Optimization Approach to Asset and Capital Structure of Companies

Erdenebat Munkhdalai^1*, Enkhbayar Jamsranjav², Oyuntsetseg Luvsandash³, Gantulga Garamdorj⁴

¹Business school, National University of Mongolia, Ulaanbaatar, Mongolia.
²School of Information Technology and Electronics, National University of Mongolia, Ulaanbaatar, Mongolia.
³Business School, National University of Mongolia, Ulaanbaatar, Mongolia.
⁴Accounting Department, University of Finance and Economics, Ulaanbaatar, Mongolia.

DOI: 10.4236/ib.2023.154023                

Abstract

Optimizing a company’s asset and capital structure has a positive effect on reducing capital costs and increasing the use of assets. In this paper, we consider the optimization problem of a company’s asset and capital structure. The proposed model is formulated as a fractional programming problem. The problem was solved numerically on “Mat-lab” using data from Mongolian mining companies. Numerical results are provided.

Keywords

Asset Structure, Capital Structure, Net Working Capital, Mongolian Mining Companies

1. Introduction

Optimizing the structure of assets is an important and complex task of financial management. The optimal asset structure is to identify working capital and non-current assets and their financing, and to ensure liquidity, solvency financial stability. This paper deals with optimizing asset and capital structure based on the relation between assets and capital by their financial ratios and characteristics.

The trade-off relationship between asset structure, capital structure and management of profitability and liquidity of asset structure provides a new research direction for us to study how to optimize enterprise asset structure (Titman, 1983) .

This is confirmed by the following studies. Capital contributes to the optimal asset structure (Koralun-Bereźnicka, 2013) . Optimal asset allocation affects capital structure through leverage (Campello & Giambona, 2013) . This shows that the optimal structure of assets and capital affects each other. High levels of non-current assets negatively affect the solvency and liquidity of assets. This will negatively affect profitability. Therefore, optimizing the amount of non-current assets is the basis for the long-term sustainability of business operations. A company with efficient working capital management can increase profitability by reducing financial costs, even if it increases the amount of capital. This suggests that working capital needs to be financed as low-interest sources as possible and that for companies, having a proper working capital structure can increase efficiency. Too much-working capital increases costs, while too little has the disadvantage of reduced solvency. Researchers such as Gebrehiwot and Wolday have argued that effective management of working capital increases the efficiency of an organization through the proper management and maintenance of working capital (Gebrehiwot & Wolday, 2006) . According to Afza and Nazir, “Effective working capital management is an integral part of an enterprise’s strategy and creates value for shareholders” (Afza & Nazir, 2008) , so effective working capital management improves profitability and thus increases business value. Therefore, companies are always trying to maintain the optimal level of working capital to increase their value (Deloof, 2003) . Researchers (Deloof, 2003) , (Koralun-Bereźnicka, 2013) , (Schilling, 1996) , (Lind, et al., 2012) views on working capital suggest that “the main purpose of working capital management is to determine the appropriate amount of working capital components, reduce the associated costs, and increase the company’s efficiency.” The structure of a company’s assets and capital varies depending on the country and industry, but it is preferable that working capital be greater than short-term liabilities. It can have a positive impact on profitability by improving solvency ratios and thereby reducing costs associated with working capital and current liabilities. The paper is organized as follows. Section 2 is devoted to the optimization model of asset and capital structure.

2. Net Working Capital Problem Formulation

We introduce the following notations and variables for asset and capital structure. (Table 1)

The following assumptions follow from the financial statements of a company.

The sum of variables is equal to one.

$x_1^A+x_2^A+x_3^A+x_4^A=1$ (1)

$y_1^L+y_2^L+y_3^L=1$ (2)

Current assets (sum of cash, receivables, inventories and expenses) are greater than current liability.

$x_1^A+x_2^A+x_3^A>y_1^L$ (3)

Non-current assets shall be more than long-term liability.

Table 1. Notations and variables.

$x_4^A>y_2^L$ (4)

Non-current assets shall be more than current assets.

$x_1^A+x_2^A+x_3^A<x_4^A$ (5)

Cash Ratio ( ${\upsilon }_1$ ): It shows the sufficiency of the entity’s marketable securities and cash to pay current liabilities. It shows the sufficiency of the entity’s marketable securities and cash to pay current liabilities. Denote by ${\upsilon }_1$ cash ratio. Then ${\upsilon }_1$ is determined as the ratio:

${\upsilon }_1=\frac{x_1^A}{y_1^L}$ (6)

Quick Ratio ( ${\upsilon }_2$ ): It indicates whether it the possible to repay current liabilities based on the collection of receivables in addition to cash and liquid securities. ${\upsilon }_2$ is defined as follows:

${\upsilon }_2=\frac{x_1^A+x_2^A}{y_1^L}$ (7)

Current Ratio ( ${\upsilon }_3$ ): Ration of the total current assets and current liabilities, which shows how current liabilities are secured by current assets. Then ${\upsilon }_3$ is defined in the following.

${\upsilon }_3=\frac{x_1^A+x_2^A+x_3^A}{y_1^L}$ (8)

Specific gravity of non-current assets ( ${\upsilon }_4$ ): It shows holding specific gravity of non-current assets from the total assets. ${\upsilon }_4$ is given by the following formula:

${\upsilon }_4=\frac{x_4^A}{x_1^A+x_2^A+x_3^A+x_4^A}$ (9)

Equity to assets ratio ( ${\upsilon }_5$ ): It defines ability to operate independently from external sources with its own funding. ${\upsilon }_5$ is defined by the following formula.

${\upsilon }_5=\frac{y_3^L}{x_1^A+x_2^A+x_3^A+x_4^A}$ (10)

Debt to Equity Ratio ( ${\upsilon }_6$ ): It shows the debt per 1 tugrik of owner’s equity. In business sectors other than the financial sector, it is preferable to have less than 1.

$\text{DebttoEquityRatio}=\frac{\text{Totaldebt}}{\text{Owner'sequity}}$

In other words,

${\upsilon }_6=\frac{y_1^L+y_2^L}{y_3^L}$ (11)

Debt to assets ratio ( ${\upsilon }_7$ ): It indicates how much of the company’s total assets are financed by external sources.

$\text{Debttoassetsratio}=\frac{\text{Totaldebt}}{\text{Totalassets}}$

This means that:

${\upsilon }_7=\frac{y_1^L+y_2^L}{x_1^A+x_2^A+x_3^A+x_4^A}$ (12)

Long-term debt-to-equity ratio ( ${\upsilon }_8$ ): High ratio means that the company has the high financial risk.

$\text{Long-term}\text{debt-to-equity}\text{ratio}=\frac{\text{Longtermliability}}{\text{Equity}}$

It can be expressed as:

${\upsilon }_8=\frac{y_2^L}{y_3^L}$ (13)

Each of variables of the specific gravity shown in Table 1 is expressed in terms of υj and other parameters. Now we can write down the above formulas as:

$x_1^A=\frac{y_3^L-{\upsilon }_5\left(x_2^A+x_3^A+x_4^A\right)}{{\upsilon }_5}$ (14)

$x_1^A={\upsilon }_1{y}_1^L$ (15)

$x_2^A={\upsilon }_2{y}_1^L-x_2^A={\upsilon }_2{y}_1^L-{\upsilon }_1{y}_1^L$ (16)

$x_3^A={\upsilon }_3{y}_1^L-x_2^A-x_1^A={\upsilon }_3{y}_1^L-{\upsilon }_2{y}_1^L+{\upsilon }_1{y}_1^L-{\upsilon }_1{y}_1^L=\left({\upsilon }_3-{\upsilon }_2\right)y_1^L$ (17)

$x_3^A=\frac{y_1^L+y_2^L-{\upsilon }_7\left(x_1^A+x_2^A+x_4^A\right)}{V_7}$ (18)

$x_4^A=\frac{V_4{x}_1^A+{\upsilon }_4{x}_2^A+\upsilon x_3^A}{1-{\upsilon }_4}$ (19)

$y_2^L={\upsilon }_8{y}_3^L$ (20)

$y_3^L={\upsilon }_5\left(x_1^A+x_2^A+x_3^A+x_4^A\right)$ (21)

Also, the variable $y_1^L$ can be expressed by ${\upsilon }_5$ , ${\upsilon }_6$ and $V_8$ ratios in the following.

$y_1^L=\frac{y_3^L}{x_1^A+x_2^A+x_3^A+x_4^A}\left(\frac{y_1^L+y_2^L}{y_3^L}-\frac{y_2^L}{y_3^L}\right)$

$y_1^L={\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)$ (22)

We can express variables $x_4^A$ via variables $x_1^A$ , $x_2^A$ , $x_3^A$ , $y_1^L$ and ${\upsilon }_4$ as follows:

$\begin{array}{c}{x}_4^A=\frac{{\upsilon }_4\left(x_1^A+x_2^A+x_3^A\right)}{1-V_4}=\frac{{\upsilon }_4\left({\upsilon }_1{y}_1^L+{\upsilon }_2{y}_1^L-{\upsilon }_1{y}_1^L+{\upsilon }_3{y}_1^L-{\upsilon }_2{y}_1^L\right)}{1-V_4}\\ =\frac{{\upsilon }_4{\upsilon }_3{y}_1^L}{1-{\upsilon }_4}=\frac{{\upsilon }_3{\upsilon }_4{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\end{array}$ (23)

Using Formulas (15)-(17), we compute a value of $y_3^L$ as:

$\begin{array}{c}{y}_3^L={\upsilon }_5\left(x_1^A+x_2^A+x_3^A+x_4^A\right)\\ ={\upsilon }_5\left({\upsilon }_1{y}_1^L+{\upsilon }_2{y}_1^L-{\upsilon }_1{y}_1^L+{\upsilon }_3{y}_1^L-{\upsilon }_2{y}_1^L+\frac{{\upsilon }_3{\upsilon }_4{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)\\ ={\upsilon }_5\left({\upsilon }_3{y}_1^L+\frac{{\upsilon }_3{\upsilon }_4{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)\\ ={\upsilon }_5\left({\upsilon }_3{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)+\frac{{\upsilon }_3{\upsilon }_4{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)\end{array}$

$\begin{array}{c}={\upsilon }_5\left(\frac{{\upsilon }_3{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)\left(1-{\upsilon }_4\right)+{\upsilon }_3{\upsilon }_4{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)\\ ={\upsilon }_5\left(\frac{{\upsilon }_3{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)\left[1-{\upsilon }_4+{\upsilon }_4\right]}{1-{\upsilon }_4}\right)\\ ={\upsilon }_5\left(\frac{{\upsilon }_3{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)\end{array}$

$y_3^L=\frac{{\upsilon }_3{\upsilon }_5^2\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}$ (24)

The variable $y_2^L$ is expressed by ${\upsilon }_8$ and $y_3^L$ as:

$y_2^L={\upsilon }_8{y}_3^L$

$y_2^L={\upsilon }_8\left(\frac{{\upsilon }_3{\upsilon }_5^2\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)$ (25)

If we substitute $x_2^A$ and $x_3^A$ into $x_2^A+x_3^A$ then we obtain:

$x_2^A+x_3^A=x_2^A={\upsilon }_2{y}_1^L-x_1^A+{\upsilon }_3{y}_1^L-x_2^A-x_1^A=$

$x_2^A+x_3^A={\upsilon }_2{y}_1^L-{\upsilon }_1{y}_1^L+{\upsilon }_3{y}_1^L-{\upsilon }_2{y}_1^L+{\upsilon }_1{y}_1^L-{\upsilon }_1{y}_1^L=\left({\upsilon }_3-{\upsilon }_1\right)y_1^L$

Now we substitute $y_1^L$ into the last expression:

$x_2^A+x_3^A=\left({\upsilon }_3-{\upsilon }_1\right){\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)$ (26)

If we substitute Formula (24), (23) and (25) into (14) then we get an expression of $x_1^A$ :

$x_1^A=\frac{\frac{{\upsilon }_3{\upsilon }_5^2\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}-{\upsilon }_5[\left({\upsilon }_3-{\upsilon }_1\right){\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)+\frac{{\upsilon }_3{\upsilon }_4{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}]}{{\upsilon }_5}$

$x_1^A=\frac{\left(\frac{{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)[{\upsilon }_3{\upsilon }_5-{\upsilon }_3{\upsilon }_5+{\upsilon }_1{\upsilon }_5+{\upsilon }_3{\upsilon }_4{\upsilon }_5-{\upsilon }_1{\upsilon }_4{\upsilon }_5-{\upsilon }_3{\upsilon }_4{\upsilon }_5]}{1-V_4}\right)}{V_5}$

Simplify the above expression to get the following expression:

$x_1^A={\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right){\upsilon }_1$ (27)

The sum $x_1^A+x_2^A$ is simplified as follows:

$x_1^A+x_2^A=V_1{y}_1^L+V_2{y}_1^L-V_1{y}_1^L=V_2{y}_1^L$

or

$x_1^A+x_2^A={\upsilon }_2{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)$ (28)

$x_3^A$ has the form:

$x_3^A=\frac{{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)+{\upsilon }_8\left(\frac{{\upsilon }_3{\upsilon }_5^2\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)-{\upsilon }_7\left({\upsilon }_2{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)+\frac{{\upsilon }_3{\upsilon }_4{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)}{1-{\upsilon }_4}\right)}{{\upsilon }_7}$ ,

$x_3^A=\frac{\frac{{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)[1-{\upsilon }_4+{\upsilon }_3{\upsilon }_5{\upsilon }_8-{\upsilon }_2{\upsilon }_7\left(1-{\upsilon }_4\right)-{\upsilon }_3{\upsilon }_4{\upsilon }_7]}{1-{\upsilon }_4}}{{\upsilon }_7}$ ,

$x_3^A=\frac{{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)[1-{\upsilon }_4+{\upsilon }_3{\upsilon }_5{\upsilon }_8-{\upsilon }_2{\upsilon }_7+{\upsilon }_2{\upsilon }_4{\upsilon }_7-{\upsilon }_3{\upsilon }_4{\upsilon }_7]}{\left(1-{\upsilon }_4\right){\upsilon }_7}$ . (29)

Net working capital (NWC): It shows the amount of fixed resources used to finance working capital.

$\text{Networkingcapital}=\text{Currentassets}-\text{Currentliability}$

$\text{NWC}=x_1^A+x_2^A+x_3^A-y_1^L$

The goal of working capital management is to maximize the net working capital.

Denote NWC by f, then it has the following expression:

$\begin{array}{l}\text{NWC}=f={\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right){\upsilon }_1+\left({\upsilon }_2-{\upsilon }_1\right){\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)\\ +\frac{{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)[1-{\upsilon }_4+{\upsilon }_3{\upsilon }_5{\upsilon }_8-{\upsilon }_2{\upsilon }_7+{\upsilon }_2{\upsilon }_4{\upsilon }_7-{\upsilon }_3{\upsilon }_4{\upsilon }_7]}{{\upsilon }_7\left(1-{\upsilon }_4\right)}\\ -{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)\end{array}$

or

$f=\frac{{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)[1-{\upsilon }_4+{\upsilon }_3{\upsilon }_5{\upsilon }_8-{\upsilon }_3{\upsilon }_4{\upsilon }_7-{\upsilon }_7+{\upsilon }_4{\upsilon }_7]}{{\upsilon }_7\left(1-{\upsilon }_4\right)}$ (30)

${\upsilon }_4=1-{\upsilon }_3{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)$

${\upsilon }_7={\upsilon }_5\ast {\upsilon }_6$

${\upsilon }_5=1-{\upsilon }_7$

${\upsilon }_4=\frac{1-{\upsilon }_5}{{\upsilon }_5}$

Taking into account (30) we can simplify NWCF as follows:

$\tilde{f}=\frac{{\upsilon }_5\left({\upsilon }_6-{\upsilon }_8\right)[1-{\upsilon }_4+{\upsilon }_3{\upsilon }_5{\upsilon }_8-{\upsilon }_3{\upsilon }_4{\upsilon }_7-{\upsilon }_7+{\upsilon }_4{\upsilon }_7]}{{\upsilon }_7\left(1-{\upsilon }_4\right)}\to \max$ (31)

We are using the objective function (31) to determine the optimal amount of assets and capital structure of Mongolian mining companies.

As of the TOP 40 mining companies of the world, other indications from the turnover ratios are “good” in terms of median and average values for each of the ratios set out in the Model Methodology for Analyzing the Financial Statements of Business Entities (Ministry of Finance Mongolia, 2015) and it leads to the conclusion that those companies source structure and capital management is optimized.

The standard (Table 2) deviation of the TOP 40 mining companies of the world is significantly lower than mining companies of Mongolia, indicating that these companies maintain a stable capital and resource structure over a long period of time.

The functional limitation conditions for achieving the net working capital target are selected as follows based on the financial stability and capital structure of the mining companies of Mongolian TOP 40 companies.

Based on the financial statements of Mongolian mining companies using Table 3 we can define the lower and upper bound for variables ${\upsilon }_j\left(j=\overline{1,8}\right)$ .

$9.23\ge {\upsilon }_3\ge 1.49$

Table 2. Standard deviations on financial ratios between Mongolia and the world’s TOP-40 mining companies.

Table 3. Functional limitation conditions for NWC.

Table 4. Optimal solutions of ratio.

Table 5. Optimal structure of asset source of Mongolian Mining Companies (%).

Source: Researcher’s calculation.

$0.72\ge {\upsilon }_4\ge 0.48$

$0.67\ge {\upsilon }_5\ge 0.14$

${\upsilon }_6\ge {\upsilon }_8$ (32)

$6.42\ge {\upsilon }_6\ge 0.70$

${\upsilon }_4\ge {\upsilon }_5{\upsilon }_8$

$0.53\ge {\upsilon }_8\ge 0.16$

Problem (31)-(32) is fractional programming and belongs to a class of global optimization.

Numerical Results

Now NWC maximization Problem (31)-(32) is solved numerically on “Matlab” based data of Mongolian mining companies.

Using the above results (Table 4) we can calculate optimal structure of assets.

Looking at the results of Table 5, it is clear that the working capital (28.46 percent of the total capital) of the mining company is less than the non-current capital. From the optimal amount of working capital (28.46%), using equation (29), the maximum amount of inventory is calculated to be 15.49% of total assets. Using the maximum amount of inventory and substituting it into equation (26), the maximum amount of receivables is calculated to be 6.16%. Working capital is defined as the sum of cash, receivables, and inventory, and the minimum amount of cash is determined to be 6.81%. Using the optimal value calculated by us for asset and resource management means that Mongolian mining companies will have good solvency and be able to use their assets effectively, as well as have less financial dependence on others.

3. Conclusions

Capital contributes to the optimal asset structure. Therefore, we propose a fractional programming problem to optimize the asset and capital structure. Companies can use this model of asset and capital structure optimization in their operations. We have optimized the asset and source optimal structure of Mongolian mining companies through function streaming to working net capital purposed value. Numerical results explain that the current assets (28.46% of total assets) of mining companies are lower than non-current assets, this situation is considered as the mining sector’s diversity.

The objective function results also show the potential maximum value of receivables value from liquid assets should be 6.16%, inventory maximum value to be 15.49%, minimum value of monetary assets to be 6.81%. Mongolian mining companies operate with a deficiency of monetary assets, private mining sector company’s receivables value and inventory value are more than optimal value, in outcome they will need to focus on these kinds of assets.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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