gallup and Wolff (2016). Equity shares by wealth percentile from
Thomson Reuters.
The second fact is that the probability of portfolio delegation increases with initial
wealth, as in Figure (4, a). Moreover, while the wealthy have been concentrating their
investments in sophisticated funds, such as hedge funds or started their own family-owned
offices, the less-wealthy have been gradually phased out from the stock market, first into
cheaper, less sophisticated intermediated products (i.e., mutual funds, ETFs) and ultimately
into riskless assets (savings, if at all) or housing. This may be optimal for them, but it has
important consequences for their stock market returns, income and wealth.
Figure 4: The Wealthy Delegate More Often... and More to Hedge Funds
a) The Wealthy Delegate More b) The Wealthy Delegate More to HFs
Source: Percentage of households delegating their wealth from SCF, for year 2013. Top 0.01% wealth
shares in the US by type of fund from Kaplan and Rauh (2013).
9
Moreover, it is important to note that managers of hedge funds have direct access to
state-of-the-art technologies for information acquisition, which managers of mutual funds or
exchange-traded funds may not (Sushko and Turner (2018)). This may explain why since
2001, high wealth investors have increased the wealth share they delegate to hedge funds
threefold, as in Figure (4, b).
The third fact is that wealthy investors achieve higher risk-adjusted returns, as in Figure
(5, a). Moreover, hedge funds also achieve higher risk-adjusted returns relative to passive
indices, as in Figure (5, b). This is suggestive of significant barriers to investing through
hedge funds, otherwise investors of all wealth levels would prefer to invest through them. In
reality, investing through a hedge fund is very expensive because there is a minimum wealth
threshold for investment, but there are also other costs and frictions related to searching and
delegating to a good hedge fund managers (see the Appendix).
Figure 5: Wealthy Investors and Hedge Funds Obtain Higher Sharpe Ratios
a) Households Sharpe Ratios by Wealth b) Fund Sharpe Ratio by Type
Legend: Sharpe ratios are risk-adjusted returns. Source: Fagereng et al. (2016) for households, Preqin and
AIMA (2018) for funds.
4 Model
4.1 Setup: Market Players, Assets, Information, and Timing
This is a model wherein investors heterogeneous in initial wealth and risk-aversion de-
cide whether to participate in the stock market or not, and whether to do so by investing
directly or by searching for an informed asset manager. Moreover, information about assets
is costly, and perfectly competitive managers charge an endogenous fee.
Investors and Managers. The economy features a continuum of investors indexed
by j, who differ in their initial wealth W0j∈[0, W max
0], a continuum of mass one of asset
managers indexed by m, who trade on behalf of groups of investors. There are also some
noise traders who make random trades in the financial market for non-strategic liquidity
reasons. This is a mathematical trick that allows private information to not be revealed in
equilibrium.
10
Assumption 1 (Participation cost)
Each investor must pay a fixed cost of participation, F > 0, to enter the stock market. If an
investor chooses not to pay it, she/he can only save through a riskless asset.
Then, each investor can either (a) invest directly in asset markets after having acquired
costly private signals, (b) invest directly in asset markets without the signals, or (c) invest
through an asset manager.
The economy also has a continuum of mass one of asset-management firms, indexed by
m. These asset-management firms are akin to family-owned offices/exclusive hedge funds
that provide tailored advice to their investors and invest according to their investors’ risk
preferences. I assume that all asset managers are informed and that this fact is common
knowledge. To invest with an informed asset manager, investors must search for and vet
managers, which is a costly activity.
Assumption 2 (Search cost)
The cost of finding an informed manager and confirming that the manager has the technology
to acquire private information (i.e., performing due diligence) is ω > 0.
The search cost ωcaptures the realistic feature that most investors spend significant
resources finding an asset manager that they trust with their money. The form of this cost
function can be generalized, but for the moment, let’s assume it is the same for all investors.
Each investor solves a portfolio choice problem to maximize a mean-variance approx-
imation of CRRA utility with a risk-aversion coefficient that declines with initial wealth.
Investors’ preferences are
max
qj
Ej(W2j)−ρ(W0j)
2V arj(W2j) (1)
where W2jis terminal wealth; W0jis initial wealth; the coefficient of absolute risk aversion
is ρ(W0j) = ρ
W0j>0, with ∂ρ(W0j)/∂W0j<0.
This preference specification can be interpreted as a local quadratic approximation of
any utility function around initial wealth, and it allows for wealth effects, similar to CRRA
utility. The heterogeneity in absolute risk aversion implies differences in the size of investors’
risky portfolios and hence different gains from investing wealth in purchases of information.
With strictly CARA preferences, investors would have collected the same amount of infor-
mation regardless of the number of shares supplied and the mass of participating investors
in equilibrium. With this CARA approximation of CRRA, however, the demand for in-
formation increases with the number of shares each investor expects to hold, reflecting the
increasing returns to scale displayed by the production of information.
11
When an investor meets an asset manager and confirms that the manager has the tech-
nology to obtain private information, they negotiate the asset management fee fj. The fee
fjis an equilibrium outcome set through Nash bargaining. For tractability, I assume that
at the bargaining stage, the manager’s information acquisition cost and the investor’s search
cost are sunk.
Assets and Information. The financial market consists of one risk-free asset in
unlimited supply, with price normalized to 1 and payoff r, and one risky asset, with price p
and a stochastic payoff z.5
z=µz+, with ∼N(0, σ2
z) (2)
Finally, the economy features a group of non-optimizing “noise traders,” who trade
for reasons independent of payoffs or prices (e.g., for liquidity or hedging reasons). This
assumption ensures that prices do not reveal the private information endogenously acquired.
Noisy traders provide a stochastic supply for the risky asset:
x=µx+v, with v∼N(0, σ2
x) (3)
Market participants know the distribution of shocks but not their realizations. Prior
to making portfolio decisions, market participants can obtain private information about the
risky payoff in the form of private signals about z. I assume that the private signal is
independent among market participants and given by
sj=z+δj, where δj∼N(0, σ2
s,j ) (4)
Assumption 3 (Information acquisition cost)
Each signal costs κ(σ−2
s,j )to acquire, and the cost function is convex and increasing in the
precision of information learned.
κ(σ−2
s,j ) = 1
2c0(σ−2
s,j )2+c1where c0>0and c1>0are strictly positive constants
The information set of an agent with no private information is Ij(z;p), and the information
set of an agent with private information is Ij(z;sj, p).
Timing. Each period is divided into two sub-periods, as shown in Figure (6).
In the first sub-period, investors decide whether to enter the stock market at a fixed
cost, F > 0, that grants access to purchasing the risky asset. Then, investors choose whether
5In the Appendix, I consider an extension with multiple risky assets (N≥2), where the payoffs are
independent of each other. The economic mechanism and the results remain unchanged, although with small
modifications to the assumptions, one can generate different results such as specialization vs. broadening of
knowledge, etc.
12
to manage their portfolios individually or delegate to an informed asset manager. To invest
with an informed asset manager, investors much search for and vet managers (i.e., perform
due diligence), which is a costly activity, ω > 0. Investors who manage their portfolios on
their own and asset managers choose how much private information to learn about the risky
asset. Learning private information costs κ > 0.
In the second sub-period, all market participants observe stock prices, learn the private
signals they have chosen to acquire, and form their portfolios of assets. Investors who have
chosen to delegate their portfolios now negotiate an asset management fee, f > 0 with their
managers. As all trading is realized and the market clears, investors get their corresponding
investment portfolios back for consumption. In the next period, investors start again with
initial wealth equal to the terminal wealth from the previous period (but they do not optimize
their decisions across periods).
Figure 6: Timing of the game
4.2 Equilibrium Concept
To solve and characterize this equilibrium, I work backwards in time, starting from the
equilibrium in financial markets, then in the market for asset management, then solving
for the managers’ endogenous information acquisition choice, and then solving for investors’
optimal participation and search decisions. Since the model involves several fixed costs, this
economy will be characterized by a threshold equilibrium. The complete solution steps and
proofs are in the Appendix. Below, I briefly outline the main steps.
An equilibrium of this noisy rational expectations economy consists of portfolio
allocations qjfor each investor type, precision levels ˆσ−2
s,m, asset prices p, asset management
fees fj, and two wealth thresholds, one for participation Wparticip
0and one for search and
delegation Wsearch
0, such that:
1. Portfolio choices, qj, solve each investor’s portfolio maximization problem, where 1[.]
denotes indicator functions for the decisions to participate and search, respectively.
This gives rise to a portfolio choice for investors who participate on their own, qparticip
j,
13
and a portfolio choice for investors who search and delegate to managers, qsearch
j.
max
qj
Ej[W2,j |Ij]−ρ(W0,j )
2V arj[W2,j |Ij] (5)
s.t. W2,j =rW0,j −1hF−1ω−fj−qj(z−rp)i(6)
2. Asset markets clear, such that the demand for the risky asset equals the stochastic
supply. Thus, the demand from participating investors and the demand from searching
investors has to equal the stochastic asset supply.
ZWsearch
0
Wparticip
0
qparticip
j+ZWmax
0
Wsearch
0
qsearch
j
| {z }
demand
=x
|{z}
supply
(7)
3. Asset management fees are the outcome of Nash Bargaining such that no investor
would like to switch status from searching for a manager or not.
max
fj
(Vsearch
j−fj−Vparticip
j)
| {z }
investor surplus
×fj
|{z}
manager surplus
(8)
4. The managers’ chosen precisions solve their endogenous information acquisition prob-
lem such that the marginal benefit of acquiring information equals the marginal cost.
max
σ−2
s,m
1
MZWmax
0
Wsearch
0
fj
| {z }
manager revenues
−κ(σ−2
s,m)
| {z }
manager costs
(9)
5. Investors optimally choose to participate (or not), and to search for a fund manager
(or not). Their decisions give rise to a wealth threshold for participation Wparticip
0and
a wealth threshold for search and delegation Wsearch
0.
max {Vnp
j, V particip
j, V search
j}(10)
4.3 Solution
Asset Market Equilibrium. Every trader invests an amount in the risky asset that is
proportional to the ratio of the expected excess return to the variance of the return given the
information set, where the factor of proportionality is the risk tolerance: 1/ρ(W0j) = W0j/ρ.
Hence, an investor with twice the wealth buys twice the number of shares, either directly or
through the asset manager.
Proposition 1 (Optimal portfolios)
The optimal portfolio is given by qdirectly
j=ˆµU
z,j −rp
ρ(W0j)ˆσU,2
z,j
for traders who trade on their own
14
uninformedly, and by qdelegate
j=ˆµI
z,j −rp
ρ(W0j)ˆσI ,2
z,j
for traders who delegate to informed managers.
I will now define some objects that will be useful going forward. Let tbe the total
risk-tolerance of all stock market participants. Let sbe the informativeness of the price
implied by aggregating the precision choices of those investors who delegate their portfolios
to asset managers, who can be informed or uninformed. Let ˜sbe the measure of informed
(i.e.,‘smart’) wealth. Let n=s−1be the total amount of noise in this economy. Note
that due to economies of scale (see the Appendix for some conditions on the parameters), a
natural equilibrium outcome is that investors do not acquire the signal directly: They either
invest individually and uninformedly or delegate to informed managers. I will highlight
weak conditions under which all realistic equilibria take this form and rule out that investors
acquire the signals on their own (see the Appendix).
t=ZWmax
0
WP articip
0
1
ρ(W0j)dj; (11)
s=ZWmax
0
WSearch
0
1
ρ(W0j)σ2
s,m
dj; (12)
˜s=ZWmax
0
WSearch
0
1
ρ(W0j)dj; (13)
n=1
s; (14)
Market clearing implies that demand for the risky asset equals its stochastic supply.
This relation gives rise to the formula for the stock price.
Proposition 2 (Asset price)
The price of the risky asset is given by rp =a+bz −cx, where
a=¯
h−1µz
σ2
z
+sµx
σ2
x;b=¯
h−1s2σ−2
x+s
t;c=¯
h−1sσ−2
x+1
t; (15)
where ¯
h=s
t+σ−2
z+s2σ−2
x
The price crucially depends on the ratio s/t, which becomes important for what is to fol-
low. This is the ratio of the mass of searching investors (who get matched with an informed
asset manager) to the mass of participating investors. Intuitively, it is the ratio between the
total amount of information in the market and the total risk-sharing in the economy.
Management Fees. The asset management fee fjis set through Nash bargaining
between an investor and a manager. The fee depends on an investor’s best outside option,
which is the larger of the utility of investing on his/her own uninformedly and the utility of
searching for another manager.
15
Definition 1 Let θbe the market inefficiency:
θ=ρ(W0j)Vdelegate
1j−Vdirectly
1j(16)
The market inefficiency records the amount of uncertainty about the asset value for an
agent, who only knows the price, relative to the uncertainty remaining when the agent knows
both the price and the private signal sj. The price inefficiency θ≥0 is a positive number.
Naturally, a higher θcorresponds to a more inefficient market, while zero inefficiency corre-
sponds to a price that fully reveals the private signal. The price inefficiency θis linked to
managers’ and investors’ value of information. It gives the relative utility of investing based
on the manager’s information (Vdelegate
1j) versus investing uninformedly (Vdirectly
1j).
The fee fjis determined through Nash bargaining, maximizing the product of the utility
gains from agreement. If no agreement is reached, the investor’s outside option is to invest
uninformedly on his/her own, yielding a utility of (rW0j−F−ω+vdirectly
1j). The utility of
searching for another manager is (rW0j−F−ω−fj+vdelegate
1j). For an asset manager, the
gain from agreement is the fee fj, as the cost of acquiring information κ(.) is sunk, and there
is no marginal cost of taking on the investor. The bargaining problem is to maximize the
surplus, which is given by (vdelegate
1j−vdirectly
1j−fj)fj. The optimality condition gives the fee
schedule for all investors j.
Proposition 3(Asset management fee)
The asset management fee is given by fj. It increases with the level of market inefficiency
and with the investor’s initial wealth.
fj=θ
2ρ(W0j)(17)
It is easy to see that the management fee increases with the market’s inefficiency, ∂ fj
∂θ =
1
2ρ(W0j)>0, and with the investor’s initial wealth, ∂fj
∂W0j=θ
2ρ>0. The fee would natu-
rally be zero if asset markets were perfectly efficient, so that investors had no benefit from
searching for an informed manager. In this setting, sophisticated asset management fees can
be construed as evidence that retail investors believe that securities markets are not fully
efficient.
Investors’ Decision to Search for Informed Managers. An investor optimally
decides to look for an informed asset manager as long as the utility difference from doing so
is at least as large as the cost of searching and paying the asset management fee:
vdelegate
1j−vdirectly
1j≥ω+fj(18)
16
Using equation (16), this translates to θ
ρ(W0j)≥ω+fj. This relation must hold with
equality in an interior equilibrium. Plugging in the equilibrium management fee, this implies
ω=θ
2ρ(W0j).
To solve for the market inefficiency, θ, one needs to first compute the expected utility.
Ex-ante utility is given by V1j=rW0j−F−ω−1(fj) + 1
2ρ(W0j)E1[( ˆµz,j −rp
√ˆσ2
z,j
)2].
The Appendix goes over this exercise step by step; for brevity, I mention here that the
ex-ante expectation of time-1 utility is
V1j= max
σ−2
s,j
rW0j−F−ω−fj(σ−2
s,m) + 1
2σ−2
s,m +σ−2
z+s2σ−2
xD−1
ρ(W0j)(19)
where D= 1
t2s
t+σ−2
z+s2σ−2
x2!E(x2) + t2σ−2
z+s2σ−2
x+ 2ts(20)
Note that the objective function in (19) captures the information choice tradeoff. Higher
precision σ−2
s,j leads to higher asset management fees fj, thereby reducing ex-ante utility. On
the other hand, higher precision increases the posterior precision ˆσ−2
z,j , which increases the
time-1 expected squared Sharpe ratio E1η2
jand thus ex-ante utility.
Proposition 4 (Benefit of learning private information)
The benefit of learning private information, D, decreases as more investors search for in-
formed asset managers (i.e., as sincreases and prices become more informative), decreases
as the total risk-tolerance of investors who participate in the stock market increases (i.e., as
tincreases), and increases with the amount of noise in the economy (i.e., as nincreases).
Proof: See the Appendix. I show that ∂D
∂s <0, ∂ D
∂t <0, and ∂ D
∂n >0.
In other words, as prices reveal more information, acquiring costly private information
becomes less beneficial. Similarly, learning costly private information pays off when aggre-
gate risk-tolerance is low and investors are more risk-averse (holding less of the risky asset).
Noting that vsearch
1j−vparticip
1j=1
2
σ−2
s,mD
ρ(W0j), the price inefficiency is given by
θ=σ−2
s,mD
2=σ−2
s,m
2 E(x2) + t2(σ−2
z+s2σ−2
x)+2st
t2s
t+σ−2
z+s2σ−2
x2!(21)
Managers’ Endogenous Information Choice. A prospective informed manager
must pay a cost κ(ˆσ−2
s,m) to acquire information about the risky asset. On the other hand,
by becoming informed, the manager can expect to get more investors. A manager chooses
how much to learn, σ−2
s,m, by equating the cost of learning to the benefit of learning.
17
I assume a quadratic form for the cost of acquiring information about the risky asset:
κ(σ−2
s,m) = 1
2c0(σ−2
s,m)2+c1. The cost increases with the precision of the information learned.
This means that more precise information is more costly to acquire.
For a manager, the benefit of learning is the fee obtained from all the investors delegating
to that manager. There is a total mass one of managers, so M= 1, and the cost of learning
is κ(σ−2
s,m). Thus, in an interior equilibrium, the manager’s marginal benefit of learning (i.e.,
fees from extra delegating investors) has to equal the marginal cost of learning (i.e., marginal
cost of acquiring private information):
max
σ−2
s,m ZWmax
0
Wsearch
0
fjdB(W0,j )−κ(σ−2
s,m) (22)
[F OC :] ˜sD
4=κ0(σ−2
s,m) =⇒σ−2
s,m =˜sD
4c0
(23)
Proposition 5 (Optimal learning)
The managers’ optimal precision choice is given by σ−2
s,m =˜sD
4c0.
A manager’s precision choice naturally decreases with the costs of acquiring information
(i.e., with c0). The more expensive information is, the lower the precision acquired. A
manager’s precision is also a concave function of sand ˜s, and a convex function of n.
Theorem 1 (Informed investing outperforms uninformed investing)
1. Informed asset managers outperform uninformed investing (before and after fees).
Vsearch
j−fj≥Vparticip
j
2. Holding fixed other characteristics, wealthier investors who delegate their portfolios
(higher W0j) earn higher expected returns (before and after fees) and pay lower per-
centage fees on average.
Proof 1See the Appendix.
The fact that informed investing outperforms uninformed investing (before and after fees)
comes from the fact that investors must have an incentive to incur search costs to find
an informed asset manager and pay the asset management fees. Thus, investors who have
incurred the search cost can effectively predict asset manager performance. These results
rationalize why wealthier investors achieve higher risk-adjusted returns in the stock market,
and they may also explain why some exclusive funds, such as hedge funds, deliver larger
outperformance even after fees. I discuss the evidence further in Section (7).
18
Wealth Thresholds. WP articip
0is the level of wealth that makes an agent indifferent
between being a non-stockholder and a passive stockholder of any risky asset. It is given by
W2|σ−2
s=0 =rW P articip
0. This can be solved explicitly from the budget constraint, as shown
in the Appendix:
WP articip
0=2rρF
[(σ−2
z+s2σ−2
x)D]−1(24)
where I plugged in ρ(W0) = ρ
W0, and Dis defined in equation (20). The wealth threshold
for participating, WP articip
0j, increases with absolute risk-aversion ρ, and with the fixed entry
fee, F.
WNotInf ormed
0is the level of wealth that makes an agent indifferent between delegat-
ing to an informed manager and not participating at all, W2|σ−2
s=σ−2
s,m =rW N otInformed
0j.
This is given by two equations. The first represents the fact that the benefit of delegat-
ing has to equal the opportunity cost of delegating. The second is the manager’s opti-
mal precision level, σ−2
s,m =˜sD
4c0. Combining the two yields an implicit equation in σ−2
s,m:
W2|σ−2
s=σ−2
s,m =rW N otInformed
0.
Finally, the last object of interest is WSearch
0, which is the level of wealth that makes an
agent indifferent between delegating to an informed manager and participating uninformedly,
W2,j|σ−2
s,j =σ−2
s,m =W2j|σ−2
s,j =0. This can be deduced from the condition of the marginally dele-
gating investor, who is exactly indifferent between delegating his/her portfolio and investing
on his/her own, Udelegate
j−ω−fj=Udirectly
j:
WSearch
0,j =4ωρ
σ−2
s,mD(25)
Theorem 2 (Two categories of equilibria depending on parameters)
There are two categories of equilibria, as shown in Figure (7).
1. Equilibrium (a) displays three types of investors: non-stockholders, direct (uninformed)
investors, and delegating (informed) investors, as shown in Figure (7)(a). The condi-
tion for this equilibrium category is given by
0< W Par ticip
0j< W NotI nformed
0j< W Search
0j< W Max
0j+∞(26)
2. Equilibrium (b) displays only two types of investors: non-stockholders and delegating
(informed) investors, as shown in Figure (7)(b). The condition for this equilibrium
category is given by
0< W Search
0j< W NotI nformed
0j< W Par ticip
0j< W Max
0j+∞(27)
19
Proof 2See the Appendix.
Figure 7: Two Configurations of Equilibria by Investor Initial Wealth
(a) Equilibrium with three investor types (b) Equilibrium with two investor types
Legend: The y-axis represents value; the x-axis represents initial wealth. rW0j= the value of not
participating in the stock market, Vparticip = the value of participating uninformedly, and Vsearch = the
value of searching for and delegating to an informed manager. Equilibrium configuration (a) has non -
stockholders, and uninformed (direct) and informed (delegating) investors. Equilibrium configuration (b)
has only non - stockholders, and informed (delegating) investors. Which type of equilibrium occurs
depends on the magnitude of the fixed costs.
For the purpose of this paper, I choose the parameters of the model in such a way as
to study only the first equilibrium category. This second equilibrium is not the focus of
this paper because it does not reflect the fact that in reality, some investors acquire stocks
independently without learning any private information about them.
In the first equilibrium, poor investors with wealth lower than WP ar ticip
0jdo not trade at
all in the risky stock. Middle-class investors with wealth higher than WP ar ticip
0jbut lower than
WSearch
0jtrade on their own without acquiring any information about any risky asset. These
uninformed traders are equivalent to indexers in the sense that they buy stocks without
learning any private information about the stocks in the index. Lastly, relatively richer
investors, whose wealth exceeds WSearch
0j, delegate their portfolios to informed professional
asset managers. These wealthy investors end up being informed through their managers and
invest informedly.
5 Comparative Statics
In this section, I perform several comparative statics. The exercise assumes innovations
in financial technology lower (1) the cost of stock market participation, (2) the cost of
information acquisition, and (3) the cost of finding an informed asset manager one at a time.
All proofs are in the Appendix.
Theorem 3 (Lower participation costs improve participation, but hurt efficiency)
As the fixed costs of stock market participation fall,
20
1. Stock market efficiency falls, and prices become less informative.
ds
dF >0
2. The overall stock market participation rate increases.
dt
dF <0
3. The equity premium and the variance of returns fall.
dEP
dF >0; dV ar
dF >0
4. Asset management fees increase. dfj
dF <0
Proof 3See the Appendix.
Figure 8: Effects of Lower Participation Costs, F
The intuition is shown in Figure (8). As more investors enter the stock market, the
wealth threshold for participation shifts to the left. But now, each participating investor
holds a smaller portfolio. The equity premium falls to clear the asset market. The market
inefficiency grows, so managers now charge higher fees.
Hence, for the marginally delegating investor, his/her value of delegating to a manager
falls, as this investor now holds fewer assets in his/her portfolio on average. The marginally
delegating investor no longer delegates but prefers to invest independently without any
private information. Thus, the wealth threshold for search and delegation moves to the
right. Overall, lower participation costs lead to more participation, but lower information,
and higher fees.
Theorem 4 (Lower information costs hurt participation, but improve efficiency)
When information processing and acquisition costs fall,
1. Stock market efficiency rises, and prices become more informative.
ds
dk <0
21
2. The overall stock market participation rate decreases.
dt
dk >0
3. The equity premium and the variance of returns rise.
dEP
dk <0; dV ar
dk <0
4. Asset management fees decrease. dfj
dk >0
Proof 4See the Appendix.
Figure 9: Effects of Lower Information Acquisition Costs, κ
The intuition is shown in Figure (9). When information costs fall, managers’ fees
fall. This encourages more investors to delegate to informed managers. So, the wealth
threshold for searching for and delegating to a manager, moves to the left, and the overall
informativeness in the economy, s, increases. Market inefficiency is now lower.
But relatively low-wealth investors exit the stock market altogether (tdecreases) because
they no longer find it profitable to participate against a larger mass of high-wealth investors,
who are now benefiting from increased information. They are driving the price up, so
the marginal participating investor exits the stock market. Thus, the wealth threshold for
participation moves to the right, in the opposite direction. Overall, lower information costs
lead to less participation, but more information, and lower asset management fees.
Theorem 5 (Lower search costs hurt participation, improve efficiency)
When investors’ cost of searching for an informed asset manager falls,
1. Stock market efficiency rises, and prices become more informative.
ds
dω <0
2. The overall stock market participation rate falls.
dt
dω >0
22
3. The equity premium and the variance of returns rise.
dEP
dω <0; dV ar
dω <0
4. Asset management fees fall. dfj
dω >0
Proof 5See the Appendix.
Figure 10: Effects of Lower Search Costs, ω
The intuition is shown in Figure (10). When search costs fall, managers’ fees fall. This
encourages more investors to delegate to informed managers. So, the wealth threshold for
finding an informed manager, moves to the left, and the overall information in the economy,
s, increases.
As with lower information costs, relatively low-wealth investors exit the stock market
altogether (tdecreases) because they no longer find it profitable to participate against a
larger measure of high-wealth investors. Thus, the wealth threshold for participation moves
to the right, in the opposite direction. Overall, lower search costs lead to less participation,
but higher information, and lower asset management fees.
6 Extensions
6.1 Exogenous information structure and free-entry for managers
I will now derive the managers’ indifference condition, assuming free entry in the in-
dustry for asset management. Let Mbe the mass of informed managers. Free-entry implies
that managers make zero profits in equilibrium. In this version I assume that the managers
can decide to acquire private information in the form of one signal, but they cannot decide
how much information to acquire (i.e, they cannot acquire multiple signals or choose their
precision). Let the cost of acquiring information be a constant κ.
The equilibrium is given by portfolios {qj}, an asset price {p}, fees {fj}, a measure
of managers {M}, and wealth thresholds Wparticip
0and Wsearch
0such that:
23
1. Portfolio choices, qj, solve each investor’s portfolio maximization problem.
max
qj
Ej[W1,j ]−ρ(W0,j )
2V arj[W1,j ] (28)
s.t. W1,j =rW0,j −F−1[ω−fj]−qj(z−rp) (29)
2. Asset markets clear.
3. Management fees are a Nash outcome.
max
fj
(Vsearch
j−Vparticip
j−fj)fj(30)
4. Free entry of managers implies zero profits.
ZWparticip
0
Wsearch
0
fjdG(W0,j )/M −κ= 0 =⇒M=sD
4κ(31)
5. Investors optimally choose to participate (or not), and search for managers (or not).
max{Vnp, V particip, V search}(32)
where Vnp =rW0,j (33)
Vparticip =rW0,j −F+(σ−2
z+s2σ−2
x)D−1
2ρ(W0,j )(34)
Vsearch =rW0,j −F−ω−fj+(σ−2
s,m +σ−2
z+s2σ−2
x)D−1
2ρ(W0,j )(35)
Proposition 6 (Number of managers)
The number of managers is:
M=sD
4κ(36)
With free entry, the equilibrium for assets and asset management is given by the man-
agers’ and delegating investors’ indifference conditions. They denote the supply (free-entry
of managers) and demand for managers (searching investors’ indifference curve). Figure (11)
shows the equilibrium in the space (s, M). In an interior equilibrium, the two lines intersect
away from (0,0).
The red line is the investors’ indifference condition for searching and delegating to an
asset manager. When (s, M ) is above and to the left of the red line, investors prefer to
search for and delegate to asset managers because managers are attractive to find due to
the limited efficiency of the asset market. When (s, M) is below and to the right of the
red line, investors prefer to be uninformed, as the costs of searching for and delegating to
an informed manager outweigh the benefits of finding one. The green line is the managers’
indifference condition toward learning about the risky asset. When (s, M) is above the green
24
Figure 11: Equilibrium for assets and asset management
Legend: The red full line is the investors’ indifference condition between investing uninformedly and
searching for an informed manager. The green line is the managers’ indifference condition, that is the
free-entry condition.
line, managers prefer not to pay for information, since too many managers are seeking to
service investors. Below the green line, managers want to become informed.
Proposition 7(Equilibrium for asset managers)
The managers’ free-entry condition (i.e., the supply of managers) is hump-shaped because of
crowding out of information.
Note that sD is a concave function in s. When the measure of searching investors in-
creases from zero, the number of informed managers also increases from zero, since managers
are encouraged to earn the fees paid by searching investors. Mdepends on both sand D(s),
which is a decreasing function of s(as shown in Proposition 4). Initially, the increase in
sdominates the decrease in D(s). However, after a point, the decrease in D(s) dominates
the increase in s, hence the hump-shaped form of M. After a certain threshold, the fees a
manager gets decrease with the number of delegating investors. This is because informed in-
vestment increases market price informativeness and reduces the value of asset management
services. Hence, when so many investors have searched and delegated their portfolios that
the reduction in the benefit of acquiring information dominates (i.e., the reduction in D(s)
dominates), additional search and delegation decreases the number of informed managers.
Theorem 6 (Lower participation costs enlarge the asset management sector)
As the fixed costs of stock market participation fall,the number of informed asset managers
grows, and asset management fees increase.
dM
dF <0; dfj
dF <0
Proof 6See the Appendix.
The intuition is that, when participation costs fall, and more investors enter the stock
market, the value of searching for an informed manager for the marginal delegating investor
25
Figure 12: Effects of Lower Participation Costs, F
(in red) falls because he/she now is making a lower return on a lower portfolio. The incentives
to delegate to an informed asset manager fall with more uninformed wealth in the economy.
This shifts the indifference condition of a searching investor to the left. A fall in the costs
of participation, F, implies a higher number of managers in equilibrium, and less informed
wealth in the economy.
Asset management fees increase because with more uninformed wealth in the economy,
relative to the informed wealth in the economy, the value of informed asset management
rises. Investors prefer to search for and delegate to informed managers because of the limited
efficiency of the asset market. Thus, each manager is going to charge higher fees.
Theorem 7 (Lower information costs enlarge the asset management sector)
As the costs of acquiring private information fall, the number of informed managers in-
creases, and asset management fees decrease.
dM
dk <0; dfj
dk >0
Proof 7See the Appendix.
Figure 13: Effects of Lower Information Costs, κ
The intuition is that, lower costs of information acquisition shift the managers’ indiffer-
ence condition up because it is now easier for a manager to acquire information. This leads
26
to a higher number of managers in the interior equilibrium and more informed wealth in the
stock market, which increases asset price informativeness.
Fees fall, particularly because there is more informed wealth now in the economy. The
asset market becomes more efficient/informative, so asset managers can no longer charge
high fees for asset management. Asset management fees have to adjust, and they do so by
falling.
Theorem 8 (Lower search costs consolidate the asset management sector)
As the costs of searching for informed asset managers fall, the number of informed managers
decreases, and asset management fees decrease too.
dM
dω >0; dfj
dω >0
Proof 8See the Appendix.
Figure 14: Effects of Lower Search Costs, ω
The intuition is that, lower search costs incentivize more investors to search for informed
managers. So their indifference condition moves to the right, leading to a smaller number of
informed asset managers in the interior equilibrium.
The number of informed asset managers decreases, as in the figure, depending on the
location of the hump in the managers’ free entry condition. In this case, because of the
conditions of economies of scale in asset management, the intersection of the two indifference
curves will always occur on the downward part of the managers’ free entry curve. The
revenues of the asset management industry will rise, because there is more informed wealth
delegated to the asset management industry, but the fee each manager makes will fall,
because the asset market becomes very efficient/informed.
As search costs continue to fall, the informed asset management industry becomes in-
creasingly concentrated, with fewer and fewer informed managers managing more and more
informed wealth in this economy. This is what Garleanu and Pedersen (2018) call an ‘effi-
ciently inefficient’ outcome.
Surprisingly, the stock market and the asset management market become almost ef-
ficient, despite the presence of costly information acquisition. This fact is driven by the
27
intuition that, as search costs decline, investors esentially share the information cost more
efficiently. Indeed, the aggregate cost for information processing is κM , which decreases
towards zero as the informed asset management industry consolidates.
6.2 Financial Technology and The Final Wealth Distribution
In this section, I do a basic simulation of the model. I start with a log-normal initial
wealth distribution and plot the intermediary and terminal wealth distributions that result
from the model when the search and information effects dominate the participation effect.
That is, I plot a counterfactual of the US capital wealth distribution in the last 20 years
where, in the early 2000s, wealth was normally distributed.
Figure (15) shows that new information technologies skew the distribution to the right,
generating fat right tails, as observed in the data. It is important to note that the wealth
distribution is not staionary in this model and it diverges over time. In a fully dynamic
model, it can be rendered stationary with the help of a modelling trick such as cohorts dying
with a finitely positive probability each period.
Figure (15) shows that, far from creating a level playing field where more readily avail-
able information simply leads to greater market efficiency, innovations in financial technology
can have the opposite impact. They can create hard-to-access opportunities for long-term
alpha generation for those players with the scale and resources to take advantage of it. These
predictions are consistent with recent empirical evidence from the United States, as I will
show in Section 7.
Figure 15: Overall Effect of Financial Technology On The Wealth Distribution
Legend: The initial wealth distribution (blue full) is log-normal. The intermediary distribution (orange
dotted after 7 periods) and the final wealth distribution (red full after 10 periods) are skewed and exhibit
long and fat right tails. The wealth distribution diverges over time in this simple model.
The model generates a thick right tail of the capital wealth distribution, as is present
in the US data, which most economic models have a hard time matching. For example,
Bewley-Aiyagari economies, which focus on precautionary savings as an optimal response to
stochastic earnings, cannot produce wealth distributions with substantially thicker right tails
(larger top shares) than the labor earnings distribution that has been fed into the model.
This is explicitly noted by DeNardi et al. (2016) and Carroll et al. (2017), and by Hubmer et
28
al. (2016), who conclude that “the wealth distribution inherits not only the Pareto tail of the
earnings distribution, but also its Pareto coefficient. Because earnings are considerably less
concentrated than wealth, the resulting tail in wealth is too thin to match the data.” Other
papers add heterogeneous lifespans in overlapping generations models (assuming death rates
independent of age) to amplify wealth inequality, but these papers imply that a significant
fraction of agents enjoy counterfactually long lifespans. Gabaix et al. (2016) and Benhabib
et al. (2017a) argue that return heterogeneity is the most plausible ingredient to obtain a
Pareto tail for the capital income distribution.
In the United States, wealth is concentrated at the very top. Data from the U.S. Census
show that between 2000 and 2011, wealth increased for those in the top two quintiles and
decreased for those in the bottom three (see Table (1)). Other statistics from the Survey of
Consumer Finance and from Saez and Zucman (2016) show that in 2013, the top 1% held
30% of total wealth, and the top 10% of families held 76% of the wealth, while the bottom
50% of families held 1%. In 2016, the top 1% held 38.6%, and the top 10% of families held
90% of the wealth, while the bottom 50% of families held 0.5%. And while the majority
of net-worth holdings is in real estate, a significant portion is also held in the stock market
(either directly or indirectly).
Table 1: Changes in net worth for US households between 2000 and 2011
Quintile Median Net Worth (2000) Change by 2011
WQ1 −$905 −566%
WQ2 $14,319 −49%
WQ3 $73,911 −7%
WQ4 $187,552 +10%
WQ5 $569,375 +11%
Source: US Census.
6.3 Generalizations
The results of the model are robust to a number of generalizations and to other sources
of heterogeneity than wealth and risk aversion. The only requirement is that the dimensions
that differentiate agents create heterogeneity in their demand for stocks. For example,
differences in information costs, differences in age and lifespan, and differences in exposure
to background risk all affect the demand for risky assets relative to bonds and generate
similar results.
It is important to clarify differences between CARA and DARA preferences (which in-
clude CRRA). With CARA utility, wealth plays no role: It is irrelevant to the decisions
to participate or to acquire information (at the intensive and extensive margins). However,
wealth is highly relevant empirically. Lusardi et al. (2017) show that the decision to partici-
pate is significantly correlated with financial wealth. The probability of participation and the
29
proportion of wealth invested in stocks increase with wealth, mean income, and education
but decrease with the variance of income.
Thus, it is important to have a setting where financial wealth matters. CRRA prefer-
ences give relevance to wealth. But because there is no closed-form solution for equilibrium
in a CRRA setting (because the price is no longer a linear function of the payoff and supply),
I resort to a CARA approximation of CRRA preferences, using a strictly decreasing absolute
risk aversion.
The effects of lower search and information costs are obtained not only in my chosen
setting but also under CARA preferences. All that matters is the presence of a margin for
participation and the coexistence of three groups of stockholders: non-stockholders, indexers,
and (delegating) informed investors. Falling search and information costs benefit wealthy
investors, who acquire more stocks, but harm direct investors who invest uninformedly, who
now face a less advantageous risk-return tradeoff.
The major difference under CARA preferences is that the fixed entry cost no longer
generates an information effect. The demand for information is unrelated to the expected
supply of shares and to the market risk tolerance and thus to the level of participation in the
stock market. This means that the variance of returns always falls with lower entry costs.
With regards to preferences for early or late resolution of uncertainty, the results in
this paper are obtained by construction because of the mean-variance preference structure.
As search and information technologies improve, investors learn more about the stochastic
payoffs earlier. As the risk moves from the consumption to the trading period, there are
larger gains from trade in earlier periods. If traders were allowed to trade before the private
signals were observed, then the ex-ante utility would be linear with wealth, and the traders
would effectively be risk-neutral. The risk premium would equal zero.
Thus, the results of this paper are obtained by construction. Lower noise (i.e., better
information) does not attract investors in the stock market in this model because better
information reduces the gains from trade in the second period. Moreover, better information
can also reduce the risk premium in the first period.
The results are also obtained in a dynamic version where the stock return consists of both
the next period’s dividend and the stock’s resale price. The difficulty lies in the variance of
returns. When information technologies improve, current asset prices reflect future earnings
and prices more closely, thereby increasing price informativeness and reducing the return
variance, as in the static model. However, the volatility of future prices also rises because
future prices reflect dividends even further into the future. But because future prices and
earnings are discounted at a risk-free rate, the former effect dominates the latter.
In the words of Campbell et al. (2001), “better information about future cash-flows
increases stock price volatility, but reduces the volatility of the stock return because news
arrives earlier, at a time when the cash-flows in question are more heavily discounted”.
30
7 Interpreting the Data Through the Lens of the Model
In this section, I will offer some suggestive evidence of the model’s predictions. I do
not claim a causal effect, because I do not perform a causal testing of the model’s predic-
tions using micro data. The macro data shown are just indicative evidence of the model’s
mechanism.
The data seem to suggest that the early 2000s were a time of a technological U-turn in
financial markets. It seems that the effect of a decline in participation costs increased the
participation rate and decreased price informativeness before 2001. The year 2001 coincides
with the emergence of electronic trading. On the other hand, the data also suggest that
access to good information technologies has become more important since 2001.
7.1 Stock Market Participation and Price Informativeness
Implication 1 (Asset Market Efficiency from Theorems 3, 4, and 5)
Stock price informativeness falls when the participation effect dominates and rises when the
search and information effects dominate.
The model suggests that as participation costs decrease, the participation rate increases
due to a boom in uninformed investing opportunities. As a consequence, stock prices become
less informative. This is what we see in the data plotted in Figure (16) prior to 2001.
In the model, improvements in data technologies make uninformed investing a less
attractive option relative to informed investing. This increases stock price informativeness
but decreases participation. This is observable in the data plotted in Figure (16) after 2001.
Figure 16: US Stock Market Participation and Price Informativeness Over Time
a) Participation b) Price informativeness
Legend: Participation (weighted) is from SCF and includes direct and indirect holdings. Bai et al. (2016)
compute stock price informativeness by running cross-sectional regressions of future cash flows on current
market prices.
The model’s predictions, however, cannot qualitatively distinguish between the effects
of lower information costs and the effects of lower search costs with the data shown thus
31
far. Hence, we need to use other macro-financial variables, such as returns, equity premia,
informed asset management industry fees and concentration to be able to assign a dominating
cost to different time periods throughout the last 40 years in the US. The data shown next
provides additional support for the economic mechanism I propose.
Implication 2 (Price Informativeness from Theorems 3, 4, and 5)
In the cross-section, the price informativeness of stocks held by high-wealth investors should
rise by more than that of stocks held by less-wealthy investors.
In the cross-section, because high-wealth investors have access to better information
technologies through privately informed asset managers, the price informativeness of the
stocks they hold should rise relative to the price informativeness of stocks held by lower-
wealth investors.
Indeed, this prediction seems to hold in the data, as shown in Figure (17). The left hand-
size panel shows the U-shaped pattern in price informativeness over time. It is negatively
correlated with the share of stocks held by the wealthiest 20%. This means that it is indeed
the wealthy who are gaining access to most of the private information acquired in financial
markets.The right hand-side panel shows the price informativeness of stocks held by high
net-worth and low net-worth investors, which diverges after 2001.
This observation elicits an interesting research question in itself: What stocks do wealthy
investors hold? Begenau et al. (2018) argues that wealthy investors hold growth stocks, whose
prices are more informative about fundamentals, because the wealthy have access to better
information acquisition and processing technologies.
Figure 17: Aggregate and Cross-Sectional Dynamics of Price Informativeness
a) U-shaped Price Informativeness b) Price Informativeness by Net-worth
7.2 Hedge Fund Industry: Number and Asset Management Fees
Implication 3 (Hedge Funds Number and Fees from Theorems 3, 4, and 5)
The model predicts that a fall in the participation cost increases the number and the fees of
32
the sophisticated informed asset management industry. A fall in information costs increase
the number of managers, but decreases fund fees and expenses. A fall in the search costs
leads to a decrease in both the number and fees charged by informed managers.
Using data from Lipper, in Figure (18), I plot the number of hedge funds entering the
US asset management industry every year. Indeed, prior to 2001, the number of hedge funds
entering the US asset management industry was exploding until it reached a peak in 2005.
After 2005, this number tapered off and starting falling (because of either exits or mergers
and acquisitions). The important takeaway is that the hedge fund industry has become more
consolidated since 2005. Hedge fund fees follow a similar pattern.
The opposing predictions for the industrial organization of the hedge fund industry is
what allows me to separate information from search frictions in the period after 2001. As
Figure (18) shows, the effects of a lower information cost were dominant only between 2001-
2005, and since 2005, the effects of lower search costs (i.e., information about managers)
have been more dominant.
Figure 18: Entry in the hedge fund industry
a) Number of Hedge Funds (US) b) HF Incentive Fees Per Rate of Return
Source: Lipper TASS Hedge Fund Database.
While my model does not assume market power in the asset management industry in
order to keep the solution steps tractable pen and paper, it predicts that, as search costs
fall, the number of informed managers falls. Perhaps a useful extension of my model would
be to assume this market power in the asset management industry. We already know from
Kacperczyk et al. (2017) that large investors with market power trade strategically in order
to obscure their private information. This suggests that not only would capital income
inequality be amplified, but market efficiency would also fall, and prices would reflect less
information than in perfectly competitive markets.
7.3 Equity Premia
Implication 4 (The Equity Premium from Theorems 3, 4, and 5)
The equity premium falls when the participation effect dominates and rises when the search
and information effects dominate.
33
The equity risk premium is the price of risk in equity markets, and it is a key input
in estimating the costs of equity and capital in both corporate finance and valuation. The
model implies a falling equity premium when the participation effect dominates and a rising
equity premium when the information effect dominates.
At the one end of the spectrum, a fall in the entry cost of holding aggregate information
constant (i.e., sis constant) results in a falling equity premium. The participation effect
operates alone. As participation rises, the equity premium and the variance of returns fall.
At the other end of the spectrum, better information technologies result in a rising equity
premium. This is because some uninformed stockholders become informed (and the price
rises), but some uninformed stockholders exit the market altogether (and the price falls).
When many uninformed stockholders exit the stock market, the equity premium rises to
compensate the investors for lower risk-sharing in the market.
Empirically estimating the equity risk premium is complicated. In the standard ap-
proach, historical returns are used. The expected risk-premium is calculated as the difference
in annual returns on stocks versus bonds over a long period. There are limitations to this
approach, many discussed by Damodaran (2019), even in developed markets like the US,
which have long periods of historical data available. The main limitation of this approach
is that it generates backwards-looking equity premia that lean heavily on assumptions of
mean reversion and past data. Thus, in the Appendix, besides the historical premium, I
also plot the implied risk premium from various models of valuation, such as a free cash flow
to equity model (FCFE) and a dividend discount model (DDM) from Damodaran (2019). .
The U-shaped pattern of the equity premium, whether historical or implied, is robust to all
these different ways of measuring the premium.
Figure 19: The Equity Premium Fell Before 2001, Then Rose After 2001
Source: Damodaran (2019), Implied Equity Risk Premium.
The theoretical implication of the model finds support in the data. Prior to 2001, the
equity premium fell from over 8% in 1982 to 0% in 2001, as shown in Figure (19). The fall
in the equity premium is simultaneous with the rise in stock market participation in the US.
However, after the start of the new millennium – which is when electronic trading and other
financial information technologies emerged – the equity premium starting rising. Note that
34
the equity premium has always been positive and has not once in the last 40 years become
negative. This always implies that when some investors lose access to the equity premium,
they become worse off.
7.4 Capital Income Inequality
Implication 5 (Returns Increase With Wealth from Theorem 1.(ii))
Risk-adjusted returns increase with wealth.
A result of endogenous information acquisition in a CRRA setting is that wealthier
investors attain higher risk-adjusted returns (see Theorem 1(ii)). This is consistent with
the empirical household finance literature (Kacperczyk et al. (2018) for the United States,
Fagereng et al. (2016) and Di Maggio et al. (2018)) for Scandinavia, and Campbell et al.
(2018) for India).
Generating capital returns that increase with wealth is not a straightforward modeling
result. Chiappori and Paiella (2011) show that relative risk-aversion is constant. So, in the
data, it is not the case that as risk-tolerant wealthy investors take on higher-risk strategies,
they achieve higher returns on wealth. In my model, returns increasing with wealth arises
through an absolute risk-aversion channel in the context of information acquisition. This
happens because information has increasing returns to scale for wealthy investors, as they
have more capital invested in the stock market anyway. The lower the absolute risk-aversion,
the higher the incentive to find a manager with more precise information, and thus, the larger
the trading payoffs.
Unfortunately, portfolio level data for the US are not easily available, but most likely,
the pattern in the distribution of Sharpe ratios in the US is similar – if not starker – than the
one in the Norwegian population. Figure (20) plots the risk-adjusted returns (i.e., Sharpe
ratios) for the five different wealth quintiles of the Norwegian population. The Sharpe ratios
for the bottom two quintiles are negative. The third wealth quintile achieves a small, positive
Sharpe ratio less than 0.5. The fourth and fifth wealth quintiles achieve much larger risk-
adjusted returns of 1.2 and 2.6, respectively.
Figure 20: Sophisticated Investors Achieve Higher Risk-Adjusted Returns
Source: Fagereng et al. (2016) compute Sharpe ratios for all Norwegian individuals.
35
Ideally, one would want to see a time series of investors’ Sharpe ratios by wealth quintile.
My model suggests that the Sharpe ratio distribution has become more unequal in the last
20 years.
Implication 6 Manager Performance from Theorem 1. (i)
Informed investing earns higher returns (before and after fees) than passive investing.
The “old consensus” in the finance literature was that the average fund manager had
no skill and that managers underperformed by an amount equal to their fees. In the last few
years, a “new consensus” has emerged. Recent empirical evidence suggests that the average
alpha after fees is not negative but actually slightly positive (Berk and Binsbergen, 2015).
Moreover, a growing body of literature shows that evidence for the average asset manager
hides significant cross-sectional variation in manager skill among mutual funds, hedge funds,
private equity, venture capital funds, etc. Theorem 1 shows that there should be significant
cross-sectional differences in returns between and within investors and managers.
Evidence on the risk-adjusted returns attained by hedge funds is provided by Preqin and
AIMA (2018), Kosowski et al. (2007), Fung et al. (2008), and Jagannathan et al. (2010); on
private equity and venture capital by Kaplan and Schoar (2005); and on single and multiple
family-owned offices by UBS Surveys. Data from Preqin and AIMA (2018) show that hedge
funds have produced more consistent and steadier returns than equities or bonds over both
the short term and the long term, as shown in Table 21. Risk-adjusted returns, represented
by the Sharpe ratio, reflect the volatility of the returns as well as the returns themselves. The
higher the ratio, the better the risk-adjusted returns. The risk-adjusted return as measured
by the Sharpe ratio is calculated by subtracting the risk-free rate (typically the return on US
treasury securities) from the fund or index performance (returns, net of fees) and dividing
this by the fund or index’s volatility. The empirical analysis is based on the returns of more
than 2,300 individual hedge funds that report to Preqin’s All-Strategies Hedge Fund Index,
an equal-weighted benchmark. This hypothesis is verified: sophisticated/informed managers
beat stock and bond indices on a risk-adjusted basis at short- and long-term horizons.
Figure 21: Sophisticated Funds Achieve Higher Risk-Adjusted Returns
Legend: Sharpe ratios for hedge fund managers, the S&P 500 equity index, and the Bloomberg-Barclays
global bond index. Source: Preqin and AIMA (2018).
36
There is more evidence that hedge funds outperform net of fees. Kosowski et al. (2007)
(p. 2551) conclude that “a sizeable minority of managers pick stocks well enough to more
than cover their costs.” In the model, this outperformance after fees is expected as compen-
sation for investors’ search costs, but it is still puzzling in the light of the “old consensus”
that all managers deliver zero outperformance after fees (or even negative performance after
fees). Kosowski et al. (2007) add that “top hedge fund performance cannot be explained by
luck, and hedge fund performance persists at annual horizons. [. . . ] Our results are robust
and neither confined to small funds nor driven by incubation bias, backhill bias, or serial
correlation.”
Data on the excess returns of family-owned offices (FO) are less systematic because
these entities are not regulated and do not have to report their financial activities to regu-
lators. However, various market surveys of their activities suggest that FOs are informed,
sophisticated asset management companies and they make annual returns of between 17%
and 35% on a non risk-adjusted basis, which seems much higher than any market index (see
the Global Family Office Report by UBS and Campden Wealth).
Implication 7 (Capital Wealth Inequality from Theorems 3, 4, and 5)
Capital income and wealth inequality decelerate with the participation effect (i.e., which fa-
cilitates higher participation which increases access to the risk premium) and accelerate with
the information and search effects (i.e., which lower participation and access to the risk
premium).
The data suggest that prior to 2001, there was a large increase in participation. In the
model, the participation/risk-sharing effect allows more investors to uninformedly access the
equity premium. Thus, in the data, we should observe a deceleration of inequality before
2001. On the other hand, after 2001, the decrease in participation coupled with the dramatic
increase in stock price informativeness suggests that the information and search effects were
more important. Thus, after 2001, the data should show an amplification of inequality with
innovations in financial technology.
Indeed, the data plotted in Figure (22) shows that between 1980 and 2001, when the
decrease in the participation cost dominated the information and search effects, capital
wealth inequality increased little, from 67.1% to 69.2% between (i.e., a 3.1% increase). After
2001, when the data suggest that the decrease in information and search costs dominated
the effect from lower participation costs, inequality rose from 69.2% to 77.2% (i.e., an 11.5%
increase).
It is important to note the limitations of this back-of-the-envelope exercise. First, it
suffers from the limitation that a static model is repeated many times, with no optimization
across time. So, it can speak about capital income inequality, but it is too stripped-down of
many other features to realistically capture dynamics in capital wealth inequality. Yet, the
innovation and contribution of this paper is about the general equilibrium effects of financial
technology on capital income inequality and participation in the presence of a margin for
(non-)participation and a margin for delegation.
37
Figure 22: Capital Wealth Inequality Before and After 2001
Legend: Capital wealth inequality is measured as the difference between the capital share of the top 10%
and that of the bottom 90%. Source: Saez and Zucman (2016).
Second, there are other effects driving up capital wealth inequality both before and after
2001: taxes, regulation, globalization, trade liberalization, and antitrust policy. My model
does not capture all of these margins. I only capture a small effect of the rise in inequality
due to the tradeoff between participation and information. This is similar to evidence by Lei
(2019), who finds that information effects alone account for only 60% of the total increase
in inequality after 1980. My model generates more inequality than that of Lei (2019) by
definition because the general equilibrium effects are more complex and the existence of a
margin for participation amplifies inequality.
8 Policy Implications
Innovations often take on lives of their own, independent of their innovators’ wishes
and intentions, and although they may be created in good faith, the old adage is that “the
road to hell is paved with good intentions.” Albert Einstein is one inventor who came to
regret his inventions, or rather, dislike their use. He initially urged Roosevelt to support
research of what would eventually become the most destructive weapon ever constructed by
mankind. Years later, he regretted this, reportedly saying, “Had I known that the Germans
would not succeed in producing an atomic bomb, I would have never lifted a finger.” With
more innovations in the present than ever before, it is important to consider the impact
of these technological advances on the wider society and carefully think about their policy
implications.
The results of my model crucially depend on the coexistence of a margin for participation
and a margin for delegation to sophisticated fund managers. Policymakers should target
these two margins to ensure access to the equity premium while ensuring equitable access
to the risk premium as well.
One direct policy implication is that policymakers should try to reduce the fixed costs
of stock market participation and facilitate universal access to the internet, phones, and
38
computers. Clearly, the fixed costs of stock market participation (i.e., time and money
spent understanding how to start trading, as well as the fixed costs of installing e-trading
applications and accessing the internet and web applications that allow small investors to
trade) have been falling over the last 40 years. They reached their lowest point with the
start of e-trading technologies. Still, while e-trading allows investors to download an app or
just trade stocks through their internet browsers, there are still opportunity costs of doing
so, such as the costs of dealing with tax forms for investing activities, of understanding
different asset classes, etc. There is only so much a policymaker can do to decrease these
costs. There is evidence, however, that computer- and internet-using households raised their
stock market participation rates substantially more than non-computer-using households
after 2000, holding fixed characteristics such as access to 401Ks (see Bogan (2008)). The
increased probability of participation was equivalent to having over $27,000 in additional
household income (or over two more years of education). A conclusion of this study is that
policymakers should ensure greater access to computers and the internet.
Another direct policy implication is related to improving the financial education of US
households through academic education, but also money management workshops, ads, etc.
Lusardi et al. (2017) show that investors who have low financial literacy are significantly
less likely to invest in stocks. This non-participation phenomenon of less sophisticated/less
wealthy/less cultured households is an important part of the potential solution to the equity
premium puzzle. Mankiw and Zeldes (1991) were among the first to make this argument.
Vissing-Jorgensen (2002) continued to stress the importance of non-participation. And lim-
ited stock market participation matters not only for capital income inequality, but also for
other macro-financial effects. For example, limited market participation can amplify the
effect of liquidity trading relative to full participation. Under certain circumstances, with
limited participation, arbitrarily small aggregate liquidity shocks can cause significant price
volatility (see Allen and Gale (1994)). Low market participation also amplifies shocks and
makes markets more volatile because aggregate risk is concentrated in fewer participating
households (see Morelli (2019)). Thus, improving the financial education of less sophisti-
cated households in order to encourage their participation in the stock market is important
for a policymaker who cares not only about equality but also about financial stability.
Policymakers should also think carefully about designing policies that not only provide
information about stocks and mutual funds but also diminish informational asymmetries
between sophisticated and less sophisticated market players. King and Leape (1987) reported
that more than a third of US households did not own stocks or mutual funds in 1987 because
they did not know enough about them.
Financial education reduces the costs of information acquisition, but it is not clear
whether it also reduces asymmetries of information between wealthy and less-wealthy in-
vestors. Since the emergence of electronic trading in 2000, many major US financial services
firms have developed a sizeable online customer base, while other companies have focused
on providing stock information and financial analysis tools. They provide financial and in-
39
vesting data on stock prices, stock trends, corporate earnings, analysts’ advice and ratings,
etc. Retail investors, especially less wealthy ones, heavily utilize e-trading platforms. So,
these firms have indeed increased the amount of investment information available, provided
easier access to the market, and decreased transaction costs. The costs of e-trades are sub-
stantially lower than those of broker-assisted trades, the competitive presence of e-trade
brokerage firms has driven down the cost of broker-assisted trades, and other rates and fees
associated with stock purchases have declined (margin rates and service fees). Policymakers
should continue to encourage these developments.
At the same time, regulation should also focus on not allowing those large/wealthy
investors to take advantage of their scale and resources to extract “excessive” private infor-
mation while small and less sophisticated retail investors struggle to acquire this information.
Since the advent of big data and machine learning technologies, asset managers have been
increasingly turning to “alternative data” sources with the aim of staying ahead of the com-
petition, fueling superior client performance, and growing their customer base. These types
of strategies have exploded in recent months, and by 2020, the industry for alternative finan-
cial data is projected to be worth $350 million, with total alternative data spending in excess
of $1.7 billion. Data have become the “new oil,” and they are businesses’ most precious re-
source (see Farboodi et al. (2019)). Regulators should think about ways of regulating data
processing, data acquisition, and data dissemination in financial markets so that everyone
has equal access to it. This could be in the form of making datasets publicly available,
offering public advice, or, if this is not possible, outright preventing access to some types
of data. This is happening in Europe with the advent of the GDPR regulations that have
recently been adopted to strengthen and standardize the protection (anonymity) of personal
data. The main driver behind this regulation lies in the problematic nature of the complex
information management system, which results in the difficulty of governing big data. Data
need to be handled appropriately, and they need to be certified and compliant with local
and national laws in respect of both privacy and security management.
This paper predicts that an increase in information need not result in greater economic
welfare because information benefits the rich and hurts the poor. From a competition policy
perspective, influential depictions of less than perfectly competitive markets demonstrate
that an increase in rivalry can enhance both competitiveness and economic welfare. In these
markets, it is held that reductions in barriers to entry and exit or information barriers cannot
retard market performance. In other words, a reduction in these barriers is expected either
to cause a fall in market prices or at least to have no effect. This perspective has led to a
competition policy “rule of thumb” that a reduction in barriers should be one of the main
objectives (rather than a means) of competition policy (Mihet and Philippon (2019)). In
this paper, I have demonstrated that consideration of the tradeoff between information and
participation raises doubts about this conclusion. I have argued that reduction in information
costs, even if coupled with a reduction in participation costs, can still decrease participation.
Stock markets are a special kind of market in that better information technologies hurt some
40
investors because they allow the very rich to generate lots of alpha and leave behind poorer,
less sophisticated investors.
Thus, more efficient markets (i.e., more informative prices) should not necessarily be
the goal of financial and securities markets regulation. I have demonstrated that there is
a substantial tradeoff between information and participation. While the model lacks some
institutional details for tractability, I have shown that increased information can decrease
participation, further exacerbating the capital income/wealth inequality problem, and this is
evident in the data. Regulation should carefully balance this tradeoff. The SEC has already
imposed various regulations limiting the access of smaller investors to hedge funds. There
are also regulations limiting hedge funds’ ability to advertise their services (see Regulation
D). What this implies is that investment in hedge funds is simply designed to cater to sophis-
ticated and/or institutional investors. This regulation may be amplifying the information–
participation tradeoff that I have exposed in this paper and actually making things worse. In
the case in which hedge funds do open up to less sophisticated investors, regulation should
ensure that these funds provide a high degree of product transparency to protect investors’
interests.
Lastly, my model has implications for the organization of the asset management in-
dustry. In the model, the overall asset management industry faces statistically decreasing
returns to scale, as a larger amount of capital with informed managers leads to more efficient
markets (i.e., lower θ), which reduces manager performance. This implication is consistent
with the evidence of Pastor et al. (2014). It would be easy to extend the model to have hetero-
geneity in asset manager size, or in asset manager sophistication in processing information.
In that case, individual managers would not face decreasing returns to scale, controlling for
industry size, and indeed, larger and more sophisticated managers would be better off on av-
erage because searching investors look for informed managers. Thus, larger managers would
perform better, which is consistent with evidence from Ferreira et al. (2012). Meanwhile,
the asset management industry’s size grows when investors’ search costs or managers’ costs
of acquiring information fall. This phenomenon is consistent with evidence from Pastor et
al. (2014), Berk and Green (2004), ˇ
Lubos Pastor and Stambaugh (2012), and Garc´ıa and
Vanden (2009). Yet, as shown above, search and information costs have different impacts on
the concentration of the asset management sector. With a fixed number of managers, when
investors’ search costs fall, the number of managers falls, while the remaining managers grow
larger. Indeed, they become so much larger that the total revenue of the industry grows.
This consolidation of the asset management industry is important for regulators, particularly
in the context where it is known from the theoretical work of Kacperczyk et al. (2017) that
players with some market power have a large impact on prices and their informativeness.
41
9 Conclusion
Financial technology has been gathering lots of attention in recent years. While there
is plenty of hype around its social impact, it is not clear yet whether it can make a true
impact in the lives of the most financially vulnerable people. That is because financial
technologies are different from other technologies. For example, information has special
economic properties, such as nonrivalry. This nonrivalry implies that production possibilities
are likely to be characterized by increasing returns to scale and monopoly effects, insights
that have profound implications for economic growth, capital returns, and income and wealth
inequality.
Wealthy investors can afford to acquire costly private information about asset managers
and stock fundamentals. Once acquired, this private information allows them to earn higher
capital returns, which in turn makes them wealthier, putting them in a better position to
acquire even more private information. This unique property of information implies that
innovations that render private information cheaper can have unintuitive effects and exter-
nalities. These innovations can make it easier for the rich to chase and achieve high returns
and pull away from less-wealthy investors, who have little access to private information.
Less-wealthy, less sophisticated investors will stop trading risky assets because of their in-
formational disadvantage. Even when they pool their information resources, they still get
outcompeted by higher-wealth investors, who can pool better private information.
In future work, I would like to test my model using cross-sectional portfolio-level data.
Scandinavian countries provide such data from tax-related forms. Unfortunately, data for
US investors are difficult to obtain. But in principle, using portfolio-level data and statistics
related to mobile phone and internet use; financial education; use of online banking, broker-
age firms; e-trading apps; and asset management offices; one can test whether the model’s
predictions for the cross-section and time series hold in the data. For example, one could
test whether the wealthy have been achieving higher Sharpe ratios over time and investing
in riskier assets, or whether poorer investors have been retrenching from risky stocks into
safer assets. I base my model on aggregate trends, but the next step would be to obtain
more micro-level details about which, what and how investors trade.
The overall growth of investment resources and competition among investors with dif-
ferent wealth levels is generally considered a sign of a well-functioning financial market. This
paper highlights how advances in financial technologies also have consequences beyond the
financial market, affecting the distribution of income.
42
References
Abis, Simona, “Man vs. Machine: Quantitative and Discretionary Equity Management,” Working Paper
July 2017. retrieved from simonaabis.com on August 1, 2018.
Acemoglu, Daron and Pascual Restrepo, “Robots and Jobs: Evidence from US Labor Markets,”
Working Paper 23285, National Bureau of Economic Research March 2017.
Admati, Anat R. and Paul Pfleiderer, “Direct and Indirect Sale of Information,” Econometrica, 1990,
58 (4), 901–928.
Aghion, Philippe, Benjamin F. Jones, and Charles I. Jones,Artificial Intelligence and Economic
Growth, University of Chicago Press, 2019.
Allen, Franklin and Douglas Gale, “Limited Market Participation and Volatility of Asset Prices,”
American Economic Review, 1994, 84 (4), 933–55.
Aron, Janine and John Muellbauer, “The Economics of Mobile Money: harnessing the transformative
power of technology to benefit the global poor,” 2019. Oxford Martin School Policy Paper.
Arrow, Kenneth J., “The Demand for Information and the Distribution of Income,” Probability in the
Engineering and Informational Sciences, 1987, 1(1), 3–13.
Autor, David, David Dorn, Lawrence F. Katz, Christina Patterson, and John Van Reenen,
“The Fall of the Labor Share and the Rise of Superstar Firms,” Working Paper 23396, National Bureau
of Economic Research May 2017.
Azarmsa, Ehsan, “Information, Participation, and Passive Investing,” Working Paper, SSRN February
2019. retrieved from SSRN on October 24, 2019.
Bai, Jennie, Thomas Philippon, and Alexi Savov, “Have financial markets become more informative?,”
Journal of Financial Economics, 2016, 122 (3), 625–654.
Begenau, Juliane, Maryam Farboodi, and Laura Veldkamp, “Big data in finance and the growth of
large firms,” Journal of Monetary Economics, 2018, 97 (C), 71–87.
Benhabib, Jess, Alberto Bisin, and Mi Luo, “Earnings Inequality and Other Determinants of Wealth
Inequality,” American Economic Review, May 2017, 107 (5), 593–97.
, Jesse Perla, and Christopher Tonetti, “Reconciling Models of Diffusion and Innovation: A Theory
of the Productivity Distribution and Technology Frontier,” Working Paper 23095, National Bureau of
Economic Research January 2017.
Berk, Jonathan B. and Richard C. Green, “Mutual Fund Flows and Performance in Rational Markets,”
Journal of Political Economy, 2004, 112 (6), 1269–1295.
Bilias, Yannis, Dimitris Georgarakos, and Michael Haliassos, “Has Greater Stock Market Partici-
pation Increased Wealth Inequality in the Us?,” Review of Income and Wealth, 2017, 63 (1), 169–188.
Bogan, Vicki, “Stock Market Participation and the Internet,” Journal of Financial and Quantitative Anal-
ysis, 2008, 43 (01), 191–211.
Bond, Philip and Diego Garcia, “The Equilibrium Consequences of Indexing,” Working Paper 2018.
Brei, Michael, Giovanni Ferri, and Leonardo Gambacorta, “Financial structure and income inequal-
ity,” CEPR Discussion Papers 13330, C.E.P.R. Discussion Papers November 2018.
Calvet, Laurent E, Laurent Bach, and Paolo Sodini, “Rich Pickings? Risk, Return, and Skill in the
Portfolios of the Wealthy,” HEC Research Papers Series 1126, HEC Paris January 2019.
Campbell, John, Martin Lettau, Burton G. Malkiel, and Yexiao Xu, “Have Individual Stocks
Become More Volatile? An Empirical Exploration of Idiosyncratic Risk,” The Journal of Finance, 2001,
56 (1), 1–43.
Campbell, John Y, Tarun Ramadorai, and Benjamin Ranish, “Do the Rich Get Richer in the Stock
Market? Evidence from India,” Working Paper 24898, National Bureau of Economic Research August
2018.
Carroll, Christopher, Jiri Slacalek, Kiichi Tokuoka, and Matthew N. White, “The distribution of
wealth and the marginal propensity to consume,” Quantitative Economics, 2017, 8(3), 977–1020.
43
Chiappori, Pierre-Andr´e and Monica Paiella, “Relative Rrisk Aversion Is Constant: Evidence From
Panel Data,” Journal of the European Economic Association, 2011, 9(6), 1021–1052.
Claessens, Stijn, Thomas Glaessner, and Daniela Klingebiel, “Electronic Finance: Reshaping the
Financial Landscape Around the World,” Journal of Financial Services Research, Aug 2002, 22 (1), 29–61.
Damodaran, Aswath, “Equity Risk Premiums (ERP): Determinants, Estimation and Implications – The
2019 Edition,” Working Paper 3378246, SSRN 2019.
Davila, Eduardo and Cecilia Parlatore, “Trading Costs and Informational Efficiency,” Working Paper
25662, National Bureau of Economic Research March 2019.
DeNardi, Maria Cristina, Giulio Fella, and Gonzalo Paz Pardo, “The Implications of Richer Earn-
ings Dynamics for Consumption and Wealth,” Working Paper 21917, NBER 2016.
Dyck, Alexander and Lukasz Pomorski, “Investor Scale and Performance in Private Equity Invest-
ments,” Review of Finance, 2016, 20 (3), 1081–1106.
Ellis, Charles D.,The Index Revolution: Why Investors Should Join It Now, US: Wiley, 2016.
Fagereng, Andreas, Luigi Guiso, Davide Malacrino, and Luigi Pistaferri, “Heterogeneity and Per-
sistence in Returns to Wealth,” Working Paper 22822, National Bureau of Economic Research November
2016.
Farboodi, Maryam, Roxana Mihet, Thomas Philippon, and Laura Veldkamp, “Big Data and
Firm Dynamics,” AEA Papers and Proceedings, May 2019, 109, 38–42.
Favilukis, Jack, “Inequality, stock market participation, and the equity premium,” Journal of Financial
Economics, 2013, 107 (3), 740 – 759.
Ferreira, Miguel A., Aneel Keswani, Ant´onio F. Miguel, and Sofia B. Ramos, “The Determinants
of Mutual Fund Performance: A Cross-Country Study*,” Review of Finance, 04 2012, 17 (2), 483–525.
Fung, William, David A. Hsieh, Narayan Y. Naik, and Tarun Ramadorai, “Hedge Funds: Perfor-
mance, Risk, and Capital Formation,” The Journal of Finance, 2008, 63 (4), 1777–1803.
Gabaix, Xavier, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll, “The Dynamics of
Inequality,” Econometrica, 2016, 84 (6), 2071–2111.
Garc´ıa, Diego and Joel M. Vanden, “Information acquisition and mutual funds,” Journal of Economic
Theory, 2009, 144 (5), 1965 – 1995.
Garleanu, Nicolae and Lasse H. Pedersen, “Efficiently Inefficient Markets for Assets and Asset Man-
agement,” The Journal of Finance, 2018, 73 (4).
Grossman, Sanford and Joseph Stiglitz, “On the Impossibility of Informationally Efficient Markets,”
American Economic Review, 1980, 70 (3), 393–408.
Hubmer, Joachim, Per Krusell, and Anthony Smith, “The Historical Evolution of the Wealth Dis-
tribution: A Quantitative-Theoretic Investigation,” NBER WP 23011, National Bureau of Economic
Research 2016.
Jagannathan, Ravi, Alexey Malakhov, and Dmitry Novikov, “Do Hot Hands Exist among Hedge
Fund Managers? An Empirical Evaluation,” The Journal of Finance, 2010, 65 (1), 217–255.
Kacperczyk, Marcin, Jaromir Nosal, and Luminita Stevens, “Investor sophistication and capital
income inequality,” Journal of Monetary Economics, 2018.
, , and Savitar Sundaresan, “Market Power and Informational Efficiency,” Technical Report 2017.
Kaplan, Steven N. and Antoinette Schoar, “Private Equity Performance: Returns, Persistence, and
Capital Flows,” The Journal of Finance, 2005, 60 (4), 1791–1823.
and Joshua Rauh, “It’s the Market: The Broad-Based Rise in the Return to Top Talent,” Journal of
Economic Perspectives, September 2013, 27 (3), 35–56.
Kasa, Kenneth and Xiaowen Lei, “Risk, uncertainty, and the dynamics of inequality,” Journal of
Monetary Economics, 2018, 94 (C), 60–78.
Katona, Zsolt, Marcus Painter, Panos N. Patatoukas, and Jean Zeng, “On the Capital Market
Consequences of Alternative Data: Evidence from Outer Space,” Working Paper 3222741, SSRN 2018.
44
King, Mervyn A. and Jonathan I. Leape, “Asset Accumulation, Information, and the Life Cycle,”
NBER Working Papers 2392, National Bureau of Economic Research, Inc September 1987.
Kosowski, Robert, Narayan Y. Naik, and Melvyn Teo, “Do hedge funds deliver alpha? A Bayesian
and bootstrap analysis,” Journal of Financial Economics, 2007, 84 (1), 229–264.
Kyle, Albert S., “Continuous Auctions and Insider Trading,” Econometrica, 1985, 53 (6), 1315–1335.
Lei, Xiaowen, “Information and Inequality,” Journal of Economic Theory, 2019, 184, 104937.
Lester, Benjamin, Ali Shourideh, Venky Venkateswaran, and Ariel Zetlin-Jones, “Market-making
with Search and Information Frictions,” Working Paper 24648, National Bureau of Economic Research
May 2018.
Lusardi, Annamaria, Pierre-Carl Michaud, and Olivia Mitchell, “Optimal Financial Knowledge
and Wealth Inequality,” Journal of Political Economy, 2017, 125 (2), 431–477.
Maggio, Marco Di, Amir Kermani, and Kaveh Majlesi, “Stock Market Returns and Consumption,”
Working Paper 24262, National Bureau of Economic Research January 2018.
Malikov, George, “Information, Participation, and Passive Investing,” Working Paper, SSRN May 2019.
retrieved from SSRN on October 27, 2019.
Mankiw, N. Gregory and Stephen Zeldes, “The consumption of stockholders and nonstockholders,”
Journal of Financial Economics, 1991, 29 (1), 97–112.
Martinez, Joseba, “Automation, Growth and Factor Shares,” 2018 Meeting Papers 736, Society for Eco-
nomic Dynamics 2018.
Mihet, Roxana and Thomas Philippon, “The Economics of Big Data and Artificial Intelligence,”
Disruptive Innovation in Business and Finance in the Digital World (International Finance Review), Oct
2019, 20, 29–43.
Morelli, Juan, “Limited Asset Market Participation and Monetary Policy,” Working Paper 2019. retrieved
from jmorelli.com on Oct 15, 2019.
Pastor, Lubos, Robert F Stambaugh, and Lucian A Taylor, “Scale and Skill in Active Management,”
Working Paper 19891, National Bureau of Economic Research February 2014.
Peress, Joel, “Information vs. Entry Costs: What Explains U.S. Stock Market Evolution?,” Journal of
Financial and Quantitative Analysis, September 2005, 40 (03), 563–594.
, “The tradeoff between risk sharing and information production in financial markets,” Journal of Eco-
nomic Theory, January 2010, 145 (1), 124–155.
Philippon, Thomas,The Great Reversal: How America Gave Up on Free Markets, Harvard University
Press, 2019.
Piketty, Thomas,Capital in the Twenty-First Century, Harvard University Press, 2014.
Preqin and AIMA, “Hedge funds have outperformed stocks and bonds on risk-adjusted basis,” 2018. Press
Release, retrieved from aima.org on April 17, 2019.
Puschmann, Thomas and Rainer Alt, “Sharing Economy,” Business & Information Systems Engineer-
ing, Feb 2016, 58 (1), 93–99.
Saez, Emmanuel and Gabriel Zucman, “Wealth Inequality in the United States since 1913: Evidence
from Capitalized Income Tax Data *,” The Quarterly Journal of Economics, 2016, 131 (2), 519–578.
Stiglitz, Joseph, “Why learning matters in an innovation economy,” Jun 2014. retrieved from The Guardian
on September 10, 2019.
Sushko, Vladyslav and Grant Turner, “The implications of passive investing for securities markets,”
BIS Quarterly Review, March 2018.
van Rooij, Maarten, Annamaria Lusardi, and Rob Alessie, “Financial literacy and stock market
participation,” Journal of Financial Economics, 2011, 101 (2), 449 – 472.
Veldkamp, Laura L., “Information Markets and the Comovement of Asset Prices,” Review of Economic
Studies, 2006, 73 (3), 823–845.
Verrecchia, Robert E, “Information Acquisition in a Noisy Rational Expectations Economy,” Economet-
rica, 1982, 50 (6), 1415–30.
45
Vissing-Jorgensen, Annette, “Towards an Explanation of Household Portfolio Choice Heterogeneity:
Nonfinancial Income and Participation Cost Structures,” 2002. FED Working Paper 8884.
Ωˇ
Lubos Pastor and Stambaugh
ˇ
Lubos Pastor and Robert F. Stambaugh, “On the Size of the Active Management Industry,” Journal
of Political Economy, 2012, 120 (4), 740–781.
46
Online Appendix
November 10, 2019
Contents
1 Solution 2
2 No investor acquires information independently 10
3 Links between ICT and poverty and inequality 18
4 Real-world search and due diligence of asset managers 20
5 New information technologies make search and due diligence easier 24
6 Returns of hedge fund and family-owned offices 26
7 Participation 28
8 Equity Premium Evidence 29
9 Price informativeness by investor size and sophistication 30
1
1 Solution
Proposition 1. The optimal portfolio is given by: qdirectly
j=ˆµU
z,j −rp
ρ(W0j)ˆσU,2
z,j
for traders who trade on their
own as uninformed, and qdelegate
j=ˆµI
z,j −rp
ρ(W0j)ˆσI,2
z,j
for traders who delegate to informed managers.
Proof. Step 1. Solve for each investor’s optimal portfolio choice
max
qj
E(W2j|Ij)−ρ(W0j)
2var(W2j|Ij) (1)
s.t. W2j=rW0j−F−ω−fj+qj[z−rp] (2)
Given that: E(W2j|Ij) = rW0j−F−ω−fj+qj[ˆµz j −rp] (3)
var(W2j|Ij) = q2
jvar(z|Ij) = q2
jˆσ2
zj (4)
The investors’ portfolio problem can be expressed as:
max
qj
rW0j−F−ω−fj+qj[ˆµzj −rp]−ρ(W0j)
2q2
jˆσ2
zj (5)
FOC: ˆµzj −rp −ρ(W0j)qjˆσ2
zj = 0 =⇒qj=ˆµzj −rp
ρ(W0j)ˆσ2
zj
(6)
Step 2. Guess and verify. Guess a linear form for the price: rp =a+bz −cx.
Bayes’ Law for the investors who delegate to informed managers:
ˆσ2,I
zj =var(z|sj, p) = σ−2
z+b2
c2σ−2
x+σ−2
sj −1
(7)
ˆµI
zj =E[zj|sj, p] = µzσ−2
z+b
cµxσ−2
x+ (z−cx
b)b2
c2σ−2
x+sjσ−2
sj
ˆσ2
zj
=µzσ−2
z+b
cµxσ−2
x+zb
cσ−2
x−xb
cσ−2
x+sjσ−2
sj
hσ−2
z+b2
c2σ−2
x+σ−2
sj i(8)
For the investors who do not delegate to informed managers, but trade the risky asset, just have
σ−2
sj = 0 and sjdisappear from the equations:
ˆσ2,U
zj =var(z|p) = σ−2
z+b2
c2σ−2
x−1
(9)
ˆµU
zj =E[zj|p] = µzσ−2
z+b
cµxσ−2
x+ (z−cx
b
b2
c2σ−2
x)
ˆσ2
zji
=
=µzσ−2
z+b
cµxσ−2
x+zb2
c2σ−2
x−xb
cσ−2
x
hσ−2
z+b2
c2σ−2
xi(10)
2
Define objects that are useful going forward. Let tbe the total risk-tolerance of investors who par-
ticipate in the stock-market (indexers and learners). Let ibe the informativeness of the price implied by
aggregating the precision choices of learning investors.
t=ZWmax
0j
WP articip
0j
1
ρ(W0j)dj (11)
s=ZWmax
0j
WLearn
0j
1
ρ(W0j)σ2
sji
dj (12)
˜s=ZWmax
0j
WSearch
0j
1
ρ(W0j)dj (13)
n=s−1(14)
Proposition 2. The price of the risky asset is given by rp =a+bz −cx, where:
Proof. Market clearing implies that:
ZWSearch
0j
WP articip
0j
qdirectly
jdj
| {z }
Directly
+ZWmax
0j
WSearch
0j
qdelegate
jdj
| {z }
Search and delegate
=x
|{z}
Supply
(15)
tµz
σ2
z
+b
c
µx
σ2
x
+ (z−cx
b)b2
c2σ−2
x+sz −rp nσ−2
z+b2
c2σ−2
x+s=x
µz
σ2
z
+b
c
µx
σ2
x+zb2
c2σ−2
x+s
t−xb
cσ−2
x−1
t=rp σ−2
z+b2
c2σ−2
x+s
t(16)
Plugging things in and given that investors who learn private information are correct on average, I
can solve for rp in terms of zand xand then match the coefficients given that a+bz −cx =rp. This
gives the price coefficients as:
a=hs
t+σ−2
z+s2σ−2
xi−1µz
σ2
z
+sµx
σ2
x(17)
b=hs
t+σ−2
z+s2σ−2
xi−1s2σ−2
x+s
t(18)
c=hs
t+σ−2
z+s2σ−2
xi−1sσ−2
x+1
t(19)
Thus the price of the asset is:
rp =hs
t+σ−2
z+s2σ−2
xi−1µz
σ2
z
+sµx
σ2
x+s2σ−2
x+s
tz−sσ−2
x+1
tx(20)
=h−1µz
σ2
z
+sµx
σ2
x+s2σ−2
x+s
tz−sσ−2
x+1
tx
where h0=σ−2
z+s2σ−2
x(21)
h(s, σ−2
sj ) = σ−2
sj +h0=hσ−2
sj +σ−2
z+s2σ−2
xi(22)
3
¯
h=h(s
t) = hs
t+σ−2
z+s2σ−2
xi(23)
Step 3. Find the indirect utility.
Plug the optimal portfolio into terminal wealth and taking its time-2 expectation and variance
W2j=rW0j−F−ω−fj+ ˆµzj −rp
ρ(W0j)ˆσ2
zj ![z−rp] (24)
then E2(W2j) = rW0j−F−ω−fj+1
ρ(W0j)
(ˆµz j −rp)2
ˆσ2
zj
(25)
var2(W2j) = var ˆµzj −rp
ρ(W0j)ˆσ2
zj ×[z−rp]!=1
ρ2(W0j)
(ˆµz j −rp)2
ˆσ2
zj
(26)
Plugging into the indirect utility (ex-ante utility) gives:
U1j=E1E2(W2j|Ij)−ρ(W0j)
2var2(W2j|Ij)
=E1"rW0j−F−ω−fj+1
ρ(W0j)
(ˆµz j −rp)2
ˆσ2
zj −1
2ρ(W0j)
(ˆµz j −rp)2
ˆσ2
zj #
=E1"rW0j−F−ω−fj+1
2ρ(W0j)
(ˆµz j −rp)2
ˆσ2
zj #
=rW0j−F−ω−fj+1
2ρ(W0j)E1η2
j(27)
where ηj=ˆµzj −rp
qˆσ2
zj
.
So we want to compute E1hη2
ji=E1(ˆµzj −rp)2
ˆσ2
zj . This is perhaps the hardest step in this entire
exercise.
E1[η2
j] =
hσ−2
sj +σ−2
z+s2σ−2
xi
t2s
t+σ−2
z+s2σ−2
x2
E(x2) + t2σ−2
z+s2σ−2
x+ 2st−1 =
=hσ−2
sj +σ−2
z+s2σ−2
xi 1
t2s
t+σ−2
z+s2σ−2
x2!E(x2) + t2σ−2
z+s2σ−2
x+ 2st
| {z }
D
−1
=hσ−2
sj +σ−2
z+s2σ−2
xiD−1 (28)
4
To finish this section off, I obtained that the ex-ante time-1 utility is:
U1j= max
σ−2
sj
rW0j−F−ω−fj+1
2
E1hη2
ji
ρ(W0j)=
= max
σ−2
sj
rW0j−F−ω−fj+1
2σ−2
sj +σ−2
z+s2σ−2
xD−1
ρ(W0j)(29)
where D= 1
t2s
t+σ−2
z+s2σ−2
x2!E(x2) + t2σ−2
z+s2σ−2
x+ 2st(30)
simplifying gives: D=1
t2¯
h2E(x2) + t2h0+ 2st(31)
Notice that the objective function in the ex-ante utility captures the information choice trade-off. Higher
precision σ−2
sj leads to higher information acquisition costs κ(σ−2
sji ) which translate into higher fees, fj(σ−2
s,j ),
thereby reducing ex-ante utility. On the other hand, higher precision σ−2
sj increases the posterior precision
ˆσ−2
zj , the time-1 expected squared Sharpe ratio E1hη2
ji, and thus ex-ante utility.
How does D, the marginal benefit of learning private information change with sand t? Differentiating
D, and remembering that ¯
h=s
t+σ−2
z+s2σ−2
xgives ∂D
∂t <0 and ∂D
∂s <0, and ∂D
∂n >0.
Proposition 3. The benefit of learning decreases as prices become more informative (ie. as sincreases),
and decreases as the total risk-tolerance of investors who participare in the stock-market increases (ie. as
tincreases). It increases with the amount of noise in the economy (ie. as nincreases).
Proof.
∂D
∂t =
∂E(x2)+t2[σ−2
z+s2σ−2
x]+2st
t2[s
t+σ−2
z+s2σ−2
x]2
∂t =
∂t−2E(x2)+[σ−2
z+s2σ−2
x]+2st−1
[st−1+σ−2
z+s2σ−2
x]2
∂t =
=[−2t−3E(x2)−2st−2]¯
h2−[t−2E(x2) + σ−2
z+s2σ−2
x+ 2st−1]2¯
h(−2st−2)
¯
h4=
=−2
t2¯
h3×h(n−1E(x2) + s)(s
t+σ−2
z+s2σ−2
x)−(t−2E(x2) + σ−2
z+s2σ−2
x+ 2st−1)ii
=−2
t3¯
h3×E(x2)σ−2
z+s2σ−2
x−s2=−2
t3¯
h3×E(x2)σ−2
z+s2(E(x2)
σ2
x−1)=
=−2
t3¯
h3×E(x2)σ−2
z+s2(σ2
x+µ2
x
σ2
x−1)=−2
t3¯
h3×E(x2)σ−2
z+s2µ2
x
σ2
x<0 (32)
and ∂D
∂s =
∂E(x2)+t2[σ−2
z+s2σ−2
x]+2st
t2[st−1+σ−2
z+s2σ−2
x]2
∂i =
∂t−2E(x2)+[σ−2
z+s2σ−2
x]+2st−1
[st−1+σ−2
z+s2σ−2
x]2
∂s =
5
=(2sσ−2
x+ 2t−1)¯
h2−[t−2E(x2) + σ−2
z+s2σ−2
x+ 2st−1]2¯
h(t−1+ 2iσ−2
x)
¯
h4=
=2
¯
h3(iσ−2
x+t−1)(h0+st−1)−[t−2E(x2) + σ−2
z+s2σ−2
x+ 2st−1](t−1+ 2iσ−2
x)=
=−2
¯
h3t−3E(x2) + st−2+ 2st−2E(x2)σ−2
x+sσ−2
xσ−2
z+s3σ−4
x+ 3s2t−1σ−2
x<0 (33)
and ∂D
∂˜s<0 (34)
and, given n=s−1, implicitly gives: ∂D
∂n >0 (35)
I define the market inefficiency, θ, as
θ= (Udelegate
1j−Udirectly
1j)ρ(W0j)
. Note that Udelegate
1j−Udirectly
1j=1
2
σ−2
s,j D
ρ(W0j), so the price inefficiency is given by:
θ=σ−2
s,j D
2=σ−2
s,j
2 E(x2) + t2σ−2
z+s2σ−2
x+ 2st
t2s
t+σ−2
z+s2σ−2
x2!(36)
Step 4. Find the asset management fees
The asset management fee fjis set through Nash bargaining between an investor and a manager,
maximizng the product of the utility gains from agreement. If no agreement is reached, the investor’s
outside option is to invest uninformed on his own yielding a utility of
(rW0j−F−ω+Udirectly
1j)
. The utility of searching for another manager is
(rW0j−F−ω−fj+Udelegate
1j)
. For an asset manager, the gain from agreement is the fee fj, as the cost of acquiring information κ(.) is
sunk, and there is no marginal cost of taking on the investor.
Proposition 4. The asset management fee is given by fj. It increases with the level of market inefficiency
and with the investor’s initial wealth.
Proof.
max
fj
(Udelegate
1j−Udirectly
1j−fj)fj(37)
6
[F OC :] (Udelegate
1j−Udirectly
1j)=2fj
fj=Udelegate
1j−Udirectly
1j
2=θ
2ρ(W0,j )(38)
Plugging θin and rearranging terms gives the following expressions for asset management fees, fj:
fj=θ
2ρ(W0,j )=θW0,j
2ρ=σ−2
s,mD
4ρ(W0,j )=σ−2
s,mDW0,j
4ρ(39)
Note that
∂fj
∂θ =1
2ρ(W0j)>0 and (40)
∂fj
∂W0j
=θ
2ρ>0 (41)
The fee would naturally be zero if asset markets were perfectly efficient, so that investors had no
benefit from searching for an informed manager. In this setting, active asset managemenet fees can be
construed as evidence that retail investors believe that security markets are not fully efficient.
Step 5. Decision to search and delegate to a manager
An investor optimally decides to look for an informed asset manager, as long as the utility difference
from doing so is at least as large as the cost of searching and paying the asset management fee:
Udelegate
1j−Udirectly
1j≥ω+fj(42)
θ
ρ(W0j)≥ω+fj(43)
ω=θ
2ρ(W0j)(44)
Equation 44 must hold with equality for the marginal investor who is indifferent between searching
(and delegating to an informed asset manager) and investing on his/her own. Plugging θinto equation
44 and rearranging terms gives:
ω=σ−2
s,mD
4ρ(W0j)=⇒ρ(W0j) = σ−2
s,mD
4ω=⇒(45)
W0j=ρ−1 σ−2
s,mD
4ω!=⇒W0j=4ρω
σ−2
s,mD(46)
We will get back to this formula once we can also substitute out the precision σ−2
s,m.
7
Step 6. Find the optimal precision.
Proposition 5. The managers’ optimal precision choice is given by σ−2
s,m =˜sD
4c0.
Proof. Each manager has the option to choose how much information to learn. Let’s assume a concrete
form for the cost of acquiring information about the stochastic asset: κ(ˆσ−2
s,m) = 1
2c0(ˆσ−2
s,m)2+c1. As
mentioned before, the cost is increasing and convex in the precision of information learned. This means
that more precise information is more costly to acquire. In addition, no manager can acquire perfect
information because that would be too costly. The managers’ problem is thus to choose the posterior
precision ˆσ−2
s,m.
For a manager, the benefit of learning is the fee obtained from all the investors delegating to that
manager. There is a total mass one of managers, and the cost of learning is κ(σ−2
z,m). Thus, in an interior
equilibrium, the manager’s marginal benefit of learning has to equal his marginal cost of learning.
max
σ−2
s,m ZWmax
0j
WSearch
0j
fjdj −κ(σ−2
s,m) = ZWmax
0j
WSearch
0j
σ−2
s,mD
4ρ(W0j
dj −κ(σ−2
s,m) (47)
[F OC :] ZWmax
0j
WSearch
0j
D
4ρ(W0j
dj =κ0(σ−2
s,m) (48)
˜sD
4=κ0(σ−2
s,m) (49)
˜sD
4=κ0(σ−2
s,m) =⇒σ−2
s,m =κ0−1˜sD
4(50)
σ−2
s,m =˜sD
4c0
(51)
Alternatively, this can be written in terms of sas σ−2
s,m =qsD
4c0.
How does precision depend on aggregate risk-tolerance, informativeness and noise?
∂ˆσ−2
s,m
∂t <0 (52)
and ∂ˆσ−2
s,m
∂s <0 in the limit (53)
∂ˆσ−2
s,m
∂κ <0 (54)
and ∂ˆσ−2
s,m
∂n >0 (55)
Step 7. Derive the wealth thresholds.
I will now first derive the level of initial wealth at which agents enter stock-markets. The level of
wealth that makes an agent indifferent betweeen being a non-stockholder and a stock-holder of any risky
8
asset is given by:
VP articip
| {z }
W2j|(σ−2
sji =0)
=VN otP articip
| {z }
rW P articip
0j
(56)
The value of not participating is given by WP articip
0j, while the value of participating can be solved explicitly
from the budget constraint in ?? plugging in σ−2
sji = 0:
rW P articip
0j−F+1
20 + σ−2
z+s2σ−2
xD−1
ρ(WP articip
0j)=rW P articip
0j
[σ−2
z+s2σ−2
xD−1]
2ρ(WP articip
0j)=F
WP articip
0j=ρinv [σ−2
z+s2σ−2
xD]−1
2F!
WP articip
0j=2ρF
[σ−2
z+s2σ−2
xD]−1(57)
where I plugged in ρ(W0j) = ρ
W0jand W0j=ρinv(ρ
W0j) and where Dis defined above.
We can go even further if we plug in ˜s:
WP articip
0j=32ρF c2
0
[16c2
0σ−2
z+ ˜s4D2σ−2
xD]−16c2
0
For the marginal delegating investor, the following holds with equality
Udelegate
j−ω−fj> Udirectly
j
. We have shown before in equation 44 that plugging θinto equation 44 and rearranging terms gives:
ω=σ−2
s,mD
4ρ(W0j)=⇒ρ(W0j) = σ−2
s,mD
4ω=⇒(58)
W0j=ρ−1 σ−2
s,mD
4ω!=⇒W0j=4ρω
σ−2
s,mD=⇒(59)
W0j=16c0ρω
˜sD2(60)
Step 8. Extension: Managers Free-Entry
Let Mbe the number of active managers.
9
Proposition 6. The number of managers is given by
M=sD
4κ0(σ−2
s,m)
Proof. For an uninformed manager to enter, the expected extra fee revenue has to cover the cost of
information, sD
4M≥κ0(σ−2
s,m)
. This condition has to hold with equality for an interior equilibrium. To simplify the algebra, assume
the cost of acquiring information is linear κ=c1σ−2
s,m. Then,
sD
4M=κ0(σ−2
s,m (61)
sD
4M=c1σ−2
s,m (62)
M=sD
4c1σ−2
s,m
(63)
Proposition 7. The managers’ condition is hump-shaped because of crowding out of information.
Proof. Notice that sD is a concave function in s. When the number of searching investors increases from
zero, the number of informed managers also increases from zero, since managers are encouraged to earn
the fees paid by searching investors. Mdepends on both sand D(s), which is a decreasing function of s.
Initially, the increase in sdominates the decrease in D(s). However, after a point, the decrease in
D(s) dominates the increase in s, hence the hump-shaped form of M.
After a certain threshold, the fees a manager gets decrease with the number of delegating investors.
this is because active investment increases market efficiency, and reduces the value of asset management
services. Hence, when so many invesots have searched and delegated their portfolios that the reduction in
the benefit of acquiring information dominates (ie. the reduction in D(s) dominates), additional search
and delegation decreases the number of informed managers.
2 No investor acquires information independently
A plausible equilibrium in one in which investors do not learn private information on their own, but prefer
to delegate their investment to an asset manager. This implies:
ui
j−κ(σ−2
s,j )≤ui
j−ω−fj(64)
κ(σ−2
s,j )≥ω+θ
2ρ(W0j)(65)
ω=κ(σ−2
s,j )−θ
2ρ(W0j)=κ(σ−2
s,j )−σ−2
s,j D
4ρ(W0j)(66)
10
So, provided that ω≥κ(σ−2
s,j )−σ−2
s,j D
4ρ(W0j), an investor prefers using an asset manager to acquiring signals
singlehandedly.
Proofs: Asset management
Theorem 1. In a general equilibrium for assets and asset management:
1. Informed asset managers outperform uninformed investing (before and after fees).
udelegate
j−fj≥udirectly
j
2. Holding fixed other characteristics, wealthier investors who delegate their portfolios (higher W0j)
earn higher expected returns (before and after fees) and pay lower percentage fees, on average.
Proof. Part 1. follows from the fact that investors who match with informed managers choose to pay the
fee and invest with the manager rather than invest directly as uninformed. We know that
θ= (Udelegate
j−Udirectly
j)ρ(W0,j ), from definition of θ, and (67)
fj=θ
2ρ(W0,j ), from Nash bargaining (68)
Substituting into what we want to prove: Udelegate
j−Udirectly
j> fjgives θ
ρ(W0,j )>θ
2ρ(W0,j ), which is
obviously true.
Note that the indifference condition for the active delegating investor is Udelegate
j−fj−Udirectly
j=ω.
The outperformance is clearly larger if the equilibrium ωis larger.
Proof. Part 2. We want to compute the expected return on the wealth invested with an active manager,
under the assumption that all managers get investors with wealth higher than Wsear ch
0,j , and have absolute
risk-aversion ρ(W0,j) and relative risk-aversion ρ.
Given total wealth under management: Wm=RWmax
0,j
Wsearch
0,j
dj, the manager invests as an agent with
absolute risk-aversion ρm=ρ
Wm. It is clear that all investors with an informed managers achieve the
same gross excess return. The expected gross return is computed as the total dollar profit per capital
invested Wmusing the fact that the aggregate position is: qdelegate =E[z|s,p]−rp
ρmvar[z|s,p]=ˆµI
z−rp
ρmˆσ2I
z. The expected
gross return in then:
RI=1
2ρmE
ˆµI
z−rp
ρmpˆσ2I
z!2
=1
2ρmEη2I(69)
RU=1
2ρ(W0,j )E
ˆµU
z−rp
ρ(W0,j )pˆσ2U
z!2
=1
2ρ(W0,j )Eη2U(70)
11
The goal is to show that RI> RU. Note that the average risk-tolerance of investors delegating with
an active manager is also larger than the tolerance of an uninformed investor, 1/ρm>1/ρ(W0,j ). There
are two reasons why it holds. The first is better information, the second is lower risk. Better information
is because
E[( ˆµI
z−rp 2> E [( ˆµI
z−rp 2
and lower risk
E
1
pˆσ2I
z!2
> E
1
pˆσ2U
z!2
The second effect is not necessary for the result. As for the first effect, it follows immediately from
Jensen’s inequality, conditional on p.
It can also be easily seen given:
Eη2I=σ−2
s,m +σ−2
z+s2σ−2
xD−1 (71)
Eη2U=σ−2
z+s2σ−2
xD−1 (72)
It is clear Eη2I> E η2Ubecause σ−2
s,mD > 0, as σ−2
s,m >0, and D > 0.
Proposition 8. Consider now the expected return of an investor with a manager, conditional on investors’
characteristics. This is increasing in initial wealth.
Proof.
E[RI|W0,j , ρ(W0,j ), ω] = pr ρω ≤W0,j θ
2σ−2
s,m +σ−2
z+s2σ−2
xD−1
2ρm(73)
where pr ρω ≤W0,j θ
2increases with W0,j .
Percentage fees for a given investors are decreasing in W0,j too, because they are a fixed multiple of
ρω
W0,j which is decreasing in W0,j.
Proposition 9. In a general eequilibrium for asset managers:
1. Managers’ returns, before and after fees, and their average investor size covary positively.
2. Manager size and expected returns, before and after fees, covary positively. Similarly, managers with
a comparative advantage in collecting information, κm≤κm0earn higher expected returns before
and after fees.
Proof. Part 1. Asset managers are identical in this framework. We want to show that cov (Rm, W m)>0.
Rewriting,
cov (Rm, W m) = cov RI, W m=cov 1
2ρmEη2I, W m= (74)
12
=cov Wm
2ρEη2I, W m=Eη2
2ρ>0 (75)
Proof. finish part 2.
When participation costs fall
How do the wealth thresholds change with F?
∂W P ar ticip
0j
∂F =2ρ
[σ−2
z+s2σ−2
xD]−1>0 (76)
while ∂W S earch
0j
∂F =−2ρF
1<0 (77)
How does the equity premium and the variance of returns change with F?
The equity premium is given by:
Eq P r =µx
¯
ht =µx
s+tσ−2
z+s2tσ−2
x
Proof. While F does not enter directly in the formula for the equity premium, it indirectly affects it by
its effect on aggregate risk tolerance. A lower entry cost F implies a higher aggregate risk tolerance t,
which translates into a lower equity premium as
∂E qP r
∂t =∂µx
¯
ht
∂t =−µx(σ−2
z+s2σ−2
x)
s+tσ−2
z+s2tσ−2
x2
| {z }
+
<0 (78)
As for the variance of returns, plugging in the coefficients:
var(z−rp) = (1 −b)2σ2
z+c2σ2
x=
= 1−s2σ−2
x+s
t
s
t+σ−2
z+s2σ−2
x!2
σ2
z+sσ−2
x+1
t2
s
t+σ−2
z+s2σ−2
xσ2
x=
=σ−2
z
s
t+σ−2
z+s2σ−2
x2+sσ−2
x+1
t2σ2
x
s
t+σ−2
z+s2σ−2
x(79)
Proof. The derivative of the variance of returns is more complex:
∂var(z−rp)
∂t =2sσ−2
z
s
t+1
σ2
z−1
σ2
x3t2
+σ2
xσ2
zst+σ2
xσ2
z−2σ2
xt−sσ2
xσ2
z
t2[(σ2
z−σ2
x)t−sσ2
xσ2
z]2=
13
=−
2σ2
x1
t+s
σ2
x
s
t+1
σ2
z−1
σ2
xt2+
sσ2
x1
t+s
σ2
x2
s
t+1
σ2
z−1
σ2
x2t2
+2s
σ2
zs
t+1
σ2
z−1
σ2
x3t2
(80)
<0, verified numerically
This expression’s sign is ambiguous to solve pen and paper and depends on the magnitude of the param-
eters. In a simulation where all variances are equal to 1, and I assume that s= 0.5, then as tincreases
(while t > 0), the variance of returns decreases. This holds numerically for different values of s.
When research costs fall
How do the wealth thresholds change with information costs PN
i=1 κ(σ−2
sji )? This is harder to calculate,
but broadly, it is equivalent to calculating the change with respect to i, bearing in mind that a larger cost
κimplies a lower i.
∂W P ar ticip
0j
∂i =
∂2ρF
PN
i=1[(σ−2
z+i2σ−2
x)Di]−N
∂i =∂2ρF [PN
i=1[σ−2
z+i2σ−2
xDi]−N]−1
∂i (81)
=−2ρF [
N
X
i=1
[σ−2
z+i2σ−2
xDi]−N]∂i2σ−2
xDi
∂i (82)
=−2ρF [
N
X
i=1
[σ−2
z+i2σ−2
xDi]−N]iσ−2
x
| {z }
positive
(2Di+i∂Di/∂i)
| {z }
likelypositiv e
<0 (83)
Remember that ∂D/∂s < 0. Signing 2Di+i∂Di/∂i and plugging in ¯
h=i
t+σ−2
z+i2σ−2
xand h0=
σ−2
z+i2σ−2
xgives:
sign(2Di+i∂Di/∂i) = sign{2
t2¯
h3[E(x2
i)¯
h+t2h0¯
h+ 2in¯
h]−2i
t2¯
h3[n−1E(x2
i)+
+i+ 2iE(x2
i)σ−2
x+it2σ−2
xσ−2
z+i3t2σ−4
x+ 3i2nσ−2
x]}=
=sign{E(x2
i)(σ−2
z−i2σ−2
x)+3inσ−2
z+t2σ−4
z+i2t2σ−2
zσ−2
x+i2}= likely positive
This means that the higher the s(the lower the cost of info acquisition ), the higher the wealth threshold
for participation.
Changes in aggregate risk tolerance t and information s:
This section goes in the Appendix. In the first equilibrium category, WSearch
0> W P articip
0. Agents with
wealth between WPar ticip
0and WSearch
0participate, but do not acquire information. Hence, the total
risk-tolerance in the economy is t=RWMax
0
WP articip
0
1
ρ(W0j)dB(W0j) and the total information in the economy is
s=RWMax
0
WSearch
0
σ−2
s,m
ρ(W0j)dB(W0j).
Define n=s−1to be the total noise in this economy.
14
Differentiating the equation that defines the aggregate amount of information yields:
ds =−1
ρ(WSearch
0j)σ−2
s,m(WS earch
0j)b(WSearch
0j)dW Search
0j
| {z }
extensive margin
+ZWMax
0
WSearch
0
dσ−2
s,m(Wj)1
ρ(Wj)dB(Wj)
| {z }
intensive margin
(84)
where the first term is differentiating the upper bound in the integral, and the second term is differentiating
the integrand.
Differentiating the expression for WSearch
0jyields:
dW Search
0j=ρ−1(WSearch
0j)
ρ−1(WSearch
0j)0−∂D
D+1
κ0(0)
∂κ0(0)
∂k dk(85)
Given that (∂D)/(∂n)>0 while (∂D)/(∂t)<0, it has to be the case that WSearch
0jis decreasing in n
and increasing in t, holding kconstant. Differentiating the information choice ?? yields:
dσ−2
s,m =1
κ00(σ−2
s,m)"1
2rρ(Wj)∂D
∂n dn +∂D
∂s ds−∂κ0(σ−2
s,m)
∂k dk#(86)
where σ−2
s,m is increasing in nand decreasing in tholding kconstant. Finally, notice that dn =−n2ds.
Putting these elements together,
Atdt +Andn =Akdk (87)
where I ≡ ZWmax
0
WSearch
0
1
2rρ2(Wj)κ00(σ−2
s,m)dB(Wj) + σ−2
s,mb(WS earch)
ρ2(Wj)Dρ−1(WSearch
0j)0>0 (88)
At≡∂D
∂t s−2I<0 (89)
An≡1 + ∂D
∂n n2I>0 (90)
and Ak≡n2ZWmax
WSearch
0
∂κ0(σ−2
s,m)
ρ(Wj)κ00(σ−2
s,m)∂ k dB(Wj) + 2rn2∂κ0(0)
Dρ−1(WSearch
0j)0∂k
>0 (91)
Differentiating the equation that defines the aggregate tolerance in the economy yields:
dt =−1
ρ(WP articip
0j)b(WP articip
0j)dW P articip
0j(92)
Plugging in the threshold for participation ?? gives:
dW P articip
0j=−2rF
[σ−2
z+s2σ−2
xD−1]2(ρ−1(WP articip
0j))0h(σ−2
z+s2σ−2
x)∂D
∂n −2D
n3σ2
xdn+
15
+ (σ−2
z+s2σ−2
x)∂D
∂t dti+2r
[σ−2
z+s2σ−2
xD−1](ρ−1(WP articip
0j))0dF (93)
Plugging back leads to
Gtdt +Gndn =GFdF (94)
where Gt≡ − (ρ−1(WP articip
0j))0
ρ−1(WP articip
0j)b(WP articip)
((σ−2
z+s2σ−2
x)D−1)2
2rF + (σ−2
z+s2σ−2
x)∂D
∂t <0 (95)
Gn≡(σ−2
z+s2σ−2
x)∂D
∂n −2D
n3σ−2
x
>0 (96)
GF≡2r
(σ−2
z+s2σ−2
x)D−1>0 (97)
This is a system of two linear equations in two unknowns dt and dn. The solution is:
dt =AnGF
∆dF −GnAk
∆dk (98)
dn =−AtGF
∆dF +GtAk
∆dk (99)
where ∆ ≡AnGt−GnAt<0.
It remains to be shown that ∆ <0. For this, note that by replacing the coefficients with their
expressions, and dropping the term that appears in every relation (ρ−1(WP
0articip))0
ρ−1(WP
0articip)b(WP
0articip)>0
−Gt
Gn
>−∂D/∂t
∂D/∂n >−At
An
(100)
It sufficies to show that
−(σ−2
z+s2σ−2
x)∂D
∂t
h(σ−2
z+s2σ−2
x)∂D
∂n −2Dσ −2
x
n3i>−
∂D
∂t
∂D
∂n
>−
∂D
∂t n2I
1 + ∂ D
∂n n2I(101)
This inequality (101) is trivially satisfied since all these terms are positive: ∂D
∂n >0, and I>0, and
Gn≡h(σ−2
z+s2σ−2
x)∂D
∂n −2Dσ −2
x
n3i>0.
From the signs of the different coefficients it follows that:
dt
dk >0 (102)
dn
dk >0 =⇒ds
dk <0 (103)
dt
dF <0 (104)
dn
dF <0 =⇒ds
dF >0 (105)
16
Thus, informativeness increases if the search and information cost fall, and decreases if the entry cost
falls. Aggregate participation levels decrease if the search or information costs fall, and rise if the entry
cost falls.
17
3 Links between ICT and poverty and inequality
Main topic Question Result Authors
ICT, growth,
poverty
What is the role of ICT
for economic growth?
Investment in ICT promotes economic
growth
Pohjola (2001), Colecchia
& Schreyer (2002)
ICT reduces production costs and in-
crease output
Vu (2011, 2013)
ICT increases employment opportuni-
ties and demand
Datta & Agarwal (2004)
What is the role of ICT
in poverty reduction?
ICT reduces poverty reduction and
are powerful tool to access education,
health and financial services
Kenny (2002), Cecchini
& Scott (2003), Shamim
(2007), Warren (2007),
Bhavnani et al. (2008),
Sassi & Goaied (2013),
Pradhan et al. (2015)
ICT, financial
inclusion
What is the role of ICT
in promotion of finan-
cial inclusion?
They suggest favorable effects of ICT
for economic growth through financial
inclusion
Kpodar & Andrianaivo
(2011)
Does ICT/mobile
banking affect the
poor ?
ICT and mobile technology promote fi-
nancial inclusion particularly in rural
areas
Kendall et al. (2010),
Sarma & Pais (2011),
Mishra & Bisht (2013)
Mobile banking improves the economic
conditions of the poor
Mbiti & Weil (2011)
Mobile money technology affects en-
trepreneurship and economic growth
positively
Beck et al. (2015)
Access to fi-
nance and
poverty
Does access to finance
lower poverty and pro-
mote household wel-
fare?
Rich and wealthy households are more
likely to have a bank account in coun-
tries with higher foreign bank presence
Beck & Brown (2011)
Access to finance has a potential to re-
duce poverty and increase employment
in low income regions
Bruhn & Love (2014)
Socio-economic conditions can be im-
proved through advancing financial in-
clusion
Alter (2015)
Financial Access and
Inequality
Show a negative correlation between fi-
nancial access (bank account) and in-
equality
Honohan (2008), Park &
Mercado Jr (2015)
ICT and stock-
market partici-
pation
What is the role of ICT
for stock-market par-
ticipation?
A positive impact of access to and use
of Computer/Internet on stock market
participation.
Bogan (2008), Servon &
Kaestner (2008)
18
Financial literacy is significantly re-
lated to financial markets participation;
it also discourages informal borrowing.
Klapper et al. (2013)
Financial literacy and schooling attain-
ment have the positive effects on house-
hold wealth accumulation. It could
have much larger benefits for individ-
uals, firms, economy and government if
they invest more in financial literacy.
Van Rooij et al. (2011),
Behrman et al. (2010),
Thomas & Spataro (2015)
Impact of ICT on en-
trepreneurship
Online banking, behavior and banking
relations help reduce perceived finan-
cial problems for the entrepreneurs; im-
proves innovation and access to credit.
Han (2008), Ayyagari et
al. (2011), Dalla Pelleg-
rina et al. (2017)
How does ICT impact
risk and insurance?
Mobile money facilitates risk-
spreading. The geographic reach
of networks can enlarge. Timely
transfers can arrest serious declines
otherwise hard to reverse. More
efficient investment decisions can be
made, improving the risk and return
trade-off.
Jack and Suri (2011),
Aron and Muellbauer
(2019)
ICT and stock-
market effi-
ciency
How does technologi-
cal progress shape fi-
nancial markets?
ICTs make markets more efficient Farboodi and Veldkamp
(2019), Garleanu and Ped-
ersen (2018)
ICTs lower trading costs and improve
price informativeness
Davila and Parlatore
(2016)
ICTs reduce search and information
costs and improve the informativeness
of prices
Benabou and Gertner
(1993)
ICT penetration which eventually
enables (in particular) under-served
groups of the society to access financial
markets.
Claessens et al. (2002),
Kpodar & Andrianaivo
(2011), Anson et al.
(2013)
19
4 Real-world search and due diligence of asset managers
Garleanu and Pedersen (2018) have a very informative discussion of real-world search and due dilligence
of asset managers that I replicate entirely here.
While the search process involves a lot of details, the main point is that the process is time consuming
and costly. For instance, there exist more funds than stocks in the United States. Many of these funds
might be charging high fees while investing with little or no real information, that claim to be active
but in fact track the benchmark, or funds investing more in marketing than their investment process.
Therefore, finding a suitable fund is not easy for investors (just like finding a cheap stock is not easy
for asset managers). Here we provide an overview of the process to illustrate the significant time and
cost related to the search process of finding an asset manager and doing due diligence, but a detailed
description of these items is beyond the scope of the paper.
The search process for finding an asset manager is costly and time-consuming. Here are some consid-
erations:
•Retail Investors Searching for an Asset Manager.
–Online Search. Some retail investors search for useful information about investing online and
may make their investment online. However, finding the right websites may require significant
search effort and, once located, finding and understanding the right information on the website
can be difficult as discussed further below.
–Walking into a Local Branch of a Financial Institution. Retail investors may prefer to invest in
person, for example, by walking into the local branch of a financial institution such as a bank,
insurance provider, or investment firm. Visiting multiple financial institutions can be time
consuming and confusing for retail investors.
–Brokers and Intermediaries. Bergstresser et al. (2009) report that a large fraction of funds are
sold via brokers and study the characteristics of these fund flows.
–Choosing from Pension System Menu. Finally, retail investors get exposure to asset management
through their pension systems. In defined contribution pension schemes, retail investors must
search through a menu of options for their preferred fund.
•Searching for the Relevant Information
–Fees. Choi et al. (2009) (p. 1405) find experimental evidence that “search costs for fees matter.”
In particular, their study “asked 730 experimental subjects to allocate $10,000 among four real
S&P 500 index funds. All subjects received the funds prospectuses. To make choices incentive-
compatible, subjects expected payments depended on the actual returns of their portfolios over
a specified time period after the experimental session. . . . In one treatment condition, we
gave subjects a one-page ‘cheat sheet’ that summarized the funds front-end loads and expense
ratios. . . . We find that eliminating search costs for fees improved portfolio allocations.”
–Fund Objective and Skill. Choi et al. (2009) (p.1407) also find evidence that investors face search
costs associated with the funds’ objectives such as the meaning of an index fund. “In a second
treatment condition, we distributed one page of answers to frequently asked questions (FAQs)
about S&P 500 index funds. . . . When we explained what S&P 500 index funds are in the
FAQ treatment, portfolio fees dropped modestly, but the statistical significance of this drop is
marginal.”
20
–Price and Net Asset Value. In some countries, retail investors buy and sell mutual fund shares
as listed shares on an exchange. In this case, a central piece of information is the relation
between the share price and the mutual fund’s net asset value, but investors must search for
these pieces of information on different websites and often they are not synchronous.
•Understanding the Relevant Information.
–Financial Literacy. In their study on the choice of index funds, Choi et al. (2009) (2010, p.
1405) find that “fees paid decrease with financial literacy.” Simply understanding the relevant
information and, in particular, the (lack of) importance of past returns is an important part
of the issue.
–Opportunity Costs. Even for financially literate investors, the non-trivial amount of time it takes
to search for a good asset manager may be viewed as a significant opportunity cost given that
people have other productive uses of their time and value leisure time.
The search and due-dilligence costs for institutional/richer investors are also extensive.
•Finding the Asset Manager: The Initial Meeting.
–Search. Institutional investors often have employees in charge of external managers. These em-
ployees search for asset managers and often build up knowledge of a large network of asset
managers whom they can contact. Similarly, asset managers employ business development
staff who maintain relationships with investors they know and try to connect with other as-
set owners, although hedge funds are subject to nonsolicitation regulation preventing them
from randomly contacting potential investors and advertising. This two-way search process
involves a significant amount of phone calls, emails, and repeated personal meetings, often
starting with meetings between the staff members dedicated to this search process and later
with meetings between the asset manager’s high-level portfolio managers and the asset owner’s
chief investment officer and board.
–Request for Proposal. Another way for an institutional investor to find an asset manager is to issue
a request for proposal (RFP), which is a document that invites asset managers to “bid” for
an asset management mandate. The RFP may describe the mandate in question (e.g., $100
million of long-only U.S. large-cap equities) and all the information about the asset manager
that is required.
–Capital Introduction. Investment banks sometimes have capital introduction (“cap intro”) teams
as part of their prime brokerage. A cap intro team introduces institutional investors to asset
managers (e.g., hedge funds) that use the bank’s prime brokerage.
–Consultants, Investment Advisors, and Placement Agents. Institutional investors often use consul-
tants and investment advisors to find and vet investment managers that meet their needs. On
the flip side, asset managers (e.g., private equity funds) sometimes use placement agents to
find investors.
–Databases. Institutional investors also get ideas regarding which asset managers to meet by looking
at databases that may contain performance numbers and overall characteristics of the covered
asset managers.
•Evaluating the Asset Management Firm.
–Assets, Funds, and Investors. An asset manager’s overall AUM, the distribution of assets across fund
types, client types, and location.
21
–People. Key personnel, overall head count information, head count by ma jor departments, and
stability of senior people.
–Client Servicing. Services and information disclosed to investors, ongoing performance attribution,
market updates, etc.
–History, Culture, and Ownership. Year the asset management firm was founded, how it has evolved,
general investment culture, ownership of the asset management firm, and whether the portfolio
managers invest in their own funds.
•Evaluating the Specific Fund.
–Terms. Fund structure (e.g., master-feeder), investment minimum, fees, high water marks, hurdle
rate, other fees (e.g., operating expenses, audit fees, administrative fees, fund organizational
expenses, legal fees, sales fees, salaries), transparency of positions, and exposures.
–Redemption Terms. Any fees payable, lock-ups, gating provisions, whether the investment manager
can suspend redemptions or pay redemption proceeds in-kind, and other restrictions.
–Assets and Investors. Net asset value, number of investors, and whether any investors in the fund
experience fee or redemption terms that differ materially from the standard ones.
•Evaluating the Investment Process.
–Track Record. Past performance and possible performance attribution.
–Instruments. Securities traded and geographical regions.
–Team. Investment personnel, experience, education, and turnover.
–Investment Thesis and Economic Reasoning. The underlying source of profit, why should the invest-
ment strategy be expected to be profitable, who takes the other side of the trade and why, and
has the strategy worked historically?
–Investment Process. Analyzing the investment process and thesis is one of the most important parts
of finding an asset manager. What drives the asset manager’s decisions to buy and sell, what
is the investment process, what data are used, how is information gathered and analyzed, what
systems are used, etc.
–Portfolio Characteristics. Leverage, turnover, liquidity, typical number of positions, and position
limits.
–Examples of Past Trades. What motivated these trades, how do they reflect the general investment
process, and how were positions adjusted as events evolved.
–Portfolio Construction Methodology. How is the portfolio constructed, positions adjusted over time,
risk measured, position limits, etc.
–Trading Methodology. Connections to broker/dealers, staffing of trading desk, whether trading desk
operates 24/7, colocation on major exchanges, use of internal or external broker algorithms,
etc.
–Financing of Trades. Prime broker relations and leverage.
•Evaluating Risk Management.
–Risk Management Team.Team members, independence, and authority.
–Risk Measures. Risk measures calculated, risk reports to investors, and stress tests.
–Risk Management. How is risk managed, what actions are taken when risk limits are breached, and
who makes the decision.
22
•Due Diligence of Operational Issues and Back Office.
–Operations Overview. Teams, functions, and segregation of duties.
–Life cycle of a Trade. What steps does a trade make as it flows through the manager’s systems. Who
can move cash and how, and what controls are in place.
–Valuation. What independent pricing sources are used, what level of portfolio manager input is there,
what controls ensure accurate pricing.
–Reconciliation. How frequently and granularly are cash and positions reconciled.
–Client Service. Reporting frequency, transparency, and other client services/reporting.
–Service Providers. The main service providers used and any major changes.
–Systems. What are the major systems with possible live system demos.
–Counterparties. Who are the main counterparties, how are they selected, and how and by whom is
counterparty risk managed.
–Asset Verification. Some large investors will ask to speak directly to the asset manager’s administrator
to verify that assets are valued correctly.
•Due Diligence of Compliance, Corporate Governance, and Regulatory Issues.
–Regulators and Regulatory Reporting. Who are the regulators for the fund, summary of recent
visits/interactions, and frequency of reporting.
–Corporate Governance. Summary of policies and oversight.
–Employee Training. Code of ethics and training.
–Personal Trading. What is the policy, recent violations, penalty for breach.
–Litigation. What litigation has the firm been involved with.
–Cybersecurity. How are IT systems and networks defended and tested.
23
5 New information technologies make search and due diligence easier
New information technologies such as Big Data, Artificial Intelligence, and Machine Learning, have re-
duced the cost of storage, computation and transformation of data (Mihet and Philippon (2020)) and
have facilitated search and matching, and due dilligence activities.
Figure 1: The price of memory hard drives over time has fallen. Source: MKomo
While the Internet of Things has had an impact for two decades now, newer information technologies
have been increasing in popularity recently. Figure 2shows that interest in these new information tech-
nologies is at an all-time high.
What sets the current digital evolution apart and could lead to qualitative changes is the combination of
Big Data with Artificial Intelligence technologies to manipulate the data and extract relevant information
that is then used for searching, replicating, transporting, tracking, or verification purposes. Lower search
costs affect prices and price dispersion.
Figure 2: Numbers represent search interest relative to the highest point on the chart for the given time. A value of 100
denotes peak popularity for the term. A value of 50 means that the term is half as popular than at the highest point. Source:
Google Trends
They affect product variety and media availability, They change matches in a variety of settings, from
labor markets (Autor 2001), to asset markets (Barber and Odean 2001), to retail markets (Borenstein
24
and Saloner 2001, and Bakos 2001) to marriage markets.
They have led to an increase in the prevalence of platform-based businesses and affected the organi-
zation of firms (Jullien 2012, and de Corniere 2016).
Data storage costs have also fallen over time. This allows new technologies to filter and extract more
information than ever before at an ever lower cost.
Artificial intelligence, for example, is increasingly used for due dilligence purposes. It can automati-
cally search through a host of unstructured documents and contracts and extract essential content within
these documents for review.
AI works just like a human researcher - except that it sorts through documents and information re-
markably faster, reducing labor and opportunity costs. While AI technology can perform more tasks in
less time, it also ensures greater accuracy in reporting.
While it is harder to obtain data on the opportunity costs of time spent searching professional asset
managers, there is more precise data on the other side of the market: the advertising industry. Google
ads, for example, are getting cheaper and cheaper, as shown in the Figure 3.
Figure 3: Year-over-year change of the average cost per click on Google ads. Source: Statista
25
6 Returns of hedge fund and family-owned offices
The ”old-consensus” in the finance literature was that the average fund manager has no skill, but a ”new
consensus” has emerged that the average hides significant cross-sectional variation in manager skill among
mutual funds, hedge funds, private equity and venture capital (Garleanu and Pedersen (2018)).
Indeed, there is plenty of evidence that managers of hedge funds, single family-owned offices, and
multi-family-owned offices earn higher returns both before and after fees. I provide more details below.
Evidence on the risk-adjusted returns attained by hedge funds is provided by Preqin and AIMA (2018),
Kosowski et al. (2007), Fung et al. (2008), Jagannathan et al. (2010), on private equity and venture capital
by Kaplan and Schoar (2005), and on single and multiple family-owned offices by UBS SURVEYS.
Data from Preqin and AIMA (2018) shows that hedge funds have produced more consistent and
steadier returns than equities or bonds over both the short term and the long term as shown in Table 2.
Risk-adjusted returns, represented by the Sharpe ratio, reflect the volatility of the returns as well as the
returns themselves. The higher the ratio, the better the risk-adjusted returns.
The risk-adjusted return as measured by the Sharpe ratio is calculated by subtracting the risk-free
rate (typically the return on US treasury securities) from the fund or index performance (returns, net of
fees) and dividing this by the fund or index’s volatility.
The empirical analysis is based on the returns of more than 2,300 individual hedge funds that report
to Preqin’s All-Strategies Hedge Fund Index, an equal-weighted benchmark. Moreover, according to my
own analysis of the data, about 32% of all hedge funds produced double-digit returns in 2017, up from
about 23% in 2016.
Table 2: Hedge funds beat stock and bond indices on a risk-adjusted basis
Horizon Expert advisors S&P 500 BB global bonds
1-year 0.65 0.40 0.18
3-year 1.37 0.98 0.09
5-year 1.58 1.46 -0.24
10-year 0.73 0.41 0.13
The table shows the Sharpe ratios for hedge fund managers, the S&P 500 equity index, and the Bloomberg-Barclays global
bond index. Source: Returns data from Preqin and AIMA (2018)
There is also evidence that hedge funds outperform even net of fees. Kosowski et al. (2007) (p. 2551)
conclude that ‘a sizeable minority of managers pick stocks well enough to more than cover their costs’.
In the model, this outperformance after fees is expected as compensation for investors’ search costs,
but it is still puzzling in the light of the ”old-consensus” that all managers deliver zero outperformance
after fees (or even negative performance after fees). Kosowski et al. (2007) add that ‘top hedge fund
performance cannot be explained by luck, and hedge fund performance persists at annual horizons (...)
26
Our results are robust and neither confined to small funds nor driven by incubation bias, backhill bias,
or serial correlation.’
Data on the excess returns of family-owned offices (FO) is less systematic because these entities are
not regulated and do not have to report their financial activities to regulators.
However, various market surveys of their activities suggest that FOs are active asset management
companies and they make annual returns of between 17% −35%, which is much higher than any passive
index (see Global Family Office Report by UBS and Campden Wealth).
Leon Cooperman, the owner of Omega Advisors and a Wall Street superstar is often quoted as saying
that ”The billionaires of this world have not become rich by chasing the S&P 500”. According to the
Economist, family-owned offices invest in high-risk, high-returns assets (consistent with the predictions
of the model).
The Economist reports that ‘FOs are embracing sectors as diverse and risky as cannabis, e-sports, and
crypto investing’. Lastly, a significant portion of FO’s portfolios consists of directly held private equity,
which is totally inaccesible to poor investors.
Family offices are generally established by attracting talented wealth and asset managers from mutual
funds. While there is no publicly available data set detailing the positions and the returns of family-owned
offices, surveys put the annual returns at an average as high as 35% per year.
The exclusive active asset management industry is subject to many frictions, however, since investors
must search for informed managers able to deliver superior returns.
27
7 Participation
Below I plot various measures of stock-market participation. The data comes from SCF, the Gallup
surveys, and Lettau et al. (2019). While the measurements differ from one series to another according to
the data source, all three time-series exhibit the inverted-U shape pattern I focus on matching.
Figure 4: US Stock market participation rates from various surveys and different ways of measuring
participation
28
8 Equity Premium Evidence
The equity premium fell before 2001, then rose with the information revolution, independently of whether
it is a historical measure or an implied measure.
Figure 5: Historical Equity Risk Premium
Source: Morningstar.
Figure 6: Implied Equity Risk Premium,
DDM method
Source: Damodoran (2019).
Figure 7: Implied Equity Risk Premium,
FCFE method
Source: Damodoran (2019).
29
9 Price informativeness by investor size and sophistication
Data sources: I follow Bai et al. (2016) to construct measures of price informativeness from 1980-2014.
I combine several firm level panel datasets, all of which are available for download on WRDS. The main
sample is Compustat accounting variables. Stock prices are obtained from CRSP. Institutional owner-
ship comes from 13-F filings that require all institutional organizations to file a report on the number of
institutional owners, the number of share issued and the percentage of outstanding shares held by each
institution (my key measure of institutional ownership). The GDP deflator used to adjust for inflation is
from the BEA. Stock prices are taken at the end of March, and accounting variables as of the end of the
previous fiscal year, typically December. This timing convention ensures that market participants have
access to the accounting variables that are used as controls.
Sample selection: I consider both the entire universe of Compustat firms and the S&P500 firms.
These firms represent more than 80 percent of the American equity market by capitalization, and they
are large-cap companies that have been around for most of the period studied. Their characteristics have
remained remarkably stable, which makes them comparable over time. In this way, I do not have to worry
about composition effects (about new firms that are very volatile and hard to price entering the market).
Moreover, I do not have to worry about firm size driving the effects, as these firms are all large in terms
of their market-capitalization.
Measure of price informativeness: Similar to Bai et al. (2016), I correct for delisting (to ensure
that the measure of price informativeness is free of survivorship bias), and for inflation (because I am
interested in real price informativeness changing over time.) The main equity valuation measure is the log-
ratio of market capitalization to total assets, logM/A. The main cash flow variable is earnings measured
as EBIT. I scale EBIT by current total assets, such that EBI T /A. Now, in a forecasting regression
for earnings with horizon h= 1,3 and 5 years, the left-side variable is EBITt+h/At. To construct the
measure of price informativeness, I run cross-sectional regressions of future earnings on current market
prices. I include current earnings and industry sector as controls to avoid crediting markets with obvious
public information. Specifically, in each year t= 1980, ..., 2014 and for every horizon h= 1,3,5, I run:
EBITi,t+h
Ai,t
=at,h +bt,hlog Mi,t
Ai,t
+ct,h
Ei,t
Ai,t
+ds
t,hIs
i,t +i,t,h
where iis the firm index and Is
i,t a sector (one-digit SIC code) indicator. These regressions give a set
of coefficients indexed by year tand horizon h. From here, price informativeness is calculated as the
predicted variance of future cash flows from market prices. I compute it here with a change of taking its
square, which gives meaningful units. From the regression above, price informativeness in year tat horizon
his the forecasting coefficient bt,h multiplied by σt(log(M/A)), the cross sectional standard deviation of
the forecasting variable logM/A in year t. This is the measure of price informativeness over time.
pVF P E t,h =bt,h ×σt(log(M/A))
Trends over time: Plotting this trend over time for the universe of Compustat firms shows a strong
U-shaped pattern of price informativeness since the 1980s. This pattern is robust across industries, even
when including or excluding finance and real estate firms.
30
Figure 8: Stock Price Informativeness Over Time
31
References
Bai, Jennie, Thomas Philippon, and Alexi Savov, “Have financial markets become more informative?,” Journal of
Financial Economics, 2016, 122 (3), 625–654.
Bergstresser, Daniel, John M. R. Chalmers, and Peter Tufano, “Assessing the Costs and Benefits of Brokers in the
Mutual Fund Industry,” The Review of Financial Studies, 05 2009, 22 (10), 4129–4156.
Choi, James J., David Laibson, and Brigitte C. Madrian, “Why Does the Law of One Price Fail? An Experiment
on Index Mutual Funds,” The Review of Financial Studies, 11 2009, 23 (4), 1405–1432.
Fung, William, David A. Hsieh, Narayan Y. Naik, and Tarun Ramadorai, “Hedge Funds: Performance, Risk, and
Capital Formation,” The Journal of Finance, 2008, 63 (4), 1777–1803.
Garleanu, Nicolae and Lasse H. Pedersen, “Efficiently Inefficient Markets for Assets and Asset Management,” The
Journal of Finance, 2018, 73 (4).
Jagannathan, Ravi, Alexey Malakhov, and Dmitry Novikov, “Do Hot Hands Exist among Hedge Fund Managers?
An Empirical Evaluation,” The Journal of Finance, 2010, 65 (1), 217–255.
Kaplan, Steven N. and Antoinette Schoar, “Private Equity Performance: Returns, Persistence, and Capital Flows,”
The Journal of Finance, 2005, 60 (4), 1791–1823.
Kosowski, Robert, Narayan Y. Naik, and Melvyn Teo, “Do hedge funds deliver alpha? A Bayesian and bootstrap
analysis,” Journal of Financial Economics, 2007, 84 (1), 229–264.
Mihet, Roxana and Thomas Philippon, “The Economics of Artificial Intelligence and Big Data,” International Finance
Review, 2020. Forthcoming.
Preqin and AIMA, “Hedge funds have outperformed stocks and bonds on risk-adjusted basis,” 2018. Press Release,
retrieved from aima.org on April 17, 2019.
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