SOCIETY OF ACTUARIES EXAM M ACTUARIAL MODELS EXAM M SAMPLE QUESTIONS

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1 SOCIETY OF ACTUARIES EXAM M ACTUARIAL MODELS EXAM M SAMPLE QUESTIONS Copyright 5 by the Society of Actuaries Some of the questions in this study note are taken from past SOA examinations. M-9-5 PRINTED IN U.S.A.

2 . For two independent lives now age 3 and 34, you are given: x q x Calculate the probability that the last death of these two lives will occur during the 3 rd year from now (i.e. q 3: 34 ). (A). (B).3 (C).4 (D).8 (E).4

3 . For a whole life insurance of on (x) with benefits payable at the moment of death: (i).4, < t δt =.5, < t.6, < t (ii) µ x () t =.7, < t Calculate the single benefit premium for this insurance. (A) 379 (B) 4 (C) 444 (D) 59 (E) A health plan implements an incentive to physicians to control hospitalization under which the physicians will be paid a bonus B equal to c times the amount by which total hospital c. claims are under 4 ( ) The effect the incentive plan will have on underlying hospital claims is modeled by assuming that the new total hospital claims will follow a two-parameter Pareto distribution with α = and θ = 3. E( B ) = Calculate c. (A).44 (B).48 (C).5 (D).56 (E).6 3

4 4. Computer maintenance costs for a department are modeled as follows: (i) The distribution of the number of maintenance calls each machine will need in a year is Poisson with mean 3. (ii) The cost for a maintenance call has mean 8 and standard deviation. (iii) The number of maintenance calls and the costs of the maintenance calls are all mutually independent. The department must buy a maintenance contract to cover repairs if there is at least a % probability that aggregate maintenance costs in a given year will exceed % of the expected costs. Using the normal approximation for the distribution of the aggregate maintenance costs, calculate the minimum number of computers needed to avoid purchasing a maintenance contract. (A) 8 (B) 9 (C) (D) (E) 4

5 5. A whole life policy provides that upon accidental death as a passenger on an airplane a benefit of,, will be paid. If death occurs from other accidental causes, a death benefit of 5, will be paid. If death occurs from a cause other than an accident, a death benefit of 5, will be paid. You are given: (i) (ii) (iii) (iv) Death benefits are payable at the moment of death. () µ = /,, where () indicates accidental death as a passenger on an airplane. ( ) µ = / 5, where () indicates death from other accidental causes. ( 3) µ = /, where (3) indicates non-accidental death. (v) δ =.6 Calculate the single benefit premium for this insurance. (A) 45 (B) 46 (C) 47 (D) 48 (E) 49 5

6 6. For a special fully discrete whole life insurance of on (4): (i) The level benefit premium for each of the first years is π. (ii) The benefit premium payable thereafter at age x is vq x, x = 6, 6, 6, (iii) Mortality follows the Illustrative Life Table. (iv) i =.6 Calculate π. (A) 4.79 (B) 5. (C) 5.34 (D) 5.75 (E) For an annuity payable semiannually, you are given: (i) Deaths are uniformly distributed over each year of age. (ii) q 69 =.3 (iii) i =.6 (iv) A 7 = 53 Calculate ( ) a. 69 (A) 8.35 (B) 8.47 (C) 8.59 (D) 8.7 (E)

7 8. For a sequence, uk ( ) is defined by the following recursion formula k (i) ( ) α k (ii) β ( k ) ( ) β( ) ( ) uk ( ) = α k + k u k for k =,, 3, q = p + i = p k (iii) u ( 7) =. k Which of the following is equal to u ( 4)? (A) A 3 (B) A 4 (C) A 4:3 (D) A 4:3 (E) A 4:3 7

8 9. Subway trains arrive at a station at a Poisson rate of per hour. 5% of the trains are express and 75% are local. The type of each train is independent of the types of preceding trains. An express gets you to the stop for work in 6 minutes and a local gets you there in 8 minutes. You always take the first train to arrive. Your co-worker always takes the first express. You both are waiting at the same station. Calculate the probability that the train you take will arrive at the stop for work before the train your co-worker takes. (A).8 (B).37 (C).5 (D).56 (E).75. For a fully discrete whole life insurance of on (4), the contract premium is the level annual benefit premium based on the mortality assumption at issue. At time, the actuary decides to increase the mortality rates for ages 5 and higher. You are given: (i) d =.5 (ii) Mortality assumptions: At issue 4 =., k =,,,...,49 k q Revised prospectively at time k q 5 =.4, k =,,,...,4 (iii) L is the prospective loss random variable at time using the contract premium. Calculate E[ LK(4) ] using the revised mortality assumption. 8

9 (A) Less than 5 (B) At least 5, but less than 5 (C) At least 5, but less than 75 (D) At least 75, but less than 3 (E) At least 3. For a group of individuals all age x, of which 3% are smokers and 7% are non-smokers, you are given: (i) δ =. (ii) (iii) (iv) (v) (vi) smoker A x =.444 non-smoker A x =.86 T is the future lifetime of (x). smoker Var a = 8.88 T non-smoker Var a = 8.53 T Calculate Var a T for an individual chosen at random from this group. (A) 8.5 (B) 8.6 (C) 8.8 (D) 9. (E) 9. 9

10 . T, the future lifetime of (), has a spliced distribution. (i) f () t follows the Illustrative Life Table. (ii) f () t follows DeMoivre s law with ω =. () kf t, t 5 (iii) ft () t =. f () t, 5 < t Calculate p 4. (A).8 (B).85 (C).88 (D).9 (E).96

11 3. A population has 3% who are smokers with a constant force of mortality. and 7% who are non-smokers with a constant force of mortality.. Calculate the 75 th percentile of the distribution of the future lifetime of an individual selected at random from this population. (A).7 (B). (C). (D).6 (E).8 4. Aggregate losses for a portfolio of policies are modeled as follows: (i) (ii) The number of losses before any coverage modifications follows a Poisson distribution with mean λ. The severity of each loss before any coverage modifications is uniformly distributed between and b. The insurer would like to model the impact of imposing an ordinary deductible, d( d b) on each loss and reimbursing only a percentage, c ( < c ), of each loss in excess of the deductible. It is assumed that the coverage modifications will not affect the loss distribution. The insurer models its claims with modified frequency and severity distributions. The modified claim amount is uniformly distributed on the interval,c( b d). Determine the mean of the modified frequency distribution. < <,

12 (A) (B) (C) (D) (E) λ λ c d λ b b d λ b b d λc b 5. The RIP Life Insurance Company specializes in selling a fully discrete whole life insurance of, to 65 year olds by telephone. For each policy: (i) The annual contract premium is 5. (ii) Mortality follows the Illustrative Life Table. (iii) i =.6 The number of telephone inquiries RIP receives follows a Poisson process with mean 5 per day. % of the inquiries result in the sale of a policy. The number of inquiries and the future lifetimes of all the insureds who purchase policies on a particular day are independent. Using the normal approximation, calculate the probability that S, the total prospective loss at issue for all the policies sold on a particular day, will be less than zero. (A).33 (B).5 (C).67 (D).84 (E).99

13 6. For a special fully discrete whole life insurance on (4): (i) The death benefit is for the first years; 5 for the next 5 years; thereafter. (ii) The annual benefit premium is P 4 for the first years; 5P 4 for the next 5 years; π thereafter. (iii) Mortality follows the Illustrative Life Table. (iv) i =.6 Calculate V, the benefit reserve at the end of year for this insurance. (A) 55 (B) 59 (C) 63 (D) 67 (E) 7 3

14 7. For a whole life insurance of on (4) with death benefit payable at the end of year of death, you are given: (i) i =.5 (ii) p 4 =.997 (iii) A4 A4 =.8 (iv) (v) A4 A4 =.433 Z is the present-value random variable for this insurance. Calculate Var(Z). (A).3 (B).4 (C).5 (D).6 (E).7 4

15 8. For a perpetuity-immediate with annual payments of : (i) The sequence of annual discount factors follows a Markov chain with the following three states: State number Annual discount factor, v (ii) The transition matrix for the annual discount factors is: Y is the present value of the perpetuity payments when the initial state is. Calculate E(Y). (A) 5.67 (B) 5.7 (C) 5.75 (D) 6.8 (E)

16 9. A member of a high school math team is practicing for a contest. Her advisor has given her three practice problems: #, #, and #3. She randomly chooses one of the problems, and works on it until she solves it. Then she randomly chooses one of the remaining unsolved problems, and works on it until solved. Then she works on the last unsolved problem. She solves problems at a Poisson rate of problem per 5 minutes. Calculate the probability that she has solved problem #3 within minutes of starting the problems. (A).8 (B).34 (C).45 (D).5 (E).59. For a double decrement table, you are given: (i) (ii) (iii) () ( τ ) x x µ () t =. µ (), t t > ( τ ) µ x () t = k t, t > q ' =.4 () x Calculate (A).45 (B).53 (C).58 (D).64 (E).73 () q x. 6

17 . For (x): (i) K is the curtate future lifetime random variable. (ii) q =.( k+ ), k =,,,, 9 x+ k Calculate Var( K 3). (A). (B). (C).3 (D).4 (E).5. The graph of the density function for losses is: f(x ) Loss amount, x Calculate the loss elimination ratio for an ordinary deductible of. (A). (B).4 (C).8 (D).3 (E).36 7

18 3. Michel, age 45, is expected to experience higher than standard mortality only at age 64. For a special fully discrete whole life insurance of on Michel, you are given: (i) The benefit premiums are not level. (ii) The benefit premium for year, π 9, exceeds P 45 for a standard risk by.. (iii) Benefit reserves on his insurance are the same as benefit reserves for a fully discrete whole life insurance of on (45) with standard mortality and level benefit premiums. (iv) i =.3 (v) V 45 =.47 Calculate the excess of q 64 for Michel over the standard q 64. (A). (B).4 (C).6 (D).8 (E). 8

19 4. For a block of fully discrete whole life insurances of on independent lives age x, you are given: (i) i =.6 (ii) A x =.495 (iii) A =.9476 x (iv) π =.5, where π is the contract premium for each policy. (v) Losses are based on the contract premium. Using the normal approximation, calculate the minimum number of policies the insurer must issue so that the probability of a positive total loss on the policies issued is less than or equal to.5. (A) 5 (B) 7 (C) 9 (D) 3 (E) 33 9

20 5. Your company currently offers a whole life annuity product that pays the annuitant, at the beginning of each year. A member of your product development team suggests enhancing the product by adding a death benefit that will be paid at the end of the year of death. Using a discount rate, d, of 8%, calculate the death benefit that minimizes the variance of the present value random variable of the new product. (A) (B) 5, (C), (D) 5, (E), 6. A towing company provides all towing services to members of the City Automobile Club. You are given: Towing Distance Towing Cost Frequency miles 8 5% miles 4% 3+ miles 6 % (i) (ii) (iii) The automobile owner must pay % of the cost and the remainder is paid by the City Automobile Club. The number of towings has a Poisson distribution with mean of per year. The number of towings and the costs of individual towings are all mutually independent. Using the normal approximation for the distribution of aggregate towing costs, calculate the probability that the City Automobile Club pays more than 9, in any given year. (A) 3% (B) % (C) 5% (D) 9% (E) 97%

21 7. You are given: (i) Losses follow an exponential distribution with the same mean in all years. (ii) The loss elimination ratio this year is 7%. (iii) The ordinary deductible for the coming year is 4/3 of the current deductible. Compute the loss elimination ratio for the coming year. (A) 7% (B) 75% (C) 8% (D) 85% (E) 9% 8. For T, the future lifetime random variable for (): (i) ω > 7 (ii) 4 p =.6 (iii) E(T) = 6 (iv) E( T t) = t.5 t, < t < 6 Calculate the complete expectation of life at 4. (A) 3 (B) 35 (C) 4 (D) 45 (E) 5

22 9. Two actuaries use the same mortality table to price a fully discrete -year endowment insurance of on (x). (i) Kevin calculates non-level benefit premiums of 68 for the first year and 35 for the second year. (ii) Kira calculates level annual benefit premiums of π. (iii) d =.5 Calculateπ. (A) 48 (B) 489 (C) 497 (D) 58 (E) 57

23 3. For a fully discrete -payment whole life insurance of, on (x), you are given: (i) i =.5 (ii) q x + 9 =. (iii) q x + =. (iv) q x + =.4 (v) The level annual benefit premium is 78. (vi) The benefit reserve at the end of year 9 is 3,535. Calculate,A x+. (A) 34, (B) 34,3 (C) 35,5 (D) 36,5 (E) 36,7 3

24 3. You are given: (i) Mortality follows DeMoivre s law with ω = 5. (ii) (45) and (65) have independent future lifetimes. Calculate e. 45:65 (A) 33 (B) 34 (C) 35 (D) 36 (E) 37 4

25 3. Given: The survival function sx Calculate µ b4g. sx b g =, x< bg b g = x di { } b g, where sx = e /, x< 4. 5 sx, 4.5 x (A).45 (B).55 (C).8 (D). (E). 33. For a triple decrement table, you are given: (i) µ x t (ii) µ x t (iii) µ x t Calculate qbg x. (A).6 (B).3 (C).33 (D).36 (E).39 bg b g = 3., t > bg b g = 5., t > bg 3 b g = 7., t > 5

26 34. You are given: (i) the following select-and-ultimate mortality table with 3-year select period: (ii) i = 3. x q x q x + q x + q x+3 x Calculate A 6 on 6., the actuarial present value of a -year deferred -year term insurance (A).56 (B).6 (C).86 (D).9 (E).95 6

27 35. You are given: (i) (ii) µ x btg = t µ x btg =., 5 t., < 5 (iii) δ = 6. Calculate a x. (A).5 (B) 3. (C) 3.4 (D) 3.9 (E) Actuaries have modeled auto windshield claim frequencies. They have concluded that the number of windshield claims filed per year per driver follows the Poisson distribution with parameter λ, where λ follows the gamma distribution with mean 3 and variance 3. Calculate the probability that a driver selected at random will file no more than windshield claim next year. (A).5 (B).9 (C). (D).4 (E).3 7

28 37. The number of auto vandalism claims reported per month at Sunny Daze Insurance Company (SDIC) has mean and variance 75. Individual losses have mean and standard deviation 7. The number of claims and the amounts of individual losses are independent. Using the normal approximation, calculate the probability that SDIC s aggregate auto vandalism losses reported for a month will be less than,. (A).4 (B).3 (C).36 (D).39 (E) For an allosaur with, calories stored at the start of a day: (i) (ii) (iii) (iv) The allosaur uses calories uniformly at a rate of 5, per day. If his stored calories reach, he dies. Each day, the allosaur eats scientist (, calories) with probability.45 and no scientist with probability.55. The allosaur eats only scientists. The allosaur can store calories without limit until needed. Calculate the probability that the allosaur ever has 5, or more calories stored. (A).54 (B).57 (C).6 (D).63 (E).66 8

29 39. Lucky Tom finds coins on his way to work at a Poisson rate of.5 coins/minute. The denominations are randomly distributed: (i) (ii) (iii) 6% of the coins are worth each % of the coins are worth 5 each % of the coins are worth each. Calculate the probability that in the first ten minutes of his walk he finds at least coins worth each, and in the first twenty minutes finds at least 3 coins worth each. (A).8 (B). (C).6 (D). (E).4 9

30 4. For a fully discrete whole life insurance of on (6), the annual benefit premium was calculated using the following: (i) i = 6. (ii) q 6 =. 376 (iii) A 6 = (iv) A 6 = 383. A particular insured is expected to experience a first-year mortality rate ten times the rate used to calculate the annual benefit premium. The expected mortality rates for all other years are the ones originally used. Calculate the expected loss at issue for this insured, based on the original benefit premium. (A) 7 (B) 86 (C) (D) 4 (E) 8 3

31 4. For a fully discrete whole life insurance of on (4), you are given: (i) i = 6. (ii) Mortality follows the Illustrative Life Table. (iii) a 4 = 77. (iv) a 5 = 757. : : 4 (v) A : = 6. At the end of the tenth year, the insured elects an option to retain the coverage of for life, but pay premiums for the next ten years only. Calculate the revised annual benefit premium for the next years. (A) (B) 5 (C) 7 (D) 9 (E) 3

32 4. For a double-decrement table where cause is death and cause is withdrawal, you are given: (i) (ii) (iii) Deaths are uniformly distributed over each year of age in the single-decrement table. Withdrawals occur only at the end of each year of age. lbg τ x = (iv) qbg x = 4. bg x x (v) d = 45. d bg Calculate p x bg. (A).5 (B).53 (C).55 (D).57 (E).59 3

33 43. You intend to hire employees for a new management-training program. To predict the number who will complete the program, you build a multiple decrement table. You decide that the following associated single decrement assumptions are appropriate: (i) (ii) (iii) (iv) Of 4 hires, the number who fail to make adequate progress in each of the first three years is, 6, and 8, respectively. Of 3 hires, the number who resign from the company in each of the first three years is 6, 8, and, respectively. Of hires, the number who leave the program for other reasons in each of the first three years is,, and 4, respectively. You use the uniform distribution of decrements assumption in each year in the multiple decrement table. Calculate the expected number who fail to make adequate progress in the third year. (A) 4 (B) 8 (C) (D) 4 (E) 7 33

34 44. Bob is an overworked underwriter. Applications arrive at his desk at a Poisson rate of 6 per day. Each application has a /3 chance of being a bad risk and a /3 chance of being a good risk. Since Bob is overworked, each time he gets an application he flips a fair coin. If it comes up heads, he accepts the application without looking at it. If the coin comes up tails, he accepts the application if and only if it is a good risk. The expected profit on a good risk is 3 with variance,. The expected profit on a bad risk is with variance 9,. Calculate the variance of the profit on the applications he accepts today. (A) 4,, (B) 4,5, (C) 5,, (D) 5,5, (E) 6,, 45. Prescription drug losses, S, are modeled assuming the number of claims has a geometric distribution with mean 4, and the amount of each prescription is 4. Calculate EbS g. + (A) 6 (B) 8 (C) 9 (D) 4 (E) 46 34

35 46. For a temporary life annuity-immediate on independent lives (3) and (4): (i) Mortality follows the Illustrative Life Table. (ii) i = 6. Calculate a 3: 4:. (A) 6.64 (B) 7.7 (C) 7.88 (D) 8.74 (E) For a special whole life insurance on (35), you are given: (i) (ii) (iii) The annual benefit premium is payable at the beginning of each year. The death benefit is equal to plus the return of all benefit premiums paid in the past without interest. The death benefit is paid at the end of the year of death. (iv) A 35 = (v) biag 35 = (vi) i = 5. Calculate the annual benefit premium for this insurance. (A) (B) 75.8 (C) 77.4 (D) (E)

36 48. Subway trains arrive at a station at a Poisson rate of per hour. 5% of the trains are express and 75% are local. The types of each train are independent. An express gets you to work in 6 minutes and a local gets you there in 8 minutes. You always take the first train to arrive. Your co-worker always takes the first express. You both are waiting at the same station. Which of the following is true? (A) (B) (C) (D) (E) Your expected arrival time is 6 minutes earlier than your co-worker s. Your expected arrival time is 4.5 minutes earlier than your co-worker s. Your expected arrival times are the same. Your expected arrival time is 4.5 minutes later than your co-worker s. Your expected arrival time is 6 minutes later than your co-worker s. 49. For a special fully continuous whole life insurance of on the last-survivor of (x) and (y), you are given: (i) b g and Tbyg are independent. xbtg = ybtg = 7., t > Tx (ii) µ µ (iii) δ = 5. (iv) Premiums are payable until the first death. Calculate the level annual benefit premium for this insurance. (A).4 (B).7 (C).8 (D). (E).4 36

37 5. For a fully discrete whole life insurance of on (), you are given: (i) P = (ii) V = 49 (iii) V = 545 (iv) V = 65 (v) q 4 =. Calculate q 4. (A).4 (B).5 (C).6 (D).7 (E).8 5. For a fully discrete whole life insurance of on (6), you are given: (i) i = 6. (ii) Mortality follows the Illustrative Life Table, except that there are extra mortality risks at age 6 such that q 6 5 =.. Calculate the annual benefit premium for this insurance. (A) 3.5 (B) 3. (C) 3. (D) 33. (E)

38 5. At the beginning of each round of a game of chance the player pays.5. The player then rolls one die with outcome N. The player then rolls N dice and wins an amount equal to the total of the numbers showing on the N dice. All dice have 6 sides and are fair. Using the normal approximation, calculate the probability that a player starting with 5, will have at least 5, after rounds. (A). (B).4 (C).6 (D).9 (E). 53. X is a discrete random variable with a probability function which is a member of the (a,b,) class of distributions. You are given: (i) P( X = )= P( X = )= 5. (ii) P( X = )= 875. Calculate PX= b 3g. (A). (B).5 (C).3 (D).35 (E).4 38

39 54. Nancy reviews the interest rates each year for a 3-year fixed mortgage issued on July. She models interest rate behavior by a Markov model assuming: (i) (ii) Interest rates always change between years. The change in any given year is dependent on the change in prior years as follows: from year t 3 to year t from year t to year t Probability that year t will increase from year t Increase Increase. Decrease Decrease. Increase Decrease.4 Decrease Increase.5 She notes that interest rates decreased from year to and from year to. Calculate the probability that interest rates will decrease from year 3 to 4. (A).76 (B).79 (C).8 (D).84 (E).87 39

40 55. For a -year deferred whole life annuity-due of per year on (45), you are given: (i) Mortality follows De Moivre s law with ω = 5. (ii) i = Calculate the probability that the sum of the annuity payments actually made will exceed the actuarial present value at issue of the annuity. (A).45 (B).45 (C).475 (D).5 (E) For a continuously increasing whole life insurance on bxg, you are given: (i) The force of mortality is constant. (ii) δ = 6. (iii) A x = 5. Calculate diai. x (A).889 (B) 3.5 (C) 4. (D) (E) 5.5 4

41 57. XYZ Co. has just purchased two new tools with independent future lifetimes. Each tool has its own distinct De Moivre survival pattern. One tool has a -year maximum lifetime and the other a 7-year maximum lifetime. Calculate the expected time until both tools have failed. (A) 5. (B) 5. (C) 5.4 (D) 5.6 (E) XYZ Paper Mill purchases a 5-year special insurance paying a benefit in the event its machine breaks down. If the cause is minor (), only a repair is needed. If the cause is major (), the machine must be replaced. Given: (i) (ii) The benefit for cause () is payable at the moment of breakdown. The benefit for cause () is 5, payable at the moment of breakdown. (iii) Once a benefit is paid, the insurance contract is terminated. (iv) µ bg b g (v) δ = 4. bg b g t =. and µ t =. 4, for t > Calculate the actuarial present value of this insurance. (A) 784 (B) 788 (C) 79 (D) 796 (E) 8 4

42 59. You are given: (i) R = e z (ii) S = e z µ bg x tdt c µ bg x t + k dt (iii) k is a constant such that S = 75. R h Determine an expression for k. b xg b xg c (A) n q /. 75q b xg b xg c (B) n. 75q / p b xg b xg c (C) n. 75p / p b xg b xg c (D) n p /. 75q b xg b xg c (E) n. 75q / q h h h h h 6. The number of claims in a period has a geometric distribution with mean 4. The amount of each claim X follows PX b = xg = 5., x = 34,,,. The number of claims and the claim amounts are independent. S is the aggregate claim amount in the period. b g. Calculate F s 3 (A).7 (B).9 (C).3 (D).33 (E).35 4

43 6. Insurance agent Hunt N. Quotum will receive no annual bonus if the ratio of incurred losses to earned premiums for his book of business is 6% or more for the year. If the ratio is less than 6%, Hunt s bonus will be a percentage of his earned premium equal to 5% of the difference between his ratio and 6%. Hunt s annual earned premium is 8,. Incurred losses are distributed according to the Pareto distribution, with θ = 5, and α =. Calculate the expected value of Hunt s bonus. (A) 3, (B) 7, (C) 4, (D) 9, (E) 35, 6. A large machine in the ABC Paper Mill is 5 years old when ABC purchases a 5-year term insurance paying a benefit in the event the machine breaks down. Given: (i) Annual benefit premiums of 6643 are payable at the beginning of the year. (ii) (iii) A benefit of 5, is payable at the moment of breakdown. Once a benefit is paid, the insurance contract is terminated. (iv) Machine breakdowns follow De Moivre s law with l x. (v) i = 6. Calculate the benefit reserve for this insurance at the end of the third year. (A) 9 (B) (C) 63 (D) 87 (E) 4 x = 43

44 63. For a whole life insurance of on x (i) The force of mortality is µ x t b g, you are given: b g. (ii) The benefits are payable at the moment of death. (iii) δ = 6. (iv) A x = 6. Calculate the revised actuarial present value of this insurance assuming µ x btg is increased by.3 for all t and δ is decreased by.3. (A).5 (B).6 (C).7 (D).8 (E) A maintenance contract on a hotel promises to replace burned out light bulbs at the end of each year for three years. The hotel has, light bulbs. The light bulbs are all new. If a replacement bulb burns out, it too will be replaced with a new bulb. You are given: (i) For new light bulbs, q =. q = 3. q = 5. (ii) Each light bulb costs. (iii) i = 5. Calculate the actuarial present value of this contract. (A) 67 (B) 7 (C) 73 (D) 76 (E) 8 44

45 65. You are given: R bg= S T µ x Calculate e 5: 5. (A) 4. (B) 4.4 (C) 4.8 (D) 5. (E) , < x < 4 5., x > For a select-and-ultimate mortality table with a 3-year select period: (i) x q x q x + q x + q x+3 x (ii) White was a newly selected life on //. (iii) White s age on // is 6. (iv) P is the probability on // that White will be alive on //6. Calculate P. (A) P <.43 (B).43 P <.45 (C).45 P <.47 (D).47 P <.49 (E).49 P. 45

46 67. For a continuous whole life annuity of on ( x ) : (i) T( x) is the future lifetime random variable for ( x ). (ii) The force of interest and force of mortality are constant and equal. (iii) a x =. 5 Calculate the standard deviation of a T( x). (A).67 (B).5 (C).89 (D) 6.5 (E) For a special fully discrete whole life insurance on (x): (i) (ii) The death benefit is in the first year and 5 thereafter. Level benefit premiums are payable for life. (iii) q x = 5. (iv) v = 9. (v) a x = 5. (vi) V x =. (vii) V is the benefit reserve at the end of year for this insurance. Calculate V. (A) 795 (B) (C) 9 (D) 8 (E) 5 46

47 69. For a fully discrete -year term insurance of on (x): (i).95 is the lowest premium such that there is a % chance of loss in year. (ii) p x =.75 (iii) p x+ =.8 (iv) Z is the random variable for the present value at issue of future benefits. b g. Calculate Var Z (A).5 (B).7 (C).9 (D). (E).3 7. A group dental policy has a negative binomial claim count distribution with mean 3 and variance 8. Ground-up severity is given by the following table: Severity Probability You expect severity to increase 5% with no change in frequency. You decide to impose a per claim deductible of. Calculate the expected total claim payment after these changes. (A) Less than 8, (B) At least 8,, but less than, (C) At least,, but less than, (D) At least,, but less than 4, (E) At least 4, 47

48 7. You own a fancy light bulb factory. Your workforce is a bit clumsy they keep dropping boxes of light bulbs. The boxes have varying numbers of light bulbs in them, and when dropped, the entire box is destroyed. You are given: Expected number of boxes dropped per month: 5 Variance of the number of boxes dropped per month: Expected value per box: Variance of the value per box: 4 You pay your employees a bonus if the value of light bulbs destroyed in a month is less than 8. Assuming independence and using the normal approximation, calculate the probability that you will pay your employees a bonus next month. (A).6 (B).9 (C).3 (D).7 (E).3 7. Each of independent lives purchase a single premium 5-year deferred whole life insurance of payable at the moment of death. You are given: (i) µ = 4. (ii) δ = 6. (iii) F is the aggregate amount the insurer receives from the lives. Using the normal approximation, calculate F such that the probability the insurer has sufficient funds to pay all claims is

49 (A) 8 (B) 39 (C) 5 (D) 6 (E) For a select-and-ultimate table with a -year select period: x p x p x + p x+ x Keith and Clive are independent lives, both age 5. Keith was selected at age 45 and Clive was selected at age 5. Calculate the probability that exactly one will be alive at the end of three years. (A) Less than.5 (B) At least.5, but less than.5 (C) At least.5, but less than.35 (D) At least.35, but less than.45 (E) At least.45 49

50 Use the following information for questions 74 and 75. For a tyrannosaur with, calories stored: (i) (ii) (iii) (iv) The tyrannosaur uses calories uniformly at a rate of, per day. If his stored calories reach, he dies. The tyrannosaur eats scientists (, calories each) at a Poisson rate of per day. The tyrannosaur eats only scientists. The tyrannosaur can store calories without limit until needed. 74. Calculate the probability that the tyrannosaur dies within the next.5 days. (A).3 (B).4 (C).5 (D).6 (E) Calculate the expected calories eaten in the next.5 days. (A) 7,8 (B) 8,8 (C) 9,8 (D),8 (E),8 5

51 76. A fund is established by collecting an amount P from each of independent lives age 7. The fund will pay the following benefits:, payable at the end of the year of death, for those who die before age 7, or P, payable at age 7, to those who survive. You are given: Mortality follows the Illustrative Life Table. (i) i =.8 Calculate P, using the equivalence principle. (A).33 (B).38 (C) 3. (D) 3.7 (E) You are given: (i) P x =. 9 (ii) nv x = 563. (iii) P xn : =. 864 Calculate P xn :. (A).8 (B).4 (C).4 (D).65 (E).85 5

52 78. You are given: (i) Mortality follows De Moivre s law with ω =. (ii) i = 5. (iii) The following annuity-certain values: a a 4 5 = = 8. 7 a = Calculate VcA 4 h. (A).75 (B).77 (C).79 (D).8 (E) For a group of individuals all age x, you are given: (i) 3% are smokers and 7% are non-smokers. (ii) The constant force of mortality for smokers is.6. (iii) The constant force of mortality for non-smokers is.3. (iv) δ = 8. F H Calculate Var a T x (A) 3. bg I K for an individual chosen at random from this group. (B) 3.3 (C) 3.8 (D) 4. (E) 4.6 5

53 8. For a certain company, losses follow a Poisson frequency distribution with mean per year, and the amount of a loss is,, or 3, each with probability /3. Loss amounts are independent of the number of losses, and of each other. An insurance policy covers all losses in a year, subject to an annual aggregate deductible of. Calculate the expected claim payments for this insurance policy. (A). (B).36 (C).45 (D).8 (E) A Poisson claims process has two types of claims, Type I and Type II. (i) The expected number of claims is 3. (ii) The probability that a claim is Type I is /3. (iii) Type I claim amounts are exactly each. (iv) The variance of aggregate claims is,,. Calculate the variance of aggregate claims with Type I claims excluded. (A),7, (B),8, (C),9, (D),, (E),, 53

54 8. Don, age 5, is an actuarial science professor. His career is subject to two decrements: (i) (ii) Decrement is mortality. The associated single decrement table follows De Moivre s law with ω =. Decrement is leaving academic employment, with bg btg µ 5 = 5., t Calculate the probability that Don remains an actuarial science professor for at least five but less than ten years. (A). (B).5 (C).8 (D).3 (E) For a double decrement model: (i) In the single decrement table associated with cause (), q bg 4 =. and decrements are uniformly distributed over the year. (ii) In the single decrement table associated with cause (), q bg 4 = 5. and all decrements occur at time.7. Calculate qbg 4. (A).4 (B).5 (C).6 (D).7 (E).8 54

55 84. For a special -payment whole life insurance on (8): (i) Premiums of π are paid at the beginning of years and 3. (ii) (iii) (iv) The death benefit is paid at the end of the year of death. There is a partial refund of premium feature: If (8) dies in either year or year 3, the death benefit is + π. Otherwise, the death benefit is. Mortality follows the Illustrative Life Table. (v) i =.6 Calculate π, using the equivalence principle. (A) 369 (B) 38 (C) 397 (D) 49 (E) For a special fully continuous whole life insurance on (65): 4. t (i) The death benefit at time t is b = e, t. t (ii) Level benefit premiums are payable for life. b g = (iii) µ 65 t., t (iv) δ = 4. Calculate V, the benefit reserve at the end of year. (A) (B) 9 (C) 37 (D) 6 (E) 83 55

56 86. You are given: (i) A x = 8. (ii) A x+ = 4. (iii) A x: = 5. (iv) i = 5. Calculate a x:. (A). (B). (C).7 (D). (E) On his walk to work, Lucky Tom finds coins on the ground at a Poisson rate. The Poisson rate, expressed in coins per minute, is constant during any one day, but varies from day to day according to a gamma distribution with mean and variance 4. Calculate the probability that Lucky Tom finds exactly one coin during the sixth minute of today s walk. (A). (B).4 (C).6 (D).8 (E).3 56

57 88. The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: b g =. x. x Fx 8. e. e, x The insurance policy pays amounts up to a limit of per claim. Calculate the expected payment under this policy for one claim. (A) 57 (B) 8 (C) 66 (D) 5 (E) A machine is in one of four states (F, G, H, I) and migrates annually among them according to a Markov process with transition matrix: F G H I F..8.. G H I.... At time, the machine is in State F. A salvage company will pay 5 at the end of 3 years if the machine is in State F. Assuming v = 9., calculate the actuarial present value at time of this payment. (A) 5 (B) 55 (C) 6 (D) 65 (E) 7 57

58 9. The claims department of an insurance company receives envelopes with claims for insurance coverage at a Poisson rate of λ = 5 envelopes per week. For any period of time, the number of envelopes and the numbers of claims in the envelopes are independent. The numbers of claims in the envelopes have the following distribution: Number of Claims Probability Using the normal approximation, calculate the 9 th percentile of the number of claims received in 3 weeks. (A) 69 (B) 7 (C) 73 (D) 75 (E) You are given: x (i) The survival function for males is sx bg=, < x < (ii) (iii) Female mortality follows De Moivre s law. At age 6, the female force of mortality is 6% of the male force of mortality. For two independent lives, a male age 65 and a female age 6, calculate the expected time until the second death. (A) 4.33 (B) 5.63 (C) 7.3 (D).88 (E)

59 9. For a fully continuous whole life insurance of : (i) µ = 4. (ii) δ = 8. (iii) L is the loss-at-issue random variable based on the benefit premium. Calculate Var (L). (A) (B) (C) (D) (E) The random variable for a loss, X, has the following characteristics: x Fx b g Eb X xg Calculate the mean excess loss for a deductible of. (A) 5 (B) 3 (C) 35 (D) 4 (E) 45 59

60 94. WidgetsRUs owns two factories. It buys insurance to protect itself against major repair costs. Profit equals revenues, less the sum of insurance premiums, retained major repair costs, and all other expenses. WidgetsRUs will pay a dividend equal to the profit, if it is positive. You are given: (i) Combined revenue for the two factories is 3. (ii) (iii) Major repair costs at the factories are independent. The distribution of major repair costs for each factory is k Prob (k) (iv) (v) At each factory, the insurance policy pays the major repair costs in excess of that factory s ordinary deductible of. The insurance premium is % of the expected claims. All other expenses are 5% of revenues. Calculate the expected dividend. (A).43 (B).47 (C).5 (D).55 (E).59 6

61 95. For watches produced by a certain manufacturer: (i) Lifetimes follow a single-parameter Pareto distribution with α > and θ = 4. (ii) The expected lifetime of a watch is 8 years. Calculate the probability that the lifetime of a watch is at least 6 years. (A).44 (B).5 (C).56 (D).6 (E) For a special 3-year deferred whole life annuity-due on (x): (i) i = 4. (ii) The first annual payment is. (iii) (iv) (v) Payments in the following years increase by 4% per year. There is no death benefit during the three year deferral period. Level benefit premiums are payable at the beginning of each of the first three years. (vi) e x = 5. is the curtate expectation of life for (x). (vii) k 3 k p x Calculate the annual benefit premium. (A) 65 (B) 85 (C) 35 (D) 35 (E) 345 6

62 97. For a special fully discrete -payment whole life insurance on (3) with level annual benefit premium π : (i) The death benefit is equal to plus the refund, without interest, of the benefit premiums paid. (ii) A 3 =. (iii) A 3 =. 88 (iv) biag 3 = 78 :. (v) a 3 = : Calculate π. (A) 4.9 (B) 5. (C) 5. (D) 5. (E) 5.3 6

63 98. For a given life age 3, it is estimated that an impact of a medical breakthrough will be an increase of 4 years in e 3, the complete expectation of life. Prior to the medical breakthrough, s(x) followed de Moivre s law with ω = as the limiting age. Assuming de Moivre s law still applies after the medical breakthrough, calculate the new limiting age. (A) 4 (B) 5 (C) 6 (D) 7 (E) On January,, Pat, age 4, purchases a 5-payment, -year term insurance of,: (i) (ii) Death benefits are payable at the moment of death. Contract premiums of 4 are payable annually at the beginning of each year for 5 years. (iii) i =.5 (iv) L is the loss random variable at time of issue. Calculate the value of L if Pat dies on June 3, 4. (A) 77, (B) 8,7 (C) 8,7 (D) 85,9 (E) 88, 63

64 . Glen is practicing his simulation skills. He generates values of the random variable X as follows: (i) He generates the observed value λ from the gamma distribution with α = and θ = (hence with mean and variance ). (ii) He then generates x from the Poisson distribution with mean λ. (iii) (iv) He repeats the process 999 more times: first generating a value λ, then generating x from the Poisson distribution with mean λ. The repetitions are mutually independent. Calculate the expected number of times that his simulated value of X is 3. (A) 75 (B) (C) 5 (D) 5 (E) 75. Lucky Tom finds coins on his way to work at a Poisson rate of.5 coins per minute. The denominations are randomly distributed: (i) 6% of the coins are worth ; (ii) % of the coins are worth 5; (iii) % of the coins are worth. Calculate the variance of the value of the coins Tom finds during his one-hour walk to work. (A) 379 (B) 487 (C) 566 (D) 67 (E)

65 . For a fully discrete -payment whole life insurance of on (x), you are given: (i) i =.6 (ii) q x+ 9 =. 54 (iii) The level annual benefit premium is 3.7. (iv) The benefit reserve at the end of year 9 is Calculate P x+, the level annual benefit premium for a fully discrete whole life insurance of on (x+). (A) 7 (B) 9 (C) 3 (D) 33 (E) For a multiple decrement model on (6): (i) µ () 6 (), t t, follows the Illustrative Life Table. ( τ ) ( ) 6 6 (ii) µ () t = µ (), t t Calculate q ( τ ) 6, the probability that decrement occurs during the th year. (A).3 (B).4 (C).5 (D).6 (E).7 65

66 4. (x) and (y) are two lives with identical expected mortality. You are given: Px = Py =. P xy = 6., where P xy is the annual benefit premium for a fully discrete insurance of on xy b g. d = 6. Calculate the premium P xy, the annual benefit premium for a fully discrete insurance of on bxyg. (A).4 (B).6 (C).8 (D). (E). 5. For students entering a college, you are given the following from a multiple decrement model: (i) students enter the college at t =. (ii) Students leave the college for failure bg b g (iii) µ t = µ t 4 bg b g µ t = 4. t < 4 b g or all other reasons (iv) 48 students are expected to leave the college during their first year due to all causes. Calculate the expected number of students who will leave because of failure during their fourth year. (A) 8 (B) (C) 4 (D) 34 (E) 4 b g. 66

67 6. The following graph is related to current human mortality: Which of the following functions of age does the graph most likely show? (A) µ bxg (B) lxµbxg Age (C) (D) lp x x l x (E) l x 7. An actuary for an automobile insurance company determines that the distribution of the annual number of claims for an insured chosen at random is modeled by the negative binomial distribution with mean. and variance.4. The number of claims for each individual insured has a Poisson distribution and the means of these Poisson distributions are gamma distributed over the population of insureds. Calculate the variance of this gamma distribution. (A). (B).5 (C).3 (D).35 (E).4 67

68 68

69 8. A dam is proposed for a river which is currently used for salmon breeding. You have modeled: (i) For each hour the dam is opened the number of salmon that will pass through and reach the breeding grounds has a distribution with mean and variance 9. (ii) The number of eggs released by each salmon has a distribution with mean of 5 and variance of 5. (iii) The number of salmon going through the dam each hour it is open and the numbers of eggs released by the salmon are independent. Using the normal approximation for the aggregate number of eggs released, determine the least number of whole hours the dam should be left open so the probability that, eggs will be released is greater than 95%. (A) (B) 3 (C) 6 (D) 9 (E) 3 9. For a special 3-year term insurance on ( x ), you are given: (i) Z is the present-value random variable for the death benefits. (ii) qx+ k=. ( k + ) k =,, (iii) The following death benefits, payable at the end of the year of death: (iv) i = 6. k b k+ 3, 35, 4, 69

70 Calculate EbZg. (A) 36,8 (B) 39, (C) 4,4 (D) 43,7 (E) 46,. For a special fully discrete -year endowment insurance on (55): (i) Death benefits in year k are given by bk = b kg, k =,,,. (ii) The maturity benefit is. (iii) Annual benefit premiums are level. (iv) kv denotes the benefit reserve at the end of year k, k =,,,. (v) V =5. (vi) 9 V =.6 (vii) q 65 =. (viii) i =.8 Calculate V. (A) 4.5 (B) 4.6 (C) 4.8 (D) 5. (E) 5.3 7

71 . For a stop-loss insurance on a three person group: (i) Loss amounts are independent. (ii) (iii) The distribution of loss amount for each person is: Loss Amount Probability The stop-loss insurance has a deductible of for the group. Calculate the net stop-loss premium. (A). (B).3 (C).6 (D).9 (E).. A continuous two-life annuity pays: while both (3) and (4) are alive; 7 while (3) is alive but (4) is dead; and 5 while (4) is alive but (3) is dead. The actuarial present value of this annuity is 8. Continuous single life annuities paying per year are available for (3) and (4) with actuarial present values of and, respectively. Calculate the actuarial present value of a two-life continuous annuity that pays while at least one of them is alive. (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 7

72 3. For a disability insurance claim: (i) The claimant will receive payments at the rate of, per year, payable continuously as long as she remains disabled. (ii) (iii) The length of the payment period in years is a random variable with the gamma distribution with parameters α = and θ =. Payments begin immediately. (iv) δ = 5. Calculate the actuarial present value of the disability payments at the time of disability. (A) 36,4 (B) 37, (C) 38, (D) 39, (E) 4, 4. For a discrete probability distribution, you are given the recursion relation Determine pb4g. bg b g pk = * pk, k =,,. k (A).7 (B).8 (C).9 (D). (E). 7

73 5. A company insures a fleet of vehicles. Aggregate losses have a compound Poisson distribution. The expected number of losses is. Loss amounts, regardless of vehicle type, have exponential distribution with θ =. In order to reduce the cost of the insurance, two modifications are to be made: (i) a certain type of vehicle will not be insured. It is estimated that this will reduce loss frequency by %. (ii) a deductible of per loss will be imposed. Calculate the expected aggregate amount paid by the insurer after the modifications. (A) 6 (B) 94 (C) 5 (D) 3 (E) For a population of individuals, you are given: (i) Each individual has a constant force of mortality. (ii) The forces of mortality are uniformly distributed over the interval (,). Calculate the probability that an individual drawn at random from this population dies within one year. (A).37 (B).43 (C).5 (D).57 (E).63 73

74 7. You are the producer of a television quiz show that gives cash prizes. The number of prizes, N, and prize amounts, X, have the following distributions: b g x Prb X xg n Pr N = n = Your budget for prizes equals the expected prizes plus the standard deviation of prizes. Calculate your budget. (A) 36 (B) 36 (C) 46 (D) 5 (E) For a special fully discrete 3-year term insurance on x (i) b g : Level benefit premiums are paid at the beginning of each year. (ii) k b k+ q x+ k,.3 5,.6,.9 (iii) i =.6 Calculate the initial benefit reserve for year. (A) 6,5 (B) 7,5 (C) 8, (D) 9,4 (E),3 74

75 75

76 9. For a special fully continuous whole life insurance on (x): (i) The level premium is determined using the equivalence principle. t (ii) Death benefits are given by bt = + ig where i is the interest rate. b (iii) (iv) L is the loss random variable at t = for the insurance. T is the future lifetime random variable of (x). Which of the following expressions is equal to L? (A) c c ν T A A x xh h c hc h (B) ν T Ax + Ax (C) c c ν T A + A x xh h c hc h (D) ν T Ax Ax (E) dv c T + A + A x xi h 76

77 . For a 4-year college, you are given the following probabilities for dropout from all causes: q q q q 3 = 5. =. = 5. =. Dropouts are uniformly distributed over each year. Compute the temporary.5-year complete expected college lifetime of a student entering the second year, e 5 :.. (A).5 (B).3 (C).35 (D).4 (E).45 77

78 . Lee, age 63, considers the purchase of a single premium whole life insurance of, with death benefit payable at the end of the year of death. The company calculates benefit premiums using: (i) mortality based on the Illustrative Life Table, (ii) i =.5 The company calculates contract premiums as % of benefit premiums. The single contract premium at age 63 is 533. Lee decides to delay the purchase for two years and invests the 533. Calculate the minimum annual rate of return that the investment must earn to accumulate to an amount equal to the single contract premium at age 65. (A).3 (B).35 (C).4 (D).45 (E).5. You have calculated the actuarial present value of a last-survivor whole life insurance of on (x) and (y). You assumed: (i) (ii) The death benefit is payable at the moment of death. The future lifetimes of (x) and (y) are independent, and each life has a constant force of mortality with µ = 6.. (iii) δ = 5. Your supervisor points out that these are not independent future lifetimes. Each mortality assumption is correct, but each includes a common shock component with constant force.. Calculate the increase in the actuarial present value over what you originally calculated. 78

79 (A). (B).39 (C).93 (D).9 (E) The number of accidents follows a Poisson distribution with mean. Each accident generates,, or 3 claimants with probabilities, 3, 6, respectively. Calculate the variance in the total number of claimants. (A) (B) 5 (C) 3 (D) 35 (E) 4 79

80 4. For a claims process, you are given: m b g r (i) The number of claims Nt, t is a nonhomogeneous Poisson process with intensity function: λ( t) = R S T, t <, t < 3, t (ii) Claims amounts Y i are independently and identically distributed random variables that are also independent of N (). t (iii) Each Y i is uniformly distributed on [,8]. (iv) The random variable P is the number of claims with claim amount less than 5 by time t = 3. (v) The random variable Q is the number of claims with claim amount greater than 5 by time t = 3. (vi) R is the conditional expected value of P, given Q = 4. Calculate R. (A). (B).5 (C) 3. (D) 3.5 (E) 4. 8

81 5. Lottery Life issues a special fully discrete whole life insurance on (5): (i) At the end of the year of death there is a random drawing. With probability., the death benefit is. With probability.8, the death benefit is. (ii) (iii) (iv) At the start of each year, including the first, while (5) is alive, there is a random drawing. With probability.8, the level premium π is paid. With probability., no premium is paid. The random drawings are independent. Mortality follows the Illustrative Life Table. (v) i = 6. (vi) π is determined using the equivalence principle. Calculate the benefit reserve at the end of year. (A).5 (B).5 (C) 3.75 (D) 4. (E) A government creates a fund to pay this year s lottery winners. You are given: (i) There are winners each age 4. (ii) (iii) (iv) Each winner receives payments of per year for life, payable annually, beginning immediately. Mortality follows the Illustrative Life Table. The lifetimes are independent. (v) i =.6 (vi) The amount of the fund is determined, using the normal approximation, such that the probability that the fund is sufficient to make all payments is 95%. 8

82 Calculate the initial amount of the fund. (A) 4,8 (B) 4,9 (C) 5,5 (D) 5,5 (E) 5,5 7. For a special fully discrete 35-payment whole life insurance on (3): (i) (ii) (iii) The death benefit is for the first years and is 5 thereafter. The initial benefit premium paid during the each of the first years is one fifth of the benefit premium paid during each of the 5 subsequent years. Mortality follows the Illustrative Life Table. (iv) i = 6. (v) A 3 =. 337 : (vi) a 3 35 = : Calculate the initial annual benefit premium. (A). (B).5 (C). (D).5 (F).3 8