# PhD Actuarial Science

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)**

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, m-thly 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.

**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**.