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 module emphasises the use of basic economic concepts to business analysis.

The first part of the module focuses on explaining a selection of basic 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. The attention is to understand the relevance of macroeconomics topics (e.g. interest rates, exchange rates, etc.) to business.

The module is self-contained to provide a basic understanding of simple 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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This module is concerned with the principles which underlie the investment and financing decision making process. Before a rational decision can be made objectives need to be considered and models need to be built. Short-term decisions are dealt with first, together with relevant costs. One such cost is the time value of money. This leads to long term investment decisions which are examined using the economic theory of choice, first assuming perfect capital markets and certainty. These assumptions are then relaxed so that such problems as incorporating capital rationing and risk into the investment decision are fully considered. The module proceeds by looking at the financing decision. The financial system within which business organisations operate is examined, followed by the specific sources and costs of long and short-term capital, including the management of fixed and working capital

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In this module we will study linear partial differential equations, we will explore their properties and discuss the physical interpretation of certain equations and their solutions. We will learn how to solve first order equations using the method of characteristics and second order equations using the method of separation of variables.

Introduction to linear PDEs: Review of partial differentiation; first-order linear PDEs, the heat equation, Laplace's equation and the wave equation, with simple models that lead to these equations; the superposition principle; initial and boundary conditions

Separation of variables and series solutions: The method of separation of variables; simple separable solutions of the heat equation and Laplace’s equation; Fourier series; orthogonality of the Fourier basis; examples and interpretation of solutions

Solution by characteristics: the method of characteristics for first-order linear PDEs; examples and interpretation of solutions; characteristics of the wave equation; d’Alembert’s solution, with examples; domains of influence and dependence; causality.

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The aim of this module is to introduce students to a set of tools that they can use to analyse macroeconomic issues in the short run and long run, and to help them understand which modelling techniques are appropriate for particular macroeconomic problems.

This module enhances the students' ability to understand the determinants of key macroeconomic variables including output, inflation, interest rates and exchange rates. Students’ technical skills are developed and they are expected to apply their knowledge to real-world policymaking and link it to financial issues.

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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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The module will aim to cover the following topics:

• the conceptual framework of financial reporting

• the financial reporting environment

• the regulation of financial reporting

• group accounting

• the International Accounting Standards Board

• content and application of International Accounting Standards as appropriate

• accounting standards

• accounting for transactions in financial statements

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This module begins with a focus on the financial system of the UK, including the major players in the markets and key interrelations. It then proceeds to cover key topics, including: advanced portfolio theory, the capital asset pricing model, arbitrage pricing theory, the implications and empirical evidence relating to the efficient market hypothesis, capital structure and the cost of capital in a taxation environment, interaction of investment and financing decisions, decomposition of risk, options and pricing, risk management, dividends and dividend valuation models, mergers and failures and evaluating financial strategies.

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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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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 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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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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Combinatorics is a field in mathematics that studies discrete, usually finite, structures, such as graphs. It not only plays an important role in numerous parts of mathematics, but also has real world applications. In particular, it underpins a variety of computational processes used in digital technologies and the design of computing hardware.

Among other things, this module provides an introduction to graph theory. Graphs are discrete objects consisting of vertices that are connected by edges. We will discuss a variety of concepts and results in graph theory, and some fundamental graph algorithms. Topics may include, but are not restricted to: trees, shortest paths problems, walks on graphs, graph colourings and embeddings, flows and matchings, and matrices and graphs.

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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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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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Sampling: Simple random sampling. Sampling for proportions and percentages. Estimation of sample size. Stratified sampling. Systematic sampling. Ratio and regression estimates. Cluster sampling. Multi-stage sampling and design effect. Questionnaire design. Response bias and non-response.

General principles of experimental design: blocking, randomization, replication. One-way ANOVA. Two-way ANOVA. Orthogonal and non-orthogonal designs. Factorial designs: confounding, fractional replication. Analysis of covariance.

Design of clinical trials: blinding, placebos, eligibility, ethics, data monitoring and interim analysis. Good clinical practice, the statistical analysis plan, the protocol. Equivalence and noninferiority. Sample size. Phase I, II, III and IV trials. Parallel group trials. Multicentre trials.

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Multivariate normal distribution, Inference from multivariate normal samples, principal component analysis, mixture models, factor analysis, clustering methods, discrimination and classification, graphical models, the use of appropriate software.

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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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Statistics methods contribute significantly to areas such as biology, ecology, sociology and economics. The real data collected does not always follow standard statistical models. This module looks at modern statistical models and methods that can be utilised for such data, making use of R programs to execute these methods.

Indicative module content: Motivating examples; model fitting through maximum likelihood for specific examples; function optimization methods; profile likelihood; score tests; Wald tests; confidence interval construction; latent variable models; EM algorithm; mixture models; simulation methods; importance sampling; kernel density estimation; Monte Carlo inference; bootstrap; permutation tests; R programs.

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