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Contingencies 1 - Actuarial Science

Unit code: MATH20962
Credit Rating: 10
Unit level: Level 2
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N



Additional Requirements

MATH20962 pre-requisites

For students on the Actuarial Science and Mathematics programme only.


The unit aims to provide a mathematical introduction to models using cashflows which depend upon survival, death and other uncertain factors.


The course covers the first part of the material required in subject CT5 of the Actuarial Profession's examinations. Techniques and concepts developed in MATH10951 & MATH20951 are extended to cover the case where the payments are uncertain in timing.

Learning outcomes

On successful completion of this module students will be able to:

  1. Use mortality models to calculate the probabilities of death/survival using both ultimate and select mortality and also calculate life expectancies.
  2. Describe the circumstances where select mortality is used and summarise in words the differences between ultimate and select mortality.
  3. Recognise simple assurance and annuity contracts, and develop formulae for the present value of the payments under these contracts and the associated means and variances of these present values.
  4. Calculate the value of the mean and variance in 3. either by hand, through the use of simple integration techniques or by the use of standard functions developed in R for this course, as is appropriate.
  5. Interpret and use correct actuarial notation.
  6. Use the relationships between the expected present values of basic contracts to calculate the expected present value for more complex insurance policies from the standard R functions developed for the course, at a point in time and over a period of years.
  7. Use the techniques from 3. to 6. above to write down the formulae for and to calculate the value of :-
  • Net and gross premiums
  • Net and gross premium reserves
  • Death Strain and mortality profit
  • Net Loss variable.
  1. Comment on and provide explanation of the relative values of the items in 7. above at a point in time and over a period of years.
  2. Provide proofs concerning the relationship between various probabilities or expected present values.


Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

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

Examination at end of semester 2, 2 hours duration, weighting 80%.


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

1. Revision of financial mathematics relevant to this course (compound interest) (1 lecture)

2. Revision of probability theory relevant to this course and general introduction to contingencies (1 lecture)

3. Introduction to mortality models : random variables Tx  and Kx  and their properties ,the probabilities of survival or death, actuarial notation, the life table, approximations for non-integer ages and select mortality (5 lectures)

4. Assurances (generally these are insurance policies that make a payment when a person dies) : explanation of what they are and the different types (whole of life, term, endowment, deferred and increasing and also depending on the precise timing of the payment made), placing a value on these policies (Expected Present Value(EPV)), calculating the present value of the underlying random variable and also the variance.(3 lectures)

5. Annuities (generally an insurance policy that makes a series of regular payments if a person remains alive):  explanation of what they are and the different types (whole of life, term,  deferred and increasing and also depending on the precise timing of the payments), placing a value on these policies (Expected Present Value), calculating the present value of the underlying random variable and also the variance. Certain important approximations are also covered.(4 lectures)

6. Explanation of the Principle of Equivalence and the calculation of net premiums. Explanation of concept of Net Loss with examples and also how it might be used to calculate premiums.(2 lectures)

7. Expenses of an insurance company and the calculation of Gross Premiums. Explanation of with profits policies and the application of bonus (a means to increase the payment made by the policy). Calculation of the related EPV's and premiums.(2 lectures)

8. Introduction to reserving and the calculation of retrospective and prospective reserves on a net premium and gross premium basis. The recursive formula that links reserves from one year to the next.(3 lectures)

9. Death strain at risk and Mortality profit.(1 lecture).

The course will be supported by the use of the software R to provide a deeper understanding and also to perform calculations.

Recommended reading

  • Core 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.
  • NL Bowers, Actuarial Mathematics, HU Gerber and JC 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..

One Tutorial a week will be held in the computer cluster when the use of R can be practised.

Study hours

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

Teaching staff

Jonathan Ferns - Unit coordinator

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