In 1950, Harry Markowitz introduced the renowned Modern Portfolio Optimization Theory. This theory operates under the assumption that, at any given time, an investor seeks to maximize the expected return of a portfolio while considering a certain level of risk, typically quantified by the standard deviation of the portfolio’s rate of return ( Fabozzi, Markowitz, & Gupta, 2011 ). Since its inception, portfolio optimization has become a widely adopted concept in finance, extensively utilized by investors and companies alike to enhance the profitability of their investment portfolios.

Every day, a vast number of individuals participate in trading equities and commodities on global exchanges, driven by the enticing profit prospects offered by the stock market. Investors often diversify their portfolios across various stocks in pursuit of attractive returns to bolster their wealth.

Nearly all publicly traded companies aspire to maximize their share prices, thereby enhancing their firm’s value and market dominance. In this context, both individuals and companies recognize the importance of comprehending the key financial and economic factors influencing their stock prices. Analyzing the behavior and interactions of these indicators aids in assessing investment risks and forecasting potential stock values with precision.

This study aims to optimize the stock price of a single stock, focusing on the relevant financial and economic indicators that impact the weekly closing price of MSFT stock. Herein, we propose an optimized predictive model leveraging response surface analysis based on the desirability function approach, a widely employed method in industry for optimizing single or multiple response processes. Response Surface Methodology (RSM) is a statistical technique that is typically used in experimental design to optimize processes and understand the relationship between several variables and an outcome of interest. While it’s not directly a mainstream tool in finance, some aspects of RSM can be applied to financial problems, especially those involving optimization and risk management.

The Response Surface Methodology (RSM) helps analyze the relationship between financial and economic indicators and a response variable to produce optimal response outcome. RSM aims to minimize or maximize the response variable(s) by achieving a desired target value of the indicators that optimize response variable.

Response Surface Methods (RSMs) are models and methodologies tailored for handling continuous treatments to either optimize or characterize the response variable ( Oehlert, 2010 ). Initially, the primary aim of RSM is to identify the optimal response. When dealing with multiple responses, it becomes crucial to determine a compromise solution that doesn’t solely optimize one response. Additionally, when working with design data constrained by limitations, the experimental design must adhere to specific constraints.

Another objective of RSM is to understand how the response changes relative to adjustments in the design variables. Typically, the analysis of the response surface can be represented graphically, allowing for visual interpretation. Through these graphs, the shape of the response surface, including hills, valleys, and ridge lines, can be observed. Consequently, the function f(X₁, X₂) can be graphed against the levels of X1 and X2 to gain insights into their effects on the response ( Montgomery, 2017 ).

The desirability function, initially outlined by Harrington (1965) and further elaborated by Derringer and Suich (1980) , stands as a significant tool for factor optimization. The Desirability Function Approach (DFA) constitutes a strategy for response optimization, where the response is optimized based on controllable input factors. By integrating both Response Surface Methodology (RSM) and the desirability approach, a more robust method emerges for achieving optimal dynamics among responses, including stock return responses (multicriteria optimization). This fusion of RSM and the desirability function gives rise to the “Desirability Optimization Methodology”.

RSM is frequently used in engineering and manufacturing industries to optimize manufacturing processes, improve product quality, and reduce production costs. Modelling and optimization of copper adsorption in solution using the response surface method through the Box-Behnken model has been studied recently in Senegal ( Ba et al., 2023 ). This method is commonly applied in industrial, microbiology, biochemistry, and chemical analysis, however to the best of our knowledge, such a method has never been applied to the plant biotechnological studies and, more specifically, to the study of the in vitro response of a plant secondary metabolism to the environmental factors ( Costa, Lourenço, & Pereira, 2011 ). In greenhouse gas emissions mainly produced by industrial power plants, In this contribution, firstly, a simulation of the process for capturing carbon dioxide from flue gas with the 1-n-hexyl-3-methylimidazolium bis(trifluoromethyl sulfonyl)amide ([hmim] [Tf2N]) ionic liquid is carried out, leading to an optimization of the system, minimizing costs and maximizing the amount of captured carbon dioxide through the response surface methodology ( Leonzio & Zondervan, 2020 ). This profile is then applied in the field of stock return optimization.

In our previous article titled “Predictive Modeling: Impact of Financial and Economic Indicators on Stock” ( Pokharel, Tetteh-Bator, & Tsokos, 2022b ), we present a real data-driven analytical model to predict the “Weekly Closing Price” (WCP) of the MSFT stock using five financial, four economic indicators and their two-way interactions and, the proposed predictive model has excellent predictive accuracy. The attributable entities that are included in the proposed optimized model have a significant relevance in the literature of finance and economics. We now proceed to identify the target values of the statistically significant indicators and their interactions used in the predictive model that maximizes the WCP of the stock using response surface analysis based on desirability function approach.

The study discusses the importance of the target values of the controllable indicators and their interactions with stock price optimization problems and presents some useful information for related stakeholders.

The procedures and the methodology of our proposed predictive model-building process are presented in Subsections 2.1, 2.2 and 2.3.

The response surface analysis based on the desirability function is a method used in industry for the optimization of multiple response processes. The desirability function, initially formulated by Harrington (1965) and further developed by Derringer and Suich (1980) , is a method employed for optimizing factors. The Desirability Function Approach (DFA) serves as a strategy for optimizing responses by transforming them into a dimensionless desirability value. This approach enables the optimization of responses in relation to controllable input factors. The values of the desirability functions lie between 0 and 1. The value 0 is attributed when the factors give an undesirable response, while the value 1 corresponds to the optimal performance for the important factors ( Derringer & Suich, 1980 ). Let $y\left(x\right)$ and $d\left(y\right)$ be the response and the desirability function, respectively. For each response $y_i\left(x\right)$, a desirability function $d_i\left(y\right)$ assigns numbers between 0 and 1 to the possible values of $Y_i$, with $d_i\left(y_i\right)=0$ representing a completely undesirable value of $y_i$ and $d_i\left(y_i\right)=1$, representing a completely desirable or ideal response value. The individual desirabilities are then combined using the geometric mean, which gives the overall desirability D, that is:

$D={\left(d_1\left(y_1\right)d_2\left(y_2\right)\cdots d_k\left(y_k\right)\right)}^{1/p}={\left[{\displaystyle \underset{i=1}{\overset{k}{\prod }}}{d}_i\left(y_i\right)\right]}^{1/p},$ (1)

where p denotes the number of responses. In this study, we want to optimize the single response AWCP where we choose p = 1.

Depending on the optimization objectives, whether a particular response $y_i$ is to be maximized, minimized, or assigned to a target value, Derringer and Suich (1980) proposed a useful class of desirability functions. Let $L_i$, $U_i$ and $T_i$ be the lower bound, upper bound, and target values of the desired response ${\hat{y}}_i$, with $L_i\le T_i\le U_i$. If the objective is to maximize the response (i.e. the larger the better), the desirability function is defined as:

$d_i\left({\hat{y}}_i\right)=(\begin{array}{ll}\mathrm{0,}\hfill & \text{for}{\hat{y}}_i\left(x\right)<L_i\hfill \\ {\left(\frac{{\hat{y}}_i-L_i}{T_i-L_i}\right)}^{\alpha }\mathrm{,}\hfill & \text{for}{L}_i\le {\hat{y}}_i\left(x\right)\le T_i\hfill \\ \mathrm{1,}\hfill & \text{for}{\hat{y}}_i\left(x\right)>T_i\hfill \end{array}$ (2)

where α is the shape parameter of the individual desirability function $d_i\left({\hat{y}}_i\right)$ and $T_i$ is interpreted as the large enough value for the response. Similarly, if the objective is to obtain the specific “target” value of the response, the two-sided desirability function is defined as:

$d_i\left({\hat{y}}_i\right)=(\begin{array}{ll}\mathrm{0,}\hfill & \text{for}{\hat{y}}_i\left(x\right)<L_i\hfill \\ {\left(\frac{{\hat{y}}_i-L_i}{T_i-L_i}\right)}^{\alpha }\mathrm{,}\hfill & \text{for}{L}_i\le {\hat{y}}_i\left(x\right)\le T_i\hfill \\ {\left(\frac{{\hat{y}}_i-U_i}{T_i-U_i}\right)}^{\beta }\mathrm{,}\hfill & \text{for}{T}_i\le {\hat{y}}_i\left(x\right)\le U_i\hfill \\ \mathrm{1,}\hfill & \text{for}{\hat{y}}_i\left(x\right)>U_i\hfill \end{array}$ (3)

And if the objective is to minimize the response, the desirability function is defined as:

$d_i\left({\hat{y}}_i\right)=(\begin{array}{ll}\mathrm{1,}\hfill & \text{for}{\hat{y}}_i\left(x\right)<T_i\hfill \\ {\left(\frac{{\hat{y}}_i-L_i}{T_i-L_i}\right)}^{\alpha }\mathrm{,}\hfill & \text{for}{T}_i\le {\hat{y}}_i\left(x\right)\le U_i\hfill \\ \mathrm{0,}\hfill & \text{for}{\hat{y}}_i\left(x\right)>U_i\hfill \end{array}$ (4)

where α and β represent the weighted parameters that define the shape of the individual desirability function $d_i\left({\hat{y}}_i\right)$. For $\alpha =\beta =1$, the shape of $d_i\left({\hat{y}}_i\right)$ is linearly increasing in the direction of $T_i$, if $\alpha <1$ and $\beta <1$, the shape of $d_i\left({\hat{y}}_i\right)$ is concave, and if $\alpha >1$ and $\beta >1$, the shape of $d_i\left({\hat{y}}_i\right)$ is convex ( Jeong & Kim, 2009 ).

We employed the Central Composite Design (CCD), a widely used approach in response surface methodology featuring a two-level full factorial design supplemented with central and axial points. Axial points are positioned at the midpoints of factor levels relative to other factors, while central points are added at the center of the design space. The CCD utilized in this study encompasses 10 continuous factors and identifies 11 significant interactions. A robust prediction model is characterized by its ability to maintain consistent and stable variance across the entire range of independent predictor variables. The contour and surface plots generated by the CCD illustrate a robust prediction model that exhibits uniform and stable variance throughout the range of independent variables.

The proposed advanced predictive model, utilizing a response surface based on the desirability function approach, is highly pertinent within applied finance and economics. To the best of our knowledge, this methodology has not been previously explored in financial literature for optimizing stock prices or returns. Developed with a strong theoretical basis in statistical concepts and financial expertise, the model has undergone rigorous testing and validation, demonstrating exceptional predictive accuracy of 98.34%.

The key findings emphasized in Subsection 3.3 underscore the model’s utility and distinction, serving as evidence of its singular contribution to the field of applied finance and economics. Our analysis revealed that the optimal or maximum value of the Weekly Closing Price (WCP) of the stock is 206.92 in the given time, achieving a desirability function score of 1. This finding suggests that the selected values for financial and economic indicators are highly effective in maximizing stock returns.

Moreover, we established a 95% confidence interval of (170.30, 243.50) and a prediction interval of (170.0, 243.80), which further supports the statistical significance of the optimal response point. With an R² of 98.34% and an adj.R² of 98.29%, alongside a high prediction accuracy of 98.2%, our analytical model provides robust validation. The response surface optimization process we devised enables the attainment of desired response values, offering flexibility for firms and investors to set their optimal targets and obtain corresponding risk factor values.

Understanding the interactive dynamics of financial and economic indicators is paramount for comprehending stock price fluctuations for risk mitigation and return optimization in investment. Effectively harnessing the proposed optimized predictive model empowers both MSFT stock investors and the company itself to achieve maximum returns by making well-informed investment decisions. The aforementioned procedure and methodology can effectively be applied to other individual stocks or sets of stocks to construct an optimal portfolio tailored to achieve desired returns, considering specified risk factors.

To the best of our knowledge, this manuscript represents our original work and meets the ethical standards set by the Committee on Publication Ethics (COPE).