# Actuarial Mathematics BSc (Hons)

- Course length: 3 years
- UCAS code: NG31
- Year of entry: 2021
- A-level requirements: AAB

## Honours Select

×This programme offers Honours Select combinations.

## Honours Select 100

×This programme is available through Honours Select as a Single Honours (100%).

## Honours Select 75

×This programme is available through Honours Select as a Major (75%).

## Honours Select 50

×This programme is available through Honours Select as a Joint Honours (50%).

## Honours Select 25

×This programme is available through Honours Select as a Minor (25%).

## Study abroad

×This programme offers study abroad opportunities.

## Year in China

×This programme offers the opportunity to spend a Year in China.

## Accredited

×This programme is accredited.

### Module details

### 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

##### Calculus I (MATH101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.3. To introduce the notions of sequences and series and of their convergence.

**Learning Outcomes**(LO1) Understand the key definitions that underpin real analysis and interpret these in terms of straightforward examples.

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

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

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

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

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

(LO7) Understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

(S1) Numeracy

##### Calculus II (MATH102)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**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**(LO1) Use Taylor series to obtain local approximations to functions

(LO2) Obtain partial derivatives and use them in several applications such as, error analysis, stationary points change of variables.

(LO3) Evaluate double integrals using Cartesian and Polar Co-ordinates.

(LO5)

##### Introduction to Finance (ACFI103)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module aims:

to introduce the students to finance.

to provide a firm foundation for the students to build on later on in the second and third years of their programmes, by covering basic logical and rational analytical tools that underpin financial decisions**Learning Outcomes**(LO1) Understand the goals and governance of the firm, how financial markets work and appreciate the importance of finance.

(LO2) Understand the time value of money.

(LO3) Understand the determinants of bond yields.

(LO4) Recognize how stock prices depend on future dividends and value stock prices.

(LO5) Understand net present value rule and other criteria used to make investment decisions.

(LO6) Understand risk, return and the opportunity cost of capital.

(LO7) Understand the risk-return tradeoff, and know the various ways in which capital can be raised and determine a firm's overall cost of capital.

(LO8) Know different types of options, and understand how options are priced.

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Teamwork

(S5) Organisational skills

(S6) Communication skills

(S7) IT skills

(S8) International awareness

(S9) Lifelong learning skills

(S10) Ethical awareness

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

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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**(LO1) Prepare basic financial statements

(LO2) Explain the context and purpose of financial reporting

(LO3) Demonstrate the use of double entry and accounting systems

(LO4) Record transactions and events

(LO5) Prepare a trial balance

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Organisational skills

(S5) Communication skills

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

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**• To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.

• To develop geometrical intuition in 2 and 3 dimensions.

• To introduce students to the concept of subspace in a concrete situation.

• To provide a foundation for the study of linear problems both within mathematics and in other subjects**Learning Outcomes**(LO1) Manipulate complex numbers and solve simple equations involving them, solve arbitrary systems of linear equations.

(LO2) Understand and use matrix arithmetic, including the computation of matrix inverses.

(LO3) Compute and use determinants.

(LO4) Understand and use vector methods in the geometry of 2 and 3 dimensions.

(LO5) Calculate eigenvalues and eigenvectors.

(S1) Numeracy

##### 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**(LO1) To know how to describe statistical data.

(LO2) To be able to use the Binomial, Poisson, Exponential and Normal distributions.

(LO3) To be able to perform simple goodness-of-fit tests.

(LO4) To be able to use an appropriate statistical software package to present data and to make statistical analysis.

(S1) Numeracy

(S2) Problem solving skills

(S3) IT skills

(S4) Communication skills

##### 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**(LO1) Explain the relationship between expenditures and national income and demonstrate how monetary and fiscal policies may be used to influence them

(LO2) Explain the behaviour of economic aggregates such as national income, inflation and unemployment over time

(LO3) Explain and assess government policy in a range of policy situations

(LO4) Explain the framework of national income accounting

(LO5) Use graphical and algebraic modelling to analyse the economy and economic policy

(LO6) Explain the interconnections between the markets for goods, money and labour

(LO7) Explain the principal influences on long-term growth and the short-run fluctuation in output around the long-run growth trend

(LO8) Locate, select and analyse information relevant to assessing the state of the economy and economic policy

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Teamwork

(S5) Organisational skills

(S6) Communication skills

(S7) International awareness

(S8) Lifelong learning skills

(S9) Ethical awareness

##### Principles of Microeconomics (ECON121)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **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**(LO1) Students will have the ability to understand, explain, analyse and solve core problems in microeconomics.

(LO2) Students will be able to practice and develop their mathematical techniques and understand the role of mathematical analysis in microeconomics.

(LO3) 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.

(LO4) Students will be able to apply their understanding of economic decision-making, optimisation and equilibrium to real world situations.

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

(S4) IT skills

### 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 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 prepare the students adequately and to develop their skills in order to be ready to sit the CM2 subject of the Institute and Faculty of Actuaries exams.

**Learning Outcomes**(LO1) Know how to optimise portfolios and calculating risks associated with investment.

(LO2) Demonstrate principles of markets.

(LO3) Assess risks and rewards of financial products.

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

##### Math273 - Life Insurance Mathematics I (MATH273)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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 CT5 subject of the Institute and Faculty of Actuaries.

**Learning Outcomes**(LO1) Be able to explain and analyze the factors that affect mortality, simple life assurance and life annuity contracts.

(LO2) Understand the concept (and the mathematical assumptions) of the future life time random variables in continuous and discrete time

(LO3) 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.

(LO4) Be able to define the survivals probabilities and the force of mortality of the (c) section of the Syllabus, 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. Understand the concept of the De Moivre, Makeham, Gompertz, Weibull and the exponential law (constant force of mortality) for modelling fractional ages, explain the basic difference between the laws above, be able to use these laws to calculate the survival probabilities of (c) of the Syllabus in numerical examples. Understand, define/calculate and derive the expected present values of all types of the life assurances of (d) of the Syllabus.

(LO5) 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.

(LO6) 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.

##### Microeconomics 1 (ECON221)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **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**(LO1) 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.

(LO2) Students will be able to think and apply themselves analytically to problems in the above-mentioned topics.

(LO3) Students will be able to gain problem solving skills using verbal, diagrammatic and mathematical methods to problems in the above topics.

(LO4) Students will be able to have a critical perspective regarding the assumptions underlying microeconomics models.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Organisational skills

(S5) communication skills

(S6) IT skills

(S7) Lifelong learning skills

##### 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**(LO1) 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.

(LO2) To derive formulae for all kinds of annuities.

(LO3) 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.

(LO4) To calculate the outstanding balance at any point in time.

(LO5) To calculate a schedule of repayments under a loan and identify the interest and capital components in a given payment.

(LO6) To calculate a missing quantity, being given all but one quantities, in a sinking fund arrangement.

(LO7) To calculate the present value of payments from a fixed interest security, bounds for the present value of a redeemable fixed interest security.

(LO8) Given the price, to calculate the running yield and redemption yield from a fixed interest security.

(LO9) To calculate the present value or real yield from an index-linked bond.

(LO10) To calculate the price of, or yield from, a fixed interest security where the income tax and capital gains tax are implemented.

(LO11) To calculate yield rate, the dollar-weighted and time weighted rate of return, the duration and convexity of a set of cash flows.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Commercial awareness

##### Financial Reporting and Finance (non-specialist) (ACFI290)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**The aim of the Financial Reporting and Finance module is to provide an understanding of financial instruments and financial institutions and to provide the ability to interpret published financial statements of non-financial and financial companies with respect to performance, liquidity and efficiency. An understanding of the concepts of taxation and managerial decision making are also introduced and developed.

**Learning Outcomes**(LO1) Describe the different forms a business may operate in;

(LO2) Describe the principal forms of raising finance for a business;

(LO3) Demonstrate an understanding of key accounting concepts, group accounting and analysis of financial statements;

(LO4) Describe the basic principles of personal and corporate taxation;

(LO5) Demonstrate an understanding of decision making tools in used in management accounting.

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

(S4) Organisational skills

(S5) Communication skills

##### Statistical Theory and Methods I (MATH263)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**75:25 **Aims**To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation.

To introduce students to an appropriate statistical software package.

**Learning Outcomes**(LO1) To have a conceptual and practical understanding of a range of commonly applied statistical procedures.

(LO2) To have developed some familiarity with an appropriate statistical software package.

(S1) Problem solving skills

(S2) Numeracy

(S3) IT skills

(S4) Communication skills

##### 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**(LO1) To have an understanding of basic probability calculus.

(LO2) To have an understanding of a range of techniques for solving real life problems of probabilistic nature.

(S1) Problem solving skills

(S2) Numeracy

#### Year Two Optional Modules

##### Introduction to 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**(LO1) After completing the module students should: - be fluent in the solution of basic ordinary differential equations, including systems of first order equations:- be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations:- be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations: - be able to solve simple first order partial differential equations: - be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.

##### Numerical Methods (MATH256)

**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

##### Operational Research: Probabilistic Models (MATH268)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce a range of models and techniques for solving under uncertainty in Business, Industry, and Finance.

**Learning Outcomes**(LO1) The ability to understand and describe mathematically real-life optimization problems.

(LO2) Understanding the basic methods of dynamical decision making.

(LO3) Understanding the basics of forecasting and simulation.

(LO4) The ability to analyse elementary queueing systems.

(S1) Problem solving skills

(S2) Numeracy

### Programme Year Three

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

#### Year Three Compulsory Modules

##### Actuarial Models (MATH376)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**1. Be able to understand the differences between stochastic and deterministic modelling.

2. Explain the need of stochastic processes to model the actuarial data

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

4. Prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT4 subject of the Institute of Actuaries.**Learning Outcomes**(LO1) 1 Understand 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).

2 The Kaplan-Meier (or product limit) estimate, the Nelson-Aalen estimate , Describe the Cox model for proportional hazards Apply the chi-square test, the standardised deviations test, the cumulative deviation test, the sign test, the grouping of signs test, the serial correlation test to testing the adherence of graduation data.

3 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.

(LO3) 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

(LO4) Understand the time series together with its applications

(S2) Problem solving skills

(S3) Numeracy

(S4) Commercial awareness

##### 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 occurring over time. To familiarise students with an important area of probability modelling.

**Learning Outcomes**(LO1) 1. Knowledge and Understanding After the module, students should have a basic understanding of:

(a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes

(b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain

(c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.

2. Intellectual Abilities After the module, students should be able to:

(a) formulate appropriate situations as probability models: random processes

(b) demonstrate knowledge of standard models (c) demonstrate understanding of the theory underpinning simple dynamical systems

3. General Transferable Skills

(a) numeracy through manipulation and interpretation of datasets

(b) communication through presentation of written work and preparation of diagrams

(c) problem solving through tasks set in tutorials

(d) time management in the completion of practicals and the submission of assessed work

(e) choosing, applying and interpreting results of probability techniques for a range of different problems.##### Financial and Actuarial Modelling in R (MATH377)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**1.To give a set of applicable skills used in practice in financial and insurance institutions. To introduce students to specific programming techniques that are widely used in finance and insurance.

2.To provide students with a conceptual introduction to the basic principles and practices of the programming language R and to give them experience of carrying out calculations introduced in other modules of their programmes.

3.To develop the abilities to set standard financial and insurance models in order to manage the risk of the cash flow of financial and insurance companies, reserve, portfolio etc.

4.To develop the awareness of statistical and numerical limitations of financial and actuarial models and to know about modern approaches to tackle these limitations.

**Learning Outcomes**(LO1) To be able to import Excel files into R.

(LO2) To know how to create and compute standard functions and how to plot them.

(LO3) To be able to define and compute probability distributions and to be able to apply their statistical inference based on specific data sets and/or random samples.

(LO4) To know how to apply linear regression.

(LO5) To be able to compute aggregate loss distributions/stochastic processes and to find the probability of ruin.

(LO6) To know how to apply Chain Ladder and other reserving methods.

(LO7) To know how to price general insurance products.

(LO8) To be able to compute binomial trees.

(LO9) To know how to apply algorithms for yield curves.

(LO10) To be able to apply the Black-Scholes formula.

(LO11) To know how to develop basic Monte Carlo simulations.

(S1) Numeracy

(S2) Problem solving skills

(S3) Communication skills

(S4) IT skills

(S5) Organisational skills

(S6) Commercial awareness

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

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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 CT5 subject of the Institute of Actuaries . Be familiar with R programming language to solve life insurance problems.

**Learning Outcomes**(LO1) Be able to explain, define and analyze the joint survival functions.

(LO2) 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.

(LO3) 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.

(LO4) 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.

(LO5) Be familiar with R solfware and uses in actuarial mathematics

(S1) Problem solving skills

(S2) Numeracy

##### 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

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

• To develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry

• To prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).

**Learning Outcomes**(LO1) After completing the module students should be able to:

(a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria's, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules.

(b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables.

(c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables.

(d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships.

(e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples.

(f) Understand and be able to use Panjer's equation when the number of claims belongs to theR(a, b, 0) class of distributions, use the Panjer's recursion in order to derive/evaluate the probability function for the total aggregate claims.

(g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group are assumed to have deterministic variables.

(h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation),

(i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process.

(j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg's inequality for exponential and mixtures of exponential claim severities.

(k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities,

(l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical examples (using convolutions).

(m) Derive Lundberg's equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes.

(n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data.

(o) Explain the difference and adjust the chain ladder method, when inflation is considered.

(p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table).

(q) Use loss ratios to estimate the eventual loss and hence outstanding claims.

(r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method). Use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor).##### Statistical Methods in Actuarial Science (MATH374)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**Provide a solid grounding in analysis of general insurance data, Bayesian credibility theory and the loss distribution concept. Provide an introduction to statistical methods for managing risk in non-life insurance and finance. Prepare the students adequately to sit for the exams of CT6 subject of the Institute of Actuaries.

**Learning Outcomes**(LO1) Be able to apply the estimation methods described in (b) of the Syllabus for thedistribution described in (a) of the Syllabus, be able to make hypothesis testingdescribed in (b) of the Syllabus for the distribution described in (a) of the Syllabus.

(LO2) Be able to estimate the parameters of the loss distributions when data complete/incompleteusing the method of moments and the method of maximum likelihood, be able to calculate the loss elimination ratio.

(LO3) Understand and use the Buhlmann model, the Buhlmann-Straub model, be able to state the assumptions of the GLM models – normal linear model, understand the properties of the exponential family.

(LO4) Be able to express the values of the life assurances in (d) of the Syllabus and the life annuities in (f) of the Syllabus in terms of the life table functions. Be able to use approximations for the evaluation of the life assurances in (d) of the Syllabus and the life annuities in (f) of the Syllabus based on a life table.

(LO5) Be able to describe the properties of a time series using basic analytical and graphical tools.

(LO6) Understand the definitions, properties and applications of well know time seriesmodels, fit time series models to practical data sets and select the suitable models, be able to perform simple statistical inference (forecasting) based on the fitted models, estimate and remove possible trend and seasonality in a time series, analyse the residuals of a time series using stationary models.

(S1) Problem solving skills

(S2) Numeracy

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**Be able to understand the stochastic modelling for different actuarial and financial problem.

Develop the necessary skills to construct asset liabilities models and to value financial derivatives.

Prepare the students to sit for the exams of CT8 subject of the Institute of Actuaries.

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

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

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

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

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

(S1) Problem solving skills

(S2) Numeracy

#### Year Three Optional Modules

##### Maths Summer Industrial Research Project (MATL391)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To acquire knowledge and experience of some of the ways in which mathematics is applied, directly or indirectly, in the workplace.

To gain knowledge and experience of work in an industrial or business environment.Improve the ability to work effectively in small groups.

Skills in writing a substantial report, with guidance but largely independently This report will have mathematical content, and may also reflect on the work experience as a whole.

Skills in giving an oral presentation to a (small) audience of staff and students.

**Learning Outcomes**(LO1) To have knowledge and experience of some of the ways in which mathematics is applied, directly or indirectly, in the workplace

(LO2) To have gained knowledge and experience of work on industrial or business problems.

(LO3) To acquire skills of writing, with guidance but largely independently, a research report. This report will have mathematical content.

(LO4) To acquire skills of writing a reflective log documenting their experience of project development.

(LO5) To have gained experience in giving an oral presentation to an audience of staff, students and industry representatives.

##### 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**(LO1) After completing the module students should be able to model problems in terms of networks and be able to apply effectively a range of exact and heuristic optimisation techniques.

##### 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**(LO1) After completing the module students should be able to:

- use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.- solve simple integral extremal problems including cases with constraints;

- classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;

- classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions.

[This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]##### International Trade (ECON335)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To develop an appreciation and understanding of basic principles determining the observed patterns of trade in the increasingly globalised world economy.

**Learning Outcomes**(LO1) Students will be able to understand the role of both absolute and comparative advantage in explaining observed patterns of trade in the global economy

(LO2) Students will be able to recognise both the strengths and limitations of the basic Ricardian approach and the Hecksher-Ohlin Theory of trade.

(LO3) Students will be able to critically evaluate the roles, achievements and failures of various international institutions in the context of international economic performance.

(LO4) Students will be able to critically understand and approach trade based on increasing returns to scale

(LO5) Students will be able to recognise the implications of different types of trade barriers and why countries protect.

(LO6) Students will develop a critical understanding of the importance and implications of different levels of economic integration in the context of NAFTA and the EU.

(LO7) Students will be able to critically evaluate trade-oriented growth strategies for less developed economies.

(S1) Problem solving skills

(S2) Teamwork

(S3) Organisational skills

(S4) Communication skills

(S5) IT skills

(S6) International awareness

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.