MODELLING INTEREST RATES USING THE COX-INGERSOLL-ROSS (CIR) AND THE VASICEK MODELS

Abstract

The purpose of this study is to model Average Monthly 182-Day Treasury
Bill Interest Rate Equivalent In Ghana using the Cox-Ingersoll-Ross and Va
sicek models. These models are examples of One-factor short rate models
because they describe interest movements driven by a particular market risk
and thus are widely used among financial institutions. The CIR and Vasicek
models provided a good fit to our real data however, the CIR model gave the
best fit to the 182-Day Treasury Bill Interest Rates as it produced the least
errors. Thus making the Cox-Ingersoll and Ross model the best fit model
for making efficient forecasts of the average monthly 182-day treasury bill
interest rates in Ghana.

Full text

KWAME NKRUMAH UNIVERSITY OF SCIENCE AND
TECHNOLOGY
MODELLING INTEREST RATES USING THE
COX-INGERSOLL-ROSS (CIR) AND THE VASICEK MODELS.
By
SOMUAH JESSIE OFOSU
TAWIAH GWENDOLYN
APPIAH MATILDA
OSEI NANCY
A RESEARCH WORK SUBMITTED TO THE DEPARTMENT OF
ACTUARIAL SCIENCE AND STATISTICS, KWAME NKRUMAH
UNIVERSITY OF SCIENCE AND TECHNOLOGY IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
BSC. ACTUARIAL SCIENCE
2022

Declaration
We hereby declare that this submission is our own work towards the award
of the BSc. Actuarial Science degree and that, to the best of our knowledge,
it contains no material previously published by another person nor material
which had been accepted for the award of any other degree of the university,
except where due acknowledgment had been made in the text.
Student Signature Date
SOMUAH JESSIE OFOSU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TAWIAHGWENDOLYN ............... ...............
APPIAHMATILDA ............... ...............
OSEINANCY ............... ...............
i

Certied by: Signature Date
Dr. GEORGE AWIAKYE-MARFO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Supervisor)
PROF. ATINUKE ADEBANJI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Head of Department)
ii

Dedication
Most importantly, we dedicate this work to God Almighty for giving us
wisdom, knowledge, and strength to see to the completion of this research
work and also to our families for their support, love, and care.
iii

Acknowledgement
Our utmost duty is to show appreciation to the following personalities, with-
out whom this work would not have become a success. First of all, we would
like to thank the Almighty God for bringing us this far with this academic
work and for His guidance throughout. Many thanks go to our supervisor,
Dr. George Awiakye-Marfo for his time, suggestions, and corrections, with-
out which this study would not have been at its best.
iv

Abstract
The purpose of this study is to model Average Monthly 182-Day Treasury
Bill Interest Rate Equivalent In Ghana using the Cox-Ingersoll-Ross and Va-
sicek models. These models are examples of One-factor short rate models
because they describe interest movements driven by a particular market risk
and thus are widely used among nancial institutions. The CIR and Vasicek
models provided a good t to our real data however, the CIR model gave the
best t to the 182-Day Treasury Bill Interest Rates as it produced the least
errors. Thus making the Cox-Ingersoll and Ross model the best t model
for making ecient forecasts of the average monthly 182-day treasury bill
interest rates in Ghana.
v

Table of Contents
Declaration i
Dedication ii
Acknowledgement iii
Abstract iv
List of Tables ix
List of Figures x
1 INTRODUCTION 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background Information . . . . . . . . . . . . . . . . . . . . . 3
1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Objectives............................. 6
1.5 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Signicance of Research . . . . . . . . . . . . . . . . . . . . . 7
1.7 Organization of The Study . . . . . . . . . . . . . . . . . . . . 8
2 Literature Review 10
vi

3 METHODOLOGY 20
3.1 Overview of Mathematical Concepts . . . . . . . . . . . . . . 20
3.1.1 Probability Space . . . . . . . . . . . . . . . . . . . . . 20
3.1.2 Wiener Process . . . . . . . . . . . . . . . . . . . . . . 21
3.1.3 Stochastic Process . . . . . . . . . . . . . . . . . . . . 21
3.1.4 Stochastic Dierential Equation . . . . . . . . . . . . . 21
3.1.5 Ito'sLemma........................ 22
3.1.6 Root Mean Squared Error . . . . . . . . . . . . . . . . 23
3.2 The Cox-Ingersoll-Ross (CIR) Model . . . . . . . . . . . . . . 23
3.2.1 The Euler Maruyama For The CIR Model . . . . . . . 27
3.3 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 The Euler Maruyama For The Vasicek Model . . . . . 30
3.4 Method Of Parameter Estimation . . . . . . . . . . . . . . . 31
3.4.1 Ordinary Least Square Estimation (OLSE) . . . . . . 31
4 ANALYSIS 34
4.1 Comparative Study Of Simulations . . . . . . . . . . . . . . . 34
4.1.1 Sensitivity Of Short Rate Simulations For The Model
Parameters ........................ 34
4.2 RealDataAnalysis........................ 38
4.2.1 A Plot Of The Monthly Average 182-Day Treasury
Bill Interest Rates In Ghana . . . . . . . . . . . . . . . 39
4.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 40
4.3.1 The Drift and Volatility Of The CIR model . . . . . . 43
4.3.2 The Drift and Volatility Of The Vasicek model . . . . 43
4.4 Finding The Best Fit Model For Our Real Data . . . . . . . . 44
vii

4.4.1 Real Data Simulations For CIR model . . . . . . . . . 44
4.4.2 Real Data Simulations For Vasicek model . . . . . . . 46
4.5 Forecasts.............................. 48
4.5.1 Forecast Simulations With CIR Model . . . . . . . . . 50
4.5.2 Forecast Simulations With Vasicek Model . . . . . . . 53
5 CONCLUSION 57
5.1 INTRODUCTION ........................ 57
5.2 SUMMARY............................ 57
5.3 RECOMMENDATIONS . . . . . . . . . . . . . . . . . . . . . 59
viii

List of Tables
4.1
Summary statistics of Average Monthly 182-Day Treasury bill
Interest Rate equivalent
..................... 38
4.2
Monthly rates for the rst year of Dataset
........... 39
4.3
Lag, Delta and Constant values computed for the CIR model
from January, 2007 to June, 2007
................ 41
4.4
Lag and Delta values computed for the Vasicek model from
January, 2007 to June, 2007
................... 42
4.5
Parameter Estimates
....................... 43
4.6
Actual Interest rates from Jun-20 to May-21
.......... 49
4.7
Parameter Estimates For Test 3(Figure 4.9 )
.......... 50
4.8
Actual Interest rates and Forecasted Interest Rates with least
errors (
0.184%
) from Jun-21 to Mar-22
............ 53
4.9
Parameter Estimates For Test 2(Figure 4.11 )
......... 53
4.10
Actual Interest rates and Forecasted Interest Rates with least
errors (
0.193%
) from Jun-21 to Mar-22
............ 56
ix

List of Figures
4.1 Sensitivity of CIR model to
α
.................. 36
4.2 Sensitivity of Vasicek model to
α
................ 36
4.3 Sensitivity of CIR model to
µ
.................. 36
4.4 Sensitivity of Vasicek model to
µ
................ 36
4.5 Sensitivity of CIR model to
σ
.................. 37
4.6 Sensitivity of Vasicek model to
σ
................ 37
4.7
Time Series plot of Interest Rate Equivalent on Time
..... 40
4.8
CIR Simulations with
α= 0.021932
,
µ= 0.18932
and
σ=
0.026568
.
............................. 45
4.9
CIR Simulations with least RMSE
6.33%
and
6.08%
..... 46
4.10
Vasicek Simulations with
α= 0.021039
,
µ= 0.189667
and
σ= 0.011137
........................... 47
4.11
Vasicek Simulations with least RMSE
6.424%
and
7.37%
. . . 48
4.12
Time Series plot of Interest Rate Equivalent on Time
. . . . 50
4.13
Forecast simulations
....................... 51
4.14
Forecast simulation with least RMSE (
0.184%
)
......... 52
4.15
Forecast simulations
....................... 54
4.16
Forecast simulation with least RMSE (
0.193%
)
......... 55
x

Chapter 1
INTRODUCTION
1.1 General Introduction
Forecasting is a technique that uses historical data as inputs to make in-
formed estimates about events whose actual outcomes have not yet been
observed. Most businesses make eective use of forecasting to determine
how to apportion their budgets or plan for expected expenses for upcoming
period of times.
Forecasting interest rates is one of the most challenging tasks in predictive -
nancial analysis [2]. This is so because it is dicult to estimate interest rates
as they continuously change and are dependent on a large number of factors
including bank lending trend schemes of countries. Ironically interest rates
also play an important role in nancial markets and impact our standard of
living.
There are several existing mathematical techniques which are used on a day
to day basis by nancial institutions to forecast interest rates. Recent ad-
1

vancements made in the area of machine learning have enabled us to predict
behaviour of complex data in nance.
Interest rates are numerical values that tell us how much our money is worth
in the future and it can also help in calculating the current value of future
money. Banks charge interest on loans, the interest rate charged is based on
the creditworthiness of the borrower and the current state of the markets.
There are several models out there that can gradually develop and forecast
interest rates. All these models have their own characteristics and method-
ologies. Short term rate models is a typical example of the ways by which
interest rates can be evolved. Interest rates are considered to follow a Markov
process whereby it is presumed that the current value of short rate is all that
is needed to predict its future values. These interest rate values also follow
a stochastic random walk. Thus this tells us that short rate models change
over time by following a stochastic process. Short term rate models can be
classied into two categories; Equilibrium Short Rate Models and No Arbi-
trage Short Rate Models.
Arbitrage free models are used to name the price of illiquid securities. This
model takes the current state of interest rate term structure as an input and
the model ensures arbitrage opportunities do not exist. Some examples of
this short term rate models are Black Derman Toy (BDT) model, HJM etc.
Equilibrium short rate models are mostly used for relative analysis. Relative
analysis tries to calculate prices of securities of a rm by comparing it with
similar rms. This concept is based on nding the points at which supply
and demand are balanced. There are large number of equilibrium short rate
2

models namely the Vasicek, Cox-Ingersoll-Ross (CIR) models, Ho Lee model,
Hull and White model etc. Once the model gradually develops interest rates,
the model is then calibrated to the initial term structure. This will be the
main focus of our project.
1.2 Background Information
Stochastic interest rate models have played an important role in nancial
markets since the 1970s. These models were gradually adopted by practi-
tioners to accurately price nancial securities, like interest rates. However
their pricing was mathematically far more demanding than equity pricing, as
a result of this the whole term structure of interest rates and its development
was required as time passed. Over the last couple of decades, practitioners
and academicians have expanded signicant eorts in the attempt to model
the behavior of interest rates.
The purpose of this project however is to perform an empirical analysis and
calibration of both the Vasicek and Cox-Ingersoll-Ross model.
The Cox-Ingersoll-Ross (CIR) model(Cox et al., 1985) was developed as a
successor to the Vasicek Interest Rate model(Vasicek, 1977) because it gained
a lot of popularity among nancial institutions due to its simplicity. This
model is used as a means by which interest rates and bond prices can be
forecasted. The Cox-Ingersoll-Ross model (CIR) is regarded as an equilib-
rium short term interest rate model. It is a mathematical formula based on
a dierential equation in which one or more of the terms is a stochastic pro-
cess. The CIR model has been regarded as the reference model for interest
3

rate modeling by both practitioners and academicians since John C. Cox,
Jonathan E Ingersoll and Stephen A. Ross created the model in 1985. The
model has been widely preferred not only because of its systematic tractabil-
ity but also because of its derivation from a general equilibrium framework
among other reasons. A well known feature of the CIR model is that it pre-
vents the occurrence of negative interest rate values.
The model is described using the following equation (1.1)
drt=α(µ−rt)dt +σ√rtdWt, ......r(0) = r0>0
(1.1)
This is a stochastic dierential equation where
Wt
is a Wiener process
having a mean of
0
and variance
dt
,
α
is the mean reversion speed, this
suggests how high or low short rates will go when they eventually revert to
the long-term mean average level.
µ
represents the long-term mean of the
short-term interest rate, and
σ
represents volatility rate.
Similar to the Vasicek model the term
α(µ−rt)
is the drift term factor,
this describes the rate of growth of interest rates over time. Now, the factor
σ√rt
prevents the interest rates from becoming negative.
The CIR model in equation
(1.1)
is mostly referred to as a one-factor time-
homogeneous model, because the parameters
σ
,
α
and
µ
are time indepen-
dent. Under this model, both the drift,
α
(
µ
-
rt
), and the volatility change
with the level of the short rate. The stochastic term has a standard deviation
which is proportional to the square root of the current short rate. This means
that as the short rate increases, it's standard deviation,
σ√rt
, increases. The
short rate under CIR model will always be strictly non-negative. As the
4

short-term falls and approaches 0, the diusion term,
r
, which satises the
condition
2ασ > σ2
causes
rt
to be strictly positive for any
t≥0
.
The Vasicek model is also a one factor short rate model as it describes inter-
est rate movements as driven by only one source of market risk. It was rst
introduced in 1977 by Oldrich Vasicek. This model was the rst to capture
mean reversion, an essential characteristic of the interest rates that set it
apart from other nancial prices. Unlike stock prices, interest rates can not
continue to climb indenitely this is due to the fact that at very high levels,
they would interrupt economic growth which inturn causes interest rates to
decline. In the sameway, interest rates do not typically fall below zero. In-
terest rates actuate within a small range and have the tendency to return
to their long-term mean value.
Vasicek's model theoretically allows for the possibility of interest rates turn-
ing negative. This is because unlike the CIR model, this model does not have
a square root multiplying the constant volatility. The model is represented
by a SDE in (1.2)
drt=α(µ−rt)dt +σdWt, ......r(0) = r0>0
(1.2)
In times of severe nancial crises, central banks use negative interest rates
which is very rare.
In the following sections we will review existing literature on CIR model
and Vasicek model, in particular its short rate mean reversion feature and
5

square root diusion process as well as look into how the parameters of both
models are estimated and how the model is calibrated to historical data.
1.3 Problem Statement
Interest rates are one of the most important categories of the national econ-
omy. Predicting how interest rates change over time can be very challenging.
However, investors and analysts have access to wide range of models such as
the Hull and White model, Ho Lee model etc, that help them to make well
informed decisions about their investments and the economy. The crucial
and practical notion in the modeling of interest rates is the short interest
rate. The Vasicek and CIR model are two very important and widely used
short rate models used in pricing of interest rate derivatives [1]. This re-
search work seeks to nd out how ecient the Cox-Ingersoll-Ross model and
the Vasicek model are in modeling the Monthly Average 182-Day Treasury
Bill Interest Rate from January, 2007 to May, 2021.
1.4 Objectives
The goal of the project is to use the Cox-Ingersoll-Ross (CIR) model and
the Vasicek model to model interest rates from observable nancial data.
The main objectives of this research work is to:
1. evaluate the eects of the individual parameters on both models,
2. make comparisons between simulated pathways obtained from both the
CIR model and the Vasicek model,
6

3. forecast monthly 182-day Treasury Bill Interest rate Equivalent in Ghana.
1.5 Research Questions
The questions we are looking forward to answering by the end of this project
are as follows;
1. What are the parameters of the CIR and Vasicek model?
2. How are the parameters of the CIR and Vasicek model estimated?
3. How dierent is the CIR model from the Vasicek model?
1.6 Signicance of Research
The ndings of this research work would go a long way to help the -
nancial market to evolve interest rates and describe interest rate movements
as driven by only one source of market risk. It could also be used to calcu-
late prices for bonds and value interest rate derivatives amongst other things.
The central bank of Ghana raised its key prime interest rate by 250bps to
17% on the 21st of March, 2022 against market expectations. As at that time
this was the biggest hike in cost of borrowing in over 20 years, causing uncer-
tainty surrounding price developments and its impact on economic activity.
Actual inatoin rate in Ghana were also expected to accelerate according to
7

the bank's governor, Ernest Addison, forecasts.
This will increase the condence of individuals in the nancial market when
they trade as it enables them to be aware of Monthly average 182-Day Trea-
sury Bill Interest Rates in Ghana and how they frequently change on a day
to day basis. Both practitioners and academicians will also be motivated to
build better frameworks from already existing knowledge where necessary.
1.7 Organization of The Study
This research work contains ve main chapters. Each of the ve chapters
has headings containing what the headings reects, namely;
Chapter 1 - Introduction;
This section provides a background that establishes the relevance for the
study within the context of previously published research.
Chapter 2 - Literature review;
This section reviews the literature. Here the emphasis is given on the incep-
tion of the model, its development, the various techniques and critics by the
eminent people and comments.
Chapter 3 - Methodology;
This section is a detailed discussion of the research process.
8

Chapter 4 - Analysis and Results;
This section includes a comprehensive description of all research results and
data.
Chapter 5 - Conclusions and Recommendations;
This section provides an interpretive critique and discussion of the results of
the study.
9

Chapter 2
Literature Review
The most widely recognized practice for modelling interest rate is the
thought that they could be represented by stochastic processes. Thus mak-
ing the theory probabilistic in nature and has also been used in the works
of (Dothan, 1990). There are vast existing literature on interest rate models
published, however most of these works have been considered as extensions
of the very few papers that have become the benchmark in this particular
study. It is well beyond the scope of our research work to present all the
theories and works done. It would however be useful to highlight some of
the main features of these papers to have a better understanding of the model.
Most recent works on the no arbitrage pricing theory were all birth from
the brilliant work of Black and Scholes(1973) [3]. They provided a solution
to the longstanding call option pricing issues which assumed that the price
of risk-less discount bond grows exponentially at the risk-less interest rate.
Merton (1973) [4] used continuous dividend in the Black-Scholes framework
10

and obtained similar pricing formula and later stated that the riskless rate
follows a Brownian motion with a drift. Vasicek in 1977 [5] used the Ornstein
 uhlenbeck process and derived an equilibrium model of discount prices. The
model has been developed on the Gaussian process and the same concept has
been used by Jamshidian(1989).
However the Cox-Ingersoll-Ross model proposed in 1985 [12] has become
one of the most widely used short-term structure models in nancial institu-
tions. This model is widely accepted because it had a positive feature which
never produces negative values. The CIR model was rst introduced to de-
scribe the price of discount zero-coupon bonds under no-arbitrage condition
which had various maturities. The idea was to generalize the Vasicek model
(1977) to the case of non-constant volatility with underlying short-term in-
terest rates assumed to be a continuous Markov process.
The CIR model and the Vasicek model have similar theoretical build ups
causing both of these models to being studied together in the past. However
academicians and researchers lost interest in the Vasicek model because of
the fact that it allows negative interest rates. Stephen J. Brown and Philip
H. Dybvig (1986) [6] undertook one of the most substantial studies on the
CIR and Vasicek models. They examined the extent to which the model in
its simplest one-factor form was descriptive of the prices of U.S Treasury Bills
from 1952 to 1983. Both authors highlighted that expected risk premiums
are reected in the term structure. This is as a result of the term structure in
the CIR model having information currently available to the nancial mar-
11

kets about the future course of events.
The Gibbons and Ramaswamy (1993) [7] model had many more advantages
in comparison with the previous studies or models. As Brown and Dybvig
(1986) failed to make use of steady or conditional state densities, this prop-
erty was extensively studied by (Gibbons and Ramaswamy, 1993). Their
work tests the Cox, Ingersoll and Ross Model of the term structure using
the steady state density of interest rates. They used Hasen's generalized
method of moments to exploit the probability distribution of the single-state
variable in CIR's model. Thus avoiding the use of aggregate consumption
of data. The ndings enables us to estimate continuous time models based
on discrete data samples. The tests show that CIR's model for index bonds
performed quite well when faced with short term Treasury bill returns. The
estimates show that the term premiums are positive and the term structure
could take several shapes. Finally this model was easy to use and understand-
able because there were no stochastic specications for aggregate price levels.
Sun (1992) came up with a discrete-time model that converges with short
term period length to the CIR model.For obvious reasons, Pearson and Sun
(1994) referred to their model as the translated CIR model. Pearson and Sun
(1994) [8] proposed an empirical framework that makes use of the conditional
density of the state variables to estimate and test a term structure namely
the CIR model using data on both discount and coupon bonds. Through
the use of the method of maximum likelihood in the estimation of parame-
ters of the CIR model they could determine the evolution of term structures
12

through time by using conditional density functions. The main advantage
their model has over previous models was that it facilitates the use of time
series information to estimate term structure model. The ndings of their
work showed that estimates solely based on bills show unreasonably large
price errors. Hence through the likelihood ratio test, they rejected the origi-
nal Cox-Ingersoll-Ross model. They then concluded that the CIR model fails
to give an adequate representation of bond prices as well as a good descrip-
tion of the Treasury market.
Up until the early 1990's a lot of researchers kept trying to describe yield
curves and term structure movements using one factor models but unfortu-
nately of these models have not yielded the required results. Due to this
shortcoming of one factor models, the development of multiple variable mod-
els started. Brennan and Schwartz (1982) and Nelson and Schaefer (1983)
were the leading characters in considering multiple state variable models for
citing factor risk premiums. But there was major breakthrough when re-
searchers decided to take current term structure exactly how its given and
developed a no arbitrage yield curve model which t perfectly with available
data. Ho and Lee (1986) [9] enhanced no arbitrage interest rate models using
a binomial-lattice method. Their model was later modied by Black, Der-
man and Troy (1990) [10] to allow one factor models of short rates to t the
current volatilities of bond yields.
Hull and White (1990) [11] expanded the Vasicek (1977) and Cox-Ingersoll
and Ross (1985) models to show that one state variable interest rate models
13

could be consistent with both current term structure of interest rates and
current volatilities of all spot or forward interest rate. Their model was also
faced with the same problems in the long term just like Vasicek and CIR
models. However Vasicek, CIR and Hull and White models are very reliable
due to the fact that their main purpose was stated to provide hedging and
pricing results than to describe term structure movements.
We will now review the one factor equilibrium Cox Ingersoll Ross (CIR)
model and and Vasicek model as well as make overviews of their primary
features since it is the main focus of our research work.
Short-term interest has always been one of the important nancial variables
in any economy. This serves as an instrument mostly targeted by central
banks used to implement monetary policies and an important indicator for
regulators and governments. It also serves as the basis of oating rate loans
for businesses. The expected values of future short-term rates may be re-
ected by longer-term interest rates. The short rate could also be a pertinent
input in determining the required returns on assets. Thus, a good short-term
interest model is of great practical relevance. Most of the well-known short
rate models make use of two very important features, namely; mean rever-
sion and constant volatility. Most of the models available allow short rates to
revert to long term means so that if the current short rate is above (below),
the long-term mean is expected to decrease (increase) towards the long-term
mean in the future. When rates are higher they become more volatile, thus,
showing that the volatility of interest rates depend on the level of interest
14

rates.
The CIR model is a one factor equilibrium model which was rst intro-
duced in 1985 by Cox, Ingersoll and Ross (Cox et al., 1985) [12]. It was
originally a linear mean reversion model which uses diusion process dier-
ent from other short rate models that existed. Their model follows a square
root diusion process which ensured that the short rate term interest rates
are always non-negative. The process follows the continuous-time framework
given in equation (2.1).
drt=α(µ−rt)dt +σ√rtdWt, r(0) = r0>0
(2.1)
where
α
represents the speed of adjustment (or mean reversion),
µ
represents
the long run mean of the short-term interest rate,
σ
represents the volatility
and
dWt
is a small random increment in the Wiener process
Wt
having mean
0 and variance dt.
In contrast to the Vasicek model (Vasicek, 1977), this models diusion term
introduces the square root of
rt
multiplying the constant volatility
σ
. This
eliminates the probability of getting interest rate values that falls below zero
or negative interest rate values. This was not an expectation until the 2008
credit squeeze, where there was massive injection of liquidity and credit fa-
cilities provided by various central banks. Also, the main drawback of the
Vasicek model is that the conditional volatility of changes in the interest rate
is constant and independent of the level of short rate,
r
. This may aect the
prices of bonds which can grow exponentially (Roger, 1996).
15

The relatively easy implementation and tractability of the model, as well
as the specic characteristic of prohibiting negative interest rates from oc-
curing, an undesirable feature under the 2008 pre-crisis assumptions, are key
elements that have made the CIR model one of the most widely used term
structure models in nance.
The Vasicek model is also a one factor equilibrium model which is quite simi-
lar to the CIR model. It was rst introduced in 1977 by Oldrich Vasicek [13].
The model has been used in the valuation of interest rate derivatives and
has also been adapted for credit markets. The Vasicek interest rate model
follows a stochastic dierential equation,
drt=α(µ−rt)dt +σdWt,
......
r(0) = r0>0
where
α
also represents the speed of adjustment (or mean reversion),
µ
rep-
resents the long run mean of the short-term interest rate ,
σ
represents the
volatility and
dWt
is a small random increment in the Wiener process
Wt
having mean 0 and variance dt.
In contrast to the CIR model, the Vasicek model allows negative interest
rate values as it does not have a square root component attached to the dif-
fusion term
σ
.
In Ghana interest rate forecasting also plays an important role in the econ-
omy of the country. Interest rates decisions in Ghana are taken by the Mon-
etary Policy Committee (MPC) of the Bank of Ghana.They provide a brief
overview of macroeconomic developments and monetary policy considera-
16

tions, released after each MPC meeting in January, May, July and November.
The MPC consists of seven members, two external members appointed by
the Board of the Bank of Ghana as well as ve members from the Bank of
Ghana including the Governor who is the Chairman. To achieve the objective
of price stability, Bank of Ghana was granted operational independence use
whichever policy tools and frameworks they deemed t to stabilise ination
and interest rates around the medium-term target.
The Bank of Ghana uses interest rate aligned with the monetary policy op-
eration. This means the Bank obtains its policy rate (MPR) and keeps the
overnight interbank rate closely aligned with the policy rate using some pol-
icy instruments such as the open market operations (OMO) instruments, the
repurchase agreements(repos), reserve requirements and term deposites.
The key instrument or tool used by the Bank of Ghana to defend its Mone-
tary Policy rate (MPR) is through overnight repo and reserve repo facilities.
This is undertaken by the Financial Markets Department (FMD) of the Bank
which controls the Interest Rate Corridor (IRC) system. This system makes
sure that overnight rates remain aligned with the MPR which in turn reduces
volatility of overnight interest rates.
Although not much research, on forecasting interest rates with the CIR and
Vasicek model in Ghana have been made or published, some researchers have
embarked on research works to model and forecast interest rates using other
models.
17

Lawence Adu Asamoah and George Adu published a work in 2016 [15] that
presents empirical analysis of determinants of bank lending rates in Ghana
using annual time series data from 1970 to 2013. They based their empirical
specications on the classical loanable funds theory of interest rate deter-
mination and Kwynesian liquidity preference theory which assumes interest
rates as an implicit function of savings and investments. Their work states
that, in the long run, bank lending rates in Ghana are positively inuenced
by nominal exchange rates and Bank of Ghana's monetary policy rate.
A study that analyses the Expectations Hypothesis (EH) using monetary
policy rate in Ghana was carried out by Boamah, Nicholas Addai in 2016 [16].
By using long-short rate spreads to forecast future variations in short-term
interest rates it tests the EH. Following works from King and Kurmann
(2002), Sargent (1979) and Campbell and Shiller (1987) their research work
deals with the linear versions of the expectation theory and tests the rela-
tionship between short an long-term interest rates. The results of the study
suggests that Ghanaian term structures partially contain information for fur-
ture variations or changes in short-term interest rates.
Trading Economics [17] makes forecasts of Ghana interest rates which they
say are driven by their own trading economics global models and calibrated
using analysts expectation.
According to their trading economics global macro models and analysts ex-
pectations they have made predictions that the interest rate in Ghana is
expected to be 22.00% by the end of this quarter and have also projected
18

that the interest rates will follow a trend around 21.00% in 2023 and 17.00%
in 2024.
In the following chapter we will delve deeper and take a detailed look into
dening the theorems, methods of estimating the parameters and the method-
ology of our research work.
19

Chapter 3
METHODOLOGY
In this chapter we will highlight some key mathematical concepts and the-
orems that are applied in both the Cox-Ingersoll-Ross and Vasicek model as
well as solve and evaluate the Stochastic Dierential Equation of both mod-
els. We will also present some methods of parameter estimation with more
emphasis on the Ordinary Least Squares Estimation (OLSE).
3.1 Overview of Mathematical Concepts
In this section we would highlight some important mathematical concepts
we may come across when dealing with our models.
3.1.1 Probability Space
A probability space is a mathematical construct that provides a formal
model of a random process. It can be dened as a space that consists of a
triple
(Ω,Σ, P )
, where
Ω
represents the sample space,
Σ
is the
σ
-algebra of
20

events and
P
represents the probability on
Σ
.
3.1.2 Wiener Process
A Wiener process
(Wt)
is a continuous-time stochastic process which can
be characterized by three factors,
1.
W0= 0
,
2.
Wt
is almost surely continuous,
3.
Wt
has independent increments, let's say for any
0< t1, t2< ... < tn
the random varibles
Wt1, Wt2−Wt1, ..., Wtn−Wtn−1
are increaments.
For any
0≤µ < t
,
Wt−Ws
are increaments with a distribution
N(0, t −s)
where
N(µ, σ2)
denoting that the mean is zero and varience is
t−s
.
3.1.3 Stochastic Process
A Stochastic process is a collection of random variables that change with
time. It is dened as group of random variables on a common probability
space
(Ω,F, P )
, where
Ω
represents a sample space,
F
is a
σ
-algebra and
P
is a probability measure. The random variables listed by some set
T
, all take
values in the same mathematical space
S
which has to be measurable with
regard to to some
σ
-algebra,
Σ
.
3.1.4 Stochastic Dierential Equation
A stochastic dierential equation is a dierential equation that yields a
solution which is a stochastic process due to the fact that one or more of its
21

terms is also a stochastic process. Assuming
{St}t≥0
is a stochastic process
and
{Wt}t≥0
is a Brownian motion which is dened in the probability space
(Ω,F, P )
, a stochastic dierential equation is an expression that can be
written in the form
dSt=µ(St, t)dt +σ(St, t)dWt
(3.1)
which is expressed in the integral form as
Zt
0
dSt=Zt
0
µ(St, t) + Zt
0
σ(St, t)dWt
St−S0=Zt
0
µ(St, t) + Zt
0
σ(St, t)dWt
St=S0+Zt
0
µ(St, t) + Zt
0
σ(St, t)dWt
(3.2)
The parameter
µ
corresponds to the deterministic part of the SDE as it
represents the drift, whereas the function
σ
represents the volatility coe-
cient which comprises of the randomness or random componet,
dWt
.
3.1.5 Ito's Lemma
Ito's Lemma is an important component of Ito Calculus used to determine
functions of a stochastic process that depend on time. Suppose
{St}t≥0
is the
stochastic process guided by a stochatic dierential equation from equation
(3.1) and let
G(s, t)∈C2(R2)
. Then
G(St, t)
becomes a new stochastic
process. Its dierential is given by the Ito's Formula as
dG(St, t) = ∂G(St, t)
∂t +µ∂G(St, t)
∂s +1
2
∂2G(St, t)
∂s2σ2dt +σ∂G(St, t)
∂s dWt
(3.3)
22

where
dt = (dWt)2
and
dt ·dWt=dt2= 0
3.1.6 Root Mean Squared Error
The Root Mean Squared Error (RMSE) is the standard deviation of resid-
uals. It is also known as prediction errors. Residuals is a measure of how
far regression lines are from data points. It is represented by the following
equation,
RMSE =rΣN
i=1(ri−ˆri)2
N
where
N
is the number of data points,
ri
is the actual observation and
ˆri
is the estimated time series.
3.2 The Cox-Ingersoll-Ross (CIR) Model
In mathematical nance, the CIR model describes the evolution of interest
rates. It is a one factor short rate model as it describes interest rate move-
ments driven by only one source of market risk. The CIR model is described
by the Stochastic Dierential Equation:
drt=α(µ−rt)dt +σ√rtdWt, ....r(0) = r0>0
(3.4)
where
Wt=W(t)t≥0
is a standard one-dimensional Wiener process,
α
is the speed of reversion. Which characterizes the velocity at which
trajectories will regroup around
µ
.
23

µ
is the long term mean level, all the future trajectories of
r
will evolve
around this mean level in the long run.
σ
is the instantaneous volatility, which measures the extent of random-
ness.
θ≡(α, µ, σ)
are the model parameters. The diusion function
σ2(rt, θ) =
rtσ2
is proportional to the interest rate
rt
and ensures that the process stays
on a positive domain.
In order to solve for the unique solution for the CIR model we rst rewrite
equation (3.4) as a SDE in the form
drt= (αµ −αrt)dt +σ√rtdWt
(3.5)
by making
αµ
=
ˆα
, we can rewrite the equation(3.5) above as
drt= (ˆα−αrt)dt +σ√rtdWt
(3.6)
Dening
f(t, r) = eαtrt
, we have
ft=αrteαt
,
fr=eαt
and
frr = 0
. Using
the Ito's formula we have
df(t, r) = ftdt +frdr +1
2frr (dr)2
(3.7)
substituting
f(t, r)
and it's partial derivatives into equation (3.7) we obtain
df(t, r) = αrteαtdt +eαtdr +1
2frr (dr)2
(3.8)
but we know that
frr
=
0
therefore (3.8) can be written as
df(t, r) = αeαtdt +eαtdr +1
2(0)(dr)2
24

df(t, r) = αrteαtdt +eαtdr
(3.9)
substituting
dr
from equation (3.6) into the equation (3.9) we obtain
df(t, r) = αrteαtdt +eαt( ˆα−αrt)dt +σ√rtdWt
df(t, r) = αrteαtdt + ˆαeαtdt −αrteαtdt +σeαt√rtdWt
df(t, rt) = ˆαeαtdt +σeαt√rtdWt
(3.10)
df(t, rt) = αµeαtdt +σeαt√rtdWt
integrating both sides of (3.10) from
0
to
t
we obtain
rt
in (3.12) by
Zt
0
df(t, rt) = Zt
0
ˆαeαsdt +Zt
0
σeαs√rsdt
[f(t, rt)−f(0, r0)] = ˆα
α(eαs−1) + σZt
0
eαs√rsdWs
(3.11)
we know that
f(t, rt) = eαtrt
, then (3.11) becomes
eαtrt−e(0)r0=ˆα
α(eαt−1) + σZt
0
eαt√rsdWs
eαtrt=r0+ˆα
α(e−αt−1) + σZt
0
e−α(s)√rsdWs
rt=r0e−αt+ˆα
α−ˆα
αe−αt+σe−αtZt
0
e−α(s)√rsdWs
rt=r0e−αt+ˆα
α(1 −e−αt) + σZt
0
e−α(t−s)√rsdWs
(3.12)
rt=r0e−αt+αµ
α(1 −e−αt) + σZt
0
e−α(t−s)√rsdWs
We now proceed to nding the conditional expectation of
rt
in (3.12).This
is the interest rate value we expect to obtain.
25

E(rt) = Er0e−αt+ˆα
α(1 −e−αt)+σE Zt
0
e−α(t−s)√rsdWs
taking
I=EhRt
0e−α(t−s)√rsdWsi
,
I= 0
because
dWs
follows a normal
distribution,
N(0, S)
, with mean zero and variance
S
E(rt) = r0e−αt+ˆα
α(1 −e−αt) + σ(0) = r0e−αt+ˆα
α(1 −e−αt)
E(rt) = r0e−αt+αµ
α(1 −e−αt)
(3.13)
Finding the Variance of (3.12), which represents the spread of values around
our expected value,
V AR(rt) = V AR r0e−αt+ˆα
α(1 −e−αt) + σZt
0
e−α(t−s)√rsdWs
Using the well known Variance formula we can solve for the variance,
V AR(rt) = E(r2
t)−(E(rt))2
we can solve for the variance as
=E"e−αtr0+αµ
α(1 −e−αt) + σeαtZt
0
e−αs√rsdWs2#−he−αtr0+αµ
α(1 −e−αt)i2
= 2(e−αtr0+αµ
α(1−e−αt)σe−αt×EZt
0
e−αs√rsdWs+σ2e−2αtEZt
0
e−αs√rsdWs
= 2(e−αtr0+αµ
α(1−e−αt)σ×e−αtZt
0
e−αs√rsE[dWs]+σ2e−2αtEZt
0
e−αsrsds
26

=σ2e−2αtEZt
0
e−αsrsds
=σ2e−2αtEZt
0
(e−αsr0+αµ
α(1 −e−αt))ds
=σ2
µr0(e−αt−e−2αt) + µσ2
2α(1 −2e−αt+e−2αt)
V AR(rt) = σ2
αr0(e−αt−e−2αt) + µσ2
2α(1 −e−αt)2
(3.14)
3.2.1 The Euler Maruyama For The CIR Model
The Euler-Maruyama method (also called the Euler method) is a method
for the approximate numerical solutions of stochastic dierential equations
(SDE). This method is an extension of the Euler method for ordinary dif-
ferential equations. The Euler-Maruyama method was named afer Leonhard
Euler who treated this method in his book (Institutionum calculi integra,
1768- 70) and Gisiro Maruyama (1995) who proposed a unique solution for
stochastic dierential equations [14].
This is one of the simplest and oldest methods for approximating dier-
ential equations. In order to make simulations we apply the Eular scheme
and discretize the basic CIR equation (3.4) and obtain
ˆrti+1 = ˆrt+α(µ−ˆrti)∆t+σp|ˆrti|√∆tZi
Where
ˆr
denotes a discretized approximation dened on time,
t1< t2<
...ti
, and
i= 1,2, ..., n −1
.
Z1, Z2...Zn−1
are standard independent Guassian
27

variables.
3.3 The Vasicek Model
The Vasicek Model is a mathematical model that describes the evolution
of interest rates. It is a one factor short-term model which was introduced by
O. Vasicek in 1977. This model also makes use of the mean-reverting process
to describe movements of interest rates following the stochastic dierential
equation
drt=α(µ−rt)dt +σdWt, ....r(0) = r0>0
(3.15)
Where similarly to the CIR model,
α
is the speed of reversion,
µ
is the long
term mean and
σ
is the instantaneouse volatility. Here the diusion term is
just
σ
.
The SDE of the Vasicek model is linear and its solution can be found us-
ing the Ito's formula. By letting a given function of a stochastic process
f(t, r) = eαtrt
,
fr=eαt
,
frr = 0
and
ft=αrteαt
. Using the Ito's formula
we have
df(t, r) = ftdt +frdr +1
2frr (dr)2
(3.16)
we solve the partial dierential equations and obtain
df(t, r) = αrteαtdt +eαtdr +1
2frr (dr)2
(3.17)
but we know that
frr
=
0
therefore (3.17) can be written as
28

df(t, r) = αeαtdt +eαtdr +1
2(0)(dr)2
df(t, r) = αrteαtdt +eαtdr
(3.18)
substituiting
drt
in (3.15) into (3.18) we have
df(t, r) = αrteαtdt +eαt((α(µ−rt)dt +σdWt))
df(t, r) = αrteαtdt +αµeαtdt −αrteαtdt +σeαtdWt
df(t, rt) = αµeαtdt +σeαtdWt
(3.19)
integrating both sides of (3.19) from
0
to
t
we obtain
rt
in (3.21) by
Zt
0
df(t, rt) = Zt
0
αµeαsdt +Zt
0
σeαsdWs
[f(t, rt)−f(0, r0)] = αµ [eαt−1] + σZt
0
eαsdWs
(3.20)
we know that
f(t, rt) = eαtrt
, then (3.19) can be written as
eαtrt−e(0)r0=αµ [eαt−1] + σZt
0
eαtdWs
eαtrt=r0+µ[eαt−1] + σZt
0
eα(s)dWs
rt=r0e−αt+µ−µeαt+σe−αtZt
0
eα(s)dWs
rt=r0e−αt+µ[1 −eαt] + σZt
0
eα(s−t)dWs
(3.21)
We can now solve for the expected value for
rt
in (3.21)
29

E(rt) = Er0e−αt+µ[1 −eαt]+σE Zt
0
eα(s−t)dWs
taking
I=EhRt
0eα(s−t)dWsi
,
I= 0
because
dWs
follows a normal distribu-
tion,
N(0, S)
, with mean zero and variance
S
E(rt) = r0e−αt+µ[1 −eαt] + σ(0)
E(rt) = r0e−αt+µ(1 −eαt)
(3.22)
Finding the Variance of (3.21) as well, we have
V AR(rt) = V AR r0e−αt+µ[1 −eαt] + σZt
0
eα(s−t)dWs
following the risk neutral probability measure, the Ito integral has a mean
of zero and the variance can directly be taken as equation (3.23)
V AR(rt) = V AR σZt
0
eα(s−t)dWs2
=σ2Zt
0
e2α(s−t)ds
V AR(rt) = σ2
2α1−e−2αt
(3.23)
3.3.1 The Euler Maruyama For The Vasicek Model
In order to make numerical simulations we can discretize the basic Vasicek
equation (3.14) by applying the Eular scheme. Considering changes in the
interest rate over a short period
∆t
we have,
∆r=α(µ−rt)∆t+σZi√∆t
30

Term structure models are additive meaning that if
rt
is the interest rate
at time
t
, then the interest rate at time
t+ ∆t
is given by
rt+∆t=rt+ ∆rt=rt+α(µ−rt)∆t+σZi√∆t
where
Z1, Z2, ..., Zn−1
are standard independent Gaussian variables,
rt
is rates
at current time
t
and
∆rt
denotes future rates discretized on time.
3.4 Method Of Parameter Estimation
There are several ways by which parameters for CIR models can be esti-
mated. In other words, we can consider how to calibrate the Cox Ingersoll
Ross interest rate model to real world data. Methods such as method of mo-
ments, the Ordinary Least Square Estimation (OLSE), Maximum Likelihood
Estimation Method (MLE), etc can all be used to estimate both CIR and
Vasicek parameters. For our work we will focus on just the Ordinary Least
Square Estimation (OLSE) for our parameter estion.
3.4.1 Ordinary Least Square Estimation (OLSE)
Ordinary Least Square Method is a linear least square method for esti-
mating parameters which are not known in a linear regression model. This
model chooses parameters by the principle of least squares, minimising the
sum squares of the gaps between observed dependent variables in a given
dataset and independent variables which have been predicted.
Using the Ordinary Least Square (OLS) Method on the discretized version
31

of (3.1) is highly recommended,
rt−rt+δt =α(µ−rt)δt +σ√rtϵt
(3.24)
where
ϵt=σδWi
is normally distributed with mean zero and variance
δt
.
For carrying out the OLS process we rewrite (3.24) as:
rt−rt+δt
p|rt−δt|=αµδt
p|rt−δt|−αp|rt−δt|δt +ϵt
(3.25)
=αµ δt
p|rt|−αp|rt|δt+ϵt
By letting
yi=rt−rt+δt
p|rt−δt|, β1=αµ, β2=−α, z1i=δt
p|rt−δt|, z2i=p|rt−δt|δt, ϵt=σδWi
(3.26)
We can now rewrite (3.25) as,
yi=β1z1i+β2z2i+ϵi
which is equivalent to
Y=Zβ +ϵ
where
Y= [y1, y2, ..., yn−1], Z ="dfracδtp|rt|, ..., δt
p|rn−1|#,hp|rt|δt, ..., p|rn−1|δti, β = [β1, β2]
and
ϵ=σ√δt
The OLSE of
β
is given as
32

ˆ
β=
arg
minβ∥Y−Zβ∥2
where
∥.∥
denotes Eclidean distance. If
ˆ
β= [ ˆ
β1,ˆ
β2]
then from the constraints in (3.26), we can estimate
α
and
µ
as
ˆα=−ˆ
β2
(3.27)
ˆµ=ˆ
β1
ˆα.
(3.28)
ˆσ=1
√δtn∥Y−Zˆ
β∥
33

Chapter 4
ANALYSIS
4.1 Comparative Study Of Simulations
The CIR and Vasicek model are two well known short rate models used
in forecasting and pricing interest rate derivatives. Some parameters play a
larger role in the forecasting process while others do not aect the process
as much. This section seeks to explore how the changes in each parameter
aect the movement of the drift terms and instantaneous volatility .
4.1.1 Sensitivity Of Short Rate Simulations For The
Model Parameters
In this section we would analyze the eect of varying or keeping param-
eters constant using both the CIR or the Vasicek model in simulations of
short rates.
Using the discretized form of both the CIR model and the Vasicek model
34

obtained from the Euler method we make simulations for both models . The
discretized version of the CIR model is,
ˆrti+1 = ˆrt+α(µ+ ˆrti)∆t+σp|ˆrti|√∆tZi
(4.1)
while the discretized version of the Vasicek model is
rt+∆t=rt+α(µ−rt)∆t+σZi√∆t
(4.2)
If we keep all other parameters the same and change the value of the
speed of reversion,
α
, by
∆α
then the discretized form of the model for CIR
is,
ˆrti+1 = ˆrt+ (α+ ∆α)(µ+ ˆrti)∆t+σp|ˆrti|√∆tZi
(4.3)
and for Vasicek,
rt+∆t=rt+ (α+ ∆α)(µ−rt)∆t+σZi√∆t
(4.4)
This causes a change of the discretized short rates in time, which is given as
(4.3) for the CIR model
∆αrt+∆t=rα
t+∆t−rt+∆t= ∆α(µ−rt)∆t
(4.5)
and for the Vasicek model (4.4)
∆αrt+∆t=rα
t+∆t−rt+∆t= ∆α(µ−rt)∆t
(4.6)
This means that as we increase the value of
α
the next value of the short
rate will increase by
∆α(µ−rt)∆t
when the current interest rate is less than
the long term mean, in the same way if
α
is reduced, the next value of the
short rate reduces by
∆α(µ−rt)∆t
when the current interest rate is greater
35

than the long term mean for both models. This shows that
α
does not really
aect interest rates in the long run, but aect the time it takes the interest
rates to revert back to the long term mean.
Figure 4.1: Sensitivity of CIR model to
α
Figure 4.2: Sensitivity of Vasicek model to
α
Figure 4.3: Sensitivity of CIR model to
µ
Figure 4.4: Sensitivity of Vasicek model to
µ
From Figure 4.1 and Figure 4.2, we observe that when
α
is increased
from (
0.1
) in
test1
to (
1
) in
test2
the variation of interest rates decreases for
both models. Therefore, the value for the speed of reversion is important in
the forecasting and pricing of nancial instruments which are aected by the
volatility of interest rates.
From Figure 4.3 and Figure 4.4, we observe that when
µ > rt
for
test1
of
36

both models, the drift term is positive, pulling the short rates up to the long
term mean. Likewise, when the
µ<rt
for
test2
the drift term becomes neg-
ative, causing
rt
to move back towards the long term mean.
The instantaneous volatility,
σ
, controls the magnitude of the random
rate of the process
(dWt)
term. The lower the value of the
σ
of the process
the less signicant its randomness, in contrast the higher the value of the
process the more signicant it randomness becomes.
Figure 4.5: Sensitivity of CIR model to
σ
Figure 4.6: Sensitivity of Vasicek model to
σ
As the value of
σ
increase in
test2
for both the CIR and Vasicek model,
the randomness of the process is much more signicant than when
σ
has
lower magnitude in
test1
. Therefore, the change in
σ
aects the short term
interest rates in both models. A change in
σ
aects the short rate simulations
in the Vasicek model more than that of the CIR model, as the Vasicek model
lacks the
√rt
in it's diusion term which decreases the eect of
σ
on the CIR
model.
37

4.2 Real Data Analysis
This research work will be centered on Average Monthly 182-Day Treasury
bill Interest Rate equivalent from January, 2007 to May, 2021. Treasury bills
are issued when the government needs money for a short period and the
interest on them are determined by some market forces. A brief summary of
the interest rate statistics of our dataset is given in Table 4.1.
Table 4.1:
Summary statistics of Average Monthly 182-Day Treasury bill
Interest Rate equivalent
Statistic Data
1 count 173
2 mean 0.1809
3 std 0.0588
4 min 0.0990
5 max 0.2886
Table 4.2 shows the data values of the Average Monthly 182-Day Treasury
Bill Interest Rate in Ghana from January, 2007 to December, 2007 .
38

Table 4.2:
Monthly rates for the rst year of Dataset
Date Interest rate(
%
) r
Jan-07 10.46 0.1046
Feb-07 10.36 0.1036
Mar-07 10.41 0.1041
Apr-07 10.31 0.1031
May-07 10.21 0.1021
Jun-07 10.22 0.1022
Jul-07 10.21 0.1021
Aug-07 10.24 0.1024
Sep-07 10.23 0.1023
Oct-07 10.27 0.1027
Nov-07 10.46 0.1046
Dec-07 10.72 0.1072
4.2.1 A Plot Of The Monthly Average 182-Day
Treasury Bill Interest Rates In Ghana
Figure 4.7 represents the time series plot of average monthly 182-day trea-
sury bill interest rate data from January, 2007 to May, 2021 .
39

Figure 4.7:
Time Series plot of Interest Rate Equivalent on Time
Ghana's average monthly interest rate equivalent after reaching its peak,
28.86%, around August of 2009, recorded a steep decline to 10.01% around
the second half of 2011. This drop was as a result of low level of ination
in the country at that time which was averaged around 8.7% as compared
to the average ination rate of 19.2% in 2009 [18] . This is due to the fact
that interest rates are a factor of supply and demand of credit, lenders will
demand higher interest rates as indemnity for the fall in purchasing power
of money recieved in the future. The higher ination rates rise, the more
interest rates are also likely to rise. The same drop is recorded around the
last quater of 2017.
4.3 Parameter Estimation
In our research work we estimate the CIR and Vasicek parameters based
on the Monthly Average 182-day Treasury Bill Interest Rate Equivalent in
Ghana form January, 2007 to May, 2021 using the Ordinary Least Square
40

Estimation method. This is done by rst converting the continuous form of
the models into a normalised form(3.24) and applying the OLSE method.
Table 4.3:
Lag, Delta and Constant values computed for the CIR model from
January, 2007 to June, 2007
Date r Lag Delta Constant
Jan-07 0.1046
Feb-07 0.1036 0.3234 -0.0031 3.0920
Mar-07 0.1041 0.3219 0.0016 3.1068
Apr-07 0.1031 0.3226 -0.0031 3.0994
May-07 0.1021 0.3211 -0.0031 3.1144
Jun-07 0.1022 0.3195 0.0003 3.1296
The delta values in Table 4.3 were obtained by substracting the interest
rate of the previous month from the interest rates of the current month and
dividing by the square root of the interest rate of the previous month.
The constant terms are to be
1
in normal circumstances but as we estimate
the model using weighted squares we adjust it by dividing by the square root
of the interest rate of the previous month.
The lagged terms are obtained by nding the square root of the interest rate
of the previous month.
41

Table 4.4:
Lag and Delta values computed for the Vasicek model from Jan-
uary, 2007 to June, 2007
Date r Lag Delta
Jan-07 0.1046
Feb-07 0.1036 0.1046 -0.0010
Mar-07 0.1041 0.1036 0.0005
Apr-07 0.1031 0.1041 -0.0010
May-07 0.1021 0.1031 -0.0010
Jun-07 0.1022 0.1021 0.0001
The lagged values in Table 4.3 are just the interest rate values of the
previous month and the delta values are given by substracting the lagged
interest rate from the current interest rate.
In estimating our parameters the OLS regression was solved by using a
LINEST function in excel. This function calculates the statistics of a straight
line that ts a given date using least squares method and returns an array
that describes the line.
The estimates of
−α
,
αµ
and
σ
are given for both models, from which
we can easily retrive our parameters (3.4.1) as shown in Table 4.5.
42

Table 4.5:
Parameter Estimates
Model
α µ σ
Cox-Ingersoll and Ross (CIR)
0.021932 0.18932 0.026568
Vasicek
0.021039 0.189667 0.011137
4.3.1 The Drift and Volatility Of The CIR model
The estimate of
−α
was a negative value and this is a good sign because
the speed of reversion,
α
has to be positive for convergence to take place.
Thus the speed of reversion,
α
, is given as
0.021932
. This implies that on
average the process will take
ln(2)/0.021932 = 31.60yrs
, which is around 2.6
months, for the rates to travel back to the long term mean .
The long term mean,
µ
, from the dataset is given as
0.18932
this means the
long run equillibrium monthly treasury bill rates from 2007-2022 is around
18.93%
. The instantaneous volatility was given as
0.026568
which is the rate
of amplitude of the randomness instant by instant.
4.3.2 The Drift and Volatility Of The Vasicek model
The estimate of
−α
was a negative value as well in this model which is a
good sign because the speed of reversion,
α
has to be positive for convergence
to take place. Thus the speed of reversion,
α
, is given as
0.021039
(26). This
implies that on average the process will take
ln(2)/0.021039 = 32.95yrs
,
which is around 2.7 months, for the rates to revert back to the long term
mean .
43

The long term mean,
µ
, is obtained from the dataset and is given as
0.189667
(27), this means the long run equillibrium monthly treasury bill rates from
2007-2022 in Ghana is around
18.97%
. The instantaneous volatility was
given as
0.011137
which is the rate of amplitude of the randomness instant
by instant.
4.4 Finding The Best Fit Model For Our Real
Data
In this section we will make simulations using the parameters obtained
from our real data against the time series plot of our interest rates for both
the CIR and Vasicek models. The Root Mean Square error (RMSE) will
be utilised to get the best forecast model for our monthly average 182-day
interest rate data.
4.4.1 Real Data Simulations For CIR model
Figure 4.8 below, show four test runs or possible pathways our interest rate
data could take based on the parameters estimated in (4.3.1) and a plot of
the real data interest rate equivalents on time represented by the blue line.
44

Figure 4.8:
CIR Simulations with
α= 0.021932
,
µ= 0.18932
and
σ=
0.026568
.
The Root Mean Squared Errors are calculated for the four test runs above
in Figure 4.8, where test 1 and test 3 give the least RMSE,
6.33%
and
6.08%
respectively. Figure 4.9 below, shows pathways of test 1 and test 3 of the
interest rates based on the least RMSE and a plot of the interest rate data
on time.
45

Figure 4.9:
CIR Simulations with least RMSE
6.33%
and
6.08%
4.4.2 Real Data Simulations For Vasicek model
Figure 4.10 shows four test runs or possible pathways our interest rate data
could take based on the parameters shown in (4.3.4) and a plot of the real
interest rate equivalents represented by the blue line based on original data.
46

Figure 4.10:
Vasicek Simulations with
α= 0.021039
,
µ= 0.189667
and
σ= 0.011137
The Mean Squared Errors are calculated for the various test runs above in
Figure 4.10, where test 2 (
6.424%
) and test 3 (
7.37%
) gave the least RMSE.
Figure 4.11 shows the two best possible pathways of the interest rates based
on the least RMSE and a plot of interest rate data on time.
47

Figure 4.11:
Vasicek Simulations with least RMSE
6.424%
and
7.37%
Comparing the calculated Root Mean Squared Errors for both the Cox-
Ingersoll-Ross model(CIR) and the Vasicek model, we can conclude that
the CIR model is the best amongst the two as it produces the least errors
(
6.33%
and
6.08%
) based on the mpnthly average interest rate data.
4.5 Forecasts
For the forecasts we made predictions, using both the Cox-Ingersoll-Ross
(CIR) model and the Vasicek model, and compared the results to the actual
rates for June, 2020 to May, 2021.
48

Table 4.6:
Actual Interest rates from Jun-20 to May-21
Date Time step Actual Interest rates
May-20 0 0.1402
Jun-20 1 0.1405
Jul-20 2 0.1405
Aug-20 3 0.1409
Sep-20 4 0.1412
Oct-20 5 0.1411
Nov-20 6 0.1412
Dec-21 7 0.1413
Jan-20 8 0.1414
Feb-21 9 0.1400
Mar-21 10 0.1378
Apri-21 11 0.1358
May-21 12 0.1354
49

Figure 4.12:
Time Series plot of Interest Rate Equivalent on Time
Figure 4.12 shows the evolution of the Average 182-Day Treasury Bill rate
for the last 12 months of the dataset, that is from June, 2020 to May, 2021.
4.5.1 Forecast Simulations With CIR Model
Based on the sample path for test 3 in Figure 4.9, forecasted paths were
obtained as shown in Figure 4.13.
Table 4.7:
Parameter Estimates For Test 3(Figure 4.9 )
Model
α µ σ
Cox-Ingersoll and Ross (CIR)
0.05606 0.1616 0.02849
Based on the estimated parameters in Table 4.7, where the mean reversion
given as
0.05606
, long term mean as
0.1616(16.16%)
and the volatility as
0.02849
, we performed forecast simulations for the last 12 months of the
50

Average Monthly 182-Day Treasury bill Interest Rate equivalent from June,
2020 to May, 2021.
Figure 4.13:
Forecast simulations
The test runs in Figure 4.13 represents the forecasted pathways. The Root
Mean Squared Errors are calculated for the four possible pathways which the
future rates could take. The errors for each of the forecasts in Figure 4.13 are
given as,
0.648%
,
0.446%
,
0.1842%
and
0.2634%
, respectively. It is observed
that the third simulated forecast produced the least errors (
0.1842%
) and
can be considered as the best possible pathway for our future rates.
51

Figure 4.14:
Forecast simulation with least RMSE (
0.184%
)
Figure 4.14 compares the forecast values form of the third test run to
the actual future interest rate equivalents, it is quite clear that the forecast
process gives us a good representation as shown in Table 4.8.
52

Table 4.8:
Actual Interest rates and Forecasted Interest Rates with least errors
(
0.184%
) from Jun-21 to Mar-22
Date Time step Actual Interest rates Forecasted Interest Rates
May-20 0 0.1402
Jun-20 1 0.1405 0.140171
Jul-20 2 0.1405 0.140149
Aug-20 3 0.1409 0.140405
Sep-20 4 0.1412 0.139044
Oct-20 5 0.1411 0.139592
Nov-20 6 0.1412 0.139359
Dec-20 7 0.1413 0.139349
Jan-21 8 0.1414 0.140231
Feb-21 9 0.1400 0.138041
Mar-21 10 0.1378 0.137524
Apr-21 11 0.1358 0.136853
May-21 12 0.1354 0.136031
4.5.2 Forecast Simulations With Vasicek Model
Based on the sample path for test 2 in Figure 4.11, forecasted paths were
obtained as shown in Figure 4.15.
Table 4.9:
Parameter Estimates For Test 2(Figure 4.11 )
Model
α µ σ
Vasicek
0.097674 0.183423 0.010957
53

Based on the estimated parameters in Table 4.9, where the mean reversion
given as
0.097674
, long term mean as
0.183423(18.34%)
and the volatility as
0.010957
, we can now perform forecast simulations for the last 12 months
of the Average Monthly 182-Day Treasury bill Interest Rate equivalent from
June, 2020 to May, 2021.
Figure 4.15:
Forecast simulations
The test run in Figure 4.15 represents the forecasted pathway. In order
to nd the best t simulated forecast path, we calcutate the Root Mean
Squared Errors for all the four possible path ways,
0.274%
,
0.312%
,
0.193%
and
0.337%
respectively. From the RMSE of the possible test runs it is
observed that the third test run produces the least errors (
0.193%
) and thus
is the best t for our data.
54

Figure 4.16:
Forecast simulation with least RMSE (
0.193%
)
Figure 4.16 compares the forecast values form of the second test run to
the actual future interest rate equivalents, we can see that it gives us a good
representation as seen in Table 4.10.
55

Table 4.10:
Actual Interest rates and Forecasted Interest Rates with least
errors (
0.193%
) from Jun-21 to Mar-22
Date Time step Actual Interest rates Forecasted Interest Rates
May-20 0 0.1402
Jun-20 1 0.1405 0.138284
Jul-20 2 0.1405 0.139768
Aug-20 3 0.1409 0.140072
Sep-20 4 0.1412 0.140195
Oct-20 5 0.1411 0.140557
Nov-20 6 0.1412 0.139888
Dec-20 7 0.1413 0.139139
Jan-21 8 0.1414 0.137426
Feb-21 9 0.1400 0.138844
Mar-21 10 0.1378 0.137265
Apr-21 11 0.1358 0.138361
May-21 12 0.1354 0.138525
56

Chapter 5
CONCLUSION
5.1 INTRODUCTION
This research work was primarily focused on the study and evaluation
of two Equilibrium Short Rate models namely the CIR and Vasicek model
and how well these models could t Monthly Average 182-Day Treasury Bill
Interest Rate Equivalent in Ghana.
5.2 SUMMARY
In the rst few sections we highlighted some literature on both models as
well as reviewing their historical backgrounds. We then went ahead to make
thorough evaluations on the stochastic dierential equations that describe
both the CIR and Vasicek model.
More light was shed on the parameters of both models and the roles they play
in the model. The mean reverting term,
α
, represents the velocity at which
57

interest rates revert back to the long term mean,
µ
, represents the long run
equilibrium of the monthly average rates, where the instantaneous volatility,
σ
, represents the magnitude of randomness of the process. The major dif-
ference between these two models is that with the CIR model there is the
presence of a square root term (
√rt
), which is absent in the diusion term of
the Vasicek model which could cause the occurrence of negative interest rates.
Comparative studies were made to evaluate the eect of the individual pa-
rameters on both models. The results of the analysis showed the similarities
in both the Vasicek and CIR model in relation to the sensitivity of the mean
reversion term and the long term mean. However the c change in the instan-
taneous volatility,
σ
, aects the prices of bonds or forecasts in the Vasicek
model more than that of the CIR model, as the square root term,
√rt
, in
the CIR model decreases the eect of
σ
on the model. Furthermore, high
magnitude of
σ
may lead to the occurrence of negative interest rates which
is not desirable in reality.
In the subsequent subsections we estimated the model parameters based on
our real dataset using the Ordinary Least Square Estimation and went fur-
ther to make real data simulations based on our estimated parameters. Four
possible paths were simulated by each of the parameters estimated from
both the Vasicek model and the CIR model where the Root Mean Squared
Errors were obtained for each of the test runs. Two test runs were then
taken for both models as they produced the least errors, (
6.33%,6.08%
) and
(
6.424%,7.37%
), for the CIR and Vasicek model respectively.
58

In the given data set it is quite clear that the CIR and Vasicek models
are similar in a number of ways and also provide a good t to our real data
however the CIR model gave us the best t to the actual path of the 182-Day
Treasury Bill Interest Rates as it produces the least errors. Thus making the
Cox-Ingersoll and Ross model the best t model of the two for making fore-
casts. After simulated forecasts for the last 12 months, Jun-20 to May-21,
were made based on the the parameters obtained from the data.
5.3 RECOMMENDATIONS
From the ndings of our research work based on our dataset it is clear that
both the Cox-Ingersoll and Ross (CIR) and the Vasicek model exhibit good
ts with the CIR model being the most preferable choice because it produced
the least errors.
Thus this model could be used to make very ecient forecasts for Monthly
Average 182-Day Treasury Bill Interest Rate Equivalent in Ghana.
Generally if the interest rates under study were far from zero, the Vasicek
model would have an edge over the Cox-Ingersoll and Ross model as the
close form solutions for the Vasicek model are easily available even for more
complex nancial interest rate derivatives. However, if interest rates under
the study are closer to zero, working with the Vasicek model might be quite
unwieldy as there may be possibilities of occurrence of negative interest rates
which are undesirable in most nancial markets.
59

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62