9. For a special fully discrete 30-payment whole life insurance on (45), you are given:
(i) The death benefit of 1000 is payable at the end of the year of death.
(ii) The benefit premium for this insurance is equal to 45 1000P for the first 15 years
followed by an increased level annual premium of π for the remaining 15 years.
(iii) Mortality follows the Illustrative Life Table.
(iv) 0.06 i =
Calculate π.
(A) 16.8
(B) 17.3
(C) 17.8
(D) 18.3
(E) 18.8
Exam M: Fall 2005 -10- GO ON TO NEXT PAGE
10. For a special fully discrete 2-year endowment insurance on (x):
(i) The pure endowment is 2000.
(ii) The death benefit for year k is ( ) 1000k plus the benefit reserve at the end of year k,
1, 2 k = .
(iii) π is the level annual benefit premium.
(iv) i = 0.08
(v) 1 0.9, 1, 2 x k p k + − = =
Calculate π.
(A) 1027
(B) 1047
(C) 1067
(D) 1087
(E) 1107
Exam M: Fall 2005 -11- GO ON TO NEXT PAGE
11. For a group of 250 individuals age x, you are given:
(i) The future lifetimes are independent.
(ii) Each individual is paid 500 at the beginning of each year, if living.
(iii) 0.369131 x A =
(iv) 2 0.1774113 x A =
(v) 0.06 i =
Using the normal approximation, calculate the size of the fund needed at inception in order to
be 90% certain of having enough money to pay the life annuities.
(A) 1.43 million
(B) 1.53 million
(C) 1.63 million
(D) 1.73 million
(E) 1.83 million
Exam M: Fall 2005 -12- GO ON TO NEXT PAGE
12. For a double decrement table, you are given:
Age ( )
x l τ ( ) 1
x d ( ) 2
x d
40 1000 60 55
41 − − 70
42 750 − −
Each decrement is uniformly distributed over each year of age in the double decrement table.
Calculate ( ) 1
41 q′ .
(A) 0.077
(B) 0.078
(C) 0.079
(D) 0.080
(E) 0.081
Exam M: Fall 2005 -13- GO ON TO NEXT PAGE
13. The actuarial department for the SharpPoint Corporation models the lifetime of pencil
sharpeners from purchase using a generalized DeMoivre model with ( ) ( ) 1 / s x x α ω = − , for
0 α> and 0 x ω ≤ ≤ .
A senior actuary examining mortality tables for pencil sharpeners has determined that the
original value of α must change. You are given:
(i) The new complete expectation of life at purchase is half what it was previously.
(ii) The new force of mortality for pencil sharpeners is 2.25 times the previous force of
mortality for all durations.
(iii) ω remains the same.
Calculate the original value of α.
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Exam M: Fall 2005 -14- GO ON TO NEXT PAGE
14. You are given:
(i) T is the future lifetime random variable.
(ii) ( ) t µ µ = , 0 t ≥
(iii) [ ] Var 100 T = .
Calculate [ ] E 10 T ∧ .
(A) 2.6
(B) 5.4
(C) 6.3
(D) 9.5
(E) 10.0
Exam M: Fall 2005 -15- GO ON TO NEXT PAGE
15. For a fully discrete 15-payment whole life insurance of 100,000 on (x), you are given:
(i) The expense-loaded level annual premium using the equivalence principle is 4669.95.
(ii) 100,000 51,481.97 x A =
(iii) :15 11.35 x a = 􀀅􀀅
(iv) 0.02913 d =
(v) Expenses are incurred at the beginning of the year.
(vi) Percent of premium expenses are 10% in the first year and 2% thereafter.
(vii) Per policy expenses are K in the first year and 5 in each year thereafter until death.
Calculate K.
(A) 10.0
(B) 16.5
(C) 23.0
(D) 29.5
(E) 36.5
Exam M: Fall 2005 -16- GO ON TO NEXT PAGE
16. For the future lifetimes of (x) and (y):
(i) With probability 0.4, ( ) ( ) T x T y = (i.e., deaths occur simultaneously).
(ii) With probability 0.6, the joint density function is
( ) ( ) , ( , ) 0.0005 T x T y f ts= , 0 40 t < < , 0 50 s < <
Calculate ( ) ( ) Prob T x T y < ⎡ ⎤ ⎣ ⎦.
(A) 0.30
(B) 0.32
(C) 0.34
(D) 0.36
(E) 0.38
Exam M: Fall 2005 -17- GO ON TO NEXT PAGE