# Actuarial Mathematics BSc (Hons)

- Course length: 3 years
- UCAS code: NG31
- Year of entry: 2022
- 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.

A programme aimed at those students who want to work in the world of insurance, financial or governmental services, where actuarial mathematics plays a key role.

We have accreditation from the Institute and Faculty of Actuaries, the professional body for actuaries in the UK and from the Institute of Mathematics and its Applications. Currently, our students can receive exemptions for CM1, CM2, CS1, CS2, CB1 and CB2 from IFoA of the professional actuarial exams.

Interested in finding out more? Tianyi and Changyi share their experiences on the course, below.

*"Every student has a academic adviser. My academic adviser holds a meeting nearly every month to give us some suggestions or help us solve the problems. He have give me some useful tips for applying for graduate study and wrote reference letter for me."*

*Zeng Changyi, Actuarial Mathematics, Y3.*

As XJTLU students will join Year 2 at The University of Liverpool, this PDF provides relevant module information for the following programme(s):

View the 2+2 Mathematical Sciences brochure.

### Programme Year Two

Actuarial mathematics prepares students to be professionals who use mathematical models to analyse and solve financial problems under uncertainty. Actuaries are experts in the design, financing and operation of insurance plans, annuities, and pension or other employee benefit plans.

#### 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**(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

##### Principles of Economics II (ECON210)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**This module aims to provide students with an opportunity to further apply mathematically core economic principles (developed in ECON127 Principles of Economics) to the economic and business environment. This will enable students to approach the Economic environment analytically in both Microeconomic and Macroeconomic approaches and prepare students to meet the requirements of the Actuarial Faculty/Institute exemptions in Economics where demonstrated to an appropriate level.

**Learning Outcomes**(LO1) Students will be able to demonstrate knowledge and critical awareness of economic theory studies, particularly with respect to the economic & business environment.

(LO2) Students will be able to apply a range of economic techniques to solve problems in the areas studied.

(LO3) Students will be able to apply microeconomic and macroeconomic theory and techniques to economic and business environment problems.

(LO4) Students will be able to understand and appreciate recent developments and methodologies in economics, such as behavioural economics.

(S1) Creative Problem Solver

(S2) Analytical (Numerate)

(S3) Commercial Awareness

(S4) Organised and able to work under pressure

(S5) IT Literate (Digitally Confident)

(S6) International Awareness

##### Statistics and Probability I (MATH253)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**Use the R programming language fluently to analyse data, perform tests, ANOVA and SLR, and check assumptions.

Develop confidence to understand and use statistical methods to analyse and interpret data; check assumptions of these methods.

Develop an awareness of ethical issues related to the design of

studies.**Learning Outcomes**(LO1) An ability to apply advanced statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) Use the R programming language fluently for a broad selection of statistical tests, in well-defined contexts.

(S1) Problem solving skills

(S2) Numeracy

(S3) IT skills

(S4) Communication skills

##### 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**(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.

(LO7) Be able to calculate premium and policy values with and without expenses.

(LO8) Be confident in using R and excel in simple life insurance problems.

##### Financial Mathematics (MATH262)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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.

##### Statistics and Probability II (MATH254)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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

##### Metric Spaces and Calculus (MATH242)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To introduce the basic elements of the theory of metric spaces and calculus of several variables.

**Learning Outcomes**(LO1) After completing the module students should: Be familiar with a range of examples of metric spaces. Have developed their understanding of the notions of convergence and continuity.

(LO2) Understand the contraction mapping theorem and appreciate some of its applications.

(LO3) Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

(LO4) Understand the inverse function and implicit function theorems and appreciate their importance.

(LO5) Have developed their appreciation of the role of proof and rigour in mathematics.

(S1) problem solving skills

##### Numerical Methods (MATH256)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**20:80 **Aims**To demonstrate how these ideas can be implemented using a high-level programming language, leading to accurate, efficient mathematical algorithms.

**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 (MATH269)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**The aims of the module are to develop an understanding of how mathematical modelling and operational research techniques are applied to real-world problems and to gain an understanding of linear and convex programming, multi-objective problems, inventory control and sensitivity analysis.

**Learning Outcomes**(LO1) To understand the operational research approach.

(LO2) To be able to apply standard methods of operational research to a wide range of real-world problems as well as to problems in other areas of mathematics.

(LO3) To understand the advantages and disadvantages of particular operational research methods.

(LO4) To be able to derive methods and modify them to model real-world problems.

(LO5) To understand and be able to derive and apply the methods of sensitivity analysis.

(LO6) To understand the importance of sensitivity analysis.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Self-management readiness to accept responsibility (i.e. leadership), flexibility, resilience, self-starting, initiative, integrity, willingness to take risks, appropriate assertiveness, time management, readiness to improve own performance based on feedback/reflective learning

### Programme Year Three

In your final year, you will cover some specialised work in advanced actuarial and financial mathematics. Subsequently, you start to study more advanced ideas in both life and non-life insurance mathematics as well as stochastic modelling, econometrics and finance. This programme is designed to prepare you for a career as an actuary, combining financial and actuarial mathematics with statistical techniques and business topics.

#### Year Three Compulsory Modules

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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.##### Life Insurance Mathematics II (MATH373)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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

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

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**Introduce the stochastic modelling for different actuarial and financial problem.

Help students to develop the necessary skills to construct asset liabilities models and to value financial derivatives, in continuous time.

Prepare the students to sit for the exams of CM2 subject of the Institute and Faculty 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

##### Mathematical Risk Theory (MATH366)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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 Insurance and Finance (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.

Demonstrate how to apply software R to solve questions

Prepare students adequately to sit for the exams in CS1 and CS2 of the Institute and Faculty of Actuaries.**Learning Outcomes**(LO1) Be able to explain concepts of Bayesian statistics and calculate Bayesian estimators.

(LO2) Be able to state the assumptions of the GLM models - normal linear model, understand the properties of the exponential family.

(LO3) Be able to apply time series to various problems.

(LO4) Understand some machine learning techniques.

(LO5) Be confident in solving problems in R.

(S1) Problem solving skills

(S2) Numeracy

##### Actuarial Models (MATH376)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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).

(LO2) 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.

(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

(S1) Problem solving skills

(S2) Numeracy

(S3) Commercial awareness

##### Financial and Actuarial Modelling in R (MATH377)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **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

#### Year Three Optional Modules

##### Maths Summer Industrial Research Project (MATH391)

**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.

##### 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.]