Faculty of ActuariesInstitute of Actuaries
EXAMINATIONS
13 September 2001 (am)
Subject 105 — Actuarial Mathematics 1
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1.
Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.Mark allocations are shown in brackets.
Attempt all 14 questions, beginning your answer to each question on aseparate sheet.
2.3.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet and this question paper.
In addition to this paper you should have availableActuarial Tables and an electronic calculator.
105 S2001
? Faculty of Actuaries? Institute of Actuaries
1
Under the Manchester Unity model of sickness, you are given the followingvalues:
sx=5
ò
10t
pxdt=0.9
Calculate the value of zx.
[2]
2
Give a formula forP21(2003) in terms of P20(2002), based on the componentmethod of population projection. Px(n)denotes the population aged x lastbirthday at mid-year n.
State all the assumptions that you make and define carefully all the symbols thatyou use. [3]
3
A life insurance company issues a policy under which sickness benefit of ￡100 perweek is payable during all periods of sickness. There is a waiting period of 1 yearunder the policy.
You have been asked to calculate the premium for a life aged exactly 30, who isin good health, using the Manchester Unity model of sickness.
Describe how you would allow for the waiting period in your calculation, giving areason for your choice of method. [3]
105 S2001—2
4
An employer recruits lives aged exactly 20, all of whom are healthy whenrecruited. On entry, the lives join a scheme that pays a lump sum of ￡50,000immediately on death, with an additional ￡25,000 if the deceased was sick at thetime of death.
The mortality and sickness of the scheme members are described by the followingmultiple-state model, in which the forces of transition depend on age only.
σxHealthy (H)ρxμxDead (D)νxSick (S)All surviving members retire at age 65 and leave the scheme regardless of theirstate of health.
abpx,t is defined as the probability that a life who is in state a at age x (a = H, S, D)
0and b=H,S,D).is in state b at age x + t (t≥
Write down an integral expression for the expected present value, at force ofinterest δ, of the death benefit in respect of a single new recruit. [3]
5
A pension scheme provides a pension of 1/60 of career average salary in respect ofeach full year of service, on age retirement between the ages of 60 and 65. Aproportionate amount is provided in respect of an incomplete year of service.At the valuation date of the scheme, a new member aged exactly 40 has anannual rate of salary of ￡40,000.
Calculate the expected present value of the future service pension on ageretirement in respect of this member, using the Pension Fund Tables in theFormulae and Tables for Actuarial Examinations. [3]
105 S2001—3
PLEASE TURN OVER
6 A life insurance company issues a special annuity contract to a male life aged
exactly 70 and a female life aged exactly 60.
Under the contract, an annuity of ￡10,000 per annum is payable monthly to thefemale life, provided that she survives at least 10 years longer than the male life.The annuity commences on the monthly policy anniversary next following thetenth anniversary of the death of the male life and is payable for the balance ofthe female’s lifetime.
Calculate the single premium required for the contract.Basis:Mortality:
Interest:Expenses:
a(55) Ultimate, males or females as appropriate8% per annumnone
[4]
The staff of a company are subject to two modes of decrement, death and
withdrawal from employment.
Decrements due to death take place uniformly over the year of age in theassociated single-decrement table: 50% of the decrements due to withdrawal
occur uniformly over the year of age and the balance occurs at the end of the yearof age, in the associated single-decrement table.
You are given that the independent rate of mortality is 0.001 per year of age andthe independent rate of withdrawal is 0.1 per year of age.
Calculate the probability that a new employee aged exactly 20 will die as anemployee at age 21 last birthday.
[4]
7
8
The following data are available from a life insurance company relating to themortality experience of its temporary assurance policyholders.θx,d
The number of deaths over the period 1 January 1998 to 30 June 2001,aged x nearest birthday at entry and having duration d at the policyanniversary next following the date of death.
Py,e(n)
The number of policyholders with policies in force at time n, aged ynearest birthday at entry and having curtate duration e at time n, wheren = 1.1.1998, 30.6.1998, 30.6.2000 and 30.6.2001.
Develop formulae for the calculation of the crude central select rates of mortalitycorresponding to the θx,d deaths and derive the age and duration to which theserates apply. State all the assumptions that you make.
[6]
105 S2001—4
(i)State the conditions necessary for gross premium retrospective and
prospective reserves to be equal. [3]
(ii)
Demonstrate the equality of gross premium retrospective and prospective
reserves for a whole life policy, given the conditions necessary for equality.
[4]
[Total 7]
9
10
A life insurance company issues a special term assurance policy to two lives agedexactly 50 at the issue date, in return for the payment of a single premium. Thefollowing benefits are payable under the contract:(i)(ii)
In the event of either of the lives dying within 10 years, a sum assured of￡100,000 is payable immediately on this death.
In the event of the second death within 10 years, a further sum assured of￡200,000 is payable immediately on the second death.
Calculate the single premium.Basis:Mortality:
Interest:Expenses:
A1967–70 Ultimate4% per annumNone
[8]
105 S2001—5
PLEASE TURN OVER
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