PAST CAS AND SOA EXAMINATION QUESTIONS ON SURVIVAL

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1 NOTES Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial Society and the Society of Actuaries. They are reproduced in this study manual with the permission of the CAS and SOA solely to aid students studying for the actuarial exams. Some editing of questions has been done. Students may also request past exams directly from both societies. We are very grateful to these organizations for their cooperation and permission to use this material. They are, of course, in no way responsible for the structure or accuracy of the manual. Exam questions are identified by numbers in parentheses at the end of each question. CAS questions have four numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the points assigned. SOA or joint exam questions usually lack the number for points assigned. W indicates a written answer question; for questions of this type, the number of points assigned are also given. A indicates a question from the afternoon part of an exam. MC indicates that a multiple choice question has been converted into a true/false question. Although we have made a conscientious effort to eliminate mistakes and incorrect answers, we are certain some remain. We are very grateful to students who discovered errors in the past and encourage those of you who find others to bring them to my attention. Please check our web site for corrections subsequent to publication.

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3 Survival Models 1 PAST CAS AND SOA EXAMINATION QUESTIONS ON SURVIVAL A. Time of Death for a Person Aged x A1. The l x column of a mortality table is calculated by the formula: l x x 2 for 0 x 100. It is desired to know the probability that a life now age 10 will survive 30 years and then die within the ensuing ten years. In which of the following ranges does this probability lie? A. <.075 B..075 but <.085 C..085 but <.095 D..095 but <.105 E..105 (80S 4 39) A2. The following mortality table is applied to a certain population. Age l x d x Age x l x d x 0 1, After several years, the table is revised to reflect the fact that mortality has increased at ages 5 and above such that only 716 lives reach age 6. However, the probability that a life age 2 will die between ages 6 and 7 is unchanged by the mortality increase. In which of the following ranges is the number of lives reaching age 7 under the revised table? A. < 600 B. 600 but < 620 C. 620 but < 640 D. 640 but < 660 E. 660 (81S 4 14) A3. A survival function is defined by: s(x) 1 x/250 for 0 x 25 s(x) (1.2)(1 x/100) for 25 < x 100 In which of the following ranges is the probability that a survivor age 10 will die between the ages 40 and 60? A. <.20 B..20 but <.24 C..24 but <.28 D..28 but <.32 E..32 (81F 4 9) A4. For a certain population, the number of lives at age x is determined by applying the following formula: l x 10, x for 0 x 121 K is the probability that a life now age 21 will die after attaining age 40, but before attaining age 57. In which of the following ranges is K? A. <.7 B..07 but <.09 C..09 but <.11 D..11 but <.13 E..13 (81F 4 15) A5. Given the following, calculate (q x+1 + q x+2 ) to the nearest q x q x q x A..15 B..20 C..25 D..27 E..30 (84S 4 B7)

4 2 Survival Models A1. l (100)(10) 2 990,000 l (100)(40) 2 840,000 l (100)(50) 2 750, p 10 l 40 l 50 l , , , A2. l' 7 l' 6 (l 7 /l 6 ) (716)(674/728) 663 Answer: E A3. s(10) 1 10/ s(40) (1.2)(1 40/100).72 s(60) (1.2)(1 60/100) q 10 s(40) s(60) s(10) A4. l 21 10, ,000 l 40 10, ,000 l 57 10, , q 21 l 40 l 57 l 21 90,000 80, , A5. 2p x+1 2 q x+1 /q x+3.171/ p x+1 2 p x q x p x+2 2 p x+1 /p x+1.855/.95.9 q x+1 + q x+2 (1 p x+1 ) + (1 p x+2 ) (1.95) + (1.90).15 Answer: A

5 Survival Models 3 A6. You are given: t q x.10, for t 0, 1, Calculate 2 p x+5. A..40 B..60 C..72 D..80 E..81 (86S 4 13) A7. t+uq x u q x+t for t 0 and u 0. (87S MC) A8. uq x+t t u q x for t 0 and u 0. (87S MC) A9. A survival function is defined by: f(t) (kt/ 2 )e -t/, t > 0, > 0 Determine k. A. 1/ 4 B. 1/ 2 C. 1 D. 2 E. 4 (88S 160 4) A10. Which of the following are equivalent to t p x? A. t u q x t+u p x B. t+u q x t q x + t+u p x C. t q x t+u q x + t p x+u D. t q x t+u q x t p x+u E. None of these expressions are equivalent to t p x. ( ) A11. Define in words the following: a. T(x) b. t u q x c. F(x). (89S 150 A1-ai-aiii 1.2) A12. Which of the following are true? 1. t u q x t p x u q x+t 2. t u q x l x+t+u l x+t l x 3. t u q x t p x t+u p x A. 1 B. 2 C. 3 D. 1,2 E. 1,3 ( ) A13. t+rp x r p x+t for t 0 and r 0. (90S MC) A14. rq x+t t r q x for t 0 and r 0. (90S MC) A15. You are given: Calculate q y. q x.04 (x + t) t, 0 t 1 (y + t) t, 0 t 1 A B C D E (90F 150 5) A16. Given s(x) [1 (x/100)] 1/2, for 0 x 100, calculate the probability that a life age 36 will die between ages 51 and 64. A. <.15 B..15 but <.20 C..20 but <.25 D..25 but <.30 E..30 (04F 3C 8 2)

6 4 Survival Models A6..1 t q x t p x t+1 p x Since 0 p x 1, this equation gives us: 1 p x.9, 5 p x.5, and 7 p x.3. 2p x+5 7 p x / 5 p x.3/.5.60 Answer: B A7. T t+u q x t q x + (1 t q x ) u q x+t u q x+t + t q x (1 u q x+t ). Since the second term must be 0, the sum of the two terms is u q x+t. A8. T t u q x ( t p x )( u q x+t ) u q x+t. A9. Since 0 f(t) 1 and f(t) (t/ 2 )e -t/ is the probability density function for an inverse exponential, k must equal 1. A10. tq x 1 t p x t uq x t p x t+u p x t+uq x 1 t+u p x Thus we get: A. ( t p x t+u p x ) t+u p x t p x 2( t+u p x ) B. (1 t+u p x ) (1 t p x ) + t+u p x t p x C. (1 t p x ) (1 t+u p x ) + t+u p x 2(t+up x ) t p x D. (1 t p x ) (1 t+u p x ) t+u p x t p x Answer: B A11. a. T(x) is the time-until-death random variable. b. t uq x is the probability (x) will die between ages (x + t) and (x + t + u). c. F(x) is the continuous distribution function of the newborn's age at death random variable. A T t p x u q x+t t p x (1 u p x+t ) t p x t+u p x 2. F The right-hand side equals t+u p x t p x, which is negative. 3. T Answer: E A13. F t+r p x t p x r p x+t rp x+t A14. T r q x+t t r q x t p x r q x+t 1 1 A15. q y 1 p y 1 exp[ y+t dt] 1 exp[2 (x + t) dt] 1 (p x ) 2 0 q y 1 (1 q y ) 2 1 (1.04) Answer: A A16. s(36) 1 36/ s(51) 1 51/ s(64) 1 64/ s(51) s(64) q 36 s(36) Answer: A

7 Survival Models 27 D. Life Table Characteristics: Expectation of Life D1. The curtate expectation of life (e x ) for a life age x is 10.9 years. The probability that a life age x dies within the next year is.05. In which of the following ranges is the curtate expectation of life in years for a life aged (x + 1)? A. < 10.0 B but < 10.5 C but < 11.0 D but < 11.5 E (80F 4 39) D2. The curtate expectation of life (e x ) for a life age x is 14.7 years. The curtate expectation of life age (x + 1) is 14.0 years. In which of the following ranges is the probability that a life age x will survive to age (x + 1)? A. <.93 B..93 but <.95 C..95 but <.97 D..97 but <.99 E..99 (81S 4 4) D3. You are given the following curtate expectations of life, e x : Age x e x In which of the following ranges is the probability that a life age 75 will survive to age 77? A..80 B..80 but <.85 C..85 but <.90 D..90 But <.95 E..95 (81S 4 39) D4. Which of the following statements is true concerning the inequality e x+1 > e x? A. This inequality cannot be true. B. The inequality is true if and only if e x+1 > p x /q x+1. C. The inequality is true if and only if e x+1 > p x /(p x+1 )(q x+1 ). D. The inequality is true if and only if e x+1 > p x+1 /q x. E. The inequality is true if and only if e x+1 > p x /q x. ( ) D5. Given the values shown below, in which of the following ranges is the probability a life aged 86 will survive one year? Assume all values are exact. x e x T x , , ,630 A. <.830 B..830 but <.835 C..835 but <.840 D..840 but <.845 E..845 ( ) D6. You are given the following: l x 75.5x, 0 x 50 l x 100 x, 50 x 100 In which of the following ranges is the curtate expectation of life at age x? A. < 57.0 B but < 57.5 C but 58.0 D but 58.5 E (83F 4 30)

8 28 Survival Models D1. p x 1 q x e x+1 e x p x px Answer: B D2. p x Answer: D e x e x D3. 2p 75 Answer: D e 75 e e 76 e D4. Since e x p x (1 + e x+1 ), we get: e x+1 e x (e x+1 )q x p x For e x+1 to be greater than e x, e x+1 must be greater than p x/ q x. Answer: E D5. l x T x / e x l 86 44,030/ ,818 l 87 34,630/3.82 9,065 p 86 l 87 /l 86 9,065/10, D6. e x (74/ / /74.5) + (50/ / /74.5) e x Answer: B

9 Survival Models 29 D7. Given the following, calculate the instantaneous absolute rate of increase of H(x) at 60. e x H(x) x + e x A..10 B..20 C..30 D..35 E..40 (84F 5 28) D8. You are given the following survival distribution; S(x) b x/a The median age is 75. Determine e 75. A. 8.3 B C D. 20 E (87S 150 1) D9. Life expectancies and death rates from one life table are denoted by superscript A. Corresponding values from another life table are denoted by superscript B. Given the following, Calculate q B 0. e B x /ea x 1.1 for x 0, 1, 2,... qa 0.05 ea 0 57 A..049 B..051 C..053 D..055 E..057 (87S 160 4) D10. Given the following force of mortality for a survival distribution, determine e 64. x (.5)(100 x) -1, 0 x < 100 A. 16 B. 18 C. 20 D. 22 E. 24 (87F 160 8) D11. Given the following information about a group of lives, what is the complete expectation of life e 39? i) The total number of years lived beyond age 38 (T 38 ) equals 95,000. ii) The total expected number of years lived between ages 38 and 39 (L 38 ) equals 2,475. iii) The central death rate at age 38 (m 38 ) equals.021. iv) The number of survivors to age 38 (l 38 ) 2,500. A. < 37.0 B but < 37.2 C but < 37.4 D but < 37.6 E ( ) D12. You are given: Calculate 0 l x (100 x).5, where 0 x 100 e 36: t tp 36 (36 + t) dt. A B C D E (88S ) D13. Define in words e x. (89S 150 A1a-v)

10 30 Survival Models D7. d/dx (x + e x ) 1 + [ e x (x) 1] e x (x) d/dx (x + e x ) (20)(.02).40. Answer: E D8. 1 S(0) b 0/a.50 S(75) b 75/a b 1 a x/ (200/3)(1 x/100)1.5 e 75 S(75) D9. e B 1 1.1eA 1 (1.1)[eA 0 (1 qa 0)] 1 q A 0 [1.1][57 (1.05)] q B eA e B 1 Answer: A 1 (1.1)(57) D10. tp 64 exp[ 0 t (.5)( t) -1 dt)] exp[.5 log (36 t)] t 0 (36 t)/36 36 e 64 0 (36 t)/36 dt (1/9)(36 t) 3/ Answer: E D11. l 39 l 38 L 38 m 38 2,500 (2,475)(.021) 2,448 T 39 T 38 L 38 95,000 2,475 92,525 e 39 T 39 /l 39 92,525/2, Answer: E 28 D12. 0 t tp 36 (36 + t) dt e 36: p t tp 36 (36 + t) dt Answer: A 28 s(64) s(36) (28)(100 64).5 (100 36).5 D13. e x is the complete expectation of life.

11 Survival Models 51 F7. A mortality table for a subset of the population with better than average health is constructed by dividing the force of mortality in the standard table by 2. The probability of an 80-year-old dying within the next year is defined in the standard table as q 80 and in the revised table it is defined as q' 80. In the standard table q Determine the value of q' 80 in the revised table. A. <.150 B..150 but <.155 C..155 but <.160 D..160 but <.165 E..165 (93S 4A 12 2) F8. You are given: i) R 1 exp[ 0 ii) S 1 exp[ (x + t) dt] [(x + t) k] dt] iii) k is a positive constant. Determine an expression for k such that S 2R/3. A. log([1 p x ] /[1 2q x /3]) B. log([1 2q x /3]/[1 p x ]) C. log([1 2p x /3]/[1 p x ]) D. log([1 q x ] /[1 2q x /3]) E. log([1 2q x /3]/[1 q x ]) (93F ) F9. You are given: i) (35 + t) for 0 t 1 ii) p iii) '(35 + t) is the force of mortality for (35) subject to an additional hazard, for 0 t 1. iv) '(35 + t) + c v) The additional force of mortality decreases uniformly from c to 0 between age 35.5 and 36. Determine the probability that (35) subject to the additional hazard will not survive to age 36. A..015 e -.25c B..015e.25c C e -c D e -.5c E e -.75c (95S ) F10. A life table for severely disabled lives is created by modifying an existing life table by doubling the force of mortality at all ages. In the original table, q Calculate q 75 in the modified table. A. <.21 B..21 but <.23 C..23 but <.25 D..25 but <.27 E..27 (96S 4A 16 2) F11. You are given the following life table information about the normal population: t l x+t 1, Assume that you insure a group of insureds who are subject to mortality 50% greater than the normal population. Determine the probability that an insured from this group will live 3 years from time t 0 (i.e., 3 p' x ). A. <.05 B..05 but <.10 C..10 but <.15 D..15 but <.20 E..20 (97F 4A 18 2)

12 52 Survival Models F7. See F2. q' q Answer: D F8. S 1 p x e k 2R/3 2q x /3 e k 1 2q x/3 p x k log( 1 2q x/3 1 q x ). Answer: E F9. (35 + t) + 2(1 t)c for.5 t q' p '.5.5 p' exp[ ((35) + c) dt] exp[ [(35 + t) + 2(1 t)c] dt] 0.5 q' 35 1 p 35 exp[.5c + (2tc t c) 2 1.5] 1 (.985)exp[.5c + 2c c c +.25c] q' 35 1 (.985)e -.75c. Answer: E F10. q 75 1 exp ( 0 1 (75 + s) ds).12 ( 0 1 (75 + s) ds) log ( 0 2(75 + s) ds) q' 75 1 exp (.25567).226. Answer: B F11. q' 0 (1.5)(1 p 0 ) (1.5)(1 700/1,000).45 q' 1 (1.5)(1 p 1 ) (1.5)(1 480/700) q' 2 (1.5)(1 p 2 ) (1.5)(1 300/480) p' x (1 q' 0 )(1 q' 1 )(1 q' 2 ) (1.45)( )(1.5625).1271.

13 Survival Models 53 F12. You are given: i) R 1 exp[ 0 t ii) S 1 exp[ 0 t X (t) dt] ( X (t) + k) dt] iii) k is a constant such that S.75R Determine an expression for k. A. log((1 q x )/(1.75q x )) B. log((1.75q x )/(1 p x )) C. log((1.75p x )/(1 p x )) D. log((1 p x )/(1.75q x )) E. log((1.75q x )/(1 q x )) (02F 3 35) (Sample M 59) F13. Individuals with flapping gum disease are known to have a constant force of mortality. Historically, 10% will die within twenty years. A new, more serious strain of the disease has surfaced with a constant force of mortality equal to 2. Calculate the probability of death in the next twenty years for an individual with this new strain. A. 17% B. 18% C. 19% D. 20% E. 21% (05F ) F14. For a group of lives aged 30, containing an equal number of smokers and nonsmokers, you are given: i) For nonsmokers, n (x).08, x 30 ii) For smokers, s (x).16, x 30 Calculate q 80 for a life randomly selected from those surviving to age 80. A..078 B..086 C..095 D..104 E..112 (05F M 32) F15. Use the Illustrative Life Table with the following values and adjustments: q * x 4q x for 67 x 68 q * 69 q 66 Values from the Illustrated Life Table are the following: 1,000q ,000q ,000q ,000q Calculate 2 2 q 66. A. <.1150 B but <.1175 C but <.1200 D but <.1225 E (08S 3L 14 2) F16. For a certain life aged 48, mortality follows the Illustrative Life Table. However, from ages 50 to 52, it is found that mortality is four times greater than the Illustrative Life Table, i.e. q 50 * 4q 50. Given the following values, calculate 1,000 4 p 48. l 48 90, l 49 90, l 50 89, l 51 88, l 52 88, A. 925 B. 925 but < 935 C. 935 but < 945 D. 945 but < 950 E. 950 (10F 3L 2 2)

14 54 Survival Models F12. S 1 p x e -k 3R/4 3q x /4 e -k 1 3q x/4 p x e k k log( 1 q x 1 3q x /4 ) Answer: A p x 1 3q x /4 F13. S(20).90 e -20 ln ln (.10536) S'(20) e -(20)(2) e -(20)(2)(.00527) q x 1 S'(20) F14. q n 80 1 e q s 80 1 e p n 80 e-(.08)(50) e p s 80 e-(.16)(50) e p n p s wn.01832/ w s.00034/ q 80 w n q n 80 + ws q s 80 (.98178)(.07688) + (.01822)(.14786) Answer: A F15. q * 69 q q * 67 (4)( ) q* 68 (4)( ) q 66 [1 q 66 ][1 q * 67 ][1 (1 q* 68 )(1 q* 69 )] 2 2q 66 [ ][ ][1 ( )( )] Answer: B F16. d 50 l 50 l 51 89, , d 51 l 51 l 52 88, , d 50 3d 50 (3)(529.89) 1, d 51 3d 51 (l 51 d 50 )/l 51 (3)(571.43)(88, ,589.67)/88, , l 52 l 52 d 50 d 51 88, , , , ,000 4 p 48 (1,000)(l 52 / l 48 ) (1,000)(85,134.35/90,456.78)

15 Survival Models 67 H28. You are given: (i) x l x 60 99, , , , , , , ,222 (ii) a q 60 assuming a uniform distribution of deaths over each year of age. (iii) b q 60 assuming a constant force of mortality over each year of age. Calculate 100,000(a b). A. 24 B. 9 C. 42 D. 73 E. 106 (13F-MLC-25)

16 68 Survival Models H28. First we note that q p q l d d l l l 60 l 63.4 l 60 l 60 Using the uniform distribution of deaths over each year of age assumption, a.k.a., UDD, we have l l UDD l , ,555 62,221.60, l l UDD l , ,333 34, Therefore, 62,221,6 34,444.1 a q UDD 99,999 On the other hand, under Constant Force (CF) l 63.4 l p 63 CF so that l 63.4 CFl 64 Similarly, l l 63 CFl l 66 Therefore, b q 60 CF e ln p 63 Constant Force 0.4 p l 64 l , , , , , , , , ,999, Finally, 100,000a b 100, Answer E.

17 Survival Models 69 I. De Moivre's Law of Mortality I1. For every 125 lives born at the same time, a mortality table shows one death each year until there are no survivors. Which of the following is the youngest age for which the probability of living to age 65 is at least 2/3? A. 25 B. 30 C. 35 D. 40 E. 45 (79F 4 17) I2. The mortality of a population is as follows: i) For 100 males born at the same time, a mortality table shows one death each year until there are no survivors. ii) For 110 females born at the same time, a mortality table shows one death each year until there are no survivors. On January 1 of a given calendar year, 150 males and 121 females are born. Which of the following is closest to the age of the group when there are exactly 5 more surviving males than females. A. 45 B. 50 C. 55 D. 60 E. 65 (79F 4 19) I3. The terminal age of a mortality table is 120, and a constant number of deaths occur each year. It is desired to know the probability that a person now age 24 will not die between the ages of 36 and 72. In which of the following ranges does this probability lie? A. <.300 B..300 but <.450 C..450 but <.600 D..600 but <.750 E..750 (80S 4 21) I4. (x) d x-2/ l x if l x is of the form k( x). ( ) I5. Mortality follows de Moivre's law and e Calculate Var(T(16)) to the nearest integer. A. 595 B. 588 C. 505 D. 472 E. 300 (84F 4 21) I6. Equal numbers of males and females are born in a population. You are given the following survival distribution functions: i) For males, s(x) 1 x/90 for 0 x 90. i) For females, s(x) 1 x/100 for 0 x 100. Calculate the force of mortality at age 60 for a cohort observed from birth. A B C D E (86S 5 A10) I7. You are given that mortality follows de Moivre's law and that e Calculate q 30. A. 1/30 B. 1/60 C. 1/61 D. 1/62 E. 1/70 (86S 4 22)

18 70 Survival Models I1. Mortality follows de Moivre's law with /3 65 p x x 35 x 125 x I2. Mortality follows de Moivre's law with 100 for males and 110 for females. Number of Males at Age x x Number of Females at Age x x Difference in Number of Males and Females ( x) ( x) 5 x 60 Answer: D I3. Mortality follows de Moivre's law with q Answer: D I4. T d x-2 k(l x-2 l x-1 ) k[ (x 2)] k[ (x 1)] k (x) k k( x) 1 x -x I5. e x tp x dt -x e x + x (2)(42) x Var[T(x)] 2 0 x t x dt t t 2 2 x -x 0 x 2 x t t p x dt ( e x ) t x t x dt ( e x ) 2 Var[T(x)] [t 2 2t 3 /3( x)] -x 0 ( x/2) 2 ( x) 2 /12 Var(T(16)) ( x) 2 /12 (100 16) 2 / Answer: B I6. 60p m 0 x /3 60p f /5 m 60 1 x /30 f /40 60 ( 60p m 0 )( m 60) + ( 60 p f 0)( f 60) 60p m p f 0 (1/3)(1/30) + (2/5)(1/40) 1/3 + 2/ Answer: A I7. See I5. 2 e x + x (2)(30) q 30 1 p 30 1 x x Answer: B 1/60.