LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 

BA 17

FIFTH SEMESTER – November 2008

ST 5404 – ACTUARIAL STATISTICS

 

 

 

Date : 12-11-08                       Dept. No.                                          Max. : 100 Marks

Time : 9:00 – 12:00

PART-A

Answer ALL the questions:                                                                                         (10×2=20 marks)

 

1. Define immediate perpetuity, perpetuity due.
2. The amount with compound interest of a certain principal at 5% pa is Rs3969. Find the  principal

when the period is 2 years.

3. Find the nominal rate p.a convertible half-yearly corresponding to an effective rate of  8% p.a.
4. The accumulated value of a certain annuity paid after 8 years at the rate of 8% is 2866.35.  Find the

present value.

5. Explain deferred annuity due.
6. Define L_x .
7. Explain A_x .
8. Explain the need for a life table.
9. What is the need for a commutation function ?
10. Expand S_x in terms of D_x  .

 

PART-B

Answer FIVE questions:                                                                                             (5×8=40 marks)

 

11. A has taken a loan of Rs2000 at rate of interest 4% pa payable half yearly. He repaid Rs.400 after

2 years,Rs.600 after a further 2 years and cleared all outstanding dues at  the end of 7 years from

the commencement of the transaction. What is the final   payment made by him?

12. Derive the formula for accumulated value and present value of annuity certain due.
13. Derive the formula for an increasing annuity .
14. Find the present value of an immediate annuity of Rs.600 p.a payable quarterly for 20 years at a

rate of 6% p.a payable half yearly.

15. Find the probability that of 2 persons A and B aged 30 and 35 respectively

i.)both die before 55.

ii.)both die after 60.

iii.)A dies before 65 while B dies after 60.

iv.)Atleast one of them survives to 70.

16. Obtain the expressions for a _{x: n} and  ( Ia )_{x : n}   .
17. Calculate the net annual premiums for sum assured of Rs.5000 for the following

assurances on (40)

a.)pure endowment assurance for 20 years.

b.)Temporary assurance for 20 years.

18. Obtain the formula for a_n and Sn.

 

 

 

PART-C

Answer TWO questions:                                                                                        (2×20=40 marks)

 

19. a) Complete the following life table.

 

Age                   l_x                d_x              q_x               L_x

10               1000000             –            .00409             –

11                    –                     –           .00370              –

12                    –                     –           .00347              –

13                    –                     –           .00342              –

 

58. b) An employee of an institution has to retire at the age of 58.A gratuity benefit of

one months salary for each year of service subject to a maximum benefit of 15

months salary is payable to an employee on retirement or death , as the case may

1. Find the probability that:

i.)full gratuity  benefit will be payable to a person aged 35, who has just now

completed 5 years of service.

ii.)the gratuity benefit will not exceed 10 months salary .

iii.)the gratuity benefit will be atleast 12  months salary.

iv.)the employee earns atleast 12months salary as gratuity benefit payable at

death.

 

20. a) A loan of Rs.5000 is to be repaid with interest at 8% p.a be means of an

immediate annuity for 10 years. Find the yearly installment. What will be the

principal and interest contained in the 5th installment? What will be the principal

outstanding immediately after the 8th payment is made?

 

1. b) Find the present value of an immediate annuity of Rs.240 p.a payable in equal

monthly installments for 10 years certain at nominal interest of 8% p.a convertible

half yearly.

 

21. a) A person aged 30 years approached a life office for special type of policy  providing for the

following benefits.

i.)Rs.1000 on death during the first 5 years.

ii.)Rs.2000 on death during the next 15 years.

iii.)survival benefit of Rs.500 at the end of the 5th year .

iv.)Further payment of Rs.2000 on survival of 20 years.

Find the annual premium assuming that the premium paying term is 20 years.

 

1. b) Derive the expression for A­_{x: n} and (IA )_{x: n} .

 

• a) A had decided to invest Rs.500 at the end of each year. He did so far 7 years.

Then there was a gap of 4 years. He could again invest Rs.500 p.a for the next 4

years beginning from the end of the 12th year. Find the amount to his credit at the

end of the 15th year assuming interest rate of 9% p.a.

 

1. b) A payment of Rs.P falls due at the end of every γ years .Find at the rate of

interest of i  p.a the present value of the payments to be paid during n years ( n is

an exact multiple of γ).

 

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 23

FIFTH SEMESTER – April 2009

ST 5404 – ACTUARIAL STATISTICS

 

 

 

Date & Time: 28/04/2009 / 1:00 – 4:00       Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

 

ANSWER ALL THE QUESTIONS                                                             (10  x 2 = 20)

 

1. Find the present value of Rs. 1000 receivable at the end of 50 years the rate of interest being 5% p.a.
2. Write the formula for finding effective rate of interest given the nominal rate of interest.
3. Write the two relations between  and .
4. Define the two types of perpetuity.
5. Explain  , .
6. Define and .
7. Explain life annuity, temporary life annuity.
8. Explain the principle of life insurance.
9. Explain the pure endowment assurance.
10. Define the office premium.

PART – B

 

ANSWER ANY FIVE QUESTIONS                                                             (5 x 8 = 40)

 

1. Two loans of Rs.500 each are made out to A three years ago and 2 years ago respectively and an interest of 6% p.a. was agreed upon.  At present A could make a repayment of Rs. 400.  He promises to clear the dues at the end of 2 years from now.  How much will he have to pay then?
2. Derive the formula for  and .
3. A has taken a loan of Rs.10,000.  He pays the level annual payment at the end of each year for 5 years.  What is the installment amount? What is the amount towards the principle in the third payment?  What is the principal outstanding at the end of 3 years?
4. Find the accumulated value of an annuity due of Rs. 400 p.a. payable in equal quarterly installments for 12 years certain at nominal rate of interest of 7% p.a. convertible half yearly.
5. Find the probability that

• a life aged 35 will die between 45 and 50
• a life aged 35 will die after 60
• a life aged 35 will die in the 10^th year from now.
• A life aged 35 will survive another 10 years.

1. On the basis of LIC table at 6% calculate the net annual premium ceasing after 15 years or at previous death for money back policy on (45) to secure the following benefits.

• 1500 on survivance to the end of 5 years.
• 1500 on survivance to the end of 10 years
• 3000 on survivance to the end of 15 years
• 6000 on death at any time within 15 years

1. Derive the formula for present value when the successive instalments form an arithmetic progression.
2. Explain the various types of life annuities.

PART – C

 

ANSWER ANY TWO QUESTIONS                                                           (2 x 20 = 40)

 

1. a). Payments of (i) Rs 50 at the end of each half-year for the first 5 years followed by (ii). Rs 50 at the end of each quarter for the next 5 years, are made into an account to which interest is credited at the rate of 9% p.a. convertible half-yearly.  Find the accumulated value at the end of 10 years.

b). Provident fund (PF) deductions are made at the rate of Rs. 200 per month.                   Find the accumulated value at the end of 10 years, at a rate of interest 10% p.a.

( 10 + 10)

20. a). Derive the formula for ,

b). A special policy provides for the following benefits.

(i). An initial sum of Rs. 10,000 with guaranteed annual additions of Rs.250                            for each years premium paid after the first if death occurs within the term                                   of assurance .

(ii). Rs 10,000 payable on survivance to the end of the term of assurance and

(iii). Free paid up assurance of Rs.10,000 payable at death after expiry of the                               term of assurance.  Calculate net annual premium under the policy on the                        life of (35) for 25 years.                                                    ( 10 + 10)

21. a). Complete the following life table

Age
35	86137	742	.00842
36	–	–	.00885
37	–	–	–
38	–	–	–
39	–	–	.00969

b). Three persons are aged 30,35,40 respectively.  Find the probability that

(i). One of them dies before 45 while the others survive to age 55.

(ii). Atleast one of them attains age 65.

(iii). All of them die after 70.

(iv). None of them reach age 70.

(v). exactly 2 of them die before 45 and one survive to age 60.    ( 10 + 10)

22. Write short notes on any three of the following

(i). Children deferred assurances  (ii). Double endowment assurances.

(iii). Equation of value                  (iv).  and

(v). Present value of perpetuities.                                         ( 10 + 10)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2012

ST 5404 – ACTUARIAL STATISTICS

 

 

Date : 27-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

ST-5404  actuarial STATISTICS                  MAX: 100 Marks     

Section – A

                                                  (Answer all the questions)                                    (10 x2 =20)

 

1. What is the present value of Rs.5,000 receivable at the end of 75 years, the rate of interest being taken as 6 % p.a?
2. Find the nominal rate p.a convertible quarterly corresponding to an effective rate of 8% p.a.
3. Show that
4. Give the formula for a_n and S_n
5. Evaluate v⁶⁰ @ 6.2%
6. Write an expression ₁₀P₄₂, _{10 │ 5}P
7. Define d
8. What is a temporary assurance?
9. What is the need for a commutation function?
10. Expand S_x in terms of  D_x

 

Section – B

                                              (Answer any five questions)                                    (5 x 8 =40)

 

11. A sum of Rs.2000 is invested at a rate of interest of 5%p.a. After 7 years, the rate of interest was changed to 5% p.a. convertible half yearly. After a further period of 3 years, the rate was again changed to 6%p.a. convertible quarterly. What is the accumulated value at the end of 15 years from the commencement?

 

12. Define the following:

• Annuity
• Immediate annuity
• Annuity due
• Deferment period

 

13. Calculate the present value of a deferred annuity payable for 10 years certain, the first payment falling due at the end of 6 years from the present time. The annuity is payable at the rate of Rs. 100 p.a. for the first 5 years and Rs.200 p.a. thereafter.

(a₅ = 4.3295,  a₁₀ = 7.7217,  a₁₅ = 10.3797)

 

14. A fund is to be set up out of which a payment of Rs.100 will be made to each person who in any year qualifies for membership of a certain profession. Assuming that 10 persons will qualify at the end of one year from now, 15 at the end of 2 years, 20 at the end of 3 years, and so on till the number of qualifiers is 50 per annum. When it will remain constant, find at 5% p.a. effective what sum must be paid into the fund now so that it sufficient to meet the outgo.

 

 

15. Derive the expression to find the present value and accumulated value of Increasing

annuity where in the successive installment form  a geometric progression.

 

16. Find the office annual premium for a capital redemption assurance policy of Rs. 3000 redeemable at the end of 20 years, assuming interest rate of 6% and a loading of 8% of office premium.

 

17. Using the LIC ( 1970 – 73 ) Ultimate table find the following  probabilities

 

• that a life aged 35 dies within 12 years
• that a life aged 40 dies not earlier than 12 years and not later than 15 years
• that a life aged 52 survive 12 years
• that a life aged 52 will not die between age 65 and 70      ( 2+2+2+2)

 

18. Explain Pure Endowment Assurance.

 

 

Section – C

                                                   (Answer any two questions)                               ( 2 x 20 =40)

 

19. Explain the various types of annuity and derive the expression for present value and accumulated value of an immediate annuity certain and deferred annuity certain.

 

20. a)A deposit annually Rs. 200 p.a. for 10 years, the first deposit being made one year from now; and after 10 years the annual deposit is enhanced to Rs. 300 p.a. Immediately after depositing the 15 payment he closes his account. What is the amount payable to him if interest is allowed at (i) 6% p.a. (ii) 9% p.a.?
21. b) What is the principle of insurance? How has endowment type assurance

emerged?

 

 

 

21. a) Fill in the blanks in the following portion of a life table

Age  X	lx	dx	qx	px
10	1000000		0.00409
11			0.00370
12				0.99653
13				0.99658
14			0.00342

 

 

 

 

 

 

 

1. b) Using commutation function based on LIC ( 1970 – 73) ultimate mortality table at 6% interest calculate for a person aged 40

 

• The present value of Whole Life Assurance of 10000
• The present value of Double Endowment Assurance of 10000

 

for 15 years term . Also calculate present value of Endowment Assurance and Pure Endowment of each for Rs. 10000 for 15 years term.

 

22. a) Explain R_x ,M_x,D_x and obtain expression for ( IA) _{x : n}

 

b)The following particulars are given:

X	25	26	27	28	29	30
lx	97380	97088	96794	96496	96194	95887
dx	292	294	298	302	307	313

 

Calculate ignoring interest and expenses:

• The value of Temporary Assurance of Rs. 1000 for 2 years for a person aged 25.
• The value of Endowment Assurance benefits of Rs. 1000 for 4 years to a person aged 25.
• The value of a Pure Endowment of Rs. 600 for a person aged 27 receivable on attaining age 30.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5404 – ACTUARIAL STATISTICS

 

 

Date : 10/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Section – A

Answer all the questions:                                                                                                     ( 10 x 2 = 20)

 

1. The amount with compound interest of a certain principal at 5% p.a. is Rs. 3969. Find the principal when period is 2 years.
2. What is meant by discount?
3. What is the effective rate p.a. corresponding to a nominal rate of 8 % p.a. convertible monthly?
4. Evaluate v⁹ s₁₃ @ 9 %

1. Define an annuity.

6. Show that
7. What is perpetuity due?
8. Define q
9. Give the expression for e
10. Write a short note on term assurance.

Section – B

Answer any five questions:                                                                                                 ( 5 x 8 =40)

 

11. The amounts for a certain sum with compound interest at a certain rate in two years and in three years are Rs. 8820 and Rs. 9261 respectively. Find the rate and sum.

1. A has taken a loan of Rs. 2000 at a rate of interest 4% p.a. payable half-yearly. He repaid Rs. 400 after 2 years, Rs. 600 after a further 2 years and cleared all outstanding dues at the end of 7 years from the commencement of the transaction. What is the final payment made by him?
2. The cash purchase price of a bike is Rs. 10,000. A company however offers instalment plan  under an immediate payment of Rs. 2000 is to be made and a series of 5 equal half-yearly payments made thereafter, the first installment being payable at the end of 6 months. If the company wishes to realize a rate of interest of 12 % convertible half-yearly in the transaction, calculate the half-yearly instalment.

14. Calculate the present value of a deferred annuity payable for 10 years certain, the first payment falling due at the end of 6 years from the present time. The annuity is payable at the rate of Rs. 100 p.a. for the first 5 years and Rs.200 p.a. thereafter.

Given (a₅ = 4.3295,  a₁₀ = 7.7217,  a₁₅ = 10.3797)

15. Derive the formula for accumulated value and present value of annuity certain due.
16. Using the LIC ultimate table find the following probabilities:

1. 1.  a life aged 35 dies within 12 years.
   2.  a life aged 40 dies not earlier than 12 years and not later than 15 years.
   3.  a life aged 2 survives 12 years

1. a life aged 52 will not die between ages 65 and 70

 

17. What are the points to be borne in mind in deciding

• Period of investigation?
• Period of selection?
• Method to be used for investigation?

18. Derive an expression for A _x:n.

Section – C

Answer any two questions:                                                                                                 ( 2 x 20 =40)

 

19. a) A has right to receive an amount of Rs.1000 at the end of 12 years from now. This right has been sold to B for a present value calculated at the rate of 8% p.a. The money thus received was invested by A in deposit account at 9% p.a. payable half yearly. After 8 years the account had to be closed and A then invested the amount available at 6% p.a. in another bank. How has A gained or lost in this transaction, as at the end of 12 years?

 

1. b) Derive an expression to find the present value for the following variable annuities:
2. Increasing annuity
3. Immediate Increasing Perpetuity

• Increasing annuity due

1. Increasing Perpetuity due
2. a) A loan of Rs. 3000 is to be repaid with interest at 6% p.a. by means of an immediate annuity for 10 years. Find the level payment. What will be the interest and principal contained in the 5^th instalment? What will be the principle outstanding immediately after the 8^th payment is made?

( 10 + 10)

1. b) In lieu of a single payment of Rs. 1000, at the present moment a person agrees to receive 3 equal payments at the end of 3 years, 6 years and 10 years respectively. Assuming a rate of interest of 6% p.a. what should be the value of each of the 3 payments? ( 10 + 10)

 

 

21. a) Write down expression for probability  in the under mentioned cases:

(i)  Life aged 25 dies between ages 60 and 65

(ii)  Of the two life aged 25 and 30, at least one life dies before attaining age 70

(iii)  Of three lives aged 40, 40 and 45, exactly two lives survive 10 years

(iv) Life aged 28 survives 12 years and dies in the 13^th, or 14^th year.

 

1. b) Fill up the blanks in the following portion of a life table:

Age  X	lx	dx	qx	px
10	1000000		0.00409
11			0.00370
12				0.99653
13				0.99658
14			0.00342

 

( 10 + 10)

22. a) A person aged 30 years has approached a life office for special type of policy providing for the following benefits:

• 1000 on death during the first 5 years
• 2000 on death during the next 5 years
• Survival benefit of Rs. 500 at the end of the 5^th year
• Further payment of Rs. 2000 on survivance to 20 years.
• An annuity of Rs. 200 per annum payable in his life time, the first such payment falling due along with the survival benefits of Rs. 2000.

 

1. b) Derive the expression for A_x and (IA) _{x : n.} ( 10 + 10)

 

 

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