This exam contains 8 pages (including this cover page) and 5 problems. Check to see if any pages are missing.

Transcription

1 Stat 475 Winter 2017 Midterm Exam 27 February 2017 Name: This exam contains 8 pages (including this cover page) and 5 problems Check to see if any pages are missing You may only use an SOA-approved calculator and a pencil or pen on this exam You are required to show your work on each problem on this exam Grade calculation errors: If I made an arithmetic mistake (I miscounted your total points) please come and see me and I will x it Regrade requests: I make every eort to grade your test (and those of your classmates) fairly If you feel I graded a portion of your test too harshly, please write an explanation on the back of the test and turn it into me by Wednesday March 15th in class Please note that to maintain fairness your entire test will be regraded, potentially resulting in a lower overall grade Problem Points Score Total: 54

2 Stat 475 Midterm Exam - Page 2 of 8 27 February You are given the following information for (70): p 70 = 0:9; 2p 70 = 0:8; 3p 70 = 0:6; 4p 70 = 0:3; 5p 70 = 0 a [3 pts] Give the PMF (probability mass function) for the random variable K 70 b [1 pt] Calculate P [K 70 > 3] c [2 pts] Calculate e 70 d [2 pts] Calculate p 71 e [2 pts] Calculate 0:7 q 70:6 under the CFM fractional age assumption Now suppose that you make a fractional age assumption such that the inverse of `x is linearly interpolated between ages, ie, for integer x and 0 s < 1; 1 `x+s = (1 s) 1`x + s 1 `x+1 : (Note: This assumption, which is rarely used in modern practice, is known as the Balducci or hyperbolic fractional age assumption) f [3 pts] Show that under this fractional age assumption, 1 sq x+s = (1 s)q x g [2 pts] Calculate 1:3 p 70:7 under this fractional age assumption

3 Stat 475 Midterm Exam - Page 3 of 8 27 February 2017 Mostly blank page to be used as extra space if needed

4 Stat 475 Midterm Exam - Page 4 of 8 27 February Let F 0 (t) = 1 e t ; where > 0 a [1 pt] Find the corresponding survival function S 0 (t) b [3 pts] Show that S 0 (t) is a valid survival function c [2 pts] Derive an expression for x under this survival model, simplifying as far as possible d [2 pts] Based on your answer to the previous part, briey comment on the form of the force of mortality function and the appropriateness of using this survival model to describe the mortality of a human population e [2 pts] Find an expression for t p x and simplify as far as possible f [3 pts] If = 0:1, calculate 15 j 5 q 10 and write a sentence that interprets this value g [3 pts] Show that e x = Hint: 1X k=0 ar k = a 1 r e 1 1

5 Stat 475 Midterm Exam - Page 5 of 8 27 February 2017 Mostly blank page to be used as extra space if needed

6 Stat 475 Midterm Exam - Page 6 of 8 27 February For a particular person age 40, you are given that a [2 pts] Calculate 10 p 40 b [3 pts] Calculate 30 p 40 c [3 pts] Calculate 30 j 10 q t = 0:03 t < 20 0:05 t 20

7 Stat 475 Midterm Exam - Page 7 of 8 27 February 2017 Mostly blank page to be used as extra space if needed

8 Stat 475 Midterm Exam - Page 8 of 8 27 February Let Z be the random variable representing the present value of benets for a 3-year term insurance with death benet payable at the end of the year of death issued to [82] You are given: The death benet is $400 if the insured dies in the rst year, $350 if the insured dies in the second year, and $300 if the insured dies in the third year Mortality is given by the following select and ultimate mortality table with a 2 year select period [x] q [x] q [x]+1 q x+2 x i = 10% a [2 pts] Calculate E[Z] b [3 pts] Calculate V ar(z) c [2 pts] Calculate P (Z < $150)

9 Stat 475 Midterm Exam - Page 9 of 8 27 February 2017 Mostly blank page to be used as extra space if needed

10 Stat 475 Midterm Exam - Page 10 of 8 27 February For Freddy, who is 60 years old, you are given the following: A 1 60:3 = 0:145; A 60:3 = 0:762; p 63 = p 64 = 0:9; i = 0:1 a [1 pt] Calculate A 1 60:3 b [1 pt] Calculate 3 p 60 and give the alternate symbol for this quantity Freddy is currently working and plans to retire at age 65 Because he has family members who depend on his income, he is considering purchasing a 5-year term policy with a death benet of $200; 000 payable at the end of the year of death to help replace his income should he die within the next 5 years c [2 pts] Calculate the EPV of this insurance d [2 pts] Calculate the probability that this policy will pay a benet Freddy's friend Jason suggests that because this policy has a low probability of ever paying a benet, Freddy would be better o using the money he would spend on this policy and instead investing it in, for example, a government insured savings account e [2 pts] Explain why Freddy should still purchase this policy, despite the fact that he will likely lose money on it (And in fact, he will lose money on average, assuming that the insurer charges more than the EPV for the policy)

11 Stat 475 Midterm Exam - Page 11 of 8 27 February 2017 Mostly blank page to be used as extra space if needed