EXAMINATIONS. 18 April 2000 (am) Subject 105 Actuarial Mathematics 1. Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE

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1 Faculty of Actuaries Institute of Actuaries EXAMINATIONS 18 April 2000 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Write your surname in full, the initials of your other names and your Candidate s Number on the front of the answer booklet. 2. Mark allocations are shown in brackets. 3. Attempt all 17 questions, beginning your answer to each question on a separate sheet. Graph paper is not required for this paper. AT THE END OF THE EXAMINATION Hand in BOTH your answer booklet and this question paper. In addition to this paper you should have available Actuarial Tables and an electronic calculator. Faculty of Actuaries 105 A2000 Institute of Actuaries

2 1 In the context of a pension scheme, explain the term prospective service benefit and state one example. [2] 2 In a select mortality investigation, θ x,r corresponds to the number of deaths aged x next birthday at entry with duration r at the policy anniversary following death. θ x,r divided by the appropriate central exposed to risk gives an estimate of µ [y]+t. Derive the values of y and t to which this estimate applies. State clearly any assumptions used. [2] 3 Mortality levels for a certain country have been studied at national and regional level. Explain the circumstances under which a particular region may have an Area Comparability Factor of 0.5. [2] 4 A 25 year annual premium endowment assurance policy was sold to a life aged 40 exact at outset. Death benefits are payable at the end of the year of death. Calculate the Zillmerised net premium reserve at the end of the tenth year per unit sum assured. Basis: Mortality: A Select Interest: 3% per annum Initial expense: 2.5% of the sum assured [3] 5 A life insurance company sells an annual premium whole life assurance policy with benefits payable at the end of the year of death. Expenses are incurred at the start of each year, and claim expenses are nil. (a) Write down a recursive relationship between the gross premium reserves at successive durations, calculated on the premium basis. Define all symbols used. (b) Explain the meaning of this formula. [3] 1 6 Calculate A 3030 : : 30 using A mortality and interest of 4% per annum. [3] 105 2

3 7 A pension scheme provides a pension of 1 of final pensionable salary for each 45 year of service, with a maximum of 2 of final pensionable salary, upon 3 retirement at age 65. Final pensionable salary is defined as average annual salary over the 3 years immediately preceding retirement. A member is now aged exactly 47 and has 14 years of past service. He earned 40,000 in the previous 12 months. Calculate the expected present value now of this member s total pension on retirement, using the symbols defined in, and assumptions underlying, the Formulae and Tables for Actuarial Examinations. [3] 8 The random variables T x and T y represent the exact future lifetimes of two lives aged x and y respectively. Let the random variable g(t) take the following values: g(t) = R S T a if max{ T, T } n n x y a if max{ T, T } > n max{ Tx, Ty } x y (i) Describe the benefit which has present value equal to g(t). [2] (ii) Express E[g(T)] as concisely as possible in the form of an annuity function. [1] [Total 3] 9 Define the term asset share in the context of a with-profit policy. [3] 10 The number of people sick with a new disease is expected to increase according to the logistic model. The initial number sick is 100,000 and it is believed that the number sick with the disease will never exceed 250,000. At the outset, sickness is assumed to grow at 5% per annum. Calculate the number of people who are sick after exactly 10 years. [3] 11 A multiple decrement table is subject to two forces of decrement α and β. Under the assumption of a uniform distribution of the independent α β decrements over each year of age, ( aq) x = 0.2 and ( aq) x = α Calculate q x and q β x. [3] PLEASE TURN OVER

4 12 An insurer sells combined death and sickness policies to healthy lives aged 35. The policies, which are for a term of 30 years, pay a lump sum of 20,000 immediately on death, with an additional 10,000 if the deceased is sick at the time of death. There is also a benefit of 3,000 per annum payable continuously to sick policyholders. There is no waiting period before benefits are payable. Annual premiums of 500 are payable continuously by healthy policyholders. The mortality and sickness of the policyholders are described by the following multiple state model, in which the forces of transition depend on age. H = healthy σ x S = sick ρ x µ x ν x D = dead gh p x, t is defined as the probability that a life aged x who is in state g(g = H, S or D) is in state h at age x + t (t 0 and h = H, S or D). The force of interest is δ. Express in integral form, using the probabilities and the various forces of transition, the expected present value of one such policy at its commencement. [4] 13 A pension scheme provides the following benefit to the spouse of a member, following the death of the member in retirement: A pension of 10,000 per annum payable during the lifetime of the spouse, but ceasing 30 years after the death of the member if that is earlier. All payments are made on the anniversary of the member s retirement. Calculate the expected present value of the spouse s benefit in the case of a female member retiring now on her 60th birthday, who has a husband aged exactly 64. Basis: a(55) Ultimate mortality at 8% per annum interest [8] 105 4

5 14 (i) Discuss the suitability of the crude death rate, the standardised mortality rate and the standardised mortality ratio for comparing (a) (b) the mortality, at different times, of the population of a given country the mortality, at a certain time, of two different occupational groups in the same population [6] (ii) The following table gives a summary of mortality for one of the occupational groups and for the country as a whole. Occupation A Whole Country Exposed Exposed Age group to risk Deaths to risk Deaths , ,000 3, , ,400,000 7, , ,000 7,100 37, ,100,000 17,700 Calculate the crude death rate, the standardised mortality rate and the standardised mortality ratio for Occupation A. [4] [Total 10] 15 An insurer issues 15 year term assurance policies to lives aged exactly 50 who have provided satisfactory answers on a basic medical questionnaire. The sum assured of 100,000 is payable at the end of the year of death during the policy term. The policy includes an option at the end of the term which allows policyholders to convert their policy to a whole life policy for the same sum assured (payable at the end of the year of death). The premiums payable for this whole life policy are the office s standard premium rates, irrespective of the health of the policyholder effecting the option. The insurer calculates annual premiums for all products using A Select mortality and 4% per annum interest, with an expense allowance of 5% of all premiums. (i) Describe: (a) (b) the North American method and the conventional method for pricing mortality options. [5] (ii) (iii) Using the conventional method calculate the extra annual premium the insurer should charge above that for a term assurance policy with no option. [5] Without performing any further calculations, describe what other considerations would arise if the option were such that the policy could be converted on the 10th or the 15th anniversary. [3] [Total 13] PLEASE TURN OVER

6 16 A life insurance company issues a 4 year unit-linked policy with a level premium of 1,000 payable annually in advance to a life aged exactly 61. The death benefit at the end of the year of death is 4,000, or the bid value of the units if greater. The maturity value is the bid value of the units. 95% of each premium is invested in units at the offer price. The bid price is 95% of the offer price. Premiums payable in the first two years are invested in capital units which are subject to a management charge of 6% per annum. Subsequent premiums are invested in accumulation units for which the management charge is 1% per annum. Management charges are deducted at the end of each year from the bid value of units before benefits are paid. Capital units are actuarially funded using factors of A calculated using 61 + t:4 t A Ultimate with 5% per annum interest for t = 0, 1, 2 and 3. The company uses the following assumptions to profit test this contract: Rate of interest on unit investments: Rate of interest on sterling fund: Mortality: Initial expenses: Renewal expenses: 8% per annum 4% per annum A Ultimate 100 plus 20% of the first premium 20 on the first policy anniversary, and increasing with inflation at 5% per annum on each subsequent anniversary (i) (ii) Using a risk discount rate of 12% per annum calculate the expected net present value of the profit on this contract. [12] Without performing any further calculations, state with reasons whether your answer in (i) would be higher or lower for each of the following, if (a) the risk discount rate were 10% per annum (b) the policyholder were aged 50 exactly (c) capital units were actuarially funded at 4% per annum [5] [Total 17] 105 6

7 17 A man aged exactly 30 effected a 35 year with profit endowment assurance for a sum assured of 50,000. Level annual premiums are payable throughout the policy term, ceasing on earlier death. The sum assured, with attaching bonuses, is payable at the end of the year of death, or on maturity. Compound reversionary bonuses vest at the end of each policy year. (i) Show that the premium (to the nearest 1) is 990 per annum using the following basis: Mortality: A Ultimate Interest: 6% per annum Expenses: Initial: 250 plus 60% of the annual premium Renewal: 2.5% of second and subsequent premiums Bonuses: 1.923% per annum [7] (ii) (iii) The random variables T x and K x represent the exact future lifetime and the curtate future lifetime of a life aged x, respectively. Using T x, K x or both, express, in stochastic form, the gross future loss random variable for this policy at duration t, where t is an integer and 0 < t < 35. Use those elements of the basis set out in part (i) as needed. Assume bonus declarations have been in line with the original bonus loadings. [3] Immediately before the 11th premium is due, and just after the 10th bonus has brought the sum assured plus accumulated bonuses to 60,000, the policyholder wishes to convert the policy to a non-profit whole life policy, with premiums of an unchanged amount payable until death. Using the mortality and interest elements of the premium basis set out in part (i), and allowing for renewal expenses of 2.5% of all future premiums as well as an alteration expense of 100, calculate the revised sum assured. [6] (iv) State one other consideration, if any, that the office should take into account before completing the alteration in (iii), and explain why they should do so. [2] [Total 18] 105 7

8 Faculty of Actuaries Institute of Actuaries EXAMINATIONS April 2000 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Faculty of Actuaries Institute of Actuaries

9 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report 1 Prospective service benefit means a benefit not dependent on either past or future service explicitly, although it may depend on total expected service. Examples include lump sum death benefit of 4 salary or spouse s pension n death in service of final salary where n is based on deceased member s 120 total potential service to NPA, including any past service. Many candidates confused prospective with future. 2 x next birthday at entry x ½ on average at entry assuming birthdays uniformly distributed over policy year. r at policy anniversary after death means exact duration r 1 at the anniversary before death (the start of the policy year rate interval for duration) and hence r ½ mid-year when the force of mortality is estimated. No assumptions are necessary. The force estimated is [x½]+r½, so y = x ½, t = r ½. 3 If its age/sex profile is such that if it experienced the same age/sex specific mortality rates as the country, then its crude death rate would be twice that of the country, i.e. the region has a much older age structure (and/or higher male proportion) than the country. 4 V a Zillmer t = 1 a a I a [ x ] tn : t [ x] tn : t [ x]: n [ x]: n a a Here 10 V = 1 (.025) a a 50:15 50:15 [40]:25 [40]: = 1 (.025) = Page 2

10 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report 5 (a) ( t V + GP e t ) (1 + i) = q x+t (S) + p x+t ( t+1 V ) where t V = gross premium time t GP = office premium e t = expenses incurred at time t i S = interest rate in premium/valuation basis = Sum Assured p x+t (q x+t ) probability life aged x + t survives (dies within) one year on premium/valuation mortality basis. (b) Income (opening reserve plus interest on excess of premium over expense, and reserve) equals outgo (death claims and closing reserve for survivors) if assumptions are borne out. A = 1 D60:60 ½A = ½ A A 30:30 : 30 D30: :30:30 30:30 60:60.04 D.04 60:60 = ½1 a30:30 1 a60: D 30: = ½1 (19.701) 1 (9.943) = ½[ (.24297)(.61758)] = Future service = past total = 32 > max of 30. Value of benefit = = 2 s (40,000) 3 s z ra 47 C65 s 46 D (40,000) = 62,033 Most candidates allowed for retirement at any age, not just 65, and many failed to notice that service exceeded 30 years so the maximum of 2/3rds applied. Page 3

11 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report 8 (i) A continuous annuity of 1 p.a. payable for a minimum of n years and continuing thereafter until the death of the survivor of x and y. (ii) E[g(T)] = a. xy: n Rather than defining asset share, some candidates discussed bonuses and policy payouts. 9 The asset share for a with-profit policy is the accumulated value of premiums less deductions plus an allocation of profits from non-profit business. The accumulation is at actual earned rates of return. The deductions include expenses, cost of benefits, tax, transfers to shareholders, cost of capital and contribution to free assets. Rather than defining asset share, some candidates discussed bonuses and policy payouts. 10 In logistic model P(t) = Ce Ht K H 1 or H CHe Ht K As t P(t) K H K = ,000 P(0) = 1 C 250,000 1 = 100,000 C = P(10) = ( ) e 250,000 (.05)(10) 1 1 = 130,904 Only a minority of candidates seemed familiar with the logistic model. 11 Under UDD in single decrement table ( aq ) = = > = = > x = qx (1 ½ qx) = qx ½qxqx = 0.2 ( aq ) > > = > = > x = qx (1 ½ qx ) = qx ½qxqx = 0.05 = > qx qx = = 0.15 q = q > x x > > > ( q 0.15) ½( q 0.15) q = 0.2 x x x > 2 > ½( q ).925q 0.15 = 0.2 x x Page 4

12 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report OR > 2 > ( qx) 1.85qx 0.1 = 0 Roots are (and q > 1 is invalid) q > x = and Alternatively, x q = x = q = = > = ( aq) x 1 ½qx and x q > = ( aq) > x 1½q = x = = Using iteration, and taking starting values in denominators of q ( aq) etc. x x 1st iteration q = x = 0.2 [1 (.5)(.05)] = q > x = 0.05 [1 (.5)(.2)] = Similarly, 2nd iteration 3rd iteration 4th iteration q = x q = x q = x =.20571, q > x = =.20573, q > x = =.20573, q > x = Hence q = x =.20573, q > x = A large number of candidates used formulae appropriate when decrements are uniform in the multiple decrement table, but the question specified that independent decrements were uniform in the single decrement tables. 12 EPV = 500 t hh 0 35, t e p dt (premiums) t hh 0 35, t 35t 20,000 e p dt (death from healthy) t hs 0 35, t 35t 30,000 e p dt (death from sick) t hs 0 35, t 3,000 e p dt (sickness income) 13 EPV 30 = 10,000 (1 60) 64 ( ) 5 tp tp v 5 t p tp tp64 v t 1 t31 30 m t f m t f m t = 10,000 5 tp64 v 5 t30 p60 tp64 v 5 tp60 t p64 v t 1 t31 t1 m m 30 f m f m = 10,000 a64:30 30 p64 v a60:94 a60:64 Page 5

13 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report m D94 a 64:30 = a a D = (1.707) = D94 p v = D = = m f m a = a 60:64 = fm : 60:94 EPV = 10,000{ ( )(1.666) 6.854} = 7,617 Very few candidates provided a satisfactory answer. Many did not attempt to deal with the term aspect of the question, and most of those who did assumed the annuity ended 30 years after retirement rather than 30 years after the pensioner s death. 14 (i) Crude death rate is heavily influenced by mortality at older ages (a) (b) OK if population structures by age and sex are reasonably stable. Therefore beware large scale emigration/immigration. Easy and practical. Not suitable age and sex distributions in occupational groups likely to vary significantly. Standardised Mortality Rate Again influenced by mortality at older ages. (a) OK to use but need age specific mortality rates at each time point. Changing population structure has no effect. (b) Copes well with age/sex variations provided age specific rates are available for occupational groups. But use of a fixed age structure may be unrepresentative of given occupation. Standardised Mortality Ratio Heavily influenced by relative mortality at older ages. (a) Fine but ensure standard rates used are same each time. Page 6

14 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report (b) Good except for possible problems gathering the data on age distributions. Use of occupational age structure maintains relevance. (ii) Occupational A Crude Rate = 235 / 37,000 = Standardised Mortality Rate = (960, ,400, , ) 10,000 3,100,000 = (3, , ,066) 3,100,000 = Standardised Mortality Ratio = 235 3,100 15, ,000 7,500 12,000 1,400,000 7,100 10, ,000 = = = Answered quite well in general, although some students tended to describe the various measures in general rather than relate them to the specific situations described. 15 (i) (a) North American Method Relies on double decrement table with explicit proportions who choose to exercise option and a special mortality table for those people post option. While theoretically accurate, it is often difficult to obtain sufficient data to estimate experience. Page 7

15 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report (b) Conventional Method Assumes all eligible lives actually take up option, and that they are subject to Ultimate mortality as opposed to Select if normal underwriting carried out. If there are many option dates etc., then the most costly from the insurers point of view is assumed. (ii) Insurer charges 1.95 (P [65]) (100,000) per annum for whole life policy i.e. (.05254)(100,000).95 = 5, p.a. At option date (age 65), the value of benefits provided is 100,000 A 65 = (100,000)(.58705) = 58,705 The insurers net liability at option date present value of benefits (present value of premiums less expenses) = 100,000 A 65 (.95)(5,530.53) a65 = 58,705 (.95)(5,530.53)(10.737) = 58,705 56, = 2, Extra premium, P, spread over term assurance policy term, is from:-.95p D65 a [50]:15 = 2, D [50] P = per annum P = (2,292.80) 2, , (.95) (11.028) (iii) The office needs to decide which option is costlier, not just in the value of the option benefit, but its impact on the overall premium required over the period to the option exercise date. In this case, it needs to compare the above option cost in premium terms plus the 15 term assurance premium to the similarly calculated extra premium for the 10 year option combined with a 10 year term insurance premium. It should then charge the higher combined premium, thereby having option cost at any date more than covered. Part (i) was well answered, but (ii) and (iii) were very poorly answered. Many candidates treated the contract as a whole life from the start making the option cost the difference between a term assurance and a whole life policy for the life aged 50. Page 8

16 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report 16 (i) q 61 = p 61 = p = 1.0 5% 0 61 A 61:4 = q 62 = p 62 = p = A 62:3 = q 63 = p 63 = p = A 63: 2 = q 64 = p 64 = p = A 64:1 = Capital unit fund fully funded Y/e fund Management Fund Year Cost of alloc. Fund b/f after 8% growth Charge 6% c/f , , , , , , , , Capital unit fund a-funded Available Needed at Extra death Management Year Cost of alloc. Fund y/e after 8% year end cost charge , , , , , , , , Premium unit fund 1% Fund Management Fund Year Cost of alloc. Fund year end charge c/f , , Death cost (using full Cap. Units) Yr 1 q 61 ( ) = Yr 2 q 62 ( ) = Yr 3 q 63 ( ) = Yr 4 q 64 ( ) = 2.18 Page 9

17 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report Sterling fund (4%) Premium less Sterling Death Management Profit Profit Year cost of alloc. Expense interest cost charge vector signature (1.86) NPV = 87.16v + 187v v v 4 = Alternative approach whereby entire death cost is charged to sterling fund is also valid, providing a-funded capital unit management charge is correspondingly increased. (ii) (a) Given the shape of the cash flows, with the positives after the negatives, a discount rate of 10% would mean larger NPV. (b) (c) Death cost would reduce, probability of being in force and hence premium income would increase, causing NPV to increase. A-funding factors would also decrease, accelerating the cash flows. Given risk discount rate (12%) > sterling fund rate this will increase NPV. At 4%, factors will be bigger, unit reserves increase and profit is deferred. Because risk discount rate exceeds sterling fund rate, NPV decreases. Generally well answered, although candidates often failed to give reasons for their correct conclusions in (ii). 17 (i) 6% Pa 30:35 = 4% 4% 1 50, :35 30: A A.025 Pa.575P 6% 30:35 Because bonuses vest at year end, maturities get an extra bonus compared to deaths in last year, and so the death benefit function is divided by (1 + bonus loading). 6% 1 D65 D 65 P.975 a 30: = ,000 A30: D30 D30 6% a 30:35 = % A 30:35 = Page 10

18 Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report D D 4% 65 4% 30 = = P( ) = ,000{ } P = = 990 p.a. (ii) Gross future loss = PV future outgo PV future income = PV future benefit payment + PV future expenses PV future premiums = G(K 30+t ) + (.025)(990) a min[ K30t 1, 35 t] (990) a min[ K30t 1, 35 t] tk30t K30t1 where G(K 30+t ) = 50,000 ( ) v.06 K30t 35 t 35 35t 50,000 ( ) v.06 K30t 35 t (iii) Reserve before alteration = reserve after alteration + cost of alteration Before 4% V = 1 4% D 65 D 65 6% 10 60,000 A40:25 (.975)(990)( a40:25 ) D40 D40 = 1 60,000 ( ) (.975)(990)(13.081) After = 23, , = 11, say 11,271 V = x 6% 6% 10 A40 (.975)(990) a40 = x (.15807) (.975)(990)(14.874) = (.15807)(x) 14,357 11,271 = (.15807)(x) 14, x = 161,498 say 161,500 (iv) The amount at risk is immediately significantly increased (by 100,000) and the term for which there is a death strain has been extended. There is a grave risk of adverse selection against the office unless it underwrites the alteration as effectively a new business case. A simple declaration of health will not suffice in this case given the size of the change of the immediate risk. Parts (i), (iii) and if attempted (iv) were well answered although most students missed the different bonus treatment needed for death benefits compared with the maturity benefit. Few candidates seemed familiar with the concept of the gross future loss as a random variable and answers to part (ii) were weak. Page 11

19 Faculty of Actuaries Institute of Actuaries EXAMINATIONS 19 September 2000 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Writeyoursurnameinfull,theinitialsofyourothernamesandyour Candidate s Number on the front of the answer booklet. 2. Mark allocations are shown in brackets. 3. Attempt all 15 questions, beginning your answer to each question on a separate sheet. Graph paper is not required for this paper AT THE END OF THE EXAMINATION Hand in BOTH your answer booklet and this question paper. In addition to this paper you should have available, Actuarial Tables and an electronic calculator. ã Faculty of Actuaries 105 S2000 ã Institute of Actuaries

20 1 Two lives, each aged x, are subject to the same mortality table. According to the mortality table and a certain rate of interest, A x =0.5andA xx =0.8. Calculate 2 A xx, using the same mortality table and interest rate. [2] 2 The following data are available in relation to a particular country and one of its regions: Age group Region A Population at 30 June 2000 (000s) Deaths in 2000 Country Population at 30 June 2000 (000s) Deaths in ,580 8, ,295 8,100 45, ,500 6, ,860 Calculate the standardised mortality ratio for region A by reference to the country as a whole. [2] 3 (i) A life insurance policy provides a benefit of 10,000 payable immediately on the death of a life (x), if (x) dies after a life (y). Express in integral form the expected present value of the benefit under this policy. [1] (ii) Set out, giving a reason, the most appropriate annuity factor to value annual premiums payable under the policy. [1] [Total 2] 4 A healthy life aged exactly 35 has a policy providing an income benefit of 50 per week payable during sickness. The benefit is not payable beyond age 60. There is no deferred or waiting period. Calculate the present value of this benefit. Basis: Mortality: English Life Table No. 12-Males Sickness: Manchester Unity Sickness Experience 1893/97 Occupation Group AHJ Interest: 4% per annum [3] 5 An annuity of 1 is payable annually in arrears while at least one of two lives, (x) and (y), is alive. Derive an expression in terms of joint-life and single life functions for the variance of the present value of the annuity. [3] 6 Describe three types of bonus that may be given to a with profits contract. [3] 105 2

21 7 In the context of a life insurance contract, explain how an asset share may be built up using a recursive formula. [3] 8 (i) On 1 January 1990 a life insurance company issued a 20-year annual premium without profits endowment assurance policy to a life then aged exactly 40, which is still in force. The sum assured of 100,000 is payable at the end of the year of death within the term of the policy, or on survival. The company values the policy using a modified net premium method, with a Zillmer adjustment. Calculate the reserve for the policy on 31 December Basis: Mortality: A Select Interest: 4% per annum Zillmer adjustment: 2% of the sum assured [3] (ii) Without carrying out any further calculations, explain how the value of the policy would differ if the company used a Zillmer adjustment of 1% of the sum assured, with the same mortality and interest assumptions. [2] [Total 5] 9 A life insurance company issues a special reversionary annuity contract. Under the contract an annuity of 10,000 per annum is payable monthly for life, to a female life now aged exactly 60, on the death of a male life now aged exactly 65, provided the male life dies within 10 years of the start date of the policy. Payments commence on the first monthly policy anniversary after the date of death. Calculate the single premium required for the contract. Basis: Mortality: a(55) Ultimate mortality, male or female as appropriate Interest: 6% per annum Expenses: none [5] 10 A pension scheme provides an ill-health retirement pension of 1/60 of Final Pensionable Salary for each year of company service, with fractions of a year to count proportionately, subject to a maximum pension of 40/60 of Final Pensionable Salary. Retirement due to ill-health may take place at any age before age 65. Final Pensionable Salary is defined as the average annual salary over the three-year period preceding retirement. Derive commutation functions to value the ill-health retirement pension for a member aged exactly 25, who has completed exactly 5 years company service to date. Define carefully all the symbols that you use. [7] 11 Describe the component method of population projection used for British Official Projections, stating carefully any assumptions that you make and defining all the symbols that you use. [7] PLEASE TURN OVER

22 12 A life insurance company issues only single premium without profit term assurance policies. The premium is to be calculated for a special 3-year term assurance for lives aged exactly 60 where the basic sum assured is 100,000, payable at the end of the year of death. This special policy carries a guaranteed insurability option that may be selected at the outset of the 3-year policy in return for the payment of an additional single premium. This option provides a guarantee to the policyholder that a further 100,000 of sum assured may be purchased, at a subsequent policy anniversary, on normal premium rates and without evidence of health. The further sum assured purchased will not itself carry any further options, and will expire at the end of the 3-year term of the original policy. A policyholder who has paid the additional single premium can subsequently decide whether or not to effect the increase in sum assured and then at which policy anniversary the first or second, but not both. The company uses the North American experience method for pricing the option. Calculate the additional single premium payable at outset for a policyholder choosing the option. Basis: Mortality: A Select, except in the case of policyholders who decide to exercise their option to increase the sum assured. For these policyholders, the mortality basis assumed to apply, from the point of increase in sum assured, is 150% of A Ultimate. Interest: 5.5% per annum Proportion of policyholders at the first anniversary who decide to increasetheirsumassuredatthatpoint:20% Proportion of policyholders at the second anniversary who decide to increasetheirsumassuredatthatpoint:20% Expenses: none [7] 105 4

23 13 A life insurance company uses the following 3-state model, to estimate the profit in respect of a 2-year combined death benefit and sickness policy issued to a healthy policyholder aged exactly 55 at inception. Healthy (H) Sick (S) Dead (D) In return for a single premium of 6,000 payable at the outset the company will pay the following benefits: 16,000 if the policyholder dies within 2 years, payable at the end of the year of death; 8,000 at the end of each of the 2 years if the policyholder is sick at those times. Let S t represent the state of the policyholder at age 55 + t, sothats 0 = H and for t =1and2,S t = H, S or D. The company uses transition probabilities defined as follows: p + = P(S t+1 = j S t = i) ij 55 t For t = 0 and 1 the transition probabilities are: HD SD p 55 + t = t SH HS p + =0.15 p 55 + t =0.75 p 55 + t =0.12 The transitions in the multiple state model are the only sources of randomness. (i) One possible outcome for this policy is that the policyholder is healthy at times 0, 1 and 2. List all the possible outcomes and the associated cash flows. [3] (ii) Calculate the probability that each outcome occurs. [5] (iii) (iv) Assuming a rate of interest of 8% per annum, calculate the net present value at time 0 of the profit for each outcome. [2] Calculate the mean and standard deviation of the net present value of the profit at time 0 for the policy. [5] [Total 15] PLEASE TURN OVER

24 14 On 1 September 1992, a life insurance company issued a whole life with profits policy to a life then aged exactly 45. The basic sum assured was 100,000. The sum assured and attaching bonuses are payable immediately on death. Level monthly premiums are payable in advance to age 85 or until earlier death. The company calculated the premium on the following basis: Mortality: Interest: Bonus loading: A Select 4% per annum % per annum compound, vesting at the beginning of each policy year Expenses: initial: 50% of the first year s premiums, incurred at the outset renewal: 5% of the second and each subsequent year s premiums, incurred at the beginning of the respective policy years. (i) Show that the monthly premium is 229, to the nearest. [7] (ii) Immediately before payment of the premium due on 1 September 2000, at the request of the policyholder, the insurance company alters the policy to a paid-up policy, with no future premiums payable. The sum assured under the policy is reduced, with no further bonuses payable. The insurance company calculates the reduced sum assured after alteration by equating prospective gross premium policy reserves immediately before and after alteration, allowing for an expense of alteration of 100. Bonuses have vested at the rate of 4% per annum compound at the beginning of each policy year from the date of issue of the policy. The company calculates prospective gross premium policy reserves for the purpose of the alteration using the following assumptions: Mortality: Interest: Expenses: Allowance for future bonuses: A Ultimate 4% per annum none none Calculate the sum assured after alteration. [6] [Total 13] 15 A life insurance company issues a 3-year unit-linked endowment assurance contract to a male life aged exactly 62 under which level annual premiums of 4,000 are payable in advance throughout the term of the policy or until earlier death. 101% 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 5.25% on capital units and 1.25% on accumulation units. Management charges are deducted at the end of each year, before death, surrender or maturity benefits are paid

25 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 10,000 or the bid value of the units allocated to the policy, if greater. On maturity, the full bid value of the units is payable. A policyholder may surrender the policy only at the end of each year. On surrender, the bid value of the accumulation units plus a proportion of the capital units is payable. The proportion of the capital units payable on surrender is determined by the year of surrender, as follows: Year of surrender Proportion of capital units paid out The life insurance company uses the following assumptions in carrying out profit tests of this contract: Mortality: A Ultimate Expenses: initial: 300 renewal: 60 at the start of each of the second and third policy years Unit fund growth rate: 9% per annum Sterling fund interest rate: 4.5% per annum Risk discount rate: 15% per annum Surrender rates: 15% of all policies still in force at the end of each of the first and second years (i) The company holds unit reserves equal to the full bid value of the accumulation units and a proportion A (calculated at 4%), of the full 62 + t:3 t bid value of the capital units, calculated just after the payment of the premium due at time t (t = 0, 1 and 2). The company holds no sterling reserves. Calculate the profit margin on the contract. [17] (ii) Assume instead that the company holds unit reserves equal to the full bid value of both the accumulation and capital units and that the company also holds sterling reserves, at the start of each policy year, equal to 10% of the annual premium. Calculate the revised profit margin on the contract. [6] [Total 23] 105 7

26 Faculty of Actuaries Institute of Actuaries EXAMINATIONS September 2000 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT ã Faculty of Actuaries ã Institute of Actuaries

27 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report 1 A x =0.5 A xx = A xx = A xx 2 A xx = A x 1 A xx = A x ½A xx =0.5 ½ 0.8 = The standardised mortality ratio (SMR) = å x å x E E c x, tmx, t c s x, t mx, t = , ,500 æ 8,347 45, ,860 ö ç 645 * * * 13,580 8,100 6,290 è ø = 30,145 32, = ò t 3 (i) 10,000 v (1 p ) p µ dt 0 where x = age of (x) y = age of (y) t y t x x+ t (ii) The premium should be payable as long as (x) is alive, while the benefit is still payable. It does not matter whether (y) is alive. The most appropriate annuity factor is, therefore: ( m) a x,where m denotes frequency of payment. 50( K35 K60) 4 Value = D 35 K 35 = K + K + K + K + K 13 13/13 26 / / / all = = K 60 = D 35 = Page 2

28 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report Value = Alternative æ D ö ç, based on value of 1 p.w. all periods, whole of life è D35 ø = Required: ( ) Var a Kxy Var ( a ) = Var( a 1) K 1 xy K xy + = K + 1 æ xy 1 v ö Var ç d è ø = 1 d 2 K + 1 xy Var( v ) = ( A A xy xy )) d = 1 ( A ( ) ) 2 x + Ay Axy Ax + Ay Axy d where 2 denotes evaluation at rate of interest i 2 +2i. Other functions are evaluated at rate of interest i. 6 The following are three types of guaranteed reversionary bonuses. The bonuses are usually allocated annually in arrears, following a valuation. Simple the rate of bonus each year is a percentage of the initial basic sum assured under a policy. The effect is that the sum assured increases linearly over the term of the policy. Compound the rate of bonus each year is a percentage of the basic sum assured and the bonuses previously added. The effect is that the sum assured increases exponentially over the term of the policy. Super compound two compound bonus rates are declared each year. The first rate (usually the lowest) is applied to the basic sum assured. The second rate is applied to the bonuses previously added. The sum assured increases exponentially over the term of the policy. The sum assured usually increases more slowly than under a compound allocation in the earlier years and faster in the later years. Page 3

29 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report 7 An asset share is evaluated for an individual policy or for a block of policies, usually for non-unit linked policies. The asset share is the accumulation of premiums less deductions associated with the contract plus an allocation of profits on non-profit business, all accumulated at the actual rate of return earned on investments. Deductions include all expenditure associated with the contract or contracts. The asset share may be built up recursively on a year-to-year basis. Initially, the asset share is zero. Each year, the cash flows including premiums received, deductions made to cover actual costs and provisions made to cover other liabilities and provision for profits allocated to the policy or group of policies are recorded. A suitable rate of return is used to accumulate the asset shares plus premiums less deductions plus profit allocations to the year-end to determine the asset share. The process is repeated for subsequent years. æ a ö a 50:10 50:10 8 (i) Reserve = 100, ,000 ç a a è ø a = :10 a = [40]:20 Reserve = 39, [40]:20 [40]:20 (ii) Using a Zillmer adjustment has the effect of reducing the policy value. Changing the Zillmer adjustment from 2% of the sum assured to 1% of the sum assured has the effect of reducing the amount of the Zillmer adjustment and hence increasing the policy value, as at 31 December æ ö (12) (12) 9 Premium = 10,000ç a a ç è m f m f ø a (12) a = :65 m f f f m a = ½( ) = Page 4

30 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report m f (12) D75 l70 a 10 = D m f 65 l60 æ ö ç a70 a 70:75 ç è f f m ø = (8.328 ½( )) = = Premium = 10,000( ) = 22, Define a service table: l x+t = no. of members aged x + t last birthday i x+t = no. of members who retire due to ill-health age x + t last birthday s x+t / s x = ratio of earnings in the year of age x + t to x + t + 1 to the earnings in the year of age x to x +1 Define z x+t = 1 (s + s + s ); 3 x 3 x 2 x 1 retiree aged exactly x + t. i a x = value of annuity of 1 p.a. to an ill-health Let (AS) be the member s expected salary earnings in the year of age 25 to 26. Assume that ill-health retirements take place uniformly over the year of age. Consider ill-health retirement between ages 25 + t and 25 + t +1,t < 35. The present value of the retirement benefits related to future service: 25++ t ½ ( ) t + ½ ( AS) z i ( + ½)( ) C z ia 25++ t ½ v 25+ t i t AS 25+ t a t ½ = s s25 v l25 D z ia where C 25 + t = z 25+t+½ v 25+t+½ i 25+t a i 25++ t ½ and s D 25 = s 25 v 25 l 25 Similarly it may be shown that the present value of the benefits is, in total: ( AS) 60 z ia z ia z ia z ia z ia ½ C25 1½ C C60 35 C C s 64 D é ë ù û 25 Page 5

31 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report = ( AS) 60 s D 25 é ½ C + 1½ C ½ C + 36½ C ½ C ë z ia z ia z ia z ia z ia z ia z ia ( ½ C ½ C64 ) + ù û ( AS) = é z ia z ia z ia z ia z ia z ia ( M25 + M M64 ) ( M60 + M M s 64 ) 60 D ë 25 [where z M ia x = 64 Σ t= 0 x z C ia z ia x+ t x ½ C ] ù û ( AS) z ia z ia = R25 R s D é ë ù û 25 where z R ia x = 64 Σ 0 t= x z M ia x+ t Similarly it may be shown that the present value of benefits related to past service is: 5( AS) 60 s D 25 z ia where M 25 = z M ia z ia å C 25 + t t= P x (n) = Survivors to n of P x 1 (n 1) + migrants during (n 1, n) who survived to be age x at n (net migrants are considered, i.e. migrants less emigrants). P 0 (n) = Births during (n 1, n) + migrants during (n 1, n) whosurvived to be age 0 at n where P x (n) is the population age x last birthday at n, wheren refers to mid-year n. Let B(n) = births during (n 1, n) M x (n) = net migrants during (n 1, n) whosurvivetobeagex last birthday at n q x ½ (n 1) = probability that a life aged last birthday x 1atn 1 diesin (n 1, n), assuming those aged x 1lastbirthdayatn 1have birthdays uniformly distributed over the calendar year. q ½ 0 (n 1) = probability that a life born in (n 1, n) diesin(n 1, n). Page 6

32 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report Then we have: P x (n) = P x 1 (n 1) (1 q x ½ (n 1)) + M x (n) P 0 (n) = B(n) (1 ½ q 0 (n 1)) + M 0 (n) Projections are carried out separately for each sex to give values P 0 (n), P 1 (n),..., P x (n),... B(n) andm n (x) are determined using separate models. Total births in (n 1, n), B(n), are projected using f f B(n) = Σ { + } ½ P ( n 1) P ( n) f ( n) x x x f where P ( n) is the number of females aged x last birthday at n. x f f { Px x Px n } ½ ( 1) + ( ) gives the average female population aged x last birthday over the year (n 1, n). f x (n) is the fertility rate over (n 1, n) for women aged x last birthday at the date of birth. The summation is taken over all ages where f x (n) >0. The sex ratio at birth has been estimated empirically to be 1.06 : 1 (males : females). This ratio is used to obtain male and female births, as follows: B m (n) = 1.06 Bn ( ) 2.06 B f (n) = B( n) 2.06 Migration numbers are estimated directly from the International Passenger Survey. Page 7

33 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report 12 Construct multiple decrement tables For those not exercising the option: Age No. alive No. of deaths , , , , For those exercising the option: Age No. alive No. of deaths 61 19, , Premiums payable: P 0 = 100,000[ v + ( ) * v ( ) *( ) * v ] = 2, P 1 = 100,000[ v +( ) * v 2 ] = 2, P 2 = 100,000[ v] = where P 0 is the premium payable at the outset, P1 is the premium payable at the first anniversary for additional cover and P 2 is the premium payable at the second anniversary for additional cover. Cost of benefits = 100, , [ v + ( * ) v + (1, *935.79) v ] =4, Value of premiums= 1 100,000 [ *100,000+2,498.85*19,866.02v+1,682.44*15,738.63v2 ] =3, Option premium= 4, , = 1, Page 8

34 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report 13 (i) Outcome Cashflow (1) HHH 6,000, 0, 0 (2) HHS 6,000, 0, 8,000 (3) HHD 6,000, 0, 16,000 (4) HSH 6,000, 8,000, 0 (5) HSS 6,000, 8,000, 8,000 (6) HSD 6,000, 8,000, 16,000 (7) HD 6,000, 16,000 indicates cashflow to policyholder (ii) Complete the set of transition probabilities: SS p + =0.8, p 55 + =0.1 HH 55 t t The probability that each outcome occurs is: Outcome Probability (1) 0.64 (2) (3) (4) 0.09 (5) (6) (7) (iii) The net present value of each outcome is: Outcome NPV of Profit (1) 6,000 (2) (3) 7, (4) 1, (5) 8, (6) 15, (7) 8, Page 9

35 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report (iv) Mean = Σ NPV liability = = 2, Variance = ΣNPV 2 Probability (Mean) 2 = 34,009, Standard deviation = 5, (i) 1.04 / = 1.03 Þ death benefits evaluated at 3% p.a. Value of death benefits = 100,000 A[45] = 100, ½ 3% A [45] Value of premiums = = 100, ½ = 42, (12) 4% Pa [45]:40 (12) a = [45]:40 N[45] N85 11 æ1 D ö 85 D[45] 24 ç D è [45] ø = æ ç 1 ö è ø = = Value of premiums = P Value of expenses = 0.45P +0.05P a [45]:20 =0.45P +0.05P = P P = P P = Monthly premium = Page 10

36 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report (ii) Sum assured = ,000 = 136, Value of reserves before alteration = 136, A (12) a 53:32 ½ 4% A 53 = 1.04 A 53 =1.04 ½ = N N 11 æ D ö ç 1 è ø (12) a = 53:32 D53 24 D53 = æ 1 ö ç è ø = = Value of reserves = 19, = SA 53 = S S = 19, S = 45, (i) Multiple decrement table age (x) d q x s q x (al) x ( ad ) d x ( ad) s x , Page 11

37 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report Unit Funds (ignoring actuarial funding) Year, t Value of Capital Units at start Premium to CUs Interest on CUs Management charge on CUs Value of CUs at end Value of Accumulation Units at start Premium to AUs Interest on AUs Management charge on AUs Value of AUs at end Surrender value of units Capital Unit Fund (allowing for actuarial funding) Year, t Actuarial funding factor Value of CUs at start Premium to CUs Interest on CUs Management charge on CUs Value of CUs at end Sterling Fund Year, t Unallocated premium Expenses Interest MC on Capital Units MC on Accumulation Units Surrender profit Extra death benefit Cost of extra allocation End of year cashflow Probability in force Discount factor Expected present value Expected p.v. of profit = Expected p.v. of premiums = = Profit margin = 5.69% Page 12

38 Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report (ii) Revised Sterling Fund (ignoring reserves) Year, t Unallocated premium Expenses Interest MC on Capital Units MC on Accumulation Units Surrender profit Extra death benefit End of year cash flow Reserves at start of year Interest on reserves Change in reserves at year end Revised cashflow Expected present value Expected present value of profit = Profit margin = 0.53% Page 13

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