Partial pseudo-triangular entropy of uncertain random variables with application to portfolio risk management
الموضوعات : فصلنامه ریاضی
1 - Department of Statistics, Risk analyst at Bank Pasargad Investment Company, Mirdamad Blvd, Tehran, Iran.
الکلمات المفتاحية: Portfolio optimization, Chance theory, Uncertain random variable, Partial entropy, Partial pseudo-triangular entropy,
ملخص المقالة :
In this paper, the concept of partial pseudo-triangular entropy as a superior measure of indeterminacy for uncertain random variables is proposed. It is first proved that partial entropy and partial triangular entropy sometimes fail to measure the indeterminacy of an uncertain random variable. Then, the concept of partial pseudo-triangular entropy and its mathematical properties are investigated. To illustrate the outperformance of partial pseudo-triangular entropy as a measure of risk, a portfolio optimization problem is optimized via different types of entropy. Furthermore, a genetic algorithm (GA) is implemented in MATLAB to solve the corresponding problem. Numerical results show that partial pseudo-triangular entropy as a quantifier of portfolio risk outperforms partial entropy and partial triangular entropy in the uncertain random portfolio optimization problem.
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Partial pseudo-triangular entropy of uncertain random variables with application to portfolio risk management
Abstract. In this paper, the concept of partial pseudo-triangular entropy as a superior measure of indeterminacy for uncertain random variables is proposed. It is first proved that partial entropy and partial triangular entropy sometimes fail to measure the indeterminacy of an uncertain random variable. Then, the concept of partial pseudo-triangular entropy and its mathematical properties are investigated. To illustrate the outperformance of partial pseudo-triangular entropy as a measure of risk, a portfolio optimization problem is optimized via different types of entropy. Furthermore, a genetic algorithm (GA) is implemented in MATLAB to solve the corresponding problem. Numerical results show that partial pseudo-triangular entropy as a quantifier of portfolio risk outperforms partial entropy and partial triangular entropy in the uncertain random portfolio optimization problem.
Keywords: Chance theory; Uncertain random variable; Partial entropy; Partial pseudo-triangular entropy; Portfolio optimization.
1. Introduction
Entropy is a quantitative measurement of indeterminacy associated with a variable. Entropy of random variables was first proposed in logarithm form (Shannon [28]). A pioneer research carried out by scholars to associate entropy with a measure of risk in portfolio optimization showed that entropy is more common and better suited in portfolio optimization than variance (Philippatos and Wilson [26]). Moreover, researches showed that entropy as a measure of risk is better than variance in wealth allocation and by using entropy instead of variance in the portfolio optimization problem, all major difficulties with Markowitz’s mean-variance portfolio optimization model can be eliminated (simonelli [31] and Mercurio et al. [25]).
In the mentioned literatures, the indeterminacy is considered under probability theory. The key assumption for using probability theory is that the probability distribution of historical data is similar to the past one and close enough to the frequency. Nevertheless, it is difficult to achieve this assumption generally. As an approach to deal with problems associated with non-random phenomena, fuzzy set theory was proposed (Zadeh [34]). As an improvement, Liu and Liu [21] presented a self-dual credibility measure for fuzzy events. Further researches by Liu [17] confirmed that using fuzzy set theory or subjective probability to model human uncertainty may lead to inaccurate results.
To better deal with non-random phenomena and in particular human uncertainty, Liu [14] founded uncertainty theory. Then, Liu [15] introduced entropy of uncertain variables in logarithm form. Since then, several scholars have been investigating entropy under uncertainty theory. Chen et al. [6] proposed the concept of cross-entropy to measure the divergence degree of uncertain variables and presented the minimum cross-entropy principle. Chen and Dai [5] proposed the maximum entropy principle for uncertain variables. Moreover, Dai and Chen [8] presented a formula to calculate the entropy of uncertain variables. As a supplement of logarithm entropy, several types of entropy for uncertain variables have been investigated by scholars (Tang and Gao [32], Yao et al. [33], Dai [9] and Abtahi et al. [1]).
In numerous situations, uncertainty and randomness may appear together in phenomena. In these situations, the concept of uncertain random variable and chance theory are used for modeling such phenomena. In order to describe such phenomena, Liu [19] proposed uncertain random variables. Liu [20] also discussed the concepts of chance distribution, expected value and variance of uncertain random variables. Then, Liu and Ralescu [22] proposed the risk index for uncertain random variables and established a formula for calculating this index. Guo and Wang [10] presented a formula to obtain variance of uncertain random variables. Liu and Ralescu [23] presented the concept of value at risk for uncertain random variables and Qin et al. [27] optimized portfolio selection problems of uncertain random returns based on value at risk models. Liu et al. [24] proposed the concept of tail value at risk for uncertain random variables and applied it to series systems, parallel systems, k-out-of-n systems, standby systems and structural system. Li et al. [13] proved some mathematical properties of tail value at risk for uncertain random variables and formulated several mean-TVaR models for hybrid portfolio optimization models.
Entropy of uncertain random variables was first proposed in logarithm form by Sheng et al. [30]. Then, Ahmadzade et al. [3] introduced a definition of partial entropy for uncertain random variables to measures how much the entropy of an uncertain random variable belongs to the uncertain variable and derived several properties. Further applications of entropy for uncertain random variables in portfolio optimization problems have been investigated by several scholars (Ahmadzadeh et al. [4], Chen et al. [7] and He et al. [11]).
As it mentioned, partial entropy and partial triangular entropy sometimes fail to measure the indeterminacy of an uncertain random variable. Therefore, in order to address this problem, in this paper, the concept of partial pseudo-triangular entropy for uncertain random variables is proposed. As an application, partial pseudo-triangular entropy as a measure of risk is applied in a portfolio optimization problem. The rest of this paper is organized as follows. In Section 2, some concepts of uncertainty theory and chance theory are reviewed. In Section 3, the concept of partial pseudo-triangular entropy for uncertain random variables together with its mathematical properties are proposed. Then in Section 4, a portfolio optimization problem based on different types of entropy are optimized via a mean–entropy model. Finally, conclusions are given in Section 5.
2. Preliminaries
This section comes with reviewing some concepts of uncertainty theory and chance theory including definition of uncertain variable, uncertainty distribution, uncertain random variable and chance distribution.
2.1. Uncertainty theory
Uncertainty theory was founded by Liu in 2007 to model human uncertainty. In lack of historical data, we should request experts to evaluate the degree of belief for the occurrence of an event. In this section, some necessary definitions and theorems in uncertainty theory are reviewed.
Assume that 
is a nonempty set and 
is a 
- algebra over
. Then, 
is named a measurable space. Each element 
in is called a measurable set. A measurable set can be considered as an event in uncertainty theory. That is, a number 
will be assigned to each event to indicate the belief degree with which we believe will happen. In order to deal with belief degree, the following axioms are suggested (Liu [14]):
Axiom 1 (Normality) For the universal set 
Axiom 2 (Duality) For any event
Axiom 3 (Subadditivity) For every countable sequence of events 
we have
Then, 
is called an uncertainty space.
Axiom 4 (Product) Let
be uncertainty spaces for
. Then, the product uncertain measure 
is an uncertain measure satisfying
where 
are arbitrarily chosen events from 
for 
respectively.
Definition 1 (Liu [14]) An uncertain variable is a function 
from an uncertainty space 
to the set of real numbers such that 
is an event for any Borel set B of real numbers.
Definition 2 (Liu [16]) For any Borel sets 
of real numbers, the uncertain variables 
are independent if
Definition 3 (Liu [16]) An uncertain variable is called normal denoted by 
if it has a normal uncertainty distribution
where
and 
.
Example 1 (Liu [16]) Let 
, then the inverse uncertainty distribution of normal uncertain variable is
Theorem 1 (Liu [16]) Let be an uncertain variable with regular uncertainty distribution
. If the expected value of exists, then
where 
is the inverse uncertainty function of with respect to 
.
Definition 4 (Liu [15]) Let be an uncertain variable with uncertainty distribution. Then, the logarithm entropy of uncertain variable
is
where 
Theorem 2 (Dai and Chen [8]) Let be an uncertain variable with inverse uncertainty distribution 
Then, the logarithm entropy of is
Definition 5 (Tang and Gao [32]) Suppose that is an uncertain variable with uncertainty distribution 
. Then, the triangular entropy of is defined by
where 
Theorem 3 (Tang and Gao [32]) Let be an uncertain variable with uncertainty distribution 
Then, the triangular entropy of is
Remark 1 Logarithm entropy and triangular entropy sometimes may fail to measure the indeterminacy of an uncertain variable.
Example 2 (Abtahi et al. [1]) Let be an uncertain variable with uncertainty distribution
and inverse uncertainty distribution
Then, the logarithm entropy and triangular entropy of 
are infinite.
As a superior measure of indeterminacy compared to logarithm entropy and triangular entropy, Abtahi et al. [1] proposed the concept of pseudo-triangular entropy for uncertain variables. They also proved that the pseudo-triangular entropy of in Example 2 is finite.
Definition 6 (Abtahi et al. [1]) Suppose that is an uncertain variable with uncertainty distribution
. Then, the pseudo-triangular entropy of is defined by
where 
Theorem 4 (Abtahi et al. [1]) Suppose that is an uncertaint variable with regular uncertainty distribution. Then, the pseudo-triangular entropy of is
Example 3 (Abtahi et al. [1]) Let 
, then the pseudo-triangular entropy of uncertain variable is
Theorem 5 (Liu [16]) Let 
be independent uncertain variables with regular uncertainty distributions
, respectively. If 
is strictly increasing with respect to 
and strictly decreasing with respect to
, then
has an inverse uncertainty distribution
2.2. Chance theory
In order to handle phenomena including both uncertainty and randomness, chance theory was proposed by Liu [19]. In chance theory, the chance space is refer to the product of
, in which is an uncertainty space and 
is a probability space. The chance measure of an uncertain random event 
=
is defined as
Liu [19] proved that a chance measure satisfies following properties:
(i) (Normality) 
(ii) (Duality) 
for any event 
(iii) (Monotonicity) 
for any real number set
.
Furthermore, Hou [12] proved that for a sequence of events 
a chance measure satisfies subadditivity as follows,
Definition 7 (Liu [19]) An uncertain random variable is a function 
from a chance space to the set of real numbers such that for any Borel set of 
of real numbers, 
is an event in.
Definition 8 (Liu [19]) Suppose is an uncertain random variable. Then the chance distribution of for any 
is defined by
Definition 9 (Sheng et al. [29]) A chance distribution 
is said to be regular if it is a continuous and strictly increasing function with respect to 
at which 
, and 
Definition 10 (Liu [19]) Let be an uncertain random variable. Then the expected value of is defined by
provided that at least one of the two integrals is finite.
Theorem 6 (Liu [20]) Let 
be independent random variables with probability distributions
, and let 
be independent uncertain variables with uncertainty distributions
, respectively. Then the uncertain random variable 
has a chance distribution
where 
is the uncertainty distribution of uncertain variable 
for any real numbers 
Theorem 7 (Ahmadzadeh et al. [2]) Let be independent random variables with probability distributions, and let be independent uncertain variables with uncertainty distributions, respectively. Suppose
, then
where 
is the inverse uncertainty distribution of uncertain variable 
.
3. Partial pseudo-triangular entropy of uncertain random variables
In this section, the concepts of pseudo-triangular entropy of uncertain random variables are proposed. Besides, some mathematical properties of pseudo-triangular entropy and a formula to calculate it via inverse uncertainty distribution are derived. We first recall the definition of entropy, partial entropy and partial triangular entropy for uncertain random variables.
Definition 11 (Sheng et al. [30]) Let be an uncertain random variable with chance distribution 
Then the entropy of is defined by
where 
Theorem 8 (Ahmadzade et al. [3]) Let be independent random variables with probability distributions and be independent uncertain variables with uncertainty distributions, respectively, and let 
be a measurable function. Also let 
be an uncertain random variable. Then, has partial entropy
where 
and 
is the uncertainty distribution of uncertain variable for any real numbers 
.
Theorem 9 (Ahmadzade et al. [3]) Let be independent random variables with probability distributions and be independent uncertain variables with uncertainty distributions, respectively, and let be a measurable function. Then, has partial entropy
where is the inverse uncertainty distribution of uncertain variable .
Theorem 10 (Ahmadzade et al. [3]) Let be an uncertain variable with uncertainty distribution function and let 
be a random variable with probability distribution function
. If
, then
Theorem 11 (Ahmadzade et al. [3]) Let 
and 
be independent random variables and let 
and 
be independent uncertain variables. Also assume that 
and
. Then, for any real numbers 
and
, we have
Theorem 12 (Ahmadzade et al. [4]) Suppose that are independent random variables, and are independent uncertain variables. Also let be an uncertain random variable. Then, partial triangular entropy of uncertain random variable is defined as follows,
where 
and is the uncertainty distribution of uncertain variable for any real numbers .
Theorem 13 (Ahmadzade et al. [4]) Let be independent random variables with probability distributions and be independent uncertain variables with uncertainty distributions, respectively, and let be a measurable function. Then, has partial triangular entropy
where is the inverse uncertainty distribution of uncertain variable .
Theorem 14 (Ahmadzade et al. [4]) Let be an uncertain variable with uncertainty distribution function and let be a random variable with probability distribution function. If, then
Theorem 15 (Ahmadzade et al. [4]) Let and be independent random variables and let and be independent uncertain variables. Also assume that and. Then, for any real numbers and, we have
Remark 2 Partial entropy and partial triangular entropy sometimes may fail to measure the indeterminacy of an uncertain random variable.
Example 4 Let be an uncertain variable with uncertainty distribution
and inverse uncertainty distribution
also let be a random variable. Consider. Since,
and 
are infinite (Abtahi et al. 2022), Theorem 10 and Theorem 14 imply that 
and
, respectively. Since, partial entropy and partial triangular entropy failed to measure the indeterminacy of uncertain random variable
, a new measure of indeterminacy for uncertain random variables will be proposed.
Definition 12 Suppose that are independent random variables, and are independent uncertain variables. Also let be an uncertain random variable. Then, partial pseudo-triangular entropy of uncertain random variable is defined as follows,
where 
and is the uncertainty distribution of uncertain variable 
.
Theorem 16 Suppose that are independent random variables, and are independent uncertain variables. Also let be an uncertain random variable. Then, the partial pseudo-triangular entropy of uncertain random variable is defined as follows,
where 
and is the inverse uncertainty distribution of uncertain variable .
Proof It is clear that 
is a derivable function with
Since,
we have,
It follows from Fubini’s theorem that
The proof is completed.
Theorem 17 Let be independent random variables with probability distributions and be independent uncertain variables with uncertainty distributions, respectively, and let be a measurable function. Then, has partial pseudo-triangular entropy
Proof According to Theorem 16 we have
Theorem 18 Let be an uncertain variable with uncertainty distribution function and let be a random variable with probability distribution function. If, then
Proof It is obvious that 
Therefore, by applying theorem 17 we have
Theorem 19 Let be an uncertain variable with uncertainty distribution function and let be a random variable with probability distribution function. If
, then
Proof It is obvious that 
Therefore, by applying theorem 16 we have
Theorem 20 Suppose that 
are independent random variables, and are independent uncertain variables. Assume
If 
is strictly increasing with respect to 
and strictly decreasing with respect to
, then 
has partial pseudo-triangular entropy
where 
is the inverse uncertainty distribution of uncertain variable 
for any real number 
Proof By applying Theorem 17, the proof of theorem is straightforward.
Theorem 21 Let and 
be independent random variables with probability distributions
and
, respectively, and let and be independent uncertain variables with uncertainty distributions
and 
respectively. Then,
(i) If 
and
,
(ii) If 
and
,
Proof of part (i) it is clear that 
and 
. By applying Theorem 20, we have
Proof of part (ii) it is clear that 
and
. By applying Theorem 20, we have
Theorem 22 Let and be independent random variables with probability distributionsand, respectively, and let and be independent uncertain variables with uncertainty distributions
and respectively. If 
and, then
Proof It is clear that and . By applying Theorem 20, we have
Theorem 23 Let and be independent random variables and also let and be independent uncertain variables. Assume that 
and 
. Then for any real numbers 
and 
, we have
Proof The theorem will be proved via three steps.
Step 1 We prove 
If 
, then the inverse uncertainty distribution of 
is
where 
is the inverse uncertainty distribution of 
. It follows from Theorem 20 that
If 
, then the inverse uncertainty distribution of is
therefore we have,
By changing variable
, we have
Thus, we have
Step 2 we prove 
Since, the inverse distribution of 
is 
, by applying Theorem 20 we have
Step 3 For any real numbers and, we have
The proof is completed.
4. Uncertain random portfolio optimization
In this section, in order to solve the portfolio optimization problem of uncertain random variables, a mean-entropy model via partial pseudo-triangular entropy is proposed. Suppose that there are 
securities with uncertain random returns
. Moreover, let
’s be investment proportions in security 
To make sure that the uncertain random portfolio risk is under control, we minimize entropy as the objective function. Moreover, we set expected value greater than some preset value
In order to optimize the portfolio optimization problem, a mean-entropy model based on partial pseudo-triangular entropy is presented as follows,
where the predetermined parameter 
is designated by investor.
By applying the expected value formula of uncertain random variables in Theorem 7, we have
Now, according to Theorem 18 and Theorem 23, the partial pseudo-triangular entropy of uncertain random variables is obtained as follows,
where 
is the pseudo-triangular entropy of uncertain variable
Thus, Model (4.1) is equivalent to the following model,
Now, in order to further investigate the outperformance of partial pseudo-triangular entropy as a quantifier of portfolio risk in comparison with partial entropy and partial triangular entropy in portfolio risk management let us consider the following example.
Example 5 Suppose there is an investment portfolio containing five securities. According to expert’s evaluation and the data from Tehran Stock Exchange, five securities are assumed to be uncertain random variables with 
depicted in Table 1. Moreover, the parameter in Model (4.2) is designated to 2 by investor.
The optimal solutions are obtained by implementing a genetic algorithm (GA) in MATLAB. The investment proportions in securities and the objective values are illustrated in Table 2. According to Table 2, the objective value for partial pseudo-triangular entropy has the lowest value amongst different types of entropy. Therefore, a portfolio based on partial pseudo-triangular entropy is less risky than partial entropy and partial triangular entropy. Furthermore, investment proportion in securities with smaller parameter
is more other securities.
Table 1 Uncertain random returns
