Mathematics with Finance BSc (Hons)

We introduce you to the foundational language and notation that are used throughout mathematics, including mathematical logic, set theory, proofs, number systems, functions, elementary number theory, symmetry groups etc. Through these diverse topics, you will gain a solid understanding of the foundational principles of mathematics, laying the groundwork for more advanced studies in the field. Our emphasis on logical reasoning and mathematical structures will equip you with the skills to construct rigorous mathematical arguments essential for all university-level mathematics. We will also have a series of workshop focusing on how to write mathematics well, how to read and comprehend mathematics, and how to learn mathematics and use feedback effectively. This will help you transition to becoming an university-level mathematician.

Topics may include:

• Introduction to basic language, concepts, methods, and results in Mathematics, aiming to help students transition to university-level mathematics, including learning key independent and group study skills.
• Exploration of mathematical language, logic, and methods of proof, including induction.
• Elementary set theory.
• Functions, including injective, surjective, and bijective functions, as well as inverses.
• Operations, binary operations, equivalence relations and properties of algebraic structures.
• Introduction to groups, permutation groups, and cyclic groups.
• Elementary number theory: congruence, hcf and lcm, Euclidean algorithm, prime numbers, Fundamental Theorem of Arithmetic
• Different types of infinity: countable and uncountable sets.

Differential equations play a pivotal role in modelling numerous mathematical, scientific and engineering problems, stretching across celestial motion dynamics, neuron interactions, cancer progression, bridge stability and financial market trends. This module serves as an introduction to the essential theory and numerical methods used in solving ordinary and partial differential equations (ODEs and PDEs) while exploring their varied applications.

In this module, we will review the essential calculus techniques, including methods of differentiation and integration, necessary to solve ODEs. We will introduce ODEs, see their applications to real-world problems and explore techniques for generating both exact and approximate solutions for ODEs. We will also give a brief introduction to PDEs and their applications.

Topics may include:

• Review of trigonometric functions, hyperbolic functions, limits and differentiation.
• Integration, techniques such as integration by parts, partial fractions, and multiple integration.
• Review of sequences and series, covering convergence and divergence.
• Exploration of complex numbers, covering axiomatic foundations, complex conjugates, loci, polar form, De Moivre's Theorem, and roots.
• Notation and classification of ordinary differential equations.
• Linear ODEs and their applications.
• Selective exploration of non-linear ODEs and their applications.
• Introduction to systems of ODEs.
• Numerical integration: Trapezoidal Rule and Simpson’s Rule.
• Numerical solutions for ODEs: Euler method, using computer code in, for example, MATLAB, or Python.
• Partial differentiation, functions of two variables.
• Brief introduction to PDEs and their applications.

In this module you will have an introduction to both analysis and linear algebra.

Analysis

What is infinity? And how can we talk about a function 'tending to a limit'? Many problems in mathematics can't be solved exactly, but we can find a series of approximations tending to a true solution. However, our intuition can start to go very wrong in these cases as various pathological functions found in the 19th century illustrate. To make sense of this, we need rigorous definitions to enable us to definitively prove results; these can then be applied to accurately solve problems.

Analysis is the rigorous underpinnings and proof of calculus – differentiation and integration. In this module, we begin by laying the foundations for analysis by studying sequences and series. We then use this to explore functions and their continuity and differentiability.

Topics may include:

• Basic properties of the real numbers.
• Sequences and Series.
• Introduction to real valued functions of one variable, including limits.
• Continuous functions and their properties.
• Continuous functions on closed interval: Boundedness Theorem, Extreme Value Theorem, Intermediate Value Theorem.
• Differentiablility and properties of differentiable functions.
• Rolle’s Theorem, Mean Value Theorem.
• L’Hôpital's rule, Taylor’s Theorem.

Linear Algebra

Linear algebra is of fundamental importance throughout science, engineering and computing and it underpins almost all higher mathematics. Its real-world applications are extensive, encompassing error-correcting codes, noise and signal analysis, facial recognition, quantum computer algorithms, search engine ranking, computer game graphics and even providing you personalised suggestions on Netflix.

In this module, we adopt a dual approach, blending theoretical exploration with a computational and practical perspective on linear algebra. On the theoretical side, we will introduce the abstract notion of a vector space and explore linear transformations between them. We will use these theoretical results to tackle various computational questions ranging from solving systems of linear equations to utilising eigenspace structure for quick matrix computation.

Topics may include:

• Introduction to vectors, Euclidean space and matrices.
• Solving systems of linear equations using matrix techniques.
• Matrices and elementary row operations.
• Generalising Euclidean space over the real numbers to vector spaces over fields.
• Linear dependence and independence, bases, dimension, subspaces.
• Linear maps/transformations between vector spaces.
• Kernel, Image and the Rank-Nullity theorem.
• Matrix determinants and inverses.
• Eigenvalues, eigenvectors and diagonalisation.

In modern life, probability, data and statistics are all around us. Computers allow us to collect ever-growing amounts of data right down to the fine details of our internet use and shopping habits. In order to make any sense of this, we need statistics to be able to analyse the data and look for hidden patterns and trends. Even so, many processes, from die rolls to the movement of the stock exchange are random and we need probability to understand them. The mathematical theory of chance and statistical inference underpins artificial intelligence and is indispensable to empirical deduction. This module introduces students to both the theoretical knowledge and applied skills required for understanding and interrogating data and modelling uncertainty in real-world scenarios.

Topics covered in this module include:

• Introduction to a set-theoretic approach to probability
• Combinatorics
• Conditional probability, independence and Bayes' Theorem
• Discrete and continuous random variables
• Probability distributions, including expectation and variance
• Introduction to commonly used distributions (e.g. Binomial, Poisson, Normal)
• An introduction to joint probability distributions
• Methods of data collection, including ethical considerations
• Introduction to non-parametric and parametric hypothesis testing
• Correlation and Regression
• Forecasting and time-series
• Confidence intervals and uncertainty in data
• An introduction to machine learning and big data
• Introduction to appropriate software tools for data analysis and interpretation

This module is designed with student's employability at the forefront, aiming to develop essential transferable skills which employers look for and value in Mathematics graduates. Programming and computational techniques are all pervasive in today's society, from the next generation of quantum-secure algorithms allowing secure communication with your bank, to using mathematical computer simulations to model real-world phenomena.

The module is split into several different parts. We will provide an in-depth introduction to algorithms and the process of translating these into computer programs, using state-of-the-art software tools, such as Python. This will provide you with a solid foundational understanding to tackle any future computational and programming challenges. You will develop important research and writing skills. You will learn to use LaTeX, an industry-standard typesetting system for produce professional scientific documents. Finally, we will also develop and practice key skills for gaining employment, such as CV building, interview techniques, and giving compelling and engaging presentations.

• Comprehension and understanding of mathematical arguments.
• How to write coherent mathematical arguments.
• Introduction to mathematical and scientific typesetting with the LaTeX package and associated editors.
• Incorporating graphs, figures, tables and bibliographic information in reports and articles, using LaTeX.
• Referencing methodologies in the Mathematical Sciences.
• Introduction to online search tools and searching the library catalogue.
• Introduce algorithms and how to translate them into computer programs.
• Introduce students to computer programming software, for example Python.
• Learn about basic programming concepts including algorithms, loops, conditional statements and functions.  Also learn about more advanced scientific functions.
• Introduction to numerical algorithms from a variety of mathematical areas.
• Working & communicating as part of a scientific team.
• Oral presentation design and delivery.
• CV building, job applications and interview training.

Choose from one of the following modules. Please note you may need GCSE or A-level qualifications in the preferred language to register, although this may not be the case for UO4107 Subsidiary Language for Beginners.

• Subsidiary Language for Beginners
• Chinese: Intermediate Language Development
• French: Intermediate Language Development
• Spanish: Intermediate Language Development
• French: Communication in Practice
• German: Communication in Practice
• Spanish: Communication in Practice

Many optimisation problems in business and industry can be expressed in the form of a linear programming problem and this is even more apparent with increasing reliance on Artificial Intelligence and Machine Learning. Businesses and industry use linear programming to determine what to make in order to maximise their profits, Amazon use it to schedule your parcels for delivery and it is also used widely in Game Theory: you can use it to beat your friends at rock-paper-scissors and other games!

In this module, we will study the theoretical background behind the linear programming methods, learn how to express real-world questions as linear programming problems and solve them by hand and using computer programs. We will also explore some other optimisation methods used in AI and Machine Learning. Topics may include:

• Canonical forms of linear programming problems.
• Theoretical considerations: relevant results from set theory and geometry.
• Integer linear programming.
• Solutions of linear programming models: Simplex and dual simplex methods, Pivot algorithm, and computer-based techniques.
• Degeneracy, cycling, duality.
• Use of a mathematical computer software package, for example, Python, Matlab, Excel etc.
• Application to logistics in transportation and assignment problems, VAM and the Hungarian algorithm.
• Game theory: zero-sum matrix games, multi-phase games.
• Non-linear optimisation techniques in Machine Learning

Stochastic processes serve as essential mathematical models for systems and phenomena exhibiting apparent randomness. Examples encompass diverse scenarios, such as the growth of a bacterial population, fluctuations in electrical current due to thermal noise, or the motion of gas molecules. The applications of stochastic processes span various disciplines, including biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, and financial markets. In order to understand such random behaviour, we introduce and study Markov chains, random walks, Brownian motion and stochastic differential equations. Through these topics, students will not only establish a robust foundation in the principles of stochastic processes but will also gain valuable insights into their diverse applications across numerous domains. The module's goal is to equip students with analytical tools essential for comprehending and modelling complex uncertainties, thereby enhancing their capacity to address real-world challenges in mathematics, statistics, and related fields.

Topics covered include:

• Brief review of Probability Theory via a Measure Theory approach.
• Martingales: Basic definitions, filtrations, stopping times.
• Doob's Martingale inequalities and Convergence Theorem.
• Markov Chains.
• Moment generating functions.
• Characteristic functions.
• Probability generating functions.

The intention of this module is to equip you with the key digital skills required to enhance the understanding of key concepts covered on this course. There will be special emphasis on the operationalisation of the concepts at the work environment, that is, how can technology be used to efficiently implement and enhance the level of productivity of the organisation. The operationalisation agenda required to be actioned from the perspective of the finance manager in using key technologies will the direction of the module. The expectation is that you will develop a broad and holistic understanding of the finance functions in the digital world. You will familiarise yourself with how technologies links various operations of the business activities of an organisation to enhance its value creation. To achieve the intentions of the module, the following indicative contents will covered:

• Data analytics (sources of data, descriptive analytics, predictive analytics and prescriptive analytics)
• Data visualisation techniques.
• Accounting software and information systems.
• Artificial intelligence and automation.
• Emerging technologies within accounting and finance
• Cybersecurity within accounting, finance and banking
• Ms Excel for accounting and finance (Analysis ToolPak and Solver, Pivot Tables-including Power Pivot and Power Query, logical functions - lookups, text functions and financial maths with Ms Excel)
• Decision trees and experimentation

The coverage of the above contents will provide a certain degree of mastery in the handling of technologies you require to become a competitive graduate. The value associated with the use of technology to implement key concepts within the accounting, finance and banking courses is immeasurable. For example, the provision of an experiential learning environment will enhance your employability skills thus making you a competitive graduate. 

This module aims to gain understating of credit risk and sustainability in the banking context. It includes understanding of financial difficulties for customers and clients leading to collections and recoveries and losses, and demonstrates the importance of policies and procedures and the PRA requirements on capital adequacy and liquidity for banks. It also covers the growing importance of the societal purpose and practices of sustainability in the banking industry including description of how the types of sustainability today change the way we operate in banking and identifying the key performances indicators in banking that reflect sustainability.

Students will be able to achieve the following learning outcomes by the end of this module:-

1. Understand how credit arrears are formed and the wider impact on the profit and loss account
2. Critically evaluate the regulatory framework within the banking sector and understand how losses, capital adequacy and liquidity have been determined to ensure compliance with PRA regulations.
3. Understand the importance of sustainability in banking including culture, conduct and ethics. 
4. Analyse the relationships between businesses and stakeholders with regards to sustainable practice
5. Interpret the principles and types of sustainability and assess how sustainability is measured within banking organisations.

Indictive contents:-

• Financial analysis of credit losses.
• Understanding arrears and losses.
• Collections and recoveries practices.
• Policies and procedures.
• Financial conduct authority policies and regulations in banking and finance including PRA, capital adequacy and stress testing.
• Why sustainable finance, ethics and conduct are important in banking.
• Business and its stakeholders.
• Principles of sustainability.
• The types of sustainability (human, social, economic, environmental).
• Finance practices and KPIs reflecting sustainability.
• Industry case studies.

This is an experiential learning opportunity that incorporates, 20 teaching contact hours/lectures to prepare for the150 contract hours where L5 students can use all their skills learned to date on an actual real-world (external business) client driven project, working in teams and produce an artefact. 

Students are also expected to undertake around 30 hours of self study.

This module not only gives them enhanced skills but the opportunity to work for a real client thus giving them a valuable CV and LInkedIn entry as work experience that can contribute to their employability portfolio. 

Students will collaborate in teams and produce full client documentation alongside a reflection of their expereince and this all give some much needed contemplation of their skills to date and how to use them. 

This module provides a structured, university-level work placement for 4, 5 or 7 weeks as one continuous block / period with a placement provider (i.e. industry apprioprate sector). It is designed to enhance your professional skills in a real-world job setting.

The placement can either be organised by you or with support from university staff.

All work placements within this module must be university-level; this means:

• Undertaking high-skilled work commensurate with level 5 study (e.g. report writing, attending meetings, delivering presentations, producing spreadsheets, writing content on webpages, social media, marketing services/products etc)
• Physically placed (albeit part of it can be hybrid) within an employer setting in one continuous block / period for 4, 5 or 7 weeks for a minimum of 140-147 hours over the course of the entire work placement

Where applicable, your existing part-time employer can be approached/used as the placement provider, if the high-skilled work.

• criterion above is fulfilled for the full duration of the placement.
• All quality assurances/agreements provided by the University are adhered to, by you and the employer.

The work placement context may not necessarily, reflect your degree discipline per se, but rather, it will give you an enriched experience to enhance your professional skills in a real-world job setting.

Quality Control is an important application of statistical theory and methods. Since the industrial revolution, factories have needed to find ways to manufacture high quality products while minimising cost to maintain a competitive edge. From monitoring an automotive production line to stop out-of-control processes, to sampling techniques in the food production industries designed to minimise food waste, quality control procedures form a key part of the modern manufacturing process.

In this module you will learn the techniques companies employ to achieve the above. Topics for this module may include:

• Introduction to quality control: summary of relevant probability theory and probability distributions.
• Methods and philosophy of statistical process control.
• Control charts for variables: mean, median and range charts, special charts.
• Control charts for attributes: p-charts, np-charts, u-charts, c-charts.
• Process capability studies.
• Acceptance control charts.
• Acceptance sampling: single, double, and sequential sampling plans, rectifying inspections.
• Modified control limits.
• Cumulative sum techniques: CUSUM Charts, V-Masks, Tabular CUSUM.

Numerical analysis is the exploration of algorithms that rely on numerical approximation; this is invaluable for the study of many complicated problems, where an exact solution is very difficult, or even impossible to obtain. Its applications extend across mathematics, physical sciences and engineering, and, in the 21st century, have expanded into life sciences, social sciences, medicine, business, and even the arts. This module serves as an introduction to the fundamental techniques and methodologies for solving mathematical problems using numerical methods and equips students with the skills needed to analyse, design, and implement numerical algorithms effectively across a broad range of mathematical problems.

The topics include:

• Solving nonlinear equations, finding roots, Newton iteration and related methods.
• Introduction to optimisation, optimisation of functions of several variables, with and without constraints.
• Interpolation and approximations: Lagrange interpolation, Hermite interpolations,
• Numerical integration and differentiation: Trapezoidal method, Simpson method.
• Solutions of ordinary differential equations: Euler method, Runge-Kutta method, multistep methods, stability, convergence.
• Solution techniques for partial differential equations including the heat equation.
• Implementation and programming, e.g., Python, MATLAB.
• Error analysis: developing an understanding of the sources of error in numerical computations and methods for analysing and controlling numerical errors.

The aim of this module is to examine strategic finance issues faced by multinational companies from an international perspective. It focuses on how these companies operate within the global financial environment. Students will learn to identify and analyze the various forms and sources of business risks that multinational companies encounter.

Specifically, the module aims to:

1. To enable students to critically evaluate the international financial environment of a multinational corporation, including financial institutions, financial markets and exchange rate systems.

2. To critically examine and evaluate the principal concepts in the theory and practice of international financial management.

3. To analyse, apply and evaluate financial strategies through application of relevant analytical tools to examine and assess major issues and developments in international financial management.

4. To provide a critical understanding of the principles of risk exposures and the management of its international financial operations

On successful completion of this module students should be able to:

1. Analyse the environment in which international financial management is undertaken and discuss the structure of a MNC.

2. Critically assess the operations of the currency and derivatives markets and theories of exchange rate determination.

3. Define, assess and apply techniques and methods to evaluate and manage exposures and risks deriving from international business.

4. Apply the principal concepts, theories and appropriate tools in international financial management and reflect upon contemporary thinking to analyse and evaluate the international financial strategies of organisations, capital structure and financing of MNCs.

5. Assess a wide body of empirical research literature on contemporary issues relating to international financial management and critically appraise it. 

The module aims to develop students' ability to apply financial models and related advanced analytical techniques to inform business decisions and to evaluate possible decision outcomes in a competitive business environment.

The project is an opportunity for you to explore deeply an area of mathematics of your choice.  This may be a topic not already covered in the degree, or you may explore in more depth a topic covered before.

You will engage with mathematical literature, seeking out material and learning for yourself, supported by regular meetings with your project supervisor.  You will develop skills of self-discipline and project management, and study the art of mathematical writing and communication.

You will negotiate with tutors the precise title and objectives for the project.