University of Manchester Contingencies 1 Lecture 3 12 February 2014 “Compound interest (recap) and life assurances ” MATH20962 Jon Ferns FIA Version 7 February 2014

Course syllabus Lecture 1Life tables (lx) and selection by the insurer 2CT4 survival models, continuous mortality, part year mortality 3Compound interest (recap) + assurances (Ax) 4Assurances (Ax) that vary each year + annuities (a x ) 5Annuities (a x ) that increase each year 6Evaluation of assurances (Ax) and annuities (a x ) 7Mid course test 8Net Premiums 9Gross Premiums 10Reserves 11Reserves / revision 12Revision

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

Compound interest revision (i.e. without mortality)

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

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) * v t * 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) * v t * Probability dies in the year ending at time (t) + Survival benefit payment(n) * v n * Probability alive (n)

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:

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

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):

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 = 1000A 40 = £ (at 4% interest) State the expected present value at an interest rate of 0% Answer: Expected present value = 1000A 40 =£1,000 (at 0% interest) Calculate the variance of the present value at 4% interest (Answer: Variance (of present value) = ( 2 A 40 – A 40 2 ) = * ( ^2) = £14,760 Therefore, Std Deviation = £121.50

Pure term assurance A 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

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

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

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

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

Formula for benefits payable immediately on death

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.