This paper studies the portfolio selection problem in hybrid uncertain decision systems. Firstly the return rates are characterized by random fuzzy variables. The objective is to maximize the total expected return rate. For a random fuzzy variable, this paper defines a new equilibrium risk value (ERV) with credibility level beta and probability level alpha. As a result, our portfolio problem is built as a new random fuzzy expected value (EV) model subject to ERV constraint, which is referred to as EV-ERV model. Under mild assumptions, the proposed EV-ERV model is a convex programming problem. Furthermore, when the possibility distributions are triangular, trapezoidal, and normal, the EV-ERV model can be transformed into its equivalent deterministic convex programming models, which can be solved by general purpose optimization software. To demonstrate the effectiveness of the proposed equilibrium optimization method, some numerical experiments are conducted. The computational results and comparison study demonstrate that the developed equilibrium optimization method is effective to model portfolio selection optimization problem with twofold uncertain return rates.
Based on mean-variance criterion, Markowitz [
The conventional portfolio methods assume that the security returns are random variables. Probability distributions of random variables are usually derived from historical data. However, in the real investment environment, the security returns often present vagueness and ambiguity. Since the seminal works of Zadeh [
In modern financial markets, there is not only randomness but also fuzziness that affects the total decision-making process. Based on the above works, this paper models the portfolio selection problem where randomness and fuzziness are considered simultaneously. Huang [
Compared with the existing literature, the main contributions of this paper consist of the following three aspects. Firstly, this paper defines a new ERV with credibility level beta and probability level alpha to measure the investment risk. The adopted risk measure quantifies the uncertainties of randomness and fuzziness simultaneously. This method shows the qualitative and quantitative analysis about the uncertainty of return rate. The proposed EV-ERV model is a useful optimization method for a practical investor. Secondly, the global optimal solution of the proposed EV-ERV model is obtained. In the case that the randomness of uncertain return rates follows normal distributions with deterministic covariance matrix, and the fuzziness is characterized by trapezoidal fuzzy variables, triangular fuzzy variables, or normal fuzzy variables, the proposed EV-ERV model is transformed into its deterministic convex programming models, which can be solved by general purpose optimization software. Thirdly, when the randomness of uncertain return rate vector follows multivariate normal distribution, the covariance matrix can reflect the interactions and correlation degrees among securities.
This paper is organized as follows. In Section
Let
Let
If
A multivariate normal distribution
Let
Let
Equation (
According to the property of equilibrium chance, in the case of
We next introduce the equilibrium risk value for a random fuzzy variable.
Let
In (
If the random fuzzy variable
With the rapid development of the economic and society, more and more investors realize the importance of portfolio optimization problem under uncertainty. They hope to use the limited funds to get the maximum benefit and, at the same time, to minimize the investment risk as much as possible. However, under the market economy, there are various uncertain factors to affect investment market. Thus, the uncertainties should be taken into account during the modeling process.
A rational investor should pursuit the maximum profit with the minimum risk. However, in real investment process, increasing the return, the investors have to tolerate greater risk; the lower risk corresponds to the less return. That is, investors need to make a trade-off between return and risk. Assume that the investor has a collection of
Let
We take ERV as the risk index to measure the investment risk. Based on Definition
Based on the notations above, if the investor wants to maximize the expected return rate, the equilibrium portfolio optimization problem in hybrid uncertain decision systems is built as the following EV-ERV model:
Introducing an additional variable
In this section, we discuss the properties of the objective function and constrains in equilibrium portfolio optimization model (
Since the return rates are denoted by the random fuzzy vector
According to [
According to [
In the above two cases, the expected value operator of fuzzy variable has linear property,
In this subsection, we will handle the following probability constraint in model (
It is assumed that random vector
If the multivariate normal distribution is nondegenerate, then it is absolutely continuous. Thus, one has
Let
Taking all the cases of
For any given
From the above analysis, if
Section
Let
Assume that
We first prove the necessity of assertion (i). By the definition of the optimistic value of fuzzy variable
We next prove the sufficiency of assertion (i). According to the definition of optimistic value, we know the credibilistic constraint
In what follows, we prove assertion (ii).
According to the property of optimistic value, the equality
It is easy to know that
Based on the above theoretical analysis, the following theorem gives the deterministic equivalent model of model (
Assume that
On the basis of Theorems
Assume that
In the case of
In this section, we specify the possibility distributions of fuzzy parameters
Let
The optimistic value of
As a consequence, for any given parameters
Triangular fuzzy variable is a special case of trapezoidal fuzzy variable. The following remark is about the result related to triangular possibility distributions.
Let
Let
Therefore, according to Theorem
In this section, we will conduct some numerical experiments to demonstrate the feasibility and effectiveness of the developed equilibrium optimization method. We first give some descriptions about our portfolio selection problem in the next subsection.
Assume that there are 20 potential risky assets for an investor. In this portfolio selection problem, the return rates have twofold uncertainty and are represented by random fuzzy variables. For the sake of presentation, we suppose that the prescribed confidence levels
Let
The distributions of trapezoidal fuzzy parameters
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In this case, model (
Firstly, we set the confidence levels
Computational results with
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Values | 0 | 0 | 0.06599 | 0.03981 | 0.03508 |
0.19412 | 0.07107 | 0.06188 | 0.07594 | 0.04273 | |
0 | 0.05974 | 0.07214 | 0.02827 | 0 | |
0.11136 | 0.04525 | 0.00917 | 0.08748 | 0 |
Secondly, considering that investors may have different attitudes towards risk, model (
The optimal solutions of equilibrium optimization model under various values of parameters.
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0.78 | 0.8 | 0.006 | 0 | 0 | 0.08332 | 0.04127 | 0.02539 | 0.03308 |
0.21540 | 0.05473 | 0.04693 | 0.07931 | 0.03955 | ||||
0 | 0.04148 | 0.07848 | 0.04074 | 0 | ||||
0.12020 | 0.05187 | 0.00402 | 0.07731 | 0 | ||||
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0.8 | 0.8 | 0.006 | 0 | 0 | 0.06599 | 0.03981 | 0.03508 | 0.03293 |
0.19412 | 0.07107 | 0.06188 | 0.07594 | 0.04273 | ||||
0 | 0.05974 | 0.07214 | 0.02827 | 0 | ||||
0.11136 | 0.04525 | 0.00917 | 0.08748 | 0 | ||||
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0.82 | 0.8 | 0.006 | 0 | 0.01039 | 0.04343 | 0.03564 | 0.04671 | 0.03271 |
0.16033 | 0.09309 | 0.08176 | 0.06673 | 0.04612 | ||||
0.00108 | 0.08616 | 0.06753 | 0.00916 | 0 | ||||
0.09753 | 0.03613 | 0.01668 | 0.10153 | 0 | ||||
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0.8 | 0.75 | 0.006 | 0 | 0 | 0.10802 | 0.04491 | 0.01061 | 0.03329 |
0.24738 | 0.02916 | 0.02402 | 0.08360 | 0.03457 | ||||
0 | 0.01334 | 0.08674 | 0.05983 | 0 | ||||
0.13391 | 0.06153 | 0 | 0.06240 | 0 | ||||
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0.8 | 0.79 | 0.006 | 0 | 0 | 0.07626 | 0.04095 | 0.02916 | 0.03302 |
0.20703 | 0.06120 | 0.05276 | 0.07790 | 0.04081 | ||||
0 | 0.04857 | 0.07578 | 0.03599 | 0 | ||||
0.11685 | 0.04931 | 0.06131 | 0.08131 | 0 | ||||
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0.8 | 0.82 | 0.006 | 0 | 0.01762 | 0.03796 | 0.03247 | 0.05033 | 0.03261 |
0.14456 | 0.09980 | 0.08373 | 0.06536 | 0.04170 | ||||
0.01818 | 0.09175 | 0.06540 | 0.00562 | 0.00613 | ||||
0.08971 | 0.03125 | 0.01620 | 0.10224 | 0 | ||||
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0.78 | 0.78 | 0.006 | 0 | 0 | 0.10076 | 0.04330 | 0.01526 | 0.03323 |
0.23748 | 0.03741 | 0.03131 | 0.08249 | 0.03619 | ||||
0 | 0.02234 | 0.08461 | 0.05375 | 0 | ||||
0.12957 | 0.05859 | 0 | 0.06693 | 0 | ||||
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0.78 | 0.78 | 0.008 | 0 | 0 | 0.07275 | 0.04044 | 0.03112 | 0.03299 |
0.20291 | 0.06426 | 0.05606 | 0.07705 | 0.04148 | ||||
0 | 0.05253 | 0.07474 | 0.03300 | 0 | ||||
0.11520 | 0.04764 | 0.00727 | 0.08355 | 0 | ||||
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0.78 | 0.78 | 0.009 | 0 | 0 | 0.05231 | 0.03833 | 0.04272 | 0.03282 |
0.17766 | 0.08354 | 0.07411 | 0.07299 | 0.04528 | ||||
0 | 0.07457 | 0.06753 | 0.01768 | 0 | ||||
0.10462 | 0.03948 | 0.01339 | 0.09579 | 0 |
Thirdly, we demonstrate the relationship between ERV and EV in our equilibrium optimization model via numerical experiments. For this purpose, we set the confidence levels
Relationship between ERV and EV under
In this subsection, we compare the proposed equilibrium optimization method with classical stochastic optimization method. For the sake of comparison, we also employ the data provided in Section
By calculation, the expected return rate vector is
The objective function is
As a consequence, our equilibrium portfolio optimization problem reduces to the following convex programming model:
To solve model (
Computational results with
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Values | 0 | 0 | 0.11689 | 0.04596 | 0 |
0.41539 | 0 | 0 | 0.01806 | 0 | |
0 | 0 | 0.08538 | 0.07634 | 0 | |
0.24198 | 0 | 0 | 0 | 0 |
To identify the influences of model parameters on solution quality, the stochastic model (
The optimal solutions of random model under various values of parameters.
Parameters | Investment ratios | Objective values | |||||
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0.78 | 0.006 | 0 | 0 | 0.09906 | 0.03658 | 0 | 0.03408 |
0.46099 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0.06700 | 0.06212 | 0 | |||
0.27424 | 0 | 0 | 0 | 0 | |||
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0.8 | 0.006 | 0 | 0 | 0.11689 | 0.04596 | 0 | 0.03398 |
0.41539 | 0 | 0 | 0.01806 | 0 | |||
0 | 0 | 0.08538 | 0.07634 | 0 | |||
0.24198 | 0 | 0 | 0 | 0 | |||
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0.82 | 0.006 | 0 | 0 | 0.12286 | 0.04955 | 0 | 0.03386 |
0.37902 | 0 | 0 | 0.04157 | 0 | |||
0 | 0 | 0.08867 | 0.08131 | 0 | |||
0.21614 | 0.01161 | 0 | 0.00927 | 0 | |||
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0.78 | 0.008 | 0 | 0 | 0.11519 | 0.04488 | 0 | 0.03399 |
0.42154 | 0 | 0 | 0.01284 | 0 | |||
0 | 0 | 0.08459 | 0.07455 | 0 | |||
0.24641 | 0 | 0 | 0 | 0 | |||
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0.78 | 0.009 | 0 | 0 | 0.12044 | 0.04822 | 0 | 0.03394 |
0.40256 | 0 | 0 | 0.02893 | 0 | |||
0 | 0 | 0.08701 | 0.08010 | 0 | |||
0.23274 | 0 | 0 | 0 | 0 |
We now compare the computational results reported in Tables
We continue to compare the computational results reported in Tables
On the other hand, under the same values of model parameters, we observe that the optimal objective values in Table
Finally, we want to point out that an investor may often encounter a hybrid uncertain environment in modern financial markets. In this situation, the investor cannot ignore the influence of fuzzy uncertainty on the solution quality. The computational results support our arguments. For example, when probability level
As a consequence, we conclude from the computational results that our equilibrium optimization method is effective in modeling practical portfolio selection problem under hybrid uncertain environment, where randomness and fuzziness are the state of affairs.
On the basis of probability and credibility measures, a new risk index called the ERV of random fuzzy variable was introduced. Under the equilibrium risk criterion, a new EV-ERV portfolio optimization model was built for portfolio selection problems, where the return rates are characterized by both probability distributions and possibility distributions. In the case where both subjective consciousness and objective factors affect the current financial markets, the developed equilibrium framework provided a novel optimization method for depicting real-life portfolio selection problem.
When the randomness of uncertain return rates follows normal distributions, the proposed equilibrium portfolio selection model was turned into an equivalent credibilistic portfolio optimization model. The convexity of the credibilistic portfolio optimization model was discussed in Theorem
We compared the proposed equilibrium optimization method with traditional stochastic optimization method via a portfolio selection problem. The computational results demonstrated that both optimization methods can provide diversified investment schemes. However, the obtained equilibrium optimal solutions are more superior in terms of diversification. That is, when the fuzziness of uncertain return rates is considered, the equilibrium optimal solution usually diversified the optimal solutions obtained by stochastic method. As a consequence, when the exact probability distributions of return rates are unavailable, the proposed equilibrium optimization method provided an effective way to model practical portfolio selection problems with hybrid uncertain return rates.
The authors declare that they have no competing interests.
This work was supported by National Natural Science Foundation of China (no. 61374184) and the Natural Science Foundation of Hebei Province (no. A2014201166).