The majority of links have weights equal to one but links with weight equal to two are also observed. In Table 1 we report summary statistics of the clusters of investors detected in the FDR networks with the community detection algorithm. The number of clusters and their size in number of investors is varying over time. The size of the clusters of investors observed is ranging from the minimum value of 2 to the maximal value of observed in Clusters of size larger than investors are observed during the period from to and in Table II of the SI shows that the number of active investors is varying by more than one order of magnitude.

In fact it is ranging from in to in The statistical test used to obtain statistically validated networks has a power that is depending on the number of nodes i. In the next subsection we investigate whether the power of the test affects our results.

The first evidence that results of statistically validated networks are not affected by the power of the test can be concluded by analyzing the ratio of validated links to the total links observed for the different calendar years. This ratio is shown in Fig. It is worth noting that although the year with the highest number of observed links is see Table II of SI and the power of the test is expected to be maximum for this year, the highest value of the ratio is observed in In order to assess the role of the power of the test in a rigorous way, we have designed the following numerical experiment.

For each year, we have drawn ten random samples of fixed size from the pool of all the links that are present in the projected network of investors. When drawing the samples we have maintained the proportion of links among investors of the different categories as observed in the original set.

For each random sample, we have obtained the corresponding statistically validated network. Figure 2 plots the ratio of validated links to total links both for the randomly selected samples blue points and for the whole system red points. The random samples used in our numerical experiment contain all 1,, links and therefore the test has the same power. The figure shows that the power of the test does not affect the estimation of the ratio of validated links to total links when we consider samples with a number of links equal to or larger than 1,, and we use the FDR correction.

Ratio of validated links as a function of the year. The samples investigated for the test contain 1,, links each. With our approach, for each calendar year we detect clusters whose investors are characterized by a high degree of similarity in their trading decisions. To quantify the similarity of collective trading decisions of different clusters we devise the following procedure.

For each cluster we compute a vector where each record counts the number of investors of the cluster that are in state b , s , or bs each trading day of the calendar year. Such a vector has approximately records the exact number depends on the exact number of trading days of the considered year.

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A simple and efficient way to highlight the main similarities present between the investment activity of different clusters is through the minimum spanning tree MST associated with the correlation coefficient matrix of all clusters of a given calendar year Mantegna, In Fig. Figure 3 is representative of the MSTs observed in different years. In the figure the first number of the cluster label is a numeric label and the second number are the last two digits of the calendar year.

Minimum spanning tree of the similarity matrix associated with the trading activity of clusters for year Each cluster is labeled by a number. Symbols in colors indicate the over-expression of one or more category of investors. Colors refer to the different categories as follows: non-financial corporations green , financial and insurance corporations red , general governmental organizations gray , non-profit organizations yellow , households cyan , and foreign organizations brown.

The size of each node is proportional to the logarithm of the number of investors that are in the cluster.

In our analysis, we also investigate whether each cluster presents an over-expression of the number of investors of a given category. The statistical test used to perform this kind of analysis is described in Tumminello et al.

## Algorithmic and high frequency trading | Mathematical finance | Cambridge University Press

It should be noted that the result of this statistical test is providing the over-expression of the number of investors belonging to a specific category with respect to a null hypothesis assuming a heterogeneous number of investors in the different categories. When the over-expression is detected, we label the symbol of the cluster with a given color. In the horizontal axis we have distinct investors of the cluster whereas we have the trading day in the vertical axis.

In the figure we label a buying day of an investor with a green spot, a red spot indicates a selling day, a white spot is used when the investor performs a buy-selling activity during the day, and a black spot indicates the absence of trading. Visual inspection of Fig. The most evident ones concern the frequency of trading, the number of investors, and the specific sequence of buy, sell, and buy-sell trading decisions.

Similarity of the profile is often related to synchronous buying green lines and selling red lines trading decisions.

## Markowitz’s Modern Portfolio Theory - What Is It & How It Works

However synchronous buying and selling decisions also have a prominent role for some clusters. We plot investors in the horizontal axis and time in vertical axis in trading days from top to bottom. A green spot indicates a primarily buying day, a red spot a primarily selling day and a white spot a buy-selling day. When no trading is performed we use a black spot. To relate investors of a cluster in a given year to investors of clusters in the successive year, we use a statistical test of the over-representation of the number of investors that are present in both clusters against a null hypothesis that takes into account the heterogeneity of the size of the clusters.

We perform the test as indicated in Marotta et al.

The test is performed on all clusters with more than five investors. For the sake of completeness, hereafter we briefly describe the procedure of the statistical test about time dynamics of clusters. The p -value is. This methodology Marotta et al. When the p -value is below a statistical threshold we connect with a directed link the clusters selected by the statistical test.

### 18.2 – Volatility based stoploss

Also in this case we are performing a multiple comparison procedure and therefore a multiple hypothesis test correction is necessary. We perform these tests for all pairs of consecutive years from to by using the control of the FDR procedure with a threshold given by. Roughly twenty percent of clusters with more than 5 investors are connected with at least one cluster of next calendar year. We address these clusters whose time evolution is validated by a statistical test as chained clusters. Several of these chained clusters have a large number of investors. In fact chained clusters comprise approximately from 20 to 50 percent of the number of investors that are present in clusters with more than 5 investors.

The details about the number of clusters and investors for both chained and all clusters are given in Table 1. When several clusters coalesce, they are characterized by a similarity between pairs of them that is typically higher than the similarity with other clusters. This is reflected in the corresponding MST where these clusters are located in a closely connected subnetwork of the tree.

Time evolution of the clusters detected in the FDR networks. Clusters are represented by a square labeled with a numerical index and the year. Chains of cluster evolution lasting several years up to 12 years are observed. Splitting and merging of clusters are also observed. Colored clusters are clusters characterized by an over-expression of one or more categories of investors.

For several chains of clusters, the dynamics of the chains present regularities with respect to the type of investors over-expressed in the clusters.

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We see that several chains of clusters show a persistent over-expression of specific categories of investors. In all clusters of the chain we observe over-expression of governmental organizations clusters labeled with gray with the additional over-expression of non-profit institutions clusters labeled with yellow color in some years. The results summarized in Fig. Moreover, some of these chains are characterized by over-expression of one or two categories of investors.

In the next section, we analyze in detail some of these chains of clusters that are simultaneously present in the market. Our analysis shows that their trading decisions are markedly different from one another, confirming the existence of groups of investors simultaneously acting in the market with different approaches and strategies for long periods of time.

We analyze 4 different attributes that are chosen to characterize different aspects of investors and of their chosen trading strategies. The attributes we consider are: i the average pairwise distance d i , j between vectors of individual trading decisions of investors belonging to a cluster or to a group of clusters. Footnote 2 ii the average value of the ratio of number of trading days coinciding with earning announcement days divided by the total number of trading days, iii the average value of the number of stocks each investor of the cluster or group of clusters is investing in, and iv the average trading frequency of investors of the cluster or group of clusters.

The trading frequency of an investor is the ratio of the number of trading days he or she performed in the considered year divided by the total number of trading days of that year. Therefore, a trading frequency equal to one indicates trading activity of an investor that performed all trading days of the year. The average distance gives us information about the degree of dissimilarity observed between the trading decisions of each pair of investors in a cluster. The average value of the ratio of earning announcement trading days provides information concerning the relevance of these special days in the trading decisions.

We consider the average number of stocks owned by investors as a proxy of their knowledge about basic financial concepts such as investment diversification. The average trading frequency tells us information about the time horizon of investors during the year.

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In Table 2 we report the year average of the yearly average value of the four attributes for each category of investors. The Table shows that the average number of stocks in the portfolio of an investor is quite different for the different categories whereas the average distance between pairs of trading and the average of the ratio of earning announcement trading days is quite similar for all categories. The average frequency of trading is also quite similar for all categories of investors with the exception of financial corporations that are showing an average frequency of trading an order of magnitude higher than the other categories.

All points and segments are provided in gray with the exception of tree groups of information referring to three specific cluster chains that are provided with crosses in color.