Heriot-Watt University. BSc in Actuarial Mathematics and Statistics. Life Insurance Mathematics I. Extra Problems: Multiple Choice

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1 Heriot-Watt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Multiple Choice These problems have been taken from Faculty and Institute of Actuaries exams. The old subject A2 corresponds roughly to Life Insurance Maths I (and Survival Models in 2nd year). In the question codes 94/1 means the first diet of exams (April) in 1994 and 96/2 means the second diet of exams (September) in If you find any errors in the questions then please let Andrew Cairns know so the the version kept on the web page andrewc/lim1/ can be updated. Please check the web version first, though. 1. (A2 94/1) Which of the following gives the most likely age last birthday at which a select life aged 21 will die, using A mortality? A 74 B 75 C 76 D (A2 94/1) A life aged 40 effects a with profit whole life assurance with sum assured of 500 plus attaching bonuses, payable immediately in death. Assuming allowance for compound bonuses of 4% per annum, vesting at the end of each policy year, which of the following gives the level annual premium payable throughout life? Basis: mortality A ultimate interest 4% per annum expenses none A B C D

2 3. (A2 94/2) Using A mortality, which of the following gives the value of 4 2q[60]? A B C D (A2 94/2) Which of the following is the value of A at 3% per annum? A B C D [30]: 15 using A mortality, with interest 5. (A2 94/2) Which of the following is equal to 15 V 25:30 1? ( 1 A D 40 M 25 M 40 (M ) 25 M 55 )(N 25 N 40 ) ( (N 25 N 55 ) 1 B D 40 M 40 M 55 (M ) 25 M 55 )(N 40 N 55 ) ( (N 25 N 55 ) M 40 M 55 + D 55 (M 55 M 55 + D 55 )(N 40 N 55 ) C 1 D 40 D 1 D 40 ( (N 25 N 55 ) M 25 M 40 + D 55 (M 25 M 55 + D 55 )(N 25 N 40 ) (N 25 N 55 ) ) ) 6. (A2 94/2) A life office issues four insurance policies to identical lives aged x. The first purchases a whole life policy, the second an endowment assurance policy with a 20-year term, the third a term assurance policy with a 20-year term and the fourth a pure endowment with a 20-year term. Each policy is effected for the same sum assured, and the office calculates reserves using the net premium method, with identical bases for the four contracts. 2

3 Which contract has the highest expected death strain in the year following issue? A the whole life policy B the endowment assurance policy C the term assurance policy D the pure endowment policy 7. (A2 95/1) On 1 January 1994 a life office held a portfolio of 200 whole life assurance policies on lives then aged 45. The policies were all issued on 1 January 1979, each with sum assured 10,000 payable at the end of the year of death. The office holds net premium reserves for these contracts, using A ultimate mortality at 4% p.a. interest. Three policyholders die during Which of the following is the actual death strain on this set of policies in 1994? A 20,241 B 24,559 C 25,066 D 30, (A2 95/1) An annual premium with-profit endowment assurance contract is issued to a life aged 40. The initial sum assured is 10,000, and the term of the policy is 20 years. The sum assured and attached bonuses are payable at the end of the year of death, or on maturity. The office declares compound reversionary bonuses. Which of the following gives the net premium policy value immediately before the 4th premium is due, given that bonuses of 3% per annum have been declared annually in advance for each of the contract? Basis: Mortality A ultimate Interest 3% per annum A 1,163 B 1,271 C 1,411 D 1, (A2 95/2) 3

4 Which of the following gives the value of A [50]:20 using A mortality with 4% interest? A B C D (A2 95/2) A life table with two year select period is being constructed. Ultimate values of l x will be the same as those for ELT 12 Males, while q [x] =0.3q x and q [x]+1 =0.6q x+1. Which of the following gives the value of l [70]? A 51,312 B 51,418 C 51,013 D 52, (A2 95/2) An impaired life age 30 is assumed to be subject to an extra risk such that the force of mortality experienced by the life is double the force of mortality of ELT12-Males at all ages. Which of the following gives the probability that the life does not survive to age 50? A B C D (A2 95/2) A 30-year with profit endowment assurance policy with a basic sum assured of 1000 is issued to a life aged 30. Simple bonuses of 3% of the basic sum assured vest at the end of each policy year if the life survives. Death benefits are paid at the end of the year of death. Which of the following formulae give the correct expected present value of the benefits? I 1 D 30 (1000M R 30 30R M D 60 ) 4

5 1 II D 30 (1000M R 31 30R M D 60 ) 1 III D 30 (970M R 30 30R M D 60 ) A I and II are correct B II and III are correct C I only is correct D III only is correct 13. (A2 96/1) Which of the following is the variance of the random variable which has expected value (IA)1 at an effective rate of interest of i per annum, υ =(1+i) 1? x:n A n 0 t2 v t tp x u xt dt ((IA) 1 x:n) 2 B n 0 t2 v 2t tp x u x+t dt ((IA) 1 x:n) 2 C n 0 t2 u 2t tp 2 xµ x+t dt ((IA) 1 x:n) 2 D n 0 t2 v 2t tp 2 xµ 2 x+tdt ((IA) 1 x:n) (A2 96/1) Which of the following gives the probability that a life aged exactly 65 will die between exact age 66 and exact age 70, assuming mortality follows the A select table. A B C D (A2 96/2) Which of the following is the value of a [57]:7 using A mortality, and an effective rate of interest of 4% per annum? A B C D

6 16. (A2 96/2) An impaired life aged 50 is assumed to be subject to an extra risk such that the force of mortality experienced by the life is equal to the force of mortality of the ELT 12 Males life table at the same age, plus a constant addition of Which of the following gives the probability that the life does not survive to age 60? A B C D (A2 96/2) A life table with a one-year select period is being constructed. Ultimate values of l x will be the same as for the A life table. During the select period the force of mortality will be one-half of the ultimate force of mortality for an individual of the same age. Which of the following is the value l [100]? A B C D (A2 97/1) A one year term assurance is issued to a life aged 50 for a sum assured of 100,000 payable at the end of the year of death. Which of the following gives the standard deviation of the present value of the term assurance, using A select mortality and 7% per annum interest? A 4993 B 5244 C 5867 D (A2 97/1) A company issues 20 identical policies to lives aged 60. The death strain at risk on each policy in the first year of the contract is 15,150. The insurer assumes that mortality follows the a(55) ultimate males mortality table. In the first year of 6

7 the contract 2 lives die. year? A 23,434 B 26,052 C 29,156 D 30,300 Which of the following gives the mortality loss in the first 20. (A2 97/1) the reserve held for a policy at duration t for a life then age 40, is t V = 15, 000, assuming A ultimate mortality and 8% p.a. interest. In the t to t + 1 no premiums are paid; expenses of 100 are incurred at time t. The benefits payable during the policy year t to t + 1 are 50,000 payable at t +1 if the life dies during the year, and 1,000 payable at t + 1 if the life survives the year. Which of the following gives the value of t+1 V? A 14,044 B 15,021 C 15,043 D 16, (A2 97/2) A life aged 40 purchases a with profit whole life assurance with sum assured 2,000 plus attaching bonuses, payable at the end of the year of death. Assuming allowance for simple bonuses of 3% per annum, vesting at the start of each policy year, which of the following gives the level annual premium payable throughout life? Basis: mortality - A ultimate interest - 4% per annum expenses - none A B C D

8 22. (A2 98/1) A life aged exactly 50 effects a with profit whole life assurance policy with sum assured of 10,000 plus attaching bonuses payable at the end of the year of death. Compound reversionary bonuses vest at the end of each policy year, provided the policyholder is still alive at the time. Which of the following gives the single premium payable at the outset? Basis: mortality: A ultimate interest: 5% per annum compound reversionary bonus rate: % of the sum assured each year A 4,697 B 4,767 C 4,788 D 4, (A2 98/2) An impaired life aged exactly 60 is subject to the mortality of a(55) males (select) with an addition of to the force of mortality. Which of the following is the expected present value at 4% p.a. of a whole of life assurance with sum assured 1,000 payable at the end of the year of death issued to the impaired life? A 297,8 B C D

9 Heriot-Watt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Profit and random variables These problems have been taken from Faculty and Institute of Actuaries exams. The old subject A2 corresponds roughly to Life Insurance Maths I (and Survival Models in 2nd year). In the question codes 94/1 means the first diet of exams (April) in 1994 and 96/2 means the second diet of exams (September) in (A2 94/2) An annuity of 1000 per annum is payable annually in arrears to a life aged 60 for a maximum of 3 years, ceasing in earlier death. Calculate the standard the deviation of the present value of the annuity, using a (55)-females ultimate mortality, and 10% per annum interest. [7] 2. (A2 98/2) A life aged exactly 50 is issued with a whole of life policy with sum assured 10,000 payable at the end of the year of death and premiums of 220 payable annually in advance. Given that A [50] evaluated at a rate of interest of 10.25% per annum is , calculate the mean and variance of the initial net present value of the policy. Basis: mortality: A (select) interest: 5% p.a. [8] 9

10 Heriot-Watt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Premiums These problems have been taken from Faculty and Institute of Actuaries exams. The old subject A2 corresponds roughly to Life Insurance Maths I (and Survival Models in 2nd year). In the question codes 94/1 means the first diet of exams (April) in 1994 and 96/2 means the second diet of exams (September) in (A2 95/1) A life offices issues a special ten year policy to a life aged exactly 55 under which the annual premium increases by 100 each year. The sum assured payable on death within ten years is 20,000 payable at the end of the year of death. On survival to the end of the ten year term the policyholder receives a refund of all the premiums paid without interest. Show that the first premium payable is Mortality: A Ultimate Interest: 4% Expenses: Initial: 150 Renewal: 4% of each premium including the first [7] 2. (A2 96/1) Prove that (IA) x:n ä x:n d(iä) x:n. [4] 3. (A2 96/1) A life office issues a deferred annuity contract to a life aged exactly 40. Premiums are payable annually in advance for 20 years or until earlier death. On survival to age 60 the annuity of 1000 per annum is payable quarterly in advance throughout life. In the event of death during the deferred period a lump sum is payable at the end of the year of death equal to the total premiums paid to date. Calculate the annual premium. 10

11 Basis: mortality before age 60 A ultimate after age 60 a(55) males ultimate interest 4% expenses nil [7] 4. (A2 96/2) A 25-year term assurance is issued to a life aged exactly 40. The sum assured of 100,000 is payable immediately on death. Level monthly premiums are payable immediately on death. Level monthly premiums are payable for the first 20 years of the contract. No premiums are payable thereafter. Calculate the monthly premium using the following basis: Mortality A select Interest 6% per annum effective Expenses Nil [6] 5. (A2 96/2) An office issues a 10-year term assurance contract to a life age 30. The sum assured, which is payable at the end of the year of death, is 150,000 in the first year of the contract, 155,000 in the second year, 160,000 in the third year, and so on, increasing by 5,000 each year. Premiums are payable annually in advance. The premium basis is: mortality: A select interest: 4% p.a. effective renewal expenses: 10 incurred at each premium date, including the first (i) Show that the premium is [5] 6. (A2 97/1) A life table with a select period of 2 years is based on rates of mortality which satisfy the following relationship: q [x r]+r = 1 3 r q x(for all values of x, and r =0,1) q 60 =0.0195, q 61 =0.0198, q 62 = and l 63 = 100, 000. (i) Calculate (a) l 62 11

12 (b) l [60]+1 (c) l [60] (ii) A select life aged 60 subject to the mortality table described in (i) above purchased a 3-year endowment assurance with sum assured 10,000. Premiums of 3,000 are payable annually throughout the term of the policy or until earlier death. The death benefit is payable at the end of the year of death. Calculate the expected value of the present value of the profit or loss to the office on the contract, assuming an effective rate of interest of 6% p.a., and ignoring expenses. [4] [Total 8] 7. (A2 97/1) A life office issues a 20 year with-profit endowment assurance policy to a life aged 40. The sum assured of 20,000 plus declared reversionary bonuses are payable immediately on death, or on survival to the end of the term. (i) Calculate the quarterly premium payable throughout the term of the policy if the office assumes that future reversionary bonuses will be declared at the rate of % of the sum assured, compounded and vesting at the end of each policy year. Basis: Mortality: A select Interest: 6% per annum Initial expenses: 114% of the first premium and 2.5% of the basic sum assured Renewal expenses: 4% of each quarterly premium, excluding the first [11] 8. (A2 97/2) An insurer issues a combined term assurance and annuity contract to a life aged 35. Level premiums are payable monthly in advance for a maximum of 30 years. On death before age 65 a benefit is paid immediately. The benefit is 200,000 on death in the first year of the contract, 195,000 on death in the second year, 190,000 on death in the third year, etc., with the benefit decreasing by 5,000 each year until age 65. On attaining age 65 the life receives an annuity of 10,000 per annum payable monthly in arrear. Calculate the monthly premium. [4] 12

13 Basis: mortality up to age 65 - A select over age 65 - a(55) ultimate females interest up to age 65-4% p.a. over age 65-6% p.a. expenses - nil [9] 9. (A2 97/2) A life aged 40 purchases a 25 year endowment assurance contract. Level quarterly premiums are payable throughout the duration of the contract. the sum assured of 100,000 is payable at maturity or at the end of the year of death. (i) Show that the quarterly premium is Basis: mortality - A select interest - 4% p.a. intitial expenses plus 60% of the gross - annual premium renewal expenses 3% of the second and subsequent quartely premium claims expenses on death; 100 on maturity 10. (A2 98/1) A life insurance company issues a special 20 year endowment assurance policy to a life aged exactly 40. The death benefit is 20,000 together with a return of 25% of the premiums paid. The payment is made 2 months after the policyholder s date of death. The survival benefit is 50,000 and is payable without delay at exact age 60. Premiums are payable annually in advance for 15 years, or until earlier death. [6] (i) Calculate the annual premium. Basis: mortality: A select interest: 4% per annum expenses: initial: 20% of first premium renewal: 3% of each premium, except the first [12] 11. (A2 98/2) An office issues a ten-year temporary increasing assurance policy payable immediately on death to a life aged exactly 50. The sum assured is 50,000 in the first 13

14 year of the policy, 55,000 in the second year of the policy, 60,000 in the third year, etc. Ignoring expenses, calculate the quarterly premium payable in advance. Basis: mortality: A (select) interest: 4% p.a. [6] 14

15 Heriot-Watt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Impaired lives These problems have been taken from Faculty and Institute of Actuaries exams. The old subject A2 corresponds roughly to Life Insurance Maths I (and Survival Models in 2nd year). In the question codes 94/1 means the first diet of exams (April) in 1994 and 96/2 means the second diet of exams (September) in (A2 94/1) An impaired life aged exactly aged exactly 55 wishes to effect a without profit endowment assurance for sum assured of 1,000 payable at the end of 10 years or at the end of the year of earlier death. Level annual premiums are payable throughout the term of the policy. Special terms are offered on the assumption that the life will experience mortality which can be represented by: for the first five years, a constant addition of to the normal force of mortality, and for the remaining five years the mortality of a life 8 years older. (i) The life office quotes a level extra premium payable throughout the term. Calculate this level extra premium. Basis: normal mortality A ultimate interest 4% per annum expenses none [9] (ii) As an alternative, the life office suggests effecting a policy providing the same death cover but giving an increased sum assured of 2000 on survival to the end of the ten years, with level annual premiums calculated using normal mortality. (a) Calculate the expected present value of the profit to the life office, on the premium basis. (b) Explain briefly why such a policy could be issued at normal rates to this impaired life. [10] [Total 19] 15

16 2. (A2 94/2) A whole life assurance, with sum assured 30,000 payable immediately on death, is issued to a life aged 45. Level monthly premiums are payable in advance for a maximum of 20 years. Calculate the monthly premium Basis: mortality A ultimate with a 5 year deduction from age up to age 65, a(55)-females ultimate from age 65. interest 4% per annum initial expenses 1% of the sum assured renewal expenses 5% of each premium after the first monthly premium [8] 3. (A2 95/1) A life office issues a whole life assurance to a life aged 40. The sum assured of 40,000 is payable immediately on death. Level premiums are payable weekly throughout the term of the contract. Calculate the weekly premium payable, using the tables provided. Basis: mortality A select rated down 5 years for 20 years, a(55) females ultimate thereafter; interest 4% per annum; expenses 40% of the annual rate of premium due at the outset of the contract plus 5% of each premium (including the first). [6] 4. (A2 95/2) An explorer aged exactly 57 has made a proposal to a life office for a whole life assurance with a sum assured of 10,000 payable at the end of the year of death. For lives accepted at normal rates, level annual premiums are payable until death under this policy. The explorer is about to undertake a hazardous expedition which will last three years. The life office estimates that during these three years the explorer will experience a constant addition of to the normal force of mortality, but after three years will experience normal mortality. The life office quotes a level extra premium payable for the first three years. Calculate this level extra premium. Basis: normal mortality A ultimate interest 3% per annum expenses none [7] 16

17 5. (A2 96/1) An impaired life aged exactly 40 wishes to effect an endowment assurance policy with a term of 25 years. The sum assured of 10,000 is payable immediately on death or at the end of 25 years if earlier. Premiums are payable monthly in advance for 25 years or until earlier death. It is assumed that the life will experience the mortality of a life ten years older for the first ten years of the contract, and normal mortality for the remaining term. (i) The life office quotes a level extra premium payable for ten years only, or until earlier death Calculate the extra monthly premium. [13] (ii) The office offers the life an alternative of paying the standard premium but with a level reduction in the sum assured for the first ten years only. Calculate the reduction. [4] Basis: standard mortality A select interest 4% per annum initial expenses 200 renewal expenses 5% of each premium including the first year s and including the extra premium [Total 17] 6. (A2 97/2) A life insurer assumes that the force of mortality of smokers at all ages is twice the force of mortality of non-smokers, which is taken from the A ultimate life table. Calculate the difference between the median future lifetimes of a non-smoker and a smoker, both aged exactly 50. [5] 7. (A2 98/1) A life insurance company issues a 20 year endowment assurance policy to an impaired life age exactly 45. The sum assured is 60,000 and death benefits are payable at the end of the year of death. The company assumes that the person will be subject to the following mortality: First ten years: standard mortality with constant addition to the force of mortality of Second ten years: standard mortality for a life three years older than the actual age 17

18 The contract is issued at the company s standard rate of premium, with premiums payable annually in advance until age 65 or earlier death. The death benefit is subject to a level debt of X for the first 10 years of the contract and 0.9X for the second 10 years. (i) Show that the standard annual premium is 2, Basis: mortality: A select interest: 4% per annum expenses: initial: 25% of the annual premium renewal: 4% of each premium, excluding the first (ii) Calculate the debt, X. [14] (iii) Without doing any further calculations, explain how the size of the debt would change if the age rating used in the second half of the term had been five rather than three years. [Total 19] 8. (A2 98/2) An impaired life aged exactly 30 suffers five times the force of mortality of a life of the same age subject to standard mortality. (i) A two-year term assurance policy is issued to both the impaired life and a standard life with a death benefit of 10,000 payable at the end of the year of death. Calculate the single premium payable for: (a) the standard life and (b) the impaired life Basis: standard mortality: A (ultimate) interest: 4% p.a. expenses: none [4] (ii) A two-year endowment assurance policy is issued to both the impaired life and a standard life with a death benefit of 100,000 payable at the end of the year of death or on survival at the end of the two years. Calculate the single premium payable for: (a) the standard life and 18

19 (b) the impaired life (iii) Comment on the relative sizes of the impaired life and standard life premiums for the policies described in (i) and (ii) above. [Total 10] 19

20 Heriot-Watt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Reserves These problems have been taken from Faculty and Institute of Actuaries exams. The old subject A2 corresponds roughly to Life Insurance Maths I (and Survival Models in 2nd year). In the question codes 94/1 means the first diet of exams (April) in 1994 and 96/2 means the second diet of exams (September) in (A2 94/1) On 1 January 1974 a life office issued a number of 25 year endowment assurance policies, with annual premiums payable throughout the term, to lives aged 40. As at 31 December 1993, a total sum assured of 100,000 remained in force. During 1993 sums assured of 2,000 became death claims, paid at the end of the year, and no policy lapsed for any other reason. The office uses net premium policy reserves, on the basis given below. Calculate the profit or loss from mortality for this group of policies for the year ending 31 December 1993 Basis: mortality A ultimate interest 4% per annum [6] 2. (A2 94/2) (i) State, and explain by general reasoning, a recursive relationship between the gross premium reserves at successive durations for an annual premium, whole life assurance, with benefit payable at the end of the year of death. Assume expenses are incurred at the start of each year. [4] (ii) Some time ago a life office issued an assurance policy to a life now aged exactly 50. 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: 20

21 reserve brought forward 4,750 annual premium 570 annual expenses 30 death benefit 20,000 mortality A ultimate interest 7% per annum Calculate the year end reserve. [Total 7] 3. (A2 96/1) On 1 January 1981 a life insurer issued a number of whole life policies to lives then aged exactly 45. Premiums for all of the contracts are payable annually in advance throughout the duration of the contract. The office holds reserves for these policies of the net premium policy values, using A ultimate mortality at an effective rate of interest of 3% per annum. Benefits are paid at the end of the year of death. The total sum assured in force at 1 January 1995 was 204,000. The total sum assured paid in death claims at 31 December 1995 was 4,000. (i) Show that the total net premium policy value as at 31 December 1995, in respect of policies in force on 1 January 1995 is 63,642. (ii) Calculate the total death strain at risk for these policies in [1] (iii) Calculate the total expected death strain. (iv) Calculate the profit or loss from mortality in [Total 8] 4. (A2 96/2) An endowment assurance issued to a life age x has a term of n years. Premiums are payable annually in advance for s years or until earlier death, where s<n. The sum assured of 1 is payable at the end of the year of death or at the end of the n year term if earlier. (i) Write down expressions at integral duration t<sfor: (a) the prospective net premium policy value; (b) the retrospective net premium policy value. 21

22 (ii) Prove that the prospective and retrospective policy values in (i) are equal. [5] [Total 7] 5. (A2 96/2) An office issues a 25-year with-profit endowment assurance policy to a life age 40. Premiums are payable quarterly in advance for 25 years or until earlier death. The sum assured is payable at the end of the year of death, or at the end of the 25-year term if earlier. The premium assumptions used by the office for this contract are: mortality: A ultimate interest: 6% per annum reversionary bonus: 1.92% p.a. compound, vesting at the start of each year premium expenses: per policy expenses: 5% of each premium (including the first) incurred at each premium payment date; the first quarterly amount is 20, with subsequent payments increasing at an assumed rate of inflation of 2.91% p.a. (i) (a) Show that the initial guaranteed sum assured for a premium of 250 per quarter year is 26,956. (b) Show that for policyholder who survives to the end of the 25-year term, the minimum rate of interest earned on the policy is % p.a. over the term of the contract. [12] (c) Explain briefly why the contract may be attractive to investors in spite of the very low minimum rate of return at maturity. (ii) During each of the first 5 years of the contract the office declares compound reversionary bonuses of 5%. The policy expenses increase at a rate of 4% per annum. (a) Calculate the prospective gross premium policy value at the end of the fifth year of the contract, using the premium basis. (b) At the fifth policy anniversary, immediately before payment of the premium then due, the policyholder wishes to alter the policy to a withoutprofit endowment assurance, with the same premium and remaining term as the original contact. Calculate the revised sum assured for the contract, using the premium basis above for the remaining term, except that there are assumed to be no further per policy expenses, and no additional costs of alteration. [7] [Total 19] 6. (A2 97/1) 22

23 (i) State the conditions which are necessary for the retrospective policy value of a contract to be equal to the prospective policy value throughout the term of the policy. (ii) Consider a term assurance contract with a term of n years and a sum assured of 1 on a life aged x, payable immediately on death. Premiums are payable annually in advance. Assuming that the conditions referred to in (i) hold, prove that the retrospective and prospective policy values are equal at any integer duration, t. Ignore expenses. [4] [Total 6] 7. (A2 97/1) A deferred annuity policy issued to a life age 50 has a 20-year deferred period, with premiums payable annually in advance throughout the deferred period, or until earlier death On death during the deferred period, the benefit payable at the end of the year of death is equal to the reserve and would have been required at that time had the life survived. On survival to age 70, a whole of life annuity of 10,000 per year is payable annually in advance. There are no expenses. (i) Show that if the annual premium is P, the reserve at duration t, t =1,2,..., 20, is P s ti where i is the effective rate of interest assumed. [6] (ii) Given that i =.06 and ä 70 =8.6, calculate P. [Total 8] 8. (A2 97/2) On 1 January 1981 a life office issued a number of 30 year pure endowment policies, to a group of lives aged 35. In each case, the sum assured was 20,000, no benefit was payable on death during the term and premiums were payable annually in advance. As at 1 January 1995, a total sum assured of 240,000 remained in force. During 1995, 2 policyholders died, and no policy lapsed for any other reason. The office calculates net premiums and net premium policy reserves on the following basis: Basis: mortality - A ultimate interest - 4% per annum 23

24 (i) Calculate the profit or loss from mortality for this group for the year ending 31 December [7] (ii) Briefly explain how the mortality profit or loss has arisen. [Total 9] 9. (A2 98/1) On 1 January 1985 a life office issued a number of 30 year pure endowment assurance contacts to lives then aged 35, with premiums payable annually in advance throughout the term or until earlier death. In each case, the only benefit was a sum assured of 20,000, payable on survival to the end of the term. During 1996, 4 policyholders died out of the 580 policyholders whose policies were in force at the start of the year. Assuming that the office uses net premium policy reserves, calculate the profit or loss from mortality for 1996 in respect of this group of policies. Basis: mortality: A ultimate interest: 4% per annum [6] 10. (A2 98/2) A life office is considering special ten-year endowment assurance policies to lives aged exactly 55. The basic sum assured is 20,000 payable at the end of the year of death or on survival to age 65. Simple reversionary bonuses are declared at the start of each year and a terminal bonus is paid on survival at age 65. In addition, at the end of each complete year that the policyholder survives, an amount of 1,000 is paid to the policyholder. Premiums are payable annually in advance throughout the term of the policy. (i) Show that the premium for the above policy is 4,284. Basis: mortality: A (select) interest: 4% p.a. expenses Intitial: 500 Renewal: 1% of each premium after the first bonuses Reversionary 5% p.a. Terminal: 50% of basic sum assured [7] (ii) 100 of the special endowment policies commenced on 1 January The office holds reserves in respect of each policy equal to the retrospective policy value calculated on the same basis as in (i). Actual bonus declarations follow 24

25 those assumed in the premium/reserving basis above. One of the policyholders died immediately after paying their first premium. No further deaths occurred during Calculate the mortality profit or loss to the office on these 100 policies for the year [7] (iii) Briefly explain how the profit or loss has arisen. [Total 17] 11. (A2 98/2) A life aged exactly 40 purchases a special premium deferred annuity. The annuity payments are to commence at age 60, and are payable monthly in advance for life. The amount of the first monthly annuity payment is to be 1,000, but once in payment the amount is to increase monthly in line with the rate of inflation. There are no death benefits payable in the event of death during the deferred period. (i) Show that the single premium is 41,706. Basis: mortality: A (select) before age 60 a(55) Males (ultimate) after age 60 interest: 6% p.a. inflation: % p.a. expenses: Initial: 500 Claim: 1% of each annuity payment [6] (ii) The office holds reserves in respect of the policy equal to the prospective gross premium policy values on the following basis: Basis: mortality A (ultimate) before age 60 a(55) Males (ultimate) after age 60 interest: 6% p.a. inflation: % p.a. expenses: Claim: 1% of each annuity payment Calculate the reserve held in respect of the policy at the end of the 10th year, assuming that the life is still alive. [Total 9] 25