# On Present Value Evaluation under the Impact of Behavioural Factors Using Oriented Fuzzy Numbers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Behavioural Essence of Present Value

- the receivables are discounted by a higher discount rate than liabilities and
- smaller amounts are discounted by a lower discount rate than large amounts.

## 3. Oriented Fuzzy Numbers—Basic Facts

**Definition**

**1.**

**Theorem**

**1.**

- FN $\mathcal{L}\left(a,b,c,d,{L}_{L},{R}_{L}\right)$ is represented by its membership function$${\mu}_{L}\left(x\right)={\mu}_{L}\left(x|a,b,c,d,{L}_{L},{R}_{L}\right),$$
- FN $\mathcal{K}\left(e,f,g,h,{L}_{K},{R}_{K}\right)$ is represented by its membership function$${\mu}_{K}\left(x\right)={\mu}_{K}\left(x|e,f,g,h,{L}_{K},{R}_{K}\right).$$

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

## 4. Oriented Fuzzy Present Value

- $\stackrel{\u02c7}{P}$ is a quoted price,
- $\left[{V}_{s},{V}_{e}\right]\subset {\mathbb{R}}^{+}$ is an interval of all possible values of PV,
- $\left[{V}_{f},{V}_{l}\right]\subset \left[{V}_{s},{V}_{e}\right]$ is an interval of all prices which do not noticeably differ from a quoted price $\stackrel{\u02c7}{P}$.

## 5. Behavioural Present Value

_{0}. If

## 6. Interval Representation of Behavioural Present Value

- P
_{min}the minimal PPV expected under the financial equilibrium condition (32), - ${P}_{max}$ the maximal PPV expected under the financial equilibrium condition (32).

- ${V}_{min}$ the minimal PPV expected for the quoted price $\stackrel{\u02c7}{P}$,
- ${V}_{max}$ the maximal PPV expected for the quoted price $\stackrel{\u02c7}{P}$.

- $\stackrel{\u02c7}{P}$ a quoted price,
- ${P}_{0}$ a balanced price,
- ${P}_{min}$ the minimal PPV expected under financial equilibrium condition (32),
- ${P}_{max}$ the maximal PPV expected under financial equilibrium condition (32),
- $\alpha $ a sentiment index.

**Example**

**1.**

- minimal PPV is ${P}_{min}=10\$$,
- maximal PPV is ${P}_{max}=60\$$.

## 7. Fuzzy Representation of Behavioural Present Value

- if the disequilibrium condition (30) is met, then rationale excludes the decrease in a quotation;
- if the disequilibrium condition (31) is met, then rationale excludes the increase in a quotation; and
- if the equilibrium condition (32) is met, then rationale cannot exclude any future quotation.

**Example**

**2.**

## 8. Behavioural Present Value Represented by Oriented Fuzzy Numbers

**Example**

**3.**

- all overvalued assets have identical BPV graphs,
- all fully valued assets have identical BPV graphs,
- all undervalued assets have identical BPV graphs.

## 9. Oriented Expected Return Determined by Behavioural Present Value

- Predicted FV ${V}_{t}$,
- Evaluated PV ${V}_{0}$.

- If O-BPV describes a subjective belief about rise in quotations, then we can anticipate a decline in the expected return rate.
- If O-BPV describes a subjective belief about fall in quotations, then we can anticipate an upturn in the expected return rate.

**Example**

**4.**

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A graphs of membership function of F-BPV (

**a**) for overvalued assets, (

**b**) for fully valued assets and (

**c**) for undervalued assets.

**Figure 2.**A graph of membership function of O-BPV predicting rise in price (

**a**) for overvalued assets, (

**b**) for fully valued assets and (

**c**) for undervalued assets.

**Figure 3.**A graph of membership function of O-BPV predicting fall in price (

**a**) for overvalued assets, (

**b**) for fully valued assets and (

**c**) for undervalued assets.

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**MDPI and ACS Style**

Piasecki, K.; Łyczkowska-Hanćkowiak, A.
On Present Value Evaluation under the Impact of Behavioural Factors Using Oriented Fuzzy Numbers. *Symmetry* **2021**, *13*, 468.
https://doi.org/10.3390/sym13030468

**AMA Style**

Piasecki K, Łyczkowska-Hanćkowiak A.
On Present Value Evaluation under the Impact of Behavioural Factors Using Oriented Fuzzy Numbers. *Symmetry*. 2021; 13(3):468.
https://doi.org/10.3390/sym13030468

**Chicago/Turabian Style**

Piasecki, Krzysztof, and Anna Łyczkowska-Hanćkowiak.
2021. "On Present Value Evaluation under the Impact of Behavioural Factors Using Oriented Fuzzy Numbers" *Symmetry* 13, no. 3: 468.
https://doi.org/10.3390/sym13030468