NEUTROSOPHIC STATISTICAL APPROACH
TO THE CAPITAL ASSET PRICING MODEL
by
Keshani Nisansala Udawatta Kankanamge
M.Sc., University of Colombo Sri Lanka, 2015
B.Sc., University of Colombo Sri Lanka, 2011

THESIS SUBMITED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
MATHEMATICS

UNIVERSITY OF NORTHERN BRITISH COLUMBIA
October 2024
© Keshani Nisansala Udawatta Kankanamge, 2024

Abstract
Neutrosophic Statistics is an emerging trend in the field of statistics, designed to
address the challenges of uncertainty. Given that the stock market is inherently uncertain, this
study aims to apply a Neutrosophic statistical approach to the Capital Asset Pricing Models.
The neutrosophic methodologies, techniques and calculations are consistently used
through out the study to address the significant criticisms of the existing model such as
unrealistic assumptions, misleading beta and single valued risk and return. The study findings
effectively capture these criticisms up to certain extent and offer customized neutrosophic
models that investors can use based on their specific investment needs. Further these results
may be able to explain the relationship between risk and return which many of the CAPM
studies struggle to justify in literature. The models’ flexibility and ability to capture the
indeterminacy may enhance the quality of the results. Additionally, this approach enhances the
reliability and accuracy of financial modeling in decision making. Not only finance, but also
these sophisticated neutrosophic methodologies can be employed in most of the real-world
scenarios with full of indeterminacy.
Netflix is a leading company in the streaming industry, and it has high trading volume
which reflects the market sentiment. Netflix stock prices, S&P500 market index, NASDAQ
market index, Gross Domestic Product (GDP) and Consumer Price Index (CPI) from 20092023 have been used to conduct 5-year and 15-year period analysis using the proposed models
in this study. The primary goal is to develop one-factor, two-factor, and three-factor
Neutrosophic models. The focus is on calculating appropriate neutrosophic beta values to
develop different Neutrosophic Models and then calculate the neutrosophic expected return
across four different scenarios.

TABLE OF CONTENT
Abstract

ii

Table of Contents

iii

List of Figures

iv

List of Tables

v

Glossary

vi

Acknowledgement

vii

1. Chapter 01
1.0. Introduction
1.1. Introduction to Neutrosophic Statistics
1.1.1. Neutrosophic Algebra
1.2. Introduction to CAPM
1.2.1. Introduction to Arbitrage Pricing Theory (APT)
1.2.2. Key Differences between CAPM and APT
1.3. Relevance of Neutrosophic Statistics to CAPM and APT
1.4. Motivation
1.5. Main Objectives

1
3
6
8
12
14
15
18
19

2. Chapter 02- Literature review
2.0. Literature review for CAPM and APT models
20
2.1. Literature review for Neutrosophic statistics to capture the indeterminacy associated
with financial decision making
27
3. Chapter 03-Data availability
3.0. Introduction to Data availability
3.1. Indeterminacy of the data set
3.2. Data sets
3.3. Charts and Graphs
4. Chapter 04-Research Methodology
4.0. Introduction to Traditional CAPM and its Limitation
4.0.1. Basic steps to calculate Traditional CAPM
4.0.2. Limitations of Traditional CAPM
4.1. Neutrosophic Statistical Approach to the traditional CAPM
4.2. Basic Techniques and Methodologies
4.2.1. Time Frame and Time interval
4.2.2. Smoothing Technique
4.2.3. Generate a neutrosophic number a+bi
4.2.4. Neutrosophic Return (NR)
4.2.5. Calculations of Factors’ Return
iii

30
30
33
43
46
47
50
51
54
56
57
58
60
62

4.2.6. Obtain Neutrosophic Beta
63
4.3. Application of Neutrosophic Methods to Calculate Neutrosophic Beta
65
4.4. Multi-Factor Models
75
4.4.1. Neutrosophic Capital Asset Pricing Model (NCAPM)
75
4.4.2. Neutrosophic Arbitrage Pricing Theory (NAPT)
76
4.4.3. Development of advanced Neutrosophic Statistical Factor Models 76
4.5. Neutrosophic Correlation Coefficient
79
4.6. Statistical Hypothesis testing technique
80
4.6.1. T- statistic for the neutrosophic models
81
4.7. Compare the different models using Mean Average Deviation (MAD)
82
5. Chapter 05 - Results and Discussion
5.0. Introduction to Results and Discussion
83
5.1. NFLX monthly β and E(Ri)
83
5.1.1. The result and discussion for the Monthly beta of NFLX stocks with S&P 500
Index
84
5.1.2. The result and discussion for the Monthly beta of NFLX stocks with NASDAQ
Index
84
5.2. Results and discussions for the Nβ1 and NE(Ri) values based on neutrosophic
calculations
85
5.2.1. Results and Discussion based on Scenarios (15-years)
86
5.2.2. Results and Discussion based on the Models (15-years)
93
5.2.3. Results and Discussions based on average Nβ1 and average NE(Ri) on
scenarios (15-years period)
96
5.2.4. Results and Discussion based on Nβ1 for 5-year periods from 2019-2023,
2017-2019 and 2009-2013
98
5.2.5. Results and Discussion based on 5-Year Neutrosophic Expected return from
2019-2023
99
5.2.6. Results and Discussion based on 5-YearAverage NE(Ri) on Scenarios and
Models from 2019-2023.
101
5.3. Results and discussion for Nβ2 and Nβ3
101
5.3.1. Neutrosophic Beta 2 (Nβ2)
102
5.3.2. Neutrosophic Beta 3 (Nβ3)
103
5.4. Correlation coefficient
104
5.5. Model Verification
105
5.5.1. Hypothesis and t-statistics for each scenario of Model 01
105
5.6. Model Accuracy of multi-factor Models for the Expected Returns
108
5.7. Further Model Applications
109
5.7.1. Results of Amazon stock based on classical and neutrosophic CAPM
109
5.7.2. Results of Apple stock based on classical and neutrosophic CAPM
111
6. Chapter 06- Conclusions, Benefits, Limitations and Future Scope
6.0. Introduction to Conclusions, Benefits, Limitations and Future Scope
6.1. Conclusions
iv

113
113

6.2. Benefits of proposed models
6.3. Limitations
6.4. Future Scope

117
118
119

7.0.References

121

8.0.Appendix
8.1.Coding Appendix

126

List of Figures
Graph 1.2.1: Total Portfolio Risk.
Graph 1.2.2: Expected Return and ߚ
Graph 1.3.1: Hourly Historical Data Set-Netflix.
Graph 3.2.1.: Netflix stock price trend.
Graph 3.2.2: Netflix stock price.
Graph 3.2.3: GDP quarterly.
Graph 3.3.1: Candlestick chart illustration from NFLX stock
Graph 3.3.2: Uptrend illustration from NFLX one day price movement
Graph 4.0.1: Netflix stock price comparison with NASDAQ and S&P500.
Graph 4.0.1.1: Undervalued and overvalued stocks
Graph 4.1.1: Monthly adjusted-closing price of the NFLX from 2009-2023
Graph 4.1.2: NFLX monthly High, Low and Adjusted Closing Prices from 2009-2023
Figure 3.3.1: Bullish and Bearish Candlesticks
Figure 3.3.2: Uptrend and Downtrend lines
List of Tables
Table 1.1.1: Crisp Numbers and Neutrosophic Numbers.
Table 1.2.2.1: Key differences in the assumptions of the CAPM and the APT models
Table 3.2.1: Details of Netflix stock.
Table 4.2.6.1: Type of data required for OLS table to calculate β for different factor models
Table 4.2.6.2: NLS table for one-factor model with respect to S&P 500 market index
Table 4.4.3.1: Different Factor models
Table 5.1.1 β and E(Ri) of NFLX with respect to S&P500.
Table 5.1.2 β and E(Ri) of NFLX with respect to NASDAQ.
Table 5.2.1.1 Scenario 01:S&P 500 Index with original data.
Table 5.2.1.2 Scenario 02: NASDAQ Index with original data.
Table 5.2.1.3 Scenario 03:S&P 500 Index with MA.
Table 5.2.1.4 Scenario 04: NASDAQ Index with MA.
Table 5.2.2.1 Nβ1 Summary for 15-year period (2009-2023).
Table 5.2.2.2 NE(Ri) Summary for 15-year period (2009-2023).
Table 5.2.3.1 Average Nβ1 for each scenario for 15-years period (2009-2023).
Table 5.2.3.1 Average NE(Ri) for each scenario for 15-years period (2009-2023).
v

Table 5.2.4 Discussion based on Nβ1 for 5-year periods from 2019-2023, 2017-2019 and
2009-2013.
Table 5.2.5.1 5-Year NE(Ri) under different Scenarios and Models from 2019-2023.
Table 5.2.6.1 5-YearAverage NE(Ri) under different Scenarios and Models from 2019-2023.
Table 5.3.1 Nβ2 values for 15-year period (2009-2023) and most resent 5 Years (2019-2023).
Table 5.3.2 Nβ3 values for 15-year period (2009-2023) and most resent 5 Years (2019-2023).
Table 5.3.1 Correlation coefficients matrix of the Factors.
Table 5.6.1 Model Accuracy measure (NMAD) for the NE(Ri) under Four Scenarios.
Table 5.7.1.1 Beta and NE(Ri) of Amazon stock based on classical CAPM.
Table 5.7.1.2 Average Nβ1 of Amazon stock based on neutrosophic calculations.
Table 5.7.1.3 NE(Ri) of AMZN stock under four scenarios and from four models (NCAPM
and NAPT).
Table 5.7.2.1 Beta and E(Ri) of Apple stock based on classical CAPM.
Table 5.7.2.2 Average E(Ri) of AAPL stock based on neutrosophic calculations.
Table 5.7.2.3 NE(Ri) of AAPL stock under four scenarios and from four models (NCAPM
and NAPT).
Glossary
AAPL: ticker symbol for Apple Inc
AMZN: ticker symbol for Amazon Inc
APT: Arbitrage Pricing Theory
CAPM: Capital Asset Pricing Model
CM: classical mean
CPI: Consumer Price Index
GDP: Gross Domestic Product
I: Indeterminacy
LL: lower limit of the return
MA: Moving Average
MAD: Mean Average Deviation
NAPT: Neutrosophic Arbitrage Pricing Theory
NASDAQ: National Association of Securities Dealers Automated Quotations
NCAPM: Neutrosophic Capital Asset Pricing Model
NFLX: ticker symbol for Netflix Inc
NLS: Neutrosophic Least Square
NM: neutrosophic mean
NR: Neutrosophic Return
NS: Neutrosophic Statistics
NE(Ri): Neutrosophic Expected Return
Nβ: Neutrosophic Beta
Nβ1: Neutrosophic Beta 1
Nβ2: Neutrosophic Beta 2
Nβ3: Neutrosophic Beta 3
OLS: Ordinary Least Square
S&P500: Standard & Poor's 500
vi

UL: upper limit of the return
Acknowledgement

My heartfelt gratitude goes to my supervisor, Dr. Pranesh Kumar, for his unwavering
encouragement, support, and guidance throughout my research journey. His expertise in
statistics and enthusiasm for the field of Neutrosophic statistics were significant sources of
inspiration, driving me to explore and contribute to this innovative area.
I am also greatly thankful to my committee members, Dr. Edward Dobrowolski and Dr. Hossein
Kazemian, for their invaluable insights, constructive feedback, and continuous support. Their
guidance was instrumental in refining my work and guaranteeing its academic rigor.
Lastly, I would like to extend my appreciation to the Faculty of Mathematics and Statistics as
well as the faculty of business and Economics at the University of Northern British Columbia
(UNBC) for providing the resources and environment necessary to carry out this research.
Thank you all for your contributions to this thesis on the Neutrosophic Statistical Approach
to the Capital Asset Pricing model (CAPM).

vii

Chapter 01
1.0 Introduction
Everything around us in the world is full of indeterminacy and these indeterminacies may
occur due to imprecise, incomplete, or unknown data values and/or size of data set.
Neutrosophic statistics offers a framework to handle these indeterminacies and uncertainties
effectively. Weather forecasting, epidemiological modeling, financial risk management,
climate change projections, supply chain management and natural disaster preparedness are
some examples of indeterminacies in real-world scenarios where neutrosophic statistics could
be applied.
Among all these fields, this research is focused to address the indeterminacy in the stock
market which is inherently uncertain. Neutrosophic concepts can be used to address the
uncertainty and indeterminacy in the stock market (Abdelfattah et al., 2024). By capturing
these complexities this study aims to develop more sophisticated version of the often-used
financial economic model called Capital Asset Pricing Model
Basically, this research directs to investigate a new way of understanding of the Capital Asset
Pricing Model (CAPM) using Neutrosophic logic and Neutrosophic Statistics. CAPM is a
widely used model in financial decision-making. In financial decision-making, indeterminacy
refers to the inherent uncertainty and ambiguity surrounding financial variables, outcomes, and
different market conditions. These uncertainties such as market volatility, market risk,
unrealistic assumptions in financial modeling, unpredictable regulatory environment,
corporate decision-making, globalization and geopolitical risks due to currency fluctuations,
set challenges for investors, financial analysts, and decision-makers. Since Neutrosophic

1

statistics provide an effective means of handling all types of indeterminacies, this study helps
to reduce the indeterminacy associate with financial market. (Smarandache,2022)
CAPM is a controversial financial decision-making method, primarily due to its unrealistic
assumptions. Time to time the researchers and financial analysts strive to address the
limitations of the CAPM framework, by modifying or developing independent variables that
are perceived to be more relevant and not adequately addressed by CAPM. For illustration
extended CAP models were introduced in different periods such as Intertemporal CAPM, Fama
French three factor model, Fama French five factor model and Arbitrage pricing theory. This
research can be introduced as a novel approach to the classical CAPM by addressing
indeterminacies in stock market and limitations of the model. By applying neutrosophic
statistical theories to CAPM, analysts can more effectively address the uncertainties and
ambiguities inherent in financial markets, allowing more robust and accurate inference in
complex and uncertain financial environments. (Smarandache,2022)
The stock market can be explained as one of the vital components of the global economy but
highly unpredictable and volatile. Indeterminacy increases the complexity of factors
influencing trend fluctuations in the stock market, making it challenging for investors and
analysts to predict and interpret market movements accurately. By addressing the
indeterminacy, decision makers can better assess the risks and uncertainties associated with
their portfolios. Further this proposed model with neutrosophic logic may help investors to
identify their risk boundaries. The stock market which is characterised by a collection of
neutrosophic random variables, presents a complex challenge for investors, financial analysts,
and researchers.

2

In this research study, we develop the neutrosophic CAP models and estimate their parameters.
Further for illustration, we use NASDAQ market Index, S&P500 market index, NFLX stock
prices, AMZN stock prices, AAPL stock prices, risk-free rates, GDP and CPI from 2009-2023.
MS Excel and python in Jupyter notebook are used for the calculations and graphing. In this
analysis, historical data sets in Yahoo finance are used to calculate Neutrosophic stock price,
Neutrosophic return, Neutrosophic beta, Neutrosophic expected returns and other considerable
factors.
1.1.Introduction to Neutrosophic Statistics
Neutrosophic logic was first introduced by Florentin Smarandache in the 1990s as a
generalization of fuzzy logic and intuitionistic logic. It allows for dealing with incomplete,
imprecise, and uncertain information. Neutrosophic statistics can be applied in various
statistical analyses and modeling tasks, including descriptive statistics, data analysis,
hypothesis testing, regression analysis, time series analysis and decision making under
uncertainty.
According to Smarandache (51), a random neutrosophic sample of size ݊ from a classical or
neutrosophic population is a sample of ݊ individuals such that at least one of them has some
indeterminacy. He explains this nicely through below example.
“Consider a random sample of 1,000 homes, in a city of over one million inhabitants, in order
to investigate how many houses have at least a laptop. One finds out that 600 houses have at
least one laptop, 300 houses don’t have any laptop, while 100 houses have each of them a
single laptop, but not working. Some of these 100 house owners tried to have their laptop

3

fixed, others said their laptops’ hard drives have crashed and it is little chance to fix them.
Therefore indeterminacy. We have a simple random neutrosophic sample of size 1000.”
Neutrosophic statistics extend beyond classical statistics by allowing the representation of
indeterminacy through sets, intervals, or approximate values, acknowledging the ambiguous,
vague, imprecise, incomplete, or unknown nature of data. This flexibility in handling
uncertainty is valuable when dealing with real-world scenarios where exact values may not be
available or feasible to obtain.
We can replace any indeterminate parameter ܽ by ܽே in neutrosophic statistics. ܽே can be
imprecise, unsure, and even completely unknown. Further ܽே can be a neighbour or an interval
that includes ܽ and it can be any set, which approximates ܽ. In worst scenario ܽே could be
unknown and in best scenario ܽே could be equals to ܽ.
In traditional statistics, sample sizes are often precise numerical values, like 10, 50, or 100.
However, in real life, most of the time we deal with approximations instead of exact numbers.
Hence instead of using exact number, it is possible to deal with sets/intervals. In neutrosophic
statistics, sample sizes can be sets or intervals, such as [5, 10] or [90, 100], representing
uncertain or imprecise information about the sample size. The sample size is [90, 100] in
neutrosophic signifies that the analyst isn't entirely certain whether these individuals fully
belong, partially belong, or do not belong to the population under study. There are several
examples of imprecise data in day-to-day life such as flock of migratory birds, trees in a jungle
and fish in a river. So, there are many of such cases when the sample size may not be exactly
known. Practically in the real world it is not possible to exactly estimate a sample or population
size of most of the events (Smarandache,2022).

4

In essence, neutrosophic samples and neutrosophic statistics provide a way to handle and
represent indeterminacy in data analysis by allowing for the use of sets, intervals, or
approximations instead of relying solely on precise numerical values.
Hence Neutrosophic Statistics can be considered as a generalization of the classical statistics
and deals with set values instead of crisp values. As an illustration the table 1.1.1 represents
crisp numbers and neutrosophic numbers separately.
Crisp Number

Neutrosophic Number

2

[0,5]

0.5

[0.1,0.6]

-3

[-5,0]

8

[0,10]

Table 1.1.1: Crisp Numbers and Neutrosophic Numbers.

The neutrosophic statistical methods can be used to interpret and organize the neutrosophic
data to discover underlying patterns. Specially, neutrosophic statistics can be used to analyze
data sets with uncertain or contradictory information. By representing data as neutrosophic
numbers, we can account for the inherent uncertainties in the data and make more reliable
statistical conclusions.
A neutrosophic number ܰ has the form: ܰ = ݀ + ݅, where ݀ is the determinate /sure part of
ܰ, ݅ is the indeterminate /unsure part of ܰ. For example, ܽ = 2 + ݅, where ݅ ∈ [0, 0.3], is
equivalent to ܽ ∈ [2, 2.3], so for sure ܽ ≥ 2 and this indicates that the determinate part of ܽ is
2, while the indeterminate part ݅ ∈ [0, 0.3] means the possibility for number ܽ to be a little
bigger than 2 but smaller than or equal to 2.3. Logically a neutrosophic number can be written
5

in different ways. Reconsider, ܽ = 2 + ݅, with ݅ ∈ [0, 0.3], or ܽ = 1 + ݅1, with ݅1 ∈ [1, 1.3], or,
in general, ܽ = ∝ + ݅∝, with ݅∝ ∈ [2−∝, 2.3−∝], and ∝ any real number. Or, in opposite way,
ܽ = 2.3 – ݅2, with ݅2 ∈ [0, 0.3], and in general, ܽ = ߚ − ݅ߚ, with ݅ߚ ∈ [ߚ – 2.3, ߚ − 2], and ߚ any
real number. (Smarandache,2014)
We can also represent data and statistical parameters using neutrosophic numbers, which
consist of three components: the degree of truth (T), the degree of indeterminacy (I), and the
degree of falsehood (F). These components represent the extent to which a particular value or
parameter is true, uncertain, or false, respectively. (Zhang et al.,2014)
1.1.1 Neutrosophic Algebra
a) Addition: For I ∈ [0, 1], let consider two neutrosophic numbers N1 =5+2I and N2=52I. The indeterminate part of N1 is 2I=[0,2] and the indeterminate part of N2 is -2I=[0,2]. Therefore, N1 +N2=5+2I+5-2I=10.
By adding N1 +N2, the indeterminate parts cancel out as 2I-2I=0. Determinate part is
equal to 10.
b) Subtraction: For I ∈ [0, 1], let N1 = 5+2I and N1 = 5-2I. The indeterminate part of N1
is 2I=[0,2] and the indeterminate part of N2 is -2I=[0,-2]. Therefore, N1-N2 = 5+2I-5+2I
= 4I
Let’s take other neutrosophic example: For I ∈ [2, 3], N3 = 6+4I and N4= 4+5I. Then,
N3-N4= 6+4I-4-5I = 2-I = 2-[2,3] = [0,-1].
c) Multiplication: For I ∈ [0, 1], let N1 =5+2I and N2=5-2I. Then the product
N1×N2=(5+2I)×(5-2I)=25-4I (I2=I, because indeterminacy × indeterminacy =
indeterminacy)
6

d) Division: For N1 =2+I and N2=4+2I, the division N1/N2=(2+I)/(4+2I)=(2+I)/[2×
(2+I)]=1/2=0.5.
Hence from one operation to another, neutrosophic statistics may diminishes the uncertainty.
(Smarandache,2022)
Difference between Classical Mean and Neutrosophic Mean
Let S = {a, b, c, d} be a sample set of four elements, such that a = 1, b = 2, c = 3, and d = 4.
In classical statistics it is assumed that all elements belong 100% to the sample, therefore S =
{a(1), b(1), c(1), d(1)}.
Whence the classical mean (CM)
‫= ܯܥ‬

(1.1 + 2.1 + 3.1 + 4.1)
= 2.5
1+1+1+1

But, in the real world, not all elements may totally (100%) belong to the sample, for example,
let’s assume the neutrosophic sample be NS = {a(1.1), b(0.2), c(0.5), d(0.3)}, which means
that the element a belongs to 110%(for example someone works overtime), b belongs only
20% to the sample, c belongs to 50%, and d belongs to 30%.
Whence the neutrosophic mean (NM) is
ܰ‫= ܯ‬

1 (1.1) + 2 (0.2) + 3 (0.5) + 4(0.3)
4.2
=
= 2.
1.1 + 0.2 + 0.5 + 0.3
2.1

Obviously, the two mean classical and neutrosophic are different, 2.5>2.0. Therefore, in this
example, CM > NM. And subsequently other statistical features such as the variance,
covariance, standard deviation, probability distribution function will also be different. The
neutrosophic mean consider the degree of membership of the elements so it is more accurate
since it reflects the real mean. (Smarandache,2022)
7

1.2. Introduction to CAPM
CAPM is a mathematical model widely used in finance for several purposes, mainly to estimate
the expected returns of assets and to evaluate investment opportunities based on the risk and
uncertainty in the market. In CAPM, risk is quantified by the beta coefficient (β), which
measures the volatility. If ߚ = 1, or ߚ > 1 or ߚ < 1, it indicates that the stock price tends to
move with the market, or the stock is more volatile than the market, or the stock is less volatile
than the market, respectively. When ߚ = 0, then the stock price does not correlate with the
market and is independent of the market movements. The stock with negative ߚ indicates
inverse movement to the market.
Risk cannot be described in the abstract, but it depends on the factors such as investor’s
preference, wealth position, time horizon and so on. Basically, risk factors fall in to two
categories as represented by graph 1.2.1.
Total Risk=Systematic Risk+ Idiosyncratic Risk

Graph 1.2.1: Total Portfolio Risk.

8

The risk, that arises from the market structure, general economic conditions and affects all
market players, is called a systematic risk. Systematic risk factors such as inflation, GDP
growth, interest rates, consumer confidence cannot diversify away. But the idiosyncratic risk
or unsystematic risk that applies to a specific firm or industry such as success or failure of R
& D, change in CEO, can be diversified away. The factor models such as CAPM focuses only
on dealing with systematic risk.
For assembling a portfolio of assets that maximizes the expected return, given a specific level
of risk, Markowitz in 1952 introduced a mathematical framework called Mean-Variance
portfolio theory. This method was developed by adding risk free rate to open a new range of
possible returns with lower levels of risk than what is possible on the efficient frontier.
Together, asset pricing and portfolio theory provided a framework to specify and measure
investment risk and to develop relationships between expected asset return. The CAPM builds
on the work of Markowitz on diversification and modern portfolio theory. The development of
the CAPM is usually credited to several different people Sharpe (1964), John Lintner (1965),
Jack Treynor (1962) and Jan Mossin (1966). Markowitz, Sharpe, and Miller received the Nobel
Prize in Economics in 1990 for their contributions to financial economics. CAPM is an
equilibrium asset pricing model, and it was developed based on several assumptions.
(Laopodis, N. T. , 2021)
The Basic Assumptions of CAPM
1. All investors are risk-averse and would take a position on the efficient frontier, focus
only on portfolio return (or mean) and the related variance (risk).
2. All investors invest for the same one period, with the same investment planning.

9

3. Investors are able to buy or sell portions from their shares of any security or a portfolio
they hold.
4. The market is frictionless such as no taxes, no inflation or no transaction costs on
purchasing or selling assets.
5. All information is publicly available, and all assets are publicly held and traded on
public exchanges.
6. Capital markets are in equilibrium, and all investments are fairly priced. Each investor
is very small and has a limited power compared to the market and cannot affect prices.
7. Investors have similar expectations; hence they choose the same distributions for the
future rates of return.
8. There is a risk-free asset, and investors can borrow and lend any amount at the riskfree rate.
Basically, these assumptions cause all firms look the same to investors and suggest an efficient
market. (Laopodis, N. T. , 2021)
In traditional CAPM, the expected return of an asset ݅ is calculated using the below formula.
‫ܴ(ܧ‬௜ ) = ܴ௙ + ߚ௜ [‫ܴ(ܧ‬௠ ) − ܴ௙ ]
(1. 1)

where ܴ௙ is the risk-free rate, ߚ௜ is the asset's beta coefficient representing its sensitivity to
market movements, and ‫ܴ(ܧ‬௠ ) is the expected return of the market portfolio.
ൣ‫ܴ(ܧ‬௠ ) − ܴ௙ ൧ is known as the risk premium/market premium.
Individual Risk Premium=Beta × Market Premium. This relationship between market beta and
asset’s risk premium is called the security market line (SML). Its slop indicates the risk

10

premium while intercept indicates the risk-free return. (Laopodis, N. T. ,2021). The graph 1.2.2
illustrate the relationship between expected return and the ߚ.

Graph 1.2.2: Expected Return and ߚ
One of the primary uses of CAPM is to estimate the expected return of an asset. Shareholders
and financial experts use CAPM to decide the appropriate expected return for individual
stocks, portfolios, or entire markets. This expected return serves as a key input in various
financial decisions, including asset allocation, portfolio management, and investment
valuation.
CAPM is also used to estimate the cost of equity capital for companies. By applying CAPM
to estimate the expected return on equity, companies can conclude the required rate of return
that investors demand for holding their equity shares. This cost of equity is a crucial input in
valuation models such as discounted cash flow analysis and in assessing the feasibility of
investment projects.

11

Portfolio managers use CAPM to construct and manage investment portfolios. By considering
the expected returns and systematic risks of individual assets, portfolio managers can optimize
portfolio allocations to achieve desired risk-return profiles. CAPM helps investors make
informed decisions about asset allocation, diversification, and risk management within their
portfolios.
CAPM provides a benchmark for evaluating the performance of investment portfolios and
individual assets. Portfolio managers compare the actual returns of their portfolios to the
expected returns predicted by CAPM to assess whether their investment decisions have
generated excess returns or underperformed the market. This performance evaluation helps
investors to assess the effectiveness of their investment strategies and to identify skilled
managers.
1.2.1 Introduction to Arbitrage Pricing Theory (APT)
CAPM is a one-factor model used to determine the expected return of a stock, based on its risk
relative to the market. But there are several other factors, which can influence the stock, return
such as economic factors, political and regulatory factors, technological changes, market
dynamics and even global events. Further to this foundational one factor model, multifactor
models have been developed over the period to handle the effect of different factors such as
Fama-French three factor model, Fama-French five factor model and APT.
To address these macroeconomic factors, APT can be considered as the one of the sophisticated
factor models. It can capture the effects of systematic risk factors. APT is a multi-factor model,
which builds up with the stochastic properties of stock returns of capital assets and several
macroeconomic factors including the market factor. APT was developed by Ross in 1976. He

12

acknowledged that there are many sources of systematic risk beyond just market risk. APT can
be considered as an extended CAPM which utilizes multiple regression analysis. APT is
naturally aligned with multiple regression analysis.
The multiple regression equation
(ܴ௜ ) = α௜ + ߚ௜ଵ [‫ܨ‬ଵ ] + ߚ௜ଶ [‫ܨ‬ଶ ] + ⋯ ߚ௜௞ [‫ܨ‬௞ ] + ε
(1.2)

Where, ܴi represents actual return of a stock, ߙi is intercept term (the expected return of the
asset when all factors are zero), βij is the factor loadings or sensitivities of asset , ‫ܨ‬j is the
factor values and ߝ is the error term which capture the idiosyncratic risk or noise.
ߚ݆݅ coefficients can be estimated using multiple regression analysis. Regression analysis helps
to understand the impact of these factor on the stock return.
Traditional APT equation can be written as below.
‫ܴ(ܧ‬௜ ) = R௜ + ߚ௜ଵ [‫ܨ‬ଵ ] + ߚ௜ଶ [‫ܨ‬ଶ ] + ⋯ + ߚ௜௞ [‫ܨ‬௞ ]
(1. 3)

Where, ߚ݅݇ represents the beta (or risk exposure) on the kth factor and F݇ is the factor risk
premium for the kth factor.
The factors are allowed to be correlated and are meant to simplify and reduce the amount of
randomness required in an analysis. When k = 1, we have a single-factor model and when k ≥
2 we have a multi-factor model. When k=1 and the single factor (F1) is the market portfolio
factor, the APT implies the CAPM.

13

If an asset has only a unit beta risk on the second factor (ߚ݅2 = 1) and zero betas on all other
factors, then its expected return will be:
‫ܴ(ܧ‬௜ ) = R௜ + [‫ܨ‬ଶ ]
(1. 4)

The ‫ܴ(ܧ‬௜ ) has increased by an extra amount to compensate for taking on the factor risk of the
second factor (F2). This is why F2 is called the factor risk premium for factor 2 or else extra
return one earns by taking 1-unit beta risk on the factor. As an example, E(RM)-Rf is market
risk premium, E(RCPI)-Rf is CPI risk premium and E(RGDP)-Rf represents GDP risk premium.
The APT makes no specific claim about what factors influence returns, or how many factors
are relevant. Both CAPM and APT models assume efficient markets and access to information,
but APT relaxes the most of the CAPM assumption.
Overall, while CAPM and APT provides a useful framework for estimating expected returns
and evaluating investment opportunities, its accuracy depends on various factors, including
model assumptions, data quality, and market dynamics. Understanding these considerations is
essential for effectively applying CAPM and extended CAP models in investment decisionmaking and financial analysis.
1.2.2. Key Differences between CAPM and APT
There are some key differences in the assumptions of the CAPM and the APT models
(Laopodis, N. T., 202). In brief APT is associated with less stringent assumptions on the
statistical distributions of asset returns compared to CAPM as illustrated by table 1.2.2.1.

14

Key difference

CAPM

APT

Number
of One factor-Market index
Factors
Market Efficiency perfectly efficient markets

Borrowing
Lending

and risk-free
lending

borrowing

Multiple factors
Does not strictly assume market
efficiency but assumes no arbitrage
and No specific assumptions about
borrowing and lending.

Investor Behavior homogeneous expectations and Allows
for
heterogeneous
and Expectations single-period horizon
expectations and multi period
analysis
Model Flexibility Less flexible, strictly defines More flexible, allows for multiple
the market portfolio as the risk and different risk factors to be
factor.
included.
Table 1.2.2.1: Key differences in the assumptions of the CAPM and the APT models

These differences highlight the individual approaches and applicability of each model in
different financial contexts. APT holds simplistic and less restrictive assumptions compared to
CAPM. This makes APT more flexible and adaptable to different financial contexts. Therefore,
this study also aims to consider the development of Neutrosophic APT, which can further
increase the effectiveness of this study.
1.3. Relevance of Neutrosophic Statistics to CAPM and APT
CAPM is a one-factor model, and that single factor is the market meanwhile APT is an
extended CAPM which can be corporate with more than one factor. The crucial point of the
CAPM or APT is to determine beta which is known as the sensitivity or risk factor. Normally
daily, weekly, quarterly, or annual beta can be calculated based on the stock market data.
However, stock market operates with full of indeterminacies. Trend may be fluctuated due to
the various macroeconomic factors contributing to the volatility and unpredictability observed
15

in stock prices. Some of the key factors which affect the trend movements are economic
indicators, monetary policies, corporate earnings, market sentiments, political events and
policy changes, global economic conditions, technological innovations, psychological factors,
natural disasters and external shocks, market structure and trading dynamics and so on.
All these factors can affect investor’s reaction and decision making, which may lead to
unpredictable trend fluctuations. This may happen in every single minute, hour, and day or at
any time, making them inherently hard to predict and analyse. All these factors are affected by
indeterminacy, and it adds another complex layer to the factors, which are affecting the trend
in stock market. As an illustration below graph 1.3.1 shows the hourly trend fluctuations of the
Netflix data set during a particular day.

Graph 1.3.1: Hourly Historical Data Set-Netflix.

Data

revisions,

measurement

errors,

unforeseen

events,

investor’s

sentiment,

interconnectedness and interdependence among economies, irrational market behaviour due to
the factor such as fear, greed, and herd mentality can lead to create uncertainty and
indeterminacy in stock market data to exhibit unpredictable trends. This unpredictable and

16

indeterminate nature of stock market highly affect the volatility of the stocks. Therefore,
understanding and addressing indeterminacy is crucial for effectively navigate the
uncertainties of the stock market and determining beta to calculate the expected return of factor
models to investors to make the informed decisions on their investments.
Recently, neutrosophic statistics is widely used as a powerful tool in decision making to
address the situation with indeterminacy (imprecise, unsure, and even completely unknown
values). Neutrosophic statistics have emerged as a compelling research frontier since around
2013, attracting significant attention from researchers across the globe (Bolos et al., 2019).
Hence to navigate these unpredictable factors in stock market, neutrosophic analysis is the
ideal approach.
The primary goals of this study are to develop and apply Neutrosophic statistics to address the
uncertainty involved in stock market data, represent the data as neutrosophic numbers, develop
a neutrosophic CAP model that account for uncertainty, extend the CAPM to capture more
than one factors (address APT model), and use forecasting techniques to make predictions
while considering uncertainty and ambiguity. This Neutrosophic approach will enable a more
comprehensive and sophisticated analysis of CAPM and extended CAPM, improving the
accuracy and reliability of decision making in complex economic environments.

17

1.4.Motivation
Neutrosophic statistics have emerged as an expanding area of research that has significant
attention from scholars worldwide. Smarandache (51) proposed a Neutrosophic logic to deal
with indeterminate and inconsistent information, simultaneously generalizing the concepts
such as classical, fuzzy, interval-valued and intuitionistic fuzzy sets which can address study
of data bases, medical diagnosis, image processing and the decision making.
Stock markets are complex and unpredictable frameworks impacted by many interrelated
financial, political, and internal factors. Stock market decisions are a significant challenge to
the investors as understanding the portfolio diversification, how much to invest in stock
markets and when to make the decision are extremely complex tasks. To navigate this
complexity, it will be a great advantage for the investors to give a neutrosophic approach for
the market analysis.
The motivation for this analysis stems from the desire to explore the criticisms, discover recent
advancements in CAPM and apply Neutrosophic Statistical techniques and methodologies to
make optimal strategies for investment decisions while addressing the indeterminacy of the
stock market and the limitations of CAPM. The financial markets and their dynamic nature
have always fascinated and motivated me to study deeper into understanding the complexities
of financial modeling and analysis. This exploration will serve as a driving force for
uncovering innovative Mathematical and Statistical approaches to asset pricing and risk
assessment in modern financial markets.

18

1.5. Main Objectives
Primary objectives of this study can be categorized as below.
➢ Apply neutrosophic statistics technique to calculate neutrosophic market returns to
capture the indeterminacy of the stock market.
➢ Apply neutrosophic statistics technique to calculate neutrosophic beta which represent
the unpredictable nature of the stock market.
➢ Develop a single factor Neutrosophic model-Neutrosophic CAPM (NCAPM) by
incorporating neutrosophic statistics, which is more sophisticated in understanding the
risks compared to traditional models.
➢ Extend the NCAPM to capture the effect of macroeconomic factors which can provide
a more comprehensive risk assessment.
o Develop 2-factor Neutrosophic models (NAPT) which can address the risk
boundaries.
o Develop a 3-factor Neutrosophic model (NAPT) to further optimize the risk
boundaries.
➢ Use these neutrosophic models to estimate neutrosophic beta and neutrosophic
expected return which can guide investors to make informed investment decisions.
➢ Verify the model Accuracy.
➢ Comparative analysis among neutrosophic models

19

Chapter 02
Literature review
2.0 Literature review for CAPM and APT models
The CAPM was introduced in the early 1960s by William Sharpe, Jack Treynor, John Lintner,
and Jan Mossin (Laopodis, N. T., 2021). It was considered as a significant milestone in modern
finance. This model introduced the initial structured approach to linking the expected return
on an investment with its associated risk.
Even though CAPM is a widely used financial model it was criticized due to some reasons
(Laubscher, 2002). Mainly CAPM is based on several assumptions that may not hold true in
the real world. One assumption of the CAPM is that the market is frictionless such as no taxes,
no inflation, or no transaction cost. But in reality, we cannot completely avoid these kind of
cost. Also, the model assume that all the investors are risk avers. But different investors have
different expectations and different risk preference levels. (Laubscher, 2002). The
assumptions, that short selling is not restricted and unrestricted risk-free borrowing and
lending, are also unrealistic (Fama & French, 2004). Furthermore, other unrealistic assumption
is that the share returns are normally distributed, but in reality, portfolio returns are
asymmetrically distributed. Hence beta is viewed as an incomplete risk measure (Ward, 2000).
However, these assumptions are often violated by behavioral biases, market inefficiencies, and
institutional constraints that influence the genuine behavior and performance of investors and
assets.
Beta coefficient is one of the key inputs of CAPM and the volatility of a stock is measured by
beta, which estimates how its returns or those of a portfolio will fluctuate concerning
20

movements in the market portfolio (Moyer, 2001; Jones, 1998 as cited by Laubscher in 2002).
Beta is calculated using the historical returns of a particular stock against the returns of a
market index, such as the S&P 500 or NASDAQ. However, beta estimation can be problematic
for several reasons. One reason is that the historical data may not be representative of the future
conditions and risks of the asset.
On the other hand, the beta value may change over time and across different market segments,
making it unstable and unreliable. Beta is specific to the data set being used to calculate. If we
use a different data set, or different time frame, the beta value will most likely be different. In
different financial website, they may give different betas for the same stock (Dybek, 2024;
Yahoo!, 2024). Not knowing how the numbers have been arrived at, it is unclear to use which
one, if any of them is accurate, resulting in uncertainty relating to the choice of beta value to
use. The regression line is used to estimate the beta and the line of best fit that minimises the
residuals. It is not a perfect fit but be the best guess using a specific data set. As the errors are
ever present, most of the time, the asset will not move in line with regression estimate. This
can be seen in the graph, which also highlights some of the outlying moves. This is a particular
risk for popular hedge fund strategies such as long, short and equity market neutral positions
(Fama & French, 1992).
Another criticism of CAPM is that it does not capture all the factors that affect share returns.
CAPM is a one factor model which capture the market factor (Fama & French, 2004).
Historically, several multivariable CAPM models have been introduced by different
researchers in different time periods at different locations all around the world, which assume
that risk is influenced by multiple factors including market returns, indeterminacy,
incompleteness, and several other factors (Assagaf, Aminullah, 2015). While multivariable
21

models represent a positive advancement in financial theory, they still have shortcomings when
applied to decision-making processes. The development of asset pricing models has been
marked by several significant contributions, each aimed at better explaining the cross-section
of stock returns.
The intertemporal capital asset pricing model -ICAPM (Merton, R. C. ,1973), Fama and
French’s three factor model (1992), Fama and French’s extended three factor model (1993),
Carhart four- factor model (1997), Fama and French five factor model (2015) and Arbitrage
pricing theory are some of the extended CAPM. Basically, such factor models have more than
one beta (Laopodis, N. T., 2021).
The ICAPM is introduced by Nobel Robert Merton in 1973.This is a consumption-based
capital asset pricing model. It extends the traditional CAPM by accounting for investors’ desire
to hedge against market uncertainties and construct dynamic portfolios over time. Instead of
one beta in CAPM, ICAPM employs multiple beta coefficients (Merton, R. C.,1973)
Especially investors use ICAPM in economic downturns. Even though this model stands as a
significant progression in financial modeling, it also has some limitations. This is recognised
as a complex model to understand and implement. ICAPM also holds some unrealistic
assumptions and challenge of accurately estimating beta coefficients over several time periods
(Fama and French, 2004).
Alternative asset pricing model is a valuable report to get a clear understanding of different
CAP models because it has carefully discussed the theories behind the different CAP models
(Graham and Stephen 2020). The reviews and evaluations are based on past literature, which
expands over 100 years in academia, financial economics research and consulting. Graham
and Stephen (2020) have examined and reviewed several articles of useful studies on CAPM.
22

They successfully described the different CAPM models such as ICAPM and CCAPM
(Consumption-based Capital Asset Pricing Model). Graham and Stephen (2020) observed the
input variable of CAPM such as beta, alpha, market premium and risk-free rate while
highlighting the conclusions of several empirical studies. Further, they outline the issues in
determination of the rate of return and discussed the significant details of implementation of
CAPM and the necessary adjustments based on prior beliefs and market conditions.
Fama and French’s Three-Factor Model (1992) expanded the traditional CAPM by adding two
additional factors to market risk. The three factors of the model are market Risk (Beta), size
and the value. The size or SMB is the return difference between small-cap and large-cap stocks
(i.e., SMB: small minus big). Value or HML is the return difference between high book-tomarket and low book-to-market stocks (i.e., HML: high minus low). This model explains a
larger portion of variations in stock returns compared to the CAPM. Strong empirical evidence
supports the presence of size and value effects in stock returns. Even though this is an extended
model it may still miss other significant factors, which are affecting returns. Another drawback
of this model is that the factors are fixed and do not account for changing market conditions
over time (Fama and French, 2004; Laopodis, N. T., 2021).
Fama and French extended their 1992 model by including changes to better capture the size
and value premiums, confirming the significance of three factors in different market
conditions. The model is Fama and French extended model (1993) and it enhanced
methodology for calculating size and value factors. This Fama and French extended model
provides consistent outperformance over the CAPM. But it still suffers from potential
exclusion of other relevant factors and fixed factor structure (Fama and French, 1993; Fama &
French, 2004 and Laopodis, N. T., 2021).
23

The Carhart Four-Factor Model (1997) is another extended factor model. Carhart extended the
Fama-French three-factor model by adding a momentum factor which explain the return
difference between stocks with high past returns and those with low past returns. Presence of
momentum accounts for the momentum effect and properly explains anomaly in stock returns.
This model delivers a more comprehensive explanation of asset returns than the three-factor
model, but momentum is harder to explain using traditional financial theories. Also, adding
more factors can sometimes lead to overfitting and less robustness across different data sets
(Carhart, 1997).
In 2015, Fama and French added two more factors, profitability and investment, to the threefactor model and suggested a Fama and French’s Five-Factor Model (2015). This model
provides a more complete framework for understanding stock returns. Empirical evidence
supports the relevance of profitability and investment factors. The model is more complex and
harder to implement. Moreover, some factors might be correlated, and presence of
multicollinearity may reduce the clarity of their individual contributions (Fama and French,
2015; Laopodis, N. T., 2021).
Arbitrage Pricing Theory (APT) was developed by Stephen Ross (1976) by extending CAPM
with multiple factors which affect the returns. It doesn't specify which factors to use, or the
numbers of factors need to use. Hence, APT offers a more flexible framework than the CAPM
or Fama-French models. Further it has simplified model assumptions compared to restricted
CAPM assumptions. specification issues and potential biases are some disadvantages of this
model (Ross, 1976; Laopodis, N. T., 2021).
Each model has its own set of strengths and weaknesses. Researchers, such as Fama and
French, Carhart and Ross made some efforts to capture the complexities of asset returns more

24

accurately. However, the choice of model should always consider the specific context, data
and information availability, and the objective of the analysis.
Multivariate test of financial models was the earliest journal in financial economics, which
discussed new approaches of financial economics (Gibbons, 1982). Gibbons discussed about
the multivariate CAPM models and tried to increase the precision of estimated risk premium
parameters. He provided the theoretical superiority to commonly used procedures and
demonstrate practical applications.
Nevertheless, due to practical challenges encountered in this model, investors are advised to
exercise caution when using the CAPM to estimate share returns and assess investment
performance (Laubscher, 2002). In the CAPM, the focus is primarily on systematic risk, which
cannot be diversified. This suggests that investors should expect proportional compensation
for assuming such risk. Laubscher (2002) thoroughly discussed the empirical studies of
CAPM, its criticisms and concluded that CAPM is a valid model but should be cautiously
applied to evaluate investment performance.
In recent literature, several studies have been conducted to determine relevance of CAPM
model. Athens Stock Exchange, focusing on weekly returns of 100 listed companies,
questioned the efficacy of CAPM (Michailidis et al., 2006). This study argued that higher risk
did not necessarily correspond to higher returns. Similar analysis has been conducted to check
the validity of CAPM using empirical evidence from Amman Stock Exchange (Alqisie and
Alqurran, 2021). The main objective was to examine whether a higher/lower risk stocks yields
higher/lower expected rate of return. They used 60 companies in AMANA Stock Exchange
from 2010-2014. The study used the monthly closing stock prices to calculate the rate of return
of each stock. They used the same methodology as Black et al. (1972) and Fama and MacBeth
25

(1973) to test the non-linearity. In this study, they concluded that there was no conclusive
evidence in support of validity of CAPM in Amman Stock Exchange (ASE) for the period
(2010 – 2014). They recommended expanding the study period and repeating the analysis by
considering several other financial and marketing indicators.
In 2015, Canadian Center of Science and Education published an article “Analysis of
Relevance Concept of Measurement of CAPM Return and Risk of Shares”, which emphasises
the relevance of CAPM by comparing alpha and beta values of earlier research models
(Assagaf, 2015). For the analysis, he used most actively traded stocks on the Indonesian Stock
Exchange (BEI) weekly during the period September 2014 to November 2014. The study has
concluded that despite these advancements made in alternative models, the CAPM concept
remain the foremost and most widely utilized method for estimating stock returns.
A review of Capital Asset pricing model by Iqbal (2011) is another interesting article which
discussed the review of foreign studies on CAPM. According to this study, ongoing debate
surrounding the dominance of APT over CAPM remains unsettled. Throughout the study, he
discussed that numerous researchers have identified strengths and weaknesses in both models.
Several empirical studies, including those by Sharpe and Cooper (1972), Foster (1978), and
Sauer and Murphy (1992), have recognised CAPM as a feasible asset-pricing model, despite
criticisms from Roll (1977, 1981), Dimson (1979), Fama and French (1992, 1996), and Davis
(2000) who have challenged the validity of CAPM tests (Iqbal, 2011). According to Iqbal
(2011), it is evident that CAPM research has provided invaluable insights into stock return
behavior across diverse global markets. CAPM is still anticipated to maintain dominance in
the capital market as a method for estimating expected returns of risky securities. The CAPM
holds considerable relevance in financial economics, with broad applications such as testing
26

asset-pricing theories, assessing the cost of capital, estimating portfolio performance, and
defining hedge ratios for index derivatives (Iqbal, 2011).
2.1. Literature review for Neutrosophic statistics to capture the indeterminacy associated
with financial decision making.
Integrating neutrosophic methodologies to address its indeterminacies and limitations can
modify CAPM. Data exactness, accuracy in expressing data and uncertainty of the closing
prices are the major issues of stock market (Jha et al., 2018). Neutrosophic soft sets can be
used to tackle the exact state of these data sets. Based on high, low, open and closing prices,
they have developed a technique to determine adjusted closing price in stock market.
Boloș et. al. (2019) modeled the return on financial assets, the financial asset risks, and the
covariance between them, with the help of triangular neutrosophic fuzzy numbers. This
neutrosophic approach of three financial assets performance indicators addressed the scenario
of uncertainty, the scenario of non-realization, and the scenario of indecision. All three
scenarios have attached performance, non-execution, or uncertainty ratios according to the
investor’s professional judgment. They checked the possibility of stratification, or the
clustering of the financial asset return values with the help of triangular neutrosophic fuzzy
numbers.
The paper entitled “Novel Single Valued Neutrosophic Hesitant Fuzzy Time Series Model:
Applications in Indonesian and Argentinian Stock Index Forecasting” by Tanuwijaya et al
(2020) can be considered as an improvement of neutrosophic time series model which absorb
the degree of the uncertainty using single-valued neutrosophic hesitant fuzzy set. Tanuwijaya

27

et al. (2020) proposed a model, which capture the uncertainty of the data movement and
simplified de-neutrosophication process with guaranteed forecast.
The study entitled “A Neutrosophic Forecasting Model for Time Series Based on First-Order
State and Information Entropy of High-Order Fluctuation” by Guan et al. (2020) presented the
concept of neutrosophic fluctuation time series (NFTS) and proposed a new financial
forecasting model based on neutrosophic soft sets. This analysis discussed the first-order
neutrosophic time series to describe the uncertainty and information fluctuation entropy to
measure the complication of historical fluctuations.
Neutrosophic portfolios were characterized by their composition of financial assets, wherein
the neutrosophic return, risk, and covariance can be quantified (Boloș et al, 2021). They
investigated the concept of neutrosophic portfolios within the framework of modern portfolio
theory. These portfolios offer insights into the probability of attaining the neutrosophic return
at both the individual asset and portfolio levels, which experience the neutrosophic risk. They
introduced two key concepts, neutrosophic covariance of financial assets and independent
neutrosophic portfolios. The proposed methodology involves a three-step approach aimed at
identifying the independent neutrosophic portfolio return, risk, and structure. The paper
includes numerical examples at each stage of the methodology to enhance understanding. The
study's findings can aid capital market investors in making informed decisions (Boloș et al.,
2021).
Abdelfattah et al. (2024) introduced a stock market movement prediction model that combines
social media data with historical stock price data. They addressed the importance of stock
market prediction in recent years and the significant influence of social media on it, using
neutrosophic logic. The results of the study revealed the superiority of the model in return and
28

Sharpe ratio scores, indicating effective ability for generating excess return and managing risk
(Abdelfattah et al., 2024).
Recently, there is a growing trend of utilizing neutrosophic statistics to address uncertainty in
the financial market in decision-making and we could find considerable number of scholarly
studies in literature, which have been dedicated to study this area.

29

Chapter 03
3.0.Introduction to Data availability
In financial decision-making, data availability and reliability are important to conduct a
comprehensive analysis. In this study identify the required data set is a challenge as there are
several factors influence on the fluctuation in the stock market. Most powerful factors need to
be carefully chosen according to the requirement of the analysis. stock prices, market indices
(S&P500 and NASDAQ), risk-free rates, GDP, and CPI are recognised as the main data set
need to be addressed carefully. the sources of these data sets, access methods, timelines and
challenges associated with those data sets are discussed through out this chapter. Accurate data
set is an advantage to make informed decisions.
3.1. Indeterminacy of the data set
Stock market, which is inherently indeterminate, displays the wealth of information for
shareholders such as stock price, price movements, dividends, trading volumes,
announcements, financial statements of the company, comparison charts and several other
important statistics such as earning per share, market volatility and growth estimates.
Based on the stock market information, analyst and financial managers help investors to take
the most desirable decisions on their portfolios and buying and selling opportunities. Future is
inherently uncertain. Unexpected things may happen in unforeseen future. Events like
economic downturns, technological disruptions, natural disasters or political instability can
significantly affect on a company performance. Not only the future events but also market
sentiment, limited information and market manipulations can highly impact on a company’s
performance.

30

Market sentiment is a one of the critical reasons which course indeterminacy in the stock
market. Now and then, the stock market is driven by emotions and psychology. News reports,
social media also can influence investor sentiment. This may cause to fluctuate prices even
without any underlying change in a value of the company.
Limited information is another factor which impact the stock price. Most of the investors do
not have access to all the available information that could impact a stock price. Management
decisions, internal developments, and upcoming regulations might not be publicly available
and that may cause to create uncertainty. The Efficient Market Hypothesis (EMH) is a theory
in financial economics that suggests stock prices reflect all available information. EMH
indicates that the market is efficient at incorporating all relevant news and data into stock
prices, making it difficult to consistently outperform the market through security selection or
market timing.
Manipulation of stock market data is another factor, which highly affects the movement of the
prices. Some people do the market manipulations with the aim of personal gains. Pump and
Dump is one type of manipulation. In this scheme individuals or groups of people artificially
increase a stock price. They mislead other investors by spreading false information, then
investor rush ups in buying those shares and price of the stock goes up. They sell their stocks
once the price rises leaving other investors with overvalued shares.
Wash Trading is also a type of market manipulation, which involves buying and selling a stock
back and forth to create a fake impression of trading activity and potentially manipulate the
price.

31

Another way to manipulate the market is known as spoofing. Investors place large orders to
buy or sell a stock, but they cancel them before execution. Their aim is to move the price in a
desired direction before placing their actual order. Even though Regulatory groups constantly
try to detect these manipulations and work to prevent them. But they remain a risk in the stock.
Apart from these kinds of uncertainties there may be several other reasons to fluctuate the
stock prices while creating indeterminacy. In stock market financial data such as stock prices,
bond yields and interest rates are observed daily (intra-day or even minute-by-minute (tick)
basis), weekly as well as monthly. Hence even basic data recording can have errors leading to
uncertainty. It is important to identify these uncertainties and manipulation in the stock market.
The financial managers and investors can use advance methodologies such as neutrosophic
statistics, which can effectively address the indeterminacy to minimise the risk of their
decisions. Neutrosophic statistic is an emerging trend to address uncertainty in the real-world
scenarios. This approach is to apply neutrosophic logic and statistical methods to the widely
used decision making models in stock market predictions.
Data sets may be downloaded from financial web sites Yahoo finance, the U.S. Bureau of
Economic Analysis (BEA), U.S. Bureau of Labour Statistics (BLS), US Department of the
Treasury, Stock Analysis from Web, Netflix official website and Kenneth & French data
library. The analysis in this thesis is based on the stock market data sets of Netflix, Amazon,
Apple, NASDAQ and S&P500 from 2009-2023 monthly basis.

32

3.2. Data sets
15-year period from 2009-2023
15-Year period is chosen to conduct a comprehensive analysis considering long-term
perspective. This long-term understanding may help to capture the uncertainty associate with
multiple economic cycles, structural changes, and long-term trends.
5-year period from 2019-2023 ,2014-2028 and 2009-2013
Short term analysis is a more sensitive analysis, and this may consider the impact of changes
in economic policies, short-term market cycles, and sector-specific movements during that
period. Specially the recent period 2019-2023 may consider the recent trends, market
conditions, and economic factors.
Stock market data
Stock market index
➢ S&P500
➢ NASDAQ Index
A stock market index is a benchmark that tracks the performance of a segment of the stock
market. It is a comparison tool, which compares the performance of different indices to
understand how various market segments are faring. This helps to evaluate overall market
trends and identify possible opportunities in the market.
Well-known stock market indices around the world can be named as Dow Jones Industrial
Average, S&P 500 and NASDAQ Composite Index. They serve as useful tools for understand

33

and direct the stock market. For this study NASDAQ and S&P 500 are used as stock market
index to analyse the stocks.
S&P 500 (^GSPC)
The S&P 500 (or Standard & Poor's 500) is an important stock market index that captures the
performance of 500 of the largest publicly traded companies in the American economy. The
companies with the largest market capitalization have a greater influence on the overall
performance of this index. The S&P 500's value fluctuates daily based on the stock prices of
the companies it includes. Usually positive trend of S&P500 indicate an economic growth
meanwhile negative trend may signal an economic challenge. This index is diversified across
different sectors including financials, healthcare, technology, industrials and many more. This
diversified exposure cause to diminish the risk and provide a solid long-term return. S&P500
is a common choice among long term investors.
NASDAQ Composite Index (^IXIC)
NASDAQ Composite Index is a stock market index that includes almost all stocks listed on
the NASDAQ stock exchange which includes more than 3000 companies in all over the world.
This is one of the most popular market indices which heavy weighting towards technologyoriented companies such as Apple, Microsoft and Amazon. But it also includes several other
companies in different sectors such as financial services, healthcare, consumer service
industries, and industrials. Hence this diversity provides a wider picture of the entire market
performance. This index is widely used by the investors particularly in tech related fields. Due
to the economic growth and technological advancement NASDAQ composite shows a strong
growth, outperforming other major indices such as Dow Jones and S&P500.Futher NASDAQ
34

is more volatile index compared to the other indies due to high concentration of technology
stocks.
Key differences between S&P500 and NASDAQ market indices
S&P500 and NASDAQ are widely followed two distinct market indices which have different
characteristics. Normally NASDAQ composite is considered as a gauge of the performance in
technology sector and S&P 500 is viewed as a key indicator in the economy in United
State.S&P500 is more diversified index compared to NASDAQ. It represents broad spectrum
of industries such as technology, healthcare, and financials, meanwhile NASDAQ heavily
weighted towards technology and biotech companies. Larger companies have a more
significant impact on the NASDAQ performance. But S&P 500 has a more balanced sector
representation compared to the NASDAQ market index. NASDAQ is more volatile due to its’
high concentration of technology and biotech firms which are more sensitive to the rapid
changes in investor sentiment and market conditions. S&P500 is less volatile with its’ more
stable and established companies. Even though all stocks listed on the NASDAQ stock
exchange are included in NASDAQ composite; to be in the list of S&P500, the companies
need to fulfil some specific requirements like a lowest market capitalization and a positive
earnings record.
Both indices have their own characteristics and unique objectives to fulfil various investor
requirements and inclinations. Netflix is a technology-based service. But to have a wider
picture, this study uses both S&P500 and NASDAQ market indices for this analysis.

35

Stock price
Stock price of a particular stock reflects the current price of a share. Price movements are
available for users in daily, weekly and monthly basis with low price, high price, open price,
closing price and adjusted closing price.
Stock: Netflix Inc. (NFLX)
This study is based on the variation of Netflix stock prices which provides entertainment
services with 270 million paid memberships. Netflix has operations in approximately 190
countries. This leading streaming service provider offers TV series, documentaries, films, and
games in different categories and languages. This entertainment company also provides
members the ability to receive streaming content through a host of internet connected devices,
including Televisions, digital video players, TV set-top boxes, and mobile devices. The
company was incorporated in 1997 and is headquartered in Los Gatos, California. Mr. Wilmot
Reed Hastings is the co founder and the executive chairman of this company.

Table 3.2.1: Details of Netflix stock.

Netflix is a publicly traded technology and internet-based company listed on the NASDAQ
stock exchange in the United States under the ticker symbol "NFLX”. This means that Netflix
is initially market traded in the United States. But investors in other countries such as Canada
36

can still trade Netflix stock through local brokerage accounts that have access to U.S. markets.
There is no separate stock market specifically for Netflix in Canada.
Netflix's stock price has an upward trend overall in 2024.But there are daily fluctuations as
illustrated in graph 3.2.1.

Graph 3.2.1.: Netflix stock price trend.

Website:
https://www.stock-analysis-on.net/NASDAQ/Company/Netflix-Inc/FinancialStatement/Income-Statement
Netflix's stock price has undergone considerable variations during the period of 15 years from
2009 to 2023.From 2009 to 2010 is a significant time period for Netflix streaming platform
because the popularity of this service was risen up during that period and at the end of 2010
the share price of the Netflix was around $25-$30.But the company’s decision to split DVD
37

and streaming services caused to show a downtrend in next two years 2011 and 2012 .However
at the end of 2012 the company recovered and achieved an upward trend. This uptrend
continued by attracting more subscribers and at the end of the 2015 the share price was over
$100.The international expansion caused to sky rocketed share price of Netflix, reaching over
$300 by mid 2018.The COVID-19 pandemic was a turning point to this online service
provider, and this caused to accelerate its demand. The Netflix stock price recorded its highest
price during the pandemic, and it was $691.69 in November 2021.Conversely stock prices
started falling down in 2022 and dropped up to $294.88 due to high competition in the market.
In 2023 it recovered slightly. At the end of the period the share price is approximately
$500.Currently

stock

prices

has

been

fluctuating

between

$600

and

$700

(Investing.com,2024) . According to the below graph in Yahoo finance the NFLX stock price
is 641.62USD and beta is 1.23 on 31st March 2024.

Graph 3.2.2: Netflix stock price.
38

Stock: Amazon.com, Inc. (AMZN)
Amazon is a well-established tech company, which has a major interest in e-commerce, cloud
computing, and digital streaming. The historical data set of Amazon from 2009-2023 has been
used to calculate the neutrosophic expected returns with the aim of testing the proposed
neutrosophic regression models.
Stock: Apple Inc. (AAPL)
Apple is a one of the biggest publicly traded companies by market capitalization. As a top
technology company, Apple stock is listed on the NASDAQ exchange. It is a key component
of the S&P 500. The stock is known for its solid financial performance, strong brand, and
consistent innovation. Investors consider factors such as product launches, earnings reports,
and market conditions when evaluating APPL stock. The historical data set of APPL from
2009-2023 has been used to calculate the neutrosophic expected returns with the aim of testing
the proposed neutrosophic regression models.
➢ Risk Free Rate
The risk-free rate refers to the hypothetical rate of return on an investment with absolutely zero
risk. In reality, every investment (even government bonds considered as the safest investments)
have some risk, like inflation which can diminish their purchasing power. Though, there are
investments that can be considered as risk-free such as government bonds. Especially shortterm bonds like treasury bills issued by governments with strong credit ratings are considered
as risk free securities. They are very safe. Risk-free investments typically offer low returns.
Investors can buy these bonds directly from government agencies or through brokerage firms.

39

There are several ways to access data relevant to risk-free rates. Government treasury bill rates
are available on government websites or financial data platforms. Several online financial
databases provide historical data on government bond yields, allowing investors to track
changes over time. These databases might require a subscription fee.
Fama-French Keneth data library is one of another easy way to download risk free rates which
are updated regularly and are available freely. This data library is a worthy resource for
researchers and finance professionals interested in testing asset pricing models and portfolio
strategies. Data are available for the factors like market risk, risk free rate, size, and book-tomarket ratio in monthly, weekly and daily basis.
Web site :(https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html )
Systematic Risk Factors (External factors)
➢ Gross Domestic Product GDP
➢ Consumer Price Index CPI
Gross Domestic Product (GDP)
GDP is a significant metric used to measure the economic stability of a country. It represents
the total monetary value of all final goods and services produced within a country in a specific
period. GDP is calculated by summing up the components such as consumer and government
spending, investment and net exports. GDP is a central indicator for policymakers, businesses,
and investors, which helps to assess economic growth. Apart from that, GDP can be used to
compare economies of different countries and make informed decisions about resource
allocation and investment strategies.

40

Generally, GDP has a positive relationship with stock prices. When GDP increases the stock
price increases accordingly and when GDP decreases normally the stock price falls. Rise in
GDP indicates the expanding economy, and this leads to increase consumer spending,
increased investments, and up-lift the economic confidence. In an economic boom, companies
generally see higher revenues and profit. This is the reason to rise in stock prices as investors
expect better future earnings. By contrast, in an economic downturn, it leads to reduce
consumer spending and lower investments. In the economic uncertainty, investors avoid invest
in stock market leading to a price decrease in the stock market. Hence, GDP is a crucial factor,
which affect the stock market fluctuations.
USA annual GPD can be downloaded from the U.S. Bureau of Economic Analysis (BEA).
One needs to register for an API key from the BEA at BEA API to download the data set.

Graph 3.2.3: GDP quarterly.

Web site: https://apps.bea.gov/api/signup/activate.html#947AB03F-061B-4141-9F4ABBE78D43A28B

41

Consumer Price Index (CPI)
The CPI represents changes in prices of all goods and services purchased for consumption by
urban households except income taxes and investment items such as stocks, bonds, and life
insurance. CPI is the most widely used economic indicator, which is a measure of inflation and
is an indicator of the effectiveness of government policy. Investors, business executives and
other private citizens use this index as a guide in making economic decisions.
CPI is a key economic indicator that measures changes in the price levels, which gauge
inflation. This factor can have a considerable influence on stock market performance through
various direct and indirect mechanisms. Increase in CPI indicates the inflation, which can lead
to lower corporate earnings. This may negatively impact stock prices. Furthermore, rising CPI
causes to reduce consumer spending and this may negatively affect companies that heavily
depend on consumer sales. This also has impact on their stock prices. On the other hand, rising
CPI can cause to raise interest rates to combat inflation. These higher interest rates can
indirectly cause to slow down the economic growth and negatively impact stock prices.
However, lower inflation can cause to improve the purchasing power of the customer. This
may positively impact on their stock prices. By contrast, falling CPI may decrease the interest
rates to stimulate economic growth and may indirectly boost investment and spending which
can positively impact stock prices.
Further, continuous high inflation can lead to uncertainty and decrease investor confidence.
This may indirectly lead to stock market volatility as stockholder seeks safer assets. Opposite
to this, deflation can be able to create a stable economic environment, which potentially boost
investor confidence.

42

Overall, the CPI is a critical factor in shaping the economic environment in which companies
operate, and investors make decisions, thereby influencing stock market performance. USA
annual CPI data can be downloaded from the U.S. Bureau of Labor Statistics (BLS) for CPI.
First, you need to register for an API key from the BLS at BLS Public Data API Signup to
download the data set.
Website: https://data.bls.gov/registrationEngine/registerkey
3.3. Charts and Graphs
This analysis is based on the data set of stock market prices. Hence Various charts and graphs
which have been used in stock market are utilized in this thesis. To get a clear understanding
of these complex trends and relationships, candlestick charts and trends lines are briefly
explained below.
Candlestick Chart
Candlestick chart is a specific type of price chart used in finance to do technical analysis for
visualizing price movements of a security over the particular period (day, week, month, etc.).
Candlesticks use bars with a rectangular body and wicks (thin lines) running from both ends
to represent the price range for a specific time. The body shows the opening and closing prices,
and the wicks show the high and low prices, and it indicates the market behaviour whether it
has a bullish or a bearish trend. Wicks extending from the top and bottom of the body
symbolize the high and low prices for the period. Long upper wicks indicate prices reaching
considerably higher than the closing price. Long lower line indicates prices dipping
significantly lower than the opening price.
43

Figure 3.3.1: Bullish and Bearish Candlesticks

By analyzing the size, color, and position of a candlestick chart, financial analysts can
identify potential trends, and they can use these trends to make significant trading decisions.

Graph 3.3.1: Candlestick chart illustration from NFLX stock

Trend Lines
A trend line is a line drawn through two or more price lines to connect price points, specially
represent the direction of the stock price movement. If the stock price increases, we call it as
an uptrend suggesting bullish market. An uptrend line can be drawn by connecting at least two
44

higher lows; meanwhile downtrend can be drawn by connecting at least two lower highs. If it
is a downtrend or the stock price is decreasing over the time, we call it as a bearish market.

Figure 3.3.2: Uptrend and Downtrend lines

Graph 3.3.2: Uptrend illustration from NFLX one day price movement
Website: https://medium.com/@satolix.io/market-trends-uptrend-downtrend-and-rangeed989df2bc8c

45

Chapter 04
Research Methodologies

4.0.

Introduction to Traditional CAPM and its Limitation

In finance, CAPM is a fundamental method to calculate the expected return of an asset. To
understand the risk of an asset relative to the overall market, analysts calculate the asset's beta,
which measures the volatility of the asset in comparison to the market. This beta can be used
to estimate the expected return of CAPM. Market return is used as the systematic risk factor
for this single factor CAPM.
For illustration, to calculate the market beta for NFLX, the returns of Netflix stock can be
compared with the returns of a stock market index such as the S&P 500(GSPC) or NASDAQ
composite index (IXIC).

Graph 4.0.1: Netflix stock price comparison with NASDAQ and S&P500.

46

4.0.1. Basic steps to calculate the Traditional CAPM (Stock: NFLX)
1. Collect historical data: Obtain historical prices for NFLX and a stock market index
(S&P 500 or NASDAQ) for the same time period (normally collect the adjusted closing
price)
2. Calculate the periodic stock returns: Compute the simple returns for Netflix stock using
the following formula:
ܴ௧(ேி௅௑) =

ܲ௧ − ܲ௧ିଵ
ܲ௧ିଵ
(4.1)

Where , Rt(NFLX) represents NFLX return at time t,Pt is NFLX share price at time t and
Pt-1 is NFLX share price in the previous time period
3. Calculate the periodic market Return: Market return can be calculated using the
following formula
R ௧(ெ஺ோ௄ா்) =

P୲(୍୒ୈ୉ଡ଼) − P୲ିଵ(୍୒ୈ୉ଡ଼)
ܲ௧ିଵ(ூே஽ா௑)
(4. 2)

Where, Rt(MARKET) is Market return at time t, Pt(INDEX) is the market index at period t
and Pt-1(INDEX) is the market index in the previous time-period
4. Calculate Beta: There are several different ways to calculate beta in traditional
CAPM
a. Calculate the covariance and the variance of the stock return with market
return.
Below formula can be used to calculate beta.

47

ߚ୒୊୐ଡ଼ =

‫ܴ(ܸܱܥ‬ேி௅௑ , ܴெ஺ோ௄ா் )
ܸ‫ܴ(ܴܣ‬ெ஺ோ௄ா் )

(4. 3)

Where RNFLX is Return of Netflix stock, RMARKET is Return of the market index,
Cov(RNFLX,RMARKET ) is covariance between Netflix stock returns and market
returns and Var (RMARKET ) is the variance of market returns.
b. Conduct regression analysis to calculate beta (excel, SPSS, STATA or manual
calculations can be used for this analysis). Here the dependent variable is the
return of NFLX, and the independent variable is the return of market index.
ܴேி௅௑ = α + ߚேி௅௑ · ܴெ஺ோ௄ா் + ε
(4. 4)

Where, α is intercept of the regression line, ߚNFLX is Slop of the regression
line, and ε is error term
c. Use the slope of the least square line to calculate beta
∑‫ݕ∑ݔ‬
݊
ߚேி௅௑ =
(∑
‫)ݔ‬ଶ
∑ ‫ݔ‬ଶ −
݊
∑ ‫ ݕݔ‬−

(4. 5)

Where, Y is stock return, X is market return and n is number of observations
5. Download risk free rate for the same period (Kenth French data library or treasurybill rates)
6. Gather all these data sets and plug them into the CAPM formula (equation 01) to
calculate the expected return for Netflix.

48

The market beta indicates the sensitivity of stock returns to changes in the overall market
returns. Therefore, beta is a valuable indicator to measure the volatility or the sensitivity of the
market.
This beta is useful to measure the expected return in CAPM.CAPM can be used to make
portfolio decisions. By analyzing the beta values and expected returns of different investments,
CAPM helps investors to choose assets with low correlation. This helps to reduce the overall
portfolio risk.
By calculating the expected return using CAPM, investor can compare it to a benchmark,
whether the stock is undervalued or overvalued. Investors and decision makers can compare
the CAPM derived expected return with analyst’s estimates or own return expectations for the
investment. Higher expected return of CAPM than required return implies that the investment
might be undervalued and potentially attractive. Lower expected return than required return
suggests the investment might be overvalued and less attractive. Undervalued and overvalued
stocks can be illustrated as the below graph 4.0.1.1.

Graph 4.0.1.1: Under values and overvalued stocks

49

4.0.2. Limitation of Traditional CAPM
CAPM can be used as a starting point, but it is vital to analyze the company's fundamentals,
market conditions, and other risk tolerances before making any move. CAPM is a widely used
financial model, but it has some criticisms and limitations. CAPM relies on several
assumptions that may not perfectly hold true in real markets (as an illustration, perfectly
rational investors, all investors hold diversified portfolios).
Accurately estimating some inputs such as the market return and beta can be challenging. For
illustration CAPM calculation are based on adjusted closing price of a particular day, week or
month. But it may not be the most accurate measurement or the top choice for the calculations.
Stock market is very complex and generates massive amounts of data at high speeds. Stock
price is fluctuated in every single minute. Hence picking up the most suitable data point to
measure the risk is challenging.
The use of single beta as a measure of risk is overly simplistic. It assumes that the risk of a
security is fully captured by its correlation with the market, which is not always the only case.
There may be several other factors, which affect the risk of a security. CAPM is a single factor
model, which considers only one source of risk (market risk) measured by beta. It does not
consider the dynamic nature of markets and ignores other factors that might affect the asset
returns such as GDP, liquidity and inflation. Another major drawback of CAPM is that it
assumes certainty around factors like expected returns and volatility (beta). But in reality, these
involve some level of uncertainty and vagueness.
Further, CAPM ignores human behaviours and assumes rational behaviour and efficient
markets. It does not account for psychological factors and market anomalies caused by

50

irrational behaviour, which are highlighted by behavioural finance. The model assumes that
beta is stable over time, but empirical evidence suggests that betas can change, making
predictions less reliable.

Apart from that, some empirical studies have found that CAPM does not completely explain
the actual returns of securities. For illustration, low beta stocks often outperform high beta
stocks contradicting CAPM's predictions.
The aim of this study is to address these limitations using relatively new statistical technique
of neutrosophic statistics, which capture almost all type of these indeterminacies to provide a
more comprehensive analysis of risk and return.
4.1. Neutrosophic Statistical Approach to the traditional CAPM
The indeterminacy of the stock market needs to be carefully addressed. Based on the risk of
the traditional CAPM, analysts estimate the expected return of an investment, but with
limitations. However, with the use of neutrosophic statistics, the theory of statistics which
generalizes classical statistics offer a potential way to address some of these limitations.
Neutrosophic sets are a Mathematical tool, which can be used to represent circumstances with
unclear, vague and inconsistent data. This study starts with converting the classical data set,
which is used to calculate traditional CAPM into the neutrosophic data sets. Focus is to use
neutrosophic techniques and methodologies which can help to minimise the indeterminacies.
Finally calculate a neutrosophic beta value and get a neutrosophic expected return.

51

These neutrosophic results would include not just a single value but a neutrosophic set number
or an interval. This approach could allow for a more nuanced understanding of risk and
uncertainty in making decisions in stock market. It will facilitate analysts and investors to take
a best investment decision based on neutrosophic expected return or neutrosophic beta.
In traditional CAPM method, adjusted closing price is used to calculate beta. The adjusted
closing price reflects the value of a share after accounting for corporate actions like stock splits
or cash dividends to closing price of the day. But the closing price is the last price at the end
of a trading day, and it does not reveal the standard value of a share. During a day share price
can be fluctuated within the high price and low price. In this study it is expected to address this
indeterminacy of the price which can be considered as a very crucial input to calculate beta
and expected return of asset pricing model. Share price should reveal the true economic value
of a security.
The graph 4.1.1 below shows the trend of monthly-adjusted closing prices of the NFLX from
2009-2023.

Graph 4.1.1: Monthly adjusted-closing price of the NFLX from 2009-2023
52

Overall NFLX depicts a significant upward trend over the entire period. From the period 20092013 NFLX trend is steady increase and during 2013-2015, there appears a sharp increase
followed by a correction. The trend between 2015 to 2021 is mostly an uptrend with some
consolidation periods and around 2021 the stock reaches to its highest price. Afterward the
stock shows a sharp downtrend in. But mid 2022-2023 period can be considered as the
recovery period. After mid 2022 the stock experience a gradual increase.
All the fluctuations of stock price during the period indicate the high competition of the
streaming industry and all the challenges and opportunities which are faced by the company.
The graph 4.1.2. represents NFLX monthly high, low and adjusted-closing prices from 20092023 and the monthly price fluctuations in between low and high prices.

Graph 4.1.2: NFLX monthly High, Low and Adjusted Closing Prices from 2009-2023

The fluctuations of three prices high, low and adjusted close prices illustrate the dynamic stock
market behaviours and investor sentiment towards NFLX stock. According to the graph, the
53

adjusted-closing prices are not continuously positioned within the expected range. Sometimes
it falls between the high and low prices but at other times, it aligns with either the lowest or
highest price of the month. For illustration all three prices are align close to each other from
2010 to 2014 and in between 2021- 2022 adjusted-closing prices position with the lower price
line of NFLX. This inconsistency can lead to an underestimation of actual volatility especially
for highly volatile stocks like NFLX.
According to the graph 4.1.2, high price and low-price lines represent the two extrema of the
stock price movement during the trading period. A monthly interval with low and high prices
[low price, high price] captures the full range of prices that a stock can fluctuate within a
month. Comparatively low and high intervals provide a more comprehensive picture of its
volatility. If we consider a daily interval, it captures the whole day price range, theoretically
encompassing all trading prices within that interval. This is an advantage over using a single
closing price, which might not reflect the full daily price movement.
The low price and high price interval represent a range of possible "true" prices within a trading
period. The interval low to high represents the indeterminacy of the price, which we use to
calculate beta. The beta value could place anywhere within the range, reflecting uncertainty
about the true relationship between the risk and the market.
However, neutrosophic statistics is an ongoing concept. Hence applying this new technique to
the CAPM is challenging.
4.2. Basic Techniques and Methodologies
This study is based on the original data sets of S&P500, NASDAQ and NFLX. To check the
effect on smoothing technique, MA of the original data set is also used with the aim of refining
54

the reliability as well as the accuracy of the results. Further, we expect to smooth the shortterm fluctuations to get a meaningful and consistent output. Altogether there are four different
scenarios with the aim of capturing the limitations of CAPM and indeterminacy in the stock
market. These four scenarios can be categorised as below.
Scenario 01:S&P 500 Index with original data
Scenario 02: NASDAQ Index with original data
Scenario 03:S&P 500 Index with Moving Average (MA)
Scenario 04: NASDAQ Index with MA
In the first two scenarios (Scenario 01 and Scenario 02) the original data set from yahoo finance
is used as the initial data set meanwhile moving average of the original data set is used as the
dataset of the second two scenarios (Scenario 03 and Scenario 04).
Scenario 01:S&P 500 Index with original data
Original NFLX monthly low and high stock prices are compared with original S&P500
market index monthly low and high prices in scenario 01.
Dependent Variable: NFLX monthly stock prices (Original data set)
Independent Variables: S&P500 market index monthly (Original data set), GDP and CPI
Scenario 02: NASDAQ Index with original data
In scenario 02, original NFLX monthly low and high stock prices are compared with original
NASDAQ market index monthly low and high prices.
Dependent Variable: NFLX monthly stock prices (Original data set)
Independent Variables: NASDAQ market index monthly (Original data set), GDP and CPI
55

Scenario 03: S&P 500 Index with MA
In scenario 03, the MA of NFLX monthly low and high stock prices are compared with the
MA of S&P500 market index monthly low and high prices.
Dependent Variable: MA of NFLX monthly stock prices
Independent Variables: MA of S&P500 market index monthly, GDP and CPI
Scenario 04: NASDAQ Index with MA
In scenario 03, the MA of NFLX monthly low and high stock prices are compared with the
MA of NASDAQ market index monthly low and high prices.
Dependent Variable: MA of NFLX monthly stock prices
Independent Variables: MA of NASDAQ market index monthly, GDP and CPI
The Methodologies and techniques associated with each scenario in this thesis are explained
in detailed below.
4.2.1. Time frame and Time interval
For this analysis, a wider time interval is considered just than a single trading day. A weekly
or monthly interval might capture a more comprehensive price movement and reduce the
influence of daily noise. Hence monthly low and high price interval [NFLX_low, NFLX_high]
of NFLX from 2009 to 2023 are used for this analysis. Similarly, we have downloaded monthly
low and high prices for stock market indices (S&P500 and NASDAQ) for the same period.
Monthly GDP, monthly CPI and monthly risk-free rates are also downloaded for the same
period. These original datasets are used for the analysis in scenario 1 and 2 while MA is used
as the “smoothing technique” for other two scenarios, scenario 3 and 4.
56

4.2.2 Smoothing Technique
It may be noted that these two extremes, i.e., high price and low price may create unnecessary
noise into the calculations. These high and low values can be influenced by temporary market
variations or outliers not necessarily reflecting the underlying risk of the security. However, it
is a challenge to eliminate all noise from the low and high interval. To reduce this unnecessary
noise impact, a smoothing technique such as “moving averages” can be applied within the
interval. This may provide a more representative picture of the trading price behavior and
potentially create a more informative interval for neutrosophic calculations. Moving average
smooth out the price fluctuations by considering an average price over a defined period (eg. 3
months or 6 moths moving average).
Moving Average (MA) is used as the smoothing technique for scenario 03 and scenario 4 in
this analysis.
Steps to calculate MA:
i)

Choose a MA Window

A larger window size smooth out more noise. For this calculation monthly stock prices are
taken as the data set. Hence, window size is chosen as 3 months to smooth out the noise.
ii)

Calculate the MA for monthly Highs and monthly Lows

For each month, we calculate the MA for the highs by taking the average of the high prices
within 3 consecutive months to create a 3-months window. Similarly, we calculate the MA for
the lows by averaging the low prices within the consecutive 3-month window. Iterate through
all monthly high and low data points.

57

iii)

Create the Smoothed Interval:

Once MA is calculated for both highs and lows, the new interval for each month will be MA
low for that month and MA high for that month [MA_low,MA_high]. Hence, MA interval for
NFLX stock and stock market indexes can be calculated for the whole period 2009-2023.
4.2.3 Generate a neutrosophic number a+bI
Neutrosophic analysis deals with sets/intervals of data when there is some indeterminacy in
data. In such cases neutrosophic analysis can be considered as a generalization of the set or
interval analysis. If sets/intervals are used in the analysis and there is no indeterminacy, then
neutrosophic analysis coincides with set/interval analysis. If there is some indeterminacy, no
matter if using only intervals, or using sets, then neutrosophic analysis is more appropriate
(Smarandache, 2015).
The stock market, by virtue of its nature, is full of vagueness, uncertainty, ambiguity,
incompleteness and contradiction. Hence in this analysis, neutrosophic techniques and
methods can be utilized to capture the indeterminacy.
For this study, in addition to the standard procedure, the intervals for NFLX stock prices and
stock market indices are converted into the neutrosophic form. This transformation is essential
because direct use of these intervals may not reduce the indeterminacy. Direct use of intervals
gives the results for interval statistics. But neutrosophic numbers can represent uncertain,
imprecise, incomplete, indeterminate, contradictory, and vague information. These two
approaches are fundamentally different (Smarandache, 2015).
The aim of this study is to give a neutrosophic statistical approach to the CAPM or develop a
neutrosophic CAPM model. The conversion from interval to neutrosophic numbers may
58

involve subjective decisions, expert’s opinion or a priori information. The applications of
neutrosophic techniques help to minimise effects of the indeterminacy.
Role of Indeterminate Component (I)
“I” is a theoretical symbol used in neutrosophic statistics to denote indeterminacy. Usually, it
is not a number but a placeholder indicating that there is an element of uncertainty associated
with the value. Indeterminate part of the neutrosophic number “b” combined with “I” to create
the indeterminacy. If “b” is greater, then the indeterminate part “bI” has a greater influence on
the overall number and it indicates higher uncertainty.
Steps to convert interval statistics to neutrosophic number a+bI
Let’s take the Interval as [L, U]; where “L” represents Lower Limit and “U” represent Upper
Limit. We need to convert this interval format in to the Neutrosophic number =a+bI; where “I”
is the indeterminacy and assume I ∈ [-1,+1].
In this study an interval [L, U] = [U, L] in the case when we do not know which one between
“L”and “U” is bigger (Smarandache, 2015).
Step 01: Determine “a”
a: this determinant part is taken as the midpoint of the interval

ܽ=

‫ܮ‬+ܷ
2
(4. 6)

As an example, if L is 3.28 and U is 4.43, then;
ܽ=

3.28 + 4.43
= 3.85
2

59

Step 02: Determine “b”
b: This indeterminate part represents the extent of the indeterminacy or the range of
uncertainty. It is taken as the half the width of the interval.

ܾ=

ܷ−‫ܮ‬
2
(4. 7)

As an example, if L is 3.28 and U is 4.43, then;
ܾ=

4.43 − 3.28
= 0.58
2

Step 03: write down interval [L, U] as a neutrosophic number a+bI
ܽ + ܾ‫= ܫ‬

‫ܮ‬+ܷ ܷ−‫ܮ‬
+
‫ܫ‬
2
2

As an example, if L is 3.28 and U is 4.43, then “a” = 3.85 and “b” = 0.58. The neutrosophic
numbers can be written as below.
ܽ + ܾ‫ = ܫ‬3.85 + 0.58‫ܫ‬
Similarly, the data sets of NFLX and stock market indexes in interval format can be
converted into neutrosophic data sets.
4.2.4 Neutrosophic Return (NR)
The neutrosophic data set can be used to calculate NR. This NR can be represented as a
neutrosophic number in the format “P+QI”.
Steps to calculate the NR
Step 01: Substitute neutrosophic values to the simple return (equation 4.1)

60

NR ୒୊୐ଡ଼ =

NP୲ − NP୲ିଵ
= P + QI
NP୲ିଵ
(4. 8)

Where, NPt = Neutrosophic Share price at time t and NPt-1 = Neutrosophic Share price at time
t-1
As an example, if NP0 = a0+b0I and NP1 = a1+b1I then the result can be denoted as;
P + QI =

aଵ + bଵ I − a଴ + b଴ I
a ଴ + b଴ I

Step 02: Determine “P”
Multiply both sides by (a0+b0I) and identify the coefficients:
[P + QI][a଴ + b଴ I] = [aଵ + bଵ ] − [a଴ + b଴ ]I
ܲܽ଴ + [ܳܽ଴ + ܾܲ଴ ]‫ ܫ‬+ ܾ଴ ܳ‫ ܫ‬ଶ = [ܽଵ − ܽ଴ ] + [ܾଵ − ܾ଴ ]‫ܫ‬
ܽ଴ ܲ + [ܽ଴ ܳ + ܾ଴ ܲ + ܾ଴ ܳ]‫ܽ[ = ܫ‬ଵ − ܽ଴ ] + [ܾଵ − ܾ଴ ]‫ܫ‬
Where, I2 =I (Smarandache,2014). When we form an algebraic system of equations by
identifying the coefficients. Therefore;
ܽ଴ ܲ = ܽଵ − ܽ଴
ܲ=

ܽଵ − ܽ଴
ܽ଴

Similarly Determine “Q”
ܽ଴ ܳ + ܾ଴ ܲ + ܾ଴ ܳ = ܾଵ + ܾ଴
[ܽ଴ + ܾ଴ ] × ܳ = ܾଵ − ܾ଴ − ܾܲ଴
substitute the value of “P”
ܽ଴ [ܽ଴ + ܾ଴ ]ܳ = ܽ଴ [ܾଵ − ܾ଴ ] − [ܽଵ − ܽ଴ ]ܾ଴

61

ܳ=

ܽ଴ [ܾଵ − ܾ଴ ] − ܾ଴ [ܽଵ − ܽ଴ ]
ܽ଴ [ܽ଴ + ܾ଴ ]

ܳ=

ܽ଴ ܾଵ − ܽଵ ܾ଴
ܽ଴ [ܽ଴ + ܾ଴ ]

Step 04: Write down the neutrosophic return
ܲ + ܳ‫= ܫ‬

ܽଵ − ܽ଴ ܽ଴ ܾଵ − ܽଵ ܾ଴
+
‫ܫ‬
ܽ଴
ܽ଴ [ܽ଴ + ܾ଴ ]

Therefore, the neutrosophic return at any time t can be introduced as
ܲ + ܳ‫= ܫ‬

ܽ௧ − ܽ௧ିଵ
ܽ௧ିଵ ܾ௧ − ܽ௧ ܾ௧ିଵ
+
‫ܫ‬
ܽ௧ିଵ
ܽ௧ିଵ [ܽ௧ିଵ + ܾ௧ିଵ ]
(4. 9)

Where, at is determinant part at time t, at-1 is determinant part at time t-1, bt is indeterminant
part at time t and bt-1 is indeterminant part at time t-1
As an example if NP0 = (3.85+0.58 I), NP1 = (4.58+0.52 I) where I=[-1,+1] then the
neutrosophic return should be;
ܲ + ܳ‫= ܫ‬

ܲ + ܳ‫= ܫ‬

ܽଵ − ܽ଴ ܽ଴ ܾଵ − ܽଵ ܾ଴
+
‫ܫ‬
ܽ଴
ܽ଴ [ܽ଴ + ܾ଴ ]

4.58 − 3.85 3.85 × 0.52 − 4.58 × 0.58
+
‫ = ܫ‬0.19 + (−0.03)‫ܫ‬
3.85
3.85[3.85 + 0.587]

Similarly, neutrosophic NFLX stock return can be calculated as a neutrosophic number “P+QI”
and neutrosophic returns for stock market index can be calculated as P/+Q/I.
4.2.5 Calculations of Factors’ Return
To incorporate GDP and CPI as the factors of neutrosophic models, return on GDP and CPI
need to be calculated which can be seen the percentage change in GDP and CPI over time.
62

Return of GDP:
ܴ஼௉ூ =

‫ܲܦܩ‬௧ − ‫ܲܦܩ‬௧ିଵ
‫ܲܦܩ‬௧ିଵ

(4.10)

Where RGDP is return on the GDP factor, GDPt is GDP in the period t and GDPt-1 is GDP in
the previous month-period t-1
Return of CPI:
ܴ஼௉ூ =

‫ܫܲܥ‬௧ − ‫ܫܲܥ‬௧ିଵ
‫ܫܲܥ‬௧ିଵ

(4.11)

Where RCPI is return on the CPI factor, CPIt is CPI in the period t and CPIt-1 is CPI in the
previous month-period t-1
4.2.6. Obtain Neutrosophic Beta (Nβ)
In traditional CAPM, slop of the ordinary least square line (OLSL)can be used to calculate
beta (β). But neutrosophic data sets need to be used instead of crisp numbers to calculate
neutrosophic beta (Nβ). As an illustration neutrosophic intervals for ∑Y, ∑X1, ∑Y2, X12 and
∑ X1Y need to be used to calculate the Nβ for the one factor model.
Steps to calculate Nβ for one factor model
Step 01: The neutrosophic stock returns of NFLX (P+QI) are converted in to the interval
format such as Y= [LL, UL], where I ∈ [-1,+1] to calculate ∑Y , ∑X1, ∑Y2,X12 ∑ X1Y .Here,
LL: lower limit of the return = P-Q ;(where I=-1)
(4.11)

UL: upper limit of the return = P+Q ;(where I=+1)
(4.12)

Step 02: Similarly neutrosophic returns of stock market index P/+Q/I are converted in to
interval format such as X1= [LL, UL] where I∈ [-1,+1]
63

Then these intervals have been used to calculate the values for the ∑[YL,YU], ∑[X1L X1U], ∑[YL2,
YU2] , ∑[X1L2 , X1U2] and ∑ [X1LYL,X1UYU] which are also in interval format. Then Nβ for one

factor model can be calculated by substituting the sums of those values to the formula (equation
4.19). The structure of the OLS table is decided by the number of factors as below. Tables for
neutrosophic calculations need to be adjusted accordingly.

Factor model

Type of data required for OLS table to calculate β

1 factor model

∑Y, ∑X1, ∑Y2, X12, ∑ X1Y.

2 factors model

∑Y, ∑X1, ∑ X2, ∑Y2, X12, ∑ X22, ∑ X1Y, ∑ X2Y, ∑ X1 X2

3 factors model

∑Y, ∑X1, ∑ X2, ∑ X3, ∑Y2, X12, ∑ X22, ∑ X32, ∑ X1Y, ∑ X2Y, ∑ X3Y, ∑ X1 X2
∑ X1 X3, ∑ X2 X3, ∑ X1 X2 X3 , ∑ X1 X2Y, ∑ X1 X3Y, ∑ X2 X3Y

Table 4.2.6.1: Type of data required for OLS table to calculate β for different factor models

As an example, the table for 1 factor model can be prepared as below.

Table 4.2.6.2: NLS table for one-factor model with respect to S&P 500 market index.

64

Similarly, NLS tables need to be prepared for 2-factor and 3-factor models to calculate the
neutrosophic beta values.
4.3. Application of Neutrosophic Methods to Calculate Neutrosophic Beta
A neutrosophic model is a model, which considers indeterminacy in data. We may not
appropriately construct an accurate classical model when the data set which describe the
physical world is incomplete, ambiguous, contradictory and unclear. In this case, to study these
kinds of environments, we need to build an approximate model. Neutrosophic statistics helps
to plot the incomplete, unclear ambiguous data and then we can design a neutrosophic
regression method. The neutrosophic linear regression and the neutrosophic least squares
regression are the most common such methods. In this study both methods are used to capture
the indeterminacy (Smarandache, 2015).
(i) General One Factor model
Simple linear regression equation with one independent variable x1;
‫ߚ = ݕ‬଴ + ߚଵ ‫ݔ‬ଵ + ߝ
(4.13)

We can rewrite this equation as
ߝ = ‫ ݕ‬− ߚ଴ − ߚଵ ‫ݔ‬ଵ
(4.14)

Minimize the sum of squared residuals RSS for the given sample of size N.
෍ ߝ ଶ = ෍(‫ ݕ‬− ߚ଴ − ߚଵ ‫ݔ‬ଵ )ଶ
(4.15)

65

Then partially differentiate the RSS function with respect to β0 and β1. For the minimum of
RSS function, we take partial derivatives equal to zero.
Then two OLS normal equation can be written as,
෍ ‫ߚ݊ = ݕ‬଴ + ߚଵ ෍ ‫ݔ‬ଵ
(4.16)

෍ ‫ݔ‬ଵ ‫ߚ = ݕ‬଴ ෍ ‫ݔ‬ଵ + ߚଵ ෍ ‫ݔ‬ଵ ଶ
(4.17)

By solving these two equations, the slop of the simple linear regression model can be obtained
as below and it is the β1 for the one variable in classical model. (Abraham & Ledolter, 2006)
∑ ‫ݔ‬ଵ ∑ ‫ݕ‬
݊
ߚଵ =
(∑ ‫ݔ‬ଵ )ଶ
ଶ
∑ ‫ݔ‬ଵ −
݊
∑ ‫ݔ‬ଵ ‫ ݕ‬−

(4.18)

But in this study neutrosophic data sets are used. Therefore, the classical formula needs to be
adjusted according to the neutrosophic statistical methodologies to calculate Neutrosophic beta
as [ܰߚ௜௅, ܰߚ௜௎ ].To calculate this interval of Nβ1, x1 and y need to be converted into the
neutrosophic intervals.
x1 can be represent as [ x1lower, x1upper] which is [x1L, x1U]
y can be represent as [ylower,yupper] which is [ yL, yU]
For the one factor neutrosophic model, Nβ1 can be represent as [Nβଵ୐, Nβଵ୙ ] and with the use
of equation 4.18 the new formula for this [Nβଵ୐, Nβଵ୙ ] can be written as below. Values of the
∑[YL,YU], ∑[X1L X1U], ∑[YL2, YU2] , ∑[X1L2 , X1U2] and ∑ [X1LYL,X1UYU] need to be calculated

66

using NLS table and substitute to the below formula to find out this neutrosophic beta. All the
calculations need to be done according to the neutrosophic logic (Smarandache, F. ,2014).

([Nβଵ୐ , Nβଵ୙ ]) =

(∑[xଵ୐ , xଵ୙ ] ∑[y୐ , y୙ ])
n
(∑[x
,
x
])ଶ
ଵ୐
ଵ୙
∑[xଵ୐ , xଵ୙ ]ଶ −
n

∑[xଵ୐ , xଵ୙ ][y୐ , y୙ ] −

(4.19)

As an illustration, we can use the NLS results in table 4.3.5.1 to calculate Nβ1 of 1-factor
model. Here we must follow the neutrosophic calculations. (Smarandache, F. ,2014 page
no:79&80)

([ܰߚଵ௅ , ܰߚଵ௎ ]) =

([1.751,1.516] × [5.794,5.534])
180
([1.751,1.516])ଶ
[0.108,0.090] −
180

[0.169,0.117] −

([ܰߚଵ௅ , ܰߚଵ௎ ]) =

[0.169,0.117] − [0.0564,0.0466]
[0.108,0.090] − [0.017,0.0128]

([ܰߚଵ௅ , ܰߚଵ௎ ]) =

[0.1225,0.0609]
[0.0951,0.0729]

([ܰߚଵ௅ , ܰߚଵ௎ ]) = [0.6403,1.6805]
(ii)Two factor model
The OLS multiple linear regression equation with two independents variable x1 and x2
‫ߚ = ݕ‬଴ + ߚଵ ‫ݔ‬ଵ + ߚଶ ‫ݔ‬ଶ + ߝ
(4.20)

where y is dependent variable, x1 and x2 are independent variables, β0 is intercept, β1 and β2
are coefficients of independent variables and ߝ is error term. Then,
ε = y − β଴ − βଵ xଵ − βଶ xଶ
(4.21)

Minimize the sum of squared residuals RSS for the given sample of size n.
67

෍ εଶ = ෍(y − β଴ − βଵ xଵ − βଶ xଶ )ଶ
(4.22)

Then partially differentiate the RSS function with respect to β0, β1 and β2. For the minimum of
RSS function, we set partial derivatives equal to zero.
Then three OLS normal equation can be written as;
෍ y = nβ଴ + βଵ ෍ xଵ + βଶ ෍ xଶ
(4.23)

෍ xଵ y = β଴ ෍ xଵ + βଵ ෍ xଵ ଶ + βଶ ෍ xଵ xଶ
(4.24)

෍ xଶ y = β଴ ෍ xଶ + βଵ ෍ xଵ xଶ + βଶ ෍ xଶ ଶ
(4.25)

The OLS normal equations constitute three linear equations with three unknowns. Solving
these equations give β coefficients as below (Abraham, B., and Ledolter, J., 2006).
β1 coefficient
∑ xଶଶ ∑ xଵ y − [∑ xଵ xଶ ∑ xଶ y]
βଵ =
∑ xଵଶ ∑ xଶଶ − [∑ xଵ xଶ ]ଶ
(4.26)

β 2 coefficient
βଶ =

∑ xଵଶ ∑ xଶ y − [∑ xଵ xଶ ∑ xଵ y]
∑ xଵଶ ∑ xଶଶ − [∑ xଵ xଶ ]ଶ
(4.27)

Where;
෍ xଵଶ = ෍ Xଵଶ −

෍ xଶଶ = ෍ X ଶଶ −

68

∑ Xଵ ∑ Xଵ
N
∑ Xଶ ∑ Xଶ
N

(i)

(ii)

෍ xଵ y = ෍ Xଵ Y −

∑ Xଵ ∑ Y
N

෍ xଶ y = ෍ X ଶ Y −

∑ Xଶ ∑ Y
N

෍ ‫ݔ‬ଵ ‫ݔ‬ଶ = ෍ ܺଵ ܺଶ −

∑ ܺଵ ∑ ܺଶ
ܰ

(iii)

(iv)

(v)

This adjustment centers the variables around their means, and it is removing the effects of the
average values of x and y variables.
Similar to the one factor model, Nβ value for 2-factor model needs to be calculated using
neutrosophic logic.
Now x1, x2 and y need to be change into the intervals.
x1 can be represent as [ x1lower, x1upper] which is [x1L, x1U]
x2 can be represent as [ x2lower, x2upper] which is [x2L, x2U]
y can be represent as [ylower,yupper] which is [ yL, yU]
There are two betas for the two-factor model. Those neutrosophic betas are [Nβଵ୐, Nβଵ୙ ] and
[Nβଶ୐, Nβଶ୙ ].Values of the ∑[YL,YU], ∑[X1L X1U], ∑[X2L X2U], ∑[YL2, YU2] , ∑[X1L2 , X1U2] , ∑[X2L2
, X2U2] , ∑ [X1LYL,X1UYU], ∑ [X2LYL,X2UYU] and ∑ [X1LX2L,X1UX2U] need to be calculated using OLS

table and substitute to the below formulas to find out these two neutrosophic betas. All the
calculations need to be done according to the neutrosophic logic. Therefore, with the use of
equations 4.26 and 4.27, Nβs for two variables can be written as below.
Nβ1 coefficient :

69

([Nβଵ୐ , Nβଵ୙ ])
=

∑[xଶ୐ , xଶ୙ ]ଶ ∑[xଵ୐ , xଵ୙ ][y୐ , y୙ ] − (∑[xଵ୐ , xଵ୙ ][xଶ୐ , xଶ୙ ] ∑[xଶ୐ , xଶ୙ ][y୐ , y୙ ])
∑[xଵ୐ , xଵ୙ ]ଶ ∑[xଶ୐ , xଶ୙ ]ଶ − (∑[xଵ୐ , xଵ୙ ] [xଶ୐ , xଶ୙ ])ଶ
(4.28)

Nβ2 coefficient :
([Nβଶ୐ , Nβଶ୙ ])
=

∑[xଵ୐ , xଵ୙ ]ଶ ∑[xଶ୐ , xଶ୙ ][y୐ , y୙ ] − (∑[xଵ୐ , xଵ୙ ][xଶ୐ , xଶ୙ ] ∑[xଵ୐ , xଵ୙ ][y୐ , y୙ ])
∑[xଵ୐ , xଵ୙ ]ଶ ∑[xଶ୐ , xଶ୙ ]ଶ − (∑[xଵ୐ , xଵ୙ ] [xଶ୐ , xଶ୙ ])ଶ
(4.29)

Where,
෍[‫ݔ‬ଵ௅ , ‫ݔ‬ଵ௎ ]ଶ = ෍[ܺଵ௅ , ܺଵ௎ ]ଶ −

෍[‫ݔ‬ଶ௅ , ‫ݔ‬ଶ௎ ]ଶ = ෍[ܺଶ௅ , ܺଶ௎ ]ଶ −

∑[ܺଵ௅ , ܺଵ௎ ] ∑[ܺଵ௅, ܺଵ௎ ]
ܰ
∑[ܺଶ௅ , ܺଶ௎ ] ∑[ܺଶ௅, ܺଶ௎ ]
ܰ

෍[‫ݔ‬ଵ௅ , ‫ݔ‬ଵ௎ ][‫ݕ‬௅ , ‫ݕ‬௎ ] = ෍[ܺଵ௅ , ܺଵ௎ ][ܻ௅ , ܻ௎ ] −

෍[‫ݔ‬ଶ௅ , ‫ݔ‬ଶ௎ ][‫ݕ‬௅ , ‫ݕ‬௎ ] = ෍[ܺଶ௅ , ܺଶ௎ ][ܻ௅ , ܻ௎ ] −

∑[ܺଵ௅ , ܺଵ௎ ] ∑[ܻ௅, ܻ௎ ]
ܰ
∑[ܺଶ௅ , ܺଶ௎ ] ∑[ܻ௅, ܻ௎ ]
ܰ

෍[‫ݔ‬ଵ௅ , ‫ݔ‬ଵ௎ ][‫ݔ‬ଶ௅ , ‫ݔ‬ଶ௎ ] = ෍[ܺଵ௅ , ܺଵ௎ ][ܺଶ௅ , ܺଶ௎ ] −

∑[ܺଵ௅ , ܺଵ௎ ] ∑[ܺଶ௅, ܺଶ௎ ]
ܰ

(i-a)

(ii-a)

(iii-a)

(iv-a)

(v-a)

This adjustment centers the variables around their means, and it is removing the effects of the
average values of X and Y variables. After substituting the data all the calculations need to be
done as per the neutrosophic logic ((Nagarajan et al., 2021)).

70

(iii)Three Factor model
The OLS multiple linear regression equation with three independents variables x1, x2 and x3
‫ߚ = ݕ‬଴ + ߚଵ ‫ݔ‬ଵ + ߚଶ ‫ݔ‬ଶ + ߚଷ ‫ݔ‬ଷ + ߝ
Where y is dependent variable, x1 ,x2,x3 are independent variables ,β0 is intercept,β1, β2, β3 are
the coefficients of independent variables and ߝ is the error term. Then
ߝ = ‫ ݕ‬− ߚ଴ − ߚଵ ‫ݔ‬ଵ − ߚଶ ‫ݔ‬ଶ − ߚଷ ‫ݔ‬ଷ
Minimize the sum of squared residuals RSS for the given sample of size N.
෍ ߝ ଶ = ෍(‫ ݕ‬− ߚ଴ − ߚଵ ‫ݔ‬ଵ − ߚଶ ‫ݔ‬ଶ − ߚଷ ‫ݔ‬ଷ )ଶ
(4.30)

Then partially differentiate the RSS function with respect to β0, β1 , β2 and β3. For the minimum
of RSS function, partial derivatives need to be equal to zero.
Then four OLS normal equations can be written as,
෍ ‫ߚ݊ = ݕ‬଴ + ߚଵ ෍ ‫ݔ‬ଵ + ߚଶ ෍ ‫ݔ‬ଶ + ߚଷ ෍ ‫ݔ‬ଷ
(4.31)

෍ ‫ݔ‬ଵ ‫ߚ = ݕ‬଴ ෍ ‫ݔ‬ଵ + ߚଵ ෍ ‫ݔ‬ଵ ଶ + ߚଶ ෍ ‫ݔ‬ଵ ‫ݔ‬ଶ + ߚଷ ෍ ‫ݔ‬ଵ ‫ݔ‬ଷ
(4.32)

෍ ‫ݔ‬ଶ ‫ߚ = ݕ‬଴ ෍ ‫ݔ‬ଶ + ߚଵ ෍ ‫ݔ‬ଵ ‫ݔ‬ଶ + ߚଶ ෍ ‫ݔ‬ଶ ଶ + ߚଷ ෍ ‫ݔ‬ଶ ‫ݔ‬ଷ
(4.33)

෍ ‫ݔ‬ଷ ‫ߚ = ݕ‬଴ ෍ ‫ݔ‬ଷ + ߚଵ ෍ ‫ݔ‬ଵ ‫ݔ‬ଷ + ߚଶ ෍ ‫ݔ‬ଶ ‫ݔ‬ଷ + ߚଷ ෍ ‫ݔ‬ଷ ଶ
(4.34)

71

These OLS standard equations constitute four linear equations with four unknowns. β
coefficients can be obtained by solving these four equations (Abraham & Ledolter, 2006;
PennState:Stat 501, Lesson 05, 2023).
Simplified symbols are used to reduce the difficulty of handling the complex formulas, as
mentioned below.
βଵ
=

Sଡ଼మ ଡ଼మ Sଡ଼య ଡ଼య Sଡ଼భ ଢ଼ − Sଡ଼మଡ଼మ Sଡ଼భ ଡ଼య Sଡ଼య ଢ଼ + Sଡ଼భ ଡ଼మ Sଡ଼య ଡ଼య Sଡ଼మଢ଼ − Sଡ଼భଡ଼మ Sଡ଼మଡ଼య Sଡ଼య ଢ଼ − Sଡ଼భଡ଼భ Sଡ଼యଡ଼య Sଡ଼మଢ଼ + Sଡ଼భ Sଡ଼మ ଡ଼య Sଡ଼మଡ଼య ଢ଼
Sଡ଼భ ଡ଼భ Sଡ଼మ ଡ଼మ Sଡ଼య ଡ଼య − [Sଡ଼భ ଡ଼భ Sଡ଼మ ଡ଼య ]ଶ + S௑భ Sଡ଼మ ଡ଼య S௑భ Sଡ଼మ ଡ଼య − Sଡ଼భ Sଡ଼భ ଡ଼మ Sଡ଼య ଡ଼య Sଡ଼య ଢ଼
(4.35)

βଶ
=

Sଡ଼భ ଡ଼భ Sଡ଼య ଡ଼య Sଡ଼మ ଢ଼ − Sଡ଼భଡ଼భ Sଡ଼మ ଡ଼య Sଡ଼య ଢ଼ + Sଡ଼భ ଡ଼మ Sଡ଼య ଡ଼య Sଡ଼భଢ଼ − Sଡ଼భଡ଼మ Sଡ଼భଡ଼య Sଡ଼య ଢ଼ − Sଡ଼మଡ଼మ Sଡ଼యଡ଼య Sଡ଼భଢ଼ + Sଡ଼మ Sଡ଼భ ଡ଼య Sଡ଼భଡ଼య ଢ଼
Sଡ଼భ ଡ଼భ Sଡ଼మ ଡ଼మ Sଡ଼య ଡ଼య − [Sଡ଼భ ଡ଼భ Sଡ଼మ ଡ଼య ]ଶ + S௑భ Sଡ଼మ ଡ଼య S௑భ Sଡ଼మ ଡ଼య − Sଡ଼భ Sଡ଼భ ଡ଼మ Sଡ଼య ଡ଼య Sଡ଼య ଢ଼
(4.36)

βଷ
=

Sଡ଼భ ଡ଼భ Sଡ଼మ ଡ଼మ Sଡ଼య ଢ଼ − Sଡ଼భଡ଼భ Sଡ଼మ ଡ଼య Sଡ଼మ ଢ଼ + Sଡ଼భ ଡ଼య Sଡ଼మ ଡ଼మ Sଡ଼భଢ଼ − Sଡ଼భଡ଼మ Sଡ଼భଡ଼య Sଡ଼మ ଢ଼ − Sଡ଼మଡ଼మ Sଡ଼యଡ଼య Sଡ଼భଢ଼ + Sଡ଼య Sଡ଼భ ଡ଼మ Sଡ଼భଡ଼మ ଢ଼
Sଡ଼భ ଡ଼భ Sଡ଼మ ଡ଼మ Sଡ଼య ଡ଼య − [Sଡ଼భ ଡ଼భ Sଡ଼మ ଡ଼య ]ଶ + S௑భ Sଡ଼మ ଡ଼య S௑భ Sଡ଼మ ଡ଼య − Sଡ଼భ Sଡ଼భ ଡ଼మ Sଡ଼య ଡ଼య Sଡ଼య ଢ଼
(4.37)

Where Sଡ଼భ =∑ x1,Sଡ଼మ =∑ x2,Sଡ଼య =∑ x3,Sଡ଼భଡ଼భ =∑x12,Sଡ଼భଡ଼మ =∑x1 x2,Sଡ଼మଡ଼మ =∑x22,Sଡ଼భଡ଼య =∑x1
x3 ,Sଡ଼మଡ଼య =∑x2 x3,Sଡ଼యଡ଼య =∑x32,Sଡ଼భଢ଼ =∑x1y,Sଡ଼మଢ଼ =∑x2y,Sଡ଼యଢ଼=∑x3y,Sଡ଼భଡ଼మଢ଼ =∑x1 x2y,
Sଡ଼మଡ଼యଢ଼ =∑x2 x3y,Sଡ଼భଡ଼యଢ଼ =∑x1 x3y
We can remove the effects of the average values of x and y variables using the equations (i),
(ii), (iii), (iv),(v) and

72

෍ xଷଶ = ෍ X ଷଶ −

∑ Xଷ ∑ Xଷ
N

෍ xଷ y = ෍ X ଷ Y −

∑ Xଷ ∑ Y
N

෍ xଵ xଷ = ෍ Xଵ X ଷ −

෍ xଶ x ଷ = ෍ X ଶ X ଷ −

෍ xଵ xଷ y = ෍ Xଵ X ଷ Y −

෍ xଶ xଷ y = ෍ Xଶ X ଷ Y −

∑ Xଵ ∑ X ଷ
N
∑ Xଶ ∑ Xଷ
N
∑ Xଵ ∑ X ଷ ∑ Y
N
∑ Xଶ ∑ Xଷ ∑ Y
N

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

For the use of neutrosophic models, βs need to be adjusted into Nβs using neutrosophic logic.
Now x1, x2, x3 and y need to be change into the intervals.
x1 can be represent as [ x1lower, x1upper] which is [x1L, x1U]
x2 can be represent as [ x2lower, x2upper] which is [x2L, x2U]
x2 can be represent as [ x2lower, x2upper] which is [x2L, x2U]
x3 can be represent as [ x3lower, x3upper] which is [x3L, x3U]
y can be represent as [ylower,yupper] which is [ yL, yU]
According to this analysis [Nβଵ୐, Nβଵ୙ ], [Nβଶ୐, Nβଶ୙ ] and [Nβଷ୐, Nβଷ୙ ] can be written down
by adjusting the formulas for β1 β2 and β3 (Smrandache, F.,2014).As an illustration Nβଵ can be
represent as below using the equation 4.35.

73

[Nβଵ୐ , Nβଵ୙ ]
= ቂቀSൣଡ଼మై, ଡ଼మ౑ ][ଡ଼మై, ଡ଼మ౑ ൧ Sൣଡ଼యై ,ଡ଼య౑ ][ଡ଼యై, ଡ଼య౑ ൧ S[ଡ଼భై ,ଡ଼భ౑ ][ଢ଼ై,ଢ଼౑] ቁ
− ൫S[ଡ଼మై ,ଡ଼మ౑ ][ଡ଼మై, ଡ଼మ౑ ] S[ଡ଼భై ,ଡ଼భ౑ ][ଡ଼యై ,ଡ଼య౑ ] S[ଡ଼యై, ଡ଼య౑ ][ଢ଼ై ,ଢ଼౑ ] ൯
+ ቀS[ଡ଼భై, ଡ଼భ౑ ][ଡ଼మై, ଡ଼మ౑ ] Sൣଡ଼యై, ଡ଼య౑ ൧[ଡ଼యై ,ଡ଼య౑ ] S[ଡ଼మై, ଡ଼మ౑ ][ଢ଼ై ,ଢ଼౑ ] ቁ
− ቀS[ଡ଼భై ,ଡ଼భ౑ ][ଡ଼మై ,ଡ଼మ౑ ] Sൣଡ଼మై ,ଡ଼మ౑ ][ଡ଼యై, ଡ଼య౑ ൧ S[ଡ଼యై ,ଡ଼య౑ ][ଢ଼ై ,ଢ଼౑ ] ቁ
− ൫S[ଡ଼భై, ଡ଼భ౑ ][ଡ଼భై, ଡ଼భ౑ ] S[ଡ଼యై, ଡ଼య౑ ][ଡ଼యై, ଡ଼య౑ ] S[ଡ଼మై, ଡ଼మ౑ ][ଢ଼ై ,ଢ଼౑] ൯
+ ቀS[ଡ଼భై, ଡ଼భ౑ ] Sൣଡ଼మై ,ଡ଼మ౑ ][ଡ଼యై, ଡ଼య౑ ൧ S[ଡ଼మై ,ଡ଼మ౑ ][ଡ଼యై, ଡ଼య౑ ][ଢ଼ై ,ଢ଼౑ ] ቁቃ
/ ൤൫S[ଡ଼భై ,ଡ଼భ౑ ][ଡ଼భై ,ଡ଼భ౑ ] S[ଡ଼మై, ଡ଼మ౑ ][ଡ଼మై, ଡ଼మ౑ ] S[ଡ଼యై ,ଡ଼య౑ ][ଡ଼యై ,ଡ଼య౑ ] ൯ – ቀS[ଡ଼భై ,ଡ଼భ౑ [ଡ଼భై ,ଡ଼భ౑ Sൣଡ଼మై ,ଡ଼మ౑ ][ଡ଼యై, ଡ଼య౑ ൧ ቁ

ଶ

+ ቀS[ଡ଼భై ,ଡ଼భ౑ ] Sൣଡ଼మై ,ଡ଼మ౑ ][ଡ଼యై, ଡ଼య౑ ൧ S[ଡ଼భై ,ଡ଼భ౑ ] Sൣଡ଼మై ,ଡ଼మ౑ ][ଡ଼యై, ଡ଼య౑ ൧ ቁ
− ቀS[ଡ଼భై ,ଡ଼భ౑ S[ଡ଼భై ,ଡ଼భ౑ ][ଡ଼మై ଡ଼మ౑ ] Sൣଡ଼యై,ଡ଼య౑ ൧ൣଡ଼యై, ଡ଼య౑ ൧ S[ଡ଼యై, ଡ଼య౑ ][ଢ଼ై ,ଢ଼౑] ቁ൨
(4.38)

Similarly, the formulas for Nβs can be obtained for [Nβଶ୐, Nβଶ୙ ] and [Nβଷ୐, Nβଷ୙ ] by
applying neutrosophic techniques to the equations 4.36 and 4.37 respectively. All the
calculations need to follow neutrosophic logic.
Below adjustment can be done to center the variables around their means, and it remove the
effect of the average values of x’s and y variables.

෍[‫ݔ‬ଷ௅ , ‫ݔ‬ଷ௎ ]ଶ = ෍[ܺଷ௅ , ܺଷ௎ ]ଶ −

∑[ܺଷ௅ , ܺଷ௎ ] ∑[ܺଷ௅, ܺଷ௎ ]
ܰ

෍[‫ݔ‬ଷ௅ , ‫ݔ‬ଷ௎ ][‫ݕ‬௅ , ‫ݕ‬௎ ] = ෍[ܺଷ௅ , ܺଷ௎ ][ܻ௅ , ܻ௎ ] −

∑[ܺଷ௅ , ܺଷ௎ ] ∑[ܻ௅, ܻ௎ ]
ܰ

෍[‫ݔ‬ଵ௅ , ‫ݔ‬ଵ௎ ][‫ݔ‬ଷ௅ , ‫ݔ‬ଷ௎ ] = ෍[ܺଵ௅ , ܺଵ௎ ][ܺଷ௅ , ܺଷ௎ ] −

(vi-a)

(vii-a)

∑[ܺଵ௅ , ܺଵ௎ ] ∑[ܺଷ௅, ܺଷ௎ ]
ܰ
(vii-a)

෍[‫ݔ‬ଶ௅ , ‫ݔ‬ଶ௎ ][‫ݔ‬ଷ௅ , ‫ݔ‬ଷ௎ ] = ෍[ܺଶ௅ , ܺଶ௎ ][ܺଷ௅ , ܺଷ௎ ] −

∑[ܺଶ௅ , ܺଶ௎ ] ∑[ܺଷ௅, ܺଷ௎ ]
ܰ
(viii-a)

74

෍[‫ݔ‬ଵ௅ , ‫ݔ‬ଵ௎ ][‫ݔ‬ଷ௅ , ‫ݔ‬ଷ௎ ][‫ݕ‬௅ , ‫ݕ‬௎ ]
= ෍[ܺଵ௅ , ܺଵ௎ ][ܺଷ௅ , ܺଷ௎ ] [‫ݕ‬௅ , ‫ݕ‬௎ ] −

∑[ܺଵ௅ , ܺଵ௎ ] ∑[ܺଷ௅, ܺଷ௎ ] ∑[‫ݕ‬௅ , ‫ݕ‬௎ ]
ܰ
(x-a)

෍[‫ݔ‬ଶ௅ , ‫ݔ‬ଶ௎ ][‫ݔ‬ଷ௅ , ‫ݔ‬ଷ௎ ][‫ݕ‬௅ , ‫ݕ‬௎ ]
= ෍[ܺଶ௅ , ܺଶ௎ ][ܺଷ௅ , ܺଷ௎ ][‫ݕ‬௅ , ‫ݕ‬௎ ] −

∑[ܺଶ௅ , ܺଶ௎ ] ∑[ܺଷ௅, ܺଷ௎ ] ∑[‫ݕ‬௅ , ‫ݕ‬௎ ]
ܰ
(xi-a)

4.4 Multi-Factor Models
Multi-factor models can be developed to capture the sensitivity of the stock market by
addressing the factors, which affect for the trend. In this study, we consider Market Index, CPI
and GDP as the factors, which affect for the share price of the NFLX stock. Based on general
CAPM and APT models, neutrosophic one-factor, two-factor and three-factor models are
developed to observe the relationships and impacts of each variable on the NFLX stock.
4.4.1Neutrosophic Capital Asset Pricing Model (NCAPM)
The main objective of this study is to develop a neutrosophic statistical approach to the CAPM
to address the uncertainty in the market. This analysis starts by converting stock prices and
stock market indices into neutrosophic numbers. Traditional methods often fail to capture the
indeterminacy of the market due to the complex interplay of several factors such as economic
indicators, unpredictable investor’s behaviors, political impacts and market sentiment. Hence
this attempt is to capture these indeterminacies by applying neutrosophic methods to build a
better model which can address this complexity. Application of neutrosophic methods into this
study of stock market offers a strong approach to address the unpredictable nature of the stock
market. This method develops the ability to model and understand the complicated financial
75

data, leading to more informed and consistent investment strategies. The new model can be
used to calculate the range of neutrosophic expected return of a stock and this interval is useful
to get a better understanding of the uncertainty of the stock market. Furthermore, this interval
captures the unpredictable stock market behaviours and fluctuations while offering a lower
limit and upper limit for these fluctuations. These two limits give a control over the risk. Hence
investors may be aware of the worst situation as well as the best-case scenario.
4.4.2 Neutrosophic Arbitrage Pricing Theory (NAPT)
APT is widely used in practice as a tool for estimating expected asset returns and their
covariance matrix. If market participants can identify the factors, which actually affect asset
returns, they can use APT to accurately estimate the expected asset returns. APT allows
investors to form a better portfolio with simplified assumptions. There is an intellectual arms
race to find the best portfolio strategies to outperform competitors. Hence this study aims to
propose extended NCAPM to capture the risk associated with macroeconomic factors like CPI
and GDP. with the purpose of offering greater flexibility on capturing the volatility or risk of
a stock. Accordingly, we focus to develop 2-factor and 3-factor neutrosophic APT models.
These new models can be introduced as extended NCAPM or NAPT models.
4.4.3. Development of advanced Neutrosophic Statistical Factor Models
Four different advanced Neutrosophic Statistical Factor Models have been developed in this
study. Independent variables can be considered as Market Index, GDP and CPI.
Model 01: one factor model
Factor 01: Market Index (S&P500 /NASDAQ)

76

This is the NCAPM which explains the relationship between the neutrosophic expected
return of a stock and neutrosophic expected return of the market index.
ܰ‫ܴ(ܧ‬௜௅, ܴ௜௎ ) = ܴ௙ + [ܰߚଵ௅, ܰߚଵ௎ ][ܰ‫ܴ(ܧ‬ெ௅ , ܴெ௎ ) − ܴ௙ ]
(4.39)

Model 02: Two factors
Factor 01: Market index (S&P500 /NASDAQ)
Factor 02: GDP
This is an extended NCAPM (or NAPT), which explains the relationship between the
neutrosophic expected return of a stock with neutrosophic expected return of market index and
GDP.
ܰ‫ܧ‬൫ܴ௜௅, ܴ௜௎ ൯ = ܴ௙ + [ܰߚଵ௅, ܰߚଵ௎ ][ܰ‫ܴ(ܧ‬ெ௅ , ܴெ௎ ) − ܴ௙ ] + [ܰߚଶ௅, ܰߚଶ௎ ] [‫ீܴ(ܧ‬஽௉ )
− ܴ௙ ]
(4.40)

Model 03: Two factors
Factor 01: Market index (S&P500 /NASDAQ)
Factor 02: CPI
This is an extended NCAPM (or NAPT), which explains the relationship between the
neutrosophic expected return of a stock with neutrosophic expected return of market index and
CPI.
ܰ‫ܧ‬൫ܴ௜௅, ܴ௜௎ ൯ = ܴ௙ + [ܰߚଵ௅, ܰߚଵ௎ ][ܰ‫ܴ(ܧ‬ெ௅ , ܴெ௎ ) − ܴ௙ ] + [ܰߚଷ௅, ܰߚଷ௎ ] [‫ܴ(ܧ‬஼௉ூ )
− ܴ௙ ]
(4.41)

77

Model 04: Three factors
Factor 01: Market index (S&P500 /NASDAQ)
Factor 02: GDP
Factor 03: CPI
This is an extended NCAPM (or NAPT), which explains the relationship between the
neutrosophic expected return of a stock with neutrosophic expected return of market index,
GDP and CPI.
ܰ‫ܧ‬൫ܴ௜௅, ܴ௜௎ ൯ = ܴ௙ + [ܰߚଵ௅, ܰߚଵ௎ ][ܰ‫ܴ(ܧ‬ெ௅ , ܴெ௎ ) − ܴ௙ ] + [ܰߚଶ௅, ܰߚଶ௎ ]ൣ‫ீܴ(ܧ‬஽௉ ) − ܴ௙ ൧
+ [ܰߚଷ௅, ܰߚଷ௎ ][‫ܴ(ܧ‬஼௉ூ ) − ܴ௙ ]
(4.42)

Where, NE(R ୧୐, R ୧୙ ) is neutrosophic expected return of the stock,NE൫R ୑୐ , R ୑୙ ൯ is
neutrosophic expected return of the market,NE[(R ୑୐ , R ୑୙ ) -Rf] is neutrosophic market risk
premium, [E(R GDP) -Rf] is GDP risk premium, [E(R CPI )-Rf] is CPI risk premium, [Nβଵ୐, Nβଵ୙ ]
is neutrosophic beta for market returns,[Nβଶ୐, Nβଶ୙ ] is neutrosophic beta for GDP and
[Nβଷ୐, Nβଷ୙ ] is neutrosophic beta for CPI.
All these four models are analysed in each of four scenarios as illustrated by table 4.4.3.1.
(S&P 500 Index with original data, NASDAQ Index with original data, S&P 500 Index with
MA and NASDAQ Index with MA)

78

Model

Factors

Model 01

Market Index

Model 02

Market Index and GDP

Model 03

Market Index and CPI

Model 04

Market Index, GDP and CPI

Table 4.4.3.1: Different Factor models

In this study, these different neutrosophic factor models are used to calculate neutrosophic
expected returns of NFLX stocks based on different factors, which are affecting stock market
behaviour. By integrating these models, investors can improve their strategies, carefully
manage risks, and make wise investment decisions to enhance their market portfolios. All these
model calculations follow neutrosophic methods and techniques.
4.5.

Neutrosophic Correlation Coefficient

The classical correlation coefficient is a crisp number between [-1, 1]. The neutrosophic
correlation coefficient is a subset of the interval [-1, 1].
▪

If the subset of the neutrosophic correlation coefficient is in the positive side of the
interval [-1,1], the neutrosophic variables ‫ ݔ‬and ‫ ݕ‬have a neutrosophic positive
correlation.

▪

If the subset of the neutrosophic correlation coefficient is in the negative side of the
interval [-1, 1], they have a neutrosophic negative correlation (Smarandache, 2015).

In this study correlation coefficients are calculated to check the collinearity of below pairs;
between Market Index and GDP/between Market Index and CPI/between GDP and CPI
79

4.6. Statistical Hypothesis testing technique
T-statistic
Hypothesis testing technique is used for one factor model and t-statistic has been used as the
hypothesis test to measure the significance of the regression coefficient in this simple linear
model. This technique is used to test the hypothesis related to the linear relationship between
the dependent variable and the independent variable. Since the regression (slope) coefficient
follows the t-distribution under the assumptions of the linear model, T-test is used in this
analysis to determine if the predictor variable is statistically significant in the model. A
statistically significant variable has a strong relationship with the dependent variable and
contributes significantly to the accuracy of the model. T-statistic can be written as;

ܶ=

ߚመଵ
ܵ
భ
൘
൫ܵ௑భ௑భ ൯మ
(4.43)

where, S2 is ∑

ఌమ
௡ିଶ

= standard error of the estimate, ߚመଵ is the estimated regression coefficient

for the predictor variable,∑ ߝ ଶ is sum of squared errors, n is number of observations and n-2 is
Degree of freedom.
The significance of relationship between response and predictor variable can be determined by
the hypothesis testing.
Null Hypothesis (H0): The null hypothesis states that there is no relationship between the
predictor variable and the response variable. Thus, in terms of the regression coefficient, the
null hypothesis is β1=0.

80

Alternative Hypothesis (Ha): The alternative hypothesis contradicts the null hypothesis. This
suggests that there is relationship between the independent and the dependent variable. It
implies that the regression coefficient β1 ≠ 0.
The observed t-statistic is compared to a critical value from the t-distribution at a given
significance level. If the absolute value of the observed t-statistic > critical value, reject the
null hypothesis at the chosen significance level. We conclude that there is evidence about the
relationship between the independent variable and the dependent variable. If the absolute value
of the t-statistic < critical value, do not reject the null hypothesis. We conclude that there is no
evidence that the fitted linear model is significant.
4.6.1.T- statistic for the neutrosophic models
In this study t-statistic is used only for the one factor neutrosophic model (equation 4.39)
because calculate t-statistic for the neutrosophic models which has more than one factors are
more complicated. Hence the equation 4.43 can be rewritten to calculate the t statistics for the
one-factor neutrosophic model as below.

(T୐୐ , T୙୐ ) =

[β෠ଵ୐ , β෠ଵ୙ ]
[S୐୐ , S୙୐ ]
భ
൘
൫Sଡ଼భై,భ౑ ,ଡ଼భై,భ౑ ൯మ
(4.44)

According to this study the Neutrosophic Null Hypothesis (NH0) and the Neutrosophic
Alternative Hypothesis (NHa) are two possible conclusions which is very similarly to the
classical statistics (Smarandache, F. ,2014).
Null hypothesis; ܰ‫ܪ‬0: Nβ1∈ [Nβ1L, Nβ1U]
Alternative hypothesis; ܰ‫ܽܪ‬: Nβ1< Nβ1L, or ܰ‫ܽܪ‬: Nβ1 > Nβ1U, or ܰ‫ܽܪ‬: Nβ1 ∉ [Nβ1L, Nβ1U]
81

Level of Significance αܰ of a neutrosophic study not necessarily a crisp number as in classical
statistics and it may be an interval. For this neutrosophic study we will assume the set of
asymptotic significance level, [0.95 ,0.99], which implies the set ߙܰ= [0.01 ,0.05]. (Villafuerte
et al., 2020).
The decision criterion is rejected NH0 if,
Min{[t − statistic] − [ctitical t value(T)]} > Max{T(1 − α୒ )}.
This means, if the minimum value of difference between the test statistic and the critical
value(Min{[t − statistic] − [ctitical t value(T)]}) is greater than the maximum critical
value(Max{T(1 − α୒ )}), then there is a clear evidence against the null hypothesis to reject it.
4.7.

Comparison of the models using Mean Average Deviation (MAD)

To compare the different models, we calculate the MAD as a major of model accuracy. The
smaller MAD value implies that the model is more accurate and better. MAD value can be
calculated by the below formula ([48],[53]):
R ୧ − E(R ୧ )
MAD = ෍ ฬ
ฬ
n
(4.45)

Hence below formula has been used to calculate the MAD for this neutrosophic study.

NMAD = ෍ ฬ

[R ୧୐ , R ୧୙ ] − [NE(R ୧୐ , R ୧୳ )]
ฬ
n
(4.46)

82

Chapter 05
Results and Discussion
5.0. Introduction to Results and Discussion
The behaviour of a particular stock in the stock market depends upon several known and
unknown factors and is full of uncertainty. The Neutrosophic approaches provide advanced
techniques to control the indeterminacy and uncertainty in the stock market. Based on these
models, investors and financial analysts can improve risk assessment and their decisionmaking in complex financial markets. In this chapter, we show the results for the proposed
neutrosophic models by considering the NFLX, AMZN and APPL stock returns.
5.1. NFLX monthly β and E(Ri)
From the classical CAPM analysis, the results of beta and expected return for NFLX stock are
given below in Table 5.1.1. and Table 5.1.2. for the S&P 500 and NASDAQ indices
respectively for the periods of 15-years and 5-years.

SP&500

15-years

5-years

β

1.199

1.512

E(Ri)

12.20%

16.66%

Table 5.1.1 β and E(Ri) of NFLX with respect to S&P500.
NASDAQ

15-years

5-years

β

1.360

1.613

E(Ri)

19.30%

22.70%

Table 5.2. β and E(Ri) of NFLX with respect to NASDAQ.

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5.1.1 The result and discussion for the Monthly beta of NFLX stocks with S&P 500 Index
Using 15 years of historical data from 2009-2023, monthly beta of NFLX stock with respect
to S&P500 market index is 1.199. This indicates that if the market price goes up by 1-unit, the
NFLX stock price is expected to go up by 1.199-units, and if the market price goes down by
1-unit, the NFLX stock price is expected to go down by 1.199-units. Further, E(Ri) is 12.20%
using classical CAPM. With respect to S&P500 index, NFLX shows a moderate risk and
balance return for 15-year period. But for 5-year period from 2019-2023, monthly beta 1.512
indicates higher volatility, and the E(Ri) is 16.66% which is comparatively higher than the 15year period. This indicates that if the market increase/decrease by 1-unit, the NFLX stock price
will increase/decrease by 1.512-units respectively.
5.1.2 The result and discussion for the Monthly beta of NFLX stocks with NASDAQ
Index
According to the NASDAQ index, monthly beta for the NFLX stock for 15-year period is
calculated as 1.360, indicating that it is 36% more volatile than the overall market NASDAQ
index. If the market price goes up by 1-unit, the NFLX stock price is expected to go up by
1.36-units. This indicates a high risk and higher return of NFLX stocks with respect to
NASDAQ market index. With the use of 5 years historical data the beta and the E(Ri) are
calculated as 1.613 and 22.70% respectively which also shows higher sensitivity to market
changes.
Varying level of economic exposure, investor sentiment, variations among risk profiles of the
two indices and two different time periods may affect the results. Conversely, two different
time periods indicate the disproportionate effect of fluctuation of interest rates, geopolitical

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events, financial crises, and industry disruptions over different periods. These outliers can
skew beta results and different beta results in different E(Ri). Moreover, a shorter timeframe
such as 5-years might capture more short-term volatility compared to 15-year period.
It is further noted that the results for 5-year monthly beta for NFLX with respect to S&P 500
in December 2023 is recorded as 1. 221.The results for 5 year-monthly beta for NFLX in July
2024 with respect to NASDAQ is recorded as 1.27 in yahoo finance website.[Source:
https://www.zacks.com/stock/chart/NFLX/fundamental/beta;https://ca.finance.yahoo.com/qu
ote/NFLX/;https://www.stock-analysis-on.net/NASDAQ/Company/Netflix-Inc/DCF/CAPM]
5.2. Results and discussions for the Nβ1 and NE(Ri) values based on neutrosophic
calculations
The results obtained using neutrosophic calculation can be presents as neutrosophic beta (Nβ1)
and neutrosophic expected return (NE(Ri)).
Firstly, the results of the original historical data sets with the neutrosophic methods are
analysed to explore and recognise its inherent characteristics and trends.Secondly, the results
with MA are examined with the expectation of advancement in the overall analysis.
Results for each of the four models under four different scenarios are discussed below. As an
illustration according to the results, neutrosophic beta 1(Nβ1 ) and NE(Ri) are represent as an
intervals such as [Nβ1L ,Nβ1U] and [NE(RiL), NE(RiU)] respectively where; Nβ1L is the lower
limit of neutrosophic beta Nβ1L is the upper limit of neutrosophic beta, NE(RiL) is the lower
limit of neutrosophic expected return and NE(RiU) is the upper limit of neutrosophic expected
return.

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5.2.1. Results and Discussion based on Scenarios (15-year period).
This study is mainly conducted under four scenarios and the results are categorised
accordingly. The results for Nβ1 and NE(Ri) for each model are given in the tables below.
Scenario 01:S&P 500 Index with original data

MODEL
MODEL 1

Nβ1L
0.794

Nβ1U
1.687

NE(RiL)
9.87%

NE(RiU)
21.79%

MODEL 2

0.780

1.692

10.09%

21.56%

MODEL 3

0.822

1.755

8.40%

24.53%

MODEL 4

1.019

1.453

13.44%

18.40%

Table 5.2.1.1 Scenario 01:S&P 500 Index with original data.

Model 1:
Under the first scenario, in model 1, Nβ1 value is within the interval [0.794, 1.687]. The interval
of NE(Ri) is [9.87%, 21.79%]. Thus, this beta and expected return show a wide range,
indicating significant uncertainty in potential returns. Accordingly, for one-unit
increase/decrease in S&P500 market index, NFLX stock can be increased/decreased by 0.794
to 1.687-units. This indicates that the stock's volatility could be anywhere from less volatile to
significantly more volatile. Accordingly, the results from NCAPM reveal that the investor can
expect a 9.87% to 21.79% return.
Model 2
Model 2 is the one of the two-factor model of this study. In this model Nβ1 takes value within
the interval [0.780, 1.692] and NE (Ri) is within [10.09%, 21.56%]. For one-unit increase in
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S&P500 market index, NFLX stock can be increased by 0.78 to 1.69. Compared to the model
1, Nβ1 and NE(Ri) are moderately similar, suggesting marginally similar uncertainty in returns
and volatilities.
Model 3
This is another two-factor model. Nβ1 is found to be within the interval [0.822, 1.755] and
NE(Ri) is shown as [8.40%,24.53%]. Among all four models, this model shows the widest
range in both Nβ1 and NE(Ri), representing the highest level of uncertainty. According to this
model NFLX potentially yield the highest return for S&P500 index but also comes with the
maximum risk under this scenario.
Model 4
Model 4 is developed to capture the effect of all three independent variables S&P500, GDP
and CPI on NFLX stock. In this model, Nβ1 is [1.019, 1.453] and NE(Ri) is [13.44%,18.40%]
which implies the shortest range of both Nβ1 and E(Ri). The narrowest and moderate ranges
of Nβ1 and NE(Ri) suggest lower uncertainty and a more stable expected return. Moreover,
Nβ1L approximately equals to one and the interval range of beta indicates that the stock is
steadily aligned with market volatility.
In brief, scenario 01, Nβ1 provides an averaged metric to compare the overall risk-adjusted
volatility of each model. The results of model 3 shows wider range and indicate higher
potential returns but also come with increased risk, while the results of model 04 express the
more stable returns with smaller risk. A relatively shorter range of Nβ1 value in the model 04
suggests consistent performance. This lower range Nβ1 implies more stability but less

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sensitivity to market changes compared to the other models. Models 02 and 03 have more
similar results with moderate risk and volatility.
Scenario 02: NASDAQ Index with original data.
MODEL

Nβ1L

Nβ1U

NE(RiL) NE(RiU)

MODEL 1

0.609

2.483

10.70%

43.52%

MODEL 2

0.590

2.568

13.49%

43.89%

MODEL 3

0.617

2.508

9.33%

47.51%

MODEL 4

1.071

1.385

22.43%

23.76%

Table 5.2.1.2 Scenario 02: NASDAQ Index with original data.

Model 1
Under the second scenario, the Nβ1 in model 1 is in the interval [0.609, 2.483]. This wider
range shows the change in volatility from low-risk scenarios to high-risk scenarios.
Nβ1L=0.609 indicates low risk meanwhile Nβ1U =2.483 indicates high risky market. It simply
captures the unpredictable nature of the stock market and reveals that NFLX stocks can be
moderately sensitive to the market changes or else it can be highly sensitive to the market
changes. The interval of NE(Ri) strongly justifies the values of the Nβ1.The NE(Ri) is
[10.70%,43.52%] and it shows that the riskiness of NFLX stock can fluctuate within lower
risk scenarios to higher risk scenarios. Overall, the model 1 has high volatility but it offers a
reasonably high average return. Comparatively, Nβ1 and NE(Ri) are higher than the model 1
in 1st scenario. This balanced risk and return make this model as a solid option for whom
seeking a mix.

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Model 2
This model has a wider Nβ1 range compared to the model 1. But lower boundary of the NE(Ri)
is slightly higher, and the upper boundary remain similar to the model 1. The interval of Nβ1
is [0.590, 2.568] meanwhile the interval of NE(Ri) is [13.49%, 43.89%]. These differences
may occur due to the effect of the 2nd factor, GDP in model 02.
Model 3
Similar to model 1, the limits of Nβ1 vary in between 0.617 and 2.508. But NE(Ri) shows the
lowest value 9.33% as well as the highest value 47.51% in this scenario. These values indicate
that model 3 gives the widest range of the expected return NE(Ri), indicating that investor can
expect the lowest return of 9.33% as well as the highest return 47.51% for this stock. By
comparing the NE(Ri) of model 02 and 03, we can have an idea about the effect of the GDP
and CPI factors on this NFLX stock movements.
Model 4
In this model, the interval of Nβ1 is [1.071, 1.385]. Hence, model 4 recorded the highest Nβ1L
which is very close to 1 and it indicates that the NFLX stock increase/decrease very similar to
NASDAQ index. Accordingly, for 1-unit increase in NASDAQ stock, NFLX stock will
increase by 1.071-units. Nevertheless, Nβ1U is the lowest Nβ1U among all. The interval of
NE(Ri) is [22.43%, 23.76%] and it shows the most narrow range of NE(Ri). But 22.43% is the
highest NE(Ri) in the lower limit and 23.76% is the lowest NE(Ri) in the upper limit. However,
among all these models, this model shows the least risky and most stable results.
Overall, extremely higher range of neutrosophic betas and neutrosophic expected returns can
be seen in this scenario. The result of each model depicts a different balance of risk and return,
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offering various investment strategies and risk tolerance levels. With the use of NASDAQ
index models 1, 2, and 3 display a wider range of Nβ1 indicating higher sensitivity to the market
volatility. Additionally, the result for the expected return is relatively higher than the scenario
01. Similar to scenario 01, model 4 shows a shorter range, indicating it is less reactive to market
changes and thus, potentially more stable.
Scenario 03:S&P 500 Index with MA.
MODEL

Nβ1L

Nβ1U

NE(RiL)

NE(RiU)

MODEL 1

0.640

1.680

7.10%

20.10%

MODEL 2

0.734

2.024

8.75%

23.32%

MODEL 3

0.778

2.011

4.60%

28.00%

MODEL 4

0.182

0.480

0.33%

4.74%

Table 5.2.1.3 Scenario 03:S&P 500 Index with MA.

Model 1
The value of Nβ1 of this model is [0.640, 1.680] and the NE(Ri) is [7.10%, 20.10%]. Compared
to model 1 in other scenarios, this is the lowest limit of Nβ1. This results implies that for 1-unit
increase/decrease in S&P500, NFLX will increase/decrease by 0.640-units to 1.680-units
respectively. Accordingly, we can see the lowest NE(Ri) interval and consideration of MA,
may be the reason.
Model 2
In model 2, intervals of Nβ1 and NE(Ri) are [ 0.734, 2.024] and [8.75%, 23.32%], respectively.
Compared to scenario 01, the upper limit is higher. This upper limit indicates that for 1-unit
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increase/decrease in NASDAQ, the NFLX stock can increase/decrease by 2.024-units.
However, this increase /decrease may vary from 0.734 units to 2.2024 units.
Model 3
The interval of Nβ1 of this model is [0.778, 2.011]. The sensitivity of this model fluctuates
from lower sensitivity to higher sensitivity. The interval of the NE(Ri) is [4.60%, 28.00%].
Through out this analysis, the smallest lower boundary for NE(Ri) is recorded under this
scenario in this model. Further the model has the highest upper boundary and the widest range
for NE(Ri) with respect to S&P500. Thus, this model is the best for individuals who aim higher
gains and can tolerate significant market fluctuations.
Model 4
This 3-factor model gives a totally different perspective. Specially, Nβ1 is unusually low. The
interval of Nβ1 is [0.182, 0.480] which is the lowest boundaries among all models. Similar to
the sensitivity, the NE(Ri) is also very low [0.33%, 4.74%]. This model, indicate the lowest
sensitivity and expected return, which is suitable for very risk-aversion investors and perfect
for very conservative investors who prioritize stability over growth.
After considering MA, the lower limit of Nβ1 compared to scenario 01, has been decreased in
each model indicating moderate sensitivity to the market changes than the 1st scenario.
Moderately wider beta ranges can be observed in models 1, 2, and 3 showing some sensitivity
to market fluctuations. These results give a clear picture of the unpredictable stock market
behaviours to the investors. The results of model 4 shows a much smaller range with lower
sensitivity to market changes while deviating from the results in other three models.

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Scenario 04: NASDAQ Index with MA.
MODEL

Nβ1L

Nβ1U

NE(RiL) NE(RiU)

MODEL 1

0.674

2.387

10.50%

38.28%

MODEL 2

0.710

2.781

12.12%

43.16%

MODEL 3

0.725

2.534

6.89%

47.79%

MODEL 4

0.193

0.470

1.06%

5.66%

Table 5.2.1.4 Scenario 04: NASDAQ Index with MA.

Model 1
This model shows quite similar results for the Nβ1 in the original NASDAQ scenario (2nd
scenario) and the Nβ1 interval is [0.674, 2.387]. The value of expected return NE(Ri) is
[10.50%, 38.28%] which offers a decent return in safer conditions and much riskier return in
risky scenarios.
Model 2
This model gives the highest upper limit for Nβ1 among all models across all scenarios. The
Nβ1 interval is [0.710, 2.781] and the NE(Ri) interval is [12.12%, 43.16%]. This model
indicates that the sensitivity of the market can change from a low-risk situation to an extreme
risky situation. This model indicates that the worst case may be 1-unit decrease in NASDAQ
stock may result in decreasing the NFLX beta by 2.781 units and vise versa.
Model 3
This model shows the highest upper limit of NE(Ri) and widest range of return among all
models across all scenarios, and it is 47.79%. The interval of the NE(Ri) may vary from lower
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12.12% to this highest value. This model suggests significant sensitivity in safe situations and
enormous sensitivity in risky scenarios. Nβ1 interval is [0.725, 2.534].
Model 4
This model-results also highly deviate from all other results except the results in scenario 3, in
model 4. This model gives minimum sensitivity, and lowest return in safe scenario as well as
risky scenarios. Nβ1 interval is [0.193, 0.470] and this is the narrowest range among all models
across all scenarios.NE (Ri) interval is [1.06%, 5.66%] which is comparatively low.
Under this scenario, the application of MA does not exhibit large deviations from all other
results except for model 4. These models show wider Nβ1 ranges. This suggests high sensitivity
to market instability. These models show slightly improved performance in certain metrics
with the introduction of the MA, while Model 4 results experience a significant decline.
5.2.2. Results and Discussion based on the Models (15-years)
In this section, we conduct a model-based discussion across all four scenarios
Scenario

Scenario 01

Scenario 02

Scenario 03

Scenario 04

MODEL

S&P500 Original
Data Set

NASDAQ
Original Data Set

S&P500 with
MA

NASDAQ
with MA

MODEL 1

[0.794,1.686]

[0.609,2.483]

[0.640,1.680]

[0.673,2.387]

MODEL 2

[0.779,1.692]

[0.589,2.568]

[0.734,2.024]

[0.709,2.780]

MODEL 3

[0.822,1.754]

[0.617,2.508]

[0.777,2.010]

[0.725,2.534]

MODEL 4

[1.019,1.453]

[1.071,1.385]

[0.180,0.475]

[0.190,0.470]

Table 5.2.2.1 Nβ1 Summary for 15-year period (2009-2023).

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Scenario
MODEL
MODEL 1
MODEL 2
MODEL 3
MODEL 4

Scenario 01

Scenario 02

S&P500 Original
NASDAQ
Data Set
Original Data Set
[9.87%,21.79%]

Scenario 03

Scenario 04

S&P500 with
MA

NASDAQ with
MA

[10.70%,43.52%] [7.10%,20.10%] [10.50%,38.28%]

[10.09%,21.56%] [13.49%,43.89%] [8.75% ,23.32%] [12.12%,43.16%]
[8.40%,24.53%]

[9.33%,47.51%]

[4.60% ,28.00%]

[6.89% ,47.79%]

[13.44%,18.40%] [22.43%,23.76%]

[0.33% ,4.74%]

[1.06%,5.66%]

Table 5.2.2.2 NE(Ri) Summary for 15-year period (2009-2023).

Model 1
This is the single factor NCAPM model. Neutrosophic expected return of NFLX stock are
calculated using this model with respect the market fluctuations. This NCAPM model indicates
varying sensitivity across scenarios with moderately wider beta ranges in some cases such as
the interval [0.609,2.483] in second scenario, representing that it captures more market
variations. With the original data set, and MA of the data set, S&P500 gives more similar
results for beta [0.794,1.686] and [0.640,1.680] respectively. On the other hand, in these two
scenarios with NASDAQ also have comparatively similar beta results.
But the results for the returns for both cases are slightly different. However, NASDAQ offers
a higher variability and potentially higher returns. This may be due to more volatile nature of
NASDAQ market index. The use of MA reduces the range of expected returns for both
S&P500 and NASDAQ, indicating a potential trade-off between stability and the magnitude
of the returns.

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Models 2
This is the one of the 2-factor NAPT models. The output for NE(Ri) of NFLX is obtained with
respect to market index and GDP. According to this model, Nβ1 in all four scenarios slightly
deviate from the model 1. This model gives the highest Nβ1U value among all four models
recording 2.780. Apparently, the results should be different from other models because, apart
from market volatility, this model 2 captures the risk associated with GDP. Evidently, the
expected returns are affected using MA, often stabilizing performance for Model 2. NASDAQ
dataset steadily shows higher returns compared to S&P 500.
Model 3
This is another two-factor NAPT model. The results for NE(Ri) of NFLX are determined with
respect to the market index and CPI. Similar to model 2, Nβ1 values in all four scenarios show
minor deviations from those in model 1. This may be due to the risk associated with CPI in
addition to market volatility. NE(Ri) is evidently affected using the MA technique, which often
stabilizes performance of model 3. This model gives the highest NE(RiL) as 47.79% in 4th
scenario. It is important to note that this model in our calculations always gave the highest
NE(RiU) in all four scenarios and hence, drive to hold the highest range of NE(Ri) among these
models.
Model 4
This is a 3-factor NAPT model, which considers the effects of three factors together (market
index, GDP and CPI) on NFLX stock. The results of this model consistently deviate from
model 1, 2 and 3. It shows narrowest Nβ1 ranges and reduces the variability and expected
returns across all scenarios. But compared to other scenarios, in scenario 01 and 02 this model
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gives more reliable results. It implies a more stable but less sensitive model. Nβ1 intervals of
these scenarios are [1.019,1.453] and [1.071,1.385]. This model also gives a fair return in
scenario 01 and 02. It implies a more stable but less sensitive model. But after considering the
smoothing technique, the model results are totally different. Nβ1 shows a huge drop compared
to other models and hence substantial decrease in NE(Ri) as well.
Among the models 1,2 and 3, the results of Nβ1 in model 1 shows the minimum upper boundary
across all scenarios. Model 04 is more stable and less reactive, providing a distinct perspective
on market risk and return. With the use of MA, all models in scenario 03 show comparatively
higher ranges of Nβ1 and NE(Ri) compared to scenario 01. Scenarios 3 and 4 tend to reduce
the range of Nβ1 values for all models. Except model 4, all other models with NASDAQ index
have resulted in the highest range of return. However, in each scenario, the results of model 4
provided a narrow range, suggesting very low volatility and high stability.
Overall, these findings suggest that each of these NCAPM and NAPT models capture different
market variations. Additionally, NASDAQ give more volatile results compared to
S&P500.These results may help investors and analyst to have a good understanding about the
relationship between risk and return.
5.2.3. Results and Discussions based on average Nβ1 and average NE(Ri) on scenarios.
(15-years)
With the aim of simplifying the results into a single representative value, as an illustration the
average (mean) of all Nβ1 and NE(Ri) are calculated. This approach reduces the complexity of
dealing with interval to have a clear picture on the correlation among the models and
corresponds to the classical statistics, which deals with the precise crisp data value.
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Nβ1
SCENARIO 1
SCENARIO 2
SCENARIO 3
SCENARIO 4

MODEL 1
1.240
1.546
1.160
1.530

MODEL 2
1.236
1.579
1.379
1.745

MODEL 3
1.289
1.563
1.394
1.630

MODEL 4
1.236
1.228
0.331
0.331

Table 5.2.3.1 Average Nβ1 for each scenario for 15-years period (2009-2023).

According to the average values in scenario 01, all models show similar average Nβ1, which
fluctuates between 1.26 to 1. 289. These results indicate a strong volatility of the stock
compared to the market index. Average Nβ1 values range from 1.228 to 1.579 in scenario 02.
According to the 3rd and 4th scenarios, the average volatility fluctuates from 0.331 to 1.394 and
0.331 to 1. 745 respectively. This lowest similar average Nβ1 values are due to the use of model
4. Except that, all other results varying among 1.160 to 1.745 showing higher volatility.
E(Ri)
SCENARIO 1
SCENARIO 2
SCENARIO 3
SCENARIO 4

MODEL 1
15.83%
27.11%
13.60%
24.39%

MODEL 2
15.83%
28.69%
16.04%
27.64%

MODEL 3
16.46%
28.42%
16.30%
27.34%

MODEL 4
15.92%
23.10%
2.53%
3.36%

Table 5.2.3.1 Average NE(Ri) for each scenario for 15-years period (2009-2023).

According to scenario 01, average NE(Ri) ranges from 15.83% to 16.46%. In scenario 02, the
average NE(Ri) is fluctuated among 23.10% to 28.69% and it indicates considerable higher
average return compared to other scenarios. This clearly indicates the difference between use
of S&P500 and NASDAQ market indices. The effect of MA technique can be observed in the
3rd and 4th scenarios. Except model 4, all the average NE(Ri) values under the scenario 3 vary
from 13.60% to 16.30%. While these in scenario 4 are between 24.39% to 27.64%. These
results slightly deviate from the results of the original data set. But it is not a huge deviation.
Model 4 predict slightly lower average NE(Ri) ranges such as 2.53% and 3.36 % in Scenario
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3 and 4, respectively. Above results suggest that different models may perceive different risk
levels and market influences. Overall, these average returns make these models attractive for
its potential returns despite moderate risk.
5.2.4 Results and Discussion based on Nβ1 for 5-year periods from 2019-2023, 2017-2019
and 2009-2013
The table below illustrates the results for Nβ1 in each scenario for each 5-year period from
2009 to 2023. This table clearly point outs the results of each scenario at different time periods
and the analysis represents the possible variations of sensitivity in each model within the 5year period.

Scenario

Scenario 01

Scenario 02

Scenario 03

Scenario 04

5-year
period
2019-2023

S&P500 with
Original Data

NASDAQ with
Original Data

S&P500 with
MA

NASDAQ with
MA

MODEL 1
MODEL 2
MODEL 3
MODEL 4
2014-2018
MODEL 1
MODEL 2
MODEL 3
2009-2013
MODEL 1
MODEL 2
MODEL 3

[0.764,1.927]
[0.728,1.924]
[0.761,1.917]
[1.674,1.785]

[0.462,2.812]
[0.418,3.051]
[0.440,2.675]
[1.605,2.119]

[1.114,1.900]
[1.421,2.769]
[1.412,2.320]
[0.664,1.106]

[0.768,2.467]
[0.813,3.298]
[0.826,2.385]
[0.406,1.089]

[1.051,1.942]
[0.941,1.865]
[0.874,1.783]

[0.768,3.202]
[0.709,3.367]
[0.752,2.945]

[0.828,1.253]
[0.667,1.167]
[0.571,1.004]

[0.890,2.483]
[0.815,2.787]
[0.817,2.329]

[0.701,1.164]
[0.751,1.229]
[0.649,1.288]

[0.695,1.561]
[0.699,1.561]
[0.639,1.866]

[0.037,1.631]
[0.234,1.961]
[-0.068,2.468]

[0.285,2.239]
[0.376,2.406]
[0.194,3.766]

Table 5.2.4 Discussion based on Nβ1 for 5-year periods from 2019-2023, 2017-2019 and
2009-2013.

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According to the recent 5-Year results, there is a rapid increase in Nβ1 intervals. Specifically,
the upper limits of most of the intervals are expanded. The upper limits of the models in
scenario 01 is raised nearly to 2.00, meanwhile the upper limit of the 2nd scenario exceeds 3.0
which indicates an extremely risky situation. With the use of MA technique, the results of first
3 models indicate most volatile situation but model 4 shows incredibly stable and less risky
result.
During 2014-2018, the NFLX stock seems to be highly volatile stock as it results in very high
Nβ1 intervals. The results may fluctuate moderately risky situation to very high level except
the Nβ1 in model 3 in the 3rd scenario.
During the 5-year period from 2009-2013, we note a moderate result with the original data.
With the smoothing technique, the results fluctuate in between lower sensitivity situation to
considerably higher sensitivity. The Nβ1 for model 3 gives slightly negative output for 3rd
scenario, indicating that 1-unit increase in the market index may cause decrease in the NFLX
stock by 0.068-units. These results may be able to give a caution to the investors to be ready
for that kind of situation as well.
5.2.5 Results and Discussion based on 5-Year Neutrosophic Expected return from 20192023
Based on the results of Nβ1 intervals in resent 5-year period, four different models give the
below NE(Ri).

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5 years
MODEL
MODEL 1
MODEL 2
MODEL 3
MODEL 4

Scenario1
Scenario2
Scenario3
Scenario4
NE(RiL) NE(RiU) NE(RiL) NE(RiU) NE(RiL) NE(RiU) NE(RiL) NE(RiU)
10.87% 25.70%
8.64% 52.38% 11.93% 19.44% 11.10% 35.09%
10.49% 23.52% 15.43% 53.62% 17.51% 27.73% 14.38% 43.17%
10.79% 25.71%
9.10% 49.53% 10.85% 23.66% 8.13% 41.05%
7.83% 13.97% -1.42% 34.01%
4.80%
8.99% -2.88% 10.32%

Table 5.2.5.1. 5-Year NE(Ri) under different Scenarios and Models from 2019-2023.

In 1st scenario, all models show relatively narrow range of expected returns NE(Ri). There is
a wider range of NE(Ri) in 2nd scenario, indicating higher uncertainty or volatility compared
to all other scenarios. The model 1 shows the widest range under this 2nd scenario, which
fluctuates from 8.64% to 52.38%. However, the model 4 of this scenario gives negative lower
bound of NE(Ri). This model predicts that the NE(Ri) can vary between -1.42% to 34.01%.
Surprisingly, use of MA technique does not show much variation during this time period as
scenarios 3 and 4 provide moderately similar results to scenario 1 and 2, respectively. Model
4 shows negative lower bound values in 4th scenario, indicating a potential loss, while other
results show considerably positive higher return.
Slightly different results can be observed in two different time periods ,5-years and 15-years.
The changes in economic policies, natural disasters, market cycles, and sector-specific
movements which happen during these two periods may caused to these deviations. Specially,
the effect of COVID-19 drives the stock performance of NFLX in to another level.

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5.2.6. Results and Discussion based on 5-YearAverage NE(Ri) on Scenarios and Models
from 2019-2023.
Average
Scenario1
Scenario2
Scenario3
Scenario4

MODEL 1
18.29%
17.17%
30.51%
32.15%

MODEL 2
17.01%
19.47%
34.52%
35.57%

MODEL 3
18.25%
17.41%
29.31%
30.19%

MODEL 4
10.90%
6.28%
16.30%
19.40%

Table 5.2.6.1 5-YearAverage NE(Ri) under different Scenarios and Models from 2019-2023.

The 5-year average returns give a simple idea about the NE(Ri) of the stock over the short
period. These results offer a clear idea about the average stock return under each scenario with
the use of each model. Under the scenario 01, the average NE(Ri) from the four models range
from 10.90% to 18.29% suggesting the effects of varying factors and different market
conditions being considered in each model. The average NE(Ri) ranges from 6.28% to 19.47%
in 2nd scenario which, gives more conservative result compared to scenario 1. In the scenario
3, the average NE(Ri) are higher, ranging from 16.30% to 34.52%. Using the smoothing
technique, results give a more bullish market scenario. The 4th Scenario is the most optimistic
scenario among the four models and NE(Ri) range from 19.40% to 35.57%. Except Model 4,
all other models give more or less similar returns in each scenario. Model 4 always give the
minimum average return.
5.3. Results and discussion for Nβ2 and Nβ3
The primary focus of this study is on the Nβ1 which explain the NFLX stock movement with
respect to the Market Index, and how does it affect the neutrosophic expected return when
using different factors and, in different time frames. In this case it is vital to accurately calculate
Nβ2 and Nβ3. Hence Nβ2 and Nβ3 are also carefully calculated, and we present the results and
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discuss those results briefly to provide a comprehensive picture of all model in each scenario
separately.
5.3.1 Neutrosophic Beta 2 (Nβ2)
Nβ2 values represent how much units may increase /decrease in NFLX stocks with respect to
the one-unit increase/decrease in GDP value. Below, we have studied factor GDP in Model 2
and Model 04 only under all four scenarios. The Nβ2 values for 15-year period (2009-2023)
and most resent 5 Years (2019-2023) are given in Table 5.3.1 below.
Nβ2
2009-2023
MODEL 2
MODEL 4
2019-2023
MODEL 2
MODEL 4

Scenario 01
Nβ2L
Nβ2U

Scenario 02
Nβ2L
Nβ2U

-0.480
-3.396

0.371
-1.168

-3.869
-7.354

-0.022
-3.466

1.155
-1.089

-3.994
-0.796

Scenario 03
Nβ2L
Nβ2U

FOR 15YEARS
1.358
-0.930
-0.770 -0.158
FOR 5 YEARS
1.629
-1.529
0.063
0.040

Scenario 04
Nβ2L
Nβ2U

0.878
0.228

-1.377
0.097

1.597
0.564

-0.164
0.101

-1.475
0.058

1.650
0.169

Table 5.3.1 Nβ2 values for 15-year period (2009-2023) and most resent 5 Years (2019-2023).

Model 02 measures the impact of market index and GDP together on NFLX stock price. It is
noted that Model 02 reflected a wider range of Nβ2 values from negative to moderately positive
values across different scenarios, indicating higher variability and uncertainty in the 15-year
period as well as for the 5-year period.
Model 04 measures the impact of combined effects of market index, GDP and CPI, on NFLX
stock price. It exhibits extreme values of Nβ2, strongly negative for the 15-year period. This
implies a strong inverse relationship with the market. Again, this provides evidence for the
sensitivity of the model on the effect of economic fluctuations during the period such as the

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effect of COVID-19. In the recent 5-year period, Model 4 displays both negative as well as
positive correlations with a generally weaker relationship compared to the 15-year period. The
values of Nβ2 demonstrate how different models and different scenarios capture unpredictable
degrees of market correlation and uncertainty over time.
5.3.2 Neutrosophic Beta 3 (Nβ3)
Measure Nβ3 represents how much units may increase /decrease in NFLX stocks with respect
to the one-unit increase/decrease in CPI respectively. In this study, we consider CPI in Model
3 and Model 04 under all four scenarios. Table 5.3.2 below provides Nβ3 values for all 15 years
and most resent 5 Years under all four scenarios.
Nβ3
2009-2023
MODEL 3
MODEL 4
2019-2023
MODEL 3
MODEL 4

Scenario 01
Nβ3L
Nβ3U
-1.073
-1.018

1.139
-0.861

-0.026
-9.194

0.062
-13.999

Scenario 02
Scenario 03
Nβ3L
Nβ3U
Nβ3L
Nβ3U
FOR 15YEARS
-0.900
2.134
-2.301
2.478
-0.957
-0.921 -1.468
-0.845
FOR 5 YEARS
-0.187
0.374
-1.780
0.176
-13.309 -2.776
0.040
0.101

Scenario 04
Nβ3L
Nβ3U
-2.609
-1.505

4.325
-1.284

-1.741
-4.561

3.360
-2.840

Table 5.3.2 Nβ3 values for 15-year period (2009-2023) and most resent 5 Years (2019-2023).

It is observed that the sensitivity of NFLX stock, in general, to the changes in CPI varies
considerably across different scenarios and models. Both models 3 and 4 have resulted in more
variability across scenarios and for different time periods. Model 3 tends to show more
negative as well as positive sensitivities with varying ranges and Nβ3 shows broader range of
values indicating higher uncertainty. This may reveal less predictability meanwhile model 4
exhibits more consistent results, especially in the 2009-2023 timeframe, with a narrow range
of values. Model 4 generally shows negative sensitivities, mainly in the more recent 5-year

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period. These negative Nβ3 values in the two models indicate that the NFLX stock return tends
to move inversely with the CPI.
According to the results, Nβ3 values are more negative. NFLX is a technology stock and some
of the technology stocks might have negative CPI betas, if they have high input costs that rise
with inflation, and they cannot easily pass these costs on to customers.
5.4. Correlation coefficient
The consistent lower NE(Ri) values from Model 4 in Scenarios 3 and 4 need to be carefully
addressed. The Model 4 predicts a narrow spread compared to other models. There may be
some reasons for these results such as multicollinearity among factors, specific factor
selection, data transformation techniques or any other reason. Hence, correlation coefficient of
each two factors is calculated to check for the multicollinearity due to correlated factors.
Table 5.3.1 provides the correlation coefficients between S&P 500 and NASDAQ financial
indices (original and with MA) and economic factors, GDP and CPI.

S&P500

NASDAQ

S&P500
with MA

NASDAQ
with MA

GDP

GDP

-0.037

-0.035

0.382

0.298

1

CPI

0.105

0.045

0.243

0.104

0.270

Factors

Table 5.3.1 Correlation coefficients matrix of the Factors.

From the above table, it is noted that the market indices with original S&P500 and NASDAQ
data set do not have a direct correlation with GDP and CPI individually as the values of
correlations are very low. When considering the MA of the S&P500 and NASDAQ, there is a
weak to moderate positive correlation. This suggests some influence of GDP on MA of
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S&P500 and MA of NASDAQ. The correlation coefficient between GDP and CPI is weakly
positive. This indicates that even though they are related, the relationship is not strong enough
in this dataset. Further, there is no sign of presence of multicollinearity.
5.5. Model Verification
In what follows now, under all four scenarios, we describe only the Model 01 fit significance
and accuracy to understand the process. In neutrosophic statistics, 2-factor or 3-factor model
verifications are more complicated. Hence, we do focus on Model 01 with one factor only.
5.5.1. Hypothesis and t-statistics for each scenario of Model 01
Test statistic: t-statistic is calculated for Model 01 in each scenario to measure the significance
of the regression coefficient in this simple linear model. We will present the results in each
scenario separately [Degrees of freedom = 178]
Critical t-value(T) at ߙܰ= [0.01 ,0.05] is [1.973,2.617]. Then,
{T(1 − α୒ )} = [1.645,2.33]
Max{T(1 − α୒ )} = 2.33
1.

Validate the results for 1st scenario :S&P 500 Index with original data

Null hypothesis: ܰ‫ܪ‬0: Nβ1 ∈ [0.794,1.686]
0Alternative hypothesis: ܰ‫ܽܪ‬: Nβ1 < 0.794, or ܰ‫ܽܪ‬: Nβ1 > 1.686 or ܰ‫ܽܪ‬: Nβ1 ∉ [0.794,1.686]
t- statistic [2.848,5.465]
Hence
{[t − statistic] − [ctitical t value(T)]} = {[2.848,5.466] − [1.973,2.617]}
{[t − statistic] − [ctitical t value(T)]} = [0.231,3.493]
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Therefore
Min{[t − statistic] − [ctitical t value(T)]} = 0.231
According to the results, Min{[t − statistic] − [ctitical t value(T)]} < Max{T(1 − α୒ )}.
Therefore, do not reject the null hypothesis for scenario 01. Hence, we can conclude that Nβ1
∈ [0.794,1.686].
2.

Validate the results for 2nd scenario: NASDAQ with original data set.

Null hypothesis: ܰ‫ܪ‬0: Nβ1 ∈ [0.609,2.483]
Alternative hypothesis: ܰ‫ܽܪ‬: Nβ1< 0.609, or ܰ‫ܽܪ‬: Nβ1> 2.483 or ܰ‫ܽܪ‬: Nβ1 ∉ [0.609,2.483]
t-statistic [3.041,8.396]
Hence
{[t − statistic] − [ctitical t value(T)]} = {[3.041,8.396] − [1.973,2.617]}
{[t − statistic] − [ctitical t value(T)]} = [0.424,6.423]
Therefore
Min{[t − statistic] − [ctitical t value(T)]} = 0.231
According to the results, Min{[t − statistic] − [ctitical t value(T)]} < Max{T(1 − α୒ )}.
Therefor do not reject the null hypothesis for scenario 02 and hence we can conclude that Nβ1
∈ [0.609,2.483].
3.

Validate the results for 3rd scenario :S&P 500 Index with MA

Null hypothesis: ܰ‫ܪ‬0: Nβ1 ∈ [0.640,1.680]
Alternative hypothesis: ܰ‫ܽܪ‬: Nβ1 < 0.640, or ܰ‫ܽܪ‬: Nβ1 > 1.680 or ܰ‫ܽܪ‬: Nβ1 ∉ [0.640,1.680]
t-statistic [2.138,4.824]

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Hence
{[t − statistic] − [ctitical t value(T)]} = {[2.138,4.824] − 1.973,2.617]}
{[t − statistic] − [ctitical t value(T)]} = [−0.479,2.851]
Therefore
Min{[t − statistic] − [ctitical t value(T)]} = |0.479|
According to the results, Min{[t − statistic] − [ctitical t value(T)]} < Max{T(1 − α୒ )}.
Therefor do not reject the null hypothesis for scenario 03 and hence we can conclude that Nβ1
∈ [0.640,1.680].
4.

Validate the results for 4th scenario: NASDAQ with MA

Null hypothesis: ܰ‫ܪ‬0: Nβ1 ∈ [0.673,2.387]
Alternative hypothesis: ܰ‫ܽܪ‬: Nβ1 < 0.673, or ܰ‫ܽܪ‬: Nβ1 > 2.387 or ܰ‫ܽܪ‬: Nβ1 ∉ [0.673,2.387]
t-statistic [3.197,7.714]
Hence
{[t − statistic] − [ctitical t value(T)]} = {[3.197,7.714] − [1.973,2.617]}
{[t − statistic] − [ctitical t value(T)]} = [0.580,5.742]
Therefore
Min{[t − statistic] − [ctitical t value(T)]} = 0.580
According to the results, Min{[t − statistic] − [ctitical t value(T)]} < Max{T(1 − α୒ )}.
Therefor do not reject the null hypothesis for scenario 02 and hence we can conclude that Nβ1
∈ [0.673,2.387].
Overall, the results for the application of the Neutrosophic Hypothesis test allowed positively
validating the result of all four scenarios of this study with a level of significance of up to 99%.

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5.6. Model Accuracy of multi-factor Models for the Expected Returns
We calculate and discuss the Neutrosophic mean absolute deviation (NMAD) measure for
predictive accuracy of the fitted models for the NE(Ri) under four scenarios in Table 5.6.1.

Scenario
Scenario 01

Scenario 02

Scenario 03

Scenario 04

Model
Model 01
Model 02
Model 03
Model 04
Model 01
Model 02
Model 03
Model 04
Model 01
Model 02
Model 03
Model 04
Model 01
Model 02
Model 03
Model 04

MAD
[0.326,0.380]
[0.328,0.377]
[0.299,0.394]
[0.294,0.409]
[0.109,0.371]
[0.105,0.344]
[0.069,0.385]
[0.254,0.306]
[0.262,0.367]
[0.229,0.351]
[0.182,0.392]
[0.415,0.435]
[0.080,0.333]
[0.031,0.317]
[0.015,0.369]
[0.406,0.427]

Length of lower boundary
to upper boundary
0.054
0.049
0.096
0.115
0.263
0.238
0.316
0.052
0.106
0.121
0.210
0.020
0.253
0.286
0.354
0.022

Table 5.6.1 Model Accuracy measure (NMAD) for the NE(Ri) under Four Scenarios.

Models 01 and 02 exhibit lower NMAD values and shorter lengths in scenario 1 and 2 with
original data, but moderate values with MA. Model 03 shows slightly higher NMAD values
and wider lengths in all four scenarios. Model 04 consistently have shorter NMAD ranges and
narrow lengths. This may indicate very low volatility and high stability across all scenarios.MA
(scenarios 3 and 4) tends to smooth out the data, resulting in lower NMAD values for Model
04 and indicating reduced volatility. Models 01, 02, and 03 also show reduced volatility with
moving averages but still exhibit moderate variability compared to Model 04.

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5.7. Further Model Applications
To test the neutrosophic model appropriateness, the Nβ1 and NE(Ri) of Amazon and Apple
stocks have been also calculated using the four new models.
5.7.1. Results of Amazon stock (AMZN) based on classical and neutrosophic CAPM
The beta and expected returns of AMZN stock under S&P index and NASDAQ index using
the classical model are presented below in Table 5.7.1.1.
(2009-2023) Period
beta
E(Ri) of AMAZN

S&P500
0.990
10.23%

NASDAQ
1.088
15.40%

Table 5.7.1.1 Beta and E(Ri) of Amazon stock based on classical CAPM.

According to the classical model E(Ri) of AMZN stock with respect to S&P500 and NASDAQ
are 10.23% and 15.40% respectively. Similar to NFLX , AMAZN also shows highest E(Ri)
with respect to highly volatile NASDAQ index.
We provide the calculation for average Nβ1 of AMZN stock based on neutrosophic calculations
across four scenarios and four models in Table 5.7.1.2.
AVERAGE Nβ1AMZN STOCK
Scenario 1
Scenario 2
Scenario 3
Scenario 4

MODEL 1
1.049
1.385
0.837
1.206

MODEL 2
1.039
1.424
0.983
1.368

MODEL 3
1.068
1.394
0.965
1.268

MODEL 4
1.541
1.482
0.202
0.235

Table 5.7.1.2 Average Nβ1 of Amazon stock based on neutrosophic calculations.

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According to the results in the above table, average Nβ1 of the models are moderately similar
in each scenario except model 04 and the NE(Ri) fluctuates accordingly. Scenario 02 gives the
highest average Nβ1 in each model meanwhile scenario 03 gives the least average Nβ1.
However, the results of model 4 shows the highest average Nβ1 in scenario 01 and 02, but the
lowest average Nβ1 in scenarioc03 and 04.
Based on the Nβ1 intervals, NE(Ri), of AMZN stock are calculate under each scenario for all
four models using the neutrosophic methods and presented as below in Table 5.7.1.3.
NE(Ri) (%)
OF AMZN
Scenario 1
Scenario 2
Scenario 3
Scenario 4

MODEL 1
[7.76,19.36]
[8.24,40.60]
[5.77,14.29]
[8.72,30.07]

MODEL 2
[7.61,19.32]
[10.79,41.30]
[6.53,15.79]
[9.77,33.21]

MODEL 3
[3.46,25.30]
[6.86,44.66]
[3.96,21.38]
[5.46,39.05]

MODEL 4
[25.92,21.15]
[31,31.17]
[0.29,4.86]
[0.70,6.20]

Table 5.7.1.3 NE(Ri) of AMZN under 04 scenarios and 04 models (NCAPM and NAPT).

Use of NCAPM and NAPT models provide the values of NE(Ri) of AMZN stock. These values
reflect inherently indeterminant nature of stock market. Under the different market conditions
and the different calculation methods in each of four scenarios, the results imply the probable
returns. Sophisticated investors can carefully use these models in trading as they are aware of
the risk boundaries. This may lead them to avoid having unexpected losses. Four different
statistical models offer distinct approaches to tackle the indeterminacy and risk, based on their
preference.
According to the results of AMZN stocks, models 1 and 2 give pretty much similar fair return
meanwhile model 3 gives most sensitive output and model 4 gives considerably narrow range
compared to the other models, specially in scenario 3 and 4.

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5.7.2. Results of Apple (AAPL) stock based on classical and neutrosophic CAPM
The beta and expected returns of AAPL stock under S&P index and NASDAQ index using the
classical model are presented below in Table 5.7.2.1.
(2009-2023) Period
Β
E(Ri)

SP500
1.063
10.92%

NASDAQ
1.064
15.07%

Table 5.7.2.1 Beta and E(Ri) of Apple stock based on classical CAPM.

The classical model gives moderately similar beta for both indices S&P500 and NASDAQ,
but the E(Ri) are 10.92% and 15.07% respectively. This difference in the expected returns may
occurred due to the different nature of the two indices.
We also provide the average Nβ1 of AAPL stock based on neutrosophic calculations below.
Average Nβ1 of
AAPL STOCK
Scenario 01
Scenario 02
Scenario 03
Scenario 04

MODEL 01
1.26
1.56
0.97
1.21

MODEL 02
1.26
1.59
1.12
1.35

MODEL 03
1.26
1.57
1.13
1.28

MODEL 04
0.34
0.43
0.21
0.20

Table 5.7.2.2 Average E(Ri) of AAPL stock based on neutrosophic calculations.

In the first scenario, the average Nβ1 values from models 1, 2 and 3 are the same 1.26 except
model 04 where it is 0.34. The models 1,2 and 3 in scenario 02 have the average values of Nβ1
in the range from 1.56 to 1.59. However, model 04 provides less volatile values of Nβ1 from
0.20 to 0.43, calculated under all scenarios. We can see a moderately similar average Nβ1 in all
models in each scenario except model 4.

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Based on the Nβ1 intervals, the neutrosophic expected returns, NE(Ri), of AAPL stock under
each scenario and four models for the (2009-2023) period have been calculated using the
neutrosophic models and presented below in Table 5.7.2.3.
NE(Ri) AAPL stock
(%)
Scenario 01
Scenario 02
Scenario 03
Scenario 04

MODEL 01
MODEL 02
[8.77,23.48] [8.77,23.33]
[8.40,46.56] [11.26,46.45]
[8.69,13.86] [9.54,16.14]
[12.14,26.30] [12.98,29.49]

MODEL 03
[2.47,28.56]
[6.68,50.54]
[7.89,19.63]
[9.82,35.10]

MODEL 04
[7.81,11.47]
[10.62,15.59]
[0.99,2.73]
[0.50,3.39]

Table 5.7.2.3 NE(Ri) of AAPL under 04 scenarios and from 04 models (NCAPM and NAPT).

AAPL stock also gives pretty much similar results to all the models in each scenario except
model 04. Generally, scenario 02 shows the largest intervals for all four models implying
highest volatility. The broadest intervals of NE(Ri) are obtained from the model 3 and
narrowest intervals of NE(Ri) are recorded from the model 4. These values of expected returns,
NE(Ri), can be considered as risk boundaries which indicate the lower expected and highest
expected returns. However, these different risk boundaries under different scenarios with
distinct models help investors to conduct a more sophisticated analysis of potential risks and
rewards under different economic scenarios. On the other hand, investors can choose the
appropriate neutrosophic model, which satisfies their requirements and may take their
investment decisions based on these risk boundaries.

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Chapter 06
Conclusions, Benefits, Limitations and Future Scope
6.0. Introduction to Conclusions, Benefits, Limitations and Future Scope
This study provides the new neutrosophic models-based approach to capture the indeterminacy
and uncertainty in the volatile stock market behaviour. The neutrosophic statistics enhance the
traditional CAPM and APT models by incorporating the neutrosophic logic, making it more
applicable to real-world scenarios where the data sets are regularly incomplete or fuzzy.
However, this neutrosophic approach also has limitations, which need to be carefully
addressed. The complexity of the neutrosophic model is one such limitation. Also, the
complexity of financial world is increasing day by day. Hence further research into these
financial models with advance technology will benefit to offer even more sophisticated
models, which can better capture the market behaviour. This attempt is to address these
benefits, limitations and future scope of these new models while concluding the outcomes.
6.1. Conclusions
The Neutrosophic logic-based approach provides advanced techniques to control the
indeterminacy and uncertainty in the stock market. Based on these models, investors and
financial analysts can improve risk assessment and their decision-making in complex financial
markets.
Each model exhibits a different balance of risk and return, offering various investment
strategies and risk tolerance levels to the investors. Even though the classical model gives a
precise value for beta, it is far more complex. In a highly competitive market, we cannot expect
a smooth flow as it is often disrupted by market manipulations, fake news, indeterminate
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events, and incomplete and vague data. Hence, based on these comprehensive results of four
models and in each scenario considered in this study, investors and financial analysts may be
able to identify their risk boundaries, and they can select their portfolios based on their
financial goals. This strategic approach will help investors to mitigate the unnecessary fear or
excessive risk taking up to a certain extent and they can make informed decisions based on the
individual risk tolerance whether risk neural, risk lover or risk averse. These models
demonstrate different levels of neutrosophic expected returns and market volatility,
encapsulating the indeterminacy and range of possible outcomes in stock performance.
Primarily, the outcome of each scenario and each model represent the indeterminacy of the
stock market. The results give a clear picture of the price fluctuations. The differences across
the four scenarios may arise due to the use of different factors, neutrosophic calculation
techniques, different market indices and different time frames.
Specially, the results obtained from NASDAQ index show more volatility compared to S&P
500. These deviations may occur due to the differences in index composition, sensitivity to
economic changes, historical performance, and inherent volatility. In general, these two indices
lead to different behaviour and performance patterns. Specially, NASDAQ market index is
more volatile than the S&P 500 index. Because of this high sensitivity, the model results with
NASDAQ may give wider prediction ranges and more significant deviations in model results,
which are reflected in the model predictions.
Moreover, the results obtained during the 15-year period are somehow different from the
results in the 5-year period. The differences highlight the uncertainty, different market
conditions and range of possible outcomes of NFLX stock, during different timeframes.
Overall, the 5-year average returns provide a quick overview of the stock's potential
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performance under various circumstances, helping investors understand the range of possible
outcomes and associated risks during that period.
According to the results of diffrent models, the range of intervals of neutrosophic expected
returns NE(Ri) fluctuates between lowest return to the highest returns indicating the adverse
condition to the more optimistic outlook, respectively. Specially, the results during the most
recent 5-year period show higher volatility and higher returns. These results may reveal the
effects of economic turnovers during that period especially the effect of the pandemic, COVID19. During this economic lockdown period, there was an extremely volatile situation of the
NFLX stock which these neutrosophic model results also reflected. The changes in consumer
behaviour during the pandemic may highly affect these results. These changes in consumer
behaviours negatively impacted on most of the companies. However, the stock prices of the
tech companies such as NFLX skyrocketed during this period, as most of the consumer
changed their market behaviour and did operations on the online platform. The lockdowns and
stay at home orders led to uplift the streaming demand and it positively affected the NFLX’s
subscriber growth and stock price. Similar to this situation the diverse nature of the results can
be observed in different time frames.
Conversely, the changes in regulations such as data privacy, content licensing and high
competition led to fluctuations in the stock price in different time frames. On the other hand,
broader economic conditions, including changes in consumer spending, interest rates, and
inflation, may affect stock prices. For instance, during economic downturns, the spending on
entertainment can decrease. Furthermore, market sentiment; advancement in AI may also
influence NFLX stock.

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The results of NE(Ri)s are affected by the Nβ1, Nβ2 and Nβ3 intervals. Nβ1 is the one of the
main factor which affects the stock market return, but the coefficients of GDP and CPI indexes
may also affected the NFLX stock in different ways. The diverse nature of the results in
different models proved it. For example, the negative coefficient of CPI sometimes positively
affected the stock price. Inflation indicates squeeze profit margins, reduce consumer spending,
higher interest rates, create uncertainty, and impact investor sentiment negatively. Deflation
can increase real purchasing power and reduce costs, potentially boosting profit margins and
consumer spending. Lower interest rates can benefit growth stocks, but persistent deflation can
signal weak demand and economic issues. The results of these models are important for
investors and analysts, for informed decision making, asset allocation, risk management, and
portfolio diversification. The results highlight the varying impacts that GDP and CPI changes
can have on NFLX stock, depending on the scenario and model used. This analysis facilitates
understanding how each model performs under various scenarios, different time frames and in
diverse conditions and how the level of ambiguity changes accordingly.
This study is an effort to capture the uncertainty associated with the systematic risk factors.
Investors continuously evaluate these risk factors in their assessment of stock’s future growth
prospects and risk profile. Hence, these models will be helpful up to a certain point to capture
the risk boundaries. Specifically, the effects of market index, GDP and CPI on NFLX has been
tested using these models and these models can be improved by switching the factors according
to the investor’s requirements.
By integrating these models, investors can privilege the benefits such as enhanced strategies
and better manage risks. All these benefits may guide them to make informed investment

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decisions. Finally, it can be concluded that this study suggests strong findings across
neutrosophic components with different data treatments, techniques, and methodologies.
6.2. Benefits of proposed models
Selecting a portfolio based on beta involves balancing stocks with different beta values to align
with investors risk tolerance and investment goals. A higher beta indicates higher risk and
potential return, while a lower beta suggests more stability. By diversifying across sectors and
periodically rebalancing the portfolio, investors can manage risk and strive for optimal returns.
Benefits for Investors:
➢ Financial markets are full of uncertain, imprecise, incomplete, and contradictory
information. These neutrosophic models can address these issues and provide useful
decision-making tool.
➢ NCAPM offer a clear formula to calculate expected return of an asset based on its
systematic risk (Nβ1) with respect to the market factor capturing the uncertainty of the
market.
➢ NAPT provide different formulas to calculate the expected return of an asset
considering macroeconomic factors, which affect asset returns. Also, it provides a more
comprehensive risk assessment based on the systematic risk (Nβ1, Nβ2 and Nβ3), while
capturing the indeterminacy in each factor.
➢ All these models help investors to be aware of the risk boundaries such as the best case
as well as the worst case.
➢ These strategic models help investors to mitigate the unnecessary fear and excessive
risk taking.
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➢ By incorporating neutrosophic statistics, these models provide a more sophisticated
understanding of risks compared to traditional models.
➢ Investors and financial analysts can use these models to assess the investment
performance against expected returns and market benchmarks.
6.3 Limitations
Classical logic can be extended to handle uncertainty, imprecision, vagueness, and
inconsistency in financial stock market using Neutrosophic logic. However, this branch of
philosophy, the neutrosophic logic and this analysis is a more complex phenomenon and has
several limitations.
➢ Neutrosophic logic introduces indeterminacy, and this is complex than the classical
logic. Neutrosophic models are challenging to develop, understand, and interpret.
➢ Neutrosophic multiple regression models are complex to interpret and solve compared
to traditional regression models. Adding the number of factors can be considered to
further improve the models but solving these equations are a big calculation hurdle.
➢ Appropriate Software need to be developed to deal with neutrosophic multiple
regression models as manual calculations are harder.
➢ Implementing efficient algorithms for neutrosophic multiple regression analysis can be
challenging, potentially leading to longer computation times and higher costs.
➢ Neutrosophic model validation is a challenge, especially for models with more than
one factor. As Neutrosophic logic is still an emerging trend, it is difficult to define
appropriate validation metrics that consider the neutrosophic components.

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➢ In this new philosophy and growing field, it suffers from lack of standardized
methodologies and techniques in financial modeling.
➢ Specialized knowledge in the filed of neutrosophic statistics need to implement these
models.
➢ These models solely focus on systematic risk factors and ignore unsystematic risk
which can be significant in investment decision making.
➢ This novel neutrosophic statistical approach which can be used to handle uncertainty
and imprecision in financial stock market analysis comes with considerable limitations.
These include increased complexity, lack of computational intensity, validation
challenges, lack of standardization, and the need of specialized expertise. Careful
consideration of these limitations is essential when dealing with this new approach in
financial modeling. Comprehensive investment strategy involves not only multi-factor
models but also combine with other fundamental and technical analysis along with
expert portfolio management practices.
6.4 Future Scope
➢ There is a never-ending competition in the stock market. Capturing the indeterminacy
is essential to compete confidently and take wise decisions. Hence, future study can be
conducted to capture the different systematic risk factors such as liquidity risk, political
risk and exchange rate risk (currency risk), which highly affect stock prices.
➢ Further studies can be conducted to capture the multicollinearity due to non-linear
relationships or partial multicollinearity to improve the model results. Hence,
multicollinearity diagnostic tools can be developed to detect the presence of
multicollinearity in a neutrosophic analysis.
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➢ As a future study, neutrosophic model verification technique can be developed.
Verification of 2-factor and 3-factor models is a main limitation of this study.
➢ Future works should be deliberated towards utilizing a wider number of systematic risk
factors by extending NAPT model to capture the effect of different factors, but solving
these neutrosophic regression equations is another major limitation of this study.

“Absolutely, there is no unique neutrosophic model to a real-world problem. And thus, there are no
exact neutrosophic rules to be employed in neutrosophic modelling. Each neutrosophic model is an
approximation, and the approximations may be done from different points of view. A model might be
considered better than others if it predicts better than others. But in most situations, a model could be
better from a standpoint, and worse from another standpoint – since a real world problem normally
depends on many (known and unknown) parameters.Yet, a neutrosophic modelling of reality is needed
in order to fastly analyse the alternatives and to find approximate optimal solutions.”
-Florin Smarandache-

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8.0.Appendix
8.1.Coding Appendix
Python Code for Graph 4.1.1
import yfinance as yf
import matplotlib.pyplot as plt
# Define the ticker symbol
ticker_symbol = 'NFLX'
# Define the start and end dates
start_date = '2009-01-01'
end_date = '2023-12-31'
# Download monthly historical data
df = yf.download(ticker_symbol, start=start_date, end=end_date, interval='1mo')
# Plotting the time series
plt.figure(figsize=(10, 4))
plt.plot(df.index, df['Adj Close'], label='Adjusted Close')
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plt.title('Adjusted Close Price for NFLX for the period 2009-2023')
plt.xlabel('Date Period')
plt.ylabel('Adjusted Close Price')
plt.legend()
plt.grid(True)
plt.xticks(rotation=45)
plt.tight_layout()
plt.show()
Python code for Graph 4.1.2
import yfinance as yf
import pandas as pd
import matplotlib.pyplot as plt

# Fetch the data for Netflix (NFLX) from Yahoo Finance
ticker = 'NFLX'
startDate = '2009-01-01'
endDate = '2023-12-31'
data = yf.download(ticker, start=startDate, end=endDate, interval='1mo')

# Keep only the necessary columns: High, Low, and Adj Close
df = data[['High', 'Low', 'Adj Close']]

# Plot the data
plt.figure(figsize=(12, 6))

plt.plot(df.index, df['High'], label='Monthly High', color='blue')
plt.plot(df.index, df['Low'], label='Monthly Low', color='green')
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plt.plot(df.index, df['Adj Close'], label='Adjusted Close', color='red')

plt.title('NFLX Monthly High, Low, and Adjusted Close Prices for the period (2009-2023)')
plt.xlabel('Date Period')
plt.xticks(rotation=45)
plt.ylabel('Price (USD)')
plt.legend()
plt.grid(True)
plt.show()

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