EXAMINATION. 6 April 2005 (pm) Subject CT5 Contingencies Core Technical. Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE

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1 Faculty of Actuaries Institute of Actuaries EXAMINATION 6 April 2005 (pm) Subject CT5 Contingencies Core Technical 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. 5. Candidates should show calculations where this is appropriate. 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 the 2002 edition of the Formulae and Tables and your own electronic calculator. CT5 A2005 Faculty of Actuaries Institute of Actuaries

2 1 Explain the difference between a profit vector and a profit signature. [2] 2 A 20-year temporary annuity-due of 1 per annum is issued to a life aged 50 exact. (a) Express the expected present value of the annuity in terms of an assurance function. (b) Hence calculate the value using the mortality table AM92 Ultimate with 4% interest. [3] 3 A life insurance company sells an annual premium whole life assurance policy where the sum assured is payable at the end of the year of death. Expenses are incurred at the start of each policy year, and claim expenses are nil. (a) Write down a recursive relationship between the gross premium provisions at successive durations, with provisions calculated on the premium basis. Define all the symbols that you use. (b) Explain in words the meaning of the relationship. [4] 4 A life insurance company issues an annuity to a life aged 60 exact. The purchase price is 200,000. The annuity is payable monthly in advance and is guaranteed to be paid for a period of 10 years and for the whole of life thereafter. Calculate the annual annuity payment. Basis: Mortality Interest AM92 Ultimate 6% per annum [4] CT5 A2005 2

3 5 A three-state transition model is shown in the following diagram: Alive Sick Dead Assume that the transition probabilities are constant at all ages with = 2%, = 4%, = 1% and = 5%. Calculate the present value of a sickness benefit of 2,000 p.a. paid continuously to a life now aged 40 exact and sick, during this period of sickness, discounted at 4% p.a. and payable to a maximum age of 60 exact. [4] 6 Calculate the probability of survival to age 60 exact using ELT15 (Males) for a life aged 45½ exact using two approximate methods. State any assumptions you make. [5] 7 A joint life annuity of 1 per annum is payable continuously to lives currently aged x and y while both lives are alive. The present value of the annuity payments is expressed as a random variable, in terms of the joint future lifetime of x and y. Derive and simplify as far as possible expressions for the expected present value and the variance of the present value of the annuity. [5] 8 A pension scheme provides a pension on ill-health retirement of 1/80 th of Final Pensionable Salary for each year of pensionable service subject to a minimum pension of 20/80 ths of Final Pensionable Salary. Final Pensionable Salary is defined as the average salary earned in the three years before retirement. Normal retirement age is 65 exact. Derive a formula for the present value of the ill-health retirement benefit for a member currently aged 35 exact with exactly 10 years past service and salary for the year before the calculation date of 20,000. [5] 9 Explain how an insurance company uses risk classification to control the profitability of its life insurance business. [5] CT5 A PLEASE TURN OVER

4 10 You are given the following statistics in respect of the population of Urbania: Males Females Age band Exposed to risk Observed Mortality rate Exposed to risk Observed Mortality rate , , , , , , , , Calculate the directly and indirectly standardised mortality rates for the female lives, using the combined population as the standard population. [6] 11 A life insurance company issues a 25-year with profits endowment assurance policy to a male life aged 40 exact. The sum assured of 100,000 plus declared reversionary bonuses are payable on survival to the end of the term or immediately on death, if earlier. Calculate the monthly premium payable in advance throughout the term of the policy if the company assumes that future reversionary bonuses will be declared at a rate of % of the sum assured, compounded and vesting at the end of each policy year. Basis: Interest Mortality Initial commission Initial expenses Renewal commission Renewal expenses 6% per annum AM92 Select 87.5% of the total annual premium 175 paid at policy commencement date 2.5% of each monthly premium from the start of the second policy year 65 at the start of the second and subsequent policy years Claim expense 2.5% of the claim amount [10] CT5 A2005 4

5 12 (i) By considering a term assurance policy as a series of one year deferred term assurance policies, show that: i A A [5] 1 1 = x: n x: n (ii) Calculate the expected present value and variance of the present value of a term assurance of 1 payable immediately on death for a life aged 40 exact, if death occurs within 30 years. Basis: Interest Mortality Expenses: 4% per annum AM92 Select None [6] [Total 11] CT5 A PLEASE TURN OVER

6 13 A life insurance company issues a 4-year unit-linked endowment assurance contract to a male life aged 40 exact under which level premiums of 1,000 per annum are payable in advance. In the first year, 50% of the premium is allocated to units and 102.5% in the second and subsequent years. The units are subject to a bid-offer spread of 5% and an annual management charge of 0.5% of the bid value of the units is deducted at the end of each year. If the policyholder dies during the term of the policy, the death benefit of 4,000 or the bid value of the units after the deduction of the management charge, whichever is higher, is payable at the end of the year of death. On surrender or on survival to the end of the term, the bid value of the units is payable at the end of the year of exit. The company uses the following assumptions in its profit test of this contract: Rate of growth on assets in the unit fund Rate of interest on non-unit fund cashflows Independent rates of mortality Independent rate of withdrawal Initial expenses Renewal expenses Initial commission Renewal commission Risk discount rate 6% per annum 4% per annum AM92 Select 10% per annum in the first policy year; 5% per annum in the second and subsequent policy years. 150 plus 100% of the amount of initial commission 50 per annum on the second and subsequent premium dates 10% of first premium 2.5% of the second and subsequent years premiums 8% per annum (i) (ii) Calculate the profit margin on the assumption that the office does not zeroise future negative cashflows and that decrements are uniformly distributed over the year. [13] Suppose the office does zeroise future negative cashflows. (a) (b) Calculate the expected provisions that must be set up at the end of each year, per policy in force at the start of each year. Calculate the profit margin allowing for the cost of setting up these provisions. [4] [Total 17] CT5 A2005 6

7 14 (i) Write down in the form of symbols, and also explain in words, the expressions death strain at risk, expected death strain and actual death strain. [6] (ii) A life insurance company issues the following policies: 15-year term assurances with a sum assured of 150,000 where the death benefit is payable at the end of the year of death 15-year pure endowment assurances with a sum assured of 75,000 5-year single premium temporary immediate annuities with an annual benefit payable in arrear of 25,000 On 1 January 2002, the company sold 5,000 term assurance policies and 2,000 pure endowment policies to male lives aged 45 exact and 1,000 temporary immediate annuity policies to male lives aged 55 exact. For the term assurance and pure endowment policies, premiums are payable annually in advance. During the first two years, there were fifteen actual deaths from the term assurance policies written and five actual deaths from each of the other two types of policy written. (a) Calculate the death strain at risk for each type of policy during (b) During 2004, there were eight actual deaths from the term assurance policies written and one actual death from each of the other two types of policy written. Calculate the total mortality profit or loss to the office in the year Basis: Interest Mortality 4% per annum AM92 Ultimate for term assurances and pure endowments PMA92C20 for annuities [13] [Total 19] END OF PAPER CT5 A2005 7

8 Faculty of Actuaries Institute of Actuaries EXAMINATION April 2005 Subject CT5 Contingencies Core Technical 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. M Flaherty Chairman of the Board of Examiners 15 June 2005 Faculty of Actuaries Institute of Actuaries

9 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report 1 The profit vector is the vector of expected end-year profits for policies which are still in force at the start of each year. The profit signature is the vector of expected end-year profits allowing for survivorship from the start of the contract. 1 A 50:20 2 (a) ä 50:20 d (b) 50: A A v p (1 A ) ( ) ä 50: d 3 (a) ' t t x t x t t 1 ( V OP e )(1 i) q ( S) p ( V ) ' where ' t V gross premium provision at time t OP = office premium et expenses incurred at time t i = interest rate in premium/valuation basis S = sum assured p x t is the probability that a life aged x + t survives one year on the premium/valuation mortality basis x t q is the probability that a life aged x + t dies within one year on the premium/valuation mortality basis Page 2

10 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report (b) Income (opening provision plus interest on excess of premium over expense, and provision) equals outgo (death claims and closing provision for survivors) if assumptions are borne out. 4 The value of a pension of 1 p.a. is (12) (12) ä ä where first term is an annuity certain (12) 10 ä 10 1 v (12) d (12) (12) (12) 10 (12) 10 ä60 ä60 ä v 60:10 10 p60ä70 p ä (12) 70 ä 70 11/ / So value of a pension of 1 p.a. is v = So annuity purchased by 200,000 is / = 16,948 Page 3

11 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report 5 The present value is ii t p 40 exp(.05 t) So value is 20 0 t exp ( ) ds 0 t t ii e p dt where ln(1.04) 20 0 t 5% t 2000 e e dt where ln(1.04) , 653 t(.05 ln(1.04)) e (.05 ln(1.04)) Require to calculate 14½ p 45½ ½ p 45½. 14 p 46 p l l (a) Assume deaths uniformly distributed so t p x. x t constant (1 ½) q45 ½ Then ½ q45½ (1 ½ q ) (1 ½.00266) So 14½ p 45½ ( ) (b) Assume that force of mortality is constant across year of age 45 to 46 ½ p 45½ e ½ 45 ln(1 q ) ln( ) ½ p 45½ e ½ So 14½ p 45½ Page 4

12 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report 7 Define a random variable T xy, the lifetime of the joint life status The expected value at a rate of interest i is a xy E( a ) T xy E 1 v T xy T xy 1 E( v ) 1 A xy The variance is var 1 v Txy 1 T var( xy v ) ( Axy ( Axy ) ) 2 2 Axy i where is at (1 ) 1 8 Past Service or t ½ i35 t v z35 t ½ a t ½ t 0 l35 v s z s M D ia Page 5

13 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report Future Service z ia M35 z ia R45 s D35 s D Insurance works on the basis of pooling independent homogeneous risks The central limit theorem then implies that profit can be defined as a random variable having a normal distribution. Life insurance risks are usually independent Risk classification ensures that the risks are homogeneous Lives are divided by risk factors More factors implies better homogeneity But the collection of more factors is restricted by The cost of obtaining data Problems with accuracy of information The significance of the factors The desires of the marketing department Age band 10 Males Females Male Female Total Total Female Total Exposed Exposed Actual Actual Actual Exposed to risk to risk deaths deaths deaths to risk Observed Mortality rate Observed Mortality rate Expected deaths using total mortality rates Expected deaths using female rates Direct Indirect Page 6

14 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report 11 Let P be the monthly premium. Then: EPV of premiums: (12) [40]:25 12Pa P (12) a a [40]:25 [40]:25 (1 25 p[40] v ) (1.06) EPV of benefits: 100,000 (1.06) { (1 ) (1 ) (1 b) 1/ q[40] b v 1 q[40] b v q[40] b v 25 p[40] b v... (1 ) } 100,000 (1 ) where b = ,000 D (1 b) 1/ 2 1 ' 65 ' A i i [40]:25 D[40] 100, / 2 (1.06) ( ) 100, where ' 1.06 i b EPV of expenses: (12) (12) [40]:25 [40]: P P ( a a ) 65[ a 1] P [40]:25 (12) a a (1 p [40]:1 [40]:1 [40] v ) 1 1 (1.06) EPV of claim expense: = Page 7

15 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report Equation of value gives P = P and P = (i) A n : t A x n x:1 t 0 n t 1 0 v 1 x t:1 t 1 t p x A x t:1 1 0 s s x t x t s A v p ds Assuming a uniform distribution of deaths, then s p x t x t s q x t 1 x t:1 x t 1 1 s x t x t 0 0 A v q ds q v ds q iv s 1 x: n n 1 t iv A v. t p x. q x t t 0 i n 1 t 0 v t 1. t px. qx t i A 1 x: n Page 8

16 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report (ii) 1 i 1 i 2 1 x: n x: n x: n var( A ) var( A ) var( A ) i 2 2 A 1 A 1 2 x: n x: n ( ( ) ) : A A v p A v A A 40 v. 40 :30 30 p 40. A v where v = 1/ x: n var( A ) ( ( ) ) ln(1.04) Expected value = i A 1 [40]: ln(1.04) 13 Annual premium Allocation % (1st yr) Risk discount rate 8.0% Allocation % (2nd yr +) Interest on investments 6.0% Man charge Interest on sterling provisions 4.0% B/O spread Minimum death benefit % % 0.50% 5.0% % prm Total Initial expense % 350 Renewal expense % 75 Page 9

17 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report (i) Multiple decrement table x d q x s q x x ( aq ) d x ( aq ) s x ( ap) t 1 ( ap ) Unit fund (per policy at start of year) yr 1 yr 2 yr 3 yr 4 value of units at start of year alloc B/O interest management charge value of units at year end Page 10

18 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report Cash flows (per policy at start of year) yr 1 yr 2 yr 3 yr 4 unallocated premium B/O spread expenses interest man charge extra death benefit end of year cashflow probability in force discount factor expected p.v. of profit premium signature expected p.v. of premiums profit margin 2.76% (ii) (a) To calculate the expected provisions at the end of each year we have (utilising the end of year cashflow figures and decrement tables in (i) above): V V (1.04) ( ap) V V V (1.04) ( ap) V V Page 11

19 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report These need to be adjusted as the question asks for the values in respect of the beginning of the year. Thus we have: Year (ap) 42 = Year (ap) 41 = Year (ap) 40 = (b) Based on the expected provisions calculated in (a) above, the cash flow for years 2, 3 and 4 will be zeroised whilst year 1 will become: = Hence the table blow can now be completed for the revised profit margin. revised end of year cash flow probability in force discount factor expected p.v. of profit profit margin 2.58% 14 (i) The death strain at risk for a policy for year t + 1 (t = 0, 1, 2 ) is the excess of the sum assured (i.e. the present value at time t + 1 of all benefits payable on death during the year t + 1) over the end of year provision. i.e. DSAR for year t + 1 S t 1V The expected death strain for year t + 1 (t = 0, 1, 2 ) is the amount that the life insurance company expects to pay extra to the end of year provision for the policy. i.e. EDS for year t + 1 q( S t 1V ) The actual death strain for year t + 1 (t = 0, 1, 2 t+1 of the death strain random variable ) is the observed value at i.e. ADS for year t + 1 ( S t 1V ) if the life died in the year t to t + 1 = 0 if the life survived to t + 1 Page 12

20 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report (ii) Annual premium for pure endowment with 75,000 sum assured given by: P PE 75, 000 D60 75, a D : Annual premium for term assurance with 150,000 sum assured given by: 150, 000A P P 2P 2P a TA EA PE 45:15 PE 45:15 150, Provisions at the end of the third year: for pure endowment with 75,000 sum assured given by: D PE 60 PE 3V 75, 000 P a D 48: , for term assurance with 150,000 sum assured given by: TA EA PE 3V 3V 3V 150, 000 A ( ) a :12 48:12 150, , , for temporary immediate annuity paying an annual benefit of 25,000 given by: IA 3V 25, 000a 58:2 25, 000( a 1) 58: , 000( a v p a 1) , (1.04) , Page 13

21 Subject CT5 (Contingencies Core Technical) April 2005 Examiners Report Sums at risk: Pure endowment: DSAR = 0 11, = 11, Term assurance: DSAR = 150, = 149, Immediate annuity: DSAR = (47, ,000) = 72, Mortality profit = EDS ADS For term assurance EDS ADS 4985 q 149, , ,340, , ,193, mortality profit = 146, For pure endowment EDS ADS 1995 q 11, , , , , mortality profit = 29, For immediate annuity EDS ADS 995 q 72, , , , , mortality profit = 39, Hence, total mortality profit = 77, END OF EXAMINERS REPORT Page 14

22 Faculty of Actuaries Institute of Actuaries EXAMINATION 7 September 2005 (pm) Subject CT5 Contingencies Core Technical 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. 5. Candidates should show calculations where this is appropriate. 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 the 2002 edition of the Formulae and Tables and your own electronic calculator. CT5 S2005 Faculty of Actuaries Institute of Actuaries

23 1 Describe what is meant by adverse selection in the context of a life insurance company s underwriting process and give an example. [2] 2 Describe how occupation affects morbidity and mortality. [3] 3 A graph of f 0 (t), the probability density function for the random future lifetime, T 0, is plotted on the vertical axis, with t plotted on the horizontal axis, for data taken from the English Life Table No. 15 (Males). You are given that f 0 (t) = t p 0 t 80 and then falls. t. You observe that the graph rises to a peak at around Explain why the graph falls at around t 80. [3] 4 Calculate the value of 1.75 p 45.5 on the basis of mortality of AM92 Ultimate and assuming that deaths are uniformly distributed between integral ages. [3] 5 A population is subject to a constant force of mortality of Calculate: (a) (b) The probability that a life aged 20 exact will die before age exact. The curtate expectation of a life aged 20 exact. [4] (12) 6 Define ä fully in words and calculate its value using PMA92C20 and 60:50:20 PFA92C20 tables for the two lives respectively at 4% interest. [5] CT5 S2005 2

24 7 A life insurance company prices its long-term sickness policies using the following three-state continuous-time Markov model, in which the forces of transition,, and are assumed to be constant: Healthy Sick Dead The company issues a particular long-term sickness policy with a benefit of 10,000 per annum payable continuously while sick, provided that the life has been sick continuously for at least one year. Benefit payments under this policy cease at age 65 exact. Write down an expression for the expected present value of the sickness benefit for a healthy life aged 20 exact. Define the symbols that you use. [5] 8 A life insurance company issues an annuity contract to a man aged 65 exact and his wife aged 62 exact. Under the contract, an annuity of 20,000 per annum is guaranteed payable for a period of 5 years and thereafter during the lifetime of the man. On the man s death, an annuity of 10,000 per annum is payable to his wife, if she is then alive. This annuity commences on the monthly payment date next following, or coincident with, the date of his death or from the 5 th policy anniversary, if later and is payable for the lifetime of his wife. Annuities are payable monthly in advance. Calculate the single premium required for the contract. Basis: Mortality PMA92C20 for the male and PFA92C20 for the female Interest 4% per annum Expenses none [9] CT5 S PLEASE TURN OVER

25 9 A life insurance company issues an annuity policy to two lives each aged 60 exact in return for a single premium. Under the policy, an annuity of 10,000 per annum is payable annually in advance while at least one of the lives is alive. (i) (ii) Write down an expression for the net future loss random variable at the outset for this policy. [2] Calculate the single premium, using the equivalence principle. Basis: Mortality PMA92C20 for the first life, PFA92C20 for the second life Interest 4% per annum Expenses ignored [3] (iii) Calculate the standard deviation of the net future loss random variable at the outset for this policy, using the basis in part (ii). You are given that a 60:60 = at a rate of interest 8.16% per annum. [4] [Total 9] CT5 S2005 4

26 10 A life insurance company issued a with profits whole life policy to a life aged 20 exact, on 1 July Under the policy, the basic sum assured of 100,000 and attaching bonuses are payable immediately on death. The company declares simple reversionary bonuses at the start of each year. Level premiums are payable annually in advance under the policy. (i) (ii) Give an expression for the gross future loss random variable under the policy at the outset. Define symbols where necessary. [3] Calculate the annual premium, using the equivalence principle. Basis: Mortality Interest Bonus loading AM92 Select 6% per annum 3% simple per annum Expenses Initial 200 Renewal 5% of each premium payable in the second and subsequent years Assume bonus entitlement earned immediately on payment of premium. [4] (iii) On 30 June 2005 the policy is still in force. A total of 10,000 has been declared as a simple bonus to date on the policy. The company calculates provisions for the policy using a gross premium prospective basis, with the following assumptions: Mortality AM92 Ultimate Interest 4% Bonus loading 4% per annum simple Renewal expenses 5% of each premium Calculate the provision for the policy as at 30 June [4] [Total 11] CT5 S PLEASE TURN OVER

27 11 A life insurance company issues a three-year unit-linked endowment assurance contract to a male life aged 62 exact under which level annual premiums of 10,000 are payable in advance throughout the term of the policy or until earlier death. 85% 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. There is an annual management charge of 1.25% of the bid value of units. Management charges are deducted at the end of each year, before death 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 20,000, or the bid value of the units allocated to the policy, if greater. On maturity, 115% of the full bid value of the units is payable. The company holds unit provisions equal to the full bid value of the units. It sets up non-unit provisions to zeroise any negative non-unit fund cashflows, other than those occurring in the first year. The life insurance company uses the following assumptions in carrying out profit tests of this contract: Mortality AM92 Ultimate Expenses Initial 600 Renewal 100 at the start of each of the second and third policy years Unit fund growth rate Non-unit fund interest rate Non-unit fund provision basis Risk discount rate 8% per annum 4% per annum AM92 Ultimate mortality, interest 4% per annum 15% per annum Calculate the profit margin on the contract. [14] CT5 S2005 6

28 12 On 1 January 2000, a life insurance company issued joint life whole life assurance policies to couples. Each couple comprised one male and one female life and both were aged 50 exact on 1 January Under each policy, a sum assured of 200,000 is payable immediately on the death of the second of the lives to die. Premiums under each policy are payable annually in advance while at least one of the lives is alive. (i) Calculate the annual premium payable under each policy. Basis: Mortality Interest PMA92C20 for the male PFA92C20 for the female 4% per annum Expenses Initial 1,000 Renewal 5% of each premium payment [5] (ii) On I January 2004, 5,000 of these policies were still in force. Under 100 of these policies only the female life was alive. Both lives were alive under the other 4,900 policies. The company calculates provisions for the policies on a net premium basis, using PMA92C20 and PFA92C20 mortality for the male and female lives respectively and 4% per annum interest. During the calendar year 2004, there was one claim for death benefit, in respect of a policy where the female life only was alive at the start of the year. In addition, one male life died during the year under a policy where both lives were alive at the start of the year. 4,999 of the policies were in force at the end of the year. Calculate the mortality profit or loss for the group of 5,000 policies for the calendar year [9] [Total 14] CT5 S PLEASE TURN OVER

29 13 Under the rules of a pension scheme, a member may retire due to age at any age from exact age 60 to exact age 65. On age retirement, the scheme provides a pension of 1/60 th of Final Pensionable Salary for each year of scheme service, subject to a maximum of 40/60 ths of Final Pensionable Salary. Only complete years of service are taken into account. Final Pensionable Salary is defined as the average salary over the three-year period before the date of retirement. The pension scheme also provides a lump sum benefit of four times Pensionable Salary on death before retirement. The benefit is payable immediately on death and Pensionable Salary is defined as the annual rate of salary at the date of death. You are given the following data in respect of a member: Date of birth 1 January 1979 Date of joining the scheme 1 January 2000 Annual rate of salary at 1 January ,000 Date of last salary increase 1 April 2004 (i) (ii) Derive commutation functions to value the past service and future service pension liability on age retirement for this member as at 1 January State any assumptions that you make and define all the symbols that you use. [12] Derive commutation functions to value the liability in respect of the lump sum payable on death before retirement for this member as at 1 January State any assumptions that you make and define all the symbols that you use. [6] [Total 18] END OF PAPER CT5 S2005 8

30 Faculty of Actuaries Institute of Actuaries EXAMINATION September 2005 Subject CT5 Contingencies Core Technical EXAMINERS REPORT Faculty of Actuaries Institute of Actuaries

31 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report In general, this examination was done well by students who were well prepared. Several questions gave difficulties particularly Question 7 and 12(ii) the latter one being very challenging. To help students comments are attached to those questions where particular points are of relevance. Absence of comments can be indicate that the particular question was generally done well. 1 Adverse selection is the manner in which lives form part of a group, which acts against a controlled process of selecting the lives with respect to some characteristic that affects mortality or morbidity. An example is where a life insurance company does not distinguish between smokers and non-smokers in proposals for term assurance cover. A greater proportion of smokers are likely to select this company in preference to a company that charges different rates to smokers and non-smokers. This would be adverse to the company s selection process, if the company had assumed that its proportion of smokers was similar to that in the general population. Other examples were credited. 2 Occupation can have several direct effects on mortality and morbidity. Occupation determines a person s environment for 40 or more hours each week. The environment may be rural or urban, the occupation may involve exposure to harmful substances e.g. chemicals, or to potentially dangerous situations e.g. working at heights. Much of this is moderated by health and safety at work regulations. Some occupations are more healthy by their very nature e.g. bus drivers have a sedentary and stressful occupation while bus conductors are more active and less stressed. Some work environments e.g. pubs, give exposure to a less healthy lifestyle. Some occupations by their very nature attract more healthy workers. This may be accentuated by health checks made on appointment or by the need to pass regular health checks e.g. airline pilots. Some occupations can attract less healthy workers, for example, former miners who have left the mining industry as a result of ill health and then chosen to sell newspapers. This will inflate the mortality rates of newspaper sellers. A person s occupation largely determines their income, which permits them to adopt a particular lifestyle e.g. content and pattern of diet, quality of housing. This effect can be positive or negative e.g. over-indulgence. Other appropriate examples were credited. 3 As t increases, t increases, but t p 0 decreases. At t = 80 approximately, the decrease in t p 0 is greater than the increase in t, hence f0 t t p 0 t decreases. A deceptively straightforward answer which many students struggled to find. The key point is to compare the 2 parameters as shown. Page 2

32 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report p p45.5 * p46 * 0.25 p47 1 q q 45 *(1 q )*( q ) *( )*(1 0.25* ) 1 0.5* * * (a) The required probability is dt e 1 e (b) The curtate expectation is k dt k e k p20 e e k 1 k 1 k 1 1 e (12) ä is the present value of 1 p.a. payable monthly in advance while two lives 60:50:20 aged 60 and 50 are both still alive, for a maximum period of 20 years. (12) (12) 20 (12) 60:50:20 60: :50 80:70 ä ä v p ä 20 ( a 11 60:50 ) v p60:50 ( a80:70 ) ( ) v ( ) Page 3

33 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report t 1 0 t t 1 20 t 0 u 21 t hh ss t u ss 7 EPV = 10,000 p * * p * e p du dt where is the force of interest t 20 hh p is the probability of a healthy life aged 20 being healthy at age 20 t 1 p20 ss t is the probability that a life who is sick at age 20 t is sick continuously for one year thereafter u 21 ss t p is the probability that a life who is sick at age 21 t is still sick at age 21 t u This question was not done well and few students obtained the whole result. Partial credits were given for correct portions. There were other potentially correct approaches which were credited provided proper definitions of symbols given. 8 Premium = 20, 000 D 10, l D l D a a a a D65 l65 D62 l65 D62 a 12 5 D D v a l l a D D v ) 67 70:67 a a a ( ) ( ) Premium = 265, , = 300,308 to nearer Page 4

34 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report 9 (i) 10, 000a A B max K60 1, K60 1 and P is the single premium. P, where A and B refer to the first and second lives A B A B (ii) P 10, 000 a60 a60 a60 * a 60 10,000* ,940 Variance = 10 d A 2 A B A A B 60 :60 60 : * * * standard deviation = 22, (i) The gross future loss random variable is 100,000 1 T20 T K20 1 K20 1 bk v I ea fv Ga where b is the annual rate of bonus I is the initial expense e is the annual renewal expense and f is the claim expense G is the gross annual premium (ii) The premium is given by Ga 20,000A 20 3,000 IA Ga G * ,000 *1.06 * ,000 *1.06 * G *( ) 1] G G Page 5

35 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report (iii) The required provision is 110,000A a 23 4,000 IA * * ,000 *1.04 * *684.49* ,000* * , , , , Unit fund Year Fund at the start of the year Premium Allocation to units Interest Management charge Fund at the end of the year Non-unit fund before provisions Year Premium margin Expenses Interest Death cost Maturity cost Management charge Profit Provision required at the start of year 3 = ( (1 p 64 )) / 1.04 = Reduced profit at the end of year 2 = *p 63 = Page 6

36 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report Revised profit vector: , ,0 p p * p62 Net Present Value = p62 2 p62 Present value of premiums = 10000* Profit margin = % Most students completed the tables satisfactorily in this question but struggled to get the revised profit vectors. Very few produced a complete result. 12 (i) Let P be the annual premium. 0.95* P* a * A m f m f 50 :50 50 :50 a a a a m f 50 :50 m f m f : m f m f m f 50 :50 50 :50 50 :50 A 1.04 * A 1.04 *(1 d * a ) *( *20.694) * P* * P 2, (ii) From part (i) the net premium is: * (1.04) * a 1 m f 50 :50 d at 4% = * (1.04) * = We require 3 provisions at end of 5 th policy year Page 7

37 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report Both lives alive * (1.04) * 1 a 0.5 m f 55 :55 a m f 50 :50 = * (1.04) = Male only alive A a m m = * (1.04) * 1 * * = Female only alive A a f f = * (1.04) * 1 * * = Mortality Profit Loss = Expected Death Strain Actual Death Strain In this case there are 4 components: (a) Both lives die during 2004 no actual claims Result = 0.5 (4900 * q m * q f 0)( * ) = (4900 * * ) ( ) = (b) Female alive at begin 2004, death during actual claim Result = 0.5 (100 * q f 1)( * ) 54 Page 8

38 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report = (100 * ) ( ) = (c) Both lives alive beginning 2004, males only die during actual claim. Here the claim cost is the change in provision from joint lives to female only surviving i.e. Result = (4900 * q * q 1)( ) m f = (4900 * * ) ( ) = (d) Both lives alive beginning 2004, females only die during 2004 no actual claims. Claim cost change in provision from joint lives to male only surviving Result = (4900 * p * q 0)( ) m f = (4900 * * ) ( ) = Hence overall total = = i.e. a mortality loss of when rounded. For part (i) assuming renewal expenses did not include the first premium (answer ) was also fully acceptable. Part (ii) was very challenging and very few students realised the extension of mortality profit/loss extended to joint life contracts involved reserve change costs on first death. Most just considered the first 2 components of the answer and in many cases failed to correctly cost this part. A few exceptional students did manage to reach the final result. 13 (i) Define a service table: l 26 t = no. of members aged 26 + t last birthday r 26 t = no. of members who retire age 26 + t last birthday sx 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 Page 9

39 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report 1 r Define z26 t s26 t 3 s26 t 2 s26 t 1 ; a 26 t = value of annuity of 1 p.a. 3 to a retiree aged exactly 26 + t. Past service: Assume that retirements take place uniformly over the year of age between 60 and 65. Retirement for those who attain age 65 takes place at exact age 65. Consider retirement between ages 26 + t and 26 + t + 1, 34 t 38. The present value of the retirement benefits related to past service: 50000*5 z r 50000*5 C t 2 z ra 26 t v 26 t r 26 t a t s 2 s v l26 D26 where z ra 26 t 1 2 r C26 t z26 t 1 v r26 ta t 2 and s 26 D26 s25.25v l 26 For retirement at age 65, the present value of the benefits is: z 65 z ra 65 v r 65 r 65 a C s v l26 D * *5 60 s 60 where z ra 65 r C65 z65v r65 a 65 Summing over all ages, the value is: 50000*5 60 z s M D ra where 39 z ra z ra M60 C26 t t 34 Future service: Assume that retirements take place uniformly over the year of age, between ages 60 and 65. Retirement at 65 takes place at exactly age 65. If retirement takes place between ages 60 and 61, the number of future years service to count is 34. If retirement takes place at age 61 or after, the number of future years service to count is 35. For retirement between ages 60 and 61, the present value of the retirement benefits is: Page 10

40 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report 34*50000 z r 34*50000 C 60 s z ra 60 v 60 i 60 a s v l26 D26 For retirement at later years, the formula is similar to the above, with 35 in place of 34. Adding all these together gives: z ra 60 35( z ra z ra C C C s 65 ) 60 D z ra = M s D 26 where z 5 M60 ra 35* z C60 ra t z C60 ra t 0 (ii) Define a service table, with l 26 t and sx t / s x defined as in part (i). In addition, define d 26 t as the number of members dying age 26 t last birthday. Assume that deaths take place on average in the middle of the year of age. The present value of the death benefit, for death between ages t 1, is t and 26 t t v 26 t s 26 d t s 26 s v l26 D * 4* s d 50000* 4* C where s d 26 t 1 C 2 26 t s26.25 t v d26 t Adding the present value of benefits for all possible years of death gives 38 s C26 d t s M 26 d s s t 0 D26 D * 4* * where 38 s d s d M 26 C26 t t 0 Examiners felt that this question was quite simple provided students constructed proper definitions and followed them through logically allowing of course for the adjusted salary scale. The above answer is one of a number possible and full credit was given for credible alternatives. Page 11

41 Subject CT5 (Contingencies Core Technical) September 2005 Examiners Report Many students, however struggled with this question despite these remarks. END OF EXAMINERS REPORT Page 12

42 Faculty of Actuaries Institute of Actuaries EXAMINATION 5 April 2006 (pm) Subject CT5 Contingencies Core Technical 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. 5. Candidates should show calculations where this is appropriate. 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 the 2002 edition of the Formulae and Tables and your own electronic calculator. CT5 A2006 Faculty of Actuaries Institute of Actuaries

43 1 It is possible to model the mortality of current active members of a pension scheme using the following three-state continuous-time Markov model, with age-dependent forces of transition x, x and x : x Active Retired x x Dead A pension scheme provides a benefit of 10,000 payable on death regardless of whether death occurs before or after retirement. Give an expression to value this benefit for an active life currently aged x. [2] 2 (i) In the context of with-profit policies, describe the super compound method of adding bonuses. [2] (ii) Suggest 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 Using the PMA92C20 table for both lives calculate: (a) 65:60 (b) 5 p 65:60 (c) 1 2q 65:65 [4] 4 State the main difference between an overhead expense and a direct expense incurred in writing a life insurance policy and give an example of each. [4] CT5 A2006 2

44 5 A life office issues term assurance policies to 500 lives all aged 30 exact with a term of 25 years. The benefit of 10,000 is payable at the end of the year of death of any of the lives into a special fund. Calculate the expected share of this fund for each survivor after 25 years. Basis: Mortality Interest AM92 Select 4% per annum [4] 6 A life office has issued for a number of years whole-life regular premium policies to a group of lives through direct advertising. Assured lives are only required to complete an application form with no further evidence of health. Outline the forms of selection that the insurer should expect to find in the mortality experience of the lives. [5] 7 (i) Show that: t p p ( ) s x t s x t x t x t s [2] (ii) Prove Thiele s differential equation for a whole-life assurance issued to a life aged x to be as follows: t V (1 V ) V Px t x t x x t t x [4] [Total 6] 8 (i) Calculate the expected present value of an annuity-due of 1 per annum payable annually in advance until the death of the last survivor of two lives using the following basis: First life: male aged 70, mortality table PMA92C20 Second life: female aged 67, mortality table PFA92C20 Rate of interest: 4% per annum [2] (ii) Give an expression for the variance of the annuity-due in terms of annuity functions. [5] [Total 7] CT5 A PLEASE TURN OVER

45 9 (i) Express fully in words: a [3] xy: n (ii) Express a as the expected value of random variables and hence show that xy: n a xy: n 1 A xy: n [4] [Total 7] 10 A 20-year special endowment assurance policy is issued to a group of lives aged 45 exact. Each policy provides a sum assured of 10,000 payable at the end of the year of death or 20,000 payable if the life survives until the maturity date. Premiums on the policy are payable annually in advance for 15 years or until earlier death. You are given the following information: Number of deaths during the 13 th policy year 4 Number of policies in force at the end of the 13 th policy year 195 (i) Calculate the profit or loss arising from mortality in the 13 th policy year. [7] (ii) Comment on your results. [2] Basis: Mortality Interest Expenses AM92 Ultimate 4% per annum none [Total 9] 11 An employer wishes to introduce a lump-sum retirement benefit payable immediately on retirement at 65 or earlier other than on the grounds of ill-health. The amount of the benefit is 1,000 for each year of an employee s service, with proportionate parts of a year counting. (i) (ii) (iii) Give a formula to value this benefit for an employee currently aged x with n years of past service, defining all terms used. [5] Using the Pension Scheme Tables from the Actuarial Formulae and Tables, calculate the value for an employee currently aged 30 exact with exactly 10 years past service. [2] Calculate the level annual contribution payable continuously throughout this employee s service to fund the future retirement benefit. [3] [Total 10] CT5 A2006 4

46 12 (i) Define the following terms without giving detailed formulae: (a) (b) (c) Crude Mortality Rate Directly Standardised Mortality Rate Indirectly Standardised Mortality Rate [3] (ii) The data in the following table are taken from data published by the Office of National Statistics in England and Wales Population Number of births Tyne and Wear Population Number of births Under 25 3,149, ,000 71,000 4, ,769, ,000 74,000 6, ,927, ,000 82,000 1,000 (a) (b) Using the population for England and Wales as the standard population calculate crude birth rates and the directly and indirectly standardised birth rates for Tyne and Wear. State an advantage of using the Indirectly Standardised Birth Rate and comment briefly on the answers you have obtained. [8] [Total 11] CT5 A PLEASE TURN OVER

47 13 A life aged 35 exact purchases a 30-year with-profit endowment assurance policy. Level premiums are payable monthly in advance throughout the duration of the contract. The sum assured of 250,000 plus declared reversionary bonuses are payable at maturity or at the end of the year of death if earlier. (i) Show that the monthly premium is if the life insurance company assumes that future simple reversionary bonuses will be declared at the rate of 2% per annum and vesting at the end of each policy year (i.e. the death benefit does not include any bonus relating to the policy year of death). Basis: mortality interest initial expenses renewal expenses claims expenses AM92 Select 4% per annum 250 plus 50% of the gross annual premium 3% of the second and subsequent monthly premiums 300 on death; 150 on maturity [7] (ii) At age 60 exact, immediately before the premium then due, the life wishes to surrender the policy. The life insurance company calculates a surrender value equal to the gross retrospective policy value, assuming the same basis as in (i) above. Calculate the surrender value using the retrospective policy value at the end of the 25 th policy year immediately before the premium then due and just after the declared bonus has increased the sum assured plus reversionary bonuses to 375,000. Assume that the life insurance company has declared a simple bonus throughout the duration of the policy consistent with the bonus loading assumption used to derive the premium in (i) above. [6] (iii) State with a reason whether the surrender value would have been larger, the same or smaller than in (ii) above if the office had used the prospective gross premium policy value, on the same basis. [1] [Total 14] CT5 A2006 6

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