© 2009 Cengage Learning/South-Western

The Time Value OfMoney

Chapter 3

2

Time Value of Money

Financial managers compare the marginal benefits and marginal cost of investment

projects.

Projects usually have a long-term horizon: timing of benefits and costs matters.

Time-value of money: A dollar received today is worth more than a dollar received in the

future.

3

Future Value

Future Value: The value of an investment made today measured at a specific future

date using compound interest.

FVn = PV x (1+r)n

Future Value

depends on:

Interest rate

Number of periods

Compounding interval

4

Future Value of $2004 years, 7% interest

0 1 2 3 4

PV = $200

End of Year

FV4 = $262.16FV4 = $262.16

FV3 = $245.01FV3 = $245.01

FV2 = $228.98FV2 = $228.98

FV1 = $214FV1 = $214

Compound interest: Interest earned both on the principal amount and on the interest

earned in previous periods.

5

Compounding

Year 1:FV1 = $214

• Earns 7% interest on initial $200

• FV1 = $200+$14 = $214

Year 2:FV2 = $228.98

• Earn $14 interest again on $200 principal

• Earns $0.98 on previous year’s interest of $14: $14 x 7% = $0.98

• FV2 = $214+$14+$0.98 = $228.98

Year 3:FV3 = $245.01

• Earn $14 interest again on $200 principal

• Earns $2.03 on previous years’ interest of $28.98: $28.98 x 7% = $2.03

• FV3 = $228.98+$14+$2.03 = $245.01

• Earn $14 interest again on $200 principal

• Earns $3.15 on previous years’ interest of $45.01: $45.01 x 7% = $3.15

• FV4 = $245.01+$14+$3.15 = $262.16

Year 4:FV4 = $262.16

6

The Power of Compound Interest

1

6

11

16

21

26

31

36

41

1 3 5 7 9 11 13 15 17 19 21 23 25

Periods

0%

10%

5%

15%

20%

7

Present Value

Present value: The value today of a cash flow to be received at a specific date in the future, assuming an opportunity to earn interest at a specified rate.

nn rPVFV 1

n

n

r

FVPV

)1(

8

Present Value of $2004 Years, 7% Interest

0 1 2 3 4

Discounting

PV = $186.92PV = $186.92

FV1 = $200FV1 = $200

Discounting: The process of calculating present values.

FV2 = $200FV2 = $200

PV = $174.69PV = $174.69

FV3 = $200FV3 = $200

PV = $163.26PV = $163.26

FV4 = $200FV4 = $200

PV = $152.58PV = $152.58

End of Year

9

The Power of Discounting

Periods

Pre

sen

t V

a lu

e o

f O

ne

Do

llar

($)

0 2 4 6 8 10 12 14 16 18 20 22 24

0.5

0.75

1.00

0.25 10%

5%

15%20%

0%

10

Future Value of Cash Flow Streams

Mixed stream

• A series of unequal cash flows reflecting no particular pattern.

n

t

tnt rCFFV

1

1

Annuity • A stream of equal periodic cash flows.

11

Future and Present Values of An Ordinary Annuity

Present Value

Present Value

0 1 2 3 4 5

$1,000 $1,000 $1,000 $1,000 $1,000

Discounting

End of Year

FutureValue

FutureValue

Compounding

12

Future Value of An Ordinary Annuity5 Years, 5.5% Interest

$1,055.00

$1,113.02

$1,174.24

$1,238.82

$1,000.00

0 1 2 3 4 5

$1,000 $1,000 $1,000 $1,000 $1,000

End of Year

08.581,5$1)1(

r

rPMTFV

n

Ordinary annuity: An annuity for which the payments occur at the end of each period.

13

Future Value of An Annuity Due5 Years, 5.5% Interest

0 1 2 3 4 5

$1,000 $1,000 $1,000 $1,000 $1,000

End of Year

Annuity due: An annuity for which the payments occur at the beginning of each period.

04.888,5$11)1(

rr

rPMTFV

n

$1,113.02

$1,174.24

$1,238.82

$1,306.96

$1,055.00

14

Present Value of Cash Flow Streams

• Mixed streams

• Annuities

• Perpetuities: cash flow streams that continue forever

n

tttr

CFPV1 1

1

15

Present Value of An Ordinary Annuity5 Years, 5.5% Interest

$947.87

$898.45

$851.61

$807.22

$1,000 $1,000 $1,000 $1,000 $1,000

End of Year

0 1 2 3 4 5

$765.13

28.270,4$

)1(

11

nrr

PMTPV

16

Present Value of An Annuity Due5 Years, 5.5% Interest

$947.87

$898.45

$851.61

$807.22

End of Year

$1,000 $1,000 $1,000 $1,000 $1,000

0 1 2 3 4 5

$1,000.00

15.505,4$1)1(

11

rrr

PMTPV

n

17

Future and Present Values of A Mixed Steam5 Years, 4% Interest

PV$5,271.7

PV$5,271.7

0 1 2 3 4 5

-$10,000 $3,000 $5,000 $4,000 $3,000 $2,000.0

Discounting

End of Year

FV$6,413.8

FV$6,413.8

Compounding- $12,166.5

$3,509.6

$5,624.3

$4,326.4

$3,120.0

$4,622.8

$3,556.0

$2,564.4

$1,643.9

$2,884.6

18

Present Value of A Perpetuity

• For a constant stream of cash flows that continues forever

1 )1(

1

ttr

PMTPV

r

PMTr

PMT

1

19

Present Value of A Growing Perpetuity

grgr

CFPV

1

0

Growing Perpetuity

CF1 = $1,000

r = 7% per year

g = 2% per year

000,20$02.007.0

000,1$0

PV

0 1 2 3 4

$1,000 $1,000(1+0.02)1 $1,000(1+0.02)2 $1,000(1+0.02)3 …

$1,000 $1,020 $1,040.4 $1,061.2

20

Compounding More Frequently Than Annually

• continuous compounding

nrn ePVFV

• The more frequent the compound period, the larger the FV!

nm

n m

rPVFV

1

• m compounding periods

21

Compounding More Frequently Than Annually

FV at end of 2 years of $125,000 at 5% interest

61.976,137$2

05.01000,125$

22

2

FV

• Semiannual compounding:

76.060,138$4

05.01000,125$

24

2

FV

• Quarterly compounding:

• Continuous compounding:

365.146,138$000,125$ 205.02 eFV

22

Stated Versus Effective Annual Interest Rates

Stated annual

rate

• The contractual annual rate of interest charged by a lender or promised by a borrower.

Effective annual

rate

• The annual rate of interest actually paid or earned, reflecting the impact of compounding frequency.

11

m

m

rEAR

1gcompoundin continuous reEAR

23

Stated Versus Effective Annual Interest Rates

Annual percentag

e rate (APR)

• The stated annual rate calculated by multiplying the periodic rate by the number of periods in one year.

Annual percentag

e yield (APY)

• The annual rate of interest actually paid or earned, reflecting the impact of compounding frequency. The same as the effective annual rate.

24

Additional Applications of Time Value

• Deposits needed to accumulate a future sum

• Loan amortization

• Implied interest or growth rates

• Number of compounding periods

25

The Time Value of Money

• Much of finance involves finding future and present values.

• The time value of money is central to all financial valuation techniques.