As we know, borrowing and lending risk-free assets arise extensively in the theory and practice of finance. However, little study has ever investigated them in fuzzy portfolio problem. In this paper, the returns of each assets are assumed to be fuzzy variables, then following the mean-variance approach, a new possibilistic portfolio selection model with different interest rates for borrowing and lending is proposed, in which the possibilistic semiabsolute deviation of the return is used to measure investment risk. The conventional probabilistic mean variance model can be transformed to a linear programming problem under possibility distributions. Finally, a numerical example is given to illustrate the modeling idea and the impact of borrowing and lending on optimal decision making.

Portfolio selection is concerned with selecting a combination of securities among portfolios containing large numbers of securities to reach the investment goal. The portfolio selection model was first formulated by Markowitz [

In the past, research has been undertaken on the assumption that future security returns can be correctly reflected by past performance and be represented by random variables. However, since the security market is so complex and the occurrence of new security is so quick, in many cases security returns cannot be accurately predicted by historical data. They are beset with ambiguity and vagueness. To deal with this problem, researchers have made use of fuzzy set theory proposed by Zadeh [

In the mature market, investors not only borrow money to expand their holdings of risky assets, but also lend to invest a portion of the portfolio in the risk-free assets such as short-term treasury securities. Recently, much attention has been focused in this area. For example, Tobin [

Though a considerable number of research papers have been published for portfolio selection problem in fuzzy environment, there are little research on fuzzy portfolio selection problem under the consideration of different interest rates for borrowing and lending. In this paper, the focus of the research is to incorporate the possibility theory into a semi-absolute deviation portfolio selection model for investors' taking into account different interest rates for borrowing and lending in fuzzy environment. The rest of the paper is organized as follows. In Section

Let us give a brief description of Markowitz's mean-variance model. Consider an investment in

However, the original result of Markowitz was derived in a discrete time, frictionless economy with the same interest rates for borrowing and lending. In reality, investors may be charged a higher interest rate for borrowing money than the interest rate for saving money. Even though many research works assume the same risk-free interest rate for borrowing and lending, the discrepancy between borrowing and lending is crucial for the operations of financial institutions.

In what follows we assume there are

It should be noted that if

Moreover, it is known that very high weighting in one asset will cause the investor to suffer from larger risk. Therefore, the upper bounds of each asset would be useful for the investor to select portfolios in reality.

Based on the above discussion, we assume that the objective of the investor is to choose a new optimal portfolio that minimizes the risk of the portfolio subject to some constraints on the expected return of the portfolio and asset holdings by adjusting the existing portfolio. Thus, the portfolio problem can be formulated as follows:

Obviously, the optimal solution of model (

Let

Based on [

Carlsson and Fullér [

Furthermore, Carlsson and Fullér [

According to the above definitions, we easily obtain the lower and upper possibilistic means, the interval-valued and crisp possibilistic mean values of the total fuzzy return as follows:

Let

In this paper, we will use possibilistic semi-absolute deviation, instead of the possibilistic variance employed by Carlsson et al. [

Next, we evaluate the possibilistic mean absolute semi-deviation with respect to the total fuzzy return. Therefore, the possibilistic semi-absolute deviation can be defined as

Based on the Theorem

Furthermore, we can obtain the crisp possibilistic semi-absolute deviation of the return associated with the portfolio

Moreover, the possibilistic mean value of the return associated with the portfolio

Analogous to Markowitz's mean-variance methodology for the portfolio selection problem, the crisp possibilistic mean value corresponds to the return while the possibilistic semi-absolute deviation corresponds to the risk. Starting from this point of view, the possibilistic portfolio model with different interest rates for borrowing and lending can be formulated as

The possibilistic portfolio model (

The problem (

Furthermore, the problem (

For

Especially, if

If

In order to illustrate our proposed effective approaches for the portfolio selection problem in this paper, we give a numerical example introduced by Markowitz in 1959 [

Sample statistics for the Markowitz's historical data.

Stock | Sample mean | SD | 5th percentile | 40th percentile | 60th percentile | 95th percentile |
---|---|---|---|---|---|---|

1 | 0.066 | 0.238 | 0.070 | 0.456 | ||

2 | 0.062 | 0.125 | 0.089 | 0.229 | ||

3 | 0.146 | 0.301 | 0.136 | 0.758 | ||

4 | 0.173 | 0.318 | 0.238 | 0.714 | ||

5 | 0.198 | 0.368 | 0.325 | 0.671 | ||

6 | 0.055 | 0.209 | 0.094 | 0.352 | ||

7 | 0.128 | 0.175 | 0.164 | 0.356 | ||

8 | 0.118 | 0.286 | 0.196 | 0.587 | ||

9 | 0.116 | 0.290 | 0.196 | 0.587 |

Possibility distributions of returns.

Stock | ||||
---|---|---|---|---|

1 | −0.011 | 0.070 | 0.273 | 0.386 |

2 | 0.052 | 0.089 | 0.227 | 0.140 |

3 | 0.018 | 0.136 | 0.211 | 0.622 |

4 | 0.161 | 0.238 | 0.468 | 0.476 |

5 | 0.062 | 0.325 | 0.491 | 0.346 |

6 | 0.094 | 0.170 | 0.258 | |

7 | 0.090 | 0.164 | 0.222 | 0.192 |

8 | 0.104 | 0.196 | 0.415 | 0.391 |

9 | 0.104 | 0.196 | 0.420 | 0.391 |

We assume that the interest rate of borrowing is 4%, the interest rate of lending is 1%, and the upper bounds for nine assets are 0.25. By solving models (

Some possibilistic efficient portfolios.

0 | 0.25 | 0 | 0.25 | 0 | 0.0123 | 0 | 0 | |

0 | 0.25 | 0 | 0.25 | 0 | 0.25 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0.22 | 0 | 0.25 | 0.25 | 0.25 | 0.25 | |

0 | 0 | 0 | 0 | 0 | 0 | 0.25 | 0.25 | |

0 | 0.25 | 0 | 0.1124 | 0 | 0 | 0 | 0 | |

0.1786 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.014 | 0.014 | |

0 | 0 | 0 | 0.1376 | 0.25 | 0.2377 | 0.25 | 0.25 | |

0 | 0 | 0 | 0 | 0.076 | 0 | 0.236 | 0.236 | |

0.8214 | 0 | 0.53 | 0 | 0.174 | 0 | 0 | 0 | |

Risk | 0.1893 | 0.1216 | 0.0696 | 0.1257 | 0.134 | 0.14 | 0.1925 | 0.1925 |

From Table

Next, in order to illustrate that borrowing and lending have effect on the optimal portfolio selection, we consider two cases, that is, portfolio selection without borrowing and lending and portfolio selection with borrowing and lending. Based on the model (

Assume that

Optimal solutions of two models (

Risk | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Model ( | 0 | 0 | 0 | 0.0629 | 0 | 0 | 0.25 | 0 | 0 | 0.6871 | 0.0388 |

Model ( | 0.25 | 0.25 | 0 | 0 | 0 | 0.25 | 0.25 | 0 | 0 | 0 | 0.1216 |

Optimal solutions of two models (

Risk | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Model ( | 0 | 0 | 0 | 0.25 | 0 | 0 | 0.25 | 0.25 | 0.0022 | 0.2478 | 0.1209 |

Model ( | 0.1146 | 0.25 | 0 | 0.25 | 0 | 0 | 0.25 | 0.1354 | 0 | 0 | 0.1370 |

From Tables

In particular, to demonstrate that different borrowing and lending interest rates also have effect on the optimal portfolio selection, we consider two special cases: (a) only lending is allowed for portfolio selection, (b) only borrowing is allowed for portfolio selection. That is to say, models (

Some possibilistic efficient portfolios with different lending interest rates (

0 | 0.02 | 0.03 | 0.05 | 0.07 | 0.09 | 0.10 | |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0.4098 | 0.3316 | 0.2294 | 0.0903 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0.8197 | 0.7844 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0.1803 | 0.2156 | 0.5902 | 0.6684 | 0.7706 | 0.9097 | 1 | |

Risk | 0.0869 | 0.0831 | 0.0802 | 0.0649 | 0.0449 | 0.0177 | 0 |

Some possibilistic efficient portfolios with different borrowing interest rates (

0 | 0.02 | 0.05 | 0.10 | 0.11 | 0.12 | 0.139 | |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 0.8769 | |

0.6375 | 0.708 | 1 | 1 | 1 | 1 | 1 | |

0 | 0 | 0 | 0.71 | 0.8297 | 0.998 | 1 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | |

1 | 1 | 0.6833 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | 0 | 1 | |

0 | 0 | 0 | 0 | 0 | 0 | 1 | |

0.6375 | 0.708 | 0.6833 | 0.71 | 0.8297 | 0.998 | 3.8769 | |

Risk | 0.2308 | 0.2446 | 0.2682 | 0.3499 | 0.3758 | 0.4124 | 0.948 |

Table

Finally, we depict a graph, as shown in Figure

A comparison of possibilistic efficient frontiers under different cases.

The fuzzy set is one of the powerful tools used to describe a uncertain environment. In this paper, we have discussed the portfolio selection problem based on the possibilistic theory under the assumption that the returns of assets are trapezoidal fuzzy numbers. We have used possibilistic mean value of the return to measure the investment return, and possibilistic semi-absolute deviation as the investment risk. We have obtained a new possibilistic mean semi-absolute deviation model for portfolio selection taking into account of different interest rates for borrowing and lending. Comparing with conventional probabilistic mean-variance model, our proposed model contains less unknown parameters and it can integrate the experts' knowledge and the managers' subjective opinions better. Numerical results have showed that our proposed model is efficient and borrowing and lending risk-free asset have great effect on the optimal portfolio selection.

Finally, for future researches, three areas are proposed: first adding other constraints of real market such as transaction costs, cardinality, and bounded constraint, second using heuristic algorithms such as artificial bee colony (ABC) algorithm to solve the proposed model and comparing its solutions with GA and PSO, and lastly, extending the proposed model to a multiperiod case.

This paper was supported by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (no. PHR201007117, PHR201108333), the Beijing Municipal Education Commission Foundation of China (no. KM201010038001, KM201110038002), the key Project of Capital University of Economics and Business (no. 2011SJZ015), the Funding Project of Scientific Research Department in the Capital University of Economics and Business.