Modeling of volatility of interest and treasury bill rates using arch / garch family models and their

47 Pages·2016·0.94 MB·English

MODELING OF VOLATILITY OF INTEREST AND TREASURY BILL RATES USING
ARCH / GARCH FAMILY MODELS AND THEIR EFFECT ON PENSION FUND

ISAAC ONCHAGA ONDIEKI
I56/76584/2014

DEPARTMENT OF ACTUARIAL SCIENCE
SCHOOL OF MATHEMATICS
UNIVERSITY OF NAIROBI

A project submitted in partial fulfillment of the requirements for the degree of
Master of Science in Actuarial Science

December 2016.
Declaration
I declare that this research project is my original work and has never been submitted in any form
as a credit academic qualification to my learning institution or any other university. It is
submitted to post graduate department of actuarial science.
Signature………………………………………………….Date……………………………………
Isaac Onchaga Ondieki.
I56/76584/2014.
DECLARATION BY THE SUPERVISOR:
This research project has been submitted with our approval of as a university supervisor
Prof. Simwa
Professor, School of Mathematics,
University of Nairobi
Signature…………………………… Date…………………………………

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ACKNOWLEDGEMENT
Foremost, I thank GOD for the far He has enabled me to have this project complete, it has not been
easy.
My sincere gratitude to my supervisors’ Prof. Simwa for his guidance availability throughout this
process and your dedication to ensure I complete my project.
Thanks to Prof. Weke for encouraging me during this process.
I thank members of my family especially my wife Margaret, my daughter Sylvia, sons Benjamin,
Shadrack and Nelson for their support during my project.

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DEDICATION
This project is dedicated to my wife Margaret Kemunto, my sons Shadrack, Benjamin and
daughter Sylvia for their support to the Completion of this project.

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LIST OF ABBREVIATIONS
AR : Autoregressive ARCH
GARCH : Generalized Autoregressive Conditional Heteroscedasticity
EVT : Extreme Value Theory
PSTECM : Panel Smooth Transition Error Correction Model.
CBK : Central Bank of Kenya.
IRA : Insurance Regulatory Authority.
CAPM : Capital Asset Pricing Method.
MPC : Monetary Policy Committee.
AR : Autoregressive
AIC : Akaike Information Criterion
ACF : Auto relation Function.
PACF : Partial Autocorrelation Function
HQIC : Hannan-Quinn information criterion
SIC : Shwartz Information Criterion
PACF : Partial Autocorrelation Function
BIC : Bayesian Information Criterion

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LIST OF SYMBOLS
Z : Identically and independent Standard Normal Random Variable.
t
ℇ : Error
t
y : Time series observations
t
y² : Squared Observations
t
ψ : Set of Observations
t
r : Return
t
μ : Constant part of return or mean
α α β Model parameters
o, 1, :
σ : Standard deviation
t
σ² : Conditional variance
t
σ : Volatility Estimate
b : Volatility Coefficient.
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TABLE OF CONTENTS
Declaration ...................................................................................................................................... ii
ACKNOWLEDGEMENT ............................................................................................................. iii
DEDICATION ............................................................................................................................... iv
LIST OF ABBREVIATIONS ......................................................................................................... v
LIST OF SYMBOLS ..................................................................................................................... vi
LIST OF TABLES: ...................................................................................................................... viii
LIST OF FIGURES: ...................................................................................................................... ix
ABSTRACT .................................................................................................................................... x
Chapter One .................................................................................................................................... 1
1.0 INTRODUCTION ............................................................................................................... 1
1.1 PENSION FUND ............................................................................................................. 1
1.2 Interest Rates: ................................................................................................................... 1
Chapter Two.................................................................................................................................... 4
2.0 LITERATURE REVIEW .................................................................................................... 5
2.1 Introduction ...................................................................................................................... 5
Chapter Three.................................................................................................................................. 8
3.0 METHODOLOGY .............................................................................................................. 8
3.5.2 Forecast of Conditional Variance in GARCH model ..................................................... 16
3.6.1 Fitting Multi-factor model ................................................................................................... 19
3.11 FORECASTING PLOTS ............................................................................................... 32
3.12 MULTIFACTOR MODEL ............................................................................................ 33
Interpretation of Findings ............................................................................................................. 35
Chapter Four ................................................................................................................................. 36
4.0 CONCLUSION .................................................................................................................. 36
5.0 REFERENCES ............................................................................................................... 37

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LIST OF TABLES:
Table 4-1: Simple and Log Return interest and Treasury rates: ................................................... 25
Table 4-2: Ljung box test for log return for interest and Treasury rates....................................... 26
Table 4-3: Criterion statistics ........................................................................................................ 29
Table 4-4: Garch (1, 1) normalized distribution ........................................................................... 30
Table 4-5: GARCH (1, 1), Student T test. .................................................................................... 31
Table 4-6: GARCH (11) GED ...................................................................................................... 31
Table 4-7: AIC table for the three Distributions: .......................................................................... 32
Table 4-8: Forecast for the next 12 months. ................................................................................. 34

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LIST OF FIGURES:
Figure 4-1: Raw data plot of interest rates vs. time in months. .................................................... 22
Figure 4-2: plot for Treasury bill rates vs. Time (months) ........................................................... 23
Figure 4-3: Combined plot for interest rates and Treasury bill rates. ........................................... 24
Figure 4-4: the autocorrelation and cross-correlation of interest and Treasury rates. .................. 24
Figure 4-5: Log returns of treasury bills vs. time in months. ....................................................... 26
Figure 4-6: plot of log returns of interest rates. ............................................................................ 27
Figure 4-7: QQ plot of GARCH (1,1) with normal Distribution. ................................................ 28
Figure 4-8: QQ plots for GARCH (1, 1), student T distribution. ................................................. 28
Figure 4-9: QQ plot for GARCH (1, 1), GED distribution ........................................................... 29
Figure 4-10: Forecast plot for next 12 Months ............................................................................. 33

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ABSTRACT
The interest rate and Treasury bill rates were converted into simple returns and modeled by use of
ARCH and GARCH (1, 1) models. The GARCH (1,1) was preferred because it allows many
parameters and considers conditional heteroscedasticity of data to assess volatility of interest rates
and Treasury rates. Volatility measures the errors made in modeling returns. It was discovered that
the average volatility is not constant but varies with time and can be forecast or predicted in both
cases. Also a multifactor model was used to investigate how the two affect pension Fund, it was
discovered that interest rates affected pension fund more than the treasury rates, and the model can
be used to project growth of the fund.

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