Effect of Increasing Horizontal Shareholding with
Index Funds on Competition and Market Prices –Investigation by Agent-Based Model –Takanobu Mizuta
SPARX Asset Management Co. Ltd.,Tokyo, Japan
Email: mizutata@gmail.com
http://www.mizutatakanobu.com/
Abstract—Recently, some empirical studies have argued that“horizontal shareholding” (or “common ownership”) lessens
competition among companies and prevents industries fromgrowing. In particular, index funds horizontally shareholding
have become the largest shareholders and has become a subjectof greater debate. In this study, I built an artificial market model,
a kind of agent-based model, and investigated the effect of increasing horizontal shareholding with index funds on competition
and market prices. My result shows that even when the holding
ratio of index funds is not that high, the funds lessen competition.
Moreover, when the value of a company successful at competition
grows, its market price grows more than the company value(overshoots) and the company becomes overvalued; then, thenumber of shareholders who encourage competition decreases
and the company loses competitive power. Alternately, when thevalue of a company unsuccessful at competition drops, its market
price falls deeper than the company value (overshoots) and thecompany becomes undervalued; then, the number of shareholders
who encourage competition increases, and the company gains
competitive power. My simulation result indicated that such amechanism balances competitive powers among corporations.
Growing index funds may possibly weaken this mechanism.

I. I NTRODUCTION
Recently, some empirical studies have argued that “horizontal shareholding” (or “common ownership”), in which
an investor holds all companies that are competitors in one
industry, lessens competition among companies and prevents
industries from growing [1]–[3]. Usually, investors encourage
a company to compete with other companies because investors
earn more when the stock price of the company rises if
the company is successfully competitive among others in theindustry. However, horizontal shareholders have different economic incentives from investors holding one company becauseeven if one company that a horizontal shareholder holds is successfully competitive, the investors always also hold the losing
companies. Therefore, they earn from the winning company
but always also take a loss from the losing companies. For
horizontal shareholders, the incentive to encourage holding
companies to compete is very weak. Moreover, it seems that
there are some cases in which they can earn more if they
make holding companies avoid competition through reducing
the prices of goods to keep the profit margin of the companies.

In particular, index funds horizontally shareholding, where amanager expects a return the same as a market index, e.g., theDow Jones industrial average, and replicates the componentsof the index, have become the largest shareholders and has
become a subject of greater debate because, recently, the assets
of index funds have increased heavily [1].
In the United States, the index fund industry is dominatedby the biggest three asset management companies, and they
constitute the largest shareholder with 40% of all listed
United States companies and hold 15% of all shares of listedcompanies [1]. This means they are horizontally shareholding
heavily. Azar et al. [2] used 14 years of quarterly panel data
to estimate that airline ticket prices in the United States are
3–7% higher because of horizontal shareholding, comparedwith a counterfactual world in which companies are separately
owned. Fichtner et al. [1] analyzed data on shareholder voting
and argued that shareholders tend to influence the managementof companies through private engagements rather than public
discussion, such as shareholder voting.
Elhauge [3] argued that recent horizontal shareholdings
are already illegal under current antitrust laws, and suchshareholding can help explain why economic inequality hasrisen in recent decades as shown by Piketty [4]1 . He also
recommended that big funds should be restricted to investingin only one company for each industry.
However, there are many arguments against this [5], [6],and there are also answers to the arguments against [7], so
no conclusion has been reached yet. Moreover, such empirical studies cannot make conclusions because they cannot
handle very complex mechanisms, such as how horizontal
shareholding affects business strategies and market prices.
In particular, they cannot directly investigate the “positive
feedback” process, as Fig. 1 shows, such as how stock tradingchanges the shareholder composition. The trading leads tochanges in the business strategy of companies, the intrinsic
value of companies (fundamental value), the stock price,and traders’ decisions. Such discussion on the mechanism of
positive feedback is very difficult when using only the results
1 The reason Piketty [4] gives for the rise in economic inequality differsfrom horizontal shareholding.

of empirical studies. Empirical studies cannot be conducted
to investigate situations that have never occurred in actual
financial markets, such as ones in which index funds are
more than present. Furthermore, because so many factors
cause price formation in actual markets, an empirical study
cannot be conducted to isolate the direct effect of changing
the distribution of investor types on price formation.
An artificial market model, which is a kind of agentbased model, is the only way to directly investigate positive
feedback. The model can also handle situations that havenever occurred, such as ones in which index funds are morethan present, and can isolate the direct effect of changing
the distribution of investor types on price formation. These
are strong advantages for an artificial market simulation. The
effects of distribution and of several changing regulations havebeen investigated by using these simulations [8]–[11].
Not only academies but also financial regulators and stock
exchanges have recently become interested in agent-basedmodels, such as artificial market models for investigating
financial markets. Indeed, an article in Science by Battiston
et al. [12] stated that “since the 2008 crisis, there has been
increasing interest in using ideas from complexity theory(using network models, agent-based models, and so on) tomake sense of economic and financial markets,” and an article
in Nature by Farmer and Foley [13] stated that “such (agentbased) economic models should be able to provide an alternative tool to give insight into how government policies could
affect the broad characteristics of economic performance, by
quantitatively exploring how the economy is likely to react
under different scenarios.”
Many studies have investigated the effects of kinds ofinvestors on price formation and the effects of several changing
regulations and rules by using artificial market simulations[10]. Indeed, the effects of market makers and index funds
were investigated by using these simulations [14]. Mizuta andHorie [15] modeled agents who reflect the characteristics of investors who trade infrequently on the basis of the fundamental
price (fundamental investors), and they investigated the effectsof active agents on market price formation and whether they
make a market more efficient or not by using the model. They
succeeded in figuring out the mechanism of how active funds
impact market prices and in proving that active funds trading
infrequently is not inconsistent with making a market efficient.
However, no simulation study has been done to investigate theeffect of increasing horizontal shareholding with index funds
on competition and market prices.
In this study, I implemented a competition model, in whichhorizontal shareholding changes the business strategy of companies and lessens competition among companies, into theartificial market model of Mizuta and Horie [15]. I investigatedthe effect of increasing horizontal shareholding with index
funds on competition and market prices.
II. A RTIFICIAL M ARKET M ODELIn this study, I expand the artificial market model of Mizuta
and Horie [15], which is not only simple enough but has also

succeeded in replicating the characteristics of fundamental
investors in real financial markets, to a two-stock market
model.
The simplicity of the model is very important for this study
because unnecessarily replicating macro phenomena leads tomodels that are over-fitted and too complex, and such models
would prevent us from understanding and discovering themechanisms that affect price formation because the number ofrelated factors would increase [16]. I explain the basic concept
for constructing our artificial market model in Appendix A.
A. Agent Model
The number of all agents is N . First, half of all agents
N/2 have one share of stock, and the other half have cash
C0 . C0 is constant for all agents. The agents who have one
share of stock always place a sell order for one share, andthe agents who have no stock always place a buy order for
one share. Therefore, the agents never have two or moreshares and never have a short position (negative number ofshares). The number of shares and buy or sell of orders are
determined automatically, so the agents should determine only
an order price. The changes in the fundamental prices with the“competition model” as mentioned in the following section
only cause there to be a correlation between two stocks, andagents determine the order prices for each stock dependently.
This simplification makes it easier to interpret the simulation
results since the types of agents only differ in terms of how anorder price is determined, and this allows us to build a modelof agents who trade infrequently. In the following sections, I
explain the rules for determining an order price for each typeof agent.
1) Fundamental Agent: I modeled fundamental agents who
reflect the characteristics of fundamental investors that trade
infrequently in real financial markets. A long time is needed
for market prices to converge with the fundamental price.
Therefore, fundamental investors must endure bad times andwait for profit patiently. Fundamental investors endure suchbad times and earn more.
The number of fundamental agents is NP . The order priceof an agent j at time t, Pot,j isPot,j = Pf exp(dσ j ± m(µj + 1)),

(1)where d and m are constant, and σ j is determined by randomvariables that follow a standard normal distribution for eachagent j. µj is determined by random variables that are
uniformly distributed in (0, 1) for each agent j. The ± gives− in the case of a buy order and + in the case of a sell order(see also Fig. 2).
The fundamental agents determine order prices Pot,j bydepending not on the latest market price P t−1 but on thefundamental price Pf , which is the intrinsic value of acompany. The agents do not know Pf but try to estimate it.
dσ j is the ratio of the difference between Pf and the estimatedfundamental price to Pf . Active funds generally want to buy(sell) at a sufficiently lower (higher) price than their estimatedfundamental prices, and such a sufficient difference in prices

is called the “margin of safety” [17]. m(µj + 1) is the ratioof the margin of safety to the estimated fundamental price.
The order price Pot,j does not depend on either the latest nor
historical market prices. This leads to the behavior of fundamental agents not depending on their profit even though they
are experiencing bad times for their investments. Therefore,
fundamental agents can represent the feature of fundamental
investors that endure bad times and wait for profit patiently.
2) Technical Agents: The number of technical agents is
NT . They adopt a momentum strategy.
The order price of an agent j at time t, Pot,j , isPot,j = P t

Pt
P t−tmj,(2)

where tmj is a natural number determined by random variables uniformly distributed in (1, tmmax ) for each agent j,
and tmmax is a constant. The technical agents decide theorder price the same as the expected price; this leads to the
expected return being ln(Pot,j /P t ). In the momentum strategy,
they expect that the expected return, ln(Pot,j /P t ), will equaljthe historical return, ln(P t /P t−tm ). This leads to equation(2).
Previous studies showed that such technical agents areneeded to replicate the price formations observed in real
financial markets [9](see also Appendix B). This is the reason
I introduced the technical agents.
3) Noise Agents: The number of noise agents is NN . The
order price of an agent j at time t, Pot,j , isPot,j = P t exp(ησ t,j ),(3)where η is constant, and σ t,j is determined by random variables that follow a standard normal distribution for each timet and agent j. In this study, I handle a stock traded at a highenough volume. I introduced noise agents to supply enoughliquidity. If there are no noise agents, order prices sometimes
are inclined to lean to one side heavily, and this leads to orders
not being matched. Also, in real financial markets, there aremany such liquidity suppliers [18].
B. Price Determination Model
After all agents determine an order price, the market price
P is determined by a “call auction” [19], where the numbersof sell/buy orders at prices lower/higher than P t are matched.
In a call market, buy and sell orders are grouped together andthen executed at specific times rather than executed one by one
continuously. The market price is determined at the crossing
point of supply and demand curves. The supply (demand)
curve is a cumulative number of orders that sellers (buyers)
want to sell over (buy under) a price.
t

C. Competition Model
Both fundamental prices Pf are changed or not changed according to their shareholder composition every time ∆t passed.
I call what fundamental price is changed the “occurrence ofcompetition.” Competition happens only when time t can bedivided by ∆t. At that time, the number of agents holding

Decision to
Trade Stock
Changing
MarketPrice

Changing
Fundamental
Price

Fig. 1.

Changing
Shareholder
Composition

Changing
Business
Strategy

Positive feedback process in horizontal shareholding

only stock 1 n1 , those holding stock 2 n2 , and those holding
both stocks nb are counted for every time from t − ∆t to t.
When the frequency at which n1 is largest among n1 , n2 , nb
is larger than those of n2 and nb , the fundamental price ofstock 1 is increased by δPf , and that of stock 2 is decreasedby δPf . When the frequency is that at which n2 is largest,
the opposite holds. When the frequency is that at which nb islargest, both fundamental prices are not changed (competition
does not occur).
This is modeling for the following phenomena. Investors
who hold only one stock earn more when the holding company
competes with another company and the fundamental pricerises. Therefore, the investors encourage the company tocompete with another company. Alternately, for investors who
hold both stocks even when one company competes withanother company and earns money when the fundamental pricerises, they always hold the stock of the other company andlose when the fundamental price decreases. Therefore, theinvestors have no economic incentive to encourage companies
to compete. As I mentioned in the introduction, that horizontal
shareholders have different economic incentives from separate
investors has become a subject of greater debate.
The reason only fundamental agents are counted in thecompetition model is reflected in the fact that, in real financial markets, only fundamental investors usually engage inthe management of companies, and other type of investors,
sometimes called “speculators,” rarely do so.
D. Index Agent
Some fundamental agents initially holding both stocks (thenumber is NF /4) never trade any stocks. We call them “index
agents.” The number of index agents is NF p . As I mentioned
in the introduction, index funds never select stocks and always
hold all stocks of the components of the index.
III. S IMULATION R ESULTS
I set NF = 400, NT = 100, NN = 1000, Pf =10000, d = 0.05, m = 0.02, tmmax = 100, η = 0.5, ∆t =100, andδPf = 2500. I compared the results of simulation
runs for NFp = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 =NF /4. I ran all simulations until t = te = 1000. I explain
the verification of these parameters in Appendix B. The model
has only the necessary minimum parameters and rarely obtains
arbitrary results.

𝑃𝑓 exp 𝑑𝜎 𝑗

True
Fundamental
Price
𝑃𝑓
Fig. 2.

Holding
One Stock

Margin of Safety
𝑃𝑓 exp 𝑚 (𝜇 𝑗 + 1)

Estimated
Fundamental
Price

Holding
No Stock

Buy
Order

No. of Index Agents/
No. of All Fundamental Agents

Order prices of fundamental agents

Market Price of Stock 1

Market Price of Stock 2

Fundamental Price of Stock 1

Fundamental Price of Stock 2

Fig. 4.

14000

Frequency of competitions for various NFp /NF

Market Price - Fundamental Price of Stock 1
Market Price - Fundamental Price of Stock 2
No. of Agents Holding only Stock 1
No. of Agents Holding only Stock 2

12000

2 Previous empirical studies showed that such overshooting occasionallyoccurs, and some previous simulation studies with artificial market models
discussed the mechanism of such overshooting [20], [21].

150

100
50

109

108

0
107

Fig. 3 shows the time evolution of market prices P t and fundamental prices Pf for each stock when no index agent exists
NFp = 0. The fundamental prices changed many times, andcompetition occurred frequently. Furthermore, the two companies won alternately, and this implies that there is a mechanismthat balances competition powers among corporations. Fig. 4
shows the frequency of competitions for various NFp /NF .
When NFp /NF > 12.5%, competition never occurred. The
result shows that even when the existing ratio of index funds
is not that large, the funds lessen competition.
Fig. 5 shows the time evolution (expanding from t = 100 −−110) of the market price minus the fundamental price of eachstock and the number of agents holding only stock 1 or 2 when
no index agent existed NFp = 0. At t = 100, competition
occurred, causing the fundamental price of stock 1 to rise andthat of stock 2 to fall. Due to the rise of the fundamental priceof stock 1, stock 1 was bought, and the number of agents
holding only stock 1 increased by time t = 102 when themarket prices of stock 1 converged with the fundamental priceof stock 1. After that, the market prices of stock 1 rose higher
and then overshot the fundamental price2 . Therefore, within

200

106

Fig. 3. Time evolution of market prices P t and fundamental prices Pf foreach stock when no index agent exists NFp = 0

250

105

Time

300

104

900

800

700

600

500

400

300

200

100

0

6000

350

103

7000

400

102

8000

3000
2500
2000
1500
1000
500
0
-500
-1000
-1500
-2000
-2500
-3000

101

9000

100

10000

Market Price - Fundamental Price

11000

No. of Separate Holding Agents

13000

Price

9
8
7
6
5
4
3
2
1
0

0.0%
2.5%
5.0%
7.5%
10.0%
12.5%
15.0%
17.5%
20.0%
22.5%
25.0%

Price

Frequency of Competitions

Sell
Order

Market𝑃𝑡

Time
Fig. 5. Time evolution (expanding from t = 100 − −110) of market priceminus fundamental price of each stock and number of agents holding only
stock 1 or 2 when no index agent exists NFp = 0

the overshooting, the number of agents holding only stock 1
was lower than that of stock 2.
As Fig. 6 shows, when the value of a company successfulat competition grows, its market price grows more thanthe company value (overshoots), and the company becomes
overvalued; then, the number of shareholders who encourage
competition decreases, and the company loses competitive
power. Alternately, when the value of a company not successful at competition drops, its market price falls deeper thanthe company value (overshoots), and the company becomesundervalued; then, the number of shareholders who encourage
competition increases, and the company gains competitive
power. My simulation result indicated that such a mechanism
balances competition powers among corporations. Growing
index funds may possibly weaken this balancing mechanism.

TABLE IS TATISTICS OF STOCK 1 WHEN NFp = 100 = NF /4

WinnerLoser

Fundamental Price

Overshoot-> Overvalued
Sell -> NO
Encouragement

MarketPrice

Buy ->

1.25%
1.29

Standard deviation of return
Kurtosis of return
Autocorrelation
coefficient forsquare return

lag
1
2
3

0.22
0.03
−0.09

Encourage
Competition

Overshoot-> Undervalued

artificial market simulations is that their outputs may, but notcertainly, occur in actual financial markets.
A PPENDIX

Fig. 6.

Mechanism balancing competition powers among corporations 3

IV. C ONCLUSION AND F UTURE W ORK
In this study, I implemented a competition model, in
which horizontal shareholding changes the business strategy ofcompanies and lessens competition among companies, in theartificial market model of Mizuta and Horie [15]. I investigatedthe effect of increasing horizontal shareholding with index
funds on competition and market prices.
The result shows that even when the holding ratio of index
funds is not that much, the funds lessen competition. Moreover, when the value of a company successful at competition
grows, its market price grows more than the company value(overshoots), and the company becomes overvalued; then, thenumber of shareholders who encourage competition decreases,
and the company loses competitive power. Alternately, whenthe value of a company unsuccessful at competition drops, its
market price falls deeper than the company value (overshoots),and the company becomes undervalued; then, the numberof shareholders who encourage competition increases, andthe company gains competitive power. My simulation result
indicated that such a mechanism balances competition powers among corporations. Growing index funds may possibly
weaken this balancing mechanism.
More detailed discussion will be for future work because
investigating horizontal shareholding both empirically andthrough simulation has only just begun. The model in thisstudy neglected many important processes, for example, thein- or out-flow of cash to index funds, so additional implementations of such processes in the model are future work.
For more detailed discussion, I should compare the simulation results with those from studies that use other methods,e.g., empirical studies and theoretical studies. An artificial
market can deal with situations that have never occurred, suchas index funds being more than present, and can isolate thedirect effect of changing the distribution of investor types on
price formation. These are strong advantages for an artificial
market simulation; however, the outputs of these simulations
may not be accurate or credible forecasts for actual markets. It
is an important role for artificial market simulations to reveal
the possible mechanisms that affect price formation throughmany runs to gain new insights; conversely, one limitation of

A. Basic Concept for Constructing ModelAn artificial market model, which is a kind of agent-basedmodel, can be used to discuss investor distributions that havenever been realized, can handle regulation changes that havenever been made, and it can isolate the pure contribution ofthese changes to price formation [8]–[11]. These are the strong
points of the artificial market simulation.
However, the outputs of this simulation would not be
accurate or credible forecasts of the actual future. The simulation needs to reveal possible mechanisms that affect priceformation through many simulation runs, e.g., searching forparameters or purely comparing the before/after of changes.
The possible mechanisms revealed by these runs provide newintelligence and insight into the effects of the changes on
price formation in actual financial markets. Other methods ofstudy, e.g., empirical studies, would not reveal such possible
mechanisms.
Indeed, artificial markets should replicate macro phenomena
existing generally for any asset and any time. Price variation,
which is a kind of macro phenomena, is not explicitly modeled
in artificial markets. Only micro processes, agents (generalinvestors), and price determination mechanisms (financial exchanges) are explicitly modeled. Macro phenomena emerge
as the outcome interactions of micro processes. Therefore, thesimulation outputs should replicate existing macro phenomena
in order to generally prove that simulation models are probablein actual markets.
However, it is not a primary purpose for an artificial market
to replicate specific macro phenomena only for a specific asset
or a specific period. An unnecessary replication of macro
phenomena leads to models that are over-fitted and too complex. Such models would prevent us from understanding anddiscovering mechanisms that affect price formation because
the number of related factors would increase.
Indeed, artificial market models that are too complex areoften criticized because they are very difficult to evaluate [9].
A model that is too complex not only would prevent us fromunderstanding mechanisms but also could output arbitrary
results by over-fitting too many parameters. It is more difficult
for simpler models to obtain arbitrary results, and these models
are easier to evaluate.
Therefore, I constructed an artificial market model that is assimple as possible and do not intentionally implement agents

to cover all the investors who would exist in actual financial
markets.
As Weisberg mentioned [16], “Modeling, (is) the indirect
study of real-world systems via the construction and analysis of models.” “Modeling is not always aimed at purely
veridical representation. Rather, they worked hard to identify
the features of these systems that were most salient to theirinvestigations.” Therefore, under different phenomena to focus
on, good models are different. Thus, my model is good only
for the purpose of this study and may be not good for otherpurposes. An aim of my study is to understand how important
properties (behaviors, algorithms) affect the investigation ofmacro phenomena and play a role in the financial system rather
than representing actual financial markets precisely.
B. Verification of Model
In many previous artificial market studies, the models wereverified to see whether they could explain stylized facts,
such as a fat-tail or volatility-clustering [8]–[11]. A fat-tailmeans that the kurtosis of price returns is positive. Volatilityclustering means that square returns have a positive autocorrelation, and this autocorrelation slowly decays as its lag
becomes longer. Many empirical studies, e.g., that of Sewell[22], have shown that both stylized facts (fat-tail and volatilityclustering) exist statistically in almost all financial markets.
Conversely, they also have shown that only the fat-tail andvolatility-clustering are stably observed for any asset and inany period because financial markets are generally unstable.
Indeed, the kurtosis of price returns and the autocorrelationof square returns are stably and significantly positive, but themagnitudes of these values are unstable and very different
depending on the asset and/or period. The kurtosis of pricereturns and the autocorrelation of square returns were observed
to have very broad magnitudes of about 1 ∼ 100 and about0 ∼ 0.2, respectively [22].
For the above reasons, an artificial market model should
replicate these values as significantly positive and within areasonable range as I mentioned. It is not essential for themodel to replicate specific values of stylized facts because thevalues of these facts are unstable in actual financial markets.
Table I lists the statistics, standard deviation of returns,
kurtosis of price returns, and autocorrelation coefficient forsquare returns of stock 1 when NFp = 100 = NF /4. Thisshows that this model replicated the statistical characteristics,
fat-tails, and volatility-clustering observed in real financial
markets.
Disclaimer
Note that the opinions contained herein are solely those of the authors anddo not necessarily reflect those of SPARX Asset Management Co., Ltd.

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