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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 do you want to receive same amount what you lent or higher than that? Obviously, higher than Rs.10,000.You already recognize that there is TIME VALUE TO MONEY!!TIME allows you the opportunity to postpone consumption and earn RETURN.

Why should money have time value?

A Rupee received today is worth more than a rupee received tomorrow.Three reasons may be attributed to the individual’s time preference for money:• preference for consumption• investment opportunities • Inflation

The time preference for money is generally expressed in percentages per annual for e.g. 8% p.a.Time value of money differs from time to time and person to person A lender/borrower has two options. Either he can go for simple interest and compound interest.

'Simple' or 'flat rate' interest is the amount of interest paid each year as a fixed percentage of the principal amount. And is calculated by using the formula

SI = Pnr If the investor does not withdraw the interest periodically, the maturity value of investment would be calculated by using the formula

FV = SI +PWhere,

o SI = Simple Interesto P = Principalo n = Maturity priodo r = interest rateo FV= Maturity value after ‘n’ years

Compound Interest•Compound Interest is the interest that is received on the original amount (principal) as well as on any interest earned but not withdrawn during earlier periods. And is calculated by 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,CI = Compound Interesto P = Principalo n = Maturity Periodo k = interest Rateo F = Total Amount after ‘n’ years

Simple Interest V/s Compound Interest

•What is the difference between Simple interest and 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 get after 5 years for simple interest and compound interest? Interpret the answer.

Time Value Adjustment

The time value of money is a way of calculating the value of a sum of money, at any time in the present or future. It allows us to calculate

oFuture Value: is the future worth of a present amount. i.e. is the investments maturity value that an investor would receive at the end of the specified period. The process of calculating future values is called as COMPOUNDING.oPresent Value: is the present worth of an amount that

will be received in the future. i.e. is the amount that needs to be invested now, at the specified rate , to get the future cash flow. The process of calculating present values is called as DISCOUNTING.

There are three types of cash flowsoSingle FlowoMultiple Flow-Uneven Series

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

oMultiple Flow-Even Series (Annuities)Cash flows occurring at the beginning of the period

[Annuity due]Cash flows occurring at the end of the period

[Regular Annuity]NOTE: A cash flow can be outflow or inflow.

An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. (say monthly, quarterly, semi-annually etc.)Types of Annuity– Regular Annuity: Payments or receipts occur at the end

of each period. examples bonds pay interest at the end.– Annuity Due: Payments or receipts occur at the

beginning of each period. examples lease rentals or mortgage. payments

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

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

Some points before we start problems….

Steps to Solve Time Value of Money Problems 1. 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

Calculation of Future Value of Single Flow 0 1 2 3 4 5

I 12% I 12% I 12% I 12% I 12%_____I 1000 FV5=?The future Value of single cash flow can be calculated by using following formula

FVn = PV(1+r)n

Where,oFVn = Future Value of Initial Cash Flow ‘n’ years oPV = Initial Cash Flowor = Annual rate of returnon = Life of Investment

Calculation of Future Value of Single Flow

In the above formula, the term (1 + r)n is the Future Value Interest Factor (FVIF) of a lump sum of Rs. 1, For example, FVIF(12%,5) =1.7623. What does it indicate?FVIF is always has a value greater than 1 for positive r. Why?What is the relationship between FV and Interest rate?Direct relationship i.e. as rate of interest increases FV also increases.FV increases as ‘r’ and ‘n’ increase. i.e. FV increases as ‘r’ and ‘n’ increase

CALCULATION OF FUTURE VALUE OF MULTIPLE FLOW-UNEVEN SERIES

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

oMultiple Flow-Uneven Series occurring at the end of the period 0 1 2 3 4 5

I 12% I 12% I 12% I 12% I 12% I 0 1000 1500 750 2000 3000 FV5=?

oMultiple Flow-Uneven Series occurring at the beginning of the period

0 1 2 3 4 5 I 12% I 12% I 12% I 12% I 12% I 1000 1500 750 2000 3000 0 FV5=? Make out the difference between two time lines.Which cash flow generate the highest FV? Why?

CALCULATION OF FUTURE VALUE OF MULTIPLE FLOW-UNEVEN SERIES

Find out future value of all cash flows individually and sum up 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’ years o 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


oRegular Annuity0 1 2 3 4 5

I 12% I 12% I 12% I 12% I 12% I 0 1000 1000 1000 1000 1000 FV5=?

oAnnuity Due 0 1 2 3 4 5 I 12% I 12% I 12% I 12% I 12% I 1000 1000 1000 1000 1000 0 FV5=?

Calculation of Future Value of Multiple Flow-Even series (Annuity)

The Future Value of Annuity can be calculated by using following Formulae

o Future Value of Regular Annuity

o Future Value of Annuity Due

NOTE: Where CF1=CF2=CF3=……=CFn=A

r1r)(1AFVA

n

n

r)(1r

1r)(1A FVAn

n

Calculation of Future Value of Multiple Flow-Even series (Annuity)

•In the above formula

oFVAn = Future Value of Annuity at the end of the ‘n’ years oA = Amount invested at the end (regular

annuity)/beginning (annuity due) of the every year for ‘n’ yearsor = Annual rate of returnon = Life of Investment

•In the above formula, the term is the Future Value Interest Factor of Annuity (FVIFA) of Rs.1.

r

1r)(1 n

For example, FVAIF(12%,5) =6.3528. What does it indicate?FVAIF is always has a value greater than ‘n’ for positive ‘r’.•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 the problem, it is assumed that the cash flow occurred at the end of the month i.e. regular annuity.

CALCULATION OF FUTURE VALUE OF MULTIPLE FLOW-EVEN SERIES (ANNUITY)

Application:o Knowing what lies in store for youo How much should you save annuallyo Annual deposit in a sinking fundo Finding the interest rateo How long should you wait

Calculation of Present Value of Single Flow 0 1 2 3 4 5 I 12% I 12% I 12% I 12% I 12%_____I PV0=? 10000 The present value of single cash flow can be calculated by using following formula

Where,oPV=Present ValueoFVn = Future Value or = Annual rate of returnon = Life of Investment

nnr)(1

1FVPV

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?What is the relationship between ‘PV’ and ‘r’?Inverse relationship i.e. as rate of interest increases FV also increases.PV increases as ‘r’ and ‘n’ decrease. i.e. PV decreases as ‘r’ and ‘n’ increase

nr)(11

Calculation of Present Value of Multiple Flow-Uneven Series


oMultiple Flow-Uneven Series occurring at the end of the period

0 1 2 3 4 5 I 12% I 12% I 12% I 12% I 12% I PV0=? 0 1000 1500 750 2000 3000

oMultiple Flow-Uneven Series occurring at the beginning of the period

0 1 2 3 4 5 I 12% I 12% I 12% I 12% I 12% I PV0=?1000 1500 750 2000 3000 0

CALCULATION OF PRESENT VALUE OF MULTIPLE FLOW-UNEVEN SERIES

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 the year

WhereoPV0 = Present Value of all cash flow ‘n’ years oCFn = Cash flow during year ‘n”or = Annual rate of returnon = Life of Investment

nn

33

22

11

0 r)(1CF.............

r)(1CF

r)(1CF

r)(1CFPV

1-nn

23

12

01

0 r)(1CF.............

r)(1CF

r)(1CF

r)(1CFPV

Calculation of Present Value of Multiple Flow-Even Series (Annuities)


oRegular Annuity 0 1 2 3 4 5 I 12% I 12% I 12% I 12% I 12% I PV0=? 0 1000 1000 1000 1000 1000

oAnnuity Due 0 1 2 3 4 5

I 12% I 12% I 12% I 12% I 12% I PV0=?1000 1000 1000 1000 1000 0

Calculation of Present Value of Multiple Flow-Even Series (Annuity)

•The Present Value of Annuity can be calculated by using following Formulae

o Present value of regular annuity

o Present value of annuity due

•NOTE: Where CF1=CF2=CF3=……=CFn=A

n

n

0 r)r(11)r1(APVA

r)(1r)r(1

1)r1(APVA n

n

0

Calculation of Present Value of Multiple Flow-Even Series (Annuity)

•In the above Formula,

oPVA = Present Value of Annuity oA = Amount deposited/invested at the end of the every

year for ‘n’ yearsok = Annual rate of returnon = Life of Investment

•In the above Formula, the term Present Value Interest Factor of Annuity

n

n

r)r(11r)(1

Calculation of Present Value of Multiple Flow-Even series (Annuity)

Application:oDetermining repayment instalmentoDetermining the periodic withdrawaloDetermining the Loan Amortization scheduleoFinding the interest rate

Look at the following problems

1. Kapoor Ltd owns a factory , the factory is expected to produce annual fixed cash flow of 78720 forever and has a cost of capital of 16.4%. What is the worth of the factory?

2. Longhorn corporation issues a security that promises to pay to pay its holders Rs 200 indefinitely. Money markets are such that investors can earn about 8 .p.a .How much can longhorn sell this special security for?

Present Value of Perpetuity

•An annuity of infinite duration is known as Perpetuity.•Present value of perpetuity

OR

•Where,o = Present value of perpetuityo A = Annual Cash flowo r = Required rate of return

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

r)(1A

r)(1A

r)(1APVA 321

rAPVA

PVA

Look at the following problem

• Seema owns a fashion clothing store that is expected to produce cash flows forever, the next cash is expected to be 50,000 and subsequent cash flows are expected to increase by 3% each year forever, the cost of capital for the shop is 9%. What is the fashion worth?

Present value of growing perpetuity

•In case of growing perpetuity the annual cash flowgrows at a constant rate of ‘g’.

•Present value of growing perpetuity

• OR

•Where,o = Present Value of PerpetuityoA = Amount receivable at the end of the every year for ‘n’ yearso r = Required rate of returnog = Growth Rate

r)(1

)g1(A.............r)(1g)1(A

r)(1g)1(A

r)(1APVA 3

2

21

g-rAPVA

PVA

DOUBLING PERIODIf you know the compound rate of return of your investment is going to earn, can you tell in how many years the investment will get doubled?It can be calculated by using below formulae.Doubling Period

o .

Out of above two, ‘Rule of 69’ gives the precise answer where as ‘Rule of 72’ gives the approximate answer.

RateInterest 7272 of Rule

RateInterest 6935.069 of Rule

Intra-Year Compounding/Discounting

•Can we go for multiple compounding/ discounting in a year?•If yes, How frequently we can go for compounding / discounting in a year?•What is the use of that?•How to calculate the future value and present value?•In two way we can find out the future value and present value

• By modifying the formula as below. r should be divided by number of frequency per annum n should be multiplied number of frequency per annum

• By taking ‘r’ as effective rate of return

Intra-Year Compounding/Discounting

•In bank most of the time we use intra year compounding/discounting•For Example,

oFixed deposits are quarterly compoundedoRecurring deposits are quarterly compoundedoAll retail loans (Personal loans, Home loans,

Educational Loans, Vehicle Loans etc.) monthly installments are monthly discounted

Impact of frequency

•You deposited Rs.10000 for 1 year which earns at an interest rate of 10%. You got three options for compounding viz. yearly, semiannually, quarterly. Which will you prefer

Impact of frequency

•Example: Consider, P=10,000, n=1year, r=10% p.a., m = 1 (annually)/2 (Semi-annually)/4 (Quarterly)

Quarterly10000.00

250.0010250256.25

10506.25262.65

10768.90269.25

11038.15

Semi-annually10000

--

50010500

--

52511025

Annually10000

------

100011000

ParticularsAmount at beginningInterest for the first QuarterAmount at the end of 3 monthsInterest for the Second QuarterAmount at the end of 6 monthsInterest for the third QuarterAmount at the end of 9 monthsInterest for the fourth QuarterAmount at the end of the year

Effective Rate of Return

•The rate of interest under annual compounding which produces the same result as that produced by an interest rate under multiple compounding. It can be calculated by using below formula

•Whereo r = effective rate of returno i= normal rate of returno m = frequency of compounding

1mi1r

m

Effective Rate of Return

•Find out the effective rate of return for the previous problem in case of annual, semi-annual and quarterly compounding.•Effective rate of return for the example which is in slide 35

Particulars Annually Semi-annually Quarterly

Normal rate of Interest

10% 10% 10%

Effective rate of interest

10% 10.25% 10.38%

Future Value 10000(1.1)=11000

10000(1.1025)=11025

10000(1.1038)=11038

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