Using Return and Risk Model for Choosing Perfect Portfolio Applied Study in Cairo Stock Exchange

Essam Al Arbed

Economic Faculty, Damascus University, Damascus, Syria.

DOI: 10.4236/ajor.2024.141002                

Abstract

Modern financial theory, commonly known as portfolio theory, provides an analytical framework for the investment decision to be made under uncertainty. It is a well-established proposition in portfolio theory that whenever there is an imperfect correlation between returns risk is reduced by maintaining only a portion of wealth in any asset, or by selecting a portfolio according to expected returns and correlations between returns. The major improvement of the portfolio approaches over prior received theory is the incorporation of 1) the riskiness of an asset and 2) the addition from investing in any asset. The theme of this paper is to discuss how to propose a new mathematical model like that provided by Markowitz, which helps in choosing a nearly perfect portfolio and an efficient input/output. Besides applying this model to reality, the researcher uses game theory, stochastic and linear programming to provide the model proposed and then uses this model to select a perfect portfolio in the Cairo Stock Exchange. The results are fruitful and the researcher considers this model a new contribution to previous models.

Keywords

Game Theory, Stochastic and Linear Programming, Perfect Portfolio, Portfolio Theory, Returns and Risks

1. The Theoretical Introduction

Recent history has shown us that many problems of our technically oriented society yield mathematical descriptions and solutions [1] .

This research is concerned with three specific fields of mathematics, stochastic programming, linear programming and game theory that offer insights into certain problems of the real world and techniques for solving some of these problems.

Game theory is a mathematical framework that is used to study decision-making in situations of strategic interaction. It is used to model and analyze situations where the outcome depends not only on the actions of an individual but also on the actions of other individuals. Game theory is widely used in many fields of science and mathematics [2] .

The methodology of this research is based on a game theory and stochastic programming model that select portfolio positions which perform well on a variety of scenarios generated through statistical modelling and optimization, necessary to reach the proposed model in this research.

The interest of accountants in quantity methods, like those of economists, goes back to the inefficiency in the concept of competitive equilibrium, which states that every decision-maker ignores the behaviour of others when making decisions. Therefore, game theory is used to analyze the actions of players in a strategic manner, and to further analyze how these players make decisions after creating a pre-image of strategies and interests of others.

Although game theory differs from the concept of competitive equilibrium, they both share the following:

1) Each player in the game should be considered rational in making decisions, should have limited preferences, and choose his perfect strategy according to these preferences.

2) The player predicts in game theory and the concept of competitive equilibrium the situation he faces and realizes to what extent he can depend on the results of his decision.

3) The player must know the suitable environment variables in order to interpret some concepts of theoretical solutions in games.

Regarding how to choose a perfect portfolio, operational research is required to a perfect solution for the problem of choosing perfect investments.

We can use an accounting information system that provides the following:

1) Information about stocks regarding prices, analysis, previous direction of prices, and predicting the returns and risks in the future.

2) Predicting and estimating returns and risks in the future by using previous data as input in the proposed model for choosing the perfect portfolio.

2. A Proposed Model

The portfolio model introduced by Markowitz [3] assumes an investor has two considerations when constructing an investment portfolio: expected return and variance in return (i.e., risk). Variance measures the variability in realized return around the expected return, giving equal weight to realizations below the expected and above the expected return.

The Markowitz model might be, mildly, criticized in this regard because the typical investor is probably concerned only with variability below the expected return, so-called downside risk. The Markowitz model requires two major kinds of information: 1) the estimated expected return for each candidate investment and 2) the covariance matrix of returns. The covariance matrix characterizes not only the individual variability of the return on each investment, but also how each investment’s return tends to move with other investments.

The proposed model is based on using game theory to formulate a linear programming model that deals with the problem of uncertain variables that do not probably exist.

It is created through the relationship between returns and risks in the portfolio. The goal of this model is to create a portfolio with different returns and risks by determining the perfect percentage of any stock in the portfolio.

We use quantity measurements to calculate the expected benefits of investments as a return or an expected average return. The measurement of risks in any investment is represented through variant (V), or standard deviation. These variables are considered to be quantity bases for the utility function of the portfolio.

2.1. The Basic Problem

The development of the mathematical model consists of translating the problem into mathematical terms, that is, into the language and concepts of mathematics.

Stochastic programming deals with a class of optimization models and algorithms in which some of the data may be subject to significant uncertainty. Such models are appropriate when data evolve over time, and decisions need to be made prior to observing the entire data stream.

This paper is dedicated to the problem of portfolio optimization through the following question:

How to distribute a limited amount of money as capital in a perfect manner between many available investments or stocks? This problem is considered the basic question in the portfolio model since the contribution of Markowitz.

2.2. The Target of the Model

Game Theory is important to enhance one’s reasoning and decision-making skills in a complex world. It is a framework for understanding choice in situations among competing players.

It can help players reach optimal decision-making when confronted by independent and competing actors in a strategic setting [2] .

The goal of a proposed model agrees with other models which are used for making decisions.

The target is:

1) Maximizing the returns of the portfolio.

2) Minimizing the risks of the portfolio.

3) Choosing stocks that provide high returns and low risks

3. Hypothesis of the Model

If the known theory does provide a complete theoretical solution to the problem, the specific answer to the problem at hand must still be calculated. It could very well be that further analysis does not provide any simplification of the problem, and only through involved computations can an estimate of the solution be made. Thus, finding a solution to a problem could mean determining a technique to approximate a solution that is financially feasible to implement within a given computer’s capabilities and provides error estimates within given tolerance limits [1] .

The model proposed is based on the following realistic hypotheses:

1) There is an uncertainty in the expected results of alternatives concerning the decision maker.

2) The alternative is embodied in choosing bonds, shares, derivatives and shares of mutual funds.

3) The previous returns are used as expected returns for the future without any possible occurrence.

4. The Proposed Model

The literature on financial optimization models dates back to the ground breaking application of Markowitz on optimizing a portfolio of financial products by concentrating on the mean return and taking the variance of the return as a measure of the risk.

In the proposed model the game consists of two players. The first player is called Returns (R), and the other is called Variance (V). Accordingly, the goal of portfolio is:

1) Maximizing the present value of expected returns.

2) Minimizing the present value of expected risks.

By matching the variables of preferences for expected risks, with the variables of preferences for expected returns, we get the perfect return value, as we do in the situation of the game theory (the value of the game).

Variances (risks) (V)

a_ij: Means the annual Dividends of stock (share—bond…).

X_i: Means the possibilities to choose alternatives (strategies x_i). In other words, it means the possibilities as percentages and substitutes (shares—bonds), regarding expected returns.

Y_i: Means the possibilities to choose alternatives (strategies e_i), as percentages regarding expected risks.

P_i: Means time periods to achieve return.

S_i: Means sort of stocks (shares—bonds…).

F: Final target (game value): means (return to portfolio).

As the target of the first player (expected return) is maximizing present value, we can rewrite the target of the first player (expected returns) as maximizing present value in the light of condition (≤) because we will maximize the return value as the following:

$\left(a_{11}{X}_1+a_{12}{X}_2+a_{13}{X}_3+a_{14}{X}_4+a_{15}{X}_5+a_{1n}{X}_n\right){\left(1+i\right)}^{-1}\le F$ (1)

$\left(a_{21}{X}_1+a_{22}{X}_2+a_{23}{X}_3+a_{24}{X}_4+a_{25}{X}_5+a_{2n}{X}_n\right){\left(1+i\right)}^{-2}\le F$

$\left(a_{31}{X}_1+a_{32}{X}_2+a_{33}{X}_3+a_{34}{X}_4+a_{35}{X}_5+a_{3n}{X}_n\right){\left(1+i\right)}^{-3}\le F$

$\left(a_{41}{X}_1+a_{42}{X}_2+a_{43}{X}_3+a_{44}{X}_4+a_{45}{X}_5+a_{4n}{X}_n\right){\left(1+i\right)}^{-4}\le F$

$\left(a_{51}{X}_1+a_{52}{X}_2+a_{53}{X}_3+a_{54}{X}_4+a_{55}{X}_5+a_{5n}{X}_n\right){\left(1+i\right)}^{-5}\le F$

$\left(a_{61}{X}_1+a_{62}{X}_2+a_{63}{X}_3+a_{64}{X}_4+a_{65}{X}_5+a_{6n}{X}_n\right){\left(1+i\right)}^{-6}\le F$

$\left(a_{n1}{X}_1+a_{n2}{X}_2+a_{n3}{X}_3+a_{n4}{X}_4+a_{n5}{X}_5+a_{nn}{X}_n\right){\left(1+i\right)}^{-n}\le F$

When we use alternatives (strategies r_i), and transfer values of returns from stochastic (possibilities) values to true values we get:

$X_1+X_2+X_3+X_4+\cdots +X_n=1$

$X_1,X_2,X_3,X_4,\cdots ,X_n\ge 0$

Then we can write the target of minimizing present value of risks in the light of (≥). As a result, we determine the expected value of the second player (expected risks) as the following:

$\left(a_{11}{Y}_1+a_{21}{Y}_2+a_{31}{Y}_3+a_{41}{Y}_4+a_{51}{Y}_5+a_{n1}{Y}_n\right){\left(1+i\right)}^{-1}\ge F$ (2)

$\left(a_{12}{Y}_1+a_{22}{Y}_2+a_{32}{Y}_3+a_{42}{Y}_4+a_{52}{Y}_5+a_{n2}{Y}_n\right){\left(1+i\right)}^{-2}\ge F$

$\left(a_{13}{Y}_1+a_{23}{Y}_2+a_{33}{Y}_3+a_{43}{Y}_4+a_{53}{Y}_5+a_{n3}{Y}_n\right){\left(1+i\right)}^{-3}\ge F$

$(a_{14}{Y}_1+a_{24}{Y}_2+a_{34}{Y}_3+a_{44}{Y}_4+a_{54}{Y}_5+a_{n4}{Y}_n){\left(1+i\right)}^{-4}\ge F$

$\left(a_{15}{Y}_1+a_{25}{Y}_2+a_{35}{Y}_3+a_{45}{Y}_4+a_{55}{Y}_5+a_{n5}{Y}_n\right){\left(1+i\right)}^{-5}\ge F$

$\left(a_{16}{Y}_1+a_{26}{Y}_2+a_{36}{Y}_3+a_{46}{Y}_4+a_{56}{Y}_5+a_{n6}{Y}_n\right){\left(1+i\right)}^{-6}\ge F$

$\left(a_{1n}{Y}_1+a_{2n}{Y}_2+a_{3n}{Y}_3+a_{4n}{Y}_4+a_{5n}{Y}_5+a_{nn}{Y}_n\right){\left(1+i\right)}^{-n}\ge F$

When we use alternatives (strategies e_i) according to the position in the first situation we get:

$Y_1+Y_2+Y_3+Y_4+\cdots +Y_n=1$

$Y_1,Y_2,Y_3,Y_4,\cdots ,Y_n\ge 0$

By dividing (1) and (2) by (F), we find that the possibility to invest in each stock equals the percentage of the expected return of this stock (x_i), which is then compared to the return on the portfolio as a whole (F).

To simplify the model, the right side will be (1) as the following:

$\left(a_{11}{X}_1/F+a_{12}{X}_2/F+a_{13}{X}_3/F+a_{14}{X}_4/F+a_{1n}{X}_n/F\right){\left(1+i\right)}^{-1}\le F/F$

$\left(a_{21}{X}_1/F+a_{22}{X}_2/F+a_{23}{X}_3/F+a_{24}{X}_4/F+a_{2n}{X}_n/F\right){\left(1+i\right)}^{-2}\le F/F$

$\left(a_{31}{X}_1/F+a_{32}{X}_2/F+a_{33}{X}_3/F+a_{34}{X}_4/F+a_{3n}{X}_n/F\right){\left(1+i\right)}^{-3}\le F/F$

$\left(a_{41}{X}_1/F+a_{42}{X}_2/F+a_{43}{X}_3/F+a_{44}{X}_4/F+a_{4n}{X}_n/F\right){\left(1+i\right)}^{-4}\le F/F$

$\left(a_{51}{X}_1/F+a_{52}{X}_2/F+a_{53}{X}_3/F+a_{54}{X}_4/F+a_{5n}{X}_n/F\right){\left(1+i\right)}^{-5}\le F/F$

$\left(a_{61}{X}_1/F+a_{62}{X}_2/F+a_{63}{X}_3/F+a_{64}{X}_4/F+a_{6n}{X}_n/F\right){\left(1+i\right)}^{-n}\le F/F$

$\left(a_{1n}{X}_1/F+a_{2n}{X}_2/F+a_{3n}{X}_3/F+a_{4n}{X}_4/F+a_{nn}{X}_n/F\right){\left(1+i\right)}^{-n}\le F/F$

For the second player (risks):

$\left(a_{11}{Y}_1/F+a_{21}{Y}_2/F+a_{31}{Y}_3/F+a_{41}{Y}_4/F+a_{n1}{Y}_n/F\right){\left(1+i\right)}^{-1}\ge F/F$ (3)

$\left(a_{12}{Y}_1/F+a_{22}{Y}_2/F+a_{32}{Y}_3/F+a_{42}{Y}_4/F+a_{n2}{Y}_n/F\right){\left(1+i\right)}^{-2}\ge F/F$

$\left(a_{13}{Y}_1/F+a_{23}{Y}_2/F+a_{33}{Y}_3/F+a_{43}{Y}_4/F+a_{n3}{Y}_n/F\right){\left(1+i\right)}^{-3}\ge F/F$

$\left(a_{14}{Y}_1/F+a_{24}{Y}_2/F+a_{34}{Y}_3/F+a_{44}{Y}_4/F+a_{n4}{Y}_n/F\right){\left(1+i\right)}^{-4}\ge F/F$

$\left(a_{15}{Y}_1/F+a_{25}{Y}_2/F+a_{35}{Y}_3/F+a_{45}{Y}_4/F+a_{n5}{Y}_n/F\right){\left(1+i\right)}^{-5}\ge F/F$

$\left(a_{16}{Y}_1/F+a_{26}{Y}_2/F+a_{36}{Y}_3/F+a_{46}{Y}_4/F+a_{n6}{Y}_n/F\right){\left(1+i\right)}^{-6}\ge F/F$

$\left(a_{1n}{Y}_1/F+a_{2n}{Y}_2/F+a_{3n}{Y}_3/F+a_{4n}{Y}_4/F+a_{nn}{Y}_n/F\right){\left(1+i\right)}^{-6}\ge F/F$

$Y_1/F+Y_2/F+Y_3/F+Y_4/F+Y_5/F+Y_6/F+Y_n/F=1/F$

Then we define new variables as the following:

$W_1=X_1/F$ , $W_2=X_2/F$ , $W_3=X_3/F$ , $W_4/F=X_4$ , $W_n=X_n/F$

$Z_1=Y_1/F$ , $Z_2=Y_2/F$ , $Z_3=Y_3/F$ , $Z_4=Y_4/F$ , $Z_n=Y_n/F$

By compensating for the first player (expected return):

$\left(a_{11}{W}_1+a_{12}{W}_2+a_{13}{W}_3+a_{14}{W}_4+a_{15}{W}_5+a_{1n}{W}_n\right){\left(1+i\right)}^{-1}\le 1$ (4)

$\left(a_{21}{W}_1+a_{22}{W}_2+a_{23}{W}_3+a_{24}{W}_4+a_{25}{W}_5+a_{2n}{W}_n\right){\left(1+i\right)}^{-2}\le 1$

$\left(a_{31}{W}_1+a_{32}{W}_2+a_{33}{W}_3+a_{34}{W}_4+a_{35}{W}_5+a_{3n}{W}_n\right){\left(1+i\right)}^{-3}\le 1$

$\left(a_{41}{W}_1+a_{42}{W}_2+a_{43}{W}_3+a_{44}{W}_4+a_{45}{W}_5+a_{4n}{W}_n\right){\left(1+i\right)}^{-4}\le 1$

$\left(a_{51}{W}_1+a_{52}{W}_2+a_{53}{W}_3+a_{54}{W}_4+a_{55}{W}_5+a_{5n}{W}_n\right){\left(1+i\right)}^{-5}\le 1$

$\left(a_{61}{W}_1+a_{62}{W}_2+a_{63}{W}_3+a_{64}{W}_4+a_{65}{W}_5+a_{6n}{W}_n\right){\left(1+i\right)}^{-6}\le 1$

$\left(a_{n1}{W}_1+a_{n2}{W}_2+a_{n3}{W}_3+a_{n4}{W}_4+a_{n5}{W}_5+a_{nn}{W}_n\right){\left(1+i\right)}^{-n}\le 1$

And by compensating for the second player (expected risks):

$\left(a_{11}{Z}_1+a_{21}{Z}_2+a_{31}{Z}_3+a_{41}{Z}_4+a_{51}{Z}_5+a_{n1}{Z}_n\right){\left(1+i\right)}^{-1}\ge 1$ (5)

$\left(a_{12}{Z}_1+a_{22}{Z}_2+a_{32}{Z}_3+a_{42}{Z}_4+a_{52}{Z}_5+a_{n2}{Z}_n\right){\left(1+i\right)}^{-2}\ge 1$

$\left(a_{13}{Z}_1+a_{23}{Z}_2+a_{33}{Z}_3+a_{43}{Z}_4+a_{53}{Z}_5+a_{n3}{Z}_n\right){\left(1+i\right)}^{-3}\ge 1$

$\left(a_{14}{Z}_1+a_{24}{Z}_2+a_{34}{Z}_3+a_{44}{Z}_4+a_{54}{Z}_5+a_{n4}{Z}_n\right){\left(1+i\right)}^{-4}\ge 1$

$\left(a_{15}{Z}_1+a_{25}{Z}_2+a_{35}{Z}_3+a_{45}{Z}_4+a_{55}{Z}_5+a_{n5}{Z}_n\right){\left(1+i\right)}^{-5}\ge 1$

$\left(a_{16}{Z}_1+a_{26}{Z}_2+a_{36}{Z}_3+a_{46}{Z}_4+a_{56}{Z}_5+a_{n6}{Z}_n\right){\left(1+i\right)}^{-6}\ge 1$

$\left(a_{1n}{Z}_1+a_{2n}{Z}_2+a_{3n}{Z}_3+a_{4n}{Z}_4+a_{5n}{Z}_5+a_{nn}{Z}_n\right){\left(1+i\right)}^{-n}\ge 1$

Maximization of $Z_1+Z_2+Z_3+Z_4+\cdots +Z_n=1/F$

According to the following constraints:

$\left(a_{11}{Z}_1+a_{21}{Z}_2+a_{31}{Z}_3+a_{41}{Z}_4+a_{51}{Z}_5+a_{n1}{Z}_n\right){\left(1+i\right)}^{-1}\le 1$

$\left(a_{12}{Z}_1+a_{22}{Z}_2+a_{32}{Z}_3+a_{42}{Z}_4+a_{52}{Z}_5+a_{n2}{Z}_n\right){\left(1+i\right)}^{-2}\le 1$

$\left(a_{13}{Z}_1+a_{23}{Z}_2+a_{33}{Z}_3+a_{43}{Z}_4+a_{53}{Z}_5+a_{n3}{Z}_n\right){\left(1+i\right)}^{-3}\le 1$

$\left(a_{14}{Z}_1+a_{24}{Z}_2+a_{34}{Z}_3+a_{44}{Z}_4+a_{54}{Z}_5+a_{n4}{Z}_n\right){\left(1+i\right)}^{-4}\le 1$

$\left(a_{15}{Z}_1+a_{25}{Z}_2+a_{35}{Z}_3+a_{45}{Z}_4+a_{55}{Z}_5+a_{n5}{Z}_n\right){\left(1+i\right)}^{-5}\le 1$

$\left(a_{16}{Z}_1+a_{26}{Z}_2+a_{36}{Z}_3+a_{46}{Z}_4+a_{56}{Z}_5+a_{n6}{Z}_n\right){\left(1+i\right)}^{-6}\le 1$

$\left(a_{1n}{Z}_1+a_{2n}{Z}_2+a_{3n}{Z}_3+a_{4n}{Z}_4+a_{5n}{Z}_5+a_{nn}{Z}_n\right){\left(1+i\right)}^{-n}\le 1$

We have also:

$X_1+X_2+X_3+X_4+X_5+\cdots +X_n=1/F$

$Y_1+Y_2+Y_3+Y_4+Y_5+\cdots +Y_n=1/F$

We go back to the first player whose target is to maximize the present value of expected return which is equal to minimizing (1/F).

Now, we can write the problem of linear programming for the first player (minimizing present value of expected returns) as the following:

Minimizing: $W_1+W_2+W_3+W_4+W_5+\cdots +W_n=1/F$

According to the following constraints:

$\left(a_{11}{W}_1+a_{12}{W}_2+a_{13}{W}_3+a_{14}{W}_4+a_{15}{W}_5+a_{1n}{W}_n\right){\left(1+i\right)}^{-1}\ge 1$

$\left(a_{21}{W}_1+a_{22}{W}_2+a_{23}{W}_3+a_{24}{W}_4+a_{25}{W}_5+a_{2n}{W}_n\right){\left(1+i\right)}^{-2}\ge 1$

$\left(a_{31}{W}_1+a_{32}{W}_2+a_{33}{W}_3+a_{34}{W}_4+a_{35}{W}_5+a_{3n}{W}_n\right){\left(1+i\right)}^{-3}\ge 1$

$\left(a_{41}{W}_1+a_{42}{W}_2+a_{43}{W}_3+a_{44}{W}_4+a_{45}{W}_5+a_{4n}{W}_n\right){\left(1+i\right)}^{-4}\ge 1$

$\left(a_{51}{W}_1+a_{52}{W}_2+a_{53}{W}_3+a_{54}{W}_4+a_{55}{W}_5+a_{5n}{W}_n\right){\left(1+i\right)}^{-5}\ge 1$

$\left(a_{61}{W}_1+a_{62}{W}_2+a_{63}{W}_3+a_{64}{W}_4+a_{65}{W}_5+a_{6n}{W}_n\right){\left(1+i\right)}^{-6}\ge 1$

$\left(a_{n1}{W}_1+a_{n2}{W}_2+a_{n3}{W}_3+a_{n4}{W}_4+a_{n5}{W}_5+a_{nn}{W}_n\right){\left(1+i\right)}^{-n}\ge 1$

At the same time the target of the second player is to minimize present value of risks, which equals maximizing value of (1/F).

Then we can write the problem of linear programming as the following:

Maximization of $Z_1+Z_2+Z_3+Z_4+\cdots +Z_n=1/F$

According to the following constraints:

$\left(a_{11}{Z}_1+a_{21}{Z}_2+a_{31}{Z}_3+a_{41}{Z}_4+a_{51}{Z}_5+a_{n1}{Z}_n\right){\left(1+i\right)}^{-1}\le 1$

$\left(a_{12}{Z}_1+a_{22}{Z}_2+a_{32}{Z}_3+a_{42}{Z}_4+a_{52}{Z}_5+a_{n2}{Z}_n\right){\left(1+i\right)}^{-2}\le 1$

$\left(a_{13}{Z}_1+a_{23}{Z}_2+a_{33}{Z}_3+a_{43}{Z}_4+a_{53}{Z}_5+a_{n3}{Z}_n\right){\left(1+i\right)}^{-3}\le 1$

$\left(a_{14}{Z}_1+a_{24}{Z}_2+a_{34}{Z}_3+a_{44}{Z}_4+a_{54}{Z}_5+a_{n4}{Z}_n\right){\left(1+i\right)}^{-4}\le 1$

$\left(a_{15}{Z}_1+a_{25}{Z}_2+a_{35}{Z}_3+a_{45}{Z}_4+a_{55}{Z}_5+a_{n5}{Z}_n\right){\left(1+i\right)}^{-5}\le 1$

$\left(a_{16}{Z}_1+a_{26}{Z}_2+a_{36}{Z}_3+a_{46}{Z}_4+a_{56}{Z}_5+a_{n6}{Z}_n\right){\left(1+i\right)}^{-6}\le 1$

$\left(a_{1n}{Z}_1+a_{2n}{Z}_2+a_{3n}{Z}_3+a_{4n}{Z}_4+a_{5n}{Z}_5+a_{nn}{Z}_n\right){\left(1+i\right)}^{-n}\le 1$

We have to note that (4) is binary to (5) when we solve one problem of linear programming, that is the basic linear programming problem, and we get the value of $Z_1,Z_2,Z_3,Z_4,\cdots ,Z_n$ .

By using the computer, we calculate the percentages of the perfect investments in the portfolio, and we get the value of the target (F) through:

$W_1+W_2+W_3+W_4+W_5+\cdots +W_n=1/F$

By going back to (3) we find that:

$X_1=F\cdot W_1,X_2=F\cdot W_2,X_3=F\cdot W_3,\cdots ,X_n=F\cdot W_n$

This means that the alternatives (percentages of stocks we choose) are ranging between 1% till 100%.

At the same time, we calculate the following:

$Y_1=F\cdot Z_1,Y_2=F\cdot Z_2,Y_3=F\cdot Z_3,\cdots ,Y_n=F\cdot Z_n$

This means that the alternatives of the second player (Risks) range between 1% and 100%.

Accordingly, we choose the percentages of high returns (Rs) of stocks, and low risks of stocks (Vs), compared with the returns and risk of portfolio (F) as a whole.

As a result, the high return on stocks represents the amount of money we have to invest in every kind of stock because it provides both a high return and a low risk simultaneously.

The researcher tested the proposed model to choose the perfect portfolio in Cairo Stock Exchange.

5. The Samples Chosen Were Active Shares of Companies Who Traded Their Shares at Cairo

Stock Exchange (Egypt) (Table 1):

Table 1. Active companies who traded their shares at Cairo Stock Exchange (Egypt) [4] [5] [6] .

1) Time period Pi from year (1) to year (10) (2014-2023).

2) The developments of the discount rate issued by Egyptian Central Bank (2014-2023) are shown in Table 2 [4] [5] [6] .

3) Discount rates and present value of one Egyptian pound after (n) year are shown in Table 3 [4] [5] [6] .

4) Data of monetary distributions of Dividends are shown in Tables 4-38 [4] [5] [6] .

5) Processing data through Linear and Integer Goal Programming and the results of the program are shown in Tables 39-41.

6. Interpreting the Output of the Program (Linear Programming Program)

1) Shares of companies represent the perfect portfolio according to the proposed model.

Table 2. Discount rate 2014-2023.

Table 3. Present value of one Egyptian pound.

Table 4. Distributions of dividends of Suez Canal Bank VAR (1).

Table 5. Distributions of dividends of Commercial International Bank (Egypt) VAR (2).

Table 6. Distributions of dividends of Oriental Weavers VAR (3).

Table 7. Distributions of dividends of Egypt Telecommunications VAR (4).

Table 8. Distributions of dividends of Edita Food Industry VAR (5).

Table 9. Distributions of dividends of Cement Egypt VAR (6).

Table 10. Distributions of dividends of Orascom Investment Holding VAR (7).

Table 11. Distributions of dividends of Ezz Steel VAR (8).

Table 12. Distributions of dividends of Abou Kir Fertilizers & Chemical VAR (9).

Table 13. Distributions of dividends of B Investment Holding VAR (10).

Table 14. Distributions of dividends of Elsewedy Electric VAR (11).

Table 15. Distributions of dividends of Egyptian Transport & Commercial Services Co. VAR (12).

Table 16. Distributions of dividends of Sidi Kerir Petrochemicals Co. VAR (13).

Table 17. Distributions of dividends of Arabian Food Industries Company VAR (14).

Table 18. Distributions of dividends of Minapharm Pharmaceutical VAR (15).

Table 19. Distributions of dividends of Glaxo Smithklin VAR (16).

Table 20. Distributions of dividends of delta for Sugar VAR (17).

Table 21. Distributions of dividends of Juhayna Food Industries VAR (18).

Table 22. Distributions of dividends of Arab Aluminum Co. VAR (19).

Table 23. Distributions of dividends of Madinat Naser for Housing and Construction VAR (20).

Table 24. Distributions of dividends of Misr Fertilizers Production Company VAR (21).

Table 25. Distributions of dividends of Pyramids Hotels & Resorts Co. VAR (22).

Table 26. Distributions of dividends of Saudi Egyptian Investments & Finance Co. VAR (23).

Table 27. Distributions of dividends of Egyptian International Pharmaceutical Industries Co. VAR (24).

Table 28. Distributions of dividends of Delta for Printing & Packaging Co. VAR (25).

Table 29. Distributions of dividends of KZ for Pesticides & Chemicals Kafr Az Zayat VAR (26).

Table 30. Distributions of dividends of Housing & Development Bank VAR (27).

Table 31. Distributions of dividends of Al Ezz Ceramics and Porcelain Co. VAR (28).

Table 32. Distributions of dividends of Egyptian Arabian Themar Securities Brokerage VAR (29).

Table 33. Distributions of dividends of Al Tawfeek for Financial Leas VAR (30).

Table 34. Distributions of dividends of Egyptian Financial & Industries Co. VAR (31).

Table 35. Distributions of dividends of Egypt Gas CO. VAR (32).

Table 36. Distributions of dividends of Faisal Islamic Bank VAR (33).

Table 37. Distributions of dividends of Ferchem Masr Fertilizers and Chemicals VAR (34).

Table 38. Distributions of dividends of Real Estate Egyptian Consortium VAR (35).

Table 39. Combined report for using computer for choosing perfect portfolio in Cairo Stock Exchange.

Objective function value = 14.78.

Table 40. Optimal solution—detailed report.

Table 41. Solution summary for using computer for choosing perfect portfolio in Cairo Stock Exchange.

C₁₅ = X₁₅ = VAR₁₅ = Minapharm Pharmaceuticals (Table 18).

Minapharm Pharmaceuticals: A leading pharmaceutical company in Egypt, Africa and the Middle East and the premier biopharmaceutical company in the region, with multiple subsidiaries in Berlin and Cairo. Minapharm commercializes over 100 life-saving and life-enhancing products ranging from small molecules with advanced galenic formulations to complex genetically engineered proteins and other advanced therapies.

C₁₇ = X₁₇ = VAR₁₇ = Delta for Sugar (Table 20).

Delta Sugar: The company engages in the manufacture and sale of beet sugar in Egypt. It offers sugar, dry beet pulp, and molasses for use in the animal feed industry. The company was founded in 1978 and is headquartered in the 6th of October City, Egypt. Delta Sugar Company is a subsidiary of Egyptian Sugar and Integrated Industries Company SAE.

2) The whole percentages of stock return represent the return of the portfolio. At the same time, they represent the perfect percentages of investments according to the model proposed as the following:

C₁₅ = X₁₅ = VAR₁₅ = the percentage of investing in a share of Minapharm Pharmaceuticals.

C₁₇ = X₁₇ = VAR₁₇ = the percentage of investing in a share of Delta for Sugar.

We calculate the percentages as follows:

$1/F=14.87$ (Table 40)

$F=1/14.87=0.0672494956$

$X_{15}=W_{15}\ast F=14.43\ast 0.0672494956=0.9704102219$ (Table 39)

$X_{17}=W_{17}\ast F=0.44\ast 0.0672494956=0.0295897781$

With regard to risks (according to the binary method in linear programming), the value of basic variables, in the perfect solution of the binary method, represents the shadow prices of slack variables in the basic method as follows:

${\text{VAR}}_{\text{7}}=Y_7=Z_7\ast F=14.04\ast 0.0672494956=0.9441829182$ (Table 40)

${\text{VAR}}_{\text{8}}=Y_8=Z_8\ast F=0.82\ast 0.0672494956=0.0551445864$

These percentages range between 1% and 100% which meet the conditions of the proposed model.

7. Conclusions

The target of this proposed model is achieved, as the risks increase by expanding the size of the portfolio because of the positive direct relationship between risks and returns.

Accordingly, we note that the increase of risks in variables (7) and (8) represents the high percentages of investments in the perfect portfolio containing (2) shares as follows:

$\left(0.9704102219,0.9441829182\right)={\text{VAR}}_{\text{15}}\left(X_{\text{15}}\right),{\text{VAR}}_{\text{7}}(\; Y\; 7\; )$

$\left(0.0295897781,0.0551445864\right)={\text{VAR}}_{\text{17}}\left(X_{\text{17}}\right),{\text{VAR}}_{\text{8}}(\; Y\; 8\; )$

Thus, the proposed hypotheses have been tested and proven. The first hypothesis states that it is difficult to estimate an expected return because of its changing features. Therefore, the researcher used the series of distributions of dividends from 2014 to 2023, and calculated the present value of these dividends. The problem was how to transfer values from stochastic values to real values. It is solved by using game theory and stochastic programming, then linear programming to calculate expected returns.

With regard to the second hypothesis, the model accepts any kind of stock, but because of the shortage of data about bonds and derivatives and mutual funds, the researcher used shares only.

In the third hypothesis, the data was tested and found to have met the conditions of the Model, where the summation of returns equals 100% and the summation of risks equals 100%. Finally, the researcher considers this model, a scientific contribution to previous models, which have dealt with choosing perfect portfolios.

Processing data through Linear and Integer Goal Programming and the results of the program are shown in interpreting the output of the program.

Acknowledgements

I am honored to Provide an original research article entitled “Using Return and Risk Model For Choosing Perfect Portfolio, Applied Study in Cairo Stock Exchange” for consideration by the Journal of Portfolio Management.

I confirm that this work is original and has not been published elsewhere, nor under any consideration for publication elsewhere nowadays. Besides the Model proposed is the contribution of a PH.D. degree in Finance and Accounting in Egypt.

The theme of this paper is how to propose a new mathematical model like that provided by Markowitz which helps in choosing a perfect portfolio and the input/output of this model, besides applying this model in reality, the researcher used the game theory with stochastic and linear programming to provide the model proposed and then used this model to select a perfect portfolio in Cairo Stock Exchange. The results were fruitful and the researcher considers this model a new scientific addition to previous models.

As your Journal is interested in Academic research, including new thoughts, I hope that my research will be up to your standards. I think it would be interesting by the readers of your journal.

Conflicts of Interest

I have no conflicts of interest to disclose.

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