2015 paper

MACQUARIE Jniversity This question paper may be retained by candidates. FORMAL EXAMINATION PERIOD: SESSION 2, NOVEMBER 2015 _EXAMINATION DETAILS: Unit Code: ACST255 and ACST859 Unit Name: Contingent Payments 1 T - S : 3 nours plus 10 minutes reading time (including reading time if applicable): © " Total No. of Questions: ¢ Total No. of Pages 8 R (including this cover sheet): » GENERAL INSTRUCTIONS TO CANDIDATES: « Candidates are required to obey all instructions provided by the Final Examination Supervisor and must refrain from communicating in any way with another student once they have entered the final examination venue. « Candidates may not write or mark the exam materials in any way during reading time. « Candidates may only access authorised materials during this examination. A list of authorised material is available on this cover sheet. « If it is alleged you have breached these rules at any time during the examination, the matter may be reported to the University Discipline Committee for determination. EXAMINATION INSTRUCTIONS: You may attempt all questions. There are 3 sections. Answer each section in a different answer booklet. Write your answers on the ruled pages of the answer booklets provided. Anything written on the unruled pages will not be marked. The number of marks allocated to each question is shown. The marks awarded for your answer will depend not only on the validity of your method and the correctness of your final answer, but also on the clarity of your solution. Errors which indicate dangerous misunderstandings may incur penalties. ' MATERIALS PERMITTED/NOT PERMITTED: Dictionaries: No dictionaries are permitted. Calculators: Non-programmable calculators that do not have text retrieval capacity are permitted. Other: You are permitted one A4 page of paper containing reference material printed on both sides. The material may be handwritten or typed. This page will be collected at the end of the exam. You will be supplied with a copy of the relevant life tables.

Section A (31 marks) Answer this section in a separate answer booklet. [9] Question 1 S Suppose that S (/)= N——— 0<1<100. Suppose also that in a life table derived from this p 0 100 PI \ survival function, /, = 100, 000. {1} (a) Calculate /.. 2] (b) Calculate the probability that a lite aged 51 will die between ages 64 and 84, . ¢ 2] (¢) Calculate ¢ . 2] (d) Assuming a force of interest of 4% per annum. derive a definite integral expression for A,. You are NOT required to evaluate your definite integral expression. [2] (e) Would @, at 4% per annum effective interest be higher, the same or lower than the correct answer to part (¢)? Do NOT perform any calculations. Explain your reasoning briefly in words. [11] Question 2 A life table has a 4 year select period. The following extract shows some ultimate mortality rates from the table. Age, x Probability of death, ¢ _ 50 0.0010 51 0.0011 52 0.0012 53 0.0013 54 0.0014 You are also gjven that

[11] Question 3 Consider a 30 year term insurance with death benefit of $100,000 payable immediately on death sold to a life aged 25 exact. Premiums are payable monthly in advance throughout life but for a maximum of 240 payments. Expenses are 20% of the premiums in the first year and 1% of all subsequent premiums. Mortality follows AM92 Select. The interest rate is 4% per annum effective. [8] (a) Calculate the monthly premium. 3 b} Find the probability that the death benefit 1s paid AND that all 240 premiums are paid. | ¥ I F P State any assumptions made i your answer. If no assumptions are needed. you should state this explicitly. Section B (34 marks) Answer this section in a separate answer booklet. Juestion 4 'onsider an insurance sold to a life aged exactly x. The insurance pays $f at the end of the year of death if the life dies between ages x+ ; exact and x+k exact. The insurance also pays $g either on survival to age x+m exact or at the end of the year of death if the life dies between ages x +/ exact and x+ m exact. Note that x, j,k,I,m are all positive integers and that j <k </ <m. You are also given that [ # g. [3] (a) Write down an expression for the present value of the benefits paid under this insurance in terms of a suitable lifetime random variable. [3] (b) Write down an expression, using actuarial notation, for the expected present value of the benefits paid under this insurance. [3] (¢) Suppose that mortality follows AM92 Ultimate and that the interest rate is 4% per annum effective. Given that x=60,f=1,¢g=2,j=5k=10,/=15 and m =20, calculate the expected present value from part (b). d) Suppose only for this part that the insurance has no survival benefit. The death benefits described above continue to apply. Find an expression, using actuarial notation for the variance of the present value of the benefits paid under this insurance. Give your answer in terms of x, f, g, j,k,/ and m. Do NOT use numerical values from part (c).