1 Time Value of Money (2)

8/13/2019 1 Time Value of Money (2)

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TIME VALUE OF MONEY


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Time Value of Money

Which one do you prefer?Eg.1 - A Rs.1000 receivable now or a year later.Obviously, Rs.1000 today.Eg.2 - Today you lent Rs.10,000 to Mr.X. After 5 years doyou want to receive same amount what you lent orhigher than that?Obviously, higher than Rs.10,000.You already recognize that there is TIME VALUE TOMONEY!!

TIME allows you the opportunity to postponeconsumption and earn INTEREST.


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Why should money have time value?

A Rupee received today is worth more than a rupeereceived tomorrow.Three reasons may be attributed to the individuals timepreference for money: preference for consumption investment opportunities InflationThe time preference for money is generally expressed byan interest rate.

A investor has two options. Either he can go for simpleinterest and compound interest.


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Compound Interest Compound Interest is the interest that is received on the

original amount (principal) as well as on any interest earnedbut not withdrawn during earlier periods. And is calculatedby using the formula

CI = P (1+r)n P The maturity value of investment would be calculated by

using the formula

FV= P (1+r)n Where,

o CI = Compound Interesto P = Principalo n = Maturity Periodo k = interest Rate

o F = Total Amount after n years


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Simple Interest V/s Compound Interest

What is the difference between Simple interestand Compound interest? How does these affect on your investment? Which one would you prefer? Why? You deposited Rs. 1000 for 5 years at 10%

interest rate. Find out how much amount you getafter 5 years for simple interest and compound

interest? Interpret the answer.


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Time Value AdjustmentThe time value of money is a way of calculating the

value of a sum of money, at any time in the present orfuture. It allows us to calculateo Future Value: is the future worth of a present amount.

i.e. is the investments maturity value that an investorwould receive at the end of the specified period. Theprocess of calculating future values is called asCOMPOUNDING.

o Present Value: is the present worth of an amount thatwill be received in the future. i.e. is the amount thatneeds to be invested now, at the specified rate , to getthe future cash flow. The process of calculatingpresent values is called as DISCOUNTING.


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Some points before we start problems. There are three types of cash flows

o Single Flowo Multiple Flow-Uneven Series

Cash flows occurring at the beginning of theperiod

Cash flows occurring at the end of the periodo Multiple Flow-Even Series (Annuities)

Cash flows occurring at the beginning of theperiodCash flows occurring at the end of the period

NOTE: A cash flow can be outflow (deposit) or inflow(receipt).


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Some points before we start problems. An Annuity represents a series of equal payments (or

receipts) occurring over a specified number ofequidistant periods. (say monthly, quarterly, semi-annually etc.)Types of Annuity

Regular Annuity: Payments or receipts occur at the end of eachperiod. Annuity Due: Payments or receipts occur at the beginning of

each period.

Examples of Annuities: Recurring deposits, PF deposits, Student Loan Payments,

Retirement Savings, Car Loan Payments, Insurance Premiums,Mortgage Payments, etc.


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Some points before we startproblems.

Steps to Solve Time Value of Money Problems1. Read Problem Thoroughly2. Create a Time Line3. Put Cash Flows and Arrows on Time Line4. Determine whether it is a PV or FV Problem5. Determine if Solution involves a Single CF, Annuity

Stream (s), or Mixed Flow6. Apply appropriate formula7. Solve the Problem


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Calculation of Future Value of Single Flow

0 1 2 3 4 5I 12% I 12% I 12% I 12% I 12%_____ I

1000 FV 5=?The future Value of single cash flow can be calculated by

using following formulaFVn = PV(1+r)n

Where,o FVn = Future Value of Initial Cash Flow n yearso PV0 = Initial Cash Flowo r = Annual rate of returno n = Life of Investment


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Calculation of Future Value of Single Flow

In the above formula, the term (1 + r) n is the FutureValue Interest Factor (FVIF) of a lump sum of Rs. 1,For example, IFVIF (12%,5) =1.7623. What does it indicate?FVIF is always has a value greater than 1 for positive r.Why?

FVIF increases as r and n increase. i.e. FV increases asr and n increase What is the relationship between FV and Interest rate?Direct relationship i.e. as rate of interest increases FV

also increasesThe solution can be find out by using Future ValueInterest Factor (FVIF) table.

FVn = PV X FVIF(r,n)


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Calculation of Future Value of Multiple Flow-UnevenSeries

Find out future value of all cash flows individuallyand sum upo When cash flows occur at the end of the year

o When cash flows occur at the beginning of the year

Where,o FVn = Future Value of all cash flow n yearso CFn = Cash flow during year n o r = Annual rate of returno n = Life of Investment

0n

3-n3

2-n2

1-n1n r)(1CF..........r)(1CFr)(1CFr)(1CFFV

1n

2-n3

1-n2

n1n r)(1CF...........r)(1CFr)(1CFr)(1CFFV


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Calculation of Future Value of Multiple Flow-Evenseries (Annuity)

The cash flow can occur either at the end of the year orbeginning of the year.

o Regular Annuity0 1 2 3 4 5I 12% I 12% I 12% I 12% I 12% I

0 1000 1000 1000 1000 1000 FV 5=?o Annuity Due0 1 2 3 4 5I 12% I 12% I 12% I 12% I 12% I

1000 1000 1000 1000 1000 0 FV 5=?Make out the difference between two time lines.Which cash flow generate the highest FV? Why?


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The Future Value of Annuity can be calculated byusing following Formulaeo Future Value of Regular Annuity

o Future Value of Annuity Due

NOTE: Where CF 1=CF2=CF3==CFn=A

r

1r)(1AFVA

n

n

r)(1r

1r)(1AFVA

n

n


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Calculation of FV of Multiple Flow-Even series(Annuity)

In the above formulao FVAn = Future Value of Annuity at the end of the n

yearso A = Amount invested at the end (regular

annuity)/beginning (annuity due) of the every yearfor n years

o k = Annual rate of returno n = Life of Investment

In the above formula, the term isthe Future Value Interest Factor of Annuity (FVIFA) ofRs.1.

r

1r)(1 n


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The solution can be found out by using Future ValueInterest Factor Annuity (FVIFA) table.o Future Value of Regular Annuity is

FVAn = A X FVIFA(r,n)o Future Value of Annuity due is

FVAn = A (1+r) X FVIFA(r,n ) NOTE: In reality, most of the cases we use regular

annuity than annuity due. If information regarding the

point of deposit/receipt of cash flow is missing in theproblem, it is assumed that the cash flow occurred at theend of the month i.e. regular annuity.

Why Regular annuity? Why not annuity due?


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Application:o Knowing what lies in store for youo How much should you save annuallyo Annual deposit in a sinking fund

o Finding the interest rateo How long should you wait


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Calculation of Present Value of Single Flow

0 1 2 3 4 5

I 12% I 12% I 12% I 12% I 12%_____ IPV0=? 10000

The present value of single cash flow can becalculated by using following formula

Where,o PV=Present Valueo FVn = Future Valueo r = Annual rate of returno n = Life of Investment

nn r)(11

FVPV


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Calculation of Present Value of Single Flow

In the above formula, the term is the Present

Value Interest Factor (PVIF) of a lump sum of Rs.1.For example, PVIF (12%,5) =0.5674. What does it indicate?PVIF is always has a value lower than 1 for positive r.

Why?PVIF increases as r and n decrease. i.e. PV decreasesas r and n increase The solution can be find out by using Present ValueInterest Factor (PVIF) table.

PV0 = FVn X PVIF(r,n)

nr)(11


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Calculation of Present Value of Multiple Flow-UnevenSeries


o Multiple Flow-Uneven Series occurring at the end of the period

0 1 2 3 4 5I 12% I 12% I 12% I 12% I 12% I

PV0=? 0 1000 1500 750 2000 3000o Multiple Flow-Uneven Series occurring at the beginning of the

period

0 1 2 3 4 5I 12% I 12% I 12% I 12% I 12% I

PV0=?1000 1500 750 2000 3000 0

C l l ti f P t V l f M lti l Fl U


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Calculation of Present Value of Multiple Flow-UnevenSeries

Find out Present value of all cash flows & sum upo When cash flow receivable at the end of the year

o When cash flow receivable at the beginning of theyear

Whereo PV

0 = Present Value of all cash flow n yearso CFn = Cash flow during year n

o r = Annual rate of returno n = Life of Investment

nn

33

22

11

0 r)(1CF

.............r)(1

CFr)(1

CFr)(1

CFPV

1-nn

23

12

01

0 r)(1CF

.............r)(1

CFr)(1

CFr)(1

CFPV


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Calculation of Present Value of Multiple Flow-EvenSeries (Annuities)


o Regular Annuity

0 1 2 3 4 5I 12% I 12% I 12% I 12% I 12% I

PV0=? 0 1000 1000 1000 1000 1000o Annuity Due

0 1 2 3 4 5I 12% I 12% I 12% I 12% I 12% I

PV0=?1000 1000 1000 1000 1000 0


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Calculation of Present Value of Multiple Flow-EvenSeries (Annuity)

The Present Value of Annuity can be calculatedby using following Formulae

o Present value of regular annuity

o Present value of annuity due

NOTE: Where CF 1=CF2=CF3==CFn=A

n

n

0

r)r(1

1)r 1(APVA

r)(1r)r(1

1)r 1(APVA n

n

0


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In the above Formula,o PVA0 = Present Value of Annuityo A = Amount deposited/invested at the end of the

every year for n years o

k = Annual rate of returno n = Life of Investment

In the above Formula, the term

Present Value Interest Factor of Annuity (PVIFA) of Rs 1.

nr)r(11)r 1( n


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The solution can be find out by using PresentValue Interest Factor Annuity (PVIFA) table.

o Present Value of Regular Annuity is

PVA0 = A X PVIFA(r,n)o Present Value of Annuity due is

PVA0 = A X PVIFA(r,n) (1+r)


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Calculation of Present Value of Multiple Flow-Evenseries (Annuity)

Application:o Period of Loan amortizationo Determining the Loan Amortization scheduleo Determining the periodic withdrawalo Finding the interest rate


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Present value of growing perpetuity In case of growing perpetuity the annual cash flow

grows at a constant rate of g. Present value of growing perpetuity

OR

Where,

o = Present Value of Perpetuityo A = Amount receivable at the end of the every year for nyears

o r = Required rate of returno g = Growth Rate

r)(1)g1(A

.............r)(1g)1(A

r)(1g)1(A

r)(1A

PVA 32

210

g-r A

PV

PV


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Intra-Year Compounding/Discounting

In bank most of the time we use intra yearcompounding/discounting For Example,

o Fixed deposits are quarterly compoundedo Recurring deposits are quarterly compoundedo All retail loans (Personal loans, Home loans, Educational

Loans, Vehicle Loans etc.) monthly installments aremonthly discounted

[Refer next chapter]


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Impact of frequency

You deposited Rs.10000 for 1 year which earns atan interest rate of 10%. You got three options forcompounding viz. yearly, semiannually,quarterly. Which will you prefer and why?


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Impact of frequency

Example: Consider, P=10,000, n=1year, r=10% p.a., m

= 1 (annually)/2 (Semi-annually)/4 (Quarterly)

Quarterly10000.00

250.0010250

256.2510506.25

262.65

10768.90269.25

11038.15

Semi-annually10000

--

50010500

-

-525

11025

Annually10000

-----

-1000

11000

ParticularsAmount at beginning

Interest for the first QuarterAmount at the end of 3 monthsInterest for the Second QuarterAmount at the end of 6 monthsInterest for the third Quarter

Amount at the end of 9 monthsInterest for the fourth QuarterAmount at the end of the year


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Effective Rate of Return

The rate of interest under annual compounding

which produces the same result as thatproduced by an interest rate under multiplecompounding . It can be calculated by usingbelow formula

Whereo r = effective rate of returno k = normal rate of returno m = frequency of compounding

1m

r 1r

m


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Effective Rate of Return Find out the effective rate of return for the previous

problem in case of annual, semi-annual andquarterly compounding. Effective rate of return for the example which is in

slide 37

Quarterly10%

10.38%

Semi-annually10%

10.25%

Annually10%10%

Particulars Normal rate of InterestEffective rate of interest


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Doubling Period If you know the compound rate of return of your

investment is going to earn, can you tell in howmany years the investment will get doubled? It can be calculated by using below formulae. Doubling Period

o .

o .

Out of above two, Rule of 69 gives the preciseanswer where as Rule of 72 gives theapproximate answer.

Inte

7272of Rule

Int

635.069of Rule

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1 Time Value of Money (2)