PhD

PhD Actuarial Science

enaidoo — Fri, 11 Sep 2020 20:18:26 +0000

PhD Actuarial Science enaidoo Fri, 09/11/2020 - 20:18

Overview

Graduates’ skills can be applied in industry as Actuarial, Statistical and Financial Analysts, just to mention a few. Such employment opportunities of the graduates of this programme will help in the design and operation of viable National Health Insurance and Pension Schemes. The financial industry of Ghana will also benefit from the skills necessary for company strategic development and planning, financial product development, pricing and valuation of a wide variety of products in various sectors of the economy. The programme prepares for an academic career in research and teaching.

Aims and Objectives

The PhD programme in Actuarial Science seeks to develop individuals with a balance in mathematical, statistical, financial, and economic theories, and their applications to practical problems. The programme is designed to provide theoretical as well as applicable education in quantitative aspects of Risk Modelling and Management, Finance and Statistics. Graduates will be capable of abstracting mathematical models for real-world problems and applying appropriate computer-based solutions to them. Graduates would also be prepared for an academic career in research and teaching.

Content of Courses for each Semester

YEAR 1 SEMESTER 1

ACTS 761: FINANCIAL MATHEMATICS (3, 0, 3)

Introduction to the mathematical theory of interest as well as the elements of economic and financial theory of interest. Topics include rates of interest and discount; simple, compound, real, nominal, effective, dollar (time)-weighted; present, current, future value; discount function; annuities; stocks and other financial instruments; definitions of key terms of modern financial analysis; yield curves; spot (forward) rates; duration; immunization; and short sales. The course will cover determining equivalent measures of interest; discounting; accumulating; determining yield rates; and amortization. Derivative pricing, fixed asset pricing, Neural networks, Valuing by expected utility.

ACTS 763: ACTUARIAL MATHEMATICS I (3, 0, 3)

ACTS 765: ACTUARIAL STATISTICS (3, 0, 3)

This course covers the Statistical Methods that provide a further grounding in mathematical and statistical techniques of particular relevance to financial work. Topics include; Bayesian Statistics: Bayesian Theorem. Prior and Posterior distributions: Determining the posterior decisions, Continuous prior distributions, Conjugate prior, improper prior distributions. The loss function: Quadratic loss, Absolute error loss, All- or-nothing loss, Some Bayesian posterior distributions. Loss Distributions: The exponential, gamma, normal, Pareto and generalised Pareto, lognormal, Weibull and Burr distributions. Estimation: The method of Moments and the maximum likelihood estimation of the exponential and gamma, the normal distribution, the Pareto and generalised Pareto, the lognormal and Weibull and Burr distributions. Goodness – of – fit test. Mixture distributions. Reinsurance: Proportional and Non – proportional reinsurance. Reinsurance arrangement: Excess of loss reinsurance – the insurer and the reinsurer. Proportional reinsurance. Particular distributions: Lognormal and normal distribution. Inflation. Estimations. Policy excess. Credibility theory: Credibility premium formula, the credibility factor. Bayesian credibility: Prior parameter distribution, likelihood function, Posterior distribution, loss function. The Poisson/gamma model with numerical illustration. The normal/normal model. The Bayesian approach to credibility. Empirical Bayes credibility theory: Model 1 and 2. Risk Model: Basic Risk Model. Collective Risk Model: Distribution functions and Convolutions. Compound Poisson, binomial and negative binomial distributions. Excess of loss insurance. Individual risk model. Ruin Theory: Surplus process, probability of ruin in continuous and discrete time process. Poisson and Compound Poisson. Probability of ruin in short term. The adjustment coefficient and Lundberg’s inequality and its application.

ACTS 767: SURVIVAL AND STOCHASTIC MODELS (3, 0, 3)

This course covers stochastic processes and survival models and their application. Topics include: Principles of actuarial modelling. Stochastic processes: Markov Chains, The two – state Markov Model, Time – homogeneous jump processes, Time – inhomogeneous Markov jump processes. Survival Models: Estimating the lifetime distributions, Proportional hazards models. The Binomial and Poisson models. Exposed to risk. Graduation and statistical tests. Methods of graduation. Prerequisite: Calculus and probability

MATH 761: MEASURE THEORY AND INTEGRATION (4, 0, 4)

In this course we will develop a proper understanding of measurable functions, measures and the Lebesgue integral. Given these concepts we will consider various concepts of convergence of measurable functions and the convergence of the corresponding integrals, changes of measures and spaces of integrable functions. A special attention will be paid to applications of Measure Theory in the Probability Theory. First, we will develop a proper understanding of probability spaces for random variables and their finite and infinite sequences. Using these concepts, we will discuss Strong Laws of Large Numbers and their applications. Changes of measures and the Radon-Nikodym Theorem will be applied to introduce a general definition of conditional expectation and to study their properties. Then we will apply this machinery to study Gaussian systems and we will introduce the so-called chaotic decompositions which provide an important tool for the Malliavin Calculus, Finance and Physics. Finally, we will introduce the weak convergence of measure, characteristic functions. We will use this theory to derive the Central Limit Theorem and we will discuss some of its applications.

YEAR 1 SEMESTER 2

ACTS 762: FINANCIAL ECONOMICS (4, 0, 4)

The course covers asset-liability models and to how value financial derivatives. These skills are also required to communicate with other financial professionals and to critically evaluate modern financial theories. Topics include: The efficient markets hypothesis, Utility theory and stochastic dominance. Measurement of investment risk. Portfolio theory. Models of asset returns. Asset pricing Models. Brownian motion and Martingales. Stochastic calculus and Ito processes. Interest rate models: Vasicek and Cox-Ingersoll-Ross bond price models, Black-Derman-Toy model binomial model matching in a given time zero yield curve and a set of volatilities: Rational Valuation of Derivative securities: Use put-call parity to determine the relationship between prices of European put and call options and to identify arbitrage opportunities. Computation of European and American options using Binomial and Black-Scholes option-pricing models, Calculation and interpretation of option Greeks, Cash flow characteristic of exotic options; Asian, Barrier, Compound, Gap and Exchange, Stock prices and Diffusion Process, Ito’s Lemma in one dimensional case and Option pricing concepts to Actuarial problems such as equity-linked insurance, Risk Management Techniques: Control of risk using the method of delta-hedging. Credit Risk. Asset Liability Management (ALM) for actuaries. Topics to be considered are Investment and valuation, general principles of asset allocation, Investment risk, Portfolio selection techniques and Investment modeling, asset and liability modeling.

ACTS 764: ACTUARIAL MATHEMATICS II (3, 0, 3)

This course covers the mathematical and probabilistic structure of life contingent ﬁnancial instruments. It introduces survival models, covers life tables and their applications, life insurance, beneﬁts, lifetime annuities. Topics include; Probability Review, Survival Distributions: Probability functions, force mortality, mortality laws, moments, percentiles and recursions, fractional ages, selected mortality. Insurance: Continuous – Moments, annual and m-thly, moments, probabilities and percentiles, recursive formulas, varying insurances, relationships between Insurance payable at moment of death and payable at end of Year. Annuities: Continuous, Expectation, annual and m-thly, variance, probabilities and percentiles, varying annuities and recursive formulas, mothly payments. Premiums: Net premiums for fully Continuous Insurances, net premiums for discrete insurances calculated from life tables, net premiums for discrete insurances calculated from formulas, net premiums paid on an m-thly basis, gross premiums, variance of future loss (Continuous), variance of future loss (discrete), probabilities and percentiles of future loss. Reserves: prospective benefit reserve, Decrements modeling and their applications to insurance and annuities, non-stochastic interest rate models to calculate present values and annuities. Models for cash flows and non-interest sensitive insurances other than traditional life insurances and annuities. Models for contract cash flows for basic universal life insurances. Models for cash flows of basic universal life insurance. Calculate the benefit reserve. Models that consider expense cash flows. Calculate an expense factor using the appropriate exposure. Calculate probabilities and moments of the present-value-of-expenses random variable based on single decrement on single life model and multiple decrements on a single life models. Modeling of expense reserve. Calculate a gross premium given expenses and benefits based on: the equivalence principle; and a return on gross profits basis. Modelling gross premium reserve. Modeling of asset share. Severity Models: compute the basic distributional quantities such as moments, percentiles, generating functions: Frequency Models, Aggregate Models: Compute relevant parameters and statistics for collective risk models, evaluate compound models for aggregate claims, compute aggregate claims distributions.

ACTS 766: PENSION MATHEMATICS (3, 0, 3)

This course focuses on fundamental issues of pension mathematics. The course content focuses on pensions system in Ghana, pension mathematics and investment of pension fund. Topics include: Introduction of pension systems in Ghana and pension plan benefits. Objectives of pension mathematics and fundamental structure. Actuarial assumptions which include decrement assumptions, salary assumptions and interest rate assumptions. General theory for funding method. Funding method used for actual pension management. Practical actuarial valuation to check appropriateness of actuarial assumptions.

ACTS 768: Statistical Modelling (3, 0, 3)

This course covers the Statistical Methods that provide a further grounding in mathematical and statistical techniques of particular relevance to financial work. Topics include; Decision Theory: zero – sum two player games-domination, the minimax criterion-saddle points, and Randomized strategies. Statistical games. Decision criteria: the minimax criterion, the Bayes criterion. Generalized linear Models: Exponential families: normal, Poisson and binomial distributions. Link functions and linear predictors, Deviance of model fitting. Run – off triangles: Estimating future claims. Projection using development factors. Chain – ladder method. The inflation adjusted chain ladder. The Bornhuetter – Fergusson Method. Time series. Monte Carlo simulation.

Programme Type

PhD

PhD Statistics

enaidoo — Fri, 11 Sep 2020 20:04:04 +0000

PhD Statistics enaidoo Fri, 09/11/2020 - 20:04

Overview

Graduate study in Statistics is very essential because it leads to some specific business positions including portfolio analysis, design studies, statistical analysis, computer simulation and software design (Data Analytics), testing, and other areas of operations research. Many laboratories, both government and private, maintain independent research staffs that include statisticians. Their work often deals with the development of new technology, including design and analysis of experiments, software development, and numerical simulation, such as weather and climate forecasting, which depends heavily on the use of supercomputers. The PhD programme in Statistics prepares students for managerial positions and also research work, leading to peer-reviewed publications as well as employment opportunity in research institutions as faculty. The diversity of applications of Statistics is an exciting and challenging, which is one reason why the demand for well-trained statisticians continues to be strong.

The programme emphasizes on the teaching of theory and principles of mathematics, statistical theory and methodology, and applications to provide the basis for meaningful practical applications. Option I (Mathematical Statistics) requires candidates to undertake research involving rigorous mathematical and statistical theories to promote knowledge in the area of Mathematical Statistics while Option II (Applied Statistics) will require candidates to develop various innovative techniques to solve real-life problems. This is in line with the Department of Mathematics’ graduate training of students to formulate abstract mathematical models for real-world problems and also design and apply appropriate computer-based solutions to real-world problems.

Aims and Objectives

The aim of the PhD programme is to produce high calibre graduates with rigorous research and analytical skills, who are well-equipped to go onto postdoctoral research, or employment in industry and private/public service that require the application of high level Statistical concept to solve problems. Graduates of the programme will:

Be competent in mainstream advanced statistical theory and modelling;
Be exposed to modern developments in Statistics;
Have the ability to design and conduct research in academic/industrial settings;
Have the ability to serve as better bridges between the academic and corporate worlds;
Have an appreciation and necessity to enable them undertake postdoctoral research in Statistics.

Content of courses for each semester

YEAR ONE SEMESTER ONE

STAT 751: Probability and Measure Theory (3, 0, 3)

The topics include: Discrete and Continuous random variables and their probability distributions; Construction of Lebesque measure on R: extension of the length of an interval; Extension of parameters (from rings of subset to algebra, uniqueness of extension, Caratheodory method using outer measure of general sets); Measure spaces and measurable functions (Definition, vector lattice properties of the space of measurable, real-valued functions on a measure space); Integration (Definition of integralbility, integration of real-valued, measurable functions defined on general space, momnotone convergence theorem, dominated convergence theorem, Fatou’s Lemma).

STAT 753: Advanced Epidemiology (2, 2, 3)

Topics include: causal inference, missing data, directed acyclic graphs, effect modification, measurement error, validity and reliability, study design, confounding and bias, diagnostic testing. The methods will be illustrated through studies of the epidemiology of both infectious and non-infectious diseases.

STAT 755: Advanced Statistical Quality Control (2, 2, 3)

Topics include: the quality of limits for process behaviour charts, autocorrelated data and process behaviour charts, degrees of freedom for process behaviour charts, process behaviour charts and chaos theory, power function for control charts, Cusum and EWMA techniques, the role of the normal distribution, the central limit theorem, precontrol and charts, the analysis of mean, manufacturing specification setting, using small amount of data for limits. Statistical software will be used to apply the techniques to real-life case studies from manufacturing and service industries.

STAT 757: Advanced Stochastic Processes (2, 2, 3)

Topics include: Random Process, Spectral representation of random processes, Poisson process, birth-death process, and renewal process, Discrete-time Markov chains, Semi-Markov processes and continuous-time Markov chains, Hidden Markov models, Filtering and prediction of random processes, Queueing and loss models.

STAT 759: Algorithm for Data Science (2, 2, 3)

Methods for organizing data (hashing, trees, queues, lists, priority queues). Streaming algorithms for computing statistics on the data. Sorting and searching. Graph models and algorithms for searching (shortest paths, and matching). Neural networks (DNNs, CNNs, and RNNs), with Tensor Flow 2.0. Dynamic programming. Linear and convex programming. Floating point arithmetic, stability of numerical algorithms, Eigenvalues, singular values, PCA, gradient descent, stochastic gradient descent, and block coordinate descent. Conjugate gradient, Newton and quasi-Newton methods. Large scale applications from signal processing, collaborative filtering, recommendations systems, etc.

STAT 761: Advanced Categorical Data Analysis (2, 2, 3)

Topics include: Linear mixed model, estimation in Gaussian mixed model, model diagnostics and variable selection, generalized linear mixed model (GLMM) for binary outcome, GLMM for multi-category nominal outcome, for ordinal outcome, GLMM for counts, likelihood based inference, generalized estimating equation, generalized least squares, GLMM diagnostics and variable selection. Statistical software will be used to apply the techniques to real-life data.

STAT 763: Advanced Statistical Inference (3, 0, 3)

Topics include: Review of likelihood functions, maximum likelihood of functions of the parameter, Wald test and confidence intervals, likelihood ratio test and confidence intervals, algorithms for maximising the likelihood (Newton-Raphson, Expectation-maximization), score function and Fisher information, quasi-likelihood, generalized estimating equations, generalised least squares, weighted least squares, model selection using likelihood.

STAT 765: Advanced Econometrics (2, 2, 3)

Topics include: Review of ordinary least squares, method of moments, maximum likelihood, heteroscedasticity and autocorrelation. Lagged dependent variable model, econometrics of panel data, generalized least squares, distributed lag model, vector error correction model. Statistical software will be used to apply the techniques to real-life data.

STAT 767: Computational Statistics and Data Science (2, 2, 3)

Topics include: Data science concepts and processes, data wrangling, data visualizations, Web analytics, predictive modelling and assessment techniques, kernel and local polynomial nonparametric regression, basis expansion and spline regression, generalized additive models, classification and regression tree, bootstrap resampling and inference, cross-validation.

YEAR ONE SEMESTER TWO

STAT 752: Advanced Survival Analysis (2, 2, 3)

Topics include: Review of survival analysis, classical survival analysis, Cox proportional model for time-dependent covariates, frailty models, cure models, Poisson models, competing risks survival analysis, multivariate survival data, joint modelling.

STAT 754: Advanced Bayesian Statistics (2, 2, 3)

Topics include: Review of modelling in classical approach, Bayesian linear models, Bayesian generalized linear models for binary, count and ordinal data, Bayesian linear mixed models and Survival models, Bayesian parameter estimation and uncertainty, model validation and variable selection. Statistical software will be used to apply the techniques to real-life data.

STAT 756: Advanced Time Series Analysis (2, 2, 3)

Topics include: Review of Nonstationary time series models, nonlinear time series models (TAR, SETAR, STAR, models), conditional volatility models (ARCH, GARCH, models), stochastic volatility models, state-space models. Model building, estimation, model validation and forecasting. Multivariate time series models and cointegration techniques. Statistical software will be used to apply the techniques to real-life data.

STAT 758: Advanced Spatial Statistics (2, 2, 3)

Topics include: spatial data manipulation and mapping, spatial descriptive statistics, spatial regression model for heterogeneous categorical outcome, spatial principal component analysis, point pattern analysis, spatial interpolation using Geostatistics. Statistical software R will be used to apply the techniques to real-life data.

STAT 760: Advanced Multivariate Analysis (2, 2, 3)

Topics include: Review of multivariate normal distribution, bivariate linear regression model, bivariate logistic regression model, extension to multivariate linear regression, multivariate logistic regression model, exploratory factor analysis, path analysis, structural equation modelling, repeated measures. Statistical software will be used to apply the techniques to real-life data.

STAT 762: Advanced Sampling Methods (2, 2, 3)

Topics include: Review of probability and non-probability sampling techniques, unequal probability sampling, two-stage sampling, two-dimensional sampling, capture-recapture sampling, randomized response sampling, nonresponse correction techniques, weighting adjustment.

STAT 764: Advanced Experimental Design and Analysis (2, 2, 3)

Topics include: Review of design of experiment, sample size planning, Statistical power, within-subject designs, mixed model, categorical outcome, nested designs, partially nested designs, repeated measures and related designs, two-level factorial and fractional factorial design. Multi-level design and analysis of variance, missing data. Statistical software will be used to apply the techniques to real-life data.

STAT 766: Artificial Neural Networks (2, 2, 3)

Introduction to artificial neural networks: Biological neural networks, Pattern analysis tasks: Classification, Regression, Clustering, Computational models of neurons, Structures of neural networks, Learning principles. Linear models for regression and classification: Polynomial curve fitting, Bayesian curve fitting, Linear basis function models, Bias-variance decomposition, Bayesian linear regression, Least squares for classification, Logistic regression for classification, Bayesian logistic regression for classification. Feed-forward neural networks: Pattern classification using perceptron, Multilayer feed-forward neural networks (MLFFNNs), Pattern classification and regression using MLFFNNs, Error back propagation learning, Fast learning methods: Conjugate gradient method, Auto-associative neural networks, Bayesian neural networks. Radial basis function networks: Regularization theory, RBF networks for function approximation, RBF networks for pattern classification. Kernel methods for pattern analysis: Statistical learning theory, Support vector machines for pattern classification, Support vector regression for function approximation, Relevance vector machines for classification and regression. Self-organizing maps: Pattern clustering, Topological mapping, Kohonen’s self-organizing map. Feedback neural networks: Pattern storage and retrieval, Hopfield model, Boltzmann machine, Recurrent neural networks.

STAT 768: Advanced Statistical Computing (2, 2, 3)

Topics include: Programming in R, writing functions and scripts, optimizing functions, creating R packages with documentations, debugging, generating pseudo random numbers, Monte Carlo simulation in parameter inference, Bootstrapping techniques, Bayesian methods and Markov chain Monte Carlo Simulation.

YEAR TWO SEMESTER ONE

STAT 851: Thesis Work and Report Writing I (0, 15, 0)

Research proposal writing and presentation.

STAT 853: Seminar I (0, 15, 0)

This is the first of four seminars organized in the Department. All students in the programme are expected to attend all seminars. Each student is expected to make his/her own presentation on a project proposal. The topic must relate to statistical issues including insurance, medicine, mortality and mobility, health outcomes, economics, policy, pension, social phenomena, mathematical finance, statistics, and other related fields with particular reference to the advancement of the statistics profession.

STAT 855 Capstone (2, 2, 3)

The capstone project will be an analysis using any statistical software tool that answers a specific scientific/business question: (1) A large and complex dataset will be provided to learners and the analysis will require the application of a variety of methods and techniques introduced in previous courses, including Computational Statistics I & II, statistical modelling as well as interpretations of these results in the context of the data and the research question. Report writing on a project to include the following sections: Motivation, problem definition, and existing approaches, Proposed solution and details of implementation, Results, conclusion, and directions for future work

YEAR TWO SEMESTER TWO

STAT 852: Thesis Work and Report Writing II (0, 15, 0)

Thesis report writing and presentation.

STAT 854: Seminar II (0, 15, 0)

This is the second in the sequel of seminar presentations. All students in the programme are expected to attend all seminars. Each student is expected to make his/her own presentation on the experiential research learning progress made on his/her research.

YEAR THREE SEMESTER ONE

STAT 951: Thesis Work and Report Writing III (0, 15, 0)

Thesis report writing and presentation.

STAT 953: Seminar III (0, 15, 0)

This is the third in the sequel of seminar presentations. All students in the programme are expected to attend all seminars. Each student is expected to make his/her own presentation on the experiential research learning progress made on his/her research.

YEAR THREE SEMESTER TWO

STAT 952: Thesis Work and Report Writing IV (0, 15, 0)

Thesis report writing and presentation.

STAT 954: Seminar IV (0, 15, 0)

This is the fourth in the sequel of seminar presentations. All students in the programme are expected to attend all seminars. Each student is expected to make his/her own presentation to discuss the findings of his/her research.

YEAR FOUR SEMESTER ONE

STAT 1051: Thesis Work and Report Writing V (0, 15, 0)

Thesis report writing and presentation.

STAT 1053: Seminar V (0, 15, 0)

This is the fourth and final in the sequel of seminar presentations. All students in the programme are expected to attend all seminars. Each student is expected to make his/her own presentation to discuss the findings of his/her research.

YEAR FOUR SEMESTER TWO

STAT 1052: Final Thesis Report (0, 15, 0)

The research is undertaken in either an applied area or theoretical development of statistical methods, after presentation of the proposal as specified in STAT 851. The final write-up of the thesis should be submitted by the end of the fourth academic year of study.

STAT 1054: Final Seminar (0, 15, 0)

Presentation of the final thesis (Viva voce)

Programme Type

PhD