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3 Exam MLC SECTION A Multiple-Choice

4 **BEGINNING OF EXAMINATION** 1. For two lives, (60) and (70), with independent future lifetimes, you are given: (i) 5p 60 = 0.92 (ii) 5p 70 = 0.88 (iii) 1000 q 65 = (iv) 1000 q 75 = Calculate 1000 q. 5 60:70 (A) 6.3 (B) 6.6 (C) 6.9 (D) 7.2 (E) 7.5 Exam MLC: Fall GO ON TO NEXT PAGE

5 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

6 2. For a double decrement table, you are given: (i) (ii) (iii) q = 0.1 (1) x q = 0.2 (2) x Each decrement is uniformly distributed over each year of age in its associated single decrement table. Calculate q (1) x. (A) (B) (C) (D) (E) Exam MLC: Fall GO ON TO NEXT PAGE

7 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

8 3. Patients are classified as Sick (S), Critical (C), or Discharged (D). Transitions occur daily according to the following transition matrix: S C D S C D Calculate the probability that a patient who is classified as Sick today will be classified as Sick three days later. (A) (B) (C) (D) (E) Exam MLC: Fall GO ON TO NEXT PAGE

9 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

10 4. You are given: (i) A = :15 (ii) A 1 35:15 = 0.25 (iii) A 35 = 0.32 Calculate A. 50 (A) 0.35 (B) 0.40 (C) 0.45 (D) 0.50 (E) 0.55 Exam MLC: Fall GO ON TO NEXT PAGE

11 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

12 5. For a 20-year deferred whole life annuity-due with annual payments of 30,000 on (40), you are given: (i) (ii) The single net premium is refunded without interest at the end of the year of death if death occurs during the deferral period. Mortality follows the Illustrative Life Table. (iii) i = 0.06 Calculate the single net premium for this annuity. (A) 91,700 (B) 93,600 (C) 95,600 (D) 97,500 (E) 99,500 Exam MLC: Fall GO ON TO NEXT PAGE

13 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

14 6. Mr. and Mrs. Peters, both age 65, purchase a contract providing the following benefits: (i) (ii) An annuity-due of R per year payable while both are alive, reducing to 0.6R per year after the death of Mr. Peters as long as Mrs. Peters is alive, and reducing to 0.7R per year after the death of Mrs. Peters as long as Mr. Peters is alive. A life insurance benefit of 100,000 payable at the end of the year of death of Mr. Peters, whether or not Mrs. Peters is alive. You are given: (i) (ii) Future lifetimes are independent. The mortality of each life follows the Illustrative Life Table. (iii) i = 0.06 (iv) The single net premium for this contract is 1,000,000. Calculate R. (A) 53,000 (B) 66,500 (C) 73,000 (D) 82,000 (E) 91,000 Exam MLC: Fall GO ON TO NEXT PAGE

15 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

16 7. For a fully discrete whole life insurance of 1 on (50), you are given: (i) (ii) Expenses of 0.20 at the start of the first year and 0.01 at the start of each renewal year are incurred. Mortality follows the Illustrative Life Table. (iii) i = 0.06 Gross premiums are determined using the equivalence principle. Calculate the variance of L, 0 the gross loss-at-issue random variable. (A) (B) (C) (D) (E) Exam MLC: Fall GO ON TO NEXT PAGE

17 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

18 8. For a special fully discrete 3-year term insurance on (75), you are given: (i) The death benefit during the first two years is the sum of the net premiums paid without interest. (ii) The death benefit in the third year is 10,000. (iii) p x x (iv) i = 0.04 Calculate the annual net premium. (A) 449 (B) 459 (C) 469 (D) 479 (E) 489 Exam MLC: Fall GO ON TO NEXT PAGE

19 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

20 9. For a special fully discrete 15-year endowment insurance on (75), you are given: (i) The death benefit is (ii) The endowment benefit is the sum of the net premiums paid without interest. (iii) d = 0.04 (iv) A = :15 1 (v) A = :15 Calculate the annual net premium. (A) 80 (B) 90 (C) 100 (D) 110 (E) 120 Exam MLC: Fall GO ON TO NEXT PAGE

21 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

22 10. For a whole life insurance of 100,000 on (45) with premiums payable monthly for a period of 20 years, you are given: (i) (ii) (iii) The death benefit is paid immediately upon death. Mortality follows the Illustrative Life Table. Deaths are uniformly distributed over each year of age. (iv) i = 0.06 Calculate the monthly net premium. (A) 150 (B) 152 (C) 154 (D) 156 (E) 158 Exam MLC: Fall GO ON TO NEXT PAGE

23 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

24 11. For fully discrete 30-payment whole life insurance policies on (x), you are given: (i) The following expenses payable at the beginning of the year: 1 st Year Years Years Years and after Per policy Percent of premium 80% 20% 10% 0% (ii) a = x (iii) a = :15 x (iv) a = :30 x (v) Annual gross premiums are calculated using the equivalence principle. (vi) The annual gross premium is expressed as kf + h, where F is the death benefit and k and h are constants for all F. Calculate h. (A) 30.3 (B) 35.1 (C) 39.9 (D) 44.7 (E) 49.5 Exam MLC: Fall GO ON TO NEXT PAGE

25 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

26 12. You are analyzing the sensitivity of some of the assumptions used in setting the premium rate for a sickness policy. You are basing your calculations on a multiple state model as diagrammed below: Healthy Sick Dead You are given: (i) (ii) (iii) Level premiums are paid continuously by Healthy policyholders. Level sickness benefits are paid continuously to Sick policyholders. There is no death benefit. Which one of the following changes to the assumptions will be certain to increase the premium rate? (A) (B) (C) (D) (E) A lower rate of interest and a higher recovery rate from the Sick state. A lower mortality rate for those in the Healthy state and a lower mortality rate for those in the Sick state. A higher mortality rate for those in the Healthy state and a higher mortality rate for those in the Sick state. A lower recovery rate from the Sick state and a lower mortality rate for those in the Sick state. A higher rate of interest and a lower mortality rate for those in the Healthy state. Exam MLC: Fall GO ON TO NEXT PAGE

27 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

28 13. For a fully discrete whole life insurance of 100,000 on (45), you are given: (i) The gross premium reserve at duration 5 is 5500 and at duration 6 is (ii) q 50 = (iii) i = 0.05 (iv) Renewal expenses at the start of each year are 50 plus 4% of the gross premium. (v) Claim expenses are 200. Calculate the gross premium. (A) 2200 (B) 2250 (C) 2300 (D) 2350 (E) 2400 Exam MLC: Fall GO ON TO NEXT PAGE

29 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

30 14. For a fully discrete whole life insurance of 100 on (x), you are given: (i) q x + 15 = 0.10 (ii) Deaths are uniformly distributed over each year of age. (iii) i = 0.05 (iv) tv denotes the benefit reserve at time t. (v) 16V = Calculate 15.6 V. (A) 49.7 (B) 50.0 (C) 50.3 (D) 50.6 (E) 50.9 Exam MLC: Fall GO ON TO NEXT PAGE

31 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

32 15. For a fully discrete 5-payment whole life insurance of 1000 on (80), you are given: (i) The gross premium is 130. (ii) q80+ k = 0.01( k + 1), k = 0,1, 2,..,5 (iii) v = 0.95 (iv) 1000A 86 = 683 (v) 3 L is the prospective loss random variable at time 3, based on the gross premium. Calculate E [ 3 L ]. (A) 330 (B) 350 (C) 360 (D) 380 (E) 390 Exam MLC: Fall GO ON TO NEXT PAGE

33 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

34 16. For a fully discrete whole life insurance of 1 on (x), you are given: (i) q x + 10 = (ii) 2 v = (iii) A x + 11 = (iv) 2 A x + 11 = (v) k L is the prospective loss random variable at time k. Calculate Var Var ( 10 L) ( L) 11. (A) (B) (C) (D) (E) Exam MLC: Fall GO ON TO NEXT PAGE

35 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

36 17. An insurer issues identical fully discrete 10-year term insurance policies of 100,000 to independent lives, all age 60. The annual premium is 1500 for each policy. You are given the following profit test assumptions: For each policy, expenses of 100 are incurred at the beginning of each of the first 2 years. q = 0.010, q = The reserve for a policy in force at the end of the first year is 400, and for a policy in force at the end of the second year is 700. i = Calculate the expected profit emerging at the end of the second year, per policy in force at the start of the second year. (A) 30 (B) 38 (C) 46 (D) 54 (E) 62 Exam MLC: Fall GO ON TO NEXT PAGE

37 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

38 18. For a Type B universal life insurance policy, you are given: Policy Year Percent of Premium Charge Annual Expense Charge Additional Death Benefit Annual Cost of Insurance (COI) Rate Annual Discount Rate for COI Annual Credited Interest Rate Annual Premium % , % 4.5% % 50 95, % 5.2% Calculate the account value at the end of year 2. (A) 5160 (B) 5180 (C) 5200 (D) 5220 (E) 5240 Exam MLC: Fall GO ON TO NEXT PAGE

39 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

40 19. For a career average pension plan with one participant currently exact age 55, you are given: (i) (ii) The annual retirement benefit is 1.75% of total earnings during the participant s years of service. Retirement occurs at age 65. The retirement benefit is payable monthly with the first payment at age 65. Current salary after the increase at age 55 is 50,000. Future salary increases will be 3% and will occur on each birthday. (iii) Total past earnings are 525,000. (iv) i = 0.04 (v) (12) a = (vi) Death is the only decrement before age 65. (vii) 10 p 55 = Calculate the actuarial present value at the participant s current age of the annual retirement benefit. (A) 129,300 (B) 134,800 (C) 140,300 (D) 145,800 (E) 151,300 Exam MLC: Fall GO ON TO NEXT PAGE

41 USE THIS PAGE FOR YOUR SCRATCH WORK EXTRA BLANK PAPER IS PROVIDED AT THE END OF THE EXAM BOOK Exam MLC: Fall GO ON TO NEXT PAGE

42 20. For a mortality table with a select period of two years, you are given: (i) (ii) x q [ x] [ x] 1 q + x 2 q + x The force of mortality is constant between integral ages. Calculate q [ 50] (A) 15.2 (B) 16.4 (C) 17.7 (D) 19.0 (E) 20.2 Exam MLC: Fall GO ON TO NEXT PAGE

43 Exam MLC SECTION B Written-Answer Exam MLC: Fall GO ON TO NEXT PAGE

44 1. (12 points) For a fully discrete whole life insurance of 100,000 on (35), you are given: (i) (ii) (iii) Policy Year Commission Rate Per Policy Expenses 1 100% % 8 11 and later 0 8 Per policy expenses are incurred at the beginning of each policy year. Mortality follows the Illustrative Life Table. (iv) i = 0.06 (v) (vi) (vii) G is the gross premium. G is calculated such that the actuarial present value of gross premiums is 110% of the actuarial present value of benefits, commissions, and per policy expenses. C is the random variable for the present value at issue of commissions. (a) (2 points) EC [ ] can be written in the form kg. Calculate k. (b) (c) (1 point) Calculate the expected present value at issue of per policy expenses. (2 points) Show that G is 1060, rounded to the nearest 10. You should calculate G to the nearest 1. (d) (3 points) Calculate Var( C ). The company charges a premium of 95 per month for a policyholder who wishes to pay monthly. Commission rates during renewal policy years are as given above for annual premium policies. * C is the present value at issue of commissions for a policy paying premiums monthly. Exam MLC: Fall GO ON TO NEXT PAGE

45 1. Continued (e) (3 points) Using the 2-term Woolhouse formula, calculate the commission rate during the first policy year for a policy paying premiums monthly, so that Ε [ C ] =Ε [ C]. Your company also sells a Type A universal life policy of 100,000 on (35) with commission rates the same as those on the whole life policy. (f) (1 point) Identify a characteristic of universal life which would suggest that commissions paid on the universal life policy would be more variable than commissions paid on the whole life policy. Support your answer. Exam MLC: Fall GO ON TO NEXT PAGE

46 2. (9 points) XYZ Company sponsors a defined benefit pension plan. Key attributes of the pension plan are: Benefits are payable as a single life annuity, paid at the beginning of each month. The annual benefit payable at age 65 is calculated as 2% of the final 3-year average salary up to 100,000 multiplied by years of service, plus 3% of the final 3-year average salary over 100,000 multiplied by years of service. The monthly benefit is the annual benefit divided by 12. You are given: (i) i = 5% (ii) (12) A 65 = Tom joined XYZ Company on January 1, 2009 at exact age 40. He is planning to retire on January 1, 2034 at age 65. Tom s salary for 2014 is 80,000 and he receives annual raises of 4% each January 1. (a) (2 points) Assuming Tom works until age 65, show that the monthly accrued benefit at age 65 is 8050, to the nearest 10. You should calculate the benefit to the nearest 1. (b) (1 point) Calculate the replacement ratio at age 65. (c) (2 points) Calculate the expected present value of the benefit at age 65. In addition to accruing a benefit in the defined benefit plan, Tom will contribute a constant percentage of his annual salary to a defined contribution plan. You are also given: (iii) Contributions are made at the start of each year, beginning on January 1, (iv) (v) Contributions earn an investment return of 7% per year. The contribution balance is converted to a monthly life annuity using assumptions (i) and (ii) above. (d) (4 points) Calculate the percentage of each year s salary that Tom needs to contribute so that his total replacement ratio is 80% at age 65. Exam MLC: Fall GO ON TO NEXT PAGE

47 3. (6 points) For (x) and (y) with independent future lifetimes, you are given that q x = 0.2 and q y = 0.1. (a) (1 point) Explain in words the meaning of the probability described by the symbol q xy. You are also given that mortality within integral ages follows a uniform distribution of deaths assumption for each of (x) and (y) individually. (b) (2 points) Sketch the graph of s p x as a function of s for 0 s 1. You should mark numerical values on each axis. (c) (3 points) Show that sqxy = sqxy + g() s q for 0 s 1, where gs () is a function xy of s that you should specify. Exam MLC: Fall GO ON TO NEXT PAGE

48 4. (8 points) A long-term care provider offers three care levels. Transitions are modeled as a Markov multiple-state model. Transitions and states are shown in the following diagram: Level 0 Care 0 Level 1 Care 1 Level 2 Care 2 Dead 3 (a) (1 point) Write down Kolmogorov s forward differential equations with the associated boundary conditions for this model for: (i) t p 10 x (ii) t p 11 x (b) 10 (3 points) Estimate 1 p 80 using Euler s forward method, a step size of h = 1 3, and the transition intensities and probabilities in the table below: t p t 80 μ 80 + t 03 μ 80 + t 10 μ 80 + t 12 μ 80 + t 13 μ 80 + t 23 μ 80 + t (c) (4 points) You are given the following annuity values at 5%: x 00 a x 01 a x 02 a x a x 10 a x 12 a x 22 a x Exam MLC: Fall GO ON TO NEXT PAGE

49 4. Continued You are also given: p80 = p80 = p80 = i = 0.05 Residents pay a service fee of 8000 per year continuously while in Level 0 or Level 1 Care. Level 2 Care costs are paid at a continuous rate of 30,000 per year for lives age 80 to 85, and 40,000 per year for lives older than 85. Ada, who is age 80, is currently in Level 0 Care. (i) (ii) Calculate the expected present value of Ada s future Level 0 and Level 1 service fees. Calculate the expected present value of Ada s future Level 2 Care costs. Exam MLC: Fall GO ON TO NEXT PAGE

50 5. (11 points) For a special 3-year term life insurance issued to (50) with a premium refund feature, you are given: (i) The death benefit is 100,000. (ii) (iii) The premium refund feature refunds the last premium payment, without interest, at the end of the 3-year term if the insured is still alive. Mortality follows the Illustrative Life Table. (iv) Pre-contract expenses are 155. (v) Commissions are 5% of each premium. (vi) The hurdle rate is 14%. (vii) The reserves of this policy have been set to: t (viii) The annual premium for this policy is t V (ix) The earned interest rates are: Year 1 Year 2 Year (a) (b) (c) (1 point) Show that the expected profit in policy year 2 for a policy in force at the start of year 2 is 37 to the nearest 1. You should calculate your answer to the nearest (4 points) Calculate the profit vector of this policy. (3 points) Calculate the profit signature and Net Present Value (NPV) of this policy. Exam MLC: Fall GO ON TO NEXT PAGE

51 5. Continued (d) (3 points) Rank from low to high the Internal Rate of Return (IRR) of the following products, explaining your order. Product A: Product B: Product C: The special 3-year term life insurance described above. A 3-year term life insurance policy with the following profit 155, 0, 0, 210 signature: [ ] The same special 3-year term life insurance as Product A, except that the reserves of the product have been set to: t t V Exam MLC: Fall GO ON TO NEXT PAGE

52 6. (10 points) For a Type A universal life policy issued to (50), you are given: (i) The face amount is 100,000. (ii) All cash flows occur at policy anniversaries. (iii) The policyholder pays an initial premium of 15,000. (iv) (v) The cost of insurance (COI) is calculated based on 120% of the mortality in the Illustrative Life Table. The interest rate for discounting the net q amount at risk, i, is The expense charge is 1% of premium. (vi) The credited interest rate for policy year 1 is 5%. (vii) The corridor factor in year 1 is 2.2. (viii) The surrender charge in policy year 1 is 5% of the premium paid. (a) (b) (1 point) Explain why the corridor factor requirement exists for universal life insurance. (5 points) (i) (ii) Calculate the COI in policy year 1 assuming there is no corridor factor requirement. Calculate the COI in policy year 1 based only on the corridor factor (as if the face amount were 0). (iii) Determine the COI in policy year 1. (c) (2 points) Calculate the account value, additional death benefit and cash surrender value at the end of policy year 1. Exam MLC: Fall GO ON TO NEXT PAGE

53 6. Continued (d) (2 points) The policy contains a no-lapse guarantee providing term insurance of 100,000 until age 70. If the expected present value of the guaranteed insurance coverage is greater than the account value, the company holds a reserve for the no-lapse guarantee equal to the difference. The expected present value is based on the Illustrative Life Table at 6% interest and no expenses. The account value at the end of policy year 10 is 10,000. Calculate the reserve for the no-lapse guarantee at the end of year 10. **END OF EXAMINATION** Exam MLC: Fall STOP

SOCIETY OF ACTUARIES. EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE WRITTEN-ANSWER QUESTIONS AND SOLUTIONS

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