The Evaluation of Mean-Detrended Cross-Correlation Analysis Portfolio Strategy for Multiple risk Assets

Abstract

The fractal characteristics of the security market were considered in portfolio strategy optimization. First, the detrended cross-correlation analysis was adopted to measure the fractal correlation of different securities. Second, the fractal correlation was embedded into the mean-variance criterion of the modern portfolio theory. Third, the mean-detrended cross-correlation analysis portfolio strategy of multiple risk assets was constructed, and the analytic solution of the strategy was given. Finally, the evaluation results revealed that the constructed the mean-detrended cross-correlation analysis portfolio strategy clearly improved investment performance, thus achieving the goal of optimizing the multiple risk asset portfolio strategy.

Keywords

portfolio strategy optimization detrended cross-correlation analysis mean-variance criterion fractal characteristic

JEL Classification:G11, G19, C53

Introduction

Money never sleeps, and ever since the advent of securities investment, people have desired to realize asset appreciation through securities investment. As a key guarantee for obtaining investment income, a securities investment strategy is of vital importance in both practical and theoretical circles. Investigating the optimization of securities investment strategies will not only provide investors with the corresponding reference for decision-making and help investors improve their investment performance, but will also facilitate the involvement of investors in the security market, ensuring that the security market performs its functions, and promotes the prosperity and development of the economy and society (Liang et al., 2020a, Zhang et al., 2019). It is also conducive to improving the theory of financial decision-making and promoting the deepening of relevant financial theoretical research. Therefore, whether from a practical perspective or a theoretical perspective, it is critical to conduct optimization research on securities investment strategies.

Furthermore, when studying the optimization problem of securities investment strategy, we must focus on the optimization problem of securities investment portfolio strategy. To be more specific, in terms of reality, because risks and returns always go hand in hand in the practical operation of securities investment, the core goal of the securities investment portfolio strategy is to maximize returns and minimize risks (Liang et al., 2020b), which caters specifically to the real needs of investors for obtaining returns while controlling risks, and it is the most widely used securities investment strategy for investors (Markowitz, 1952; Sotiropoulos & Rutterford, 2019). Therefore, optimization of the securities investment portfolio strategy is particularly promising. Theoretically, the portfolio problem marks the beginning of modern finance and is the most widely studied problem in financial literature (Van et al., 2017). Studying the optimization of securities investment portfolio strategy is not only beneficial to the improvement of the modern portfolio theory (MPT) initiated by Markowitz, which is to realize asset appreciation and control risks by considering the intrinsic correlation between securities and using diversified investment, but is also conducive to the advancement and development of related financial theory research. Therefore, the study of investment portfolio strategy optimization is of a particular theoretical value (Fabozzi et al., 2002; Puerto et al., 2020). In the theoretical research and practical operation of optimizing securities investment strategies, it is especially important to study the optimization issue of securities investment portfolio strategies.

Given the significance of researching the optimization of securities investment portfolio strategy, many scholars optimized it different perspectives and some literature review papers written by scholars can be considered specifically (Huang, 2017; Kalayci et al., 2019). In general, the relevant achievements are primarily focused on the following aspects: first, the securities investment portfolio strategy is optimized from the perspective of improving the return or risk measurement (Guo et al., 2019; Nesaz et al., 2019); second, optimize the securities investment portfolio strategy from the perspective of multi-period dynamic investment (Kamali et al.,2019; Zhang et al., 2012); third, the securities investment portfolio strategy is optimized from the perspective of considering transaction costs and weight constraints (Quek & Atkinson, 2017; Xu et al., 2016); fourth, optimize the securities investment portfolio strategy from the perspective of considering investor behavior deviations (Omid et al., 2019; Pfiffelmann et al., 2016); fifth, MPT has been applied in other fields such as horticultural systems and crop insurance (Paut et al., 2019; Walters & Preston, 2018); sixth, the investment portfolio strategy is optimized under background risks, such as income fluctuation and health (Baptista, 2012; Xu et al., 2015). Undoubtedly, these studies have improved MPT and deepened the study of financial theoretical research, and it has also reflected the necessity of optimizing securities investment portfolio strategies at the same time.

Although there have been many achievements in studying the optimization of securities investment portfolio strategy, there are some defects that cannot be overlooked among the existing achievements, and the most important is that the actual fractal characteristics of the security market have not been considered (Quang, 2014). To be specific, first, the correlation among the assets in the securities investment portfolio determines the risk of the securities investment portfolio strategy. The correlation between assets must be accurately surveyed to ensure that the securities portfolio strategy constructed can achieve risk diversification (Clements et al., 2019; Medo et al., 2009). Second, the real characteristics of financial markets must be factored into financial analysis. To accurately survey the correlation between assets and ensure the effectiveness of the constructed securities portfolio strategy, the actual characteristics of the securities market must be considered (Cont, 2001; Wu et al., 2018). Finally, a large body of empirical evidence indicates that real security markets follow the fractal distribution under the fractal market hypothesis rather than the normal distribution under the efficient market hypothesis (Goddard and Onali, 2009; Ikeda, 2017). Currently, there is neither a simple deterministic correlation relationship nor a simple stochastic correlation relationship among assets, albeit a complex hybrid correlation relationship that contains randomness in certainty and certainty in randomness. The correlation between time series has statistical self-similarity under different observation intervals, showing a power-law relationship and long memory in time. However, when developing securities investment portfolio strategies, the existing relevant results are based on either a simple deterministic correlation relationship or a simple stochastic correlation relationship, which are far from the actual situation, making it difficult to obtain accurate results and ultimately affecting the effectiveness of the securities portfolio strategy (Liem et al., 2016). Consequently, the existing related results have significant defects because they do not consider the fractal correlation among assets (Karp & Van, 2019; Lin et al., 2011).

Furthermore, by introducing the portfolio strategy of detrended cross-correlation analysis (DCCA, noting that it is not DCC which is the abbreviation of Dynamic Conditional Correlation), some scholars have empirically tested the fractal characteristics of the securities market and built a portfolio model based on it, which has also empirically verified its effectiveness (Tang & Zhu, 2018; Zhou, 2008). Undoubtedly, this initiative has a pioneering and guiding effect to a certain extent, providing a new perspective for both practitioners and researchers; however, there are still some limitations. First, in building the portfolio model, the weight allocation of different assets was restricted, namely, short-selling was restricted. Thus, the rationality of the asset allocation ratio cannot be guaranteed, and the results obtained are not universal; second, only a single time scale was considered, and the investors’ preferences for different time scales were not taken into account, which not only cannot guarantee the accurate measurement of fractal correlation but also cannot meet the realistic demands of investors’ diversity in the fractal market; third, only two types of assets were considered as samples of risk assets, which cannot meet the actual needs of investors to conduct investment activities with multiple assets. To compensate for the weaknesses of existing studies and promote the validity of securities portfolio strategies (Huy et al., 2016), it is not only important to bring the fractal correlation characteristics of securities into the research framework of portfolio investment strategy optimization, but also to combine the actual preferences of investors and consider multiple risk assets to meet the realistic needs of investor diversity in the fractal market. This is the difference between this study and the relevant literature on theoretical model building, model solving, empirical analysis, and application value, which adds further depth to it and supplements it (Chun et al., 2020; Tang & Zhu, 2018; Zhou, 2008).

To incorporate the fractal correlation between securities to optimize the securities investment portfolio strategy, the natural solution is to construct an index that can survey the fractal correlation and embed them into the mean-variance criterion of MPT to build the securities investment portfolio strategy instead of the survey that cannot review the fractal correlation. In the selection of methods, combined with the above mentioned (Chun et al., 2020; Zhou, 2008), DCCA can characterize the correlation among time series at multiple time scales, which is a useful tool for describing fractal correlation and has been widely used in many fields (Iqbal et al., 2020; Podobnik & Stanley, 2008). Compared with the VaR, CVaR, and risk surveys built on this basis, DCCA can better deal with the nonlinear problem of security price when the real security market is a fractal market, and the risk measurement index constructed according to it can better adapt to the nonlinear and non-normal complex financial environment, and the result will be more convincing and universal. Therefore, in a real situation where fractal correlation characteristics exist among securities, the fractal correlation index can be constructed to survey the price fluctuations among securities according to DCCA, combine the degree of investor preference for different time scales, and optimize the securities investment portfolio strategy on this basis to cover the shortage of existing research (Supanantaroek et al., 2017).

To sum up, under the constraint that the real securities market is a fractal market, this article will construct the index to survey the fractal correlation between securities prices according to DCCA, and embed the constructed index into the reward-risk criteria of MPT to construct a Mean-DCCA securities investment portfolio strategy (hereinafter referred to as the M-D portfolio). The motivation of this paper is to further improve the effectiveness of portfolio strategy and provide decision-making references for portfolio investors to better control risks and obtain returns. The contribution of this study lies in incorporating the fractal correlation characteristics that actually exist between securities into the research framework of portfolio strategy optimization and comprehensively considering multiple time scales and investors’ preferences for different time scales, which not only compensates for the weaknesses of the existing study and promotes the validity of the securities investment portfolio strategy, but also improves the cross-integration of MPT and fractal statistical analysis and is conducive to promoting the development of econophysics. The structure of the article is as follows: Introduction is the introduction, Mean-DCCA portfolio Model Construction describes the Mean-DCCA securities investment portfolio strategy construction of multiple risk assets, The Evaluation of Mean-DCCA portfolio Strategy presents the evaluation and validity analysis of the Mean-DCCA securities investment portfolio strategy of multiple risk assets, and Conclusion and Outlook presents the conclusion and outlook.

Mean-DCCA portfolio Model Construction

The role of correlation on the portfolio

The goal of constructing the securities investment portfolio strategy is to realize the maximization of return and minimization of risk, which is equivalent to achieving the minimum risk while the return is established or the maximum return while the risk is established, referred to as realizing the mean-variance criterion (RRC) (Giorgi et al., 2011; Markowitz, 1952). According to the specific return measurement and risk measurement used, RRC can also be given some specific names, such as the mean-variance criterion, mean-VaR criterion, mean-CVaR criterion, and so on (Guo et al., 2019; Wang et al., 2020). In fact, no matter what measurements of return and risk are used, the correlation among assets in a securities investment portfolio has an important impact on the realization of RRC. Considering that people often use expectation and variance as the return measurement and risk measurement of securities (Kalayci et al., 2019; Merton, 1972), the mean-variance securities investment portfolio is taken as an example to illustrate the important role of correlation in realizing RRC.

According to MPT, when constructing the mean-variance securities investment portfolio strategy, it can be assumed that the securities investment portfolio $P$ is composed of $m$ kinds of securities ${Y_{i}}_{i = 1}^{m}$ . We denote the weight $ω_{i}$ , expected $E (r_{i})$ , and variance $V (r_{i})$ , respectively, as, the optimal investment weight, expected return, whereas the risk of securities $Y_{i}$ and the covariance $C (r_{i}, r_{j})$ is the correlation between the return sequences of securities $Y_{i}$ and $Y_{j}$ . Obviously, $V (r_{i}) = C (r_{i}, r_{i})$ is established, that is, the risk of security $Y_{i}$ is the correlation between itself and itself, which is a special correlation. To ensure that the securities investment portfolio $P$ meets RRC and realizes the risk minimization under the given return level $u$ , $ω_{i}$ should be solved according to the objective function and constraint conditions, as illustrated in equation (1) below. For the sake of brevity, $r_{i}$ can be directly used instead of $E (r_{i})$ to represent the expected return of asset $Y_{i}$ , then the analytical solution of $ω_{i}$ can be obtained as established in equation (2) by using the Lagrange multiplier method.

{\begin{cases} O b j e c t i v e f u n c t i o n : \min \frac{1}{2} V (r_{P}) = \min \frac{1}{2} \sum_{i, j = 1}^{m} ω_{i} ω_{j} C (r_{i}, r_{j}) \\ C o n s t r a i n t c o n d i t i o n : \sum_{i = 1}^{m} ω_{i} = 1 a n d \sum_{i = 1}^{m} ω_{i} E (r_{i}) = u \end{cases}

(1)

{\begin{matrix} ω_{i} = \frac{u (\sum_{i, j = 1}^{m} θ_{i j} \sum_{j = 1}^{m} θ_{i j} r_{j} - \sum_{i, j = 1}^{m} θ_{i j} r_{j} \sum_{j = 1}^{m} θ_{i j}) + (\sum_{i, j = 1}^{m} θ_{i j} r_{j} r_{i} \sum_{j = 1}^{m} θ_{i j} - \sum_{j = 1}^{m} θ_{i j} r_{j} \sum_{i, j = 1}^{m} θ_{i j} r_{j})}{\sum_{i, j = 1}^{m} θ_{i j} r_{j} r_{i} \sum_{i, j = 1}^{m} θ_{i j} - {(\sum_{i, j = 1}^{m} θ_{i j} r_{j})}^{2}} \\ [\begin{matrix} θ_{11} & θ_{12} & \dots & θ_{1 m} \\ θ_{21} & θ_{22} & \dots & θ_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ θ_{m 1} & θ_{m 2} & \dots & θ_{m m} \end{matrix}] = {[\begin{matrix} C (r_{1}, r_{1}) & C (r_{1}, r_{2}) & \dots & C (r_{1}, r_{m}) \\ C (r_{2}, r_{1}) & C (r_{2}, r_{2}) & \dots & C (r_{2}, r_{m}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ C (r_{m}, r_{1}) & C (r_{m}, r_{2}) & \dots & C (r_{m}, r_{m}) \end{matrix}]}^{- 1} \end{matrix}

(2)

From equation (2), the optimal investment weight

ω_{i}

is chiefly influenced by the covariance between securities. Moreover, from equation (1), the correlation among assets, namely, the accuracy of the covariance measurement, is critical to whether the securities portfolio can achieve RRC and reflect the diversification effect (Kalayci et al., 2019; Markowitz, 1991). Thus, the correlation among assets is a pivotal factor that influences the validity of the securities portfolio. To ensure that the constructed securities investment portfolio strategy can achieve risk diversification and asset appreciation, the corresponding measurements must be constructed to precisely survey the correlation among assets.

From the foregoing, the real securities market is indeed a fractal market, and fractal correlation characteristics indeed exist among securities price fluctuations resulting in a complex hybrid correlation relationship that contains randomness in certainty and certainty in randomness. At this time, if the time scale changes, the correlation among assets will also change, and the traditional covariance as the correlation measurement among assets cannot guarantee the accuracy of the results obtained. Simultaneously, because the risk of security prices is a special case of the fractal correlation among security prices, and variance is another sense of covariance, namely, the correlation of the fluctuation of security price itself, hence, the traditional variance as the risk measurement of security price also cannot obtain exact results. Thus, as a result of the fractal correlation characteristic among security prices, the mean-variance securities portfolio (hereinafter referred to as the M-V portfolio) has certain defects due to the inability of covariance to precisely survey the fractal correlation among assets and the inability of variance to precisely survey the asset risk, which in turn influences the accuracy of the allocation of optimal weights of assets in the portfolio, and render the constructed portfolio strategy invalid.

DCCA correlation indicators construction

Given that the DCCA can accurately describe the correlation between security prices under multiple time scales, it is a powerful tool for depicting fractal correlation (Ferreira et al., 2019; Iqbal et al., 2020). Therefore, to precisely survey the fractal correlation characteristics among assets in the securities investment portfolio under the fractal market and construct an effective securities investment portfolio strategy to overcome the inability of traditional correlation surveys in the M-V portfolio strategy to accurately depict the disadvantages of the fractal correlation among assets, DCCA was considered in the research on the mean-variance securities investment portfolio strategy as follows. According to DCCA, the index of fractal correlation of price fluctuations between securities was constructed to substitute the traditional covariance or traditional variance. Subsequently, the securities investment portfolio was optimized on this basis to compensate for the weaknesses of the existing research. Specifically, for the return sequences ${r_{i} (t)}_{t = 1}^{n}$ and ${r_{j} (t)}_{t = 1}^{n}$ of any two risky assets, the major steps of DCCA are as follows:

First, we calculate the cumulative deviation sequences ${\tilde{r_{i}} (T)}_{T = 1}^{n}$ and ${\tilde{r_{j}} (T)}_{T = 1}^{n}$ of the return series ${r_{i} (t)}_{t = 1}^{n}$ and ${r_{j} (t)}_{t = 1}^{n}$ according to equation (3), we then divide the cumulative deviation sequences ${\tilde{r_{i}} (T)}_{T = 1}^{n}$ and ${\tilde{r_{j}} (T)}_{T = 1}^{n}$ into $n_{s} = int (\frac{n}{s})$ non-overlapping subsequences with length $s$ from beginning to end, respectively. There may be a situation where the length of the time series divided by s is not an integer. This problem will not completely cause a waste of information, because the existence of those s value which can be divided evenly by the cumulative deviation sequences. Meanwhile, if $s$ cannot be divided evenly by the cumulative deviation sequences, equal distance division should not be required, in order to make full use of the sequence information. Obviously, when $T \in {f s - s + 1, \dots, f s}$ , ${\tilde{r_{i}} (T)}$ and ${\tilde{r_{j}} (T)}$ are the elements of the $f^{t h}$ subsequence, where $s$ represents the time scale and is usually valued in the interval $[3, int (0.25 n)]$ , which is the length of the subsequence selected when observing the time series (Podobnik & Stanley, 2008; Zhao, 2017).

{\begin{cases} \tilde{r_{i}} (T) = \sum_{t = 1}^{T} [r_{i} (t) - \frac{1}{n} \sum_{t = 1}^{n} r_{i} (t)] \\ \tilde{r_{j}} (T) = \sum_{t = 1}^{T} [r_{j} (t) - \frac{1}{n} \sum_{t = 1}^{n} r_{i} (t)] \end{cases}

(3)

Second, for

k \in {f s - s + 1, \dots, f s}

, denote

R_{f}^{i} (k) = \sum_{T = f s - s + 1}^{k} \tilde{r_{i}} (T)

and

R_{f}^{j} (k) = \sum_{T = f s - s + 1}^{k} \tilde{r_{j}} (T)

, and use the least square method to fit

{R_{f}^{i} (k)}

and

{R_{f}^{j} (k)}

to obtain the local trend functions

{\bar{R}}_{f}^{i} (k)

and

{\bar{R}}_{f}^{j} (k)

Finally, calculating the covariance $F_{i j} (s)$ of each subsequence of the return series ${r_{i} (t)}_{t = 1}^{n}$ and ${r_{j} (t)}_{t = 1}^{n}$ after the trend is eliminated by using equation (4). In equation (4), the sum may have negative values. In this sense, by treating the negative value in the square root, we can treat it by using the properties of imaginary numbers in the field of complex numbers. Obviously, $F_{i j} (s) = F_{j i} (s)$ holds. Empirical evidence indicates that the value of $s$ in interval $[3, int (0.25 n)]$ can ensure the stability and non-zero of $F_{i j} (s)$ (Podobnik & Stanley, 2008; Zhao, 2017).

F_{i j} (s) = {{(s)}^{- 1} \sum_{f = 1}^{n s} \sum_{k = f s - s + 1}^{f s} [R_{f}^{i} (k) - {\bar{R}}_{f}^{i} (k)] {[R_{f}^{j} (k) - {\bar{R}}_{f}^{j} (k)]}}^{0.5}

(4)

Through the calculation steps of DCCA above, it can be found that

F_{i j} (s)

will change if the time scale

s

changes. In fact, when fractal correlations exist among assets, there is a power-law relationship between

F_{i j} (s)

and

s

, that is,

F_{i j} (s) \sim s^{H}

is true, in which

H

is the generalized scale index.

Thus, using DCCA to survey the correlation among assets can be achieved by adjusting the time scale $s$ to survey the correlation between risk assets $Y_{i}$ and $Y_{j}$ at different time scale observation intervals. In addition, considering $F_{i j} (s)$ at more time scales, $s$ helps to obtain a more precise measurement of correlations between assets, which can improve the insufficient accuracy of traditional covariance extrapolation. When the return series ${r_{i} (t)}_{t = 1}^{n}$ and ${r_{j} (t)}_{t = 1}^{n}$ are identical, DCCA degenerates into detrended fluctuation analysis (DFA) (Peng et al., 1995), which is devoted to analyzing whether a single time series has fractal fluctuation characteristics. At this point, $F_{i j} (s)$ can be denoted by $F_{i i} (s)$ or $F_{j j} (s)$ , which is devoted to surveying the fluctuation risk of the same return series at different time scales of $s$ .

The above analysis indicates that using $F_{i j} (s)$ in DCCA to replace traditional covariance or using traditional variance to survey the correlation among assets can achieve the purpose of accurately characterizing the fractal correlation characteristics existing between security prices.

Building the Mean-DCCA portfolio of multi-risk assets

Subsequently, $F_{i j} (s)$ can be combined with the mean-variance criterion of MPT to overcome the above-mentioned drawbacks of the traditional mean-variance securities portfolio strategy. Based on this, the $F_{i j} (s)$ in DCCA is introduced into the research of the mean-variance securities portfolio strategy as follows. According to DCCA, an index that surveys the fractal correlation characteristics of price fluctuations between securities is constructed to replace traditional covariance or traditional variance. As depicted above, the M-D investment portfolio strategy is constructed to optimize the securities investment portfolio strategy and compensate for the weaknesses of existing research.

To correspond to the mean-variance securities portfolio strategy of multi-risk assets constructed by equation (1), it is still assumed that portfolio $P$ is composed of $m$ kinds of risk assets ${Y_{i}}_{i = 1}^{m}$ while building the M-D securities investment portfolio strategy of multi-risk assets, $F_{i j} (s)$ is used to replace the covariance to survey the correlation among assets $Y_{i} \in {Y_{i}}_{i = 1}^{m}$ and $Y_{j} \in {Y_{i}}_{i = 1}^{m}$ , when $i = j$ , the volatility risk of asset $Y_{i}$ or $Y_{j}$ can be surveyed according to $F_{i j} (s)$ . Noticeably, under the M-D securities investment portfolio strategy, the assets’ weight will be influenced by the time scale $s$ . At this point, it is advisable to set that the investment weight of asset $Y_{i}$ is $ω_{i} (s)$ in time scale $s$ . Then, under time scale $s$ , the expected return $E_{P} (s)$ of portfolio $P$ can be presented in equation (5) below.

E_{P} (s) = \sum_{i = 1}^{m} ω_{i} (s) E (r_{i})

(5)

Meanwhile, considering that the risk of portfolio

P

is influenced by the fluctuation of assets, the correlation between assets, and the investment weight, the

F_{i j} (s)

in DCCA surveys the correlation or volatility risk of assets

Y_{i}

and

Y_{j}

under time scale

s

. Therefore, after considering the multi-fractal correlation characteristics between asset prices, the risk

V_{P} (s)

of portfolio

P

can be calculated according to the following equation (6).

V_{P} (s) = \sum_{i, j = 1}^{m} ω_{i} (s) ω_{j} (s) F_{i j} (s)

(6)

Thus, according to equations (5) and (6), constructing the portfolio

P

on the basis of M-D at different time scales

s

, that is,

E_{P} (s)

and

V_{P} (s)

are substituted into the RRC, and the value of

ω_{i} (s)

is determined to make portfolio

P

meet the RRC, thus the objective function and constraint conditions as displayed in equation (7) are established. Similar to the solution method of equation (1), for the sake of simplicity,

r_{i}

is still used instead of

E (r_{i})

to signify the expected return of asset

Y_{i}

. Then, the Lagrange multiplier method can be used to obtain the analytic solution of

ω_{i} (s)

, as presented in equation (8) below under the time scale

s

{\begin{matrix} O b j e c t i v e f u n c t i o n : \min V_{P} (s) = \min \sum_{i, j = 1}^{m} ω_{i} (s) ω_{j} (s) F_{i j} (s) \\ C o n s t r a i n t c o n d i t i o n : \sum_{i = 1}^{m} ω_{i} (s) = 1 a n d \sum_{i = 1}^{m} ω_{i} (s) E (r_{i}) = u \end{matrix}

(7)

{\begin{matrix} ω_{i} (s) = \frac{u \sum_{j = 1}^{m} θ_{i j} (r_{j} \sum_{i, j = 1}^{m} θ_{i j} - \sum_{i, j = 1}^{m} θ_{i j} r_{j}) + \sum_{j = 1}^{m} θ_{i j} (\sum_{i, j = 1}^{m} θ_{i j} r_{j} r_{i} - r_{j} \sum_{i, j = 1}^{m} θ_{i j} r_{j})}{\sum_{i, j = 1}^{m} θ_{i j} r_{j} r_{i} \sum_{i, j = 1}^{m} θ_{i j} - {(\sum_{i, j = 1}^{m} θ_{i j} r_{j})}^{2}} \\ [\begin{matrix} θ_{11} & θ_{12} & \dots & θ_{1 m} \\ θ_{21} & θ_{22} & \dots & θ_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ θ_{m 1} & θ_{m 2} & \dots & θ_{m m} \end{matrix}] = {[\begin{matrix} F_{11} (s) & F_{12} (s) & \dots & F_{1 m} (s) \\ F_{21} (s) & F_{22} (s) & \dots & F_{2 m} (s) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ F_{m 1} (s) & F_{m 2} (s) & \dots & F_{m m} (s) \end{matrix}]}^{- 1} \end{matrix}

(8)

The solution given by equation (8) above is only the optimal investment weight of assets under a single time scale

s

. In the fractal market, the correlation between security price fluctuations will change as time scale

s

changes, namely, the correlation between security price fluctuations will be different under different time scale observation intervals. Therefore, it is necessary to further calculate the optimal investment weight of asset

{Y_{i}}_{i = 1}^{m}

under multiple time scales based on equation (8) to ensure the validity of the constructed portfolio strategy. For this reason, it can be assumed that the set composed of multiple time scales that investors tend to use is

S \subseteq {3, \dots, int (0.25 n)}

s_{d}

, and

s_{u}

are the maximum and minimum elements of the set

S

, respectively, so that investors can obtain

s_{u} - s_{d} + 1

groups of optimal investment weights according to equation (8). If it is assumed that the investor’s relative preference for time scale

s \in S

α_{i} (s)

, that is, the investor’s relative preference for the optimal investment weight

ω_{i} (s)

under time scale

s \in S

α_{i} (s) \in [0,1]

, and

\sum_{s \in S} α_{i} (s) = 1

holds, then according to equation (9), the weighted value

ω_{i} (S)

of the optimal investment weights of

s_{u} - s_{d} + 1

groups can be acquired on the basis of the relative preference.

ω_{i} (S) = \sum_{s \in S} α_{i} (s) ω_{i} (s)

(9)

From equation (9), it can be seen that

ω_{i} (S)

comprehensively considers all time scales in set

S

and is the investment weight acquired after combining

s_{u} - s_{d} + 1

time scales and investors’ preference for time scales, which not only fully considers the fractal correlation characteristics between securities price fluctuations and conforms to the real environment of the securities market as a fractal market, but also fully reflects the investor preference for time scales and meets the objective needs of investor diversity. Therefore, under the constraint of fractal characteristics, it is more reasonable to regard

ω_{i} (S)

as the optimal investment weight of asset

{Y_{i}}_{i = 1}^{m}

than

ω_{i} (s)

under the M-D portfolio strategy. Thus, this study takes

ω_{i} (S)

as the optimal investment weight of risk assets under the M-D securities portfolio strategy.

The above analysis indicates that the constructed M-D portfolio strategy of multi-risk assets uses the $F_{i j} (s)$ in DCCA to survey the correlation and volatility between securities prices, which can capture the fractal correlation between securities better than that of the M-V portfolio strategy. At the same time, compared to the M-V portfolio strategy of multi-risk assets, the optimal investment weight $ω_{i} (S)$ under the M-D portfolio strategy synthetically considers multiple time scales, which are more refined and more easily adapted to the reality that the securities market has fractal characteristics. Consequently, the M-D portfolio strategy of multi-risk assets surmounts the weaknesses of the M-V portfolio strategy, which is beneficial for promoting the validity of the portfolio strategy and achieving better investment performance.

The Evaluation of Mean-DCCA Portfolio Strategy

Fractal feature analysis of the sample

From the above theoretical analysis, it can be observed that in the case of the reality that the security market is a fractal market, the M-D portfolio strategy of multi-risk assets constructed based on DCCA can compensate for the weaknesses of the M-V portfolio strategy, which is beneficial for promoting the validity and investment performance of the portfolio. To avoid the theory being divorced from reality, the following analysis takes the M-V portfolio strategy as the benchmark portfolio strategy and compares and analyzes the performance of the two portfolio strategies to further test the validity of the M-D portfolio strategy of multi-risk assets. In the empirical analysis, this study takes the daily closing prices of all six industry indexes of the Shanghai Stock Exchange as a sample of risky assets, which include the industrial index (Ind.), business index (Bus.), real estate index (Rea.), utility index (Uti.), composite index (Com.), and the financial index (Fin.) The sample interval is from January 1, 2004, to January 1, 2020. As there are still non-systemic risks in the industry indexes, which can reflect the diversification of the investment portfolio strategy, it is reasonable to consider the daily closing price of the industry index as a sample. In the following text, for convenience, we might as well directly treat the six industry indexes as an asset. At the same time, the sample interval contains complete market conditions (bull market, bear market, and shock market), which can fully reflect the validity of the portfolio strategy under different market conditions, hence, the selection of the sample interval is representative.

To verify that the security market is not subject to a normal distribution under the efficient market hypothesis, albeit, it may have fractal characteristics; Figure 1 below indicates the daily rate series of the six industry indexes in the sample interval. Graphically, the return rate series has the phenomenon of fluctuation agglomeration, suggesting a fractal fluctuation characteristic and the return series may not obey the normal distribution.

Figure 1.

The fluctuation of the daily rate series.

Subsequently, we applied Kolmogorov–Smirnov (K-S) test, Jarque–Bera (J-B) test, and Shapiro–Wilk (S-W) test to empirically test the normality of the return series. All the test’s results are showed in Table 1. Additionally, Hurst’s exponent is showed in Table 2.

Table 1.

Normality test results of daily return series.

Index	K-S stat.	J-B stat.	S-W stat.
Ind.	0.077	2585.899	0.944
Bus.	0.071	1743.721	0.955
Rea.	0.067	1055.039	0.962
Uti.	0.086	3729.021	0.929
Com.	0.080	3313.371	0.936
Fin.	0.079	1839.553	0.948

Table 2.

DCCA test results of daily return series.

Index	H	Index	H	Index	H
Ind. v.s. Ind.	0.945	Ind. v.s. Rea.	0.952	Bus. v.s. Fin.	0.979
Bus. v.s. Bus.	0.977	Ind. v.s. Uti.	0.951	Rea. v.s. Uti.	0.956
Rea. v.s. Rea.	0.951	Ind. v.s. Com.	0.947	Rea. v.s. Fin.	0.978
Uti. v.s. Uti.	0.953	Ind. v.s. Fin.	0.965	Rea. v.s. Com.	0.952
Com. v.s. Com.	0.947	Bus. v.s. Rea.	0.970	Uti. v.s. Com.	0.953
Fin. v.s. Fin.	0.976	Bus. v.s. Uti.	0.969	Uti. v.s. Fin.	0.969
Ind. v.s. Bus.	0.966	Bus. v.s. Com.	0.966	Com. v.s. Fin.	0.963

As showed in Table 1, the adjoint probability of all industry index return series in K-S test, J-B test and S-W test is 0.000, less than 0.01 and 0.05, that means no evidence to support the sample sequence obeys the normal distribution at the significance level of 1% and 5%. The non-normality of the return series of six industry indexes is verified. All scaling indices $H$ in Table 2 are calculated according to the step of DCCA, means there is a power-law relationship $F_{i j} (s) \sim s^{H}$ between $F_{i j} (s)$ and $s$ . The DCCA test shows that the return series of six industry indexes do have fractal fluctuation correlation. So, there is a realistic basis to bring the fractal correlation between assets into the portfolio model, and it is necessary to construct mean DCCA combination model.

Effectiveness analysis of Mean-DCCA portfolio

Meanwhile, considering that the pros and cons of the portfolio strategy are chiefly determined by the testing effect of strategies out of a sample. And to reflect the effect of strategy optimization, the performance of the optimized strategy needs to be better than that of the pre optimized strategy in most cases to overcome the survival bias (Li et al., 2020; Wu et al., 2021). Hence according to natural years, this study divides the entire sample interval into 16 subintervals and uses the data of each year in the sample range as the “estimated sample interval” to build M-D and M-V portfolios, then it uses the data of the “estimated sample interval” for the following year as the “predicted sample interval” to look into the performance of the two portfolios constructed, namely, there are 15 estimated sample intervals and 15 predicted sample intervals. Furthermore, the M-D and M-V portfolio strategies of the six industry indexes can be constructed according to equations (2) and (8), respectively. This study combines the given returns $u \in {0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40}$ commonly used by portfolio investors and analyzes the performance of the two portfolio strategies under eight types of given returns $u$ , in order to fully evaluate the effectiveness of the Mean-DCCA through back test under multiple different $u$ conditions.

From the previous analysis, it can be observed that when constructing the M-D portfolio strategy according to equation (8), it is necessary to set the investor preference degree $α_{i} (s)$ for time scale $s$ . In this regard, given that there are approximately 242 trading days per year, and according to the first step of DCCA, the time scale is generally considered as $S \subseteq {3, \dots, int (0.25 n)}$ , thus $S = {3, \dots, 60}$ is taken as the time scale set frequently used by investors. Simultaneously, in combination with the actual situation of the investors, their preferences for the length of the time scale can be approximately split into three types: no obvious preference for different time scales (Type 1), more preference for shorter time scales (Type 2), and more preference for longer time scales (Type 3). Accordingly, we can take $α_{i} (s) = {(s_{u} - s_{d} + 1)}^{- 1} = 58^{- 1}$ , $α_{i} (s) = (s_{u} + s_{d} - s) {(\sum_{s \in S} s)}^{- 1}$ , and $α_{i} (s) = s {(\sum_{s \in S} s)}^{- 1}$ , respectively, for $α_{i} (s)$ . $α_{i} (s) = {(s_{u} - s_{d} + 1)}^{- 1} = 58^{- 1}$ indicating that investors have no special preference for any time scale; they give the same preference weight to each time scale, corresponding to type 1; $α_{i} (s) = (s_{u} + s_{d} - s) {(\sum_{s \in S} s)}^{- 1}$ represents investors who prefer the shorter time scale, and they give higher preference weight to the shorter time scale, corresponding to type 2; $α_{i} (s) = s {(\sum_{s \in S} s)}^{- 1}$ represents investors who prefer the longer time scale and give higher preference weight to the longer time scale, corresponding to type 3.

Thus, under the M-D and M-V portfolio strategies, for each given return and estimated sample interval, the optimal investment weights of six industry indexes under the M-V and M-D portfolio strategies can be calculated according to equations (2) and (8), the M-V and M-D portfolio models can be built, and the performance differences of the two portfolio strategies constructed in the sample prediction interval can be calculated. Because there are eight different values for the given return $u$ , eight M-V and eight M-D portfolios of a given preference type can be obtained in each sample estimation interval, and a total of 120 M-V portfolios and 120 M-D portfolios with a given preference type can be obtained in the 15 estimated sample intervals.

To analyze the validity of the M-D portfolio strategy, Tables 3 to 5, respectively, count the excess return rates of 120 M-D portfolios under three given preference types relative to the M-V portfolios, that is, the differences in the return rate of the M-D portfolios in the sample prediction interval minus that of the M-V portfolios in the sample range. From Tables 3 to 5, it can be observed that under eight kinds of given returns, there are 68, 79, and 67 cases in which the excess return rate is greater than zero, accounting for 56.67%, 65.83%, and 55.83%, respectively. In other words, the proportions of the M-D portfolio return rate under the three given preference types are higher than those of the M-V portfolio, which are 56.67%, 65.83%, and 55.83%, respectively. It can be noted that the M-D portfolio strategy is better than the M-V portfolio strategy.

Table 3.

The excess return rate of the M-D portfolio under Type 1 relative to the M-V portfolio.

Interval	u=0.05	u=0.10	u=0.15	u=0.20	u=0.25	u=0.30	u=0.35	u=0.40
2005	−1.452	−1.392	−1.331	−1.270	−1.210	−1.149	−1.088	−1.028
2006	4.729	4.663	4.596	4.530	4.464	4.398	4.332	4.265
2007	0.077	0.060	0.042	0.025	0.008	−0.009	−0.027	−0.044
2008	−0.079	−0.079	−0.079	−0.080	−0.080	−0.080	−0.081	−0.081
2009	−5.394	−5.770	−6.146	−6.522	−6.898	−7.274	−7.650	−8.026
2010	0.712	0.710	0.707	0.705	0.702	0.699	0.697	0.694
2011	0.413	0.433	0.453	0.473	0.492	0.512	0.532	0.552
2012	−0.014	0.053	0.121	0.188	0.256	0.323	0.391	0.459
2013	−0.024	−0.033	−0.041	−0.050	−0.059	−0.068	−0.077	−0.086
2014	−1.602	−1.786	−1.969	−2.152	−2.336	−2.519	−2.702	−2.885
2015	0.741	0.791	0.842	0.892	0.942	0.993	1.043	1.093
2016	−1.117	−1.141	−1.166	−1.191	−1.215	−1.240	−1.265	−1.289
2017	1.746	1.718	1.689	1.661	1.633	1.605	1.577	1.549
2018	0.006	0.011	0.015	0.020	0.024	0.028	0.033	0.037
2019	0.167	0.188	0.210	0.231	0.253	0.274	0.295	0.317

Table 4.

The excess return rate of the M-D portfolio under Type 2 relative to the M-V portfolio.

Interval	u=0.05	u=0.10	u=0.15	u=0.20	u=0.25	u=0.30	u=0.35	u=0.40
2005	0.450	0.506	0.562	0.618	0.674	0.730	0.786	0.842
2006	6.881	6.779	6.677	6.574	6.472	6.370	6.268	6.166
2007	0.085	0.072	0.060	0.048	0.036	0.023	0.011	−0.001
2008	−0.066	−0.066	−0.066	−0.067	−0.067	−0.068	−0.068	−0.068
2009	−3.565	−3.870	−4.175	−4.480	−4.785	−5.090	−5.395	−5.700
2010	0.537	0.536	0.535	0.534	0.532	0.531	0.530	0.528
2011	0.213	0.218	0.223	0.228	0.232	0.237	0.242	0.247
2012	0.026	0.085	0.144	0.204	0.263	0.323	0.382	0.441
2013	−0.222	−0.238	−0.253	−0.268	−0.283	−0.298	−0.313	−0.328
2014	−0.718	−0.820	−0.921	−1.023	−1.125	−1.226	−1.328	−1.430
2015	0.661	0.709	0.758	0.806	0.855	0.903	0.951	1.000
2016	−0.719	−0.735	−0.750	−0.766	−0.781	−0.797	−0.813	−0.828
2017	0.811	0.798	0.785	0.771	0.758	0.745	0.732	0.718
2018	0.008	0.012	0.016	0.020	0.023	0.027	0.031	0.035
2019	0.753	0.832	0.911	0.990	1.070	1.149	1.228	1.307

Table 5.

The excess return rate of the M-D portfolio under Type 3 relative to the M-V portfolio.

Interval	u=0.05	u=0.10	u=0.15	u=0.20	u=0.25	u=0.30	u=0.35	u=0.40
2005	−3.355	−3.290	−3.225	−3.159	−3.094	−3.029	−2.963	−2.898
2006	2.577	2.546	2.516	2.486	2.456	2.425	2.395	2.365
2007	0.069	0.047	0.024	0.002	−0.020	−0.042	−0.064	−0.086
2008	−0.092	−0.092	−0.092	−0.092	−0.093	−0.093	−0.093	−0.094
2009	−7.222	−7.669	−8.116	−8.563	−9.010	−9.457	−9.904	−10.351
2010	0.887	0.883	0.880	0.876	0.872	0.868	0.864	0.860
2011	0.613	0.648	0.683	0.718	0.753	0.787	0.822	0.857
2012	−0.054	0.021	0.097	0.173	0.249	0.324	0.400	0.476
2013	0.175	0.172	0.170	0.167	0.164	0.161	0.159	0.156
2014	−2.487	−2.752	−3.017	−3.282	−3.547	−3.812	−4.076	−4.341
2015	0.821	0.873	0.925	0.978	1.030	1.082	1.135	1.187
2016	−1.515	−1.548	−1.582	−1.616	−1.649	−1.683	−1.717	−1.750
2017	2.680	2.637	2.594	2.551	2.508	2.465	2.422	2.379
2018	0.004	0.009	0.014	0.020	0.025	0.030	0.035	0.040
2019	−0.420	−0.456	−0.492	−0.528	−0.565	−0.601	−0.637	−0.673

Mean-DCCA portfolio evaluation based on reward-risk ratio

Tables 3 to 5 assess that the M-D portfolio strategy is indeed superior to the M-V portfolio strategy overall only in view of the return rate, and can achieve the goal of portfolio strategy optimization and the improvement of investment performance. However, similar results can still be obtained by using risk-adjusted performance measurements such as the reward–risk ratio to test the validity of the M-D portfolio. Therefore, to further verify that the M-D portfolio strategy is better than that of the M-V portfolio strategy in realizing the mean-variance criterion, the differences in the reward-risk ratio between the M-D and M-V portfolios are listed in Tables 6 to 8 below.

Table 6.

The difference of the reward-risk ratio of the M-D portfolio under Type 1 relative to the M-V portfolio.

Interval	u=0.05	u=0.10	u=0.15	u=0.20	u=0.25	u=0.30	u=0.35	u=0.40
2005	−167.561	−23.402	68.761	124.426	156.386	173.544	181.651	184.305
2006	129.884	290.401	443.879	590.833	733.180	873.062	1012.223	1151.666
2007	−1036.776	−1095.370	−1155.865	−1218.137	−1282.022	−1347.311	−1413.746	−1481.008
2008	620.783	602.797	583.110	561.663	538.417	513.364	486.536	458.021
2009	−525.042	−485.731	−451.726	−422.047	−395.937	−372.800	−352.166	−333.655
2010	1329.561	1307.984	1282.804	1254.104	1222.028	1186.785	1148.640	1107.911
2011	1807.962	1647.402	1491.760	1349.380	1222.877	1112.040	1015.464	931.379
2012	−121.408	−101.838	−26.015	64.133	147.366	215.538	267.325	304.395
2013	1363.519	1284.562	1171.541	1047.834	927.433	817.150	719.407	634.293
2014	−3197.652	−2869.553	−2498.865	−2150.090	−1848.471	−1597.386	−1391.375	−1222.718
2015	−170.749	−168.563	−165.228	−160.762	−155.217	−148.677	−141.257	−133.092
2016	369.482	454.696	524.731	580.960	624.955	658.336	682.669	699.401
2017	−4522.293	−4849.341	−4762.492	−4401.233	−3921.776	−3428.551	−2972.409	−2570.719
2018	757.986	687.102	610.108	534.002	465.038	407.404	362.622	329.816
2019	58.776	46.490	39.152	34.395	31.069	28.584	26.625	25.017

Table 7.

The difference of the reward-risk ratio of the M-D portfolio under Type 2 relative to the M-V portfolio.

Interval	u=0.05	u=0.10	u=0.15	u=0.20	u=0.25	u=0.30	u=0.35	u=0.40
2005	441.818	531.839	575.669	588.413	582.302	565.649	543.650	519.409
2006	−177.377	−50.319	65.840	171.363	268.029	358.044	443.542	526.399
2007	−841.735	−892.886	−946.114	−1001.338	−1058.440	−1117.257	−1177.576	−1239.126
2008	601.504	581.787	560.235	536.794	511.441	484.191	455.118	424.365
2009	−436.862	−408.860	−383.956	−361.711	−341.754	−323.772	−307.502	−292.723
2010	1395.221	1376.285	1353.538	1327.049	1296.950	1263.436	1226.760	1187.234
2011	1719.777	1553.105	1392.644	1246.576	1117.355	1004.622	906.840	822.122
2012	−19.166	0.193	63.179	133.919	196.119	244.710	279.817	303.483
2013	1535.161	1464.268	1353.637	1228.430	1103.964	988.009	883.659	791.470
2014	−3470.242	−3110.306	−2709.619	−2335.008	−2011.754	−1742.666	−1521.634	−1340.362
2015	−93.662	−94.014	−93.213	−91.243	−88.131	−83.947	−78.796	−72.806
2016	263.386	349.235	420.179	477.534	522.826	557.635	583.495	601.827
2017	−4267.801	−4558.830	−4434.793	−4035.761	−3518.669	−2988.793	−2497.922	−2064.417
2018	769.477	681.639	585.560	489.892	403.027	331.265	277.612	241.633
2019	433.980	374.090	327.657	290.869	261.139	236.688	216.269	198.986

Table 8.

The difference of the reward-risk ratio of the M-D portfolio under Type 3 relative to the M-V portfolio.

Interval	u=0.05	u=0.10	u=0.15	u=0.20	u=0.25	u=0.30	u=0.35	u=0.40
2005	−96.884	39.413	124.037	172.530	197.717	208.516	210.691	207.839
2006	1149.962	1462.200	1777.214	2086.438	2376.970	2630.282	2823.885	2936.362
2007	−1187.115	−1249.249	−1313.019	−1378.280	−1444.849	−1512.499	−1580.953	−1649.880
2008	639.428	623.172	605.368	585.960	564.907	542.190	517.817	491.842
2009	−520.034	−480.837	−446.978	−417.461	−391.516	−368.544	−348.070	−329.713
2010	1271.684	1248.342	1221.461	1191.134	1157.516	1120.822	1081.326	1039.357
2011	1816.714	1657.731	1503.387	1362.069	1236.429	1126.286	1030.264	946.615
2012	−191.604	−186.996	−113.094	−15.032	81.860	165.856	233.170	284.108
2013	1841.340	1748.619	1617.906	1474.331	1333.128	1201.955	1083.783	979.043
2014	−3124.874	−2803.039	−2437.902	−2094.001	−1796.651	−1549.313	−1346.601	−1180.862
2015	−194.985	−192.225	−188.314	−183.277	−177.168	−170.077	−162.118	−153.430
2016	435.117	519.664	588.957	644.384	687.531	720.031	743.459	759.270
2017	−4683.407	−5019.649	−4942.113	−4590.226	−4120.136	−3636.199	−3189.183	−2796.364
2018	778.018	727.029	670.188	612.387	557.779	509.032	467.127	431.691
2019	−333.197	−276.164	−233.253	−200.308	−174.504	−153.916	−137.213	−123.458

From Tables 6 to 8, there are 75, 77, and 67 cases where the difference of the reward–risk ratio of the M-D portfolio and the M-V portfolio under the three given preference types is greater than zero, accounting for 62.50%, 64.16%, and 55.83%, respectively. As the reward–risk ratio surveys the cumulative rate of return obtained from taking a unit of risk, the larger the reward–risk ratio is, the more effective the portfolio is in maximizing returns, minimizing risks, and more in line it will be with the mean-variance criterion. Thus, Tables 6 to 8 indicate that the M-D portfolio strategy meets the mean-variance criterion more than the M-V portfolio strategy, and the M-D portfolio strategy continues to be superior to the benchmark portfolio strategy under the risk-adjusted performance measurement index, thus the M-D portfolio strategy is valid.

Robustness evaluation of Mean-DCCA

Although Tables 3 to 8 set the investor preference degree $α_{i} (s)$ for time scale $s$ when verifying the validity of the M-D investment portfolio, similar results can be obtained when the investor preference degree for the time scale takes other values. This means that M-D has good robustness when investors’ preferences for different time scales are taken into account. Considering that there are mostly short-term investors in the market, here we will take the time scale that investors prefer to be $S = {3, \dots, 31}$ as an example for illustration; thus, when $s \in S$ is established, we take $α_{i} (s) = 29^{- 1}$ , otherwise we take $α_{i} (s) = 0$ . For the sake of intuition and simplicity, Figure 2 below only presents the cumulative return series of the M-D portfolio under eight kinds of given returns and the cumulative return series of the benchmark portfolio in the sample range, which are denoted by M-D and M-V, respectively.

Figure 2.

The cumulative return series of the M-D portfolio under eight kinds of given returns and the cumulative return series of the benchmark portfolio.

Figure 2 clearly indicates that there are 112 cases in which the cumulative return rate of the M-D portfolio is greater than that of the benchmark portfolio under eight kinds of given returns $u$ , accounting for 93.33% of the total 120 cases. Under the long-term implementation of the M-D portfolio strategy and the benchmark portfolio strategy, the superiority of the M-D portfolio relative to the benchmark portfolio will be more obvious, and it can also reflect that the M-D portfolio strategy is more robust than the benchmark portfolio strategy and can be used by investors for a long time in practice. At the same time, the validity of the M-D portfolio strategy is further verified under other preference conditions and thus the robustness of the M-D portfolio strategy is verified.

To summarize, if there are fractal characteristics of the securities market, whether it is based on the return rate as the performance measurement or the reward–risk ratio as the performance measurement, the M-D portfolio strategy is generally better than that of the M-V portfolio strategy and will be more stable and more in line with the mean-variance criterion. Thus, the empirical results are consistent with the theoretical analysis. The empirical analysis further indicates that the M-D securities portfolio strategy that considers the fractal characteristics of the securities market is effective and can overcome the weaknesses of the mean-variance securities portfolio strategy.

Conclusion and Outlook

To accurately survey the correlation among assets, this study incorporates the fractal correlation characteristics of assets into the research framework of the portfolio strategy when the security market is a fractal market. Based on the use of DCCA to obtain indexes for measuring the fractal correlation between securities, the obtained fractal correlation measurement indexes are embedded in the reward-risk criteria of the MPT, the Mean-DCCA portfolio strategy for multiple risk assets is built, and the analytical solution of the portfolio strategy is given. Theoretical analysis indicates that the Mean-DCCA portfolio strategy synthetically considers multiple time scales, which are more refined than the M-V portfolio strategy of multiple risk assets and surmounts the drawbacks of the M-V portfolio strategy. Meanwhile, the fractal correlation index is more suitable to capture the correlation between assets than the traditional covariance, and it has more stable extrapolation robustness. Therefore, the mean-DCCA portfolio combination constructed by using fractal correlation index helps to make up for the stability of covariance matrix and its inverse matrix when calculate the weight. The empirical analysis also found that, compared to the M-V portfolio strategy, the Mean-DCCA portfolio strategy can improve investment performance and meet the RRC better, thus achieving the goal of optimizing the portfolio strategy of multiple risk assets.

In the case that the security market is a fractal market, this study positions DCCA into the research framework of the portfolio optimization model, and constructs the Mean-DCCA portfolio model of multiple risk assets under different time scales. Compared with previous studies, this model was no longer limited to two kinds of assets, albeit through the allocation of investment weight to a variety of assets it produces the diversification effect of investment, so as to fully reflect the diversification of investment and realize asset appreciation. In addition, this study further introduces investors’ preferences for different time scales, which can fully reflect investors’ preferences for different time scales and meet the objective needs of investors’ diversity. The related research results were not only beneficial for providing a new research perspective for scientific research workers but were also helpful in providing a reference for practice workers to make decisions. In the future, we can refer to the ideas of this paper and combine it with the multi-fractal detrended cross-correlation analysis (MF-DCCA) to make a more detailed description of the fractal volatility among security prices, that is, incorporate different volatility ranges into the framework of subsequent studies; using MF-DCCA correlation coefficient to substitute for DCCA correlation coefficient or traditional covariance, the Mean-MF-DCCA portfolio model of various risk assets can be further constructed and the multi-validity of the model was tested. At the same time, the Mean-DCCA portfolio model can also be brought into feedback trading strategies to further explore relevant applications of the Mean-DCCA portfolio model and expand the research in this paper. Furthermore, although the samples obtained in this paper have been able to evaluate the effectiveness of the Mean-DCCA portfolio model, we can further take the multinational stock market as samples for evaluation, to better provide decision-making reference for multinational investors.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (71903017), Ministry of Education in China Project of Humanities and Social Sciences (17YJC790168), and the National Social Science Foundation of China (17BJY188).

ORCID iDs

Xi Zhang

Xu Wu

Zhonglu Chen

Author Biographies

Xi Zhang is a Lecturer in Business School at Chengdu University of Technology. Her research interests include quantitative investment strategy, behavioral finance, risk management, empirical finance, and securities market microstructure.

Xu Wu is an Associate Professor in Business School at Chengdu University of Technology. His research interests include financing dilemma and quantitative investment strategy. He has published many papers in North American Journal of Economics and Finance, Journal of Computational and Applied Mathematics, Economic Computation and Economic Cybernetics Studies and Research, Nonlinear Dynamics, etc.

Linlin Zhang is a postgraduate student in Business School at Chengdu University of Technology. Her research interest is empirical finance and securities market microstructure.

Zhonglu Chen is a PHD student in the School of Economics & Management at Southwest Jiaotong University. His research interests include economic forecasting and corporate governance. He has published in journal such as: Resources Policy, North American Journal of Economics and Finance, International Journal of Finance & Economics, etc.

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