北美精算师(SOA)考试P 2000 November年真题
SOA真题November20XXCourse8E第4页-精算师考试.doc
SOA真题November2004Course8E第4页-精算师考试9.Continued(c)(4points)DescribewaystohedgetheliabilitiesoftheV Aandthepractical considerations associated witheach.(d)(2points) ExplainthefactorsGALshould considerindecidingto hedgeitsV A liabilities.(e)(3 points) The Chief Actuary has calleda meeting tofirstyear performance oftheV A product. You suspectshewantsto discontinue future sales. Outlineresponse which: supports continued salesand addresses anyproduct shortcomings. COURSE 8:Fall2004-12GOONTONEXT PAGE Enterprise Risk Management Segment Afternoon Session Question10 pertainstotheCase Study. 10.(10 points)AsthenewCFOof Zoolander, youhave been askedto determinewhether thenew Enhanced Savings Annuity (ESA) product shouldbe launched as scheduled. You havemetwithproduct teamandother executives involved withthe product andhave received the following additional information:productwillbesoldtomiddle-income individuals. Theproduct involvesacomplex investment andtax strategy.The investment strategyistotarget high returns with mostoftherisk being bornebytheindividual. The earnings model usedbythe product development team indicatesahighROE basedonthemarketing area’s projected sales volume. Pricing was basedon estimated expenses usingthe same projected sales volumeinthe earnings model.The productis expectedto generateahighdaily volumeof transactions,willbe administered usingthenewTripleP system. Marketing materials havebeen developed and ordered.Regulations require detailed quarterly statements tobesenttoeach participant. Duetothe complexity oftheproduct therearestrict disclosure requirements to prospective clientsandthe company mustattest annually thatthesalesforcehasbeen properly trained. Thetax authority hasnotissued clarification regarding thetax implicationsthis productbutis expectedtodosoinseveral months. Basedonthis information: (a)points) Identify areasof concern withtheESA product development processtodate,and propose ERMbest practices to address these areasof concern.(b)(4 points) Determine the operational risk exposures toZoolander associated with launching the ESA,and categorize as people, process,or technology.(c)(2 points) Describecomponents ofan operational risk assessment reportthatcanbeusedto evaluate therisks associatedthis product. COURSE 8:Fall 2004-13-GOONTO NEXT PAGE Enterprise RiskManagement Segment Afternoon Session11.(5points)Themulti-line, multi-national insurance company, Titanic Life, recently establishedaRisk Management Committee (RMC)uponthe recommendation ofitsinternal auditing team.TheCEOofTitanicischairing the RMC. Following thefirsttwo monthly meetings, he concluded thatthe meeting agendas andtheitself werenotvery fruitful, sincethemajor risks were handled wellbyeach respective operationalAsthenewly hiredCROof Titanic, youhavebeen askedtojustifythe continuation ofRMC. (a) Explainthe benefitsofa centralized risk management function andthe elements requiredforits successful implementation.(b)IdentifythetypesofriskthatapplytoTitanicLifeasamulti-line,multi-national insurance company which would warranta centralized risk management function. 12.(5points)Youareaconsultant to Zambro, Inc.,aBBB-rated oilandgas company. The company isrunbyagroupsenior managers who participateinan incentive compensation programthathasgranted thema significant amountstock options. Zambro believesithas valuable information regarding futureoilpricesandis considering eitherselective hedging strategy basedonthis information orafull-cover hedging strategy. Asanexpertinmanagement youhavebeen askedtoprepareareportforsenior management on hedgingasamanagement tool.(a) Explainthemajor costs associated with higher variability of corporate earnings andcashand describe howrisk management can reduce eachofthese costs. (b) Explain how Zambro’s creditand compensation program couldeach influencethe company’s choiceofahedging strategy. COURSE 8:Fall-14-GOONTONEXT PAGE Enterprise Risk Management Segment Afternoon Session 13.(15Exponential Lifehas recently suffered excessive credit lossesinits corporate bond holdings. Exponential Lifehasthatits systemof setting credit exposure limitsis inadequate andhas assigned youto exploretools with whichto better manage its credit risk position. You have extracted the following datacommercial creditrisk models: Year(t)Risk-free ZeroRate1-Year Forward Credit Spreads,tYears ForwardFSt) Cumulative risk-neutral EDFs()Qt13.00%0.90%2.00%23.00%1.26%4.00%Using theKMV approach, determine the credit spreadofa2-year corporate bond withafacevalueof $100 that pays annual coupons of 5%, assuming a40% recovery rate. Show your work.(b)Usingthe Reduced-Form approach, determinethecredit spreadofa2-year corporate bondwithafacevalueof $100 that pays annual coupons of 5%, assuming a40% recovery rate. Show your work.。
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 2001 November年真题
is independent of the number of accidents that occur in all other months.
Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.
(A) (B) (C) (D) (E)
0.0000 0.0004 0.0027 0.0064 0.3679
Course 1, November 2001
9
9.
Among a large group of patients recovering from shoulder injuries, it is found that 22% visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.14 the probability that a patient visits a physical therapist.
(–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
AMC 美国数学竞赛 2000 AMC 10 试题及答案解析
USA AMC 10 20001In the year , the United States will host the InternationalMathematical Olympiad. Let , , and be distinct positive integerssuch that the product . What's the largest possiblevalue of the sum ?SolutionThe sum is the highest if two factors are the lowest.So, and .2Solution.3Each day, Jenny ate of the jellybeans that were in her jar at thebeginning of the day. At the end of the second day, remained. Howmany jellybeans were in the jar originally?Solution4Chandra pays an online service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but inJanuary her bill was because she used twice as much connecttime as in December. What is the fixxed monthly fee?SolutionLet be the fixed fee, and be the amount she pays for the minutes she used in the first month.We want the fixed fee, which is5Points and are the midpoints of sides and of . Asmoves along a line that is parallel to side , how many of the fourquantities listed below change?(a) the length of the segment(b) the perimeter of(c) the area of(d) the area of trapezoidSolution(a) Clearly does not change, and , so doesn'tchange either.(b) Obviously, the perimeter changes.(c) The area clearly doesn't change, as both the base and itscorresponding height remain the same.(d) The bases and do not change, and neither does the height,so the trapezoid remains the same.Only quantity changes, so the correct answer is .6The Fibonacci Sequence starts with two 1s andeach term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in thet units position of a number in the Fibonacci Sequence?SolutionThe pattern of the units digits areIn order of appearance:.is the last.7In rectangle , , is on , and and trisect. What is the perimeter of ?Solution.Since is trisected, .Thus,.Adding, .8At Olympic High School, of the freshmen and of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?There are five times as many sophomores as freshmen.There are twice as many sophomores as freshmen.There are as many freshmen as sophomores.There are twice as many freshmen as sophomores.There are five times as many freshmen as sophomores.Let be the number of freshman and be the number of sophomores.There are twice as many freshmen as sophomores.9If , where , thenSolution, so ...10The sides of a triangle with positive area have lengths , , and .The sides of a second triangle with positive area have lengths , ,and . What is the smallest positive number that is not a possiblevalue of ?SolutionFrom the triangle inequality, and . The smallestpositive number not possible is , which is .11Two different prime numbers between and are chosen. When theirsum is subtracted from their product, which of the following numbers could be obtained?Two prime numbers between and are both odd.Thus, we can discard the even choices.Both and are even, so one more than is a multipleof four.is the only possible choice.satisfy this, .12Figures , , , and consist of , , , and nonoverlapping unitsquares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?SolutionSolution 1We have a recursion:.I.E. we add increasing multiples of each time we go up a figure.So, to go from Figure 0 to 100, we add.Solution 2We can divide up figure to get the sum of the sum of the firstodd numbers and the sum of the first odd numbers. If you do not see this, here is the example for :The sum of the first odd numbers is , so for figure , there areunit squares. We plug in to get , which ischoice13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?SolutionIn each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.In the second column from the right, there are two spaces for pegs. One of them is in the same row as the corner peg, so there is only one remaining choice left for the yellow peg in this column.By similar logic, we can fill in the yellow pegs as shown:After this we can proceed to fill in the whole pegboard, so there is onlyarrangement of the pegs. The answer is14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were , ,, , and . What was the last score Mrs. Walter entered?The sum of the first scores must be even, so we must choose evensor the odds to be the first two scores.Let us look at the numbers in mod .If we choose the two odds, the next number must be a multiple of ,of which there is none.Similarly, if we choose or , the next number must be amultiple of , of which there is none.So we choose first.The next number must be 1 in mod 3, of which only remains.The sum of the first three scores is . This is equivalent to in mod.Thus, we need to choose one number that is in mod . is the onlyone that works.Thus, is the last score entered.15Two non-zero real numbers, and , satisfy . Which of thefollowing is a possible value of ?SolutionSubstituting , we get16The diagram shows lattice points, each one unit from its nearestneighbors. Segment meets segment at . Find the length ofsegment .SolutionSolution 1Let be the line containing and and let be the line containingand . If we set the bottom left point at , then ,, , and .The line is given by the equation . The -intercept is, so . We are given two points on , hence we cancompute the slope, to be , so is the lineSimilarly, is given by . The slope in this case is ,so . Plugging in the point gives us , so is theline .At , the intersection point, both of the equations must be true, soWe have the coordinates of and , so we can use the distanceformula here:which is answer choiceSolution 2Draw the perpendiculars from and to , respectively. As it turnsout, . Let be the point on for which ., and , so by AA similarity,By the Pythagorean Theorem, we have ,, and . Let , so, thenThis is answer choiceAlso, you could extend CD to the end of the box and create two similar triangles. Then use ratios and find that the distance is 5/9 of the diagonal AB. Thus, the answer is B.17Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?SolutionConsider what happens each time he puts a coin in. If he puts in a quarter, he gets five nickels back, so the amount of money he has doesn't change. Similarly, if he puts a nickel in the machine, he gets five pennies back and the money value doesn't change. However, if he puts a penny in, he gets five quarters back, increasing the amount of money he has by cents.This implies that the only possible values, in cents, he can have are the ones one more than a multiple of . Of the choices given, theonly one is18Charlyn walks completely around the boundary of a square whose sides are each km long. From any point on her path she can seeexactly km horizontally in all directions. What is the area of theregion consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?SolutionThe area she sees looks at follows:The part inside the walk has area . The part outside thewalk consists of four rectangles, and four arcs. Each of the rectangles has area . The four arcs together form a circle with radius .Therefore the total area she can see is, which rounded to the nearest integeris .19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square isSolutionLet the square have area , then it follows that the altitude of one ofthe triangles is . The area of the other triangle is .By similar triangles, we haveThis is choice(Note that this approach is enough to get the correct answer in the contest. However, if we wanted a completely correct solution, we should also note that scaling the given triangle times changes eachof the areas times, and therefore it does not influence the ratio ofany two areas. This is why we can pick the side of the square.)20Let , , and be nonnegative integers such that .What is the maximum value of ?SolutionThe trick is to realize that the sum is similarto the product .If we multiply , we get.We know that , therefore.Therefore the maximum value of is equal tothe maximum value of . Now we will find this maximum.Suppose that some two of , , and differ by at least . Then thistriple is surely not optimal.Proof: WLOG let . We can then increase the value ofby changing and .Therefore the maximum is achieved in the cases where is arotation of . The value of in this case is . And thus the maximum of is.21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?I. All alligators are creepy crawlers.II. Some ferocious creatures are creepy crawlers.III. Some alligators are not creepy crawlers.SolutionWe interpret the problem statement as a query about three abstract concepts denoted as "alligators", "creepy crawlers" and "ferocious creatures". In answering the question, we may NOT refer to reality -- for example to the fact that alligators do exist.To make more clear that we are not using anything outside the problem statement, let's rename the three concepts as , , and .We got the following information:▪If is an , then is an .▪There is some that is a and at the same time an .We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are s, but only Johnny is a "meets both conditions, but the first statement is false.We CAN conclude that the second statement is true. We know that there is some that is a and at the same time an . Pick one suchand call it Bobby. Additionally, we know that if is an , then is an. Bobby is an , therefore Bobby is an . And this is enough toprove the second statement -- Bobby is an that is also a .We CAN NOT conclude that the third statement is true. For example, consider the situation when , and are equivalent (represent thesame set of objects). In such case both conditions are satisfied, but the third statement is false.Therefore the answer is .22One morning each member of Angela's family drank an -ouncemixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?SolutionThe exact value "8 ounces" is not important. We will only use the fact that each member of the family drank the same amount.Let be the total number of ounces of milk drank by the family and the total number of ounces of coffee. Thus the whole family drank a total of ounces of fluids.Let be the number of family members. Then each family memberdrank ounces of fluids.We know that Angela drank ounces of fluids.As Angela is a family member, we have .Multiply both sides by to get .If , we have .If , we have .Therefore the only remaining option is .23When the mean, median, and mode of the list arearranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?SolutionAs occurs three times and each of the three other values just once,regardless of what we choose the mode will always be .The sum of all numbers is , therefore the mean is .The six known values, in sorted order, are . From this sequence we conclude: If , the median will be . If , themedian will be . Finally, if , the median will be .We will now examine each of these three cases separately.In the case , both the median and the mode are 2, therefore wecan not get any non-constant arithmetic progression.In the case we have , because. Therefore our three values inorder are . We want this to be an arithmetic progression. From the first two terms the difference must be . Therefore thethird term must be .Solving we get the only solution for this case: .The case remains. Once again, we have ,therefore the order is . The only solution is when , i. e., .The sum of all solutions is therefore .24Let be a function for which . Find the sum of allvalues of for which .SolutionIn the definition of , let . We get: . Aswe have , we must have , in other words.One can now either explicitly compute the roots, or use Vieta'sformulas. According to them, the sum of the roots ofis . In our case this is .(Note that for the above approach to be completely correct, we should additionally verify that there actually are two distinct real roots. Thisis, for example, obvious from the facts that and .)25In year , the day of the year is a Tuesday. In year , theday is also a Tuesday. On what day of the week did the ofyear occur?SolutionClearly, identifying what of these years may/must/may not be a leap year will be key in solving the problem.Let be the day of year , the day of year andthe day of year .If year is not a leap year, the day will bedays after . As , that would be a Monday.Therefore year must be a leap year. (Then is days after .)As there can not be two leap years after each other, is not a leapyear. Therefore day is days after . We have. Therefore is weekdays before , i.e., is a.(Note that the situation described by the problem statement indeed occurs in our calendar. For example, for we have=Tuesday, October 26th 2004, =Tuesday, July 19th, 2005 and=Thursday, April 10th 2003.)。
2000年10月TOEFL试题含答案
2000年10月TOEFL试题含答案2000年10月TOEFL试题Section One: Listening ComprehensionPart A1. (A) She has had the man's calculator since Thursday.(B) The man's calculator is broken.(C) The man may use her calculator.(D) She'll return the man's calculator onThursday.2. (A) Buy a different kind of medicine.(B) See a doctor.(C) Take a second pill.(D) Avoid taking any medication.3. (A) He'll go running after his study group meeting.(B) He doesn't agree with the woman about the weather.(C) He doesn't like to go running.(D) He'll go with the woman this afternoon.4. (A) Another friend commented on his haircut too.(B) The woman has mistaken him for another person.(C) He decided to try a new barbershop.(D) A different person cut his hair this time.5. (A) The man shouldn't be surprised at how busy he is.(B) The man should leave more time for his studies.(C) The man should try to find a different job.(D) The bookstore will be hiring more people.6. (A) The woman should get more sleep.(B) The woman may be sicker than she realizes.(C) He isn't sick.(D) He doesn't think the woman is sick.7. (A) The art museum isn't open today.(B) The number 42 bus doesn't run on Mondays.(C) The man should wait for the number' 42 bus.(D) She has never taken the bus to the art museum.8. (A) The man should buy a jacket instead of a suit.(B) The green jacket doesn't fit as well as the blue one.(C) The style of clothing is more important than the color.(D) The man looks better in blue.9. (A) The woman will get to her class on time.(B) The woman should go to a different counter.(C) He doesn't like sandwiches very much.(D) He's having trouble deciding what to eat.10. (A) Her sister's train is late.(B) Her sister will visit in three months.(C) She'll have to leave without her sister.(D) She's eager to see her sister.11. (A) She's pleased they were invited.(B) Susan gave them the wrong directions.(C) They'll probably be late for dinner.(D) Susan's house is probably nearby.12. (A) Buy some orange juice for the woman.(B) Borrow some money from the woman.(C) Drive the woman to the store.(D) Pay back money the woman lent him.13. (A) She hasn't worn the dress in a long time.(B) She doesn't like the dress very much.(C) She intends to give the dress to her sister.(D) She doesn't remember where her sister bought the dress.14. (A) She never cleans the apartment.(B) She's doing a report with her roommate.(C) She's too busy to clean the apartment.(D) She doesn't like sharing an apartment.15. (A) He'll try to finish the novel tonight.(B) He liked the novel very much.(C) He doesn't remember where he put the novel.(D) He's looking forward to the next literature assignment.16. (A) He doesn't like to wake up early in the morning.(B) The woman seems unusually sad.(C) There's no special reason for his good mood.(D) He wasn't in a good mood when he woke up.17. (A) Get a ride to the station with the woman.(B) Take the woman to the station.(C) Borrow the woman's car to go to the station.(D) Drive his car instead of taking the train.18. (A) Review the assignment by himself.(B) Wait a few minutes before trying to phone John again.(C) Ask one of John's housemates about the assignment.(D) Go over to John's house.19. (A) He won't vote for the woman.(B) He may also run for class president.(C) The woman already asked him for his vote.(D) The woman should ask his roommate to vote for her.20. (A) She isn't sure that the author's ideas would work.(B) The author isn't an expert in economics.(C) She has a better theory about the economy.(D) The author spends too much time arguing about details.21. (A) She doesn't agree with the man.(B) The man doesn't need an official grade report.(C) Official copies of grades used to be cheaper.(D) The man should go to a different office.22. (A) Take her bicycle to the repair shop.(B) Leave her bicycle outside.(C) Go to work when it stops raining.(D) Check to make sure the garage is dry.23. (A) Others should hear about the man's accomplishment.(B) The man should avoid talking about his accomplishment.(C) The man's parents helped him gel the scholarship.(D) The man's parents already told her about his scholarship.24. (A) The course is too difficult,(B) The professor changed his mind.(C) The final exam was cancelled.(D) The woman misunderstood the professor.25. (A) The coffee used to taste better.(B) He's surprised that the woman drinks coffee.(C) He'd rather drink something other than coffee.(D) The coffee tastes the same as before.26. (A) Come back later in the day.(B) Join the staff meeting. .(C) Wait for the pool to open.(D) Wait for the competition to begin.27. (A) He'd like to go to the theater Friday night.(B) He already has a ticket for the Friday night performance.(C) He doesn't think he can exchange his ticket.(D) He rarely goes to the movies.28. (A) She took a history class last year.(B) She doesn't trust the man's opinion.(C) She probably won't take any history classes.(D) She didn't like her sociology professor.29, (A) The other job wouldn't have paid for her tuition.(B) The woman should have taken the other job offer.(C) The woman should get an advanced degree.(D) Paid tuition is only a small benefit.30. (A) The man should have signed her up for the class.(B) The man needs to pay more attention in class.(C) She warned the man not to take an early morning class.(D) She thought the chemistry class was difficult.PartB31. (A) To return some business books.(B) To apply for a new library card.(C) To check out some books from the library(D) To find out where the art books are located.32. (A) The library assistant thinks he has an overdue book.(B) The books he needs have been checked out by someone else.(C) The library assistant is unable to locate the books that he needs.(D) A library notice was sent to him at his previous address.33. (A) To explain why he had difficulty finding the library.(B) To explain why he couldn't have borrowed library books in June.(C) To explain why he doesn't yet have a library card,(D) To explain why he needs assistance in locating a book.34. (A) The man has mistakenly received someone else's books.(B) The man changed his major from art to business.(C) The man recently moved off campus.(D) There are two students named Robert Smith.35. (A) See if he is related to any of the students.(B) Apply for a job as a library assistant.(C) Use his middle name.(D) Use a different library.36. (A) Its similarities to previous architecture.(B) Its impressive and distinctive features.(C) Methods used in its construction.(D) How it was preserved for later generations.37. (A) Public market days.(B) Races and sporting events.(C) Processions of priests.(D) Speeches by politicians.38. (A) It was removed by an invading army.(B) It broke off when part of the hall collapsed.(C) It was cut away to let banners pass through the entrance.(D) It was later used in building another temple.39. (A) Its lighting.(B) Its sound quality.(C) Its air circulation.(D) Its stability in an earthquake.Part C40. (A) The relationship between physics and philosophy.(B) Ancient Greek beliefs about matter and motion.(C) The effects of Aristotle's philosophy on current theories of physics.(D) Aristotle's use of fire in scientific experiments.41. (A) Earth.(B) Water.(C) Air.(D) Fire.42. (A) Pulling and pushing motions.(B) Throwing motions.(C) Planetary motions.(D) Natural downward or upward motions.43. (A) It's pushed away from Earth by fire.(B) It's trying to return to its natural resting place.(C) It's attracted to other planets.(D) Its main substance is water.44. (A) To solicit volunteers for Turtle Watch.(B) To give an assignment to a biology class.(C) To warn students not to hurt green turtles.(D) To describe the nesting and hatching activities of the green turtle.45. (A) The lights attract predators.(B) They need to save electricity.(C) The baby turtles are attracted to light.(D) The volunteers use lights for signals.46. (A) Write a report about their activities.(B) Attend make-up classes with Dr.Webster.(C) Help find turtle eggs before they hatch.(D) Spend two hours working for the project.47. (A) How people in rural areas preserved food.(B) The construction of icehouses.(C) An important industry in the nineteenth century.(D) How improvements in transportation affected industry.48. (A) Modem technology for the kitchen.(B) Improved transportation systems.(C) Industrial use of streams and rivers.(D) Increased temperatures in many areas.49. (A) Only wealthy families had them.(B) They were important to the ice industry.(C) They were built mostly on the east coast.(D) They are no longer in common use.50. (A) To keep train engines cool.(B) To preserve perishable food.(C) To store ice while it was being transported.(D) To lift blocks of ice from frozen lakes and ponds.Section Two: Structure and Written Expression1. The role of the ear is acoustic disturbances into neural signals suitable for transmission to the brain.(A) to code(B) so that coded(C) coded(D) it coding2. The imagist movement in poetry arose during the second decade of the twentieth century----against romanticism,(A) when a revolt(B) as a revolt(C) a revolt was(D) that a revolt3. Virtually species have biological clocks that regulate their metabolism over a 24-hour period.(A) all there are(B) all(C) all are(D) they all4. According to United States criminal law, insanity may relieve a person from the usual legal consequences(A) what his or her acts have(B) of his or her acts are(C) of his or her acts(D) what of his or her acts5. In addition to a place where business deals are made, a stock exchange collects statistics, publishes price quotations, and sets rules and standards for trading.(A) being(B) it is(C) that which(D) where is6. The first inhabitants of the territories ------Canada came across the Bering Strait and along the edge of the Arctic ice.(A) make up that now(B) make up now that(C) that make up now(D) that now make up7. need for new schools following the Second World War that provided the sustained thrust for the architectural program in Columbus, Indiana.(A) Since the(B) To be the(C) The(D) It was the8. The soybean contains vitamins, essential minerals, high percentage of protein.(A) a(B) and a(C) since a(D) of which a9. Hail is formed when a drop of rain is carried by an updraft to an altitude where -----to freeze it.(A) is the air cold enough(B) the air cold enough(C) the cold enough air(D) the air is cold enough10. Geometrically, the hyperbolic functions are related to the hyperbola,------the trigonometric functions are related to the circle.(A) just as(B) same(C) similar to(D) and similar11. , Kilauea is one of the world's mostactive volcanoes, having erupted dozens of times since 1952.(A) The big island of Hawaii's location(B) Locates the big island of Hawaii(C) Located on the big island of Hawaii(D) On the big island of Hawaii's location12. Not until the eighteenth century the complex chemistry of metallurgy(A) when scientists began to appreciate(B) did scientists begin to appreciate(C) scientists who were beginning to appreciate(D) the appreciation of scientists began13. 1810, water-powered textile manufacturing arrived in New Hampshire with the founding of a company in Manchester that manufactured cotton and wool.(A) Early(B) In the early(C) As early as(D) When early14. The settings of Eudora Welty's stories may be rather limited, but--------about human natureis quite broad.(A) exposes(B) exposes that(C) she exposes(D) what she exposes15. Lichens grow extremely well in very cold parts of the world -------- plants can survive.(A) where few other(B) few others(C) where do few others(D) there are few others16．The pear tree has simple, oval leaves that are smoother and shinier than them of theA B C D apple.17．In the orbit of a planet around the Sun, the point closest to the Sun is called it theA B C D perihelion.18．In the early 1900’s, Roy Harris created and promoted a distinctly American style of A B Cclassical music and greatly influenced a number of composer in the United States.D19．The eighteenth century witnessed the emergence of North American ports,Aparticular Boston, New York, and Philadelphia, as major commercial centers within theB C D British empire.20．Guitarlike instruments have exist since ancient times, but the first written mentionA B Cof the guitar itself is from the fourteenth century.D21．The law of biogenesis is the principle what all living organisms are derived from aA B Cparent or parents.D22．Onyx is a mineral that can be recognized its regular and straight parallel bands ofA B Cwhite, black, or brown.D23．There are as many as 200 million insects for every human beings, and in fact theirA Btotal number exceeds that of all other animals taken together.24．Native to South America and cultivated there for thousands of years, the peanutA Bis said to have introduced to North America by early explorers.C D25．Originally canoes were made by the hollowing out of logs and used were for combatA B Cas well as transport.D26．Among the symptoms of measles, which takes about twelve days to incubate, are a highA B C fever, swelling of glands in the neck, a cough, and sensitive to light.D27．Ice crystals in a glacier tends to melt and recrystallize within a brief moment of travel A B C Don a downhill glide.28．Photograph was revolutionized in 1851 by the introduction of the collodion processA B Cfor making glass negatives.D29．The piano is a stringed musical instrument in which the strings are strike byA B Cfelt-covered hammers controlled by a keyboard.D30．The sounds used in human languages to create meaning consist of small variation inA Bair pressure can be sensed by the ear.C D31．The mountains, especially the Rocky Mountains, formerly constituted a seriouslyA Bbarrier to east-west trade in British Columbia.C D32．Telescope are frequently used in astronomy to collect light from a celestial object,A Bbring the light into focus, and producing a magnified image.C D33．Diamond is the hardest known substance, so diamond can be cut only by anotherA B C D diamonds.34．There are about 350 species and subspecies of birds in danger of become extinct,A Bwith a large number of them, 117 in all, found on oceanic islands.35．The nineteenth-century romantic movement in art was partially a reaction to whatA B Cwas perceived as overemphasis on reasonable and order in neoclassicism.D36．Like triglycerides, cholesterol is a type of fat that is both consumed in the diet butA B C D manufactured by the body.37．Both the United States silver dollar and half-dollar, first minted in 1794, had a figureA B Cof Liberty on one side and a eagle on the reverse side.D38．For an advertisement to be effective, its production and placement must to be basedA B Con a knowledge of human nature and a skilled use of the media.D39．While photosynthesis in green plants, light energy is captured and used to convertA B Cwater, carbon dioxide, and minerals into oxygen and energy-rich organic compounds.D40．The Democratic Party, the most oldest existing political party in the United States,A Bhas played a vital role in the nation’s history.C DSection Three: Reading ComprehensionQuestions 1-10One area of paleoanthropological study involves the eating and dietary habits of hominids, erect bipedal primates—including early humans. It is clear that at some stage of history, humans began to carry their food to central places, called home bases, where itLine was shared and consumed with the young and other adults. The use of home bases is a(5) fundamental component of human social behavior; the common meal served at a common hearth is a powerful symbol, a mark of social unity. Home base behavior does not occur among nonhuman primates and is rare among mammals. It is unclear when humans began to use home bases, what kind of communications and social relations were involved, and what the ecological and food-choice contexts of the shift were. Work on early tools,(10) surveys of paleoanthropological sites, development and testing of broad ecological theories, and advances in comparative primatology are contributing to knowledge about this central chapter in human prehistory.One innovative approach to these issues involves studying damage and wear on stone tools. Researchers make tools that replicate excavated specimens as closely as possible(15) and then try to use them as the originals might have been used, in woodcutting, hunting, or cultivation. Depending on how the tool is used, characteristic chippage patterns andmicroscopically distinguishable polishes develop near the edges. The first application of this method of analysis to stone tools that are 1.5 million to 2 million years old indicates that, from the start, an important function of early stone tools was to extract highly(20) nutritious food—meat and marrow-from large animal carcasses. Fossil bones with cut marks caused by stone tools have been discovered lying in the same 2-million-year-old layers that yielded the oldest such tools and the oldest hominid specimens (including humans) with larger than ape-sized brains. This discovery increases scientists' certainty about when human ancestors began to eat more meat than present-day nonhuman(25) primates. But several questions remain unanswered: how frequently meat eating occurred; what the social implications of meat eating were; and whether the increased use of meat coincides with the beginnings of the use of home bases.1. The passage mainly discusses which of the following aspects of hominid behavior?(A) Changes in eating and dietary practices(B) The creation of stone hunting tools(C) Social interactions at home bases(D) Methods of extracting nutritious food from carcasses2. According to the passage, bringing a meal to a location to be shared by many individuals is(A) an activity typical of nonhuman primates(B) a common practice among animals that eat meat(C) an indication of social unity .(D) a behavior that encourages better dietary habits3. The word "consumed" in line 4 is closest in meaning to(A) prepared(B) stored(C) distributed(D) eaten4. According to paragraph 2, researchers make copies of old stone tools in order to(A) protect the old tools from being worn out(B) display examples of the old tools in museums(C) test theories about how old tools were used(D) learn how to improve the design of modern tools5. In paragraph 2, the author mentions all of the following as examples of ways in which early stone tools were used EXCEPT to(A) build home bases(B) obtain food(C) make weapons(D) shape wood6. The word "innovative" in line 13 is closest in meaning to(A) good(B) new(C) simple(D) costly7. The word "them" in line 15 refers to(A) issues(B) researchers(C) tools(D) specimens8. The author mentions "characteristic chippage patterns" in line 16 as an example of(A) decorations cut into wooden objects(B) differences among tools made of various substances(C) impressions left on prehistoric animal bones(D) indications of wear on stone tools9. The word "extract" in line 19 is closest in meaning to(A) identify(B) remove(C) destroy(D) compare10. The word "whether" in line 26 is closest in meaning to(A) if(B) how(C) why(D) whenQuestions 11-20In seventeenth-century colonial North America, all day-to-day cooking was done in the fireplace. Generally large, fireplaces were planned for cooking as well as for warmth. Those in the Northeast were usually four or five feet high, and in the South, they wereLine often high enough for a person to walk into. A heavy timber called the mantel tree was(5) used as a lintel to support the stonework above the fireplace opening. This timber might be scorched occasionally, but it was far enough in front of the rising column of heat to be safe from catching fire.Two ledges were built across from each other on the inside of the chimney. On these rested the ends of a "lug pole" from which pots were suspended when cooking. Wood(10) from a freshly cut tree was used for the lug pole, so it would resist heat, but it had to be replaced frequently because it dried out and charred, and was thus weakened. Sometimes the pole broke and the dinner fell into the fire. When iron became easier to obtain, it was used instead of wood for lug poles, and later fireplaces had pivoting metal rods to hang pots from. (15) Beside the fireplace and built as part of it was the oven. It was made like a small, secondary fireplace with a flue leading into the main chimney to draw out smoke. Sometimes the door of the oven faced the room, but most ovens were built with the opening facing into the fireplace. On baking days (usually once or twice a week) a roaring fire of "oven wood," consisting of brown maple sticks, was maintained in the oven until its(20) walls were extremely hot. The embers were later removed, bread dough was put into the oven, and the oven was sealed shut until the bread was fully baked.Not ai! baking was done in a big oven, however. Also used was an iron "bake kettle," which looked like a stewpot on legs and which had an iron lid. This is said to have worked well when it was placed in the fireplace, surrounded by glowing wood embers, with more(25) embers piled on its lid.11. Which of the following aspects of domestic life in colonial North America does the passage mainly discuss?(A) Methods of baking bread(B) Fireplace cooking(C) The use of iron kettles in a typical kitchen(D) The types of wood used in preparing meals12. The author mentions the fireplaces built in the South to illustrate(A) how the materials used were similar to the materials used in northeastern fireplaces(B) that they served diverse functions(C) that they were usually larger than northeastern fireplaces(D) how they were safer than northeastern fireplaces13. The word "scorched" in line 6 is closest in meaning to(A) burned(B) cut(C) enlarged(D)bent14. The word "it" in line 6 refers to(A) the stonework(B) the fireplace opening(C) the mantel tree(D) the rising column of heat15. According to the passage, how was food usually cooked in a pot in the seventeenth century?(A) By placing the pot directly into the fire(B) By putting the pot in the oven(C) By filling the pot with hot water(D) By hanging the pot on a pole over the fire16. The word "obtain" in line 12 is closest in meaning to(A) maintain(B) reinforce(C) manufacture(D) acquire17. Which of the following is mentioned in paragraph 2 as a disadvantage of using a wooden lug pole?(A) It was made of wood not readily available.(B) It was difficult to move or rotate.(C) It occasionally broke.(D) It became too hot to touch.18. It can be inferred from paragraph 3 that, compared to other firewood, "oven wood" produced(A) less smoke(B) more heat(C) fewer embers(D) lower flames19. According to paragraph 3, all of the following were true of a colonial oven EXCEPT:(A) It was used to heat the kitchen every day.(B) It was built as part of the main fireplace.(C) The smoke it generated went out through the main chimney.(D) It was heated with maple sticks.20. According to the passage, which of the following was an advantage of a "bake kettle"?(A) It did not take up a lot of space in the fireplace.(B) It did not need to be tightly closed.(C) It could be used in addition to or instead of the oven.(D) It could be used to cook several foods at one time.Questions 21-29Butterflies are among the most extensively studied insects—an estimated 90 percent of theworld's species have scientific names. As a consequence, they are perhaps the best group of insects for examining patterns of terrestrial biotic diversity and distribution. Butterflies alsohave a favorable image with the general public. Hence, they are an excellent group for(5) communicating information on science and conservation issues such as diversity.Perhaps the aspect of butterfly diversity that has received the most attention over the past century is the striking difference in species richness between tropical and temperate regions.For example, in 1875 one biologist pointed out the diversity of butterflies in the Amazon when he mentioned that about 700 species were found within an hour's walk, whereas the total (10) number found on the British islands did not exceed 66, and the whole of Europe supported only321. This early comparison of tropical and temperate butterfly richness has been well confirmed.A general theory of diversity would have to predict not only this difference betweentemperate and tropical zones, but also patterns within each region, and how these patterns vary (15) among different animal and plant groups. However, for butterflies, variation of species richnesswithin temperate or tropical regions, rather man between them, is poorly understood. Indeed, comparisons of numbers of species among the Amazon basin, tropical Asia, and Africa are still mostly "personal communication" citations, even for vertebrates, In other words, unlike comparison between temperate and tropical areas, these patterns are still in the documentation(20)phase.In documenting geographical variation in butterfly diversity, some arbitrary, practicaldecisions are made. Diversity, number of species, and species richness are used synonymously; little is known about the evenness of butterfly distribution. The New World butterflies makeup the preponderance of examples because they are the most familiar species. It is hoped that (25) by focusing on them, the errors generated by imperfect and incomplete taxonomy will be minimized.21. Which aspect of butterflies does the passage mainly discuss?(A) Their physical characteristics(B) Their names(C) Their adaptation to different habitats(D) Their variety22. The word "consequence" in line 2 is closest in meaning to(A) result(B) explanation(C) analysis(D) requirement23. Butterflies are a good example for communicating information about conservation issues because they(A) are simple in structure(B) are viewed positively by people(C) have been given scientific names(D) are found mainly in temperateclimates24. The word "striking" in line 7 is closest in meaning to(A) physical(B) confusing(C) noticeable(D) successful25. The word "exceed" in line 10 is closest in meaning to(A) locate(B) allow(C) go beyond(D) come close to26. All of the following are mentioned as being important parts of a general theory of diversity EXCEPT(A) differences between temperate and tropical zones(B) patterns of distribution of species in each region(C) migration among temperate and tropical zones(D) variation of patterns of distribution of species among different animals and plants。
course2_1101
November 2001Course 2Interest Theory, Economics and Finance Society of Actuaries/Casualty Actuarial Society1.Ernie makes deposits of 100 at time 0, and X at time 3 . The fund grows at a force ofinterest2100ttδ=, t > 0 .The amount of interest earned from time 3 to time 6 is X. Calculate X.(A)385(B)485(C)585(D)685(E)7852.The production of a good requires two inputs, labor and capital. At its current level ofdaily output, a competitive firm employs 100 machine hours of capital and 200 laborhours. The marginal product of machine hours is 10 units. The marginal product of labor hours is 5 units. The rental rate, or “price,” of capital is 20 per machine hour.If the firm minimizes its costs, what is the hourly wage rate?(A) 2.5(B) 5.0(C)10.0(D)20.0(E)40.0st year, a country's output increased 2.6%. The country’s capital stock increased4.0%while its labor hours increased 2.0%. The labor share of total income was 70%.What was the country’s total factor productivity growth last year?(A)–0.8(B) 0.0(C) 2.3(D) 5.2(E) 6.04.Consider a project lasting one year. The initial outlay is 100,000 at the beginning of theyear and the expected inflow is 120,000 at the end of the year. The opportunity cost of capital for the project is 20%, the borrowing rate is 8%, and the marginal tax rate is 35%.Calculate the adjusted present value if the company borrows 54% of the project’srequired investment.(A) 800(B)1260(C)1400(D)3150(E)35005.Mike buys a perpetuity-immediate with varying annual payments. During the first 5years, the payment is constant and equal to 10 . Beginning in year 6, the payments start to increase. For year 6 and all future years, the current year’s payment is K% larger than the previous year’s payment.At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50 .Calculate K, given K < 9.2 .(A) 4.0(B) 4.2(C) 4.4(D) 4.6(E) 4.86. A 10-year loan of 2000 is to be repaid with payments at the end of each year.It can be repaid under the following two options:(i)Equal annual payments at an annual effective rate of 8.07% .(ii)Installments of 200 each year plus interest on the unpaid balanceat an annual effective rate of i .The sum of the payments under option (i) equals the sum of the payments under option (ii) .Determine i .(A)8.75%(B)9.00%(C)9.25%(D)9.50%(E)9.75%7.The demand curve for Product X is steep, whereas the demand curve for Product Y is almost flat.The supply curves for the two products are identical. Equilibrium price and quantity are thesame for the two products. Consider a 5% excise tax on both products.Which of the following statements about the impact of the tax is FALSE?(A)Equilibrium quantity will decrease less for Product X than for Product Y .(B)Consumers face a larger price increase for Product X than for Product Y .(C)The tax burden experienced by producers is larger for Product Y than for Product X .(D)The government collects more taxes from Product Y than from Product X .(E)In each market, the economic incidence of the tax would be the same if the 5%excise tax were replaced by a 5% sales tax.8. A corporation is considering an investment in one of two potential projects. Each projectrequires an initial investment of 5000 .Project X will produce cash flows of 300 at the end of each 6-month period. The cashflows are expected to continue forever. The first cash flow is expected 6 months after the initial investment.Project Y will have a single cash flow of Z, which will be received exactly 5 years after the initial investment.The IRR on both projects is the same.Calculate the profitability index on Project Y, using an annual effective interest rate of10% .(A) 8.5%(B) 9.4%(C)10.3%(D)11.2%(E)12.1%9. A loan is amortized over five years with monthly payments at a nominal interest rate of9% compounded monthly. The first payment is 1000 and is to be paid one month from the date of the loan. Each succeeding monthly payment will be 2% lower than the prior payment.Calculate the outstanding loan balance immediately after the 40th payment is made.(A)6751(B)6889(C)6941(D)7030(E)734410.Five self-interested colleagues are dining together at a restaurant where desserts cost 5.0each. If dining alone, one of the colleagues would be willing to pay 3.5 for a dessert.Two of the colleagues would be willing to pay 4.5 for a dessert while the other two would be willing to pay 5.5 . The restaurant will not give individual checks, so the colleagueshave agreed to split the bill evenly.If the five colleagues each act out of their own self-interest, how many desserts will beordered?(A)0(B)2(C)3(D)4(E)511.In Country X, the effect of a one-unit increase in income is to increase consumption by0.70 and to decrease net exports by 0.10 . Changes in income do not affect tax payments.What is the government expenditure multiplier in Country X?(A)0.91(B) 1.43(C) 2.50(D) 3.33(E) 5.0012.To accumulate 8000 at the end of 3n years, deposits of 98 are made at the end of each ofthe first n years and 196 at the end of each of the next 2n years.The annual effective rate of interest is i . You are given (l + i)n = 2.0 .Determine i .(A)11.25%(B)11.75%(C)12.25%(D)12.75%(E)13.25%13.Marvin has the following newspaper excerpt of option listings:Strike January April ClosingPrice Vol Last Vol Last PricePfizer Pfizer Philip Morris Philip Morris CallCallPutPut25305080185777520.50141839245941152226266363Assuming an option contract is for 100 shares, and no transaction costs, which of the following is worth the most at market closing?(A)Selling his holding of 2 Philip Morris January put contracts at a strike price of 80(B)Selling his holding of 50 shares of Philip Morris stock(C)Selling his holding of 30 Pfizer April call contracts at a strike price of 30(D)Exercising his 35 Pfizer April call contracts at a strike price of 25 and instantlyselling the stock(E)Exercising his 5000 Pfizer April call contracts at a strike price of 30 and instantlyselling the stock14.Firms in a particular industry purchase inputs in perfectly competitive markets and selltheir output in a perfectly competitive market. The current market price is 3.50 per unit.The minimum long-run average total cost is 4.00 .Over time, what will happen to the number of firms in this industry and the equilibrium price?(A)The number of firms will decrease, and price will rise.(B)The number of firms will decrease, and price will fall.(C)The number of firms will increase, and price will rise.(D)The number of firms will remain unchanged, but they will raise their prices.(E)The number of firms will remain unchanged, but they will lower their costs.15.Which of the following will lead to a decrease in the money supply?(A)The central bank reduces its bond portfolio.(B)The central bank reduces reserve requirements.(C)The central bank reduces the discount rate.(D)The demand for currency declines.(E)There is an exogenous increase in market interest rates.16.Olga buys a 5-year increasing annuity for X .Olga will receive 2 at the end of the first month, 4 at the end of the second month, and for each month thereafter the payment increases by 2 .The nominal interest rate is 9% convertible quarterly.Calculate X .(A)2680(B)2730(C)2780(D)2830(E)288017.An investor is considering opening one of two equally risky franchises, Shop X orShop Y. This investor has collected the following information:Investment at time zero ExpectedSalesProbability ofExpected SalesExpected Annual CashFlow beginning at time 1 high0.6perpetuity of 120Shop X300low0.4perpetuity of 40high0.5perpetuity of 100 Shop Y200low0.5perpetuity of 50 The expected net present value of investing in Shop X is 800 .What is the expected net present value of investing in Shop Y?(A)550(B)616(C)682(D)710(E)73818.Preventing market entry is one of the key motivations behind the actions of a monopolist.Which of the following actions by a monopolist is least likely to prevent entry?(A)Spending considerable amounts on research and development(B)Hiring the industry's best and brightest people(C)Lobbying policy makers to prevent the removal of tariffs(D)Advertising to shift the industry demand curve out(E)Lowering the price below the monopoly price and providingcustomers good quality products19.If there is an imbalance between supply and demand, firms might choose to adjust thequantity produced rather than the price of the output.When would firms most likely make this choice?(A)When the period is short-run(B)When output is perishable(C)When price elasticity of demand is known precisely(D)When consumers are informed quickly of price changes(E)When all output is homogeneous and markets are perfectly competitive20.You are given the following information about the activity in two different investmentaccounts:Account KFund value ActivityDate before activity Deposit WithdrawalJanuary 1, 1999100.0July 1, 1999125.0XOctober 1, 1999110.02XDecember 31, 1999125.0Account LFund value ActivityDate before activity Deposit WithdrawalJanuary 1, 1999100.0July 1, 1999125.0XDecember 31, 1999105.8During 1999, the dollar weighted return for investment account K equals the timeweighted return for investment account L, which equals i .Calculate i .(A)10%(B)12%(C)15%(D)18%(E)20%21.In a perfectly competitive market, the market supply and demand functions are:Supply:P = 4QDemand:P = 100 – Qwhere P is price and Q is quantity.The total variable cost function (TVC) for an individual firm is:TVC = 76q + q2where q is the quantity produced by the firm.What is the output for the firm?(A) 0(B) 2(C) 8(D)20(E)3822. A firm’s market value balance sheet is as follows:The risk-free rate of interest is 3.5%, b equity is 1.2, b debt is 0.2, and the return on the market portfolio is 14.4% .Calculate the company’s cost of capital.(A) 5.7%(B) 7.2%(C)10.0%(D)12.2%(E)16.6%23.Suppose purchasing power parity exists between the United States and the UnitedKingdom. The current inflation rates are 3.0% in the U.S. and 5.0% in the U.K.The current nominal exchange rate is 2.0 dollars per pound.What is the expected exchange rate one year from now?(A) 1.20(B) 1.96(C) 2.00(D) 2.04(E) 3.3324.David can receive one of the following two payment streams:(i)100 at time 0, 200 at time n, and 300 at time 2n(ii)600 at time 10At an annual effective interest rate of i, the present values of the two streams are equal.Given v n = 0.75941, determine i .(A) 3.5%(B) 4.0%(C) 4.5%(D) 5.0%(E) 5.5%25. A stock currently has a price of 45.00 and pays no dividends. One year from now, thereis a 50% probability that the price of the stock will be 30.00 and 50% that it will begreater than 40.00 .The risk-free rate is 4% .Calculate the price of a one-year European call option with an exercise price of 40.00 .(A) 4.81(B) 6.35(C) 9.81(D)10.00(E)11.3526.Joe's budget line is defined by the equation y = –0.5x + 18, where y is the quantityof meat and x is the quantity of bread.If Joe's income increases, what will happen to the slope and the y-intercept of his budget line?(A)Both the slope and the y-intercept will increase.(B)The slope will increase while the y-intercept will remain unchanged.(C)The slope will decrease while the y-intercept will remain unchanged.(D)The slope will remain unchanged while the y-intercept will decrease.(E)The slope will remain unchanged while the y-intercept will increase.27. A man turns 40 today and wishes to provide supplemental retirement income of 3000 atthe beginning of each month starting on his 65th birthday. Starting today, he makesmonthly contributions of X to a fund for 25 years. The fund earns a nominal rate of 8% compounded monthly.Each 1000 will provide for 9.65 of income at the beginning of each month starting on his 65th birthday until the end of his life.Calculate X.(A)324.73(B)326.89(C)328.12(D)355.45(E)450.6528.Payments are made to an account at a continuous rate of (8k + tk), where 010t≤≤ .Interest is credited at a force of interest δt =1 8t+.After 10 years, the account is worth 20,000 . Calculate k .(A)111(B)116(C)121(D)126(E)13129.Suppose a consumer’s income increases and, at the same time, the price of X decreases.Further suppose that in the new situation the consumer purchases less X than before.Which of the following must be true?(A)X is a normal good.(B)X is an inferior good.(C)X is a Giffen good.(D)X is a luxury good.(E)X is a good for which the compensated demand curveis steeper than the uncompensated demand curve.30.Nate intends to invest in two different stocks, X and Y .Stock X has an expected return of 10% and a standard deviation of Z .Stock Y has an expected return of 20% and a standard deviation of 1.5Z .After investing in both stocks, the expected return on Nate’s two-stock portfolio is 12% and the standard deviation is Z .Calculate the correlation between the returns on Stocks X and Y .(A)0.50(B)0.53(C)0.56(D)0.60(E)0.6331.You have decided to invest in two bonds. Bond X is an n -year bond with semi-annual coupons, while bond Y is an accumulation bond redeemable in2nyears. The desired yield rate is the same for both bonds. You also have the following information:Bond X• Par value is 1000 .• The ratio of the semi-annual bond rate to the desired semi-annual yield rate, ri,is 1.03125 .• The present value of the redemption value is 381.50 .Bond Y• Redemption value is the same as the redemption value of bond X .• Price to yield is 647.80 .What is the price of bond X?(A)1019(B)1029(C)1050(D)1055(E)107232. A monopoly faces the following demand and marginal cost functions:Demand:P= 10 – QMarginal Cost:MC = 3Qwhere P is price, Q is quantity, and MC is marginal cost.Now suppose the monopolist is “broken up” by a federal judge such that the marginalcost function becomes the competitive supply function.What will be the difference between the equilibrium price in the competitive market and the monopoly price?(A)0.0(B)0.5(C) 1.0(D) 1.5(E) 2.033. A company’s common stock is currently selling for 25 per share. All of the financialanalysts following the firm are surprised when the company unexpectedly announces that it expects its future economic income to be lower after the next quarter. Assume that the stock market is semi-strongly efficient.How should this news affect the stock price?(A)The price should not change at all.(B)The price should not change until the next quarter.(C)The price should fall immediately to adjust for theexpected slowing earnings growth.(D)The price should fall gradually over the next quarter.(E)The price should go up following the announcement.34.You are given the following data from the national income and product accountsof a country:AccountPersonal Consumption Expenditure 4.5Gross private domestic investment 2.1Producer’s durable equipmentand nonresidential structures 1.0Residential structures0.7Changes in business inventories0.4Exports 1.1Imports0.9Government purchases of goods and services 3.1What is the Gross Domestic Product of this country?(A) 9.5(B) 9.7(C) 9.9(D)11.6(E)12.035.At time t = 0, Sebastian invests 2000 in a fund earning 8% convertible quarterly,but payable annually.He reinvests each interest payment in individual separate funds each earning 9% convertible quarterly, but payable annually.The interest payments from the separate funds are accumulated in a side fund that guarantees an annual effective rate of 7%.Determine the total value of all funds at t = 10 .(A)3649(B)3964(C)4339(D)4395(E)448536.Jack has an equally weighted portfolio of stocks X and Y . The beta of his portfoliois 0.9 . Jill has an equally weighted portfolio of stocks X, Y, and Z . The beta of stock Z is 1.2, the Treasury bill rate of return is 6%, and the expected return on the marketportfolio is 14.4% .What is the expected risk premium on Jill’s portfolio?(A) 6.0%(B)7.6%(C)8.4%(D)8.8%(E)10.1%37. A company’s dividend per share is expected to grow indefinitely at a rate of 5% per year.Suppose the current stock price is 500 and the next annual dividend, payable one yearfrom now, is 10 . Assume the opportunity cost of capital is constant.Three investors, Alex, Bill, and Carl, each invest in the company. Alex invests for oneyear, Bill invests for two years, and Carl invests for three years.Who expects the highest annualized rate of return?(A)Alex(B)Bill(C)Carl(D)Alex, Bill, and Carl all have the same expected rate of return.(E)Not enough information is given here to answer the question.38. A competitive industry is composed of identical firms.Which of the following statements about long-run equilibrium in the industry are true?I.Firms’ accounting profits are equal to their opportunity costs.II.Firms produce the quantity at which average costs are minimized.III.Firms produce the quantity at which marginal costs are minimized.(A)II only(B)I and II only(C)I and III only(D)II and III only(E)I, II, and III39.Suppose there is a simultaneous increase in the money supply and an increase ingovernment purchases.Based on the IS-LM model, what will be the effect on real output and interest ratesin the short run?(A)Real output will decrease, and interest rates will increase.(B)Real output will increase, and interest rates will increase.(C)Real output will increase, and interest rates will decrease.(D)Real output will increase, but the effect on interest rates cannot be determined.(E)The effect on real output cannot be determined, but interest rates will increase.40. A company stock is currently trading at 50 . Over the next year, this stock will eitherincrease in value by 10% or decrease by x%. The risk-free rate is 4%. The value of a one-year European put option for this stock at an exercise price of 50 is 1.28 .Calculate x.(A)0(B)2(C)4(D)6(E)841.Linda consumes two goods, X and Y. At the point of equilibrium for X, Linda'suncompensated demand curve for X has a larger negative slope than does hercompensated demand curve for X. At the point of equilibrium for Y, Linda'suncompensated demand curve for Y has a positive slope while her compensated demand curve for Y has a negative slope.What does this information tell us about X and Y?(A)Both X and Y are normal goods.(B)Neither X nor Y is a normal good.(C)X is a normal good, but Y is an inferior good.(D)Y is a normal good, but X is an inferior good.(E)X is a substitute good, but Y is a complementary good.42. A copier costs X and will have a salvage value of Y after n years.(i)Using the straight line method, the annual depreciation expense is 1000 .(ii)Using the sum of the years digits method, the depreciation expense in year 3 is 800 .(iii)Using the declining balance method, the depreciation expense is 33.125% of the book value in the beginning of the year.Calculate X .(A)4250(B)4500(C)4750(D)5000(E)525043. A firm has an annual dividend yield of 7.5% and a constant dividend growth rate of 3%per year. It also has five-year bonds outstanding that have an annual coupon rate of 8% and are selling at par. The firm has a 40% marginal tax rate and a debt-to-assets ratioof 0.30 .Calculate the firm’s after-tax weighted average cost of capital.(A) 6.9%(B)7.8%(C)8.8%(D)9.2%(E)9.8%44.Which of the following statements is most consistent with the predictions of the rationalexpectations model?(A) A fully anticipated monetary policy will have no effect on either the level of realincome or the price level.(B)In the short run, unanticipated policy might affect the aggregate price level, butnot real income.(C)Expected and actual inflation will always differ by a constant, predictable amount.(D)The unemployment rate will differ from the natural rate of unemployment onlywhen actual inflation differs from expected inflation.(E)Monetary policy is preferred to fiscal policy for permanently changing the level ofreal income.45. A manufacturer buys a machine for 20,000. The manufacturer estimates that the machinewill last 15 years. It will be depreciated using the constant percentage method with anannual depreciation rate of 20%.At the end of each year, the manufacturer deposits an amount into a fund that pays 6%annually. Each deposit is equal to the depreciation expense for that year.How much money will the manufacturer have accumulated in the fund at the end of15 years?(A)29,663(B)34,273(C)36,329(D)38,509(E)46,25046.Two lawyers, Smith and Jones, each practice two types of law, criminal law and divorcelaw. To prepare a brief, it takes Smith 14 hours in a criminal case and 16 hours in adivorce case. It takes Jones 20 hours in a criminal case and 18 hours in a divorce case.Suppose Smith and Jones merged their practices and each handled only the cases inwhich they possessed a comparative advantage. Further suppose that prior to the merger they each handled one criminal and one divorce case a week and that their new firmhandles two of each case a week.How many hours would Smith and Jones gain in efficiency from the merger—i.e., how much less time per week would they collectively require to handle their caseload?(A)0 hours(B) 2 hours(C) 4 hours(D) 6 hours(E)8 hours47.Project P requires an investment of 4000 at time 0 . The investment pays 2000 at time 1and 4000 at time 2 .Project Q requires an investment of X at time 2 . The investment pays 2000 at time 0 and 4000 at time 1 .Using the net present value method and an interest rate of 10%, the net present values of the two projects are equal.Calculate X .(A)5400(B)5420(C)5440(D)5460(E)548048.The after-tax earnings and dividends of a company are expected to increase at a constantrate. The market capitalization rate is 15.5% and is expected to stay constant. Details of the company’s financial statements for the year 2000 are as follows:After-tax earnings 150Dividends 45Average book equity1000Calculate the expected dividend yield, y.(A) 4.5%y≤(B) 4.5% 6.5%y≤<(C) 6.5%8.5%<y≤(D)8.5%10.5%<y≤(E)10.5%y<49. A company has one plant that uses technology from 1990 to manufacture good X. It hasrecently developed a new patented technology to manufacture good X and plans to builda new plant to use its improved technology.Which of the following statements about the company’s economic rents are correct?I.If the company has some degree of monopoly power, its economic rents willbe temporary.II.The company expects to receive economic rents during the term of its patent.III.When the company predicts its economic rents, it will need to consider the impact on its original plant of building the new plant.(A)I only(B)II only(C)III only(D)I and III only(E)II and III only。
北美精算考试试题
北美精算考试试题1. The probability that a visit to a primary care physician’s (PCP) office results in neither lab work nor referral to a specialist is 35% . Of those coming to a PCP’s office, 30% are referred to specialists and 40% require lab work.Determine the probability that a visit to a PCP’s office results in both lab work and referral to a specialist.(A) 0.05(B) 0.12(C) 0.18(D) 0.25(E) 0.352. A study of automobile accidents produced the following data:An automobile from one of the model years 1997, 1998, and 1999 was involved in an accident.Determine the probability that the model year of this automobile is 1997 .(A) 0.22(B) 0.30(C) 0.33(D) 0.45(E) 0.503. The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100 printers, how much should it expect to pay in refunds?(A) 6,321(B) 7,358(C) 7,869(D) 10,256(E) 12,6424. Let T denote the time in minutes for a customer service representative to respond to 10 telephone inquiries. T is uniformly distributed on the interval with endpoints 8 minutes and 12 minutes. Let R denote the average rate, in customers per minute, at which the representative responds to inquiries.Which of the following is the density function of the random variable R on the interval(A)12/5(B) 3 (C) (D) (E)5. Let T1 and T2 represent the lifetimes in hours of two linked components in an electronic device. The joint density function for T1 and T2 is uniform over the region defined by 0 <= t1<= t2<=L where L is a positive constant.Determine the expected value of the sum of the squares of T1 and T2 .(A)L2/3(B)L2/2(C)2 L2/3(D) 3 L2/4(E) L26. Two instruments are used to measure the height, h, of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056h . The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation 0.0044h . Assuming the two measurements are independent random variables, what is the probability that their average value is within 0.005h of the height of the tower?(A) 0.38(B) 0.47(C) 0.68(D) 0.84(E) 0.907. An insurance company’s monthly claims are modeled by a continuous, positive randomvariable X, whose probability density function is proportional to (1 + x)-4 ,where 0 < x Determine the company’s expected monthly claims.(A)1/6(B)1/3(C)1/2(D) 1(E) 38. A probability distribution of the claim sizes for an auto insurance policy is given in thetable below:What percentage of the claims are within one standard deviation of the mean claim size?(A) 45%(B) 55%(C) 68%(D) 85%(E) 100%9. The total claim amount for a health insurance policy follows a distributionwith density function The premium for the policy is set at 100 over the expected total claim amount.If 100 policies are sold, what is the approximate probability that the insurancecompany will have claims exceeding the premiums collected?(A) 0.001(B) 0.159(C) 0.333(D) 0.407(E) 0.46010. An insurance company sells two types of auto insurance policies: Basic and Deluxe. The time until the next Basic Policy claim is an exponential random variable with mean two days. The time until the next Deluxe Policy claim is an independent exponential random variable with mean three days. What is the probability that the next claim will be a Deluxe Policy claim?(A) 0.172(B) 0.223(C) 0.400(D) 0.487(E) 0.50011. A company offers a basic life insurance policy to its employees, as well as a supplemental life insurance policy. To purchase the supplemental policy, an employee must first purchase the basic policy.Let X denote the proportion of employees who purchase the basic policy, and Y the proportion of employees who purchase the supplemental policy. Let X and Y have the joint density function f(x,y) = 2(x + y) on the region where the density is positive. Given that 10% of the employees buy the basic policy, what is the probability that fewer than 5% buy the supplemental policy?(A) 0.010(B) 0.013(C) 0.108(D) 0.417(E) 0.50012. Let C be the curve defined by x = sin t + t and y = cos t – t,Find an equation of the line tangent to C at (0, 1) .(A) y = 1(B) y = 1 + 2x(C) y = 1 – 2x(D) y = 1 –x(E) y = 1 –0.5x13. For a certain product priced at p per unit, 2000 – 10p units will be sold.Which of the following best represents the graph of revenue, r, as a function of price, p ?(A) (B) (C) (D) (E)14. A virus is spreading through a population in a manner that can be modeled by thefunction where A is the total population, g(t) is the number infected at time t, and B is a constant.What proportion of the population is infected when the virus is spreading the fastest?(A)1/3(B)1/2(C)2/3(D)3/4(E) 115. In a certain town, the rate of deaths at time t due to a particular disease is modeled by What is the total number of deaths from this disease predicted by the model?(A) 243(B) 370(C) 556(D) 1,111(E) 10,00016. The total cost, c, to a company for selling n widgets is c(n) = n2 + 4n + 100 . The price per widget is p(n) = 100 – n .What price per widget will yield the maximum profit for the company?(A) 50(B) 76(C) 96(D) 98(E) 10017. An insurance company has 120,000 to spend on the development and promotion of a new insurance policy for car owners. The company estimates that if x is spent on development and y is spent on promotion, then policies will besold.Based on this estimate, what is the maximum number of policies that the insurance company can sell?(A) 3,897(B) 9,000(C) 11,691(D) 30,000(E) 90,00018. An insurance policy reimburses dental expense, X, up to a maximum benefit of 250 . The probability density function for X is: where c is a constant.Calculate the median benefit for this policy.(A) 161(B) 165(C) 173(D) 182(E) 25019. In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from _2.5 years to 2.5 years. The healthcare data are based on a random sample of 48 people. What is the approximate probability that the mean of the rounded ages is within 0.25 years of the mean of the true ages?(A) 0.14(B) 0.38(C) 0.57(D) 0.77(E) 0.8820. Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0,12) . Given X = x, Y is uniformly distributed on the interval (0, x) . Determine Cov(X, Y) according to this model.(A) 0(B) 4(C) 6(D) 12(E) 2421. A ball rolls along the polar curve defined by r = sin . The ball starts at = 0 and ends at Calculate the distance the ball travels.(A) (B) (C) (D) (E)22. An actuary determines that the annual numbers of tornadoes in counties P and Q are jointly distributed as follows:Calculate the conditional variance of the annual number of tornadoes in county Q, giventhat there are no tornadoes in county P .(A) 0.51(B) 0.84(C) 0.88(D) 0.99(E) 1.7623. An insurance policy is written to cover a loss X where X has density function The time (in hours) to processa claim of size x, where 0 _ x _ 2, is uniformly distributed on the interval from x to 2x .Calculate the probability that a randomly chosen claim on this policy is processed in three hours or more.(A) 0.17(B) 0.25(C) 0.32(D) 0.58(E) 0.8324. An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims.If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?(A) (B) 1(C) (D) 2(E) 425. An advertising executive claims that, through intensive advertising, 175,000 of a city’s 3,500,000 people will recognize the client’s product after one day. He further claims that product recognition will grow as advertising continues according to the relationship an+1 = 0.95an +175,000, where an is the number of people who recognize the client’s product n days after advertising begins. If the advertising executive’s claims are correct, how many of the city’s 3,500,000 people will not recognize the client’s product after 35 days of advertising?(A) 552,227(B) 561,468(C) 570,689(D) 581,292(E) 611,88626. The bond yield curve is defined by the function y(x) for 0 < x _ 30 where y is the yield on a bond which matures in x years. The bond yield curve is a continuous, increasing function of x and, for any two points on the graph of y, the line segment connecting those points lies entirely below the graph of y . Which of the following functions could represent the bond yield curve?(A) y(x) = a a is a positive constant(B) y(x) = a + kx a, k are positive constants(C) , k are positive constants(D) y(x) = , k are positive constants(E) y(x) = a + k log(x + 1) a, k are positive constants27. A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold.P(X = 0, Y = 0) =1/6 P(X = 1, Y = 0) =1/12 P(X = 1, Y = 1) =1/6 P(X = 2, Y = 0) =1/12P(X = 2, Y = 1) =1/3 P(X = 2, Y = 2) =1/6 What is the variance of X ?28. Inflation is defined as the rate of change in price as a function of time. The figure below is a graph of inflation, I, versus time, t . Price at time t = 0 is 100 . What is the next time at which price is 100 ?(A) At some time t, t (0, 2) .(B) 2(C) At some time t, t (2, 4) .(D) 4(E) At some time t, t (4, 6) .29. An investor buys one share of stock in an internet company for 100 . During the first four days he owns the stock, the share price changes as follows (measured relative to theprior day’s price): If the pattern of relative price movements observed on the first four days is repeated indefinitely, how will the price of the share of stock behave in the long run?(A) It converges to 0.00 .(B) It converges to 99.45 .(C) It converges to 101.25 .(D) It oscillates between two finite values without converging.(E) It diverges to .30. Three radio antennas are located at points (1, 2), (3, 0) and (4, 4) in the xy-plane. In order to minimize static, a transmitter should be located at the point which minimizes the sum of the weighted squared distances between the transmitter and each of the antennas. The weights are 5, 10 and 15, respectively, for the three antennas. What is the x-coordinate of the point at which the transmitter should be located in order to minimize static?(A) 2.67(B) 3.17(C) 3.33(D) 3.50(E) 4.0031. Let R be the region bounded by the graph of x2 + y2 = 9 .Calculate(A) (B) (C) (D) (E)32. A study indicates that t years from now the proportion of a population that will beinfected with a disease can be modeled by Determine the time when the actual proportion infected equals the average proportion infected over the time interval from t = 0 to t = 3 .(A) 1.38(B) 1.50(C) 1.58(D) 1.65(E) 1.6833. A blood test indicates the presence of a particular disease 95% of the time when thedisease is actually present. The same test indicates the presence of the disease 0.5% ofthe time when the disease is not present. One percent of the population actually has thedisease.Calculate the probability that a person has the disease given that the test indicates the presence of the disease.(A) 0.324(B) 0.657(C) 0.945(D) 0.950(E) 0.99534. An insurance policy reimburses a loss up to a benefit limit of 10 . The policyholder’sloss, Y, follows a distribution with density function:What is the expected value of the benefit paid under the insurance policy?(A)1.0(B) 1.3(C) 1.8(D) 1.9(E) 2.035. A company insures homes in three cities, J, K, and L . Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are:MJ(t) = (1 – 2t)-3 MK(t) = (1 – 2t)-2.5 ML(t) = (1 – 2t)-4.5 Let X represent the combined losses from the three cities.Calculate E(X3) .(A) 1,320(B) 2,082(C) 5,760(D) 8,000(E) 10,56036. In modeling the number of claims filed by an individual under an automobile policyduring a three-year period, an actuary makes the simplifying assumption that for all integers , where pn represents the probability that the policyholder files n claims during the period.Under this assumption, what is the probability that a policyholder files more than one claim during the period?(A) 0.04(B) 0.16(C) 0.20(D) 0.80(E) 0.9637. Let S be the surface described by f(x,y) = arctany/x Determine an equation of the plane tangent to S at the point(A) (B) (C) (D) (E)38. An insurance policy is written to cover a loss, X, where X has a uniform distributionon [0, 1000] .At what level must a deductible be set in order for the expected payment to be 25% of what it would be with no deductible?(A) 250(B) 375(C) 500(D) 625(E) 75039. An insurance policy is written that reimburses the policyholder for all losses incurred up to a benefit limit of 750 . Let f(x) be the benefit paid on a loss of x .Which of the following most closely resembles the graph of the derivative of f ?(A) (B) (C) (D) (E)40. A company prices its hurricane insurance using the following assumptions:(i) In any calendar year, there can be at most one hurricane.(ii) In any calendar year, the probability of a hurricane is 0.05 .(iii) The number of hurricanes in any calendar year is independentof the number of hurricanes in any other calendar /doc/fb5f1bbcc77da26925c5b09e.html ing the company’s assumptions, calculate the probability that there are fewer than 3 hurricanes in a 20-year period.Course 1 May 2000 Answer Key1. A 21. B2. D 22. D3. D 23. A4. E 24. D5. C 25. D6. D 26. E7. C 27. B8. A 28. C9. B 29. A10. C 30. B11. D 31. D12. E 32. D13. E 33. B14. B 34. D15. C 35. E16. B 36. A17. C 37. B18. C 38. C19. D 39. C20. C 40. E。
soa北美精算师第二门FMTIA样题一
soa北美精算师第⼆门FMTIA样题⼀The In?nite Actuary’sJoint Exam2/FMSample Exam1by James Washer,FSA,MAAAlast updated-August14,2009Take this sample exam under strict exam conditions.Start a timer for3hours and stop imme-diately when the timer is done.Do not stop the clock when you go to the bathroom.Do not look at your notes.Do not look at the answer key.This exam contains35questions.Do not spend too much time on any one question.Choose the best available answer for each question.1.Which of the following is not a way to create a40-45-50butter?y?A.buy40-strike call,write two45-strike calls,buy50-strike callB.buy40-strike put,write two45-strike puts,buy50-strike putC.buy40-strike put,write45-strike call,write45-strike put,buy50-strike callD.buy40-strike call,write45-strike call,write45-strike put,buy50-strike putE.all of the above will create a40-45-50butter?y2.Letδt=14+t,0≤t≤15What is the?rst year for which the e?ective rate of discount is less than12.5%?A.3B.4C.5D.6E.73.A bond will pay a coupon of100at the end of each of the next three years and will pay the facevalue of1000at the end of the three-year period.The bond’s modi?ed duration when valued using an annual e?ective interest rate of20%is X.Calculate X.A.2.25B.2.61C.2.70D.2.77E.2.894.You are given the following table of interest rates:CalendarYear of PortfolioOriginal RatesInvestment Investment Year Rates(in%)(in%)y i y1i y2i y3i y4i y5i y+519928.258.258.408.508.508.3519938.508.708.758.909.008.6019949.009.009.109.109.208.8519959.009.109.209.309.409.1019969.259.359.509.559.609.3519979.509.509.609.709.70199810.0010.009.909.80199910.009.809.7020009.509.5020019.00A person deposits1000on January1,1997.Let the following be the accumulated value of the 1000on January1,2000.P:under the investment year methodQ:under the portfolio yield methodCalculate P+Q.A.2575B.2595C.2610D.2655E.27005.A loan is repaid with10annual payments.The?rst payment occurs one year after the loan.The?rst payment is100and each subsequent payment increases by10.The annual e?ective rate of interest is5%.The amount of principal repaid in the4th payment is X.Determine X.A.71B.76C.80D.84E.916.A1000par value10-year bond that pays9%coupons semiannually is purchased for X.Thecoupons are reinvested at a nominal rate of7%convertible semiannually.The bond investor’s nominal annual yield rate convertible semiannually over the10-year period is9.2%.Determine X.A.924B.987C.1024D.1386E.14427.Bill writes a$100-strike call option on stock XYZ with6months to expiration for a premium of$7.24.The risk-free rate is5%convertible semiannually.For what rage of prices at expiration does Bill make a pro?t?A.[0,92.58)B.(92.58,∞)C.[0,107.42)D.(107.42,∞)E.[0,107.60)8.10deposits of$2000are made every other year with the?rst deposit made immediately.Theresulting fund is used to buy a perpetuity with payments made once every3years following the pattern X,4X,7X,10X,...The?rst perpetuity payment is made3years after the last deposit of$2000.The annual e?ective rate of interest is6%.Determine X.A.408B.458C.471D.512E.6039.John buys a perpetuity-due with annual payments that are adjusted each year for in?ation.The?rst payment is100.In?ation is3%for years1-5and2%thereafter.Calculate the price of the perpetuity if the yield rate is an e?ective6%per annum.A.2750B.2760C.2770D.2780E.279010.Given the following information about the treasury market:Term Coupon Price10%96.6220%X30%88.90It is known that the2-year forward rate is4.5%.Calculate X.A.87.65B.89.70C.92.90D.93.45E.95.5011.A20-year bond is priced at par and pays R%coupons semiannually.The bond’s duration is 13.95years.Determine R.A.2B.3C.4D.5E.612.Which of the following is not true?A.An asset insured with a?oor is equivalent to investing in a zero-coupon bond and buying a call option on the asset.B.A short position insured with a cap is equivalent to writing a zero-coupon bond and buying a put option on the asset.C.A covered written call is equivalent to investing in a zero-coupon bond and writing a put option on the asset.D.A covered written put is equivalent to writing a zero-coupon bond and writing a call option on the asset.E.All of the above are true.13.A fairly smart actuary(also know as an FSA)is o?ered the following rates on a loan:1.X%nominal annual rate of interest compounded monthly2.X%nominal annual rate of discount compounded monthly3.X%annual e?ective rate of interest4.X%annual e?ective rate of discount5.X%constant force of interestWhich rate does the FSA take?A.1B.2C.3D.4E.514.An annuity pays1at the beginning of each year for n /doc/894fe700eff9aef8941e067b.html ing an annual e?ective interestrate of i,the present value of the annuity at time0is8.55948.It is also known that(1+i)n=3.172169.Find the accumulated value of the annuity immediately after the last payment.A.27.152B.28.456C.29.324D.30.765E.31.97315.Deposits are made at the beginning of every month into a fund earning a nominal annual rateof6%convertible monthly.The?rst deposit is100and deposit increase2%every year.In other words,deposits1-12are100,deposits13-24are100×1.02=102,deposits25-36are 100×1.022=104.04,and so on.Calculate the fund balance at the end of10years.A.16,569B.16,893C.17,257D.17,770E.17,85916.On January1a fund has a balance of$100.Sometime during the year a withdrawal of$20ismade.Immediately before the withdrawal the fund balance is$110.At year-end the balance is $95.If the time weighted and dollar weighted rates for the year are equal,then in what month was the$20withdrawal made?A.JuneB.JulyC.AugustD.SeptemberE.October17.A common stock pays annual dividends at the end of each year.The earnings per share inthe year just ended were J.Earnings are assumed to grow10%per year in the future.The percentage of earnings paid out as a dividend will be0%for the next5years and50%thereafter.Calculate the theoretical price of the stock to yield the investor21%.A.5J(1.1)4B.5J(1.1)5C.5J(1.1)6D.10J(1.1)5E.10J(1.1)618.You are the CFO of In?nite Life.In?nite Life only has one liability of$5000due in8years.In?nite Life uses a nominal rate of6%convertible semiannual to discount all liability cash?ows.You call up your favorite bond broker and ask him what bonds he has for sale today.Your broker says he has5-year and10-year bonds.Both bonds are priced to yield6%convertible semiannually.The5-year bond pays6%coupons semiannually and the10-year bond is a zero-coupon bond.The bonds can be bought in any face amount.What face amount of the5-year bond should you buy in order to meet the?rst two conditions of immunization?A.777B.888C.999D.1111E.2222。
SOA真题November2003Course8V第5页-精算师考试.doc
SOA真题November2003Course8V第5页-精算师考试20.(6points)AfinancialinstitutionhassoldshortthreeEuropeancalloptionsonastockwiththefollowing characteristics:Stock price:100Strikeprice:110V olatility:20%Expiry:2yearsContract size:1,000 sharesDividends: noneRisk-free rate:5%(a)Points) Describe and evaluate strategies the financial institution canuse tomanage theriskofthis portfolio.(b)Point) Calculate delta, thetaand gammaforthe portfolio.(c) (1Point) Verifythe calculations ofdelta,andgammabyshowingthattheresultsfortheportfoliosatisfytheBlack-Scholes-Merton differential equation.COURSE 8:Fall-17-GOONTONEXT PAGEInvestmentAfternoon Session21.(6points)AmodelofthetypeGARCH(1,1) isbeingusedtoset volatilityfor integervalues ofstrikepriceforanoption pricing modelthatisbasedona lognormaldistribution. Youhavethefollowing informationonthemodelforaspecificunderlyingasset:Aweightof10%isgiventoavarianceof4%thatis reachedasymptotically asthestrike price increases toward infinity. The model assumes returns area constant 20%.A weight of80%isgiventothe variance determined fortheprior strikeprice. The varianceis9%forastrikepriceof0.(a) Calculate the change in volatility for eachof strike pricesof1and2assuming thevariance forastrike priceof0 decreases from9%to8%.(b) Assess whether themodelthegiven parameters ismore appropriate for anequity optionora currency option. Support your answerinterpreting theresulting implications forthe volatility smile.(c) Describe howto determine a maximum likelihood estimateforabovemodel,assuming the distributionofreturns conditionedonthevarianceisnormal.COURSE 8:Fall2003--GOONTONEXT PAGEInvestmentAfternoon Session22.(7points)YouwouldliketocalibrateanewMonteCarlo pricing modelby comparingthe modelpricetotheBlack-Scholes priceofaEuropeancalloptionanon-dividendpayingstock.TheEuropeancalloptionandtheunderlyingstockhavethe followingcharacteristics:Current stockvalue100Expected growthrateofthestockprice=10%V olatility =25%Exercise price=95Timetomaturity6monthsRisk-free interestrate=6%Black-Scholes priceoftheoption=11.37Assume thatthe percentage changethestockpriceisnormally distributed.(a) Calculatethecontrol variate technique adjustment using stratified samplingon3intervals.(b) Describe additional techniques thatcanbeusedtoimprovethe efficiencyof theMonte Carlo method.(c) DescribehowtheMonteCarlomethodcanbeadaptedtocalculatetheprice ofAmerican options.COURSE 8:Fall2003-19-GOONTONEXT PAGEInvestmentAfternoon Session23.(6points)Youaregiventhefollowinginformationaninterestratecollar withquarterly resets.Assume quarterly compounding unless otherwise indicated.Notional amount=100,000,000Strike interestforcap=7.0%Strike interest rateforfloor=4.5%Time to expiration =2 yearsTime t3-monthinterestratestartingattimetZero-couponinterestratetotimet (continuouscompounding)9 months5.0%4.49%12months5.2%months 5.3%4.77%(a) Compare thecollartoa portfolio ofbond options with equivalent payoffs.(b) Calculate,using Black’s model, thepriceofthe floorlet that prevents the interestrate from being lower thanthestrikepricefor3months startinginoneyear.Assume thatthe volatilityofthe underlying three-month rate20%.(c)AssessthesuitabilityofBlack’smodelforvaluingoptions embeddedinamortgage backed security.COURSE 8:2003-20- STOPInvestmentMorning Session24.(3points)Youhavebeenaskedtoreviewthetradingcostsoffollowing stock:Time Price Number ofshares traded(Previous closeis441/8)1:40pm441/810001:42441/420001:45 44 5001:46 437/8 50001:50 433/4 20001:51 437/8 4001:54 441/4 6000(a) Explain whymeasurement oftrading costsis important.(b) Calculate theprice impactoftradingfortheabovestockthe“moneyflow”systemasadaptedbyBirinyi.***ENDOF EXAMINATION***。
SOA真题November2003Course8F第4页-精算师考试.doc
SOA真题November2003Course8F第4页-精算师考试11.(8points)Youarethepricingactuaryforamulti-nationalreinsurancecompany,ABCRe.WXYLifehas approached you regarding a transaction thatwould efficiently utilizeWXY Life’s expiring taxlosscarryforwards (TLCF).Note the following: WXYLifeisratedBBB-byawellknown credit rating agency. NoneofWXY Life’s existing business is currently reinsured. Local regulations prohibit WXYLifefrom takinga reservecreditfor reinsurance. ABCRedoesnot currently havea relationship withWXY Life.Below isWXY Life’sprojected statutory income:WXY Life’s Projected Statutory IncomeYear Pre-tax Statutory Income Expiring TLCF2003 -2102004 22005 42006 5200782009 82010 82011 820128You have decidedto proposea modified coinsurance structure effective December 31,2003the following features: Retention equals 25%ofall existing business. Therisk charge equals 4%ofbeginning ofyear Outstanding Surplus Account (OSA). The modified coinsurance interest rate equals LIBOR+3%.WXYLifecan recapture theblockof business oncetheOSA balance reaches zero.Theinitial reinsurance allowanceefficiently usethe expiring TLCF.(a) (5 points) Assumeall reinsurance accounts are settledattheendeach calendaryear andLIBORequals4% throughouttheprojection period.Basedonthe aboveprojection, determine whenLifecan recapturetheblockof business. Showyour work.COURSE 8:Fall2003-11-GOTONEXT PAGEFinanceAfternoon Session11. Continued(b)(2points)Commentonthe appropriatenessofthelevelofthecharge beingoffered byABCRe.(c)(1point) Identifythe conditions that shouldbe includedinthisfinancialreinsurance treatyinordertogainstateregulator acceptance.12.(5points)SpencerLifeisamedium-sizedlifeinsurance corporationwith approximately$100 millioninassetsunder management.TheChief Investment Officerhasaskedyoutoevaluatetax-structured dealofferedbyan investment broker.The following detailshave been provided: Proceedsofthe investmentgiventotheFederalHousing Corporation, agovernment-sponsored corporationthatbuildsandrentslow-income housing.Theinvestmenta quarterly coupon of LIBOR +100 basis points. 50%ofthe interest income from theinvestment is deductible for federaltax purposes. Alternative investments thatarefully taxable andhave similar creditcurrentlyyield 125basispointsover LIBOR.The minimum investmentis$1million. Current LIBORis1.50%.Federal TaxRateis35%.If Spencer decidesto participate inthe investment, an interest rateswapbeusedtoimprovethematchbetweenthecashflowsfromtheinvestmentandthecorporation’sfixed-rate liabilities.(a)Analyzepotential benefitsandcostsof participating inthetax-structured deal.(b) Recommend howtousetheFAS133accountingrulestominimizenetincomevolatility.(c)Describetheimpactonthefinancialstatementandthe documentationrequirementsyour recommended approachinpart(b).COURSE 8:Fall2003-12-GOONTONEXTPAGEFinanceAfternoon Session13.(17points)NougatInc.isasmallregionalproducerofchocolate confectionaries. ThefounderandCEOthecompany recentlyretiredandplacedthecompanyupforsale.Butterscotch Inc.,Caramel Inc.,andFudgeareall confectionary companies thatplanto makebidsfor Nougat. Butterscotch hasbeenanational manufacturerofavarietyofcandybarsforover100years. Butterscotch is100% ownedbyaprivate trust.Caramelisamaturemulti- national conglomerate thathasbeeninsomeaspectof theconfectionary businessforover300 years.Overthepast decade management hasbeen pursuing growth throughasmall numberof strategicacquisitions. FudgewasstartedfiveyearsagoafterFudge’s founder cashedoutofhis internetcompany andwasinspiredafterreadingaboutayoungboy’striptoa chocolatefactory. ThefinancialpresshascalledFudge“innovating –not imitating–a hipcompany thatistakingashotat bringing growth backtothechocolate business’bottom lineandwaist-line”.Assumethefollowing:Allphysicalassetsareownedby corporations. Capitalmarketsare frictionless. Thereareno corporateorpersonal income taxes,securities canbe purchasedorsold costlesslyinstantaneously, andthereare nobankruptcy costs. Corporations canissueonlytwotypesof securities, riskyequityrisk-free debt.Both individuals and corporations can borroworlendattherisk-free interestInvestorshave homogeneous expectations aboutthefuturestreamof corporateprofits. Thereisnogrowth,soallflow streamsare perpetuities. All corporations canbe classified intooneofseveral “equivalent return classes”suchthatthereturnsonsharesofallfirmsinthatclassare proportionalto,and perfectlycorrelated with,otherfirmsofthatclass.Youareprovidedwiththefollowing information:Nougat Butterscotch Caramel FudgeExpected return10%10%10%Expected operating profits100,000 200,000 200,000 200,000Debt outstanding@5%interest1,000,000 100,000SharesStock50,00040,00020,00038,000Financing plansforacquisitionofNougatN/A100%equity50%debt,50%equityNot yetdeterminedCOURSEFall2003-13-GOONTONEXT PAGEFinanceAfternoon。
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(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.
What is the 95th percentile of actual losses that exceed the deductible?
(A) (B) (C) (D) (E)
600 700 800 900 1000
November 2000
9
Course 1
10.
Let S be a solid in 3-space and f a function defined on S such that:
How many of the company’s policyholders are young, female, and single?
(A) (B) (C) (D) (E)
280 423 486 880 896
November 2000
3
Course 1
4.
A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease present and 0 for disease not present. Let X denote the disease state of a patient, and let Y denote the outcome of the diagnostic test. The joint probability function of X and Y is given by:
Course 1
4
Form 00B
5.
An equation of the line tangent to the graph of a differentiable function f at x = 0 is y = 3x + 4 .
Determine lim
x f ( x) . x → 0 sin(2 x )
8.
An insurance company can sell 20 auto insurance policies per month if it charges 40 per policy. Moreover, for each decrease or increase of 1 in the price per policy, the company can sell 1 more or 1 less policy, respectively. Fixed costs are 100 . Variable costs are 32 per pololders, 3000 are young, 4600 are male, and 7000 are married. The policyholders can also be classified as 1320 young males, 3010 married males, and 1400 young married persons. Finally, 600 of the policyholders are young married males.
MX(t) =
1 . (1 − 2500t )4
Determine the standard deviation of the claim size for this class of accidents.
(A) (B) (C) (D) (E)
1,340 5,000 8,660 10,000 11,180
8
Form 00B
9.
An insurance company sells an auto insurance policy that covers losses incurred by a policyholder, subject to a deductible of 100 . Losses incurred follow an exponential distribution with mean 300 .
According to these models, by how much will the value of the first asset exceed the value of the second asset after 25 years?
(A) (B) (C) (D) (E)
4050 4835 5035 5718 6000
What is the conditional probability that V exceeds 40,000, given that V exceeds 10,000 ?
(A) (B) (C) (D) (E)
0.08 0.13 0.17 0.20 0.51
November 2000
7
Course 1
P(X = 0, Y = 0) = 0.800 P(X = 1, Y = 0) = 0.050 P(X = 0, Y = 1) = 0.025 P(X = 1, Y = 1) = 0.125
Calculate Var(Y X = 1) .
(A) (B) (C) (D) (E)
0.13 0.15 0.20 0.51 0.71
Vn = 1.10Vn–1 for n = 1, 2, 3, . . .
The value Wn of the second asset after n years can be modeled by the relationship:
Wn = Wn–1 + 0.20W0 for n = 1, 2, 3, . . .
November 2000 Course 1
Society of Actuaries/Casualty Actuarial Society
1.
A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260 .
∫∫∫ f ( x, y, z )dV
S
=
5
∫∫∫ (4 f ( x, y, z ) + 3)dV =
S
47
Calculate the volume of S .
(A) (B) (C) (D) (E)
2 5 7 9 14
Course 1
10
Form 00B
11.
An actuary determines that the claim size for a certain class of accidents is a random variable, X, with moment generating function
If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car?
Course 1
6
Form 00B
7.
A group insurance policy covers the medical claims of the employees of a small company. The value, V, of the claims made in one year is described by V = 100,000Y where Y is a random variable with density function k (1 − y ) 4 f ( y) = 0 where k is a constant. for 0 < y < 1 otherwise,