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CS/APR 2003/ASC475

UNIVERSITI

  TEKNOLOGI

  MARA

FINAL  EXAMINATION

COURSE

COURSE CODE

DATE

TIME

FACULTY

SEMESTER

PROGRAMME/CODE

ACTUARIAL

  MATHEMATICS 2

ASC475

23 APRIL

 2003

3 HOURS  2.15  p.m -  5.15 p.m)

Information Technology

  and

  Quantitative Sciences

December 2002 -

 May

 2003

Bachelor

 of

 Science Hons) Actuarial Science)

 / CS 222

INSTRUCTIONS

  TO

  CANDIDATES

  1

1.   This question paper consists  of  eight   (8)  questions.

2.

  Answer

 ALL

  questions

3.

  Answ er to each question should start on a new page.

4.   Do not  bring  any

 ma terial into

 the

 examination room unless permission

  is

 given

 by the

invigilator

5.

  Please  check

 to

 make sure that this exam ination pack consists

 o f:

i)   the Question Paper

ii)   an Answer Booklet  -   provided  by the  Faculty

iii)  an  Illustrative Service T able  -   provided  by the  Faculty

iv)   an  Illustrative Life Table  -   provided  by the  Faculty

DO NOT TURN THIS PAGE UNTIL YOU  ARE TOLD TO DO SO

This  examination paper cons ists

 of

printed pages

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Answer

 ALL

 Q uestions.

1.   a)

  Define

 in words

 each

 of the

 following:

i)

  T(xy)

«)  n / u Q x y

b)   XYZ Company has just   purchased   two new   machines,   A and B,

  with

independent future  lifetimes.

  Each

  machine   has its own De   Moivre   survival

pattern.

 Machine

 A has a 12-year

  maximum

 future

 lifetime

 and the other a 10-

year

 maximum  future  lifetime.

Calculate the expected time  until  both  machines

 fail.

(1 2  marks)

2. Given :

i) (x) and (y) are

 independent

  lives,

ii) 5 is the  force  of

 interest,

iii)  Mt)  =

 m

  f o r t >0 ,

iv )

  u,

x

(t) =

 u,

2

  for t > 0   ,

v)   5,  u,.,,

  n

2

 are constants.

a)   Express  each

 of the

 following

  in

 terms

 of

  u,

1

 ,   ja

2

 and 8

xy

ii)

  A

x y

b)   A special   fully   continuous  whole   life   insurance of  RM1000  is   payable   on the

last-survivor of (x) and   (y).   Premiums are payable   until   the first   death.

Assuming

  8

 =

 0.03 and

  |a

x

(t)

 =

 2u.

y

(t)

 =

 0.06,   calculate

  the

  level   annual

benefit

  premiums

 for this  insurance.

(14

 marks)

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3.

4.

a)

b)

a)

b)

In

 a

 double  decrement  model,

 the

 joint  probability density function

of  T(x)

 and J is given by:

, j

=

 1-t

2

, j = 2t 1-t)

0 < t < 1 , j = 2

Determine:

i)  E[T(x)]

John,

 age 40, is an

 actuarial  science

 professor.  His   career  is

 subject

 to two

decrements:

Decrement

  1 is

 mortality which

 follows  De

 Moivre s

  law

 with  o

  =

 100.

Decrement 2 is leaving  academic  employment,

 with

 ût)=

 0.05,

t>0.

Calculate the  probability that  John  remains  an  actuarial  science

 professor

 for

at  least  five  but  less  than ten

 years.

(14 marks)

In

 a  multiple  decrement  model, for  each cause  j, there is an associated  single

decrement model.  Explain briefly

 the difference

 between

 q

(

x

j)

and

 q ̂.

For a

 triple  decrement  model,

 you are

 given

 the  following:

i)

X

63

64

65

gd

Hx

0.020

0.025  J

0.030

q <

2)

0.030

0.035

0.040

q *

3)

0.250

0.200

0.150

i i qgU

  0.02716

iii)

  Assum e that  each decrement has a constant

 force

 over each year of

age.

Calculate

  1

(10  marks)

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5.   a)

  Describe

 briefly  the difference between how   retirement

 benefits

 are

determined

 under

i)  defined-contribution  plans

ii) defined-benefit

 plans

b)   In a

 defined-benefit pension

 plan,  the  retirement   benefit is

  100

 per  month  per

year of service.  Retirement   is allowed  beginning  at age 53 and  occurs  only  on

birthdays

 of

 employees. There

 are no  pre-retirement  decrements.

You are given the

 following

 data:

Age

53

54

55

q

 (r)

Hx

0.15

0.20

1.00

o(12)

a

x

12

11

10

An  employee  was hired at  exact  age 25 and is  currently exact  age 33.

Assuming i =

 0.10,  calculate

  the actuarial present  value  of the

 projected

retirement benefit  for this employee.

(1 4  marks)

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6.

  An employee  was

 hired

  12  years  ago. He is  currently age 35 and is

 entitled

 to

withdrawal

 and

 retirement benefits.

 The

 earliest retirement

 age is 60. He can

withdraw

 anytime  before reaching age 60.

 Withdrawal

 benefits  are in the form of an

annuity with

 payments deferred

 until

  the employee

  reaches

 age 60. The withdrawal

benefit

 is at the

  rate

 o f

 0.03  times

 the

  total  salary  paid over

 the

 whole   career

 to the

employee.

If

  his

 current

 annual  earning   is

 RM60.000

 and his  total  past  salary   is

 RM400,000,

give  an

 expression

 for the actuarial

 present value

 of the  withdrawal

 benefits

 in the

form

 of definite

 integrals.

 You should

 define

 all symbols

 used.

(8  marks)

7.   A

 fully discrete

 whole

 life  insurance

 of 1,000 is

 issued

 to

 (60).

 The

 contract

 premium

is

 35. Cash values are the  minimum  based on the

  1980

 Non-Forfeiture

  law.

 On the

basis  of the Illustrative   Life  Table

 with

  interest at 6  percent,

 answer

 the  following:

a)

  Suppose

 that the policy is

 converted

 to a reduced

 paid-up

 insurance  at the

end of the fifth year after  policy

 issue.

  If

 there

 is no

 policy  loan  outstanding,

calculate the reduced paid-up  amount  of  insurance   that can be

 purchased

 by

the  fifth  year

 cash

 value.

c)  Suppose that the policy is kept in  force  but there are premiums  default  after

the  fifth

 year.

 Show  that  th e   maximum  number  of  contract premiums that  can

be paid

 after

 the fifth  year   using  the automatic  premium  loan  provision  is

three. Assume

 that the

 policy  loan interest rate

 is 10 percent.

(14

 marks)

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8.

  A fully  discrete 3-year  endowment insurance of  5,000 is  issued   to (x)  payable  b y  level

annual contract premiums,   G.  There   are  only   two causes of  decrements:  deaths   (d)

and  withdrawals (w).  Cash  values are

 paid

 at the end of  year  of  withdrawal.

You  a re  given  the  following:

i)

k

0

1

2

d

x+k

20

50

150

W

x

+k

250

120

0

c

k

0.5

0.1

0.1

e

k

60

20

20

k

+

iCV

500

2,000

5,000

ii)

iii)

iv)

v)

|<

X

T

=   1.000

c

k

  is the  fraction  of the

 contract  premium

 paid at

 time

 k for expenses

e

k

 is the  amount  of per policy expenses paid at  time   k.

i =

 0.10

a)

  Using

 the above  symbols,  write an expression

 for

3

 AS.

b)  If G = 2,250,  determine  whether

  3

 AS >

 3

 CV .

(1 4

 marks)

END OF

 QUESTION PAPER

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