CT5 Study Guide 2014

CT5: Study Guide Page 1

The Actuarial Education Company © IFE: 2014 Examinations

2014 Study Guide

Subject CT5

Introduction This Study Guide contains all the information that you will need before starting to study Subject CT5 for the 2014 exams. Please read this Study Guide carefully before reading the Course Notes, even if you have studied for some actuarial exams before. When studying for the UK actuarial exams, you will need a copy of the Formulae and Tables for Examinations of the Faculty of Actuaries and the Institute of Actuaries, 2nd Edition (2002). These are often referred to as simply the yellow Tables and are available separately from the Publications shop of the Institute and Faculty of Actuaries. You will also need a “permitted” scientific calculator from the list published in the Student Handbook. Please check the list carefully, since it is reviewed each year. You will find the list of permitted calculators and a link to the Publications shop on the profession’s website at www.actuaries.org.uk.

Contents Section 1 The Subject CT5 course structure Page 2 Section 2 ActEd study support Page 3 Section 3 How to study to pass the exams Page 12 Section 4 Frequently asked questions Page 16 Section 5 Core Reading and the Syllabus Page 17 Section 6 Syllabus Page 20

Page 2 CT5: Study Guide

© IFE: 2014 Examinations The Actuarial Education Company

1 The Subject CT5 course structure

There are four parts to the Subject CT5 course. The parts cover related topics and have broadly equal lengths. The parts are broken down into chapters. The following table shows how the parts, the chapters and the syllabus items relate to each other. The end columns show how the chapters relate to the days of the regular tutorials. This table should help you plan your progress across the study session.

Part Chapter Title No of pages

Syllabus objectives

Half day

2 full days

3 full days

1

1 Life assurance contracts 53 (i) 1-4, 6

1

1

1

2 Life annuity contracts 34 (i) 1, 3-4,

6-7

3 The life table 49 (i) 2, 6,

(ii) 1-3, 7

4 Evaluation of assurances and annuities

28 (i) 5-6, 8

(ii) 4-5

2

5 Net premiums and reserves 70 (i) 9, (ii) 6, (iii)

2 6 Variable benefits and with-profit

policies 48 (iv)

2

7 Gross premiums and reserves for fixed and variable benefit contracts

58 (v)

3

8 Simple annuities and assurances involving two lives

31 (vi)

3

2

9 Contingent and reversionary benefits

57 (vi)

10 Profit testing 51 (viii) 1-4

3

11 Determining reserves using profit testing

40 (viii) 5

4

12 Competing risks 21 (vii)

4

13 Multiple decrement tables 34 (viii) 6

14 Pension funds 71 (viii) 7-9

15 Mortality, selection and standardisation

58 (ix)



2 ActEd study support

Successful students tend to undertake three main study activities:

1. Learning – initial study and understanding of subject material

2. Revision – learning subject material and preparing to tackle exam-style questions

3. Rehearsal – answering exam-style questions, culminating in answering questions at exam speed without notes.

Different approaches suit different people. For example, you may like to learn material gradually over the months running up to the exams or you may do your revision in a shorter period just before the exams. Also, these three activities will almost certainly overlap. ActEd offers a flexible range of products to suit you and let you control your own learning and exam preparation. The following table shows the products that ActEd produces. Note that not all products are available for all subjects.

LEARNING

Course Notes

LEARNING &REVISION

Q&A Bank

X Assignments

Combined

Materials Pack (CMP)

X Assignment

Marking

Tutorials

Online Classroom

REVISION

Flashcards

REVISION & REHEARSAL

Revision Notes

ASET

Revision Tutorials

REHEARSAL

Mock Exam A

Additional Mock Pack

(AMP)

Mock / AMP Marking

The products and services available for Subject CT5 are described below.



“Learning” products

Course Notes

The Course Notes will help you develop the basic knowledge and understanding of principles needed to pass the exam. They incorporate the complete Core Reading and include full explanation of all the syllabus objectives, with worked examples and short questions to test your understanding. Each chapter includes the relevant syllabus objectives, a chapter summary and, where appropriate, a page of important formulae or definitions.

“Learning & revision” products

Question and Answer Bank

The Question and Answer Bank provides a comprehensive bank of questions (including some past exam questions) with full solutions and comments. The Question and Answer Bank is divided into five parts. The first four parts include a range of short and long questions to test your understanding of the corresponding part of the Course Notes. Part five consists of 100 marks of exam-style questions. X Assignments

The four Series X Assignments (X1 to X4) cover the material in Parts 1 to 4 respectively. Assignments X1 and X2 are 80-mark tests and should take you two and a half hours to complete. Assignments X3 and X4 are 100-mark tests and should take you three hours to complete. The actual Subject CT5 examination will have a total of 100 marks. Combined Materials Pack (CMP)

The Combined Materials Pack (CMP) comprises the Course Notes, the Question and Answer Bank and the Series X Assignments. The CMP is available in eBook format for viewing on a range of electronic devices. eBooks can be ordered separately or as an addition to paper products. Visit www.ActEd.co.uk for full details about the eBooks that are available, compatibility with different devices, software requirements and printing restrictions.



CMP Upgrade The CMP Upgrade lists all significant changes to the Core Reading and ActEd material so that you can manually amend last year’s study material to make it suitable for study for this year. The upgrade includes replacement pages and additional pages where appropriate. If a large proportion of the material has changed significantly, making it inappropriate to include all changes, the upgrade will still explain what has changed and if necessary recommend that students purchase a replacement CMP or Course Notes at a significantly reduced price. The CMP Upgrade can be downloaded free of charge from our website at www.ActEd.co.uk. X Assignment Marking

We are happy to mark your attempts at the X assignments. Marking is not included with the Assignments or the CMP and you need to order it separately. You can submit your scripts by email, fax or post. Your script may be marked electronically, in which case you will be able to download your marked script via a secure link on the internet. Otherwise your marked script will be returned to you in the post. Don’t underestimate the benefits of doing and submitting assignments:

Question practice during this phase of your study gives an early focus on the end goal of answering exam-style questions.

You’re incentivised to keep up with your study plan and get a regular, realistic assessment of progress.

Objective, personalised feedback from a high quality marker will highlight areas on which to work and help with exam technique.

In a recent study, it was found that students who attempt more than half the assignments have significantly higher pass rates.



Series Marking Series Marking applies to a specified subject, session and student. If you purchase Series Marking, you will not be able to defer the marking to a future exam sitting or transfer it to a different subject or student. We typically send out full solutions with the Series X Assignments. However, if you order Series Marking at the same time as you order the Series X Assignments, you can choose whether or not to receive a copy of the solutions in advance. If you choose not to receive them with the study material, you will be able to download the solutions via a secure link on the internet when your marked script is returned (or following the final deadline date if you do not submit a script). If you are having your attempts at the assignments marked by ActEd, you should submit your scripts regularly throughout the session, in accordance with the schedule of recommended dates set out in information provided with the assignments. This will help you to pace your study throughout the session and leave an adequate amount of time for revision and question practice. The recommended submission dates are realistic targets for the majority of students. Your scripts will be returned more quickly if you submit them well before the final deadline dates. Any script submitted after the relevant final deadline date will not be marked. It is your responsibility to ensure that scripts are received by ActEd in good time. Marking Vouchers Marking Vouchers give the holder the right to submit a script for marking at any time, irrespective of the individual assignment deadlines, study session, subject or person. Marking Vouchers can be used for any assignment. They are valid for four years from the date of purchase and can be refunded at any time up to the expiry date. Although you may submit your script with a Marking Voucher at any time, you will need to adhere to the explicit Marking Voucher deadline dates to ensure that your script is returned before the date of the exam. The deadline dates are provided with the assignments. If you live outside the UK you must ensure that your last script reaches the ActEd office earlier than this to allow the extra time needed to return your marked script.



Tutorials

ActEd tutorials are specifically designed to develop the knowledge that you will acquire from the course material into the higher-level understanding that is needed to pass the exam. ActEd runs a range of different tutorials at various locations, including online. Full details are set out in ActEd’s Tuition Bulletin, which is sent regularly to all students based in the UK, Eire and South Africa and is also available from the ActEd website at www.ActEd.co.uk. Regular and Block Tutorials In preparation for these tutorials, we expect you to have read the relevant part(s) of the Course Notes before attending the tutorial so that the group can spend time on exam questions and discussion to develop understanding rather than basic bookwork. You can choose one of the following types of tutorial:

Regular Tutorials (two or three days) spread over the session.

A Block Tutorial (two or three consecutive days) held two to eight weeks before the exam.

Online Classroom

The Online Classroom acts as either a valuable add-on to a face-to-face tutorial or a great alternative to a tutorial, particularly if you’re not based in the UK or near a tutorial venue. At the heart of the Online Classroom in each subject is a comprehensive, easily-searched collection of over 100 tutorial units. These are a mix of:

teaching units, helping you to really get to grips with the course material, and

guided questions, enabling you to learn the most efficient ways to answer questions and avoid common exam pitfalls.

The best way to discover the Online Classroom is to see it in action. You can watch a sample of the Online Classroom tutorial units on the ActEd website at www.ActEd.co.uk.



Flashcards

Flashcards are a set of A6-sized cards that cover the key points of the subject that most students want to commit to memory. Each flashcard has questions on one side and the answers on the reverse. We recommend that you use the cards actively and test yourself as you go. Flashcards are available in eBook format for viewing on a range of electronic devices. eBooks can be ordered separately or as an addition to paper products. Visit www.ActEd.co.uk for full details about the eBooks that are available, compatibility with different devices, software requirements and printing restrictions.

“Revision & rehearsal” products

Revision Notes

ActEd’s Revision Notes have been designed with input from students to help you revise efficiently. They are suitable for first-time sitters who have worked through the ActEd Course Notes or for retakers (who should find them much more useful and challenging than simply reading through the course again). The Revision Notes are a set of nine A5 booklets – perfect for revising on the train or tube to work. Each booklet covers one main theme of the course and includes:

Core Reading with a set of integrated short questions to develop your bookwork knowledge

relevant past exam questions with concise solutions from the last ten years

detailed analysis of key past exam questions, and

other useful revision aids.

ActEd Solutions with Exam Technique (ASET)

The ActEd Solutions with Exam Technique (ASET) contains ActEd’s solutions to the previous four years’ exam papers, ie eight papers, plus comment and explanation. In particular it will highlight how questions might have been analysed and interpreted so as to produce a good solution with a wide range of relevant points. This will be valuable in approaching questions in subsequent examinations. A “Mini-ASET” will also be available in the summer session covering the April Exam only.



Revision Tutorials

Revision Tutorials are intensive one-day face-to-face tutorials or half-day online tutorials in the final run-up to the exam. They give you the opportunity to practise interpreting and answering past exam questions and to raise any outstanding queries with an ActEd tutor. These courses are most suitable if you have previously attended Regular Tutorials or a Block Tutorial in the same subject. Details of how to apply for ActEd’s tutorials are set out in our Tuition Bulletin, which is available from the ActEd website at www.ActEd.co.uk.

“Rehearsal” products

Mock Exam A

Mock Exam A is a 100-mark mock exam paper and is available for students as a realistic test of their exam preparation. It is based on Mock Exam A from last year but it has been updated to reflect any changes to the Syllabus and Core Reading. Additional Mock Pack (AMP)

The Additional Mock Pack (AMP) consists of two further 100-mark mock exam papers – Mock Exam B and Mock Exam C. This is ideal for students who are retaking and have already sat Mock Exam A, or for those who just want some extra question practice. Mock / AMP Marking

We are happy to mark your attempts at Mock Exam A or the mock exams included within the AMP. The same general principles apply as for the X Assignment Marking. In particular:

Mock Exam Marking is available for Mock Exam A and it applies to a specified subject, session and student

Marking Vouchers can be used for Mock Exam A or the mock exams contained within the AMP; please note that attempts at the AMP can only be marked using Marking Vouchers.



Recall that:

marking is not included with the products themselves and you need to order it separately

you can submit your scripts by email, fax or post

your script may be marked and returned to you electronically, or marked and returned by post.

Queries and feedback

From time to time you may come across something in the study material that is unclear to you. The easiest way to solve such problems is often through discussion with friends, colleagues and peers – they will probably have had similar experiences whilst studying. If there’s no-one at work to talk to then use ActEd’s discussion forum at www.ActEd.co.uk/forums (or use the link from our home page at www.ActEd.co.uk). Our online forum is dedicated to actuarial students so that you can get help from fellow students on any aspect of your studies from technical issues to study advice. You could also use it to get ideas for revision or for further reading around the subject that you are studying. ActEd tutors will visit the site from time to time to ensure that you are not being led astray and we also post other frequently asked questions from students on the forum as they arise. If you are still stuck, then you can send queries by email to [email protected] (but we recommend that you try the forum first). We will endeavour to contact you as soon as possible after receiving your query but you should be aware that it may take some time to reply to queries, particularly when tutors are away from the office running tutorials. At the busiest teaching times of year, it may take us more than a week to get back to you. If you have many queries on the course material, you should raise them at a tutorial or book a personal tuition session with an ActEd tutor. Information about personal tuition is set out in our current brochure. Please email [email protected] for more details. If you find an error in the course, please check the corrections page of our website (www.ActEd.co.uk/Html/paper_corrections.htm) to see if the correction has already been dealt with. Otherwise please send details via email to [email protected] or send a fax to 01235 550085.



Each year ActEd tutors work hard to improve the quality of the study material and to ensure that the courses are as clear as possible and free from errors. We are always happy to receive feedback from students, particularly details concerning any errors, contradictions or unclear statements in the courses. If you have any comments on this course please email them to [email protected] or fax them to 01235 550085. The ActEd tutors also work with the profession to suggest developments and improvements to the Syllabus and Core Reading. If you have any comments or concerns about the Syllabus or Core Reading, these can be passed on via ActEd. Alternatively, you can send them directly to the Institute and Faculty of Actuaries’ Examination Team by email to [email protected].



3 How to study to pass the exams

The CT Subject exams

The Core Reading and exam papers for these subjects tend to be very technical. The exams themselves have many calculation and manipulation questions. The emphasis in the exam will therefore be on understanding the mathematical techniques and applying them to various, frequently unfamiliar, situations. It is important to have a feel for what the numerical answer should be by having a deep understanding of the material and by doing reasonableness checks. Subjects CT2 and CT7 are more “wordy” than the other subjects, including an “essay-style” question or two in Subject CT7. As a high level of mathematics is required in the courses it is important that your mathematical skills are extremely good. If you are a little rusty you may wish to consider buying the Foundation ActEd Course (FAC) available from ActEd. This covers all of the mathematical techniques that are required for the CT Subjects, some of which are beyond A-Level (or Higher) standard. It is a reference document to which you can refer when you need help on a particular topic. You will have sat many exams before and will have mastered the exam and revision techniques that suit you. However it is important to note that due to the high volume of work involved in the CT Subjects it is not possible to leave all your revision to the last minute. Students who prepare well in advance have a better chance of passing their exams on the first sitting. Unprepared students find that they are under time pressure in the exam. Therefore it is important to find ways of maximising your score in the shortest possible time. Part of your preparation should be to practise a large number of exam-style questions under timed exam conditions as soon as possible. This will:

help you to develop the necessary understanding of the techniques required

highlight the key topics, which crop up regularly in many different contexts and questions

help you to practise the specific skills that you will need to pass the exam. There are many sources of exam-style questions. You can use past exam papers, the Question and Answer Bank (which includes many past exam questions), assignments, mock exams, the Revision Notes and ASET.



Overall study plan

We suggest that you develop a realistic study plan, building in time for relaxation and allowing some time for contingencies. Be aware of busy times at work, when you may not be able to take as much study leave as you would like. Once you have set your plan, be determined to stick to it. You don’t have to be too prescriptive at this stage about what precisely you do on each study day. The main thing is to be clear that you will cover all the important activities in an appropriate manner and leave plenty of time for revision and question practice. Aim to manage your study so as to allow plenty of time for the concepts you meet in this course to “bed down” in your mind. Most successful students will probably aim to complete the course at least a month before the exam, thereby leaving a sufficient amount of time for revision. By finishing the course as quickly as possible, you will have a much clearer view of the big picture. It will also allow you to structure your revision so that you can concentrate on the important and difficult areas of the course. A sample CT subject study plan is available on our website at:

www.ActEd.co.uk/Html/help_and_advice_study_plans.htm It includes details of useful dates, including assignment deadlines and tutorial finalisation dates.

Study sessions

Only do activities that will increase your chance of passing. Try to avoid including activities for the sake of it and don’t spend time reviewing material that you already understand. You will only improve your chances of passing the exam by getting on top of the material that you currently find difficult. Ideally, each study session should have a specific purpose and be based on a specific task, eg “Finish reading Chapter 3 and attempt Questions 1.4, 1.7 and 1.12 from the Question and Answer Bank”, as opposed to a specific amount of time, eg “Three hours studying the material in Chapter 3”. Try to study somewhere quiet and free from distractions (eg a library or a desk at home dedicated to study). Find out when you operate at your peak, and endeavour to study at those times of the day. This might be between 8am and 10am or could be in the evening. Take short breaks during your study to remain focused – it’s definitely time for a short break if you find that your brain is tired and that your concentration has started to drift from the information in front of you.



Order of study

We suggest that you work through each of the chapters in turn. To get the maximum benefit from each chapter you should proceed in the following order: 1. Read the Syllabus Objectives. These are set out in the box on page 1 of each

chapter. 2. Read the Chapter Summary at the end of each chapter. This will give you a useful

overview of the material that you are about to study and help you to appreciate the context of the ideas that you meet.

3. Study the Course Notes in detail, annotating them and possibly making your own

notes. Try the self-assessment questions as you come to them. Our suggested solutions are at the end of each chapter. As you study, pay particular attention to the listing of the Syllabus Objectives and to the Core Reading.

4. Read the Chapter Summary again carefully. If there are any ideas that you can’t

remember covering in the Course Notes, read the relevant section of the notes again to refresh your memory.

It’s a fact that people are more likely to remember something if they review it several times. So, do look over the chapters you have studied so far from time to time. It is useful to re-read the Chapter Summaries or to try the self-assessment questions again a few days after reading the chapter itself. You may like to attempt some questions from the Question and Answer Bank when you have completed a part of the course. It’s a good idea to annotate the questions with details of when you attempted each one. This makes it easier to ensure that you try all of the questions as part of your revision without repeating any that you got right first time. Once you’ve read the relevant part of the notes and tried a selection of questions from the Question and Answer Bank (and attended a tutorial, if appropriate) you should attempt the corresponding assignment. If you submit your assignment for marking, spend some time looking through it carefully when it is returned. It can seem a bit depressing to analyse the errors you made, but you will increase your chances of passing the exam by learning from your mistakes. The markers will try their best to provide practical comments to help you to improve. To be really prepared for the exam, you should not only know and understand the Core Reading but also be aware of what the examiners will expect. Your revision programme should include plenty of question practice so that you are aware of the typical style, content and marking structure of exam questions. You should attempt as many questions as you can from the Question and Answer Bank and past exam papers.



Active study

Here are some techniques that may help you to study actively. 1. Don’t believe everything you read! Good students tend to question everything

that they read. They will ask “why, how, what for, when?” when confronted with a new concept, and they will apply their own judgement. This contrasts with those who unquestioningly believe what they are told, learn it thoroughly, and reproduce it (unquestioningly?) in response to exam questions.

2. Another useful technique as you read the Course Notes is to think of possible

questions that the examiners could ask. This will help you to understand the examiners’ point of view and should mean that there are fewer nasty surprises in the exam room! Use the Syllabus to help you make up questions.

3. Annotate your notes with your own ideas and questions. This will make you

study more actively and will help when you come to review and revise the material. Do not simply copy out the notes without thinking about the issues.

4. Attempt the questions in the notes as you work through the course. Write down

your answer before you refer to the solution. 5. Attempt other questions and assignments on a similar basis, ie write down your

answer before looking at the solution provided. Attempting the assignments under exam conditions has some particular benefits:

It forces you to think and act in a way that is similar to how you will behave in the exam.

When you have your assignments marked it is much more useful if the marker’s comments can show you how to improve your performance under exam conditions than your performance when you have access to the notes and are under no time pressure.

The knowledge that you are going to do an assignment under exam conditions and then submit it (however good or bad) for marking can act as a powerful incentive to make you study each part as well as possible.

It is also quicker than trying to write perfect answers.

6. Sit a mock exam four to six weeks before the real exam to identify your

weaknesses and work to improve them. You could use a mock exam written by ActEd or a past exam paper.



4 Frequently asked questions

Q: What knowledge of earlier subjects should I have? A: The Course Notes are written on the assumption that students have studied

Subjects CT1 and CT3. Q: What level of mathematics is required? A: The level of maths you need for this course is broadly A-level standard.

However, there may be some symbols (eg the gamma function) that are not usually included on A-level syllabuses. You will find the course (and the exam!) much easier if you feel comfortable with the mathematical techniques used in the course and you feel confident in applying them yourself. If you feel that you need to brush up on your mathematical skills before starting the course, you may find it useful to study the Foundation ActEd Course (FAC) or read an appropriate textbook. The full Syllabus for FAC, a sample of the Course Notes and an Initial Assessment to test your mathematical skills can be found on our website at www.ActEd.co.uk.

Q: What calculators am I allowed to use in the exam? A: Please refer to the Profession’s website for the latest advice.

Q: What should I do if I discover an error in the course? A: If you find an error in the course, please check our website at:

www.acted.co.uk/Html/paper_corrections.htm

to see if the correction has already been dealt with. Otherwise please send details via email to [email protected] or send a fax to 01235 550085.

Q: What do I need to do to pass the exam? A: Work extremely hard and practise as many questions as you can.



5 Core Reading and the Syllabus

Core Reading

The Syllabus for Subject CT5, and the Core Reading that supplements it, has been written by the Institute and Faculty of Actuaries to state the requirements of the examiners. The relevant individual Syllabus Objectives are included at the start of each course chapter and a complete copy of the Syllabus is included in Section 6 of this Study Guide. We recommend that you use the Syllabus as an important part of your study. The purpose of Core Reading is to give the examiners, tutors and students a clear, shared understanding of the depth and breadth of treatment required by the Syllabus. In examinations students are expected to demonstrate their understanding of the concepts in Core Reading. Examiners have the Core Reading available when setting papers. Core Reading deals with each syllabus objective. Core Reading covers what is needed to pass the exam but the tuition material that has been written by ActEd enhances it by giving examples and further explanation of key points. The Subject CT5 Course Notes include the Core Reading in full, integrated throughout the course. Here is an excerpt from some ActEd Course Notes to show you how to identify Core Reading and the ActEd material. Core Reading is shown in this bold font.

Note that in the example given above, the index will fall if the actual share price goes below the theoretical ex-rights share price. Again, this is consistent with what would happen to an underlying portfolio. After allowing for chain-linking, the formula for the investment index then becomes:

I t

N P

B t

i t i ti( )

( )

, ,

where Ni t, is the number of shares issued for the ith constituent at time t;

B t( ) is the base value, or divisor, at time t.

This is Core Reading

This is ActEd text



Core Reading accreditation

The Institute and Faculty of Actuaries would like to thank the numerous people who have helped in the development of this material and in the previous versions of Core Reading.

Changes to the Syllabus and Core Reading

The Syllabus and Core Reading are updated as at 31 May each year. The exams in April and September 2014 will be based on the Syllabus and Core Reading as at 31 May 2013. We recommend that you always use the up-to-date Core Reading to prepare for the exams.

The Institute and Faculty of Actuaries’ Copyright

All study material produced by ActEd is copyright and is sold for the exclusive use of the purchaser. The copyright is owned by Institute and Faculty Education Limited, a subsidiary of the Institute and Faculty of Actuaries. Unless prior authority is granted by ActEd, you may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material. You must take care of your study material to ensure that it is not used or copied by anybody else. Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the Institute and Faculty of Actuaries or through your employer. These conditions remain in force after you have finished using the course.

Past exam papers

You can download some past exam papers and Examiners’ Reports from the profession’s website at www.actuaries.org.uk.

Further reading

The exam will be based on the relevant Syllabus and Core Reading and the ActEd course material will be the main source of tuition for students.



However, some students may find it useful to obtain a different viewpoint on a particular topic covered in Subject CT5. The following list of further reading for Subject CT5 has been prepared by the Institute and Faculty of Actuaries. This list is not exhaustive and other useful material may be available. Actuarial mathematics. Bowers, Newton L et al. – 2nd ed. – Society of Actuaries, 1997. xxvi, 753 pages. ISBN: 0 938959 46 8. The analysis of mortality and other actuarial statistics. Benjamin, Bernard; Pollard, John H. – 3rd ed. – Faculty and Institute of Actuaries, 1993. 519 pages. ISBN 0 90106626 5. Life contingencies. Neill, Alistair. – Heinemann, 1977. vii, 452 pages. ISBN 0 434 91440 1. (This text is no longer in print, but has been used as a textbook under earlier education strategies. You should find it relatively easy to borrow a copy from a colleague. Alternatively, you can borrow it from the library at Napier House.)

Life insurance mathematics. Gerber, Hans U. – 3rd ed. – Springer. Swiss Association of Actuaries, 1997. 217 pages. ISBN 3 540 62242 X. Modern actuarial theory and practice. Booth, Philip M et al. – Chapman & Hall, 1999. xiii, 716 pages. ISBN 0 8493 0388 5.

Calculators

Please refer to www.actuaries.org.uk for the latest advice on which calculators are permitted in the exams.



6 Syllabus

The full Syllabus for Subject CT5 is given here. To the right of each objective are the chapter numbers in which the objective is covered in the ActEd course. Aim The aim of the Financial Mathematics course is to provide a grounding in financial mathematics and their simple applications. Links to other subjects Subject CT2 – Finance and Financial Reporting: develops the use of the asset

types introduced in this subject. Subject CT4 – Models: develops the idea of stochastic interest rates. Subject CT5 – Contingencies: develops some of the techniques introduced in

this subject in situations where cashflows are dependent on survival. Subject CT7 – Business Economics: develops the behaviour of interest rates. Subject CT8 – Financial Economics: develops the principles further. Subjects CA1 – Actuarial Risk Management, CA2 – Model Documentation,

Analysis and Reporting, and the Specialist Technical and Specialist Applications subjects: use the principles introduced in this subject.



Objectives On completion of the course the candidate will be able to: (i) Define simple assurance and annuity contracts, and develop formulae for the

means and variances of the present values of the payments under these contracts, assuming constant deterministic interest. (Chapters 1 to 4)

1. Define the following terms:

whole life assurance

term assurance

pure endowment

endowment assurance

critical illness assurance

whole life level annuity

temporary level annuity

premium

benefit

including assurance and annuity contracts where the benefits are deferred.

2. Define the following probabilities: n|mqx , n|qx and their select equivalents

n|mq[x]+r , n|q[x]+r .

3. Obtain expressions in the form of sums for the mean and variance of the

present value of benefit payments under each contract above, in terms of the curtate random future lifetime, assuming that death benefits are payable at the end of the year of death and that annuities are paid annually in advance or in arrear, and, where appropriate, simplify these expressions into a form suitable for evaluation by table look-up or other means.

4. Obtain expressions in the form of integrals for the mean and variance of

the present value of benefit payments under each contract above, in terms of the random future lifetime, assuming that death benefits are payable at the moment of death and that annuities are paid continuously, and, where appropriate, simplify these expressions into a form suitable for evaluation by table look-up or other means.



5. Extend the techniques of 3. and 4. above to deal with the possibility that premiums are payable more frequently than annually and that benefits may be payable annually or more frequently than annually.

6. Define the symbols xA , :x nA , 1:x nA , 1

:x nA , xa , :x na , :x nmaΩ , xa ,

:x na ,

:m x na and their select and continuous equivalents. Extend the annuity

factors to allow for the possibility that payments are more frequent than annual but less frequent than continuous.

7. Derive relations between annuities payable in advance and in arrear, and

between temporary, deferred and whole life annuities. 8. Derive the relations 1x xA da= - , : :1x n x nA da= - , and their select and

continuous equivalents. 9. Define the expected accumulation of the benefits in 1., and obtain

expressions for them corresponding to the expected present values in 3., 4., and 5. (note: expected values only).

(ii) Describe practical methods of evaluating expected values and variances of the

simple contracts defined in objective (i). (Chapters 1 to 4)

1. Describe the life table functions xl and xd and their select equivalents

[ ]x rl + and [ ]x rd + .

2. Express the following life table probabilities in terms of the functions in

1.: n xp , n xq , |n m xq and their select equivalents [ ]n x rp + , [ ]n x rq + ,

| [ ]n m x rq + .

3. Express the expected values and variances in objective (i) 3. in terms of

the functions in 1. and 2. 4. Evaluate the expected values and variances in objective (i) 3. by table

look-up, where appropriate, including the use of the relationships in objectives (i) 7. and 8.

5. Derive approximations for, and hence evaluate, the expected values and

variances in objective (i) 4. in terms of those in objective (i) 3. 6. Evaluate the expected accumulations in objective (i) 9.



7. Describe practical alternatives to the life table which can be used to obtain the evaluations in 4., 5., and 6.

(iii) Describe and calculate, using ultimate or select mortality, net premiums and net

premium reserves of simple insurance contracts. (Chapter 5)

1. Define the net random future loss under an insurance contract, and state the principle of equivalence.

2. Define and calculate net premiums for the insurance contract benefits in

objective (i) 1. Premiums and annuities may be payable annually, more frequently than annually, or continuously. Benefits may be payable at the end of the year of death, immediately on death, annually, more frequently than annually, or continuously.

3. State why an insurance company will set up reserves. 4. Describe prospective and retrospective reserves. 5. Define and evaluate prospective and retrospective net premium reserves

in respect of the contracts in objective (i) 1., with premiums as in (iii) 2.

6. Show that prospective and retrospective reserves are equal when calculated on the same basis.

7. Derive recursive relationships between net premium reserves at annual

intervals, for contracts with death benefits paid at the end of the year of death, and annual premiums.

8. Understand Thiele’s differential equation, satisfied by net premium

reserves for contracts with death benefits paid at the moment of death, and premiums payable continuously.

9. Define and calculate, for a single policy or a portfolio of policies (as

appropriate):

death strain at risk

expected death strain

actual death strain

mortality profit



(iv) Describe the calculation, using ultimate or select mortality, of net premiums and net premium reserves for increasing and decreasing benefits and annuities.

(Chapter 6)

1. Extend the techniques of (ii) to calculate the expected present value of an annuity, premium, or benefit payable on death, which increases or decreases by a constant compound rate. Calculate net premiums and net premium reserves for contracts with premiums and benefits which vary as described.

2. Define the symbols ( )xIA , ( )xIa , and ( )xIa and their select equivalents.

3. Calculate the expected present value of an annuity, premium or benefit

payable on death, which increases or decreases by a constant monetary amount. Calculate net premiums and net premium reserves for contracts with premiums and benefits which vary as described.

4. Define with profits contract.

5. List the types of bonus that may be given to with profits contracts.

6. Calculate net premiums and net premium reserves for with profits

contracts. (v) Describe the calculation of gross premiums and reserves of assurance and

annuity contracts. (Chapter 7)

1. List the types of expenses incurred in writing a life insurance contract. 2. Describe the influence of inflation on the expenses listed in 1. 3. Define the gross future loss random variable for the benefits and

annuities listed in (i) 1. and (iv). 4. Calculate gross premiums using the future loss random variable and the

equivalence principle. Premiums and annuities may be payable annually, more frequently than annually, or continuously. Benefits may be payable at the end of the year of death, immediately on death, annually, more frequently than annually, or continuously.

5. Calculate gross premiums using simple criteria other than the

equivalence principle.



6. Calculate gross premium prospective reserves using the future loss random variable.

7. Define and calculate the gross premium retrospective reserve. 8. State the conditions under which, in general, the prospective reserve is

equal to the retrospective reserve allowing for expenses. 9. Prove that, under the appropriate conditions, the prospective reserve is

equal to the retrospective reserve, with or without allowance for expenses, for all standard fixed benefit and increasing/decreasing benefit contracts.

10. Derive a recursive relation between successive annual reserves for an

annual premium contract, with allowance for expenses, for standard fixed benefit contracts.

(vi) Define and use straightforward functions involving two lives. (Chapters 8 and 9)

1. Extend the techniques of objectives (i)–(v) to deal with cashflows dependent upon the death or survival of either or both of two lives.

2. Extend the techniques of 1. to deal with functions dependent upon a fixed

term as well as age. (vii) Describe methods which can be used to model cashflows contingent upon

competing risks. (Chapter 12)

1. Explain how the value of a cashflow, contingent upon more than one risk, may be valued using a multiple-state Markov Model.

2. Derive dependent probabilities from given transition intensities, using the

Kolmogorov equations introduced in Objective (vii) of the Models Subject.

3. Derive transition intensities from given dependent probabilities.

(viii) Describe the technique of discounted emerging costs, for use in pricing,

reserving, and assessing profitability. (Chapters 10, 11, 13 and 14)

1. Define unit-linked contract. 2. Evaluate expected cashflows for whole life, endowment and term

assurances, annuities, and unit-linked contracts.



3. Profit test simple annual premium contracts of the types listed in 2. and determine the profit vector, the profit signature, the net present value, and the profit margin.

4. Show how the profit test may be used to price a product. 5. Show how the profit test may be used to determine reserves. 6. Describe the construction and use of multiple decrement tables, including

the relationships with associated single decrement tables. 7. Use multiple decrement tables to evaluate expected cashflows dependent

upon more than one decrement, including:

pension benefits

other salary related benefits

health and care insurance

8. Describe practical alternatives to the multiple decrement table which can be used to obtain the evaluations in 7.

9. Extend the techniques of 3., 6., and 7. to evaluate expected cashflows

contingent upon risks other than human life. (ix) Describe the principal forms of heterogeneity within a population and the ways

in which selection can occur. (Chapter 15)

1. State the principal factors which contribute to the variation in mortality and morbidity by region and according to the social and economic environment, specifically:

occupation

nutrition

housing

climate/geography

education

genetics



2. Define and give examples of the main forms of selection:

temporary initial selection

class selection

time selection

spurious selection

adverse selection

3. Explain how selection can be expected to occur amongst individuals taking out each of the main types of life insurance contracts, or amongst members of large pension schemes.

4. Explain why it is necessary to have different mortality tables for different

classes of lives.

5. Explain how decrements can have a selective effect.

6. Explain the theoretical basis of the use of risk classification in life insurance.

7. Explain the impact of the availability of genetic information on risk

classification in life insurance.

8. Explain the concept of a single figure index and its advantages and disadvantages for summarising and comparing actual experience.

9. Define the terms crude mortality rate, directly standardised and indirectly

standardised mortality rate, standardised mortality ratio, and illustrate their use.


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CT5 Study Guide 2014