BSc Sem VI 2022 2023 Apr 2023 STATISTICS ACTURIAL SCIENCE Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 (a) () Define the term force of By making suitable assumptions prove that py = 6 marks
   • (ii) Define central death rate (m,,). Show that p, = 4
   • (b) () Define the term curate expectation of life and complete expectation of life(e? ). Further in usual notations, show that — 8
   • (1) ey = ai) If 1, = 100 — x then find probability of person with age 10 years will survive 5 more years 2
   • (p) () State and prove Makeham’s first law of mortality. 6
   • (ii) Explain the following terms- 4
   • (1) Select mortality
   • (2) Aggregate mortality
   • (q) Prove that- The average age at death of those who die in the age group x to x + nis given by, x + 10
2. Q2 (a) Define nominal rate of interest and effective rate of interest (i). Further in usual notations, show that =m ["/(1 +i) - 1] 10 marks
   • (b) Obtain an expression for present value and accumulated value of immediate annuity of Rs. 1 p.a. when successive payments are in Arithmetic progression. (Assume rate of interest is i per unit per 10
   • (p) () Obtain an expression for present value of deferred immediate annuity certain and in usual notations show that = where m is deferment period. (Assume rate of interest is i per unit per annum) 7
   • (ii) Tp usual notations show that - 3
   • (q) Define the following terms- 10
   • (1) perpetuity Hence obtain expression for present value of deferred perpetuity due and deferred immediate perpetuity of Rs.1 p.a. with deferment period of m years. (Assume rate of interest is i per unit per annum)
   • (a) Define Deferred immediate life annuity. Obtain the expression for the present value of deferred immediate life annuity of Rs. | p.a. toa person aged x in terms of commutation functions 10
   • (b) Define life annuity due. Obtain the expression for present value of life annuity due of Rs. | p.a. on a life now aged x for years in terms of 10
   • (p) Define the term temporary life annuity. How is it different form annuity certain? Obtain the expression for the present value of immediate temporary life annuity of Rs. 1 p.a. to a person aged x years in terms of commutation functions 10
   • (q) Obtain the expression for the present value of increasing life annuity when the payments are made at the end of each year to a person aged 10
   • x. Write the result in terms of commutation functions
3. Q4 (a) Obtain the expression for the level annual premium for temporary assurance in terms of commutation functions after deriving all the 10 marks
   • (b) Compute the present value of Pure endowment assurance plan. Derive it’s the level annual premium in terms of commutation functions 10
   • (p) Obtain the present value of special endowment assurance plan. Derive the expression for level annual premium in terms of commutation functions for this plan 10
   • (q) Derive the expression for the level annual premium for whole life assurance in terms of commutation functions after deriving all the 10
4. Q5 Attempt ANY TWO sub-questions
   • (a) Prove that- Probability that among three live all aged x years, the first death will occur in year is given by (tPx 6
   • (ii) State uses of life tables. 4
   • (b) Obtain an expression for accumulated value of immediate annuity of Rs. | p.a. payable p times a year for n years certain. (Assume rate of interest is 1 per unit per annum) 10
   • (c) Derive the expression for the present value of deferred life annuity due of Rs. 1 p.a. payable to a person aged x in terms of commutation 10
   • (d) Explain the concept of limited payment assurance with the help of endowment assurance plan. Derive the level annual premium for the same in terms of commutation functions 10

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