北美精算师(SOA)考试P 2001 November年真题
精算师考试试题 (2)
Faculty of Actuaries Institute of ActuariesEXAMINATIONS7 September 2001 (pm)Subject 102 — Financial MathematicsTime allowed: Three hoursINSTRUCTIONS TO THE CANDIDATE1.Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.2.Mark allocations are shown in brackets.3.Attempt all 12 questions, beginning your answer to each question on aseparate sheet.Graph paper is not required for this paper.AT THE END OF THE EXAMINATIONHand in BOTH your answer booklet and this question paper.In addition to this paper you should have availableActuarial Tables and an electronic calculator.ã Faculty of Actuaries1 A 91-day government bill provides the purchaser with an annual effective rate ofreturn of 5%. Determine the annual simple discount rate at which the bill is discounted.[2]2 A particular share is expected to pay a dividend of d 1 in exactly one year.Dividends are expected to grow by g per annum effective every year thereafter.The share pays annual dividends. Let V 0 be the present value of the share and r be the investor’s required annual effective rate of return.Show that V 0 = 1d r g −.[3]3An asset has a current price of 100p. It will pay an income of 5p in 20 days’ time.Given a risk-free rate of interest of 6% per annum convertible half-yearly and assuming no arbitrage, calculate the forward price to be paid in 40 days.[4]4An annuity is paid half-yearly in arrears at a rate of £1,000 per annum, for 20years. The rate of interest is 5% per annum effective in the first 12 years and 6%per annum convertible quarterly for the remaining 8 years.Calculate the accumulation of the annuity at the end of 20 years.[4]5An investor purchases a bond, redeemable at par, which pays half-yearly couponsat a rate of 8% per annum. There are 8 days until the next coupon payment and the bond is ex-dividend. The bond has 7 years to maturity after the next coupon payment.Calculate the purchase price to provide a yield to maturity of 6% per annum effective.[4]6(1 + i t ) follows a log normal distribution where i t is the rate of interest over agiven time period beginning at time t . The parameters of the distribution are µ = 0.06 and σ2 = 0.0009.Calculate the inter-quartile range for the accumulation of 100 units of money over the given time period, beginning at time t .[6]7(i) The annual effective forward rate applicable over the period from t to t + ris defined as f t,r where t and r are measured in years. If f 0,1 = 8%, f 1,1 = 7%,f 2,1 = 6% and f 3,1 = 5%, calculate the gross redemption yield at the issue date from a 4-year bond, redeemable at par, with a 5% coupon payable annually in arrears.[7](ii)Explain why the gross redemption yield from the 4-year bond is higher than the 4-year forward rate f 3,1.[2][Total 9]8 A fast food company is considering opening a new sales outlet. The initial cost ofthe outlet would be £1,000,000 incurred at the outset of the project. It is expected that rents of £40,000 per annum would have to be paid quarterly in advance for 10 years, increasing after ten years to £48,000 per annum. The net revenue (sales minus costs, other than rent) from the venture is expected to be £100,000 for the first year and £200,000 for the second year. Thereafter, the net revenue is expected to grow at 3% per annum compound so that it is £206,000 in the third year, £212,180 in the fourth year and so on. The revenue would be received continuously throughout each year. Twenty years after the outset of the project, the revenue and costs stop and the project has no further value.Calculate the internal rate of return from the project.[11]9(i)Prove that =nn n a Da i −.[3](ii) A bank makes a loan to be repaid by instalments paid annually in arrears.The first instalment is 20, the second is 19 with the payments reducing by1 per annum until the end of the 10th year after which there are no further payments. The rate of interest charged by the lender is 6% per annum effective.(a)Calculate the amount of the loan.(b)Calculate the interest and capital components of the first payment.(c) Calculate the amount of capital repaid in the instalment at the end of the 8th year.[8][Total 11]10An investor purchased a bond with exactly 20 years to redemption. The bond, redeemable at par, has a gross redemption yield of 6%. It pays annual coupons,in arrears, of 5%. The investor does not pay tax.(i) Calculate the purchase price paid for the bond.[3](ii)After exactly ten years, immediately after payment of the coupon then due, this investor sells the bond to another investor. That investor paysincome and capital gains tax at a rate of 30%. The bond is purchased bythe second investor to provide a net rate of return of 6.5% per annum.(a) Calculate the price paid by the second investor.(b) Calculate the annual effective rate of return earned by the firstinvestor during the period for which the bond was held.[10][Total 13] 11The force of interest, δ(t), is:δ(t)= 0.05 for 0 < t≤ 10,= 0.006t for 10 < t≤ 20= 0.003t + 0.0002t2 for 20 < t(i) Calculate the present value of a unit sum of money due at time t = 25.[7](ii) Calculate the effective rate of interest per unit time from time t = 19 to time t = 20.[3](iii) A continuous payment stream is paid at the rate of e−0.03t per unit time between time t = 0 and time t = 5. Calculate the present value of thatpayment stream.[4][Total 14]12(i)(a) In the context of a stream of future receipts paid at discrete times,let volatility be defined as the proportionate change in the presentvalue of a payment stream per unit change in the force of interest,for small changes in the force of interest. Prove that thediscounted mean term is equal to the volatility.(b)If volatility is now defined as the proportionate change in thepresent value of a payment stream per unit change in the annualeffective rate of interest, for small changes in the annual effectiverate of interest, find the relationship between the discounted meanterm and volatility.[5](ii) A life insurance company manages a small annuity fund. Payments are expected to be made from the fund of £1,000,000 per annum at the end ofyears 1 to 10 and £1,500,000 at the end of each of the following 10 years.Assets are held in two types of bonds. The first is a zero coupon bondredeemable in 10 years’ time. The second is a bond which pays an annual coupon of g% per annum in arrears and is redeemable at par at the end of19 years. £10,000,000 nominal of the zero coupon bond have beenpurchased.Find the nominal amount of the coupon paying bond which must bepurchased and the rate of coupon which is received from the bond if theinsurance company is to equalise the present values and discounted mean terms of its assets and liabilities at an effective rate of interest of 5% perannum.[12] (iii)If the present value and discounted mean term of the assets and liabilities are equalised, state the third condition which is necessary for theinsurance company to be immunised from small, uniform changes in therate of interest.[2][Total 19]。
北美精算考试试题
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 be sold.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 constants 27. 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 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。
北美精算师(SOA)考试 FM 2001 May 年真题
1.
Which of the following statements is an implication of the semi-strong form of the Efficient Market Hypothesis?
(A) (B) (C) (D) (E)
Course 2
12
Form 01A
9.
Which of the following statements regarding consumer goods in the marketplace is true?
(A) (B) (C) (D) (E)
The quantity demanded of an inferior good decreases as its price decreases. The intersection of a consumer’s Engel curve and demand curve represents how much the consumer will buy given a specific income level. When the price of a normal good decreases, the increase in quantity due to the income effect cannot be greater than the substitution effect. The compensated demand curve for a normal good will be steeper than the uncompensated demand curve. The income elasticity of demand is equal to the slope of the Engel curve.
精算师考试试题 (5)
Faculty of Actuaries Institute of ActuariesEXAMINATIONS13 September 2001 (am)Subject 105 — Actuarial Mathematics 1Time allowed: Three hoursINSTRUCTIONS TO THE CANDIDATE1.Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.2.Mark allocations are shown in brackets.3.Attempt all 14 questions, beginning your answer to each question on aseparate sheet.Graph paper is not required for this paper.AT THE END OF THE EXAMINATIONHand in BOTH your answer booklet and this question paper.In addition to this paper you should have availableActuarial Tables and an electronic calculator.ã Faculty of Actuaries1Under the Manchester Unity model of sickness, you are given the following values:=5xs1 0=0.9 t xp dtòCalculate the value ofxz. [2]2Give a formula for21(2003)P in terms of20(2002)P, based on the component method of population projection. ()xP n denotes the population aged x last birthday at mid-year n.State all the assumptions that you make and define carefully all the symbols that you use. [3]3 A life insurance company issues a policy under which sickness benefit of £100 perweek is payable during all periods of sickness. There is a waiting period of 1 year under the policy.You have been asked to calculate the premium for a life aged exactly 30, who isin good health, using the Manchester Unity model of sickness.Describe how you would allow for the waiting period in your calculation, giving a reason for your choice of method. [3]4An employer recruits lives aged exactly 20, all of whom are healthy whenrecruited. On entry, the lives join a scheme that pays a lump sum of £50,000immediately on death, with an additional £25,000 if the deceased was sick at the time of death.The mortality and sickness of the scheme members are described by the following multiple-state model, in which the forces of transition depend on age only.All surviving members retire at age 65 and leave the scheme regardless of their state of health.,ab x t p is defined as the probability that a life who is in state a at age x (a = H, S, D )is in state b at age x + t (0 and ,,)t b H S D ≥=.Write down an integral expression for the expected present value, at force of interest δ, of the death benefit in respect of a single new recruit. [3]5 A pension scheme provides a pension of 1/60 of career average salary in respect ofeach full year of service, on age retirement between the ages of 60 and 65. A proportionate amount is provided in respect of an incomplete year of service.At the valuation date of the scheme, a new member aged exactly 40 has an annual rate of salary of £40,000.Calculate the expected present value of the future service pension on age retirement in respect of this member, using the Pension Fund Tables in the Formulae and Tables for Actuarial Examinations. [3]6 A life insurance company issues a special annuity contract to a male life agedexactly 70 and a female life aged exactly 60.Under the contract, an annuity of £10,000 per annum is payable monthly to thefemale life, provided that she survives at least 10 years longer than the male life.The annuity commences on the monthly policy anniversary next following thetenth anniversary of the death of the male life and is payable for the balance ofthe female’s lifetime.Calculate the single premium required for the contract.Basis:Mortality:a(55) Ultimate, males or females as appropriateInterest:8% per annumExpenses:none [4]7The staff of a company are subject to two modes of decrement, death and withdrawal from employment.Decrements due to death take place uniformly over the year of age in theassociated single-decrement table: 50% of the decrements due to withdrawaloccur uniformly over the year of age and the balance occurs at the end of the year of age, in the associated single-decrement table.You are given that the independent rate of mortality is 0.001 per year of age and the independent rate of withdrawal is 0.1 per year of age.Calculate the probability that a new employee aged exactly 20 will die as anemployee at age 21 last birthday. [4]8The following data are available from a life insurance company relating to the mortality experience of its temporary assurance policyholders.,x dθThe number of deaths over the period 1 January 1998 to 30 June 2001, aged x nearest birthday at entry and having duration d at the policyanniversary next following the date of death.,()y eP n The number of policyholders with policies in force at time n, aged y nearest birthday at entry and having curtate duration e at time n, wheren = 1.1.1998, 30.6.1998, 30.6.2000 and 30.6.2001.Develop formulae for the calculation of the crude central select rates of mortality corresponding to the,x dθ deaths and derive the age and duration to which these rates apply. State all the assumptions that you make.[6]9(i)State the conditions necessary for gross premium retrospective and prospective reserves to be equal. [3] (ii)Demonstrate the equality of gross premium retrospective and prospective reserves for a whole life policy, given the conditions necessary for equality.[4][Total 7]10 A life insurance company issues a special term assurance policy to two lives agedexactly 50 at the issue date, in return for the payment of a single premium. The following benefits are payable under the contract:(i)In the event of either of the lives dying within 10 years, a sum assured of£100,000 is payable immediately on this death.(ii)In the event of the second death within 10 years, a further sum assured of £200,000 is payable immediately on the second death.Calculate the single premium.Basis:Mortality:A1967–70 UltimateInterest:4% per annumExpenses:None [8]11 A life insurance company sells term assurance policies with terms of either 10 or20 years.As an actuary in the life office, you have been asked to carry out the first review of the mortality experience of these policies. The following table shows thestatistical summary of the mortality investigation. In all cases, the central rates of mortality are expressed as rates per 1,000 lives.All policies10-year policies20-year policiesAge Numberin forceCentralmortalityrateNumberin forceCentralmortalityrateNumberin forceCentralmortalityrate–246,991 1.086,0130.86978 2.12 25–446,462 2.055,438 1.741,024 3.68 45–645,81513.264,94211.5587322.94 65–3,05175.702,57071.5348197.70 Total22,31918,9633,356(i)Calculate the directly standardised mortality rate and the standardisedmortality ratio separately in respect of the 10-year and 20-year policies.In each case, use the “all policies” population as the standard population.[6](ii)You have been asked to recommend which of these two summary mortality measures should be monitored on a regular basis.Give your recommendation, explaining the reasons for your choice. [3][Total 9]12 A life insurance company offers an option on its 10-year without profit termassurance policies to effect a whole life without profits policy, at the expiry of the 10-year term, for the then existing sum assured, without evidence of health.Premiums under the whole life policy are payable annually in advance for thewhole of life, or until earlier death.(i)Describe the conventional method of pricing the mortality option, statingclearly the data and assumptions required. Formulae are not required.[3](ii) A policyholder aged exactly 30 wishes to effect a 10-year without profits term assurance policy, for a sum assured of £100,000.Calculate the additional single premium, payable at the outset, for theoption, using the conventional method.The following basis is used to calculate the basic premiums for the termassurance policies.Basis:Mortality:A1967–70 SelectInterest:6% per annumExpenses:none [4](iii)Describe how you would calculate the option single premium for the policy described in part (ii) above using the North American method, statingclearly what additional data you would require and what assumptions youwould make. [4](iv)State, with reasons, whether it would be preferable to use theconventional method or the North American method for pricing themortality option under the policy described in part (ii) above. [3][Total 14]13(i)On 1 September 1996, a life aged exactly 50 purchased a deferred annuity policy, under which yearly benefit payments are to be made. The firstpayment, being £10,000, is to be made at age 60 exact if he is then alive.The payments will continue yearly during his lifetime, increasing by1.923% per annum compound.Premiums under the policy are payable annually in advance for 10 yearsor until earlier death.If death occurs before age 60, the total premiums paid under the policy,accumulated to the end of the year of death at a rate of interest of 1.923%per annum compound, are payable at the end of the year of death.Calculate the annual premium.Basis:Mortality: before age 60:A1967–70 Ultimateafter age 60:a(55) Males UltimateInterest:6% per annumExpenses: initial:10% of the initial premium, incurredat the outsetrenewal:5% of each of the second andsubsequent premiums, payable at thetime of premium paymentclaim:£100, incurred at the time of paymentof the death benefit[9](ii)On 1 September 2001, immediately before payment of the premium then due, the policyholder requests that the policy be altered so that there is nobenefit payable on death and the rate of increase of the annuity inpayment is to be altered. The premium under the policy is to remainunaltered as is the amount of the initial annuity payment.The life insurance company calculates the revised terms of the policy byequating gross premium prospective reserves immediately before andafter the alteration, calculated on the original pricing basis, allowing foran expense of alteration of £100.Calculate the revised rate of increase in payment of the annuity. [7][Total 16]14 A life insurance company issues a 3-year unit-linked endowment assurancecontract to a male life aged exactly 60 under which level annual premiums of£5,000 are payable in advance throughout the term of the policy or until earlier death. 102% of each year’s premium is invested in units at the offer price.The premium in the first year is used to buy capital units, with subsequent years’premiums being used to buy accumulation units. There is a bid-offer spread in unit values, with the bid price being 95% of the offer price.The annual management charges are 5% on capital units and 1% on accumulation units. Management charges are deducted at the end of each year,before death, surrender or maturity benefits are paid.On the death of the policyholder during the term of the policy, there is a benefit payable at the end of the year of death of £12,000 or the bid value of the units allocated to the policy, if greater. On maturity, the full bid value of the units is payable.The policy may be surrendered only at the end of the first or the second policy year. On surrender, the life insurance company pays the full bid value of the accumulation units and 80% of the nominal bid value of the capital units,calculated at the time of surrender.The company holds unit reserves equal to the full bid value of the accumulation units and a proportion, 60:3t t A +−(calculated at 4% interest and A1967-70 Ultimate mortality), of the full bid value of the capital units, calculated just after thepayment of the premium due at time t (t = 0,1 and 2). The company holds no sterling reserves.The life insurance company uses the following assumptions in carrying out profit tests of this contract:Mortality:A1967–70 Ultimate Expenses:initial:£400renewal:£80 at the start of each of the second and third policy years Unit fund growth rate:8% per annum Sterling fund interest rate:5% per annum Risk discount rate:15% per annum Surrender rates:20% of all policies still in force at the end of each of the first and second yearsCalculate the profit margin on the contract.[18]。
北美精算师(SOA)考试P 2000 November年真题
(A) (B) (C) (D) (E)
208 260 270 312 374
November 2000
1
Course 1
2.
An investor purchases two assets, each having an initial value of 1000 . The value Vn of the first asset after n years can be modeled by the relationship:
0.06 0.16 0.19 0.22 0.25
Course 1
12
Form 00B
13.
An actuary believes that the demand for life insurance, L, and the demand for health insurance, H, can be modeled as functions of time, t:
What is the approximate probability that there is a total of between 2450 and 2600 claims during a one-year period?
(A) (B) (C) (D) (E)
0.68 0.82 0.87 0.95 1.00
Course 1
2
Form 00B
3.
An auto insurance company has 10,000 policyholders. Each policyholder is classified as
(i) (ii) (iii)
young or old; male or female; and married or single.
北美精算师(SOA)考试P 2001 November年真题
is independent of the number of accidents that occur in all other months.
Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.
(A) (B) (C) (D) (E)
0.0000 0.0004 0.0027 0.0064 0.3679
Course 1, November 2001
9
9.
Among a large group of patients recovering from shoulder injuries, it is found that 22% visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.14 the probability that a patient visits a physical therapist.
(–1)n n
− an = 1 2n n
Course 1, November 2001
11
11.
A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is
北美精算师(SOA)考试 FM 2001 November 年真题和注解
25.
E In order to solve for the price of a call option, first the price of a put option must be determined and then the put-call parity formula is used. Price of put option: [(0.5 x 0) + (0.5 x 10)] / 1.04 = 5/1.04 = 4.81 Price of call option = Price of put option + current stock price – present value of exercise price Price of call option = 4.81 + 45 – 40/1.04 = 11.35
Course 2 Solutions
1
November 2001
5.
A ¬9.2 + 10 ⋅V Present value = 10 ⋅ a5
1+ k 20 = 0.092 − k 0.84 = 21k k = 0.04
5 9.2
(1 + k ) ( 6.44 )(1 + k ) ∑ (1.092) = 38.70 + 0.092 − k = 167.50 t =1
3.
B
∆Y ∆N ∆K − (α N ) − (1 − α N ) = 2.6 − (0.7)(2.0) − (1 − 0.7)(4.0) = 2.6 − 1.4 − 1.2 = 0.0 Y N K
4.
C APV = Base-case NPV + PV tax shield 120, 000 0.35 × 0.08 × 0.54(100, 000) = −100, 000 + + = 0 + 1,400 = 1,400 . 1.2 1.08
Spring 2001 Exam
Determine b .
(A)
5 ln 6 5 5 ln 6 6 5 6 ln 6 5 7 ln 6 6 ln 5
(B)
(C)
(D)
(E)
May 2001
7
Course 1
2.
A stock pays annual dividends. The first dividend is 8 and each dividend thereafter is 7% larger than the prior dividend.
Let m be the number of dividends paid by the stock when the cumulative amount paid first exceeds 500 .
Calculate m .
(A) (B) (C) (D) (E)
23 24 25 26 27
May 2001
What is the expected excess of premiums over claims, given that the husband survives at least ten years?
(A) (B) (C) (D) (E)
350 385 397 870 897
May 2001
May 2001
12
Course 1
7.
A joint density function is given by kx f ( x, y ) = 0 where k is a constant. for 0 < x < 1, 0 < y < 1 otherwise,
What is Cov(X,Y) ?
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is independent of the number of accidents that occur in all other months.
Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.
K , for N = 1, . . . , 5 and K a constant. These N
are the only possible loss amounts and no more than one loss can occur.
Determine the net premium for this policy.
Course 1, November 2001
4
4.
Upon arrival at a hospital’s emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year: (i) (ii) (iii) (iv) (vi) (vii) 10% of the emergency room patients were critical; 30% of the emergency room patients were serious; the rest of the emergency room patients were stable; 40% of the critical patients died; 10% of the serious patients died; and 1% of the stable patients died.
Let C1 = X+Y denote the size of the combined claims before the application of a 20% surcharge on the hospital portion of the claim, and let C2 denote the size of the combined claims after the application of that surcharge.
(A) (B) (C) (D) (E)
0.01 0.12 0.23 0.29 0.41
Course 1, November 2001
12
12.
Let f be a function such that f ′(0) = 0 . The graph of the second derivative f ′′ is shown below.
S = 175 L 3/2 A 4/5.
Which of the following statements is true?
(A) (B) (C) (D) (E)
S increases at an increasing rate as L increases and increases at a decreasing rate as A increases. S increases at an increasing rate as L increases and increases at an increasing rate as A increases. S increases at a decreasing rate as L increases and increases at a decreasing rate as A increases. S increases at a decreasing rate as L increases and increases at an increasing rate as A increases. S increases at a constant rate as L increases and increases at a constant rate as A increases.
Given that a patient survived, what is the probability that the patient was categorized as serious upon arrival?
(A) (B) (C) (D) (E)
0.06 0.29 0.30 0.39 0.64
(A) (B) (C) (D) (E)
0.0000 0.0004 0.0027 0.0064 0.3679
Course 1, November 2001
9
9.
Among a large group of patients recovering from shoulder injuries, it is found that 22% visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.14 the probability that a patient visits a physical therapist.
(–1)n n
− an = 1 2n n
Course 1, November 2001
11
11.
A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is
Determine the probability that a randomly chosen member of this group visits a physical therapist.
(A) (B) (C) (D) (E)
0.26 0.38 0.40 0.48 0.62
Course 1, November 2001
10
10.
Let {an} be a sequence of real numbers.
For which of the following does the infinite series
1 a + converge? ∑ n
n n =1
∞
(A) (B) (C) (D) (E)
an = 1 an = 1 n an = 12 n an =
(A) (B) (C) (D) (E)
0.031 0.066 0.072 0.110 0.150
Course 1, November 2001
6
6.
Let C be the curve defined by the parametric equations x = t2 + t, y = t2 , y) dA = 6 .
Determine
∫∫
R
[4f(x, y) – 2] dA .
(A) (B) (C) (D) (E)
12 18 20 22 44
Course 1, November 2001
3
3.
Sales, S, of a new insurance product are dependent upon the labor, L, of the sales force and the amount of advertising, A, for the product. The relationship can be modeled by
7
7.
Let X denote the size of a surgical claim and let Y denote the size of the associated hospital claim. An actuary is using a model in which E(X) = 5, E(X 2) = 27.4, E(Y) = 7, E(Y 2) = 51.4, and Var(X+Y) = 8 .
F ( t ) = te − t
where t is time elapsed, in weeks, since the study began.
What is the minimum fraction of patients exhibiting severe symptoms between the end of the first week and the end of the seventh week of the study?
November 2001 Course 1
Mathematical Foundations of Actuarial Science
Society of Actuaries/Casualty Actuarial Society
1.
An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is 0.44 .