Optimization of investment portfolios of Chinese commodity futures market based on complex networks

Futures trading in developing countries is now attracting more attention since investors may easily generate more excess return compared to the markets in developed countries, especially in Chinese market. In this paper, we analyzed the relationship between the centrality of commodity in the Chinese commodity futures market network and the optimal weight of each commodity in a portfolio, empirically examined the market systemic factors and commodity idiosyncratic factors that affect the centrality of commodity, and evaluated the effect of network structure on the optimization of commodity portfolio selection under the mean-variance framework. We found that the commodities with high network centrality are often related to industrial products with high volatility and small portfolio weights. We put forward a kind of commodity futures investment strategy based on this network and results showed that cumulative yield is better than other benchmark portfolios. The main contribution of this paper is to apply complex network theory to optimize futures portfolio selection by establishing the relationship between portfolio weight and commodity centrality in Chinese market, which is still under explored.


Introduction and literature review
Markowitz laid the foundation of modern investment portfolio theory, which jointly explained the mechanism of asset diversification and constructed the effective frontier of portfolio from the two aspects of expected return and portfolio variance [1].In this framework, investors only consider the mean and variance of the return distribution in a group of assets for optimal allocation.Some studies have pointed out that Markowitz's framework underperforms expectations outside the sample, mainly due to the large estimation of the expected return vector [2] and covariance matrix [3] error, leading to the well-documented error-maximization property discussed by Michaud and Michaud [4].Furthermore, there is evidence that simpler equal-weighted portfolio rules outperform even mean-variance and some more complex portfolio methods [5].
Since Mantegna [6] pioneered research on the complex network of the stock market, network research on financial markets has gradually intensified, and research using graph theory and network analysis as tools has emerged [7][8][9], the common method of constructing the network is minimum spanning tree [10], threshold cut [11], planar maximum filter graph [7], wavelet correlation [12] and etc.Despite the novel insights gained from these network-related papers, the results of most studies are still descriptions of network structures or use network structures as intermediate results of other studies, lacking specific applications in the portfolio selection process.In recent years, research on financial markets based on complex networks has deepened and expanded to research on investment portfolios [10,[13][14][15] pointed out how the topological positions of stocks affect the performance of the constructed portfolio, and the performance of the marginal stock portfolio is better than that of the core stock portfolio.Peralta and Zareei [16] established the connection between complex networks and optimal portfolios, and pointed out that under the 'mean-variance' framework, the centrality of stock market network stocks has a negative correlation with their optimal weights.Li et al [17] took the US S&P 500 and China's Shanghai and Shenzhen 300 constituent stocks as samples, constructed a portfolio of 10-30 stocks, and adopted edge nodes, randomly selected nodes, and centrality node strategies to investigate performance, finding that edge node performance outperforms index benchmark returns and other strategic returns.Mandac et al [18] determined optimal hedging ratios and portfolio weights for commodity investors and portfolio managers based on risk transfer networks among stocks, bonds, and commodities.The study found that for equity market volatility investors, shorting energy futures can achieve the highest hedging effect, while for bond and U.S. dollar volatility investors, shorting metals futures can achieve the highest hedging effect.Chen et al [19] constructed a complex network based on correlations of abnormal stock returns and highlighted centrality is a useful tool to detect inter-industry correlation and find high returns.Xiao et al [20] also selected futures portfolio based on some centralities based on complex network theory.These references all point out that complex network could be a potential option which could facilitate portfolio construction.
However, there is no research involving network topology as an effective tool to optimize the portfolio selection process in the commodity futures market, especially for China's commodity futures market, so the relevant research is still blank.The commodity futures market is an important part of the derivatives market and even the capital market.The global commodity futures market has experienced violent fluctuations in the post-financial crisis period, and the risk of transmission of fluctuations among various commodity prices has continued to rise.Co-movements in commodity prices during periods of market booms and busts can increase systemic financial risk [21], posing challenges for regulators and investors seeking effective hedging strategies and optimal portfolios.Commodity futures have a low correlation with traditional financial assets and are an important asset allocation category, and their price trends have been closely watched by policymakers and analysts [21].Financialization has become an important commodity in recent years.A new attribute of the market may increase the degree of correlation between commodities [21,22], endowing commodity prices with new characteristics of linkage.Therefore, a deep understanding of the price relationship and commodity portfolio optimization in the commodity futures market has very important theoretical significance and application value for market regulators, arbitrage hedgers and speculative traders in the commodity futures market.On the one hand, it helps to understand the root cause of the linkage between commodities and the investment decision-making mechanism of market participants, which is of great benefit to market supervision; on the other hand, by understanding the correlation structure between commodity prices and constructing a portfolio that can obtain stable returns is beneficial for traders to obtain excess returns.In view of the possible relationship mechanism between the centrality of commodity futures price volatility network and the weight of the optimal commodity portfolio, the study of the relationship mechanism between the two has two meanings.First, we can deepen and expand the application depth of volatility network research, extending from the empirical description of volatility network to investment management; second, we may discover low-correlation, low-risk commodity clusters, and discover commodity combinations that can effectively diversify risks.
The purpose of this paper is to establish the connection between the complex network structure of the commodity futures market and the process of investment portfolio selection in Chinese futures market.There are several market differences between China and USA market, like trading varieties, trading hours, etc. which are most important features making this study quite unique.Firstly, China's commodity futures markets implement a so called 'penetrating regulation' system, that is, in principle, the exchange can monitor every futures account opened through a futures company broker, while this is not the same in USA.Secondly, China's futures market is dominated by retail investors, while the US futures market is dominated by various institutional investors, such as funds, banks and so on.This difference in investor structure may have an impact on trading behavior and price volatility in the market.
Taking the intraday minute-level high-frequency data of the Chinese commodity futures market as a sample, the volatility network of commodity futures prices based on the minimum spanning tree structure is established to establish modern investment theory and theoretical mechanism of the application of graph theory in the commodity futures market.This paper empirically tests the impact of commodity category and market factors on the centrality of commodity networks, constructs investment portfolios according to the differences in commodity centrality, and uses the relationship between network centrality and commodity portfolio centrality to configure commodity portfolios.The key answer is in the commodity futures market.Above all, whether the results of network analysis can be used as an effective investment tool to improve the performance of investment portfolio is a key issue.Finally, this paper uses out-of-sample data and transaction costs such as adding handling fees and slippage shocks to test the conclusions of this paper.The research results of this paper have important implications for the policy discussion of market risk and the risk management of commodity investment portfolio during the period of market boom and bust.
The research questions and research results of this paper are innovative to a certain extent, and it is the expansion and deepening of the literature on the application of modern portfolio theory and network analysis in financial markets.First of all, this paper studies the optimal commodity portfolio problem from the perspective of network analysis, and considers the commodity portfolio as a network to solve the asset allocation problem, in which each commodity futures contract is a node of the commodity network, and the link represents the commodity futures price-yield relationship structure.This paper establishes the connection between modern investment portfolio theory and China's commodity futures market network analysis, and theoretically proves that there is a positive correlation between the market performance of the optimal portfolio based on the commodity network and the weight of the optimal commodity portfolio.The optimal weights from the 'mean-variance' framework can be interpreted as the optimal trade-off between the system risk and the idiosyncratic risk of the commodity.Secondly, this paper establishes an investment strategy based on the centrality of commodity networks, showing that there is a negative relationship between the optimal portfolio weights and the centrality of commodities in the commodity futures market network.In the commodity network, commodities with high centrality are deeply embedded in the network constructed based on correlation, because these commodities are key and core nodes and have a great influence on the market.
The rest of this paper is organized as follows.The second part introduces the modern investment portfolio and Markowitz's 'mean-variance' framework, as well as the relationship between the centrality of the commodity futures market network and the weight of the optimal commodity portfolio; the third part is data description and descriptive statistics; the fourth part examines the relationship mechanism between the commodity network centrality and the optimal portfolio weight; the fifth part expounds the investment strategy based on network analysis; the final part is the conclusion.

Mechanism of relationship between commodity portfolio weight and network centrality
The concept of network centrality in financial markets comes from complex network analysis.The purpose of establishing a commodity futures market network is to quantitatively evaluate the influence and importance of commodity nodes in a given network.This paper conducts research according to the following steps: (1) Construct the price volatility network of China's commodity futures market using intraday minute-level data; (2) Calculate static and dynamic network topology indicators, such as eigenvector centrality; (3) Propose the commodity futures market network theoretical assumptions of centrality and optimal commodity portfolio weight; (4) Propose a commodity portfolio optimization strategy based on network centrality; (5)Test the commodity portfolio based on network centrality under the ideal scenario without transaction costs and the scenario with transaction costs.Flowchart 1 demonstrate our full procedure.

Commodity futures market network centrality and optimal commodity portfolio weight
In this paper, G = {N, Ω} is used to represent a network consisting of a group of commodity futures as nodes, that is, N = {1, 2, . . . . . ., N} and a group of links ω connecting node pairs Commodity Futures Market Network.If there is a link between nodes i and j, it is expressed as When the node j in Ω ij and the node i G is a directed network when there is a causal connection among them.Since (i, j ∈ w) does not necessarily mean (j, i ∈ w)), there is a possibility of Ω ̸ = Ω T .If it is unweighted network, then Ω ij ∈ {0, 1}, so the network only exists whether there is an association relationship.When there is strength in the interaction between nodes in Ω ij , it is called a weighted network.In this study, the focus of our research is the relationship between the centrality of commodity network and investment portfolio.Therefore, this paper uses commodity futures price correlation to establish the static and dynamic correlation matrix between commodities, and constructs the adjacency matrix accordingly.We are not very concerned about the causality of price transmission between commodities.So we use undirected graphs and non-causal networks.
In network analysis, centrality is a key indicator used to describe network topology.Bonacich [23] proposed eigenvector centrality, which has become the core centrality indicator in network analysis.This part proposes the relationship between the eigenvector centrality of the commodity futures market network and the optimal commodity portfolio weight.The eigenvector centrality v i of a node i is defined as the proportional sum of the centralities of its neighbors.Newman and Girvan [24] gave the mathematical definition of eigenvector centrality, assuming that the undirected weighted network G = {N, Ω}, N is the node set, and Ω is the adjacency matrix.The eigenvector centrality v i of node i in G is proportional to the ith component of the eigenvector and the largest eigenvalue λ 1 corresponding to the ith component Ω of the eigenvector, the mathematical form is as follows: The greater the centrality of the eigenvector of node i in the network, it reflects that the number of nodes linked by the node is larger or it is linked with other high centrality nodes.Expressing formula (1) in matrix form gives λv = Ωv, indicating that the center vector v is given by the eigenvector Ω corresponding to the largest eigenvalue λ.

Review of the main achievements of modern portfolio theory
The pioneering research of Markowitz laid the cornerstone of modern portfolio theory, and constructed investment portfolios under the benchmark framework of modern portfolio theory.The following briefly reviews two fundamental results of modern portfolio theory: the minimum variance and mean variance investment rules.The following briefly reviews two fundamental results of modern portfolio theory: the minimum variance and mean variance investment rules.Suppose there are n kinds of risk assets, the expected return vector is µ, and the covariance matrix is Σ = [σ ii ].Consider the problem of finding the optimal portfolio weight vector ω, so that the variance of the portfolio conforming to ω T 1 = 1 is the smallest.The 'mean-variance' strategy is introduced by the mean-variance model.This strategy mainly solves the following two problems: one is to obtain the maximum expected return given a certain risk; the other is to minimize the risk given the maximum expected return.Compared with the 'minimum variance' strategy, the 'mean-variance' strategy considers both the return and risk of the portfolio.For the 'minimum variance' strategy: Solve formula (2) with the optimization method to obtain the combination weight of the optimal 'minimum variance' strategy: Use Ω to represent the correlation matrix of income, and use ∆ to represent the diagonal matrix whose ith main diagonal element is σ i = √ σ ii , then the relationship between Ω and Σ can be written as Σ = ∆Ω∆.Then formula (3) is re-expressed according to the correlation matrix as follows: For the 'mean-variance' investment rule, this paper introduces a risk-free asset with return r f , defines the excess return of asset i as r e i = r i − r f and the expected vector of excess return is u e .Under the given level of expected excess return R e of the portfolio, the problem of finding the optimal portfolio weight that minimizes the variance of the portfolio is expressed as: Equation ( 5) is the standard 'mean-variance 'strategy.Since investors' wealth can be partially allocated to risk-free assets, and risk-free assets are allowed to be sold short, formula (5) does not contain the constraint condition w T > 0. The optimal 'mean-variance' combined weight is obtained by the following formula: Or in the form of matrix: where

Theoretical assumptions of network centrality and optimal commodity portfolio weight in commodity futures market
This paper establishes a theoretical relationship between commodity futures price networks and commodity portfolios.For N commodities, given the return correlation matrix Ω, the commodity futures market network CMN = (N, Ω) can be obtained, where N is the network node, and the adjacency matrix transformed by Ω.Since the assignment of a node to itself has no meaning in a market network, the main diagonal of Ω is assumed to be 0.This paper extends the research idea of Peralta and Zareei [16] to the commodity futures market, and divides the characteristics of each commodity into two dimensions: system and individual.The system dimension is represented by the centrality of the eigenvector of the commodity in the volatility network, which is used to measure the position of a commodity in the volatility network, measured by the centrality of the commodity in the market network.The individual dimension specifically refers to the market factors of the commodity.These factors have the individual characteristics of the commodity, including the standard deviation of yield, Sharpe ratio, and etc.The optimal weight under the 'mean-variance' framework is actually the best trade-off between systemic factors and individual factors in the volatility network of each commodity.Assume CMN = (N, Ω), where {v 1 , v 2 , . . ., v n } and {λ 1 , λ 2 , . . ., λ n } represent the set of eigenvectors and eigenvectors of Ω, then the optimal weight in ( 7) can be written: where In equation ( 8), μe M is the weighted average of the return rate and the inverse of the standard deviation of the Sharpe ratio, which can be interpreted as variables at the market level.Starting from the first item in formula (8), which is the individual dimension of the commodity, it can be considered that in the commodity portfolio, a commodity with a lower return standard deviation or Sharpe ratio corresponds to a higher portfolio weight, and vice versa corresponds to a lower portfolio weight.The second item reflects the relationship between the system factors of commodities and the optimal weights.There is a negative correlation between the optimal commodity combination weight and the centrality of commodities in the network.Therefore, this paper puts forward the following assumptions under the 'mean-variance' model: Hypothesis 1.The weight of the optimal commodity portfolio is related to the systematic factors of the commodity, and the network centrality of commodity futures price volatility is inversely proportional to the weight of the optimal commodity portfolio.
The mathematical expression of hypothesis 1 can be described as follows.Assume CMN = (N, Ω), where {v 1 , v 2 , . . ., v n } and {λ 1 , λ 2 , . . ., λ n } represent the set of eigenvectors and eigenvectors of Ω, then: The economic implications of the above theoretical assumptions are very intuitive.Commodities with higher network centrality have a smaller proportion in the investment portfolio, while commodities with lower centrality have a larger proportion.It is easy to understand that if a certain commodity has a higher centrality in the commodity volatility network, it means that it has a greater impact on the market.If there are more commodities with higher centrality in the portfolio, it is likely to destroy the benefits of portfolio diversification, resulting in greater variance in the portfolio.
According to the modern portfolio theory, a better portfolio is one that has the least risk for a given return or the greatest return for a given risk.Obviously, at the same level of return, choosing assets with a smaller standard deviation of return is in line with this theory.At the same time, the Sharpe ratio is the ratio of the difference between the expected rate of return and the risk-free rate of return of the investment portfolio to the standard deviation of the expected rate of return of the investment portfolio.Based on the Sharpe ratio definition, obviously the higher the ratio, the better the portfolio.Therefore, the optimal portfolio weights towards assets with higher Sharpe ratios.Therefore, we have hypothesis 2. Hypothesis 2. Under the 'mean-variance' framework, the weight of the optimal commodity portfolio is related to the individual market factors of each commodity, inversely proportional to the volatility of the commodity, and positively proportional to the Sharpe ratio.The formal description is as follows: For μe M ∈ R, r = mv, we have In summary, the theoretical logic analysis above shows that investors who want to find the optimal commodity portfolio are more inclined to hold commodities with low-centered, low-volatility and with a high Sharpe ratio.This result is similar to Pozzi et al [15]'s view that the optimal portfolio strategy based on the US stock exchange market allocates more weight to low-center stocks and less weight to high-center stocks.The difference is that the study of this paper extends this mechanism to China's commodity futures market.

Data explanation and descriptive statistics
As of December 2021, there are a total of 61 contracts in China's commodity futures market.We obtained the daily frequency and intraday 5 min frequency closing price and trading volume of 47 commodity futures price indices from JoinQuan.The reason we choose only 47 futures is because these 47 futures contracts are most active, while the others are not.And these inactive contracts are not as important as those 47 futures in real economy.The sample data is from 9 December 2019 until 30 December 2021.Since a single futures contract has an expiration date, splicing contracts to construct a continuous price series is one of the thorny problems in basic data processing.Usually adjacent futures contracts are rolled forward, that is, only holding futures contracts with the latest expiration date to construct a continuous time series of futures prices is a common method [25].However, in practice, such splicing contracts will still encounter 'price gaps' , which is not the best solution.In order to avoid the problem of data discontinuity caused by contract splicing, we directly use the commodity price indices provided by JoinQuant to generate continuous time series futures prices, thus avoiding the disadvantages of splicing contracts in different ways, while maintaining the maximum value of each commodity trend structure.
We use intraday 5 min high-frequency data to construct and verify a network-based commodity portfolio strategy, the main reasons are as follows.The listing time of each commodity futures contract is different.If daily low-frequency data is used, it is difficult to unify the starting date of the sample.In the case of uniform start dates, the use of high-frequency data provides a larger sample of data.Due to the difference in commodity trading time, some contracts have night trading, the trading time is 9:00-11:30 in the morning, 1:30-3:00 in the afternoon, and 9:00-11:00 in the evening, but some contracts do not have night trading, so we delete the night market data and missing value samples.Each trading day has a total of 45 5 minute highest price, opening price, lowest price, closing price, trading volume and open interest data.All commodity futures include eight precious and non-ferrous metal futures (gold, silver, copper, aluminum, zinc, lead, nickel, tin), six ferrous metal futures (rebar, iron ore, hot rolled coil, stainless steel, silicon iron, manganese silicon), six types of energy futures (coal, coking coal, thermal coal, crude oil, fuel oil, bitumen), two types of light industry futures (glass, pulp), 11 types of chemical futures (rubber, plastic, Purified terephthalic acid, polyester staple fiber, polyvinyl chloride, ethylene glycol, methanol, polypropylene, styrene, urea, soda ash), ten kinds of grain oil futures (corn, soybean, starch, soybean meal, soybean oil, rapeseed meal, japonica rice, rapeseed oil, palm oil, canola oil), three types of soft commodity futures (cotton, white sugar,), three types of agricultural and sideline product futures (eggs, apples, dates).Figure 1(a) draws a box diagram of the Sharpe ratio, which gives the risk-adjusted return distribution.It is easy to find that the distribution of the Sharpe ratio of each commodity in the entire commodity network is not constant, and the influence of changes with the centrality distribution.Although not significantly different in mean, the distribution of Sharpe ratios shrinks with greater centrality, indicating greater risk for high centrality commodities.Figure 1(b) is the normalized price trend of each commodity.

Empirical test on the relationship between network centrality and optimal commodity combination weight
One of the main application scenarios for studying commodity futures market networks is to provide new insights into commodity portfolios from the perspective of complex networks, and to find commodity portfolios that can effectively diversify risks.The mainstream quantitative strategies in the commodity futures market are trend-following strategies and arbitrage strategies.How to optimize the weight of commodity portfolios in the strategies to diversify risks is a crucial issue.
This part conducts empirical research on the theoretical hypothesis of the relationship mechanism between the commodity futures price network and the commodity portfolio weight proposed in the second part above.Figure 2(a) shows the relationship between the network centrality (X-axis), Sharpe ratio (Y-axis) of each commodity and the commodity combination weight (Z-axis) under the 'mean-variance' model in the commodity network.The two respectively represent the system dimension and the individual dimension of the commodity in the commodity network.It is easy to see that the commodity with a high weight in the investment portfolio has a small centrality and a large Sharpe ratio.Figure 2(b) shows another set of relationships, commodity network centrality (X-axis), return standard deviation (Y-axis) and commodity portfolio weight (Z-axis).For a few commodities, the portfolio weight is still inclined to low centrality and low volatility commodities.Overall, the plot shows a relationship sloping to the upper left, suggesting that optimal weights under the mean-variance framework are associated with high Sharpe ratios and low yield volatility, with commodities with a high center tending to correspond to higher systematic and individual risk, so the funds in the commodity portfolio are mainly allocated to the commodities in the upper left corner of the chart, and greater capital allocation is given to low-risk commodity groups with low centrality and low volatility.
More in-depth, the market information embedded in commodity centrality can be used to explain and identify the optimal weight driving factors in commodity combinations.In this part, we first use the  cross-sectional data to test the hypotheses 1 and 2 proposed in the second part above, and establish the following econometric equation: w * i,mv in equation ( 9) are the weights of different commodities; the network centrality, standard deviation of returns of different commodities and Sharpe ratio are represented by v i , σ i and SR i .
Table 1 shows the cross-sectional OLS estimation results.Portfolio weights under the 'mean-variance' framework are linearly dependent on their volatility network centrality, return standard deviation, and Sharpe ratio.Since the weight of the investment portfolio is relatively concentrated, this paper uses logarithmic processing to reduce the effect of heteroscedasticity through linear transformation.Commodity weights are negatively correlated with commodity network centrality and positively correlated with Sharpe ratio.The signs of the two explanatory variables are in line with assumptions.The centrality of commodities is statistically significant only at the 90% confidence level, and the Sharpe ratio is at the 99% confidence level.Significantly, the return standard deviation is statistically insignificant although the sign is in line with expectations.The regression results show that if the commodity allocation strategy is guided by the 'mean-equation' model and the commodity network relationship, commodity allocation to low centrality, high Sharpe ratio and low standard deviation of return is the main mode of sample performance.Specifically, there is an inverse relationship between commodity network centrality and portfolio weight, so hypothesis 1 is verified.There is a positive relationship between the Sharpe ratio and the portfolio weight, but the inverse relationship between the return standard deviation and the portfolio weight is not statistically significant, so hypothesis 2 is weakly supported.
In addition to the static conclusions of the cross-sectional regression, a more detailed dynamic analysis of the portfolio selection can be carried out through the panel regression method.Since this paper uses intraday high-frequency data, it is possible to construct a network on each trading day and calculate the statistical characteristics of the daily network, such as eigenvector centrality.This approach avoids the disadvantages of subjective window size selection when using overlapping sliding time windows to construct dynamic networks on daily low-frequency data.This paper first constructs two centrality indicators, the dynamic average eigenvector centrality v t and the dynamic weighted eigenvector centrality v r,t of the entire sample of 47 commodities.The specific algorithm is as follows: In this paper, π t and ρ t are used to denote the cross-correlation of v it and σ it and v it and SR it with time t, respectively.Considering the mean-variance strategy, when π t > 0 commodities with lower volatility tend to be matched with commodities with less centrality.In this case, it is a better choice to allocate funds to such commodities.When π t < 0, commodities with low network centrality correspond to commodities with high volatility.At this point, the optimal portfolio rule should adjust the portfolio to include more 'key commodities' with high centrality.For ρ t < 0 commodities with the highest Sharpe ratios correspond to commodities with lower centralities, indicating the choice of 'marginal commodities' as optimal portfolio choices.When ρ t > 0, the funds of the portfolio weight should be allocated to commodities with higher allocation centrality.Due to the noise of the two indicators, this paper uses the Hodrick-Prescott method which is often used in macroeconomics to filter the indicators and enter the regression equation.
Table 2 reports the panel mixed OLS estimates, noting that all coefficients are highly statistically significant.The negative coefficient α 1 indicates that a higher v t promotes the portfolio to favor low-centrality commodity allocation.Due to the deeper embedded network of high-centrality commodities, it reduces the risk of high-centrality commodity allocation to avoid portfolio concentration.The sign of α 2 is negative, which can be interpreted as after controlling v t , adding commodities with low centrality in the commodity combination will bring benefits.Such commodities are marginal commodities in the network, and configuring marginal commodities in the combination is conducive to decentralization risk.The coefficient α 3 of π t is negative, indicating that the larger the value of π t , there is no trade-off between the system dimension and the individual dimension of the commodity.On the contrary, the coefficient α 4 is positive, indicating that the trade-off between the system dimension and the individual dimension will lead to a larger ρ t , and the investment portfolio will make vmv,t by moving to the central node.
Under M-V theory, the goal is to minimize total risk based on expected return which are arbitrarily set by investors.As we explain above, the key nodes (futures) are the most volatile, so it is not surprising that the weight to which allocated are small.The larger the centrality, the smaller the weight.The weak positive relationship between Sharpe ratio and centrality is not surprising either, since Sharpe ratio reflects risk adjusted return.If we allocate more weights to futures with less centrality, then the total volatility will decease.Mathematically, holding everything else constant, the Sharpe ratio should go up, indicating a inverse relationship.However, the risk adjusted return will not increase correspondingly because there is no free lunch in market.The less the risk, the less total return.So investors are always facing trade off between risk and return.M-V theory only sheds light on the positive relationship of risk and return but cannot teaches investors about making premium without taking risk.If investors want high return, they must take risk as price.

Commodity portfolio optimization strategy based on network centrality
DeMiguel et al [5] pointed out that the 'mean-variance' strategy is not always superior to the 'equal-weight strategy' , and even in some cases, the 'mean-variance' strategy is considered to be inferior to the 'equal-weight strategy' .Therefore, the 'equal weight strategy' and the 'mean-variance' strategy can be regarded as two reasonable benchmarks for evaluating portfolio performance.Inspired by the relationship between commodity network centrality and combination weight in the previous section, this paper proposes an investment strategy based on commodity network centrality.The core logic of the strategy is to select commodity groups according to the centrality ranking of commodities in the network, and evaluate its performance as a portfolio in the sample against the strategy.
Specifically, first of all, according to the network centrality grouping of commodities, this paper constructs a portfolio of commodities with different centralities, including the 'core group' composed of the 15 commodities with the highest centrality and the 'peripheral group' composed of the 15 commodities with the lowest centrality, and the 'full variety' group composed of all commodities in the sample; secondly, by constructing a strategy backtesting framework, dynamic backtesting is carried out on the investment portfolios composed of the above groups.In quantitative investment strategy backtesting, the investment strategy captures the logic used to make asset allocation decisions when backtesting.As the backtesting progresses, each strategy periodically updates its portfolio weights based on past market conditions.In quantitative investment strategy backtesting, the investment strategy captures the percentage of available capital that is used to represent the asset weights invested in each asset at the time the backtest is performed, with each element in the weight vector corresponding to the corresponding element in the asset price list.Due to the advantages of using intraday high-frequency data in this paper, the paper uses the trading day as a unit to adjust the configuration of the investment portfolio, that is, the weight distribution of commodities, and evaluate the performance of each trading day strategy; finally, this paper analyzes the above three groups of commodities portfolios which are constructed on the basis of daily rebalancing rules, are three types of optimization models, namely 'equal weight strategy' , 'mean-variance' strategy and 'maximum Sharpe ratio strategy' for comparative analysis.The trading rules are: calculate the initial weight of the commodity portfolio under the three strategies based on the 45 five-minute closing price data on the first trading day, buy at the opening of the second trading day, and close the position before the market closes on that day.The weights of the three types of portfolios will be recalculated on the same day as the basis for the combination configuration of the next trading day, and will continue to roll forward.
Figure 3 and table 3 show that without including transaction costs, the three groups of commodity portfolios can obtain positive expected returns under the three types of combination strategies.Specifically, the largest sharp strategy in the 'all varieties group' achieved the highest cumulative return of 78.92% among all commodity groups and strategies.In the 'peripheral group' composed of low centrality commodities, the cumulative returns of the three types of strategies are similar, and they all exceed the corresponding three types of strategies in the high centrality commodity group.At the same time, under the 'equal weight strategy' and 'mean-variance' strategy, the cumulative return of the 'peripheral group' is the highest among the three groups of commodities.This result proves once again from the perspective of the practical significance of trading strategies that giving more capital allocation weights to 'peripheral commodities' with low network centrality in commodity portfolios can improve trading performance.In terms of the volatility of portfolio returns, the 'peripheral group' is significantly lower than the 'full variety' group and the 'core group' composed of commodities with high centrality.However, the 'peripheral group' commodities do not perform well in all aspects.For example, in terms of maximum drawback, which refers to the maximum return rate drawback range when the portfolio net value reaches the lowest point during the sample period, the maximum drawback of the 'peripheral group' exceeds 30%, while the average drawback of the three types of strategies in the 'core group' is less than 13%, and the average drawback of the 'full variety' group is about 13%.This indicator reflects the worst possible situation of the portfolio and is an important risk indicator in quantitative investment.The 'peripheral group' has a larger drawback, which reflects that the stability of the commodity portfolio's profitability and the ability to resist risks are worse than those of the other two groups of commodity portfolios.The main reason for the excessive drawback is that since this paper uses the rule of adjusting positions every trading day and 'buy and hold' until the closing before the market close, the prices of some 'peripheral commodities' have fallen sharply during the sample period, resulting in 'peripheral commodities' portfolio net worth declines.
In order to evaluate the robustness of the strategy, this paper introduces transaction fees and other impact costs on the basis of the ideal state of the strategy,as shown in figure 4 and table 4. In commodity futures  trading, the transaction cost includes two parts, the transaction fee stipulated by the exchange and the impact cost, such as slippage, etc.This paper sets the ratio of the sum of the two items to the commodity margin to be 8/10 000 as the total transaction cost.In terms of total rate of return, the three groups of commodities only make profits in the same-weight strategy, and the strategy of the 'peripheral group' achieves the highest return with the smallest loss.In terms of the maximum drawback, similar to the case without transaction costs, the drawback of the 'peripheral group' is the largest among the three groups of commodities.
To sum up, based on the network centrality of the commodity combination, no matter in the ideal scenario without transaction costs or the actual scenario with transaction costs, the combination ability of selecting the 'peripheral group' commodity structure with less centrality according to the centrality of the commodity network obtain excess returns.The research results of this paper may unearth the 'network centrality factor' of China's commodity futures market.

Conclusion
The commodity futures market is an important part of China's derivatives market and even the financial market.In the commodity futures market, speculation, hedging and arbitrage are the three most important types of transactions.Commodity portfolio optimization is a very important issue in speculation and arbitrage.This paper analyzes the relationship mechanism between the centrality of each commodity in the commodity futures market network and the optimal weight in the commodity portfolio, and empirically tests the market systemic factors and commodity individual factors that affect the centrality of commodities.On this basis, this paper proposes a commodity investment strategy based on commodity network centrality.
The research in this paper finds that commodities with higher network centrality are often related to black and energy commodities and industrial products whose volatility are high, and commodities with higher centrality correspond to lower portfolio weights.In this paper, commodities are grouped according to the centrality of the network.The investment portfolio constructed by peripheral commodities with low network centrality has a higher cumulative return than the core commodity portfolio with high centrality and the return rate of the entire market commodity portfolio, but its stability and anti-risk ability are poor.The research results of this paper are of great significance.First of all, the logical mechanism between the commodity futures market network topology and commodity portfolio is theoretically established; secondly, in terms of application, the method based on the network centrality combination optimization strategy proposed in this paper is to use the commodity futures market network as a tool to discover 'network centrality factor' in commodity futures markets.This paper suggests that in the targeted speculative trading and arbitrage trading in the commodity futures market, actively establish the connection between portfolio weight and commodity centrality, mine low-correlation, low-risk commodity clusters, and find commodity portfolios that can effectively diversify risks.

Figure 1 .
Figure 1.Four-quantile Sharpe ratio (left) and commodity standardized price (right) grouped according to commodity eigenvector centrality.

Figure 2 .
Figure 2. The relationship between volatility, Sharpe ratio and commodity network centrality.

Figure 3 .
Figure 3.The cumulative return performance of strategies based on commodity network centrality grouping (without transaction costs).

Figure 4 .
Figure 4.The cumulative return performance of strategies based on commodity network centrality grouping (including transaction costs).

Table 1 .
Regression results of the relationship between the optimal portfolio weight and individual dimensions.

Table 2 .
Regression results of the relationship between the optimal portfolio weight and individual dimensions.Robust t statistics are in parentheses, * * * p < 0.01, * * p < 0.05, Note: * p < 0.1.

Table 3 .
Comprehensive performance of strategies based on commodity network centrality grouping (excluding transaction costs).

Table 4 .
Comprehensive performance of strategies based on commodity network centrality grouping (including transaction costs).