# The Development of a Fuzzy Logic System in a Stochastic Environment with Normal Distribution Variables for Cash Flow Deficit Detection in Corporate Loan Policy

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## Abstract

**:**

## 1. Introduction

## 2. State of the Art

- It is the first financial management tool that uses an artificial intelligence technique for bank lending issues;
- The Mamdani fuzzy logic system uses stochastic fuzzy variables, which allows for the identification of probable lending periods in which credit reimbursement risk may occur;
- It examines the lending risk for future periods during the reimbursement period according to two important economic indicators, namely, EBITDA and changes in working capital $(\Delta {W}_{K})$. The values of these two indicators are also probabilistic values;
- It allows for identification of cash-flow deficits for future periods of time underlying the credit reimbursement risk during the lending period;
- It allows preventive measures to be taken in order to avoid or even mitigate bank lending risks.

## 3. The EBITDA and Available Cash-Flow Rule, and the Influence of Stochastic Elements

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

- $\frac{EBIT\left(t\right)}{EBIT\left({t}_{n-1}\right)}$. has a log-normal distribution;
- $\mathrm{ln}\frac{EBIT\left(t\right)}{EBIT\left({t}_{n-1}\right)}$ is normally distributed;
- $\mathrm{ln}\frac{EBIT\left(t\right)}{EBIT\left({t}_{n-1}\right)}$, with random, independent and finite variables.

**Theorem**

**2:**

**Proof:**

- $\frac{\partial CFA}{\partial t}=0$;
- $\frac{\partial CFA}{\partial EBIT}=\frac{1}{EBIT}$;
- $\frac{{\partial}^{2}CFA}{\partial EBI{T}^{2}}=-\frac{1}{EBI{T}^{2}}$.

## 4. Cash Flow Deficits during Lending Period

## 5. Identification of Fuzzy Stochastic Variables

**Definition**

**2.**

**Definition**

**3.**

- (1)
- The variable EBIT = (x, a, b) is defined by the membership function of the form:$${\mu}_{EBIT}\left(z\right)=\{\begin{array}{c}1-\frac{x-z}{a},\mathrm{for}x-a\le z\le x\\ 1-\frac{z-x}{b},\mathrm{for}x\le z\le x+b\\ 0,\mathrm{otherwise}\end{array}$$
- (2)
- The interval [a, b] is defined by the variable probability of occurrence of the variable$\left(EBI{T}_{z}\right)$which is given by the relation:$EBIT({t}_{0}){e}^{\left(\mu -\frac{1}{2}{\sigma}^{2}\right)t-\alpha \sigma \sqrt{t}}\le EBIT\left(z\right)\le {e}^{\left(\mu -\frac{1}{2}{\sigma}^{2}\right)t+\alpha \sigma \sqrt{t}}EBIT({t}_{0})$;
- (3)
- The number of membership degrees for the stochastic fuzzy triangular variable$\left(EBI{T}_{z}\right)$is located in the range:$\mu \left(EBIT({t}_{0}){e}^{\left(\mu -\frac{1}{2}{\sigma}^{2}\right)t-\alpha \sigma \sqrt{t}}\right)\le \mu \left(EBIT\left(z\right)\right)\le \mu \left({e}^{\left(\mu -\frac{1}{2}{\sigma}^{2}\right)t+\alpha \sigma \sqrt{t}}EBIT({t}_{0})\right)$;
- (4)
- $EBIT\left(t\right)$is a random variable with the distribution density function$f\left(EBIT\right)=\frac{1}{\sqrt{2\pi {\sigma}^{2}}}{e}^{-\frac{{\left(EBIT-\mu \right)}^{2}}{2{\sigma}^{2}}}$and with the distribution function written as a primitive of the distribution density function${{\displaystyle \int}}_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi {\sigma}^{2}}}{e}^{-\frac{{\left(EBIT-\mu \right)}^{2}}{2{\sigma}^{2}}}dt=1.$

## 6. Elaboration of the Mamdani Fuzzy Logic System in a Stochastic Environment

**The existing problem**: Most companies have difficulties in assessing the risks involved in banking lending policy. This is because the factors influencing these kinds of risks are extremely complex and diversified. It is not known how the EBITDA will fluctuate or how the working capital will evolve $(\Delta {W}_{K})$ over the lending period to prevent any financial difficulties caused by the lack of available cash-flow from which credit rates and interest are reimbursed. Thus, a financial management tool that detects bank lending periods in which the company may face financial difficulties in credit reimbursement must be identified. The credit reimbursement risk arises in the form of cash-flow deficits that occur during the bank lending period and lead to the impossibility of repaying bank loans and interest, with immediate consequences on the company’s operating activity. The financial management tool which is developed within this study will allow companies to set preventive measures over bank lending periods to avoid as much as possible the credit reimbursement risk mentioned above.

**Suggested solution:**For the detection of the cash-flow deficit risk during the lending period, as well as to establish preventive measures to mitigate this risk, a Mamdani fuzzy logic system with stochastic fuzzy variables has been developed. These variables allow for the study of reimbursement risk at certain probabilistic value ranges, in order to determine whether this risk may occur during these intervals. The output variable (CFA) indicates whether there is a probability of cash-flow deficits during these intervals.

**Definition**

**4.**

- The input variables of the fuzzy logic system are stochastic, normally distributed, and have the estimated values determined by the relation:$X({t}_{0}){e}^{\left(\mu -\frac{1}{2}{\sigma}^{2}\right)t-\alpha \sigma \sqrt{t}}\le X\left(t\right)\le {e}^{\left(\mu -\frac{1}{2}{\sigma}^{2}\right)t+\alpha \sigma \sqrt{t}}X({t}_{0})$, with the corresponding fuzzy set${\mu}_{X}:X\to \left[0,1\right]$;
- The fuzzy logic system rules are formed on the variation intervals of the fuzzy input stochastic variables and from the fuzzy modeling of the rules, with the output variable also being a stochastic and fuzzy variable;
- The rule-based inference operation is a max-min type, specific to the Mamdani fuzzy logic system;
- The fuzzy defuzzification of the result after the fuzzy inference operation is obtained with the Center of Aria (COA) method or the centroid defuzzification method. The results obtained using this type of defuzzification are sufficiently linear to obtain a control curve without sudden variations.

**Step 1. Input variables identification:**In the input variables category are included the variables that directly affect the available cash-flow and those considered the variables with direct impact on the cash flow deficit, being EBIT(t) and $\Delta {W}_{K}$, being.

**Step 2. Identification of the stochastic fuzzy output variable:**the stochastic fuzzy output variable is the available cash-flow whose membership function is of the form:

**Step 3. The Probabilistic Fuzzy Rule Base:**This is constructed starting from the linguistic values assigned to them so as to set the CFA values to influence the company’s cash flow deficits during the lending period. The resulting rule basis is shown in Table 1.

**Step 4. Defuzzification:**The final step in the probabilistic fuzzy logic system is the determination of the output variable, the defuzzification of the fuzzy result obtained after the fuzzy inference operation. This operation is known in the literature as the centroid defuzzification, or the Center of Aria (COA) method. The results obtained by this type of defuzzification are sufficiently linear in order to obtain a control curve without sudden variations.

- For continuous variables:$$\Delta CF{A}^{\ast}=\frac{{{\displaystyle \int}}_{-CFA}^{+CFA}\Delta CFA{\mu}_{\Delta CF{A}^{\ast}}\left(\Delta CFA\right)d\Delta CFA}{{{\displaystyle \int}}_{-\%}^{+\%}{\mu}_{\Delta CF{A}^{\ast}}\left(\Delta CFA\right)d\Delta CFA}$$
- For discrete variables:$$\Delta CF{A}^{\ast}=\frac{{{\displaystyle \sum}}_{-CFA}^{+CFA}\Delta CFA{\mu}_{CF{A}^{\ast}}\left(\Delta CFA\right)}{{{\displaystyle \sum}}_{-CFA}^{+CFA}{\mu}_{\Delta CF{A}^{\ast}}\left(\Delta CFA\right)}$$

## 7. Implementation of the Probabilistic Fuzzy Logic System for Detecting Cash-Flow Deficiencies in Bank Lending Policies

## 8. Fuzzy Logic Simulation in a Stochastic Environment

## 9. Conclusions

- The system allows for the study of the behavior of input variables for future periods of time, making it easy to anticipate situations of financial difficulty within a company;
- The system allows for fuzzy modeling of the evolution and behavior of the output variables on which the financial decisions are based. Values of output variables are determined for future time periods by probabilistic calculations, which makes it possible to identify the anticipation of financial risks. Consequently, the company can take measures to mitigate or even avoid risks;
- The system helps to substantiate the company’s financial decisions by anticipating the development of exogenous factors with high risk, which have an unfavorable influence on the company’s economic performance.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**The membership function for earnings before interest and taxes (EBIT(t)) in probabilistic environment.

**Figure 3.**The membership function for change in working capital ($\Delta {W}_{K}$) in a probabilistic environment.

**Figure 4.**The membership function for the output variable available cash-flow (CFA) in a probabilistic environment.

**Figure 6.**Probabilistic available cash-flow according to the fuzzy logic system and the membership function.

EBIT | Low (L) | Medium (M) | High (H) | |
---|---|---|---|---|

$\mathbf{\Delta}{\mathit{W}}_{\mathit{K}}$ | ||||

Low (L) | Low (L) | Medium (M) | Very high (VH) | |

Medium (M) | Low (L) | Medium (M) | High (H) | |

High (H) | Very low (VL) | Low (L) | Medium (M) |

Year (Lending Period) | Cash-Flow Requirements (CFR) | Available Resources $\left({\mathit{R}}_{\mathit{a}}\right)$ | Cash-Flow Deficit Formula |
---|---|---|---|

Year 1 | $CFR\left(1\right)=\left(\frac{IK}{R{P}_{c}}+Ir\left(IK-1\times {R}_{c}\right)\right){\left(1+{r}_{a}\right)}^{1}$ | ${R}_{a}\left(1\right)=\frac{IK}{EL}{\left(1+{r}_{a}\right)}^{1}$ | $CFD\left(1\right)=\left(\frac{IK}{R{P}_{c}}+Ir\left(IK-1\times {R}_{c}\right)-\frac{IK}{EL}\right){\left(1+{r}_{a}\right)}^{1}$ |

Year 2 | $CFR\left(2\right)=\left(\frac{IK}{R{P}_{c}}+Ir\left(IK-2\times {R}_{c}\right)\right){\left(1+{r}_{a}\right)}^{2}$ | ${R}_{a}\left(2\right)=\frac{IK}{EL}{\left(1+{r}_{a}\right)}^{2}$ | $CFD\left(2\right)=\left(\frac{IK}{R{P}_{c}}+Ir\left(IK-2\times {R}_{c}\right)-\frac{IK}{EL}\right){\left(1+{r}_{a}\right)}^{2}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

Year m | $CFR\left(m\right)=\left(\frac{IK}{R{P}_{c}}+Ir\left(IK-m\times {R}_{c}\right)\right){\left(1+{r}_{a}\right)}^{m}$ | ${R}_{a}\left(m\right)=\frac{IK}{EL}{\left(1+{r}_{a}\right)}^{m}$ | $CFD\left(m\right)=\left(\frac{IK}{R{P}_{c}}+Ir\left(IK-m\times {R}_{c}\right)-\frac{IK}{EL}\right){\left(1+{r}_{a}\right)}^{m}$ |

Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

EBIT ** | 1000 | 1050 | 1025 | 1010 | 1075 | 1005 | 1040 | 1090 | 1055 | 1050 |

**Table 4.**$\Delta {W}_{K}$ evolution resulting from the company’s operating activity over a 10 month period *.

Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

$\Delta {{W}_{K}}^{\ast \ast}$ | 500 | 550 | 515 | 520 | 505 | 525 | 540 | 590 | 525 | 510 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Boloș, M.-I.; Bradea, I.-A.; Delcea, C.
The Development of a Fuzzy Logic System in a Stochastic Environment with Normal Distribution Variables for Cash Flow Deficit Detection in Corporate Loan Policy. *Symmetry* **2019**, *11*, 548.
https://doi.org/10.3390/sym11040548

**AMA Style**

Boloș M-I, Bradea I-A, Delcea C.
The Development of a Fuzzy Logic System in a Stochastic Environment with Normal Distribution Variables for Cash Flow Deficit Detection in Corporate Loan Policy. *Symmetry*. 2019; 11(4):548.
https://doi.org/10.3390/sym11040548

**Chicago/Turabian Style**

Boloș, Marcel-Ioan, Ioana-Alexandra Bradea, and Camelia Delcea.
2019. "The Development of a Fuzzy Logic System in a Stochastic Environment with Normal Distribution Variables for Cash Flow Deficit Detection in Corporate Loan Policy" *Symmetry* 11, no. 4: 548.
https://doi.org/10.3390/sym11040548