Looking for online learning materials for this unit?
Course materials are available in Blackboard

Contingencies 2

Unit code: MATH39522
Credit Rating: 10
Unit level: Level 3
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N



Additional Requirements

MATH39522 pre-requisites


Provide further mathematical instruction in models using cashflows which depend upon survival, death and other uncertain factors.


This unit is a continuation of MATH20962 (Contingencies 1) and covers the remaining part of the material required in subject CT5 of the Actuarial Profession's examinations.

In Contingencies 1 we looked at one life who may be in one of two states, alive or dead.

In Contingencies 2 we will consider assurances and annuities where payments are contingent on the lives of two people (e.g. an assurance that pays out when the second of two people dies or an annuity that only pays if two people are both alive). We will also consider models with more than two states (e.g. (i) alive and healthy or (ii) alive but ill or (iii) dead).

We will also look again at the recursive formula which links reserves but, in Contingencies 2, will use this to analyse profits. We will finally study mortality a bit more.



Learning outcomes

On completion of this course unit students will be able to: 

  1. Develop formula for and calculate expected present values (using either the standard R functions used in the course, by hand or using simple integration techniques) for assurances and annuities for joint life policies, last survivor policies and also policies where the order in which death might occur is a factor. Use these to calculate premiums and reserves for policies.
  2. Write down integrals for the expected present value of benefits subject to competing risks (e.g. retirement and death) defined by a Markov model.
  3. Recognise the difference between the forces of transition, dependent rates of transition and independent rates of transition and be able to calculate one from another.
  4. Use the forces and rates of transition described in 3. correctly in the context of both Multi State Modelling and the development and application of a Multiple Decrement Tables to calculate expected present values.
  5. Develop formula to calculate the expected present value of defined benefit pension schemes and use the standard functions developed in R to calculate their value.
  6. Develop formula for and calculate the profit vector, profit margin and Internal Rate of Return for both conventional and Unit Linked policies.
  7. Describe why insurers zeroise their reserves and demonstrate how to do this.
  8. Describe the factors affecting mortality and also the different forms of mortality selection.
  9. Write down the formula and calculate values for a number of single figure indices developed during the course which are used to compare the mortality experience between different populations.



Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

Two short tests under examination conditions, each worth 10%.

End of semester examination: two hours weighting 80%



This unit explores some further simple financial topics from a mathematical point of view.

1. Annuities and Assurances involving two lives. Single and joint life function. Last survivor functions. Present values of joint life and last survivor annuities, contingent assurances , reversionary annuities and annuities payable m times per year. [5 lectures]

2. Competing Risks. Multiple State modelling. Valuing benefits contingent upon competing risks. [3 lectures]

3. Multiple decrement tables and their relationship to single decrement tables. [2 lectures]

Multiple decrement service tables. Salary-related pensions benefit and contributions. Non-life contingencies [4 lectures]

Discounted emerging cost techniques. Unit-linked contracts. Expected cashflows. Profit tests and profit criteria. Product pricing [4 lectures]

Determining reserves by zeroising negative cashflows [2 lectures]

4. heterogeneity. Factors affecting mortality, selection and the need for different mortality tables. Risk-classification. Single figure indices. [2 lectures]



Recommended reading

  • Subject CT5 Contingencies Produced by the Actuarial Profession.
  • D.C.M Dickson, M.R Hardy and H.R Waters, Actuarial Mathematics for Life Contingencies.
  • Actuarial Mathematics N.L. Bowers, H.U. Gerber and J.C. Hickman, Society of Actuaries, 1997.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 22 hours
  • Tutorials - 22 hours
  • Independent study hours - 56 hours

Teaching staff

Jonathan Ferns - Unit coordinator

▲ Up to the top