# BSc Actuarial Science and Mathematics

Year of entry: 2022

## Course unit details:Contingencies 1 - Actuarial Science

Unit code MATH20962 10 Level 2 Semester 2 Department of Mathematics No

### Overview

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.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Probability 1 MATH10141 Pre-Requisite Compulsory
Probability 2 MATH20701 Pre-Requisite Compulsory
Financial Mathematics for Actuarial Science 1 MATH10951 Pre-Requisite Compulsory
Financial Mathematics for Actuarial Science 2 MATH20951 Pre-Requisite Compulsory
MATH20962 pre-requisites

For students on the Actuarial Science and Mathematics programme only.

### Aims

The unit aims to provide a mathematical introduction to models using cashflows which depend upon survival, death and other uncertain factors and forms the grounding for pricing and financially managing life assurance and pension products.

### 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 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.

### Syllabus

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.

### 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.

Examination at end of semester 2, 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..

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

• 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

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

### Teaching staff

Staff member Role
Jonathan Ferns Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.