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

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

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: A1967 70 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 A1967 70 mortality and interest of 4% per annum. [3] 105 2

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 = 0.05. α Calculate q x and q β x. [3] 105 3 PLEASE TURN OVER

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

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 20 34 15,000 52 960,000 3,100 35 49 12,000 74 1,400,000 7,500 50 64 10,000 109 740,000 7,100 37,000 235 3,100,000 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 A1967 70 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] 105 5 PLEASE TURN OVER

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 A1967 70 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 A1967 70 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

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: A1967 70 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

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

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]:25 11.671 11.671 = 1 (.025) 17.180 17.180 = 0.30368 Page 2

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 reserve @ 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 6 1 30:30:30 30:30 60:60.04 D.04 60:60 = ½1 a30:30 1 a60:60 1.04 D 30:30 1.04 =.04 2487.2117.04 ½1 (19.701) 1 (9.943) 1.04 10236.789 1.04 = ½[1.75773 (.24297)(.61758)] =.0461 7 Future service = 18 + 14 past total = 32 > max of 30. Value of benefit = = 2 s (40,000) 3 s z ra 47 C65 s 46 D47 2 4.28 35846 (40,000) 3 4.18 15778 = 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

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 =.05 250,000 P(0) = 1 C 250,000 1 = 100,000 C = 0.000006 P(10) = (.000006) 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 > + 0.15 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

Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report OR > 2 > ( qx) 1.85qx 0.1 = 0 Roots are 1.85 1.85 0.4 2 2 0.05573 (and q > 1 is invalid) q > x = 0.05573 and Alternatively, x q = x = 0.20573 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)] =.205128 q > x = 0.05 [1 (.5)(.2)] =.055556 Similarly, 2nd iteration 3rd iteration 4th iteration q = x q = x q = x =.20571, q > x =.05571 =.20573, q > x =.05573 =.20573, q > x =.05573 Hence q = x =.20573, q > x =.05573 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 30 @ t hh 0 35, t e p dt (premiums) 30 @ t hh 0 35, t 35t 20,000 e p dt (death from healthy) 30 @ t hs 0 35, t 35t 30,000 e p dt (death from sick) 30 @ t hs 0 35, t 3,000 e p dt (sickness income) 13 EPV 30 = 10,000 (1 60) 64 ( 30 60 60 ) 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

Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report m D94 a 64:30 = a a D 64 94 64 = 7.616 16.4 (1.707) 5844.0 = 7.611 D94 p v = D = 16.4 5844.0 = 0.002806297 m 30 64 30 64 f m a = 1.666 a 60:64 = 6.854 fm : 60:94 EPV = 10,000{7.611 + (.002806297)(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

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 = 0.00635 Standardised Mortality Rate = (960,000 + 1,400,000 52 15000 74 12000 + 740,000 109 ) 10,000 3,100,000 = (3,328 + 8,633.33 + 8,066) 3,100,000 = 0.00646 Standardised Mortality Ratio = 235 3,100 15,000 960,000 7,500 12,000 1,400,000 7,100 10,000 740,000 = 235 48.44 64.29 95.95 = 235 208.68 = 1.126 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

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 = 5,530.53 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 56,412.20 = 2,292.80 Extra premium, P, spread over term assurance policy term, is from:-.95p D65 a [50]:15 = 2,292.80 D [50] P = 102.43 per annum P = (2,292.80) 2,144.1713 4,581.3224 (.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

Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report 16 (i) q 61 =.016 013 56 p 61 =.983 986 44 p = 1.0 5% 0 61 A 61:4 =.82703 q 62 =.017 749 72 p 62 =.982 250 28 p = 0.983 986 1 61 A 62:3 =.86624 q 63 =.019 654 64 p 63 =.980 345 36 p = 0.966 521 2 61 A 63: 2 =.90792 q 64 =.021 743 10 p 64 =.978 256 90 p = 0.947 524 3 61 A 64:1 =.95238 Capital unit fund fully funded Y/e fund Management Fund Year Cost of alloc. Fund b/f after 8% growth Charge 6% c/f 1 902.50 974.70 58.48 916.22 2 902.50 916.22 1,964.21 117.85 1,846.36 3 1,846.36 1,994.07 119.64 1,874.43 4 1,874.43 2,024.38 121.46 1,902.92 Capital unit fund a-funded Available Needed at Extra death Management Year Cost of alloc. Fund b/f @ y/e after 8% year end cost charge 1 746.39 806.10 793.67 1.96 10.47 2 781.78 793.67 1,701.49 1,676.35 3.02 22.12 3 1,676.35 1,810.46 1,785.17 1.75 23.54 4 1,785.17 1,927.98 1,902.92 25.06 Premium unit fund 1% Fund Management Fund Year Cost of alloc. Fund b/f @ year end charge c/f 3 902.50 974.70 9.75 964.95 4 902.50 964.95 2,016.85 20.17 1,996.68 Death cost (using full Cap. Units) Yr 1 q 61 (4000 916.22) = 49.38 Yr 2 q 62 (4000 1846.36) = 38.23 Yr 3 q 63 (4000 1874.43 964.95) = 22.81 Yr 4 q 64 (4000 1902.92 1996.68) = 2.18 Page 9

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 1 253.61 300.00 (1.86) 49.38 10.47 87.16 87.16 2 218.22 20.00 7.93 38.23 22.12 190.04 187.00 3 97.50 21.00 3.06 22.81 33.29 90.04 87.03 4 97.50 22.05 3.02 2.18 45.23 121.52 115.14 NPV = 87.16v + 187v 2 + 87.03v 3 + 115.14v 4 = 206.37 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 50,000 1 1 30:35 30:35 250 1.01923 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 30:35.575 = 250 + 50,000 A30:35 1.01923 D30 D30 6% a 30:35 = 15.019 4% A 30:35 =.27483 Page 10

Subject 105 (Actuarial Mathematics 1) April 2000 Examiners Report D D 4% 65 4% 30 = 2144.1713 10433.31 =.20551 P(14.0685) = 250 + 50,000{.06801 +.20551} P = 989.87 = 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 (1.01923) v.06 K30t 35 t 35 35t 50,000 (1.01923) 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 ) 1.01923 D40 D40 = 1 60,000 (.40005.30690).30690 1.01923 (.975)(990)(13.081) After = 23,897.55 12,626.44 = 11,271.11 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) 14,357 + 100 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

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

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 2000 0 39 645 350 13,580 8,347 40 59 450 2,295 8,100 45,360 60+ 385 27,500 6,290 489,860 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

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 1999. Basis: Mortality: A1967 70 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] 105 3 PLEASE TURN OVER

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: A1967 70 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 A1967 70 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

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 =0.08 55 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] 105 5 PLEASE TURN OVER

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: A1967 70 Select 4% per annum 0.97087% 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: A1967 70 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. 105 6

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 1 0.85 2 0.90 3 1 The life insurance company uses the following assumptions in carrying out profit tests of this contract: Mortality: A1967 70 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

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

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report 1 A x =0.5 A xx =0.8 1 2A xx = A xx 2 A xx = A x 1 A xx = A x ½A xx =0.5 ½ 0.8 = 0.1. 2 The standardised mortality ratio (SMR) = å x å x E E c x, tmx, t c s x, t mx, t = 350 + 2,295 + 27,500 æ 8,347 45,360 489,860 ö ç 645 * + 450 * + 385 * 13,580 8,100 6,290 è ø = 30,145 32,899.93 = 0.9163. 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 / 26 52 /52 104 / all 35 35 35 35 35 = 462592 + 143625 + 154161 + 179711 + 716291 = 1656380 K 60 = 970852.7 D 35 = 23986 Page 2

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report Value = 1429.02 Alternative æ D ö ç, based on value of 1 p.w. all periods, whole of life è D35 ø 60 50 69.056 129.405 = 1428.99 5 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 ) = 1 2 2 2 ( A A xy xy )) d = 1 ( 2 2 2 2 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

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 1 2,000 ç a a è ø a = 8.207 50:10 a = 13.772 [40]:20 Reserve = 39,216.24 [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 1999. æ ö (12) (12) 9 Premium = 10,000ç a 65 60 10a 65 60 ç è m f m f ø a (12) a = 65 60 60 60:65 m f f f m a = 10.996 ½(7.753 + 7.335) = 3.452 Page 4

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report m f (12) D75 l70 a 10 = 65 60 D m f 65 l60 æ ö ç a70 a 70:75 ç è f f m ø = 6809 780683 (8.328 ½(4.9 + 4.525)) 17857 897001 = 0.381307 0.870326 3.616 = 1.200 Premium = 10,000(3.452 1.2) = 22,520. 10 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 25 25++ t ½ = s s25 v l25 D25 60 60 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 C26... 35 C60 35 C61... 35 C s 64 D é + + + + + + ë ù û 25 Page 5

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report = ( AS) 60 s D 25 é ½ C + 1½ C +... + 35½ C + 36½ C +... + 39½ C ë z ia z ia z ia z ia z ia 25 26 60 61 64 z ia z ia ( ½ C60... 4½ C64 ) + ù û ( AS) = é z ia z ia z ia z ia z ia z ia ( M25 + M26 +... M64 ) ( M60 + M61 +... + 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 60 60 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 25 30 z ia å C 25 + t t= 0. 11 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

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

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 60 100,000 669.90 61 79,464.08 770.94 62 62,954.51 1,117.42 For those exercising the option: Age No. alive No. of deaths 61 19,866.02 477.19 62 35,147.46 935.79 Premiums payable: P 0 = 100,000[0.0066990v + (1.006699) *0.00970168v (1.006699) *(1.00970168) *0.01774972v 3 2 + ] = 2,987.67 P 1 = 100,000[0.00970168v +(1.00970168) * 0.01774972v 2 ] = 2,498.85 P 2 = 100,000[0.01774972v] = 1682.44 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 = 100,000 100,000 2 3 [ 669.90v + (770.94 + 2 * 477.19) v + (1,117.42 + 2*935.79) v ] =4,730.567 Value of premiums= 1 100,000 [2987.67*100,000+2,498.85*19,866.02v+1,682.44*15,738.63v2 ] =3,696.12. Option premium= 4,730.57-3,696.12 = 1,034.45 Page 8

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) 0.096 (3) 0.064 (4) 0.09 (5) 0.012 (6) 0.018 (7) 0.08 1.000 (iii) The net present value of each outcome is: Outcome NPV of Profit (1) 6,000 (2) 858.711 (3) 7,717.421 (4) 1,407.407 (5) 8,266.118 (6) 15,124.829 (7) 8,814.815 Page 9

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report (iv) Mean = Σ NPV liability = 6000 0.64 858.711 0.096 +... = 2,060.36 Variance = ΣNPV 2 Probability (Mean) 2 = 34,009,436.35 Standard deviation = 5,831.76 14 (i) 1.04 / 1.0097087 = 1.03 Þ death benefits evaluated at 3% p.a. Value of death benefits = 100,000 A[45] = 100,000 1.04 ½ 3% A [45] Value of premiums = = 100,000 1.04 ½ 0.42060 = 42,892.952 (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 = 99744.168 1095.1562 11 241.28824 æ ç 1 ö 5680.3705 24 è 5680.3705 ø = 17.366651 0.438864 = 16.92779 Value of premiums = 16.92779P Value of expenses = 0.45P +0.05P a [45]:20 =0.45P +0.05P 17.366651 = 1.31833P 16.92779P = 42892.952 + 1.31833P P = 2747.882 Monthly premium = 228.99 Page 10

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report (ii) Sum assured = 1.04 8 100,000 = 136,856.91 Value of reserves before alteration = 136,856.91 A (12) 53 228.99 12a 53:32 ½ 4% A 53 = 1.04 A 53 =1.04 ½ 0.4226 = 0.430969 N N 11 æ D ö ç 1 è ø (12) 53 85 85 a = 53:32 D53 24 D53 = 60363.851 1095.1562 11 241.28824 æ 1 ö 4020.9326 24 ç 4020.9326 è ø = 14.74004 0.43083 = 14.30921 Value of reserves = 19,661.094. = SA 53 = S 0.430969 S 0.430969 + 100 = 19,661.094 S = 45,388.63 15 (i) Multiple decrement table age (x) d q x s q x (al) x ( ad ) d x ( ad) s x 62 0.01774972 0.15 100,000 1774.972 14733.754 63 0.01965464 0.15 83491.274 1640.991 12277.542 64 0.02174310 0 69572.741 1512.727 0 65 68060.014 Page 11

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report Unit Funds (ignoring actuarial funding) Year, t 1 2 3 Value of Capital Units at start 0 3963.790 4093.703 Premium to CUs 3838 0 0 Interest on CUs 345.42 356.741 368.433 Management charge on CUs 219.630 226.828 234.262 Value of CUs at end 3963.790 4093.703 4227.874 Value of Accumulation Units at start 0 0 4131.127 Premium to AUs 0 3838 3838 Interest on AUs 0 345.42 717.221 Management charge on AUs 0 52.293 108.579 Value of AUs at end 0 4131.127 8577.769 Surrender value of units 3369.222 7815.460 12805.643 Capital Unit Fund (allowing for actuarial funding) Year, t 1 2 3 Actuarial funding factor 0.89097 0.92528 0.96154 Value of CUs at start 0 3667.616 3936.259 Premium to CUs 3419.543 0 0 Interest on CUs 307.759 330.085 354.263 Management charge on CUs 195.683 209.879 225.252 Value of CUs at end 3531.618 3787.822 4065.270 Sterling Fund Year, t 1 2 3 Unallocated premium 580.457 162 162 Expenses 300 60 60 Interest 12.621 4.59 4.59 MC on Capital Units 195.683 209.879 225.252 MC on Accumulation Units 0 52.293 108.579 Surrender profit 23.927 15.218 0 Extra death benefit 114.813 40.902 0 Cost of extra allocation 113.546 123.692 162.604 End of year cashflow 284.329 219.297 277.817 Probability in force 1 0.834913 0.695727 Discount factor 0.869565 0.756144 0.657516 Expected present value 247.243 138.445 127.088 Expected p.v. of profit = 512.776 Expected p.v. of premiums = 4000 2.25208 = 9008.323 Profit margin = 5.69% Page 12

Subject 105 (Actuarial Mathematics 1) September 2000 Examiners Report (ii) Revised Sterling Fund (ignoring reserves) Year, t 1 2 3 Unallocated premium 162 162 162 Expenses 300 60 60 Interest 6.21 4.59 4.59 MC on Capital Units 219.630 226.828 234.262 MC on Accumulation Units 0 52.283 108.579 Surrender profit 87.602 60.199 0 Extra death benefit 107.141 34.89 0 End of year cash flow 55.881 411.01 449.449 Reserves at start of year 400 400 400 Interest on reserves 18 18 18 Change in reserves at year end 66.035 66.683 400 Revised cashflow 260.084 95.693 467.449 Expected present value 226.160 60.412 213.835 Expected present value of profit = 48.087 Profit margin = 0.53% Page 13

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 11 April 2001 (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 14 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 A2001 Institute of Actuaries

1 In the context of a unit-linked contract, state a key reason for the use of actuarial funding of capital units. [2] 2 In the context of a pension fund, state what is meant by a transfer value. [2] 3 Under the Manchester Unity model of sickness, you are given the following values: t p x = 1.05t2 (0 t 1) z x + t= 0.1 (0 t 1) Calculate s x. [3] 4 Some time ago, a life office issued an assurance policy to a life now aged exactly 55. Premiums are payable annually in advance, and death benefits are paid at the end of the year of death. The office calculates reserves using gross premium policy values. The following information gives the reserve assumptions for the policy year just completed. Expenses are assumed to be incurred at the start of the policy year. Reserve brought forward at the start of the policy year 12,500 Annual premium 1,150 Annual expenses 75 Death benefit 50,000 Mortality A1967 70 ultimate Interest 5.5% per annum Calculate the reserve at the end of the policy year. [3] 5 Life insurance company A calculates paid-up policy values for endowment assurance policies by applying the net premium reserve as a single premium at the time of the alteration. It holds net premium reserves based on A1967 70 ultimate mortality and 3% per annum interest. Life insurance company B calculates its corresponding paid-up values by reducing the sum assured to (t/n) times the original sum assured, where n is the original term of the policy and t is the number of premiums which have been paid at the time of the calculation of the paid-up sum assured. The sum assured is paid at the end of the term or the end of the year of death, if earlier. Premiums are payable annually in advance. Identify, showing your calculations, which company pays the higher paid-up sum assured after 15 years, immediately before payment of the 16 th premium, for a 25-year endowment assurance policy originally taken out by a life then aged exactly 40. [4] 105 A2001 2

6 A life office prices sickness insurance contracts using the following three state model in which the forces of transition depend on age: healthy σ x ρ x sick µ x υ x dead Level premiums are payable continuously. Benefits are payable continuously during periods of sickness. There is no death benefit, and the contracts have a deferred period of three months and include a waiver of premiums during periods of benefit payment. Reserves are always positive under the normal premium basis. State briefly, with reasons, what effect the following changes will have on the premium (certain increase, certain decrease, not certain), if the same net present value of profit is to be achieved: (a) (b) an increase in the death rate from the sick state together with an increase in the rate of transition from the healthy state to the sick state a fall in the death rate from the sick state together with a fall in the rate of transition from the sick state to the healthy state. [4] 7 A population is subject to two modes of decrement, α and β, between ages x and x + 1. In the single decrement tables t p α x = x x + t 2 and t p β x = 3 x x + t where 0 t 1. Write down an integral expression for( aq) α x. Hence or otherwise obtain an expression for this probability in terms of x only. [6] 105 A2001 3 PLEASE TURN OVER

8 The following data relate to a population projection being carried out using the component method, and specifically give information about the female population of the country. P(x, t) = population at 1 January 2001+ t aged x last birthday, x, t = 0, 1, 2,.. M(x, t) = estimated net number of emigrants from the population during the year 2001 + t, aged x last birthday at 1 January 2001 + t q(x, t) = independent probability that a life who attains exact age x during the year 2001 + t, dies during that year The following is a selection from the available data. x P(x,0) q(x,0) q(x,1) M(x,0) M(x,1) 54 728,610.0121.0115 37,013 31,461 55 700,369.0136.0129 35,868 30,126 56 678,123.0152.0144 34,312 28,994 57 620,975.0170.0161 31,179 24,943 Calculate, from the information given, the projected number of females aged 57 last birthday at 1 January 2003. State any assumptions you make. [6] 9 Define each of the following terms and give one example of each: (a) class selection (b) selective decrement (c) spurious selection [6] 10 A life office has just sold a single premium deferred pension policy to a lady aged exactly 45. This policy guarantees to pay a cash sum of 200,000 on her 60 th birthday, which must be used to buy a whole of life annuity at that time. The policy also carries an annuity option whereby the policyholder can elect to receive a pension of 15,000 per annum payable monthly in advance from the same date, until her death. The office invests the single premium such that the value of related assets on the policyholder s 60 th birthday will be normally distributed with a mean value of 250,000 and a standard deviation of 50,000. It also believes that the annual interest rate, i, which will be available on the policyholder s 60 th birthday is a random variable where i =.04 with probability.25.06 with probability.50.08 with probability.25 The distribution of the value of assets on the policyholder s 60 th birthday is independent of this annual interest rate, i. 105 A2001 4

Calculate the probability that the value of assets on the policyholder s 60 th birthday is less than the cost of providing the annuity benefit, assuming the policyholder is alive at age 60. Basis for annuity rates: Mortality: a(55) female ultimate Interest: i per annum Expenses: Nil [8] 11 An annuity of 40,000 per annum is payable annually in arrear in respect of two lives both aged 40. The first payment is deferred until the end of the year in which the first of the two lives dies, and Payments continue until 5 years after the death of the survivor. Assume the two lives are independent with respect to mortality. Calculate the expected present value of this annuity. Basis: A1967 70 ultimate mortality 4% per annum interest [10] 12 Two life offices operating in the same economy have maintained the following records in respect of their male assured lives data: deaths, subdivided by age last birthday at the preceding policy anniversary, and duration at the preceding policy anniversary in force, each 1 January, subdivided by age next birthday at issue, and calendar year of policy issue (i) (ii) Describe how you would calculate select forces of mortality, defining carefully all symbols you use and stating necessary assumptions. State clearly the ages and durations to which the resultant rates would apply. [8] Discuss briefly the advantages and disadvantages of pooling the data of the two companies to form one mortality rate estimate for each combination of age and duration. [3] [Total 11] 105 A2001 5 PLEASE TURN OVER

13 A life insurance company sells with-profit whole life policies, with the sum assured payable immediately on the death of the life assured and with premiums payable annually in advance ceasing with the policyholder s death or on reaching age 65 if earlier. The company markets two versions of this policy, one with simple reversionary bonuses and the other with compound reversionary bonuses. In both cases the bonuses are added at the end of the policy year. The company prices the products using the following basis: Mortality A1967 70 select Interest 4% per annum Expenses initial 250 renewal 2% of second and subsequent premiums claim 150 at termination of contract Bonuses simple 6% of basic sum assured per annum compound 4% of accumulated sum assured and bonuses per annum (i) (ii) (iii) Write down an expression for the gross future loss at the point of sale for each of these policies, assuming they are sold to a life aged x exact (x < 65) at outset. Write the expression in terms of functions of the random variables T x and K x, which represent the exact future lifetime and the curtate future lifetime of (x), respectively. [5] Calculate the gross premium required for each of the two policies for a sum assured of 200,000 and a life aged 40 exact at outset, using the equivalence principle. [8] After 10 years, bonuses totalling 90,000 have been declared for the compound reversionary bonus contract. Calculate the net premium reserve for that policy at that time, using A1967 70 ultimate mortality and interest of 4% per annum. [4] [Total 17] 105 A2001 6

14 A life office issues an endowment assurance with a term of five years to a life aged exactly 55. The sum assured is 100,000, payable at the end of the five years or at the end of the year of death if earlier. Premiums are payable annually in advance throughout the term of the policy. The office assumes that initial expenses will be 300, and renewal expenses, which are incurred at the beginning of the second and subsequent years of the policy, will be 30 plus 2.5% of the premium. The funds invested for the policy are expected to earn 7.5% per annum, and mortality is expected to follow the A1967 70 select life table. The office holds net premium reserves, using A1967 70 ultimate mortality and interest of 4% per annum. The office sets premiums so that the net present value of the profit on the contract is 15% of the annual premium, using a risk discount rate of 12% per annum. (i) Calculate the premium. [12] (ii) Without carrying out any further calculations, state with brief reasons what the effect on the premium would be in each of the following cases: (a) The reserves are calculated using a lower rate of interest. (b) The office uses a risk discount rate of 15%. (c) Mortality is assumed to be A1967 70 ultimate. [6] [Total 18] 105 A2001 7

Faculty of Actuaries Institute of Actuaries EXAMINATIONS April 2001 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report Examiners Comments The overall standard of scripts was somewhat disappointing, especially with regard to answers offered for Questions 10, 11 and 12 where candidates were required to apply principles from the syllabus to a problem that they may not have seen before. It is also obvious that many candidates did not read the question asked, or at least did not address the specifics of the question in their solution. Comments on the individual questions follow. Question 3 A surprisingly high number of candidates used the half-year approximation to work out the integral without any justification, and a number omitted the factor of 52.18 Question 4 Many students used q55 and not q54 to calculate the death cost Question 7 The majority of candidates could not quote any of the acceptable integrals and a further number did not know how to obtain an expression for the force of the decrement from the available data. Question 8 Common errors included adding rather than subtracting the net number of emigrants and not applying a survival factor to these emigrants. Clearly students did not read the question correctly. Question 9 Generally answered well, although a number of candidates seemed unaware of selective decrements. Question 10 This question was answered very poorly. Far too many candidates averaged the interest rate as a first step and calculated the cost of annuity benefits at the expected interest rate (6%). A further number then overlooked the option when finalising their solutions. Page 2

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report Question 11 This question was answered poorly by many candidates. Most candidates who made progress favoured the alternative answer shown below. While a number of students identified the reversionary element correctly, few were able to correctly handle the 5- year element. Question 12 While many candidates had a general understanding of the construction of select mortality rates, most had trouble applying it with the data given. Only a minority of candidates appreciated that the in force data would have to be manipulated in order to get a central exposed to risk which corresponded with the data for deaths. Question 13 Part (ii) was generally well done. The commonest errors were to limit the benefit term to 25 years and to ignore that it was paid immediately on death. Part (i) was not answered well as students seemed to confuse random variables and their expected values. Part(iii) proved the most difficult. Many students calculated a gross premium reserve, others correctly omitted allowance for expenses and future bonuses but used the gross premium from (ii) and finally most of those who could correctly compute a net premium did so using select mortality. Question 14 This was generally answered quite well. A significant minority failed to recognise it as a profit test question, and others constructed the reserves using the gross premium. In part (ii), most students identified the correct effect on the NPV of the profit ( a reduction) but wrongly assumed that the premium must also reduce. Page 3

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report 1 The office can more closely match income and outgo. The initial strain caused by high initial expenses is reduced by capitalising the higher management charges from the capital unit fund. 2 When a member leaves a pension scheme with an entitlement to deferred benefits, they may elect in lieu to have a cash payment made by the scheme to either a new scheme or an individual pension policy. 3 s x = 1 52.18 tpx zx+ t dt 0 = 1 2 t dt = 5.218 0 52.18 (1.05 ) (0.1) t.05t 3 3 1 0 = (5.218).05 1 3 = 5.131 4 In general: ( t V + P E) (1 + i) = (q x+t ) (S) + (p x+t ) ( t+1 V) Here (12,500 + 1,150 75) (1.055) = (q 54 ) (50,000) + (p 54 ) ( t+1 V) q 54 =.00755572 p 54 =.99244428 t+1 V = = 1 {(13,575)(1.055) (.00755572)(50000)}.99244428 1 {14,321.62 377.79} = 14,050.99244428 Page 4

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report a55:10 5 Company A: 15 V = 1 40:25 a 40:25 PUP SA = 15 V40:25 A 55:10 = 1 8.371 17.169 = 0.5124 = 0.5124 0.75619 = 0.6777 Company B: t = 15 n = 25 PUP SA = t n = 15 25 = 3 5 = 0.6 Therefore Company A provides the higher PUP SA. 6 (a) Uncertain healthy to sick transition rate increase causes reduced premium income and higher claims but increased sickness death rate leads to reduced claims. Depends on interaction of both effects. (b) Definite increase Both effects mean people are sick for longer. Therefore higher premiums will be needed to meet higher claims. 7 ( aq) α x = 1 α t x µ x + t ( ap ) ( a ) dt = 0 1 0 α β α t x t x x+ t ( p )( p )( µ ) dt = 1 α α β t x x t t x ( p µ + ) 0 pdt But t p α α d x µ x + t = ( t p α x ) dt t p α x µ α x + t = d x dt x + t 2 = ( 1) ( 2) (x 2 ) (x + t) 3 = 2x 2 3 ( x + t) ( aq) α x = 1 2 3 2x x. dt = 2x 5 1 0 3 3 0 ( x + t) ( x + t) 1 ( x + t) 6 dt = 2x 5 1 1 5 ( x + t) 5 1 0 = 5 2 x 1 5 5 ( x + 1) Page 5

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report 8 We need P(57, 2) so we need to project P(55, 0) for 2 years, allowing appropriately for emigrants. 2001 P(55, 0) = 700369 all have their 56 th birthday during 2001 use q(56, 0) P(56, 1) = (700,369) (1.0152) (35,868) [1 (½) (.0152)] assuming net emigration is spread uniformly across year = 689723.4 35,595.4 = 654,128.0 2002 P(57, 2) = (654,128) (1.0161) (28,994) [1 (½) (.0161)] using (q 57, 1 ) = 643,596.5 28,760.6 = 614,835.9 = 614,836 9 (a) Class selection: Refers to a factor affecting relative mortality which is a permanent feature, e.g. age, sex, smoking status etc. (b) Selective decrement: When lives grouped by one decrement experience different levels of another decrement, e.g. ill health retirers usually experience heavier mortality than other scheme members or retired members of similar age/sex or marriage/mortality. (c) Spurious selection: An investigation wrongly suggests that a certain selection is present when it is not. It usually results from unrecognised heterogeneity in the data, with perhaps changing proportions of lives subject to different underlying mortality rates, e.g. occupational differences being the underlying cause of regional mortality effects or changing sex mix leading to wrongly attributing mortality rate progressions to temporary initial selection. Page 6

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report (12) 10 a 60 = a 60 + 13 24 @ 4% =.542 + 13.294 = 13.836 @ 6% =.542 + 10.996 = 11.538 @ 8% =.542 + 9.294 = 9.836 annuity per annum available is @ 4%: (200,000 13.836) = 14,455 p.a. @ 6%: (200,000 11.538) = 17,334 p.a. @ 8%: (200,000 9.836) = 20,333 p.a. so the guaranteed minimum annuity option will only be chosen if i =.04 if i =.06 or.08 (occurs with probability 0.75), the office will only make a loss if assets at age 60 < 200,000. The guarantee has value at vesting of (15,000) a 60 (12) = (15,000) (13.836) = 207,540 so the insurer makes a loss if i =.04 (with prob =.25) if assets at age 60 < 207,540 Probability of loss = (.75) [Prob (Assets < 200,000)] + (.25) [Prob(Assets < 207,540)] = (.75) A 250,000 200,000 250,000 P < 50,000 50,000 where A ~ N(250,000, 50,000 2 ) A 250,000 207,540 250,000 + (.25) P < 50,000 50,000 = (.75) [P(z < 1)] + (.25) [P(z <.8492)] = (.75) [1.84134] + (.25) [1.80234] = (.75) (.15866) + (.25) (.19766) =.118995 +.049415 =.16841 = 0.17 Page 7

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report 11 From considering only the second condition, the first five payments are certain to occur: EPV = 5 t 40,000 v = 40,000a5 t= 1 Payments thereafter will be made if either life was alive five years earlier, with probability t 5 p40:40 EPV = 40,000 v t 5 p40:40 t= 6 Setting n = t 5 ( t = n + 5) we get t EPV = 40,000 n+ 5 v n p = 40:40 n= 1 5 40,000v v p n= 1 n n 40:40 = 5 40,000 v a 40:40 However, the first condition means that no payment occurs on any date when both lives are still alive, which occurs with probability t p 40:40. t The previous values are too high by 40,000 v n p40:40 = 40,000 a 40:40. t= 1 Therefore the total expected present value is given by: 5 5 40,000[ a + v ( a ) a ] 40:40 40:40 = 5 5 40,000[ a + v ( a + a a ) a ] 40 40 40:40 40:40 = 40,000 a 5 2N41 N N + v D D D 5 41:41 41:41 40 40:40 40:40 = (2)(125,015.43) 109,071.05 109,071.05 40,000 4.45182 + (.82193) 6,986.4959 6,794.7238 6,794.7238 = 40,000[4.45182 + (.82193) {(2)(17.89387) (16.05232)} 16.05232] = 40,000[4.45182 + (.82193) (19.73542) 16.05232] = (40,000)(4.62063) = 184,825 Page 8

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report Alternative solution: The first condition is the sum of two reversionary annuities: (40,000)(2)( a 40 40 ) = (80,000)( a40 a40:40 ) or is the difference between a last survivor annuity and joint life annuity: (40,000)( a a 40:40 40:40 ) = (80,000)( a40 a40:40 ) The second condition pays a 5 year annuity certain, with the first payment at the end of the year of death of the survivor: (40,000)( a ) A 5 40:40 Using A = 1 da = 1 d(2 a 40:40 40:40 40 a 40:40 ) evaluating and summing the 2 elements leads to the same solution as above. 12 (i) Force of mortality Deaths Corresponding central exposed to risk In this case let θ y,r = number of deaths where y r = age last birthday at previous policy anniversary = duration at previous policy anniversary both are policy year rate intervals θ y, r c E y, r estimates µ [y+½ r]+r+½ At the start of the policy year rate interval lives are aged y last birthday, with duration in force of exactly r years. Assuming birthdays are spread uniformly over the policy year, this gives an average exact age at the start of the rate interval of y + ½, or y + ½ r at the start of the policy. The duration in force half-way through the rate interval (appropriate for force of mortality) is clearly r + ½. For the central exposed to risk, initially we define n P x,t = number of lives in force on 1.1.n where x t = age next birthday at issue = calendar year of issue Page 9

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report x + n t = age next birthday at following policy anniversary x + n t 1 = age last birthday at following policy anniversary x + n t 2 = age last birthday at previous policy anniversary and also n t 1 = duration at previous policy anniversary For correspondence with deaths we need n P y+2+t n, n r 1 and n+1 P y+1+t n, n r and the appropriate central exposed to risk for calendar year n is: n c y, r E = ½( n P y+2+t n, n r 1 + n+1 P y+1+t n, n r ) assuming all movements (new business, deaths, lapses etc.) are spread evenly throughout the calendar year. Then µ [y+½ r]+r+½ = n θ yr, c ney, r summing the central exposed to risk over the years of the study. Alternative method for (i) It is probably easier to actually restructure in force as follows: At each census date, calculate r = census year calendar year of issue Let y = x + r age y 2 last birthday at preceding policy anniversary duration r 1 at preceding policy anniversary Redefine in force as n P y,r = no. of lives in force on 1.1.n where y = age next birthday at following policy anniversary r = duration at following policy anniversary Page 10

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report Appropriate central exposed to risk for calendar year n is n c y, r E = ½( n P y+2, r+1 + n+1 P y+2, r+1 ) assuming n P y,r varies linearly over the calendar year (rest of part (i) solution is as above) (ii) Pooling the date will give rise to more credible estimates of true underlying mortality rates, since greater exposure means lower variance. However, one must be wary of heterogeneity in the data from the two offices: e.g. differing geographical coverage differing underwriting standards different distribution or target market etc. 13 (i) Simple bonus version: L = 250 + (S[1 + (.06)K x ] + 150) v Tx { P (.98) a +.02 P min[1 + K,65 x } Compound bonus version: L = 250 + ( S K [(1.04) 150) { (.98).02 min[1,65 } x Tx + v P a + P + K x x x (ii) Equivalence principle E[L] = 0 Also we shall assume E[T] * E[K] + ½ Simple bonus: [ x ] + 1 250 + ( S + 150) A[ x] + (.06 S) ( IA) [ x] + 1 D[ x ] """"""""" """""""""! more easily valued as (.94S + 150) A +.06 S( IA) [ x] [ x] D P (.98) a +.02 x = [ x]:65 Page 11

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report In this case: R 250 + (1.04) ½ [40] {(.94) (200,000) + 150} A[40] + (.06) (200,000) D[40] P[(.98) a +.02] = [40]:25 ½ 57,705.359 250 (1.04) + 188,150 (.27284) + (12,000) 6,981.5977 = P[(.98) (15.609) +.02] 250 + (1.04) ½ [51,334.85 + 99,184.22] = P[15.31682] P = 153,749.94 15.31682 = 10,038 p.a. Compound bonus: * ½ A[40] 250 + (1.04) 200,000 + 150A 1.04 [ x ] = 15.31682P i b * at 1 + b i.e. 0% ½ 200,000 (250) + (1.04) + (150) (.27284) 1.04 = 15.31682P 250 + (1.04) ½ [192,307.69 + 40.926] = 15.31682P P = 196,407.87/15.31682 = 12,823p.a. (iii) Net Premium Reserve for WP policies (i) (ii) allows for accrued bonuses only net premium ignoring any bonuses 10 V = 290,000 A50 (NP) a 50:15 A40 where NP = 200,000 a 40:25 = ½ (200,00) (1.04) (.27331) 15.599 = 3,573.60 p.a. Page 12

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report 10 V = (290,000) (1.04) ½ (.38450) (3,573.60) (10.995) = 113,713.23 39,291.73 = 74,421.50 a 56:4 14 (i) Reserves 1V = 1 55:5 a 55:5 = 1 3.720 4.547 = 0.18188 Similarly V 2 =.37189 V 3 =.57115 V 4 =.78007 V = 0 assuming all claims paid in cash flow outgo 5 Also q [55] =.00447362 p [55] =.99552638 0 p [55] = 1 q [55]+1 =.00625190 p [55]+1 =.99374810 p 1 [55] =.995526 q 57 =.01049742 p 57 =.98950358 p 2 [55] =.989302 q 58 =.01168566 p 58 =.98831434 p 3 [55] =.978917 q 59 =.01299373 p 59 =.98700627 p 4 [55] =.967478 Opening Closing Profit Year Premium Expense Reserve Interest Claim Reserve Vector 1 P 300 0.075P 22.5 447.36 18106.63 1.075P 18876.49 2 P.025P+30 18188.073125P+1361.8 625.19 36956.50 1.048125P 18061.89 3 P.025P+30 37189.073125P+2786.9 1049.74 56515.44 1.048125P 17619.28 4 P.025P+30 57115.073125P+4281.4 1168.57 77095.44 1.048125P 16897.61 5 P.025P+30 78007.073125p+5848.3 100000 0 1.048125P 16174.70 Profit Signature NPV of Profit Signature 1 1.075P 18,876.49 v =.95982P 16,854.01 2 1.043446P 17,981.08 v 2 =.83183P 14,334.41 3 1.036912P 17,430.79 v 3 =.73805P 12,406.89 4 1.026027P 16,541.36 v 4 =.65206P 10,512.33 5 1.014038P 15,648.67 v 5 =.57539P 8,879.48 3.75715P 62,987.12 NPV =.15P = 3.75715P 62,987.12 P = 62,987.12 (3.75715.15) = 17,462 Page 13

Subject 105 (Actuarial Mathematics 1) April 2001 Examiners Report (ii) (a) Lower interest rate larger reserves Larger reserves profit deferred Profit deferred NPV reduced (since risk rate (12%) > earned (7.5%)) profit falls below 15% premium need to increase premium to re-establish 15% margin (b) (c) Higher risk rate lower NPV etc. need higher premium to meet profit requirement Ultimate mortality more death claims earlier claims (NPV claims increases) and reduced premium income NPV falls need to increase premium Page 14

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 13 September 2001 (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 14 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 S2001 Institute of Actuaries

1 Under the Manchester Unity model of sickness, you are given the following values: s x =5 1 t 0 pdt=0.9 x Calculate the value of z x. [2] 2 Give a formula for P 21 (2003) in terms of P 20 (2002), based on the component method of population projection. Px ( 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 per week 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 is in 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] 105 S2001 2

4 An employer recruits lives aged exactly 20, all of whom are healthy when recruited. On entry, the lives join a scheme that pays a lump sum of 50,000 immediately 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. Healthy (H) σ x Sick (S) ρ x µ x ν x Dead (D) All surviving members retire at age 65 and leave the scheme regardless of their state of health. ab xt, 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 ( t 0 and 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 of each 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] 105 S2001 3 PLEASE TURN OVER

6 A life insurance company issues a special annuity contract to a male life aged exactly 70 and a female life aged exactly 60. Under the contract, an annuity of 10,000 per annum is payable monthly to the female life, provided that she survives at least 10 years longer than the male life. The annuity commences on the monthly policy anniversary next following the tenth anniversary of the death of the male life and is payable for the balance of the female s lifetime. Calculate the single premium required for the contract. Basis: Mortality: a(55) Ultimate, males or females as appropriate Interest: 8% per annum Expenses: none [4] 7 The 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 the associated single-decrement table: 50% of the decrements due to withdrawal occur 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 an employee at age 21 last birthday. [4] 8 The following data are available from a life insurance company relating to the mortality experience of its temporary assurance policyholders. θxd, The 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 policy anniversary next following the date of death. Pye, ( 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, where n = 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 θ xd, deaths and derive the age and duration to which these rates apply. State all the assumptions that you make. [6] 105 S2001 4

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 aged exactly 50 at the issue date, in return for the payment of a single premium. The following benefits are payable under the contract: (i) (ii) In the event of either of the lives dying within 10 years, a sum assured of 100,000 is payable immediately on this death. 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 Ultimate Interest: 4% per annum Expenses: None [8] 105 S2001 5 PLEASE TURN OVER

11 A life insurance company sells term assurance policies with terms of either 10 or 20 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 the statistical summary of the mortality investigation. In all cases, the central rates of mortality are expressed as rates per 1,000 lives. All policies 10-year policies 20-year policies Age Number in force Central mortality rate Number in force Central mortality rate Number in force Central mortality rate 24 6,991 1.08 6,013 0.86 978 2.12 25 44 6,462 2.05 5,438 1.74 1,024 3.68 45 64 5,815 13.26 4,942 11.55 873 22.94 65 3,051 75.70 2,570 71.53 481 97.70 Total 22,319 18,963 3,356 (i) (ii) Calculate the directly standardised mortality rate and the standardised mortality 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] 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] 105 S2001 6

12 A life insurance company offers an option on its 10-year without profit term assurance 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 the whole of life, or until earlier death. (i) (ii) Describe the conventional method of pricing the mortality option, stating clearly the data and assumptions required. Formulae are not required. [3] 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 the option, using the conventional method. The following basis is used to calculate the basic premiums for the term assurance policies. Basis: Mortality: A1967 70 Select Interest: 6% per annum Expenses: none [4] (iii) (iv) Describe how you would calculate the option single premium for the policy described in part (ii) above using the North American method, stating clearly what additional data you would require and what assumptions you would make. [4] State, with reasons, whether it would be preferable to use the conventional method or the North American method for pricing the mortality option under the policy described in part (ii) above. [3] [Total 14] 105 S2001 7 PLEASE TURN OVER

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 first payment, 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 by 1.923% per annum compound. Premiums under the policy are payable annually in advance for 10 years or 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 Ultimate after age 60: a(55) Males Ultimate Interest: 6% per annum Expenses: initial: 10% of the initial premium, incurred at the outset renewal: claim: 5% of each of the second and subsequent premiums, payable at the time of premium payment 100, incurred at the time of payment of 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 no benefit payable on death and the rate of increase of the annuity in payment is to be altered. The premium under the policy is to remain unaltered as is the amount of the initial annuity payment. The life insurance company calculates the revised terms of the policy by equating gross premium prospective reserves immediately before and after the alteration, calculated on the original pricing basis, allowing for an expense of alteration of 100. Calculate the revised rate of increase in payment of the annuity. [7] [Total 16] 105 S2001 8

14 A life insurance company issues a 3-year unit-linked endowment assurance contract 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, A (calculated at 4% interest and A1967-70 Ultimate 60 + t:3 t mortality), of the full 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. The life insurance company uses the following assumptions in carrying out profit tests of this contract: Mortality: A1967 70 Ultimate Expenses: initial: 400 renewal: Unit fund growth rate: Sterling fund interest rate: Risk discount rate: Surrender rates: 80 at the start of each of the second and third policy years 8% per annum 5% per annum 15% per annum 20% of all policies still in force at the end of each of the first and second years Calculate the profit margin on the contract. [18] 105 S2001 9

Faculty of Actuaries Institute of Actuaries EXAMINATIONS September 2001 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report Examiners Comments Overall the standard of attempts was lower than the examiners would have expected. There was evidence that many candidates spent too much time on the earlier questions, with consequent time problems later on. Questions 1,2,3,4,9 and 11 were well answered. In question 5, career average salary was not dealt with well. Candidates had difficulty with year-end decrements in question 7 and with the duration in question 8. Question 9(i) was poorly answered, although it was a standard question. Question 10 was the most poorly attempted, with few candidates scoring more than half marks. There was an ambiguity in this question: the benefit payable on the second death could have been interpreted as 200,000 or 300,000. Candidates were given credit for either approach. Many candidates did not give sufficient detail in their answers to question 12(i). Question 13(ii) was poorly attempted and the answers to question 14 were not as strong as one would have expected for a fairly standard question. Page 2

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report 1 s 52.18 p z dt x t x xt 0 1 z x 1 52.18 lxtzxtdt 0 sx * lx sx 5 1 1 1 lxtdt lxtdt 0 0 tpxdt 0 5.56 0.9 2 The required formula is: P (2003) P (2002)(1 q (2002)) M (2003) 21 20 201 2 21 q 20 (2002) 1 2 is the probability that a life aged 20 last birthday at mid-year 2002 dies between mid-year 2002 and mid-year 2003, assuming those aged 20 last birthday at mid-year 2002 have birthdays uniformly distributed over the calendar year. M (2003) 21 denotes the number of migrants entering the population during midyear 2002 and mid-year 2003 who survive to be aged 21 last birthday at mid-year 2003. The formula is applied separately to males and females. 3 To allow for the fact that benefit cannot be paid for at least one year, the sickness benefit could be valued using the factor K K D 31 x 100 30, where x is the ceasing age for benefits. However, the factor K 31 is not accurate as it takes into account sickness of all durations, whereas a new policyholder aged 30 cannot experience sickness of all durations from age 31. For this reason and because the numerical effect is not significant, I would use the factor K 30 rather than K31 in the above formula. 4 The required expression is 45 0 t HH HS 20, t 20t 20, t 20t 25,000 e {2 p. 3 p. } dt Page 3

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report 5 The value of the benefits is s 39 1 2 ra 40 40,000 R 40,000 1,758,471 63,816.35 60 s D 60 (3.48 3.58).5,204 40 1 2 6 The required single premium is given by f D m f 70 70 f 10 (12) 70 10 p60v a a a 70 70 :70 D60 10000 f m f 3571.2 10000 7.308 5.106 8858.7 = 8,876.90 7 Construct a multiple decrement table. Age No. alive No. deaths No. withdrawals over year No. withdrawals at year end 20 100000 97.50 4997.5 4745.25 21 90159.75 87.9058 At age 20, no. of deaths = 100000*0.001(1-0.5*0.05) = 97.50 no. of withdrawals over year = 100000*0.05*(1-0.5*.001) = 4997.5 no. of withdrawals at year end = 100000*(1-0.05)*(1-0.001)* 0.05 = 4745.25 Required probability = 87.9058/100000 = 0.00087906. 8 Define a census taken at time t after the start of the period of investigation ' (1.1.98), Pxd, () t, of those lives having a policy in force at time t, who were x nearest birthday at entry and will be duration d on the policy anniversary next following time t. The central exposed to risk is then given by t3.5 c ' xd, xd, t 0 E P () t dt Page 4

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report ' Assuming that Pxd, () t varies linearly between the census dates the integral can be approximated by 1 * 1 P 0 P 1 *2 P P 2 1 *1 P 2 P 3 2 2 2 2 ' ' 1 ' 1 ' 1 ' 1 ' 1 xd, xd, 2 xd, 2 xd, 2 xd, 2 xd, 2 However, the census data have been recorded according to age x nearest birthday at entry and curtate duration d at time t. The following formula may be written: ' xd, xd, 1 P () t P () t. Substituting this into the equation above gives P 1 1 1 xd, 1 Pxd, 1 2 Pxd, 1 2 Pxd, 1 2 xd, 1 2 xd, 1 2 c Exd, 1 1 1 2 2 2 1 *1 P 1 1 2 P 3 xd, m E xd, c xd, * 0 *2 2 2 estimates mx d 1 because the average age at entry is x assuming birthdays are uniformly distributed over the policy year and the exact duration at the start of the rate year of death is d 1 for all lives (no assumptions are necessary). 9 (i) Gross premium retrospective and prospective reserves will be equal if: The mortality and interest rate basis is the same for the retrospective and prospective reserves and is the same as that used to determine the gross premium at the date of issue of the policy. The same expenses (excluding the initial expenses) are valued in the retrospective and prospective reserves and also the expenses valued in the retrospective reserves are the same as those used to determine the original gross premium. The gross premium valued in the retrospective and prospective reserves is that determined on the original basis using the equivalence principle. (ii) The prospective reserves at time t are given by SA ea fa Ga (a) ( m) ( m) xt xt xt xt where S is the sum assured e is the annual rate of renewal expenses f is the claim expense G is the annual rate of gross premium Page 5

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report The retrospective reserve at time t is given by Dx Ga 1 ( m) 1 xt : SA xt : I ea xt : fa xt : x t (b) D where I is the additional initial expense. The original gross premium is given by Ga SA I ea fa 0.. (c) ( m) ( m) x x x x Dx ( ) ( ) Add Ga m m x SA x I ea x fa x, which is identically 0, to (a). D x t Combining terms, e.g. D D x xt expression for the retrospective reserve. D Ga Ga Ga gives (b), the ( m) ( m) x ( m) x xt xt : Dxt 10 The value of benefit (a) and (b) is A 1 1 100000 2A A 50:10 1 2 1.04 1 da 50:50 :10 50:50 :10 1 2 50:50:10 D D 60 50 D D 60:60 50:50 a 2a a 50:50:10 50:10 50:50:10 a N N 59513.103 24729.51 4354.5857 50:50 60:60 50:50:10 D50:50 a 50 :10 D D D D 50 8.207 2855.5942 4597.0607 60 50:50 2487.2117 4354.5857 60:60 0.621178 0.57117 7.98781 M 1767.5555 1477.0842 1.04 1 1 2 50 M 60 1 2 A 1.04 0.064438 50:10 D 4597.0607 the required premium is 50 Page 6

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report 2 100000 2 * 0.064438 1 1.04 1 0.0384622 *8.207 7.98781 2 * 0.621178 0.57117 = 13,369.55 Alternative solution The benefits payable may be regarded as a sum of 200,000 payable on either death, less a sum of 100,000 payable on the first death. the value of the benefits is: 200000* 2A 100000 1 50:10 1 A 50:50:10 A 1 50 :10 0.064438 A 1 50:50:10 1 2 1.04 1 da 50:50:10 D D 60:60 50:50 1 2 1.04 1 0.038462 * 7.98781 0.57117 0. 124011 the required premium is 200000*2*0.064438-100000*0.124011 = 13,374.10 (The difference in the two answers is due to rounding) Page 7

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report 11 (i) The directly standardised rate (DSR) is given by DSR E x x cs xt, mxt, E cs xt, The standardised mortality ratio (SMR) is given by SMR x x E E c xt, mxt, c s xt, mxt, For 10-year policies: 6.991 * 0.86... DSR 13.56053 6.991 6.013 * 0.86... SMR 0.920149 6.013 *1.08... For 20-year policies: 6.991 * 2.12... DSR 21.06187 6.991 0.978 * 2.12... SMR 1.424669 0.978 *1.08... Note: In each of the above the DSR is expressed as the number of deaths per 1,000. (ii) I would favour the standardised mortality ratio. The directly standardised mortality rate requires m xt, to be recorded for each age group, for the 10- year and 20-year policies separately. The data may not be readily available. The SMR requires the number of deaths in each age and policy group only to be recorded: these data should be easily recorded. 12 (i) In pricing the mortality option using the conventional method, the actuary pricing the option assumes: that all lives eligible to take up the option will do so, and that the mortality experience of those who take up the option will be the Ultimate experience which corresponds to the Select experience that would have been used as a basis if underwriting had been completed as normal when the option was exercised The mortality basis used is not usually assumed to change over time, so that the only data required are the Select and Ultimate mortality tables used in the original pricing basis. In pricing the mortality option, the actuary values the premium income assuming that the premium payable at the end of the ten years is calculated using Select rates according to the original premium basis and Page 8

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report values the premiums assuming Select rates apply only from the date of issue of the original policy. The actuary values the liabilities similarly. The difference in the present value of the premium income and benefit liability per policy originally issued gives the additional option single premium, per policy issued. (ii) Whole life premium payable = 100000P 40 Whole life premium which should be paid according to the actuary s basis = 100000P40 Option premium = present value of the difference in premiums = 10 l40 100000 P40 P40v a40 l 30 1 33542.311 100000 0.01063 0.01058 *14.874 41.18 10 1.06 33828.764 (iii) I would require the following data: an estimate of the probability of those reaching age 40 as policyholders, who exercise the option a multiple decrement table to describe the mortality and other relevant decrements (such as surrender) of those who exercise the option, commencing at age 40 the basis on which the whole life premium payable is to be calculated: this would normally be assumed to based on the 1967-70 Select mortality, similar to the premium basis set out I would calculate the present value of the additional liability, using the multiple decrement table from age 40 and allowing for the probability of exercise of the option and A1967-70 Select mortality before age 40. I would then calculate the whole life premium payable and also the present value of the whole life premiums payable, similarly to the method used to calculate the additional liability. The difference between the two values, per term assurance policy issued, would be the option premium. Page 9

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report (iv) The more accurate method is the North American method. However, I would favour the conventional method for the following reasons. There may not be sufficient data available to apply the North American method. If the policy were being sold in a market where the conventional method was generally used for pricing, then there would be adequate experience of the use of the method in the market. Even if sufficient data were available in respect of the North American method, they might not be appropriate for pricing the portfolio concerned, particularly if the pricing were being done when the business was first issued. 13 (i) Present value of annuity payments: 1.01923 1.06 1.04 D D 6% annuity payments are valued at 4% p.a. 1 1.06 60 4% 60 Value = 100000 a 100000 11.625 1 50 60 10 l l 50 1 30039.787 10000 12.625 64,821.99 10 1.06 32669.855 Present value of death benefits: Present value of death benefit at age 50 t 1.923% t P s t v 6% 1.01923 0.01923, where P is the annual premium. t 1 1.01923 t t 1.01923 1 Pv v P t 4% 6% 1.06 0.01923 the present value of the death benefits is 1.01923 0.01923 P A 4 1 % 50:10 A 6 1 % 50:10 A D 2855.5942 0.68436 0.063182 4597.0607 4% 60 1 A 50:10 50:10 D50 6% 1 l 30039.787 A1 A A 0.25736 0.55839* *0.39136 50:10 32669.855 60 50 10 60 1.06 l50 =0.056421 Page 10

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report the present value of the death benefits is 1.01923 0.01923 0.063182 0.056421P 0. 358347P Present value of premiums less expenses: 0.95Pa 0.05P 100A 1 50:10 50:10 0.95 P*7.599 0.05P100*0.056421 7.16905P 5.6421 7.16905P 5.6421 64821.99 0. 358347P P 9,518.49 (ii) At the date of alteration: Present value of annuity payments before alteration 6% D60 4% 1 l 60 10000 a60 10000 (11.625 1) 10000 * 0.708453 *12.625 5 D 1.06 l 55 55 89442.19 Present value of death benefit before alteration 1.01923 Ps A P A A 0.01923 1.923% 4% 4% 6% 1 1 1 5 55:5 55:5 55:5 1.01923 P*5.295953*0.045886 P0.045886 0.043249 0.01923 0.382777P 3643.459 Present value of annuity payments after alteration =89442.19+3643.459-100 = 92985.649 Present value of annuity payments before alteration, based on a rate of interest of 6% after age 60 10.813 * 89442.19 76605.02 12.625 Page 11

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report Estimated interest rate underlying annuity payments after alteration 92985.649 89442.19 0.04 * 0.02 0.034479 89442.19 76605.02 Revised rate of escalation 1.06 1.034479 1 2.467% 14 Multiple decrement table Age d q x ( al ) d x ( ad ) d x ( ad) s x 60 0.014432 100000 1443.246 19711.35 61 0.016014 78845.4 1262.596 15516.56 62 0.01775 62066.25 1101.658 12192.92 Probabilities of survival Age tpx 60 1 61 0.788454 62 0.620662 Unit fund (ignoring actuarial funding) Year 1 2 3 Fund brought forward 0 4970.97 5100.215 Premiums allocated to CU 4845 0 0 Interest 387.6 397.6776 408.0172 Management charge 261.63 268.4324 275.4116 Fund carried forward 4970.97 5100.215 5232.821 Fund brought forward 0 0 5180.274 Premiums allocated to AU 0 4845 4845 Interest 0 387.6 802.0219 Management charge 0 52.326 108.273 Fund carried forward 0 5180.274 10719.02 Surrender values 3976.776 9260.446 15951.84 Unit fund (with actuarial funding) Year 1 2 3 Actuarial funding factor 0.890605 0.925148 0.961538 Fund brought forward 0 4598.885 4904.053 Premiums allocated to CU 4314.979 0 0 Interest 345.1983 367.9108 392.3242 Management charge 233.0089 248.3398 264.8189 Page 12

Subject 105 (Actuarial Mathematics 1) September 2001 Examiners Report Fund carried forward 4427.169 4718.456 5031.558 Sterling fund Year 1 2 3 Unallocated premium 685.021 155 155 Expenses 400 80 80 Interest 14.251 3.75 3.75 Management charge 233.0089 300.6658 373.0918 Mortality charge 109.2946 33.64881 0 Surrender profit 88.7785 125.6126 0 Additional allocation 135.3906 146.0999 201.2624 Fund at year end 376.3742 325.2797 250.5794 Present value of profit = 623.4689 Present value of premiums = 10774.61 Profit margin = 6.19% Page 13

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 9 April 2002 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination details as requested on the front of your answer booklet. 2. You must not start writing your answers in the booklet until instructed to do so by the supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 14 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, with any additional sheets firmly attached, and this question paper. In addition to this paper you should have available Actuarial Tables and your own electronic calculator. Faculty of Actuaries 105 A2002 Institute of Actuaries

1 (i) Explain what is meant by the following expression: 3 4 q [40] 1. (ii) Calculate its value using A1967 70 mortality. [3] 2 Two lives, each aged exactly 35, are independent with respect to mortality and are each subject to a constant force of mortality of 0.02. Calculate the value of the following expression for these lives: 1 10 20 q 35:35 [4] 3 (i) Define the term Total Fertility Rate and explain the difference between rates calculated on a cohort basis and a period basis. [3] (ii) In the context of population projection, state, with a reason, which basis is preferable. [2] [Total 5] 4 Due to a downturn in the economy, the numbers unemployed in a certain country are expected to increase. The current number unemployed is 100,000, and this is expected to rise towards but not exceed 300,000 following the logistic growth model. The initial rate of increase in unemployment will be 25% per annum. Calculate how long it will take for the unemployed population to reach 200,000. [5] 5 A life insurance company sold a number of 4-year single premium policies with a guaranteed amount payable at maturity. The closest matching investment available was a 5-year zero-coupon bond. Interest rates at the time of the insurance company selling the policies and investing the money in the bonds were 5.25% effective per annum. The office invested all the premiums received in these assets. The insurance company guaranteed a return of 5.0% per annum at maturity. On death, the return was not guaranteed but the company promised to pay out the full market value of the related asset immediately at the date of death. If the distribution of 1 + i is log-normal with parameters = 0.05 and = 0.01, and mortality follows English Life Table No. 12 Males, calculate the probability that the office makes a loss on a policy sold to someone aged exactly 56. You should assume that the company sells the matching asset at the time of any claim. (Remember that if X~lognormal with parameters and, then log e (X) is normally distributed with mean and standard deviation.) [5] 105 A2002 2

6 A retiring employee aged exactly 60 is given a choice between the following two pensions: Pension A is payable annually in arrear throughout the pensioner s lifetime, with at least 4 payments guaranteed to be made. The first payment is 20,000 and payments increase by 0.9709% per annum compound thereafter. Pension B is payable annually in arrear, with an initial payment of 13,000. Each subsequent payment increases by 1,000 and payments cease immediately on death. Calculate the expected present value of each pension using the following basis: Mortality: A1967 70 Select Interest: 4% per annum [6] 7 Describe how nutrition and education affect mortality. [6] 8 A mortality investigation of pensioners who retired due to ill health is being undertaken to investigate if there are any initial temporary select effects. Data in respect of deceased pensioners are categorised as follows: x: age last birthday at death r: curtate number of years between retirement and death (i) (ii) Estimates of [y]+t are made by dividing the death data by its corresponding central exposed to risk. Derive the values of y and t in terms of x and r, stating clearly any assumptions you need to make. [3] The following data are also available in respect of one pensioner: Date of birth 1 August 1936 Date of retirement 1 November 1998 Date of death 1 July 2001 Calculate the contribution of this individual to each of the appropriate central exposed to risk figures corresponding to the available summary data. [3] [Total 6] 105 A2002 3 PLEASE TURN OVER

9 A life insurance company sells 4-year unit-linked endowment assurance contracts to males aged 61 exact. Premiums of 1,000 are payable annually in advance. Capital units are bought by the premium in the first year, and accumulation units are bought thereafter. 102% of each year s premium is invested in units at the offer price. There is a bid-offer spread in unit values, with the bid price being 95% of the offer price. Capital units bear a management charge of 5% per annum of the bid value of the fund, and this charge is deducted at the end of every year. The death benefit under the policy is paid at the end of the year of death, and is the full bid value of units under the policy, after deduction of relevant management charges. The pricing actuary assumes that fund growth will be 7.5% per annum and that mortality experience will follow A1967 70 Select. He is contemplating using part of the management charge for actuarial funding of capital units. The actuarial factor at duration t would be A [61] t:4 t at a rate of interest of 4% per annum, with mortality as above. Assuming non-unit fund growth is 5% per annum, and ignoring expenses, calculate the non-unit fund cash flow for the first year of the policy if: (a) the full value of capital units is held (b) only the actuarially funded value of capital units is held [6] 10 The following 3 state model is used to price various sickness policies. The forces of transition, and depend only on age. x H: healthy S: sick x x x D: dead The following probabilities are defined: ij xt, p is the probability that a life aged x in state i will be in state j at age x + t; ii xt, p is the probability that a life aged x in state i will remain in state i until age x + t; t ij x, z p is the probability that a life aged x in state i will be in state j at age x + t, having been in state j for period z. 105 A2002 4

Using these probabilities and / or forces of transition, write down an expression for the expected present value of each of the following sickness benefits for a life currently aged 35 and healthy. The constant force of interest is. (a) (b) (c) 1,000 per annum payable continuously while sick, but all benefits cease at age 65 1,000 per annum payable continuously while in the sick state for any continuous period in excess of a year. However, any benefit period is limited to 5 years payments, but the number of possible benefit periods is unlimited 1,000 per annum payable continuously throughout the first period of sickness only [6] 11 (i) A life insurance company prices endowment assurance policies allowing for mortality, expenses and interest. For surrenders, it wants to base the values it is prepared to pay on gross premium reserves, using the premium basis unchanged except for the interest rate. If it is to make a profit on surrenders, state in what direction it should change the interest rate element of the basis, if the reserves are: (a) (b) retrospective prospective Give reasons for your answers. You should assume that experience is the same as the premium basis. [4] (ii) A policyholder aged exactly 60 has 5 years remaining on his endowment policy which has a sum assured of 100,000 payable immediately on death, or maturity, whichever occurs first. He can no longer afford to pay any further premiums. He is offered a choice of: (a) a surrender value of 41,000 (b) a paid up sum assured of 54,000 (c) a whole life policy, without future premiums, with a death benefit, payable immediately on death, of 100,000 Show which he should choose, assuming he wants the one with the highest expected present value of benefits. Basis: Mortality: A1967 70 Ultimate Interest: 6% per annum [4] [Total 8] 105 A2002 5 PLEASE TURN OVER

12 A small employer decides to set up a pension scheme for his 2 employees, who are described by the following details: Age (exact) Past service (years) Expected salary in next year ( ) 30 5 25,000 35 6 20,000 The scheme will provide a pension of 1/60 th of pensionable salary for each year of service (fractions of a year counting proportionally) on retirement for any reason. Pensionable salary is the average annual salary earned in the final 36 months of employment. The employer meets the full cost of the scheme. The contribution rate is determined by equating the expected present value of the total scheme liabilities to the expected present value of contributions. Contributions are calculated to be a constant percentage of the total salaries of the members at any time. (i) (ii) Using the symbols defined in, and assumptions underlying, the Formulae and Tables for Actuarial Examinations, calculate the contribution rate required for the scheme. Ignore the possibility of new members joining the scheme. [8] Immediately after the scheme is set up, a new employee joins the company and pension scheme. She is aged exactly 40, and will earn 30,000 in the next year. The employer decides to maintain the contribution rate determined in part (i) and to apply it to the new total salaries. Determine whether the funding rate is sufficient to meet the liabilities of the extended membership. [3] [Total 11] 13 100 people aged exactly 50 are each sold a 15-year endowment assurance policy with sum assured 100,000. The premiums are paid annually in advance, and the sum assured is paid on maturity or at the end of the year of earlier death. The life insurance company s assumptions are: Mortality: Interest: A1967 70 Ultimate, and the lives are independent with respect to mortality 6% per annum Expenses: Initial: 300 Renewal: 2.5% of each premium, including the first Let P be the gross annual premium. (i) State the gross future loss random variable for one policy at the outset. [3] 105 A2002 6

(ii) Using your answer to part (i) or otherwise, evaluate, in terms of P, (a) (b) the mean and variance of the loss (in present value terms) for a single policy at outset the mean and variance of the loss (in present value terms) for the entire portfolio at outset. [7] Note: A at 12.36% per annum = 0.20426 50:15 (iii) Show what values the gross annual premium P can take if the company requires that the probability it incurs a loss (in present value terms) on the entire portfolio has to be less than 2.5%. Use the Normal approximation. [4] [Total 14] 14 A life insurance company issues a number of 3-year term assurance contracts to lives aged exactly 60. The sum assured under each contract is 200,000, payable at the end of the year of death. Premiums are payable annually in advance for the term of the policy, ceasing on earlier death. The company carries out profit tests for these contracts using the following assumptions: Initial expenses: 200 plus 35% of the first year s premium Renewal expenses: 25 plus 3% of the annual premium, incurred at the beginning of the second and subsequent years Mortality: Investment return: Risk discount rate: Reserves: A1967 70 Ultimate 7% per annum 15% per annum One year s office premium (i) (ii) (iii) Show that the office premium, to the nearest pound, is 4,396, if the net present value of the profit is 25% of the office premium. [10] Calculate the cash flows if the company held zero reserves throughout the contract, using the premium calculated in part (i). [2] Explain why the company might not hold reserves for this contract and the impact on profit if they did not hold any reserves. [3] [Total 15] 105 A2002 7

Page 1 Faculty of Actuaries Institute of Actuaries REPORT OF THE BOARD OF EXAMINERS ON THE EXAMINATIONS HELD IN April 2002 Subject 105 Actuarial Mathematics 1 Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. K Forman Chairman of the Board of Examiners 11 June 2002 Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report EXAMNINER S COMMENTS The overall standard of scripts was better than in recent sittings. However answers were very disappointing for questions 5, 10 and 13(iii) in particular, where the question posed a problem not seen in recent examinations. It is also clear that many candidates statistical knowledge or understanding is not up to the standard required. Finally candidates are urged to read the questions carefully. In many cases the answers for questions 5, 8, 9, 10 and 12 omitted elements asked for or added details not required for the question. Comments on individual questions follow after the solution to each question. Page 2

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report 1 3 4 q [40] 1 is the probability that a life now aged 41, who entered the population of interest a year ago subject to select mortality at that time, will survive for 3 more years, and die during the following 4, when aged between 44 and 48. = ( 3 p [40] 1 )( 4 q 44 ) for a 2 year select table l l = 44 [40] 1 l l l. 44 48 44 l l l = 44 48 [40] 1 = (33,309.271 32,934.221) / 33,484.739 = 0.0112006 Comment on Question 1 Well answered, with only a small minority of candidates mixing up the survival and death periods. 1 2 10 20 q 35:35 = (.5)( 10 20 q 35:35 ) =.5[ 10 p 35:35 {1 20 p 45:45 }] =.5[ 10 p 35 2 30 p 35 2 ] =.5[(e.2 ) 2 (e.6 ) 2 ] =.5(.67032.30119) =.1846 Comment on Question 2 Answers were generally of a reasonable standard. The commonest errors related to the factor of.5, and a range of errors in evaluating the required integrals. 3 Total fertility rates summarise the age specific fertility rates f x (i.e. the ratio of births to population of women aged x generating them). The summation is over all ages for which f x > 0, often taken as 15-49. Cohort: fertility rates are summed (over a period of time) for women born in a specified period e.g. all those born in the same calendar year Period: fertility rates are summed at a point of time (e.g. the rates experienced in one calendar year) for women of different ages Cohort rates are generally preferred for their greater stability and their smooth rate of change over time or Period rates are quicker and easier to obtain, and therefore suitable for immediate use Other sensible reasons also gained credit. Comment on Question 3 Good standard, although some candidates mixed up cohort and period rates while others provided formulae that dealt with numbers of births rather than fertility rates. Page 3

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report 4 Logistic model {1/P(t)}{dP(t)/dt} = kp(t) or also P(t) = [Ce t + (k/)] 1 or = /[Ce t + k] Current rate: k100,000 =.25 Limiting population: k300,000 = 0 leading to k = (.25/200,000) = 1/800,000 and =.375 (=3/8) We want t such that P(t) = 200,000 From t = 0 (now) 100,000 = [C + (1/300,000)] 1 so C = 2/300,000 or 0.00000666667 200,000 = [(2/300,000) e.375t + (1/300,000)] 1 (1/200,000) (1/300,000) = (2/300,000) e.375t e.375t = (1/4) so using logs t = 3.70 years Comment on Question 4 Overall standard was quite good, although a surprising number of candidates did not seem to know the logistic model at all. The commonest error was to use = 0.25. 5 Insurance company received P so guaranteed maturity benefit = [(1.05) 4 ] * P = 1.21550625P The company invests P @5.25% so is due to receive 1.2915479P in 5 years. On death, the office breaks even because it pays out exactly the value of asset available. This occurs with probability 4 q 56 = (1 [l 60 /l 56 ]) = 0.0690 At maturity (t = 4) office loses money only if yields at the time are j such that {1.2915479P / (1 + j)} < 1.21550625P i.e (1 + j) > 1.06256 Prob (1 + j > 1.06256) for lognormal (1 + j) = Prob (z > [Ln 1.06256 0.05] / 0.01) from standard normal = Prob (z >1.07) = 1.85769 = 0.14231 Maturity occurs with probability 1.0690 =.9310 so the overall probability of a loss is 0.9310 * 0.14231 = 0.1325 = 0.13 Comment on Question 5 Very poor standard of answers. Many made no reasonable attempt. Of those who did, some tried to calculate a surrender profit or loss, even though this was clearly zero. Many tried to calculate the value of the zero coupon bond at the end of 4 years (one year short of redemption) by considering the distribution of (1+i) 4 and accumulating rather than using the distribution of 1+i directly and discounting. Page 4

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report 6 Pension A EPV = (20,000/1.009709) (a + v 4 4 4 p [60] a 64 ) @ (1.04/1.009709) = 3% = (20,000/1.009709)[3.7171 + (.88849)(.947214)(11.962 1)] = 256,363 Pension B EPV = 12,000 a [60] + 1,000(Ia) [60] = 12,000 a [60] + (1,000S [60]+1 /D [60] ) = 12,000[12.710 1] +1,000 [307,254.58/2,815.3028] = 249,657 Comment on Question 6 Well answered. Common mistakes in A were not getting initial level correct (missing divisor of 1.009709) and using 4% interest for the deferred period until life annuity commences. In B, many evaluated (Ia) [60] using S [60]. 7 Nutrition influences morbidity and (in longer term) mortality. Lack of nutrition leads to general weakening of body Poor quality increases risk of disease / hinders recoveries. Excessive / inappropriate can lead to obesity and associated diseases (e.g. hypertension, heart disease). This can arise from social factors e.g. ready processed food / fast food etc. Poor / lack of nutrition can arise from adverse economic circumstances. Education (covering formal and also general awareness from public health campaigns). It influences awareness of elements of healthy lifestyle. This can affect behaviour in many areas e.g. nutrition / diet; personal health and hygiene; awareness of effects of tobacco, alcohol, drugs; Education level will also have a bearing on income level, occupation, standard of housing and general lifestyle, all of which are themselves regarded as influencers of mortality.. Other reasonable points also received credit. Comment on Question 7 Well answered overall. Some candidates were inclined to repeat the same point rather than identifying distinct influences on mortality. 8 (i) Age last birthday = x at start of rate interval in which dies Curtate duration = r at start of rate interval in which dies No assumptions needed x +.5 at mid-point of interval, with duration r +.5 so we are estimating [x+.5(r+.5)]+r+.5 = [xr]+r+.5 This does require an assumption of an even spread of retirements over the year of age, because we have no other information about ages at entry (we can only deduce that they can range from x r + 1 to x r 1). Page 5

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report (ii) Retired aged 62 years and 3 months. Dies aged 64 years and 11 months so total exposure is 2 years 8 months. (62,0) 9 months (63,0) 3 months (63,1) 9 months (64,1) 3 months (64,2) 8 months Comment on Question 8 Very poorly answered, especially part (ii). Some otherwise correct answers omitted assumptions completely while others gave standard assumptions e.g. policy anniversaries spread evenly over the year of age when there are no policies (only retirements). Overall, the understanding of the different rate intervals and the associated assumptions seems confused. In part(ii), many candidates calculated the total exposure incorrectly, including in some cases not even calculating the age at death correctly. 9 (a) Capital units no actuarial funding Year Cost of Fund at end Management Fund at end investment before m.c. Charge 1 969 1,041.67 52.08 989.59 Non-unit fund Year Premium less cost of allocation Interest Death cost Management charge Cashflow 1 31 1.55 0 52.08 84.63 (b) A funding factors A 0.85697 [61]:4 A 0.89045 [61] from (M 1:3 [61]+1 M 65 + D 65 ) / D [61]+1 = (1,337.8829 1,258.7316 + 2,144.1713) / 2,541.7641 Page 6

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report A funded capital unit fund Year Cost of investment Fund available at end Fund needed at end Management Charge 1 830.40 892.68 881.18 11.50 Non-unit fund Year Premium less cost of allocation Interest Death cost Management charge Cashflow 1 169.60 8.48 0.78 11.50 188.80 The death cost is q [61] *(full capital unit fund A funded capital unit fund @ t = 1) i.e. 0.00723057*(989.59-881.21) Comment on Question 9 Handled very well. Errors, where they occurred, were to include a death cost in (a), use of the wrong funding factor at t=1 and incorrect calculation of the death cost in (b). Some candidates completed a full profit test for each year of the contract, wasting valuable time. 10 (a) EPV = 1,000 (b) EPV = 1,000 or 30 0 t HS 35, t e p dt 6 HS t 0 1 t 35, z e p dzdt HH t 6 r e p 0 35t 35, e t 1 SS 35t, r 1,000 p drdt HH t r SS 0 35, t 0 (c) EPV = 1,000 e p 35 t e p35 t, rdrdt or HH t SS e p 35 a p 0 t r 0 35 t, r ( 35 t r 35, t 1,000 ) drdt 35tr Comment on Question 10 This was a testing question that was not answered well at all. Most attempted (a) but often got it wrong, while very few candidates made any real attempt at (b) and (c), even though (b) in particular just required direct use of a formula given in the appropriate Core Reading. Where an attempt was made at (b) or (c), candidates often used the benefit ceasing age for (a), although none applied in (b) or (c). Page 7

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report 11 (i) The retrospective and prospective reserves equal each other on the premium basis. We want the SV calculation to result in a lower reserve. Retrospective reserve needs to be done at a smaller interest rate, as it is accumulating past excess premiums over claims/ expenses. Prospectively, the interest rate needs to be higher than the premium basis, so that the discounting of the excess of future outgo (claims / expenses) over premium income results in a lower answer. (ii) SV = 41,000 PUPSA = 54,000 A 60:5 = 54,000((1.06).5 1 1 { A A } A ) where 60:5 60:5 60:5 1 A = v 5 60:5 5 p 60 = (.747258)(27,442.681/30,039.787) =.68265 EPV of PUPSA = 54,000(1.02956{.75477.68265} +.68265) = 40,873 Whole Life option 100,000 A 60 = (1.06).5 (.39136) = 40,293 So SV is best Comment on Question 11 Well answered. Those who got (i) wrong often wrestled with reserving formulae rather than considering the underlying concept needed. In part (ii), a surprisingly large number of candidates overvalued the paid up option by multiplying the entire endowment factor by 1.06 0.5 rather than just the death element. Page 8

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report 12 EPV of past pensions: (n/60)(sal)( z M x ia + z M x ra )/ s D x EPV future pensions: (1/60)(Sal)( z R ia x + z R ra x )/ s D x EPV of contributions @ 1% of salary: (.01)(Sal)( s N x )/ s D x age salary past service s D x z ia M x z ra M x z ia Rx z ra Rx s N x 30 25,000 5 28,043 8,636 88,345 231,941 2,915,486 540,020 35 20,000 6 22,276 8,513 88,345 188,977 2,473,760 417,224 EPV past pension EPV future pension EPV cont. 1%sal 7,204.78 46,764.89 4,814.21 8,696.18 39,844.63 3,745.95 Total 15,900.96 86,609.52 8,560.16 Total Liability = 15,900.96+86,609.52 = 102,510.48 Contribution rate needed = 102,510.48/8,560.16 = 11.98% of salary New employee Age salary past service s D x z ia Rx z ra Rx s N x 40 30,000 0 18,629 147,045 2,032,033 317,121 EPV past pension EPV future pension EPV cont. 1%sal 0.00 58,486.18 5,106.89 Contribution needed = 58,486.18 / 5,106.89 = 11.45%. Therefore the contribution rate of 11.98% established for the original 2 members is more than that required to meet the costs of the new entrant, and the scheme is in surplus. Comment on Question 12 Answered very well, but a disappointing number of candidates overlooked the ill-health retirement benefits. Page 9

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report 13 All values at t = 0 (i) Future loss random variable = min( K501,15) 100,000v 300.975Pa min( K501,15) = min( K501,15) 1 100,000v 300.975P v min( K501,15) d (ii) (a) Mean for single policy just take expected value of random variable X = EV one policy = 100,000 A + 300.975 P 50:15 = (100,000)(.44395) + 300 (.975)P(9.823) = 44,695 9.577425P a 50:15 Y = Variance one policy Variance = (using 2nd form of loss r.v.) min( K 1,15) = [100,000 + (.975P/d)] 2 Var ( v 50 2 2 = [100,000 + (.975P/d)] 2 ( A [ A ] ) 50:15 50:15 ) where the 2 superscript denotes at i 2 + 2i = [100,000 + (17.225P)] 2 (.007168397) or Standard Deviation = (100,000 + 17.225P)(.084666) (b) 100 policies Mean = 100X Variance = 100Y assuming the lives are independent or Standard Deviation = 10 Std Dev above Page 10

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report (iii) Using Central Limit Theorem (n = 100) we can assume normality of portfolio loss. We want Prob (loss > 0) <.025 Prob ([loss mean]/ Std Dev > [0 mean]/std Dev) <.025 Prob (z > mean/std Dev) <.025 This means that ( mean / Std Dev) > 1.96 or (mean/ std Dev) < 1.96 (100)(44,695 9.577425 P) < (1.96) (84,666 + 14.5837P) P 463,5445/929.158448 = 4,988.86 say 4,989 Comment on Question 13 A very mixed standard. It is clear some candidates do not have a good understanding of the difference between a random variable and its expectation, at least in this context. Common errors in (i) were to use assurance or life annuity functions, to miss the +1 in the K x +1 terms or to give a profit (rather than loss) random variable. In (ii), very few got the variance correct for a single policy, usually not making the conversion of the annuity into (1-v n )/d format used in the model solution, and then missing the cross-product or covariance term between the benefit and premium random variables. A surprising number of candidates missed the independence of lives within the portfolio and therefore concluded that the variance of the portfolio was 100 2 times the variance of one policy. In (iii), of the few candidates who attempted this part, many started with considering a loss < 0, when the loss has to be > 0 to be a loss. Page 11

Subject 105 (Actuarial Mathematics 1) April 2002 Examiners Report 14 (i) age q x p x t1 p x 60 0.01443246 0.98556754 1 61 0.01601356 0.98398644 0.98556754 62 0.01774972 0.98225028 0.96978510 Year Prem. Expense Opening reserve Interest Death claim Closing reserve Profit vector Profit signature NPV 1 P 0.35P +200 0 0.0455P 14 2,886.49 0.985568P 0.290068P 3,100.49 0.290068P 3,100.49 0.252233P 2,696.08 2 P 0.03P +25 P 0.1379P 1.75 3,202.71 0.983986P 1.123914P 3,229.46 1.107693P 3,182.85 0.837575P 2,406.69 3 P 0.03P +25 P 0.1379P 1.75 3,549.94 0 2.107900P 3,576.69 2.044210P 3,468.62 1.344101P 2,280.67 1.929443P 7,383.44 Therefore 1.929443P 7383.44 =.25P Premium = 4,396.36 = 4,396. (ii) (iii) If we use this premium, and ignore reserves, the cash-flows per policy in force at the start of each year are (43, 1,334, 986). As the cash flows in years 2 and 3 are all positive, there is no need to establish reserves at the end of any year. In such a scenario, the profits emerge earlier and because the discount rate exceeds the earned rate of interest, the NPV increases. Comment on Question 14 Answered well overall. The most common error was mishandling of reserves. A disappointing number of students started from a commutation function approach when a cashflow model was needed. In (iii), a number of candidates made the general statement that it was not necessary to hold reserves for term assurance contracts because the probability of death was low, without any reference to the specifics of the cashflows in this case. Page 12

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 10 September 2002 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination details as requested on the front of your answer booklet. 2. You must not start writing your answers in the booklet until instructed to do so by the supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 14 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, with any additional sheets firmly attached, and this question paper. In addition to this paper you should have available Actuarial Tables and your own electronic calculator. Faculty of Actuaries 105 S2002 Institute of Actuaries

1 Explain in words what is meant by the logistic model for projecting the size of a population. Write down a differential equation whose solution gives a formula for the size of a population based on the logistic model. Define carefully all symbols that you use. [3] 2 Describe how option pricing techniques may be used to determine the value of the guarantee under a deferred annuity policy with a guaranteed minimum annuity. [3] 3 Define each of the following terms and give an example of each in life assurance business: (a) class selection (b) spurious selection (c) adverse selection [3] 4 Explain what is meant by the following, in the context of with profit life insurance contracts: (a) earned asset share (b) retrospective valuation reserve [4] 5 A life insurance company issues a whole life assurance policy to a life aged exactly 60, paying a sum assured S, together with attaching bonuses, immediately on death. Compound bonuses are added annually in advance. Premiums under the policy are payable annually in advance, ceasing at exact age 85 (the last premium is payable at age 84 exact) or on earlier death. Write down an expression for the net future loss random variable at outset for this policy. Define carefully all the symbols that you use. [4] 6 A life insurance company issues a temporary annuity policy to two independent lives, each aged exactly 60. The annuity of 10,000 per annum is payable quarterly in arrears, while at least one of the lives is alive. The annuity is payable for a maximum of 10 years. Calculate the single premium. Basis: mortality: A1967 70 Ultimate interest: 4% per annum expenses: ignore [4] 105 S2002 2

7 Two lives, (x) and (y), are assumed to be independent with respect to mortality and are each assumed to be subject to a constant force of mortality of 0.01. Calculate the probability that (x) dies more than 10 years after (y). [4] 8 Calculate 2 A [20]:[20] using A1967 70 mortality and interest of 4% per annum. [4] 9 Members of a pension scheme are subject to two modes of decrement namely death (d) and withdrawal (w). The following assumptions are made in respect of the two decrements: Independent rate d qx is A1967 70 Ultimate; w x Independent rate q is 0.05 per annum at age 20 last birthday and increases by 5% at each successive age attained. (For example, the annual rate of withdrawal at age 20 + t last birthday is 0.05 1.05 t ); the decrements are statistically independent; each decrement is uniformly distributed in its single decrement table. Calculate the probability that a new entrant aged exactly 20 will withdraw from the scheme at age 22 last birthday. [5] 10 (i) Define, giving a formula, the term Standardised Mortality Ratio. Define all the symbols that you use. [2] (ii) Show how the Standardised Mortality Ratio may be expressed as a weighted average, setting out clearly what function is averaged and what the weights are. [3] [Total 5] 105 S2002 3 PLEASE TURN OVER

11 A pension scheme provides the following deferred benefits for a member aged 55 exact, who leaves service before Normal Pension Age (NPA), which is age 65 exact. (a) (b) A deferred pension of 10,000 per annum, payable monthly in advance from NPA. The pension is payable for a minimum of 60 monthly payments. The pension increases monthly in deferment and payment at the effective rate of 3.846% per annum compound. On the death of the member after NPA, a dependant s pension of 50% of the member s pension entitlement at the date of death. The pension is payable monthly in advance beginning on the first day of the month following the date of the member s death, or the fifth anniversary of the member s NPA, if later and increases monthly in payment at the effective rate of 3.846% per annum compound. Calculate the expected present value of the deferred benefits. Basis: Mortality: A1967 70 Ultimate Interest: 8% per annum Proportion with dependants: 90% of members have dependants at the date of retirement Age difference: members are the same age as their dependants (assume that females are treated exactly the same as males) Expenses: none [11] 12 A life insurance company issues a disability insurance contract to a healthy life aged exactly 30. Under the contract, a benefit of 20,000 per annum is payable weekly in the event of disability. The benefit continues to be payable during disability, until the policyholder recovers or reaches age 65. The benefit increases continuously in payment at the rate of 3% per annum compound. There is no waiting period or deferred period. Premiums continue to be payable during periods of disability. Disability benefit payments are valued using rates of claim inception and termination. (i) (ii) Describe the method of valuing disability benefit payments under this contract, setting out the data required. [6] Derive commutation functions for valuing the benefits payable under the contract, stating clearly any assumptions that you make and defining carefully all the symbols that you use. [7] [Total 13] 105 S2002 4

13 A life insurance company issues a long-term care contract to a healthy life aged 50 exact. Under the contract, the life insurance company will pay the costs of long-term care while the policyholder satisfies the conditions for payment. The conditions for payment are assessed each year on the policy anniversary, just before payment of the premium then due. If the policyholder satisfies the conditions, the full annual amount of the benefit payable is paid immediately. Regular premiums are payable annually in advance under the policy until death and are waived during periods of benefit payment. For those lives needing care at 100% of maximum, the current payment on the policy anniversary is 50,000. The company uses the following data in respect of the expected proportions of lives at each age needing care at different expected cost levels, for pricing the long-term care contract. Exact age Proportion needing care at 50% of maximum Proportion needing care at 100% of maximum 51 70 0.01 0.01 71 85 0.04 0.06 86+ 0.08 0.10 Basis: Mortality: Interest: Benefit inflation: Expenses: A1967 70 Ultimate 6% per annum Maximum payment at 100% care level at policy anniversary t(t = 1, 2,..) = 50,000 (1.019231) t 10% of each premium (i) Write down an expression for the expected present value of benefits (including the waiver of premium benefit) at outset for the contract. Define carefully all the symbols that you use. [4] (ii) Calculate the annual premium payable under the contract. [10] [Total 14] 105 S2002 5 PLEASE TURN OVER

14 A life insurance company issues a 2-year unit-linked endowment assurance contract to a male life aged exactly 63, under which level annual premiums of 6,000 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 the second year s premium 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% of the bid value of capital units and 1% of the bid value of 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 10,000, or the full bid value of the units allocated to the policy, if greater. On maturity, the full bid value of the units is payable. The policyholder may surrender the policy only at the end of the first policy year. The surrender value is equal to 87% of the bid value of the capital units. The life insurance company uses the following assumptions in carrying out profit tests of this contract: Mortality: A1967 70 Ultimate Surrender: 10% of policies then in force, occurring at the end of the first policy year Expenses: Initial: 500 Renewal: 100 at the start of the second policy year Unit fund growth rate: 8% per annum Non unit fund interest rate: 4% per annum Risk discount rate: 15% per annum (i) Calculate the net present value on this contract, assuming that the company holds unit reserves equal to the full bid value of the accumulation units and capital units. [12] (ii) Assume that the company holds unit reserves equal to the full bid value of the accumulation units and a proportion, A 63 t:2 (calculated at 4% and t A1967 70 Ultimate mortality), of the full bid value of the capital units (t = 0, 1). Calculate the net present value on the contract. [9] (iii) Explain what the effect would be on the answers in parts (i) and (ii) if the mortality assumption were changed to mortality of A1967 70 Select. [2] [Total 23] 105 S2002 6

Faculty of Actuaries Institute of Actuaries EXAMINATIONS September 2002 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. K G Forman Chairman of the Board of Examiners 12 November 2002 Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report EXAMINERS COMMENTS The overall standard of attempts by candidates was high. A number of questions were answered very well. The more challenging questions were less well answered, such as Questions 2,6,7, 12 and 13, with evidence of lack of proper preparation. A common mistake, which was also a feature of previous examinations, was to misread some of the questions. Detailed comments are given after the solution to each question. Page 2

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report 1 A logistic model for projecting the size of a population is a model under which an initial rate of growth for the population is assumed to decrease over time in proportion to the size of the population. The model may be expressed in the form 1 dp t P t dt kp t, where P(t) is the size of the population at time t is the rate of growth, a constant, >0 k is a constant, >0. Full credit was given for the solution set out. However, a fuller treatment would be to assume that the initial rate of growth,, may also be negative, in which case may be assumed to either increase or decrease over time in proportion to the size of the population. Few candidates gave this fuller treatment. The question was answered well in general. A number of candidates were vague in the definition, omitting the point that the initial rate of growth decreased over time in proportion to the size of the population. 2 A guaranteed annuity rate corresponds to a call option on the bonds that would be necessary to ensure that the guarantee was met, i.e. at an exercise price that generated the required fixed rate of return. Alternatively, it can be modelled by an option to swap floating rate returns at the option date for fixed rate returns sufficient to meet the guaranteed option. It is difficult to ensure that the whole investment fund corresponds to a single option traded in the market. An approximation is possible by using options written on indices. At the date of policy issue, all guarantees will be out of the money, i.e. they will have no intrinsic value because current market rates are more than sufficient to meet the guarantees, but will have a time value that is the result of the views of many investors ( the market ) of the present value of the likely future costs of the option. Thus the market price of a suitable option produces a way of pricing the guaranteed annuity rates. This question was poorly answered. Many candidates described other techniques rather than option pricing techniques. 3 (a) Class selection is the process whereby lives are divided into separate groups, within which mortality or morbidity is homogenous, where each group is Page 3

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report specified by a category or class of a particular characteristic of the population. An example in life assurance business is the use of individual rating factors which produce mortality differences, e.g. smoking status. (b) (c) Spurious selection is the process whereby lives are divided into separate groups, within which mortality or morbidity is homogenous, where the differences in mortality or morbidity are due to factors other than those used to form the groups. An example is a change in underwriting over time leading to mortality improvements, where such improvements are ascribed to the passage of time. Adverse selection is the process whereby lives are divided into groups that tend to act against a controlled selection process imposed on the groups, in respect of mortality or morbidity. An example is where smokers will tend to select policies from a life office that does not use smoking status as a rating factor. This question was generally well answered. Credit was given for all reasonable examples. Some candidates confused self-selection with adverse selection in part (c). 4 (a) The asset share for a with profit contract is the accumulation of premiums less deductions associated with the contract plus an allocation of profits on nonprofits business, all accumulated at the actual rate of return earned on investments. The deductions include expenses, claims, cost of capital and transfer to shareholder funds, if relevant and are based on actual experience. (b) The retrospective valuation reserve is the expected accumulation of past premiums received, less expected expenses and benefits including any reversionary or interim and terminal bonuses included in past claims. This question was well answered. Some candidates omitted the allocation from non-profit business in part (a). 5 The net future loss random variable is given by K 60 1 T 1 60 S b v Pa. min( K 1,25) b is the annual rate of future bonus 60 K60, T 60 are the curtate and complete future lifetimes of a life aged 60 P is the annual premium This question was well answered. A common error was the inclusion of K 60 rather than K 1. 60 Page 4

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report 6 The required single premium is given by 4 4 4 10000a 10000 2a a 60:60:10 60:10 60:60:10 4 3 D70 3 60 70 a a a 60:10 8 D 8 60 11.551 1516.9972 2855.5942 7.957 7.500 3 3 8 8 a a a 60:60:10 1039.0172 8.943 5.498 2487.2117 6.865 4 3 D70:70 3 60:60 70:70 8 D 8 60:60 3 3 8 8 the single premium is 10000* 2 *7.500 6.865 81, 350 This question was relatively poorly answered. Many candidates struggled with the correct evaluation of the annuity factors. 7 The required probability is p pdt, where p exp 0.01ds t 0 t yyt t10 x t x 0 0.01t 0.1 0.01t e *0.01* e dt 0 0.02t 0.1 e 0.01* e * 0.02 0.1 0.5* e 0.45242 0 This question was answered less well than the examiners had expected, with many candidates setting out the initial integral expression incorrectly. Page 5

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report 2 1 8 A[20]:[20] A[20] A[20]:[20] A A 20 1 [20] A 2 [20]:[20] A 0.13312 1 da 20: 20 20: 20 = 1 0.038462*21.509 = 0.17272 2 A [20]:[20] 0.13312 0.5*0.17272 = 0.04676 This question was well answered by most candidates. 9 Independent rates of withdrawal: Age attained Rate 20 0.05 21 0.0525 22 0.055125 Probability of survival to age 22 = l22 34029.283 0.95*0.9475* 0.95*0.9475* 0.898568 l 34088.257 20 22 22 1 2 22 2 w w 1 d 1 aq q q 0.055125* 1 * 0.00079739 0. 055103 Required probability = 0.898568*0.055103 = 0.049514 This question was well answered. Some candidates used w q 22 in place of w aq 22. 10 (i) The Standardised Mortality Ratio is the ratio of the actual deaths in a population compared with the expected deaths, based on standard mortality rates. The formula is x x E E c xt, mxt, c s xt, mxt,, where E c xt, is the central exposed to risk in the population between ages x and x + t Page 6

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report m xt, is the central rate of mortality for the population between ages x and x + t s m is the central rate of mortality for a standard population between ages x xt, and x + t (ii) The Ratio may be written in the form x E c s xt, mxt, s x E m c s xt, mxt, xt, m xt, which is the weighted average of the age-specific mortality differentials between the population being studied and the standard population. i.e. m s xt, m xt, weighted by the expected deaths in the population being studied based on standard mortality. c s i.e. Ext, m, xt Part (i) was well answered in general. A common error was not basing expected deaths on the mortality of a standard population. Part (ii) was very poorly answered, with few candidates obtaining full marks. 11 (a) The net rate of interest is 4% per annum in deferment and payment. The value of the deferred pension in payment is 10000* D65 * 12 D70 12 5 * a a70 D55 D65 Page 7

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report D D a a 65 55 12 5 D D 70 65 12 70 2144.1713 0.585109 3664.5684 i * a 1.021537* 4.4518 4.5477 (12) 5 d 1516.9972 0.707498 2144.1713 8.957 0.458 8.499 Required value = 10000*0.585109*(4.5477+(0.707498*8.499)) = 61792 (b) The value of the pension on death in retirement is D 0.9*0.5*10000* D a l l (12) 70 70 70 65 a Required value = 70 70 a 70 65 55 a * D D 70:70 70 65 l * l 23622.102 0.8607797 27442.681 70 65 * a (12) 70 70 l 1 l 70 65 7.957 5.498 2.459 * a 0.9*0.5*10000*0.585109* 0.707498* 0.8607797*2.459 10.8607797 *8.499 = 6147 Total value = 67,939 Attempts at valuing the benefits in part (a) were reasonable in general. Attempts at valuing the benefits in part (b) were very poor. 12 70 12 (i) The approach uses two double decrement tables. One table relates to healthy policyholders and decrements of falling sick and dying. Recovery and subsequent rates of sickness are allowed for in the table. The table is used to calculate probabilities of surviving to be a healthy policyholder at age 30+t, al30 al t 0 t 35. The table is also used to calculate the dependent initial 30 rate of falling sick at age 30+t, aq 30 t, 0 t 35. The rate aq 30 t is called the inception rate for disability. The second table relates to policyholders receiving disability benefits and has decrements of recovery from disability and dying (while disabled). The survival probabilities from this double decrement table are used, together with an appropriate interest rate, to determine the present value at the date of becoming disabled of a disability annuity of 20,000 per annum, increasing in payment continuously at the rate of 3% per annum compound and payable Page 8

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report according to the policy conditions until the policyholder dies while disabled, recovers or reaches age 65. The probabilities of surviving as a healthy policyholder to age 30+t, the inception rates for disability at age 30+t and the disability annuity payable from age 30+t are calculated for each value of t and are integrated or summed over the range 0 t 35. The data required are those set out in the two multiple decrement tables above, for ages from 30 to 65. (ii) The value of a disability benefit of 1 p.a. payable weekly to a healthy life now aged 30 exact is t35 t0 al al, a al 30t 30 i 30t 30t i t i 20000* 30t a v a dt, where 30t are based on a double decrement table for healthy policyholders described in part (i) above and gives the number surviving to age 30 + t while healthy and the force of inception of disability at age 30 + t. a30 i t is a continuous annuity based on the second double decrement table described above, evaluated at rate of interest i, where 1 i i 1 1.03 Assume that lives becoming disabled in 30 t,30 t 1 do so on average at age 30 t 1 and the integral is approximated by i ad 30 t35 t 2 i 30 t 1 2 i ad v 20000* a30 1 30t t t0 30 al v 30 2, where is the number of lives becoming disabled at age 30 + t last birthday in the first decrement table described in part (i) above. Page 9

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report Define commutation functions as follows 1 2 ia i 30 i 30 t 30t 30 t C ad v a 30 30 30 t ia ia 30 C30 t t0 D al v M So the value is approximated by M 20000* D ia 30 30 Poorly answered in general. The evidence was that many candidates were not well prepared for this topic, with some candidates not attempting the question and others giving very poor answers. For a well-prepared candidate, the question should have been relatively straightforward and a small number of candidates did achieve high marks. 1 2 13 (i) The expected present value of benefits at outset is given by 50 c t 4% 2 6% 2 D 50t c c D 50t c I50tL P I50t t1 D50 c1 t1 D50 c1 using A1967-70 Ultimate, I is the proportion of policyholders needing care at exact age 50+t, at benefit level c 1, 2 Benefit levels: c 1 is the benefit level at 50% of maximum c 2 is the benefit level at 100% of maximum 1 2 L 25, 000, L 50, 000 P is the annual premium Page 10

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report (ii) Exact age Proportion needing care at 50% of maximum Proportion needing care at 100% of maximum Total proportion Expected payment per current life 51-70 0.01 0.01 0.02 750 71-85 0.04 0.06 0.10 4000 86+ 0.08 0.10 0.18 7000 Present Value of Long Term Care Benefits (all calculated at 4% interest): Exact age (x to x+ t) D 50 = 4597.0607 Expected payment per current life A ( N N )/ D x x t 1 50 Value B A*B 51-70 750 12.37730 9282.98 71-85 4000 2.44004 9760.16 86+ 7000 0.18574 1300.18 Total Present Value 20343.32 Present value of Waiver Payments (all calculated at 6% interest) Present value is = P*{(0.02*(D 51 +D 52 +.+D 70 )+0.1*(D 71 +D 72 +.+D 85 ) +0.18*(D 86 +D 87 + )}/D 50 This can be rewritten as: PV = P*{0.02*a 50 +0.08*v 20 *(l 70 /l 50 )*a 70 +0.08*v 35 *(l 85 /l 50 )*a 85 } = P*(0.02*12.120+0.08*0.31180*0.723055*7.018 +0.08*0.13011*0.20712*3.297) = 0.376084*P So the final premium equation allowing for expenses is: 0.9*P* a 50 (at 6%) = 20343.32+0.376084*P i.e. P = 20343.32/(0.9*13.1200.376084) i.e. P = 1,779.52 The examiners expected this to be a challenging question. It required the application of basic actuarial techniques to pricing a product that was probably relatively unfamiliar to most candidates. A number of candidates performed very well, achieving full, or near to full, Page 11

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report marks. However, the majority of candidates performed poorly, with part (i) being better answered than part (ii). 14 (i) Multiple decrement table x d q x s qx al x ad d x ad s x 63 0.01965464 0.1 100000 1965.464 9803.454 64 0.0217431 0 88231.08 1918.417 0 65 86312.67 Unit Fund Year, t 1 2 Value of Capital units at start 0 5965.164 Premium to Capital units 5814 0 Interest on Capital units 465.120 477.213 Management charge on CUs 313.956 322.119 Value of Capital units at end 5965.164 6120.258 Value of Accumulation units at start 0 0 Premium to Accumulation units 0 5814 Interest on Accumulation units 0 465.120 Management charge on Aus 0 62.791 Value of Accumulation Units at end 0 6216.329 Total value of units 5965.164 12336.587 Surrender value 5189.693 0 Non-unit Fund Unallocated premium 186 186 Expenses 500 100 Interest -12.56 3.44 MC on Capital units 313.956 322.119 MC on Accumulation units 0 62.791 Surrender profit 76.023 0 Extra death benefit 79.303 0 End of year cash flow -15.884 474.350 Probability in force 1 0.882311 Profit signature -15.884 418.52 Discount factor 0.869565 0.756144 Expected present value -13.812 316.465 Net present value 302.65 Page 12

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report (ii) Unit Fund Year, t 1 2 Actuarial funding factor 0.92528 0.96154 Value of Capital units at start 0 5735.744 Premium to Capital units 5379.578 0 Interest on Capital units 430.366 458.860 Management charge on CUs 290.497 309.730 Value of Capital units at end 5519.447 5884.873 Total value of units 5519.447 12101.202* Surrender value 5189.693 0 *including accumulation units Non-unit Fund Unallocated premium 620.422 186 Expenses 500 100 Interest 4.817 3.440 Surrender profit 32.327 0 Extra death benefit 88.064 5.118 MC on Capital units less cost of additional allocation 99.656 79.463 MC on Accumulation units 0 62.791 End of year cash flow 169.158 226.576 Probability in force 1 0.882311 Profit signature 169.158 199.91 Discount factor 0.869565 0.756144 Expected present value 147.094 151.161 Net present value 298.25 (iii) If A1967-70 Select mortality were used in the profit tests instead of A1967-70 Ultimate mortality, the cost of the extra death benefit would decrease and, separately, the profit signature would increase. The effect of these two factors would be to increase the net present value of profit in part (i) and part (ii). Part (i) was generally well answered. A surprising number of candidates calculated the probability of being in force for year 2 incorrectly. A number of candidates achieved full marks for part (ii). However, in general this part was less well answered than part (i). Common errors were the incorrect calculation of the surrender profit, extra death benefit and management charge on capital units less the charge of additional allocation. Page 13

Subject 105 (Actuarial Mathematics 1) September 2002 Examiners Report Part (iii) was well answered. A number of candidates described the effect of changing the mortality basis from Ultimate to Select in the actuarial funding factors, in addition to the two factors given in the solution. Full credit was also given for this approach. Page 14

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 9 April 2003 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination details as requested on the front of your answer booklet. 2. You must not start writing your answers in the booklet until instructed to do so by the supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 14 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, with any additional sheets firmly attached, and this question paper. In addition to this paper you should have available Actuarial Tables and your own electronic calculator. Faculty of Actuaries 105 A2003 Institute of Actuaries

1 In the context of Manchester Unity sickness functions, state the relationship between 26 26 26 26 s 26 and z 26. [2] 2 (i) In the context of with profit policies, describe the super compound method of adding bonuses. [2] (ii) Give a reason why a life insurance company might use the super compound method of adding bonuses as opposed to the compound method. [1] [Total 3] 3 Under a policy issued by a life insurance company, the benefit payable on death, at the end of the year of death, is a return of premiums paid without interest. A level premium of 1,500 is payable annually in advance, throughout the term of the policy. For a policy in force at the start of the tenth year, you are given the following information: Reserve at the start of the year, 9 V: 11,300 Reserve at the end of the year per survivor, 10V: 13,200 Probability of death during the year: 0.04 Rate of interest earned: 5% p.a. Calculate the profit expected to emerge at the end of the tenth year per policy in force at the start of that year. Ignore expenses and all decrements other than death. [3] 4 Compare the use of the component method and the logistic mathematical modelling method for projecting the size of the population in a certain country. [4] 5 A researcher into international mortality experience is interested in comparing death rates in different countries by cause of death (cancer, heart disease, accidents etc.). An initial study compares crude death rates by cause of death for each country, and indicates a wide range of experience among the different countries. (i) (ii) Comment on the approach of using crude rates for this comparison, indicating any advantages and disadvantages of this method. [2] Suggest an alternative approach which addresses any shortcomings identified in (i). You should assume any data required are available. [2] [Total 4] 105 A2003 2

6 A life insurance company sells disability insurance contracts, under which the benefit is 100 per week, payable while a life insured is alive, disabled and aged not more than 65. It calculates premiums and reserves using the inception rate / disability annuity methodology. Calculate the expected present value of future benefit payments for the following two policyholders: (a) (b) A 45 year old who is healthy at the valuation date, and whose policy has a deferred period of one year. The value should take into account all possible future periods of sickness claims. A 55 year old who has been receiving benefit payments for the last two years. The value should allow only for the remaining payments under the current sickness claim. Basis: Interest: 6% per annum Morbidity & Mortality: S(ID) in the Actuarial Tables [4] 7 A life insurance company issues a reversionary annuity policy to a male and a female, both of whom are aged exactly 60. The annuity commences immediately on the death of the first of the lives to die and is payable subsequently while the second life is alive, for a maximum period of 20 years after the commencement date of the policy. The annual amount of the annuity is 10,000 and is payable continuously. Calculate the single premium for the policy. Basis: Mortality: Interest: PMA92C20 for the male life and PFA92C20 for the female life. The lives are independent with respect to mortality. 4% per annum Expenses: Initial: 300 incurred at the outset Annuity: 2% per annum of the annuity payment, incurred continuously while the annuity is being paid [7] 105 A2003 3 PLEASE TURN OVER

8 An insurer sells a special 3-year single premium non-profit term assurance policy for an initial sum assured of 250,000. This policy includes an option such that the policyholder can double the sum assured at the end of the second year of the policy by paying an additional premium at that time, based on normal mortality rates, without evidence of health. All death benefits are payable immediately on death. The company uses the North American method for pricing this policy. Calculate the premiums payable by a female life aged exactly 55 at the outset who does take up the option. Basis: Normal mortality: ELT15 (Females) Mortality of those who exercise the option: Interest: Expenses: Proportion of policyholders who exercise the option: 300% of ELT15 (Females) 5% per annum None 40% of those alive on the second policy anniversary [8] 9 (i) Explain why a life insurance company might need to set up non-unit reserves in relation to a unit-linked assurance contract. [3] (ii) A ten-year contract has the following profit signature before non-unit reserves are set up: (1, 0, +1, -2, +1, +1, 0, 1, 0, +1) If positive non-unit reserves are set up to zeroise negative cash flows, write down the revised profit signature. You should ignore interest. [2] (iii) State the advantages of cash flow techniques for product pricing compared with traditional commutation functions. [3] [Total 8] 105 A2003 4

10 A member of a pension scheme is aged exactly 40, having joined the scheme at age exactly 22. He earned 30,000 in the immediately preceding 12 months. Final pensionable salary is defined as the annual average earnings over the three years immediately prior to retirement. Normal Retirement Age is a member s 65th birthday. Using the functions and symbols defined in, and assumptions underlying, the Example Pension Scheme Table in the Actuarial Tables, calculate the expected present value of each of the following: (i) A pension on ill-health retirement of two-thirds of final pensionable salary. [3] (ii) A pension on retirement at any stage on grounds other than ill-health of oneeightieth of final pensionable salary for each year of service (fractions of a year counting proportionately), subject to a maximum of 40 years. [3] (iii) A lump sum on retirement at any age for any reason of 50,000. [3] [Total 9] 11 In a select mortality investigation, tabulations of in force populations are available for a certain class of business, in the following 2 ways: On each of 1 January 2000, 2001 and 2002, P x,t is available where x and t are defined as: Method x t A Age last birthday Curtate duration B Age next birthday at issue plus calendar year of census minus calendar year of issue Duration at policy anniversary during year of census Two different tabulations of deaths in each of the years 2000 2002 are also available, y,r where y and r are defined as: Method y r 1 Age last birthday at policy anniversary prior to death death 2 Age last birthday at death Duration at policy anniversary following Curtate duration at death These data are to be used to estimate select forces of mortality. For each tabulation of deaths: (i) (ii) Determine the ages and durations to which these estimates apply, stating all assumptions you make. [6] Indicate which of the tabulations of census data gives the best match to each of the tabulations of deaths and write down an appropriate approximation to the required exposed to risk. State all assumptions you make. [4] [Total 10] 105 A2003 5 PLEASE TURN OVER

12 You are the actuary of a life insurance company which issued 5,000 with-profit endowment assurance policies to lives then aged exactly 40 on 1 January 2002. Each policy had an original sum assured of 100,000 and a term of 20 years, with annual premiums of 4,300 payable in advance throughout the term, ceasing on earlier death or discontinuance. You are given the following information, most but not all of which is needed to calculate asset shares: The office holds net premium prospective reserves for in force policies based on AM92 Ultimate mortality and 4% per annum interest. On death, policies receive the original sum assured plus previously declared reversionary bonuses and any applicable terminal bonuses. The claim payment is made at the end of the calendar year of death. On discontinuance within the first two years, policies receive a surrender value equal to 25% of premiums paid. The surrender value is payable at the end of the calendar year of discontinuance. On 31 December 2002, the office declared a reversionary bonus of 2% of the original sum assured for all policies fully in force on that date (i.e. not including any policies terminating during 2002 for reason of death or surrender). On 31 December 2002, the office also declared a terminal bonus for death claims which arose in the previous 12 months whereby the total death benefit payable is 125% of the original sum assured plus 125% of any attaching reversionary bonuses. Expenses incurred were 15.0 million on 1 January 2002. During 2002, 4 policyholders died and 200 discontinued. The office earned interest of 6.5% on its assets during 2002. The company uses actual death claims when calculating asset shares and ignores all other factors affecting profit or expenses not given above. (i) Calculate the asset share per in force policy on 31 December 2002. [7] (ii) State with reasons which information given is not required for your calculation in (i). [3] [Total 10] 105 A2003 6

13 (i) Describe the benefit whose present value is shown below. T x and T y are the complete future lifetimes of two lives aged x and y respectively: T ( ) 100,000 y g T v if Tx T gt ( ) 0 otherwise y [2] (ii) The policy in (i) was originally paid for by a single premium at outset. The policyholders, who are both still alive, now request that the benefit be modified immediately to be paid on the earlier death of either life. Calculate the level premium payable annually in advance from now until the first death of either life if the policy is amended in the manner requested. Basis: Mortality: (x) subject to force of mortality of 0.02 (y) subject to force of mortality of 0.03 (x), (y) independent with respect to mortality Interest: force of interest of 0.04 Renewal expenses: 2.5% of all premiums payable from the alteration date Alteration expenses: 100 [8] (iii) State, with reasons, any actions the life insurance company should undertake before proceeding with the alteration described in (ii). [2] [Total 12] 105 A2003 7 PLEASE TURN OVER

14 A life insurance company sells 4-year decreasing term assurance policies, with level premiums payable annually in advance for the term of the policy, but ceasing on earlier death. The initial sum assured is 200,000 decreasing by 50,000 at each policy anniversary and the death benefit is payable at the end of the year of death. The company allows for the following when calculating premiums: Initial expenses: Renewal expenses: Mortality: Interest: 300 plus 25% of the annual premium 30 per annum plus 2.5% of annual premium, incurred at the time of payment of the second and subsequent premiums AM92 Select 4% per annum (for all rates needed) For a male aged exactly 60 at outset: (i) (ii) (iii) (iv) Write down the gross future loss random variable at the outset of the policy. [3] Calculate the office premium using commutation functions, setting the expected value of the gross future loss random variable to zero. [4] Calculate the office premium using a discounted cash flow projection, assuming no withdrawals, ignoring reserves and using the same profit criterion as in (ii). [6] Without further calculation explain the effect of: (a) (b) allowing for the setting up of reserves in the calculation in part (iii) having set up the reserves in (iv)(a), increasing the discount rate to 10% per annum [3] [Total 16] 105 A2003 8

Faculty of Actuaries Institute of Actuaries REPORT OF THE BOARD OF EXAMINERS April 2003 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. J Curtis Chairman of the Board of Examiners 3 June 2003 Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report Overall Comments The standard this year was generally good, slightly improved from last year. Candidates seemed to cope well with the new areas that were examined this time (mainly questions 6 and 12). However, the following areas continue to prove the most difficult for candidates:- estimation of select forces of mortality (question 11), mortality options (question 8), and contingent assurances / reversionary annuities (question 7 and 13(ii)), despite the questions asked being very standard for these topics. Comments for individual questions follow after each question which we hope will assist students. 1 26 26 26 26 1 26 26 26 26 0 t 26 26 0.5 26 s z p dt ( z p ) Well answered. The main error, if one was present, was to confuse the exact and approximate relationships. 2 (i) The super compound bonus method is a method of allocating annual bonuses under which two bonus rates are declared each year. The first rate, usually the lower, is applied to the basic sum assured and the second rate is applied to the bonuses added in the past. (ii) The sum assured and bonuses increase more slowly than under other methods for the same ultimate benefit, enabling the office to retain surplus for longer and thereby providing greater investment freedom. This method also rewards longer standing policyholders and discourages surrenders, relative to other methods. Very well answered overall. In part (ii), other reasons, where valid, were accepted. 3 The death benefit in year 10 is 15,000 Profit emerging per policy in force at the start of the year is: ([ 9 V + P]*1.05) (15,000*0.04) ([1 0.04]* 10 V) = ([11,300 + 1,500]*1.05) (15,000*0.04) (0.96*13,200) = 168 Well answered. Two common errors recurred, using a wrong death benefit (usually nine times the premium) and omitting the survival probability of 0.96 for closing reserves. 4 The component method builds up recursively year on year, allowing explicitly for each of the 3 key elements: births, deaths and net emigration. Each of these can be modelled separately to incorporate changing trends, although to do so relies on detailed data and / or assumptions, usually split by age and sex. Page 2

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report The logistic model is easy to apply, but is restricted in the variation it can allow for a population, relying on 2 parameters which give a limiting population and an initial growth rate, which reduces as population increases. The model does not lend itself to understanding mechanisms of population changes. In reality, growth varies over time in a different manner and most recent projections using the logistic and similar models have tended to overestimate the population. Also well answered. Occasionally candidates gave extremely lengthy and detailed descriptions of the two methods, too much for the marks available, while at the same time overlooked the comparison of the two approaches, which was the main thrust of the question asked. 5 (i) Crude rates are easily calculated, relying only on total population at risk and total deaths for each cause of death in this case. However, the relative results for different countries can vary widely if the death rate for a certain cause of death (a) varies by age as most do and (b) population structures vary by age between countries. Differences in the crude rates for a cause of death would then be confounded with differences in population structures. (ii) The rates could be standardised. Direct standardisation is best, whereby each countries actual age-specific death rates are applied to a common population. Any reasonable standard population could be chosen, but where possible it should have some relevance to the study e.g. a European study could standardise according to the population in Europe sorted by age as follows: Exm x Directly standardised death rate for cause A for a given country = s E s c A x Where s Ex c is the central exposed to risk at age x in the standard population A and mx is the central mortality rate from cause A at age x in the country in question. Generally very well answered, especially part (i). While many candidates did not relate their answers to the specific question which concerned a cause of death study and wrote about mortality rates generally, this was accepted by the examiners. In part (ii), alternative suggestions were also accepted, where justified. x c x 6 (a) (0.242488)(100)(52.18) using HS(1/ all) 45 a = 1,265.30 SS (b) (5.4952)(100)(52.18) using a 55,2 = 28,673.95 Very well answered. The only common error was the omission of the 52.18 factor. Candidates seemed clearly familiar with the new examination tables. Page 3

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report 7 Value 10,000*1.02I 300 where m f m f 2* 60:20 60:20 60:60:20 I a a a m l 6,953.536 a a v 60:20 a 15.132 (0.456387) (7.006) 12.869 9,826.131 1 20 80 1 60 2 80 l 2 60 f l 7,724.737 a a v a 16.152 (0.456387) (8.489) 13.113 60:20 9,848.431 1 20 80 1 60 2 80 l 2 60 f m 1 20 l80 l80 m f 1 60:60 2 80:80 l 2 60 l60 m f 6,953.536 7, 724.737 a a v a 13.590 (0.456387) (5.357) 12.233 60:60:20 9,826.131 9,848.431 I = 12.869+13.113-2*12.233=1.516 Value =10000*1.02*1.516+300 Premium = 15,763 This question caused considerable problems to candidates. Common errors were to only allow for one reversion (usually on death of male), omit the factor of 2 for joint life annuity, use a factor of 0.98 instead of 1.02 for expenses, or assuming that the annuity ran for 20 years from the first death. A surprisingly high proportion of candidates used erroneous formulae to convert annuities from annually in advance to continuous, often dividing by 1.04 0.5. This is a basic actuarial function which is given in the examination tables. 8 Let the full single premium at commencement = P The premium (based on normal mortality) payable at the time of exercising the option on the 2 nd anniversary = (1.05) 0.5 250,000q 57 v 250,000v 0.5 d l 57 57 250,000v 0.5 554 93,583 1,444.30 Therefore the premium required at duration 2, if the option is exercised, is 1,444.30 Thus equating the expected present value of all premium income with the expected present value of all claims, we get: Page 4

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report P (0.4) p 2 l P (0.4) l 57 55 55 2 v (1,444.30) (1.05) 2 v (1,444.30) (1.05) 250,000( q 250,000 ( d l 3d 93,583 2 0.5 250,000 P (0.4) v (1,444.30) (1.05) (450v 499v 94,532 94,532 P 518.75 6,278.54 leading to P 5,759.79 0.5 0.5 55 v q 55 55 1 v d Alternative approach based on non-option policy 250,000A 55 56 v v 2 2 2 p 55 2 [(.6) q 57 v 3 ) 57 v 3 3(554) v (.4)(2)(3q If the policy were a simple 3-year term assurance without any options, the single premium at commencement would be: 1 55:3 (1.05) (1.05) (1.05) 0.5 0.5 0.5 250,000[ q 250,000 ( d l 55 55 v p 55 55 v d q 56 56 250,000 (450v 499v 94,532 v 2 v 2 2 d 2 57 p v 55 3 554v q ) 3 57 v 3 ] ) 3,684.80 To allow for the option, the initial single premium needs to be increased by: 2 0.5 * *.4 p v {1.05 [250,000( q q ) v 250,000( q q ) ]} 0 2 55 57 57 57 57 v * q57 represents the mortality of optioners post-option = 3q 57 (The 1 st term in square brackets represents the extra mortality of optioners on the original SA, and the 2 nd term represents the extra mortality on the additional SA over and above that paid for by the normal rates premium paid at the time of exercising option, t=2) 3 0.5 0.5 4d57 3 (0.4)( 2p55) v (1.05) 250,000(4q57 ) (0.4)(1.05) 250,000 v l (0.4)(1.05) 0.5 4 *554 250,000 v 94,532 3 2,075.00 The total single premium at outset = 3,684.80+2,075.00 = 5,759.80 (same as above, allowing for rounding) The premium payable by policyholders at t=2 when exercising their option is (unchanged from original solution): 0.5 0.5 d57 0.5 554 (1.05) 250,000q57v 250,000v 250,000v 1,444.30 l 93,583 57 This proved the most difficult question for students, with few fully correct answers. A number of candidates seemed to misread the question and tried to calculate the cost of the option (instead of the premiums) while others treated the policy as annual premium. Many students calculated the basic premium for a policy with no option and tried to calculate the additional premium required for the option so the examiners have provided an alternative solution along these lines. 55 3 ) 57 v 3 )]) Page 5

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report 9 (i) To zeroise future negative cash flows. The office must meet all future outgo (additional to unit liabilities) e.g. death claims in excess of units, expenses, maturity guarantees. It can take credit for future income to the non-unit fund but cannot assume recourse to future capital. If there are negative cash flows, we cannot assume that they will be met from subsequent positive cash flows or future capital (lapse risk, regulations). They are future losses which we need to reserve for now. With adequate non-unit reserves established, the minimum expected cash flow in future years, allowing for release of reserves, is zero, hence the zeroisation of cash flows. (ii) (2, 0, 0, 0, +1, 0, 0, 0, 0, +1) (iii) Cash flow approach is more flexible in general and allows for clarity of thought and ease of presentation of results Allows for complex policies (varying benefits, options) Permits variable or stochastic premium basis e.g. interest basis Best (often only) approach for multiple state model situations Allows amount and timing of cash flows to be observed Provides net cash flows useful for investment strategy Allows for explicit amount of profit to be calculated. Makes explicit allowance for cost of capital Only way to calculate non-unit reserves Facilitates repeating with altered basis for sensitivity testing (once spreadsheet or program set up) Generally well answered, especially part (ii). Some candidates only gave examples of outgo in part (i), without considering offsetting income while in part (iii) some candidates tended to concentrate on only one reason. z ia 2 s40 M40 2 7.814 58, 094 10 (i) (30, 000) (30, 000) 47,527.51 3 s s D 3 7.623 25,059 39 40 (ii) 30, 000 80 s M R R s z ra z ra z ra 40 18 40 40 62 s 39 D40 30, 000 7.814 (18)(128, 026) 2,884, 260 159, 030 80 7.623 25, 059 77,153.73 Page 6

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report i r M40 M 40 369 782 (iii) 50,000 50,000 17,945.12 D 40 3, 207 Well answered throughout. The commonest mistakes related to omitting the salary adjustment, treating (i) as service-related, omitting the factor for age 62 in the future service part of (ii). In (iii), some candidates used annuity functions and / or omitted one of the types of retirement. 11 (i) We are estimating [x]+t From 1 y,r y is policy year rate interval and lives are aged between y and y + 1 at the start of the interval in which death occurs, giving an average age at the policy anniversary before death of y +.5, assuming an even spread of birthdays over the policy year. r is also a policy year rate interval, and is the same as a duration of r 1 years exact at the policy anniversary before death, without assumption. The age at entry is y +.5 (r 1) = y r + 1.5 and the duration midway through the rate interval (needed for the duration when estimating forces of mortality) is r +.5 = r.5 so we are estimating [yr+1.5]+r.5. No further assumptions are required. From 2 y,r y is age last birthday at death giving a life year rate interval, with lives y exact at the start of the interval without assumptions needed. r is again a policy year rate interval, giving duration r years exact at the policy anniversary before death, without assumption. The average age at entry is y r, but we must assume an even spread of birthdays over the policy year because the 2 rate intervals are not the same (the age at entry could range from y r 1 to y r + 1 based on the information we have) and the duration midway through the rate interval is r +.5 so we are estimating [yr]+r+.5. (ii) Census A gives a life year for age, with y last birthday, and a policy year for duration with r curtate. Census B gives y next birthday at next policy anniversary, which is also y 2 last birthday at previous policy anniversary. It also gives duration r at policy anniversary following census or r curtate at census. For the 1 y,r deaths, census B fits perfectly but we just need to be careful with age labels. To get y last birthday at previous policy anniversary, and r curtate, we need P y+2,r. Page 7

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report The approximate exposed to risk is estimating [yr+1.5]+r.5 1 B 2000 B 2001 1 B 2002 P 2 y2, r Py2, r P 2 y2, r for For the 2 y,r deaths, census A fits perfectly. To get y last birthday, and r curtate, we need P y,r. The approximate exposed to risk is estimating [yr]+r+.5 1 A 2000 A 2001 1 A 2002 P 2 yr, Pyr, P 2 yr, for We assume that P x,t.varies linearly between census dates. Generally not well answered, especially part (ii). In (i), some candidates based their answer on the census data rather than on the death data, listed standard assumptions regardless of if they applied here. Others, having defined the age and duration labels correctly did not define the force of mortality at all or incorrectly. In part (ii), while many students correctly matched the censuses to the death tabulations, almost none got the correct age / duration labels for census B matched with deaths method 1. There was a slight discrepancy in the question between the number of years of death data (3) and the time period spanned by the censuses (2). This was not central to any of the answers required, but the examiners accepted all valid interpretations / assumptions made by students in this regard. 12 (i) Death claims in 2002 get SA, no reversionary bonus, and terminal bonus = 125,000 Discontinuances in 2002 get 0.25*4,300 = 1,075 2002 money flows: Premium income: 5000*4,300 = 21,500,000 Expenses: 15,000,000 Balance: 6,500,000 Interest during 2002 @ 6.5%: 422,500 Balance @ 31/12/2002 before claims: 6,922,500 Death claims 2002: 4*125,000 500,000 Surrender claims: 200*1,075 215,000 Total funds 31/12/2002: 6,207,500 No. of policies in force 31/12/2002: 5000 4 200 4,796 Page 8

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report (ii) 6, 207,500 Asset share per policy in force at 31/12/2002 = = 1,294 4,796 The basis for net premium reserves and the 2002 reversionary bonus declaration were the unnecessary items. Neither affected the cash flows during 2002 nor therefore the year end asset share. Well answered, especially as this was the first time an asset share calculation had appeared. The main error was to allow for reserves in some way. Some students tried to do the calculation per policy sold but this usually led to errors. 13 (i) A (contingent) whole life assurance with benefit of 100,000 paid immediately on the death of (y) providing it occurs after (x) s death (ii) Reserve before alteration t t x t y yt 0.04 t.02 t.03 t.07 t.09t V = 100, 000 e (1 p ) p dt = 100, 000 e (1 e ).03e dt 3, 000 e dte dt 0 0 0 = 1 1 3, 000 9,523.81.07.09 Reserve post alteration: 100,000A (0.975P) xy a xy xy t t x t y xt yt 0 A e p p ( ) dt.04 t.02 t.03 t.09t e e e (0.05) dt 0.05 e dt 0 0 1.05 0.555556.09 t.04 t.02 t.03 t.09t xy t x t y 0 0 0 e a e p p e e e e 11.6186 0.09 e 1 0.09 Page 9

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report or alternatively 0 e.09t.09 1 1 i a (at i e 1 0.094174) 11.6186 d i Reserve before = Reserve after + alteration expense 9,523.81 = (100,000)(.555556) P(.975)(11.6186) + 100 so P = 4,072.32 p.a. (iii) Both lives should be underwritten at this time. The proposed change increases the probability of claim payout by the insurer substantially with regard to life y. Previously if y was worse than assumed mortality, it was a margin for the office, but now the office is at immediate risk in relation to y. The risk with regard to x is similar to that before the alteration as regards the likelihood of a claim arising, but because the claim would now be paid immediately on x s death, the present value could increase significantly. Part (i) was well answered. In part (ii), many candidates made a good effort but many omitted or could not calculate the pre-alteration reserve. In part (iii), many candidates made general comments about underwriting without explaining why in the context of this particular alteration. 14 (i) Gross future loss random variable (GFL r.v.) = K 1 ( v [60] ){(200,000 (50,000)( K [60] )} 300 30 a K P(.975 a [60] K[60] 1.225) for K [60] < 4 or 300 30 a P(.975 a.225) for K [60] 4 3 4 (ii) E(GFL r.v.) = 0 1 1 [60]:4 [60]:4 [60]:4 [60]:4 250, 000 A 50, 000( IA ) 300 30( a 1) P (.975 a.225) 1 [60]:4 A M[60] M64 400.74 372.69 0.0318547 D 880.56 [60] 1 R[60] R64 4M64 7380.215813.76 4(372.69) ( IA) 0.08595666 [60]:4 D 880.56 [60] Page 10

Subject 105 (Actuarial Mathematics 1) April 2003 Examiners Report a [60]:4 N[60] N64 12475.24 9186.74 3.734555 D 880.56 [60] leading to 7,963.68 4,297.83 + 300 + 82.04 = 3.41619P P = 1,184.91 (iii) q [60] 0.005774 p [60] 0.994226 0p [60] 1 q [60]+1 0.00868 p [60]+1 0.99132 1p [60] 0.994226 q 62 0.010112 p 62 0.989888 2p [60] 0.985596 q 63 0.011344 p 63 0.988656 3p [60] 0.97563 Year Prem Expense Interest Claim Cash flow Profit Signature NPV 1 P 0.25P300 0.03P12 1154.8 0.78P1466.8 0.78P1466.8 0.75P1410.38 2 P 0.025P30 0.039P1.2 1302 1.014P1333.2 1.008145P1325.5 0.932087P1225.5 3 P 0.025P30 0.039P1.2 1011.2 1.014P1042.4 0.999394P1027.39 0.888458P913.342 4 P 0.025P30 0.039P1.2 567.2 1.014P598.4 0.989289P583.817 0.845648P499.049 Total NPV = 3.416193P 4,048.28 So P = 1,185.03 (same as above except for rounding due to use of commutation functions) (iv) (a) Profit is deferred but as earned interest and risk discount rate are equal, there is no impact on NPV or premium. (b) Profit is deferred but because the discount rate exceeds earned rate, NPV falls and premium would have to increase to satisfy the same profit criterion. Parts (ii) and (iii) were handled well throughout, with only the death benefit element of part (ii) causing any difficulty. In part (i), a number of students gave the expectation of the random variable, and among those who did give a random variable many omitted the select notation and / or struggled with the benefit element. In part (iv), many gave correct answers for (b), but in (a) very few students recognised that there would be no impact on the premium because the earned interest rate equalled the discount rate. Page 11

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 16 September 2003 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination details as requested on the front of your answer booklet. 2. You must not start writing your answers in the booklet until instructed to do so by the supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 14 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, with any additional sheets firmly attached, and this question paper. In addition to this paper you should have available Actuarial Tables and your own electronic calculator. Faculty of Actuaries 105 S2003 Institute of Actuaries

1 A life insurance company issues a number of 3-year unit-linked policies to lives each aged 40 exact. The year-end non-unit fund cash flows ( NUCF ) t, per policy in force at the start of policy year t, are as follows (in s): Year (t) 1 2 3 ( NUCF) t 100 100 150 Non-unit fund reserves are to be set up at each year end for each policy then in force to zeroise future negative cash flows. Calculate the adjusted value of ( NUCF ) t at the end of year 1, assuming that interest is earned on reserves at the rate of 5% per annum and that the mortality basis is AM92 Select. [3] 2 Describe four benefit options that may be available to an individual member of a pension scheme who leaves the scheme before normal pension age. [4] 3 A life insurance company issues a disability insurance policy to a healthy life aged exactly 45. The benefits under the policy are as follows. There is no waiting period, but there is a deferred period of one year. A benefit of 10,000 per annum is payable continuously while the policyholder is sick, after the completion of the deferred period. The benefit is payable until the policyholder reaches age 65, dies or recovers. Premiums are waived while the policyholder is in receipt of benefit payment. Level annual premiums are payable continuously under the policy until age 65 or the policyholder s earlier death. Calculate the annual premium. Basis: Sickness: S(ID) Tables Mortality: Interest: ELT(15) Males 6% per annum Expenses: Initial: 60% of the annual rate of premium Regular: 50 per annum, assumed incurred continuously in all years of the policy, including periods of sickness Claim: 1.5% of sickness benefit payments made to the policyholder [4] 105 S2003 2

4 Describe the calculation of a surrender value for a without-profit endowment assurance policy, under which level annual premiums are payable monthly in advance and cease on earlier death or surrender and the sum assured is payable immediately on death. Give formulae, defining carefully all the symbols that you use. [5] 5 A life insurance company issues a term assurance policy to a life aged 55 exact for a term of 10 years. The sum assured is payable immediately on death. The sum assured is given by 100,000 (1 0.05 t) t = 0,1, 2...,9. where t denotes the curtate duration in years since the inception of the policy. Level premiums are payable monthly in advance for a period of 10 years or until earlier death. The life insurance company calculates the premium using the equivalence principle. Calculate the annual premium. Basis: Mortality: AM92 Select Interest: 4% per annum Expenses: None [5] 6 A pension scheme provides a pension of 1/60 of Final Pensionable Salary for each year of scheme service upon retirement for any reason. Fractional years of service count proportionately. Final Pensionable Salary is defined as the average annual salary in the three years immediately prior to retirement. Members are required to contribute continuously at the rate of 5% of salary. You are given the following data in respect of Member A as at 1 January 2003: Age: 50 exact Annual rate of salary: 50,000 Using the data in the Actuarial Tables, calculate, in respect of Member A: (i) The expected present value of future contributions payable. [3] (ii) The expected present value of the pension benefits on retirement for any reason based on future service. [2] [Total 5] 105 S2003 3 PLEASE TURN OVER

7 A life insurance company uses the following 3-state model, to calculate premiums for a 3-year sickness policy issued to healthy policyholders age 50 exact at inception. Healthy (H) Sick (S) Dead (D) In return for a single premium of P payable at the outset the company will pay a benefit of 10,000 at the end of each of the 3 years if the policyholder is sick at that time. Let S t represent the state of the policyholder at age 50 t, so that S 0 = H and for t 1, 2 and 3, St = H, S or D. The life insurance company uses transition probabilities defined as follows: ij 50 t = ( t1 = t = ) p PS j S i For t 0, 1 and 2 the transition probabilities are: HD SD SH HS p50 t = 0.05 p 50 t = 0.15 p 50 t = 0.80 p 50 t = 0.1 The life insurance company calculates P as the expected present value of the benefit payments, assuming interest at 6% per annum and expenses of 5% of P. Calculate P. [5] 8 A life insurance company issues 10-year unit linked policies to lives aged exactly 50. Premiums paid in the first two years of the policies are applied to purchase capital units, with premiums in subsequent years being applied to purchase accumulation units. The management charge on the capital unit fund is 5% of the bid value of the units, deducted at the end of each policy year. The management charge on the accumulation unit fund is 1% of the bid value of the units, deducted at the end of each policy year. The life insurance company wishes to use actuarial funding assuming a rate of interest of 3% per annum. In calculating the actuarial funding factors, the life insurance company assumes that mortality is constant, with qx = 0.001 for 50 x 60. The life insurance company ignores surrenders. 105 S2003 4

(i) (ii) Calculate the actuarial funding factor to be applied at the end of the third year of a policy. [4] The life insurance company is considering using a higher rate of interest for actuarial funding factors. It wishes to assume the same mortality basis and to ignore surrenders in calculating the revised actuarial funding factors. Describe how you would determine the maximum rate of interest it would be prudent to use in calculating the actuarial funding factor to be applied at the end of the third year of the policy. Set out the considerations you would take into account. [5] [Total 9] 9 You are a consulting actuary to a client who wishes to invest 1m now to provide an immediate income for his partner and himself in retirement. Both the client and his partner are aged 60 exact. The client wishes to provide a payment annually in advance each year while either he or his partner is alive. He wishes the amount of the payment to be t I(1.05) t = 0,1, 2... where I denotes the amount of the initial payment and t denotes the curtate duration in years since the inception of the policy. The client further requests that he wishes the amount of the initial payment I to be such that the capital of 1m is at least 95% likely to be sufficient to provide the required payments and he asks you to advise what the maximum value of the initial payment I should be. In carrying out the calculations, you assume that the only source of random variation is the future mortality of the client and his partner. Calculate the required value of I based on the following assumptions. Mortality: Rate of future investment returns: Expenses: The client and his partner are independent with respect to mortality and are each subject to the mortality of PMA92C20. 6% per annum none [9] 105 S2003 5 PLEASE TURN OVER

10 A life insurance company offers an option on its 10-year level term assurance policies to effect a whole life without profits policy, for the sum assured, without evidence of health. The option may be exercised once only, either on the fifth anniversary of the policy or at the expiry of the 10-year term. If the option is exercised on the fifth policy anniversary, the term assurance policy ceases immediately. The sums assured under the 10-year term assurance policy and under the whole life policy are both payable immediately on death. A single premium, inclusive of the option premium, is payable at the outset under the term assurance policy and level premiums under the whole life policy are payable annually in advance until death. The premiums under the whole life policy are calculated using the company s normal annual premium basis. (i) (ii) Describe the conventional method of pricing the mortality option, stating clearly the data and assumptions required. [4] A policyholder aged exactly 45 wishes to effect a 10-year without profits term assurance policy, for a sum assured of 200,000. Calculate the total single premium payable under the term assurance policy, using the conventional method to calculate the option premium. The following basis is used to calculate the basic term assurance premium: Basis: Mortality: AM92 Select Interest: 4% per annum Expenses: none [5] [Total 9] 11 On 1 January 2000, a life insurance company issued an endowment assurance policy to a life aged exactly 50 for a term of 10 years. Under the policy, a sum assured of 100,000 is payable on survival to age 60 exact or at the end of the year of death on earlier death. Level premiums are payable annually in advance for 10 years or until earlier death. On 1 January 2003, the policy is still in force and the life insurance company calculates on a prospective basis both the gross premium reserve and the net premium reserve for the policy at this date, using the assumptions shown below. The same assumptions were used to calculate the gross premium at inception as follows: Mortality: AM92 Ultimate Interest: 4% per annum Expenses: Initial: 300 incurred at the outset Renewal: 5% of each premium (i) Calculate the gross premium reserve as at 1 January 2003. [3] (ii) Calculate the net premium reserve, with Zillmer adjustment, as at 1 January 2003. Identify clearly the Zillmer adjustment. [2] 105 S2003 6

(iii) (iv) Explain why the net premium reserve with Zillmer adjustment calculated in part (ii) might be used in preference to the net premium reserve with no Zillmer adjustment, calculated as at 1 January 2003, using the same assumptions. [2] Assume instead that the life insurance company calculated the gross premium reserve as at 1 January 2003 using a rate of interest of 3.5% per annum following a general fall in market interest rates, with all other assumptions unchanged. Assume also that the net premium reserve with a Zillmer adjustment, calculated in part (ii), is unchanged. State, giving a reason, whether you consider it appropriate to use this unchanged net premium reserve with a Zillmer adjustment for reserving purposes. [2] [Total 9] 12 A life insurance company issues a two-year without-profit policy to a member, aged exactly 50, of a certain club. The policy provides the following benefits: (a) on death as a member during two years, a sum of 10,000 (b) on withdrawal from the club within two years, a return of 75% of premiums paid without interest (c) on survival as a member to the end of two years, the sum of 5,000 Death and withdrawal benefits are payable at the end of the year of death or withdrawal respectively and the survival benefit is payable on the maturity date of the policy. There are no decrements from membership of the club other than death or withdrawal. A premium of 3,000 is payable annually in advance under the policy for 2 years or until earlier death or withdrawal. Calculate the net present value of the profit under the policy to the life insurance company. Basis: Mortality: the independent rate of mortality is that of AM92 Select Withdrawal: the independent rate of withdrawal is 5% per annum Rate of decrements: Mortality and withdrawal occur uniformly throughout each policy year in the respective associated single decrement tables. Expenses: 150 incurred at outset Rate of interest: 5% per annum Reserves: Ignore Risk discount rate: 15% per annum [9] 105 S2003 7 PLEASE TURN OVER

13 A life insurance company issues a special annuity policy to a male and a female life, both aged exactly 60. Under the policy, an annuity is payable annually in arrear for a maximum of 4 years, ceasing on the first death of the two lives. The first payment under the policy is 10,000 and subsequent payments increase by 1.9231% per annum compound. (i) Calculate the standard deviation of the present value of benefits under the annuity policy. Basis: Mortality: The male and the female lives are independent with respect to mortality and are subject to the mortality of PMA92C20 and PFA92C20 respectively. Interest: 6% per annum [8] (ii) State, with reasons, whether the standard deviation would be higher, lower or the same if the annuity were to cease on the second death of the two lives, other conditions remaining unchanged. [2] [Total 10] 14 A life insurance company uses the following multiple-state model for pricing and valuing annual premium long-term care contracts, which are sold to lives that are healthy at outset. 0: Healthy 1: Claim level 1 2: Claim level 2 3: Dead Under each contract, the life company will pay the costs of long-term care while the policyholder satisfies the conditions for payment. These conditions are assessed every year on the policy anniversary, just before payment of the premium then due. If the policyholder satisfies the conditions, the annual amount of the benefit payable is paid immediately. A maximum of four benefit payments may be made under the policy, after which time the policy expires. The policy also expires on earlier death. Premiums are payable annually in advance under the policy until expiry, and are waived if a benefit is being paid at a policy anniversary. 105 S2003 8

For lives at claim level 1, benefits of 60% of the maximum level are paid, while lives at claim level 2 receive 100% of the maximum level. The current maximum level is 50,000 per annum and is expected to increase by 6% per annum compound in the future. ij p x is the probability that a life aged x in state i will be in state j at age x+1 and the insurer uses the following probabilities for all values of x: p p 00 x 11 x 0.87 0.6 p p 01 x 12 x 0.1 0.3 p p 02 x 22 x 0.0 0.6 (i) Calculate the annual premium under the contract. Basis: Interest: 6% per annum Expenses: 7.5% of each premium [9] (ii) A policyholder has already received two benefit payments at level 1, and is about to receive a third benefit instalment. Calculate the reserves the office should hold for this policy immediately after the benefit payment is made, if the policyholder is assessed as entitled to either: (a) (b) benefit at level 1 = 42,000 per annum benefit at level 2 = 70,000 per annum Reserve basis: Transition probabilities: as given Interest: Benefit inflation: 5% per annum Inflation of the maximum benefit level of 7% per annum. [5] [Total 14] 105 S2003 9

Faculty of Actuaries Institute of Actuaries REPORT OF THE BOARD OF EXAMINERS September 2003 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. J Curtis Chairman of the Board of Examiners 11 November 2003 Faculty of Actuaries Institute of Actuaries

Faculty of Actuaries Institute of Actuaries EXAMINATIONS September 2003 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report Overall Comments The standard of answering overall was at a lower level than the examiners expected. Candidates found particular difficulty with questions 4, 8, 9, 13 and 14. Attempts at questions 9 and 13 in particular were generally unsatisfactory. In relation to the other questions many candidates performed well. Individual comments follow after each question and we hope that these will be of assistance to students. 1 p[ 40] = 0.999212 p [ 40] + 1 = 0.999038 150 The reserve required per policy in force at the end of year 2, 2 V = = 142. 857 1.05 The cost of this, at the end of year 2, per policy in force at the start of year 2 = p [ ] 142. 720 40 + 1 * 2 V = The adjusted value of ( NUCF ) = 100 142.720 = 42. 720 2 42.720 The reserve required per policy in force at the end of year 1, 1 V = = 40. 686 1.05 The cost of this, at the end of year 1, per policy in force at the start of year 1 = p [ ] V 40. 654 40 * 1 = The adjusted value of ( NUCF ) = 100 40.654 59. 35 1 = The question was well answered in general. A number of candidates used incorrect mortality rates. 2 Return of the member s contributions Under this option, the total of the member s contributions are returned, with or without interest. This option is available normally only after a short period of service. There is likely to be a tax charge on the sum paid to the member. A deferred pension payable from normal pension age This option provides for the member to receive, from the scheme the member is leaving, a pension payable from normal pension age. The pension is normally based on the number of years service to the date of leaving and final pensionable salary at Page 3

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report the date of leaving. The basic amount of the deferred pension is increased each year, from the date of leaving to normal pension age, by a revaluation rate. An immediate pension from the date of leaving This option provides an immediate pension payable from the scheme, from the date of leaving. This option is normally restricted to members close to normal pension age. The pension can be calculated in a number of ways: a common method is to determine the pension amount as that which is actuarially equivalent to the deferred pension the member would otherwise have received. A transfer cash equivalent The transfer cash equivalent is an amount determined by the scheme actuary as a fair assessment of the present value of the deferred pension and other benefits given up by the member leaving the scheme. The transfer cash equivalent may be paid to a new scheme that the member is joining, or to a special individual policy that a member can effect for this purpose with a life insurance company. This question was well answered in general. Some candidates just listed the benefit options, whereas use of the word Describe required a fuller treatment. 3 Let P be the annual premium. P is given by HS( 1/ all) HS( 1/ all) P( a a 45:20 ) = 10000*1.015* a + 0.6P+ 50a 45:20 45:20 45:20 ( ) P 11.299 0.242488 = 10150*0.242488 + 0.6P+ 50*11.299 P = 289.41 Overall this question was answered well. Some candidate had difficulty with valuing the waiver benefit. 4 The retrospective policy value is determined, using a basis that reflects the experience of the policy and takes account of the cost of surrender. The formula for the policy value is as follows: Dx ( 12) ( 12) { Ga SA } 1 xt : xt : I ea fa xt : 1 xt : C, where D + x t x is the age of policyholder at inception Page 4

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report t G S I e f C is the policy duration at which the surrender value is being calculated is the annual office premium is the sum assured are the initial expenses, in excess of the regular expenses occurring each year are the regular annual expenses are the additional expenses that occur when the contract terminates are the surrender expenses The prospective policy value is calculated using a basis that reflects the future expected investment earnings, future expected expenses and future expected mortality experience of the surrendering policyholders, less the cost of surrender. The formula is as follows: SA ea ( 12) fa Ga ( 12) + + C x+ tn : t x+ tn : t x+ tn : t x+ tn : t Additional definition: n is the original term of the policy. A table of surrender values by policy duration is produced. The surrender value at a particular duration is usually a blend of the retrospective and prospective policy values, subject to a minimum of zero. Generally, the retrospective policy value is given a greater weighting at earlier durations and the prospective value is given a greater weighting at later durations. Other considerations, such as the asset share and marketing influences, are also generally taken into account. Where possible, the surrender value should be less than the asset share. Marketing considerations may mean adjusting surrender values upwards. Most candidates did not answer this question well. The examiners view was that this was a standard theoretical question and well-prepared candidates should have scored reasonably. Very few candidates mentioned both prospective and retrospective reserves; most formulae given were not fully correct; and very few candidates dealt with the considerations set out in the final part of the solution. Page 5

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report 5 Let P be the annual premium. P is given by (12) 1 1 = 95000 + 5000( ) [55]:10 [55]:10 [55]:10 Pa A IA (12) D 65 689.23 a = a 0.458 1 = 8.228 0.458 1 = 8.056 [55]:10 [55]:10 D[ 55] 1104.05 D 1 0.5 65 =1.04 A [55]10 [55]10 D[55] A 0.5 ( ) = 1.04 0.68354 0.624274 = 0.060439 1 0.5 R R 10* M ( IA) =1.04 * [55]:10 D [55] 65 65 [55] 0.5 9482.75 5441.07 10*363.82 = 1.04 * = 0.372692 1104.05 95000*0.060439 + 5000*0.372692 P = = 944.04 8.056 Candidates attempted this question well in general. There were some minor errors in the formulae and numerical calculations. Page 6

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report 6 (i) Salary at age 50 exact salary earned between age 49.5 and 50.5, assuming that the salary increase was given at age 49.5. s 49.5 = 0.5*(9.031+ 9.165) = 9.098 Value of future contributions s N50 = 0.05*50000* s * D 49.5 50 163638 = 2500* = 25, 036.40. 9.098*1796 (ii) Value of future retirement benefits z ra z ia R50 R50 50000 + 50000 1604000 + 363963 = * = * = 100,365.26. 60 s D 60 9.098*1796 49.5 50 The solution given is based on the assumption that Member A s salary was increased 6 months before the valuation date. The examiners gave full credit for any other sensible assumption so long as the assumption was stated. For example, assuming that the salary had just been increased, s 50 would be used in place of s 49. 5. Candidates answered the question well, in general. 7 The remaining transition probabilities are: p HH SS 50+ t p50+ t = 0.85 = 0.05 Probability of being sick at t = 1 = 0.1 Probability of being sick at t = 2 HH HS HS SS = p 50 p 51 + p 50 p 51 = 0.85*0.1+ 0.1*0.05 = 0.09 Probability of being sick at t = 3 = HH HH HS HH HS SS HS SS SS HS SH HS p50 p51 p52 + p50 p51 p52 + p50 p51 p52 + p50 p51 p52 = 0.85*0.85*0.1+0.85*0.1*0.05+0.1*0.05*0.05+0.1*0.8*0.1 = 0.08475 Page 7

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report P is given by 2 3 ( 0.1v + 0.09v 0.08475 ) 0.95P = 10000* + v P = 2,585.23 Most candidates scored well on this question, with many getting full marks. 8 (i) The actuarial funding factor is given by A 53:7 at a rate of interest of 3% and mortality given by q = 0.001 53 x 60. x A = 1 da 53:7 53:7 53:7 2 6 a = 1+ 0.999 v+ (0.999 v) +... + (0.999 v) 1 (0.999 v) = 1 0.999v 7 = 6.39873 0.03 A = 1 *6.39873 = 0.81363 53:7 1.03 (ii) In assessing the maximum rate of interest, I would make a prudent estimate of the level of the company s future renewal expenses (including renewal commissions) and express this as a regular percentage of the projected bid values of the funded capital and accumulation units, say i%. I would use discounted cash flow techniques to calculate i. Conventionally, the rate i% tends to be the management charge used for accumulation units, 1% in this case. In practice, we might tend to increase the 3% interest rate to 4% (5%-1%). Mathematically, however, the maximum rate of interest is ( 5% i %) ( 100% 5% ). In this case, assuming i = 1%, this would give a maximum theoretical rate of 4.21%. In assessing whether this would be prudent to use, I would compare the funded value of capital units at the end of the third year using the revised actuarial funding factor with the surrender value of capital units at that time. The funded value should not be less than the surrender value. A further check should be made to ensure that this remains the case at all subsequent policy durations. Page 8

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report I would also consider whether the mortality assumption was appropriate for calculating the actuarial funding factor. The assumed level of mortality should not be lighter than that prudently expected for the group of policyholders. Otherwise the company would be anticipating future management charges it might not receive. Part (i) was not well answered. Many candidates did not show that the actuarial funding factor as the present value of an endowment benefit. Credit was given for variations from the solution set out: if a candidate assumed that the amount of the management charge being pre-funded was 3% per annum and used a rate of 0.03 interest of for the present value of the endowment benefit, credit was given; if a 0.95 candidate assumed that the death benefit was payable immediately on death rather than at the end of the year of death in the calculation of the present value of the endowment benefit, credit was also given. Part (ii) caused particular difficulties. Few candidates mentioned the use of discounted cash flow techniques or the considerations set out in the final two paragraphs of the solution. 9 Let t be the future lifetime of the joint status. For the payments to be exactly 95% likely to be sufficient, since the lives are independent with respect to mortality, the value of t is given by t p 0.05 60:60 = 2 p p = 0.05 t 60 t 60:60 2 60 t p60 ( p ) 2 + 0.05= 0 t 2 ± 4 4*0.05 t p60 = = 0.02532 or 1.9747 2 p = 0.02532 t 60 60 l60+ t = 0.02532 39 < t < 40 l Therefore, for the payments to be at least 95% likely to be sufficient, there must be at least 40 payments. Alternative derivation that there must be at least 40 payments Page 9

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report For payments to be at least 95% likely to be sufficient, t is given by t q 60:60 0.95 2 ( q ) 0. 95 t 60 q 60 0.97468 t p 60 0.02532 t l 60 = 9826.131 l60+t 248.798 t 40 I is given by 1.06 1000000 = Ia i = 1 = 0.9524% 40 1.05 a = 33.44892 40 I = 29,896 This was the most poorly answered of all the questions, with few candidates gaining many marks. The question was based on a practical application of standard joint life mortality and the examiners would have expected candidates to have performed much better. 10 (i) Under the conventional method, the premiums that should be charged and the premiums that will be charged for the new policy or policies that the policyholder can opt to take are determined. The present value of the differences between the premiums is then calculated and this is the present value of the cost of the option. Where there is more than one option, the present value of one option only is taken into account: the option chosen is the one that gives the highest present value of the differences in premiums. In carrying out the calculations, the following assumptions are made: all lives eligible to take up the option will do so; the mortality experience of those who take up the option will be the Ultimate experience which corresponds to the Select experience that would have been used as a basis if underwriting had been completed as normal when the option had been exercised. Page 10

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report The mortality basis used is not usually assumed to change over time, so the only data required are the Select and Ultimate mortality rates used in the original pricing basis. (ii) The present value of the differences in premiums are as follows: Option exercised at the fifth anniversary Present value = = D 50 A A 50 [ 50] 200000 a D[ 45] a50 a [ 50] 1366.61 0.5 0.32907 0.32868 200000* *1.04 *17.444 1677.42 17.444 17.454 50 = 96.10 Option exercised at the tenth anniversary Present value = D 55 A A 55 [ 55] 200000 a D[ 45] a55 a [ 55] 55 1105.41 0.5 0.38950 0.38879 = 200000* *1.04 *15.873 1677.42 15.873 15.891 = 154.62 The cost of the option is the greater value, i.e., 154.62 The basic single premium is given by M[ 45] M 0.5 1 0.5 P= 200000*1.04 A = 200000*1.04 * [45]:10 D[ 45] 55 0.5 462.68 430.55 = 200000*1.04 = 3,906.75 1677.42 The total single premium = 3,906.75 + 154.62 = 4,061.37. Candidates performed well on this question in general. In part (ii) there is a subtle point that if the 5 year option is taken then a release of the Term Assurance reserve would take place. The Examiners did not expect students to cover this and the solution is based on this assumption. A few candidates did point this out and due credit was allowed within the total marks in these cases. Page 11

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report 11 (i) The original gross premium is given by 0.95Pa = 300 + 100000A 50:10 50:10 a 50:10 = 8.314 A 50:10 = 0.68024 P = 8, 650.47 The gross premium reserve = 100000A 0.95*8650.47* a 53:7 53:7 a 53:7 = 6.166 A 53:7 = 0.76286 Gross premium reserve = 25,614.14 (ii) Net premium reserve with Zillmer adjustment a a 100000 1 300* a a = 53:7 53:7 50:10 50:10 = 25,835.94 222.49 = 25,613.45 222.49 is the Zillmer adjustment. (iii) (iv) The net premium reserve with Zillmer adjustment equals the gross premium reserve calculated in part (i) (subject to rounding errors). If the insurance company actuary is satisfied that there are sufficient margins in the gross premium reserve then the net premium reserve with Zillmer adjustment would be adequate. In addition, the use of the net premium reserve with Zillmer adjustment compared with the use of the reserve without adjustment would reduce the company s funding requirements. If the life insurance company s actuary decided that the gross premium reserve using 4% interest was no longer adequate given the fall in market interest rates and that 3.5% interest should be used, this would give a higher value for the gross premium reserve. The net premium reserve calculated in part (ii) was equal to the gross premium reserve using 4% interest and this net premium reserve would not be adequate. Page 12

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report Many of the well prepared performed well on this question. A surprising number of candidates showed a lack of understanding of a Zillmer adjustment. 12 The multiple decrement table is as follows. Age ( x ) ( al ) x ( ad ) d x ( ad ) w x 50 100000 192.17 4995.07 51 94812.76 252.55 4734.16 52 89826.04 Values for the multiple decrement table are calculated from formulas of the following type: 1 ( 2 ) d d w x x x ( aq) = q 1 q d d x x x ( ad) = ( al) *( ad) d w x+ 1 = x x x ( al) ( al) ( ad) ( ad) The profit test is set out as follows. Year 1 2 Premium 3000 3000 Expenses 150 Interest 142.5 150 Death benefit 19.217 26.637 Withdrawal benefit 112.389 224.692 Survival benefit 4737.02 Cash flow 2860.894 1838.349 Probability in force 1 0.94813 Discounted cash flow 2487.734 1317.954 Net present value 1,169.78 Candidates performed well on this question in general. Where errors occurred, they were mostly in respect of the multiple decrement table. A number of candidates did not use a cash flow approach which is what the Examiners were expecting. Page 13

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report 13 (i) With i = 0.06 and payments increasing at the rate of 1.9231% per annum, we can value at 4%, but we must make the initial payment = 10000/1.019231. age l x (male) l x (female) k kp xy Pr(K xy = k) 60 9826.131 9848.431 0 1 0.004504 61 9802.048 9828.163 1 0.995496 0.005364 62 9773.083 9804.173 2 0.990132 0.006361 63 9738.388 9775.888 3 0.98377 0.007514 64 9696.99 9742.64 4 0.976257 0.976257 k a min( k,4) a min( k,4) 2 E[x] 2 E[x ] 0 0 0 0 0 1 0.961538 0.924556 0.005158 0.00496 2 1.886095 3.557353 0.011998 0.02263 3 2.775091 7.70113 0.020851 0.057863 4 3.629895 13.17614 3.543709 12.86329 3.581717 12.94875 Variance = 12.94875 (3.581717) 2 = 0.120052 Std Dev: (0.120052) 0.5 = 0.346485 Std Dev for this annuity is (10000/1.019231)*0.346485=3399.48 Alternative solution With i = 0.06 and payments increasing at the rate of 1.9231% per annum, we can value at 4%, but we must make the initial payment = 10000/1.019231. We require 1 v Var( a ( )) = Var( a ( ) 1) = Var min K xy,4 min K xy + 1,5 1 2 = d 2 ( A ( A ) 2 ) 60:60:5 60:60:5 ( K xy + 1,5) d min age l x (male) l x (female) k Pr(K xy = k) 60 9826.131 9848.431 0 0.004504 61 9802.048 9828.163 1 0.005364 62 9773.083 9804.173 2 0.006361 63 9738.388 9775.888 3 0.007514 Page 14

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report 64 9696.99 9742.64 4 0.976257 k + 1 k + 1 k v4 % * Pr( K xy = k) v 8.16% * Pr( K xy = k) 0 0.0043308 0.0041642 1 0.0049593 0.0045852 2 0.0056549 0.0050272 3 0.0064230 0.0054904 4 0.8024121 0.6595242 0.8237801 0.6787912 2 2 ( ) = 0. = A 0.6786136 82378 60:60:5 1 Variance = ( 0.6787912 0.6786136) 0. 12005 d = 2 4 % Std Dev: (0.12005) 0.5 = 0.34648 Std Dev for this annuity is (10000/1.019231)*0.34648=3399.43 (ii) If the annuity were a last survivor annuity, the standard deviation would be smaller. The chances of both lives dying during the 4 years would be much lower, so more annuities would be payable for 4 years, with a consequent reduction in the deviation from the average present value of the annuity payments. This question was very poorly answered in general. Many candidates were unable to make any reasonable attempt. The examiners had expected the question to be challenging, but not to the extent experienced. 2 alternative solutions are given which the Examiners hope will assist. Page 15

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report 14 (i) With no recovery to the healthy state, premiums are payable only until the first claim. 00 00 t px t px = = (0.87) t EPV premiums P{1 + 0.87v+(0.87v) 2 + (0.87v) 3 +...} = 5.578947P Valuing the benefit from the point when the first claim arises, we get the following probabilities: the first claim payment will be at level 1; the second claim payment will be at level 1 with probability 0.6 and level 2 with probability 0.3; the third claim payment will be at level 1 with probability 2 0.6 = 0.36 and at level 2 with probability 0.6*0.3 + 0.3*0.6 = 0.36 ; the fourth claim payment will be at level 1 with probability 0.6 3 = 0.216 and at level 2 with probability 0.6 * 0.3* 0.6 + 0.6 * 0.6 * 0.3 + 0.3* 0.6 * 0.6 = 0.324. If the first claim is in n years time, the expected present value will be n n 50000*0.6*1.06 * v. With v at 6%, this is 30,000 for all n. Similarly the present value of any level 2 claim will be 50,000, so we can ignore interest in valuing claims. The EPV of all claims at the point of the first claim payment arising is therefore: 30, 000*(1 + 0.6 + 0.36 + 0.216) + 50, 000*(0 + 0.3+ 0.36 + 0.324) = 114, 480 Finally the probability that the first claim occurs at the end of year 1 is 0.1, at the end of year 2 is (0.87)*(0.1), at the end year 3 is (0.87) 2 *(0.1) and in general at the end of year n is (0.87) n 1 *(0.1). The probability of a claim is therefore 2 0.1 0.1*(1 + 0.87 + 0.87 +...) = = 0.76923 0.13 The EPV of all claims = (0.76923)*(114,480) = 88,061.45 Page 16

Subject 105 (Actuarial Mathematics 1) September 2003 Examiners Report The equation of value is: (1 0.075)*5.578947*P = 88,061.45 P = 17,064.43 (a) If the third instalment is at level 1, then the fourth claim will be at level 1 with probability 0.6, or at level 2 with probability 0.3. However, interest and claim inflation no longer cancel, so the reserve immediately after paying the third claim is: V 1.07 1.07 = 42, 000* *(0.6) + 70, 000* *(0.3) = 47, 080 1.05 1.05 (b) If the third instalment is at level 2, then the fourth can only be at level 2, and will occur with probability 0.6. This gives the following reserve value V 1.07 = 70, 000* *(0.6) = 42,800 1.05 This question was also not answered well. Many candidates valued the policies as four-year policies only and many also failed to appreciate that interest could be ignored in valuing claims in part (i) after which the question became much easier to complete. Few candidates made reasonable attempts at part (ii). Page 17

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 21 April 2004 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination details as requested on the front of your answer booklet. 2. You must not start writing your answers in the booklet until instructed to do so by the supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 14 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, with any additional sheets firmly attached, and this question paper. In addition to this paper you should have available Actuarial Tables and your own electronic calculator. Faculty of Actuaries 105 A2004 Institute of Actuaries

1 (a) Give a formula for the Area Comparability Factor, defining all terms you use. (b) Explain the role of this Factor in calculating standardised mortality rates, indicating any advantages it has over other available methods. [3] 2 A life insurance company uses the following model for pricing and valuing sickness and other contracts. x 1: Healthy 2: Sick x x x 3: Dead ab x, t p is the probability that a life now aged x and in state a will be in state b when aged x + t aa x, t p is the probability that a life now aged x and in state a will remain continuously in state a until age x + t Explain what is represented by each of the following integrals: (a) 65 0 x 12 x, t 12, 000e p dt t (b) 30 30 0 0 t ( t r) 11 22 35, t x t 35 t, r 10, 000e p p drdt [3] 3 Explain the main differences in approach between the conventional and North American methods for pricing mortality options in life assurance contracts. [4] 105 A2004 2

4 In a certain country, the population has reached a stationary size, and there is no immigration or emigration. Women between the ages of 20 39 inclusive are regarded as being of childbearing age and mortality in this age range is zero. In the past every woman had a new baby on each of her 21 st, 26 th, 31 st and 36 th birthdays. From 1 January 2004, a change in birth patterns means that every woman is expected to have a new baby on each of her 23 rd, 28 th, 33 rd and 38 th birthdays. During the transition from one pattern to the other, it is expected that every woman will still have 4 babies, with a gap of at least 5 years between consecutive births. Calculate the Total Fertility Rate for: (a) the calendar year 2003 (b) the calendar year 2004 (c) women born in 1962 (d) women born in 1982 [4] 5 (a) Explain what is meant by 2 nq [ x][ y] (b) Evaluate 2 25 q[40][40] assuming both lives are subject to AM92 mortality. [4] 6 In a select mortality investigation, x,r corresponds to the number of deaths aged x nearest birthday at death with duration r at the policy anniversary preceding death. x,r divided by the appropriate central exposed to risk gives an estimate of [y]+t. Derive the values of y and t to which this estimate applies, stating any assumptions used. [5] 7 The future lifetimes of two individuals aged x and y are independent, and subject to constant forces of mortality of 0.02 and 0.03 respectively. (i) Calculate the probability that their first death occurs after 3 years and before 8 years from now. [3] (ii) Calculate the probability that their second death occurs after 3 years and before 8 years from now. [3] [Total 6] 105 A2004 3 PLEASE TURN OVER

8 A company issues a block of 5-year single premium investment policies to lives each aged 60 exact at commencement of a policy. It guarantees simple annual reversionary bonuses of 8% per annum of the single premium, with the possibility of a terminal bonus at maturity. The death benefit is 5 times the single premium. All premiums received under this policy are invested in an asset class where 5-year returns have a normal distribution with a mean of 50% and standard deviation of 25%. The company intends to declare terminal bonuses on maturity such that the proceeds of the policy are the greater of the guaranteed amount and 90% of the underlying asset value. Calculate the probability that: (a) (b) the insurer makes a loss on a particular policy. a policyholder receives a terminal bonus. Basis: Mortality: ELT15 (Females) Expenses: Ignore [6] 9 A retirement benefits scheme provides a lump sum retirement benefit equal to 3/80ths of the salary rate at retirement for each completed year of service in the scheme. Fractions of a year do not get credit. Retirement can occur at any age after attaining age 60 but not later than a member s 65 th birthday. Calculate the total service liability for the lump sum benefit in respect of a member aged 63 exact on the valuation date who has exactly 30 years of past service and is earning 40,000 per annum. Basis: Interest: 6% per annum Salary increases: Nil Independent mortality rates: PMA92Base Independent retirement rates: Age 63 last birthday 10% Age 64 last birthday 6% State any other assumptions you rely on. [6] 10 List the main categories of costs incurred by life insurance companies, giving an example of each, and indicating the manner in which they are usually allowed for in calculating premiums. [8] 105 A2004 4

11 (i) In the context of Manchester Unity Sickness Tables, state the meaning of: (a) (b) the force of sickness zx the annual rate of sickness sx [2] (ii) An insurance sickness policy provides combined endowment and sickness benefits. The sickness benefit is 200 per week for the first 26 weeks of sickness, 150 per week for the next 26 weeks and 100 per week thereafter while sickness lasts. All sickness payments cease on a policyholder s 65 th birthday. There are no waiting or deferred periods. The endowment part of the policy pays 10,000 immediately on the death of the policyholder or on survival to age 65. Premiums are waived during periods of sickness. Calculate the level premium per annum payable continuously by a new policyholder aged 35. Premiums are payable to age 65 but cease on earlier death. Basis: Sickness: Mortality: Interest: Expenses: S(MU) ELT 15 (Males) 4% per annum Nil [7] [Total 9] 105 A2004 5 PLEASE TURN OVER

12 On 1 January 1993, a life insurance company issued a number of 25-year without profit endowment assurance policies to lives then aged 35 exact. Level premiums were payable annually in advance throughout the term of the policy, ceasing on the earlier death of the life assured. The sum assured was payable on survival to the end of the term, or at the end of the year of death, if earlier. Premiums and reserves were calculated on the following basis: Mortality: AM92 Select Interest: 6% per annum Expenses: 60% of the first premium 5% of each premium excluding the first Calculate, as at 31 December 2003, the profit or loss for the calendar year 2003 in respect of these policies, given the following information: The total sums assured in force on 1 January 2003 were 50,000,000. The total death claims occurring during 2003 and paid on 31 December 2003 were 200,000. During 2003, policies with sums assured of 2,500,000 were surrendered. Surrender values, paid on 31 December 2003, were calculated as the retrospective reserve using the above basis, but with interest at 4% per annum. During 2003, policies with sums assured of 1,000,000 (before alteration) were made paid up with effect from 31 December 2003. Paid-up sums assured were calculated on a proportionate basis, namely the original sum assured * t/25 where t is the number of premiums actually paid. The company incurred expenses of 100,000 on 1 January 2003. The company earned a total return of 7% on its assets during 2003. Ignore tax, and assume that reserves for paid-up policies ignore future expenses. [10] 105 A2004 6

13 A life insurance company issues a policy to male lives aged 45 exact, providing the following benefits: A decreasing term assurance with a death benefit, which is payable immediately on death, of 200,000 in the first year, 190,000 in the second year thereafter reducing by 10,000 each year until the benefit is 10,000 in the 20 th year, with cover ceasing at age 65. An annuity of 25,000 per annum, increasing by 2,000 each year, where the first payment is made on the policyholder s 65 th birthday, and continues annually for life thereafter. The policy is paid for by level quarterly premiums payable in advance for 20 years, ceasing on earlier death. Calculate the premium, using the equivalence principle. Basis: Mortality: AM92 Select Interest: 4% per annum Expenses: Initial: Renewal: 200 plus 35% of the premiums paid in the first year 5% of all subsequent premiums and 40 per annum, increasing by 4% per annum compound, on each policy anniversary Claim: Death: 250*(1.04) t where t is the exact duration of the policy at death, measured in years with fractions counting Annuity: 2% of annuity payments [14] 105 A2004 7 PLEASE TURN OVER

14 (i) Under a 4-year unit-linked policy issued to a male aged 60 exact, the following non-unit cash flows, NUCF t, (t = 1,2,3,4) are obtained at the end of year t per policy in force at the start of year t. Year t 1 2 3 4 NUCF t 400 210 190 450 Mortality follows AM92 Select. (a) Show that the annual internal rate of return lies between 5% and 6%. (b) (c) If the rate of interest earned on non-unit reserves is 7.5% per annum, calculate the reserves required at times t = 1, 2 and 3 in order to zeroise future negative cash flows. Without doing any further calculations, explain what effect the zeroisation of future negative cash flows in part (b) above will have on the internal rate of return relative to that in (a) above. [7] (ii) A unit-linked endowment policy with an annual premium of 5,000 and a term of 2 years is to be issued to a male life aged 60 exact. 97.5% of each premium will be allocated to units at the offer price. The units will be subject to a bidoffer spread of 4%. At the end of each year a management charge of 1% of the bid value of the units will be deducted from the unit fund. If the policyholder dies during the term of the contract the office will pay out the greater of 40,000 and the bid value of the units at the end of the year of death (after the deduction of the management charge). The company carries out all profit test calculations on the contract using the following basis: Mortality: AM92 Select Rate of growth on assets in the unit fund: 9% per annum Rate of interest on non-unit fund cash flows: 6% per annum Expenses: 250 at time 0; 50 at time 1 Risk discount rate: 12% per annum (a) (b) (c) (d) If the policyholder dies in the second year of the contract, calculate the amounts of the non-unit fund cash flows in both of the years of the contract. Hence calculate the net present value of the profit assuming that the policyholder dies during the second year of the contract. The policyholder could also die in the first year, or survive to the end of the term of the contract. Calculate the net present value of the profit for each of these two events. Hence or otherwise, calculate the expected net present value of the profit under this contract. [11] [Total 18] END OF PAPER 105 A2004 8

Faculty of Actuaries Institute of Actuaries EXAMINATIONS April 2004 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report In general terms this was a relatively straightforward paper of standard questions with the possible exceptions of Questions 4 and 8. It was well done by the well prepared students. The Examiners noted, however, that many students appeared unprepared for this examination and often their marks were well short of the required pass mark resulting in overall a disappointing pass ratio. 1 (a) ACF x s c s c s Ex, t mx, t Ex, t mx, t x s c c Ex, t Ex, t x x where E c x, t : Central exposed to risk in population being studied between ages x and x + t s c x, t E : Central exposed to risk in standard population between ages x and x +t s m x, t : central rate of mortality either observed or from a life table in standard population for ages x to x + t (b) When multiplied by the crude death rate for the population or area under consideration, the ACF provides a standardised mortality rate ( the indirectly standardised rate ). This approach is often favoured when data required by other methods, usually local age-specific mortality rates, are unavailable. Question 1 was generally well done although clearly many students could not remember the standard formula. 2 (a) Expected present value of a benefit of 12,000 p.a. payable continuously to a life now aged x and healthy whenever x is sick, with the benefit ceasing at age 65 (b) EPV of a benefit of 10,000 p.a. payable continuously to a life now aged 35 and healthy throughout their first period of sickness, ceasing at age 65 in any event This question was done reasonably well. In part (b) of the question there was an erroneous symbol x in the formula which should have been 35. The examiners gave full credit for using either x or 35 in the answer above. Page 2

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report 3 Conventional assumes all eligible lives exercise the option, experience ultimate mortality according to some table and pay premiums on the option policy based on select mortality from the same table as if underwriting took place at the time of commencement of the option policy. The North American approach assumes that only a certain proportion of eligible lives exercise the option. Opters and non-opters are subject to different mortality levels. This is normally achieved by having a double decrement table of mortality / exercise of option for original policyholders and also a mortality table for post-option mortality for those who exercise the option. Question done well. Credit was given for other appropriate comments. 4 Total Fertility Rate = fx where f x is age specific fertility rate at age x. x For calendar years, we use the period rate approach where we sum the f x s observed in that year. For women born in a calendar year, we sum across the f x s observed over their lifetime, each x coming from the rate observed in the calendar year in which they were aged x. (None of this is required from the student, it is just explanation for the following results). Up to the end of 2003, f x = 1 for x = 21, 26, 31, 36 and f x = 0 otherwise. Therefore, the answer to (a) and (c) = 4, seeing as all relevant births occur before the change at the end of 2003. From 1 January 2004, f x = 1 for x = 23, 28, 33, 38 and f x = 0 otherwise. The answer to (d) is also 4. They will have babies when they are aged 21 (in 2003), 28, (in 2010), 33 (in 2015) and 38 (in 2020). The answer to (b) is zero. There are no women who will have babies in 2004. Those aged 23, 28, 33, and 38 all had babies during 2002 when aged 21, 26, 31 and 36, and therefore will not have their next baby until 2009 when they are aged 28, 33 and 38 respectively. This question was not done well and many students failed to understand the concept of a Total Fertility Rate attempting often to construct probabilities. The solution above is a full one. One mark was awarded for each part if the student just wrote down the correct numerical answer. Page 3

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report 5 (a) 2 n [ x][ y] q represents the probability that a select life, now aged y, will die within n years, having been predeceased by a select life now aged x (b) l * *{( ) } *{(1 ) } 2 1 1 2 1 65 2 25 q[40][40] 25q 2 [40][40] 2 25 q[40] 2 l[40] = 1 8,821.2612 2 2 9,854.3036 *{(1 ) } 0.005495 Question was done reasonably well. 6 x nearest birthday at death x ½ at start of rate interval (life year from x ½ to x + ½) during which life dies or x at mid-point when the force of mortality is estimated. No assumption necessary. r at policy anniversary preceding death means exact duration r at the anniversary before death (the start of the policy year rate interval for duration) and hence r + ½ mid-year. No assumption necessary The average age at entry [y] is therefore [(x ½) r], but we must assume an even spread of birthdays over the policy year because the two rate intervals are not the same type and therefore not coincident. (Based on the information we have the age at entry could range from (x ½) (r + 1) to (x + ½) (r) i.e. x r 1½ to x r + ½, on average x r ½.) Therefore we get an estimate of [x r ½]+r+½ Well prepared students scored well on this question. For full marks all comments regarding assumptions needed to be stated. 7 (i) q p p p * p p * p 3 5 xy 3 xy 8 xy 3 x 3 y 8 x 8 y 3 3 8 8 exp[ 0.02 dt]*exp[ 0.03 dt] exp[ 0.02 dt]*exp[ 0.03 dt] 0 0 0 0 3 8 0.15 0.4 dt dt e e 0 0 exp[ 0.05 ] exp[ 0.05 ].8607.6703 0.1904 Alternatively, the joint life status has constant hazard rate 0.02+0.03 = 0.05 giving a probability of the first death occurring between time 3 and 8: 8 8 3 5 xy 3 t xy x t: y t 3 0.05t 0.15 0.4 q p dt 0.05e dt e e 0.1904 Page 4

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report (ii) q p p ( p p p * p ) ( p p p * p ) 3 5 xy 3 xy 8 xy 3 x 3 y 3 x 3 y 8 x 8 y 8 x 8 y Alternatively,.06.09 0.15.16.24 0.40 ( e e e ) ( e e e ) (.9418.9139.8607) (.8521.7866.6703).9950.9685 0.0265 3 5 xy 3 5 x 3 5 y 3 5 xy 8 3 0.02t 0.03t 0.05t q q q q (0.02e 0.03e 0.05 e ) dt 0.0265 Although this question was a simple application of probabilities it was surprisingly not done well overall. 8 (a) Insurer makes a loss if either the policyholder dies or the asset value s 5-year return is less than 40% for survivors Probability of loss = q p * [ 40 50] = 5 q 60 5 p 60 * [ 0.4] 5 60 5 60 25 p 5 60 l l 65 60 87, 093 0.94942877 91, 732 [ 0.4] 1 [0.4] 1 0.65542 0.34458 Probability of loss = 0.05057123 (0.94942877)*(0.34458) 0.3777 (b) Terminal bonus is received if both the policyholder is alive and the asset value exceeds 155.556% of single premium Probability of terminal bonus = p *(1 [ 55.556 50]) = 5 p 60 *(1 [.2222]) 5 60 25 [0.2222] 0.58792 interpolating linearly between values for 0.22 and 0.23 Probability of terminal bonus = 0.94942877*(1-.58792)=0.3912 This question was done very poorly overall. Even though the question defined a Normal Distribution very few students appreciated how to apply this in this case. 9 Dependent decrement rates from independent using: r r 1 d x x 2 x ( aq) q (1 q ) etc. assuming a uniform distribution of decrements in the single decrement tables. Page 5

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report Age q r q d (aq) r (aq) d (ap) 63 0.1 0.009189 0.099541 0.00873 0.89173 64 0.06 0.010604 0.059682 0.010286 0.930032 Probability of retiring at 64 last birthday = (.89173)*(.059682) = 0.05322 Probability of retiring at 65 = (.89173)*(.930032) = 0.829338 Assuming that those retiring age 63 or 64 last birthday can be represented as retiring on average half-way through the year then we get Age Benefit Discount Probability EPV 63 45,000 0.971286 0.099541 4,350.73 64 46,500 0.916307 0.05322 2,267.61 65 48,000 0.889996 0.829338 35,429.16 Total 42,047.50 Despite the definitions given in the question, many students failed to appreciate that they needed to use dependent decrements and produced an answer based merely on independent decrements. Limited credit was given for a solution based on independent decrements and to score well the dependent approach was necessary. 10 Category Example Pricing Initial Commission Allow for directly, usually premium related Marketing, promotional Underwriting / Processing proposal / Issue of policy documentation Per policy on estimated volumes Usually per policy, although some elements might be tied to other driver (e.g. medical expenses might be sum assured related) Renewal commission Allow for directly, usually premium related Claim administration Calculation and payment of benefit Per policy per annum, allow for inflation Per policy, allow for inflation Page 6

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report Overhead Central services e.g. IT, legal Per policy per annum The solution above are the main items the Examiners were seeking. The question was open to wide interpretation as the word Costs was used as opposed to Expenses. Thus the Examiners gave full credit within the total marks for other valid references. Allowing for this the question was well done. 11 (i) (a) the force of sickness zx is the probability that a person aged exactly x is sick at that moment (b) the annual rate of sickness s x is the expected number of weeks sickness that a life aged exactly x will experience in the year of age x to x + 1 (ii) P Pa A a a a a 52.18 HS(0/ 26) HS(26/ 26) HS(52/ all) HS(0/ all) 35:30 10, 000 35:30 200 35 150 35 100 35 35 using S(ID) notation and where all sickness benefit functions are understood to terminate at age 65. Using values from S(MU) tables, and noting that A35:30 1 a35:30 we get P*16.979 10, 000{(1 (0.039221)(16.979)} P 200(10.813 1.931) 150(2.203) 100(2.972 11.859) (29.778) 52.18 P(16.979-0.571) = 3,340.72+(2,548.80+330.45+1,483.10) = 7,703.07 P = 469.47 per annum (Theoretically, some adjustment to the age should be made to reflect the fact that a healthy 35 year-old cannot receive the 2 nd / 3 rd levels of benefits immediately, but the usual adjustments are approximate and have only minor influence on the result, so are ignored here.) Question done well by well prepared students. Page 7

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report 12 Premium per 100,000 given by: 100, 000A 0.95 Pa.55P [35]:25 [35]:25 100, 000*(.24198) P*[(0.95)(13.392) 0.55] P 1,987.94 Reserve per 100,000 for fully in force policy at 31 12 2002 given by: V 100, 000A 0.95Pa 10 45:15 45:15 10 V (100, 000)(.42556) (0.95)(1,987.94)(10.149) 23,389.18 Reserve per 100,000 for fully in force policy at 31 12 2003 given by: V 100, 000A 0.95Pa 11 46:14 46:14 11 V (100, 000)(.45028) (0.95)(1,987.94)(9.712) 26, 686.47 SV at 31 12 2003 per 100,000 SA given by D [35] 1 11 SV Pa P A [35]:11 [35]11 : D46 (0.95.55 100, 000 ) @4% a [35]:11 N [35] 46 D [35] N 52, 662.65 29,905.96 2,507.02 9.0772 1 [35]:11 A M [35] 46 D [35] M 481.53 460.84 2,507.02 0.0082528 11 SV 2,507.02 [{1,987.94}*{(.95*9.0772) 0.55} {100,000*0.0082528}] 23,690.44 1, 611.07 Cost of PUPs at 31 12 2003 per 100,000 SA given by: 11 11PUPV A 25 46:14 ( )100, 000 44, 000*0.45028 19,812.32 Total funds available at 31 12 2003 before paying any claims or setting up reserves: [{500 * (P+ 10 V)} 100,000] * (1.07) = 13,469,759.20 = A Total claims paid on 31 12 2003 = (200,000 + 25 * 11 SV) = 792,261 = B Total closing reserves required: (500 2 25 10) 11 V + 10 * 11 PUPV = 12,553,958.81 = C Profit for 2003 = A B C = 123,539.39 Page 8

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report Question not done well and very few complete answers were presented 13 Equivalence principle EPV premiums = EPV benefits + EPV expenses Let P = quarterly premium EPV premiums: D 4Pa 4 [ (1 )] (4) 3 65 P a [45]:20 [45]:20 8 D[45] 3 689.23 8 1,677.42 4 P[13.785 (1 )] 54.2563P EPV death benefit: 1 1 I A [45]:20 [45]:20 210, 000A 10, 000( ) 0.5 M[45] M65 0.5 R[45] R65 20M65 210,000*(1.04) [ ] 10,000*(1.04) [ ] D D [45] [45] 0.5 462.68 363.82 0.5 13,987.39 5, 441.07 20*363.82 210,000*(1.04) [ ] 10,000*(1.04) [ ] 1, 677.42 1, 677.42 12, 621, 61 7, 720.60 4,901.01 D65 EPV annuity: (23, 000a 2, 000( Ia) D [45] 65 65 (0.410887)({23, 000*12.276} {2, 000*113.911}) 209, 622.20 EPV expenses: Death claim: 250 * 20 q [45] = 250 * (1 0.90030) = 24.92 Annuity.02 * EPV annuity = 4,192.44 Premium related: ) Other: (4) (4) Pa [45]:20 [45]:1 (0.05)(4 Pa ) (0.30)(4 ) [45] 1 (0.05)(54.2563 P) (0.3)(4 P)(1 [1 ]) 8 D[45] 2.712815P 1.182171P 3.8950P 3 D 160 40 @ 0% 160 40({1 } l a e {1 e }) [45]:20 65 [45] 65 l[45] 160 40[35.282 (0.90030)(17.645)] 935.85 Page 9

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report 54.2563P = 4,901.01 + 209,622.20 + 24.92 + 4,192.44 + 3.8950P + 935.85 50.3613P = 219,676.42 Hence P = 4,362.01 Quarterly Premium is 4,362 to nearer whole unit. Very few complete answers were presented but many well prepared students did successfully complete a number of parts. For the item of Renewal Expenses, credit was also given if the student took the alternative approaches of: 1. Assuming that this particular expense applied throughout life. 2. If the inflation escalator of 4% applied from year 3 rather than year 2 14 (i) (a) Year t q [60]+t 1 p [60]+t 1 t 1 p [60] NUCF t Profit signature NPV @ 5% NPV @ 6% 1 0.005774 0.994226 1 400 400.00 380.95 377.36 2 0.008680 0.991320 0.99423 210 208.79 189.38 185.82 3 0.010112 0.989888 0.98560 190 187.26 161.77 157.23 4 0.011344 0.988656 0.97563 450 439.03 361.19 347.76 Total 7.85 1.01 Because there is a change in sign in NPV between 5% and 6%, there must be a solution to NPV = 0 for an interest rate between 5% and 6%. (b) 3V = 0 since policy has positive cash flow in year 4. V = 190 / 1.075 = 176.74 2 Clearly 1 V = 0 since 210 (NUCF 2 ) > p [60]+1 * 2 V (i) (c) It will increase it. The rate on non-unit reserves exceeds the IRR so in this case the deferral of profits, by introducing reserves, will increase NPV and IRR. (Usually the discount rate exceeds the non-unit rate of return and allowing for reserves would then reduce NPV and IRR) Page 10

Subject 105 (Actuarial Mathematics) April 2004 Examiners Report (ii) These preliminary calculations, while also an alternative way to get the answer required in (ii)(d), are presented here as background calculations for (ii)(a), (b) and (c). Unit fund Fund Year Cost of allocation brought forward Interest Mgmt. charge Fund at end t a t b t c t d t e t 5000*.975*.96 e t-1.09*(a t +b t ).01*(a t +b t +c t ) a t +b t -c t -d t 1 4,680.00 0.00 421.20 51.01 5,050.19 2 4,680.00 5050.19 875.72 106.06 10,499.85 Non-unit fund Year Unallocated premium Expense Interest Death cost Mgmt. charge Cash flow Prof Sig. NPV t f t g t h t i t j t k t l t m t 5,000 a t.06*(f t g t ) q [60]+t 1 *(40,000 e t ) d t f t g t +h t i t + j t t 1 p [60] *k t 1.12 t * l t leading to 1 320.00 250.00 4.20 201.80 51.01 76.59 76.59 68.38 2 320.00 50.00 16.20 256.06 106.06 136.20 135.41 107.95 Total 39.57 (a) Yr 1: 320 250 +4.20 + 51.01 = 125.21 Yr 2: 320 50 +16.20 + 106.06 (40,000 10,499.85) = 29,107.89 (b) NPV = 125.21v 29,107.89v 2 = 23,092.84 (@12%) (c) Die Yr 1 Cash flow: 320 250+4.20+51.01 (40,000 5,050.19) = 34,824.60 NPV = 34,824.6v = 31,093.39 (@12%) Survive: Yr 1 = 125.21 Yr 2 = 320 50 + 16.20 + 106.06 = 392.26 NPV = 125.21v + 392.26v 2 = 424.50 (@12%) (d) NPV for contract = 31,093.39q [60] 23,092.84p [60]q[60]+1 + 424.50 2 p [60] = 39.57 (same NPV as in preliminary calculations above in non-unit cash flows) A very straightforward question done very well by well prepared students many of whom scored virtually full marks. END OF EXAMINERS REPORT Page 11

Faculty of Actuaries Institute of Actuaries EXAMINATIONS 22 September 2004 (am) Subject 105 Actuarial Mathematics 1 Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination details as requested on the front of your answer booklet. 2. You must not start writing your answers in the booklet until instructed to do so by the supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 13 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, with any additional sheets firmly attached, and this question paper. In addition to this paper you should have available Actuarial Tables and your own electronic calculator. Faculty of Actuaries 105 S2004 Institute of Actuaries

1 A life insurance company issues an annuity policy to two lives aged 65 and 62 exact in return for a single premium. Under the policy an annuity of 10,000 per annum is payable monthly in advance while at least one of the lives is alive. Calculate the single premium. Basis: Mortality: PMA92C20 in respect of the life aged 65 exact PFA92C20 in respect of the life aged 62 exact Interest: 4% per annum Expenses: none [3] 2 A member of a pension scheme is aged 50 exact, having joined the scheme at age 30 exact. His current salary is 50,000 per annum. Final pensionable salary is defined as the annual average earnings over the three years immediately prior to retirement. Normal Retirement Age is a member s 65 th birthday. Salary increases take place six months before the member s birthday. Using the functions and symbols defined in, and the assumptions underlying, the Example Pension Scheme Table in the Actuarial Tables, calculate the expected present value of the following: A pension on retirement at any stage on grounds of ill health of one-sixtieth of final pensionable salary for each year of service, with fractions of a year counting proportionately. [4] 3 A life insurance company issues a policy to a life aged 50 exact. The policy provides the following sickness benefit: 100 per week for the first two years of sickness, reducing to 50 per week thereafter during sickness. Sickness benefit ceases at age 65, or on earlier recovery or death. There is no waiting or deferred period. Level premiums under the policy are payable weekly in advance until age 65 or until earlier death. Any premiums falling due during periods of sickness are waived. Calculate the weekly premium. Basis: Mortality: ELT 15 (Males) Sickness Table: S(ID) in the Actuarial Tables Interest: 6% per annum Expenses: 5% of each premium (Expenses continue even when premiums are waived) [5] 105 S2004 2

4 (i) Describe the use of risk classification by life insurance companies in underwriting life assurance policies. [2] (ii) State two limitations to the use of risk classification and explain how life insurance companies deal with these limitations. [3] [Total 5] 5 A life insurance company issued a non-profit term assurance policy to a life aged x exact at the outset, with a term of 20 years. Under the policy, the sum assured of 100,000 is payable at the end of the year of death. Premiums under the policy are level and payable monthly in advance for 20 years, or until earlier death. The company values the policy at duration t years using a gross premium prospective policy value, t V. Derive algebraically the relationship between t V and t 1V. Define all the symbols that you use, where necessary. [6] 6 On 1 January 2001, a life insurance company issued a 10-year joint life non-profit term assurance policy to two lives aged 50 exact. Under the policy, the sum assured of 500,000 is payable immediately on the death of the first of the lives to die. Premiums of 1,000 per annum are payable annually in advance for 10 years, or until the first death of the lives assured. On 31 December 2003 the policy is still in force. Calculate the gross premium prospective policy value at this date, using the following valuation assumptions: Mortality: PMA92C20 for the first life and PFA92C20 for the second life Interest: 4% per annum Expenses: Renewal: 3% of each premium Claim: 200 on payment of a claim [6] 105 S2004 3 PLEASE TURN OVER

7 A double decrement table is to be constructed from two single decrement tables. The modes of decrement are and. For each of the single decrement tables you are given 2 x t = x. x l l t d and 3 x t = x. x l l t d for 0 t 1 where i l x = the number of lives in the single decrement table i at age x exact (i =, ) i d x = the number of decrements over [x, x + 1] in the single decrement table i (i =, ) (i) Show that t px. x t = 2tq x for 0 t 1 where t p i x = the probability that a life aged x exact survives t years i x t = the force of decrement by cause i at age x + t i q x = the probability that a life aged x exact becomes a decrement by cause i over [x, x + 1] in the single decrement table for cause i (i =, ). [3] (ii) Hence or otherwise show that the dependent initial rate of decrement at age x exact due to cause is: 2 aq = qx 1 q x 5 x [3] [Total 6] 105 S2004 4

8 A life insurance company issues 10-year non-profit term assurance policies, for a sum assured S, to lives aged x exact. It offers an option on the policies to effect, either on the fifth policy anniversary or at the expiry of the 10-year term, a whole life nonprofit policy for the same sum assured, without evidence of health. Premiums under the term assurance policies are payable annually in advance for 10 years, or until earlier death, or until the fifth policy anniversary, if the option is then exercised. Premiums under the whole life policy are payable annually in advance for the whole of life. The sums assured under the term assurance and whole life policies are payable at the end of the year of death. An additional single premium is charged at the outset under the term assurance policy for the mortality option. The company uses the North American method for pricing options. Give formulae for calculating the additional single premium charged at outset for the mortality option. You may ignore expenses. Define all the symbols that you use, where necessary. [8] 9 A life insurance company sells with profit whole life policies, with the sum assured and attaching bonuses payable immediately on the death of the life assured and with level premiums payable annually in advance ceasing with the policyholder s death or on reaching age 65 if earlier. Simple reversionary bonuses vest under the policies at the end of each year. The company prices the product using the following basis: Mortality: AM92 Select Interest: 4% per annum Expenses: Initial: 250 Renewal: 2% of second and subsequent years premiums Claim: 150 at termination of contract Bonuses: Simple: 6% of basic sum assured per annum (i) (ii) Write down an expression for the gross future loss at the point of sale for one of these policies, assuming it is sold to a life aged x exact (x < 65) at the outset. Write the expression in terms of functions of the random variables T [x] and K [x], which represent the exact future lifetime and the curtate future lifetime of (x) respectively. [3] Calculate the gross premium required for one of these policies for a sum assured of 200,000 and issued to a life aged 40 exact at the outset, using the equivalence principle. State any assumptions you make. [6] [Total 9] 105 S2004 5 PLEASE TURN OVER

10 The following data are available from a life insurance company, relating to the mortality experience of its term assurance policyholders. x, d The number of deaths over the period 1 January 2000 to 30 September 2003, aged x nearest birthday at entry and having exact duration d at the next policy anniversary following the date of death. Py, e( 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, where n = 1.1.2000, 30.9.2000, 30.9.2002 and 30.9.2003. (a) (b) Develop formulae for the calculation of the crude select forces of mortality corresponding to the x, d deaths. Derive the age and duration to which these estimates apply. Assume that all months are of equal length. State all other assumptions that you make. [11] 11 A special 3-year endowment assurance policy provides that the death benefit payable at the end of year of death is 10,000 plus the endowment assurance net premium reserve for that year that would have been held had death not occurred. 10,000 is payable on survival to the end of the 3 years. On the basis set out below, use a discounted cash flow method to calculate the level annual premium payable in advance for a life aged 57 exact. The requirement is that at the discount rate defined below the value of the annual emerging surpluses should sum to zero. Basis: Mortality: Expenses: Reserves: AM92 Select for experience and reserves 20% of the first annual premium 5% of subsequent premiums Value as a normal endowment assurance for a 3-year term on a net premium basis using a valuation rate of interest of 4% per annum. Ignore the effect on reserving of the extra death benefit defined above. Interest earnings: 7% per annum on cash flow Discount rate: 10% per annum Ignore tax and any other items. [12] 105 S2004 6

12 A pension scheme provides the following benefits in respect of a former male member of the scheme who has just left service: (a) (b) A pension to him for life of 10,000 per annum if he survives to age 65: the pension commences on his 65 th birthday and is guaranteed payable for five years in any event. A spouse s pension of 5,000 per annum, commencing immediately on the death of the former member before his 65 th birthday and payable for life to the spouse. A spouse s pension is payable on death in deferment if the former member is married at the date of death. Pensions are payable monthly in advance. Pensions in payment and deferment are increased monthly in arrears at the effective rate of 2.8846% per annum. The former member is now aged 62 exact. You are not given any information as to whether he has a spouse. Calculate the expected present value of these benefits using the following basis: Basis: Valuation rate of interest: Mortality in deferment and in retirement: 7% per annum PMA92C20 for the former member and PFA92C20 for his spouse Proportion of former members with a spouse at each age up to age 65: 90% Age difference of spouses: Females are exactly 3 years younger than their husbands Assume that death before retirement occurs at the mid-point of the year of age in respect of each year of age. [12] 105 S2004 7 PLEASE TURN OVER

13 You are a member of a committee responsible for monitoring the trend in assured lives mortality rates. You have been presented with the following ratios of actual to expected mortality rates on the basis of a standard table constructed twenty years ago ( Standard Table A ) and the total expected deaths over the period 2000 2003 based on this table. Age Ratio of Actual to Expected Total Expected Deaths (000 s) Mortality Rates 2000 2001 2002 2003 2000 2001 2002 2003 15 44 1.80 2.00 10 10 45+ 0.90 0.80 20 20 You have also been given details of the exposed to risk data in the two age groups 15 44 and 45+ corresponding to Standard Table A. The exposed to risk data are described as Standard Population A. (i) (ii) (iii) (iv) Define, giving a formula, the term Standardised Mortality Ratio. Define all the symbols that you use. [2] Show how the Standardised Mortality Ratio may be expressed as a weighted average. Describe the function averaged and the weights. [3] Calculate the Standardised Mortality Ratios for the periods 2000 2001 and 2002 2003 with reference to Standard Table A, using the data presented. [2] The committee measured the change in mortality between the periods 2000 2001 and 2002 2003 by calculating a Comparative Mortality Factor (CMF) for each period. This factor was calculated as r 1 r, where 2 r1 r2 was the expected number of deaths for the period obtained by applying the observed mortality rates to Standard Population A is the expected number of deaths in Standard Population A over a two-year period based on Standard Table A The CMF was 0.95 for the period 2001 2001 and 0.99 for the period 2002 2003, which led the committee to conclude that mortality was deteriorating. (a) (b) Explain the difference between the results of your calculation of the Standardised Mortality Ratios in part (iii) and these CMF figures. (Hint: Express the CMF figures as weighted averages.) State, giving a reason, which set of figures you think provides the better results. 105 S2004 8

(c) Comment on the conclusion of the committee that mortality was deteriorating. [6] [Total 13] END OF PAPER 105 S2004 9

Faculty of Actuaries Institute of Actuaries EXAMINATIONS September 2004 Subject 105 Actuarial Mathematics 1 EXAMINERS REPORT Faculty of Actuaries Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report In general, well prepared candidates did well on this examination which contained reasonably standard questions. Indeed some students scored high marks testifying to the fairly straightforward nature of the paper. The Examiners noted however that there were many candidates who were just not well prepared for the examination and this resulted in a large number being quite a few marks below the required pass level. Questions without further comment below were those that were in general done well by candidates. 1 The premium is given by: P 10000a (12) 65:62 12 11 65:62 65 62 65:62 24 a a a a = 13.666 + 15.963 12.427 0.458 = 16.744 P = 167,440 2 The expected present value is given by: 50000 0.5*60* s s * D 49 50 50 20 z ia z ia M50 R50 50000 0.5*60* 9.031 9.165 *1796 20* 45392 363963 64,861 3 Let P be the weekly premium. P is given by HS 0/ all HS 2/ all 50 50 50:15 52.18* 100 P a 50a 0.95*52.18 P* a 100 P *0.456447 50*0.184025 0.95* P*9.516 P 4.25 Page 2

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report 4 (i) Risk classification is used as an underwriting tool by life insurance companies. The company divides policyholders into different risk groups according to factors that affect mortality. The company s expectation is that policyholders in the same risk group are homogeneous with respect to mortality risk. The groups are defined by the use of rating factors, e.g., age, sex, smoking habit. (ii) In theory the company should add rating factors to its underwriting system until the all mortality differences are fully accounted for, apart from random variation. In reality, the ability of prospective policyholders to provide accurate responses to questions and the cost of collecting information limit the extent to which rating factors can be used. In addition, from a marketing point of view, proposers are anxious that the process of underwriting should be straightforward and speedy. In setting underwriting terms, companies compromise between the conflicting requirements of risk classification and marketing and use a limited number of rating factors. It is important for a company not to omit a significant rating factor that is used by other companies in the market: otherwise, there would be a risk of selection against the company. Credit was given for other suitable points and description. 5 Let P =monthly premium G = annual equivalent premium (=12P) e = annual regular expenses f = claim expenses 1 12 tv ' 100000 f A G e a x t:20 t x t:20 t 1 1 x t:20 t x t x t x t 1:20 t 1 A vq vp A and 12 12 12 x t:20 t x t:1 x t x t 1:20 t 1 a a vp a 1 12 12 tv ' 100000 f vqx t vpx t A G e a vp x t 1:20 t 1 x t:1 x ta x t 1:20 t 1 12 1 x t x t:1 x t x t 1:20 t 1 100000 f vq G e a vp 100000 f A G e a 12 x t 1:20 t 1 = 12 x t x t:1 x t t 1 100000 f vq G e a vp V ' 12 x t:1 V ' G e a 1 i q 100000 f t x t Page 3

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report = px t t 1 * V ' Many students attempted to just write down the relationship which was not satisfactory. To score well the relationship had to be derived from 1 st principles and the nature of the monthly premium effect clearly brought out. 6 The gross premium prospective policy value is given by: 500, 200A 1, 000*0.97* a 53:53:7 1 53:53:7 53:53:7 m f l60 l60 7 53:53 * * * m f 60:60 l53 l53 a a v a 9826.131 9848.431 1 16.716 * * *14.090 9922.995 9934.574 1.04 16.716 0.745975*14.09 = 6.205 7 m f 1/ 2 60 60 7 1 (1.04) * 1 l 53:53:7 * l m f * 53:53:7 l53 l53 A da v = (1.04) 1/2 *(1 0.038462 * 6.205 0.745975) = 0.015676 the gross premium policy value is: 500,200 * 0.015676 1,000 * 0.97 * 6.205 = 1822 to nearer Page 4

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report 7 (i) Let 2 lx t lx t d x and 3 lx t lx t dx 2 t x 1 x p t q and 2 t qx t qx p t t x t px x t 2tqx (ii) Therefore x 1 aq p p dr r x r x x r 0 = 1 0 3 1 r q 2rq dr x x = 1 q 2r 2r q dr x 0 4 x = 5 2 2r qx r q 5 x 1 0 = q x 2 1 5 q x This question was done very poorly and few candidates derived satisfactory answers. 8 The expected present value of the benefits is given by 9 d 2 w ad ad x t t 1 x 5t 1 5t S v v A al al t 0 x t 1 x x 5t, (I) where A S p q v x t x x t t 0 t 1 The expected present value of the premium income is given by 2 w ad x 5t 1 5t x x:10 x 5t x 5t t 1 al x P a P v a, (II) where 9 a ap v and x:10 t x t 0 t t x t pxv t 0 a Page 5

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report P x is the premium for the term assurance and Px 5 or P x 10 is the premium for the whole life assurance at the date on which the option is effected. The additional single premium is given by (I) (II). A double decrement table is constructed for all lives that effect the term assurance policy, with decrements of death and exercising the option, with the following definitions: ad d x, the number of decrements due to death aged x last birthday; ad w x 4 and ad w x 9, the number of decrements due to exercise of the option at the fifth policy anniversary and at the expiry of the 10-year term respectively; and al x, the number of lives aged exactly x in the double decrement table. t ap x al al x t x The dashed functions represent the mortality of those who have exercised the option. The above solution is just one of a number of possible approaches and credit was given to candidates whose chosen method showed clear definitions. It was not totally necessary to adopt a multiple decrement approach as movements took place at discrete points and again credit was given for other methods. T[ x] 9 (i) L 250 S 1 0.06 K 150 v 0.98 Pa 0.02P (ii) Equivalence principle E L 0 [ x] min[1 K,65 x] [ x] Assume E T E K 1 2 250 0.94S 150 A 40 0.06S IA = 40 0.98Pa 0.02P 40 :25 1 2 A 40 IA [40] 250 1.04 0.94 200, 000 150 0.06 200, 000 = 0.98Pa 0.02P 40 :25 1 2 250 1.04 188,150* 0.23041 12, 000 *7.95835 Page 6

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report = P 0.98*15.887 0.02 1 250 1.04 2 43351.64 95500.2 P 15.58926 P 9, 099.32 10 x, d is classified as x nearest birthday at entry and duration d at policy anniversary following death. Define a census taken at time t after the start of the period of investigation (1.1.2000), P ' x, d t, of those lives having an in force policy at time t, who were aged x nearest birthday at entry and will be duration d on the policy anniversary following time t. The Central Exposed to Risk is then given by t 3.75 c x, d x, d t 0 E P ' t. dt Then assuming that P ' x, d t varies linearly between the census dates (1.1.2000, 30.9.2000, 30.9.2002, 30.9.2003) the integral can be approximated by 1 * 3 P ' 0 P ' 0.75 2 4 x, d x, d 1 *2 P ' 0.75 P ' 2.75 2 x, d x, d However the censuses P ' x, d Px, d 1 *1 P ' 2.75 P ' 3.75 2 x, d x, d t have not been recorded. The recorded censuses t have lives classified by x nearest birthday at entry and curtate duration d at time t. We can write P t P t ' x, d x, d 1 Substituting into the previous formula gives an expression for the required Central Exposed to Risk. Then: x, d E x, d c x, d estimates x d 0.5 because the average age at entry is x assuming birthdays are uniformly distributed over the policy year, and the exact duration at the mid-point of the rate year (policy year) of deaths is d 0.5 for all lives (no assumptions are necessary). This question was generally done well by well prepared students but many did not appreciate the relatively straightforward triangulation method. Page 7

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report 11 If S t is surplus in year t per policy in force at begin year t then: ( t 1 V+P E t )*1.07 = q t (10000 + t V) + (1 q t )* t V + S t Where t V etc is relevant reserve, P the required premium, E t is expenses for year t and q t the relevant mortality for year t So S t = (P E t )*1.07 + t 1 V *1.07 t V 10000* q t We need to sum t-1 p [x] *S t *v t at 10% for t = 1, 2, 3 and set to zero. 1 V = 10000*(1 a / a ) = 10000*(1 (1+v*l [57] 1:2 [57]:3 59 /l [57]+1 )/2.873) = 10000*(1 1.956/2.873) = 3191.79 2 V=10000*(1 1/2.873) = 6519.32 and 3V=10000 using 4% interest. The following table can now be completed: Year end t 1 2 3 Prem-Expense 0.8*P 0.95*P 0.95*P t 1 V [57] 0 3191.79 6519.32 10000*q [57]+t 1 41.71 61.80 71.40 Interest 0.056*P 0.0665*P+223.43 0.0665*P+456.35 t V [57] 3191.79 6519.32 10000.00 S t 0.856P 3233.50 1.0165*P 3165.90 1.0165*P 3095.73 t-1 p [57] 1.00000 0.99583 0.98967 t-1 p [57]* S t 0.856*p 3233.50 1.0123*P 3152.70 1.006*P 3063.75 Therefore: (0.856*P 3233.5)*v+(1.0123*P 3152.70)*v 2 +(1.006*P 3063.75)*v 3 =0 at 10% i.e. 2.3706*P=7846.92 P = 3,310.10 Very few students produced a full answer here. Although most solutions attempted were as above, it was also acceptable to take the 3 rd year reserve as zero i.e. assuming the 10000 maturity value had been paid. This approach would have given a numerical answer of 3287.7 Page 8

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report 12 The inflation rate of 2.8846% p.a. combined with the valuation rate of 7% p.a. means that all benefits can be valued at 4% per annum effective. The expected present value of the member s pension is given by Benefit a l65 1 10000* * * a l 1.04 62 12 3 65:5 l a a v a 12 12 70 5 12 * * 65:5 5 70 l65 at i 4% a 12 5 d i 12 * a 1.021537* 4.4518 4.5477 5 l l 70 65 9238.134 9647.797 0.957538 v 5 0.82193 12 a 70 11.562 0.458 11.104 a 12 65:5 13.287 the expected present value is given by 9647.797 10000* *0.889*13.287 9773.083 116,607 Benefit b The expected present value of the spouse s pension on death before retirement is given by 2 1 d 5000 * h * * t 0.5 1.04 t 0 62 t 12 62 t 0.5 a59 t 0.5 l62 12 a 59.5 0.5* 16.982 16.652 0.458 16.359 Page 9

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report Similarly, (12) a 60..5 16.024 (12) a 61.5 15.679 the value is given by 34.694 41.398 5000* 0.980581*0.9* *16.359 0.942866*0.9* *16.024 9773.083 9773.083 49.193 0.906602*0.9* *15.679 9773.083 = 866 the total expected present value is 116,607+866 = 117,473. 13 (i) The Standardised Mortality Ratio is the ratio of the actual deaths in a population compared with the expected deaths, based on standard mortality rates. The formula is x x E c x, tmx, t c s x, t x, t E m, where E c x, t mx, t s m x, t is the central exposed to risk in the population between ages x and x t is the central rate of mortality for the population between ages x and x t is the central rate of mortality for a standard population between ages x and x t Page 10

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report (ii) The Ratio may be written in the form x E c s x, t mx, t s x E m c s x, t mx, t x, t m x, t which is the weighted average of the age-specific mortality differentials between the population being studied and the standard population. i.e. mx, t s m, x, t weighted by the expected deaths in the population being studied based on standard mortality. c s i.e. Ex, t m, x t (iii) The SMR for 2000 2001 is 1.8*10 0.9*20 1.2 30 The SMR for 2002 2003 is 2*10 0.8*20 1.2 30 (iv) (a) A formula for the CMF is x x s c E x, t m x, t s c s Ex, t mx, t which may be written in the form x s c s Ex, t mx, t s m s c s Ex, t mx, t x x, t m x, t. This is simply a weighted average of mx, t s m, x, t Page 11

Subject 105 (Actuarial Mathematics 1) September 2004 Examiners Report weighted by s c s x, t x, t E m. The differences between the SMR and CMF figures indicates that the Standard Population A and the observed population have different proportions in the two age ranges. As the CMF<SMR, this indicates that Standard Population A is more heavily weighted to the older age group. (b) (c) In my opinion, use of the SMR gives better results for comparing the population in each of the two periods. The mortality experience in the two periods is compared using Standard Population A exposed to risk in the CMF calculations and the observed population exposed to risk in the SMR calculations. Standard Population A appears to have a significantly different composition from the observed population. Therefore, using the Standard Population A exposed to risk in the weight calculations could introduce differences in the results which have nothing to do with underlying mortality differences. Use of the observed population exposed to risk removes this difficulty and results should be more reliable. I disagree with the committee s conclusion. The SMR figures indicate that the mortality experience has not changed between 2000 2001 and 2002 2003. In part (iv) other acceptable comments were given credit. END OF EXAMINERS REPORT Page 12