Chapter 3:Time Value of Money (TVM)or: Discounted Cash Flow

(DCF)

Professor ThomsonFin 3014

2

Albert Einstein

• Professor Einstein was asked near the end of his career what his greatest discovery was. He replied . . .

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Time Value of Money

• TVM is the foundation of the “House of Value” that we are building this semester

• The fundamental equation of finance can be stated as:

...)1()1()1( 3

32

21

10

i

CF

i

CF

i

CFCFNPV

4

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 concerned with adjusting for different timing of benefits and

costs.

5

The fundamental concept

• Value next period = Value this period + interest

earned• Interest earned is the interest rate

this period times the Value deposited

Value

Time---+----------------------------------+---

Vt Vt+1

t t+1

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Basic difference equation

•Vt+1 = Vt + it+1* Vt

Vt+1 = Value next period (period t+1)

Vt = Value this period

it+1 = Interest rate over period from t to t+1

By algebra we can write:

•Vt+1 = Vt (1+ it+1)

7

See handwritten notes

• Note: The handwritten notes and examples that follow go hand in hand.

8

Example 3.1

• You deposit $105 into a bank that pays 7% interest per year. How much will you have after 1 year?

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Example 3.2

• You deposit $100 into a bank that pays 5% interest per year. How much will you have after 1 year?

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Example 3.3

• You deposit $100 into a bank that pays 5% interest the first year, and 7% interest the second year. How much will you have after 2 years?

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Example 3.4

• You deposit $100 into a bank that pays 5% interest the first year, and 7% interest the second year. How much have you earned in interest over the two years?

• If you withdraw $5 after the first year, and then withdraw your balance after the second year, how much will you have earned in interest over the two years?

12

Defn. Compound Interest

• Compound interest is earning interest on your original deposit, plus earning interest on interest

• Note: Interest on interest (and thus compound interest) begins in the second period.

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Example 3.5

• You purchase a security for $100 that earns 5% the first year, 7% the second year, and loses 4% the third year. What is the value of your investment after 3 years?

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Example 3.6

• You deposit $200 into a bank that pays 7% interest per year. How much will you have after 4 years if you don’t withdraw any?

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Future Value (for a single sum with a constant interest rate)

The value of a lump sum or stream of cash payments at a future point in time:

FVN = PV x (1+i)N

Future Value

depends on:

Interest rate

Number of periods

Compounding interval

16

Future Value of $200 (4 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

Compounding: the process of earning interest in each successive year

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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

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Calculator hints• Clear the calculator before new

problems (Use the C ALL)• Make sure:

– The desired number of decimal places are displayed•Set using DISP followed by entering a

digit

– You have the correct payments (periods) per year •Set by typing a number then press

P/YR

•Check by holding down C ALL

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Calculator hints (continued)

BEGIN indicator is not displayed, unless you are told this problem has beginning of period cash flows– Set using BEG/END

If you have a comma where you should have a decimal point (European notation) then toggle to decimal by:

– Toggle using ./,

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Notation when using Calculator• P/YR = 1 (indicate the periods per year)• FV(PV= – 200, N= 4, I/YR=7) = 262.16• Order of inputs does not matter• Negative sign for PV indicates a cash

outflow• N = number of periods• I/YR = stated annual interest rate• The last button one pushes is what you

want to solve for – in this case FV.

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1

6

11

16

21

26

31

36

41

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

The Power of Compound Interest

Fu

ture

Val

ue

of

On

e D

oll

ar (

$ )

Periods

0%

10%

5%

15%

20%Note: If you are given any point on one of these curves, you should be able to determine any other point

22

What if one of Jesus’ disciples was an investor, and you were his heir?• A newly discovered gospel suggests

a disciple put $1 in the First Bank of Galilee which promised to pay 5% per year. How much is that deposit worth today? (Assume deposit was in year 25 AD, so N = 2006 – 25 = 1981 yr.)

• FV(PV=-1, I/YR=5, N=1981) = = 9.46 * 1041 or approx. 1042

23

How much is that?

• Is that more than Bill Gates has?• Bill Gates has give or take $50 Billion

(in orders of magnitude say 1011)• World population:6,520,566,380 See

http://www.census.gov/main/www/popclock.html

• If you divided the account among the world population we would each have about: 1032 which would make Bill Gates poor.

• Do you get Einstein’s point?

24

Example 3.7

• You deposit $250 into a bank that pays 7% per year. At the end of 3 years, you withdraw $100, and leave the rest in the bank. How much will you have in your account after 8 years?

25

What about solving for PV?

• Often what we want to know is the value in today's framework of money we will receive in the future

• We call this process discounting• We use discounting to compute the

Present Value (PV)• Use some algebra: Take the future

value equation, and divide through by (1+i)N

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Present Value

Today's value of a lump sum or stream of cash payments received at a future point in time:

NN iPVFV 1

N

N

i

FVPV

)1(

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Example 3.8

• A savings bond will pay you $200 on your 27th birthday, which is 4 years away. Assuming a discount rate of 7%, what is the present value of this payment?

28

By financial calculator

• P/YR = 1• PV(FV=200, N=4, I/YR=7) = – 152.58

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Present Value of $200 (4 Years, 7% Interest )

0 1 2 3 4

Discounting

PV = $186.92PV = $186.92

FV1 = $200FV1 = $200

Discounting: the process of converting a future cash flow into a present value

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

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The Power of High Discount Rates

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%

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Example 3.9

• Tuition and fees for a semester was about $1800 five years ago, and now is about 3100. What is the annual rate of increase?

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Example 3.10 Stradivarius Violin Example• According the the April 3, 1998 Wall

Street Journal, musician George Kress sold his Stradivarius violin for $1.58 Million. He had purchased said violin in 1958 for $24,000. What rate of return did he earn on this violin.

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Example 3.11 Or, invest in Stock Market• A popular stock market index had a

value of 25.3 in 1958, and 2347 in 1998. If George had simple put his $24000 into this index, what would his investment be worth today? What rate of return would he have earned? Would this have been better than buying the Stradivarius violin?

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From WSJ April 12, 2007 (P D7)• Record price for Strad (2006): $3,544,000• The “Solomon - ex Lambert” sold in April

2007 for $2,700,000 (had been purchased for about $43,000 in 1972 (noted to not be one of the best Strads). From 1972 to the end of 1996, the overall stock market went up about 46.41 times (including reinvestment of all dividends).

• What rate of return on the Strad and stock market was realized over this period?

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Example 3.12

• UTSA tuition has been increasing at about 10% each year. How long will it take for tuition to double? (Will you graduate before it doubles?)

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The rule of 72

• To find the period for an investment, growing at rate i, takes to double, use the rule of 72

%

72

teInterestRaublePeriodToDo

Example: Let i = 10% per year

years2.710

72

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Illustration of Rule of 72

0

5

10

15

20

25

3 4 5 6 7 8 9 10 11 12 13 14 15

Interest Rate %

Per

iod

to D

ou

ble

.

Rule of 72

Exact

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Dealing with Multiple Cash Flows

• There are two basic options for dealing with multiple cash flows:

1. Treat each CF individually, and sum up the results

– Must use this approach if the CF’s are not equal in amount or are irregularly spaced

2. Create new formulas to use when the CF’s are equal and regularly spaced

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Example 3.13 – Non equal CF’sWhat is the present value (using a 5%

discount rate) of the following CF’s?

Year 1 2 3 4

CF 100 50 0 325

What is the PV using a 10% rate?

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Solving using the CF and NPV keys

• The calculator has the following programmed:

...)1()1()1( 3

32

21

10

i

CF

i

CF

i

CFCFNPV

One accesses this function by entering a series of Cash Flows, with the first cash flow assumed to be a period 0. After all cash flows are entered, enter the I/YR and press the Shift NPV sequence to display the NPV

NPV indicates Net Present Value, where net usually means to subtract off the initial CF

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Calculator Solution• P/YR = 1• Enter the CF’s

t CF

0 0

1 100

2 50

3 0

4 325

•Type 5, press I/YR

•Press NPV

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Example 3.14 – FV of Non equal CF’s

What is the future value (using a 5% discount rate) of the following CF’s?

Year 1 2 3 4

CF 100 250 550 325

What is the FV using a 10% rate?

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Regularly spaced equal payments

• Perpetuity – A series of equally spaced, equal amount cash flows that continue in perpetuity (i.e. forever). The first payment is assumed to be one period from now.

• Annuity – A series of equally spaced, equal amount cash flows that end at time period N.

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If you deposit $100 in a bank that pays 5% per year, how much interest will you have earned after one year?Answer: $5If you withdraw the $5 and and reinvest your $100, how much interest will you have after one additional year? $5. Note: You could do this forever. In other words, you can create a payment of $5 to yourself, every year, forever, by depositing $100 into a bank that pays 5% per year.

Creating a Perpetuity

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You could write this statement down as a formula that would be: PMT = i * Original depositThe original deposit is made today, so it is a Present Value, so we can write: PMT = i * PVUsing algebra we can write this as a Present Value formula: PV = PMT/iThis is the present value of a perpetuity.NOTE: The first PMT is at period 1, (i.e. it is one period from today).

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Example 3.15 – A perpetuity

• KindaCheep Life Insurance Co. is offering you an investment policy that will pay you and your heirs $10 per year forever. If your required return on this investment is 5%, how much will you pay for the policy?

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Example 3.16: Delayed perpetuity• EvenCheeper Life Insurance Co. is

offering you an investment policy that will pay you and your heirs $10 per year forever, with your first payment 4 years from today. If your required return on this investment is 5%, how much will you pay for the policy?

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Example 3.17: PV of Annuity

• What is the present value of receiving $10 per year for 3 years if the discount rate is 5%?

• You could discount term by term using of PV of a single cash flow formula and add up the individual present values

• However, this can be computed as:• 200.00 – 172.77 = 27.23• Why?

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Diagram to derive annuity formula

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Formula for PV of Annuity

• It’s the present value of a perpetuity minus the present value of a delayed perpetuity

Ni

iPmt

i

PmtPV

)1(

/

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Solving by Financial Calculator• We use a “new” key, the PMT key

P/YR=1PV(PMT=-10, I/YR=5, N=3) =

27.23

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FV of an Annuity

• To compute the FV of an Annuity, you can simple multiply the PV by (1+i)N

• Or, use your financial calculator where the last key you press is FV

• What about the FV of a perpetuity? – It is undefined as you are accumulating

forever

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Example 3.18: FV of AnnuitySaving for Retirement• What is the FV of investing $3600 per year

for 40 years if you invest at 6%, if your first investment is 1 year from today?

P/YR = 1FV(PMT=-3600, N=40, I/YR=6) =

$557,143.08

• Is this enough to retire on?

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Habit 2: Begin with the end in mind

• How much money do you want to spend in retirement? How long do you want to have income in retirement?

• Example: I want to have an income of $36000 per year for 25 years. At 6%, how much money do I need in my retirement fund?

P/YR=1PV(PMT=36000, N=25, I/YR=6) = -460,200.82

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If you die after 25 years, how much do your heirs get?• The balance in your retirement account will

always be the present value of the remaining payments. If you have $557,143.08 in your account, and if you withdraw 36000 in each year, we can compute the number of years you can get such payments (still assuming a 6% interest rate).

P/YR = 1N(PV=-557143.08, PMT=36000,

I/YR=6)=45.29

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How large is the inheritance

• Your heirs could continue to withdraw money for 20.29 years.

P/YR = 1PV(PMT=36000, N=20.29, I/YR=6) = 416,051.92• This number is quite high, because during

retirement your were mainly living off your interest

• First year interest amount in retirement was:

$557,143.08*0.06 = 33,428.58

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•Which perhaps explains why Mark Twain said, “Thrift if a virtue – especially in ones ancestors”

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Future Value and Present Value 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

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Future Value of Ordinary Annuity(End of 5 Years, 5.5% Interest Rate)

0 1 2 3 4 5

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

$1,238.82

$1,174.24

$1,113.02

$1,055.00

$1,000.00

End of Year

08.581,5$1)1(

r

rPMTFV

n

How is annuity due different ?

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Future Value of Annuity Due(End of 5 Years, 5.5% Interest Rate)

0 1 2 3 4 5

End of Year

04.888,5$11)1(

rr

rPMTFV

n

$1,306.96

$1,238.82

$1,174.24

$1,113.02

$1,055.00

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

Annuity due: payments occur at the beginning of each period

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

End of Year

$947.87

$898.45

$851.61

$807.22

0 1 2 3 4 5

$765.13

Present Value of Ordinary Annuity(5 Years, 5.5% Interest Rate)

28.270,4$)1(

11

nrr

PMTPV

63

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

End of Year

$947.87

$898.45

$851.61

$807.22

0 1 2 3 4 5

Present Value of Annuity Due(5 Years, 5.5% Interest Rate)

15.505,4$1)1(

11

rrr

PMTPV

n

64

Example 3.18 revised: FV of Annuity - Saving for Retirement• Original Version: What is the FV of investing $3600

per year for 40 years if you invest at 6% if your first investment is 1 year from today?

• Revised version: What is the FV of investing $300 per month for 480 month if you invest at 0.5% if your first investment is 1 month from today?

P/YR = 1FV(PMT=-300, N=480, I/YR=0.5) = $597,447.22Why is this about $40,300 more than the

original? Higher interest rate (.5% per month is more

than 6% per year because compounding starts in month 2 rather than year 2. Also, higher annual payment – Why?

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1

6

11

16

21

26

31

36

41

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

The Power of Compound Interest

Fu

ture

Val

ue

of

On

e D

oll

ar (

$ )

Periods

0%

10%

5%

15%

20%Note: If you are given any point on one of these curves, you should be able to determine any other point

66

Example 3.19 Interest rate for period other than 1 year.

If you invest one dollar at 5% per year, how much interest will you have earned after:

• A) 2 years (This is the 2 year interest rate for a 5% per year rate)

• B) 2 months (This is the 2 month interest rate for a 5% per year rate)

67

Example 3.20

Interest Earned After 1 Year: Invest $1:• If you earn 0.11 interest after 3 years,

how much did you earn after 1 year?• This takes a interest rate earned after

some period and converts to the interest rate earned each year. This is called the EAR – Effective Annual Rate (or EAY – Effective Annual Yield).

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Example 3.21

Interest Earned After 1 Year: Invest $1:

• If you earn 0.03 interest after 2 months, how much would you earn in 1 year

• Once again, this is an example of computing an EAR.

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Effective Annual Rate: EAR

• The EAR, takes an interest rate earned over any period of time, and computes the interest you would earn after one year.

• The EAR is the amount of interest you earn on an investment in one year, per one dollar invested.

• EAR = (1+i)m-1, where m=number of periods per year. It is the FVIF for a one year period, minus your original $1 investment

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Compounding Intervals

Yrm

N m

APRPVFV

1

m compounding periods per year

The more frequent the compounding period, for a given stated rate, the larger the FV!

Note: APR is the stated interest rate. APR/m is the rate used for compounding, i

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Compounding More Frequently Than Annually

76.060,138$4

05.01000,125$

24

2

FV

– For quarterly compounding, m equals 4:

61.976,137$2

05.01000,125$

22

2

FV

– For semiannual compounding, m equals 2:

FV at end of 2 years of $125,000 deposited at 5% stated interest (APR)

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Compounding More Frequently Than Annually

42.145,138$365

05.01000,125$

2365

2

FV

– For daily compounding, m equals 365:

67.117,138$12

05.01000,125$

212

2

FV

– For monthly compounding, m equals 12:

FV at end of 2 years of $125,000 deposited at 5% stated interest (APR)

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Continuous Compounding

• In extreme case, interest compounded continuously:

FVN = PV x (e r x Yr)

FV at end of 2 years of $125,000 at 5 % annual interest, compounded continuously:

365.146,138$000,125$ 205.02 eFV

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Handling multiple periods per year using the HP10BIIFV at end of 2 years of $125,000 deposited at 5% interest

Annual: P/YR=1FV(PV=-125000, I/YR=5, N=2) = 137,812.50

SemiAnnual: P/YR=2FV(PV=-125000, I/YR=5, N=4) = 137,976.61

Monthly: P/YR=12FV(PV=-125000, I/YR=5, N=24) = 138,117.67

Daily: P/YR=365

FV(PV=-125000, I/YR=5, N=730) = 138,145.42

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How the HP10BII calculator treats multiple periods per year• The interest rate the HP10BII calculator uses for

performing its calculations is :

Example: What is the future value of $125 after 2 years at 5% compounded monthly?

P/YR = 12, FV(PV=-125, N=24, I/YR=5) = 138.12

Note: To solve using years, must use the EAR as the interest rate: FV = 125*(1.052379)2 = 138.12

YRP

YRI

/

/

12.13812

05.01125

/

/1

24

N

YRP

YRIPVFV

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The Stated Rate versus the Effective Rate Rate per period = r/m

11

m

m

APREAR

Effective rate: the annual rate actually paid or earned

Stated rate: the contractual annual rate charged

by lender or promised by borrowerThis is called the APR: Annual Percentage

RateAPR = (r/m) * m = r

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• FV of $100 at end of 1 year, invested at

8% stated annual interest (APR),

compounded:

– Annually: FV = $100 (1.08)1 = $108.00

– Semiannually: FV = $100 (1.04)2 = $108.16

– Quarterly: FV = $100 (1.02)4 = $108.24

The Stated Rate versus the Effective Rate

Stated rate (APR) of 8% does not change.What about the effective rate?

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Effective Rates: Always Greater Than or Equal to Stated Rates

%00.811

08.01

1

EAR

• For annual compounding, effective = stated

%16.812

08.01

2

EAR

• For semiannual compounding

%24.814

08.01

4

EAR

• For quarterly compounding

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Computing the EAR using the HP10BII

The key for computing the EAR is EFF%

To compute the EAR for 8% compounded quarterly

P/YR=4 EFF% (I/YR=8) = 8.2432

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Annual vs. Quarterly Compounding

100.00 108.00+-----------------------------------------------+Yr 0 i=8% Yr 1

100.00 102.00 104.04 106.12 108.24

+-----------+----------+----------+----------+Q0 Q1 Q2 Q3 Q4 i=2

%i=2%

i=2% i=2%

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Example 3.18 revised again: FV of Annuity - Saving for Retirement• Original Version: What is the FV of investing $3600 per year for

40 years if you invest at 6% if your first investment is 1 year from today? $557,143.08

• Revised version: What is the FV of investing $300 per month for 480 month if you invest at 0.5% if your first investment is 1 month from today? $597,447.22

• What if the original version was at the EAR of the revised version, and had the same yearly investment?

• EAR = (1.005)12 – 1 = 0.061678 = 6.1678%• $300 per month, invested a .5% per month for one

year P/YR=1 FV(PMT=-300,N=12, I/YR=.5) = 3,700.67Invest this amount for 40 years: P/YR = 1FV(PMT=-3700.67, N=40, I/YR=6.1678) = $597,450.27

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Reminder• Interest rate and time period must always be in

agreement!!!!• 1% per month is not the same as 12% per year• 1% per month is equivalent to 12.6825% (the

EAR) per year because of compounding.• 12% compounded monthly is a common way to

say that you are earning 1% per month. In this case, 12% is the APR, or stated rate, but you earn (or pay) at the rate of 12.6825%.

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PV of $100 received one year from today, discounted at 12% compounded monthlyTwo approaches:

1. Use the monthly rate and discount for 12 months.

100/(1.01)12 = 88.74, or by FinCalc

P/YR=1, PV(N=12,I/YR=1,FV=100), or

P/YR=12, PV(N=12,I/YR=12,FV=100)

2. Use the correct yearly discount rate, the EAR, and discount for 1 year.

100/(1.126825) = 88.74, or by FinCalc

P/YR=1, PV(N=1,I/YR=12.6825,FV=100)

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Restating our Formulas

• FV = PV(1+APR/m)N

• Where:APR = annual interest rate as APR (I/YR)

m = number of periods per year (P/YR)

N = Number of periods

N = Number of years x m

• FV = PV(1+APR/m)(YR*m)

i as used in our formulas is APR/m

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Example 3.22 Financing a car• You have $2500 to put down on a

new car that you have bargained to $24962 including Tax, Title, License, and all fees. Two financing options are:

• 36 months at 6.9%• 72 months at 7.7%• What is your monthly loan payment

for each option?• How much do you pay into interest

for each option?

86

Example 3.23 Cash back or lower interest rate

• You require a loan of $20,000 on your car purchase and can either have a 1.8% finance rate for 48 months or $1000 cash back. If you choose the cash back, it will cost you 5.7% for 48 months? Which is the best alternative? Why?

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Example 3.24 Getting rid of plastic hubcaps• Kristy Turney got rid of her plastic

hubcaps just 4 days after buying her “new used car.” She received a new set of 1800 wheels by agreeing to pay $57 per week for 52 weeks.

• How much interest will Ms Turney pay?

• What is the APR and the EAR for this “loan”?

90

Example 3.25 Social Security versus personal investment accountYou are currently age 22 and will invest

12.3% of your (constant) gross income through age 67 then withdraw through your life expectancy

Social Security Actuarial Assumptions– Age 22 Male will live to 74– Age 22 Female will live to 79

• How much of your salary will your investment account replace?

91

Social Security or Invest 12.3%

0

1

2

3

4

5

6

3% 4% 5% 6% 7%

Rate of Return

Sal

ary

Rep

lace

men

t F

acto

r

Male Factor

Female Factor

Social Security Actuarial AssumptionsAge 22 Male will live to 74Age 22 Female will live to 79Contribute from Age 22 through Age 67

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Social Security Benefits Retire 2045

0%

10%

20%

30%

40%

50%

60%

70%

80%

16470 36600 58560 90000

Wages in 2005

Current Benefits

Insolvent

Progressive Index

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Support for Social Security Changes

0%

10%

20%

30%

40%

50%

60%

70%

Raisepayroll tax

cap

Limitbenefits for

wealthy

Allowprivate

accounts

IncreasePayroll tax

rate

Slowgrowth ofbenefits

Raiseretirement

age

Under 55

55 and Older

94

Example 3.26.

• Your friendly loan shark makes you a “2 for 4 or I knock at your door” offer, which means he will loan you $200 today, but you must pay him back $400 in one month, or else you get the knock at your door. What is the APR and EAR of this financial transaction?

95

Example 3.27

• You have $100 to invest and the following options are available. Invest at 5% per year for three years, or invest at 6% for two years. For you to have an equal amount in your account after three years, at what rate would you have to invest in the third year, if you chose the two year investment?

96

Example 3.28 Skipping Perpetuity

• What is the present value of receiving $100 every 3 months if the discount rate is 12% compounded monthly?

97

Example 3.29 Skipping Perpetuity

• What is the present value of receiving $100 one year from now, $200, two years from now, $100 three years from now, $200 four years from now, and so on, in perpetuity if the stated rate is 5% compounded annually?

98

Example 3.30 – Non equal CF’s(Example 3.13, revised). What is the

present value (using a 5%, compounded quarterly discount rate) of the following CF’s?

Year 1 2 3 4

CF 100 50 0 325

99

Present Value of 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

100

Example 3.31

• A growing perpetuity. Even Better Assurance Company offers you a perpetuity whose payment will grow at 3% every year. The first payment of $1500 will be one year from now. For a 9% discount rate, what is the present value of this growing perpetuity?

Much Of Finance Involves Finding Future And

(Especially) Present Values

Central to all financial valuation

techniques

Techniques used by investors and firms

alike