corp finance

Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Richard A Brealey Stewart C MyersFranklin Allen Pitabas Mohanty


Topics Covered

• Future Values and Present Values• Looking for Shortcuts—Perpetuities and

Annuities• More Shortcuts—Growing Perpetuities and

Annuities• How Interest Is Paid and Quoted


Present and Future Value


Future Values

Future Value of Rs.100 = FV

trRsFV )1(100.


Future Values

Example - FV

What is the future value of Rs.100 if interest is compounded annually at a rate of 7% for two years?

49.114.)07.1(100.

49.114.)07.1()07.1(100.2 RsRsFV

RsRsFV

trRSFV )1(100.


0

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years

FV

of

Rs.

100

0%

5%

10%

15%

Future Values with Compounding

Interest Rates


Present Value

1factordiscount =PV

PV=ValuePresent

C


Present Value

Discount Factor = DF = PV of Rs.1

Discount Factors can be used to compute the present value of any cash flow.

DFr t

1

1( )


• The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. Also, you can reverse the prior example.

10049.1142)07.1(1

22

PV

CDFPV

Present Value


0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years

PV

of

Rs.

100

0%

5%

10%

15%

Present Values with Compounding

Interest Rates


Valuing an Office BuildingStep 1: Forecast cash flows

Cost of building = C0 = 370,000

Sale price in Year 1 = C1 = 420,000

Step 2: Estimate opportunity cost of capitalIf equally risky investments in the capital marketoffer a return of 5%, then

Cost of capital = r = 5%


Valuing an Office Building

Step 3: Discount future cash flows

Step 4: Go ahead if PV of payoff exceeds investment

000400051000420

11 ,).(

,)( r

CPV

00030

000370000400

,

,,NPV


Net Present Value

r

C

1C=NPV

investment required-PV=NPV

10


Risk and Present Value• Higher risk projects require a higher rate of

return• Higher required rates of return cause lower

PVs

000,400.051

420,000PV

5%at Rs.420,000 C of PV 1


Risk and Present Value

000,400.051

420,000PV

5%at Rs.420,000 C of PV 1

000,375.121

420,000PV

12%at Rs.420,000 C of PV 1


Risk and Net Present Value

Rs.5,000

370,000-75,0003=NPV

investment required-PV=NPV


Net Present Value Rule• Accept investments that have positive

net present value

Example

Use the original example. Should we accept the project given a 10% expected return?

000,30.1.05

420,000+-370,000=NPV Rs


Rate of Return Rule• Accept investments that offer rates of return in

excess of their opportunity cost of capital

Example

In the project listed below, the foregone investment opportunity is 12%. Should we do the project?

13.5%or .135370,000

370,000420,000

investment

profitReturn


Multiple Cash Flows

For multiple periods we have the Discounted Cash Flow (DCF) formula

tt

r

C

r

C

r

CPV)1()1()1(0 ....2

21

1

T

tr

Ct

tCNPV1

)1(00


Net Present Values

Present Value

Year 0

20,000/1.12

420,000/1.122

Total

= Rs.17,900

= Rs.334,800

= - Rs.17,300

Rs.20,000

- Rs.370,000

Year0 1 2

Rs. 420,000

-Rs.370,000


Short Cuts• Sometimes there are shortcuts that make it

very easy to calculate the present value of an asset that pays off in different periods. These tools allow us to cut through the calculations quickly.


Short Cuts

Perpetuity - Financial concept in which a cash flow is theoretically received forever.

PV

Cr

luepresent va

flow cashReturn


Short Cuts

Perpetuity - Financial concept in which a cash flow is theoretically received forever.

r

CPV 1

0

ratediscount

flow cash FlowCash of PV


Present ValuesExample

What is the present value of Rs.1 billion every year, for all eternity, if you estimate the perpetual discount rate to be 10%??

billion 10.10.0bil Rs.1 RsPV


Present Values

Example - continued

What if the investment does not start making money for 3 years?

billion 51.7.31.101

10.0bil Rs.1 RsPV


Short CutsAnnuity - An asset that pays a fixed sum each year for a

specified number of years.

r

CPerpetuity (first payment in year 1)

Perpetuity (first payment in year t + 1)

Annuity from year 1 to year t

Asset Year of Payment

1 2…..t t + 1

Present Value

trr

C

)1(

1

trr

C

r

C

)1(

1


Example

Tiburon Autos offers you “easy payments” of $5,000 per year, at the end of each year for 5 years. If interest rates are 7%, per year, what is the cost of the car?

Present Values

5,000Year

0 1 2 3 4 5

5,000 5,000 5,000 5,000

20,501NPV Total

565,307.1/000,5

814,307.1/000,5

081,407.1/000,5

367,407.1/000,5

673,407.1/000,5

5

4

3

2

Present Value at year 0


Short CutsAnnuity - An asset that pays a fixed sum each

year for a specified number of years.

trrrC

1

11annuity of PV


Annuity Short Cut

Example

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?


Annuity Short Cut

Example - continued

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

10.774,12$

005.1005.

1

005.

1300Cost Lease 48

Cost


Annuity Short Cut

Example

The state lottery advertises a jackpot prize of Rs.295.7 million, paid in 25 installments over 25 years of Rs.11.828 million per year, at the end of each year. If interest rates are 5.9% what is the true value of the lottery prize?

000,600,152.

059.1059.

1

059.

1828.11ValueLottery 25

RsValue


FV Annuity Short CutFuture Value of an Annuity – The future value of an

asset that pays a fixed sum each year for a specified number of years.

r

rC

t 11annuity of FV


Annuity Short Cut

Example

What is the future value of Rs.20,000 paid at the end of each of the following 5 years, assuming your investment returns 8% per year?

332,117.

08.

108.1000,20 FV

5

Rs


Constant Growth Perpetuity

gr

CPV

1

0

g = the annual growth rate of the cash flow



gr

CPV

1

0

NOTE: This formula can be used to value a perpetuity at any point in time.

gr

CPV t

t 1



Example

What is the present value of Rs.1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and a constant growth rate of 4%?

billion 667.16.04.10.

10

Rs

PV


Perpetuities

A three-year stream of cash flows that grows at the rate g is equal to the difference between two growing perpetuities.


Effective Interest Rates

Annual Percentage Rate - Interest rate that is annualized using simple interest.

Effective Annual Interest Rate - Interest rate that is annualized using compound interest.



ExampleGiven a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?



ExampleGiven a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

12.00%or .12=12 x .01=APR

12.68%or .1268=1-.01)+(1=EAR

r=1-.01)+(1=EAR12

12


Web Resources

Click to access web sitesClick to access web sites

Internet connection requiredInternet connection required

www.smartmoney.com

http://finance.yahoo.com

www.in.gov/ifa/files/TollRoadFinancialAnalysis.pdf

www.mhhe.com/bma

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