北美精算师(SOA)考试 FM 2001 May 年真题

Faculty of Actuaries Institute of ActuariesEXAMINATIONS7 September 2001 (pm)Subject 102 — Financial MathematicsTime allowed: Three hoursINSTRUCTIONS TO THE CANDIDATE1.Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.2.Mark allocations are shown in brackets.3.Attempt all 12 questions, beginning your answer to each question on aseparate sheet.Graph paper is not required for this paper.AT THE END OF THE EXAMINATIONHand in BOTH your answer booklet and this question paper.In addition to this paper you should have availableActuarial Tables and an electronic calculator.ã Faculty of Actuaries1 A 91-day government bill provides the purchaser with an annual effective rate ofreturn of 5%. Determine the annual simple discount rate at which the bill is discounted.[2]2 A particular share is expected to pay a dividend of d 1 in exactly one year.Dividends are expected to grow by g per annum effective every year thereafter.The share pays annual dividends. Let V 0 be the present value of the share and r be the investor’s required annual effective rate of return.Show that V 0 = 1d r g −.[3]3An asset has a current price of 100p. It will pay an income of 5p in 20 days’ time.Given a risk-free rate of interest of 6% per annum convertible half-yearly and assuming no arbitrage, calculate the forward price to be paid in 40 days.[4]4An annuity is paid half-yearly in arrears at a rate of £1,000 per annum, for 20years. The rate of interest is 5% per annum effective in the first 12 years and 6%per annum convertible quarterly for the remaining 8 years.Calculate the accumulation of the annuity at the end of 20 years.[4]5An investor purchases a bond, redeemable at par, which pays half-yearly couponsat a rate of 8% per annum. There are 8 days until the next coupon payment and the bond is ex-dividend. The bond has 7 years to maturity after the next coupon payment.Calculate the purchase price to provide a yield to maturity of 6% per annum effective.[4]6(1 + i t ) follows a log normal distribution where i t is the rate of interest over agiven time period beginning at time t . The parameters of the distribution are µ = 0.06 and σ2 = 0.0009.Calculate the inter-quartile range for the accumulation of 100 units of money over the given time period, beginning at time t .[6]7(i) The annual effective forward rate applicable over the period from t to t + ris defined as f t,r where t and r are measured in years. If f 0,1 = 8%, f 1,1 = 7%,f 2,1 = 6% and f 3,1 = 5%, calculate the gross redemption yield at the issue date from a 4-year bond, redeemable at par, with a 5% coupon payable annually in arrears.[7](ii)Explain why the gross redemption yield from the 4-year bond is higher than the 4-year forward rate f 3,1.[2][Total 9]8 A fast food company is considering opening a new sales outlet. The initial cost ofthe outlet would be £1,000,000 incurred at the outset of the project. It is expected that rents of £40,000 per annum would have to be paid quarterly in advance for 10 years, increasing after ten years to £48,000 per annum. The net revenue (sales minus costs, other than rent) from the venture is expected to be £100,000 for the first year and £200,000 for the second year. Thereafter, the net revenue is expected to grow at 3% per annum compound so that it is £206,000 in the third year, £212,180 in the fourth year and so on. The revenue would be received continuously throughout each year. Twenty years after the outset of the project, the revenue and costs stop and the project has no further value.Calculate the internal rate of return from the project.[11]9(i)Prove that =nn n a Da i −.[3](ii) A bank makes a loan to be repaid by instalments paid annually in arrears.The first instalment is 20, the second is 19 with the payments reducing by1 per annum until the end of the 10th year after which there are no further payments. The rate of interest charged by the lender is 6% per annum effective.(a)Calculate the amount of the loan.(b)Calculate the interest and capital components of the first payment.(c) Calculate the amount of capital repaid in the instalment at the end of the 8th year.[8][Total 11]10An investor purchased a bond with exactly 20 years to redemption. The bond, redeemable at par, has a gross redemption yield of 6%. It pays annual coupons,in arrears, of 5%. The investor does not pay tax.(i) Calculate the purchase price paid for the bond.[3](ii)After exactly ten years, immediately after payment of the coupon then due, this investor sells the bond to another investor. That investor paysincome and capital gains tax at a rate of 30%. The bond is purchased bythe second investor to provide a net rate of return of 6.5% per annum.(a) Calculate the price paid by the second investor.(b) Calculate the annual effective rate of return earned by the firstinvestor during the period for which the bond was held.[10][Total 13] 11The force of interest, δ(t), is:δ(t)= 0.05 for 0 < t≤ 10,= 0.006t for 10 < t≤ 20= 0.003t + 0.0002t2 for 20 < t(i) Calculate the present value of a unit sum of money due at time t = 25.[7](ii) Calculate the effective rate of interest per unit time from time t = 19 to time t = 20.[3](iii) A continuous payment stream is paid at the rate of e−0.03t per unit time between time t = 0 and time t = 5. Calculate the present value of thatpayment stream.[4][Total 14]12(i)(a) In the context of a stream of future receipts paid at discrete times,let volatility be defined as the proportionate change in the presentvalue of a payment stream per unit change in the force of interest,for small changes in the force of interest. Prove that thediscounted mean term is equal to the volatility.(b)If volatility is now defined as the proportionate change in thepresent value of a payment stream per unit change in the annualeffective rate of interest, for small changes in the annual effectiverate of interest, find the relationship between the discounted meanterm and volatility.[5](ii) A life insurance company manages a small annuity fund. Payments are expected to be made from the fund of £1,000,000 per annum at the end ofyears 1 to 10 and £1,500,000 at the end of each of the following 10 years.Assets are held in two types of bonds. The first is a zero coupon bondredeemable in 10 years’ time. The second is a bond which pays an annual coupon of g% per annum in arrears and is redeemable at par at the end of19 years. £10,000,000 nominal of the zero coupon bond have beenpurchased.Find the nominal amount of the coupon paying bond which must bepurchased and the rate of coupon which is received from the bond if theinsurance company is to equalise the present values and discounted mean terms of its assets and liabilities at an effective rate of interest of 5% perannum.[12] (iii)If the present value and discounted mean term of the assets and liabilities are equalised, state the third condition which is necessary for theinsurance company to be immunised from small, uniform changes in therate of interest.[2][Total 19]。

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]。

May 2001Course 3**BEGINNING OF EXAMINATION**1.For a given life age 30, it is estimated that an impact of a medical breakthrough will be anincrease of 4 years in e o30, the complete expectation of life.Prior to the medical breakthrough, s(x) followed de Moivre’s law with ω=100as thelimiting age.Assuming de Moivre’s law still applies after the medical breakthrough, calculate the new limiting age.(A)104(B)105(C)106(D)107(E)1082.On January 1, 2002, Pat, age 40, purchases a 5-payment, 10-year term insurance of100,000:(i)Death benefits are payable at the moment of death.(ii)Contract premiums of 4000 are payable annually at the beginning of each year for5 years.(iii)i = 0.05(iv)L is the loss random variable at time of issue.Calculate the value of L if Pat dies on June 30, 2004.(A)77,100(B)80,700(C)82,700(D)85,900(E)88,0003.Glen is practicing his simulation skills.He generates 1000 values of the random variable X as follows:(i)He generates the observed value λ from the gamma distribution with α=2 andθ=1 (hence with mean 2 and variance 2).(ii)He then generates x from the Poisson distribution with mean λ.(iii)He repeats the process 999 more times: first generating a value λ, thengenerating x from the Poisson distribution with mean λ.(iv)The repetitions are mutually independent.Calculate the expected number of times that his simulated value of X is 3.(A) 75(B)100(C)125(D)150(E)1754.Lucky Tom finds coins on his way to work at a Poisson rate of 0.5 coins per minute.The denominations are randomly distributed:(i)60% of the coins are worth 1;(ii)20% of the coins are worth 5;(iii)20% of the coins are worth 10.Calculate the variance of the value of the coins Tom finds during his one-hour walk to work.(A)379(B)487(C)566(D)670(E)7685.For a fully discrete 20-payment whole life insurance of 1000 on (x), you are given:(i)i = 0.06(ii)q x+=19001254 .(iii)The level annual benefit premium is 13.72.(iv)The benefit reserve at the end of year 19 is 342.03.Calculate 1000 P x+20, the level annual benefit premium for a fully discrete whole life insurance of 1000 on (x+20).(A)27(B)29(C)31(D)33(E)356.For a multiple decrement model on (60):(i)µ6010()(),,t t ≥ follows the Illustrative Life Table.(ii)µµτ6060120()()()(),t t t =≥Calculate 1060q ()τ, the probability that decrement occurs during the 11th year.(A)0.03(B)0.04(C)0.05(D)0.06(E)0.077. A coach can give two types of training, “ light” or “heavy,” to his sports team before agame. If the team wins the prior game, the next training is equally likely to be light or heavy. But, if the team loses the prior game, the next training is always heavy.The probability that the team will win the game is 0.4 after light training and 0.8 after heavy training.Calculate the long run proportion of time that the coach will give heavy training to the team.(A)0.61(B)0.64(C)0.67(D)0.70(E)0.738.For a simulation of the movement of a stock’s price:. and(i)The price follows geometric Brownian motion, with drift coefficient µ=001..variance parameter σ2=00004(ii)The simulation projects the stock price in steps of time 1.(iii)Simulated price movements are determined using the inverse transform method.(iv)The price at t = 0 is 100.(v)The random numbers, from the uniform distribution on ,are 0.1587 and 0.9332, respectively.(vi)F is the price at t = 1; G is the price at t = 2.Calculate G – F.(A)1(B)2(C)3(D)4(E)59.(x) and (y) are two lives with identical expected mortality.You are given:P P==0.1x yP xy=006., where P xy is the annual benefit premium for a fully discreteb g.insurance of 1 on xyd=006.Calculate the premium P xy, the annual benefit premium for a fully discrete insurance of 1b g.on xy(A)0.14(B)0.16(C)0.18(D)0.20(E)0.2210.For students entering a college, you are given the following from a multiple decrementmodel:(i)1000 students enter the college at t =0.(ii)Students leave the college for failure 1b g or all other reasons 2b g .(iii) µµ1b g b gt = 04≤≤t µ2004b g b g t =. 04≤<t (iv) 48 students are expected to leave the college during their first year due toall causes.Calculate the expected number of students who will leave because of failure during their fourth year.(A)8(B)10(C)24(D)34(E) 4111.You are using the inverse transform method to simulate Z, the present value randomvariable for a special two-year term insurance on (70). You are given:(i)(70) is subject to only two causes of death, with(ii)Death benefits, payable at the end of the year of death, are:During year Benefit for Cause 1Benefit for Cause 2110001100211001200.(iii)i=006(iv)For this trial your random number, from the uniform distribution on , is 0.35.(v)High random numbers correspond to high values of Z.Calculate the simulated value of Z for this trial.(A) 943(B) 979(C)1000(D)1038(E)106812.You are simulating one year of death and surrender benefits for 3 policies. Mortalityfollows the Illustrative Life Table. The surrender rate, occurring at the end of the year, is 15% for all ages. The simulation procedure is the inverse transform algorithm, with low random numbers corresponding to the decrement occurring. You perform the following steps for each policy:(1)Simulate if the policy is terminated by death. If not, go to Step 2; if yes, continuewith the next policy.(2)Simulate if the policy is terminated by surrender.The following values are successively generated from the uniform distribution on ,0.3,0.5,0.1,0.4,0.8,0.2,0.3,0.4,0.6,0.7,….You are given:Policy #Age DeathBenefitSurrenderBenefit1100101029125203962015 Calculate the total benefits generated by the simulation.(A)30(B)35(C)40(D)45(E)5013.Mr. Ucci has only 3 hairs left on his head and he won’t be growing any more.(i)The future mortality of each hair followsk x q k k =+=0110123.,,,,b gand x is Mr. Ucci’s age (ii) Hair loss follows the hyperbolic assumption at fractional ages.(iii)The future lifetimes of the 3 hairs are independent.Calculate the probability that Mr. Ucci is bald (has no hair left) at age x +2.5.(A) 0.090(B) 0.097(C) 0.104(D) 0.111(E)0.11814.The following graph is related to current human mortality:Which of the following functions of age does the graph most likely show?(A)µx b g(B)l xxµb g(C)l px x(D)lx(E)lx215.An actuary for an automobile insurance company determines that the distribution of theannual number of claims for an insured chosen at random is modeled by the negativebinomial distribution with mean 0.2 and variance 0.4.The number of claims for each individual insured has a Poisson distribution and themeans of these Poisson distributions are gamma distributed over the population ofinsureds.Calculate the variance of this gamma distribution.(A)0.20(B)0.25(C)0.30(D)0.35(E)0.4016. A dam is proposed for a river which is currently used for salmon breeding. You havemodeled:(i)For each hour the dam is opened the number of salmon that will pass through andreach the breeding grounds has a distribution with mean 100 and variance 900.(ii)The number of eggs released by each salmon has a distribution with mean of 5 and variance of 5.(iii)The number of salmon going through the dam each hour it is open and the numbers of eggs released by the salmon are independent.Using the normal approximation for the aggregate number of eggs released, determinethe least number of whole hours the dam should be left open so the probability that10,000 eggs will be released is greater than 95%.(A)20(B)23(C)26(D)29(E)3217.For a special 3-year term insurance on ()x , you are given:(i) Z is the present-value random variable for the death benefits.(ii) q k x k +=+0021.()k =0, 1, 2(iii)The following death benefits, payable at the end of the year of death:k b k +10300,0001350,0002400,000(iv) i =006.Calculate E Z b g.(A) 36,800(B) 39,100(C) 41,400(D) 43,700(E)46,00018.For a special fully discrete 20-year endowment insurance on (55):(i)Death benefits in year k are given by b k k =−21b g, k = 1, 2, …, 20.(ii) The maturity benefit is 1.(iii) Annual benefit premiums are level.(iv) k V denotes the benefit reserve at the end of year k , k = 1, 2,…, 20.(v) 10=5.0V (vi) 19=0.6V(vii) q 650.10=(viii)i =0.08Calculate 11.V (A) 4.5(B) 4.6(C) 4.8(D) 5.1(E)5.319.For a stop-loss insurance on a three person group:(i)Loss amounts are independent.(ii)The distribution of loss amount for each person is:Loss Amount Probability00.410.320.230.1(iii)The stop-loss insurance has a deductible of 1 for the group.Calculate the net stop-loss premium.(A) 2.00(B) 2.03(C) 2.06(D) 2.09(E) 2.1220.An insurer’s claims follow a compound Poisson claims process with two claims expectedper period. Claim amounts can be only 1, 2, or 3 and these are equal in probability.Calculate the continuous premium rate that should be charged each period so that theadjustment coefficient will be 0.5.(A) 4.8(B) 5.9(C) 7.8(D) 8.9(E)11.8e the following information for questions 21 and 22.The Simple Insurance Company starts at time t=0 with a surplus of S=3. At thebeginning of every year, it collects a premium of P=2. Every year, it pays a randomclaim amount:Claim Amount Probability of Claim Amount00.1510.2520.5040.10Claim amounts are mutually independent.If, at the end of the year, Simple’s surplus is more than 3, it pays a dividend equal to theamount of surplus in excess of 3. If Simple is unable to pay its claims, or if its surplusdrops to 0, it goes out of business. Simple has no administrative expenses and its interestincome is 0.21.Determine the probability that Simple will ultimately go out of business.(A)0.00(B)0.01(C)0.44(D)0.56(E) 1.0021-22.(Repeated for convenience) Use the following information for questions 21 and 22.The Simple Insurance Company starts at time t=0 with a surplus of S=3. At thebeginning of every year, it collects a premium of P=2. Every year, it pays a randomclaim amount:Claim Amount Probability of Claim Amount00.1510.2520.5040.10Claim amounts are mutually independent.If, at the end of the year, Simple’s surplus is more than 3, it pays a dividend equal to theamount of surplus in excess of 3. If Simple is unable to pay its claims, or if its surplusdrops to 0, it goes out of business. Simple has no administrative expenses and its interestincome is 0.22.Calculate the expected dividend at the end of the third year.(A)0.115(B)0.350(C)0.414(D)0.458(E)0.55023. A continuous two-life annuity pays:100 while both (30) and (40) are alive;70 while (30) is alive but (40) is dead; and50 while (40) is alive but (30) is dead.The actuarial present value of this annuity is 1180. Continuous single life annuitiespaying 100 per year are available for (30) and (40) with actuarial present values of 1200 and 1000, respectively.Calculate the actuarial present value of a two-life continuous annuity that pays 100 while at least one of them is alive.(A)1400(B)1500(C)1600(D)1700(E)180024.For a disability insurance claim:(i) The claimant will receive payments at the rate of 20,000 per year, payable continuously as long as she remains disabled.(ii)The length of the payment period in years is a random variable with the gamma distribution with parameters αθ==21and .(iii) Payments begin immediately.(iv)δ=005.Calculate the actuarial present value of the disability payments at the time of disability.(A) 36,400(B) 37,200(C) 38,100(D) 39,200(E)40,00025.For a discrete probability distribution, you are given the recursion relationp k kp k b gb g=−21*,k = 1, 2,….Determine p 4b g.(A) 0.07(B) 0.08(C) 0.09(D) 0.10(E)0.1126. A company insures a fleet of vehicles. Aggregate losses have a compound Poissondistribution. The expected number of losses is 20. Loss amounts, regardless of vehicle type, have exponential distribution with θ=200.In order to reduce the cost of the insurance, two modifications are to be made:(i) a certain type of vehicle will not be insured. It is estimated that this willreduce loss frequency by 20%.(ii) a deductible of 100 per loss will be imposed.Calculate the expected aggregate amount paid by the insurer after the modifications.(A)1600(B)1940(C)2520(D)3200(E)388027.An actuary is modeling the mortality of a group of 1000 people, each age 95, for the next three years.The actuary starts by calculating the expected number of survivors at each integral age byl p k k k 95951000+==,1, 2, 3The actuary subsequently calculates the expected number of survivors at the middle of each year using the assumption that deaths are uniformly distributed over each year of age.This is the result of the actuary’s model:Age Survivors 95100095.58009660096.548097--97.528898--The actuary decides to change his assumption for mortality at fractional ages to the constant force assumption. He retains his original assumption for each k p 95.Calculate the revised expected number of survivors at age 97.5.(A) 270(B) 273(C) 276(D) 279(E)28228.For a population of individuals, you are given:(i)Each individual has a constant force of mortality.(ii)The forces of mortality are uniformly distributed over the interval (0,2).Calculate the probability that an individual drawn at random from this population dies within one year.(A)0.37(B)0.43(C)0.50(D)0.57(E)0.6329-30.Use the following information for questions 29 and 30.You are the producer of a television quiz show that gives cash prizes. The number of prizes, N , and prize amounts, X , have the following distributions:n Pr N n =b gx Pr X x =b g10.800.220.21000.710000.129.Your budget for prizes equals the expected prizes plus the standard deviation of prizes.Calculate your budget.(A) 306(B) 316(C) 416(D) 510(E)51829-30.(Repeated for convenience) Use the following information for questions 29 and 30.You are the producer of a television quiz show that gives cash prizes. The number of prizes, N , and prize amounts, X , have the following distributions:n Pr N n =b gx Pr X x =b g10.800.220.21000.710000.130.You buy stop-loss insurance for prizes with a deductible of 200. The cost of insurance includes a 175% relative security load.Calculate the cost of the insurance.(A) 204(B) 227(C) 245(D) 273(E)35731.For a special fully discrete 3-year term insurance on x b g:(i)Level benefit premiums are paid at the beginning of each year.(ii)k b k+1q x k+0200,0000.031150,0000.062100,0000.09(iii)i=0.06Calculate the initial benefit reserve for year 2.(A) 6,500(B) 7,500(C) 8,100(D) 9,400(E)10,30032.For a special fully continuous whole life insurance on (x ):(i)The level premium is determined using the equivalence principle.(ii)Death benefits are given by b i t t=+1b g where i is the interest rate.(iii)L is the loss random variable at t =0 for the insurance.(iv)T is the future lifetime random variable of (x ).Which of the following expressions is equal to L ?(A)νTx x A A −−c hc h1(B)νT x x A A −+c hc h1(C)νTx x A A −+c hc h1(D)νT x x A A −−c hc h1(E)v A A Tx x ++d ic h133.For a 4-year college, you are given the following probabilities for dropout from allcauses:q q q q 0123015010005001====....Dropouts are uniformly distributed over each year.Compute the temporary 1.5-year complete expected college lifetime of a student entering the second year, e o115:..(A)1.25(B)1.30(C)1.35(D)1.40(E) 1.4534.Lee, age 63, considers the purchase of a single premium whole life insurance of 10,000with death benefit payable at the end of the year of death.The company calculates benefit premiums using:(i)mortality based on the Illustrative Life Table,(ii)i = 0.05The company calculates contract premiums as 112% of benefit premiums.The single contract premium at age 63 is 5233.Lee decides to delay the purchase for two years and invests the 5233.Calculate the minimum annual rate of return that the investment must earn to accumulate to an amount equal to the single contract premium at age 65.(A)0.030(B)0.035(C)0.040(D)0.045(E)0.05035.You have calculated the actuarial present value of a last-survivor whole life insurance of1 on (x) and (y). You assumed:(i)The death benefit is payable at the moment of death.(ii)The future lifetimes of (x) and (y) are independent, and each life has a constant force of mortality with µ=006...(iii)δ=005Your supervisor points out that these are not independent future lifetimes. Eachmortality assumption is correct, but each includes a common shock component withconstant force 0.02.Calculate the increase in the actuarial present value over what you originally calculated.(A)0.020(B)0.039(C)0.093(D)0.109(E)0.16336.The number of accidents follows a Poisson distribution with mean 12. Each accidentgenerates 1, 2, or 3 claimants with probabilities 12, 13, 16, respectively.Calculate the variance in the total number of claimants.(A)20(B)25(C)30(D)35(E)4037.For a claims process, you are given:(i) The number of claims N t t b g m r ,≥0 is a nonhomogeneous Poisson process withintensity function:λ(),,,t t t t =≤<≤<≤R S |T |10121232(ii) Claims amounts Y i are independently and identically distributed random variablesthat are also independent of N t ().(iii) Each Y i is uniformly distributed on [200,800].(iv) The random variable P is the number of claims with claim amount less than 500 by time t = 3.(v) The random variable Q is the number of claims with claim amount greater than 500 by time t = 3.(vi)R is the conditional expected value of P , given Q = 4.Calculate R.(A)2.0(B)2.5(C)3.0(D)3.5(E)4.038.Lottery Life issues a special fully discrete whole life insurance on (25):(i)At the end of the year of death there is a random drawing. With probability 0.2,the death benefit is 1000. With probability 0.8, the death benefit is 0.(ii)At the start of each year, including the first, while (25) is alive, there is a random drawing. With probability 0.8, the level premium π is paid. With probability0.2, no premium is paid.(iii)The random drawings are independent.(iv)Mortality follows the Illustrative Life Table..(v)i=006(vi)π is determined using the equivalence principle.Calculate the benefit reserve at the end of year 10.(A)10.25(B)20.50(C)30.75(D)41.00(E)51.2539. A government creates a fund to pay this year’s lottery winners.You are given:(i)There are 100 winners each age 40.(ii)Each winner receives payments of 10 per year for life, payable annually, beginning immediately.(iii)Mortality follows the Illustrative Life Table.(iv)The lifetimes are independent.(v)i = 0.06(vi)The amount of the fund is determined, using the normal approximation, such that the probability that the fund is sufficient to make all payments is 95%.Calculate the initial amount of the fund.(A)14,800(B)14,900(C)15,050(D)15,150(E)15,25040.For a special fully discrete 35-payment whole life insurance on (30):(i)The death benefit is 1 for the first 20 years and is 5 thereafter.(ii)The initial benefit premium paid during the each of the first 20 years is one fifthof the benefit premium paid during each of the 15 subsequent years.(iii)Mortality follows the Illustrative Life Table.(iv)i =006.(v)A 3020032307:.=(vi)&&.:a 303514835=Calculate the initial annual benefit premium.(A)0.010(B)0.015(C)0.020(D)0.025(E)0.030**END OF EXAMINATION**COURSE 3: MAY 2001- 41 –STOPCOURSE 3MAY 2001MULTIPLE-CHOICE ANSWER KEY。

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。

2001年真题与解析The government is to ban payments to witnesses by newspapers seeking to buy up people involved in prominent cases 1 the trial of Rosemary West.In a significant 2 of legal controls over the press, Lord Irvine, the Lord Chancellor, will introduce a 3 bill that will propose making payments to witnesses 4 and will strictly control the amount of 5 that can be given to a case 6 a trial begins.In a letter to Gerald Kaufman, chairman of the House of Commons Media Select Committee, Lord Irvine said he 7 with a committee report this year which said that self regulation did not 8 sufficient control. 9 of the letter came two days after Lord Irvine caused a 10 of media protest when he said the 11 of privacy controls contained in European legislation would be left to judges 12 to Parliament.The Lord Chancellor said introduction of the Human Rights Bill, which 13 the European Convention on Human Rights legally 14 in Britain, laid down that everybody was 15 to privacy and that public figures could go to court to protect themselves and their families.“Press freedoms will be in safe hands 16 our British judges,” he said.Witness payments became an 17 after West was sentenced to 10 life sentences in 1995. Up to 19 witnesses were 18 to have received payments for telling their stories to newspapers. Concerns were raised 19 witnesses might be encouraged to exaggerate their stories in court to 20 guilty verdicts.1. [A] as to [B] for instance [C] in particular [D] such as2. [A] tightening [B] intensifying [C] focusing [D] fastening3. [A] sketch [B] rough [C] preliminary [D] draft4. [A] illogical [B] illegal [C] improbable [D] improper5. [A] publicity [B] penalty [C] popularity [D] peculiarity6. [A] since [B] if [C] before [D] as7. [A] sided [B] shared [C] complied [D] agreed8. [A] present [B] offer [C] manifest [D] indicate9. [A] Release [B] Publication [C] Printing [D] Exposure10. [A] storm [B] rage [C] flare [D] flash11. [A] translation [B] interpretation [C] exhibition [D] demonstration12. [A] better than [B] other than [C] rather than [D] sooner than13. [A] changes [B] makes [C] sets [D] turns14. [A] binding [B] convincing [C] restraining [D] sustaining15. [A] authorized [B] credited [C] entitled [D] qualified16. [A] with [B] to [C] from [D] by17. [A] impact [B] incident [C] inference [D] issue18. [A] stated [B] remarked [C] said [D] told19. [A] what [B] when [C] which [D] that20. [A] assure [B] confide [C] ensure [D] guarantee试题解析1. 【答案解析】D考查知识点: 逻辑关系＋短语辨析文章第一句话结构比较复杂，也比较重要。

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。

Faculty of Actuaries Institute of ActuariesEXAMINATIONS10 September 2001 (am)Subject 402 — UK Fellowship Life InsurancePaper OneYou must answer this subject only,you may not attempt another subject in the 400 series.Time allowed: Three hoursINSTRUCTIONS TO THE CANDIDATE1.You have 15 minutes at the start of the examination in which to read thequestions. You are strongly encouraged to use this time for reading onlybut notes may be made. You then have three hours to complete the paper.2.You must not start writing your answers in the booklet until instructed todo so by the supervisor.3.Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.4.Mark allocations are shown in brackets.5.Attempt all 7 questions, beginning your answer to each question on aseparate sheet.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 Actuaries 402(1)—S2001 (21.2.02)ã Institute of Actuaries1For each of the following contracts:(a) a term assurance(b)an immediate annuity(c) a long term sickness insurance contract(i)Describe, with reasons, at what point underwriting would take place.[4](ii) Describe how the company might alter the terms offered new business if a new fatal disease were to emerge.[4][Total 8]2At the pricing date, a unit-linked life fund of a UK life insurance company has net unrealised chargeable gains.(i)Describe how an allowance for deferred tax on these gains would becalculated when pricing the units on an offer basis. [6](ii)Describe how your answer would differ if the units were being priced on a bid basis.[2][Total 8]3(i)State the elements that make up the shareholders’ interest in a life insurance company.[2] (ii)Discuss why the company might wish to analyse the change in theshareholders’ interest.[3](iii)State the sources of profit into which the change in the value of the shareholders’ interest might be analysed.[4][Total 9] 4 A UK proprietary life insurance company sells a full range of conventional andunitised business. The Appointed Actuary has just conducted the annualsupervisory valuation for the purposes of demonstrating solvency.(i)Discuss the advantages and disadvantages of using the net premiumvaluation method for valuing the conventional with profits business.[5]The managing director is concerned that the company's free asset ratio hasdeclined from the previous year, despite a large rise in the FTSE-100 index.(ii)State with reasons why the free asset ratio may have declined. [4] (iii)Discuss why the managing director might be concerned about the fall in the free asset ratio. [4][Total 13]402(1) S2001—25 A country that currently has the same regulatory regime as the UK isconsidering some possible amendments to its regulations for life insurancecompanies.Discuss the following suggestions:(i)At least 30% of all assets backing non-linked liabilities are to be investedin government stocks.[7](ii)The required minimum margin calculation is now to be calculated as 1% of all long-term fund liabilities.[5](iii)All long-term liabilities must be valued using a basis which will be defined each year by the Government Actuary.[4][Total 16] 6 A UK mutual life insurance company last reviewed its premium rates forconventional with profits endowment assurances in 1985. At that time the main elements of the pricing basis were:Interest9.0% p.a.Mortality A67/70 rated down 1 year for males and 5 years for femalesInitial Expenses£80 per policyRenewal Expenses£15 per policyInitial Commission 2.5% of the annual premium for each year of the term paidat outsetRenewal Commission 2.5% of each premiumTaxation37.5% tax on interest less expensesInflation7.5%p.a.Lapses nilBonus loading 4.0% p.a. compound bonusA compound reversionary bonus of 5.0% was declared at the end of 1985. Thisrate was maintained until 1991 when a switch to a super-compound bonus of3.5% on the sum assured and 6% on attaching bonuses was made. Bonus ratessteadily reduced, and at the end of 2000 the declaration was 1.5% on the sumassured and 3.5% on attaching bonuses. This type of policy is still marketed.(i)Explain why the 1985 declaration might have been different from thebonus loading and indicate how the difference might have been financed.[2](ii)Discuss why premium rates might have been left unchanged for over 15 years.[5] (iii)Discuss why the switch from compound to super-compound reversionary bonuses might have been made.[3] (iv)Describe the investigations that would be carried out to determine an appropriate level of reversionary bonus and discuss the factors that wouldlead to a change in premium rates being recommended. You do not needto describe how investigations into the company’s mortality, expense orpersistency experience would be carried out.[13][Total 23]402(1) S2001—3PLEASE TURN OVER7 A UK proprietary life insurance company has written a wide range of lifeassurance and pensions business, including individual term assurance, for many years. It is now considering launching a new group life contract for up to 100lives, where premiums are paid by the sponsoring employer and which pays alump sum benefit on the death of an employee. In addition, there is an option to extend the cover to pay a spouse’s/dependant's pension on the death of theemployee. The product will be distributed via independent intermediaries.(i)Describe how the company will determine the mortality assumption in thepricing basis.[8] (ii)Describe how the company will determine the other assumptions required to calculate the value of new business. [15][Total 23]402(1) S2001—4。

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May2002Course6第3页-精算师考试5.Youaregiventhefollowinginformationfora15-yearcallablebond:annualcouponrate:9%payable semi-annually price:95.32 effective duration:3.17 convexity measure(C):(67.31)CVVVVy22002bgCalculate thepriceofthebond aftera50basis point increase in interest rates.(A) 93.65(B) 93.97(C)95.32(D)96.67(E)96.99COURSE 6:MAY2002GOONTONEXT PAGEMORNING SESSION6.Youaregiventhefollowinginformation with respecttoasingle-period securities model:SPS010101113301101122bg LNMMMOQPPPDetermine thevalueofPwhichmakesthemodelarbitrage-free.(A)14(B)15(C)16(D)17(E)18COURSEMAY2002GOONTONEXT PAGEMORNING SESSION7-16. Eachofquestions7through16consistsoflists.Inthelistattheleftaretwo items,lettered XandY.Inthelisttherightare three items, numbered I,II,andIII.ONE ofthe lettered itemsis relatedsomewayto EXACTLY TWOofthe numbered items.Indicate therelated itemsusingthe following answercode:Lettered ItemIsRelatedto Numbered Items(A)XIandIIonly(B)XIIandIIIonly(C)IandIIonly(D) YIandIII only(E) The correct answer isnot given by(B),(C)or (D).7. X. Asian call options I. Payoffs depend onthe average priceof theunderlying asset during thelifeofthe option.Y. Lookback call options II. Guarantees the purchase ofthe assetatlowestpriceduringthelifeofthe option.III. Canuse averagesforthe exerciseprice.8. X.Cliquet optionGuaranteed exchange-rate contracts.Y. Quanto optionII.Aseriesofstandardcalloptions thatpaystheannual increasetheunderlying assets.III. Thestrikeresetsatthe beginning ofeach year.COURSE 6:MAY2002GOONTONEXT PAGEMORNING SESSION9.X.InterestratecorridorI.Thepurchaseofacapatonestrikerateandsaleofafloorata lowerstrike rate.Y. Interest ratecollarII.The purchase ofaatone strikerate andthesaleof another capat ahigher strike rate.III. Sometimes described asswappinginto abond.10. X.Zero-coupon convertible bondI.Sacrifice yieldY. Putable convertible bondII.Greater credit riskIII.premium11. X. Modified duration I.Allowsfor changing cashflows asinterest rates change.Y. Effective duration II.notallowforchanging cashflowsasinterestrates change.III.Notan appropriate measure forcallable bonds.COURSE 6:2002GOONTONEXT PAGEMORNING SESSION7-16. Eachofquestions7through16consistsoftwoInthelistattheleftaretwo items,lettered XandY.Inthelistatrightare three items, numbered I,II,andIII.ONE ofthe lettered itemsis relatedinwayto EXACTLY TWOofthe numbered items.Indicate therelated itemsusingthe following answer code:LetteredIs Relatedto Numbered Items(A) XIandIIonly(B)XIIandIIIonly(C)YIandIIonly(D) YIandIII only(E) The correct answer isnot given by (A),(B),(C) or。

精算师考试练习题SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETYEXAM FM FINANCIAL MATHEMATICSEXAM FM SAMPLE QUESTIONSCopyright 2005 by the Society of Actuaries and the CasualtyActuarial Society Some of the questions in this study note are taken from past SOA/CAS examinations. FM-09-05 PRINTED IN U.S.A. 11/08/04 2 These questions are representative of the types of questions that might be asked of candidates sitting for the new examination on Financial Mathematics (2/FM). These questions are intended to represent the depth of understanding required of candidates. The distribution of questions by topic is not intended to represent the distribution of questions on future exams.11/08/04 31.Bruce deposits 100 into a bank account. His account is credited interest at a nominal rate of interest of 4% convertible semiannually.At the same time, Peter deposits 100 into a separate account. Peter’s account is credited interest at a force of interest of .After 7.25 years, the value of each account is the same.Calculate .(A) 0.0388(B) 0.0392(C) 0.0396(D) 0.0404(E) 0.041411/08/04 42.Kathryn deposits 100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective interest rate of i. The accumulated amount in the account at the end of 40 years is X, which is 5 times the accumulated amount in the account at the end of 20 years.Calculate X.(A) 4695(B) 5070(C) 5445(D) 5820(E) 619511/08/04 53.Eric deposits 100 into a savings account at time 0, which pays interest at a nominal rate of i, compounded semiannually.Mike deposits 200 into a different savings account at time 0, which pays simple interest at an annual rate of i.Eric and Mike earn the same amount of interest during the last 6 months of the 8th year. Calculate i.(A) 9.06%(B) 9.26%(C) 9.46%(D) 9.66%(E) 9.86%11/08/04 64.John borrows 10,000 for 10 years at an annual effective interestrate of 10%. He can repay this loan using the amortization method with payments of 1,627.45 at the end of each year. Instead, John repays the 10,000 using a sinking fund that pays an annual effective interest rate of 14%. The deposits to the sinking fund are equal to 1,627.45 minus the interest on the loan and are made at the end of each year for 10 years. Determine the balance in the sinking fund immediately after repayment of the loan.(A) 2,130(B) 2,180(C) 2,230(D) 2,300(E) 2,37011/08/04 75.An association had a fund balance of 75 on January 1 and 60 on December 31. At the end of every month during the year, the associationdeposited 10 from membership fees. There were withdrawals of 5 on February 28, 25 on June 30, 80 on October 15, and 35 on October 31.Calculate the dollar-weighted (money-weighted) rate of return for the year.(A) 9.0%(B) 9.5%(C) 10.0%(D) 10.5%(E) 11.0%11/08/04 86.A perpetuity costs 77.1 and makes annual payments at the end of the year. The perpetuvtw paws 1 at the end of wear 2, 2 at the end of wear 3, …., n at the end of year (n+1). After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%.Calculate n.(A) 17(B) 18(C) 19(D) 20(E) 2111/08/04 97.1000 is deposited into Fund X, which earns an annual effective rate of 6%. At the end of each year, the interest earned plus an additional 100 is withdrawn from the fund. At the end of the tenth year, the fund is depleted.The annual withdrawals of interest and principal are deposited into Fund Y, which earns an annual effective rate of 9%.Determine the accumulated value of Fund Y at the end of year 10.(A) 1519(B) 1819(C) 2085(D) 2273(E) 243111/08/04 108.You are given the following table of interest rates:Calendar Year Portfolioof Original RatesInvestment Year Rates (in %) Investment (in %)i1y i2y i3y i4y i5y y iy+51992 8.25 8.25 8.4 8.5 8.5 8.351993 8.5 8.7 8.75 8.9 9.0 8.61994 9.0 9.0 9.1 9.1 9.2 8.851995 9.0 9.1 9.2 9.3 9.4 9.11996 9.25 9.35 9.5 9.55 9.6 9.359.7 1997 9.5 9.5 9.6 9.71998 10.0 10.0 9.9 9.81999 10.0 9.8 9.72000 9.5 9.59.0 2001A person deposits 1000 on January 1, 1997. Let the following be the accumulated value of the 1000 on January 1, 2000:P: under the investment year methodQ: under the portfolio yield methodR: where the balance is withdrawn at the end of everyyear and is reinvested at the new money rate Determine the ranking of P, Q, and R.P Q R (A)(B) P R Q(C) Q P R(D) R P Q(E) R Q P11/08/04 119.A 20-year loan of 1000 is repaid with payments at the end of each year. Each of the first ten payments equals 150% of the amount of interest due. Each of the last ten payments is X.The lender charges interest at an annual effective rate of 10%.Calculate X.(A) 32(B) 57(C) 70(D) 97(E) 11711/08/04 1210.A 10,000 par value 10-year bond with 8% annual coupons is bought at a premium to yield an annual effective rate of 6%.Calculate the interest portion of the 7th coupon.(A) 632(B) 642(C) 651(D) 660(E) 66711/08/04 1311.A perpetuity-immediate pays 100 per year. Immediately after thefifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. The annual effective rate of interest is 8%.Calculate X.(A) 54(B) 64(C) 74(D) 84(E) 9411/08/04 1412.Jeff deposits 10 into a fund today and 20 fifteen years later. Interest is credited at a nominal discount rate of d compoundedquarterly for the first 10 years, and at a nominal interest rate of 6% compounded semiannually thereafter. The accumulated balance in the fund at the end of 30 years is 100.Calculate d.(A) 4.33%(B) 4.43%(C) 4.53%(D) 4.63%(E) 4.73%11/08/04 1513.Ernie makes deposits of 100 at time 0, and X at time 3. The fund grows at a force of interest2 t t , t > 0.100The amount of interest earned from time 3 to time 6 is also X. Calculate X.(A) 385(B) 485(C) 585(D) 685(E) 78511/08/04 1614.Mike buys a perpetuity-immediate with varying annual payments.During the first 5 years, 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) 4.811/08/04 1715.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%.Installments of 200 each year plus interest on the unpaid balance at an annual effective (ii)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%11/08/04 1816.A loan is amortized over five years with monthly payments at a nominal interest rate of 9% compounded monthly. The first payment is 1000 and is to be paid one month from the date of the loan. Eachsucceeding 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) 734411/08/04 1917.To accumulate 8000 at the end of 3n years, deposits of 98 are made at the end of each of the 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%11/08/04 2018.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 eachmonth 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) 288011/08/04 2119.You are given the following information about the activity in two different investment accounts:Account KFund value ActivityDate Deposit Withdrawal before activityJanuary 1, 1999 100.0125.0 July 1, 1999 X2X October 1, 1999 110.0December 31, 1999 125.0Account LActivity Fund valueDate Deposit Withdrawal before activityJanuary 1, 1999 100.0X July 1, 1999 125.0December 31, 1999 105.8During 1999, the dollar-weighted (money-weighted) return for investment account K equals the time-weighted return for investment account L, which equals i. Calculate i.(A) 10%(B) 12%(C) 15%(D) 18%(E) 20%11/08/04 2220.David can receive one of the following two payment streams:(i) 100 at time 0, 200 at time n, and 300 at time 2n600 at time 10 (ii)At an annual effective interest rate of i, the present values of the two streams are equal. Given vn = 0.76, determine i.(A) 3.5%(B) 4.0%(C) 4.5%(D) 5.0%(E) 5.5%11/08/04 2321.Payments are made to an account at a continuous rate of (8k + tk), where0 t 10 .1 Interest is credited at a force of interest δt = . 8 tAfter 10 years, the account is worth 20,000.Calculate k.(A) 111(B) 116(C) 121(D) 126(E) 13111/08/04 2422.You have decided to invest in Bond X, an n-year bond with semi-annual coupons and the following characteristics:Par value is 1000.r The ratio of the semi-annual coupon rate to the desired semi-annual yield rate, , is 1.03125. iThe present value of the redemption value is 381.50.Given vn = 0.5889, what is the price of bond X?(A) 1019(B) 1029(C) 1050(D) 1055(E) 107211/08/04 2523.Project P requires an investment of 4000 at time 0. The investment pays 2000 at time 1 and 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.The net present values of the two projects are equal at an interest rate of 10%. Calculate X.(A) 5400(B) 5420(C) 5440(D) 5460(E) 548011/08/04 2624.A 20-year loan of 20,000 may be repaid under the following two methods:i) amortization method with equal annual payments at an annual effectiverate of 6.5%ii) sinking fund method in which the lender receives an annual effectiverate of 8% and the sinking fund earns an annual effective rate of j Both methods require a payment of X to be made at the end of each year for 20 years. Calculate j.(A) j , 6.5%(B) 6.5% < j , 8.0%(C) 8.0% < j , 10.0%(D) 10.0% < j , 12.0%(E) j > 12.0%11/08/04 2725.A perpetuity-immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments. Brian's share of the present value of the original perpetuity is 40%, and Jeff's share is K.Calculate K.(A) 24%(B) 28%(C) 32%(D) 36%(E) 40%11/08/04 2826.Seth, Janice, and Lori each borrow 5000 for five years at a nominal interest rate of 12%, compounded semi-annually.Seth has interest accumulated over the five years and pays all the interest and principal in a lump sum at the end of five years.Janice pays interest at the end of every six-month period as it accrues and the principal at the end of five years.Lori repays her loan with 10 level payments at the end of every six-month period. Calculate the total amount of interest paid on all three loans.(A) 8718(B) 8728(C) 8738(D) 8748(E) 875811/08/04 2927.Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits 100 into his bank account, and Robbie deposits 50 into his. Each account earns the same annual effective interest rate.The amount of interest earned in Bruce's account during the 11th year is equal to X. The amount of interest earned in Robbie's account during the 17th year is also equal to X. Calculate X.(A) 28.0(B) 31.3(C) 34.6(D) 36.7(E) 38.911/08/04 3028.Ron is repaying a loan with payments of 1 at the end of each yearfor n years. The amount of interest paid in period t plus the amount of principal repaid in period t + 1 equals X. Calculate X.n t v 1 + (A)in t (B) 1 + vd1 + vn ti (C)1 + vn td (D)1 + vn t (E)11/08/04 3129.At an annual effective interest rate of i, i > 0%, the present value of a perpetuity paying `10 at the end of each 3-year period, with thefirst payment at the end of year 3, is 32. At the same annual effective rate of i, the present value of a perpetuity paying 1 at the end of each 4-month period, with first payment at the end of 4 months, is X.Calculate X.(A) 31.6(B) 32.6(C) 33.6(D) 34.6(E) 35.611/08/04 3230.As of 12/31/03, an insurance company has a known obligation to pay $1,000,000 on 12/31/2007. To fund this liability, the company immediately purchases 4-year 5% annual coupon bonds totaling $822,703 of par value. The company anticipates reinvestment interest rates to remain constant at 5% through 12/31/07. The maturity value of the bond equals the par value.Under the following reinvestment interest rate movement scenarios effective 1/1/2004, what best describes the insurance company’s profit or (loss) as of 12/31/2007 after the liability is paid?Interest Interest Rates Rates Drop Increase by ?% by ?%(A) +6,606 +11,147(B) +14,418 (14,757)(C) (18,911) +19,185(D) (1,313) +1,323(E) Breakeven Breakeven 11/08/04 3331.An insurance company has an obligation to pay the medical costs fora claimant. Average annual claims costs today are $5,000, and medical inflation is expected to be 7% per year. The claimant is expected tolive an additional 20 years. Claim payments are made at yearly intervals, with the first claim payment to be made one year from today.Find the present value of the obligation if the annual interest rate is 5%. (A) 87,932(B) 102,514(C) 114,611(D) 122,634(E) Cannot be determined11/08/04 3432.An investor pays $100,000 today for a 4-year investment that returns cash flows of $60,000 at the end of each of years 3 and 4. The cashflows can be reinvested at 4.0% per annum effective.If the rate of interest at which the investment is to be valued is5.0%, what is the net present value of this investment today?(A) -1398(B) -699(C) 699(D) 1398(E) 2,62911/08/04 3533.You are given the following information with respect to a bond: par amount: 1000term to maturity 3 yearsannual coupon rate 6% payable annuallyTerm Annual Spot InterestRates7% 12 8%3 9%Calculate the value of the bond.(A) 906(B) 926(C) 930(D) 950(E) 100011/08/04 3634.You are given the following information with respect to a bond: par amount: 1000term to maturity 3 yearsannual coupon rate 6% payable annuallyTerm Annual Spot InterestRates7% 12 8%3 9%Calculate the annual effective yield rate for the bond if the bond is sold at a price equalto its value.(A) 8.1%(B) 8.3%(C) 8.5%(D) 8.7%(E) 8.9%11/08/04 3735.The current price of an annual coupon bond is 100. The derivative of the price of the bond with respect to the yield to maturity is -700. The yield to maturity is an annual effective rate of 8%.Calculate the duration of the bond.(A) 7.00(B) 7.49(C) 7.56(D) 7.69(E) 8.0011/08/04 3836.Calculate the duration of a common stock that pays dividends at the end of each year into perpetuity. Assume that the dividend is constant, and that the effective rate of interest is 10%. (A) 7(B) 9(C) 11(D) 19(E) 2711/08/04 3937.Calculate the duration of a common stock that pays dividends at the end of each year into perpetuity. Assume that the dividend increases by 2% each year and that the effective rate of interest is 5%.(A) 27(B) 35(C) 44(D) 52(E) 5811/08/04 4038.Eric and Jason each sell a different stock short at the beginning of the year for a price of 800. The margin requirement for each investor is 50% and each will earn an annual effective interest rate of 8% on his margin account.Each stock pays a dividend of 16 at the end of the year. Immediately thereafter, Eric buys back his stock at a price of (800 - 2X), and Jason buys back his stock at a price of (800 + X).Eric’s annual effective yield, i, on the short sale is twice Jason’s annual effective yield. Calculate i.(A) 4%(B) 6%(C) 8%(D) 10%(E) 12%11/08/04 4139.Jose and Chris each sell a different stock short for the same price. For each investor, the margin requirement is 50% and interest on the margin debt is paid at an annual effective rate of 6%.Each investor buys back his stock one year later at a price of 760. Jose’s stock paid a dividend of 32 at the end of the year whileChris’s stock paid no dividends. During the 1-year period, Chris’s return on the short sale is i, which is twice the return earned by Jose.Calculate i.(A) 12%(B) 16%(C) 18%(D) 20%(E) 24%11/08/04 4240.Bill and Jane each sell a different stock short for a price of 1000. For both investors, the margin requirement is 50%, and interest on the margin is credited at an annual effective rate of 6%.Bill buys back his stock one year later at a price of P. At the endof the year, the stock paid a dividend of X.Jane also buys back her stock after one year, at a price of (P –25). At the end of the year, her stock paid a dividend of 2X.Both investors earned an annual effective yield of 21% on theirshort sales. Calculate P.(A) 800(B) 825(C) 850(D) 875(E) 90011/08/04 4341. *On January 1, 2005, Marc has the following options for repaying a loan: Sixty monthly payments of 100 beginning February 1, 2005. A single payment of 6000 at the end of K months.Interest is at a nominal annual rate of 12% compounded monthly. The two options havethe same present value.Determine K.(A) 29.0(B) 29.5(C) 30.0(D) 30.5(E) 31.0*Reprinted with permission from ACTEX Publications11/08/04 4442. *You are given an annuity-immediate with 11 annual payments of 100 and a final payment at the end of 12 years. At an annual effectiveinterest rate of 3.5%, the present value at time 0 of all payments is 1000.Calculate the final payment.(A) 146(B) 151(C) 156(D) 161(E) 166* Reprinted with permission from ACTEX Publications.11/08/04 4543. *A 10,000 par value bond with coupons at 8%, convertible semiannually, is being sold 3 years and 4 months before the bond matures. The purchase will yield 6% convertible semiannually to the buyer. The price at the most recent coupon date, immediately after the coupon payment, was 5640.Calculate the market price of the bond, assuming compound interest throughout. (A) 5500(B) 5520(C) 5540(D) 5560(E) 5580* Reprinted with permission from ACTEX Publications.11/08/04 4644. *A 1000 par value 10-year bond with coupons at 5%, convertible semiannually, is sellingfor 1081.78.Calculate the yield rate convertible semiannually.(A) 1.00%(B) 2.00%(C) 3.00%(D) 4.00%(E) 5.00%* Reprinted with permission from ACTEX Publications.11/08/04 4745.You are given the following information about an investment account:Date Value Immediately DepositBefore DepositJanuary 1 10July 1 12 XX December31Over the year, the time-weighted return is 0%, and the dollar-weighted (money- weighted) return is Y.Calculate Y.(A) -25%(B) -10%(C) 0%(D) 10%(E) 25%11/08/04 4846.Seth borrows X for four years at an annual effective interest rate of 8%, to be repaid with equal payments at the end of each year. The outstanding loan balance at the end of the third year is 559.12.Calculate the principal repaid in the first payment.(A) 444(B) 454(C) 464(D) 474(E) 48411/08/04 4947.Bill buys a 10-year 1000 par value 6% bond with semi-annual coupons. The price assumes a nominal yield of 6%, compounded semi-annually.As Bill receives each coupon payment, he immediately puts the money into an account earning interest at an annual effective rate of i.At the end of 10 years, immediately after Bill receives the final coupon payment and the redemption value of the bond, Bill has earned an annual effective yield of 7% on his investment in the bond.Calculate i.(A) 9.50%(B) 9.75%(C) 10.00%(D) 10.25%(E) 10.50%11/08/04 5048.A man turns 40 today and wishes to provide supplemental retirement income of 3000 at the beginning of each month starting on his 65th birthday. Starting today, he makes monthly contributions of X to a fund for 25 years. The fund earns a nominal rate of 8% compounded monthly.On his 65th birthday, each 1000 of the fund will provide 9.65 of income at the beginning of each month starting immediately and continuing as long as he survives. Calculate . X(A) 324.73(B) 326.89(C) 328.12(D) 355.45(E) 450.6511/08/04 5149.Happy and financially astute parents decide at the birth of their daughter that they will need to provide 50,000 at each of their daughter’s 18th, 19th, 20th and 21st birthdays to fund her college education. They plan to contribute X at each of their daughter’s 1st through 17th birthdays to fund the four 50,000 withdrawals. If they anticipate earning a constant 5% annual effective rate on their contributions, which the following equations of value can be used to determine X, assuming compound interest?1 2 1 4 ， v ，....v.1075] , 50,000[v， ...v (A)X[v ] .05 .05 .05 .051 3 (B) X[(1.05)16 ， (1.05)15 ，...(1.05)] , 50,000[1，...v ] .053 (C) X[(1.05)17 ， (1.05)16 ，...1] , 50,000[1，...v ] .05] 1 3 (D) X[(1.05)17 ， (1.05)16 ，...(1.05)] , 50,000[1，...v .051 (E) X[(1， v ，....v.1075] , 50,000[v.1085 ] .05，...v.205211/08/04 5250.A 1000 bond with semi-annual coupons at i(2) = 6% matures at par on October 15, 2020. The bond is purchased on June 28, 2005 to yield the investor i(2) = 7%. What is the purchase price?Assume simple interest between bond coupon dates and note that: Date Day of the YearApril 15 105June 28 179October 15 288(A) 906(B) 907(C) 908(D) 919(E) 92511/08/04 53The following information applies to questions 51 thru 53.Joe must pay liabilities of 1,000 due 6 months from now and another 1,000 due one year from now. There are two available investments:a 6-month bond with face amount of 1,000, a 8% nominal annual coupon rate convertible semiannually, and a 6% nominal annual yield rate convertible semiannually; anda one-year bond with face amount of 1,000, a 5% nominal annual coupon rate convertible semiannually, and a 7% nominal annual yield rate convertible semiannually 51.How much of each bond should Joe purchase in order to exactly (absolutely) match the liabilities?Bond I Bond II1 .97561 (A)(B) .93809 1(C) .97561 .94293(D) .93809 .97561(E) .98345 .9756111/08/04 54The following information applies to questions 51 thru 53.Joe must pay liabilities of 1,000 due 6 months from now and another 1,000 due one year from now. There are two available investments:a 6-month bond with face amount of 1,000, a 8% nominal annual coupon rate convertible semiannually, and a 6% nominal annual yield rate convertible semiannually; anda one-year bond with face amount of 1,000, a 5% nominal annual coupon rate convertible semiannually, and a 7% nominal annual yield rate convertible semiannually 52.What is Joe’s total cost of purchasing the bonds required toexactly (absolutely) match the liabilities?(A) 1894(B) 1904(C) 1914(D) 1924(E) 193411/08/04 55The following information applies to questions 51 thru 53.Joe must pay liabilities of 1,000 due 6 months from now and another 1,000 due one year from now. There are two available investments:a 6-month bond with face amount of 1,000, a 8% nominal annual coupon rate convertible semiannually, and a 6% nominal annual yield rate convertible semiannually; anda one-year bond with face amount of 1,000, a 5% nominal annualcoupon rate convertible semiannually, and a 7% nominal annual yield rate convertible semiannually 53.What is the annual effective yield rate for investment in the bonds required to exactly (absolutely) match the liabilities?(A) 6.5%(B) 6.6%(C) 6.7%(D) 6.8%(E) 6.9%11/08/04 5654.Matt purchased a 20-year par value bond with semiannual coupons at a nominal annual rate of 8% convertible semiannually at a price of 1722.25. The bond can be called at par value X on any coupon date starting at the end of year 15 after the coupon is paid. The price guarantees that Mattwill receive a nominal annual rate of interest convertible semiannually of at least 6%.Calculate X.(A) 1400(B) 1420(C) 1440(D) 1460(E) 1480.11/08/04 5755.Toby purchased a 20-year par value bond with semiannual coupons at a nominal annual rate of 8% convertible semiannually at a price of 1722.25. The bond can be called at par value 1100 on any coupon date starting at the end of year 15. What is the minimum yield that Toby could receive, expressed as a nominal annual rate of interest convertible semiannually?(A) 3.2%(B) 3.3%(C) 3.4%(D) 3.5%(E) 3.6%11/08/04 5856.Sue purchased a 10-year par value bond with semiannual coupons at a nominal annual rate of 4% convertible semiannually at a price of 1021.50. The bond can be called at par value X on any coupon date starting at the end of year 5. The price guarantees that Sue will receive a nominal annual rate of interest convertible semiannually of at least 6%. Calculate X.(A) 1120(B) 1140(C) 1160(D) 1180(E) 120011/08/04 5957.Mary purchased a 10-year par value bond with semiannual coupons at a nominal annual rate of 4% convertible semiannually at a price of 1021.50. The bond can be called at par value 1100 on any coupon date starting at the end of year 5. What is the minimum yield that Mary could receive, expressed as a nominal annual rate of interest convertible semiannually?(A) 4.8%(B) 4.9%(C) 5.0%(D) 5.1%(E) 5.2%11/08/04 60。