ANNUITIES
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DEFINATION:
An annuity is a series of equal payments made at EQUAL interval of time. This interval usually a year but it may be a half-year, a quarter- year, a month and so on.
Examples: Payment for rent, premium of life insurance, recurring deposits in a bank etc.
The interval between two successive payments is called PAYMENT PERIOD. The interval between the beginning of the first payment period and the end of the last payment period is called the TERM or STATUS of the annuity and is measured in years.
The sum payable in each payment period is called PERIODIC PAYMENT or PERIODIC RENT and the total sum payable in a year is called ANNUAL RENT.
TYPES:
A) ANNUITY CERTAIN: An annuity in which payments begin and end at fixed dates is...
B) PERPETUITY: An annuity in which payments begin at a fixed date but continue for ever is...
C) CONTINGENT: It an annuity in which the payments are dependent on some conditions.
*According to their time of payment
A) ORDINARY or IMMEDIATE ANNUITY: If the payment are made at the end of each payment period is...
B) ANNUITY DUE: If the payment are made at the beginning of each payment period,
C)* DEFERRED ANNUITY: The payment of which commence after a certain period. If an IMMEDIATE ANNUITY is deferred for m years, the first payment is to be made at the end of (m+1)th year, but if an ANNUITY DUE is referred for m years, the first payment is to be made at the beginning of (m+1)th year.
**NOTE
Unless otherwise stated or implied
A) annuity will mean ordinary or immediate annuity and
B) payment period will be one year.
FORMULA USED
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1) The amount of an immediate annuity or, simply, annuity is given by
M= A/i{(1+ i)ⁿ -1}
Here,
A is paid at the end of every year for n years.
n is number of years
i is rate/100.
NOTE: If the number of payments made in a year be m and the total amount paid in a year be A,
If the payment is made at the end of half yearly i.e, twice a year, then
M= A/i {1+ i/2)²ⁿ - 1}
REMEMBER: A is always the total payment made in a year.
2) The amount of an annuity due is:
M= (1+ i). A/i {(1+ i))ⁿ -1}
NOTE: If the number of payments made in a year be m and the total payment paid in a year be A, then
M= (1+ i/2) A/i {(1 + i/2)²ⁿ -1}
* PRESENT VALUE OF AN ANNUITY
The present value of an annuity is the sum of the present values of all the payment of the annuity.
V= A/i {1- (1+ i)⁻ⁿ}.
NOTE: If the payment is made m times a year and the total payment in a year be A, then
V= A/i {1- (1+ i/2)⁻²ⁿ}
** The present value of an annuity due , when paid beginning of each year.
V= (1+ i)A/i {1- (1+ i))⁻ⁿ}
NOTE: If the payment is made m times a year and the total payment in a year be A,
V= (1+ i/2)A/i {1- (1+ i/2))⁻ⁿ}
** The present value of an immediate perpetuity or, simply, perpetuity is given by:
V= A/i
EXERCISE -1
A) 1000 B) 1257.80. c) 1500 D) 1698.30
2) Find, correct to nearest rupee, the amount of an annuity of ₹100 in 20 years allowing compound interest 9/2%, Given log 1.04= 0.0 191163 and log 24 117= 1.3823260.
A) 3100 B) 3400 C) 3137. D) 4200
3) Find the amount and the present value of annuity of ₹150 for 12 years, reckoning compound interest at 3.5 % per annum.
A) 1200,1430
B) 2190.28, 1449.50.
C) 3000, 320 D) none
4) What sum will buy an annuity of ₹1050 payable for 4 years, the rate of interest 3.5 % per annum compound ?
A) 3846. B) 4300 C) 5200 D) 6300
5) find the amount of the annuity of ₹150 in half yearly installments for 15 years at 4% p.a. interest also payable half-yearly.
A) 2041.30 B) 3041.25.
C) 4320 D) 6700.43
6) calculate the amount and present value of annuity of annual value of ₹400 payable at the end of each of 3 months for 5 years at 4% C. I compounded quarterly.
A) 2000, 1500 B) 2201, 1804.
C) 2100, 1370 D) 3200, 1400
7) A man decided to deposit ₹300 at the end of each year in a bank which pays 3% p.a.(compound interest). If the installments are allowed to accumulate, what will be the total accumulation at the end of 15 years ?
A) 5500 B) 5560 C) 5580 D) 6000
8) If the present value of an annuity for 10 years at 6% p.a is ₹2500, find the annuity.
A) 300 B) 350 C) 340. D) 400
9) If the present value of an annuity for 25 years at 5% per annum is 50000, find the annuity.
A) 1000 B) 1046.90 C) 1500 D) n
10) what annuity should be paid for 10 years if 10,000 is paid now, the first payment of annuity being made in one year time and the subsequent payment in yearly installments at the end of each subsequent year ? compound interest is to be taken at 5% p.a
A) 1100.94 B) 1200.94
C) 1294.91 D) none
11) A loan of ₹40000 is to be paid back by 30 equal installments. Find the amount of each such installment to cover principle and compound interest at 4% per annum.
A) 2316. B) 2512 C) 2814 D) N
12) A loan of ₹10000 is to be repaid in 30 equal installments of ₹ P. find, P if the compound interest charge is at the rate of 4% p.a. (Annuity is an annuity immediate) given (1.04)³⁰=3.2434
A) 500 B) 525.30 C) 578.40. D) n
13) A loan of ₹40000 is to be repaid equal installments consisting of principal and interest due in course of 30 years. find the amount of each installment reckoning Interest @4%.
A) 2121 B) 2315. C) 2400 D) 2500
14) A company borrows ₹10000 on condition to repay it with compound interest 5% p.a. annual installments of ₹1000 each. In how many years will the Debt be paid off ?
A) 10.2yrs B) 11.2yrs C) 14.2yrs D) n
15) S. Roy borrows ₹20000 at 4% compound interest and agrees to pay both the principal and the interest in 10 equal installments at the end of each year. find the amount of these installments.
A) 2000.15 B) 2300.75
C) 2470.15. D) 2550.75
16) A loan of Rs4000 is to be repaid on equal half-yearly installments in four years. If the rate of compound interest be 10% per annum. find the value of each instalment.
A) 618.80. B) 720.30
C) 830.60 D) 900.50
17) A firm borrows ₹1000 on condition to repay it with compound interest at 4% per annum by annual installments of rupees 100 each. In how many years will the debt be paid off.
A) 13.1 years . B) 14.3 yrs
C) 15yrs D) none
18) A person borrows ₹4000 on the condition that he will repay the money with compound interest at 5% per annum 6 equal annual installments, the first one being payable at the end of 1st year. find the value of each installment.
A) 441.44 B) 666.66
C) 787.71. D) 991.91
19) A Overdraft of ₹50000 is to be paid back in equal installment over a period of 10 years. find the value of this Payment reckoning compound interest at 5% p.a
A) 6,499. B) 6785 C) 7500 D) N
20) A motorcycle is purchased on installment basis, such that ₹3400 is to be paid on the signing of the contract and four yrly installments of ₹2400 each payable at the end of 1st ,2nd, 3rd and 4th years. If interest is charged at 8% per annum. what would be the cash down price? Given that (1.08)⁴= 1.36
A) 11341.18. B) 11911.50
C) 12651.50 D) 12000.35
21) A man buys a house for ₹40000 on the following conditions. He will pay ₹10000 cash down and the balance in 10 equal annual installments, the first to be paid one year after the date of purchase. Calculate the amount of each installment, compound interest being calculated at the rate of 5% p.a.
A) 3885. B) 4338 C) 4600 D) n
22) S. Roy wishes to purchase a printing machinery valued ₹17000. He is prepared to pay now ₹9000 and the balance in 8 equal annual installments, if interest is calculated at 7/2% p.a., how much should he pay annualy
A) 2137 B) 1167. C) 2100 D) 3200
23) A government constructed housing flat cost ₹136000; 40 % of which is to be paid at the time of possession and the balance, reckoning compound interest @9%p.a., is to be paid in 12 equal annual installments. find the amount of each such instalment. Given 1/(1.09)¹²= 0.3558.
A) 10000 B) 11400. C) 12000 D) n
24) A person deposits his whole fortune of ₹20000 in a Bank at 5% compound interest and settles to withdraw ₹1800 per year for his personal expenses. if he begins to spend from the end of the first year and goes on spending at this rate, find the time he will be ruined his savings.
A) 12 B) 14 C) 16 D) 17th year.
25) On his 48th birthday, a man decides to make a gift of ₹5000 to a hospital on his 60th birthday. He decides to save this amount by making equal, annual payments up to and including his 60th birthday to a fund which gives 7/2% compound interest, the first payment being made at once. Calculate the amount of each annual payment (answer to the nearest paise).
A) 219.90 B) 310.90. C) 400 D) n
26) A person retires at the age of 60. He is entitled to get a pension of ₹200 per month payable at the end of every six months. He is expected to live up at the age of 70. if the rate of interest be 12% per annum payable half-yearly, what single sum he is entitled to get at the time of his retirement equivalent to his pension? (Given (1/(1.06)²⁰= 0.3119).
A) 12762 B) 13762. C) 21000 D)n
27) find the sum of money received by a pensioner at 58 if he wants to commute his annual pension of ₹1200 for present payment when compound interest is reckoned at 4% p.a., and the expectation of his life is assessed at 10 years only.
A) 9000 B) 9500 C) 9717. D) n
28) A man retires at the age of 60 years and is employer gifts him pension of ₹3600 a year paid in half yearly installments for the rest of his life. If the expectation of his life is taken to be 10 years and interest is @6% per annum payable half-yearly, determine the present value of the pension.
A) 25000 B) 26724. C) 30000 D)n
29) A sinking fund is to be created for the purpose of replacing some machinary worth ₹100000 after 20 years. How much money should be set aside each year out of the profits for the sinking fund, if the rate of Compound interest is 5% per annum.
A) 3000 B) 3021. C) 3500 D) n
30) A sinking fund is created for replacing some machinery worth ₹ 54000 after 25 years. The scrap value of the machine at the end of the period is ₹4000. How much should be set aside from profit each year for the sinking fund when the rate of compound interest is 5%p.a.
A) 1046.90 . B) 12000 C) 13900 d) n
31) A machine costs a company ₹65000 and its effective life is estimated to be 25 years. A sinking fund is created for replacing the machine at the end of his life, when its scrap realises a sum of ₹2500 only. Calculate what amount should be provided every year from the profits earned for the sinking fund, If it accumulates 7/2% p.a. compound (given that (1.035)²⁵= 2 358)
A) 15000 B) 16000 C) 1610.82. D) n
32) A sinking fund to created for the redemption of debentures of ₹100000 at the end of 25 years. how much money should be set aside out of profits each year for the sinking fund, if the investment can earn interest at 4% per annum ?
A) 2408.09. B) 2500 C) 3200 D) n
33) A company buys a machine for ₹1 lakh. Its estimated life is 12 years and scrap value is ₹5000. what amount is to be retained every year from the profits and allowed to accumulate at 5% Compound interest for buying new machine at the same price after 12 years.
A) 5959.85. B) 6000 C) 6500 D) n
34) An equal sum is provided out of profit by the company each year for the replacement of a machinery after 15 years. Present cost of the machinery is ₹90000 and the cost of machine at the time of time of Replacement is 20% more than the present cost. find the sum if it can earn interest at 5% per annum compound interest.
A) 5000. B) 6000 C) 7000 D) n
35) An equal sum is set aside every year for 25 years to pay off a debenture issue of ₹1 lakh. If the fund accumulates at 2% per annum compound interest, find the value of this annual payment. Given that the amount of annuity of rupees 1 p.a. for 25 years at 2% per annum is equals to 32.03030
A) 3122. B) 4122 C) 5122 D) 5000
36) What sum should be invested every year at 8% per annum compound interest for 10 years, to replace plant and machinery, which is expected to cost then 20% more than its present cost of ₹50000?
A) 3212 B) 4121 C) 4142. D) n
37) the cost of a machine is ₹80000. The estimated scrap value of the machine at the end of its life time of 10 years is ₹₹12000. find the amount of each equal annual installments to be deposited at 9% per annum compound interest annually just sufficient to meet the cost of new machine after 10 years assuming an increase of 40% of the price of the machine then. The first installment is to be paid at the end of the first year. (Given (1.09)¹⁰ = 2..368)
A) 6500 B) 6578.35. C) 4500 D) N
39) A man propose to make an endowment on 1st July 1969 to the university by depositing a sum of money to its banking account stipulating
a) the payment of scholarship of ₹1000 p.a. for 10 years
b) the award of book prizes to the value of ₹500 every year for 20 years. the money deposited together with compound interest @ 5% p.a is to exhaust itself at the end of said 20 years. assuming that the payments and the awards stipulated are made at 1st July every year commencing with the year 1970, determine the amount of endowment required to be made on 1st July 1669 for this purpose.
A) 12000 B) 13000 C) 13975 D) n
40) A man aged 40 years take an insurance policy for ₹50000 for which he is expected to make equal annual payment of ₹ x to the insurance Company commencing now and going on until his death. If the expectation of life of a man aged 40 years is 25 years, find the value of x if the insurance company agrees to pay interest at 4% p.a. compound.
A) 1000 B) 1157.78 C) 1221.31 D) n
41) The annual rent of a perpetuity is ₹4000. find its value, the interest being compounded at 5% per annum .
A) 60000 B) 70000 C) 80000 D) 90000
42) the value and the annual rent of a perpetuity are ₹15000 and ₹900 respectively. find the rate of compound interest..
A) 4% B) 5% C) 6% D) 7%
43) The annual rent of a freehold estate is ₹1000. if the rate of compound interest be 10% per annum. find the price of the estate.
A) 10000. B) 11000
C) 12000 C) 13000
44) the price of freehold estate is ₹50000. If the rate of interest be 9% per annum. what would be the property was worth 12500 if the annual rent?
A) 4000 B) 4500 C) 5000 D)5500
45) A free-hold property was worth ₹12500. If the annual rent of the property be ₹1000, find the rate per cent per annum.
A) 8%. B) 9% C) 10% D) 11%
46) The annual subscription for the membership of a club is ₹₹240 and a person may become a life member by paying ₹8000 at a time. find the rate percent per annum .
A) 3% . B) 4% C) 5% D) 8%
47) which is better--an annuity of ₹150 to last for 10 years or the reversion of a freehold estate of ₹ 79.20 p.a., to commence 7 years hence, the rate of interest being 5% per annum.
A) the first one . B) 2nd one
C) both. D) no comment
48) For endowing an annual scholarship of ₹12000 a man wishes to make 3 equal annual contributions. the first award of the scholarship is to be made three years after the last of his 3 Compounded annually.
A) 1300000 B) 150000.
C) 200000 D) none
49) how many years purchase should be given for a freehold estate, if 5% interest be desired
A) 10 B) 20. C) 30 D) 40 yrs
50) The number of years purchase of a property is 12. reckoning compound interest at 6% per annum find the nearest to the duration of lease.
A) 22. B) 24 C) 26 D) 30 years
51) how many years purchase should be given for a free home estate, if 3% compound interest be desired?
A) 33.33 years. B) 44.44yrs
C) 55.55yrs D) 22.22 yrs
52) what perpetuity can be purchased by investing ₹15000 at 5/2% per annum compounded intrest?
A) 375 . B) 475 C) 575 D) 675
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