# Actuarial Mathematics BSc (Hons)

- Course length: 3 years
- UCAS code: NG31
- 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 Actuarial Maths 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 Reposting and Finance

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

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

Subject **CT5 Contingencies**: Life Insurance Mathematics I & Life Insurance Mathematics II

Subject **CT6 Statistical Methods**: Mathematical Risk Theory & Statistical Methods in Actuarial Science

Subject **CT7 Economics**: Principles of Microeconomics, Principles of Macroeconomics, Microeconomics I & International Trade.

Subject **CT8 Financial Economics**: Financial Mathematics II, Security Markets & Stochastic Modelling in Insurance and Finance

#### Year One Compulsory Modules

##### 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

##### 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

##### 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.

##### Principles of Microeconomics (ECON121)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**This module aims to provide students with a clear foundation of the purpose, scope and topics of microeconomic analysis. Students will develop their ability to think critically and analytically, and understand how to frame real world problems in an economic model. This module forms the starting point for all future courses in Microeconomics.

This module also emphasizes the role of mathematics in economics.

**Learning Outcomes**Students will have the ability to understand, explain, analyse and solve core problems in microeconomics.

Students will be able to practice and develop their mathematical techniques and understand the role of mathematical analysis in Microeconomics.

Students will be able to familiarise themselves with the principles of using an ''economic model'' and how to model individual decision-making for both consumers and producers.

Students will be able to apply their understanding of economic decision-making, optimisation and equilibrium to real world situations.

##### Principles of Macroeconomics (ECON123)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**40:60 **Aims**The aims of this module are:

- To complement and build on Principles of Microeconomics and to provide a foundation for further studies in macroeconomics
- To introduce concepts and theories of economics which help understand changes in the macroeconomic environment
- to explain and analyse the formulation of government macroeconomic policy

**Learning Outcomes**Explain the relationship between expenditures and national income and demonstrate how monetary and fiscal policies may be used to influence them

Explain the behaviour of economic aggregates such as national income, inflation and unemployment over time

Explain and assess government policy in a range of policy situations

Explain the framework of national income accounting

Use graphical and algebraic modelling to analyse the economy and economic policy

Explain the interconnections between the markets for goods, money and labour

Explain the principal influences on long-term growth and the short-run fluctuation in output around the long-run growth trend

Locate, select and analyse information relevant to assessing the state of the economy and economic policy

##### 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 choose the equivalent of eight modules. Please note that we regularly review our teaching so the choice of modules may change.

Please note that along with the compulsory modules, two modules in Life Insurance and Financial Reporting & Finance must be taken.

#### Year Two Compulsory Modules

##### 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.

##### Microeconomics 1 (ECON221)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**This module, in accordance with Microeconomics 2, aims to provide a solid foundation of intermediate level microeconomic theory. It develops and extends three of the topics introduced in Principles of Microeconomics, namely, Consumer Theory, Producer Theory and General Equilibrium. It prepares the students for the more advanced modules in the second and third year like Microeconomics 2 and Game Theory.

**Learning Outcomes**Students will be able to demonstrate a thorough understanding of the core concepts and models used in consumer theory, producer theory and general equilibrium and an ability to apply these to arange of markets and settings. Students will be able to think and apply themselves analytically to problems in the above-mentioned topics.

Students will be able to gain problem solving skills using verbal, diagrammatic and mathematical methods to problems in the above topics.

Students will be able to have a critical perspective regarding the assumptions underlying microeconomics models.

##### 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.

##### Life Insurance Mathematics I (MATH273)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**Provide a solid grounding in the subject of life contingencies for single life, and in the subject of the analysis of life assurance and life annuities, including pension contracts.

Provide an introduction to mathematical methods for managing the risk in life insurance.

Develop skills of calculating the premium for a certain life insurance contract, including allowance for expenses and profits.

Prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CM1 subject of the Institute and Faculty of Actuaries.**Learning Outcomes**Be able to explain and analyze the factors that affect mortality, simple life assurance and life annuity contracts.

Understand the concept (and the mathematical assumptions) of the future life time random variables in continuous and discrete time, Be able to define the survivals probabilities and the force of mortality, explain these types of probabilities and the force of mortality intuitively, be able to calculate the different types of the survival probabilities in theoretical and numerical examples. Also modelling of fractional ages

Understand, define/calculate and derive the expected present values of all types of the life assurances of (d) of the Syllabus. Derive relations between life assurances both in continuous and discrete time, be able to use recursive equations for the calculation of the expected present value of different types of life assurances, calculate the variance of the present values for basic forms of life assurances.

Be able to derive the distributions and the moment/variance of the aforementioned future lifetimes, be able to make graphs of these future life times.

Be able to calculate premium and policy values with and without expenses

Be confident in using R and excel in simple life insurance problems

##### 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

##### 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.

#### Year Two Optional Modules

##### 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.

##### 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.

##### 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.

##### 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.

##### Microeconomics 1 (ECON221)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**This module, in accordance with Microeconomics 2, aims to provide a solid foundation of intermediate level microeconomic theory. It develops and extends three of the topics introduced in Principles of Microeconomics, namely, Consumer Theory, Producer Theory and General Equilibrium. It prepares the students for the more advanced modules in the second and third year like Microeconomics 2 and Game Theory.

**Learning Outcomes**Students will be able to demonstrate a thorough understanding of the core concepts and models used in consumer theory, producer theory and general equilibrium and an ability to apply these to arange of markets and settings. Students will be able to think and apply themselves analytically to problems in the above-mentioned topics.

Students will be able to gain problem solving skills using verbal, diagrammatic and mathematical methods to problems in the above topics.

Students will be able to have a critical perspective regarding the assumptions underlying microeconomics models.

##### 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.

##### Measure Theory and Probability (MATH265)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**The main aim is to provide a sufficiently deep introduction to the measure theory and to the Lebesgue theory of integration. In particular, this module aims to provide a solid background for the modern probability theory which is essential for Financial Mathematics.

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

· master the basic results about measures, measurable functions, Lebesgue integrals and their properties;

· to understand deeply the rigorous foundations of the probability theory;

· to know certain applications of the measure theory to probability and financial mathematics.

##### 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

##### Microeconomics 2 (ECON222)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **Aims**- This module, following on from with Microeconomics 1, aims to provide a solid foundation of intermediate level microeconomic theory.
- The module uses the theoretical foundations developed in the first semester and aims to extend the application of the skills acquired to more advanced topics such as welfare economics.
- This module also aims to prepare students for the more advanced modules in the third year by introducing topics such as asymmetric information and game theory.

**Learning Outcomes**Have a thorough understanding of the core concepts and models used in Welfare Economics, Asymmetric Information, and Game Theory. To prepare students to think and apply themselves to analyse a range of problems in the three areas mentioned above.

To develop problem solving skills using verbal, diagrammatic and mathematical methods to problems in the above topics.

To deepen a critical perspective regarding the assumptions underlying microeconomics models.

##### 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

### Programme Year Three

In addition to compulsory modules, choose two modules from the indicative list below.

#### Year Three Compulsory Modules

##### 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.##### 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

##### Life Insurance Mathematics II (MATH373)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**Provide a solid grounding in the subject of life contingencies for multiple-life, and in the subject of the analysis of life assurance, life annuities, pension contracts, multi-state models and profit testing.

Provide an introduction to mathematical methods for managing the risk in life insurance.

Analyze problems of pricing and reserving in relation to contracts involving several lives.

Prepare the students to sit for the exams of CM1 subjects of the Institute and Faculty of Actuaries.

Be confident in solving life insurance problems using R programming language and excel.

**Learning Outcomes**Be able to explain, define and analyze the joint survival functions.

Understand the concept (and the mathematical assumptions) of the joint future life time random variables in continuous and discrete time and monthly. Be able to derive the distributions and the moment/variance of the joint future lifetimes.

Be able to define the survivals probabilities/death probabilities of either or both two lives, explain these types of probabilities and the force of interest intuitively, be able to calculate the different types of the survival/death probabilities in theoretical and numerical examples. Understand, define and derive the expected present values of different types of the life assurances and life annuities for joint lives, be able to calculate the expected present values of the joint life assurances and life annuities in theoretical and numerical examples.

Understand the emerging costs for both traditional insurance and equity-linked insurance

Be confident using R and excel in actuarial mathematics

##### Statistical Methods in Actuarial Science (MATH374)

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

Provide good knowledge in time series including applications.

Provide an introduction to machine learning techniques.

Apply software R to solve questions

Prepare students adequately to sit for the exams in CS2 and CS1 of the Institute and Faculty of Actuaries.

**Learning Outcomes**Be able to explain concepts of Bayesian statistics and calculate Bayesian estimators

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

Be able to apply time series to varous problems.

Understand some machine learning techniques

Be confident in solving problems in R

##### Stochastic Modelling in Insurance and Finance (MATH375)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**1. To understand stochastic modelling for different actuarial and financial problems.

2. Develop the necessary skills to construct asset liabilities models and to value financial derivatives, in continuous time.

3. Prepare the students to sit for the exams in CM2 subject of the Institute and Faculty of Actuaries.**Learning Outcomes**Understand the continuous time log-normal model of security prices, auto-regressive model of security prices and other economic variables (e.g. Wilkie model). Compare them with alternative models by discussing advantages and disadvantages. Understand the concepts of standard Brownian motion, Ito integral, mean-reverting process and their basic properties. Derive solutions of stochastic differential equations for geometric Brownian motion and Ornstein-Uhlenbeck processes.

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

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

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

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

##### 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

#### Year Three Optional Modules

##### Industrial Organisation (ECON333)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To apply the tools of microeconomics to the analysis of firms, markets and industries in order to understand the nature and consequences of the process of competition. These tools will also be applied to the evaluation of relevant government policy. This will extend knowledge and skills of microeconomic analysis by covering recent advances in theory as well as empirical analysis of relevant microeconomic topics.

**Learning Outcomes**Students will be able to use economic principles, concepts and techniques to discuss and analyse government policy and economic performance with reference to standard frameworks in Industrial Organisation. Students will be able to apply standard frameworks, including verbal, graphical, mathematical and statistical representations of economic concepts and models, to explain and evaluate the effects of a range of competitive behaviours by firms and how they are influenced by economic incentives.

Students will be able to analyse current issues and problems in business and industry.

Students will be able to compare, contrast and critically evaluate alternative schools of thought in Industrial Organisation with reference to empirical evidence.

Students will be able to conduct competent applied economic research by locating, selecting and analysing information relevant to the study of Industrial Organisation.

Students will be able to communicate effectively in writing and in accordance with a report specification

##### International Trade (ECON335)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**85:15 **Aims**To develop an appreciation and understanding of basic principles determining the observed patterns of trade in the increasingly globalised world economy.

**Learning Outcomes**After successful completion of this module, students will be able to:

recognise both the strengths and limitations of the basic Ricardian approach

understand the determinants of the terms of trade, in both theory and practice

recognise the implications of the observed behaviour of the terms of trade for both developed and less developed economies

appreciate the relationships between globalisation, economic performance

critically evaluate the roles, achievements and failures of various international institutions in the context of international economic performance

##### Competition and Regulation (ECON337)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To apply the tools of microeconomics to the analysis of firms, markets, consumers and regulators in order to understand the nature and consequences of the process of competition and regulation. These tools will also be applied to the evaluation of competition and government policy. This will extend knowledge and skills of microeconomic analysis by covering recent advances in theory as well as empirical analysis of relevant microeconomic topics.

**Learning Outcomes**Use economic principles, concepts and techniques to discuss and analyse: - the power and limitations of competition as a force for market regulation;

- government policy to regulate market power to protect firms, consumers and employees.

Apply standard frameworks, including verbal, graphical, mathematical and statistical representations of economic concepts and models, to explain and evaluate the effects of a range of behaviours by firms and regulators and how they are influenced by economic incentives.

Identify and analyse current issues and problems in regulation and propose solutions.

Compare, contrast and critically evaluate regulation of different industries and market failures.

Communicate effectively orally and in writing and in accordance with project specifications

Conduct independent research in applied economics

Deliver a professional quality formal presentation by exhibiting: clarity and appropriate pace; logical structure; credibility; effective use of visual aids/technology

Identify problems

Analyse problems

Offer viable solutions

Use economic principles, concepts and techniques to discuss and analyse government policy and economic performance.

##### Behavioural Finance (ACFI311)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**Provide students with knowledge and understanding of theoretical and empirical limitations of and challenges to the efficient markets hypothesis;

Present the psychological foundations of Behavioural Finance and how they impact upon investors’ rationality and arbitrage;

Provide the opportunity for students to critically evaluate behaviourally induced market puzzles;

Present key behavioural trading patterns from a theoretical perspective and outline their empirical design.

**Learning Outcomes**(LO1) Understand the key concepts and issues in behavioural finance

(LO2) Possess a good command of the key theoretical and empirical literature in behavioural finance

(LO3) Have practiced skills of problem-solving and critical thinking

(LO4) Be familiar with the main implications (theoretical and empirical) of Behavioural Finance findings

(LO5) Construct and present critical evaluations of key academic papers in Behavioural Finance

(LO6) Development of key skills in terms of writtencommunication (eg by completing formative class questions and assignment;through a summative assignment and examination)

(LO7) Developmentof key skills in oral presentation (eg through group work)

(LO8) Development of key skills in terms of class participation

(LO9) Development of key skills in terms of planning and time-management (eg preparing for classes; observing assignment deadlines),problem-solving, critical thinking and analysis, numeracy (eg by applyingtheir extant quantitative knowledge in understanding behavioural financeissues) and initiative (eg searching relevant literature and information inpreparation of seminars and summative assignment)

(S1) Teamwork

(S2) Organisational skills

(S3) Communication skills

(S4) IT skills

(S5) International awareness

(S6) Lifelong learning skills

(S7) Ethical awareness

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

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

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

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

**Learning Outcomes**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.]

##### 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.

##### 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.

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.