Mathematics and Accounting and Finance - BSc (Hons)

This is an introductory module to introduce students to the role and evolution of accounting

Topics to be covered may include: single entry accounting; double entry bookkeeping; financial reporting conventions; recording transactions and adjusting entries; principal financial statements; institutional requirements; auditing; monetary items; purchases and sales; bad and doubtful debts; inventory valuation; non-current assets and depreciation methods; liabilities; sole traders and clubs, partnerships, companies; capital structures; cash flow statements; interpretation of accounts through ratio analysis; problems of, and alternatives to, historical cost accounting.

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This module introduces students to economics in its two main components, microeconomics and macroeconomics. The module is designed to explain the main ways in which economists think about economic problems faced by individuals, firms, markets and governments.

The first part of the module focuses on explaining a selection of microeconomic topics including, the behaviour of individuals and firms; demand and supply of goods and services and determination of prices; costs in the short and long term and market structures. The second part aims to introduce the core of macroeconomic topics; for instance, macroeconomic objectives and trade-offs; unemployment; inflation; international trade; balance of payments and exchange rates; and the main types of economic policies that are implemented by governments. Overall, the application of economics to contemporary issues illustrates how economic analysis can be used to understand the different parts of the economy and to inform and evaluate policy interventions that support a range of different economic outcomes.

The module is self-contained to provide a basic understanding of economic concepts and debates. It is a suitable module for students interested in taking economics further, either as part of another degree programme or as part of a future professional qualification.

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This module serves as an introduction to algebraic methods and linear algebra methods. These are central in modern mathematics, having found applications in many other sciences and also in our everyday life.

Indicative module content:

Basic set theory, Functions and Relations, Systems of linear equations and Gaussian elimination, Matrices and Determinants, Vector spaces and Linear Transformations, Diagonalisation, Orthogonality.

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This module introduces widely-used mathematical methods for functions of a single variable. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage.

Basic notation for sets and number systems including complex numbers (a+ib representation only). Standard functions: trig functions, polynomials, rational functions, exponentials and logarithms.

Single variable calculus: Differentiation, including product and chain rules; Fundamental Theorem of Calculus (statement only), elementary integrals, change of variables, integration by parts, differentiation of integrals with variable limits.

Curve sketching: graphs of elementary functions, maxima, minima and points of inflection, asymptotes.

Algebra of matrices and vectors; addition, multiplication, transposes, inner-products.

Row reduced echelon form, solving linear systems (homogeneous and inhomogeneous).

Inverse of a matrix.

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This module introduces widely-used mathematical methods for vectors and functions of two or more variables. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Vectors: Cartesian coordinates; vector algebra; scalar, vector and triple products (and geometric interpretation); straight lines and planes expressed as vector equations; parametrized curves; differentiation of vector-valued functions of a scalar variable; tangent vectors; vector fields (with everyday examples)

Partial differentiation: Functions of two variables; partial differentiation (including the chain rule and change of variables); maxima, minima and saddle points; Lagrange multipliers

Integration in two dimensions: Double integrals in Cartesian coordinates; plane polar coordinates; change of variables for double integrals; line integrals; Green's theorem (statement – justification on rectangular domains only).

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Introduction to Probability. Concepts of events and sample space. Set theoretic description of probability, axioms of probability, interpretations of probability (objective and subjective probability).

Theory for unstructured sample spaces. Addition law for mutually exclusive events. Conditional probability. Independence. Law of total probability. Bayes' theorem. Permutations and combinations. Inclusion-Exclusion formula.

Discrete random variables. Concept of random variable (r.v.) and their distribution. Discrete r.v.: Probability function (p.f.). (Cumulative) distribution function (c.d.f.). Mean and variance of a discrete r.v. Examples: Binomial, Poisson, Geometric.

Continuous random variables. Probability density function; mean and variance; exponential, uniform and normal distributions; normal approximations: standardisation of the normal and use of tables. Transformation of a single r.v.

Joint distributions. Discrete r.v.'s; independent random variables; expectation and its application.

Generating functions. Idea of generating functions. Probability generating functions (pgfs) and moment generating functions (mgfs). Finding moments from pgfs and mgfs. Sums of independent random variables.

Laws of Large Numbers. Weak law of large numbers. Central Limit Theorem.

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Introduction to R and investigating data sets. Basic use of R (Input and manipulation of data). Graphical representations of data. Numerical summaries of data.

Sampling and sampling distributions. ?² distribution. t-distribution. F-distribution. Definition of sampling distribution. Standard error. Sampling distribution of sample mean (for arbitrary distributions) and sample variance (for normal distribution) .

Point estimation. Principles. Unbiased estimators. Bias, Likelihood estimation for samples of discrete r.v.s

Interval estimation. Concept. One-sided/two-sided confidence intervals. Examples for population mean, population variance (with normal data) and proportion.

Hypothesis testing. Concept. Type I and II errors, size, p-values and power function. One-sample test, two sample test and paired sample test. Examples for population mean and population variance for normal data. Testing hypotheses for a proportion with large n. Link between hypothesis test and confidence interval. Goodness-of-fit testing.

Association between variables. Product moment and rank correlation coefficients. Two-way contingency tables. ?² test of independence.

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The module provides an understanding of the role of management accounting in the current global scenario and develops key skills in relation to cost accumulation and determination for decision-making. Areas that will be covered are:

Identify what is management accounting and how it differs from financial accounting. Appreciate who are the users of management accounting information and how management accountants can suit their information needs for the creation of customer and shareholder value in a complex and rapidly changing international context.

Understand the different typologies of costs that can be used for decision-making purposes and how cost behaviour has a significant impact on management accounting reports. Appreciate why different costs must be used for different decisions.

Analyse the relationship between the cost structure of a business and the level of production needed to achieve the desired level of profit for the said business. Apply this knowledge to the preparation of the optimal production plan for single and multi-product businesses. Appreciate the impact of any changes in the original assumptions on the forecasted profit for a business.

Calculate the cost of products/services considering all costs involved. Allocate costs to products under different internationally recognised costing systems and understand how the choice of a costing system is linked to the activity performed by a business. Understand the differences between different methodologies of cost calculation and their impact of on decision-making.

Core areas of the syllabus are:

• Management accounting and management accountants in an international context

• Cost terms and purposes

• Cost-volume-profit analysis

• Costing systems

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This module is designed to build upon financial accounting topics taught in previous modules and assess them at a more advanced level. It will also introduce topics, not previous taught. Areas that will be covered are:

The conceptual and regulatory framework for financial reporting – The need for a conceptual framework and the characteristics of useful information. Define what is meant by 'recognition' in financial statements and applying the recognition criteria to assets/liabilities and income/expenses.

Look at why an international regulatory framework is needed over a national regulatory framework. Review the work of the International Accounting Standards Board in setting international accounting standards and how they are moving to harmonised global accounting standards using a principles based rather than a rules based framework.

Describe the concept of a group as a single economic unit and explain and apply the definition of a subsidiary within relevant accounting standards. Prepare basic consolidated financial statements using these concepts.

Distinguish between tangible and intangible non-current assets. Review methods of valuation/revaluation including impairment of assets.

Account for current and deferred taxation within financial statements.

Account for the translation of foreign currency transactions at the reporting date.

Core areas of the syllabus are:

• A conceptual framework for financial reporting

• A regulatory framework for financial reporting

• Financial statements using historic cost and current value accounting

• Business combinations

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This module will introduce the financial system, the markets within the system, various instruments and key concepts. It provides an overview of the roles of financial intermediaries, as well as the fundamental products that they trade. The module will include an historical consideration of the markets, as well as the investigation of current developments, to allow understanding of inter-relationships within the wider economy. An introduction to various financial securities will provide contexts for focus on key concepts of Finance.

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This module builds on MAST4014 Calculus and Differential Equations. We will cover advanced methods for solving ordinary differential equations and introduce, and learn to solve, partial differential equations. We will explore the properties of these differential equations and discuss the physical interpretation of certain equations and their solutions.

Introduction to ODEs: review of ordinary differentiation; what is an ODE; qualitative methods for ODEs; stationary points and stability.

Series methods for ODEs: solution methods for Cauchy-Euler equations; properties of power series; the Frobenius method.

Introduction to linear PDEs: review of partial differentiation; first and second order linear and partial differential equations; simple models that lead to the heat equation, Laplace's equation and the wave equation; the principle of superposition; initial and boundary conditions.

Fourier series: properties of Fourier series, derivation of coefficients, orthogonality of the Fourier basis.

Separable PDEs: The method of separation of variables; simple separable solutions of the heat equation and Laplace’s equation; examples and interpretation of solutions.

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Constructing suitable models for data is a key part of statistics. For example, we might want to model the yield of a chemical process in terms of the temperature and pressure of the process. Even if the temperature and pressure are fixed, there will be variation in the yield which motivates the use of a statistical model which includes a random component. In this module, students study linear regression models (including estimation from data and drawing of conclusions), the use of likelihood to estimate models and its application in simple stochastic models. Both theoretical and practical aspects are covered, including the use of R.

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Probability: Joint distributions of two or more discrete or continuous random variables. Marginal and conditional distributions. Independence. Properties of expectation, variance, covariance and correlation. Poisson process and its application. Sums of random variables with a random number of terms.

Transformations of random variables: Various methods for obtaining the distribution of a function of a random variable —method of distribution functions, method of transformations, method of generating functions. Method of transformations for several variables. Convolutions. Approximate method for transformations.

Sampling distributions: Sampling distributions related to the Normal distribution — distribution of sample mean and sample variance; independence of sample mean and variance; the t distribution in one- and two-sample problems.

Statistical inference: Basic ideas of inference — point and interval estimation, hypothesis testing.

Point estimation: Methods of comparing estimators — bias, variance, mean square error, consistency, efficiency. Method of moments estimation. The likelihood and log-likelihood functions. Maximum likelihood estimation.

Hypothesis testing: Basic ideas of hypothesis testing — null and alternative hypotheses; simple and composite hypotheses; one and two-sided alternatives; critical regions; types of error; size and power. Neyman-Pearson lemma. Simple null hypothesis versus composite alternative. Power functions. Locally and uniformly most powerful tests.

Composite null hypotheses. The maximum likelihood ratio test.

Interval estimation: Confidence limits and intervals. Intervals related to sampling from the Normal distribution. The method of pivotal functions. Confidence intervals based on the large sample distribution of the maximum likelihood estimator – Fisher information, Cramer-Rao lower bound. Relationship with hypothesis tests. Likelihood-based intervals.

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This module is an introduction to the methods, tools and ideas of numerical computation. In mathematics, one often encounters standard problems for which there are no easily obtainable explicit solutions, given by a closed formula. Examples might be the task of determining the value of a particular integral, finding the roots of a certain non-linear equation or approximating the solution of a given differential equation. Different methods are presented for solving such problems on a modern computer, together with their applicability and error analysis. A significant part of the module is devoted to programming these methods and running them in MATLAB.

Introduction: Importance of numerical methods; short description of flops, round-off error, conditioning

Solution of linear and non-linear equations: bisection, Newton-Raphson, fixed point iteration

Interpolation and polynomial approximation: Taylor polynomials, Lagrange interpolation, divided differences, splines

Numerical integration: Newton-Cotes rules, Gaussian rules

Numerical differentiation: finite differences

Introduction to initial value problems for ODEs: Euler methods, trapezoidal method, Runge-Kutta methods.

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This module covers aspects of Statistics which are particularly relevant to insurance. Some topics (such as risk theory and credibility theory) have been developed specifically for actuarial use. Other areas (such as Bayesian Statistics) have been developed in other contexts but now find applications in actuarial fields. Indicative topics covered by the module include Bayesian Statistics; Loss Distributions; Reinsurance and Ruin; Credibility Theory; Risk Models; Ruin Theory; Generalised Linear Models; Extreme Value Theory. This module will cover a number of syllabus items set out in Subjects CS1 and CS2 – Actuarial Statistics published by the Institute and Faculty of Actuaries.

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Formulation/Mathematical modelling of optimisation problems

Linear Optimisation: Graphical method, Simplex method, Phase I method, Dual problems,

Transportation problem.

Non-linear Optimisation: Unconstrained one dimensional problems, Unconstrained high dimensional problems, Constrained optimisation.

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This module introduces the basic ideas to solve certain ordinary differential equations, like first order scalar equations, second order linear equations and systems of linear equations. It mainly considers their qualitative and analytical aspects. Outline syllabus includes: First-order scalar ODEs; Second-order scalar linear ODEs; Existence and Uniqueness of Solutions; Autonomous systems of two linear first-order ODEs.

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The security of our phone calls, bank transfers, etc. all rely on one area of Mathematics: Number Theory. This module is an elementary introduction to this wide area and focuses on solving Diophantine equations. In particular, we discuss (without proof) Fermat's Last Theorem, arguably one of the most spectacular mathematical achievements of the twentieth century. Outline syllabus includes: Modular Arithmetic; Prime Numbers; Introduction to Cryptography; Quadratic Residues; Diophantine Equations.

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This module will cover the following topics:

• The historical development of auditing

• The nature, importance, objectives and underlying theory of auditing

• The philosophy, concepts and basic postulates of auditing

• The regulatory and socio-economic environment within which auditing process takes place

• Auditing implications of agency theories of the firm

• Auditing implications of the efficient markets hypothesis

• The statutory and contractual bases of auditing, including auditing regulation and auditors' legal duties and liabilities

• Truth and fairness in financial reporting

• Materiality and audit judgement

• Audit independence

• The nature and causes of the audit expectation gap

• Auditors' professional ethics and standards

• Audit quality control, planning, programming, performance, supervision and review

• The nature and types of audit evidence

• Principles of internal control

• Systems based auditing and the nature and relationship of compliance and substantive testing

• The audit risk model and statistical sampling

• Audit procedures for major classes of assets, liabilities, income and expenditure

• Audit reporting.

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This module is designed to build upon financial accounting topics taught in previous modules and assess them at a more advanced level. It will also introduce topics, not previous taught.

The following is an indicative list of topics to be covered:

• Accounting for complex transactions in financial statements

• Analysing and interpreting financial statements

• CSR

• Preparation of financial statements including those for complex groups

• Content and application of International Accounting Standards, as appropriate.

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This module is designed to explain the operation and scope of the UK tax system and the obligations of taxpayers and the implications of non-compliance. Areas covered are as follows:

The UK tax system including the overall function and purpose of taxation in a modern economy, different types of taxes, principle sources of revenue law and practice, tax avoidance and tax evasion.

Income tax liabilities including the scope of income tax, income from employment and self-employment, property and investment income, the computation of table income and income tax liability the use of exemptions and reliefs in deferring and minimising income tax liabilities.

National insurance contributions including the scope of national insurance, class 1 and 1A contributions for employed persons, class 2 and 4 contributions for self-employed persons.

Introduction to chargeable gains including the scope of taxation of capital gains, the basic principles of computing gains and losses, the computation of capital gains tax payable by individuals and minimising tax liabilities arising on the disposal of capital assets,

Principles of Inheritance Tax and the use of exemptions and reliefs in deferring and minimising inheritance tax liabilities.

The obligations of taxpayers and/or their agents including the systems for self-assessment and the making of returns, the time limits for the submission of information, claims and payment of tax the procedures relating to enquiries, appeals and disputes, penalties for non-compliance.

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A synopsis of the curriculum

The module will aim to cover the following topics:

• The UK tax system including the overall function and purpose of taxation in a modern economy, different types of taxes, principal sources of revenue law and practice, tax avoidance and tax evasion.

• Income tax liabilities including the scope of income tax, income from employment and self-employment, property and investment income, the computation of taxable income and income tax liability, the use of exemptions and reliefs in deferring and minimising income tax liabilities.

• Corporation tax liabilities including the scope of corporation tax, profits chargeable to corporation tax, the computation of corporation tax liability, the use of exemptions and reliefs in deferring and minimising corporation tax liabilities.

• Chargeable gains including the scope of taxation of capital gains, the basic principles of computing gains and losses, gains and losses on the disposal of movable and immovable property, gains and losses on the disposal of shares and securities, the computation of capital gains tax payable by individuals, the use of exemptions and reliefs in deferring and minimising tax liabilities arising on the disposal of capital assets.

• National insurance contributions including the scope of national insurance, class 1 and 1A contributions for employed persons, class 2 and 4 contributions for self-employed persons.

• Value added tax including the scope of VAT, registration requirements, computation of VAT liabilities.

• Inheritance tax and the use of exemptions and reliefs in deferring and minimising inheritance tax liabilities. Introduction to international tax strategy, implementation, compliance and defence. An understanding of principles of normative ethics in business and in taxation from local and global perspectives.

• The obligations of taxpayers and/or their agents including the systems for self-assessment and the making of returns, the time limits for the submission of information, claims and payment of tax, the procedures relating to enquiries, appeals and disputes, penalties for non-compliance.

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This module will cover the following topics:

- Features of debt instruments and risks associated with investing in these instruments

- Debt and money markets (participants, operations, trading activities)

- Fixed-income instruments (Government bonds, corporate bonds, credit ratings, high-yield bonds, international bonds, mortgage-backed securities, etc.)

- Money market instruments (Treasury bills, commercial paper, repurchase agreements, bills of exchange, etc.)

- Fixed-income valuation (traditional approach, arbitrage-free approach, yield measures, volatility measures)

- Term-structure of interest rates and classic theories of term structure, derivation of zero-coupon yield curve

- General principles of credit analysis (credit scoring, credit risk modelling, etc.)

- Fixed-income portfolio construction and management strategies (portfolio's risk profile, managing funds against a bond market index).

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This module will examine how Excel can be used for financial data analysis.

A brief revision of each financial concept will be presented. The syllabus will typically cover:

Introduction to Excel:

• Basic functions, mathematical expressions

Data Analysis with Excel:

• Data analysis, charts, solver, goal seek, pitot tables and pivot charts

Financial Valuation:

• Applications of time value of money

• Applications of capital budgeting techniques in Excel (IRR, NPV, Scenario Analysis, Monte Carlo simulation)

• Company Valuation Models

Portfolio Analysis and Security Pricing:

• Portfolio models, calculations of efficient portfolios, variance-covariance matrix

• Beta coefficient estimations and security market line

• Bond Valuations

• Binomial option pricing, Black-Scholes model.

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This module is concerned with International Investment Banks’ products and strategies that involve the description and analyses of the characteristics of more commonly used financial derivative instruments such as forward and future contracts, swaps, and options involving commodities, interest, and equities markets. Modern financial techniques are used to value financial derivatives. The main emphasis of the module is on how International Investment Banks value, replicate, and arbitrage the financial instruments and how they encourage their clients to use derivative products to implement risk management strategies in the context of corporate applications.

In particular, students will first cover the topics related to forward, futures and swap contracts. They will then be introduced to options and various strategies thereof. Valuing options using Black-Scholes model and binomial trees is also an important part of the module. The important finance concepts of no-arbitrage and risk-neutral valuation and their implications for pricing financial derivatives are also covered in the module. This will help students to learn the techniques used in valuing financial derivatives and hedging risk exposure.

Successful completion of the module will provide a solid base for the student wishing to pursue a career in International Investment Banking and Treasury Management. The students will have the knowledge of essential techniques of risk management and financial derivative trading.

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The module begins with motivations for risk management in general and then covers the practice of risk management. In particular, students are introduced to the current thinking on governance and regulatory systems, followed by industry practices for managing certain common types of risk. Critical evaluation of these practices is incorporated where applicable.

Topics covered in this module include:

- Introduction to general risk management theory, how and why it generates value

- A taxonomy of risks, including Market Risk, Credit Risk, Liquidity Risk, Operational Risk, Model Risk, Regulatory Risk, Legal/Contract Risk, Tax Risk, Accounting Risk, and Political Risk.

- Introduction to Governance and Regulation

- Standard measures of risk

- Risk measurement for security portfolios

- Hedging techniques using financial derivatives

- Evaluation of hedging performance

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Discrete mathematics has found new applications in the encoding of information. Online banking requires the encoding of information to protect it from eavesdroppers. Digital television signals are subject to distortion by noise, so information must be encoded in a way that allows for the correction of this noise contamination. Different methods are used to encode information in these scenarios, but they are each based on results in abstract algebra. This module will provide a self-contained introduction to this general area of mathematics.

Syllabus: Modular arithmetic, polynomials and finite fields. Applications to

• orthogonal Latin squares,

• cryptography, including introduction to classical ciphers and public key ciphers such as RSA,

• "coin-tossing over a telephone",

• linear feedback shift registers and m-sequences,

• cyclic codes including Hamming,

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Most differential equations which arise from physical systems cannot be solved explicitly in closed form, and thus numerical solutions are an invaluable way to obtain information about the underlying physical system. The first half of the module is concerned with ordinary differential equations. Several different numerical methods are introduced and error growth is studied. Both initial value and boundary value problems are investigated. The second half of the module deals with the numerical solution of partial differential equations. The syllabus includes: initial value problems for ordinary differential equations; Taylor methods; Runge-Kutta methods; multistep methods; error bounds and stability; boundary value problems for ordinary differential equations; finite difference schemes; difference schemes for partial differential equations; iterative methods; stability analysis.

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Linear PDEs. Dispersion relations. Review of d'Alembert’s solutions of the wave equation. Review of Fourier transforms for solving linear diffusion equations.

Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.

Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.

General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.

Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).

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Bayes Theorem for density functions; Conjugate models; Predictive distribution; Bayes estimates; Sampling density functions; Gibbs and Metropolis-Hastings samplers; Stan and Python; Bayesian hierarchical models; Bayesian model choice; Objective priors; Exchangeability; Choice of priors; Applications of hierarchical models.

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This module is designed to cover: Ethics and compliance of data science. Impact of international regulations. Appropriate handling of data. Simple random sampling. Sampling for proportions and percentages. Estimation of sample size. Stratified sampling. Systematic sampling. Cluster sampling. Data streams. Finding frequentist items. Estimating the number of distinct elements. Sparse recovery. Weight-based sampling. Real time analytics. Network data: Density, clustering coefficient, centrality and degree distribution.

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In this module we study the fundamental concepts and results in game theory. We start by analysing combinatorial games, and discuss game trees, winning strategies, and the classification of positions in so called impartial combinatorial games. We then move on to discuss two-player zero-sum games and introduce security levels, pure and mixed strategies, and prove the famous von Neumann Minimax Theorem. We will see how to solve zero-sum two player games using domination and discuss a general method based on linear programming. Subsequently we analyse arbitrary sum two-player games and discuss utility, best responses, Nash equilibria, and the Nash Equilibrium Theorem. The final part of the module is devoted to multi-player games and cooperation; we analyse coalitions, the core of the game, and the Shapley value.

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• Scalar autonomous nonlinear first-order ODEs. Review of steady states and their stability; the slope fields and phase lines.

• Autonomous systems of two nonlinear first-order ODEs. The phase plane; Equilibra and nullclines; Linearisation about equilibra; Stability analysis; Constructing phase portraits; Applications. Nondimensionalisation.

• Stability, instability and limit cycles. Liapunov functions and Liapunov's theorem; periodic solutions and limit cycles; Bendixson's Negative Criterion; The Dulac criterion; the Poincare-Bendixson theorem; Examples.

• Dynamics of first order difference equations. Linear first order difference equations; Simple models and cobwebbing: a graphical procedure of solution; Equilibrium points and their stability; Periodic solutions and cycles. The discrete logistic model and bifurcations.

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Background material: multivariate normal distribution, inference from multivariate normal samples

Indicative module content:

• Principal component and factor analysis, latent variable model, clustering and classification methods

• Likelihood-based analysis such as maximum likelihood, EM algorithm, optimisation, confidence interval construction

• Simulation and sampling methods, bootstrap, permutation tests

• Model building including tests such as the Wald test

• R programming including real-world applications in areas such as biology, ecology, sociology and economics to data that does not always follow standard statistical models.

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Introduction: Principles and examples of stochastic modelling, types of stochastic process, Markov property and Markov processes, short-term and long-run properties. Applications in various research areas.

Random walks: The simple random walk. Walk with two absorbing barriers. First–step decomposition technique. Probabilities of absorption. Duration of walk. Application of results to other simple random walks. General random walks. Applications.

Discrete time Markov chains: n–step transition probabilities. Chapman-Kolmogorov equations. Classification of states. Equilibrium and stationary distribution. Mean recurrence times. Simple estimation of transition probabilities. Time inhomogeneous chains. Elementary renewal theory. Simulations. Applications.

Continuous time Markov chains: Transition probability functions. Generator matrix. Kolmogorov forward and backward equations. Poisson process. Birth and death processes. Time inhomogeneous chains. Renewal processes. Applications.

Queues and branching processes: Properties of queues - arrivals, service time, length of the queue, waiting times, busy periods. The single-server queue and its stationary behaviour. Queues with several servers. Branching processes. Applications.

In addition, level 7 students will study more complex queuing systems and continuous-time branching processes.

This module will cover a number of syllabus items set out in Subject CS2 published by the Institute and Faculty of Actuaries. This is a dynamic syllabus, changing regularly to reflect current practice.

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Stationary Time Series: Stationarity, autocovariance and autocorrelation functions, partial autocorrelation functions, ARMA processes.

ARIMA Model Building and Testing: estimation, Box-Jenkins, criteria for choosing between models, diagnostic tests for residuals of a time series after estimation.

Forecasting: Holt-Winters, Box-Jenkins, prediction bounds.

Testing for Trends and Unit Roots: Dickey-Fuller, ADF, structural change, trend-stationarity vs difference stationarity.

Seasonality and Volatility: ARCH, GARCH, ML estimation.

Multiequation Time Series Models: transfer function models, vector autoregressive moving average (VARM(p,q)) models, impulse responses.

Spectral Analysis: spectral distribution and density functions, linear filters, estimation in the frequency domain, periodogram.

Simulation: generation of pseudo-random numbers, random variate generation by the inverse transform, acceptance rejection. Normal random variate generation: design issues and sensitivity analysis.

This module will cover a number of syllabus items set out in Subject CS2 published by the Institute and Faculty of Actuaries. This is a dynamic syllabus, changing regularly to reflect current practice.

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There is no specific mathematical syllabus for this module; students will chose a topic in mathematics, statistics or financial mathematics from a published list on which to base their coursework assessments (different topics for levels 6 and 7). The coursework is supported by a series of workshops covering various forms of written and oral communication. These may include critically evaluating the following: a research article in mathematics, statistics or finance; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

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