Topic outline

  • Announcements

    Collapse all Expand all

  • Week 1 : Loss Distributions

    Activities
    • Study 'Getting started with R'
    • See Slides for Week 1 
    • Take your own notes during lectures

    • Attempt Worksheet 1 before the next week seminar session

    Topic
      Loss distributions
      Moment generating functions (MGFs) & Statistical distributions
      Estimation: Method of moments, Maximum likelihood estimation, Method of percentiles
      Goodness-of-fit

  • Week 2 : Reinsurance & Risk Models

    Activities
    • Work with R
    • See Slide Week 2
    • Attend lectures
    • Take your own notes during lectures

    • Attempt Worksheet 2 before the next week seminar session

    Topic

      Reinsurance
      Excess of loss reinsurance
      Proportional reinsurance
      Estimation when the sample is censored
      Policy excess
      Risk models, Features of a general insurance product, Models for short term insurance contracts



  • Week 3: Risk Models (continued)

    Activities
    • Work with R 
    • See Slide for Week 3
    • Attend lectures
    • Take your own notes during lectures

    • Attempt Worksheet 3 before the next week seminar session

    Topic

      Collective risk models
      Distribution functions
      Moments of compound distributions, The law of total expectation, The law of total variance, The mean, The variance
      Moment generating functions
      The compound Poisson distribution, Sums of independent compound Poisson random variables
      The compound Binomial distribution

  • Module Description

    This module leads to IFoA exemptions from subjects CS2 and CM2.

    Academic Content

    This module covers:

    • Loss distributions, with and without risk sharing.

    • Compound distributions and their applications in risk modelling.

    • Introduction to copulas.

    • Introduction to extreme value theory.

    • Ruin theory.

    • Run-off triangles.

     

    Disciplinary Skills

    At the end of this module, students should be able to:

    • Estimate parameters of a loss distribution.

    • Construct models appropriate for short term insurance contracts in terms of the number of claims and the amounts of individual claims.

    • Describe how a copula can be characterised as a multivariate distribution function.

    • Recognise extreme value distributions, calculate and interpret various measure of tail weights.

    • Calculate probabilities of the number of events in a given time interval, waiting times and ruin.

    • Describe and apply techniques for analysing a delay (or run-off) triangle.

     

    Attributes

    At the end of this module, students should have developed with respect to the following attributes:

    • Acquire and apply knowledge in a rigorous way.

    • Grasp the principles and practices of their field of study.

    • Acquire substantial bodies of new knowledge.

    • Use quantitative data confidently and competently.

    • Use information for evidence-based decision-making and creative thinking.

    • In this module we will use distributions such as exponential, gamma and normal to model insurance claims. These statistical distributions are called loss distributions and play a central role in the daily operations of insurance companies. We shall also learn how to construct models appropriate for short term insurance contracts in terms of the number of claims and the amounts of individual claims.

      For this, we start the module with a study of loss distributions, with and without reinsurance. We then study compound distributions and their applications in risk modelling. The module then introduces the concepts of copulas and extreme value theory. Finally, we study topics related to ruin theory and look at how insurance companies estimate their liabilities using run-off triangles.


  • Teaching Arrangements

    COURSE MATERIALS 

    All lecture slides, coursework questions and solutions are available on QMplus. The final exam and the two assessed courseworks are based on them.

    LECTURES

    These are the core learning experience of the module and will take place in person on campus at the times indicated, unless otherwise announced. Your engagement with the lectures is therefore crucial to your success.

    Please prepare for each lecture by reading the slides and reviewing the slides from previous weeks to be up-to-date. 

    COURSEWORK

    In addition to revising lectures, it is vital that you work on exercises to consolidate the material. Exercise sheets will be published every week focusing either on paper-based or programming problems. Solutions are discussed in weekly tutorials. The exercises provide a good starting point for further self-study on days where no lectures and tutorials are taking place. Please note that the coursework is not marked.

    TUTORIALS

    During the tutorial sessions (also called seminars) we will primarily discuss the exercise sheet of the previous week. You will join a group and present the coursework solutions. You also have the opportunity to ask general questions on the material.

    WORK TIME 

    According to recent studies most of the learning process happens outside lectures and tutorials, so it is indispensable that you allocate enough time to work on the module. The module is worth 15 credits, which translates to 150 hours of work. In addition to lectures and tutorials you should expect to invest around 10 hours or one full day in the module during lecture weeks.


    • Add information here.

  • Where to Get Help


    • There will undoubtedly be times during the term when you get stuck with revising the material or solving exercises. This is normal and an important part of the learning process. Nevertheless, if you feel overwhelmed or simply need some pointers to help you understand some of the concepts, there are various ways to reach out.

      Who to contact for what:

      • Student forum: it is encouraged that you discuss the material and exercises with your fellow students. You can post any kinds of questions in the student forum. The module organiser will also monitor any posts and answer queries.
      • Module organiser and tutor: are always willing to help either during the office hour or by email. Please contact us if you need any kind of more personal support/advice or have questions that you don't want to post in the forum. Your queries might be answered through the forum if they are of interest to all students in the class.
      • For problems beyond that: QMUL  counselling service   https://www.qmul.ac.uk/welfare/

  • Week 4: Risk Models (continued)

    Activities
    • Work with R
    • See Slide for Week 4
    • Attend lectures (Use 'Online Course Room' if you are not able to attend in person)
    • Take your own notes during lectures

    • Attempt Worksheet 4 before the next week seminar session

    Topic

    Aggregate claims distributions under proportional reinsurance
    Aggregate claims distributions under excess of loss reinsurance
    The individual risk model, Assumptions, Differences compared with the collective risk model, Mean and variance of aggregate claims in the individual risk model, Special case
    Parameter variability, Variability in a heterogeneous portfolio, Variability in a homogeneous portfolio

  • Week 5: Extreme Value Theory

    Activities
    • Work with R
    • See Slide for Week 5
    • Attend lectures
    • Take your own notes during lectures

    • Attempt Worksheet 5 before the next week seminar session

    Topic

      Extreme Value Theory: Extreme events, Key idea
      Generalised Extreme Value (GEV) distribution: Fréchet-type, Weibull-type and Gumbel-type GEV distribution
      Generalised Pareto Distribution (GPD): Threshold exceedances, Calculating threshold exceedances using R, GPD
      Measures of tail weight:
      1) Existence of moments
      2) Limiting density ratios, Plotting the limiting density ratios using R
      3) Hazard rate, Example: Pareto distribution, Plotting hazard rate using R
      4) Mean residual life, Example: Pareto distribution, Plotting mean residual life using R

  • Week 6: Copulas

    Activities
    • Work with R
    • See Slide for Week 6
    • Attend lectures
    • Take your own notes during lectures

    • You have Assessed Coursework 1

    Topic

      Marginal and joint distribution
      Association, concordance, correlation and tail dependence
      Copulas, Definition, Sklar’s theorem, Expressions of tail dependence and survival copula, Types of copula functions
      Fundamental copulas, Independence (or product) copulas, Co-monotonic (or minimum) copulas, Counter-monotonic (or maximum) copulas
      Explicit copulas, Archimedean copulas, Gumbel copula, Clayton copula, Frank copula
      Implicit copulas, Gaussian copula, Student’s t copula
      Choosing a suitable copula function

  • Week 7: Reading Week

    There will be lectures on Monday (a swap for Easter Monday) but no tutorial during week 7. You need to submit your first assessed coursework based on R during this week.

    See Assessment 1 in the Tab Assessments to access coursework questions, deadline and submission portal.

  • Week 8: Ruin Theory

    Activities
    • See Slide for Week 8
    • Attend lectures
    • Take your own notes during lectures

    • Attempt Worksheet 8 before the next week seminar session

    Topic

        Ruin theory
        The surplus process
        The probability of ruin in continuous time
        The probability of ruin in discrete time
        The Poisson process, Time to the first claim, Time between claims
        The compound Poisson process, Mean, variance and MGF
        Premium security loadings
        Lundberg’s inequality, Pictorial view, Interpretation, R as a function of the loading factor θ
        The adjustment coefficient, When individual claims are exponentially distributed, An upper bound for R, A lower bound for R, Summary of upper and lower bounds for R



  • Week 9: Ruin Theory (continued)

    Activities
    • See Slides for Week 9
    • Attend lectures
    • Take your own notes during lectures

    • Attempt Worksheet 9 before the next week seminar session

    Topic

        The effect of changing parameter values
        A formula for ψ(U) when X is exponential
        ψ(U, t ) as a function of t
        Ruin probability as a function of initial surplus
        Ruin probability as a function of loading factor
        Ruin probability as a function of the Poisson parameter
        Concluding remarks
        Valuing basic guarantees using simulation

  • Week 10: Run-off Triangles

    Activities
    • See Slide for Week 10
    • Attend lectures
    • Take your own notes during lectures

    • Attempt Worksheet 10 before the next week seminar session

    Topic

        Run-off triangles
        Types of reserves
        Presentation of claims data
        Estimating future claims
        Projections using development factors, Arithmetic average, Weighted average

        A statistical model for run-off triangles, Notation, Notes

        The chain ladder method

        The inflation-adjusted chain ladder method, Dealing with past inflation, Dealing with future inflation, Outstanding claims reserve



  • Week 11: Run-off Triangles (continued)

  • Week 12 - Revision + discussion of last year's exam paper

  • Assessment information

    • The module will be assessed by:

      • Final exam in the May/June examination period: 60% weighting. All materials and courseworks discussed in the module is examinable unless otherwise stated. Please note that the exam will be an on-campus three-hour exam. You will be allowed a non-programmable calculator. No notes will be allowed, and it is totally closed-form.
      • Assessed coursework 1 to be completed in Week 7 : 20% weighting. You will complete an analysis with the software R.
      • Assessed coursework 2 to be completed in Week 12 : 20% weighting. You will complete an analysis with the software Excel.

      If you fail the module based on these assessments, you will have the opportunity to obtain a pass mark by resitting the final exam during the Late Summer Examination Period (typically August).

      You will find written feedback on the Assessed Coursework on QMPlus before the final exam. Please feel free to join the office hours or contact the module organisers if you want to discuss the result.

      Past exam papers can be found below.

  • Assessed Coursework 1 - R-based

    Available: 12:00 noon, 26 February (Monday), 2024

    Deadline: 12:00 noon, 5 March (Tuesday), 2024


  • Assessed Coursework 2 - Excel-based

    Available: 11:00 am, 25 March (Monday), 2024

    Deadline: 12:00 noon, 8 April (Monday), 2024


  • Past Exam Papers

  • Q-Review


  • Online Reading List