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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.17 No.4 pp.632-641
DOI : https://doi.org/10.7232/iems.2018.17.4.632

# Development of Robust Random Variable for Portfolio Selection Problem

Alireza Ghahtarani, Majid Sheikhmohammady*, Amir Abbas Najafi
Faculty of Industrial and systems Engineering, Tarbiat Modares University, Tehran Province, Tehran
Department of Industrial Engineering, K.N. Toosi University of Technology, Tehran
* Corresponding Author, E-mail: msheikhm@modares.ac.ir
November 28, 2017 July 20, 2018 September 17, 2018

## ABSTRACT

In this paper, mathematical modeling is developed for portfolio selection problem under uncertainty circumstances with regard to a robust stochastic variable. Two popular and common approaches in the area of modeling uncertainty are robust optimization and stochastic programming. These two methods are used with different considerations in mathematical modeling, but each one has a limitation. Stochastic programming assumes a static distribution function with static parameters over time for non-deterministic data, and robust optimization considers an indeterminate parameter in a uniform interval around nominal values. Using combination of these two methods can help us to eliminate their drawbacks. For this purpose, the concept of a robust stochastic variable has been developed in this research. This variable enables distribution of the uncertainty parameter to vary over time, and its mean, varies from one period to another; in fact, the parameter of the mean of uncertain probable distribution. The risk measure of CVaR, which allows changes in mean of uncertainty from time to time, is used to implement the proposed approach. As a numerical example, the actual data of Tehran Stock Exchange is used for a year as one-month periods. The practical results of this research show that the developed method can properly overcome the shortcomings of the previous methods.

## 1. INTRODUCTION

Investors have always looked for ways to enhance one’s regular income through good investing. There are two main criteria which must be considered in investing:

• 1) Maximum possible return

• 2) The measure of stability

Portfolio selection aims to find the best portfolio which would achieve the best result based on the criteria that mentioned above.

Markowitz, 1952 was the pioneer researcher that proposed a mathematical model for portfolio selection problem. He proposed the best portfolio that maximizes expected return and minimizes variance as risk measure.

Markowitz, 1959, extended his model with objective of maximizing expected return (mean) and minimizing a risk measure. The general constraint for this problem is that the fractional amount invested in each of the securities should add up to one with the additional requirement that each of the fractions is non-negative. As mentioned above, the portfolio selections problem is a bi-criteria mathematical model.

Many authors have proposed different risk measures to quantify risk of portfolio. Konno and Yamazaki, 2017, proposed mean absolute deviation as a risk measure. Rockafellar and Uryasev, 2000 modeled conditional value at risk in a linear programing. Other developments on risk measures and portfolio selection problem include (Mansini et al., 2003;Michalowski and Ogryczak, 2001;Kellerer et al., 2000;Chiodi et al., 2003;Papahristodoulou and Dotzauer, 2004;Nielsen, 2018;Olszewski and Vohra, 2014).

In classical mathematical programming, parameters are exact and there is no assumption about uncertainty of parameters. However, uncertainty of parameters exists in real word problems. Stochastic programming is a method that considers parameter uncertainty in mathematical model. Stochastic programming models have been proposed by (Dantzig, 1955;Charnes and Cooper, 1959).

In stochastic programming, parameters are random with a given probability distribution. This method is widely used by researchers in portfolio selection problem. For more details we refer to (Abdelaziz et al., 2007;Abdelaziz and Masri, 2010;Bertocchi et al., 2006;Dai et al., 2014;Şakar and Köksalan, 2013).

Another approach to consider uncertainty of parameters is robust optimization. The first approach to this method was proposed by Soyster, 1973. The results of Soyster’s model are too conservative in the sense that we give up too much of optimality for the nominal problem in order to ensure robustness. The second approach of robust optimization was proposed by Ben-Tal and Nemirovski (Ben-Tal and Nemirovski, 2000).

Their attitude is less conservative than soyster’s approach, but more complicated. Another development in robust optimization was made by Bertsimas and Sim (Bertsimas and Sim, 2004). Their approach is conservative controllable. The robust counterpart of linear programming remains linear programming.

We now refer some practical models which employed robust optimization for application in finance. Ghaoui et al. (2003) proposed a robust portfolio model under uncertainty of covariance matrix witch is developed by semi-definite programming (SDP) (Ghaoui et al., 2003). This model considers worst case value-at-risk. Tütüncü and Koenig (2004) developed a robust portfolio optimization problem formulated in a quadratic program (QP). Kawas and Theile (2011) constructed a log robust portfolio model to consider the heavy tailed property of stock prices (Kawas and Thiele, 2011). Moon and Yao (2011) developed a robust mean absolute deviation model for portfolio optimization (Moon and Yao, 2011). Quaranta and Zaffaroni (2008) extended a robust optimization of conditional value at risk (Quaranta and Zaffaroni, 2008). For other works we refer to (Chen and Tan, 2009;Ling and Xu, 2012;Ghahtarani and Najafi, 2013)

This paper reports on the development of a robust optimization in a chance constraint portfolio selection problem. This research uses CVaR as a risk measure and develops Mean-CVaR by chance constraint method. Here, normal distribution is assumed for expected return. But the mean of normal distribution is considered uncertain. We use robust optimization to consider this uncertainty. The major contribution of this research is to consider uncertainty of parameters by employing robust optimization and stochastic programming.

This paper is structured as follows: the second section explains the novelty and characteristic of problem the third section explians mean-CVaR portfolio selection problem. The fourth section proposes random robust variable for Mean-CVaR portfolio selection problem. In the fifth section, numerical results are examined; we finally present the concluding remark.

## 2. NOVELTY AND CHARACTERISTICS OF THE PROBLEM

The main innovation of this research is development of the random robust variable to be employed in the portfolio selection problem.

In the chance constraint approach in within the stochastic programming, an attempt has been made to minimize the probability of a violation of a constraint. On the other hand, for this purpose, for the technical coefficients of a model, the probability distribution function is usually assumed to be normal distribution. In the utilized normal distribution, which is in fact a hypothesis for the model coefficients, the mean nominal value is considered, and its value is assumed to be certain. Both the distribution function is considered as technological coefficients and the mean of this distribution is assumed to be nominal, and of course, its value is assumed to be a static value, while in the real world, both distribution and the mean value in the determined time interval changes as the circumstances varies.

In the robust optimization approach for uncertain parameter, especially in the Bertsimas and Sim approach, a uniform interval is considered around nominal mean, while in the real world, the distribution is not often uniform. On the other hand a nominal amount is assumed to be certain. Robust optimization focuses on a particular parameter.

We use two above mentioned approaches together. In fact, with utilization of chance constraint approach, the proposed model can be developed. In this case, the mean of uncertain variable is considered as given and nominal. Then, for the purpose of considering the uncertainty for this variable, the robust optimization approach is used to develop the uncertainty of the parameter mean in a uniform interval.

This innovation may cause the development of mathematical models to be closer to the real world under more uncertain circumstances.

Moreover this initiative is useful in models that, parameters face numerous changes over time and the distribution mean of their technical coefficient change a lot over time. Also, the proposed model would be useful for models that have different time periods and the model parameters’ circumstances vary greatly from one time period to another one.

This initiative applies to all math models, but it is undoubtedly more applicable to portfolio selection models, because of the uncertainty and the periodicity of these models. In this research, a well-known and widely used risk measure for portfolio selection, the conditional value at risk measure is employed. The reason for choosing this measure is that, for a periodic basis (for example, monthly periods), the mean of losses beyond assurance is considered as a risk. For example, if the periods are monthly, for each month, if the mean of losses are more than the acceptable loss, they are calculated as a risk. But, due to turbulence and fluctuations in financial markets, in any period i.e. a month, the mean of return distribution can be different. As a result, there is a new distribution with a new average parameter during each period when the risk is calculated. In such situations, if chance constraint approach is used for modeling, for all periods of time, a distribution with a parameter is used. As explained, it is not a true assumption as the main parameters of this distribution change over time.

However, if the initiative presented in this study is used, it is possible to consider uncertainty in the random distribution parameter and mote the possibility of change over time. In this case, the mentioned problem is removed

The proposed concept developed in this study is the robust random variable that can be used in a variety of mathematical models with the introduced characteristics. This approach is used for all financial risk measures except for CVaR. The reason for choosing this measure is just the vast utilization of this measure in today's world. Its expansion in uncertainty conditions can increase its efficiency.

Therefore, the CVaR model is first introduced in this paper. The reason for this introduction is to specify the characteristics of this model, especially the calculation of the periodical deviations from determined reliability level, and the characteristic, which is well-versatile with the concept of a random robust variable. In the next stage, the CVaR is developed through utilization of stochastic variable and chance restraints. But, as mentioned before, such consideration of uncertainty for these circumstances of the model which changes monthly is deficient. Therefore, the robust optimization approach is used in the next stage. Before implementing the robust optimization in parameter of the expected return’s mean, the convex proof of the model is presented. This proof is not necessary for other models, in other words without the model being convex, the introduced variable can be used to develop the model. However, the convexity of the model increases the calculation merits. At last, the developed model is solved in the real world market.

The proposed model in this study is a fundamental research as in the utilized modeling, the convex mathematical programming is used, while, the characteristics of the used model provide the possibility of using real data from Tehran’s stock exchange to solve the model. For this purpose, one needs to calculate each share’s return. On the other hand, LINGO software is used to solve the problem. The used data is related to 10 random shares in 12 monthly periods.

## 3. MEAN-CVaR PORTFOLIO SELECTION

We now introduce mean-CVaR portfolio selection problem. Rockafellar and Ursayev (Rockafellar and Uryasev, 2000) established this risk measure. CVaR quantifies expected loss corresponding to certain confidence level.

Let f(x) is the loss function of the portfolio. Given a confidence level B, CVaR is the expected value of all (1−B)% losses:(1)

$C V a R α ( X , η ) = η + ( 1 − B ) − 1 ∫ ​ ε ∈ R n [ f ( x , ε ) − η ] + p ( ε ) d ε$
(1)

$η − V a R ε − Random Variable Z + = max { z , o }$
(2)

Rockafellar and ursayev reformulate CVaR as a linear programming. Since rX denotes the portfolio return matrix, –rX represents losses. The mean-CVaR portfolio selection problem is as follow:(3)

$min η + 1 ( 1 − B ) S ∑ i = 1 S ( y i )$
(3)

S.t(4)(5)(6)(7)

$y i ≥ ∑ [ ( − r i j X j ) − η ] : i = 1 , 2 , ... , 5$
(4)

$y i ≥ 0 i = 1 , 2 , ... , 5$
(5)

$X ′ μ = E 0$
(6)

$∑ i = 1 n X i = 1$
(7)

$X ≥ 0$
(8)

The formulation (1) to (8) is a linear programming that can be solved efficiently.

## 4. ROBUST RANDOM VARIABLE FOR PORTFOLIO SELECTION PROBLEM

### 4.1. Chance Constrained Mean-CVaR Portfolio Selection

Chance constraint is an approach within stochastic programming paradigm developed by Charnes and Cooper (Charnes and Cooper, 1959). Portfolio selection under chance constraint aims to minimize risk measure under the condition that the probability of a portfolio’s rate of return is greater than the expected rate of return be equal or greater than a confidence level. For n securities and their mean return ε as a random variable, the chance constraint for expected return E. is as follow:(9)

$Pr { X ' ε ≥ E 0 } ≥ α C$
(9)

αC is confidence level. We assume that ε has a normal distribution N(μ, C), and C is positive definite symmetric matrix:

If we assume ε ~ (μ , C) then $X ′ ε = ∑ i = 1 n X i ε i ~ N ( E ( x ) , σ ( x ) )$

where $E ( x ) = X ′ μ$ and $σ ( x ) = X ′ C X$

U is a standard normal if we have:

$u = ∑ i = 1 n X i ε i − E ( x ) σ ( x )$

The chance constraint is as follows:(10)(11)(12)(13)(14)(15)(16)

$Pr { X ′ ε ≥ E 0 } ≥ α C$
(10)

$Pr { X ′ ε − E ( x ) σ ( x ) ≥ E 0 − E ( x ) σ ( x ) } ≥ α C$
(11)

$Pr { u ≥ E 0 − E ( x ) σ ( x ) } ≥ α C$
(12)

$Pr { u ≤ E ( x ) − E 0 σ ( x ) } ≥ α C$
(13)

$φ ( E ( x ) − E 0 σ ( x ) ) ≥ α C$
(14)

$E ( x ) − E 0 σ ( x ) ≥ φ − 1 ( α C )$
(15)

$⇒ − E ( x ) + E 0 + σ ( x ) φ − 1 ( α C ) ≤ 0$
(16)

We call (17) to (19) constraint set A:(18)

$∑ i = 1 n X i = 1$
(17)

$− X ′ μ + E 0 + X ′ C X φ − 1 ( α C ) ≤ 0$
(18)

$X ≥ 0$
(19)

The formulation (17) to (19) is convex. The convexity of the above set is prooved by Tang et al (Tang et al., 2001)(20)(21)(22)

$D = { X | − E ( x ) + E 0 + σ ( x ) φ − 1 ( α C ) ≤ 0 }$
(20)

$− E ( x ) + E 0 + σ ( x ) φ − 1 ( α C ) ≤ 0$
(21)

$⇒ E ( x ) − E 0 ≥ σ ( x ) φ − 1 ( x )$
(22)

E(x) is a linear function and for any λ∈(01) and any X1, X2D we have:(23)(24)(25)(26)

$E ( λ X 1 ) − λ E 0 ≥ λ σ ( X 1 ) φ − 1 ( α C )$
(23)

$E ( ( 1 − λ ) X 2 ) − ( 1 − λ ) E 0 ≥ ( ( 1 − λ ) σ ( X C ) ) φ − 1 ( α C )$
(24)

$E ( λ X 1 + ( 1 − λ ) X 2 ) = λ E ( X 1 ) + ( 1 − λ ) E ( X C )$
(25)

$E ( λ X 1 + ( 1 − λ ) X 2 ) − E 0 = λ E ( X 1 ) + ( 1 − λ ) E ( X 2 ) − E 0 ≥ λ σ ( x 1 ) φ − 1 ( α C ) + ( ( 1 − λ ) σ ( x C ) ) φ − 1 ( α C ) = ( λ σ ( X 1 ) + ( 1 − λ ) σ ( X 2 ) ) φ − 1 ( α C )$
(26)

σ(x) is a strict convex function on D, for λ∈(01) :(27)

$λ σ ( x 1 ) + ( 1 − λ ) σ ( X C ) ≥ σ ( λ X 1 + ( 1 − λ ) X C )$
(27)

So we have:(28)(29)

$E ( λ X 1 + ( 1 − λ ) X C ) − E 0 ≥ ( λ σ ( X 1 + ( 1 − λ ) σ X 2 ) ) φ − 1 ( α C ) ≥$
(28)

$≥ ( σ ( λ x 1 + ( 1 − λ ) X 2 ) φ − 1 ( X c ) ∈ D$
(29)

D is a convex set, A is also convex set. Based on the above formulation, the mean-CVaR chance constraint is as follows:(31)(32)(33)(34)

$min η + 1 ( 1 − B ) S ∑ i = 1 S ( y i )$
(30)

$y i ≥ ∑ j = 1 n [ ( − r i j X j ) − η ] : i = 1 , 2 , ... , 5$
(31)

$y i ≥ 0$
(32)

$− X ′ μ + E 0 + X ′ C X φ − 1 ( α C ) ≤ 0$
(33)

$∑ j = 1 n X j = 1$
(34)

$X ≥ 0$
(35)

The formulation (30) to (35) is mean -CVaR chance constraint. The main assumption for above formulation is normal distribution for rate of return. Later we relax this condition and assume that μ is uncertain. Then, we use robust optimization to consider this uncertainty.

### 4.2. Robust Random Variable

Now, we develop a robust optimization for chance constrained model to consider uncertainty. We use the Bertsimas and Sim’s approach.

Consider the chance constraint assumed as nominal problem. Let J be the set of coefficients aj and jJ are uncertain coefficients. $a ˜ j , j ∈ J$ takes values according to $[ a j − a ^ j , a j + a ^ j ]$, this interval has symmetric distribution. This approach introduces Γ, not necessarily integer that takes values in $[ 0 , | j | ]$. Γ is the price of robustness that adjusts the robustness against the level of conservatism.

In order to reformulate nominal problem to robust formulation, we have to consider the formulations (36) to (43).

$min η + 1 ( 1 − B ) S ∑ i = 1 S ( y i )$
(36)

$y i ≥ ∑ j = 1 n [ ( − r i j X j ) − η ] ∀ i$
(37)

$y i ≥ 0$
(38)

$− X ′ μ + E 0 + X ′ C X φ − 1 ( α C ) + max { ∑ j ∈ s μ ^ j w j + ( Γ − [ Γ ] μ ^ t w t } ≤ 0 { S ∪ { t } | S ≤ j , | S | = [ Γ ] , t ∈ J \ S }$
(39)

$∑ j = 1 n X j = 1$
(40)

$X ≥ 0$
(41)

$− W j ≤ X j ≤ W j$
(42)

$W ≥ 0$
(43)

If Γ is chosen integer, then:

$B ( x , Γ ) = max { ∑ j ∈ s μ ^ j | x j | } { s | s ⊆ j , | s | = Γ }$
(44)

When Γ = 0 then B(x, Γ) = 0 and the constraint is equivalent to nominal constraint. If $Γ = | j |$ then we have soyster’s method.

To reformulate the above function as a linear optimization the following proposition is explained mathematically:

For a given vector x*:

$B ( x * , Γ ) = max { s ∪ { t } | s ⊆ J , | s | = [ Γ ] , t ∈ J \ s } { ∑ j ∈ s μ ^ j | x J * | + ( Γ i − [ Γ i ] ) μ ^ t | x j * | }$

is equal to objective function of the following optimization problem:(45)(46)(47)

$B ( x * , Γ ) = ∑ j ∈ J μ ^ j | x j * | Z j$
(45)

$∑ j ∈ J z j ≤ Γ$
(46)

$0 ≤ z j ≤ 1 ∀ j ∈ J$
(47)

The dual form of above formulation is as follows:(48)

$min ∑ j ∈ J P j + Γ z$
(48)

S.t(49)(50)(51)

$Z + P j ≥ μ ^ j | x j * | ∀ j ∈ J$
(49)

$Z + P j ≥ μ ^ j | x j * | ∀ j ∈ J$
(50)

$Z ≥ 0$
(51)

Based on strong duality theorem, since primal problem is feasible and bounded for all $Γ ∈ [ 0 , | j | ]$, thus the dual problem is also feasible and bounded. Since B(X*, Γ) is equal to objective function of primal problem B(X*, Γ) is equal to dual objective function. By substituting the dual problem in nominal problem, we have the following reformulation for robust optimization.(53)(54)(55)(56)(57)(58)(59)(60)(61)

$min η + 1 ( 1 − B ) S ∑ i = 1 S ( y i )$
(52)

$y i ≥ ∑ j = 1 n [ ( − r i j X j ) − η ] ∀ i$
(53)

$y i ≥ 0$
(54)

$− X ′ μ + E 0 + X ′ C X φ − 1 ( α C ) + Z Γ + ∑ j ∈ J P j ≤ 0$
(55)

$Z + P j ≥ μ ^ j w j ∀ j ∈ J$
(56)

$− W j ≤ X j ≤ W j ∀ j$
(57)

$P j ≥ 0 ∀ j$
(58)

$W j ≥ 0 ∀ j$
(59)

$Z ≥ 0$
(60)

$∑ j = 1 n X j = 1$
(61)

$X j ≥ 0 ∀ j$
(62)

Formulation (52) to (62) is a robust optimization. This model considers parameter uncertainty in chance constraint.

## 5. NUMERICAL RESULTS AND DISCUSSION

In this section, we examine numerical results of our proposed model. We use real data from New York financial market. The data comes from New York stock exchange between April, 2012 and April 1, 2013 for 10 stocks. The stocks that we use in this case study is from:

Amazon, bank of America, bank of Montreal, Exxon Mobil, face book, FedEx, ford, general electric, general motors and yahoo.

Decision variables X1 to X10 indicate the fraction of the above mentioned stokes respectively.

Table 1 shows monthly rate of returns for each stock.

For this problem, we design 400 runs. For all run the number of scenarios s = 12. However, different β, Г and E0 are considered. Table 2 shows the results of numerical example for the confidence level (β = 0.5). This table shows the objective function for different level of price of robustness:

Chart 1 shows the different levels of objective function based on different level of price of robustness and different E0

Chart 2 shows how the objective function (risk measures) increases by increase of price of robustness.

Rates of return of portfolio in β = 0.5 are shown in Table 3:

Chart 3. illustrates to what extent the rate of return is sensitive to price of robustness.

Chart 4 shows that by increasing the price of robustness, objective function (risk measure) increases and portfolio rate of return increases as well. In all levels, when price of robustness is equal to zero it means that there is no uncertainty in model. In other words, robust counterpart problem when Γ= 0 changes to nominal model. So, we can compare the result of our model with nominal problem. When Γ= 0 This comparison between nominal problem and robust counterpart problem indicates that the objective function of our model is worse than nominal problem but the portfolio rate of return in our model is better than nominal problem. These changes in the results are mainly due to the fact that stocks rate of return are not considered fixed and could change in different intervals.

Chart 3 shows that risk increases as the price of robustness increases. However, the slope of risk decreases. Moreover, risk converges to some specific value in all levels of expected returns.

In contrast, Chart 4 illustrates that the slope of rate of return increases as the expected rate of return and price of robustness increases.

The results for β = 0.05 are shown in Table 4 and Table 5.

This model is a convex programming and considers uncertainty with two different approaches. The computational results in all tables with different beta show that by increase of price of robustness the objective function goes worse but rate of return of portfolio increases.

## 6. CONCLUSIONS

This paper developed a new robust model for chance constraint portfolio selection problem. The portfolio selection problem we use is mean-CVaR which is developed by stochastic programming. The proposed model is tested by real world data. The proposed model considers uncertainty of parameters in a constraint. Some parameters in this model are uncertain. We used a combinational model which employs stochastic programming as well as robust optimization to take into account uncertainty. The results show that by increase of price of robustness, both the objective function and the portfolio rate of return increase. The results show the vigor of the proposed model in comparison with deterministic model. Comparison between nominal problem and robust counterpart problem indicates that the objective function of our model is worse than nominal problem but the portfolio rate of return in our model is better than nominal problem.

## Figure

Different level of objective function.

Objective function for different E0.

rate of return of portfolio for different levels of price of robustness.

rate of return for different E0.

## Table

Summary of real world data for stocks monthly rate of return

The value of objective function for β = 0.5

Portfolio rate of return for β = 0.5

Objective function for β = 0.05

Portfolio rate of return for β = 0.05

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