SOCIETY OF ACTUARIES EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE SOLUTIONS Copyright 2013 by the Society of Actuaries The solutions in this study note were previously presented in study note MLC-09-08 and MLC-09-11. They have been edited for use under the 2014 learning objectives and textbook. Most solutions are mathematically the same as those in MLC-09-11. Solutions whose wording has changed are identified with a * before the question number. The only question which has changed mathematically is 300. Questions 310-318 are new. Some of the questions are taken from past SOA examinations. No questions from published exams after 2005 are included except 310-313, which come from exams of 2012 or 2013. The November 2006 Exam M and May 2007, May 2012, November 2012, May 2013 and November 2013 Exam MLC are available at http://www.soa.org/education/exam-req/syllabus-study-materials/edu-multiple-choiceexam.aspx. The average time allotted per multiple choice question will be shorter beginning with the Spring 2014 examination. Some of the questions here would be too long for the new format. However, the calculations, principles, and concepts they use are still covered by the learning objectives. They could appear in shorter multiple choice questions, perhaps with intermediate results given, or in written answer questions. Some of the questions here would still be appropriate as multiple choice questions in the new format. The weight of topics in these sample questions is not representative of the weight of topics on the exam. The syllabus indicates the exam weights by topic. April 2014 changes: In Solutions 122B and 194, instances of subscripts of the form u:v were changed to be just u or v as appropriate. Further April 2014 changes: Solution to 175 has a typo corrected. Solution to 248 is the negative of the previous solution. MLC-09-13 PRINTED IN U.S.A. Question #1 Answer: E 2 q30:34 2 p30:34 3 p30:34 2 p30 0.9 0.8 0.72 2 p34 0.5 0.4 0.20 2 p30:34 0.72 0.20 0.144 2 p30:34 0.72 0.20 0.144 0.776 3 p30 0.72 0.7 0.504 3 p34 0.20 0.3 0.06 3 p30:34 0.504 0.06 0.03024 3 p30:34 0.504 0.06 0.03024 0.53376 2 q30:34 0.776 0.53376 0.24224 Alternatively, 2 q30:34 2 q30 2 q34 2 q30:34 b g 2 p30q32 2 p34q36 2 p30:34 1 p32:36 = (0.9)(0.8)(0.3) + (0.5)(0.4)(0.7) – (0.9)(0.8)(0.5)(0.4) [1-(0.7)(0.3)] = 0.216 + 0.140 – 0.144(0.79) = 0.24224 Alternatively, 2 q30:34 3 q30 3 q34 2 q30 2 q34 1 3 p30 1 3 p34 1 2 p30 1 2 p34 1 0.504 1 0.06 1 0.72 1 0.20 0.24224 (see first solution for MLC-09-13 2 p30 , 2 p34 , 3 p30 , 3 p34 2 ) Question #2 Answer: E 1000 Ax 1000 Ax1:10 10 Ax 10 1000 e0.04t e0.06t (0.06)dt e 0.4 e 0.6 e 0.05t e 0.07 t (0.07)dt 0 0 10 1000 0.06 e0.1t dt e1 (0.07) e0.12t dt 0 0 0.10 t 10 0.12 t 1000 0.06 e0.10 e1 (0.07) e0.12 0 0 1 1 e1 0.07 1000 0.06 0.12 e 0.10 1000 0.37927 0.21460 593.87 Because this is a timed exam, many candidates will know common results for constant force and constant interest without integration. For example Ax1:10 10 E x Ax e 1 10 Ex 10 With those relationships, the solution becomes 1000 Ax 1000 Ax1:10 10 Ex Ax 10 0.06 0.07 0.06 0.04 10 0.06 0.04 10 1000 e 1 e 0.07 0.05 0.06 0.04 1000 0.60 1 e1 0.5833 e1 593.86 MLC-09-13 3 Question #3 Answer: D 1 dt 20 1 100 5 0.07 t e 0 20 7 7 2 1 1 bt vt t px x t dt e0.12t e 0.16t e 0.05t dt e 0.09t dt 0 20 20 0 1 100 0.09t 5 e 0 9 20 9 E Z bt v t t px x t dt e0.06 t e 0.08 t e 0.05 t 0 0 E Z 2 0 2 5 5 Var Z 0.04535 9 7 Question #4 Answer: C Let ns = nonsmoker and s = smoker k= q xb kg ns pxb kg q xb gk s px k 0 .05 0.95 0.10 0.90 1 .10 0.90 0.20 0.80 2 .15 0.85 0.30 0.70 1 ns A x:2 ns v qx 1 0.05 1.02 A x:2 1 s v 1 1.02 qx s ns px v2 1 1.022 ns ns 0.95 0.10 0.1403 v2 0.10 qx 1 1 1.02 2 px s qx 1 s 0.90 0.20 0.2710 A1x:2 weighted average = (0.75)(0.1403) + (0.25)(0.2710) = 0.1730 MLC-09-13 4 s Question #5 Answer: B x x1 x 2 x 3 0.0001045 t px e0.0001045t 1 APV Benefits e t 1,000,000 t px x dt 0 2 e t 500,000 t px x dt 0 3 e 200,000 t px x dt 0 1,000,000 0.0601045t 500,000 0.0601045t 250,000 0.0601045t e dt e dt e dt 2,000,000 0 250,000 0 10,000 0 27.5 16.6377 457.54 *Question #6 Answer: B 1 EPV Benefits 1000 A40:20 EPV Premiums a40:20 k E401000vq40k k 20 k E401000vq40k k 20 Net premiums Equivalence principle 1 1000 A40:20 k E401000vq40k a40:20 k E401000vq40k k 20 20 1 1000 A40:20 / a40:20 161.32 0.27414 369.13 14.8166 0.27414 11.1454 5.11 1 While this solution above recognized that 1000P40:20 and was structured to take advantage of that, it wasn’t necessary, nor would it save much time. Instead, you could do: MLC-09-13 5 EPV Benefits 1000 A40 161.32 EPV Premiums = a40:20 a40:20 20 E40 k E60 1000vq60 k k 0 20 E40 1000 A60 14.8166 0.27414 11.1454 0.27414 369.13 11.7612 101.19 11.7612 101.19 161.32 161.32 101.19 5.11 11.7612 Question #7 Answer: C ln 1.06 0.53 0.5147 0.06 1 A70 1 0.5147 a70 8.5736 d 0.06 /1.06 0.97 a69 1 vp69a70 1 8.5736 8.8457 1.06 A70 A70 i a69 2 a69 2 1.00021 8.8457 0.25739 2 8.5902 m 1 works well (is closest to the exact m Note that the approximation ax ax 2m answer, only off by less than 0.01). Since m = 2, this estimate becomes 1 8.8457 8.5957 4 Question #8 - Removed Question #9 - Removed MLC-09-13 6 Question #10 Answer: E d = 0.05 v = 0.95 At issue 49 A40 v k 1 k q40 0.02 v1 ... v50 0.02v 1 v50 / d 0.35076 k 0 and a40 1 A40 / d 1 0.35076 / 0.05 12.9848 so P40 1000 A40 350.76 27.013 a40 12.9848 E 10 L K40 10 1000 A50 Revised P40 a50 549.18 27.013 9.0164 305.62 Revised where Revised A50 and 24 v k 1 k q50 Revised k 0 Revised a50 1 A Revised 50 0.04 v1 ... v 25 0.04v 1 v 25 / d 0.54918 / d 1 0.54918 / 0.05 9.0164 *Question #11 Answer: E Let NS denote non-smokers and S denote smokers. The shortest solution is based on the conditional variance formula Var X E Var X Y Var E X Y Let Y = 1 if smoker; Y = 0 if non-smoker 1 AxS E aT Y 1 axS 1 0.444 5.56 0.1 1 0.286 Similarly E aT Y 0 7.14 0.1 E E aT Y E E aT 0 Prob Y=0 E E aT 1 Prob Y=1 7.14 0.70 5.56 0.30 6.67 MLC-09-13 7 E E aT Y 7.142 0.70 5.562 0.30 2 44.96 44.96 6.672 0.47 E Var aT Y 8.503 0.70 8.818 0.30 Var E aT Y 8.60 Var aT 8.60 0.47 9.07 Alternatively, here is a solution based on Var(Y ) E Y 2 E Y , a formula for the variance of any random variable. This can be 2 transformed into E Y 2 Var Y E Y which we will use in its conditional form E aT 2 2 2 NS Var aT NS E aT NS Var aT E aT 2 E aT 2 E aT E aT S Prob S E aT NS Prob NS 0.30axS 0.70axNS 0.30 1 AxS 0.70 1 A NS x 0.1 0.1 0.30 1 0.444 0.70 1 0.286 0.30 5.56 0.70 7.14 0.1 1.67 5.00 6.67 E aT 2 E aT 2 S Prob S E aT 2 NS Prob NS 2 0.30 Var aT S E aT S 0.70 Var aT NS E aT NS 2 2 2 0.30 8.818 5.56 0.70 8.503 7.14 11.919 + 41.638 = 53.557 Var aT 53.557 6.67 9.1 2 MLC-09-13 8 Alternatively, here is a solution based on aT 1 vT 1 v Var aT Var vT Var since Var X constant Var X T since Var constant X constant Var vT 2 2 Ax Ax 2 Var X 2 2 This could be transformed into 2Ax 2 Var aT Ax2 , which we will use to get 2 Ax NS and 2AxS . Ax E v 2T 2 E v 2T NS Prob NS E v 2T S Prob S 2 2Var aT NS Ax NS Prob NS 2 2 Var aT S AxS Prob S 0.01 8.503 0.2862 0.70 0.01 8.818 0.4442 0.30 0.16683 0.70 0.28532 0.30 0.20238 Ax E vT E vT NS Prob NS E vT S Prob S 0.286 0.70 0.444 0.30 0.3334 Ax Ax 2 Var aT MLC-09-13 2 2 0.20238 0.33342 9.12 0.01 9 Question #12 - Removed Question #13 Answer: D Let NS denote non-smokers, S denote smokers. Prob T t Prob T t NS Prob NS P rob T t S P rob S 1 e0.1t 0.7 1 e0.2t 0.3 1 0.7e0.1t 0.3 e0.2t S0 (t ) 0.3e0.2 t 0.7e0.1t Want tˆ such that 0.75 1 S0 tˆ or 0.25 S0 tˆ ˆ ˆ ˆ 2 0.25 0.3e0.2t 0.7e0.1t 0.3 e0.1t ˆ 0.7e0.1t ˆ Substitute: let x e0.1t 0.3x2 0.7 x 0.25 0 This is quadratic, so x 0.7 0.49 0.3 0.25 4 2 0.3 x 0.3147 ˆ e0.1t 0.3147 MLC-09-13 so tˆ 11.56 10 *Question #14 Answer: A At a constant force of mortality, the net premium equals the force of mortality and so 0.03 . 0.03 2 Ax 0.20 2 2 0.03 0.06 Var 0 L where A 2 Ax Ax a 2 2 0.20 13 0.06 0.09 0.03 1 0.09 3 2 2 a 0.20 1 1 0.09 Question #15 - Removed *Question #16 Answer: A 1000 P40 A40 161.32 10.89 a40 14.8166 a 11.1454 1000 20V40 1000 1 60 1000 1 247.78 14.8166 a40 20V 5000P40 (1 i) 5000q60 21V P60 247.78 (5)(10.89) 1.06 5000 0.01376 255 1 0.01376 [Note: For this insurance, 20V 1000 20V40 because retrospectively, this is identical to whole life] Though it would have taken much longer, you can do this as a prospective reserve. The prospective solution is included for educational purposes, not to suggest it would be suitable under exam time constraints. 1000P40 10.89 as above 1 1000 A40 4000 20 E40 A60:5 1000P40 5000P40 20 E40 a60:5 1 A60 5 E60 A65 0.06674 where A60:5 MLC-09-13 11 20 E40 5 E60 a65 a40:20 a40 20 E40 a60 11.7612 a60:5 a60 5 E60 a65 4.3407 1000 0.16132 4000 0.27414 0.06674 10.89 11.7612 510.89 0.27414 4.3407 0.27414 0.68756 9.8969 161.32 73.18 128.08 64.79 1.86544 22.32 Having struggled to solve for , you could calculate above) calculate 21V recursively. 20V 20V prospectively then (as 1 4000 A60:5 1000 A60 5000 P40 a60:5 5 E60 a65 4000 0.06674 369.13 5000 0.01089 4.3407 22.32 0.68756 9.8969 247.86 (minor rounding difference from 1000 20V40 ) Or we can continue to 21V 21V prospectively 1 5000 A61:4 1000 4 E61 A65 5000P40 a61:4 4 E61 a65 where 4 E61 l65 4 7,533,964 v 0.79209 0.73898 l61 8,075, 403 1 A61:4 A61 4 E61 A65 0.38279 0.73898 0.43980 a61:4 0.05779 a61 4 E61 a65 10.9041 0.73898 9.8969 3.5905 21V 5000 0.05779 1000 0.73898 0.43980 5 10.89 3.5905 22.32 0.73898 9.8969 255 Finally. A moral victory. Under exam conditions since prospective net premium reserves must equal retrospective net premium reserves, calculate whichever is simpler. MLC-09-13 12 Question #17 Answer: C Var Z 2 A41 A41 2 A41 A40 0.00822 A41 (vq40 vp40 A41 ) A41 (0.0028 /1.05 0.9972 /1.05 A41) A41 0.21650 2 A41 2 A40 0.00433 2 A41 v 2 q40 v 2 p40 2 A41 2 A41 (0.0028 /1.052 0.9972 /1.052 2 A41 0.07193 Var Z 0.07193 0.216502 0.02544 Question #18 - Removed Question #19 - Removed MLC-09-13 13 2 A41 ) *Question #20 Answer: D x x1t x 2t 0.2 xt x 2t x 2t 0.8 xt '1 qx k 3 1 '1 px 1 1 e 0.2 k t 2 dt 0 1 e 0.2 k 3 0.04 ln 1 0.04 / 0.2 0.2041 k 0.6123 2 2 qx px x 2t dt 0.8 2 0 t 2 0 0.8 2 qx 0.8 1 2 px 2 t px xt dt ( ) xt d t 0 2 px e 2 e k t2 d t 0 8 k e 3 8 0.6123 3 e 0.19538 2 2 qx 0.8 1 0.19538 0.644 MLC-09-13 14 Question #21 Answer: A k min(k ,3) f(k) f k min(k ,3) 0 1 2 3+ 0 1 2 3 0.1 (0.9)(0.2) = 0.18 (0.72)(0.3) = 0.216 1-0.1-0.18-0.216 = 0.504 0 0.18 0.432 1.512 f k min k ,3 0 0.18 0.864 4.536 2.124 5.580 E min( K ,3) 2.124 E min K ,3 2 5.580 Var min( K ,3) 5.580 2.1242 1.07 Note that E min( K ,3) is the temporary curtate life expectancy, ex:3 if the life is age x. Question #22 Answer: B 0.1 60 0.08 60 e e 2 0.005354 S0 (60) 0.1 61 0.08 61 e e 2 0.00492 0.00492 q60 1 0.081 0.005354 S0 (61) MLC-09-13 15 2 *Question #23 Answer: D Let q64 for Michel equal the standard q64 plus c. We need to solve for c. Recursion formula for a standard insurance: 20V45 19V45 P45 1.03 q64 1 20V45 Recursion formula for Michel’s insurance 20V45 19V45 P45 0.01 1.03 q64 c (1 20V45 ) The values of 19V45 and 20V45 are the same in the two equations because we are told Michel’s net premium reserves are the same as for a standard insurance. Subtract the second equation from the first to get: 0 1.03 (0.01) c(1 20V45 ) c (1.03) 0.01 1 20V45 0.0103 1 0.427 0.018 MLC-09-13 16 Question #24 Answer: B K is the curtate future lifetime for one insured. L is the loss random variable for one insurance. LAGG is the aggregate loss random variables for the individual insurances. AGG is the standard deviation of LAGG . M is the number of policies. L v K 1 aK 1 1 v K 1 d d 1 Ax E L Ax ax Ax d 0.75095 0.24905 0.025 0.082618 0.056604 Var L 1 d 2 2 Ax Ax2 2 E LAGG M E L 0.082618M Var LAGG M Var L M (0.068034) AGG 0.260833 M L E LAGG E LAGG Pr LAGG 0 AGG AGG AGG 0.082618M Pr N (0,1) M 0.260833 0.082618 M 0.260833 M 26.97 1.645 minimum number needed = 27 MLC-09-13 0.025 2 1 0.09476 0.24905 0.068034 0.056604 17 Question #25 Answer: D 1 v K 1 for K 0,1, 2,... d Z 2 Bv K 1 for K 0,1,2,... Z1 12,000 Annuity benefit: Death benefit: 1 v K 1 Bv K 1 d 12,000 12,000 K 1 B v d d Z Z1 Z 2 12,000 New benefit: 2 12,000 K 1 Var( Z ) B Var v d 12,000 Var Z 0 if B 150,000 . 0.08 In the first formula for Var Z , we used the formula, valid for any constants a and b and random variable X, Var a bX b2Var X Question #26 Answer: A xt: y t xt y t 0.08 0.04 0.12 Ax xt / xt 0.5714 Ay y t / y t 0.4 Axy xt: y t / xt: y t 0.6667 axy 1/ xt: y t 5.556 Axy Ax Ay Axy 0.5714 0.4 0.6667 0.3047 Premium = 0.304762/5.556 = 0.0549 MLC-09-13 18 *Question #27 Answer: B P40 A40 / a40 0.16132/14.8166 0.0108878 P42 A42 / a42 0.17636/14.5510 0.0121201 a45 a45 1 13.1121 E 3 L K42 3 1000 A45 1000P40 1000P42 a45 201.20 10.89 12.1213.1121 31.39 Many similar formulas would work equally well. One possibility would be 1000 3V42 1000P42 1000P40 , because prospectively after duration 3, this differs from the normal net premium reserve in that in the next year you collect 1000P40 instead of 1000P42 . *Question #28 Answer: E E min T ,40 40 0.005 40 32 2 32 t f t dt 40 f t dt 40 w 0 40 t f t dt t f t dt 40 .6 w w 0 40 86 tf t dt w 40 tf t dt 54 w 40 t 40 f t dt 54 40 .6 50 w e 40 40 MLC-09-13 S0 40 .6 19 *Question #29 Answer: B d 0.05 v 0.95 Step 1 Determine p x from Kevin’s work: 608 350vpx 1000vqx 1000v2 px px1 qx1 608 350 0.95 px 1000 0.95 1 px 1000 0.9025 px 1 608 332.5 px 950 1 px 902.5 px px 342 / 380 0.9 Step 2 Calculate 1000 Px:2 , as Kira did: 608 350 0.95 0.9 1000 Px:2 1 0.95 0.9 299.25 608 489.08 1000 Px:2 1.855 The first line of Kira’s solution is that the expected present value of Kevin’s net premiums is equal to the expected present value of Kira’s, since each must equal the expected present value of benefits. The expected present value of benefits would also have been easy to calculate as 1000 0.95 0.1 1000 0.952 0.9 907.25 Question #30 Answer: E Because no premiums are paid after year 10 for (x), 11Vx Ax11 One of the recursive reserve formulas is 10V h 1V hV h 1 i bh1qxh 32,535 2,078 1.05 100,000 0.011 35,635.642 px h 0.989 35,635.642 0 1.05 100,000 0.012 36,657.31 A 11V x 11 0.988 MLC-09-13 20 Question #31 Answer: B t The survival function is S0 (t ) 1 : Then, x t ex and t px 1 2 x 105 45 e45 30 2 105 65 e65 20 2 e45:65 40 0 t p45:65dt 40 60 t 0 FG H 60 40 t dt 40 1 60 40 2 1 3 60 40 t t t 60 40 2 3 IJ K 40 0 1556 . e 45:65 e45 e 65 e 45:65 30 20 1556 . 34 In the integral for e45:65 , the upper limit is 40 since 65 (and thus the joint status also) can survive a maximum of 40 years. Question #32 Answer: E 4 S0 (4) / S0 (4) d i e4 / 100 1 e4 / 100 e4 / 100 1 e4 / 100 e4 1.202553 100 e4 MLC-09-13 21 Question # 33 Answer: A L ln px big OP qb g LM ln e OP qxbi g qxb g M M ln p b g P x M ln e-b g P bi g N qxb g Q x N Q bi g b g x b g x b1g x b 2 g x b 3g 15 . qxb g 1 e b g 1 e1.5 =0.7769 b0.7769gb g b0.5gb0.7769g 2 qxb2 g b g 15 . 0.2590 Question # 34 Answer: D 22 A 60 v 3 B B q 60 2 B pay at end live of year 3 2 years in year 3 v4 pay at end of year 4 2 p 60 1 1.03 3 3p 60 then die q60 3 live then die 3 years in year 4 1 0.09 1 0.11 0.13 1 1.03 4 1 0.09 1 0.11 1 0.13 0.15 0.19 MLC-09-13 22 Question # 35 Answer: B ax ax:5 5Ex ax 5 a x:5 5 Ex 1 e 0.07b5g 4.219 , where 0.07 for t 5 0.07 e 0.07b5g 0.705 a x5 1 12.5 , where 0.08 for t 5 0.08 b gb g ax 4.219 0.705 12.5 13.03 Question #36 Answer: D 1 2 px p' x p' x 0.8 0.7 0.56 1 qx ln p ' 1 x q since UDD in double decrement table ln p x x ln 0.8 0.44 ln 0.56 0.1693 1 0.3 qx 0.1 0.3qx 1 0.053 1 0.1qx To elaborate on the last step: Number dying from cause 1 between x 0.1 and x 0.4 1 0.3 qx 0.1 Number alive at x 0.1 Since UDD in double decrement, 1 lx 0.3 qx lx 1 0.1qx MLC-09-13 23 *Question #37 Answer: E The net premium is oL 1 1 0.04 0.04333 ax 12 v T (0.04333 0.0066)aT 0.02 0.003aT 1 vT v T 0.04693 0.02 0.04693 0.04693 v T 1 0.02 Var o L Var v T 2 0.04693 1 0.1(4.7230) 0.4723 Question #38 Answer: D 0.7 0.1 0.2 T 0.3 0.6 0.1 0 0 1 0.52 0.13 0.35 T 0.39 0.39 0.22 0 0 1 2 Actuarial present value (APV) prem = 800(1 + (0.7 + 0.1) + (0.52 + 0.13)) = 1,960 APV claim = 500(1 + 0.7 + 0.52) + 3000(0 + 0.1 + 0.13) = 1800 Difference = 160 Question # 39 - Removed MLC-09-13 24 Question # 40 Answer: D Use Mod to designate values unique to this insured. b b g g b gb g 1000b0.36933 / 111418 . . g 3315 a60 1 A60 / d 1 0.36933 / 0.06 / 106 . 111418 . 1000 P60 1000 A60 / a60 d i Mod Mod Mod A60 v q60 p60 A61 d b b i gb g 1 01376 . 0.8624 0.383 0.44141 106 . g Mod a Mod 1 A60 / d 1 0.44141 / 0.06 / 106 . 9.8684 d Mod Mod E 0 LMod 1000 A60 P60a60 i b g 1000 0.44141 0.03315 9.8684 114.27 *Question # 41 Answer: D The prospective reserve at age 60 per 1 of insurance is A60 , since there will be no future premiums. Equating that to the retrospective reserve per 1 of coverage, we have: s40:10 A60 P40 P50Mod s50:10 20 k40 10 E50 1 a40:10 a A40 A40 :20 Mod 50:10 A60 P50 a40 10 E40 10E50 10 E50 20 E40 0.36913 016132 . 7.70 7.57 0.06 P50Mod 14.8166 0.53667 0.51081 0.51081 0.27414 b gb g 0.36913 0.30582 14.8196 P50Mod 0.21887 1000 P50Mod 19.04 MLC-09-13 25 Alternatively, you could equate the retrospective and prospective reserves at age 50. Your equation would be: A50 a50:10 P50Mod 1 A40 a40:10 A40:10 a40 10 E40 10 E40 1 where A40 A40 10 E40 A50 :10 b gb g 016132 . 0.53667 0.24905 0.02766 d . 7.70 0.02766 ib g 14016132 .8166 0.53667 0.53667 . b1000gb014437 g 19.07 0.24905 P50Mod 7.57 1000 P50Mod 7.57 Alternatively, you could set the expected present value of benefits at age 40 to the expected present value of net premiums. The change at age 50 did not change the benefits, only the pattern of paying for them. A40 P40 a40:10 P50Mod 016132 . 10 E40 a50:10 . FG 016132 I H 14.8166JK b7.70g d P ib0.53667gb7.57g b1000gb0.07748g 19.07 Mod 50 1000 P50Mod 4.0626 Question # 42 Answer: A d xb2g qxb2g lxb g 400 b g d xb1g 0.45 400 180 qx b2 g d xb2 g 400 0.488 l b g d b1g 1000 180 x x pxb2g 1 0.488 0.512 Note: The UDD assumption was not critical except to have all deaths during the year so that 1000 - 180 lives are subject to decrement 2. MLC-09-13 26 Question #43 Answer: D Use “age” subscripts for years completed in program. E.g., p0 applies to a person newly hired (“age” 0). Let decrement 1 = fail, 2 = resign, 3 = other. 1 1 1 Then q0b g 14 , q1b g 15 , q2b g 13 q0b2g 15 , q1b2g 13 , q2b2g 18 q0b3g 110 , q1b3g 19 , q2b3g 14 b gb gb g This gives p0b g 1 1 / 4 1 1 / 5 1 1 / 10 0.54 b gb gb g pb g b1 1 / 3gb1 1 / 8gb1 1 / 4g 0.438 So 1b g 200, 1b g 200 b0.54g 108 , and 1b g 108 b0.474g 512 . p1b g 1 1 / 5 1 1 / 3 1 1 / 9 0.474 2 0 1 q2b1g log p2b1g / log p2b g q2b g 2 c h / logb0.438g 1 0.438 b0.405 / 0.826gb0.562g q2b1g log 2 3 0.276 d2b1g l2b g q2b1g 512 . 0.276 14 b gb g Question #44 - Removed MLC-09-13 27 Question #45 Answer: E ex For the given life table function: x 2 1 k qx x x 1 Ax k b 1 x 1 k 1 v x k b a x Ax ax v k 1 k qx x 1 Ax d e50 25 100 for typical annuitants ey 15 y Assumed age = 70 A70 a30 0.45883 30 a70 9.5607 500000 b a70 b 52, 297 Question #46 Answer: B 10 E30:40 10p30 10 d p v id p v ib1 i g b E gb E gb1 i g b0.54733gb0.53667 gb179085 . g p40 v10 10 10 30 10 10 10 40 10 10 30 10 40 0.52604 The above is only one of many possible ways to evaluate which should give 0.52604 a30:40:10 a30:40 10 E30:40 a3010:4010 b g b gb g b13.2068g b0.52604gb114784 . g a30:40 1 0.52604 a40:50 1 7.1687 MLC-09-13 28 10 10 p30 10 p40 v , all of *Question #47 Answer: A Equivalence Principle, where is annual net premium, gives 1000 A35 ( IA)35 a35 d 1000 A35 1000 0.42898 (1199143 . 616761 . ) a35 IA 35 b gi 428.98 582382 . 73.66 We obtained a35 from a35 1 A35 1 0.42898 1199143 . d 0.047619 Question #48 - Removed Question #49 Answer: C xy x y 014 . 0.07 Ax Ay 0.5833 0.07 0.05 xy 014 . 014 . 1 1 Axy 0.7368 and axy 5.2632 xy 014 . 0.05 019 . xy 014 . 0.05 P Axy axy Ax Ay Axy MLC-09-13 axy b g 2 0.5833 0.7368 0.0817 5.2632 29 Question #50 Answer: E 20V P20 1 i q40 1 21V 21V b0.49 0.01gb1 ig 0.022b1 0.545g 0.545 b1 ig b0.545gb1 0.022g 0.022 0.50 111 . 21V P20 1 i q41 1 22V 22V . g q b1 0.605g 0.605 b0.545.01gb111 41 q41 0.61605 0.605 0.395 0.028 Question #51 Answer: E 1000 P60 1000 A60 / a60 b gb g 1000bq p A g / b106 . p a g c15 b0.985gb382.79gh / c106 . b0.985gb10.9041gh 33.22 1000 v q60 p60 A61 / 1 p60 v a61 60 60 61 60 61 Question #52 - Removed MLC-09-13 30 Question #53 Answer: E g ln(0.96) 0.04082 x02t: y t 0.04082 0.01 0.03082 h ln(0.97) 0.03046 x01t: y t 0.03046 0.01 0.02046 5 5 pxy 5 pxy00 exp x01t: y t x02t: y t x03t: y t dt e 5(0.06128) 0.736 0 Question #54 Answer: B Transform these scenarios into a four-state Markov chain, as shown below. State from year t – 3 to year t – 2 from year t – 2 to year t – 1 0 Decrease (D) Decrease (I) Probability that year t will decrease from year t -1 0.8 1 Increase Decrease 0.6 2 Decrease Increase 0.75 3 Increase Increase 0.9 0.80 0.00 0.20 0.00 0.60 0.00 0.40 0.00 The transition probability matrix is 0.00 0.75 0.00 0.25 0.00 0.90 0.00 0.10 Note for example that the value of 0.2 in the first row is a transition from DD to DI, that is, the probability of an increase in year three after two successive years of decrease. The requested probability is that of starting in state 0 and then being in either state 0 or 1 after two years. That requires the first row of the square of the transition probability matrix. It can be obtained by multiplying the first row of the matrix by each column. The result is [0.64 0.15 0.16 0.05]. The two required probabilities are 0.64 and 0.15 for a total of 0.79. The last two values in the vector do not need to be calculated, but doing so provides a check (in that the four values must sum to one). Alternatively, the required probabilities are for the successive values DDID and DDDD. The first case is transitions from state 0 to 2 and then 2 to 1 for a MLC-09-13 31 probability of 0.2(0.75) = 0.15. The second case is transitions from 0 to 0 and then 0 to 0 for a probability of 0.8(0.8) = 0.64 for a total of 0.79. MLC-09-13 32 Question #55 Answer: B lx x 105 x t p45 l45t / l45 (60 t ) / 60 Let K be the curtate future lifetime of (45). Then the sum of the payments is 0 if K 19 and is K – 19 if K 20 . F 60 K IJ 1 60 K b40 39...1g b40gb41g 13.66 60 2b60g 20 a45 1 GH 60 K 20 Hence, c h c Prob K 19 13.66 Prob K 32.66 h b g ProbbT 33g Prob K 33 since K is an integer 33p45 l78 27 l45 60 0.450 MLC-09-13 33 Question #56 Answer: C Ax 2 Ax d IAi z E 0s x zd 0 0.4 x 0.25 0.04 2 z 0 s Ax ds Ax ds ib g e0.1s 0.4 ds F e I b0.4gG H 01. JK 0.1s 0 0.4 4 01 . Alternatively, using a more fundamental formula but requiring more difficult integration. c IA h x z z 0 0 bg b0.04g e t t px x t e t dt t e 0.04 t 0.04 z 0 0.06t dt t e 0.1t dt (integration by parts, not shown) t 1 0.04 e 0.1 t 0 01 . 0.01 0.04 4 0.01 FG H MLC-09-13 IJ K 34 Question #57 Answer: E Subscripts A and B here distinguish between the tools and do not represent ages. We have to find e AB eA eB FG 1 t IJ dt t t H 10K 20 z z FGH z FGH 10 0 IJ K t t2 1 dt t 7 14 7 0 7 e AB 2 10 1 t 7 7 0 49 0 49 35 . 14 IJ FG1 t IJ dt z FG1 t t t IJ dt K H 10K H 10 7 70K 2 7 0 t2 t2 t3 t 20 14 210 7 0 7 49 49 343 2.683 20 14 210 10 5 5 e AB e A e B e AB 5 35 . 2.683 5817 . Question #58 Answer: A xt 0.100 0.004 0.104 t pxb g e0.104t Expected present value (EPV) = EPV for cause 1 + EPV for cause 2. z 5 b z g b 5 g 2000 e0.04t e0.104t 0100 . dt 500,000 e0.04 t e0.104 t 0.400 dt 0 0 2000 0.10 500, 000 0.004 e0.144t dt 5 0 MLC-09-13 35 2200 0.144 5 1 e 7841 0.144 Question #59 Answer: A R 1 px qx S 1 px e k 1 1 1 1 e 0 xt dt k dt xt k dt 0 since e 0 e 0 e 0 1 xt dt k dt So S 0.75R 1 px e k 0.75qx 1 0.75qx px px 1 qx ek 1 0.75qx 1 0.75qx e k k ln LM 1 q OP N1 0.75q Q x x MLC-09-13 36 Question #60 Answer: C A60 0.36913 2 d 0.05660 A60 0.17741 and 2 2 A60 A60 0.202862 Expected Loss on one policy is E L 100,000 A60 d d 2 Variance on one policy is Var L 100,000 2 A60 A60 d On the 10000 lives, E S 10,000E L and Var S 10,000 Var L 2 The is such that 0 E S / Var S 2.326 since 2.326 0.99 10,000 100,000 A60 d d 2.326 2 100 100,000 2 A60 A60 d 100 100,000 0.36913 d d 2.326 100,000 0.202862 d 0.63087 36913 d 0.004719 100,000 d 0.63087 36913 471.9 0.004719 d d 36913 471.9 d 0.63087 0.004719 59706 59706 d 3379 MLC-09-13 37 Question #61 Answer: C 0V 1 i 1000 1V 1V q75 1V 1.05 1000q75 Similarly, 2 V 1V 1.05 1000q76 3V 2V 1.05 1000q77 1000 3V 1.053 1.052 1.05 1000 q75 1.052 1000 1.05 q76 1000 q77 * 1000 1000 1.052 q75 1.05q76 q77 1.05 1.05 3 2 1.05 1000 x 1 1.052 0.05169 1.05 0.05647 0.06168 3.310125 1000 1.17796 355.87 3.310125 * This equation is an algebraic manipulation of the three equations in three unknowns 1V , 2V , . One method – usually effective in problems where benefit = stated amount plus reserve, is to multiply the 1V equation by 1.052 , the 2V equation by 1.05, and add those two to the 3V equation: in the result, you can cancel out the 1V , and 2V terms. Or you can substitute the 1V equation into the 2 V equation, giving 2 V in terms of , and then substitute that into the 3V equation. Question #62 Answer: D A281:2 z 2 t e 1 72 dt 0 a28:2 3V d i FG IJ H K b g 1 1 e2 0.02622 since ln 106 . 0.05827 72 71 1 v 19303 . 72 500,000 A281:2 6643 a28:2 = 287 MLC-09-13 38 Question #63 Answer: D Let Ax and a x be calculated with x t and 0.06 Let Ax* and ax* be the corresponding values with x t increased by 0.03 and decreased by 0.03 ax 1 Ax 0.4 6.667 0.06 ax ax * xs 0.03 ds 0.03t * 0 e dt Proof: ax 0 e t 0 e t x s ds 0.03t 0.03t 0 e e t e 0 x s ds 0 e 0.06 t dt ax Ax* 1 0.03 ax* 1 0.03 ax b gb 1 0.03 6.667 g 0.8 MLC-09-13 39 dt Question #64 Answer: A bulb ages Year 0 1 0 1 2 3 10000 0 1000 9000 100+2700 900 280+270+3150 2 3 0 0 6300 0 0 0 The diagonals represent bulbs that don’t burn out. E.g., of the initial 10,000, (10,000) (1-0.1) = 9000 reach year 1. (9000) (1-0.3) = 6300 of those reach year 2. Replacement bulbs are new, so they start at age 0. At the end of year 1, that’s (10,000) (0.1) = 1000 At the end of 2, it’s (9000) (0.3) + (1000) (0.1) = 2700 + 100 At the end of 3, it’s (2800) (0.1) + (900) (0.3) + (6300) (0.5) = 3700 1000 2800 3700 105 . 105 . 2 105 . 3 6688 Expected present value Question #65 Answer: E e25:25 z z 15 0 t p25dt 15 p25 z 10 0t p40 dt F IJ e e dt G e z H Kz 1 L1 1 e i e M d1 e d .04 N.05 15 15 .04 t 0 .04 ds 0 0 .60 .60 112797 . 4.3187 15.60 MLC-09-13 10 .05t dt .50 iOPQ 40 # replaced 1000 2800 3700 Question #66 Answer: C 5 p 60 1 e1 q je1 q jb1 q gb1 q gb1 q g b0.89gb0.87gb0.85gb0.84gb0.83g 60 1 60 2 63 64 65 0.4589 Question # 67 Answer: E 1 0.08 0.04 12.50 ax Ax 0.5 1 2 Ax 2 3 e j Var aT 2 Ax Ax2 2 1 1 3 4 52.083 0.0016 S.D. 52.083 7.217 MLC-09-13 41 *Question # 68 Answer: D v 090 . d 010 . Ax 1 dax 1 010 . 5 0.5 b gb g Net premium 5000 Ax 5000vqx ax b5000gb0.5g 5000b0.90gb0.05g 455 5 Net premium reserve at the end of year 10 1 0.2 1 ax 10 ax ax 10 ax 10 4 5 b gb g a b5000gb0.6g b455gb4g 1180 Ax 10 1 dax 10 1 010 . 4 0.6 10V 5000 Ax 10 x 10 Question #69 Answer: D v is the lowest premium to ensure a zero % chance of loss in year 1 (The present value of the payment upon death is v, so you must collect at least v to avoid a loss should death occur). Thus v = 0.95. 2 E Z vqx v 2 px qx 1 0.95 0.25 0.95 0.75 0.2 bg d i b g 0.3729 b g b g 2 4 E Z v qx v px qx 1 0.95 0.25 0.95 0.75 0.2 2 2 4 0.3478 b g d i c b gh Var Z E Z2 E Z MLC-09-13 2 b g 2 0.3478 0.3729 0.21 42 Question #70 Answer: D Expected present value (EPV) of future benefits = 0.005 2000 0.04 1000 /1.06 1 0.005 0.04 0.008 2000 0.06 1000 /1.062 47.17 64.60 111.77 EPV of future premiums 1 1 0.005 0.04 /1.06 50 1.9009 50 95.05 E 1 L K55 1 111.77 95.05 16.72 Question #71 - Removed Question #72 Answer: A Let Z be the present value random variable for one life. Let S be the present value random variable for the 100 lives. E ( Z ) 10 e t e t dt 5 10 e ( )5 2.426 FG IJ e b g H 2 K F 0.04IJ de i 11233 10 G . H 0.16K Var b Z g E d Z i c E b Z gh d i 2 5 E Z 2 102 0.8 2 2 2 11233 . 2.4262 5.348 bg bg VarbSg 100 Varb Zg 534.8 E S 100 E Z 242.6 F 242.6 1645 . F 281 534.8 MLC-09-13 43 Question #73 Answer: D Prob{only 1 survives} = 1-Prob{both survive}-Prob{neither survives} b ge 13p50 3p 50 13p50 13p 50 b gb gb gb j gb gb g b gb g 1 0.9713 0.9698 0.9682 0.9849 0.9819 0.9682 1 0.912012 1 0.93632 0.912012 0.936320 0.140461 Question # 74 - Removed Question #75 - Removed Question # 76 Answer: C This solution applies the equivalence principle to each life. Applying the equivalence principle to the 100 life group just multiplies both sides of the first equation by 100, producing the same result for P. EPV Prems P EPV Benefits 10q70v 10 p70q71v 2 Pp70 p71v 2 P b10gb0.03318g b10gb1 0.03318gb0.03626g Pb1 0.03318gb1 0.03626g 108 . 108 . 2 0.3072 0.3006 0.7988 P P 0.6078 3.02 0.2012 (EPV above means Expected Present Value). MLC-09-13 44 108 . 2 *Question #77 Answer: E Level net premiums can be split into two pieces: one piece to provide term insurance for n years; one to fund the reserve for those who survive. Then, Px Px1:n Px:n1 nV And plug in to get b gb g 0.090 Px1:n 0.00864 0.563 Px1:n 0.0851 Another approach is to think in terms of retrospective reserves. Here is one such solution: V Px Px1:n sx:n n Px Px1:n aE x:n n P P P P Px Px1:n x ax:n 1 x:n ax:n 1 x x:n 1 x:n e j 0.563 0.090 Px1:n / 0.00864 b gb g Px1:n 0.090 0.00864 0.563 0.0851 MLC-09-13 45 Question #78 Answer: A b g ln 105 . 0.04879 Ax x t 0 x 0 px x t e t dt 1 t e dt for the given mortality function x 1 a x x From here, many formulas for the reserve could be used. One approach is: Since A50 a50 50 1 A50 18.71 0.3742 so a50 12.83 50 1 A40 19.40 0.3233 so a40 13.87 60 60 0.3233 P A40 0.02331 13.87 reserve A50 P A40 a50 0.3742 0.0233112.83 0.0751. A40 a60 Question #79 Answer: D Ax E vTx E vTx NS Prob NS E vTx S Prob S 0.03 0.6 0.70 0.30 0.03 0.08 0.06 0.08 0.3195 Similarly, 2 Ax F H Var aT I b gK x MLC-09-13 2 FG 0.03 IJ 0.70 FG 0.06 IJ 0.30 01923 . . H 0.03 016 H 0.06 016 . K . K Ax Ax2 2 01923 . 0.31952 141 .. 0.082 46 Question #80 Answer: B 2 q80:84 2 q80 2 q84 2 q80:84 0.5 0.4 1 0.6 0.2 0.15 1 0.1 0.5 0.4 0.2 0.15 (1 0.6 0.1) = 0.10136 Using new p82 value of 0.3 0.5 0.4 1 0.3 0.2 0.15 1 0.1 0.5 0.4 0.2 0.15 (1 0.3 0.1) = 0.16118 Change = 0.16118 – 0.10136 = 0.06 Alternatively, 2 p80 0.5 0.4 0.20 3 p80 2 p80 0.6 0.12 2 p84 0.20 0.15 0.03 3 p84 2 p84 0.10 0.003 2 p80:84 2 p80 2 p84 2 p80 2 p84 since independent 3 p80:84 0.20 0.03 0.20 0.03 0.224 3 p80 3 p84 3 p80 3 p84 0.12 0.003 0.12 0.003 0.12264 2 q 80:84 2 p80:84 3 p80:84 0.224 0.12264 0.10136 Revised 3 p80 0.20 0.30 0.06 3 p 80:84 0.06 0.003 0.06 0.003 0.06282 2 q 80:84 0.224 0.06282 0.16118 change 0.16118 0.10136 0.06 Question #81 - Removed MLC-09-13 47 Question #82 Answer: A 5 b g pb1g pb2g p50 5 50 5 50 FG 100 55IJ e b gb g H 100 50K b0.9gb0.7788g 0.7009 0.05 5 Similarly 10 FG 100 60IJ e b g b g H 100 50 K b0.8gb0.6065g 0.4852 b g p50 0.05 10 b g pb g pb g 0.7009 0.4852 5 5 q50 5 50 10 50 0.2157 Question #83 Answer: C Only decrement 1 operates before t = 0.7 b g b0.7g qb1g b0.7gb010 . g 0.07 1 0.7 q40 40 Probability of reaching t = 0.7 since UDD is 1-0.07 = 0.93 Decrement 2 operates only at t = 0.7, eliminating 0.125 of those who reached 0.7 b gb g b2g 0.93 0125 q40 . 011625 . MLC-09-13 48 Question #84 Answer: C e j 1 2 p80v 2 1000 A80 FG H 1 0.83910 106 . 2 vq80 v 3 2 p80q82 2 2 IJ 665.75 FG 0.08030 0.83910 0.09561IJ K . g 2b106 . g H 2b106 K 3 b g b b167524 . g 665.75 g 174680 . 665.75 0.07156 397.41 Where 2 p80 3,284 ,542 0.83910 3,914 ,365 b gb g Or 2 p80 1 0.08030 1 0.08764 0.83910 *Question #85 Answer: E At issue, expected present value (EPV) of benefits bt vt t p65 0 65 t dt 1000 e0.04t e 0.04t t p65 65 t dt 0 1000 0 t p65 65 t dt 1000 q65 1000 EPV of premiums a65 FG 1 IJ 16.667 H 0.04 0.02K Net premium 1000 / 16.667 60 2V z z 0 0 b g b2u v u u p67 65 2 u du a67 b g b gb g 1000 e0.04b 2uge0.04u u p67 65 2 u du 60 16.667 1000e0.08 z 0 u b g p67 65 2 u du 1000 108329 . q67 1000 108329 . 1000 8329 . MLC-09-13 49 Question #86 Answer: B (1) ax:20 ax:20 120 Ex (2) ax:20 (3) Ax:20 (4) Ax A1x:20 20 Ex Ax 20 1 Ax:20 d A1x:20 Ax:201 b gb g 0.28 A1x:20 0.25 0.40 A1x:20 018 . Now plug into (3): Ax:20 018 . 0.25 0.43 ax:20 Now plug into (2): Now plug into (1): b 1 0.43 1197 . 0.05 / 105 . g ax:20 1197 . 1 0.25 1122 . Question #87 - Removed Question #88 Answer: B ex px pxex 1 px ex 8.83 0.95048 1 ex 1 9.29 ax 1 vpx v2 2 px .... a x:2 1 v v 2 2 px ... ax:2 ax vqx 5.6459 5.60 0.0459 v 1 0.95048 0.0459 v 0.9269 1 i 1 0.0789 v MLC-09-13 50 Question #89 Answer: E One approach is to enumerate the possible paths ending in F and add the probabilities: FFFF – 0.23 = 0.008 FFGF – 0.2(0.8)(0.5) = 0.080 FGFF – 0.8(0.5)(0.2) = 0.080 FGHF – 0.8(0.5)(0.75) = 0.300 The total is 0.468. An alternative is to use matrix multiplication. The desired probability is the value in the upper left corner of the transition probability matrix raised to the third power. Only the first row needs to be evaluated. The first row of the matrix squared is [0.44 0.16 0.40 0.00], obtained by multiplying the first row by each column, in turn. The first row of the matrix cubed is obtained by multiplying the first row of the squared matrix by each column. The result is [0.468 0.352 0.080 0.100]. Note that only the first of the four calculations in necessary, though doing the other three and observing that the sum is 1 provides a check. Either way, the required probability is 0.468 and the actuarial present value is 500v3 (0.468) 500(0.9)3 (0.468) 171 . Question #90 – Removed MLC-09-13 51 Question #91 Answer: E 1 1 75 60 15 1 1 3 1 60F 85 60 15 5 25 60M t 10 t 1 25 t M p65 1 t F p60 Let x denote the male and y denote the female. e x 5 mean for uniform distribution over 0,10 e y 12.5 mean for uniform distribution over 0,25 10 t t e xy 1 1 dt 0 10 25 10 7 t2 1 t dt 0 50 250 t 7 2 t3 t 100 750 10 0 10 7 1000 100 100 750 10 7 exy ex e y exy 5 MLC-09-13 4 13 3 3 25 13 30 75 26 1317 . 2 3 6 52 *Question #92 Answer: B 1 3 1 2 Ax 2 5 Ax Let P ( Ax ) denote the net premium. c h P Ax 0.04 F P c A hI A A Var b Lg G 1 j H JK e F 0.04 IJ FG 1 FG 1IJ IJ G1 H 0.08 K H 5 H 3K K F 3I F 4 I G J G J H 2 K H 45K 2 x 2 2 x x 2 2 2 1 5 *Question #93 Answer: A Let be the net premium Let kV denote the net premium reserve at the end of year k. For any n, nV 1 i q25n n1V p25n n1V n1V Thus 1V 0 V 1 i 2V 3V 1V 1 i 1 i 1 i s2 2V 1 i s2 1 i s3 By induction (proof omitted) nV sn For n 35, nV a60 (expected present value of future benefits; there are no future premiums) a60 s35 a60 s35 For n 20, MLC-09-13 20V a s20 60 s20 s 35 53 Alternatively, as above nV 1 i n1V Write those equations, for n 0 to n 34 0 : 0V 1 i 1V 1: 1V 1 i 2V 2 : 2V 1 i 3V 34 : 34V 1 i 35V 34 k Multiply equation k by 1 i and sum the results: 0V 1 i 35 1V 1 i 34 2V 1 i 33 34V 1 i 34 33 32 34V 1 i 35V 1V 1 i 2V 1 i 3V 1 i For k 1, 2, 0V 35k , 34, the k V 1 i 1 i 35 1 i 35 1 i 34 terms in both sides cancel, leaving 1 i 35V Since 0V 0 s35 35V a60 (see above for remainder of solution) MLC-09-13 54 *Question #94 Answer: B Note: The symbol x t: y t can be ambiguous. Here it means that both lives were alive at ages x and y and the last survivor status is intact at time t (that is, at least one of the two lives is alive). This meaning matches the wording of the question, which does not use the symbol. Then, t q y t p x x t t q x t p y y t x t: y t t q x t p y t px t q y t px t p y For (x) = (y) = (50) 5010.5:5010.5 where 10.5 p50 10.5 q50 10 2 l60 l61 12 8,188,074 8,075, 403 0.90848 l50 8,950,901 1 10.5 p50 0.09152 p50 10.5 1 10.5 q50 10 p50 q60 2 0.09152 0.91478 0.01376 2 10.5 q50 10.5 p50 2 10.5 p50 2 0.09152 0.90848 2 0.908482 8,188,074 0.91478 8,950,901 p50 50 10.5 10 p50 q60 since UDD Alternatively, 10t p50 10 p50 t p60 10t p50:50 10 p50 2 t p60 2 10t p 50:50 2 10 p50 t p60 10 p50 2 t p60 2 2 10 p50 1 tq60 10 p50 1 tq60 since UDD 2 2 Derivative 2 10 p50 q60 2 10 p50 1 tq60 q60 2 Derivative at 10 t 10.5 is 2 0.91478 0.01376 0.91478 1 0.5 0.01376 0.01376 0.0023 2 10.5 p 50:50 2 10.5 p50 10.5 p50 2 2 0.90848 0.90848 2 0.99162 (for any sort of lifetime) MLC-09-13 dp dt 0.0023 0.0023 p 0.99162 55 0.0023 Question #95 Answer: D xt x1t x 2t 0.01 2.29 2.30 P P vt t px x 2t dt 50,000 vt t px x1t dt 50,000 vt t px xt dt 2 2 0 2 0.1t 2.3t 0 P P e e 0 2 2 0.1t 2.3t 2.29dt 50,000 e 0 2 2.4 1 e P 1 2.29 2.4 P = 11,194 e 0.01dt 50,000 e0.1t e2.3t 2.3dt 2 2 2.4 1 e 50000 0.01 2.4 2.3 e 2 2.4 *Question #96 Answer: B ex px 2 px 3px ... 1105 . b g Annuity v 3 3 p x 1000 v 4 4 p x 1000 104 . ... . g 1000b104 k 3 k v k px k 3 1000v 3 k px k 3 F 1 IJ 1000v be 0.99 0.98g 1000G H 104 . K 3 3 x Let = net premium. e j 1 0.99v 0.98v 2 8072 2.8580 8072 2824 MLC-09-13 56 9.08 8072 2.4 Question #97 Answer: B b g b ge 1 a30:10 1000 A30 P IA 30 10 :10 10 A30 j 1000 A30 b g 1 a30:10 IA 3010 1010 A30 : b g b 1000 0102 . 7.747 0.078 10 0.088 g 102 6.789 15.024 Question #98 Answer: E For the general survival function S0 (t ) 1 FG1 t IJ dt H 30K L t OP Mt N 2b 30g Q e 30 z t ,0t , 30 0 2 30 0 30 2 100 30 35 2 Prior to medical breakthrough 100 e 30 After medical breakthrough e 30 e30 4 39 so e30 39 30 2 108 Question #99 Answer: A L 100, 000v 2.5 4000a3 @5% = 77,079 MLC-09-13 57 Question #100 Answer: D accid 0.001 total 0.01 other 0.01 0.001 0.009 Expected present value 500,000 e0.05t e0.01t 0.009 dt 0 10 50,000 e0.04t e0.05t e0.01t 0.001 dt 0 0.009 0.001 500, 000 100, 000 0.06 0.02 Question #101 Removed Question #102 Answer: D 1000 20V 1000 Ax 20 ax 20 1000 19V 20 Px 1.06 qx 19 1000 px 19 342.03 13.72 1.06 0.01254 1000 369.18 0.98746 1 0.36918 11.1445 0.06 /1.06 so 1000 Px 20 1000 MLC-09-13 Ax 20 369.18 33.1 ax 20 11.1445 58 Question #103 Answer: B k 1 k xt dt 2 xt dt e 0 e 0 k px x1t dt e 0 k 2 k px where k px is from Illustrative Life Table, since follows I.L.T. 2 10 6,616,155 0.80802 8,188,074 6,396,609 0.78121 8,188,074 p60 11 p60 10 1 b g p b g p b g q60 10 60 11 60 b 10 p60 g b 2 11 p60 g 2 from I.L.T. 0.80802 2 0.781212 0.0426 Question #104 Answer: C Ps 1 d , where s can stand for any of the statuses under consideration. a s as 1 Ps d 1 6.25 01 . 0.06 1 axy 8.333 0.06 0.06 ax ay axy axy ax ay axy 6.25 6.25 8.333 4167 . Pxy 1 0.06 018 . 4.167 MLC-09-13 59 Question #105 Answer: A z b g g j 48 d 0b g 1000 e b 0.04 gt 0.04 dt 1 0 1000 1 e b 0.04 e e b 0.04 g 0.952 b 0.04 ln 0.952 g 0.049 0.009 z b g d 3b1g 1000 e 0.049t 0.009 dt 4 3 1000 0.009 b0.049 gb3g b0.049 gb4 g e e 7.6 0.049 e j Question #106 Answer: B This is a graph of lx x . b g x would be increasing in the interval 80,100 . The graphs of lx px , l x and lx2 would be decreasing everywhere. Question #107 Answer: B Variance v30 15 px 15 qx Expected value v15 15 px v30 15 px 15 qx 0.065 v15 15 px v15 15qx 0.065 15 qx 0.3157 Since is constant 15 q x px 15 1 px 15 0.6843 px 0.975 qx 0.025 MLC-09-13 60 *Question #108 Answer: E 1 2 11V I 11V II 1 2 10V 11V I 10V I 0 II p 1 i II 11V II qx 10 1000 px10 x 10 p 1 i x 10 10V I qx 10 1000 px 10 10V II II 101.35 8.36 p 1 i x 10 1.06 1 0.004 98.97 Question #109 Answer: A 2 EPV (x’s benefits) v k 1bk 1 k px q x k k 0 b g b gb g b gb gb g 1000 300v 0.02 350v 2 0.98 0.04 400v 3 0.98 0.96 0.06 36,829 *Question #110 Answer: E denotes the net premium V EPV future benefits - EPV future premiums 1 0.6 0.326 108 . . q65 10 10V 108 11V p65 19 b gb g b gb g . g b010 . gb10g b5.0 0.326gb108 1 010 . =5.28 MLC-09-13 61 Question #111 Answer: A Expected present value Benefits 0.8 0.110,000 0.8 0.9 0.097 9,000 1, 239.75 1.062 1.063 0.8 0.8 0.9 1, 239.75 P 1 1.062 1.06 P 2.3955 P 517.53 518 Question #112 Answer: A 1180 70a30 50a40 20a30:40 b gb g b gb g 1180 70 12 50 10 20a30:40 a30:40 8 a30:40 a30 a40 a30:40 12 10 8 14 100a30:40 1400 Question #113 Answer: B 0 0 a at f (t )dt 1 e0.05t 1 tet dt 0.05 (2) 1 0.05 zb te t i te 1.05t dt LM b g FG IJ H K N 1 L F 1 I O . M1 G . JK PP 185941 0.05 NM H 105 Q 1 t 1 t 1 et e 1.05t 2 0.05 105 . 105 . 2 20,000 185941 . 37,188 MLC-09-13 62 OP Q 0 Question #114 Answer: C Event x0 Prob 0.05 0.95 0.10 0.095 0.95 0.90 0.855 x 1 x2 Present Value 15 15 20 /1.06 33.87 15 20 /1.06 25/1.062 56.12 E X 0.0515 0.095 33.87 0.855 56.12 51.95 E X 2 0.0515 0.095 33.87 0.855 56.12 2813.01 2 2 2 Var X E X 2 E X 2813.01 51.95 114.2 2 2 Question #115 Answer: B Let K be the curtate future lifetime of (x + k) kL 1000v K 1 1000 Px:3 aK 1 When (as given in the problem), (x) dies in the second year from issue, the curtate future lifetime of x 1 is 0, so 1L 1000v 1000 Px:3 a1 1000 279.21 1.1 629.88 630 The premium came from A Px:3 x:3 ax:3 Ax:3 1 d ax:3 Px:3 279.21 MLC-09-13 1 d ax:3 ax:3 1 d ax:3 63 Question #116 Answer: D Let M = the force of mortality of an individual drawn at random; and T future lifetime of the individual. Pr T 1 n E Pr T 1 M z zz zd 0 0 0 bg Pr T 1 M f M d 0 2 1 2 s 1 2 e t dt d 1 e i 21 du 21 d2 e 2 056767 . Question #117 Answer: E For this model: 1/ 60 1 (1) 40(1)t ; 40 20 1/ 40 0.025 1 t / 60 60 t 1/ 40 1 (1) 40(2)t ; 40 20 1/ 20 0.05 1 t / 40 40 t 40 20 0.025 0.05 0.075 MLC-09-13 i 21 d1 e i 1 64 2 *Question #118 Answer: D Let net premium Expected present value of benefits = 0.03 200,000 v 0.97 0.06 150,000 v 2 0.97 0.94 0.09 100,000 v 3 b gb g b gb gb g b gb gb gb 5660.38 7769.67 6890.08 20,32013 . Expected present value of net premiums ax:3 b gb g 1 0.97v 0.97 0.94 v 2 2.7266 20,32013 . 7452.55 2.7266 1V . g b200,000gb0.03g b7452.55gb106 1 0.03 1958.46 Initial reserve, year 2 = 1V = 1958.56 + 7452.55 = 9411.01 Question #119 Answer: A Let denote the premium. b g L bT v T aT 1 i T v T aT 1 aT E L 1 ax 0 L 1 aT 1 b g aT ax 1 ax d ax 1 v T i ax v T 1 ax v T Ax ax 1 Ax MLC-09-13 65 g Question #120 Answer: D (0, 1) (1, 0.9) (1.5, 0.8775) (2, 0.885) tp1 1 2 t 1 p1 (1 01 . ) 0.9 2 p1 0.9 1 0.05 0.855 b gb g b g since uniform, 1.5 p1 0.9 0.855 / 2 0.8775 e1:1.5 area between t 0 and t 1.5 1 0.9 0.9 0.8775 (1) (0.5) 2 2 0.95 0.444 1.394 MLC-09-13 66 Alternatively, e1:1.5 1.5 t 0 1 0.5 0 0 p1dt t p1dt 1 p1 1 t p2 dt 0.5 (1 0.1t )dt 0.9 (1 0.05t )dt 0 0 1 0.5 0.1t 2 0.05t 2 t 0.9 t 2 0 2 0 0.95 0.444 1.394 Question #121 Answer: A b g 10,000 A63 112 . 5233 A63 0.4672 Ax 1 A64 b g Ax 1 i qx px . g 0.01788 b0.4672gb105 1 0.01788 0.4813 A65 . g 0.01952 b0.4813gb105 1 0.01952 0.4955 Single gross premium at 65 = (1.12) (10,000) (0.4955) = 5550 b1 ig MLC-09-13 2 5550 5233 i 5550 1 0.02984 5233 67 Question #122A Answer: C Because your original survival function for (x) was correct, you must have x t 0.06 x02t: y t x03t: y t x02t: y t 0.02 x02t: y t 0.04 Similarly, for (y) y t 0.06 x01t: y t x03t: y t x01t: y t 0.02 x01t: y t 0.04 The first-to-die insurance pays as soon as State 0 is left, regardless of which state is next. The force of transition from State 0 is x01t: y t x02t:y t x03t:y t 0.04 0.04 0.02 0.10 . With a constant force of transition, the expected present value is 0.10 t 00 01 02 03 0.05t 0.10t 0 e t pxy (xt:y t xt:y t xt:y t )dt 0 e e (0.10)dt 0.15 MLC-09-13 68 Question #122B Answer: E Because (x) is to have a constant force of 0.06 regardless of (y)’s status (and vice23 versa) it must be that 13 x t y t 0.06 . There are three mutually exclusive ways in which both will be dead by the end of year 3: 1: Transition from State 0 directly to State 3 within 3 years. The probability of this is 3 0.02 0.10t 0.3 0 t p dt 0 e 0.02dt 0.10 e 0 0.2(1 e ) 0.0518 2: Transition from State 0 to State 1 and then to State 3 all within 3 years. The probability of this is 3 3 0 t 00 xy 3 03 x t : y t 0.10t 3 0.10 t pxy00 x01t: y t 3t p13 0.04(1 e 0.06(3t ) )dt x t dt e 0 3 0.04 e 0 0.4(1 e 0.10 t 0.3 e )e 0.18 0.04 t 0.18 e (1 e 0.04 0.10t 0.04e0.18 0.04t e e 0.10 0.04 0.12 3 0 ) 0.00922 3: Transition from State 0 to State 2 and then to State 3 all within 3 years. By symmetry, this probability is 0.00922. The answer is then 0.0518 + 2(0.00922) = 0.0702. MLC-09-13 69 Question #122C Answer: D Because the original survival function continues to hold for the individual lives, with a constant force of mortality of 0.06 and a constant force of interest of 0.05, the expected present values of the individual insurances are Ax Ay 0.06 0.54545 , 0.06 0.05 Then, Axy Ax Ay Axy 0.54545 0.54545 0.66667 0.42423 Alternatively, the answer can be obtained be using the three mutually exclusive outcomes used in the solution to Question 122B. 1: 0 e0.05t t pxy00 x03t: y t dt e0.05t e0.10t 0.02dt 2 and 3: 0 0 0.02 0.13333 0.15 13 e0.05t t pxy00 x01t: y t e0.05 r r p11 x t : y t x t r : y t r drdt 0 0 0 e0.05t e0.10t 0.04 e0.05 r e0.06 r 0.06drdt 0.04 0.06 0.14545 0.15 0.11 The solution is 0.13333 + 2(0.14545) = 0.42423. The fact that the double integral factors into two components is due to the memoryless property of the exponential transition distributions. MLC-09-13 70 Question #123 Answer: B 5 q35:45 5 q35 5 q45 5 q35:45 5 p35q40 5 p45q50 5 p35:45q40:50 b g p q p q p p b1 p p g b0.9gb.03g b0.8gb0.05g b0.9gb0.8g 1 b0.97gb0.95g 5 p35q40 5 p45q50 5 p35 5 p45 1 p40:50 5 35 40 5 45 50 5 35 5 45 40 50 0.01048 Alternatively, 6 p35 5 p35 p40 0.90 1 0.03 0.873 6 p45 5 p45 p50 0.80 1 0.05 0.76 5 q35:45 5 p35:45 6 p35:45 5 p35 5 p45 5 p35:45 6 p35 6 p45 6 p35:45 5 p35 5 p45 5 p35 5 p45 6 p35 6 p45 6 p35 6 p45 0.90 0.80 0.90 0.80 0.873 0.76 0.873 0.76 0.98 0.96952 0.01048 Question #124 – Removed Question #125 - Removed MLC-09-13 71 Question #126 Answer: E Let Y = present value random variable for payments on one life S Y present value random variable for all payments E Y 10a40 148166 . dA 2 Var Y 10 2 40 d 2 A40 d i 2 ib g 2 100 0.04863 016132 . 106 . / 0.06 2 70555 . E S 100 E Y 14,816.6 Var S 100 Var Y 70,555 Standard deviation S 70,555 265.62 By normal approximation, need E [S] + 1.645 Standard deviations = 14,816.6 + (1.645) (265.62) = 15,254 *Question #127 Answer: B Initial net premium Where e e 1 5 A30 4 A30 :20 j 5a30:35 4a30:20 b g b g b g 5 010248 . 4 0.02933 5 14.835 4 11959 . 0.5124 011732 . 0.39508 0.015 74.175 47.836 26.339 b g j 1 A30 A30:20 A30:201 0.32307 0.29374 0.02933 :20 and a30:20 1 A30:20 d 1 0.32307 11959 . 0.06 106 . FG IJ H K Comment: the numerator could equally well have been calculated as A30 4 20E30 A50 = 0.10248 + (4) (0.29374) (0.24905) = 0.39510 MLC-09-13 72 Question #128 Answer: B 0.75 b gb g px 1 0.75 0.05 0.9625 0.75 b gb g p y 1 0.75 .10 0.925 0.75 qxy 1 0.75 pxy b p gd p i since independent = 1- b0.9625gb0.925g 1 0.75 x 0.75 y 01097 . *Question #129 Answer: D Let G be the gross premium. Expected present value (EPV) of benefits = 100,000A35 EPV of premiums = Ga35 EPV of expenses = 0.1G 25 2.50 100 a35 Equivalence principle: Ga35 100,000 A35 0.1G 25 250 a35 A35 0.1G 275 a35 0.9G 100,000 P35 275 G 100,000 G 100 8.36 275 0.9 1234 MLC-09-13 73 Question #130 Answer: A The person receives K per year guaranteed for 10 years Ka10 8.4353K The person receives K per years alive starting 10 years from now 10 a40 K b g Hence we have 10000 8.435310 E40a50 K Derive 10 E40: 1 A40 A4010 : 10 E40 Derive a50 b 10 E40 1 A40 A4010 : A50 gA 50 0.30 0.09 0.60 0.35 1 A50 1 0.35 16.90 .04 d 104 . Plug in values: 10,000 8.4353 0.60 16.90 K c b gb gh 18.5753K K 538.35 MLC-09-13 74 Question #131 Answer: D 11 STANDARD: e25:11 11 0 t t2 1 dt t 10.1933 2 75 0 75 e z 1 MODIFIED: p25 0.1ds 0 e.1 0.90484 1 10 t e25:11 t p25 dt p25 1 dt 0 0 74 IJ K F t I 1 e e Gt 01 . H 2 74JK 0.95163 0.90484b9.32432g 9.3886 z 1 0.1t 0 e dt e0.1 0.1 0.1 z FGH 10 0 1 t dt 74 2 10 0 Difference =0.8047 *Question #132 Answer: B Comparing B & D: Prospectively at time 2, they have the same future benefits. At issue, B has the lower net premium. Thus, B has the higher reserve. Comparing A to B: Consider them retrospectively. At issue, B has the higher net premium. Until time 2, they have had the same benefits, so B has the higher reserve. Comparing B to C: Visualize a graph C* that matches graph B on one side of t=2 and matches graph C on the other side. By using the logic of the two preceding paragraphs, C’s reserve is lower than C*’s which is lower than B’s. Comparing B to E: Reserves on E are constant at 0. MLC-09-13 75 Question #133 Answer: C Since only decrements (1) and (2) occur during the year, probability of reaching the end of the year is 1 2 p60 b g p60 b g 1 0.01 1 0.05 0.9405 b gb g Probability of remaining through the year is 1 2 3 p60 . 0.84645 b g p60 b g p60 b g 1 0.01 1 0.05 1 010 b gb gb g Probability of exiting at the end of the year is b3g 0.9405 0.84645 0.09405 q60 Question #134 - Removed Question #135 Answer: D EPV of regular death benefit zb 0 gd ib gd zb 0 gd ib gd i 100000 e- t 0.008 e- t dt i 100000 e-0.06t 0.008 e-0.008t dt b g 100000 0.008 / 0.06 0.008 11,764.71 EPV of accidental death benefit zb 30 0 gd ib gd i zb 30 0 gd ib 100000 e-0.06t 0.001 e-0.008t dt 100 1 e-2.04 / 0.068 1,279.37 Total EPV 11765 1279 13044 MLC-09-13 gd i 100000 e- t 0.001 e- t dt 76 Question #136 Answer: B b gb g b gb g g b gb g l 60 .6 .6 79,954 .4 80,625 80,222.4 b gb l 60 1.5 .5 79,954 .5 78,839 79,396.5 0.9 q 60 .6 80222.4 79,396.5 80,222.4 0.0103 Question #137 - Removed Question #138 Answer: A b g qb1g qb2g 0.34 q40 40 40 1 2 1 p40 b g p40 b g 0.34 1 0.75 p40 b 2g p40 b 2g 0.88 q40 . y b 2g 012 q41 b2g 2 y 0.24 b gb g l b g 2000b1 0.34gb1 0.392g 803 b g 1 0.8 1 0.24 0.392 q41 42 MLC-09-13 77 Question #139 Answer: C bg Pr L ' 0 0.5 Pr 10,000v K 1 ' aK 1 0 0.5 From Illustrative Life Table, 47 p30 0.50816 and 48 p30 .47681 Since L is a decreasing function of K, to have Pr L 0 0.5 means we must have L 0 for K 47. b g b g b g Highest value of L for K 47 is at K = 47. b g L at K 47 10,000 v 471 a471 b g 609.98 16.589 b g L 0 609.98 16.589 0 609.98 36.77 16.589 Question #140 Answer: B b g Prb K 1g p p 0.9 0.81 0.09 Prb K 1g p 0.81 . .81 2.72 2.4715 EbY g = .1 1.09 187 . .81 2.72 6.407 E dY i = .1 1 .09 187 VARbY g 6.407 2.4715 0.299 Pr K 0 1 px 01 . 1 x 2 x 2 x 2 2 2 2 2 MLC-09-13 78 Question #141 Answer: E E Z b Ax since constant force Ax /( ) E(Z) b 0.02 b b/3 0.06 Var Z Var b v T b 2Var v b 2 2 Ax Ax2 2 b 2 2 1 4 b2 b2 10 9 45 2 Var Z E Z 4 b b2 45 3 4 1 b b 3.75 45 3 *Question #142 Answer: B In general, for any premium P and corresponding loss at issue random variable L, 2 Var L 1 P 2 Ax Ax2 b g b gc Here P h 1 1 0.08 0.12 ax 5 b g FGH ..1208IJK c A A h .5625 F b.12gIJ c A A h and Varb L *g G1 H .08 K b1 g b0.5625g .744 So Var b L *g b1 g E L * A .15a 1 a b .15g 1 5b.23g .15 E L * Varb L *g .7125 2 So Var L 1 2 2 x x 2 5 4 2 x 2 x 15 2 8 12 2 8 x MLC-09-13 x x 79 Question #143 - Removed Question #144 Answer: B Let l0b g number of students entering year 1 superscript (f) denote academic failure superscript (w) denote withdrawal subscript is “age” at start of year; equals year - 1 p0b g 1 0.40 0.20 0.40 l2b g 10 l2b g q2b f g q2b f g 01 . b g q2b wg q2b g q2b f g 10 . 0.6 01 . 0.3 l1b gq1b f g 0.4 l1b g 1 q1b f g q1b wg e j q1b f g 0.4 1 q1b f g 0.3 e q1b f g j 0.28 0.2 14 . p1b g 1 q1b f g q1b wg 1 0.2 0.3 0.5 b g q b w g p b g q b w g p b g p b g q b w g w 3 q0 0 0 1 0 1 2 b gb g b gb gb g 0.2 0.4 0.3 0.4 0.5 0.3 0.38 MLC-09-13 80 Question #145 Answer: D e25 p25 (1 e26 ) N M e26 e26 due to having the same 1 N M M 0.05 p25 exp 25 0.1(1 t )dt p25 e 0 t N N M M e25 p25 (1 e26 ) e 0.05 p25 (1 e26 ) 0.951e25 9.51 Question #146 Answer: D E[YAGG ] 100 E[Y ] 100(10, 000)ax 100(10, 000) 1 Ax 10, 000, 000 b10,000g 1 c A A h b10,000g b0.25g b016 . g 50,000 Y Var Y 2 2 2 b x 2 x g AGG 100 Y 10 50,000 500,000 0.90 Pr LM F E Y N AGG AGG 1282 . OP Q 0 F E YAGG AGG F 1282 . AGG E YAGG b g F 1282 . 500,000 10,000,000 10,641,000 MLC-09-13 81 Question #147 Answer: A 1 A30:3 1000vq30 500v 2 1 q30 250v3 2 q30 2 3 1 1.53 1 1.61 1 1.70 1000 500 0.99847 250 0.99847 0.99839 1.06 1000 1.06 1000 1.06 1000 1.4434 0.71535 0.35572 2.51447 1 1 1 1 2 0.00153 1 a30:1 1 2 1 2 1 2 q30 0.97129 1 2 2 2 1.06 1 1 0.97129 0.999235 2 2 0.985273 2.51447 Annualized premium 0.985273 2.552 2 2.552 2 1.28 Each semiannual premium Question: #148 Answer: E b DAg 1 80:20 q80 .2 q80 .1 eb g j 1 20vq80 vp80 DA 8119 : bg b g b g b g b g b gb 2.9b12.225g 12.267 20 .2 .8 1 DA 8119 : 106 . 106 . 13 106 . 4 1 DA 8119 12.225 : .8 1 DA80 20v .1 v .9 12.225 :20 13 106 . Question #149 - Removed MLC-09-13 82 g Question #150 Answer: A t 1 100 x t t t px exp ds exp ln(100 x s) 0 0 100 x s 100 x e50:60 e50 e60 e50:60 50 e50 50 e60 40 0 50 t 1 t2 dt 50t 25 50 50 2 0 40 0 40 t 1 t2 dt 40t 20 40 40 2 0 40 e50:60 40 0 2 40 2000 90t t 1 t3 50 t 40 t 2 dt dt 2000 t 45 t 14.67 0 2000 2000 3 0 50 40 e50:60 25 20 14.67 30.33 Question #151 Answer: C Ways to go 0 2 in 2 years 0 0 2; p 0.7 0.1 0.07 0 1 2; p 0.2 0.25 0.05 0 2 2; p 0.11 0.1 Total = 0.22 Binomial m = 100 q = 0.22 Var = (100) (0.22) (0.78) = 17 MLC-09-13 83 Question #152 Answer: A For death occurring in year 2 0.3 1000 EPV 285.71 1.05 For death occurring in year 3, two cases: (1) State 2 State 1 State 4: (0.2 0.1) = 0.02 (2) State 2 State 2 State 4: (0.5 0.3) = 0.15 0.17 Total EPV = 0.17 1000 154.20 1.052 Total. EPV = 285.71 + 154.20 = 439.91 Question #153 - Removed *Question #154 Answer: C Let denote the single net premium. 1 30 a35 A35 :30 30 a35 1 1 A35 :30 A e 35:30 j 1 A35 a65 :30 1 1 A35 :30 b.21.07g9.9 b1.07g 1.386 .93 1.49 MLC-09-13 84 Question #155 Answer: E 0.4 p0 .5 e z 0.4 0 e F e2 x jdx LM e22x OP.4 N Q0 e 0.8 .4 F F e 2 1I H K e .4 F .5 e.4 F .6128 bg ln .5 .4F .6128 .6931 .4F .6128 F 0.20 *Question #156 Answer: C 9V P 1.03 qx9b 1 qx9 10V qx 9 b 10V 10V 3431.03 0.02904 872 10V 10V 327.97 b b 10V 10V 872 327.97 1199.97 1 1 0.03 P b d 1200 14.65976 1.03 ax = 46.92 9V net premium reserve at the start of year ten – P = 343 – 46.92 = 296.08 Question #157 Answer: B d = 0.06 V = 0.94 Step 1 Determine p x 668 258vpx 1000vqx 1000v 2 px px 1 qx 1 668 258 0.94 px 1000 0.94 1 px 1000 0.8836 px 1 668 242.52 px 940 1 px 883.6 px px 272/ 298.92 0.91 MLC-09-13 85 Step 2 Determine 1000 Px:2 668 258 0.94 0.91 1000 Px:2 1 0.94 0.91 220.69 668 479 1000 Px:2 1.8554 Question #158 Answer: D 1 1 1 100,000 IA 40:10 100,000 v p40 IA 41:10 10 v10 9 p41 q50 A40:10 100,000 see comment 8,950,901 10 0.99722 0.16736 9, 287, 264 0.00592 100,000 1.06 1.0610 0.02766 100,000 =15,513 1 A40 10 E40 A50 Where A40:10 0.16132 0.53667 0.24905 0.02766 Comment: the first line comes from comparing the benefits of the two insurances. At each of age 40, 41, 42,…,49 IA40:10 provides a death benefit 1 greater than 1 IA141:10 . Hence the 1 A40:10 term. But IA41:10 provides a death benefit at 50 of 10, 1 while IA40:10 provides 0. Hence a term involving 9 q41 9 p41 q50 . The various v’s 1 and p’s just get all expected present values at age 40. MLC-09-13 86 Question #159 Answer: A 10001V 1 i qx 1000 10001V 40 80 1.1 qx 1000 40 qx 88 40 0.05 960 1 AS G expenses 1 i 1000qx px 100 0.4100 1.1 1000 0.05 1 0.05 60 1.1 50 16.8 0.95 Question #160 Answer: C 1 At any age, px e0.02 0.9802 1 1 qx 1 0.9802 0.0198 , which is also q x , since decrement 2 occurs only at the end of the year. Expected present value (EPV) at the start of each year for that year’s death benefits = 10,000*0.0198 v = 188.1 px 0.9802*0.96 0.9410 Ex px v 0.941 v 0.941*0.95 0.8940 EPV of death benefit for 3 years 188.1 E40 *188.1 E40 * E41 *188.1 506.60 MLC-09-13 87 Question #161 Answer: B 40 e30:40 t p30dt 0 30 t dt 30 0 40 t t2 2 30 40 0 800 30 27.692 40 95 Or, with a linear survival function, it may be simpler to draw a picture: 0 p30 1 40 30 70 e30:40 area =27.692 40 40 p30 p30 0.3846 1 40 p30 2 70 0.3846 30 95 t p30 65 t 65 Var E T E T 2 MLC-09-13 2 88 Using a common formula for the second moment: Var T 2 t t px dt ex2 0 2 65 0 t 65 t t 1 dt 1 dt 0 65 65 2 2* 2112.5 1408.333 65 65/ 2 2 1408.333 1056.25 352.08 Another way, easy to calculate for a linear survival function is Var T t 2 t px x t dt 0 0 t t px x t dt 1 1 65 t dt t dt 0 65 65 65 2 0 t3 3 65 65 0 t2 2 65 65 0 2 2 2 1408.33 32.5 352.08 2 With a linear survival function and a maximum future lifetime of 65 years, you probably didn’t need to integrate to get E T30 e30 32.5 Likewise, if you realize (after getting 95 ) that T30 is uniformly distributed on (0, 65), its variance is just the variance of a continuous uniform random variable: 2 65 0 Var 352.08 12 Question #162 Answer: E 218.15 1.06 10,000 0.02 31.88 1 0.02 31.88 218.151.06 9,000 0.021 77.66 2V 1 0.021 1V MLC-09-13 89 Question #163 Answer: D ex ey t px 0.95 0.952 ... k 1 0.95 19 1 0.95 exy pxy 2 pxy ... 1.02 0.95 0.95 1.02 0.95 0.95 ... 2 1.02 0.95 1.02 0.95 0.95 ... 1 0.952 exy ex ey exy 28.56 2 4 2 2 9.44152 Question #164 - Removed Question #165 - Removed Question #166 Answer: E ax e0.08t dt 12.5 0 3 0.375 8 3 2 Ax e0.13t 0.03 dt 0.23077 0 13 2 1 2 aT Var aT 2 2 Ax Ax 400 0.23077 0.375 6.0048 Ax e0.08t 0.03 dt 0 Pr aT ax aT Pr aT 12.5 6.0048 1 vT Pr 6.4952 Pr 0.67524 e0.05T 0.05 ln 0.67524 Pr T Pr T 7.85374 0.05 e0.037.85374 0.79 MLC-09-13 90 Question #167 Answer: A 5 0.05 5 p50 e e0.25 0.7788 1 5 q55 5 1 0.030.02 t 55 dt 0.02 / 0.05 e0.05t t e 0 0.4 1 e 0.25 = 0.0885 Probability of retiring before 60 5 p50 5 q55 = 0.7788*0.0885 = 0.0689 MLC-09-13 1 91 5 0 Question #168 Answer: C Complete the table: l81 l80 d80 910 l82 l81 d81 830 (not really needed) e x ex 1 2 1 since UDD 2 e x e x 1 2 l l l 1 e x 81 82 83 l80 2 e 1 l l l Call this equation (A) 80 2 80 81 82 e 1 l l Formula like (A), one age later. Call this (B) 81 2 81 82 Subtract equation (B) from equation (A) to get 1 1 l81 e80 l80 e81 l81 2 2 910 8.5 0.51000 e81 0.5 920 e81 8000 460 910 8.21 920 MLC-09-13 92 Alternatively, and more straightforward, 910 0.91 1000 830 p81 0.902 920 830 p81 0.912 910 p80 e80 1 q 80 p 80 1 e81 2 1 where q80 contributes since UDD 2 1 8.5 1 0.91 0.91 1 e81 2 e81 8.291 1 e81 q81 p81 1 e82 2 1 8.291 1 0.912 0.912 1 e82 2 e82 8.043 1 e81 q81 p81 1 e82 2 1 1 0.902 0.902 1 8.043 2 8.206 Or, do all the recursions in terms of e, not e , starting with e80 8.5 0.5 8.0 , then final step e81 e81 0.5 MLC-09-13 93 Question #169 Answer: A T p x t 0 1 2 3 0.7 0.7 px vt vt t px 1 0.7 0.49 1 0.95238 0.90703 – 1 0.6667 0.4444 t 2 From above ax:3 vt t px 2.1111 t 0 a 1000 2Vx:3 1000 1 x 2:1 ax:3 1 1000 1 526 2.1111 Alternatively, Px:3 1 d 0.4261 ax:3 1000 2Vx:3 1000 v Px:3 1000 0.95238 0.4261 526 You could also calculate Ax:3 and use it to calculate Px:3 . MLC-09-13 94 Question #170 Answer: E Let G be the gross premium. Expected present value (EPV) of benefits = 1000A50 . EPV of expenses, except claim expense = 15 1 a50 EPV of claim expense = 50A50 (50 is paid when the claim is paid) EPV of premiums = G a50 Equivalence principle: Ga50 1000 A50 15 1 a50 50 A50 1050 A50 15 a50 G a50 For the given survival function, a50 1 1.0550 1 50 k 1 50 k A50 v ( l l ) v (2) 0.36512 50k 1 50k 100 l50 k 1 50 0.05(50) k 1 1 A50 a50 13.33248 d Solving for G, G = 30.88 Question #171 Answer: A 4 0.05 4 p50 e 0.8187 10 8 0.05 10 p50 e 0.6065 0.04 8 p60 e 0.7261 18 p50 10 p50 8 p60 0.6065 0.7261 0.4404 414 q50 4 p50 18 p50 0.8187 0.4404 0.3783 MLC-09-13 95 Question #172 Answer: D a40:5 a40 5 E40 a45 14.8166 0.73529 14.1121 4.4401 a40:5 100,000 A45 v5 5 p40 IA40:5 1 100,000 A45 5 E40 / a40:5 IA40:5 1 100,000 0.20120 0.73529 / 4.4401 0.04042 3363 Question #173 Answer: B Calculate the probability that both are alive or both are dead. P(both alive) = k pxy k px k p y P(both dead) = k qxy k qx k q y P(exactly one alive) = 1 k pxy k qxy Only have to do two year’s worth so have table Pr(both alive) Pr(both dead) Pr(only one alive) 1 0 0 (0.91)(0.91) = 0.8281 (0.09)(0.09) = 0.0081 0.1638 (0.82)(0.82) = 0.6724 (0.18)(0.18) = 0.0324 0.2952 0.8281 0.6724 0.1638 0.2952 1 0 EPV Annuity 30, 000 20, 000 80, 431 0 1 2 0 1.05 1.05 1.051 1.052 1.05 1.05 Alternatively, 0.8281 0.6724 2.3986 1.05 1.052 0.91 0.82 ax:3 a y:3 1 2.6104 1.05 1.052 EPV 20,000 ax:3 20,000 a y:3 10,000 axy:3 axy:3 1 (it pays 20,000 if x alive and 20,000 if y alive, but 10,000 less than that if both are alive) 20,000 2.6104 20,000 2.6104 10,000 2.3986 80, 430 MLC-09-13 96 Question #174 Answer: C Let P denote the gross premium. 0 axIMP 0 P ax e t e t dt e0.05t dt 20 E L 10 axIMP e P 0.03t 0.02t e dt e 0.0310 0.0210 e 0 e 0.03t 0.01t e dt 0 l e0.5 e0.5 23 0.05 0.04 E L 23 20 3 E L 3 15% P 20 Question #175 Answer: C 1 A30:2 1000vq30 500v 2 1 q 30 2 1 1 1000 0.00153 500 0.99847 0.00161 1.06 1.06 2.15875 Initial fund 2.15875 1000 participants = 2158.75 Let Fn denote the size of the fund at the end of year n. F1 2158.75 1.07 1000 1309.86 F2 1309.86 1.069 500 900.24 Expected size of the fund at end of year 2 = 0 (since the amount paid was the single net premium). Difference is 900. MLC-09-13 97 Question #176 Answer: C 2 Var Z E Z 2 E Z E Z t t t 0 v b 0 E Z 2 px xt dt e0.08t e0.03t e0.02t 0.02 dt 0 0.02 2 0.02 e0.07t dt 7 0.07 0 2 vt bt t px xt dt 0 e 0.02 e0.12t xt dt 2 0 Var Z 1 2 7 6 2 12 0.05t 2 1 e0.02t 0.02 dt 6 1 4 0.08503 6 49 *Question #177 Answer: C 0.1 8 311 1.1 0.1 1 6 511 1.1 From Ax 1 d ax we have Ax 1 Ax10 Ax Ax i Ax 3 0.1 0.2861 11 ln 1.1 Ax 10 5 0.1 0.4769 11 ln 1.1 Let P be the net premium. 10V Ax10 P ax10 0.2861 0.4769 6 8 = 0.2623 There are many other equivalent formulas that could be used. MLC-09-13 98 Question #178 Answer: C 100,000 e0.06t e0.001t 0.001dt Regular death benefit 0 0.001 100,000 0.06 0.001 1639.34 Accidental death 100,000 e0.06t e0.001t 0.0002 dt 10 0 10 20 e0.061t dt 0 1 e0.61 20 149.72 0.061 Expected Present Value 1639.34 149.72 1789.06 Question #179 Answer: C 00 p61 560 / 800 0.70 01 p61 160 / 800 0.20 10 p61 0, once dead, stays dead 11 p61 1, once dead by cause 1, stays dead by cause 1 00 01 11 p61 p61 p10 61 p61 0.70 0.20 0 1 1.90 MLC-09-13 99 Question #180 Answer: C The solution requires determining the element in row 2 column 3 of the matrix obtained by multiplying the first three transition probability matrices. Only the second row needs to be calculated. After one period, the row is [0.0 0.7 0.3]. For the second period, multiply this row by the columns of Q1 . The result is [0.07 0.49 0.44]. For the third period, multiply this row by the columns of Q2 . The result is [0.1645 0.3465 0.4890] with the final entry providing the answer. An alternative is to enumerate the transitions that result in extinction. Label the states S (sustainable), E (endangered) and X (extinct). The possible paths are: EX – 0.3 EEX – 0.7(0.2) = 0.14 EEEX – 0.7(0.7)(0.1) = 0.049 Total = 0.489 Note that if the species is not extinct by time 3, it will never become extinct. Question #181 Answer: B Pr(dies in year 1) p 02 0.1 Pr(dies in year 2) p 00 p 02 p 01 p12 0.8(0.1) 0.1(0.2) 0.10 Pr(dies in year 3) p 00 p 00 p 02 p 00 p 01 p12 p 01 p11 p12 p 01 p10 p 02 0.095 EPV (benefits) 100, 000[0.9(0.1) 0.92 (0.10) 0.93 (0.095)] 24, 025.5 Pr(in State 0 at time 0) 1 Pr(in State 0 at time 1) p 00 0.8 Pr(in State 0 at time 2) p 00 p 00 p 01 p10 0.8(0.8) 0.1(0.1) 0.65 EPV ($1 of premium) 1 0.9(0.8) 0.92 (0.65) 2.2465 Net premium = 24,025.5/2.2465 = 10,695. Question #182 - Removed Question #183 - Removed MLC-09-13 100 Question #184 Answer: B 1000P45a45:15 a60:15 15 E45 1000 A45 1000 A45 a45 15 E45 a60 a60 15 E60 a75 15 E45 1000 A45 a45 201.20 14.1121 0.72988 0.5108111.1454 14.1121 11.1454 0.68756 0.39994 7.2170 0.72988 0.51081 201.20 where 15 Ex was evaluated as 5 Ex 10 Ex5 14.2573 9.9568 3.4154 201.20 17.346 Question #185 Answer: A 1V 0 V 1 i 1000 1V 1V qx 2V 1V 1 i 2000 2V 2V qx1 2000 1 i 1000q 1 i 2000q 2000 1.08 1000 0.1 1.08 2000 0.1 2000 x x 1 1027.42 MLC-09-13 101 Question #186 Answer: A Let Y be the present value of payments to 1 person. Let S be the present value of the aggregate payments. E Y 500 ax 500 Y Var Y 1 Ax 5572.68 d 5002 1 d2 2 Ax Ax2 1791.96 S Y1 Y2 ... Y250 E S 250 E Y 1,393,170 S 250 Y 15.811388 Y 28,333 S 1,393,170 F 1,393,170 0.90 Pr S F Pr 28,333 28,333 F 1,393,170 Pr N 0,1 28,333 0.90 Pr N 0,1 1.28 F 1,393,170 1.28 28,333 =1.43 million Question #187 Answer: A bg q41 1 q b1g 1 p b1g 1 e p b g j 41 41 41 bg q41 1 2 l41 l40 d40 d40 1000 60 55 885 1 2 d41 l41 d41 l42 885 70 750 65 q41 1 750 p41 885 750 1 1 q41 885 MLC-09-13 q41 65 135 65 135 0.0766 102 Question #188 Answer: D t S0 (t ) 1 d t log S0 (t ) dt t ex t x 1 dt 1 x x 0 1 e0new old new new 2 old 1 2 1 1 old old 2 1 9 0new old 4 4 Question #189 Answer: C Constant force implies exponential lifetime Var T E T E T 2 2 0.1 2 1 1 2 2 100 2 E min T ,10 t 0.1e .1t dt 10 0.1e .1t dt 10 0 te 10 .1t 10e 1 .1t 10 0 1 10e .1t 10e 10e 10 10e 10 1 10 1 e1 6.3 MLC-09-13 103 Question #190 Answer: A % premium amount for 15 years Gax:15 100,000 Ax 0.08G 0.02Gax:15 x 5 5ax Per policy for life 4669.95 11.35 51,481.97 0.08 4669.95 0.02 11.35 4669.95 x 5 5ax ax 1 Ax 1 0.5148197 16.66 d 0.02913 53,003.93 51,481.97 1433.67 x 5 83.30 4.99 x 5 x 9.99 The % of premium expenses could equally well have been expressed as 0.10G 0.02G ax:14 . The per policy expenses could also be expressed in terms of an annuityimmediate. MLC-09-13 104 Question #191 Answer: D For the density where Tx Ty , Pr Tx Ty 40 y y 0 x 0 0.0005dxdy 40 y 0 50 y 0.0005 x dy 0 50 y 40 40 50 0 y 40 0.0005 ydy 40 x0 0.0005dxdy y 40 0.0005 y 2 2 40 0 0.02 y 0.0005 x 40 dy 0 0.02 dy 50 40 0.40 + 0.20 = 0.60 For the overall density, Pr Tx Ty 0.4 0 0.6 0.6 0.36 where the first 0.4 is the probability that Tx Ty and the first 0.6 is the probability that Tx Ty . Question #192 Answer: B The conditional expected value of the annuity, given , is 1 . 0.01 The unconditional expected value is 0.02 1 0.01 0.02 ax 100 d 100 ln 40.5 0.01 0.01 0.01 0.01 100 is the constant density of on the interval 0.01,0.02 . If the density were not constant, it would have to go inside the integral. MLC-09-13 105 Question #193 Answer: E Recall ex x 2 ex:x ex ex ex:x ex:x t t dt 1 1 x y x 0 Performing the integration we obtain x ex:x 3 2 x ex:x 3 2 2a 2 3a 3 2 7a 3 3 2 3a 2 a k 3 3 3.5a a k 3.5a 3a (i) (ii) k 5 The solution assumes that all lifetimes are independent. Question #194 Answer: B Although simultaneous death is impossible, the times of death are dependent as the force of mortality increases after the first death. There are two ways for the benefit to be paid. The first is to have (x) die prior to (y). That is, the transitions are State 0 to State 2 to State 3. The EPV can be written with a double integral 0 e0.04t t pxy00 x02t: y t e0.04 r r p y22t y23t r drdt 0 0 0 e0.04t e0.12t 0.06 e0.04 r e0.10 r 0.10drdt 0.06 0.10 0.26786 0.16 0.14 By symmetry, the second case (State 0 to State 1 to State 3) has the same EPV. Thus the total EPV is 10,000(0.26786+0.26786) = 5,357. MLC-09-13 106 Question #195 Answer: E Let k p0 Probability someone answers the first k problems correctly. 2 p0 0.8 0.64 4 p0 0.8 0.41 2 p0:0 2 p0 0.642 0.41 4 p0:0 0.41 0.168 2 p0:0 2 p0 2 p0 2 p0:0 0.87 4 p0:0 0.41 0.41 0.168 0.652 2 4 2 2 Prob(second child loses in round 3 or 4) = 2 p0:0 4 p0:0 = 0.87-0.652 = 0.218 Prob(second loses in round 3 or 4 second loses after round 2) = 2 p0:0 4 p0:0 2 = p0:0 0.218 0.25 0.87 Question #196 Answer: E If (40) dies before 70, he receives one payment of 10, and Y = 10. The probability of this is (70 – 40)/(110 – 40) = 3/7 If (40) reaches 70 but dies before 100, he receives 2 payments. Y 10 20v30 16.16637 The probability of this is also 3/7. If (40) survives to 100, he receives 3 payments. Y 10 20v30 30v60 19.01819 The probability of this is 1 – 3/7 – 3/7 = 1/7 E Y 3/ 7 10 3/ 7 16.16637 1/ 7 19.01819 13.93104 E Y 2 3/ 7 102 3/ 7 16.166372 1/ 7 19.018192 206.53515 Var Y E Y 2 E Y 12.46 Since everyone receives the first payment of 10, you could have ignored it in the calculation. MLC-09-13 2 107 Question #197 Answer: C 2 E Z v k 1bk 1 k 0 k px qx k v 300 0.02 v 2 350 0.98 0.04 v3 400 0.98 0.96 0.06 36.8 2 E Z 2 v k 1bk 1 k 0 2 k px qx k v 2 300 0.02 v 4 350 0.98 0.04 v 6 4002 0.98 0.96 0.06 2 2 11,773 Var Z E Z 2 E Z 2 11,773 36.82 10, 419 Question #198 Answer: A Benefits + 3 0 Le 1000v Expenses 0.20G 8 0.06G 2 v 0.06G 2 v2 at G 41.20 and i 0.05, 0L for K 2 770.59 MLC-09-13 108 – Premiums G a3 Question #199 Answer: D P 1000P40 235 P 1 i 0.015 1000 255 255 [A] 255 P 1 i 0.020 1000 272 272 [B] Subtract [A] from [B] 20 1 i 3.385 17 1 i Plug into [A] 20.385 1.01925 20 235 P 1.01925 0.015 1000 255 255 235 P 255 11.175 1.01925 P 261.15 235 26.15 1000 25V 272 26.151.01925 0.0251000 1 0.025 286 MLC-09-13 109 *Question #200 Answer: A 1.2 1 1 0.8 0.6 0.5 0.4 0.4 0.2 0 0 10 x Give n 0 s x 1 55 q35 20 55 q15 20 55 q15 55 q35 1 20 30 15 0.70 Linear Interpolation 40 50 Give n 25 60 35 0.50 0.48 Linear Interpolation S0 90 0.16 32 1 0.6667 S0 35 0.48 48 S0 35 S0 90 0.48 0.16 32 0.4571 S0 15 0.70 70 0.4571 0.6856 0.6667 MLC-09-13 110 70 Give n 75 0.4 80 90 0 100 Given 90 100 0.16 Linear Interpolation 0 Alternatively, 20 55 q15 55 q35 20 p15 55 q35 55 q35 20 p15 S0 35 S0 15 0.48 0.70 0.6856 *Question #201 Answer: A S0 (80) 1 * e ^ 0.16*50 e ^ 0.08*50 0.00932555 2 S0 (81) 1 * e ^ 0.16*51 e ^ 0.08*51 0.008596664 2 p80 S0 81 / S0 80 0.008596664 / 0.00932555 0.9218 q80 1 0.9218 0.078 Alternatively (and equivalent to the above) For non-smokers, px e0.08 0.923116 50 px 0.018316 For smokers, px e0.16 0.852144 50 px 0.000335 So the probability of dying at 80, weighted by the probability of surviving to 80, is 0.018316 1 0.923116 0.000335 1 0.852144 = 0.078 0.018316 0.000335 MLC-09-13 111 Question #202 Answer: B x 40 41 42 l x 2000 d x 20 d x 60 1920 1840 30 40 50 because 2000 20 60 1920 ; 1 2 1920 30 50 1840 Let premium = P 1840 2 2000 1920 EPV premiums = v v P 2.749 P 2000 2000 2000 30 2 40 3 20 EPV benefits = 1000 v v v 40.41 2000 2000 2000 40.41 P 14.7 2.749 MLC-09-13 112 Question #203 Answer: A 10 a30 e0.08t e0.05dt 10 Ex e0.08t e0.08t dt 0 10 0.13t e 0 dt e 0 1.3 0.16 0 e e0.13t 10 e0.16t e1.3 0.13 0 0.16 1.3 1.3 e 1 e 0.13 0.13 0.16 = 7.2992 dt 0 A30 e0.08t e0.05t 0.05 dt e1.3 e0.16t 0.08 dt 10 0 0 1 e e 0.05 0.08 0.16 0.13 0.13 0.41606 1.3 1.3 = P A30 A30 0.41606 0.057 a30 7.29923 1 1 a40 0.08 0.08 0.16 A40 1 a40 1 0.08/ 0.16 0.5 10V A30 A40 P A30 a40 0.5 0.057 0.14375 0.16 *Question #204 Answer: C Let T be the future lifetime of Pat, and K be the curtate future lifetime. L 100,000 vT 1600 aK 1 0 T 10 100,000vT 1600 a10 10 t 20 1600a10 Minimum is 1600a10 12,973 MLC-09-13 20<t when evaluated at i = 0.05 113 Question #205 - Removed Question #206 Answer: A Pax:3 EPV (stunt deaths) 2500 2486 /1.08 2466 / 1.08 2 4 /1.08 5/ 1.082 6 / 1.08 3 P 500000 2500 2500 P 2.77 2550.68 p 921 *Question #207 Answer: D x2 1 dx S x dx 30 10, 000 30 0 2 S0 30 30 1 100 x3 80 x 30, 000 30 0.91 33.833 0.91 37.18 80 e30:50 80 Question #208 Answer: B A60 v p60 A61 q60 1/1.06 0.98 0.440 0.02 0.42566 a60 1 A60 / d 1 0.42566 / 0.06 /1.06 10.147 1000 10V 1000 A60 1000 P50 a60 425.66 10.147 25 172 MLC-09-13 114 Question #209 Answer: E Let Y65 present value random variable for an annuity due of one on a single life age 65. Thus E Y65 a65 Let Y75 present value random variable for an annuity due of one on a single life age 75. Thus E Y75 a75 E (X) 50 2 a65 30 1 a75 100 9.8969 30 7.217 1206.20 Var (X) 50 22Var Y65 30 1 Var Y75 200 13.2996 30 11.5339 3005.94 2 where Var Y65 and Var Y75 1 d2 1 d2 2 2 2 A65 A65 2 A75 A75 1 0.05660 1 0.05660 2 2 0.23603 0.43982 13.2996 0.38681 0.59149 2 11.5339 95th percentile E X 1.645 Var X 1206.20 1.645 54.826 1296.39 Question #210 Answer: C a e t e t dt 0 EPV 50, 000 1 0.5 1 1 0.5 d 100, 000 ln 1 ln 0.5 1 0.045 1 100,000 ln 0.045 0.5 65,099 Question #211 - Removed Question #212 - Removed MLC-09-13 115 Question #213 - Removed Question #214 Answer: A Let be the net premium at issue. 10,000 A45:20 a45:20 0.20120 0.25634 0.43980 0.25634 10,000 14.1121 0.25634 9.8969 297.88 The expected prospective loss at age 60 is 10,00015V45:20 10,000 A60:5 297.88 a60:5 10,000 0.7543 297.88 4.3407 6250 1 0.36913 0.68756 0.4398 0.06674 where A60:5 A60:51 0.68756 A60:5 0.06674 0.68756 0.7543 a60:5 11.1454 0.68756 9.8969 4.3407 1 (Reduced Amount) A60:51 After the change, expected prospective loss = 10,000 A60:5 Since the expected prospective loss is the same 6250 10,000 0.06674 Reduced Amount 0.68756 Reduced Amount = 8119 MLC-09-13 116 Question #215 Answer: D Ax Ax1:5 5 Ex Ax15 :7 12 Ex Ax12 where 5 0.040.02 0.7408 5 Ex e 0.04 Ax1:5 1 0.7408 0.1728 0.04 0.02 7 0.050.02 0.6126 7 Ex 5 e 0.05 Ax 1 5 :7 1 0.6126 0.2767 0.05 0.02 12 Ex 5 Ex 7 Ex 5 0.7408 0.6126 0.4538 0.05 Ax12 0.625 0.05 0.03 Ax 0.1728 0.7408 0.2767 0.4538 0.625 = 0.6614 Question #216 Answer: A EPV of Accidental death benefit and related settlement expense = 0.004 89.36 2000 1.05 0.004 0.04 0.05 0.04 EPV of other DB and related settlement expense = 1000 1.05 446.81 0.094 EPV of Initial expense = 50 3 EPV of Maintenance expense = 31.91 0.094 100 EPV of future premiums 1063.83 0.094 EPV of 0 L 89.36 446.81 50 31.91 1063.83 445.75 MLC-09-13 117 Question #217 Answer: C One approach is to enumerate all the paths from state 1 to state 3 and evaluate their probabilities. The paths are: 113 – (0.8)(0.05) = 0.04 123 – (0.15)(0.05) = 0.0075 133 – (0.05)(1) = 0.05 The sum is 0.0975. Alternatively, the required probability is in row 1 column 3 of the square of the transition probability matrix. Multiplying the first row of the matrix by each column produces the first row of the square, [0.6475 0.255 0.0975]. While only the third calculation is required, doing all three provides a check as the numbers must sum to one. The number of the 50 members who will be in state 3 has a binomial distribution, so the variance is 50(0.0975)(0.9025) = 4.40. MLC-09-13 118 Question #218 Answer: C 0.6 0.3 0.1 Q0 0 0 1 0 0 1 0.36 0.18 0.46 Q0 Q1 0 0 1 0 0 1 0 0.108 0.892 Q0 Q1 Q2 0 0 1 0 0 1 Note that after four transitions, everyone must be in state 2. For premiums, the probability is the sum of the first two entries in row 1 (with probability 1 that a premium is paid at time 0). Then, APV Premiums 1 0.9v 0.54v2 0.108v3 2.35 . For benefits, the probabilities are the second entry in row 1. Then, APV Benefits 4 0.3v 0.18v2 0.108v3 2.01 Difference = 2.35 – 2.01 = 0.34 Note that only the first row of each of the products needs to be calculated. It is also possible to enumerate the transitions that produce the required events, though for this problem the list is relatively long: Premium at time 1: 00 (0.6) 01 (0.3) Total = 0.9 Premium at time 2: 000 (0.6)(0.6) = 0.36 001 (0.6)(0.3) = 0.18 Total = 0.54 Premium at time 3: 0001 (0.6)(0.6)(0.3) = 0.108 Benefit at time 1: 01 (0.3) Benefit at time 2: 001 (0.6)(0.3) = 0.18 Benefit at time 3: 0001 (0.6)(0.6)(0.3) = 0.108 MLC-09-13 119 Question #219 Answer: E 0.25 1.5 qx 0.25 px 1.75 px Let be the force of mortality in year 1, so 3 is the force of mortality in year 2. Probability of surviving 2 years is 10% 0.10 px px 1 e e3 e4 ln 0.1 0.5756 4 0.25 1 px e 4 0.5756 0.8660 1.75 px px 0.75 px1 e e 1.5 qx 0.25 0.251.5 qx 0.25 px 0.25 3 3 4 e 13 0.5756 4 px 1.75 px 0.866 0.154 0.82 0.866 0.25 px Question #220 Answer: C 500 1 500(110 x) 110 x 2 1 xS xS 2 110 x xNS lxS l0S 110 x [see note below] 2 Thus S t p20 NS t p25 MLC-09-13 0.1540 S l20 90 t t S l20 902 2 NS l25 85 t t NS 85 l25 120 e20:25 85 0 t p20:25dt 85 pS 0 t 20 t NS p25 dt 2 90 t 85 t dt 0 90 2 85 85 85 1 2 90 t 90 t 5 dt 688,500 0 1 85 90 t 3 dt 5 85 90 t 2 dt 0 688,500 0 85 90 t 4 5 90 t 3 1 688,500 4 3 0 1 156.25 208.33 16, 402,500 1, 215,000 688,500 = 22.1 [There are other ways to evaluate the integral, leading to the same result]. The S0 ( x) form is derived as S0 ( x) e x 2 dt 0 110t The l x form is equivalent. Question #221 Answer: B a30:20 a30:10 10 E30 a40:10 15.0364 8.7201 10 E30 8.6602 15.0364 8.7201 /8.6602 0.72935 Expected present value (EPV) of benefits = 1 1 1000 A40:10 2000 10 E30 A50:10 10 E30 16.66 2 0.72935 32.61 64.23 EPV of premiums = a30:10 2 0.72935 a40:10 8.7201 2 0.72935 8.6602 21.3527 64.23/ 21.3527 3.01 MLC-09-13 121 e 2ln 110t x 0 110 x 110 2 *Question #222 Answer: A 15V P25 s25:15 1 P25:15 1 A25:15 15 E25 P25:15 P25:151 (this is the retrospective reserve calculation) 0.05332 0.05107 0.00225 0.05107 P25:151 1 A25:15 15 E25 a25:15 15 E25 a25:15 1 A25:15 / a25:15 15 E25 1 A25:15 / a25:15 1 s25:15 0.00225 0.04406 0.05107 0.01128 0.22087 0.05107 25,00015V 25,000 0.22087 0.04406 25,000 0.17681 4420 P25 s25:15 There are other ways of getting to the answer, for example writing A: the retrospective reserve formula for 15V . B: the retrospective reserve formula for the 15th net premium reserve for a 15-year term insurance issued to (25), which = 0 Subtract B from A to get 1 P25 P25:15 s25:15 15V Question #223 Answer: C ILT: We have p70 6,396,609/ 6,616,155 0.96682 2 p70 6,164,663/ 6,616,155 0.93176 e70:2 0.96682 0.93176 1.89858 CF: 0.93176 2 p70 e2 0.03534 Hence e70:2 p70 2 p71 e e2 1.89704 DM: Since l70 and 2 p70 for the DM model equal the ILT, therefore l72 for the DM model MLC-09-13 122 also equals the ILT. For DM we have l70 l71 l71 l72 l71 DM 6,390,409 Hence e70:2 6,390,409 / 6,616,155 6,164,663/ 6,616,155 1.89763 So the correct order is CF < DM < ILT You could also work with p’s instead of l’s. For example, with the ILT, p70 1 0.03318 0.96682 2 p70 0.96682 1 0.03626 0.93176 Note also, since e70:2 p70 2 p70 , and 2 p70 is the same for all three, you could just order p70 . Question #224 Answer: D l60 1000 l61 1000 0.99 0.97 0.90 864.27 d60 1000 864.27 135.73 d60 135.73 3 ln 0.9 ln 0.99 0.97 0.9 135.73 0.1054 98.05 0.1459 l62 864.27 0.987 0.95 0.80 648.31 d61 864.27 648.31 215.96 d61 215.96 3 ln 0.80 ln 0.987 0.95 0.80 215.96 d61 98.05 167.58 265.63 So d60 3 MLC-09-13 3 123 0.2231 167.58 0.2875 Question #225 Answer: B t t p40 e0.05t p50 60 t / 60 50t 1/ 60 t 10 0 t p40:50 50t dt 10 e 10 0.05t 0 1 e0.05t dt 60 60 0.05 0 20 1 e0.5 0.13115 60 Question #226 Answer: A Actual payment (in millions) q3 1 1 q3 3 5 2 6.860 1.1 1.1 0.30 0.5 0.60 0.30 0.10 0.333 0.60 0.5 0.333 Expected payment = 10 7.298 2 1.1 1.1 Ratio 6.860 94% 7.298 Question #227 Answer: E At duration 1 Kx 1 >1 1L v Px1:2 Prob qx 1 0 Px1:2 1 qx 1 So Var 1 L v2qx1 1 qx1 0.1296 MLC-09-13 124 That really short formula takes advantage of Var a X b a 2Var ( X ) , if a and b are constants. Here a = v; b Px1:2 ; X is binomial with p X 1 qx1 . Alternatively, evaluate Px1:2 0.1303 0.9 0.1303 0.7697 if K x 1 1 L 0 0.1303 0.1303 if K x 1 1L E 1 L 0.2 0.7697 0.8 0.1303 0.0497 E 1 L2 0.2 0.7697 0.8 0.1303 0.1320 2 2 Var 1 L 0.1320 0.0497 0.1295 2 Question #228 Answer: C P Ax Ax Ax Ax 0.1 13 0.05 ax 1 Ax 1 Ax 1 13 P Ax Var L 1 1 0.05 1 5 0.10 2 2 2 2 2 Ax Ax2 Ax Ax2 Ax Ax2 0.08888 Var L 1 2 2 Ax Ax2 16 1 0.08888 45 0.1 2 1 4 0.1 0.1 2 MLC-09-13 125 Question #229 Answer: E Seek g such that Pr aT 40 g 0.25 aT is a strictly increasing function of T. Pr T 40 60 0.25 since Pr aT 40 60 p40 100 40 0.25 120 40 a60 0.25 g a60 19.00 Question 230 Answer: B 0.05 A51:9 1 da51:9 1 7.1 0.6619 1.05 11V 2000 0.6619 100 7.1 613.80 10V P 1.05 11V q50 2000 11V 10V 100 1.05 613.80 0.011 2000 613.80 10V 499.09 where q50 0.00110 0.001 0.011 Alternatively, you could have used recursion to calculate A50:10 from A51:9 , then a50:10 from A50:10 , and used the prospective reserve formula for 10V . MLC-09-13 126 Question #231 Answer: C 1000 A81 1000 A80 1 i q80 1000 A81 689.52 679.80 1.06 q80 1000 689.52 q80 720.59 689.52 0.10 310.48 q80 0.5q80 0.05 1000 A80 1000vq80 vp80 1000 A81 1000 0.05 0.95 689.52 665.14 1.06 1.06 Question #232 Answer: D l x 776 752 d x 8 8 42 43 d x 16 16 1 2 1 2 l42 and l43 came from lx 1 lx d x d x 2000 8v 8v 2 1000 16v 16v 2 EPV Benefits = 776 76.40 776 752v EPV Premiums 34 34 1.92 = 65.28 776 2V 76.40 65.28 11.12 MLC-09-13 127 *Question #233 Answer: B pxx 1 qxx 0.96 px 0.96 0.9798 px1:x1 1 qx1:x1 0.99 px1 0.99 0.995 ax:3 1 vpx v 2 2 px 1 0.9798 0.9798 0.995 1.05 1.052 2.8174 0.96 0.96 0.99 2.7763 1.05 1.052 6000 axx:3 axx:3 1 vpxx v 2 2 pxx 1 EPV = 2000 ax:3 2000 ax:3 4000 2.8174 6000 2.7763 27,927 MLC-09-13 128 Question #234 Answer: B t 1 1 1 px x t qx 0.20 t 2 2 px 1 tqx 1 0.08t t 3 3 px 1 tqx 1 0.125t qx 1 1 0t 1 1 2 3 1 1 px x t dt t px t px t px xt dt 0 1 0.08t 1 0.125t 0.20 dt 1 0 1 0.2 1 0.205t 0.01t 2 dt 0 1 0.205t 2 0.01t 3 0.2 t 2 3 0 0.01 0.2 1 0.1025 0.1802 3 MLC-09-13 129 Question #235 Answer: B 1 d w G 0.1G 1.50 1 1.06 1000q40 2.93 q40 AS 1 q40 q40 d w 0.9G 1.50 1.06 1000 0.00278 2.93 0.2 1 0.00278 0.2 0.954G 1.59 2.78 0.59 0.79722 1.197G 6.22 2 d w 1 AS G 0.1G 1.50 1 1.06 1000q41 2 CV q41 AS 1 q41 q41 d w 1.197G 6.22 G 0.1G 1.501.06 1000 0.00298 2 CV 0 1 0.00298 0 2.097G 7.721.06 2.98 0.99702 2.229G 11.20 2.229G 11.20 24 G 15.8 MLC-09-13 130 Question #236 Answer: A 5 1 2 4 AS G 1 c4 e4 1 i 1000qx 4 5 CV q x 4 AS 1 qx4 qx 4 1 2 396.63 281.77 1 0.05 7 1 i 90 572.12 0.26 1 0.09 0.26 657.311 i 90 148.75 0.65 694.50 657.311 i 90 148.75 0.65 694.50 690.18 1.05 657.31 i 0.05 1 i Question #237 - Removed Question #238 – Removed MLC-09-13 131 Question #239 Answer: B Let P denote the annual premium The EPV of benefits is 25,000 Ax:20 25,000(0.4058) 10,145 . The EPV of premiums is Pax:20 12.522P The EPV of expenses is 25, 000 (15 3) 3ax:20 1, 000 0.20 P 0.6261P 194.025 12 37.566 0.8261P 243.591 0.25 0.05 P 0.05 P ax:20 2.00 0.50 0.50 ax:20 Equivalence principle: 12.522 P 10,145 0.8261 P 243.591 10,389.591 12.522 0.8261 888.31 P Question #240 Answer: D Let G denote the premium. Expected present value (EPV) of benefits = 1000A40:20 EPV of premiums = G a40:10 EPV of expenses 0.04 0.25 G 10 0.04 0.05 G a40:9 5a40:19 0.29G 10 0.09G a40:9 5a40:19 0.2G 10 0.09G a40:10 5a40:19 (The above step is getting an a40:10 term since all the answer choices have one. It could equally well have been done later on). MLC-09-13 132 Equivalence principle: G a40:10 1000 A40:20 0.2G 10 0.09G a40:10 5 a40:19 G a40:10 0.2 0.09 a40:10 1000 A40:20 10 5 a40:19 G 1000 A40:20 10 5 a40:19 0.91a40:10 0.2 Question #241 - Removed Question #242 Answer: C 11 AS 10 AS G c10 G e10 1 i 10,000qxd10 11CV qxw10 1 qx10 qx10 d w 1600 200 0.04 200 70 1.05 10,000 0.02 1700 0.18 1 0.02 0.18 1302.1 0.8 1627.63 MLC-09-13 133 *Question #243 Answer: E The net premium reserve at the end of year 9 is the certain payment of the benefit one year from now, less the premium paid at time 9. Thus, it is 10,000v – 76.87. The gross premium reserve adds expenses paid at the beginning of the tenth year and subtracts the gross premium rather than the net premium. Thus it is 10,000v + 5 + 0.1G – G where G is the gross premium. Then, 10, 000v 76.87 (10, 000v 5 0.9G) 1.67 0.9G 81.87 1.67 0.9G 83.54 G 92.82 Question #244 Answer: C 4 AS 3 AS G c4G e4 1 i 1000qxd3 4 CVqxw3 1 qx 3 qx3 d w Plugging in the given values: 4 AS 25.22 30 0.02 30 5 1.05 1000 0.013 75 0.05 1 0.013 0.05 35.351 0.937 = 37.73 MLC-09-13 134 With higher expenses and withdrawals: 4 AS revised 25.22 30 1.2 0.02 30 5 1.05 1000 0.013 75 1.2 0.05 1 0.013 1.2 0.05 48.51.05 13 4.5 0.927 33.425 0.927 = 36.06 4 AS 4 AS revised 37.73 36.06 = 1.67 Question #245 Answer: E Let G denote the gross premium. EPV (expected present value) of benefits 100010 20 A30 . EPV of premiums Ga30:5 . EPV of expenses 0.05 0.25 G 20 first year 0.05 0.10 G 10 a 30:4 years 2-5 10 5 a 35:4 years 6-10 (there is no premium) 0.30G 0.15G a30:4 20 10 a30:4 10 5 a 30:5 0.15G 0.15Ga30:5 20 10 a30:9 (The step above is motivated by the form of the answer. You could equally well put it that form later). Equivalence principle: Ga30:5 100010 20 A30 0.15G 0.15G a30:5 20 10 a30:9 G MLC-09-13 1000 10 20 A30 20 10 a 30:9 20 10 a30:9 1 0.15 a30:5 0.15 1000 10 20 A30 0.85 a30:5 0.15 135 Question #246 Answer: E Let G denote the gross premium EPV (expected present value) of benefits 0.1 3000 v 0.9 0.2 2000 v 2 0.9 0.8 1000v 2 300 360 720 1286.98 2 1.04 1.04 1.042 EPV of premium = G EPV of expenses 0.02G 0.03G 15 0.9 2 v 0.05G 16.73 Equivalence principle: G 1286.98 0.05G 16.73 1303.71 G 1372.33 1 0.05 Question #247 Answer: C EPV (expected present value) of benefits = 3499 (given) EPV of premiums G 0.9 G v G 0.9G 1.8571G 1.05 EPV of expenses, except settlement expenses, 25 4.510 0.2 G 0.9 10 1.5 10 0.1G v 0.9 0.85 10 1.5 10 v2 0.9 25 0.1G 0.765 25 1.05 1.052 =108.78+0.2857G 70 0.2G MLC-09-13 136 Settlement expenses are 20 110 30 , payable at the same time the death benefit is paid. 30 So EPV of settlement expenses EPV of benefits 10,000 0.003 3499 10.50 Equivalence principle: 1.8571G 3499 108.78 0.2857G 10.50 3618.28 G 2302.59 1.8571 0.2857 Question #248 Answer: D a50:20 a50 20 E50 a70 13.2668 0.23047 8.5693 11.2918 0.06 A50:20 1 d a50:20 1 11.2918 1.06 0.36084 Expected present value (EPV) of benefits 10,000A50:20 3608.40 EPV of premiums 495a50:20 5589.44 EPV of expenses 0.35 495 20 1510 0.05 495 5 1.50 10 a50:19 343.25 44.75 11.2918 1 803.81 E(L) = EPV benefits + EPV expenses – EPV premiums = 3608.40 + 803.81 – 5589.44 = –1177.23 MLC-09-13 137 Question #249 Answer: B 1 1 0 0 q1xy t pxy x t dt t px t p y x t dt 1 qx e0.25t dt (under UDD, t px x t qx ) 0 1 0.125 qx (4e 0.25t ) qx (4)(1 e 0.25 ) 0.8848qx 0 qx 0.1413 Question #250 Answer: C 2 p[11x ]1 p[11x ]1 p[11x ] 2 p[12x ]1 p[21x ]2 0.1 0.1 0.1 0.2 0.7 0.7 0.3 0.4 2 3 2 3 0.75(0.7333) 0.25(0.3333) 0.6333 Note that Anne might have changed states many times during each year, but the annual transition probabilities incorporate those possibilities. Questions #251-260 – Removed Question #261 Answer: A The insurance is payable on the death of (y) provided (x) is already dead. E ( Z ) Axy2 e t t qx t p y y t dt 0 e 0.06 t 0 (1 e 0.07 t )e 0.09 t 0.09dt 0.09 e 0.15t e 0.22t dt 0 1 1 0.09 0.191 0.15 0.22 MLC-09-13 138 Question #262 Answer: C t px 95 x t 1 , x t , 95 x 95 x t t p y e t Pr(x dies within n years and before y) n 0 t n 0 px t p y x t dt n 95 x t t 1 1 1 e n t e dt e dt 95 x 95 x t 95 x 0 (95 x) Question #263 Answer: A 0.25 2 q30.5:40.5 0.25 0 t 0.25 0.4 0.6t dt 1 0.5(0.4) 1 0.5(0.6) 0 p30.5 30.5 t t q40.5 dt 0.4(0.6) t 2 0.8(0.7) 2 0.25 0.0134 0 Question #264 – Removed Question #265 Answer: D t t t 2 px exp 5rdr e 2.5t 0 t 2 p y exp rdr e 0.5t 0 1 1 0 0 1 qx1: y t p y t px x t dt e 0.5t e 2.5t 5tdt 5 e 3t tdt 2 2 0 1 5 2 5 e 3t (1 e 3 ) 0.7918 6 6 0 MLC-09-13 139 2 Question #266 Answer: B 5 G 5 0 1 t t2 1 dt 30 25 30(25)(2) 0 60 H 5 p80:85 10 p80:85 GH 25 20 20 15 200 4 30 25 30 25 750 15 1 16 17 0.2833 60 60 60 Question #267 Answer: D t t S0 (t ) exp (80 x) 0.5 dx exp 2(80 x) 0.5 exp 2((80 t ) 0.5 800.5 ) 0 0 F S0 (10.5) exp 2(69.50.5 800.5 ) 0.29665 S0 (10) 0.31495 S0 (11) 0.27935 G S0 (10.5)exp [0.31495(0.27935)]0.5 0.29662 F G 0.00003 Question #268 Answer: A 4 4 E ( Z ) 500 0.2(1 0.25t )dt 1000 0.25(0.2t )dt 0 0 500(0.2) t 0.125t 2 1000(0.25) (0.1t 2 ) 4 4 0 0 100(4 2) 250(1.6) 600 Question #269 Answer: A 10 q30:50 10 q30 10 q50 (1 e0.5 )(1 e0.5 ) 0.1548 MLC-09-13 140 Question #270 Answer: C e30:50 e30 e50 e30:50 e30 e50 e 0.05t dt 20 0 e30:50 e 0.10t dt 10 0 e30:50 20 20 10 30 Question #271 Answer: B 0 0 1 A30:50 e t t p30 t p50 30t dt e0.03t e0.05t e0.05t 0.05dt 0.05 0.3846 0.13 Question #272 Answer: B T30:50 has the exponential distribution with parameter 0.05 + 0.05 = 0.10 and so its mean is 10 and its variance is 100. Question #273 Answer: D Cov[T30:50 , T30:50 ] e30 e30:50 e50 e30:50 (20 10)(20 10) 100 See solution to Question #270 for the individual values. Question #274 Answer: E V ( 2V 3 )(1 i3 ) qx 2 (b3 3V ) 3 96 (84 18)(1.07) qx 2 (240 96) qx 2 13.14 /144 0.09125 MLC-09-13 141 Question #275 Answer: A ( 3V 4 )(1 i4 ) qx 3b4 (96 24)(1.06) 0.101(360) 101.05 px 3 0.899 V 4 Question #276 Answer: D Under UDD: 0.5qx 3 0.5(0.101) 0.0532 0.5 qx 3.5 1 0.5qx 3 1 0.5(0.101) Question #277 Answer: E V v 0.5 0.5 p3.5 4V v 0.5 0.5 q3.5b4 3.5 1.060.5 (0.9468)(101.05) 1.060.5 (0.0532)(360) 111.53 Question #278 Answer: D 1 10 p30:40 1 10 p30 10 p40 1 60 50 2 70 60 7 Question #279 Answer: A 10 2 q30:40 10 0 t p30 30t (1 t p40 )dt 10 0 1 t 50 dt 0.0119 70 60 70(60) Question #280 Answer: A 20 10 t p30 30t t q40 dt 20 10 20 10 t p40 40t t q30 dt 20 1 t 1 t 1 400 100 1 400 100 dt dt 0.0714 10 70 60 60 70 70 2(60) 60 2(70) MLC-09-13 142 Question #281 Answer: C 140, 000 30 t 0 p30 30t t p40 dt 180, 000 30 0 t p40 40 t t p30 dt 30 1 70 t 1 60 t dt 180, 000 dt 0 70 60 0 60 70 1 602 302 1 702 402 140, 000 180, 000 115, 714 70 2(60) 60 2(70) 140, 000 30 Question #282 Answer: B P 20 0 t p30 t p40 dt P 20 0 70 t 60 t P 20 dt 4200 130t t 2 dt 0 70 60 4200 P [4200(20) 130(200) 8000 / 3] 14.444P 4200 Question #283 Answer: A Note that this is the same as Question 33, but using multi-state notation rather than multiple-decrement notation. The only way to be in State 2 one year from now is to stay in State 0 and then make a single transition to State 2 during the year. 1 1 1 p t p dt e 02 x 0 00 x MLC-09-13 02 x t 0 (0.3 0.5 0.7) t e1.5t 1 0.5dt 0.5 (1 e1.5 ) 0.259 1.5 0 3 143 Question #284 Answer: C m 1 m2 1 ( 80 ) . Then, 2m 12m2 2 1 22 1 4 1 42 1 (2) (4) a80 a80 a80 ( ) a ( 80 ) 80 80 2(2) 12(2) 2 2(4) 12(4) 2 8.29340 8.16715 (1/ 4) (3 / 8) [(3 / 48) (15 /192)]( 80 ) (m) Woolhouse’s formula to three terms is a80 a80 0.00125 0.015625( 80 ) 80 0.08 (2) a80 a80 1/ 4 (3 / 48)(0.08) 8.5484. The answer is: (12) a80 a80 12 1 122 1 11 143 ( 80 ) 8.54840 (0.08) 8.08345. 2 2(12) 12(12) 24 1728 Question #285 Answer: B 1 | a70:2 v 2 2 p70 v 3 3 p70 B x t c ( c 1) At ln c t p70 e 2 p70 e0.0002(2) (0.9754483)1.211 0.9943956 3 p70 e 0.0002(3) (0.9754483)1.3311 0.9912108 1 e e 0.000003 70 t 1.1 (1.1 1) 0.0002 t ln(1.1) e e 0.0002t (0.9754483)1.1 1 t | a70:2 0.9943956 /1.052 0.9912108 /1.053 1.7582. Question #286 Answer: E 1 A50:2 vq50 v 2 p50 q51 px e B x c ( c 1) ln c e 0.000005 x 1.2 (0.2) ln(1.2) e 0.0000054848(1.2) 50 p50 0.951311 p51 0.941861 1 A50:2 0.048689 /1.03 0.951311(0.058139) /1.032 0.09940. MLC-09-13 144 *Question #287 Answer: E The reserve at the end of year 2 is 10,000vq52 3 where 3 is the net premium in year 3. From the equivalence principle at time zero: 10, 000(vq50 v 2 p50 q51 v 3 p50 p51q52 ) 0.5 3 0.5 3vp50 3v 2 p50 p51 0.95 0.95(0.94) 0.05 0.95(0.06) 0.95(0.94)(0.07) 10, 000 0.5 0.5 3 2 3 1.04 1.04 1.04 1.04 2 1.04 1563.4779 1.78236 3 3 877.1953 V 10, 000(0.7) /1.04 877.1953 204.12 2 *Question #288 Answer: C The net premium in year one is vq50 . By the equivalence principle: 10, 000(vq50 v 2 p50 q51 v3 p50 p51q52 ) 10, 000vq50 (vp50 v 2 p50 p51 ) 0.95(0.06) 0.95(0.94)(0.07) 0.95 0.95(0.94) 10, 000 2 1.043 1.042 1.04 1.04 1082.708665 1.739090 622.57 V 10, 000(0.07) /1.04 622.57 50.51. 2 Note that this is a full preliminary term reserve without that term being used. Question #289 Answer: E t-th year 0* 1 2 3 V P E I EDB E tV Pr 0 0 700 700 0 14500 14500 14500 1000 100 100 100 0 864 906 906 0 14000 15000 16000 0 690.2 689.5 0 -1000 573.8 316.5 6 t 1 t 1 px 1.000 1.000 0.986 0.971 -1000.0 573.8 312.1 5.8 NPV 1000 573.8v 312.1v2 5.8v3 216.08 using a 10% discount rate. *The 1000 at time 0 is neither accumulated nor discounted. The value is treated as occurring at the end of time 0 and not as occurring at the beginning of year 1. *Question #290 MLC-09-13 145 Answer: C Expected present value of future profits 140v 0.95(135) p67 v2 (0.95)2 (130) 2 p67v3 314.09 at a 10% discount rate. *Question #291 Answer: B Solution Profit (6, 000 700 10)(1.06)1/12 0.002(200, 000) 0.0015(50, 000 15, 000) 0.9965(6, 200) 46.7. *Question #292 Answer: C 245 p40 274 and 300 2 p40 395 Present value of expected premiums: 1000[1 (245 / 274)(1/1.12) (300 / 395)(1/1.122 )] 2403.821. Present value of expected profits: 400 150 /1.12 245 /1.122 300 /1.123 142.775. PV Profit / PV premium = 5.94% Question #293 Answer: B P[1 0.97(0.98624) 0.92(0.98624)(0.98499)] 100, 000[0.97(0.01376) 0.92(0.98264)(0.01501) 0.87(0.98264)(0.98499)(0.01638)] P 1431.74. MLC-09-13 146 Question #294 Answer: A Let IF equal 1 if the index drops below its current level in the next year and equals 0 otherwise. E[ X N | I F 1] N (1000)vqx 48.54 N E[ X N | I F 0] 0 E[ X N ] 0.1(48.54 N ) 0 4.854 N Var[ X N | I F 1] (1000v) 2 Nqx (1 qx ) 44, 773.31N Var[ X N | I F 0] 0 E[Var ( X N | I F )] 0.1(44, 773.31N ) 0 4, 477.33 N Var[ E ( X N | I F )] 0.1(48.54 N ) 2 0 (4.854 N ) 2 212.05 N 2 Var[ X N ] 4, 477.33 N 212.05 N 2 Var ( X 10 ) 10 lim 25.69 Var ( X N ) N lim N N 25.69 14.56 11.13. 4, 477.33 N 212.05 N 2 212.05 14.56 N *Question #295 Answer: E The actuarial present value of the death benefit is ( s62 )(4)d 62(2) v 0.5 ( s63 )(4)d 63(2) v1.5 ( s64 )(4)d 64(2) v 2.5 100, 000 s62l62 100, 000 3.589(4)(213)v 0.5 3.643(4)(214)v1.5 3.698(4)(215)v 2.5 3.589(52,860) 4,585. MLC-09-13 147 Question #296 Answer: A Policy 1: AV371 [ AV36 G E (100, 000 AV371 ) 1/12 q63 /1.004](1.004) ( AV36 G E )1.004 100, 000 1/12 q63 AV371 1/12 q63 [( AV36 G E )1.004 100, 000 1/12 q63 ] / (1 1/12 q63 ) Policy 2 AV372 [ AV36 G E 100, 000 1/12 q63 /1.004](1.004) ( AV36 G E )1.004 100, 000 1/12 q63 Because the starting account value, G and E are identical for both policies: AV361 / AV362 1/ (1 1/12 q63 ) 1/ [1 (1/12)(0.01788)] 1.0015. *Question #297 Answer: E The recursive formula for the account values is: AVk 1 ( AVk 0.95G 50)1.06 (100,000 AVk 1 )q50k . This is identical to the recursive formula for net premium reserves for a 20-year term insurance where the net premium is 0.95G – 50. Because the net premium reserve is zero after 20 years, using this premium will ensure that the account value is zero after 20 years. Therefore, 0.95G 50 100, 000 1 A50:20 a50:20 A50 20 E50 A70 0.24905 0.23047(0.51495) 100, 000 1154.55 a50 20 E50 a70 13.2668 0.23047(8.5693) G = (1154.55+50)/0.95 = 1,267.95. MLC-09-13 148 Question #298 Answer: A 2 5, 000 E ( Z ) (5, 000 /1.030) (0.005) (1 0.005)(0.006) 1.030(1.032) 2 2 2 5, 000 (1 0.005)(1 0.006)(0.007) 1.030(1.032)(1.035) 392,917 Question #299 Answer: E Note: This solution used the derivatives at 4.50 and 4.75. The question did not specify whether this should be done as a forward or a backward recursion. Either would have been fine. Using the derivatives at 4.75 and 5.00 produces an answer of 922.3. See sample questions 306 and 307, where the derivatives to use are specified. 44.75 0.00004(1.144.75 ) 0.002847 44.5 0.00004(1.144.5 ) 0.002780 1000 E ( 4.75 L) 0.25{0.04 E ( 4.75 L) 150 150(0.05) 0.002847[10, 000 100 E ( 4.75 L)]} E ( 4.75 L) {1000 0.25[150 150(0.05) 0.002847(10, 000 100)]} / [1 0.25(0.04 0.002847)] 961.27 961.27 E ( 4.5 L) 0.25{0.04 E ( 4.5 L) 150 150(0.05) 0.002780[10, 000 100 E ( 4.5 L)]} E ( 4.5 L) {961.27 0.25[150 150(0.05) 0.002780(10, 000 100)]} / [1 0.25(0.04 0.002780)] 922.795 Question #300 Answer: B Assumed profit = 800[(83.30 87(1 0.03))(1.05) (0.0035)(10,000) (1 0.0035)(141.57)] 0 . This is expected because it should be zero with gross premium reserves and the gross premium reserve assumptions Actual profit = 800[(83.30 87(1 0.025))(1.04) (0.0025)(10,000) (1 0.0025)(141.57)] 6907.14. Gain from all sources = 6,907.14 – 0 = 6,907.14. MLC-09-13 149 Question #301 Answer: E The death benefit is max(100,000,1.3 85,000 110,500) 110,500 . The withdrawal benefit is 85,000 1,000 84,000 . (death) (withdrawal) ( 9 AS P E )(1 i ) 110,500q79 84, 000q79 AS 10 (death) (withdrawal) 1 q79 q79 (75, 000 9, 000 900)(1.08) 110,500(0.01) 84, 000(0.03) 1 0.01 0.03 89, 711. Note – companies find asset share calculations less useful for universal life policies than for traditional products. This is because policyholders who choose different premium payment patterns have different asset shares. However, for any pattern of premiums, an asset share can be calculated. Question #302 Answer: B 10V ( 9V G E )(1 i ) 1000qx(death) 10 CV (1 qx(death) )qx(withdrawal) 9 9 9 (death) (withdrawal) 1 qx(death) (1 q ) q 9 x 9 x 9 (115 16 3)(1.06) 1000(0.01) 110(1 0.01)(0.10) 128.83. 1 0.01 (1 0.01)(0.10) Expected deaths = 1000(0.01) = 10; actual deaths = 15. Expected withdrawals = 1000(1 – 0.01)(0.1) = 99; actual withdrawals = 100. Gain from mortality and withdrawals is equal to (Expected deaths – Actual deaths)(Death benefit – End of Year Reserve) + (Expected withdrawals – Actual withdrawals)(Withdrawal benefit – End of Year Reserve) = (10 – 15)(1000 – 128.83) + (99 – 100)(110 – 128.83) = -4337. Question #303 Answer: C Because there are no cash flows at the beginning of the year, the only item earning interest is the reserve from the end of the previous year. The gain is 1000(10,994.49)(0.05 – 0.06) = –109,944.90 MLC-09-13 150 Question #304 Answer: B Expenses are incurred for all who do not die. Because gain from mortality has not yet been calculated, the anticipated experience should be used. Thus, expenses are assumed to be incurred for 1000(1 – 0.01) = 990 survivors. The gain is 990(50 – 60) = -9,900. Question #305 Answer: E The tenth reserve is (d ) 9V (1 i ) (1 q74 )(1000 50) V 10 (d ) 1 q74 10,994.49(1.06) (1 0.01)(1050) 1 0.01 V 10, 721.88. 10 V 10 Note that reserves are prospective calculations using anticipated experience. There were 2 more deaths than expected. For each extra death, an annuity benefit is not paid, an expense is not paid and a reserve does not have to be maintained. Thus, the saving is 1000 + 60 + 10,721.88 = 11,781.88. The total gain is 2(11,781.88) = 23,563.76. Because the gain from expenses has already been calculated, the actual value is used. As an aside, the total gain from questions 303-305 is -96,281. The actual profit can be determined by first calculating the assets at the end of the year. Begin with 1000(10,994.49) = 10,994,490. They earn 5% interest, to accumulate to 11,544,215. At the end of the year, expenses are 60 for each of 988 who did not die, for 59,280. Annuity benefits of 1000 are paid to the same 988 people, for 988,000. Assets at the end of the year are 11,544,215 – 59,280 – 988,000 = 10,496,935. Reserves must be held for the 988 continuing policyholders. That is, 988(10,721.88) = 10,593,217. The difference, 10,496,935 – 10,593,217 = -96,282. Note also that gain is actual profit minus expected profit. If reserves are gross premium reserves, as in this problem, expected profit = 0 and thus total gain = actual profit. Problem 300 illustrates a case where total gain is different from actual profit. MLC-09-13 151 Question #306 Answer: E d tV G ( tV ) t (bt tV ) dt At t = 4.5, 4.5V 25 0.05( 4.5V ) 0.02(4.5)(100 4.5V ) 16 0.14( 4.5V ) Euler’s formula in this case is 5.0V 4.5V (5.0 4.5) 4.5V . V' t Because the endowment benefit is 100, 5.0V 100 and thus, 100 4.5V 0.5( 4.5V ) 4.5V 0.5[16 0.14( 4.5V )] 8 1.07( 4.5V ) V 85.981. Similarly, 4.5V 4V (4.5 4.0) 4V and 4.0V 25 0.05( 4.0V ) 0.02(4.0)(100 4.0V ) 17 0.13( 4.0V ) 85.981 4.0V 0.5( 4.0V ) 4.0V 0.5[17 0.13( 4.0V )] 8.5 1.065( 4.0V ) 4.5 V 72.752. 4.0 Note that if smaller step sizes were used (which would be inappropriate for an exam question, where the step size must be specified), the estimate of the time 4 reserve converges to its true value of 71.96. Question #307 Answer: A d tV G ( tV ) t (bt tV ) dt At t = 5.0, 5.0V 25 0.05( 5.0V ) 0.02(5.0)(100 5.0V ) 15 0.15( 5.0V ) V' t Because the endowment benefit is 100, 5.0V 100 and thus, 5.0V 15 0.15(100) 30 . Euler’s formula in this case is 4.5V 5.0V (4.5 5.0) 5.0V 100 0.5(30) 85. Similarly, 4.0V 4.5V (4.0 4.5) 4.5V and 4.5V 25 0.05( 4.5V ) 0.02(4.5)(100 4.5V ) 16 0.14( 4.5V ) 16 0.14(85) 27.9 . 4.0V 85 (4.0 4.5)(27.9) 71.05. Note that if smaller step sizes were used (which would be inappropriate for an exam question, where the step size must be specified), the estimate of the time 4 reserve converges to its true value of 71.96. MLC-09-13 152 Question #308 Answer: A t t 0 s ds e 0 (0.05 0.02 s ) ds e (0.05t 0.01t 2 ) t p0 e 1 p0 e0.06 0.94176 4 APV 80(1 0.25t )e t t p0 t dt 0 f (1) 80(1 0.25)e0.05 (0.94176)(0.05 0.02) 3.7625. Question #309 Answer: D Let P be the net premium. There are three ways to approach this problem. The first two are intuitive: The actuarial present value of the death benefit of 1000 is 1000 10| A40 . The return of net premium benefit can be thought of as two benefits. First, provide a ten-year temporary annuity-due to everyone, with actuarial present value Pa40:10 . However, those who live ten years must then return the accumulated value of the premiums. This forms a pure endowment with actuarial present value Ps10 10 E40 . The total actuarial present value of all benefits is 100010| A40 Pa40:10 Ps10 10 E40 . Setting this equal to the actuarial present value of net premiums ( Pa40:10 ) and solving gives 1000 10| A40 Pa40:10 Ps10 10 E40 Pa40:10 1000 10| A40 Ps10 P 1000 10| A40 s10 10 E40 1000 10 E40 A50 1000 A50 . s10 10 E40 s10 The second intuitive approach examines the reserve at time 10. Retrospectively, premiums were returned to those who died, so per survivor, the accumulated premiums are only the ones paid by the survivors, that is Ps10 . There are no other past benefits so this is the reserve (it is easy to show this using a recursive formula) Prospectively, the reserve is the actuarial present value of benefits (there are no future premiums), or 1000A50 . Setting the two reserves equal to each other produces the premium: 1000A50 . P s10 MLC-09-13 153 The final approach is to work from basic principles: APV (net premiums) = Pa40:10 . 9 9 k APV benefits = 1000 10| A40 P v k j 1 k p40 (1 i) j 1 j| q40 k k 0 j 0 In that double sum, k is the time the premium is paid, with probability k p40 that it is paid. j is the curtate time, from when the premium was paid, until death. The amount of premium refunded, with interest is P(1 i) j 1 . The probability, given that it was paid, that it will be refunded at time j+1 is j | q40 k . The interest discount factor, from the date of refund to age 40 is v j k 1 vk+j+1. Then, 9 9k 9 9k P v k j 1 k p40 (1 i ) j 1 j| q40 k P v k k p40 j| q40 k k 0 j 0 k 0 j 0 P v k k p40 10 p40 P a40:10 10 p40 a10 9 k 0 Setting APV net premiums = APV benefits: Pa40:10 1000 10| A40 P(a40:10 10 p40 a10 ) P 10 p40 a10 1000 10| A40 P 10 E40 (1000 A50 ) v10 10 p40 (1000 A50 ) 1000 A50 . p a p a s 10 40 10 10 40 10 10 The numerical answer is 249.05 P 17.83. 13.97164 MLC-09-13 154 Question #310 Answer: B Final average salary 50, 000 (1.04) 26 (1.04) 25 (1.04) 24 133,360.2 3 Annual retirement benefit 0.017(27)(final average salary)(0.85) 52,030 Note that the factor of 0.85 is based on an interpretation of the 5% reduction as producing a factor of 1 – 3(0.05) = 0.85. This is the approach used in AMLCR. An alternative approach is to use 0.953. Upon rounding, the same answer choice results. Question #311 Answer: E Retirement Benefit 20% Age at Retirement 63 64 65 Salary 253,094 263,218 273,747 Interest discount e0.05(3) e0.05(4) e0.05(5) Prob 0.06 0.05 0.43 Years of Service 0.6 0.8 1.0 Reduction 0.40 0.20 0.0 Sum Total APV 4,705 6,896 91,674 103,275 APV above = Salary Interest discount prob (20% Years of service) (1 reduction) Death Benefit Age at Death 60 61 62 63 64 Salary 225,000 234,000 243,360 253,094 263,218 Interest discount e0.05 e 0.05(2) e 0.05(3) e 0.05(4) e0.05(5) prob 0.01 0.01 0.01 0.01 0.01 sum total 2,140 2,117 2,095 2,072 2,050 10,474 Total APV of both benefits = 103,275 + 10,474 = 113,749. MLC-09-13 155 Question #312 Answer: B a 81 a 80 1 1 i / p 80 5.89 1 1.06 / 1 0.077 5.616 A81 1 d a 81 1 0.06/1.06 5.616 0.6821 Modified renewal net premium 1000 0.6821/ 5.616 121.5 A90 1 d a 90 1 0.06/1.06 3.65 0.7934 V FPT 1000 0.7934 3.65 121.5 350 10 Note: Other reserve formulas could also be used, including 1000 1 a 90 / a 81 . MLC-09-13 156 Question #313 Answer: E Gross premium = G 2000 2000 Ga 45 2000 A45 1 20 0.5 10 a 45 0.20G 0.05Ga 45 1000 1000 0.95a G 22 45 0.20 G 2000 A45 22 11a 45 2000 A45 22 11a 45 0.95a 45 0.20 11 2000(0.20120) 22 11(14.1121) 43.89 0.95(14.1121) 0.20 There are two ways to proceed. The first is to calculate the gross premium reserve and the benefit reserve and take the difference. 2000 A45 2000(0.2012) The net premium is 28.52. a45 14.1121 The net premium reserve is 2000 A55 28.52a55 2000(0.30514) 28.52(12.2758) 260.17. The gross premium reserve is 2000 A55 [0.05(43.89) 0.5(2000 /1000) 10]a55 43.89a55 2000(0.30514) (13.19 43.89)(12.2758) 233.41. Expense reserve is 233.41 – 260.17 = –27 The second is to calculate the expense reserve directly based on the pattern of expenses. The first step is to determine the expense premium. The present value of expenses is [0.05G 0.5(2000 /1000) 10]a45 0.20G 1.0(2000 /1000) 20 13.1945(14.1121) 30.778 216.98. The expense premium is 216.98/14.1121=15.38 The expense reserve is the expected present value of future expenses less future expense premiums, that is, [0.05G 0.5(2000 /1000) 10]a55 15.38a55 2.1855(12.2758) 27 There is a shortcut with the second approach based on recognizing that expenses that are level throughout create no expense reserve (the level expense premium equals the actual expenses). Therefore, the expense reserve in this case is created entirely from the extra first year expenses. They occur only at issue so the expected present value is 0.20(43.89)+1.0(2000/1000)+20 = 30.778. The expense premium for those expenses is then 30.778/14.1121 = 2.181 and the expense reserve is the present value of future non-level expenses (0) less the present value of those future expense premiums, which is 2.181(12.2758) = 27 for a reserve of –27. MLC-09-13 157 Question #314 Answer: B 1000 A40 161.32 10.89. a40 14.8166 2.78 The valuation premium in year 1 is 1000q40v 2.62. 1.06 Let P 2 denote the valuation premium in years 2-20. By the equivalence principle: 1000 A40 2.62 P2 (a40:20 1) 10.89 20| a40 The valuation premium in years 21 and later is where 20| a40 20 E40 a60 0.27414(11.1454) 3.0554 and a40:20 a40 20| a40 14.8166 3.0554 11.7612. Then, 161.32 2.62 (11.7612 1) P 2 10.89(3.0554) P 2 11.66 V mod 1000 A55 11.66a55:5 10.89 5| a55 15 where 5| a55 0.70810(11.1454) 7.8921 a55:5 12.2758 7.8921 4.3837 V mod 305.14 11.66(4.3837) 10.89(7.8921) 168.08 15 Question #315 Answer: D px 0.9, 2 px 0.81 Let v(k ) denote the present value at time 0 of a payment of 1 at time k. Thus, 1 1 1 v(1) and v(2) . 1 year 1 rate 1 year 1 rate 1 year 2 rate This notation is used in Chapter 11 of ALMCR. There all the rates are known based on the initial yield curve. Here the v(k ) are conditional upon the future interest rates. For the up/up path, the year 1 rate is 0.07 and the year 2 rate is 0.09. Then, v(1) 0.93458, v(2) 0.85741, ax:3 1 1 px v(1) 2 px v(2) 1 1 0.9(0.93458) 0.81(0.85741) 2.53562. For the other paths: v(1) Path Year 1 rate Year 2 rate Up/down Down/up Down/down MLC-09-13 0.07 0.03 0.03 0.05 0.05 0.01 0.93458 0.97087 0.97087 158 v(2) ax:3 0.89008 0.92464 0.96126 2.56209 2.62274 2.65240 The unconditional expected present value is the weighted average of the conditional ones: Y 100, 000[0.75(0.75)(2.53562) 0.75(0.25)(2.56209) 0.25(0.75)(2.62274) 0.25(0.25)(2.65240)] 256, 422. Question #316 Answer: C E40 (1 q40 ) /1.04 (1 0.00278) /1.04 0.95887 If i becomes 0.06 starting at age 41, APV benefits 1000q40v 1 E40 (1000) A41 1 1000(0.00278) /1.04 0.95877(168.69) 164.41 APV premiums of P P(1 1 E40 a41 ) P[1 0.95887(14.6864)] 15.0823P If i remains at 0.04, 1 A40 1 0.27345 a40 18.8911 (0.04 /1.04) 0.03846 Unconditional APV of benefits 0.75(273.45) 0.25(164.41) 246.19 Unconditional APV of premiums 0.75(18.8911P) 0.25(15.0823P) 17.9389 P By the equivalence principle, P 246.19 /17.9389 13.72 Question #317 Answer: D Profit ( 20V G E )(1 i ) 10, 000q 21V (1 q) (2, 750 200 20)(1.07) 10, 000(0.01) 2,920(0.99) 144.30 Dividend 0.8(144.30) 115.44 MLC-09-13 159 Question #318 Answer: B Profit ( 20V G E )(1 i ) (Total face)q 21V (1 q) (3, 600 200 20)(1.07) (10, 000 2, 000)(0.01) 3,800(0.99) 162.60 Dividend 0.8(162.60) 130.08 130.08 Reversionary bonus amount 296.31 0.439 296.31 Compound reversionary bonus rate 0.025 10, 000 2, 000 MLC-09-13 160
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