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Uncertainty theory based multiple objective meanentropyskewness stock portfolio selection model with transaction costs
Journal of Uncertainty Analysis and Applications volume 1, Article number: 16 (2013)
Abstract
Purpose
The aim of this paper is to develop a meanentropyskewness stock portfolio selection model with transaction costs in an uncertain environment.
Methods
Since entropy is free from reliance on symmetric probability distributions and can be computed from nonmetric data, it is more general than others as a competent measure of risk. In this work, returns of securities are assumed to be uncertain variables, which cannot be estimated by randomness or fuzziness. The model in the uncertain environment is formulated as a nonlinear programming model based on uncertainty theory. Also, some other criteria like shortand longterm returns, dividends, number of assets in the portfolio, and the maximum and minimum allowable capital invested in stocks of any company are considered. Since there is no efficient solution methodology to solve the proposed model, assuming the returns as some special uncertain variables, the original portfolio selection model is transformed into an equivalent deterministic model, which can be solved by any stateoftheart solution methodology.
Results
The feasibility and effectiveness of the proposed model is verified by a numerical example extracted from Bombay Stock Exchange, India. Returns are considered in the form of trapezoidal uncertain variables. A genetic algorithm is used for simulation.
Conclusions
The efficiency of the portfolio is evaluated by looking for risk contraction on one hand and expected return and skewness augmentation on the other hand. An empirical application has served to illustrate the computational tractability of the approach and the effectiveness of the proposed algorithm.
Introduction
The Markowitz [1] formulation of modern portfolio theory has been the most impactmaking development in mathematical finance management to date. Since returns are uncertain in nature, the allocation of capital in different risky assets to minimize the risk and to maximize the return is the main concern of it.
In most of the significant works on portfolio selection, the firstorder moment of return distribution about the origin, i.e., the mean, quantifies the return, and the secondorder moment about the mean, i.e., the variance, quantifies the risk. Consideration of variance as risk is erroneous as it equally suggests penalties for up and down deviations from the mean. To face this problem, Markowitz [2] recommended semivariance, a downside risk measure. Another alternative definition of risk is the probability of an adverse outcome [3]. The popular risk measure value at risk [4, 5] is in fact an alternative expression of the definition by Roy [3]. Different authors like Philippatos and Wilson [6], Philippatos and Gressis [7], Nawrocki and Harding [8], Simonelli [9], Huang [10], Qin et al. [11], and Bhattacharyya et al. [12] used entropy as an alternative measure of risk to replace the variance proposed by Markowitz [1]. Uncertainty causes loss and so investors dislike uncertainty. Since entropy is a measure of uncertainty, it is used to measure risk. Entropy is more general than others as an efficient measure of risk because entropy is free from reliance on symmetric distributions and can be computed from nonmetric data.
One of the important theoretical difficulties of these studies is that they assume that asset returns are normally distributed or the utility function is quadratic or that the higher moments are irrelevant to the investors' decision. However, some experimental studies show that portfolio returns are generally not normally distributed. As a result, a natural extension of the meanvariance model is to add the skewness as a factor for consideration in portfolio management. The importance of higher order moments in portfolio selection was suggested by Samuelson [13]. However, considerations of skewness in portfolio selection problem were started by 1990 and were done by Lai [14], Konno and Suzuki [15], Chunhachinda et al. [16], Liu et al. [17], Prakash et al. [18], Briec et al. [19], Yu et al. [20], Li et al. [21], Bhattacharyya et al. [22], Bhattacharyya and Kar [23, 24], Bhattacharyya [25], and others. Consideration of a meanentropyskewness model in portfolio selection problem is introduced by Bhattacharyya et al. [12]. They have constructed three portfolio selection models in fuzzy environment using the credibility theory approach.
In most of the abovementioned research works on portfolio selection, the common assumptions are that the investor has enough historical data and that the situation of asset markets in the future can be reflected with certainty by asset data in the past. However, it cannot always be made with certainty. Basically, the usual feature of a financial environment is uncertainty. Mostly, it is realized as risk uncertainty and is modeled by stochastic approaches. However, the term uncertainty has a second aspect vagueness (imprecision or ambiguity), which can be modeled by fuzzy methodology. In this respect, to tackle the uncertainty in the financial market, stochasticfuzzy and fuzzystochastic methodologies are extensively used in portfolio modeling. Authors like Konno and Suzuki [15], Leon et al. [26], Vercher et al. [27], Bhattacharyya et al. [12, 22], Dey and Bhattacharyya [28], etc. used fuzzy numbers to replace uncertain returns of securities, and they define portfolio selection as a mathematical programming problem in order to select the best alternative. Huang [29] measures portfolio risk by credibility measure and proposed two credibility theorybased meanvariance models. Huang [30] also proposed a meansemivariance model for describing the asymmetry of fuzzy returns. She extends the risk definition of variance and chance to a random fuzzy environment and formulates optimization models where security returns are fuzzy random variables.
So, in attempts dealing with portfolio selection problems, randomness and fuzziness are considered as the two basic types of uncertainty contained in security returns. It has become a common practice that when security returns cannot be reflected by historical data, fuzzy variables can be used to show experts' knowledge and estimation of security returns. However, illogicality will come into view if fuzzy variables are used to describe the subjective estimation of security returns. For example, a stock return is considered as a triangular fuzzy variable ξ = (−0.2, 0.3, 0.7). Using possibility theory (or credibility theory), the return is exactly 0.3 with belief degree 1 in possibility measure (or 0.5 in credibility measure). However, this conclusion is unacceptable because the belief degree of exactly 0.3 is almost 0. In addition, the return being exactly 0.3 and not exactly 0.3 has the same belief degree in either possibility measure or credibility measure, which implies that the two events will happen equally likely. This conclusion is quite astonishing and hard to accept.
Again, philosophically, though randomness and fuzziness are two basic types to represent uncertain phenomena, in real life, there are some situations where uncertainty behaves neither randomly nor fuzzily. For example, the occurrence chance of a security price falling in the interval of [100, 110] is 30%, and the occurrence chance of the security price in the interval of [110, 120] is 20%. Then what is the occurrence chance of the security price in the interval of [100, 120]? A survey shows that some people believe that the occurrence chance should be somewhere that is not less than 30% but not greater than 50%. In this case, the security price is neither random nor fuzzy. Recently, Liu [31] proposed an uncertain measure and developed an uncertainty theory, which can be used to handle subjective imprecise quantity. Much research works have been done on the development of uncertainty theory and related theoretical works. Though some considerable amounts of publications have been done in the field of uncertainty theory, not much work has been done in the portfolio selection problem. Huang [32] proposed a meanrisk model for uncertain portfolio selection. Yan [33] found out the deterministic forms of meanvariance portfolio section models corresponding to different special uncertain variables like rectangular uncertain variable, triangular uncertain variable, trapezoidal uncertain variable, and normal uncertain variable. In this study, security returns are considered as uncertain variables, which are characterized by identification functions, and instead of possibility/credibility measure, uncertain measure is used to handle the uncertain events.
Not all the relevant information for an investment decision can be confined in terms of explicit return, risk, and skewness. By capturing additional and alternative decision criteria, a portfolio that is dominated with respect to expected return, skewness, and risk may frame for the shortfall in these three important factors by a very good act on one or several other criteria. As a result, portfolio selection models that consider more criteria than the standard expected return and variance objectives of the Markowitz model have become well liked. Ehrgott et al. [34] proposed a model having five criteria, viz., shortand longterm return, dividend, ranking, and risk, and used a multicriteria decision making approach to solve the portfolio selection problem. Fang et al. [35] proposed a portfolio rebalancing model with transaction costs based on fuzzy decision theory considering three criteria: return, risk, and liquidity.
The main focus of this paper is to propose a meanentropyskewness portfolio selection framework with transaction cost having returns in the form of uncertain variables. In addition, it incorporates some useful constraints in the model to make the model more realistic. In addition, this paper provides a real application by using data from Bombay Stock Exchange (BSE), where we consider returns as trapezoidal uncertain variables.
The rest of the paper is organized as follows. We review the necessary knowledge about uncertainty variables and develop some essential results in the ‘Uncertainty theory: related topics’ section. In the ‘Meanentropyskewness model formulation’ section, a triobjective meanentropyskewness portfolio selection model is formulated with constraints on shortterm and longterm returns, dividends, number of assets in the portfolio, and the maximum and minimum allowable capital invested in stocks of any company. The model is then converted into a singleobjective constrained optimization problem with weights over mean, skewness, and entropy. To solve the proposed optimization problem, we provide a genetic algorithm in the ‘Genetic algorithm’ section. In the ‘Case study: Bombay Stock Exchange’ section, a case study from Bombay Stock Exchange is done to illustrate the method. The same section also contains a comparative study with other relevant models. Finally, in the last section, some concluding remarks are specified.
Uncertainty theory: related topics
In this paper, the concept of uncertainty theory has been introduced in the field of stock portfolio selection. This section contains only those definitions and theorems on uncertainty theory which are directly used for the formation of this article. The concepts of uncertain measure, uncertain variable, uncertain space, first and second identification functions, rectangular uncertain variable, triangular uncertain variable, exponential uncertain variable, bellshaped uncertain variable, linear uncertain variable, zigzag uncertain variable, normal uncertain variable, lognormal uncertain variable, and others would be useful to understand the backbone of the article and can be obtained from Liu [31].
Definition 1. A trapezoidal uncertain variable is defined to be the uncertain variable which is fully determined by the fourtuple (a, b, c, d) of crisp numbers with a < b < c < d, and whose first identification function is
Definition 2. The uncertainty distribution Φ : ℝ → [0, 1] of an uncertain variable $\stackrel{\u2323}{\xi}$ is defined by
Definition 3. An uncertain variable $\stackrel{\u2323}{\xi}$ is said to have an empirical uncertainty distribution if
and is denoted by ϵ(x _{1}, α _{1}, x _{2}, α _{2}, …, x _{ n }, α _{ n }) , where x _{1} < x _{2} < … < x _{ n } and 0 ≤ α _{1} ≤ α _{2} ≤ … ≤ α _{ n } ≤ 1.
Example 1. The trapezoidal uncertain variable $\stackrel{\u2323}{\xi}=\left(a,b,c,d\right)$ follows the empirical uncertain distribution given by
Definition 4. The uncertain variables ${\stackrel{\u2323}{\xi}}_{1},{\stackrel{\u2323}{\xi}}_{2},\dots ,{\stackrel{\u2323}{\xi}}_{n}$ are said to be independent if
for Borel sets B _{ 1 } , B _{ 2 } ,…, B _{ n } of real numbers. Here M denotes the uncertain measure.
Definition 5. Let $\stackrel{\u2323}{\xi}$ be an uncertain variable. Then the expected value of $\stackrel{\u2323}{\xi}$ is given by
provided that at least one of the two integrals is finite.
Definition 6. Let $\stackrel{\u2323}{\xi}$ be an uncertain variable. Then the entropy of $\stackrel{\u2323}{\xi}$ is given by
Definition 7. Let $\stackrel{\u2323}{\xi}$ be an uncertain variable with finite expected value e. Then the variance and skewness of $\stackrel{\u2323}{\xi}$ are respectively given by
Example 2. If $\stackrel{\u2323}{\xi}=\left(a,b,c,d\right)$ is a trapezoidal uncertain variable then
Theorem 1. Let ${\stackrel{\u2323}{\xi}}_{1},\stackrel{\u2323}{\xi}$ be two independent uncertain variables with finite expected values. Then for any real numbers a and b, we have
Theorem 2. Let ${\stackrel{\u2323}{r}}_{i}=\left({a}_{i},{b}_{i},{c}_{i},{d}_{i}\right),\left(i=1,2,\dots ,\phantom{\rule{0.5em}{0ex}}n\right)$ be n independent uncertain trapezoidal variables and let x _{ i } (i = 1, 2,…, n) be n real variables. Then
Proof. As ${\stackrel{\u2323}{r}}_{i}=\left({a}_{i},{b}_{i},{c}_{i},{d}_{i}\right),\left(i=1,\phantom{\rule{0.15em}{0ex}}2,\dots ,n\right)$ are n independent uncertain trapezoidal variables and x _{ i } (i = 1, 2,…, n) are n real variables, we have
Hence, ${\stackrel{\u2323}{r}}_{1}{x}_{1}+{\stackrel{\u2323}{r}}_{2}{x}_{2}+\dots +{\stackrel{\u2323}{r}}_{n}{x}_{n}$ is a trapezoidal uncertain variable. Combining the above result with the results obtained in Example 2, we are with the theorem.
Meanentropyskewness model formulation
In this section, we will first describe the assumptions and notations used in the construction of the paper. Then the objective functions of the models will be constructed in the next subsection. In the third subsection, we will discuss the constraints used in our portfolio selection model. The fourth subsection will include three different mathematical models for different situations.
Assumptions and notations
Let us consider a financial market with n risky assets offering uncertain returns. An investor allocates his wealth among these risky assets.
For the i th risky asset (i = 1, 2,…, n), let us use the following notations:
x _{ i } = portion of the total capital invested in i th security
${\stackrel{\u2323}{p}}_{i}$ = uncertain variable representing the closing price of the i th security at present
${\stackrel{\u2323}{\mathit{p}}}_{\mathit{i}}^{\text{'}}$ = uncertain variable representing the estimated closing price of the i th security in the next year
d _{ i } = the estimated dividends in the next year
${\stackrel{\u2323}{\mathit{r}}}_{\mathit{i}}=\frac{{\stackrel{\u2323}{\mathit{p}}}_{\mathit{i}}^{\text{'}}+{\mathit{d}}_{\mathit{i}}{\stackrel{\u2323}{\mathit{p}}}_{\mathit{i}}}{{\stackrel{\u2323}{\mathit{p}}}_{\mathit{i}}}$ = uncertain variable representing the return of the i th security
R _{ i } ^{(12)} = the average 12 month performance
R _{ i } ^{(36)} = the average 36 month performance
k _{ i } = the constant transaction cost per change in a proportion, k _{ i } ≥ 0
Formulation of objective functions
It is impossible to predict future returns of stocks in any budding security market. The arithmetic mean of historical data is in general considered as the expected return of securities, which yield us a crisp value. However, for this technique, two main problems need to be solved. Firstly, if historical data for a long period are considered, the influence of earlier historical data is the same as that of recent data, whereas recent data of a security is more important than the earlier historical data. Secondly, if the historical data of a security are not adequate, due to the lack of information, the estimations of the statistical parameters are not adequate. For these reasons, the expected return of a security is considered here as an uncertain variable instead of the crisp arithmetic mean of historical data. Similarly, in an uncertain environment, the risk (entropy) and skewness cannot be predicted exactly. Therefore, the entropy and skewness are also considered here as uncertain variables.
Let us consider the transaction cost c _{ i } to be a Vshaped function of the difference between a given portfolio ${x}^{0}=\left({x}_{1}^{0},{x}_{2}^{0},\dots ,{x}_{n}^{0}\right)$ and a new portfolio x = (x _{1}, x _{2},…, x _{ n } ) and is incorporated explicitly into the portfolio return. Thus, the transaction cost of i th risky asset can be expressed as
Hence the total transaction cost is
The expected return of portfolio x = (x _{1} , x _{2},…, x _{ n } ) with transaction cost is thus given by
The entropy of portfolio x = (x _{1} , x _{2},…, x _{ n } ) is given by
The skewness of portfolio x = (x _{ 1 } , x _{ 2 } ,…, x _{ n } ) is given by
We consider the portfolio selection problem as a triobjective optimization problem. As discussed earlier, the objectives we consider are
Construction of the constraints
For the portfolio x = (x _{1} , x _{2},…, x _{ n } ), the expected shortterm return is expressed as
For the portfolio x = (x _{ 1 } , x _{ 2 } ,…, x _{ n } ), the expected longterm return is expressed as
Since investors plan their asset allocation on shortterm, longterm, or both cases, they should prefer a portfolio having at least a minimum shortterm, longterm, or both types of return. For that reason, we consider the following two constraints:
where ς and τ will be allocated by the investor.
Dividend is the payment made by a company to its shareholders. It is the portion of corporate profits paid out to the investors. For the portfolio x = (x _{1}, x _{2},…, x _{ n } ), the annual dividend is expressed as
Clearly, investors would like to have a portfolio that yields them a high dividend. Keeping in mind this fact, we propose the following constraint:
where d will be allocated by the investor.
The wellknown capital budget constraint on the assets is presented by
The maximum and minimum fractions of the capital budget being allocated to each of the assets in the portfolio depend upon factors like price relative to the asset in comparison with the average of the price of all the assets in the chosen portfolio, minimal lot size that can be traded in the market, the past performance of the price of the asset, information available about the issuer of the asset, trends in the business of which it is a division, etc. That is, an investor will have to look upon a host of the basics affecting the commerce. Different investors having different views may allocate the same overall capital budget differently.
Let the maximum fraction of the capital that can be invested in a single asset i be M _{ i }. Then
Let the minimum fraction of the capital that can be invested in a single asset i be m _{ i }. Then
The investor would like to pick up the assets among all the assets in a given set that in his subjective estimate are likely to yield the greatest performance. Thus it is not necessary that all the assets in the given set may configure in the portfolio. Investors can thus consider the number of assets they can effectively handle in a portfolio.
Let the number of assets held in a portfolio be k. Then
As no short selling is considered, we have
If X is the set of feasible portfolios, then we have,
Weighted portfolio selection model formulation
The portfolio selection model is thus formulated as
To convert the above triobjective optimization problem into a preferencebased singleobjective optimization problem, let us consider three singleobjective optimization problems optimizing separately the three objectives of the model subject to the constraints of the problem. The optimum values as well as the values of the remaining objective functions in each of the three cases are calculated. Considering all the three problems, let the minimum values of the three objectives be Re^{min}, En^{min}, and Sk^{min}, respectively. Also, let the maximum values of the three objectives be Re^{max}, En^{max}, and Sk^{max}, respectively. Then the above triobjective portfolio selection model is transformed into the following model:
where w _{ 1 } , w _{ 2 } , and w _{ 3 } are weights or preferences to the objectives Re(x), En(x), and Sk(x), respectively. w _{ 1 } , w _{ 2 } , and w _{ 3 } will be allocated by the investor.
Genetic algorithm
After development of the genetic algorithm (GA) by Holland in 1975, it has been extensively used/modified to solve complex decision making problems in different fields of science and technology. A GA normally starts with a set of potential solutions (called initial population) of the decision making problem under consideration. Individual solutions are called chromosome. Crossover and mutation operations happen among the potential solutions to get a new set of solutions, and the process continues until terminating conditions are encountered. The following functions and values are adopted in the proposed GA to solve the problem [36]. The different parameters on which this GA depends are the number of generation (MAXGEN), population size (POPSIZE), probability of crossover (PCROS), and probability of mutation (PMUTE).
Chromosome representation
An important issue in applying a GA is to design an appropriate chromosome representation of solutions of the problem together with genetic operators. Traditional binary vectors used to represent the chromosome are not effective in many nonlinear problems. Since the proposed model is highly nonlinear, hence, to overcome the difficulty, a realnumber representation is used. In this representation, each chromosome V _{ i } is a string of n number of genes G _{ ij } (i = 1, 2,…, POPSIZE, j = 1, 2,…, n) where these n number of genes respectively denote n number of decision variables x _{ j }.
Initial population production
For each chromosome V _{ i }, every gene G _{ ij } is randomly generated between its boundary (LB _{ j } , UB _{ j }) where LB _{ j } and UB _{ j } are the lower and upper bounds of the variables x _{ j } (j = 1, 2,…, n and i = 1, 2,…, POPSIZE), respectively.
Evaluation
Evaluation function plays the same role in GA as that the environment plays in natural evolution. Now, evaluation function (EVAL) for chromosome V _{ i } is equivalent to the objective function f (x _{1} , x _{2},…, x _{ n } ). The following are the steps of evaluation:

1.
Find EVAL(V _{ i } ) = f (x _{1}, x _{2},…, x _{ n }), where the genes G _{ ij } represent the decision variable x _{ j } , j = 1, 2,…, n and f is the objective function.

2.
Find total fitness of the population: $F={\displaystyle \sum _{i\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}1}^{\mathrm{POPSIZE}}\mathrm{EVAL}\left({V}_{i}\right)}.$

3.
The probability p _{ i } of selection for each chromosome V _{ i } is determined by the formula ${p}_{i}=\frac{1}{F}\mathrm{EVAL}\left({V}_{i}\right).$

4.
Calculate the cumulative probability Y _{ i } of selection for each chromosome V _{ i } by the formula ${Y}_{i}={\displaystyle \sum _{j\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}1}^{i}{p}_{i}}.$
Selection
The selection scheme in GA determines which solutions in the current population are to be selected for recombination. Many selection schemes, such as stochastic random sampling roulette wheel selection, have been proposed for various problems. In this paper, we adopt the roulette wheel selection process. This roulette wheel selection process is based on spinning the roulette wheel POPSIZE times each time we select a single chromosome for the new population in the following way:

(a)
Generate a random (float) number r between 0 and 1.

(b)
If r < Y _{ 1 } , then the first chromosome is V _{1} ; otherwise, select the i th chromosome V _{i} (2 ≤ i ≤ POPSIZE) such that Y _{ i − 1} ≤ r < Y _{ i }.
Crossover
A crossover operator is mainly responsible for the search of new strings. Crossover operates on two parent solutions at a time and generates offspring solutions by recombining both parent solution features. After selection of chromosomes for new population, the crossover operator is applied. Here, the arithmetic crossover operation is used. It is defined as a linear combination of two consecutive selected chromosomes V _{ m } and V _{ n } , and resulting offspring's ${V}_{m}^{\text{'}}$ and ${V}_{n}^{\text{'}}$ are calculated as
where c is a random number between 0 and 1.
Mutation
A mutation operator is used to prevent the search process from converging to local optima rapidly. It is applied to each single chromosome V _{ i }. The selection of a chromosome for mutation is performed in the following way:

1.
Set i ← 1.

2.
Generate a random number u from the range [0, 1].

3.
If u < PMUTE, then we select the chromosome V _{ i }.

4.
Set i ← i + 1.

5.
If i ≤ POPSIZE, then go to step 2. Then the particular gene G _{ ij } of the chromosome V _{ i } selected by the abovementioned steps is randomly selected. In this problem, the mutation is defined as ${G}_{\mathit{ij}}^{\mathrm{mut}}$ random number from the range (LB_{ j }, UB_{ j }).
Termination
If the number of iteration is less than or equal to MAXGEN, then the process goes on; otherwise, it terminates.
Proposed GA procedure
Start
{
t ← 0
while (all constraints are not satisfied)
{
initialize Population (t)
}
evaluate Population (t)
while(not terminate  condition)
{
t ← t + 1
select Population (t) from Population (t − 1)
crossover and mutate Population (t)
evaluate Population (t)
}
print optimum result
}.
Case study: Bombay Stock Exchange (BSE)
Bombay Stock Exchange is the oldest stock exchange in Asia with a rich heritage of over 133 years of existence. What is now popularly known as BSE was established as ‘The Native Share & Stock Brokers' Association’ in 1875. It is the first stock exchange in India which obtained permanent recognition (in 1956) from the Government of India under the Securities Contracts (Regulation) Act (SCRA) 1956. With demutualization, the stock exchange has two of world's prominent exchanges, Deutsche Borse and Singapore Exchange, as its strategic partners. Today, BSE is the world's number one exchange in terms of the number of listed companies and the world's fifth in handling of transactions through its electronic trading system. The companies listed on BSE command a total market capitalization of US$1.06 trillion as of July 2009.
The BSE index, SENSEX, is India's first and most popular stock market benchmark index. SENSEX is tracked worldwide. It constitutes 30 stocks representing 12 major sectors. It is constructed on a ‘freefloat’ methodology, and is sensitive to market movements and market realities. Apart from SENSEX, BSE offers 23 indices, including 13 sectoral indices.
Case study
We have taken monthly share price data for 60 months (March 2003 to February 2008) of just five companies which are included in the BSE index. Though any finite number of stocks can be considered, we have taken only five stocks to reduce the complexity of representation.
The Table 1 shows the stocks along with their returns in the form of trapezoidal uncertain numbers, the average shortterm returns, the average longterm returns, and the dividends. We also have k _{ i } = 0.001. We consider, ${x}_{i}^{0}$ = 0 for i = 1, 2, 3, 4, 5.
Example
With respect to the above data, we consider the following triobjective portfolio selection model:
Solution
To solve the above example, the GA is used with the parameters POPSIZE = 50, PCROS = 0.2, PMUTE = 0.2, and MAXGEN = 100. A realnumber presentation is used here. In this representation, each chromosome x is a string of m (here, m = 5) number of genes; these represent decision variables. For each chromosome x, every gene (here, x _{ 1 } , x _{ 2 } , x _{ 3 } , x _{ 4 } , x _{ 5 }) is randomly generated between its boundaries until it is feasible. In this problem, arithmetic crossover and random mutation are applied to generate new offsprings.
As discussed in the ‘Weighted portfolio selection model formulation’ section, optimizing the three single objectives Re(x), En(x), and Sk(x) separately subject to the constraints in (4), we obtain the minimum and maximum values of the objectives with the same parameters. In each case, only the best solution is considered.
With reference to model (3), the problem (4) is transformed into the following model:
For different preassigned values of w _{1} , w _{2} , and w _{3} , the above problem is solved. We have considered only the best solutions. The solutions obtained are shown in Table 2.
In case 1, where an investor gives same importance to all the three objectives, the portfolio states that the investor should invest 45%, 45%, and 10% of the money to the second, third, and fourth stocks, respectively. In case 3, where the importance is given towards minimization of risk, the investor should invest 39.6%, 18.8%, and 41.6% of the total money to the second, fourth, and fifth stocks, respectively. Similarly, we can explain the other two cases.
In case 2, where more importance is given to return, the investor gets a return of 0.03957813 which is higher than that of the other three cases {0.03893750, 0.03586275, and 0.03830000}. In case 3, where more importance is given to risk, the investors' risk (0.09320499) is lower than in all other cases {0.09551172, 0.09846223, and 0.09487761}. Similarly, in case 4, we get the best result for skewness. In case 1, where equal importance is given to all objectives, the outputs are intermediate. We represent the portfolios obtained in cases 1, 2, 3, and 4 graphically in Figure 1.
Some questions may arise on the appropriateness of the portfolios obtained in Table 2 under different circumstances (cases 1, 2, 3 and 4). For example, question may arise on the absence of reliance energy in all the obtained portfolios. To explain that, the individual mean, entropy and skewness of the stocks are calculated by Example 2 and are shown in Table 3. It is seen that reliance energy has the lowest rerun among the five stocks. It is also possessing negative skewness. So, the absences of reliance energy on the portfolios in cases 1, 2 and 4 are obvious. In case 3, where more importance is given to entropy, the selected portfolio contains L&T, Tata steel and Bhel. Tata steel and Bhel are the two stocks with lowest risks. Again, though L&T has a higher risk, it also has very high return. So, the portfolio in case 3 is not compromising too much towards entropy and is maintaining the characteristic of multiobjective optimization. This is also to note that if the constraint x _{ i } ≥ 0.1y _{ i } is not considered, then some of the portfolios would contain nonzero x _{1} .
Comparative study
We compare the results in Table 2 with other relevant literature to demonstrate how the results from the proposed technique compare with the literatures of uncertainty theory in the portfolio selection problem. Thus, the models in [33], [37], and [38] which apply uncertainty theory in portfolio selection are considered with the same data set as that in Table 1. We also used the following set of constraints (X) for each case:
We also used the following set of constraints (X) for each case:
Model of Yan
We considered the following model [33]:
Here E stands for mean (return) and V stands for variance (risk). The solution is shown in Table 4.
Model of Ning et al
We considered the following model [37]:
Here E stands for mean (return) and TvaR stands for tail value at risk. The solution is shown in Table 5.
Model of Liu and Qin
We considered the following model [38]:
Here E stands for mean (return) and SAD stands for semiabsolute deviation (risk). The solution is shown in Table 6.
In the discussions done in the first and second sections, we see that using entropy as a measure of risk/uncertainty is analytically better than the other conventional measures. Again, if we compare Tables 4, 5, and 6 with Table 2, we see that the performance of the proposed model is clearly at par or better than the established models.
Conclusions
This paper has introduced a new framework of meanentropyskewness portfolio selection problem with transaction cost under the constrains on shortand longterm returns with transaction costs, dividends, number of assets in the portfolio, and the maximum and minimum allowable capital invested in stocks. Uncertainties of future return of stocks are characterized by uncertain variables. The efficiency of the portfolios is evaluated by looking for risk contraction on one hand and expected return and skewness augmentation on the other hand. An empirical application has served to illustrate the computational tractability of the approach and the effectiveness of the proposed algorithm. A comparative study with other relevant literatures proves the usefulness of the proposed model. In addition to the GA, some other metaheuristic algorithms such as tabu search, simulated annealing, ant colony optimization, and particle swam optimization may be employed to solve the nonlinear programming problem.
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Keywords
 Uncertainty modeling
 Meanentropyskewness portfolio selection model
 Uncertain variables
 Trapezoidal uncertain variable
 Genetic algorithm