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School of Mathematics

# MATH20962 - 2011/2012

General Information
• Title: Contingencies 1
• Unit code: MATH20962
• Credit rating: 10
• Level: 2
• Pre-requisite units: MATH10951, MATH20962, MATH20702
• Co-requisite units: None
• This course unit may only be taken by students on the Actuarial Science and Mathematics degree programme
• School responsible: Mathematics
• Members of staff responsible: Harry Harper FIA
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## Unit specification

### Aims

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

### Brief description

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.

### Intended learning outcomes

On successful completion of this module students will be able to

* Retain knowledge and demonstrate understanding of the topics in this course unit. In particular :
• Define simple assurance and annuity contracts, and develop formulae for the means and variances of the present values of the payments under these contracts, as well as derive premiums for them;
• be comfortable using actuarial notation;
• derive relationships between expected present values of different contracts;
• understand the concept of, and use, a life table and a select life table;
• calculate and demonstrate understanding of net and gross premium reserves, as well as death strain and mortality profit; and
• be able to apply concepts learnt in the course to other applications not necessarily involving payments dependent on whether a person is alive or not.
* Have the basic knowledge and a set of tools and methods that can be used
• in subsequent course units;
• (together with MATH39522) to gain exemption from the Actuarial Profession CT5 examination;
• in a career involving mathematical and/or actuarial topics.

### Syllabus

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

1. Introduction to contingencies generally. Introduction to mortality, and probability of survival/death. Patterns of mortality. The life table. (2 lectures)
2. Introduction to stochastic mortality (CT4 topic). Tx, Kx, Fx, μx, ex including derivations where necessary. (2 lectures)
3. Approximations to the life table at non-integer ages. Select mortality. (2 lectures)
4. Annuities and assurances. definitions; benefits. (1 lecture). Valuation of these contracts from first principles (mean and variance) (3 lectures) including approximations.
5. Introduction of formulae and tables books. (tutorial)
6. Increasing/decreasing benefits (2 lectures) (including introduction to commutation functions)
7. Introduction to Premiums & Net Premium notation. Equivalence principle. Calculation of net premiums. Introduction to 'basis'.  (2 lectures)
8. Introduction to Net Premium Reserves, Prospective Reserves, profit. (2 lectures)
9. Retrospective reserves. Proof of equality of prospective and retrospective reserves. Recursive reserves. (2 lectures)
10. Death strain at risk, Mortality Profit, Thiele's differential equation. (2 lectures)
11. Bonuses (1 lecture) Expenses (1 lecture) including application to Gross Premiums & Reserves.

### Textbooks

• Core Reading : Subject CT5, Contingencies. Produced by the Actuarial Profession
• NL Bowers, Actuarial Mathematics, HU Gerber and JC Hickman, Society of Actuaries, 1997

### Learning and teaching processes

Two lectures per week in weeks 1-11 of semester 2. (22 lecture hours)
Feedback tutorial in weeks 2-12. (11 tutorial hours)
Private Study (67 hours)

### Assessment

Coursework Assignment. Will include questions involving short reports. weighting 20%,
Examination at end of semester 2, 2 hours duration, weighting 80%.

## Arrangements

Online course materials are available for this unit.