A new measure of indeterminacy for uncertain variables with application to portfolio selection

1 Introduction

In probability entropy is a measure for characterize indeterminacy of a random variable. Pal and Pal [11] proposed the concept of exponential entropy for random variables with regard to features of Shanon entropy. After that, Kvalseth [8] two different families of exponential entropies that are one-parameter generalizations of exponential entropy. Furthermore, Zografos and Nadarajah [13] introduced the concept of survival exponential entropy based on Pal’s entropy. Also, Amini et al. [4] discussed about some properties of exponential entropy.

In practice, we do not have any sample sometimes, such as financial affairs, new market in portfolio selection and so on. In these situations, we have to reply on expert’s belief. Therefore, Liu [9] established the uncertainty theory via mathematical principles. After that, Liu [10] introduced the concept of logarithm entropy for uncertain variables. In order to characterize difference between two uncertainty distributions, Chen et al. proposed the concept of cross entropy for uncertain variables. It is mentioned that several authors devoted to the field of entropy of entropy of uncertain variables, for instance [1–3, 7]. Since logarithm entropy proposed by Liu [10] may fail to measure the uncertainty of an uncertain variable, by using bounded function such as exponential function evaluated by an uncertainty distribution, we can derive a new device for measuring uncertainty. The new device (exponential entropy of an uncertain variable) is bounded and consequently can be used to compare uncertainty of two uncertain variables. However, we can not compare two uncertain variables with infinite logarithm entropy. Since, exponential entropy of uncertain variables takes finite values, it can be applied to image processing, pattern recognition and portfolio selection.

In this paper, we present the concept of exponential entropy for uncertain variables and investigate its properties by invoking the inverse of uncertainty distributions. The rest of this paper is organized as follows. In Section 2, we propose the concept of exponential entropy for uncertain variables. Section 3 provides an approach for computation of exponential entropy via the inverse of uncertainty distributions. As an application of exponential entropy, portfolio selection problems of uncertain returns are optimized via mean-exponential models. Finally, some conclusions are obtained in Section 4.

2 Exponential entropy

Pal and Pal [11] introduced the concept of exponential entropy for a continuous random variable as follows: EH ( X ) = E [ exp ( 1 - f ( X ) ) ] = ∫ f ( x ) e 1 - f ( x ) d x , where, X is a continuous random variable and f (x) is its corresponding density function. As a special case, exponential entropy for a Bernoulli random variable X with parameter p is equal to EH ( X ) = pe 1 - p + ( 1 - p ) e p By inception of exponential entropy for Bernoulli random variable, we can propose the concept of exponential entropy for an uncertain variable as follows.

Definition 1. Suppose τ is an uncertain variable with uncertainty distribution Φ (x). Exponential entropy of τ is EH ( τ ) = ∫ ( Φ ( x ) e 1 - Φ ( x ) + ( 1 - Φ ( x ) ) e Φ ( x ) ) d x = ∫ T ( Φ ( x ) ) d x , where T (t) = te^1-t + (1 - t) e ^t , for 0 ≤ t ≤ 1 .

Remark 1. In other point of view, Rao et al. [12] introduced the concept of cumulative residual entropy as follows: CRE ( X ) = - ∫ - ∞ ∞ F ¯ ( x ) ln F ¯ ( x ) d x , where F ¯ ( . ) is the survival function of random variable X. Similarly, Di Crescenzo and Longobardi [6] proposed the concept of cumulative entropy of random variable X as follows: CE ( X ) = - ∫ - ∞ ∞ F ( x ) ln F ( x ) d x , where F (.) is the cumulative distribution function of random variable X. Di Crescenzo and Longobardi [6] showed the relation between cumulative and residual cumulative entropy as follows: CE ( X ) + CRE ( X ) = - ∫ - ∞ ∞ h ( x ) d x , where h ( x ) = - [ F ( x ) ln F ( x ) + F ¯ ( x ) ln F ¯ ( x ) ] is the partition entropy of X evaluated at x. By inception of above formula, we can propose exponential entropy of random variables based on mixture of distribution functions and survival functions as follows: EH ( X ) = ∫ ( F ( x ) e F ¯ ( x ) + F ¯ ( x ) e F ( x ) ) d x . This remark shows the relation between exponential entropy of uncertain and random variables.

The following theorem implies an approach for computing exponential entropy of uncertain variables via the inverse uncertainty distribution.

Theorem 1. Suppose τ is an uncertain variable with uncertainty distribution Φ (x) . Then EH [ τ ] = - ∫ 0 1 ( ( 1 - α ) e 1 - α - α e α ) Φ - 1 ( α ) d α .

Proof 1. It is obvious that T ( Φ ( x ) ) = ∫ 0 Φ ( x ) T ′ ( α ) d α = - ∫ Φ ( x ) 1 T ′ ( α ) d α . where T (α) = αe^1-α + (1 - α) e ^α and T′ is derivative of T. Thus, exponential entropy of τ is equal to EH [ τ ] = ∫ - ∞ ∞ T ( Φ ( x ) ) d x = ∫ - ∞ 0 ∫ 0 Φ ( x ) T ′ ( α ) d α - ∫ 0 ∞ ∫ Φ ( x ) 1 T ′ ( α ) d α . Therefore, Fubini’s theorem concludes that EH [ τ ] = ∫ 0 Φ ( 0 ) ∫ Φ - 1 ( α ) 0 T ′ ( α ) d x d α - ∫ Φ ( 0 ) 1 ∫ 0 Φ - 1 ( α ) T ′ ( α ) d x d α = - ∫ 0 Φ ( 0 ) Φ - 1 ( α ) T ′ ( α ) d α - ∫ Φ ( 0 ) 1 Φ - 1 ( α ) T ′ ( α ) d α = - ∫ 0 Φ ( 0 ) Φ - 1 ( α ) T ′ ( α ) d α = - ∫ 0 1 ( ( 1 - α ) e 1 - α - α e α ) Φ - 1 ( α ) d α .

Theorem 2. If τ is an uncertain variable with uncertainty distribution Φ (x) and c is a constant, then EH [ τ + c ] = EH [ τ ] .

Proof 2. It is clear that inverse uncertainty distribution for uncertain variable τ + c is equal to Φ^-1 (α) + c. Thus, Theorem 1 implies that EH [ τ + c ] = - ∫ 0 1 T ′ ( α ) ( Φ - 1 ( α ) + c ) d α = - ∫ 0 1 T ′ ( α ) Φ - 1 ( α ) d α + c ∫ 0 1 T ′ ( α ) d α = EH [ τ ] + c ( T ( 1 ) - T ( 0 ) ) = EH [ τ ] + c ( 1 - 1 ) = EH [ τ ] .

Theorem 3. Suppose τ₁ and τ₂ are independent uncertain variables with uncertainty distributions Φ₁ and Φ₂, respectively. For any non negative real numbers a₁ and a₂, we have EH [ a 1 τ 1 + a 2 τ 2 ] = a 1 EH [ τ 1 ] + a 2 EH [ τ 2 ] .

Proof 3. It is obvious that inverse uncertainty distribution of uncertain variable a₁τ₁ + a₂τ₂ is equal to a 1 Φ 1 - 1 + a 2 Φ 2 - 1 . Therefore, by invoking Theorem 1, we have EH [ a 1 τ 1 + a 2 τ 2 ] = ∫ 0 1 ( a 1 Φ 1 - 1 ( α ) + a 2 Φ 2 - 1 ( α ) ) T ′ ( α ) d α = a 1 ∫ 0 1 Φ 1 - 1 ( α ) T ′ ( α ) d α + a 2 ∫ 0 1 Φ 2 - 1 ( α ) T ′ ( α ) d α = a 1 EH [ τ 1 ] + a 2 EH [ τ 2 ] . Above theorem is useful and applicable in optimization of portfolio selection.

3 Monte Carlo simulation for exponential entropy

By using Theorem 1, we can write exponential entropy of an uncertain variable via expectation of a function of standard uniform random variables as follows: EH [ τ ] = - E [ Φ - 1 ( U ) ( ( 1 - U ) e 1 - U - U e U ) ] , where U is a standard uniform random variable, i.e. U ∼ U (0, 1) . Therefore, we use Monte Carlo simulation for calculating exponential entropy of uncertain variable τ as follows:

Step 1: Generate u₁, u₂, ⋯ , u _N from standard uniform distribution, randomly.

Step 2: Compute Φ^-1 (u _i )  ((1 - u _i ) e^{1-u _i}  - u _i e ^{u _i} ) for i = 1, 2, ⋯ , N .

Step 3: Consider - 1 N ∑ i = 1 N Φ - 1 ( u i ) ( ( 1 - u i ) e 1 - u i - u i e u i ) , as an approximation for exponential entropy.

Example 1. Suppose τ is an uncertain variable such that τ ∼ L ( 3 , 7 ) . By using Monte Carlo simulation for exponential entropy, we have EH [ τ ] = - E [ Φ - 1 ( U ) ( ( 1 - U ) e 1 - U - U e U ) ] = - ∫ 0 1 ( 3 ( 1 - α ) + 7 α ) ( ( 1 - α ) e 1 - α - α e α ) d α = 1.746255 .

Example 2. Suppose τ is an uncertain variable such that τ ∼ N ( 0 , 1 ) . By using Monte Carlo simulation for exponential entropy, we have EH [ τ ] = - E [ Φ - 1 ( U ) ( ( 1 - U ) e 1 - U - U e U ) ] = - ∫ 0 1 ( 3 π log α 1 - α ) ( ( 1 - α ) e 1 - α - α e α ) d α = 1.453195 .

Example 3. Suppose τ is an uncertain variable such that τ ∼ LOGN ( 0 , 1 ) . By using Monte Carlo simulation for exponential entropy, we have EH [ τ ] = - E [ Φ - 1 ( U ) ( ( 1 - U ) e 1 - U - U e U ) ] = - ∫ 0 1 ( exp ( 3 π log α 1 - α ) ) ( ( 1 - α ) e 1 - α - α e α ) d α = 2.5743 .

4 Portfolio selection of uncertain returns

In many situations, we face with several new markets. Since, we have not enough data to predict empirical probability distribution functions, we consider the returns as uncertain variables. Therefore, we invite experts to determine the uncertainty distributions corresponding to uncertain variables. In this section, we optimize portfolio selection based on mean-entropy models. Based on the investor’s view, we introduce the following portfolio selection models. When upper bound of exponential entropy of returns is known, the investor will prefer a portfolio with large expected value. { max x i E [ x 1 ξ 1 + x 2 ξ 2 + ⋯ + x n ξ n ] subject to : EH [ x 1 ξ 1 + x 2 ξ 2 + ⋯ + x n ξ n ] ≤ λ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , By using Theorem 1, we can write above model as the following crisp model. { max x i x 1 E [ ξ 1 ] + x 2 E [ ξ 2 ] + ⋯ + x n E [ ξ n ] subject to : x 1 EH [ ξ 1 ] + x 2 EH [ ξ 2 ] + ⋯ + x n EH [ ξ n ] ≤ λ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , When upper bound of expectation of returns is known, the investor will prefer a portfolio with small partial semi entropy. { min x i EH [ x 1 ξ 1 + x 2 ξ 2 + ⋯ + x n ξ n ] subject to : E [ x 1 ξ 1 + x 2 ξ 2 + ⋯ + x n ξ n ] ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , where, δ is known. By using Theorem 1, we can write above model as the following crisp model. { min x i x 1 EH [ ξ 1 ] + x 2 EH [ ξ 2 ] + ⋯ + x n EH [ ξ n ] subject to : x 1 E [ ξ 1 ] + x 2 E [ ξ 2 ] + ⋯ + x n E [ ξ n ] ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n ,

Example 4. Consider five securities with uncertain returns τ _i , i = 1, 2, ⋯ , n shown in Table 1. By using Monte Carlo simulation, we have EH [τ₁] =0.34592, EH [τ₂] =0.4079673, EH [τ₃] =0.3142211, EH [τ₄] =0.2008193 and EH [τ₅] =0.2182818. We want to optimize the portfolio selection via mean-exponential entropy. First, we maximize expectation subject to known upper bound of exponential entropy.

{ max x i 0.032 x 1 + 0.039 x 2 + 0.031 x 3 + 0.03 x 4 + 0.05 x 5 subject to : 0.34 x 1 + 0.4 x 2 + 0.31 x 3 + 0.2 x 4 + 0.21 x 5 ≤ 0.85 x 1 + x 2 + . . . + x 5 = 1 , x i ≥ 0 , i = 1 , 2 , . . . , 5 ,

Now, by solving the crisp optimization problem, we obtain the optimal solutions as Table 2. Also, the expected value of the total returns is 0.05.

Now, We want to minimize the exponential entropy of total returns with constrained expected value.

{ min x i 0.34 x 1 + 0.4 x 2 + 0.31 x 3 + 0.2 x 4 + 0.21 x 5 subject to : 0.032 x 1 + 0.039 x 2 + 0.031 x 3 + 0.03 x 4 + 0.05 x 5 ≥ 0.04 . x 1 + x 2 + . . . + x 5 = 1 , x i ≥ 0 , i = 1 , 2 , . . . , 5 ,

Now, by solving the crisp optimization problem, we obtain the optimal solutions as Table 3. Also, the exponential entropy of the total returns is 0.205.

In this paper, the concept of exponential entropy for an uncertain variable was proposed. By invoking the inverse uncertainty distributions, we obtained a formulas for calculating the exponential entropy of an uncertain variable via Monte Carlo simulation. As an application of exponential entropy, portfolio selection problems with uncertain returns were optimized via mean-exponential entropy models.