The Optimization of the Portfolio Selection Based on AC-CVaR Model—Evidence from China’s Privately Offering Funds


Based on the traditional mean-variance model research framework, this paper creatively proposes a model of using AC algorithm to forecast returns (adds the similar model of mature capital market to the training set), using CVaR as the risk index, and applies it to China’s Privately Offering funds allocation. The empirical analysis shows that the AC-CVAR model is significantly better than other comparative models in terms of risk diversification and portfolio return. The proposed model has filled in the blank of research of portfolio in the field of China’s privately offering funds.

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Ju, L. , Yu, N. and Ma, M. (2022) The Optimization of the Portfolio Selection Based on AC-CVaR Model—Evidence from China’s Privately Offering Funds. Open Journal of Business and Management, 10, 3564-3579. doi: 10.4236/ojbm.2022.106175.

1. Preface

In recent years, China’s capital market becomes more and more efficient. Asset allocation has become a research focus among scholars as the continuous enrichment of investment tools. At present, most of the research focuses on allocation of publicly offering funds. Jiang Xiaoquan and Ding Xiuying (2007) have researched the efficiency of the asset allocation of China market by establishing panel data model; Song Jiangli (2018) discusses the operation mode and development path of public offering funds in China combined with the actual situation of Chinese market; Li Yan (2018) has systematically and comprehensively studied the strategy and application of public offering funds in his paper.

The scale of China’s Privately Offering funds was 20.39 trillion yuan at the end of July, 2022 from statistics in Fund Industry Association. In this paper China’s Privately Offering funds mainly refer to funds privately raised and invested in stocks, bonds, futures, options and other financial products listed and traded in the exchange. China’s privately offering funds have been “sunshine” for 7 years in which the number and scale of privately offering funds have grown explosively, but there is little research on privately offering funds allocation. Wang Hongjun (2018) put forward suggestions on the allocation of privately offering funds in China from two aspects that are strategy design and underlying fund screening by drawing on the successful development experience of overseas FUNDS, but the research needs to be further explored. In this paper we combined with the current situation of privately offering Funds in China, further study the features of privately offering funds strategy index in wind which is a widely used data supplier, pioneering using Analog Complexing (AC) to predict the yield of index based on which we build a model which called AC-CVaR derived from mean-variance model applied to the privately offering funds configuration field, by which new ideas and enlightenments for China privately offering funds investment are expected.

2. Research Review

2.1. Mean-Variance Model Extended Research

Harry Markowitz published the paper Portfolio selection in 1952 (Markowitz, 1952) in which the mean value is proposed to describe the expected return and the variance to measure the risk for the first time, and then the famous portfolio model “mean-variance model” is established. The model framework is of great theoretical significance, but according to the previous research, there are many shortcomings in the practical application, specifically as follows:

1) The model assumption is too ideal, such as requiring the yield of risk assets to obey a normal distribution;

2) In terms of risk measurement, the variance is taken as the risk estimate, and both extreme returns and losses may have the same impact, which does not accord with the subjective feelings of investors. Most investors only regard the losses as real risks;

3) In terms of income forecast, it is not accurate to predict the yield of each future period with the historical yield;

The existing quantitative measurement method of asset risk can be roughly divided into two categories. One method emphasizes the deviation degree of the actual rate of return relative to the expected return rate by measuring the certainty of the deviation, of which representative indicators are variance and standard deviation; the other method emphasizes the maximum possible loss at a given confidence level and a certain holding period, and measures the risk by calculating the expected maximum loss, including VaR (Value at Risk) and CVaR (Conditional VaR). As a risk measure index, VaR considers more about the risks brought by the downside change of asset prices, which is more in line with the subjective feelings of investors. After being proposed, it is widely used in the field of financial risk measurement. In the field of portfolio optimization, domestic and foreign scholars have proposed to introduce VaR into the traditional mean-variance model and build the mean-VaR model. Nevertheless, VaR, Beder (1995) pointed VaR that is insufficient for two reasons: firstly, VaR does not satisfy the axiom of consistency; secondly, VaR is insufficient to measure the tail loss. Rockafellar and Uryasev (1999) proposed CVaR (Conditional VaR) based on the VaR method. CVaR measures the mean of the tail loss, and calculates the loss exceeding the conditional expectation of the VaR part. The effectiveness of using VaR and CVaR instead of variance as measurement of risk has been greatly verified by the academic community, and Tian & Huang (2004), Gao and Li (2005), Liu Junshan (2007) found that CVaR, as a risk measure, is more suitable for portfolio optimization than VaR.

The traditional method of mean-variance model is to calculate the sample mean of yield in a certain period. The prediction accuracy of this method is poor. Since then, some scholars have proposed several improvement methods, such as local integral mean method, moving average method, and exponential smoothing method. In addition to improving the metric method, domestic and foreign scholars have found that volatility models can well describe the properties of financial time series with thick tail peaks and conditional heteroscedasticity. Engle (1982), Bollerslev (1986) and Nelson (1991) proposed ARCH model, GARCH model and EGARCH model to describe the financial data successively. Many domestic scholars have applied the GARCH family model to the portfolio optimization problem, Wu Zhenxiang et al. (2006) Using the t-GARCH model to estimate the conditional probability distribution of certain future yield rate of a single asset, Guo and Xu (2009) used EGARCH model to estimate the distribution of individual assets. Moreover, since it is difficult to effectively predict expected yields, Fisher Black and Robert Litterman (1992) put forward to a classical improvement model Black-Litterman model (abbreviated as B-L model). The biggest contribution of this model is to predict the rate of yield. In order to reduce the estimation error, the Bayesian estimation with subjective information is added. Nowadays, the B-L model is widely used in the field of investment, but there are some problems, such as being difficult to obtain expert views, having subjectivity, and having unrepeatable verification.

2.2. Improved Mean-Variance Model Based on AC-CVaR

This paper applies the AC algorithm to predict the yield of the privately offering funds index. The AC algorithm also becomes a similar body synthesis algorithm. The algorithm can classify, cluster and predict complex objects, and is a wide-ranging sequence pattern recognition method. The algorithm was firstly and successfully used to predict meteorological aspects, and later extended to management decision-making, economic prediction and other fields. In recent years, scholars have improved the inductive and self-organized data mining method and the selection program to enhance the application scope and effect of the algorithm. AC algorithm has its unique advantages over the usual parameter model: when predicting the output variables, it does not need to estimate the input variables in advance, that is, the prediction results are completely calculated through the known data information. Scholars have made a lot of exploration on the application of AC algorithm, Zhang Zhiyong and He Changzheng (2005) applied AC algorithm to predict stock changes, after comparing AC algorithm with the hierarchical clustering method; Yao Yi et al. (2010) applied the AC algorithm to the prediction of China Economy-Environment Comprehensive Development Index; Xuehong Li et al. (2011) predicted electricity consumption in Gansu based on AC algorithm; Li Qiumin (2013) combined GMDH algorithm (packet data processing algorithm) and AC algorithm modeling is used to predict the fluctuation of the exchange rate market, reflecting the wide application of A C algorithm.

Based on the research framework of the traditional mean-variance model, this paper breakthrough proposed a model combining AC algorithm and CVaR risk index, and applied it to the study of privately offering funds combination optimization in China. When using the AC algorithm for modeling, considering that China’s capital market is constantly mature and its similarity with foreign developed capital markets will become higher and higher, the similar model of mature foreign markets is creatively added as a training sample to improve the accuracy of the future return prediction of the privately offering funds index. In the solution of the AC-CVaR model, by using Liu Xiaomao et al. (2003, 2005) as reference, the proposed method can not only avoid various assumptions that do not conform to the actual market situation, but also improve the solution efficiency. Finally, the results of the AC-CVaR are compared with other classical models to demonstrate the superiority and utility of the AC-CVaR model.

3. Model Introduction

3.1. Mean-Variance Model

Traditionally, mean-variance model takes the yield rate and the covariance of different assets as the estimate of the expected return rate and the expected return risk respectively. At the same time, combined with the utility non-difference curve of investors, the weight of each asset is determined by solving the non-linear optimization model, so as to calculate the optimal allocation parameters of different risk asset combinations.

The specific model is:

ω = argmax ω ( ω T μ λ 2 ω T Σ ω ) (1)

s .t . { ω T 1 = 1 ω 0 ( shortsellinglimited )

Among them, ω is the weight vector of the asset, ω is the asset weight vector determined after optimization, μ is the expected return vector of the asset, Σ is the variance covariance matrix of the asset, λ is the risk aversion coefficient, and 1 is the column vector of all 1 elements. The following models will be used to illustrate these symbols.

When solving the mean-variance model, firstly we get the effective frontier of asset allocation by inputting the expected return and expected risk of various risky assets, and then solve the optimal allocation point from the tangent of the effective frontier through the optimization model. However, the optimal allocation points found by the model often cannot meet the investment expectations of investors. In view of this problem, the utility function can be transferred into a constraint condition, so that the asset allocation effect will be closer to the investment target. One of the most representative types is: target return model, target risk model.

The target benefit constraint optimization formula is:

ω = argmin ω ( ω T Σ ω ) (2)

s .t . { ω T μ = μ 0 ω T 1 = 1 ω 0

Among them, μ0 is the asset allocation target income value.

The target risk constraint optimization formula is:

ω = argmax ω ( ω T μ ) (3)

s .t . { ω T Σ ω = σ 0 ω T 1 = 1 ω 0

Among them, σ0 is the asset allocation target risk value.

3.2. Risk Measures

Regarding the risk measures, the VaR model, proposed by the international research institute G30 in its research report, Practice and Rules for Derivatives (1993), is widely recognized. The VaR model is the maximum loss value of an investor’s portfolio positions held during a holding period at a given confidence level.

The mathematical expression for the VaR model is:

Prob ( X VaR ( α ) ) = α (4)

Prob ( X VaR ) represents the probability that the loss value X of the portfolio does not exceed the VaR value given the confidence level α.

The CVaR model can be understood as the average of tail loss, including tail information of the loss above the threshold VaR. Its mathematical expression is:

CVaR = E [ X | X > VaR ( α ) ] (5)

Among them, X represents the loss value of the portfolio.

Compared with the traditional VaR model, the advantages of CVaR model are mainly reflected in the following three aspects: Firstly, CVaR satisfies the axioms of consistency, that is, to meet the secondary additivity, monotony, translation invariance and positive homogeneity (Lin & He, 2003); Secondly, tail risk of loss exceeding VaR is considered; thirdly, VaR value can be obtained while calculating the CV aR value, which provides a method of calculating VaR value. Therefore, it is more appropriate to apply CVaR model in studies of portfolio optimization.

3.3. AC Algorithm

AC algorithm assumes that pattern state of time series during a period repeats in some form, and that the current development trend has historically one or more similar samples, in common sense. In this way, we can combine the samples from historically similar periods through transformation and combination to infer them and predict the future trends in the current state (He, 2005).

When applying the AC algorithm for the prediction, the implementation procedure is as follows:

1) Generate pending mode:

For a given sequence of real-value dimensions with observations, generate a pattern defined as an inclusion, row table from the row: N m x t = { x 1 t , , x m t } ( t = 1 , 2 , , N ) x t i k P k ( i )

P k ( i ) = [ x 1 i x l i x m i x 1 , i + j x l , i + j x m , i + j x 1 , i + k 1 x l , i + k 1 x m , i + k 1 ] k × m (6)

k is called the pattern length ( i = 1 , 2 , , N k + 1 ) , and different parameters k will lead to different models. Initially compare all selected modes P k ( i ) ( i = 1 , , l , , N k + 1 ) and reference modes P R and find similar modes to study. When making the prediction, the AC algorithm combines the extension of the historical similar mode as the future prediction of the reference mode. Therefore, when this method is used to predict, prediction interval is exactly the extension length of the reference mode, and the nearest sample mode before the prediction starting point is selected to reference, i.e. P R = P k ( N k + 1 ) .

2) Transform the pending mode:

x l , i + j For a certain reference pattern of length k , there may be one or several similar patterns of length k in the data sample. But since the system is dynamic, similar patterns in different periods may have different means and standard variances. In order to measure the similarity between these patterns, it is necessary to transform the patterns to the same reference point to make them comparable. The transformation between the modes generally adopts the linear transformation, and the transformed mode is as follows:

T i [ P k ( i ) ] = [ x 1 i x l i x m i x 1 , i + j x l , i + j . x m , i + j x 1 , i + k 1 x l , i + k 1 x m , i + k 1 ] (7)

Among them, x l , i + j = a 0 l i + a 1 l i x l , i + j , j = 0 , 1 , , k 1 ; i = 1 , 2 , , N k + 1 ; l = 1 , 2 , m , the parameters a 0 l i are the state difference between the reference modes and the similar modes P k ( i ) , and the parameters a 1 l i are the uncertain factors. Taking the reference mode x i j ( i = N k + 1 , N k + 2 , , N ; j = 1 , 2 , m ) as the benchmark, use the least squares method to estimate for each mode P k ( i ) to be selected, obtain unknown weights a 0 l i , a 1 l i , and calculate the sum of error squares of the pattern similarity.

3) Select the most similar pattern:

Transform all modes to the same benchmark, followed by identifying similarity between patterns, namely pattern similarity. Mode similarity can be measured by calculating the distance between the modes. The ith mode P k ( i ) to be selected corresponds to the reference mode P R . The similarity s i definition formula is as follows: i

s i = 1 d i (8)

i Among them, d i is the distance between the ith mode P k ( i ) to be selected and the reference mode P R , the formula is:

d i = 1 k + 1 j = 0 k 1 r = 1 m ( x r , i + j x r , N k + j + 1 ) 2 (9)

After the pattern similarity is calculated, the similar pattern set is selected according to the similarity size and the actual usage requirements.

4) The set of similar patterns selected by the above steps makes the N-step length prediction, and the prediction results of each pattern are combined to get the final prediction results.

3.4. AC-CVaR Model

Using the AC algorithm to predict the yield rate, and the CVaR as the risk constraint indicator, the portfolio optimization model can be expressed as:

ω = argmin ω ( CVaR ( α ) ) (10)

s .t . { ω T μ A C = μ 0 ω T 1 = 1 ω 0

where CVaR ( α ) represents CVaR value at the confidence level α , and μ AC is the vector of return on asset predicted using the AC algorithm.

4. Empirical Study

4.1. Data Selection and Processing

Empirically, the eight Wind privately offering funds indexes are selected as the underlying asset pool, which are respectively multi-strategy, stock strategy, bond strategy, stock market neutrality, management futures, macro strategy, event-driven, and arbitrage strategy privately offering funds index. Privately offering funds index series is an index system launched by Wind that comprehensively reflects China’s privately offering funds market, based on its extensive experience of international and domestic privately offering funds index compilation, using its own advantages in the field of financial data services and combining with the characteristics of the Chinese market. Based on the historical backtracking number of the components of the private equity index, the base date is set at December 31, 2014 and the basis point at 1000 points. The point update frequency is weekly, and the last trading day of each week is updated on the last trading day of the previous week.

When performing yield prediction using the AC algorithm, to increase the accuracy of the prediction, in addition to the historical time-series data of the eight indices themselves, multi-dimensional data were also added into the training set, specifically including: China stock series index (Shanghai Composite Index, Shenzhen Composite Index, CSI 300 Index, CSI 500 Index, and Kechuang 50 Index), China Futures Series Index (Nanhua Commodity, Nanhua Agricultural Products, Nanhua Energy Chemical, Nanhua Industrial Products, Nanhua Metals), International important Index (Brazil I BOVESPA Index, Dow Jones Industrial Average, NASDAQ Index, S&P 500, UK FTSE 100, France C AC40, Germany DAX, Nikkei 225, Korea Composite Index, Hong Kong Hang Seng Index, Australia S&P 200, India S ENSEX30, Russia RTS, Taiwan Weighted Index).

The data was selected from weekly closing price from January 1, 1990 to July 4, 2021. The time interval of the index that has not appeared yet was taken as empty value, and the rest was processed by post filling with the price index. The data source for the empirical analysis is Wind Information Co., Ltd which is widely used by nearly all Chinese financial institutions for its accuracy and timeliness. The main implementation software is Spyder (python3.8).

4.2. Normal Distribution Characters of Indexes

The traditional mean-variance model assumes that the yield of various risky assets in the portfolio meets the characters of normal distribution. So the first step is to compute the 3.2 Normal distribution characters of each major index. The Kolmogorov-Smirnov test (KS test) will be used to detect whether the returns of the indexes follows a normal distribution, and the results are shown in Table 1.

The results show that only among the eight strategy indexes, the Market Neutral Strategies index accepts the original hypothesis when the confidence level is 95%, that is, the yield sequence basically follows a normal distribution. But the yield of the other indexes does not meet the demands of normal distribution. As can be seen from the above tests, the yield sequence of most privately offering funds indexes does not obey the normal distribution, and the reliability of the combination optimization method such as the mean-variance model method is questionable. In the paper, we find other ways such as AC-CVaR model which has a low demand for data quality to make a reasonable portfolio.

4.3. Empirical Analysis

Experimental design and related parameters description:

1) Assets: Eight Wind privately offering funds index in China.

2) Empirical period is from January 1, 2015 to July 4, 2021. The data sets of January 2015 were used as the training sample without adding into the back-test period, and the latest data was gradually added as the training sample thereafter.

3) Position adjustment frequency: On the last trading day of each month, the model calculates the new weights for different asset.

Table 1. Normality tests of the eight major privately offering funds strategy index.

Note: The closer the statistic w-value is to 0, the better the sample data and the standard normal distribution fit. If the p-value is greater than the significance level (usually 5%), the null hypothesis is accepted and the overall sample is to obey the normal distribution is considered.

4) Minimize the risk model including mean-variance model, mean-VaR model and mean-CVaR model. With other conditions are the same, but different methods are used to measure risk, and different empirical results are compared to prove the superiority of CVaR in measuring risk of privately offering funds index.

5) Given the constraint of target return, minimize the risk model including mean-CVaR model, GARCH-CVaR model and AC-CVaR model. When other conditions are the same and CVaR is selected as the way to measure the risk. Compared with the empirical results, the superiority of AC algorithm in predicting returns in privately offering funds index is proved.

4.3.1. Empirical Comparison of Different Risk Measures

To choose more reasonable way to measure risk, Wu Shinong and Chen Bin (1999) proposed the three risk methods including variance, Downside-Risk and VaR in the paper “the application efficiency”. They transformed the model into the same coordinate axis and comparing the advantages of them. But the views is refuted by Li Jian (2000) who pointed out that Wu Shinong’s article discussed both methods and conclusions, and believed that it is unreasonable to compare the application efficiency of different models with the standard of “various effective boundaries in the coordinate axis”. In order to select the indicators to measure the risk, a unified standard is constructed to compare different indicators, which provide better reference for privately offering funds selecting when making investment decisions. The following empirical judgment is based on the return of the way. Specifically, if a certain method allows a portfolio to get more returns at the same risk over the same period, it is determined that this indicator is better. The way can more comprehensively and profoundly depict the risk, and more in line with the investors’ subjective feelings of suffering risk.

The mean-variance model, mean-VaR model and mean-CVaR model are respectively used for portfolio optimization in the following paper, and the Monte Carlo simulation method is used to solve the optimal allocation weight of risk assets under the minimum variance, minimum VaR (95% confidence level) and minimum CVaR (95% confidence level). The empirical result is shown in Figures 1-3 below. The minimum variance constrain leads to the stable changes as the numbers of the samples increase compared with the other two models which are minimum VaR constraint and minimum CVaR constraint. That is to say minimum VaR constraint and minimum CVaR constraint are more sensitive to the market. Figure 4 shows the comparison of the net value of the portfolio under different way of risk constraints. Table 2 shows the comparison of the risk-return performance of the investment portfolio under different risk index constraints.

As can be seen from the diagrams of position adjustment, the weight of various risk under VaR and CVaR index constraint is reasonable and the risk dispersion is effective. But the weight under the variance constraint is nearly unchanged and lacks flexibility, which mainly focusing on the four types of privately offering index assets corresponding to bond strategy, stock market neutrality, management futures, and arbitrage strategy.

Figure 1. Adjustments under the minimum variance constraint.

Figure 2. Adjustments under the minimum VaR constraint.

Figure 3. Adjustments under constraints of Minimum CVaR.

Figure 4. Comparison of the portfolio under risk index constraints. X-axis represents the number of the samples, Y-axis represents the net value.

Table 2. Comparison of portfolios under different risk index constraints.

From the perspective of risk and return, the portfolio under the CVaR constraint is significantly better than that of the combination under the variance and VaR constraints. There is no obvious difference between the variance and VaR constraint in the return performance. The Maximum Drawdown of the combination under the CVaR constraint is also well controlled and the Calmar ratio is greatly improved.

In general, CVaR is used to measure risk more considering the risks brought by the downside of the asset, which is more consistent with the subjective feelings of investors. The empirical result also shows that it is useful for the optimization of privately offering fund portfolio, which providing a higher return and a lower risk than other constraints. Therefore, it is more appropriate to minimize CVaR as a risk constraint in the privately offering funds index configuration model.

4.3.2. Empirical Comparison of Return Forecasting Method

Based on the empirical results above, the mean-CVaR model, GARCH-CVaR model and AC-CVaR model are used to conduct portfolio optimization in the research of privately offering funds configuration. Specifically, GARCH (1, 1) is selected to describe the yield sequence, and AC algorithm is applied to select the similar mode with the same characteristics in the training set, so as to predict the return of the next period A C-CVaR model. The target annualized yield of the portfolio is set at 12% in the model, and the above three models are applied to optimize and solve the optimal allocation weight of risk assets under the constraint of minimum CVaR (confidence level of 95%). The result is as shown in Figures 5-7 below. Figure 8 shows the comparison of the net value performance of the optimal asset allocation portfolio under the different models, and Table 3 shows the comparison of the risk-return performance of the investment portfolios under the different models.

As can be seen from the previous position change chart, these three models are effective for risk dispersion. The average turnover rate of mean-CVaR model is 8.06% compared with 8.37% in GARCH-CVaR model and 10.25% in AC-CVaR model which shows that the AC-CVaR model has relatively higher turnover rate and is more sensitive to market information.

Figure 5. Month position adjustments under the mean-CVaR model.

Figure 6. Month position adjustments under GARCH-CVaR model.

Figure 7. Month position adjustments under the AC-CVaR model.

Figure 8. Net value of portfolios under different models. X-axis represents the number of the samples, Y-axis represents the net value.

Table 3. Comparison of portfolios under different models.

From the perspective of risk-return performance, the return performance of AC-CVaR model is higher than the other two models, and the maximum drawdown is smaller than other two models. Sharp ratio and Calmar ratio are also greatly improved in AC-CVaR model reflecting the superiority of AC-CVaR model in privately offering funds configuration.

In conclusion, it is more appropriate to use the AC-CVaR model to optimize the investment portfolio in the study of the privately offering funds allocation problem in China.

5. Conclusion and Outlook

This paper mainly explores the portfolio optimization problem of privately offering funds in China, innovatively proposes a model combining AC algorithm and CVaR index. The empirical results verify the superiority of this model in risk diversification and improving the yield, which has great practical significance.

The innovations of this paper are as follows: 1) Use the AC algorithm to predict the yield of the privately offering funds, and add the similar mode of funds in the mature capital market in the training model sample as the training sample to improve the accuracy of the prediction. As a developing country, using AC model to find similar paths and modes in foreign mature capital markets, as a reference to predict the yield of domestic privately offering fund index, is in line with the law of actual capital market. 2) The empirical result shows that it is reasonable to use CVaR as an index to measure the risk of China’s privately offering fund index, which has an important value for measuring the risk of China’s privately offering fund index. 3) Empirically show that the performance of AC-CVaR model is significantly better than other comparative models in terms of risk dispersion and portfolio yield. The proposal of this model fills the gap in the field of privately offering funds allocation in China. However, the model proposed in this paper is worth further study in improving the AC algorithm and optimizing the model solution process, which is the direction in the future study.

Conflicts of Interest

The authors declare no conflicts of interest.


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