Heriot-Watt University BSc in Actuarial Mathematics and Statisticsandrewc/lim1/problems2.pdf · 2000. 11. 13. · A company issues 20 identical policies to lives aged 60. The death

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 oldsubject A2 corresponds roughly to Life Insurance Maths I (and Survival Models in 2ndyear). In the question codes 94/1 means the first diet of exams (April) in 1994 and 96/2means the second diet of exams (September) in 1996.

If you find any errors in the questions then please let Andrew Cairns know sothe the version kept on the web pagehttp://www.ma.hw.ac.uk/∼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 lifeaged 21 will die, using A1967-70 mortality?

A 74

B 75

C 76

D 79

[2]

2. (A2 94/1)

A life aged 40 effects a with profit whole life assurance with sum assured of £500plus attaching bonuses, payable immediately in death. Assuming allowance forcompound bonuses of 4% per annum, vesting at the end of each policy year, whichof the following gives the level annual premium payable throughout life?

Basis: mortality A1967-70 ultimateinterest 4% per annumexpenses none

A £25.45

B £25.95

C £26.46

D £26.99

1

[3]

3. (A2 94/2)

Using A1967-70 mortality, which of the following gives the value of 4|2q[60]?

A 0.03895

B 0.04226

C 0.04286

D 0.04763

[2]

4. (A2 94/2)

Which of the following is the value of A[30]:

1

15using A1967-70 mortality, with interest

at 3% per annum?

A 0.54529

B 0.54546

C 0.63033

D 0.63053

[3]

5. (A2 94/2)

Which of the following is equal to 15V1

25:30?

A 1D40

(M25 −M40 − (M25 −M55)(N25 −N40)

(N25 −N55)

)

B 1D40

(M40 −M55 − (M25 −M55)(N40 −N55)

(N25 −N55)

)

C 1D40

(M40 −M55 + D55 − (M55 −M55 + D55)(N40 −N55)

(N25 −N55)

)

D 1D40

(M25 −M40 + D55 − (M25 −M55 + D55)(N25 −N40)

(N25 −N55)

)

[3]

6. (A2 94/2)

A life office issues four insurance policies to identical lives aged x. The first pur-chases a whole life policy, the second an endowment assurance policy with a 20-yearterm, the third a term assurance policy with a 20-year term and the fourth a pureendowment 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 basesfor the four contracts.

2

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

[3]

7. (A2 95/1)

On 1 January 1994 a life office held a portfolio of 200 whole life assurance policieson lives then aged 45. The policies were all issued on 1 January 1979, each withsum assured £10,000 payable at the end of the year of death. The office holds netpremium reserves for these contracts, using A1967-70 ultimate mortality at 4% p.a.interest.

Three policyholders die during 1994. Which of the following is the actual deathstrain on this set of policies in 1994?

A £20,241B £24,559C £25,066D £30,000

[3]

8. (A2 95/1)

An annual premium with-profit endowment assurance contract is issued to a lifeaged 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 thefollowing gives the net premium policy value immediately before the 4th premiumis due, given that bonuses of 3% per annum have been declared annually in advancefor each of the contract?

Basis: Mortality A1967-70 ultimateInterest 3% per annum

A £1,163B £1,271C £1,411D £1,735

[3]

9. (A2 95/2)

3

Which of the following gives the value of A[50]:20 using A1967-70 mortality with 4%interest?

A 0.4845

B 0.5000

C 0.5015

D 0.5066

[3]

10. (A2 95/2)

A life table with two year select period is being constructed. Ultimate values of `x

will be the same as those for ELT 12 Males, while q[x] = 0.3qx and q[x]+1 = 0.6qx+1.

Which of the following gives the value of `[70]?

A 51,312

B 51,418

C 51,013

D 52,013

[3]

11. (A2 95/2)

An impaired life age 30 is assumed to be subject to an extra risk such that the forceof mortality experienced by the life is double the force of mortality of ELT12-Malesat all ages.

Which of the following gives the probability that the life does not survive to age 50?

A 0.00296

B 0.10579

C 0.10875

D 0.23318

[3]

12. (A2 95/2)

A 30-year with profit endowment assurance policy with a basic sum assured of £1000is issued to a life aged 30. Simple bonuses of 3% of the basic sum assured vest atthe end of each policy year if the life survives. Death benefits are paid at the endof the year of death. Which of the following formulae give the correct expectedpresent value of the benefits?

I 1D30

(1000M30 + 30R30 − 30R60 − 1900M60 + 1900D60)

4

II 1D30

(1000M30 + 30R31 − 30R60 − 1870M60 + 1900D60)

III 1D30

(970M30 + 30R30 − 30R60 − 1870M60 + 1900D60)

A I and II are correct

B II and III are correct

C I only is correct

D III only is correct

[2]

13. (A2 96/1)

Which of the following is the variance of the random variable which has expectedvalue (IA)1

x:nat an effective rate of interest of i per annum, υ = (1 + i)−1?

A∫ n0 t2vt

tpxuxtdt− ((IA)1x:n)2

B∫ n0 t2v2t

tpxux+tdt− ((IA)1x:n)2

C∫ n0 t2u2t

tp2xµx+tdt− ((IA)1

x:n)2

D∫ n0 t2v2t

tp2xµ

2x+tdt− ((IA)1

x:n)2

[2]

14. (A2 96/1)

Which of the following gives the probability that a life aged exactly 65 will diebetween exact age 66 and exact age 70, assuming mortality follows the A1967-70select table.

A 0.07837

B 0.08848

C 0.10619

D 0.11831

[3]

15. (A2 96/2)

Which of the following is the value of a[57]:7 using A1967-70 mortality, and an effec-tive rate of interest of 4% per annum?

A 5.758

B 5.762

C 6.067

D 6.424

[2]

5

16. (A2 96/2)

An impaired life aged 50 is assumed to be subject to an extra risk such that theforce of mortality experienced by the life is equal to the force of mortality of theELT 12 Males life table at the same age, plus a constant addition of 0.04. Whichof the following gives the probability that the life does not survive to age 60?

A 0.18483

B 0.40813

C 0.41273

D 0.59187

[3]

17. (A2 96/2)

A life table with a one-year select period is being constructed. Ultimate values oflx will be the same as for the A1967-70 life table. During the select period the forceof mortality will be one-half of the ultimate force of mortality for an individual ofthe same age. Which of the following is the value l[100]?

A 55.468

B 94.380

C 53.565

D 91.142

[3]

18. (A2 97/1)

A one year term assurance is issued to a life aged 50 for a sum assured of £100,000payable at the end of the year of death. Which of the following gives the standarddeviation of the present value of the term assurance, using A1967-70 select mortalityand 7% per annum interest?

A 4993

B 5244

C 5867

D 6452

[3]

19. (A2 97/1)

A company issues 20 identical policies to lives aged 60. The death strain at riskon each policy in the first year of the contract is £15,150. The insurer assumesthat mortality follows the a(55) ultimate males mortality table. In the first year of

6

the contract 2 lives die. Which of the following gives the mortality loss in the firstyear?

A 23,434

B 26,052

C 29,156

D 30,300

[2]

20. (A2 97/1)

the reserve held for a policy at duration t for a life then age 40, is tV = £15, 000,assuming A1967-70 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 + 1if the life dies during the year, and £1,000 payable at t + 1 if the life survives theyear.

Which of the following gives the value of t+1V ?

A 14,044

B 15,021

C 15,043

D 16,128

[2]

21. (A2 97/2)

A life aged 40 purchases a with profit whole life assurance with sum assured £2,000plus attaching bonuses, payable at the end of the year of death. Assuming allowancefor simple bonuses of 3% per annum, vesting at the start of each policy year, whichof the following gives the level annual premium payable throughout life?

Basis: mortality - A1967-70 ultimateinterest - 4% per annumexpenses - none

A £54.30

B £55.16

C £55.98

D £56.03

[3]

7

22. (A2 98/1)

A life aged exactly 50 effects a with profit whole life assurance policy with sumassured 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 thepolicyholder is still alive at the time. Which of the following gives the singlepremium payable at the outset?

Basis: mortality: A1967-70 ultimateinterest: 5% per annumcompound reversionary bonus rate: 1.94175% of the sum

assured each year

A £4,697

B £4,767

C £4,788

D £4,813

[3]

23. (A2 98/2)

An impaired life aged exactly 60 is subject to the mortality of a(55) males (select)with an addition of 0.03774 to the force of mortality. Which of the following is theexpected 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 £371.9

C £635.4

D £673.8

[3]

8




Extra Problems: Profit and random variables


1. (A2 94/2)

An annuity of £1000 per annum is payable annually in arrears to a life aged 60 for amaximum of 3 years, ceasing in earlier death. Calculate the standard the deviationof 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 £10,000payable at the end of the year of death and premiums of £220 payable annually inadvance.

Given that A[50] evaluated at a rate of interest of 10.25% per annum is 0.12495,calculate the mean and variance of the initial net present value of the policy.

Basis: mortality: A1967-70 (select)interest: 5% p.a.

[8]

9




Extra Problems: Premiums


1. (A2 95/1)

A life offices issues a special ten year policy to a life aged exactly 55 under which theannual premium increases by £100 each year. The sum assured payable on deathwithin ten years is £20,000 payable at the end of the year of death. On survival tothe end of the ten year term the policyholder receives a refund of all the premiumspaid without interest.

Show that the first premium payable is £934.36

Mortality: A1967-70 UltimateInterest: 4%Expenses: Initial: £150

Renewal: 4% of each premium including the first

[7]

2. (A2 96/1)

Prove that (IA)x:n − ax:n − d(Ia)x:n.

[4]

3. (A2 96/1)

A life office issues a deferred annuity contract to a life aged exactly 40. Premiumsare payable annually in advance for 20 years or until earlier death. On survival toage 60 the annuity of £1000 per annum is payable quarterly in advance throughoutlife. In the event of death during the deferred period a lump sum is payable at theend of the year of death equal to the total premiums paid to date.

Calculate the annual premium.

10

Basis: mortality before age 60 A1967-70 ultimateafter 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 payableimmediately on death. Level monthly premiums are payable for the first 20 yearsof the contract. No premiums are payable thereafter. Calculate the monthlypremium using the following basis:

Mortality A1967-70 selectInterest 6% per annum effectiveExpenses 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 ofthe 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. Thepremium basis is:

mortality: A1967-70 selectinterest: 4% p.a. effectiverenewal expenses: £10 incurred at each premium

date, including the first

(i) Show that the premium is £148.38.

[5]

6. (A2 97/1)

A life table with a select period of 2 years is based on rates of mortality which satisfythe following relationship:

q[x−r]+r =1

3− r× qx(for all values of x, and r = 0, 1)

q60 = 0.0195, q61 = 0.0198, q62 = 0.0200 and l63 = 100, 000.

(i) Calculate

(a) l62

11

(b) l[60]+1

(c) l[60]

[4]

(ii) A select life aged 60 subject to the mortality table described in (i) above pur-chased 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 earlierdeath. 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 theoffice on the contract, assuming an effective rate of interest of 6% p.a., andignoring expenses.

[4]

[Total 8]

7. (A2 97/1)

A life office issues a 20 year with-profit endowment assurance policy to a life aged40. The sum assured of £20,000 plus declared reversionary bonuses are payableimmediately on death, or on survival to the end of the term.

(i) Calculate the quarterly premium payable throughout the term of the policy ifthe office assumes that future reversionary bonuses will be declared at the rateof 1.92308% of the sum assured, compounded and vesting at the end of eachpolicy year.

Basis: Mortality: A1967-70 selectInterest: 6% per annumInitial expenses: 114% of the first premium and

2.5% of the basic sum assuredRenewal 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 £200,000on 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 £5,000each year until age 65.

On attaining age 65 the life receives an annuity of £10,000 per annum payablemonthly in arrear.

Calculate the monthly premium.

12

Basis: mortality up to age 65 - A1967-70 selectover 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 quarterlypremiums 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 £704.61.

Basis: mortality - A1967-70 selectinterest - 4% p.a.intitial expenses - £250 plus 60% of the gross

- annual premiumrenewal expenses 3% of the second and subsequent

quartely premiumclaims expenses - £500 on death; £100 on maturity

[6]

10. (A2 98/1)

A life insurance company issues a special 20 year endowment assurance policy to alife 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.

(i) Calculate the annual premium.

Basis: mortality: A1967-70 selectinterest: 4% per annumexpenses: 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 immedi-ately on death to a life aged exactly 50. The sum assured is £50,000 in the first

13

year of the policy, £55,000 in the second year of the policy, £60,000 in the thirdyear, etc.

Ignoring expenses, calculate the quarterly premium payable in advance.

Basis: mortality: A1967-70 (select)interest: 4% p.a.

[6]

14




Extra Problems: Impaired lives


1. (A2 94/1)

An impaired life aged exactly aged exactly 55 wishes to effect a without profitendowment assurance for sum assured of £1,000 payable at the end of 10 years or atthe end of the year of earlier death. Level annual premiums are payable throughoutthe term of the policy.

Special terms are offered on the assumption that the life will experience mortalitywhich can be represented by:

for the first five years, a constant addition of 0.009569 to the normal force ofmortality, 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 A1967-70 ultimateinterest 4% per annumexpenses none

[9]

(ii) As an alternative, the life office suggests effecting a policy providing the samedeath cover but giving an increased sum assured of £2000 on survival to the endof 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 thepremium basis.

(b) Explain briefly why such a policy could be issued at normal rates to thisimpaired life.

[10][Total 19]

15

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 amaximum of 20 years. Calculate the monthly premium

Basis: mortality A1967-70 ultimate with a 5 year deduction from ageup to age 65, a(55)-females ultimate from age 65.

interest 4% per annuminitial expenses 1% of the sum assuredrenewal 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 weeklythroughout the term of the contract.

Calculate the weekly premium payable, using the tables provided.

Basis: mortality A1967-70 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 plus5% 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 lifeassurance 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 deathunder this policy.

The explorer is about to undertake a hazardous expedition which will last threeyears. The life office estimates that during these three years the explorer willexperience a constant addition of 0.02871 to the normal force of mortality, but afterthree years will experience normal mortality. The life office quotes a level extrapremium payable for the first three years.

Calculate this level extra premium.

Basis: normal mortality A1967-70 ultimateinterest 3% per annumexpenses none

[7]

16

5. (A2 96/1)

An impaired life aged exactly 40 wishes to effect an endowment assurance policywith a term of 25 years. The sum assured of £10,000 is payable immediately ondeath or at the end of 25 years if earlier. Premiums are payable monthly in advancefor 25 years or until earlier death.

It is assumed that the life will experience the mortality of a life ten years older forthe 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 untilearlier death

Calculate the extra monthly premium.

[13]

(ii) The office offers the life an alternative of paying the standard premium butwith a level reduction in the sum assured for the first ten years only.

Calculate the reduction.

[4]

Basis: standard mortality A1967-70 selectinterest 4% per annuminitial expenses £200renewal 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 theforce of mortality of non-smokers, which is taken from the A1967-70 ultimate lifetable.

Calculate the difference between the median future lifetimes of a non-smoker and asmoker, both aged exactly 50.

[5]

7. (A2 98/1)

A life insurance company issues a 20 year endowment assurance policy to an im-paired life age exactly 45.

The sum assured is £60,000 and death benefits are payable at the end of the yearof death.

The company assumes that the person will be subject to the following mortality:

First ten years: standard mortality with constant additionto the force of mortality of 0.009569

Second ten years: standard mortality for a life three yearsolder than the actual age

17

The contract is issued at the company’s standard rate of premium, with premiumspayable annually in advance until age 65 or earlier death. The death benefit issubject to a level debt of £X for the first 10 years of the contract and £0.9X for thesecond 10 years.

(i) Show that the standard annual premium is £2,260.04.

Basis: mortality: A1967-70 selectinterest: 4% per annumexpenses: initial: 25% of the annual

premiumrenewal: 4% of each premium,

excluding the first

[3]

(ii) Calculate the debt, £X.

[14]

(iii) Without doing any further calculations, explain how the size of the debt wouldchange if the age rating used in the second half of the term had been five ratherthan three years.

[2]

[Total 19]

8. (A2 98/2)

An impaired life aged exactly 30 suffers five times the force of mortality of a life ofthe same age subject to standard mortality.

(i) A two-year term assurance policy is issued to both the impaired life and astandard life with a death benefit of £10,000 payable at the end of the year ofdeath.

Calculate the single premium payable for:

(a) the standard life and

(b) the impaired lifeBasis: standard mortality: A1967-70(ultimate)

interest: 4% p.a.expenses: none

[4]

(ii) A two-year endowment assurance policy is issued to both the impaired life anda standard life with a death benefit of £100,000 payable at the end of the yearof death or on survival at the end of the two years.

Calculate the single premium payable for:

(a) the standard life and

18

(b) the impaired life[3]

(iii) Comment on the relative sizes of the impaired life and standard life premiumsfor the policies described in (i) and (ii) above.

[3]

[Total 10]

19




Extra Problems: Reserves


1. (A2 94/1)

On 1 January 1974 a life office issued a number of 25 year endowment assurancepolicies, with annual premiums payable throughout the term, to lives aged 40. Asat 31 December 1993, a total sum assured of £100,000 remained in force. During1993 sums assured of £2,000 became death claims, paid at the end of the year, andno 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 yearending 31 December 1993

Basis: mortality A1967-70 ultimateinterest 4% per annum

[6]

2. (A2 94/2)

(i) State, and explain by general reasoning, a recursive relationship between thegross premium reserves at successive durations for an annual premium, wholelife assurance, with benefit payable at the end of the year of death. Assumeexpenses 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 exactly50. Premiums are payable annually in advance, and death benefits are paid atthe end of the year of death. The office calculates reserves using gross premiumpolicy values; the following information gives the reserve assumptions for thepolicy year just completed; expenses are assumed to be incurred at the startof the policy year:

20

reserve brought forward £4,750annual premium £570annual expenses £30death benefit £20,000mortality A1967-70 ultimateinterest 7% per annum

Calculate the year end reserve.

[3]

[Total 7]

3. (A2 96/1)

On 1 January 1981 a life insurer issued a number of whole life policies to lives thenaged exactly 45. Premiums for all of the contracts are payable annually in advancethroughout the duration of the contract. The office holds reserves for these policiesof the net premium policy values, using A1967-70 ultimate mortality at an effectiverate 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 sumassured 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, inrespect of policies in force on 1 January 1995 is £63,642.

[2]

(ii) Calculate the total death strain at risk for these policies in 1995.

[1]

(iii) Calculate the total expected death strain.

[2]

(iv) Calculate the profit or loss from mortality in 1995.

[3]

[Total 8]

4. (A2 96/2)

An endowment assurance issued to a life age x has a term of n years. Premiumsare payable annually in advance for s years or until earlier death, where s < n. Thesum assured of £1 is payable at the end of the year of death or at the end of the nyear term if earlier.

(i) Write down expressions at integral duration t < s for:

(a) the prospective net premium policy value;

(b) the retrospective net premium policy value.[2]

21

(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. Thesum assured is payable at the end of the year of death, or at the end of the 25-yearterm if earlier. The premium assumptions used by the office for this contract are:

mortality: A1967-70 ultimateinterest: 6% per annumreversionary bonus: 1.92% p.a. compound, vesting at the start of each yearpremium expenses: 5% of each premium (including the first)per policy expenses: incurred at each premium payment date; the first quarterly

amount is £20, with subsequent payments increasing at anassumed rate of inflation of 2.91% p.a.

(i) (a) Show that the initial guaranteed sum assured for a premium of £250 perquarter 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 0.5912% p.a. over theterm of the contract.

[12]

(c) Explain briefly why the contract may be attractive to investors in spite ofthe very low minimum rate of return at maturity.

(ii) During each of the first 5 years of the contract the office declares compoundreversionary bonuses of 5%. The policy expenses increase at a rate of 4% perannum.

(a) Calculate the prospective gross premium policy value at the end of thefifth year of the contract, using the premium basis.

(b) At the fifth policy anniversary, immediately before payment of the pre-mium then due, the policyholder wishes to alter the policy to a without-profit endowment assurance, with the same premium and remaining termas the original contact. Calculate the revised sum assured for the con-tract, using the premium basis above for the remaining term, except thatthere are assumed to be no further per policy expenses, and no additionalcosts of alteration.

[7][Total 19]

6. (A2 97/1)

22

(i) State the conditions which are necessary for the retrospective policy value ofa contract to be equal to the prospective policy value throughout the term ofthe policy.

[2]

(ii) Consider a term assurance contract with a term of n years and a sum assuredof £1 on a life aged x, payable immediately on death. Premiums are payableannually in advance. Assuming that the conditions referred to in (i) hold,prove that the retrospective and prospective policy values are equal at anyinteger 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, withpremiums payable annually in advance throughout the deferred period, or untilearlier death

On death during the deferred period, the benefit payable at the end of the year ofdeath is equal to the reserve and would have been required at that time had the lifesurvived.

On survival to age 70, a whole of life annuity of £10,000 per year is payable annuallyin 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 sti where i is the effective rate of interest assumed.

[6]

(ii) Given that i = .06 and a70 = 8.6, calculate P .

[2]

[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 benefitwas payable on death during the term and premiums were payable annually inadvance.

As at 1 January 1995, a total sum assured of £240,000 remained in force. During1995, 2 policyholders died, and no policy lapsed for any other reason.

The office calculates net premiums and net premium policy reserves on the followingbasis:

Basis: mortality - A1967-70 ultimateinterest - 4% per annum

23

(i) Calculate the profit or loss from mortality for this group for the year ending31 December 1995.

[7]

(ii) Briefly explain how the mortality profit or loss has arisen.

[2]

[Total 9]

9. (A2 98/1)

On 1 January 1985 a life office issued a number of 30 year pure endowment assur-ance contacts to lives then aged 35, with premiums payable annually in advancethroughout the term or until earlier death. In each case, the only benefit was asum 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 werein force at the start of the year.

Assuming that the office uses net premium policy reserves, calculate the profit orloss from mortality for 1996 in respect of this group of policies.

Basis: mortality: A1967-70 ultimateinterest: 4% per annum

[6]

10. (A2 98/2)

A life office is considering special ten-year endowment assurance policies to livesaged exactly 55. The basic sum assured is £20,000 payable at the end of the yearof death or on survival to age 65. Simple reversionary bonuses are declared at thestart of each year and a terminal bonus is paid on survival at age 65. In addition, atthe end of each complete year that the policyholder survives, an amount of £1,000is paid to the policyholder. Premiums are payable annually in advance throughoutthe term of the policy.

(i) Show that the premium for the above policy is £4,284.

Basis: mortality: A1967-70(select)interest: 4% p.a.expenses Intitial: £500

Renewal: 1% of each premium after the firstbonuses Reversionary 5% p.a.

Terminal: 50% of basic sum assured

[7]

(ii) 100 of the special endowment policies commenced on 1 January 1997. Theoffice holds reserves in respect of each policy equal to the retrospective policyvalue calculated on the same basis as in (i). Actual bonus declarations follow

24

those assumed in the premium/reserving basis above. One of the policyholdersdied immediately after paying their first premium. No further deaths occurredduring 1997.

Calculate the mortality profit or loss to the office on these 100 policies for theyear 1997.

[7]

(iii) Briefly explain how the profit or loss has arisen.

[3]

[Total 17]

11. (A2 98/2)

A life aged exactly 40 purchases a special premium deferred annuity. The annuitypayments 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 inpayment 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: A1967-70(select) before age 60a(55) Males (ultimate) after age 60

interest: 6% p.a.inflation: 1.9231% 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 grosspremium policy values on the following basis:

Basis: mortality A1967-70(ultimate) before age 60a(55) Males (ultimate) after age 60

interest: 6% p.a.inflation: 1.9231% 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.

[3]

[Total 9]

25

Documents

Heriot-Watt University BSc in Actuarial Mathematics and Statisticsandrewc/lim1/problems2.pdf · 2000. 11. 13. · A company issues 20 identical policies to lives aged 60. The death