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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

We propose a continuous maximum entropy method to investigate the robust optimal portfolio selection problem for the market with transaction costs and dividends. This robust model aims to maximize the worst-case portfolio return in the case that all of asset returns lie within some prescribed intervals. A numerical optimal solution to the problem is obtained by using a continuous maximum entropy method. Furthermore, some numerical experiments indicate that the robust model in this paper can result in better portfolio performance than a classical mean-variance model.

Since Markowitz’s pioneering work [

Although the mean-variance framework is still widely accepted and used, there are several challenges that are needed to be overcome. One of the major challenges is that optimal portfolios are sensitive to the estimation errors of mean and variance. Best and Grauer [

In order to deal with the sensitivity of optimal portfolio to input data, many attempts have been made to develop new techniques such as optimization and parameter estimation. One of the techniques is a Bayesian approach to create stable expected returns. Black and Litterman [

This paper proposes a continuous maximum entropy method to investigate the robust optimal portfolio selection problem for the market with transaction costs and dividends. Our focus will be on the following three cases. Firstly, all of the asset expected returns lie within some specified intervals which can be estimated by their historical return rates. This means that the errors in estimates of expected returns are considered by investors. It should be pointed out that our model only concerns this error, which is the major factor of estimation risk in the mean-variance framework. Secondly, we consider more complicated market situations than those in [

The paper proceeds as follows: In Section 2, the robust portfolio model considering the market with transaction costs and dividends is discussed. Section 3 provides a numerical solution to the problem by using the maximum entropy method. Section 4 presents some numerical experiments with our model. Finally, concluding remarks and suggestions for future work are given in Section 5.

Our focus here is to introduce the formulation of robust portfolio problem in a market with transaction costs and dividends. Therefore, we start with some notations and assumptions.

Assume that an investor allocates his initial wealth to _{g}_{0} is the tax rate of marginal basic income. _{f}_{s}_{i}_{i}_{ij}_{ij}_{n}_{×}_{n}_{i}

Next, we give the following assumptions:

_{ij}_{n}_{×}_{n} is strictly positively definite.

_{i} is located in a known interval. That is, r_{i}_{i}_{i}_{i}_{i}

_{1}(_{2}(

To make things easier, it is useful to define the following concepts. The net return is the total returns of expected return and dividends after paying the basic income tax and capital income tax. The pre-tax profit represents the total returns of expected return and dividends excluding the commission and stamp tax. The after-tax profit is the value of pre-tax profit after paying the capital income tax.

On the basis of the above notations and assumptions, we are now ready to formulate the robust portfolio model in the market with transaction costs and dividends.

Generally speaking, the stamp tax can be eliminated as a pre-tax expense in the period. However, the commission can not be ruled out since it is considered as investment cost. Therefore, the total pre-tax profit of a portfolio can be expressed by

Hence, the after-tax profit of a portfolio can be obtained by the following expression:

Substituting expressions _{1}(_{2}(

Define

and

Then the after-tax profit of a portfolio can be expressed as:

It is obvious that _{i}

Based on the mean-variance framework, in our model we attempt to maximize the minimization over expected returns subject to the constraint that all of the asset expected returns lie within some specified intervals. Therefore, it can be formulated by the following min-max problem:

where the parameter

In this section, we are devoted to finding the optimal solution to Problem

We first consider the following min-max optimal problem:

where ^{n}_{1}] × [0, _{2}] × ⋯ [0, _{m}^{m}

We define

Obviously, Problem

According to the results by Huang and Shen [

where λ(^{∗}(

Then, the Lagrange function of Problem

where

Denote

where

According to variational principle, it yields

Since

It is known from

Thus, it has the following system of equations:

From the first equation of

Substituting

Since exp{

Hence, in view of

Substituting

It follows from

Define

Obviously, _{p}

where

Define

Our main objective is to prove the convergence of _{p}

Without loss of generality, we set

_{p}

Since

then we obtain

Moreover, we can deduce that

Hence, we arrive at

Thus, _{p}

Based on the discussion of subsection 3.1, now we aim to determine the solution of Problem

then Problem

where

It can be seen that min_{x}_{∈}_{Ω} _{p}

Let

According to Theorem 4, Problem

For the sake of convenience, denote

Therefore, Problem

Substituting both

Obviously, Problem

In order to test the performance of robust mean-variance model, we consider a real portfolio which selects six stocks of historical data from Shanghai Stock Exchange. Original data of six stocks from each week’s closing prices from January in 2009 to May in 2014. As a result, the covariance matrix of six stocks is as follows:

The week’s expected returns and their estimated intervals are listed in

The classical mean-variance model under the same market situation of this paper can be formulated as follows:

where the notations of mathematical symbols are the same as this paper.

Now, we provide the market performances for the classical mean-variance model and the robust one. Let’s assign the following parameters: _{g}_{s}_{f}

By using the subroutine “fmincon”in Matlab, we obtain the optimal strategies, efficient points and efficient frontier of classical mean-variance model in the following

By using the maximum entropy method and subroutine ‘fmincon’ in Matlab, we obtain the optimal strategies, efficient points and efficient frontier of robust mean-variance model in the following

Next, we put the efficient frontiers of the robust mean-variance model and classical mean-variance model in the same coordinate plane.

The robust mean-variance model without transaction costs can be expressed as follows:

where the notations of mathematical symbols are the same as this paper.

By using the maximum entropy method and subroutine “fmincon” in Matlab, we obtain the optimal strategies, efficient points and efficient frontier of robust mean-variance model without transaction costs in the following

Comparing

This paper provides a maximum entropy method to investigate the robust optimal portfolio selection problem for the market with transaction costs and dividends. For avoiding the sensitivity of optimal portfolio to input data such as expected return and variance, we restrict the asset expected return to lie within a specified interval. By maximizing the minimization over expected returns, we naturally establish our robust portfolio model. We find that this problem can be solved by using a continuous maximum entropy method. The numerical experiments indicate that the robust portfolio model is achieved at relatively good performance than the classical mean-variance ones. In addition, we consider the market with transaction costs and dividends, which is an important concern for investors. The research on multi-period robust models and other risk measures instead of variance under this market circumstance is left for future work.

We acknowledge the contributions of Fundamental Research Funds for the Central Universities (No.2012QNB19) and Natural Science Foundation of China (No.11101422, 11371362 and 71173216).

Yingying Xu established the model and provided numerical experiments. Zhuwu Wu proposed an approach to solve the problem. Long Jiang and Xuefeng Song provided the theoretical analysis and economic explanations. All authors have read and approved the final published manuscript.

The authors declare no conflict of interest.

The efficient frontier to classical mean-variance model.

The efficient frontier to robust mean-variance model.

The efficient frontiers to the robust and classical mean-variance model.

The efficient frontier to robust mean-variance model without transaction costs.

The week’s expected returns of six stocks.

Code | 000581 | 002041 | 600362 | 600252 | 600406 | 600021 |

Mean | 0.00785 | 0.005028 | 0.005744 | 0.001903 | 0.001422 | 0.00222 |

The estimated intervals of week’s expected returns.

Code | 000581 | 002041 | 600362 | 600252 | 600406 | 600021 |

Range | (0.0061, 0.0109) | (0.0038, 0.0076) | (0.0040, 0.0088) | (0.0005, 0.0052) | (0.0004, 0.0040) | (0.0011, 0.0052) |

The optimal strategies and efficient points corresponding to risk aversion

( | ||
---|---|---|

20 | (0.1726, 0.1344, 0.1083, 0.0347, 0.1173, 0.4327) | (0.0006631, 0.002608) |

35 | (0.1507, 0.1297, 0.1011, 0.0426, 0.1274, 0.4485) | (0.0006581, 0.002487) |

50 | (0.1415, 0.1273, 0.0968, 0.0441, 0.1320, 0.4583) | (0.0006567, 0.002433) |

65 | (0.1363, 0.1257, 0.0941, 0.0443, 0.1341, 0.4655) | (0.0006561, 0.002401) |

80 | (0.1332, 0.1243, 0.0931, 0.0474, 0.1364, 0.4656) | (0.0006559, 0.002382) |

100 | (0.1308, 0.1238, 0.0926, 0.0488, 0.1374, 0.4666) | (0.0006557, 0.002370) |

The optimal strategies and efficient points corresponding to risk aversion

( | ||
---|---|---|

20 | (0.1762, 0.1335, 0.1047, 0.0184, 0.1191, 0.4481) | (0.0006635, 0.003011) |

35 | (0.1501, 0.1293, 0.0988, 0.0370, 0.1284, 0.4564) | (0.0006579, 0.002864) |

50 | (0.1423, 0.1262, 0.0958, 0.0409, 0.1316, 0.4632) | (0.0006567, 0.002816) |

65 | (0.1369, 0.1247, 0.0934, 0.0422, 0.1338, 0.4690) | (0.0006562, 0.002785) |

80 | (0.1330, 0.1238, 0.0915, 0.0443, 0.1350, 0.4724) | (0.0006559, 0.002762) |

100 | (0.1312, 0.1231, 0.0926, 0.0477, 0.1371, 0.4683) | (0.0006557, 0.002752) |

The optimal strategies and efficient points corresponding to risk aversion

( | ||
---|---|---|

20 | (0.1580, 0.1299, 0.1006, 0.0308, 0.1255, 0.4551) | (0.001345, 0.004979) |

35 | (0.1410, 0.1257, 0.0943, 0.0397, 0.1315, 0.4678) | (0.001340, 0.004839) |

50 | (0.1355, 0.1238, 0.0940, 0.0459, 0.1352, 0.4656) | (0.001339, 0.004794) |

65 | (0.1313, 0.1243, 0.0924, 0.0485, 0.1358, 0.4677) | (0.001338, 0.004764) |

80 | (0.1301, 0.1235, 0.0924, 0.0486, 0.1372, 0.4682) | (0.001338, 0.004753) |

100 | (0.1283, 0.1231, 0.0920, 0.0494, 0.1381, 0.4691) | (0.001337, 0.004740) |