MSc.MF Theses and Dissertations

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An Inclusive pension model for Kenya’s informal sector with late entries and early exit rates
(Strathmore University, 2019) Lagat, Cherono Asumpta
The purpose of this study is three-fold: first, we develop 1'1 pension model that uses preretirement mobile phone airtime expenditures to accumulate the pension fund. Secondly, we · calculate the exit and entry rates into the comprehensive pension scheme. Finally, we determine the expenditure patterns experienced post-retirement and use these patterns to advise on the daily amount required to be charged per minute above the current rate in order to facilitate a comfortable post-retirement life. The data utilized in this study was retrieved from various secondary sources. Inflation and interest rates data -were retrieved from Kenya's Central Bank database. The entry and exit data into informal pension schemes was retrieved from Eagle Africa the administrators of Mbao Pension scheme the largest informal pension scheme in Kenya. The mortality rates were retrieved from the World Health Organization and the life expectancy from ·world Atlas, Lancet and World Life Expectancy. Pre-retirement data was retrieved from November 2013 from an integrated survey on land ownership and tenure, provision, access and control of basic services, asset ownership, financial resources, evictions and demolition of houses, as well as thirty-two key informant interviews with informal small-scale service provider’s facilitated by Strathmore University. The inflation and interest rates were forecasted using ARIMA (1,9,5)-GARCH (0,1) model while the backward entry and exit data points were simulated in R. Our results show that an unemployed Kenyan spends approximately KShs. 2, 000.37 a month considering inflation this amount will translate KShs. 4, 025.45 to maintain the same life style post-retirement assuming the person joins the scheme at 18 and exits at the age of 55. Given the expenditure pre-retirement of this group of people, it will require them to be charged KShs. 3.41 per minute above the current rate in order to raise an amount sufficient to sustain their lifestyle post-retirement.
Analysis of risk measures in portfolio optimization for the Uganda Securities Exchange
(Strathmore University, 2021) Birungi, Criscent
For the most recent years, risk has become one of the essential parameters in portfolio optimization problems. Today most practitioners and researchers in portfolio optimization have used variance as a standard risk measure. This approach has been found subjective. The Markowitz (1952) mean-variance model considered variance as an adequate portfolio risk measure, and asset returns are multivariate normally distributed and that investors have a quadratic utility function which is subjective too. Other risk measures have been suggested to overcome the limitations of the mean-variance model. This paper analyzes which portfolio optimization models can better explain the optimal portfolio performance (high return, low risk) for the Uganda Security Exchange(USE). We compare Mean-Variance (MV), Mean Absolute Deviation (MAD), Robust Portfolios and Covariance Estimation Models( The Shrinked Mean-Variance (SMV) Models & Alternative Covariance Estimator (ACE) Models ) and Mean-Conditional Value-at-Risk (Mean-CVaR) models in terms of the risk and performance. Portfolios were developed by employing the MV, MAD, SMV, ACE and Mean-CVaR models. For the computed monthly returns and price data (February 2010 to January 2021) for USE selected stocks, we considered the results show that Mean-CVaR and ACE portfolios have the highest performance ratio compared to other models. We find that VaR is the best risk measure for portfolio optimization for the USE since it has lower values across all models than other risk measures. It is vital to consider all the available risk measures for a regulator or practitioner to make a good decision since using one can be subjective; as seen in our results, different risk measures yield different results.
Application of long-short term memory Deep Neural Network in financial forecasting
(Strathmore University, 2019) Wanyonyi, Watua Peter
The goal of this research was to apply Long-Short Term Memory Deep Neural Networks in financial forecasting. In order to predict the financial data, we used long-short term model and we compared its performance to ARIMA-GARCH hybrid model. In the study we used the ARIMA-GARCH time series model and studied its limitations in time series forecasting. We then introduced Deep Neural Network model (DNN) so as to improve accuracy which was tested on different financial datasets. Lastly we compared the results of the models employed using the root mean square error (RMSE) and p-value; LSTM had RMSE of 0.09989178 while the ARIMA-GARCH had RMSE of 0.0178. It was then concluded that the long-short term memory (LSTM) model, which is one of the DNN models, had significantly better than the ARIMA-GARCH Hybrid model in prediction/forecasting financial returns on FTSEIOO and S&PSOO indices.
Bad beta, good beta and stochastic volatility in an inter-temporal asset pricing model for the Kenyan stock market
(Strathmore University, 2018) Kimundi, Gillian Nduku
The study seeks to investigate whether bad beta (sensitivity to cash-ow news), good beta (sensitivity to discount rate news) and volatility news are significantly priced in the Kenyan stock market. A comparison of the 3 models is done: 2-beta pricing model (with cash-ow news and discount rate news as risk factors), a 3- beta model (including volatility news) and a 4-beta model (including covariation risk in cash-ow and discount rate news). The findings from the study suggest that news terms related to cash flows, discount rates, volatility and the covariation of cash-ow news and discount rate news are all significantly priced in the Kenyan Market. There is evidence that Kenyan investors are highly risk averse, more so towards cash-ow news, than they are to discount rate news. Similarly, the premium charged for volatility news is just as high as that attached to cash flow news. Investors also attach a significant but relatively smaller premium to the risk due to covariation between cash-ow news and discount rate news.
Calibration of vasicek model in a hidden markov context: the case of Kenya
(Strathmore University, 2017) Chelimo, John Kigen
This dissertation calibrates the Vasicek term-structure model to the evolution of interest rate dynamics in Kenya. This is done for both a single-state and a multi-state model using state estimated under a Hidden Markov Model (HMM). The findings of this paper provide a starting point for the management of the risk posed by interest rate-dependent instruments.The Vasicek model is calibrated using monthly observations of the 91-day treasury bill rate from September 1994 to July 2014 as a proxy for the short rate. Key results show an increase in the mean reversion parameter with an increase in the number of states, suggesting higher stability of states. The volatility is observed to move independently of the level of the interest rate,supporting the idea that risk is not necessarily a function of the level of the interest rate but rather related to the inherent variability of rates in a particular state. Findings from this parameter estimation provide support for interest rate models that incorporate regime switches.
A Comparative modelling of price dynamics of Certified Emission Reductions using diffusion processes: a case study of the European Energy Exchange
(Strathmore University, 2021) Kariuki, Evalin Wanjiru
In this study, the price dynamics of Certified Emission Reductions were forecasted by comparing and acquiring the most consistent and accurate forecast model using the diffusion processes: Geometric Brownian Motion, Vasicek Model and Mean Square Root Process. The assumption of each model with drift and diffusion component were investigated focusing on the Certified Emission Reductions prices traded within European Energy Exchange (EEX) between the years of 2012-2020.The forecasted prices were compared to the actual to evaluate the validity of the models. Based on the research findings, Bayesian information criterion, which determined the goodness of fit to assess the performance of the model with respect to how well it explains the data, shows that vasicek model is the most preferred model among the three models since it has the lowest Bayesian information criterion (BIC). This study compared the accuracy of the models, Geometric Brownian Motion, Vasicek Model and Mean Square Root Process, in terms of forecasting the price dynamics of CERs, and concluded the vasicek most is the most accurate among the three models since the predictive power is high toward a specified time frame proven by the lower value of mean absolute error and Mean Absolute Percentage Error.
A Comparative study on mathematical models for interest rate dynamics: a Kenyan case study
(Strathmore University, 2021) Maina, Hudson Mwangi
This dissertation calibrates equilibrium one-factor short-term interest rate models to the evolution of interest rate dynamics in Kenya. The aim of the study is to find out which one-factor short-rate model best captures the dynamics of the short-term interest rate in Kenya. Additionally, the study aims to evaluate the relationship between conditional volatility of interest rate changes and the level of interest rate. The findings of this study provide a basis for valuation of contingent claims and hedging of interest rate risk. The data used in the study was obtained from the Central Bank of Kenya (CBK) website 1 for the period between January 2005 to July 2016. Since the short-term interest rate is unobservable in the market the 91-day Treasury Bill (TB) rate was used as its proxy. The Generalized Method of Moments (GMM) estimation technique was used to obtain the parameters for all the models under study. Key results showed that there is weak evidence of mean reversion for all the models evaluated. Furthermore, it was established that there exists a positive relationship between interest rate volatility and the level of interest rate. The best performing model from the study is determined to be the Chan, Karolyi, Longstaff and Sanders (CKLS) model which allows the volatility of interest rate changes to be highly dependent on the level of the interest rate. This model also has the best volatility forecasting ability among the models under study. It is therefore recommended to interest rate policy makers for use in their work.
Comparison of survival analysis approaches to modelling credit risks
(Strathmore University, 2019) Mungasi, Sammy Monyoncho
Credit risk is a critical area in finance and has drawn considerable research attention. As such, survival analysis has widely been used in credit risk, in particular, to model debt's time to default mechanisms. In this study, we revisit different survival analysis approaches as applied in credit risk defaulters' data and assess their performance in light of the Kenyan context. In practice, inconsistency in the validity of credit risk models used by many companies when predicting and analysis of loan default is a common phenomenon that occurs unexpectedly. Loan defaults often cause major loses to creditors' and can be of great benefit if quantified correctly in advance by using correct models. Here, we address the unbiasedness, analysis, and comparison of survival analysis approaches, particularly, the models of credit risk. We carry out data analysis using the Cox proportional hazard model and its extensions as well as the mixture cure and non-cure model. We then compare the results systematically by investigating the most efficient awl preferable model that produces best estimates in the Kenyan real data, sets. Results show the Cox Proportional Hazard (Cox PH) model is more efficient in the analysis of Kenyan real data set compared to the frailty, the mixture cure, and non-cure model.
Construction of a Financial Inclusion Index for Kenya
(Strathmore University, 2021) Wafula, Kelly Akuku
This study constructed a Financial Inclusion Index (FII) to measure access to, availability of and usage of financial services in Kenya using data collected from IMF reports for the period 2013 – 2019. A two-stage principal component analysis (PCA) method was applied in constructing the FII that was found to satisfy the Kaiser-Meyer-Olkin (KMO) measure to both the indicators and the dimensions. The research offers ideas for policy-making by highlighting the contributions of the variables to the dimensions, subsequently, the contributions of the dimensions to the index. Therefore, the FII can act as an analytical tool for surveillance of the variables for a more inclusive financial system.
Consumer credit risk modelling using machine learning algorithms: a comparative approach
(Strathmore University, 2019) Nyangena, Brian Okemwa
Consumer credit risk scoring involves the assessment of the risk that is associated with a customer that applies for credit. The ability to confidently identify customers that will repay the credit and those that will not is therefore, an important aspect for any institution. The purpose of this study is to model consumer credit risk using machine learning models and compare the results to the traditional logistic model. The aim is to identify whether there is improved performance in the classification of default among customers when machine learning algorithms are used. Additionally, the study aims to identify how different customer characteristics affects their default experience. The data used was obtained from Kenya Metropol between 2014-2017 and had customer details such as age, loan amount, marital status and sex among others, during this period. 5 models are used to model the default experience namely: Logistic regression, Random Forest, Support Vector Machine, Gradient Boosting and Multi-layer Perceptron Neural Network. The efficiency of the models was assessed using the following metrics; Accuracy, Precision, Recall, F1-score and Precision-Recall curve. Due to the imbalanced nature of credit data set, the F1-score, which is a weighted average of the Precision and Recall, was eventually used as the metric to determine the best model for credit scoring. The findings showed that Random Forest performed the best, having an F1-score of 0.307. The machine learning algorithms outperformed the logistic model and showed an improved performance in the classification of default, especially in identifying false positives. It was also established that male customers had a higher default probability, younger customers were more likely to default and single customers defaulted more than married customers
Day of the week effect on stock returns and volatility in emerging and frontier markets in Africa
(Strathmore University, 2021) Mbonoka, Faith Mbula
The study examines the day of the week effect on average stock returns and compares the daily price volatilities in emerging and frontier markets in Africa. The study focuses on eight key stock markets in Africa’s emerging and frontier markets of Nigeria, Botswana, Egypt, Tunisia, South Africa, Kenya, Mauritius and Morocco for the period May 11th, 2018 to May 12th, 2021. Average stock returns generated from daily closing price indices are regressed against a measure of volatility while controlling for trading day effect. From the findings based on a static panel data regression analysis, there is a Monday effect evident across both the emerging and frontier markets. In addition, there is also evident statistically significant differences in the volatility patterns across the days of the week in the emerging and frontier markets. In particular, Thursday records the highest average volatility and Monday the least in the emerging markets, while Friday and Tuesday show the highest and lowest volatilities in the frontier markets respectively. The research findings contribute to empirical literature on risk-return analyses generating useful insights for investment decision making. Based on these findings, investors will be able to make better investment selections based on return and risk; possibly prompting them to consider day of the week in their trading strategies.
Determination of optimal public debt ceiling for Kenya using stochastic control
(Strathmore University, 2018) Kithinji, Millicent
Public debt is a key economic variable. It is the totality of public and publicly guaranteed debt owed by any level of government to either citizens or foreigners or both. Due to recent debt crises in developed countries such as Portugal, Italy, Ireland, Greece and Spain, debt control has become a key important fiscal policy of every government. In this study, we applied a formula proposed by (Cadenillas and Aguilar, 2015) to find out the optimal public debt ceiling for Kenya. We made modification to subjective variables in the explicit formula and used the formula to find the optimal public debt ceiling for Kenya. We illustrate that it is prudent for that government to use a fiscal policy that maintains the debt ratio under an optimal debt ceiling.
Dynamic portfolio optimization using reinforcement learning
(Strathmore University, 2021) Yegon, Donald Kibet
This study uses machine learning in the development of a dynamic investment strategy for portfolio optimization. We aim to explore the efficiency of this approach over a passively managed portfolio and assess the whether transaction costs erode the gains in the dynamically managed portfolio. To this end we explore the application of recurrent reinforcement learning for optimal asset allocation of a portfolio consisting of stock prices for six companies in different sectors. We develop an environment based on monthly historic prices of these stocks and a re-balancing agent that acts on the environment. The risk and return factors of the individual stock are taken as the state of the environment. Using a modified version of Sterling Ratio as the performance measure, we select model parameters through direct recurrent reinforcement learning from historical data and test the efficacy of the strategy on unseen data. From the analysis we find that the regularly re-balanced portfolio out performs the market portfolio based on buy and hold strategy based on both the terminal wealth and the risk adjusted return measure.
The Effect of the fluctuation of the Chinese Yuan on the returns of stocks traded in the Kenyan, Ugandan and Tanzanian markets
(Strathmore University, 2018) Kibathi, Leonie Njeri
This paper investigated the relationship between Kenya, Tanzanian and Ugandan ex-change rates and the returns of three stocks traded in all three markets. The exchange rates analyzed were from the three countries versus the Chinese Yuan. The data was maintained at weekly intervals and the time period was from January 2012 to December 2017. In this study, both the exchange rate and the stock returns data were found to be non-normally distributed. A unit root test (Augmented Dickey-Fuller) found that both time series were stationary at level form. A test into the causal relationship between the two variables by the Granger Causality test showed that there was a unidirectional relationship between stock returns and the exchange rates that run from the stock returns to the exchange rates. Understanding the flow of influence between exchange rates and stock market returns is essential as the two variables have become important aspects in trading markets. The information about this relationship between stock market returns and exchange rates would help investors to invest prudently by reducing their exposure to risk.
Efficiency of the markov regime switching GARCH Model in modelling volatility for tea prices
(Strathmore University, 2018) Maiyo, Mathew Kiplimo
This study examines the ability of the Markov Regime Switching GARCH model, in comparison with the univariete GARCH models, in modelling and forecasting price volatility of the tea traded at the Mombasa Tea Auction within some time horizon. The study uses weekly data, from 2010 to 2017, to analysis regime switching in volatility and provides an in-sample and out-of-sample forecast. Volatility regime switching is first modelled with a Markov switching framework. In-sample and out-of-sample forecasts of volatility using competing MRS-GARCH models and the single regimes GARCH models are then provided. Comparison of in-sample forecast is done on the basis of goodness-of-fit and the comparison of the out-of-sample forecasts is done on the basis of forecast accuracy, using the statistical loss function. The results show that the MRS-GARCH models can remove the high persistence of GARCH models. This shows the priority of MRS-GARCH models and provides evidence of regime clustering. In out-of-sample forecast perfomance, the MRS-GARCH models were better than the single regime GARCH model. However, this superioirity fades for longer time horizon.
An Empirical evaluation of alternative asset allocation policies for emerging and frontier market investors in Africa
(Strathmore University, 2021) Okwaro, Douglas Job
Despite forming an integral part of literature and practitioner knowledge, Markowitz based optimization has been shown to suffer severe drawback of estimation errors and sensitivity to input parameters when implemented in practice. The best diversification methods from the perspective of a private investor in real-life situations still remains largely unsolved. Most of the potential diversification benefits so far have primarily been analyzed for internationally diversified stock portfolios, with focus on the special viewpoint of U.S investors. Studies have suggested that the Mean-Variance optimization can be robustified by the use of robust covariance estimators other than the sample covariance that relies on the classical Maximum Likelihood Estimator. Using a portfolio formed from 2 Emerging Market and 5 Frontier Market indices in Africa, this study sought to compare the performance of the traditional Mean-Variance model against the performance of the Mean-Variance optimization model robustified with the Orthogonalized Gnanadesikan-Kettenring, Minimum Covariance Determinant, Minimum Volume Ellipsoid and shrink estimators, with an aim of recommending the best model applicable to the African emerging and frontier markets investors. The robustified models were found to indeed have better characteristics in terms of gross returns, annualized returns and net portfolio returns over time compared to the traditional Mean-Variance optimization model.
Empirical performance of alternative risk measures in portfolio selection - the case of South African stock market
(Strathmore University, 2021) Macharia, Richard
Portfolio selection is the process of apportioning capital to a finite number of assets given the wider set of all investment options. The decision of best combination of assets to invest in is the subject of debate among practitioners and researchers alike. Individuals face a multitude of constraints when making allocation decisions thus their patterns of investing are wildly different. However, economists have studied asset price patterns for long enough to be able to pick out aggregate patterns and develop a theory of decision making: Utility Theory.
Estimating Heston stochastic volatility model parameters
(Strathmore University, 2022) Musya, Martin
The Black Scholes model is widely favoured for pricing derivatives such as the European put and call options. This model while having the benefit of ease of application has some restrictive assumptions. First there is the assumption that volatility of asset returns is constant. This assumption is easily violated in the volatility smile that is widely documented in literature as well as observed in option market data, another assumption is that asset returns are normally distributed. The assumption of normal distribution is reasonable for long term horizons but not for shorter horizons. Market are rarely if ever complete. There always exists informational asymmetry where some investors know more about the market than others. It is also well known that a single asset is insufficient to hedge away risk. The Heston Model improves on previous assumption of constant variance (homoskedasticity) by allowing correlation between volatility and the price of the underlying. This research sought to estimate the Heston model parameters over various periods using maximum likelihood method. This was to compare the performance of both the Heston model and the Black Scholes model in periods of market uncertainty and in relatively stable periods where markets are performing well. The three periods, also called epochs, in this study are during the 2008 financial crisis, the Covid-19 pandemic and the relatively stable years between the two periods. The data for this thesis was obtained from the S&P 500 volatility index (VIX). Both the option data and the underlying data was available. A¨ıt-Sahalia (2002) constructed the sequence of approximations to the transition probability for a diffusion process. The first step involved standardizing the diffusion function using the Lamperti transform in order to remove some state dependent term in order to get rid of boundary conditions and correlation structures resulting in simple diffusion terms. This allowed for parameter estimation for the differentiable unit diffusion function. After this, the pseudo normalized” increment for random variable of the diffusion function was obtained and maximum likelihood estimation was done on this transformed variable to get rid of the issue of transition probabilities getting peaked when the step size gets small. The state variables for the Heston diffusion function were those of the underlying process and the diffusion process. The option price and that of the underlying in the data were first organized into a matrix. The determinant of that matrix which was the Jacobian term was then used to estimate the Heston and Black-Scholes parameters. A hypothesis test on the output for the three periods showed that at the 95% significance level the Heston model parameters are significant unlike the Black-Scholes parameters for all periods under study. The Heston model performed better even in times of financial instability such as during the 2008 crisis and the Covid-19 pandemic.
Exotic derivatives pricing using copula-based martingale approach
(Strathmore University, 2018) Muganda, Brian Wesley
This study examines the pricing of bivariate exotic derivatives, namely: capped spread option and bivariate digital options, using martingale approach and pair copulae formulations. Pair copulae is used to capture the joint distribution of asset price process and varying dependence structure rather than the univariate marginal distribution used in pricing univariate options. Unique payoff conditions for these exotic options are developed and the prices of these exotic options are obtained under the best fitting pair copulae. We then assess the sensitivity of the exotic option prices to the copula parameter, by formulating a `dependence delta' and `dependence gamma' formula obtained by application of chain rule de-composition to the copula derivative to have h-function and density function representation. Data from 2012 to 2018 from the NYSE of Equity ETFs and Bond ETFs of Frontier Markets, Emerging Market and Developed Markets to construct 10 pair combination of Equity and Bond ETFs as underlyings for the bivariate exotic options. The findings reveal that the t-copula captures best the dependence between the 10 pair combinations of underlyings. The prices of the bivariate exotic options are affected by the strength of the dependence of underlyings. Emerging and Developed market equity ETFs combination are more sensitive to changes in copula parameter. However, emerging market equity ETF and Developed market bond ETF exhibit lower downside dependence and have lower dependence delta. Dependence gamma is generally of similar strength and signage as the dependence delta.
Frontier stock market linkages: an African perspective
(Strathmore University, 2018) Onyango, Christine Amanda
Volatility modelling in the multivariate case is becoming an important area of study as the world becomes increasingly more integrated and as barriers to entry in frontier markets come down. Understanding how frontier markets in the African region behave in contrast to those in developed markets is vital in driving portfolio allocation decisions as well as regulatory interventions.In this study we investigate the co-movements of the stock indices of four African countries, Nigeria, Morocco, Mauritius and Kenya using various multi-variate volatility models in relation to those of South Africa and the United Kingdom. We also fit a Kalman filter to the data set and examine the goodness of fit of the two approaches. For the Multi-variate models we fit an Exponentially Weighted Moving Average (EWMA) model, two specifications of Dynamic Conditional Correlation (DCC) models as well as a multivariate volatility model based on Cholesky Decomposition. We use a dynamic linear specification of the Kalman filter to allow for time-varying variances, and generate forecasts. Empirical results show that the diagnostic tests with upper tail trimming reject the EWMA model while both specifications of the DCC model as well as a multivariate model based on Cholesky decomposition is found to be adequate. Kalman filters also provide adequate modelling for each return series on the basis of assessment of residuals.

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