Optimal Investment Portfolios for Internet Money Funds Based on LSTM and La-VaR: Evidence from China

Abstract

The rapid development of Internet finance has impacted traditional investment patterns, and Internet money funds (IMFs) are involved extensively in finance. This research constructed a long short-term memory (LSTM) neural network model to predict the return rates of IMFs and utilized the value-at-risk (VaR) and liquidity-adjusted VaR (La-VaR) methods to measure the IMFs’ risk. Then, an objective programming model based on prediction and risk assessment was established to design optimal portfolios. The results indicate the following: (1) The LSTM model results show that the forecast curves are consistent with the actual curves, and the root-mean-squared error (RMSE) result is mere 0.009, indicating that the model is suitable for forecasting data with reliable time-periodic characteristics. (2) With unit liquidity cost, the La-VaR results match the actuality better than the VaR as they demonstrate that the fund-based IMFs (FUND) have the most significant risk, the bank-based IMFs (BANK) rank 2nd, and the third-party-based IMFs (THIRD) rank 3rd. (3) The programming model based on LSTM and the La-VaR can meet different investors’ preferences by adjusting the objectives and constraints. It shows that the designed models have more practical significance than the traditional investment strategies.

1. Introduction

Over the past three decades, the financial market in China has experienced unprecedented development. The proliferation of financial products and the upgrade of Internet technology have led to a growing trend toward Internet finance. Internet money funds (IMFs), the most widely used Internet finance products in China, have rapidly moved into the fast lane of development, stimulating the transformation of traditional financial investment methods. Therefore, the research on IMF investment portfolios to maximize profits has become a crucial flashpoint.

IMFs are the money market funds released by fund companies on Internet platforms and gather the idle funds of individual investors and then invest them with fund management companies (Ma et al. (2021) [1]). IMFs have the advantages of convenience and reasonable return rates, attracting a wide range of investors. Although the returns of IMFs in America and Europe are not so outstanding, at the moment, the IMF market in China is still booming. It is worth performing a profound study of the investment portfolios with the IMFs in China. Therefore, our research mainly discusses China’s IMFs. As a typical representative of IMFs in China, Yu’E Bao’s fund size reached 111,019 billion dollars by the end of December 2021. In just a few years, Internet transactions have become the primary medium for financial transactions and have triggered the explosion of the Chinese IMF industry. However, fast-developing Internet finance brings many uncertainties to online fund transactions, such as security risk with third-party platforms and potential loopholes in online trading. These pose challenges to IMF transaction management.

The existing investment methods face the challenges of high liquidity products and personalized investors. However, the investors need tools to predict the IMF return and determine risks to make proper portfolios. Therefore, this research comprehensively investigated different aspects related to IMFs, such as the prediction of return rates and risk assessment to balance investors’ expectations and as a basis to design the ideal IMF investment portfolios.

We first constructed a long short-term memory (LSTM) neural network model to predict the future return rates of IMFs, then utilized the value-at-risk (VaR) and liquidity-adjusted VaR (La-VaR) methods to calculate the risk of IMFs. Finally, we established an objective programming model to design the target portfolio with IMFs. The contributions and innovations are summarized as follows:

• We design an LSTM model to predict the return values in the next three months based on the IMFs data in China from the previous periods. To ensure the accuracy of the prediction, we first take the existing dataset as the training and test set and continuously conduct training and testing on this set interval. Compared with the GARCH model, the LSTM prediction model we constructed to forecast the return rates of IMFs has the advantage of predicting the large-scale and long-term data with high accuracy and robustness.
• We discuss the method of the La-VaR calculation. The La-VaR method we proposed to calculate the risk of IMFs can take the high liquidity of IMFs into account. The La-VaR method can be modified to apply to financial fields such as stocks and have broader practical significance than the traditional ones.
• We put forward the objective programming model combining deep learning algorithms and the liquidity factor, which is better than the existing portfolio models. The parameters can be adjusted based on the investors’ personalized preferences. It provides a practical method for investors to design portfolios and manage their assets.

The remainder of this paper is organized as follows: Section 2 summarizes the relevant literature. In Section 3, we construct an LSTM model to forecast the IMFs’ return rates, propose the VaR and La-VaR methods to measure the IMFs’ risks, and establish the objective programming model for portfolio design. In Section 4, we study actual IMF data in depth and conduct an empirical analysis to predict the return rates by the GARCH and LSTM models and assess the risks by the VaR and La-VaR methods. The portfolio optimization design is described in Section 5, and the conclusions are presented in Section 6.

2. Literature Review

Portfolio design and management comprise a complex process involving financial forecasting, risk measurement, portfolio optimization, and other multi-faceted aspects (Huang, 2017, [2]; Paiva, 2019, [3]). It is also necessary to continuously adjust portfolios according to market changes combined with product information and relevant policy information to maximize benefits and make better investments (Masmoudi, 2017, [4]; Zhao et al., 2019, [5]). The portfolio theories of references (Ma et al., 2021, [1]; Tian et al., 2019, [6]; Iryna et al., 2019, [7]) provided crucial theoretical guidance for investment research and the theoretical basis for the effective design of optimal portfolios.

The return rates of IMFs are crucial to carrying out investment portfolio design. With the development of computer network technology and the wide application of machine learning theory, applying neural networks to forecast financial data has become the latest hot spot. Zhang et al. (2019) [8] and Bruce et al. (2019) [9] constructed neural network models for data prediction. The former researched financial trends, while the latter used auto-regressive neural networks and sentiment predictors for stock price prediction. Convolutional neural networks (CNNs) were used by both Muhammad et al. (2020) [10] and Chen et al. (2020) [11]. Muhammad et al. (2020) [10] also adopted a recurrent neural network (RNN) to predict financial trading signals. Chen et al. (2020) [11] applied deep learning technology to conduct quantitative research on financial investments. Wang et al. (2019) [12] proposed a motifs-based dynamic Bayesian networks (m_DBNs) model and introduced the multi-DBN model to improve prediction accuracy. Later, they proposed a high-precision and robust CNN model for predicting the online reliability of ample data space. Sadik et al. (2019) [13] constructed a news-augmented GARCH model to forecast the volatility of stock returns. Lei et al. (2021) [14] used text mining and the LSTM model to forecast and analyze the stock volatility of high-frequency financial data. Chen et al. (2022) [15] improved the LSTM model based on modified K-means for stock price prediction. The research showed that the LSTM model was better than the RNN model for the mid- and long term, and deep learning algorithms had quite scientificity and feasibility in financial time series analysis.

Comparing GARCH, CNN, and RNN, the GARCH model has a certain hysteresis quality. CNN tends to fall into the optimal local solution instead of the global one by the gradient descent algorithm, and RNN suffers from gradient explosion and gradient disappearance when dealing with long-range data, which reduces the accuracy of long-term prediction (Xue et al., 2020, [16]). The LSTM model is more suitable for training and predicting the time series data with strong regularity and periodicity. It can reflect the nonlinear relationship of the time series data and solve the gradient disappearance or gradient explosion problem of long series data by introducing the concept of gates (Chen et al., 2022, [15]; Yang et al., 2019, [17]; Wang et al., 2021, [18]). Thus, the LSTM model has the advantage of effectively solving the forgetting problem of the long-term series memory, is well applied to the prediction of the cyclical long time-span yield series, and can effectively solve the problem of gradient disappearance.

The measurement of the risk of IMFs is also a vital issue for portfolio design. Wang et al. (2019) [19] designed the risk measurement method and applied it to effectively assess the Open-source Software Ecosystem’s health. In the financial field, the VaR is often used in risk management and measurement (Teplova et al., 2019, [20]). Liquidity risk will cause errors in calculating the risk of financial products. In this regard, Muela et al. (2017) [21] added liquidity risk to risk estimation through the VaR based on the spread of bonds. Febi et al. (2018) [22] explained the effect of liquidity risk on bond yield spread. Fiza et al. (2019) [23] and Hung et al. (2020) [24] believed that liquidity characteristics have a significant influence on financial markets. Fiza et al. (2019) [23] constructed a simplified VaR model to explore the liquidity characteristics of the stock market in China, while Hung et al. (2020) [24] researched the influence of liquidity on the accuracy and efficiency of the VaR method from the portfolio level.

According to Masmoudi et al. (2018) [25], Mehlawat et al. (2020) [26], and Zhai et al. (2021) [27], it is of great importance to carry out objective planning for portfolios with risks and liquidity. They analyzed programming models for portfolios and evaluated the performance of portfolio design. Meanwhile, Zhai et al. (2021) [27] applied the uncertainty criterion to make the results of portfolio design more reliable.

The research of [13,14,15] showed that GARCH and LSTM can be used as practical tools to predict the yield of financial products, including stocks and bonds, but whether the prediction models can be applied to IMFs’ return forecast remains to be discussed. In addition, return prediction must consider risk factors, and the research of [21,22,23,24] stressed the critical impact of liquidity risk on the return of financial products and how to evaluate the liquidity factor in IMF risk. The application of liquidity risk results in portfolio research also needs further exploration. Besides, the literature [1,2,3,4,5,6,7,25,26,27] already proved the necessity of portfolio design, but whether the portfolio design of different IMFs can also follow similar portfolio ideas still needs further proof.

This research mainly addressed the following questions: Are prediction models such as GARCH and LSTM applicable to IMF forecast analysis in China’s IMF market? How do we incorporate liquidity risk into the IMF risk measurement process? How can the portfolio of different IMFs be designed using the LSTM model and considering liquidity risk?

By addressing the above questions, we conducted IMF return prediction with LSTM and GARCH, redesigned a La-VaR model to assess IMFs’ liquidity risk, and provided the optimal investment portfolio thinking for IMFs.

Therefore, we selected 18 representative IMFs for research, applied the LSTM model to forecast the future return rates, and utilized the La-VaR method, which introduces the liquidity factor to accurately measure the risk of the highly liquid IMFs. Furthermore, we constructed an objective programming model to design investment portfolios.

3. Models and Methods

To design the investment portfolio of IMFs more efficiently, the LSTM model was introduced to predict the return rates of the selected IMFs in future periods. The VaR and La-VaR models were used to calculate the risk with the liquidity factor taken into account. Subsequently, an objective programming model was constructed to design the target portfolio based on the obtained return rates and risk values of the IMFs.

3.1. LSTM Neural Network Prediction Model

The portfolio design needs the forecast return rates of the selected IMFs. Since the IMF return rates are highly cyclical (Ma et al., 2021, [1]), if we adopt the traditional step-by-step forecast method, the errors in the results will gradually accumulate. The longer the prediction period, the larger the difference between actual values. The errors of the prediction results will gradually become more prominent and deviate from the actual situation. Therefore, we constructed an LSTM model to forecast the return rates in the next three months.

The LSTM model can introduce the state of the information sets and use the “gate” concept to add the memory modification modules to the algorithm (Cao et al., 2020, [28]). The “gate” function filters the information, allowing certain information through. This study adds the forgetting gate, the input gate, and the output gate for filtering information. Based on the research of Chen et al. (2022) [15], the structure of the memory unit is visualized in Figure 1.

In Figure 1, $\otimes$ represents matrix multiplication and $\oplus$ represents matrix addition. A two-arrow merge means vector merger, and a one-arrow split means vector replication. For example, the upper $h_t$ indicates the output data of the output gate, while the right $h_t$ is the same as the former, but indicates the transfer to the next memory cell.

The forgetting gate reflects the need to “forget” some information in forecasting to “remember” new information. The forgetting gate vector $f_t$ is defined as follows:

$$f_t=\sigma \left(W_f\cdot \left[h_{t-1},x_t\right]+b_f\right),$$

(1)

where $\sigma$ is a sigmoid function ranging from 0 to 1; it controls the opening and closing of the forgetting gate. $W_f$ represents the weight matrix. $h_{t-1}$ represents the output state vector at time $t-1$. $x_t$ represents the input vector at time $t$. $b_f$ is the offset vector.

The input gate is responsible for selecting the new yield values. The vector formula of the yield information received $\widetilde{C_t}$ at the current moment $t$ is

$$\widetilde{C_t}=\tanh (W_c\cdot \left[h_{t-1},x_t\right]+b_c),$$

(2)

where tanh is a hyperbolic tangent function and other parameter definitions are similar to the forgetting gate.

Since it is impossible to remember amounts of data at a time, the yield values must be remembered selectively. The input gate vector $i_t$ is defined as

$$i_t=\sigma \left(W_i\cdot \left[h_{t-1},x_t\right]+b_i\right),$$

(3)

where the parameter definitions are similar to the forgetting gate.

Therefore, the information vector $C_t$ to be put into the information set at time $t$ can be defined as

$$C_t=f_t\times C_{t-1}+i_t\times \widetilde{C_t}.$$

(4)

The final input information vector is determined by the forgetting gate and the input gate jointly.

The output gate is responsible for outputting the required forecast values. The output gate vector $o_t$ is defined as

$$o_t=\sigma \left(W_o\cdot \left[h_{t-1},x_t\right]+b_o\right),$$

(5)

where the parameter definitions are similar to the forgetting gate and the input gate.

Therefore, the yield prediction output vector $h_t$ after selection is

$$h_t=o_t\times \tanh (C_t).$$

(6)

3.2. Risk Measurement Methods for IMFs

3.2.1. VaR Method

VaR represents the maximum possible loss of a financial institution or asset portfolio in a certain period at a specific confidence level (Chen et al., 2019, [29]). It is defined as

$$\mathrm{Prob}(\Delta P<\mathrm{VaR})=\alpha ,$$

(7)

where $\Delta P$ is the loss of a financial institution or asset portfolio during the holding period and $\alpha$ is the confidence level.

Assuming that the price volatility of the portfolio is only related to market factors, the volatility of each IMF is set as the risk factor ${\sigma }_i$ in the VaR model; it is expressed as

$${\sigma }_i=\sqrt{\frac{{\displaystyle {\displaystyle \sum }_{t=1}^N}{(r_{it}-{\mu }_i)}^2}{N-1}},$$

(8)

where $r_{it}$ is the return rate of ${\mathrm{IMF}}_i$ on day $t$, ${\mu }_i$ is the average return rate of ${\mathrm{IMF}}_i$, and $N$ is the number of days of ${\mathrm{IMF}}_i$.

Thus, the VaR value can be calculated as follows:

$${\mathrm{VaR}}_i=P_0\cdot \left(\alpha \cdot {\sigma }_i\cdot \sqrt{\Delta t}\right),$$

(9)

where the confidence level is taken to be 90%, corresponding to 1.65 standard deviations, that is $\alpha =1.65$. Since all the data in this study are daily, $\Delta t=1$ was set.

3.2.2. Liquidity Risk Assessment Method

Due to the liquidity characteristics of IMFs, we introduced a unit liquidity cost (ULC), and constructed a La-VaR model based on the bid-ask spread.

The redemption rule of IMFs has the feature that the redemption quantity of the day reaching the upper limit will lead to delayed redemption. The larger the bid-ask spread, the less liquid the assets; the smaller the bid-ask spread, the more liquid the assets. The bid-ask spread of the intraday data can reflect the liquidity of the asset position (Ma et al., 2022, [30]). Thus, the differences in return rates due to delayed redemption are defined as liquidity cost, which is used as the standard for liquidity risk assessment of IMFs.

Among the numerous asset allocations of IMFs, cash holdings are the most liquid part and can directly fulfill users’ redemption requests. Thus, cash holdings can represent each IMF’s maximum daily redemption amount. Based on the Wind Database, we use the demand deposit of IMFs to describe the most liquidity asset. According to Bangia et al. (2001) [31], the relative spread of returns is defined as

$$S_t=\frac{\left|r_{t+1}-r_t\right|\cdot \left(1-a\right)\cdot v}{\left(r_{t+1}+r_t\right)/2},$$

(10)

where $S_t$ is the relative spread of IMF returns on day $t$, $r_{t+1}$ and $r_t$, respectively, represent the return rates of IMFs on day $t+1$ and day $t$, $a$ is the ratio of cash holdings to the IMF size, and $v$ is the IMF size.

Due to the IMF sizes varying widely, the relative spreads of IMF returns need to be standardized. The ${\mathrm{ULC}}_i$ of ${\mathrm{IMF}}_i$ can be expressed as

$${\mathrm{ULC}}_i=\left[r_t\left(\overline{S_t^{*}}+Z_{\alpha }{\sigma }_{S^{*}}\right)\right],$$

(11)

where $\overline{S_t^{*}}$, ${\sigma }_{S^{*}}$, and $Z_{\alpha }$ denote the mean, volatility, and $\alpha -$ quantile level of the standardized unit relative spreads of IMFs, respectively.

Therefore, the La-VaR of IMFs is

$${\mathrm{La}\text{-}\mathrm{VaR}}_i={\mathrm{VaR}}_i+{\mathrm{ULC}}_i=P_0\cdot \left(\alpha \cdot {\sigma }_i\cdot \sqrt{\Delta t}\right)+\left[r_t\left(\overline{S_t^{*}}+Z_{\alpha }{\sigma }_{S^{*}}\right)\right].$$

(12)

3.3. Objective Programming Model

Assume that investors select IMFs primarily based on return rates and risks in the financial market, without interference from other external factors. Since each investor has a different risk appetite, his/her acceptable losses also vary. For the programming model, net profit maximization and investment risk minimization are usually set as the objectives. Hence, we applied the representative risk values to fix the risk level, transforming the dual-objective into a single-objective problem.

Investors are individuals who pursue investment returns. Although investors have different degrees of pursuing high returns, their investment purpose is to gain revenue. We took profit maximization as the goal, and the objective function formula is expressed as

$$\mathrm{Max}R={\displaystyle \sum }_{i=1}^n{A}_i{r}_i{y}_i,$$

(13)

where $R$ is the total return of the IMFs, $A_i$ is the investment amount on ${\mathrm{IMF}}_i$, $r_i$ is the return rate of ${\mathrm{IMF}}_i$, and $y_i$ is a 0–1 variable; it determines whether to buy ${\mathrm{IMF}}_i$ or not.

As different investors hold different amounts of capital, which makes portfolio design complicated, we set the amount of investment owned to be fixed as $W$, and the amount can be adjusted by modifying its value. Moreover, different investors have different risk tolerance, and we limited the ability through constraints. In addition, the maximum number of IMFs held was set to avoid the practical problems caused by the excessive diversification of IMFs. Thus, the constraint conditions are expressed as

$$\{\begin{array}{c}{\displaystyle {\displaystyle \sum }_{i=1}^n}{A}_i{y}_i\le W,\\ {\displaystyle {\displaystyle \sum }_{i=1}^n}{A}_i{v}_i{y}_i<kW, 0\le k\le 1,\\ {\displaystyle {\displaystyle \sum }_{i=1}^n}{y}_i\le m.\end{array}$$

(14)

${\displaystyle {\displaystyle \sum }_{i=1}^n}{A}_i{y}_i\le W$ represents that the investor’s holdings are no more than the total amount of investment. $W$ is the investor’s holding wealth.

${\displaystyle \sum _{i=1}^n{A}_i{v}_i{y}_i}<kW$ represents that the acceptable investment loss for the investor is less than $k$ of the capital held. $v$ is the risk value of each IMF. $k$ is a constant between 0 and 1, indicating the acceptable level of risk.

${\displaystyle {\displaystyle \sum }_{i=1}^n}{y}_i\le m$ represents that the number of IMFs held by the investors is limited. Due to the limitation, the investor is allowed to hold no more than $m$ IMFs at a time.

4. Return Rates’ Prediction and Risk Assessment of IMFs

Based on the LSTM model and the La-VaR method established above, each IMF’s seven-day annualized return rate for the next three months was predicted, and the VaR and La-VaR values of each IMF were obtained to measure the liquidity risk. Finally, the investment portfolios were designed by constructing an objective programming model.

4.1. Sample Data Selection and Processing

4.1.1. Data Selection

We divided the IMFs into three categories: third-party-based IMFs (THIRD), bank-based IMFs (BANK), and fund-based IMFs (FUND). THIRD mainly refers to IMFs docked by third-party payment institutions; BANK refers to IMFs docked by banks; FUND refers to IMFs docked by fund companies (Ma et al., 2021, [1]). We selected six representative IMF products in each category (see

Table 1. Basic information of the IMF samples.

Category	IMFs	Fund Companies	Fund Code
THIRD	Tianhong Yu’E Bao	Ant Financial (Hangzhou, China)	000198
	Jiashi Huoqianbao A	JD Finance (Beijing, China)	000581
	Yifangda Yilicai A	Tencent Financial Technology (Shenzhen, China)	000359
	Huitianfu Quan’E Bao	Tencent Financial Technology (Shenzhen, China)	000397
	Guangfa Tiantianhong A	Finance SN (Nanjing, China)	000389
	Yifangda Tiantianlicai A	Du Xiaoman Finance (Beijing, China)	000009
BANK	Zhongyin Huoqi Bao	Bank of China (Beijing, China)	000539
	Jianxin Xianjintianli A	China Construction Bank (Beijing, China)	000693
	Nongyinhuili Honglirijie A	Agricultural Bank of China (Beijing, China)	000907
	Ping’an Rizengli A	Ping An Bank (Shenzhen, China)	000379
	Zhaoshang Zhaoqian Bao A	China Merchants Bank (Shenzhen, China)	000588
	Gongyin Xinjinhuobi A	Industrial and Commercial Bank of China (Beijing, China)	000528
FUND	Zhonghai Huobi A	Zhonghai Fund (Shanghai, China)	392001
	Huaxia Xianjin Zengli A	China Asset Management (Beijing, China)	003003
	Nanfang Tiantianbao A	China Southern Fund (Shenzhen, China)	004970
	Jianxin Tiantianyi A	CCB Principal Asset Management (Beijing, China)	003391
	Yinhe Yinfu A	Galaxy Fund Management (Shanghai, China)	150005
	Guangfa Qiandaizi A	GF Fund Management (Zhuhai, China)	000509

).

In order to ensure consistency and continuity, we selected the seven-day annualized return rates and the ratio of bank deposits to fund scale of the 18 IMFs from 1 January 2018 to 15 March 2022, as the research samples. As the IMFs do not generate returns on Sundays, the return series are null on Sundays, so removing the null values from the return data is necessary. Excluding invalid data, there are 1236 observations of each IMF, totaling 22,248 observations. The data are from the Wind database.

4.1.2. Data Analysis

We conducted a descriptive analysis to compare the return rates in each category. The mean values reflect the average of the seven-day annualized return rates, and the standard deviation values reflect the variation. The skewness and kurtosis indicate whether the dataset is symmetrical or asymmetrical and whether the peak is high or low, respectively.

Table 2. Descriptive statistics of return rate series.

Category	Fund Code	Mean (%)	Min. (%)	Max. (%)	Standard Deviation	Skewness	Kurtosis
THIRD	000198	2.4274	1.311	4.394	0.7030	1.180	0.744
	000581	2.7491	1.531	5.153	0.7721	1.049	0.247
	000359	2.6865	1.594	4.690	0.8056	1.053	−0.440
	000397	2.6802	1.644	4.806	0.8054	1.082	−0.058
	000389	2.6399	1.410	5.580	0.7669	1.265	1.094
	000009	2.6072	1.387	4.386	0.7414	1.009	0.068
BANK	000539	2.6799	1.599	4.634	0.7551	1.121	0.335
	000693	2.7315	1.653	4.569	0.7365	0.939	0.131
	000907	2.5082	1.372	4.548	0.7574	1.049	0.257
	000379	2.5745	1.468	4.451	0.7464	0.917	−0.151
	000588	2.6020	1.423	4.593	0.8069	1.049	−0.009
	000528	2.7291	1.435	4.829	0.8208	0.717	−0.468
FUND	392001	2.3955	0.974	10.774	1.052	2.586	12.778
	003003	2.6239	1.049	5.438	0.9246	1.174	0.660
	004970	2.5665	1.297	8.264	0.8674	2.691	9.774
	003391	2.8199	1.308	6.901	0.7975	1.357	2.745
	150005	2.4049	1.388	4.736	0.6402	1.201	1.151
	000509	2.6228	1.509	4.796	0.7341	1.100	0.344

presents the descriptive statistics of IMFs.

Table 2 indicates that the mean return rates of IMFs in different categories range from 2.3955% to 2.8199%. The mean return rates for the THIRD and BANK series were higher than the FUND series, above 2.60%. However, those for the FUND series vary widely, making the yield sequence unstable. The standard deviation for the THIRD series is 0.7657, the BANK series is 0.7705, and the FUND series is 0.8360. The values are closely related to the degree of data dispersion, and a more considerable value indicates that the data are more dispersed.

Accordingly, it can be concluded that the THIRD series has higher and less volatile yields with relatively stable data sequences. The BANK series has moderate yields and volatility, while the FUND series has more significant volatility and poorer stability.

4.1.3. Data Processing

Since the return intervals of different IMFs are different, to obtain a better fitting effect and prevent training divergence, the training data need to be normalized into a dimensionless form with zero mean and unit variance before LSTM prediction. The standardized return rate $r_{it}^{*}$ is as follows:

$$r_{it}^{*}=\frac{r_{it}-{\mu }_i}{{\sigma }_i},$$

(15)

where $r_{it}$ is the return rate of ${\mathrm{IMF}}_i$ on day $t$, ${\mu }_i$ is the mean return of ${\mathrm{IMF}}_i$, and is the standard deviation of ${\mathrm{IMF}}_i$ and the volatility mentioned above. Save the relevant data, and de-standardize the yield data after prediction.

4.2. The Forecast of the IMFs’ Return Rates

4.2.1. The LSTM Forecast Model

We built an LSTM model for IMF yield prediction and input each IMF’s seven-day annualized return rate data from 1 January 2018 to 15 March 2022, into the model. The return rates are predicted the next day at a time, and the data obtained will be input into the model as variables to predict the next day’s data. We selected the model training set and test set in a ratio of 4:1 and evaluated the prediction effect of the model.

Typical IMFs from each category were selected as examples: Tianhong Yu’E Bao (000198) in the THIRD series, Jianxin Xianjintianli A (000693) in the BANK series, and Guangfa Qiandaizi A (000509) in the FUND series. Figure 2 and Figure 3 show the model prediction results.

In Figure 2, the blue curve represents the training set and the red curve represents the test set, which reflects the forecast results of the return series. In Figure 3, the blue curve represents the actual return results and the red curve represents the forecast return results, reflecting the consistency between the prediction and actual results.

The figures indicate that: (1) For the THIRD series, the difference between the LSTM prediction results and the actual results is quite small. The predicted return trend is nearly the same as the actual one. The two curves essentially match, and the prediction effect is quite good. (2) For the BANK series, the prediction results show a consistent trend with the actual results, and the curves match correspondingly. Thus, the accuracy of prediction is exceptional. (3) For the FUND series, the coincidence between the prediction results and the actual data is high, but the errors are relatively larger than the THIRD and the BANK series.

Besides, we also evaluated the effect using the metric of the root-mean-squared error (RMSE) to express the accuracy, which measured the deviation between the observed values and the true values. The result was 0.009, indicating that the model predicted commendably. In general, the LSTM model has a tremendous effect on predicting IMF returns in all three categories. It predicts the return rates effectively, and the prediction results can be applied to represent the actual data in the subsequent three months.

4.2.2. The GARCH Forecast Model

We also built a GARCH model for IMF yield prediction and still input each IMF’s seven-day annualized return data from 1 January 2018 to 15 March 2022 into the model. Similarly, we selected the model training set and test set in the ratio of 4:1 and evaluated the prediction effect of the model.

Table 3. The GARCH forecast results of the test set (%).

Fund Code	16 May 2021	17 May 2021	18 May 2021	19 May 2021	14 March 2022	15 March 2022
000198	2.1184	2.1169	2.1153	2.1137	1.7287	1.7273
000581	2.2483	2.2463	2.2656	2.2640	2.0040	2.0030
000359	2.1411	2.1380	2.1351	2.1320	1.3133	1.3107
000397	2.1656	2.1673	2.1741	2.1731	2.2338	2.2341
000389	2.3569	2.3509	2.3436	2.3405	2.1094	2.1085
000009	2.1316	2.1385	2.1381	2.1366	1.7065	1.7049
000539	2.2898	2.2754	2.2728	2.2693	1.7741	1.7723
000693	2.2826	2.2783	2.2760	2.2753	2.0355	2.0346
000907	2.0518	2.0621	2.0523	2.0547	1.7647	1.7635
000379	1.9300	1.9276	1.9189	1.9263	1.6228	1.6216
000588	2.2894	2.2856	2.2881	2.2908	1.8087	1.8070
000528	2.1967	2.2113	2.1824	2.1934	2.3864	2.3874
392001	1.7179	1.7835	1.9161	1.9105	4.7084	4.7265
003003	2.1287	2.1259	2.1128	2.1238	2.1298	2.1297
004970	2.1872	2.1729	2.1733	2.1699	1.5647	1.5626
003391	2.4911	2.4933	2.4987	2.5131	5.6847	5.7033
150005	2.0680	2.0567	2.1247	2.1213	2.0023	2.0019
000509	2.1333	2.1178	2.1099	2.1497	1.9064	1.9052

shows the forecast results of the test set.

The RMSE value of the GARCH model is 0.3669. By comparison, we found that its RMSE value is larger than that of LSTM, which indicates that the LSTM prediction results are more accurate. The LSTM model is more suitable for forecasting the return series. Therefore, we decided to use the LSTM model to predict the return rates of the IMFs.

4.2.3. The Forecast Result

We chose the one-layer LSTM as the main network, set the input vector as the existing yield sequence, the hidden layer unit as 200, and the output vector as the predicted yield sequence, and adopted the RMSE as the loss function. To balance the training time and model accuracy and prevent gradient explosion, the maximum number of iterations was set as 250, the gradient threshold was set as one, and the initial learning rate was 0.005, which was changed to 0.001 after 125 iterations.

All 18 IMFs were trained with the LSTM model, and the de-normalized return rates’ series for the following three months from 16 March 2022 to 7 June 2022 (excluding Sundays for a total of 72 days) were obtained.

Table 4. The forecast results of the seven-day annualized return rates (%).

Fund Code	16 March 2022	17 March 2022	18 March 2022	19 March 2022	6 June 2022	7 June 2022
000198	2.1279	2.1242	2.1226	2.1154	2.0605	2.0583
000581	2.2749	2.2643	2.2752	2.2707	2.0886	2.0814
000359	2.1610	2.1551	2.1512	2.1424	2.1025	2.1023
000397	2.1447	2.1561	2.1466	2.1291	2.0862	2.0915
000389	2.3730	2.3790	2.3673	2.3473	2.2417	2.1238
000009	2.0797	2.0980	2.1197	2.1336	2.0368	2.0309
000539	2.3199	2.3163	2.3098	2.3011	2.0877	2.0876
000693	2.3470	2.3302	2.3113	2.2870	2.3718	2.3571
000907	2.0808	2.0787	2.0656	2.0538	2.0203	2.0171
000379	2.0889	2.0437	1.9420	1.8772	2.0833	2.1080
000588	2.2483	2.2336	2.2493	2.2519	2.1281	2.0504
000528	2.1780	2.1710	2.2191	2.1948	1.9923	2.0568
392001	2.0603	2.0348	1.9988	1.9864	1.5255	1.5325
003003	2.1932	2.1546	2.1421	2.1249	1.8717	1.9768
004970	2.1869	2.1852	2.1831	2.1903	2.0936	2.1111
003391	2.5267	2.5142	2.5027	2.4805	2.5464	2.5150
150005	2.0727	2.0287	2.0358	2.0943	2.0210	2.0323
000509	2.4972	2.2861	2.1333	2.1000	1.9855	1.9864

shows the forecast results.

The forecast values from Day 5 to Day 70 are not shown in detail here. Data from Table 4 can be used to reflect the future return of the corresponding IMFs and provide references for the investment portfolio.

4.3. The VaR and La-VaR Calculation

4.3.1. The Calculation of VaR

Each IMF’s VaR and the average VaR within each category in the next three months were obtained based on Equation (9).

Table 5. The VaR and La-VaR values of IMFs.

Category	Fund Code	IMFs	VaR Value	Average VaR Value	La-VaR Value	Average La-VaR Value
THIRD	000198	Tianhong Yu’E Bao	−0.02372	−0.02299	−0.02849	−0.08018
	000581	Jiashi Huoqianbao A	−0.02235		−0.14510
	000359	Yifangda Yilicai A	−0.02156		−0.03850
	000397	Huitianfu Quan’E Bao	−0.02322		−0.19811
	000389	Guangfa Tiantianhong A	−0.02365		−0.03975
	000009	Yifangda Tiantianlicai A	−0.02341		−0.03112
BANK	000539	Zhongyin Huoqi Bao	−0.02282	−0.02312	−0.03605	−0.06360
	000693	Jianxin Xianjintianli A	−0.02484		−0.03657
	000907	Nongyinhuili Honglirijie A	−0.02214		−0.04245
	000379	Ping’an Rizengli A	−0.02083		−0.05314
	000588	Zhaoshang Zhaoqian Bao A	−0.02382		−0.03983
	000528	Gongyin Xinjinhuobi A	−0.02426		−0.17352
FUND	392001	Zhonghai Huobi A	−0.01591	−0.01737	−0.03627	−0.05630
	003003	Huaxia Xianjin Zengli A	−0.02138		−0.03768
	004970	Nanfang Tiantianbao A	−0.01957		−0.08814
	003391	Jianxin Tiantianyi A	−0.01301		−0.09340
	150005	Yinhe Yinfu A	−0.02181		−0.03698
	000509	Guangfa Qiandaizi A	−0.01253		−0.04533

illustrates the VaR calculation results.

According to Table 5, the VaR value for the FUND series (−0.01737) is relatively larger, indicating that the series has the highest risk. The VaR values for the BANK series (−0.02312) and the THIRD series (−0.02299) are relatively lower, indicating that they have the relatively lower risk.

For the IMFs, Jianxin Xianjintianli A (000693), Gongyin Xinjinhuobi A (000528), and Zhaoshang Zhaoqian Bao A (000588) have the lowest VaR values, and their risks are relatively small. The VaR values of Guangfa Qiandaizi A (000509), Jianxin Tiantianyi A (003391), and Zhonghai Huobi A (392001) are the largest, which indicates that their risks are more significant than the others.

For the VaR value of each IMF, a larger value indicates a higher risk. It is necessary to consider whether to invest and the amount of investment carefully when investing in the high-risk IMFs.

4.3.2. The Calculation of La-VaR

The La-VaR formula (see Equation (12)) introduces the impact of liquidity risk based on the VaR. Because the ratio of bank deposits to IMF is the quarterly data, which will bring inaccuracy to the measurement by prediction once in the next three months, we designed a rolling prediction method to predict the La-VaR values in the next three months based on the calculated La-VaR values of the previous periods.

In order to ensure the accuracy of the prediction, we calculated the RMSE (0.1157), which indicated that the method could be used to predict the future La-VaR values. According to Equation (12), the La-VaR values of each IMF and the average values within the categories were obtained (see Table 5).

Among them, the La-VaR value for the THIRD series (−0.08018) is the lowest, suggesting that the THIRD series has the lowest risk and is suitable to invest in. The La-VaR value for the BANK series (−0.06360) is relatively lower, indicating that the risk is moderately low. The La-VaR value for the FUND (−0.05630) is the largest, representing that these IMFs have the highest risk.

For the IMFs, the La-VaR values of Huitianfu Quan’E Bao (000397), Gongyin Xinjinhuobi A (000528), and Jiashi Huoqianbao A (000581) are the lowest, and their risks are comparatively low. The La-VaR values of Tianhong Yu’E Bao (000198), Yifangda Tiantianlicai A (000009), and Zhongyin Huoqi Bao (000539) are the largest, and their risks are comparatively high.

Liquidity is an essential feature of IMFs. The convenient and fast transaction mode makes IMFs highly liquid. Different categories of IMFs have different characteristics and liquidity, bringing new considerations to IMF investment on the Internet.

By comparing the VaR and La-VaR values, the La-VaR calculation results illustrate that the FUND series has the highest risk, while the THIRD series has the lowest risk, and their values differ significantly. These results are slightly different from those of the VaR. The VaR results show that the difference between the three categories is minor, and the values are small. Therefore, the La-VaR calculation method can be a good measurement of the liquidity risk of IMFs. This result is consistent with the conclusions of Li et al. (2021) [32] and Chakrabarti et al. (2021) [33]. They both believe that the funds are highly liquid and should attract attention in financial research. Since liquidity is a prominent feature of IMFs, we should take the liquidity factor into account when measuring the risk of IMFs, to assess and avoid loss more comprehensively.

The risks of the different IMF categories are different for the THIRD, the BANK, and the FUND series. Among them, the THIRD series has the lowest liquidity risk, related to its flexible and convenient transaction form and the “T+0” transaction mode. The BANK series has a medium liquidity risk, closely connected with its prominent trading volume and strict long-term investment range. The FUND series has the highest liquidity risk, concerning its bond assets and the uneven fund companies. Furthermore, we discussed in detail why IMFs’ liquidity risk differs from series to series.

For the THIRD series, it is the most prevalent and renowned category due to its flexibility, convenience, and security. It is convenient to store idle deposits for daily expenditures, and the transaction volume is not strictly regulated. Thus, it is widely applied in consumers’ daily lives. At the same time, the “T+0” transaction mode of the THIRD series means it can be purchased and redeemed on the same day. The “T+0” mode makes the transactions more convenient and flexible. Besides, the THIRD series commonly connects with “big-name” fund companies and has an extensive product scale. The yields are almost all higher than those of fixed deposits. Therefore, we can make the purchase and redemption conveniently, and the transaction volume can be set flexibly, making it characterized by high liquidity. By introducing the liquidity risk, the THIRD series has the lowest La-VaR values (−0.08018) and differs significantly from the VaR result (−0.02299), indicating that the liquidity factor is highly influential.

For the BANK series, it is the traditional category of IMFs for the public, and its trading mode has been mature. Banks can issue IMFs independently or cooperatively and have the advantages of capital scale and public trust. The BANK series supports a large transaction scale with little restriction on investment. Because of the presence of banks, the investment in the BANK series is believed to be more secure. However, the investment mode with a long period and the “T+1” transaction mode reduce the liquidity of the IMFs. The asset the investors apply for redemption can only be received on the next day, so the timeliness of IMF access is not as high as in the THIRD series. This makes it hard to solve unexpected problems in time. This denotes that the enormous transaction volume, the time limit of long-term investment, and the “T+1” transaction mode make the BANK series have a higher liquidity risk than the THIRD series. Thus, the La-VaR value of the BANK series (−0.06360) is slightly higher than that of the THIRD series and much lower than the VaR value (−0.02312), indicating that, though its liquidity risk is relatively a little higher, it remains stable.

For the FUND series, it is generally considered to have a high risk and a relatively high return, which attracts numerous investors. Compared with the THIRD and the BANK series, the investment products of the FUND series are mainly bond assets, with a long time and large amounts required, bringing uncertainties to the stability of IMFs. The long-term bond investment period also results in poor liquidity for the IMFs in this category. Moreover, fund companies have different risk-control capabilities and product ranges, so the FUND series has the highest risks. The imperfect product allocation, long and fixed investment period with high risk, and different risk response capabilities of varying fund companies all pose significant challenges to the liquidity and stability of the FUND series. Accordingly, the FUND series has the highest VaR value (−0.05630) and La-VaR value (−0.01737).

Consequently, the La-VaR results are more consistent with the actual situation than the VaR and suitable for representing the risks of each IMF and the different IMF series. Introducing the liquidity factor provides a more accurate risk measurement of IMFs and matches the actual situation remarkably. Therefore, it is appropriate to quantify the risk effect of each IMF by the La-VaR method.

5. Investment Portfolio Design for Sample IMFs

We established an objective programming model with a fixed risk level to transform a dual-objective investment scheme into a single-objective one. The model can set parameters according to the preferences of various investors, rather than calculating fixed mathematical parameters, so that the model has more flexibility and practical significance. We took the average forecast three-month return rates from the LSTM model as the indicative yields and the La-VaR values as the liquidity risk indicator values. The investors were limited to holding capital of RMB 1 million, and the number of IMFs held cannot be more than 10.

The objective function is as follows:

$$\mathrm{Max}R={\displaystyle \sum }_{i=1}^{18}{A}_i{r}_i{y}_i,$$

(16)

where $R$ is the total return of the IMFs, $A_i$ is the investment amount on ${\mathrm{IMF}}_i$, $r_i$ is the return rate of ${\mathrm{IMF}}_i$, and $y_i$ is a 0–1 variable; it determines whether to buy ${\mathrm{IMF}}_i$ or not.

The constraint conditions are as follows:

$$\{\begin{array}{c}{\displaystyle {\displaystyle \sum }_{i=1}^{18}}{A}_i{y}_i\le W,\\ {\displaystyle {\displaystyle \sum }_{i=1}^{18}}{A}_i{v}_i{y}_i<kW,0\le k\le 1,\\ {\displaystyle {\displaystyle \sum }_{i=1}^{18}}{y}_i\le m.\end{array}$$

(17)

The parameters can be modified personally based on the preferences of different investors, which makes the model have great practical application value in financial activities. We define $W$ as RMB 1 million, $k$ as 0.15, $v$ as the positive VaR or La-VaR value, and $m$ as 10. Thus, the conditions indicate that the investor’s holdings are no more than RMB 1 million, the acceptable loss for the investor is less than 15% of the capital, and the number of IMFs held by the investor is allowed to be no more than 10 at a time.

According to the risk diversification theory, choosing different IMFs can diversify investment risks. Thus, the IMFs invested in each category should be more than one.

Based on the defined programming model and the risk diversification theory, we calculated the typical portfolio results with the LSTM model, the VaR method, the LSTM model, and the La-VaR method (see

Table 6. The investment portfolio design results with LSTM and the VaR.

Variable	Value	Variable	Value	Variable	Value	Variable	Value
y1	0.0000	y10	0.0000	A1	0.0000	A10	0.0000
y2	0.0000	y11	0.0000	A2	0.0000	A11	0.0000
y3	0.0000	y12	0.0000	A3	0.0000	A12	0.0000
y4	0.0000	y13	0.0000	A4	0.0000	A13	0.0000
y5	1.0000	y14	0.0000	A5	32.0949	A14	0.0000
y6	0.0000	y15	0.0000	A6	0.0000	A15	0.0000
y7	1.0000	y16	1.0000	A7	26.0520	A16	41.8531
y8	0.0000	y17	0.0000	A8	0.0000	A17	0.0000
y9	0.0000	y18	0.0000	A9	0.0000	A18	0.0000

and

Table 7. The investment portfolio design results with LSTM and the La-VaR.

Variable	Value	Variable	Value	Variable	Value	Variable	Value
y1	0.0000	y10	0.0000	A1	0.0000	A10	0.0000
y2	0.0000	y11	0.0000	A2	0.0000	A11	0.0000
y3	0.0000	y12	1.0000	A3	0.0000	A12	26.7527
y4	1.0000	y13	0.0000	A4	67.0211	A13	0.0000
y5	0.0000	y14	0.0000	A5	0.0000	A14	0.0000
y6	0.0000	y15	0.0000	A6	0.0000	A15	0.0000
y7	0.0000	y16	1.0000	A7	0.0000	A16	6.2262
y8	0.0000	y17	0.0000	A8	0.0000	A17	0.0000
y9	0.0000	y18	0.0000	A9	0.0000	A18	0.0000

). The return rates, the VaR and La-VaR values for each IMF were substituted into the programming model to obtain the optimal investment portfolios. Table 6 is the portfolio designed with the LSTM and VaR values, and Table 7 is the portfolio designed with the LSTM and La-VaR values. We can find that different risks correspond to different portfolio results.

Table 6 indicates that the portfolio should invest RMB 320,949 in the fifth IMF (Guangfa Tiantianhong A (000389)), RMB 260,520 in the seventh IMF (Zhongyin Huoqi Bao (000539)), and RMB 418,531 in the sixteenth IMF (Jianxin Tiantianyi A (003391)). The portfolio manifests that investors favor the high-return FUND IMFs (RMB 418,531) and the prosperous THIRD IMFs (RMB 320,949), and the effect of risk is underestimated as the difference between the VaR values is small within 0.015.

Table 7 indicates that the portfolio should invest RMB 670,211 in the fourth IMF (Huitianfu Quan’E Bao (000397)), RMB 267,527 in the twelfth IMF (Gongyin Xinjinhuobi A (000528)), and RMB 62,262 in the sixteenth IMF (Jianxin Tiantianyi A (003391)). The portfolio manifests that investors favor the THIRD IMFs (RMB 670,211) and the BANK IMFs (RMB 267,527), and the liquidity risk is differentiated in different categories of IMFs with the degree of impact varying significantly.

It turns out that the programming model we designed has strong stability and adaptability to the investors’ preferences. Apart from that, we provided the extended out-of-sample data from 16 March 2022 to 5 August 2022 to measure the effects and performance of the model based on the LSTM model and the La-VaR method. The portfolio results are presented in

Table 8. The investment portfolio design results of the out-of-sample data.

Variable	Value	Variable	Value	Variable	Value	Variable	Value
y1	0.0000	y10	0.0000	A1	0.0000	A10	0.0000
y2	0.0000	y11	0.0000	A2	0.0000	A11	0.0000
y3	0.0000	y12	0.0000	A3	0.0000	A12	0.0000
y4	0.0000	y13	0.0000	A4	0.0000	A13	0.0000
y5	1.0000	y14	0.0000	A5	63.6622	A14	0.0000
y6	0.0000	y15	0.0000	A6	0.0000	A15	0.0000
y7	1.0000	y16	1.0000	A7	24.1836	A16	12.1542
y8	0.0000	y17	0.0000	A8	0.0000	A17	0.0000
y9	0.0000	y18	0.0000	A9	0.0000	A18	0.0000

.

Table 8 indicates that the portfolio should invest RMB 636,622 in the fifth IMF (Guangfa Tiantianhong A (000389)), RMB 241,836 in the seventh IMF (Zhongyin Huoqi Bao (000539)), and RMB 121,542 in the sixteenth IMF (Jianxin Tiantianyi A (003391)). The portfolio manifests that investors favor the THIRD IMFs (RMB 636,622) and the BANK IMFs (RMB 241,836), and the investment distribution is similar to the portfolios of the former three months in Table 7, which points out that the changes in the portfolio over a continuous period are gradual rather than drastic. The results emphasize that the model can provide investors with the ideal portfolios in different time intervals, which is in accordance with the investors’ expectations and makes the model broadly applicable.

It can be seen that applying LSTM and the La-VaR can be a good reflection of the demand preferences of investors. The investor of the second portfolio with LSTM and the La-VaR belongs to the prudent and steady type who holds a relatively conservative investment philosophy and prefers a low-risk and low-return investment approach that protects the principal. The THIRD and the BANK series can relatively maintain capital security and stability, so they are excellent choices for investors who prefer quick assets.

By multiple personalized modifications of the model parameters, we identified that investors have different preferences regarding their investment portfolios. By adjusting the programming model’s objective function and constraint conditions based on the real values of the investors’ actual situation such as the capital they hold and the investment loss they can accept, we can obtain the ideal portfolios of investors with their preferences. Our conclusions tie in well with the conclusions of Wang et al. (2020) [34]. Wang et al. (2020) [34] proved that investment behavior is affected by the investors’ sentiment. The sentiment varies from person to person as the investors have various preferences, then the people ready to invest hold diverse attitudes towards risks and returns. Investors tend to be either aggressive, steady, or conservative, and different types of investors have different attitudes when investing. Depending on the preferred characteristics of different investors, we can adjust the parameters of the programming model, such as the investment amount or the acceptable level of risk, to obtain portfolios that meet the various investors’ expectations.

Aggressive investors have a more radical and bold investment philosophy and prefer to put more money in the high-risk category to achieve the desired returns when investing. They prefer high-return investment products and pay more attention to yield than risk in portfolio design. Thus, FUND products that deliver relatively higher returns seem to be the most suitable choice for them, and they will invest a considerable amount of money into the FUND series.

Steady investors pay more attention to seeking a balance between higher returns and lower risks and accept fewer risks than aggressive investors. They are eager to find the IMFs that can bring a certain amount of income and have relatively few risks to invest in. They mainly hope to make a reasonable investment in high-risk and low-risk IMF to maintain a balance. Steady investors seeking stability tend to prefer the more diverse portfolios to balance returns and risks.

Conservative investors are more concerned about the safety of the available capital, pursuing an inevitable return based on ensuring low risks. They prefer IMFs that guarantee the principal and the lowest risks. Gaining income is not their main priority, and they tend to be cautious in choosing the safest investment option while trying to ensure the safety of their principal. Thus, they will choose the safer and more secure IMF categories, such as the BANK and the THIRD series.

At the same time, to reduce the adverse effects of risks, investors of different types tend to choose different categories of IMFs according to risk diversification theory to minimize the impact of potential risks. The investment scale and number of diverse categories reflect the investors’ priorities to a large extent. Different investors set their personalized parameters in the model, and their portfolio results differ significantly. Investors who seek higher returns focus on the FUND series and invest less in other series to reduce risks, while investors who seek lower risks focus on the THIRD series and invest less in other categories to improve their returns.

6. Conclusions

We constructed an LSTM model to predict the future IMF return rates and utilized the La-VaR method to measure the liquidity risk of the representative IMFs. On this basis, we designed an objective programming model for designing portfolios and set reasonable assumptions for investments. The following conclusions were obtained:

• Firstly, an LSTM model can be constructed to accurately forecast the future return rates of IMFs. The prediction curve of the model test set matches the absolute values commendably, and the RMSE value is 0.009. By comparing with the GARCH model, it was found that the prediction results obtained from the LSTM model are in excellent agreement with the actual data, and the model is suitable for predicting the future return rates of the IMFs. Adopting a deep complex network can avoid the problem of falling into local optimal solutions and solve the gradient explosion or gradient disappearance problem of long time series data. Thus, it is well applied to the prediction of IMF data with reliable time-periodic characteristics.
• Secondly, the VaR and La-VaR methods apply to measuring the risk value of each IMF. Since the IMFs are highly liquid, the risk values calculated by the La-VaR introducing the liquidity factor are more realistic. The BANK series has the moderate risk due to its large trading volume and strict long-term investment range, and the THIRD series has the lowest risk values due to its flexible and convenient transaction form and the “T+0” transaction mode, while the FUND series has the highest risk values due to its bond assets and the uneven fund companies.
• Thirdly, an objective programming model for an investment portfolio was designed based on the return rates and risk values. Meanwhile, different investors have different investment preferences, and the appropriate portfolio objectives and constraints can be set based on their preferences. Therefore, the programming model with deep learning algorithms helps ensure more reliable profits and provides personalized portfolios for various investors, offering it more practical significance than the traditional portfolio strategies.

We designed the LSTM and La-VaR models for return prediction and risk assessment to design portfolios for various investors. Still, there are limitations to this research. Firstly, owing to limited data availability, this research only used 18 IMFs’ data in China. Future research can be extended to IMFs from America and Europe and make detailed comparisons with more samples of IMFs in China. Secondly, this research used the GARCH and LSTM models for forecasting. These models can be updated with deep learning and combined ensemble forecasts with expert judgment in future research. Thirdly, the measure for liquidity risk in this study is additive. It is essential to conduct future research with multiplicative measurement.