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Table 1: Answer Key: Place all your answers here for considerationQ# Answer
1 f
2 a
3 a
4 b
5 c
6 a
7 a
Michigan State UniversityDepartment of MathematicsSTT 456 Spring 13TEST 4Name (Print): SolutionsName (Sign):
• Case 1: A five year policy with annual cash flows issued to a life (x)produces the profit vector
Pr =(−360.98 149.66 14.75 273.19 388.04 403.00
)(1)
where Pr0 is the profit at time 0 and Prt for t ∈ {1, 2, ..., 5} is theprofit at time t per policy in force at time t− 1.
The survival model used in the profit test is given by
qx+t = 0.0085 + 0.0005t. (2)
• Case 2 Consider a 10−year term insurance issued to a life aged 60.The details of the policy are as follows. The sum insured, denotedS = 100000, payable at the end of the year of death. Level annualpremiums, denoted P = 1500 are payable throughout the term.
The profit test basis is
– Interest: 5.5% per year effective on all cash flows.
– Initial expenses: 400 plus 20% of the first premium.
– Renewal expenses: 3.5% of premiums.
– Survival model: q60+t = 0.01 + 0.001t for t ∈ {0, 1, ..., 9}
The reserve basis is
– Interest: 4% per year effective on all cash flows.
– Survival model: q60+t = 0.011 + 0.001t for t ∈ {0, 1, ..., 9}.– The Net Premium is used to calculate the corresponding policy
values.
Assume that
– 1V = 410.05
– NPV (0.1) = 124.48
– P..a60:10 = 9684.
2
1. For Case 1, calculate Π5
(a) 360.98
(b) 149.66
(c) 14.62
(d) 268.43
(e) 377.66
(f) 388.29
(g) None of the above.
(h) Not enough info to compute
Answer: (f)
Π5 = 4pxPr5
= pxpx+1px+2px+3Pr5
= (1− qx)(1− qx+1)(1− qx+2)(1− qx+3)Pr5
= (1− 0.0085)(1− 0.0090)(1− 0.0095)(1− 0.01)(403.00)
= 388.294
(3)
3
2. For Case 1, calculate the NPV for this policy using a risk discount rateof 15% per year.
(a) 365.69
(b) 487.88
(c) 388.29
(d) None of the above.
(e) Not enough info to compute.
Answer: (a)
NPV(r) =
n∑k=0
Πk
(1 + r)k=
n∑k=0
k−1pxPrk(1 + r)k
= −360.98 +149.66
1 + r+
(0.9915)(14.75)
(1 + r)2+
(0.9915)(0.9910)(273.19)
(1 + r)3
+(0.9915)(0.9910)(0.9905)(388.04)
(1 + r)4
+(0.9915)(0.9910)(0.9905)(0.9900)(403.00)
(1 + r)5
= −360.98 +149.66
1 + r+
14.62
(1 + r)2+
268.43
(1 + r)3+
377.66
(1 + r)4+
388.29
(1 + r)5
. ·. NPV(0.15) = 365.69(4)
4
3. For Case 1, calculate the IRR for this policy.
(a) 42.7196%
(b) 49.7196%
(c) 52.7196%
(d) 62.7196%
(e) None of the above.
(f) Not enough information to compute.
Answer: (a)
0 = NPV(IRR)
. ·. 0 = −360.98 +149.66
1 + r+
14.62
(1 + r)2+
268.43
(1 + r)3+
377.66
(1 + r)4+
388.29
(1 + r)5
(5)
Substitution immediately leads to
0 = −360.98+149.66
1.427196+
14.62
(1.427196)2+
268.43
(1.427196)3+
377.66
(1.427196)4+
388.29
(1.427196)5
(6)
5
4. For Case 2, analyzing the net cash flows for the 10-year term insurancereturns Π0 =
(a) −1000
(b) −700
(c) 0
(d) 100
(e) None of the above
(f) Not enough information to compute.
Answer: (b).
Initial expenses are
Pr0 = Π0 = −(400 + 0.2P ) = −700. (7)
6
5. For Case 2, analyzing the emerging surplus, per policy in force at startof year, for the 10-year term insurance returns Pr1 =.
(a) 96.55
(b) 126.55
(c) 176.55
(d) 256.55
(e) None of the above.
(f) Not enough information to compute.
Answer: (c).
• Initially, the expenses are 400 + 0.2(1500) = 700.
• This upfront expense is incorporated into the initial profit (orlack thereof!)
• From t = 0 to t = 1, it follows that after this initial expense ispaid there is no renewal expense and so E1 = 0.
Therefore
Pr1 =(0V + P − E1
)(1 + r)− Sqx − 1V px
=(
0 + 1500− 0)
(1.055)− (100000)(0.01)− (410.05)(0.99)
= 176.55
(8)
7
6. For Case 2, the profit margin with a risk discount rate of 10% is
(a) 1.29%
(b) 3.29%
(c) 5.29%
(d) 8.29%
(e) None of the above.
(f) Not enough information to compute.
Answer: (a).
Profit Margin =NPV (0.1)
P..a60:10
=124.48
9684= 0.0129.
(9)
8
7. True or False A risk is diversifiable if we can eliminate it (relative toits expectation) by increasing the number of policies in the portfolio.
(a) True
(b) False
Answer: (a) True.
Scratch Paper
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Scratch Paper
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