IJMwaniki.  2019.  On long-term-memory volatility and asymmetry in TOP40 and NSE20 index log returns. ECONOMICS AND INTERNATIONAL FINANCE. AbstractWebsite

This article investigates the presence of long term memory of returns in south African and Kenyan financial markets over a period 1995-2010. Empirical results indicate significant presence of linear autocorrelation of order three for NSE20 index and autocorrelation of order one for TOP40 index. There is strong evidence of changing variance for both indices in addition to more autocorrelation between absolute returns for both markets. Theoretical autocorrelation function is fitted and parameters estimated. Different ARCH type models are conditioned on normal distribution and A-PGARCH model based on absolute daily returns seems to significantly outperform four other models (TGARCH, GARCH, GARCH-M and GJR-GARCH) in modeling the changing variance and volatility asymmetry in the two emerging markets.

IJMwaniki.  2019.  Geometric Brownian Motion assumption and the generalized hyperbolic distribution on modeling returns. Journal of advances in applied mathematics. 4 (3):103-111. AbstractWebsite

Generalized hyperbolic distribution and some of its subclasses like normal, hyperbolic and variance gamma distributions are used to fit daily log returns of eight listed companies in Nairobi Securities Exchange and Montréal Exchange. EM-based maximum likelihood estimation procedure is used to estimate parameters of the model. Kernel densities and empirical distribution of data are compared. The goodness of fit statistics of proposed distributions are used to measure how well model fits the data. Empirical results show that Generalized hyperbolic Distribution seems to improve partially, the geometric Brownian assumption on modeling returns of the underlying process, both in a developed and emerging market. Both markets seem to have different stochastic time

IJMwaniki.  2019.  Modeling heteroscedastic, skewed and leptokurtic returns in discrete time. Journal of Applied Finance & Banking. 9(5):1-14. AbstractWebsite

Popular models of finance, fall short of accounting for most empirically found stylized features of financial time series data, such as volatility clustering, skewness and leptokurtic nature of log returns. In this study, we propose a general framework for modeling asset returns which account for serial dependencies in higher moments and leptokurtic nature of scaled GARCH filtered residuals. Such residuals are calibrated to normal inverse Gaussian and hyperbolic distribution. Dynamics of risky assets assumed in Black Scholes model, Duans GARCH model and other benchmark models for contract valuation, are shown to be nested in the the proposed framework

IJMwaniki.  2019.  On Heteroscedastic, Skewed and Leptokurtic Log Returns and Spectral Density of Standardized Residuals. Journal of Advances in Economics and Finance. 4 (May 2019):79-90. AbstractWebsite

A search for a distribution which adequately describes the dynamics of log returns has been a subject of study for many years. Empirical evidence has resulted in stylized facts of returns. Arguably, in this study, the three components of returns, mean equation part, the changing variance. part and the resulting residuals are determined and their corresponding parameters estimated within the proposed framework. Spectral density analysis is used to trace the seasonality component.
inherent in the standardized residuals. Empirical data sets from eight different indexes and common
stock are applied to the model, and results tabulated in support of the resulting framework.


Njeru, K, NA K, IJMwaniki.  2018.  Forecasting future customer call volumes: A case study. International Journal on Future Revolution in Computer Science & Communication Engineering. 4 (6):12-16. AbstractWebsite

Forecasting future volumes of customer calls in call centers has proved to be a tedious and challenging task. This study, using time series analysis proposes two adequate ARIMA (p, d, q) models that are suitable to forecast two volumes of customer calls, IVR Hits Volumes and Offered Call volumes. 1472 times series data points from date 01/01/2014 to 11/01/2018 were obtained from a call center based in Kenya on the two variables of interest (IVR Hits Volumes and Offered Call volumes). The appropriate orders of the two models are picked based on the examination of the results of the ACF and PACF plots. The AIC criterion is used to select the best model for the data. The best ARIMA model for log IVR Hits volumes is ARIMA (5, 1, 3) with and the best ARIMA model for log Offered Call Volumes is ARIMA (6, 1, 3) with. The two models are recommended to model and forecast the daily arrival volumes of customer call data. The obtained forecast will be used in providing insights for appropriate workforce management

Metto, MK'etich, Anthony K, Ivivi M.  2018.  Influence of Teacher Qualification on Students’ Performance in Mathematics in the Uasin Gishu County. THE INTERNATIONAL JOURNAL OF HUMANITIES & SOCIAL STUDIES. 6(5):1-6. AbstractWebsite

Mathematics is a very crucial subject in the school curriculum especially being a compulsory subject since the adoption of 8-4-4 system in Kenya. Mathematics is rated a very important subject because it makes a man methodical or systematic which depends on teacher’s qualification. The study sought to identify the effect of teacher qualification on students’ performance in Mathematics in the Uasin Gishu County. The study used descriptive survey design. The target population for this study consists of teachers of mathematics of two hundred and sixteen secondary schools in Uasin Gishu County. A sample of forty two secondary schools was purposively selected from the two hundred and sixteen schools. Simple random sampling technique was used to select 126 teachers. Primary data collected using questionnaires. Data collected was analyzed using inferential statistics using the Statistical Package for Social Sciences (SPSS). There was a positive significant relationship between teacher’s qualification and mathematics performance. From the linear regression model, (R2 = .092) showing that teachers qualification account for 9.2% variation in mathematics performance. The teacher’s qualification positively influenced the mathematics performance in secondary school. The Ministry of Education should offer an opportunity for further teacher professional training through workshops, seminars and short courses


IJMwaniki.  2017.  On skewed leptokurtic returns and pentanomial lattice option valuation via minimal entropy martingale measure. Cogent Economics & Finance. 5(1):1-16. AbstractWebsite

This article develops, a lattice-based approach for pricing contingent claims when parameters governing the logs of the underlying asset dynamics are modelled by generalized hyperbolic distribution and normal inverse Gaussian distribution. The pentanomial lattice is constructed using a moment matching procedure. Moment generating functions of generalized hyperbolic distribution and normal inverse Gaussian distribution are utilized to compute probabilities and jump parameters under historical measure P. Minimal entropy martingale measure (MEMM) is used to value European call option with a view of comparing the results with some of the existing benchmark models such as Black Scholes model. Empirical data from S&P500 index, RUTSELL2000 index and RUI1000 index are used to demonstrate how the model works. There is a significant difference especially for long term maturity (six months and above) type of contracts, the proposed model outperform the benchmark model, while performing poorly at short term contracts. Pentanomial NIG models seem to outperform the other models, especially for long-dated maturities.

Mwambora, SK, IJMwaniki.  2017.  Estimation of Waiting Times for the Three Transient States of HIV Infection in Kenya. International Journal of Mathematics and Physical Sciences Research. 5(1):73-76. AbstractWebsite

The methods that were employed in this project analyzed HIV data. The aim was to evaluate the evolution of HIV positive patients to bring out some significant factors associated with this pathology. Many clinical situations can be described in terms of the conditions that individuals can be in (states), how they can move among such states (transitions), and how likely such moves are (transition probabilities). State transition models were, therefore, best suited to analyze this decision problem. Transition probabilities from states 1, 2, and 3 into state 4 increased as time progressed. The estimated total length of stay in state 1 was longer than state 2 and 3 respectively.


RJ, S, IJMwaniki.  2016.  Mixed Poisson distribution in explicit form and their properties. The 3rd Eastern Africa universities mathematics programme (EAUMP) conference. Advances of Mathematics and its applications. . , Makere university Kampala Uganda
Okeyo, J, IJMwaniki, Ngare P.  2016.  Modelling Inflation Rate Volatility in Kenya Using Arch-Type Model Family. Research journal of finance and accounting.. 7(23):10-17. AbstractWebsite

This paper describe the empirical study based on financial time series modelling with special application to modelling inflation data for Kenya. Specifically the theory of time series is modelled and applied to the inflation data spanning from January 1985 to April 2016 obtained from the Kenya National Bureau of Statistics. Three Autoregressive Conditional Heteroscedastic (ARCH) family type models (traditional ARCH, Generalized ARCH (GARCH), GJR GARCH and the Exponential GARCH (EGARCH)) models were fitted and forecast to the data. This was principally because the data were characterized by changing mean and variance. The outcome of the study revealed that the ARCH –family type models, particularly, the EGARCH (1, 1) with generalized error distribution (GED) was the best in modelling and forecasting Kenya’s monthly rates of inflation. The study recommends that governments, policy makers interested in modelling and forecasting monthly rates of inflation should take into consideration Heteroscedastic models since it captures the volatilities in the monthly rates of inflation

Ntwiga, Pweke, MManene, IJMwaniki.  2016.  MODELING TRUST IN SOCIAL NETWORK. International Journal of mathematical archive. 7(2):64-68. AbstractWebsite

We rely on trust in our day to day interactions and activities with each other. It is not easy to estimate it but we offer a
simple and powerful method for estimating trust levels of agents in a social network using data from the agents’
reputation matrix. The reputation resultant method (RRM) is based on the mean values of the reputation rating matrix
and the reputation resultant matrix. Reputation ratings are derived from the agents’ peer to peer ratings and the
resultant reputation data is the relative reputation ratings by the agents. A comparison is made between the results of
Singular value decomposition (SVD) and our new method, the RRM. The two methods offer results that are highly
comparative with the RRM being simple, powerful and easy to understand and implement.

Aduda, J, Weke P, Ngare P, Mwaniki J.  2016.  Financial time series modeling of trends and patterns in the energy markets. Journal of Mathematical Finance. 7(2):64-68. AbstractWebsite

Precise recognition of a time series path is important to policymakers, statisticians, economists, traders, hedgers and speculators alike. The correct time series path is also a key ingredient in pricing models. This study uses daily futures prices of crude oil and other distillate fuels. This paper considers the statistical properties of energy futures and spot prices and investigates the trends that underlie the price dynamics in order to gain further insights into possible nuances of price discovery and energy market dynamics. The family of ARMA-GARCH models was explored.
The trends depict time-varying variability and persistence of oil price shocks. The return series conform to a constant mean model with GARCH variance.

Kalovwe, SK, Mwaniki IJ.  2016.  Modeling Stock Returns Volatility of the Nairobi Securities Exchange Index and Other Indices. Journal of Advanced Statistics. 1(2):87-93. AbstractWebsite

This paper seeks to model daily, weekly and monthly stock indices returns using GARCH
(1,1) model which is expected to reproduce most of the stylized facts of financial time series data which,
in most cases, are found in different types of market. In addition, the distributional behavior of returns
as the data changes from daily through to monthly returns is investigated by performing the JB and
K-S tests. The results indicate evidence of volatility clustering, leverage effects, Gaussianity and
leptokurtic distribution in the stock returns. A key observation is that the monthly returns of the three
indices follow a Gaussian distribution (i.e. as the data changes from daily through to monthly returns
it follows a normal distribution)

Ntwiga, DB, Weke P, Manene M, Maniki I.  2016.  Trust and Distrust: A Reputation Ratings Approach. International Advanced Research Journal in Science, Engineering and Technology. 3(2):111-114. AbstractWebsite

Agents’ reputation ratings in a social network form a real-valued matrix which is discounted with singular
value decomposition (SVD) to estimate the trust and distrust levels of agents. SVD eliminates noise as future expected
trust and distrust are based on current reputation ratings. A discounting of 20 percent is optimal, further discounting
does not improve error reduction. Reputation and trust are closely related. Distrust is different from trust and
reputation. Distrust is similar to trust negation, and trust is similar to distrust negation


Ssebugenyi, CS, Mwaniki IJ, Konlack VS.  2012.  On the Minimal Entropy Martingale Measure and Multinomial Lattices with Cumulants. : Taylor & Francis Abstract

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