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Portfolio selection problem introduced by Markowitz has been one of the most important research fields in modern finance. In this paper, we propose a model (least squares support vector machines (LSSVM)-mean-variance) for the portfolio management based on LSSVM. To verify the reliability of LSSVM-mean-variance model, we conduct an empirical research and design an algorithm to illustrate the performance of the model by using the historical data from Shanghai stock exchange. The numerical results show that the proposed model is useful when compared with the traditional Markowitz model. Comparing the efficient frontier and total wealth of both models, our model can provide a more measurable standard of judgment when investors do their investment.

Portfolio is the combination of securities such as foreign exchange, stocks, and other market instruments. Stock investment has become very common for household investors to involve in the stock market. Investors used many technical methods to minimize risk and optimize return. Among the methods, Markowitz model developed by Harry Markowitz in 1952 had serious practical limitations due to complexities involved in compiling the variance, covariance, expectation, standard deviation of each asset to other assets in the portfolio. In recent years, many works have been done by scholars to make the portfolio theory more efficient. In [

However, there are few scholars who used the methods of machine learning to modify the Markowitz model. As we know, the return rate in the mean-variance model refers to the historical return rate, which can also be called historical volatility. Historical volatility refers to the standard deviation of the underlying asset price changes over the past period of time, which represents the past volatility law. The actual volatility in the trading point cannot be determined, but can only be predicted with historical volatility and current market information. In this paper, we predict the actual volatility with historical volatility by using machine learning. As one of important applications of the machine learning, LSSVM has been used to deal with various financial problems such as stock price prediction [

In our study, we apply LSSVM regression model to traditional Markowitz model and an efficient result is achieved by our proposed model when we compare the efficient frontier and total wealth of both models.

The contents of this paper are as follows: in Section

Modern portfolio theory was first introduced by Markowitz [

Support vector machine (SVM) has been successfully applied for financial problems, especially in time series forecasting. LSSVM is the least squares formulation of SVM and was developed by Pelckmans [

In this section, we give a description of applying LSSVM to mean-variance model. We first select a portfolio and then calculate the returns of the assets in the portfolio. As mentioned above, in the LSSVM model, we take the matrix of assets’ returns as the training sets, by the process of Section

We select a portfolio consistimg of three assets which are chosen from Shanghai stock market. To do a buy-and-sell test, we use the historical data for the stock “-zgyh-”, “-nyyh-”, and “-jtyh-” from August 09, 2018, to October 26, 2018. We take the data from August 09, 2018, to October 25, 2018, into mean-variance model and LSSVM-mean-variance model. There are 50 data in total. In the LSSVM model, we divide the data into a training set with 39 data and a test set with 10 data. Then, we compare the performance with the two models on October 26, 2018. To do a buy-and-hold test, we use the historical data for the stock “-zgyh-”, “-nyyh-”, and “-jtyh-” from March 10, 2017, to March 12, 2018. We take the data of closing price every 5 days; then, there are 50 data in total. In the LSSVM model, we also divide the data into a training set with 39 data and a test set with 10 data. Then, we compare the performance with the two models on March 19, 2018. For the calculation process, MATLAB R2016a will be used.

From the stocks data chosen in Section

Candlestick chart for “-zgyh-”, “-nyyh-”, and “-jtyh-” from August 09, 2018, to October 26, 2018. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article. The data sources were downloaded from the web site “http://quotes.money.163.com/stock”.)

The body in the candlestick usually consists of an opening price and a closing price; the price excursions below or above the body are called the wicks. For a stock during the time interval represented, the wick contains the lowest and highest prices, as well as the body contains the opening and closing prices. The red body of a candlestick indicates the security has a higher closed price than it opened, the opening price at the bottom and the closing price at the top. The green body of a candlestick indicates the security has a lower closed price than it opened, the opening price at the top and the closing price at the bottom.

Now we select the real historical data of the stocks “-zgyh-”, “-nyyh-”, and “-jtyh-” from August 09, 2018, to October 25, 2018. Taking the closing data to the calculation of return rate for every day, the total number of data is 50. Then, we get 49 return rate data for each asset. The return rate of “-jtyh-” is shown in Figure

Proportion of each asset for traditional mean-variance model.

Investment proportion combination | Proportion of “-zgyh-” | Proportion of “-nyyh-” | Proportion of “-jtyh-” |
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1 | 0.7994 | 0.0000 | 0.2006 |

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2 | 0.6483 | 0.0000 | 0.3517 |

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3 | 0.4972 | 0.0000 | 0.5028 |

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4 | 0.3461 | 0.0000 | 0.6539 |

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5 | 0.1950 | 0.0000 | 0.8050 |

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6 | 0.0635 | 0.0349 | 0.9016 |

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7 | 0.0000 | 0.0349 | 0.9016 |

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8 | 0.0000 | 0.4608 | 0.5392 |

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9 | 0.0000 | 0.7304 | 0.2696 |

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10 | 0.0000 | 1.0000 | 0.0000 |

Return rate of mean-variance model and LSSVM mean-variance model for (a) “-zgyh-”, (b) “-nyyh-”, and (c) “-jtyh-”.

As a comparison, we calculate the proportion of each asset for LSSVM-mean-variance model by using the LSSVM regression. We take the return rate mentioned above to the LSSVM model described in Section

Then, we take regression return rate to Markowitz model. Table

Proportion of each asset for LSSVM-mean-variance model.

Investment proportion combination | Proportion of “-zgyh-” | Proportion of “-nyyh-” | Proportion of “-jtyh-” |
---|---|---|---|

1 | 0.9668 | 0.0000 | 0.0332 |

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2 | 0.6654 | 0.0000 | 0.3346 |

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3 | 0.4173 | 0.0305 | 0.5521 |

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4 | 0.2670 | 0.1171 | 0.6159 |

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5 | 0.1167 | 0.2037 | 0.6796 |

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6 | 0.0000 | 0.3094 | 0.6906 |

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7 | 0.0000 | 0.4821 | 0.5179 |

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8 | 0.0000 | 0.6547 | 0.3453 |

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9 | 0.0000 | 0.8274 | 0.1726 |

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10 | 0.0000 | 1.0000 | 0.0000 |

Each investment proportion combination in the table responds to a maximum return for a given level of risk as measured by the standard variance. The points are constituted by mean and standard variance forming an efficient frontier.

As seen in Figure

Efficient portfolio frontier of mean-variance model and LSSVM mean-variance model.

According to the investment proportion combinations shown in Tables

Total wealth for each investment proportion combination of two models.

As the proposed model conducted by using buy-and-sell strategy, we get a satisfied result. However, the data set we selected is small and between summer and autumn, which makes people think that the above results have specific seasonality. To address this concern, we select a long data set covering all seasons of the year, from March 10, 2017, to March 12, 2018. The candlestick charts for three stocks are shown in Figure

Candlestick chart for “-zgyh-”, “-nyyh-”, and “-jtyh-” from March 10, 2017, to March 12, 2018. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article. The data sources were downloaded from the web site “http://quotes.money.163.com/stock”.)

The return rate of “-jtyh-” for buy-and-hold for 5-day strategy is shown in Figure

Proportion of each asset for mean-variance model for buy-and-hold for 5-day strategy.

Investment proportion combination | Proportion of “-zgyh-” | Proportion of “-nyyh-” | Proportion of “-jtyh-” |
---|---|---|---|

1 | 0.0924 | 0.0000 | 0.9076 |

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2 | 0.2957 | 0.0000 | 0.7043 |

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3 | 0.4409 | 0.0302 | 0.5288 |

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4 | 0.4531 | 0.1296 | 0.4172 |

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5 | 0.4654 | 0.2290 | 0.3056 |

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6 | 0.4776 | 0.3284 | 0.1940 |

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7 | 0.4898 | 0.4279 | 0.0823 |

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8 | 0.4410 | 0.5590 | 0.0000 |

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9 | 0.2205 | 0.7795 | 0.0000 |

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10 | 0.0000 | 1.0000 | 0.0000 |

Return rate of mean-variance model and LSSVM mean-variance model for buy-and-hold for 5-day strategy of (a) “-zgyh-”, (b) “-nyyh-”, and (c) “-jtyh-”.

As seen in Figure

Efficient portfolio frontier of mean-variance model and LSSVM mean-variance model for buy-and-hold for 5-day strategy.

Then, we take regression return rate to Markowitz model. Table

Proportion of each asset for LSSVM-mean-variance model for buy-and-hold for 5-day strategy.

Investment proportion combination | Proportion of “-zgyh-” | Proportion of “-nyyh-” | Proportion of “-jtyh-” |
---|---|---|---|

1 | 0.4498 | 0.0000 | 0.5502 |

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2 | 0.5634 | 0.0301 | 0.4065 |

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3 | 0.5718 | 0.1123 | 0.3159 |

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4 | 0.5803 | 0.1944 | 0.2253 |

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5 | 0.5887 | 0.2766 | 0.1347 |

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6 | 0.5971 | 0.3588 | 0.0441 |

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7 | 0.5134 | 0.4866 | 0.0000 |

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8 | 0.3422 | 0.6578 | 0.0000 |

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9 | 0.1711 | 0.8289 | 0.0000 |

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10 | 0.0000 | 1.0000 | 0.0000 |

According to the investment proportion combinations shown in Tables

Total wealth for each investment proportion combination of two models for buy-and-hold for 5-day strategy.

To illustrate that this result is not caused by the specific period we selected, we calculate the total wealth of each day in 15 days from August 30, 2018, to September 19, 2018 for the two models according to the former 15 days. The calculation steps are taken as same as the above process. We set the total wealth of mean-variance model as

Total wealth difference of two models.

Machine learning over the last few years has resulted in a potential opportunity for investors to invest in the financial market with a smarter and profitable way. Combining machine learning technology with financial investment, it can entirely change the way we make investment decisions. This paper gives an overview of how the two technologies can be combined into a powerful tool and proposes the LSSVM-mean-variance algorithm with the aim of maximizing return for a given level of risk as measured by the variance of returns. The efficiency of the proposed method is measured by empirical data, namely, efficient frontier and total wealth. Comparing the efficient frontier and total wealth of both models, the curve of mean-variance model is always below the proposed model. This shows that our model has a higher yield under the same risk and has more total wealth under each combination; our model performs a more measurable standard of judgment when investors do their investment. We confirm the efficiency through the strategy both buy-and-sell and buy-and-hold. The encouraging performance shows that our proposed method may become a promising model for the context of study and the results indicate a positive opportunity to be explored in the future.

Data and source program codes in this paper are available upon request from the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The first author (Jian Wang) was supported by the China Scholarship Council (201808260026).