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Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved

Lecture

3

Time Value of Money

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Slide 2

Key Concepts and Skills

Be able to compute the future value and/or

present value of a single cash flow or

series of cash flows Be able to compute the return on an

investment

Understand perpetuities and annuities

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Slide 3

Session Outline

Valuation: The One-Period Case

The Multiperiod Case

Compounding PeriodsSimplifications

What Is a Firm Worth?

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Slide 4

The One-Period Case:

Future Value If you were to invest Rs.10,000 at 5-percent interest

for one year, your investment would grow toRs.10,500.

Rs. 500 would be interest (Rs.10,000 .05)

Rs. 10,000 is the principal repayment (Rs.10,000 1)

Rs. 10,500 is the total due. It can be calculated as:

Rs.10,500 = Rs.10,000(1.05)

The total amount due at the end of the investmentis call the Future Value (FV).

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Slide 5

Future Value

In the one-period case, the formula forFV

can be written as:

FV= C0(1 + r)

Where C0 is cash flow today (time zero), and

ris the appropriate interest rate.

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Slide 6

Present Value

If you were to be promised Rs.10,000 due in oneyear when interest rates are 5-percent, your

investment would be worth how much in todays

rupees.

05.1000,10.81.523,9. RsRs !

The amount that a borrower would need to set

aside today to be able to meet the promised

payment of Rs.10,000 in one year is called thePresent Value (PV).

Note that Rs.10,000 = Rs.9,523.81(1.05).

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Slide 7

Present Value

In the one-period case, the formula forPVcan be written as:

r

CPV

!

1

1

Where C1 is cash flow at date 1, and

ris the appropriate interest rate.

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Slide 8

Net Present Value The Net Present Value (NPV) of an

investment is the present value of the

expected cash flows, less the cost of theinvestment.

Suppose an investment that promises to

pay Rs.10,000 in one year is offered forsale for Rs.9,500. Your interest rate is 5%.

Should you buy?

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Slide 9

Net Present Value

81.23.81.523,9.500,9.

05.1

000,10.500,9.

RsNPV

RsRsNPV

RsRsNPV

!!

!

The present value of the cash inflow is greater

than the cost. In other words, the Net Present

Value is positive, so the investment should be

purchased.

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Slide 10

Net Present ValueIn the one-period case, the formula forNPVcan

be written as:

NPV= Cost+ PV

If we had notundertaken the positive NPVproject

considered on the last slide, and instead invested our

Rs.9,500 elsewhere at 5 percent, ourFVwould be

less than the Rs.10,000 the investment promised,and we would be worse off in FVterms :

Rs.9,500(1.05) = Rs.9,975 < Rs.10,000

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Slide 11

The Multiperiod Case

The general formula for the future value ofan investment over many periods can bewritten as:

FV= C0(1 + r)T

Where

C0 is cash flow at date 0,

ris the appropriate interest rate, and

Tis the number of periods over which the cashis invested.

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Slide 12

Future Value

Suppose a stock currently pays a

dividend of Rs.1.10, which is expected to

grow at 40% per year for the next five

years.

What will the dividend be in five years?

FV= C0(1 + r)T

Rs.5.92 = Rs.1.10(1.40)5

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Slide 13

Future Value and Compounding

Notice that the dividend in year five,

Rs.5.92, is considerably higher than the

sum of the original dividend plus fiveincreases of 40-percent on the original

Rs.1.10 dividend:

Rs.5.92 > Rs.1.10 +5[Rs.1.10.40] =

Rs.3.30

This is due to compounding.

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Slide 14

Future Value and Compounding

0 1 2 3 4 5

10.1.Rs

3)40.1(10.1. vRs

02.3.Rs

)40.1(10.1. vRs

54.1.Rs

2)40.1(10.1. vRs

16.2.Rs

5)40.1(10.1. vRs

92.5.Rs

4)40.1(10.1. vRs

23.4.Rs

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Slide 15

Present Value and Discounting

How much would an investor have to set

aside today in order to have Rs.20,000

five years from now if the current rate is

15%?

0 1 2 3 4 5

Rs.20,000PV

5)15.1(

000,20.53.943,9.

RsRs !

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Slide 16

How Long is the Wait?

If we deposit Rs.5,000 today in an account paying

10%, how long does it take to grow to Rs.10,000?

T

rCFV )1(0 v!

T

RsRs )10.1(000,5.000,10. v!

2000,5.

000,10.)10.1( !!

Rs

RsT

)2ln()10.1ln( !T

years27.7

0953.0

6931.0

)10.1ln(

)2ln(!!!T

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Slide 17

Assume the total cost of a college education will beRs.50,000 when your child enters college in 12years. You have Rs.5,000 to invest today. What rateof interest must you earn on your investment tocover the cost of your childs education?

What Rate Is Enough?

TrCFV )1(0 v!

12)1(000,5.000,50. rRsRs v!

10000,5.

000,50.)1( 12 !!

Rs

Rsr

12110)1( ! r

2115.12115.1110121

!!!r

About 21.15%.

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Slide 18

Multiple Cash Flows

Consider an investment that pays Rs.200

one year from now, with cash flows

increasing by Rs.200 per year through

year 4. If the interest rate is 12%, what is

the present value of this stream of cash

flows?

If the issuer offers this investment forRs.1,500, should you purchase it?

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Slide 19

Multiple Cash Flows

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

Present Value < Cost Do Not Purchase

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Slide 20

Compounding Periods

Compounding an investment m times a

year forTyears provides for future value of

wealth:Tm

m

rCFV

v

v! 10

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Slide 21

Compounding Periods

For example, if you invest Rs.50 for 3

years at 12% compounded semi-

annually, your investment will grow to

93.70.)06.1(50.

2

12.150. 6

32

RsRsRsFV !v!

v!

v

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Slide 22Effective Annual Rates of

Interest

A reasonable question to ask in the above

example is what is the effective annual

rate of interest on that investment?

The Effective Annual Rate (EAR) of interest is the

annual rate that would give us the same end-of-

investment wealth after 3 years:

93.70.)06.1(50.)2

12.1(50. 632 RsRsRsFV !v!v! v

93.70.)1(50. 3 RsEARRs !v

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Slide 23

Effective Annual Rates of

Interest

So, investing at 12.36% compounded

annually is the same as investing at 12%

compounded semi-annually.

93.70.)1(50. 3 RsEARRsFV !v!

50.

93.70.)1( 3

Rs

Rs

EAR !

1236.150.

93.70.31

!

!

Rs

RsEAR

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Slide 24

Effective Annual Rates of Interest

Find the Effective Annual Rate (EAR) of an18% APR loan that is compounded

monthly.

What we have is a loan with a monthlyinterest rate rate of 1%.

This is equivalent to a loan with an annual

interest rate of 19.56%.

1956.1)015.1(12

18.11 12

12

!!

!

vmn

m

r

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Slide 25

Simplifications

Perpetuity

A constant stream of cash flows that lasts forever

Growing perpetuity

A stream of cash flows that grows at a constant rateforever

Annuity

A stream of constant cash flows that lasts for a fixed

number of periods

Growing annuity

A stream of cash flows that grows at a constant rate

for a fixed number of periods

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Slide 26

Perpetuity

A constant stream of cash flows that lasts forever

0

1

C

2

C

3

C

.! 32 )1()1()1( r

C

r

C

r

C

PV

r

CPV!

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Slide 27

Perpetuity: Example

What is the value of a British consol that

promises to pay 15 every year for

ever?The interest rate is 10-percent.

0

1

15

2

15

3

15

15010.

15!!PV

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Slide 28

Growing Perpetuity

A growing stream of cash flows that lasts forever

0

1

C

2

C(1+g)

3

C(1+g)2

.

v

v

!3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

C

PV

gr

CPV

!

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Slide 29

Growing Perpetuity: Example

The expected dividend next year is Rs.1.30, anddividends are expected to grow at 5% forever.

If the discount rate is 10%, what is the value of

this promised dividend stream?

0

1

Rs.1.30

2

Rs.1.30(1.05)

3

Rs.1.30(1.05)2

00.26.05.10.

30.1.Rs

RsPV !

!

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Slide 30

AnnuityA

constant stream of cash flows with a fixedmaturity

0 1

C

2

C

3

C

T

r

C

r

C

r

C

r

CPV

)1()1()1()1(32

! .

-

!

Trr

CPV

)1(

11

T

C

.

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Slide 31

Annuity: Example

If you can afford a Rs.400 monthly car payment,how much car can you afford if interest rates are7% on 36-month loans?

0 1

Rs.400

2

Rs.400

3

Rs.400

59.954,12.)1207.1(

11

12/07.

400.36

RsRs

PV !

-

!

36

Rs.400

.

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Slide 32

What is the present value of a four-year annuity of Rs.100

per year that makes its first payment two years from today if

the discount rate is 9%?

22.297.09.1

97.327.

0 Rs

Rs

PV !!

0 1 2 3 4 5

Rs.100 Rs.100 Rs.100 Rs.100Rs.323.97Rs.297.22

97.323.)09.1(

100.

)09.1(

100.

)09.1(

100.

)09.1(

100.

)09.1(

100.4321

4

1

1 RsRsRsRsRsRs

PV

t

t!!!

!

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Slide 33

Growing Annuity

A growing stream of cash flows with a fixedmaturity

0 1

C

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

v

v

!

.

-

!

T

r

g

gr

CPV

)1(

11

.

2

C(1+g)

3

C(1+g)2

T

C(1+g)T-1

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Slide 34

Growing Annuity: Example

A defined-benefit retirement plan offers to pay Rs.20,000per year for 40 years and increase the annual payment by3% each year. What is the present value of the retirementplan if the discount rate is 10%?

0 1

Rs.20,000

57.121,265.10.1

03.11

03.10.

000,20.40

RsRs

PV !

-

!

.

2

Rs.20,000(1.03)

40

Rs.20,000(1.03)39

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Slide 35

Growing Annuity: Example

Youareevaluating an income generating property. Netrentisreceivedattheendofeachyear. Thefirstyear'srentis

expectedtobeRs.8,500, andrentisexpectedtoincrease 7%

eachyear. Whatisthe presentvalueoftheestimatedincome

st

ream

o

vert

hefi

rst

5 years

if

t

hedis

cou

nt

rat

eis

12%?

0 1 2 3 4 5

500,8.Rs

!v )07.1(500,8.Rs!v 2)07.1(500,8.Rs

095,9.Rs 65.731,9.Rs

!v 3)07.1(500,8.Rs

87.412,10.Rs

!v 4)07.1(500,8.Rs

77.141,11.Rs

Rs.34,706.26

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Slide 36

What Is a Firm Worth?

Conceptually, a firm should be worth the

present value of the firms cash flows.

The tricky part is determining the size,timing, and riskof those cash flows.