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Topics Covered Today

Future Values and Present Values

Looking for ShortcutsPerpetuities and

Annuities

More ShortcutsGrowing Perpetuities and

Annuities

How Interest Is Paid and Quoted

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Present and Future Value

Present Value

Value today of a

future cash

flow.

Future Value

Amount to which an

investment will grow

after earning interest

How do we get from one to another?

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Future Values

FV r t

$100 ( )1

Example - FV

What is the future value of $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

FV

FV

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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

FVo

f$100

0%

5%

10%

15%

Future Values with Compounding

Interest Rates

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Present Value (PV)

Discount Factor = DF = PV of $1

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

DF r t 1

1( )

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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

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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

PVo

f$1

00

0%

5%

10%

15%

Present Values with Compounding

Interest Rates

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Valuing an Office Building

Step 1: Forecast cash flows

Cost of building = C0 = 370,000

Sale price in Year 1 = C1 = 420,000

Step 2: Estimate opportunity cost of capital

If equally risky investments in the capital market

offer a return of 5%, then

Cost of capital = r = 5%

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Valuing an Office Building

Step 3: Discount future cash flows

Step 4: Go ahead if PV of payoff exceeds investment

000400051000420

1

1

,).(,

)(

r

C

PV

00030

000370000400

,

,,NPV

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Net Present Value

r

C

1

C=NPV

investmentrequired-PV=NPV

10

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Risk and Present Value

Higher risk projects require a higher rate ofreturnCENTRAL POINT OF FINANCE

Higher required rates of return cause lower

PVs

000,400.051

420,000PV

5%at$420,000CofPV 1

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Risk and Present Value

000,400.051

420,000PV

5%at$420,000CofPV 1

000,375.121

420,000PV

12%at$420,000CofPV 1

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Risk and Net Present Value

$5,000

370,000-75,0003=NPV

investmentrequired-PV=NPV

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Net Present Value Rule

Accept investments that have positivenet present value

ExampleUse the original example. Should we accept the

project given a 10% expected return?

000,30$1.05

420,000+-370,000=NPV

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Rate of Return Rule

Accept investments that offer rates of returnin 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

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Multiple Cash FlowsFor multiple periods we have the

Discounted Cash Flow (DCF)formula

tt

rC

rC

rCPV )1()1()1(0 ....2

21

1

T

tr

C

t

tCNPV1

)1(00

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Net Present Values

Present Value

Year 0

20,000/1.12

420,000/1.122

Total

= $17,900

= $334,800

= - $17,300

$20,000

- $370,000

Year0 1 2

$ 420,000

-$370,000

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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 calculationsquickly.

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Short Cuts

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

PVCr

luepresent va

flowcashReturn

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Short Cuts

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

r

CPV1

0

ratediscount

flowcashFlowCashofPV

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Present ValuesExample

What is the present value of $1 billion every year, for all eternity, if you

estimate the perpetual discount rate to be 10%??

billion10$10.0

bil$1 PV

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Present ValuesExample - continued

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

billion51.7$31.101

10.0bil$1 PV

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Short Cuts

Annuity- An asset that pays a fixed sum each year for

a specified number of years.

r

CPerpetuity (firstpayment 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

t

rr

C

)1(

1

t

rr

C

r

C

)1(

1

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Example

Tiburon Autos offers you easy payments of $5,000 per year, at the endof each year for 5 years. If interest rates are 7%, per year, what is the

cost of the car?

Present Values

5,000

Year0 1 2 3 4 5

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

20,501NPVTotal

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 Valueat year 0

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Short Cuts

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

trrr

C1

11annuityofPV

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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?

trrr

C

1

11annuityofPV

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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.

1300CostLease

48

Cost

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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

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FV Annuity Short Cut

Future Value of an Annuity

The future value of anasset that pays a fixed sum each year for a

specified number of years.

r

rC

t

11annuityofFV

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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.

108.1000,20FV

5

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Constant Growth Perpetuity

gr

CPV

10

g = the annual growth

rate of the cash flow

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Constant Growth Perpetuity

gr

CPV

10

NOTE: This formula can be

used to value a perpetuity at

any point in time.

gr

CPV tt

1

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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%?

billion667.16$04.10.

10

PV

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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.

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Effective Interest Rates

example

Given a monthly rate of 1%, what is the Effective

Annual Rate(EAR)? What is the Annual Percentage

Rate (APR)?

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Effective Interest Rates

example

Given a monthly rate of 1%, what is the Effective

Annual Rate(EAR)? What is the Annual Percentage

Rate (APR)?

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

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

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

12

12

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Web Resources

Click to access web sitesInternet connection required

www.smartmoney.com

http://finance.yahoo.comwww.in.gov/ifa/files/TollRoadFinancialAnalysis.pdf

www.mhhe.com/bma

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