Lecture Notes MTH5124 Actuarial Mathematics I

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0 Prologue 5,0 1 What is an actuary 5,0 2 About this course 5. 0 3 About these notes 6,0 4 Life Tables 6,0 5 Books and tables 7. 0 6 Acknowledgements 7,1 Compound interest 11,1 1 Two types of interest 11. 1 1 1 Simple interest 12,1 1 2 Compound interest 12. 1 2 Nominal and effective interest rates 14,1 2 1 Accumulation factor 14.
1 2 2 Nominal interest rates 15,1 2 3 Effective interest rates 17. 1 3 Force of interest 18,1 3 1 Time dependent interest rates 18. 1 3 2 Force of interest 19, 1 3 3 Special case of constant force of interest 21. 1 4 Rates of discount 23, 1 4 1 Relation between nominal rates of discount and interest 24. 1 5 Discounting Cash Flows or Present Values 26,1 5 1 Discrete cash flows 27.
1 6 Annuities certain introduction 28,1 7 Annuities certain more variations 31. 1 8 Continuous cash flows 35, 1 8 1 Continuous cash flow with variable force of interest 36. 1 9 Repayment of Loans 37,1 9 1 Schedule of payments 37. 1 9 2 Consolidating loans 39,1 10 Investment project appraisal 40. 1 11 Fixed Interest Securities and Other Investments 43. 1 11 1 Fixed Interest Securities 44,1 11 2 Cash including Treasury Bills 47.
1 11 3 Inflation Linked Bonds and Real Returns 48,1 11 4 Equities 52. 1 11 5 Property 54,1 11 6 Taxation 56,2 Life tables and life table functions 59. 2 1 Lifetime as a random variable 59,2 2 Basic life table functions 61. 2 3 Force of mortality 66,2 4 Analytical laws of mortality 68. 2 5 The expectation of life 69,2 6 Interpolation for fractional ages 72.
2 7 Select mortality 78,3 Life insurance and related functions 85. 3 1 Introduction to life assurance 85,3 2 Whole life assurance 87. 3 3 Whole life annuities payable annually 93,3 4 Policies of duration n 97. 3 5 p thly payments 102, 3 6 Loss policy value prospective reserve and surrender values 106. 3 7 Retrospective accumulation and reserves 111, 3 8 Importance of variance normal approximation 114.
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 peo. ple 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 gen. eral 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 Actuar. ies 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. The idea of this course is to introduce you to some of the basic mathematical ideas used in actuarial. work MTH5124 is designed for second or third year undergraduates and assumes a background. in 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 distributions. 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 compound. interest is added to a loan investment You will learn how to calculate accumulations given. the force of interest or related parameters and how to deal with series of payments in. A quick search with Google yields a variety of other humorous and not so humorous definitions as well as more. useful career descriptions, Formal prerequisites are Calculus II and Introduction to Probability contact the lecturer if you have difficulty.
meeting these, particular annuities certain and perpetuities You will also learn about the most common. 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. 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 grand. mother 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 Copies will be provided in the examination. Full details of mortality tables published by the CMI can be found at. https 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. ELT17 can be found at, https 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 Gar13, J J McCutcheon W F Scott An Introduction to the Mathematics of Finance Butterworth. cited in these notes as MS86, D C M Dickson Mary R Hardy Howard R Waters Actuarial Mathematics for Life Con. tingent Risks Cambridge University Press,cited in these notes as DHW13.
A Neill Life Contingencies Heinemann,cited in these notes as Nei77. N L Bowers H U Gerber J C Hickman D A Jones and C J Nesbitt Actuarial Mathe. matics Society of Actuaries,cited in these notes as BGH 97. 0 6 Acknowledgements, These notes have been produced by Jim Webber from notes for the predecessor module MTH6100. Actuarial Mathematics 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 the MTH6100 notes. Also thanks to Dr D Stark and Dr W Just for their additions and amendments to the notes since. the original draft In her original introduction Dr Harris gave credit to the previous notes of Prof. B Khoruzhenko and also the work of another previous lecturer Dr L Rass. The changes I have made have been limited but 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 a baule qmul ac uk. Bibliography, Ber89 J Bernoulli Tractatus de seriebus infinitis manuscript 1689. BGH 97 N L Bowers H U Gerber J C Hickman D A Jones and C J Nesbitt Actuarial. Mathematics Society of Actuaries Schaumburg 1997, DHW13 D C M Dickson M R Hardy and H R Waters Actuarial Mathematics for Life.
Contingent Risks Cambridge University Press Cambridge 2013. Gar13 S J Garrett An Introduction to the Mathematics of Finance Butterworth Heinemann. Oxford 2013, Hal93 E Halley An estimate of the degrees of mortality of mankind drawn from the curious. tables of the births and funerals at the city of Breslaw with an attempt to ascertain the. price of annuities upon lives Philosophical Transactions 17 596 610 1693. MS86 J J McCutcheon and W F Scott An Introduction to the Mathematics of Finance. Butterworth Heinemann Oxford 1986, Nei77 A Neill Life Contingencies Heinemann London 1977. Pac94 L Pacioli Summa de Arithmetica Venice 1494,Compound interest. The material in this Chapter is covered very well in Gar13 It is also covered in Chapters 1 4 of. 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 lender receives a. kind of reward for lending the money you can also think of this as the charging 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 to the lender investor is the interest The time dependent value of the investment 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. For the present we assume that interest rates are constant in time and that there is no dependence. 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 of time which. was longer or shorter than one year In that case we would need to distinguish between simple. interest and compound interest,1 1 1 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 In general the accumulated amount is after one time. Accumulation P iP P 1 i 1 1, After n time units you obtain interest of niP and thus.
Accumulation P niP P 1 ni 1 2,where P principal invested. i rate of interest,n duration of investment loan, Expression 1 2 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. N L Bowers H U Gerber J C Hickman D A Jones and C J Nesbitt Actuarial Mathe matics Society of Actuaries cited in these notes as BGH 97 0 6 Acknowledgements These notes have been produced by Jim Webber from notes for the predecessor module MTH6100 Actuarial Mathematics a module which covered a very similar syllabus Great credit and thanks to Dr Rosemary Harris for her

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