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					                         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.

SHOW ALL YOUR WORK.


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
          thereafter. In addition, in the event of death, all premiums are
          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
              Year                     Premium
                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)

				
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