Looking at microeconomic issues relating to markets and firms, and macroeconomic issues relating to money, banking and monetary policy, this module helps you to analyse economic issues from a business perspective. It demonstrates why economic concepts and principles are relevant to business issues by applying introductory economic theory to a range of issues that affect economic aspects of the business environment. Particular emphasis is given to interpreting the economic behaviour of individuals and firms, using theory to interpret events and evaluate policies.

This module covers project evaluation methods as well as risk, return and the cost of capital, including the capital asset pricing model. Corporate financing, including dividend policy and capital structure, options, and working capital management will also be investigated.

This module covers project evaluation methods, risk, return and the cost of capital (including the capital asset pricing model), corporate financing (including dividend policy and capital structure) and an introduction to options.

Probability provides the theoretical basis for statistics and is of interest in its own right.

Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions. Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas.

Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level.

Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.

Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series.

Further possible topics include Stirling's Formula, infinite products and Fourier series.

Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.

The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.

Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.

In this module, students will develop their ability to solve complex mathematical problems using computers. They will learn basic programming techniques, functions and syntax within Python, good programming practice, as well as how to code simple algorithms. Within this module, students will undertake a large programming project in which they will explore a mathematical topic from a range of choices, such as cryptography, error-correcting codes, or finite group theory, allowing them to explore an area of computational mathematics that interests them. Upon successful completion of the module, students will be able to write and debug functions and programs in Python, understand the benefits and limitations of using computers to solve mathematical problems, and produce a report that incorporates text, mathematics, and Python code.

In this module, students will develop their ability to solve complex mathematical problems using computers. They will learn basic programming techniques, functions and syntax within R, good programming practice, as well as how to code simple algorithms. Within this module, students will undertake a large programming project in which they will explore a mathematical modelling topic. The project itself will consolidate learning from the module, and will answer questions about real-world systems such as epidemics or gravitational bodies. Upon successful completion of the module, students will be able to write and debug functions and programs in R, understand the benefits and limitations of using computers to solve mathematical problems, and produce a report that incorporates text, mathematics, and R code.

In this module, students will develop their ability to solve complex mathematical problems using computers. They will learn basic programming techniques, functions and syntax within R, good programming practice, as well as how to code simple algorithms. Within this module, students will undertake a large programming project in which they will explore a mathematical topic from a range of choices, such as Markov chains or Monte Carlo simulations, allowing them to explore an area of computational statistics that interests them. Upon successful completion of the module, students will be able to write and debug functions and programs in R, understand the benefits and limitations of using computers to solve mathematical problems, and produce a report that incorporates text, mathematics, and R code.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the discrete mathematics and computing strands of mathematics, such as Ramsey's theorem, the birth of computer science, and Game theory. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the dynamics and analysis strands of mathematics, such as continued fractions, branching processes, and Kepler's laws. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the geometry and algebra strands of mathematics, such as symmetry in music, error-correcting codes, and counting equivalent patterns. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

This module provides a detailed analysis of four key Finance paradigms:

1. decision making under uncertainty, including utility theory,
2. state preference theory and arbitrage pricing
3. capital asset pricing and market equilibrium,
4. option pricing.

Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.

They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the modelling and probability strands of mathematics, such as Markov chains, beyond the Central Limit Theorem, and epidemic modelling. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes’ theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.

Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.

Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.

This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.

Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?

The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.

First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.

Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

The concept of generalised linear models (GLMs), which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored. The response variable may be classified as quantitative (continuous or discrete, i.e. countable) or categorical (two categories, i.e. binary, or more than categories, i.e. ordinal or nominal).

Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.

Much of contemporary statistics and AI assumes that data follows a fixed pattern for all time. However we know that change happens, and it is critically important that methods can detect and adapt to these changes. Efficient change-detection can either be a tool to recognise that something has changed in an underlying system (e.g. in medical or financial data collected through time) or an important component of a larger system that enables efficient model-fitting and prediction “in between” the changes. Lancaster University is a world-leading centre of excellence in developing methods to detect changes in data, with our methods deployed in domains such as retail, finance, medicine, space exploration, environmental science, cybersecurity and engineering.

This module introduces students to changepoint detection motivated by its applicability in a variety of different settings. Changepoints are sudden, and often unexpected shifts in the behaviour of a physical, biological, industrial or financial process. Students will understand the importance of identifying these shifts, and the module will cover the fundamental concepts within changepoint detection, the key approaches to At Most One Change (AMOC), and multiple changepoint identification. Students will explore key challenges around model selection concepts and techniques through a variety of models used in real-world applications, including changes in mean, variance, and regression. Successful completion of this module will see students be able to understand the importance of changepoint methodology, communicate technical ideas, critically evaluate approaches to solve problems, and make conclusions based on evidence in relation to a real-world problem.

This course equips students with the knowledge to apply corporate finance theory to real-world situations. It builds on and extends the concepts covered in the basic financial management courses and introduces advanced topics in Corporate Finance. The major topics covered include capital budgeting, capital structure, corporate valuation, real options, equity financing for startups, IPOs, leasing, short term financing, merger and acquisitions, and corporate governance

This module introduces students to two classes of continuous-time, dynamic compartmental models that describe the time evolution of natural systems such as epidemics, biological reactions in a cell, and the interactions between populations of predators and prey in an environment: Markov jump processes (MJPs) and ordinary differential equations (ODEs).

Students will examine model formulation in numerous real-world scenarios, delve into some tractable and partially-tractable examples, and use numerical simulation as a general method for visualising and classifying behaviour. Students will also examine stationary distributions and large n limits in order to further their understanding of MJPs, and find and examine stationary points and their stability to get to grips with ODEs, allowing students to justify their use within a variety of real-world processes. Upon successful completion of this module, students will be able to construct a mathematical model using real-world evidence.

The objective of this module is to offer a practical introduction to the workings of today’s financial markets and institutions built on a theoretical base. Moving beyond the descriptions and definitions provided by other textbooks and UK university courses in the field, this module encourages students to understand the connection between the theoretical concepts and their real-world applications.

Topics include foreign exchange, stock markets, bond markets, derivatives, central banks' monetary policy and financial crises. We also look at how trading in financial markets takes place. This module prepares students for successful careers in the financial services industry or successful interactions with financial institutions, whatever their future careers.

This module provides knowledge that is important to those concerned with financial management in a multinational setting. Areas covered include the relationships between exchange rates, interest rates and inflation rates, forward, futures and options markets, and corporate exchange rate risk management.

The aim of this module is to equip students with the tools necessary to enable them to make the core investment management decisions that managers face on a daily basis as well as the knowledge as to where they can find the information necessary to apply those tools. This course is an introduction to investment analysis, with emphasis on the pricing of equity securities. This course covers fundamental concepts and key issues in asset pricing; modern portfolio theory and its applications; equilibrium theories of asset pricing; portfolio performance evaluation; mutual funds and hedge funds. It provides an entry point to advanced-level subjects and foundational knowledge on the valuation and arbitrage of investment assets.

Cryptography is both a vital and thriving area of research, whose central aim is to provide robust methods of keeping sensitive data secure, validated and authenticated. This is clearly a highly impactful area, especially in an age where much of our sensitive data is being stored online. Without solid security protocols in place, hackers can freely access your data and use it for criminal gain (e.g. stealing money from your online banking, accessing/modifying your sensitive health data, selling your personal data on the cyber black market or using it to commit fraud).

This module provides an introduction to mathematical cryptography, introducing students to a range of methods of encryption, from the more classical forms (such as Caesar Shift, Vigenère and Enigma), to more contemporary methods of encrypting data (eg. RSA, El Gamal and Knapsack). During this module, students will consider the advantages, disadvantages, and efficiency of the different cryptographical approaches. They will also explore examples of potential attacks on these hard problems (e.g. Trial Division, Fermat’s method, Pollard rho, Dixon, Baby-step giant-step, Index calculus), and demonstrate attacks in situations where bad key generation or implementation has occurred.

The module will culminate in a brief introduction to Post-Quantum Cryptography, the theory of lattices, tough problems, and how these lead to schemes that are considered impenetrable to both classical and quantum computers. Successful completion of this module will see students be able to formulate real-world problems in mathematical formality, and recognise the applications of cryptography to other mathematical fields, such as Algebra and Number Theory.

Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.

An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise high-dimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.

This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.

Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and Black-Scholes models.

Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:

• Conditional expectation
• Filtrations
• Martingales
• Stopping times
• Brownian motion
• Black-Scholes formula

Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.

In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the real-world; improve their written and oral communication skills; and develop their critical thinking.

The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.

Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data.

They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.

This module is designed as an introduction to econometric and time series methods for the analysis and research of financial assets and capital markets relationships. The focus will be on the analysis of financial data and econometric methods for modelling of financial time series, risk management and forecasting. The key objectives are to i) explain how econometric methods can be used to learn about the behaviour of financial assets; ii) endow students with practical experience of analysing financial data useful for research and practical work in the quantitative finance industry using basic statistical software packages; and iii) endow students with the relevant quantitative skills for advanced studies in MSc programmes.

Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module.

As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time (initial analyses focused on whether there are betting strategies for a gambler that would ensure they won).

The focus will then be on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). Students will discover how to analyse Markov processes, and how they are used to model queues and populations.

Modern statistics is characterised by computer-intensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas.

Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application.

Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.

Lastly,students will spend time looking at Change-Point Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.