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

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1 Stat 475 Winter 207 Midterm Exam 27 February 207 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 5th 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 207 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 [ pt] Calculate P [K 70 > 3] c [2 pts] Calculate e 70 d [2 pts] Calculate p 7 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 < ; `x+s ( s) `x + s `x+ : (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, s q x+s ( s)q x g [2 pts] Calculate :3 p 70:7 under this fractional age assumption Solution:

3 Stat 475 Midterm Exam - Page 3 of 8 27 February 207 (b) P [K 70 > 3] 0:3 (a) k P [K 70 k] (c) There are a couple dierent ways to nd e 3 : (i) We can use the formula for a generic discrete RV and the pmf above: E[K 3 ] 0(0:) + (0:) + 2(0:2) + 3(0:3) + 4(0:3) 26 (ii) We can use the formula we derived for e x : e 70 X k (d) p 70 p 7 2 p 7 ) p 7 0:89: kp 70 p p p p p :9+0:8+0:6+0:3+0+ 2:6 (e) 0:7 q 70:6 0:7p 70:6 ( 0:4 p 70:6 )( 0:3 p 7 ) CF (p 70 ) 0:4 (p 7 ) 0:0742 (f) sq x+s `x+ `x+s 0:3 CF `x+ `x+s `x+ ( s) `x + s `x+ ( s) `x+ + s `x+ `x+ `x [( s) p x + s ] ( s) p x s s ( s) p x ( s)( p x ) ( s)q x (0:9) 0:4 (0:89) 0:3 CF (g) :3 p 70:7 ( 0:3 p 70:7 )(p 7 ) ( 0:3q 70:7 )(p 7 ) Bal ( (0:3)q 70 )(p 7 ) Bal ( (0:3)(0:))(0:89) 0:8633

4 Stat 475 Midterm Exam - Page 4 of 8 27 February Let F 0 (t) e t ; where > 0 a [ 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:, calculate 5 j 5 q 0 and write a sentence that interprets this value g [3 pts] Show that e x Hint: X ar k a r k0 e

5 Stat 475 Midterm Exam - Page 5 of 8 27 February 207 Answer: (a) S 0 (t) F 0 (t) e (b) t (i) lim t! S 0(t) lim t! e t 0 (ii) S 0 (0) e 0 d (iii) dt S 0(t) ( )e t The rst term is negative and the second term is positive, so the product is negative for all t > 0, meaning that S 0 (t) is a non-increasing function of t Then we have shown that S 0 (t) is a valid survival function d dx (c) x S 0(x) ( )e t S 0 (x) e t (d) The force of mortality is a constant (it does not vary by age), making it inappropriate to model human populations; in human populations, the force of mortality will tend to increase with age (e) tp x S x (t) S 0(x + t) S 0 (x) F 0(x + t) F 0 (x) e (x+t) e (x) e t (f) 5 j 5 q 0 5 p 0 5 q 25 5 p 0 ( 5p 25 ) e (0:)(5) ( e (0:)(5) ) 0:08779 This is the probabiity that a person age 0 dies between the ages of 25 and 30 (g) e x X X k k kp x X k e k e k " X k0 e k # e e e e e e e e

6 Stat 475 Midterm Exam - Page 6 of 8 27 February For a particular person age 40, you are given that 0:03 t < t 0:05 t 20 a [2 pts] Calculate 0 p 40 b [3 pts] Calculate 30 p 40 c [3 pts] Calculate 30 j 0 q 40 Solution: (a) A survival function S x (t) for a person age x must meet the following three criteria: (i) S x (0) (ii) lim t! S x (t) 0 (iii) S x (t) is a non-increasing function of t R 0 (b) 0 p 40 e 0 40+t dt e 0:03tj (c) (d) 30p 40 e e e e e : R t dt R t dt R t dt R 20 R 0 0:03 dt 30 0:05 dt 20 0:6 0: j 0 q p 40 0 q 70 30j 0 q 40 (0:33287) e 30j 0 q 40 (0:33287) e 0:5 30j 0 q 40 (0:33287) (0:39346) R :05 dt

7 Stat 475 Midterm Exam - Page 7 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]+ q x+2 x i 0% a [2 pts] Calculate E[Z] Answer: E[Z] (363:64 0:208) + (289:25 0:73) + (225:39 0:4) + (0 0:478) 5746 b [3 pts] Calculate V ar(z) Answer: E[Z 2 ] (363:64 2 0:208) + (289:25 2 0:73) + (225:39 2 0:4) + (0 2 0:478) 49; 4:72 V ar(z) E[Z 2 ] E[Z] 2 49; 4:72 (57:46) 2 24,34807 c [2 pts] Calculate P (Z < $50) Answer: P (Z < $50) 0:478 from the PMF above

8 Stat 475 Midterm Exam - Page 8 of 8 27 February For Freddy, who is 60 years old, you are given the following: A 60:3 0:45; A 60:3 0:762; p 63 p 64 0:9; i 0: a [ pt] Calculate A and give the alternate symbol for this quantity 60:3 Answer: A 60:3 3E 60 A 60:3 A 0:762 0: :3 b [ pt] Calculate 3 p 60 Answer: 3 E 60 A 0:67 60:3 v3 3p 60 so that 3 p 60 (0:67)(:) 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 i Answer: EP V ha 60:3 + 3E 60 q 63 v + 3 E 60 p 63 q 64 v [0: ] d [2 pts] Calculate the probability that this policy will pay a benet Answer: The probability this policy will pay a benet is 5 q 60 5p 60 3 p 60 p 63 p 64 0:822(0:9) 2 0:6652 so that 5 q 60 5p 60 0: 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) Answer: It may still be sensible to buy the insurance, even if he loses money in most cases, and even loses money on average, because doing so hedges his mortality risk In this case, this is the risk that he will die in the next 5 years and his family will not have the income from him that they need Purchasing the insurance helps to mitigate the nancial loss associated with him dying in the next 5 years If he were to put the money in the bank and then die soon thereafter, his family would only have maybe $55; 000 or $60; 000, which may not be enough