# Mathematics with Finance BSc (Hons)

- Course length: 3 years
- UCAS code: G1N3
- Year of entry: 2020
- Typical offer: A-level : AAB / IB : 35 / BTEC : Applications considered

## Honours Select

×This programme offers Honours Select combinations.

## Honours Select 100

×This programme is available through Honours Select as a Single Honours (100%).

## Honours Select 75

×This programme is available through Honours Select as a Major (75%).

## Honours Select 50

×This programme is available through Honours Select as a Joint Honours (50%).

## Honours Select 25

×This programme is available through Honours Select as a Minor (25%).

## Study abroad

×This programme offers study abroad opportunities.

## Year in China

×This programme offers the opportunity to spend a Year in China.

## Accredited

×This programme is accredited.

### Module details

### Programme Year One

The Mathematics with Finance degree has been accredited by the UK Actuarial Profession, which means that students can obtain exemptions from some of the subjects in the Institute and Faculty of Actuaries’ examination system.

All exemptions will be recommended on a subject-by-subject basis, taking into account performance at the University of Liverpool.

Further information can be found at the actuarial profession’s website actuaries.org.uk website.

**Core Technical Stage**

Exemptions are based on performance in the relevant subjects as listed below.

Subject **CT1 - Financial Mathematics**: Financial Mathematics I & II

Subject **CT2 - Finance & Financial Reporting**: Introduction to Financial Accounting, Introduction to Finance & Financial Reporting and Finance

Subject **CT3 - Probability & Mathematical Statistics**: Statistical Theory I & II

Subject **CT4 - Models**: Applied Probability & Actuarial Models

#### Year One Compulsory Modules

##### Calculus I (MATH101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.3. To introduce the notions of sequences and series and of their convergence.

**Learning Outcomes**(LO1) Differentiate and integrate a wide range of functions;

(LO2) Sketch graphs and solve problems involving optimisation and mensuration

(LO3) Understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

(S1) Numeracy

##### Calculus II (MATH102)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**- To discuss local behaviour of functions using Taylor’s theorem.

- To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals.

**Learning Outcomes**(LO1) Use Taylor series to obtain local approximations to functions

(LO2) Obtain partial derivatives and use them in several applications such as, error analysis, stationary points change of variables.

(LO3) Evaluate double integrals using Cartesian and Polar Co-ordinates.

##### Math103 - Introduction to Linear Algebra (MATH103)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**• To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.

• To develop geometrical intuition in 2 and 3 dimensions.

• To introduce students to the concept of subspace in a concrete situation.

• To provide a foundation for the study of linear problems both within mathematics and in other subjects**Learning Outcomes**(LO1) Manipulate complex numbers and solve simple equations involving them solve arbitrary systems of linear equations

(LO2) Understand and use matrix arithmetic, including the computation of matrix inverses

(LO3) Compute and use determinants

(LO4) Understand and use vector methods in the geometry of 2 and 3 dimensions

(LO5) Calculate eigenvalues and eigenvectors and, if time permits, apply these calculations to the geometry of conics and quadrics.

(LO6) calculate eigenvalues and eigenvectors and, if time permits, apply these calculations to the geometry of conics and quadrics

(S2) Numeracy

##### Mathematical It Skills (MATH111)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**•To acquire key mathematics-specific computer skills.

•To reinforce mathematics as a practical discipline by active experience and experimentation, using the computer as a tool.

•To illustrate and amplify mathematical concepts and techniques.

•To initiate and develop problem solving, group work and report writing skills.

•To initiate and develop modelling skills.

•To develop team work skills.**Learning Outcomes**(LO1) After completing the module, students should be able to tackle project work, including writing up of reports detailing their solutions to problems.

(LO2) After completing the module, students should be able to use computers to create documents containing formulae, tables, plots and references.

(LO3) After completing the module, students should be able to use mathematical software packages such as Maple and Matlab to manipulate mathematical expressions and to solve simple problems.

(LO4) After completing the module, students should be able to better understand the mathematical topics covered, through direct experimentation with the computer.

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

(S4) IT skills

(S5) Teamwork

(S6) Adaptability

(S7) Leadership

(S8) Mathematical modelling skills

##### Introduction to Statistics (MATH162)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**•To introduce topics in Statistics and to describe and discuss basic statistical methods.

•To describe the scope of the application of these methods.**Learning Outcomes**(LO1) To know how to describe statistical data.

(LO2) To be able to use the Binomial, Poisson, Exponential and Normal distributions.

(LO3) To be able to perform simple goodness-of-fit tests.

(LO4) To be able to use an appropriate statistical software package to present data and to make statistical analysis.

(S1) Numeracy

(S2) Problem solving skills

(S3) IT skills

(S4) Communication skills

##### Introduction to Financial Accounting (ACFI101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**15:85 **Aims**To develop knowledge and understanding of the underlying principles and concepts relating to financial accounting and technical proficiency in the use of double entry accounting techniques in recording transactions, adjusting financial records and preparing basic financial statements.

**Learning Outcomes**(LO1) Prepare basic financial statements

(LO2) Explain the context and purpose of financial reporting

(LO3) Demonstrate the use of double entry and accounting systems

(LO4) Record transactions and events

(LO5) Prepare a trial balance

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Organisational skills

(S5) Communication skills

##### Introduction to Finance (ACFI103)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module aims:

to introduce the students to finance.

to provide a firm foundation for the students to build on later on in the second and third years of their programmes, by covering basic logical and rational analytical tools that underpin financial decisions**Learning Outcomes**(LO1) Understand the goals and governance of the firm, how financial markets work and appreciate the importance of finance.

(LO2) Understand the time value of money.

(LO3) Understand the determinants of bond yields.

(LO4) Recognize how stock prices depend on future dividends and value stock prices.

(LO5) Understand net present value rule and other criteria used to make investment decisions.

(LO6) Understand risk, return and the opportunity cost of capital.

(LO7) Understand the risk-return tradeoff, and know the various ways in which capital can be raised and determine a firm's overall cost of capital.

(LO8) Know different types of options, and understand how options are priced.

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Teamwork

(S5) Organisational skills

(S6) Communication skills

(S7) IT skills

(S8) International awareness

(S9) Lifelong learning skills

(S10) Ethical awareness

#### Year One Optional Modules

##### Newtonian Mechanics (MATH122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems.

Learning Outcomes:After completing the module students should be able to analyse real world problems

involving:- the motions of bodies under simple force systems

- conservation laws for momentum and energy

- rigid body dynamics using centre of mass,

angular momentum and moments of inertia**Learning Outcomes**(LO1) the motions of bodies under simple force systems

(LO2) conservation laws for momentum and energy

(LO3) rigid body dynamics using centre of mass, angular momentum and moments

(LO4) oscillation, vibration, resonance

(LO5) oscillation, vibration, resonance

(S1) Representing physical problems in a mathematical way

(S2) Problem Solving Skills

##### Numbers, Groups and Codes (MATH142)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**- To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.

- To give an appreciation of the utility and power of group theory as the study of symmetries.

- To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.

**Learning Outcomes**(LO1) Be able to apply the Euclidean algorithm to find the greatest common divisor of a pair of positive integers, and use this procedure to find the inverse of an integer modulo a given integer.

(LO2) Be able to solve linear congruences and apply appropriate techniques to solve systems of such congruences.

(LO3) Be able to perform a range of calculations and manipulations with permutations.

(LO4) Recall the definition of a group and a subgroup and be able to identify these in explicit examples.

(LO5) Be able to prove that a given mapping between groups is a homomorphism and identify isomorphic groups.

(LO6) To be able to apply group theoretic ideas to applications with error correcting codes.

(LO7) Engage in group project work to investigate applications of the theoretical material covered in the module.

### Programme Year Two

In the second and subsequent years of study, there is a wide range of modules. Each year you will take the equivalent of eight modules. Please note that we regularly review our teaching so the choice of modules may change. In addition to the compulsory modules below, you will choose one optional module.

#### Year Two Compulsory Modules

##### Corporate Financial Management for Non-specialist Students (ACFI213)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**The aim of the module is to provide an introduction to financial markets and to contextualise the application of mathematical techniques.

**Learning Outcomes**(LO1) Students will be equipped with the tools and techniques of financial management

(LO2) Students will be able to interpret and critically examine financial management issues and controversies.

(LO3) Students will attain the necessary knowledge to underpin the more advanced material on Quantitative Business Finance.

(S1) Commercial awareness

(S2) Organisational skills

(S3) Problem solving skills

(S4) IT skills

(S5) International awareness

(S6) Numeracy

##### Financial Reporting and Finance (non-specialist) (ACFI290)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**The aim of the Financial Reporting and Finance module is to provide an understanding of financial instruments and financial institutions and to provide the ability to interpret published financial statements of non-financial and financial companies with respect to performance, liquidity and efficiency. It also provides understanding in selected topics on financial management i.e. raising finance, capital structure, cost of capital and company restructuring, dividend policy, investment appraisal techniques and evaluation of projects, financial risk assessment; an understanding of the concepts of taxation and managerial decision making are also introduced and developed.

**Learning Outcomes**(LO1) Describe the different forms a business may operate in;

(LO2) Describe the principal forms of raising finance for a business;

(LO3) Demonstrate an understanding of key accounting concepts, group accounting and analysis of financial statements;

(LO4) Describe the basic principles of personal and corporate taxation;

(LO5) Demonstrate an understanding of decision making tools in used in management accounting.

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Organisational skills

(S5) Communication skills

##### Financial Mathematics (MATH262)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide an understanding of basic theories in Financial Mathematics used in the study process of actuarial/financial interest.

To provide an introduction to financial methods and derivative pricing financial instruments in discrete time set up.

To gain understanding of some financial models (discrete time) with applications to finance/insurance industry.

To prepare the students adequately and to develop their skills in order to be ready to sit the CM2 subject of the Institute and Faculty of Actuaries exams.

**Learning Outcomes**(LO1) To understand the assumptions of the Capital Asset Pricing Model (CAPM), to be able to explain the no riskless lending or borrowing and other lending and borrowing assumptions, to be able to use the formulas of CAPM, to be able to derive the capital market line and security market line.

(LO2) To be able to describe the Arbitrage Theory Model (APT) and explain its assumptions as well as perform estimating and testing in APT.

(LO3) To be able to explain the terms long/short position, spot/delivery/forward price, understand the use of future contracts, describe what a call/put option (European/American) is as well as be able to create graphs and explain their payouts, describe the hedging for reducing the exposure to risk, to be able to explain arbitrage, understand the mechanism of short sales.

(LO4) To be able to explain/describe what arbitrage is, what the risk neutral probability measure is, as well as to be able to use (and perform calculation) the binomial tree for European and American style options.

(LO5) To understand the role of Filtrations, martingales and option pricing in discrete time.

(LO6) To be able to discuss the theories of financial market behaviour.

(LO7) To be able to perform calculations using the risk measures.

(S1) Problem solving skills

(S2) Adaptability

(S3) Numeracy

(S4) Commercial awareness

##### Math201 - Ordinary Differential Equations (MATH201)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**75:25 **Aims**•To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.

•To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.

**Learning Outcomes**(LO1) To understand the basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions.

(LO2) To know the elementary techniques for the solution of ODEs.

(LO3) To understand the idea of reducing a complex ODE to a simpler one.

(LO4) To be able to solve linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix.

(LO5) To understand a range of applications of ODE.

(S1) Problem solving skills

(S2) Numeracy

##### Theory of Interest (MATH267)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**This module aims to provide students with an understanding of the fundamental concepts of Financial Mathematics, and how the concepts above are applied in calculating present and accumulated values for various streams of cash flows. Students will also be given an introduction to financial instruments, such as derivatives and the concept of no-arbitrage.

**Learning Outcomes**(LO1) To understand and calculate all kinds of rates of interest, find the future value and present value of a cash flow and to write the equation of value given a set of cash flows and an interest rate.

(LO2) To derive formulae for all kinds of annuities.

(LO3) To understand an annuity with level payments, immediate (or due), payable m-thly, (or payable continuously) and any three of present value, future value, interest rate, payment, and term of annuity as well as to calculate the remaining two items.

(LO4) To calculate the outstanding balance at any point in time.

(LO5) To calculate a schedule of repayments under a loan and identify the interest and capital components in a given payment.

(LO6) To calculate a missing quantity, being given all but one quantities, in a sinking fund arrangement.

(LO7) To calculate the present value of payments from a fixed interest security, bounds for the present value of a redeemable fixed interest security.

(LO8) Given the price, to calculate the running yield and redemption yield from a fixed interest security.

(LO9) To calculate the present value or real yield from an index-linked bond.

(LO10) To calculate the price of, or yield from, a fixed interest security where the income tax and capital gains tax are implemented.

(LO11) To calculate yield rate, the dollar-weighted and time weighted rate of return, the duration and convexity of a set of cash flows.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Commercial awareness

##### Statistical Theory and Methods I (MATH263)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **Aims**To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation. The emphasis is on applications.

**Learning Outcomes**(LO1) To have a conceptual and practical understanding of a range of commonly applied statistical procedures.

(LO2) To have developed some familiarity with an appropriate statistical software package.

(S1) Problem solving skills

(S2) Numeracy

(S3) IT skills

(S4) Communication skills

##### Statistical Theory and Methods II (MATH264)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory.

**Learning Outcomes**(LO1) To have an understanding of basic probability calculus.

(LO2) To have an understanding of a range of techniques for solving real life problems of probabilistic nature.

(S1) Problem solving skills

(S2) Numeracy

#### Year Two Optional Modules

##### Mathematical Models: Microeconomics and Population Dynamics (MATH227)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**•To provide an understanding of the techniques used in constructing, analysing, evaluating and interpreting mathematical models.

•To do this in the context of two non-physical applications, namely microeconomics and population dynamics.

•To use and develop mathematical skills introduced in Year 1 - particularly the calculus of functions of several variables and elementary differential equations.**Learning Outcomes**(LO1) After completing the module students should be able to: - Use techniques from several variable calculus in tackling problems in microeconomics. - Use techniques from elementary differential equations in tackling problems in population dynamics. - Apply mathematical modelling methodology in these subject areas. All learning outcomes are assessed by both examination and course work.

##### Metric Spaces and Calculus (MATH241)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**To introduce the basic elements of the theory of metric spaces and calculus of several variables.

**Learning Outcomes**(LO1) After completing the module students should: Be familiar with a range of examples of metric spaces. Have developed their understanding of the notions of convergence and continuity. Understand the contraction mapping theorem and appreciate some of its applications. Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map. Understand the inverse function and implicit function theorems and appreciate their importance. Have developed their appreciation of the role of proof and rigour in mathematics.

(S1) Problem solving skills

##### Introduction to Methods of Operational Research (MATH261)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**The aims of the module are to develop an understanding of how mathematical modelling and operational research techniques are applied to real-world problems and to gain an understanding of linear and convex programming, multi-objective problems, inventory control and sensitivity analysis.

**Learning Outcomes**(LO1) To understand the operational research approach.

(LO2) To be able to apply standard methods of operational research to a wide range of real-world problems as well as to problems in other areas of mathematics.

(LO3) To understand the advantages and disadvantages of particular operational research methods.

(LO4) To be able to derive methods and modify them to model real-world problems.

(LO5) To understand and be able to derive and apply the methods of sensitivity analysis.

(LO6) To understand the importance of sensitivity analysis.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Self-management readiness to accept responsibility (i.e. leadership), flexibility, resilience, self-starting, initiative, integrity, willingness to take risks, appropriate assertiveness, time management, readiness to improve own performance based on feedback/reflective learning

##### Introduction to Data Science (COMP229)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:0 **Aims**1. To provide a foundation and overview of modern problems in Data Science. 2. To describe the tools and approaches for the design and analysis of algorithms for da-ta clustering, dimensionally reduction, graph reconstruction from noisy data. 3. To discuss the effectiveness and complexity of modern Data Science algorithms. 4. To review applications of Data Science to Vision, Networks, Materials Chemistry.

**Learning Outcomes**(LO1) describe modern problems and tools in data clustering and dimensionality reduction,

(LO2) formulate a real data problem in a rigorous form and suggest potential solutions,

(LO3) choose the most suitable approach or algorithmic method for given real-life data,

(LO4) visualise high-dimensional data and extract hidden non-linear patterns from the data.

(S1) Critical thinking and problem solving - Critical analysis

##### Operational Research: Probabilistic Models (MATH268)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce a range of models and techniques for solving under uncertainty in Business, Industry, and Finance.

**Learning Outcomes**(LO1) The ability to understand and describe mathematically real-life optimization problems.

(LO2) Understanding the basic methods of dynamical decision making.

(LO3) Understanding the basics of forecasting and simulation.

(LO4) The ability to analyse elementary queueing systems.

(S1) Problem solving skills

(S2) Numeracy

##### Securities Markets (ECON241)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**75:25 **Aims**This module seeks to provide an understanding of:

the role of securities markets in the economy;

their basic mechanics and technical features;

the valuation of financial assets;

the operational and allocative efficiency of the market.**Learning Outcomes**(LO1) Appreciate the central role of securities markets in the economy

(LO2) Understand and apply appropriate economic theory to market organisation

(LO3) Display an understanding of the usefulness of portfolio theory and the approaches to the valuation of financial assets

(LO4) Read the financial press and appreciate issues relating to the study of the securities markets

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Commercial awareness

(S5) Teamwork

(S6) Organisational skills

(S7) Communication skills

(S8) IT skills

(S9) International awareness

(S10) Lifelong Learning Skills

(S11) Ethical Awareness

##### Introduction to the Methods of Applied Mathematics (MATH224)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**•To provide a grounding in elementary approaches to solution of some of the standard partial differential equations encountered in the applications of mathematics.

•To introduce some of the basic tools (Fourier Series) used in the solution of differential equations and other applications of mathematics.

**Learning Outcomes**(LO1) After completing the module students should: - be fluent in the solution of basic ordinary differential equations, including systems of first order equations:- be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations:- be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations: - be able to solve simple first order partial differential equations: - be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.

##### Math256 - Numerical Methods (MATH256)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide an introduction to the main topics in Numerical Analysis and their relation to other branches of Mathematics

**Learning Outcomes**(LO1) To strengthen students’ knowledge of scientific programming, building on the ideas introduced in MATH111.

(LO2) To provide an introduction to the foundations of numerical analysis and its relation to other branches of Mathematics.

(LO3) To introduce students to theoretical concepts that underpin numerical methods, including fixed point iteration, interpolation, orthogonal polynomials and error estimates based on Taylor series.

(LO4) To demonstrate how analysis can be combined with sound programming techniques to produce accurate, efficient programs for solving practical mathematical problems.

(S1) Numeracy

(S2) Problem solving skills

#### Year Three Compulsory Modules

##### Quantitative Business Finance (ACFI314)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**This module aims to provide students with a fundamental understanding of the core theoretical and empirical aspects involved in corporate finance. In particular, the aims are that students will:

Understand aspects of theories in corporate finance;

Become familiar with a range of mathematical techniques commonly employed in corporate finance with particular emphasis on bond valuation, stock valuation, firm valuation and assessing the probability that the firm will default on its debt obligations;

Be aware that all mathematical models, which are dependent on a set of underlying assumptions, have limitations in the sense that the answer to a particular problem might change once the underlying assumptions change.

**Learning Outcomes**(LO1) Understand the principles of bonds and stocks valuation

(LO2) Understand how credit rating agencies assign credit rating scores to bonds

(LO3) Develop an understanding of issues involved in capital budgeting under uncertainty, market efficiency

(LO4) Understand portfolio theory, asset pricing models (CAPM, APT) and portfolio management

(LO5) An ability to analyse financial data in order to derive the optimal capital structure of firms

(LO6) Understand how option pricing theory can be used to firm valuation and assess the probability that a firm will default on its debt obligations

(LO7) An ability to analyse data in order to calculate Value at Risk as a single number summarising the total risk in a portfolio of financial assets.

(LO8) Understand the principles and practices involved in leasing, mergers and acquisitions

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Communication skills

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for ‘‘dynamic" events occurring over time. To familiarise students with an important area of probability modelling.

**Learning Outcomes**(LO1) 1. Knowledge and Understanding After the module, students should have a basic understanding of:

(a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes

(b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain

(c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.

2. Intellectual Abilities After the module, students should be able to:

(a) formulate appropriate situations as probability models: random processes

(b) demonstrate knowledge of standard models (c) demonstrate understanding of the theory underpinning simple dynamical systems

3. General Transferable Skills

(a) numeracy through manipulation and interpretation of datasets

(b) communication through presentation of written work and preparation of diagrams

(c) problem solving through tasks set in tutorials

(d) time management in the completion of practicals and the submission of assessed work

(e) choosing, applying and interpreting results of probability techniques for a range of different problems.##### Numerical Analysis for Financial Mathematics (MATH371)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**1. To provide basic background in solving mathematical problems numerically, including understanding of stability and convergence of approximations to exact solution.

2. To acquaint students with two standard methods of derivative pricing: recombining trees and Monte Carlo algorithms.

3. To familiarize students with implementation of numerical methods in a high level programming language.

**Learning Outcomes**(LO2) Ability to analyse a simple numerical method for convergence and stability

(LO3) Ability to formulate approximations to derivative pricing problems numerically.

(LO4) Ability to program matlab for pricing options

(LO5) Awareness of the major issues when solving mathematical problems numerically.

(S1) Problem solving skills

(S2) Numeracy

##### Financial and Actuarial Modelling in R (MATH377)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**1.To give a set of applicable skills used in practice in financial and insurance institutions. To introduce students to specific programming techniques that are widely used in finance and insurance.

2.To provide students with a conceptual introduction to the basic principles and practices of the programming language R and to give them experience of carrying out calculations introduced in other modules of their programmes.

3.To develop the abilities to set standard financial and insurance models in order to manage the risk of the cash flow of financial and insurance companies, reserve, portfolio etc.

4.To develop the awareness of statistical and numerical limitations of financial and actuarial models and to know about modern approaches to tackle these limitations.

**Learning Outcomes**(LO1) To be able to import Excel files into R.

(LO2) To know how to create and compute standard functions and how to plot them.

(LO3) To be able to define and compute probability distributions and to be able to apply their statistical inference based on specific data sets and/or random samples.

(LO4) To know how to apply linear regression.

(LO5) To be able to compute aggregate loss distributions/stochastic processes and to find the probability of ruin.

(LO6) To know how to apply Chain Ladder and other reserving methods.

(LO7) To know how to price general insurance products.

(LO8) To be able to compute binomial trees.

(LO9) To know how to apply algorithms for yield curves.

(LO10) To be able to apply the Black-Scholes formula.

(LO11) To know how to develop basic Monte Carlo simulations.

(S1) Numeracy

(S2) Problem solving skills

(S3) Communication skills

(S4) IT skills

(S5) Organisational skills

(S6) Commercial awareness

#### Year Three Optional Modules

##### Stochastic Theory and Methods in Data Science (MATH368)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **Aims**1. To develop a understanding of the foundations of stochastics normally including processes and theory.

2. To develop an understanding of the properties of simulation methods and their applications to statistical concepts.

3. To develop skills in using computer simulations such as Monte-Carlo methods

4. To gain an understanding of the learning theory and methods and of their use in the context of machine learning and statistical physics.

5. To obtain an understanding of particle filters and stochastic optimisation.

**Learning Outcomes**(LO1) Develop understanding of the use of probability theory.

(LO2) Understand stochastic models and the use statistical data.

(LO3) Demonstrate numerical skills for the understanding of stochastic processes.

(LO4) Understand the main machine learning techniques.

##### Maths Summer Industrial Research Project (MATL391)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To acquire knowledge and experience of some of the ways in which mathematics is applied, directly or indirectly, in the workplace.

To gain knowledge and experience of work in an industrial or business environment.Improve the ability to work effectively in small groups.

Skills in writing a substantial report, with guidance but largely independently This report will have mathematical content, and may also reflect on the work experience as a whole.

Skills in giving an oral presentation to a (small) audience of staff and students.

**Learning Outcomes**(LO1) To have knowledge and experience of some of the ways in which mathematics is applied, directly or indirectly, in the workplace

(LO2) To have gained knowledge and experience of work on industrial or business problems.

(LO3) To acquire skills of writing, with guidance but largely independently, a research report. This report will have mathematical content.

(LO4) To acquire skills of writing a reflective log documenting their experience of project development.

(LO5) To have gained experience in giving an oral presentation to an audience of staff, students and industry representatives.

##### Applied Stochastic Models (MATH360)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of stochastic model building for 'dynamic' events occurring over time or space. To enable further study of the theory of stochastic processes by using this course as a base.

**Learning Outcomes**(LO1) To understans the theory of continuous-time Markov chains.

(LO2) To understans the theory of diffusion processes.

(LO3) To be able to solve problems arising in epidemiology, mathematical biology, financial mathematics, etc. using the theory of continuous-time Markov chains and diffusion processes.

(LO4) To acquire an undertanding of the standard concepts and methods of stochastic modelling.

(S1) Problem solving skills

(S2) Numeracy

##### Derivative Securities (ACFI310)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**This course provides an introduction to derivative securities. Alternative derivative securities like forwards, futures, options, and exotic derivative contracts will be discussed. This incorporates detailing the properties of these securities.

Furthermore, a key aim is to outline how these assets are valued. Also the course demonstrates the use of derivatives in arbitrage, hedging and speculation. Finally, practical applications of derivatives and potential pitfalls are discussed.

The class is run as a discussion based forum and students are expected to read all necessary materials prior to each session.

**Learning Outcomes**(LO1) Students will be able to describe the principles of option pricing.

(LO2) Students will be able to compare and contrast alternative fair valuation techniques for pricing derivative instruments.

(LO3) Students will be able to explain the biases in option pricing models.

(LO4) Students will be able to apply an appropriate pricing model to a variety of contingent claim securities.

(LO5) Students will be able to recognize the trading strategy appropriate to expected future market conditions.

(LO6) Students will be able to derive and apply evolving models of derivative options to effectively manage risk transfer and assess their behaviour in the face of volatile financial and economic conditions.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Commercial awareness

(S5) Teamwork

(S6) Organisational skills

(S7) Communication skills

(S8) IT skills

(S9) International awareness

(S10) Lifelong learning skills

(S11) Ethical awareness

##### Finance and Markets (ACFI341)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**The module builds on the foundations of the existing finance modules and aims to give students a solid grounding in terms of understanding the recent global financial crisis and a wide range of risk management tools available to financial managers. Particular emphasis is placed on the issue of risk measurement. The following types of risk will be analysed extensively:

Interest rate risk;

Market risk;

Credit risk;

Liquidity risk;

Capital adequacy, and;

Sovereign risk.

The class is run as a discussion based forum and you are expected to read all necessary materials prior to each session.

**Learning Outcomes**(LO1) Understand how risk managementcontributes to value creation

(LO2) Understand how theglobal market for credit operates

(LO3) Explain the causes ofthe recent global credit crisis

(LO4) Overview the risksfacing a modern corporation

(LO5) Analyse the effectsof interest rate volatility on risk exposure

(LO6) Examine market risk,which results when companies actively trade bonds, equities and othersecurities

(LO7) Examine how creditrisk adversely impacts a financial institution’s profits

(LO8) Analyse the problemscreated by liquidity risk

(LO9) Familiarize with theconcept of capital adequacy and also with the Basel Accords

(LO10) Examine severalaspects of sovereign lending and the underlying risks

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Teamwork

(S5) Organisational skills

(S6) Communication skills

(S7) IT skills

(S8) International awareness

(S9) Lifelong learning skills

(S10) Ethical awareness

##### Further Methods of Applied Mathematics (MATH323)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**•To give an insight into some specific methods for solving important types of ordinary differential equations.

•To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.

•To build on the students'' existing knowledge of partial differential equations of first and second order.

**Learning Outcomes**(LO1) After completing the module students should be able to:

- use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.- solve simple integral extremal problems including cases with constraints;

- classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;

- classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions.

[This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]##### Linear Statistical Models (MATH363)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**- To understand how regression methods for continuous data extend to include multiple continuous and categorical predictors, and categorical response variables.

- To provide an understanding of how this class of models forms the basis for the analysis of experimental and also observational studies.

- To understand generalized linear models.

- To develop skills in using the computer package SPSS.

**Learning Outcomes**(LO1) Be able to understand the rationale and assumptions of linear regression and analysis of variance.

(LO2) Be able to understand the rationale and assumptions of generalized linear models.

(LO3) Be able to recognise the correct analysis for a given experiment.

(LO4) Be able to carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

(LO5) Be able to carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

(LO6) Be able to perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.

(S1) Be able to perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.

##### Mathematical Economics (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**· To explore, from a game-theoreticpoint of view, models which have been used to understand phenomena in whichconflict and cooperation occur.

· To see the relevance of the theorynot only to parlour games but also to situations involving humanrelationships, economic bargaining (between trade union and employer, etc),threats, formation of coalitions, war, etc..

· To treat fully a number ofspecific games including the famous examples of "The Prisoners'' Dilemma"and "The Battle of the Sexes".

· To treat in detail two-personzero-sum and non-zero-sum games.

· To give a brief review of n-persongames.

· In microeconomics, to look atexchanges in the absence of money, i.e. bartering, in which two individualsor two groups are involved. To see how the Prisoner''s Dilemmaarises in the context of public goods.

**Learning Outcomes**(LO1) After completing the module students should: Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

.Be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

·Be able to solve mathematically a variety of standard problems in the theory of games.

·To understand the relevance of such solutions in real situations.##### Mathematical Risk Theory (MATH366)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**•To provide an understanding of the mathematical risk theory used in the study process of actuarial interest

• To provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities)

• To develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry

• To prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).

**Learning Outcomes**(LO1) After completing the module students should be able to:

(a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria's, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules.

(b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables.

(c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables.

(d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships.

(e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples.

(f) Understand and be able to use Panjer's equation when the number of claims belongs to theR(a, b, 0) class of distributions, use the Panjer's recursion in order to derive/evaluate the probability function for the total aggregate claims.

(g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group are assumed to have deterministic variables.

(h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation),

(i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process.

(j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg's inequality for exponential and mixtures of exponential claim severities.

(k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities,

(l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical examples (using convolutions).

(m) Derive Lundberg's equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes.

(n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data.

(o) Explain the difference and adjust the chain ladder method, when inflation is considered.

(p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table).

(q) Use loss ratios to estimate the eventual loss and hence outstanding claims.

(r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method). Use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor).##### Measure Theory and Probability (MATH365)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**The main aim is to provide a sufficiently deepintroduction to measure theory and to the Lebesgue theory of integration. Inparticular, this module aims to provide a solid background for the modernprobability theory, which is essential for Financial Mathematics.

**Learning Outcomes**(LO1) After completing the module students should be ableto:

(LO2) master the basic results about measures and measurable functions;

(LO3) master the basic results about Lebesgue integrals and their properties;

(LO4) to understand deeply the rigorous foundations ofprobability theory;

(LO5) to know certain applications of measure theoryto probability, random processes, and financial mathematics.

(S1) Problem solving skills

(S2) Logical reasoning

##### Networks in Theory and Practice (MATH367)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**•To develop an appreciation of network models for real world problems.

•To describe optimisation methods to solve them.

•To study a range of classical problems and techniques related to network models.

**Learning Outcomes**(LO1) After completing the module students should be able to model problems in terms of networks and be able to apply effectively a range of exact and heuristic optimisation techniques.

##### Theory of Statistical Inference (MATH361)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options.

**Learning Outcomes**(LO1) To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

(LO2) To acquire an understanding of the blossoming area of Bayesian approach to inference.

(S1) Problem solving skills

(S2) Numeracy

##### Econometrics 1 (ECON212)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**Econometrics is concerned with the testing of economic theory using real world data. This module introduces the subject by focusing on the principles of Ordinary Least Squares regression analysis. The module will provide practical experience via regular laboratory session. This module also aims to equip students with the necessary foundations in econometrics to successfully study more advanced modules such as ECON213 Econometrics II, ECON311 Methods of Economic Investigation: Time Series Econometrics and ECON312 Methods of Economic Investigation 2: Microeconometrics.

**Learning Outcomes**(LO1) Reinforce the understanding of fundamental principles of statistics, probability and mathematics to be used in the context of econometric analysis

(LO2) Estimate simple regression models with pen and paper using formulae and with the econometric software EViews7

(LO3) Understand the assumptions underpinning valid estimation and inference in regression models

(LO4) Formulate and conduct intervals of confidence and tests of hypotheses

(LO5) Evaluate the impact that changes in the unit of accounts of variables and changes in the functional form of equations may have upon the results of OLS and their interpretation

(LO6) Assess the goodness of results by means of appropriate tests and indicators

(LO7) Assess predictions

(LO8) Extend analysis to the context of multiple linear regression

(LO9) Use EViews7 to estimate simple linear regression models and multiple linear regression models

(S1) Problem solving skills

(S2) Numeracy

(S3) IT skills

The programme detail and modules listed are illustrative only and subject to change.

#### Teaching and Learning

Your learning activities will consist of lectures, tutorials, practical classes, problem classes, private study and supervised project work. In Year One, lectures are supplemented by a thorough system of group tutorials and computing work is carried out in supervised practical classes. Key study skills, presentation skills and group work start in first-year tutorials and are developed later in the programme. The emphasis in most modules is on the development of problem solving skills, which are regarded very highly by employers. Project supervision is on a one-to-one basis, apart from group projects in Year Two.

#### Assessment

Most modules are assessed by a two and a half hour examination in January or May, but many have an element of coursework assessment. This might be through homework, class tests, mini-project work or key skills exercises.