Modelling The Exchange Rate Market Volatility Of Kenya Shilling Against The Us Dollar

Background

Different countries represent their currency relative to other countries’ currency. This representation is what is known as an exchange rate. Well known exchange rates in Kenya includes the Kenya shilling against either the US dollar, the Euro, the pound or the Japanese yen. Various countries employ varying mechanisms in an attempt to ensure stability of their currency, through finding out a suitable exchange rate for their economy. This entails the application of volatility models as tools for derivative asset pricing and risk management, and also coming up with relevant monetary and fiscal policies. The foreign exchange rate is one of the key macroeconomic variable as it plays a major role in international trade, and in making investment decisions. It's however faced with challenges one of which is the volatility of exchange rates. This volatility is the driving force behind the increase in exchange rate risk. The non constant variance in financial markets and the uncertainties in stock prices and returns drove financial analysts to model and establish an explanation for the behaviour of exchange rates returns and volatility with the aid of time series econometric models. The suitable models for changing variance includes the autoregressive conditional heteroscedasticity (ARCH) models and the generalised autoregressive conditional heteroscedasticity (GARCH) family models. These models were developed by Engle, R. F( 1982) and extended by Baillie, R. T and Bollorslev (2002).

Problem Statement

The dollar is the legal tender in the United States. It is the most commonly used currency by international tourists in Kenya and the major trading foreign currency in Kenya. The exchange rate for the Kenyan shilling relative to the US dollar has weakened over the past decade. In 2009, the exchange rate had reached 74. 3216 shillings per US dollar, but over the next several years it has weakened to a rate of 106. 199 KES/USD in October 2011. This just goes to show how volatile the KES/USD exchange rate has been. These fluctuations in the US dollar against the Kenya shilling are attributed to factors such as imports, exports and political instability. Imports and political instability weaken the Kenya shilling while strengthening the US dollar. The exports on the other hand strengthens the Kenya shilling thereby weakening the US dollar. As was highlighted above in the background of the study, these fluctuations affects to a great extent the investors in the international markets, exporters, importers, corporate decision makers, the government and also the individual traders. Most of these consumers of exchange rates calculate their next move on a daily basis. Our research will therefore major in modelling the volatility behaviour of KES/USD exchange rate and obtain a model that can best accommodate all characteristics of the KES/USD exchange rates in the case of daily data. This model will also be used in forecasting future exchange rates.

Justification

The findings of this study will be of great importance to the government of Kenya, investors, importers, exporters, foreign exchange bureaus, tourists, and other researchers who are the end users of the exchange rate data. For the government, with the aid of the identified model, they will be able to model and forecast the volatility of the exchange rates and hence employ the relevant measure to manage their exposure in the short run. Investors; one of the factors that informs the investment decisions, evaluation of earnings and budgeting for investors is the behaviour of exchange rates. Therefore identifying a suitable model for the daily data will help them model and forecast the expected behaviour of exchange rates. Importers, exporters and forex bureaus with the use of the suitable model will easily asses the conditions prevailing in the market prior to making their transactions. Tourists will as well be able to predict the probable future behaviour of exchange rates and hence budget accordingly. This research will also benefit other researchers in the finance field as well as scholars as it is going to be a starting point for further research. They will base their discussions and empirical literature review on this.

General Objective

To model the market volatility of KES/USD foreign exchange rate.

Specific Objectives

1. To find out the model(s) that can best capture daily volatility of KES/USD exchange rate.
2. To establish the ARCH effects of exchange rate market volatility.
3. To forecast exchange rates using the model that will be identified.

Methodology

In this chapter we will describe the steps that will be taken during the study.

Population and sample

A population is a well-defined or set of people, group of things or households, events, elements, and services that are being studied (Ngechu, 2004). This definition ensures that population of interest is homogenous. In this research we will use non-probabilistic sampling techniques. Our population is Kenya’s KES/USD daily foreign exchange rates. The sample size will consist of data commencing from 1st October 2009 to 10th October 2018 which will be obtained from https: //m. investing. com/currencies/usd-kes-historical-data Log returns of the sample data will be modelled. This is because log returns are more mathematically tractable compared to simple returns. It is expressed as: rt= log( FEt / FEt-1)Where: FEt- Current daily Foreign Exchange RateFEt-1- Previous daily Foreign Exchange Rate. rt- daily Foreign Exchange log return

Methods of Data Analysis

It’s known from empirical studies that financial data poses common feature known as stylized facts. With respect to the stylized facts exhibited by financial time series data i. e. exchange rates data. We will major in the ARCH and GARCH models as our tools for modelling ARCH effects in daily market volatility. Franke and Hafner( 2008) argued that stock, exchange rates, interest rates and other financial time series have stylized facts and that the ARCH class of models can account for these features effectively. The stylized facts portrayed include: i. Fat tails-The unconditional distribution of returns has fatter tails than is expected from a normal distribution. Bogard (2015) defined fat tails as situations that can be described by probability distributions with heavy mass in tails. According to Bogard (2015), a data distribution with negative kurtosis has thinner or shorter tails than the normal distribution while a data distribution with positive kurtosis has fatter tails. ii. Volatility clustering- Refers to the serial correlation of volatility of returns. That is, large changes (large positive or negative returns) tend to follow large changes while small changes follow small changes. (Wang,Gelder,Vrijling and Ma,2005)iii. Aggregated normality- This basically means that as the frequency or returns lengthens, then distributions of the returns approaches a normal distribution. The Jarque Bera (JB) test will be used to test normality. This test is named after a combination of Carlos Jarque and Anil K. Bera 1980. It is represented as:

JB= Where: n= the no of returns

S= the skewness of the returns

K= the kurtosis of the returns

The hypothesis to be tested at 5% significance level is:

H0: The daily returns are normally distributed

H1: The returns are not normally distributed

The decision criterion shall be rejecting the null hypothesis of normality if the P-value of the JB statistic is less than the significance level. iv) Stationarity- this implies that the mean function and the variance remains constant over time. One data deviates from the mean; it tends to revert back to it. A test for stationarity that has become widely popular over the past several years is the unit root test (Gujarati and Porter 2009). The Augmented Dickey-Fuller (ADF) test for a unit root will be used to test for stationarity. This test was developed by David Dickey and Wayne Fuller (1979). Omolo (2014) argues that it is useful to quantify the evidence of non-stationarity in data generating mechanism and this can be achieved by hypothesis testing. We will test the following hypothesis at 5% level of significance. H0: The test results are non-stationaryH1: The returns are stationaryThe sample ADF statistic will be compared with the tabulated critical value and the more negative it will be then the stronger the rejection of the hypothesis at 5% level of significance. The original data if found to be non- stationary will be differenced to make them stationary. This is so as to get rid of spurious regression problem that may arise from regressing a non-stationary time series on one or more non-stationary time series (Gujarati and Porter 2009). v) Leverage effect- that is changes in stock returns tend to negatively correlate with changes in volatility (Rossi, 2013) In presenting the ARCH or GARCH models, we will specify the mean equation and the variance equation.

Selecting the mean equation

The models usually considered in mean equation selection includes:

• Autoregressive model (AR)
• Moving Average Model (MA)
• Autoregressive moving average model (ARMA)
• Autoregressive integrated moving average model (ARIMA)

Autoregressive Model (AR)

This model foretells future behaviour with respect to past behaviour. The linear autoregressive process of order p (AR(p) is defined as : Xt= + +………+ + MOVING AVERAGE MODEL (MA )This model sums up the white noise error. Its general form that is MA (q) is given by Xt=β0+ β1+β2+……………+βPWhere α0 is a constant Represents the white noiseHere, the process observed depends on previous ’s. MA (q) can define correlated noise structure in our data and goes beyond the traditional assumption where errors are independent and identically distributed (iid).

Autoagressive Moving Average Model. (ARMA).

This model combines both the AR and MA components. Its advantage over the pure AR and MA models is that being a relative parsimonious model, it can provide a better representation of a time series( Davidson and Mackinnon,1993). The ARMA process of order p and q is defined as; = + + A time series Xt follows an ARMA (1,1) model if it takes the form;= ++β0+ β1

Autoagressive Integrated Moving Average (ARIMA) model.

The AR, MA, ARMA model discussed above assumes that time series data are stationary i. e. the conditional variance given the past doesn’t change. This is usually not the case for financial time series. An example is the daily exchange rates which at times may seem unusually volatile. In order to model a non-stationary time series using ARMA (p,q) models, there is need to difference them first. The ARMA ( p, q) models fitted to differenced time series data is what is referred to as ARIMA ( p, d, q) model where ; P represents the number of AR terms, d -order of differentiation q-the number of moving average terms. We will use the auto. arima () command in R to identify a model that will be optimal for the daily exchange rates data. It is from the optimal model's mean equation that we will extract the residual component. The residual component with then be squared and used to test for the ARCH effects.

ARCH Effects Test

An arch effect refers to a non- linear phenomenon of the variance behaviour. A time series is said to possess an arch effect if it exhibits conditional heteroskedasticity or autocorrelation in the squared series. To carry out an ARCH test, we will obtain the autocorrelation function (ACF) plots and the partial autocorrelation function(PACF) plots. Moreover, we will perform the Lagrange multiplier(LM) test and the Ljung-Box test. The ACF of a process refers to the plot of the autocorrelation coefficient against lag k where k is the distance between observation Xt and Xt-1(Box,Jenkins and Reinsel, 2008). we will use the acf () command in R to obtain the plot. The PACF measures correlation between observations after controlling the correlations at intermediate lags. We will use the pacf () command in R to obtain the plot. The Lagrange Multiplier(LM) test was proposed by Breush(1978) and Godfrey(1978). Its test statistic is TR2 where T is the number of observations while R2 is the coefficient of determination. It follows a chi-squared distribution with p degrees of freedom. To perform this test in R, ArchTest () command is used. Our hypothesis at a 5% significance will be;H0: daily returns do not exhibit ARCH effects. H1: daily returns exhibit ARCH effects. Our decision criteria will be to reject the null hypothesis if the p value will be less than the 5% significance level. The Ljung box test was suggested by Ljung and Box (1979) as an improvement of Box and Pierce (1970) q-test. Box. Test () command in R is used for Ljung Box test. Our hypothesis at 5% significance level will be;H0: daily returns do not exhibit ARCH effects. H1: daily returns exhibit ARCH effects. Our decision criteria will be to reject the null hypothesis if the p value will be less than the 5% significance level. Having performed the ARCH effects tests, we will fit various models which include: i. ARCH Modelii. GARCH Modeliii. E-GARCH ModelARCH ModelIt was proposed by Engle, R. F. , 1982. It is formally a regression model with the conditional volatility as the response variable and the past lags of the squared returns as the covariates consider heteroskedacity as the variance to be modeled. The generalized representation of an ARCH model; ARCH(q) is expressed as

= + Where: q is the number of lagged terms. = Constant = ARCH component = Residual terms. The simplest form of the ARCH model which was proposed by (Engle, 1982) is the ARCH (1) model. Its representation is;= +However, ARCH(q) models possess shortcomings despite being successfully applied to the linear and non-normal series. Some of the weaknesses of using the ARCH model to estimate the market volatility include:

1. The model assumes that positive and negative shock have the same effects on volatility because it depends on the square of the previous shock.
2. The ARCH model is rather restrictive.
3. The ARCH model does not provide any view insight for understanding the source of variation in time series.
4. The ARCH model is likely to over predict.

GARCH Model

The Generalized Autoregressive Conditional Heteroskedastic Model (GARCH) model is used to keep track of variations in the volatility through time. The model was proposed by Bollerslev, T, (1986) to counteract the drawbacks of the Engle’s ARCH model. Unlike the simple ARCH models, the GARCH models allow the variance to evolve over time in a much more general way (Greene, 2012). Having only 3 parameters, it makes it possible for an infinite number of squared roots to influence the conditional variance (volatility).