Table Of ContentMTH5124: Actuarial Mathematics I
School of Mathematical Sciences
Notes corresponding to undergraduate lecture course
· Semester A 2017/18 ·
This version published on 17 September 2017.
Contents
0 Prologue 1
0.1 What is an actuary? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 About this course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.3 About these notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.4 Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.5 Books and tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 Compound interest 5
1.1 Two types of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Nominal and effective interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Force of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Rates of discount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Discounting Cash Flows or Present Values . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Annuities-certain: introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 Annuities-certain: more variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.8 Repayment of Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.9 Investment project appraisal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.10 Fixed Interest Securities and Other Investments . . . . . . . . . . . . . . . . . . . . 45
1.10.1 Fixed Interest Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.10.2 Cash including Treasury Bills . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.10.3 Inflation Linked Bonds and Real Returns . . . . . . . . . . . . . . . . . . . 49
1.10.4 Equities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.10.5 Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.10.6 Taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.11 Time weighted rate of return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2 Life tables and life-table functions 65
2.1 Lifetime as a random variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Basic life-table functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.3 Force of mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4 Analytical laws of mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5 The expectation of life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.6 Interpolation for fractional ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7 Select mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
i
3 Life insurance and related functions 95
3.1 Introduction to life assurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 Whole-life assurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3 Whole-life annuities payable annually . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4 Policies of duration n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.5 p-thly payments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.6 Loss, policy value,prospective reserve and surrender values . . . . . . . . . . . . . . 121
3.7 Retrospective accumulation and reserves . . . . . . . . . . . . . . . . . . . . . . . . 126
3.8 Importance of variance, normal approximation . . . . . . . . . . . . . . . . . . . . 129
A Glossary of actuarial symbols 134
Bibliography 136
ii
Chapter 0
Prologue
0.1 What is an actuary?
“An actuary is someone who expects everyone to be dead on time.” is one definition you can
find on the internet.1 In fact, actuaries do much more than deal with the probabilities of people
dying! In short, they use financial and statistical theories to quantify and manage risk in all
areas of business. Traditionally actuaries specialize in consultancy, investment, life and general
insurance or pensions. However, their analytical skills are also increasingly valuable in other
areas—after all, “risk management” is of wide importance in financially turbulent times. Actu-
aries are highly-regarded and (well-paid!) professionals. For more details of career paths, etc.,
see http://www.actuaries.org.uk/becoming-actuary/.
0.2 About this course
Theideaofthiscourseistointroduceyoutosomeofthebasicmathematicalideasusedinactuarial
work. MTH5124isdesignedforsecondorthirdyearundergraduatesandassumesabackgroundin
basic Calculus and Probability.2 All practicalities about the course itself (timetable, coursework,
assessment details), etc., can be found on the course QM+ website
http://qmplus.qmul.ac.uk/course/view.php?id=767.
Important announcements and corrections will also be posted there.
The course uses familiar mathematical concepts (e.g., geometric series, probability distribu-
tions) but in an unfamiliar context. I hope you will be interested to see how maths you already
know can be used effectively in the “real world”. However, one of the problems in applying maths
to a different field is that you often have to learn new terminology, notation, etc. in order to
communicate with specialists in that area. This is certainly the case here; indeed you will soon
be introduced to a whole zoo of complicated actuarial symbols and new vocabulary. It is crucial
for success that you learn this “new language” so that the familiar mathematical objects do not
become lost in the fog of unfamiliar actuarial terminology.
The course has three main “chapters”:
1. Compound interest: Here we will cover various (interrelated) ways to quantify how com-
poundinterestisaddedtoaloan/investment. Youwilllearnhowtocalculateaccumulations
1A quick search with Google yields a variety of other humorous, and not-so-humorous, definitions as well as
more useful career descriptions.
2FormalprerequisitesareCalculusIIandIntroductiontoProbability;contactthelecturerifyouhavedifficulty
meeting these.
1
0.3 About these notes 2
given the force of interest (or related parameters) and how to deal with series of payments,
inparticularannuities-certainandperpetuities. Youwillalsolearnaboutthemostcommon
forms of investments and how to value investments using compound interest.
2. Life tables and life-table functions: Life tables are the actuary’s basic tool. You will
learn how to interpret them in order to find various probabilities related to life and death.
3. Life insurance and related functions: Here we will combine material from the first two
chapters to deal with situations involving payments (with compound interest) whose value
and/or timing may depend on a person’s survival or death! Life insurance is the classic
example here.
0.3 About these notes
These notes will cover the material in roughly the same order as the lectures but their style
will probably be slightly different. In particular, the printed notes may contain some longer
explanations and extra examples which time prevents me covering in class. The lectures, however
will be important for emphasizing the main points and giving exam tips—I strongly recommend
that you attend and take your own notes!
To help you with revision, all important actuarial terms are printed in bold. You must
be sure to understand what these mean, both in everyday language (could you explain them to
your grandmother or your next-door neighbour?) and in terms of the associated mathematical
formulation. The notes will also contain a number of “examples” and “exercises”. For the former
you will find full details of the working; for the latter (usually) only the answers. A good way to
check your own understanding would be to read the text and try the associated exercises (contact
me if you have any difficulty getting the stated answers). I intend the unstarred exercises to
correspond roughly to the Key Objectives for the course (i.e., everyone should be able to do
them); starred exercises will be somewhat harder and should be attempted by those aiming for a
high grade.
0.4 Life Tables
The examples in this document are based on the following life tables:
• English Life Table No 17,
• AMC00 mortality table for assured lives.
These are available on the QM+ page for MTH5124. These tables will also be used for the
2017/18 exams; copies will be provided in the examination.
For years 2015/16 and earlier, alternative life tables were used:
• English Life Table No 12,
0.5 Books and tables 3
• A67/70 mortality table for assured lives.
These are also available on the QM+ page for MTH5124.
FulldetailsofmortalitytablespublishedbytheCMIcanbefoundathttps://www.actuaries.org.uk/learn-
and-develop/continuous-mortality-investigation/cmi-mortality-and-morbidity-tables.
The CMI is a subsidiary of the Institute and Faculty of Actuaries, and was previously known
as The Continuous Mortality Investigation. CMI tables relate to the mortality experience of life
insurance policyholders and the members of pension schemes.
The English Life Tables represent the mortality experience of the population of England and
Wales. Tables are published by the Office of National Statistics (ONS); further information on
ELT17canbefoundathttps://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/lifeexpectancies/bulletins/englishlifetablesno17/2015-
09-01.
0.5 Books and tables
The course is designed to be fairly self-contained and does not follow any one textbook. However,
you may find the following useful for background reading:
• S. J. Garrett An Introduction to the Mathematics of Finance (Butterworth-Heinemann);
– cited in these notes as Garrett (2013).
• J.J.McCutcheon&W.F.ScottAnIntroductiontotheMathematicsofFinance(Butterworth-
Heinemann);
– cited in these notes as McCutcheon and Scott (1986).
• D. C. M. Dickson, Mary R. Hardy & Howard R. Waters Actuarial Mathematics for Life
Contingent Risks (Cambridge University Press);
– cited in these notes as Dickson, Hardy, and Waters (2013).
• A. Neill Life Contingencies (Heinemann);
– cited in these notes as Neill (1977).
• N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones and C. J. Nesbitt Actuarial
Mathematics (Society of Actuaries);
– cited in these notes as Bowers et al. (1997).
0.6 Acknowledgements
These notes have been derived from the notes originally produced for MTH6100 Actuarial Math-
ematics, a module which covered a very similar syllabus.
Great credit and thanks to Dr Rosemary Harris for her excellent work in producing the
original draft of these notes. Also thanks to Dr D. Stark and Dr W. Just for their additions and
amendments to the notes since the original draft.
0.6 Acknowledgements 4
In her original introduction, Dr Harris gave credit to the previous notes of Prof. B. Kho-
ruzhenko and also the work of another previous lecturer, Dr L. Rass.
ThechangesIhavemadehavebeenlimited. Asnotedearlier,Ihavemigratedtomoremodern
mortality tables which has resulted in material reductions in the size of calculated life insurance
premiums and corresponding increases in the present value of life annuity functions. I have also
included material relating to fixed interest securities and other investment classes. This will
allow students to have a better understanding of how the mathematics of Chapter 1 relates to
the biggest asset class held by insurance companies and pension funds.
I take full responsibility for this edition of the notes and the mistakes and contradictions that
students will inevitably find. Please alert me to any mistake that you find by sending an email
to: [email protected].
Chapter 1
Compound interest
The material in this Chapter is covered very well in (Garrett (2013)). It is also covered in Chapters
1–4 of (McCutcheon and Scott (1986).)
1.1 Two types of interest
Often in the course of daily life and business, people need (or choose) to borrow money. For
example, you might have a student loan or, later in life, need a mortgage or business loan. On
the other hand, if you happen to have “spare” money you can lend it to a bank, for example, by
investing in a savings account or fixed term bond. In general, in all these situations the money
lenderreceivesakindof“reward”forlendingthemoney; youcanalsothinkofthisasthecharging
of “rent” for the use of the money.
To be more specific the original loan/investment is called the capital (or principal) and the
“reward”tothelender/investoristheinterest. Thetime-dependentvalueoftheinvestment, i.e.,
the original loan plus the interest, is known as the accumulated amount (or accumulation).
Interest is expressed as a rate in two senses: per unit capital and per unit time. In practice,
the interest rate is often quoted in percent and usually, but not always the basic time unit is one
year. Note that “p.a.” is often used as an abbreviation for per annum (i.e., each year). To avoid
confusion you should always state the basic time period when giving an interest rate.
The interest rate on a transaction is affected by various factors including:
• the market rate for similar loans;
• the risk involved in the use to which the borrower puts the money (cf. mortgage loan rates
and unsecured personal loan rates);
• the anticipated rate of appreciation or depreciation in the value of the currency in which
the transaction is carried out (e.g., in times of high inflation the interest is higher).
Forthepresentweassumethatinterestratesareconstantintimeandthatthereisnodependence
on the sum invested. We will later relax the first of these assumptions but the second will remain
throughout the course.
Now let’s consider a concrete example—a savings account with an interest rate of 5% p.a..
In other words, one gets a return of £5 in one year for each £100 invested. Suppose you were
to invest £200 then you could close the account after one year and withdraw £210 made up of
the principal (£200) and the interest (£10). But what if the account were kept open for a period
5
1.1 Two types of interest 6
of time which was longer (or shorter) than one year? In that case we would need to distinguish
between simple interest and compound interest.
Simple interest
For the bank account considered above then, in the case of simple interest, £10 would be added
each year to your original deposit of £200. The general formula for the accumulated amount is
Accumulation = C(1+ni) (1.1)
where C = principal invested;
i = rate of interest;
n = duration of investment/loan.
Expression (1.1) applies for all non-negative values of n. The normal commercial practice in
relation to fractional periods of a year is to pay interest on a pro rata basis (i.e., proportional
to the time the account is open). For an account of duration of less than one year it is usual to
allow for the actual number of days the account is held.
What happens when n is greater than 1 year? Imagine investing the £200 of our example for
two years (again at 5% p.a. simple interest) then after these two years the accumulation will be
£200(1+2×0.05) = £220.
Is there a way to make more money?
Yes, you can close this account (Let’s call it account A.) after one year, at which time you will
withdraw £210 [see (1.1)]. Then place this sum on deposit in a new account, say B. When you
close account B after one further year, the sum withdrawn will be £210(1+1× 0.05)=£220.50.
In other words, you will have gained the princely sum of 50p!
The difference here is that, effectively, account B pays interest on the interest already earned.
In general, banks do not want people to be frequently opening and closing accounts. (Whilst you
might not bother to do that to gain 50p you probably would for £50,000!) This is one reason
why, forperiodsgreaterthanoneyear, mostbankaccountsdonotpaysimpleinterestbutinstead
the interest is compounded...
Compound interest
In this case interest is paid on the previous interest accrued1 but not withdrawn and the general
formula for the accumulated amount is modified to
Accumulation = C(1+i)n. (1.2)
1Here “interest accrued” simply means interest already earned / applied to the account.
1.1 Two types of interest 7
To see this, we argue that if A is the accumulation after n time units then A = A (1 +
n n n−1
i) = A (1 + i)2 = ... = A (1 + i)n, where A is the capital invested initially, i.e. C. This
n−2 0 0
argument obviously holds for integer n; the validity of (1.2) for non-integer n will be established
in Section 1.3.
Exercise 1.1.1: Effect of compound interest
Convince yourself that applying compound interest according to (1.2) has the same effect as closing an
account with simple interest after each time period and reinvesting the money in an account paying the
same interest.
Exercise 1.1.2: Compound interest formula
Write a formal proof of (1.2) for integer n using induction.
From now on, “interest” always means compound interest unless explicitly stated otherwise.
However, simple interest calculations (which you should be able to do in your head!) can some-
timesbeausefultoolforcheckingthatcompoundinterestresultsareintherightballpark. Simple
interest is a good approximation for compound interest when n and i are small since in that case
(1+i)n ≈ 1+ni.
Example 1.1.1: Low interest rates
Suppose you invest £1,000 for three years in a bank account which pays interest annually at 1.5% p.a..
What is the difference in accumulation if the bank pays compound interest compared to simple interest?
Solution
For simple interest then, using (1.1),
Accumulation=£1000×(1+3×0.015)=£1045.00. (1.3)
On the other hand, if interest is compounded, then we use (1.2):
Accumulation=£1000×(1+0.015)3 =£1045.68. (1.4)
[Unless stated otherwise, you should always give monetary answers rounded to the nearest penny.] So the
accumulation is only 68p greater if compound interest is paid.
In real-life there are many different kinds of transactions such as:
• payment of a series of premiums throughout a given time period in return for a lump sum
at the end of the period;
• mortgage loans, i.e., loans which are made for the specific purpose of house purchase (The
property to be purchased usually acts as security for the loan);
• fixed interest securities, such as regular income payments throughout a given time period
and an additional lump sum at the end of the period in return for a one-off payment.
Knowledge of compound interest can be useful in comparing the merits of such transactions.
Description:Notes corresponding to undergraduate lecture course. · Semester A . aries are highly-regarded and (well-paid!) professionals These notes have been derived from the notes originally produced for MTH6100 Actuarial Math-.
