Chapter 3

The Time Value of Money

© 2005 Thomson/South-Western

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Time Value of Money

The most important concept in finance

Used in nearly every financial decisionBusiness decisionsPersonal finance decisions

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Cash Flow Time Lines

CF0 CF1 CF3CF2

0 1 2 3k%

Time 0 is todayTime 1 is the end of Period 1 or the beginning of Period 2.

Graphical representations used to show timing of cash flows

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100

0 1 2 Year

k%

Time line for a $100 lump sum due at the end of Year 2

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Time line for an ordinary annuity of $100 for 3 years

100 100100

0 1 2 3k%

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Time line for uneven CFs - $50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3

100 50 75

0 1 2 3k%

-50

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The amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate.

Future Value

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Future Value

Calculating FV is compounding!

Question: How much would you have at the end of one year if you deposited $100 in a bank account that pays 5 percent interest each year?

Translation: What is the FV of an initial $100 after 3 years if k = 10%?

Key Formula: FVn = PV (1 + k)n

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Three Ways to Solve Time Value of Money Problems

Use EquationsUse Financial CalculatorUse Electronic Spreadsheet

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Solve this equation by plugging in the appropriate values:

Numerical (Equation) Solution

nn k)PV(1FV

PV = $100, k = 10%, and n =3

$133.100)$100(1.331

$100(1.10)FV 3n

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Financial calculators solve this equation:

There are 4 variables (FV, PV, k, n).

If 3 are known, the calculator will solve for the 4th.

Financial Calculator Solution

nn k)PV(1FV

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First: set calculator to show 4 digits to the right of the decimal placeTo enter “Format” register:

Type: 2nd, period See: DEC (decimal) = (varies)Type: 4, enterSee: DEC = 4

Exit Format register: hit CE/CSee 0.0000

Financial Calculator Solution

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INPUTS

OUTPUT

3 10 -100 0 ? N I/YR PV PMT FV

133.10

Here’s the setup to find FV:

Clearing automatically sets everything to 0, but for safety enter PMT = 0.

Set: P/YR = 1, END

Financial Calculator Solution

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Spreadsheet SolutionSet up Problem Click on Function Wizard

and choose Financial/FV

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Spreadsheet SolutionReference cells:

Rate = interest rate, k

Nper = number of periods interest is earned

Pmt = periodic payment

PV = present value of the amount

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Present Value

Present value is the value today of a future cash flow or series of cash flows.

Discounting is the process of finding the present value of a future cash flow or series of future cash flows; it is the reverse of compounding.

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100

0 1 2 310%

PV = ?

What is the PV of $100 due in 3 years if k = 10%?

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INPUTS

OUTPUT

3 10 ? 0100

N I/YR PV PMT FV

-75.13

Financial Calculator Solution

VITAL: Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.

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If sales grow at 20% per year, how long before sales double?

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INPUTS

OUTPUT

? 20 -1 0 2N I/YR PV PMT FV

3.8

GraphicalIllustration:

01 2 3 4

1

2

FV

3.8

Year

Financial Calculator Solution

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Future Value of an Annuity

Annuity: A series of payments of equal amounts at fixed intervals for a specified number of periods.

Ordinary (deferred) Annuity: An annuity whose payments occur at the end of each period.

Annuity Due: An annuity whose payments occur at the beginning of each period.

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PMT PMTPMT

0 1 2 3k%

PMT PMT

0 1 2 3k%

PMT

Ordinary Annuity Versus Annuity Due

Ordinary Annuity

Annuity Due

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100 100100

0 1 2 310%

110

121

FV = 331

What’s the FV of a 3-year Ordinary Annuity of $100 at 10%?

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Financial Calculator Solution

INPUTS

OUTPUT

3 10 0 -100 ?

331.00

N I/YR PV PMT FV

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Present Value of an Annuity

PVAn = the present value of an annuity with n payments.

Each payment is discounted, and the sum of the discounted payments is the present value of the annuity.

26248.69 = PV

100 100100

0 1 2 310%

90.91

82.64

75.13

What is the PV of this Ordinary Annuity?

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We know the payments but no lump sum FV,so enter 0 for future value.

Financial Calculator Solution

INPUTS

OUTPUT

3 10 ? 100 0

-248.69

N I/YR PV PMT FV

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100 100

0 1 2 310%

100

Find the FV and PV if theAnnuity were an Annuity Due.

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ANNUITY Due: Switch from “End” to “Begin”Method: (2nd BGN, 2nd Enter)

Then enter variables to find PVA3 = $273.55.

Then enter PV = 0 and press FV to findFV = $364.10.

Financial Calculator Solution

INPUTS

OUTPUT

3 10 ? 100 0

-273.55

N I/YR PV PMT FV

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What is the PV of a $100 perpetuity if k = 10%?

You MUST know the formula for a perpetuity:

PV = PMT k

So, here: PV = 100/.1 = $1000

31250 250

0 1 2 3k = ?

- 846.80

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250 250

You pay $846.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year. What interest rate will you earn on this investment?

Solving for Interest Rates with Annuities

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Financial Calculator Solution

INPUTS

OUTPUT

4 ? -846.80 250 0

7.0

N I/YR PV PMT FV

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What interest rate would cause $100 to grow to $125.97 in 3 years?

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What interest rate would cause $100 to grow to $125.97 in 3 years?

INPUTS

OUTPUT

3 ? -100 0 125.97

8%

N I/YR PV PMT FV

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Uneven Cash Flow Streams

A series of cash flows in which the amount varies from one period to the next:Payment (PMT) designates constant

cash flows—that is, an annuity stream.Cash flow (CF) designates cash flows

in general, both constant cash flows and uneven cash flows.

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0

100

1

300

2

300

310%

-50

4

90.91

247.93

225.39

-34.15530.08 = PV

What is the PV of this Uneven Cash Flow Stream?

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Financial Calculator Solution

In “CF” register, input the following: CF0 = 0 C01 = 100 F01 = 1 C02 = 300 F01 = 1 C03 = 300 F01 = 1 C04 = -50 F01 = 1

In “NPV” Register: Enter I = 10% Hit down arrow to see “NPV = 0” Hit CPT for compute See “NPV = 530.09” (Here NPV = PV.)

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Semiannual and Other Compounding Periods

Annual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added once a year.

Semiannual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added twice a year.

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Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated k constant? Why?

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If compounding is more frequent than once a year—for example, semi-annually, quarterly, or daily—interest is earned on interest—that is, compounded—more often.

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated k constant? Why?

LARGER!

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0 1 2 310%

100133.10

0 1 2 35%

4 5 6

134.01

1 2 30

100

Annually: FV3 = 100(1.10)3 = 133.10.

Semi-annually: FV6/2 = 100(1.05)6 = 134.01.

Compounding Annually vs. Semi-Annually

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kSIMPLE = Simple (Quoted) RateSimple (Quoted) Rate

kPER = Periodic Rate Periodic Rate

EAR = Effective Annual RateEffective Annual Rate

APR = Annual Percentage RateAnnual Percentage Rate

Distinguishing Between Different Interest Rates

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kSIMPLE = Simple (Quoted) RateSimple (Quoted) Rate

*used to compute the interest paid per period*stated in contracts, quoted by banks & brokers*number of periods per year must also be given*Not used in calculations or shown on time lines

Examples:8%, compounded quarterly8%, compounded daily (365 days)

kSIMPLE

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Periodic Rate = kPer

kPER: Used in calculations, shown on time lines.

If kSIMPLE has annual compounding, then kPER = kSIMPLE

kPER = kSIMPLE/m, where m is number of compounding periods per year.

Determining m: m = 4 for quarterly m = 12 for monthly m = 360 or 365 for daily compounding

Examples: 8% quarterly: kPER = 8/4 = 2% 8% daily (365): kPER = 8/365 = 0.021918%

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APR = Annual Percentage RateAnnual Percentage Rate = kSIMPLE periodic rate X

the number of periods per year

APR = ksimple

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EAR = Effective Annual RateEffective Annual Rate

* the annual rate of interest actually being earned

* The annual rate that causes PV to grow to the same FV as under multi-period compounding.

* Use to compare returns on investments with different payments per year.

* Use for calculations when dealing with annuities where payments don’t match interest compounding periods .

EAR

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How to find EAR for a simple rate of 10%, compounded semi-annually

Hit 2nd then 2 to enter “ICONV” register:

NOM = simple interest rateType: 10, enter, down arrow twice

C/Y = compounding periods per yearType: 2, enter, up arrow (or down arrow twice)

EFF = effective annual rate = EARType: CPT (for compute)See: 10.25

POINT: Any PV would grow to same FV at 10.25% annually or 10% semiannually.

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Continuous Compounding

The formula is FV = PV(e kt) k = the interest rate (expressed as a decimal) t = number of years

Calculator “workaround” Store 9999999999 (as many nines as possible) in

your calculator under STO + 9 Then, for N: N = # of years times “RCL 9” Then, for I: I = simple interest divided by “RCL 9”

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Continuous Compounding

Question: What is the value of a $1,000 deposit invested for 5 years at an interest rate of 10%, compounded continuously?

INPUTS

OUTPUT

5* 10/ RCL9 RCL9 -1000 0 ?

1648.72

N I/YR PV PMT FV

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Fractional Time Periods

0 0.25 0.50 0.7510%

- 100

1.00

FV = ?

What is the value of $100 deposited in a bank at EAR = 10% for 0.75 of the year?

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Fractional Time Periods

0 0.25 0.50 0.7510%

- 100

1.00

FV = ?

What is the value of $100 deposited in a bank at EAR = 10% for 0.75 of the year?

INPUTS

OUTPUT

0.75 10 -100 0 ?

107.41

N I/YR PV PMT FV

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Amortized Loans Amortized Loan: A loan that is repaid in equal

payments over its life. Amortization tables are widely used for home

mortgages, auto loans, business loans, retirement plans, and so forth to determine how much of each payment represents principal repayment and how much represents interest. They are very important, especially to homeowners!

Financial calculators (and spreadsheets) are great for setting up amortization tables.

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Task: Construct an amortization schedule for a $1,000, 10 percent loan that requires three equal annual payments.

PMT PMTPMT

0 1 2 310%

-1,000

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PMT PMTPMT

0 1 2 310%

-1000

INPUTS

OUTPUT

3 10 -1000 ? 0

402.11

N I/YR PV PMT FV

Step 1: Determine the required payments

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Hit 2nd, PV to enter “Amort Register”

For 1st principal payment, 1st interest payment, and 1st year remaining balance, enter:

P1 = 1P2 = 1Down arrow

See: Bal = -697.88, down arrowPRN = 302.11INT = 100

Enter “Amort” Register

56Interest declines, which has tax implications.

Step 2: Create Loan Amortization Table

YR Beg Bal PMT INT Prin PMT End Bal

1 $1000.00 $402.11 $100.00 $302.11 $697.88

2 697.88 402.11

* Rounding difference

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Hit 2nd, PV to enter “Amort Register”

For 2nd principal payment, 2nd interest payment, and 2nd year remaining balance, enter:

P1 = 2P2 = 2Down arrow

See: Bal = -365.56, down arrowPRN = 332.32INT = 69.79

Enter “Amort” Register

58Interest declines, which has tax implications.

Step 2: Create Loan Amortization Table

YR Beg Bal PMT INT Prin PMT End Bal

1 $1000.00 $402.11 $100.00 $302.11 $697.89

2 697.89 402.11 69.79 332.32 365.57

3 365.57 402.11

Total

* Rounding difference

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Hit 2nd, PV to enter “Amort Register”

For 3rd principal payment, 3rd interest payment, and 3rd year remaining balance, enter:

P1 = 3P2 = 3Down arrow

See: Bal = 0, down arrowPRN = 365.56INT = 36.65

Enter “Amort” Register

60Interest declines, which has tax implications.

Step 2: Create Loan Amortization Table

YR Beg Bal PMT INT Prin PMT End Bal

1 $1000.00 $402.11 $100.00 $302.11 $697.89

2 697.89 402.11 69.79 332.32 365.57

3 365.57 402.11 36.55 365.56 .01*

Total 1206.33 206.34 1000.00

* Rounding difference

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1. Review Chapter 3 materials

2. Do Chapter 3 homework3. Prepare for Chapter 3 quiz3. Read Chapter 4

Before Next Class