Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
ETC2430
Actuarial Statistics
Exam
Part 1
Version A
Question 1.A:
(i) (True/False) For compound interest, interest itself earns interest, whereas for efective interest,
interest credited to an account does not itself earn further interest.
(i.a) True.
(i.b) False. The answer is False, as compound and efective interests are the same.
(ii) Let i be the efective interest rate and let d be the equivalent efective discount rate. Which of the following is true?
(ii.a) i = 1 - d.
(ii.b) i = -d.
(ii.c) None of the above. This is the correct answer.
(iii) For an efective interest i = 3% pa, which of the following is the present value of an increasing annuity-certain payable for 60 years in arrears?
(iii.a) 610.7282. This is the correct answer.
(iii.b) 142.7175.
(iii.c) 923.1992.
(iii.d) None of the above.
Question 2.A:
A cash-flow consists of an immediate annuity payable monthly in arrears where the payments are 1, 000 pa for the first 6 years and 2, 000 pa for the next 4 years. Suppose that the interest rate is 6% pa convertible monthly during the first five years, and 12% pa convertible monthly during the remaining five years.
(i) Which of the following is the correct expression for the time-0 present value of the first five payments?
(i.a) 1, 000a .167781%. This is the correct answer.
(i.b) 1, 000a .
(i.c) 1; 000a5
, evaluated at 6.167781%.
(ii) Which of the following is the numerical time-0 present value of the first five payments?
(ii.a) 4; 310.463. This is the correct answer.
(ii.b) 4; 193.207.
(ii.c) 4; 226.985.
(iii) If, everything else remaining unchanged, the frequency of payments changes from monthly to quarterly, will the time-0 present value of the first five payments be higher, lower, or unchanged?
(iii.a) Lower. This is the correct answer.
(iii.b) Higher.
(iii.c) Unchanged.
(iv) Which of the following is the correct expression for the time-0 present value of the remaining five payments?
(iv.a) 1; [email protected]% × (a
12)@12.6825% + [email protected]% a
.6825% ). This is the correct an-swer.
(iv.b) 1; [email protected]% × (a
12)@12% + [email protected]% a
).
(iv.c) 1; 000u5@6% × (a
12.6825% + [email protected]% a
12.6825% )
(iv.d) 1; 000u5@12% × (a
12% + 2u1@12% a
12% ).
(v) Which of the following is the numerical time-0 present value of the remaining five payments?
(v.a) 4; 859.386. This is the correct answer.
(v.b) 4; 597.880.
(v.c) 3; 777.364.
(v.d) 7; 235.901.
(vi) What is the amount of the level annuity payable continuously for 10 years having the same present value as the overall cash-flows of this annuity.
(vi.a) 1; 289.265. This is the correct answer.
(vi.b) 1; 320.677.
(vi.c) 1; 345.655.
(vi.d) 1; 332.335.
(vii) If, everything remaining unchanged, the level annuity payable continuously from the previous question was payable for more than 15 years instead of 10 years, will the amount of the annuity be higher, lower, or unchanged.
(vii.a) Lower. This is the correct answer.
(vii.b) Higher.
(vii.c) Unchanged.
(viii) Assuming that the present value of the first five payments is evaluated at the rate 6% pa convertible monthly, we are looking for the interest rate j% pa convertible monthly for the remaining five years such that the time-0 present value of the first five payments is equal to the time-0 present value of remaining five payments. Which of the following equations will give the correct value of j?
(viii.a) Time-0 present value of the first five payments evaluated at 6% pa convertible monthly
- Time-0 present value of the remaining five payments evaluated at j% pa convertible monthly = 0. This is the correct answer.
(viii.b) Time-0 present value of the first five payments evaluated at 6% pa convertible monthly + Time-0 present value of the remaining five payments evaluated at j% pa convertible monthly = 0.
(ix) In this question, you are expected to determine the numerical value of j from (vi). In par- ticular, assuming that the present value of the first five payments is evaluated at the rate 6% pa convertible monthly, what is the interest rate convertible monthly for the remaining five years such that the time-0 present value of the first five payments is equal to the time-0 present value of remaining five payments?
(ix.a) 16.84610%. This is the correct answer.
(ix.b) 18.20965%.
(ix.c) 15.11801%. (ix.d) 17.21081%.
Question 3.A:
The force of interest at time t, where t is measured in years, is given by:
δ(t) = {
0.058 − 0.002t, for 0 ≤ t < 4,
0.015t − 0.01, for 4 ≤ t < 8,
0.11, for 8 ≤ t.
(i) Which of the following is the expression of the present value at time 0 of a payment of 1, 000 at time 12?
(i.a) 1, 000e- 
δ(s)ds. This is the correct answer.
(i.b) 1, 000 R 0 12 e − R t 12 δ(s)ds .
(i.c) 1, 000 ∫0(12) e- ∫0(t) δ(s)ds.
(ii) Which of the following is the present value at time 0 of a payment of 1, 000 at time 12?
(ii.a) 376.8153. This is the correct answer.
(ii.b) 976.
(ii.c) 144.2799.
(ii.d) None of the above.
(iii) Which of the following is the annual efective rate of interest that is equivalent to this variable force of interest from time 0 to time 12?
(iii.a) 
δ(s)ds - 1. This is the correct answer.
(iii.b) e- 
δ(s)ds - 1.
(iii.c) 
δ(s)ds.
(iii.d) ,0(12) δ(s)ds.
(iv) Which of the following is the monthly efective rate of interest that is equivalent to this variable force of interest from time 0 to time 12?
(iv.a) 0.68007989%. This is the correct answer.
(iv.b) 0.01687135%.
(iv.c) 1.35352274%.
(iv.d) -0.67548604%.
(v) Which of the following is the present value at time 0 of a continuous payment stream of 100e0.05t payable from time 15 to time 20?
(v.a) 247.7339. This is the correct answer.
(v.b) 87.37882.
(v.c) 161.78251.
(v.d) None of the above.