The document discusses compounding techniques and calculating future values. It defines compound interest as interest earned on the initial principal sum becoming part of the principal. It then provides formulas and examples for calculating the future value of a single cash flow (lump sum) and the future value of a series of cash flows (annuity), including ordinary annuities where payments occur at the end of each period and annuities due where payments occur at the beginning of each period. Non-annual compounding and sinking funds are also briefly discussed.

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Compounding Techniques
Compounding concept
means the interest
earned on the initial
principal sum becomes
a part of the principal
or initial sum at the end
of the compounded
period.
Year 1 2 3
Beginning
Amount 1000 1050 1102.5
Interest rate 5% 5% 5%
Amount of
interest 50 52.5 55.125
Beginning
Principal 1000 1050 1102.5
Ending
Principal 1050 1102.5
1157.6
25

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Future Value
The compounding technique is used to find out the
FUTURE VALUE of a present money.
It can further be explained with reference to:
• The future value of a single cash flow (Lump sum
amount)
• The Future value of a series of Cash Flows
(Annuity)

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(i) The FV of Single Cash Flow
FormulaFormula
FVFV = PVPV (1+r)n
FV
is Future Value
PV is Present Value
r is the interest rate
n is time period
• If you deposited Rs 55,650 in a bank, which was
paying a 15 per cent rate of interest on a ten-year time
deposit, how much would the deposit grow at the end
of ten years?

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• The general form of equation for calculating the
future value of a lump sum after n periods
may, therefore, be written as follows:
• The term (1 + r)n
is the compound value
factor (CVF) of a lump sum of Re 1, and it
always has a value greater than 1 for positive i,
indicating that CVF increases as r and n
increase.
FVn = PV x CVFr,n

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We will first find out the compound value factor
at 15 per cent for 10 years which is 4.046.
Multiplying 4.046 by Rs.55,650, we get Rs
225,159.90 as the compound value:
FV= 55,650 X CVF 10, .15
= 55,650 X 4.046
= Rs. 225159.90

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Non-Annual Compounding
Compounding is not always annually it may be half- yearly,
quarterly, monthly. So in this case compounding can be done
be using the following formula.
FV = PV(1+r/m)mn
m is the number of time compounding is done in a year
n is the time period.
Compounding Period No of period (m)
Annually 1
Half- Yearly 2
Quarterly 4
Monthly 12
Note: More frequently the compounding is made, the faster is the growth
in the FV

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(ii) Future Value of series of cash flow
(Annuity)
AnAn AnnuityAnnuity represents a series of payments (or
receipts) occurring over a specified number of
equidistant periods
Types of Annuities
• Ordinary AnnuityOrdinary Annuity: Payments or receipts
occur at the end of each period.
• Annuity DueAnnuity Due: Payments or receipts occur
at the beginning of each period

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Examples of Annuities
•Student Loan Payments
• Car Loan Payments
• Insurance Premiums
• Mortgage Payments
• Retirement Savings

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0 1 2 3
$100 $100 $100
(Ordinary Annuity)
EndEnd of
Period 1
EndEnd of
Period 2
Today EqualEqual Cash Flows
Each 1 Period Apart
EndEnd of
Period 3
PARTS OF ANNUITY

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PARTS OF ANNUITY DUE
0 1 2 3
$100 $100 $100
(Annuity Due)
BeginningBeginning of
Period 1
BeginningBeginning of
Period 2
Today EqualEqual Cash Flows
Each 1 Period Apart
BeginningBeginning of
Period 3

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Note
The future value of an ordinary annuity
can be viewed as occurring at the endend
of the last cash flow period,
whereas the future value of an annuity
due can be viewed as occurring at the
beginningbeginning of the last cash flow period

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Formula for ordinary Annuity
• As it is clear now that Annuity is a fixed payment (or
receipt) each year for a specified number of years. If
you rent a flat and promise to make a series of
payments over an agreed period, you have created an
annuity.
• The term within brackets is the compound value factor
for an annuity of Re 1, which we shall refer as CVAF.
Fn = A x CVAFn,r
(1 ) 1n
n
i
F A
i
 + −
=  
 

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Example of ordinary Annuity
FVAFVA33 = $1,000(1.07)2
+
$1,000(1.07)1
+ $1,000
= $1,145 + $1,070 + $1,000
= $3,215$3,215
$1,000 $1,000 $1,000
0 1 2 33 4
$3,215 = FVA$3,215 = FVA33
7%
$1,070
$1,145
Cash flows occur at the end of the period

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Formula For Annuity Due
(1 ) 1n
n
i
F A
i
 + −
=  
 
(1+i)

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FVADFVAD33 = $1,000(1.07)3
+
$1,000(1.07)2
+ $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440$3,440
$1,000 $1,000 $1,000 $1,070
0 1 2 33 4
$3,440 = FVAD$3,440 = FVAD33
7%
$1,225
$1,145
Cash flows occur at the beginning of the period

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Sinking Fund
• Sinking fund is a fund, which is created out of fixed
payments each period to accumulate to a future sum
after a specified period. For example, companies
generally create sinking funds to retire bonds
(debentures) on maturity.
• The factor used to calculate the annuity for a given
future sum is called the sinking fund factor (SFF).