Present Value.ppt

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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

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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

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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

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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(

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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

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Present Values

PVs can be added together to evaluate multiple cash flows.

PV C

r

C

r

1

12

21 1( ) ( )....

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Present Values

PVs can be added together to evaluate multiple cash flows.

88.26521 )0771(200

)07.1(100

PV

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Present Values

Present Value

Year 0

100/1.07

200/1.0772

Total

= $93.46

= $172.42

= $265.88

$100

$200

Year0 1 2

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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

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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

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Present Values

Example - continued


011,25$

249,290000,320907.2

238,95000,100952.1

000,170000,1700.10Value

Present

Flow

Cash

Factor

DiscountPeriod

205.11

05.11

TotalNPV

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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


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Short Cuts

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

PV

Cr

luepresent va

flow cashReturn

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Short Cuts

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

r

CPV 1

0

ratediscount

flow cash FlowCash of PV

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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$1 0.0b i l $ 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 (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

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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

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Short Cuts

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

trrrC

1

11annuity of PV

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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?

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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

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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

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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.

108.1000,20 FV

5

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Constant Growth Perpetuity

gr

CPV

1

0

g = the annual growth rate of the cash flow

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gr

CPV

1

0

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

gr

CPV t

t 1

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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

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Perpetuities

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

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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

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Simple and Compound Interest

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

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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.

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Compound Interest

02468

1012141618

Number of Years

FV

of

$1

10% Simple

10% Compound

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Compound Interest

Example

Suppose you are offered an automobile loan at an APR of 5% per year. What does that mean, and what is the true rate of interest, given monthly payments?

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Compound Interest

Example - continued

Suppose you are offered an automobile loan at an APR of 5% 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

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Assumptions in Valuation

An Ideal Capital Market:1. Capital markets are frictionless2. All market participants share

homogenous expectationsAll market participants are atomisticFirm’s investment program is fixed and

knownThe financing is fixed

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Worked Example Your father is 50 years old and will retire in 10 years. He expects to

live for 25 years after he retires, until he is 85. He wants a fixed retirement income that has the same purchasing power at the time he retires as RM40,000 has today. (The real value of his retirement income will decline annually after he retires.)

His retirement income will begin the day he retires,10 years from today; and he will then receive 24 additional annual payments. Annual inflation is expected to be 5 percent. He currently has RM100,000 saved, and he expects to earn 8 percent annually on his savings. How much must he save during each of the next 10 years (end of year deposits) to meet his retirement goal?

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Worked Example 1. Will save for 10 years, and then receive payments for 25 years. How much must he deposit at the end of each of the next 10 years?

2. Wants payments of $40,000 per year in today’s dollars for first payment only. Real income will decline. Inflation will be 5%. Therefore, to find the inflated fixed payments, we have this time line:

0 5 10 | | | 40,000 FV = ?

Enter N = 10, I/YR = 5, PV = -40000, PMT = 0, and press FV to get FV = $65,155.79. 3. He now has $100,000 in an account that pays 8%, annual compounding. We need to find the FV of the $100,000 after 10 years. Enter N = 10, I/YR = 8, PV

= -100000, PMT = 0, and press FV to get FV = $215,892.50.

4. He wants to withdraw, or have payments of, $65,155.79 per year for 25 years, with the first payment made at the beginning of the first retirement year. So, we have a 25-year annuity due with PMT = 65,155.79, at an interest rate of 8%. Set the calculator to “BEG” mode, then enter N = 25, I/YR = 8, PMT = 65155.79, FV = 0, and press PV to get PV = $751,165.35. This amount must be on hand to make the 25 payments.

5. Since the original $100,000, which grows to $215,892.50, will be available, we must save enough to accumulate $751,165.35 - $215,892.50 = $535,272.85. So, the time line looks like this: Retires 50 51 52 59 60 61 83 84 85 | | | • • • | | | • • • | | | $100,000 PMT PMT PMT PMT -65,155.79 -65,155.79 -65,155.79 -65,155.79 + 215,892.50 - 751,165.35 = PVA(due) Need to accumulate -$535,272.85 = FVA10

6. The $535,272.85 is the FV of a 10-year ordinary annuity. The payments will be deposited in the bank and earn 8% interest. Therefore, set the calculator to “END” mode and enter N = 10, I/YR = 8, PV = 0, FV = 535272.85, and press PMT to find PMT = $36,949.61.

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Worked Example 4. He wants to withdraw, or have payments of, $65,155.79 per year for 25 years, with

the first payment made at the beginning of the first retirement year. So, we have a 25-year annuity due with PMT = 65,155.79, at an interest rate of 8%. Set the calculator to “BEG” mode, then enter N = 25, I/YR = 8, PMT = 65155.79, FV = 0, and press PV to get PV = $751,165.35. This amount must be on hand to make the 25 payments.

5. Since the original $100,000, which grows to $215,892.50, will be available, we must save enough to accumulate $751,165.35 - $215,892.50 = $535,272.85.

So, the time line looks like this: Retires 50 51 52 59 60 61

83 84 85 | | | • • • | | | • • •

| | | $100,000 PMT PMT PMT PMT -65,155.79 -65,155.79 -65,155.79 -

65,155.79 + 215,892.50 - 751,165.35 = PVA(due) Need to accumulate -$535,272.85 = FVA10

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Worked Example

The $535,272.85 is the FV of a 10-year ordinary annuity. The payments will be deposited in the bank and earn 8% interest. Therefore, set the calculator to “END” mode and enter N = 10, I/YR = 8, PV = 0, FV = 535272.85, and press PMT to find PMT = $36,949.61.

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Cases & Readings Case: The Battle for Value, 2004: Fed EX Corp. versus United Parcel

Services, Inc.,

(Refer to Pages 53-73 of the Book, Case Studies in Finance: Managing for Corporate Value Creation, by Robert F. Bruner, Kenneth M. Eades and Michael J. Schill, 6th Edition, McGraw-Hill International Edition,

2010). Articles: 1. An Islamic Perspective on the economics of Discounting in project

evaluation, Muhamad Anas Al Zarqa, Chapter 6, in the book, An Introduction to Islamic Economics & Finance, Edited by Sheikh Ghazali Sheikh Abod, Syed Omar Syed Agil, and Aidit Ghazali, CERT Publications, 2005

2. Time value from Islamic Perspective, Muhamad Akram Khan, Chapter 7, in the book, An Introduction to Islamic Economics & Finance, Edited by Sheikh Ghazali Sheikh Abod, Syed Omar Syed Agil, and Aidit Ghazali, CERT Publications, 2005.

Present Value.ppt

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