北美精算师考试官方样题2015-12-exam-fm-syllabus

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]。

ACTUARIES/CASUALTY ACTUARIAL SOCIETYOFSOCIETYEXAM FM FINANCIAL MATHEMATICSEXAM FM SAMPLE QUESTIONSCopyright 2005 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study note are taken from past SOA/CAS examinations.11/08/04 2U.S.A.FM-09-05INPRINTED11/08/04 3These 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 41.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 δ.After 7.25 years, the value of each account is the same.Calculate δ.0.0388(A)0.0392(B)0.0396(C)0.0404(D)0.0414(E)11/08/04 52.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.4695(A)5070(B)5445(C)5820(D)6195(E)11/08/04 63.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.9.06%(A)9.26%(B)9.46%(C)9.66%(D)9.86%(E)11/08/04 74.John borrows 10,000 for 10 years at an annual effective interest rate 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.2,130(A)2,180(B)2,230(C)2,300(D)2,370(E)11/08/04 85.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 association deposited 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%10.0%(C)10.5%(D)11.0%(E)11/08/04 96.A perpetuity costs 77.1 and makes annual payments at the end of the year.The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, …., n at the endof year (n+1). After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%.Calculate n.17(A)18(B)19(C)20(D)21(E)11/08/04 107.1000 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 10.1519(A)1819(B)2085(C)2273(D)2431(E)11/08/04 1111/08/04128.You are given the following table of interest rates: Calendar Year of Original Investment Investment Year Rates (in %) PortfolioRates(in %) yi 1y i 2y i 3y i 4y i 5y i y +5 1992 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 9.351997 9.5 9.5 9.6 9.7 9.71998 10.0 10.0 9.9 9.81999 10.0 9.8 9.72000 9.5 9.52001 9.0A 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 method Q :under the portfolio yield method R : where the balance is withdrawn at the end of everyyear and is reinvested at the new money rateDetermine the ranking of P , Q , and R .(A)P Q R >> (B) P R Q >>(C) Q P R >>(D) R P Q >>(E)R Q P >>9.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 1310.A 10,000 par value 10-year bond with 8% annual coupons is bought at a premiumto yield an annual effective rate of 6%.Calculate the interest portion of the 7th coupon.632(A)642(B)651(C)660(D)(E)66711/08/04 1411.A perpetuity-immediate pays 100 per year. Immediately after the fifth 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.54(A)64(B)74(C)84(D)94(E)11/08/04 1512.Jeff deposits 10 into a fund today and 20 fifteen years later. Interest is credited at a nominal discount rate of d compounded quarterly 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.4.33%(A)4.43%(B)4.53%(C)4.63%(D)4.73%(E)11/08/04 1611/08/041713.Ernie makes deposits of 100 at time 0, and X at time 3. The fund grows at a force of interest2100t t δ=, t > 0.The amount of interest earned from time 3 to time 6 is also X .Calculate X .(A) 385(B) 485(C) 585(D) 685(E) 78514.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 of167.50.Calculate K, given K < 9.2.4.0(A)4.2(B)4.4(C)4.6(D)4.8(E)11/08/04 1815.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 balance at 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.8.75%(A)9.00%(B)9.25%(C)9.50%(D)9.75%(E)11/08/04 1916.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. Each succeeding monthly payment will be 2% lower than the prior payment.Calculate the outstanding loan balance immediately after the 40th payment is made.6751(A)6889(B)6941(C)7030(D)7344(E)11/08/04 2017.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.11.25%(A)11.75%(B)12.25%(C)12.75%(D)13.25%(E)11/08/04 2118.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.2680(A)2730(B)2780(C)2830(D)2880(E)11/08/04 2219.You are given the following information about the activity in two different investment accounts:Account KFundvalue ActivityWithdrawal Date beforeactivity DepositJanuary 1, 1999 100.0July 1, 1999 125.0 XOctober 1, 1999 110.0 2XDecember 31, 1999 125.0Account Lvalue ActivityFundactivity DepositWithdrawal Date beforeJanuary 1, 1999 100.0July 1, 1999 125.0 XDecember 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.10%(A)12%(B)(C)15%11/08/04 23(D)18%20%(E)11/08/04 2420.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 = 0.76, determine i.3.5%(A)4.0%(B)4.5%(C)5.0%(D)5.5%(E)11/08/04 2521.Payments are made to an account at a continuous rate of (8k + tk), where010t≤≤ .Interest is credited at a force of interest δt =18t+.After 10 years, the account is worth 20,000.Calculate k.(A)111(B)116(C)121(D)126(E)13111/08/04 2622.You have decided to invest in Bond X, an n-year bond with semi-annual coupons and the following characteristics:• Par value is 1000.• The ratio of the semi-annual coupon rate to the desired semi-annual yield rate,ri, is 1.03125.• The present value of the redemption value is 381.50.Given v n = 0.5889, what is the price of bond X?(A)1019(B)1029(C)1050(D)1055(E)107211/08/04 2723.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.5400(A)5420(B)5440(C)5460(D)5480(E)11/08/04 2824.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 jBoth methods require a payment of X to be made at the end of each year for 20 years. Calculate j.(A) j≤ 6.5%j≤ 8.0%<6.5%(B)<j≤ 10.0%8.0%(C)<j≤ 12.0%10.0%(D)(E) j > 12.0%11/08/04 2925.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.24%(A)28%(B)32%(C)36%(D)40%(E)11/08/04 3026.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.8718(A)(B)87288738(C)8748(D)(E)875811/08/04 3111/08/04 3227.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.28.0(A)31.3(B)34.6(C)36.7(D)38.9(E)11/08/04 3311/08/043428.Ron is repaying a loan with payments of 1 at the end of each year for n years. The amount of interest paid in period t plus the amount of principal repaid in period t + 1 equals X .Calculate X .(A)1 + n t v i − (B)1 + n t v d − (C)1 + v n −t i (D)1 + v n −t d (E)1+ v n −t29.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 the first 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.31.6(A)32.6(B)33.6(C)34.6(D)35.6(E)11/08/04 3530.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?InterestRates Dropby ½% Interest Rates Increase by ½%(A) +6,606 +11,147(B) (14,757) +14,418(C) (18,911) +19,185(D) (1,313) +1,32311/08/04 36(E) Breakeven Breakeven11/08/04 3731.An insurance company has an obligation to pay the medical costs for a claimant. Average annual claims costs today are $5,000, and medical inflation is expected to be 7% per year. The claimant is expected to live 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 3832.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 cash flows can be reinvested at 4.0% per annum effective.If the rate of interest at which the investment is to be valued is 5.0%, what is the net present value of this investment today?(A) -1398(B) -699(C) 699(D) 1398(E) 2,62911/08/04 3933.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 InterestRates1 7%2 8%3 9%Calculate the value of the bond.(A) 906(B) 926(C) 930(D) 950(E) 100011/08/04 4011/08/04 4134.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 InterestRates1 7%2 8%3 9%Calculate the annual effective yield rate for the bond if the bond is sold at a price equal to its value.(A) 8.1%(B) 8.3%(C) 8.5%(D) 8.7%11/08/04 42(E) 8.9%11/08/04 4335.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 4436.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 4537.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 4638. – 44. skipped11/08/04 5245.You are given the following information about an investment account:ImmediatelyDate ValueDepositBefore DepositJanuary 1 10July 1 12 XXDecember31Over 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%11/08/04 53(E) 25%11/08/04 5446.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 5547.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 56。

北美精算考试试题1. The probability that a visit to a primary care physician’s (PCP) office results in neither lab work nor referral to a specialist is 35% . Of those coming to a PCP’s office, 30% are referred to specialists and 40% require lab work.Determine the probability that a visit to a PCP’s office results in both lab work and referral to a specialist.(A) 0.05(B) 0.12(C) 0.18(D) 0.25(E) 0.352. A study of automobile accidents produced the following data:An automobile from one of the model years 1997, 1998, and 1999 was involved in an accident.Determine the probability that the model year of this automobile is 1997 .(A) 0.22(B) 0.30(C) 0.33(D) 0.45(E) 0.503. The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100 printers, how much should it expect to pay in refunds?(A) 6,321(B) 7,358(C) 7,869(D) 10,256(E) 12,6424. Let T denote the time in minutes for a customer service representative to respond to 10 telephone inquiries. T is uniformly distributed on the interval with endpoints 8 minutes and 12 minutes. Let R denote the average rate, in customers per minute, at which the representative responds to inquiries.Which of the following is the density function of the random variable R on the interval(A)12/5(B) 3 (C) (D) (E)5. Let T1 and T2 represent the lifetimes in hours of two linked components in an electronic device. The joint density function for T1 and T2 is uniform over the region defined by 0 <= t1<= t2<=L where L is a positive constant.Determine the expected value of the sum of the squares of T1 and T2 .(A)L2/3(B)L2/2(C)2 L2/3(D) 3 L2/4(E) L26. Two instruments are used to measure the height, h, of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056h . The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation 0.0044h . Assuming the two measurements are independent random variables, what is the probability that their average value is within 0.005h of the height of the tower?(A) 0.38(B) 0.47(C) 0.68(D) 0.84(E) 0.907. An insurance company’s monthly claims are modeled by a continuous, positive randomvariable X, whose probability density function is proportional to (1 + x)-4 ,where 0 < x Determine the company’s expected monthly claims.(A)1/6(B)1/3(C)1/2(D) 1(E) 38. A probability distribution of the claim sizes for an auto insurance policy is given in thetable below:What percentage of the claims are within one standard deviation of the mean claim size?(A) 45%(B) 55%(C) 68%(D) 85%(E) 100%9. The total claim amount for a health insurance policy follows a distributionwith density function The premium for the policy is set at 100 over the expected total claim amount.If 100 policies are sold, what is the approximate probability that the insurancecompany will have claims exceeding the premiums collected?(A) 0.001(B) 0.159(C) 0.333(D) 0.407(E) 0.46010. An insurance company sells two types of auto insurance policies: Basic and Deluxe. The time until the next Basic Policy claim is an exponential random variable with mean two days. The time until the next Deluxe Policy claim is an independent exponential random variable with mean three days. What is the probability that the next claim will be a Deluxe Policy claim?(A) 0.172(B) 0.223(C) 0.400(D) 0.487(E) 0.50011. A company offers a basic life insurance policy to its employees, as well as a supplemental life insurance policy. To purchase the supplemental policy, an employee must first purchase the basic policy.Let X denote the proportion of employees who purchase the basic policy, and Y the proportion of employees who purchase the supplemental policy. Let X and Y have the joint density function f(x,y) = 2(x + y) on the region where the density is positive. Given that 10% of the employees buy the basic policy, what is the probability that fewer than 5% buy the supplemental policy?(A) 0.010(B) 0.013(C) 0.108(D) 0.417(E) 0.50012. Let C be the curve defined by x = sin t + t and y = cos t – t,Find an equation of the line tangent to C at (0, 1) .(A) y = 1(B) y = 1 + 2x(C) y = 1 – 2x(D) y = 1 –x(E) y = 1 –0.5x13. For a certain product priced at p per unit, 2000 – 10p units will be sold.Which of the following best represents the graph of revenue, r, as a function of price, p ?(A) (B) (C) (D) (E)14. A virus is spreading through a population in a manner that can be modeled by thefunction where A is the total population, g(t) is the number infected at time t, and B is a constant.What proportion of the population is infected when the virus is spreading the fastest?(A)1/3(B)1/2(C)2/3(D)3/4(E) 115. In a certain town, the rate of deaths at time t due to a particular disease is modeled by What is the total number of deaths from this disease predicted by the model?(A) 243(B) 370(C) 556(D) 1,111(E) 10,00016. The total cost, c, to a company for selling n widgets is c(n) = n2 + 4n + 100 . The price per widget is p(n) = 100 – n .What price per widget will yield the maximum profit for the company?(A) 50(B) 76(C) 96(D) 98(E) 10017. An insurance company has 120,000 to spend on the development and promotion of a new insurance policy for car owners. The company estimates that if x is spent on development and y is spent on promotion, then policies will besold.Based on this estimate, what is the maximum number of policies that the insurance company can sell?(A) 3,897(B) 9,000(C) 11,691(D) 30,000(E) 90,00018. An insurance policy reimburses dental expense, X, up to a maximum benefit of 250 . The probability density function for X is: where c is a constant.Calculate the median benefit for this policy.(A) 161(B) 165(C) 173(D) 182(E) 25019. In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from _2.5 years to 2.5 years. The healthcare data are based on a random sample of 48 people. What is the approximate probability that the mean of the rounded ages is within 0.25 years of the mean of the true ages?(A) 0.14(B) 0.38(C) 0.57(D) 0.77(E) 0.8820. Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0,12) . Given X = x, Y is uniformly distributed on the interval (0, x) . Determine Cov(X, Y) according to this model.(A) 0(B) 4(C) 6(D) 12(E) 2421. A ball rolls along the polar curve defined by r = sin . The ball starts at = 0 and ends at Calculate the distance the ball travels.(A) (B) (C) (D) (E)22. An actuary determines that the annual numbers of tornadoes in counties P and Q are jointly distributed as follows:Calculate the conditional variance of the annual number of tornadoes in county Q, giventhat there are no tornadoes in county P .(A) 0.51(B) 0.84(C) 0.88(D) 0.99(E) 1.7623. An insurance policy is written to cover a loss X where X has density function The time (in hours) to processa claim of size x, where 0 _ x _ 2, is uniformly distributed on the interval from x to 2x .Calculate the probability that a randomly chosen claim on this policy is processed in three hours or more.(A) 0.17(B) 0.25(C) 0.32(D) 0.58(E) 0.8324. An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims.If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?(A) (B) 1(C) (D) 2(E) 425. An advertising executive claims that, through intensive advertising, 175,000 of a city’s 3,500,000 people will recognize the client’s product after one day. He further claims that product recognition will grow as advertising continues according to the relationship an+1 = 0.95an +175,000, where an is the number of people who recognize the client’s product n days after advertising begins. If the advertising executive’s claims are correct, how many of the city’s 3,500,000 people will not recognize the client’s product after 35 days of advertising?(A) 552,227(B) 561,468(C) 570,689(D) 581,292(E) 611,88626. The bond yield curve is defined by the function y(x) for 0 < x _ 30 where y is the yield on a bond which matures in x years. The bond yield curve is a continuous, increasing function of x and, for any two points on the graph of y, the line segment connecting those points lies entirely below the graph of y . Which of the following functions could represent the bond yield curve?(A) y(x) = a a is a positive constant(B) y(x) = a + kx a, k are positive constants(C) , k are positive constants(D) y(x) = , k are positive constants(E) y(x) = a + k log(x + 1) a, k are positive constants27. A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold.P(X = 0, Y = 0) =1/6 P(X = 1, Y = 0) =1/12 P(X = 1, Y = 1) =1/6 P(X = 2, Y = 0) =1/12P(X = 2, Y = 1) =1/3 P(X = 2, Y = 2) =1/6 What is the variance of X ?28. Inflation is defined as the rate of change in price as a function of time. The figure below is a graph of inflation, I, versus time, t . Price at time t = 0 is 100 . What is the next time at which price is 100 ?(A) At some time t, t (0, 2) .(B) 2(C) At some time t, t (2, 4) .(D) 4(E) At some time t, t (4, 6) .29. An investor buys one share of stock in an internet company for 100 . During the first four days he owns the stock, the share price changes as follows (measured relative to theprior day’s price): If the pattern of relative price movements observed on the first four days is repeated indefinitely, how will the price of the share of stock behave in the long run?(A) It converges to 0.00 .(B) It converges to 99.45 .(C) It converges to 101.25 .(D) It oscillates between two finite values without converging.(E) It diverges to .30. Three radio antennas are located at points (1, 2), (3, 0) and (4, 4) in the xy-plane. In order to minimize static, a transmitter should be located at the point which minimizes the sum of the weighted squared distances between the transmitter and each of the antennas. The weights are 5, 10 and 15, respectively, for the three antennas. What is the x-coordinate of the point at which the transmitter should be located in order to minimize static?(A) 2.67(B) 3.17(C) 3.33(D) 3.50(E) 4.0031. Let R be the region bounded by the graph of x2 + y2 = 9 .Calculate(A) (B) (C) (D) (E)32. A study indicates that t years from now the proportion of a population that will beinfected with a disease can be modeled by Determine the time when the actual proportion infected equals the average proportion infected over the time interval from t = 0 to t = 3 .(A) 1.38(B) 1.50(C) 1.58(D) 1.65(E) 1.6833. A blood test indicates the presence of a particular disease 95% of the time when thedisease is actually present. The same test indicates the presence of the disease 0.5% ofthe time when the disease is not present. One percent of the population actually has thedisease.Calculate the probability that a person has the disease given that the test indicates the presence of the disease.(A) 0.324(B) 0.657(C) 0.945(D) 0.950(E) 0.99534. An insurance policy reimburses a loss up to a benefit limit of 10 . The policyholder’sloss, Y, follows a distribution with density function:What is the expected value of the benefit paid under the insurance policy?(A)1.0(B) 1.3(C) 1.8(D) 1.9(E) 2.035. A company insures homes in three cities, J, K, and L . Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are:MJ(t) = (1 – 2t)-3 MK(t) = (1 – 2t)-2.5 ML(t) = (1 – 2t)-4.5 Let X represent the combined losses from the three cities.Calculate E(X3) .(A) 1,320(B) 2,082(C) 5,760(D) 8,000(E) 10,56036. In modeling the number of claims filed by an individual under an automobile policyduring a three-year period, an actuary makes the simplifying assumption that for all integers , where pn represents the probability that the policyholder files n claims during the period.Under this assumption, what is the probability that a policyholder files more than one claim during the period?(A) 0.04(B) 0.16(C) 0.20(D) 0.80(E) 0.9637. Let S be the surface described by f(x,y) = arctany/x Determine an equation of the plane tangent to S at the point(A) (B) (C) (D) (E)38. An insurance policy is written to cover a loss, X, where X has a uniform distributionon [0, 1000] .At what level must a deductible be set in order for the expected payment to be 25% of what it would be with no deductible?(A) 250(B) 375(C) 500(D) 625(E) 75039. An insurance policy is written that reimburses the policyholder for all losses incurred up to a benefit limit of 750 . Let f(x) be the benefit paid on a loss of x .Which of the following most closely resembles the graph of the derivative of f ?(A) (B) (C) (D) (E)40. A company prices its hurricane insurance using the following assumptions:(i) In any calendar year, there can be at most one hurricane.(ii) In any calendar year, the probability of a hurricane is 0.05 .(iii) The number of hurricanes in any calendar year is independentof the number of hurricanes in any other calendar /doc/fb5f1bbcc77da26925c5b09e.html ing the company’s assumptions, calculate the probability that there are fewer than 3 hurricanes in a 20-year period.Course 1 May 2000 Answer Key1. A 21. B2. D 22. D3. D 23. A4. E 24. D5. C 25. D6. D 26. E7. C 27. B8. A 28. C9. B 29. A10. C 30. B11. D 31. D12. E 32. D13. E 33. B14. B 34. D15. C 35. E16. B 36. A17. C 37. B18. C 38. C19. D 39. C20. C 40. E。

soa北美精算师第⼆门FMTIA样题⼀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。

北美精算师考试试卷
考试科目：精算学基础
考试时间：3小时
注意事项：
1. 请在答题卡上填写您的姓名和考试编号。

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3. 考试结束后，试卷和答题卡需一并上交。

第一部分：选择题（共40分）
1-10题：精算数学基础
- 1. 以下哪个不是精算学中常用的数学工具？
A. 微积分
B. 概率论
C. 线性代数
D. 统计分析
- 2. 以下哪个公式用于计算年金的现值？
A. PV = P * (1 + r)^n
B. PV = P * (1 - r)^n
C. PV = P / (1 + r)^n
D. PV = P * (1 - r^n) / r
...（此处省略其他选择题）
第二部分：简答题（共30分）
11题：请简述精算师在保险产品设计中的作用。

12题：解释什么是风险管理和精算师如何在此领域发挥作用。

...（此处省略其他简答题）
第三部分：计算题（共30分）
13题：给定一个年利率为5%的普通年金，每年支付1000美元，支付期为10年，请计算该年金的现值。

14题：一个保险公司面临两种风险：损失风险和投资风险。

请根据给定的数据，计算该公司的总风险价值。

...（此处省略其他计算题）
结束语：
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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]。

2015北美精算师考试考前综合辅导(3)·SOA精算考试课程及大纲-高级教育阶段高级教育阶段(2门课程)：课程1 精算模型应用说明：该课程向学员介绍了建立精算模型实际考虑的因素。

要求学员具备基础课程的知识。

课程的一部分内容对所有的学员都是相同的。

要强调的是学员要重视与其从事的领域相关的技术和问题，这些内容因学员从事的领域而异。

模型可用于精算科学的许多方面，如定价/费率拟定，保险利益设计，资产/负债/资本管理，资产和负债估价，动态偿付能力检验。

不论模型用于哪一方面，其步骤是相同的。

该课程将概述这些步骤。

完成课程的学习后，学员将学会建模的过程并且用来解决一些问题。

主要内容和概念：模型设计或选择，数据输入分析，数据输出分析，结果的比较、检验和反馈。

课程2(a) 高级精算实务-金融说明：该课程研究金融中的高级内容。

学员在完成课程的学习后可加深在一些领域如：金融机构的资本管理，公司财务，金融风险管理(工具和技术)，征税原理，期权定价理论和应用，发展金融战略等的知识和技能。

该课程帮助学员培养在保险公司、储蓄和信用机构、银行等金融机构从事金融和财税工作所应具备的技能。

课程的一些研究领域如期权和套利定价理论和应用，金融原理，资本结构和投资组合管理要求有严格的数学基础，目的是让学员能将这些原理应用于广阔的经济环境。

综合了数学、金融和经济知识及前面课程的技能将使学员在金融机构的管理起到不可代替的作用。

课程2(b) 高级精算实务-团体人寿险;个人和团体健康险说明：该课程对精算原理在团体人寿险和以个人及团体险形式提供下列保障：残疾收入，牙科支出，医疗和长期护理费用的险种中的应用进行深入的探讨。

该课程内容将包括提供这些保障的系统：保险公司，兰十字/兰盾组织，公众健康组织，会员优先服从组织，健康维护组织及Physician 医院组织。

课程2(c) 高级精算实务-健康管理计实务说明：该课程能为精算原理在医疗及牙科服务领域中的费用提供深入而有效的方法。

2015年北美精算师考试考前综合辅导(1)-精算师考试小编整理“2015年北美精算师考试考前综合辅导(1)”更多精算师考试复习指导信息，请关注精算师考试。

查看汇总：2015年北美精算师考试考前综合辅导汇总金牌证书：FSA(北美精算师资格认证)。

代表人物：李振华现为中宏人寿保险公司助理总经理。

考证费用：8门课共计3500美元，加上考试期间短期研讨会2000元左右，合计考试费用3万元。

每年年费100余美元。

考证步骤：先在精算学会/协会注册为学生会员，参加一系列课程考试，成为准精算师，然后再参加几门高级课程，并参加短期职业培训、答辩等，才能最终成为精算师，一般要花5～7年时间证书“含金量”：考证前为保险公司精算部门工作人员，取得证书4年后担任中宏保险的CFO兼总精算师。

内地执业一般年薪为40～100万元，最高可超300万元。

“在内地，精算师的身价之所以比较高，完全是因为供需不平衡引起的。

和其它任何商品一样，供不应求直接导致了精算师的薪酬处于一个较高的价位。

”现任中宏人寿保险公司助理总经理，曾经担任该公司CFO兼总精算师的李振华先生在日前接受记者采访时谈到。

“物以稀为贵”，李振华用一个朴素的经济学原理阐释了“精算师”这个在外人看来颇有几分神秘色彩的职业状态。

人才稀缺身价惊人据2004年召开的“第五届中国精算年会”统计数字表明，目前全国共有中国精算师43人，准精算师79人。

此外，由于本土精算师培养从1999年才起步，因此大多数保险机构聘用的总精算师都是有具有北美或英国精算师资格认证的国际人才。

“根据美国精算协会SOA的统计，目前长期在中国内地执业的FSA(fellowsofSOA)大约有70人左右，其中本地成长起来的FSA不足一半;而英国精算师FIA(fellowsofIOA)人数甚至还不到两位数。

”作为SOA的正式会员，李振华介绍道。

与现有较少的人才储备对比鲜明的是，随着我国保险业的深入开放和大力发展，专家预测在未来五到十年内，中国精算师的市场需求缺口在4000～5000名左右。

2015年北美精算师考试考前综合辅导(4)-精算师考试小编整理“2015年北美精算师考试考前综合辅导(4)”更多精算师考试复习指导信息，请关注精算师考试。

查看汇总：2015年北美精算师考试考前综合辅导汇总·SOA精算考试课程及大纲基本教育阶段说明：这门课程包括利息理论，中级微观经济学和宏观经济学，金融学基础。

在学习这门课程之前要求具有微积分和概率论的基础知识。

主要内容及概念：利息理论，微观经济学，宏观经济学，金融学基础课程3：随机事件的精算模型说明：通过这门课程的学习，培养学员关于随机事件的精算模型的基础知识及这些模型在保险和金融风险中的应用。

在学习这门课程之前要求熟练掌握微积分、概率论和数理统计的相关内容。

建议学员在通过课程1和课程2后学习这门课程。

主要内容及概念：保险和其它金融随机事件，生存模型，人口数据分析，定量分析随机事件的金融影响课程4：精算建模方法说明：该课程初步介绍了建立模型的基础知识和用于建模的重要的精算和统计方法。

在学习这门课程之前要求熟练掌握微积分、线性代数、概率论和数理统计的相关内容。

主要内容及概念：模型-模型的定义-为何及如何使用模型-模型的利弊-确定性的和随机性的模型-模型选择-输入和输出分析-敏感性检验-研究结果的检验和反馈方法-回归分析-预测-风险理论-信度理论课程5-精算原理应用说明：这门课程提供了产品设计，风险分类，定价/费率拟定/建立保险基金，营销，分配，管理和估价的学习。

覆盖的范围包括金融保障计划,职工福利计划，事故抚恤计划，政府社会保险和养老计划及一些新兴的应用领域如产品责任，担保的评估，环境的维护成本和制造业的应用。

该课程的学习材料综合了各种计划和覆盖范围以展示精算原理在各研究领域中应用的一致性和差异性。

为了鼓励这种学习方法，该课程在研究各精算课题，如定价等时考虑该课题在各领域中的应用而不是相反。

主要内容及概念：计划和产品设计，风险分类原理和技术，精算原理和实务在定价、费率拟定、建立保险基金及传统和新兴的应用领域中的应用，营销、分配和管理，负债和保险基金评估的精算技术课程6-投资和资产管理说明：该课程是用于投资和资产负债管理领域的精算原理的拓展。