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MATH20962 Jon Ferns FIA Jon.ferns@btinternet 07789 487 194 Version 7 February 2014

University of Manchester Contingencies 1 Lecture 3 12 February 2014 “Compound interest (recap) and life assurances ”. MATH20962 Jon Ferns FIA Jon.ferns@btinternet.com 07789 487 194 Version 7 February 2014. Course syllabus. Financial maths revision (i.e. without mortality).

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MATH20962 Jon Ferns FIA Jon.ferns@btinternet 07789 487 194 Version 7 February 2014

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  1. University of ManchesterContingencies 1Lecture 312 February 2014“Compound interest (recap) andlife assurances ” MATH20962 Jon Ferns FIA Jon.ferns@btinternet.com 07789 487 194 Version 7 February 2014

  2. Course syllabus

  3. Financial maths revision (i.e. without mortality) Present Value of Premiums = the Sum of: Premium(t) * v ^ t Present Value of Benefits = the Sum of: Benefit payment(t) * v ^ t where v = 1 / (1+i) Premiums are usually paid in advance (single or regular) Benefits are usually paid in arrears • Why does the insurer want paying in advance? • So it can invest the money from all policyholders, before it has to pay anything out, i.e. cashflow requirements are minimised • So it does not use up its shareholders’ capital (remember that the return on shareholders money = profit / capital used) • So the policyholder is not tempted to lapse the contract

  4. Compound interest revision (i.e. without mortality)

  5. Compound interest revision (without mortality) • Derive the formula on the previous page from first principles • Compound interest revision questions (done on the board)

  6. Insurance company pricing - Bringing in mortality (the probability of a payment being made) Premiums will only be paid if still alive at the point the premium would be paid Expected Present Value of Premiums = Sum of: Premium(t) * vt * Probability alive (t) The contract might specify that benefits are due on death, or on survival to the end of the contract (on both) Expected Present Value of Benefits = Sum of: Death benefit payment(t) * vt * Probability dies in the year ending at time (t) + Survival benefit payment(n) * vn * Probability alive (n)

  7. Life Assurances (life cover) and Endowments “Life Assurance”: = An amount is paid on death during the specified term “Endowment”: = An amount is paid on survival to the end of the specified term “Endowment Assurance”: = An amount is paid on death during the specified term or on survival to the end of the term Expected Present Value:

  8. Whole of life assurance (life cover)A payment of £1 at the end of year of death, with no upper age limit Whole of life assurance (life cover)A payment of £1 at the instant of death, with no upper age limit

  9. Variance for a whole of life assurance (life cover) Know: So, Variance of the Present Value of £1 at end of year of death (whole of life):

  10. Example: looking up the expected value and variance of a whole of life assurance, Ax, in the table book Calculate the expected present value of £1,000 paid at end of year of death (whole life) for a male aged 40, by assuming: mortality = “Assured Males 1992” ultimate mortality table “(AM92)” AM92 is a mortality table produced from a survey of “insurance company” policyholder mortality, for Males, experience gathered from an exercise centred 1992, from life Assurance policies State the expected present value at an interest rate of 4%Answer: Expected present value = 1000A40 = £230.56 (at 4% interest) State the expected present value at an interest rate of 0%Answer: Expected present value = 1000A40 =£1,000 (at 0% interest) Calculate the variance of the present value at 4% interest (Answer: Variance (of present value) = 10002.(2A40 – A402) = 10002 * (0.06792-0.23056^2) = £14,760 Therefore, Std Deviation = £121.50

  11. Pure term assuranceA payment of £1 at the end of the year of death, during the term n. There is no survival payment, i.e. this is pure life cover

  12. Pure term assuranceA payment of £1 at the end of the year of death, during the term n. There is no survival payment, i.e. this is pure life cover

  13. Pure endowment“Payment of £1 on survival to “n”. Nothing on death”

  14. Endowment assuranceA payment of £1 at end of year of death.. OR.. £1 on surviving to time n.

  15. Endowment assuranceA payment of £1 at end of year of death.. OR.. £1 on surviving to time n. Variance of the Present Value

  16. Example: looking up term assurances and endowments in the table book

  17. Formula for benefits payable immediately on death

  18. Quiz • 3% chance of an employee leaving each year. Employees get £1000 if they “survive in employment” for 20 years. What is the expected present value at interest 8%pa? Write down the appropriate assurance symbol. • A man aged 40, buys a whole life assurance paying £10,000 at the end of year of death (the “sum assured” = £10,000). Write out the formula for and calculate the expected present value and standard deviation using AM92 ultimate mortality and 4%pa interest.

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