# 10

Document Sample

```					                         Actuarial Mathematics Test

Chapters 3, 4, 5, and 6

January 19, 2000
10:00 a.m. - 1:00 p.m.

Each question is worth 5 points for an exam total of 150 points.

1.    You are given:

(i)     t px   = (0.8)t, t  0,

(ii)    lx+2 = 6.4

Calculate Tx+1.

2.    If l65 = 100 and l66 = 80, calculate ½q65½ under the UDD, constant force and
hyperbolic assumptions.

3.    You are given a select and ultimate mortality table where:

(i)     The select period is 2 years with q[x] = 80% x qx and q[x]+1 = 90% x
qx+1,

(ii)    Deaths are uniformly distributed over each year of age in the select
period,

(iii)   The ultimate period of the table follows de Moivre's law with  = 90.

Calculate the average remaining future lifetime of [60].
4.   You are given:

(i)     5+t) = , 0  t  1,

(ii)    p35 = .985,

(iii)   '(35+t) is the force of mortality for (35) subject to an extra hazard, 0
 t  1,

(iv)    '(35+t) =  + 5%, 0  t  0.5,

(v)     The additional 5% force of mortality decreases uniformly from 5% to
0% between (35.5) and (36).

Determine the probability that (35), subject to the extra hazard, will not
survive to (36).

5.   You are given for 0  x  10:

(10  x) 2
s ( x) 
100
Calculate the average number of years lived between ages 1 and 2 by those
who die between those two ages.

6.   You are given for 0  x  100:

4
 ( x) 
100  x
Calculate the average number of years lived between (40) and (60) by those
who are currently (30).
7.   For a current type of refrigerator, you are given:

x
(i)     s ( x)  1         , 0  x 

o
(ii)    e0  20
For a proposed new type, with the same  , the new survival function is:

s* ( x)  1, 0  x  5

x
s* ( x)         , 5 x 
 5
Calculate the increase in life expectancy at time 0.

8.   You are given:

(i)     The force of mortality is a constant, 

(ii)      1,

(iii)   33q33   = 0.0030

Calculate 1000 
9.    A whole life insurance of 1 with benefits payable at the moment of the
death of (x) includes a double-indemnity provision. This provision
pays an additional death benefit of 1 for death by accidental means. S
is the net single premium for this insurance.

A second whole life insurance of 1 with benefits payable at the
moment of the death of (x) includes a triple-indemnity provision. This
provision pays an additional death benefit of 2 for death by accidental
means. T is the net single premium for this insurance.

You are given:

(i)     1% is the force of mortality by accidental means,

(ii)    5% is the force of mortality by other means,

(iii)   7% is the force of interest.

Calculate T - S.

10.   For a 10-year deferred whole life insurance of 1 payable at the moment of
death on a life (35), you are given:

(i)     Z is the present value random variable for this insurance.

(ii)    6% is the force of mortality.

(iii)   10% is the force of interest.

Determine the 90th percentile of Z.
11.   For a certain age at issue the following premiums are available for an n-year
endowment insurance policy:

(i)     The net single premium of a 1000 endowment, with return of
the net single premium on death during the endowment period,
is 500.

(ii)    The net single premium of a 1000 endowment which pays
double the face amount if and only if the insured survives n
years, also with return of the net single premium on death
during the endowment period, is 800.

What is the net single premium for a 1500 n-year endowment, without a
return of the net single premium or double the death benefit?

12.   An Alumni Association has 50 members, each age x. You are given:

(i)     All lives are independent.

(ii)    Each member makes a single contribution of R to establish a fund.

(iii)   The probability is 0.95 that the fund will be sufficient to pay a death
benefit of 1000 to each member.

(iv)    Benefits are payable at the moment of death.

(v)     A x  0.06

2
(vi)        A x  0.01

(vii) Assuming a normal approximation, Pr(Z < 1.645) = 95%.

Calculate R.
13.   For a 25-year term insurance of 1 on (25), you are given:

(i)     Benefits are payable at the moment of death.

(ii)    Z is the present value random variable at issue of the benefit payment.

(100  x)
(iii)   s(x) =              , 0  x  100
100

(iv)    i = 0.10

Calculate 1000 Var (Z).

14.   For a 20-year pure endowment of 1 on (x), you are given:

(i)     Z is the present value random variable at issue of the benefit payment

(ii)    Var (Z) = 0.05 E(Z)

(iii)   20   p x  0.65

Calculate i.
15.   For a special whole life insurance on (0), you are given:

(i)     Benefits are payable at the moment of death.

200 0  t  65
(ii)    bt  
100 t  65

(iii)    0 (t )  0.03, t  0

0.01, 0  t  65
(iv)    t  
 0.02, t  65

Calculate the actuarial present value at issue of this insurance.

16.   For a washing machine sold by Company ABC, you are given:

(i)     The purchase price of the washing machine is 600.

(ii)    Company ABC provides a ten year warranty.

(iii)   In the event of failure of the washing machine at time t, for t  10,
Company ABC will refund (1 – 0.1t) times the purchase price.

(iv)    The refund is payable at the moment of failure.

(v)     Washing machines are subject to a constant force of failure of 0.02.

(vi)      0.08

Calculate the actuarial present value, at the time of purchase, of the
warranty.
17.   A fully discrete 40-year endowment insurance of 1 is issued to each of
n people age x. You are given:

(i)     The net single premium that was mistakenly charged was
the net single premium for a fully discrete 40-year term
insurance of 1 on (x),

(ii)    d = 0.057,

(iii)   A1:40  0.257 ,
x

(iv)    Ax:40  0.268 ,

(v)     2   Ax:40  0.0745 ,

(vi)    Losses are independent and the total portfolio of losses is
assumed to follow the normal distribution, where
Pr(Z > -1.645) = 95%,

(vii) The probability of a positive loss on the total portfolio is 95%.

Calculate n.
18.   For a 10-year deferred whole life insurance of 1 payable at the moment of
death on a life (35), you are given:

(i)     Z is the present value random variable for this insurance.

(ii)    6% is the force of mortality.

(iii)   10% is the force of interest.

Determine the median of Z.

19.   For a 2-year term insurance of 10 on (60), you are given:

(i)     Benefits are payable at the end of the year of death,

(ii)    Z is the present-value random variable for this insurance,

(iii)   i = 0.05,

(iv)    The following extract from a select-and-ultimate mortality table:

x           l[x]     l[x]+1        1x  2      x+2
60         78,900    77,200       75,100         62
61         76,400    74,700       72,500         63
62         73,800    72,000       69,800         64

Calculate Var (Z).
20.   You are given:

(i)     q x  0.20 ,

(ii)    q x 1  0.40 ,

(iii)   i = 0.12,

(iv)    Deaths are uniformly distributed over each year of age.

2   1
Calculate A x:2 .

21.   For a 20-year term insurance on (x), you are given:

(i)     (1)(x+t) = 5% where (1) represents death by accidental means,

(ii)    (2)(x+t) = 10% where (2) represents death by other means,

(iii)   The benefit is 2 if death occurs by accidental means and 1 if death
occurs by other means,

(iv)    Benefits are payable at the moment of death,

(v)     i = 0.

Calculate the net single premium for this insurance.
22.   You are given:

(i)     1000 (IA)50 = 4996.75,

(ii)    1000 vq50 = 5.58,

(iii)   1000 A51 = 249.05,

(iv)    i = 6%.

Calculate 1000 (IA)51.

23.   For a fully continuous 30-year term insurance of 1 on (50), you are
given:

(i)       0.05 ,

(ii)    Mortality follows De Moivre’s law with   120
th
(iii)   P , the 20 percentile premium for this insurance, is the
smallest annual premium that the insurer can charge such that
the probability of a positive financial loss is at most 20%

Calculate P .
24.   For a 5-year certain and life annuity of 1 on (x), payable continuously,
you are given:

(i)     Y is the present-value random variable,

(ii)    FY ( y) is the distribution function for Y

(iii)    x (t )  0.01, t  0

(iv)      0.04

Calculate FY (30)  FY (5).

25.   For a special fully discrete 5-year term insurance on (20), you are
given:

(i) The death benefit is 2000 during the first 3 years and 1000
refunded without interest,

(ii) Mortality follows De Moivre’s law with   100 ,

(iii)   i = 0,

(iv)    Benefit premiums are payable according to the following
schedule:

Policy                    Benefit
1                         P
2                         2P
3                         3P
4 and                        0
later

Calculate P.
26.   For a fully continuous whole life insurance of 1 on (50), you are
given:

(i) The loss random variable L is defined as follows:
L  v T  [ P ( A50 )  c] a T , T  0, c is a constant

(ii)        50 (t )  0.06, t  0 ,

(iv)         0.010 ,

(v)       E[L] = -0.1875.

Calculate E[L 2]

27.   You are given:

(i)         x (t )  0.01, t  0 ,

(ii)         0.05

(iii)     T is the future lifetime of (x)

Calculate the probability that a T will exceed a x:20
28.   For a fully continuous whole life insurance of 1 on (x), you are given:

(i)     Z is the present-value random variable at issue of the death
benefit,

(ii)    L is the loss-at-issue random variable

(iii)   Premiums are determined using the equivalence principle

Var ( Z )
(iv)               0.36
Var ( L)

(v)     a x  10

Calculate P( A x ).

29.   For a 3-year temporary life annuity-due on (30), you are given:

x
(i)     s ( x)  1       , 0  x  80 ,
80

(ii)    i = 0.05,

aK 1 , K  0,1,2,


(iii)   Y   
a3 , K  3,4,5,...


Calculate Var(Y)

30.   You are given:

(i)     a4)  17.287 ,
(

(ii)    Ax = 0.1025

(iii)   Deaths are uniformly distributed over each year of age

(
Calculate ax4)

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 16 posted: 11/2/2011 language: English pages: 14