TIME VALUE OF MONEY
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FUTURE VALUE OF AN ANNUITY
• What is Annuity?
An annuity is a stream of constant cash flows
(payments or receipts) occurring at regular
intervals of time.
Example: premium payments of a life
insurance policy.
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FUTURE VALUE OF AN ANNUITY
• For example: You deposit Rs. 1,000 annually at
the end of the year in a bank for 5 years and
your deposits earn a compound rate of 10%.
What will be the value of this series of
deposits (an annuity) at the end of 5 years?
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FUTURE VALUE OF AN ANNUITY
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FUTURE VALUE OF AN ANNUITY
= Rs.1,000(1.10)4+Rs.1,000(1.10)3+
Rs.1,000(1.10)2 + Rs.1,000(1.10) + Rs.1000
= Rs.1,000(1.464)+Rs.1,000(1.331)+
Rs.1,000(1.21) + Rs.1,000(1.10) + Rs.1000
= Rs. 6,105
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FUTURE VALUE OF AN ANNUITY
• Formula
FVAn = A [(1 + r)n – 1] / r OR A (FVIFAr,n)
FVAn : future value of an annuity which has a duration of n periods
A: constant periodic flow
r: Interest rate per period
n: duration of the annuity
[(1 + r)n – 1] / r = future value interest factor for an annuity
(FVIFAr,n)
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FUTURE VALUE OF AN ANNUITY
• Solve with formula
• You deposit Rs. 1,000 annually at the end of the
year in a bank for 5 years and your deposits earn a
compound rate of 10%. What will be the value of
this series of deposits (an annuity) at the end of 5
years?
FVAn = A [(1 + r)n – 1] / r OR A (FVIFAr,n)
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FUTURE VALUE OF AN ANNUITY
• Example 1: What lies in store for you
You decide to deposit Rs. 30,000 annually at
the end of each year in Public Provident
Fund(PPF) Account for 30 years which is
fetching you a rate of return of 8%p.a.What
will be the accumulated amount in your PPF
A/c at the end of 30 years?
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FUTURE VALUE OF AN ANNUITY
• Solution:
Rs. 30,000 (FVIFA 8%,30yrs)
=Rs. 30,000 (1.08)30 - 1
.08
=Rs. 30,000 [113.283]
= Rs. 3,398,490
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FUTURE VALUE OF AN ANNUITY
• Example 2: How much should you save
annually?
You want to buy a house after 5 years when it
is expected to cost Rs. 2million. How much
should you save annually (at the end of each
year) if your savings earn a compound return
of 12 %?
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FUTURE VALUE OF AN ANNUITY
• Solution:
FVAn = A [(1 + r)n – 1] / r OR A (FVIFAr,n)
Rs. 2000,000= A [(1 + 0.12)5 – 1]
0.12
Rs. 2000,000 = A x 6.353
A = Rs. 2000,000
6.353
= Rs. 314,812
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FUTURE VALUE OF AN ANNUITY
• Annual Deposit in a Sinking Fund
Example:
Futura Limited has an obligation to redeem
Rs.500 million Debentures 6 years hence. How
much should the company deposit annually in
a sinking fund account wherein it earns 14%
interest, to cumulate Rs.500 million in 6 years
time?
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FUTURE VALUE OF AN ANNUITY
• Solution:
FVAn = A [(1 + r)n – 1] / r
Rs. 500 million= A [(1 + 0.14)6 – 1]
0.14
Rs. 500 million = A x 8.536
A = Rs. 500 million
8.536
Annual sinking fund deposit= Rs. 58.575 million
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FUTURE VALUE OF AN ANNUITY
• Finding the Interest Rate
A finance company advertises that it will pay a
lumpsum of Rs. 8,000 at the end of 6 years to
investors who deposit annually Rs. 1,000 for 6
years. What interest rate is implicit in this
offer?
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FUTURE VALUE OF AN ANNUITY
• Solution:
Step One- Find FVIFAr,6 for this contract:
FVAn = A [(1 + r)n – 1] / r OR A (FVIFAr,n)
Rs. 8,000 = Rs. 1,000 X FVIFAr,6
FVIFAr,6 = Rs. 8,000
Rs. 1,000
= 8.000
8 = FVIFAr., 6
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FUTURE VALUE OF AN ANNUITY
• Step two- Look at the FVIFAr,n table and read
the row corresponding to 6 years until you find
a value close to 8.000. Doing so, we find that
FVIFA12%,6 = 8.115
So we conclude that the interest rate is slightly
below 12%.
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FUTURE VALUE OF AN ANNUITY
• How long should you wait?
You want to take up a trip to the moon which
costs Rs. 10,00,000.You can save annually Rs.
50,000 to fulfill your desire. How long will you
have to wait if your savings earn an interest of
12%?
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FUTURE VALUE OF AN ANNUITY
• Solution: The future value of an annuity of Rs.50,000
that earns 12% is equated to Rs. 1,000,000.
FVAn = A [(1 + r)n – 1] / r OR A (FVIFAr,n)
1,000,000 = Rs. 50,000 X FVIFA n=?,12%
FVIFA = 20
Look for 20 in FVIFA table at r=12%
Value is 20.655 at n= 11 years
So answer would be between 10 & 11 years
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FUTURE VALUE OF AN ANNUITY
• Ordinary Annuity and Annuity Due
Ordinary Annuity
When the cash flows occur at the end of each period,
the annuity is called an Ordinary Annuity or a
Deferred Annuity.
Note : the first cashflow starts after one year from now
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FUTURE VALUE OF AN ANNUITY
• Annuities Due
When cash flows starts at the beginning of the first period,
such an annuity is called an annuity due.
– Example : lease for an apartment where lease payments are due
at the beginning of the month, Insurance premium for a life
insurance policy
– Note : the first cashflow occurs today!
– Annuity due value = Ordinary annuity value x (1 + r)
This applies for both present and future values. So, two steps
are involved in calculating the value of an annuity due.
Step 1: Calculate the present value or future as though it were
an ordinary annuity.
Step 2 : Multiply your answer by (1 + r). 20
FUTURE VALUE OF AN ANNUITY DUE
For example: You deposit Rs. 1,000 annually at the
beginning of the year in a bank for 5 years and your
deposits earn a compound rate of 10%. What will be
the value of this series of deposits (an annuity) at the
end of 5 years?
1000 1000 1000 1000 1000 ?
0 1 2 3 4 5 FVA = 6105 (1.10) = 6715.5
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PRESENT VALUE OF AN ANNUITY
• Suppose you expect to receive Rs. 1,000
annually for 3 years, each receipt occurring at
the end of the year. What is the present value
of this stream if the discount rate is 10%?
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PRESENT VALUE OF AN ANNUITY
• Solution:
Rs. 1,000 (1/1.10) + Rs. 1,000 (1/1.10)2 + Rs.
1,000 (1/1.10)3
= Rs. 1,000 x 0.9091 + Rs. 1,000 x 0.8264 Rs.
1,000 x 0.7513
= Rs. 2,486.8
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PRESENT VALUE OF AN ANNUITY
• Formula
PVAn = A [{1 – (1/(1 + r)n}/ r] OR A(PVIFAr,n)
PVAn = the present value of an annuity which has a
duration of n periods
A = constant periodic flow (Annuity)
r = discount rate
n = no. of periods
[{1 – (1/(1 + r)n}/ r] = present value interest factor for
an annuity(PVIFAr,n)
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PRESENT VALUE OF AN ANNUITY
• Solve with formula
Suppose you expect to receive Rs. 1,000
annually for 3 years, each receipt occurring at
the end of the year. What is the present value of
this stream if the discount rate is 10%?
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PRESENT VALUE OF AN ANNUITY
How much can you borrow for a car?
You can afford to pay Rs.12,000 per month for
3 years towards a new car. A car finance co
can lend @ 1.5 per cent per month for 36
months. How much can you borrow?
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PRESENT VALUE OF AN ANNUITY
• Solution:
PVAn = A [{1 – (1/(1 + r)n}/ r]
= 12,000 x 1 – 1/(1+0.015)36
0.015
= 12000 x 27.66 = 3,31,920
You can, therefore, borrow Rs.3,31,920 to buy the car.
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PRESENT VALUE OF AN ANNUITY
Example : ABC Ltd. has borrowed Rs 30,00,000
from Canbank Home Finance Ltd. to finance
purchase of a house for 15 years. The rate of
interest on such loans is 24% per annum.
Compute the amount of annual instalments.
30,00,000 = A*PVIFA(24%, 15y)
30000000 = A* 4.0013
A = 7,49,756
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Loan amortization schedule
• Determining the Loan Amortisation Schedule
Most loans are repaid in equal periodic
instalments (monthly,quarterly or annually)
which cover interest as well as principal
repayment. Such loans are referred to as
amortised loans
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Loan amortization schedule
• Determining the Loan Amortisation Schedule
For an amortised loan we would like to know
(a) the periodic instalment payment and
(b) the loan amortisaton schedule showing the
breakup of the periodic instalment payments
between the interest component and the
principal repayment component.
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Loan amortization schedule
• Determining the Loan Amortisation Schedule
Example: A firm borrows Rs. 1,000,000 at an
interest rate of 15% and the loan is to be
repaid in 5 equal instalments payable at the
end of each of the next 5 years. Calculate
annual installment payment. Prepare the loan
amortisation schedule.
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PRESENT VALUE OF AN ANNUITY
• Solution:
PVAn = A [{1 – (1/(1 + r)n}/ r]
Loan amount = A x PVIFA n=5,r=15%
1,000,000 = A x [{1 – (1/(1 + 0.15)5}/ 0.15]
1,000,000 = A x [{1 – 0.497}/ 0.15]
1,000,000 = A x 3.3522
A = Rs. 298,312
So annual instalments will be Rs 298312
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