ABSTRACT
Developing an optimal investment strategy for pension administrators is one of the major problems in recent times in Nigeria. In this work an investment model of a pension plan participant (PPP) under the defined contribution (DC) scheme was developed. The PPP was assumed to have a deterministic portion of his salary contributed on a monthly basis into the pension fund; part of this wealth is invested in a risk free asset and the remaining part in a risky asset. The risky asset is considered to be a Heston volatility model and the risk appetite of the PPP is assumed to be a constant relative risk aversion (CRRA) utility function. The Hamilton Jacobi Bellman (HJB) equation for the obtained model is derived and an approximate closed form solution is obtained for the resulting partial differential equation using the Prandtl asymptotic matching. Furthermore, the stochastic differential equation of the wealth process is transformed into a stochastic difference equation. The difference equation so transformed is matched to a standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to capture the impact of news in the market, while the wavelet analysis is used to capture the impact of noise on the volatility of the stock market. Historical data from the Nigerian Stock Market is used for empirical analysis in a concrete setting. Our results indicate that the Nigerian stock market is driven by noise at the low period and driven by information at the high period.
TABLE OF CONTENTS
Title Page i
Declaration ii
Certification iii
Dedication iv
Acknowledgement v
Table of Contents vii
List of Tables xi
List of Figures xii
Abstract xiv
Abbreviation xv
CHAPTER 1: INTRODUCTION
1.1 Background of the Study 1
1.2 Statement of the Problem 3
1.3 Aim and Objectives of the Study 5
1.4 Justification of the Study 5
1.5 The Purpose of the Study 6
1.6 The Significance of the Study 6
1.7 The Scope and Limitations of the Study 6
1.8 Definition of Terms 6
1.8.1 Stochastic process 7
1.8.2 Independent and identically distributed
(IID) noise 7
1.8.3 Probability measure 7
1.8.4 Random variable. 7
1.8.5 Time series 8
1.8.6 Stationary 8
1.8.7 Auto-covariance function (ACVF) and
autocorrelation function (ACF) 8
1.8.8 White noise 8
1.8.9 Random walk 9
1.8.10 AR(1) process 9
CHAPTER 2:
LITERATURE REVIEW
2.1 Defined Contribution Pension Plan 10
2.2 The Nigerian Stock Exchange 11
CHAPTER 3:
MATERIALS AND METHODS
3.1 Fourier Transform (FT) 14
3.2 Short-time Fourier Transform 15
3.3 Wavelet Analysis Applications 16
3.3.1 Haar wavelet 19
3.3.2 The Morlet wavelet 20
3.3.3 Mexican hat wavelet 20
3.4 Wavelet De-noising 21
3.5 Continuous Wavelet Transform (CWT) 22
3.5.1 Steps for creating a CWT: 22
3.5.2 Practical tips 23
3.6 Discrete wavelet transform (DWT) 24
3.6.1 Advantages of discrete wavelet transform 24
3.7 The Pyramid Algorithm 25
3.8
Wavelet Power Spectrum (WPS) 26
3.9 Cross-wavelet
transform(XWT) and cross-wavelet power (XWP) 26
3.10 Wavelet Coherency 27
3.11 GARCH Process: 27
3.11.1 GARCH
28
3.11.2 GARCH
29
3.12 Other GARCH Extensions 30
3.12.1 EGARCH model 30
3.12.2 PARCH model 30
3.12.3 TARCH model 31
3.12.4 GJR (Glosten, Jagannathan and Runkle) GARCH 31
3.12.4 Distributional assumptions 32
3.13 Stochastic Differential Equation and
Stochastic Difference Equation 33
3.14 Convergence of Stochastic Difference
Equations
34
3.15 Distributional Uniqueness 39
CHAPTER 4:
RESULTS AND DISCUSSIONS
4.1 Optimal Investment Strategy Using the
Hamilton Jacobi Bellman Equation: 40
4.2.1 The underlying asset dynamics 41
4.2.2 The salary process and contribution dynamics 42
4.2.3 The wealth process 42
4.2.4 The
admissible portfolio strategy 43
4.2.5 The optimal controls and value function 44
4.2.6
Solving the PDE 45
4.2.7
Optimal investment policy 59
4.3.6 Transforming the differential equation to a
difference equation model 50
4.3.7 Data 53
4.4 Wavelet Analysis 73
4.4.1 The Morlet wavelet 74
4.4.2 Continuous
wavelet transform (CWT) 74
4.4.4 Discrete wavelet
transform (DWT) 75
CHAPTER 5:
CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion 89
5.2 Recommendations 91
5.3 Contributions to knowledge 92
Publication
from the dissertation 93
References 94
LIST OF TABLE
1 Multi-fund pension structure 2
2 The values of θ, k and ξ as π changes 56
3 Summary statistics for the stock prices 60
4 Summary statistics for the stock price
returns 60
5 Summary statistics for the stock price
denoised return 60
6 Parameter estimates of the GARCH (1, 1)
models 71
7 Parameter estimates of the GARCH (2, 1)
models 71
8 Parameter estimates of the GJR (1, 1) models 72
9 Parameter estimates of the EGARCH (1, 1)
models 72
10 Parameter estimates of the PGARCH models 73
11 Parameter estimates of the TARCH models 73
LIST OF FIGURES
1 Multi-Fund Pension Structure 3
3 Optimal Portfolio and Optimal Contribution 20
2 Haar wavelet and Morlet wavelet 48
4 Plot of k against π for UBNr, PZr, UACNr,
and FMILLr 57
5 Plot of θ against π for UBNr, PZr, UACNr,
and FMILLr 58
6 Plot of ξ against π for UBNr, PZr, UACNr,
and FMILLr 59
7 The density Kernel plot for PZr and UBNr 61
8 The density Kernel plot for UBNr and FMillr 62
9 Time Series Plot for The Stock Price, Returns, and The Denoised Return of PZ 63
10 Time Series Plot for the Stock Price,
Returns, and The Denoised Return of UBN 64
11 The Time Series Plot for The Stock Price,
Returns, and The Denoised Return of UACN
65
12 Time Series Plot for The Stock Price, Returns
and The Denoised Return of FMill
66
13 The ACF, PACF, and the Q-Q Plot of PZr 67
14 The ACF, PACF, and the Q-Q Plot of UBNr 68
15 The ACF, PACF, and the Q-Q Plot of UACNr 69
16 The ACF, PACF, and the Q-Q Plot of FMillr 70
17 CWT of PZr, PZrd, UBNr and UBNrd 76
18 CWT OF UACNr, UACNrd, FMillr, and FMillrd 77
19 Wavelet Coherence (WTC) of PZrUBNr,
PZrUBNrdb, UBNrUACNr, and UBNrUACNrdb 78
20 Wavelet Coherence (WTC) of PZrFMillr,
PZrFMillrdb, PZrUACNr, and PZrUACNrdb
79
21 Wavelet Coherence (WTC) of UBNrUACNr, UBNrUACNrdb,
UBNrFMillr, and UBNrUACNrdb 80
22 Wavelet Coherence (WTC) of UACNrFMillr,
UACNrFMillrdb, Cross Wavelet (XWT) of PZrUBNr, and PZrUBNrdb 81
23 Cross Wavelet (XWT) of PZrUACNr, PZrUACNrdb,
PZrFMillr, and PZrFMillrdb 82
24 Cross Wavelet (XWT) of PZrUACNr, PZrUACNrdb,
PZrFMillr, and PZrFMillrdb 83
25 Conditional Variance Forecast 84
26 Conditional Variance
Forecast (Continued) 85
27 Scalogram 86
28 Scalogram (Continued) 87
ABBREVIATION
ACF-Auto correlation function
ACVF-Auto covariance function
ANOVA-Analysis of Variance
ARCH- Autoregressive Conditional
Heteroskedasticity
ARMA- Autoregressive Moving Average
BSE-Bombay Stock Exchange Index
CAPM-Capital Asset Pricing Model
CEE-Central and Eastern European
CRRA- Constant relative risk aversion
CWT-Continuous wavelet Transform
DC- Contribution Scheme
DWT- Discrete wavelet Transform
EGARCH- Exponential Generalized
Autoregressive Conditional Heteroskedasticity
Fmill-Flour Mill of Nigeria Ltd
FT- Fourier Transform
GARCH- Generalized Autoregressive
Conditional Heteroskedasticity
GED-Generalized Error Distribution
GJR- Glosten, Jaganannthan and Runkle
GARCH
HJB- Hamilton Jacobi Equation
iid -Independent and identically
distributed noise
IPI- Industrial Production Index
KSE-Khartoum Stock Exchange
MODWT- Maximum Overlap Discrete wavelet
Transform
MRA- Multi Resolution Analysis
NSE- National Stock Exchange Index
PACF- Partial Autocorrelation Function
PDE-Partial Differential Equation
PFA-Pension Fund Administrator
PPP- Pension plan participant
STFT-Short time fourier transform
TARCH- Threshold Autoregressive
Conditional Heteroskedasticity
UBN- Union Bank of Nigeria
UBN-Union Bank of Nigeria
VaR- Value at Risk
WPS- Wavelet power spectrum
WTC- Wavelet Coherence
XWP- Cross wavelet power
XWT- Cross wavelet Transform
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND OF THE STUDY
The
defined contribution (DC) pension plan is a pension model that has a
pre-decided pace of contribution from the Pension Plan Participant (PPP) while
the reward accruable to the PPP relies upon the arrival on speculation of the
pension wealth. The DC plan is completely financed, privately overseen, and
there is an external administration of the assets and resources. (Okonkwo et al., 2018)
The
pension fund administration in Nigeria is riddled with a lot of challenges
threatening the long-term sustenance of the pension fund. Some of these factors
threatening the pension fund are the government dipping its hand into the
pension fund in the face of its rising debt profile, the policy somersault of
the Federal government, the volatility of the stock market, etc. Also of
importance are the unwillingness of some employers to join the scheme. As of September 2018, nine state governments
are yet to join the contributory pension fund. Six of these states have drafted
the bill and two states have the bill in the house of assembly. Out of those
who have joined the fund, only seven state governments are faithful in funding
the RSA of their workers. (Punch, 2015).
The
Multi-fund is also known as the life-cycle investment for the PPPs was
introduced by PENCOM to suit the risk appetite and the investment horizon of
each investor concerning the varying phase of their career. The multi-fund
structure is made up of four funds: RSA fund I, RSA fund II, RSA fund III, and RSA
fund IV (which is strictly for the retirees). The age of the investor as
well as his or her risk appetite to invest in risky assets determines the fund
an investor will be placed. (Premium Pension Limited, 2019)
The risky
assets allow the investor to make a high rate of return that is determined by
market forces. It however comes with attendant high risk. Thus they are
characterized by higher reward as well as higher risk. The risky assets include
but are not limited to ordinary shares, real estate, infrastructure funds, and
private equity funds.
Table 1 Multi-Fund Pension Structure
|
|
FUND I
|
FUND II
|
FUND III
|
FUND IV
|
|
Age
|
Less than 50 years
|
Less than 50 years
|
50 years and above
|
50 years and above
|
|
Risk Appetite
|
High appetite for risk
|
Medium appetite for risk
|
Low appetite for risk
|
Low appetite for risk
|
|
Allocation to Risky
Assets
|
20%-75%
|
10%-55%
|
5%-20%
|
0%-10%
|
|
Default Option
|
Not default
|
Default
|
Default
|
Not default
|
|
Membership
|
Only on demand. It
cannot be available
for anyone who has
retired, exited the
scheme or attained
the age of 50 and
above.
|
Normal for
pension plan
participants
who have not
attained the
age of 50 years
of age.
|
Normal for
pension plan
participants
who have
attained the
age of 50 years
and above.
|
Only for pension
plan participants
who are retirees
or those who are
exiting the
contributory
scheme.
|
Figure 1 Multi-Fund Pension Structure
From
Table 1 and Figure 1, it is seen that at the entry point, if the PPP has a high
risk appetite, he goes to fund 1, else he goes to fund II. At the age of fifty,
the PPP has a choice to resign he goes to fund IV, else he goes to fund III,
until when he is due for retirement, he moves to fund IV.
The PPP and the PFA (Pension Fund Administration)
want the optimal investment strategy that maximizes a pre-determined utility
function. Several works have been done in this area, see for instance, (Okonkwo
et al., 2018), and (Chang et al., 2019)
1.2 STATEMENT OF THE PROBLEM
The PPP and
PFA have the herculean challenge of investing in a myriad of volatile assets
with the sole aim of optimizing returns and minimizing loss. The investor
desires to have every available information about the market. These include the
market volatilities, the impact of noise, the movement of the stock price in
time both for the short term and the long term horizon. However, most models in
the literature capture only some aspect of these features; hence the need for
an approach that more adequately arms the investor.
In this
work, we derived a model that captures the wealth process of a PPP in a defined
contributory pension scheme. The model we derived is
Equation (1.2.1) is the equation of the
wealth process which is assumed to follow the Heston volatility model (1.2.2). We
also derived the optimal investment strategy for the PPP from the wealth
process using the HJB equation and solving the resulting PDE. Using the Euler
discretization method, we transform the stochastic differential equation
(1.2.1) and (1.2.2) into a stochastic difference equation. We use historical
data of some selected stocks in the Nigerian Stock Market to analyze their
volatilities using the GARCH model and wavelet transform.
1.3 AIM AND
OBJECTIVES OF THE STUDY
This
research aims to derive the model of the wealth process of a PPP in a defined
contributory pension scheme and also generate the optimal investment strategy
with the assumption that the wealth process is a Heston volatility model.
The
objectives of this research are:
i. To
derive the optimal investment strategy for a PPP in a defined contributory
pension scheme assuming Heston volatility.
ii. To transform the stochastic
differential equation model into a stochastic difference equation model.
iii. To study the volatility of the stock
price return using the GARCH model.
iv. To study the volatility of the stock price
return using wavelet analysis.
iv. To
compare the volatility result obtained from the GARCH and wavelet transform
models.
1.4 JUSTIFICATION
OF THE STUDY
This work
introduces the GARCH model and the wavelet analysis into the optimal investment
strategy of a participant in a defined contributory pension scheme. This is a
novel idea and it captures more properly the dynamics of the stock market than
the traditional methods. It also bridges the gap between the theoreticians who
are more at home with the stochastic differential equation and the practitioners
who prefer the stochastic difference equation.
The
purpose of this work is to: derive the model for the wealth process of a PPP in
a defined contributory pension scheme assuming Heston volatility, obtain the
optimal investment strategy for the model, and study the volatility of the
invested wealth using GARCH and wavelet transform. The resulting model and
procedure will provide a tool for portfolio selection for a PPP investing his
wealth in a stock market.
1.6 THE SIGNIFICANCE OF THE STUDY
a. This work presents useful tools that will help the PPP, PFA’s, investors,
like banks, Government, firms, foreign investors, and individuals to:
i.
Make the right choices in creating and managing their portfolio.
ii.
Make the right decision on when to hold or when to sell stocks.
iii.
Understand the volatility dynamics of the market in the time and
frequency domain.
iv.
Understand the impact of noise in the market volatility.
1.7 THE SCOPE AND LIMITATIONS OF THE STUDY
This work
covers the development of the optimal investment strategy for a PPP in a
defined contributory pension scheme as well as the application of Wavelet
transform and GARCH models to study the volatility of a portfolio invested in the
Nigerian stock market.
The work
is limited to a single investor who makes only the mandatory contribution.
1.8 DEFINITION
OF TERMS
1.8.1 Stochastic
process
1.8.2 Independent
and identically distributed (IID) noise
1.8.4 Random
Variable.
1.8.5 Time series
1.8.6 Stationary
1.8.7 Autocovariance
Function (ACVF) and Autocorrelation Function (ACF)
1.8.8 White noise
1.8.9 Random walk
The stochastic process Xt
follows a
random walk if it can be represented as 
with
a constant
and white
noise ϵt. If c is not zero, then the variables 
have
a nonzero mean. We call it a random walk with a drift.
1.8.10 AR(1) Process
The stochastic process
follows an
autoregressive process of the first order, written AR(1) process, if 
with a constant parameter
.
Click “DOWNLOAD NOW” below to get the complete Projects
FOR QUICK HELP CHAT WITH US NOW!
+(234) 0814 780 1594
Login To Comment