Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved....

Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 9

The Time Value of

Money

1

Chapter 9- Learning Objectives Identify various types of cash flow patterns (streams) that are

observed in business. Compute (a) the future values and (b) the present values of

different cash flow streams, and explain the results. Compute (a) the return (interest rate) on an investment (loan)

and (b) how long it takes to reach a financial goal. Explain the difference between the Annual Percentage Rate

(APR) and the Effective Annual Rate (EAR), and explain when each is more appropriate to use.

Describe an amortized loan, and compute (a) amortized loan payments and (b) the balance (amount owed) on an amortized loan at a specific point during its life.


2

Time Value of Money

The principles and computations used to revalue cash payoffs at different times so they are stated in dollars of the same time period

The most important concept in finance used in nearly every financial decisionBusiness decisionsPersonal finance decisions


3

Cash Flow Patterns

Lump-sum amount – a single payment paid or received in the current period or some future period

Annuity - A series of equal payments that occur at equal time intervals

Uneven cash flow stream – multiple payments that are not equal and do not occur at equal intervals


4

Cash Flow Time Lines


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0 1 2 36%

Time:

PV = 100Cash Flows: FV = ?

Time 0 is todayTime 1 is the end of Period 1 or the beginning of Period 2, and so forth.

Graphical representations used to show timing of cash flows

Future Value

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.


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Future Value

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

7

FVn = FV1 = PV + INT

= PV + PV(r)

= PV (1 + r)

= $100(1 + 0.05) = $100(1.05) = $105


What’s the FV of an initial $700 after three years if r = 10%?


8

FV = ?

0 1 2 310%

700

Finding FV is Compounding

Future Value


9

After 1 year:FV1 = PV + Interest1 = PV + PV (r)

= PV(1 + r)= $700 (1.10)= $770.00.

After 2 years:FV2 = PV(1 + r)2

= $700 (1.10)2

= $847.00.

After 3 years: FV3 = PV(1 + r)3

= $700 (1.10)3

= $931.70

In general, FVn = PV (1 + r)n

Three Ways to Solve Time Value of Money Problems

Use EquationsUse Financial CalculatorUse Electronic Spreadsheet


10

Numerical (Equation) Solution


11

PV = $700, r = 10%, and n =3

Financial Calculator Solution


12

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



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INPUTS

OUTPUT

3 10 -700 0 ? N I/Y PV PMT FV

=931.70

Here’s the setup to find FV:

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

Set: P/YR = 1, END

Spreadsheet Solution


14

According to the equation shown in cell C8, the input values must be entered in a specific order: I/Y, N, PMT, PV, and PMT type (not used for this problem).

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


15

Solve for n:

FVn = 1(1 + r)n

2 = 1(1.20)n

The numerical solution is somewhat difficult.

Graphical Illustration


16

0 1 2 3 4

1

2

3.8

Year

FV



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INPUTS

OUTPUT

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

3.8

Future Value of an Annuity

Annuity: A series of payments of equal amounts at equal 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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Ordinary Annuity Versus Annuity Due


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PMT PMTPMT

0 1 2 3r%

PMT PMT

0 1 2 3r%

PMT

Ordinary Annuity

Annuity Due

FV of a 3-year Ordinary Annuity of $400 at 5%


20

Numerical Solution


21



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INPUTS

OUTPUT

3 5 0 -400 ?

=1261.00

N I/Y PV PMT FV



23

FV of a 3-year Annuity Due of $400 at 5%


24

Numerical Solution


25



26

INPUTS

OUTPUT

3 5 0 -400 ?

=1,324.05

N I/Y PV PMT FV

BGN



27

N = 3I/Y = 5.00%PV = $0.00

PMT = -$400.00PMT type = 1 (0 = ordinary annuity; 1 = annuity due)

FV = ?

The equation used to solve for FV in cell B8

Values that correspond to thecell reference in cell C8

FVA(DUE) = $1,324.05 =FV(B2,B1,B4,B3,B5 =FV(0.05,3,-400,0,1)

Future Value of an Uneven Cash Flow


28

Numerical Solution


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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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What is the PV of $935 due in three years if r = 10%?


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Numerical Solution


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Solve FVn = PV (1 + r )n for PV:



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INPUTS

OUTPUT

3 10 ? 0935

N I/Y PV PMT FV

-702.48

Either PV or FV must be negative. HerePV = -702.48. Invest $702.48 today, take out $935 after 3 years.



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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


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What is the PV of this Ordinary Annuity?


36

Numerical Solution


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

INPUTS

OUTPUT

3 5 ? 400 0

= -1,089.30N I/Y PV PMT FV



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Present Value of an Annuity Due


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1,143.76 = PVA (DUE)3

400 400400

0 1 2 305%

400.00380.95362.81

(1.05) x

(1.05) x

(1.05) x

Value of Each Deposit Today (Year 0)

Numerical Solution


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