北美精算师(SOA)考试 FM 2001 May 年真题
精算师考试试题 (2)
Faculty of Actuaries Institute of ActuariesEXAMINATIONS7 September 2001 (pm)Subject 102 — Financial MathematicsTime allowed: Three hoursINSTRUCTIONS TO THE CANDIDATE1.Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.2.Mark allocations are shown in brackets.3.Attempt all 12 questions, beginning your answer to each question on aseparate sheet.Graph paper is not required for this paper.AT THE END OF THE EXAMINATIONHand in BOTH your answer booklet and this question paper.In addition to this paper you should have availableActuarial Tables and an electronic calculator.ã Faculty of Actuaries1 A 91-day government bill provides the purchaser with an annual effective rate ofreturn of 5%. Determine the annual simple discount rate at which the bill is discounted.[2]2 A particular share is expected to pay a dividend of d 1 in exactly one year.Dividends are expected to grow by g per annum effective every year thereafter.The share pays annual dividends. Let V 0 be the present value of the share and r be the investor’s required annual effective rate of return.Show that V 0 = 1d r g −.[3]3An asset has a current price of 100p. It will pay an income of 5p in 20 days’ time.Given a risk-free rate of interest of 6% per annum convertible half-yearly and assuming no arbitrage, calculate the forward price to be paid in 40 days.[4]4An annuity is paid half-yearly in arrears at a rate of £1,000 per annum, for 20years. The rate of interest is 5% per annum effective in the first 12 years and 6%per annum convertible quarterly for the remaining 8 years.Calculate the accumulation of the annuity at the end of 20 years.[4]5An investor purchases a bond, redeemable at par, which pays half-yearly couponsat a rate of 8% per annum. There are 8 days until the next coupon payment and the bond is ex-dividend. The bond has 7 years to maturity after the next coupon payment.Calculate the purchase price to provide a yield to maturity of 6% per annum effective.[4]6(1 + i t ) follows a log normal distribution where i t is the rate of interest over agiven time period beginning at time t . The parameters of the distribution are µ = 0.06 and σ2 = 0.0009.Calculate the inter-quartile range for the accumulation of 100 units of money over the given time period, beginning at time t .[6]7(i) The annual effective forward rate applicable over the period from t to t + ris defined as f t,r where t and r are measured in years. If f 0,1 = 8%, f 1,1 = 7%,f 2,1 = 6% and f 3,1 = 5%, calculate the gross redemption yield at the issue date from a 4-year bond, redeemable at par, with a 5% coupon payable annually in arrears.[7](ii)Explain why the gross redemption yield from the 4-year bond is higher than the 4-year forward rate f 3,1.[2][Total 9]8 A fast food company is considering opening a new sales outlet. The initial cost ofthe outlet would be £1,000,000 incurred at the outset of the project. It is expected that rents of £40,000 per annum would have to be paid quarterly in advance for 10 years, increasing after ten years to £48,000 per annum. The net revenue (sales minus costs, other than rent) from the venture is expected to be £100,000 for the first year and £200,000 for the second year. Thereafter, the net revenue is expected to grow at 3% per annum compound so that it is £206,000 in the third year, £212,180 in the fourth year and so on. The revenue would be received continuously throughout each year. Twenty years after the outset of the project, the revenue and costs stop and the project has no further value.Calculate the internal rate of return from the project.[11]9(i)Prove that =nn n a Da i −.[3](ii) A bank makes a loan to be repaid by instalments paid annually in arrears.The first instalment is 20, the second is 19 with the payments reducing by1 per annum until the end of the 10th year after which there are no further payments. The rate of interest charged by the lender is 6% per annum effective.(a)Calculate the amount of the loan.(b)Calculate the interest and capital components of the first payment.(c) Calculate the amount of capital repaid in the instalment at the end of the 8th year.[8][Total 11]10An investor purchased a bond with exactly 20 years to redemption. The bond, redeemable at par, has a gross redemption yield of 6%. It pays annual coupons,in arrears, of 5%. The investor does not pay tax.(i) Calculate the purchase price paid for the bond.[3](ii)After exactly ten years, immediately after payment of the coupon then due, this investor sells the bond to another investor. That investor paysincome and capital gains tax at a rate of 30%. The bond is purchased bythe second investor to provide a net rate of return of 6.5% per annum.(a) Calculate the price paid by the second investor.(b) Calculate the annual effective rate of return earned by the firstinvestor during the period for which the bond was held.[10][Total 13] 11The force of interest, δ(t), is:δ(t)= 0.05 for 0 < t≤ 10,= 0.006t for 10 < t≤ 20= 0.003t + 0.0002t2 for 20 < t(i) Calculate the present value of a unit sum of money due at time t = 25.[7](ii) Calculate the effective rate of interest per unit time from time t = 19 to time t = 20.[3](iii) A continuous payment stream is paid at the rate of e−0.03t per unit time between time t = 0 and time t = 5. Calculate the present value of thatpayment stream.[4][Total 14]12(i)(a) In the context of a stream of future receipts paid at discrete times,let volatility be defined as the proportionate change in the presentvalue of a payment stream per unit change in the force of interest,for small changes in the force of interest. Prove that thediscounted mean term is equal to the volatility.(b)If volatility is now defined as the proportionate change in thepresent value of a payment stream per unit change in the annualeffective rate of interest, for small changes in the annual effectiverate of interest, find the relationship between the discounted meanterm and volatility.[5](ii) A life insurance company manages a small annuity fund. Payments are expected to be made from the fund of £1,000,000 per annum at the end ofyears 1 to 10 and £1,500,000 at the end of each of the following 10 years.Assets are held in two types of bonds. The first is a zero coupon bondredeemable in 10 years’ time. The second is a bond which pays an annual coupon of g% per annum in arrears and is redeemable at par at the end of19 years. £10,000,000 nominal of the zero coupon bond have beenpurchased.Find the nominal amount of the coupon paying bond which must bepurchased and the rate of coupon which is received from the bond if theinsurance company is to equalise the present values and discounted mean terms of its assets and liabilities at an effective rate of interest of 5% perannum.[12] (iii)If the present value and discounted mean term of the assets and liabilities are equalised, state the third condition which is necessary for theinsurance company to be immunised from small, uniform changes in therate of interest.[2][Total 19]。
北美精算师真题course1
Calculate the percentage of the group that watched none of the three sports during the last year.
(A) (B) (C) (D) (E)
24 36 41 52 60
Course 1
6
Form 03A
2.
c f ( x ) − dg ( x ) . x →0 f ( x ) − g ( x )
(A) (B) (C) (D) (E)
0 cf ′ ( 0 ) − dg ′ ( 0 ) f ′ ( 0) − g′ ( 0)
f ′ ( 0) − g′ ( 0) c−d c+d
Course 1
8
Form 03A
(A) (B) (C) (D) (E)
0.07 0.29 0.38 0.42 0.57
May 2003
9
Course 1
5.
An insurance company examines its pool of auto insurance customers and gathers the following information:
Each of the graphs below contains two curves.
Identify the graph containing a curve representing a function y = f ( x ) and a curve representing its second derivative y = f ′′ ( x ) .
Calculate the rate at which the value of the investment is changing after 8 years.
精算师考试试题 (5)
Faculty of Actuaries Institute of ActuariesEXAMINATIONS13 September 2001 (am)Subject 105 — Actuarial Mathematics 1Time allowed: Three hoursINSTRUCTIONS TO THE CANDIDATE1.Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.2.Mark allocations are shown in brackets.3.Attempt all 14 questions, beginning your answer to each question on aseparate sheet.Graph paper is not required for this paper.AT THE END OF THE EXAMINATIONHand in BOTH your answer booklet and this question paper.In addition to this paper you should have availableActuarial Tables and an electronic calculator.ã Faculty of Actuaries1Under the Manchester Unity model of sickness, you are given the following values:=5xs1 0=0.9 t xp dtòCalculate the value ofxz. [2]2Give a formula for21(2003)P in terms of20(2002)P, based on the component method of population projection. ()xP n denotes the population aged x last birthday at mid-year n.State all the assumptions that you make and define carefully all the symbols that you use. [3]3 A life insurance company issues a policy under which sickness benefit of £100 perweek is payable during all periods of sickness. There is a waiting period of 1 year under the policy.You have been asked to calculate the premium for a life aged exactly 30, who isin good health, using the Manchester Unity model of sickness.Describe how you would allow for the waiting period in your calculation, giving a reason for your choice of method. [3]4An employer recruits lives aged exactly 20, all of whom are healthy whenrecruited. On entry, the lives join a scheme that pays a lump sum of £50,000immediately on death, with an additional £25,000 if the deceased was sick at the time of death.The mortality and sickness of the scheme members are described by the following multiple-state model, in which the forces of transition depend on age only.All surviving members retire at age 65 and leave the scheme regardless of their state of health.,ab x t p is defined as the probability that a life who is in state a at age x (a = H, S, D )is in state b at age x + t (0 and ,,)t b H S D ≥=.Write down an integral expression for the expected present value, at force of interest δ, of the death benefit in respect of a single new recruit. [3]5 A pension scheme provides a pension of 1/60 of career average salary in respect ofeach full year of service, on age retirement between the ages of 60 and 65. A proportionate amount is provided in respect of an incomplete year of service.At the valuation date of the scheme, a new member aged exactly 40 has an annual rate of salary of £40,000.Calculate the expected present value of the future service pension on age retirement in respect of this member, using the Pension Fund Tables in the Formulae and Tables for Actuarial Examinations. [3]6 A life insurance company issues a special annuity contract to a male life agedexactly 70 and a female life aged exactly 60.Under the contract, an annuity of £10,000 per annum is payable monthly to thefemale life, provided that she survives at least 10 years longer than the male life.The annuity commences on the monthly policy anniversary next following thetenth anniversary of the death of the male life and is payable for the balance ofthe female’s lifetime.Calculate the single premium required for the contract.Basis:Mortality:a(55) Ultimate, males or females as appropriateInterest:8% per annumExpenses:none [4]7The staff of a company are subject to two modes of decrement, death and withdrawal from employment.Decrements due to death take place uniformly over the year of age in theassociated single-decrement table: 50% of the decrements due to withdrawaloccur uniformly over the year of age and the balance occurs at the end of the year of age, in the associated single-decrement table.You are given that the independent rate of mortality is 0.001 per year of age and the independent rate of withdrawal is 0.1 per year of age.Calculate the probability that a new employee aged exactly 20 will die as anemployee at age 21 last birthday. [4]8The following data are available from a life insurance company relating to the mortality experience of its temporary assurance policyholders.,x dθThe number of deaths over the period 1 January 1998 to 30 June 2001, aged x nearest birthday at entry and having duration d at the policyanniversary next following the date of death.,()y eP n The number of policyholders with policies in force at time n, aged y nearest birthday at entry and having curtate duration e at time n, wheren = 1.1.1998, 30.6.1998, 30.6.2000 and 30.6.2001.Develop formulae for the calculation of the crude central select rates of mortality corresponding to the,x dθ deaths and derive the age and duration to which these rates apply. State all the assumptions that you make.[6]9(i)State the conditions necessary for gross premium retrospective and prospective reserves to be equal. [3] (ii)Demonstrate the equality of gross premium retrospective and prospective reserves for a whole life policy, given the conditions necessary for equality.[4][Total 7]10 A life insurance company issues a special term assurance policy to two lives agedexactly 50 at the issue date, in return for the payment of a single premium. The following benefits are payable under the contract:(i)In the event of either of the lives dying within 10 years, a sum assured of£100,000 is payable immediately on this death.(ii)In the event of the second death within 10 years, a further sum assured of £200,000 is payable immediately on the second death.Calculate the single premium.Basis:Mortality:A1967–70 UltimateInterest:4% per annumExpenses:None [8]11 A life insurance company sells term assurance policies with terms of either 10 or20 years.As an actuary in the life office, you have been asked to carry out the first review of the mortality experience of these policies. The following table shows thestatistical summary of the mortality investigation. In all cases, the central rates of mortality are expressed as rates per 1,000 lives.All policies10-year policies20-year policiesAge Numberin forceCentralmortalityrateNumberin forceCentralmortalityrateNumberin forceCentralmortalityrate–246,991 1.086,0130.86978 2.12 25–446,462 2.055,438 1.741,024 3.68 45–645,81513.264,94211.5587322.94 65–3,05175.702,57071.5348197.70 Total22,31918,9633,356(i)Calculate the directly standardised mortality rate and the standardisedmortality ratio separately in respect of the 10-year and 20-year policies.In each case, use the “all policies” population as the standard population.[6](ii)You have been asked to recommend which of these two summary mortality measures should be monitored on a regular basis.Give your recommendation, explaining the reasons for your choice. [3][Total 9]12 A life insurance company offers an option on its 10-year without profit termassurance policies to effect a whole life without profits policy, at the expiry of the 10-year term, for the then existing sum assured, without evidence of health.Premiums under the whole life policy are payable annually in advance for thewhole of life, or until earlier death.(i)Describe the conventional method of pricing the mortality option, statingclearly the data and assumptions required. Formulae are not required.[3](ii) A policyholder aged exactly 30 wishes to effect a 10-year without profits term assurance policy, for a sum assured of £100,000.Calculate the additional single premium, payable at the outset, for theoption, using the conventional method.The following basis is used to calculate the basic premiums for the termassurance policies.Basis:Mortality:A1967–70 SelectInterest:6% per annumExpenses:none [4](iii)Describe how you would calculate the option single premium for the policy described in part (ii) above using the North American method, statingclearly what additional data you would require and what assumptions youwould make. [4](iv)State, with reasons, whether it would be preferable to use theconventional method or the North American method for pricing themortality option under the policy described in part (ii) above. [3][Total 14]13(i)On 1 September 1996, a life aged exactly 50 purchased a deferred annuity policy, under which yearly benefit payments are to be made. The firstpayment, being £10,000, is to be made at age 60 exact if he is then alive.The payments will continue yearly during his lifetime, increasing by1.923% per annum compound.Premiums under the policy are payable annually in advance for 10 yearsor until earlier death.If death occurs before age 60, the total premiums paid under the policy,accumulated to the end of the year of death at a rate of interest of 1.923%per annum compound, are payable at the end of the year of death.Calculate the annual premium.Basis:Mortality: before age 60:A1967–70 Ultimateafter age 60:a(55) Males UltimateInterest:6% per annumExpenses: initial:10% of the initial premium, incurredat the outsetrenewal:5% of each of the second andsubsequent premiums, payable at thetime of premium paymentclaim:£100, incurred at the time of paymentof the death benefit[9](ii)On 1 September 2001, immediately before payment of the premium then due, the policyholder requests that the policy be altered so that there is nobenefit payable on death and the rate of increase of the annuity inpayment is to be altered. The premium under the policy is to remainunaltered as is the amount of the initial annuity payment.The life insurance company calculates the revised terms of the policy byequating gross premium prospective reserves immediately before andafter the alteration, calculated on the original pricing basis, allowing foran expense of alteration of £100.Calculate the revised rate of increase in payment of the annuity. [7][Total 16]14 A life insurance company issues a 3-year unit-linked endowment assurancecontract to a male life aged exactly 60 under which level annual premiums of£5,000 are payable in advance throughout the term of the policy or until earlier death. 102% of each year’s premium is invested in units at the offer price.The premium in the first year is used to buy capital units, with subsequent years’premiums being used to buy accumulation units. There is a bid-offer spread in unit values, with the bid price being 95% of the offer price.The annual management charges are 5% on capital units and 1% on accumulation units. Management charges are deducted at the end of each year,before death, surrender or maturity benefits are paid.On the death of the policyholder during the term of the policy, there is a benefit payable at the end of the year of death of £12,000 or the bid value of the units allocated to the policy, if greater. On maturity, the full bid value of the units is payable.The policy may be surrendered only at the end of the first or the second policy year. On surrender, the life insurance company pays the full bid value of the accumulation units and 80% of the nominal bid value of the capital units,calculated at the time of surrender.The company holds unit reserves equal to the full bid value of the accumulation units and a proportion, 60:3t t A +−(calculated at 4% interest and A1967-70 Ultimate mortality), of the full bid value of the capital units, calculated just after thepayment of the premium due at time t (t = 0,1 and 2). The company holds no sterling reserves.The life insurance company uses the following assumptions in carrying out profit tests of this contract:Mortality:A1967–70 Ultimate Expenses:initial:£400renewal:£80 at the start of each of the second and third policy years Unit fund growth rate:8% per annum Sterling fund interest rate:5% per annum Risk discount rate:15% per annum Surrender rates:20% of all policies still in force at the end of each of the first and second yearsCalculate the profit margin on the contract.[18]。
北美精算师(SOA)考试P 2000 November年真题
(A) (B) (C) (D) (E)
208 260 270 312 374
November 2000
1
Course 1
2.
An investor purchases two assets, each having an initial value of 1000 . The value Vn of the first asset after n years can be modeled by the relationship:
0.06 0.16 0.19 0.22 0.25
Course 1
12
Form 00B
13.
An actuary believes that the demand for life insurance, L, and the demand for health insurance, H, can be modeled as functions of time, t:
What is the approximate probability that there is a total of between 2450 and 2600 claims during a one-year period?
(A) (B) (C) (D) (E)
0.68 0.82 0.87 0.95 1.00
Course 1
2
Form 00B
3.
An auto insurance company has 10,000 policyholders. Each policyholder is classified as
(i) (ii) (iii)
young or old; male or female; and married or single.
北美精算师(SOA)考试 FM 2001 November 年真题和注解
25.
E In order to solve for the price of a call option, first the price of a put option must be determined and then the put-call parity formula is used. Price of put option: [(0.5 x 0) + (0.5 x 10)] / 1.04 = 5/1.04 = 4.81 Price of call option = Price of put option + current stock price – present value of exercise price Price of call option = 4.81 + 45 – 40/1.04 = 11.35
Course 2 Solutions
1
November 2001
5.
A ¬9.2 + 10 ⋅V Present value = 10 ⋅ a5
1+ k 20 = 0.092 − k 0.84 = 21k k = 0.04
5 9.2
(1 + k ) ( 6.44 )(1 + k ) ∑ (1.092) = 38.70 + 0.092 − k = 167.50 t =1
3.
B
∆Y ∆N ∆K − (α N ) − (1 − α N ) = 2.6 − (0.7)(2.0) − (1 − 0.7)(4.0) = 2.6 − 1.4 − 1.2 = 0.0 Y N K
4.
C APV = Base-case NPV + PV tax shield 120, 000 0.35 × 0.08 × 0.54(100, 000) = −100, 000 + + = 0 + 1,400 = 1,400 . 1.2 1.08
Spring 2001 Exam
Determine b .
(A)
5 ln 6 5 5 ln 6 6 5 6 ln 6 5 7 ln 6 6 ln 5
(B)
(C)
(D)
(E)
May 2001
7
Course 1
2.
A stock pays annual dividends. The first dividend is 8 and each dividend thereafter is 7% larger than the prior dividend.
Let m be the number of dividends paid by the stock when the cumulative amount paid first exceeds 500 .
Calculate m .
(A) (B) (C) (D) (E)
23 24 25 26 27
May 2001
What is the expected excess of premiums over claims, given that the husband survives at least ten years?
(A) (B) (C) (D) (E)
350 385 397 870 897
May 2001
May 2001
12
Course 1
7.
A joint density function is given by kx f ( x, y ) = 0 where k is a constant. for 0 < x < 1, 0 < y < 1 otherwise,
What is Cov(X,Y) ?
北美精算师(SOA)考试P 2001 November年真题
are the only possible loss amounts and no more than one loss can occur.
Determine the net premium for this policy.
(–1)n n
− an = 1 2n n
Course 1, November 2001
11
11.
A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is
7
7.
Let X denote the size of a surgical claim and let Y denote the size of the associated hospital claim. An actuary is using a model in which E(X) = 5, E(X 2) = 27.4, E(Y) = 7, E(Y 2) = 51.4, and Var(X+Y) = 8 .
Given that a patient survived, what is the probability that the patient was categorized as serious upon arrival?
(A) (B) (C) (D) (E)
course3_0501
May 2001Course 3**BEGINNING OF EXAMINATION**1.For a given life age 30, it is estimated that an impact of a medical breakthrough will be anincrease of 4 years in e o30, the complete expectation of life.Prior to the medical breakthrough, s(x) followed de Moivre’s law with ω=100as thelimiting age.Assuming de Moivre’s law still applies after the medical breakthrough, calculate the new limiting age.(A)104(B)105(C)106(D)107(E)1082.On January 1, 2002, Pat, age 40, purchases a 5-payment, 10-year term insurance of100,000:(i)Death benefits are payable at the moment of death.(ii)Contract premiums of 4000 are payable annually at the beginning of each year for5 years.(iii)i = 0.05(iv)L is the loss random variable at time of issue.Calculate the value of L if Pat dies on June 30, 2004.(A)77,100(B)80,700(C)82,700(D)85,900(E)88,0003.Glen is practicing his simulation skills.He generates 1000 values of the random variable X as follows:(i)He generates the observed value λ from the gamma distribution with α=2 andθ=1 (hence with mean 2 and variance 2).(ii)He then generates x from the Poisson distribution with mean λ.(iii)He repeats the process 999 more times: first generating a value λ, thengenerating x from the Poisson distribution with mean λ.(iv)The repetitions are mutually independent.Calculate the expected number of times that his simulated value of X is 3.(A) 75(B)100(C)125(D)150(E)1754.Lucky Tom finds coins on his way to work at a Poisson rate of 0.5 coins per minute.The denominations are randomly distributed:(i)60% of the coins are worth 1;(ii)20% of the coins are worth 5;(iii)20% of the coins are worth 10.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)670(E)7685.For a fully discrete 20-payment whole life insurance of 1000 on (x), you are given:(i)i = 0.06(ii)q x+=19001254 .(iii)The level annual benefit premium is 13.72.(iv)The benefit reserve at the end of year 19 is 342.03.Calculate 1000 P x+20, the level annual benefit premium for a fully discrete whole life insurance of 1000 on (x+20).(A)27(B)29(C)31(D)33(E)356.For a multiple decrement model on (60):(i)µ6010()(),,t t ≥ follows the Illustrative Life Table.(ii)µµτ6060120()()()(),t t t =≥Calculate 1060q ()τ, the probability that decrement occurs during the 11th year.(A)0.03(B)0.04(C)0.05(D)0.06(E)0.077. A coach can give two types of training, “ light” or “heavy,” to his sports team before agame. If the team wins the prior game, the next training is equally likely to be light or heavy. But, if the team loses the prior game, the next training is always heavy.The probability that the team will win the game is 0.4 after light training and 0.8 after heavy training.Calculate the long run proportion of time that the coach will give heavy training to the team.(A)0.61(B)0.64(C)0.67(D)0.70(E)0.738.For a simulation of the movement of a stock’s price:. and(i)The price follows geometric Brownian motion, with drift coefficient µ=001..variance parameter σ2=00004(ii)The simulation projects the stock price in steps of time 1.(iii)Simulated price movements are determined using the inverse transform method.(iv)The price at t = 0 is 100.(v)The random numbers, from the uniform distribution on ,are 0.1587 and 0.9332, respectively.(vi)F is the price at t = 1; G is the price at t = 2.Calculate G – F.(A)1(B)2(C)3(D)4(E)59.(x) and (y) are two lives with identical expected mortality.You are given:P P==0.1x yP xy=006., where P xy is the annual benefit premium for a fully discreteb g.insurance of 1 on xyd=006.Calculate the premium P xy, the annual benefit premium for a fully discrete insurance of 1b g.on xy(A)0.14(B)0.16(C)0.18(D)0.20(E)0.2210.For students entering a college, you are given the following from a multiple decrementmodel:(i)1000 students enter the college at t =0.(ii)Students leave the college for failure 1b g or all other reasons 2b g .(iii) µµ1b g b gt = 04≤≤t µ2004b g b g t =. 04≤<t (iv) 48 students are expected to leave the college during their first year due toall causes.Calculate the expected number of students who will leave because of failure during their fourth year.(A)8(B)10(C)24(D)34(E) 4111.You are using the inverse transform method to simulate Z, the present value randomvariable for a special two-year term insurance on (70). You are given:(i)(70) is subject to only two causes of death, with(ii)Death benefits, payable at the end of the year of death, are:During year Benefit for Cause 1Benefit for Cause 2110001100211001200.(iii)i=006(iv)For this trial your random number, from the uniform distribution on , is 0.35.(v)High random numbers correspond to high values of Z.Calculate the simulated value of Z for this trial.(A) 943(B) 979(C)1000(D)1038(E)106812.You are simulating one year of death and surrender benefits for 3 policies. Mortalityfollows the Illustrative Life Table. The surrender rate, occurring at the end of the year, is 15% for all ages. The simulation procedure is the inverse transform algorithm, with low random numbers corresponding to the decrement occurring. You perform the following steps for each policy:(1)Simulate if the policy is terminated by death. If not, go to Step 2; if yes, continuewith the next policy.(2)Simulate if the policy is terminated by surrender.The following values are successively generated from the uniform distribution on ,0.3,0.5,0.1,0.4,0.8,0.2,0.3,0.4,0.6,0.7,….You are given:Policy #Age DeathBenefitSurrenderBenefit1100101029125203962015 Calculate the total benefits generated by the simulation.(A)30(B)35(C)40(D)45(E)5013.Mr. Ucci has only 3 hairs left on his head and he won’t be growing any more.(i)The future mortality of each hair followsk x q k k =+=0110123.,,,,b gand x is Mr. Ucci’s age (ii) Hair loss follows the hyperbolic assumption at fractional ages.(iii)The future lifetimes of the 3 hairs are independent.Calculate the probability that Mr. Ucci is bald (has no hair left) at age x +2.5.(A) 0.090(B) 0.097(C) 0.104(D) 0.111(E)0.11814.The following graph is related to current human mortality:Which of the following functions of age does the graph most likely show?(A)µx b g(B)l xxµb g(C)l px x(D)lx(E)lx215.An actuary for an automobile insurance company determines that the distribution of theannual number of claims for an insured chosen at random is modeled by the negativebinomial distribution with mean 0.2 and variance 0.4.The number of claims for each individual insured has a Poisson distribution and themeans of these Poisson distributions are gamma distributed over the population ofinsureds.Calculate the variance of this gamma distribution.(A)0.20(B)0.25(C)0.30(D)0.35(E)0.4016. A dam is proposed for a river which is currently used for salmon breeding. You havemodeled:(i)For each hour the dam is opened the number of salmon that will pass through andreach the breeding grounds has a distribution with mean 100 and variance 900.(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, determinethe least number of whole hours the dam should be left open so the probability that10,000 eggs will be released is greater than 95%.(A)20(B)23(C)26(D)29(E)3217.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) q k x k +=+0021.()k =0, 1, 2(iii)The following death benefits, payable at the end of the year of death:k b k +10300,0001350,0002400,000(iv) i =006.Calculate E Z b g.(A) 36,800(B) 39,100(C) 41,400(D) 43,700(E)46,00018.For a special fully discrete 20-year endowment insurance on (55):(i)Death benefits in year k are given by b k k =−21b g, k = 1, 2, …, 20.(ii) The maturity benefit is 1.(iii) Annual benefit premiums are level.(iv) k V denotes the benefit reserve at the end of year k , k = 1, 2,…, 20.(v) 10=5.0V (vi) 19=0.6V(vii) q 650.10=(viii)i =0.08Calculate 11.V (A) 4.5(B) 4.6(C) 4.8(D) 5.1(E)5.319.For a stop-loss insurance on a three person group:(i)Loss amounts are independent.(ii)The distribution of loss amount for each person is:Loss Amount Probability00.410.320.230.1(iii)The stop-loss insurance has a deductible of 1 for the group.Calculate the net stop-loss premium.(A) 2.00(B) 2.03(C) 2.06(D) 2.09(E) 2.1220.An insurer’s claims follow a compound Poisson claims process with two claims expectedper period. Claim amounts can be only 1, 2, or 3 and these are equal in probability.Calculate the continuous premium rate that should be charged each period so that theadjustment coefficient will be 0.5.(A) 4.8(B) 5.9(C) 7.8(D) 8.9(E)11.8e the following information for questions 21 and 22.The Simple Insurance Company starts at time t=0 with a surplus of S=3. At thebeginning of every year, it collects a premium of P=2. Every year, it pays a randomclaim amount:Claim Amount Probability of Claim Amount00.1510.2520.5040.10Claim amounts are mutually independent.If, at the end of the year, Simple’s surplus is more than 3, it pays a dividend equal to theamount of surplus in excess of 3. If Simple is unable to pay its claims, or if its surplusdrops to 0, it goes out of business. Simple has no administrative expenses and its interestincome is 0.21.Determine the probability that Simple will ultimately go out of business.(A)0.00(B)0.01(C)0.44(D)0.56(E) 1.0021-22.(Repeated for convenience) Use the following information for questions 21 and 22.The Simple Insurance Company starts at time t=0 with a surplus of S=3. At thebeginning of every year, it collects a premium of P=2. Every year, it pays a randomclaim amount:Claim Amount Probability of Claim Amount00.1510.2520.5040.10Claim amounts are mutually independent.If, at the end of the year, Simple’s surplus is more than 3, it pays a dividend equal to theamount of surplus in excess of 3. If Simple is unable to pay its claims, or if its surplusdrops to 0, it goes out of business. Simple has no administrative expenses and its interestincome is 0.22.Calculate the expected dividend at the end of the third year.(A)0.115(B)0.350(C)0.414(D)0.458(E)0.55023. A continuous two-life annuity pays:100 while both (30) and (40) are alive;70 while (30) is alive but (40) is dead; and50 while (40) is alive but (30) is dead.The actuarial present value of this annuity is 1180. Continuous single life annuitiespaying 100 per year are available for (30) and (40) with actuarial present values of 1200 and 1000, respectively.Calculate the actuarial present value of a two-life continuous annuity that pays 100 while at least one of them is alive.(A)1400(B)1500(C)1600(D)1700(E)180024.For a disability insurance claim:(i) The claimant will receive payments at the rate of 20,000 per year, payable continuously as long as she remains disabled.(ii)The length of the payment period in years is a random variable with the gamma distribution with parameters αθ==21and .(iii) Payments begin immediately.(iv)δ=005.Calculate the actuarial present value of the disability payments at the time of disability.(A) 36,400(B) 37,200(C) 38,100(D) 39,200(E)40,00025.For a discrete probability distribution, you are given the recursion relationp k kp k b gb g=−21*,k = 1, 2,….Determine p 4b g.(A) 0.07(B) 0.08(C) 0.09(D) 0.10(E)0.1126. A company insures a fleet of vehicles. Aggregate losses have a compound Poissondistribution. The expected number of losses is 20. Loss amounts, regardless of vehicle type, have exponential distribution with θ=200.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 willreduce loss frequency by 20%.(ii) a deductible of 100 per loss will be imposed.Calculate the expected aggregate amount paid by the insurer after the modifications.(A)1600(B)1940(C)2520(D)3200(E)388027.An actuary is modeling the mortality of a group of 1000 people, each age 95, for the next three years.The actuary starts by calculating the expected number of survivors at each integral age byl p k k k 95951000+==,1, 2, 3The actuary subsequently calculates the expected number of survivors at the middle of each year using the assumption that deaths are uniformly distributed over each year of age.This is the result of the actuary’s model:Age Survivors 95100095.58009660096.548097--97.528898--The actuary decides to change his assumption for mortality at fractional ages to the constant force assumption. He retains his original assumption for each k p 95.Calculate the revised expected number of survivors at age 97.5.(A) 270(B) 273(C) 276(D) 279(E)28228.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 (0,2).Calculate the probability that an individual drawn at random from this population dies within one year.(A)0.37(B)0.43(C)0.50(D)0.57(E)0.6329-30.Use the following information for questions 29 and 30.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:n Pr N n =b gx Pr X x =b g10.800.220.21000.710000.129.Your budget for prizes equals the expected prizes plus the standard deviation of prizes.Calculate your budget.(A) 306(B) 316(C) 416(D) 510(E)51829-30.(Repeated for convenience) Use the following information for questions 29 and 30.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:n Pr N n =b gx Pr X x =b g10.800.220.21000.710000.130.You buy stop-loss insurance for prizes with a deductible of 200. The cost of insurance includes a 175% relative security load.Calculate the cost of the insurance.(A) 204(B) 227(C) 245(D) 273(E)35731.For a special fully discrete 3-year term insurance on x b g:(i)Level benefit premiums are paid at the beginning of each year.(ii)k b k+1q x k+0200,0000.031150,0000.062100,0000.09(iii)i=0.06Calculate the initial benefit reserve for year 2.(A) 6,500(B) 7,500(C) 8,100(D) 9,400(E)10,30032.For a special fully continuous whole life insurance on (x ):(i)The level premium is determined using the equivalence principle.(ii)Death benefits are given by b i t t=+1b g where i is the interest rate.(iii)L is the loss random variable at t =0 for the insurance.(iv)T is the future lifetime random variable of (x ).Which of the following expressions is equal to L ?(A)νTx x A A −−c hc h1(B)νT x x A A −+c hc h1(C)νTx x A A −+c hc h1(D)νT x x A A −−c hc h1(E)v A A Tx x ++d ic h133.For a 4-year college, you are given the following probabilities for dropout from allcauses:q q q q 0123015010005001====....Dropouts are uniformly distributed over each year.Compute the temporary 1.5-year complete expected college lifetime of a student entering the second year, e o115:..(A)1.25(B)1.30(C)1.35(D)1.40(E) 1.4534.Lee, age 63, considers the purchase of a single premium whole life insurance of 10,000with 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 = 0.05The company calculates contract premiums as 112% of benefit premiums.The single contract premium at age 63 is 5233.Lee decides to delay the purchase for two years and invests the 5233.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)0.030(B)0.035(C)0.040(D)0.045(E)0.05035.You have calculated the actuarial present value of a last-survivor whole life insurance of1 on (x) and (y). You assumed:(i)The death benefit is payable at the moment of death.(ii)The future lifetimes of (x) and (y) are independent, and each life has a constant force of mortality with µ=006...(iii)δ=005Your supervisor points out that these are not independent future lifetimes. Eachmortality assumption is correct, but each includes a common shock component withconstant force 0.02.Calculate the increase in the actuarial present value over what you originally calculated.(A)0.020(B)0.039(C)0.093(D)0.109(E)0.16336.The number of accidents follows a Poisson distribution with mean 12. Each accidentgenerates 1, 2, or 3 claimants with probabilities 12, 13, 16, respectively.Calculate the variance in the total number of claimants.(A)20(B)25(C)30(D)35(E)4037.For a claims process, you are given:(i) The number of claims N t t b g m r ,≥0 is a nonhomogeneous Poisson process withintensity function:λ(),,,t t t t =≤<≤<≤R S |T |10121232(ii) Claims amounts Y i are independently and identically distributed random variablesthat are also independent of N t ().(iii) Each Y i is uniformly distributed on [200,800].(iv) The random variable P is the number of claims with claim amount less than 500 by time t = 3.(v) The random variable Q is the number of claims with claim amount greater than 500 by time t = 3.(vi)R is the conditional expected value of P , given Q = 4.Calculate R.(A)2.0(B)2.5(C)3.0(D)3.5(E)4.038.Lottery Life issues a special fully discrete whole life insurance on (25):(i)At the end of the year of death there is a random drawing. With probability 0.2,the death benefit is 1000. With probability 0.8, the death benefit is 0.(ii)At the start of each year, including the first, while (25) is alive, there is a random drawing. With probability 0.8, the level premium π is paid. With probability0.2, no premium is paid.(iii)The random drawings are independent.(iv)Mortality follows the Illustrative Life Table..(v)i=006(vi)π is determined using the equivalence principle.Calculate the benefit reserve at the end of year 10.(A)10.25(B)20.50(C)30.75(D)41.00(E)51.2539. A government creates a fund to pay this year’s lottery winners.You are given:(i)There are 100 winners each age 40.(ii)Each winner receives payments of 10 per year for life, payable annually, beginning immediately.(iii)Mortality follows the Illustrative Life Table.(iv)The lifetimes are independent.(v)i = 0.06(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%.Calculate the initial amount of the fund.(A)14,800(B)14,900(C)15,050(D)15,150(E)15,25040.For a special fully discrete 35-payment whole life insurance on (30):(i)The death benefit is 1 for the first 20 years and is 5 thereafter.(ii)The initial benefit premium paid during the each of the first 20 years is one fifthof the benefit premium paid during each of the 15 subsequent years.(iii)Mortality follows the Illustrative Life Table.(iv)i =006.(v)A 3020032307:.=(vi)&&.:a 303514835=Calculate the initial annual benefit premium.(A)0.010(B)0.015(C)0.020(D)0.025(E)0.030**END OF EXAMINATION**COURSE 3: MAY 2001- 41 –STOPCOURSE 3MAY 2001MULTIPLE-CHOICE ANSWER KEY。
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1.
Which of the following statements is an implication of the semi-strong form of the Efficient Market Hypothesis?
(A) (B) (C) (D) (E)
Course 2
12
Form 01A
9.
Which of the following statements regarding consumer goods in the marketplace is true?
(A) (B) (C) (D) (E)
The quantity demanded of an inferior good decreases as its price decreases. The intersection of a consumer’s Engel curve and demand curve represents how much the consumer will buy given a specific income level. When the price of a normal good decreases, the increase in quantity due to the income effect cannot be greater than the substitution effect. The compensated demand curve for a normal good will be steeper than the uncompensated demand curve. The income elasticity of demand is equal to the slope of the Engel curve.
May 2001
5
Course 2
2.
The free rider problem is considered a major reason that public goods are provided by the government rather than by the private market.
Calculate K .
(A) (B) (C) (D) (E)
24% 28% 32% 36% 40%
May 2001
9
Course 2
6.
Suppose a constant-cost, competitive industry is in long-run equilibrium. Now suppose the government imposes an annual licensing fee as a requirement for firms to produce in the industry.
May 2001
7
Course 2
4.
A 20-year loan of 20,000 may be repaid under the following two methods:
i) ii)
amortization method with equal annual payments at an annual effective rate of 6.5% sinking fund method in which the lender receives an annual effective rate of 8% and the sinking fund earns an annual effective rate of j
Calculate the total amount of interest paid on all three loans.
(A) (B) (C) (D) (E)
8718 8728 8738 8748 8758
May 2001
11
Course 2
8.
The following is given for Company X:
Authorized number of shares Common stock ($1 par value) Additional paid-in capital Retained Earnings Treasury Stock
600,000 $ 450,000 $1,000,000 $5,000,000 $ 150,000
As a result of this fee, what will happen to the quantity supplied in the market and the quantity supplied by an individual firm in the long run?
(A) (B) (C) (D) (E)
Course 2
6
Form 01A
3.
Suppose the economy is entering the contractionary phase of a business cycle.
Which of the following is an indicator of this downturn in economic activity?
The company plans to issue an additional 100,000 shares at $3 per share.
Which of the following is true?
(A) (B) (C) (D) (E)
Authorized number of shares will be 700,000 . Common stock will be $750,000 . Additional paid-in capital will be $1,200,000 . Retained earnings will be $5,200,000 . Treasury stock will be $50,000 .
Market price reflects all information. Prices slowly adjust over time to incorporate past information. Past price data should have predictive power for stock returns. Actively managed portfolios cannot consistently outperform the market. The correct answer is not given by (A), (B), (C), or (D).
(A) (B) (C) (D) (E)
A decrease in the unemployment rate An increase in the number of new building permits for private housing units An increase in stock prices An increase in delivery lags An increase in business inventories
The quantity supplied in the market will increase, and the quantity supplied by an individual firm will increase. The quantity supplied in the market will increase, but the quantity supplied by an individual firm will decrease. The quantity supplied in the market will decrease, but the quantity supplied by an individual firm will not change, as some firms go out of business. The quantity supplied in the market will decrease, and the quantity supplied by an individual firm will decrease. The quantity supplied in the market will decrease, but the quantity supplied by an individual firm will increase.
Course 2
10
Form 01A
7.
Seth, Janice, and Lori each borrow 5000 for five years at a nominal ed semi-annually. Seth has interest accumulated over the five years and pays all the interest and principal in a lump sum at the end of five years. Janice pays interest at the end of every six-month period as it accrues and the principal at the end of five years. Lori repays her loan with 10 level payments at the end of every six-month period.
Both methods require a payment of X to be made at the end of each year for 20 years.