# Mathematics with Finance BSc (Hons)

- Course length: 3 years
- UCAS code: G1N3
- Year of entry: 2022
- A-level requirements: AAB

## 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

**Due to the impact of COVID-19 we are changing how the course is delivered.**

### 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**50:50 **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) Understand the key definitions that underpin real analysis and interpret these in terms of straightforward examples.

(LO2) Apply the methods of calculus and real analysis to solve previously unseen problems (of a similar style to those covered in the course).

(LO3) Understand in interpret proofs in the context of real analysis and apply the theorems developed in the course to straightforward examples.

(LO4) Independently construct proofs of previously unseen mathematical results in real analysis (of a similar style to those demonstrated in the course).

(LO5) Differentiate and integrate a wide range of functions;

(LO6) Sketch graphs and solve problems involving optimisation and mensuration

(LO7) 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**0:100 **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.

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

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**45:55 **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.

(S1) 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 Using R (MATH163)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. Use software R to display and analyse data, perform tests and demonstrate basic statistical concepts.

2. Describe statistical data and display it using variety of plots and diagrams.

3. Understand basic laws of probability: law of total probability, independence, Bayes’ rule.

4. Be able to estimate mean and variance.

5. Be familiar with properties of some probability distributions and relations between them: Binomial, Poisson, Normal, t, Chi-squared.

6. To perform simple statistical tests: goodness-of-fit test, z-test, t-test.

7. To understand and be able to interpret p-values.

8. To be able to report finding of statistical outcomes to non-specialist audience.

9. Group work will help students to develop transferable skills such as communication, the ability to coordinate and prioritise tasks, time management and teamwork.

**Learning Outcomes**(LO1) An ability to apply statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) An ability to understand, communicate, and solve straightforward problems related to the theory and derivation of statistical methods covered in the module's syllabus.

(LO3) An ability to understand, communicate, and solve straightforward theoretical and applied problems related to probability theory covered in the syllabus.

(LO4) Use the R programming language fluently in well-defined contexts. Students should be able to understand and correct given code; select appropriate code to solve given problems; select appropriate packages to solve given problems; and independently write small amounts of code.

##### 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

##### Introduction to Study and Research in Mathematics (MATH107)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**This module addresses what it means to be a mathematician, as an undergraduate and beyond that into academia or industry, and prepares students to succeed as such. It aims to:

- bridge the gap in language and philosophy between A-level and (more rigorous) University mathematics;

- equip students with the basic tools they need for their mathematical careers;

- enable students to take responsibility for their learning and become active learners;

- familiarise students with mathematics research as conducted within the department;

- build students' confidence in handling various forms of mathematical communication.**Learning Outcomes**(LO1) Foundational knowledge of objects, processes, logic and reasoning required for university level mathematics.

(LO2) Awareness of the nature of mathematics at University and beyond, and the implications of this for themselves.

(LO3) Demonstrate proactive engagement in the student's own learning.

(LO4) Development of skills for mathematical communication (including mathematics proofs).

##### Theory of Interest (MATH167)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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.

To teach students:

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.

To derive formulae for all kinds of annuities.

To understand an annuity with level payments, immediate (or due), payable monthly, (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.

To calculate the outstanding balance at any point in time.

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

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

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

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

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

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

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

**Learning Outcomes**(LO1) Ability to understand, communicate, and solve straightforward problems and calculated quantities in the theory of interest.

(LO2) Ability to apply concepts and methods of theory of interest to well defined contexts, and interpret results.

### 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**90:10 **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. 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

##### Statistics and Probability I (MATH253)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**Use the R programming language fluently to analyse data, perform tests, ANOVA and SLR, and check assumptions.

Develop confidence to understand and use statistical methods to analyse and interpret data; check assumptions of these methods.

Develop an awareness of ethical issues related to the design of

studies.**Learning Outcomes**(LO1) An ability to apply advanced statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) Use the R programming language fluently for a broad selection of statistical tests, in well-defined contexts.

(S1) Problem solving skills

(S2) Numeracy

(S3) IT skills

(S4) Communication skills

##### Differential Equations (MATH221)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **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

##### Financial Mathematics (MATH262)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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 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) Know how to optimise portfolios and calculating risks associated with investment.

(LO2) Demonstrate principles of markets.

(LO3) Assess risks and rewards of financial products.

(LO4) Understand mathematical principles used for describing financial markets.

##### Statistics and Probability II (MATH254)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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

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

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **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**First Semester **Exam:Coursework weighting**50:50 **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

##### Metric Spaces and Calculus (MATH242)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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.

(LO2) Understand the contraction mapping theorem and appreciate some of its applications.

(LO3) Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

(LO4) Understand the inverse function and implicit function theorems and appreciate their importance.

(LO5) Have developed their appreciation of the role of proof and rigour in mathematics.

(S1) problem solving skills

##### Numerical Methods (MATH256)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**20:80 **Aims**To demonstrate how these ideas can be implemented using a high-level programming language, leading to accurate, efficient mathematical algorithms.

**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

##### Operational Research (MATH269)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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

### Programme Year Three

Choose 4 further modules (2 from semester 1 and 1 from semester 2) of which at least 3 must be MATH modules

#### Year Three Compulsory Modules

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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.##### Stochastic Modelling in Insurance and Finance (MATH375)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**Introduce the stochastic modelling for different actuarial and financial problem.

Help students to develop the necessary skills to construct asset liabilities models and to value financial derivatives, in continuous time.

Prepare the students to sit for the exams of CM2 subject of the Institute and Faculty of Actuaries.

**Learning Outcomes**(LO1) Understand the continuous time log-normal model of security prices, auto-regressive model of security prices and other economic variables (e.g. Wilkie model). Compare them with alternative models by discussing advantages and disadvantages. Understand the concepts of standard Brownian motion, Ito integral, mean-reverting process and their basic properties. Derive solutions of stochastic differential equations for geometric Brownian motion and Ornstein-Uhlenbeck processes.

(LO2) Acquire the ability to compare the real-world measure versus risk-neutral measure. Derive, in concrete examples, the risk-neutral measure for binomial lattices (used in valuing options). Understand the concepts of risk-neutral pricing and equivalent martingale measure. Price and hedge simple derivative contracts using the martingale approach.

(LO3) Be aware of the first and second partial derivative (Greeks) of an option price. Price zero-coupon bonds and interest–rate derivatives for a general one-factor diffusion model for the risk-free rate of interest via both risk-neutral and state-price deflator approach. Understand the limitations of the one-factor models.

(LO4) Understand the Merton model and the concepts of credit event and recovery rate. Model credit risk via structural models, reduced from models or intensity-based models.

(LO5) Understand the two-state model for the credit ratings with constant transition intensity and its generalizations: Jarrow-Lando-Turnbull model.

(S1) Problem solving skills

(S2) Numeracy

##### Numerical Analysis for Financial Mathematics (MATH371)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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 familiarise students with sample generating methods, including acceptance-rejection and variance reduction, and its application in finance**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 generate a sample for a given probability distribution and its use in finance

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

(S1) Problem solving skills

(S2) Numeracy

##### Statistical Methods in Insurance and Finance (MATH374)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **Aims**Provide a solid grounding in GLM and Bayesian credibility theory.

Provide good knowledge in time series including applications.

Provide an introduction to machine learning techniques.

Demonstrate how to apply software R to solve questions

Prepare students adequately to sit for the exams in CS1 and CS2 of the Institute and Faculty of Actuaries.**Learning Outcomes**(LO1) Be able to explain concepts of Bayesian statistics and calculate Bayesian estimators.

(LO2) Be able to state the assumptions of the GLM models - normal linear model, understand the properties of the exponential family.

(LO3) Be able to apply time series to various problems.

(LO4) Understand some machine learning techniques.

(LO5) Be confident in solving problems in R.

(S1) Problem solving skills

(S2) Numeracy

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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

##### Maths Summer Industrial Research Project (MATH391)

**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.

##### Econometrics 1 (ECON212)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **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 EViews8

(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

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

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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**40:60 **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 an appropriate statistical software package.

**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 an appropriate statistical software package.

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

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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.

##### Measure Theory and Probability (MATH365)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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

##### Derivative Securities (ACFI310)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **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

##### Game Theory (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur. To see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc. To treat fully a number of specific games including the famous examples of "The Prisoners' Dilemma" and "The Battle of the Sexes". To treat in detail two-person zero-sum and non-zero-sum games. To give a brief review of n-person games. In microeconomics, to look at exchanges in the absence of money, i.e. bartering, in which two individuals or two groups are involved.To see how the Prisoner's Dilemma arises in the context of public goods.

**Learning Outcomes**(LO1) To extend the appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

(LO2) To be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

(LO3) To be able to solve mathematically a variety of standard problems in the theory of games and to understand the relevance of such solutions in real situations.

##### Applied Stochastic Models (MATH360)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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 understand the theory of continuous-time Markov chains.

(LO2) To understand 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 understanding of the standard concepts and methods of stochastic modelling.

(S1) Problem solving skills

(S2) Numeracy

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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.

##### Mathematical Risk Theory (MATH366)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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).

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.