Mean-Variance-CvaR Model of Multiportfolio Optimization via Linear Weighted Sum Method

We propose a new approach to optimizing portfolios to mean-variance-CVaR (MVC) model. Although of several researches have studied the optimal MVCmodel of portfolio, the linear weighted summethod (LWSM) was not implemented in the area. The aim of this paper is to investigate the optimal portfolio model based on MVC via LWSM. With this method, the solution of the MVC model of portfolio as the multiobjective problem is presented. In data analysis section, this approach in investing on two assets is investigated. AnMVCmodel of the multiportfolio was implemented inMATLAB and tested on the presented problem. It is shown that, by using three objective functions, it helps the investors to manage their portfolio better and thereby minimize the risk and maximize the return of the portfolio. The main goal of this study is to modify the current models and simplify it by using LWSM to obtain better results.


Introduction
In financial activity, a portfolio is a set of assets that allows somebody to choose from several choices in investing.The main problem in this area is to find out the optimal combination to distribute a given fund on a set of existing assets.Maximization of expected return and minimization of risk are two main aims of this problem [1].
Our goal is to develop the MVC model for portfolio with the weighted coefficients in which this approach is adaptive of the weighted sum method (WSM).One of the aims is to offer a practical method for the construction and optimize the proposed model, in order to produce the Pareto optimal solution.There are several researches that have been done on the optimal mean-variance-CVaR (MVC) model of portfolio but no much studies have been carried out in the area by using the linear weighted sum method (LWSM).This approach will give better insight into the multiportfolio process and the position of investing via finding the feasible solution set.
The paper is organized into five sections.In the next section, this paper presents the properties of multiobjective portfolio optimization based on mathematical approach and MVC model of portfolio optimization.Then, MVC model of portfolio based on LWSM will be presented.After that, the empirical example and some challenges will be shown to clarify the discussion.For this aim, an MVC model of the multiportfolio was implemented in MATLAB and tested on the presented problem.Finally, results and some future works are presented in the conclusion.

MVC Model of Portfolio
We face making many decisions in the real world.There are numerous methods to optimize them.The aim of this paper is to find the Pareto optimal solutions.However, in the theoretical problem cases, if Pareto set normally cannot be solved by an algorithm, then we try to approximate the Pareto sets [2].
For example, the multiobjective model for five objective functions  1 , . . .,  5 and some constraints can be shown by following model [3]: For  = 1, . . ., . ( MVC uses the three parameters (the expected value (), the variance ( 2 ), and the CVaR (conditional value at risk) at a specified confidence level  ∈ (0, 1)) for better modeling.The aim of the proposed model is to give a developed method for the solution [4].
We can obtain a better model of portfolio by replacing the three indexes instead of two parameters as usual so that it increases the efficiency of the model [5].Several studies on portfolio optimizations via mathematical approaches have been investigated in the literature.To start the construction of this model, we consider some variables, terms, and objective functions as the following nomenclature [6].
Let  be the number of assets that are available;   = return (as a random variable) depending on a decision vector  that belongs to a feasible set ;   = amount of investing in ith asset; Ω = the feasible set of solutions or search space;  = a solution of problem;   the expected mean of the th asset; 2 (  ) = the variance belong to   ;  the number of assets to invest ( ≤ ).
We consider the MVC model based on Aboulaich et al. method with following formula [4]: We note that, for random variable, the value of variance is calculated by the following formula: To calculate the variance, let   =  1  1 + ⋅ ⋅ ⋅ +     , so we have where   = cov(  ,   ).In addition, in the model above, preferring the random variable   than random variable   is as the following conditions: in which CVaR is calculated according to following theorem.
Proposition 1 (CVaR calculation and optimization).Let   be a random variable depending on a decision vector  that belongs to a feasible set  and  ∈ (0, 1).Consider the function where [] + =  for  ≥ 0 and [] + = 0 for  < 0.Then, according to a function of V,   is finite and continuous (hence convex) and In addition, the set consisting of the values of V for which the minimum is attained, denoted by   (), is a nonempty, closed, and bounded interval.Minimizing   with respect to  ∈  is equivalent to minimizing   with respect to (, V) ∈ : In addition, a pair ( * , V * ) minimizes the right-hand side if and only if  * minimize the left-hand side and V * ∈   ( * ).  (  ) is convex with respect to  and   (, V) is convex with respect to (, V).
Thus, if the set  of feasible decision vectors is convex (which is the case for the basic version of the portfolio selection problem), and even if we impose a further lower limit on the expected return, minimizing CVaR is a convex optimization problem [7].
On the other expression, MVC model denotes as the following: where variables = ( 1 , . . .,   ) ∈  ⊂   , V ∈ , and set of feasible decision vectors denotes via  that is a convex set.In the model above, there is one multiobjective system for portfolio optimization that consists of three objective functions.The first objective function is to minimize the conditional value at risk for portfolio optimization.The second objective function denotes minimizing the expected value of return of the portfolio.And the last objective function considers minimizing the variance of returns.This procedure shows that we can manage our capital to invest in several assets [5].

MVC Model of Portfolio Based on LWSM
This method was introduced by Zadeh in 1963 and it is one of the main ways of solving the MO (see [8]).Three types of this method are used for multiobjective portfolio optimization (MPO) in previous research.They are weighted sum method, weight quadratic method, and weight quadratic variation.One of the main ways of weighting methods is the weighted sum method.The aim of weighting method is the optimization of the objective functions that they arranged by linear combination (weighted sum).Different efficient solutions can be found by changing the weights of the objective functions [9].
The weighted sum method changes the MO problem with a single model of mathematical optimization problem.In the model of this method sum weighting coefficient   multiply each objective function   to make the structure of the objective function as follows: (note that normalization of those coefficients is not necessary) With convexity supposition, if   > 0, ∀ = 1, . . ., , then the solution of the above system is Pareto optimal.This means that if that system is convex then any Pareto optimal solution can be found [10].There are three criteria to measure the weight.They are subjective preference of the decision makers, the variance measure, and the independence of criteria.Usually two methods can be used to achieve this aim: the equal weights and the rank-order weights [11,12].
Example 2 (see [13]).Consider that a multiobjective problem includes three criteria, which are denoted by the same unit, and two alternatives.According to Table 1 let  1 = 0.10,  2 = 0.05, and  3 = 0.30.The values of   are assumed to be as follows: Therefore, the decision matrix for this MCDM problem is as follows.
So the optimal case in the minimization situation is alternative A 1 because it has the lowest WSM score.We can write the alternative ranking as follows: A 1 < A 1 (where "<" denotes "better than").Now, we formulate the portfolio revision problem with transaction costs as a standard mathematical optimization problem with the mean-risk framework.We will employ the following notation.
: number of risky assets : vector with all entries equal to ones.
Consider the optimization problem with the normalized single objective as follows: where the weights are denoted by   , and And the objective function of th objective function   () after normalization is denoted by  0  (),  = 1, .Also we call   (),  () inequality and equality constraints, respectively.With LWSM approach, we have Therefore, normalized objective functions have the following form: in which case the floating-point values   are evaluated in the following way: Obviously, in many practical problems, the objective functions are represented by various measure units (e.g., if  1 is measured in kilos,  2 in seconds, etc.).For this reason the objective function normalization is required.It is now obvious that the coefficients have values of the segment [0, 1].Denote that we now have the linear programming problem Maximize: The following theorem gives the practical criteria for the detection of some Pareto optimal solutions of problem (2.4).Yu et al. [14] presented mean-CVaR model of portfolio optimization based on LWSM in which they assumed that the return of the portfolio is based on multi-t-distribution and they obtained the model for as follows: There are some conditions for coefficients that are used in WSM.For example, the sum of them equal 1 and they are strictly positive.In addition to LWSM they must be normalized.With this idea, the multiportfolio optimization problem is as follows: where  1 ,  2 , and  3 are strictly positive and  is as the above assumptions.Since all objective functions on MVC model are convex, we can define the model above to find the solution as the Pareto optimal set [15].For instance, if we consider  1 = 1,  2 = 0, and  3 = 0 the problem turns to minimize the variance of the portfolio [5].Now, according to (P2) as the multiobjective optimization problem, LWSM will be used to solve it as in the following procedure.
Step 1. Construct the problem based on min ( ()) = min ( 1  2 () −  2  () +  3 CVaR) .(24) There are several positions of investing for investor who can manage his/her budget which is related to the amount of  1 ,  2 , and  3 .This note creates several cases for investors to increase the return or decrease the risk of a portfolio.The risk (or return) has an inverse (direct) relation with those coefficients.The LWSM leads to following procedure for MVC model of portfolio.
Step 2. Consider min 1 (  ) −  2 (  ) where ,  are the belief degree and specific parameter, respectively.In this paper, it is considered that  = 0.5.In addition, according to ∑  =1   = 1 we have 1−( 1 +  2 ) =  3 ; we rewrite problem of (18) as follows: where the belief degree of  set is equal to 0.95.In addition, in CVaR formula after linearization [16,17] we have and (,   ) is the loss function which can be defined [16] by (,   ) = −    .We can rewrite (19) according to the assumptions above as follows: where   is return of th asset.In the model above, we consider  = 2 and we have (29)

Empirical Research and Challenges
In order to evaluate the model above to find the portfolio optimization based on LWSM we consider two assets whose mean return  and covariance matrix  are as the following data (data from [14,16]):  = ( 0.06760987 0.034191 ) ,  = ( 0.058884101 0.000464043 0.000464043 0.0003643663 ) . (30) According to the data above and the model of ( 29) we have where Now, we consider this proposed model for  1 = 0.25,  2 = 0.25.Therefore, we have (33) In order to compare the performance of mean-variance model and MVC model of multiportfolio optimization, the minimum functions are demonstrated via MATLABsoftware, as Figures 1 and 2 show the results of LWSM method for two models, respectively.Comparative performance was measured using a MAT-LAB software, and performance was shown in Figures 1 and  2 which are called the mean-variance and MVC model of multiportfolio, respectively.
In summary, we compare the performance of MVC model of multiportfolio with LWSM technique and other models like mean-variance approaches.The mean-variance minimization function and the MVC model are as described in Figures 1 and 2, respectively.
Table 2 is arranged to illustrate the results of MVC model of portfolio and refer to two assets by Matlab by this approach.
As shown in the above results, we tested two assets via proposed model and we presented the results of that as Table 2 which included 20 results from our proposed model by MATLAB software.It showed, except for two obvious cases ( 1 = 0 and  2 = 1;  1 = 1 and  2 = 0), that the minimum situation of MVC model of multiportfolio will occurr on  1 = 0.01 and  2 = 0.99.The results illustrate that the proposed method is better for investing a small amount of assets such as the above two assets.Despite of the several advantages, this method has some disadvantages which one of them is the weights  1 ,  2 , and  3 that are hard to understand [5].The other one is how to use the Simplex method to solve the weighted sum problem.The solution of this problem is using other techniques (e.g., interior-point methods) [10].

Conclusion
In this investigation, we consider three objective functions to model portfolio optimization with LWSM.In addition, we propose a linear weighted sum method to solve the MCV model of portfolio optimization.Also, with empirical example and MATLAB software, we evaluated proposed model.Our example includes two assets for investing.However, it is flexible to choose more assets and find the Pareto optimal according to procedures that are used in this paper.
The contribution of this study is a presentation of MVC model of portfolio optimization based on LWSM.The empirical examples have shown that using the three objective functions to make a powerful multi-portfolio model.We have described the MVC model of multiportfolio optimization that it is extended of the model presented by Yu et al. [14].In addition, we showed that the MVC model generally covers both mean-variance and mean-CVaR models of portfolio optimization (with changing the coefficients of   ,  = 1, 3).
Furthermore, this paper has given new insight into the multiobjective decision making process via WSM.A hybrid of this method and other techniques such as interior-point methods can be a good idea for future researchers.
To sum up, with using three objective functions, it helps the investors to manage their portfolio better and thereby minimize the risk and maximize the return of the portfolio.Also, using LWSM dues to modify the current models and simplify it to obtain better results.The same approach can also be used for other multiobjective optimization model and investing applications.

Table 1 :
Situation of two assets.

Table 2 :
Results of MVC model of portfolio refer to two assets (see Section 4) by MATLAB.