o13 Introduction to Actuarial Science Matthias Winkel1 University of Oxford MT 2002 1Departmental lecturer at the Department of Statistics, supported by the Institute of Actuaries o13 Introduction to Actuarial Science 16 lectures MT 2002 and 16 lectures HT 2003 Aims This course is supported by the Institute of Actuaries. It is designed to give the under- graduate mathematician an introduction to the (cid:12)nancial and insurance worlds in which the practising actuary works. Students will cover the basic concepts of risk management models for mortality and sickness, and for discounted cash (cid:13)ows. In the (cid:12)nal examina- tion, a student obtaining at least an upper second class mark on paper o13 can expect to gain exemption from the Institute of Actuaries’ paper 102, which is a compulsory paper in their cycle of professional actuarial examinations. Synopsis Fundamental nature of actuarial work. Use of generalised cash (cid:13)ow model to describe (cid:12)nancial transactions. Time value of money using the concepts of compound interest and discounting. Present values and the accumulated values of a stream of equal or unequal payments using speci(cid:12)ed rates of interest and the net present value at a real rate of interest, assuming a constant rate of in(cid:13)ation. Interest rates and discount rates in terms of di(cid:11)erent time periods. Compound interest functions, equation of value, loan repayment, project appraisal. Investment and risk characteristics of investments. Simple compound interest problems. Price and value of forward contracts. Term structure of interest rates, simple stochastic interest rate models. Single decrement model, present values and the accumulated values of a stream of payments taking into account the probability of the payments being made according to a single decrement model. Annuity functionsandassurance functionsforasingledecrement model. Liabilitiesunderasimple assurance contract or annuity contract. Reading All of the following are available from the Publications Unit, Institute of Actuaries, 4 Worcester Street, Oxford OX1 2AW Subject 102: Financial Mathematics. Core reading 2003. Faculty & Institute of (cid:15) Actuaries 2002 J J McCutcheon and W F Scott, An Introduction to the Mathematics of Finance, (cid:15) Heinemann 1986 P Zima and R P Brown, Mathematics of Finance, McGraw-Hill Ryerson 1993 (cid:15) H U Gerber, Life Insurance Mathematics, Springer 1990 (cid:15) N L Bowers et al, Actuarial mathematics, 2nd edition, Society of Actuaries 1997 (cid:15) Contents 1 Introduction 7 1.1 The actuarial profession . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The generalised cash (cid:13)ow model . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Actuarial science as examples in the generalised cash (cid:13)ow model . . . . . 9 2 The theory of compound interest 11 2.1 Simple versus compound interest . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Time-dependent interest rates . . . . . . . . . . . . . . . . . . . . . . . . 13 3 The valuation of cash (cid:13)ows 15 3.1 Accumulation factors and consistency . . . . . . . . . . . . . . . . . . . . 15 3.2 Discounting and the time value of money . . . . . . . . . . . . . . . . . . 16 3.3 Continuous cash (cid:13)ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Constant discount rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Fixed-interest securities and Annuities-certain 19 4.1 Simple (cid:12)xed-interest securities . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Securities above/below/at par . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 pthly paid interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 Securities with pthly paid interest . . . . . . . . . . . . . . . . . . . . . . 21 4.5 Annuities-certain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.6 Perpetuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.7 Annuities and perpetuities payable pthly and continuously . . . . . . . . 22 5 Mortgages and loans 23 5.1 Loan repayment schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Equivalent cash (cid:13)ows and equivalent models . . . . . . . . . . . . . . . . 24 5.3 Fixed, discount, tracker and capped mortgages . . . . . . . . . . . . . . . 26 6 An introduction to yields 27 6.1 Flat rates and APR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 The yield of a cash (cid:13)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.3 General results ensuring the existence of the yield . . . . . . . . . . . . . 29 3 4 Contents 7 Project appraisal 31 7.1 A remark on numerically calculating the yield . . . . . . . . . . . . . . . 31 7.2 Comparison of investment projects . . . . . . . . . . . . . . . . . . . . . 32 7.3 Investment projects and payback periods . . . . . . . . . . . . . . . . . . 32 7.4 Funds and weighted rates of return . . . . . . . . . . . . . . . . . . . . . 33 8 Taxation and in(cid:13)ation 35 8.1 Fixed interest securities and running yields . . . . . . . . . . . . . . . . . 35 8.2 Income tax and capital gains tax . . . . . . . . . . . . . . . . . . . . . . 36 8.3 In(cid:13)ation indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9 In(cid:13)ation models and real interest 39 9.1 Modelling in(cid:13)ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 9.2 Constant in(cid:13)ation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.3 In(cid:13)ation adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 10 Uncertain payment and probabilistic models 43 10.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 10.2 Notation and introduction to probability . . . . . . . . . . . . . . . . . . 43 10.3 Fair premiums and risk under uncertainty . . . . . . . . . . . . . . . . . 45 11 Corporate bonds and uncertain payment 47 11.1 Uncertain payment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 11.2 Pricing of corporate bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 49 12 Uncertain investment projects and risk 51 12.1 Pricing of equity shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 12.2 Examples: Comparison of investment projects . . . . . . . . . . . . . . . 52 12.3 Individual risk models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 12.4 Pooling reduces risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 13 Life insurance: the single decrement model 55 13.1 Uncertain cash (cid:13)ows in life insurance . . . . . . . . . . . . . . . . . . . . 55 13.2 Conditional probabilities and the force of mortality . . . . . . . . . . . . 56 13.3 The curtate future lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . 57 13.4 Insurance types and examples . . . . . . . . . . . . . . . . . . . . . . . . 58 14 Life insurance: premium calculation 59 14.1 Residual lifetime distributions . . . . . . . . . . . . . . . . . . . . . . . . 59 14.2 Actuarial notation for life products . . . . . . . . . . . . . . . . . . . . . 60 14.3 Lifetables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 14.4 Life annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 14.5 Multiple premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Contents 5 15 Some elements of General Insurance 63 15.1 Premium principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 15.2 The Central Limit Theorem and an example . . . . . . . . . . . . . . . . 64 16 Summary: it’s all about Equations of Value 67 16.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 16.1.1 Basic notions used throughout the course . . . . . . . . . . . . . . 67 16.1.2 Deterministic applications . . . . . . . . . . . . . . . . . . . . . . 68 16.1.3 Applications with uncertaincy . . . . . . . . . . . . . . . . . . . . 68 16.2 Equations of value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 16.3 Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 16.4 Hilary Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 16.5 Assignment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A A 1967-70 Mortality table 71 Lecture 1 Introduction This introduction is two-fold. First, we give some general indications on the work of an actuary. Second, we introduce cash (cid:13)ow models as the basis of this course and a suitable means to describe and look beyond the contents of this course. 1.1 The actuarial profession Actuarial Science is an old discipline. The Institute of Actuaries was formed in 1848, (the Faculty ofActuaries in Scotland in 1856), but the profession is much older. An important root is the construction of the (cid:12)rst life table by Sir Edmund Halley in 1693. However, this does not mean that Actuarial Science is oldfashioned. The language of probability theory was gradually adopted between the 1940s and 1970s. The development of the computer has been re(cid:13)ected and exploited since its early days. The growing importance and complexity of (cid:12)nancial markets currently changes the profession. Essentially, the job of an actuary is risk assessment. Traditionally, this was insurance risk, life insurance and later general insurance (health, home, property etc). As typically enourmous amounts of money, reserves, have to be maintained, this naturally extended to investment strategies including the assessment of risk in (cid:12)nancial markets. Today, the Faculty and Institute of Actuaries claim in their slogan yet more broadly to make \(cid:12)nancial sense of the future". To become an actuary in the UK, one has to pass nine mathematical, statistical, economic and (cid:12)nancial examinations (100 series), an examination on communication skills (201), an examination in each of the (cid:12)ve specialisation disciplines (300 series) and for a UK fellowship an examination on UK speci(cid:12)cs of one of the (cid:12)ve specialisation disciplines. This whole programme takes normally at least three or four years after a mathematical university degree and while working for an insurance company. This course is an introductory course where important foundations are laid and an overview of further actuarial education and practice is given. The 101 paper is covered by the second year probability and statistics course. An upper second mark in the examination following this course normally entitles to an exemption from the 102 paper. Two thirds of the course concern 102, but we also touch upon material of 103, 104, 105 and 109. 7 8 Lecture 1: Introduction 1.2 The generalised cash (cid:13)ow model The cash (cid:13)ow model systematically captures cash payments either between di(cid:11)erent parties or, as we shall focus on, in an in/out way from the perspective of one party. This can be done at di(cid:11)erent levels of detail, depending on the purpose of an investigation, the complexity of the situation, the availability of reliable data etc. Example 1 Look at the transactions on a worker’s monthly bank statement Date Description Money out Money in 01-09-02 Gas-Elec-Bill $21.37 04-09-02 Withdrawal $100.00 15-09-02 Telephone-Bill $14.72 16-09-02 Mortgage Payment $396.12 28-09-02 Withdrawal $150.00 30-09-02 Salary $1022.54 Extracting the mathematical structure of this example we de(cid:12)ne elementary cash (cid:13)ows. De(cid:12)nition 1 A cash (cid:13)ow is a vector (t ;c ) of times t 0 and amounts c IR. j j 1(cid:20)j(cid:20)m j j (cid:21) 2 Positive amounts c > 0 are called in(cid:13)ow. If c < 0, then c is called out(cid:13)ow. j j j (cid:0) Example 2 The cash (cid:13)ow of Example 1 is mathematically given by j t c j t c j j j j 1 1 -21.37 4 16 -396.12 2 4 -100.00 5 28 -150.00 3 15 -14.72 6 30 1022.54 Often, the situation is not as clear as this, and there may be uncertainty about the time/amount of a payment. This can be modelled using probability theory. De(cid:12)nition 2 A generalised cash (cid:13)ow is a random vector (T ;C ) of times T 0 j j 1(cid:20)j(cid:20)M j (cid:21) and amounts C IR with a possibly random length M IN. j 2 2 Sometimes, in fact always in this course, the random structure is simple and the times or the amounts are deterministic, or even the only randomness is that a well speci(cid:12)ed payment may fail to happen with a certain probability. Example 3 Future transactions on a worker’s bank account j T C Description j T C Description j j j j 1 1 -21.37 Gas-Elec-Bill 4 16 -396.12 Mortgage payment 2 T C Withdrawal? 5 T C Withdrawal? 2 2 5 5 3 15 C Telephone-Bill 6 30 1022.54 Salary 3 Here we assume a (cid:12)xed Gas-Elec-Bill but a varying telephone bill. Mortgage payment and salary are certain. Any withdrawals may take place. For a full speci(cid:12)cation of the generalised cash (cid:13)ow we would have to give the (joint!) laws of the random variables.