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Time Value of

Money

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Obviously, Rs10,000 today.

You already recognize that there isTIME VALUE TO MONEY!!

The In terest Rate

Which would you preferRs10,000

today or Rs10,000 in 5 years?

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TIMEallows you the oppor tun i tytopostpone consumption and earn

INTEREST

Arupee todayrepresents a greater realpurchas ing powerthan a rupee a year

hence

Why TIME?

Why is TIMEsuch an important

element in your decision?

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Time Value Adjustment

Two most common methods ofadjusting cash flows for time value ofmoney:

Compoundingthe process ofcalculating future valuesof cashflows and

Discountingthe process ofcalculating present valuesof cashflows.

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Types o f In teres t

Compound Interest

Interest paid (earned) on any previousinterest earned, as well as on the

principal borrowed (lent).

Simple Interest

Interest paid (earned) on only the original

amount, or principal borrowed (lent).

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Simp le In terest Formu la

Formula SI = P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Period

n: Number of Time Periods

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SI = P0

(i)(n)

= Rs1,000(.07)(2)

= Rs140

Simp le In terest Example

Assume that you deposit Rs1,000in an

account earning 7%simple interest for

2years. What is the accumulatedinterestat the end of the 2nd year?

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FV = P0+ SI

= Rs1,000+ Rs140

=Rs 1,140

Future Value is the value at some futuretime of a present amount of money, or a

series of payments, evaluated at a given

interest rate.

Simp le In terest (FV)

What is the Future Value (FV) of the

deposit?

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The Present Value is simply the

Rs 1,000you o r ig inal ly deposi ted.

That is the value today!

Present Value is the current value of afuture amount of money, or a series of

payments, evaluated at a given interest

rate.

Simp le In terest (PV)

What is the Present Value (PV) of the

previous problem?

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Assume that you deposit Rs 1,000at a compound interest rate of 7%

for 2 years.

Futu re Value

Sing le Depos it (Graph ic)

0 1 2

Rs 1,000

FV2

7%

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FV1 = P0(1+i)1 = Rs 1,000(1.07)

= Rs 1,070

FV2 = FV1(1+i)1

= P0 (1+i)(1+i) = Rs1,000(1.07)(1.07)

= P0(1+i)2

= Rs1,000(1.07)2

= Rs1,144.90

You earned an EXTRA Rs 4.90in Year 2 with

compound over simple interest.

Future Value

Sing le Depos it (Formu la)

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FV1= P0(1+i)1

FV2= P0(1+i)

2

General Future Value Formula:

FVn= P0(1+i)n

or FVn= P0(FVIFi,n)

General Fu tu re

Value Formu la

etc.

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Reena wants to know how large her deposit ofRs 10,000today will become at a compound

annual interest rate of 10%for 5 years.

Problem

0 1 2 3 4 5

Rs10,000

FV5

10%

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Solut ion

Calculation based on general formula:

FVn = P0(1+i)n

FV5 = Rs10,000(1+0.10)5

= Rs 16,105.10

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We will use the Rule-of-72.

Doub le Your Money!!!

Quick! How long does it take to double

Rs 5,000 at a compound rate of 12%

per year (approx.)?

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Doubling Period = 72 / Interest Rate

6 years

For accuracy use theRule-of-69

.

Doubling Period

=0.35 +(69 / Interest Rate)

6.1 years

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Assume that you need Rs 1,000in 2 years.Lets examine the process to determinehow much you need to deposit today at a

discount rate of 7% compounded annually.

0 1 2

Rs 1,000

7%

PV1PV0

Present Value

Sing le Depos it (Graph ic)

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PV0= FV2/ (1+i)2 = Rs 1,000/ (1.07)2

= FV2

/ (1+i)2 = Rs 873.44

Present Value

Sing le Depos it (Formu la)

0 1 2

Rs 1,000

7%

PV0

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PV0= FV1/ (1+i)1

PV0= FV2/ (1+i)2

General Present Value Formula:

PV0= FVn/ (1+i)n

or PV0= FVn(PVIFi,n)

General Presen t

Value Formu la

etc.

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Reena wants to know how large of adeposit to make so that the money willgrow to Rs 10,000 in 5 yearsat a discount

rate of 10%.

Problem

0 1 2 3 4 5

Rs 10,000

PV0

10%

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Calculation based on general formula: PV0 = FVn/ (1+i)

n

PV0 = Rs 10,000/ (1+0.10)5= Rs 6,209.21

Prob lem Solu t ion

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Types o f Annui t ies

Ordinary Annuity: Payments or receipts

occur at the endof each period.

Annuity Due: Payments or receipts

occur at the beginningof each period.

An Annu i tyrepresents a series of equal

payments (or receipts) occurring over a

specified number of equidistant periods.

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Parts of an Annui ty

0 1 2 3

Rs 100 Rs 100 Rs 100

(Annuity Due)

Beginning of

Period 1

Beginning of

Period 2

Today Equal Cash Flows

Each 1 Period Apart

Beginning of

Period 3

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FVAn= R(1+i)n-1 + R(1+i)n-2 +

... + R(1+i)1+ R(1+i)0

Ordinary Annu i ty -- FVA

R R R

0 1 2 n n+1

FVAn

R= Periodic

Cash Flow

Cash flows occur at the end of the period

i% . . .

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Example o f an


Rs1,000 Rs1,000 Rs1,000

0 1 2 3 4

7%


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FVA3=1,000(1.07)2 +1,000(1.07)1 + 1,000(1.07)0

= 1,145+1,070+1,000=Rs 3,215

Example o f an


Rs1,000 Rs1,000 Rs1,000

0 1 2 3 4

Rs3,215 =

FVA3

7%

Rs1,070

Rs1,145


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General Formu la fo r Calcu lat ing

Fu tu re Value of an Ord inary

Annu i ty

AiAiAFVAn nn ...)1()1( 21

i

i

A

n 1)1(

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FVADn= R(1+i)n + R(1+i)n-1 +... + R(1+i)2+ R(1+i)1

= FVAn (1+i)

Annu ity Due -- FVAD

R R R R R

0 1 2 3 n-1 n

FVADn

i% . . .

Cash flows occur at the beginning of the period

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FVAD3= 1,000(1.07)3 +1,000(1.07)2 + 1,000(1.07)1

= 1,225+1,145+1,070

=Rs 3,440

Example o f an

Annu ity Due -- FVAD

1,000 1,000 1,000 1,070

0 1 2 3 4

Rs 3,440 =FVAD3

7%

Rs1,225

Rs1,145


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PVAn= R/(1+i)1 + R/(1+i)2

+ ... + R/(1+i)n

Ordinary Annui ty -- PVA

R R R

0 1 2 n n+1

PVAn

R= Periodic

Cash Flow

i% . . .


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Example o f an

Ordinary Annu i ty -- PVA

Rs1,000 Rs1,000 Rs1,000

0 1 2 3 4

7%


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PVA3= 1,000/(1.07)1

+1,000/(1.07)2 +1,000/(1.07)3

=934.58 + 873.44 + 816.30=2,624.32

Example o f an

Ordinary Annu i ty -- PVA

Rs1,000 Rs1,000 Rs1,000

0 1 2 3 4

Rs 2,624.32 = PVA3

7%

934.58

873.44

816.30


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nn

i

A

i

A

i

A

PVA)1(

...)1()1( 2

n

n

ii

iA)1(

1)1(

General Formu la fo r Calcu lat ing

Present Value o f an Ord inaryAnnu i ty

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PVADn= R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1

= PVAn (1+i)

Annu ity Due -- PVAD

R R R R

0 1 2 n-1 n

PVADn

R: Periodic

Cash Flow

i% . . .


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PVADn= 1,000/(1.07)0 + 1,000/(1.07)1 +

1,000/(1.07)2 = Rs 2,808.02

Example o f an

Annu ity Due -- PVAD

1,000.00 1,000 1,000

0 1 2 3 4

2,808.02 = PVADn

7%

934.58

873.44


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Reena will receive the set of cashflows below. What is the Present

Value at a discount rate of 10%?

Mixed Flows Examp le

0 1 2 3 4 5

600 600 400 400 100

PV0

10%

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Solut ion

0 1 2 3 4 5

600 600 400 400 100

10%

545.45

495.87

300.53273.21

62.09

Rs 1677.15 = PV0of the Mixed Flow

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General Formula:

FVn= PV0(1 + [i/m])mn

Or=PV0* PVIFi/m,m*n

n: Number of Years

m: Compounding Periods per Year i: Annual Interest Rate

FVn,m: FV at the end of Year n

PV0: PV of the Cash Flow today

Sho rter Discoun t ing Per iods

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Reena has Rs1,000to invest for 1 yearat an annual interest rate of 12%.

Example

Annual FV = 1,000(1+ [.12/1])(1)(1)

= 1,120

Semi FV = 1,000(1+ [.12/2])(2)(1)

= 1,123.6

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Effective vs. Nominal Rate of

InterestRs. 1000 Rs.1123.6

So,Rs. 1000 grows @ 12.36% annually

Effective Rate of Interest

r = 1 + i/mm

- 1

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Basket Wonders (BW) has a Rs1,000 CDat the bank. The interest rate is 6%compounded quarterly for 1 year.

What is the Effective Annual InterestRate (EAR)?

Problem

EAR = ( 1 +6%/ 4)4- 1= 1.0614 - 1 = .0614 or

6.14%!

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Perpetuity

A perpetuity is an annuity with aninfinite number of cash flows.

The present value of cash flows

occurring in the distant future is veryclose to zero.

At 10% interest, the PV of Rs 100

cash flow occurring 50 years fromtoday is Rs 0.85!

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Present Value of a

Perpetuity

nn

i

A

i

A

i

APVA

)1(...

)1()1( 2

When n=

PVperpetuity= [A/(1+i)][1-1/(1+i)]

= A(1/i) = A/i

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Present Value of a

PerpetuityWhat is the present value of a perpetuityof Rs270 per year if the interest rate is

12% per year?

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Present Value of a

Perpetuity

PV Aiperpetuity

Rs2700.12

Rs 2250

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1. Calculate the payment per period.

2. Determine the interestin Period t.

(Loan balance at t-1) x (i% / m )3. Computeprincipal payment in Period t.

(Payment- interestfrom Step 2)

4. Determine ending balance in Period t.(Balance- pr inc ipa l payment f rom Step 3)

5. Start again at Step 2 and repeat.

Steps to Amort izing a Loan

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Reena is borrowing Rs10,000 at a compoundannual interest rate of 12%. Amortize the loan

if annual payments are made for 5 years.

Amort izing a Loan Example

Step 1: Payment

PV0 = R(PVIFA i%,n)

Rs10,000 = R(PVIFA 12%,5)

Rs10,000 = R(3.605)

R= Rs10,000/ 3.605 = Rs2,774

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End ofYear

Payment Interest Principal EndingBalance

0

12

3

4

5

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End of

Year

Payment Interest Principal Ending

Balance

0 --- --- --- Rs10,000

1 Rs2,774 Rs1,200 Rs1,574 8,426

2 2,774 1,011 1,763 6,663

3 2,774 800 1,974 4,689

4 2,774 563 2,211 2,478

5 2,775 297 2,478 0

Rs13,871 Rs3,871 Rs10,000

[Last Payment Slightly Higher Due to Rounding]

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Usefu lness of Amort izat ion

2. Calculate Debt Outstanding -- The

quantity of outstanding debt

may be used in financing theday-to-day activities of the firm.

1. Determine Interest Expense --

Interest expenses may reduce

taxable income of the firm.

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EXERCISE

Ashish recently obtained a

Rs.50,000 loan. The loan carries

an 8% annual interest. Amortizethe loan if annual payments are

made for 5 years.

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SOLUTION50000 5 0.08

12523

TIME PAYMENT INTERESTPRINCIPAL AMOUNTOUTSTANDING

0 50000

1 12523 4000 8523 41477

2 12523 3318 9205 32272

3 12523 2582 9941 22331

4 12523 1786 10737 11594

5 12522 928 11594 0

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EXERCISE

Compute the present value of thefollowing future cash inflows,

assuming a required rate of 10%:Rs. 100 a year for years 1through 3, and Rs. 200 a year

from years 6 through 15.

ANS: 1011.75

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Solution

100 100 100 200 200 200

0 1 2 3 6 7 15

248.70

i% . . .


. . .

1228.9

763.05

1011.75

Till 5th

year

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