Investigation of the rule for investment diversification at the time of a market crash using an artificial market simulation

Article

Abstract

As financial products have grown in complexity and level of risk compounding in recent years, investors have come to find it difficult to assess investment risk. Furthermore, companies managing mutual funds are increasingly expected to perform risk control and thus prevent assumption of unforeseen risk by investors. A related revision to the mutual fund legal system in Japan led to establishing what is known as “the rule for investment diversification” in December 2014, without a clear discussion of its expected effects on market price formation having taken place. In this paper, we therefore, used an artificial market to investigate its effects on price formation in financial markets where investors must follow the rule at the time of a market crash that was caused by the collapse of the asset fundamental price. As results, we found that, in a two-asset market where investors had to follow the rule for investment diversification, when the fundamental price of one asset collapsed and its market price also collapsed, the other asset market price also fell.

Keywords

Artificial market Multi-agent based simulation The rule for investment diversification Leverage Financial market 

JEL Classification

C63 - Computational techniques Simulation modeling · G18 · Government Policy and Regulation · G01 · Financial crises 

1 Introduction

Financial products have grown in complexity and level of risk compounding in recent years. For example, mutual funds have come to choose various assets, some of which may have high risk, and there may be some funds whose performances depend on those of the particular assets that the funds hold. Therefore, investors have come to find it difficult to assess investment risk. Companies managing mutual funds are increasingly expected to perform risk control and thereby prevent assumption of unforeseen risk by investors who hold mutual funds. In Japan, The Securities Mutual Fund Law was revised in 2013, and the rule for investment diversification was established in December 2014 (Sectional Committee 2009; The Investment 2014).

The rule for investment diversification is a holding weight limitation for mutual funds; that is, mutual funds are prohibited from holding a greater weight of each stock than some limit.1 For example, the rule specifies that mutual funds have to keep each stock to 10% or less of the mutual funds’ net asset value (NAV). The rule for investment diversification is applied not to individual stocks, but to mutual funds. Thus, the mutual fund manager must act to ensure that mutual funds do not hold each stock in more weight than the weight limit set by the regulator. On the other hand, individual investors and institutional investors who do not own mutual funds are not affected by the rule for investment diversification.

The purpose of the rule for investment diversification is to restrict the investment risk of investors who hold mutual funds. When the number of mutual fund components is small (i.e., the fraction of each stock in the mutual fund is large), if the value of one of them decreases sharply due to bankruptcy or scandal, investors who hold the mutual fund suffer large losses. However, when the number of mutual fund components is large, even if the price of one of them collapses due to bankruptcy, the damage to investors is limited. As you can see from the previous explanation, the purpose of the rule for investment diversification is not to stabilize the market. Thus, there has been no clear discussion on whether the rule for investment diversification can stabilize the market.

There have been many empirical studies on diversified investment. Cremers et al. Cremers and Petajisto (2009) showed that the funds with the highest Active Share 2 significantly outperform their benchmark indexes, while the non-index funds with the lowest Active Share underperform. On the other hand, there has been no clear discussion on the effects of market price formation when restrictions on diversified investments are imposed. An empirical study cannot isolate the pure contribution of these regulations to price formation because many kinds of traders can effect price formation in actual markets.

One way of analyzing how particular transactions influence the market is to use an artificial market. An artificial market is a multi-agent based model of financial markets (Chiarella et al. 2009; Chen et al. 2012; Cristelli 2014; Mizuta et al. 2014). Each of the agents is assigned a specific trading (i.e., buying and selling) rule and then set to trade financial assets as an investor. The market can then be observed to see how the agents behave. At the same time, by modeling and incorporating certain restrictions on the market side (e.g., limitations to ensure market stability and efficiency such as short selling regulations), it is possible to examine how investors behave and also what kinds of effects their behaviors induce in the market.

Studies on artificial markets to investigate market regulations have had some success in market analysis in recent years (Yagi et al. 2010; Mizuta et al. 2014); however, the only study on the effects of the rule for investment diversification on the market using an artificial market has been Nozaki et al. (2015). Specifically, Nozaki et al. investigated market volumes, the efficiency of markets, and investors’ performance in the following two stable (i.e., the fundamental prices of all assets are constant) markets. In one market, all investors must follow the rule for investment diversification, and in the other one, all investors do not follow the rule for investment diversification. For these markets, they obtained the following three results: (1) the more strict the rule for investment diversification is, the less the market volume is; (2) the efficiency of a market where all investors (i.e., mutual fund managers in the real world) must follow the rule for investment diversification is not always worse than that of a market where all investors do not follow the rule for investment diversification; and (3) the investors’ performances in a market where all investors must follow the rule for investment diversification are not always worse than those in the market where all investors do not follow the rule for investment diversification.

On the other hand, we investigate the difference between (1) the market price transitions of two assets whose fundamental prices are constant in both a market where agents must follow the rule for investment diversification and in a market where agents do not follow the rule for investment diversification and (2) the market price transitions of an asset whose fundamental price falls and those of the other asset whose fundamental price is stable in both the markets where agents must follow the rule for investment diversification and in a market where agents do not follow the rule for investment diversification. Our study supposes the following situation in real financial markets. If one of the companies that issues a stock in the mutual fund becomes bankrupt, its financial results are much worse than the market forecast, or it causes a scandal or has serious accidents, then the fundamental price of only this company falls sharply and the market price also collapses. However, even if the above types of troubles occur for a particular company, the impact on the economy is not great, so the fundamental prices and market prices of other companies’ stocks do not decrease sharply. For such a situation, we assess the influence of the rule for investment diversification on the markets. Previewing our results, we found that when the fundamental price of one asset collapsed and its market price also collapsed, the other asset market prices also fell in a market where all investors must follow the rule for investment diversification.3

The structure of our paper is as follows. First, we explain our proposed artificial market used in this study in Sect. 2. Specifically, the order process and the learning process of our artificial market are explained in Sects. 2.1 and 2.2, respectively. In Sect. 2.3, we model the rule for investment diversification in our artificial market. Next, we perform some simulations using our artificial market and analyze the results in Sect. 3. In Sect. 3.1, we assess the validity of our artificial market. In Sect. 3.2, we observe market price transitions of two assets whose fundamental prices are constant in a market where agents must follow the rule for investment diversification and in a market where agents do not follow the rule for investment diversification. In Sect. 3.3, we also observe the market price transitions of an asset whose fundamental price falls and those of the other asset whose fundamental price is stable in these same two markets. Finally, we discuss our conclusions regarding this study and our future work in Sect. 4.

2 An artificial market model

As many artificial market models have many parameters, it has been pointed out that these models are unnecessarily intricate and difficult to evaluate (Chen et al. 2012). The validity of a model is evaluated in terms of its ability to reproduce stylized facts such as a fat tail and volatility clustering obtained by empirical analysis (LeBaron 2006). However, in most cases, neither the precision of the reproduction of a stylized fact nor the number of kinds of stylized facts that can be reproduced increase when the model is made more complicated. As models are made more complicated, their analytic validation and simulation become more difficult. Thus, Chen et al. (2012) suggested that as simple a model as possible that can still reproduce stylized facts should be adopted.

Chiarella et al. (2009) succeeded in structuring a simple agent model which reproduced the statistical properties of the kinds of long-term price fluctuations observed in empirical analyses. Their model was not intended to completely reproduce a real market, but was built to be as simple as possible while still being able to achieve the goal of the analysis, and did not cover all cases, such as investors becoming bankrupt.

Mizuta et al. (2014) added a learning process to the model of Chiarella et al. and reproduced large-scale market confusion such as a bubble or a financial crisis in their artificial market model. Their model also includes investors following different investment strategies, such as fundamental investments or technical investments. Furthermore, no investor sticks to a single strategy when making investment decisions, but rather switches between strategies according to market price information.

The model built in the present study is based on this model.

2.1 The order process

In the proposed model, only two risk assets are available for trading. The mechanism for determining prices is a continuous double auction (continuous trading session) method in which both the seller and the buyer present prices, and if the offered prices from the buyer and seller agree, a trade is immediately concluded at that price.

There are n agents, each of which places an order in sequence, from agent \(j = 1\) through \(j = 2, 3, 4, \ldots \). When the final agent \(j = n\) has placed an order, the first agent, agent \(j = 1\), places another order at the next instant. The time t is incremented by 1 each time an agent places an order. Thus, the process moves one step forward even when a trade does not occur and this new order is placed on the order book. This is referred to as the tick time.

The order prices of agent j by transaction are determined as shown below. The rate of change of the price expected by agent j at time t (the expected return) \({r_e}_{j,k}^t\) is given by
$$\begin{aligned} {r_e}_{j,k}^t = \frac{1}{w_{1,j,k} + w_{2,j,k} + u_{j,k}}\left( w_{1,j,k}\log \frac{{P_f}^t_k}{P_{k}^{t-1}} + w_{2,j,k}{r_h}_{j,k}^{t-1} + u_{j,k}\epsilon _{j,k}^t\right) \end{aligned}$$
(1)
where \(w_{i,j,k}\) is the weight of the i-th term at time t for asset k and agent j, and is set as a uniformly distributed random error between 0 and \(w_{i, \mathrm{max}}\) at the start of the simulation; it is then varied using the learning process described later. Furthermore, \(u_{j,k}\) is the weight of the third term for agent j and is set as a uniformly distributed random number between 0 and \(u_\mathrm{max}\) at the start of the simulation, and then remains constant; \({P_f}^t_k\) is the fundamental price; \(P_{k}^t\) is the market price for asset k at time t and is set to the most recent price if no trading is occurring (\(P_{k}^t = {P_f}^t_k\) at \(t = 0\)); \(\epsilon _{j,k}^t\) is a random error term for agent j and asset k at time t, and is a normally distributed random number with zero mean and standard deviation \(\sigma _{e}\); and \({r_h}_{j,k}^t\) is the past return measured by agent j, given by \({r_h}_{j,k}^t = \log (P_{k}^t/P_{k}^{t-\tau _j})\), in which \(\tau _j\) is a uniformly distributed random number from 1 to \(\tau _\mathrm{max}\) that is set at the start of the simulation.

The first term of the right-hand side of Eq. (1) represents the fundamental strategy, which reflects that an agent expects a positive (negative) return when the market price is lower (higher) than the fundamental price. The second term shows the technical strategy, which reflects that an agent expects a positive (negative) return when the historical return is positive (negative). The third term represents noise.

Based on the expected return \({r_e}_{j,k}^t\), the expected price \(P_{k}^t\) is found by the equation
$$\begin{aligned} {P_e}^t_{j,k} = P^{t-1}_{k}\exp ({r_e}^{t-1}_{j,k}) \end{aligned}$$
(2)
The order price \({P_o}_{j,k}^t\) is set as a uniformly distributed random number from \({P_e}_{j,k}^t - P_d\) to \({P_e}_{j,k}^t + P_d\), where \(P_d\) is a constant.4 The minimum unit for the price limit is \(\delta {P}\), and fractional values smaller than this are rounded down. The choice between buying and selling is determined by the relative sizes of the expected price \({P_e}_{j,k}^t\) and the order price \({P_o}_{j,k}^t\). That is,
  • An agent issues a buy order for one share if \({P_e}_{j,k}^t>{P_o}_{j,k}^t\),

  • An agent issues a sell order for one share if \({P_e}_{j,k}^t<{P_o}_{j,k}^t\).

The mechanism for determining the price in this model is a continuous double auction. As a result, if there are some sell (buy) order prices in the order book that are lower (higher) than the agent’s buy (sell) order price, then the agent’s order is immediately matched to the lowest sell order (highest buy order) in the order book. If there are no such orders in the order book, the order does not match any other order and remains in the order book. The remaining orders in the order book are canceled at \(t_c\).

2.2 The learning process

Previous studies using an artificial market implemented various kinds of learning processes. For example, agents switched strategies and/or tuned their strategy parameters based on their performance, market price, etc. Arthur et al. (1997); Lux and Marchesi (1999); Nakada and Takadama (2013). The learning process in the present study is implemented to switch the strategy between fundamental and technical strategies. We modeled the learning process as follows. Learning is performed by each agent immediately before the agent issues an order. The expected return \({r_e}_{1,j,k}^t=\log ({P_f}_k^t/P_{k}^t)\) is set for only the fundamental strategy, and the expected return \({r_e}_{2,j,k}^t={r_h}_{j,k}^t\) is set only for the technical strategy. When \({r_e}_{i,j,k}^t\) and \({r_l}_k^t=\log (P_k^t/P_k^{t-t_l})\) have the same sign, \(w_{i,j}\) is updated as follows:
$$\begin{aligned}&w_{i,j,k}{\leftarrow } w_{i,j,k} + k_l{r_l}_k^t q_j^t (w_{i,\mathrm{max}} - w_{i,j,k}) \end{aligned}$$
(3)
When \({r_e}_{i,j,k}^t\) and \({r_l}_k^t\) have opposite signs, \(w_{i,j}\) is updated as follows:
$$\begin{aligned}&w_{i,j,k}{\leftarrow } w_{i,j,k} - k_l{r_l}_k^t q_j^t w_{i,j,k} \end{aligned}$$
(4)
where \(k_l\) is a constant and \(q_j^t\) is a uniformly distributed random number between 0 and 1 assigned to each agent j at each time.

Separately from the process for learning based on past performance, \(w_{i,j,k}\) is reset with a small probability m. That is, it is set again using a uniformly distributed random number from 0 to \(w_{i, \mathrm{max}}\). This means that learning is random, and this, in combination with learning based on performance, allows objective modeling of the situation in which agents find the weights of strategies by trial and error.

2.3 The rule for investment diversification model

In this model, we implement a leverage limitation and the rule for investment diversification. The leverage limitation is defined as follows:
$$\begin{aligned} \sum _{k=1}^{2}|P_{k}^t{\times }S_{j,k}^t|&{\le } v {\times } NAV_{j}^{t} \end{aligned}$$
(5)
where \(S_{j,k}^t\) is the quantity of asset k possessed by agent j at time t. Note that agent j takes a long position when \(S_{j,k}^t > 0\) and agent j takes a short position when \(S_{j,k}^t < 0\). Furthermore, v is the leverage ratio and is set as \(v=1\) in this model. If \(C_{j}^{t}\) is the quantity of cash possessed by agent j at time t, then \(NAV_{j}^{t}\), the net asset value of agent j at time t, is given as follows:
$$\begin{aligned} NAV_{j}^{t} = \sum _{k=1}^{2}(P_{k}^t {\times } S_{j,k}^t + C_{j}^{t}) \end{aligned}$$
(6)
Next, the rule for investment diversification is defined as follows:
$$\begin{aligned} \frac{|P_{k}^t{\times }S_{j,k}^t|}{NAV_{j}^{t}} {\le } w_\mathrm{dir} \end{aligned}$$
(7)
where \(w_\mathrm{dir}\) is called the ratio of the rule for investment diversification and is an upper limit on the ratio of the amount of asset k that agent j can own at t to \(NAV_j^t\). When agent j’s order of asset k at t matches an order in the order book but Formula (7) is not satisfied, then this order by agent j is canceled. When Formula (7) is not satisfied for some reason (e.g., when \(P^t_k\) rises much more than \(P^{t-1}_k\)), then agent j is forced to continue to send an order of asset k until Formula (7) is satisfied. The reason that the agent continues to send an order of the asset is as follows. In the real world, when the amount of an asset that is owned exceeds the upper limit on the ratio, if mutual fund managers do not return the amount of the asset to less than the upper limit within a month, they violate the law. Thus, mutual fund managers must send orders of the asset until the amount of the asset does not exceed the upper limit on the ratio.

Hereinafter, we refer to such an order as a rule contravention resolution order.5 We further differentiate this by referring to the order as a rule contravention resolution buy order (RCRBO) if an agent sends a buy order and a rule contravention resolution sell order (RCRSO) if an agent sends a sell order.

3 Simulation results and discussions

We set the parameters as follows: \(n = 1{,}000\), \(k=2\), \(w_{1,\mathrm{max}}=1\), \(w_{2,\mathrm{max}}=10\), \(u_{\mathrm{max}}=1\), \(\tau _{\mathrm{max}}=10{,}000\), \(\sigma _{\epsilon }=0.03\), \(P_d=1{,}000\), \(t_c=10{,}000\), \(t_l=10{,}000\), \(k_l=4\), and \(m=0.01\). We ran simulations to t = 1,000,000.

3.1 Validation of our artificial market

One way of validating an artificial market is to replicate a fat tail and volatility clustering. As many empirical studies have mentioned (Sewell 2006; Cont 2001), a fat tail and volatility clustering appear in actual markets, therefore, we set our artificial market parameters to replicate these features.
Table 1

Stylized facts of an artificial market where the fundamental prices of two assents are stable

\(w_\mathrm{dir}\)

Asset 1

Asset 2

No rule

0.75

0.5

0.25

No rule

0.75

0.5

0.25

Kurtosis

4.72

4.91

5.12

5.87

4.71

4.82

5.42

5.40

Autocorrelation coefficients for square returns

 Lag

  1

0.12

0.12

0.11

0.09

0.11

0.11

0.12

0.10

  2

0.09

0.10

0.10

0.08

0.10

0.10

0.10

0.08

  3

0.08

0.08

0.08

0.07

0.08

0.07

0.08

0.07

  4

0.06

0.06

0.07

0.06

0.06

0.06

0.07

0.06

  5

0.05

0.06

0.06

0.05

0.05

0.05

0.06

0.04

  6

0.05

0.04

0.05

0.05

0.04

0.05

0.05

0.04

Table 1 shows statistics for stylized facts that are averages of 100 simulation runs, for which we calculated the price returns at intervals of 100 time units. All the following figures also use averages of 100 simulation runs. Table 1 shows that both kurtosis and autocorrelation coefficients for square returns with several lags are positive, which means that all runs replicate a fat tail and volatility clustering. This shows that the model replicates long-term statistical characteristics observed in real financial markets.

3.2 The case when the fundamental prices of asset 1 and asset 2 are constant

This section describes simulations for the case in which the fundamental prices of assets 1 and 2 are constant (\({P_f}^t_1 = {P_f}^t_2 = 10{,}000\)). The initial quantity of cash possessed by agent j\(C_j^{0}\) is 100,000 and the initial quantity of asset possessed by agent j\(S_{j,k}^0\) is 0.

Figure 1 shows the price transitions for a market where all agents do not follow the rule for investment diversification (no rule) and a market where all agents must follow the rule (\(w_\mathrm{dir}\) = 0.25, 0.5, and 0.75).
Fig. 1

Asset 1 market price transition when the fundamental price of asset 1 is stable

As can be seen from Figure 1, the market price in the market where agents must follow the rule fluctuates with the fundamental price in much the same way as in the market where agents do not follow the rule. No significant difference is evident. The asset 2 case is not shown, but the same results occur.

There are two likely reasons for these results. One possible reason is that there are few contraventions of the rule. From the market where all agents do not follow the rule for investment diversification, we see that the market prices of both assets are essentially stable. Accordingly, even if the rule of investment diversification is followed, few opportunities for contravention of the rule would arise, and therefore the volume of rule contravention resolution orders would also be low, resulting in a relatively small impact on market price. The other possible reason is that there is an equilibrium between RCRBOs and RCRSOs. In this simulation, the initial quantities of the two assets possessed are 0, so we can assume that the number of agents contravening the rule regarding buying and possessing assets and the number of agents contravening the rule regarding short selling assets become approximately equal. Therefore, since RCRSOs and RCRBOs issued by these agents offset each other, no essential difference in price becomes evident.

3.3 The case when the fundamental price of asset 1 drops sharply

In this section, we examine the price transition of the two assets separately in a market where all agents do not follow the rule and in a market where all agents must follow the rule, when the fundamental price of asset 1 drops sharply (i.e., \({P_f}^t_2 = 10{,}000\) from period \(t=0\) to \(t=1{,}000{,}000\); \({P_f}^t_1 = 10{,}000\) from period \(t=0\) to \(t=100{,}000\) and \({P_f}^t_1 = 7{,}000\) from period \(t=100{,}001\) to \(t=1{,}000{,}000\)), as well as the internal mechanism of the market in this case. In this simulation, the initial quantity of cash \(C_{j}^{0}\) is 50, 000, the initial quantity of assets \(S_{j,k}^0\) is 48, and the ratio of the rule for investment diversification \(w_\mathrm{dir}\) is 0.5, as we investigate simply the effect of a crash of the asset 1 fundamental price on the asset 2 price formation.

Figures 2 and 3 show the price transitions of both assets 1 and 2 in a market where all agents do not follow the rule for investment diversification and in a market where all agents must follow the rule for investment diversification, respectively, when there is a sharp drop only in the fundamental price of asset 1. As shown, in both markets, after a sharp drop in the fundamental price of asset 1, the market price of asset 1 converges to a new fundamental price; however, whereas it converges without overshooting in the market where all agents do not follow the rule for investment diversification, it overshoots the fundamental price in the market where all agents must follow the rule for investment diversification.
Fig. 2

Price transitions of a market where all agents do not follow the rule for investment diversification when the fundamental price of asset 1 declines

Fig. 3

Price transitions of a market where all agents must follow the rule for investment diversification when the fundamental price of asset 1 declines

The reason that the market price of asset 1 in the market where all agents must follow the rule for investment diversification overshoots after a sudden drop in the fundamental price of asset 1 is likely as follows. During trading after the sudden drop in the fundamental price of asset 1, some agents violate the rule for investment diversification because the execution of their buy orders for asset 1 in the order book leads to an increase in the quantity of asset 1 held by them and the ratio of the amount of asset 1 that they own to their NAV exceeds \(w_\mathrm{dir}\). In order to resolve the rule contravention, these agents issue RCRSOs, which are market orders as already explained, for asset 1. Thus, the market price of asset 1 tends to decrease. After the market price of asset 1 has continued to decrease for some time, many agents issue ordinary sell orders on the expectation of a negative return, under the influence of technical strategies. This results in an overshooting of the market price of asset 1.

To verify the correctness of this hypothesis, we examined the trading volume for three kinds of new orders (RCRSO, ordinary sell order, and buy order as determined by Eq. (1)) by agents between periods 100,000 and 200,000 (Fig. 4). Figure 4 shows that after a sudden drop in the fundamental price, the number of RCRSOs gradually increases over time. Thus, the hypothesis seems correct. In practice; however, when the fundamental price drops suddenly, buy orders in the order book may be canceled, so this result cannot be said to apply to real-world markets with certainty.
Fig. 4

Trading volume of asset 1 between periods 100,000 and 200,000

Figure 3 shows that the market price of asset 2 declines along a similar trajectory as that of asset 1, ultimately converging to a value of approximately 8,700.

The reason why the market price drop of asset 2 is coupled to a price drop of asset 1 could be as follows.

First, when the price of asset 1 starts to drop sharply, the agents’ NAVs decrease, to the point that the amounts of asset 2 held by many agents contravene the rule for investment diversification. To resolve these contraventions, many agents start to issue RCRSOs of asset 2, leading to a fall in the market price of asset 2. For a concrete example, consider Fig. 5. We set \(w_\mathrm{dir}\) to 0.5. Assume that an agent’s NAV is 1,000,000, and the amounts of asset 1 and asset 2 that the agent holds are both 50,000, before the sharp fall in the market price of asset 1 (see Fig. 5 left). When the market price of asset 1 falls by 30%, the agent’s NAV drops to 850,000, and the amount of asset 1 that the agent holds drops to 350,000 (see Fig. 5 right). However, since the amount of asset 2 that the agent holds remains unchanged, it now exceeds 50% of the agent’s NAV, contravening the rule for investment diversification. To resolve this situation, a RCRSO of asset 2 is issued for 75,000 units. When a large number of agents take this action at the same time, the market price of asset 2 drops sharply. When the drop in the market price of asset 1 subsides and RCRSOs of asset 2 continue to match buy orders in the book for some time, the rule contraventions relating to asset 2 will also gradually be eliminated. At the same time, RCRSOs will also decline and the market price of asset 2 will stop falling. After this point, the influence of market fundamentals kicks in. That is, asset 2 will be considered comparatively cheap, and agents come to expect positive returns and send buy orders of asset 2, resulting in a rise in the market price.
Fig. 5

Overview of sending a rule contravention resolution order when the market price of an asset (asset 1) falls

Of these above hypotheses about the price transition of asset 2, we first tested the correctness of the proposed explanation for why the drop in the market price of asset 2 follows the same trajectory as that of the market price of asset 1. Here, we examined the trading volume for the three kinds of new orders by agents, as listed previously, from immediately after the fundamental price of asset 1 drops sharply until the drop in the market price of asset 2 subsides (see Fig. 6). Figure 6 reveals that for some time after the fundamental price of asset 1 falls, a large volume of RCRSOs are executed for asset 2. We can also see that the drop in the market price of asset 2 continues until approximately period 200,000, and that the volume of RCRSOs peaks around period 120,000, after which it declines. The likely reason for this is that as trading proceeds, many of the agents with amounts of asset 2 that contravened the rule for investment diversification gradually resolve their rule contraventions. Note that even after the volume of RCRSOs decreases, the market price of asset 2 continues falling for some time. This is probably the result of many agents coming to expect negative returns under the influence of technical strategies and sending ordinary sell orders, which are executed eventually.
Fig. 6

Trading volume of asset 2 between periods 100,000 and 200,000

Next, we discussed the reason for the market price of asset 2 converging to a value that was higher than the minimum market price of asset 2. We examined the volumes of the three kinds of new orders by agents, as listed previously, from the time that the market price of asset 2 reaches its minimum value to the time that the price converges to a constant value (see Fig. 7). Figure 7 reveals that the time between the market price of asset 2 reaching its minimum value and that of the market price stabilizing (between periods 200,000 and 300,000) closely matches the time when the number of ordinary buy orders is higher than the total number of RCRSOs and ordinary sell orders. Moreover, after the market price stabilizes (after period 300,000), the number of ordinary buy orders and the total number of RCRSOs and ordinary sell orders are about the same.
Fig. 7

Trading volume of asset 2 between periods 200,000 and 500,000

This result suggests that our hypothesis about the price transition of asset 2 is correct. However, the result that the market price of asset 2 converges to a price lower than the fundamental price would probably not be observed in a real market. This is because all agents in this simulation must follow the rule for investment diversification, whereas a real-world market also includes many investors who need not follow this rule. The fact that the market price of asset 2 drops in Fig. 2 seems to support the previous statement. Therefore, further study is needed to understand what happens when the model is modified so that not only agents that must follow the rule for investment diversification, but also agents that need not follow the rule (e.g., individual investors or institutional investors who do not own mutual funds) participate in the market. Moreover, a detailed verification regarding the convergence of the market price of asset 2 to a value of around 8,700 is a task for future study.

4 Conclusion

In this study, we investigated the impact of the rule for investment diversification on markets when one asset price drops sharply, using artificial market simulations. More specifically, we conducted simulations for the case when the fundamental prices of asset 1 and asset 2 are constant and when only the fundamental price of asset 1 drops sharply in the course of trading. The results showed that when the fundamental prices of asset 1 and asset 2 are constant, there is no evident difference in price transition between a market in which only agents that do not follow the rule for investment diversification participate and a market in which only agents that must follow the rule for investment diversification participate. In addition, we confirmed that when the fundamental price of asset 1 drops sharply, the market price of asset 1 in a market in which only agents that must follow the rule for investment diversification participate reaches the fundamental price after the sharp drop and then overshoots it. We also confirmed that in a market in which agents that apply an investment diversification rule participate, the drop in the market price of asset 2 follows a trajectory parallel to the fall in the market price of asset 1.

Some further questions that need to be explored are as follows. First, how can it be arranged that orders by agents contravening the rule for investment diversification do not remain in the order book, and how can agents that need not follow the rule for investment diversification participate in the market in such a way that their orders remain in the order book. Next, we wish to determine whether changing the initial quantity of cash or initial quantity of assets held by agents affects the market influence of the rule for investment diversification. Then, we will assess the effects on price formation in financial markets where investors must follow the rule at the time of a market crash that is due to a price deviation from the fundamental price. Finally, we will assess the influence of the rule for investment diversification on the stability of the market.

Footnotes

  1. 1.

    The rule for investment diversification applies to investment trusts in Japan and mutual funds in Europe, but most investment trusts in Japan are financial products for individual investors. These trusts are similar to mutual funds in Europe. Therefore, we call them mutual funds herein for simplicity.

  2. 2.

    Active Share can be easily interpreted as the “fraction of the portfolio that is different from the index”.

  3. 3.

    The crash occurs not only due to the sudden decrease in the fundamental price but also due to the price deviation from the fundamental price, such as a flash crash due to an erroneous order. When a crash caused by a price deviation from the fundamental price occurs, the market price converges to the fundamental price relatively quickly; i.e., it returns to the market price before the crash. Therefore, although mutual funds may violate the rule for investment diversification temporarily, there is a high possibility that the violation of the rule for investment diversification will be resolved automatically in the medium and long term. On the other hand, if a crash caused by the sudden decrease in the fundamental price occurs, then the market price converges to the new fundamental price, i.e., a price lower than the market price before the crash. Therefore, the mutual fund manager must act to ensure that mutual funds do not violate the rule for investment diversification. For this reason, we investigated the effects on price formation in financial markets where investors must follow the rule for investment diversification at the time of a market crash that was caused by the collapse of an asset fundamental price.

  4. 4.

    Investors do not always place orders according to the expected prices, because their investment behaviors tend to be affected by market conditions and they determine their order prices as higher or lower than the expected prices by taking into account market conditions. To express such investment behaviors, we set the order price uniformly and randomly. In real financial markets, there are many limit orders in the order book. When a trade does not occur, the new order is placed in the order book; i.e., it becomes a limit orders in the order book.

  5. 5.

    The rule contravention resolution order is a kind of market order.

Notes

Acknowledgements

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

Disclaimer

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

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Copyright information

© Japan Association for Evolutionary Economics 2017

Authors and Affiliations

  1. 1.Faculty of Information TechnologyKanagawa Institute of TechnologyAtsugiJapan
  2. 2.Course of Information and Computer SciencesGraduate School of Kanagawa Institute of TechnologyAtsugiJapan
  3. 3.SPARX Asset Management Co., Ltd.TokyoJapan

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