# Mathematics with Finance BSc (Hons)

- Course length: 3 years
- UCAS code: G1N3
- Year of entry: 2019
- 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 exemption 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.

**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**80:20 **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**differentiate and integrate a wide range of functions;

sketch graphs and solve problems involving optimisation and mensuration

understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

##### 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**use Taylor series to obtain local approximations to functions;

obtain partial derivaties and use them in several applications such as, error analysis, stationary points change of variables

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**80:20 **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**manipulate complex numbers and solve simple equations involving them

solve arbitrary systems of linear equations

understand and use matrix arithmetic, including the computation of matrix inverses

compute and use determinants

understand and use vector methods in the geometry of 2 and 3 dimensions

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

##### 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**After completing the module, students should be able to

- tackle project work, including writing up of reports detailing their solutions to problems;

- use computers to create documents containing formulae, tables, plots and references;

- use mathematical software packages such as Maple and Matlab to manipulate mathematical expressions and to solve simple problems,

- better understand the mathematical topics covered, through direct experimentation with the computer.

After completing the module, students should have developed their team working skills.

##### 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**

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

oscillation, vibration, resonance

##### 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**To know how to describe statistical data.

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

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

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

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

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **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**Prepare basic financial statements

Explain the context and purpose of financial reporting

Demonstrate the use of double entry and accounting systems

Record transactions and events

Prepare a trial balance

##### Introduction to Finance (ACFI103)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**Understand the goals and governance of the firm, how financial markets work and appreciate the importance of finance.

Understand the time value of money

Understandthe determinants of bond yields

Recognizehow stock prices depend on future dividends and value stock prices

Understandnet present value rule and other criteria used to make investment decisions

Understand risk, return and the opportunity cost of capital

Understandthe risk-return tradeoff, and know the various ways in which capital can beraised and determine a firm''s overall cost of capital

Knowdifferent types of options, and understand how options are priced

### 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**Students will be equipped with the tools and techniques of financial management

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

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

##### 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**Describe the different forms a business may operate in; Describe the principal forms of raising finance for a business;

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

Describe the basic principles of personal and corporate taxation;

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

##### 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**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.

To know the elementary techniques for the solution of ODEs.

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

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

To understand a range of applications of ODE.

##### 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 financial/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.

**Learning Outcomes**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.

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

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.

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.

To understand the role of Filtrations, martingales and option pricing in discrete time.To be able to discuss the theories of financial market behaviour.

To be able to perform calculations using the risk measures.

##### 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**To have a conceptual and practical understanding of a range of commonly applied statistical procedures.

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

##### 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** To have an understanding of basic probability calculus.

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

##### 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**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 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.

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.

#### 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**1. To provide an understanding of the techniques used in constructing, analysing, evaluating and interpreting mathematical models.

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

3. To use and develop mathematical skills introduced in Year 1 - particularly the calculus of functions of several variables and elementary differential equations.

**Learning Outcomes**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**90:10 **Aims**To introduce the basic elements of the theories of metric spaces and calculus of several variables.

**Learning Outcomes**To become familiar with a range of examples of metric spaces.

To develop an understanding of the notions of convergence and continuity.

To understand the contraction mapping theorem and appreciate some of its applications.

To become familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

To understand the inverse function and implicit function theorems and appreciate their importance.

To develop an appreciation of the role of proof and rigour in mathematics.

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

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**- Appreciate the operational research approach.
- Be able to apply standard methods to a wide range of real-world problems as well as applications in other areas of mathematics.
- Appreciate the advantages and disadvantages of particular methods.
- Be able to derive methods and modify them to model real-world problems.
- Understand and be able to derive and apply the methods of sensitivity analysis.

**Learning Outcomes**Appreciate the operational research approach.Be able to apply standard methods to a wide range of real-world problems as well asapplications in other areas of mathematics.

Appreciate the advantages and disadvantages of particular methods.

Be able to derive methods and modify them to model real-world problems.Understand and be able to derive and apply the methods of sensitivity analysis. Appreciate the importance of sensitivity 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**The ability to understand and describe mathematically real-life optimization problems.

Understanding the basic methods of dynamical decision making.

Understanding the basics of forecasting and simulation.

The ability to analyse elementary queueing systems.

##### 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**Appreciate the central role of securities markets in the economy Understand and apply appropriate economic theory to market organisation

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

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

##### 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**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.

##### Numerical Methods (MATH266)

**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 afundamental understanding of the core theoretical and empirical aspectsinvolved in corporate finance. In particular, the aims are that students will:

- Understand aspects of theoriesin corporate finance.
- Become familiar with a rangeof mathematical techniques commonly employed in corporate finance withparticular emphasis on bond valuation, stock valuation, firm valuation andassessing the probability that the firm will default on its debt obligations.
Be aware that all mathematical models, which are dependenton a set of underlying assumptions, have limitations in the sense that the answerto a particular problem might change once the underlying assumptions change.

**Learning Outcomes**Understand the principles of bonds and stocks valuation

Understand how credit rating agencies assign credit rating scores to bonds

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

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

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

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

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.

Understand the principles and practices involved in leasing, mergers and acquisitions

##### 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 occuring over time.

To familiarise students with an important area of probability modelling.

**Learning Outcomes** 1. Knowledge and Understanding:

After completing 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;

(d) understanding of Markov jump processes.2. Intellectual Abilities:

After completing 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 Skill:

(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**Ability to analyse a simple numerical method for convergence and stability

Ability to formulate approximations to derivative pricing problems numerically.

Ability to program matlab for pricing options

Awareness of the major issues when solving mathematical problems numerically.

##### Time Series and Its Applications in Economics (MATH372)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**1.

Give students an understanding of econometric time-series methodology.

2.

Give students an understanding of important extensions include volatility models of financial time-series and multivariate (multiple equations) models such as vector error correction and related co-integrating error correction models.

3.

Present interesting applications that econometric time-series methodology can be applied.

**Learning Outcomes**To be able to specify and demonstrate the distributional characteristics of a range of time series models

To be able to estimate appropriate models of financial and economic time series for the purposes of forecasting and inference

To be able to apply univariate and multivariate model selection and evaluation methods

To be able to accommodate conditional heteroskedasticity, unit roots and cointegration in economic and financial time series analysis

#### Year Three Optional Modules

##### 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**Reinforce the understanding of fundamental principles of statistics, probability and mathematics to be used in the context of econometric analysis Estimate simple regression models with pen and paper using formulae and with the econometric software EViews7

Understand the assumptions underpinning valid estimation and inference in regression models

Formulate and conduct intervals of confidence and tests of hypotheses

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

Assess the goodness of results by means of appropriate tests and indicators

Assess predictions

Extend analysis to the context of multiple linear regression

Use EViews7 to estimate simple linear regression models and multiple linear regression models

##### 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**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**100:0 **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 familiarity with the computer package SPSS.

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

understand the rationale and assumptions of linear regression and analysis of variance.

· understand the rationale and assumptions of generalized linear models.

· recognise the correct analysis for a given experiment.

· carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

· carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

· perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.

##### 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**After completing the module students should

. be able to model problems in terms of networks.

· 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**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**After completing the module students should be ableto:

master the basic results about measures and measurable functions;

master the basic results about Lebesgue integrals and their properties;to understand deeply the rigorous foundations ofprobability theory;

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

##### Derivative Securities (ACFI310)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**This courseprovides an introduction to derivative securities. Alternative derivative securities likeForwards, Futures, Options, and Exotic Derivative Contracts will bediscussed. This incorporates detailingthe properties of these securities. Furthermore, a key aim is to outline how these assets are valued. Also the course demonstrates the use ofderivatives in arbitrage, hedging and speculation. Finally, practicalapplications of derivatives and potential pitfalls are discussed.

The class is runas a discussion based forum and students are expected to read all necessarymaterials prior to each session.

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

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

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

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

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

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.

##### 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 (i) interest rate risk

(ii) market risk

(iii) credit risk

(iv) liquidity risk

(v) capital adequacy and

(vi) 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**Understand how risk managementcontributes to value creation Understand how theglobal market for credit operates

Explain the causes ofthe recent global credit crisis

Overview the risksfacing a modern corporation

Analyse the effectsof interest rate volatility on risk exposure

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

Examine how creditrisk adversely impacts a financial institution’s profits

Analyse the problemscreated by liquidity risk

Familiarize with theconcept of capital adequacy and also with the Basel Accords

Examine severalaspects of sovereign lending and the underlying risks

##### Mathematical Economics (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**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.

##### 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** To understans the theory of continuous-time Markov chains.

To understans the theory of diffusion processes.

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.

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

##### 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**To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

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

##### 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,

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

Tto 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 CM2 and CS2 of the Institute and Faculty of Actuaries.

**Learning Outcomes**(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 the R(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).

(s) Recognise extreme value distributions and calculate various measures of tail weight

##### Actuarial Models (MATH376)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **Aims**1 Understand the differences between stochastic and deterministic modelling

2 Be able to explain the need for stochastic processes to model the actuarial data

3 Be able to perform model selection depending on the outcome from a model.

4 Be able to forecast mortality rates

5 Be confident applying R software

6 Prepare students adequately and to develop their skills in order to be ready to sit the exams of CS2 subject of the Institute and Faculty of Actuaries.**Learning Outcomes** Use Markov processes to describe simple survival, sickness and marriage models, and describe other simple applications. Derive an appropriate Markov multi-state model for a system with multiple transfers, derive the likelihood function in a Markov multi-state model with data and use the likelihood function to estimate the parameters (with standard errors). The Kaplan-Meier (or product limit) estimate, the Nelson-Aalen estimate. Describe the Cox model for proportional hazards. Apply the chi-square test, the stardardised deviations test, the cumulative deviation test, the sign test, the grouping of signs test, teh serial correlation test to testing the adherence of graduation data

Understand the connection between estimation of transition intensities and exposed to risk (central and initial exposed to risk). Apply exact calculation of the central exposed to risk

Be able to forecast mortality rates using different models

Estimate actuarial models (e.g. lifetime distribution, mortality modes...) using R software

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.