This document analyzes the exchange rates of the Nigerian naira against the US dollar, British pound, and Euro from May 2015 to April 2020 using a mean reversion model. Unit root tests show the exchange rate data is stationary, meaning it reverts to a long-run mean over time. Regression analysis reveals the rate of mean reversion, long-run mean, and volatility for each currency. The US dollar has the shortest half-life, indicating it will revert fastest to its long-run mean and be the cheapest currency going forward, followed by the Euro. The study recommends using the mean reversion model to simulate future exchange rates and that investors invest more in the dollar and euro than the pound.
A Comparative Study between Time Series and Neural Network for Exchange Rate ...iosrjce
IOSR Journal of Computer Engineering (IOSR-JCE) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of computer engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in computer technology. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Comparative Study between Time Series and Neural Network for Exchange Rate ...iosrjce
IOSR Journal of Computer Engineering (IOSR-JCE) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of computer engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in computer technology. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Name ;- SECOND EXAM SPRING 20201. Mark Price the marketi.docxgemaherd
Name ;
- SECOND EXAM SPRING 2020
1. Mark Price the marketing manager for Speakers needs to find which variable most affects the demand for a line of speakers. He is uncertain whether the price of speakers of the advertising expenditures drive the sale of speakers. He plans to use the regression analysis to determine the relative impact price an advertising. He used 12 years of data which is given below. The output from his regression analysis also give:
Sales
(000)
Price Per Unit
Advertising
($000)
400
280
600
700
215
835
900
211
1100
1300
210
1400
1150
215
1200
1200
200
1300
900
225
900
1100
207
1100
980
220
700
1234
211
900
925
227
700
800
245
690
Regression Statistics
Multiple R
0.8550
R Square
0.7310
Adjusted R Square
0.6712
Standard Error
146.6234
Observations
12
ANOVA
df
SS
MS
F
Significance F
Regression
2
525718.3339
262859.1669
12.2269
0.0027
Residual
9
193485.9161
21498.4351
Total
11
719204.2500
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Intercept
2191.34
826.08
2.65
0.03
322.61
Price
-6.91
2.92
-2.37
0.04
-13.51
Advertising
0.33
0.24
1.36
0.21
-0.21
a. Interpret the output from the regression analysis given above – is this a good regression model to forecast sales
b. Evaluate the regression model. Also, comment on the sample size (observations)
c. Write the regression equation showing the relationship between Sales versus Advertising and Price
d. Determine whether Price or Advertising has more impact on the forecast of Sales
e. Predict average yearly speaker sales the price was $300 per unit and Mark is planning to spend $900 thousand in Advertising.
2. Assume the network and data as follows:
a. Construct the network diagram
b. Indicate the critical path when normal activity times are used
c. Explain the procedure you would use to crash this project if you had a penalty cost per week above 15 weeks as well you had indirect costs per week
3. Ace Steel Mill estimates the Demand for steel in millions of tons per year as follows
Millions of t tons
Probability
10
10
12
25
14
30
16
20
18
15
a. If capacity is set at 18 Million tons what is the capacity cushion
b. What is the probability of Idle capacity
c. What is the average utilization of the plant at 18 million ton capacity
d. If it costs $8 million per ton of lost business and $80 million to build a million ton of capacity how much capacity should be built to minimize the total cost
4. Explain the reasons for
a. Carrying large levels of inventories
b. Carrying Small Levels of Inventories
5. We discussed in class how MRP (Materials Planning System) works we examined an example of the a Table with four legs and a leg assembly. We discussed the following charts.
Please explain how the Bill of Materials Explosion takes place in these charts
6.
7. Explain the following as discussed in class
Thompson manufacturing produces industrial scales for the electronics ind.
Name ;- SECOND EXAM SPRING 20201. Mark Price the marketi.docxroushhsiu
Name ;
- SECOND EXAM SPRING 2020
1. Mark Price the marketing manager for Speakers needs to find which variable most affects the demand for a line of speakers. He is uncertain whether the price of speakers of the advertising expenditures drive the sale of speakers. He plans to use the regression analysis to determine the relative impact price an advertising. He used 12 years of data which is given below. The output from his regression analysis also give:
Sales
(000)
Price Per Unit
Advertising
($000)
400
280
600
700
215
835
900
211
1100
1300
210
1400
1150
215
1200
1200
200
1300
900
225
900
1100
207
1100
980
220
700
1234
211
900
925
227
700
800
245
690
Regression Statistics
Multiple R
0.8550
R Square
0.7310
Adjusted R Square
0.6712
Standard Error
146.6234
Observations
12
ANOVA
df
SS
MS
F
Significance F
Regression
2
525718.3339
262859.1669
12.2269
0.0027
Residual
9
193485.9161
21498.4351
Total
11
719204.2500
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Intercept
2191.34
826.08
2.65
0.03
322.61
Price
-6.91
2.92
-2.37
0.04
-13.51
Advertising
0.33
0.24
1.36
0.21
-0.21
a. Interpret the output from the regression analysis given above – is this a good regression model to forecast sales
b. Evaluate the regression model. Also, comment on the sample size (observations)
c. Write the regression equation showing the relationship between Sales versus Advertising and Price
d. Determine whether Price or Advertising has more impact on the forecast of Sales
e. Predict average yearly speaker sales the price was $300 per unit and Mark is planning to spend $900 thousand in Advertising.
2. Assume the network and data as follows:
a. Construct the network diagram
b. Indicate the critical path when normal activity times are used
c. Explain the procedure you would use to crash this project if you had a penalty cost per week above 15 weeks as well you had indirect costs per week
3. Ace Steel Mill estimates the Demand for steel in millions of tons per year as follows
Millions of t tons
Probability
10
10
12
25
14
30
16
20
18
15
a. If capacity is set at 18 Million tons what is the capacity cushion
b. What is the probability of Idle capacity
c. What is the average utilization of the plant at 18 million ton capacity
d. If it costs $8 million per ton of lost business and $80 million to build a million ton of capacity how much capacity should be built to minimize the total cost
4. Explain the reasons for
a. Carrying large levels of inventories
b. Carrying Small Levels of Inventories
5. We discussed in class how MRP (Materials Planning System) works we examined an example of the a Table with four legs and a leg assembly. We discussed the following charts.
Please explain how the Bill of Materials Explosion takes place in these charts
6.
7. Explain the following as discussed in class
Thompson manufacturing produces industrial scales for the electronics ind ...
Criteria Mark 1. Professional Presentation 1 · Well organiz.docxfaithxdunce63732
Criteria / Mark
1. Professional Presentation/ 1
· Well organized, neatly presented, word processed, spelling, grammar, and punctuation
· Name and student number as well as lab partner/s name, date and time of experiment
· In-text referencing, reference list at end of report
2. Title and Abstract/ 1
· Title: concise title directly above abstract (even if you have it on the cover page already)
· Abstract: Brief description of the experiment, the major results, associated uncertainties, comparison with theoretical results/expected results.
3. Introduction and Aims/ 1
· A brief paragraph describing key physics and relevant background
· At the end of the introduction, state the aims of experiment
4. Procedure and Apparatus/ 0.5
· Describe the equipment used, you may include labeled sketches, drawing, or photographs of apparatus
· Describe how you conducted the experiment. Remember to use past tense, you are telling the reader what you have done.
5. Risk Assessment/ 0.5
· Use a risk assessment matrix (under lab section of Blackboard) to:
(i) Consider the health and safety risk of this experiment to yourself and others in the laboratory.
(ii) Determine what measures you would put in place to mitigate/reduce these risks where possible.
· Attach the risk assessment at the end of your report after the references list
6. Data, Critical Analysis and Uncertainties / 2
· Data presented in numbered table format with proper titles and headings ie. “Table 1: Description of Some Tabulated Data”
· Figures to presented with number headings and with proper titles and headings ie. “Figure 1: Graph of some Data”
· You should write text which refers to/describes data in tables and plots
· Show calculations of key parameters and calculations of uncertainties where necessary
· Present data in scientific notation where applicable
· Use appropriate SI units – eg. metres (m), seconds (s), metres per second (m/s).
7. Summary and Discussions /2
· Brief summary the experiment and aims
· Describe the results and put them in context with the expected or theoretical values – do they match? If not, why? You may need to do some research for this part.
· Identify the major source(s) of uncertainty and how they contribute to the final uncertainty
· Statements describing ways to improve the precision and/or accuracy of the final result by improving the method employed as well as considering alternative equipment
· Unnecessary repetition of method or results will not be considered as discussion
8. Answers to specific questions /2
MOD001072
MANAGING THE ECONOMY
Weeks 7-8
Classical model- small open economy
*
Weeks 7-12
The three topics
WKS 7-8
CLASSICAL MODEL‘Long run’ –flexible pricesOpen economy
WKS 9-10
IS-LM [‘Keynesian’] MODEL‘Short run’ – fixed pricesOpen economy
WKS 11-12
INFLATIONARY EXPECTATIONSAdaptive expectationsRational expectations
HOLIDAY BREAK
*
WEEKS 7-8 SUMMARY
CLASSICAL MODEL0. Classical models –basic features, closed v.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Name ;- SECOND EXAM SPRING 20201. Mark Price the marketi.docxgemaherd
Name ;
- SECOND EXAM SPRING 2020
1. Mark Price the marketing manager for Speakers needs to find which variable most affects the demand for a line of speakers. He is uncertain whether the price of speakers of the advertising expenditures drive the sale of speakers. He plans to use the regression analysis to determine the relative impact price an advertising. He used 12 years of data which is given below. The output from his regression analysis also give:
Sales
(000)
Price Per Unit
Advertising
($000)
400
280
600
700
215
835
900
211
1100
1300
210
1400
1150
215
1200
1200
200
1300
900
225
900
1100
207
1100
980
220
700
1234
211
900
925
227
700
800
245
690
Regression Statistics
Multiple R
0.8550
R Square
0.7310
Adjusted R Square
0.6712
Standard Error
146.6234
Observations
12
ANOVA
df
SS
MS
F
Significance F
Regression
2
525718.3339
262859.1669
12.2269
0.0027
Residual
9
193485.9161
21498.4351
Total
11
719204.2500
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Intercept
2191.34
826.08
2.65
0.03
322.61
Price
-6.91
2.92
-2.37
0.04
-13.51
Advertising
0.33
0.24
1.36
0.21
-0.21
a. Interpret the output from the regression analysis given above – is this a good regression model to forecast sales
b. Evaluate the regression model. Also, comment on the sample size (observations)
c. Write the regression equation showing the relationship between Sales versus Advertising and Price
d. Determine whether Price or Advertising has more impact on the forecast of Sales
e. Predict average yearly speaker sales the price was $300 per unit and Mark is planning to spend $900 thousand in Advertising.
2. Assume the network and data as follows:
a. Construct the network diagram
b. Indicate the critical path when normal activity times are used
c. Explain the procedure you would use to crash this project if you had a penalty cost per week above 15 weeks as well you had indirect costs per week
3. Ace Steel Mill estimates the Demand for steel in millions of tons per year as follows
Millions of t tons
Probability
10
10
12
25
14
30
16
20
18
15
a. If capacity is set at 18 Million tons what is the capacity cushion
b. What is the probability of Idle capacity
c. What is the average utilization of the plant at 18 million ton capacity
d. If it costs $8 million per ton of lost business and $80 million to build a million ton of capacity how much capacity should be built to minimize the total cost
4. Explain the reasons for
a. Carrying large levels of inventories
b. Carrying Small Levels of Inventories
5. We discussed in class how MRP (Materials Planning System) works we examined an example of the a Table with four legs and a leg assembly. We discussed the following charts.
Please explain how the Bill of Materials Explosion takes place in these charts
6.
7. Explain the following as discussed in class
Thompson manufacturing produces industrial scales for the electronics ind.
Name ;- SECOND EXAM SPRING 20201. Mark Price the marketi.docxroushhsiu
Name ;
- SECOND EXAM SPRING 2020
1. Mark Price the marketing manager for Speakers needs to find which variable most affects the demand for a line of speakers. He is uncertain whether the price of speakers of the advertising expenditures drive the sale of speakers. He plans to use the regression analysis to determine the relative impact price an advertising. He used 12 years of data which is given below. The output from his regression analysis also give:
Sales
(000)
Price Per Unit
Advertising
($000)
400
280
600
700
215
835
900
211
1100
1300
210
1400
1150
215
1200
1200
200
1300
900
225
900
1100
207
1100
980
220
700
1234
211
900
925
227
700
800
245
690
Regression Statistics
Multiple R
0.8550
R Square
0.7310
Adjusted R Square
0.6712
Standard Error
146.6234
Observations
12
ANOVA
df
SS
MS
F
Significance F
Regression
2
525718.3339
262859.1669
12.2269
0.0027
Residual
9
193485.9161
21498.4351
Total
11
719204.2500
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Intercept
2191.34
826.08
2.65
0.03
322.61
Price
-6.91
2.92
-2.37
0.04
-13.51
Advertising
0.33
0.24
1.36
0.21
-0.21
a. Interpret the output from the regression analysis given above – is this a good regression model to forecast sales
b. Evaluate the regression model. Also, comment on the sample size (observations)
c. Write the regression equation showing the relationship between Sales versus Advertising and Price
d. Determine whether Price or Advertising has more impact on the forecast of Sales
e. Predict average yearly speaker sales the price was $300 per unit and Mark is planning to spend $900 thousand in Advertising.
2. Assume the network and data as follows:
a. Construct the network diagram
b. Indicate the critical path when normal activity times are used
c. Explain the procedure you would use to crash this project if you had a penalty cost per week above 15 weeks as well you had indirect costs per week
3. Ace Steel Mill estimates the Demand for steel in millions of tons per year as follows
Millions of t tons
Probability
10
10
12
25
14
30
16
20
18
15
a. If capacity is set at 18 Million tons what is the capacity cushion
b. What is the probability of Idle capacity
c. What is the average utilization of the plant at 18 million ton capacity
d. If it costs $8 million per ton of lost business and $80 million to build a million ton of capacity how much capacity should be built to minimize the total cost
4. Explain the reasons for
a. Carrying large levels of inventories
b. Carrying Small Levels of Inventories
5. We discussed in class how MRP (Materials Planning System) works we examined an example of the a Table with four legs and a leg assembly. We discussed the following charts.
Please explain how the Bill of Materials Explosion takes place in these charts
6.
7. Explain the following as discussed in class
Thompson manufacturing produces industrial scales for the electronics ind ...
Criteria Mark 1. Professional Presentation 1 · Well organiz.docxfaithxdunce63732
Criteria / Mark
1. Professional Presentation/ 1
· Well organized, neatly presented, word processed, spelling, grammar, and punctuation
· Name and student number as well as lab partner/s name, date and time of experiment
· In-text referencing, reference list at end of report
2. Title and Abstract/ 1
· Title: concise title directly above abstract (even if you have it on the cover page already)
· Abstract: Brief description of the experiment, the major results, associated uncertainties, comparison with theoretical results/expected results.
3. Introduction and Aims/ 1
· A brief paragraph describing key physics and relevant background
· At the end of the introduction, state the aims of experiment
4. Procedure and Apparatus/ 0.5
· Describe the equipment used, you may include labeled sketches, drawing, or photographs of apparatus
· Describe how you conducted the experiment. Remember to use past tense, you are telling the reader what you have done.
5. Risk Assessment/ 0.5
· Use a risk assessment matrix (under lab section of Blackboard) to:
(i) Consider the health and safety risk of this experiment to yourself and others in the laboratory.
(ii) Determine what measures you would put in place to mitigate/reduce these risks where possible.
· Attach the risk assessment at the end of your report after the references list
6. Data, Critical Analysis and Uncertainties / 2
· Data presented in numbered table format with proper titles and headings ie. “Table 1: Description of Some Tabulated Data”
· Figures to presented with number headings and with proper titles and headings ie. “Figure 1: Graph of some Data”
· You should write text which refers to/describes data in tables and plots
· Show calculations of key parameters and calculations of uncertainties where necessary
· Present data in scientific notation where applicable
· Use appropriate SI units – eg. metres (m), seconds (s), metres per second (m/s).
7. Summary and Discussions /2
· Brief summary the experiment and aims
· Describe the results and put them in context with the expected or theoretical values – do they match? If not, why? You may need to do some research for this part.
· Identify the major source(s) of uncertainty and how they contribute to the final uncertainty
· Statements describing ways to improve the precision and/or accuracy of the final result by improving the method employed as well as considering alternative equipment
· Unnecessary repetition of method or results will not be considered as discussion
8. Answers to specific questions /2
MOD001072
MANAGING THE ECONOMY
Weeks 7-8
Classical model- small open economy
*
Weeks 7-12
The three topics
WKS 7-8
CLASSICAL MODEL‘Long run’ –flexible pricesOpen economy
WKS 9-10
IS-LM [‘Keynesian’] MODEL‘Short run’ – fixed pricesOpen economy
WKS 11-12
INFLATIONARY EXPECTATIONSAdaptive expectationsRational expectations
HOLIDAY BREAK
*
WEEKS 7-8 SUMMARY
CLASSICAL MODEL0. Classical models –basic features, closed v.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
1. www.ajms.com 18
ISSN 2581-3463
RESEARCH ARTICLE
Analysis of Nigerian Naira Exchange Rates against US Dollar, British Pounds, and
Euro Currency Using Mean Reverting Model
Joy C. Nwafor1
, Bright O. Osu2
, Kingsley Uchendu3
1
Department of Mathematics, AkanuIbiam Federal Polytechnic, Unwana, Nigeria, 2
Depaertment of
Mathematics, Abia state University, Uturu, Nigeria, 3
Department of Statistics, Michael Okpara University of
Agriculture, Umudike, Nigeria
Received: 15-08-2021; Revised: 15-10-2021; Accepted: 15-11-2021
ABSTRACT
This work is aim at analyzing Nigerian Naira exchange rate against American Dollar, British Pounds,
and the Euro currency. Monthly average exchange rates from May 2015 to April 2020 were used for the
study. Augmented Dickey-Fuller was used to determine the presence of mean reversion in the data. The
solution to the model is gotten using Ito’s formula. Mean reversion rate, long-run mean, and volatility
were calculated from the historic data using Microsoft Excel. The concept of half-life was used to
determine the time it will take for the rates to revert to their long-run mean. The data analysis reveals
that the data are stationary. The analyses also reveal that American Dollar will be the cheapest currency
followed by Euro in the near future. Based on the result, the researchers recommend mean reverting
model in modeling exchange rates. The researchers also recommend that investors should invest using
Dollar or Euro more than they use pounds
Key words: Exchange rates, mean reversion models, half-lives
Address for correspondence:
Bright O. Osu,
osu.bright@abiastateuniversity.edu.ng
INTRODUCTION
In the past few years, the price of dollar has
become a source of concern to every Nigerian.
Investors, traders, civil servants, and even
peasant farmers have all felt the impact of the
rise and fall of the Nigerian currency against
other countries’ currencies especially within
the past 2 years. Nevertheless, the Nigerian
government on their own part has developed
a lot of policies to help fight the situation but
it seems that all the policies work for some
times and then crashed. Noko[1]
has it that when
there are fluctuations in exchange rate, various
economic activities are usually affected such as;
the purchasing power, balance of payment, price
of goods and services, import structure, export
earning, government revenue, and external
reserve among others. Ayodele[2]
asserted that
exchange rate has a negative impact on the GDP
because as it increases, the economic growth is
negatively affected, while inflation rate exerts
a positive impact on GDP, indicating that firms
are more willing to produce when inflation rate
is high and vice versa.
It has now become imperatively necessary that
the naira exchange rate against these currencies be
analyzed to determine the trend and also provide
a simulation equation that will guide investors on
the way forward, that is when to either invest or
hold their peace[3,4]
.
Mean reverting processes are naturally attractive
to model commodity prices since they embody
the economic argument that when prices are
“too high,” demand will reduce and supply will
increase, producing a counter-balancing effect[5]
.
When prices are “too low” the opposite will occur,
again pushing prices back toward some kind of
long-term mean[6]
. Mean reverting processes are
also useful for modeling other processes, observed
or unobserved, such as interest rates or commodity
“convenience yield.” Hence, this work is aimed
at using this mean reverting model to analyze the
behavior of naira exchange rate against the three
major foreign currencies[7,8]
.
2. Nwafor, et al.: Analysis of Nigerian Naira Exchange Rates against US Dollar
AJMS/Oct-Dec-2021/Vol 5/Issue 4 19
Statement of the problem
The exchange rate is perhaps one of the most
widelydiscussedtopicsinNigeriatoday.Thisisnot
surprising given its macro-economic importance
especially in a highly import dependent economy
as Nigeria[9]
.
Since September 1986, when the market
determined exchange rate system was introduced
through the second tier foreign exchange market,
the naira exchange rate has exhibited the features
of continuous depreciation and instability[10]
.
This instability and continued depreciation of the
naira in the foreign exchange market have resulted
in declines in the standard of living of the populace,
increased cost of production which also leads to cost
push inflation[11]
. It has also tended to undermine
the international competitiveness of non-oil exports
and make planning and projections difficult at both
micro and macro levels of the economy. A good
number of small and medium scale enterprises
have been strangled out as a result of low dollar/
naira exchange rate and so many other problems
resulting from fluctuations in exchange rates[12]
.
Hence, this work is aimed at using the mean
reversion model to analyze the behavior of
naira exchange rate with the aim of developing
simulation equations that will help to determine
future exchange rates in the country.
MATERIALS AND METHODS
Data
The data that are used for this study are monthly
average exchange rates of dollar, pounds, and euro
from May 2015 to April 2020 (60 observations)
gotten from the archives of the central bank of
Nigeria (CBN). In the case of dollar, IFEM and
DAS are ignored while BDC is used. This is
because BDC is the one available to customers at
any point in time.
The Model
Mean reversion model is given by
dSt
=λ(μ–St
)dt+σSt
dBt
(1)
Where
• Bt
is a Brownian Motion, so dBt
~ N(0√dt)
• λ is the speed of mean reversion
• µ is the “long run mean” to which the process
tends to revert
• σ as usual is a measure of the process volatility.
The mean that is the first moment of the model is
given by (see[13,14]
)
Es(t)=μ+(S0
–μ)e–λt
which is equal to μ as t → ∞
While the variance is given by
Var s t e t
( ) = −
( )
−
σ
λ
λ
2
2
2
1 which is equal to
σ
λ
2
2
ast → ∞
Data analysis
Testing for mean reversion
Equation (1) can be discretized as follows
∆St
=β0
+β1
St–1
+σSt–1
εt
, εt
~N(0,1)(2)
Where β0
=λμ∆t, β1
=–λ∆t and ∆t
n
=
1
. This implies
that observations of the exchange rates through
time can be considered as observations of the
linear relationship between ∆St
and St
in the
presence of noise σSt–1
εt
.G[15]
.
To test for evidence of mean reversion,
Augmented Dickey-Fuller test (ADF) is used to
perform unit root test. The Dickey-Fuller version
used is the type one version that is constant and
no trend[16]
. The Microsoft Excel real statistics
regression analysis tool is used to determine the
calculated tau, while the tabular tau is gotten
from ADF table.
The following formal hypothesis is tested
H0
: β = 0 (series is non-stationary, i.e., unit root)
H1
: β 0 (series is stationary, i.e., mean reverting)
Rate of mean reversion, long-run mean, and
volatility
The mean reversion rate, long-run mean, and the
volatility are estimated from the historic data using
excel regression tool. In[17]
the Mean reversion rate
(λ)= negative of the slope
Mean reversion level/the long-run mean =
intercept divided by mean reversion rate
Volatility is gotten using STEYX function in excel
which is the residual standard deviation. All these
were gotten using Microsoft Excel regression
tool.
Half life
This is the average time that will be taken for the
rates to revert half way to its long-term level or
long-run mean. From equation (1), let us consider
the ODE
dst
=λ(μ–st
)dt which gives
st
=μ+(S0
–μ)e–λt
(3)
3. Nwafor, et al.: Analysis of Nigerian Naira Exchange Rates against US Dollar
AJMS/Oct-Dec-2021/Vol 5/Issue 4 20
Hence, the half-life can be deduced thus:
s
S e
t
t
1
2
0
2
−
− =
−
( )
µ
µ λ
hence, we have that
t1
2
2
=
ln
Where λ is the rate or speed of mean reversion.
Simulation equations and predictions
To get the simulation equation, we make use of the
fact that equation (2) can further be
simplified as St
=St–1
+β0
+β1
St–1
+σSt–1
εt
which in
turn gives
St
=β0
+(1+β1
+σεt
) St–1
(4)
Equation (3) is then used for the simulation
equation.
For predictions, the expected value, that is, the
mean is used.
RESULTS
Solution to the model
To solve equation (1), we first derive the Ito’s
formula thus
Let f(s, t) be a continuous and differentiable
function. Then, by Taylor’s series expansion for
two variables:
df s t
f
s
ds
f
s
ds
,
( ) =
∂
∂
+
∂
∂
( )
1
2
2
2
2
(5)
but, ds=λ(μ–s)dt+σsdB
hence, (ds)2
=(λ(μ–s)dt+σsdB)2
= (λμdt–λsdt+σsdB)(λμdt–λsdt+σsdB)
=λ2
μ2
(dt)2
–2λ2
μs(dt)2
+2λμσsdtdB–2λs2
σdtdB+λ2
s2
(dt)2
+σ2
s2
(dB)2
Since (dB)2
→dt as dt →0, we have that (ds)2
=σ2
s2
dt(Xeurong (2007)
Hence,
df s t
f
s
dt sdt sdB
f
s
s dt
,
( ) =
∂
∂
− +
( )+
∂
∂
λµ λ σ σ
1
2
2
2
2 2
=
∂
∂
−
∂
∂
+
∂
∂
+
∂
∂
λµ λ σ σ
f
s
dt s
f
s
dt s
f
s
dB s
f
s
dt
1
2
2 2
2
2
Finally, we say that:
df s t s
f
s
dB
f
s
s
f
s
s
f
s
dt
,
( ) =
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
σ λµ λ σ
1
2
2 2
2
2
(6)
We then proceed to the main solution thus:
Taking f = ln s hence;
∂
∂
=
f
s s
1
and
∂
∂
= −
2
2 2
1
f
s s
so
substituting we get
d s s
s
dB
s
s
s
s
s
dt
s
ln
( ) = + − + −
= − −
σ λµ λ σ
λµ
λ
σ
1 1 1 1
2
1
2
2 2
2
2
+
dt dB
σ
Hence,
d lns
s
dt dB dt
( ) = + − +
λµ
σ λ
σ 2
2
(7)
Now, stochastically integrating both sides from 0
to t, we have
∫ ∫ ∫ ∫
= + − +
d lns
s
dt dB dt
( )
λµ
σ λ
σ 2
2
lns
s
dt B t
t
= + − +
∫ ( )
λµ σ λ
σ
1
2
2
= − +
+ +
( ) ∫
λ
σ
σ λµ
2
2
1
t B
s
dt
t
= − +
+ + − +
+
( ) ∫
λ
σ
σ λµ λ
σ
σ
2 2
2 2
t B dB dt
t s
t
t
( )
Hence,
ln s t B
t s B B
t
t s
= − +
+ +
− +
−
( )+ −
( )
( )
( ) ( )
∫
λ
σ
σ
λµ λ
σ
σ
2
2
2
2
ds
Now, taking the exponent of both sides and letting
s(0) = s0
, we have
S S exp t B
exp t
t t
t
( ) ( )
= − +
+
+
− +
−
∫
0
2
0
2
2
2
λ
σ
σ
λµ λ
σ
s
s B B ds
t s
( )+ −
( )
( ) ( )
σ
(8)
Equation (8) is the solution to the equation.
4. Nwafor, et al.: Analysis of Nigerian Naira Exchange Rates against US Dollar
AJMS/Oct-Dec-2021/Vol 5/Issue 4 21
Unit root test[18]
Table 1 shows the unit root test of the naira
exchange rate of the three currencies at 5% level
of significance
H0
: The exchange rates has unit root
From Table 1, the test statistic for all the currencies
is smaller than the critical value; hence, the null
hypothesis is rejected in all[19]
. Therefore, we
conclude that the exchange rates are stationary,
that is, mean reverting
Rate of mean reversion, long-run mean, and
volatility
Using Excel 2007 and making use of the data
stated above, the mean reversion rate, long-run
mean, and volatility of the exchange rates of the
three currencies are given in Table 2.
Half-lives
The half-lives of the exchange rates of the three
currencies are as follows
Dollar
H t
( )
ln .
.
.
1
2
2 0 69315
0 1
6 9315
= = =
Pounds
H t
( )
ln .
.
.
1
2
2 0 69315
0 04
17 32875
= = =
Euro
H t
( )
ln .
.
.
1
2
2 0 69315
0 04
17 32875
= = =
Simulation equations and predictions
The simulation equation is gotten using equation
(3.3)
Dollar
St
=β0
+(1+β1
+σεt
)St–1
β λµ
0 0 1 380 85
1
60
0 6348
= = × × =
∆t . . .
β λ
1 0 1
1
60
0 0017
= − = − × = −
∆t . .
Volatility(σ)=0.06
Hence, the simulation equation is
St
=0.6348+(1–0.0017+0.06εt
) St–1
=0.6348+(0.9983+11.93εt
) St–1
Where εt
is a random number.
Pounds
St
=β0
+(1+β1
+σεt
) St–1
β λµ
0 0 04 433 47
1
60
0 28898
= = × × =
∆t . . .
β λ
1 0 04
1
60
0 00067
= − = − × = −
∆t . .
Volatility(σ)=0.03
Hence, the simulation equation is
St
=0.28898+(1–0.00067+0.03εt
) St–1
=0.28898+(0.9993+0.03εt
) St–1
Where εt
is a random number.
Euro
β λµ
0 0 04 398 32
1
60
0 2655
= = × × =
∆t . . .
β λ
1 0 04
1
60
0 00067
= − = − × = −
∆t . .
Volatility(σ)=0.03
Hence, the simulation equation is
St
=0.2655+(1–0.00067+0.03εt
) St–1
=0.2655+(0.9993+0.03εt
) St–1
Where εt
is a random number.
Predictions
Using the mean, we can predict the naira exchange
rate of the currencies. Hence, in the next 40 months
Table 1: Unit root test for exchange rates
Currency Test statistic Critical value (5%)
Dollar –2.94451 –2.891
Pounds –3.1894 –2.891
Euro –3.2789 –2.891
Table 2: Rate of mean reversion, long‑run mean, and
volatility
Currency Mean reversion
rate(λ)
Long‑run
mean
Volatility
Dollar 0.1 380.85 0.06
Pounds 0.04 433.47 0.03
Euro 0.04 398.32 0.03
5. Nwafor, et al.: Analysis of Nigerian Naira Exchange Rates against US Dollar
AJMS/Oct-Dec-2021/Vol 5/Issue 4 22
taking S0
to be value for April 2020, we have the
following:
Dollar
ES(t)
=μ+(S0
–μ)e–λt
Taking the values in table above, we have
ES(40)
= 380.85+(420.15–380.85)e–0.1×40
= + ( )×
380 85 39 3 1
2 718284
. .
.
= +
380 85
39 3
54 5980
.
.
.
=381.5698
≅N381.57K
Pounds
ES(t)
=μ+(S0
–μ)e–λt
Furthermore, taking the values in table above, we
have
ES(40)
=433.47+(447.92–433.47)e–0.04×40
= + ( )×
433 47 14 45 1
2 718281 6
. .
. .
= +
433 47
14 45
4 9530
.
.
.
≅N436.39K
Euro
ES(t)
=μ+(S0
–μ)e–λt
Furthermore, taking the values in table above, we
have
ES(40)
=398.32+(392.12–398.32)e–0.04×40
= + −
( )×
398 32 6 2 1
2 718281 6
. .
. .
= −
398 32
6 2
4 9530
.
.
.
=397.0682
≅N397.6K
DISCUSSION
FromTable 1,onecanobservethattheexchangerates
of Nigerian naira as against all the three currencies
are stationary.This means that mean reversion model
can be used to model the exchange rates
The result in Table 2 shows that though the
exchange rates are stationary, their mean
reversion rates are small especially for pounds
and euro.
The implication of the half-lives is that apart
from noise term, it will take dollar 6.9315 months
to revert to half of its long-run mean of 380.85.
Hence, the dollar exchange rate will take
approximately 14 months to go back to a rate very
close to N380:85k per dollar.
Furthermore, it will take pounds 17.32875 months
to revert back to half its mean exchange rate
of 433.47. Hence, it will take approximately
35 months for the exchange rate to go back to a
rate very close to N433:47k per pound.
Moreover, it will also take Euro 17.32875 months
to revert back to half its long-run mean exchange
rate of 398.32. Hence, Euro currencies will also
take approximately 35 months to go back to the
exchange rate of N398:32k per Euro.
From predictions, we will observe that, in the near
future, dollar will have the cheapest exchange rate
followed by euro, then pounds.
RECOMMENDATIONS
Based on the results, the researcher recommends
as follows:
i. Mean reversion model should be used in
modeling exchange rate since it best models
the concept
ii. Investors should make use of dollar and Euro
currencies instead of the pounds since dollar
and Euro will always be cheaper than the
pounds
iii. Since exchange rates follow mean reversion
process, investors are advised to hold their
peace whenever it is on the high side since
what goes up will surely come down.
CONTRIBUTIONS TO KNOWLEDGE
i. To the best of our knowledge, mean reversion
modelhasnotbeenusedbyanyotherresearcher
to model exchange rates; hence, this work has
shown that mean reversion model can be used
to model exchange rates
ii. Equation (5) that is the extension of Ito’s
formula for mean reversion model is developed
by the researcher
iii. The step-by-step method of solving mean
reversion model has not been done by any
other researcher
6. Nwafor, et al.: Analysis of Nigerian Naira Exchange Rates against US Dollar
AJMS/Oct-Dec-2021/Vol 5/Issue 4 23
iv. Simulation equation and predictions are
developed for the three currencies.
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