BSc Actuarial Science and Mathematics / Course details

Year of entry: 2024

Course unit details:
Contingencies 2

Course unit fact file
Unit code MATH39522
Credit rating 10
Unit level Level 3
Teaching period(s) Semester 2
Offered by Department of Mathematics
Available as a free choice unit? No


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.




Unit title Unit code Requirement type Description
Contingencies 1 - Actuarial Science MATH20962 Pre-Requisite Compulsory
MATH39522 pre-requisites


The course aims to provide further mathematical instruction in models using cashflows which depend upon survival, death and other uncertain factors. These have practical application in more complex forms of life assurance, health insurance and pension provision than was seen in MATH20962.


Learning outcomes

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

  1. Develop formula and associated probabilities¿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. Apply these to various settings found in practice. 

  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. 



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 more frequently than annually.

2. Competing Risks. Multiple State modelling. Valuing benefits contingent upon competing risks. 

3. Multiple decrement tables and their relationship to single decrement tables. 

Multiple decrement service tables. Defined benefit pensions benefit and contributions. Non-life contingencies. 

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

Determining reserves by zeroising negative cashflows




Assessment methods

Method Weight
Other 20%
Written exam 80%

One short, in-class test worth 20% and an optional non-credit bearing mock test also held in class.

End of semester examination: weighting 80%

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.

Recommended reading

Essential: Long form Course notes are provided. These are available on Black Board and cover all the material needed for the course.

Recommended: If you wish to read around the course and access some further examples then go to:

D.C.M. Dickson, M.R. Hardy and H.R. Waters, Actuarial Mathematics for Life Contingencies.

Study hours

Scheduled activity hours
Lectures 24
Tutorials 17
Independent study hours
Independent study 59

Teaching staff

Staff member Role
Jonathan Ferns Unit coordinator

Additional notes

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates

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