Exam M Fall 2005
FINAL ANSWER KEY
Question # Answer Question # Answer
1 C 21 E
2 C 22 B
3 C 23 E
4 D 24 E
5 C 25 C
6 B 26 E
7 A 27 E
8 D 28 D
9 B 29 A
10 A 30 D
11 A 31 A
12 A 32 A
13 D 33 B
14 C 34 C
15 A 35 A
16 D 36 A
17 D 37 C
18 D 38 C
19 B 39 E
20 B 40 B
Exam M: Fall 2005 -1- GO ON TO NEXT PAGE
**BEGINNING OF EXAMINATION**
1. For a special whole life insurance on (x), you are given:
(i) Z is the present value random variable for this insurance.
(ii) Death benefits are paid at the moment of death.
(iii) ( ) 0.02, 0 xt t µ = ≥
(iv) 0.08 δ=
(v) 0.03, 0 t
tb e t = ≥
Calculate ( ) Var Z .
(A) 0.075
(B) 0.080
(C) 0.085
(D) 0.090
(E) 0.095
Exam M: Fall 2005 -2- GO ON TO NEXT PAGE
2. For a whole life insurance of 1 on (x), you are given:
(i) Benefits are payable at the moment of death.
(ii) Level premiums are payable at the beginning of each year.
(iii) Deaths are uniformly distributed over each year of age.
(iv) 0.10 i =
(v) 8 x a = 􀀅􀀅
(vi) 10 6 x a + = 􀀅􀀅
Calculate the 10th year terminal benefit reserve for this insurance.
(A) 0.18
(B) 0.25
(C) 0.26
(D) 0.27
(E) 0.30
Exam M: Fall 2005 -3- GO ON TO NEXT PAGE
3. A special whole life insurance of 100,000 payable at the moment of death of (x) includes a
double indemnity provision. This provision pays during the first ten years an additional
benefit of 100,000 at the moment of death for death by accidental means.
You are given:
(i) µ τ
x t t b gb g= ≥ 0 001 0 . ,
(ii) µx t t 1 0 0002 0 b gb g= ≥ . , , where µx
1 b g is the force of decrement due to death by
accidental means.
(iii) δ= 006 .
Calculate the single benefit premium for this insurance.
(A) 1640
(B) 1710
(C) 1790
(D) 1870
(E) 1970
Exam M: Fall 2005 -4- GO ON TO NEXT PAGE
4. Kevin and Kira are modeling the future lifetime of (60).
(i) Kevin uses a double decrement model:
x ( )
x l τ ( ) 1
x d ( ) 2
x d
60 1000 120 80
61 800 160 80
62 560 − −
(ii) Kira uses a non-homogeneous Markov model:
(a) The states are 0 (alive), 1 (death due to cause 1), 2 (death due to cause 2).
(b) 60 Q is the transition matrix from age 60 to 61; 61 Q is the transition matrix
from age 61 to 62.
(iii) The two models produce equal probabilities of decrement.
Calculate 61 Q .
(A)
1.00 0.12 0.08
0 1.00 0
0 0 1.00
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
(B)
0.80 0.12 0.08
0.56 0.16 0.08
0 0 1.00
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
(C)
0.76 0.16 0.08
0 1.00 0
0 0 1.00
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
(D)
0.70 0.20 0.10
0 1.00 0
0 0 1.00
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
(E)
0.60 0.28 0.12
0 1.00 0
0 0 1.00
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
Exam M: Fall 2005 -5- GO ON TO NEXT PAGE