In this study, we proposed definition of order
The purpose of portfolio selection is to allocate optimal assets to purchase stocks. In general, to achieve this goal, returns are considered as random variables. The uncertainty variables are used as returns in the absence of historical data. Accordingly, the use of entropy in portfolio selection is considered for measuring the uncertainty of the uncertain variables. As an application of this concept, we can consider the generalized entropy maximization model and cross entropy minimization model under the uncertainty environment.
Portfolio selection is a very extensive topic in finance, which has been introduced to this field in the 1950’s, and the interest in it does not seem to be subsiding. Markowitz [1] was the first scientist to introduce the modern portfolio selection theory. Later, many researchers revised or developed the model with new methods or elements to improve the results. A typical extension was suggested by Philippatos and Wilson [2], who employed entropy to measure the risk of the portfolio selection. After that, several important works were published in this topic, for example see Zhou et al. [3] and Yu et al. [4].
Fuzzy entropy is a way to characterize the uncertainty on the possible values of fuzzy variables, which has been studied by many researchers such as Bhandari and Pal [5], De Luca and Termini [6], and Liu [7]. Within the framework of credibility theory, Li and Liu [8] presented an entropy for fuzzy variable. Li and Liu [9] proposed the maximum entropy. Based on the concept of fuzzy entropy, Li et al. [10] proposed the maximum optimization model by minimizing the uncertainty of the fuzzy objective under certain expected constraints. Further, Qin et al. [11] established credibilistic cross-entropy minimization models for portfolio optimization with fuzzy returns in the framework of credibility theory. Bhattacharyya et al. [12] introduced the cross-entropy, mean, variance, skewness model. Cross entropy was used to quantify the level of dispersion for the fuzzy return. Yari et al. [13] presented the Renyi entropy-mean-variance maximization and Renyi cross entropy-mean-variance minimization models for portfolio selection with fuzzy return under the credibility theory framework. Kar et al. [14] introduced a fuzzy bi-objectives portfolio model with objectives “fuzzy VaR ratio” and “fuzzy Sharpe ratio”. They tested the performance of the model with different evolutionary algorithms.
In the situations where historical data is not available, another feasible way is to estimate returns using expert opinion based on their subjective evaluation under the uncertainty theory (Liu [15]). Recently, portfolio selection problems have been studied under uncertainty conditions. By using the uncertainty theory, several researchers, including Qin et al. [11], who formulated the uncertain counterpart of mean-variance model, Liu and Qin [16], Huang and Qiao [17], Yao and Ji [18], took the security returns as uncertain variables. In order to measure the uncertainty of a variable, entropy was provided by Liu [19] under the uncertainty theory. Subsequently, some properties of entropy for uncertain variables were investigated by Dai and Chen [20]. In order to address the divergence of uncertain variables via uncertain distribution, Chen et al. [21] proposed cross entropy of uncertain variables. Bhattacharyya et al. [22] showed that their developed mean-entropy-skewness model in an uncertain environment is more effective in comparison with other earlier proposed models.
This paper focuses on the portfolio optimization with uncertain returns subject to expert evaluation. Section 2 reviews some important concepts and measures of uncertain variables and explain their implications in portfolio selection. Section 3 proposes order υ entropy and order υ cross entropy for uncertain variables. In Section 4, we establish order υ entropy maximization and order υ cross entropy minimization models for portfolio optimization with uncertain returns and give numerical examples. We demonstrate that using different values of υ can lead to a higher diversification of the asset allocations. Finally, some conclusions are listed in Section 5.
The uncertainty theory, introduced by Liu [15] is a branch of mathematics that studies the behavior of human uncertainty. In this section, we review some basic concepts about uncertain measures and uncertain variables, which are related with this paper.
Let Γ be a nonempty set and be a σ–algebra over Γ. Each element is called an event. In order to indicate the chance that Λ will happen, Liu [23] proposed the following four axioms to ensure that ℳ{Λ} satisfying certain mathematical properties.
If ξ is an uncertain variable, then its uncertainty distribution is define as follows:
where x ∈ R.
In uncertainty theory, Liu and Liu [24] defined the expected value and variance of ξ as follows:
Provided that at least one of the two integrals is finite.
If ξ = Z(a, b, c) is a zigzag uncertain variable with following uncertainty distribution
where a, b, c are real numbers with a < b < c. Further, the zigzag uncertain variable has an expected value
Let ξ be a uncertain variable with uncertain distribution Φ. Li and Liu [8] presented the following definition for uncertain entropy
where S (t) = −t ln t − (1 − t) ln (1 − t).
Let ξ and η be uncertain variables. Then the cross entropy of ξ from η is defined as
where
where Φ_{ξ} and Φ_{η} are the respective distribution functions of uncertain variables ξ and η.
The entropy concept was introduced by Shanon [25] and Renyi [26] defined the order υ entropy of a probability distribution (p_{1}, p_{2}, ..., p_{n}) as
Suppose that ξ is an uncertain variable with uncertain distribution Φ. Then its order υ entropy is defined by
where
The order υ cross-entropy of ξ and η can be written as:
Let ξ is zigzag uncertain variable with uncertainty distribution ξ = Z(a, b, c). Then the order υ entropy of ξ is
Considering
By the changes of variable technique, Theorem 1 can be easily proved. Therefore, the theorem is proved, and solving this integral numerically for different values of υ, (υ >0) is possible.
Cross entropy was first proposed by Kullback and Leibler [27] to measure the difference between two probability distribution. Renyi [26] defined the order υ cross entropy of a probability distribution (p_{1}, p_{2}, ..., p_{n}) as
Let ξ and η be two uncertain variables. Then, the order υ cross entropy of ξ and η is defined as
where
It is easy to verify that T(s, t) is strictly convex with respect to (t, s) and attains it minimum value 0 on the line s = t, also for any 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1, we have T(s, t) = T(1 − s, 1 − t).
The order υ cross-entropy of ξ and η can be written as
Let ξ and η are zigzag uncertain variables with uncertainty distribution ξ = Z(a, b, c) and η = Z(d, b, e), (d ≤ a < b < c ≤ e), respectively. Then the order υ cross-entropy of ξ and η is
Considering
In particular if the uncertainty distributions of ξ and η are Z(2, 3, 4) and Z(1, 3, 5), respectively, using the theorem above, we get D[ξ; η] = 0.14, when υ → 2.
In this section, Kapur and Kesavan [28] entropy maximization and cross-entropy minimization model is extended to the portfolio optimization with uncertain returns. Let ξ_{i} be the ith return of the security, and x_{i} is the proportion of capital allocated for the i-th security, where i = 1, 2, ..., n. Let ξ_{1}, ξ_{2}, ..., ξ_{n} be the uncertain variables in the uncertain space (Γ, ,ℳ). Then, the total return from the investment is ξ_{1}x_{1} + ξ_{2}x_{2} + · · · + ξ_{n}x_{n}, which is an uncertain variable. Then, order υ entropy-mean-variance model is presented as follows:
where r_{0} is the predetermined expected return and d_{0} is the predetermined risk for the portfolio.
Suppose that each security return is a zigzag variable denoted by ξ_{i} = Z(a_{i}, b_{i}, c_{i}) (i = 1, 2, ..., n). Then the model
Note that in uncertain environment, E[ξ_{1}x_{1} +ξ_{2}x_{2} + ...+x_{n}ξ_{n}] ≠ x_{1}E[ξ_{1}]+x_{2}E[ξ_{2}]+...+x_{n}E[ξ_{n}] for uncertain variables ξ_{1}, ξ_{2}, ..., ξ_{n}. However, the inequality will become equality when ξ_{1}, ξ_{2}, ..., ξ_{n} are independent. Further, we assume that security returns are all zigzag uncertain variables, denoting the return of security i by ξ_{i} = Z(a_{i}, b_{i}, c_{i}). It follows that the portfolio return
Since the cross entropy is a common method for measuring the degree of divergence of uncertain variables, we formulate different cross entropy minimization model for portfolio optimization. Suppose that η is a prior uncertain investment return for an investor. Then, the mean-variance-order υ cross entropy model is presented as follows:
Suppose that each security return is a zigzag variable denoted by ξ_{i} = Z(a_{i}, b, c_{i}) (i = 1, 2, ..., n). Let the prior investment return be η = (d, b_{i}, e). Then model
The theorem can be easily proved taking into account the relations used in proving Theorem 3. Solving the integral using numeric methods is possible for different values of υ. In the rest of the this section, minimization order υ cross entropy-mean-variance models with 3 different υ are applied to the data from Qin and Yao [29], who had used them to illustrate the application of uncertain mean-lower partial moment model. Assume that an investor plans to invest his fund among to securities. Further, all the future returns of the securities are assumed to be zigzag uncertain variable. We apply model
Tables 1
In the present study, we compared the applicability of two models, order υ entropy-mean-variance and order υ cross entropy-mean-variance, for portfolio selection under the uncertainty set. We showed that there is no difference between the two models in portfolio optimization. It was also presented that lowering the values of the υ parameter in the order υcross entropy-mean-variance minimization results in more diversified portfolio selection. We may also conclude that using different values of υ in proposed models for portfolio optimization would affect the decision of an investor to allocate his capital to purchase various securities.
No potential conflict of interest relevant to this article was reported.
Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
E-mail: alireza.sajedi@srbiau.ac.ir
E-mail: Yari@iust.ac.ir