Chapter 3 Principles PrinciplesofCorporateFinance Concise Edition How To Calculate Present Values Slides by Matthew Will Copyright © 2009 by The McGraw-Hill.

Chapter 3 PrinciplesPrinciples

ofof

CorporateCorporate

FinanceFinance

Concise Edition

How To Calculate Present Values

Slides by

Matthew Will

Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

McGraw Hill/Irwin

3- 2

Topics Covered

Valuing Long-Lived AssetsLooking for Shortcuts – Perpetuities and

AnnuitiesMore Shortcuts – Growing Perpetuities and

AnnuitiesCompound Interest & Present Values

3- 3

Present Values

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

DFr t

1

1( )

1

11 1 r

CCDFPV

3- 4

Present Values

Example

You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

572,2$2)08.1(3000 PV

3- 5

Present Values

Example

You have the opportunity to purchase the baseball hit by Barry Bonds to break Hank Arron’s home run record (home run # 756). You estimate this baseball will be worth $2,000,000 when you retire at the end of twenty years. If you expect a 12% return on your investment, how much will you pay for the baseball ?

334,207$20)12.1(

000,000,2 PV

3- 6

Present Values

Replacing “1” with “t” allows the formula to be used for cash flows that exist at any point in time

tt

t r

CCDFPV

)1(

3- 7

Present Values

Example

You will receive $200 risk free in two years. If the annual rate of interest on a two year treasury note is 7.7%, what is the present value of the $200?

42.172$2)077.1(200 PV

3- 8

Present Values

PVs can be added together to evaluate multiple cash flows.

PV C

r

C

r

1

12

21 1( ) ( )....

3- 9

Present Values

PVs can be added together to evaluate multiple cash flows.

88.26521 )0771(200

)07.1(100

PV

3- 10

Present Values

Present Value

Year 0

100/1.07

200/1.0772

Total

= $93.46

= $172.42

= $265.88

$100

$200

Year0 1 2

3- 11

Present Values

Given two dollars, one received a year from now and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%.

87.

83.

2

1

)07.1(00.1

2

)20.1(00.1

1

DF

DF

3- 12

Present Values

Example

Assume that the cash flows from the construction and sale of an office building is as follows. Given a 5% required rate of return, create a present value worksheet and show the net present value.

000,320000,100000,170

2Year 1Year 0Year

3- 13

Present Values

Example - continued

Assume that the cash flows from the construction and sale of an office building is as follows. Given a 5% required rate of return, create a present value worksheet and show the net present value.

011,25$

249,290000,320907.2

238,95000,100952.1

000,170000,1700.10Value

Present

Flow

Cash

Factor

DiscountPeriod

205.11

05.11

TotalNPV

3- 14

Present Values

Present Value

Year 0

-170,000

-100,000/1.05

320,000/1.052

Total = NPV

-$170,000

= -$170,000

= $95,238

= $290,249

= $25,011

-$100,000

+$320,000

Year0 1 2

Example - continued

Assume that the cash flows from the construction and sale of an office building is as follows. Given a 5% required rate of return, create a present value worksheet and show the net present value.

3- 15

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.

3- 16

Short Cuts

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

PV

Cr

luepresent va

flow cashReturn

3- 17

Short Cuts

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

r

CPV 1

0

ratediscount

flow cash FlowCash of PV

3- 18

Present Values

Example

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

billion 10$10.0bil $1 PV

3- 19

Short Cuts

Annuity - 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

3- 20

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

3- 21

Short Cuts

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

trrrC

1

11annuity of PV

3- 22

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?

3- 23

Annuity Short Cut

Example - continuedYou 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

3- 24

Annuity Short Cut

Example

The state lottery advertises a jackpot prize of $295.7 million, paid in 25 installments over 25 years of $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

Value

3- 25

FV Annuity Short Cut

Future 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

3- 26

Annuity Short Cut

Example

What is the future value of $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

3- 27

Constant Growth Perpetuity

gr

CPV

1

0

g = the annual growth rate of the cash flow

3- 28

Constant Growth Perpetuity

gr

CPV

1

0

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

gr

CPV t

t 1

3- 29

Constant Growth Perpetuity

Example

What is the present value of $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

PV

3- 30

Perpetuities

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

3- 31

Compound Interest

i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183

3- 32

Simple and Compound Interest

The value of a $100 investment earning 10% annually.

3- 33

Compound Interest

Compound interest versus simple interest. The top two ascending lines show the growth of $100 invested at simple and compound interest. The longer the funds are invested, the greater the advantage with compound interest. The bottom line shows that $38.55 must be invested now to obtain $100 after 10 periods. Conversely, the present value of $100 to be received after 10 years is $38.55.

3- 34

Compound Interest

The same story as the previous chart, except that the vertical scale is logarithmic. A constant compound rate of growth means a straight ascending line. This graph makes clear that the growth rate of funds invested at simple interest actually declines as time passes.

3- 35

Compound Interest

02468

1012141618

Number of Years

FV

of

$1

10% Simple

10% Compound

3- 36

Compound Interest

Example

Suppose you are offered an automobile loan at an APR of 6% per year. What does that mean, and what is the true rate of interest, given monthly payments?

3- 37

Compound Interest

Example - continued

Suppose you are offered an automobile loan at an APR of 6% per year. What does that mean, and what is the true rate of interest, given monthly payments? Assume $10,000 loan amount.

%1678.6

78.616,10

)005.1(000,10PmtLoan 12

APR

3- 38

Web Resources

www.bankrate.com

www.money.cnn.com

www.quicken.com

www.smartmoney.com

Click to access web sitesClick to access web sites

Internet connection requiredInternet connection required