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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

1157.625

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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 FlowFormulaFormula

FVFV = PVPV (1+r)n

FV is Future ValuePV is Present Valuer is the interest raten 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 CompoundingCompounding 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 1Half- Yearly 2Quarterly 4Monthly 12Note: 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)

An An 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 1EndEnd of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

EndEnd ofPeriod 3

PARTS OF ANNUITY

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PARTS OF ANNUITY DUE

0 1 2 3

$100 $100 $100

(Annuity Due)BeginningBeginning of

Period 1BeginningBeginning of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3

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NoteThe 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

iF 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 3 3 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

iF 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 3 3 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).