Why do Active Funds that Trade Infrequently Makea Market more Efficient?
– Investigation using Agent-Based Model
Takanobu Mizuta
SPARX Asset Management Co. Ltd.,Tokyo, Japan
Email: mizutata@gmail.com
http://www.mizutatakanobu.com/

Abstract—Since managers of active funds choose stocks thatare expected to raise their prices on the basis of the fundamental
value, many argue that active funds discover the fundamental
value and make a market more efficient. However, it has notbeen clear whether actual active funds make a market more
efficient or not. It has been shown that active funds that trade
infrequently earn more. At first glance, infrequent trades seemto not impact and change market prices and this leads to market
prices not converging with the fundamental price. Therefore, it isimportant to discuss whether active funds that trade infrequently
make a market more efficient or not, and if so, we should
investigate the mechanism of how they do so. In this study,
we built a model of investors who trade infrequently in an
artificial market model, and we investigated effects of theseinvestors on market prices and whether they make a market more
efficient by using the model. The results indicate that such active
investors trade frequently in the rare situation that the market
becomes unstable and inefficient due to the market price moving
away from the fundamental price. These trades, occurring only
at a necessary time, impact the market prices and lead themconverging with the fundamental price. This leads preventing themarket from becoming more unstable and less efficient. Though
the trading volume of fundamental investors is low throughout
whole period, the volume increases greatly only when a market
becomes less efficient, and these trades then make the market
efficient. An increasing market volatility makes the order pricesof speculators (technical investors) move further away from thefundamental price, and this leads to amplifying market volatility
more excessively. It is possible that the orders of active investors
prevent this amplification. This also implies that money movingfrom active funds to passive funds leads a market to become lessefficient.

I. I NTRODUCTION
There are two types of portfolio management strategies forfunds that invest in stocks and/or bonds, “active” managed
funds, where a manager chooses stocks expected to raise theirprices and invests in them, and “passive” managed funds,
where a manager expects a return the same as a market index,e.g., the Dow Jones industrial average and S&P 500, andreplicates the components of the index. Recently, the assets ofpassive funds have increased; however, those of active funds
have decreased because, as some empirical studies [1], [2]have argued, the average return of active funds lost that ofSadayuki Horie
Nomura Research Institute, Ltd.,Tokyo, Japan

passive funds, and recent regulations have made fund sellers
accountable for fund costs,1 especially in the United States[3].
Since active funds invest in stocks on the basis of the intrinsic value of companies (fundamental value), there are manyarguments made that they discover the fundamental value andthis leads to market prices converging with the fundamental
price (make a market more efficient); therefore, they play animportant role in allocating capital, which is an importantfunction in capitalism [4]. Furthermore, some claim that therise in passive funds threatens to fundamentally undermine
the entire system of capitalism and market mechanisms thatfacilitate an increase in the general welfare [5].
However, it has not been clear whether actual active funds
make a market more efficient or not and how much the assetsof active funds are needed to make a market more efficient.
In addition, an empirical study [6] showed that active funds
that trade infrequently earn more. This implies that the assetsof these funds will increase, while those of active funds thatfrequently trade will decrease. At first glance, infrequent trades
seem to not impact and change market prices and thus notlead to market prices converging with the fundamental price.
Therefore, it is important to discuss whether active funds that
trade infrequently make a market more efficient or not, and if
so, we should investigate the mechanism of how they do so.
In fact, there are opposing arguments. On the one side,
since active funds that perform well measure fundamental
value more precisely, these funds make a market more efficient even though they trade infrequently [7], [8], and on
the other side, funds that trade infrequently do not make amarket more efficient [9]. Anyways, the mechanism by whichfunds that trade infrequently make a market more efficient ispoorly understood, although several mechanisms have beensuggested. An empirical study [10] showed that the volumeof active funds varies over time and that funds earn when thevolume is larger, which implies the mechanism of how funds
1 Since passive funds need not research companies, costs of passive fundare cheaper than those of active funds [1], [2].

No Fundamental Agent
10 Fundamental Agents
500 Fundamental Agents

11,000Market Price

10,50010,000
9,500
9,000
8,500

0
600
1200
1800
2400
3000
3600
4200
4800
5400
6000
6600
7200
7800
8400
9000
9600

8,000
Time
Fig. 1. Time evolution of market prices P t when number of fundamental
agents NF = 0, 100, 500.

Frequency
Trading Volume of Fundamental Agents

180

1400

160

1200

140
120

1000

100

800

80

600

60
40

200

20

0

0
9500
9540
9580
9620
9660
9700
9740
9780
9820
9860
9900
9940
9980
10020
10060
10100
10140
10180

400

Trading Volume

1600

Frequencythat trade infrequently make a market more efficient.
Such discussion on the mechanism between the micromacro feedback of certain types of investors is very difficult
only using the results of empirical studies. Empirical studies
cannot be conducted to investigate situations that have neveroccurred in actual financial markets, such as ones in which
passive investors are more than present. Furthermore, because
so many factors cause price formation in actual markets, anempirical study cannot be conducted to isolate the direct
effect of changing the distribution of investor types on priceformation.
An artificial market, which is a kind of agent-based model,
can handle situations that have never occurred, such as onesin which passive investors are more than present, and canisolate the direct effect of changing the distribution of investor
types on price formation. These are strong advantages foran artificial market simulation. The effects of the distribution
and of several changing regulations have been investigated byusing artificial market simulations [11]–[14].
Not only academies but also financial regulators and stock
exchanges have recently become interested in agent-basedmodels such as artificial market models to investigate financial
markets. Indeed, an article in Science by Battiston et al. [15]stated that “since the 2008 crisis, there has been increasing
interest in using ideas from complexity theory (using networkmodels, agent-based models, and so on) to make sense of
economic and financial markets.”, and an article in Nature
by Farmer and Foley [16] stated that “such (agent-based)
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 some kinds ofinvestors on price formation and the effects of several changing
regulations and rules by using artificial market simulations, forexample, leveraged ETFs [17], high-frequency traders (HFTs)[18]–[22], arbitrage traders between markets that have different latencies [23], market impacts [24], [25], financial market
crashes [26]–[28], price variation limits [29]–[31], frequent
batch auctions [32], dark pools [33]–[35], investor networks
and herding [36], the increasing speed of order matchingsystems on financial exchanges [37], market efficiency [38],and the rules for investment diversification [39].
Indeed, the effects of market makers and passive funds were
investigated by using artificial market simulations [40]. However, the reason active investors who trade infrequently affectmarket prices and make a market more efficient has not been
investigated. In the first place, investors who trade infrequently
have not been modeled in artificial market models.
Therefore, in this study, we built a model of agents who
trade infrequently in the artificial market model and investigated the effects of active investors who trade infrequently onmarket prices and whether they make a market more efficientby using the model.

Market Price
Fig. 2.
Distribution of frequencies of market price ranges and trading
volume of fundamental agents for various market price ranges when numberof fundamental agents NF = 500.

II. A RTIFICIAL M ARKET M ODEL
We modeled agents who reflected the characteristics ofinvestors who trade infrequently. The simplicity of the model isvery important for this study because unnecessarily replicating
macro phenomena leads to models that are over-fitted and toocomplex, and such models would prevent us from understanding and discovering the mechanisms that affect price formation
because the number of related factors would increase. We
explain the basic concept for constructing our artificial market
model in “Basic Concept for Constructing Model” in the
Appendix.
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TABLE I
M ARKET INEFFICIENCY Mie FOR VARIOUS NUMBERS OF FUNDAMENTAL AGENTS NF .

Market Inefficiency (Mie )0
7.7%

Number of Fundamental Agents (NF )
20
50
100
200
300
2.2%
2.4% 1.4% 0.7% 0.7%

10
3.0%

400
0.9%

500
0.6%

TABLE II
T RADING VOLUME SHARE ( TOTAL VOLUME OF A TYPE OF AGENTS / TOTAL VOLUME OF ALL TYPE OF AGENTS ) FOR VARIOUS NUMBER OF FUNDAMENTAL
AGENTS NF .
Trading Volume Share
Noise Agents (NN = 1000)
Technical Agents (NT = 100)
Fundamental Agents

0
97.3%
2.7%
—

10
97.2%
2.8%
0.002%

Number of Fundamental Agents (NF )
20
50
100
200
300
97.2%
97.0%
97.0%
96.9%
96.7%
2.8%
3.0%
3.0%
3.1%
3.3%
0.002%
0.006% 0.010% 0.011% 0.014%

400
96.8%
3.2%
0.018%

500
96.9%
3.1%
0.023%TABLE IIIAVERAGE OF PROFITS OF AGENTS FOR VARIOUS NUMBERS OF FUNDAMENTAL AGENTS NF .
Average of Profits
Noise Agents (NN = 1000)
Technical Agents (NT = 100)
Fundamental Agents

0-0.06%
0.64%—

10-0.07%-0.09%
8.36%

Number of Fundamental Agents (NF )
20
50
100
200
300
-0.13%-0.18% -0.30% -0.52% -0.48%
0.24%-0.31% -0.62% -0.80% -2.54%
5.18%
4.22%
3.61%
3.02%
2.44%

Average of Absolute Returns
Trading Volume of Fundamental Agents

120

100
80
60
40

Trading Volume

160
140

500
-0.85%-0.91%
1.87%

explain the rules for determining an order price for each typeof agent.
1) Fundamental Agents: The number of fundamental
agents is NF . The order price of an agent j at time t, Pot,j is

180

Average of Absolute Returns

0.20%
0.18%
0.16%
0.14%
0.12%
0.10%
0.08%
0.06%
0.04%
0.02%
0.00%

400
-0.59%-1.53%
1.85%

20
9500
9540
9580
9620
9660
9700
9740
9780
9820
9860
9900
9940
9980
10020
10060
10100
10140
10180

0

Market Price
Fig. 3. Distribution of frequencies of average of absolute returns and trading
volume of fundamental agents for various market price ranges when numberof fundamental agents NF = 500.

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 orderfor one share. Therefore, the agents never have two 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
order prices.
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, wePot,j = Pf exp(dσ j ± mµj ),j(1)where d and m are constant, and σ 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.
The fundamental agents determine order prices Pot,j bydepending not on the latest market price P t−1 but on the fundamental price Pf , which is the intrinsic value of a company.
The agents do not know Pf but try to estimate it. dσ j is a ratioof the difference between Pf and the estimated fundamental
price to Pf . Fundamental investors generally want to buy(sell) at a sufficiently lower (higher) price than their estimatedfundamental prices, and such a sufficient difference of prices
is called the “margin of safety” [41]. mµj is the ratio of themargin of safety to the estimated fundamental price.
2) Technical Agents: The number of technical agents is
NT . A half of the technical agents NT /2 adopts a momentum
strategy, and the other half adopts a contrarian strategy.
The order price of a momentum strategy agent j at time t,
Pot,j isPt
Pot,j = P t t−tmj .
(2)
P
The technical agents decide order price same as an expected
price, this leads an expected return is ln(Pot,j /P t ). In the
momentum strategy, they expect that the expected return,jln(Pot,j /P t ) equals the historical return, ln(P t /P t−tm ). This
leads the equation (2).

That of a contrarian strategy agent isj

Pot,j = P t−tm ,(3)where tmj is a natural number determined by random variables uniformly distributed in (1, tmmax ) for each agent j, andtmmax is a constant. In the contrarian strategy, they expectthat the expected return, ln(Pot,j /P t ) equals the inverses ofjhistorical return, − ln(P t /P t−tm ). This leads the equation(3).
Previous studies showed that such technical agents areneeded to replicate price formations observed in real financial
markets [12](see also “Verification of Model” in Appendix.).
This is the reason we 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 ),(4)where η is constant, σ t,j is determined by random variables
that follow a standard normal distribution for each time t andagent j. In this study, we handle a stock traded at a highenough volume. We introduce noise agents to supply enoughliquidity. If there are no noise agents, order prices sometimes
incline to one side heavily, and this leads to orders not being
matched. Also, in real financial markets, there are such manyliquidity suppliers [42].
4) Note on Modeling Passive Investors: In this study,
passive agents are not introduced. Passive funds, where amanager expects a return the same as a market index andreplicates the components of the index, trade only when they
are subscribed to or redeemed, or when names included ina tracking index are changed. In this study, we do not deal
with these trades, so passive agents never trade in this model.
This means that this study cannot handle cases in which only
the number of passive investors increases. This means that adecrease in the number of fundamental agents NF corresponds
to money moving from active funds to passive funds because
passive agents never trade. Therefore, we can interpret thisstudy as an investigation into price formation in the case ofmoney moving from active funds to passive funds.
B. Price Determination Model
After all agents determine orders, buy or sell and anorder price, the market price P t is determined by a “callauction” [43] where the numbers of sell/buy orders at prices
lower/higher than P t are matched. In a call market, buy andsell orders are grouped together and then executed at specifictimes rather than executed one by one continuously. The
market price is determined at the crossing point of supply anddemand curves. The supply(demand) curve is a cumulative
number of orders that sellers(buyers) want to sell over(buyunder) a price.
III. S IMULATION R ESULTS
Specifically, we set NT = 100, NN = 1000, C0 = Pf =10000, d = 0.1, m = 0.1, tmmax = 100, and η = 0.1. We ran
simulations until t = te = 10000. We explain the verificationof these parameters in “Verification of Model” in Appendix.
The model has only the necessary minimum parameters andhardly obtains arbitrary results
We compared several statistical values of the simulation
runs for NF = 0, 10, 20, 50, 100, 200, 300, 400, and 500 not
only under other parameters that were fixed but also thesame random number table. As mentioned in section II-A-4, adecrease in the number of fundamental agents NF corresponds
to money moving from active funds to passive funds because
passive agents never trade. Therefore, a decrease in NF meansthat the number of fundamental agents operating as active
funds decreases and that of passive agents operating as passive
funds increases.
We introduce the parameter “market inefficiency” Mie fordirectly measuring market efficiency,e
1 ∑|P t − Pf |,te t=1
Pf

t

Mie =

(5)

where || is an absolute value. Mie is always greater than zero,and Mie = 0 means a market is perfectly efficient2 . The larger
the Mie , the less efficient the market3 .
Figure 1 shows the time evolution of market prices P twhen the number of fundamental agents was NF = 0, 100,and 500. The higher the number of fundamental agents, themore efficient the market became because P t oscillated ina narrower range near the fundamental price Pf = 10000.
Table I lists the market inefficiency Mie for various NF andshows that the higher the NF and lower the Mie , the more
efficient the market became. This indicates the possibility
that a decrease in the number of active investors makes amarket less efficient, and this implies that money moving from
active funds to passive funds leads to a market becoming lessefficient. As we mentioned in section II-A-4, note that thisstudy cannot deal with the case in which only the number ofpassive investors increases.
Table II lists the trading volume share (total volume of a
type of agent/total volume of all types of agents) for various
NF . The trading volume of fundamental agents was muchsmaller than those of other type agents.
To discuss why fundamental agents whose trading volume
is low make a market more efficient, we show two figures.
Figure 2 shows a distribution of the frequencies of market
price ranges and the trading volume of fundamental agents forvarious market price ranges, and Fig. 3 show a distribution of
the frequencies of the average of the absolute returns and the
trading volume of fundamental agents for various market priceranges; both figures are for the case of NF = 500.
2 Even though we calculated market inefficiency, we did not intend to
discuss an efficient market hypothesis. In our model, the market is not efficientbecause of the existence of technical agents.
3 This index is sometimes used in experimental financial studies of people,
in which this market inefficiency is sometimes called “relative absolutedeviation” (RAD) [44]. Many indications for measuring market efficiency
have been proposed [45]. A feature of Mie is that it is calculated directly byusing a fundamental price Pf , which is never observed in empirical studies.
We can also use Mie in simulation and experimental studies because we canexactly define Pf .

Figure 2 indicates that the fundamental agents traded very
near P t = 9800, far from Pf (solid line), though the frequency
at P t = 9800 was low (dashed line). This means that thefundamental agents traded frequently when the market became
inefficient as P t moved farther away from Pf .
Figure 3 indicates that the absolute return near P t = 9800
was large (dashed line) and that market volatility (dispersionof returns) increased. In short, the fundamental agents tradedfrequently when market prices sharply declined and market
volatility was excessive. This trading behavior is consistent
with a previous empirical study [10]. However, this mechanismof making a market more efficient is inconsistent with themechanisms argued by [7], [8].
These results indicate that the fundamental agents tradedfrequently in the rare situation that a market becomes unstable
and inefficient due to the market price getting away from thefundamental price. These trades, occurring only at a necessarytime, impact the market prices and lead them converging withthe fundamental price. This leads to preventing the market
from becoming more unstable and less efficient. Though thetrading volume of fundamental agents was low throughout thewhole period, the volume increased a lot only when the market
became less efficient, and these trades then made the market
efficient. It is implied that money moving from active funds topassive funds leads a market to become less efficient because
these trades of active funds decrease.
From equation (2), an increasing market volatility makes theorder prices of momentum strategy agents move further awayfrom Pf , and this leads to amplifying market volatility more
excessively. It is possible that the orders of fundamental agents
prevent this amplification. Future work should investigate thedetails of this mechanism.
Table III lists the averages of the profits of agents for various
NF . At the simulation end time t = te , we evaluated the priceof the stock the agents held at Pf . The averages of the profits
for the fundamental agents was higher than those of the othertypes of agents. The higher NF , the fundamental agents earn
less, but this amount is higher than zero. This suggests that
there are more opportunities for fundamental agents to earn
when the market becomes less efficient.

TABLE IVS TATISTICS WHEN NF = 0
0.22%
2.37

Standard deviation of return
Kurtosis of return

Autocorrelation
coefficient forsquare return

lag
1
2
3
4
5

0.18
0.14
0.13
0.15
0.12

when the market became less efficient, and these trades thenmade the market efficient. This trading behavior is consistent
with a previous empirical study [10]. However, this mechanismof making a market more efficient is inconsistent with themechanisms argued by [7], [8].
From equation (2), an increasing market volatility makes theorder prices of momentum strategy agents move further awayfrom Pf , and this leads to amplifying market volatility more
excessively. It is possible that the orders of fundamental agents
prevent this amplification. Future work should investigate thedetails of this mechanism.
It is implied that money moving from active funds to passive
funds leads a market to become less efficient because theseorders of active funds decrease. Note that this study cannot
handle the case in which only the number of passive investors
increases.
The averages of the profits for fundamental agents werehigher than those of the other types of agents. The higher thenumber of fundamental agents NF , the less the fundamental
agents earn less, but this amount is higher than zero. Thissuggests that there are more opportunities for fundamental
agents to earn when the market becomes less efficient.

IV. C ONCLUSION AND F UTURE W ORKThe case in which only the number of passive investors
increases is also important future work. This study cannot deal
with this case since passive agents never traded in this model.
However, actual passive funds trade when a fund is subscribedor redeemed, or when names included in a tracking index arechanged. Therefore, investigating the effects of these trades isimportant.

In this study, we built a model of agents who trade infrequently in an artificial market model, and we investigated theeffects of active investors who trade infrequently on market
prices and whether they make a market more efficient by using
the model.
Simulation results indicated that the fundamental agentstraded frequently in the rare situation that the market becomes
unstable and inefficient due to the market price moving further
away from the fundamental price. These trades, occurring only
at a necessary time, impacted the market prices and lead themconverging with the fundamental price. This lead to preventing
the market from becoming more unstable and less efficient.
Though the trading volume of fundamental agents was low
throughout the whole period, the volume increased a lot only

For more detailed discussions, we should compare the simulation results to 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, such as
passive investors 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
artificial market simulations is that their outputs may, but notcertainly, occur in actual financial markets.

A PPENDIX
Basic Concept for Constructing ModelAn artificial market, which is a kind of agent-based model,
can be used to discuss investor distributions that have neverbeen realized, can handle regulation changes that have neverbeen made, and can isolate the pure contribution of thesechanges to price formation [11]–[14]. 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 will give us
new intelligence 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 macro phenomena existing
generally due to prove that simulation models are probable inactual 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 [12].
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, we constructed an artificial market model thatis as simple as possible and do not intentionally implement
agents to cover all the investors who would exist in actual
financial markets.
Verification of Model
In many previous artificial market studies, the models wereverified to see whether they could explain stylized facts, suchas a fat tail or volatility clustering [11]–[14]. A fat tail meansthat the kurtosis of price returns is positive. Volatility clustering means that the square returns have positive autocorrelation,and this autocorrelation slowly decays as its lag becomes
longer. Many empirical studies, e.g., that of Sewell [46], haveshown that both stylized facts (fat tail and volatility clustering)
exist statistically in almost all financial markets. Conversely,
they also have shown that only the fat tail and volatility
clustering are stably observed for any asset and in any period
because financial markets are generally unstable.
Indeed, the kurtosis of price returns and the autocorrelationof the square returns are stably and significantly positive, but
the magnitudes of these values are unstable and very different
depending on the asset and/or period. The kurtosis of pricereturns and the autocorrelation of the square returns wereobserved to have very broad magnitudes of about 1 ∼ 100
and about 0.01 ∼ 0.2, respectively [46].
For the above reasons, an artificial market model should
replicate these values as significantly positive and within areasonable range as we mentioned. It is not essential for themodels to replicate specific values of stylized facts because thevalues of these facts are unstable in actual financial markets.
Table IV lists the statistics, standard deviation of returns,
kurtosis of price returns, and the autocorrelation coefficientfor square returns when NF = 0. This shows 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. andNomura Research Institute, Ltd.

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