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Session Details
This session covers the following:
• Meaning of annuity
• Types of annuity
• Ordinary annuity- future and present value
• Application of ordinary annuity for various
purpose
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Meaning of Annuity
Annuity is a series of equal payments made at usually equal interval of time or in other word
an annuity is a periodic amount of money that is paid at regular intervals
The concept of annuity can be used to turn the one time lump sum payment into payments of
smaller denomination in future time period for our convenience.
Examples of annuity:
• Instalment payment – leasing agreement, Repayment of loan
• Regular deposits to saving bank account
• Insurance premium
• Hire purchase agreements etc.
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Types of annuity
Annuity
Ordinary
Annity
Annuity Due
Deferred
annuity
Perpetuity
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Ordinary annuity
Ordinary annuity is defined as the annuity where the payment is made at the end of
each payment interval.
Ordinary annuity is series of payments which possess the following characteristics:
1) Amount of payment should be same
2) Interval time between the payment should be same
3) Payment should be made at the end of each period
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Ordinary annuity
Future value of ordinary annuity – refers to the value that is
compounding till the end of its term/duration or it is also
defined as the sum of future value of all the periodic payments
at the end of annuity.
Present value of ordinary annuity – refers to the sum of
discounted value of each periodic payment at the given rate of
interest. Also termed as capital value.
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Future value of ordinary annuity
The formula for the future value of an ordinary annuity
• OR
• Where,
=Future value of annuity at the end of each period
• R= Regular payment at the end of each payment interval
= interest rate per period
= number of period for which annuity will lasts
= tabulated value of future value of Interest rate factor annuity (FVIFA)
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Derivation of Future value
of ordinary annuity will be
given in the next slide
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Present value of ordinary annuity
The formula for the Present value of an ordinary annuity
P
• OR
P
• Where,
=Future value of annuity at the end of each period
• R= Regular payment at the end of each payment interval
= interest rate per period
= number of period for which annuity will lasts
= tabulated value of present value of Interest rate factor annuity (PVIFA)
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Derivation of Present value of
ordinary annuity given in the
next slide
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ILLUSTRATIVE PROBLEMS
1) At six-month interval, A deposited ₹2000 in a saving account which credit
interest at 10% p.a. compounded semi-annually. The first deposit was made
when A’ s son was six-month-old and the last deposit was made when his son
was 8 years old. The money remained in the account and was presented to the
son on his 10
th
birthday. How much did he receive?
2) Mr. X purchases a house for ₹2, 00,000. He agrees to pay for the house in 5
equal installments at the end of each year. If the money is worth 5% p.a.
effective, what would be size of each investment? In case X makes a down
payment of ₹50, 000 what would be the size of each installment?
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Amortization of loan
Amortization refers to the repayment of loan through a fixed repayment schedule in regular
payment over a period of time. Each payment includes both interest on the outstanding
amount of loan and principal amount.
This is done by applying the concept of present value of an annuit y. Suppose a loan of
Rs. A has been taken at a interest rate i % which is to be repaid in n regular payment , payable
at the end of each payment interval , then the value of R regular installment can be obtained as
follows:
• A
• A
• R = A /
A loan is amortized if both the capital and interest are paid by a sequence
of periodic payments.
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ILLUSTRATIVE PROBLEM
3. Mr. X took a loan of ₹80,000 payable in 10 semiannual
installments, rate of interest being 8% p.a. compounded
semiannually, find:
• The amount of each installment;
• Loan outstanding after 4
th
payment;
• Interest component of 5
th
payment; and
• Loan repaid after four payments.
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LEASING DECISIONS
Once a firm has evaluated the economic viability of an asset as an investment and
accepted/selected the proposal, it has to consider alternate methods of financing the
investment
the firm may consider leasing of the asset rather than buying it. Hence, lease financing
decisions relating to leasing or buying options primarily involve comparison between the
cost of debt-financing and lease financing.
Evaluation of lease financing decisions involves the following steps:
• (i) Calculate the present value of net-cash flow of the buying option, called NPV (B).
• (ii) Calculate the present value of net cash flow of the leasing option, called NPV (L)
• (iii) Decide whether to buy or lease the asset or reject the proposal altogether by applying the following criterion:
• (a) If NPV (B) is positive and greater than the NPV (L), purchase the asset.
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Capital expenditure decisions
In capital expenditure decisions a company has to
make choice between two machines, both can be used
to improve operation by saving in labour costs.
Given the time value of money , we can use the concept
of annuity to determined the net annual savings of
each machine and then decide which machine to buy.
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ILLUSTRATIVE PROBLEM
4) Machine A costs ₹ 10,000 and has a useful life
of 8 years. Machine B costs ₹8000 and has a useful
life of 6 years. Suppose machine a generates an
annual labour saving of ₹ 2000 which machine B
generate an annual saving of ₹1800. Assuming the
time value of money is 10% p.a., find which
machine is preferable?
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Valuation of Bond
A bond is generally a security for a debt, in which the person who is issuing
holds a debt against the person who has taken the loan and thereby is
obliged to pay the interest and the principal amount.
Usually, bonds are issued for longer periods which are usually greater than
one year and which upon maturity, will be paid upon the principal amount
(redemption value) or the periodical interest.
The process of determining these bonds is called bond valuation. It is used
to determine the theoretical price or fair price or intrinsic price of the
bonds.
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THE FORMULAE FOR COMPUTING THE VALUE OF BOND
• Where,
• D =the periodic dividend payment
= the yield rate per period
• RV= the redemption price
• v= the purchase price
= number of period before redemption
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5) A ₹1000 bond paying annual dividends at the
8.5% will be redeemed at par at the end of 10
years. Find the purchase price of this bond if the
investor wishes a yield rate of 8%.
Illustrative Problem
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CONTINUOUS COMPOUNDED ANNUITY
Continuous compounding is compounding that is constant. One way some try to visualize
the concept of continuous compounding is that is fluid, constantly compounding moment by
moment, as opposed to daily, monthly, quarterly, or annually.
The future value of annuity with continuous compounding formula is the sum of future cash
flows with interest. The sum of cash flows with continuous compounding can be shown as
• This is considered a geometric series as the cash flows are all equal. The common ratio for this example is
. To
solve this continuous compounding series summation will be denoted by integration. So the formula will be
• Similarly,
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ILLUSTRATIVE PROBLEMS
6) An annuity of ₹500 p.a. is flowing continuously for 10
years. Find its future value if the rate of interest is 10%
compounded continuously
7) Find the capital value of a uniform income stream of ₹ R
per year for m years, reckoning interest continuously at
100r% per year. What will be the result if income is forever?
B.Com. (Hons.) IInd Year
Business Mathematics, Section A and H Asha Rani
Questions for Ordinary Annuity
Ques:1. At six-month interval, A deposited ₹2000 in a saving account which
credit interest at 10% p.a. compounded semi-annually. The first deposit was made
when A’ s son was six-month-old and the last deposit was made when his son
was 8 years old. The money remained in the account and was presented to the son
on his 10
th
birthday. How much did he receive?
Ques:2. An annuity of ₹500 p.a. is flowing continuously for 10 years. Find its
future value if the rate of interest is 10% compounded continuously.
Ques:3. Mr. X deposits in his son’s account ₹1000 times his son age at the end
of each birthday. Find the balance accumulated at the 10
th
birthday, if the rate of
interest is 10% p.a. compounded annually.
Ques:4. A man requires ₹ 2, 00,000 to purchase a house after 5 years. He has an
opportunity to invest the fund in an account which can earn 6% p.a. compounded
quarterly. Find how much be deposited at the end of each quarter so as to have
the required amount at the end of 5 years.
Ques:5. Mr. X purchases a house for ₹2, 00,000. He agrees to pay for the house
in 5 equal installments at the end of each year. If the money is worth 5% p.a.
effective, what would be size of each investment? In case X makes a down
payment of ₹50, 000 what would be the size of each installment?
Ques:6. What should be the monthly sales volume of a company if it desires to
earn 12% annual returns convertible monthly on its investment of ₹ 2,00,000?
Monthly costs are ₹3, 000. The investment will have eight-year life with no scrap
value?
Ques:7. Mr. X sells his old car for ₹ 100,000 to buy a new one costing ₹ 2,58,000.
He pays ₹ x cash and balance by payment of ₹7000 at the end of each mount for
18 months. If the rate of interest is 9% compounded monthly, find x.
Ques:8. Find the capital value of a uniform income stream of ₹ R per year for m
years, reckoning interest continuously at 100r% per year. What will be the result
if income is forever?
Ques:9. According to an investment proposal, an initial investment of ₹ 1,00,000
is expected to yield a uniform income stream of ₹ 10,000 p.a. if the money is
worth 8% p.a. compounded continuously, what is the expected payback period,
i.e. after what time, the initial investment will be recovered?
Ques:10. If the present value and amount of an ordinary annuity of ₹1 p.a. for n
years are ₹8.1109 and ₹12.0061 respectively, Find the rate of interest and the
value of n without consulting the compound interest table.
B.Com. (Hons.) IInd Year
Business Mathematics, Section A and H Asha Rani
Ques:11. Mr. X took a loan of ₹80,000 payable in 10 semiannual installments,
rate of interest being 8% p.a. compounded semiannually, find:
1) The amount of each installment;
2) Loan outstanding after 4
th
payment;
3) Interest component of 5
th
payment; and
4) Loan repaid after four payments.
Ques:12. Mr. M borrowed ₹10, 00,000 from a bank to purchase a house and
decided to repay by monthly equal installment in 10 years. The bank charges
interest at 9% compounded monthly. The bank calculated his EMI as ₹12, 668.
Find the principal and the interest paid in Ist and IInd year.
Ques:13. Machine A costs ₹ 10,000 and has a useful life of 8 years. Machine B
costs ₹8000 and has a useful life of 6 years. Suppose machine a generates an
annual labour saving of ₹ 2000 which machine B generate an annual saving of
₹1800. Assuming the time value of money is 10% p.a., find which machine is
preferable?
Ques:14. Find the purchase of a ₹ 1000 bond, redeemable at the end of 10 years
at ₹1100 and paying annual dividends at 4% if the yield rate is to be 5% p.a.
effective.
Ques:15. A ₹1000 bond paying annual dividends at the 8.5% will be redeemed
at par at the end of 10 years. Find the purchase price of this bond if the investor
wishes a yield rate of 8%.
