Assignment 7 - Questions PDF

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Assignment 7...

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9

MTH5124

Actuarial Mathematics I 2017/18

Assignment 7

For submission in week 8

Solutions to question 1 are to be completed as groupwork by groups of 3 students in the same tutorial class and to be submitted at the appropriate tutorial or to Box 3 in the reception of the School of Mathematical Sciences in the Queens Building. All submissions must include the day and time of your tutorial group and names, IDs and signatures for all students in the group. This is required in order to record your engagement with this module on QM+. Final Coursework Submission Dates Tutorial Group

Latest Submission Time

Monday 10 am Tuesday 11am Friday 1pm

Thursday 23rd November 10 am Friday 24th November 10 am Monday 27th November 5 pm

1 *This question to be submitted for assessment and feedback (a) Assuming that the cumulative distribution function for the future lifetime of a newborn be given x 0.25 ) for 0 ≤ x ≤ 120, determine: by FX (x) = 1 − (1 − 120 (i) The probability that a new born dies before the age of 65. (ii) The probability that an 65 year old survives to age 90. (iii) Then probability that a life aged 40 dies between the ages of 6441 and 76 21 . (iv) The force of mortality at age 86 12 . (v) The complete expectation of life at age 50. (vi) Estimate the curtate expectation of life at age 50. (b) A survival model follows Makeham’s Law: µx = A + Bcx Show that under Makeham’s Law: t px

where s = e−A and g = e

−B ln c

for x ≥ 0 x

t

= st gc (c −1)

(1)

(2)

.

R x HINT: Remember that cx dx = ( lnc c ).

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(c)

(i) Explain in words the meaning of the symbol 10 q[49]+1 and use the table AMC00 Select to evaluate it to 5 decimal places. (ii) Assuming mortality is given by the AMC00 table determine the following: i. 5 p[60]+1 ii. 7.5 p[50] iii. 1|10.25 q[40] (iii) Dr Life is selling a new ”Rejuvenating Face Cream” with almost miraculous properties. Determine the complete expectation of life for a 50 year old male who starts to use this fabulous product on the following assumptions: • There is a five year select period for individuals starting to use the product. During this period the force of mortality is given by µ[x]+t = 0.0003t for 0 ≤ t ≤ 5. • After the select period, ultimate mortality is equal to that of a life 15 years younger using the mortality table ELT17 Males.

2 Billy Bones retires through ill health on his fiftieth birthday. With the exception of the first two years after retirement, his mortality is the mortality given in table ELT17 Male. The chance he survives the first year after retirement is 21 and the chance that he then survives the second year is 34 . (a) Find Billy’s complete (non-curtate) further expectation of life on retirement. (b) Find the probability that he is still alive a year after retirement but dies within the following eighteen months.

3

(a) Consider the probability that a man of age 45 survives 6 more months, but dies before he reaches age 50. (i) State the actuarial symbol corresponding to this probability. (ii) Using linear interpolation, use English Life Table No. 17 – Male to estimate this probability to 5 decimal places. (b) Dr Gabelhauser retires on his 60th birthday due to ill health. He is assumed to be subject to a constant force of mortality of 0.2931 for one year after retirement and then to the mortality of English Life Table No. 17 – Male. (i) Find the probability that Dr Gabelhauser survives from retirement to age 61. Give your answer to 4 decimal places. (ii) Find Dr Gabelhauser’s curtate expectation of further life at age 60. Give your answer to 2 decimal places.

4

(a) Use linear interpolation on s(x + t) for 0 < t < 1 to find an approximation for n+t dx where x and n are positive integers and 0 < t < 1. (b) Consider a group of 1,000 newborn girls. Calculate the expected number of these newborns who die within 18 months after reaching their 60th birthday, assuming the mortality rates given in table ELT17 Females.

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