# Resource page for ST345: Life Contingencies

### ***Important Information for Students Enrolling 2018/19***

- This is
**a limited time module:**The Institute and Faculty of Actuaries (IFOA) is currently re-designing its syllabus and re-structuring its exams. Prior to 2017/18, Warwick did not offer ST345 Life Contingencies, or exemption from CT5, but is running this module for a short period of time (ie will not be offered beyond 2020) while the IFOA transitions between its old and new exam structure. This is so our currently enrolled students are able to apply for equivalent exemptions under the new exam structure. - This is
**run so as to offer the IFOA CT5 exemption:**To offer this exemption we are entirely constrained by the IFOA. In particular, there is a fixed body of material provided by the IFOA which is entirely examinable (see below "core reading" for details) and the class tests and exam will be of standard and setting similar to the IFOA CT5 exam (see below for ST345 and CT5 past papers). - There is
**no guarantee you will get the exemption:**The average mark of students taking this module in 2017/18 was 62.8%, and the threshold set by the IFOA for exemption was 65%. This equated to roughly 40% of the enrolled students achieving the CT5 exemption. - This is a
**reading course:**It is not possible to cover the volume of material examinable within a single module. As such, lectures will be flexible and attempt to cover the material as broadly and intuitively as possible, while focussing on challenging topics as arise during class. I will attempt to make it as interesting as possible. It is expected students will present assigned material. In-lecture interruptions and feedback are positively welcomed. - There will be
**no typeset notes or lecture capture:**Given the volume of material, IFOA constraints, and module life-span there will be no typeset notes produced, there are core resources which are fully typeset and cover the entire syllabus (below). The purpose of the lectures is solely to help students read, understand, and contextualise the core reading. Indeed, preparation for the exam could be done entirely independently. During lectures students should try and concentrate and follow my prepared lecture with this in mind, rather than taking notes. There will be no lecture capture, but notes presented on the visualiser will be uploaded to this website weekly afterwards. **A note on lecture capture:**At the University and College Union (UCU) Higher Education Sector Conference 2018 two motions (HE25/HE34) resulted on adoption of strong guidance on lecture-capture. In summary, it concluded lecture-capture should not be made compulsory, academics should retain rights to lectures, and the use of lecture-capture should not be used for disciplinary matters. Guidance from the Warwick branch of UCU is not to opt-in to lecture-capture as university departments were using lecture-capture to break the strike (2018) and the University of Warwick held the position that “lecture-capture can be shared with any Warwick student at any time (including to strike-break during industrial action) in perpetuity”. Whether lecture capture is used is entirely at the discretion of the lecturer, and to be entriely transparent there will be no lecture capture for ST345 while this guidance from the UCU is in effect.- This will be
**unlike other actuarial modules at Warwick:**The pace we encounter new material and concepts in ST345 will be (necessarily) a particular challenge. Unlike students studying specialist actuarial science degrees at other universities, who are able to spend considerable time covering contextual and technical background prior to attempting the material for CT5, we will be doing this in addition within a single module. - I am
**happy to offer career advice:**

### Class Times & Key Dates:

**Lecture 1:**Mondays, 11:00 -- 13:00, Term 2, MSB0.08 Mathematics and Statistics Building.**Lecture 2:**Mondays, 17:00 -- 18:00, Term 2, MSB0.08 Mathematics and Statistics Building.**Lecture 3:**Tuesdays, 16:00 -- 17:00, Term 2, MSB0.08 Mathematics and Statistics Building.**Example Classes:**Each week time from the above four scheduled lectures will be alloted to examples.**Additional ST345 Office Hours:**10:30 -- 12:00, Tuesday 5th February 2019 (Murray Pollock).**Class Test 1:**16:00 -- 17:00, Tuesday 5th February 2019, MSB0.08 Mathematics and Statistics Building.**Class Test 2:**16:00 -- 17:00, Tuesday 12th March 2019, MSB0.08 Mathematics and Statistics Building.**Revision Class:**10:00 -- 12:00, Wednesday 9th May 2019, MSB0.08 Mathematics and Statistics Building.**Final Exam:**Time TBC (Term 3), Location TBC.

### Resources:

- Murray Pollock's Office Hours (lecturer for weeks 1 - 5, term 2)

**NB:**MP will hold additional office hours, exclusively for ST345 students, from 10:30 -- 12:00 on Tuesday 5th February 2019.

- Sigurd Assing's Office Hours (lecturer for weeks 6 - 10, term 2)
- IFOA CT5 Resource Page (including Past Exams and Syllabus, IFOA Recommended Reading)
- Formulae and Tables - You will require a copy for this course, available on loan from the UG Support Office.
- CT5 "Core Reading" Notes - Available directly from the IFOA, or at cost from the UG Support Office.
- Background Reading - Actuarial Methods for Life Contingent Risks, Solutions Manual
- Lecture Content - The current plan is for the "core reading" to be covered as follows (although as detailed above is subject to alteration):
- Survival Models
- Life Tables
- Simple Assurances / Annuities
- Premiums, Expenses and Profit
- Variable Assurances / Annuities
- Frequency of Assurances / Annuities
- With Profits Policies
- Reserving
- Policy Alteration
- Profit Testing
- Joint Lives (as per single life)
- Multiple Decrement Models (including Disability and Sickness)
- Pension Fund Mathematics
- Unit Linked Contracts

### Class Tests and Exam:

**Assessment:**Each of the two class tests will ask "Typical Exam Questions" (see above), and it is expected that all questions be answered to obtain full marks. Each of the class tests will be marked out of 20 marks, will last 45 minutes, and is worth 15% of your overall module mark. The final exam will be marked out of 70 marks, will last 2 hours, and is worth 70% of your overall module mark. Again, all questions need to be answered to obtain full marks. Questions are grouped into Part I and Part II questions, Part I referring to week 1-5 material, and Part II to week 6-10 material.**Class Test 1:**Will assess material covered up until the end of Week 4 (U1-5 and U10 have been entirely covered and are examinable, except from U4/6.2-U4/6.4), and will be held during Week 5 Lecture 3 (16:00 -- 17:00, Tuesday 5th February 2019).

**2017/18 Class Test 1 and Model Solutions****2018/19 Class Test 1 and Model Solutions (see cohort feedback / discussion below)**

**Class Test 2:**Will assess material of Unit 6, of the Core Reading of CT5, only, and will be held during Week 10 Lecture 3 (16:00 -- 17:00, Tuesday 12th March 2019).**Final Exam:**Will assess all material covered, except Unit 6, and will be held during Term 3 (Time and Date TBC).**Specimen questions:**IFOA CT5 Resource Page (including Past Exams and Syllabus).

### Lectures:

***NOTE:**This is a reading course - the core reading (above) is entirely examinable, the lectures are merely helping you to read (in a pedagogical fashion) the core reading. The class tests and final exam will be comprised of questions similar in style to those of the IFOA (above)***NB: UX/Y.Z**- denotes Unit X of the CT5 Core Notes, Section Y.Z - You are expected to go away and read this!**NB: QY/M/ZZ**- denotes a past IFOA exam, Year Y Month M, Question Z - you need to be well practised!**Week 1: An Introduction**- Week 1: Covered (in order) U1/1.1-1.3, U2/1-3, U10/1-5, U2/6 (beginnings of).
- Week 1 (Illustrative) Example: Make or break (The Economist, 15/12/18).
- Week 1 Typical Exam Questions: For instance, Q2015/4/2, Q2015/4/4a)-b), Q2015/4/7, Q2015/4/8, Q2015/10/1a)-1b), Q2015/10/2, Q2015/10/4, Q2016/4/1, Q2016/4/3a), Q2016/10/4a), Q2016/10/6.

**Week 2: EPV of simple Benefits, Premiums and Expenses. Profit Criteria.**- Week 2: Covered (in order) U1/1-6, U2/5, (The beginnings of) U3/5, U4/1-3
- Types of Insurance: MAS
- Week 2 (Illustrative) Example 1: Money Supermarket Life Insurance Comparison (see covariates!).
- Week 2 (Illustrative) Example 2: Coping with the 100-year-life society (The Economist, 17/11/18)
- Week 2 (Illustrative) Example 3: Death and the salesman (The Economist, 31/3/18)
- Week 2 Typical Exam Questions: For instance, Q2015/4/1, Q2015/4/6, Q2015/4/9i), Q2015/4/13i), Q2015/10/3, Q2016/4/3c), Q2016/4/5, Q2016/4/10, Q2016/10/1, Q2016/10/5.

**Week 3: Increasing / decreasing annuities and assurances, m-thly payments, and with-profits policies.**- Week 3: Covered (in order) U1/6.4, U4/1-3, U1/7-8 and an initial look at U4/5.
- Week 3 (Illustrative) Example 1: See-through wrapper (The Economist, 22/12/18)
- Week 3 (Illustrative) Example 2: Pick of the menu (The Economist, 12/1/19)
- Week 3 (Illustrative) Example 3: No-deal Brexit may leave millions unable to claim on insurance policies (Financial Times, 15/1/19)
- Week 3 (Illustrative) Example 4: Zurich Life Insurance Company Policy (Key Features)
- Week 3 Typical Exam Questions: For instance, Q2015/4/4c), Q2015/4/10i)-ii), Q2015/10/1c, Q2015/10/5, Q2015/10/8, Q2016/4/3b), Q2016/4/6, Q2016/10/4b), Q2015/4/12, Q2016/4/12i), Q2016/10/9

**Week 4: Reserving, Policy Alteration, and an introduction to Profit Testing.**- Week 4: Covered "Reserving" from U1-5 (in particular [U1/9, U3/5 + U5/5]). We only briefly covered the key concepts, you ought to familiarise yourself with the detail as presented in the Core Notes. We also covered U3/8 and U10/10.
- Week 4 (Illustrative) Example: The Equitable Life Assurance Society / BBC Timeline
- Week 4 Typical Exam Questions: For instance, Q2015/4/9ii), Q2015/4/13, Q2016/4/2, Q2016/4/12, Q2016/4/13, Q2016/10/12. Note that Reserving appears extensively in questions primarily assessing later material.

**Class Test 1 Reminder:**Note that at this point U1-5 and U10 have been entirely covered and are examinable, except from U4/6.2-U4/6.4 which is best looked in conjunction with the `Unit Linked Contracts' still to be covered.*SSLC Feedback:**Feedback was solicited during week 4 for the first half of ST345. Many thanks to all who took the time to respond. A summary of the feedback, along with my response, can be found here.***Week 5 Part 1: Profit Testing (for single lives, note not examinable in class test 1).**- Week 5: Covered "Profit Testing" for (single lives) from U9. Note that we
U9/1, U9/2 Example 2.***did not cover*** - Week 5 Typical Exam Questions: For instance, Q2016/4/7, Q2016/4/13, Q2016/9/2. Note, a number of other questions arise upon covering unit-linked contracts.
- Week 5 (Illustrative) Example 1: The spy who hired me (The Economist, 5/1/19) - Extra, not covered in class
- Week 5 (Illustrative) Example 2: Where it's due (The Economist, 3/5/18) - Extra, not covered in class
- Week 5 (Illustrative) Example 3: Demographic warrior (The Economist, 17/11/18) - Extra, not covered in class

- Week 5: Covered "Profit Testing" for (single lives) from U9. Note that we
**Week 5 Part 2: Class Test Revision.**- Briefly reviewed strategies, and key questions from 2017 IFOA CT5 examinations in preperation for class test.
- Week 1-4 2017 Typical Exam Questions: Q2017/4/1, Q2017/4/3-5, Q2017/4/7-8, Q2017/4/11-12, Q2017/9/1-6, Q2017/9/11-12

**Week 5 Part 3: Class Test 1.****Class Test 1 and Model Solutions**.**Cohort feedback:**Summary comments and summary marks being released now (and your individual marks being released through data reveal from the undergraduate support office) come with the usual caveat that all marks released now for all modules are provisional until the end-of-year scaling meeting and the end-of-year exam board.**Overall: Average - 13.5/20 (67.5%);**Overall excellent marks. Some of you *didn’t attempt questions*, particularly the ones which asked for intuition and definitions. If you *put something down* I can try to be as generous as possible, but nothing equals nothing. If you are not sure of the answer *make an attempt at it*, if you put something sensible down I will be generous. Some minor parts of questions asked for bookwork not covered explicitly in class, but appearing directly in the core notes - this is a gentle reminder not to rely on my notes as your single source of course material, please at the very least read the core notes (and I would strongly encourage you to read the suggested textbook for this course as it really is a fantastic book).**Q1): Average - 4.4/6 (73%);**This is actuarial “bread-and-butter”, and some of you tripped up. In general you need to be familiar with concisely transcribing from words to actuarial notation to words any type of contingent event and cashflow type you encounter. Once in appropriate actuarial notation try to find a number of different ways of tackling the question (perhaps a number of different formulae using the tables, or thinking about non-table mortality questions and how they may be approached).**Q1a): Average - 1/1 (100%);**Well done!**Q1b): Average - 1/1 (100%);**Well done spotting my trick question! As noted by everyone, although the life table does not extend to the age required, all lives are dead at an earlier age (and they can (hopefully) not be revived…).**Q1c): Average - 1/2 (50%);**Take extreme care when applying approximations to ensure they are appropriate in the setting you are in. Most of you incorrectly assumed the difference between a life m-thly annuity in arrears and in advance was 1. It is 1/m as the first cashflow is the one omitted which is 1/m-th of the yearly cashflow.**Q1d): Average - 1.4/2 (70%);**Generally well done. Recall you need to be familiar with a number of approaches to solve a problem, particularly here as at 6% AM92 does not have the required commutation functions. I awarded partial marks to people where possible who started with commutation functions and got stuck, particularly if across question 1 they made other very minor mistakes.**Q2): Average - 5.8/8 (72.5%);**This question *looked* very intimidating, but part a was specifically written to encourage you to think about the relationship before diving into the proof with both feet. Overall this question was very well done for those who were not overly intimidated by the recursive formula and made a start. Although 3 marks were given for part a and 5 for part b, I was generous on balance across the question if for instance the explanation wasn’t entirely convincing but you made very minor mistakes in the derivation.**Q2a): Average - 2.4/3 (80%);**Well done on the whole. The majority of you got that there was a decomposition into the first and subsequent years. Some of you didn’t make explicit enough mention of the requirement to make an adjustment for the incrementation of the increasing term assurance starting at 2 from year 2 by decomposing it.**Q2b): Average - 3.4/5 (68%);**Although well done, a number of you made silly mistakes with the limits of summations, and the notational form of the increasing term assurance. A number of you “meshed” the answer together with what was being asked by starting on the LHS and expanding, and then doing the same on the RHS and trying to equate. This is of course fine, but this approach makes me suspicious and leads me to double check you aren’t missing steps (so make sure you write them all down).**Q3): Average - 3.3/6 (55%);**Despite being largely core note book work, this was overall the question people struggled on the most. Let this be a gentle reminder to read the core notes (however terse they are) prior to the final exam…**Q3a): Average - 1.4/2 (70%);**Please don’t answer a question asking for a definition using the words you are trying to define…**Q3b): Average - 1.1/3 (37%);**I was slightly harsh here on https://echo360.org.uk/lesson/G_5e2a1bff-0f2d-4a0b-859b-d907b7bd152f_89d85e11-4b1e-45ea-ac6f-0a4dc8223c94_2019-02-18T11:01:00.000_2019-02-18T12:59:00.000/classroombalance (see part c) as many of you listed types of expense, not the categorisation that appears in the core notes.**Q3c): Average - 0.8/1 (80%);**I was extremely generous here. There is an explicit approach given in the core notes for accounting for the overhead expenses of an insurer, but (partially as I’m unconvinced there is an “industry standard” on this matter) I gave the mark to anyone who put down an answer which would be a sensible accounting for the expense.

**Week 6: warm-up on continuous time Markov chains.**- LECTURE. First I gave an outline of my part of the course, see outline. Then I started to motivate the use of continuous time Markov chains. Unfortunately lecture capture hasn't been set-up properly, yet, so I made a few pictures of what was on the board, see Lecture 11 February.
- TUTORIAL. Worked example about a life assurance covering two lives. A summary of the calculations can be found here, but there is a little question left open on the last slide. I would like to encourage the interested student to find a better lower bound.

**Week 7: finishing Unit 7.**- LECTURE. Please go to the ST345 moodle page for the recorded lecture (click on Lecture Capture on the right-hand side). However, the most important result was the sufficient condition for the smallness of of the probability at the bottom of this Screenshot.
- TUTORIAL. First, have a look at this Screenshot for a better lower bound w.r.t. the example discussed last time. Second, when I derived the EPV of the death benefit, I made a mistake---see Correction. Otherwise, we discussed how the easy permanent disability model can be made more complex by adding the possibility of an earthquake which leads to a situation, where the straight forward modelling attempt would give a Markov chain Y(t) whose transition probabilities would not be smooth enough for standard theory (sufficient condition of Monday's lecture would be violated, for eaxmple). The trick to overcome this problem is to introduce artificial states. Using the new model with artificial states, write down all EPVs needed to find the premium for the wanted policy, and find the involved transition probabilities in terms of the given intensities by solving the corresponding Kolmogorov forward equations.

**Week 8: finishing Unit 8.**- LECTURE. We covered defined benefit schemes, Unit 8, Section 5, and touched defined contribution schemes, too. Defined contribution schemes are only mentioned in the CT5 core reading, students should consult Chapter 9 in the Dickson-Hardy-Waters book for further details.
- TUTORIAL. We discussed an example of how a promised annuity could be covered by an investment strategy. I ran a little bit out of time, and so the last bit of my calculations was not recorded---please find a summary of the missing bit here. I edited this pdf file, and parts of my editing could disappear when printing the file. Students should double check any printed version of the file reviewing the file using Acrobat, or other PDF readers.

**Week 9: how to read a paper.**Please get hold of the paper 'Nonparametric Inference for a Family of Counting Processes' in*The Annals of Statistics*1978, Vol.6, No. 4, 701--726, an electronic copy of which is available from the Library.- LECTURE---recording available.
- TUTORIAL. Worked examples on two lives.

**Week 10: merging Unit 7 and ST338.**- LECTURE---recording available.

**Week 10: Class Test 2.**Students were asked to answer 3 questions. Questions 1 and 3 were standard questions on two lifes taken from past IFoA exam papers, and Question 3 had even been discussed during the tutorial held in Week 9. Since students were allowed to use the CT5 Core Reading during the Class Test, the expectation was that they would score high on Questions 1 and 3, and high scores on Questions 1 and 3 would be enough for getting a 2:1 overall, without even touching Question 2. Question 2 was designed to test for distinction. Good students should be able to score something on part (a), and the occasional student might have an idea on part (b). Unfortunately my expectations were only partly met---the average performance was 11.3/20 (56.5%). However. I am rather satisfied with the mark distribution, counting four 1st class, one 2:1, three 2:2 performances, and two fails. I think students who engaged with the course performed better, on average, just check yourself once the marks come out.**Q1(a)-1/2(50%);**Shows that quite some students are still weak when it comes to expressing things in terms of random variables. Some students lost marks because they assumed that the annuity would pay out continuously though it says in the text that it pays annually, at the beginning of each year.**Q1(b)-2.1/3(70%);**Understanding of expectations of random variables is obviously much better---well done. Interestingly, some students swapped from continuous payment to annuity-due when taking expectations, so they got it right, eventually.**Q2(a)-1/3(33.3%);**Hoped that more students would have had an idea. Still two students got full marks, and three scored some marks. The task was to analyse a slightly difficult sequence of events, and then to split the result into an event which is likely, and one which is less likely, subject to the constraint that the more likely event should be computable. The skill required was intuition on how likely certain lifetime-events are, and students should have developed this intuition during the course including Murray's part. Maybe, after the class test, you would enjoy the policy I constructed. It's a pretty good insurance for old rich women against being married just because of their money. I should sell it to Hollywood actresses ☺.**Q2(b)-0/4(0%);**Please see the solution. Some students tried, but I think that they took expectations too early. My advice is to work with indicator random variables as long as one can---said this in the tutorials many times.**Q3-7.2/8(90%);**This question was discussed in last week's tutorial, but I am glad that students understood the rather complex set-up. Some students miscalculated some of the numbers, but when I saw that the idea was clear to them, I ignored this. All in all, seven students obtained full marks---well done.- Solutions

- 18/19: Response Part II
- 18/19: Response Part I
*17/18: Final (not taken, strike)*- 17/18: Initial Response

ST345 Grade Distribution (not available as <10 students)