Analysis of the Impact of Leveraged ETF
Rebalancing Trades on the Underlying Asset
Market Using Artificial Market Simulation
Isao Yagi1 and Takanobu Mizuta2
1

Kanagawa Institute of Technology, Kanagawa, Japan
iyagi2005@gmail.com
2
SPARX Asset Management Co., Ltd. Tokyo, Japanmizutata@gmail.com
http://www.geocities.jp/muzuta ta/

Abstract. Financial markets occasionally become highly volatile, as aresult of a financial crisis or other factors. Previously, index futures trading and program trading have been singled out as direct causes of market
destabilization, but more recently it has been suggested that leveraged
ETFs (funds aimed at amplifying several-fold the movement of a priceindex such as the Nikkei Stock Average or underlying assets) rebalancing
trades may also be a factor. This study uses a financial market simulation (artificial market) constructed virtually on a computer to assess theimpact of leveraged ETF rebalancing trades on the underlying assets
market. Analysis results showed that a larger amount of the managed
assets of leveraged ETFs corresponds to a higher volatility of the underlying securities market. They also demonstrated that leveraged ETF
trading can destroy the underlying assets market, if the leveraged ETF
trading impact on the market is greater than that of ordinary volatilityof the underlying assets.

1

Introduction

Over recent years, there have been repeated periods of high market volatility asa result of events such as the Global Financial Crisis and the Greek financial
crisis. It was widely suggested that the direct cause of this market instability
may have been index trading (e.g., the Nikkei 225 futures and S&P 500 futures)
or automated (program) trading. More recently, however, leveraged exchangetraded fund (ETF) rebalancing trading has also been put forward as a possible
cause for such instability.
A leveraged ETF is an ETF that aims at achieving a rate of return several times greater than that of the underlying asset (e.g., Nikkei 225 ). These
kinds of ETFs invest in financial market indexes, so they are generally less risky
than individual stocks, but through their use of leveraging, they are able toprovide high returns, so the volume of assets under their management has dra-

matically expanded3 . Leveraged ETFs have to execute rebalancing trades on adaily basis—that is, buy the underlying asset when its price goes up and sell it
when it goes down—in order to maintain their level of leverage (i.e., to maintainthe net asset value of the ETF at a predetermined multiple of the underlying
asset exposure). For this reason, it is thought that this kind of trading causes
destabilization in the prices of underlying assets[1]4 .
To give an example, suppose a leveraged ETF based on the Nikkei 225 (Nikkei
Stock Average) aims at achieving a return that is several times higher than thatof the Nikkei average itself by means of trading in Nikkei futures. The rebalancing
trades of this leveraged ETF are likely to be executed between the close of tradingof the stock market and the close of trading of the Nikkei futures market5 . The
suggestion is that volatility becomes high during this period, resulting in market
instability. It is feared that due to the price comovement between the Nikkei
225 and Nikkei futures, whenever a large volume of trading in Nikkei futures
drives up volatility, the volatility of the Nikkei 225 will rise similarly, resulting
in market instability.
A number of previous studies have tried to demonstrate the impact of leveraged ETFs on the underlying index. Cheng et al.[2] suggested that if the baseindex of an ETF moves 1% in one day, then the volume of rebalancing trading of a leveraged ETF may account for 16.8% of total trading in the index
at the close of trading on that day. It has also been reported that volatility
tends to become much higher than normal in times of economic recession andafter a significant slump in share prices, and that this tendency increases withincreases in the volume of rebalancing trading[3]. Since leveraged ETFs havebecome increasingly popular investment products in recent years, categoricallystating that leveraged ETFs are the cause of increasing market volatility on thebasis of these findings is not easy. On the other hand, Deshpande et al.[4] concluded that since leveraged ETFs accounted for a mere 0.0079% of trading in
the S&P 500, their impact on the market is limited. Furthermore, Trainor Jr.[5]examined the impact of leveraged ETF rebalancing on the volatility of the S&P
500 by comparing the volatility before the close of trading on the previous day
to the volatility after the start of trading on the following day but was unable
to draw any firm conclusions.
3

In fact, the ETFs trade financial market index futures as underlying assets, as wecannot invest financial market indexes directly.
4
Many Japanese and American leveraged ETFs invest financial market index futures
and OTC options as underlying assets, respectively. Issuers of OTC options holdthe index futures on behalf of leveraged ETFs and trade the futures to maintaintheir level of leverage. Therefore, even if the leveraged ETFs invest OTC options
as underlying assets, the performance impacts on futures price formations as wellas when the leveraged ETFs invest futures as underlying assets. The movementof financial market indexes such as the Nikkei 225 and S&P 500 is similar to the
movement of the index futures due to the trade of arbitrage traders. Thus, wheneverthe volatility of index futures increases, the volatility of the indexes also increases.
5
The closing time of the stock market is 15:00 and that of the Nikkei futures is 15:15.

Thus, previous studies that have focused on verification have come to very
diﬀerent conclusions regarding the impact of leveraged ETFs on the market. In
addition, the various phenomena and factors involved in financial markets aremany and complex, making it diﬃcult to extract and analyze isolated factors ina verification analysis.
In light of this, one way of analyzing financial markets is to use an artificial
market. An artificial market is a financial market multi-agent system that is simulated on a computer[6–8]. Agents are each assigned their own trading method
and made to engage in trading financial assets as investors. It is then possible
to observe how the market behaves. Conversely, by modeling and incorporating
certain market-side restrictions (e.g., mechanisms to enhance market stability oreﬃciency), it is possible to observe how these aﬀect the behavior of investors andhow in turn this behavior impacts the market. Up to now, however, no study onleveraged ETFs based on this kind of market simulation has been conducted.
In view of all this, this study attempts to verify the impact of leveraged
ETF balancing on the movement of underlying asset prices by implementing
a leveraged ETF trading model in the artificial market developed by Yagi etal.[9]6 . The results show a greater value of assets managed by the leveraged
ETF corresponds to a greater volume of rebalancing trading in the underlying
asset and a higher volatility in the market. They also demonstrate that leveraged
ETF trading has the potential to destroy the market of the underlying asset if
it has a greater impact on the market than volatility.

2

Artificial Market ModelIn this study, we constructed a model based on the artificial market model ofYagi et al.[9]. In this model only one risk asset, corresponding to an underlying
security, is traded. The method used by the model to determine market prices
is described, followed by a description of the investor agents participating in thesimulated market.
2.1

Method for Determining Market Prices

The general investor agents described in Sect. 2.2 determine the order price andnumber of shares to order according to their own method and then issue orders
accordingly. Trading in the market is conducted by matching the sell orders
and buy orders of all the agents at period t. On the buyer side, the priority oftransactions is in order from agents oﬀering the highest order price; on the sellerside, the highest priority is given to agents oﬀering the lowest order price. If
a buyer’s order price exceeds or matches the order price of a seller, a trade is
executed. This method of determination is generally referred to as the itayose(bid/oﬀer) method.
On the other hand, leveraged agents trading underlying assets in order tooperate a leveraged ETF conduct rebalancing trading at the close of market
6

In this study, a leveraged ETF has an index future as underlying assets.

trading in order to achieve a return on the underlying asset that is several times
higher than the prevailing return on the underlying asset, as explained above. To
simulate the trading method of these leveraged agents, we extended the above
mentioned itoyose method as follows. First, a tentative price is determined forthe underlying asset at the present time, by tentatively matching the orders ofgeneral investors. The leveraged agents calculate their needed asset exposure
based on this tentative price and then place an order for a certain number ofassets (shares) corresponding to the diﬀerence between their needed exposureand their current exposure. Finally, the underlying share price is determinedby matching the orders of the leveraged agents with the orders of the generalinvestor agents mentioned above.
2.2

AgentsThere are four types of agents participating in the market: fundamentalists,
chartists, noise traders, and leveraged agents. All agents except leveraged agentsare referred to as general investor agents.
Fundamentalists Fundamentalists predict current market prices based on theoretical prices7 and control the number of shares that they hold to maximize their
Net Asset Value (NAV) (= (expected stock price) × (the number of shares) +(their cash)). The theoretical price is a given value.The initial theoretical price is
set to 10,000. Let the expected asset price of agent i at period t be P̃i,t , and let thetheoretical price be Pt . Then, P̃i,t is determined according to N ((1 + ϵi,t )P t , (α(1+ϵi,t )Pt )2 ), where ϵi,t denotes the degree of bullishness8 of i at t according to
N (0, σϵ2 ), and α is a coeﬃcient that describes the degree of spread of expected
prices. Let Ci,t−1 denote the amount of cash that i holds before a transaction att, qi,t−1 denote the number of shares that i holds before a transaction at t, and
Pt−1 denote the market price at t − 1. The assets of agent i before a transaction
at t areWi,t−1 = Ci,t−1 + Pt−1 · qi,t−1 .
(1)
Let the number of shares held by i at t that maximizes the subjective expectedutility under condition (1) be q̃i,t . Then, q̃i,t is as follows:q̃i,t =

(1 + ϵi,t )Pt − Pt−1a(α(1 + ϵi,t )Pt )2Note that constant a(> 0) is a coeﬃcient of risk aversion and the larger thevalue of a is, the smaller the number of shares that fundamentalists hold to avoid
risk. Fundamentalists attempt to determine their trading rules based on q̃i,t . If
7

A theoretical price is calculated by using the value of a company, which is intrinsic
or contained in the company itself.
8
Since a bullish agent expects a higher price than the theoretical price, this value ispositive for a bullish agent. However, for a bearish agent, who expects a lower pricethan the theoretical price, this value is negative.

Table 1. Rebalancing to maintain daily leverage ratio for a 2x Leveraged ETF
Underlying Asset

2x Underlying Asset

2x Leveraged ETF(6)Needed (7) Change in (8) # of Needed
Period (1) Value (2) Return (3) Return (4) Exposure (5) NAV (leverage) Exposure
Exposure Underlying Asset(×(2))(×(3) ( = (4)/(5)) (= (5)×2) (= (6)-(4))(= (7)/(1))
0
100
2,000
1,000 (2)
2,000
1
110
10%
20%
2,200
1,200 (1.83)
2,400
200
1.81
2
99
-10%
-20%
2,160
960 (2.25)
1,920
-240
-2.42

Table 2. Volatility and stylized facts for each trial
Ratio of initial cash of leveraged agent
to initial cash of general investor agents
No participation ×1 ×10 ×100 ×1,000 ×1,500 ×1,600 ×1,700 ×1,800 ×1,900 ×2000 ×5000 ×10000
Volatility(×10−2 )1.13
1.16 1.18 1.40 1.47
1.39
6.42
29.5
31.8
13.5
22.3 39.0
39.1
Kurtosis
4.64
4.35 4.34 2.56 6.29
19.2
15.8
3.41
131
10.4
8.07 -1.87 -1.87
Lag
1
0.518
0.507 0.493 0.379 0.391 0.562 0.481 -0.488 -0.522 0.137 -0.111 -0.997 -0.997
Autocorrelation 2
0.707
0.709 0.704 0.722 0.637 0.452 0.480 0.789 0.959 0.400 0.310 0.999 0.999
of the squared 3
0.521
0.513 0.500 0.416 0.345 0.313 0.266 -0.554 -0.543 0.144 0.149 -0.997 -0.997
rate of return 4
0.647
0.650 0.643 0.656 0.519 0.213 0.281 0.725 0.939 0.214 0.0318 0.999 0.999
5
0.528
0.522 0.510 0.440 0.314 0.129 0.110 -0.601 -0.560 0.0691 0.280 -0.997 -0.997

q̃i,t > qi,t−1 , i places an order to buy q̃i,t − qi,t−1 shares at P̃i,t . If q̃i,t < qi,t−1 , iplaces an order to sell qi,t−1 − q̃i,t shares at P̃i,t . If q̃i,t = qi,t−1 , i does not tradeat period t.
Chartists Chartists trade based on the moving averages of stock prices. They
consist of market followers and contrarians. Let ni,t be a number that satisfies<
1<= ni,t = 25. Let the ni,t period moving average at t used by agent i be
M At,ni,t =

ni,t
1 ∑
Pt−j ,
ni,t j=1

and let ∆M At,ni,t = M At,ni,t − M At−1,ni,t . Market followers place orders basedon the following rules.
Tshares at price (1 +– If ∆M At,ni,t > 0, then i places an order to buy qi,tαT )Pt−1 .
T– If ∆M At,ni,t < 0, then i places an order to sell qi,tshares at (1 + αT )Pt−1 .
– If ∆M At,ni,t = 0, then i does not trade.

Contrarians place orders based on the following rules.
T– If ∆M At,ni,t > 0, then i places an order to sell qi,tshares at (1 + αT )Pt−1 .
T– If ∆M At,ni,t < 0, then i places an order to buy qi,tshares at (1 + αT )Pt−1 .
– If ∆M At,ni,t = 0, then i does not trade.

<
Let the initial ni,t be a random number that satisfies 1 <= ni,t = 25, and let αT2T
be a random number distributed according to N (µTP , (σPT ) ). Further, let qi,tbe
2a random number distributed according to N (µTV , (σVT ) ).
Noise Traders Noise traders place orders to buy, sell, and wait with equalN
probabilities. At t, agent i places an order to buy qi,tshares at price (1+αt )Pt−1 ,2NNwhere qi,tand αN are random numbers distributed according to N (µN
V , (σV ) )2

NN
and N (µNP , (σP ) ), respectively. At t, i places an order to sell qi,t shares at(1 + αt )Pt−1 .

Leveraged Agents Leveraged agents are agents that manage leveraged assets
with the aim of achieving L times the return of the underlying asset. If the shareprice increases or decreases, the leverage value9 of the leveraged shares of eachagent will also change. Thus, leveraged agents must conduct rebalancing tradesin the asset such that the rate of return on their holding NAV is L times therate of return of the underlying asset10 .
Suppose, for example, that a leveraged agent who holds an amount of 1,000
in cash is trading in an asset A priced at 100 with a leverage of 2x (refer to Table1). To begin, the leveraged agent purchases 20 shares of A (assuming the cash
deficit of 1,000 is borrowed). At this point, the leveraged agent’s NAV is 1,000(= 100 × 20 - 1000) and the leverage is 2 (= (100 × 20)/(100 × 20 - 1000).
If the share price rises to 110, then the leveraged agent’s NAV becomes 1,200(= 110 × 20 - 1000), and the return achieved on the shares is 2 (= ((1200 1000)/1000)/((110 - 100)/100)). Now, however, the leverage has dropped to 1.83(= (110 × 20)/(110 × 20 - 1000)), so the agent must purchase 1.81 additional
shares to maintain a leverage of 2.
Leveraged agents that conduct trading as described above are modeled asfollows.
The time period in which a trade is made in this artificial model is expressed
as t. Let the tentative share price be determined solely by the trading of generalinvestor agents at period t be Pt′ , the market price of the share at t be Pt , and thenumber of shares and cash held by agent i at period t be Sti and Cti , respectively.
In addition, trading is started under conditions that satisfy the target leverage
L. In other words, the initial number of shares S0i is given by S0i = L · C0i /P0 ,and the initial cash C0i is given by C0i = C0i − P0 · S0i . At period t, the leveraged
agent’s NAV N AVti and leverage Lit of agent i are expressed as follows:
N AVti = Pt′ · Sti + Cti
9Leverage = (Underlying asset exposure) / (Leveraged agents’ holding NAV), where
Underlying asset exposure = (Asset price) × (total number of assets held), and NAV= (Asset price) × (total number of assets held) + Cash.
10
This is because each time the asset price changes, if the leverage is not corrected tomaintain its value of L, the rate of return on total asset exposure will no longer be
L times the rate of return on their holding NAV.

Lit =

Pt′ · Sti
N AVtiThe number of shares ordered by agent i at period t, Vti , is expressed by thefollowing equation11 . Note that if the number of shares ordered turns out to be
fractional, it is rounded oﬀ.
Vti = ⌊

(L − 1)Pt′ · Sti + L · Cti⌋
Pt′

The orders of this leveraged agent can only be executed if matched with otheroﬀers. This means that in the case of buy orders, the order price has to be higher
than that of any general investor agent, and in the case of sell orders, the orderprice must be lower than that of another general investor agent. When N AVti is
below 0, the agent is assumed to be bankrupt, and after returning to the initialstate, the agent can participate in the market again from the next time period(after bankruptcy).

2.3

Modeling of an Agent Influenced by the Trading Rules ofHigh-performance Agents

After a transaction at period t, all agents evaluate their performances. Eachagent whose performance is lower than that of any other agent attempts tolearn the trading rules of the high-performance agents. Fundamentalists learn
the degree of bullishness and Chartists learn the period moving average. However, noise traders do not evaluate their performance or learn the trading rulesof other agents. In actual markets, there are some participants who attempt
to improve their performance further by trial and error, although their performances are comparatively high. To model them, a genetic algorithm whose
selection is elitism is applied to learning the trading rules. Thus, comparatively
high-performance agents who do not rank in the top NH (= 20)% also improve
trading rules.
Agents cannot change their trading strategies to other strategies, because theratio of the three strategies is fixed. For example, a low-performance fundamentalist attempts to learn the trading rule of a high-performance fundamentalist,
not that of a high-performance chartist.
Let the ratio of change in the asset of an agent i from period t − 1 to t be
Ri,t = Wi,t /Wi,t−1 and let the mean ratio of change in the assets of i in the past∑N
N periods be Ri,t = j=1 Ri,t−(j−1) /N .
11

Vti is determined by the following equation.
L=

Pt′ (Sti + Vti )′i
Pt (St + Vti ) + (Cti − Pt′ · Vti )Table 3. Correlation between the initial cash of the leveraged agent and market impact
Ratio of initial cash of leveraged agent
to initial cash of general investor agents
No participation
×1
×10
×100 ×1,000 ×1,500 ×1,600(1) No. of orders
0.0
0.76543 7.5734 71.5162 88.3309 90.4669 6341.36(2) MI(×10−2 )0.0
0.000505 0.00493 0.0396 0.0539 0.0785
2.18(3) Volatility(×10−2 )1.13∗1
1.16
1.18
1.40
1.47
1.39
6.42
(4) Rate(= (2)/(*1 in (3)))
0.00
0.000445 0.00434 0.0349 0.0476 0.0692
1.92
×1,700×1,800 ×1,900 ×2,000 ×5,000 ×10,000(1) No. of orders
57616.9
72786.2 23741.6 47638.40 256003 512158(2) MI(×10−2 )
12.5
13.2
4.59
8.15
16.5
16.5
−2
(3) Volatility(×10 )
29.5
31.8
13.5
22.3
39.0
39.1
(4) Rate(= (2)/(*1 in (3)))
11.0
11.6
4.05
7.19
14.5
14.5

Fundamentalists When Ri,t ranks in the bottom NL % of all fundamentalists,
ϵi,t is changed with probabilitypi =Rank of performance of agent i.
Total number of fundamentalists

An agent i′ whose Ri′ ,t ranks in the top NH % of all fundamentalists is chosen
at random, and ϵi,t+1 = ϵi′ ,t is set
To implement a change in the trading rules of agents who would like toimprove their performance, for agents whose Ri,t does not rank in the top NH %,their degree of bullishness is changed with probability Pϵ at random.
Chartists When Ri,t ranks in the bottom NL % of all chartists, the moving
average period, ni,t , is changed with the following probability.
pi =Rank of performance of agent i.
Total number of chartists

An agent i′ whose Ri′ ,t ranks in the top NH % of all chartists is chosen at random,and ni,t+1 = ni′ ,t is set.
To implement a change in the trading rules of agents who would like to improve their performance, for agents whose Ri,t does not rank in the top NH %,their trading rules (market followers and contrarians) and/or their moving average periods are changed with probability Pn at random, under the restrictionthat their moving average periods must fall within [1, 25].

3

Experiment

In this study, we determined the price movement in an underlying asset market
and the volatility of the rate of return for diﬀerent values of the initial cash heldby leveraged agents, in order to assess the impact of leveraged ETF balancing
trades on underlying assets.

3.1

Experiment environment

We ran experiments with the initial cash amount of leveraged agents set to 1, 10,100, 1,000, 1,500, 1,600, 1,700, 1,800, 1,900, 2,000, 5,000, and 10,000 times thatof general investor agents. Other parameter settings for the simulated market are
as follows. The number of general investor agents was set to 10,000, and thesegeneral investors comprised 4,500 fundamentalists, 4,500 chartists, and 1,000
noise traders. The initial number of assets was set to 1,000,000. The number ofleveraged agents was set to 1. The
0.02,√ agent parameters were set as α =√ a=N0.001, σϵ = 0.1,√µTP = 0, σPT = 0.02, µTV = 2, σVT = 0.1, µN
0.02,
P = 0, σP =
N
=
2,
σ
=
µN
0.1,
N
=
5,
N
=
N
=
20%,
and
P
=
P
=
0.05.
H
L
ϵ
n
V
V
The experiment was configured to run over 10,000 time periods, but samplingof market volatility was only performed for the latter 5,000 periods (periods 5,001
to 10,000). In addition, the trading of the leveraged agent was only commenced
from period 3,001, to allow suﬃcient time for the trading of general investor
agents to stabilize.
3.2

Validity of model

On initiating the simulation, we verified the validity of the artificial market model
proposed by this study. As a method of verification, we determined whether thetwo typical statistical properties (stylized facts) below, obtained previously byverification analysis[10], were satisfied.
– The kurtosis of the rate of return is positive.
– The autocorrelation of the squared rate of return is positive.
Table 2 shows the stylized facts for the case in which only general investor
agents participate in asset trading. These results show that the stylized facts are
satisfied for the proposed model, so we can conclude that the model serves as avalid simulation environment.

4

Experiment results and discussion

The second and lower columns of Table 2 show the rate of return volatility of theasset market for diﬀerent values of the initial cash of the leveraged agent. These
results reveal that a higher amount of initial cash corresponds to a higher degreeof volatility, which can be explained as follows. If the tentative price Pt′ is higher
than the market price in the previous time period, Pt−1 , then the leveraged agent
issues a buy order to increase the number of shares; if Pt′ is lower than the market
price Pt−1 , then the agent issues a sell order to reduce the number of shares held.
At this time, the leveraged agent may buy the underlying asset at a very highprice or sell it at a very low price in order to secure the number of shares needed.
In addition a greater amount of initial cash implies a greater quantity of sharesis necessary to restore the leverage, and so a higher likelihood that shares arebought or sold at an unforeseen price. Accordingly, the market impact of thisbuying and selling by the leveraged agent appears to be an increase in volatility.

Table 3 shows the correlation between the leveraged agent’s initial amountof cash and market impact, MI. (Note that here, the market impact at periodt, MIt , is given by MIt = |Pt − Pt′ |/Pt , whereas MI denotes the average market
impact for each trial. ) These results reveal that a larger amount of initial cash
held by the leveraged agent corresponds to a greater average number of shares
being ordered per transaction. From this we can conclude that volatility also
increases.
Furthermore, the results shown in Table 2 also reveal that if the initial cash
becomes excessively high (in this experiment, when the initial cash of the leveraged agent is 1,700 times or more greater than that of a general investor agent),
the autocorrelation of the squared rate of return becomes negative. Although
this phenomenon cannot be observed in actual markets, this result suggests thatthe market of the asset would have been destroyed. The point to take note ofhere is that MI becomes greater than the market volatility when the leveraged
agent does not participate. This suggests that if the initial cash amount is large
and leveraged agents impact the market more than ordinary volatility, a market
may be destroyed. For example, Table 2 and Table 3 show that the market may
be destroyed when MI is greater than seven times the market volatility, because
the autocorrelation of the squared rate of return of the leveraged agent whose
initial cash is 2,000 times that of a general investor agent is negative.

5

Conclusion

The aim of this study was to use an artificial market simulation to verify the
impact of leveraged ETF rebalancing trades on the price movement of market
assets. It was found that a higher amount of initial cash held by a leveraged
ETF corresponds to a higher volume of rebalancing trading in the underlying
asset and a greater volatility in the market for that asset. Furthermore, it wasobserved that if the leveraged ETF trading impact on the market is greater thanthat of ordinary volatility, then the market may be destroyed. A further challenge
to take up is a detailed investigation to estimate what level of leveraged ETF
trading would have suﬃcient impact to destroy a market.

DISCLAIMER
It should be noted that the opinions contained herein are solely those of theauthors and do not necessarily reflect those of SPARX Asset Management Co.,Ltd.

Acknowledgment
This research was supported by JSPS KAKENHI Grant Number 15K01211.

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